Handbook of Power Systems Engineering with Power Electronics Applications

Preface

This book is a revision of ‘Handbook of Power System Engineering’ originally published in 2007. Further to various additional revisions on previous chapters 1–24, new chapters 25–28 for power electronics applications have been prepared. The preface for the original version is first quoted.

This book deals with the art and science of power systems engineering for those engineers who work in electricity-related industries such as power utilities, manufacturing enterprises, engineering companies, or for students of electrical engineering in universities and colleges. Each engineer's relationship with power system engineering is extremely varied, depending on the types of companies they work for and their positions. We expect readers to study the characteristics of power systems theoretically as a multi-dimensional concept by means of this book, regardless of readers' business roles or specialties.

We have endeavoured to deal with the following three points as major features of the book:

First, as listed in the Contents, the book covers the theories of several subsystems, such as generating plants, transmission lines and substations, total network control, equipment-based local control, protection, and so on, as well as phenomena ranging from power (fundamental) frequency to lightning and switching surges, as the integrally unified art and science of power systems. Any equipment in a power system network plays its role by closely linking with all other equipment, and any theory, technology or phenomenon of one network is only a viewpoint of the profound dynamic behaviour of the network. This is the reason why we have covered different categories of theories combined in a single hierarchy in this book.

Secondly, readers can learn about the essential dynamics of power systems mostly through mathematical approaches. We explain our approach by starting from physically understandable equations and then move on to the final solutions that illustrate actual phenomena, and never skip explanations or adopt half-measures in the derivations.

Another point here is the difference in meaning between `pure mathematically solvable' and `engineering analytically solvable'. For example, a person (even if expert in transient analysis) cannot derive transient voltage and current solutions of a simple circuit with only a few LCR constants connected in series or parallel because the equational process is too complicated, except in special cases. Therefore only solutions of special cases are demonstrated in books on transient analysis. However, engineers often have to find solutions of such circuits by manual calculation. As they usually know the actual values of LCR constants in such cases, they can derive `exact solutions' by theoretically justified approximation. Also, an appropriate approximation is an important technique to find the correct solution. Readers will also find such approximation techniques in this book.

Thirdly, the book deals with scientific theories of power system networks that will essentially never change. We intentionally excluded descriptions of advanced technologies, expecting such technologies to continue to advance year by year.

In recent years, analytical computation or simulation of the behaviour of large power system or complicated circuits has been executed by the application of powerful computers with outstanding software. However, it is quite easy to mishandle the analysis or the results because of the number of so many influential parameters. In this book, most of the theoretical explanation is based on typical simple circuits with one or two generators and one or two transmission lines. Precise understanding of the phenomena in such simple systems must always be the basis of understanding actual large systems and the incidents that may occur on them. This is the reason why power system behaviour is studied using small models.

The new chapters 25–28 are arranged for power electronic applications but from four different viewpoints. These are: the theory of induction generators/motors (chapter 25), fundamental characteristics of various power electronic devices (chapter 26), power electronic circuits and control theories (chapter 27) and finally various applications of power electronics focusing on power system engineering and some industrial load applications (chapter 28). The author intended to describe these four different layered subjects all together in this book, because, the author believes, most of existing books for power electronics applications usually discusses only two or three subjects, omitting the other closely related ones. In particular, chapter 25 for induction machines may be helpful for readers who are already familiar with power electronic applications.

Yoshihide Hase

Kawasaki-city, Japan

15 August, 2012

Acknowledgements

This book contains the various experiences and knowledge of many people. I am deeply indebted to these people, although I can only humbly acknowledge them in a general way.

Also, I wish to acknowledge all my former colleagues and friends who gave me various opportunities to work and study together over many years throughout my engineering career.

I would also like to deeply thank Laura Bell, Liz Wingett, Clarissa Lim, Stephanie Loh of John Wiley & Sons, Ltd, and in particular Peter Mitchell who led the staff of the second edition and Simone Taylor who led the staff of the original edition for their constant encouragement. Finally, I wish to sincerely acknowledge Sara Barnes and Mary Malin who worked on the editing and amendments, and Sharib Asrar for his help typesetting, both hard tasks essential to my work.

About the author

Yoshihide Hase was born in Gifu Prefecture, Japan, in 1937. After graduating in electrical engineering from Kyoto University, he joined the Toshiba Corporation in 1960 and took charge of various power system projects, both at home and abroad, including the engineering of generating station equipment, substation equipment, as well as power system control and protection, until 1996. During that time, he held the positions of general manager, senior executive of technology for the energy systems sector, and chief fellow. In 1996, he joined Showa Electric Wire & Cable Company as the senior managing director and representative director and served on the board for eight years. He was a lecturer at Kokushikan University for five years since 2004. He was the vice president of the IEEJ (1995–96) and has been bestowed as a honorary member. He was also the representative officer of the Japanese National Committee of CIGRE (1987–1996) and has been bestowed as a distinguished member of CIGRE.

The author's address: hasey@jcom.home.ne.jp

Introduction

‘Utilization of fire’, ‘agricultural cultivation’ and ‘written communication’: these three items are sometimes quoted as the greatest accomplishments of humankind. As a fourth item, ‘social structures based on an electrical infrastructure’, which was created by humans mostly within the twentieth century, may be added.

Within the last hundred years, we have passed through the era of ‘electricity as a convenient tool’ to the point where electricity has become an inevitable part of our infrastructure as a means of energy acquisition, transport and utilization as well as in communication media. Today, without electricity we cannot carry out any of our living activities such as ‘making fire’, ‘getting food and water’ ‘manufacturing tools’, ‘moving’, ‘communicating with others’, and so on. Humans in most parts of the world have thus become very dependent on electricity. Of course, such an important electrical infrastructure means our modern power system network.

A power system network can be likened to the human body. A trial comparison between the two may be useful for a better understanding of the essential characteristics of the power system.

First, the human body is composed of a great many subsystems (individual organs, bones, muscles, etc.), and all are composed in turn of an enormous number of minute cells. A power system network of a large arbitrary region is composed of a single unified system. Within this region, electricity is made available in any town, public utility, house and room by means of metal wires as a totally integrated huge network.

Generating plants, substations and transmission lines; generators, transformers, switchgear and other high-voltage equipment; several types of control equipment, protection equipment and auxiliary equipment; control and communication facilities in a dispatching or control centre; and the various kinds of load facilities – all these are also composed of a very large number of small parts or members. Individual parts play their important roles by linking with the rest of the network system. Human operators at any part of the network can be added as important members of the power system. We might say that a power system network is the largest and greatest artificial system ever produced by people in the modern era.

Secondly, the human body maintains life by getting energy from the external environment, and by processing and utilizing this energy. New cellular tissue is consequently created and old tissue is discarded. In such a procedure, the human body continues to grow and change.

A power system can be compared in the same way. A prerequisite condition of a power system network is that it is operated continuously as a single unified system, always adding new parts and discarding old ones. Since long-distance power transmission was first established about a hundred years ago, power systems have been operating and continuing to grow and change in this way, and, apart from the failure of localized parts, have never stopped. Further, no new power system isolated from the existing system in the same region has ever been constructed. A power system is the ultimate inheritance succeeded by every generation of humankind.

Thirdly, humans experience hunger in just a few hours after their last meal; their energy storage capacity is negligible in comparison with their lifetimes. In a power system such as a pumped-storage hydro-station, for example, the capacity of any kind of battery storage system is a very small part of the total capacity. The power generation balance has to be maintained every second to correspond to fluctuations or sudden changes in total load consumption. In other words, ‘simultaneity and equality of energy generation and energy consumption’ is a vital characteristic of power system as well as of human body.

Fourthly, humans can continue to live even if parts of the body or organs are removed. At the other extreme, a minute disorder in cellular tissue may be life-threatening. Such opposites can be seen in power systems.

A power system will have been planned and constructed, and be operated, to maintain reasonable redundancy as an essential characteristic. Thus the system may continue to operate successfully in most cases even if a large part of it is suddenly cut off. On the contrary, the rare failure of one tiny part, for example a protective relay (or just one of its components), may trigger a kind of domino effect leading to a black-out.

Disruption of large part of power system network by ‘domino-effect’ means big power failure leaded by abrupt segmentation of power system network, which may be probably caused by cascade trips of generators caused by total imbalance of power generation and consumption which leads to ‘abnormal power frequency exceeding over or under frequency capability limits (OF/UF) of individual generators’, ‘cascade trips of generators caused by power stability limits, Q-V stability limits or by any other operational capability limits’, ‘cascade trips of trunk-lines/stations equipment caused by abnormal current flow exceeding individual current capacity limits (OC), or by over or under voltage limits (OV/UV)’, ‘ succeeding cascade trips after fault tripping failure due to a breaker set back or caused by mal-operation of a protective relay’ and so on, and may be perhaps caused as of ‘these composite phenomena’. These nature of power systems is the outcome that all the equipment and parts of the power system, regardless of their size, are closely linked and coordinated. The opposites of toughnees with well redundancy and delicacy are the essential nature of power systems.

Fifthly, as with the human body, a power system cannot tolerate maltreatment, serious system disability or damage, which may cause chronic power cuts, and moreover would probably causes extremely fatal social damages. Recovery of a damaged power system is not easy. It takes a very long time and is expensive, or may actually be impossible. Power systems can be kept sound only by the endeavours of dedicated engineers and other professional people.

Sixthly, and finally, almost as elaborate as the human body, all the parts of power system networks today (including all kinds of loads) are masterpieces of the latest technology, based on a century of accumulated knowledge, something which all electrical engineers can share proudly together with mechanical and material engineers. Also all these things have to be succeeded to our next generations as the indispensable social structures.

Chapter 1

Overhead Transmission Lines and Their Circuit Constants

In order to understand fully the nature of power systems, we need to study the nature of transmission lines as the first step. In this chapter we examine the characteristics and basic equations of three-phase overhead transmission lines. However, the actual quantities of the constants are described in Chapter 2.

1.1 Overhead Transmission Lines with LR Constants

1.1.1 Three-Phase Single Circuit Line without Overhead Grounding Wire

1.1.1.1 Voltage and Current Equations, and Equivalent Circuits

A three-phase single circuit line between a point m and a point n with only L and R and without an overhead grounding wire (OGW) can be written as shown in Figure 1(a). In the figure, rg and Lg are the equivalent resistance and inductance of the earth, respectively. The outer circuits I and II connected at points m and n can theoretically be three-phase circuits of any kind.

Figure 1.1 Single circuit line with LR constants

All the voltages Va, Vb, Vc and currents Ia, Ib, Ic are vector quantities and the symbolic arrows show the measuring directions of the three-phase voltages and currents which have to be written in the same direction for the three-phases as a basic rule to describe the electrical quantities of three-phase circuits.

In Figure 1.1, the currents Ia, Ib, Ic in each phase conductor flow from left to right (from point m to point n). Accordingly, the composite current Ia+Ib+Ic has to return from right to left (from point n to m) through the earth–ground pass. In other words, the three-phase circuit has to be treated as the set of ‘three-phase conductors + one earth circuit’ pass.

In Figure 1(a), the equations of the transmission line between m and n can be easily described as follows. Here, voltages V and currents I are complex-number vector values:

(1.1)

Substituting into , and then eliminating

(1.2)

Now, the original Equation (1.1) and the derived Equation (1.2) are the equivalent of each other, so Figure 1.1(b), showing Equation (1.2), is also the equivalent of Figure 1.1(a).

Equation (1.2) can be expressed in the form of a matrix equation and the following equations are derived accordingly (refer to Appendix B for the matrix equation notation):

(1.3)

(1.4)

Now, we can apply symbolic expressions for the above matrix equation as follows:

(1.5) equation

where

(1.6)

Summarizing the above equations, Figure 1.1(a) can be described as Equations (1.3) and (1.6) or Equations (1.5) and (1.6), in which the resistance rg and inductance Lg of the earth return pass are already reflected in all these four equations, although Ig and are eliminated in Equations (1.5) and (1.6). We can consider Figure 1.1(b) as the equivalent circuit of Equations (1.3) and (1.4) or Equations (1.5) and (1.6). In Figure 1.1(b), earth resistance rg and earth inductance Lg are already included in the line constants Zaa, Zab, etc., so the earth in the equivalent circuit of Figure 1.1(b) is ‘the ideal earth’ with zero impedance. Therefore the earth can be expressed in the figure as the equal-potential (zero-potential) earth plane at any point. It is clear that the mutual relation between the constants of Figure 1.1(a) and Figure 1.1(b) is defined by Equation (1.4). It should be noted that the self-impedance Zaa and mutual impedance Zab of phase a, for example, involve the earth resistance rg and earth inductance Lg.

Generally, in actual engineering tasks, Figure 1.1(b) and Equations (1.3) and (1.4) or Equations (1.5) and (1.6) are applied instead of Figure 1.1(a) and Equations (1.1) and (1.2); in other words, the line impedances are given as Zaa, Zab, etc., instead of Zaag, Zabg. The line impedances Zaa, Zbb, Zcc are named ‘the self-impedances of the line including the earth–ground effect’, and Zab, Zac, Zbc, etc., are named ‘the mutual impedances of the line including the earth–ground effect’.

1.1.1.2 Measurement of Line Impedances

Let us consider how to measure the line impedances taking the earth effect into account.

As we know from Figure 1.1(b) and Equations (1.3) and (1.4), the impedances Zaa, Zab, Zac, etc., can be measured by the circuit connection shown in Figure 1.2(a).

Figure 1.2 Measuring circuit of line impedance

The conductors of the three-phases are grounded to earth at point n, and the phase b and c conductors are opened at point m. Accordingly, the boundary conditions can be adopted for Equation (1.3):

(1.7)

Therefore the impedances Zaa, Zab, Zac can be calculated from the measurement results of , and Ia.

All the impedance elements in the impedance matrix of Equation (1.7) can be measured in the same way.

1.1.1.3 Working Inductance

Figure 1.2(b) shows the case where the current I flows along the phase a conductor from point m to n and comes back from n to m only through the phase b conductor as the return pass. The equation is with boundary conditions equation

(1.8a)

Therefore

(1.8b)

Equation (1.8b) indicates the voltage drop of the parallel circuit wires a, b under the condition of the ‘go-and-return-current’ connection. The current I flows out at point m on the phase a conductor and returns to m only through the phase b conductor, so any other current flowing does not exist on the phase c conductor or earth–ground pass. In other words, Equation (1.8b) is satisfied regardless of the existence of the third wire or earth–ground pass. Therefore the impedance Zaa – Zab as well as Zbb – Zba should be specific values which are determined only by the relative condition of the phase a and b conductors, and they are not affected by the existence or absence of the third wire or earth–ground pass. Zaa – Zab is called the working impedance and the corresponding Laa – Lab is called the working inductance of the phase a conductor with the phase b conductor.

Furthermore, as the conductors a and b are generally of the same specification (the same dimension, same resistivity, etc.), the impedance drop between m and n of the phase a and b conductors should be the same. Accordingly, the working inductances of both conductors are clearly the same, namely .

The value of the working inductance can be calculated from the well-known equation below, which is derived by an electromagnetic analytical approach as a function only of the conductor radius r and the parallel distance sab between the two conductors:

(1.9)

This is the equation for the working inductance of the parallel conductors a and b, whose deriving process is shown in the section 1.3.1 as of theory of electromagnetism. The equation shows that the working inductance Laa – Lab for the two parallel conductors is determined only by the relative distance between the two conductors sab and the radius r, so it is not affected by any other conditions such as other conductors or the distance from the earth surface.

The working inductance can also be measured as the value (1/2)V/I by using Equation (1.8b) .

1.1.1.4 Self- and Mutual Impedances Including the Earth–Ground Effect Laa, Lab

Now we evaluate the actual numerical values for the line inductances contained in the impedance matrix of Equation (1.3).

The currents Ia, Ib, Ic flow through each conductor from point m to n and Ia+Ib+Ic returns from n to m through the ideal earth return pass. All the impedances of this circuit can be measured by the method of Figure 1.2(a). However, these measured impedances are experimentally a little larger than those obtained by pure analytical calculation based on the electromagnetic equations with the assumption of an ideal, conductive, earth plane surface.

In order to compensate for these differences between the analytical result and the measured values, we can use an imaginary ideal conductive earth plane at some deep level from the ground surface as shown in Figure 1.3.

Figure 1.3 Earth–ground as conductor pass

In this figure, the imaginary perfect conductive earth plane is shown at the depth Hg, and the three imaginary conductors are located at symmetrical positions to conductors a, b, c, respectively, based on this datum plane.

The inductances can be calculated by adopting the equations of the electromagnetic analytical approach to Figure 1.3.

1.1.1.4.1 Self-Inductances Laa, Lab Lcc

In Figure 1.3, the conductor a (radius r) and the imaginary returning conductor are symmetrically located on the datum plane, and the distance between a and is . Thus the inductance of conductor a can be calculated by the following equation which is a special case of Equation (1.9) under the condition :

(1.10a)

Conversely, the inductance of the imaginary conductor (the radius is Ha, because the actual grounding current reaches up to the ground surface), namely the inductance of earth, is

(1.10b)

Therefore,

(1.11)

Lbb, Lcc can be derived in the same way.

Incidentally, the depth of the imaginary datum plane can be checked experimentally and is mostly within the range of . On the whole Hg is rather shallow, say 300 – 600 m in the geological younger strata after the Quaternary period, but is generally deep, say 800 – 1000 m, in the older strata of the Tertiary period or earlier.

1.1.1.4.2 Mutual Inductances Lab, Lbc, Lca

The mutual inductance Lab can be derived by subtracting Laa from Equation (1.11) and the working inductance (Laa – Lab) from Equation (1.9):

(1.12a)

Similarly

(1.12b)

where , and so on.

Incidentally, the depth of the imaginary datum plane would be between 300 and 1000 m, while the height of the transmission tower ha is within the range of 10–100 m (UHV towers of 800–1000 kV would be approximately 100 m or less). Furthermore, the phase-to-phase distance Sab is of order 10 m, while the radius of conductor r is a few centimetres (the equivalent radius reff of EHV/UHV multi-bundled conductor lines may be of the order of 10–50 cm).

Accordingly,

(1.13)

Then, from Equations (1.9), (1.11) and (1.12),

(1.14) equation

1.1.1.4.3 Numerical Check

Let us assume conditions

Then calculating the result by Equation (1.11) and (1.12),

If , then . As is contained in the logarithmic term of the equations, constant values Laa, Lab and so on are not largely affected by , neither is radius r nor reff as well as the phase-to-phase distance sab. Besides, 0.1 and 0.05 in the second term on the right of Equations (1.9)–(1.12) do not make a lot of sense.

Further, if transmission lines are reasonably transpositioned, can be justified so that Equation (1.3) is simplified into Equation (2.13) of Chapter 2.

1.1.1.5 Reactance of Multi-Bundled Conductors

For most of the recent large-capacity transmission lines, multi-bundled conductor lines (n = 2 – 8 per phase) are utilized as shown in Figure 1.4. In the case of n conductors (the radius of each conductor is r), Laag of Equation (1.10a) can be calculated from the following modified equation:

(1.15a)

Figure 1.4 Overhead double circuit transmission line

Refer the Supplement 1 for the introduction of equivalent radius of a multi-bundled conductors.Since the self-inductance Lg of the virtual conductor given by Equation (1.10b) is not affected by the adoption of multi-bundled phase a conductors, accordingly

(1.15b)

1.1.1.5.1 Numerical Check

Using TACSR = 810 mm² (see Chapter 2), 2r = 40 mm and four bundled conductors (n = 4), with the square allocation w = 50 cm averaged distance

(1.16)

The equivalent radius req = 24.7 cm is 12.4 times r = 2.0 cm, so that the line self-inductance Laa can also be reduced by the application of bundled conductors. The mutual inductance Lab of Equation (1.12a) is not affected by the adoption of multi-bundled conductor lines.

1.1.1.6 Line Resistance

Earth resistance rg in Figure 1.1(a) and Equation (1.2) can be regarded as negligibly small. Accordingly, the so-called mutual resistances rab, rbc, rca in Equation (1.4) become zero. Therefore, the specific resistances of the conductors ra, rb, rc are actually equal to the resistances raa, rbb, rcc in the impedance matrix of Equation (1.3).

In addition to the power loss caused by the linear resistance of conductors, non-linear losses called the skin-effect loss and corona loss occur on the conductors. These losses would become progressionally larger in higher frequency zones, so they must be major influential factors for the attenuation of travelling waves in surge phenomena. However, they can usually be neglected for power frequency phenomena because they are smaller than the linear resistive loss and, further, very much smaller than the reactance value of the line, at least for power frequency.

In regard to the bundled conductors, due to the result of the enlarged equivalent radius req, the dielectric strength around the bundled conductors is somewhat relaxed, so that corona losses can also be relatively reduced. Skin-effect losses of bundled conductors are obviously far smaller than that of a single conductor whose aluminium cross-section is the same as the total sections of the bundled conductors.

1.1.2 Three-Phase Single Circuit Line with OGW, OPGW

Most high-voltage transmission lines are equipped with OGW (overhead grounding wires) and/or OPGW (OGW with optical fibres for communication use).

In the case of a single circuit line with single OGW, the circuit includes four conductors and the fourth conductor (x in Figure 1.5) is earth grounded at all the transmission towers. Therefore, using the figure for the circuit, Equation (1.3) has to be replaced by the following equation:

(1.17a)

Figure 1.5 Single circuit line with OGW

Extracting the fourth row,

Substituting Ix into the first, second and third rows of Equation (1.17a),

(1.18)

This is the fundamental equation of the three-phase single circuit line with OGW in which Ix has already been eliminated and the impedance elements of the grounding wire are slotted into the three-phase impedance matrix. Equation (1.18) is obviously of the same form as Equation (1.3), while all the elements of the rows and columns in the impedance matrix have been revised to smaller values with corrective terms etc.

The above equations indicate that the three-phase single circuit line with OGW can be expressed as a impedance matrix equation in the form of Equation (1.18) regardless of the existence of OGW, as was the case with Equation (1.3). Also, we can comprehend that OGW has roles not only to shield lines against lightning but also to reduce the self- and mutual reactances of transmission lines.

1.1.3 Three-Phase Double Circuit Line with LR Constants

The three-phase double circuit line can be written as in Figure 1.6 and Equation (1.19) regardless of the existence or absence of OGW:

(1.19)

Figure 1.6 Three-phase double circuit line with LR constants

In addition, if the line is appropriately phase balanced, the equation can be expressed by Equation (2.17) of Chapter 2.

1.2 Stray Capacitance of Overhead Transmission Lines

1.2.1 Stray Capacitance of Three-Phase Single Circuit Line

1.2.1.1 Equation for Electric Charges and Voltages on Conductors

Figure 1.7(a) shows a single circuit line, where electric charges qa, qb, qc [C/m] are applied to phase a, b, c conductors and cause voltages Va, Vb, Vc [V], respectively. The equation of this circuit is given by

(1.20a)

Figure 1.7 Stray capacitance of single circuit line

The inverse matrix equation can be derived from the above equation as

(1.20b)

Here, and are inverse matrices of each other, so that unit matrix; refer to Appendix B).

Accordingly,

(1.20c)

where are the coefficients of the potential and are the electrostatic coefficients of static capacity.

Modifying Equation (1.20b) a little,

(1.21)

then

(1.22)

with qa, qb, qc [C/m], Vb, Vb, Vc [V] and

(1.23)

Equations (1.22) and (1.23) are the fundamental equations of stray capacitances of a three-phase single circuit overhead line. Noting the form of Equation (1.22), Figure 1.7(b) can be used for another expression of Figure 1.7(a): Caa, Cbb, Ccc are the phase-to-ground capacitances and Cab = Cba, Cbc = Ccb, Cca = Cac are the phase-to-phase capacitances between two conductors.

1.2.1.2 Fundamental Voltage and Current Equations

It is usually convenient in actual engineering to adopt current [A] instead of charging value , and furthermore to adopt effective (rms: root mean square) voltage and current of complex-number V, I instead of instantaneous value .

As electric charge q(t) is the integration over time of current i, the following relations can be derived:

(1.24)

Equation (1.22) can be modified to the following form by adopting Equation (1.24) and by replacement of etc.:

(1.25)

Therefore

(1.26a)

or, with a small modification,

(1.26b)

This is the fundamental equation for stray capacitances of a three-phase single circuit transmission line. Also Figure 1.7(c) is derived from one-to-one correspondence with Equation (1.26).

1.2.1.3 Coefficients of Potential (paa, pab), Coefficients of Static Capacity (kaa, kab) and Capacitances (caa, cab)

The earth surface can be taken as a perfect equal-potential plane, so that we can use Figure 1.8, in which the three imaginary conductors are located at symmetrical positions of conductors a, b, c, respectively, based on the earth surface plane. By assuming electric charges and , per unit length on conductors a, b, c, and respectively, the following voltage equation can be derived:

Figure 1.8 Three parallel overhead conductors

Equations for Vb, Vc can be derived in the same way. Then

(1.27)

where

. Refer the section 1.3.2 for the deriving process as of the Equation (1.27) as of theory of electromagnetism.

The equation indicates that the coefficients of potential (paa, pab, etc.) are calculated as a function of the conductor’s radius r, height (ha, hb, hc) from the earth surface, and phase-to-phase distances (sab, sac, etc.) of the conductors. paa, pab, etc., are determined only by physical allocations of each phase conductor (in other words, by the structure of towers), and relations like are obvious.

In conclusion, the coefficients of potential (paa, pab, etc.), the coefficients of static capacity (kaa, kab, etc.) and the capacitance (Caa, Cab, etc.) are calculated from Equations (1.27), (1.20) and (1.23), respectively. Again, all these values are determined only by the physical allocation of conductors and are not affected by the applied voltage.

1.2.1.4 Stray Capacitances of Phase-Balanced Transmission Lines

Referring to Figure 1.8, a well-phase-balanced transmission line, probably by transposition, can be assumed. Then

(1.28)

(1.29) equation

Accordingly, Equation (1.20) can be simplified as follows:

(1.30)

and from Equation (1.23)

(1.31)

and from Equation (1.27)

(1.32)

Substituting ps, pm from Equation (1.32) into Equation (1.31),

(1.33)

In conclusion, a well-phase-balanced transmission line can be expressed by Figure 1.9(a) and Equation (1.26b) is simplified into Equation (1.34), where the stray capacitances Cs, Cm can be calculated from Equation (1.33):

(1.34)

Figure 1.9 Stray capacitances of single circuit overhead line (well balanced)

Incidentally, Figure 1.9(a) can be modified to Figure 1.9(b), where the total capacitance of one phase is called the working capacitance of single circuit transmission lines, and can be calculated by the following equation:

(1.35)

Refer the Supplement 1 for the introduction of equivalent radius of a multi-bundled conductors.

1.2.1.4.1 Numerical Check

Taking the conditions conductor radius , averaged phase-to-phase distance and average height then by Equations (1.33) and (1.35), we have

1.2.2 Three-Phase Single Circuit Line with OGW

Four conductors of phase names a, b, c, x exist in this case, so the following equation can be derived as an extended form of Equation (1.26a):

(1.36a)

where , because OGW is earth grounded at every tower. Accordingly,

(1.36b)

This matrix equation is again in the same form as Equation (1.26b). However, the phase-to-ground capacitance values (diagonal elements of the matrix C) are increased (the value of Cax is increased for the phase a conductor, from ).

1.2.3 Three-Phase Double Circuit Line

Six conductors of phase names a, b, c, A, B, C exist in this case as is shown in Figure 1.10, so the following equation can be derived as an extended form of Equation (1.26a):

Figure 1.10 Stray capacitance of double circuit line (well balanced)

(1.37a)

Then

(1.37b)

It is obvious that the double circuit line with OGW can be expressed in the same form.

The case of a well-transposed double circuit line is as shown in Figure 1.9(b):

(1.38)

Above, we have studied the fundamental equations and circuit models of transmission lines and the actual calculation method for the L, C, R constants. Concrete values of the constants are investigated in Chapter 2.

1.3 Working Inductance and Working Capacitance

The Equation (1.9) for working inductance and Equation (1.35) for working capacitance as well as Equation (1.27) for capacitive induced voltage were briefly shown in the previous sections. Now we introduce these equations and examine what these equations mean from the physical viewpoint of electromagnetism.

1.3.1 Introduction of Working Inductance

1.3.1.1 Introduction of Self-Inductance Laa of a Straight Conductor

As is shown in Figure 1.11, one conductor a (radius r) is laid out straight in an area of permeability ( is permeability in vacuum space and is relative permeability and in vacuum space). If current i flows through conductor a, concentric circular magnetic paths are composed in a conductor section as well as in outer space, and the central point O of the conductor a is also the central point of induced concentric magnetic paths. The concentric magnetic paths in the outer space of the conductor a is examined first. A thin concentric magnetic ring path at point x from O with length and width dx can be imaged. The magnetic resistance R of the ring path is proportional in the length of the ring path and is inversely proportional in the sectional . Namely,

(1.39a) equation

where :the permeability of the ring path

: permeability in vacuum space ( by MKS rational unit system)

: relative permeability ( in vacuum space)

Figure 1.11

The reason that is in MKS rational unit system is discussed later in section 1.3.4. If current is flowed through the conductor (or if electromotive force is charged in the conductor), flux is produced through the ring path with sectional depth dx and

(1.39b) equation

The linking flux number is

(1.39c) equation

Therefore the total linking flux of the space from the conductor surface (radius ) to point S is

(1.39d)

Next, linking flux number in the conductor section is examined. If current is flowed through the conductor, the current within space of diameter is

(1.40a) equation

The intensity of electric field at the ring path with length and width dx which is x distant from point O in the radial direction is:

(1.40b) equation

The flux density is:

(1.40c)

where is the relative permeability of the conductor

The flux at the x distant ring path with dx width is:

(1.40d)

The turn number of the conductor within circle of radius x can be considered , then the linking flux number is

(1.40e)

As the result of all the above Equations (1.39d)(1.40e), total linking flux numbers which is produced by current i of the conductor a and interlink with the current i itself between the area of conductor a to outer space point S is:

(1.41a)

As the definition of inductance is the linking flux number per 1A, or then

(1.41b) equation

or

(1.41c)

This is the self inductance of the conductor a, and the equation correspond with Equation (1.10a).

1.3.1.2 Introduction of Working-Inductance Laa – Lab of two Conductors

In next, working inductance of two conductors a and b is examined. (refer Figure 1.11(b)).

Two conductors a and b (radius r) are lay out in parallel with distance S and the current go out on the conductor a and come back from b, or current flows in a and current flows in b. Now, we image an arbitrary point , which is distant from a and distant from b, and the point y is far distant from both conductors a and b, namely, .

Current of conductor a produces concentric flux of conductor a and all these flux interlink with the current i, so that linking flux number is given by Equation (1.41a). That is again,

(1.42a)

Next, current of conductor b produces concentric flux of conductor b. Among these flux, linking flux to which current i of conductor a links with can be calculated by accumulating from S to . That is,

(1.42b)

The total linking flux number of current i of conductor a is the sum of , and reminding

(1.42c)

The definition of inductance is linking flux numbers per 1Ampere, that is then

(1.42d)

Now, we have introduced general equation of working inductance .

The Equation (1.42d) is modified a little by putting as of MKS rational unit system.

(1.42e)

This is the working inductance of two conductors lay out through three dimensional vacuum space, and is of course the same with Equation (1.9). In case of vacuum space or air space and is the permeability of aluminum or copper and is .

1.3.2 Introduction of Working Capacitance

Now referring to Figure 1.12(a), we introduce working capacitance of two parallel conductors a and b (radius r) with the same lay out of that in the previous section. Supposing the case in that the conductor a is charged by and b is charged by , and the condition of point y is examined which is , distant from the conductors a and b. Because the conductor radius r is quite small it can be presumed that the charges +q and -q are allocated at the center pin points of the conductors a and b. The intensity of electric field at point y caused by of conductor a and caused by of conductor b are:

(1.43) equation

where : permittivity of the circuit field

Figure 1.12

(1.44)

The electric potential at the mid-point which is the same distance from the two conductors (the point of ) should be obviously zero, then,

(1.45a)

then

(1.45b) equation

where

(1.46) equation

The Equation of surface potential v of the conductor a is given by as a special case of (1.45a).

(1.47a) equation

(1.47b)

The capacitance from conductor a (or b) to the zero potential plane (neutral plane) at the mid-point of conductors a and b is given by:

(1.48a) equation

Applying MKS rational unit system by Equation (1.46) and decimal logarithm,

(1.48b)

The Equations (1.45a)(1.47a)(1.48a) explain natural physics whose forms are not affected by selection of any measuring unit system, and Equations (1.45b)(1.47b)(1.48b) are the expression by MKS rational unit system based on Equation (1.46).

Now, let us compare the Figure 1.12(a) and (b). The potential of neutral plane g is zero, so that the plane can be equated with earth ground, and therefore Figure 1.12(a) and (b) are equivalent of each other. In other words, theory of transmission line can be treated by a set of real conductor a with charge +q and imaginary conductor α with charge -q. Needless to say Equation (4.18b) corresponds to Equation (1.35).

Furthermore, if we change space distance , from the conductors a and α but by keeping as of constant value, of Equation (1.45a,b) should be kept unchanged. So Equation (1.45) gives equipotential lines as is shown in Figure 1.12(b).

1.3.3 Special Properties of Working Inductance and Working Capacitance

The equation of working inductance and working capacitance were introduced in the previous section. These are again:

(1.42d) equation

(1.48a) equation

Also permeability and permittivity were explained through the deriving process, and these are again, by our MKS rational unit system:

(1.49)

Now, let us examine furthermore about the above equations. The right side second term of Equation (1.42d) is of linking flux number in narrow conductor section, so that it can be ignored when phenomena of wide space is examined. Then, working inductance Laa – Lab and working capacitance Ca relate of each other as follows:

In case of vacuum space and , then

(1.50)

By MKS unit system

(1.51)

Now it was found that always comes to which takes constant value c0 unconditionally. From the physical viewpoint, if current flow through a straight conductor lay out in three dimensional vacuum space, it would be accompanied by magnetic line with permeability and electric line of force with permittivity . Furthermore, takes constant value c0 unconditionally.

In fact, Equation (1.50)(1.51) are the climax of the conclusion which was presented by James C Maxwell in 1873 in his famous paper (refer Coffee break 5). The constant c0 is of a value with dimension of `distance/time' or `velocity'. With these conclusive equations, Maxwell presumed as follows

i. electromagnetic wave would exist and it can propagate through `vacuum space without `ether',

ii. The propagating velocity of the wave is always constant value , and it would be 300,000km/sec if it is measured by MKS rational unit system. This was the time that electromagnetic wave was discovered theoretically by Maxwell. He also presumed by analogy that light from the sun must be also a kind of wave having the same velocity 300,000km/sec.

1.3.4 MKS Rational Unit System and the Various MKS Practical Units in Electrical Engineering Field

1.3.4.1 MKS Rational Unit System

We discuss about fundamentals of MKS rational unit system as the last subject of this chapter.

The velocity of electromagnetic wave c0 is an universal unchanged constant, and the value is if measured by MKS unit system. In next, c0 is the unchanged value 300,000km/sec, so that is also unchanged value. In other words, we can freely determine either one of or as methods of unit system selection although is unchanged value. Namely, if one value is given to one of and as its definition, the another should be defined dependently to satisfy the above equation.

Therefore, and are defined as follows by MKS rational unit system.

(1.52a) equation

(1.52b) equation

(1.52c) equation

Now, we go back to the historical story of MKS rational unit system.

Famous Coulomb's laws for force by electric charge and for force by magnetic pole can be described by Gaussian unit system and by MKS rational unit system as follows:

(1.53)

where and are defined by Equation (1.52a,b) by MKS rational unit system.

In order to compare both unit systems, we imagine a hollow sphere as shown in Figure 1.13. If the radius is r, the surface area is regardless of unit system. If electric charge = 1.0 is placed at the center point of the sphere, the electric line of force would be radiated uniformly towards the sphere surface. Now, we are free to count the total numbers of the radiated line of force. And then, the number is counted as 1(one) by Gaussian unit system, 4π by CGS(cm, gr, sec) rational unit system, by MKS rational unit system.

Figure 1.13

By Gaussian unit system, the expression of Coulomb's law is simple; however the number of line of force per unit area would become . By CGS unit system, total number is 4π, and then the number per unit area at the surface is which means we can count the line of force per unit area by the equation without 4π. Generally by CGS rational unit system, we can escape from the inconvenience of 4π or by removing them in the related equations in counting various physical quantities, while on the other hand 4π or are always included in equations based on Gaussian unit system.

MKS rational unit system has the same concept with the CGS rational unit system except that m instead of cm and kg instead of g are adopted. In conclusion, and are defined by Equation (1.52b)(1.52c) by MKS rational unit system because of the above reason.

Hereunder is an comparison of MKS rational unit system and CGS rational unit system in regard with force F and energy,

(1.54)

The digits number are different by times for force and by times for energy.

1.3.4.2 Practical MKS Units for Electrical Engineering Physics

A conspectus of various electrical practical units is explained in brief as the last part of this chapter.

The meter unit system was established in 1875 and then unit system based on three fundamental units m, kg, sec were popularized all over the world. In 1951, Ampere was added as the forth fundamental unit, and the expanded MKSA unit system was authorized, which means various units for electrical physics were officially combined with various units for Newton physics. After this year, Kelvin(K) for temperature and Candela(cd) for light intensity were added, and then in 1960, the International unit system (SI: International System of Units) was established which includes seven fundamental units as shown in Table 1.1. This is today's Expanded MKS unit system. All other units except these seven units are defined dependently as the derived units from seven fundamental units. Further, useful derived units are defined with proper unit names. As an example, the unit for electric charge ±q is counted as time-integration of Ampere then having unit value of Ampere · sec. Therefore new unit name Coulomb is defined for the derived unit Ampere · sec. In other words is a derived unit defined with proper unit name. Table 1.2 shows various derived units having proper defined unit names in electrical physics.

Table 1.1 Fundamental units by International unit system (SI)

Table 1.2 Definition of various derived units in electrical physics.

1.4 Supplement: Proof of Equivalent Radius for a Multi-bundled Conductor

The equivalent radius of a multi-bundled conductor in Equations (1.15a) and (1.35 ) can be proved as follows.

1.4.1 Equivalent Radius for Inductance Calculation

One phase n-bundled conductor is examined where (n: number of conductors, r: radius of each conductor, w: averaged distance between two conductors, h: height above ground level. As all the elemental conductors are well balanced, the equation bellow is derived as of analogy to Equation (1.3).

(1)

If the voltage and current of the bundled-conductor are v and i, the voltage and current of each elemental conductor is v and , then.

(2)

Then we have

(3) equation

(4a) equation

(4b) equation

If the above bundled-conductor is equivalent with a single conductor with radius and arranged at the same height h, and is charged with the same v and i,

(5) equation

(6) equation

As the Equation (3) and (5) should be equal, then

(7) equation

therefore

then,

(8)

therefore

(9) equation

This is the same with Equation (1.15a).

1.4.2 Equivalent Radius of Capacitance Calculation

If the voltage and charge of a n-bundled conductor is v and +q, the charge of each elemental conductor is . Then the following equation is derived in analogy with Equation (1.27).

(10)

If the above n-bundled conductor is equivalent to a single conductor with radius and the same height h, and is charged with the same v and +q,

(11) equation

Comparing the both equations, the equation below is derived.

(12) equation

This is the same with Equation (9), and of course with Equation (1.35 ).

Now above all, inductance as well as capacitance of multi-bundled conductors can be calculated by applying equivalent radius given by Equation (9) or (12). This is the proof of Equation (1.15a) and (1.35 ).

Coffee Break 1: Electricity, its Substance and Methodology

The new steam engine of James Watt (1736–1819) ushered in the great dawn of the Industrial Revolution in the 1770s. Applications of the steam engine began to appear quickly in factories, mines, railways, and so on, and the curtain of modern mechanical engineering was raised. The first steam locomotive, designed by George Stephenson (1781–1848), appeared in 1830.

Conversely, electrical engineering had to wait until Volta began to provide ‘stable electricity’ from his voltaic pile to other electrical scientists in the 1800s. Since then, scientific investigations of the unseen electricity on one hand and practical applications for telegraphic communication on the other hand have been conducted by scientists or electricians simultaneously, often the same people. In the first half of the nineteenth century, the worth of electricity was recognized for telegraphic applications, but its commercial application was actually realized in the 1840s. Commercial telegraphic communication through wires between New York and Boston took place in 1846, followed at Dover through a submarine cable in 1851. However, it took another 40 years for the realization of commercial applications of electricity as the replacement energy for steam power or in lighting.

Chapter 2

Symmetrical Coordinate Method (Symmetrical Components)

The three-phase circuit generally has four electric conducting passes (phase a, b, c passes and an earth pass) and these four electric passes are closely coupled by mutual inductances L and mutual capacitances C. Therefore phenomena on any pass of a three-phase circuit cannot be independent of phenomena on the other passes. For this reason, the three-phase circuit is always very complicated, even for smaller system models. Furthermore, rotating machines including generators cannot be treated as adequate circuit elements to be combined with transmission line or transformers. Accordingly, the analysis of three-phase circuits by straightforward methods is practically impossible, even for only small models. Symmetrical components is the vital method to describe transmission lines, solid-state machines, rotating machines and combined total power systems as ‘precise and simple circuits’ instead of ‘connection diagrams’ by which circuit analysis can be conducted. Surge phenomena as well as power frequency phenomena of total networks or partial three-phase circuits cannot actually be solved without symmetrical components regardless of the purposes of analysis or the size of the networks.

In this chapter, the essential concept of the symmetrical coordinate method is examined first, followed by a circuit description of three-phase transmission lines and other equipment by symmetrical components.

2.1 Fundamental Concept of Symmetrical Components

It should be noted that the direct three-phase analytical circuits of power systems cannot be obtained even for a small, local part of a network, although their connection diagrams can be obtained. First, mutual inductances/mutual capacitances existing between different phases (typically of generators) cannot be adequately drawn as analytical circuits of phases a, b, c. Furthermore, the analytical solution of such circuits, including some mutual inductances or capacitances, is quite hard and actually impossible even for smaller circuits. In other words, straightforward analysis of three-phase circuit quantities is actually impossible regardless of steady-state phenomena or transient phenomena of even small circuits. The symmetrical coordinate method can give us a good way to draw the analytical circuit of a three-phase system and to solve the transient phenomena (including surge phenomena) as well as steady-state phenomena of any size.

The symmetrical coordinate method (symmetrical components) is a kind of variables transformation technique from a mathematical viewpoint. That is, three electrical quantities on a, b, c phases are always handled as one set of variables in the a–b–c domain, and these three variables are then transformed into another set of three variables named positive (1), negative (2) and zero (0) sequence quantities in the newly defined 0–1–2 domain. An arbitrary set of three variables in the a–b–c domain and the transformed set of three variables in the 0–1–2 domain are mathematically in one-to-one correspondence with each other. Therefore, the phenomena of a–b–c phase quantities in any frequency zone can be transformed into the 0–1–2 domain and can be observed, examined and solved from the standpoint in the defined 0–1–2 domain. Then the obtained behaviour or the solution in the 0–1–2 domain can be retransformed into the original a–b–c domain.

It can be safely said that the symmetrical coordinate method is an essential analytical tool for any kind of three-phase circuit phenomenon, and inevitably utilized in every kind of engineering work of power systems. Only symmetrical components can provide ways to obtain the large and precise analytical circuits of integrated power systems including generators, transmission lines, station equipment as well as loads.

Figure 2.1 shows the concept of such a transformation between the two domains in one-to-one correspondence. One set of a, b, c phase currents (or phase voltages ) at an arbitrary point in the three-phase network based on the a–b–c domain is transformed to another set of three variables named (or ) in the 0–1–2 domain, by the particularly defined transformation rule. The equations of the original a–b–c domain will be changed into new equations of the