Design A Robust Power System Stabilizer on SMIB Using Lyapunov Theory

Design A Robust Power System Stabilizer on SMIB Using Lyapunov Theory Yin Li, Stuent Member, IEEE, Lingling Fan, Senior Member, IEEE Abstract This paper proposes a robust power system stabilizer (PSS) using Lyapunov Stability theory introuce in Optimization & Control in Power System class at University of South Floria. This research examines the robustness of the novel PSS in the synchronous generator connecte to an infinite bus (SMIB) system uner multiple operation conitions. After eriving the state space matrices of the linear moel of the system, the gains of robust PSS are estimate using Lyapunov function in the linear matrix inequality (LMI). The moel analyze in this paper is an electromagnetic moel with automatic voltage regulator (EMTAVR). Then, EMTAVR moel with the robust PSS is analyze an compare with the conventional PSS. In the simulation part, the nonlinear simulation is conucte in MATLAB/Simulink to compare the conventional PSS an novel PSS. Inex Terms SMIB, EMTAVR, PSS, Lyapunov theory, LMI. I. INTRODUCTION Power system stabilizer (PSS) is use to introuce an auxiliary signal to automatic voltage regulator (AVR). When power output from a generator is at high level, electromechanical oscillation will be ominant with a fast exciter an large gain of the amplifier in AVR which make the poles move to right half plane (RHP) easily []. The function of conventional PSS is to change the root locus to avoi poles moving to RHP by aing one pole an two zeros of the system transfer function. However, its esign is base on a linearize moel of a certain operating conition. It may not work well when the operating conition changes. The objective of this paper is to esign a PSS that can work for a range of operating conitions, so the novel PSS or robust PSS has much better robustness. In aition, the selecte values for robust PSS gains can be estimate using CVX toolbox of MATLAB rather than manually selecting like conventional PSS. Different than conventional PSS, robust PSS only contains a state feeback gain matrix, K, which has the same size as the system outputs. It is assume that K is existing using linearquaratic regulator (LQR). Then, Lyapunov stability theory which is introuce in Optimization & Control in Power System class is use to write a Lyapunov function with K in (LMI). Finally, the existing K can be estimate using CVX toolbox when each of linear matrix inequality constraint which is corresponing to each operating conition is satisfie. Before esign robust PSS, the EMTAVR moel of SMIB system will be linearize in state space in Section II. Section Y. Li an L. Fan are with Dept. of Electrical Engineering, University of South Floria, Tampa FL 3362. Email: linglingfan@usf.eu. III oes not only introuce the esign of robust PSS, but also compare it with conventional PSS uner several operating conitions. The linear analysis results will be also valiate by nonlinear simulations which are teste in MATLAB/Simulink in simulation part. II. LINEAR PLANT MODEL Synchronous generator connecte to an infinite bus shown in Fig. is the typical power system in Chapter 7 of a classic textbook Power System Analysis [] use in EEL 6936 Power Systems II at University of South Floria. It is normally use to analyze the effect of AVR an PSS on the power system. The rotor of the synchronous generator is salientpole an its amper is assume to be ignore, so i D = i Q =. Except stator resistance, r s, all of resistances an inuctances of the rotor an stator are consiere, r f, L f, L, an L q. The scripts, an q, mean the components in q frame. X is the impeance of the corresponing inuctance. Certainly, the transmission line impeance, X L, is inclue for linear analysis. SG E a j2x L j2x L Fig.. A generator connecte to an infinite bus. A. EMTAVR moel V Accoring to the linear moel of SMIB system erive in [] using steay state analysis, the EMTAVR moel in this paper has four state variables,,, E a, an E f, because the ynamics of the exciter is consiere, T e. Therefore, Fig. 2 presents the block iagram of the linear EMTAVR moel. H an D are the inertial an amping of the synchronous generator while the gains from k to k 6 (k =T ) are calculate if the initial real power an reactive power, P m an Q, are provie. P m an Q can etermine the transient stator voltage, E a, an the phase angle of the stator voltage,, using ().

2 ΔVref Δ Vt Δve ke Tes ΔEf k5 k6 k3 k3t os Δ E a Fig. 2. Block iagram of EMTAVR moel. P m = P e = E a V Q = E a V k4 k2 ΔPM= 2HsD T Δ s Δ ) sin(2) q ) ( sin() V 2 2 ( cos() V 2 cos 2 () sin2 () q Then, using (2) from [2] to calculate the initial values of k to k 6 to buil linear moel after obtaining E a an. ( ) k = E a V cos() V 2 cos(2) q k 2 = V sin() k 3 = k 4 = k 5 = V ( k 6 = X LV aq V a B. State Space X V aq V a sin() X qv a q V a cos() Because Lyapunov function is the function of the state space, the state space matrices of EMTAVR moel, A, B, C, an D, shoul be erive. Accoring to its block iagram shown in Fig. 2, A an B matrices can be foun easily if is consiere as the input. t E a E f = T 2H k4 T o k5k A D 2H k2 2H T o T e T e k 3T o T e k6k A ) } {{ } A k A Te }{{} B () (2) E a E f U (3) Normally, the feeback of input on the output is neglecte, D =. Due to the limitation of Lyapunov function which will be mentione in next section, C matrix has to be a unit matrix. Hence, four state variables are also the outputs. Y = E (4) a E f }{{} C III. ROBUST PSS USING LYAPUNOV THEORY Although AVR makes the SMIB system transfer more power, the power transfer level () of EMTAVR moel is relative low, so the system will become unstable even if is still small [2]. Power system stabilizer (PSS) is the general metho to increase the power transfer level. A. Conventional PSS The conventional PSS makes the EMTAVR moel stable by changing the root locus of the system. There are one pole an two zeros ae shown in Fig. 3. Δ Vt ΔEf ke k3 k2 ΔVref Δve Tes k3t os Δ E a ΔVc γs 2 τs PSS k5 k6 ΔPM= k4 2HsD Fig. 3. The block iagram of EMTAVR moel with conventional PSS. The parameters of conventional PSS, τ an γ, are esigne by manually tuning, so the selecte values of τ an γ cannot make sure that the system is always stable uner a lot of conitions like power changing or faults occurring. Therefore, the robustness of conventional PSS is limite. For example, in this paper, there are twentyfive ifferent combinations of real power an reactive power, P m an Q, generate by the synchronous machine. In aition, the fault on the transmission line is consiere, so its impeance will be ouble after the fault base on Fig.. Hence, the total number of conitions is fifty an liste in Table I. The system stability can be use to verify the effect of PSS on EMTAVR moel. For linear analysis, the eigenvalues of A matrix is one kin of metho to etermine the system stability. Ten conitions liste in Table I are selecte ranomly to test the system stability incluing EMTAVR moel an EMTAVRPSS moel. The elements in the corresponing A matrices are calculate using P m an Q liste in Table I an the system parameters liste in Table V. The eigenvalues of corresponing A matrices are foun using MATLAB coe an liste in Table II. It is observe that each of A matrices for Con an Con7 has two eigenvalues incluing positive real parts. It means that the system is unstable uner Con an Con7. To fin the effect of conventional PSS on EMTAVR moel, the four ominant eigenvalues of EMTAVR moel integrate T Δ s Δ

3 TABLE I 5 CONDITIONS. X L =.4 Q(p.u.) P(p.u.).8.5.5.8.28 Con Con2 Con3 Con4 Con5.26 Con6 Con7 Con8 Con9 Con. Con Con2 Con3 Con4 Con5.8 Con6 Con7 Con8 Con9 Con2.6 Con2 Con22 Con23 Con24 Con25 X L =.2 Q(p.u.) P(p.u.).8.5.5.8.28 Con26 Con27 Con28 Con29 Con3.26 Con3 Con32 Con33 Con34 Con35. Con36 Con37 Con38 Con39 Con4.8 Con4 Con42 Con43 Con44 Con45.6 Con46 Con47 Con48 Con49 Con5 TABLE II EMTAVR: EIGENVALUES OF A UNDER DIFFERENT CONDITIONS. Eig, 2 Eig 3,4 Con 2.3626 ± 5.2994i.977 ± 4.7385i Con7.3975 ± 5.6237i.326 ± 5.232i Con5.84 ± 4.564i.565 ± 8.352i Con7.8392 ± 5.965i.4257 ± 4.992i Con22.7872 ± 6.89i.4777 ± 3.736i Con28.934 ± 3.8556i.52 ± 8.9879i Con34.593 ± 3.8652i.2723 ± 9.464i Con39.23 ± 3.969i.32 ± 9.7459i Con43.97 ± 3.645i.464 ± 8.2878i Con5.35 ± 3.987i.266 ± 9.4927i oes not have poles or zeros, but the pure gain matrix, K. If the state space of a specific system is known (5), { ẋ = Ax Bu (5) y = Cx A state feeback gain matrix, K, is esigne to generate output static feeback to replace the input using LQR (6). Substituting (6) into ẋ function of (5), u = Ky = KCx (6) ẋ = Ax BKCx = (A BKC)x (7) It is easily to fin that A matrix becomes ABKC, so the system will be stable if a existing K is foun to make the real parts of eigenvalues of (A BKC) negative. In this paper, K is a X4 matrix because of (3) an (4). Therefore, each of output or state variable has a corresponing gain to generate the static feeback shown in Fig. 4. The sum of feeback is name as the compensate voltage,, an is ae to the input. Δ Vt ΔEf ke k3 k2 ΔVref Δve Tes k3t os Δ E a ΔVc KC k5 k6 ΔPM= k4 2HsD T Δ s Δ by conventional PSS are foun an liste in Table III.. an. are selecte for γ an τ. Although the system becomes stable uner Con7 by aing conventional PSS, the limite robustness of conventional PSS cannot make the system stable uner Con. TABLE III EMTAVRCONVENTIONAL PSS: EIGENVALUES OF A UNDER DIFFERENT CONDITIONS. Eig, 2 Eig 3,4 Con 2.795 ± 5.398i.99 ± 4.778i Con7.899 ± 5.766i.2979 ± 5.637i Con5.997 ± 4.537i.345 ± 8.3666i Con7.75 ± 6.82i.484 ± 4.29i Con22.6996 ± 6.2538i.499 ± 3.588i Con28.926 ± 3.786i.4377 ± 9.965i Con34.459 ± 3.865i.2342 ± 9.4882i Con39.92 ± 3.8548i.2659 ± 9.822i Con43.955 ± 3.5458i.497 ± 8.3936i Con5.295 ± 3.898i.72 ± 9.5335i B. Robust PSS Design The novel PSS using Lyapunov theory has a much better robustness because its gains are estimate by MATLAB CVX toolbox rather than manually tuning. The novel robust PSS Fig. 4. The block iagram of EMTAVR moel with robust PSS. To fin K, the Lyapunov function nees to be expresse in LMI. (A BKC) T P P (A BKC) βp (A BKC) T P (A BKC)P β A T C T K T B T P AP P KCP P P A T P C T K T B T AP BKCP P β (8) where P is semiefinite an has the same size as A matrix; β is a constant which is greater than zero. If X = P an Y = KCP, (8) will be rewritten as: XA T Y T B T AX BY Xβ (9) X CVX toolbox of MATLAB is employe to fin the existing X an Y to satisfy LMI constrain. Each of LMI constrain is corresponing to each operating conition. After X an Y are foun, the gain matrix of robust PSS can be can calculate using K = Y X C. If the robustness of PSS nees to be increase, multiple conitions of the system shoul be consiere because one gain matrix, K, foun by CVX can satisfy all of LMI constrains. It means that more conitions consiere causes the better robustness of PSS. The following coe is using CVX toolbox to fin K.

4 cvx_begin sp variable X(n,n) symmetric variable Y(,n) X*A Y *B A*XB*YX*beta <= X*A2 Y *B A2*XB*YX*beta <=... X>=eye(n) cvx_en KC=Y*inv(X) where n presents the size of A matrix. In Lyapunov function, P matrix must be symmetric, so the constrain of symmetric shoul be ae to X matrix. Base on (3) an (4), only A matrix is change corresponing to ifferent conitions because the exciter of the system is normally not variable. Each conition has one specific set of initial values to generate a specific A matrix using (2), so how many A matrices is equal to how many consiere conitions. Due to C is the unit matrix (4), K = KC = Y X. After running CVX coe, the state feeback gain matrix is estimate: KC =.e3 *.8 7.636.6.4 After applying the robust PSS to EMTAVR moel, the eigenvalues of new A matrices, A BKC, are calculate an liste in Table IV. Compare with the eigenvalues liste in Table??, the real parts of all eigenvalues in Table IV are negative besies eigenvalues of Con an Con7. Therefore, it is conclue that the robust PSS has the better robustness on the stability of EMTAVR moel. TABLE IV EMTAVRROBUST PSS: EIGENVALUES OF (A BKC) UNDER DIFFERENT CONDITIONS. Eig, 2 Eig 3,4 Con 4.24 &9.356 8.7674 ±53.8i Con7 6.5566 ± 2.68i 8.9842 ±53.798i Con5 3.4965 ± 7.9962i 2.443 ±53.654i Con7 6.2844 ± 2.6568i 9.2564 ±53.578i Con22 5.935 ± 2.8572i 9.6273 ±53.474i Con28 8.6645 ± 4.66i 7.43 ±53.32i Con3 6.287 ± 7.99i 9.4988 ±53.37i Con39 7.43 ± 7.674i 8.5662 ±53.953i Con43 7.527 ± 5.352i 8.5547 ±53.3i Con5 3.4588 ± 9.2725i 2.2486 ±53.455i IV. SIMULATION RESULTS FROM NONLINEAR MODEL The nonlinear EMTAVR moels with conventional PSS an robust PSS are esigne an simulate in MAT LAB/Simulink using the ifferential equations from [], [2]. The selecte γ an τ an the estimate K are applie to the corresponing nonlinear moels to verify the robustness of two PSS. Fig. 5 presents the screen shot of the nonlinear EMTAVRPSS moel. The techniques for builing nonlinear moel are etaile presente in [3] an are also applie in variety of system [4], [5]. For example, the vector feature technique is employe. Three of four state variables, E a,, an, are arrange in a vector an generate by integrating their erivatives from Embee MATLAB Function which contains the ifferential equations. The initial values for four state variables shoul be calculate to make the flat run before the step change. When the k A T es flat run, the output of the exciter,, is zero, so the initial value of E f is not require. The synchronous generator is assume at the rate spee, so the initial value of is p.u. Therefore, only the initial values of E a an requires to be calculate base on P m an Q from conitions an system parameters liste in Table V. A. Conventional PSS TABLE V PARAMETERS OF THE SYSTEM. Parameter Value(p.u.) V inf Spee, H 5 D.3 T e.8s K A 5 X, X q,.7 r f, X f.5,. X L, X.2,.2 Compare with linear analysis, the system stability of a nonlinear moel can be etermine by the step response, so there was a step change,.p.u., ae to the input,. Corresponing to the unstable conitions analyze in Section III, Con an Con7 were selecte to calculate the initial values; then, start to run the nonlinear moels with the flat run. The step change happene at s, so the step responses of five measurements in EMTAVRconventional PSS moel uner Con an Con7 appeare after s shown in Fig. 6 an Fig. 7. The five measurements are terminal voltage,, compensator voltage from PSS,, stator voltage angle,, generator frequency,, an generate real power, P e Fig. 7 showe that the system became stable after several secons while Fig. 6 presente a unstable system because the step responses became larger an larger rather than approaching a constant. It verifie that the conventional PSS cannot make the system stable uner both of operation conitions. Moreover, one cycle of the oscillation wave in Fig. 7 was aroun.2s, so the oscillation frequency, osc, was 5.24ra/s. The nonlinear moel was verifie to have the same ynamics of its corresponing linear moel because the imaginary part of eigenvalues of Con7 in Table III was the same as osc. B. Robust PSS In the linear analysis section, the system is always stable after applying robust PSS to EMTAVR moel, even if it is uner Con. The simulation results of nonlinear

5 Fig. 5. The nonlinear EMTAVRPSS moel base on SMIB systems is esigne in MATLAB.86.855.85.9.8.7..965.96.955.97.96 9 2 3 4 5 6 7 8 9 2.95 9 2 3 4 5 6 7 8 9 2 x 3. 2 8.995.99.5 9 2 3 4 5 6 7 8 9 2 75 7 9 2 3 4 5 6 7 8 9 2.5.9995 9 2 3 4 5 6 7 8 9 2.28.26.24 9 2 3 4 5 6 7 8 9 2 Fig. 6. Con. The step responses of SMIB system with conventional PSS uner Fig. 7. Con7. The step responses of SMIB system with conventional PSS uner EMTAVRrobust PSS moel verifie this point. The simulation results were plotte in Fig. 8 an Fig. 9. It was observe that all of step responses were stable after the step change regarless of Con an Con7. Furthermore, robust PSS has the faster response spee an smaller oscillation. It can be prove by Table III an Table IV because the eigenvalues in Table IV have larger amping an are further away from the imaginary axis. V. CONCLUSION A novel robust power system stabilizer (PSS) is propose in this paper an its effect on the stability of SMIB system is analyze. Compare with the conventional PSS, the parameter or gain matrix of the robust PSS are estimate by CVX coe rather than selecte manually. Because the gain matrix can be estimate base on multiple operating conitions, this kin of PSS has the much better robustness. CVX toolbox is a goo tool to fin the state feeback gain matrix of Lyapunov

6.86.855.85.8484.8484.8483 x 3 5 5 88.9 88.895 88.89.2799.2798.28.2797 function expresse in LMI. Certainly, the SMIB system shoul be linearize to erive its state space matrices before the estimation. By comparing the eigenvalues, the effect of robust PSS on the linear EMTAVR is presente. The simulation results from the corresponing nonlinear moels gave a strong support to the results from the linear analysis. Although this robust PPS is only applie to a simple power system in this paper, it has a huge potential on the complex an variable power systems ue to its goo robustness. REFERENCES [] A. Bergen an V. Vittal, Power systems analysis (secon eition), 999. [2] Y. Li an L. Fan, Determine power transfer limits of an smib system through linear system analysis with nonlinear simulation valiation, in North American Power Symposium (NAPS), 25, Oct 25, pp. 6. [3] Z. Miao an L. Fan, The art of moeling an simulation of inuction generator in win generation applications using highorer moel, Simulation Moelling Practice an Theory, vol. 6, no. 9, pp. 239 253, 28. [4] L. Fan, R. Kavasseri, Z. L. Miao, an C. Zhu, Moeling of figbase win farms for ssr analysis, Power Delivery, IEEE Transactions on, vol. 25, no. 4, pp. 273 282, 2. [5] L. Fan, Z. Miao, S. Yuvarajan, an R. Kavasseri, Hybri moeling of figs for win energy conversion systems, Simulation Moelling Practice an Theory, vol. 8, no. 7, pp. 32 45, 2. Fig. 8. The step responses of SMIB system with robust PSS uner Con..965.96.955.955.9549 x 3 5 5 74 73.995 73.99.26.2598 Fig. 9. the step responses of SMIB system with robust PSS uner Con7.