PSO Algorithm-Based Optimal Tuning of PSS for Damping Improvement of Power Systems

Transcription

1 PSO Algorthm-Base Optmal unng of for Dampng Improvement of Power Systems A. Jallvan, Senor member, IACSI, an M. Davran Keshavarz Abstract In ths paper a metho base on the Partcle Swarm Optmzaton (PSO algorthm s presente for tunng Power System Stablzer ( parameters. In the propose metho, base on the optmzaton of a sutable objectve functon, optmal values for controllng parameters nclung lea-lag compensator tme constants as well as the controller gan are calculate. he employe objectve functon s the ampng rato of egenvalues corresponng to system crtcal moes obtane from the analyss of the lnearze moel of system aroun the operatng pont. Controllable an crtcal moes of the system are entfe usng moal controllablty an observablty crtera. he propose algorthm s apple to a sngle machne power system an for varous operatng contons. Smulaton results prove the capablty of the propose algorthm n ampng mprovement of power system. Inex erms controllablty, observablty, partcle swarm optmzaton, power system stablzer, root locus. I. INRODUCION A power system s consste of generators, loas, transformers, an transmsson lnes. A sturbance n the power system wll cause electromechancal oscllatons an hence, system varables wll start to oscllate. hese varables may nclue system voltage, freuency, loa angles of generators, or other parameters of the system. Stablzng these parameters s of great mportance n power system stablty []. Controllers esgne wth classc methos, espte ther smplcty, are only sutable for specfc operatng contons of the system an wll not work for many unprecte stuatons [-]. herefore, when the operatng pont of the system vares, oscllatons may not be ampe satsfactorly an even the system may become unstable. hs s ue to the fact that parameters settng of the controller whch s sutable for a pont of system operatng conton wll not be satsfactory for other ponts. Although aaptve control methos elmnate ths rawback to a great extent [5], they are rather complex an costly. hus, the necessty for esgnng a smple controller that can make the system stable aganst the sturbances n a we range of operatng contons wll be of great value. Once the structure of controller s specfe, t can be sa that the metho for parameter settng s not that much mportant snce Manuscrpt receve August 8,. A. Jallvan s wth the Electrcal Engneerng Department, Zanjan Unversty, Zanjan, Iran (phone: ; fax: ; e-mal: ajallvan@ znu.ac.r. M. Davran Keshavarz., was wth the Electrcal Engneerng Department, Zanjan Unversty, Zanjan, Iran (e-mal: mavrank@yahoo.com. 556 an accurate esgn means the accurate settng of parameters. In applcatons n whch a smple, fast, an a sutable settng for at least several ponts of system operatng contons are neee, the effectveness of classc methos may be restrcte. Optmzaton methos whch are base on ranom search algorthms are very sutable for solvng complex problems that are ffcult or even mpossble to be solve usng conventonal mathematcal methos such as graent [6]. Power system stablty s one of these problems as solvng fferental euatons of power system n orer to nvestgate ts stablty s an ntrcate problem. Applcaton of such methos n solvng complcate problems has recently been n the focus of researchers' attenton [7-9]. For nstance, n [9] the mult-objectve Genetc algorthm has been utlze for tunng the controller parameters. In some applcatons, the steay state error of system factors, whch cause the system to become stable, s mnmze by the algorthm []. he paper s organze as follows: n Secton II, lnear moel of a power system s explane an scusse along wth mathematcal euatons. In Secton III, moal analyss s explane. Next, controllablty an observablty concepts are brefly revewe n Secton VI. Partcle Swarm Optmzaton s stue n Secton V an the objectve functon s presente an explane n Secton VI. Next, the propose algorthm s apple to the consere system an smulaton results are scusse n Secton VII. Fnally, conclusons are presente n Secton VIII. II. LINEAR MODEL OF A POWER SYSEM In ths paper, a power system consste of a sngle machne connecte to the nfnte bus (SMIB s consere where ts parameters an sngle lne agram are presente n the Appenx an Fg., respectvely. L L Fg.. Sngle machne power system connecte to the nfnte bus he thr orer moel s consere for the generator that nclues two fferental euatons for rotor electro-mechanc oscllatons an one euaton representng the nternal voltage of generator [, ]. δ = ω ( ω ( b L

2 ω = m e ω M ( P P D( o ( E ( x x E E = F ( ( Where,,, an are lag compensator tme constants; w s the flter tme constant; an K pss s the constant gan of the controller. w s selecte large enough so that only the varatons are permtte to pass. In the above-mentone euatons, δ an ω are mechancal egree an rotor spee, respectvely; ω b s the reference spee; M s the rotor nerta constant; D s the rotor frcton constant; P e an P m are electrcal output power an mechancal nput power, respectvely; E s the nternal voltage of the machne; E f s the exctng voltage; τ' o s the tme constant of open crcut exctaton; X an X' are the steay state an transent reactances of the machne on axs, respectvely; an s the stator current along axs. he relatonshp between output electrcal power an the machne voltages an currents s as follows: P = v + v e Where v, v,, an are voltage an current components of the machne along an axes, respectvely, an are obtane usng the ntal contons of the system. he IEEE-S generator wth exctaton system an self-controlle voltage together wth s shown n Fg.. E F Δω (s H U U mn U + Vref + V t K + s A Fg.. IEEE-S exctaton system together wth. E f ( In ths system, the voltage exctaton system s as follows: = [ K A A ( v erf v + u t E f ] Where V t an V ref are termnal voltage an reference voltage, respectvely; u pss s the stablzer sgnal; an K A as well as A are transent state gan an exctaton tme constant, respectvely. Moreover, the euatons of voltages are: (5 III. MODEL ANALYSIS he lnearze moel of system n the state space s utlze to entfy the oscllatory moes an to set the parameters. he euatons of the lnearze moel n the state space are gven by []: Δ x = AΔx + BΔu ( Where Δx s the column vector of state varables wth the egree of n; A an B are state an nput matrxes wth consstent mensons, respectvely; an Δu s the nput vector wth the length of r. In ths system, the state varables an nput vectors are as follows: Δx = Δu [ Δδ Δω ΔE ΔE f ] = [ ΔP Δv ] m ref ( Varous methos may be employe to lnearze the system. he lnearzaton of euatons ( to (8 n orer to get to euaton ( s escrbe thoroughly n [, ]. Here, t s assume that the system s lnearze aroun the operatng pont an the state space realzaton s avalable. he system egenvalues are obtane usng the characterstc euaton of matrx A an from (: et( λ I A = ( here exsts a column egenvector φ for each nvual egenvalue λ satsfyng the followng euaton: Aφ = λ φ ( Where egenvalue corresponng to the egenvalue φ s the rght egenvector corresponng to the λ ; an n the smlar way, the left egenvector ψ λ s gven by: t v = v + v v = x (6 (7 ψ = λψ ( A Moal matrxes are obtane employng rght an left egenvectors. v = E x he classc moel of the power system controller nclues two lea-lag compensatng blocks as well as a flter an a gan [8-]. H ΔU ( s = Δω = K w + s W + s + s + s + s (8 (9 Φ = [ φ Ψ = [ ψ φ ψ φ ] n n ψ ] (5 he rght egenvector ncates the strbuton of each oscllatory moe n the system state varables. In other wors, t shows n whch varables of the system each oscllatory moe s vsble. he left egenvector together wth the prmary state values specfes the ampltue of each 557

3 oscllatory moe. A left egenvector nclues nformaton about the controllablty of moes. For each complex egenvalue λ = σ ± jω, ampng rato n percent s efne as follows: σ ξ = (6 σ + ω he moes havng less than % ampng rato are known as crtcal moes [] an the controller shoul be esgne n a way that ther ampng rato s mprove by changng the locatons of these egenvalues. However, conserng a certanty margn s necessary ue to uncertantes an sturbances. Hence, the ampng rato less than 5% s consere n esgnng an the PSO algorthm s employe n ths paper to obtan ths goal. IV. CONROLLABILIY AND OBSERVABILIY In orer to control an oscllatory moe usng a sutable feeback, the controller output shoul have mpact on that moe an on the other han, that moe shoul be observable n that feeback. he behavor of the oscllatory moe s mrrore n the feeback sgnal. In orer to nvestgate ths case, the controllablty an observablty crtera are employe whose matrxes are efne as follows []: B C m m = Φ B = C Φ (7 If the corresponng row of a moe n B m s zero, that moe wll not be controllable an f the corresponng column of a moe n C m s zero, that moe wll not be observable. If a moe s not controllable an observable, the feeback between nput an output wll not have any effects on t []. V. PARICLE SWARM OPIMIZAION ALGORIHM Partcle Swarm Optmzaton (PSO s a populaton base stochastc optmzaton technue evelope by Dr. Kenney an Dr. Eberhart n 995, nspre by socal behavor of br flockng or fsh schoolng [6, 5, an 6]. It s base on exchangng nformaton among the partcles n a network. In general, three topologes can be consere for the partcles n a network known as star, rng, an rotform topologes. hers structures are presente n Fg. Fg.. he topologes utlze n the PSO algorthm. In the star topology each partcle s able to assocate wth all of the partcles n the swarm an each partcle wants to 558 follow the populaton best movement. In the rng topology, each partcle assocate wth ts n neghborng partcles an wants to follow the movement of the best neghborng partcle. In the rotform topology, all of the partcles relate wth a partcle calle canoncal partcle. hs topology s base on travelng geese whch obey ther leaer. Base on the three above-mentone topologes for partcles, Global best, Invual best, an Local best algorthms can be consere for optmzatons whch employ star, rng, an rotform topologes, respectvely. Global best an Local best algorthms are the most common ones for optmzatons. he former ntens to search n vast areas an the latter ntens to search n small areas an exstng ponts n the neghborhoo. hese two algorthms are structurally smlar to each other an they are sometmes use n a mxe form. In ths paper, the Global best algorthm s employe to solve the optmzaton problem. In ths algorthm, a populaton consstng of L nvuals from X vectors s consere. he X vector has n partcles each referre as a unknown varable. Frst, the swarm s ranomly gven values an next, the partcles are steere towars the objectve,.e., fnng the mnmum pont. he poston of each partcle s ts corresponng X vector wth the value of the ftness functon n the assocate poston. Durng the algorthm process, the best experence of each partcle an ts corresponng poston can be save. he best experence of the th partcle s calle pbest an ts corresponng poston s known as x pbest. In a smlar way, the best experence exstng among all of the partcles an ther corresponng postons are calle gbestan x gbest, respectvely. In movng towars the mnmum pont, the velocty of each partcle an ts upate poston s entfe by (8 an (9, respectvely. v ( t + = C[ ϕ v ( t + r C ( x x ( t pbest + r C ( x x ( t] gbest x ( t + = x ( t + v ( t + (8 (9 Where: v (t : he current velocty of th partcle; v ( t + : he next velocty of th partcle; φ: nerta weght; r : cogntve factor; r : socal factor; C: contracton factor; C an C : acceleraton constants. he flowchart of the algorthm s shown n Fg.. he algorthm starts wth a ranomly generate populaton. he success of the algorthm to a great extent epens on correct settng of parameters such as problem menson, number of partcles, number of teratons, mum velocty of partcles, contracton factor, nerta weght, an acceleraton constants.

4 F(p > pbest Yes pbest = F(p xpbest = x (t F(p > pbest Yes gbest = F(p xgbest = x (t ν (t + = C[ ϕν (t + rc (x pbest x (t + rc (x gbest x (t] No Ranomly Generaton of Intal Populaton CalculateFtnessof Partcles n her Current Posstons x (t + = x (t + ν (t + Iteraton = Max Iteraton No Yes Fg.. Flow chart of the PSO algorthm. Results A change n these parameters wll affect the algorthm senstvty an accuracy to a great eal. he values for the above-mentone parameters are all expermental an there s not theoretcal metho for ther accurate calculaton. Conserng an upper boun for the velocty wll cause the partcles not to jump wth a hgh velocty n the search area an thus, the space s more accurately assesse n orer to fn more esrable areas an furthermore, the algorthm vergence ue to hgh veloctes s prevente. When the velocty vector of each partcle s upate, the followng constrant s apple: f : V ( t > V f : V ( t < V V ( t = V V ( t = V ( he contracton factor controls the effect of velocty n upatng the partcles postons conserng the parameters lmts. he nerta weght controls the effect of prevous velocty on the current velocty. Large values of ths parameter wll cause a larger search n the search space an ts smaller values wll lea n focusng on a smaller area. r an r are ranom numbers n the range (, an C as well as C are postve numbers. Due to a stuy on the effect of C an C on the partcles movement paths, the followng constrant s propose as a necessary conton for the algorthm convergence: C + C ( VI. OBJECIVE FUNCION Selectng the objectve functon s the man part of the PSO algorthm. he purpose of the algorthm s to mprove the stablty as well as ampng mprovement of the power system. Any functon that can acheve ths goal may be selecte as the ftness functon. In some researches the steay state error of those system factors n whch oscllaton ampng wll lea n system stablty are utlze []. hese factors, epenng on the controller type, may be the actve power flow from the transmsson lne, bus voltage or current or relatve spee of the generator. Inee, the man goal s to ncrease the ampng rato of oscllatory an controllable moes n the system. Every settng whch can acheve ths goal can optmze the power system stablty. On the other han, the PSO algorthm s able to fn ths soluton by searchng n a space that s efne for parameters. Hence, mzng the ampng rato of crtcal oscllatory moes ( ξ < 5% s selecte as the objectve functon whch s efne as follows: J = mn( ξ K p n Subject to:,mn,mn,mn,mn,mn K K,,,,, ( ( In the above euatons, p s the number of selecte operatng ponts an n s the number of egenvalues wth a relatve ampng less than 5%. In the optmzaton stage several operatng ponts of the system can be utlze so that the controller woul work sutably over a we range of operatng contons. Euaton ( expresses the selecte ranges of controllng parameters. he sutable selecton of these ranges s of great mportance n gettng to an optmal settng. In orer to specfy the permtte ranges, the root locus metho can be apple. VII. SIMULAION RESULS AND DISCUSSION he propose algorthm s apple to the presente system n Fg. an the parameters are calculate. In orer to get an accurate result as well as a fast convergence of the algorthm, the values of the aforementone parameters are consere as summarze n able (. ABLE I HE VALUES OF PSO PARAMEERS Dmenson 5 Number of partcles 5 Number of teratons 5 Maxmum velocty Inerta weght.75 r A ranom number n the range (, r A ranom number n the range (, C.7 C..9 C.6 Four operatng ponts are selecte for the optmzaton nclung one: nomnal contons; two an three: a % ncrease an a % ecrease n the system nomnal loa, respectvely; an four: sconnectng the transmsson lne L. he varaton of nomnal loa or system structure wll 559

5 cause the system operatng contons to change for lnearzaton. Generally, the more the number of such ponts, the more the valty of obtane results. he uncontrolle system has a crtcal moe wth a ampng less than 5% as shown n Fg. 5. jω 8 6 Conventonal unng PSO unng Dampng rato=5% 8 jω 6 Dampng rato=5% Crtcal moe σ Fg. 7. Close loop egenvalues movement wth PSO settng ( compare to conventonal metho settng ( σ Fg. 5. Egenvalues locus of the uncontrolle system. he root locus presents a sutable crteron to know the approxmate ranges of the parameters. For example, Fg. 6 shows the movement of system egenvalues for the varaton of constant gan K from zero to. As mentone before, stochastc searchng algorthms are able to search n a efne space an fn the optmum ponts. Fg. 8 shows the algorthm convergence for the consere objectve functon. In general, the accuracy an convergence rato of the PSO algorthm s more than other optmzaton algorthms as t s observe n Fg jω Dampng rato=.5 J Iteratons σ Fg. 6. System root locus for K varatons from zero (* to (. Applyng the algorthm to the consere system, the parameters of the transfer functon are calculate. he results are summarze n able (. Fg. 8. he PSO convergence base on the objectve functon. In orer to better assess the capabltes of the esgne stablzer, tme-base smulaton of the system s utlze uner sturbance. In the frst step, a hgh ampltue sturbance s consere, whch s a three-phase fault occurre n t=s on the bus wth a uraton of cycles. After two cycles, lnes L an L are sconnecte from two ens. Fgs. 9 an show the machne spee oscllatons base on the reference spee n pu an ts loa angle n egrees, respectvely. ABLE II CALCULAED PARAMEERS USING HE PSO ALGORIHM K W he algorthm performance s n a way that when the mnmum ampng rato gets the esrable value ( ξ.5 n the mum teraton, the algorthm wll stop. Fg. 7 epcts the close loop egenvalues of ths system compare to the conventonal metho settng as well as uncontrolle system. he valty of the PSO-base settng s completely clear from Fg. 7. he notceable pont n Fg. 7 s the secon egenvalue whch not only has not move to the postve se of the real axs, but also t has move to the negatve se of the axs. Rotor Spee Devaton Δω [pu] PSO unng Conventonal unng me[s] Fg. 9. he machne spee oscllatons base on the reference spee n pu for a three-phase fault n bus. 56

6 Loa Angle δ [eg] 5 PSO unng Conventonal unng me[s] Fg.. he machne loa angle n egrees for a three-phase fault n bus. In the prevous step, the nput mechancal power of the generator was assume to be constant. In the secon stage the system s affecte by a step sturbance n the nput mechancal power an t s smulate conserng a % ncrease n the mechancal power n t=s for 5 ms. Fgs. an epct the rotor spee evaton an output electrcal power of the generator, respectvely. he results of ths stage reveal that the PSO settng for ths type of sturbance oes not ffer a lot wth the conventonal settng metho. Rotor Spee Devaton Δω [pu] x PSO unng Conventonal unng me[s] Fg.. he generator spee evaton for a step ncrease n the mechancal nput power. Output Electrcal Power [pu] PSO unng Conventonal unng me[s] Fg.. he generator output actve power oscllatons for a step ncrease n the mechancal nput power. VIII. CONCLUSION In ths paper, tunng of Power System Stablzer ( 56 parameters s stue for a sngle machne system by means of PSO algorthm. he objectve functon s efne base on the system lnear moel analyss aroun the operatng pont an the objectve s to fn the crtcal oscllatory an controllable moes an next, to ncrease ther ampng. he algorthm crtera n orer to fn these moes are controllablty an moal observablty matrxes as well as a ampng rato less than 5%. As the algorthm s employe for varous operatng contons, the obtane results wll be val for a vast range of operatng contons. Comparng the smulaton results of the propose algorthm wth those of a conventonal-tune controller reveal that the PSO-base settng works better even for hgh ampltue sturbances such as short crcuts. APPENDIX All of the parameters as well as values of ntal contons for system smulaton are n pu otherwse t wll be mentone. Genarator : H =.7 s, X r s P o e =.96, =.9s, =., =.875, D =, X o X f = 6 Hz Rate Values :.8Kv e =.68 =.5 X =.7, =.5 X =. Q =.5, δ Excter : K A =, A =. ransmsson Lne : X L = j., ransforme r : Conventonal : V =.997 Vbase = KV.7 + j.8 K =, w =., = =.5 o MVA X L, L S =, bsee = j.567 = MVA =, =.5 = 5. REFERENCES [] A. D. DelRosso, C. A. Canzares an V. M. Dona, A Stuy of CSC Controller Desgn for Power System Stablty Improvement, "Influence of harmoncs on power strbuton system protecton," IEEE rans. Power System, vol. 8, no., Nov.. [] A. D. DelRosso, C. A. Canzares an V. M. Dona, A Stuy of CSC Controller Desgn for Power System Stablty Improvement, IEEE rans. Power System, vol. 8, no., Nov.. [] C. Y. Chung, K. W. Wang, C.. se, an R. Nu, Power System Stablzer ( Desgne by Probablstc Senstvty Inexes(PSIs, IEEE rans. Power System, vol. 7, no., Aug.. [] N,Yang,Q.Lu,J.D,MacCalley, CSC Controller Desgne for Dampng Interarea Oscllatons, IEEE rans. Power System, vol., no., Nov [5] Wenyan. Gu, System ampng mprovement usng aaptve power system stablzer, n Proc IEEE Canaan Conference on Electrcal an Computer Engneerng, pp [6] Rany L. Haupt, Sue Ellen Haupt, Practcal Genetc Algorthms, n Eton, John Wlly & Sons Inc.,. [7] M. Dubey an A. Dubey, Smultaneous Stablzaton of Mult-machne Power System Usng Genetc Algorthm Base Power System Stablzers, n Proc. 6 st Internatonal IEEE Power Engneerng Conf., pp. 6-. [8] M. A. Abo, Robust esgn of mult-machne power system stablzers usng smulate annealng, IEEE rans. Energy Converson, vol. 5, no., pp. 97-, Sep..

7 [9] Y. L. Abel-Mag, Optmal mult-objectve esgn of robust power system stablzers usng genetc algorthms, IEEE rans. Power System, vol. 8, no., pp. 5-, Aug.. [] M. A. Abo, Desgn of an SACOM-base ampng stablzers usng genetc algorthms, presente at the IEEE Power Engneerng Socety General Meetng, 6. [] P.Kunur, Power System Stablty an Control," McGraw Hll, New York, 99. [] P.M.Anerson an A.A.Foua, Power System Control an Stablty, IEEE Press, 99. [] IEEE Recommene Practce for Exctaton System Moels for Power System Stablty Stues, IEEE Stanar [] R. K. Sakovc, an P.Anersson, Applcaton of FACS evces for ampng of power system oscllatons, n Proc 5 IEEE Power ech. Conference, Russa, pp [5] M.Clerc, Partcle Swarm Optmzaton, ISE Lt., New Port Beach, CA, 6. [6] X. F. Xe, W. J. Zhang, an Z. L. Yang, A Dsspatve Partcle Swarm Optmzaton, n Proc Congress on Evolutonary Computaton, Hawa, USA, pp. 56-6,. Abolfazl Jallvan was born n aketan, Iran, 97. He receve B. Sc. n Electrcal an Electronc Engneerng from Electrcal an Computer Engneerng Faculty of Shah Behesht Unversty, Iran, n 995. He receve M. Sc. an PhD egrees from Electrcal an Computer Engneerng faculty of abrz Unversty, Iran, n Power Engneerng an Control Engneerng n 998 an 5, respectvely. In 6 he jone the Electrcal Engneerng Department of Zanjan Unversty, Iran, as an assstant professor where he s ean of faculty now. Hs man research nterests nclue the Hybr Control Systems, Petr Nets, Intellgent Control, Moelng an Control of Power Electronc Converters, Control an Stablzaton of Power Systems, Applcaton of Intellgent methos n Power Systems an so on. He has over 65 papers n journals an conferences. Dr Jallvan s a member of the Insttute of Electrcal an Electronc Engneers (IEEE. Also he s a senor member of Internatonal Assocaton of Computer Scence an Informaton echnology (IACSI. Morteza Davran Keshavarz was born n ehran, Iran, 98. He receve B. Sc. n Electrcal Engneerng from Engneerng Faculty of Islamc Aza Unversty, ehran South Branch, Iran, n 5. He receve M. Sc. egree from Engneerng Faculty of Zanjan Unversty, Iran, n Power Engneerng n 8. In 8 he jone the Saba Power & Inustres Holng Company, Iran, as the Power Plants Optmzaton Projects Manager. Hs man research nterests nclue the Control an Stablzaton of Power Systems, Grounng an Lghtnng Protecton systems, Intellgent Optmzaton Methos an Power Plant Operaton an Control. He has several publshe papers n journals an conferences. 56