Analysis and Design of PSS Based on Sliding Mode Control Theory for SMIB

Transcription

1 Analysis an Design of P Base on liing Moe Control Theory for MIB Amin Rajabi, Ghazanfar hahgholian, Bahram Karimi, Mohamma Reza Yosefi Department of Electrical Engineering Islamic Aza University Najafaba Branch Esfahan, Iran amin_rajabi@sel.iaun.ac.ir, shahgholian@iaun.ac.ir, bahram-karimi@aut.ac.ir, mr-yousefi@iaun.ac.ir Abstract This paper present a new metho for esign of power system stabilizer (P base on sliing moe control (MC technique. The control objective is to enhance stability an improve the ynamic response of the single machine infinite bus (MIB system. In orer to test effectiveness of the propose scheme, simulation will be carrie out to analyze the small signal stability characteristics of the system about the steay state operating conition following the change in reference mechanical torque an also parameters uncertainties. For comparison, simulation of a conventional control P (lea-lag compensation type will be carrie out. The main approach is focusing on the control performance which later proven to have the egree of shorter reaching time an lower spike. Keywor- power system stabilizer (P; single-machine infinite-bus (MIB; sliing moe control (MC. I. INTRODUCTION In recent years, consierable efforts have been mae to enhance the ynamic stability of power systems. Moern voltage regulators an ecitation systems with fast response can be use to improve the transient stability by increasing the synchronizing torque of a machine. However they may have a negative impact on the amping of rotor swing. In orer to reuce this unesirable effect an improve the system ynamic performance, it is useful to introuce supplementary signal to increase the amping. One of the cost effective solution to this problem is fitting the generators with a feeback controller to inject a supplementary signal at the voltage reference input of the automatic voltage regulator to amp the oscillations. This is evice known as a P [-]. Various control metho have been propose for P esign to improve overall system performance. Among these, conventional P of lea-lag compensation type have been aapte by most utility companies because of their simple structure, fleibility an ease of implementation. The power system is a highly comple system an the system equations are nonlinear an the parameters can vary ue to noise an loa fluctuation. However, the performance of conventional stabilizer can be consierably egrae with the change in the operation conition. In aition if some changes occur in AVR parameters, there will be great changes in system conitions. Therefore the conventional stabilizer won't have a goo performance in action [7-9]. Now there are many stuies on P in power systems that contain P optimal placement, P coorination an using more effective metho in P esigning [0]. In recent contet using of optimal control theory [], aaptive controllers [] an some techniques such as artificial neural networks [] an genetic algorithm [, ] are performe. A nonlinear aaptive back-stepping controller esign base on the fourth orer power system moel incluing the unknown parameters for multi-machine power systems propose in []. In [7] the ynamic characteristics of the propose P base on synergetic control theory are stuie in a typical single-machine infinite-bus power system an compare with the cases with a conventional P an without a P. liing moe control is one of the main metho employe to overcome the uncertainty of the system. This controller can be applie very well in presence of both parameter uncertainties an unknown nonlinear function such as isturbance. liing moe control technique has been use to control robots, motors, mechanical systems, etc an assure the esire behavior of close loop system [8]. liing moe controllers rely on high spee switching to achieve the esire output tracking. This high spee switching phenomenon is calle chattering. The high frequency components of the chattering are unesirable because they may ecite un-moele high frequency plant ynamics which coul cause system instabilities. This chattering can be eliminate by choosing a bounary layer in sliing surface. This paper is organize as follows. ection II presents the ynamic moel of a MIB system. In section III the mathematical moel of the single machine infinite bus system is transforme into a form that facilitates the esign of nonlinear control schemes. Then the sliing moe controller is propose. Conclusions are rawn in section V. The controller is valiate using non-linear moel simulation. II. DYNAMIC MODEL A single machine infinite bus system consisting of a synchronous generator with loa, connecte through transmission lines to a very large network that can be approimate by an infinite bus. An infinite bus is a source of constant frequency /0/$ IEEE IPEC 00 99

2 an voltage either in magnitue an angle. Although the case of the MIB is not a true representation of the real power system, but it is hope that by analyzing such single machine case can help in the esign of sliing moe control technique for multi-machine power systems. A schematic representation of this system is shown in Figure. Generator T.Line Infinite bus Transformer T.Line Fig. : ingle machine connecte to a large system through transmission lines The etaile nonlinear moel of a single machine infinite bus system is a sith orer moel. However this moel is usually reuce to a generalize one-ais nonlinear thir orer moel. The equations escribing a thir orer moel of a single machine infinite bus system can be written as: Where: δ = ω K D ω ( ω + (P E q = (E F E q F = = c o E q = E = k u F + + E q T T + + L L VE q Pe = sin( δ m P V cos( δ an δ is the rotor angle of the generator (raians, ω is the spee of the rotor of the generator (raian/sec, ω o is the synchronous machine spee (raian/sec, K D is the amping constant (pu, H is the inertia constant (sec, P m is the mechanical input power of the generator (pu, P e is the active electrical power elivere by the generator (pu, E q is the EMF of the q-ais of the generator (pu, E' q the transient EMF in the q-ais of the generator (pu, E F is the equivalent EMF in the ecitation wining of the generator (pu, T' o is the -ais transient short circuit time constant (sec, K C is the gain of the ecitation amplifier, u F is the control input of the ecitation amplifier with gain K C, is the total irect reactance of the system (pu, X' is the total transient reactance of the system (pu, X is the -ais reactance of the generator (pu, X' is the -ais transient reactance of the generator (pu, X T is the reactance of the transformer (pu, X L is the reactance e ( ( of the transmission line (pu an V is the infinite bus voltage (pu. The states of the system choice as follows: = δ = ω ω = E Hence the system state vector will be: q o ( T ( t = [ ] ( Also the control input u is taken to be: u k c = u f ( Whit a view to clear presentment of nonlinear equations of the system, efine the following constants of the generator: o o K D = P o m ( = V H = Therefore, using ( through ( an (, the equations escribing the single machine infinite bus system can be written as: = = + sin( + sin( = + cos( + u V V + Also esire values of the system states implemente with, an.therefore the esire system state vector will be: ( (7 D = [ ] (8 The control input which enables the system to achieve the esire states is enote by u. In aition the eviations of the rotor angle from its esire value take as output of the system. Hence: y = (9 Therefore using (7, the values of erive as follows:, an to be 99

3 + = 0 sin( cos( III. u u sin( DEIGN CONTROLLER + = 0 (0 The objective of this section is to esign a controller base on sliing moe theory for a single machine infinite bus system so that regulate the states of the MIB system to their esire values an maintain the stability of the system in operation point an uncertainty an also increase the rate of oscillation amping. The equations (7 an (9 those escribing the MIB system are highly nonlinear. Therefore, in first step, to facilitation esign of nonlinear controller a change of variable z=t( is consiere, such that: z = z = z = + sin( + sin( + ( If z converges to zero as t, then converges to D as t. For sin( 0, the inverse of the transmission given in ( is: = z + = z = /( sin(z + (z z sin((z + The conition sin( 0 means that: ( = δ nπ n = 0, ±, ±, ( whereas, the operating region of rotor angle is in ( 0, π, hence this conition is always satisfie in operation region. However if rotor angle is not in ( 0, π, then synchronism will be lost. The equations of the MIB system can be written as function of the new variable such that: Where: z = z z = z z = f (z + G(zu y = z ( an ( + z z f (z = + sin((z + z cos((z + + z cot(z + G(z + z z sin((z + ( = sin(z + ( In the original coorinate, the functions (z = f ( an G an (z = G ( are: + cos( + + sin( f ( = sin( ( + cos( sin( G f cos( (7 ( = sin( (8 The moel of the synchronous generator will be use for esigning the sliing moe controller. Then the esigne controller will be transforme into the original coorinate using =T - (z that given in (. The secon step of the sliing moe control esign process is the esign of the sliing surface. The sliing surface for the MIB system is as follows: = y + ρ (9 y + ρ y = z + ρz + ρz Where coefficients ρ an ρ are positive scalars an are chosen to obtain the esire transient response of the output of the system. The switching surface can be written as a function of, an such that: = + sin( + sin( + ρ + ρ ( + (0 Note that the choice of the switching surface guarantees that the output of the system converges to zero as t on the sliing surface (X=0. The thir step of the propose sliing moe controller process is to esign the control function that provies the motion on the sliing surface, such that: u = (f (z + ρz + ρz + ηsign(z + ρz + ρz G(z ( That η is a positive scalar an etermine by esigner. For eamination the sliing moe eistence with respect to the time, it follows that: 99

4 Therefore: Hence: = y + ρ ( ηsign( y + ρy = f (z + G(zu + ρz + ρz = f (z + ρ z + ρ z + ( f (z ρ z ρ z ηsign(z + ρ z ηsign(z + ρ z + ρ z + ρz ( ηsign(s η < 0 ( Therefore the ynamics of guarantees that < 0. ince riven to zero in a finite time, the output y=z is governe after such finite amount of time by the secon orer ifferential equation y + ρ y + ρy = 0. Thus the output y=z will converge to zero as t because ρ an ρ are positive scalars. ince z converges to zero as t. Then z an z will also converge to zero as t. Therefore it can be conclue that the propose sliing moe controller guarantees the asymptotic convergence of z to zero as t. The controller function given can be written in the original coorinate as follow: where: u = + + sin( sin( sin( = + ρ + sin( ( ( + ρ sin( + + ( + cos(sin( cos( cos( ( ρ + + ρ ( ηsign( sin( + sin( + ( ( Therefore, the propose controller given by ( an ( when applie to the MIB system that given in (7 an (9 guarantees the asymptotic convergence of to D as t. The propose controller is confronte with the problem of chattering which is unesirable in practice. This chattering can be eliminate by choose a bounary layer of with ε, in (=0 such that the iscontinuous control function given in ( is rewritten as: u = + + ( ρ sin( {( ( ( + cos( cos( ηsat( } ε + sin( + ρ sin( + + sin( cos( (7 where ε>0 form bounary layer in the vicinity of sliing surface. The coefficients ρ an ρ are ientifie to provie esire eigenvalue placement of the ifferential equation (9. The settling time (t is sec. The natural frequency (ω n is calculate by ω n 0/t =0. Using the stanar secon orer homogeneous equation s +.ωns + ωn = 0 in ITAE criterion, the coefficients are chosen to obtain the esire transient response of the output ynamics as ρ = an ρ =00. IV. IMULATION REULT The propose sliing moe control scheme given by ( an (7, is applie to the single machine infinite bus system given by (7 an (9. The controlle system is simulate using MATLAB. The performance of propose control scheme (MCP will be compare to the performance of a conventional controller (AVR+P an with the system without P (NOP. Two ifferent cases are consiere for simulation purposes. A. Change in mechanical torque The nominal parameters of the synchronous generator are use. The system is in steay state. A 0% step change in the reference mechanical torque is assume at t=7. sec. In this case the eciter gain K A is equal to 00. The responses of the angle of the generator δ, spee eviation an electrical power P e, when the sliing moe controller is use are shown in Figure, Figure an Figure respectively. elta(ra Time(sec Fig. : The rotor angle of the generator using MCP (case Also the responses of the angle of the generator δ when the sliing moe controller, AVR+P controller an without any controller are use, are shown in Figure 7. It can be seen that the response of MCP converges to constant value earlier than AVR+P controller. Figure 8 an Figure 9 show the responses of the spee eviation an electrical power when 997

5 lip P e (p.u elta(ra lip Time(sec Fig. : The spee eviation using MCP (case Time(sec Fig. : The electrical power using MCP (case NOP CP MCP Time(sec Fig. : The rotor angle of the generator (case NOP CP MCP Time(sec Fig. : The spee eviation (case P e (p.u NOP CP MCP Time(sec Fig. 7: The electrical power (case the sliing moe controller, AVR+P controller an without any controller are use, respectively. Again it can be seen from these two figures that the best response are obtaine when the MCP is use. B. Change in eciter gain This case is use to inicate the robustness of propose controller to change in the one of the parameters of the system. Hence, the eciter gain K A, is change from 00 to 00. The responses of the angle of the generator δ when the sliing moe controller, AVR+P controller an without any controller are use are shown in Figure 0. It can be seen that the response of MCP converges to constant value earlier than AVR+P controller. elta(ra NOP CP MCP 0. 0 Time(sec Fig. 8: The rotor angle of the generator (case Also the responses of the spee eviation an electrical power when the sliing moe controller, AVR+P controller an without any controller are use are shown in Figure an Figure respectively. Again it can be seen from these two figures that the best response are obtaine when the MCP is use. However, the simulation results inicate that the propose sliing moe controller work well when applie to the MIB system. Moreover, the simulation results show that the propose controller is robust to parameter uncertainties an to the isturbance. In aition, the sliing moe controller gave better results than the conventional AVR+P controller. 998

6 lip P e (p.u NOP CP MCP 0 Time(sec Fig. 9: The spee eviation (case 0.7 NOP CP MCP 0 Time(sec Fig. 0: The electrical power (case V. CONCULION Whereas power system is a highly comple system an the system equations are nonlinear an the parameters can vary ue to noise an loa fluctuation, it's essential that use a controller that can maintain the stability of the system an provies goo amping enhancement an also have a goo performance, when occur changes in system operation conitions. Accoring to non-linear simulation results of MIB system, it is foun that the propose controller work well an are robust to change in parameters of the system an to isturbance acting on the system an also inicate that the sliing moe controller achieves a better performance than the conventional P. REFERENCE [] M. Bouhamia, M.A. Denai, Robust stabilizer of electric power generator using H with placement constraints, Jou. of Elec. Eng., Vol., No.7-8, pp.7-8, 00. [] A. houlaie, M. Bayati-Poueh, G. hahgholian, Damping torsional torques in turbine generator shaft by novel P base on genetic algorithm an fuzzy logic, Jour. of Trans. on Elec. Tech. (JTET, Vol., No., pp.-0, Winter 009. [] G. hahgholian, M. Arezooman, H. Mahmooian, Analysis an simulation of the single machine infinite bus power system stabilizer an parameters variation effects, IEEE/ICIA, pp.7-7, Nov [] Y.J. Lin, ystematic approach for the esign of a fuzzy power system stabilizer, IEEE/POWERCON, pp.77-7, Nov. 00. [] V. Banal, B. Banyopahyay, "Robust ecentralise output feeback sliing moe control technique-base power system stabiliser (P for multimachine power system", IET Con. The. an App., Vol., No., pp.-, ep [] A.G.E. Abera, B. Banyopahyay, "Digital reesign of sliing moe control with application to power system stabilizer", IEEE/IECON, pp.-9, Orlano, Nov [7] V. Banal, B. Banyopahyay, A.M. Kulkarni, "Decentralize sliing moe control technique base power system stabilizer for multimachine power system", IEEE/CCA, pp.-0, Toronto Canaa, Aug. 00. [8].I. afie, M. Mhah, A.R. Hasimah, A. Wahab, H.M. Yusri, "liing moe control power system stabilizer for single machine connecte to infinite bus", IEEE/PECON, pp.-, Dec [9] J. Faiz, G. hahgholian, M.Arezooman, Analysis an simulation of the AVR system an parameters variation effects, IEEE/POWRENG, 0-, April 007. [0] G. Cheng, L.Q. Zhan, "imultaneous coorinate tuning of P an FACT amping controllers using improve particle swarm optimization", IEEE/APPEEC, pp.-, March 009. [] R. Tiako, K.A. Folly, "Impact of weighting coefficients on the convergence rate of the steepest escent algorithm use for the esign of an optimal power system stabilizer (OP", IEEE/AUPEC, pp.-, yney, NW, Dec []. Zhang, F.L. Luo,"An improve simple aaptive control applie to power system stabilizer", IEEE Tran. on Pow. Elec., Vol.0, No., pp.9-7, Feb [] C.J. Chen, T.C. Chen, H.J. Ho, C.C. Ou, "P esign using aaptive recurrent neural network controller", IEEE/ICNC, Vol., pp.77-8, Tianjin, Dec []. Pana, N.P. Pahy, Application of genetic algorithm for P an FACT base controller esign, Inte. Jou. of Com. Met., Vol., No., pp.07-0, 008. [] R. Prahan,. Pana, "Application of Genetic Algorithm base P for two-area AGC system in eregulate scenario", IEEE/NABIC, pp.07-, Coimbatore, Dec [].. Lee,.Y. Li, J.K. Park, "Nonlinear aaptive back-stepping controller esign for power system stabilizer in multi-machine power systems", IEEE/AACC, pp.0-09, Washington UA, June 008. [7] Z. Jiang, "Design of a nonlinear power system stabilizer using synergetic control theory", Elec. Pow. ys. Res., Vol.79, No., pp/8-8, June 009. [8] K.D. Young, V.I. Utkin, U. Ozguner, "A control engineer's guie to sliing moe control", IEEE Trans. on Con. ys. Tec., Vol.7, No., pp.8-, Aug. 00. [9] Y. Lee, Y.B. htessel, "Comparison of a feeback linearization controller an sliing moe controllers for a permanent magnet stepper motor", IEEE/T, pp.8-, Baton Rouge, Mar./April