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Electric Power Systems Research, 14 (1988) 155 - 166 155 Transient Response of Synchronous and Asynchronous Machines to Asymmetrical Faults in an Unbalanced Network V. OMAR ZAMBRANOand ELHAM B. MAKRAM Clemson University, Clemson, SC 29634-0915 (U.S.A.) RONALD G. HARLEY University of Natal, Durban 4001 (South Africa) (Received October 10, 1987) SUMMARY This paper investigates the transient stabil- ity response of interconnected synchronous and asynchronous machines in an unbalanced system when subjected to asymmetrical shunt faults. The method involves the phase frame representation of the line and the full model simulation of the synchronous generator. The imbalance of the untransposed line and shunt loads is reflected in polyphase impedance matrices. Asymmetrical shunt faults are represented by a polyphase admittance matrix. Thus, the method proposed in this investigation allows a study to be made of the transient stability response of a system containing interconnected AC machines to a series of temporary or permanent shunt faults occurring anywhere in the system; the result of several case studies is presented. A significant finding is that by neglecting imbalance the swing rotor angle peaks are higher, thereby 'failing to safety'. The running- up of the motor can cause the second peak of the rotor angle swing curve to be higher than the first peak and still the machine goes back to the prefault conditions. In such cases, i.t is no longer correct to judge the system's stability only on the traditional 'first swing cycle' of the rotor angle. INTRODUCTION The transient stability of a power system containing rotating synchronous and asyn- chronous machines is usually evaluated by considering only a symmetrical disturbance applied to a balanced system. Imbalance refers not only to unbalanced loads, but also to the case of energizing the three-phase bus with unbalanced voltages or the use of an unbalanced distribution network [1-3]. Un- balanced currents can cause problems in a power system [4]. It has been shown that significant current imbalance exists during unbalanced disturbances somewhere in the network, such as asymmetrical shunt and/or series faults [5, 6]. Stability studies of sys- tems having several interconnected AC machines, static and dynamic loads, and three-phase transmission lines cannot assume constant speed for each machine. On the other hand, to find the transient response after a disturbance has occurred somewhere in the system, a step-by-step integration process is required. In addition, to find the initial conditions for voltages and currents in the system a three-phase load flow study must be performed. Because of the number of dif- ferential equations needed on each machine, and the initial conditions required, model simulation soon becomes cumbersome and computer time consuming if the system is not small. In the early days stability calculations were carried out for generators with fixed excita- tion, assuming the field flux linkages to be constant [7], and having a second-order nonlinear differential equation for the mechanical motion (the well-known 'equal area criterion'). Damping could be allowed for by a damping coefficient multiplied by slip. Such a mathematical model did not represent the electrical transients in a ma- chine's winding. Later methods [8, 9] were developed to allow for changing field flux linkage in order to represent the effects of voltage regulators. For this grew the need to represent not only the transient field 0378-7796/88/$3.50 © Elsevier Sequoia/Printed in The Netherlands
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Electric Power Systems Research, 14 (1988) 155 - 166 155

Transient Response of Synchronous and Asynchronous Machines to Asymmetrical Faults in an Unbalanced Network

V. OMAR ZAMBRANO and ELHAM B. MAKRAM

Clemson University, Clemson, SC 29634-0915 (U.S.A.)

RONALD G. HARLEY University of Natal, Durban 4001 (South Africa)

(Received October 10, 1987)

SUMMARY

This paper investigates the transient stabil- ity response o f interconnected synchronous and asynchronous machines in an unbalanced system when subjected to asymmetrical shunt faults. The method involves the phase frame representation o f the line and the full model simulation o f the synchronous generator. The imbalance o f the untransposed line and shunt loads is reflected in polyphase impedance matrices. Asymmetr ical shunt faults are represented by a polyphase admittance matrix. Thus, the method proposed in this investigation allows a s tudy to be made o f the transient stability response o f a system containing interconnected AC machines to a series o f temporary or permanent shunt faults occurring anywhere in the system; the result o f several case studies is presented. A significant f inding is that by neglecting imbalance the swing rotor angle peaks are higher, thereby 'failing to safety'. The running- up o f the motor can cause the second peak o f the rotor angle swing curve to be higher than the first peak and still the machine goes back to the prefault conditions. In such cases, i.t is no longer correct to judge the system's stability only on the traditional 'first swing cycle' o f the rotor angle.

INTRODUCTION

The transient stability of a power system containing rotating synchronous and asyn- chronous machines is usually evaluated by considering only a symmetrical disturbance applied to a balanced system. Imbalance refers not only to unbalanced loads, but also

to the case of energizing the three-phase bus with unbalanced voltages or the use of an unbalanced distribution network [ 1 - 3 ] . Un- balanced currents can cause problems in a power system [4]. It has been shown that significant current imbalance exists during unbalanced disturbances somewhere in the network, such as asymmetrical shunt and/or series faults [5, 6]. Stability studies of sys- tems having several interconnected AC machines, static and dynamic loads, and three-phase transmission lines cannot assume constant speed for each machine. On the other hand, to find the transient response after a disturbance has occurred somewhere in the system, a step-by-step integration process is required. In addition, to find the initial conditions for voltages and currents in the system a three-phase load flow study must be performed. Because of the number of dif- ferential equations needed on each machine, and the initial conditions required, model simulation soon becomes cumbersome and computer time consuming if the system is not small.

In the early days stability calculations were carried out for generators with fixed excita- tion, assuming the field flux linkages to be constant [ 7 ] , and having a second-order nonlinear differential equation for the mechanical motion (the well-known 'equal area criterion'). Damping could be allowed for by a damping coefficient multiplied by slip. Such a mathematical model did not represent the electrical transients in a ma- chine's winding. Later methods [8, 9] were developed to allow for changing field flux linkage in order to represent the effects of voltage regulators. For this grew the need to represent not only the transient field

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current, but also the transient in the three- phase stator windings ABC. However, the nonlinear differential equations describing the ABC phase currents contain the products of variables as well as time-varying inductances which are a function of the rotor position. Then Park [10] proved that the ABC equa- tions can be transformed into the so-called two-axis or d,q equations with constant inductances; moreover, if the speed is as- sumed constant, the d,q equations are linear and analytical solutions are possible.

In most power system transient stability studies, the system loads are represented by shunt impedances. This method is accepted for hand calculations and for AC network analyzer studies. Loads such as induction motors are widely used owing to their versa- tility, dependability and economy. Individual machines of 10 MW in size are no longer a rarity. In many cases, they represent as much as 60% of the system load [11]. Induction motors, of course, do not behave as static impedances, and methods [7] have been developed for taking into account more correctly the proper variation of the im- pedance as a function of the slip. Recently, several large concentrations of induction motor loads have been installed on power systems. It has thus seemed advisable during stability studies to take the load character- istics [12] into account in a still more refined manner. In particular, it has been necessary to take into account the load inertia of the induction machines so that the mechanical transients will 'be considered properly. In some cases, these transients are of importance in investigations on generator swing response. One representation that can be made is to use the induction motor steady-state equivalent circuit. This permits the calculation of elec- trical torque at each point in a step-by-step swing curve calculation taking into account the mechanical transients, but assuming that the electrical time constants are negligible.

Because of the relatively small effective rotor time constant of many induction motors this representation is, in many cases, perfectly valid and reasonable. In some cases, however, where the induction motor rotor resistance is small or the external reactance to the motor is specially large so that the rotor effective time constant is too large to neglect, it has been desirable also to include the rotor

electrical transients. Certain studies have been made which include the stator electrical transients [13]. However, this has been shown to be unnecessary and impractical in an overall system stability study involving several machines.

A recent investigation has shown the effects of unbalanced systems as well as the transient response to asymmetrical faults of an induction motor [6]. This investigation has simulated the induction motor connected to an infinite bus through a three-phase transmis- sion line.

This paper reports on the transient response to asymmetrical faults of synchronous and asynchronous machines connected to an unbalanced system (Fig . 1). The method involves the phase frame representation of the line and the full model simulation of the synchronous generator. Several case studies for permanent and temporary shunt faults give as results that the traditional three line to ground fault is the most severe shock to a network. In addition, a second higher rotor angle swing peak does not guarantee ultimate state instability.

INFINITE BUS

©

Vab¢l

L.

I b

I c

Fig. 1. Network configuration.

1 labcg

cm

STATIC LL. ~ i "1" LOAD

L - X

THEORY

The line is simulated by a 3 × 3 impedance matrix containing off-diagonal terms to represent mutual coupling between phases. The shunt fault and shunt load are also simulated by 3 X 3 impedance matrices with suitable elements according to the boundary conditions [14]. Details of the network equations are given in Appendix A.

The synchronous and asynchronous ma- chines are simulated by their positive-, nega- tive-, and zero-sequence AC equivalent circuits and these are transformed by the

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Fortesque sequence transformation at the machine terminals into ABC phase variables as explained in Appendices B and C. The induced voltage inside the synchronous machine is assumed to be balanced under all network conditions and thus contains only a positive-sequence voltage component E" , the so-called 'voltage behind the sub-transient reactance'; the negative- and zero-sequence equivalent circuits therefore have no voltage sources. During the simulation of any distur- bance E" is updated as a function of the flux linkages at each step of the step-by-step integration process; electric transient effects in both the stator and the network are ne- glected. The speed of the asynchronous machine is also updated at each step of the step-by-step integration process to represent the variation of the slip of the machine.

The voltage equation at bus 2 is obtained as a function of the voltage at bus 1 (infinite bus) and the parameters of line, machines, shunt static loads and dynamic loads. The equations that describe the system in detail are explained in Appendix D.

RESULTS

In order to evaluate the effects of asym- metrical faults on the unbalanced system in Fig. 1, the motor was started up with the generator steadily supplying power to the unbalanced shunt load across the unbalanced line to an infinite bus. The motor parameters used are in Appendix B, and the generator parameters in Appendix C. In addition, the system is subjected to various faults with unbalanced shunt load. The system param- eters appear in Appendix A.

Two types of line configuration are con- sidered: firstly, a balanced triangular spacing of 5.33 ft between phases; secondly, with dimensions D a b = 2 ft, D b c = 6 ft and Dac = 8 ft. This yields a line matrix Zab c with the same diagonal but different off-diagonal values for the two types of lines. For each of the transient tests, the prefault initial conditions were taken to be a balanced three- phase infinite bus voltage of 1 p.u. on each phase and a unit power factor load of 0.4 p.u. per phase at the infinite bus. The unbalanced line and load were then temporarily replaced by an equivalent balanced system to find the

157

initial values of rotor angle 8 and the voltage E" in Fig. C-l, which were then used as initial conditions, together with unit balanced infinite bus voltages, in the equations of the unbalanced system to redefine the initial values of the unbalanced current and voltage in the network. The line length was chosen such that in the case of the unbalanced line its positive-sequence impedance was (0.0714 + j0.3622) p.u. This gives an initial rotor angle ~ of about 64 ° between generator rotor and the infinite bus phase A voltage. All the fault simulations in this paper were applied for 200 ms at the middle of the line (X = 0.5L), where L is the length of the line, as shown in Fig. 1. However, the fault can be simulated anywhere in the system.

Motor start-up To study the effects of unbalanced net-

works and shunt faults during the running-up on the transient response of the machines in the system shown in Fig. 1, the following cases are presented.

(1) Balanced and unbalanced network To study the effects of unbalanced feeders

with no shunt load when running up the motor, two case studies (balanced and un- balanced feeder) are shown in Fig. 2. Curves D1 and D2 show the variation of the genera- tor rotor angle 5 versus time with and without equal mutual coupling between the feeder's phases. Similar plots were obtained as a result of unbalancing the feeder and the shunt load. An imbalance on the load was introduced by concentrating the load on phase B with an admittance of (0.21 + j0.13) p.u., which gives a lagging power factor of 0.8. The shunt

60

c~ 50

4o

, t 2 ~ ' , , ; - 0 1 ~ 4 5

TIME. SEC

Fig. 2. G e n e r a t o r r o t o r angle w h i l e t h e m o t o r is running up: D1, balanced network; D2, unbalanced network.

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158

admittance for a balanced load was (0.07 + j0.043) p.u. on each phase.

Results also show that as the induction motor runs up, the power taken by the motor from the generator rises and goes through a peak at about 1.8 s. If the motor rotor were locked, then like a passive network being switched onto the generator, it would give a smaller second peak on the generator rotor angle 5. However, its running-up is a second disturbance which continues until it reaches full speed. During this time it is a nonlinear load which goes through a maximum and causes the second peak of the rotor angle 6 to be larger than the first since the motor drains a lot of power from the generator for about 1 - 2 s.

By neglecting imbalance of the feeder and/ or assuming the shunt load to be balanced, D1 has larger peaks than D2 in Fig. 2, thereby 'failing to safety'. In multimachine systems it is difficult to give physical explanations for the phenomenon beyond about the first few hundred ms since both machines are nonlinear.

Figure 3, curve TM1, shows the electric motor torque for a balanced network. Curve TM2 in the same Figure shows the effect of unbalancing the network. The results ob- tained with these two case studies show that unbalanced feeder and/or shunt load create an unbalanced motor terminal voltage which increases the positive-sequence torque by a larger amount than the increase for the negative-sequence torque. This arises because the imbalance not only creates a negative- sequence voltage, but also increases the positive-sequence voltage of the balanced feeder. Owing to the nature of the induction motor and the increase of the positive-

10

~ o . s

_~0.4

~ 0 . 2 o.o TN1

- - 1 - - , i , i , i , i , - - T ~ - T

0 .0 0 .5 1.0 1.5 2 .0 2 .5 3 0

TIME, SEC

Fig. 3. Induction motor torque during the running- up: TM1, balanced network; TM2, unbalanced net- work.

sequence voltage, the motor produces much more positive-sequence torque than the increase due to the negative-sequence voltage.

The motor torque dips for the first 250 ms because the generator internal voltage is dropping (E" in Fig. C-l(a)) because of flux decays. Finally, both curves (TM1 and TM2) settle at the steady equilibrium equal to the load torque of almost 0.7 p.u.

(2) S h u n t faul ts on an unbalanced n e t w o r k The plots shown in Fig. 4 (curves TM3 and

TM4) were obtained for two case studied during the running-up of the motor. After running up the motor for 500 ms, a shunt fault was applied for 200 ms and then the fault was cleared out. TM3 shows the electric motor torque when a single line to ground fault was applied at the middle of the line for 200 ms. TM4 shows the electric motor torque for a three line to ground fault under the same conditions as TM3. These are multiple faults or disturbances since the fault occurs while the motor starts up. Obvi- ously, the 3LGF along the line almost cuts the motor off completely (TM4) from the synchronous generator, hence the motor torque is virtually zero. The SLGF (TM3) shows, however, that substantial power is still reaching the induction motor while the fault is on.

Figure 5 shows the rotor angle when the motor starts up and a shunt fault is applied. Curve D3 is the result for an SLGF and D4 for a 3LGF. This Figure shows that in the case of an SLGF curve (D3) the first peak is lower than the second peak; this simply illustrates that whether the second peak is larger than the first depends on generator and motor conditions and the type of disturbance.

1.0

~ 0 . 8

~ 0 .6

~ 0 . 4

c~

~ 0 . 2

00

0 .0 0,,5 1.0 1.5 2 0 3 .0

TIME. S;F£.

Fig. 4. Induction motor torque during the running- up and temporary shunt faults: TM3, SLGF (asym- metrical fault); TM4, 3LGF (symmetrical fault).

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110 .

90.

~ 70.

~ 50-

30-

',.,j" " , o , , / ' " ,

i i i i i l i i i

0 1 2 3 4 5 6 7 8

TIME. SEC

Fig. 5. Generator rotor angle during the motor running-up and temporary shunt faults: D3, SLGF (asymmetrical fault); D4, 3LGF (symmetrical fault).

Both curves (D3 and D4) eventually con- verged, but they may not for a different type of generator or motor and/or system prefault conditions.

Motor in equilibrium state; temporary shunt faults

To study the effects of unbalanced net- works and shunt faults on the transient response of the machines in Fig. 1, different types of faults were simulated for 200 ms at the middle of the feeder.

(1) Base case: balanced fault (3LGF), unbalanced network The well-known symmetrical three-phase

line to ground fault was applied to the system in Fig. 1 at the middle of the feeder. The generator had an unbalanced static shunt load of admit tance (0.21 + j 0 . 1 3 ) p . u . con- centrated on phase B. The resultant swing curve of the generator ro tor angle appears as curve D7 in Fig. 6 and shows a first peak of about 77 ° before settling back to the prefault

80

o 7O

60

-.c 50

o 4O

30

f \ / / ~"~--D7

/ \ / - - \ ITx~ \ / \ / -

/.~' :", \ I - ~ ] ~ : ~ - ~ . \ ._-.,~- . . . . . . -.,%. • . . • ...--.._~. _, , ~::-~.,.__~

\ /

' 0 .'5 ' ~ v , , 0 . 0 1.0 1 5 2.0 2.5 3.0

TIME, SEC

Fig. 6. Generator rotor angle in an unbalanced faulted network (temporary fault): D5, SLGF on phase A; D6, 2LGF on phases A and B; D7, 3LGF on phases A, B and C; D8, 2LF between phases A and B.

159

1.0

~o.e

_~0.4

~ 0 . 2

0.0

TH6 /-P~.. ~ \ TH7

I I

0'0 05 ,:0 ,'5 TIME, SEC

Fig. 7. Induction motor torque in an unbalanced faulted network (temporary faults): TM5, SLGF on phase A; TM6, 2LGF on phases A and B; TM7, 3LGF on phases A, B and C; TM8, 2LF between phases A and B.

steady condition. The corresponding electric motor torque is shown in Fig. 7 (TM7).

(2) Asymmetrical two line to ground fault (2LGF) Curve D6 in Fig. 6 is the generator ro tor

angle swing curve after a 2LGF between phases A and B at X = 0.5L on an unbalanced line with an unbalanced static shunt load. Its values are much less than for D7, as expected. Curve TM6 in Fig. 7 is the corre- sponding induct ion motor response to this type of fault.

(3) Asymmetrical single line to ground fault (SLGF) The SLGF fault occurs by far the most

of ten of all fault types and ref. 14 states that this makes up 70% of all faults on lines and feeders with the 3LGF, 2LGF and 2LF contributing 5%, 10% and 15% respectively. Curve D5 in Fig. 6 is the generator ro tor angle swing curve after an SLGF on phase A of an unbalanced line with the static shunt load of (0.21 + j 0 . 1 3 ) p . u . concentrated on phase B. Curve TM5 in Fig. 7 is the corre- sponding induction motor electric torque after an SLGF. Curves D 5 - D 8 and TM5- TM8 clearly show the validity of the tradi- t ionally accepted practice tha t the three-phase line to ground fault presents the most severe test of the system's stability.

(4) Asymmetrical line to line fault (2LF) A 2LF was applied across phases A and B

of an unbalanced line at the same fault location and with the static shunt load of (0.21 + j0.13) p.u. concentrated on phase B,

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160

and the resultant swing curve is D8 in Fig. 6. Curve TM8 is the corresponding induction motor response to a 2LF. As was expected, this type of fault is less severe than tha t for a 2LGF at the same location.

Motor in equilibrium state; permanent shunt faults

These case studies are a direct extension of those above by simulating the case where the shunt fault is never cleared out by any source of error in the protection system. The generator rotor angle swing curves (D9 - D12) in Fig. 8 clearly show all four cases to be unstable. Curve D9 corresponds to an SLGF, curve D10 to a 2LGF, curve D l l to a 3LGF and curve D12 to a 2LF. They confirm the results obtained in Fig. 6, namely that the order of severity rises in the sequence SLGF, 2LF, 2LGF and 3LGF.

Figure 9 shows the electric motor torque for the types of faults shown in Fig. 8. These last two Figures show that for a permanent

150

120

90

60

Vl l_~l/J i ~DIO /

I I~D12

,ii

- - i i i i i ~ -

0 2 4 5 8 1 TIME. SEC

Fig. S. Generator rotor angle in an unbalanced faulted network (permanent faults): D9, SLGF on phase A; D10, 2LGF on phases A and B; D l l , 3LGF on phases A, B and C; D12, 2LF between phases A and B.

0.8

i0 .5 ,.A

0.4

0.2.

0.0.

,~--TH12

~ I 1 " %

o' 2' T I M E , SEC

8 ~ 10

Fig. 9. Induction motor torque in an unbalanced faulted network (permanent faults): TMg, SLGF on p h a s e A; TM10, 2LGF on phases A and B; TMl l , 3LGF on phases A, B and C; TM12, 2LF b e t w e e n phases A and B.

3LGF, 2LGF, and 2LF both machines go unstable. The generator starts pole-slipping and the motor runs down to standstill. The SLGF illustrates that the machines could almost remain stable and probably would if their prefault loading had been lower, e.g. infinite bus power 0.2 instead of 0.4 and even the mechanical torque connected to the motor shaft 0.2 instead of 0.6.

CONCLUSION

This paper has investigated the transient stability of synchronous and asynchronous machines connected by an unbalanced net- work to an infinite bus supply and subjected to a variety of asymmetrical line to ground and line to line shunt faults. Results of several case studies have been used to evaluate the effects of untransposed lines and unbalanced load. Line imbalance has been shown to have a significant effect on the transient response of synchronous and asynchronous machines, as well as the type of shunt fault. In addition, it was confirmed that the traditional three line to ground fault is the most severe shock to a network. The running up of the induc- tion motor can cause a second higher rotor angle swing peak, and still the synchronous generator comes back to its prefault steady- state operating conditions. Therefore, it is not correct to judge a system's stability by the well-known 'first swing cycle' for asym- metrical types of faults since it does not guarantee instability in such conditions.

REFERENCES

1 W. H. Kertsing and S. A. Seeker, A program to study the effects of mutual coupling and un- balanced loading on a distribution system, IEEE PES Winter Meeting, New York, 1975, Paper No. CC 75 047-6.

2 M. A. Wortman, D. L. Allen and L. L. Grigsby, Techniques for the steady state representation of unbalanced power systems, Part I: A systematic building block approach to network modeling, IEEE Trans., PAS-104 (1985) 2807 - 2814.

3 R. W. Fisher, Voltage unbalance on a three-phase distribution system, Southeastern Electric Ex- change 1977 Annu. Conf., Engineering and Operation Division, New Orleans, April, 1977.

4 R. G. Harley and M. A. Tshabalala, Induction motor behaviour in t h e p r e s e n c e of unbalanced voltages, Trans. S. Af t . Inst. Electr. Eng., 76 (1985) 51 - 55.

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5 E. B. Makram, G. G. Koerber and K. C. Kruempel, An accurate computer method for obtaining boundary conditions in faulted power systems, IEEE Trans., PAS-101 (1982) 3252 - 3260.

6 R. G. Harley, E. B. Makram and E. G. Duran, The effects of unbalanced networks and unbalanced faults on induction motor transient stability, IEEE PES Summer Meeting, San Francisco, 198 7, Paper No. 87 SM 607-5.

7 S. B. Crary, Power System Stability, Vol. 2, Chapman and Hall, London, 1947.

8 C. Concordia, Steady state stability of synchro- nous machine as affected by voltage regulator characteristics, Trans. AIEE, 63 (1944) 215 - 220.

9 J. G. Miles, Analysis of overall stability of multi- machine power systems, Proc. Inst. Electr. Eng., PartA, 109 (1962) 203 - 211.

10 R. H. Park, Two-reaction theory of synchronous machines, Trans. AIEE, 48 (1929) 716.

i l D.J . Limebeer and R. G. Harley, Subsynchronous resonance of single-cage induction motors, Proc. Inst. Electr. Eng., Part B, 128 (1981) 33 - 42.

12 D. F. Shankle, Transient stability studies, I: Synchronous and induction machines, Trans. AIEE, Part III-B, 73 (1954) 1563 - 1577.

13 F. G. Maginnies and N. R. Schultz, Transient performance of induction motors, Trans. AIEE, 63 {1944) 641 - 646.

14 P. Anderson, Analysis o f Faulted Power Systems, Iowa State University Press, Ames, IA, 1981.

15 B. Adkins and R. G. Harley, The General Theory o f Alternating Current Machines, Chapman and Hall, London, 1975.

APPENDIX A

F e e d e r i m p e d a n c e e q u a t i o n

The impedance matrix of any three-phase line can be obtained in general for any confi- guration, as shown in Fig. A-1 as an example:

Z'aa = r a + r d + jcok ln(De/Ds)

t = Z~b = Z'o¢ = Zgg

Z'ab = r d + jcok in(De~Dab) = Z~a

Z~¢ = r d + jcok ln(De/Dbc) = Z'cb

(A-l)

(A-2)

(A-3)

Oab .I. Obc ~l

-- "F -I O . O

a b ~D c bg

g Fig. A-1. Unbalanced feeder configuration.

161

Z'¢~ = rd + jcok l n ( D J D c a ) = Z '~ (A-4) W P Zag = r d + jcok ln(De/Dag ) = Zg a (A-5) ! ¢

Zbg = r d + jcok ln(De/Dbg ) = Zg b (A-6) t P

Zcg = r d + jcok In(De~Dog ) = Zgc (A-7)

.where

ra = rb = rc (A-8)

Equations (A-l) - (A-8) are in &2/unit length and Dsa =D,b = D,c = Ds is the con- ductor self geometric mean distance in ft. D e is a function of both earth resistivity p (taken to be 100 [2 m) and the frequency f, and is defined by the relation [14]

De = 2 1 6 0 ( p / f ) i n = 2790 ft

rd = 1.588 × 10-3f ~2/mile

for f = 60 Hz

(A-9)

(A-10)

The constant cok is 0.121 34 when the unit of length is in miles and co = 2~f.

The result of the above eqns. (A-l) - (A-10) leads to a primitive 4 × 4 impedance matrix which can be reduced by partitioning to compute the 'Z~bc matrix for the feeder as

Zabc = Zba Zbb Zbc (A-11)

Zca Zeb Zcc

F e e d e r p a r a m e t e r s

Length of feeder Type of conductor

Size of conductor Number of strands Geometric mean radius Resistance ra Diameter of individual

strand Outside diameter

0.6 p.u. aluminum steel re-

inforced 795 000 54 0.0368 ft 0.1190 [2/mole 0.1214 in.

1.093 in.

S h u n t fau l t s

The shunt fault admittance can be simu- lated at any location along the feeder by choosing suitable values for the different elements of the matrix Yfabc to define the type of fault [6].

Consider the case where the fault in the network in Fig. A-2 is a single line to ground fault on phase A. The resistor R~a represents the arc resistance due to a fault to ground.

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162

/ I a

INFINITE TERMINALS

sos ~ i b ]Rfa

Vabc I 2'7 7 Vabc 2

L. ~l. h3 r X ~1 ~ n - X ~I

Fig. A-2. Single line to ground fault (SLGF) represen- tation.

In general the fault admittance matrix is given by

[i :1 Yfabc = 1/Rfb

0 1/Rfc

0 :] = 0 Yfb (A-12)

0 0 Yfc

Suitably high or low values for Rfa, Rfb and Rfc represent faults ranging from a single line to ground (SLGF), two lines to ground (2LGF), and three lines to ground (3LGF). For example, in a single line to ground fault on phase A, the elements Yfb and Yfc in eqn. (A-12) become zero.

APPENDIX B

I n d u c t i o n m o t o r m o d e l The induction motor is simulated by its

positive- and negative-sequence AC equivalent circuits as shown in Fig. B-1.

lp RmpI2p XZp R2p s

(a)

In RmnI2n X2n

R2n 2-S Ix °

(b)

Fig. B-1. Induction motor AC equivalent circuit: (a) positive-sequence circuit; (b) negative-sequence cir- cuit.

The sequence component variables of voltage, current and impedance are from time to time transformed by the Fortesque se- quence transformation into ABC phase variables when connecting the machine to the network. The sequence components of the voltage at bus 2 are given as

i oI li i lira21 LVt j a 2 a LV 2j

or

VtOpn -- A-lVabc2

(B-l)

where a is the complex operator a = 1/120 ° = --0.5 + j0.866. Values of Vabc2 are given by eqn. (D-12). The current I2p in Fig. B-l(a) is given by

I2p = ( V t p / Z p ) [ Z m p / ( Z m p -}-Z2p)] (8-2)

where Zp is the positive-sequence input i m p e d a n c e ; Z m p and Z2D are the positive- sequence magnetizing and rotor impedances. The positive-sequence torque is

Tep = 112p 12R2p/S (B-3)

In a similar way, the following expressions are found for the negative-sequence circuit:

I2n = ( V t n / Z n ) [ Z m n / ( Z m n + Z 2 n ) ] ( B - 4 )

Ten = J I ~ J 2 R ~ / ( 2 - - s) (8-5)

The equation for motion of the rotor is given by

pw m = (Tep -- Ten -- TL)/(2H/OJz) (B-6)

where H is the inertia constant in s, co z the synchronous speed in tad s -1, T L the load torque in p.u., and corn the shaft speed in rad s -1.

For the purpose of this paper the motor load is assumed to be a pump or a fan with a typical characteristic of

TL = A + BOOm 2 (B-7)

where A sets the initial zero-speed load torque and B sets the final equilibrium speed where motor torque is equal to load torque.

Equations (B-l)-(B-7) represent a non- linear system and they can only be solved by a numerical step-by-step integration process.

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I n d u c t i o n m o t o r p a r a m e t e r s The motor 's mechanical load parameters

are A = 0.1 and B = 0.6 p.u., which gives an initial steady-state motor torque of about 0.65 p.u. The motor parameters used were (all in p.u.)

Rlp = R in ---- 0.0210 Xlp = Xln = 0.0490

Rmp -- Rmn = 0.0 X m p = Xmn = 3.0380

R2D = 0.057 X2p = 0.1320

R2n = 0.057 X2. = 0.1320

f = 60 Hz H = 0.3 s

APPENDIX C

S y n c h r o n o u s g e n e r a t o r m o d e l The sequence component voltages are

obtained from the node 2 phase voltages Vabc2 as in eqn. (B-l):

Vtopn = A-1Vabc2 (C-I)

The positive-sequence generator current Igp in Fig. C-l(a) is given by

Igp = (E" - - V tp ) / (Ra + jX~) (C-2)

the positive-sequence power developed by the generator is given by

Pup = Re(E"Igp*) (C-3)

I qp

Vtp

! (a)

gin

! (h)

Iqo

Vto

1 (c)

,,x/~,c.....rv-~ R a X "d

E "

l fNfNt'N

Ra Xn

Ra Xo

Fig. C-1. Synchronous machine AC equivalent cir- cuits: (a) positive-sequence circuit; (b) negative- sequence circuit; (c) zero-sequence circuit.

163

and the torque associated with the developed power is

Tgp = Pgp(toz/to,) (C-4)

where toz is the synchronous speed in rad s -I and ton the instantaneous speed in tad s -1.

Similarly, the following expressions apply to the negative- and zero-sequence circuits which are shown in Fig. C-1:

Ign = - - V t n / ( R a + R ~ +jXn)

/gO = --Yto/(Ro + jX0)

Tgn = [Ignl:/R2n Tgo--IIgol~/Ro

(c-5)

(c-6)

(c-7)

(c-8) The equation for the motion of the rotor is given by

Ptom = (Tpm -- Tgp -- Tgn -- T g o ) / ( 2 H / t o z )

(C-9) where H is the inertia constant in s, and Tpm the prime mover driving torque.

For the purpose of this investigation, the prime mover output power Pm is assumed to remain constant at its pre<listurbance value during disturbances; hence,

Tpm = P p m ( t o z / t o m ) (C-10)

Finally, the load angle 5 is found from integrating

p6 = tom -- coz (C-11)

Equations (C-1) - (C-11) can be solved by a numerical step-by-step integration process.

It is well known that a synchronous ma- chine with damper circuits can be represented in the two-axis theory by three electrical circuits on the d-axis (D, F, KD) and two on t h e q-axis (Q, KQ). Based upon the sign conventions and the per unit system of ref. 15, the voltage equations for the five circuits can be expressed as follows in terms of d and q variables when the d and q axes are attached to the minor and major reluctance axes of the rotor respectively:

ed = Raid + P~d + tokI /q (C-12)

eq = Rai q + pxlJq - - ( . O ~ I / d (C-13)

e~ = R~i~ + pk~ (C-14)

ekd = Rkdikd + P~kd (C-15)

ekq = Rkqikq + pxISkq (C-16)

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164

The five flux linkages are given by

-xI, d

xiYkd

xpf

Xpq

XIIkq

"Ldd Lmd

Lind Lkkd

= Lind Lind

0

"id l ikd

× i~

iq

.ikq

For the purpose P~d and p~q or

Lind

Lind

Lffd I I

ImLqq I I J Lmq I

0

L m q

Lkkq

( c - 1 7 )

of this investigation the 'stator electric transient'

terms in eqns. (C-12) and (C-13) are ne- glected, which then become

e d = cOOjq + Rai d (C-18)

eq = --09~f d + Rai q (C-19)

Now, returning to eqn. (C-17), ikd, ikq and i~ can be eliminated to find

tt. . -b ¢OOXPd = Xd~d + ¢o0(Xd -- Xa) X~d Xkd /

(C-20) kXJkq f t . ?t

(,#O0kI/q -- Xq$q 4- (,~oo(X q - - X a ) - - (C-21) X k ~

Replacing these values for XI/d and ~q in

(C-22)

eqns. (C-18) and (C-19) yields

e d = Rai d + (1 -- s)co0~ q I t • t!

= R a i d + Xq/q 4" e d

e q = R a i q - - (1 -- S)O)OXI'td I t . r l

= R a i q - - X d l d "b e q

where

s = (¢o - ¢ o 0 ) / ¢ o 0

" = " c o ( x ; - X a ) - - e d - - S Z q l q +

(C-23)

~I/kq (C-24)

X k q

eq = sX~id - - ¢o(X~ - - Xa) + Xk d

( C - 2 5 )

The two voltages e~ and e~ are known [15] as the axes components of the voltage behind the subtransient reactance.

The following set of equations is obtained for the time rate of change of rotor flux linkages by eliminating ikd, ikq and if f rom eqns. (C-14) -(C-16), and defining new

f fl ?! parameters Tdo, Tdo and Tuo as described in ref. 15:

P~kd /(X~ Xa) id ~ f

= - - - + ( x : ~ - X a ) [ (z) 0 X f d

1 (C-26) - q ' k d T~'o

i__~q _ ~kq ,, (C-27) pxI/kq = Xmq co O Tqo

+ f [ *"1 • X --Xa-x + pxp~ = uf [ ~d -- X-"-'~ \ md (,z)--O Xkd ]

( . )]1 - - ~ 1 + Xd - -Xa Xmd (C-28)

XkdXfd ~ o

In order to represent the machine as an element of an AC network and thereby avoid a d,q analysis of the network, it is customary

H -~ Z H to assume at this stage that Xd =Xq {zero subtransient saliency). This allows the machine to be considered as a phasor voltage E" behind a subtransient reactance X ' , and the equations are redefined as follows:

ed = Raid + X" iq + e~ (C-29) It

eq = Rai q - - X" id + eq (C-30)

XI/kq e~ = - - s X ' i q + ¢ o ( X " - - Xa) (C-31)

X k q

eq = s Z " i d - - c o ( X " - - Xa) Xfd Xkd ]

(C-32)

P~kd [(X~ Xa)

I

id = - - - + ( x : ~ - - X a )

t COo X~d

I (C-33) - ~I'kd T~o

i--2% -- ~kq ,, (C-34) P~kq = Xmq coo Tqo

[ X " - - X a ( x id 4" X m d ~I/kd 1 POdt = Uf + [--~-dd __ Xa ~dco ° Xk d ]

- - ~ l + - - Xmd ~ (C-35) XkdX~d Tdo

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Equations (C-30) and (C-31) can now be rewritten as a single phasor equation of the form (in which E, E" and I are phasors)

E = RaI + j X " I + E" (C-36)

where

t l • tr • I t

E " = E d + JEq = (e i - - j eq) /V/2 (C-37)

I = I d + jlq = (id - - j iq) /X/~ (C-38)

For the unbalanced conditions described in this paper, the variables in eqns. (C-36)- (C-38) are related to those in eqns. (C-1) and (C-2) as follows:

E = Vtp and I = Igp

provided Vtp and Igp have coordinates within a d,q frame of reference attached to the rotor of the machine.

At each step of the integration process, the positive-sequence current is therefore used to find id and iq which are substituted into eqns. (C-33)- (C-35) which in turn are inte- grated to yield the flux rotor linkages; these, together with i d and iq, a r e then used in eqns.

Ft (C-31) and (C-32) to update e l and eq and therefore E" in eqn. (C-37). This model of the synchronous machine therefore uses a 'voltage behind a subtransient reactance' which gets updated at every step of the solution.

Generator parameters Based on the per unit system of ref. 15, the

generator parameters are as follows:

?l R~ = 0.0 Xq = 0.294

Rf = 0.000 984 Xkq = 0.0545

X d = 2.0 Rkq = 0.013 35

Z m d -- 1.780 T~o = 5.311

X~ = 0.392 T~' o = 0.053 ?P

Xi = 0.294 Tqo = 0.338

Xfd = 0.22 Xo = 0.1475

X k d ----- 0.13 R 0 = 0.0

Rkd = 0.015 R ~ = 0.06

Xq -- 2.0 H = 5.0 s

Xmq = 1.780 f = 60.0 Hz

165

APPENDIX D

Net equations The abc voltage vector at node 2 in Fig. 1

is given by

Vabc2 -- Vfabc - - Zabc2/abc2 (D-I)

where Zabc2 represents the feeder impedance matrix for the line length L - -X , I~bc2 repre- sents the current in it and Vf~bc the voltages at the fault location. I~b¢2 in turn is related to the synchronous generator, induction motor and shunt load currents by

Iabc2 =/abcg + IabcL +/abcm (3-2)

Similarly,

Vfabc = Vabc I - - Zabc I/abc I

where

(D-3)

Zabcl and Iabcl apply to line length X, and

labcl = labc2 + lfabc (D-4)

where

Ifabc = Yfabc Vfabc (D-5)

Now labcg, Iabcm and I~b~L can in turn be defined in terms of the generator, the induc- tion motor and the shunt load admittances and their terminal voltages as follows:

labcm = YabcmVabc2 (D-6)

labcL = YabcLVabc 2 (D-7)

Iabcg = Yabcg[ Vabc2 - - Vabcg] (D-S)

Combining eqns. (D-1)-(D-8) yields the voltage Vfabc at the fault location as

[U + Mlf ] Vfabc

= Vabcl + MlgVabcg -- [Mlg + Mira + MIL]

X ~/ab c 2 (D-9)

where

Mlf = ZabclYfabc

Mlg = ZabclYabcg

Mlm = Zabc IYabcm

M1L = ZabclYabcL

U = identi ty matrix

The final equation of Vfabc is

Vfabc = JV~bcl + MlgV~bcg - - KV~bc2 (3 -10 )

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166

where

J = [U + ZabciYfabc] -I

K = Zabcl[Yabcg + YabcL + Yabcm]

By combining eqns. (D-9) and (D-10),

V~bc2 = [U + JK + Q]-l[JV~bcl

+ (JM lg + M2g)Vabcg]

or

Vabc2 = NJVabcl + NRVabcg

where

(D-11)

(D-12)

M2g = Zabc2Yabcg

M2L = Zabc2YabcL

N= [U+JK+Q]-I

Q = Zabc2[Yabcg + YabcL + Yabcm]

R = JMlg + M2g

Imbalance in the shunt load can be repre- sented by substituting suitable values of YawL, YbbL, and YccL in eqn. (D-7). For a balanced load these three admittances are equal and for no shunt load whatsoever they are all three equal to zero.