IEEE Transactions on Power Systems,Vol.11,No.4,November 1996 1957 DECENTRALIZED NONLINEAR OPTIMAL EXCITATION CONTROL O.Lu,Senior Member,IEEE,Y.Sun, Z.Xu.T.Mochizuki Department of Electrical Engineering Department of Electrical Engincering Tsinghua University,Beijing,China Kyushu Institute of Technology Kitakyushu,Japan Abstract -A design method to lay emphasis on differential connection,meritorious results was acquired by [11], which geometric approach for decentralized nonlinear optimal excitation proposed a decentralized nonlinear excitation control strategy. control of multimachinc systems is suggested in this paper.The However,the important issue of optimality of the control systcm has control law achieved is implemented via purely local measurements. not been involved in that paper.And it would be more valuable to the Moreover,it is independent of the parameters of power networks. industrial application,if the feedback variables (,i,E and Simulations are performed on a six-machinc system.It has been E)appeared in the expression of the control strategy of [11]can be demonstrated that the nonlinear optimal excitation control could adapt measured directly and conveniently. to the conditions under large disturbances.Besides,this paper has The optimality of the controller developed in this paper is verificd, verified that the optimal control in the sensc of LOR principle for the and the general expression of the proposed nonlinear control law is linearized system is equivalent to an optimal control in the sense of a implemcnted by purcly local and direct measurements,so that it is quasi-quadratic performance index for the primitive nonlinear control convenient for industrial applications.The simulation results on a 6- system. machine system show the validity of the proposed approach and the acquired control law. 1.INTRODUCTION 2.LINEARIZATION VIA FEEDBACK Excitation control of large generators is one of the most effective and economical techniques for improving dynamic performance and The linearization approach plays an important role in designing large disturbance stability of power systems.Because of this,the various nonlinear control systems.The succinct presentation of that development of excitation control both in theoretical and practical approach related directly to excitation control system design being respects has drawn considerable attention.From the PID excitation used and developed in this paper will be given as follows. control to supplementary control (PSS),to lincar optimal/suboptimal Consider a multivariable affine nonlinear system cxcitation control,valuablc contributions have been made by an extensive literature [1-5].The supplementary excitation control =fX)+g1(X)4+2(X) technology has widely been employed in many power systems of y1=w1(X) (1) many countries. '2=w2(X) The design approach of the various types of controllers mentioned where XER"is a statc vector,f(X)andg(X),i=1,2,are c vector above arc bascd on an approximately linearized model implemented at a fixed cquilibrium point of one-machine infinite-bus system.We fields on R",U=,is the control vector(here m=2);y:is know that a power system contains a large number of generator sets i-th output,and w(X)is a smooth function having relative degree set and possesses nonlinearity,so that we would better explore a design rr2}at a equilibrium point Xo If r=r+rn,and we sct approach which is based on the nonlinear model directly in order to 华1(X)=w1(X) further enhance the transient stability of power systems. (X)=Lw(X) In the past decade,the theory of nonlinear control systems with special emphasis on the differential geometric approach has been developed systematically,and it has proven to be an effective means of 9,(X)=w,(X) design of control systems described by affinc nonlinear mathematical 4(X)=w2X) models [6,7].In particular,lincarization techniqucs have been (X)=LW2(X) successfully applied to various nonlinear control systems,such as flight control systems [8]and robotic control systems[9].The first ,(X)=w2(X) literature applying that approach to power systems was published in 1989 [10].Subsequently some valuable research results dealing with where L'w,denotes the rth order Lie derivative of w:along f.Then power system nonlinear control have been reported in a series of the mapping that we have is as follows litcrature (11-14].It scems that a new field of study-nonlinear control of power systems is coming into being. 2=φ(X)=【9,…,9-14,…,4-19,92(X) The rescarchers and engineers who deal with power system control In such a condition,the dynamic description of system (1)in the new projects have been addressing themselves to a class of problems for a coordinates Z will be long time.The problems are of this type:How to achicve a general 名1=名 expression of excitation control strategy for each generator in a r1t180 multimachine system which is:first,optimal for the original nonlinear power system model;secondly,decentralized,meaning the control -2=1 law is implemented only by local measurements;thirdly,independent -1=-1 of the parameters of networks;lastly,valid for large disturbance case. It is significant and worthwhile to solve that type of problems.In that 玄=气1 (2) -3=n2 96 WM 260-0 PWRS A paper recommended and approved by the IEEE 州二之=2H Power System Engineering Committee of the IEEE Power Engincering Society for presentation at the 1996 IEEE/PES Winter Meeting,January 21- n-1=4(☑+b11(Z)1+b12(O2 25,1996,Baltimore,MD.Manuscript submitted January 31,1994;made =马(Z☑+b21(☑+b22(Z) available for printing January 15,1996. The compact form of(2)can be written as i=DZ =a(Z)+b(Z)U 0885-8950/96/$05.00©1996IEEE
IEEE Transactions on Power Systems, Vol. 11, No. 4, November 1996 DECENTRALIZED NONLINEAR OPTIMAL EXCITATION CONTROL Q. Lu , Senior Member, IEEE, I.: Sun, Z. Xu, i? Mochizuki Department of Electrical Engineering minghua University, Beijing, China Department of Electrical Engineering Kyushu Institute of Technology, Ritakyushu, Japan Abstract - A design method to lay emphasis on differential geometric approach for decentralized nonlinear optimal excitation control of multimachine systems is suggested in this paper. The control law achieved is implemented via purely local measurements. Moreover, it is independent of the parameters of power networks. Simulations are performed on a six-machine system. It has been demonstrated that the nonlinear optimal excitation control could adapt to the conditions under large disturbances. Besides, this paper has verified that the optimal control in the sense of LQR principle for the linearized system is equivalent to an optimal control in the sense of a quasi-quadratic performance index for the primitive nonlinear control system. 1. INTRODUCTION Excitation control of large generators is one of the most effective and economical techniques for improving dynamic performance and large disturbance stability of power systems. Because of this, the development of excitation control both in theoretical and practical respects has drawn considerable attention. From the PID excitation control to supplementary control (PSS), to linear optimal / suboptimal excitation control, valuable contributions have been made by an extensive literature [ 1-51. The supplementary excitation control technology has widely been employed in many power systems of many countries. The design approach of the various types of controllers mentioned above are based on an approximately linearized model implemented at a fixed equilibrium point of one-machine infinite-bus system. We know that a power system contains a large number of generator sets and possesses nonlinearity, so that we would better explore a design approach which is based on the nonlinear model directly in order to further enhance the transient stability of power systems. In the past decade, the theory of nonlinear control systems with special emphasis on the differential geometric approach has been developed systematically, and it has proven to be an effective means of design of control systems described by affine nonlinear mathematical models [ 6,7]. In particular, linearization techniques have been successfully applied to various nonlinear control systems, such as flight control systems [SI and robotic controlsystems [9]. The first literature applying that approach to power systems was published in 1989 [lo]. Subsequently some valuable research results dealing with power system nonlinear control have been reported in a series of literature [ll-141. It seems that a new field of study -nonlinear control of power systems is coming into being. The researchers and engineers who deal with power system control projects have been addressing themselves to a class of problems for a long time. The problems are of this type: How to achieve a general expression of excitation control strategy for each generator in a multimachine system which is: first, optimal for the original nonlinear power system model: secondly, decentralized, meaning the control law is implemented only by local measurements; thirdly, independent of the parameters of networks; lastly, valid for large disturbance case. It is significant and worthwhile to solve that type of problems. In that 96 Wh4 260-0 PWRS A paper recommended and approved by the IEEE Power System Engineering Committee of the LEEE Power Engineering Society for presentation at the 1996 IEEWES Winter Meeting, January 21- 25, 1996, Baltimore, MD. Manuscript submitted January 31, 1994; made available for printing January 15, 1996. 1957 connection, 'meritorious results was acquired by [ll], which proposed a decentralized nonlinear excitation control strategy However, the important issue of optimality of the control system has not been involved in that paper. And it would be more valuable to the industrial application, if the feedback variables (6, id, i, , E; and E;) appeared in the expression of the control strategy of [ll] can be measured directly and conveniently. The optimality of the controller developed in this paper is verified, and the general expression of the proposed nonlinear control law is implemented by purely local and direct measurements, so that it is convenient for industrial applications. The simulation results on a 6- machine system show the validity of the proposed approach and the acquired control law. 2. LINEARIZATION VIA FEEDBACK The linearization approach plays an important role in designing various nonlinear control systems. The succinct presentation of that approach related directly to excitation control system design being used and developed in this paper will be given as follows. Consider a multivariable affine nonlinear system ri: = f(X) + Pl(X) U1 + gz(X) U, Y1 = Wl(X) (1) Yz = where XER" is a statevector, f(X) and g,(X), i = 1,2, are Cm vector fields on R", U=[U,, ..., u,IT is the controlvector(here m=2); yl is 1-th output, and w,(X) is a smooth function having relative degree set {rl, rz) at a equilibrium point X,. If r=rI+ rz=n, and we set q1 (XI = Wl(X) q2 (X) = yJl(X) 7-4 (XI = WZ(X) %(X) =Lf%(X) q(X) = L;-Iw1(x) %AX) = L71W2(X) ... where Lfrw, denotes the rth order Lie derivative of w, along$ Then the mapping that we have is as follows T z = Q(X) = [ R , , %,-1 ; 74, ' 3 %*-I ; 'p., 1 W,I (XI In such a condition, the dynamic description of system (1) in the new coordinates Z will be i1 =3 0885-8950/96/$05.00 0 1996 IEEE
1958 where D∈R(r-2)×is a constant matrix Lw(X) 010-· 00 5w2(X) 00 a(仲(X)= 001.00 1-2 … D= 00.010 Lwn(X) 0-,01000 ·.00 0·0100--3 0·0·01 %Lw…Lw2 and b(中(X)= X 州X) 142(J w2区)x=② Wm Lelf wm…wn Now,we define that y in (8)is the optimal input in the sense of b(Z)= w()w() Linear Quadratic Riccati (LQR)principle for the linearized system (6). Lewn(X)w)x四 That implies that in (8) By settingin(3) V=-K*Z=-K*中(X) (9) a(Z)+b(Z)U=V (4) where K*is an optimal constant gain matrix which is deduced from the Riccati algebraic equation corresponding to the linear system(6). we have the nonlinear control law for system (1)as follows By substituting (9)into (8),the nonlinear control law for the u-[-bo团+w original system(1)can be finally stated in the form U*=b(仲X)(a(中(X)+K*φ(X) (10) =b'(φ(X)(-a(pX)+V) (5) Since it has been assumed previously that the relative degree 3.DISCUSSION OF THE OPTIMALITY r=r1+r2=n,the matrix b(φ(X)》must be nonsingular at X。So,the What we have recognized from the statements of the previous expression of the control law (5)holds.Thus,the system (1)has scction is that the main points of the tcchnique of feedback been transformed to a linear and controllable system in the normal lincarization for affine nonlincar control systems is to discover an form as appropriate coordinates transformationZ=(x)and a state feedback i=AZ+BV (6) (5)such that the nonlinear system can be transformed into a linear and where AER"x"and BERx2 are in the Brunovsky normal forms as controllable one (6)in the new coordinates Z.Then for the acquired linear system (6)the LOR design method can be used.In view of (3) A-B]8-9 and(4)we know that the equation(6)can be rewritten as Now,the remaining task is to find the "output"functions w.(X)and =DZ 2-V (11) w(X),such that the relativedegreer=r equals n.In view of previous statement of deduction,we know that the vector field where z1∈R",Z2∈Rm.DeRm-侧is a constant matrix,and W(X)=[w(X),wX)]will be the solution of the partial differential VER"is an exteral input vector.For the linear system in the equation set as follows: normal form(6)or(11),a quadratic performance index 4g)=0 i=1,2k=0,1,5-2 (7) Z'oZ+VRv d (12) L3w,(8)=0 In addition,the solution of(7)w,and w should satisfy the condition is taken into account in which Q∈Rxn and R∈R#×mare semi- that the matrix b((X))is nonsingular at X. positive and positive definite weighting matrices,respectively.For In general,there is little chance of obtaining an analytical solution system(11)and index(12),by using the LOR principle,we know that of(7).In fact,in this paper we shall arrive at the vector field W(X) the optimal input V*=-RBTP*Z,where P*ERx is a positive not by solving the partial differential cquation set(7)directly,but by definite solution of the following Riccati cquation composing solutions of the ordinary differential equations and a series AP+PA-PBRB'P+Q=0 (13) of transformation steps.The algorithm for achieving the vector ficld Substituting V*=-RBTP*Z and Z=(x)into (8),we get the W(X)is implicit in the process of the verification of the Theorem 1 of nonlinear control law (10)repeatedly given as follows the paper [7]and it will not be repeated here. The calculations illustrated above in the case of a system with two U*=-b((X)a(φ(X)+RBpφ(X) (14) inputs and outputs can be extended without much difficulty to more Since the performance index(12)does not make much sense to us general affine nonlinear systems.And it is easily realized that the (we do not know what the physical meaning of the parameter V is), control law will be represented in the form as follows the problem needing to be discussed and verified is whether U*in (14)can be an optimal control of the system under somc substantive bφ()(-a中X)+V) performance index.Lct us answer that question. (8) From (3)we know DZ (15) where a(Z☑+b(ZU
D= X=#-l(Z) - - 01 0 00 00 0 0 1 .. 0 0 -r1-2 00 010 0 01 000 00 0 " 0 10 0 --n-3 -0 0 01- By setting in (3) a(Z) + b(Z)U = V (4) we have the nonlinear control law for system (1) as follows U = [ 2 ] = b-'(Z) (-a(Z) + V) = b-'($(X)) (-a($(X>) t V) (5) Since it has been assumed previousIy that the relative degree r = rl+ r2 = n, the matrix b($(X)) must be nonsingular at X,. So, the expression of the control law (5) holds. Thus, the system (1) has been transformed to a linear and controllable system in the normal form as Z=AZtBV (6) where A ERnX and B €Rnx are in the Brunovsky normal forms as I ... ... ... ... 1 Now, we define that V in (8) is the optimal input in the sense of Linear Quadratic Riccati (LQR) principle for the linearized system (6). That implies that in (8) where K* is an optimal constant gain matrix which is deduced from the Riccati algebraic equation corresponding to the linear system (6). By substituting (9) into (8), the nonlinear control law for the original system (1) can be finally stated in the form V = -K* Z = -K* $(X) (9) U* = -b-' ($@)I (a($@)) t K* $(XI) (10) 3. DISCUSSION OF THE OPTIMALITY What we have recognized from the statements of the previous section is that the main points of the technique of feedback linearization for affine nonlinear control systems is to discover an appropriate coordinates transformation Z= $ (X) and a state feedback (5) such that the nonlinear system can be transformed into a linear and controllable one (6) in the new coordinates Z. Then for the acquired linear system (6) the LQR design method can be used. In view of (3) and (4) we know that the equation (6) can be rewritten as (11) &=DZ i2=V where zlERRm, z2ERrn. D€R(n-m)Xn is a constant matrix, and V€Rm is an external input vector. For the linear system in the normal form (6) or (ll), a quadratic performance index rW Now, the remaining task is to find the "output" functions w,(X) and w,(X), such that the relative degree r = rl+ r2 equals n. In view of previous statement of deduction, we know that the vector field W(X) = [w,(X), w2(X)IT will be the solution of the partial differential equation set as follows: J1= (ZTQZ t VTRV\ dt In addition, the solution of (7) w1 and w2 should satisfy the condition that the matrix b($ (X)) is nonsingular at X,. In general, there is little chance of obtaining an analytical solution of (7). In fact, in this paper we shall arrive at the vector field W(X) not by solving the partial differential equation set (7) directly, but by composing solutions of the ordinary differential equations and a series of transformation steps. The algorithm for achieving the vector field w(X) is implicit in the process of the verification of the Theorem 1 of the paper [7] and it will not be repeated here. The calculations illustrated above in the case of a system with two inputs and outputs can be extended without much difficulty to more general affine nonlinear systems. And it is easily realized that the control law will be represented in the form as follows = b-' ($(X)) (-a($(X)) t V) (8) where is taken into account in which Q€RnXn and R€RmXm are semipositive and positive definite weighting matrices, respectively. For system (11) and index (12), by using the LQRprinciple, we know that the optimal input V*=-R'BTP*Z, where P*ERnXn is a positive definite solution of the following Riccati equation A~P + PA - PBR-'B~P + Q = o (13) Substituting V*=-K~BTP*Z and Z=$(X) into (8), we get the nonlinear control law (10) repeatedly given as follows U * = -br' ($ (X)) [a(+ (X)) + R-'BTP * $ (X)) (14) Since the performance index (12) does not make much sense to us (we do not know what the physical meaning of the parameter V is), the problem needing to be discussed and verified is whether U* in (14) can be an optimal control of the system under some substantive performance index. Let us answer that question. From (3) we know i=[ DZ j a(Z) t b(Z)U
1959 We consider the performance index 4.DYNAMIC MODEL AND CONTROL LAW The Model (z'Qz+2R方d (16) We consider a dynamic model of a multimachine power system with m generators,each generator described by third-order dynamic model where [5],which is given by R-10R] 6=叫 (25-1) For system (15),the problem becomes to minimize functional subject to constraint (15).In the light of optimal principle it is known -0- (25-2) that in order to have optimal control U.,the following so called Bellman equation must be fulfilled. 0*品 (25-3) where P and E are the nonlinear functions of the state variables, zu,9z9=0 which are given by (17) w(0)=0 Pa-GuE+Ed Byj Eisin(-) j=1,j where Hamilton function H=zo2+52z+肠u DZ (18) E←-Ea+xa-taXBF4Fgo6-》 And in (25-3)V is the control variable.is considered as and w is a positive definite function in Z. constant by neglecting the influencc of magnetic saturation [5,10-14]. Solving the equation From the equations(25-1)to (25-3),we get the state cquation for 肥-0 an /-machine system in the nonlinear affine form (it is nonlinear with (19) respect to X,and linear to u,) we get the optimal control U being in the form as X(0=fX()+g1(X()两)+…+8m(X(O(0 (26) u.=-bi☑hc☑+Ro,z where (20) X0-EEeEmi,2,5w;d,4,,dn Substituting (20)into (17),and-assuming w(Z)=ZPZ/2 (P is a [1 0 positive definite constant matrix),we have Ta -first 1 DZ second 0 QZ+ZPR0.ZZPR0.IPZ-0 0 81)= 82凶= 4-m-th Ta Equation above can be reexpressed in the equivalent form : 2m iZ0z-ZPR.Z+ZF-0 (21) 0 0 0 Thus the determination of P is equivalent to solve the following Riccati cquation The Control Law It can be verified that the system(26).satisfies the necessary and BP+P3-P0RP+0-0 (22) sufficient conditions of the Theorem 1 in (7],which implies that system(26)is globally linearable in the state space via state feedback. It can be seen that (13)and(22)are the same Riccati equation, The steps involved in the verification for Theorem 1 in [7]are able to therefore,they must have the same solution on P.By substituting lead us to find the appropriate coordinates transformation Z =(X) w=ZTP*Z/2 and Z=(X)into (20),the optimal control law U can be for system(26)expressed as follows acquired in the form as 3=0-00: U#=-b(φ(X)(a(Φ(X)+R0,IP*中X) (23) for i=1,,m Comparing(14)with(23)leads us to the conclusion that mH=4-0 (27) U*=U* 2m牧=% That result shows that the nonlinear control U*obtained in terms of and also to have the implicit function expression of control vector U in minimizing the performance index J for linear system (6)is exactly the form (4),which is substantively expresscd by (considering equivalent to the optimal control U acquired by means of minimizing PeiEgilg and Pei-EE) the performance index functional (we call it as a quasi-quadratic performance index) M:Taoy (ZQZ+R)dr As a result,it is clear that the nonlincar control law given in (10)is an (28) optimal control strategy in the sense of the quasi-quadratic Thus,we obtain the excitation control strategy for each generator performance index which is given by A14-M上 4==1-Tw (29) where T=(x/x)To
1959 4. DYNAMIC MODEL AND CONTROL LAW The Model We consider a dynamic model of a multimachine power system with m generators, each generator described by third-order dynamic model [5], which is given by ai = U, (25-1) We consider the performance index 7= il (ZTQZ t ZTRZ) dt W where For system (15), the problem becomes to minimize functional r subject to constraint (15). In the light of optimal principle it is known that in order to have optimal control U*, the following so called Bellman equation must be fulfilled. aw az minH(Z, U, -, t) = 0 w(0) = 0 where Hamilton function and w is a positive definite function in Z. Solving the equation we get the optimal control U* being in the form as a(Z) t R-l[O, I] - az (20) Substituting (20) into (17), and assuming w(Z)=ZTPZ/2 (P is a positive definite constant matrix), we have lZTQZ 2 t ZTP[(tl R-'[O, ']PZ+ZTP[-R-lEI,pz] = 0 Equation above can be reexpressed in the equivalent form lZTQZ 2 - ZTP[:] R-'[O, I]PZ+ZTP[! ]Z = 0 (21) Thus the determination of p is equivalent to solve the following Riccati equation It can be seen that (13) and (22) are the same Riccati equation, therefore, they must have the same solution on P*. By substituting w=ZTP*Z/2 and Z=+(X) into (20), the optimalcontrol law U* can be acquired in the form as U, = -b-Y+(X)) @($(XI) + R-9, Ilp*+(X)) (23) Comparing (14) with (23) leads us to the conclusion that That result shows that the nonlinear control U * obtained in terms of minimizing the performance index J for linear system (6) is exactly equivalent to the optimal control U * acquired by means of minimizing the performance index functional (we call it as a quasi-quadratic performance index) U* =U* W J= il (ZTQZ + ZTRZ) dt AS a result, it is clear that the nonlinear control law givcn in (10) is an optimal control strategy in the sense of the quasi-quadratic performance index * 00 9 WO q = - P, --(U, (t) -- U())- -Pel 3% M, El (t) =-lo + 1v TdOi " TdOi fr qc (25-2) (25-3) where Pel and E which are given by are the nonlinear functions of the state variables, 4' ni ]=1,]#' Eqi = t (xdi -X'di )( B$'q'- 2 B1lE'g cOs(s~-s~)> And in (25-3) Vfi is the control variable. xB is considered as constant by neglecting the influence of magnetic saturation [S, 10-141. From the equations (25-1) to (25-3), we get the state equation for an rn-machine system in the nonlinear affine form (it is nonlinear with respect to X, and linear to U,) where XCS = f(X(t)) + g*(X(t))u,(t) + + &(X@)k(t) (26) The Control Law It can be verified that the system (26) satisfies the necessary and sufficient conditions of the Theorem 1 in [7], which implies that system (26) is globally linearable in the state space via state feedback. The steps involved in the verification for Theorem 1 in [7] are able to lead us to find the appropriate coordinates transformation Z = Q, (X) for system (26) expressed as follows z, = bz - a,, z,+,=@,-@~ for z=1, ,m (27) zZm+1= 01 and also to have the implicit function expression of control vector U in the form (4), which is substantively expressed by (considering .. per Eql Iql and pa = E', Iql + E', zqz 1 (28) Thus, we obtain the excitation control strategy for each generator which is given by where T, = (xIdl /xdJ T,
1960 Now,in the new coordinates Z the cquivalent linear system can be written as P =+1 M:Tai 4d+3.94m+2.&- (34) 克洲=3m (30) 2m+1=为 Remarks 1)The control law proposed in(34)is of optimality in the scnse of 衫mV the quasi-quadratic performance index for ith-generator in vicw of LOR method,according to the designers'engineering expericnce and the results of dynamic simulations of power systems,it oz+iRid is appropriate to choose the weighting matrix Q=diag(5,10,0)in accordance withZ=【△8，△w,A]T and R=l,theni the optimal v:(v*)can be calculated out 7=-44h-3.91侧-2.84@ s{gA+9m4of+9贴44+rd (31) As a matter of fact, Substituting(31)into(29),the control law is achieved as follows 44费4织： M 4ad+3.940,+(2.8- So,the performance index could be comprehended as M = a4d+4+d+4h (32) Since in(32)the variables E and I are not convenient to be 2)The control law (34)looks complicated,but it is not difficult to measured directly,the following transformation is rcquired.From implement with the software of a microcomputer.And it has powersystem thcory onc knows that substantially been implemented in Fengman Hydropower Station of Eot=Vu+xdi Qei/Vu (33) the Northeast Power System.and a thermal power station in the By.substituting P“Egre(R。=Egg+(化g-xk，since the Inner Mongolia,China.The "STD-Bus"has been used as the hardware of the controllers quantity of the second term is much less than the first one,it could be 3)All of the variables (P V and appearing in the neglected)and (33)into(32),we finally acquire the nonlinear optimal expression of the control law are local measurements,so that it really excitation control (NOEC)law which is given by (for P0) is a decentralized control strategy. 0.015 0.003+0.03 0.002+0.03 13 0.015+0.10 j0.73 f0.73 100 /0.02 j0.018 0.01 410 15 j0.002 j0.002 i0.03 0.003+j0.013 20 14 g 0.0210.086 0.0170.07 j1.993 2,87+i147 3.76+j2.2 L201 j0.022 0.72+0.47 2 5.0t2.9 0.037+0.18 2 21 0.058+j022 0.015+j0.06 0.01+j0.04 2.25+1.69 0.86+j0.66 43+j0.26 0.02 0.7+j0.5 i0.064 i0.038 Fig.1 A 6-machinc system,Basc Power =100 MVA (From [10]courtesy of IEEE,1988)
1960 Now, in the new coordinates Z the equivalent linear systcm can be written as 21 =2,11+1 zzlll = %I (30) %"+l v1 ..... .... . In view of LQR method, according to thc designers' engineering expcrience and thc results of dynamic simulations of power systcrns, it is appropriate to choose the wcighting matrix Q=diag(5,10,0) in accordagcc with Z = [AS, Aw, &GI]' and R = 1, then the optimal ill (v*,) can be calculated out v *i = - ,( A ai dt - 3.9 A wi -2.84 &, (31) Substituhg (31) into (29, the control law is achieved as follows (32) Since in (32) the variablcs E,, and I are not convenicnt to be measured directly, the following transformation is required. From power system theory one knows that 4' Eq[ = '<L + xdi ai/ %I (33) By substituting pCi- I (p, = E' I + (X -x'Ji& sincc the quantity of the second term is much less than the first one, it could bc neglected) and (33) into (32), we finally acquirc the nonlincar optimal excitation control (NOEC) law which is givcn by (for Per # 0) 41 4' 44 4 4' Reniarks 1) Thc control law proposed in (34) is of optimality in thc sensc of thc quasi-quadratic performancc index for ith-generator As a matter of fact, w dIji=0dpei ML So, the performance indcx could be comprehcndcd as .i) 4 = (% A4 + 9, A.)," +93 4, + 94 dl 2) Thc control law (34) looks complicated, but it is not difficult to implement with the software of a microcomputer. And it has substantially bcen implemcnted in Fengman Hydropower Station of the Northeast Power System and a thermal powcr station in the Inncr Mongolia, China. The "STD-Bus" has been uscd as thc hardware of thc controllers. 3) All of thcvariablcs (P,,, Qe,, V,l and A[d,) appcaring in thc cxprcssion of the control law arc local mcasurcments, so that it rcally is a dcccntralizcd control strategy. 0 007+/ 0 03 11 -- -- 13 0 015+j 0 10 j073 j 0 02 t- 0 0 7 ;: I7 0 - 0 -jooo2 -1 0 002 9 20 14 0 003+j 0 013 8 - 0017+j007 LB L9 2 87+j 1 4 - 3 lG+j 10022 2 2 - - 072+j047 2 0 e : -7 6 b- 'TO 0 0 Vl 0 0 0 037+j 0 18 0 015+10 06 0 01+~ 0 04 2 26+j 1 69 0 86+j 0 66 ] 0 012 07+j05 3 0 064 3+ 4+ I 18 4 7+/ 0 26 j 0 018 51 Fig. 1 4 6-machinc systcrn, Base Powcr = 100 MVA ( From [lo] courtesy of IEEE, 1088)
1961 4)In view of (34),we see that the control law is independent of the parameters of the network,so it may possesses the adaptability to the variations in parameters and structure of networks. 5)The control law is valid for the global state space (except Pe=0), so that it could be adapted to a wide range of operating conditions, especially to improve large disturbance stability of multimachine systems However,the effectiveness of the optimal control has to be limited by the real boundary values of the excitation voltages and 80 currents ) 120 5.SIMULATION RESULTS 160 A 6-machine system shown in Figurc 1 is employed for disturbance 200 simulation to display the fact of NOEC's advantages over other types 0.0 2.0 3.0 4.0 of linear control in strengthening the capability of oscillation tite(second）周 attenuation and improving large disturbance stability of a power Fig.2 Dynamic responses of the system with PSS system. In Figure 1,No.6 machinc is a synchronous condenser and No.1 80 generator itself actually represents an equivalent of a large power system.The transmission line and load data are shown in Figure 1; the rest are included in Appendix 1. In order to survey the difference of effectiveness of the different types of excitation controllers we made comparisons among the following control configurations. First,No.2~No.5 generators are the PSS-cquipped machines. The PSSs are designed strictly in the light of (2]and [5],and the PSS transfer function of each generator is given in Appendix 2.1. 12 Secondly,the above-mentioned generators are equipped with linear optimal control designed by applying LOR approach [5].The linear 160 optimal control strategy (actually it could only be a suboptimal control 200 for a generator in a multimachine system)for cach generator is C.D 1.0 2.0 3.0 4.0 expressed in Appendix 2.2. time (second) Thirdly,the gencrators are cquipped with the nonlinear optimal Fig.3 Dynamic responses of the system with LOEC excitation controllers proposed in this paper. A thrcc-phasc temporary short circuit occurs at No.11 bus (see Figure 1)for 0.09 seconds.The simulation results are shown in Figures 2,3 and 4,which indicate the rotor angle responses for the 2. three control configurations respectively.We can see from Figure 4 that under the large disturbance the system remains stablc by cmploying the nonlincar optimal excitation control strategy expressed 90 in (34).Whereas the system incorrigibly faults out of synchronism soon after (in above two seconds)the fault occurs,if the linear PSS or LOEC is used to take the place of NOEC. 120 6.CONCLUSIONS 160 1)A new design method for excitation control of generators in a multimachine power system which is based on the fundamentals .200 10 20 3.0 40 of the theory of nonlinear control systems is proposed in this paper. time (second) 2)It has been verificd in this paper that the presented nonlinear Fig.4 Dynamic responses of the system with NOEC control law is of optimality in the sense of quasi-quadratic performance index 3)The control strategy possesses these properties:decentralization, mechanical power,assumed to be constant,in per unit independence of the parameters of networks,and adaptation for a wide E internal voltage,in per unit range of operating conditions,especially for large disturbance terminal voltage of a generator,in per unit attenuation. internal transient voltage,in per unit 4)It is effective to implement the control law proposed in this paper voltage of the ficld circuit of a gencrator,the control variable by the software of a microcomputer,STD-Bus for instance. in per unit d-axis current,in per unit GLOSSARY OF SYMBOLS g-axis current,in per unit damping constant,in per unit rotor angle,in radian,A6=6-60 M inertia coefficient of a generator set,in seconds rotor speed,in rad./scc.,A field circuit time constant,in seconds f frequency of terminal voltage of a gencrator,in per unit Yi=G+jB admittancc of line-j,in per unit P active clectrical power,in per unit Y=G+jB self-admittance of bus i,in per unit reactive power,in per unit Xd d-axis synchronous reactance of a generator,in per unit
1961 4) In view of (34), we see that the control law is independent of the parameters of the network, so it may possesses the adaptability to the variations in parameters and structure of networks. 5) The control law is valid for the global state space (except Pe=O), so that it could be adapted to a wide range of operating conditions, especially to improve large disturbance stability of multimachine systems ( However, the effectiveness of the optimal control has to be limited by the real boundary values of the excitation voltages and currents ). 5. SIMULATION RESULTS A 6-machine system shown in Figure 1 is employed for disturbance simulation to display the fact of NOEC's advantagcs over other types of linear control in strengthening the capability of oscillation attenuation and improving large disturbance stability of a power system. In Figure 1, No.6 machinc is a synchronous condenser and No.1 generator itself actually represents an equivalent of a large power system. The transmission line and load data are shown in Figure 1; the rest are included in Appendix 1. In order to survey the difference of effectiveness of the different types of excitation controllers we made comparisons among the following control configurations. First, No.2-No.5 generators are the PSS-equipped machines. The PSSs are designed strictly in the light of [2] and [5], and the PSS transfer function of each generator is given in Appendix 2.1. Secondly, the above-mentioned generators are equipped with linear optimal control designed by applying LQR approach [SI. The linear optimal control strategy (actually it could only be a suboptimal control for a generator in a multimachine system) for each generator is expressed in Appendix 2.2. Thirdly, the generators are equipped with the nonlinear optimal excitation controllers proposed in this paper . A three-phase temporary short circuit occurs at No.11 bus (see Figure 1) for 0.09 seconds. The simulation results are shown in Figures 2, 3 and 4, which indicate the rotor angle responses for the three control configurations respectively. We can see from Figure 4 that under the large disturbance the system remains stable by employing the nonlinear optimal excitation control strategy expressed in (34). Whereas the system incorrigibly faults out of synchronism soon after (in above two seconds) the fault occurs, if the linear PSS or LOEC is used to take the place of NOEC. 6. CONCLUSIONS 1) A new design method for excitation control ofgenerators in a multimachine power system which is based on the fundamentals of the theory of nonlinear control systems is proposed in this paper. 2) It has been verified in this paper that the presented nonlinear control law is of optimality in the sensc of quasi-quadratic performance index. 3) The control strategy possesses these properties: decentralization, independence of the parameters of networks, and adaptation for a wide range of operating conditions, especially for large disturbance attenuation. 4) It is effective to implement the control law proposed in this paper by the software of a microcomputcr, STD-Bus for instance. GLOSSARY OF SYMBOLS S o f P, Q, rotor anglc, in radian, AS=b-SO rotor speed, in rad.iscc., wo= 2nf Ao=w-oo frequency of terminal voltage of a generator, in per unit active electrical power, in per unit reactive power, in per unit 0' 80 I I 00 10 20 30 40 time (second) Fig. 2 Dynamic responses of the system with PSS 00 10 20 30 40 time (second) Fig. 3 Dynamic responses of the system with LOEC 80 40 0 8 -40 F-. OJ M .... 2 -80 E - 120 - 160 - 200 E 00 10 20 70 40 time (second) Fig. 4 Dynamic responses of the system with NOEC P, E, V, E', V' mechanical power, assumed to be constant, in per unit internal voltage, in per unit terminal voltage of a generator, in per unit internal tramient voltage, in per unit voltagc of the field circuit of a gencrator, the control variable in per unit d-axis current, in per unit q-dxis current, in per unit damping constant, in per unit incrtia coefficient of a generator set, in seconds field circuit time constant, in seconds I, Iq D M T, Ye = GI/ +jBe Yil = GI, +jB,, x, admittance of line id, in per unit self-admittance of bus i, in per unit d-axis synchronous reactance of a generator, in per unit
1962 q-axis reactance of a generator,in per unit 2.Excitation Control Laws d-axis transient rcactancc,in per unit 2.1 PSS transfer functions input signal is 40) ACKNOWLEDGMENTS generator 2: G2(s)= 15.483x 1+0.43s 1+0.01g1+3s1+0.045s This work was supported by grants from the NSF,China and the gencrator 3: G3)= 63.35 3 1+0.425 /2 Kyushu Electric Power Company,Japan. 1+0.01s1+3s11+0.1553 14338 1+0.34s 12 REFERENCES generator 4: G4)= 1+0.01s1+3x11+0.126s [1]F.P.deMello and C.Concordia,'Concepts of synchronous 25.63x【1+0.436 /3 machine stability as a affected by excitation control',IEEE Trans. generator 5:Gs(s)= 1+0.0151+3s1+0.21s Power Appar.Syst.pp.316-329,April,1969. The output limitation of each PSS is +5.0% [2]F.P.deMello,P.J.Nolan,T.F.Laskowski,and J.M.Undrill, Coordinated application of stabilizers in multimachine power 2.2 Linear Optimal Control Laws systems,IEEE Trans.Power Appar.Syst.pp.892-901, gencrator2:V2=-73.84'2-35.64Pe2+5.84aM2 May/June,1980. [3]J.H.Anderson,The control of a synchronous machine using generator3:V3=-69.24'3-23.34P.3+6.14w3 optimal control theory',Proc.IEEE 90,pp.25-35,1971. generator4:4=-62.9A4-10.0APe4+2.34w4 [4]Y.N.Yu and H.A.M.Moussa, 'Optimal stabilization of generator 5: Vs=-58.44Vs-8.04Pes+1.64 multimachinc systems',IEEE Trans.Power Appar.Syst pp.1174-1182,May/June,1972. Qiang Lu (SM'85)was born in Anhui, [5]Y.N.Yu,Electric Power System Dynamics,Academic Press, China,on April 19,1937.He graduated from 1983,pp.95-137. the Graduate Student School of Tsinghua [6]A.Isidori,Nonlinear Control Systems,2nd cdition,Springer- University,Beijing,in 1963 and joined the Verlag,Berlin-New York,1989,pp.234-391. faculty of the same University. [7]D.Cheng,T.J.Tarn and A.Isidori,'Global external linearization of From 1984 to 1986 he was a visiting nonlinear systems via feedback',IEEE Trans.on Automatic scholar and a visiting professor in Washington Control,Vol.AC-30,No.8,pp.808-811,Aug.,1985 University,St.Louis and Colorado State [8]G.Meyer and L.Cicolani,'Application of nonlinear system University,Ft.Collins,respectively.From inverses to automatic flight control design',NATO AGARD- 1993 to 1995 he was a visiting professor of Kyushu Institute of AG251,1980,Pp10.1-10.29. Technology (KIT),Japan.He is now a professor of Dept.of [9]T.J.Tarn,A.K.Bejczy,A Isidori,and Y.L.Chen,Nonlinear Elcctrical Engineering,Tsinghua University,Beijing.Since 1991 he feedback in robot arm control,Proc.23rd,IEEE Conf.Dec has been elected to be an academician of Chinese Academy of Contr.,Las Vegas,1984. Sciences.Currently,he is responsible for nonlinear control of power [10]Q.Lu and Y.Sun,Nonlinear stabilizing control of multimachine system studies in Tsinghua University,China. systems',IEEE/PES Vol.4,No.1,pp.236-241,February,1989. [11]J.W.Chapman,M.D.Ilic,C.A.King,L.Eng,and H.Kanfman, Yuanzhang Sun was born in Hunan,China,on September 21, 'Stabilizing.a multimachine power system via decentralized 1954.He received the B.S.degree from Wuhan Irrigation and feedback linearizing excitation control',IEEE.Trans.on Power Electrical Institute,China,the M.S.degree from Electric Power Systems,Vol.8,No.3,pp.830-838,Aug.,1993. Rescarch Institute (EPRD),China,and Ph.D.degree from Tsinghua [12]Y.Wang,D.J.Hill,R.H.Middlcton and LGao,Transient University,Beijing,in 1978,1982,and 1988,respectively,all in stability enhancement and voltage regulation of power systems' electrical engineering. IEEE Trans.on Power Systems,Vol.8,No.2,pp.620-627,May, From 1982 to 1985 he was with the EPRI,China,working in the 1993. Power System Research Department.Now he is an associate [13]S.Kaprielian,K.Clements and J.Turi,'Applications of exact professor of Tsinghua University. linearization techniques for steady-state stability enhancement in a weak AC/DC system',IEEE Trans.on Power Systems,Vol.7, Zheng Xu was born in Shanghai,China on N0.2,pp.536-543,May,1992 September 16,1960.He reccived the B.S. [14]F.K.Mak and M.D.Ilic,Towards most effective control of degree from Shanghai Jiaotong University, reactive power reserve in electric machines',10th Power System China,in 1983,and M.S.degree and Ph.D Computation Conference,pp.359-367.Graz.Austria,Aug..1990 degree from Kyushu Institute of Technology, Japan,in 1989 and 1993,respectively,all in APPENDIX electrical engineering.From 1983 to 1985 he was with Shanghai Jiaotong University, 1.Generator and Excitation Data China,working in Dept.of Electrical No. M D Engineering.He is now a visiting assistant professor of Kyushu Xd xd To Institute of Technology. 0.0150.0150.015 140.823.0 9.0 0.321 0.321.0.0382 30.03.0 8.375 Takuro Mochizuki was born in Oita,Japan 0.1380.03960.0396 79.5 3.0 7.24 0.77 0.770.121 on June 25,1933.He received the B.S. 15.623.0 6.2 M.S.and Ph.D degrees from Kyushu 0.3060.3060.048 39.2 3.0 6.2 University,Japan,in 1956,1958 and 1973 6 1.633 1.6330.197 2.623.0 6.92 respectively.From 1958 he has bcen with All in per unit except M and Tao in scconds Kyushu Institute of Technology as a lecturer, an associate professor and a professor
1962 x xld q-axis reactance of a generator, in per unit d-axis transient reactance, in per unit Y ACKNOWLEDGMENTS This work was supported by grants from the NSF, China and the Kyushu Electric Power Company, Japan. REFERENCES [1] F.P.deMello and C.Concordia, 'Concepts of synchronous machine stability as a affected by excitation control', IEEE Trans. Power Appar. Syst. pp.316-329, April, 1969. [2] F.P.deMello, P.J.Nolan, T.F.Laskowski, and J.M.Undrill, 'Coordinated application of stabilizers in multimachine power systems, IEEE Trans. Power Appar. Syst. pp.892-901, May/June, 1980. [3] J.H.Anderson, 'The control of a synchronous machine using optimal control theory', Proc. IEEE 90, pp.25-35,1971. [4] Y.N.Yu and H.A.M.Moussa, 'Optimal stabilization of multimachine systems', IEEE Trans. Power Appar. Syst. pp. 1174-1 182, MayiJune, 1972. [5] Y.N.Yu, Electric Power SyJtem DynamzcJ, Academic Press, [6] A.Isidori, Nonlinear Control SyJtemJ, 2nd edition, SpringerVerlag, Berlin-New York, 1989, pp.234-391. [7] D.Cheng, T.J.Tarn and A.Isidori, 'Global external linearization of nonlinear systems via feedback', IEEE Trans. on Automatic Control, Vol. AC-30, No.8, pp.808-811,Aug., 1985. [8] G.Meyer and L.Cicolani, 'Application of nonlinear system inverses to automatic flight control design', NATO AGARDAG251, 1980, pp10.1-10.29. [9] T.J.Tarn, A.K.Bejczy, A Isidori, and Y.L.Chen, 'Nonlinear feedback in robot arm control', Proc. 23rd, IEEE Conf.Dec. Contr., Las Vegas, 1984. [lo] Q. Lu and Y.Sun, 'Nonlinear stabilizing control of multimachine systems', IEEE/PES Vo1.4, No.1, pp.236-241, February, 1989. [11] J.W.Chapman, M.D.Ilic, C.A.King, L.Eng, and H.Kanfman, 'Stabilizing a multimachine power system via decentralized feedback linearizing excitation control', IEEE. Trans. on Power Systems, Vo1.8, No.3, pp.830-838,Aug., 1993. [12] Y.Wang, D.J.Hi11, R.H.Middleton and L.Gao, 'Transient stability enhancement and voltage regulation of power systems', IEEE Trans. on Power Systems, vol.8, No.2, pp.620-627, May, 1993. [13] SKaprielian, K.Clements and J.Turi, 'Applications of exact linearization techniques for steady-state stability enhancement in a weak ACDC system', IEEE Trans. on Power Systems, Vo1.7, No.2, pp.536-543, May, 1992 [14] F.K.Mak and M.D.Ilic, 'Towards most effective control of reactive power reserve in electric machines', 10th Power System Computation Conference, pp 359-367, Graz, Austria, Aug., 1990 1983, pp.95-137. APPENDIX 1. Generator and Excitation Data NO. xd Xq x'd M D Td 1 0.015 0.015 0.01.5 140.82 3.0 9.0 2 0.321 0.321 0.0382 30.0 3.0 8.375 3 0.138 0.0396 0.0396 79.5 3.0 7.24 4 0.77 0.77 0.121 15.62 3.0 6.2 5 0.306 0.306 0.048 39.2 3.0 6.2 6 1.633 1.633 0.197 2.62 3.0 6.92 All in per unit except M and Tdo in seconds 2. Excitation Control Laws 2.1 Pss transfer functions ( input signal is Aw ) generator 2 : G s 15.48- 2( ) = 1t0.01s 1+3s generator4: G s *~(ltv38r_)~ 4( ) = 1t0.01s 1+3s 1t0.126~ The output limitationof eachPSS is L5.0%. 2.2 Linear Optimal Control Laws generator2: generator 3 : generator 4 : generator5: Vfj=-58.4AKj-8.0APejt 1.6Aw5 V,=-73.8 AV,,-35.6 Ape,+ 5.8 Amz VB = - 69.2 AV,, - 23.3 APe3 + 6.1 Am3 V, = - 62.9 AV,,- 10.0 APe4 + 2.3 Am4 Qiang Lu (SM'85) was born in Anhui, China, on April 19, 1937. He graduated from the Graduate Student School of Tsinghua University, Beijing, in 1963 and joined the faculty of the same University. From 1984 to 1986 he was a visiting scholar and a visiting professor in Washington University, St. Louis and Colorado State University, Ft. Collins, respectively. From 1993 to 1995 he was a visiting professor of Kyushu institute of Technology (KIT), Japan. He is now a professor of Dept. of Electrical Engineering, Tsinghua University, Beijing. Since 1991 he has been elected to be an academician of Chinese Academy of Sciences. Currently, he is responsible for nonlinear control of power system studies in Tsinghua University, China. Yuanzhang Sun was born in Hunan, China, on September 21, 1954. He received the B.S.degree from Wuhan Irrigation and Electrical Institute, China, the M.S. degree from Electric Power Research Institute (EPRI), China, and Ph.D. degree from Tsinghua University, Beijing, in 1978, 1982, and 1988, respectively, all in electrical engineering. From 1982 to 1985 he was with the EPRI, China, working in the Power System Research Department. Now he is an associate professor of Tsinghua University. Zheng Xu was born in Shanghai, China on September 16, 1960. He received the B.S. degree from Shanghai Jiaotong University, China, in 1983, and M.S degree and Ph.D degree from Kyushu Institute of Technology, Japan, in 1989 and 1993, respectively, all in electrical engineering. From 1983 to 1985 he was with Shanghai Jiaotong University, China, working in Dept. of Electrical Engineering. He is now a visiting assistant professor of Kyushu Institute of Technology. Takuro Mochizuki was born in Oita, Japan on June 25, 1933. He received the B.S., M.S. and Ph.D degrees from Kyushu University, Japan, in 1956, 1958 and 1973, respectively. From 1958 he has been with Kyushu Institute of Technology as a lecturer, an associate professor and a professor