A new adaptive power system stabilizer using a Lyapunov design technique 0 AbuI-Haggag Ibrahim and A M Kamel Department of Electrical Power Engineering, Faculty of Engineering, Cairo University, Cairo. Egypt

A technique for designing an adaptive power system stabilizer is presented which improves the dynamic stability of power systems by effectively increasing the damping torque of the synchronous generator in the system. The proposed adaptive stabilizer is optimal in the sense of minimizing a specifiedperformance condition. A character- istic feature of using a Lyapunov technique is that it leads to stable adaptation loops. The dynamic responses following a step disturbance by digital simulations are obtained by means of three types of stabilizers: the conventional power system stabilizer, a frequency-response-based optimal adaptive stabilizer and the proposed Lyapunov stabilizer. Simulation results on a typical single-machine/infinite-bus system illustrate the superiority of the proposed technique.

Keywords: on-line control strategies, dynamic stability, L yapunov method

I. I n t r o d u c t i o n Considerable efforts have been devoted to improving the dynamic stability of power systems through the use of power system stabilizers (PSSs) 1-6. The power system stabilizer usually employed by the utility industry is a lead-lag network using the speed signal as input. The fundamental concept for the design of such a PSS is to compensate the phase lag introduced by the voltage regulator and exciter so that a supplementary damping torque component which is in phase with the rotor speed is generated 1 . This supplementary signal can be employed to enhance the dynamic stability of the system.

Several methods have been reported in the literature to provide the damping torque required for improving the dynamic stability. These include approaches which employ a conventional PSS to generate a supplementary signal 6, as well as approaches which employ a linear optimal stabilizer using the theory of linear optimal regulators 3'6'7. However, such methods produce fixed parameter stabilizers which are tuned for a single operating point and give satisfactory performance only for that condition. Variations in the system operating

Received: December 1987; revised September 1988

conditions such as loading, adding or disconnecting generators and lines would require a readjustment of the PSS parameters.

Adaptive power system stabilizers provide an attractive alternative for supplying appropriate damping torque components under various system operating conditions. Several papers illustrating the use of different adaptive control techniques for excitation systems have appeared over the past few years 8-1°. These techniques produce self-tuning controllers based on pole assignment. The difficulties of applying such methods to the power system stabilization problem may be attributed to a variety of reasons, among which are11:

• the large number of parameters that may have to be adjusted simultaneously while assuring the stability of the overall adaptive system;

• the lack of exact information regarding the variation of the power system parameters;

• the noise present in the measurement of the plant outputs.

Developments in model reference adaptive control 12 15 using the method of Lyapunov have enabled engineers to overcome some of these difficulties and have resulted in schemes that may be attractive in practical situations.

In this paper a new approach for the adaptive stabilization of power systems is presented. The approach is based on a model reference adaptive control scheme using a Lyapunov synthesis technique. Examples for the single-machine/infinite-bus system are given to illustrate the effectiveness of the proposed stabilizer. Moreover, simulation results are presented and compared with those obtained by using the conventional stabilizer and the frequency-response-based optimal adaptive stabilizer. The main features of the proposed stabilizer are as follows.

• The overall system is asymptotically stable for different operating conditions.

• The controller structure may easily be changed to the conventional lead-lag structure with the controller parameters adjusted via simple adaptive loops.

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• The desired performance of the power system may be altered at will according to the specific needs simply by changing the model parameters. This does not require any modification to the controller structure or to the parameter adaptation mechanism.

• The results obtained are better than those for the conventional stabilizer and the frequency-response- based optimal adaptive stabilizer in the sense of dynamic performance.

II. N o m e n c l a t u r e B 1, B2, B 3 positive weights D differentiation operator, D = d/dt E difference between model and plant outputs efd equivalent excitation voltage e'q q-axis component of voltage behind

transient reactance G +jB terminal load admittance Go(s) controller transfer function 3 performance index K, a~, a 2 model parameters K 1 K 6 constants of the linearized model of

synchronous machines K c, h 1, h 2 adjustable parameters of model reference

stabilizer K e voltage regulator gain Kq, " L ' q constants of wash-out circuit K V, a l l , a2~ system coefficients M inertial coefficient, M = 2H MRAC model reference adaptive control PSS power system stabilizer r + jX c tie-line impedance s Laplace operator T model torque T~ energy conversion torque T m mechanical input T, torque component proportional to eq V Lyapunov function V o infinite bus voltage Vre f armature reference voltage V S stabilizer output V, terminal voltage A linearized incremental quantity 3 torque angle vo angular velocity ~)o synchronous frequency, COo = 377 ~o* angular velocity output of wash-out circuit

summation operator ~1, T2 parameters of lead controller ~o d-axis transient open-circuit time constant z~ voltage regulator time constant.

III. P o w e r s y s t e m m o d e l The system configuration for the single-machine/infinite- bus system is shown in Figure 1. The linearized incremental model of the system, including the voltage regulator and the exciter, is shown in Figure 21'2'6 . The relations in the block diagram apply to a two-axis machine representation with a field circuit in the direct axis but without amortisseur effects, The parameters K 1 K 6 are functions of machine and system impedances as well as the operating points.

The stabilization problem is that of designing a

stabilizer which provides a supplementary stabilizing signal to increase the damping torque of the system. Three types of stabilizers will be examined.

II1.1 The conven t iona l p o w e r s y s t e m stabilizer The conventional fixed parameter stabilizer may be expressed by the following transfer function"2:

G(s )=AVs ( s )_ Kqs l + z l s (1) A~o(s) l + ~ q s l + ~ 2 s

The first factor on the right of (1) is used to wash out the compensating effect after a time Zq. The second factor is a simple lead compensator having rl >z2. The input used for the stabilizer is the speed signal &o.

111.2 The frequency-response-based optimal adaptive stabilizer This stabilizer is designed in such a way as to minimize the phase difference between the speed signal Aco and the torque component AT S developed through the stabilizer loop 1°. The controller has the same first-order lead transfer function as that given in (1), except that this time the parameters ~ and z" 2 are not fixed. This controller introduces a phase lead component to compensate for the phase lags introduced by the exciter and the machine field time constants. Since the amount of phase lead required to stabilize the power system must be adjusted for different operating conditions, the need for an adaptive scheme to tune the controller parameters arises. The adaptive laws for adjusting the parameters ~1 and ~2 were obtained by Zein El-Din et al.l° through the minimization of a cost function J of the form

J = • e2(•) (2) ( o = o ) 1

where e is a measure of the phase difference between the speed signal and the stabilizing torque signal. Moreover, the frequency range (o~1, co2) is chosen as (4, 12) rad s -1, which covers all electromechanical modes of concern I. The minimization of the cost function given by (2) was

V r X e ~ Vo

I

i t B

Figure 1. System configuration for single machine to infinite bus

128 Electrical Power & Energy Systems

AVref I I ,-- I

AV s

K 3

I +sK3T~O

~ 4

Ae'q . v _ ~

AC0* Kq s l, ~ I +S-Cq

1 ] ~

I A~

2Hs °_ol s I

.( +

A6

i " ~ t

Figure 2. Linearized incremental model of synchronous machine with an exciter and a stabilizer

further simplified by fixing the parameter ~2 at a relatively low value. This resulted in the linear least-squares estimate for zl given by

12

T1 -- 12 (3)

where f(~o) is a function of frequency as well as of the system parameters. Therefore the value of the function f(m) changes as the operating conditions are varied, and consequently the controller parameter z 1 is adaptively adjusted.

IV. Proposed adaptive stabilizer via Lyapunov design The adaptive power system stabilizer proposed in this paper employs a model reference approach. The survey paper by Landau 14 provides an exhaustive list of references on the general model reference adaptive control (MRAC) problem. In this paper, however, we deal specifically with the MRAC problem solution using Lyapunov's second method, making a function V of the system variables a Lyapunov function 13.

The basic scheme of a MRAC system is given in Figure 3. The reference model gives the desired response of the adjustable system, and the task of adaptation is to minimize a function of the difference between the output of the adjustable system and that of the reference model. This is done by the adaptation mechanism that modifies the parameters of the adjustable system (more specifically, the controller parameters). The disturbances shown in

0 Reference m model

Disturbances

" / _1 Adjustable - J system

Parameter / adaptation [

Adaptation _ mechanism

Figure 3. Basic configuration of a model reference adaptive control system

Figure 3 signify changes in the system operating conditions. One of the most important advantages of this type of adaptive system is its high speed of adaptation 14.

IV.1 PSS design procedures The relations shown in the block diagram of Figure 2 for a single machine supplying an infinite bus have been treated previously by the concept of small perturbations on the fundamental synchronous machine equations. With the exception of K3, which is only a function of the ratio of

Vol 12 Number 2 April 1990 129

A~ sK q A~*

1 +S~q

s2 +als +a2

Model

K c ] + \ "

Cont ro l le r

K v

s2 +alls +a21

Process

hlS + h 2

Plant

m

_ IATs

E

Figure 4. MRAC scheme for power system stabilization

impedances, all other parameters change with loading, making the dynamic behaviour of the machine quite different at different operating points. In practical power systems K4/K e is much smaller than K 5, and thus the effect of K 4 is usually neglected in stabilizer design 1. Consequently, the transfer function Gp(s), in the absence of the controller, becomes

Gp(s) = ATe(s) _ K v (4) Aco*(s) s2 +alxs+a21

where K V, a~ i and a21 are coefficients determined by the operating conditions and are related to the system parameters as follows:

K e K 2 K v - ?

TeTdo

1 1 all ------+-- (5) t

Ze Zdo K3

1 + K e K 3 K 6

a21 - - TeZ~oK 3

Using D-operator notation, where D =d/dt , the uncon- trolled process in (4) may be expressed as

(D 2 + a~ ~D + a 21)ATs = Kv Aco* (6)

One possible MRAC scheme for the above process is shown in Figure 4. The equation of the controlled process (the plant) then becomes

[D2 + (all +Kvhl)D+(a21 +Kvh2)]ATs=KvK¢ A~*

(7)

The governing equation of the linear model shown in Figure 4 is given by

(D 2 + aiD + a2)AT= K A(o* (8)

where AT is the model output. In the frequency domain the relationship between the

torque (AT) and speed (Am*) signals is

AT K A(.D-- ~ ( j ( / ) ) = (a 2 _ 092) 2 Jr- ( a l t o ) 2 [ ( a 2 - - c o 2 ) - - j a l c o ] (9)

Since our primary objective is to ensure that the torque and speed signals remain in phase over the frequency range of interest (co~ ~ co 2 rad s- ~), the model parameters a t and a 2 are chosen in such a way as to make the real part of (9) greater than the imaginary part. Moreover, by choosing a 2 > OJ 2 we gaurantee that the two signals will not be in phase opposition.

Thus, to ensure that the plant behaves in the same manner as the model, suitable adjustments of the controller parameters K¢, h~ and h 2 m u s t be made.

The first step in the design technique is to derive an equation in terms of the error E, and since

E = A T - A T , (10)

the required equation is

(D 2 + aiD + a2)E = (K-- KcKv)Aco*

+(all + Kvh l -a l )D AT s

+ (a21 +KvhE-a2)AT s

= g 1A(.o* + Gt2D AT s + o~ 3 AT s (11)

130 Electrical Power & Energy Systems

where

oq = K - KcK v

(x 2 = a l l +Kvh 1 - a 1 (12

0~ 3 = a 2 1 +Kvh 2 - a 2

If the variables

Yl =E, y2=DE

are introduced, the error equation may be written as

.91 =Y2 (13)

- 9 2 = - - a l Y 2 - - a 2 Y l + °~1 Aog* + c~2D A T s + o~ 3 AT s

Now, Lyapunov's second method may be applied by choosing a Lyapunov function of the form

- 2 0~2 ~2 T H ~1 2 (14)

V = y y + ~ - 1 + ~ 2 q'B3

where H is the Hermite matrix of the equation

(D2 +alD+a2)E=O

yV= (y~ Y2) and B 1, B 2, B 3 are positive constants. To ensure the negative semidefiniteness of the derivative of the Lyapunov function, the following set of parameter adjustment equations are utilized:

• B1Z K¢ = ~ - A~o*

fh = -B2~ZD AT~ (15) K,

B3Z /~2 - AZ K~ s

where Z =al y2 =aiDE.

The above choice of parameter adjustments results in the following derivative of the Lyapunov function:

V= - - 2 Z 2 (16)

Thus, if the adaptive loops of (15) are employed, I? is negative-semidefinite. Further, if each of the gains B1, B2, B 3 is positive, and if the model is governed by a stable characteristic equation (i.e. the principal minors of H are positive-definite), then V is positive-definite and the system is stable 13

V. A n a l y t i c a l re su l t s A single-machine/infinite-bus system is considered with the following given data:

K e = 25 X d = 1.5

re = 0.05 Xq = 0.3 t t

"rdo = 6 X d = 0.3

H e = 1.5 Xc =0.1

Ka = 60 Zq = 3

A comparison study of the stabilization of the power system has been made for three types of stabilizers: the conventional stabilizer, the frequency-response-based optimal adaptive stabilizer and the proposed adaptive stabilizer via Lyapunov design. Moreover, four different operating conditions have been considered in the study to highlight the adaptive nature of the proposed stabilizer. These operating conditions correspond to the reference case and to the three change cases illustrated in Table 1. In all these cases the dynamic responses following a 0.05 p.u. step change in the mechanical torque AT m have been compared and are plotted in Figures 5 8.

V.1 The conventional power system stabilizer using speed signal as input A conventional fixed parameter lead stabilizer is derived based on asymptotic plots for two extreme operating conditions. The transfer function of the conventional PSS is given by

1 +0.154s Go(s) = 1 +0.033s (17)

V.2 The frequency-response-based optimal adaptive stabilizer This stabilizer has the same form as that of the conventional type. The parameter "c 2 is fixed at an arbitrary small value of 0.01 and the linear least-squares estimate of the parameter Zl is evaluated through the

Table 1. Operating conditions

Reference Change Change Change case case I case II case III PG = 1.0 p.u. PG = 0.5 p.u. P~ = 0.5 p.u. P~ = 1.0 p.u. Qc= 0 .5 p.u. QG=0.5 p.u. QG= - 0 . 5 p.u. Q~=0.5 p.u., Xe=0.3 p.u.

K 1 1.01 1.4 1.47 0.816 K 2 1.149 1.15 2.0 0.612 K 3 0.36 0.25 0.25 0.33 K 4 1.47 1.4 2.4 0.76 K 5 --0.097 --0.054 --0.035 --0.0916 K 6 0.419 0.255 0.18 0.46

Vol 12 Number 2 April 1990 131

A speed

- 0 . 0 0 1 -

< <

/ ~ / ~ Angle

- ~ ..._... _ _ _

0.0! I i i i ~ " ~ ""'" . . . . . . " . . . . . . . Conventional ~ "~" ~- ' ' .......... Optimal

Lyapunov I I I i I

0 1 2 Time (s)

Figure 5. Dynamic responses for reference case

0.001

-0.001

< <

0.04

~I I ~ Speed

Angle

.... Lyapunov l I I I ~ I

0 1 2 Time (s)

Figure 6. Dynamic responses for c h a n g e case I

adjustment equation (3). Table 2 summarizes the values obtained for r~ under the different operating conditions.

IV.3 The model reference adaptive stabilizer using Lyapunov synthesis As was mentioned earlier in the introduction to this paper , the M R A C structure may easily be changed to that of a lead lag controller. The transfer function in this case becomes

s2 +a~s+a2~ Gc(s)=Kc S2 + (a lx +Kvh1)s+(a21 + K v h 2)

(18)

where K c, h 1 and h 2 are the adjustable controller parameters obtained from the adjustment equations (15), and K v, a l l and a2a are system coefficients determined by the operating conditions and are obtained by use of

0.015

. . . - -. 0

i//'\.,, Angle

I ..... ~ C . . ~ . ~ ~= ..... ......

0.0.' f .."" --'"

Lyapunov

I I t 1 ~ I

T i m e Is}

Figure 7. D y n a m i c responses for c h a n g e case II

0.003

>

0.06

C o n v e n t i o n a l /~ .......... Opt ima l , \ - L y a p u n o v

~'~\\ A S p e e d

'"."~_~\\ . ~ ¢

A Angle

i I i I l I 2 3

T i m e (s)

Figure 8. D y n a m i c responses for c h a n g e case I I I

Table 2. Estimates of ~1 under various operating conditions

Reference Change Change Change case case I case II case III

zl 1.014 0.206 0.983 0.405

1 3 2 Electr ical P o w e r & Energy S y s t e m s

Table 3. Estimates of model reference, adaptive controller parameters

Reference Change Change Change case case I case II case III

h 1 0.23 0.25 0.14 0.42 h 2 501 493 297 968

equation (5). Table 3 gives a list of the controller parameters h~ and h z for the different change cases.

Figure 5 shows the dynamic responses of A6 and Ato when the reference system is subjected to a 0.05 p.u. step change in the mechanical torque AT m. Notice also that in Figure 5 the responses obtained from using the three types of stabilizers are all included for comparison purposes. Figures 6-8 show the responses for the other three change cases. The responses for the adaptive model reference Lyapunov stabilizer were obtained after the parameters h 1 and h z had converged to the values depicted in Table 3.

I t can be seen from the figures that the adaptive model reference Lyapunov stabilizer yields better system performance than the other stabilizers.

VI. Conclusions A technique for the stabilization of power systems is presented by using a model reference adaptive technique employing Lyapunov synthesis. The main feature of the model reference adaptive stabilizer is that it guarantees the asymptotic stability of the system for different operation conditions.

Simulation results indicate that the proposed model reference adaptive stabilizer provides a new means for increasing the damping torque of the system and therefore enhancing the dynamic stability of power systems.

VII. References 1 d e l e l l o , F P and Concordia, C 'Concepts of synchronous

machine stability as affected by excitation control' IEEE Trans. Power Appar. & Syst. Vol PAS-88 (1 969) pp 31 6-329

2 Anderson, P i and Fouad, A A Power System Contro land Stabil i ty Iowa State University Press, Ames, (1977)

3 Uoussa, H A M and Yu, Y N 'Optimal power system stabilization through excitation and/or governor control' IEEE Trans. Power Appar. & Syst. Vol. PAS-91 (1 972) pp 1166 1174

4 deMel lo , F P, Nolan, P J, Laskowski, T F and Undr i l l , J M 'Coordinated application of stabilizers in multimachine power systems" IEEE Trans. PowerAppar. & Syst. Vol PAS-99 (1 980) pp 892-901

5 Schleif, F R, Hunkins, H D, Mart in, G E and Hattan, E E 'Excitation control to improve power line stability' IEEE Trans. PowerAppar. & Syst. Vol PAS-87 (1 968) pp 1426 1434

6 Chan, W C and Hsu, Y Y 'An optimal variable structure stabilizer for power system stabilization' IEEE Trans. Power Appar. & Syst. Vol PAS-102 (1 983) pp 1738-1 746

7 Yu, Y N and Siggers, C 'Stabilization and optimal control signals for a power system' IEEE Trans. Power Appar. & Syst. Vol PAS-90 (1 971 ) pp 1469-1481

8 Ghosh, A, Ledwich, G, Mal ik , O P and Hope, G S "Power system stabilizer based on adaptive control techniques' IEEE Trans. Power Appar. & Syst. Vol PAS-103 (1984) pp1983 1989

9 Ledwich, G 'Adaptive excitation control' Proc. lEE Vol 126 (1 979) pp 249 253

10 Zein El-Din, H M, Ibrahim, 0 A-H and E l - i a r s a f a w y , M 'Adaptive power system stabilizer for synchronous machines using speed signal' Proc. 7th IFAC Conf. on Digi tal Computer Applications, Vienna (1 984) pp 277 282

11 Narendra, K S and Kudva, P 'Stable adaptive schemes for system identification and control Part I' IEEE Trans. SMC Vol SMC-4 (1 974) pp 542-551

12 Parks, P C "Liapunov redesign of model reference adaptive control systems' IEEE Trans. Automatic Control Vol AC-11 (1 966) pp 362-367

13 Shackcloth, B "Design of model reference control systems using a Lyapunov synthesis technique" Proc. IEEE Vol 114 (1967) pp299 302

14 Landau, I D 'A survey of model reference adaptive techniques--theory and applications' Automatica Vol 10 (1 974) pp 353-379

15 Ibrahim, O A-H and Solirnan, M A'Bootstrap identification via Lyapunov design techniques' Adv. Modul. & SimuL Vol 3 (1 985) pp 51-64

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