Comput.& E/ect.~ Vol.7, pp. 205-210 ~PersamonPressLtd.,1980. PrintedinGreatBritain
0045-79G6/80/Og01-0205/$02.00/0
DESIGN OF DYNAMIC OUTPUT FEEDBACK CONTROLLER FOR POWER SYSTEM STABILIZATION LIM CHOO MIN Ngee Ann TechnicalCollege,ElectricalEngineeringDepartment,Singapore2159
and CHOI SAN SHING University of Singapore, Electrical Engineering Department, Singapore 05 ! I
(Receivedfor publication 2 July 1980) Abstract--A computational method of designing dynamic compensators for stabilization of power systems is described. The control strategy only employs feedback from the available system output variables. A design procedure which would result in a stable closed-loop system over wide operating points is also included.
INTRODUCTION
The use of a high-gain, fast-acting voltage regulator in the excitation system of a large synchronous generator to improve system transient stabilityis well established[I].However, it has been shown that under certain loading conditions [2],the voltage regulator could introduce negative damping into the network. As a result, even a minor disturbance could set off sustained oscillationin the system. Such a phenomenon has been reported in [3,4]. One effective way of damping the oscillationis to design a supplementary controller in the excitation control loop of the generator set so as to left shift the system dominant eigenvalues. This approach has been considered by several researchers, see Moussa et al.[5],Habibullah et al.[6], and Mansour[7]. In [5] and [6], pole shifting is achieved with the aid of an optimal control law while in [7], a linear programming technique is used. The control strategies described in [5,6] necessitate the feeding back of all the system state variables but it has been recognised recently that not all these natural system states are directly measurable [8]. Other researchers have also proposed various design techniques for the power system stabilizer via classical control theory. Based on the concept of synchronizing and damping torque, De Mello and Concordia[2] have proposed a frequency domain technique to the design problem while Bollinger and Laha[9] have made use of the classical root locus methods. Although considerable insight into the design problem can be gained from these techniques, they are not particularlysuitable for use on large systems. In this paper a design procedure, adapted with some modifications from that shown in [I0], is described and its use to achieve dominant-poles shifting in power systems is demonstrated. Unlike the work of [5] and [6],the resulting control law only requires the feeding back of the available system output variables. Moreover, the structure of the controller is completely general and no constraints are imposed in any of the controller matrices. Consequently, system stabilizationis obtained with dynamic compensators that are of low orders. A design procedure which would result in a stable power system over wide operating conditions has also been included. ANALYSIS
Consider a linear system with the state equation i=Ax+Bu } y = Cx
(I)
where x is an n-vector of system state variables, u is an m-vector of system input and y is an 205
206
LlldCHO0MINand C.oi SANSmr~G
r-vector of system output. It is desired to synthesize a dynamic system with input y, which will generate an appropriate control law u to shift the dominant eigenvalues of (1) to the left of the complex plane. Such a controller, of order p, say, may be represented as ks = Pxs + Ny ] I"
u = Hxs + G y J
(2)
where x, is the p-order dynamic controller state variables and P, N, H and G are constant matrices of appropriate dimensions. The composite closed-loop system may be described by
00] 0,])
x = (A + BFC)i
where
[:,], [A
p
]
(3)
To aid the shifting of the system q dominant eigenvalues, a least-square criterion of the form
J= ~
ISiA(Ai)[2=
hrh
i=l
h = [S~, Re{A(A,)}, $1, Im{A(A;)}. . . . .
(4)
&Re{atAq)}, Sjm{atAq)}] T is chosen where A(,~) = 0 is the characteristic equation of system eqn (3), Si a positive constant and ,~; the new desired dominant eignevalues. Thus the design problem is to select the controller parameter F which minimizes J subject to system eqn (3). SOLUTION TECHNIQUE Any standard optimization algorithm may be used to find solutions to the numerical problem. Storing F as a m r × ! column vector and defining the m r × 2 q - s matrix N, where s is the number of real desired poles, as ~h r N = c~F
(5)
then the well-known Gauss-Newton iteration formula minimizing J w.r.t F is given by
8F =
- ( N N T ) -' N h _N(NrN)-lh
if m r if m r
<- 2 q - s > 2 q - s.
(6)
The cost J and N are computed from A ( A ) = A "+p + al A " + p - I + • • • + a.+p
(7)
~F = A"+P-I/+
, ~ " + P - 2 & l + " • • + ~b.+p-i
where at, a2. . . . . a,+p and ,#,, 4'2. . . . . tb,+p , are readily obtained using the Leverrier's algorithm [l 1].
Design of dynamic output feedback controller for power system stabilization
207
An algorithm for the minimization of J is given in Fig. 1. To ensure system stability over a wide operating range, the algorithm also requires the computation of closed-loop eigenvalues at different operating points. Should any of these eigenvalues lie near or to the right of the imaginary axis of the complex plane, the controller is redesigned by further left shifting the unstable dominant poles. Repeated failures to stabilize the closed-loop system under wide operating conditions would indicate that the order of the compensator is too low. Therefore, p may be increased by 1 and the procedure repeated as outlined above. Note that for a controllable, observable system (1), all n +p poles of the closed-loop system (3) may be arbitrarily assigned using a compensator of order p
min(vo- 1, v~ I)
=
-
where Voand vc are the observability and controllability indices of (1)[12]. EXAMPLE
Consider a single-machine infinite-bus power system. It has been demonstrated that a 3rd order machine model with a one-time constant voltage regulator is good enough for controller design purposes[5, 8]. The linearised equation for such a model may be written in the form of (1) where x [AS Aw AE,~AEH]r and y [Aw Avt] r. For practical realisation of the controller, the output of the system is taken to be either the machine speed (Aw) and/or terminal voltage deviations (Av,)[8]. Note that the machine terminal voltage is actually a linear combination of the machine rotor angle (A~) and (AE~)[2]. =
=
J
Read data Select I
I Accord, I Acc
i
I
I I
Set iteration count IC=O
I
J Compute J , N and SF
old.
No
I
Compute system poles X over wide operoting conditions
:.]_
,x.inew x Iold
'e,J
Display F and J
E3
I
1 Stop
I
Note: E1 . £2 and E3 ore certain preass~jned tolerances Fig. I. An algorithm for power systems stabilization over wide operating conditions.
208
LIM CHO0 M1N and CHOI SAN SHING
Using the numerical data from [6], the following system matrices are obtained
A
0
001[
- 22.5 -0.086 85.7
0 - 47.4 0 ,B = 0 - 0.195 0.129 - 882 - 20 103
[0 C:
-0.086
1
,0 0 0.822
and the system open-loop eigenvalues are 0.25-j4.96, -10.4-+./3.31, indicating an unstable system. It is desired to stabilize the system over a wide-operating range with low order controller by left shifting the system dominant poles. A sample of the results obtained are summarised in Tables 1 and 2, where only the dominant eigenvalues are included. In each of the three cases considered, the closed-loop system is stable. The three controllers shown in Table 1 are next implemented on a 6th order non-linear model of the single-machine infinite-bus system simulated on a digital computer, in order to test the effectiveness of each of the controllers designed. A three-phase short-circuit fault is assumed on one of the two transmission lines which connect the machine and the infinite-bus. The fault clearing sequence is exactly like that shown in [6]. Figure 2 shows the response of the closed-loop system under nominal load conditions while Fig. 3 shows the rotor angle swing under heavy and light load conditions. Clearly, all three supplementary controllers obtained can stabilize the unstable open-loop system over wide operating points. CONCLUSION
A numerical technique for pole shifting is shown and its use to achieve dominant eigenvalues shift of the power system has been demonstrated. The method enables the design of physically realisable controllers since only the system measurable output variables are feedback. Stabilization of the system over wide operating points can also be attained by following the design procedure outlined above. Table I. Output feedback controllers' matrices and eigenvalues--power system under nominal load conditions System at nominal load; Case Output vector, yT
p
1
law Avt]
0
2
[Awl
I
3
[AwAvt]
desired dominant poles - 2.25 -j4,96 F Closed-loop dominant poles [0.6818 - 8.39] [0.2412 0.7219] [-5.00 - 19.8]
[0.2204 -0.501 1 [4.00 0
-0.501] -15.5 1
- 2.25 -+j4.% - 2.26_ j4.96
-2'23-+j4"97
Table 2. System dominant eigenvalues over wide operating conditions Open-loop dominant poles System at heavy load load 0.676 - j3.40
System at light load - 0.009 _+j5.23
Closed-loop dominant poles Case 1 2 3
Heavy load condi-Lightload conditions tions - 2.42 _+j2.30 - 1.45 ± j5.54 - 1.17 +_j3.33 - 1.57 - j5.56 - 1.59-+j3.17 -1.50+-J 5.55
Design of dynamic output feedback controller for power system stabilization
209
(rod)
/~S
~!\ ,,-',~----'--~-~
o
,L
,'
I 1.0
1 2.0
-
tfsec)
(a)
A~
(rod/sec)
0
L
I 1.0
I 2,0
-
t(sec)
(b)
UE (P.U.)
I
I 2.0
1.0
t(se¢)
(c)
Hg. 2. System responses under nominal load with different controllers: controller 1 2 . . . . ; controller 3 ..... . CAEE VoL 7, No. 3--E
; controller
210
Llu C.oo MIN hnd C,ol SAN S.ING A~
(rod)
1 2.0
1.0
t(sec)
(a)
AS (rod)
tO O
f~.
, , ' 7 - :.-z- - . ".x / /
',.%/
6
I Z0
10
t (sec)
(b) Fig. 3. Machine torque angle swing under (a) heavy load conditions; (b) light load conditions; controller I ~; controller 2 . . . . ; controller 3 ..... .
REFERENCES 1. P. M. Anderson and A. A. Souad, Power System Control and Stability, Vol. I. Iowa State University Press (1977). 2. F. P. De Mello and C. Concordia, Concepts of synchronous machines stability as affected by excitation control. IEEE Trans Power Apparatus and Sys. PAS-U, 316-329 (1969). 3. P. A. Rusche, D. L. Hackett, D. H. Baker, G. E. Gareis and P. C. Kraus, Investigation of the dynamic oscillations of the Ludington pumped storage plant. IEEE Trans Power Apparatus and Sys. PAS-95, 1854--1862 (1976). 4. R. H. Millan, J. A. Mendoza, C. Cardoza and A. de Lima, Dynamic stability and power system stabilizers analysis and tests on the Venezuelen system. IEEE Trans Power Apparatus and Sys. PAS-96, 855-862 (1977). 5. H. A. M. Moussa and Y. N. Yu, Optimal power system stabilization through excitation and/or governor control. IEEE Trans Power Apparatus and Sys. PAS-91, 1166-1174 (1972). 6. B. Habibullah and Y. N. Yu, Physically realizable wide range optimal controllers for power systems. IEEE Trans Power Apparatus and Sys. PAS-93, 1498-1506 (1974). 7. M. O. Mansour, Hiarchical control of interconnected power systems. Ph.D. Thesis, Purdue University, School of Engineering (1975). 8. J. H. Anderson, M. A. Hutchison, W. J. Wilson, M. A. Zohdy and J. D. Aplevich, Microalternator experiments to verify the physical realizability of simulated optimal controllers to associated sensitivity studies. IEEE Trans Power Apparatus and Sys. PAS-97, 649-658 (1978). 9. K. Bollinger and A. Laha, Power system stabilizer design using root locus methods. IEEE Trans PowerApparatus and Sys. PAS-94, 1458--1484 (1975). 10. H. R. Sirisena and S. S. Choi, Optimal pole placement in linear multivariable systems using dynamic output feedback, Int. J. Cont. 21, 661-671 (1975). 11. S. Barnett and C. Storey, Matrix Methods in Stability Theory, Nelson (1972). 12. F. M. Brasch Jr., and J. B. Pearson, Pole placement using dynamic compensators, IEEE Trans Auto. Control AC-15, 34--43 (1970).