Computer Applications in power systems

CHAPTER-2
LOAD FLOW ANALYSIS
2.1 Introduction:
Load Flow or Power Flow is the solution for the Power System under static conditions of
operation. Load Flow studies are undertaken to determine:
1. The line flows
2. The bus voltages and system voltage profile
3. The effect of changes in circuit configuration, and incorporating new circuits on system
loading
4. The effect of temporary loss of transmission capacity and (or) generation on system loading
and accompanied effects
5. The effect of in-phase and quadrature boost voltages on system loading.
6. Economic system operation
7. system transmission loss minimization
8. Transformer tap settings for economic operation and
9. Possible improvements to an existing system by change of conductor sizes and system
voltages.
For the purpose of load flow studies, a single phase representation of the power network is used
since the system is generally balanced. When systems had not grown to the present size,
networks were simulated on network analyzers for power flow studies. These analyzers are of
analogue type, scaled down miniature models of power systems with resistances, reactance’s,
capacitances, autotransformers, transformers, loads, and generators. The generators are just
supply sources operating at a much higher frequency than 50Hz to limit the size of the
components. The loads are represented by constant impedances. Meters are provided on the
panel board for measuring voltages, currents, and powers. The load flow solution is obtained
directly from measurements for any system simulated on the analyzer.
With the advent of the modern digital computer possessing large storage and high speed, the
mode of load flow studies have changed from analog to digital simulation. A large number of
algorithms are developed for digital power flow solutions. Some of the generally used methods
are described in this chapter. The methods basically distinguish between themselves in the rate of
convergence, storage requirement and time of computation. The loads are generally represented
by constant power.
In the network at each bus or node there are four variables viz.
(i) Voltage magnitude

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(ii) Voltage phase angle

(iii) Real power and
(iv) Reactive power.
Out of these four quantities two of them are specified at each bus and the remaining two are
determined from the load flow solution. To supply the real and reactive power losses in lines
which will not be known till the end of the power flow solution, a generator bus. called slack or
swing bus is selected. At this bus, the generator voltage magnitude and its phase angle are
specified so that the unknown power losses are also assigned to this bus in addition to balance of
generation if any. Generally, at all other buses, voltage magnitude and real power are specified.
At all load buses the real and the reactive load demands are specified. Table 2.1 illustrates the
types of buses and the associated known and unknown variables.
Table 2.1
Bus Specified variables Computed variables
Slack - bus Voltage magnitude and its phase angle Real and reactive powers
Generator bus Magnitudes of bus voltages and real powers Voltage phase angle and reactive
(PV - bus or voltage (limit on reactive powers) power.
controlled bus)
Load bus Real and reactive powers Magnitude and phase angle of
bus voltages

2.2 Network Modeling:

Transmission plant components are modeled by their equivalent circuits in terms of inductance,
capacitance and resistance. Each unit constitutes an electric network in its own right and their
interconnection constitutes the transmission system.
Among the many alternative ways of describing transmission systems to comply with
Kirchhoff's laws, two methods—mesh and nodal analysis—are normally used. Nodal analysis
has been found to be particularly suitable for digital computer work, and is almost exclusively
used for routine network calculations.
The nodal approach has the following advantages.
• The numbering of nodes, performed directly from a system diagram, is very simple.
• Data preparation is easy.
• The number of variables and equations is usually less than with the mesh method for power
networks.
• Network crossover branches present no difficulty.
• Parallel branches do not increase the number of variables or equations.

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• Node voltages are available directly from the solution, and branch currents are easily calculated.
• Off-nominal transformer taps can easily be represented.
The power system will be considered as a balanced system in which the transmission lines and
loads are balanced (the impedances are equal in all the three phases) and the generator produces
balanced three phase voltages (magnitudes are equal in all the 3 phases while the angular
difference between any two phases is 120 degree). For the purpose of load flow studies, a single
phase representation of the power network is used since the system is generally balanced
Basically, an AC transmission system consists of
i) Synchronous generator
ii) Transformers
iii) Transmission lines and
iv) Loads
We will look into the network models of synchronous generators, Transformers, Transmission
lines and loads.
Interconnection of two or more power system components (Generators, Transformers,
Transmission lines and loads) is called bus.
Transformer Model
For power system steady-state and fault studies, generally the exciting current of the transformer
is neglected as it is quite low compared to the normal load current flowing through the
transformer. Therefore, a two winding transformer connected between buses ‘i’ and ']' is
represented by its per unit leakage impedance as shown in figure below

Loads
Loads can be classified into three categories; i) constant power, ii) constant impedance and iii)
constant current. However, within the normal operating range of the voltage almost all the loads
behave as constant power loads.
As the objective of the AC power flow analysis is to compute the normal steady-state values of
the bus voltages, the loads are always represented as constant power loads. Hence, at any bus 'k'
(say), the real and reactive power loads are specified as 100 MW and 50 MVAR (say)
respectively.

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Transmission line Model

In a transmission grid, the transmission lines are generally of medium length or of long length. A
line of medium length is always represented by the nominal- n model as shown in Figure, where
z is the total series impedance of the line and Bc is the total shunt charging susceptance of the
line. On the other hand, a long transmission line is most accurately represented by its distributed
parameter model.
However, for steady-state analysis, a long line can be accurately represented by the equivalent- π
model, which predicts accurate behavior of the line with respect to its terminal measurements
taken at its two ends. The equivalent-π model is shown in Figure

Synchronous Generator: Concept of Injected Power and current

As the name suggests, the injected power(P)/ (current) indicates the power (current) which is fed
'in' to a bus. To understand this concept, let us consider Figure (a), here a generator is connected
at bus 'k' supplying both real and reactive power to the bus and thus, the injected real and
reactive power are taken to be equal to the real (reactive) power supplied by the generator. The
corresponding injected current is also taken to be equal to the current supplied by the generator.
On the other hand, for a load connected to bus 'k' (as shown in Fig.(b)), physically the real
(reactive) power consumed by the load flows away from the bus and thus, the injected real
(reactive) power is taken to be the negative of the real (reactive) power consumed by the load.
Similarly, the corresponding injected current lk is also taken as the negative of the load current. If
both a generator and a load are connected at a particular bus (as depicted in Fig(c)), then the net
injected real (reactive) power supplied to the bus is equal to the generator real (reactive) power
minus the real (reactive) power consumed by the load. Similarly, the net injected current in this
case is taken to be the difference of the generator current and the load current.

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2.3 Basic Nodal Method:

In the nodal method as applied to power system networks, the variables are the complex node
(bus bar) voltages and currents, for which some reference must be designated. In fact, two
different references are normally chosen: for voltage

Simple network showing nodal quantities

magnitudes the reference is ground, and for voltage angles the reference is chosen as one of the
bus bar voltage angles, which is fixed at the value zero (usually). A nodal current is the net
current entering (injected into) the network at a given node, from a source and/or load external to
the network. From this definition, a current entering the network (from a source) is positive in
sign, while a current leaving the network (to a load) is negative, and the net nodal injected
current is the algebraic sum of these. One may also speak in the same way of nodal injected
powers S = P + jQ.
Figure above gives a simple network showing the nodal currents, voltages and powers. In the
nodal method, it is convenient to use branch admittances rather than impedances. Denoting the
voltages of nodes k and i as Ek and Ei respectively, and the admittance of the branch between
them as yki, then the current flowing in this branch from node k to node i is given by

Let the nodes in the network be numbered 0,1,...,«, where 0 designates the reference node
(ground). By Kirchhoff's current law, the injected current /k must be equal to the sum of the
currents leaving node k, hence

Since E0 = 0, and if the system is linear,

If this equation is written for all the nodes except the reference, i.e. for all bus bar in the case of a
power system network, then a complete set of equations defining the network is obtained in
matrix form as

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The nodal admittance matrix in equations (2.3.4) or (2.3.5) has a well-defined structure, which
makes it easy to construct automatically. Its properties are as follows.
• Square of order n x n.
• Symmetrical, since Yki = Yik.
• Complex.
• Each off-diagonal element Yki is the negative of the branch admittance between nodes k and i,
and is frequently of value zero.
• Each diagonal element Ykk is the sum of the admittance of the branches which terminate on
node k, including branches to ground.
• Because in all but the smallest practical networks very few nonzero mutual admittances exist,
matrix Y is highly sparse.

2.4 One Line Diagram(OLD)/ Single Line Diagram(SLD):

In practice, electric power systems are very complex and their size is unwieldy. It is very
difficult to represent all the components of the system on a single frame. The complexities could
be in terms of various types of protective devices, machines (transformers- generators, motors,
etc.). their connections (star, delta, etc.). etc. Hence, for the purpose of power system analysis, a
simple single phase equivalent circuit is developed called, die one line diagram (OLD) or the
single line diagram (SLD). An SLD is this, the concise form of representing a given power system.
It is to be noted that a given SLD will contain only such data that are relevant to the system
analysis/study tinder consideration. For example, the details of protective devices need not be
shown for load flow analysis nor it is necessary to show the details of shunt values for stability
studies.
Symbols used for SLD
Various symbols are used to represent the different parameters and machines as single phase
equivalents on the SLD,. Some of the important symbols used are as listed in the table of Figure
below:

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TABLE OF SYMBOLS FOR USE ON SLDS

Example of OLD/SLD
Consider for illustration purpose, a sample example power system and data as under:
Generator 1: 30 MVA, 10.5 KV, X"= 1.6 ohms,
Generator 2:15 MVA, 6.6 KV, X"= 1.2 ohms,
Generator 3: 25 MVA, 6.6 KV, X"= 0.56 ohms,
Transformer 1 (3-phase): 15 MVA, 33/11 KV, X=15.2 ohms/phase on HT side,
Transformer 2 (3-phase): 15MVA, 33/6.2 KV, X=16.0 ohms/phase on HT side,
Transmission Line: 20.5 ohms per phase,
Load A: 15 MW, 11 KV, 0.9 PF (lag); and
Load B: 40 MW, 6.6 KV, 0.85 PF(lag).
The corresponding SLD incorporating the standard symbols can be shown as in figure 2
Below

It is observed here, that the generators are specified in 3-phase MVA, L-L voltage and per phase
Y-equivalent impedance, transformers are specified in 3-phase MVA, L-L voltage

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transformation ratio and per phase Y-equivalent impedance on any one side and the loads are
specified in 3-phase MW, L-L voltage and power factor.
2.5 Impedance Diagram:
The impedance diagram on single-phase basis for use under balanced conditions can be easily
drawn from the SLD. The following assumptions are made in obtaining the impedance diagrams.
Assumptions:
1. The single phase transformer equivalents are shown as ideals with impedances on appropriate
side (LV/HV).
2. The magnetizing reactance’s of transformers are negligible.
3. The generators are represented as constant voltage sources win! series resistance or reactance,
4. The transmission lines are approximated by their equivalent ^-Models,
5. The loads are assumed to be passive and are represented by a series branch of resistance or
reactance and
6. Since the balanced conditions are assumed, the neutral grounding impedances do not appear
in the impedance diagram.
Example system
As per the list of assumptions as above and with reference to the system, of figure 2, the
impedance diagram can be obtained as shown in figure 3.

2.6 Reactance Diagram:

With some more additional and simplifying assumptions, the impedance diagram can be
simplified farmer to obtain the corresponding reactance diagram. The following are the
assumptions made.
Additional assumptions:

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> The resistance is often omitted during the fault analysis. This causes a very* negligible error
since, resistances are negligible
> Loads are Omitted
> Transmission line capacitances are ineffective &
> Magnetizing currents of transformers are neglected.
Example system
As per the assumptions given above and with reference to the system of figure 2 and figure 3, the
reactance diagram can be obtained as shown in figure 4.

2.7 Per Unit Quantities:

During the power system analysis, it is a usual practice to represent current, voltage, impedance,
power, etc., of an electric power system in per unit or percentage of the base or reference value
of the respective quantities. The numerical per unit (pu) value of any quantity is its ratio to a
chosen base value of the same dimension. Thus a pu value is a normalized quantity with respect
to the chosen base value.
Definition: Per Unit value of a given quantity is the ratio of the actual value in any given unit to
the base value in the same unit. The percent value is 100 times the pu value. Both the pu and
percentage methods are simpler than the use of actual values. Further, the main advantage in
using the pu system of computations is that the result that comes out of the sum, product,
quotient, etc. of two or more pu values is expressed in per unit itself.
In an electrical power system, the parameters of interest include the current, voltage, complex
power (VA), impedance and the phase angle. Of these, the phase angle is dimensionless and the
other four quantities can be described by knowing any two of them. Thus clearly, an arbitrary
choice of any two base values will evidently fix the other base values.
Normally the nominal voltage of lines and equipment is known along with the complex power
rating in MVA. Hence, in practice, the base values are chosen for complex power (MVA) and
line voltage (KV). The chosen base MVA is the same for all the parts of the system. However,
the base voltage is chosen with reference to a particular section of the system and the other base
voltages (with reference to the other sections of the systems, these sections caused by die

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presence of the transformers! are then related to the chosen one by the turns-ratio of the
connecting transformer.
If Ib is the base current in kilo amperes and Vb. the base voltage in kilovolts, then the base MVA
is, Sb = (VbIb). Then the base values of current & impedance are given by

On the other hand die change of base can also be done by first converting the given pu
impedance to its ohmic value and then calculating its pu value on the new set of base values.
Merits and Demerits of pu System
Following are the advantages and disadvantages of adopting the pu system of computations in
electric power systems:
Merits:
> The pu value is the same for both 1 -phase and & 3-phase systems
> The pu value once expressed on a proper base, will be the same when refereed to either side
of the transformer. Thus the presence of transformer is totally eliminated.
> The variation of values is in a smaller range 9nearby unity)- Hence the errors involved in pu
computations are very less.
> Usually the nameplate ratings will be marked in pu on the base of the name plate ratings, etc.
De Merits:
> If proper bases are not chosen, then the resulting pu values may be highly absurd (such as 5.8
pu. -18.9 pu. etc.). This may cause confusion to the user. However, this problem can be avoided

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by selecting the base MVA near the high-rated equipment and a convenient base KV in any
section of the system.
Per Unit Impedance/ Reactance Diagram:
For a given power system with all its data with regard to the generators, transformers,
transmission lines, loads, etc.. it is possible to obtain the corresponding impedance or reactance
diagram as explained above. If the parametric values are shown in pu on the properly selected
base values of the system, then the diagram is referred as the per unit impedance or reactance
diagram. In forming a pu diagram, the following are the procedural steps involved:
1. Obtain the one line diagram based on the given data
2. Choose a common base MVA for the system
3. Choose a base KV in any one section (Sections formed by transformers)
4. Find the base KV of all the sections present
5. Find pu values of all the parameters: R.X, Z, E, etc.
6. Draw the pu impedance reactance diagram.
2.8 Formation of YBUS and ZBUS
The performance equations of a given power system can be considered in three different frames
of reference as discussed below:
Frames of Reference:
Bus Frame of Reference: There are b independent equations (b = no. of buses) relating the bus
vectors of currents and voltages through the bus impedance matrix and bus admittance matrix:

Branch Frame of Reference: There are b independent equations (b = no. of branches of a

selected Tree sub-graph of the system Graph) relating the branch vectors of currents and voltages
through the branch impedance matrix and branch admittance matrix:

Loop Frame of Reference: There are b independent equations (b = no. of branches of a selected
Tree sub-graph of the system Graph) relating the branch vectors of currents and voltages through
the branch impedance matrix and branch admittance matrix:

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YBUS formation using Rule of Inspection

Consider the 3-node admittance network as shown in figure. Using the basic branch relation: I
= YV, for all the elemental currents and applying Kirchhoff's Current Law principle at the nodal
points, we get the relations as under:

These are the performance equations of the given network in admittance form and they can be
represented in matrix form as:

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2.9 Basic Power Flow Equations:

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Equations (18) and (19) are known as the basic load flow equations. It can be seen that for any ith
bus, there are two equations. Therefore, for a 'n'-bus power system, there are altogether '2n' load-
flow equations.
Now, from equations (18) and (19) it can be seen that there are four unknown variables (Vi, δi, Pi,
and Qi) associated with the ith bus. Thus for the 'n'-bus system, there are a total of '4n' variables.
As there are only '2n' equations available, out of these '4n' variables, '2n' quantities need to be
specified and remaining '2n' quantities are solved from the '2n' load-flow equations. As '2n'
variables are to be specified in a 'n' bus system, for each bus, two quantities need to be specified.
For this purpose, the buses in a system are classified into three categories and in each category;
two different quantities are specified as described below.

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PQ Bus: At these buses loads are connected and therefore, these buses are also termed as load
buses. Generally the values of loads (real and reactive) connected at these buses are known and
hence, at these buses Pi and Qi are specified (or known). Consequently, Vi and θi need to be
calculated for these buses
PV Bus: Physically, these buses are the generator buses. Generally, the real power supplied by
the generator is known (as we say that the generation is supplying 100 MW) and also, the
magnitude of the terminal voltage of the generator is maintained constant at a pre-specified value
by the exciter (provided that the reactive power supplied or absorbed by the generator is within
the limits). Thus, at a PV bus, Pi and Vi are specified and consequently, Qi and θi need to be
calculated, these buses are also termed as voltage controlled buses
Slack Bus: To calculate the angles θi (as discussed above), a reference angle (θi = 0) needs to be
specified so that all the other bus voltage angles are calculated with respect to this reference
angle. For this generator, Vi and θi(= 0) are specified and the quantities Pi and Qi are calculated.
Basic Power flow Equations (18) and (19) represent a set of simultaneous, non-linear, algebraic
equations. As the set of equations is non-linear, no closed form, analytical solution for these
equations exist. Hence, these equations can only be solved by using suitable numerical iterative
techniques. For solving the load flow problem, various iterative methods exist. These are:
1. Gauss-Seidel method
2. Newton Raphson (polar coordinate) technique
3. Newton Raphson (rectangular coordinate) technique
4. Fast-decoupled load flow
2.10 Newton Raphson (polar coordinate) technique
Newton Raphson method is more efficient and practical for large power system. Main advantage
of this method is that the number of iterations required to obtain the solution is Independent of
the size of the system and computationally is very fast

The above basic load flow equations constitute a set of Non-linear algebraic equations. In terms
of independent variables, voltage magnitude in per unit and phase angle in radians. There will be
two equations (Pi and Qi equation) for each load bus(PQ Bus) and one equation for PV bus.
Initially we assume a flat voltage profile i.e magnitude of Vi=1 and δi=0 for all PQ buses

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For voltage controlled buses (P,V specified)δ set equal to 0.

Expanding Eqns. 18 & 19 in Taylor's series about the initial estimate and neglecting higher order
terms, we get

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Flow Chart of Newton-Raphson Load Flow( Polar co-ordinates) Technique

2.11 Fast-Decoupled Load-Flow(FDLF) Technique:
An important and useful property of power system is that the change in real power is primarily
governed by the changes in the voltage angles, but not in voltage magnitudes. On the other hand,
the changes in the reactive power are primarily influenced by the changes in voltage magnitudes,
but not in the voltage angles. Under normal steady state operation, the voltage magnitudes are all
nearly equal to 1.0.
♦ Practical power transmission lines have high X/R ratio.

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• Real power changes are less sensitive to voltage magnitude changes and are most sensitive to
changes in phase angle Δδ.
•Similarly, reactive power changes are less sensitive to changes in angle and are mainly
dependent on changes in voltage magnitude.
•Therefore the Jacobian matrix in Eqn.(21) can be written as

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Flowchart of Fast decoupled Load Flow Technique

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