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A self-tuning automatic voltage regulator
Electric Power Systems Research, 2 ( 1 9 7 9 ) 199 - 213 © Elsevier S e q u o i a S.A., L a u s a n n e - - P r i n t e d in t h e N e t h e r l a n d s
199
A Self-Tuning Automatic Voltage Regulator
M. A. H. S H E I R A H
Ain-Shams University, Cairo (Egypt) O. P. M A L I K a n d G. S. H O P E
The University of Calgary, Calgary (Canada) (Received J u n e 19, 1 9 7 9 )
SUMMARY
A self-tuning automatic voltage regulator (AVR) for a synchronous generator is presented. The regulator proposed improves the system stability; it is simple and can handle stochastic load changes. The algorithm for the proposed AVR combines a leastsquares estimator with a minimum variance control strategy computed from an estimated model. It is shown that if the parameter estimates converge, the control law obtained is in fact the minimum variance control law that would be computed if the parameters of the system were known. The algorithm proposed has been tested by simulation and also by implementation on a minicomputer. Results show that, in general, the system performance is improved with a self-tuning regulator.
1. I N T R O D U C T I O N
Common procedure in designing power system controllers is to subject the power system model to step changes in load or voltage reference and observe the output under the influence of the controller. These off-line simulation studies using deterministic linear models are performed with the system linearized around a fixed operating point, the system parameters being assumed constant. Systems are, in general, non-linear and their parameters depend upon the operating conditions. Further, they are continuously subjected to load variations of random magnitude and duration, rather than to step changes. In view of this it would be more appropriate to:
(i) consider the problem as a stochastic control problem, and (ii) have controllers with dynamically changing coefficients to suit the varying operating conditions of the system. Because of problems in implementation using analog devices, controllers are built using coefficients that cannot be changed dynamically. However, some of these problems can be overcome by using digital devices. The use of digital computers for real-time control and operation of power systems has been increasing gradually [1, 2]. Initial applications of computers in power systems tended to be at the higher levels of system operation. However, recent developments in digital technology have made the use of digital devices as real-time controllers feasible at the generating level [3 - 5]. Use of digital devices for real-time control makes implementation of sophisticated control policies not only possible but also economically feasible [6- 8]. This paper describes an algorithm to perform the basic function of an automatic voltage regulator (AVR) considering the generating unit as a stochastic system. The proposed regulator has been tested on a generating unit modelled by a 9th-order set of simultaneous non-linear differential equations simulating a real-time process. Results of the simulation studies are described. Encouraging results obtained from the simulation studies prompted the authors to implement the proposed regulator. The regulator has been implemented using a minicomputer as a real-time controller and tested on-line to control the field excitation of a generating unit physical model in the labora-
200 tory. Initial test results have verified the results of the simulation studies. An illustrative set of test results is also included in the paper.
2. CONTROL STRATEGY The control inputs for a stochastic system are generally based on observations of some of the variables which describe the system. Because of various uncertainties, for example noise, unpredictable load variations, e t c . , these observations are imperfect. Further, the control policy that is utilized must be based on a p r i o r i knowledge of the system characteristics and on the time history of the input and o u t p u t variables. For such a system it is possible to compute the control based on minimizing the variance of an o u t p u t variable [9]. Variance is a measure of the deviation of a variable from its nominal value. An electric generating unit is a complex non-linear system. It may be modelled by a linear low-order discrete model with timevarying parameters. A recursive least-squares parameter estimation technique is used in this work to estimate the time-varying parameters of the discrete model. Based on this model, which tracks the operating conditions of the system, control is c o m p u t e d using a minimum variance strategy. Thus this process is the equivalent of a controller with dynamically changing coefficients and is referred to as the self-tuning regulator. The basic theory of a self-tuning regulator and its application as the equivalent of a conventional AVR with auxiliary stabilizing signal is described below.
3. SELF-TUNING REGULATOR Suppose that the system to be controlled can be represented by the following singleinput single-output stochastic discrete model: y(t) + a l y ( t - = blu(t--k--1)
+
. . . + bnu(t--k--n)
1)
+
. . . + cne(t
y ( t + k + 1) + a l y ( t ) =flo[u(t) +fllu(t-+ flzu(t--l)]
+ --
+ . . . + amY(t--m
+ 1)
1) + . . . +
+ e(t + k + l)
(2)
where m = n and l -- n + k --- 1. The parameters ~1 . . . . , ~m, ~1 . . . . . flz must now be identified. Go, the choice of which is not critical, is assumed to be known in order to avoid difficulty arising from the use of feedback [10]. The disturbance e ( t ) is a moving average of order k of the driving noise e ( t ) . The model of eqn. (2) can be obtained by writing eqn. (1) at time t + k + 1, for all ci = 0, and substituting the values of y ( t + k}, y(t + k -- 1), . . . , y ( t + 1) using eqn. {1) written at t i m e s t + k, t + k - - 1 . . . . , t + 1, respectively. As shown in Appendix 1, in this case the minimum variance strategy is 1
u(t) = ~[~ly(t)
1) + . . . + a , y ( t - - n )
+ ),[e(t) + c l e ( t - -
instant, k is a pure time delay of the system, n is the order of the system, ~ is a constant, (e(t)} is a sequence of independent, normal (0, 1) random variables, y ( t - - i), u ( t - - i) a n d e( t - - i) are the variables at time t - - iT, and T is the sampling period. The minimum variance control strategy for the system of eqn. (1) is given by eqn. (A116) which is developed in Appendix 1. The control is calculated using a model with known parameters for the system to be controlled. Since most systems are complex, non-linear, and time-varying, general models that aze valid for all operating conditions cann o t be obtained. Hence, to use a minimum variance strategy, a model has to be postulated and the parameters of such a model identified in real-time each sampling period. As shown in Appendix 1, implementation of eqn. (A1-16) requires the solution of the identity (A1-7) at each sampling instant, which is very time-consuming. To obtain an easy-to-implement minimum variance strategy, the model of eqn. (1) may be replaced by
+... + amy(t--m
--fllu(t--1)--.
. .--fltu(t--l)
+ 1)] --
(3)
n)]
(1) where y(t) and u ( t ) are the o u t p u t and input of the model, respectively, t is the sampling
A comparison of eqns. (A1-16) and (3) shows that use of the model structure (eqn. (2)) simplifies the computation of the control strategy.
201
In the self-tuning regulator, therefore, the parameters a l , . • .,a~, and ~1, • • .,#, in the predictive model (eqn. (2)) are estimated online at any sampling instant t. The estimated values ~1, • • . , 6m and/~t, • • .,/~l°f these parameters are then used to calculate the minimum variance control strategy (eqn. (3)). To estimate the parameters of the model in eqn. (2), the model m a y be rewritten at the instant t in the following form: r'(t)0
=
(4)
regulator (eqn. (13)) is the same as the minimum variance regulator. Since the system modelled by eqn. (2) with the regulator (eqn. (13)) and the least-squares parameter estimator (eqn. (8)) is described by a set of non-linear time-dependent stochastic difference equations, it is extremely difficult to provide a general p r o o f of convergence for the proposed algorithm [12]. However, it was verified that the algorithm did in fact converge for all simulation and experimental studies that were performed.
where F(t) = [ - - y ( t - - k - - 1), . . . . - - y ( t -- k -- m), ~ou(t--k--
2) . . . .
4. PROBLEM F O R M U L A T I O N
,~ou(t--k--l--1)]'
0 =
(5) (6)
~(t) = y ( t ) - - ~ o u ( t - - k - - 1)
(7)
and [ ]' denotes the transpose of a matrix. Several methods can be used to obtain an estimate #(t) for the vector parameters 8 [ 1 1 ] . Use of the recursive least-squares technique gives {](t) = ~(t -- 1) + K ( t ) [ ~ ( t ) - - r ' ( t ) ~ ( t -- 1)] (8)
The correction vector, K ( t ) , is obtained by the recursive relations K ( t ) = P(t -- 1)l'(t)[1 + F'(t)P(t --
1)r'(t)l-1 (9)
and P(t) = P(t -- 1) - - K ( t ) F ' ( t ) P ( t - - 1)
(10)
where P(t) is the covariance of the error in the estimates. To start the algorithm, the following initial conditions m a y be used:
0(0) =
(11)
0
P(0) = qI (12) where q is a large number and I is the identity matrix. Once 0 is determined, u ( t ) can be obtained as
1 u ( t ) = ~o [ 6 1 ( t ) y ( t ) + . . .
1)] --
+ 6m(t)y(t--m+
- - ~ l ( t ) u ( t - - 1) - - . . .
-- ~ t ( t ) u ( t - - 1) (13)
If the vector parameters #(t) converge to constant value vectors, then the self-tuning
The generating unit with the proposed selftuning A V R is shown in Fig. 1. The field voltage et is given by ef = e~n + ef s
(14)
where etn is the nominal value of the excitation voltage and efs is the self-tuning regulator output, ifn is the corresponding nominal value of the field current at steady state, e f , / R f , and it is the actual field current. In Fig. 1, V is the terminal voltage of the synchronous machine and Vr is the desired value of this voltage. ~ is the p o w e r angle and p6 = d 5 / d t is the speed error. T o represents the load disturbance. The objectives of this work are two-fold: (i) to damp the mechanical oscillations of the machine during transients, and (ii) to keep the terminal voltage of the machine as close as possible to the reference voltage. At the same time it is desired that the field current return to the nominal value as soon as the system returns to steady state. Selection of a suitable objective function and proper weighting coefficients would allow the desired relative importance to be given to the above t w o objectives. Mathematically, it is required to obtain a control signal ets which minimizes the variance of y s ( t ) defined by eqn. (A1-9). Considering the desired objective above, y s ( t ) has been taken to be Ys(S) = p l A i f ( t ) + pg.p~(t) + p s A V ( t )
(15)
where Pl, P2 and Ps are constants, Ai,(t) = if(t) -- if. (t) and
(16)
202 T
I cf
efn
Cenerating Unit
lif
Vr
us
--~
On-line Identifier h Ifn Computation ~ _ _ _ _
Ys = p~Aif * p2p~ + P3~V
Self-tuning Regulator
Fig. 1. Generating unit with proposed control scheme.
A V ( t ) = Vr(t) -- V(t)
(17)
The choice of P2 larger than p3 in eqn. (15) will give higher priority to the damping of mechanical oscillations and preservation of stability over voltage regulation. By modifying their relative values, one can reverse the priorities. For best overall performance the individual variances of Air(t), p~(t) and A V ( t ) must be minimized. If these three quantities are independent, minimization of the variance of y , ( t ) automatically yields individual minimum variances for if(t), p6 (t) and A V(t). However, these quantities are not independent. Thus the parameters p 1, P 2 and p a must be selected such that the minimum variance of y , ( t ) is close to the sum of the minimum variances of p l i A ( t ) , p z p 6 (t) and paA V(t). The system to be controlled is a complex non-linear system. It is possible to obtain a slowly time-varying linear model of the system about an arbitrary operating point. Let this model be of the following form:
y , ( t + k + 1) + alY,(t) + . . . + a m y , ( t - - m
+ 1)
= ~0[us(t) + ~ l U , ( t - - 1 ) + . . . +
+ ~zUs(t--/)] + e(t + k + 1)
(18)
where a l , • • • , am and ~1, • • •, ~z are slowly time-varying parameters, and ~o is a constantvalue parameter. k, m and l must be selected such that the model of eqn. (18) is a good approximation for the actual system. The parameters of this model can be estimated using the recursive least-squares technique (eqns. (8) (10)). The control signal, Us(t), can then be obtained as
1 Us(t) = ~o [&l(t)y,(t) + . . . + & m ( t ) y , ( t - - r n + 1)]
-~l(t) u(t-- 1)--...--~t(t)u(t--l)
(19) and
e%(T) = Gus(t)
t<<. r < t + 1
(20)
where G is a constant gain.
5. SYSTEM SIMULATION
A generating unit connected to an infinite bus through a short transmission line has been simulated by a 9th-order non-linear model.
203 The model and parameters used are given in Appendices 2 and 3, respectively. The synchronous machine model takes into account the stator transients and damper windings on the direct and quadrature axes. The turbine and governor are, however, represented by simplified models. The ultimate objective of these studies is to implement in real-time the proposed regulator using a minicomputer. The simulation studies therefore have been performed on an HP 21MX minicomputer which will also be used for implementation. For the studies described below, a sampling period of 50 ms was selected. In the discrete time-varying model of eqn. (18), rn has been taken equal to 3, k equal to zero, l equal to 2 and /30 equal to unity. Thus, the system model (eqn. (2)) becomes y,( t + I) +~I y,(t) + ~2 y,( t -- 1) + a3Y,( t -- 2)
= u , ( t ) + {Jl u , ( t - - 1) + ~2 U s ( t - - 2) + e(t + 1)
(21) where ys(t) = pIAi~(t) + P2P6(t) + p s A V ( t )
(22)
6. RESULTS OF SIMULATION STUDIES
The studies described in this section concentrate on the effect of the parameters Pl, P2 and P3 (eqn. {22)). Notice that to ensure the system will work at the same operating point if the load returns to its nominal value after a load disturbance, T
nominal value after a load disturbance, P l must be greater than zero. The following values for p 1, P 2 and p3 were considered in these studies {these values are based on the p.u. system used): (i) P l = 1.0, P2 = 1.0 and P3 = 1.0; (ii) Pl = 1.0, p2 = 1.0 a n d p s = 0.0; (iii) Pl = 1.0, p2 = 0.0 a n d p 3 = 1.0. In the first ease both p6 and ~ V have equal weight. The limiting values of P3 and P2 in the second and third cases, respectively, were chosen intentionally to investigate the effect of speed stabilization on system performance. Case (ii), in which the effect of terminal voltage variation is neglected, gives maximum weight to the speed stabilizing signal and would be expected to provide improved system stability during large system transients. On the other hand, case (iii), with no speed stabilizing signal, would be expected to result in minimum variations in the terminal voltage at the expense of system stability during large system disturbances. The most appropriate values for p 2 and p 3 would depend upon a particular system and lie between 0.0 and 1.0. The choice of Pz and P3 between 0.0 and 1.0 would yield results that lie between those of the second and third cases. (iv) A conventional A V R with a c o m m o n l y used stabilizer based on ref. 13 was considered. The system is shown in Fig. 2. The A V R used has the form [13] ely _ K, V 1 + TrP
i
V 6 P~ p.
I
Generating Unit
L_
ef t
e ~i f v
f
Stabilizer ^VR
Fig. 2. Conventional AVR with stabilizer.
/
t_ I-
=,j f
k-fl
Vr
(23)
204
where K, 0.5 and T, 0.02. The stabilizer used is =
6. ooi
=
p.u.
TOHC2~,',~" 5.56i
eft
Kp(1
_
p5
(1
+
+ Tip)
T2P)(1
+
2
Tap) 2
(24)
6.20 XI.E- O]
where K = 0.01, T1 = 0.125, T2 = 3 and Ta = 0.05.
i
i
4,80
i
4. q6
7/?4L s
6.1. Step change in reference voltage Remits for a 5% step change in the reference voltage are shown in Fig. 3. The selftuning regulator with p 1 = P 2 = P s = 1 gives considerably improved performance over the conventional regulator with stabilizer.
Fig. 4. T o r q u e d i s t u r b a n c e %.0
OEL TA
output.
rad
-5.2
NI
-S. 4
1°° t ERROR ]N TERMINAL V©L.
p.u.
Xl. E-g:
-5.6
• 600 I
-5.8 / L,~JL s -4.
0 ~ ,000
-3.0
• .Ou~o0
1.)0
2.'40 ' 5. BO
4.80
&00'
(a)
.
. . 2.. qO
DELTA
.
.
f.. 80.
.
.
.
. . 7.20
.
.
. . . 9.50
12.0
(i)
rad
5.2 ¸ -3. zl Xl.E-OI
-2. 0
-5. 6
[?EL TA
-2.4
rad
(i)
-L8
TIME s -2, 8
4..°ooo L~
Xl. E-Of
-L 2
-3.0
-Z 5
2.~o
IJEL TA
4.'8~
7. aO
9. 50
~-~O
7. 20
9. 60
la. 0
(ii)
rad
-5.2
TIME s
-4.0 )0 ' 1.'20
2, t0
S. 60
4. BO 6.00
-3. t XL E-Of
(b)
-5.5
Fig. 3. Terminal voltage (a) and load angle response (b) to a step change in reference voltage: curves (i), self-tuning regulator; curves (ii),conventional voltage regulator.
-5.8
6.2. Step changes in torque The generating unit was subjected to load disturbances of the form shown in Fig. 4. Changes in the torque from the nominal value (0.5 p.u.) were considered to be step changes with random magnitude and random duration. Power angle 6 and speed error p5 for all four cases are shown in Figs. 5 and 6, respectively. A close examination of these Figures shows that, from the point of view of stabili-
TIME s
-~.Oo~
2. 40
/
12. 0
(iii)
f
f
TIME 9. 5o
s
la. o
(iv)
Fig. 5. Power angle (rad) response to torque input for the four cases studied.
205 5.00 P DE," ]-~4
p
3. ooj:f ERROR I/V TERMINAL V©L.
rad/s
3.00+
;.B0!
I. O0 ':,
• 6001.
-.6of
p.u.
V
-l.B i
I
t
J
Z I~X~: s
TIME
s
(i) 5.00
(i) S. O0
ERROR I N TERMINAL
P ~ E L ,TA rad/s
5.00
1.80
l. OO
• 600
XI. E-O
XI.E-Oi
-I. 0
-.60
-5.0
VOL.
p,u.
-I.B
TIME -5. 0
2.'~0
4.80
7.20
9.60
s
12.0
TIME s (ii)
-S. 0 C
---~.~o
S.OOT
5. O0
P DELTA
i ERROR I N TERMINAL
rad/s
3.00
1.801
l. OO
• 6o0
~.6o 72.7
4.'BO ' 7.'ao
(ii)
VOL. p.u.
Xl. E-Ol
-1.0
-.60
.....
-].B I
-3.0
!~
TIME s -5. 0~
,0
'
2.'40
'
4.'80
'
7.'20 - '
9.~
12.0'
(iii)
-~.°o.
TIME s "
2:~-o
~
~.~o-
7.~o
+
~.'6o-~T~.o'
(iii)
5.00
P Z2ELTA rad/s 5.00
XI, E 0:
t :?°j,.A
-i.0
:'. [o-I ,"" v................
L00
-~.0
B
TIME s -5. o C o
'
2.'4o
'
4. eo
7.~o
'
~.'6o
72.0'
(iv)
................. ,..,^ .......... A.,..
.............
............................ vv,,'
1 t
-$'PO~OO '
TIME s 2.40
'
q,'O0 '
7.20
'
9.60
'12.0'
(iv)
Fig. 6. Speed (rad/s) response to torque input for the four cases studied.
Fig. 7. Error in terminal voltage (p.u.) in response to torque input for the four cases studied.
ty, the self-tuning regulator of case (ii) gives the best results. This is understandable because the regulator of case (ii) is m o s t responsive to speed variations. The disturbance applied, being a torque disturbance, is reflected in the speed variations. However, it minimizes the speed deviations at the cost of voltage deviations, as is evident from Fig. 7 which shows the error in terminal voltage. For minimization of the error in terminal voltage alone, case (iii) is the best. Case (i), where the
variance of all three quantities, i.e. i~, p5 and A V, is included, gives satisfactory results for both speed and voltage variations.
6.3. Quantitative evaluation of performance Qualitative results only can be obtained from time-response curves of the type given in Figs. 5 - 7. At times it becomes difficult to draw clear conclusions from such curves. For a quantitative evaluation and comparison of the four cases shown in Figs. 5 - 7, estimates
206 for the variances of AS, p8 and AV are calculated. A5 is given by Aa(t) = 6d(t ) - - 6 ( t )
(25)
where 5 a is the steady-state value of 6 corresponding to each step change in load disturbance. Table 1 shows the relative values of these variances with respect to the reference case (case (iv)), i.e.
V~{x(t)) = E{x~(t))/E{x~(t)), i = 1 , . . . , 4 (26) where i represents the case number. From Table 1 we see that satisfactory results can be obtained from the self-tuning regulator. Table 1 also shows that case (ii) gives the best performance when the damping of machine oscillations and the preservation of stability is of prime concern, but the voltage regulation deteriorates in this case. Case (iii) shows exactly the opposite results because terminal voltage regulation has the highest priority in this case. Out of these four studies, case (i) gives the best overall performance, showing that satisfactory results can be obtained from the self-tuning regulator. By changing the relative values of P2 and P3 it is possible to have the desired priorities to suit the goals of a particular system.
6.4. Field voltage Total field voltage e~ for the above four cases is shown in Fig. 8. We may notice that the excursions in total field voltage with the self-tuning regulator (cases (i) - (iii)) are in general much smaller than that for case (iv).
6.5. Identified model parameters Identified model parameters for case (i) are shown in Fig. 9. The parameters converge to constant values very quickly for each step change. A very similar trend was observed for the other two cases. This shows that the selftuning regulator is close to the minimum variance regulator.
7. IMPLEMENTATION AND EXPERIMENTAL RESULTS Software programs, used in simulation studies, for the identification and control algorithms described in § § 3 and 4 were suitably modified and implemented on an HP 21MX minicomputer for on-line tests. The minicomputer was used as a real-time controller for a laboratory micromachine set. The alternator was synchronized to the main laboratory power supply through a short transmission line represented by series R and L. The load on the alternator was varied by changing the armature current of the driving DC motor. The experimental set-up is shown in Fig. 10. Two channels of an eight-channel analogto-digital converter system with ten-bit resolution were used to supply the computer with the values of if(t) and AV(t) at each sampling instant. An eight-bit digital-to-analog converter was used to provide the field system with the desired value of ef(t) after its calculation in real-time in the computer. The normal operating conditions were selected as:
TABLE 1 Estimates of variances relative to the reference case (iv) Case
Yr {~a(t))
Yr(pa(t))
V~{~Y(t))
0.8966
0.8075
0.9212
0.7241
0.6721
1.1719
1.2414
1.2381
0.7969
1.0000
1.0000
1.0000
(i) Pl -- P2 -- P3 = 1.0 Self-tuning voltage regulator with stabilizer (ii) Pl = P2 = 1.0, P3 = 0.0 Self-tuning stabilizer (iii) Pl = P3 = 1.0, P2 = 0 Self-tuning voltage regulator (iv) AVR with stabilizer
207 5.00 -. O0
/: IELD VOL TAEE
p.u.
&O0
IDEIVTIF IED MODEl PARAME TESS
..
=62
l. OO -l. 2
-I. 0
×l, E-O~
k.---a.___ -S. 0 -& 5 TIME s -S.I OOO •
2.40
4.8O ~
-'00 1 f IELD V©L.TAEE -. 62
12.0
(i)
-5. 0
~ 4 . ' 8 o
'
7.'2o
'
m'6o
'~2. o'
Fig. 9. T h e i d e n t i f i e d m o d e l p a r a m e t e r s f o r case (i). Note: Actual units for various parameters depend u p o n t h e i n d i v i d u a l variables w i t h w h i c h t h e y are associated.
p.u.
-1.2 XI. E-O~ - -
-1.3 -& 5 Tlivi~ s
-S. 1~ -. O0
Z. 40
FIELD
~ -
7.20
3.'~o
la.o
(ii)
Machine response to step changes in load with a self-tuning regulator is shown in Fig. 11. The self-tuning regulator had the following parameters: m=3,
V©L TAEE p.u.
l =2,
=62
Pl = 0.1,
-1.2
#0 = 1.0
Xl. E-Oi
-2. 5
TIME s
! 0~
2.40
' 4.80 ' %-TO
3.6B
12.~
(iii)
0
t iv)
-.00 -. 62
-1.2 XL E-O
-1.3 -2. 5
-L .lOBo
P2 = 0, ps = 1.0
[eqn.(18)] [eqn. (15)] [eqn. (18)]
Sampling period = 100 ms
-1.3
-~"
k=0
2. 40
4.8~
7. 20
& 60 ~
Fig. 8. T o t a l field v o l t a g e in p . o . in r e s p o n s e t o torque input for the four cases studied.
(i) reference voltage V~ = 1.0 p.u (229 V line-to-line) (ii) load current Is = 0.4 p.u. (4 A) (iii) field current If = 0.65 p.u. (2.6 A) (iv) field voltage E~ = 0.65 p.u. (v) limits on ~ e f = +0.325 p.u. The following step changes in load of 20 s duration each were applied to the machine: (a) change Is from 0.4 p.u. to 0.64 p.u. and back to 0.4 p.u. (b) change I s from 0.4 p.u. to 0.56 p.u. and back to 0.4 p.u.
It can be seen from Fig. 11 that the voltage is regulated in less than 0.5 s. The supply to which the machine was connected caused small changes in the terminal voltage. Figure 11 shows that continuous regulation was obtained for these changes also. Selection of P2 = 0 in the self-tuning regulator means that the stabilizing effect of speed has been neglected. For a valid comparison of the system performance, a simple voltage regulator (Fig. 12) without a speed stabilizer [ 13 ] was selected. Machine response to the same load changes b u t with the simple regulator is shown in Fig. 13. It can be seen that the oscillations with this regulator are larger than with the self-tuning regulator and a steady-state error results in the terminal voltage. Although the experimental test results included in this Section are only preliminary results, they do look promising. A more extensive series of tests is planned in order to study the effect of the following elements: (i) sampling period, (ii) order of the system model, (iii) weighting function in the identification algorithm, (iv) relative values of coefficients P l, P2 and P3.
208
T.L.
Vdc
220
1 )
220idc-
. - Lab main power supply
I DT I CI
2],~
HP miniccmputer
Load changes Recorder Fig. 10. Experimental set-up.
IL P
~W
Ir p
%,
o
~
))
)~
w
B
m
m
~)
a
w
N u T~, I ~ ) ~
Fig. 11. System response to load changes with self-tuning regulator.
r
I
Vr )v~
0.52 l+0.1s
Fig. 12. Block diagram for a simple AVR.
Syr thronou$
machine
209
i
II
, I
o
s
I
i i
i
l
i'
I-t'-
4~
|o
Fig. 13. System response with simple AVR.
Results of these studies will be reported in the near future.
8. CONCLUSIONS
A self-tuning automatic voltage regulator for synchronous generators has been proposed. This regulator has the advantage of minimizing the speed error which improves the system stability. The field voltage, which is the control variable, changes slowly around the nominal value. It has been shown by simulation studies that the self-tuning regulator can give improved system performance in comparison with the conventional automatic voltage regulator. In the above studies the load variations were considered to be stochastic in nature. The computation time
for identification and control is small enough to make this regulator suitable to implement for real-time applications. The proposed algorithm has been implemented on a minicomputer. With the minicomputer acting as a real-time controller controlling the field voltage of an alternator, initial test results look promising. Based on these test results, though the authors feel that the results obtained by simulation studies can be achieved, additional tests are planned.
ACKNOWLEDGEMENTS
The authors wish to acknowledge the financial support of the National Research Council of Canada.
APPENDIX 1
M i n i m u m variance r e g u l a t o r
Consider the following single-input single-output stochastic discrete model: y(t) + aly(t--
1) + . . . + a , y ( t - - n )
= blu(t--k
- - 1) + . . .
+ k[e(t) + c l e ( t - Equation (A1-1) can also be expressed as A(q-1)y(t)
+ b,u(t--k--n)
1) + . . . + c , e ( t - - n ) ]
= B ( q - 1 ) u ( t - - k - - 1) + ~ C ( q - 1 ) e ( t )
+
(AI-1) (A1-2)
where q-1 is the backward shift operator, i.e. q - l y ( t ) = y ( t - - 1) and A ( q - 1 ) = 1 + a l q -1 + . . . + a , q -~
(A1-3a)
B ( q - 1 ) = b l + b 2 q -1 + . . .
(A1-3b)
C ( q - 1 ) = 1 + c l q -1 + . . .
+ b,q -'+1 + cnq-"
(A1-3c)
The polynomials B ( q - 1 ) a n d C ( q - 1 ) are assumed to have all their zeros inside the unit circle. This
210
assumption is necessary to derive the m i n i m u m variance control law [ 9]. At the instant t + k + 1, eqn. (A1-2) can be written as
y(t + k + 1) - B(q-1) u(t) + ~ C(q-1) e(t + k + 1) A(q -1 ) A-~)
(A1-4)
Let us define the following two polynomials: F(q -1) = 1 + f l q -1 + . . . + fkq -k G(q -1) = go + gl q-1 + . . . + g~ -1 q-n+1
(A1-5) (A1-6)
where F(q-1) and G(q-1) satisfy the following identity:
C(q-1) = A(q-1)F(q-i) + q-h-1G(q-1) Substitution of eqn. (A1-7) in eqn. (A1-4) and rearrangement of the terms gives B(q-1)F(q -1) G(q -1) y(t + k + 1) = ~,F(q-~)e(t + k + 1) + u(t) + - - y ( t ) C(q -1) C(q -1)
(A1-7)
(A1-8)
A m i n i m u m variance regulator is a regulator that minimizes the variance V of the o u t p u t y(t), given by
V = S(y2(t)}
(A1-9)
where E denotes the expectation operator. Let u(t) be an arbitrary function of y(t), y(t -- 1 ) , . . . , and u(t -- 1), u(t -- 2 ) , . . . . Then, from eqn. (A1-8), E{y2(t + k + 1)} = E{[hF(q-1)e(t + k + 1)] 2} +
+ E 2XF(q-1)e(t + k + 1 t
C-(q-~
u(O + ~C(q_1) y ( t
(AI-IO)
Since e(t + 1) . . . . , e(t + k), e(t + k + 1) are independent of y(t), y(t -- 1 ) , . . . and u(t), u(t -- 1), • . . , the last term in eqn. (AI-10) can be written as
]B(q-1)F(q-1) u(t)+ G(q-1) y(t)l f E{2hF(q-1)e(t + k + 1)t ~ C(q_l------) = 2hE {F( q-1 ) e ( t + k + l )} E{ B(q-1 )F( q-1) u (t)+ G(q-1-----~)y(t) f = 0 C(q -I) C(q -I)
(AI-ll)
Also E {e(t)} = 0 for all values of t. Equation (A1-10) can be reduced to
E(y2(t + k + 1)} = E([;~F(q-1)e(t + k + 1)] 2} + + E l[ s ( q - 1 ) f ( q - 1 ) t
E(y2(t+k+l)}
~
=~,2[l+f~ +...+f~]
G(q-1) y(t)
+
+ E {] B(q-1)F(q-1) G(q-~) y(t) L C~) u(t) + C(q - -~ ) Since
(A1-12)
u(t) + C(q - - -1)
]21
(Al-13)
E{e2(t)} = 1, it is clear that
E{y2(t + k + 1)}~> ),2[1 + f2 + . . .
+ f~]
(A1-14)
The m i n i m u m of the variance of the o u t p u t is obtained when the equality in eqn. (A1-14) holds. The equality holds for
211
B(q-1)F(q-1)u(t) + G(q-S)y(t) = 0
(A1-15)
Then the desired m i n i m u m variance control law is V(q -1) u(t) = -- B ( q _ l ) F ( q _ l ) y(t)
(A1-16)
where G ( q - 1 ) and F ( q - 1 ) are determined from the identity (A1-7). If the m i n i m u m variance regulator is used to control the system represented by eqn. (A1-1), then the o u t p u t is a moving-average process of order k, i.e. y(t) = k F ( q - 1 ) e ( t )
(A1-17)
If the parameters of the model (A1-1) are unknown, then they must be identified at each sampling instant and the control strategy (A1-16) implemented. To implement this control strategy it is necessary to solve the identity (A1-7) at each sampling instant. Owing to this difficulty, it is more convenient to replace eqn. (AI-1) at time t + k + 1 by [14] y ( t + k + 1) + a l y ( t ) + . . . + a m Y ( t - - m + 1) = 130[u(t) +~3iu(t-- 1) + . . . + ~ z u ( t - - l ) ] + e(t + k + 1)
(A1-18)
where m = n and l = n + k -- 1. The parameters a l . . . . , am and J30, 131, • • . , fit are obtained as functions of the parameters of the model (AI-1). The disturbance e(t) is a moving average of order k of the driving noise e(t). Equation (A1-18) can be written in the following form: y ( t + k + 1) = - - A ( q - 1 ) y ( t ) + B ( q - 1 ) u ( t ) + e(t + k + 1)
(Al-19)
where A(q -1 ) = al
+ a2q -1 + • • • +
am q-m+1
B(q - 1 ) =~011 +/~lq -1 + • . . + ~zq-t]
(A1-20)
(A1-21)
Proceeding as before, the minimum variance control strategy can be shown to be given by A ( q -1) u(t) - S ( q _ l ) y ( t )
(A1-22)
or 1
u(t) = ~o [al y ( t ) + . . .
+ a m Y ( t - - m + 1)] -- ~3xu ( t - - 1) - - . . . -- ~zu(t - - I)
(A1-23)
APPENDIX 2
System model The general equations which express the relationship between the various voltages, currents and flux linkages in a synchronous machine based on the equivalent circuits shown in Fig. A2-1 [15] are given below, and a simplified governor action [16] is represented: P $ d = laPid + LmdP(ia + ikd + if)
(A2-1)
0 = lkdPikd + rkaikd +LmdP(ia + ikd + if)
(A2-2)
ef = (rf + lfp)if - - IkdPikd - - rkdika
(A2-3)
P~d
=Em
sin 5 - - (ra + r e ) i d ~ v ~ q
p~q
=Em
cos~
--(r a +re)i q +V~d
--Xeiq +Xei d
(A2-4) (A2-5)
v = Wo --p5
(A2-6)
w° p2~ = 2H
(A2-7)
(To--Te)+G
212
__id j._
i
G
I rkd P~d id+ikd+i I~
ikdI
>
!f
Ld = l a + Lind
L = ~ +Lmq l~d= 2-kd+ Lind Lkq= 2kq+ Lmq q
f ~fII~fp
ikq
p~q iq+ikq I!
[
1
(a)
a
hmqP
(b)
Fig. A2-1. Equivalent circuits: (a) direct axis; (b) quadrature axis.
Te -- P, co 0/v
(A2-8)
Re = ~ ( $ q i d -- ~diq)
(A2-9)
where G is the g o v e r n o r effect, given by b G = ap5 + ~ p6 l + Tgp
(A2-10)
The a b o v e e q u a t i o n s were r e w r i t t e n in a f o r m suitable f o r simulation o n a digital c o m p u t e r using a general p u r p o s e s i m u l a t i o n program.
APPENDIX 3
Simulated system parameters
la
= 0 . 0 0 0 3 1 p.u.
L i n d = 0 . 0 0 2 9 6 p.u.
Lmq = 0 . 0 0 1 6 6 p.u.
/kd
= 0.00007 p.u.
lkq
= 0 . 0 0 0 0 7 p.u.
l~
= 0 . 0 0 0 5 9 8 p.u.
r~
= 0 . 0 0 0 8 9 p.u.
ra
= 0 . 0 0 7 p.u.
rkd
= 0 . 0 1 1 5 9 p.u.
rkq
= 0 . 0 1 1 5 9 p.u.
re
= 0 . 0 2 4 p.u.
Xe
= 0 . 1 1 5 p.u.
H
= 3.64 kW s / k V A
¢o0
= 377 rad/s
e%
= 0 . 0 0 1 5 5 p.u.
Em
= 1 . 4 1 4 2 1 p.u.
Vr
= 1.459p.u.
a
= --0.5
b
= --20
Tg
= 0.25 s
REFERENCES 1 M. W. Jervis, On-line computers in power stations. IEE Rev., 119 (8R) (!972) 1052 - 1076. 2 T. E. DyLiacco, Real-time computer control of power systems. Proc. IEEE, 62 (1974) 884 - 891. 3 K. J. Runtz, A. S. A. Farag, D. W. Huber, G. S. Hope and O. P. Malik, Digital control scheme for a generating unit. IEEE Trans., PAS-92 (1973) 478 - 483.
40. P. Malik, G. S. Hope and A. A. M. EIGhandakly, On-line adaptive control of synchronous machine excitation. Conference Record, IEEE lOth PICA Conference, Toronto, May 1977, pp. 59 - 67. 5 0 . P. Malik, G. S. Hope and R. Ramanujam, Real-time model reference adaptive control of synchronous machine excitation. IEEE PES Winter Meeting, New York, January 29 February 3, 1978, Paper No. A78 297-4.
213 6 P. A. L. Ham, Electronics in the control of turbine generators. Electron. Power, 24 (1978) 365 369. 7 L. R. Johnstone and C. R. Marsland, Distributed computer control for power station plant. Electron. Power, 24 (1978) 374 - 378. 8 F. C. Schweppe, Power systems "2000": hierarchical control strategies. IEEE Spectrum, 15 (1978) 42 - 47. 9 K. J. A~trom, Introduction to Stochastic Control Theory, Academic Press, New York, 1970. 10 K. J. Astrom and B. Wittenmark, Problems of identification and control. J. Math. Anal. Appl., 34 (1971) 99 - 113. 11 J. M. Mendel, Discrete Techniques of Parameter Estimation, Marcel Dekker, New York, 1973. -
12 K. J. ~kstrSm and B. Wittenmark, On self-tuning regulators. Automatica, 9 (1973) 185 - 199. 13 F. P. deMello and C. Concordia, Concepts of synchronous machine stability as affected by excitation control. IEEE Trans., PAS-88 (1969) 316 329. 14 K. J. i~strSm, U. Borisson, L. Ljung and B. Wittenmark, Theory and application of selftuning regulators. Automatica, 13 (1977) 457 476. 15 B. Adkins, The General Theory of Electrical Machines. Chapman and Hall, London, 1964. 16 O. P. Malik and B. J. Cory, Study of asynchronous operation and resynchronization of synchronous machines by mathematical models. Proc. Inst. Electr. Eng., 113 (1966) 1977 - 1990. -
199
A Self-Tuning Automatic Voltage Regulator
M. A. H. S H E I R A H
Ain-Shams University, Cairo (Egypt) O. P. M A L I K a n d G. S. H O P E
The University of Calgary, Calgary (Canada) (Received J u n e 19, 1 9 7 9 )
SUMMARY
A self-tuning automatic voltage regulator (AVR) for a synchronous generator is presented. The regulator proposed improves the system stability; it is simple and can handle stochastic load changes. The algorithm for the proposed AVR combines a leastsquares estimator with a minimum variance control strategy computed from an estimated model. It is shown that if the parameter estimates converge, the control law obtained is in fact the minimum variance control law that would be computed if the parameters of the system were known. The algorithm proposed has been tested by simulation and also by implementation on a minicomputer. Results show that, in general, the system performance is improved with a self-tuning regulator.
1. I N T R O D U C T I O N
Common procedure in designing power system controllers is to subject the power system model to step changes in load or voltage reference and observe the output under the influence of the controller. These off-line simulation studies using deterministic linear models are performed with the system linearized around a fixed operating point, the system parameters being assumed constant. Systems are, in general, non-linear and their parameters depend upon the operating conditions. Further, they are continuously subjected to load variations of random magnitude and duration, rather than to step changes. In view of this it would be more appropriate to:
(i) consider the problem as a stochastic control problem, and (ii) have controllers with dynamically changing coefficients to suit the varying operating conditions of the system. Because of problems in implementation using analog devices, controllers are built using coefficients that cannot be changed dynamically. However, some of these problems can be overcome by using digital devices. The use of digital computers for real-time control and operation of power systems has been increasing gradually [1, 2]. Initial applications of computers in power systems tended to be at the higher levels of system operation. However, recent developments in digital technology have made the use of digital devices as real-time controllers feasible at the generating level [3 - 5]. Use of digital devices for real-time control makes implementation of sophisticated control policies not only possible but also economically feasible [6- 8]. This paper describes an algorithm to perform the basic function of an automatic voltage regulator (AVR) considering the generating unit as a stochastic system. The proposed regulator has been tested on a generating unit modelled by a 9th-order set of simultaneous non-linear differential equations simulating a real-time process. Results of the simulation studies are described. Encouraging results obtained from the simulation studies prompted the authors to implement the proposed regulator. The regulator has been implemented using a minicomputer as a real-time controller and tested on-line to control the field excitation of a generating unit physical model in the labora-
200 tory. Initial test results have verified the results of the simulation studies. An illustrative set of test results is also included in the paper.
2. CONTROL STRATEGY The control inputs for a stochastic system are generally based on observations of some of the variables which describe the system. Because of various uncertainties, for example noise, unpredictable load variations, e t c . , these observations are imperfect. Further, the control policy that is utilized must be based on a p r i o r i knowledge of the system characteristics and on the time history of the input and o u t p u t variables. For such a system it is possible to compute the control based on minimizing the variance of an o u t p u t variable [9]. Variance is a measure of the deviation of a variable from its nominal value. An electric generating unit is a complex non-linear system. It may be modelled by a linear low-order discrete model with timevarying parameters. A recursive least-squares parameter estimation technique is used in this work to estimate the time-varying parameters of the discrete model. Based on this model, which tracks the operating conditions of the system, control is c o m p u t e d using a minimum variance strategy. Thus this process is the equivalent of a controller with dynamically changing coefficients and is referred to as the self-tuning regulator. The basic theory of a self-tuning regulator and its application as the equivalent of a conventional AVR with auxiliary stabilizing signal is described below.
3. SELF-TUNING REGULATOR Suppose that the system to be controlled can be represented by the following singleinput single-output stochastic discrete model: y(t) + a l y ( t - = blu(t--k--1)
+
. . . + bnu(t--k--n)
1)
+
. . . + cne(t
y ( t + k + 1) + a l y ( t ) =flo[u(t) +fllu(t-+ flzu(t--l)]
+ --
+ . . . + amY(t--m
+ 1)
1) + . . . +
+ e(t + k + l)
(2)
where m = n and l -- n + k --- 1. The parameters ~1 . . . . , ~m, ~1 . . . . . flz must now be identified. Go, the choice of which is not critical, is assumed to be known in order to avoid difficulty arising from the use of feedback [10]. The disturbance e ( t ) is a moving average of order k of the driving noise e ( t ) . The model of eqn. (2) can be obtained by writing eqn. (1) at time t + k + 1, for all ci = 0, and substituting the values of y ( t + k}, y(t + k -- 1), . . . , y ( t + 1) using eqn. {1) written at t i m e s t + k, t + k - - 1 . . . . , t + 1, respectively. As shown in Appendix 1, in this case the minimum variance strategy is 1
u(t) = ~[~ly(t)
1) + . . . + a , y ( t - - n )
+ ),[e(t) + c l e ( t - -
instant, k is a pure time delay of the system, n is the order of the system, ~ is a constant, (e(t)} is a sequence of independent, normal (0, 1) random variables, y ( t - - i), u ( t - - i) a n d e( t - - i) are the variables at time t - - iT, and T is the sampling period. The minimum variance control strategy for the system of eqn. (1) is given by eqn. (A116) which is developed in Appendix 1. The control is calculated using a model with known parameters for the system to be controlled. Since most systems are complex, non-linear, and time-varying, general models that aze valid for all operating conditions cann o t be obtained. Hence, to use a minimum variance strategy, a model has to be postulated and the parameters of such a model identified in real-time each sampling period. As shown in Appendix 1, implementation of eqn. (A1-16) requires the solution of the identity (A1-7) at each sampling instant, which is very time-consuming. To obtain an easy-to-implement minimum variance strategy, the model of eqn. (1) may be replaced by
+... + amy(t--m
--fllu(t--1)--.
. .--fltu(t--l)
+ 1)] --
(3)
n)]
(1) where y(t) and u ( t ) are the o u t p u t and input of the model, respectively, t is the sampling
A comparison of eqns. (A1-16) and (3) shows that use of the model structure (eqn. (2)) simplifies the computation of the control strategy.
201
In the self-tuning regulator, therefore, the parameters a l , . • .,a~, and ~1, • • .,#, in the predictive model (eqn. (2)) are estimated online at any sampling instant t. The estimated values ~1, • • . , 6m and/~t, • • .,/~l°f these parameters are then used to calculate the minimum variance control strategy (eqn. (3)). To estimate the parameters of the model in eqn. (2), the model m a y be rewritten at the instant t in the following form: r'(t)0
=
(4)
regulator (eqn. (13)) is the same as the minimum variance regulator. Since the system modelled by eqn. (2) with the regulator (eqn. (13)) and the least-squares parameter estimator (eqn. (8)) is described by a set of non-linear time-dependent stochastic difference equations, it is extremely difficult to provide a general p r o o f of convergence for the proposed algorithm [12]. However, it was verified that the algorithm did in fact converge for all simulation and experimental studies that were performed.
where F(t) = [ - - y ( t - - k - - 1), . . . . - - y ( t -- k -- m), ~ou(t--k--
2) . . . .
4. PROBLEM F O R M U L A T I O N
,~ou(t--k--l--1)]'
0 =
(5) (6)
~(t) = y ( t ) - - ~ o u ( t - - k - - 1)
(7)
and [ ]' denotes the transpose of a matrix. Several methods can be used to obtain an estimate #(t) for the vector parameters 8 [ 1 1 ] . Use of the recursive least-squares technique gives {](t) = ~(t -- 1) + K ( t ) [ ~ ( t ) - - r ' ( t ) ~ ( t -- 1)] (8)
The correction vector, K ( t ) , is obtained by the recursive relations K ( t ) = P(t -- 1)l'(t)[1 + F'(t)P(t --
1)r'(t)l-1 (9)
and P(t) = P(t -- 1) - - K ( t ) F ' ( t ) P ( t - - 1)
(10)
where P(t) is the covariance of the error in the estimates. To start the algorithm, the following initial conditions m a y be used:
0(0) =
(11)
0
P(0) = qI (12) where q is a large number and I is the identity matrix. Once 0 is determined, u ( t ) can be obtained as
1 u ( t ) = ~o [ 6 1 ( t ) y ( t ) + . . .
1)] --
+ 6m(t)y(t--m+
- - ~ l ( t ) u ( t - - 1) - - . . .
-- ~ t ( t ) u ( t - - 1) (13)
If the vector parameters #(t) converge to constant value vectors, then the self-tuning
The generating unit with the proposed selftuning A V R is shown in Fig. 1. The field voltage et is given by ef = e~n + ef s
(14)
where etn is the nominal value of the excitation voltage and efs is the self-tuning regulator output, ifn is the corresponding nominal value of the field current at steady state, e f , / R f , and it is the actual field current. In Fig. 1, V is the terminal voltage of the synchronous machine and Vr is the desired value of this voltage. ~ is the p o w e r angle and p6 = d 5 / d t is the speed error. T o represents the load disturbance. The objectives of this work are two-fold: (i) to damp the mechanical oscillations of the machine during transients, and (ii) to keep the terminal voltage of the machine as close as possible to the reference voltage. At the same time it is desired that the field current return to the nominal value as soon as the system returns to steady state. Selection of a suitable objective function and proper weighting coefficients would allow the desired relative importance to be given to the above t w o objectives. Mathematically, it is required to obtain a control signal ets which minimizes the variance of y s ( t ) defined by eqn. (A1-9). Considering the desired objective above, y s ( t ) has been taken to be Ys(S) = p l A i f ( t ) + pg.p~(t) + p s A V ( t )
(15)
where Pl, P2 and Ps are constants, Ai,(t) = if(t) -- if. (t) and
(16)
202 T
I cf
efn
Cenerating Unit
lif
Vr
us
--~
On-line Identifier h Ifn Computation ~ _ _ _ _
Ys = p~Aif * p2p~ + P3~V
Self-tuning Regulator
Fig. 1. Generating unit with proposed control scheme.
A V ( t ) = Vr(t) -- V(t)
(17)
The choice of P2 larger than p3 in eqn. (15) will give higher priority to the damping of mechanical oscillations and preservation of stability over voltage regulation. By modifying their relative values, one can reverse the priorities. For best overall performance the individual variances of Air(t), p~(t) and A V ( t ) must be minimized. If these three quantities are independent, minimization of the variance of y , ( t ) automatically yields individual minimum variances for if(t), p6 (t) and A V(t). However, these quantities are not independent. Thus the parameters p 1, P 2 and p a must be selected such that the minimum variance of y , ( t ) is close to the sum of the minimum variances of p l i A ( t ) , p z p 6 (t) and paA V(t). The system to be controlled is a complex non-linear system. It is possible to obtain a slowly time-varying linear model of the system about an arbitrary operating point. Let this model be of the following form:
y , ( t + k + 1) + alY,(t) + . . . + a m y , ( t - - m
+ 1)
= ~0[us(t) + ~ l U , ( t - - 1 ) + . . . +
+ ~zUs(t--/)] + e(t + k + 1)
(18)
where a l , • • • , am and ~1, • • •, ~z are slowly time-varying parameters, and ~o is a constantvalue parameter. k, m and l must be selected such that the model of eqn. (18) is a good approximation for the actual system. The parameters of this model can be estimated using the recursive least-squares technique (eqns. (8) (10)). The control signal, Us(t), can then be obtained as
1 Us(t) = ~o [&l(t)y,(t) + . . . + & m ( t ) y , ( t - - r n + 1)]
-~l(t) u(t-- 1)--...--~t(t)u(t--l)
(19) and
e%(T) = Gus(t)
t<<. r < t + 1
(20)
where G is a constant gain.
5. SYSTEM SIMULATION
A generating unit connected to an infinite bus through a short transmission line has been simulated by a 9th-order non-linear model.
203 The model and parameters used are given in Appendices 2 and 3, respectively. The synchronous machine model takes into account the stator transients and damper windings on the direct and quadrature axes. The turbine and governor are, however, represented by simplified models. The ultimate objective of these studies is to implement in real-time the proposed regulator using a minicomputer. The simulation studies therefore have been performed on an HP 21MX minicomputer which will also be used for implementation. For the studies described below, a sampling period of 50 ms was selected. In the discrete time-varying model of eqn. (18), rn has been taken equal to 3, k equal to zero, l equal to 2 and /30 equal to unity. Thus, the system model (eqn. (2)) becomes y,( t + I) +~I y,(t) + ~2 y,( t -- 1) + a3Y,( t -- 2)
= u , ( t ) + {Jl u , ( t - - 1) + ~2 U s ( t - - 2) + e(t + 1)
(21) where ys(t) = pIAi~(t) + P2P6(t) + p s A V ( t )
(22)
6. RESULTS OF SIMULATION STUDIES
The studies described in this section concentrate on the effect of the parameters Pl, P2 and P3 (eqn. {22)). Notice that to ensure the system will work at the same operating point if the load returns to its nominal value after a load disturbance, T
nominal value after a load disturbance, P l must be greater than zero. The following values for p 1, P 2 and p3 were considered in these studies {these values are based on the p.u. system used): (i) P l = 1.0, P2 = 1.0 and P3 = 1.0; (ii) Pl = 1.0, p2 = 1.0 a n d p s = 0.0; (iii) Pl = 1.0, p2 = 0.0 a n d p 3 = 1.0. In the first ease both p6 and ~ V have equal weight. The limiting values of P3 and P2 in the second and third cases, respectively, were chosen intentionally to investigate the effect of speed stabilization on system performance. Case (ii), in which the effect of terminal voltage variation is neglected, gives maximum weight to the speed stabilizing signal and would be expected to provide improved system stability during large system transients. On the other hand, case (iii), with no speed stabilizing signal, would be expected to result in minimum variations in the terminal voltage at the expense of system stability during large system disturbances. The most appropriate values for p 2 and p 3 would depend upon a particular system and lie between 0.0 and 1.0. The choice of Pz and P3 between 0.0 and 1.0 would yield results that lie between those of the second and third cases. (iv) A conventional A V R with a c o m m o n l y used stabilizer based on ref. 13 was considered. The system is shown in Fig. 2. The A V R used has the form [13] ely _ K, V 1 + TrP
i
V 6 P~ p.
I
Generating Unit
L_
ef t
e ~i f v
f
Stabilizer ^VR
Fig. 2. Conventional AVR with stabilizer.
/
t_ I-
=,j f
k-fl
Vr
(23)
204
where K, 0.5 and T, 0.02. The stabilizer used is =
6. ooi
=
p.u.
TOHC2~,',~" 5.56i
eft
Kp(1
_
p5
(1
+
+ Tip)
T2P)(1
+
2
Tap) 2
(24)
6.20 XI.E- O]
where K = 0.01, T1 = 0.125, T2 = 3 and Ta = 0.05.
i
i
4,80
i
4. q6
7/?4L s
6.1. Step change in reference voltage Remits for a 5% step change in the reference voltage are shown in Fig. 3. The selftuning regulator with p 1 = P 2 = P s = 1 gives considerably improved performance over the conventional regulator with stabilizer.
Fig. 4. T o r q u e d i s t u r b a n c e %.0
OEL TA
output.
rad
-5.2
NI
-S. 4
1°° t ERROR ]N TERMINAL V©L.
p.u.
Xl. E-g:
-5.6
• 600 I
-5.8 / L,~JL s -4.
0 ~ ,000
-3.0
• .Ou~o0
1.)0
2.'40 ' 5. BO
4.80
&00'
(a)
.
. . 2.. qO
DELTA
.
.
f.. 80.
.
.
.
. . 7.20
.
.
. . . 9.50
12.0
(i)
rad
5.2 ¸ -3. zl Xl.E-OI
-2. 0
-5. 6
[?EL TA
-2.4
rad
(i)
-L8
TIME s -2, 8
4..°ooo L~
Xl. E-Of
-L 2
-3.0
-Z 5
2.~o
IJEL TA
4.'8~
7. aO
9. 50
~-~O
7. 20
9. 60
la. 0
(ii)
rad
-5.2
TIME s
-4.0 )0 ' 1.'20
2, t0
S. 60
4. BO 6.00
-3. t XL E-Of
(b)
-5.5
Fig. 3. Terminal voltage (a) and load angle response (b) to a step change in reference voltage: curves (i), self-tuning regulator; curves (ii),conventional voltage regulator.
-5.8
6.2. Step changes in torque The generating unit was subjected to load disturbances of the form shown in Fig. 4. Changes in the torque from the nominal value (0.5 p.u.) were considered to be step changes with random magnitude and random duration. Power angle 6 and speed error p5 for all four cases are shown in Figs. 5 and 6, respectively. A close examination of these Figures shows that, from the point of view of stabili-
TIME s
-~.Oo~
2. 40
/
12. 0
(iii)
f
f
TIME 9. 5o
s
la. o
(iv)
Fig. 5. Power angle (rad) response to torque input for the four cases studied.
205 5.00 P DE," ]-~4
p
3. ooj:f ERROR I/V TERMINAL V©L.
rad/s
3.00+
;.B0!
I. O0 ':,
• 6001.
-.6of
p.u.
V
-l.B i
I
t
J
Z I~X~: s
TIME
s
(i) 5.00
(i) S. O0
ERROR I N TERMINAL
P ~ E L ,TA rad/s
5.00
1.80
l. OO
• 600
XI. E-O
XI.E-Oi
-I. 0
-.60
-5.0
VOL.
p,u.
-I.B
TIME -5. 0
2.'~0
4.80
7.20
9.60
s
12.0
TIME s (ii)
-S. 0 C
---~.~o
S.OOT
5. O0
P DELTA
i ERROR I N TERMINAL
rad/s
3.00
1.801
l. OO
• 6o0
~.6o 72.7
4.'BO ' 7.'ao
(ii)
VOL. p.u.
Xl. E-Ol
-1.0
-.60
.....
-].B I
-3.0
!~
TIME s -5. 0~
,0
'
2.'40
'
4.'80
'
7.'20 - '
9.~
12.0'
(iii)
-~.°o.
TIME s "
2:~-o
~
~.~o-
7.~o
+
~.'6o-~T~.o'
(iii)
5.00
P Z2ELTA rad/s 5.00
XI, E 0:
t :?°j,.A
-i.0
:'. [o-I ,"" v................
L00
-~.0
B
TIME s -5. o C o
'
2.'4o
'
4. eo
7.~o
'
~.'6o
72.0'
(iv)
................. ,..,^ .......... A.,..
.............
............................ vv,,'
1 t
-$'PO~OO '
TIME s 2.40
'
q,'O0 '
7.20
'
9.60
'12.0'
(iv)
Fig. 6. Speed (rad/s) response to torque input for the four cases studied.
Fig. 7. Error in terminal voltage (p.u.) in response to torque input for the four cases studied.
ty, the self-tuning regulator of case (ii) gives the best results. This is understandable because the regulator of case (ii) is m o s t responsive to speed variations. The disturbance applied, being a torque disturbance, is reflected in the speed variations. However, it minimizes the speed deviations at the cost of voltage deviations, as is evident from Fig. 7 which shows the error in terminal voltage. For minimization of the error in terminal voltage alone, case (iii) is the best. Case (i), where the
variance of all three quantities, i.e. i~, p5 and A V, is included, gives satisfactory results for both speed and voltage variations.
6.3. Quantitative evaluation of performance Qualitative results only can be obtained from time-response curves of the type given in Figs. 5 - 7. At times it becomes difficult to draw clear conclusions from such curves. For a quantitative evaluation and comparison of the four cases shown in Figs. 5 - 7, estimates
206 for the variances of AS, p8 and AV are calculated. A5 is given by Aa(t) = 6d(t ) - - 6 ( t )
(25)
where 5 a is the steady-state value of 6 corresponding to each step change in load disturbance. Table 1 shows the relative values of these variances with respect to the reference case (case (iv)), i.e.
V~{x(t)) = E{x~(t))/E{x~(t)), i = 1 , . . . , 4 (26) where i represents the case number. From Table 1 we see that satisfactory results can be obtained from the self-tuning regulator. Table 1 also shows that case (ii) gives the best performance when the damping of machine oscillations and the preservation of stability is of prime concern, but the voltage regulation deteriorates in this case. Case (iii) shows exactly the opposite results because terminal voltage regulation has the highest priority in this case. Out of these four studies, case (i) gives the best overall performance, showing that satisfactory results can be obtained from the self-tuning regulator. By changing the relative values of P2 and P3 it is possible to have the desired priorities to suit the goals of a particular system.
6.4. Field voltage Total field voltage e~ for the above four cases is shown in Fig. 8. We may notice that the excursions in total field voltage with the self-tuning regulator (cases (i) - (iii)) are in general much smaller than that for case (iv).
6.5. Identified model parameters Identified model parameters for case (i) are shown in Fig. 9. The parameters converge to constant values very quickly for each step change. A very similar trend was observed for the other two cases. This shows that the selftuning regulator is close to the minimum variance regulator.
7. IMPLEMENTATION AND EXPERIMENTAL RESULTS Software programs, used in simulation studies, for the identification and control algorithms described in § § 3 and 4 were suitably modified and implemented on an HP 21MX minicomputer for on-line tests. The minicomputer was used as a real-time controller for a laboratory micromachine set. The alternator was synchronized to the main laboratory power supply through a short transmission line represented by series R and L. The load on the alternator was varied by changing the armature current of the driving DC motor. The experimental set-up is shown in Fig. 10. Two channels of an eight-channel analogto-digital converter system with ten-bit resolution were used to supply the computer with the values of if(t) and AV(t) at each sampling instant. An eight-bit digital-to-analog converter was used to provide the field system with the desired value of ef(t) after its calculation in real-time in the computer. The normal operating conditions were selected as:
TABLE 1 Estimates of variances relative to the reference case (iv) Case
Yr {~a(t))
Yr(pa(t))
V~{~Y(t))
0.8966
0.8075
0.9212
0.7241
0.6721
1.1719
1.2414
1.2381
0.7969
1.0000
1.0000
1.0000
(i) Pl -- P2 -- P3 = 1.0 Self-tuning voltage regulator with stabilizer (ii) Pl = P2 = 1.0, P3 = 0.0 Self-tuning stabilizer (iii) Pl = P3 = 1.0, P2 = 0 Self-tuning voltage regulator (iv) AVR with stabilizer
207 5.00 -. O0
/: IELD VOL TAEE
p.u.
&O0
IDEIVTIF IED MODEl PARAME TESS
..
=62
l. OO -l. 2
-I. 0
×l, E-O~
k.---a.___ -S. 0 -& 5 TIME s -S.I OOO •
2.40
4.8O ~
-'00 1 f IELD V©L.TAEE -. 62
12.0
(i)
-5. 0
~ 4 . ' 8 o
'
7.'2o
'
m'6o
'~2. o'
Fig. 9. T h e i d e n t i f i e d m o d e l p a r a m e t e r s f o r case (i). Note: Actual units for various parameters depend u p o n t h e i n d i v i d u a l variables w i t h w h i c h t h e y are associated.
p.u.
-1.2 XI. E-O~ - -
-1.3 -& 5 Tlivi~ s
-S. 1~ -. O0
Z. 40
FIELD
~ -
7.20
3.'~o
la.o
(ii)
Machine response to step changes in load with a self-tuning regulator is shown in Fig. 11. The self-tuning regulator had the following parameters: m=3,
V©L TAEE p.u.
l =2,
=62
Pl = 0.1,
-1.2
#0 = 1.0
Xl. E-Oi
-2. 5
TIME s
! 0~
2.40
' 4.80 ' %-TO
3.6B
12.~
(iii)
0
t iv)
-.00 -. 62
-1.2 XL E-O
-1.3 -2. 5
-L .lOBo
P2 = 0, ps = 1.0
[eqn.(18)] [eqn. (15)] [eqn. (18)]
Sampling period = 100 ms
-1.3
-~"
k=0
2. 40
4.8~
7. 20
& 60 ~
Fig. 8. T o t a l field v o l t a g e in p . o . in r e s p o n s e t o torque input for the four cases studied.
(i) reference voltage V~ = 1.0 p.u (229 V line-to-line) (ii) load current Is = 0.4 p.u. (4 A) (iii) field current If = 0.65 p.u. (2.6 A) (iv) field voltage E~ = 0.65 p.u. (v) limits on ~ e f = +0.325 p.u. The following step changes in load of 20 s duration each were applied to the machine: (a) change Is from 0.4 p.u. to 0.64 p.u. and back to 0.4 p.u. (b) change I s from 0.4 p.u. to 0.56 p.u. and back to 0.4 p.u.
It can be seen from Fig. 11 that the voltage is regulated in less than 0.5 s. The supply to which the machine was connected caused small changes in the terminal voltage. Figure 11 shows that continuous regulation was obtained for these changes also. Selection of P2 = 0 in the self-tuning regulator means that the stabilizing effect of speed has been neglected. For a valid comparison of the system performance, a simple voltage regulator (Fig. 12) without a speed stabilizer [ 13 ] was selected. Machine response to the same load changes b u t with the simple regulator is shown in Fig. 13. It can be seen that the oscillations with this regulator are larger than with the self-tuning regulator and a steady-state error results in the terminal voltage. Although the experimental test results included in this Section are only preliminary results, they do look promising. A more extensive series of tests is planned in order to study the effect of the following elements: (i) sampling period, (ii) order of the system model, (iii) weighting function in the identification algorithm, (iv) relative values of coefficients P l, P2 and P3.
208
T.L.
Vdc
220
1 )
220idc-
. - Lab main power supply
I DT I CI
2],~
HP miniccmputer
Load changes Recorder Fig. 10. Experimental set-up.
IL P
~W
Ir p
%,
o
~
))
)~
w
B
m
m
~)
a
w
N u T~, I ~ ) ~
Fig. 11. System response to load changes with self-tuning regulator.
r
I
Vr )v~
0.52 l+0.1s
Fig. 12. Block diagram for a simple AVR.
Syr thronou$
machine
209
i
II
, I
o
s
I
i i
i
l
i'
I-t'-
4~
|o
Fig. 13. System response with simple AVR.
Results of these studies will be reported in the near future.
8. CONCLUSIONS
A self-tuning automatic voltage regulator for synchronous generators has been proposed. This regulator has the advantage of minimizing the speed error which improves the system stability. The field voltage, which is the control variable, changes slowly around the nominal value. It has been shown by simulation studies that the self-tuning regulator can give improved system performance in comparison with the conventional automatic voltage regulator. In the above studies the load variations were considered to be stochastic in nature. The computation time
for identification and control is small enough to make this regulator suitable to implement for real-time applications. The proposed algorithm has been implemented on a minicomputer. With the minicomputer acting as a real-time controller controlling the field voltage of an alternator, initial test results look promising. Based on these test results, though the authors feel that the results obtained by simulation studies can be achieved, additional tests are planned.
ACKNOWLEDGEMENTS
The authors wish to acknowledge the financial support of the National Research Council of Canada.
APPENDIX 1
M i n i m u m variance r e g u l a t o r
Consider the following single-input single-output stochastic discrete model: y(t) + aly(t--
1) + . . . + a , y ( t - - n )
= blu(t--k
- - 1) + . . .
+ k[e(t) + c l e ( t - Equation (A1-1) can also be expressed as A(q-1)y(t)
+ b,u(t--k--n)
1) + . . . + c , e ( t - - n ) ]
= B ( q - 1 ) u ( t - - k - - 1) + ~ C ( q - 1 ) e ( t )
+
(AI-1) (A1-2)
where q-1 is the backward shift operator, i.e. q - l y ( t ) = y ( t - - 1) and A ( q - 1 ) = 1 + a l q -1 + . . . + a , q -~
(A1-3a)
B ( q - 1 ) = b l + b 2 q -1 + . . .
(A1-3b)
C ( q - 1 ) = 1 + c l q -1 + . . .
+ b,q -'+1 + cnq-"
(A1-3c)
The polynomials B ( q - 1 ) a n d C ( q - 1 ) are assumed to have all their zeros inside the unit circle. This
210
assumption is necessary to derive the m i n i m u m variance control law [ 9]. At the instant t + k + 1, eqn. (A1-2) can be written as
y(t + k + 1) - B(q-1) u(t) + ~ C(q-1) e(t + k + 1) A(q -1 ) A-~)
(A1-4)
Let us define the following two polynomials: F(q -1) = 1 + f l q -1 + . . . + fkq -k G(q -1) = go + gl q-1 + . . . + g~ -1 q-n+1
(A1-5) (A1-6)
where F(q-1) and G(q-1) satisfy the following identity:
C(q-1) = A(q-1)F(q-i) + q-h-1G(q-1) Substitution of eqn. (A1-7) in eqn. (A1-4) and rearrangement of the terms gives B(q-1)F(q -1) G(q -1) y(t + k + 1) = ~,F(q-~)e(t + k + 1) + u(t) + - - y ( t ) C(q -1) C(q -1)
(A1-7)
(A1-8)
A m i n i m u m variance regulator is a regulator that minimizes the variance V of the o u t p u t y(t), given by
V = S(y2(t)}
(A1-9)
where E denotes the expectation operator. Let u(t) be an arbitrary function of y(t), y(t -- 1 ) , . . . , and u(t -- 1), u(t -- 2 ) , . . . . Then, from eqn. (A1-8), E{y2(t + k + 1)} = E{[hF(q-1)e(t + k + 1)] 2} +
+ E 2XF(q-1)e(t + k + 1 t
C-(q-~
u(O + ~C(q_1) y ( t
(AI-IO)
Since e(t + 1) . . . . , e(t + k), e(t + k + 1) are independent of y(t), y(t -- 1 ) , . . . and u(t), u(t -- 1), • . . , the last term in eqn. (AI-10) can be written as
]B(q-1)F(q-1) u(t)+ G(q-1) y(t)l f E{2hF(q-1)e(t + k + 1)t ~ C(q_l------) = 2hE {F( q-1 ) e ( t + k + l )} E{ B(q-1 )F( q-1) u (t)+ G(q-1-----~)y(t) f = 0 C(q -I) C(q -I)
(AI-ll)
Also E {e(t)} = 0 for all values of t. Equation (A1-10) can be reduced to
E(y2(t + k + 1)} = E([;~F(q-1)e(t + k + 1)] 2} + + E l[ s ( q - 1 ) f ( q - 1 ) t
E(y2(t+k+l)}
~
=~,2[l+f~ +...+f~]
G(q-1) y(t)
+
+ E {] B(q-1)F(q-1) G(q-~) y(t) L C~) u(t) + C(q - -~ ) Since
(A1-12)
u(t) + C(q - - -1)
]21
(Al-13)
E{e2(t)} = 1, it is clear that
E{y2(t + k + 1)}~> ),2[1 + f2 + . . .
+ f~]
(A1-14)
The m i n i m u m of the variance of the o u t p u t is obtained when the equality in eqn. (A1-14) holds. The equality holds for
211
B(q-1)F(q-1)u(t) + G(q-S)y(t) = 0
(A1-15)
Then the desired m i n i m u m variance control law is V(q -1) u(t) = -- B ( q _ l ) F ( q _ l ) y(t)
(A1-16)
where G ( q - 1 ) and F ( q - 1 ) are determined from the identity (A1-7). If the m i n i m u m variance regulator is used to control the system represented by eqn. (A1-1), then the o u t p u t is a moving-average process of order k, i.e. y(t) = k F ( q - 1 ) e ( t )
(A1-17)
If the parameters of the model (A1-1) are unknown, then they must be identified at each sampling instant and the control strategy (A1-16) implemented. To implement this control strategy it is necessary to solve the identity (A1-7) at each sampling instant. Owing to this difficulty, it is more convenient to replace eqn. (AI-1) at time t + k + 1 by [14] y ( t + k + 1) + a l y ( t ) + . . . + a m Y ( t - - m + 1) = 130[u(t) +~3iu(t-- 1) + . . . + ~ z u ( t - - l ) ] + e(t + k + 1)
(A1-18)
where m = n and l = n + k -- 1. The parameters a l . . . . , am and J30, 131, • • . , fit are obtained as functions of the parameters of the model (AI-1). The disturbance e(t) is a moving average of order k of the driving noise e(t). Equation (A1-18) can be written in the following form: y ( t + k + 1) = - - A ( q - 1 ) y ( t ) + B ( q - 1 ) u ( t ) + e(t + k + 1)
(Al-19)
where A(q -1 ) = al
+ a2q -1 + • • • +
am q-m+1
B(q - 1 ) =~011 +/~lq -1 + • . . + ~zq-t]
(A1-20)
(A1-21)
Proceeding as before, the minimum variance control strategy can be shown to be given by A ( q -1) u(t) - S ( q _ l ) y ( t )
(A1-22)
or 1
u(t) = ~o [al y ( t ) + . . .
+ a m Y ( t - - m + 1)] -- ~3xu ( t - - 1) - - . . . -- ~zu(t - - I)
(A1-23)
APPENDIX 2
System model The general equations which express the relationship between the various voltages, currents and flux linkages in a synchronous machine based on the equivalent circuits shown in Fig. A2-1 [15] are given below, and a simplified governor action [16] is represented: P $ d = laPid + LmdP(ia + ikd + if)
(A2-1)
0 = lkdPikd + rkaikd +LmdP(ia + ikd + if)
(A2-2)
ef = (rf + lfp)if - - IkdPikd - - rkdika
(A2-3)
P~d
=Em
sin 5 - - (ra + r e ) i d ~ v ~ q
p~q
=Em
cos~
--(r a +re)i q +V~d
--Xeiq +Xei d
(A2-4) (A2-5)
v = Wo --p5
(A2-6)
w° p2~ = 2H
(A2-7)
(To--Te)+G
212
__id j._
i
G
I rkd P~d id+ikd+i I~
ikdI
>
!f
Ld = l a + Lind
L = ~ +Lmq l~d= 2-kd+ Lind Lkq= 2kq+ Lmq q
f ~fII~fp
ikq
p~q iq+ikq I!
[
1
(a)
a
hmqP
(b)
Fig. A2-1. Equivalent circuits: (a) direct axis; (b) quadrature axis.
Te -- P, co 0/v
(A2-8)
Re = ~ ( $ q i d -- ~diq)
(A2-9)
where G is the g o v e r n o r effect, given by b G = ap5 + ~ p6 l + Tgp
(A2-10)
The a b o v e e q u a t i o n s were r e w r i t t e n in a f o r m suitable f o r simulation o n a digital c o m p u t e r using a general p u r p o s e s i m u l a t i o n program.
APPENDIX 3
Simulated system parameters
la
= 0 . 0 0 0 3 1 p.u.
L i n d = 0 . 0 0 2 9 6 p.u.
Lmq = 0 . 0 0 1 6 6 p.u.
/kd
= 0.00007 p.u.
lkq
= 0 . 0 0 0 0 7 p.u.
l~
= 0 . 0 0 0 5 9 8 p.u.
r~
= 0 . 0 0 0 8 9 p.u.
ra
= 0 . 0 0 7 p.u.
rkd
= 0 . 0 1 1 5 9 p.u.
rkq
= 0 . 0 1 1 5 9 p.u.
re
= 0 . 0 2 4 p.u.
Xe
= 0 . 1 1 5 p.u.
H
= 3.64 kW s / k V A
¢o0
= 377 rad/s
e%
= 0 . 0 0 1 5 5 p.u.
Em
= 1 . 4 1 4 2 1 p.u.
Vr
= 1.459p.u.
a
= --0.5
b
= --20
Tg
= 0.25 s
REFERENCES 1 M. W. Jervis, On-line computers in power stations. IEE Rev., 119 (8R) (!972) 1052 - 1076. 2 T. E. DyLiacco, Real-time computer control of power systems. Proc. IEEE, 62 (1974) 884 - 891. 3 K. J. Runtz, A. S. A. Farag, D. W. Huber, G. S. Hope and O. P. Malik, Digital control scheme for a generating unit. IEEE Trans., PAS-92 (1973) 478 - 483.
40. P. Malik, G. S. Hope and A. A. M. EIGhandakly, On-line adaptive control of synchronous machine excitation. Conference Record, IEEE lOth PICA Conference, Toronto, May 1977, pp. 59 - 67. 5 0 . P. Malik, G. S. Hope and R. Ramanujam, Real-time model reference adaptive control of synchronous machine excitation. IEEE PES Winter Meeting, New York, January 29 February 3, 1978, Paper No. A78 297-4.
213 6 P. A. L. Ham, Electronics in the control of turbine generators. Electron. Power, 24 (1978) 365 369. 7 L. R. Johnstone and C. R. Marsland, Distributed computer control for power station plant. Electron. Power, 24 (1978) 374 - 378. 8 F. C. Schweppe, Power systems "2000": hierarchical control strategies. IEEE Spectrum, 15 (1978) 42 - 47. 9 K. J. A~trom, Introduction to Stochastic Control Theory, Academic Press, New York, 1970. 10 K. J. Astrom and B. Wittenmark, Problems of identification and control. J. Math. Anal. Appl., 34 (1971) 99 - 113. 11 J. M. Mendel, Discrete Techniques of Parameter Estimation, Marcel Dekker, New York, 1973. -
12 K. J. ~kstrSm and B. Wittenmark, On self-tuning regulators. Automatica, 9 (1973) 185 - 199. 13 F. P. deMello and C. Concordia, Concepts of synchronous machine stability as affected by excitation control. IEEE Trans., PAS-88 (1969) 316 329. 14 K. J. i~strSm, U. Borisson, L. Ljung and B. Wittenmark, Theory and application of selftuning regulators. Automatica, 13 (1977) 457 476. 15 B. Adkins, The General Theory of Electrical Machines. Chapman and Hall, London, 1964. 16 O. P. Malik and B. J. Cory, Study of asynchronous operation and resynchronization of synchronous machines by mathematical models. Proc. Inst. Electr. Eng., 113 (1966) 1977 - 1990. -