Strategy for the control of multiple series compensators in the enhancement of interconnected power system sta bi I ity

K.L.Lo and Y.J.Lin

Abstract: A multiple controlled series compensator (CSC) control strategy to enhance power system stability is presented. The proposed control strategy consists of three different stages, each of whch has a different purpose and approach. The effectiveness of the proposed control strategy is tested on a multimachne power system associated with two CSCs. Computer simulation results indcate that CSC regulation using the proposed control strategy can reinforce power system stability.

1 Introduction

The progress of power electronic techniques has enabled electrical engineers to implement high rating power switch- ing in a rapid manner. This advancement has led to a tech- nology whch utilises power electronics based devices and alters the way that power is delivered. FACTS (flexible AC transmission systems) technology, whch utilises modern fast acting thyristors or other semiconductor means of con- trol rather than traditional mechanical switches, can be used to guide power flow in networks and to increase their loading capabilities to thermal limits [l]. In their applica- tions, electrical engineers will inevitably encounter stability criteria before approaching the line thermal limit. This means that FACTS can affect the stability limit. Many researchers have consequently explored this benefit by vir- tue of many FACTS devices, including the static var com- pensator (SVC) [24], controlled series compensators (CSC) [5], thyristor controlled phase shifter (TCPS) [6], thyristor controller breaking resistor (TCBR) [13] and unified power flow controller (UPFC) [7].

These FACTS devices are rapid switchmg devices like many other fast stability enhancement instruments, such as governor and excitation control systems. With fast speed modulation via FACTS devices, it is essential to exercise great caution, otherwise some unwanted adverse effects may render more harm than good if a proper co-ordination does not exist. Engineers and researchers have learned the important factor of co-ordination and much attention has been paid to this issue. In [l 11, a control scheme using eigen-analysis for multiple CSCs was developed to reinforce power system stability, and the same objective was also achieved by using the quasioptimal control method for ver- satile FACTS devices 19, 10, 121.

It is of interest to note that, however, most researchers start their analyses with the linearisation of the study sys-

0 IEE, 1999 IEE Proceeu'ings online no. 1999o060 DOL lO.I049/ipgtd: 19990060 Paper fmt received 7th May and in revised form 8th October 1998 The authors are with the Department of Electronic and Electrical Engmering, University of Strathclyde, UK

tem around an operating point and then, based on ths lin- earised system model, they used linear system theory to carry out controller designs. Ths is acceptable when only small-scale perturbations are assumed in power systems, since system dynamics will oscillate in the vicinity of the operating point and one can accordingly treat such oscilla- tions as linear dynamics. Hence linear controllers should be able to cope with such situations. In the event that power systems experience severe disturbances, the system dynam- ics will swing far away from the operating point and behave much more nonlinearly. As a result, linear control- lers become vulnerable. Several examples will be examined later in this paper to support this statement. A more sophisticated control strategy is needed.

This paper aims to develop an advanced control strategy for multiple CSCs and will be illustrated by computer sim- ulation results.

2 System models

The dynamics equations of each generator, together with the automatic voltage regulator (AVR), are expressed as follows

6 = w - w R (1)

where S, w, eq', eFD represent the rotor angle in radian, speed in radian per second, transient emf on q-axis in per unit and excitation voltage in per unit, respectively; pe, pn,, vt and vREF are the per unit output electrical power, input mechanical power, terminal voltage and reference voltage, respectively. w,, H, T,, TA, KA denote the synchronous speed in radian per second, inertia constant, direct axis field circuit time constant, excitation system time constant and excitation system gain, respectively. xd, xrd and id are the reactance, transient reactance and current on the d-axis, all

149 IEE Proc.-Gener. Tronsm. Distrib., Vol. 146, No. 2, Murch 1999

in per unit. The voltage and current across the d- and q- axis are given as

v d = i,x, (5)

( 6 ) vq = eh - i d X L

.---. .__. stagel b maximum

regulation

where vd, vq, iq, represent the voltage on d-axis, voltage and current on g-axis, respectively. Furthermore, the output electrical power pe and the terminal voltage vt can be expressed by

Pe = i d v d + iqv9 (7) ::I: stage2 :I: stage3

* FLC linear co-ordination controller

vt = .\/vi + vp

W R = 2 r f (9) f denotes power system frequency.

ferential equation: The dynamics of a CSC are modelled by a first-order dif-

where xc, Tc and U represent the compensating impedance, CSC time constant and input signal of CSC, respectively.

The network equations can be expressed as

where In and V,, represent the nodal current and voltage vector, and Yn is the admittance matrix.

3 Proposed control strategy

Two fundamental concepts are highlighted here before introducing the proposed control strategy. The first relates to the cause of power systems instability which can be understood from eqn. 2. When a disturbance occurs in the network and some of the transmission lines are temporarily out of service, the generated electrical energy could not be entirely delivered. The imbalance between the mechanical input and electrical output energy could lead to power sys- tems instability.

The second concept relates to the role of the CSC in power system operations. CSC regulation alters the net- work parameters - line impedance. This means that differ- ent network parameters are obtained from different CSC regulations. With these two concepts, the proposed control strategy can then be established. For example, consider that a severe fault occurs in the power systems and that some of the transmission lines are out of service; one may apply a specific CSC regulation so that the new network parame- ters would allow more electric power to be transferred from the generator. In other words, by increasing the transmitted electric power from this accelerating generator, its accelera- tion is reduced. Likewise, if the generator is decelerating, one may apply another CSC regulation and obtain another set of network parameters whch are able to reduce the electric power transmitted and thereby reduce the decelera- tion.

The proposed control strategy consists of three stages as illustrated in Fig. 1. The first stage aims to pull back the first swing of any generator in the network. The basic crite- rion for the first swing stability is that the acceleration energy should be counterbalanced. Hence in the first stage, as soon as a fault happens, the new network parameters will be established so that the electric power transmitted from each generator in the grid would be as much as possi- ble. This can be illustrated in the test system shown in

150

Fig. 2. In the first stage, both CSCs are adjusted to their maximum negative values to draw more electrical power from generators 2 and 3 when a fault occurs in the grid. This stage wdl be held until one of the generators starts to swing back and then the control strategy wdl enter the sec- ond stage.

91 31 G3 I

load csc2 e load load *-

GI (large system)

Fig.2 Multimachine power system

When one of the generators in the power system starts to swing back, it indicates that ths generator is decelerating. The network parameters should be modified to reduce the electric power transfer of this generator but keep the other accelerating generators increasing their electric power trans- fer, some generators need to transmit more electric power but others do not. Such a dilemma needs to be overcome with proper CSC co-ordination and it depends to a great extent on CSC locations as well as each generator’s dynam- ics. Different CSC locations and generator dynamics require different co-ordination.

In this paper, the co-ordination will be performed through the fuzzy logic controller (FLC). A detailed mathe- matical description of a general FLC can be found in [5]. The input signals of this FLC are power system informa- tion, such as the generator’s angle deviation and speed that can be used to judge whether the generators concerned are accelerating or decelerating, and the output signals are used to regulate CSCs.

The co-ordination principles will reflect on fuzzy rules. For instance, consider the test system shown in Fig. 2; if generator 2 needs to increase the electrical power transfer but generator 3 needs to reduce it, then the CSCl adjust- ment should be negative so as to reduce the line 7-8 imped- ance to allow more electrical power delivered from generator 2, while CSC2 adjustment should be positive so as to increase the line 6-9 impedance to prevent more elec- trical power being delivered from generator 3.

However, this stage might encounter a problem in the speedy transmission of generator speed and angle signals to the location of the FLC and then assigning a control signal to each CSC. To overcome this, a distributed computing technique developed in [14] could be used to achieve such a fast information exchange. In addition, for large systems, the co-ordination rules can be established from a network equivalent system. It should also be noted that the mem- bershp functions and fuzzy rules used are derived through extensive computer simulation study and are selected heu- ristically.

When all generator oscillations are successively decaying to within a certain lirmt, the control strategy will enter the third stage. At this moment, as oscillations become smaller,

IEE Proc.-Gener. Transm. Distrib.. Vol. 146. No. 2, March 1999

it is possible to treat them as if they were triggered by small disturbances, and linear system control theory is applicable. Hence, in the thrd stage the control strategy will follow lin- ear controllers. Advantages of using linear controllers include the avoidance of high frequency switchng of the CSC and ensuring the damping performance of the remain- ing odlations. In this stage, co-ordination is constructed by means of the design of offline hea r controllers. The design of such linear controllers can be performed by the genetic algorithms developed by the authors in [SI.

4 involving multiple CSCs

In ths Section, the proposed control strategy will be tested on the power system shown in Fig. 2. Generator 1 is set as a reference bus and represents a large system; hence it can be treated as an infinite bus. Generators 2 and 3 are equipped with an automatic voltage regulator (AVR). A CSC was installed in line 9-6 and 7-8, respectively. The system data are given in the Appendix (Section 7.1, Tables 3-7). The dynamics equations used in this study are eqns. 1-4 for generator modelling and eqn. 7 for CSC modehg.

A linear control scheme using pole placement is first introduced. ms linear control scheme is also used as the third stage in the proposed control strategy. To start with, we linearise the test system around the operation point, which results in the following form:

Case study - multimachine power systems

X = A X + B U (12)

Y = C X (13) where A’, Y, U denote the system state, output and input variable vector, respectively and A, B, C are constant matrices with appropriate dimensions. The state variable vector X is defined as A[& w, et: eFD2 S, q et: eFD3 xcl xdT. They represent state variables of generators 2 and 3, and CSC 1 and 2. U is defined as A[ul u-JT, which is the input vector for CSC 1 and 2. Computations of matrix A and B are derived in the Appendix. In this case, they are

A =

B =

0 377

-0.19 0

-0.36 0

-2 0

0 0

0.11 0

0.14 0

1.80 0

0 0

0 0

0 0

0 0

0 0

0 0

2 0 0 2 - -

0 0 0 0

--0.25 0 0.04 0

--0.56 0.17 0.10 0

-24.26 -2 -1.43 0

0 0 0 377

0.10 0 -0.34 0

0.18 0 -0.43 0

-7.57 0 -6.37 0

0 0 0 0

0 0 0 0

0 0 0 0

0.03 0 0.03 0.07

0.11 0 0.04 0.12

-7.25 0 2.21 3.03

0 0 0 0

-0.4 0 0.10 -0.12

-0.65 0.17 0.11 -0.17

-21.82 -2 3.89 0.54

0 0 - 2 0

0 0 0 - 2

IEE Proc.-Gener. Trimsm. Distrib., Vol. 146, No. 2, March 1999

The eigenvalues of matrix A of these open loop systems are listed in the first column of Table 1. AU of them have nega- tive real parts but two sets of poor damping ratio eigenval- ues, 4 .29 2 j11.74 and -0.22 2 j7.95, can be treated as dominant poles because they are near the imaginary axis.

Table 1: Eigenvalues of normal load demand case ~

Open loop Closed loop

-0.29 2 jll.74 4 . 2 2 2 j7.95 -0.54 j7.47

-1.01 2j1.85 -1.02 2 j1.85

-1 .OO 2 jll .43

-1.08 2 0.96

-2, -2 -4.00, -1 1.29 -1.08 5 0.95

The input signal for each CSC is the active power deviation AP on the line where the CSC is installed. AP is the differ- ence between the active power measured under disturbance and measured under steady state. Hence the output equa- tion is like eqn. 13 with Y being A[P7-8 P9.6]T, and matrix C is

1 0.49 0 0.50 0 0.99 0 1.18 0 -1.82 -0.77 -1.14 0 -1.6 0 0.88 0 0.75 0 0.88 2.41

c = [ The output feedback gain K can be decided by the method introduced in [SI:

The closed loop eigenvalues are listed in the second column of Table 1. They suggest that the damping performance of this system has improved. The dominant poles 4 .29 2 j 1 1.74 and 4.22 2 j7.95, have been moved to -1 .OO 2 j 1 1.43 and 4 .54 2 j7.47. They are now further to the left in the complex plane and have higher damping ratios. To test the effectiveness of the proposed control scheme, a three- phase ground fault is assumed to occur near bus 7 for a duration of 12 cycles and is removed afterwards.

4 r

a

01 1 1

0 1 2 time, s

b Fi .3 rroyers a G2 b G3

N o d loading operatwn while CSCS were regulated by linear con-

151

The swing curves illustrate that the system was not stable under such a fault. They also suggest that the linear con- trolled CSC regulations cannot maintain system stability, even though aU eigenvalues have negative real parts.

4 2

Fig. 8 control strategjz “ P ? S h p 9 4

Notnral loadiig operution when CSCs were reghied by the proposed

Using the proposed control strategy, Fig. 6 shows that the swing curves of generators 2 and 3 have become stable after a period of oscillation. Figs. 7 and 8 give the CSCs regulations and power flow across regulated lines reapec- tively. The changes in the control stages are recorded in Fig, 11. Figs. 9 and 10 give the acderation of the genera- tors and the input vector respectively. The FLC used in stage two in this case is described in the Appendix.

-1 1 a

b -1

h u2

l r

Fig. 9 coritro/ sirutegy U ace2 b acc3

Ivunnul lmuiiiig opratlon w51m CSC’ ~ v r e reguhted by the proposed

Control stage

Table 2: Eigenvalues of higher load demand case

Open loop Closed loop

-0.27 kj12.06 -1.Orj11.74

-0.26 * j6.92 -0.58 .c j6.33 -0.98 3 j2.39 -0.99 f jZ.42

-1.10t 1.32 -1.09 -C 1.34

-2, -2 -4.49, -10.8

Consider another operating condition in which the active power load demands at bus 5, 8, and 9 increase from 125,

IEE Prur.-Gt’ner. Trurtsn?. Dislrrb., Vol. 146, No. 3, March 1999 1 5 3

90 and IOOMW to 175, 130 and 150MW, respectively. Also, the reactive power load demands increase from 50,30 and 35MVAR to 75, 60 and 80MVAR, respectively, while the generation at generators 2 and 3 increase from 163 and 8 5 M W to 240 and 160MW. Such operation will burden the transmission systems. In this case, matrix A and C become

A = 0 377

-0.17 0

-0.34 0

3.34 0

0 0

0.15 0

0.14 0

2.99 0

0 0

0 0 r

0 0 0 0 0 0 0 0

-0.27 0 0.06 0 0.02 0 0.06 0.14

-0.58 0.17 0.10 0 0.10 0 0.08 0.24

-24.98 -2 0.43 0 -6.74 0 1.82 0.04

0 0 0 3 7 7 0 0 0 0

-0.06 0 -0.35 0 -0.44 0 0.23 -0.14

0.16 0 -0.42 0 -0.67 0.17 0.24 -0.25

-8.09 0 0.18 0 -21.1 -2 3.19 7.57

0 0 0 0 0 0 - 2 0

0 0 0 0 0 0 0 - 2 1

0.39 0 0.53 0 1.01 0 1.31 0 -3.65 -1.15

-1.03 0 -1.73 0 1.08 0 0.06 0 1.59 4.09

The eigenvalues of matrix A of the open loop systems are listed in the first column in Table 2. Similarly, two sets of dominant poles, 4 .28 2 j12.06 and -0.29 2 j6.92, are read- ily identified.

a

" 0 1 2

time, s b

Fi .I2 trolers a G2 b G3

Higher loudmg operation when CSCs were regulated by linear con-

The controller gains are chosen as

0.06

0.04

0.02 c .- C

$ 0

2 -0.02

-0.04

-0.06

0.15 r a

-0.15L Fi . I 3 trolers a CSCl b CSC2

H i g h l&g oprutwn when CSCs were reguluted by linear con-

2-

1 - c c

ii Q

g 1 m time, s

a = I a

L

2- I

1-

* .- C

ii 6 0-,

g o n time, s

E

Q 1

1 2 1

b -

-1 -

-2 J Fi .I4 trolers a P7-8 b p9-6

Higher l&g operutwn when CSCs were regulated by linear con-

154 IEE Proc.-Gener. Transm. Disfrib., Vol. 146, No. 2, March 1999

3r same type of fault at the same location but is removed after six cycles, the swing curves given in Fig. 12 suggests that if the CSCs are regulated by the linear controllers, it is still impossible to maintain stability. Figs. 13 and 14 give the corresponding CSC regulations and h e flows respectively. However, the results in Figs. 1 >20 illustrate that the stabil- ity of both generators are improved by the use of the pro- posed control strategy.

a

1

a

1

0 5 10 time, s

b Fig. 15 control strategy a G2. b G3

Higher loudmg operation when CSCS were regulated by proposed

0-

time, s b

Fig.17 control strategy a p7-89 b p94

Higher lo&g operation when CSCS were regulated by proposed

-0.151~ a

-1 L a

n -0.15L

Fig. 16 control strategy a CSC1. b CSC2

Higho loading operation when CSCs were regulated by proposed

Then the closed loop system eigenvalues are shown in the second column in Table 2. Two sets of dominant poles have now moved further to the left and higher damping ratios are expected. When the system is subjected to the

-1 L Fig. 18 control strategv a acc2, b acc3

Higher loading operutwn when CSCs were regulated by proposed

1 ss IEE Proc.-Gener. Transm. Distrib., Vol. 146, No. 2, March 1999

It is interesting to observe that the oscillations in Fig. 6 are much severe than those in Fig. 15. This is because the duration of fault in the former case is longer, even though the loads in the latter case have been increased.

-1 L a

Fig. 19 control strategy a ul b 3

Higher loafing operation when CSCs were reguluted by proposed

0 5 10

time, s Fig.20 control strategy Control stage

Higher locldylg operutwn when CSCs were regulated by proposed

5 Conclusions

This paper develops a new control strategy for multiple CSC devices in the enhancement of power system stability. The findings of ths study indicate that CSC control scheme based on linear system theory may fail to maintain stability during a major disturbance. Even if all eigenvalues of the studied system are in the stable region, it is still impossible to guarantee stability from the standpoint of linear system analysis. The proposed control strategy is capable of over- coming ths weakness. This is because the proposed control strategy takes into account the inherent characteristics of

156

the electric power system. The effectiveness of the proposed method has been demonstrated from case studies.

6

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

7

7.

References

HINGORANI, N.G.: ‘FACTS-flexible AC transmission system’. IEE 5th international conference on AC and DC nower trummission. Lon- don, 1991, pp. 1-7 LIOU, K.L., and HSU, Y.Y.: ‘Damping of generator oscillations using static vdr compensators’, IEEE Truns. AES, 1986, 22, (5). pp. 605317 CHENG, C.H., and HSU, Y.Y.: ‘Damping of generator oscillations using an adaptive static var compensator’, IEEE Trans. PWR-S, 1992, 7, (2), pp. 718-725 ZHAO, Q., and JING, J.: ‘Robust SVC controller for improving power systems damping’, IEEE Trans. PWR-S, 1995, 10, (4). pp. 1927-1932 LO, K.L., LIN, Y.J., TRECAT, J., and CRAPPE, M.: ‘Improvement of FACTS incorporated power systems stability performance through fuzzv control’. 3rd IPEC Proceedines. Sineanore. 1997. nD. 160-165 EDRIS, A.: ‘Enhancement of first~swing’s~ability using ‘a high-speed phase shifter’, IEEE Trans. PWR-S, 1991, 6, (3), pp. I 113-1 118 ZHU, X., VILATHGAMUWA, M., and CHOI, S.S.: ‘A robust H-

Proceed- inf control of an unified power flow controller’,. 3rd IPEC ings, Singapore, 1997, pp. ‘397402 LO, K.L., and LIN, Y.J.: ‘Improvement of power systems damping stability using proportional plus derivative output feedback controlled series compensator’. Technical report, Power Systems Research Group, Department of Electronic and Electrical Engineering, Univer- sity of Strathclyde, 1998 SCALA, M.L., SBRIZZAI, R., TORELLI, F., and TROVATO, M.: ‘Enhancement of interconnected power system stability using a control strategy involving static phase shifted, Inf. J. Electr. Power Energy Syst , 1993, 15, (6), pp. 387-396 BRUCOLI, M., TROELLI, F., and TROVATO, M.: ‘A decentral- ised control strategy for dynamic shut VAR compensator in intercon- nected power systems’, IEE Proc. C, 1985, 132, (5), pp. 229-236 CHEN, X.R., PAHALAWATHTHA, N.C., ANNAKKAGE, V.D., and KUMBLE. C.S.: ‘Outout feedback TCSC controllers to imurove damping of meshed muldmachine power systems’, IEE Pr& C, Gener. Transm. Distrib.. 1997, 144, (3), pp. 243-248 CHEN, H., and ANDERSSON, G.: ‘A versatile approach for the control of FACTS equipment in multimachine power systems’, Int. J. Electr. Power Energy Syst., 1995, 17, (3), pp. 215-221 WANG, Y., MOHLER, R.R., MITTELSTADT, W.A., and MARATUKLAM, D.J.: ‘Variable structure breaking resistor control in a multimachine power system’, IEEE Trum. PWR-S, 1994, 9, (3), pp. 1557-1 562 NGAN, H.W., DAVID, A.K., and LO, K.L.: ‘Multi-terminal HVDC system control using a distributed computing technique’. IEE 5th international conference on AC and DC power transmission, London, 1991, pp. 114119 LIN, Y.J.: ‘Design of fuzzy logic controller for FACTS. PhD disxrta- tion, University of Strathclyde, 1998

Appendix

I Multimachine power systems data (all on a 100-MVA base)

Table 3: Generator data

Generator X, Xd xd’ H Td,,’ K, T, 2 0.8654 0.8958 0.1198 6.4 5.9 25 0.5

3 1.2578 1.3125 0.1813 3.01 5.89 25 0.5

Table 4 Bus data (normal operation)

P Q P Q

(MW (MW (MW (MW Bus Voltage generation generation demand demand

1.04LO

1.025L0.161

1.025L0.080

1.026L-0.039

0.996L-0.070

1.013L-0.064

1.026L0.065

1.01 6L0.013

1.032L0.034

71.6

163

85

0

0

0

0

0 0

27 0 0

6.7 0 0

-10.9 0 0

0 0 0

0 125 50

0 90 30

0 0 0

0 100 35

0 0 0

IEE Proc-Genrr Trunym. Distrib., Vol. 146. No. 2. March 1999

Table 5: Bus data (higher power transfer operation) on a 100-MVA base

P Q P Q

( M W (MW (MW (MW Bus Voltage generation generation demand demand

1 1.04LO 69.52 111.90 0 0 2 1.005L0.332 240 77.18 0 0 3 1.005L0.245 160 45.62 0 0 4 0.979L-0.039 0 0 0 0 5 0.912L4.074 0 0 175 75 6 0.931L4.056 0 0 130 60 7 0.968L0.172 0 0 0 0 8 0.936L0.095 0 0 150 80 9 0.983L0.144 0 0 0 0

Table 6 CSC data - ~~

CSC Tcsc (s) Min (per unit) Max (per unit)

1 0.5 -0.0504 0.0504 2 0.5 -0.119 0.1 19

Table 7: Transmission line data

Line Impedance Susceptance

1-4 2-7 3-9 4-5 4-6 5-7 6-9 7 4 8-9

j0.0576 j0.0625 j0.0586 0.010+i0.085 0.017+j0.092 0.032+j0.161 0.039+j0.170 0.0085+j0.072 0.01 19+j0.1008

0 0 0 j0.088 j0.079 j0.153 j0.179 j0.0745 j0.1045

7.2 FLC in stage two Two different FLCs are used in this stage. The first is used to determine, for each generator, whether it needs accelera- tion or deceleration force. Linguistic variables used in this FLC include LN (large negative) SN (small negative), SP (small positive) and LP (large positive). The input signals are the rotor angle deviation from steady state value and the rotor speed of each generator, denoted by 6, and w, respectively. Output signal ace indicates that this generator requires acceleration or deceleration force. The membership functions of the input and output signals are shown in Figs. 21 and 22.

LN SN SP LP

-1.0 -0.8 -0.7 -0.1 0.1 0.7 0.8 1.0 6850 (degree)

a

LN SN SP LP

-1.0 -0.8 -0.7 -0.1 0.1 0.7 0.8 1.0 0*30 (radkec)

b Fig.27 Mmbershipjimctionsjor input 6d rmd w

LN SN SP LP

-1.0 -0.8 -0.7 -0.3 -0.2 0.2 0.3 0.7 0.8 1.0

acc Fig.22 Membershipfunction~for output ucc

The fuzzy rules of this FLC are given in Table 8. For example, Rule 1 indicates IF 6, is LP AND w is LP THEN acc is LN

Table 8 Fuzzy rules for FLCl

LN SN SP LP

w LP SP SN LN LN’

SP LP SP SN LN

SN LP SP SP LN

LN LP LP SP SN

Finally, the defuzzification methodology is that of the centroid of area (COA). The formula for the COA can be found in [ 151.

The second FLC is used to decide the input signal for each CSC. The input signals of ths FLC are the outputs of the previous one, i.e. ucc of each generator. The output signals of thls FLC are for the CSCl and CSC2 control signals, u1 and y. The membership functions used to repre- sent the input and output signals are depicted in Figs. 23a and b.

LN SN SP LP

-1 .o -0.3 0 0.3 1 .o acc2andacc3

a

SP LP LN SN .

-1 .o -0.3 0 0.3 1 .o U, and u2

b Fig.24 M d e r s h i p functions for ucc2 miucc3. U , &U,

The fuzzy d e s are listed as follows: 1. IF acc2 IS LN AND arc3 IS LN THEN ul IS LN AND u2 IS LN 2. IF arc2 IS LP AND ace3 IS LP THEN u l IS LP AND U2 IS LP 3. IF acc2 IS LP AND arc3 IS LN THEN u l IS LP AND u2 IS LN 4. IF acc2 IS LN AND acc3 IS LP THEN ul IS LN AND U 2 IS LP 5. IF arc2 IS SN AND ace3 IS LNTHEN u l IS SN AND u2 IS LN 6. IF arc2 IS SP AND arc3 IS LP THEN ul IS SP AND U2 1s LP

IEE Proc.-Gener. Tnmsm. Distrib., Vol. 146, No. 2, March 1999 I57

7. IF acc2 IS SP AND acc3 IS LN THEN ul IS SP AND u2 IS LN 8. IF acc2 IS SN AND acc3 IS LP THEN ul IS SN AND U2 IS LP 9. IF acc2 IS LN AND acc3 IS SN THEN u l IS LN AND u2 IS SN 10. IF acc2 IS LP AND acc3 IS SP THEN ul IS LP AND U 2 IS SP 11. IF acc2 IS LP AND acc3 IS SN THEN u l IS LP AND u2 IS SN 12. IF acc2 IS LN AND acc3 IS SP THEN ul IS LN AND u2 IS SP 13. IF acc2 IS SN AND acc3 IS SN THEN u1IS SN AND u2 IS SN 14. IF acc2 IS SP AND acc3 IS SP THEN u l IS SP AND U2 IS SP 15. IF acc2 IS SP AND acc3 IS SN THEN ul IS SP AND u2 IS SN 16. IF acc2 IS SN AND acc3 IS SP THEN ul IS SN AND U 2 IS SP In short, these fuzzy rules can be expressed as ul = acc2 and u2 = acc3 The defuzzification method is the same, COA.

7.3 Derive linear state space equation The principle equations used for the study are listed below according to the requirement of eqns. 12 and 13. The detailed derivation is given in [15]. Network equation For a n-generator-m-load power system, the network equa- tion can be written as

where IGxy, VGxy, VLxy denote the vectors of injected gener- ator current, generator terminal voltage and load terminal voltage, respectively, which are all based on a common x-y axis co-ordinate. YGG, YGL, YLG, YL, denote the appropri- ate admittance matrices with appropriate dimensions. Generator equation The linearised state equations for n generators can be expressed as

XG = A ~ X G + A ~ A I G ~ ~

AvGdq = E; -k XGdqAIGdq

(15)

(16) where XG = A[6, Wi ebi e F D i

T * * * 6, wn ehn e F D n ]

A , and A, are block diagonal matrices with

11 A1 = diag

2nf 0 0

0

0 0

CSC equation Also assume that f CSC devices are considered; their line- arisation models can be written as

Xc = A3Xc + B1U (17) where

T T X C = A [xCl . . . z C f ] , U = A [uI . . . ~ f 3

A3 = diag { [-L]}, TCi B1 = diag { [&I} State space equation To drive the state space representation in eqn. 12, one can use the method introduced in [ll] to obtain the following relation:

= A4E; + A56 + A6Xc

AIGdq = M I X (19)

(18) The vectors E;, 6 and Xc are state variables. By properly rearranging eqn. 18, one can obtain

where x = [ X G XCIT

Combining eqns. 15, 17 and 19, one obtains the h e a r sys- tem state equation

X = A X + B U (20) where

A = diag { A I , A3) + A21CI1, B = [N B1IT and N is a zero matrix with n x f dimension. Output equation Start with the network equation in eqn. 14, one can obtain

in which VLxy = y 2 I G x y (21)

Yz = -YL;YLGYG1 Differentiate both sides in eqn. 21:

(22 ) From eqn. 22, one can derive the deviation of current, active and reactive power in the network. The current flows from node m to node n can be written as

~ ~ L z Y = AY2IGzy + & ~ I G ~ ~ = M2X

Imn = -YNmn(vm - vn) (23)

Arm, = - a Y ~ m n ( ~ m - v,) - Y~mn(avm - AV,) (24)

Furthermore, the active and reactive power flow along h e m-n can be expressed as

in which the superscript '*' denotes the conjugate complex symbol. Using eqns. 23 and 25, one can compute the devia- tions of voltage and current at first and, in turn, obtain the deviations of active and reactive power individually in the network.

Hence, the deviation of current Imn is

Smn = pmn + jQmn = VmIGn (25)

I58 IEE Proc.-Gener. Transm. Distrib., Vol. 146, No. 2, March 1999