Discrete mode power system stabilisers

Transcription

1 Dicrete mode power ytem tabilier M.L. Kothari J. Nanda K. Bhattacharya Indexing term: Power ytem tabilier. Dicrete control, Dynamic tability Abtract: The paper preent a comprehenive approach for the deign of dicrete mode conventional power ytem tabilier (PSS) conidering a machine infinite bu ytem, uing the ISE technique. Invetigation reveal that the ampler and zero order hold (ZOH) need to be modelled for ampling frequencie le than 2 time the Nyquit frequency and that, beyond 2 time the Nyquit frequency, the ampler and ZOH may be neglected from the mathematical model. A comprehenive enitivity analyi reveal that the dicrete mode conventional PSS i quite robut and it nominal optimum parameter need not be reet following ±2 change in inertia contant H, field open circuit time contant T' i, line reactance x e, AVR gain K A or loading P and Q from their nominal value. 1 Introduction High initial repone, high gain excitation ytem equipped power ytem tabilier (PSS) have been extenively ued in modern power ytem a an effective mean of enhancing the overall ytem tability. Input ignal uch a rotor peed, bu frequency, electrical power and accelerating power which can be ued to derive the tabiliing ignal working through the reference of automatic voltage regulator (AVR) have been propoed in the literature [1-6]. In almot all application, implementation of the PSS ha been olid tate analogue component. Lee et al. [7] have decribed the experience of uing power ytem tabilier for large thermal unit at Ontario Hydro baed on direct meaurement of turbine generator haft peed (commonly known a the Delta- Omega tabilier). They have decribed the pecial peed ening arrangement to mitigate the problem of excitation of torional mode. A uitably deigned torional filter in the tabiliing path i ued a an added precaution againt the poibility of torional excitation. During the lat decade, the deviation in equivalent rotor peed (Ao).,) derived from haft peed and integral of change in terminal electrical power ha been ued [7, 8] a input ignal to PSS. Stabilier equivalent Paper 9596C (PI 1), firt received 3rd February 1992 and in revied form 16th April 1993 The author are the Department of Electrical Engineering, Indian Intitute of Technology, Delhi, New Delhi 11 16, India rotor peed (Aw eq ) a input ignal are known a Delta- P-Omega tabilier. Delta-P-Omega tabilier are free from the inherent limitation of Delta-Omega tabilier. DeMello ha uggeted [S, 6, 9] that the problem of excitation of torional mode Delta-Omega tabilier can completely be eliminated if peed i derived from the meaurement of the frequency of a voltage yntheied from machine terminal voltage and current, intead of being meaured by a tachometer. It may be noted that the tructure of the PSS remain unchanged, irrepective of the input ignal to PSS (i.e. AOJ or Att> eq or frequency of yntheied voltage ignal). In recent literature, the application of adaptive PSS ha been propoed [1-15] to counteract the problem of variation in the ytem parameter and operating condition. However, for the reliation of uch adaptive elftuning PSS, online identification of ytem parameter, obervation of ytem tate and computation of feedback gain in a hort ampling period are needed. A an alternative to a elf-tuning PSS, a variable tructure PSS ha been propoed in the literature [16] to counteract the problem of variation of ytem parameter and operating condition. Kundur et al. [8] have demontrated that, fixed parameter PSS, it i poible to atify the ytem requirement for a wide range of ytem condition, thu undermining the need for complex adaptive or variable tructure PSS. Mot of the work pertaining to the deign of PSS deal continuou time PSS. In the modern day, digital intrument are ued for meauring the ytem variable uch a peed, voltage, terminal power, current etc. It i thu obviou that input ignal to PSS are available in a dicrete form. A brief literature review preented above how that no attempt ha been made to deign either dicrete mode Delta-Omega or Delta-P-Omega tabilier. The main objective of the preent work are a follow. (i) To preent a ytematic approach for deigning a Delta-Omega power ytem tabilier operating in (a) a continuou mode and (b) a dicrete mode, uing the ISE technique. The performance of both continuou mode PSS and dicrete mode PSS are alo compared. (ii) To tudy the effect of variation of ampling period on the election of optimum dicrete mode PSS and it impact on ytem dynamic performance. (iii) To etablih a cutoff ampling frequency in term of a multiple of Nyquit frequency, below which the ampler and ZOH need be modelled and beyond which it i divorced from the mathematical model. (iv) To perform enitivity analyi to highlight the effect of variation of operating load and ytem parameter on the performance of the optimal dicrete mode PSS. 523

2 2 Sytem invetigated The ytem invetigated comprie a ynchronou generator connected to an infinite bu through a long tranmiion line. The IEEE type-1 excitation ytem model [17], neglecting aturation of the exciter and voltage limit of amplifier output, ha been conidered. A conventional PSS compriing a pair of lead lag network peed deviation a input ignal, i aumed. 3 Small perturbation tranfer function block diagram The mall perturbation tranfer function block diagram of the ytem, relating the pertinent variable of electrical torque, peed, angle, terminal voltage, field voltage and flux linkage, i hown in Fig. 1. Thi linearied model i obtained by lineariing the nonlinear differential equation of the ytem around a nominal operating point. The tranfer function of the PSS i given a U() _ where T w i the wah out time contant, H-T^ are PSS time contant and K c i the PSS gain. 4 PSS performance objective In the analyi and control of ytem tability, two ditinct type of ytem ocillation are uually recognied [8]. One type i aociated unit at a generating tation winging wrt the ret of the power ytem. Such ocillation are referred to a 'local plant mode ocillation'. The frequencie of thee ocillation are typically in the range.8 to 2. Hz. The econd type of ocillation i aociated the winging of many machine in one part of the ytem againt machine in other part. Thee AUJ I _ 1-.- T» I Awd power ytem tabilier APS1 _ KeO-t^T]) APS^ 1+T2 Aoo 2gf -AS AE<d 1 AVf K A '+«T do K 3 Ke«TE 1<-tTA KF t+tp i IEEE typei excitation ytem Fig. 1 Small perturbation tranfer Junction model of ingle generator connected to infinite bu through tranmiion line The input ignal to the PSS i obtained through a ampler and a zero order hold circuit. Conidering haft peed deviation Aa> a the input ignal and Aco d a the output ignal from ZOH, the tranfer function relating Act) and Aco, i given by 1 -e~ where T i the ampling period. For ampling period T < 1, the tranfer function of the ZOH may be approximated to 1-e The ZOH thu introduce a phae lag a well a a gain in the PSS path. The phae lag and gain contributed by ZOH depend on the ampling period and the frequency of ocillation. 524 (1) (2) are referred to a 'interarea mode' ocillation, and have frequencie in the range of.1 to.7 Hz. The baic function of the PSS i to add damping to both type of ytem ocillation. The overall excitation control ytem (including PSS) i deigned o a to: (i) maximie the damping of the local plant mode a well a interarea mode ocillation out compromiing the tability of other mode; (ii) enhance ytem tranient tability; (iii) not adverely affect ytem performance during major ytem upet which caue large frequency excurion; (iv) minimie the conequence of excitation ytem malfunction due to component failure. 5 Primary conideration for the election of PSS parameter Phae lead compenation The PSS tranfer function hould have an appropriate phae lead characteritic to compenate for the phae lag between the exciter input and electrical torque. The phae characteritic to be compenated change ytem condition. Therefore a compromie mut be made and a characteritic acceptable for a deired range of frequencie (normally.1 to 2. Hz) and different ytem condition i elected. IEE PROCEEDINGS-C, Vol. 14, No. 6, NOVEMBER 1993

3 52 Stabiliing ignal wahout The ignal wah out function i a high pa filter which remove DC ignal and, out it, teady change in peed would modify the terminal voltage. Kundur et al. [8] have conidered the effect of the variation of T w on the phae lead characteritic of the PSS and out tranient gain reduction (TGR). They find that the change in phae lead characteritic i inignificant for T w > 1, over a wide range of frequency of ocillation. Thu T w = 1 ha been choen for the preent invetigation. 5.3 Stabilier gain The tabilier gain hould be et to a value which reult in atifactory damping of the critical ytem mode out compromiing the tability of other mode, or tranient tability, and which doe not caue exceive amplification of tabilier input ignal noie. 5.4 Stabilier output limit To retrict the level of generator terminal voltage fluctuation during tranient condition, limit are impoed on PSS output. 6 Dynamic model in tate pace form The dynamic model in tate pace form i obtained from the tranfer function block diagram of Fig. 1 a x = Ax + Tp (4) where tate vector x = [Aa> AS AE', AE fd AV R AV E Aco d AP,, AP, 2 l/] r and perturbation vector p = [AT B AV nl Y The tate and perturbation matrice A and F, repectively, are given in Appendix 11. It i een that the ytem matrix i a function of ytem parameter, operating point and the ampling period T, while the perturbation matrix depend on ytem parameter only. In eqn. 4, x() = and p i aumed to be a tep perturbation vector remaining contant during the dynamic of the proce. A coordinate tranformation in the tate pace i applied to eliminate the perturbation term, Vp of eqn. 4, by defining: Mt) = x(t)-*ao) (5) where JC(OO) i the teady-tate value of the tate vector x(t). Eqn. 4 reduce to the form: where Ax\t) x(oc) i evaluated from eqn. 4 i.e. x(oo) = A~ l Yp, ince Mao) = 7 Analyi 7.1 Evaluation of K contant of the ytem The initial d-q axi current and voltage component and torque angle needed for evaluating the K contant are obtained from the teady-tate equation given in Appendix 12 uing the ytem data given in Appendix 13. Thee 1EE PROCEEDINGS-C, Vol. 14, No. 6, NOVEMBER 1993 (6) (7) (8) are given a follow: V d =.545 p.u. I i =.9372 p.u. fi^, = p.u. V q =.8413 p.u. /, =.3487 p.u. 5 = K o =.8246 p.u. The evaluated K contant, uing relationhip given in Appendix 14 are Ki = X 2 = K 3 =.36 K 4 = K 5 = K 6 = Comparion of ytem dynamic performance and out PSS, in continuou mode The dynamic model in tate pace form PSS operating in continuou mode i obtained by deleting the ZOH block from the tranfer function block diagram reulting in a ninth-order model tate vector x = [Aco AS AE; AE ti AV R AV E AP tl AP, 2 t/] r The ytem matrice A and F are appropriately modified Optimiation of PSS parameter uing ISE technique: The PSS parameter to be optimied are the time contant ^-7^ and gain K c. A low value of T 2 = 7i =.5 i choen from conideration of the phyical realiation [2, 18]. 7V = 1 i choen a dicued in Section 5.2. Conidering two identical cacade connected lead lag network for PSS, T, = T 3, and hence the problem reduce to the optimiation of K c and T, only. The integral of quared error (ISE) technique i ued for optimiation of the parameter of the PSS. The choice of a uitable performance index i extremely important for the deign of PSS. A quadratic performance index given by [a Am 2 + A3 2 ] dt (9) i conidered. The tate variable A& and Ah are penalied to obtain the deired performance, a i a relative weighting factor. The performance index in eqn. 9 may be expreed a J = Jo where ix T Qx\ dt Q = diag [a 1 J i evaluated by uing the relation J = x T ()Px() ] (1) (11) where x() i the initial tate defined in eqn. 7 while P i a poitive definite ymmetric matrix obtained by olving Lyapunov' equation: A T P + PA -Q (12) The value of a i varied over a wide range (from to 4 x 1 3 ) to undertand it effect on ytem dynamic performance. When at i 4 x 1 3, it i een that the contribution of f ia& 2 dt and Ah 2 dt to J are equal. However, the ytem dynamic performance PSS, obtained conidering a = wa found to be the bet. In view of the above, a i et equal 525

4 to zero reulting in a imple performance index of the form: f \S2 dt (13) Fig. 2 how the plot of J veru K c for everal value of T,. Optimum value of K c and T, correponding to minimum of J, (./ ), are K? = 47. and Tf = P-.8pu. Q-.6p.u. 6 O=55.6 T, = Fig. 2 J =f(kc)for different T, Table 1 how the eigenvalue of the ytem and out the PSS. It i een that the damping ratio, for the electromechanical mode of ocillation, identified by.2 65 Table 1: Eigenvalue of the y*tem and out PSS Eigenvalu rad/ Hz Without PSS ±> * A * / With optimum PSS ±j\ 1.37 (/ff ±> T, -.35 ) * /4jjg ±> computing the undamped frequency of ocillation from the relation <, = (t/ki/^) 1 ' 2 (14) ha increaed from.14 out PSS to.ss32 PSS, implying enhanced ytem damping PSS. Thi i further ubtantiated by plotting the dynamic repone for a 1 tep increae in mechanical input and out PSS (Fig. 3). It i important to examine whether the optimum PSS obtained in continuou mode work atifactorily in the actual dicrete mode. To anwer thi, dynamic repone are obtained for everal ampling period T, an optimum continuou mode PSS parameter etting for 1 tep increae in mechanical input (Fig. 4). For the ample problem olved, it i een that, a T increae, the dynamic repone gradually deteriorate and ultimately.12 '..8 3 < n K -w n \ \ i ; l!' > ;':' ii'<\ A ' H 1 1»/? \H W M "i/ ' Fig. 3 Dynamic repone and out PSS a foraa) b foraj out PSS optimum PSS K' time, b Fig. 4 Dynamic repone for everal T the optimum continuou mode PSS etting a optimum continuou mode, Kf - 47.rf-.351 T -.2 l,kc- 47.,T1-J5 T -.5 t,kc- 47.,7,-351 T -.1,Kc-47.,7, -.35 b optimum continuou mode K} - 4.7, Tf -.35 r Kc T, -.35 > T-J,X c- 47., T, -.35 IEE PROCEEDINGS-C, Vol. 14, No. 6, NOVEMBER 1993

5 the ytem become untable for value of T >.2S. However, the degradation in dynamic performance i not very ignificant for T <.S. I it deirable to optimie the PSS parameter conidering actual dicrete time dynamic model of the PSS? To anwer thi pertinent quetion, it i extremely important to explore the optimum PSS etting in the dicrete mode for credible ampling period uing the mathematical model (eqn. 6). 7.3 Effect of variation of ampling period T on optimum etting of dicrete mode PSS parameter Optimum dicrete mode PSS parameter (Kf and Tf) are obtained uing the ISE technique by minimiing the performance index J (eqn. 13) for everal value of T. Table 2 how the optimum value of thee PSS parameter and the correponding J* for everal value of ampling period. Table 2: Optimum PSS parameter for everal value of 7* Sampling period, r Optimum PSS K'c ? J* x1" x1"' x 1-" x1-' x1* x1" 17.57x1-' x1"' Examining Table 2, the following inference are made: (i) The optimum gain etting KJ decreae while the optimum phae lead time contant TJ increae increae in ampling period. T. Decreae in K and increae in Tf increae in T i needed to compenate for the additional phae lag and gain contributed by the preence of ZOH in the PSS path. (ii) J* gradually increae a the ampling time T i increaed. Thi implie a gradual deterioration in the ytem dynamic repone increae in T. Fig. 5-8 how the dynamic repone for Aco dicrete mode PSS for T =.2,.5 and.1 and.15, repectively, conidering: (i) correponding dicrete optimum PSS; and (ii) optimum continuou mode PSS (KJ = 47. and TJ =.35 ). Examining the repone, it i een that the continuou mode optimum PSS parameter are acceptable actual dicrete mode PSS for T <.5 (i.e. equal to or.12.8 Fig. 6 Dynamic repone for T =.5 dicrete and continuou mode optimum PSS continuou mode optimum, X 47., dicrete mode optimum, X? - 39., 7/f = time, Fig. 7 Dynamic repone for T =.1 dicrete and continuou mode optimum PSS continuou mode optimum, KJ - 47., Tf -.35 dicrete mode optimum,!:; -31., Tf cl.4 3 A Fig. 8 Dynamic repone for T =.15 dicrete and continuou mode optimum PSS continuou mode optimum, K 47., dicrete mode optimum, XJ - 25., o time, Fig. 5 Dynamic repone for T = O2 wit* dicrete and continuou mode optimum PSS continuou mode optimum, XJ 47,, dicrete mode optimum, XJ -, Tf -.36 IEE PROCEED1NGS-C, Vol. 14, No. 6, NOVEMBER 1993 greater than about 2 time the Nyquit frequency, Fig. 5 and 6). It can thu be inferred that, for ampling frequencie beyond 2 time the Nyquit frequency, the ampler and ZOH may be divorced from the model. However, for ampling frequencie le than 2 time the Nyquit frequency, the dynamic performance of the dicrete mode PSS it correponding optimum K c and 527

6 T t i quite uperior to the repone correponding to optimum K c and T t parameter for the continuou mode PSS, thu ignifying the need to model the ampler and ZOH in the dicrete model. Thi i clearly revealed from two ample cae ampling period T =.1 (nearly 11 time the Nyquit frequency) and for T =.15 (nearly 8 time the Nyquit frequency) (Fig. 7 and 8). It can thu be concluded that the nominal optimum PSS parameter need not be reet following a 2 increae in the nominal value of H. 8 Senitivity analyi Although conceptually the optimum PSS parameter for the nominal condition would naturally change owing to change in ytem/operating parameter, it i worth exploring whether it i at all deirable to reet the optimum nominal PSS parameter from practical conideration. Thi i the main thrut of enitivity analyi in our work. A ampling period of T =.2 ha been conidered for the preent analyi. 8.1 Effect of change in inertia contant H Table 3 give the optimum PSS parameter (K? and Tf) and the correponding performance indice for a +2 change in H from it nominal value. It i een that the Table 3: Effect of change in H H (nominal) 6. K'r * KJ anotf K =, 7? = PSS parameter change ±2 change in H. For example, H = 4. (a reduction by 2 from nominal value), K = 38.2, Tf =.37 and the correponding J* i 93.1 a againt K$ =, Tf =.35 and J* = 1. for H = 5.. I it neceary to reet the PSS parameter from it nominal optimum, K? = and Tf =.36 to the deired optimum, K? = 38.2 and Tf =.37? To anwer thi, the percentage value of the performance index J i tabulated for H = 4. conidering nominal optimum PSS (Kf = and 7? =.35 ). It i een that the difference in J* and J for 2 decreae in H i quite inignificant. Thi i alo confirmed by plotting the dynamic repone for H 4. the deired optimum PSS and the nominal optimum PSS (Fig. 9). The repone do not differ much for all practical purpoe. Hence it may be inferred that there i no need to reet the nominal PSS parameter following 2 decreae in H from it nominal value. Similarly, for if = 6. (an increae by 2 from the nominal value), K? = 49.4 and TJ =.36 and the correponding value of J* i I it deirable to reet the PSS parameter from the nominal optimum to the deired optimum when H i changed from 5. to 6.? To anwer thi, the percentage J i evaluated H = 6., optimum PSS and i found to be equal to It can thu be noted that the two percentage value of J* = 19.5 and J = are quite cloe to one another, and hence are uppoed to provide more or le imilar dynamic performance. Thi i alo confirmed by plotting the actual dynamic repone for H = 6. the deired optimum and nominal optimum PSS parameter (Fig. 9). The repone are quite cloe to one another for all practical purpoe time, b Fig. 9 Effect of ±2 change in H on dynamic repone for Am a Kc -, T, (nominal optimum) Kf , TJ -.37,H-4. b Kc -, 7\ (nominal optimum) Kf -49.4, rj =.36, H- 6. The above approach of analyi i applied to tudy the effect of change in other ignificant parameter a dicued below. 82 Effect of change in field open circuit time contant T M Table 4 how the variation in optimum PSS parameter and correponding performance indice for ±2 change in T' i from it nominal value. Examining Table 4 Table 4: Effect of change in 7^, (nominal) * K*c and 7f *S = 7T = it i een that the difference in J* and J for ±2 change in T' iq i quite inignificant. Dynamic repone obtained for T M = 4.8 (2 decreae from it nominal value) and T' i = 7.2 (2 increae from it nominal value) conidering correponding optimum PSS and nominal optimum PSS reveal that there i no need to reet the nominal optimum PSS parameter to thendeired optimum value following ±2 change in T' l. 1EE PROCEEDINGS-C, Vol. 14, No. 6, NOVEMBER 1993

7 8.3 Effect of change in line reactance x. Table 5 how the variation in optimum PSS parameter and the percentage J value variation in x e by ±2 from it nominal value. Analying the value of J* and J, it may be inferred that there i no need to reet the parameter of the nominal optimum PSS following ±2 excurion in x e from it nominal value. Thi i alo confirmed by plotting the appropriate dynamic repone. Table 5: Effect of change in x. p.u (nominal).48 K' * K* and T> K = 7» = Effect of change in A VR gain K A The effect of change in AVR gain K A by ±2 from it nominal value on the parameter of optimum PSS and percentage value of J* and J i hown in Table 6. Examining Table 6 it i inferred that the nominal optimum PSS i quite robut and need not be reet following a ±2 change in K A from it nominal value. Table 6: Effect of change ink, K'c (nominal) * K* and 7* AT? = Ff = Effect of change in loading condition. P and Q Change in optimum PSS parameter following ±2 change in P and Q are given in Table 7 and 8, repectively. The correponding percentage value of J* and J are alo tabulated. Analying the data, it i inferred that the nominal optimum PSS i quite robut and need not be reet following an excurion of ±2 in P and Q from their nominal value. Thi i alo confirmed by plotting the appropriate dynamic repone. Table 7: Effect of change in P p p.u (nominal) Table 8: Effect of change in Q Q p.u (nominal).72 Af? * K and 7? ' ATJ and 7? /TJ = rf = AT? = 7* = Concluion (i) A dynamic model of the ytem in tate pace form conidering dicrete mode conventional power ytem tabilier i developed. (ii) A comprehenive approach for obtaining the optimum value of the dicrete mode PSS parameter uing ISE technique ha been preented. (iii) Invetigation reveal that, for ampling frequencie greater than about 2 time the Nyquit frequency, the effect of ampler and ZOH can comfortably be divorced from the mathematical model. (iv) A comprehenive enitivity analyi reveal that the dicrete mode conventional PSS i quite robut and it nominal optimum parameter need not be reet following ± 2 change in H, T'^, x,, K A, P and Q. 1 Reference 1 HEFFRON, W.G., and PHILLIPS, R.A.: Effect of modern amplidyne voltage regulator on under-excited operation of large turbine generator', A1EE Tran., 1952, PAS-71, pp DEMELLO, F.P, and CONCORDIA, C: 'Concept of ynchronou machine tability a affected by excitation control', IEEE Tran., 1969, PAS-IK, pp KEAY, F.W., and SOUTH, W.H.: 'Deign of a power ytem tabilizer ening frequency deviation', IEEE Tran., 1971, PAS-98, pp WATSON, W, and MANCHUR, G.: 'Experience upplementary damping ignal for generator tatic excitation ytem', IEEE Tran., 1973, PAS-92, pp BAYNE, J.P, LEE, D.C, and WATSON, W.: 'A power ytem tabilizer for thermal unit baed on derivation of accelerating power', IEEE Tran^ 1977, PAS-96, pp DBMELLO, F.P., HANNET, L.N., and UNDRILL, J.M.: 'Practical approache to upplementary damping from accelerating power', IEEE Tran., 1978, PAS-97, pp LEE, D.C, BEAULIEU, R.E., and SERVICE, J.R.R.: 'A power ytem tabilizer uing peed and electrical power input: deign and field experience', / Tran., 1981, PAS-1M, pp KUNDUR, P, KLEIN, M, ROGERS, GJ, and ZYWNO, M.S.: 'Application of power ytem tabilizer for enhancement of overall ytem tability', IEEE Tran^ 1989, PWRS-4, pp DEMELLO, F.P., HANNET, L.N., PARKINSON, D.W., and CZUBA, C.S.: 'A power ytem tabilizer deign uing digital control', IEEE Tram., 1982, PAS-11, pp PAHALAWATHTHA, N.C, HOPE, G.S., and MALIK, O.P.: 'Multivariable elf-tuning power ytem tabilizer imulation and implementation tudie', / Tran., 1991, EC-4, pp CHENG, SJ., MALIK, O.P., and HOPE, G.S.: 'Damping of multimodal ocillation in power ytem uing a dual-rate adaptive tabilizer'. / 7><m., PWRS-3, pp LIM, CM.: 'A elf-tuning tabilizer for excitation or governor control of power ytem', IEEE Tran^ 1989, EC-4, pp CHENG, S.J., CHOW, Y.S, MALIK, O.P, and HOPE, G.S.: 'An adaptive ynchronou machine tabilizer', IEEE Tran., 1986, PWRS-1, pp MAO, C.X., MALIK, O.P, HOPE, G.S, and FAN, J.: 'An adaptive generator excitation controller baed on linear optimal control', / Tran., 199, EC-S, pp WU, C.J., and HSU, Y.Y.: 'Deign of elf-tuning PID power ytem tabilizer for multimachine power ytem', / Tran., 1988, PWRS-3, pp HSU, Y.Y, and CHAN, W.C.: 'Application of liding mode to digital power ytem tabilizer deign', Electr. Much. Power Syt., 1986, vol. 11, pp IEEE Committee Report: 'Computer repreentation of excitation ytem', IEEE Tran., 1968, PAS-87, pp FLEMING, RJ., MOHAN, M.A., and PARVATISAM, K.: -Selection of parameter of tabilizer in multi-machine power ytem', / Tran., 1981, PAS-11, pp Appendix The linear tate pace model of the ytem i given by x = Ax + Tp IEE PROCEEDINGS-C, Vol. 14, No. 6, NOVEMBER

8 where x = [Aa> A«5 AE', AV R AV E A<o, AJ>,, &P, 2 Al/] T -1 =r /lwhere A 9 1 Aag ' 1_ T T 2K c T t T t T T A = 2/C c r,/t 2 [iff [p KJT A -=i oj T E T r T A 2_ ~ T _2 L T -2K C T, TT 2 A T -1 T w T 2 T W T -1 ~T\~ E' t = voltage proportional to direct axi flux linkage S angle between quadrature axi and infinite bu V o = infinite bu voltage, = open circuit terminal voltage Subcript mean teady-tate value Appendix The teady-tate value of d-q axi voltage and current component for the machine infinite bu ytem for the nominal operating condition are given below [2]. Thee are expreed a function of teady-tate terminal voltage ^ and teady-tate real and reactive load current In and IQO, repectively. where 53 -UKo- I r r. - 1 QO x«) 2 x, + x.) - uxj x e - I Q r = direct and quadrature axi component of armature current = direct and quadrature axi component of terminal voltage 13 Appendix The following are the nominal parameter of the ytem and the operating condition ued for the ample problem invetigated. All data are given in per unit of value, except that H and time contant are in econd. Generator: H = 5. ri = 6. x,, = 1.6 xi =.32 x, = 1.55 IEEE Type I excitation ytem: K A = 5. K E = -.5 K T =.5 Tranmiion line: x, =.4 r«=. Operating condition: T A =.5 T E =.5 T F =.5 P =.8 g =.6 Ko = 1 /=5Hz 14 Appendix The contant K t -K 6 are evaluated uing the relation given below [2] conidering zero external reitance (i.e. 1EE PROCEEDINGS-C, Vol. 14, No. 6, NOVEMBER 1993

9 r e = for the ample problem invetigated). KA = x d x' d Vn in i X. + XA V O in d K - K 2 K 6 f^_^o x, + x' d V, o K" F COS ' - 7T7 K 2 Ko in IEE PROCEEDINGS-C, Vol. 14, No. 6, NOVEMBER