Optimization Design of Multivariable Control in Multimachine Power Systems

Copyright

©

I FAC P,,"'er S, stc m, ",1<1

I'o\\'e r Plant Control. Beijing. 1986

OPTIMIZA TION DESIGN OF MUL TIV ARIABLE CONTROL IN MULTIMACHINE POWER SYSTEMS Qiu Changtao, Wang Yupeng, Deng Jixiang and Xu Lijie

x.

E. China IlISlill/le of Eleclric POWI'/' t ; llgiIlPerillg. ji/iall Cily. PF!.C

Abstract A new algorithm and model for the optimization design of mul tivariable control in multimachine power sys ·t ems are presented in this paper. This method exploits the modified random search app~oach to di~ect ­ ly optimize the output feedback gains, without solving Riccati equation. Using this approach, we can design the optimal control with all state fe edbac~ of the entire system, or the suboptimal control with partial state feedback of the entire system, or the optimal decentralized control with any state feedback of the local individual machine in the system. It has been employed for the parame ters design of an optimal excitation control ler (OEC) installed at a power station in N.W. power system of China. Re sults show that this method can effectively search out the optimal parameters of the controller, hence much improve dynamic performance and highly enhance the steady stability limit of the system. A comparative study of the OEC (from solving Riccati equation) and the decentralized local excitation control (DLEC) has been completed; and it is shown that both dynamic performances are very close. This means that the DLEC is a practical, valuable and perspective way for the power system control. Keywords. Optimal control : Optimization design; Multivariable control; Multimachine power system; Optimal excitation control; Dec entralized local excitation control; Random search approach. INTRODUCTION

Based on the theory of linear optimal control. Levine and Athans (1970) proposed an analytical algorithm for solving the pro ·· blem , i.e. to obtain the local feedbac~ gains K by solving Levine - Athans matrix equations. But the difficulty has not been overcome yet , because of the poor conver gence in the process of solving the non linear matrix equations. Using only one

This paper presents a new algorithm and model for the optimization design of multivariable control in multimachine electric power systems. The algorithm exploits the modified random search approach (Qiu, Zhao, and Song 1982) to optimize the output feedback gains directly, without solving Ricca ti equation. This is ·the continuing work and advanced development and extension in multimachine systems of the paper pres ented by Qiu, Wang and Deng

feeaback variable from each generator Zhou (1981) solved the equations by means of Newton's approach. But its algorithm is too complicated, in addition, many measurable signals of each generator are not yet being used. By avoiding analytical solution method, Qiu, Wang and Deng (1984) presented a new approach for directly optimizing the feedback gains K based on minimizing the performance index, subject to some cons traints. This approach has simple and flexible characteristics, and would be an effective method for the optimal control design. The steady characteristics of load and the saturation effect in generators has been considered in the model of multimachine system. It can be used for the computation of the power system with or with out infinite bus. By exploiting this algorithm and model, a software package for optimization design of controller has b een completed, and employed for the parameters design of an OEC installed at a hydro power station in N.W. power system of China. Results show that the proposed approach is practical, flexible and adaptive. It can be used not only for the optimal parameter design of OEC, but also for the suboptimal excitation control (SOEC), or the decentralized local excitation control

tI984). Theoretically, any dynamical system for which we wish to have the optimal dynamic performance , must have a control with all state feedback of the entire system. In multimachine power systems, however, various power stations are located far apart from each other; hence, it is very difficul~ for a power station to get the state variables from other stations. This makes the optimal excitation controller (OEC) with all state feedback extremely difficult to be implemented in multimachine power sys ·tems. In order to make the theoretica l investigation possible to practical application in power system, electric engineers and experts have turned their attention from all state feedback of the entire system t o the measurable partial output feedback of the system, and/or the decentralized local feedback of individual machines. The crux for implementing the optimal control is to find an efficien·t method of design for obtaining the optimal local feedbac~ gains K conveniently. 44 1

442

Qiu Changtao et (1/ .

(DLEC) with any state variables of any individual machine. Computation shows that this approach can effectively and easily search out the optimal feedback K of the controller.

where V is the real symmetric positive-definite solution of Lyapunov matrix equa tion

A comparative study between OEC and DLEC has been carried out. The eigenvalues and dynamic performance of both controllers are very close. This means that DLEC would be a practical, valuable and perspective way for power system control.

In order to eliminate the dependence of J on initial state X(O), we take the initial state X(O) as a set of random variables with numeric characteristics of zero average value and the variance matrix being as

(10 )

(11 )

PERFORMANCE INDEX AND OPTIMIZATION ALGORITHM The linearized state equation in multimachine power system is X=AX+BU

(1)

The relationship between the output variables and the state variables is ¥ =CX

(2)

where X is state vector. The state variables ,,3, """" .. E'q , .Efd are taken from individual generator; ¥ is output vector. ,.Pe,,,W, "Vt, AEfd, are chosen as the output variables; A is nxn system matrix; B is nxm control matrix; U is control vector; C is lxn output matrix. The quadratic performance index is T

e>

J

T

(3)

J=lj2 0 (X QX+U RU)dt For a controller, the feedback gain K being given, then we have U=-K¥

where K is the output feedback gain matrix in the form 1

kp1 k~ kV1~l L _ _ _ _ _

I kp2

0

1

kw2 kV2 kF21

-r - - - -

r.1

-t -1kPm

o

k,.,m kv.;;kF-

I From E q. (1), E q.

(3), and Eq. (4), we have

X=A1 X

(5)

J = lj2J.\TQ1Xdt

(6 )

where A1 =A-BKC

(7)

Q1 =Q+CTKTRKC

(8 )

From Eq. (8), if Q is positive-definite (or positive-semidefinite) and R is positive-definite then CTKTRKC is positive-definite (or positive-semidefinite), hence, QJ must be positive-definite (or positivesemidefinite). The conditions which Q and R should satisfy here are exactly what the Q and R must satisfy in the linear optimal control theory. Thus we can completely assure that Q1 is positive-definite (or positive-semidefinite). Substituting Q1 into Eq. (3), and solving Lyapunov equation, we obtain a new expression of the performence index as following; J = lj2X T (O)VX(O)

where notation {) represents diagonal matrix , qii is the diagonal elements of the weight1ng metrix Q in the performance index (3). The expectation value of performance index J is j =E[XT(O)VX(O)] = ~Vi' D[X(O)] 1,]=1 J that is •

n

Vii

J = ljW ~(-) (12) 1 =1 qii I n general, under the constraint of Eq. (10) for obtaining the random performance index Eq. (12), we proceed by the following routine: first, building a set of nonlinear matrix equations based on the linear optimal control theory and solving the matrix equations, then we obtain the optimal feedback gain K (Levine and Athans, 1970 ; Lu and colleague, 1982). But in multimachine system, the convergence is so poor in the solution process. This is the difficulty not easy to overcome. Using optimization techniques, by minimizing performence index Eq. (12), subject to some constraints, we can directly search out the feedback gain K without solving Levine - Athans equation and Riccati equation. Thus the complicated optimal control problem becomes a simple and intuitive problem of optimization. Besides the simple , easy and feasible characteristics, the optimization approach for designing control , also has the advantages of flexible and adaptive charicteristics. For example, as an optimal controller being designded we can optionally choose any combination of signals as the state vector, according to what we need and prefer. No matter whether the optimal control with all (or partial) state feedback of the entire system , or the decentralized control with all (or partial) local state feedback of individual machi ·· nes being designed , there is no trouble as we encounter with the analytical method. For optimizing the cost function (12). there are many efficient optimization ' techniques available, such as pattern search method. Powell's conjugated direction method ; Complex approach and random search approach (Qiu, 1984) etc, all of them are efficient techniqes. Owing to simplicity and the characteristics of easily getting rid of the local optimum, the modified random search method would be the better one of them. This is why we exploit the modified random search method in practice . The flow chart of designing optimal feedback K is shown in Fig. 1.

44 3

Optimization Design MATHEMATICAL MODEL OF MULTIMACHINE POWER SYSTEM

The system with infinite bus can be treated as below. From Eq. (17), we have

For the system described in Eq. (1), we consider the following reducible criteria:

V; ZI Expanding the equation, we have

(a). Ignoring the resistance of stator windings; (b). Ignoring the flux linkage variation in all windings except in the excitation winding, The electroaagnetic transient equation of gena re tors see Appendix

(18)

where V 'm, I In are the terminal voltage vector and current vector respectively in the system without infinite bus. From Eq. (18), we obtain (19)

Linearizing Eq. (19), we have 6 VO:

; ZmAi 'm

(20)

where Z," ; Z'mm-Z.lt.,Z;~Z':'m Equation (20) is the linearized network equation without infinite bus. After transforming the coordinate system and solving Eq. (13), Eq. (20), we have the relationship between voltages and currents on d-q axis. Substituting Eq. (20) into Eq. (14), Eq. (15) and Eq. (16), we have p

0

JF P .. WF

[POE;,

r

PAEfdF

{Wol

0

0

.. dF

Ad

AuJ

AE

0

<>

EJ

0

EE

{f~dO

FJ

0

FE

{ -1

-1'1\

IN F

"E'qF " Efd

0

+

0

U 0

t:~

( 21) Where notation [ I represents diagonal matrix, suffix F denotes F-vector. The elements of the submatrics in A are shown in appendix. Fig. 1. The flow chart of optimizing K using Random Search Approach. The rotor motion equations are

EXAMPLE OF DESIGNING OEC AND DLEC IN MULTIMACHINE POWER SYSTEM ( 13)

2HPW; Mm-Me-DW

(14)

The electromagnetic torque equation is

We exploit the IEEE model of excitation regulation system 1

KA

KA

PE f d ; -T~Efd +'f;,:-Vt - T AUE

A system with five machines, shown in Fig. 2. is exploited for the study of OEC design. Choosing .. Pe , .. W, "V t , AVF ( .. Efd) from individual machine itself as feedback signals, we call it "decentralized local control", (DLEC). The design result has been compared to that OEC (with all state feedback of the entire system).

(16)

Equations (13) to (16) are the differential and algebraic equations used for describing the mechanical-electromagnetic transient process of OEC design. After eliminating the free buses, we obtain the network equation as hYV

When the matrix Y in Eq . (17) being formed the steady characteristics of load can be counted with updating the diagonal terms of matrix Y by the differential admittance, i.e.

.0159+ jO.1154

o Fig. 2 A system with five machines.

Qiu Changtao et al. Both controllers designed under the same weig hts of output variables given as Qp= 1.0, Qw =5000, Qv =250, QE= O.OOOOOI, and R=O.Ol. Genera~o~~ate

TABLE - - - - -1 Ce. Xd 1# .3652 2/1 .0796 3/1 .0796 4# .49

---

Xd .1496 .0194 .2992 .1693

Tdo 2H(s) rI'e(s XQ .2588 7. 72 24.8 0.7 .0547 5.0 131.09 0.7 . 5176 7.72 12.4 0.1 .3367 4.76 12.42 1.06

6/1

P+jQ 2.0+jO.22 11.0+ j5. 63 1.0+jO.l05 1. 5+ jO. 055 -2.86+j2.0

We e xploit the modified random search approach which consists of two optimization ste ps (Qiu, 1982; zhao, 1981): the first step is to apply statistical trial approach to search a better initial point. The iterations of search is set as 200 times. The second step is to apply the random direction aproach to search the optimum point. After 4 iterations, we obtain the optimal feedback gain K, while the per formance index J in Eq. (12) reaches to the minimum. The upper and lower bounds used for the statistical trial, the step length in the random direction search,. and the optimal feedback K are given in table 2.

TABLE 2 Range of feedback gains for search and optimized result Lower bound Upper bound Step length for statis - for statis· for random result tics trial tics trial direction search kpl

2.0

20.0

5.0

83.95

kWl

-2500.0

-1500.0

100.0

- 1508

kVl

50.0

150.0

10.0

98.99

kFl

0.5

1.5

0.1

1.41

kp2

2.0

20.0

5.0

79.99 -3357

kw2

-4000.0

-3000.0

100.0

kV2

150.0

310.0

20.0

174

kF2

1.0

2.0

0.2

1. 95

kP3

2.0

20.0

5.0

12.28

kuJ3

-2000.0

-1000.0

100.0

-1918

kV3

20.0

100.0

10.0

k F3

0.1

0.5

kp4

20.0

50.0

kw 4

0.0

100.0

k V4

5.0

50.0

k F4

0.2

0.7

~ABLE

0.04 4.0

0.18

5.0

36.75 0.54

_::'~l~.--~ 1/

/

\-

/-\_

- =:II\~~,

/

, _ .... 1'

\

........

,

,

'_,

1 ----~~

2

3

t(sec)

Fig. 3 Dynamic response of A Pe 3 (stepchange in Power of 0.05 p.u.)

t~~~~ste~

cl.osed system with DLEC

·005

-18.15 -16 .:!:jI5.5

-2·5

-16.8.:!:jI6.2

-16.6.:!:jI8.4

-8.8.:!:j9. 1

-7.7.:!:j6.68

-0.16.:!:j4.45

-1.02.:!:j7.4

-3. 25.:!:j I 7.1

-0.0044.:!:j1.56

-3.03.:!:j6.3

-4.3.:!:jl1.4

- 0. 89.:':jl. 39

-1. 2.:!:j4. 78

-1.6.:!:5. 16

-0. 795.:!:j 1. 04

I

86.6

-0 .299.:!:j7.25

-0.66.:!:jl.08

The dynamic responces of the system are shown in Fig. 3, Fig. 4 and Fig. 5 as 5% step-change reference power is applied to generator #3. The open loop system is un stable. After putting the controller in, n< matter the OEC of the DLEC, the power ~ re · turns to steady state within two seconds, and oscillates only two cycles during this period. The speed w returns to its stable value after only two or three cycles oscillation as well. The duration of transient process is about two seconds. Results in Fig. 3 and Fig. 4 show that the effect of DLEC and the effect of OEC are nearly the same.

41.4

-5.17 -0 . 225.:!:j6.68

From eigenvalues shown in table 3, the eigenvalues of the colsed system with OEC and the eigenvalues of the close loop system with DLEC are almost the same; thereby, the dynamical performances of them are very similar.

I

0.05

Closed system with OEC

-9-:67- ----- -

I

10.0

3 The eigenvalues of

Open system

45.38 I

closea loov system with all state feedback of the entire system (from solving Riccati equation), and the eigenvalues of closed loop system with the decentralized local control. In the open loop condition, the dominant eigenvalues being -0.0044+jl.56, are a pair of weakly damping mechanical oscillation modes mainly affected by the power angles (Ad) of machines #1, #2, and #3 · Both OEC and DLEC can much improve the damping modes, and the real part of them would shift away from the point near the imaginary axis. The dynamic performance of the system is much improved, and the system stability is highly enhanced. In like manner, the other mechanical oscillation modes, - 0.299+j2.75, mainly affected by the speed (..w) of machine #3; -0.16+j4.45, by the power angle (hd) of machine #1, have also been much improved to be as strong damping and stable modes. This means that the dynamical performance and the ability of anti-disturbance in the system are much improved. Thereby the power limit of the system is highly enhanced.

/

I

... ,

~\

I

\ \

\

I

\

I

\ \

I

, 3

\

,

.... ,

I

I

2

\

t(sec)

\

I \

I

\

J \

-0. 99.:!:jO. 83

-3. 16.:!:j3.03 _...L-_ _ _ __

-0.97.:!:jo.819 -0.668.:!:j2.8

Table 3 records the eigenvalues of the open loop system, the eigenvalues of the

I

\

J

\

I

\ \

....

,

J I

Fig. 4 Dynamic response of .c.u 3 (stepchange in power of 0.05 p.u.)

445

Optimization Design

III

Fi g . 3 Fig . 4 andf'ig . 5 DLEC c l osed

- - - - - - OEe

.5

\

--------

0pP. ll

TABLE 4 Power limit of system with DLEC on machine #3

J. oop

c l osed J oap loop

\

Open loop Gen. #3

I

Increment

0.254

1. 38

1. 13

1. 617

2. 74

1. 13

--- - - - -- - - - - - -- - - - Line from 8 to 9

\

ClOsed loop

If more machines in the system are equipped with OEC or DLEC, the power limit of related transmission lines and other lines will also be enhanced.

Fig 5

Dynamic response of 4Efd 3 (stepchange in power of 0.05 p.u.)

The reason why the dynamic performance of the system can be much improved when generators are equipped with OEC, is that the multivariable control can make the excitation voltage quickly rise and quickly lower as there occurs any perturbation, so tbat the system damping would be enforced strongly in time. From the dynamic response of AV F (AEfd), in Fig. 5, we can clearly observe that, in open loop condition, the variation of "VFJ is small (the highest amplitude even only 0.05 p.u.). In close loop condition, the variation of "V F 3 is much enhanced, the highest amplitude increases to 0.58 and -0.4 p.u. Hence, the optimal excitation controller has a good effect on power system stability. For the same reason, if we design DLEC under the same performance index Eq. (3), the controller has a good effect as the OEC. However, because the feedback of DLEC has fewer output variables, the control effect is slightly worse than OEC.

(B) Only Machine #3 With OEC Using algorithm and model proposed in this paper, we install the DLEC on machine #3 only, and take the local output (4 Pe , "IV , "Vt , AVF ) as feedback signals, then we obtain the optimal feedback gain K as following kp=13.55 k..,=-1008 kv =32.35 kf'=0.29 1 The dominant eigenvalues are improved from -0.0044+j1.56 (in open loop case) to -0.27 +j1.9 (in closed loop case), so that the stability is much improved. When the output power of machine #3 increases with regulating factor 0.05 (in open loop case) and 0.02 (in closed loop case) starting from P =O, the terminal voltage remaining at 1.05 p.u, the steady stability limit of machine #3 and the transmission power of line 8 to line 9 are calculated and shown in table 4.

CONCLUSSION 1. The mathematical model and algorithm presented in this paper are flexible, practical and adaptive for the optimization design of multivariable control in multimachine power systems. It can be used for designing the optimal control with all state (or partial state) feedback of the entire system, or the optimal decentralized local control with all states(or partial state) of individual generator. Prac tical computation indicates that the approach can effectively search out the optimal parameters of various controllers. 2. The mathematical model proposed here can be counted with the steady characteristics of load, the saturation effect in generators and the features of the system, i.e. with or without infinite bus. Hence it has a wide range of application and is convenient for use. 3. The DLEC has the advantages of fewer feedback signals and channels, and convenient for measuring signals. The effect upon dynamic performance and the increace of stability limit are very close to that of OEC. Therefore, it is a practical, valuable and perspective way for multimachine system control. REFERENCES Qiu, C.T., J.H. Song, and D. Z. Zhao (1981). Optimal design of integrated controller of large steam turbine generator. J of N. E. China Inst. of EPE. No. 1, 30-42. - - - - - - - - - - - - - --Qiu, C. T., D. Z. Zhao, and J. H. Song (1982). Determination of state weighting Matrix for optimal linear control by means of optimization techniques. :! ournal_o.f-.-!.!'lctr i c~Engineer i'!!L.P_~ fhin~L-N'p. 1. Qiu, C. T. (1982J. Optimization design of electrical machines using random search approach. Journal of Noretheast f.hin_a Insti tute- O:f. _ _E_~_c_t!"J.£....!'~W.!!I_EE..&: i!t_e~_ri'!!L.~_.~27-34-=-

Qiu. C. T., D. Z. Zhao, and Y.P. Wang (1983). Optimal parameter design of power system stabilizer (PSS). Academic Meeting On Automation Organized B~ CHPI, KunMin~~i~ Qiu, C. T., Y. P. Wang, and J. X. Deng (1984). Optimization design of multivariable control in power systems by directly optimizing output feedback gains. ~ademic Meetin~~emo~~on-: ~rol and Remote Sensing, Organized~ CAUl & CASI, ChongQing, China. Qiu, C. T., Y. Wang (1984). Study of optimization design for mul~ivariable con-

Qiu Changtao et al.

446

trol in power systems. Academic MeetinJL.Q!l_ .~_mo~_e_~ntrol and _ReJiiOte--se-ns,:i,,'!!L......9£Kani_zed By_ CAl!,Iu &CA§..!.,_Shong.= .2i_n!L... "Cl:!.:i,n~ . Lu, Q. Z.H. Wang, and Y. D. Han (1982). In T. F. Fan (Ed.) , QI>_t_ima}__c.o!'tro.!""oJ: ~r~~~missi~n_~L~te~, China Science Press. Levine, W. S., and M. Athans (1970) On the de t ermination of the optimal constant output feedback gain for linear multivariable system . IEEE. Trans. Vol. AC-

.1_5-,- .44-4.!:.

- - - _ .- - ---" - "

Zhou, X. X (1981). Determination of The feedback parameters for optimal stabil i ty cont rol in multimachine power system. Electric Network techn~ue of Ch i ~}i.~: __ JT~4.:'- ---Yu , Y. N., H. A. M. Moussa (1972). Optimal stabilization of a multimachine syst e m . g :_E..§_ ,-"T.!:.?E..~ ._.l'"A_S ._ !'i!-.Yl:JJ!_n~_,_ Xi-An Jiao Tung University of China (1978) Pow e ~_ Syst e m Computation. WR & EP publication, China.

C

---

--- --

Construct The Coefficient matrix A, C In State Equ a tion and Output Equation. From el e ctromagnetic transi e nt equation of gen e rators, We hav e

AVXY 2F-vector; Yxy 2F x 2F matrix Transforming Eq. (28) into d-q axis, we have the linearize d network equations of generators on d-q axis as below; (29) wher e ~Idq is 2F vector consisting of daxis and q-axis currents of each generator, AVdq is 2F vector consisting of d - axis and q-axis voltages of each generator. Form Eq.

(22) and Eq.

(29), we have

""Vd q = ~K I

(30)

Y'" = [ - y* ]

dq

Y~q

is obtained by substituting the following Bs into Y matrix; 11

(22)

From Eq. (IS) and Eq. (22), we have the linearized el e ctromagnet torque equation of gene rator (23)

ii

K

I

(24)

Ks =Ido+Vqo/Xq Le ·t the deviation of mechanical torque Me be zero. Leanearizing the Eq.(14) , we have (25)

From above equations, we have the linearized equation of synchronous machine

(26)

where

KF is the saturation factor for considering the saturation effect in generator

ii

d

=K .. ; dq

p

-

(A~q] X, d

where suffix F denotes F-vector; dq

K 2 =Vdo/Xd

q

11

B* =B" tl/X'(i)

K

whe r e

Kp = I+(b/a) Eq~n-l)

2F-vector;

xy

B~ .= B~ _- I/X (i)

AIq = ~Vd/Xq

K4 =Iqo-V do/Xd

where AI

where

APPENDIX

AId = (AEq-AVq)/Xb

Suppose there are F machines in the system. After eliminating the free buses, we linearize Eq. (17) and obtain AIxy =Yxy,t>.Vxy (28)

=1

From Eq. tain the in A and equation

DO

-Y

dq

(23), Eq. (26), and (30), we obelements of individual submatrix the coefficient matrix C in output (2).