A Comparison between a Conventional Power System Stabilizer (PSS) and Novel PSS Based on Feedback Linearization Technique

Transcription

1 J. Basic. Appl. Sci. Res., ()9-99,, TextRoa Publication ISSN Journal of Basic an Applie Scientific Research A Comparison between a Conventional Power System Stabilizer (PSS) an Novel PSS Base on Feeback Linearization Techniue *Babak Bayati Chaloshtori, Sai Hoghoughi Isfahani, Navi Reza Abjai 3,,3 Department of Electrical Engineering, Faculty of Engineering, Shahrekor University, Shahrekor, Iran ABSTRACT Relying on input-output linearization techniue, a power system stabilizer (PSS) was moele using nonlinear control metho in this stuy. Similarly, two ifferent synchronous generator moels, a 3 r orer moel excluing ampers an a 6 th orer moel incluing three amper winings, were employe to moel the PSS. Results of the evelope moels on the single machine infinite bus power system upon occurrence of three phase fault in the transmission line were analyze using MATLAB software. For the first stage, results of the simulation process emonstrate better an faster performance of the nonlinear techniue base PSS which is relie on the input-output linearization metho. Likewise, these results suggeste superior performance of the moele power system stabilizer base on the synchronous generator 6 th orer moel to the 3 r orer moel. KEY WORDS: feeback linearization, power system stabilizer, synchronous generator moel. INTRODUCTION Ascertaining the exact moel of the generator before any type of moeling is highly necessary because of heavy epenency of the PSS performance on the moel of the generator [, ]. Because of the significant role of the generator moel in power systems stuies, initially it has been attempte to use of the synchronous generator moel in association with its ampers, i.e. a moel which covers effects of the transient state uring moeling of the synchronous generator [3]. Then, the input-output linearization control metho, as a moeling metho to the nonlinear control area, given the nonlinear behavior of the power system an particularly the synchronous generator was use to moel the PSS. Subseuently, using input-output linearization techniue in a single-machine infinite-bus power system the PSS was moele in association with the generator 3 r orer moel, i.e. excluing ampers which it inicates transient state of the generator an then simulation results were obtaine. Afterwars, the same stages were repeate consiering ampers, i.e. inclusion of the transient state of the generator which is the 6 th orer moel of the generator, an results of the simulation were compare with the preceent moe. Subseuently, results of two previous stages were compare with the results obtaine from the conventional PSS.. Synchronous generator moel Generally, synchronous generator behaves very complicately, so, achieving a very etaile moel is actually impossible for such generators; however, moeling of the synchronous generator is very essential for analysis of the power system. Hence, researchers have sought for various moels of the synchronous generator all the time in orer to use of it to either analyze or moel power systems. 8 th orer moel is a fairly complete an etaile moel of the synchronous generator which has been generalize to analyze its behavior. It inclues six electrical nonlinear ynamic euations an two mechanical nonlinear ynamic euations. The orer of this moel for salient pole synchronous generator is 7 which has a lesser amper wining at axis to the roun pole synchronous generator 8 th orer moel [,,3]. Although, it is a complete an etaile moel, it is harly use for power system relate moeling proceures because of its increase state variables as well as voluminous measurements, hence esigners prefer to use of more low-orer moels for moeling purposes[4]. Two moels of the roun pole synchronous generators calle 3 r orer moel an 6 th orer moel are introuce through the next sections of this paper... Synchronous Generator 3 r orer Moel Consiering the three state variables E, ω r, δ, effect of the amper winings was ignore uring the esigning of this moel. Likewise, transient EMF of -axis E, has been suppose constant. If E is constant in this moel, then amping inuce by ey currents of the rotor boy has been ignore, as well. If there is not any wining parallel to the perpenicular axis, which moels the effect of rotor boy, then we have X = X, E = [,]. *Corresponing Author: Babak Bayati Chaloshtori, Department of Electrical Engineering, Faculty of Engineering, Shahrekor University, Shahrekor, Iran. bbayati@yahoo.com 9

2 Chaloshtori, The preominant euations of the roun-pole synchronous generator 3 r orer moel are summarize as follows []: δ = ω b ω r ω r = (T H m - T e - K D ω r ) E = [-E 3 T - (X - X ) I + E f ].. Synchronous Generator 6 th orer Moel In this moel, for the roun pole synchronous generator incluing 6 state variables δ ω r, E, E, an, a generator with series subtransient EMF, an, an subtransient reactance, X" an X", which their names are moele consiering amper winings of an axes [,]. The sole ifference between this moel an the complete one (8 th orer moel) is that in this moel ω r = ω s, an also electromotive forces of the transformer are unerestimate for voltage euations of armature. The preominant euations of the roun-pole synchronous generator 6 th orer moel are summarize as follows []: δ = ω b ω r 4 ω r = (T H m - T e - K D ω r ) E = [-E T - (X - X ) I + E f ] E = [-E T + (X - X ) I ] = [- T" - (X X" ) I + E ] = [- T" - (X X" ) I + E ] 3. Conventional Power System Stabilizer (CPSS) The architecture of the conventional power system stabilizer (CPSS) is shown in the fig. []. Block iagram of the CPSS has 3 blocks: Fig : Block iagram of the conventional power system stabilizer (CPSS) a) Phase compensation block which provies proper phase lea property in orer to compensate the phase lag between the exciter input an electric torue of the generator. b) The washout block which acts as a high pass filter with time constant, T w, which is enough to perform the task. c) Gain block which ascertains the generate amping inuce by the PSS. CPSS has the following conversion function []: V s (s) ω r (s) = K s st w st w st st Fig. shows the structure of the single machine infinite bus power system incluing PSS [4]. Fig : Single machine infinite bus power system with PSS configuration 9

3 J. Basic. Appl. Sci. Res., ()9-99, The mentione power system contains synchronous generator, exciter an transmission line connecte to the infinite bus. PSS control signal is applie to AVR(Automatic Voltage Regulator) inputs. 4. Moeling of PSS using nonlinear control techniue base on the input-output linearization metho 4.. Moeling Non Linear PSS (NLPSS) incluing synchronous generator 3 r orer moel The moeling of the single machine infinite bus power system is evelope through moeling of the synchronous generator with three ynamic euations using input-output linearization techniue[8,9]. The preominant euations of the single machine infinite bus power system incluing synchronous generator 3 r orer moel are summarize as follows []: δ = ω b ω r ω r = (T H m - T e - K D ω r ) E = [-E T - (X - X ) I + E f ] E f = [ -E T f +K A (V ref +V t -u)] A Where, electrical torue (T e ) is (R s + R L =): T e = V I + V I = E I + ( X - X ) I I An V t = V + V = E R s I X I + (X I R s I ) Similarly, values of I, an I are measure using algebraic euations of the stator an the network: 6 (R s + R L )I + ( X + X L )I = E + V cos δ X + X L I ( R s + R L )I = V sin δ The following euations are obtaine by replacing values of T e, I an I in euations - 4 an ignoring resistance of the stator an transmission line an replacing the numerical values of the parameters, which are liste in the appenix : δ = 34.6 ω r 7 ω r = -.5 E sin δ.3 sin δ +.43 T m E = -.3 E -. cos δ +.5 E f E f = - E f + 8 V ref - 8 V t + 8 u V t =. 68E. 3 cos δ + (. 73 sin δ) If rotor velocity ω r is etermine as our output, then we have: y = ω r Regaring input-output linearization techniue, y is ifferentiate in orer to get a stage in which the input (u) is appeare at the measurement [5,6,7]. y = ω r = f 3 y = ω r = f y (3) = ω (3) r = f 3 ( 5 sin δ ) u Where, preominant euations of f, f, an f 3 are summarize in the appenix. Accoring to the input-output linearization techniue, the input v is selecte as follows [6]: v = f 6 3 ( 5 sin δ ) u u = (f 5 sin δ 3 - v) Then, we have: ω (3) r = v 7 By selecting v as: v = ω (3) r - K e - K e - K 3 e Where, error signal,e is efine as: e ω r - ω r Where, ω inicates the reference spee, so we have: e (3) + K 3 e + K e + K 3 e = By selecting x = e, x = e, x 3 = e we have:

4 x x = x 3 K K K 3 x x Chaloshtori, X = AX x 3 In the case of selecting proper k, k an k 3, it will be possible to inicate that (3) is stable asymptotically, i.e. (e )an hence (ω r ω r )which is esire for the problem [5]..4. Moeling NLPSS Incluing Synchronous Generator 6 th orer Moel In this section, relying on the input-output linearization techniue synchronous generator 6 th orer moel incluing 6 ynamic euations, two mechanical an 4 electrical ynamic euations, are use to moel the PSS of the single machine infinite bus power system. The relevant euations of the power system are liste here []: δ = ω b ω r 3 ω r = (T H m - T e - K D ω r ) E = [-E T - (X - X ) I + E f ] E = [-E T + (X - X ) I ] = [- T" - (X X" ) I + E ] = [- T" - (X X" ) I + E ] E f = T A [ -E f +K A (V ref +V t -u)] Where, electric torue (T e ) is (R s + R L =) : T e = V I + V I = I + I + ( X" X" ) I I An V t = V + V = R s I X" I + R s I + X" I Similarly, values of I an I are measure using algebraic euations of the stator an the network: 4 ( R s + R L )I + (X" + X L )I = + V cos δ X" + X L I ( R s + R L )I = + V sin δ The following euations are obtaine by replacing values of T e, I an I in euations (3-4) an ignoring resistor of the stator an transmission line an replacing the numerical values of the parameters base on the selective moel: δ = ω r 4 ω r = -.63 sin δ cos δ -.8sin δ T m 4 E = -.5 E +.44 cos δ E f E = - E +.33 sin δ -.33 E = cos δ E = sin δ E E f = - E f + 8 V ref - 8 V t + 8 u V t = sin δ + ( cos δ) If rotor velocity, ω r, is selecte as our output, then we have: y = ω r Now with regaring input-output linearization techniue was introuce in section (4-) have y = ω r = f 4 y = ω r = f 5 y (3) = ω (3) r = f 6 y (4) = ω (4) r = f 7 + ( sin δ ) u Where, euations of f 4 to f 7 are summarize in the appenix. Accoring to the input-output linearization techniue the input,v can be selecte as follows: v = f 7 + ( sin δ ) u u = ( v - f sin δ 7 )

5 J. Basic. Appl. Sci. Res., ()9-99, Then we have: ω (4) r = v By selecting v as: v = ω (4) r - K e - K e - K 3 e - K 4 ω (3) r We have: e (4) + K 4 e (3) + K 3 e + K e + K e= Where, by selecting x = e, x = e, x 3 = e, x 4 = e (3) we have: x x x 3 x 4 = x x x X = AX 3 K K K 3 K 4 x 4 If proper values are selecte for k to k 4 as poles of the close loop system are lai on an appropriate location in the left sie of jω axis, then error signal will move towars zero (e ) [5] Simulation an results observation Steam turbine moel, Governor, exciter, roun pole synchronous generator, three phase generator, transmission line, loa an the infinite bus have been consiere for the simulation of the single machine infinite bus power system. MATLAB software was employe to conuct the simulation. Performance of the CPSS an NLPSS incluing synchronous generator 3 r an 6 th orer moels were analyze an compare at the presence of 3-phase fault of the transmission line. Three phase fault in the transmission line occurs within. secons an then is remove within.3 secons. (a) elta ( Wr ) (b) 94

6 Chaloshtori, Delta ( e g r e e ) (c) Fig 3: Results of simulation with CPSS; a) A scheme of simulation of the single machine infinite bus power system incluing CPSS, b) rotor spee variation iagram ( ω r ) before an after occurrence of three phase fault, c)torue-angle curve (δ) before an after fault occurrence (a).8.6 elta ( Wr ) (b) 95

7 J. Basic. Appl. Sci. Res., ()9-99, 5 4 Delta ( e g r e e ) (c) Fig 4: Results of simulation with NLPSS along with generator 3 r orer moel ; a) A scheme of simulation of the single machine infinite bus power system incluing NLPSS, b) rotor spee variation iagram ( ω r ) before an after occurrence of three phase fault, c)torue-angle curve (δ) before an after fault occurrence (a).4.3 elta ( Wr ) (b) 96

8 Chaloshtori, Delta ( e g r e e ) (c) Fig 5: of simulation with NLPSS along with generator 6 th orer moel ; a) A scheme of simulation of the single machine infinite bus power system incluing NLPSS, b) rotor spee variation iagram ( ω r ) before an after occurrence of three phase fault, c)torue-angle curve (δ) before an after fault occurrence 6. Comparison an Conclusion They explicitly inicate superiority of nonlinear control moeling metho base on input-output linearization metho to the classical linear control moeling metho; it is because of this fact that naturally synchronous generator acts nonlinearly, hence some key factors in escription of behavior of the generator are unerestimate in the case of linearization of the generator moel aroun a given operating point. (a) 97

9 J. Basic. Appl. Sci. Res., ()9-99, (b) Fig 6: comparison between two nonlinear metho : (a) comparison between ω r results, (b) comparison between δ results By comparing the figures obtaine from the carrie out simulations, it is observe when input-output linearization controlling metho is use an 6 th orer moel of synchronous generator is taken into account amping velocity Δω r has been ecrease severely with respect to the other two moes. In aition, the over shoot has been ecrease in this moe. Comparing two moeling methos, which are similar in terms of the application of input-output linearization metho base nonlinear control metho an are ifferent base on selection of the moel of the generator, inicates superiority of consiering the higher orer moel for the generator uring the PSS. By comparing the three figures that shown in figure 6, obtaine from the results for the angle of torue (δ), when CPSS is use, it is observe that the amount of reaches 5 which results in falling of transmission power of the generator. When NLPSS is use an the 3 r orer of generator is taken into account, δ reaches45, which inicates that the esign metho base on input-output linearization is superior to the classical esign metho. Taking the 6 th orer moel for synchronous generator into account, the obtaine results are improve once again, as the amount of δ reaches 6. It inicates that by taking higher orer moel for generator in the power system stabilizer esign increases the efficiency of power system stabilizer an conseuently, it improves the performance of synchronous generator. 7. REFERENCES [] P.Kunur. "Power System Stability an Control", Mc. Graw Hill,994. [] Machowiski, J., J.W.Bialek, an J.Bumby,"Power System Dynamics an Stability",st En.,Wiley,Englan. PP:484,998 [3] K.R.Paiyar, Power System Dynamics,Stability an Control, John wiley & sons(asia) Pte Lt, book, 996. [4] P. W. Sauer an M. A. Pai., Power system Dynamics an Stability, 998, Prentice-Hall Inc. [5] A. Isiori, Nonlinear Control Systems An Introuction. New York: Springer-Verlag, 989. [6] J.J.E. Slotine, Applie NonLinear Control, Prentice-Hall, book,99. [7] H.K. Khalil, Nonlinear Systems, Prentice-Hall pub., book,996. [8] J.W. Chapman, M.D. Ilic, C.A. King, L. Eng., H. Kaufman, Stabilizing a Multimachine Power System Via Decentralize Feeback Linearizing Excitation Control, IEEE Trans. On Power Systems, Vol. 8,No.3, Aug.993, p.p [9] S.S. Sharif, Nonlinear Power System Stabilizer Design Techniue, CCECE, IEEE 995,p.p Appenix 8. A Synchronous Generator Parameters S B = 5 MVA, V B = Kv, f s = 5 Hz H= 3.5 MWsec MVA, K D= 98

10 Chaloshtori, Stator Resistance (pu) : R s =.3 Reactances (pu) : X =.8, X =.3, X" =.3, X =.76, X =.65, X" =.5 Time Constants (sec) : T = 8, T" =.3, T =, T" =.7 8.B. Evaluation of the employe function in the text using the parameters given in appenix A, the functions f to f 7 are obtaine as follows : f.5e sin.3 sin. 43T m f f 3.48E sin.875 E sin 47.7E Cos.6Cos.5 sin f (3.6 Cos.536sin 5.8CosSin 3.6SinCos )E 9.43(Cos Cos) 3.53E Sin.48Sin 3.7Sin4 ( 633.Sin E Sin ( 6.74E Cos 8.75Cos )T (5Sin ) V f 4 m ref f ) (.38Sin.8Cos)E (5Sin ) V.63 sin.5868 cos.8sin Tm f 4.98 sin.75 sin 3.79 Cos 5 5.4E' sin.7 E' Cos.7 5 cos sin.e'.48e'.3 cos f 5. sin Cos E' sin 48.98E' cos 39.6 cos 8.4 sin 38 cos 6.6 sin cos.7.9 " cos (.sin cos cos.7.8 sin cos ) E 8 E cos " (.8 cos cos.9 sin sin ) 5sin.sin 4 (45.7 sin 449.5Cos 35.5) w 7 sin ( 7.E " sin 7.8 cos.6 cos ) (.375E ".68sin ) E f sin 95.8 T m cos 59.83E' sin cos 53.7 E " E' E' 7.47E' sin 3.77E' Cos 449. sin E' cos 373.6E' sin sin 44.8 (.65E' sin.3e sin sin cos cos ) E' (663.5sin 45.cos.6sin ) (9.45E' 4.43.E " 43.8 ) Cos sin ( sin 7.67 cos.545w sin 64.5E' 64.87) sin (539.4E' 497 )cos (.6E' ) sin sin ( E' 9.38 ) cos cos ( ) sns cos (566.8E' 44.48E' ) sin.6cos 4.Sin4 3 ( 697E' 5.59E' )cos 3 ( E' (.39.4E' E' E' )cos (67.39 cos 93.8 sin 58E' sin 78.34E' cos 9.67 sin t y E' ) sin cos 9.34 sin sin 7.3 cos.3) T m (.6 E' Sin 6.3 cos ) Ef (. 544Sin )( Vref Vt) 99