2023 11 20 Libro Power System Harmonic Analysis Arrillaga1997
2023 11 20 Libro Power System Harmonic Analysis Arrillaga1997
2023 11 20 Libro Power System Harmonic Analysis Arrillaga1997
POWER SYSTEM
HARMONIC ANALYSIS
John Wiley & Sons (Asia) Pte Ltd. 2 Clementi Loop #02-01,
JinXing Distripark, Singapore 129809
A catalogue record for this book is available from the British Library
Preface xi
1 Introduction 1
1.1 Power System Harmonics
1.2 The Main Harmonic Sources
1.3 Modelling Philosophies 2
1.4 Time Domain Simulation 3
1.5 Frequency Domain Simulation 3
1.6 Iterative Methods 4
I. 7 References 5
2 Fourier Analysis 7
2.1 Introduction 7
2.2 Fourier Series and Coefficients 7
2.3 Simplifications Resulting from Waveform Symmetry 10
2.4 Complex Form of the Fourier Series 13
2.5 Convolution of Harmonic Phasors 15
2.6 The Fourier Transform 17
2.7 Sampled Time Functions 19
2.8 Discrete Fourier Transform 20
2.9 Fast Fourier Transform 24
2.10 Transfer Function Fourier Analysis 26
2.11 Summary 31
2.12 References 31
3 Transmission Systems 33
3.1 Introduction 33
3.2 Network Subdivision 33
3.3 Frame of Reference used in Three-Phase System Modelling 35
3.4 Evaluation of Transmission Line Parameters 37
3.4.1 Earth Impedance Matrix [Zel 37
3.4.2 Geometrical Impedance Matrix [Zg] and Admittance Matrix [Yg] 39
3.4.3 Conductor Impedance Matrix [Zc] 41
3.5 Single Phase Equivalent of a Transmission Line 46
3.5.1 Equivalent PI Models 46
vi CONTENTS
Index 365
PREFACE
The subject of Power System Harmonics was first discussed in a book published by
J. Wiley & Sons in 1985 which collected the state of the art, explaining the presence
of voltage and current harmonics with their causes, effects, standards, measurement,
penetration and elimination. Since then, the increased use of power electronic devices
in the generation, transmission and utilisation of systems has been accompanied by a
corresponding growth in power system harmonic problems.
Thus, Power System Harmonic Analysis has become an essential part of system
planning and design. Many commercial programmes are becoming available, and
CIGRE and IEEE committees are actively engaged in producing guidelines to
facilitate the task of assessing the levels of harmonic distortion.
This book describes the analytical techniques, currently used by the power
industry for the prediction of harmonic content, and the more advanced algorithms
developed in recent years.
A brief description of the main harmonic modelling philosophies is made in
Chapter I and a thorough description of the Fourier techniques in Chapter 2.
Models of the linear system components, and their incorporation in harmonic
flow analysis, are considered in Chapters 3 and 4. Chapters 5 and 6 analyse the
harmonic behaviour of the static converter in the frequency domain. The remaining
chapters describe the modelling of non-linearities in the harmonic domain and their
use in advanced harmonic flow studies.
The authors would like to acknowledge the assistance received directly or
indirectly from their present and previous colleagues, in particular from E. Acha,
G. Bathurst, P. S. Bodger, S. Chen, T. J. Densem, J. F. Eggleston, B. J. Harker,
M. L. V. Lisboa and A. Medina. They are also grateful for the advice received from
J. D. Ainsworth, H. Dommel, A. Semylen and R. Yacamini. Finally, they wish to
thank Mrs G. M. Arrillaga for her active participation in the preparation of the
manuscript.
1
INTRODUCTION
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
2 I INTRODUCTION
(3) Large static power converters and transmission system level power electronic
devices.
The first category consists mainly of single-phase diode bridge rectifiers, the power
supply of most low voltage appliances (e.g. personal computers, TV sets, etc.). Gas
discharge lamps are also included in this category. Although the individual ratings
are insignificant, their accumulated effect can be important, considering their large
numbers and lack of phase diversity. However, given the lack of controllability, these
appliances present no special simulation problem, provided there is statistical
information of their content in the load mix.
The second category refers to the arc furnace, with power ratings in tens of
megawatts, connected directly to the high voltage transmission network and
normally without adequate filtering. The furnace arc impedance is randomly
variable and extremely asymmetrical. The difficulty, therefore, is not in the
simulation technique but in the variability of the current harmonic injections to be
used in each particular study, which should be based on a stochastic analysis of
extensive experimental information obtained from measurements in similar existing
installations.
As far as simulation is concerned, it is the third category that causes considerable
difficulty. This is partly due to the large size of the converter plant in many
applications, and partly to their sophisticated point on wave switching control
systems. The operation of the converter is highly dependent on the quality of the
power supply, which is itself heavily influenced by the converter plant. Thus the
process of static power conversion needs to be given special attention in power
system harmonic simulation.
If the derivation of harmonic sources and harmonic flows could be decoupled, the
theoretical prediction would be simplified. Such an approach is often justified in
assessing the harmonic effect of industrial plant, where the power ratings are
relatively small. However, the complex steady state behaviour of some system
components, such as an HVdc converter, require more sophisticated models either in
the frequency or time domains.
As with other power system studies, the digital computer has become the only
practical tool in harmonic analysis. However, the level of complexity of the computer
solution to be used in each case will depend on the economic consequences of the
predicted behaviour and on the availability of suitable software.
(1) The derivation, form and accuracy of the non-linear equations used to describe
the system steady state.
(2) The iterative procedure used to solve the non-linear equation set.
1. 7 References
1. Fourier, J B J (1822). Theorie Analytique de Ia Chaleur (book), Paris.
2. Arrillaga, J, Bradley, D and Bodger, P S, (1985). Power System Harmonics, J Wiley & Sons,
London.
3. Chuah, L D and Lin P M, (1975) Computer-aided Analysis of Electronic Circuits,
Englewood Cliffs, Prentice Hall, NJ.
4. Kuh, E Sand Rohrer, R A, (1965). The state variable approach to network analysis, Proc
IEEE.
5. Balabanian, N, Bickart, T A and Seshu, S, (1969). Electrical Network Theory, John Wiley
& Sons, New York.
6. Arrillaga, J, Arnold, C P and Harker, B J, (1983). Computer Modelling of Electrical Power
Systems, J Wiley & Sons, London.
7. Kulicke, B. (1979). Digital program NETOMAC zur Simulation Elecktromechanischer
und Magnetischer Ausleighsvorgange in Drehstromnetzen, Electrizitatwirstschaft, 78,
S.l8-23.
8. Dommel, H W, Yan, A and Wei Shi, (1986). Harmonics from transformer saturation,
IEEE Trans, PWRD-1(2) 209-215.
9. Aprille, T J, (1972). Two computer algorithms for obtaining the periodic response of non-
linear circuits, Ph.D Thesis, University of Illinois at Urbana Champaign.
6 I INTRODUCTION
10. Usaola, J (1990). Regimen permanente de sistemas electricos de potencia con elementos
no lineales mediante un procedimiento hibrido de analisis en los dominios del tiempo y de
Ia frecuencia. Doctoral Thesis, Universidad Politecnica de Madrid.
11. Yacamini, R and de Oliveira, J C, (1980). Harmonics in multiple converter systems: a
generalised approach, lEE Proc B, 127(2), 96-106.
12. Arrillaga, J, Watson, N R, Eggleston, J F and Callaghan, CD, (1987). Comparison of
steady state and dynamic models for the calculation of a.c.jd.c. system harmonics, Proc
lEE, 134C(1), 31-37.
13. Carpinelli, G. eta!., (1994). Generalised converter models for iterative harmonic analysis
in power systems, Proc lEE General Transn. Distrib, 141(5), 445-451.
14. Callaghan, C and Arrillaga, J, (1989), A double iterative algorithm for the analysis of
power and harmonic flows at ac-dc converter terminals, Proc lEE, 136(6), 319-324.
15. Smith, B, eta!., (1995). A Newton solution for the harmonic phasor analysis of ac-dc
converters, IEEE PES Summer Meeting 95, SM 379-8.
16. Callaghan, C and Arrillaga, J, (1990). Convergence criteria for iterative harmonic analysis
and its application to static converters, ICHPS IV, Budapest, 38--43.
2
FOURIER ANALYSIS
2.1 Introduction
Fourier analysis is the process of converting time domain waveforms into their
frequency components [1].
The Fourier series, which permits establishing a simple relationship between a time
domain function and that function in the frequency domain, is derived in the first part
of this chapter and its characteristics discussed with reference to simple waveforms.
More generally, the Fourier Transform and its inverse are used to map any
function in the interval -oo to oo in either the time or frequency domain, into a
continuous function in the inverse domain. The Fourier series, therefore, represents
the special case of the Fourier Transform applied to a periodic signal.
In practice, data is often available in the form of a sampled time function,
represented by a time series of amplitudes, separated by fixed time intervals of
limited duration. When dealing with such data a modification of the Fourier
Transform, the Discrete Fourier Transform, is used. The implementation of the
Discrete Fourier Transform, by means of the Fast Fourier Transform algorithm,
forms the basis of most modern spectral and harmonic analysis systems. The FFT is
also a powerful numerical tool that enables the Harmonic Domain description of
non-linear devices to be implemented in either the frequency or time domain,
whichever is appropriate. The development of the Fourier and Discrete Fourier
Transforms is also examined in this chapter along with the implementation of the
Fast Fourier Transform.
The main sources of harmonic distortion are power electronic devices, which
exercise controllability by means of multiple switching events within the fundamental
frequency waveform. Although the standard Fourier method can still be used to
analyse the complete waveforms, it is often advantageous to subdivide the power
electronic switching into its constituent Fourier components; this is the transfer
function technique, which is also described in this chapter.
with a magnitude:
A 11 = Va 11
2 +b/
and a phase angle
For a given function x(t), the constant coefficient, a00 can be derived by integrating
both sides of equation (2.1) from - T /2 to T /2 (over a period T), i.e.
The Fourier series of the right-hand side can be integrated term by term, giving
J
-T/2
0
-T/2
L
T/2 x(t)dt = a JT/2 dt + 00 [ a JT/2 cos (2nnt)
n=l
11
-T/2
T
-T/2
- - dt] . (2.4)
dt + b11 JT/2 sin (2nnt)
T
The first term on the right-hand side equals Ta 0 , while the other integrals are zero.
Hence, the constant coefficient of the Fourier series is given by
T/2
a0 = 1/T J x(t)dt, (2.5)
-T/2
which is the area under the curve of x(t) from -T/2 to T/2, divided by the period of
the waveform, T.
The a11 coefficients can be determined by multiplying Equation (2.1) by
cos(2nmt/T), where m is any fixed positive integer, and integrating between -T/2
and T/2, as previously, i.e.
T/ 2 JT/ 2 [
00
(2nmt)
J -T/ 2 x(t)cos ----y;- dt= -T/ 2 a0 +~ [
a11 COS (2nnt)
T +b11 sin (2nnt)]]
T
cos ( r
2nnt)
dt
(2.6)
2.2 FOURIER SERIES AND COEFFICIENTS 9
= Go
T/ 2. cos ( -
J-Tj2 - dt + L
2nmt) 00
[
an JT/
1
cos (-
2nnt)
- X
T n=t -T/2 T
The first term on the right-hand side is zero, as are all the terms in b11 since
sin(2rr.nt/T) and cos(2mnt/T) are orthogonal functions for all n and m.
Similarly, the terms in a11 are zero, being orthogonal, unless m = n. In this case,
Equation (2. 7) becomes
2 1 2 2
J T/Z. x(t)cos ( n; t)dt = a11 JT/ cos ( ;t)dt
-T;2 -T/2
(2.8)
=-
4nnt)
an JT/2 cos ( - an JT/2 dt.
- dt+-
2 -T/2 T 2 -T/2
The first term on the right-hand side is zero while the second term equals anT/2.
Hence, the coefficients an can be obtained from
G 11
2 JT/ 2 x(t)cos
= T ( 2nnt) dt
T for n = 1 -+ oo. (2.9)
-T/2
To determine the coefficients b11 , Equation (2.1) is multiplied by sin(2mnt/T) and, by
a similar argument to the above
G0 = -21
7r.
J"-n x(wt)d(wt), (2.11)
J"
1 -n x(wt)cos(nwt)d(wt),
an=~ (2.12)
bn =-1
7r.
J"-n x(wt) sin(nwtd(wt), (2.13)
so that
+ L[a
00
an 2JT/
=y. 0
2
x(t)cos (2nnt)
T dt+y. 2J 0
-T/ 2 x(t)cos
(2nnt)
T dt, (2.15)
b11 =~ r/ 2
X(t) sin ( 2; !) dt + ~
1
rT/ 2 x(t) sin ( 2; t ) dt. (2.16)
2JT/ 2
an =y. 0
x(t)cos (2nnt)
T dt+y.21°+T/ 2 x(-t)cos (-2nnt)
-T- d(-t)
(2.17)
2 JT/2 [x(t) + x(-t) ] cos (2nnt)
=T T dt.
0
Similarly,
Odd symmetry:
4 IT/ 2
bn = T 0
x(t)sin ( T
2nnt) dt. (2.19)
The Fourier series for an odd function will, therefore, contain only sine terms.
Even symmetry:
In this case
and
2.3 SIMPLIFICATIONS RESULTING FROM WAVEFORM SYMMETRY 11
G11 = T4JT/2
0
x(t)cos (2nnt)
T dt. (2.20)
The Fourier series for an even function will, therefore, contain only cosine terms.
Certain waveforms may be odd or even depending on the time reference position
selected. For instance, the square wave of Figure 2.1, drawn as an odd function, can be
transformed into an even function simply by shifting the origin (vertical axis) by T/2.
Halfwave symmetry:
i.e. the shape of the waveform over a period t + T /2 to t + T is the negative of the
shape of the waveform over the period t tot+ T/2. Consequently, the square wave
function of Figure 2.1 has halfwave symmetry with t = - T /2.
Using Equation (2.9) and replacing (t) by (t + T/2) in the interval ( -T/2, o)
J
O+T/2
an= T
2JT/l x(t)cos ( T
0
2nnt) 2
dt + T x(t + Tj2)cos
( 2nn(t +T T /2)) dt
-T/2+T/2
2 JT/ 2x(t)
= T . [cos (2nnt)
T - T
cos (2nnt + nn )] dt (2.22)
0
2nnt
cos ( T+nn ) =-cos (2nnt)
T
X (t)
--- r-- k-
-t----~--~-TT~--~T~--+-~----~-
/2 12
- --k ~ ---
and
4 JT/ 2x(t)cos
a,= T 0
2nnt) dt.
(T (2.23)
a, =0.
Similarly,
4 J·T/ 2
b, = T x(t)sin
2nnt) dt
(T for n odd,
0 (2.24)
=0 for n even.
Thus, waveforms which have halfwave symmetry, contain only odd order
harmonics.
The square wave of Figure 2.1 is an odd function with halfwave symmetry.
Consequently, only the b, coefficients and odd harmonics will exist. The expression
for the coefficients taking into account these conditions is
4 (T
b, = T8 JT/
0
x(t)sin 2nnt) dt, (2.25)
4 k /;r · -
~(4k/.T )
~ (4k /;r )
0 11 311 5~
t t
711 • • ----.I
911 1111
(2.27)
Thus, each harmonic component of a real valued signal can be represented by two
half amplitude contra-rotating vectors as shown in Figure 2.3, such that
e jnwt + e -jnwt
cos (nwt) = 2 , (2.29)
Instantaneous
amplitude
Figure 2.3 Contra-rotating vector pair producing a varying amplitude (pulsating) vector
14 2 FOURIER ANALYSIS
e jnwt - e -jnwt
sin (nwt) = 21 (2.30)
(2.31)
where
n>O
Cn =-1
n
J"
-n
x(wt)e-Jnwt d(wt),
0
(2.32)
C0 = -21
n
J"
-n
x(wt) d(wt). (2.33)
If the time domain signal x(t) contains a component rotating at a single frequency
nf, then multiplication by the unit vector e-i 2nft, which rotates at a frequency -nf,
annuls the rotation of the component, such that the integration over a complete
period has a finite value. All components at other frequencies will continue to rotate
after multiplication by e-i 2mift, and will thus integrate to zero.
The Fourier Series is most generally used to approximate a periodic function by
truncation of the series. In this case, the truncated Fourier series is the best
trigonometric series expression of the function, in the sense that it minimizes the
square error between the function and the truncated series. The number of terms
required depends upon the magnitude of repeated derivatives of the function to be
approximated. Repeatedly differentiating Equation (2.32) by parts, it can readily be
shown that
(2.36)
Finally, the phase shifted sine terms can be represented as peak value phasors by
setting
(2.37)
so that
=L
00
which does not contain negative frequency components. Note that the de term
becomes
(2.40)
In practice, the upper limit of the summation is set to n11 , the highest harmonic
order of interest.
if k~m
(2.43)
otherwise.
(2.46)
In practice, the discrete convolution can be evaluated faster using FFT methods.
2.6 THE FOURIER TRANSFORM 17
The expression for the time domain function x(t) which is also continuous and of
infinite duration, in terms of X(f) is then:
Using Equations (2.51) to (2.55), the inverse Fourier transform can be expressed in
terms of the magnitude and phase spectra components.
x(t)
-t --~-T--+------;!---!-
12 12
i.e. the function is continuous over all t but is zero outside the limits (- T /2, T /2).
Its Fourier transform is
(2.57)
X( f)=~ sin(n/1)
(2.58)
= KT[sin(nf1)].
nfT
The term in brackets, known as the sine function, is shown in Figure 2.5.
While the function is continuous, it has zero value at the points f = n/T for
n = ±1, ± 2, ... and the side lobes decrease in magnitude as 1/T. This should be
compared to the Fourier series of a periodic square wave which has discrete
frequencies at odd harmonics. The interval 1/T is the effective bandwidth of the
signal.
2.7 SAMPLED TIME FUNCTION 19
L
00
X( f)= x(nt,)e-j2nfntt. (2.59)
n=-oo
The frequency domain spectrum, shown in Figure 2.7, is periodic and continuous.
X(t)
X(f)
,,.- . , ...
,,
I '
''
''
'\
= 1/fs J
fs/2
x( t) X(f) e j2nfntl df (2.60)
-fs/2
and
N-1
x(tn) = L X(.f/J ej2nkn/N. (2.62)
k=O
Both the time domain function and the frequency domain spectrum are assumed
periodic as in Figure 2.8, with a total of N samples per period. It is in this discrete
form that the Fourier Transform is most suited to numerical evaluation by digital
computation.
Consider equation (2.61) rewritten as
N-1
X(.f/,) = 1/Nl..:x(tn)Wkn. (2.63)
n=O
where W = e-iln/N.
X (t)
X (f)
Over all the frequency components, Equation (2.63) becomes a matrix equation.
X(fo) 1 1 1 x(to)
X(f1) 1 w w" wN-1 x(t 1)
= l/N
X(fk) 1 w" JV'<2 JV'<{N-1) x(tk)
or in a condensed form
w= e-j2n/8
n .. n
= cos 4 - J sm 4 .
As a consequence
Wl=-W=l
w1 = -W5 = (1~2- jl/~2)
W2 = -Wi = -j
W= -W7 = -(1/~2+jl/~2).
These can also be thought of as unit vectors rotated through ±0°, ±45°, ± 90°
and ± 135°, respectively.
Further, WS is a complete rotation and hence equal to 1. The value of the elements
of JVkn for kn > 8 can thus be obtained by subtracting full rotations, to leave only a
fraction of a rotation, the values for which are shown above. For example, if k = 5
and n = 6, then kn = 30 and w 30 = W 3 X 8+6 = W 6 = j.
Thus, there are only 4 unique absolute values of wkn and the matrix [Wkn], for the
case N = 8, becomes
22 2 FOURIER ANALYSIS
w -J w3 -1 -W -W3
1 -j -1 j 1 -j -1
1 w3 j w -1 -W3 -J -W
-1 1 -1 -1 1 -1
-W -j -W3 -1 w j w3
J -1 -j 1 j -1 -j
-W3 J -W -1 w3 -J w
With regard to equation (2.64) for the Discrete Fourier Transform and the matrix
[Wk"] it can be observed that for the rows N/2 toN, the rotations applied to each
time sample are the negative of those in rows N /2 to 1. Frequency components
above k = N /2 can be considered as negative frequencies, since the unit vector is
being rotated through increments greater than n between successive components. In
the example of N = 8, the elements of row 3 are successively rotated through -n/2.
The elements of the row 7 are similarly rotated through -3nj2; or in negative
frequency form through n/2. More generally, a rotation through
I Sampling
interval
,./f\., /]\-,
(b)
,/[\, /l'\ ,/[\,
t
I I
I ! \
I ~·
\
'' ''
' ' 'I '
',_ '' '' ' _, ' ' '·' ' '
·~ ·' '
(c)
lsamplingl
Figure 2.10 The effect of aliasing: (a) x(t) = k; (b) x(t) = k cos 2nnft. For (a) and (b) both
signals are interpreted as being de. In (c) the sampling can represent two different signals with
frequencies above and below the Nyquist or sampling rate
X (f)
-f--+-------1------+-----
fc
Figure 2.11 Frequency domain characteristics of an ideal low pass filter with cut-off
frequency j~
-1
-1
-1
w
-W
-1
-1
-1
-1
As previously stated, each factor matrix has only two non-zero elements per row,
the first of which is unity.
The re-ordering of the [Wk"] matrix results in a frequency spectrum which is also
re-ordered. To obtain the natural order of frequencies, it is necessary to reverse the
previous bit-reversal.
In practice, a mathematical algorithm implicitly giving factor matrix operations is
used for the solution of an FFT [8].
Using N = 2m, it is possible to represent n and k by m bit binary numbers such
that:
For N = 8:
and
where n2, n 1, no and k 2 , k 1, k 0 are binary bits (n2, k2 most significant and no, ko least
significant).
Equation (2.63) can now be re-written as:
1 I I
X(k 2,k 1,k0 ) = L L L 1/ N x (n2,nbno) W. (2.68)
n2 =0 n 1=0 no=O
26 2 FOURIER ANALYSIS
L
I
A 1(k 0 ,n 1 ,n0 ) = 1/Nx(n 2 ,n 1 ,n 0 )Uf~kon2, (2.69)
112=0
I
A 2(k 0• k I' n0) = "
~
A I (k 0• n J, n0 )W2(ko+2kll"J ' (2.70)
'IJ=O
L
I
A3(ko,k"k2 ) = Aik0 ,k"n0 ) w<ko+ 2k, +4k2ln 0 . (2.71)
no=O
From Equation (2.71) it is seen that the A 3 coefficients contain the required X(k)
coefficients but in reverse binary order.
Order of A 3 in binary form is k 0 k 1k 2 .
Order of X(k) in binary form is k 2k 1k 0 •
Hence
Binary Reversed
A3 (011) X(llO) X(6)
A3 (100) X(lOO) X(l)
A3(101) X(lOl) X(5).
(2.72)
I'P
a
c
4
where'¥ is 0, 120 and 240 degrees, referring to phases a, band c, and Y'I'dc and Y'I'ac
are the transfer function to de voltage and ac current, respectively.
By way of illustration, Figure 2.13 shows the six pulse ideal converter transfer
function with a steady converter firing angle, related to each phase of the described
voltage waveform, which written as a Fourier series is
2v'3 1
Y'¥ = -L:)±)-cos[m(W01 -IY. 0 - '¥)], (2.74)
n m m
In general, the switched functions V'¥ and Ide in Equations (2.72) and (2.73) will
contain any number of harmonics, i.e.
Ya Ya
,--------, 2
]}
I
I Va I
---- ··--'
Va
fi
e ~----------~~----------~9
I -1
,----------'
I Ji
-1
I
I._ _ _ _ _ _ _ _ _ J -2
: .
fi ~- ---- ---- _:
The spectra of the de voltage and ac current waveforms will then result from the
multiplication of Expressions (2. 74) by either (2. 75) or (2. 76).
An alternative to the multiplication of the component functions in the time
domain is their convolution in the frequency domain. This alternative is used to
calculate converter harmonic cross-modulation in Chapter 8.
The transfer function approach is essential to the derivation of the cyclo-converter
frequency components, since in this case the frequency spectra of the output voltage
and input current waveforms are related to both the main input and output
frequencies. These waveforms contain frequencies which are not integer multiples of
the main output frequency.
Each output phase of the basic cycloconverter is derived from a three-phase
system via a 'positive' and a 'negative' static converter, as shown in Figure 2.14 [11].
By expressing the switching function as a phase-modulated harmonic series, a
general harmonic series can be derived for the output voltage (or input current)
waveform in terms of the independent variables.
By way of illustration, the quiescent voltage waveform of the positive converter
shown in Figure 2.15, is given by
and
+ -
'\ \ '\ io
T, Ta ~ Ts
T2' T.'
cl T'6
I
A cl I
B Load c
c 8
'\ '\ '\ A
r. Te T2 vP vN
Ts' TJ' r,·
if cl cl
- +
J
~-------- J
------ J
Quiescent voltage
of positive
converter
Figure 2.15 Derivation of voltage waveforms of the positive converter for quiescent (ct = 90°)
operation
since the phase modulation of the firing angles of the positive and negative
converters is equal but of opposite sign.
Moreover, it can be shown [11] that the optimum output waveform, i.e. the
minimum r.m.s. distortion, is achieved when the firing angle modulating function is
derived by the 'cosine wave crossing' control. Under this type of control the phase of
firing of each thyristor is shifted with respect to the quiescent position by
(2.78)
Voltage of line
VNsin ej
Wanted component
- - - - - - - - - -
I
of output voltage
-~- -
I I
1 I
I I
-+!"' I-
I 't'O I
i
A
= I
-r---------- -j
+
-----------~'~~J
r - -,
Current in input line A I I
Figure 2.16 Derivation of the input line current of a cycloconverter. The input line current is
shown in the bottom part of the figure as a continuous line for a single-phase load and as a
broken line for a three-phase load
(2.80)
FN(8 0 ) = ~- ~ [ sin(8 0 + c/Y 0 ) +~sin 3(8 0 + c/Y 0 ) +~sin 5(8 0 + c/Y 0 ) + ... ]. (2.82)
2.11 Summary
The main Fourier concepts and techniques relevant to power system harmonic
analysis have been described. These included the basic Fourier series, the Fourier
Transform and its computer implementation in the form of the Fast Fourier
Transform.
A Fourier-domain-based transfer function concept has also been introduced for
the analysis of power electronic waveforms resulting from complex controls and
multiple periodic switchings. The effectiveness of this technique will become
apparent in Chapters 5 and 8.
2.12 References
1. Fourier, J B J, (1822). Tluiorie Analytique de la Chaleur (book).
2. Kreyszig, E, (1967). Advanced Engineering Mathematics, John Wiley and Sons Inc, 2nd
Edition.
3. Kuo, F F, (1966). Netll'ork Analysis and Synthesis, John Wiley and Sons, Inc.
4. Brigham. E 0, (1974). The Fast Fourier Transform, Prentice-Hall, Inc.
5. Cooley, J Wand Tukey, J W, (1965). 'An algorithm for machine calculation of complex
Fourier series', Math Computation, 19, 297-301.
6. Cochran, W T, eta!, (1967). What is the fast Fourier Transform. Proc IEEE, 10, 1664-
1677.
7. Bergland, G D, (1969). A guided tour of the fast Fourier Transform. IEEE Spectrum, July,
41-42.
8. Bergland, G D, (1968). A fast Fourier Transform algorithm for real-values series.
Numerical Analysis, 11(10), 703-710.
32 2 FOURIER ANALYSIS
9. Stemmler, H, (1972). HVdc back to back interties on weak a.c. systems, second harmonic
problems and solutions, CIGRE Symposium, 09-87, no 300-08, 1-5.
10. Wood, A R, (1993). An analysis of non-ideal HVdc converter behaviour in the frequency
domain, and a new control proposal, Ph.D. Thesis, University of Canterbury, New
Zealand.
11. Pelly, B R, (1971). Thyristor Phase Controlled Converters and Cycloconverters, Wiley
Interscience, New York.
3
TRANSMISSION SYSTEMS
3.1 Introduction
As the main vehicle of harmonic propagation, the transmission system must be
accurately represented to predict the levels of waveform distortion throughout the
power system. The following steps are used in the derivation of a multi-phase
transmission system model:
• Definition of the components of the transmission system and their separation into
homogeneous elements; typical elements in this context are an untransposed
section of the transmission line, a cable, a series impedance and a shunt
admittance.
• Selection of the location of observation points. If standing waves are to be
displayed then observation points must be inserted at intervals of less than one
tenth of a wavelength at the highest frequency of interest. Element data is then
partitioned so that the observation points occur at the junctions between the
component elements.
• Provision of element type data and those parameters necessary for the
determination of the elements' electrical characteristics, such as the conductor
type, their arrangement, earth resistivities, etc.
• Derivation of reduced equivalent impedance (admittance) matrices for the
frequencies of interest.
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
34 3 TRANSMISSION SYSTEMS
(a)
[ [A][ B)]
[C][D]
(b)
• A subsystem is the unit into which any part of the system may be divided such
that no subsystem has any mutual coupling between its constituent branches and
those of the rest of the system.
• The smallest unit of a subsystem is a single network element.
• The subsystem unit is retained for input data organisation. Data for any
subsystem is input as a complete unit, the subsystem admittance matrix is
formulated and then combined in the total system admittance matrix.
• Subsystem admittance matrices may be derived by finding, for each section, the
ABCD or transmission parameters.
This procedure involves an extension of the usual two-port network theory to
multi-two-port networks. Current and voltages are now matrix quantities as defined
in Figure 3.1. The dimensions of the parameter matrices correspond to those of the
section being considered, i.e. three, six, nine or twelve for one, two, three or four
mutually coupled three-phase elements, respectively. All sections must contain the
same number of mutually coupled three-phase elements, ensuring that all the
parameter matrices are of the same order and that the matrix multiplications are
executable. Uncoupled elements need to be considered as coupled ones with zero
coupling to maintain correct dimensions for all matrices.
For the case of a non-homogeneous line with n different sections:
(3.1)
It must be noted that in general [A] =f. [D] for a non-homogeneous line.
Once the resultant ABCD parameters have been found the equivalent nodal
admittance matrix for the subsystem can be calculated from
3.3 FRAl\IE OF REFERENCE USED IN THREE-PHASE SYSTEM MODELLING 35
-[Ysd
[Yssl + [Y1d -[Y12J
-[Y21J [Y11 ] + [Y22J
[!abc] = [IJ,,J,]T
(3.5)
[Valn·l = [Va Vb VcJT
l
and
[y""
Yah
[Yahc] = Yba Yhb Ym
J'bc (3.6)
Yea Ycb J'cc
36 3 TRANSMISSION SYSTEMS
By the use of the symmetrical components transformation the three coils of Figure
3.2 can be replaced by three uncoupled coils. This enables each coil to be treated
separately with a great simplification of the mathematics involved in the analysis.
The transformed quantities (indicated by subscripts 012 for the zero, positive and
negative sequences respectively) are related to the phase quantities by
If the original phase admittance matrix [Yahc] is in its natural unbalanced state
then the transformed admittance matrix [ Y0 !2] is full. Therefore, current flow of one
sequence will give rise to voltages of all sequences, i.e. the equivalent circuits for the
sequence networks are mutually coupled. In this case, the problem of analysis is no
simpler in sequence components than in the original phase components.
From the above considerations it is clear that the asymmetry inherent in
transmission systems cannot be studied with any simplification by using the
symmetrical component frame of reference.
With the use of phase coordinates the following advantages become apparent:
(1) Any system element maintains its identity.
(2) Features such as asymmetric impedances, mutual couplings between phases and
between different system elements, and line transpositions are all readily considered.
(3) Transformer phase shifts present no problem.
Thus phase components are normally retained throughout the formation and
solution of the admittance matrices in the following sections, while sequence
components are used as an aid to interpretation of results.
Moreover, it will be shown in later chapters that iterative solutions involving static
converters can be more efficient in sequence components due to the absence of zero
sequence currents at the converter terminals.
where
[Zc] is the internal impedance of the conductors (Q.km- 1),
[Zg] is the impedance due to the physical geometry of the conductor's arrangement
(Q.km- 1),
[Ze] is the earth return path impedance (Q.km- 1), and
[Yg] is the admittance due to the physical geometry of the conductor (Q- 1km- 1).
In multiconductor transmission all primitive matrices (the admittance matrices of
the unconnected branches of the original network components) are symmetric and,
therefore, the functions that define the elements need only be evaluated for elements
on or above the leading diagonal.
The impedance due to the earth path varies with frequency in a non-linear fashion.
The solution of this problem, under idealised conditions, has been given in the form
of either an infinite integral or an infinite series [2].
38 3 TRANSMISSION SYSTEMS
where
;:c E [ZcJ
J(r,B) = W{la {P(r,B) +JQ(r,B)}
n
1• ..
I)
-_r:Pa p Du
Du = V,....(/-7;_+_17_)_
2 -+-c£2---ii for i #j
Du = 2h; for i = j
du
8;1 = arctan 1
- - 1- for i #j
. 1; + 1j
Bu =0 fori= j
co= 2nf(rad.s- 1)
h; = height of conductor i (m)
du = horizontal distance between conductors i and j (m)
!la =permeability of free space= 4 nx 10- 7 H m- 1
p = earth resistivity (Q.m).
where p = 1/ y'jw,u 0 a is the complex depth below the earth at which the mirroring
surface is located.
An alternative and very simple formulation has been recently proposed by Acha
[4], which for the purpose of harmonic penetration yields accurate solutions when
compared to those obtained using Carson's equations. The following alternative
formulation is used for the real and imaginary components of equation (3.14):
P= Se- fer (3.17)
Q= Ue- VeIn r (3.18)
where the se, te, ue and Ve coefficients are derived from accurate curve fitting of
Carson's equations. For the calculation of line parameters for practical tower
geometries, ground conductivities and frequencies of interest, r = 2 appears a
reasonable maximum value to be considered, e.g. r < 1.9 for p = 100 Q m,
f = 3000Hz, and d =120m. Larger values of r are required only for calculating
inductive coupling to distant cables. Coefficients calculated at steps of 0.5 in r
produce very accurate results, except for the first section which is subdivided into
two, i.e. r < 0.20 and 0.20:;:; r < 0.50. Moreover, the exercise is only valid for a
particular value of angle e, but fittings at 15 degree intervals, with linear
interpolation in-between have been found to be sufficiently accurate. The coefficients
are given in Tables 3.1 to 3.4.
Once the values of r and 8 have been computed, the nearest values in the tables are
selected and inserted in Equations (3.17) and (3.18).
An example of the curve fitting approach and its comparison with Dubanton 's
solution is illustrated in Figure 3.3.
The error criteria used here is the difference between Carson's result and the
approximate values of the real and imaginary part, relative to the magnitude of the
Carson impedance, i.e.
where
Ep and EQ =coefficients of error for the P and Q terms
Rc and Xc =resistance and reactance calculated using Carson's equation
RF and XF =resistance and reactance calculated using curve fitting
IZcl =magnitude of the Carson impedance
= jR~+X~
If the conductors and the earth are assumed to be equipotential surfaces, the
geometrical impedance can be formulated in terms of potential coefficients theory.
40 3 TRANSMISSION SYSTEMS
The self-potential coefficient '¥;; for the ith conductor and the mutual potential
coefficient '¥ iJ between the ith and jth conductors are defined as follows,
where r; is the radius of the ith conductor (m) while the other variables are as defined
earlier.
Potential coefficients depend entirely on the physical arrangement of the
conductors and need only be evaluated once.
For practical purposes the air is assumed to have zero conductance and
This term accounts for the internal impedance of the conductors. Both resistance and
inductance have a non-linear frequency dependence. Current tends to flow on the
surface of the conductor, this skin effect increases with frequency and needs to be
computed at each frequency. An accurate result for a homogeneous nonferrous
conductor of annular cross-section involves the evaluation of long equations based
on the solution of Bessel functions, as shown in Equation (3.23).
(3.23)
42 3 TRANSMISSION SYSTEMS
where
Xe =jJjw,UoO"c re
xi= jJjw,uoO"c ri
re = external radius of the conductor (m)
ri =internal radius of the conductor (m)
] 0 = Bessel function of the first kind and zero order
J~ = derivative of the Bessel function of the second kind and zero order
The Bessel functions and their derivatives are solved, within a specified accuracy,
by means of their associated infinite series. Convergence problems are frequently
encountered at high frequencies and low ratios of conductor thickness to external
radius i.e. (re - ri)/re, necessitating the use of asymptotic expansions.
3..:1 EVALUATION OF TRANSl\HSSION LINE PARAMETERS 43
0
----------
r 0' 15c 30G 45 75 90
Ep
------
c c
":v'" (i) ":v'"
Q_ Q_
2 -1 2 -1 (ii)
w (ii) w
-3 -3
-5 L_~~-~~-~~ -5 L_~~~~~-~~
0 2 4 10 12 0 4 6 10 12
r Parameter r Parameter
Figure 3.3 Relative errors in the calculation of the self-impedance of an earth return
conductor. (i) Fitting technique, (ii) Dubanton's method
A new closed form solution has been proposed based on the concept of complex
penetration, Semlyen [3]; unfortunately errors of up to 6.6% occur in the region of
interest.
To overcome the difficulties of slow convergence of the Bessel function approach
and the inaccuracy of the complex penetration method at relatively low frequency,
44 3 TRANSMISSION SYSTEMS
an alternative approach based upon curve fitting to the Bessel function formula has
been proposed by Acha [4]. Equation (3.24) is used to approximate the internal
impedance.
(3.24)
where
Zc E [ZJ
c= /J7R:c
Rdc =direct current resistance of the conductor ( O.km- 1)
Sc,tc,Uc,uc = curve fitting coefficients.
A maximum value of c = 300 and thickness to radius ratios of between 1.0 and 0.4
are considered in the derivation of the coefficients listed in Table 3.5, where linear
interpolation can be used.
Based on a similar criteria as that used for the case of ground impedances, an
assessment of the errors introduced by the curve fitting approach shows, in
3.4 EVALUATION OF TRANSMISSION LINE PARAMETERS 45
RESISTANCE INDUCTANCE
~u 2 ~ 2
u
:;; :;;
Q_
e
Q_
..:
e -2 -2
w w
-6 -6
Figure 3.4 Relative errors in the calculation of conductor impedance. (i) Fitting technique
(ii) Complex penetration method
50r-------------------------------------,
"""§ 40
:;;
g
.g; 30
.3 B
·c
Ol
~ 20
Q) A
g3' 10
0 r---.
11 13 15 17 19 21 23 25
Order of harmonic
Figure 3.5 The effect of skin effect modelling: curve A skin effect included; curve B, no skin
effect
Figure 3.4, a maximum error of 2% for the real and reactive components of the
internal impedance for a thickness ratio of 0.5.
The matrix [Zc] is diagonal, and normally computations for one phase conductor
and one earth-wire are sufficient.
For long lines, skin effect resistance (Rae/ Rdc) ratios and their effect on the
resonant voltage magnitudes are important. Because the series resistance of a
transmission line is a small component of the series impedance when the
transmission line is not at resonance, the harmonic voltages, shown in Figure 3.5,
do not change to any significant extent when skin effect is included. At resonance the
series resistance and shunt conductance become the dominant system components.
Changes in the series resistance magnitude change the voltage peaks but do not
affect the resonant frequency.
In Figure 3.5 the voltage calculated with skin effect is, at resonance, 50% higher
than without skin effect; these results correspond with a Rae/ Rdc ratio of 2. In a
single-phase model without ground return the ratio of voltages at resonance, with
46 3 TRANSMISSION SYSTEMS
and without skin effect, is the same as the skin effect ratio. In a three-phase model
the presence of shunt conductance and series resistance coupling between phases,
and the different resonant frequencies of the phases, reduces the resonant peak
voltages compared with single-phase modelling.
Skin effect is also taken into account in modelling the earth return as a conductor.
The depth of penetration of the earth currents decreases with an increase in
frequency or a decrease in earth resistivity. The series inductance decreases as a result
of these changes.
As an alternative to the rigorous analysis described above, power companies often
use approximations to the skin effect by means of correction factors. Typical
corrections in current use by the NGC(UK) and EDF(France) are given in Table 3.6.
The case of a single (phase) line is considered first to introduce the various concepts
involved in the simplest possible way, before attempting a generalization to the more
practical case of multiconductor transmission.
A transmission line consists of distributed inductance and capacitance, illustrated
in Figure 3.6, which represent the magnetic and electrostatic conditions of the line,
and resistance and conductance which represent the line losses.
Under perfectly balanced conditions, three-phase transmission lines can be
represented by their single-phase positive sequence models and nominal PI circuits.
For inclusion into an admittance matrix, it is necessary to use the admittances
between busbars, and from busbars to earth as in Figure 3.7.
3.5 SINGLE PHASE EQUIVALENT OF A TRANSMISSION LINE 47
X
6x
X= I
Figure 3.6 Distributed parameter transmission line. V, voltage; I, current; Z', series
impedance per unit distance; Y' shunt admittance per unit distance; l, transmission line length
Figure 3.7 Admittance model of transmission line, where GL + jBL = 1/(RL + jXL),
Yc = 1/jXc, XL= coL, and Xc = 1/wC. G, conductance; B, susceptance; X. reactance; co,
frequency (in radian per second)
For long lines a number of PI models are connected in series to improve the
accuracy of voltages and currents, which are affected by standing wave effects. For
example, a three-section PI model provides an accuracy to 1.2% for a quarter
wavelength line (a quarter wavelength corresponds with 1500 and 1250 km at 50 and
60 Hz, respectively).
As the frequency increases, the number of nominal PI sections to maintain a
particular accuracy increases proportionally, e.g. a 300 km line requires 30 nominal
PI sections to maintain the 1.2% accuracy for the 50th harmonic. However, near
resonance the accuracy departs significantly from an acceptable value.
The computational effort can be greatly reduced and the accuracy improved with
the use of an equivalent PI model derived from the solution of the second order
linear differential equations describing wave propagation along transmission lines
[6]. With reference to Figure 3.6
d 2 V(x)= Z 'Y'V()
X d2 !(~) = Y' Z' J(x) (3.25)
d 0
x~ dx
where Z' = r + j2nfL is the series impedance per unit length and Y' = g + j2nfC is
the shunt admittance per unit length.
The solution of wave Equations (3.25) at a distance x from the sending end of the
line is:
48 3 TRANSMISSION SYSTEMS
where y = -JZ!Yi = oc + jf3 is the propagation constant and V; and V,. the forward
and reverse travelling voltages, respectively.
Depending on the problem in hand, e.g. if the evaluation of terminal quantities
only is required, it may be more convenient to formulate a solution using two-port
matrix equations.
Considering a homogeneous multiconductor line of length l,
Vs = V; + V,. (3.28)
Is= (1/Z0 )(V;- V,.) for x = 0 (3.29)
where Z0 = VZ'JYi is the characteristic impedance
and
V R = exp( -yl) V; + exp(yl) V,. (3.30)
IR = (1/Z0 )[exp(-yl)V;- exp(yl)V,.] for x = l. (3.31)
The following relationships can be written for the circuit of Figure 3.8
z
s R
vs y1 ,I VR
0
Z=Z0 sinhy I
Y,=Y,= i; tanh~
Equating the last two equations with Equations (3.34) and (3.35)
1 + ZY1 = 1 + ZY2 = cosh(y/) (3.40)
Z = Z 0 sinh(yl) (3.41)
I I
I I
-200 I I
\ I
\ I
I I I
-300 \
\
I I
I I
\
I I
\ I I
-400 '.!..../
Figure 3.10 Impedance versus frequency for the equivalent PI model (skin effect included)
30r---~--------------------------------------------~
\
\
\
5 \
\
E ''
'
::J
~20
<J)
' '
u
c '-..... ..._ .::_Zo coth af, asymptote
--- ---
0
...........
a3a. 15
.f
10
The lower asymptote is small in value and slowly increases with frequency, while
the upper asymptote decreases from an infinite value as frequency increases. For
large frequencies these two asymptotes approach a value equal to the characteristic
impedance.
Due to the standing wave effect of voltages and currents on transmission lines, the
maximum value of these are likely to occur at points other than at the receiving end
or sending end busbars. These local maxima could result in insulation damage,
overheating or electromagnetic interference. It is thus important to calculate the
maximum values of currents and voltages along a line and the points at which these
occur.
Knowing the receiving end current and voltage, for each harmonic frequency, the
current and voltage at any point on the line can be calculated for each frequency by
using the following equations [8]:
where xis the distance from the receiving end, IRis the receiving end current, VR is
the receiving end voltage, and ZR = VR/IR· The points on the transmission line at
which these are maximum, are obtained by considerations of the currents and
voltages as forward (incident) and backward (reflected) travelling waves with respect
to the receiving end.
For example. consider the current Equation (3.45). The incident current at the
receiving end is
52 3 TRANSMISSION SYSTEMS
(3.48)
The angles associated with these currents, at any point along the line, are given by
e+ = e"k + f3x, (3.49)
where et
8R. are the angles of the current at the receiving end.
The current will be a maximum for e+ equal to e-. Thus
e"k + f3x = eR - f3x (3.50)
or
eR- e"k (3.51)
X= 2{3
The current will also have local maximum at intervals of one half wavelength
along the line.
While the total r.m.s. voltage and current (over the fundamental and all
harmonics) are of greatest importance, the location of the maximum total r.m.s.
voltage and current will most likely be dominated by that harmonic which is closest
to a resonant frequency of the system.
(a) (b)
Figure 3.12 (a) Three-phase transmission series impedance equivalent and (b) three-phase
transmission shunt impedance equivalent
3.6 MUL TICONDUCTOR TRANSMISSION LINE 53
With respect to Figure 3.12, the following equation can be written for the series
impedance equivalent of phase a:
where
(3.53)
and substituting
(3.54)
giVes
or
(3.58)
and writing similar equations for the other phases and earth wire, the following
matrix equation results:
Usually we are interested only in the performance of the phase conductors, and it
is more convenient to use a three-conductor equivalent for the transmission line.
This is achieved by writing matrix equation (3.59) in partitioned form as follows:
(3.60)
From (3.60)
From Equations (3.60) and (3.62), and assuming that the earth wire is at zero
potential,
(3.63)
where
(3.64)
With reference to Figure 3.12(b), the potentials of the line conductors are related to
the conductor charges by the matrix equation [9]
(3.65)
(3.66)
where [P~bJ is a 3 x 3 matrix which includes the effects of the earth wire. The
capacitance matrix of the transmission line of Figure 3.12 is given by
(3.67)
ybb
Jk
Half shunt Half shunt
admittance yaa
Jk
(a)
r/
Zaa Zab Zac
i k
~
Zca Zeb Zec
--- L__
vfr ~
(b)
(c)
Figure 3.13 Lumped PI model of a short three-phase line series impedance: (a) full circuit
representation; (b) matrix equivalent; (c) using three-phase compound admittances
[Z]-1 + [Y]/2
[~~~] [
-[Z]-1 (3.68)
6 X 1 6x6 6X1
This forms the element admittance matrix representation for the short line
between busbars i and k in terms of 3 x 3 matrix quantities.
56 3 TRANSMISSION SYSTEMS
When two or more transmission lines occupy the same right of way for a
considerable length, the electrostatic and electromagnetic coupling between those
lines must be taken into account.
Consider the simplest case of two mutually coupled three-phase lines. The two
coupled lines are considered to form one subsystem composed of four system
busbars. The coupled lines are illustrated in Figure 3.14, where each element is a
3 x 3 compound admittance and all voltages and currents are 3 x 1 vectors.
The coupled series elements represent the electromagnetic coupling while the
coupled shunt elements represent the capacitive or electrostatic coupling. These
coupling parameters are lumped in a similar way to the standard line parameters.
With the admittances labelled as in Figure 3.14, and applying the rules of linear
transformation for compound networks, the admittance matrix for the subsystem is
defined as follows:
It is assumed here that the mutual coupling is bilateral. Therefore Y21 = yT 12 , etc.
The subsystem may be redrawn as in Figure 3.15. The pairs of coupled 3 x 3
compound admittances are now represented as a 6 x 6 compound admittance. The
matrix representation is also shown. Following this representation and the labelling
3.6 MUL TICONDUCTOR TRANSMISSION LINE 57
r;; Yzz
6x6
[~ 1 y33
T
y34
[~,]
y55
T
y56
[~zl
6X1 y34 y44 6X6 Ys6 y66 6X6 6 X 1
(a)
[~:) [Y.,~r:]
Figure 3.15 A 6 x 6 compound admittance representation of two coupled three-phase lines:
(a) 6 x 6 matrix representation; (b) 6 x 6 compound admittance representation
of the admittance block in the figure, the admittance matrix may be written in terms
l
of the 6 x 6 compound coils as
Busbar A
-IA1
-
JB1 Busbar B
::I -- [A2
- I::
1s2
not distinct then the admittance matrix as derived from Equation (3.70) must be
modified. This is considered in the following section.
The admittance matrix as derived above must be reduced if there are different
elements in the subsystem connected to the same busbar. As an example, consider
two parallel transmission lines as illustrated in Figure 3.16.
The admittance matrix derived previously related the currents and voltages at the
four busbars A1, A2, B1 and B2. This relationship is given by
(3. 71)
(3.72)
similarly
(3.73)
(3.74)
The required matrix equation relates the nodal injected currents, /A and IB, to the
voltages at these busbars. This is readily derived from Equation (3.71) and the
conditions specified above. This is simply a matter of adding appropriate rows and
columns, and yields
(3. 75)
where [YAB] is the required nodal admittance matrix for the subsystem.
It should be noted that the matrix in Equation (3.71) must be retained as it is
required in the calculation of the individual line currents.
3.6 l\IUL TICONDUCTOR TRANSMISSION LINE 59
There is no direct way of calculating sinh or tanh of a matrix, thus a method using
eigenvalues and eigenvectors, called modal analysis is employed [11].
The basis of such a method for a multiconductor line is as follows: Consider the
second order linear differential equations describing wave propagation along a single
transmission line (section 3.5.1). These can be expanded for the multiconductor line
in the form of Equation (3.77), where the matrices are of order m, the number of
phases involved:
(3.77)
(3.78)
It should be noted that in this case the matrix products [Z'][Y'] and [Y'][Z'] are
not equal, except in special cases [11]. By transforming phase voltages to modal
voltages, and by choosing the proper transformation matrix [Tv], Equation (3.75)
can be changed to
(3.79)
where [A] is now a diagonal matrix; the elements of [A] are the eigenvalues of the
matrix product [Z'][Y'], and the transformation matrix [Tv] is the matrix of
eigenvectors of that matrix product. Equation (3.78) can be diagonalized as well,
with the same diagonal matrix [.1], i.e.
(3.80)
With the diagonalized Equations (3.79) and (3.80), an m-phase line can now be
studied as if it consisted of m single-phase lines, similar to the symmetrical
component approach, except that the zero, positive and negative sequence networks
now become the mode-l, mode-2 and mode-3 networks. In each mode, the single-
phase long line series impedance and shunt admittance of Figure 3.6 are used. The
propagation constant of each mode is simply
where}'; is the ith eigenvalue or ith element in [A]. The modal series impedance and
shunt admittance are not directly available, but must be computed from
(3.82)
with both modal matrices being diagonal. [Y;,odel may no longer be purely imaginary
even though only shunt capacitance is modelled. This will depend on how the
transformation matrices were normalised. For steady-state analysis at one particular
frequency, this causes no problems. Once Zseries and Yshunt have been calculated for
each mode, the representation in phase quantities is easily obtained by transforming
back, with
(3.83)
becoming the values of the equivalent PI model which will accurately represent the
untransposed line.
In expanded form, the following are expressions for the series impedance and
shunt admittance of the equivalent PI model [12]:
[Z]
EPM
= l[Z'][M] [sinhy/]
yf
[M]_ 1 (3.84)
where lis the transmission line length, [Z]EPM is the equivalent PI series impedance
matrix, [M] is the matrix of normalised eigenvectors,
sinhy 1 /
0 0
Ytl
sinhy 2l
0 0
[ sinhyl] hl (3.85)
yl
sinh}jl
0 0
Yi
and y1 is the jth eigenvalue for J/3 mutually coupled circuits. Similarly
Order of harmon1c
Figure 3.17 Comparison of the equivalent and nominal PI transmission line models at 5Hz
intervals: curve A, three sections; curve B, six sections; curve C, equivalent PI
7.58m 7.58m
:J
• •
Cascading of nominal PI circuits requires a large number of sections for long lines
at higher frequencies, to achieve acceptable accuracy. The equivalent PI model
avoids the problems of determining the number of sections needed and round-off
error that accumulates in this situation.
Computer derivation of the correction factors for conversion from the nominal PI
to the equivalent PI model, and their incorporation into the series impedance and
shunt admittance matrices, is carried out as indicated in the structure diagram of
Figure 3.19. The LR2 algorithm of Wilkinson and Reinsch [13] is used for accurate
calculations in the derivation of the eigenvalues and eigenvectors.
I Colculale equivalent PI
senes 1mpedonce and
I
shunt odm1ttonce matrices
I I I I
Form motri~ Calculate the Calculate the Calculate
product IY1rz'J eigenvalues and solut1on for dio~anol correction
/2 /3
v.1 ~!- ~L-
-
I /6 -V6 r--
5
~· r----- 1
where y;,. is the mutual admittance between primary coils, y;;, is the mutual
admittance between primary and secondary coils on different cores, and y~; is the
mutual admittance between secondary coils.
If a tertiary winding is also present, the primitive network consists of nine (instead
of six) coupled coils and its mathematical model will be a 9 x 9 admittance matrix.
The interphase coupling can usually be ignored (e.g. the case of three single-phase
separate units) and all the primed terms are effectively zero.
The connection matrix [C] between the primitive network and the actual
transformer buses is derived from the transformer connection.
By way of example consider the Wye G-Delta connection of Figure 3.21. The
following connection matrix applies:
3.7 THREE-PHASE TRANSFORMER MODELS 63
vo
p
v, 1 0 0 0 0 0 vap
Vz 0 1 0 0 0 0 ~
v3 0 0 0 0 0 vcp
(3.88)
v4 0 0 0 1 -1 0 VAs
Vs 0 0 0 0 -1 VB
s
v6 0 0 0 -1 0 Vi
or
[Y]NODE =
If the primitive admittances are expressed in per unit the upper right and lower left
quadrants of matrix (3.91) must be divided by .J3 and the lower right quadrant by 3.
Then, in the absence of interphase coupling the nodal admittance matrix equation of
the Wye G-delta connection becomes:
64 3 TRANSMISSION SYSTEMS
tE ~
ypp yps
s ysp Yss s
[I P J ['s]
~ /
[',I r [ ypp J
[ yps J
....-----...
[ ysp]
[Yss] r[, :
Figure 3.22 Two-winding three-phase transformer as two coupled compound coils
where y is the transformer leakage admittance in per unit, bearing in mind the
modification suggested earlier to take into account the increase of resistance with
frequency.
In general, any two-winding three-phase transformer may be represented by two
coupled compound coils as shown in Figure 3.22 where [Ysp] = [Yps]T.
If the parameters of the three phases are assumed balanced, all the common three-
phase connections can be modelled by three basic submatrices. The submatrices
[Ypp],[Yps], etc, are given in Table 3.7 for the common connections in terms of the
following matrices:
Shunt reactors and capacitors are used in a transmission system for reactive power
control. The data for these elements are usually given in terms of their rated
megavolt-amps and rated kilovolts. The equivalent phase admittance in per unit is
calculated from these data.
The coupled admittances to ground at bus k are formed into a 3 x 3 admittance
matrix as shown in Figure 3.23, and this reduces to the compound admittance
representation indicated. The admittance matrix is incorporated directly into the
system admittance matrix, contributing only to the self-admittance of the particular
bus.
While provision for off-diagonal terms exists, the admittance matrix for shunt
elements is usually diagonal, as there is normally no coupling between the
components of each phase.
Consider, as an example, the three-phase capacitor bank shown in Figure 3.24. A
3 x 3 matrix representation similar to that for a line section is illustrated.
[t:bc]
k
'•l,,tk
c
"k
"lk
b
>1.., lka
1
[v:bc]
yaa Yab yac
[v:bc]
Yba ybb Ybc [Ykk]
Figure 3.23 Representation of a shunt element: (a) coupled admittance, (b) admittance
matrix, (c) compound admittance
66 3 TRANSMISSION SYSTEMS
1/ iXc
i
1/ iXc
-L- -L... _L,_
1/ iXc
I
Figure 3.24 Representation of a shunt capacitor bank
,a
I
~
lb
I
,c
I ~
(a)
(b)
(c)
Figure 3.25 Representation of a series element: (a) coupled admittances: (b) admittance
matrix; (c) compound admittance
3.9 UNDERGROUND AND SUBMARINE CABLES 67
V']=[[U]
[ I, [Y,"] [U]
]x[Vr]
-Ir
(3.93)
Series elements are connected directly between two buses and for modelling purposes
they constitute a subsystem in the network subdivision.
A three-phase coupled series admittance between two busbars i and k is shown in
Figure 3.25, as well as its reduced nodal admittance matrix (Figure 3.25(b)) and
compound admittance (Figure 3.25(c)).
The series capacitor, used for transmission line reactance compensation, is an
example of an uncoupled series element; in this case the admittance matrix is
diagonal. For a lumped series element, the ABCD parameter matrix equation is:
(3.95)
where
Loop 1
Loop 2
Loop 3
X
Extemal
Similarly
and
where
z~heath-mutual : mutual impedance (per unit length) of the tubular sheath between
inside loop 1 and the outside loop 2.
z~rmour-mutual : mutual impedance (per unit length) of the tubular armour between
the inside loop 2 and the outside loop 3.
Z 'insulation · f1 l ~"outside
= JW ~ n-- in n;m (3.100)
2n rinside
3.9 UNDERGROUND AND SUBMARINE CABLES [14,15] 69
If the insulation is missing, e.g. between armour and earth, then Z' insulation = 0.
The internal impedances and the mutual impedance of a tubular conductor are a
function of frequency, and can be derived from Bessel and Kelvin functions.
I W.f..l
Zwhe-mutuat = 2n mq mr D (3.10lc)
where
mr= JK 1-1 s 2
(3.102)
~ (3.1 03)
mq= vl.\.~
with
K = Sn.l0- 4 ff..l,.
(3.104)
R'de
S=Cj_ (3.105)
r
q = inside radius
r = outside radius
R~c = de resistance in Qjkm
The only remaining term is Z~arth-inside in Equation (3.97) which is the earth return
impedance for underground cables, or the sea return impedance for submarine cables.
The earth return impedance can be calculated approximately with equation (3.101a)
by letting the outside radius go to infinity. This approach, also used by Bianchi and
Luoni [15] to find the sea return impedance is quite acceptable considering the fact
that sea resistivity and other input parameters are not known accurately.
70 3 TRANSMISSION SYSTEMS
Equation (3.95) is not in a form compatible with the solution used for overhead
conductors, where the voltages with respect to local ground and the actual currents
in the conductors are used as variables. Equation (3.95) can easily be brought into
such a form by introducing the appropriate terminal conditions, namely with
VI = Vcore - Vsheath /1 =/core
V2 = V.,heath - Vannour /2 = /core + r,heath
and V3 = Varmour h = /core + /sheath + /armour
Equation (3.95) can be rewritten as
(3.106)
where
l[ l[ l
where c; = 2n £0 er/l11 (rjq). Therefore, when converted to core, sheath and armour
quantities,
dlcore/dx Y]
-Y; 0
Vcore
- [ df,healhjdx = - Y'l Y' + Y2 - Y2 Vshealh (3.109)
dlarmour / dx 0 -Y2 Y2 + Y) Varmour
where Y] = jwh. If, as before, Vsheath = Varmour =zero, equation (3.109) reduces to
(3.110)
Therefore, for frequencies of interest, the cable per unit length harmonic
impedance, Z', and admittance. Y', are calculated with both the zero and positive
3.10 EXAMPLES OF APPLICATION OF THE MODELS 71
NGC 400, 275 (Based on 2.5 sq.in. h): 1.5 0.74 R 1 (0.267 + 1.073Vh)
conductor at 5 in. spacing
between centres)
132 h):2.35 R 1 (0.187 + 0.532Vh)
EDF 400. 225 h):2 0.74 Rl (0.267 + 1.073Vh)
150, 90 h):2 R 1 (0.187 + 0.532 vii)
sequence values being equal to the Z in Equation (3.107), and the Y' in Equation
(3 .11 0), respectively.
In the absence of rigorous computer models, such as described above, power
companies often use approximations to the skin effect by means of correction
factors. Typical corrections used by the NGC(UK) and EDF(France) are given in
Table 3.8.
A 230 km 220 k V line of flat configuration is used as the first test system; the
parameters of this line are shown in Figure 3.27. A three-dimensional graphic
representation is used to provide simultaneous information of the harmonic levels
along the line. At each harmonic (up to the 25th harmonic). one per unit positive
sequence current is injected at the Islington end of the line. The voltages caused by
this current injection are, therefore, the same as the calculated impedance, i.e. V +
gives Z++• V_ gives Z+- and V 0 gives Z+o (the subscripts +. -. 0 refer to the
positive. negative and zero sequences, respectively).
Figures 3.28, 3.29 and 3.30 illustrate the effect of two extreme cases of line
termination (at Kikiwa), i.e. the line open-circuited and short-circuited, respectively.
The difference in harmonic magnitudes along the line are due to standing wave
effects and shifting of the resonant frequencies caused by line terminations.
Figure 3.28 indicates the existence of high voltage levels at both ends of the open-
circuited line at the half wavelength frequencies. The 25th harmonic clearly
illustrates the standing wave effect, with voltage maxima and minima alternating at
quarter of the wavelength intervals.
At any particular frequency, a peak voltage at a point in the line will indicate the
presence of a peak current of the same frequency at a point about a quarter
wavelength away. This is clearly seen in Figure 3.29.
When the line is short-circuited at the extreme end, the harmonic current
penetration is completely different. as shown in Figure 3.30(a). The high current
levels at the receiving end of the line are due to the short-circuit condition. Figure
3.11 shows that the resonant maxima decrease as frequency increases. However, this
72 3 TRANSMISSION SYSTEMS
7. 58m 7.58m
Figure 3.27
•-:7" /. -;/. -;7'//7/7- : r7/ ~/'/7 ~/'7/: 5
m1
Conductor information for the Islington to Kikiwa line: conductor type, Zebra
( 54/3.18 + 7/3 .18); length, 230 km; resistivity, 100 Q m
Order of hormon1c
Figure 3.28 Positive sequence voltage versus frequency along the open-ended Islington to
Kikiwa line
Order of harmonic
Figure 3.29 Positive sequence current along the open-ended line for a 1 per unit positive
sequence current injection at Islington
does not appear to be the case in Figure 3.30(a). The reason is that the points plotted
correspond only to harmonic frequencies and resonances do not fall exactly on these
frequencies; i.e. the peak-magnitudes at non-harmonic frequencies can be greater
than the values plotted in the figure.
20
18
16
14
12
10
8
6
4
2
0
(a)
~::> 20
18
~
~ 16 ~
14
OJ
::> 12 1
D
c 10 l
C!'
0 8t
E 6/
c
~
u
::> ~~
0
(b)
10
09
08 lslrnglon
07
0.6
0.5
04
03
02
0.1
00
1 5 9 13 17 21 25
Order of harmonic
(c)
Figure 3.30 Sequence currents along the short circuited line for a 1 per unit positive sequence
current injection at Islington: (a) positive sequence current; (b) negative sequence current; (c)
zero sequence current
w
0.
18
aJ
D
:J 15
c 12
0'
0
E 9
aJ
0'
6
0
3
0
> 0
L__L__"---_____L--_____j _ _ _ L ____ _J_ _ _ _ _ L _ _ _ _ L _
Figure 3.31 Three-phase resonant frequencies of the Islington to Kikiwa line with a 1 per unit
positive sequence current injection (skin effect included)
frequency. However, the use of the three-phase algorithm to model the Islington-
Kikiwa unbalanced transmission line shows that the resonant frequencies are
different for each phase. In this case, the spread of frequencies can be seen from
Figure 3.31 to be approximately 6 Hz.
The different magnitudes of the resonant frequencies (up to 30%) of the three
phases, partly explains the problems encountered with correlating single-phase
modelling and measurements on the physical network. The results clearly indicate
that harmonics in the transmission system are unbalanced and three-phase in nature.
Figure 3.32 Line geometry of a double circuit line. Length. 167 km: earth resitivity. I 00 Q m:
two conductors per bundle; bundle spacing, 0.45 m; conductor. 30/3.71 + 7;3.71
3.10 EXAMPLES OF APPLICATION OF THE MODELS 75
(a)
10
4
-;::-
c::>
a; 2
:;
u"'
2 0
c
0
..
0
E
u
§ 10 (b)
u
"'
Cl.
.E 8
9 11 13 15 17 19 21 23 25
Order of harmonic
Figure 3.33 Sequence impedance magnitude versus frequency: (a) double circuit coupled line;
(b) two single circuit lines
zero sequence injections, respectively. The figure also displays the coupling between
the positive sequence and the other sequence networks, i.e. Z+- and Z+o·
Results for the case of a coupled line are illustrated in Figure 3.33(a) and those of
two single circuit lines in Figure 3.33(b ).
The magnitudes and resonant frequencies of the Z++ and Z+o impedances are not
affected by the modelling of mutual coupling. However, the level of Z+- has
changed substantially at resonance showing appreciable imbalance. Moreover, the
magnitude and resonant frequency of Zoo is very different in the two cases.
Robinson [20] reported that telephone interference caused by zero sequence
currents did not coincide with high levels of power system harmonics. This can partly
be explained by the different resonant frequencies of the Z++ and Zoo observed.
It is generally accepted that, for practical distances, the effect of line asymmetry
can be eliminated by the use of phase transpositions dividing the line into three, or
multiples of three equal lengths. Accordingly, transpositions are often used in long
distance transmission as a means of balancing the fundamental frequency
impedances of the line.
In fundamental frequency studies the effect of transpositions is generally
accounted for by averaging the distributed parameters of the three transposed
sections and using them in a single nominal or equivalent PI-circuit. Such a method,
however, assumes that the line geometry is perfectly symmetrical at all points,
whereas the transpositions occur at two discrete distances, at different points on the
standing wave.
The series impedance and shunt equivalent matrices are combined into one
l
admittance matrix that represents the transposed section, i.e.
where
The admittance parameters for the individual sections are then transformed into
A', B', C', D' parameters, such that they can be cascaded, i.e.
(3.112)
(3.113)
where Is, IR, Vs and VR are vectors of a size determined by the number of coupled
conductors.
Applying a partial inversion algorithm to Equation (3.113), the following matrix
of inverse hybrid parameters is obtained.
(3.114)
or
(3.115)
Two different cases are of interest and will be used in the following sections. The
first relates to a harmonic voltage excited open-ended line, specified as Vs = 1 p.u.
3.10 EXAMPLES OF APPLICATION OF THE MODELS 77
f
line line
I I--[1Rl
~I V5=1 IR=O Vs=O IR=1
(a) (b)
Figure 3.34 Diagram of terminal conditions (a) voltage source and open-ended line; (b)
current source and short-circuited line. Reproduced from [21] by permission of lEE
and IR = 0. This case produces the highest voltage harmonic levels and must,
therefore, be considered for design purposes. The second important case is the
harmonic current excited short-circuited ended line, specified as Vs = 0 and
IR = 1 p.u. which is more likely to be of practical interest. These two cases are
illustrated by the simplified diagrams of Figures 3.34(a) and (b), respectively.
Figure 3.35 Test Line. Details of the test line: The test line, shown in Figure 3.35, is of flat
configuration without earth wires and the main parameters are: Nominal voltage= 500 kV;
Conductor type: Panther (30/3.00 + 7/3.00 ACSR); Resistivity= 100 Ojm; Equal distances
between transpositions and the natural impedance matrix. Reproduced from [21] by
permission of lEE
78 3 TRANSMISSION SYSTEMS
12
10
j
<i
.&a
·c:-"
g>6
.
E
0\4
~
0
>
_-:"': .. ····
1-------""-
050 350 650 950 1250 1550
distance, km
a
12
10
"<i
.&a
-~c
g'6
_84
.
E
0
>
Figure 3.36 Fundamental frequency three-phase voltages at the end of the test line (open-
circuited) versus line distance. Reproduced from [21] by permission of lEE
(a) without transpositions (b) with transpostions
--R __ y
------ y ------ B
........ B ........ R
The test line is fed from 1 p.u. voltage sources at fundamental and harmonic
frequencies. It is realised that the presence of I p.u. harmonic voltage sources is
unrealistic, but such a figure provides a good reference for comparability between
the effects at different frequencies. The expected harmonic voltage levels are likely to
be about 1-3% of the fundamental and, therefore, the results plotted in later figures
should be scaled down proportionally.
30
:; 24
.,.ci.
-~ 18
c
0>
.
0
E 12
2
0
> 6
distance, km
a
24
§, 12
..
0
E
1\
.8 6 / ~
0
>
Figure 3.37 Three-phase third harmonic voltages at the end of the test line (open-circuited)
versus line distance). See Key for Figure 3.36. Reproduced from [21] by permission of lEE
wavelength (i.e. 750 km at 50 Hz). For distances approaching the quarter wavelength,
the transposed line produces considerably greater imbalance than the untransposed.
Although such transmission distances are impractical without compensation, the
results provide some indication of the behaviour to be expected with shorter lines at
harmonic frequencies. Such behaviour is exemplified in Figure 3.37 which
corresponds to the case of a line excited by 1 p.u. 3rd harmonic voltage. However,
the results plotted in Figure 3.37 obtained at 50 km intervals, are not sufficiently
discriminating around the points of resonance. Thus the region of resonant distances
has been expanded in Figure 3.38 to illustrate more clearly the greatly increased
imbalance caused by the transpositions. The resonant peaks of the three phases
occur at very different distances, e.g. Figure 3.37(b) shows 50 km diversity between
the peaks. Therefore, for a given line distance the resonant frequencies will vary, thus
increasing the risk of a resonant condition.
It is also interesting to note the dramatic voltage amplification which occurs for
electrical distances equal to the first quarter wavelength. Figure 3.38 shows a peak of
35 per unit for the 3rd harmonic when the line is 500 km long and the 5th harmonic
peak (not shown) reaches 45 per unit at about 300 km.
Figure 3.37(a) and (b) also show the effect of attenuation with distance, i.e. the
considerable reduction of the peaks at resonant distances at the odd quarters of
wavelength other than the first. Such attenuation is caused by the series and shunt
resistive components of the equivalent PI-model.
80 3 TRANSMISSION SYSTEMS
30 30
::i24
ci
,; ...
..,·/\ \
\
.:'1
,
u
.I .
.318
·c ....... \
"'
0
E I
I \
.\
~ 12 I .,_
.'"'
:l ··\~.
~
~~oo~--4~3~o----~4s~o~--7
49~o~--=s2~o----~ss~o
distance, km distance-, km
(a) (b)
Figure 3.38 Results of Figure 3.37 expanded in the region of resonance. For Key see Figure
3.36. Reproduced from [21] by permission of lEE
Figure 3.39(a) Standing waves along a line of quarter wavelength (i) voltage wave (ii) current
wave. Reproduced from [21] by permission of lEE
Figure 3.39(b) Third harmonic standing waves along a line of three-quarter wavelengths. (i)
voltage wave (ii) current wave. Reproduced from [21] by permission of lEE
3.10 EXAMPLES OF APPLICATION OF THE MODELS 81
Line Loaded If an ideal (uncoupled and unattenuated) line is loaded with its
characteristic impedance, the sending end voltage will be sustained throughout the
line, provided that the phase angle difference between the sending and receiving end
voltages is kept below 45o (or 750 km at 50 Hz). To assess the effectiveness of
transpositions with loading, the test line was loaded with its characteristic impedance
calculated at 50 Hz. It must be noted that in a coupled multiconductor line such
impedance is a matrix, of which only the diagonal elements are being used for the
loading. Furthermore, the three diagonal elements are different and are also
frequency dependent. We cannot therefore expect to see the uniform 1 p.u. voltage
predicted by conventional theory.
Results for the fundamental frequency, plotted in Figure 3.40, illustrate that the
effectiveness of transpositions is limited to distances up to about 750 km. For longer
lines, similarly to the open-ended line case, transpositions are not effective, although
the per unit voltage imbalance of the loaded transposed line (Figure 3.40(b)) is
greatly reduced as compared with that of the open line (Figure 3.36(b)).
Up to the first quarter wavelength the effect of natural (fundamental frequency)
loading on the harmonic voltages is very similar to the fundamental frequency. For
this particular loading conditions the effectiveness of transpositions is limited to
distances of about 350 and 200 km for the 3rd and 5th harmonics, respectively.
Beyond those distances the transposed lines produce higher levels of imbalance.
Subsequent harmonic peaks are seen to reduce rapidly with loading. By way of
example, the 5th harmonic voltages without and with transpositions are shown in
Figures 3.41(a) and (b), respectively.
The harmonic behaviour of a loaded transmission line without and with
transpositions is illustrated in Figures 3.42(a) and (b), respectively. This figure
displays the variation of 5th harmonic voltage at the receiving end of a 250 km line
with one per unit voltage injection at the sending end. The level of imbalance of the
untransposed line (Figure 3.42(a)) shows a gradual increase up to about the natural
load (1 p.u. admittance) and very little change thereafter. In contrast, Figure 3.42(b)
1.6 1.6
~
c. ~
,; 1.2
]
·c
--~
a; 1.2
~
·c:
~----~·~~=-=~~ . . .. ···················
g' 1.8
E
~
Jl"' 0.4
."'
go.e
E
.8 0.4
~ ~
~~0--~3~50~--~65~0--~9~50~~1~25~0--~1~550 ~~0--~3~50~--6~5~0--~9~50~~1~25~0---1~550'
distance, km distance, km
(a) (b)
Figure 3.40 Fundamental frequency three-phase voltages at the end of the test line (loaded
with the characteristic impedance). For Key see Figure 3.36. Reproduced from [21] by
permission of lEE
82 3 TRANSMISSION SYSTEMS
1.6 1.6
s
-- ........ _____ __,....--- "' '
0
Q_
d. - ........
~1.2 /-~ .& 1.2 .:-,.;:;,
~ .. / ,'!
c
'-
go. a "'E 0.8
0
E
~
5!"' 0.4
0
>
0 5~0----~35~0----76~50~--~9~50~--~,2~50~--~,5~50 ~Lo----~35~o----~6~5o~--~9~5o~--~,2~5o~--~,5~5o
distance-, km distance, km
(a) (b)
Figure 3.41 Three-phase fifth harmonic voltages at the end of the test line (loaded with the
characteristic impedance). For Key see Figure 3.36. Reproduced from [21] by permission ofiEE
Figure 3.42 Three-phase fifth harmonic voltages at the end of a 250 km test line versus
loading admittance (referred to the characteristic admittance). For Key see Figure 3.36.
Reproduced from [21] by permission of IEE
illustrates a dramatic increase in the voltage imbalance as the load reduces from the
natural level (1 p.u. admittance) to the open circuit condition. A qualitative
justification for this behaviour has been made in Figure 3.39. As the line load
increases above the natural level, Figure 3.42(b) shows that effectiveness of the
transposition increases.
Considering the relatively insignificant levels of harmonic voltage excitation
expected from a well-designed system, the resulting voltage distortion in a
transposed or untransposed load line is not expected to cause problems, except
when the line is lightly loaded. With harmonic current excitation the situation may
be quite different, and its effect is examined next.
Effect of Transpositions with Current Excitation The main cause of power system
harmonic distortion is the large static power converter, such as used in HYde
transmission and in the metal reduction industry. Because of their large de
smoothing inductance compared to the ac system impedance, static converters can
be considered as current sources on the ac side and voltage sources on the de
side [25].
3.10 EXAMPLES OF APPLICATION OF THE MODELS 83
3.2 3.2
~ ~
c. c.
.,; 2.4 .g 2.4
"
"0
"
·~ 1.6
E
c
g1.6
E
Io.a
~
..'l"' 0.8
~ ~
Figure 3.43 Three-phase third harmonic voltages caused by I p.u. third harmonic current at
the point of harmonic current injection. For Key see Figure 3.36. Reproduced from [21] by
permission of lEE
Thus, the harmonic modelling of a long transmission line feeding a static converter
is basically that of Figure 3.34(b), i.e. a harmonic current source at the receiving end
of the line with the sending end shorted to ground through a relatively low
impedance.
The harmonic voltages at the point of current harmonic injection follow the same
pattern as those of the open circuit line with harmonic voltage excitation. This is
clearly illustrated in Figure 3.43 for a case of 3rd harmonic current injection.
Similarly to the voltage excited open line, substantial voltage distortion results when
the line length is close to a quarter wavelength, although the imbalance caused by
transpositions is less pronounced in the case of current injection. Figure 3.43
indicates that even 1% of harmonic current injection can produce 3 or 4% voltage
harmonic content at the point of harmonic current injection, which is above the
levels normally permitted by harmonic legislation.
As the harmonic order increases, the line experiences higher levels of voltage
distortion. For example, the case of 5th harmonic current injection, illustrated in
Figure 3.44 shows a peak voltage of about 4.5% in one of the phases. However, in
this case the transposed line is seen to reduce considerably the harmonic peaks for
the quarter wavelength distance line.
4.0
~
1!
t·
.I
~ 3.2 3.2
a. l(
.,; I(
".~2.4 !i
l I
c
"'E
0
~1.6
g
0
>
0.8
distance, km
(b)
Figure 3.44 Three-phase fifth harmonic voltages caused by 1 p.u. third harmonic current at
the point of harmonic current injection. For Key see Figure 3.36. Reproduced from [21) by
permission of lEE
84 3 TRANSMISSION SYSTEMS
While the 5th harmonic current is normally eliminated by filters, this is not the
case for non-characteristic orders like the 3rd, which will then distort the supply
waveform and, in the absence of equidistant firing control, may increase further the
production of 3rd harmonic current [26].
where
-J
0 0
[jhowoC 0 hw 0 Lc
-j
[Yc] = 0 jhowoCc 0 l [YL] = 0
hw 0 Lc
0 (3.117)
0 0 jh 0 W 0 Cc
-J
0 0
hw 0 Lc
v· v·· v
'
~4
Figure 3.45 Equivalent circuit of VAR compensated transmission line
3.10 EXAMPLES OF APPLICATION OF THE MODELS 85
Harmonic Voltage Excitation The test line is a 1000 km of the same configuration
as in section 3.10 (see Figure 3.35).
The addition of shunt inductive compensation effectively increases the
characteristic impedance and thus reduces the load that causes the optimum voltage
profile.
For the positive sequence shunt admittance values of the test system, a standard
load flow programme was used to derive the optimal discrete shunt inductances
required to provide a practically constant voltage along the line at the fundamental
frequency.
However, the addition of shunt inductance isolates the line from ground (reducing
its ability to act as a low pass filter) and thus reduces its ability to dampen
harmonics.
The results, plotted in Figure 3.46 correspond to an open-ended line and show that
while the fundamental frequency voltage profile is good, the line performance at
harmonic frequencies is worse than without compensation. In particular, the level of
the receiving end voltage for second harmonic injection has increased dramatically.
In the absence of compensation, the natural load of the line under consideration is
approximately 950 MW, but the maximum nominal loading planned is 1450 MW,
i.e. 1.5 times the natural load.
For this loading condition Figure 3.47 shows the effect of a combined
compensation scheme, consisting of shunt and series capacitors. It is noted that
shunt capacitors tend to amplify harmonic distortion at the compensation points,
while having the opposite effect elsewhere.
Harmonic Current Excitation In this case, a one percent harmonic current was
injected at the receiving end of the line.
The effect of shunt inductive compensation in the harmonic behaviour of the
unloaded line is shown in Figure 3.48; again the second harmonic shows the greater
amplification.
The results of combining series and shunt capacitive compensation for the case of
a heavily-loaded line are shown in Figure 3.49.
The magnitudes of the harmonic voltages for the loaded line are smaller than
those of an open-ended line.
86 3 TRANSMISSION SYSTEMS
6.0
z
5.0
4.0
3.0
2.0
1.0
0
1 2 4 5
y
----x=l
4 ------x=2
- - -x=3
.... ----
-·-·-x=4
3 -·-·-·-·-X= 5
-z
Figure 3.46 Fundamental and harmonic voltage levels along the unloaded line with shunt
inductive compensation and 1 per unit voltage at the sending end. x- harmonic order; y -
voltage magnitude; ;;-line position w.r.t. point of harmonic injection
----x=l
3 ------x=2
- - -x==3
-----x~4
-·-·-·-·-X= 5
2
- ------------------
Figure 3.47
oL----L----~--~-----L----L----L----~--~
-z
Standing waves along the heavy-loaded line with series as well as shunt
capacitive compensation
3.10 EXAMPLES OF APPLICATION OF THE MODELS 87
- - - - - - x=2
6 - - - x=J
-·-·-x=4
-·-·-·-·- X= 5 -----------
4
Figure 3.48
compensation
-z
Harmonic voltage levels along the unloaded line with shunt inductive
------x=2
6 - - -x=3
-·-·-x=4
-·-·-·-·-X= 5
Figure 3.49
-z
Harmonic voltage levels along the loaded line with series as well as shunt
capacitive compensation
Due to the limited number of phases and switching devices, the de output voltages at
converter stations contain considerable ripple. Under perfectly symmetrical ac
supply and switching conditions, the voltage ripple consists only of twelve pulse
related harmonics. In practice, however, ac system imbalance and asymmetrical
firing may lead to other frequencies being present in the de voltage waveforms.
It is, therefore, necessary to derive the full spectrum of harmonic admittances of
the de link. For generality the test system, based on the New Zealand system,
contains overhead lines and submarine cables and each of them must be represented
by the frequency dependent models derived in earlier sections. The New Zealand
HYde link, illustrated in Figure 3.50, consists of six major subsystems, (ii), (v), (vi),
(vii), (viii) and (ix) and three auxiliary components (iii), (iv) and (x). The distances of
the main transmission components are:
88 3 TRANSMISSION SYSTEMS
(i) (i)
Derivation of Parameters [29] Considering the perfectly balanced self and mutual
impedance of the line, the HV de scheme is best analysed using sequence networks (of
positive and zero sequence). With reference to the circuit diagram of Figure 3.52, the
positive sequence current is defined as the average current flowing from node 1 to
node 2, i.e.
(3.118)
and the zero sequence component is the average current flowing into the network,
and returning by some other path, i.e.
The relationships between the phase and sequence components of current and
voltage are:
(3.120)
3.10 EXAMPLES OF APPLICATION OF THE MODELS 89
I I
1 3
t
"':"
vJ v3
~ v2 V4 ~
- I
2 I
4
_.;,.
(3.121)
where
(3.122)
(3.123)
(3.124)
Therefore
(3.126)
Similarly
(3.127)
The impedance and admittance matrices for each of the sections must be
transformed in ABCD parameter matrices in order to cascade the section. For the
circuit of Figure 3.53, the ABCD parameter transformation equations are as follmvs:
3.10 EXAMPLES OF APPLICATION OF THE MODELS 91
A= 1 + Y2Z
B=Z
(3.128)
c= + y2 + z yl y2
yl
D = 1 + Y1Z.
For the situation under consideration, the scalar quantities Z, Y1 and Y 2 must be
replaced with the appropriate matrices, i.e.
Y1 = Y2 = O.S[YplwseJ
(3.129)
Z = [ZphaseJ·
The final form of the ABCD parameter matrix for a particular section 1s,
therefore:
The sections may then be cascaded by simply multiplying their respective ABCD
matrices together, i.e.
[A]
[ [C) [B]] [[A 1] [B 1] ] [[A"] [B"]] (3.132)
[D) = [C 1] [D 1] X ... X [C) [D"] .
[Y1Jl = [D][Br 1
[Yn] = [C]- [D][Br 1[A]
(3.133)
[Y2d = -[Br 1
[Yn] = [Br 1[A].
yl2 yl3
y22 y23
(3.134)
y32 y33
y42 y43
Thus the transmission network of Figure 3.52 may be represented by the following
equation:
In order to calculate the impedance seen from the sending end terminals, the 4 x 4
admittance matrix must first be reduced to a 2 x 2 matrix by eliminating the
receiving end voltages and currents. Since the receiving end converter can be
approximated by a voltage source, it appears as a short circuit to harmonic
frequencies. It may, therefore, be assumed that
l
This leads to the 2 X 2 admittance matrix in terms of /1' vl and h v2 only:
Y11 - Ya Y12 - Yb
Y'= [ (3.136)
Y21 - Yc Y22 - Yd
y _ (Y13 + Y14)(Y31 + Y41)
where
a - (Y33 + Y34 + Y43 + Y44)
Moreover, if the midpoint between the two poles at the far end is earthed, then
v3 = v4 = 0, and the above matrix simplifies to:
3.10 EXAMPLES OF APPLICATION OF THE MODELS 93
(3.137)
Impedance Plots Figure 3.54 compares the impedance plots, as seen from the
Benmore terminal, obtained both with and without ancillary components. As can be
readily seen, the effect of including the ancillary components is quite dramatic at
high frequencies, shifting all the resonances to the left, and markedly altering the
magnitude of the very first peak.
Figure 3.55 compares the same impedance plots as seen from the Haywards end.
Although the resonances are again shifted to the left, the effect of the cable, as
discussed in the previous section, tends to mask out the standing wave effects
occurring in the SI inland line section. Of note here, however, is the 600Hz damping
circuit, with the resonant point modified by the surrounding components. This
resonant point may also be observed from the Benmore end, in Figure 3.54 although
its magnitude is reduced by the masking effect of the cable.
10000
~
~
0
9000 I,, II II
I
II
~ 8000 II II
...
::>
..... ,, II II
~
'"
::E
7000
,, ,, I I
~ 6000
,, I
0::
II I II
"""'g_ 5000 I I
.s / \;
4000
I
3000
I II
2000 y \f
1000
0
0 100 200 300 400 500 600 700
Frequency (Hz)
10000
~
E
.c 9000
0
-o"'
...,;:l
8000
·~
"""'"
7000
::E
II
"'
u 6000
"'"
"C II
~ 5000
~ II
4000
I
3000
2000
1000
0
0 100 200 300 400 500 600 700
Frequency (Hz)
Figure 3.55 Harmonic impedances seen from Haywards
- - with ancillary components
------ without ancillary components
3.11 Summary
Most of the chapter has been devoted to the multiconductor transmission line, as the
most influential component in harmonic analysis.
A frequency-dependent equivalent PI model has been described suitable for
computer implementation. The formulation of earth path and conductor impedances
taking into account skin effect has been carried out and then used to derive simpler
solutions based on tabulated coefficients.
Frequency-dependent models have also been derived for the transformers and
VAR compensating equipment. These models, combined with the line equivalent Pis
are used to derive the network admittance matrix.
A detailed analysis of high voltage underground/submarine transmission systems
has also been made, based on current cable technology.
Several practical examples of application of the models to ac and de transmission
have been included.
3.12 References
1. Fortescue, C L, (1918). Method of symmetrical co-ordinates applied to the solution of
polyphase networks. Trans. AlEE, 37(2), 1027-1140.
3.12 REFERENCES 95
., Carson, J R, (1926). Wave propagation in overhead wires with ground return. Bell System
Technical Journal, 5, 539-554.
3. Deri, A, et a!. (1981). The complex ground return plane, a simplified model for
homogeneous and multi-layer earth return. IEEE Transactions on Power Apparatus and
Systems, PAS-100, 3686--3693.
4. Acha, E (1988). Modelling of power system transformers in the complex conjugate
harmonic space. Ph.D. Thesis, University of Canterbury, New Zealand.
5. Semlyen, A and Deri, A, (1985). Time domain modelling of frequency dependent three-
phase transmission line impedance. IEEE Transactions on Power Apparatus and Systems,
PAS-104, 1549-1555.
6. Kimbark, E W, (1950). Electrical Transmission of Power and Signals, John Wiley, New
York.
7. Elgerd, 0, (1971). Electric Energy Systems Theory: An Introduction, McGraw Hill, New
York.
8. Shultz, R D, Smith, R A and Hickey, G L, (1983). Calculation of maximum harmonic
currents and voltages on transmission lines, IEEE Trans, PAS-102, 817-821.
9. Chen, M S and Dillon, WE, (1874). Power system modelling, Proc. lEE, 62, 901.
10. Arrillaga, J, Arnold C P and Harker, B J, (1983). Computer Modelling of Electrical
Power Systems, John Wiley & Sons Ltd, London.
11. Galloway, R H, Shorrocks, W Nand Wedepohl, L M, (1964). Calculation of electrical
parameters for short and long polyphase transmission lines, Proc. lEE, 111, 2051-2059.
12. Bowman, W I and McNamee, J M, (1964). Development of equivalent PI and T matrix
circuits for long untransposed transmission lines, IEEE Trans, PAS-84, 625-632.
13. Wilkinson, J Hand Reinsch, C, (1971). Handbook for Automatic Computations, Vol II,
Linear Algebra, Springer-Verlag, Berlin.
14. Dommel, H W (1978). Line constants of overhead lines and underground cables, Course
E.E 553 notes, University of British Columbia.
15. Bianchi, G and Luoni, G, (1976). Induced currents and losses in single-core submarine
cables, IEEE Trans, PAS-95, 49-58.
16. Densem, T J, (1983). Three phase power system harmonic penetration, Ph.D. Thesis,
University of Canterbury, New Zealand.
17. Arrillaga, J, Densem, T J and Harker, B J, (1983). Zero sequence harmonic current
generation in transmission lines connected to large converter plant, IEEE Trans, PAS-
102, 2357-2363.
18. Hesse, M H, (1966). Circulating currents in parallel untransposed multicircuit lines. I
Numerical evaluations. IEEE Trans, PAS-85, 802-811.
19. EHV Transmission Line Reference Book, Edison Electric Institute (1968), New York.
20. Robinson, G H, (1966). Harmonic phenomena associated with the Benmore-Haywards
HVdc transmission scheme, New Zealand Engineer, 21, 16--29.
21. Arrillaga, J, eta/., (1986). Ineffectiveness of transmission line transpositions at harmonic
frequencies, Proc. lEE, 123C(2), 99-104.
22. Semlyen, A, Eggleston, J F and Arrillaga, J, (1987). Admittance matrix model of a
synchronous machine for harmonic analysis, IEEE Trans, PWRS-2(4), 833-840.
23. Arrillaga, J and Duke, R M, (1979). Thyristor-controlled quadrature boosting, Proc lEE,
126(6), 493-498.
24. Stemmler, H and Guth, G, (1982). The thyristor-controlled static phase-shifter, Brown
Boveri Rev, 69(3), 73-78.
25. Arrillaga, J, (1983). High Voltage Direct Current Transmission, Peter Peregrinus Ltd,
London.
26. Ainsworth, J D, (1967). Harmonic instability between controlled static converters and ac
networks, Proc. lEE, 114, 949-957.
96 3 TRANSMISSION SYSTEMS
4.1 Introduction
When the calculation of the harmonic sources can be decoupled from the analysis
of harmonic penetration a direct solution is possible. In such case, the expected
voltage levels (or the results of a fundamental frequency load flow) are used to
derive the current waveforms of the non-linear components. A Fourier analysis is
then applied to obtain the harmonic currents injected by each non-linear
component into the linear system.
The simplest model involves a single harmonic source and performs a single phase
harmonic analysis. This model is commonly used to derive the system harmonic
impedances at the point of common coupling as required in filter design.
In general, the network may contain several harmonic sources and may be
unbalanced. The derivation of the harmonic voltages and currents will, therefore,
require a three-phase harmonic flow solution.
Most power system non-linearities manifest themselves as harmonic current
sources, but sometimes harmonic voltage sources are used to represent the distortion
background present in the network prior to the installation of the new non-linear
load; moreover, some power electronic applications apply voltage rather than
current distortion. A comprehensive algorithm of general applicability should have
the following capabilities:
• Provide graphical interfaces for the specification and display of the system to be
analysed, and for the post-processing of the information obtained from the
analysis.
98 4 DIRECT HARMONIC SOLUTIONS
Yn y.,
~~ Y;, yik Y;n
[Yj] = (4.1)
where
(4.2)
Figure 4.1 shows a case of two three-phase harmonic sources and an unbalanced
ac system. The current injections, i.e. lu, - hh and !411 - h1, can be unbalanced in
magnitude and phase angle.
The system harmonic voltages are calculated by direct solution of the linear
equation
(4.3)
where [Yh] is the system admittance matrix.
4.2 NODAL HARMONIC ANALYSIS 99
(4.4)
[h] = [Y3] [V3]
[h] = [Y2J (V2l·
The injected currents at most ac busbars will be zero, since the sources of the
harmonic considered are generally from static converters. To calculate an admittance
matrix for the reduced portion of a system comprising of just the injection bus bars,
the admittance matrix is formed with those buses, at which harmonic injection
occurs, ordered last. Advantage is taken of the symmetry and sparsity of the
admittance matrix [2], by using a row ordering technique to reduce the amount of
off-diagonal element build-up. The matrix is triangulated using Gaussian
elimination, down to but excluding the rows of the specified buses.
The resulting matrix equation for an n-node system with n - j + 1 injection
points is
0 vj
0
0 V;-1
Y;; Yin (4.5)
I; vi
0
Yni Ynn
In Vn
As a consequence, Ii ... In remain unchanged since the currents above these in the
current vector are zero. The reduced matrix equation is
(4.6)
and the order of the admittance matrix is three times the number of injection
bus bars. The elements are the self- and transfer-admittances of the reduced system as
100 4 DIRECT HARMONIC SOLUTIONS
viewed from the injection busbars. Whenever required, the impedance matrix may be
obtained for the reduced system by matrix inversion.
Reducing a system to provide an equivalent admittance matrix, as viewed from a
specific bus, is an essential part of filter design.
By making I 1 = Ioo per unit, h = l-120s per unit, h = 1120° per unit, the matrix
equation
(4.7)
(4.8)
(4.9)
(4.1 0)
The harmonic currents injected by the harmonic voltage sources are then found by
solving:
(4.11)
With this formulation some extra processing is required to obtain the reduced
admittance matrix, which is not generated as part of the solution.
4.3 HARMONIC IMPEDANCES 101
For the purpose of determining the network harmonic admittances the generators
and transformer can be modelled as a series combination of resistance and inductive
reactance, i.e.
1
Y 117 - for the transformer (4.13)
- R.Jh + jX1h
·c:
"to
a.
0.31
.....
2 4 6 8 10 12 14
Frequency ( Hz x 50 )
The harmonic impedances seen from primary transmission system buses are greatly
affected by the degree of representation of the distribution system and consumers'
loads fed radially from each busbar.
A typical simplified dominant configuration of a distribution feeder is shown in
Figure 4.3. Generally, the bulk of the load fed from distribution feeders is located
behind two transformers downstream. Thus, to calculate the harmonic impedances
seen from the high voltage primary transmission side it may be sufficient to use a
discrete model of the composite effect of many loads and distribution system lines
and transformers at the high voltage side of the main distribution transformers;
typically the 110 kV in a system using 400 and 220 kV transmission.
The aggregate nature of the load makes it difficult to establish models based
purely on theoretical analysis. Attempts to deduce models from measurements have
been made [3, 4] but lack general applicability. Utilities should be encouraged to
develop data base of their electrical regions, with as much information as possible to
provide accurate equivalent harmonic impedances for future studies.
The following guidelines are recommended for the derivation of distribution
feeder equivalents [5].
220kV
60kV 60kV llkV
I I I
Figure 4.3 Typical distribution feeder
4.3 HARMONIC IMPEDANCES 103
(2) Various models of predominantly motive loads have been suggested using
resistive-inductive equivalents, their differences being often due to the
boundary of system representation. A detailed analysis of the induction
motor response to harmonic frequencies, leading to a relatively simple model,
is described in Section 4.3.3.
(3) Modelling the power electronic loads is a more difficult problem because,
besides being harmonic sources, these loads do not present a constant R, L, C
configuration and their non-linear characteristics cannot fit within the linear
harmonic equivalent model. The presence of system non-linearities has been
discussed earlier. In the absence of detailed information the power electronic
loads are often left open-circuited when calculating harmonic impedances.
However, their contribution to the harmonic current flow may have to be
considered when the power ratings are relatively high, such as arc furnaces,
railway locomotives, etc.
An alternative approach to explicit load representation based on detailed
information, is the use of empirical models derived from measurements.
In studies concerning mainly the transmission network the loads are usually
equivalent parts of the distribution network, specified by the consumption of active
and reactive power. In this situation the load model A suggested by reference [6] can
be used. The A model is a parallel connection of inductive reactance and resistance
whose values are:
v2 v2
R = (4.15)
X= j (O.lh + 0.9)Q -;-::c-::-:----::--::-:--=-
(O.lh + 0.9)P
(4.16)
Xml = XI + X2 = Xo (4.17)
(4.18)
4.3 HARMONIC IMPEDANCES 105
1000
I
I
I
I
I
I .
I I
I n._l 11
I .I ,
11/i
l If~
tl rl
~ I
.~~ ,A N
I' )I
I
z I
100 f--------1
I
IV
1\ •
II
0 I lr .'I 'I
h ~ I IWI I _'j II
m
:J I ir I \If I \_\ I'
J 1 I \If \\ I
. l I
I I l1 II ~
I
y
I I ,\ I
I
I
!
I
I
II
i J I
\VII
I rl
I
~I
I
I
I
I
10
I I I I
= ~ ~ ~ 1~ 1= 1~ 1~ 1~ ~
f [Hz]
where
At harmonic frequencies:
(4.19)
(4.20)
106 4 DIRECT HARMONIC SOLUTIONS
90
80
'\
\
70
,,
60
j\' ,- ' I I
(\
II
~
I
I I
I I ~
I I
II\.
50 I
.il
I ~ I I I
40 '
30
J, I I J
I
~
I
I I
I
I
I I I
,Ill I I I
20 I
I I
I
~ I:
I
I I I
10 I I I II I
I
I~:~: ~~I
u :; ~
I
,I
0 I II I I
I
e -10 I II
'lil'
I'
'
I
I
I
I I
I
I ' I
~v I I I
I
J
I
:~
I I
-20 I I
I
I I
I I
I \
I
:I
I
-30 \{ I
-40 I I
I
I
iI I
I
I I
I
I
I
-50
i
I I I '.t \II I
I
I
I
-60
I ,I
I
-70
II I I I 1~ I I
I I I l'-"-1 I
-80 ' I
-90
I
J
0 200 400 600 800 1 000 1200 1400 1600 1 BOO 2000
f [Hz]
Figure 4.5 Load effect on network harmonic impedance phase. Reproduced from [9] by
permission of CIGRE
where
ka,kh = correction factors to take into account skin effect in the stator
and rotor, respectively.
sh = apparent slip at the superimposed frequency, i.e.
(4.21)
co,.
I.e. sf ~ 1 - - for the positive sequence harmonics
' hws
w,.
sh ~ 1 + -- for the negative sequence harmonics
hws
4.3 HARMONIC IMPEDANCES 107
-r, xmh
~·
s
x,
kb =(±h-I)"
As the system harmonic admittances vary with the network configuration and load
patterns, large amounts of data are generated.
Considering the large number of studies involved in filter design, it is prohibitive
to represent the whole system with the same degree of detail for every possible
operating condition. The detail of components representation depends on their
relative position with respect to the harmonic source, as well as their size in
comparison with the harmonic source.
Any local plant components such as synchronous compensators, static capacitors
and inductors etc., will need to be explicitly represented.
108 4 DIRECT HARMONIC SOLUTIONS
As the high voltage transmission system has relatively low losses, it is also
necessary to consider the effect of plant components with large (electric) separation
from the harmonic source. It would thus be appropriate to model accurately at least
all the primary transmission network (i.e. using the models described in Chapter 3).
Moreover, due to the standing wave effect on lines and cables, a very small load
connected via a line or cable can have a dramatic influence on the system response at
harmonic frequencies.
It is recommended to consider the loads on the secondary transmission network in
order to decide whether these should be modelled explicitly or as an equivalent
circuit. If these loads are placed directly on the secondary side of the transformer,
their damping can be overestimated when using simple equivalents.
Increasing network complexity results in a greater number of resonant
frequencies. By way of an illustration [9], Figure 4. 7 shows the harmonic impedance
at a converter bus of a primary (400 kV) system with either 25, 232 or 1682 buses
included; the 25 bus case includes the nearest 400 kV lines terminated by equivalent
circuits plus the transformers and large generators in this area. The continuous thick
line shows the same information when the network representation consists of 1682
buses which include the complete 400 kV, 220 kV and 110 kV networks plus the
generators down to the 1 MV A size; however, it must be emphasized that the number
of buses is not the only relevant criterion for increased accuracy. The considerable
differences observed are due to the hand made formation of the small network while
the large network is produced automatically from the network data base without any
equivalencing. Because modern computers can handle the larger network in
reasonable times, the larger representation must be recommended as it gives
accurate results at any point in the network and only one model for the whole
network has to be maintained.
1000
--
fl.
.r
--
£.L.
.
---
l ./1
,A fA\·· lAf
1\
E
J:; A
~--,~ [}. .,,., ~ ... ~/JJ - '
'
0 100 --I-I --.:
,,
N
r ll '-
I ''f l1
v
l l~
10 ~
0 200 400 600 800 1000 1200 1400 1600 1800 2000
t [Hz}
Figure 4.7 Effect of size of system representation. Reproduced from [9] by permission of
CIGRE
4.3 HARMONIC IMPEDANCES 109
90MW
54MVAR
Figure 4.8 The lower South Island of New Zealand test system
110 4 DIRECT HARMONIC SOLUTIONS
zaa zac
150
-50
-250 -2t!IO
3!50 zbb
1!50
-eo
350 ~50
z ca zcc
150
-50
In the past, an impedance circle [11], as shown in Figure 4.10 encompassing all
evaluated harmonic impedances, was used for all harmonics together with computer
search techniques which maximized voltage distortion at a specified busbar. This
approach leads to unduly pessimistic filter designs, particularly at low order
harmonics. Besides, such an approach requires considerable computing and
engineering time which is often not available at the tendering stage.
The Annular Sector Concept The annular sector approach, illustrated in Figure
4.11, restricts the geometric area applicable to each harmonic by setting upper and
lower limits to the magnitude and phase of the harmonic impedance. Taking into
account all the relevant operating conditions, a comprehensive scatter plot is
produced for each harmonic on the impedance plane; all these points are then
encompassed by two circles and a sector and the resulting values of Z 1• Z 2, 01 and 02
are tabulated.
4.3 HARMONIC IMPEDANCES 111
X
X/Rmaxlnd
X/RmaxCap
Figure 4.10 Traditional boundary for ac network impedance. Reproduced from [9] by
permission of CIGRE
This approach was used in the design of the filters attached to the expansion of the
New Zealand HV de link and the information obtained is shown in Table 4.1.
The Discrete Polygon Concept In this case a distinction is made between low and
high harmonic orders. At the lower harmonics discrete points are obtained for the
different operating conditions as for the annular sector. Encompassing these points
by a polygon results in a set of polygons for each harmonic of interest.
At higher harmonic frequencies, e.g. 14th to 49th, the scatter of the R ± jX values
and hence the boundary of the encompassing polygons would become increasingly
large. Additionally, the impedances would begin to extend into the capacitive region
of the impedance plane. From detailed information of the particular system involved
it is possible to decide on the use of a realistic outer boundary with a single
geometrical shape without introducing an unacceptable degree of pessimism into the
filter design studies.
A computer technique is then used to search each polygon in turn to evaluate the
system impedance which maximizes voltage distortion at, or current injection into,
the point of common coupling.
This approach was used in the design of the ac harmonic filters on the 2000 MW
Cross-Channel HVdc scheme [4]. Individual search areas were defined for harmonics
1-13 as shown in Figures 4.12, 4.13 and 4.14 for 24 defined operating conditions as
follows:
In the case of the Cross-Channel scheme, a circle of centre 750 +joO. and radius
7500. (as shown in Figure 4.15) was considered sufficient to encompass all possible
impedance loci derived from the 24 operating conditions considered. These figures
indicate that the first harmonic to exhibit a resonance condition is the 13th, whereas
a generalized impedance circle approach would have allowed even low order
harmonics (2nd, 3rd) to exhibit resonance.
In this particular application a further refinement was introduced. Having chosen
the particular worst (resonance) condition from the polygon search areas, the
remaining system impedances for harmonic numbers 2 to 25 were chosen from a
number of tables of harmonic impedance, from the column which included the
impedance closest to the resonant impedance. For harmonic numbers greater than
25, the network impedance was chosen from the impedance circle of Figure 4.15 to
maximise the voltage distortion at each harmonic.
The calculated R ± jX values used in the polygons are the equivalent Thevenin
impedances of the entire network reduced to the Sellindge 400 kV busbar. These
include the harmonic impedances of individual plant items such as transmission
lines, generators, transformers, etc.
It must be understood that the quantitative impedance plots used in this scheme
cannot be taken as typical and used as a default option in other schemes. For
instance, in cases of ac networks with long EHV or UHV lines the first resonant
frequency may even occur below the second harmonic.
The discrete polygon approach provides a realistic way of representing the ac
network for the purposes of ac filter design. It avoids the pessimism of a
generalized approach using a single search area, and provides a technique which
provides acceptably quick solution times for the highly iterative task of filter
design.
4.3 HARMONIC IMPEDANCES 113
2 47.6 87 19.7 82
3 37.2 77 26.6 30
4 85.7 78 45.5 49
5 71.2 61 29.2 17
6 70.1 58 30.8 29
7 66.60 78 47.6 40
8 114.9 78 55.7 54
9 97.3 63 70.9 24
10 156.9 63 81.1 26
11 168.9 55 109.6 -38
12 93.6 48 24.1 -10
13 121.8 79 51.8 56
14 198.38 55 129.7 -19
15 117.7 43 35.5 -10
16 99.7 51 43.1 24
17 97.7 75 54.7 53
18 140.9 78 84.6 68
19 304.4 82 146.0 68
20 604.2 76 236.4 10
21 657.9 52 163.0 -69
22 291.2 0 65.9 -39
23 128.3 86 37.4 -35
24 146.5 23 72.6 0
25 204.8 71 62.0 58
26 341.9 59 157.8 51
27 525.1 52 200.3 1
28 1319.6 46 381.7 -61
29 460.9 -49 179.0 -72
30 97.0 0 56.0 -34
31 431.3 43 162.8 -34
32 449.8 -14 118.2 -53
33 372.6 5 107.6 -54
34 333.0 -22 82.4 -69
35 130.5 25 41.6 -5
36 238.2 8 102.8 -37
37 368.5 -13 210.3 -65
38 209.8 -0 136.8 -33
39 172.2 -40 114.4 -52
40 169.9 -73 88.7 -76
41 143.1 6 66.2 -12
42 149.6 -69 70.5 -79
43 145.0 0 69.2 -70
44 105.9 -53 52.0 -70
45 73.3 -58 37.5 -70
46 74.9 29 37.1 11
47 136.5 -55 90.6 -63
48 102.2 -65 41.8 -71
49 55.1 -60 28.4 -69
50 43.1 -2 35.6 -22
114 4 DIRECT HARMONIC SOLUTIONS
The main tasks of a direct solution are shown in Figure 4.16. Blocks (i) to (iii)
describe the input data requirements, which will be single or multiphase depending
on the information required. Block (iv) is commonly used to assess the voltage
harmonic content at the point of common coupling (PCC) of a new non-linear load,
100
90
80
70
60
j0 50
x
40
30
20
10
0
10 20 30 40 50
R(ohms)
Figure 4.12 Harmonic impedances for harmonic order 2 to 5. Reproduced from [9] by
permission of CIGRE
4.4 COMPUTER IMPLEMENTATION 115
200
180
160
140
-;;;-
120
E
.£
~ 100
X
80
60
40
20
0
20 40 60 80 100 120
R(ohms)
Figure 4.13 Harmonic impedances for harmonic order 6 to 9. Reproduced from [9] by
permission of CIGRE
300
250
200
E 150
...c
0
x 100
50
-50
0 40 80 120 160 200 240 280 320 360
R(ohms)
Figure 4.14 Harmonic impedances for harmonic order 10 to 13. Reproduced from [9] by
permission of CIGRE
as required for filter design, on the assumption that there are no other harmonic
injections in the system.
In single phase studies data entry is normally manual and the harmonic
parameters for each component are scaled from the fundamental frequency
parameters; therefore only one admittance matrix is constructed and solved at a
time. Since the structure of the harmonic admittance matrices is the same at all
frequencies, the elements only varying in size, the sparse techniques only require
updating the array containing the non-zero elements (but not the pointers), to form
the next harmonic admittance.
116 4 DIRECT HARMONIC SOLUTIONS
800
600
400
200
-;;;-
E
...c 0
0
x
-200
-400
-600
-BOO
Figure 4.15 Harmonic impedances for harmonic order 13, and envelope of harmonic
impedance locii for harmonic order 14 to 49. Reproduced from [9] by permission of CIGRE
When the effect of unbalance, and particularly the presence of zero sequence
current, needs to be determined, there is no alternative to three-phase modelling with
accurate representation of the frequency dependence of the transmission lines;
however, scaled versions of the fundamental frequency parameters are still used for
loads, generators and transformers.
Figure 4.17 shows the data files and data flow between a typical set of component
programs used in three-phase harmonic analysis. Before describing these programs
in detail it will be helpful to make some general observations.
The calculation of the electrical parameters of lines and cables takes up a
significant amount of computation. Moreover, the harmonic impedances and levels
alter with varying loading and configuration and many studies are normally
required. Therefore, the frequency dependent parameters of all components are
stored in a file so that they can be used as required for many simulations.
The electrical parameters of branches are calculated for all frequencies and stored in
a data file, and all the harmonic admittance matrices are constructed simultaneously.
This data is read sequentially once and the electrical parameters are added to the
appropriate admittance matrix as it is read. As all admittance matrices are constructed
simultaneously, memory constraints become important. The amount of memory
required limits the size of the arrays and this, in turn, constrains either the size of the
system or the number of harmonics to be modelled. This problem could be avoided if
each admittance matrix was read and solved individually. However, in the three-phase
case, this would require complicated file handling to allow only the correct frequency
data to be incorporated, and file rewind and reread for each harmonic considered.
GIPS Gathering system data used to be a time consuming process, which involved
entering numbers in specified columns of a text file. This process is simplified by a
graphical entry program such as GIPS (graphical interface for power systems). This
l____ Direct harmonic analysis - --- - -~
.(....
:,_.
Specify: I Read shunt Read transmis- Calculate ! Input harmonic Solve [I]= Calculate n
0
- frequency I capacitors, sion line and reduced system I current and [Y][V] for branch currents 2:::
"1::1
range I transformers, cable harmonic I voltage sources unknown for all c
....,
-harmonic I generators, data for all admittance I voltages and frequencies and I'T1
;;o
impedance I induction frequencies matrices by I currents at all output the
<'
busbars or II motors, simp
. Ie and include in matrix reduc- !
frequencies. results ;:;
r
harmonic source 1senes harmonic tion. Invert and I Output phase 1'1'1
information elements, filters admittance output the ! harmonic 2:::
1'1'1
1
1 matnces
I I I
Dat3
Harmonic Current
Injections & Harmonic
Voltage Sources
Programs
program collects the physical geometry of transmission lines and cables, as well as
the electrical parameters of the various components involved, and writes out files for
other data processing programs.
The use of windows facilities makes the graphical interface easy and intuitive to
use, due to the users familiarity with the windows environment. Menus are activated
with the help of push buttons across the top of the screen, and the various power
system components are displayed by icons on the left-hand side of the screen, with
the active icon in a depressed state.
Due to the limited amount of information that can be displayed on the screen, it is
essential that the drawing area is not limited by the bounds of the screen. This is
achieved by providing panning and zooming features. Moreover, due to the size and
complexity of a modern power system, it is impractical to have all the system
permanently on display, as this hinders visual interpretation of the information.
Instead, it is possible to view a part of the system, the selection terms made by
voltage level, area, or other user specified criteria.
It is impossible to draw a straight line with the mouse without moving slightly off
track; instead, the use of auto-routing and line straightening features allows the lines
to be entered quickly and perfectly straight.
Writing drivers for hardware is a tedious and time consuming task. Fortunately,
there are now commercially available graphics libraries, and a simple call to a
subroutine in the graphics library allows selection of the appropriate built-in drivers.
Moreover, new hardware devices can be accommodated quickly and easily. The
4.4 COMPUTER IMPLEMENTATION 119
purchase of an updated version of the graphics library allows immediate use of new
printers, graphics cards and plotters, with no changes required by the graphical
interface program. Another advantage is that the graphics library is available for
different platforms, hence allowing portability.
Easy entry of power system data is achieved by using pop-up windows, called
FORMs, which appear when a component is selected for editing. The FORM is a
table with the left column being a protected field identifying the data required, and
the right column being available for the user to enter the data. A help field in the
FORM also gives useful information to the user. In a power system there are
normally many components that are identical, and entering their parameters
individually would be laborious, therefore the ability to copy data from one to
another with a few key pushes is an important feature.
The modelling of multiple transmission lines with different terminations is
achieved by using multi-node junctions. Without these the graphical interface
program assumes that all circuits are terminated on the same busbar or junction.
With multi-node junctions the FORMs for all branches will have additional pop-up
FORMs allowing specification of internal node connections. It is also necessary to
modify the system diagram and data to carry out repetitive studies with individual
components taken out of service or reinstated, without re-entering the data. This is
simply achieved by selecting an out of service command, which is indicated by a
colour change in the selected component. While the component FORMs allow
system data modifications with great ease, modifying the system diagram is slightly
more complicated. Any component can be deleted at any time and the connectors
linking the deleted component to a busbar or junction will be deleted also.
Components can be picked up and moved around. However, connectors will not
move with them, they must be deleted and re-entered. As there is no data associated
with connectors this is a trivial task. Dragging and rubber-banding are only used in
drawing transmission lines, cables and connectors, as the extra code needed to
extend their use to other components is not warranted.
Although the graphical interface could have been added directly to the harmonic
analysis software, it is desirable to keep it separate, as shown in Figure 4.17. This is
because the software code required to draw the system and handle all the data
requires a large amount of memory by itself. If incorporated as part of the analysis
program it would further restrict the memory available and hence the size of system
that can be analysed. Another benefit is that it is easier to maintain groups of smaller
programs than one large monolithic program.
GIPS stores all the data and drawing instructions collected from the User in one
file under a name specified by the User. A typical screen is shown in Figure 4.18.
When data entry is completed GIPS creates the three files required for harmonic
analysis (default names are LINE.DAT, CABLE.DAT and COMP.DAT).
The geometry and conductor codes of the transmission lines are stored in the
LINE.DAT files, an example of which for the system drawn in Figure 4.18 is:
MANAPOURI220 INVERCARG220
0 0 1 1 1 1 1 1 152.90 50 51 0.00 12.50 31 220 1 2 1 10.1967
6.47 0.00 0.00 0.00 -6.47 0.00 4.61 5.91 RYB RYB ASS
120 4 DIRECT HARMONIC SOLUTIONS
INVERCARG220 TIWAI-~220
0 0 I I I I I I 24.30 50 51 0.00 12.50 0 220 I 9 I 1.1973
7.20 0.00 0.00 0.00 -7.20 0.00 0.00 0.00 RYB RYB ASS
TIWAI-~220 MANAPOURI220
0 0 I I I I I I 175.60 50 51 0.00 12.50 0 220 I 9 I 5.1972
7.20 0.00 0.00 0.00 -7.20 0.00 0.00 0.00 RYB RYB ASS
Similarly, the geometry and conductor loads of the cables are stored m the
CABLE.DAT files, an example of which is:
54/3.90+ 19/2.34 ACSR PHEASANT 35.10 628 14.20 0.04491 825 1000 1140 1270 1200 1320 n
0
54/3.18+7/3.18 ACSR ZEBRA 28.58 400 11.55 0.07009 610 750 845 940 890 980 ~
30/3.71 +7/3.71 ACSR GOAT 25.96 300 10.70 0.09010 525 640 725 805 765 840 "0
c:::
30/3.00 + 7/3.00 ACSR PANTHER 20.98 200 8.76 0.13790 405 495 560 620 590 650
...,
m
30/2.59 + 7/2.59 ACSR WOLF 18.14 150 7.47 0.18450 330 400 455 505 475 525 ~
26/2.57 + 7/2.00 ACSR PARTRIDGE 16.31 125 6.61 0.21320 300 365 410 460 435 480 ~
"0
26/2.54+7/1.91 ACSR COYOTE 15.88 125 6.40 0.22120 295 360 405 450 425 470 r'
BRAHMA 18.14 100 3.17 0.29170 270 330
m
16/2.86+ 19/2.48 ACSR 370 415 390 430 ~
7/4.39+7/1.93 ACSR HYENA 14.58 100 2.48 0.30700 255 315 355 395 m
6/4.72+ 1/4.72 ACSR HARE 14.17 100 2.44 0.30800 255 310 350 390
370
370
410
405
...,z
DOG 14.17 100 2.44 310
>
...,
6/4.72+7/1.57 ACSR 0.30570 255 350 390 370 405
PIGEON 12.75 80 1.83 0.33800 220 270 0
6/4.25 + 1/4.25 ACSR 300 340 320 355 z
12/2.59 + 7/2.59 ACSR SKUNK 12.95 60 1.83 0.45690 190 235 265 295 275 305
6/3.66+ 1/3.66 ACSR MINK 10.97 60 1.49 0.47290 180 220 250 280 265 290
37/3.66 COPPER COPPER 25.60 600 9.83 0.04600 720 880 995 1110 1050 1160
61/2.62 COPPER COPPER 23.55 500 9.09 0.05450 650 790 895 995 940 1040
37/2.62 COPPER COPPER 18.31 300 7.03 0.09000 470 570 645 720 680 750
continued
......
N
......
....
N
N
MEGA CABLE 5 2
0.001 0.022 0.0395 0.044 0.0475 0.0583
0.0653
1.68E-081. 2.20E-071. 1.8E-07 1.
4.1 0. 1. 2.3 0. 1.
1. 0. 1.
SUPER CABLE 5 3
0.001 0.022 0.043 0.044 0.0475 0.0583
0.0653
1.68E-081. 2.20E-071. 1.8E-07 1.
4.1 .5 1. 2.3 0. 1.
1. 0. 1.
SUPER CABLE2 5 4
0.001 0.022 0.043 0.044 0.0475 0.0583
0.0653
1.68E-081. 2.20E-071. 1.8E-07 1.
4.1 .5 1. 2.3 0. 1.
1. 0. 1.
SUPER CABLE2 5 5
0.001 0.022 0.043 0.044 0.0475 0.0583
0.0653
1.68E- 081. 2.20E-071. 1.8E-07 1.
4.1 .5 1. 2.3 0. 1.
1. 0. 1.
FILE END
The cable entry number is to enable cable.exe to use a data input file
created by GIPS as GIPS cannot at present pass the cable type strings
through to the output files. The cable entry numbers do not need to
be in order as Cable.exe scans the entire file.
Cable Type Case Number Cable Entry Number
Cond-Inner(m) Cond-Outer Sheath-Inner Sheath-Outer Armour-Inner Armour-Outer
Cable-Radius
Con-res( ohm-m)Con-permeabil She-resistivity She-permeabil Arm-resistivity She-permeabil
c2s dielectr c2s loss angle c2s permeabil s2a dielectric s2a loss angle s2a permeabil
a2ext dielec a2ext loss ang a2ext permea
Case I = Conductor and insulator
Case 2 = Conductor, insulator and sheath
Case 3 = Conductor, insulator, sheath and insulator
Case 4 = Conductor, insulator, sheath, insulator and armour
Case 5 = Conductor, insulator, sheath, insulator, armour and insulator
124 4 DIRECT HARMONIC SOLUTIONS
TL The Transmission Line program uses data contained in two files; the line
geometry and conductor code file created by GIPS (default name LINE.DAT) and a
look-up table giving details of the conductors (called CONDUCT.DAT). The
CONDUCT.DAT file is shown in Table 4.2.
Taking into account line geometry and conductor types, this program calculates
the electrical parameters of the transmission lines at all relevant frequencies using the
equivalent PI model. The results are used to create an output file called TL.DAT. As
described in Chapter 3, the TL program models skin effect and earth return as well
as different three-phase line geometries with one or more earth wires.
Where a transmission line consists of several sections with different geometry, TL
calculates the electrical parameters for each section independently. The harmonic
analysis programs read each section data and convert it to ABCD parameters; then
the sections are cascaded by multiplying them together to obtain one equivalent PI
for the complete length. To save space TL.DAT is an unformatted file and hence a
sample of it cannot be shown.
1LL 2150.
11 0.01085802 0.101948251 2 0.00705311 0.041111611 3 0.00705355 0.03455950
21 0.00705311 0.04111161 2 2 0.01085814 0.10194774 2 3 0.00705311 0.04111161
31 0.00705355 0.03455950 3 2 0.00705311 0.04111161 3 3 0.01085802 0.10194825
11 0.00000497 0.085401431 2 0.00000161 -0013720431 3 0.00000234-0.00557466
21 0.00000161 -0.01372043 2 2 0.00000465 0.08724183 2 3 0.00000161 -0.01372043
31 0.00000234 -0.00557466 3 2 0.00000161 -0.01372043 3 3 0.00000497 0.08540142
TIWAI--220 MANAPOURI220
21 25
1LL 2 50.
11 0.04225127 0.255678421 2 0.01687885 0.109818171 3 0.01687820 0.09412731
21 0.01687885 0.10981817 2 2 0.04226027 0.25566909 2 3 0.01687885 0.10981817
31 0.01687820 0.09412731 3 2 0.01687885 0.10981817 3 3 0.04225127 0.25567842
11 0.00013120 0.206362291 2 0.00000516 -0.032995901 3 0.00002781 -0.01331885
21 0.00000516 -003299590 2 2 0.00013033 0.21077853 2 3 0.00000516 -0.03299590
31 0.00002781 -0.01331885 3 2 0.00000516-003299589 3 3 0.00013120 0.20636232
1LL 2 100.
11 0.05609922 0.482902381 2 0.03093567 0.196274791 3 0.03093806 0.16552891
21 0.03093567 0.19627479 2 2 0.05612547 0.48284081 2 3 0.03093567 0.19627479
31 0.03093806 0.16552891 3 2 0.03093567 0.19627479 3 3 0.05609922 0.48290238
11 0.00070339 0.41735372 1 2 0.00015925 -0.06572535 1 3 0.00026892 -0.02597622
2 1 0.00015925 -0.06572535 2 2 0.00067195 0.42608768 2 3 0.00015925 -0.06572535
31 0.00026892 -0.02597622 3 2 0.00015925 -0 06572534 3 3 0.00070339 0.41735381
1LL 2 150.
11 0.06525607 0.680698991 2 0.04056496 0.263320981 3 0.04057887 0.21875937
21 0.04056496 0.26332098 2 2 0.06529872 0.68051624 2 3 0.04056496 0.26332098
31 0.04057887 0.21875937 3 2 0.04056496 0.26332098 3 3 0.06525607 0.68069899
11 0.00207110 0.637876331 2 0.00071223 -0.097959131 3 0.00101316 -0.03732961
21 0.00071223 -0.09795913 2 2 0.00192903 0.65073633 2 3 0.00071223 -0.09795912
31 0.00101316-003732961 3 2 0.00071223 -0.09795912 3 3 0.00207110 0.63787627
END DATA
HARM_Z This program reads the SYS.DA T file produced by INTER and creates
a series of files with the information required for the analysis of the system harmonic
impedances. The specific files produced are:
• ZFULL.DAT, with the values of the system harmonic impedances seen from a
specified bus.
• ZSEQ.DAT, with the values of the sequence components of the system harmonic
impedances seen from a specified bus.
• ZRED.DAT, with the values of the system harmonic impedances seen from a
specified bus and reduced to only phase values assuming balanced currents.
• HYCONV.DAT, with the values of the system admittance matrix seen from a
specified bus.
HARM_AC This program also reads the SYS.DAT file, and creates a series of files
with the information required to carry out current and voltage harmonic penetration
studies. The specific files produced are:
-!.-! COi\iPUTER IMPLEMENTATION 127
• HVC.DA T. with the values of the harmonic voltages (in phases R, Y, Bas well as
their sequence components) present at each of the system buses.
• HLC.DAT, with the values of the harmonic currents (in phases R, Y, Bas well as
their sequence components) present in the system
HARMAC HARMAC creates the same files that both HARM_Z and HARM_AC
create. Although HARMAC only allows harmonic current injections, and cannot
handle as large a system as HARM_Z and HARM_AC, it is ideal for implementing
more sophisticated models requiring the calculation of the system harmonic
admittance and harmonic penetration in the same study.
An example is the accurate assessment of HVdc system harmonics, with the
converter represented as a harmonic Norton equivalent, the Norton admittance
being itself a function of the ac system harmonic impedance. In this case, a two-pass
approach is adopted, the first pass being the derivation of the system harmonic
impedances seen from the converter terminal bus, the second pass is a harmonic
penetration study incorporating the converter's harmonic admittances (derived in
Chapter 10).
4.4.4 Post-Processing
z-mat For a specified branch obtains the values of the harmonic impedances,
resistances and reactances, as well as the impedance locus.
v-mat Contains the phase and sequence values of the harmonic voltages at all the
network busbars.
i-mat Contains the phase and sequence values of the harmonic currents of each
line.
128 4 DIRECT HARMONIC SOLUTIONS
MATLAB is then run, incorporating the output GIPS software such that a menu
of options is displayed. The post-processing software should have the following
features:
• Perform a variety of display formats
• 2-D plots (X-Y plots)
• 3-D plots
• Loci plots
• Scatter plots.
• View harmonics of selected parts of the system with selectable ranges.
• Plot and compare data from different files (simulations).
• View both the phase and sequence quantities.
• Calculate the profile of harmonics along the branches.
• Apply curve fitting and interpolation techniques.
• Plot both equally spaced data and X-Y pairs.
• Perform simple calculations on the harmonic data.
• Have flexibility to drive various printers/plotters.
• Be easy to use, maintain and update.
Although the 3-D plots are useful in getting an overview of the harmonic levels
throughout the system it is difficult to get quantitative information from them. The
2-D plots are more informative in this respect. Harmonic impedance information can
be displayed in the form of impedance loci, as explained in Section 4.3.5.
Figures 4.19 to 4.22 offer a selection of graphic displays illustrating the post
processing facilities described above.
4.5 Summary
Assuming perfectly linear network components, other than those specified as the
distortion sources, this chapter has described the state of the art for the analysis of
currents and voltages in the network at the frequencies injected by the distorting
source or sources. Although the purpose of the book is harmonic analysis, the direct
solution described is equally applicable to sub-harmonic and inter-harmonic
frequencies.
Because of their greater influence at frequencies other than the fundamental, the
modelling of transmission lines and cables has been given special consideration in
Chapter 3.
The computer implementation has been divided in two parts, to achieve more
efficient solutions for their respective applications. These are HARM_Z, the
calculation of harmonic impedances from any selected bus and HARM_AC, the
4.5 SUMMARY 129
·~ 1.0
:l
...
~u TI~~
"' Invercarg 220
] 0.6 Invercarg 033
·~ Manapouri 2014
fJ' 0.4 Manapouri 1014
8 Manapouri 220
~ 0.2 Roxburgh 1011
Roxburgh 220
~ 0 .Ll.J....l..J....l..J....l..J....1.::!::o<::I....I...L.J...L.J...U....U....U....!...2J
5 / Roxburgh 011
~ 9 13 17 21
Order ofharmonic
Figure 4.19 Three-dimensional plot of the positive sequence harmonic voltage magnitudes at
each busbar of the system of Figure 4.8
20 A
3'
c:: 18 ------ B
::
... 16 - --c ~
II
"
.Eo 14
-·-·-·-·- D II
II
"
II
'<:1 12 II
.e 10
II
II
~
I I
8 I I
e 6
I I
I
"'
""
I
4
~"'
\
>
' ' '~-
3 5 7 9 11 13 15 17 19 21 23 25
(a) Order ofhannonic
(b)
Two-dimensional plots of harmonic voltage magnitude (a) and phase (b) for the
Figure 4.20
system of Figure 4.8 at the lnvercargill busbar.
A- with Roxboro ugh open circuited. B- with Roxboro ugh short-circuited.
C-with Roxborough with load and generation
penetration of harmonic currents and voltages, also known as harmonic flow. The
former constitute the basis of filter design while the latter are needed to assess the
harmonic levels throughout the system. Both algorithms can be used by themselves,
as described by the direct solution in this chapter, or as part of more elaborate
iterative solutions as described in later chapters.
130 4 DIRECT HARMONIC SOLUTIONS
jx (Q)
350
300
250
-200
20
-250
Figure 4.21 Impedance locus plot of the test system of Figure 4.8 seen from Tiwai
100 ------------------
-
0
2:S
75 -----------------
""'
l
'o
50 -----------------
0
z 25
0
0
...., nn
0.6
Jl 1.2 1.8 2.4 3.0 3.6 4.2 4.8
Harmonic current (A)
4.6 References
1. Arrillaga, J, Bradley, D A, and Bodger, P S, (!985). Power System Harmonics, John Wiley
& Sons. London.
2. Zollenkopf, K (!960). Bifactorization-basic computational algorithm and programmmg
techniques, Conference on large sets of sparse linear equations. Oxford.
4.6 REFERENCES 131
5.1 Introduction
The direct harmonic analysis described in Chapter 4 requires information about the
harmonic sources; this is obtained either from field measurements or from Fourier
analysis of the expected current waveform of the various non-linear components. As
the main individual contributor to power system harmonic distortion, the three-
phase bridge converter requires special consideration in this respect.
Under balanced and undistorted ac terminal voltage and perfect de current
conditions, a p-pulse converter can be regarded as a linear frequency modulator
interconnecting the ac and de systems. Three main sets of frequencies are involved,
one equal to npf (f being the fundamental frequency of the ac system and n an
integer) on the de voltage waveform and two equal to np{± l on the ac current
waveforms.
In practice, the above ideal conditions never exist. There is always some
asymmetry in the parameters of the plant components and in the operating
conditions. The delay and commutation angles may all be different and as a result
additional harmonics will appear on the ac and de sides of the converter. Also,
subharmonic and inter-harmonic frequencies often occur in non-synchronous
interconnections.
Thus, a more general converter model is needed to simulate the variety of
characteristic and non-characteristic frequencies involved.
After discussing the harmonic behaviour of the ideal converter, this chapter
analyses the ac-dc converter in the frequency domain taking into account ac voltage
distortion, de current distortion and the effect of the converter controller.
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
134 5 A C-DC CONVERSION- FREQUENCY DOMAIN
converter unit. A number of six-pulse bridges with their supply voltage phase-shifted
by appropriate transformer connections are joined either in series or parallel on the
de side, depending on whether high voltage or high current is required.
Generally, these ac-dc converters have considerably more inductance on the de
side than on the ac side and the converter acts like a source of harmonic voltage on
the de side and of harmonic current on the ac side.
For the three-phase bridge configuration the orders of the harmonic voltages are
k = 6n and the corresponding de voltage waveforms are illustrated in Figure 5.2.
v,
(c)
I I
/1 I I • '.
)' 11 1 ,\ '.
'I A I • \ '
:I
1I I
I
I
I
:
I
\
\
\
\
I I \
I I
I I
I I I
Figure 5.2
CiJ
Six-pulse converter de voltage waveforms: (a) at the positive terminal: (b) at the
negative terminal; (c) between output terminals
5.2 CHARACTERISTIC CONVERTER HARMONICS 135
The repetition interval of the waveform shown in Figure 5.2(c) is n/3, and it
contains the following three different functions with reference to voltage crossing C 1:
vd = .J2 T~ cos [ wt + ~] + ~.J2 v;. sin [wt] = .v; r/; cos [wt] for rx < wt < rx + fi, (5.2)
n
for rx + Jl < wt < 3. (5.3)
where v;. is the (commutating) phase to phase r.m.s. voltage, and rx and f1 the firing
and commutation angles respectively.
From Equations 5.1, 5.2 and 5.3, the following expression is obtained for the
r.m.s. magnitudes of the harmonic voltages of the de voltage waveform:
Figure 5.3 illustrates the use of Equation 5.4 to derive the variation of the sixth
harmonic as a percentage of v;.0 , the maximum average rectified voltage, which for
the six-pulse bridge converter is 3-}2J;;:jn. These curves and equations show some
interesting facts. Firstly, for rx = 0 and f.1 = 0, Equation 5.4 reduces to
(5.5)
or
(5.6)
glVlng 4.04, 0.99 and 0.44% voltage distortion for the sixth, 12th and 18th
harmonics, respectively.
Generally, as rx increases, harmonics increase as well, and for rx = n/2 and 11 = 0,
(5.7)
2v'3 ( 1 1 1
i 11 =---:;;- Id cos wt - Scos 5wt +?cos 7wt - TI cos 11 wt
The three-phase currents are shown in Figure 5.4(b), (c) and (d), respectively. Some
useful observations can now be made from Equation 5.8:
• the absence of triplen harmonics;
• the presence of harmonics of orders 6n ± 1 for integer values of n;
• the r.m.s. magnitude of the fundamental frequency is
(5.9)
(5.10)
(a)
Figure 5.4 Six-pulse bridge waveforms: (a) phase to neutral voltages; (b)-(d) phase currents
on the converter side; (e) phase current on the system side with Delta- Y transformer
2.J3 ( 1 1 I
ia =---:;-Id coswt + 5cos 5wt - 7cos 7wt- 11 cos 11wt
+ 01 cos 13wt + 17
1 cos 17wt - 1 cos 19wt- ... )
19 (5.11)
This series only differs from that of a star-star connected transformer by the sign of
harmonic orders 6n ± 1 for odd values of n, i.e. the fifth, seventh, 17th, 19th, etc.
138 5 A C-DC CONVERSION- FREQUENCY DOMAIN
1/j3
_..,. 1/J3 ,.
0
' Time
Figure 5.5 Time domain representation of a six-pulse waveform with delta-star transformer
connection
Twelve-pulse configurations consist of two six-pulse groups fed from two sets of
three-phase transformers in parallel, with their fundamental voltage equal and
phase-shifted by 30°; a common 12-pulse configuration is shown in Figure 5.6.
Moreover, to maintain 12-pulse operation the two six-pulse groups must operate
with the same control angle and, therefore, the fundamental frequency currents on
the ac side of the two transformers are in phase with one another.
The resultant ac current is given by the sum of the two Fourier series of the star-
star (Equation 5.8) and delta-star (Equation 5.11) transformers, i.e.
. 2-J3 ( 1 1
(z 12 ) = 2------:;:-- Id cos wt - U cos 11 wt + U cos 13wt
This series only contains harmonics of order 12n ± 1. The harmonic currents of
orders 6n ± 1 (with n odd), i.e. n = 5, 7, 17, 19, etc, circulate between the two
converter transformers but do not penetrate the ac network. The time domain
representation of the 12-pulse waveform is shown in Figure 5.7(a) and the
corresponding frequency domain representation in Figure 5.7(b).
Six-pulse bridges
Converter
Bus bar
(a)
1.0
0.8
- 0.6
~
- 0.4
0.2
0~------------~~~----------~--~~
1113 23 25
Frequency (x fundamental frequency)
(b)
Figure 5.7 (a) Time domain representation of the 12-pulse phase current and (b) frequency
domain representation of 12-pulse operation
In the last section, the use of two transformers with a 30° phase-shift has been shown
to produce 12-pulse operation. The addition of further appropriately shifted
transformers in parallel provides the basis for increasing pulse configurations.
For instance, 24-pulse operation is achieved by means of four transformers with
15° phase shifts and 48-pulse operation requires eight transformers with 7.Y phase-
140 5 A C-DC CONVERSION~ FREQUENCY DOMAIN
shifts. Although theoretically possible, pulse numbers above 48 are rarely justified
due to the practical levels of distortion found in the supply voltage waveforms, which
can have as much influence on harmonic generation as the theoretical phase-shifts.
Similarly to the case of the 12-pulse connection, the alternative phase-shifts
involved in higher pulse configurations require the use of appropriate factors in the
parallel transformer ratios to achieve common fundamental frequency voltages on
their primary and secondary sides.
The theoretical harmonic currents are related to the pulse number (p) by the
general expression pn ± 1 and their magnitudes decrease in inverse proportion to the
harmonic order. Generally, harmonics above the 49th can be neglected as their
amplitude is small.
Considering the limited inductance of the motor armature winding and the larger
variation of firing angle, the constant de current assumption of the large size
converters cannot be justified in the case of de drives.
The de load can be represented as an equivalent circuit which in its simplest form
includes resistance, inductance and back e.m.f. With sinusoidal supply voltage
V,, sin (wt), the following equation applies:
. ( ) di
V,, sm wt = Ri + L dt +E (5.13)
where
wL
¢ = arctanR (5.15)
(a)
I
I I I 1
I I I I
I I I I
I I (b)
I 1
I I
I I I I IB
8 6 I 18 7 1 8
wt= 0
. = -Vm
z ,.~.
- { cos'l'cos 2n r~.) --+
( wt---'1' E [ --cos'l'cos(8
E ,.1.
1-'l')e
,.1. -R(wt-n/3-8 I )/wL} J
R 3 V,n Vm
(5.19)
Application of Fourier analysis to these current pulses indicates that the fifth
harmonic can reach peak levels of up to three times those of the rectangular
waveshape with the same fundamental component.
An approximate solution [2] on the assumption of a negligible commutation
overlap is shown in Figure 5.9.
t 1 t 1
t t
Jd -+-Direct- I,;- "f'V";..TI,
1 '•- f\:f\. I, --ttJ··
'·T - -·T
T current
waveforms
T t T t T t5th
'
Sign reversal
~~7th
Figure 5.9 Harmonic content of supply current for six-pulse converter with finite inductive
load
.
lc = Vc [COS IX- coswt]
r;;; (5.20)
v2Xc
where Xc is the reactance (per phase) of the commutation circuit, which is largely
determined by the transformer leakage reactance. At the end of the commutation
ic = Id and wt =IX+ Jl, and Equation 5.20 becomes
Id =
Vc
r;;; [cos IX -cos (IX+ fl)] (5.21)
v2Xc
and this expression applies for Y.. < wt < Y.. + p. The rest of the positive current pulse
is defined by
2n
for Y.. +p < wt < Y.. +3 (5.23)
and
2n 2n
for Y.. +3 < wt < Y.. +3 + p
(5.24)
The negative current pulse still possesses half-wave symmetry and, therefore, only
odd-ordered harmonics are present. These can now be expressed in terms of the
delay(firing) and overlap angles; the magnitude of the fifth harmonic related to the
fundamental component is illustrated in Figure 5.10. In summary, the existence of
system impedance is seen to reduce the harmonic content of the current waveform.
The effect of commutation overlap on the de voltage waveform is illustrated in
Figure 5.3.
20
19
.....-
0
c:
"'
E
0
18
"0
c
.2
0
"'0'0 17
c:
"'~.,
c.
0 16
"'
0
....."'
15
=00
14 0,___ __J1o'-----:'z'::-o---::3'::-0----:40
Figure 5.10 Variation of fifth harmonic current in relation to angle of delay and overlap
144 5 A C-DC CONVERSION- FREQUENCY DOMAIN
(5.25)
where 1/J = 0, 120, and 240 degrees for phases a, b and c, N is the converter
transformer ratio (converter to ac system side), and v</1 are the three phase voltages.
Y</Jdc has values between -1 and 1, where 1 signifies a connection of the de side
positive bus to the phase in question, -1 signifies a connection of the de side negative
bus to the phase in question, and 0 indicates no connection. By assigning the transfer
function a value of 0.5 for the two commutating phases the de voltage is correctly
represented during the commutation process.
The ac current in each phase can be defined by Equation 5.26
(5.26)
where ide is the de side current, and Y</Jac is similar to Y</Jdc, except that during the
commutation period the ac current rises or falls in a continuous manner. In this
analysis it is approximated by a linear transfer of the de side current from one phase
to the next.
Both transfer functions are built up by the summation of a basic function (no
commutation period, steady firing angle), a firing angle variation function, and a
commutation function. The process is demonstrated graphically in Figure 5.11, in
which the dotted line represents the basic transfer function, the dashed line the
function revised to include a firing angle variation of Lla, and the solid line the
function further revised to include the effect of a commutation period. Breaking up
the transfer functions in this way allows the frequency spectra to be more easily
written.
The firing angle variation function is characterised as a set of pulses, with fixed
leading edges and variable trailing edges. For Y</Jdc, the commutation function
comprises a set of rectangular pulses, of which the leading edges match the firing
angle variation, and trailing edges vary somewhat differently. For Y</Jac· the
5.3 FREQUENCY DOMAIN MODEL 145
~de
1
-1
(a) Transfer function to de voltage
~a ll1
+-*--+e
I
-1
(b) Transfer function to ac current
commutation function comprises a set of sawtooth pulses, of which the leading and
trailing edges match the firing angle variation. When the spectrum of this waveform
is written, the current-time area of the commutation function has the dominant
effect. An effective commutation period duration fll is defined, such that the area of
the Y'¥ac commutation function matches the area of the true commutation
waveform. In addition, a small variable triangular pulse is added to account for
the variation in this area consequential to the ac voltage, de current, or firing angle
variations.
If the fundamental frequency positive sequence ac voltage component at the
converter terminals is described in the form
(5.28)
146 5 AC-DC CONVERSION- FREQUENCY DOMAIN
at the instant of firing, and the end of the commutation period defined by the
expressiOn
(5.29)
also at the instant of firing, the frequency spectrum of the transfer function YI/Jdc,
derived from spectra developed in [5], and Appendices II and III can be written
2J3 (
J0 (mba) + 1 + 2J0(mba) sm
. (m~to m~t0 2n)
2 ) / - 2-
Yl{!dc(t) = -2)±) 2m cos[m(w 0 t - a0 -1/1)]
n m
+----;-- ~~
./3""""' m cos [(m + 11k)w0 t- m(a0 + J.to) + 11 ( (Jke- k~t 0 - 2 -
~(±) Jn(mbe) n) mljl ]
~~
./3""""'
+----;-- ~(±) l
11
m cos [(m - 11k)w0 t- m(a0
(mbe) + J.to) - 11 ( (Jke - k~t0 n)
+ 2 - mljl ]
./3""""'
+----;-- ~ Jn(mbrx)
~ ~(±) m cos [(m + 11k)w0 t- ma0 + 11 ( (Jkrx- 2 - n) mljl ]
~~
./3""""'
+- ~(±) Jn(mbrx) cos [(m - 11k)w0 t- 1110!0 - 11 ( (Jicrx n)
+- - mljl ] (5.30)
n m n=l 111 2
For simplicity this is written in a form that assumes that both the commutation
period variation and the firing angle variation are at the frequency kwo.
The frequency spectrum of the transfer function Y l{!ac can be similarly written
Yl/l(t) = 2../3
n
2)±)( 1- Jo(111ba)
m 2m
+ Jo(111brx) 2 sin (111Jlo)/- 111 ~to/ 2) x
m 111Jlo 2
2 ./3 oo J b
2 . [(m
sm
+ 11k)11 1 J
cos[m(w0 t-a 0 -ljl)]+(-z=z=c±) nCmoJ 2 x
n m n=l 111 (m + 11k)~tJ
5.3 FREQUENCY DOMAIN MODEL 147
7 . [ (m - nk)f..l 1 ]
')
+ ( _J} L _L(±) 1 (mh,)
oc
11
~Sill
2 X
n m n=I m (m - nk)J..li
+ J3 bd"""'
n 2 ~ ~
m n=-oc,
~ (±)(m + I
nk) sin [ (m +
(m+nk)J..l 1
4
:k)f..li] 12 ln(mba) x
m
where bd and bkd define a current-time correction term such that the change in the
effective commutation period duration is
(5.33)
at the instant of firing, and x approximates the average angular position of the
correction pulse, being 1 for an inverter and 2 for a rectifier.
These spectra contain both characteristic and non-characteristic harmonics. As the
de voltage comprises a summation of these transfer functions multiplied by the three
ac voltages, the ac current comprises the de current multiplied by the appropriate
transfer function, non-characteristic frequencies in the transfer function will lead to
non-characteristic frequencies in the converter currents and voltages. It is necessary
to examine the transfer function more closely, via the control and commutation
process, to determine these spectra.
The spectrum of the transfer functions depends on the variation of the commutation
period parameters, and this variation must be described. The commutation circuit
itself is simple, as shown in Figure 5.12. Writing the circuit equations and integrating
from the time of firing wot; to wot, gives
(5.34)
where Vj is the peak single phase ac voltage at the converter transformer primary and
Xc is the converter transformer leakage reactance referred to the converter
transformer secondary.
Equating the area of the equivalent sawtooth pulse of duration 11 1 to the
integration of the commutation current between ()(o and ()(o + Jlo results in the
following term
(5.36)
Sensitivity to firing angle From Equation 5.35 it is apparent that variation in the
firing angle will cause a variation in the commutation period duration.
Differentiating Equation 5.35 with respect to the firing angle, keeping the de
current and ac voltage undistorted, yields the following
sin(()(0 )
-r============~-1 (5.37)
2
1- ( COS()(o-
2XJd
r::i
)
v 3NT-J
Differentiating Equation 5.36 with respect to the firing angle yields, with a small
signal assumption,
(5.38)
Sensitivity to de current Letting the de current distortion tend towards zero, solving
Equation 5.34 for a constant firing angle and an undistorted commutating voltage.
5.3 FREQUENCY DOMAIN MODEL 149
and differentiating with respect to the de current distortion at the instant of firing
results in
Integrating the ac current over the commutation period yields the current-time area
of the commutation current waveform, and differentiating with respect to the de
current at the instant of firing yields the sensitivity of the commutation period
. current-time area to the de current. However, some of this sensitivity is already
accounted for by the unmodulated transfer function spectrum, which must be
subtracted to yield
(5.41)
where
(5.42)
(5.43)
The frequency of the voltage that interferes with the commutation process is less
than the positive sequence frequency on the three phase ac voltage, by the
fundamental frequency.
Taking into account the variation of the ac voltage waveform over the
commutation period, assuming the distortion level tends towards zero, and keeping
150 5 AC-DC CONVERSION- FREQUENCY DOMAIN
a constant de current and firing angle, allows the following relationship between the
commutation period duration and ac voltage distortion
-llo ] k2 sm 2
. (kilo); oc 0 + kllo/2 - n/2
2X Id ] 2 llo
I - [ cos(oc 0 ) - .J3 c
3NVj
(5.44)
consider the case of a converter under constant current control. For a controller
of the proportional/integral (PI) type, the transfer function for de current to
firing angle is
1
G,(k) = p + :----k
T (5.48)
J wo
where P is the proportional gain, and T is the integral time constant, such that
G1(k) is in radians per kiloampere. The transfer function of the current transducer
will be significant at the higher frequencies, and should be included in this
equation.
The transfer functions Yljldc and Yljlac, in conjunction with the firing angle and
commutation period variability terms allow the prediction of voltage waveform
distortion on the de side of the converter, and current waveform distortion on the
ac side of the converter. The characteristic distortion levels are little changed
from the simplified analysis. As becomes apparent, many frequencies are
generated as a result of just one distortion source. However the most significant
frequencies are limited to a set of three, being one on the de side, at frequency
kwo and two on the ac side, being at frequencies (k + 1)w0 in positive sequence
and (k- 1)wo in negative sequence. Approximation to these three frequencies
leads to the three port model [4], and verification is limited to these terms.
Extension is made to a twelve pulse converter by doubling the magnitude of the
transfer function and using the terms for m = 1, 11, 13, 23,25 etc.
To verify the predictions of the model, and establish the importance of the
described mechanisms, comparative results are gained by another method. In this
case, a time domain dynamic simulation program based on the EMTP
formulation [3] is used. An HVdc rectifier is modelled, based on the CIGRE
benchmark HVdc test system [6] described in Appendix VI. With infinite ac and
de side busbars, single distorting frequencies are injected on the firing angle
order, the ac side voltage, and the de side current. After a short settling period, a
Fast Fourier Transform is made of the required variables, and the results
compared in the frequency domain. In all the figures, the harmonic transfer
obtained from dynamic simulation, the predicted transfer not allowing for
commutation period variation, and the prediction allowing for commutation
period variation are shown.
Sensitivity to firing angle Expanding Equation 5.25 out over 1/J, and making the
reasonable approximation that ln[(m- 1)b"']/(m- 1) = ln(mbrx)/m = ln[(m + 1)brx]/
(m + 1), yields the non-characteristic de voltage distortion
152 5 A C-DC CONVERSION- FREQUENCY DOMAIN
I
6v'3~[ln(b(J.)cos ( ao+2
vd= N~----;-~ nn) ln<>ke-nflo ] x
nn)ln<>ka+ln(be)cos\o+flo+2
(
cos(nkw0 t) } + {N~----;- ~ ~
6v'3" ~ [Jn(mb(J.) .
m sm(a0 )~ +
/MY..
ln(mbe)
m sm(a ~ - (m
. 0 +flo) I nuke + nk )flo ] . [ (m + nk)w0 t- ma0
x sm nn J) +
-
2
---(a)
250
(b)
- - - - - - - - (c)
---(a)
50 - - - - -- - - (c)
4 5 6 7 ·•o;---;----:;----~---74--;-5--:-.--:-7--:---o-~
Firing angle harmonic Firing angle harmonic
Figure 5.13 CIGRE rectifier harmonic transfer, firing angle modulation to de voltage. (a)
dynamic simulation (b) prediction without commutation period variation (c) prediction with
commutation period variation
5.3 FREQUENCY DOMAIN MODEL 153
1.2 12
c 1 c 1
"'
~ ~
"V! 0.8
a.
"'a.
~08
E E
;;:"'
.2 0.6 gos (a)
---(a) "'
0.4 0.4 (b)
(b)
0.2 - - - - - - - - (c)
o.21 -------- (c)
oo':----;---c~-~4:;----!-5~6::----!--7--=---~10 0
0 4 5 6 7 10
Firing angle harmonic Firing angle harmonic
-<J.S ---(a)
1.5
,c (b)
·35
.{)5 - - - - - - - - (c)
·4
-4 5 -~6 --:7:-------c---;-_j
s0l_-,--..,;----c:----~---=-
4 ·1
Firing angle harmonic
10 0
' 5 6 7
Firing angle harmonic
10
Figure 5.14 CIGRE rectifier harmonic transfer, firing angle modulation to +ve and -ve
sequence ac current. For key see Figure 5.13
predicted and measured distortions resulting from a 3° peak to peak firing angle
modulation. Only the lowest order components are shown.
A particular feature of this spectrum is that the magnitude of the harmonics
generated decrease relatively slowly with m. The commutation period variation has a
strong effect on the resulting distortion, both in magnitude and phase. The spikes on
the curves occur as the higher order components were neglected.
Sensitivity to ac voltage The contributions to the converter de side voltage from the
ac side voltage can be divided into two; the direct transfer due to the unmodulated
transfer function, and the indirect transfer via the commutation period modulation.
154 5 A C-DC CONVERSION- FREQUENCY DOMAIN
Taking the direct transfer portion, and expanding out over l/1, form = 12, 24, 36
etc. gives, for a positive sequence ac voltage
6vJ
vd=N { ~-;-cos (2
/lo ) cos [ (k-l)wot+oco+2+bk
/lo J}
6v'3" cos((m- 1)!10 /2)
-N [ ---;-~~ m-l
I. { Jlo)
cosl(m+k-l)w 0 t-(m-l\oc 0 + 2 +bk
JJ
vd = N { -n-cos
Vk6v'3 (/lo)
2 cos [ (k + l)w 0 t - ( oc0 + 2/lo) + bk ]}
~6v'3" cos((m-
-N [ -n-~ m-l
1)!10 /2) I. { Jlo)
cosl(m-k-l)w0 t-(m-l\oc 0 + 2 -bk
J)
V/c6v'3"cos((m+
+ N [- -- ~
n
1)!10 /2) cos ~(m + k + l) w 0 t - (m + l)(oc 0 +/lo)
m+ 1
- +
2
~
uk J)
111
(5.51)
Sensitivity to de current There are two ways the de current variation affects the ac
side current, directly from transfer via the unmodulated transfer function, and
5.3 FREQUENCY DOMAIN MODEL 155
J____ _ 2.5
1.5f-
I
l ---(a)
1.5 ---(a)
I
J (b)
(b)
- - - - - - - - (c)
0.5
- - - - - - - - (c)
4 5 6 7 10 00 4 5 6 7 10
DC voltage harmonic DC voltage harmonic
.().1 ---(a)
w
~-0.2
(b)
- - - - - - -- (c)
g_
I
l
:§
lll-0.3
ill
"'
s:;
·::1
0.15 (a)
0.1 (b)
--- -1
O.OS - - - - - - - - (c)
.().6
0 4 5 6 7 10
DC voltage harmonic
' 5 6 7 10
DC voltage harmonic
2.J3"' Ik 2 . ( -
lac, =N {--L.,..(±)---sm - cos [ (m+k)w 0 t-m ( a0 +-
mflo) flo) +6~c-ml/J]}
'' n 111 m 111flo 2 2
2.J3"' h 2 . ( -;-cos
+N {--L.,..(±)---sm 117flo) [ (m-k)w 0 t-m ( a0 +-
flo) -6k-mlj; ] }
n m m l11flo . _ 2
(5.52)
form = I, 11, 13, 23, etc. The spectrum from the second mechanism can be defined
by substituting the commutation period variation term from Equations 5.33 and 5.40
into Yl/Jac• and multiplying by h
For the test case there are two ways that de current distortion affects the converter
de side voltage. Firstly and most simply the commutation period modulation directly
156 5 A C-DC CONVERSION- FREQUENCY DOMAIN
reflects a distortion onto the de voltage. This can be defined by substituting the
commutation period variation from Equations 5.29 and 5.39 into equation 5.49.
Secondly, the de current distortion that is directly transferred through the
converter passes through the converter transformer leakage reactance. This in turn
generates a voltage distortion at the valve terminals, which is transferred back onto
the de side by the converter transfer function. To determine these terms
Equation 5.52 is multiplied by the frequency dependent impedance of the converter
transformers (for a 12-pulse converter, the leakage reactance of both transformers in
parallel) to yield an ac voltage distortion, which is in turn passed back to the de side
by substitution of all the terms into Equations 5.50 and 5.51 as appropriate. This
results in a proliferation of terms, most of which are not significant. The term of
most interest, that at the original distorting frequency, is as follows
(5.53)
5.3.4 Discussion
Each of the plotted graphs demonstrate several important points concerning transfer
of distortion through the ac-dc converter. One is the importance of the dynamic
variation of the commutation period on transfer of waveform distortion. This is
shown by the the difference between the curves showing the transfer function
predictions including and excluding commutation period variation. For the case of
firing angle modulation to de voltage, shown in Figure 5.13, the commutation period
duration variation was at least as important as its absolute duration, and both have a
significant effect. For other transfers, the commutation period variation affects the
distortion transfer by up to 20%.
Reasonable agreement is demonstrated between the transfer function predictions
and the time domain simulation verification. Agreement between such different
approaches to converter modelling implies that the linearizations made in the
transfer function model are justifiable. The validity of the linearizing assumptions
decreases with increasing distortion magnitude and frequency, which can be seen in
the loss of accuracy with increasing frequency in almost all the predictions.
Some disagreement is exhibited between the frequency domain calculated and the
time domain simulated results at integer multiples of the fundamental frequency.
Integer harmonics have an associated spectrum that includes higher order terms that
reflect back to the same frequency. These terms are most significant for the firing
angle and commutation period variation effects, as the magnitudes of the higher
order terms from this mechanism of transfer are far greater than the higher order
terms from the more direct transfer of distortion through the converter. These terms
5.-1 THE CONVERTER FREQUENCY DEPENDENT EQUIVALENT 157
0.7 0.91
0.8
~0.6
~0.5
I:: ---
~ ~051
~0.4
l'l ~OAf
~=I
~0.3 ---(a)
0.2
0.1
(b)
-----(a) 0.6
(b)
~.02 ~ 0.5
'6 '6
g_ -03 - - - - - - - - (c) 1:! I
:: ~0.4
------ 1
---1
:2 :2
<I) ..(),4
m
i(l
"'
w -----(a)
~0.3
&. -0 5 D..
0.2 (b)
0.1 - - - - - - - - (c)
~~~~-~-7--~.-~.-~.~~,~~--~__j ~~~--~~--~.--~,--~.--~,~~--~_j\0
DC current harmonic DC current harmonic
Figure 5.16 CIGRE rectifier harmonic transfer, de current to + ve and -ve sequence ac
current. For key see Figure 13
were not included in the frequency domain predictions. Interestingly, many of the
higher order terms undergo a magnitude and phase transformation that makes the
final harmonic transfer magnitude and phase dependent. This dependence makes the
accurate prediction or simulation of integer harmonic transfer through the ac-dc
converter substantially more difficult.
Although the effect of unbalance in the three phase supply is not discussed, the
analysis is still directly applicable, by dividing the unbalance into positive, negative,
and zero sequence components. Zero sequence components have no effect.
For the purpose of this section, the ac and de systems around the converter are
reduced to their equivalents, with the ac system modelled by a frequency dependent
Thevenin source, and the de system modelled by a frequency dependent Norton
source. Cross-coupling between phases is not accounted for. The equivalents are
illustrated in Figure 5.17(a).
A primary goal of this section is to reduce the converter and its associated ac
system, or the converter and its associated de system to a Thevenin or Norton
equivalent. With the accompanying system equivalents, the harmonic interactions
around the converter can be simply defined. This is illustrated in Figure 5.17(b)
and (c).
The equivalents developed, due to the non-linear nature of the converter, will have
cross-coupling between frequencies, and can be written in matrix notation.
The harmonic relationships for the converter, ac and de systems shown are written
in the following form
(5.56)
(5.57)
where
and
(5.59)
The converter from the de side can be viewed as an impedance with a voltage
source, and from the ac side can be viewed as an admittance with a current
source. If the harmonic sources on the de side are zero, then from the ac side the
converter will appear as an admittance matrix, and if the harmonic sources on
the ac side are zero, from the de side the converter will appear as an impedance
matrix. This needs to be interpreted very carefully. Given only one external
harmonic source, many harmonics will be generated. However, all these
harmonics are approximately linearly related to the originating harmonic, and
the response of the converter to the original harmonic, at the original harmonic,
remains approximately linear. This is why the relationships can be represented as
impedances or admittances.
The elements of these matrices can be derived by numerical methods [4], or by the
relationships developed in Section 5.3, as is done here.
160 5 AC-DC CONVERSION- FREQUENCY DOMAIN
6v'3
a 1 =-;-cos (f.lo
2 )! rxo + f.A.o/2 (5.69)
(5.70)
(5.71)
(5.72)
a5 =2v'3 2 .
- -sm ( -f.lo ) / -rx 0 - f.lo/2 (5.73)
n f.lo 2 l__-"----'-"-'-
(5.74)
(5.75)
(5.76)
(5.77)
(5.78)
GtJ =
[ NVt
-~
1 - [ cos(rxo)- v'3NVt J
2XJd
2 ]
X
2 . ((k+ l)f.lo)]!L_.rxo'-+---"(k_+_l-'-)'--f.A.o=/_2_-_n'--/2
[ -(k+1)fJ.o
c - - . , - - sm (5.79)
2
I[
162 5 A C-DC CONVERSION- FREQUENCY DOMAIN
ai2 = [
-flo
2XJd ]2 ] X[ (k-21)flo SID. ((k-l)flo)J]
2
NV1 1- cos(1X 0 ) - r:;
v3Nf?t
(5.81)
sin( 1Xo)
a,4 = ---;========::::::= (5.82)
2
1 - ( COS IXo- 2XJd
M
)
v3NV1
2
[ """"(k:-_-1.,...)!-lo . ((k-1)p 0
SID 2
)]}l
L__-_IXo"-+---"(k_-_1-'-')P'-'o"-/_2_+_n...:./_2 (5.84)
(5.86)
These terms depend on the actual and effective commutation periods, defined as
follows
2XJd ] - IXo
Po =cos -1 [ cos IXo - J}NTj (5.87)
J3Nf?t . .
/11 = 2po - X I [p 0 cos(czo) + SID(1X 0 ) - sm(1X 0 + p 0 )] (5.88)
(' d
5.-+ THE CONVERTER FREQUENCY DEPENDENT EQUIVALENT 163
\Vhere Ti is the peak single phase ac voltage at the converter transformer primary, Xc
is the converter transformer leakage reactance referred to the converter transformer
secondary, and Id is the average de current.
a19, azo and an depend on the converter firing angle controller, and must be user
specified. They are specified here for the type of control described in section 5.3.3.
For the constant current control
(5.89)
(5.90)
(5.91)
These equations describe the interaction as shown in Figure 5.18. This is similar to
the three port network model proposed by [4]. Linear superposition of the different
effects is assumed, which means for example that commutation period modulation
due to ac voltage does not affect the commutation period enough to significantly
modify the effect of de current modulation of the commutation period. This
assumption is reasonable for small distortion levels. Interactions at higher order
frequencies also exist, but at insignificant levels when their reflections back onto the
original frequency are considered.
The frequencies depicted in Figure 5.18 are of primary importance. In a similar
way to the commutation period variation, firing angle modulation will be mainly
confined to the frequency one harmonic less than the positive sequence ac side
distortion, one harmonic more than the ac side negative sequence distortion, and the
same frequency as the de side distortion.
AC DC
equivalents Converter equivalent
Firing angle
modulation
End of commutation
period modulation
..._,_Duration of commutation
period modulation
Harmonic
k
The interrelationship between the three harmonic sequences around the converter
expressed by Equations 5.60-5.68 can be reduced and rewritten by a 3 x 3 matrix as
follows [8]
(5.92)
If no ac side distorting sources are allowed, Equations 5.66 and 5.67 can be rewritten
(5.94)
(5.95)
The solution of these combined with the Equation Set 5.92 is straightforward, and if
the following assignments are made
bgZacnZacp
A = -aZacp + ----:---------:---=----'--
1 + hZacn
biZacn
B=c-----
1 + hZacn
C = -:-:----:::e_g_Z_a,---o--,-'Z_c__,lcJ_?,---
(1 + dZacp)(l + dZacn)
D f
= ---"---- eiZacn
(5.96)
1 + dZacp
450
BOO
400
700
600 (b)
,'
(a)
' (c)
~~--~2~---7.-----78----~8~--~10
DC frequency (multiple of fundamental)
500 3
• 5
• 7
DC frecuency (multiple of fundamental)
8
• 10
(a) 1.4
(a)
1.2
1.5 (b).
0.8
0.6
0.4
0.2
2 4 6 8 10
DC frequency (multiple of fundamental)
-o.20
3 4 5
• 7
DC frequency (multiple of fundamental)
8
' 10
Vdch _ _ (
-[dch-
AD + B)
1- C
(5.97)
In Figure 5.19 the measured and predicted impedances are shown for the CIGRE
model rectifier and inverter. The rectifier is operating in constant current control,
and the inverter is in minimum gamma control.
In each figure curve (a) represents the converter impedance measured from a
dynamic simulation, curve (b) the impedance prediction without firing angle
variation, and curve (c) the full impedance prediction. Useful agreement is obtained
for both the current controlled rectifier and the minimum gamma controlled
inverter. Numerical noise, and higher order terms in the simulated results give the
jagged appearance of the plotted curve.
166 5 A C-DC CONVERSION- FREQUENCY DOMAIN
If no de side distorting source and no ac side negative sequence distorting source are
allowed, Equations 5.67 and 5.68 can be rewritten
(5.98)
and
(5.99)
The solution of these equations combined with Equation Set 5.92 is again
straightforward, and if the following assignments are made
A= a- bgZacn
1 + hZacn
B = C- biZacn
1 + hZacn
C = d- egZacn
1 + hZacn
D =f- eiZacn (5.100)
1 + hZacn
then the positive sequence admittance of the ac side of the converter can be written
lacp = C+ DA (5.101)
Vacp Zdch- B
(5.102)
and
(5.103)
The solution of these equations combined with Equation Set 5.92 1s agam
straightforward, and if the following assignments are made
5.4 THE CONVERTER FREQUENCY DEPENDENT EQUIVALENT 167
A = b- aeZa'1'
1 + dZacp
B=c- afZacp
1 + dZacp
C = h- geZacp
1 + dZacp
D= i_ gfZacp (5.104)
1 + dZacp
then the negative sequence admittance of the ac side of the converter can be written
(5.105)
The converter de side impedance comprises three components, being the ac side plus
transformer impedance at the higher positive sequence frequency, the ac side plus
transformer impedance at the lower negative sequence frequency, scaled approxi-
mately by (6jn) 2 N 2 , and a final more complicated contribution from the transfer
function variation terms. Just the scaled sum of the ac side impedances gives a good
approximation at all but the lowest frequencies.
Whenever there is significant smoothing reactance on the de side of the converter,
this component dominates the converter ac side impedance. The converter
impedance at harmonic order k can then be simplified to a multiple of the de side
impedance, i.e.
1600 700
1400
600
1200 (b)
500
1000
EBOO
J::
300 (b)
0 600
400 200
200 100
%~--~---7--~6~--~---,~0--~12 %~--~---7--~.~--~---1~0--~12
Harmonic Harmonic
1.5
1.5
0.5
0.5
"' 0
"
:g"' -0.5
a:
·1
·0 5
-1.5
·10L -------,----~--~6:-----:-------:10;-0------!12
" 2~o--___,:----7-----:,~--~--~10--~12
Harmonic
Harmonic
Figure 5.20 CIGRE rectifier ac side harmonic impedance, with firing angle control
C would be approximately (n/6) 2 .(I I Ni, but as is visible in Figure 5.20, transfer
function modulation can have a substantial effect. For a 500 MW HYde model [6],
Zx x C has a value of 44.0 i68.75"0. The zero sequence impedance is dependent
only on the transformer connections, and the presence of a delta winding.
high pass) are also connected at this bus. It is assumed that a new non-linear load is
proposed for connection at Invercargill, requiring local filtering. It is, therefore,
necessary to calculate harmonic impedance information viewing the system from
Invercargill, as an aid to ac filter design.
Figures 5.22 and 5.23 show the system impedance and reactance. respectively,
calculated with models described in Chapter 4 (graphs (i) and (iv)) and those derived
in this section (graphs (ii) and (iii)). The detailed and simplified cross-coupling model
give very similar results. This can be attributed to a high de side impedance due to
the presence of a large de smoothing reactor. Greater differences between these two
models would be expected for a small de side impedance.
As expected, near the harmonic filters tuned frequencies, the results are the same
regardless of the converter model due to the dominance of the harmonic filters.
However, outside these frequencies there are large differences between the models,
particularly in the frequency range 100Hz to 500Hz. This is very significant since it
is in this frequency range where important uncharacteristic harmonic resonances
may occur.
5.5 Summary
Algebraic predictions of distortion transfer around a controlled 12-pulse ac-dc
converter have been developed using frequency-domain defined transfer functions,
and verified for low-order terms by comparison with results obtained by a time-
domain technique. It has been established that to model waveform distortion
transfer through an ac-dc converter with reasonable accuracy, it is necessary to
include the influence of waveform distortion on the dynamics of the commutation
period.
The transfer function analysis has also been used to calculate the harmonic
impedances of ac-dc converters and the results have been used to assess the validity
170 5 A C-DC CONVERSION- FREQUENCY DOMAIN
900
800
700
(i) No Converter
en (ii) ~. Simple
(iii) . Two Pass
~ 600 (iv) ::::::PO
Q
0)
""0
.~ 500
c
Ol
"'
20) 400
0 r' I
I \
'
c
"'g;_300
""0
·I
_s \
\
200 I
I
100 I
-\
'..._-I
?
"
0
0 200 400 600 800 1000 1200 1400
Frequency (Hz)
=·
800,------,-------,------~-------r------,-------.-------,
(i) No Converter
(ii) Simple
(iii) . Two Pass
600 (iv) --PO
400
en 200
E
.r::
Q
~ 0
c
u"'
£"' -200
-400
-600
-800L-------L-----~------~-------L------~------~----~
0 200 400 600 800 1000 1200 1400
Frequency (Hz)
of other. less rigorous models. The current practice of either ignoring the presence of
ac-dc converters, or using load-flow information, when calculating network
harmonic impedances has been shown to be highly inaccurate at all frequencies in
the absence of local filters. A simplified method derived from the rigorous analysis,
on the other hand, has been shown to provide reasonable results for the frequency
range of interest to harmonic filtering.
5.6 References
1. Arrillaga, J, Bradley, DA and Bodger, PS, (1985). Power System Harmonics. John Wiley &
Sons, Chichester, UK.
2. Dobinson, LG, (1975). Closer accord on harmonics, Electronics and Power, 21, 567-572.
3. Dommel, HW, (1969). Digital computer simulation of electromagnetic transients in single
and multiphase networks, IEEE Transactions on Power Apparatus and Systems, PAS-88(4),
388-399.
4. Larson, EV, Baker, DH and Mciver, JC, (1989). Low order harmonic interaction on acjdc
systems, IEEE Transactions on Power Delivery, 4(1), 493-501.
5. Schwarz, M, Bennett, WR and Stein, S, (1966). Communication Systems and Techniques,
McGraw-Hill, New York.
6. Szechtman, M, Weiss, T and Thio, CV, (1991). First benchmark model for hvdc control
studies, Electra, 135, 55-75.
7. Wood, AR, (1993). An analysis of non-ideal HYde convertor behaviour in the frequency
domain, and a new control proposal. PhD thesis, University of Canterbury, NZ.
8. Wood, AR and Arrillaga, J, (1994). The frequency dependent impedance of an hvdc
converter, ICHPs Conference, 21-24 Sept. 1994, Bologna, Italy, September.
6
HARMONIC INSTABILITIES
6.1 Introduction
Ac-dc systems with low short circuit ratios (SCR) often experience problems of
instability in the form of waveform distortion. The low SCR indicates a high ac
system impedance, whose inductance may resonate with the reactive compensation
capacitors and the harmonic filters installed at converter terminals. These resonant
frequencies can be low, possibly as low as the second harmonic. The resonances can
be excited under certain operating conditions or in the event of fault, and the small
initial distortion may develop to an instability. Instability related to the interaction
of harmonics (or any frequencies) has been customarily referred to as harmonic
instability.
The problem of harmonic instability in ac-dc systems was first identified in
relation to the individual valve firing control [1] and an alternative control principle,
the phase locked oscillator [2], was adopted for new installations. Since then, other
forms of harmonic instability have been identified, involving complementary
resonances, composite resonances, cross-modulation and transformer core satura-
tion.
The problem has in the past been discovered during or after the system had been
commissioned and usually occurs only under certain unfavourable conditions.
Ideally, the problem should be detected at the early planning stage to introduce
counter-measures and to avoid costly modifications later.
The term complementary resonance has been used to describe the situation where
a parallel resonance at a harmonic on the ac side is closely coupled to a series
resonance on the de side, via the three port frequency transforming characteristics of
a converter. Although, strictly speaking, this does not imply an instability, a small
remote injection can excite harmonic levels sufficient to compromise the operation of
the converter.
Cross modulation is used to describe the frequencies generated when two ends of a
de link are operating at slightly different frequencies. The low frequencies generated
can excite generator or motor shaft mechanical oscillation modes, which can
similarly lead to shut-down of the plant.
Composite resonance takes an overall view of the converter embedded in its ac and
de systems, and looks for a true instability that requires only a very small excitation
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
174 6 HARMONIC INSTABILITIES
for the instability to grow in a self-sustaining manner. This may or may not include a
contribution from converter transformer core magnetic saturation.
This chapter examines composite resonance without a core saturation contribu-
tion from a simple circuit analysis approach, and also directly analyses the instability
that incorporates a converter transformer core saturation contribution.
Convener DC system
equivalent equivalent
where
176 6 HARMONIC INSTABILITIES
(6.2)
Using this relationship, any de side converter impedance and composite resonance
damping factor can be specified, within controller limitations.
The limitation that a controller cannot see into the future, coupled with the phase
delay inherent in the added impedance function fz, limits the range of achievable
impedances. In Figure 6.2, the frequency dependent transfer function of fz is plotted
for the CIGRE benchmark test system rectifier [5] which shows clearly how a
significant phase delay is introduced.
A feature of an uncontrolled converter de side impedance is that while a low
resistance is possible, a negative resistance is extremely unlikely. Simple constant
current control will degrade the damping of distortion by adding a negative
resistance at some frequencies, and a design goal should be to minimise the
degradation near the composite resonant frequencies.
- - - Magnitude
- - - - - - Phase
.§'"'
3
-9- 2
"u
§
'0
.\
j" "'c
0 0
---- ---- :a"
.• ____ ... "..
'\
-100 -1
-200 -2
\''
'
'
-300 -3
-400
0 2 3 4 5 6 7 8 9 10
Harmonic
Figure 6.2 Transfer function for impedance contribution of constant current control
6.~ COMPOSITE RESONANCE-A CIRCUIT APPROACH 177
Table 6.1 Constant current control gains for test case examples
is presented where a high pass filter is placed in parallel with the PI control path,
having a characteristic such that its gain can be written
Gs 2 /wr
HP= " (6.3)
1 +2(/w 1 +s2 /w!
where G = 0.9 rad/kA, ( = 0.3 radjs, and w 1 = 200n radjsec. This was chosen to
increase the damping factor at the composite resonance, without affecting the
transient response at low frequencies. The de current transducers have a time
constant of 1 ms, and the inverter is operating in minimum gamma control.
Dynamic simulations are run for each case, in two cases involving a three phase
fault on the inverter ac busbar for 70 ms to excite the composite resonance. The
control gains for all three cases are given in Table 6.1.
Figure 6.3 shows the calculated real and reactive components of the series
composite resonant circuit for examples 1 and 3. The underdamped resonant point,
indicated by a zero reactive component and a low resistance, occurs at about 1.5
times the fundamental frequency, or 75 Hz. This frequency is mainly defined by the
ac and de networks, with small changes in constant current control gain having little
effect.
The constant current control gain has a strong effect on the composite resonance
damping factor through its effect on the circuit resistance. Table 6.1 gives the
selected constant current control gains and the calculated composite resonance
damping factors. Rather higher gains than would normally be employed are chosen
to show light positive and negative damping. The effect of the high pass filter in
example 3 is to increase the circuit resistance at the composite resonance, at the cost
of introducing a significant negative resistance at a higher frequency and an
additional, but well damped, composite resonance at about 180Hz. However, both
these features are less critical than the original lightly damped composite resonance.
The results of dynamic simulation runs are shown in Figure 6.4. Example I
demonstrates the composite instability predicted by the analysis, and example 2
shows the lightly damped resonance also predicted by the analysis. Example 3
represents the response for the example I system including the additional high pass
filter control block. Because the high pass filter block allows a higher PI gain to be
used, transient overcurrents can be substantially reduced.
Table 6.2 presents both the calculated and simulated time for the waveform
distortion to halve (for positive damping) or double (for negative damping).
178 6 HARMONIC INSTABILITIES
''
1000
I•
'
, '
.
!!
'
l 0
I
I
-500
..
II I \
\I
f
Reoislai"C4
- - - - - - - - Reec:tanoo
•
-1000
'
0 5 g 10
...............
(a) Example 1
1000 ''
I I
I
l I
I
- - - - Relistanc<o
:'. '\.'
' I
-500
I
- - - - - - - - Reec:tanoo
-looooL---~--~--~--~--~s~--7---~--~.-----g~~~o
Hannonic
(b) Example 2
Figure 6.3 Composite circuit series impedance
Agreement is shown for all the examples. The resonant frequency for the
simulation is at 67.5 Hz, some 7.5 Hz lower than predicted. The explanation for this
is unclear, although it could arise from approximations in the transfer function
coefficients or the estimated impedance of the inverter, or in the dynamic simulation.
6.2 C0.\1POSITE RESONANCE-A CIRCUIT APPROACH 179
0.0000 02000 0.4000 0.6000 0.8000 1.0000 12000 1.4000 1.6000 1.8000 2.0000
Time(s)
Example I
Current (pu)
1.3980 .,..-----,-----,,..---..,-----,-----,c-r----,---,...---..,-----,------,
--~----t---------~---····--·---t·····- - --------:····-------i----------·--:·····--······----+-----------
---t-·
1.1184
O.S3BB -i-------t·---j----; - --t---·--t----r--·-·-··----t---··
·-·----:------··-r-·---;----r----;---·----r----r---···--:---··-·-··-
0.5592 -··---~
02796 - - - --r----:--····-----r·-----!----t----!-·----:--l···----·---l---·--··-
0.0000
0.0000 02000 0.4000 0.6000 0.8000 1.0000 12000 1.4000 1.6000 1.8000 2.0000
Time (s)
Example2
0.0000 02000 0.4000 0.6000 0.8000 1.0000 1.2000 1.4000 1.6000 1.8000 2.0000
Time (s)
Example 3
6.2.3 Discussion
The dynamic simulation indicates that a predicted instability rises only to a certain
level and then stabilizes. As the firing angle modulation amplitude rises, its effect
does not rise proportionally, due to the non-linearity of the effect of firing angle
modulation on the converter. The exception to this is when there is another source of
waveform distortion, such as valve commutation failure or transformer core
saturation, both of which render this analysis inadequate.
The algebraic analysis indicated that an additional control block that reduces the
control gain around the composite resonance would improve the system response. A
control block was incorporated to do this, which damped the instability whilst
keeping the high gain and, therefore, high speed PI control. This control solution
was not optimised.
The frequency domain analysis presented takes into account the inverter de side
impedance, but with a very crude model of the minimum gamma control response,
which is of importance below the fundamental frequency. The control system of the
converter at the other end of the de link must be accounted for if it is effective at the
frequency in question. Typically, a minimum gamma controller generates a
significant negative resistance which may contribute to a lightly damped low
frequency sub-harmonic in the de link transient response.
It should be noted that the equations used in this analysis can be converted into a
controlled system block diagram format, so that more traditional control design
tools can be used.
The mechanism of ac-dc harmonic interaction has already been discussed in Chapter
5 and is summarised in Figure 6.5. The presence of a harmonic distortion at k times
fundamental frequency on the de side of a 12-pulse HVdc converter will produce on
the ac side, positive sequence harmonics of orders k + 1, 13 ± k, 25 ± k, 12n + 1 ± k;
n = 3, 4, ... , and negative sequence harmonics of orders k- 1, 11 ± k, 23 ± k,
12n- 1 ± k; n = 3, 4 ... These harmonic sequences are reflected back to the de side
as the kth harmonic and various high order harmonics of 12 ± k, 24 ± k, 12n ± k;
n = 3, 4 ... Among these harmonics, the most significant terms are the first order kth
harmonic on the de side, and the positive sequence k + 1 and negative sequence k - I
harmonics on the ac side. The higher harmonics are an order of magnitude smaller
than the lower order harmonics. Therefore, for most analyses. particularly those
6.3 TRANSFORMER CORE RELATED HARMONIC INSTABILITY 181
AC side DC side
Positive sequence harmonics
k+1
13±k
25±k
12n+1±k, n=3,4, ...
k
12±k
24±k
Negative sequence harmonics 12n±k. n=3,4, ..
k-1
11±k
23±k
12n-1±k, n=3,4, ...
with small distortion levels, it is reasonable to ignore the contribution from high
order harmonics. Figure 6.5 gives a representation of the harmonic orders that can
be expected on each side of an HYde converter.
With the high order harmonics ignored, the presence of a 2nd harmonic distortion
on the de side will result in a positive sequence 3rd harmonic and a negative sequence
fundamental frequency component on the ac side.
The interaction mechanism can be extended to non-harmonic frequencies. For
instance, if there is a distortion near the fundamental frequency such as at 51 Hz on
the de side, the distortions on the ac side would be near the second harmonic
(lOl Hz) for the positive sequence component and near de (l Hz) for the negative
sequence component. As the frequency of the de side distortion approaches
fundamental frequency, the lower corresponding frequency component on the ac
side, which is in the negative sequence format, will be approaching 0 Hz, i.e.
approaching de.
If the de side distortion is exactly at fundamental frequency, the negative sequence
component on the ac side is true de, but with different levels in the three phases,
which is customarily regarded as 'unbalanced de' generated by the converter.
However, the sum of the de distortions in the three phases will be zero, and these
distortions can be written mathematically in a negative sequence format as follows:
c
Figure 6.6 The form of negative sequence de produced by HV de converter
AC side DC side
Positive sequence
Fundamental
second harmonic
frequency voltage
voltage distortion Convertor
( AC side r-----, distortion
second harmonic
---r-J
switching
impedance action .~
r
Po sitive sequence Ideal transformer
DC side
fundamental "'
second harmonic -< ~~ frequency
current distortion L_..., impedance
J
I __
Convertor ~
Transformer - switching
I
Fundamental
Mu ltitude of core Majority of negative action frequency current
dis tort10ns at saturation sequence de current distortion
many frequencies distortion
,----...,
A small part of negat1ve
sequence de current distortion
-----~Ideal transformer
The techniques used to analyse this instability can be grouped into the three
categories of direct frequency domain [6,8], iterative frequency domain [7,9], and
184 6 HARMONIC INSTABILITIES
time domain simulation [10]. Due to the complicated nature of the interaction of the
non-characteristic harmonics around the converter, the two latter numerical
methods have been more popular than the linear approximation approach. This
preference is further enhanced by advances in computer technology, enabling
complicated and numerically intensive algorithms to be implemented in a short time.
However, the direct frequency domain technique provides greater insight into the
mechanism of the interaction and should lead to more effective control solutions.
The following sections introduce a direct frequency domain method [8], which has
been used to reveal various characteristics of the ac-dc systems susceptible to this
instability [11].
To analyse the instability mechanism as shown in Figure 6.7, the ac-dc system is
simplified to the equivalent circuit of Figure 6.8 consisting of a converter block
interconnecting the ac and de side impedances at the relevant frequencies. On the ac
side, the positive sequence second harmonic current Iacp generated by the converter
flows into the ac system second harmonic impedance Zacp• while the negative
sequence de IaCil flows into a parallel circuit of transformer magnetizing inductance
Lm and ac side impedance at the low frequency of Iacn variation. Assuming that the
ac side impedance remains fairly constant around 0 Hz, the impedance can be
simplified to the ac system resistance at 0 Hz of Racn· On the de side, the presence of
fundamental frequency voltage distortion causes an equivalent current distortion to
flow through the de side impedance.
The converter block accounts for the transfer of ac side voltage distortion to the de
side and the de side current distortion to the ac side. In reference [8], this operation of
the converter in the frequency domain is represented with a linearised transfer
function describing the conduction and non-conduction periods of the thyristors.
Considering only the most significant low order harmonics, these interactions have
been analysed in Section 5.4.1 with the help of a 3 x 3 matrix, which is reproduced in
/acp
Positive sequence
second harmonic
Fundamental
zacp /2+ vacp frequency
- -
IUCII
li
Negative sequence
v<.lch zclch
de harmonic
Ia.
Racn vacn
Equation (6.5). The elements within the matrix describe the amplitude change and
phase shift introduced during the transfer of distortions from either sides of a
converter. The response of the converter controller and the signal transducers is
included in the matrix.
Vdch]
[ Iacp = [a
d
b
e
C] . [Vacp]
f Vacn . (6.5)
lacn g h I ldch
From Figure 6.7, the contribution from the transformer saturation to the
instability feedback loop comes in the form of an additional positive sequence
second harmonic current resulting from the saturation. This effect is modelled as a
positive sequence second harmonic current injection lz+ into the second harmonic
part of the equivalent circuit. The depth of the saturation is calculated according to
the amount of negative sequence de from the converter that is flowing into the
transformer magnetizing inductance Lm (i.e. / 0 _).
The transformer saturation related to this instability can be regarded as
asymmetrical saturation since the transformer is only saturated in one-half of a
fundamental cycle, as shown in Figure 6.9. As the transformer is coming in and out
of saturation, the magnetising inductance Lm is non-linear, but it is only in
saturation for a short period of each cycle as indicated by the width of the distorted
magnetising current pulse. Therefore, it is reasonable to assume Lm as the
unsaturated magnetising reactance, as indeed it has this value most of the time.
Under the worst case scenario, the transformer magnetisation flux is taken as
reaching the limits of the non-saturated part of its magnetisation characteristic, and
the magnetisation characteristic of Imag/flux is approaching infinity in the saturated
region, as shown in Figure 6.9. Algebraic analysis has shown that under this
condition, there is a one-to-one linear relationship between the resulting positive
sequence second harmonic current lz+ and the level of saturating negative sequence
time
time
''!'"''"
'-- fa
dc/0 _ [8]. However, converter transformers are usually over-designed and there is always
a considerable margin before reaching the saturation region. Moreover, the actual
Imag/flux ratio in the saturation region is far from infinite, and therefore, the relationship
between h+ and / 0 _ will realistically be less than one, depending on the magnetization
characteristic of the transformer, i.e.
12+ = -X.lo -, O<X < l. (6.6)
The reference point for phase is, as per the converter analysis, at the peak of the red
phase fundamental voltage component.
Dynamic simulations are utilised to approximate this ratio X. Using a three-phase
star-star transformer, a series of negative sequence de currents are injected into the
transformer secondary while measuring the amount of positive sequence second
harmonic flowing into the primary. The amplitude of the negative sequence de
current is ramped up in stages and the corresponding amplitude of the positive
sequence second harmonic current is plotted against it as shown in Figure 6.10. The
de injections are determined as percentages of the transformer rated magnetizing
current, making the extent of the transformer saturation which is determined by the
de flux levels correspond directly to the level of injected currents. The measured
values show a nearly linear relationship and by applying linear approximation, the
slope of the curve is calculated as the ratio X for that particular transformer.
Figure 6.10 shows that at saturation levels below 40-50%, the measured values are
less than the linear approximated values. The non-linear effect at low saturation
levels is determined by calculating the value of X from the slope between the
measured h+ and the injected / 0 _ at each saturation level. The various values of X at
different low saturation levels are summarised in Table 6.3 alongside the linear
approximated values.
Table 6.3 also depicts the influence of the transformer knee point voltage level on
the ratio X. Transformers with higher knee point voltage will be less prone to
saturation and, hence, result in lower harmonic contribution. The value of the ratio
70
60
50
e"'
- 40
0
c. 30
20
10
I o. (% of I mag)
X will also vary according to other characteristics of the transformer, including its
levels of magnetising current and saturated reactance.
Therefore, depending on the category of core saturation instability (i.e.
spontaneous or kick-started) under study, a series of simulations need to be
undertaken on the particular transformer under analysis, and the calculated
relationship linearized and applied to the model to predict the system stability.
With the 3 x 3 matrix describing the converter operation, and the current injection
modelling the transformer saturation effect, the stability of the system can be
determined from the roots of the system characteristic equations. By expressing one
of the concerned distortions, such as Iacn, in the exponential vector form "'';:~ .e-<cx+iPlt
with "'';:~ as the initial condition, the distortion will decay away resulting in stable
systems if the real part of the root -a is negative (i.e. a is positive). This term a has
been defined as the Saturation Stability Factor and used to indicate the susceptibility
of an ac-dc system to the development of core saturation instability [8].
This Saturation Stability Factor technique has been verified against dynamic
simulations [8], but for completeness, a further validation is presented here. The
parameters of the CIGRE benchmark model [3] were modified to create an unstable
test case [12]. The resultant Saturation Stability Factor was lowered to become
negative at -0.152 and the presence of instability was verified by the EMTDC
dynamic simulation results of Figure 6.11. The system was run for one second to
achieve a steady state, and then a fundamental frequency modulation was added to
the rectifier firing angle order. This modulation caused harmonic distortions on both
sides of the converter, including the negative sequence de, which began to saturate
the transformer. The system was subjected to this modulation for 0.5 second, and its
stability assessed by the growth or decay of the harmonic distortions after the
external stimulus is being removed.
The gradual rise of the distortion in the de current indicates the onset of the
instability. As the instability develops, the level of transformer saturation increases
which tends to accelerate the development of the instability as indicated by the large
increase in the amplitude of the saturating negative sequence de after 3.5 seconds.
The waveforms of the transformer magnetising current show that the three phases
188 6 HARMONIC INSTABILITIES
DC Current
... - ....j - -'- -1- - ,_ - 1- - .j... - -' - __j - _,- -I- - :.... - .... - A - -j
0.25 ~ - ._- -~- ...J- -'- -~-- ' , _ - ~ ·- l . - ..J- ....I- -'- -1-- L- .I...-_..- J
000
05 1.0 1.5 2.0 2.5 l.O l.5 40 45
nme (seconds)
200 ·- -·
150 L - ... - .. - ~ - -•- -- 1- - r- - r - .,. - 1 - -,- -1- - r- - !"' - ..,. -
c 100 ·- - ... - .. - -1 - _,- -1- - ...... - .... - ... - - - _,- -1- - 1- - .... -
so . . . - l... ..... -- - - -I- -1- - ._ - ._ - ..J. - -I - -1- -
0 '
05 , 0 , 5 2.0 2.5 3.0 l.5 40 4.5
Time (seconds)
~ l~ =~ ~ ~ ~ =~ =~ ~ ~· ~ ~ :~ ~ :~ ~ t ~ i ~ ~ ~~~~·~ ~ !
·1 .00 -
.• so _ _
- :__ -
____;_~
.: - - - - I- - - - - - l - j - _j - -'- -!- - - - - - !
"'!
lE ~- = = = =:=Ii ~·;s;7=e';.:·;\_=B
. I
~~re;•;g*{;"~§' =~c =<= =_= =~-
1,;..;_1~~~~ • .
£ ~~ :...
_1 00 .....
-
_
'- - " - - - -·
!... _ .!.. _ .J __ . __ r __ '- _ :__ !.... _ 1. _ _ _ _ _ _ 1 _ _
- -'- -"· ~""! - ijllji q~~lffi'Blll " ~J.~'HI"U______ 1_ _ _ _ ~ _
l
;
-\ S0 I I 1 ___j
0.5 1.0 15 2.0 2.5 3.0 J5 '0 ' 5
Time (seconds)
Figure 6.11 EMTDC simulation results of the modified CIGRE HVdc benchmark model
are not de-biased to the same offset level, and there is a pseudo-sinusoidal variation
alongside the increase in the saturating de. This observation further validates the
vector form of solution (e-cxC+iPlt) proposed by the Saturation Stability Factor
approach that the distorting harmonic sequences not only vary in their magnitude
but also rotate over time. In this test system, the vector rotation term of f3 was found
to be positive at 0.188, which implies that the negative sequence de vector of Figure
6.6 will rotate in the clockwise direction, forcing the de component in the
magnetizing current to vary in the phase order of A, B, C, as shown by the
simulation results.
The direct frequency domain approach described above possesses great potential for
the evaluation of the properties exhibited by systems prone to this type of instability.
Its minimal computational burden and hence quick solution provides an effective
6.3 TRANSFORMER CORE RELATED HARMONIC INSTABILITY 189
way of investigating the system characteristics under such unstable conditions. With
the use of linear approximation and a direct solution, each individual factor can be
easily altered to unravel its particular effect on the instability mechanism.
Furthermore, by considering several factors simultaneously, it is possible to find
out the dominant factor contributing to the build up of the instability. Analysis using
the Saturation Stability Factor, described in [11] has revealed that a vulnerable HYde
rectifier system is likely to have the following impedance profile:
On the other hand, a susceptible invertor system will possess opposite reactive
characteristic with inductive de side impedance at fundamental frequency, and
capacitive ac side second harmonic impedance. A high 0 Hz resistance is also
observed at the unstable invertor station but the two reactive components have the
dominant role in determining the system stability.
The common use of HV de back to back interties to interconnect large and weak ac
networks has resulted in low order resonances at the converter terminals, making
them prone to core saturation instability. However, with comparable network sizes
at both the rectifier and invertor ends, this harmonic instability is most likely to
occur only at one end of the scheme. This is due to the opposite reactive
requirements of the impedances for the instability to occur at either ends. Moreover,
the high resistance at the unstable end will be reflected onto the de side as additional
damping which tends to stabilise the opposite end system. Therefore, for the back-to-
hack scheme, it is necessary to consider the consequential impact on the stability of
the remote end system when undertaking any modification at the local end.
Besides the system impedances, the stability of the ac-dc system is strongly
dependent on the response of the converter controller. Considering the stringent
reactive requirements for the instability to develop, the onset of this harmonic
instability almost certainly involves a destabilising contribution from the converter
controller. This suggests the possibility of preventing the onset of the instability
through proper tuning of the converter controller.
the frequencies related to this instability, it usually affects the system response at
other frequencies as well. The design of such preventative measures has to ensure
that other system requirements or constraints are still met after the modifications.
These actions can be broadly regarded as passive measures.
On the other hand, active measures can be applied to stabilise the system when
the development of the instability is detected. This type of solution has been used
to prevent core saturation instability in existing schemes, with some sensing
instruments estimating the level of transformer saturation and appropriate action
taken in accordance with the extent of the saturation. Active measures should be
designed to function at certain limited range of frequencies without altering the
system response significantly under normal operating conditions. This will allow
the system to be operated as usual, but with the added security of some stabilizing
action when the instability is suspected.
Due to the great differences in the characteristics of the various HVdc systems, it
is difficult to pinpoint which is the best solution to counter this instability. The
high dependency of the system stability on the properties of the HVdc scheme
suggests that the most appropriate solution for one system may not suit the others.
Moreover, each HVdc scheme usually has its own unique requirements and
restrictions that have to be fulfilled. It is therefore necessary to undertake
independent analysis for different systems or for a similar system under different
operating conditions.
The addition of a high pass filter to the converter controller has been found to be
effective for this particular test case, but had to be properly tuned to avoid exciting a
composite resonance at about 70Hz [12] With the high pass filter, the system
Saturation Stability Factor was evaluated to be positive at 0.168 indicating stability,
which is confirmed by the dynamic simulation results of Figure 6.12.
6.4 Summary
The linearized direct frequency domain model, described in Chapter 5, has been used
to simulate the converter and a composite resonance concept has been introduced,
DC CurTent
:.:: ' ' ' ' ' ' ' ' I • '
-
Time (seconds)
0 .
0.5 1.0 1.5 2.0 2.5 3.0 3.5 40 4.5
Time (seconds)
Figure 6.12 EMTDC simulation results with the addition of an auxiliary high pass filter to
the converter controller
6.5 REFERENCES 191
involving the converter, converter controL and ac and de side harmonic impedances
as an integrated whole. The ac-dc system has been represented as an electrical
network.
It has been shown that the converter control system is a critical factor in
determining system waveform stability, and that the dominant frequency is not
limited to integer multiples of the fundamental frequency.
A description of the interactions of three frequencies around the converter has
helped in predicting the system susceptibility to converter transformer core
saturation instability.
Due to the sensitive dependence of the system stability on numerous parameters, it
is necessary to undertake detailed analysis on the particular installation in order to
reach the most appropriate solution.
Although in the simple example a constant current control is used, the set of
equations can be manipulated in a similar way to analyse the effect of any control
strategy. Further to this, the design of special control blocks to manage composite
resonances at higher frequencies may be undertaken.
The technique developed here should prove a useful tool in optimizing converter
control systems, with particular regard to recovery from system transients and
avoidance of harmonic instabilities.
6.5 References
1. Ainsworth, JD, (1967). Harmonic Instability Between Controlled Static Converters and
a.c. Networks, lEE Proceedings, 114(7), 949-957.
2. Ainsworth, JD, (1968). The Phase Locked Oscillator-A New Control System for
Controlled Static Converters, IEEE Transaction on Power Apparatus and Systems, PAS-
87(3), 859-865.
3. Szechtman, M, Wess T and Thio, CV, (1991). First benchmark model for HVdc control
studies, Electra, 135, 55-75.
4. Ainsworth, JD, (1977). Core Saturation Instability in the Kingsnorth HV-d.c. Link, Paper
presented to CIGRE study committee No.l4, Winnipeg, Canada.
5. Chand, J and Tang, D, (1987). Experience with Resonances and Oscillations in the Nelson
River HVdc System, HVdc System Operating Conference, Winnipeg, Canada.
6. Stemmler, H, (1987). HVdc Back-to-back Interties on Weak a.c. Systems, Second
Harmonic Problems, Analyses and Solutions, CIGRE Symposium, 09-87, Paper No. 300-
08, 1-5, Boston.
7. Hammad, AE, (1992). Analysis of Second Harmonic Instability for the Chateauguay
HVdc/SVC Scheme, IEEE Transactions on Power Delivery,. 7(1), 410-415.
8. Chen, S Wood, AR and Arrillaga, J, (1996). HVdc Converter Transformer Core
Saturation Instability: A Frequency Domain Analysis, lEE Proceedings-Generation;
Transmission; Distribution, 143(1), 75-81.
9. Yacamini, R and de Oliveira, JC, (1980). Instability in HVdc Schemes at Low Integer
Harmonics, lEE Proceedings Pt. C, 127(3), 179-188.
10. Burton, RS, (1994). Report on Harmonic Effects on HVdc Control and Performance,
CEA No. 337 T 750, Prepared by Manitoba HVdc Research Centre.
11. Chen, S, Wood, AR and Arrillaga, J, (1996). A Direct Frequency Domain Investigation
of the Properties of Converter Transformer Core Saturation Instability, Accepted for
192 6 HARMONIC INSTABILITIES
7.1 Introduction
In the models described in Chapter 4 the generators and transformers have been
represented by frequency dependent impedances at harmonic frequencies.
In practice, design restrictions and cost effectiveness will result in some distortion
of the machine's internal emfs. Moreover, the use of iron cores makes the
electromagnetic coupling between windings of rotating machines and transformers
non-linear due to saturation. Also, the frequency conversion process of the non-ideal
rotating machine will react to the presence of system asymmetry or distortion by
returning other frequency components.
Under certain operating conditions some of these effects may influence the
network harmonic content and need to be represented in the analysis.
As in the case of the ac/dc converter, discussed in Chapters 5 and 6, machine non-
linearities interact with the network parameters and operating conditions and, in
general, iterative rather than direct solutions are required. The iterative methods are
discussed in Chapter 9 but the machine models involved in the solution are described
here, with particular regard to the synchronous generator and transformer. These are
described in a Harmonic, rather than Frequency, Domain where the interaction
between all the relevant harmonics is solved in a common unified algorithm.
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
194 7 MACHINE NON-LINEARITIES-HARMONIC DOMAIN
8
ae
8
The asymmetry of the winding distribution and structure of the rotor, coupled to
some unbalance or distortion in the stator currents. create harmonic mmfs in the
rotor which in turn induce harmonic voltages in the stator.
7.2 SYNCHRONOUS MACHINE 195
Figure 7.2 Response of a salient pole generator to the presence of a harmonic current
196 7 MACHINE NON-LINEARITIES-HARMONIC DOMAIN
The variables s and t in Equation (7 .1) represent closed damper windings in axes d
and q, respectively. In the analysis to follow, it is assumed that the applied field
voltage vr contains no harmonics and, therefore, this variable has been set to zero in
Equation (7.1c). The harmonic domain linearization process requires a phasor
representation of Equations (7.1). At a particular frequency h, the required variables
of Equations (7.1) are described in phasor form as follows [6]:
(7.2)
p=jhw (7.4)
Writing Equations (7.1) in phasor form and solving for dq quantities yields
(7.5)
(7.6)
-sin wt] [
coswt
vd,]
Vq,
(7.7)
Phasors Vdh and Vqh can be expressed in the trigonometric form of Equation (7. 7) as
7.2 SYNCHRONOUS MACHINE 197
where
Similar expressions apply for cos wt and sin wt. Substitution of Equations (7 .8)
and (7.9) into (7.7) yields, when combined into phasors of the form given by
Equation (7.3), the following equations
M*
N* M*
N* 2M*
[C] =! N* N (7.12c)
2M N
M N
M
where
198 7 MACHINE NON-LINEARITIES-HARMONIC DOMAIN
1.
M=N*=[ -J j]
1
(7.13)
To derive [Y~p] the general matrix equation is first written for all harmonics as
(7.14)
where
(7.17)
where
Bo = MYdqoN/2 (7.19e)
B1 = MYdqlN/2 (7 .19f)
A direct admittance transformation from the two-phase afJ reference to the three-
phase abc cannot be achieved because the conversion matrix has non-square form,
thus precluding inversion. This problem is solved by adding to [Ya[J] a zero sequence
diagonal matrix of order equal to the harmonic spectrum analysed and also adding
to the transformation matrix, defined as T 11 for a particular harmonic h, a third row
of equal numerical constants. The augmented matrix [Y,[J] has the form
(7.20)
The zero sequence component satisfies the condition of having zero contribution
or being uncoupled from the machine :xfJ and dq components [5]. With the
incorporation of the zero sequence component the following relationships can be
defined for any harmonic h
200 7 MACHINE NON-LINEARITIES-HARMONIC DOMAIN
(7.2la)
(7.2lb)
where
[~
3
1
-:1
T
1
3
~~]1
3
2
(7.22)
(7.23)
Dropping the subscript h and substituting Equations (7.22) into Equation (7.23)
leads to
(7.24)
where
(7.25)
In Equation (7.24) the terms abc are sequentially accommodated for each
harmonic h in blocks of order 3. A different representation is required for the
harmonic domain solution where the complete harmonic spectrum (positive and
negative harmonics) is used for each phase. As an example, considering three
harmonics and a de term, the current and voltage vectors of equation (7.24) have the
form
(7.26b)
and the structure of matrix [Yabc], with dots indicating non-zero elements, is
illustrated in Figure 7.3.
a h c
-l -2 -1 -3 -2 -1 0 -.l -2 -1 0
-3
_ 2 1r"~--~-r_,--+--+_,~-r_,~+--t--r-~_,r-,_~~+--+--r-,_~l
-llr"~--~~_,~+--+_,~-r_,~+--+~r--r_,~-r_,~+--+~r--r~l
-3
•
-2
-1
3
-l
-2
-1
0
Z!i; and z=i: are defined as the open circuit harmonic impedances while z;;- 2 and
Z~1 2 are defined as the open circuit conversion impedances.
Figure 7.4 shows that the magnitude of the (self) harmonic inductance is very close
to that of the commonly used approximation.
15
(i)
:i 10
8
"'c::
(J
"'
"0
"
~ 5
(ii)
(iii)
5 15 25 35 45
Harmonic order
Figure 7.4 Magnitude of the hydrogenerator harmonic impedances. (i) Z!i: and Z~j,.
.. ) z+lh+2J ( ... ) 2 -ch-2J
(11 -h 111 0 +h
202 7 MACHINE NON-LINEARITIES-HARMONIC DOMAIN
very inductive, i.e. their phase angles are higher than 87o at frequencies higher than
the fundamental. However, experimental information obtained from negative
sequence injection tests indicates that in practice the power factors are higher. The
difference is expected to be largely due to the presence of eddy currents.
Generally, the effect of saturation is taken into account by iteratively modifying the
self- and mutual inductances on the dq axes [8].
Saturation depends on the total mmf rather than on separate contributions of the
d and q axes [9].
Using the original dq differential equations a phase co-ordinate model is
developed here to obtain a Norton harmonic equivalent to represent magnetic
7.2 SYNCHRONOUS MACHINE 203
0.2
0.1
()In,•
2 3 4 5 6 7 8 9 10 11 12 13
Harmonic order
2 4 6 7 8 9 10 11 12 13
Harmonic order
~
E 0.10
·a
~ 0.05
:::8
2 3 4 5 6 7 9 10 11 12 13
Harmonic order
zoo~----------P_o_s_h_i\_'e_-_se_q~u_e_n_c_e_h_a_rm
__o_ru_·c_·v_o_l_ta~g~e_s__________~
100
ll)
~ -100
2 3 4 6 7 8 9 10 II 12 13
Harmonic order
Figure 7.5 Comparison of time and harmonic domain solutions for generator harmonic
voltages with harmonic current injection
The stator magnetic and electrical parts are represented by the state equations
describing the non-linear behaviour of an ideal inductor, i.e.
v = p¢, (7.27)
i = /(¢). (7.28)
L1 V = jhwi1¢ (7.29)
(7.30)
where
(7.31)
(7.32)
where
Equation (7.32) represents the Norton harmonic equivalent for the generator
stator combining together its magnetic and electrical parts. iN is a vector of Norton
harmonic currents and [Hg] is a square harmonic matrix of Norton magnetic
admittances.
The Norton harmonic equivalent representing the effect of saturation is combined
with the machine abc quantities according to the winding connections. As an
example, the structure of the resulting matrix [Yabc] + [Hg] is shown in Figure 7.6 for
the case when the machine windings use a grounded Star configuration.
Figure 7.6 illustrates that explicit cross-coupling exists between even and odd
harmonics in phases a, b and c. However, even harmonics are only present if
hysteresis effects are taken into account, if they are externally excited or if the
magnetising characteristic is asymmetrical on its positive and negative regions. For
single-valued symmetrical magnetizing curves, only odd harmonics are produced and
only cross-coupling between these harmonics takes place.
7.2 SYNCHRONOUS MACHINE 205
a b c
-3 -2 -1 -3 -2 -1 -3 -2 -1 0
-3
-2 !--"---!--=---t--'+-"-----+--+-----ll---=-t-t-"-1---t-+--+----l r--r--r--r--r--r~r-~
-1 1---~-=--t---t--'~--t--1-----ll
o r--r~r--r~r--r~~~r--~-r--r--r--r--r~r--r~r--r~r--r~r-~
-3
-2
-I
0 1---1--=-t---t--'+--~~~r--1--~--=--~~~~-r~r--l--~--~~--~-~--~l
I
-3
-2
-1
(7 .34)
(7 .35)
206 7 MACHINE NON-LINEARITIES-HARMONIC DOMAIN
The currents I(+) and Ig(+) can be written in terms of currents I and Ig as
Ic+l = TI (7.37a)
where
fundamental h
a a2
[T] =!3 a2 a 0 (7.38)
a a2
0 0
Substitution of Equations (7.37) into Equation (7.36) and re-arranging terms leads
to the following matrix expression
(7.39)
Substitution of Equation (7.40) into Equation (7.34) finally leads to the following
matrix equation
(7.41)
Ll l.l
0.6 0.6
O.l O.l
.,
..""
~ -0.4 -0.4
-0.9 -0.9
-1.4 +--~---.,------,------,----+- -u
0 12 16 20
Time ( msec. )
Figure 7.8 Voltage waveforms across unbalanced load; generator represented with x;;
~
LS / - -;f 16
E
~ 0.6 t:_ 06
~~--~~'--~--~~~
ci.
.
., ' / E
5-0.4 t -OA
~ ~
t
] '7--
--21..44 ., / -u
-=~!------,---~---~---,.----+-~ -2.4
12 to 20
Time ( rnsec. )
Figure 7.9 Voltage waveforms across unbalanced load; detailed generator model without
saturation
7.3 Transformers
When determining the harmonic impedances of a network or performing harmonic
penetration studies (Chapter 4) from given harmonic current injections, the
transformers were represented by linear impedances, i.e. their magnetization non-
linearities were ignored on the assumption that the transformers normally operate
within the linear region of the magnetizing characteristics. However, transformers
208 7 MACHINE NON-LINEARITIES-HARMONIC DOMAIN
5.0
4.0
(a)
~
.,
0>
3.0
..,"'
~ 2.0
>
1.0
7.0
6.0
5.0 (b)
;;;
.,
0> 4.0
..,"'
0
> 3.0
2.0
1.0
Figure 7.10 Variation of third harmonic voltage with transmission line length. (a) Negative
sequence, (b) Positive sequence
are designed to operate very close to the limit of the linear characteristics and, even
under small over-excitation, their contribution to the harmonic content is often
important.
A more accurate model of the transformer for harmonic studies is developed in
this section, describing first the case of a single phase transformer and then extending
the model to multi-limb multi-winding units.
Each magnetizing curve can be stored in the computer as a set of points(¢- i), such
that each flux value impressed in the magnetizing characteristic numerically provides
the corresponding current value. However, a significant number of points is required
for an accurate solution, although at the expense of increased computation time.
Alternatively, the experimental magnetizing curves can be analytically approxi-
mated.
7.3 TRANSFORMERS 209
(7.42)
where
The term ~¢ allows the modulation of the knee region. When this is not required
~¢ = 0. The solution of the hyperbolic function in the first quadrant leads to the
following expression for the magnetizing current
where
A =m 1m 2
B = m1(b2- ¢) + m2(b,- ¢)
c= ¢2 - ¢(bl + b2 + ~)
Four significant digits in the specified variables m,
and m2 have been found
sufficient to ensure an accurate determination of the magnetizing current values.
where
L
00
L
00
11i = 11i,ejhwt
h=-oo
L
00
c = J'(ljJ(t)) = C; ejkwt
i=-00
210 7 MACHINE NON-LINEARITIES-HARMONIC DOMAIN
·O
L
magnetizing currents
(7 .46)
The linearization IS about a point (l/Jo ,io) in the space of complex conjugate
harmonics, where
(7 .47)
/1i=i-i 0
•
(7.48)
(7 .49)
7.3 TRANSFORMERS 211
-1
l.jJ
j
Figure 7.12 Norton equivalent circuit
FFT i ,i", I,
+
r' = ( ik+l- ik_.)/("f;k+l -'lh-1)
~
FFT -- c ,c· ,[H]
(7.51)
at any junction.
For the case of the three-limb transformer shown in Figures 7.14 and 7.15, the
winding flux values </> 1 to 4> 3 can be calculated from Faraday's law <f>s = 1/ N f v5 .dt).
Therefore, the magnetic equivalent circuit will have eight unknown variables, i.e.
three magnetizing currents i,, i2 and i3 and five fluxes, </> 4 to ¢ 8. Five basic equations
can be derived from the first magnetic circuit law,
i, =f,(¢,)+!6(¢6) (7.52)
i2 =f2(4>2)+f?(</>7) (7.53)
i3 =!3(4>3) + fs(</>s) (7.54)
i, - i2 = j, (</>,) +f4(</>4)- f2(4>2) (7.55)
i2- i3 =!2(</>2) + fs(</>s)- 13(</>3) (7.56)
7.3 TRANSFORMERS 213
where the branch mmf drop across each branch reluctance has been represented by
the corresponding magnetizing characteristic i = f (¢). The other three necessary
equations are obtained from the second magnetic circuit law
where the subscript b denotes the base values and the magnetic reluctance Rkb is the
derivative of the magnetizing characteristic h, =fk(rfJ 1J with respect to the base flux
rPkb(Rkb = f'(rPkb), k = 1,2, 0 0 °, 8) 0 Adding Equations (7057) to (7059) to this set of
linearized equations, and expressing them in matrix form, with the magnetizing
currents and rjJ 4 to r/J 8 as functions of the winding fluxes, leads to
il R!brP! -a!
i2 R2brP2- a2
i3 R3brP3- a3
rP4 R!brP! - R2b¢2 - a4
[H] (7065)
cPs R2brP2 - R3b¢3 - as
rP6 rP!
rP7 rP2
cPs rP3
where
-R6h
-R7b
-Rsb
-1 -R4h
[H] = -1 -Rs,
-1 -1
and
Equation (7065) produces the following expressions for the magnetizing currents
7.3 TRANSFORMERS 215
R 1¢ 1 -a 1
R2¢2- a2
R3¢3- a3
R 1¢ 1 - R2¢2- a4
(7.66)
R2¢2 - R3¢3 -as
¢,
¢2
¢3
where the matrix [d] comprises the first three rows of [Hr' (3 x 8). These matrix
equations can be re-arranged in the form
(7.67)
or
where s= 1, 2, 3.
The interface with the external electrical system is made by relating the
magnetizing currents to the winding voltages. Using harmonic phasors and the per
unit form, Faraday's law is expressed by
V5 = j hw¢,.. (7.69)
where
[Y] = [K]Diag{jhw}- 1
and
where s= I, 2, 3.
The elements of matrix [K] are defined below
The full harmonic model is obtained with the incorporation of the magnetically
coupled core contribution, represented by [Y] and i11, in Equation (7.70). The per unit
magnetizing admittance is often distributed equally between all terminals [13], i.e.
halved and placed at both sides of the leakage admittance. This approach has been
used here for the treatment of the self and mutual terms of the Norton admittance
matrix [Y] and the Norton current injections, described by ins·
However, a more rigorous distribution between primary and secondary windings
at each frequency can be derived from the Steinmetz 'exact' equivalent circuit, i.e.
I* I lzexactl
[(stm;::, == ~
I
== prim (7. 71)
I Imprim I lzexact*
sec
I
where
and superscript * is used to indicate parameters referred to the primary side. The
subdivision of magnetizing current components needs to take into account the
impedances of the source (primary system) and load (secondary system); thus the
K51111 ~ ratio will, in general, be different for each harmonic and the magnetizing
harmonic current injections of the primary and secondary will be very different
under near-resonance conditions.
The application of the linear transformations given in Section 3.7 allows the
determination of full harmonic electromagnetic models for multilimb power
transformers. Generalized models for transformers in star-star and grounded star-
delta configurations are represented by the harmonic nodal matrix Equations (7.72)
and (7.73). Models for other transformer connections can be directly obtained from
these basic equations. Figures 7.16 and 7.17 illustrate their associated harmonic
lattice equivalents.
A B a b
The magnetizing characteristics associated with the winding, yoke branches and
zero-sequence flux paths are represented by curves 1, 2 and 3 in Figure 7.19.
For verification purposes, the test system was also analysed in the time domain,
using the EMTDC program [15] with a 50 ps time step.
The results of the simulations involve the first 15 harmonics and convergence was
achieved to a tolerance of 0.0001 (p.u.) for the harmonic voltages. These are shown
in Figures 7.20(a) and 7.20(b) for the magnetizing current waveforms and their
218 7 MACHINE NON-LINEARITIES-HARMONIC DOMAIN
Connection: Star-Star
!A- aln!
Yt + IXJ'31 Yt + IXJ'32 Yl + IXJ'33
Is- a/,12
Ic- al113
-y,- Y/ 0 Yt+ Y/
fa - fJJ111 + {J/113
I, + fJJnl - fJ/112
I, + f3In2 - f3In3
y,+y/' -y,- Y/ 0
IN+ ctlnl +
ctln2 + all13
0 J'1 + Y7 -y,- Y/
Connection: Star-Delta
7.3 TRANSFORMERS 219
r"I
~
r"
- I
-yl- 2y/ Y1- 2y'(
-:XYtt - :XY12
-:xyl3
-y, y'( -y,- 2y/ y,-2y'(
-:xy2l - :X)'22
-:xy23
y'( -y, -.Vi- 2y/ y-
I 2v"
~ I
-:XJ'3l - :XJ'32
VA
-i)(J'33
-yi- 2y/
Vs
Y/" + f3Yt2 y/" + f3yl3 Yt-2y'(
Vc
-fJYll - fJY12 - f3Yt3
-yi- 2y'('
Va
Yt + fJY22 Y/" + f3Yn Yt- 2y/'
-f3y2l - fJY22 - f3y23
vb
y'(' + f3y33 Yt-2Y'J -y,- 2y'f'
vc
Yt + f3Y32 VN
- f3Y3t - f3Y32 - f3y33
Yt- 2y/' y,-2y'J -3y, + 6y/' -3yl + 6y"
v/1
+aJ'11 + :XJ'12 + :XJ't3
+:xy2l + ay22 + :XJ'23
+ay31 + ay32 + :XJ'33
-yi- 2y/" -y,- 2y/" -3yl + 6y/" +3y, + 6yf"
-f3yl2 - fJY22 -f3Yt3 - fJJ'23 +fJYll + f3Yt2 + f3Yu
-f3Y32 -f3Y33 +fJY21 + fJY22 + f3y23
+f3Y3t + f3Y32 + f3y33
(7.72)
12
2
--
.. --- ,,
3
~
><
~0.6
•.. ,,
~
I
02 1
I
I
I
•.;-,--;.;';_,:------;;._,
0'o~-----:._:-,-~.-:;2;---;:•.>~-.:-._,:---;;.'";_s--:
current(p.u.)
0.030
0.10
/\
:' ''' 0.025
0.05 '
ci -; 0.020
s s 0,015
=
"~
=
t:
u" u" 0.010
-0.05
/ '' 0.016
''
"'
ci
s... sci 0.012
..
! '
! '
I
~
JS" JS" 0.008
i
i
> ' >
-0.5 ;
''
fl
0.004
-1.0 .. - ...
0
0 0.004 0.008 0.012 0.016 0.020 11 13 15
Time(sec) Harmonic order
(c) (d)
Figure 7.20 HDA simulations-Example
spectra, respectively. Due to the saturation and magnetic coupling between the three-
phases, the three-limb transformer magnetizing currents have two maxima within
one half-wave. The second and smaller one is not very significant in this example due
to the slight transformer saturation considered. The corresponding voltage
7.5 REFERENCES 221
waveforms at the transformer primary side and their harmonic spectra are illustrated
in Figure 7.18(c) and 7.18(d). Although the voltage distortions caused by the
magnetizing currents are within 1.0% of the fundamental, it should be pointed out
that the individual harmonic levels permitted by legislation are usually of this order.
Of course, the magnitude of the harmonic voltages are affected by the overvoltage
levels and by the frequency-dependent network impedances. By increasing the
infinite bus source to 1.2 (p.u.), or the series reactance to 1.0 (p.u.), distortion levels
of up to 4% are calculated.
7.4 Summary
The harmonic domain, an alternative frame of reference to represent the process of
frequency conversion in the presence of non-linearities, has been introduced. All the
relevant harmonics are explicitly and simultaneously represented, including
frequency-dependent factors such as skin effect and, therefore, this method provides
greater accuracies than time domain simulation. It is also more efficient
computationally since it avoids the long runs of time domain simulation needed to
reach the steady state.
In the harmonic domain the order of the network matrix admittance is equal to
the number of nodes times the number of phases multiplied by twice the number of
harmonics (when positive and negative frequencies are used).
This chapter has only discussed the synchronous generator and transformer non-
linearities. In the case of the synchronous machine, the harmonic Norton equivalents
include the effect of rotor saliency and machine saturation. The Norton equivalent of
the transformer represents the effect of saturation in the various magnetic branches
of the multi-limb configuration.
Because of their greater contribution to the harmonic content, the static converters
are given detailed consideration in the following chapters.
7.5 References
1. Park, R H, (1929). Two-Reaction Theory of Synchronous Machines; Generalized Method
of Analysis-Part I, AlEE Transactions, 48(3), 716-730.
2. Park, R H, (1933). Two-Reaction Theory of Synchronous Machines-Part II, AlEE
Transactions, 52(2), 352-355.
3. Clarke, E, (1950). Circuit Analysis of A-C Power Systems- Vol II, John Wiley & Sons,
London.
4. Hwang, H H, (1965). Unbalanced Operations of A.C. Machines, IEEE Transactions on
Power Apparatus and Systems, PAS-84(11), 1054--1066.
5. O'Kelly, D and Simmons, S, (1968). Introduction to Generalized Electrical Machine Theory,
McGraw-Hill, London.
6. Semlyen, A, Eggleston, J F and Arrillaga, J, (1987). Admittance Matrix Model of a
Synchronous Machine for Harmonic Analysis, IEEE Transactions on Power Systems,
PWRS-2(4), 833-840.
7. Medina, A, (1992). Power Systems Modelling in the Harmonic Domain. PhD Thesis,
University of Canterbury, New Zealand.
222 7 MACHINE NON-LINEARITIES-HARMONIC DOMAIN
8. Anderson, PM and Fouad, A A, (1977). Power System CQntrol and Stability, The IOWA
State University Press, USA.
9. Brandwajn, V, (1980). Representation of Magnetic Saturation in the Synchronous
Machine Model in an Electro-Magnetic Transients Program, IEEE Transactions on Power
Apparatus and Systems, PAS-99(5,) 1996-2002.
10. Semlyen, A and Castro, A, (1975). A Digital Transformer Model for Switching Transient
Calculations in Three-Phase Systems, 9th PICA Conference, New Orleans, Lousiana, 121-
126.
11. Acha, E, (1988). Modelling of Power System Transformers in the Complex Conjugate
Harmonic Space. PhD Thesis, University of Canterbury, New Zealand.
12. Lisboa, M L V, Enright, W and Arrillaga, J, (1995). Harmonic and Time Domain
Simulation of Transformer Magnetisation Non-Linearities, Proc IPENZ Conf, 2, 72-77.
13. Dommel, H W (1975). Transformer Models in the Simulation of Electromagnetic
Transients, 5th Power Systems Computation Conference, Cambridge, England, 1, 16.
14. Dick, E P and Watson, W, (1981). Transformer Models for Transient Studies Based on
Field Measurements, IEEE Transactions on Power Apparatus and Systems, PAS-100(1),
106-110.
15. The EMTDC Users Manual, Manitoba Hydro, Canada, 1988.
8
AC-DC CONVERSION-
HARMONIC DOMAIN
8.1 Introduction
Using the transfer function approach, Chapter 5 has described a general linearized
solution of the converter for small levels of distortion. The transfer functions, by
means of modulation theory, have been expressed in terms of switching instants that
are themselves modulated as a result of applied distortions. The modulation of the
switching instants and the transfer function shapes involve approximations valid for
small levels of distortion, and low order harmonic, sub-harmonic, and interharmonic
frequencies.
Most of the approximations made in previous chapters can be removed by
modelling the converter in the time domain, but at the expense of solution speed,
since time domain simulations must run until transients have decayed. Thus the
motivation for this chapter is the need for greater accuracy while retaining
computational efficiency.
In this chapter a general set of non-linear equations is derived to describe
harmonic transfer through the ac-dc converter in the steady state. The proposed
formulation convolves periodic sampled quantities in the harmonic domain with
their sampling functions, so that no Fourier transform is required, resulting in
substantial computational savings. The sampling functions are defined in terms of
the exact switching instants, which are obtained as part of the over-all iterative
procedure that accurately models the effect of ac voltage and de current distortion
on the valve conduction periods. As described here, the model takes one cycle of the
ac voltage as the fundamental, and so only harmonics are analysed. However, an
extension to the steady state over several cycles would allow inter-harmonics to be
solved.
The following are important considerations that must be taken into account in a
complete harmonic model of the converter:
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
224 8 AC-DC CONVERSION-HARMONIC DOMAIN
• There is a voltage drop across the commutating impedance, due to phase current
harmonics.
• The commutating impedances may be unbalanced.
• There is a conduction voltage drop across the valves.
• The firing instants are a function of the controller and periodic converter variables
(for example the de ripple and terminal voltage harmonics).
• The commutation current is affected by the ac voltage and de current harmonics.
• Each commutation ends when the instantaneous commutation current is equal to
the instantaneous de current. The de ripple therefore affects the overlap angles.
• The de voltage is a function of the unbalanced and distorted ac voltages, the
irregular firing and overlap angles, and the voltage drop across the commutating
impedance.
• The ac phase currents are a function of the irregular firing and overlap angles, the
commutation currents, and also of the de current harmonics.
The commutation circuit to be analysed is that of Figure 8.1, where Va, Vb, Ic and Id
are sums of harmonic phasors. In this diagram phase 'a' is commutating off, whilst
phase 'b' is commutating on. The commutation ends when Ic =let. Note that Ic is
always the commutating 'on' current waveform, not the current in phase 'c'. In this
case it is the current commutating on in phase 'b', i.e. it is equivalent to i" during the
commutation period. Assuming the periodic steady state, and summing voltage
drops around the commutation loop at harmonic order k:
(8.1)
8.2 THE COMMUTATION PROCESS 225
jkXaldk - Vabk
(8.2)
jk(Xa +X;,) .
(8.3)
where
(8.4)
(8.5)
Equation 8.5 completes the commutation analysis for a bridge connected to a star
connected source via an inductance. This equation is suitable for modelling the
connection to an unbalanced star-gjstar connected transformer, if the leakage
reactances and terminal voltages are referred to the secondary side after scaling by
off-nominal tap ratios on the secondary or primary windings. This issue is addressed
fully in section 8.6.
Xc
ic
~ +
c ld
ib
va
where for an inductance Xk = kX1. This set of equations is readily solved to yield the
de voltage during the commutation, and the commutation current itself:
(8.7)
(8.9)
where
controller
current transducer
G !--------':,. alpha
Jd
1 + jwT order
current order
Figure 8.3 Current controller
With reference to Figure 8.4, it can be seen that firing occurs when the elapsed angle
from the equidistant timing reference is equal to the instantaneous value of the alpha
order, i.e. a = 8; - /3;. The equidistant timing references are represented by
/3; = (i- l)n/3. The firing mismatch equation is therefore:
(8.11)
This analysis of the firing process is also valid for a bridge connected via a star-g/
delta bridge to the ac system, in which case, the equidistant timing references should
be advanced by 30o. The constant component of the alpha order, a0 , cannot be
solved for directly, as the PI control has a pole at zero frequency. However in the
steady state, the average delay angle, a0 , takes on a value that causes the de
component of the de current to be equal to the current order. This requirement is
easily expressed as a mismatch equation that has a zero crossing at the current order:
(8 .12)
angle
Figure 8.4 Method of finding the firing instants. The timing instants are assumed perfectly
equidistant (rr./3)
8.4 DC-SIDE VOLTAGE 229
where Vtird is the constant forward voltage drop through a group. This equation
states that the de voltage, when applied to the de system, causes the current order to
flow in the de system. The de voltage is obtained from Equation 8.3. Note that
Equation 8.12 is not a function of ct 0 , the average delay angle does however feature in
the firing angle mismatch Equations 8.11. When the converter is solved in Chapter 9,
the average alpha order emerges from the Newton solution.
A similar equation to Equation 8.12 can also be written to describe a constant
power control:
(8.13)
This equation has a zero crossing at a value of average de current which causes the
power order to be satisfied.
During normal conduction the positive and negative rails of the de side are directly
connected to different phases of the ac terminal via the commutating reactance in
each phase. The kth harmonic component of the de voltage is therefore:
(8.14)
230 8 A C-DC CONVERSION- HARMONIC DOMAIN
A B c e b 0 + eqn
where the subscripts + and - refer to the phases connected to the positive and
negative de rails respectively. During a commutation on the positive rail, analysis of
Figure 8.1 yields:
(8.15)
and
(8 .16)
for a commutation on the negative rail. In these equations e refers to phase ending
conduction. b to a phase beginning conduction, and o to the other phase. The
subscript p refers to the conduction interval being described.
From the known conduction pattern in each of the twelve states, Equations 8.14,
8.15 and 8.16 are used to assemble the twelve samples of the de voltage. These
samples are summarized in Table 8.1.
(8.17)
8.4 DC-SIDE VOLTAGE 231
c
l
B
Figure 8.5 Representative linear circuit for a particular conduction period with a delta
connected source
(8.18)
(8.19)
(8.20)
(8.21)
where for the commutation periods (i.e. p = 1, 3, 5, 7, 9, 11) the subscripts {e, b, o}
are a permutation of {a, b, c} according to which phases are involved in the
commutation. A similar result holds for a commutation on the negative rail.
During a normal conduction period all three phases of the voltage source
contribute to the de voltage. Figure 8.5 shows the representative linear circuit of a
particular conduction period. This circuit is analysed by first writing nodal and loop
equations at harmonic k:
samprmg f unction p
'I\
....,
.;
21t
(8.23)
As for the star connected source, the solution for the de voltage samples is
generalised over all twelve conduction periods into a matrix of coefficients of the de
and ac sources, ie
(8.24)
(8.25)
(8.26)
otherwise
8.4 DC-SIDE VOLTAGE 233
sample (p) aP bp
1 (}j 4>1
2 4>1 fh
3 82 4>z
4 4>z 83
5 fh 4>3
6 4>3 84
7 84 4>4
8 4>4 8s
9 8s 4>s
10 4>s 86
11 86 4>6
12 4>6 81
Since the end of one conduction interval is the beginning of the next, all of the
trigonometric evaluations are used in two consecutive sampling functions, thus
halving the number of calculations. The de voltage can now be written as:
12
vd = L
p=i
vdp ® 'PP" (8.27)
if k~l
(8.28)
otherwise.
"h 2nh
vdp ® 'Pp = L L vdp" ® 'Pp,.
k=i 1=0
(8.29)
234 8 AC-DC CONVERSION- HARMONIC DOMAIN
This equation generates voltage harmonic components of order above n1z which are
discarded. By using Equations 8.27, 8.28, and 8.29, the kth harmonic phasor
component of vd is
(8.30)
(8.31)
and similarly for one of the other two phases. The third phase must always be the
negative sum of the first two, since there is no path for zero sequence into a bridge.
This leads to a total of 8 convolutions to calculate the three phase currents. As
evident in Equation 8.32, the periodic samples for the phase current calculation are
just the de side current, and the commutation currents derived in section 8.2. The
calculation of the phase current flowing into the transformer primary is addressed in
the next section. The derivation of a phase current is illustrated graphically in
Figure 8.8.
'0" '
"
'',
' '0" '
~----------------------~N·~\1 ~
------------1 cycle----------
Figure 8.7 Construction of the de voltage, and validation against time domain solution
'
'
'
'
-- -------I '
.A
'
,/ ''
''
\
'
''
1\. :'" ' ,A
'V"'
I
'
J
convolved de current \ '
~
Figure 8.8 Construction of the phase current, and validation against time domain solution
The outcome of this analysis is a transfer model of the transformer; the primary
currents are related directly to the secondary currents, and the secondary voltage to
the primary voltage. This is easily achieved for the star-g/star connection, shown as a
single line diagram in Figure 8.9.The transformer and thyristor resistances have been
referred to an equivalent ac side resistance, Rae:
(8.33)
8.6 PHASE CURRENTS ON THE SYSTEM SIDE 237
The leakage reactance has been referred to an equivalent on the secondary side:
(8.34)
Since all impedance has been removed from the transformer and incorporated into
either the ac system or commutation circuit, the secondary voltage is now written as
if it were independent of the current through the transformer:
(8.35)
(8.36)
These equations are repeated over all harmonics, and all three phases.
The star-gjdelta connected transformer is considerably more difficult to model,
and in fact requires two separate analyses for transfers from the star to delta side and
vice versa. As shown in Figure 8.1 0, the transfer from star to delta is primarily
concerned with setting up the delta connected source for the voltage sampling and
commutation analyses. The secondary side delta connected voltage source is scaled
by:
(8.37)
(8.38)
The ,.J3 scaling for the delta winding does not affect the transfer of thyristor
resistance through the transformer, since it is not connected in delta. Thus, the
referred ac system resistance is the same as Equation 8.33.
Calculation of the primary side phase currents in terms of the secondary currents
is complicated by the circulating current in the delta winding. If the transformer is
unbalanced, some of this appears as a positive or negative sequence current on the
primary side.
238 8 A C-DC CONVERSION- HARMONIC DOMAIN
(a) star-g-delta
~------------,
Rac 8~
~
(c) secondary to primary
Figure 8.10 (a) equivalent circuit for star-g/delta transformer. (b) transfer from star to delta.
(c) transfer from delta to star
Y=--- (8.40)
R,+JX
1
Cl.=- (8.41)
al
f3 = __I__ . (8.42)
../3a2
Equation 8.39 is used to calculate the primary current, by assuming that Vp and ! 5
are known, and eliminating V 5 . The admittance matrix in Equation 8.39 is not
8.6 PHASE CURRENTS ON THE SYSTEM SIDE 239
invertible, as the delta winding is floating; there are an infinite number of possible
potentials of the delta winding which are consistent with a given current injection
into the transformer. One such potential is that obtained by grounding phase 'c' on
the secondary so that Vsc = 0. This permits the removal of the last row and column
from Equation 8.39:
lpa Vpa
lpb VPb
fpc = [~ ~] Vpc
'
(8.43)
I sa Vsa
lsb Vsb
where:
(8.44)
(8.45)
(8.46)
(8.47)
8.7 Summary
A set of equations have been derived which describe steady-state relationships
between variables relevant to a twelve-pulse controlled rectifier, with ac and de
system representation. The equations model the effect of unbalance in the ac and de
systems, harmonic sources in both systems, and unbalance in the converter
transformers. The firing and commutation processes have also been modelled in
detail, as have the current and power control constraints, and the current control
response to harmonic ripple on the de side.
Harmonic transfer through the converter has been analysed using a convolution
method that avoids the problems of aliasing associated with numerical FFT
calculations, or the complexity of Fourier analysis. An additional advantage of the
convolution analysis described here is that all of the equations are differentiable
when decomposed into real and imaginary components, a feature that enables a
straighforward implementation of a Newtons method solution in Chapter 9.
8.8 References
1. Arrillaga, J, (1983). High Voltage Direct Current Transmission, Vol. 6. Peter Peregrinus Ltd,
London, UK.
2. Ainsworth, JD, (1968). The phase locked oscillator-a new control system for controlled
static converters, IEEE Transactions on Power Apparatus and Systems, 87(3); 859-865.
3. Smith, BC, et al., (1995). A steady state model of the ac-dc converter in the harmonic
domain, lEE Proceedings Generation, Transmission and Distribution, 142(2); 109-118.
4. Smith, BC, (1996). A harmonic domain model for the interaction of the HVdc convertor
with ac and de systems. PhD thesis, University of Canterbury, New Zealand.
9
ITERATIVE HARMONIC
ANALYSIS
9.1 Introduction
It has already been explained in earlier chapters that the analysis of harmonic
interactions between the network linear and non-linear components requires detailed
models of these components as well as an iterative solution. The main non-linearities
of synchronous generators and transformers have been analysed in Chapter 7 and
those of the three-phase static converter in Chapters 5, 6 and 8. These models are
used as the main components of the iterative algorithms discussed in this chapter.
Considering the power ratings and complex controllability of HVde converters much
of this chapter is devoted to them, although the techniques discussed are equally
relevant to other power electronic devices.
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
242 9 ITERATIVE HARMONIC ANALYSIS
In general, the solution diverges when the de system harmonic impedance is large
and the commutating reactance small [4]. Divergence can sometimes be avoided, or
convergence improved by inserting a fictitious reactance pair between the converter
transformer primary and the filter bus. The inserted pair consists of a series
combination of a reactance, and its negative, with the midpoint voltage being the
new voltage to be used as the commutating voltage [5]. The reactance value is chosen
to cancel the ac system reactance and to increase the commutating reactance.
However, the reactance pair cannot cancel high impedance in the ac system due to
parallel resonances; instead, a matched impedance pair can be chosen to mirror quite
closely the frequency dependence of the ac system impedance, and consists of a
simple RLC network [6,7].
The main problem with the matched impedance pair method is the added
complexity. Selection of the RLC network is by no means straightforward, and the
resulting commutation process is formidably difficult to solve. An additional
requirement is a separate ac terminal for every six-pulse group, as the matched
impedance pair can only be placed in one series path.
More recent work has been directed toward improving the solution method itself,
rather than improving the fixed point iterative technique.
in the time domain. For such devices, both the current injection and the Norton
admittance can be calculated by an elegant procedure involving an excursion into the
time domain. At each iteration, the applied voltage harmonics are inverse Fourier
transformed to yield the voltage waveshape. The voltage waveshape is then applied
point by point to the static voltage/current characteristic, to yield the current
waveshape. By calculating the voltage and current waveshapes at 2n equi-spaced
points, a FFT is readily applied to the current waveshape, to yield the total harmonic
injection.
To calculate the Norton admittance, the waveshape of the total derivative,
d/ di(t) dt di(t)jdt
---- (9.2)
dV dt dv(t) dv(t)jdt'
9.3 THE METHOD OF NORTON EQUIVALENTS 243
is calculated by dividing the point by point changes in the voltage and current
waveshapes. Fourier transforming the total derivative yields columns of the Norton
admittance matrix; in this matrix all the elements on any diagonal are equal, i.e. it
has a Toeplitz structure. The Norton admittance calculated in this manner is actually
the Jacobian for the source.
The complex conjugate solution, applied to the transformer magnetization non-
linearity in Chapter 7, is used here as a basis for an iterative solution combining the
linear and linearized components as illustrated in the flow diagram of Figure 9.1.
At each iteration the fluxes in all magnetic branches are required to adequately
evaluate the harmonic Norton equivalents. The winding fluxes can be determined
directly from the transformer terminal voltages, but the remaining fluxes can only be
obtained by solving the magnetic circuit. Some of the equations are non-linear and,
therefore, the solution has to be achieved through an iterative procedure. Matrix
equation 7.65 expresses this set of equations in a linearized form, and a Newton-
Raphson solution is possible with the Jacobian defined by matrix [H].
,------------------------------------------------ -,
Calculate the remaining branch ftuxes
using the Newton flux distribution a
iterative procedure
No
End
·~
I
Figure 9.2 Test system to assess the effect of transformer magnetization characteristics.
x = 0.1 p.u., c = 0.9 p.u., R = 1.0 p.u.
The starting values for each iteration can be those obtained from the flux
distribution derived in the previous iteration, except at the first iteration when the
starting values have to be derived from an approximation, such as the linear
distribution suggested in the EMTP manual [8].
A simple three-phase system for the application of the linear and linearized
algorithm is shown in Figure 9.2, with a multi-limb transformer.
Figure 9.3(a) illustrates the waveforms of the current flowing through the
inductance and 9.3(b) their corresponding harmonic content. The transformer
Phase A currents
0.40r--------------,
=OJ
~ -0.5
u
0
- - - HDA
I::::
u
0.10
- - - - - li.'\l.TDC
0.05
0 n.l
0.004 0.008 0.012 0.016 0.020 1 7 9 11 13 15
Time (sec) Harmonic order
(a) (b)
Phase A voltages
1.5r--------------, 0.10
0 HDA
1.0 - - - HDA • EMTDC
0.08
- - - - - EMTDC
';' 0.5 ::i
5
f
50.06
a;
bll
0 ol
~ 0.04
;;.. -0.5 ;;..
-1.0 0.02
0.004 0.008
0
n.l
0.012 0.016 0.020 1 5 9 11 13 15
Time (sec) Harmonic order
(c) (d)
Figure 9.3 Comparison of time and harmonic domain solutions for the test system
9.3 THE METHOD OF NORTON EQUIVALENTS 245
primary side voltage waveforms and harmonic content are shown in Figures 9.3(c)
and 9.3(d) (only phase A results are shown). Almost a perfect match is achieved for
the voltage waveforms, the difference between the two solutions is indistinguishable
in Figure 9.3(c). In this case, the third harmonic voltage is nearly 7%.
The results indicate very good agreement with the time domain solutions,
especially considering the difficulty of the case analysed. The maximum magnitude
differences are 0.003 (p.u.) for the voltages and 0.009 for the currents, both occurring
at the third harmonic. The solution is achieved in four iterations.
However, the ac-dc converter does not fall in the above category defined by
Equation 9.1. Instead, the V-1 relationships result from many interdependent factors
such as phase and magnitude of each of the ac voltage and de current harmonic
components, control system functions, firing angle constraints, etc.
The converter Norton admittance is not Toeplitz, and, in general, contains of the
order of n2 different elements, as opposed to the n elements obtained from a FFT.
This is illustrated in Figure 9.4, for the ac side of a twelve pulse converter. The
admittance clearly displays a lattice structure related to the twelve pulse switching
action of the converter.
The problem with the method of Norton equivalents is that the converter is really
an interface between the ac and de systems, with only the ac system represented in
the overall solution process. If the converter controller is modelled, a separate
iterative procedure is required to solve the converter interaction with the de system at
each iteration.
0 10 20 30 40 50 60
Order of Applied Harmonic Voltage Distortion
Figure 9.4 Structure of the Norton admittance at a solution with 0.85% negative sequence
fundamental voltage distortion at the converter terminal
246 9 ITERATIVE HARMONIC ANALYSIS
(9.3)
where 11/, 11 Vd, 11 V, 11/d are vectors of harmonic perturbations and Vd, Id are de side
quantities. The ABCD matrix, therefore, links harmonics of different orders on both
sides of the converter. Only positive order harmonics are used in the formulation and
the ABCD matrix is obtained by harmonic perturbations of a time domain
simulation. However, to be fully general, both positive and negative harmonics must
be included, or the matrix should be of Cartesian components.
If the effects of control and commutation variations are neglected, the converter is
linear in the harmonic domain, and the 11 symbol in Equation 9.3 can be dropped.
An iterative approach is possible between a direct solution of the ac-dc system
interaction described by an ABCD matrix and an update to the commutation
duration [10, 11 ]; at each iteration the ABCD matrix is updated by evaluating the
harmonic domain convolutions of terminal voltage and direct current spectra with
switching functions.
However, a decoupled solution similar to this model, iterating between a linear
solution of the ac-dc system interaction, and a Newton solution for the switching
angles has been found to diverge in cases where even order harmonic sources are
present in the ac system [12].
There is thus a motivation to linearize switching angle variation with terminal
harmonic variation, and develop a full Newton solution, incorporating more than
just electrical equivalents. This approach is considered in the rest of the chapter.
YgD
Rae
lftiter X
v vD /:a
-
share the same ac terminal voltage. A solution for this system is developed by using
the inter-relationships of Chapter 8 to derive a set of simultaneous mismatch
equations, the simultaneous zero of which corresponds to the desired steady state
solution. This non-linear problem is solved by Newton's method with sparsity, and
validated against a time domain simulation of a test system to the steady state.
The test system of Figure 9.5 will be described by the following quantities:
• V, the three phase harmonic series converter terminal voltage.
• I~, If the phase currents flowing into the secondary windings of the star-g/star
and star-gjdelta transformers, respectively.
• F£, F~;. the firing angle mismatches of the thyristors in the bridges attached to the
star-g/delta and star-g/star transformers, respectively.
• Fg;, F~;· the end of commutation mismatches in the bridges attached to the star-g/
delta and star-g/star transformers, respectively.
248 9 ITERATIVE HARMONIC ANALYSIS
(9.4)
where the square brackets denote a three phase quantity. The ac system impedance,
[Ycc]k" 1, is calculated by inverting the sum of the admittances attached to the
converter bus. These are the filter admittance, YJilter, and the Thevenin source
admittance, Yet· This equation will be represented over all phases and harmonics by:
(9.5)
The converter terminal voltage is then related to the equivalent transformer primary
voltages by the voltage drops through the equivalent commutating resistances:
Chapter 8 described how to calculate the de voltage across a six pulse bridge
attached to either a star or delta connected ac source, with inductive source
impedance. Using the transformer models of Section 8.6, the primary voltages, V~
and V~, can be transformed into equivalent star or delta connected inductive sources
on the secondary side. The de voltages across each group are then added to obtain
the total de voltage, and the constant forward voltage drop associated with each
group is subtracted from the total de side voltage direct component. The calculation
of the de voltage is again represented in functional notation:
The de voltage is functionally dependent on the switching angles since they define the
limits of the convolution analysis used to calculate the de side voltage across each
9.5 NEWTON'S METHOD 249
group. The de current harmonics are present in the calculation of the voltage samples
which are convolved with the sampling functions. The de voltage, when applied to
the de system, yields the de side current. For example
where Vr"d is the constant forward voltage drop through a group. Summarizing for
any topology:
The de current, switching angles, and primary transformer voltage can be used to
calculate the transformer secondary phase currents by applying the analysis of
Section 8.5 to each bridge:
The transformer analysis of Section 8.6 then describes how the primary currents are
obtained:
Implicit equations have been derived in Chapter 8 for the converter switching angles,
the power control, and the average delay angle. These equations, written in the form
of mismatch equations which equal zero at the solution, are summarized below and
in Table 9.L
lc~o 1 f7(Vdo)
Iss 300 fs(V~, ldk• ldo• ()js, ¢()
JDs 300 f9(V~, ldk• ldo• ef, ¢f)
[S
p 300 fio(l~)
JDp 300 fn(V~,If)
Fs 6 fdV~. I""' e(, ¢()
4>
FD 6 fi3(v~. Id"' ef, ¢f)
4>
Fff 6 fi4(/dk• [dO• ()(, txo)
FR 6 fisUdk· !,10, ef, cxo)
Fs !t6(Vdo• ldo)
F,o !17( Vdo· Ido)
V = f1 (/ft. Jj?)
= f, (flO (Iff), f11 (Vf?, If))
=fiUioUs(V~, ldk• !do• Bf, ¢f)), f11(Vj?, j9(Vj?, ldlo !do• ef, ¢f)))
=fiUio(fs(f2(V), ldk• /,10, ef, ¢f)), jj,(f}(V), f9(f3(V), !din !do. 8f, ¢f)))
=f,s(V, ldlo ldo• 8f, ¢f, 8f, ¢f) (9.26)
The new function, f 18 , is a composition of several functions which describe how the
de current and terminal voltage, together with the switching angles are used to
calculate the primary phase currents. The primary phase currents are then injected
into the ac system impedance to yield the terminal voltage. The relationship
(9.27)
9.5 NEWTON'S METHOD 251
(9.28)
Equation 9.28 is called the voltage mismatch equation, and when decomposed into
phases, harmonics, and rectangular components, yields 300 real equations. A similar
type of mismatch equation can be constructed on the de side:
ldk =f6(Vdk)
=f6U4(Vf,, v;;, Id,, ef, ef, ¢f, ¢f))
=.f6(f4(f2(V), .f3(V), Idl• ef, ef, ¢f', ¢f))
=.fi9(V, Idk· ef, ¢f, ef, ¢f) (9.29)
(9.30)
Equation 9.30 yields 100 equations when decomposed into harmonic real and
imaginary components, and is a composition of functions that describe the
calculation of the de voltage, and its application to the de system model, to yield
the de current. A further 26 mismatch equations are obtained in a similar manner,
related to the converter controller and the switching instants:
A smaller system of 426 simultaneous mismatch equations in 426 variables has now
been developed, and is summarized in Table 9.2. The reduced set of variables to be
solved for consists of the ac terminal voltage, the de current, the switching angles,
and the average delay angle.
252 9 ITERATIVE HARMONIC ANALYSIS
It would be possible to solve for a different set of variables, however those chosen
have the advantage of being less distorted. In particular, the de side terminal voltage
is less distorted than the phase currents, and the de side current is less distorted than
the de voltage. In fact, a fundamental frequency ac-dc load flow will give a very
reasonable estimate of the fundamental voltage component on the ac side, and the de
current component on the de side.
The interaction of the converter with the ac system has been specified in terms of
terminal voltage mismatch. This requires the injection of phase currents into the ac
system impedance to obtain a voltage to be compared with the estimated terminal
voltage.
The ac system interaction can also be expressed in terms of a current mismatch.
The estimated terminal voltage is applied to the ac system admittance to obtain
phase currents that are compared with phase currents calculated by the converter
model using the estimated voltage:
Note that the current mismatch is still expressed in terms of the same variables as the
voltage mismatch. The current mismatch has the advantage that it doesn't require
the system admittance to be inverted, a possible difficulty if it has a high condition
number. The current mismatch is also the preferred method of modelling the
interaction with a purely inductive de system, such as the unit connection, as the
system admittance will decrease with increasing harmonic order.
Only the voltage mismatch is implemented here, as the ac system admittance is
usually invertible, and the ac system impedance is typically much less than one per
unit. A hybrid mixture of voltage and current mismatches at different harmonic
orders would be the most robust and versatile approach to take; the voltage
mismatch would be used at all harmonics where the system impedance is less than
one per unit, otherwise the current mismatch would be used. In the hybrid mismatch
method, the converter interacts with a reasonably strong ac system at all harmonic
orders, and no admittance or impedance is larger than one per unit.
9.5 NEWTON'S METHOD 253
The de current mismatch, Flit· defining the interaction with the de system, can also
be written as a de voltage mismatch, F 1·d· This mismatch is obtained by injecting the
estimated de current into the de system impedance and comparing the resulting
voltage with the calculated de voltage:
Fr·d = y/dk - j4(f2(V), J3(V), ldk, ef, ef, ¢f·, ¢f) (9.39)
dk
The mismatch equations and variables of Section 9.5.1 are a mixture of real and
complex valued. Newton's method is implemented here entirely in terms of real
valued equations and variables. All complex quantities are therefore converted into
real form by taking the real and imaginary components. A decomposition into real
form is required, since the de voltage and de current mismatch equations are not
differentiable in complex form. Newton's method is implemented by first assembling
the variables to be solved for into a real vector X:
(9.40)
(9 .41)
Given an initial estimate of the solution, XO, Newton's method is an iterative process
for finding the solution vector, x•, that causes the mismatch vector to be zero:
F(xe) = 0. (9.42)
with convergence deemed to have occurred when some norm of the residual vector
F(XN) is less than a preset tolerance. Newton's method is not guaranteed to
converge, but convergence is likely if the starting point is close to the solution.
Central to Newton's method is a Jacobian matrix, JN, of partial derivatives. For a
system of 426 equations, the Jacobian is 426 elements square, as it contains the
partial derivative of every mismatch function, with respect to every variable. This is
illustrated for the twelve pulse converter functions and variables in Figure 9.6, for a
system with constant current control.
oR{Fva}
l
N
oR{Fva} oR{Fva} oR{Fva} oR{Fva} oR{Fva} oR{Fva} oR{Fva} oR{Fva} oR{Fva}
k k k k k k k k k k 0 ....
til
oR{V~J ai{ v,~,} oR{V~,} ai{V~,} oR{V,~,} ai{ V,i,} aR{I"/11} ai{I"111 } a¢; 38;
oi{Fva} oi{Fva} oi{Fva} oi{Fva} oi{Fva} ai{Fva} oi{Fva} oi{Fva} oi{Fva} oi{Fva}
k k k k k k k k k k
0
oR{Vg,} ai{V~,} aR{ v~,} ai{V~,} aR{ v,;,} ai{V,~} aR{!"/11} oi{I""'} a¢; 38;
oR{Fvd oR{Fvh} oR{Fvb} oR{Fvb} oR{Fvd oR{Fvh} oR{Fv~>} aR{Fvb} oR{Fv~>} oR{Fv"}
k k k k k k k k k k
0
oR{ Vg,} ai{Vg,} aR{V~,} ai{ v,~,} oR{V,i,} ai{ v~,} oR{Idm} ai{Id,} a¢; 38;
ai{Fvh} ai{Fvd ai{Fv+} ai{Fvh} ai{Fvh} ai{Fvd ai{Fv"} ai{Fvh} ai{Fvh} oi{Fv"} 'C)
k k k k k k k k k k
0
oR{Vg,} ai{Vg,} oR{V~,} ai{ v~,} aR{ v,~,} ai{V~,} 3R{fd111 } ai{!dm} a¢; 38; ::j
tT1
oR{Fvc} oR{Fvc} oR{Fvc} oR{Fvc} aR{Fvc} oR{Fvc} oR{Fvc} oR{Fvc} oR{Fvc} oR{Fvc} ?:l
k k k k k k k k k k
0 ...,
)>
aR{ v;;,} ai{Vg,} aR{ v~,} ai{V~} oR{V,i,} ai{ V,i,} oR{Idm} ai{Id111 } a¢; 38; ':;2
tT1
J = 1 ai{Fv;;} ai{Fvd
k
oi{Fvc}
k
oi{Fvc}
k
oi{Fvc}
k
oi{Fvc}
k
oi{Fvc}
k
oi{Fvc}
k
oi{Fvc}
k
oi{Fvc}
k ::r:
0 )>
oR{V,~,} ai{Vg,} oR{V,~} ai{ v;;,} oR{ Vii,} ai{V,~} oR{/"111 } aiUc~111 } a¢; 38; ?:l
$::
oR{Fldk} oR{F!dk} oR{F!dk} oR{Fldk} oR{Fldk} 0
oR{Fidk} oR{Fld"} oR{Fid~c} oR{Fidk} oR{FM,)
0 z
oR{Vf~} ai{ v,~,} aR{ v,~} ai{V/;,} oR{V,~,} ai{ v,;;} aR{!d111 } oi{Idm} a¢; 38; (=)
)>
ai{Fid~cl ai{FJd"} ai{Fldk} ai{FJ""} oi{Fidr,} ai{Fld"} ai{FJ"k} ai{Fld"} ai{F1c~k} ai{FJdk} z
0 )>
oR{ Vg,} ai{ vg,} aR{ v;;,} ai{V/;,} aR{ v,;,} ai{ V,i,} oR{Idm} ai{Idm} a¢; 38; r'
--<
[/]
aF1,i 3Fq,i aF,t>i aF1,i aF,h aFq,i 3Fq,i aF1,i aF,I>; 3Fq,i v;
0
aR{V,~,} ai{Vg,} aR{ v~,} ai{V~,} aR{V:;,} ai{ v,;,} oR{Ic~m} ai{I"") a¢; 38;
3Fo; 3Fo; 3Fa; 3Fo; 3Fo; 3F11; 3Fe; 3Fo; aFo; aF0
_,
0
oR{Vg,} oi{ ViA} aR{ v,~,} ai{ v,~,} oR{V,i,} ai{ v,~,} oR{Idm} oi{Idm} 38; arx;
ai{Fcxo} ai{Fcxo} ai{Fcxo} oi{Fcxo} ai{Fcxo} ai{Fxol ai{F,0 } ai{Fcxo} oi{Fao} ai{Fcxo}
0
oR{V,',',} ai{V,~,} aR{V~,} ai{ v~,} aR{ v,;,} ai{ v,;;} oR{I"m} ai{Id,J a¢; 38;
There are two methods for obtaining the Jacobian elements; numerical partial
differentiation, and the evaluation of analytically derived expressions for the partial
derivatives. The numerical method is used here to validate the analytic expressions
for the Jacobian elements. Numerical calculation of the Jacobian has the advantage
of ease of coding, but is quite slow. Each column of the Jacobian requires an
evaluation of all the mismatch equations, and the resulting calculation is only an
approximation to the partial derivative. The numerical Jacobian is obtained by
sequentially perturbing each element of X, and calculating the change in all the
mismatches: Ju = !J.Fi/ !J.Xi. Provided !J.Xi is small enough, this gives a good
approximation to the Jacobian.
The analytical method of calculating the Jacobian matrix requires considerable
effort to obtain all the partial derivatives in analytic form, but is exceptionally fast.
Frequently, the amount of computation required to calculate the analytic Jacobian is
of the same order as that required to calculate the complete set of mismatches just
once. For the converter system of 426 mismatch equations, the analytic Jacobian can
be calculated about twenty times faster than the numerical Jacobian. The numerical
Jacobian for the test system has been plotted in Figure 9.7, for a solution up to the
thirteenth harmonic. This Jacobian was calculated at the solution, and so represents
a linearization of the system of equations in Table 9.2 around the converter
operating point.
Referring to Figure 9.7, the elements of the Jacobian have been ordered in blocks
corresponding to the three phases of terminal voltage, the de current, the end of
commutation angles. the firing angles, and the average delay angle. The blocks
associated with interactions between the de current harmonics, and the ac voltage
harmonics comprise the ac-dc partition, which is 104 elements square. All other
parts of the Jacobian are called the s11·itching terms. Within the ac-dc partition,
elements have been arranged within each block in ascending harmonic order, with
the real and imaginary parts of each harmonic alternating. Each block in the ac-dc
partition is therefore 26 elements square in Figure 9.7, but 100 elements square for
a solution to the fiftieth harmonic.The Jacobian displays several important
structural features:
A: The test system contains a parallel resonance in the ac system at the second
harmonic. This leads to rows of large terms in the Jacobian aligned with the
second harmonic terminal voltage mismatch (the resonance terms).
E: The average delay angle mismatch, since it relates to the average de current, is
extremely sensitive to changes in the fundamental terminal voltage. There is
256 9 ITERATIVE HARMONIC ANALYSIS
also some sensitivity to harmonics coupled to the fundamental; i.e. the 11th
and 13th harmonics on the ac side.
F: As would be expected, there is strong coupling between the switching angles
and the switching mismatches.
20
18
16
. · / { : ·El
__ ' __
.. . .
:
... .
.
.· . . .
mismatches variables
Figure 9.7 Numerically calculated Jacobian for the test system; 13 harmonics
9.5 NEWTON'S METHOD 257
Figure 9.8 Effect of variation in firing angle versus end of commutation angle
0
' •• 0·.1' • rl•
,0
Fva
20
0
•,I'
0 0
Fvh 40
Fva
20
FVb 40
60
FVc
80
~d
100
Fq,
Fe 120
Fa
0 20 40 60 80 100 120
Va vb Vc Id <I> ea.
Figure 9.10 Scan structure of the sparse Jacobian: 13 harmonics
9.5 NEWTON'S METHOD 259
0.9
0.8
.g
.a 0.7
·a
~0.6
:E.., 0.5
::>
]0.4
.D
<o.3
variables
In this section the application of Newton's method to the case at hand is described in
detail. Several issues are addressed that have not yet been discussed. Of particular
importance is the method of determining a suitable starting point for the Newton
method, the updating of the Jacobian matrix, the sparse solution of the linear
Jacobian system, and the stopping criteria for the iterative process. These points are
illustrated in the flow diagram for the solution (Figure 9.12), where it can be seen
that a two stage process is employed to calculate the starting point. A first estimate
of the converter is obtained by using a classical analysis, followed by a Newton
solution of the switching system, with no harmonics. If the switching system
converges, a full harmonic solution follows, after which the results are printed to
output files.
Initialization An initial estimate for the converter delay angle is obtained from the
equation:
2-3~ 3X
Vdo = IvthJI cos()(- -!do, (9.45)
n n
260 9 ITERATIVE HARMONIC ANAL YS!S
ignoring voltage magnitude drop through the ac system impedance. The de voltage is
estimated from the voltage drop through the de system and the de source:
VdO = E +{to
y (9.46)
dO
9.5 NEWTON'S METHOD 261
3J2
Vdo =--I Vrhl[cos !Y.. +cos (!Y.. + .u)] (9.47)
n
These angles are then used to assemble the individual firing and end of commutation
angles:
These calculations yield a very rough estimate of the converter switching angles,
which is subsequently improved substantially by a Newton solution of the switching
system.
The Switching System The purpose of the switching system is to solve the
relationships between the fundamental terminal voltage, the de current, and the
switching angles for both bridges. The switching system is thus a complete model of
the 12-pulse converter in the presence of constant terminal voltage and de current
harmonics, since the harmonic quantities appear as constant parameters. The
mismatch equations and partial derivatives for the twelve pulse switching system
have all been derived previously. The set of equations to be solved in the switching
system (for constant current control) are:
A flow diagram for the switching system is shown in Figure 9.12, part (b), while
the structure of the switching Jacobian can be seen in Figure 9.13. Those
elements corresponding to the partial derivatives of voltage mismatch with respect
to end of commutation angle have been set equal to zero, since they are always
insignificant.
This Jacobian is quite sparse, and always has the same sparsity structure,
however it is of an intermediate size, being too small for a general purpose sparse
solution, and yet large enough to be significant. In the case of an interharmonic
model, the switching Jacobian would be of size 24n + 8, where n is the number of
cycles over which the steady state is defined. It is therefore worthwhile to develop
an ad hoc sparse solution of the switching Jacobian, and the best way to do this
is to employ a partitioning method.
262 9 ITERATIVE HARMONIC ANALYSIS
e v1
•.
ao I dO
0
• ....
•. • ... .
•
• ..• ..• .•• .• • .• •
••
•••
•
. • •• •• •• •• • •
•.
•
••.
.. •.
. • • B_!. •• .••
•
10
Iii
.D A .r-
E
c" •
l5 15
1;
.,."
'§"
.
~ 20
9
~
•
c ' • ••• D v,
Figure 9.13 Sparsity structure of the switching Jacobian matrix (power control)
With reference to Figure 9.13, the partitioning method exploits the fact that the
top left hand part of the switching Jacobian, A, is almost diagonal, and can easily be
reduced to the identity matrix. It is then straightforward to create a reduced system
of size 8 x 8 by multiplying the rectangular cross coupling partitions, C and B, to
give D- CB. In this case the cross coupling matrix multiplication is very fast, since
the variables have been ordered so that most of the nonzero elements of each of the
cross coupling partitions correspond to the zeros of the other. Using this method,
the linear system can be solved in approximately 1000 flops, instead of some 10 000.
Indexing overheads have been virtually eliminated by storing partial derivatives
directly into specific vectors, and then using ad hoc code for the partitioning method.
The reduced 8 x 8 system is solved by LU decomposition for the terminal voltage,
average delay angle, and de current order updates. These are then backsubstituted to
find the switching angle updates.
The switching Jacobian is updated every iteration of the Newton method, and
convergence has been found to be rapid and robust. The convergence criteria for the
switching system is:
9.5 NEWTON"S METHOD 263
IFvii <0.001
lVII
IFill <0.001
III I
IFaol <0.001
lfctol
IF!dol <0.001
IPI
IFe I < 5 X w-s
I
(9.51)
The Harmonic Solution The system of harmonic phasor and switching angle
equations is quite large (426 elements square), and at each iteration of Newton's
method, a linear system this size must be solved for the update vector:
F(XN) = JNyN_ This step represents the bulk of the computation required in
Newton's method, and so techniques for speeding up the overall solution are
concerned with details of the Jacobian linear system, and its solution method. The
Jacobian has been made sparse, and it is essential that this sparsity is exploited in an
efficient manner.
Three types of sparse linear solver have been implemented. One of these, the
sparse symmetric bifactorization [14] method was found to be unsuitable, as it
requires the Jacobian to be diagonally row dominant. Although the Jacobian has a
large diagonal, it is not diagonally row dominant. The method of Zollenkopf
pivots for sparsity, not numerical stability, and does not yield the correct solution
when applied to the Jacobian system. The Zollenkopf method is essentially
optimized for solving admittance matrix systems, which are necessarily symmetric
in structure, and diagonally row dominant. The two other sparse solvers that have
been implemented are an asymmetric sparse bifactorization that pivots for a
compromise between numerical stability and sparsity, and the iterative conjugate
gradient method [15]. The sparse bifactorization employed is the yl2m solver from
the netlib [16]. Both methods have been found to be satisfactory, but suited to
different types of solution algorithm.
264 9 ITERATIVE HARMONIC ANALYSIS
(9.52)
9.5 NEWTON'S METHOD 265
Note that convergence tolerance for the complex mismatches, Fv, FJ, F1d is expressed
in terms of the magnitude of the mismatch. This means that the error in an estimated
value for a variable , for example Vll, is smaller than 0.1% of its own length. The
advantage of a relative mismatch of this type, is that it treats all harmonics equally.
However to prevent an attempt to converge to an absolute error of zero for
harmonics that are not present (e.g. even harmonics in some cases), this convergence
test is only applied to harmonics that have a size larger than 10- 5 per unit. These
convergence tolerances can be made tighter to obtain more accurate solutions if
necessary. A relative convergence tolerance of 2 x w- 6 has been used in the
impedance calculations of chapter 10.
The tolerance set for the switching mismatches of 5 x w- 8 corresponds to
1.4 x 10- 4 degrees at the 50th harmonic, or 0.1 nsec. Another type of convergence
tolerance that can be used, is to calculate a norm of the real mismatch vector; for
example, the 1-norm:
n
lXI, = ~)Xil (9.53)
i=l
A tolerance of 1X1 1 < 10- 5 is suitable for general purpose use. This type of
convergence test is fast and easy to apply, but does not imply that all harmonics have
converged to a satisfactory accuracy.
The model has been verified against time domain simulation of the test system
described in Appendix VI, using the program PSCAD/EMTDC. The steady state
solution was obtained by simulating for one second, with a time step of 20 flS, and
then obtaining waveforms over one cycle for subsequent comparison with the
harmonic domain solution. The results of four tests are given here, comparing the de
voltage and ac phase current waveforms and spectra, since these quantities are the
most distorted. The tests are designed to highlight any modelling, and convergence
deficiencies. The tests carried out were:
Test I: A base case solution with no harmonic sources in the ac system.
Test 2: The Thevenin voltage source in the ac system was distorted by 5% positive
sequence second harmonic. This excites the complementary ac-dc system
resonance, leading to noncharacteristic harmonics, and a high degree of
interaction between converter switching angles and the ac-dc harmonics.
Test 3: The leakage reactance of the phase 'b' star-gjdelta transformer was
increased from 0.18 to 0.3 per unit. This imbalance causes the generation
of odd harmonics, and a relatively large coupling to the zero sequence,
which is illustrated in Figure 9.18.
Test 4: A 0.1 per unit resistance was placed in series with the star-gjdelta
transformer, and the secondary tap changer of that transformer was set to
1.1 p.u. Convergence with such a large series resistance indicates that the
266 9 ITERATIVE HARMONIC ANALYSIS
"0 0.3
.a
·abQ 0.2 EMTDC Harmonic Domain
-------
c<S ~
E 0.1
0
0 5 10 15 20 25 30 35 40
harmonic order
DC voltage, Test One
200
~
"'
<1.)
2 100
bQ
<1.) ·m
~ 0
<1.)
_g"' -100
c.
-200
0 5 10 15 20 25 30 35 40
harmonic order
d 0.12
0.06
~ ~
=
0.0 0.04
"'E o.o2
0
0 5 10 15 20 25 30 35 40
harmonic order
"'f::!
II ~
<1.)
100
~
bQ
~
<1.)
m m
lif
0
Q.)
"'
~ -100
c.
-200
0 5 10 15 20 25 30 35 40
Figure 9.14 Comparison of time and harmonic domain solutions for phase currents and de
voltage spectra: Base Case
9.5 NEWTON"S METHOD 267
::?1
c
~ 0
::::
u
-1
-2
-3L---~--~--~--~----~--~--~--~--~--~
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
time (sec)
~
~500
0
>
450
400L---~--~---L--~--~----L---~--~---L---J
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
time (sec)
Figure 9.15 Comparison of time and harmonic domain solutions for phase current and de
voltage waveforms: Base Case
there is a close match between the time and harmonic domain solutions. The time
domain solution yielded small residual non-characteristic harmonics in the spectra,
which were suppressed in the phase graphs. The de voltage waveform was generated
from the harmonic domain solution by plotting the Fourier series for each de voltage
sample during the appropriate interval, rather than inverse transforming the de
voltage spectra. This eliminates Gibbs phenomena associated with the step changes
in voltage, but gives overly sharp voltage spikes. These are not present in the time
domain solution due to modelling of the snubber circuits, which limit the d Vjdt. The
time domain derived de voltage waveform is therefore more rounded. Clearly, if an
accurate time domain waveform was required from the harmonic domain solution, it
would be necessary to post process the waveshape using knowledge of the snubber
circuit time domain response. The comparison of the waveshapes indicates that all
the switching angles are correct.
When a second order harmonic voltage source was placed in the ac system, the
composite resonance was excited, resulting in non-characteristic harmonics.
Referring to Figures 9.16 and 9.17, odd harmonics are present on the de side, and
268 9 ITERATIVE HARMONIC ANALYSIS
~1
~
~ 0
!;;
u
-1
-S0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 O.Q18 0.02
time (sec)
g,
.flsoo
0
>
450
4000 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
time (sec)
Figure 9.16 Comparison of time and harmonic domain solutions for phase current and de
voltage waveforms: Test 2
.,0)
0.3
2
·a 0.2 EMTDC Hannonic Domain
~ ~
OJ)
"'E 0.1
hannonic order
DC voltage, Test Two
200
---.
"'
0)
~
OJ)
0)
~
0)
"'
~ -100
c..
-200
0 5 10 15 20 25 30 35 40
hannonic order
0.0 0.04
"'E o.o2
0
0 5 30 35 40
~
0)
"'
~ -100
c..
-200
0 5 10 15 20 30 35 40
Figure 9.17 Comparison of time and harmonic domain solutions for phase currents and de
voltage spectra: Test 2
- 30 0.002 0.004 0.006 0.008 0.01 0.012 0.014 O.Q16 0.018 0.02
time (sec)
DC voltage, Test Three
6oo.---~---.---.---.----.---,----.---.---.----,
~
<!)
Oil
.:3
0 450
>
400'-----~--~--~--~----~--~--~--~--~--__J
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 O.Q16 0.018 0.02
time (sec)
Zero sequence ac current, Test Three
0.2 .---.----.---,..-----.----.------..,---....-----.----,------,
-0.2 '----~--~---'------'----'-----'---....._____,____---l._ __ .
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 O.Q18 0.02
time (sec)
Figure 9.18 Comparison of time and harmonic domain solutions for phase current and de
voltage waveforms: Test 3
to converge. The only situation in which this has been observed is when the
composite resonance is excited by a very large (0.3 p.u.) second harmonic source.
The constant Jacobian method is therefore likely to be faster in any realistic case.
It is evident from Table 9.3 that convergence is slowed by the low harmonic order
composite resonance, and by the presence of a large commutating resistance. The
second harmonic composite resonance is particularly difficult for the constant
Jacobian method, as there is a higher coupling between low order harmonics and the
switching angles. This is evident in the convergence of Test 5, which required 21
iterations. For more realistic systems, convergence in eight iterations using a
9.6 DIAGONALIZING TRANSFORMS 271
'0 0.3
a
·a EMTDC Harmonic Domain
....---------
0.2
~
OJ)
s
oj
0.1
0
0 40
harmonic order
DC voltage, Test Three
200
~
"'
...
<!)
<!)
OJ)
<!)
:;::.
~
~ -100
0..
-200
0 5 10 15 20 25 30 35 40
harmonic order
::i 0.12
s
oj
0.02
0
0 40
"'
...
<!)
<!)
OJ)
<Ll
3
<!)
"'
~ -100
0..
-200
0 5 10 15 20 25 30 35 40
Figure 9.19 Comparison of time and harmonic domain solutions for phase currents and de
voltage spectra: Test 3
constant Jacobian might be expected. The execution times listed in Table 9.3 are for
a Sun Sparcstation IPX.
-3L_~~~~~~--~==~--~~~~~
0 0.002 0.004 0.006 0.008 O.Q1 0.012 0.014 O.Q16 O.Q18 0.02
time (sec)
1;1,
_g0 500
;>
450
400L---~--~---L--~--~----L---~--J----L--~
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 O.D18 0.02
time (sec)
Figure 9.20 Comparison of time and harmonic domain solutions for phase current and de
voltage waveforms: Test 4
iteration of the Newton solution. The objective is to improve the sparsity of the
Jacobian matrix by transforming to a new system of mismatches and variables which
is more diagonal.
Balanced against the improved Jacobian sparsity of this method, is the extra
calculation overhead associated with calculating the transform at each iteration. It is
therefore not feasible to completely diagonalize the Jacobian matrix by calculating
its eigenvalues, primarily because the matrix of eigenvectors which describes the
diagonalizing transform is the same size as the Jacobian matrix, but full.
Applying either the sequence or dqO transforms to quantities on the ac side of the
converter yields a considerable improvement in Jacobian sparsity, with insignificant
transformation overheads. Under either of these transformations, the zero sequence
component is completely diagonalized, unless the star-gjdelta transformer is
unbalanced. Additional frequency coupling terms between the ac and de sides are
also removed in the sequence transform, since a harmonic k on the de side couples
mainly to harmonics k + 1 in positive sequence, and k- 1 in negative sequence on
the ac side.
9.6 DIAGONALIZING TRANSFORMS 273
------
0.2
~
b()
"'
E 0.1
0
0 5 10 35 40
harmonic order
DC voltage, Test Four
200
'""'
"'0
~
b()
0
:s.
0
] "' -100
p.
-200
0 5 10 15 20 25 30 35 40
harmonic order
- 200 ~o----~5~---7,~o----~,5~--~2~0~--~2=5----~3o~---3~5-----4~o
Figure 9.21 Comparison of time and harmonic domain solutions for phase currents and de
voltage spectra: Test 4
Application of the dqO transform is not warranted since in the steady state, the
dqO transform is just a sequence transform followed by a rotation of the positive
sequence into the direct and quadrature axes, and a rotation of the negative sequence
into the conjugate of these axes. Unlike the synchronous machine, where the direct
and quadrature axes are aligned with the rotor, there is no such preferred phase
reference for the converter. It would therefore be necessary to choose a direct axis at
every harmonic, which introduced the least coupling between the direct and
quadrature axes quantities and other variables.
274 9 ITERATIVE HARMONIC ANALYSIS
to•
tO~t~--~--~--~4--~5--~6--~7--~8--~---t~O--~tt
Iteration number
Figure 9.22 Convergence with the switching terms updated each iteration
to'
to•
t5 20 25
Iteration number
Table 9.3 Convergence and performance of the solution a) updating switching terms, b)
constant Jacobian
In the existing model, interaction with the de system is specified by summing the
de voltage harmonics across each bridge, and applying the resulting voltage
harmonics to the de system linear model. The resulting harmonic current,
(9.54)
should be equal to the de current harmonic ripple, Idk· The de side current mismatch
equations can therefore be written;
(9.55)
A similar equation is applied on the ac side, by summing the phase currents from
each bridge and injecting them into the ac system. This yields voltage harmonics,
(9.56)
that should equal the estimated converter terminal voltage harmonics, Vk. The ac
side voltage mismatch equations can therefore be written
(9.57)
(9.58)
[aFw] _
~-I-T-T--1 -[aV'J (9.60)
aw av
ax = -T-[aV'
[aFw] ax J , X E {Idk• 8;n , c/J;,
n 8; , c/J; } Y Y
aF,] = [aF,]
[aw av -_ T
1
, x E
D D y
{Id'" 8; , ¢; , 8; , ¢; }
y
The resulting Jacobian matrix is plotted in the bottom of Figure 9.24, and shows a
greater degree of sparsity than that for the phase components, shown in Figure 9.7.
This is primarily due to the absence of any coupling between the zero sequence ac
voltage harmonics and any of the other converter variables.
. . ... :
100-J~'a
... ~ cjl 9
40 ....!!!__} 1d
20 4 __:::_) v+
o ---.1 V variables
Vo -
Table 9.5 Convergence and performance of the converter model, b) constant unified
Jacobian, d) Constant sequence components Jacobian
Test No. CPU time Main
(seconds) Iterations
lb 11.1 6 0.1712
ld 10.9 5 0.0851
2b 14.8 11 0.2562
2d 15.0 12 0.2973
3b 12.3 7 0.1130
3d 11.5 6 0.1056
278 9 ITERATIVE HARMONIC ANALYSIS
Jacobian is held constant. Integration of the converter model with a load flow
therefore requires that the load flow be reformulated in Cartesian components,
with no decoupling in the Jacobian matrix.
In summary, the complete set of mismatch equations are;
• PQ bus. The real and reactive power in each phase are as specified.
• PV bus. The positive sequence voltage magnitude, and real power in the positive
sequence are as specified. Kirchoffs current law is applied for the negative and
zero sequence current, so that the current shunted by the machine balances that
flowing out of the network.
• Slack bus. The real and imaginary parts of the positive sequence voltage are as
specified. Kirchoffs current law is applied for the zero and negative sequence
currents.
9.8 Summary
After a brief review of currently available iterative methods for harmonic analysis,
the inter-relationships of Chapter 8 have been used in this chapter to describe a
Newton solution of a 12-pulse rectifier with ac and de system representation. A
functional description of inter-dependent quantities has been used to assemble a
280 9 ITERATIVE HARMONIC ANALYSIS
Variable Types
V- V+ ld cjl 9 a
PQ
PQ
PQ
PV 20
Converter
PQ
PQ
Slack
---4---~---~----~-~-
1 I I I
~ Fvo 60
"c..
?' I
..<: I
80
~ Fv. I I
iil I I
~
100
FV+ I I
120
Fld
140
~
160
Fa I I
l',o 20 40 60 80 100 120 140 160
0
Figure 9.25 Sparsity of the Jacobian matrix for an integrated load flow and converter model
reduced set of mismatch equations suitable for use in Newton's method. The
Newton's method solution has been described, and the Jacobian matrix of partial
derivatives used in this method analysed.
The Jacobian matrix can be made sparse by setting small elements to zero. Among
the methods of solving the sparse Jacobian system, a direct sparse bifactorization
method is preferred when the Jacobian is held constant. Despite requiring more
iterations, fastest convergence is obtained when the Jacobian is held constant.
Additional sparsity and compatibility with other component models, is obtained
by solving for sequence components voltages on the ac side. Integration of the
converter model with a reformulation of the three-phase load flow has been
described.
The description of the system in terms of functions affords a modular
implementation of the model. For example, the functions that describe transfer
through the converter transformer could readily be extended to cover other
transformer connections. The Jacobian matrix itself retains the same structure, even
9.9 REFERENCES 281
9.9 References
1. Yacamini, R and de Oliveira, JC, (1980). Harmonics in multiple convertor systems: a
generalized approach, lEE Proceedings Pt.B, 127(2), 96-106.
2. Yacamini, R and de Oliveira, JC, (1986). Comprehensive calculation of convertor
harmonics with system impedances and control representation, lEE Proceedings PT. B,
133(2), 95-102.
3. Reeve, J and Baron, JA, (1971). Harmonic interaction between hvdc convertors and ac
power systems, IEEE Transactions on Power Apparatus and Systems, 90(6), 2785-2793.
4. Callaghan, C and Arrillaga, J, (1989). A double iterative algorithm for the analysis of
power and harmonic flows at ac-dc terminals, Proc. lEE, 136(6), 319-324.
5. Callaghan, CD and Arrillaga, J, (1990). Convergence criteria for iterative harmonic
analysis and its application to static convertors, Proc. Inti. Conf on Harmonics in Power
Systems (IEEE), Budapest, 38--43.
6. Carbone, R, et al., (1992). Some considerations on the iterative harmonic analysis
convergence, Proc. Inti. Conf on Harmonics in Power Systems (IEEE), Atlanta.
7. Carpinelli, G, Gagliardi, F, Russo, M and Villacci, D, (1994). Generalised converter
models for iterative harmonic analysis in power systems, Proceedings of the lEE Gener.
Transm. Distrib., 141(5), 445--451.
8. Dommel, HW (ed.), (1986). Electromagnetic Transients Program Manuai/EMTP Theory
Book. Bonneville Power Administration.
9. Larson, EV, Baker, DH and Mciver, JC, (1989). Low order harmonic interaction on acfdc
systems, IEEE Transactions on Power Delivery, 4(1), 493-501.
10. Jalali, SG and Lasseter, RH, (1994). A study of nonlinear harmonic interaction between a
single phase line-commutated converter and a power system, IEEE Transactions on Power
Delivery, 9(3), 1616-1624.
11. Rajagopal, N and Quaicoe, JE, (1993). Harmonic analysis of three-phase acfdc converters
using the harmonic admittance method, 1993 Canadian Conference on Electrical and
Computer Engineering, Vancouver, BC, Canada, 1, 313-316.
12. Smith, BC, (1996). A harmonic domain model for the interaction of the HYde convertor
with ac and de systems. PhD thesis, University of Canterbury, New Zealand.
13. Szechtman, M, Weiss, T and Thio, CV, (1991). First benchmark model for hvdc control
studies, Electra, 135, 55-75.
14. Zollenkopf, K, (1970). Hi-factorization-basic computational algorithm and programming
techniques, In Confernce on Large Sets of Sparse Linear Equations, 75-96 Oxford.
15. Press, WH, eta!., (1992). Numerical Recipes in Fortran, The Art Of Scientific Computing.
Cambridge University Press, 2nd edition.
16. Accessible on the World Wide Web at http://netlib.att.com/netlib/y12m/index.html
10
CONVERTER HARMONIC
IMPEDANCES
10.1 Introduction
The representation of ac-dc converters as frequency-dependent equivalents has been
discussed in Chapter 5 (Section 5.4), as well as its application to filter design. A more
accurate approach, based on the converter models described in Chapter 8, is used
here to derive the converter harmonic impedances.
A widely used and effective technique for analysing nonlinear devices is to
linearise their response around an operating point. If the nonlinearity is a voltage-
current relationship, the linearization yields an impedance. This type of linearization
is particularly relevant to the converter, as the converter impedance can be combined
with the ac and de system impedances to analyse resonances, harmonic transfers, and
harmonic magnification factors.
An example of its application to the calculation of the de side impedance of a
converter, including the effect of ac side impedance at coupled harmonics, has been
described by Bahrman [1]. Another reported application relates to a GIC induced
5th harmonic resonance [2), which could only be explained by combining the
impedance of the converter with that of the ac system.
When linearizing general nonlinear devices to an equivalent impedance,
representation by a single complex number is not possible. Instead the use of either
complex or real valued matrices is necessary [3]. The fact that the complex value of
an impedance can depend upon the phase angle of the current flowing through it,
and still be linear, is not widely appreciated. The linear transfer from firing angle
modulation to direct current at the 6th harmonic has been shown to be phase
dependent [4], a fact that also applies to the converter linearized impedance [5].
Central to Newton's method for solving nonlinear systems is the Jacobian matrix,
which represents a linearization of the system of equations at every iteration.
Although the Jacobian is typically held constant during the solution, it can be re-
calculated at convergence. It is therefore possible to use the Jacobian matrix to
calculate equivalent impedances for the converter, or indeed any other linearized
relationship. The advantage of calculating impedances in this way is that the effect of
control, switching instant variation, unbalance, and system impedances are
automatically taken into account.
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
284 10 CONVERTER HARMONIC IMPEDANCES
where T is the sequence transform matrix. Next, a small pos1t1ve sequence 7th
harmonic voltage source is added to the ac system Thevenin voltage source, and the
converter model is re-converged. The perturbed positive sequence 7th harmonic ac
terminal quantities are then calculated, as above:
A value of 7th harmonic impedance is then obtained from the ratio of change in
voltage, to change in current:
Z +- Vi;,- Vjb
(10.5)
7 - + + .
17p- 17b
The small source yields a good approximation to the 'slope' converter impedance,
whereas the 0.12 p. u. distortion resulting from the large source is larger than would
be encountered in practice.
Forty phase angles, from 0 to 2n, of voltage distortion were applied at both
magnitudes, and at every harmonic order from 2 to 37, in positive and negative
sequence. The resulting impedance loci have been plotted in Figure 10.1.
Convertor impedance
. EB34
"'
E
.<:
'i'
::::>
t:
"'
0.
EB17
<='
"'c: lll16
11000 1&15
500 7
To obtain accurately the value of small current and voltage perturbations, the
convergence tolerance for the model was reduced to:
and thresholding was removed for the impedance harmonic under consideration.
Since the system was the same for every perturbation, the Jacobian was held
constant for all 5760 runs. The main features to be observed in Figure 10.1 are the
phase dependence of the converter impedance at six-pulse characteristic harmonics,
and the independence of impedance to the magnitude of applied voltage. At the 12-
pulse characteristic harmonics, a large value of distortion leads to an epicyclic phase
dependent locus, due to the coupling of these harmonics to the average de current,
and hence control action. At all other harmonics and sequences, the converter
impedance is essentially completely linear over the range of magnitudes likely to be
encountered. In practice, there will also be filters at the characteristic and high order
harmonics, and high levels of voltage distortion will not be encountered.
Despite a uniform progression of voltage phase angles applied to the 5th and 7th
harmonics, the associated circular loci display a clustering of impedance points near
the origin. This is a consequence of the perturbation method used, rather than the
impedance itself, and is best explained by considering the locus of the returned
current perturbations, as plotted in Figure 10.2 for the positive sequence 7th
harmonic.
As the applied voltage phase progresses uniformly through 2n, the returned
current is determined by a uniform double rotation around the admittance locus.
The admittance locus is not concentric with the origin, and so the returned current is
small only for those few points on the admittance locus close to the origin. This
results in the clustering of points on the extrema of the elliptical current locus. When
points on the impedance locus are calculated, this clustering effect is compounded by
the radial distance from the origin on the impedance plane, since the current locus is
clustered at large values of current, which correspond to small impedances. If the
voltage perturbation method is used to calculate the admittance locus instead, the
points are equi-spaced.
In relation to the earlier discussion on phase dependence, it is evident that the
converter impedance can be described by complex numbers at all harmonics, apart
from h = 6n ± I. At these harmonics the impedance should be a tensor
representation, or a coupling to the conjugate of the applied voltage. The generation
10.2 CALCULATION OF THE CONVERTER IMPEDANCE 287
Several authors, most notably Larson [6], have noted that from the point of view of
the ac system, the converter presents a stable and quite linear set of interrelationships
between harmonics and sequences. For example, the application of a positive
sequence voltage distortion at harmonic order k + 1, leads to the injection of
harmonic currents (11 ± k), (23 ± k), etc. in negative sequence, and (1 ± k), (13 ± k),
(25 ± k) etc. in positive sequence. The difference terms are always phase conjugated,
unless the harmonic would be negative, in which case the sequence is reversed. In fact
the analysis of Wood [7] predicts many other multiple reflections that usually. but
not always, decay with order (for example 12m± nk ± 1). This 'numerology of the
converter' is succinctly summarized by a lattice-like diagram of connections between
harmonics and sequences, as shown in Figure 10.3, for n = 1.
Points on Figure 10.3 marked with a '+ ', can be represented by a complex
admittance. Points marked by a '0' represent couplings between positive and
negative harmonics, and can be represented either by a circular admittance locus
centered on the origin in the complex admittance plane, or by a tensor (the
impedance tensor concept is described in Appendix V). Points with both markers, at
lattice vertices, indicate that the total current returned consists of two components,
related to the voltage applied, and its conjugate. Such points correspond to a circular
locus in the complex plane shifted away from the origin. Figure 10.13 is a plot of
impedances along the diagonal of Figure 10.3, with circular loci occurring at the six-
pulse characteristic harmonics. Also marked on Figure 10.3 is the special case of a
10.2 CALCULATION OF THE CONVERTER IMPEDANCE 289
Figure 10.3 Principal harmonic currents returned by a twelve pulse converter in response to
an applied voltage distortion. '+ '- current phase related to phase of applied voltage, '0' -
current phase related to conjugate of applied voltage
av
----
a.fts (10. 7)
ax ax
where .ftR is the calculated terminal voltage, obtained by injecting the phase currents
into the ac system impedance:
(10.8)
consequently
Those rows of the Jacobian that contain partial derivatives of the voltage mismatch
are therefore easily modified to contain partial derivatives of the phase currents with
respect to the system variables (terminal voltage, de current, switching angles). The
matrix equation defined by the new matrix, J', is now:
aI aI aI aI aI
AI AV
av aid a¢ ae acto
aFid aFid aF1d aFid aFid
AF!d Aid
av aid a¢ ae acto
aFq, aFq, aFq, aFq, aFq,
AFq, A¢ (10.10)
av aid a¢ ae acto
aFo aFo aFo aFo aF0
AF0 Ae
av aid a¢ ae acto
aFIY. 0 aFIY.0 aFrxo aFrto aF'10
AF'10 Acto
av aid a¢ ae acto
At the converter solution, all the mismatches are equal to zero. When a harmonic
voltage perturbation is applied, it is required that all the mismatches in
equation 10.10 remain zero, i.e. AF = 0. Partitioning J' around I, and V,
equation 10.10 can be written:
(10.11)
Eliminating AX':
(10.12)
The large matrix Dis not actually inverted, instead columns, y; of n- 1 Care obtained
from solutions to the linear system
(10.13)
where c; are columns of C. This procedure is much faster as it avoids the matrix
multiplication and storage of n- 1, and the LU decomposition of Dis only calculated
once.
Since I and V have been decomposed into real valued components, the matrix
Yep = A - BD- 1Cis a second rank tensor. It is actually a phase components version
of the lattice network, connecting harmonics and phases, rather than harmonics and
sequences. The phase components cross coupling tensor has been plotted in
Figure 10.4, for harmonic interactions up to the 21st harmonic. Only admittances
larger than 0.005 p.u. have been retained, and only magnitudes have been plotted.
The tensor is 126 elements square (2 x 3 x 21).
Application of the sequence components transform yields the same tensor in
sequence components, plotted in Figure 10.5. In sequence components the lattice
tensor clearly contains far fewer terms, and the zero sequence part has been
10.2 CALCULATION OF THE CONVERTER IMPEDANCE 291
Figure 10.4 Phase components lattice tensor calculated at the solution of Test!
diagonalized by the transform. The lattice tensor has also been calculated around
the operating points of Test 2 in Figure 10.6, and Test 3 in Figure 10.7. The
presence of 2nd harmonic terminal voltage distortion in Figure 10.6 has increased
the number of significant cross harmonic couplings. As expected, unbalance of the
star-g/delta transformer in Figure 10.7 has resulted in substantial coupling to the
zero sequence, due to sequence transformation by that transformer. As in Test 2,
the unequal commutation periods and voltage samples lead to many more
significant admittance terms.
Ignoring now the zero sequence, since there is usually little or no coupling to it
in the converter, the lattice diagram of Figure 10.3 is verified against the lattice
tensor calculated above. Each · +' or '0' in Figure 10.3 represents four real
elements in the lattice tensor, corresponding to variations in the real and imaginary
parts of the returned current distortion in response to variations in the real and
imaginary parts of the applied voltage distortion (which may be at a different
harmonic and sequence). These four elements constitute a tensor cross coupling
term that can be represented by a circular locus in the complex admittance plane.
By considering sequentially every such cross coupling tensor, Figure 10.8 is the
result of applying the following plotting rules to the lattice tensor calculated at the
solution of Test 1:
292 10 CONVERTER HARMONIC IMPEDANCES
.,
~2
.2
<(
0
120
+VB !Pe~.
120
+ve Seq.
Figure 10.5 Sequence components lattice tensor calculated at the solution of Test I
If the center of the locus is farther than 0.005 p.u. from the origin, plot a'+'.
2 If the radius of the circular locus is greater than 0.005, plot a '0'.
3 If a'+' has been plotted, and the radius is greater than 1% of the distance of
the center of the locus from the origin, plot a '0' as well.
4 If an '0' has been plotted, and the distance of the center of the locus from the
origin is greater than 1% of the radius of the locus, plot a '+' as well.
The purpose of the plotting rules is to sift out very small admittances, but to still
show the nature of admittances that have been retained. The result is a classification
of the crosscoupling tensors into direct '+ ', and phase conjugating '0' admittances,
that essentially recreates the lattice diagram of Figure 10.3 in Figure 10.8. However,
if the same process is applied to the lattice tensor calculated at the solution of Test 2,
the result is a lattice diagram, Figure 10.9, with a considerable amount of fill-in. A
more realistic case is plotted in Figure 10.10, where the ac system source was
distorted by 1% negative sequence fundamental, resulting in 0.8% negative sequence
10.::! CALCULATION OF THE CONVERTER IMPEDANCE 293
Figure 10.6 Sequence components lattice tensor calculated at the solution of Test 2
(10.14)
with the ABDC parameters calculated by means of the perturbation of time domain
simulations, or perturbation of a switching function approach. The ABCD matrix
(or tensor in positive frequency analysis), can be calculated directly from the
Jacobian matrix by modifying the de mismatch derivatives to be derivatives of the de
voltage, and then performing a Kron reduction to eliminate the switching angle
perturbations, as for the lattice tensor above. Once again however, this is of limited
294 10 CONVERTER HARMONIC IMPEDANCES
Figure 10.7 Sequence components lattice tensor calculated at the solution of Test 3
use in ac-dc system harmonic analysis, smce the full interaction can be solved
quickly by the converter model.
In the next section, the lattice tensor is combined with the ac system tensor
admittance, and then used to calculate the impedance of the converter at selected
harmonics and sequences.
&
Q)
en
Q)
>
I
&
Q)
en
Q)
>
+
0 10 20 0 10 20
Applied Voltage Distortion
~
I
10 20 30 40 50 60
Order of Applied Harmonic Voltage Distortion
Figure 10.10 Lattice diagram calculated at a solution with 0.85% negative sequence
fundamental voltage distortion at the converter terminal
harmonic couplings, simply by duplicating every bus at every harmonic. If there are
phase conjugating admittances in the network, it would be necessary to create
additional buses representing conjugate current injections and voltages.
Alternatively, creating vectors of the real and imaginary parts of voltage and
current harmonic phasors, the system admittance matrix becomes a real valued
tensor. The system tensor can include the linearization of any device in the system,
including voltage controlled buses, converters, and load flow buses. In general then,
the system tensor, when Kron reduced to a single three phase bus, will be a matrix
2 x 3 x n11 square. In phase components, the system tensor derived from a
conventional equivalent complex admittance, will be block diagonal, with each
6 x 6 block being the three phase tensor corresponding to each three phase harmonic
admittance. Assuming that there is no cross coupling between harmonics in the ac
system, the connection of the converter to the ac system may be visualized as in
Figure 10.11.
In this figure a harmonic current is injected into the three phase harmonic k node.
In general currents will flow at every harmonic order into the ac system, causing
harmonic voltages that influence the harmonic voltage at the kth node. The ac
system has been disconnected from the converter at the kth harmonic, since we want
to calculate the converter impedance only. The nodal equation corresponding to this
scenario is:
[ [0] ] (10.15)
11h
10.2 CALCULATION OF THE CONVERTER IMPEDANCE 297
I=O I=O
The matrices A, B, C, and D are obtained by moving the six rows and columns of the
lattice tensor associated with harmonic order k., to the end of the tensor, and then
partitioning off these last six rows and columns. The matrix A is therefore of size
6(nh- 1) x 6(nh- 1). A current injected at any of the three phase harmonic nodes
marked in Figure 10.11 will see the system and converter tensor admittances in
parallel, consequently the system 6 x 6 equivalent admittance tensors are added as a
block diagonal matrix to partition A of the converter lattice tensor. The system
admittance at the k.th harmonic has been disconnected from the converter, and so is
not added to the D partition.
Applying the Kron reduction to Equation 10.15, yields the equivalent 6 x 6, k.th
harmonic three phase tensor admittance at the converter ac terminal:
where the Yi are the columns of (A+ diag(Ys;s))- 1B. Sequence transforming Yck. it is
possible to apply another Kron reduction to obtain a single sequence equivalent. The
situation here is more complicated than in Figure 10.11, as in general the ac system
298 10 CONVERTER HARMONIC IMPEDANCES
[ A/~ ]
A/7
= A[AV7
AV~ ] + BAV+
7
(10.19)
AI+= c[AV~]
Av-
7
+DAV+ 7
(10.21)
7
Equation 10.23 indicates that the positive sequence 7th harmonic admittance of the
converter is obtained by reducing the ac system admittance to a zero and negative
sequence equivalent, adding this to the converter three sequence equivalent, and
Kron reducing to the positive sequence. Inverting Y;j and multiplying by the
impedance base yields the impedance tensor in ohms. For the rectifier end of the
CIGRE benchmark (Appendix VI), at the Test 1 solution,
z+ _ [465.39 -306.22Jn
c7 - 719.96 93.47 ·
6x6 6x6
'
'' ~~~=0 ''
'
' '
''
'
a ' b
'' ~v~ f ';2 -
I' ~17=0
A
'' B
' ''
''
''
j_ I'';~I+7
' I:N-!
------------~------ 7 ' ------------~------
'' '
c ' d c ''' D
' ~v+! I: AV; ''
' 7 '
Figure 10.12 Linearized connection of the converter to an ac system at the 7th harmonic
10.2 CALCULATION OF THE CONVERTER IMPEDANCE 299
Repeating these calculations for every harmonic from 2 to 37, in positive and
negative sequence, the resulting circular impedance loci have been plotted in
Figure 10.13. Data points obtained by the perturbation method have also been
plotted on this diagram.
The method developed here for calculating the converter impedance is somewhat
circuitous, as it is not necessary to first calculate the lattice tensor, and then reduce to
a single harmonic equivalent. For example to calculate the 7th harmonic impedance
from the Jacobian, it would be possible to retain voltage mismatches for every
Convertor impedance
2000
positive sequence
•sa .Y31B34
11132 o -Jacobian Reduction
.031 + - Perturbation
. 11130
~ +29
1112911126
lte
1500
02~27 • 1119'24
negative sequence
11123 A 11125
y23
~
11121 11121
11120
019 11119
11118
·•17
11118
G
11116
lll14
11
11112
11110
500 7
Figure 10.13 Intervalidation of the analytic and perturbation methods of calculating the
converter impedance
300 10 CONVERTER HARMONIC IMPEDANCES
harmonic except the 7th, and then Kron reduce the Jacobian to the 7th harmonic
partition. This method would be faster for calculating a single harmonic impedance,
but slower for calculating many. Another method would be to repeatedly Kron
reduce to small lattice tensors covering a limited range of frequencies. Typically we
are interested in harmonics below the 11th, and could therefore retain voltage
mismatches in the Jacobian above the lOth harmonic, reduce to a ten harmonic
lattice tensor, and then sequentially reduce this tensor to harmonic equivalents. This
method would offer a (49/9) 3 = 161-fold improvement in speed for the calculation of
each harmonic, after the initial calculation of the lattice tensor, but would give the
same results. Nevertheless, by obtaining exact agreement with the perturbation
method, the lattice tensor itself, and the nodal analysis utilizing that tensor, have
been validated.
The motivation for developing a sparse Kron reduction, is the assumption that small
elements in the Jacobian will have little, or no effect on the calculated admittance.
Having calculated the full Jacobian matrix at the converter solution, elements larger
than a preset tolerance are copied into sparse storage arrays. The Kron reduction to
a single harmonic can then be performed using the same sparse routines as were
employed in the Newton solution. Since this is a feasibility study only, the method
will be implemented for the de side impedance, as it is single phase.
The starting point is to calculate the full and unmodified Jacobian matrix at the
converter solution. Next, elements larger in absolute size than rJ, are retained and
copied into sparse storage. The full Jacobian is stored in a two dimensional array,
and 17 = 0.001 initially. The sparse Jacobian is represented by a list of elements, row
indices, and column indices. In order to perform a Kron reduction, rows and
columns associated with harmonic k of the de current must be identified and
removed from the list. This is illustrated in Figure 10.14, where only parts 'a', 'b', 'c'
C D
IC'=-=-====-11•1•1 • 1
Figure 10.14 Elimination of rows and columns from the sparse Jacobian associated with de
harmonic k
10.2 CALCULATION OF THE CONVERTER IMPEDANCE 301
and 'd' of the Jacobian are retained and stored in a new sparse matrix 'A'. This
process requires that the row indices of elements in partitions 'c' and 'd' be
decremented by two, and similarly for the column indices of elements in partitions 'b'
and 'd'. For a de side harmonic of order k,. the rows and column indices to be
removed are 6nh + 2k- 1, and 6nh + 2k. The matrices 'B', 'C', and 'D' are copied
directly from the full Jacobian, and so are themselves full.
The Kron reduction consists in calculating D- CA- 1B, with A sparse in this case.
For a de side harmonic impedance, B contains only two columns, and letting bi
represent sequentially the first and second columns of B, the solution to the linear
equation
(10.25)
will be the corresponding columns of A- 1B. This sparse linear equation is readily
solved by first calculating the sparse LU decomposition of A, and then solving with
the two right hand sides bi. The remaining calculation in the sparse Kron reduction is
a straightforward matrix multiplication. The Kron reduction yields the matrix
D- cA- 1B, which is a reduction of the Jacobian to a 2 x 2 matrix consisting of the
partial derivatives of the de current mismatches of order k with respect to de ripple of
order k. The de mismatch is the calculated de voltage, applied to the de system to
yield the de current, and then subtracted from the estimated de current. The de side
impedance is therefore:
- -1 --1
Zdk =-[D-CA B- I]Ydk (10.26)
where the de side admittance, Y dk, has been written as a tensor. The multiplication
by Yd1, and subtraction of I, reverse the Jacobian calculations whereby the de
mismatch partial derivatives are written in terms of the calculated de voltage partial
derivatives.
The de side impedance calculated by the sparse method may contain errors due to .
the large number of small Jacobian elements that are neglected. The effect of the
small terms on the impedance can be determined, without sacrificing sparsity, by
means of the iterative refinement method. When the linear system described by
equation 10.25 is solved, the error due to the Jacobian terms less than 17 is determined
by multiplying the solution Xi by the full matrix JA:
(10.27)
AtJ.xi = r, (10.28)
DC Side Impedance
500 +11
+10
400
+9
300 .+.E)
100 +2 <15
-100
-200
Figure 10.15 Calculated de side impedances of the CIGRE rectifier using the sparse Kron
reduction technique, with and without refinement
De Side Impedance
1500
3 4 5 6 7 8 9 10
Harmonic Order
2 3 4 5 6 7 8 9 10
Harmonic Order
Figure 10.16 Comparison of harmonic and frequency domain solutions to the CIGRE
rectifier de side impedance.'+' and '0' mark the range of the complex locus at each harmonic
10.2 CALCULATION OF THE CONVERTER IMPEDANCE 303
BOO
11
700
600
300
200
100
0~----L------L----~------~-----L----~------
-100 0 100 200 300 400
Real part (Ohms)
Figure 10.17 Variation in the negative sequence impedance of the CIGRE rectifier as the
third harmonic positive sequence terminal voltage distortion is increased from 0 to 0.12 p.u
requires n2 flops. The solution of the linear system is less than n2 , since A has already
been reduced.
The calculated de side impedances for the CIGRE rectifier are plotted as loci in
Figure 10.15 for harmonics 1 to 12. Figure 10.15 indicates that in this case the
neglected Jacobian terms have had virtually no effect on the de side impedance, as
the loci have been plotted twice, first with no refinement, and then with one iteration
of the refinement method.
The sparse calculation of the de side impedance is validated against a direct
frequency domain calculation in Figure 10.16 for harmonics up to the lOth. The
frequency domain calculation of converter impedances is described in detail in
Chapter 5, and was there validated against time domain calculations of the converter
de side impedance using the perturbation method.
304 10 CONVERTER HARMONIC IMPEDANCES
900
13
800
300 ............. .
200 ..
. . . . . . . . . . . . . . . .. . .
.
100
oL--L-----L----~~~~------~--~L-----~-----
-100 0 100 200 300 400 500
Real part (Ohms)
Figure 10.18 Variation in the positive sequence impedance of the CIGRE rectifier as the fifth
harmonic negative sequence terminal voltage distortion is increased from 0 to 0.12 p.u
10.3 VARIATION OF THE CONVERTER IMPEDANCE 305
800
600
"'E
.r:
9
"'
1ii 400
c.
~
-~
.E
Figure 10.19 Variation in the positive sequence impedance of the CIGRE rectifier as the
current order is decreased from 2000A to 200A. Harmonics 2 to 6
oL---~------L------L----~------L------L-----
0 200 400 600 BOO 1000
Real part (Ohms)
Figure 10.20 Variation in the positive sequence impedance of the CIGRE rectifier as the
current order is decreased from 2000 A to 200 A. Harmonic 7
analysis of Chapter 5 predicts a sine function envelope for the converter impedance
magnitude as a function of commutation angle. Figures 10.19 to 10.22 plot the
variation in the converter positive sequence impedance loci as a function of the
current order. The current order was reduced in one hundred steps from 2000 A to
200 A, which resulted in a variation of firing angle from 14° to 50°, and
commutation duration from 23° to 0.7°. The large variation in firing angle was
due to a combination of factors; there was no tap change control, the inverter was
represented by a constant de voltage source, and since there was no reduction in
reactive power compensation, the fundamental component of the terminal voltage
increased. Clearly, a comprehensive study of the converter impedance variation
would require additional steady state control equations to be integrated with the
Jacobian matrix. Nevertheless, the procedure for calculating the converter
impedance would remain the same, and the results obtained demonstrate the
converter impedance is a strong function of the operating state.
10.4 SUMMARY 307
200A :
1800
1600
"'E
8
d, 1200
t:
g_
~
'"
<::
-~
-1000
600
Figure 10.21 Variation in the positive sequence impedance of the CIGRE rectifier as the
current order is decreased from 2000 A to 200 A. Harmonics 8 to 12
10.4 Summary
The representation of phase dependent admittances by second rank tensors has been
described, and interpreted geometrically as a circular admittance locus on the
complex plane. The Jacobian of the converter model is used to directly calculate
tensor admittances for the converter on the ac side. A nodal analysis of the cross
coupling converter lattice tensor attached to an ac system is applied to obtain
equivalent impedances at the converter bus. The impedances thus obtained are
verified against impedances calculated by a perturbation study at the converter bus.
The de side impedances have also been obtained and verified against those obtained
from a frequency domain converter model.
308 10 CONVERTER HARMONIC IMPEDANCES
2000 ......... ..
200
Figure 10.22 Variation in the positive sequence impedance of the CIGRE rectifier as the
current order is decreased from 2000 A to 200 A. Harmonic 13
10.5 References
1. Bahrman, MP, eta!., (1986). De system resonance analysis, IEEE Transactions on Power
Delivery, PWRD-2(1), 156--164.
2. Dickmander, DL, eta!., (1994). Acfdc harmonic interactions in the presence of gic for the
Quebec-New England phase II hvdc transmission, IEEE Transactions on Power Delivery,
9(1), 68-78.
3. Semlyen, A, Acha, E and Arrillaga, J, (1988). Newton-type algorithms for the harmonic
phasor analysis of non-linear power circuits in periodical steady state with special reference
to magnetic non-linearities, IEEE Transactions on Power Delivery, 3(3), 1090-1097.
4. Persson, EV, (1970). Calculation of transfer functions in grid controlled convertor systems,
lEE Proceedings, 177(5), 989-997.
5. Wood, AR, Smith, BC and Arrillaga, J, (1995). The harmonic impedance of an hvdc
converter, 6th European Conference on Power Electronics and Applications (EPE 95),
Seville, 1039-44.
6. Larson, EV, Baker, DH and Mciver, JC, (1989). Low order harmonic interaction on acfdc
systems, IEEE Transactions on Power Delivery, 4(1), 493-501.
7. Wood, AR and Arrillaga, J, (1995). Composite resonance; a circuit approach to the
waveform distortion dynamics of an hvdc converter, IEEE Transactions on Power Delivery,
10(4), 1882-1888.
Appendix I
EFFICIENT DERIVATION OF
IMPEDANCE LOCI
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
312 APPENDIX I
80
60
]"'
0
Q
'--'
~
~
40
CJ
s=
CJ
~
II)
~
0 20 40 60 80 100
Resistance, R(f) ohms
Figure 1.1 Impedance of a 220 kV network at 100% (Benmore, South Island, New Zealand)
away. Ideally then, it might be argued that the impedances should be calculated at
frequencies {fk. k = 1, 2, ... } such that 1/Jk+l -1/!k =a, where 1/Jk = 1/f(fk) is the angle
between the tangent to the locus at the frequency fk and the R-axis (or some other
fixed line) and a represents a constant angle. This approach would guarantee that
small loops are given as much attention as large loops. For example, if the true locus
were a circle and if a were fixed at 45°, then the locus would be summarized by eight
equally spaced points around the circumference, irrespective of the radius of the
circle.
The change in tangent angle, 111/fk = 1/Jk+l -1/Jk> as the frequency changes fromfk
to fk+ 1 is referred to as the winding angle. If the estimated 111/fk is too large the
impedance should be evaluated again at some intermediate point, say
f = !CA +fk+,).
The diagram in Figure 1.2 represents a magnified view of a small portion of the
impedance locus. Suppose that the impedance Z(f) = R(f) + jX(f) has already
been calculated at each of two frequencies fk. ik+l· Then the angle through which
the tangent to the curve winds as the locus is traversed from Zk =
Z(f) to Zk+l = Z(fk+!) can be estimated by evaluating the impedance Z(r) at an
intermediate point r in the interval [fk. fk+d· If () denotes the angle between the
chords Z(r)- Zk and Zk+l - Z(r) and if the locus has constant curvature over the
interval fk ~ f ~ ik+l (i.e. it is a circular arc), then some simple geometry shows
that the tangent winds through an angle 111/f = 2(). Thus, it can be hoped that the
winding angle is estimated well by 2(), if the curvature does not vary too much over
I.2 WINDING ANGLE CRITERION 313
Figure 1.2 Estimating the winding angle, /::;.'lj;k. using the chord angle(}
the interval in question. Moreover, for a smooth locus (at least differentiable) the
mean value theorem can be invoked twice to show that the winding angle must be
at least 8 on an interval that contains 1: and which is a strict subset of the interval
[/k, ik+Il· In any event, if the angle 8 is not small then clearly more points are
required if the locus is to be summarized well.
The diagram in Figure 1.2 is appropriate only if 8 < 90°. This situation is easily
detected by calculating the chord lengths a, band c. If a2 + b2 < c2 then the angle 8 is
acute and A.l/fk can be estimated by 28; the smaller this angle the more accurate is the
approximation. If a2 + b2 ~ c2 then it is not even worth calculating the angle 8,
clearly the points Zk. Z(1:) and Zk+l are spaced too far apart. In this case, each of the
sub-intervals [/k, 1:] and [1:, fk+Il can be processed separately by evaluating the
impedance again at an intermediate frequency and estimating the winding angles on
each new sub-interval.
The calculation of the chord lengths is not wasted when 8 < 90°, because the
lengths a, b and c can be used to get estimates of the winding angles Al/fk+l and
A.I/Jk+Z on each of the sub-intervals. Again, some simple geometry provides the
estimates:
A.I/Jk+l =arcsin(~ sin 8)'
and it is straightforward to verify that these estimates satisfy Al/lk+l + A.l/fk+Z = 28, as
expected. In fact, the angle 8 can also be calculated from the chord lengths since it
satisfies the equation.
that th~y can be used to get error estimates whenever the interval is subdivided and
the winding angle subsequently recalculated. If two successive estimates agree within
a pre-specified tolerance and if the best estimate indicates that the winding angle is
sufficiently small (again to within a pre-specified tolerance), then there should be no
need to subdivide further. This is the basis of the adaptive scheme.
The impedance is evaluated initially at the two end points of the interval in
question before the first call. The length of the chord between these two initial points
is also required together with an estimate of the winding angle over the entire length
of the locus. Any value exceeding 180° will suffice initially since this value is re-
estimated by the routine. A tolerance parameter is then required to be set before
entry to the subroutine. Each subsequent call of the subroutine causes the impedance
to be evaluated at the midpoint of the frequency interval. This effectively produces
two sub-intervals. The chord lengths corresponding to each of these intervals are
calculated and if the angle between them is not acute or if this angle is positive
(corresponding to an anti-clockwise rotation), then the subroutine is called again
immediately on each of the sub-intervals. The subroutine is also called again if either
the initial estimate of the winding angle (passed in the parameter list) or the new
estimate of the winding angle calculated y the subroutine exceeds the given tolerance.
To prevent a possibly infinite recursion when the true locus is not differentiable the
subroutine is not called if the range of a frequency interval is too small. Moreover, as
an extra safeguard, the subroutine is also called automatically if the range is
considered to be too large.
In practice, the user provides values for minimum and maximum frequency
increments and the algorithm effectively varies the actual increment between these
bounds in a way that ultimately causes the winding angle between adjacent points to
be approximately constant.
The algorithm was applied to the power system locus of Figure I.l, and the results
are displayed in Figure I.3. It can be seen that the adaptive scheme has worked quite
well on this curve. In the region near 250 Hz the curve has been sampled at the
minimum increment because the orientation is anti-clockwise and the algorithm tries
hard to discover an extra loop that is not there. The algorithm has also started to
subdivide more finely in the region near 1100 Hz but the given tolerances have
prevented the algorithm from trying too hard here. A parametric cubic spline fitted
through the points is displayed in Figure I.4 which cannot be distinguished by eye
from the true curve of Figure I.l; moreover, the number of samples used is an order
of magnitude lower than in the original plot.
Although the adaptive scheme provides a perfect match, the number of sample
points required is still unnecessarily high. Further savings can be made by relaxing
the equal curvature criterion to reduce the number of samples in regions of high
curvature. This can be achieved by fixing the winding angle to suit the region of large
frequencies and increasing its value in inverse proportion to the cord between
successive samples.
As the discrete harmonic impedances are always required, these points provide the
initialisation of the adaptive scheme. The magnitudes of the outer radius and largest
cord between successive harmonics can be used to decide the value of the winding
angle to be used as a reference. The magnitude of the cords between samples can
then be used to decide the winding angle.
1.2 WINDING ANGLE CRITERION 315
80
0 0 0
0
0
oOOo0
60 ooO 0
00
] 0
0
8 0
1250Hz
~ 0
~
40 c
u 0
:::~ 0
...u /
0
~:~
0
~ <e>
<l) 0
0:: 0 0
20 0 o9Jo9o
0 0
0 0
0
0
P 50Hz
0
0 of 0
0
0 0
0
0 20 40 60 80 100
Resistance, R(t) ohms
80
60
00
8
.a
0
Q
'--'
~
~
40
u
:::~
...u 250
800
~
<l)
0::
20
0 20 40 60 80 100
Resistance, R(t) ohms
For each selected configuration two discrete samples are initially derived for
each harmonic of interest, one at the maximum expected frequency (i.e.fmax. h)
and the other at the minimum frequency ifmin. h).
2 The interval between !max . h and /min . h is then subjected to the modified
adaptive sampling criterion and a cubic spline curve fitted to the resulting
points (taking advantage of the previously derived harmonic contour); it is
expected that in most cases an extra intermediate point will provide sufficient
information.
1.3 Reference
1. Dominguez, M, et a!., (1994). ·An adaptive scheme for the derivation of harmonic
impedance contours, IEEE Transactions on Power Delivery, 9(2), 879-886.
Appendix II
PULSE POSITION MODULATION
ANALYSIS
E(t)
p qQ . ~ ~ mp + nq
= ln- lP sm(qt) + ~ nf=oo mn Jn
(mnQ)
-----p- [
cos (mp + nq)t + lnrr] (II.l)
t
p
~
tp
~
rT-T/2 rT+T/2
t Figure 11.1
t
PPM generation waveforms
j
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
318 APPENDIX II
w 0 bkw0 . (k ~ )
E ( t ) = 2 n +~sm w 0 t+b~c
~
+~ ~ (m+nk)w 0 J,(mb)
~ cos
[cm+nk) _ nn]
w 0 t+nb~c-- (II.4)
m=l n=-cc n rn 2
This statement of the PPM spectrum forms the basis of the analysis of the non-
ideal converter transfer function spectra.
Ya
(a) e
-I
w0 bkw0 .
Fiet) =hi-+ hi-2-smekw0 t + b~c)
2n n
~ ~
+hi f,:'] nf-'oo em+ nnk)w 0 Jnemb) [
m cos mewot- T;)
nn]
+ nkw 0 t + nbk- 2 ell.5)
2-J}
L Jn(mb)
00
F(t)=-L: (±)(m+nk)w0 - - .
n m n=-oo
m
form= 1, 5, 7, 11 etc.
The next step is to calculate the Fourier transform of the rectangular pulse. The
Fourier transform of a rectangular pulse of duration T and height A, centred around
time t = T /2, is
e11.7)
Applying this to the calculated spectrum, and extending to three phases, yields
FvJt) = ~ ~ ntoo(±) ln;~b) cos [em+ nk)w0 t- m(a0 + f.lo) + n(ok- kf.lo- ~)- mtjJ]
./3"
+--;- L..,.. ~ ln(mb)
L)±)----;-cos [ (m- nk)w0 t - m(a0 + Jlo)- n ( ?h- k11 0 +
2 n) - ]
mt/1
m n=i
Ya
(a)
-I
(b)
-I
The sum of the four spectra, approximating each pulse with a delta function, is
described in Equation II.6.
The Fourier transform for the triangular pulse of duration T and height A is
(II.13)
w
Li
2w0 . (f1.
1 [ 1 --+-sm
G(w)=-
2 fl.J w
-
2w0
1W) n -f1. 1
-
2 2w0
W]
- (II.14)
Applying this to Equation II.6 and extending to three phases yields the spectrum of
the commutation function.
(II.l5)
2v'3 "'(± ) -
+--L..., - 2 sin(mf1. 1 /2) cos [ mw 0 t-m ( a0 +2
J 0 (mb) fl.J ) -m'l'
''']
n 111 rn m11 1
(\over)
322 APPENDIX II
"'
' ''
''
''
''
''
''
(a)
''
(b)
(c)
~--Ill--~
The position of the pulse is dependent strongly on the firing angle variation b.()(, and
only weakly on the variation in commutation period duration b..J1 1 • If b..f1 1 is assumed
to be small, equation II.l8 can be reduced to
8=()(0 +b..()(+ ~ (II.l9)
At the inverter, the true commutation period ends somewhat later than the
effective commutation period. The centralized angular position of the correction
pulse will be before the end of the true commutation period, and as an
approximation the end of the effective commutation period is chosen. The relevant
equation can be written
(II.20)
A variable xis set, such that x = 1 at the inverter, and x = 2 at the rectifier.
The duration of the waveform is fairly close to one commutation period. It is
modelled initially as a dirac delta function, which is later converted to a symmetrical
triangular pulse of duration Jli.
From Equations II.l7 and II.l8, it can be seen that the area of the pulse is
dependent only on the modulation of the end of the commutation period relative to
the beginning of the commutation period.
If b..f1 1 is held constant such that the area of the associated pulse is 1, and firing
angle modulation such that b,.()( = b cos(kw 0 t + 15~c) is applied, the resulting impulse
train has the PPM spectrum, derived from Equation II.6 as follows
2-J3
F,~(t) = -
n
LL 00
ln(mb)
(±)(m + nk)w0 - - x
m
m n=-oo
PI ) + n ( b1c -kp
cos [ (m + nk)w 0 t - m ( ()( 0 + ~
JX JX
n) - -n- mljf] (II.21)
- 1- -
2 2
(II.22)
324 APPENDIX II
Choosing the appropriate dimensions for a triangular pulse of duration 11 1 yields the
Fourier transform
(I1.23)
4w 0
~
[
(II.24)
If flJ1 1 is now allowed to vary as per flJ1 1 = b1 cos(kwot + bk! - kfld .JX), which is the
commutation period modulation referred to the centre of the correction pulse, this
can be substituted directly into Equation II.24 to yield
(II.25)
(II.26)
I
The remaining terms expand out to the following
2
. (m +nk)JJ. 1]
"~(±)(m +nk)
[
F. (t) = J3 bl Sill 4 ) Jn(mb) x
"' n 2 ~~ (m+nk)JJ.1 m
4
.l [
~
+cos (m + nk- k 1)w0 t - m (a0 +J1.- kJJ. 1
1 ) + n ( Jk----
./X ./X
n)2
- Jkl +kJJ.-1- - - mlj;
Jx
n2 ]}
(m - nk)JJ. 1] ) 2
+ J3 bl "~(±)(m- nk) Sill 4 Jn(mb) x
n 2 ~~ (m-nk)JJ. 1 m
4
~
+cos (m- nk- k 1)w0 t- m (a0 +J1.1)
- - n (Jk -kJJ.
./X Jx 2 ./X 2
n)
-1 +- - c5kl +kJJ.-1- - - mlj; n ]}
(II.27)
for m = 1, 5, 7, 11 etc. The most significant terms of this series relate to n = 1 and
are at the frequencies (m + k + k1)wo, (m + k- kl)wo, (m - k + k1)wo, and
(m- k- k 1)w0 . These frequencies are at low levels.
Only the terms listed in Equation II.26 are carried on in the main text.
11.5 Reference
1. Schwarz, M, Bennett, WR and Stein, S, (1966). Communication Systems and Techniques.
McGraw-Hill.
Appendix III
PULSE DURATION
MODULATION ANALYSIS
t -_ -h - -cos
E () hQ h Loo [( - 1)"'
(qt ) +- - 10 (mnQ)]
-- sin(mpt)
2 2P nm=! P m
h~~ln(mnQ/P) . [( ) nn]
- - ~~ sm mp+nq t+J
nm=! n=! m -
h~~ln(mnQ/P).
-- ~ ~ sm [( mp - nq ) t nn]
+- (III.1)
n m=! n=! rn 2
where pis the angular sweep frequency, which will now be known as w 0 .
Redefining the PDM in terms of the pulse angular position (radians) as
b cos(kwot +!h), results in the substitution
-Ph
Q=- (111.2)
n
and
q=kw0 (III.3)
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
328 APPENDIX III
1'
p
~ Threshold
1'p
~ +------1~
rT-T/2 r.:I+~/2
h hb h~ m sin(mw0 t)
E(t) = -2 + -2 cos(kw0 t + bk) +- L.)( -1) - J 0 (mb)]
n nm=l m
h~~Jn(mb)
-- . [ nn]
~~--sm (m+nk)w 0 t+nbk--
n~1=1 m 2
h~~Jn(mb)
-- . [ nn]
~ ~--sm (m- nk)w 0 t- nbk-- (III.4)
nm=l n=l m 2
In the following applications, only the change in spectrum that results from an
existing wave being modulated needs to be considered. The spectrum of the
unmodulated pulse train should be subtracted from that of the modulated pulse train
to define the harmonic contribution of the modulation.
The unmodulated pulse train has the spectrum
(III.5)
hb h~ sin(mw0 t)
E(t) = 2n cos(kw0 t + bk) + ~ £:1[1 - J 0 (mb)] m
h~~Jn(mb) . [
- ~ £:1 f::J------;;;- sm (m + nk)w0 t + nbk - 2nn]
h~~Jn(mb)
-- . [ nn]
~~--sm (m-nk)w 0 t-nbk - - (III.6)
n~1=1 m 2
III.2 FIRING ANGLE MODULATION 329
This statement of the PDM spectrum forms the basis of the analysis of firing angle
modulation applied to the ideal model, and of natural modulation to the
commutation period.
h; Lee [1
_ h;b (k
F; ( t ) --cos w 0 t + uk~ ) +- ( b)] sin(mw 0 (t- T;))
-10 m
2n n m=! m
h;~~l11 (mb) . [ nn]
-; f,:j ~~sill mw0 (t- T;) + nkw0 t + nok- 2
h; ~ ~111 (mb) . [ nn]
-; f,:j ~~sill mw0 (t- T;) - nkw0 t - nok - 2 (III. 7)
Ya
(a)
-!
(b)
-!
Summing over i and extending to three phases results in the following spectrum
,J3" J 0 (mb)- 1
Fv1 (t) = -L._.,(±) cos[m(w 0 t- a0 -1/1)]
n m
m
~~
2,J3"
+-;- 1-cos (m + nk)w0 t- ma 0 + n bk- 2 -
J,(mb)
~(±)-n- [ ( n) ]
mlj;
+-;- L._., ~
2,J3" L._., ( ±) -Jn(mb) cos [ (m - (~o + n) - J
- -
111
nk)w0 t - ma 0 - n k
2 mlj;
m n=l
(III.8)
form=1,5,7 ....
This is the spectrum of frequencies in the six-pulse converter transfer function that
result from applying firing angle modulation of a = a0 + b cos(kw0 t + b~c).
111.3 Reference
1. Schwarz, M, Bennett, WR and Stein, S, (1966). Communication Systems and Techniques.
McGraw-Hill.
Appendix IV
DERIVATION OF THE
JACOBIAN
In this Appendix, the full Jacobian matrix is derived for a six-pulse rectifier attached
to the ac system via a star-gjstar transformer. The elements of the Jacobian are the
partial derivatives of the mismatch functions with respect to the variables that are
being solved for by Newton's method. An important distinction is made in Newtons
method between functions and variables. A variable is never a function of something
else. A function is only a function of the variables being solved for, and never
another mismatch function. The variables that are being solved for are defined to be
Vk, yt, Vk:, Idk' ¢i, ei, oco. Thus Idk' the de current harmonics, are defined not to be a
function of Vk. the terminal voltage harmonics. In the analysis to follow, any other
quantity is either a function of these variables or a constant.
In finding the partial derivative of a mismatch function with respect to one of
these variables, all other variables are held constant. Thus the chain rule is never
applied to give a derivative of one variable with respect to another. For example, in
the de current mismatch equations, the partial derivative with respect to the average
delay angle is always zero. Intuitively this is incorrect, as we expect a change in delay
angle to change all the de harmonics. However, in terms of the converter mismatch
equations, this requires several applications of the chain rule to quantities that have
been defined as variables, not functions. The overall network of interdependency
between all the converter quantities is only fully represented in the inversion of the
Jacobian matrix, and in the definition of the mismatch functions.
All of the partial derivatives obtained below have been confirmed by comparison
with the numerical partial derivative obtained from the corresponding mismatch
equation. That is;
aF F(x +Ax) - F(x)
- "'---,----- (IV.l)
ax Ax
Power System Harmonic Analysis Jos Arrillaga, Bruce C Smith, Neville R Watson and Alan R Wood
© 1997 John Wiley & Sons, Ltd. Published 1997 by John Wiley & Sons, Ltd.
332 APPENDIX IV
First, the partial derivative of the terminal voltage mismatch with respect to terminal
voltage variation is obtained. In this analysis all other variables are held constant.
The commutation current, sampled de current, and phase currents are all functions,
and require an application of the chain rule. The phase voltage mismatch equation,
which represents the interaction of the calculated phase currents with the ac system
impedance is
oc =F fJ or k =F m
(IV.3)
(IV.S)
(IV.6)
Since the ac phase currents are calculated by sampling the de and commutation
currents, the effect of phase voltage variation on phase current is determined from
the effect of phase voltage variation on the sampled commutation currents alone.
IV. I VOLTAGE MISMATCH PARTIAL DERIVATIVES 333
This is because the de current and switching angles are variables which are held
constant, while the commutation currents are functions of the terminal voltage. Four
commutations contribute to one cycle of ac current, consequently
(IV.8)
where ~i is the ith commutation current, lei, sampled over the commutation period.
CrJ.th E [ -1, 0, 1] is a coefficient matrix that defines whether for the ith commutation
the phase r:t current is a function of the phase f3 terminal voltage, and if so, in which
sense. C is defined in Table IV.l where i indexes each set of data:
The commutation current is given by
+L
nh
lei( t) = D leik ejkwt, (IV.9)
k=i
where
(IV.10)
(IV.ll)
In Equation IV.ll, e and b are subscript functions of i, and refer to the phases ending
and beginning conduction respectively. Sampling the commutation current requires
a convolution with the relevant sampling function:
(IV.l2)
Table IV.l. The coefficient matrix Co:6; which specifies the dependence between commutation
current i, terminal voltage phase, and ac current phase
[J
a b c
a {-1, 0, -1, -1, 0, -1} {0, 0, 1, 0, 0, 1} {1, 0, 0, 1, 0, 0}
ex b {0, 0, 1, 0, 0, 1} {0, -1, -1, 0, -1, -1} {0, 1, 0, 0, 1, 0}
c {1, 0, 0, 1, 0, 0} {0, 1, 0, 0, 1, 0} {-1,-1,0,-1,-1,0}
334 APPENDIX IV
Setting Icio = D, expanding the convolution, and taking the kth component yields
(IV.l3)
alei
aR{ J 6}
Ill
= 0 V /, m such that l #- m and l > 0.
ai
+ (
31.·
Clm I.Jl*
aR{Vb} (2i-1Jm+k
)* + { -aR{Vo}
aJ.
m
Clm I.Jl
.(2i-llk-m'
(IV.l4)
m Clm I.Jl* .
3R{V,~} (2z-llm-k'
Similarly,
+( aJ
Clm I.Jl*
)* + - ar{v~}'Pc2i-l)lc-m' m~kl
{
ale;
m (IV.l5)
ai{Vo}
m
(2i-l)m+k aJ.
Clm
ai{V,~,}
I.Jl*
(2i-llm-k' m>k
The calculations required to evaluate Equations IV.l4 and IV.l5 can be
approximately halved by using the Cauchy Reimann equations for the partial
derivatives of complex functions. For an analytic function
aF(z) . 3F(z)
(IV.l6)
3I{z} = J 3R{z}.
aF(z)* . 3F(z)*
3I{z} = -J 3R{z} . (IV.l7)
parts. The remaining partial derivatives, aic;0 jaR{V~n} and aici.,/aR{V~}, are
obtained from equations IV.lO and IV.ll;
(IV.l8)
(IV.l9)
(IV.20)
aR{F } R { _zaa
-----"v'~~- aika - zCJ./3 aikf3 - zrxr aik" } (IV.21)
aR{I~ } - k aR{I~ } k aR{I~ } k aR{I } '
~
(IV.22)
(IV.23)
(IV.24)
336 APPENDIX IV
Table IV.2. Coefficient matrix Ef defining the contribution of the commutation currents to
each phase current. i is the commutation number
rx
a b c
1 1 0 -1
2 0 -1 1
3 -1 1 0
i
4 1 0 -1
5 0 -1 1
6 -1 1 0
The phase currents are composed of commutation currents and the de current
sampled on the ac side:
6
lrx. = L)Eflci ® 'Pz;-J] + ld ® '~"rx.· (IV.25)
i=l
m~k]·
m~k
(IV.27)
Expanding the convolution in the second term of Equation IV.26 gives the kth
component of the sampled de current:
(IV.28)
IV.! VOLTAGE MISMATCH PARTIAL DERIVATIVES 337
where fd~ is the phase a sample of the de current. This equation can be differentiated
to yield:
(IV.29)
Applying the above analysis for variations in the imaginary part of Id, gives
(
8/
Clm tp*
)* -
{
8/ci
8I{Jm }'P(2i-l)k-m'
dm m~k]
+ 8I{I }
dm
C2 i-llm+k + 81Clm· tp* m"?;k '
8I{l } (2i-llm-k,
dm
(IV.30)
and
(IV.31)
These equations are then substituted into an equation analogous to Equation IV.26
and hence into Equations IV.22 and IV.24. The remaining partial derivatives,
8lci0 /8R{ldm} and 8lc;,,I8R{ld,}, are obtained from Equations IV.lO and IV.ll:
(IV.32)
81.
cl 0 . eO' {L
~ike;}
8I{I } = -JR - L - ' (IV.33)
dm eb
8/cim Le
(IV.34)
8R{Id,J Leb
A variation in the end of commutation angle, ¢ 1, affects the (2h- l)th and 2hth
sampling functions. This affects all harmonics of the ac side sampled commutation
currents, and de current. Combining these two effects into the phase current
338 APPENDIX IV
variation, and injecting into the ac system yields the variation in the terminal voltage
mismatch;
(IV.35)
(IV.36)
A variation in ¢" affects only the sampling of the hth commutation current in the
first term of Equation IV.37. Expanding only this convolution from the summation
yields
(IV.38)
+ 6~(Ichz a'¥(2h-l)z+k)
a¢
*_~(I
6 cht
a'¥C2h-llk-z)}
a¢ , k > 0.
1=1 h 1=0 h
(IV.39)
where
The compound sampling function in the second term of Equation IV.37 is affected
by variation in four of the six end of commutation angles. This is because the
transfer of de current to the ac side is defined by two conduction periods per cycle;
the beginning and end of each conduction period corresponding to an end of
commutation angle. The effect of a variation in ¢~z therefore depends upon whether
it corresponds to the beginning or end of a positive or negative conduction period for
phase rx. This information is already collated as the coefficient matrix -Ea,·
IV. I VOLTAGE MISMATCH PARTIAL DERIVATIVES 339
A similar analysis to the above for the second term of Equation IV.37 results in
(IV.41)
The transfer of de current to the ac side of the converter is defined entirely with
reference to the end of commutation angles. The effect of a variation in the firing
instants on the phase currents can therefore be obtained by analysing just the
sampling of the commutation currents. The partial derivative of the ac voltage
mismatch with respect to a firing angle variation is
(IV.42)
Noting that a change in (}h affects Icho and 'P(2h-1), applying the product rule to the
expansion of the convolution leads to
This completes the analysis of the terminal voltage mismatch equation partial
derivatives, as there is no dependence upon a0 , the only remaining variable.
340 APPENDIX IV
(IV.45)
Finding the partial derivatives of this equation is therefore mainly concerned with
Vdk' which is a function of all the converter variables except IXo.
(IV.46)
(IV.47)
(IV.48)
(IV.49)
The only terms in this equation which are a function of V~ are the twelve pre-
convolved de voltage phasors Vd11 • There are three equations which define the twelve
pre-convolved de voltage phasors;
(IV.51)
(IV.52)
IV.2 DIRECT CURRENT PARTIAL DERIVATIVES 341
(IV.53)
for a commutation on the negative rail. Differentiating these three equations with
respect to v~ gives 36 possible values for avd;,;av~ according to the sample
number i, and the phase of 6. The required partial derivatives are summarized in
Table IV.3 by reference to the partial derivatives of Equations IV.51, IV.52,
and IV.53 listed below:
avdiz = 1
aR{VtJ
avdi[ = -1 (IV.54)
an{Vn
avdi[ Lb
aR{V/} Lb +Le
aVc~;l Le
aR{Vfl - Lb + Le
avc~;,
-----'- = -1 (IV.55)
aR{V[}
_av_d_._i,_ = 1 (IV.56)
aR{V/}
for a commutation on the negative rail. The Cauchy Reimann equations can be used
to give the partial derivative with respect to the imaginary part of the voltage
variation as j times that listed above.
342 APPENDIX IV
e b 0 + equation
1 A c B {IV.55)
2 A B {IV.54)
3 c B A {IV.56)
4 A c (IV.54)
5 B A c (IV.55)
6 B c {IV.54)
7 A c B (IV.56)
8 B A {IV.54)
9 c B A (IV.55)
10 c A (IV.54)
11 B A c (IV.56)
12 c B (IV.54)
The partial derivatives in Equations IV.54, IV.55, and IV.56 are then substituted
into the partial derivatives of Equation IV. 50 which are given below:
(IV.57)
(IV.58)
Apart from being the most significant term in the de current mismatch equation, the
de ripple affects the commutation currents, and also causes a voltage drop through
the commutating reactance. These last two effects mean that the de voltage is a
function of the de current ripple. It is therefore necessary to obtain partial derivatives
of the de voltage harmonics in a similar manner to that undertaken already for the
IV.2 DIRECT CURRENT PARTIAL DERIVATIVES 343
k=Fm
(IV.59)
k=m
(IV.60)
k=Fm
(IV.61)
k=m
(IV.62)
m~k]'
m?;k
(IV.63)
Similarly,
m~kl'
m?;k
(IV.64)
The partial derivatives avdim/iii{Idm} are obtained from Equations IV.51, IV.52,
and IV.53:
aVdim .k (L + L )
aR.{I } = -J w + - (IV.65)
dm
during normal conduction, and
(IV.66)
344 APPENDIX IV
during any commutation. Again, the imaginary partial derivatives are obtained by
the Cauchy Reimann equations. The correct phase subscripts can be obtained from
Table IV.3. This completes the linearized dependence of the de mismatch upon de
ripple variation.
1
L
12
vd = 2j vdi ® '¥;. (IV.67)
i=i
In this equation, only two of the twelve'¥; are functions of ¢h· Table IV.4 shows that
'¥ 2h and '¥ 2h-i are functions of ¢h· Equation IV.50 can therefore be differentiated to
yield:
(IV.68)
aR{Fik} _ {- avdk}
a¢h - R Ydk a¢h , (IV.69)
and
(IV.70)
Substituting Equation IV.68 into IV.69 and IV.70 gives the required partial
derivative of de ripple mismatch with respect to end of commutation.
The partial derivatives of de ripple mismatch with respect to firing angle are obtained
in an exactly similar manner. The result is:
(IV.71)
and
(IV.72)
where
-L nh [
vd(Zh-z),
( a'¥ Zh-i/-k )
ae
*) J* - Lk vd(2h-i) 1
T 2h-lk-/
an1
ae
1=0 h 1=0 h
+ ~ vd(2h-l),
nh ( a'¥ Zh-1 1 k )
aeh -
*+ ~
nh [
vd(Zh-l),
( rzh-! 1 k )
an1
aeh -
*) ]*) · (IV.73)
(IV.74)
346 APPENDIX IV
(IV.75)
(IV.76)
(IV.77)
(IV.78)
and
The partial derivatives in these equations are obtained from Equations IV. 75 to IV. 77
as
(IV.80)
aicim -1
(IV.81)
BR{ V~} - jmw(Le + Lb)'
(IV.82)
aJcim - -j
(IV.83)
BI{ V~} - jmw(Le + Lb) .
(IV.84)
and
(IV.85)
The partial derivatives in these equations are obtained from Equations IV.75 to IV.77
as
aD Le eikB;
(IV.86)
aR{Idm} Le + Lb ,
a/Clm -Le
(IV.87)
aR{Idm} Le + Lb,
a/Clm -jLe
(IV.89)
ai{Idm} Le+Lb
This analysis assumes that the commutation is on the positive rail. A similar analysis
holds for a commutation on the negative rail, but with -Idm substituted into
equation IV.74.
This partial derivative gives the effect on the 'residual' commutating-off current at
the end of the commutation, if the end of commutation is moved. It is obtained
simply by differentiating Equation IV.74 with respect to ¢, to yield
(IV.90)
348 APPENDIX IV
The de offset to the commutation, D, is a function of the firing instant, and so the
only effect of I.J; on F¢ 1 is through D. Differentiating the expression for D,
Equation IV.75, gives the required partial derivative.
(IV.91)
(IV.92)
where
(IV.93)
oFei- 1 (IV.94)
arxo - '
(IV.95)
(IV.96)
(IV.97)
where
(IV.98)
IV.5 AVERAGE DELAY ANGLE PARTIAL DERIVATIVES 349
(IV.99)
(IV.lOO)
Thus Fa 0 is a function of all the converter variables, with the exception of IXo itself.
Analysis of the partial derivatives of Equation IV.99 is similar to that for the partial
derivatives of the de ripple mismatch.
Differentiating the mismatch Equation IV.99 with respect to an arbitrary phase and
harmonic of ac voltage yields
(IV.lOl)
and
(IV.102)
(IV.103)
and
(IV.104)
which when substituted back into Equations IV.lOl and IV.102 give the required
partial derivatives. The remaining partial derivatives, aVd;"'jaR.{V!} etc., have
already been obtained in Equations IV.51, IV.52, and IV.53, and by reference to
Table IV.3.
350 APPENDIX IV
Variation in the de ripple affects the de voltage samples in a similar manner to that of
a variation in the terminal voltage above. Differentiating the mismatch
Equation IV.99 with respect to an arbitrary phase and harmonic of de current
ripple yields
(IV.105)
and
(IV.l06)
(IV.107)
and
(IV.108)
which when substituted back into Equations IV.l05 and IV.l06 give the required
partial derivatives. The remaining partial derivatives, aVd;,jaR{ld,} etc., have
already been obtained in Equations IV.65, and IV.66.
The effect of a variation in the end of commutation is to modify the sampling of the
de voltage sections in Equation IV.lOO. Differentiating Equation IV.99 with respect
to ¢h yields
(IV.l09)
This requires the partial derivative avdo/a¢h· Differentiating Equation IV.lOO yields
where use has been made of Table IV.4 to determine the only two sampling functions
that are affected by ¢h·
IV.5 AVERAGE DELAY ANGLE PARTIAL DERIVATIVES 351
The effect of a change in the firing angle on the average delay angle mismatch
equation is similar to that for a change in the end of commutation angle. The
sampling of the de voltage sections is modified, and this changes the average de
voltage. The analysis carried out above for the end of commutation variation is also
valid in this case, with only the two affected sampling functions, as determined from
Table IV.4 being different. The result is that
(IV.l12)
This completes the derivation of the partial derivatives required for the Jacobian
matrix, as the average delay angle mismatch equation is not a function of the average
delay angle.
Appendix V
THE IMPEDANCE TENSOR
V .1 Impedance Derivation
The impedance tensor is a convenient framework for the linearization of the
converter in the steady state. It is used in Chapter 10 to derive a linearized converter
equivalent impedance for use in the analysis of ac-dc system interactions.
A power system component can be represented by a voltage controlled current
source: I = F( V), where in general I and V are vectors of harmonic phasors. The
function F, is a complex vector function, and may be non-linear, and non-analytic. If
Fis linear, it may include linear cross-coupling between harmonics, and may be non-
analytic, i.e. harmonic cross coupling and phase dependence do not imply non-
linearity in the harmonic domain.
The linearized response of F to a single applied harmonic may be calculated by
taking the first partial derivatives in rectangular coordinates:
(V.l)
(V.2)
[a
b
-b]
a '
354 APPENDIXV
which is a type of matrix isomorphic with the complex number field. The
linearization then becomes
aF (V.5)
AI= avAV.
(V.6)
I 1 = Y, V1 + Y2 VT (V.7)
h = Y, V2 + Y2 v *2 (V.8)
V3 = aV, +bV2 (V.9)
then
h = Y, V3 + Y2 v~ (V.lO)
= Y 1(aV1 + bV2) + Y 2(aV1 + bV2)* (V.ll)
= a Y1 V1 + bY, V2 + a Y 2VT + b Y2 v; (V.l2)
=ai, +bh. (V.13)
Y=!_ (V.14)
v
Y 1V + Y 2V*
(V.l5)
v
= Y1 + I Y2l L( L Y2 - 2 L V) (V.l6)
The complex admittance twice traces a circular locus centered at Y1 in the clockwise
direction, as the angle of the applied voltage is varied through 2n. By the inverse
mapping, Z = 11 Y, also traces a circular locus, but in the anti-clockwise direction.
Next, it will be shown that any current injection, when linearized, can be written in
the form
(V.17)
(V.18)
V.l IMPEDANCE DERIVATION 355
Equating the matrices on the right hand sides of equations V.l8 and V.l, the real and
imaginary parts of Y, and Y2 are readily expressed in terms of the partial derivatives:
(V.l9)
(V.20)
This is the nodal analysis used in the transformer model of Chapter 7. IfF is non-
analytic, Y2 =!= 0, and the complex admittance will be a circular locus. For positive
frequency nodal analysis, the real matrix of partial derivatives is retained, and is
henceforth called the admittance or impedance tensor:
y= [Yu (V.21)
Y21
The nodal analysis is then performed in real components using a Cartesian vector
representation of voltage and current phasors, and a second rank tensor
representation of admittances and impedances. Tensors are widely used in physics
to represent a relationship between vectors that is invariant under rotation of the
coordinate axes. In this case, the coordinate axes are the real and imaginary
components of the complex voltage phasor, which rotate under a shift in phase
reference. The components of the voltage therefore transform like the elements of a
vector under this rotation. Since the current is a vector function of the voltage, the
matrix of partial derivatives can be written as the direct product of the gradient
operator with the current vector, yielding a second rank tensor
(V.22)
Although the elements of the tensor admittance are modified by a change in phase
reference, they transform in such a way that the relationship described by the tensor,
dependence of current on voltage, is invariant. When the tensor admittance is
356 APPENDIXV
extended to three phases and multiple harmonics, the dimension will be 6nh, but the
rank of the tensor will still be two.
(V.23)
(V.24)
Separating out the real and imaginary parts of the complex impedance:
Since the objective is to obtain a phase dependent locus, the polar transformation,
IR = l/lcose (V.29)
h =Ill sine (V.30)
is applied, to yield:
The square magnitude of the current evidently cancels from equations V.31, resulting
in a phase dependent impedance:
where for the sake of brevity, the following notation has been defined:
s~sin8 (V.33)
c~cos8 (V.34)
(V.35)
s2 =1-c2 (V.36)
s4 = 1 - 2c 2 + c4 (V.37)
s 3c = sc- sc 3 (V.38)
s2c2 = c2- c4 (V.39)
(V.40)
where a and b, are the coordinates of the circle center, and r is the radius. The
problem is to determine the values of a, b and r, since ZR and Z 1 have been defined
parametrically in terms of 8, the angle of the current injection. A lucky guess gave:
(V.41)
(V.42)
(V.43)
RHS=
RHS=
[z211 - 2z 11 "- 22 - (z 12 + z21 )2 + z222 + z221 + 2z21 "- 12 - (z22 - z 11 )2 + z212]c4
+ [2zll (z12 + z21)- 2zn(zl2 + z21) + 2z21 (zn - z 11 ) + 2zn(z 22 - z 11 )]sc 3
(V.46)
The trigonometric terms all cancel, leaving only a constant term, which shows
that all points on the locus are the same distance, r, from the point (a+ jb). It
remains only to show that for a full 2n range of applied current angles, all points
on the circular locus are visited by the impedance. This is done by subtracting the
V.2 PHASE DEPENDENT IMPEDANCE 359
locus position, (a+ jb), from the complex impedance locus. Considering first the
real part:
ZR- a= ZJJ cos 2 0 + !<zl2 + Z2J) sin20 + z22 sin2 0- !{zu + z22)
= Z11 COS 2 0 +! (z12 + Z21) Sin 20 + Z22 - Z22 COS 2 0-! (zu + Z22)
= (zu - Z22) COS 2 0 +! (z12 + ZzJ) sin 20- !{zu - Zzz)
=! (zu - Z22) +! (zu - z22) cos 20 + !(z12 + z21) sin 20-! (zu - z22)
where y == - tan- 1((z!2 + Z2J)/(zu - z22)). The analysis for the imaginary part is
similar, yielding:
(V.48)
Equations V.47 and V.48 indicate that as the angle of the applied current is increased
from zero to 2n radians, the circular impedance locus is traversed twice in a counter
clockwise direction, starting from a point on the locus that makes an angle y radians
to the real axis. This is illustrated in Figure V.l below.
'
'
'
'' '
'
b ----____ 1,~~----
J).
Real
Figure V.l Complex impedance locus for a tensor impedance. The locus point rotates
counter clockwise twice, starting from the angle y, as the current injection ranges in angle from
0 to 2n
360 APPENDIXV
Since the impedance tensor is a matrix, it may not be invertible, in which case the
determinant is zero. However
and the tensor is invertible if and only if the complex circular locus does not intersect
the origin. The x-axis intercepts of the circular locus correspond to a resistive
impedance, so that
Zl=Rl. (V.50)
The eigenvalues of Z are therefore the values of resistance where the locus intersects
the x-axis. The characteristic equation is
(V.51)
=0, (V.52)
which is a quadratic in R, and it is easy to show that a real solution for R is only
possible if r2 > b2 , i.e. the circular locus is close enough to the x-axis. If the x-axis
intercepts are considered as being a type of resonance, then the eigenvectors of Z are
those currents that have the correct angle to excite the resonance.
Appendix VI
TEST SYSTEMS
3.737
0.5968 2.5 2.5 0.5968
2160.6
+
·1~29.76
0.1364 6 ·685 26.0
E
74.28
261.87
·1~685
83.32
Figure Vl.l Rectifier end of the ClORE benchmark model. Components values in Q, H, and 11F
362 APPENDIX VI
1.5,--~---~------'--.,.----,
500
400 0.5
~300
0
200 -0.5
-1
-1.5'--~-~-~-~-~-~:--_J
6 8 10 12 14 0 6 8 10 12 14
Harmonic multiple Hannonic multiple
1.57'
5000
4000
0.5
~3000
0
-0.5
2000
-1
1000
-1.5
- 2o'---:--~~-~6:---~-~10_ __,12
4 6 8 10 12
Harmonic multiple Harmonic multiple