Predictive control of synchronous generator: a multiciterial optimization approach

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1 Preictive control of synchronous generator: a multiciterial optimization approach Marián Mrosko, Eva Miklovičová, Ján Murgaš Abstract The paper eals with the preictive control esign for nonlinear systems. The resulting control system performances epen on the choice of the control esign parameters, namely the preiction horizon, the control horizon an the control input penalization. For an optimal choice these parameters a genetic algorithm has been applie. The theory of multicriterial optimization has been use to combine several control obectives in one obective function. The propose approach has been evaluate using an example of the 259MVA synchronous generator of the nuclear power plant Mochovce (EMO. Keywors: preictive control, synchronous generator, nonlinear moel, genetic algorithm, multicriterial optimization Introuction The synchronous generator is an integral part of power systems so there is an interest to esign the controller with the best possible control performances. A number of synchronous generator control algorithms were propose in the past, but only few of them have been implemente in inustry. From the control theory point of view the synchronous generator is a nonlinear system with variable properties that can be linearize in an operating point, so the control algorithms base on the linear control theory can be use. The synchronous generator control obectives can be characterize as follows: - To ensure a minimal voltage steay state offset. - To obtain an acceptable voltage settling time. - To obtain the esire active power oscillation amping. The first two reuirements can be satisfie using the PI controller. However, this controller can not ensure the esire amping of transient processes. For this reason the special circuits calle the power system stabilizer are ae to the synchronous generator control systems. It is avantageous to satisfy all of above reuirements using one control algorithm. One of the possibilities is to use the preictive control algorithms. The preictive control has become popular over the past twenty years as a powerful tool in the feeback control for solving many problems for which other control approaches were prove to be ineffective (Clarke et. Al., 989. One of these methos is escribe in this paper. The common feature of these methos is, that from ata measure in past time Y ( k = { y(,..., y( k } an U ( k = { u(,..., u( k } one or several values of plant output is preicte. Values preicte in this way are also function of future manipulate variables u ( k, u ( k +,... Control strategy is then efine by minimization of functional with the loss function efine as a sum of ifferences between values of reference signal an a values of preictive output (preiction of control error an the penalization of manipulate variable. There are many approaches of preictive control, which vary from each other by the number of preicte values, use functional an also by limiting conitions for control strategy. Generalize preictive control will be introuce in the next section. The paper is organize as follows. Firstly, esign of the preictive control, genetic algorithm an multicriterial optimization are briefly introuce in the section 2. The case stuy is then iscusse in Section 3. Section 4 inclues experimental results. A summary an conclusions are given in Section 5.. Generalize preictive control. Plant moel Preiction itself can be realize only on the base of known moel of plant. This moel oul accurately enough escribe ynamics of a real plant. Consier that the control system is escribe by ARMAX moel C ( ( ( ( ( z y k = B z u k + ε ( k A z where u ( k is control input an ( k y is output value. This moel allows to incorporate the internal moel of state isturbances so that the control esign can ensure the offsetfree performance..2 Basic principle of preictive control The preictive control algorithm uses a moving horizon, as own in Figure. Base on the plant moel, the size of the future output values epening on the changes of control input is to be preicte. The control input is generate so that the esire ( 7

2 output characteristics are achieve. The plant behavior for steps ahea is preicte in the time k. The leastsuares metho is usually use to achieve the optimal output value behavior. In this metho the constraints of the control input, the control input increment an also of the process output are usually consiere. PAST FUTURE y (k y ( k + y ( k + 2 u(k u(k+ Target u(k+ch k k+ k+2 k+ch k+ Fig. Basic principle of preictive algorithm.3 GPC esign The control obective is to minimize in a receing horizon sense the following cost function J pc with 2 + [ ~ ( ( ] [ ( ] 2 y k + r k + + ρ u k + = (2 : begin of the preictive horizon : en of the preictive horizon ch : control horizon ρ 0 : penalization of control input. It is suppose, that future values of reference signal are r k +, for =,2,... known ( The GPC control esign then consists in performing the following three steps:. Preiction of output ~ y( k + for =,2,..., which is function of future controlle inputs u ( k + for =,2,..., is calculate a few steps ahea. 2. Optimal seuence of future control inputs accoring to uaratic criteria J pc with a function of loss is calculate. 3. Only the first component u ( k out of this seuence of control inputs will be use for control, an the whole process will be repeate in the next sampling perio. Parameters of uaratic criteria (2 an aitional constraint parameters can be chosen as follows: : = + if plant elay time is known, otherwise =. : is selecte so as the essential part of time response of the controlle system is inclue in steps. ch : ch = is selecte for stable an ampe systems, otherwise ch will be eual to a number of unstable poles, or poles close to the stability bounary. ρ : ρ = 0 is chosen in most cases. Less ynamic control is obtaine by increasing ρ, but at the interest of regulation process of lower uality. The right choice of these preictive control parameters is not simple, reuires practice with the controlle plant an epens on expectations from the control. The GPC control can be implemente using the general linear control law ( D( z u( k + R( z y( k = T( z r( k + S z i.e. the control input in the step k is obtaine as follows ( ( ( ( ( T z R( z r k + S z D z S( z D( z ( k u k = y (4 where ( z g.f ( z R (5 = ( z = C( z + g.z.h ( z S (6 ( z = C( z = + T g.z (7 T T gr : terms of the first line of matrix [ G G + ρi ] G G : matrix of imension [ + ;ch]. Polynomials F( z, H( z an ( z following polynomial euations E ( z A( z + z F ( z ( z B( z C( z G ( z + z H ( z ch (3 G are solutions of the = (8 E = (9 This approach has been use in our implementation. (k + r(k T(z + u(k + y(k SD(z Plant - R(z SD(z Fig.2 Simulation scheme of preictive control 2. Genetic optimization A choice of preictive control parameters is not efinite, therefore genetic algorithm is chosen. In this case cost function is minimize f i ( x r y min = (0 The aim is to minimize the cost function by signal reference tracking f an also by isrturbance regulation f Multicriterial optimization There are often many ifferent aspects which oul be consiere when optimizing problems are solve. It is obvious, that several aspects (partial obective functions have to be consiere by optimization metho. It is so calle multicriterial optimization. To connect partial obective functions into one obective function, consiering that partial 8

3 obective functions have ifferent weights, can be an option how to eal with several criteria. F (x w f, x = {x, x..., x } ( = N i i i(x 2, The only remaining problem is an appropriate choice of weights, which efines significance of partial obective functions f i ( x. In the case that their values are incomparable they have to be synchronize. Each person can have ifferent reuests on thuality of system an therefore the values of weight coeficients are also ifferent. Implementation of a ominant principle has been a significant contribution in solving multicriterial problems. It is associate with a term Pareto optimality set. In the case of solving problem of multicriterial optimization it hols that solution x ominate over solution y (or solution y is ominate of solution x, if i =,2,...,n, f (x f (y i i n an after substitution of (4 we obtain: e ( e i + e i + ( x x ii p = (6 During the operation of large power systems it is necessary to ensure an effective oscillation amping process. ' ( x x ( x x e f ' ' ( x ( x x x x x i i u together f (x < =,2,..., n, where f (y ' e e u Not ominate components create pareto optimal set of solutions. It is not possible to make a ecision which component from this set is the best. Each person can choose from the set a solution which is the best accoring own priority. It is not a target to fin only one solution using present approach of solving multicriterial problem, but fin all or maority of not ominate solution. Flowline of all not ominate solutions is calling Paret s front. The iea of Pareto optimality was use by several authors (Golberg, 989, (Louis an Rawlins, Synchronous generator moel The synchronous generator moel has been erive an escribe in many papers. In our paper the synchronous generator moel of 5 th orer will be consiere (MACHOWSKI, 997. The machine motion euation: ω& = M δ& = ω ω 0 ( p p D ω s b m e = ω Euation escribing the electromagnetic processes: T0 e& = e e + i( x x T0 e& = e e + i( x x T e& = e e + i ( x x (2 (3 Fig.3 Generator euivalent circuits 5. Experimental results The propose algorithm of the synchronous generator control has been verifie using an example of the 259MVA synchronous generator of the nuclear power plant Mochovce (EMO in Slovakia (Murgaš, In our paper the synchronous generator has been escribe by the nonlinear moel of 5 th orer (2 - (3. Because the GPC synthesis is base on the plant moel transfer function, it is necessary to ientify the ARX moel of this nonlinear system at the operating point using the least-suares metho. The system input is the fiel voltage v f an the system output is the terminal voltage v t. The operating point correspons to v t = p.u. The perioic suare signal with amplitue 0,05p.u. an freuency 0,0Hz has been use as the input signal. The ientifie moel transfer function is as follows: T (z f 2 3 0,00348z 0,00247z =, 2 3,535z + 0,2595z + 0,2696z T VZ = 0,25s (7 Meaning of symbols is presente in Appenix. In this moel the screening effect of the rotor boy eycurrents in th-axis is neglecte, so that x = x an e = 0. This moel reverts to the classical five wining moel with armature transformer emfs neglecte. v v e R = e x x i R i (4 The synchronous generator active power can be escribe as follows: p 2 2 ( v i + v i + ( i i R = (5 e + Fig.4 Solutions 9

4 Each point in this figure represents epenency of fitness function f 2 on fitness function f. First solution (where = 0, ch = 2, ρ = 0,0 is selecte from the set of right solutions, where the value of the fitness function f is the smallest, which means this solution is the best one for the reference signal tracking. We choose also the last solution (where = 7, ch = 4, ρ =, in which fitness function f 2 is the smallest an this solution is the best one for active power oscillation amping. The resulting control system performances of these two solutions are own in Fig. 5 an Fig. 6. Pel enotes the first solution controlle variable an Pel 2 enotes the last solution controlle variable. Table Right solutions set ch ρ f f ,0 0,844 0, ,0 0,88 0, ,0 0,2442 0, , 0,2666 0, , 0,292 0, , 0,2924 0, , 0,3208 0, , 0,3899 0, ,3932 0, ,4284 0, ,4469 0, ,52 0, ,5659 0, ,6085 0, ,6248 0, ,643 0, ,73 0, ,756 0, ,7253 0, ,73 0, ,7366 0, ,804 0, ,855 0, ,8639 0, ,8847 0,02 Fig.6 Time responses of the active power P el Fig.7 Time responses of the fiel voltage v f It is obvious from time responses, that Pel is better in a signal reference tracking than Pel 2, which is on the other sie better in active power oscillation amping. Results of reference signal tracking as well as of active power oscillation amping epen on chosen criteria. Conclusion In conclusion, the plant control is mostly affecte by the change of the preictive horizon. The faster reference signal tracking is in most cases achieve by reucing, which means that values of fitness function f are the lowest, however values of fitness function f 2 are the highest. It ows that these two criteria perform against each other. Therefore the multicriterial optimization with a ominant principle has been applie. It is up to esigner to choose a solution out of right solution set (Fig. 4., which is the most suitable for him. This is an universal approach, suitable for any plant. It is also necessary to choose a solution with respect to uality of the reference signal tracking an the active power oscillation amping. Fig.5 Time responses of the terminal voltage v t an its esire value r Acknowlegement This work has been supporte by the Slovak Research an Development Agency uner the contract APVV

5 References [] CLARKE, D.W., C. MOHDANI an P.S. TUFFS (989, Generalize Preictive Control. Part I: The Basic Algorithm, Part II: Extensions an interpretations. Automatica, vol.23, No2, [2] GOLDBERG, D. E. (989. Genetic Algorithms in Search, Optimisation, an Machine Learning, Aison- Wesley Publiing Company, Inc. [3] LOUIS, S. J., G. J. E. RAWLINS (993. Pareto optimality, GA-easiness an eception, In: 5th conf. ICGA, [4] MACHOWSKI, J., BIALEK, J.W., BUMBY, J. R. (997. Power system ynamics an stability, Chichester, Englan. [5] MURGAŠ, J., I. HEJDA (993. Aaptive control of technological process, Publiing Company STU in Bratislava. Appenix - Symbols ω - angular velocity of the generator (in electrical raians ω s - synchronous angular velocity in electrical raians ω rotor spee eviation M - inertia coefficient p m mechanical power supplie by a prime mover to a generator p e D - electromagnetic air-gap power - amping coefficient T 0, T0 - open-circuit -axis transient an subtransient time constants T 0, T0 - open-circuit -axis transient an subtransient time constants i, i - currents flowing in the fictitious - an -axis armature coils - steay-state emf inuce in the fictitious -axis armature coil proportional to the fiel wining self-flux linkages e - transient emf inuce in the fictitious -axis armature coil proportional to the flux linkages of th-axis coil representing the soli steel rotor boy e - transient emf inuce in the fictitious - axis armature coil proportional to the fiel wining flux linkages e - subtransient emf inuce in the fictitious -axis armature coil proportional to the total -axis rotor flux linkages e - subtransient emf inuce in the fictitious -axis armature coil proportional to the total -axis rotor flux linkages x,x, x - total -axis synchronous, transient an subtransient reactance between (an incluing the generator an the infinite busbar x,x, x - total -axis synchronous, transient an subtransient reactance between (an incluing the generator an the infinite busbar Appenix 2 Parameters of the synchronous generator T0 = 7,7 T2 = 0,04 T2 = 0,23 T = 0,035 x x x =,99 = 0,267 = 0,3 Ing. Marián Mrosko x =,82 x = 0,204 x t = 0,4 x = 0,4 an Information Technology Institute of Control an Inustrial Informatics 82 9 Bratislava marian.mrosko@stuba.sk oc. Ing. Eva Miklovičová, PhD. an Information Technology Institute of Control an Inustrial Informatics 82 9 Bratislava eva.miklovicova@stuba.sk prof. Ing. Ján Murgaš, PhD. an Information Technology Institute of Control an Inustrial Informatics 82 9 Bratislava an.murgas@stuba.sk l