Suboptimal control of large scale power systems using decomposition techniques

SUBOPTIMAL CONTROL OF LARGE SCALE POWER SYSTEMS USING DECOMPOSITION TECHNIQUES J. D. Delacour*, M. Darwish** andJ. Fantin* *Laboratoire d'Automatique et d'Analyse des Systemes du C.N.R.S., 7, Avenue du Colonel Roche, 31400 Toulouse, France **Laboratoire d'Automatique et d'Analyse des Systemes du C.N.R.S., 7, Avenue du Colonel Roche, 31400 Toulouse, France and with Department of Electronics and Communication Engineen'ng, Cairo University, Egypt

The solution of linear regulator problem for large scale power systems is a difficult task, since it involves the solution of a matrix Riccati equation consisting of n (n+1)/2 coupled nonlinear equations for nth order system. In this paper a suboptimal control strategy is presented in which the system is decomposed to N- " € -coupled" subsystems. In this way the solution of matrix Riccati equation is approximated by a trucated Mac-Laurin series which requires less computations than the original solution of the problem. The applicability of the method is illustrated by an example consisting of three power plants.

1. INTRODUCTION Electric power system is usually subjected to per tubations of different natures and it is necessary to bring it back to steady state in an optimal manner [1, 2 ] , this is the role of the system r~ulators which include speed and voltage regulators [3J • But due to the rise in the demand of energy sources, the complexity of power systems is increased, hence the problem of optimal regulators will be a difficult task. In order to overcome the problems of high dimensionality and complexity of power systems, several methods has been suggested [4, 5 ] In these methods either an optimal solution is obtained through decomposition-coordination techniques of hierarchical control (which requires large information exchange between the decomposed subsystems and the coordinator) or a suboptimal solution is obtained through partitioning the system state vector (which is less accurate). In [6] a method for computing near optimal regulators for linear system with quadratic performance indices based on approximating matrix Riccati equation by a truncated series is developed. In this paper a generalization of this method is illustrated as applied to large scale power systems. Finally the method is demonstrated by an example consisting of three power plants, each plant is modelled as a fourth order system (third order synchronous machine and a first order exciter regulator system) and the suboptimal trajectories obtained indicate the efficiency of this approach when compared with the optimal solution of the problem.

where Q is (n x n ) positive semidefinite matrix and R is (m x m) positive definite matrix. Using Pontryagins' maximum principle the control signal u can be given by [9 ] u =_R-

1 BT K X

(3)

where K is (n x n) symmetric matrix and is the solution of the matrix nonlinear Riccati equation. AT K + KA - K (BR- 1 BT) K + Q = 0 (4) Equation (4) represents a set of n (n+1)/2 coupled nonlinear scalar equations. For small n, e.g. n ~ 4, the solution of equation (4) presents no difficulty to obtain, but for higher n, this solution can prove to be a formidable job, the computation time varies roughly in proportion to n 3 , therefore it is desirable to develop methods by which a suboptimal solution can be obtained where the computational savings and reduction in complexity result in an acceptable performance. 3. DESIGN PROCEDURE FOR THE DETERMINATION OF SUBOPTIMAL CONTROL

In this approach, the system is decomposed to what is called N- " E. -coupled" subsystems by decomposing the system equations (1) and (2) as follows [6]:

2. OPTIMAL REGULATOR PROBLEM We consider time invariant, dynamical systems described by a linearized model of the form [7, BJ :

x

= AX +

Bu

(1)

where X is (n x 1) state vector, u is (m x 1) control vector, A is (n x n) state distribution matrix and B is (n x m) control distribution matrix.

where E is a scalar parameter such that (O~E. ~1) Aij and B .. are (n. x n.) and (ni x mj) submatrices re spectivel9~ and wh~re J

The control u is chosen to minimize a quadratic cost function of the form

n =

J = 1/2

J:(XT

Q X + uT R u)

dt

(2)

499

~ i = 1

N

m

L j =

1

J. D. Delacour, M. Darwish and J. Fantin

500

in this decomposition the off-diagonal submatrices are smaller in some sense than the diagonal submatrices., and : Qll

_,

Q

I

{

N

_! Q12 - - - - - EQ 1N] _.......

K .. A .. +€ ~J

I I

_

t

The matrix Riccati equation (10), consists of N(N+l)/2 coupled non linear matrix equation of which the general equations is given by JJ

N

T

~

~~ ~J

1=1

f~l - - - - - - - - .::--QNN

N

IJ

IFi

N

L

L

1=1 (i

k=l

+

T Al' KIj ~

Kil A · + A .. K .. +E L

Ih

I

-_--.

L 1=1

o

K' k SkI KIJ· + E QiJ' ~ , 1 ~ i, j

(11)

~ N)

and N

o

.. L K ' l CIIK l , + a ~~ ~ 1=1 ~

Q .. and R .. are (n. x n.) and (m. x m.) submatrices r~~pectiv~1y. ~ J J J The optimal feedback gain K, solution of equation (4) will be function of E , hence K = K

(

€)

E)I

+E:

dK(Et ~E

E~

+ ••• + Em 'bmK(E)

m!

~

~mE.

I

(7)

E~

Then we define a suboptimal control as

~=o

~mK( €.)

Let € = 0

equation (11) reduces to N

T

Kij Ajj + Aii K .. ~J

(i

F

' €=o

I

dK(E)! dE. E:=O'

Since (13) is a homogenous equation with final value (14) zero, then Kij = 0 (i F j,1~i, j ~ N) N

Kii Aii + Aii Kii - ~ KilCllKli+Qii

I (9)

I

~1

- -

- - - - -

- -

o

(1 ~ i ~ N)

(15)

equation (15) represents a set of n (n +1/2 scalar i i equations hence K(E)\€=oiS given by : ([l)

Kll K22 K(€)IE=o

=1

'-

,,

,

@

l)..N

K12 - - - - - - -

Kll [

(13)

f=o

To find these matrices, we partition the gain matrix K (E:) as follows

)=

o

Cll K · IJ

~

~i, j ~N)

j,

(in this paper a second order approximation P of the optimum gain matrix K is investigated which is a good approximation for the general solution).

K(

L. K' l 1=1

T

(8)

and the problem will be to find K(E)\

d m €:

Calculation of K (€)

From equation (12) we obtain

1 u = - R- BT P X

••• ,

~ N)

where SkI and C are as given in Appendix 1. ll

(6)

This approach is based on approximating the optimal gain K by a gain P (which is obtainable from much simpler calculations) which is a truncated Mac_Laurin series expansion of the solution K ( E ) for fixed E, the calculation becomes simpler for € = 0, hence we ha_ ve : P = K(

(1 ~ i

(12)

l

(16)

equation (16) can be obtained in solving separately equation (15) for 1 ~ i ~ N, this greatly reduces or even eleminates convergence difficulties.

-

Calculation of the first derivative Then using the matrices A, B, Q, R, K in the decompo_ sed form, the matrix Riccati equation will be :

Let

d

K (€)

d

= KI

KC€:) \ €

dE.

=0

(17)

"OE

by differentiating equations (11) and (12) with respect to € and then substituting for = 0, we obtain

K~. (A .. -C .. K .. )+(AT._K .. C .. )K! .+D .. =0 ~J

JJ

JJ JJ

(ifj, 1

~i,

~~

~~

~~

~J

~J

j~N)

(18)

the term Dij is given Appendix 2 and I

T

I

Kii (Aii-CiiKii) + (Aii-KiiCii)Kii = 0 (1 ~ i

~ N)

(19)

equation (19) is homogenous with final value zero , hence I

Kii = 0

(1

~ i

~ N)

using equations (18) and (20), we get

(20)

501

Suboptimal control of large scale power systems

d

d€)1

dE

-K 1 (=0

I

0

I

K12 ------K 1N I

I

E=o

I

(21 )

0 , - - - - - - K2N

K21

,

I I

."

."

I

~1

.....

I I

.... ....

Conclusion

I

---------- 0

Calculation of second derivative 2

d2 ~ (E) ~ E.

I

E.:o

Let d K(E) K2 (22) "dZE. each term of the matrix equation is obtained by differentiating equations (11) and (12) twice with respect to E , then substituting € = 0, hence we obtain 2

From the results we can see that decomposition in two subsystems gives results very near to the optimal solu tion, and the three subsystem decomposition are slightly far from the global solution but it still a good approximation to the optimal one.

A suboptimal control strategy for large scale power system has been developed, in which the system is decomposed to N " E-coupled" subsystems. By this technique the solution of the matrix Riccati equation will be easily obtained. The potential application of the technique to three plant power systems indicate its efficiency when compared with the optimal solution. APPENDIX Determination of SkI' C ll

T

Kij(Ajj-CjjKjj)+(Aii-Kii

Starting from equation (10) SkI has the form (i

F

(23)

j, 1 ~ i, j ~ N)

Skl=

and 2

T

2

o

Kii (Aii-CiiKii)+(Aii-KiiCii)Kii+ Zii (1 ~ i

(1

(24)

~ N)

Kl l ." K12 -----K 1N = K

21

=

(i= 0

[

2

2

2

I I . " '....

I I

II

"

J

(25)

' __, :

2

.....

2. N -1 <::.... ~ Bki Rii

T B

li +

i=l iFk,l

F k, 1

~

l,k

~

€

-

1

(BkkRkk

N)

(27)

and

(1 ~ 1 ~ N)

(28)

where

2

KNl -------~KNN

(29)

For E. = 0 4. APPLICATION TO LARGE SCALE POWER SYSTEM

then

Modeling of the system

(V

The system considered, consists of three power plants, the first is a thermal unit while the second and third are hydro units, a fourth plant is considered to represent the infinite system equivalents.Fig.l shows, the interconnections between these plants. The individual plant is modelled by a fourth order system, a third order synchronous machine which represented by the state variables 'PF (field flux linkage), ~ (torque angle) and uV (angular velocity) and a voltage regulator which represented by first order system (when the time constant TE of the solid state exciter is neglected) and represented by the state variable VF (equivalent excitation voltage) as shown in Fig. 2. Then the overal system model is given by

l,k

~ N)

(30)

~ 1 ~ N)

(31)

APPENDIX 2 Determination of the first derivative KI Differentiating equations (11 ) and (12) with respect to €., we get "

I T I K, , Ajj +Aii Kij + q

N

N

L K'lA1 J,+ L 1=1 1=1 1

lh

T Ali Klj +

1#

N

(26)

X=AX+Bu

(1

1 ~

k,

ELL 1=1 lh

where X is (12 x 1) state vector T

u is (3 x 1) control vector, u = [u ,u ,u ] E1 E 2 E3 A is (12x12) matrix and B is (12x3) matrix, the numerical values of A and B are shown in Appendix 4.

~ [~«kSklKlj+KikS~lKlj+KikSklK~j)1

Determination of suboptimal control

(ih,

The suboptimal control technique described above is applied to the system shown in Fig. 1, and which represented by equation (26) • The system is decomposed in two and three subsystems and the trajectories are obtained using this technique, then compared with the optimal trajectories, some of the results obtained are illustrated in Figs. (3-8).

N

N

1 ~i,j ~N)

(32)

J. D. Delacour, M. Darwish and J. Fantin

S02

+ KikS!lKli+KikSklK{i)]

(1

~ i ~

N)

(33)

2

I d SkI where SkI = ~ I N -1 T -1 T -1 T Skl=2€ (Bk , ,R" Bli)+Bkk ~k BIk+Bkl Rll Bll i=l 1. 1.1. i;l!k,l I

Sll

2€

~

~

k,l

N)

(i ;I! j, 1

k;l!l

(1 ~ i, j

~

o (43)

N)

T

2

o

~ i

~ N)

(44)

where : N Yij = 2

~ N,

i,j

Kii (Aii-CiiKii)+(Aii-KiiCii) Kii+Z ii (1

(35)

Hence for € = 0 , T KI, Kij(Ajj-CjjKjj)+(Aii- KiiC ii ) 1.) + Dij

~

2

T

Blk ~k B1k

2

and (34)

-1

LN k=l

T

Kij(Ajj-CjjKjj)+(Aii-KiiCii)Kij + Yij

L.

(k;l! 1, 1

Hence for € = 0, we have

~{
0 (36)

i ;I! j)

l;1!i,j

'" (L.

T

2K" 1.1. k=l k;l!i,j

where

N

I

I

(45)

B'k)K, ,-2 LK'lCllK1' J )) 1=1 1. ) l;I!i, j

and (38)

APPENDIX 3

N N ..,T I I ...-' -1 Z1.'1., =2 L-(A .. K .. +K, ,A .. )-2 L- K'k(BkkR kk j=l J1.)1. 1.J J1. k=l 1. j;l!i k;l!i -1

~

E

N

1=1

1h

l;I!j

2

N +€ [ K'lA , + 1=1 1. 1 ) 1=1

L.

L.

l;I!j

T2] A ,K, 11. 1)

L. 1=1

1

-.L.. L

)

2

N L.

N 2 LK ,

I

N TIN L. A, ,K, ,+ E. L j=l )1.)1. j=l j;l!i

j# -1

-1

I

K" (B, ,R~, Bli +B i1 Rll Bll )K Ii 1.1. 1.1. 1.1.

(46)

(47)

A

where

2

+A~i KH + All =

T 2 N[N 2 A, ,K , , L(K'kSk1Kl' + J1. J1. 1=1 k=l 1. 1.

L

'1

1.0

-0.266

-2.75 0

-2.78

-1.36

0

0

-0.037 -0. 1

-4.95

0

-55.5

-0.039

0

-0.087

0

1. 11

l-O.'"

00

j;l!i

[-0.158 0.'" (1 ~ i ~ N)

S kl

-

Numerical values of the matrices A and B

(39)

KHA , , +2 K, ,A,,+ € ,Aj' 1.1. j=l 1.))1. j=l 1.) 1. j;l!i

2

1.1.

J= 0

;I! j, 1 ~ i,j~N)

where

T

K"

APPENDIX 4

N[N 1=1 k=l

and

2

N

I

l;1!i

KikSk1K~j

(i

I

1#

I I I '2 I I + 2KikSklKlj+2KikSklKlj+KikSk1Klj+2KikSklKlj +

+

N

2

N

1=1

N

l;l!i

Differentiating equations (32) and (33) ,with respect to £, we obtain N

T

Bk,R" B, ,)K, ,-2 LK'lCIIK1,-2K" 'C" 1. 1.1. 1.1. 1.1. 1=1 1. 1. 1.1.j~ 1.)

Determination of the second derivative K2

(40)

2

0 SkI N = --2 = 2

"dE:

L.

-1

T

(Bk 1.,R1.'1.' Bl 1.') i=l i;l!k,l

(k;l!l, 1 ~ k,l ~ N) 2 ;lSll N S11 = d€.2 2 Blk

E k;l!l

-1

~k

(41)

0

0

0

0.222

0

8.17

~-0.46 o.on

0

-0.25

0

2.8

0

0

0

0

17.5

0.02

0 LO.924 (42

0.00' -0.011

1

0 0.004

O.O"j

-0.02

l

Suboptimal control of large scale power systems

0.0"

0

00' 003] -0.015

-1.62

0

0

0

-2.43

0

1.37

-0.034 -0.005

-1.6

[0." -1.9 0 -3.1

-1.8

9.3

0

0

0

-56

-0.12

0.032

0

0.46

0.0' -1

0

1.49

-0.04

0

0

0

o

0.1?

0

-6.78 l-O.OO' A31 =

0

[

l

-1.24

A = 32

A33=

-0.028

0

o

0

0.498

-0.017

0

0.22

0

1.7

0

0

-0.07

0

6.37

r·'" 0

1

(Plant Thermal)

(Plant Hydro)

'---~III'

-0.011 003 _0. -2.37

70.1

1

0

0 0

-3.4

M. DARWISH, J. FANTIN, A. TITLI, "Optimum load frequency control of large scale power systems using decomposition-aggregation techniques" proposed to IEEE Trans. on Systems Man and Cybernetics (SMC). M. DARWISH, J.D. DELACOUR, J. FANTIN, "Sensitivity analysis of optimal regulators with application to large scale power systems", Proc. of the 7th Annual Pittsburgh Conference on Modeling and Simulation, Pittsburgh, USA, April 26-27, 1976.

_0.0~231

-1.2 -20

[8J

_ 0.09

0

-54.4

]

o

0.083 -10.1

0

O.OU -2.1

M. DARWISH, J.D. DELACOUR, J. FANTIN, "Modelisation d'un reseau electrique de grande dimension" Technical Report, nO 76T02, LAAS, France, Jan. 1976.

0.002

29.8

0

0

P.V. KOKOTOVIC, W.R. PERKINS, J.B. CRUZ, G. D'ANS, "E-coupling methods for near-optimum design of large scale linear systems", Proc. lEE, vol. 116, n° 5, May 1969, pp. 889-892.

0.121

0

-1.1

503

-21

-0.017 (Infinite Network)

(Plant Hydro)

and

BT

"

(admittances or .. in p .... )

[:

36.1

0

0

0

0

0

0

0 78.9

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1000

0

:J

(48)

Figure 1 - Three plants power system

REFERENCES

Y.N.YU, K. VONGSURIYA, L.N. WEDMAND, "Application of an optimal control theory to a power system" IEEE Trans. PAS, Vol. PAS-89, nO 1, Janv. 1970, pp. 55-62.

l!.V, (tf'r"'lTlincl voltoge) Input

Voltogl!'

Fihf''''

Rp-gulator

E.J. DAVISON, N.S. RAO, "The optimal output feedback control of a synchronous machine", IEEE Winter Power Meeting, New-York, Janv. 31-Feb. 5, 1971, pp. 2123-2134. H.A. MOUSSA, "Linear optimal stabilization and representation of multimachine power systems", Ph.D. Dissertation, University of British Calumbia, Canada, 1971. J.D. DELACOUR, M • DARWISH, J. FANTIN, "Control strategies for large scale power systems" to be published in Int. J. COntrol.

[5J

K.D. WALL, K.S. P. KUMAR, "Aggregation, decoupling and optimum control", IFAC Conference on Multivariable control system, Dusseldorf, Oct. 1971.

Figure 2 - Typical exciter-voltage regulator system