55

Co-ordination of thyristor-controlled VAr compensator and auxiliary controller of excitation

system for small signal stability enhancement

Coordination des compensateurs statiques à thyristors et des commandes auxiliaires des systèmes d'excitation en vue d'améliorer la

stabilité en présence de faibles fluctuations de signal

By R.M. HamOUda, Department of Electrical Engineering, University of Windsor, Windsor, Ontario, M.R. IrâVanï, Department of Electrical Engineering, University of Toronto, Toronto, Ontario and R. HaCKam, Department of Electrical Engineering,

University of Windsor, Windsor, Ontario.

Technical advantages of co-ordinating thyristor-controlled static VAr compensators (SVC) and power system stabilizers (PSS) for stability enhancement during small signal dynamics are presented. The technical feasibility of the co-ordinated scheme for simultaneous damping of inertial and torsional oscillatory modes in a series-capacitor-compensated system is demonstrated. The investigations show that the co-ordinated scheme not only increases the overall system damping, but also prevents undesirable interactions between oscillatory modes. New control schemes for SVC and PSS, based on multi-input control concepts, are given. The studies are performed on system-1 of the second IEEE benchmark model, using an eigenvalue analysis method. The analytical results are verified by digital computer simulation studies, using the Bonneville Power Administration's Electromagnetic Transients Program (EMTP).

La communication présente les avantages techniques de la coordination des compensateurs statiques à thyristors et des stabilisateurs de puissance des systèmes d'excitation des groupes turbo-générateurs, en vue d'améliorer la stabilité en présence de faibles fluctuations de signal. Les résultats ont établi la faisabilité technique de cette méthode coordonnée pour l'amortissement simultané des modes oscillatoires inertiels et de torsion dans un système à compensation par condensateurs série. Les recherches ont en outre montré que la méthode coordonnée permet non seulement d'accroître l'amortissement global du système, mais aussi de prévenir les interactions nuisibles entre les modes oscillatoires. De nouvelles méthodes de commande de compensateurs statiques et de stabilisateurs de puissance fondées sur une approche de commande à entrées multiples, sont présentées. Les études ont été réalisées sur le système 1 du second modèle de référence de ΓΙΕΕΕ, à l'aide d'une méthode d'analyse des valeurs propres. Les résultats analytiques ont été vérifiés par des études de simulation numérique sur ordinateur, à l'aide du programme sur les transitoires électromagnétiques BPA.

Introduction

As a result of small signal disturbances in series-capacitor-compensated power systems, each turbine-generator (T-G) set experiences oscillatory modes in the subsynchronous frequency range (0.5-55 Hz). The number of oscillatory modes and their frequencies depend upon the mechanical structure of the T-G set and the system parameters. The oscillatory mode with the lowest frequency (0.5-2.5 Hz) represents the T-G set oscillations versus the electrical system and is referred to as the inertial mode (mode zero, hunting mode)' of oscillation. The inertial mode is the well-known oscillatory mode in power system dynamics. The other oscillatory modes (6-55 Hz) represent the relative oscillations of the masses of each T-G set versus each other, and are referred to as the torsional oscillatory modes. 1

An SVC is conventionally used in a power system for voltage regulation 2 a n d / o r damping the inertial oscillatory modes . 3 SVC has also been proposed for damping torsional oscillations in series-capacitor-compensated AC networks. 4 " 6 Modulat ion of a generator excitation voltage through auxiliary control loop(s) of the excitation system has been extensively utilized for mitigation of low-frequency oscillations in power systems. 7 The auxiliary control loops of the excitation system are referred to as power system stabilizers (PSS). PSS can also be used as a supplementary device for damping torsional oscillatory modes . 8

This paper presents a novel concept for simultaneous damping of inertial and torsional modes by co-ordinating PSS and auxiliary controllers of SVC. In the proposed scheme, the SVC maintains its original voltage stabilization function in the system while providing damping for the oscillatory modes. New designs using multi-input signals for the PSS and the SVC auxiliary controller are given. Modal speeds are used as the feedback signals.

This investigation reveals that as compared with cases where either the SVC or the PSS alone is used to damp the system oscillations, the co-ordinated SVC-PSS provides significantly higher damping for the oscillatory modes. Another merit of the proposed scheme is that by utilizing modal speeds as the feedback signals, the undesirable interaction of the oscillatory modes is prevented. The study is performed on system-1 of the second IEEE benchmark model, using an eigenvalue analysis technique. The analytical results are verified by a digital computer simulation study, using the E M T P .

Studied system

Power system The system under study is system-1 of the second IEEE

benchmark model 9 , which is composed of a 600-MVA steam T-G set connected to an infinite bus through a 500-kV series-capaci­tor-compensated line (see Figure 1). The shaft system of the T-G set

Can. J. Elect. & Comp. Eng., Vol. 14 No. 2, 1989

56 C A N . J. ELECT. & COMP. ENG., VOL. 14 NO. 2, 1989

Figure 1: One-line diagram of the system studied.

Figure 2a: Block diagram of the generator excitation system.

IT S T P I I [••«le!»-* ι . I

1 I . S T P E J

I 1 * S T P E | *

" 1

Figure 2b: Block diagram of power system stabilizer (PSS).

has three torsional modes at frequencies 24.65 Hz, 32.39 Hz and 51.10 Hz. The first two torsional modes become unstable as a result of interaction between the T-G set and the compensated line. The system data is given in Reference 9.

Excitation system The IEEE type-1 excitation system (see Figure 2a) is used . 1 0 The

main control signal of the excitation system is the terminal voltage error Ve. The excitation system maintains the terminal voltage within ± 6 % of the steady-state value. The auxiliary control Vse (PSS signal) is added to the main control signal to damp the system oscillatory modes. Vse is the output signal of the multiple-channel PSS of Figure 2b. It can be expressed as

1 + ST^

+ \k, V STpl

'\ + STpl

•ι*

•111 + STpJ Δω 2 . (1)

Feedback signals Δω 0 , Δωχ and Δω 2 are the T-G modal speeds and are phase- and gain-adjusted through the multi-channel PSS to provide damping for their corresponding oscillatory modes.

Thyristor-controlled static VAr compensator (SVC) The SVC used in this study is composed of a thyristor-controlled

reactor (TCR) in parallel with a fixed capacitor (FC) connected to the generator bus (see Figure 3a). The main function of the SVC is to

assist the excitation system in maintaining the terminal voltage by using the terminal voltage error signal bVt. The SVC auxiliary control signal Vss is derived from the modal speeds through the multi-channel feedback system of Figure 3b. It can be expressed as

Li + st;6J 1 L 5 3 i + sts1u\ + sts9i

χ

Each modal speed is properly phase- and gain-adjusted to provide adequate damping for its corresponding oscillatory mode. The function of the limiter blocks in Figures 2b and 3b is to prevent the auxiliary controllers from affecting the system operation when the amplitudes of the auxiliary signals are larger than the prespecified values (large disturbances).

Analysis methods

Eigenvalue method A n eigenvalue method is used to examine the stability of the

oscillatory modes and to co-ordinate the SVC and the PSS. For eigenvalue studies, the equations of the system components are developed in the generator d-q frame. Park's equations are used to represent the generator electrical system with two damper windings on the q-axis and one damper winding and one field circuit on the d-axis, as shown in Equation (3):

{V} = { X } + [ * ] { / } + ^ [ β ] { λ > (3)

where

{V}

{ λ } :

[R] :

(0 =

= { E / ^ O O ^ O } 7

diag[/?y, — Ra, Ru, Rg, — Ra> Rkq]

Figure 3a: One-line diagram of the SVC unit.

Figure 3b: Block diagram of the SVC auxiliary controller.

H A M O U D A / I R A V A N I / HACKAM: SMALL SIGNAL STABILITY E N H A N C E M E N T 57

igure 4: Equivalent model of the SVC for eigenvalue studies.

ÔG is the generator speed, and

[ β ] is given in Appendix A.

The term {λ} is related to {/} by Equation (4):

{λ} = [L]{i) (4)

vhere

[L] = diag[[LJ, [Lq]].

rhe definitions of [1^] and [Lq] are also given in Appendix A.

A spring-mass model is used to represent the mechanical system )f the T-G set:

[Η](β'] + [D]{Ô) + [Κ]{θ] = {Tm} - {Te} (5)

where

[H], [D] and [K] are matrices of moments of inertias, damping and stiffness respectively, as given in Reference 11 :

{0} = [BHPeLPeGeEXC]T

{Tm} = [T%PT?P00)T

{Te} = [ 0 0 7 $ ) ] r .

The generator air-gap torque TG couples Equations (3) and (5) and is given by Equation (6):

η = -3{i}T[L)T[Q]T(i}

where η is the number of pole pairs.

(6)

Based on the block diagrams in Figures 2a and 2b, the dynamics of the excitation system including the PSS are given by Equation (7):

SAEf = KAVse - AES - AVt + AEr - —AEf

1 Kf SAEo = -AEc + — SAEf.

Tf Tf (7)

The step-up transformer and the transmission lines are represented by their equivalent lumped parameters, and their mathematical models in the generator d-q frame are given by Equation (8):

[A,J{MC) = [Λ,,ΚΔχ,} + [5,,]{Δ«,}

where

(8)

[i4,J, [Ati] and [Bt{] are given in Appendix B, and

{Axc} = [Μίώ Μίφ Mdh Mqh Aucdi Aucd]T

{Aut} = [AVtd9AVtq,AeG9Ao>G]T.

For the eigenvalue studies, the SVC is modelled as a controlled voltage source behind a fixed reactor 3 (see Figure 4). In Figure 4 the parameters are

Ta = 0.02s

Ka = 1.0 (9)

F(a) = [(2a - sin(2a))/*r - 1]

where a is the thyristor firing angle.

The SVC auxiliary control signal Vss (see Figure 3b) is added to the main control signal in the equivalent model of Figure 4. The mathematical representation of the SVC and its main and auxiliary controllers is given by Equation (10):

(Axr) = lAR){Axr} + [BR]{Aur] (10)

where

[AR] and [BR] are given in Appendix C,

and

{ Δ ^ } = [AVtd9 AVtqi Aifo Aiqn Aa]T

{Aur} = [Mcd9 Aicq9 Αωθ9 AVSS, AEr]T.

Modal speeds, Δωο, Δω! and Δω 2 , which are used as the auxiliary control signals for the SVC and the PSS, are calculated from deviations of the speeds of the T-G masses using Equation (11):

{Avm} = [Ν]'ι{Αω) (11)

where

{Aum} = [ΔωοΔωιΔω 2 Δω 3 ] Γ

{Δω} = [ Δ ω ^ ρ Δ ω ^ ω ^ ω ^ ] 7

[Ν] is a matrix of eigenvectors of [H]~1 [K].

For the system under consideration, is given in Appendix D.

Equations (3), (5), (7), (8) and (10), which represent the overall dynamics of the system, are arranged into* a set of first-order ordinary differential equations, linearized about an operating point and rearranged in the general state-space form of Equation (12):

< Δ * } = [A]{AX} + [B][AEr\ (12)

where [Λ] is the system matrix. The general structure of [A] is given in Appendix E.

The eigenvalues of [A ] reveal the system behaviour during small signal disturbances. Eigenvalues with positive real parts corre­spond to the oscillatory modes which their amplitudes grow with time, and represent unstable modes. Co-ordination between PSS and the auxiliary controllers of the SVC is achieved by deter­mination of values of Tp9 Kp9 Ts and Ks (see Figures 2b and 3), such that the real parts of the eigenvalues of [A ] corresponding to the system oscillatory modes have the desired values. The following aspects should be considered in the co-ordination process of the PSS and the auxiliary controller of the SVC:

1. The parameters of the co-ordinated SVC-PSS should minimize any undesirable interaction between the system oscillatory

58 C A N . J. ELECT. & C O M P . ENG. , VOL. 14 N O . 2, 1989

modes. This matter is a major consideration for a system in which the frequencies of the oscillatory modes are close.

2. The parameters of the co-ordinated SVC-PSS must not result in an undesirable system response during contingency periods where either the SVC or the PSS is out of service.

Based on the above-mentioned considerations, a set of values for the PSS and the SVC controllers is determined as given in Appen­dix F .

Digital time simulation method Although eigenvalue analysis provides an accurate prediction

regarding the overall stability of the system, it cannot provide an accurate picture of the system time response. Therefore, to verify the eigenvalue analysis results and to obtain an accurate time response of the system, a t ime-domain analysis based on nu­merical solution of the system nonlinear differential equations, is carried out. The E M T P is used for the digital t ime simulation s tud ies . 1 1 ' 1 3 " 1 5

For the simulation studies, the generator electrical system is represented in the Park's d-q frame with two damper windings on the q-axis and one damper winding and one field winding on the d-axis. The rotating system of the turbine-generator is represented by a spring-mass system. 1 1 The step-up transformer is modelled as a three-phase transformer including the magnetization branches. The saturation characteristics of the generator and the step-up transformer are included in the system model. The transmission lines are represented by their lumped parameters. The SVC is represented bynhe detailed models of the thyristor valves a n d their snubber circuits. The SVC controller, the excitation system and the PSS are modelled in detail according to Figures 2a, 2b, 3a and 3b, using the T A C S 1 6 program of the E M T P

Study results

The results from the eigenvalue analysis and the simulation studies are obtained on the basis of the following initial conditions and assumptions:

Table 1 Studied cases

C O U N T E R M E A S U R E CASE #

WÊBÊÊtÊÊÊÊÊÊÊM C O U N T E R M E A S U R E CASE # Δω, Δ ω 2

PSS

1

PSS 2 *

PSS 3 *

PSS

4 * * *

llllllllillli

SVC

HHI * llllllllillli

SVC 6

llllllllillli

SVC 7 *

llllllllillli

SVC

8 * * *

C O - O R D I N A T E D SVC-PSS

9 *

C O - O R D I N A T E D SVC-PSS

10 * C O - O R D I N A T E D

SVC-PSS 11

C O - O R D I N A T E D SVC-PSS

12 * *

1. The generator delivers 0.90 p.u. power to the system. 2. The steady-state voltages at the sending- and receiving-end

buses are 1.05 p.u. and 1.0 p.u., respectively. 3. In all cases, the SVC voltage regulator (main controller) is in

service. T o consider the worst condition regarding system oscillations, the mechanical damping of the T-G shaft system is assumed to be zero. The dynamics of the governor system are neglected and the input mechanical power to the turbines is assumed to be constant.

4.

5.

Table 1 presents the cases studied and their corresponding control signals where either the SVC or the PSS or the co-ordinated SVC-PSS is used to damp the system oscillations.

N o countermeasure in service Figure 5 presents the real par t of the eigenvalues corresponding

to the first three oscillatory modes when the compensation level is changed from 0% to 100%. The PSS and SVC auxiliary controller are disabled. Figure 5 indicates that the inertial mode (mode zero), curve a, is stable, and its stability is further enhanced by increasing the compensat ion level. Curves b and c show that the first and second torsional modes are unstable for wide ranges of compensa­tion levels.

The time responses in Figure 6 were obtained from the EMTP and show the changes in rotor speed deviation (àcoG) and shaft

Figure 5: Real part of the eigenvalues corresponding to the shaft oscillatory modes (no countermeasure in service).

Figure 6: The system time response as a result of a small signal disturbance (no countermeasure in service).

H A M O U D A / I R A V A N I / H A C K A M : S M A L L S I G N A L STABILITY E N H A N C E M E N T 59

torque in the segment LP-G. The PSS and the SVC auxiliary controller are out of service. The disturbance which excites the oscillations is a sudden change in the compensation level (Xc/X2\ from 35% to 52%. As Figure 6 indicates, the disturbance excites the inertial mode and the first torsional mode. The inertial mode is stable and is positively damped after a few oscillations; however, the first torsional mode is unstable and its ampli tude grows. The results presented in Figures 5 and 6 agree favourably.

Stability enhancement of the inertial mode Figure 7 shows the real part of the eigenvalue corresponding to

the inertial mode (mode zero) when Δ ω 0 alone is used as the feedback signal for the SVC, the PSS and the co-ordinated SVC-PSS

0 20 40 60 80 100 JT o.o Γ I 1 1 1 1 1

3

OA

Figure 7: Real part of the eigenvalues corresponding to the inertial mode (mode zero): (a) No countermeasure in service, (b) SVC (Δω0) in service, (c) PSS (Δω0) in service, (d) Co-ordinated SVC-PSS (Δω0) in service.

to d a m p the inertial mode (study cases 1, 5 and 9, respectively). Al though mode zero is stable even with no countermeasure in service (curve a), curves b, c and d indicate that the SVC, the PSS and the co-ordinated SVC-PSS can noticeably increase the damping of mode zero and further enhance the system stability. Utilization of the modal speed Δ ω 0 ensures that there will be no interaction between the oscillatory modes. Table 2 compares the system eigenvalues corresponding to the oscillatory modes (study cases 1,5 and 9) with the case where no countermeasure is in service. Table 2 shows that the eigenvalues of the torsional modes are not affected when mode zero is damped by the SVC, PSS and the co-ordinated SVC-PSS.

0 5

a / \

0 0

- 1 0 c c υ

LU ' d \ \ <+» Ο

a) no countermeasure \\ S _ 2 0

a)

c? b) PSS ( Δ ω χ ) \ \ / S «- c) SVC ( Δ ω ^ \ V / / rt

SVC ( Δ ω ^ \ V / / υ OC

d) SVC ( Δ ω ^ + PSS ( Δ ω ρ \ /

- 3 0 ι ι ι ι » 1 0 20 40 χ 60 80 100

*2

Figure 8: Real part of the eigenvalues corresponding to thefirst torsional mode: (a) No countermeasure in service, (b) PSS (Δω;) in service, (c) SVC (Ao3}) in service, (d) Co-ordinated SVC-PSS (àuj) in service.

Table 2 Effect of using Δω0 as the stabilizing control signal to damp the inertial mode (xc/xi = 52%)

Case

Modes N o Countermeasure Case # 1 Case # 5 Case # 9

TORSIONAL M O D E S

(3) - 0 . 0 0 6 1 ± 7321.19 - 0 . 0 0 6 1 ± 7 3 2 1 . 1 9 - 0 . 0 0 6 1 ± 7'32L19 - 0 . 0 0 6 2 ± 7321.19

TORSIONAL M O D E S (2) - 0 . 0 1 3 9 zt 7203.57 - 0 . 0 1 4 0 ± 7203.57 - 0 . 0 1 4 0 ± 7203.57 - 0 . 0 1 4 1 ± 7203.57 TORSIONAL M O D E S

(1) +0 .4780 ± : j'155.25 +0.4898 ± 7155.26 +0 .4824 zt 7155.26 + 0.4944 ± 7155.26

Inertial (0) - 0 . 4 5 7 2 ± y'9.935 - 2 . 0 2 5 6 ± : /10.39 - 1 . 0 2 6 2 dt 7*9.9011 - 2 . 6 4 1 3 ± 7*10.32

Table 3 Effect of using Ao)t as the stabilizing control signal to damp the first torsional mode (χ€/χ% = 52%)

^ ^ < ^ ^ Case

Modes N o Countermeasure Case # 2 Case # 6 Case # 1 0

TORSIONAL MODES

(3) - 0 . 0 0 6 1 ± /321.19 - 0 . 0 0 6 1 ± 7321.19 - 0 . 0 0 6 1 ± 7321.19 - 0 . 0 0 6 1 ± 7321.19

TORSIONAL MODES (2) - 0 . 0 1 3 9 ± 7 2 0 3 . 5 7 - 0 . 0 1 3 9 ± 7203.57 - 0 . 0 1 4 2 ± 7203.57 - 0 . 0 1 4 2 ± 7 2 0 3 . 5 7 TORSIONAL MODES

(1) +0 .4780 ± yl55.25 +0 .1421 ± 7155.25 - 2 . 1 7 5 4 ± 7155.31 - 2 . 5 9 2 0 ± 7*155.31

Inertial (0) - 0 . 4 5 7 2 ± y9.935 - 0 . 4 4 4 0 ± j 10.026 - 0 . 4 7 9 0 ± 7*9.929 - 0 . 4 6 6 6 ± 7IO.O2I

60 C A N . J. ELECT. & C O M P . ENG. , VOL. 14 N O . 2, 1989

Damping the first torsional mode Figure 8 demonstrates the damping effect of the PSS, the SVC

and the co-ordinated SVC-PSS on the first torsional mode (study cases 2 ,6 and 10) when Δω! is used as the control signal. Curve b in Figure 8 shows that the PSS has a small damping effect on the first torsional mode. Curve c confirms that the SVC can effectively damp the first mode. The damping effect of the co-ordinated SVC-PSS (Figure 8, curve d) is larger than the SVC and the PSS for the whole compensation range. Table 3 indicates that the eigenvalues of the other modes are not affected when Δω! is used as the feedback signal to damp the first torsional mode; therefore, no interaction among the oscillatory modes is excited.

Damping the second torsional mode Figure 9 presents the results of study cases 3, 7 and 11, when Δ ω 2

is used as the control signal to damp the second torsional mode. Curves b and c indicate that the PSS and SVC can provide adequate damping for the second torsional mode; however, the system damping is noticeably increased (curve d) when the co-ordinated SVC-PSS is utilized. Comparison of eigenvalues in Table 4 reveals that utilization of Δω 2 as the feedback signal does not result in modal interaction.

effectively enhance the stability of the inertial mode. Curve c indicates that the second oscillatory mode is positively damped in the whole compensation range. Curve b indicates that the first torsional mode is effectively damped for those compensation levels corresponding to its maximum instability (about 52%).

Figure 11 presents the system time response to a small signal disturbance, corresponding to study cases 4, 8 and 12. The disturbance is a result of a sudden change in the compensation level from 35% to 52%. The disturbance excites the inertial mode as well as the first torsional mode. Case 4 demonstrates that the PSS can noticeably assist in damping the inertial mode, while its damping effect on the first torsional mode is insignificant. The SVC can provide damping for both the inertial and the torsional mode (case 8). As compared with the PSS (case 4) and the SVC (case 8), the co-ordinated SVC-PSS can noticeably increase the positive damping for both oscillatory modes (case 12). The E M T P results of Figure 11 agree favourably with the eigenvalue results shown in Figure 10.

Conclusions

Damping all oscillatory modes Figure 10 presents the real parts of eigenvalues of the inertial and

the first two torsional modes when the co-ordinated SVC-PSS is used to damp the three oscillatory modes (study case 12). The feedback signals for the co-ordinated SVC-PSS are Δω 0 , Δω! and Δω 2 . Curve a in Figure 10 shows that the co-ordinated SVC-PSS can

A practical approach for damping inertial and torsional oscillations of power systems, based on co-ordination of PSS and the auxiliary controller of SVC, was presented. The technical feasibility of the proposed approach was demonstrated by eigenvalue analysis and digital computer simulation methods. New designs for PSS and the SVC auxiliary controller based on

0.00

a) no c o u n t e r m e a s u r e

b) PSS ( Δ ω 2 )

c) SVC ( Δ ω 2 )

d) SVC ( Δ ω 2 ) • PSS ( Δ ω 2 )

1 . 00

20 60 80 100

Figure 9: Real part of the eigenvalues corresponding to the second torsional mode: (a) No countermeasure in service, (b) PSS (Δω2) in service, (c) SVC (Δω2) in service, (d) Co-ordinated SVC-PSS (Δω2) in service.

0.0

1.0

3.0

60 L_

100

~2

a ) mode (0) b ) n o d e (1) c ) mode (2)

Figure 10: Real part of the eigenvalues corresponding to the inertial and the torsional modes. Co-ordinated SVC-PSS (Δω0 + Δω ; + Δω2) is in service.

Table 4 Effect of using Δω2 as the stabilizing control signal to damp the second torsional mode (xc/xt - 52%)

^s***n"-*n^' Case

Modes N o Countermeasure Case # 3 Case # 7 Case # 1 1

T O R S I O N A L M O D E S

(3) - 0 . 0 0 6 1 ± 7321.19 - 0 . 0 0 6 1 ± 7*321.19 - 0 . 0 0 6 1 ± 7321.19 - 0 . 0 0 6 1 ± . 7*321.19

T O R S I O N A L M O D E S m - 0 . 0 1 3 9 ± 7*203.57 - 0 . 0 5 3 8 ± 7*203.59 - 0 . 8 6 8 0 ± : 7*203.57 - 0 . 9 0 7 9 ± 7*203.58 T O R S I O N A L M O D E S

(1) +0 .4780 d= 7155.25 +0 .4783 ± 7155.25 +0 .4869 ±7*155.25 +0 .4872 ± : 7155.25

Inertial (0) - 0 . 4 5 7 2 ±J93$5 - 0 . 4 5 4 0 ± : 7*9.956 - 0 . 4 6 9 0 db 7*9.935 ||j^l§||§^plll

</> 0 D 3 « > C CJ M UJ ο U rt α

« ce

ι

V) 4) 3 cd > G Ο oo

•H ω ο •J u ce α. ce 4» ο.

H A M O U D A / I R A V A N I / H A C K A M : SMALL S I G N A L STABILITY E N H A N C E M E N T 61

No Countermeasure

SECONDS

a

Case 4

0 0 C . 6 0 1 . 2 0 1 . 8 0 2 . 4 0 3 . 0 0 SECONDS

? -<3 Ο

H" <=θ1

ο: e V / » : 2ο\*/"ι : β ο ' ^ ' ί ^ ^ 3 ' : 0 0 SEC0N0S

Case 8

0 0 0 . 6 0 1 . 2 0 Ι . Θ 0 2 . 4 0 3 . 0 0 SECONDS

20 1 . 8 0 2 . 4 0 3 . 0 0 SECONDS

Case 12

00 0 . 6 0 1 . 2 0 1 . 8 0 2 . 4 0 3 . 0 0 SECONDS

Appendix Β

Figure 11: Damping effects of the PSS, SVC and the co-ordinated SVC-PSS on the oscillatory modes. Case 4: PSS (Δω0 + Δω; + Δω2) /.s in service. Case 8: S VC (Δω0 + Δω; + Δω2) is in service. Case 12: Co-ordinated SVC-PSS (Δω0 + Δω7 + Δω2) is in service.

multi-input control systems were given. The investigation results show that:

1. As compared with the cases where the SVC or the PSS alone are used to damp system oscillations, the co-ordinated SVC-PSS is significantly more effective in counteracting power system oscillations.

2. PSS can assist in damping the inertial mode while its damping effect on the torsional modes is insignificant. SVC can provide damping for the inertial mode as well as the torsional modes.

3. Utilization of the modal speeds as the feedback signals in the co-ordinated SVC-PSS scheme prevents modal interactions.

4. In the co-ordinated SVC-PSS, the SVC maintains its original function, namely, voltage stabilization, while it provides damping for the system oscillatory modes.

Appendix A

[Q] [0] [02]

[ f i l ] [0] Γ 1 '

0 0 0 0 , [β ΐ ] = - IQ2]T = 0 - 1 0 0

0 0 0 0

- ί ! " ο

_ ί ι " ο

_ * 2

χ\2

1

1

3 3 =

Β,,

where

R t s Xts * i - Χχ

x t s R t s x\ * 1

R2 - x 2 ~ R\2 x\2 1

x2 R2 ~ X\2 ~ R\2 1

ω0 Xc - u0 xc <*o

ω0 Xc ~ "ο Xc - <*0

- 1 u0 cos eG

x t s . x \ . ltqo lq\§

<*o °>o

- 1 - U0 sin eG

x t s · x \ -hdo "· *</10

ω ο ω ο

_ Χ* . . *1 , lq20 + lq\0

°>o <*0

x t \ . , x2 .

ld\Q "l ld2Q ^o ^o

UCqo

~~ Ucdo

Xts — Xt + Xs, Rt + Rs

M2 X\ ~f" ^2» -^12 — -^1 -^2

Appendix C

îo 0

- < o 0

« o 0 ~ c o ) XR

«oO _ Q )

·*/? - ω 0

<*0CTVtq0

XR

TaVt0

1

Τ

_ Lf ~Laf Lfkd Lg ~Lag Lgkq lLd\ - Lqf ~ L d Lakd > \Lq] = Lag ~Lq L a k q

Lfkd ~Lakd L k d ^ [_Lgkq "Lakq L k q

62 C A N . J. ELECT. & C O M P . ENG. , VOL. 14 N O . 2, 1989

where

C0(x) = (2a - sin(la))/7r - 1.0

CT(x) = (2 - 2 cay(2a)) /7T

^ 0

V o

~'dr0

Ka

Appendix D

0.09279 0.5778

0.31102 0.49796

-0.43026 0.75254

-0.00119 0.01626

0.32686

-0.80076

-0.31807

-0.07737

0.00257

-0.00823

-0.00420

0.06230

Appendix Ε

Appendix F

D a t a of the PSS and the SVC auxiliary controller in the co-ordinated SVC-PSS scheme:

= 0.0600 TPI = 3.0000 TP2 = 0.1000 TP* = 0.0050

= 0.6000 Tpâ, — 1.0000 TPS = 0.0050 TP6 = 0.0046

Kp3 = 0.6000 1.0000 TP* = 0.0050 TP9 = 0.0010

Ks\ = 0.1200 TsX = 3.0000 Ts2 = 0.1000 Ts3 = 0.1200

Ks2 = 0.4000 1.0000 Ts5 = 0.0020 Ts6

= 0.0100

&s3 = 0.1500 Tsl = 1.0000 τ Λ = 0.0010 Ts9 = 0.0027

A u A\2 A \ l A\S Network

A2\ A22 A23 A21 A2S SVC

A32 A33 A 34 A35 A3S Stator

Currents

AA2 A43 A44 A4S Rotor

Current

A52 A55 A56 A51 A5S Excitation

A66 Α6Ί A6S PSS

Λ 78 Angles

AS3 AS4 AS7 ^88 Velocities

Excitation System Data :

KA = 200.0 TA = 0.3575 KE = 1.0000 TE = 0.0000

Kf = 0.0529 Tf = 1.0

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