I~ U T T E R W Q R T H I N E M A N N

Electrical Power & Energy Systems, Vol. 17, No. 5, pp. 313-323, 1995 Copyright © 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0142-0615(95)00004-6 0142-0615/95/$10.00+0.00

Performance improvement of a conventional power system stabilizer

M Saidy* and F M Hughes Department of Electrical Engineering and Electronics, UMIST, P.O. Box 88, Manchester M60 1QD, UK

A conventionalpower system stabilizer (PSS), based on a speed input signal, can provide good performanee over the operating power range of a generator. However, its per- formance deteriorates significantly when the connection of the generator with the grid system becomes weak. In this paper, an augmented PSS is proposed which extends performance capabilities into the weak tie-line case. In addition to extending operating regime capabilities, the augmented controller, through its contribution at low frequencies, can also improve the damping of the inter- area mode of oscillation and increase transmission system capability.

Keywords: excitation control systems, generator dynamic model, frequency response eharacteristics

I. I n t r o d u c t i o n In the early days of interconnected power systems, the most common form of instability between interconnected generators was loss of synchronism, monotonically, in the first few seconds following a fault, due to lack of synchronizing power. This type of stability is essentially caused by the nonlinear nature of the dynamics of the interconnected generators 1 .

Automatic voltage regulators (AVRs) operating through the generators' excitation systems, have the effect of increasing the synchronizing power between the interconnected generators. However, they have the secondary effect of reducing the damping powers and make the system more prone to oscillatory instability 2. A major approach in combatting the lack of damping is to introduce a supplementary control loop, commonly in the form of power system stabilizer 3'4.

*This paper was written when Dr M. Saidy was a research visitor at UMIST. Currently, Dr Saidy works as a senior power system analyst at WestingHouse Systems Ltd, Advanced Power Application Depart- ment, Langley Road, P.O. Box 41, Chippenham SN15 1J J, UK. Received 12 May 1994; revised 8 December 1994; accepted 19 December 1994

The performance of the conventional PSS, with well tuned parameters, can be good over a reasonable range of system operating conditions. However, with speed as an input signal, the influence of the PSS on the power system decreases when the tie-line connection of the generator with the grid system becomes weak. Also, the PSS damping power contribution to the inter-area modes of oscillation is greatly reduced.

With the aim of extending dynamic performance contribution into the low system frequency region, addi- tional compensation loops have been proposed 5. The augmented compensation has been designed using open loop models, identified from measured or simulated data 6.

In this paper an additional loop, based on a speed input signal, is proposed to enhance the performance of a conventional PSS under weak tie-line conditions. The design of this additional control loop is based on the inherent generator and the excitation system dynamic characteristics and does not require a priori knowledge or measurements of the system open loop characteristics.

The frequency response and the time simulation results obtained show how the augmented conventional PSS improves overall performance by counteracting the reduction in the total system synchronizing and damping powers in cases of weak tie-line connections.

The augmented controller, through its contribution at low frequencies, can also improve the damping of the inter-area mode of oscillation and increase transmission system capability.

II. Design of a power system stabilizer A schematic block diagram of a generator, excitation system and power system stabilizer (based on a speed error signal) is shown in Figure 1.

A phase compensation approach is usually employed for design of the PSS. At the natural mechanical fre- quency of oscillation, or local mode frequency, the phase lead of the PSS is chosen to compensate for the combined lag of the excitation system and the generator relating A Vt-ref to AP e. The PSS then provides a component of electrical power in phase with speed and therefore directly

313

314 Performance improvement of a conventional PSS: M. Saidy and F. M. Hughes

Vt-ref Excitation

System

Upss

Ef• Generator and Load

Power ~ AW System

Stabiliser

Figure 1. Schematic diagram of excitation control system with power system stabiliser

contributes to the system damping at that frequency. The magnitude of this damping component is determined by the chosen compensator gain.

The basic damping and control characteristics of a generator with a conventional power system stabilizer can be demonstrated from the consideration of the situation where the generator feeds an infinite busbar. A block diagram transfer function model of this system is given in Figure AI.1 of Appendix 1, and the generator parameters employed are given in Appendix A2.1. The transfer function model has been described by Saidy and Hughes in Reference 7, and can be used to facilitate PSS design. Using the phase compensation approach a PSS, of the form shown in Figure 2, was designed which gives rise to the frequency responses shown in Figure 3. The design was based on the normal, strong interconnection situa- tion (where the tie-line reactance is 0.2p.u.), with the generator operating under rated power conditions. The responses were obtained by incorporating very large inertia in the model so that the power term generated is solely the excitation system contribution.

Figure 3 shows the responses APe/A Vt_ref, A Vt_ref/A& and APe/Aw. It can be seen from the Bode diagrams that the design, which aims to provide exact phase com- pensation at 8 rad/s, also provides an electrical power component closely in phase with speed variation over the range 4 to 9 rad/s. In the phasor diagram of Figure 4a, at the frequency of concern, the power component is along the Aa; direction and therefore directly contributes damping power.

III. Generator natural damping In terms of the block diagram transfer function model of Figure A1.1, a measure of the contribution of the d- and q-axis circuits to synchronizing and damping power, due to oscillations in rotor angle, can be determined from the frequency responses of APed/A6 and APeq/A6. The frequency responses shown in Nyquist form in Figure 5 correspond to the full power, strong tie-line conditions of the previous section. The coordinate of power in the A~ direction (axis of imaginary part) gives the damping power component, and the coordinate in the A6 direction (axis of real part) gives the synchronizing power component.

(a + T2s) AV, (1 + ras)

It can be seen that, at the natural frequency of 8 rad/s for the conditions considered, the major damping power contribution is derived from the q-axis damper. The responses show that the q-axis damping power reduces markedly as the frequency of oscillation reduces over the range 20 to 0.6 rad/s.

The d-axis damping power contribution remains fairly constant over the range 20 to 3 rad/s, and rises as the frequency reduces further to 0.6 rad/s. At low frequen- cies, for the operating conditions considered, the d-axis contribution to damping becomes greater than that of the q-axis.

The synchronizing power contributions of both d- and q-axis elements considered are negative over the entire frequency range. However both these contributions are minor compared with the main synchronizing power component given by K 1A6.

IV. Inf luence on characterist ics due to changes in system impedance The influence of a change from a strong to a weak tie- line connection on the basic synchronizing and damping power characteristics will now be demonstrated. It is assumed that a change in system configuration occurs which results in an increase in the tie-line impe- dance from 0.2 to 1.0p.u. The generator operating condition is maintained at the nominal full power level considered previously. In terms of the natural frequency of oscillation the weak tie-line situation gives rise to a natural frequency of oscillation of approximately 3.2 rad/s.

IV.1 PSS contribution The Bode diagrams of Figure 6 show that the change in tie-line impedance produces significant change in the frequency response characteristics. The PSS designed for the nominal condition no longer provides the required phase compensation, and excessive phase shift results for oscillation frequencies of 8 rad/s and less. This swings the power vector into the second quadrant in the A6-A~ plane so that a negative synchronizing power is now produced as demonstrated in Figure 4b.

Observation of the gain plot also reveals that the increase in impedance greatly reduces the gain of APe/A Vt_re f.

(1 + T2S) T,,,s [ Aw ( l + r , s ) ~ - ~ ( l + T ~ , s ) 7-

Figure 2. Block diagram of a conventional PSS control loop with speed input signal ~-1 = 0.0595, T2 = 0.3, Tw = lOsec, Ks = 10

Performance improvement of a conventional PSS." M. Saidy and F. M. Hughes 31 5

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-10~10_ 1 ~ ; , ; ; ; ; J 10 o

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Figure 3. Frequency response characteristics of APe/AVt_re t ( . . . . . ), AVt_ref/Aw (- - - --) and APe/Acv ( strong tie-line

The combined effect of the changes in phase and gain result in a marked reduction in the damping power provided by the PSS.

IV.2 Generator contribution The increase in system impedance causes an even more dramatic change in the generator natural damping char- acteristics. Whilst it leads to only a minor reduction in the magnitude of the frequency response plot APed~A6 a s

shown in Figure 7a, the q-axis response APcq/A6 almost disappears entirely as demonstrated in Figure 7b. This is due to the increase in load angle to almost 90 ° , making V~q o very small. In terms of the transfer function model, this results in the values of coefficients K4q and KEO becoming almost zero.

A~ Aw

(a)

A6

(b)

Figure 4. Direction of power component contributed by PSS (a) strong tie-line, (b) weak tie-line

) for

Consequently, for the weak tie-line condition, the natural damping of the generator is drastically reduced.

V. Augmented PSS It can be seen from the previous assessment that the

0 . 5

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Figure 5. Frequency response characteristics AP e d/A6(a) and AP e q/A6 (b) for strong tie-line

of

316 Performance improvement of a conventional PSS. M. Saidy and F. M. Hughes

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, , i 2 i i i . . . . i . • . i i i i i , , i , , 10 0 10 ~ 10 2

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Figure 6. Frequency response characteristics of APe~AFt_re f ( . . . . . ), A Vt_ref / Aa~ ( - - - ) and Ape/ Aw ( weak tie-line

) for

change to a weak tie-line condition presents a situation where an increased damping contribution from the PSS is sorely needed. Unfortunately, it is a situation where a conventional PSS, designed for nominal conditions, can contribute little.

A significant increase in the gain of the PSS cannot be considered as a solution to the problem. Although this does have the effect of increasing the damping power, the increase in the negative synchronizing power produced is detrimental to overall system performance and stability. The reduction in total synchronizing power due to a gain increase serves to reduce further the natural frequency of oscillation. At this lower frequency, the excess phase lead increases and causes the power vector to swing further into the second quadrant so that the effective improve- ment in damping power is reduced.

Ways of augmenting a PSS will now be considered so that improved performance can be achieved under weak tie-line conditions.

V.1 Idealized control Excitation control can only influence the d-axis contri- bution to damping performance. An approach which can be adopted when aiming to improve performance is to add an additional loop which seeks to co-ordinate the contributions of the d-axis and the PSS.

The d-axis elements of the block diagram model of Figure A1.1 can be re-expressed as shown in Figure 8.

The transfer function Paux(S) of the auxiliary loop provides a path for the adjustment of the field voltage AEfd and hence AEq and APed.

For ease of analysis it will be assumed that g e x ( S ) = l/aex(S ).

This makes AEfd = A Up.

If now Paux(S) is chosen such that

Paux(S) - K4d (cT~o - T~'o)S aK 3 -(-f ~_-T~oS ) (1)

Then the total feedback AU = AUp + AUd is given by

Au_ -K4d [(I +cT os) aK3 (1 + 7~'oS ) (1 + T~d~os) j A r

- - - - g 4 d m t ~ ( 2 )

aK3 Hence, the combination gives rise to the transfer function relationship

--K4d(1 + T~ros)K2 APe°re(s) = APed/A6 = (1 + aTOneS)(1 + K3T~oS ) (3)

The Nyquist frequency response of this transfer function is shown in Figure 9, superimposed on the response for the d-axis contribution alone. It can be seen that response is dominated by the pole associated with the field winding, and closely approximates that of a single lag transfer function.

Performance improvement of a conventional PSS. M. Saidy and F. M. Hughes 317

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Figure 7. Frequency response characteristics of APe d/A~ (a) and AP e q/A~ (b) for weak tie-line

A considerable improvement is seen in the syn- chronizing power provided over the relevant frequency range. At 8rad/s, where the damping provision of the basic PSS loop is adequate, positive synchronizing power is provided by the auxiliary loop. At 3 rad/s a slight increase in the damping power is provided, along

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Figure 9. Frequency response characteristics of APe d/A~. ( a ) Without augmented control loop. (b) With augmented control loop

with positive contribution to synchronizing power. It can be seen from Figure 9 that over the frequency ranges 8-3 rad/s, the combined response of the d-axis rotor circuits and the additional control loop is essentially in phase with speed variations as aimed for in the design approach.

V.2 Proposed augmented control loop In the previously considered idealized situation the existence of the inverse of the closed loop excitation system was assumed to be attainable. Since this would be a non-proper, not to mention highly complex transfer function, this is not possible. However, a simplified approximate inverse transfer function can be derived which is appropriate for the control aims.

For higher frequencies, the attenuation introduced by the generator field between excitation variations and terminal voltage variations makes the voltage feedback signal a minor component of the error signal of the excitation control system. This being the case the closed loop response of the excitation control system is closely approximated by its open loop response.

The open loop transfer function of the excitation system with transient feedback compensation as shown

au~

~] K,,(,) [-F

7 • a(l+Td'°s)K3

l (l+aTJIos)(l+K3T~o s)

AU~

A~ AP~

a K 3 (I÷T~'s)

I P.u=(") I- V

Figure 8. Schematic diagram of combined d-axis rotor circuits and auxiliary control loop

31 8 Performance improvement of a conventional PSS." M. Saidy and F. M. Hughes

(1 + T is) ~ _ ~ AV, (1 + T;s)

Figure 1 0. Block diagram of the auxiliary control loop with

in Figure AI.1, has the approximate closed loop inverse transfer function given in equation (4)

(1 + Tis ) (4) K a ( 1 + Tfs)

where Ti = Ka * Kf + Ta + Tf. In addition, to avoid any interference with the steady

state regulation characteristics of the AVR, a wash-out term also needs to be employed. The wash-out term of the basic power system stabilizer loop can be used for this.

The proposed additional loop would then have the transfer function:

Faux(S ) = K4d • (cTfao - T~'o) * 27rfo • _ _ ( 1 + ris ) aK 3 K a (1 + Tfs)

1 Tws , , , / x ~ (5)

(1 + T~i'oS ) (1 + Tws ) where sA6 is replaced by 27rfoAw.

The transfer function Faux(S) can be represented by the block diagram shown in Figure 10, in which Ks = Kaa/aK3 presents a term, which changes with the system operating condition and Kc = (cT~o- T~'o)* 27rfo/Ka is constant, as it depends only on the generator parameters and the AVR gain Ka;

1/(1 + T~'oS ) low-pass filter;

Tws/(1 + Tws) washout term.

With this scheme, at an oscillation frequency of 8 rad/s, the approximated inverse transfer function Kex (s) closely approximates that of the closed loop excitation system.

Hence, as in the idealized situation, the added loop has little influence on damping at this frequency, but does provide a positive contribution to synchronizing power.

As oscillation frequency reduces, the attenuation of the excitation control loop due to the generator field winding becomes less, and consequently the terminal voltage feedback becomes much more influential. At very low frequencies, the high gain forward loop path leads to the closed loop transfer A E q / A Vt_re f approximating to l / K 6.

At a frequency of the order of 3 rad/s the approximated inverse transfer function Kex(S) provides phase lead compensation which is in excess of the phase lag of the actual closed loop excitation system. Consequently, phase lead in excess of the idealized situation is provided. This, however, is not detrimental to performance, as the power contribution of the added loop becomes more closely aligned in phase with speed variations and serves to increase the system damping.

This being the case the gain employed no longer needs to be restricted to the exact value formulated for the idealized situation, and can be increased to provide the desired level of damping. Gain increase in the additional loop is not detrimental to performance, as it serves to increase the synchronizing power at the oscillation fre- quency associated with the strong tie-line condition, and serves to increase damping at the oscillation frequency associated with the weak tie-line situation. Hence, the augmented controller is beneficial to performance for both strong and weak tie-line conditions and therefore

(1 + T~'oS) (1 + Tws)

speed input signal

presents the capability of providing improved power system stabilizer operation over a wider range of condi- tions than the conventional scheme.

The responses of the d-axis power contributions due to generator windings and the additional loop are shown in Figures 1 la and 1 lb, for the cases of strong and weak tie- lines respectively. A gain of Ks = 10 was employed for these responses.

Vl. Performance assessment in the t ime domain

VI.1 Test system A performance comparison of the conventional PSS control scheme with and without the auxiliary control loop has been carried out on the test system shown

(a)

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Performance improvement of a conventional PSS: M. Saidy and F. M. Hughes

schematically in Figure A2.1 of Appendix 2. It comprises a generator connected to a large network via a step-up transformer and reactive tie lines with intermediate load bus. The large power network is represented as an infinite busbar connection. Load disturbances can be applied by switching in specified load admittances at the centre bus.

In the simulation studies the generator was represented via the continuous nonlinear model defined in Appendix A2.2 s. It was assumed that the generator was of the salient pole type, typical[ of those used in hydro stations, having the parameters given in Appendix A2.1.

A static excitation system with fast response and exciter time constant T~x -- 0.03 s is employed as shown in Figure A2.2 of Appendix 29'1°.

The solution of the system differential equations in the simulation studies is performed using a Runge Kutta 4th order method with an integration time step of 5 ms.

The performance capabilities of the improved conven- tional PSS is assessed for a load disturbance at the centre busbar.

The AVR parameters used were designed to provide well damped terminal voltage response under open circuit conditions as per standard industrial practice 11. The conventional PSS parameters employed are: T 2 = 0.3s, ~-1 = 0.0595s, which were obtained during the design procedure using the linearized system model under full generator load condition and a strong tie- line connection with the rest of the power system. In both the main PSS and the auxiliary loop, washout time constants of T w = 10..~ and gain of K s = 10 were employed.

VI.2 Load disturbance performance The generator operating condition considered is with full load, having lagging power factor P6 = 0.8p.u., QG = 0.6p.u. The transformer and tie-line reactances were Xtr + XI1 = 0.1 p . u . , 212 = 0.1 p . u . The load distur- bance was simulated by an admittance Yd whose value was equal to (0.2-0.12j)p.u. This can be considered as the operating condition with a relatively strong connec- tion between the generator and the grid system.

The responses of Figure 12 show that the performance of the conventional PSS without and with the auxiliary control loop is good with an increase in the frequency of power oscillations due to the synchronizing power com- ponent contributed by the auxiliary loop as shown in Figure 12c. Whereas, both cases display an insignificant overshoot in the terminal voltage response as depicted in Figure 12a.

VI.3 Performance with weak transmission line When the transmission line between the generator and the rest of the system is long and consequently the transmis- sion line impedance is large, the stability margins of operation are reduced.

The responses shown in Figure 13 are for the condition where the transmission line impedances are:

Xtr + Xu = 0.5 p.u. and X12 ~- 0.5 p.u. (1)

The generator initial output is again PG = 0.8p.u. and Q6 = 0.6 p.u. and a disturbance is applied by switch- ing in the admittance Yd = 0.2 -- 0.12j p.u. at the centre busbar.

The conventional PSS designed to provide well- damped power oscillations under generator full load and strong tie-line conditions, exhibits oscillatory,

321

poorly damped voltage and power responses as shown in Figures 13a and 13b by the dashed lines when a load disturbance is applied.

The high impedance line case results in greatly reduced synchronizing torque for the generator and consequently a reduction in the natural frequency of rotor angle oscillation. At this lower frequency, the conventional PSS provides excessive phase lead and consequently a significant reduction in its damping power contribution, and increased negative synchronizing power.

The auxiliary loop frequency response is such that it still provides significant damping power contribution at the lower oscillation frequency and hence contributes significantly to the system damping as the responses of Figure 13 show.

VII. Conclusion A conventional power system stabilizer improves system damping by designing it to provide a generator power component in phase with speed at a particular oscillation frequency. Consequently if during operation the network configuration changes, such that the effective generator interconnection becomes weak, then the resulting reduc- tion in natural frequency of oscillation incurs a signifi- cantly reduced performance capability. With the addition of the proposed auxiliary loop, which has a frequency response characteristic that maintains good damping power contribution despite significant reduction in the natural frequency, control capability is significantly extended.

The augmented control scheme proposed, which consists simply of an additional loop in parallel with conventional stabilizer, greatly enhances control per- formance under weak tie-line conditions.

The benefits brought by the auxiliary control loop can be summarized as follows.

• It contributes positive damping power at the lower frequency of the inter-area mode of oscillation without adversely affecting the performance at the higher local mode frequencies.

• It contributes a positive synchronizing power compo- nent to enhance the electromagnetic coupling of the generator with the grid. This is beneficial, especially, in the case of weak tie-line connection.

• The amount of damping power contribution to the inter-area mode of oscillation can be varied by the auxiliary control loop gain based on the system damp- ing requirements.

• The auxiliary control loop is not expensive compared with other means of system stability augmentation such as FACTS.

• It can easily be implemented. The conventional PSS can be modified to incorporate it and it can be turned ON/OFF whenever it is needed.

In general, the improved conventional PSS control scheme can reduce the constraints imposed on the power transfer capabilities of the overhead transmission lines due to dynamic instability reasons. It can therefore push the dynamic limits of the transmission lines closer to the boundaries dictated by the thermal constraints and economical considerations.

IX. Acknowledgements Dr Saidy, gratefully, acknowledges the financial

322 Performance improvement of a conventional PSS: M. Saidy and F. M. Hughes

support provided by the H A R I R I Foundation during his postgraduate studies at the Department of Electrical Engineering and Electronics, University of Manchester, Institute of Science and Technology.

He also wishes to thank the Committee of Vice- Chancellors and Principals of the Universities of the United Kingdom for the Overseas Research Students Award granted to him.

X. References 1 Roger, G J and Kundur, P 'Small signal stability of power

systems' in Eigenanalysis and frequency domain methods for system dynamic performance IEEE Technical Report 90TH0292-3-PWR (1989)

2 Lee, D C and Kundur, P 'Advanced excitation system con- trols for power system stability enhancement' CIGRE paper 38-01 (August 1986)

3 Larsen, E V and Swann, D A 'Applying power system stabiliser, Parts l, 2, 3,' IEEE Trans. PAS, Vol PAS-100 (June 1981)

4 IEEE Tutorial course on power system stabilisation via excitation control, 81EH0 175-0 PWR

5 Grondin, R, Kamwa, I, Soulieres, L, Potviu, J and Cham- pagne, R 'An approach to PSS design for transient stability improvement through supplementary damping of the common low frequency' IEEE Trans. Power Syst Vol 8 No 3 (1993) 954-963

6 Demello, F P, Czuba, J S, Rushe, P A and Willis, J 'Devel- opments in application of stabilising measures through excitation control' CIGRE, Paris, Paper 38-05 (1986)

7 Saidy, M and Hughes, F M 'A transfer function block diagram form of a generator with damper windings' Paper submitted to IEE Proc. C Vol 141 No 6 (November 1994) 599-608

8 Hammous, T J and Winning, D J 'Comparisons of synchro- nous machine models in the study of the transient behavior of electrical power systems' Proc. IEEE, Vol 118 No 10 (1971)

9 IEEE Committee report 'Computer representation of excita- tion systems' IEEE Trans. Power Appar. Syst. Vol PAS-87 (1968) 1460-1464

10 IEEE Committee report 'Excitation system models for power system stability studies' IEEE Trans. Power Appar. Syst. Vol PAS-10 No 2 (1981) 494 509

11 Erinmez, I A 'Generator excitation system performance, requirements arising from grid system considerations', Colloquium organized by IEE, 28 January 1992

Appendix 1

A1.1 Parameters of Figure A 1.1

(Xq + x,) Tq ~=bT~ro where b - - ( X q + X l )

2 2 V~cdo V~cqo

K~ (X, + Xk') ~/do V~qo + (X, + X+')

V~cdo Vocqo K2 Xl + X,d,. K2d XI + X'~'

(x'~ + x~) (x'd' + x~) K 3 -- (X d -[- Xl ) , a -- (X~ -[- X l ) '

Iqo Vocdo

( x ~ - x ~ ' ) C - -

( X d - X'~')

,~ ( X d - X~') (Xq - X'~') K4d =V~d° -~d + XI) ~ K4q = V°°q° ~q ~- Xl)

K s = Vto [. (X ,+ X;') ---~to (Xl+X~')

K6- Vt.o Xl K6d = Vtdo Xl Vto (XI-{-X~') ' Vto ( X I + X ~ ' )

Appendix 2

A2.1 Salient pole machine parameters

Direct axis parameters Synchronous reactance Xd = 1.445 p.u. Transient reactance X~ = 0.316 p.u. Subtransient reactance X~ = 0.179 p.u. Transient time constant T~o = 5.26 s Subtransient time constant T~'o = 0.028 s

Quadrature axis parameters Synchronous reactance Xq ---- 0.959 p.u. Subtransient reactance Xq ' -- 0.162 p.u. Subtransient time constant Tqto = 0.159 s Stator resistance R a = 0 Inertia constant H = 4.27 s Nominal frequencyfo = 50 Hz

A2.2 System differential and algebraic equations

dE'q 1 d t - T~o [Efd -- (Xd -- X~d)Id -- Eq] (A2.1)

" _dffq (A2.2) dF.q _ 1 [F~'. - (X'~ - X'd'lZd -- Eg] -~ d t dt Z~o

Figure A1.1. Block diagram transfer function model of generator including damper windings

Performance improvement of a conventional PSS. M. Saidy and F. M. Hughes 323

v, X,, X.

\

/ /

A2.1. Genera to r c o n n e c t e d Figure

I I

to an inf ini te

dE~ 1 dt - T" [(Xq - Xq)Iq - E~]

qo

dw _ 7rfo [Pm - Po] dt H d~ d t = ~ - 27rf°

V t : (Vd 2 -]- v 2 ) l / 2

Vd = JE~-F Xqt, Iq

V . = E ' ~ - X ' ~ ' , I d

e~ = Vd * Id + Vq * Iq

A2.3 AVR type l s 9'I0

- I

K. (i + sTo)

s K f ( 1 + s T f )

B i d . , ~ = 5.5 V~

Figure A2.2. AVR type 1 S: Ta = 0.03 s; Ka = 198; Tf = 0.371 s; Kf = 0.01

(A2.3)

(A2.4)

(A2.5)

(A2.6)

(A2.7)

(A2.8)

(A2.9)