229 Nonlinear Excitation Control in Electrical Power Systems Xin Zhang and Jin Jiang Department of Electrical and Computer Engineering University of Western Ontario London Ontario,N6A 5B9 Canada Abstract-Based on differential geometric theory,a feedback linearization was addressed in detail in [2-4]. systematic design of a nonlinear excitation controller is However,only recently,feedback linearization developed for a third order single machine infinite bus techniques have been used to design the nonlinear (SMIB)power system.The design approach is to controller for electrical power systems.The first transform the original nonlinear system into an application of nonlinear control in power systems was equivalent linear system.Furthermore,this approach reported in [5].Nonlinear controller design were has been extended to the design of a nonlinear considered in [6-7]for SMIB,[8-9]for multi-machine systems. controller for multimachine power systems.One of the advantages of the nonlinear controller for multimachine power system lies in the fact that the MIMO nonlinear In this paper,a nonlinear controller is first developed system is decoupled under the proper state for a SMIB,and then extended to multi-machine power transformation.Also,the obtained nonlinear controller systems.The formula for the nonlinear control law are can be implemented via local measurements.Finally presented explicitly.Finally,the performance of the the performance of the nonlinear controller is examined nonlinear controller is evaluated by simulation. by simulation when it is used with high order power II.Controller design for SMIB Power System systems. In the paper,the nonlinear controller is first designed I.Introduction for a SMIB system represented by a third order equation (1),which is derived from the sixth order by using an Electric power systems are highly nonlinear in nature integral manifold approach [10]. due to nonlinear synchronous generators and loads.The interconnected large electric power systems are being do expanded for more reliable and economic generation dt =0-06 and transmission of power energy.Also,the rapid growth in the load demand reduced the system stability = margin.As a consequence,the nonlinear characteristics d 2H of the system dominate more and more heavily on the system behavior.Even though the well developed linear control theory has a long history of many successful applications in industry,it is very difficult to deal with such nonlinearities in electric power systems.The xa-xa) general practice is to linearize the power system at a dt Tao v,cos6] x specific operating point,and then design a linear controller for the linearized system.Of course,such a For simplicity,let's define linear controller is unable to compensate for large variations in the system operating conditions.On the x1=δ，x2=0-06,x3=eg other hand,nonlinear controllers can handle nonlinearities in the large range operation directly,if Po= @b Vr (2) properly designed. 器BR= 2Hx Twenty years ago,the feedback linearization technique P2= was successfully applied to flight control systems[1]. The differential geometric theory associated with the 0-7803-4314-X/98/$10.00©1998IEEE
229 Nonlinear Excitation Control in Electrical Power Systems Xin Zhang and Jin Jiang Department of Electrical and Computer Engineering University of Westem Ontario London Ontario, N6A 5B9 Canada Abstract - Based on differential geometric theory, a systematic design of a nonlinear excitation controller is developed for a third order single machine infinite bus (SMIB) power system. The design approach is to transform the original nonlinear system into an equivalent linear system. Furthermore, this approach has been extended to the design of a nonlinear controller for multimachine power systems. One of the advantages of the nonlinear controller for multimachine power system lies in the fact that the MIMO nonlinear system is decoupled under the proper state transformation. Also, the obtained nonlinear controller can be implemented via local measurements. Finally, the performance of the nonlinear controller is examined by simulation when it is used with high order power systems. I. Introduction Electric power systems are highly nonlinear in nature due to nonlinear synchronous generators and loads. The interconnected large electric power systems are being expanded for more reliable and economic generation and transmission of power energy. Also, the rapid growth in the load demand reduced the system stability margin. As a consequence, the nonlinear characteristics of the system dominate more and more heavily on the system behavior. Even though the well developed linear control theory has a long history of many successful applications in industry, it is very difficult to deal with such nonlinearities in electric power systems. The general practice is to linearize the power system at a specific operating point, and then design a linear controller for the linearized system. Of course, such a linear controller is unable to compensate for large variations in the system operating conditions. On the other hand, nonlinear controllers can handle nonlinearities in the large range operation directly, if properly designed. Twenty years ago, the feedback linearization technique was successfully applied to flight control systems[ 11. The differential geometric theory associated with the feedback linearization was addressed in detail in [2-41. However, only recently, feedback linearization techniques have been used to design the nonlinear controller for electrical power systems. The first application of nonlinear control in power systems was reported in [5]. Nonlinear controller design were considered in [6-71 for SMIB, [8-91 for multi-machine systems. In this paper, a nonlinear controller is first developed for a SMIB, and then extended to multi-machine power systems. The formula for the nonlinear control law are presented explicitly. Finally, the performance of the nonlinear controller is evaluated by simulation. 11. Controller design for SMIB Power System In the paper, the nonlinear controller is first designed for a SMIB system represented by a third order equation (I), which is derived from the sixth order by using an integral manifold approach [lo]. =W-Wb d6 dt - For simplicity, let’s define x, = 6, x2 = o -ob, x3 = el 0-7803-4314-X/98/$10.00 0 1998 IEEE
230 The general steps to design the nonlinear controller are shown below: 1.Calculate the vector fields 14,P 1(x:-x) g,adg,..,ad g (10) P3= T。 y u=EFD 2.Check the controllability and involutivity Hence,Eqn.(1)can be rewritten in terms of the vector The controllability matrix is expressed as: fields fx)and g(x). =f(x)+g(x)u (3) g,ad 8,ad,'g (11) In order to proceed with the feedback linearization,it is 3.Find T (x) necessary to review some basic notations and concepts from differential geometry [2]. a=0, aT ≠0 (12) Definition 1 (Lie derivative):The Lie derivative of a 8x2 a0, ax real-valued function h with respect to the vector field f From (12)it is easy to see that the simplest solution to is defined as the above equation is T (x)=x,then the nonlinear (x) state transformation can be expressed as L,h=Vh.f (4) dx z=T(x)=T LT LT (13) f(x) 4.Derive the control law Definition 2 (Lie bracket):The Lie bracker of two v-L'T vector fields f and g is denoted by adg and defined u=a(x)+B(x)v= (14) L,, as Then,under the nonlinear state transformation (13)and ad 8=f.8=Vgf-Vf g (⑤ the control law (14),the original nonlinear system (3) Repeated Lie bracketfand g can then be defined can be transformed into a simple linear system: recursively as i1=z2 adg=g i2=; (15) (6 ad,g=[,ad,g]i=l,2… i3=v Definition 3 (Relative degree):A single-input single- 5.Controller design Using the state equation (15),we can easily design output nonlinear system is said to have relative degree r controller for either stabilization or tracking purpose. in a region U if,for all xeU,LL,'h(x)=0 With the equation: (0≤i<r-l)and V=-k131-k222-k323 (16) LL,h(x)≠0. one can place the poles of the closed-loop system to (7) desired locations by choosing the proper feedback gains Definition 4 (Complete integrability):A linearly such that the original nonlinear system is stabilized. independent set of vector fieldsf,f2,f is III.Feedback Linearization of MIMO Systems said to be completely integrable if,and only if,there exist n-m scalar functions (h,h2,,h-m In practice,the power system consists of many generators.The technique of feedback linearization for satisfying the system of partial differential equations SISO systems can not be used directly for MIMO Vh·f;=0 (8) systems.Fortunately,the concepts developed for SISO where1≤i≤n-m,I≤j≤m,and the gradients systems can be extended to MIMO systems.However, Vh;are linearly independent. the corresponding definitions and theorems for feedback Definition 5 (Involutivity):A linearly independent set linearization of MIMO systems will be more complex than those for SISO systems of vector fields [f,f2,,f is said to be involutive if,and only if,there exist such that In our study of the multi-machine system,we will restrict our analysis to the systems with the same U.ef,小-a=(.因 寸i,j(9) numbers of inputs and outputs,which can be described in the following state space form:
230 Hence, Eqn.(l) can be rewritten in terms of the vector fieldsflx) and g(x). In order to proceed with the feedback linearization, it is necessary to review some basic notations and concepts from differential geometry [2]. Definition 1 (Lie derivative): The Lie derivative of a real-valued function h with respect to the vector field f is defined as x = f(x)+ g(x)u (3) Definition 2 (Lie bracket): The Lie bracket of two vector fields f and g is denoted by adfg and defined as ad,g=[f4]=Vgf-W (5) Repeated Lie bracket f and g can then be defined recursively as Definition 3 (Relative degree): A single-input singleoutput nonlinear system is said to have relative degree r in a region u if, for all XEU, L,L,'h(x) = O (0 5 i < I - 1) and L,L,'-'h(x) # 0. (7) Definition 4 (Complete integrability): A linearly independent set of vector fields [ f, , f, , - -., f, } is said to be completely integrable if, and only if, there exist n-m scalar functions {hl, h,, -1.) ha-,} satisfying the system of partial differential equations where 15 i I n - m, 1 I j 5 m, and the gradients Vh; are linearly independent. Definition 5 (Involutivity): A linearly independent set of vector fields (f,, f2, ---, f, } is said to be involutive if, and only if, there exist allk such that Vh, . f, =O (8) 'd i, j (9) The general steps to design the nonlinear controller are shown below: 1. Calculate the vector fields g, ad,g, ..., adf"-'g (10) 2. Check the controllability and involutivity The controllability matrix is expressed as: 3. Find TI (x) From (12) it is easy to see that the simplest solution to the above equation is TI ( x ) = x1 , then the nonlinear state transformation can be expressed as z = T(x) = [TI L,T, Lf2TIr (13) 4. Derive the control law v-Lf3T, u = a(')+ P(')V = (14) Then, under the nonlinear state transformation (13) and the control law (14), the original nonlinear system (3) can be transformed into a simple linear system: Vf2 i, = 2, i, = z3 (15) 2, =v 5. Controller design Using the state equation (15), we can easily design controller for either stabilization or tracking purpose. With the equation: one can place the poles of the closed-loop system to desired locations by choosing the proper feedback gains such that the original nonlinear system is stabilized. III. Feedback Linearization of MIMO Systems In practice, the power system consists of many generators. The technique of feedback linearization for SISO systems can not be used directly for MIMO systems. Fortunately, the concepts developed for SISO systems can be extended to MIMO systems. However, the corresponding definitions and theorems for feedback linearization of MIMO systems will be more complex than those for SISO systems. v =-k,z, -k,z2 -k3z3 (16) In our study of the multi-machine system, we will restrict our analysis to the systems with the same numbers of inputs and outputs, which can be described in the following state space form:
231 主=f0x)+28,(x4 [y=,,v (23) -1 It is interesting to note that the system input-output y1=h,(x) (17) relation becomes decoupled and linear under the above control law (21). IV.Simulations y=h (x) In the previous sections,the nonlinear controller is For convenience,Let's define developed for power systems.It has been shown u=tu....u theoretically that the design of a nonlinear controller amounts to the design of a linear controller based on a y=[y... (18) linear system which is obtained from feedback g(x)=81(x),,8n(x)] linearization.In this section,the performance of the h(x)=[h,(x),,hn(x】 nonlinear control strategy is illustrated.It should be mentioned that the sixth order power system model is which is in a more compact form used in the simulation.The nonlinear controller is 元=f(x)+g(x)H employed to stabilize the system and track the desired (19) y=h(x) operating point and load angles. Similar to the SISO system,the relative degree in MIMO systems with m outputs can be defined by m In Fig.1,the system becomes stable within seconds integers. under the disturbance of an increase of infinite bus Definition 6 (Relative degree):The system is said to voltage from 1.0 to 1.2.Furthermore,the operating point after 2.5 seconds is very close to the values before have a relative degree,..)at a point xo if the the occurrence of the disturbance.In order to see the following two conditions are satisfied for all x in a maximum ability of the nonlinear controller to stabilize neighborhood of x: the sixth order system,we increase the disturbance from ·L,L,*h(x)=01≤i,j≤mk<n:-1 1.0 to 2.4.From Fig.2,the nonlinear control is still able to stabilize the full order system.But,the disturbance 「LL,4-h(x)…L.Lh( larger than 2.4,the system becomes unstable quickly. Similarly for the tracking cases,the results are presented A(x)= … in Figs 3 and 4.The tracked load angle are 0.7 radian L.L."h (x) and the 1.2 radian,respectively.All the graphs clearly illustrate the ability of handling the control problem is nonsingular at o over a large range operation. We can obtain the following multi-machine system extended from the third order SMIB system with the Stabilization of full ordar model with 7415] form: (20) 05 15 25 The formula for nonlinear control law for MIMO system has the form of u=a(x)+B(x)v: [4,,.了=A(x,-L,f,，。-,f (21) 25 Applying the nonlinear transformation and the control law (21)results in a linear and controllable system represented by the m sets of third order equations: 讲= 15 25 站=z对 (22) =v; Fig.1 Stabilization of the full order nonlinear model On the other hand,a simple relation between the input v and the output y becomes:
23 1 [y;3', ..., y?']T =[VI, .--, V,lT ... Y, =h,(x) u = [u,, ..., U,IT g(x) = [g,(x), -*-1 g,(x)l h(x) = [h,(x), ---, h,(x)l For convenience, Let's define : Y =[YI'.*-I Y,IT (1 8) which is in a more compact form i = f(x) + g(x)u Y = h(x) (19) Similar to the SISO system, the relative degree in MIMO systems with m outputs can be defined by m integers. Definition 6 (Relative degree): The system is said to have a relative degree { r, , . . . , r,,, } at a point x if the following two conditions are satisfied for all x in a neighborhood of x : L,,L,'~,(~)=o I<i,j<m k<ri-l 1 Lgt L, 't-lhl (x) . . . Lgm L, 'I-%, (x ) ... ... is nonsingular at x We can obtain the following multi-machine system extended from the third order SMIB system with the form: i'= fl'(X), fl(x), fgi(x)+,. ui ]- [ (20) The formula for nonlinear control law for MIMO system has the form of u = a(x) + p(x)v : (21) Applying the nonlinear transformation and the control law (21) results in a linear and controllable system represented by the m sets of third order equations: .. i; = 2; i; = 2; i; =vi (22) .. On the other hand, a simple relation between the input v and the output y becomes: It is interesting to note that the system input-output relation becomes decoupled and linear under the above control law (21). IV. Simulations In the previous sections, the nonlinear controller is developed for power systems. It has been shown theoretically that the design of a nonlinear controller amounts to the design of a linear controller based on a linear system which is obtained from feedback linearization. In this section, the performance of the nonlinear control strategy is illustrated. It should be mentioned that the sixth order power system model is used in the simulation. The nonlinear controller is employed to stabilize the system and track the desired operating point and load angles. In Fig. 1, the system becomes stable within seconds under the disturbance of an increase of infinite bus voltage from 1.0 to 1.2. Furthermore, the operating point after 2.5 seconds is very close to the values before the occurrence of the disturbance. In order to see the maximum ability of the nonlinear controller to stabilize the sixth order system, we increase the disturbance from 1.0 to 2.4. From Fig. 2, the nonlinear control is still able to stabilize the full order system. But, the disturbance larger than 2.4, the system becomes unstable quickly. Similarly for the tracking cases, the results are presented in Figs 3 and 4. The tracked load angle are 0.7 radian and the 1.2 radian, respectively. All the graphs clearly illustrate the ability of handling the control problem over a large range operation. Stabilization of full order model with k+20 74 151 il :OS5 0 u, 0.5 1 1-1 1.5 2 2.5 3 3.5 4 Time(secs) 1r l a x - 8377 - I 2 0 0.5 1 1.5 2 2.5 3 3.5 4 Time(=-) I 0.9 Et -,, , ,I 0 0.5 1 1.5 2 2.5 3 3.5 4 Time(-) $0.85 Fig. 1 Stabilization of the full order nonlinear model
232 Stabilization of tul order model with k(120741] V.Conclusions The nonlinear controller design procedure for a third order SMIB power system is developed using the Timo(socs) 2.5 feedback linearization.The same approach has been 386 extended to the nonlinear controller design for multi- machine systems.The stabilization and tracking ability of the nonlinear controller for a full order system are 3/0 05 .5 examined.Satisfactory results are obtained. References [1]G.Meyer and L.Cicolani,Application of nonlinear system inverses to automatic flight control design,NATO AGARDAG251,1980,Pp.234-391.· 15 2.5 3 [2]A.Isidori,Nonlinear Control Systems An introduction. 2Edition,Springer-Verlag,NewYork,1989. Fig.2 Stabilization of the full order system under lager [3]J.J.Slotine and W.Li,Applied Nonlinear Control, disturbance Prenctice Hall,New Jersey,1991. [4]H.Nijmeijer and A.J.Van Der Schaft,Nonlinear Dynamical Control systems,Springer-Verlag,NewYork. Tracking cf full order model with k-l120 74 15) 1990. [5]Q.Lu and Y.Sun,Nonlinear Stabilizing Control of Multimachine Systems,IEEE Transactions on Power .5 253 35 Systems.Vol.4,No.1.February,1989,pp.236-241. [6]W.Mielczarski and A.M.Zajaczkowski,Nonlinear Field 77 Voltage Control of a Synchronous Generator using Feedback Linearization,Automatica.Vol.30.No.10. 1994,pp.1625-1630. [7]D.C.Kennedy,Control of a Synchronous Generator by Input-Output Feedback Linearization,University of Waterloo,Ontario,Canada,1995. [8]Q.Lu and Y.Sun et al,Decentralized Nonlinear Optimal Excitation Control,IEEE Transactions on Power Systems Vol.11.No.4,November,1996,pp.1957-1962. 05 15 25 35 [9]J.W.Chapman and M.D.Ilic et al,Stabilizing a Multimachine Power System via Decentralized Feedback Fig.3 Tracking the operating point of the full order Linearization Excitation Control,IEEE Transactions on nonlinear model Power Systems,Vol.8,No.3,August,1993,pp.830-839. [10]P.W.Sauer,S.Ahmed-zaid and P.V.Kokotovic,An Tracking ol tul order mod6lwt格n2o7415别 internal Manifold approach to Reduced Order Dynamic Modeling of Synchronous Machines,IEEE Transactions on Power Systems,Vol.3,No.1,February,1988,pp.17- 23. 2.6 15 Fig.4 Tracking the large operating point (load angle is 1.2 radian)
232 ml.;l P 0 0.5 1 1.5 2 2.5 3 3.5 4 5 0.5 Time(secs) - $370 < 0 0.5 1 15 2 2.5 3 35 4 l"(Secs) d 0 0.5 1 1.5 2 2.5 3 35 4 a, oi lime(secs) Fig. 2 Stabilization of the full order system under lager disturbance Tracking of full order model wi?h k+20 74 151 30.81 1 < Ti! 0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 Ttmelsecs) .. - B 377.2 P z 37/0 0.5 1 1.5 2 2.5 3 3.5 4 Tme(secs) z 37/0 0.5 1 1.5 2 2.5 3 3.5 4 Tme(secs) ,//:I 0 0.5 1 1.5 2 2.5 3 3.5 4 0.8 Time(secs) Fig. 3 Tracking the operating point of the full order nonlinear model Tracklngoffuliodermodefwmi~[~2074 151 fl.5, 1 f 3 0.5 li 0 0.5 1 15 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Tme(ssa) V. Conclusions The nonlinear controller design procedure for a third order SMlB power system is developed using the feedback linearization. The same approach has been extended to the nonlinear controller design for multimachine systems. The stabilization and tracking ability of the nonlinear controller for a full order system are examined. Satisfactory results are obtained. References [l] G. Meyer and L. Cicolani, Application of nonlinear system inverses to automatic flight control design, NATO [2] A. Isidori, Nonlinear Control Systems - An introduction, 2nd Edition, Springer-Verlag, NewYork, 1989. [3] J.J. Slotine and W. Li, Applied Nonlinear Control, Prenctice Hall, New Jersey, 1991. [4] H. Nijmeijer and A.J. Van Der Schaft, Nonlinear Dynumical Control systems, Springer-Verlag, NewYork, 1990. [5] Q. Lu and Y. Sun, Nonlinear Stabilizing Control of Multimachine Systems, IEEE Transactions on Power Systems, Vol. 4, No. 1, February, 1989, pp. 236-241. [6] W. Mielczarski and A.M. Zajaczkowski, Nonlinear Field Voltage Control of a Synchronous Generator using Feedback Linearization, Automatica, Vol. 30, No. 10, [7] D.C. Kennedy, Control of a Synchronous Generator by Input-Output Feedback Linearization, University of Waterloo, Ontario, Canada, 1995. [8] Q. Lu and Y. Sun et al, Decentralized Nonlinear Optimal Excitation Control, IEEE Transactions on Power Systems, Vol. 11, No. 4, November, 1996, pp. 1957-1962. [9] J.W. Chapman and M.D. Ilic et al, Stabilizing a Multimachine Power System via Decentralized Feedback Linearization Excitation Control, IEEE Transactions on Power Systems, Vol. 8, No. 3, August, 1993, pp. 830-839. [lO]P.W. Sauer, S. Ahmed-zaid and P.V. Kokotovic, An internal Manifold approach to Reduced Order Dynamic Modeling of Synchronous Machines, IEEE Transactions on Power Systems, Vol. 3, No. 1, February, 1988, pp. 17- 23. AGARDAG251, 1980, pp. 234-391. . 1994, pp. 1625-1630. Fig. 4 Tracking the large operating point (load angle is 1.2 radian)