A Particle Swarm approach for the Design of Variable Structure Stabilizer for a Nonlinear Model of SMIB System

Transcription

1 A Partcle Swarm approach for the Desgn of Varable Structure Stablzer for a Nonlnear Moel of SMIB System NAI A AL-MUSABI**, ZAKARIYA M AL-HAMOUZ*, HUSSAIN N AL-DUWAISH* ** The Petroleum Insttute, Electrcal Engneerng Program, Abu Dhab, PO BOX: 533 UNITED ARAB EMIRATES, ABU DHABI * Kng Fah Unversty of Petroleum an Mnerals, Electrcal Engneerng Department, Dhahran 36, SAUDI ARABIA Abstract: - There s a vv tren n engneerng optmzaton problems towars the aopton of heurstc optmzaton algorthms to arrve at optmal solutons Ths s manly ue to the smplcty of these algorthms an the great cut own of complcate mathematcal manpulatons that are requre n other optmzaton theory methos Ths paper emonstrates the applcaton of an teratve heurstc optmzaton algorthm, namely, Partcle Swarm Optmzaton (PSO), n the esgn of varable structure stablzers for a nonlnear sngle machne nfnte bus system (SMIB) Two versons of PSO, namely the nerta weght metho of upatng the veloctes (PSO-w) an constrcton factor metho (PSO-cf) are apple n the optmal esgn of the stablzer The success of the PSO approach s supporte by smulaton results that confrm the attanment of the stablzer control objectves Key-Wors: - Varable Structure control, Power system stablzer, Partcle Swarm Optmzaton Introucton An mportant problem n stablty of power systems s the exctaton control of synchronous machnes The sgnfcance of exctaton control nuce researchers to stuy an esgn new control methos for the problem such as Proportonal-Integral-Dervatve (PID) exctaton control, Power System Stablzers (PSS), Lnear Optmal/Sub-Optmal Exctaton Control (LOEC), Nonlnear Optmal Exctaton Control (NOEC), Aaptve an Intellgent Control [-5] In recent years, Power System Stablzers (PSS) were usually use to enhance the ampng of power oscllatons cause by several types of small sturbances n a power system The conventonal lea-lag compensaton s aopte by most esgners ue to ts smple structure an easy mplementaton [6] LQR, Neural Networks, an Fuzzy logc are some of the other esgn methos propose for PSS [7-9] Furthermore, a PSS esgn base on Varable Structure law s reporte n [-3] Robustness an goo transent response are some of the attractve features of VSC However, the swtchng feeback gans of the VSC were not prevously chosen by a systematc way Furthermore, a VSC that operates satsfactory over a we range of operatng pont was propose n []ca However, the feeback gans were agan chosen by emprcal experments In [], a VS PSS was propose, for lnear moel of synchronous machne, whch operates over a we range of operatng ponts by usng a neural network to aapt the feeback gans of the controller For each operatng pont, the feeback gans were chosen by Genetc Algorthms In ths paper, a nonlnear moel of synchronous machne has been stue an a VSC s esgne for t In conventonal esgn methos, nonlnear transformaton technques are use before lnear system theory s apple to the system The new esgn metho utlzes teratve heurstc optmzaton technques (PSO) an proves a smpler an more systematc esgn Nonlnear SMIB Moel The nonlnear moel of a sngle machne nfnte bus system s shown n Fgure [3] The machne has an AC/DC converter, a slcon-controlle rectfer, for ae control purposes The ynamcs of the system are escrbe by the followng equatons: δ = ω () ωb ω [ ] () = Pm Pac KPc D ω H Pc = (cos( β ) Rc I ) I (3) I = / L(cos( β ) Rc I ) (4) P m = α P + v (5) m

2 Where, δ : rotor angle of the machne n electrcal raans relatve to the center of mass ω : rotor angular velocty n raans per secon wth respect to synchronous spee H : nerta constant n secons D : ampng coeffcent n secons - P m : per unt mechancal power P ac : per unt AC power P c : per unt power store n the converter ω = 377 ra/s K= B α : tme constant of governor/turbne or mechancal power actuator v: the corresponng nput I : DC current through converter R c : per unt commutatng resstance X = X + X t + X l, P ac = (E E /X)snδ The ynamc equatons can be put nto state form by the followng efntons: x = δ, x = I, x 3 =ω, x 4 = P m The two control nputs are u = cos(β) an u = v δ cos β Ths gves: k =, k = , k 3 = 87857, k 4 = 66667, an k 5 = The control objectve n [3] s to erve the machne from a perturbe state to a esre equlbrum pont an mantan t there Ths objectve nvolves the followng goals: ) Operatng the machne at the rate frequency, e x 3 must be zero at equlbrum ) keepng the c current x at zero 3) Delverng a specfe amount of AC power to the bus an ths efnes a esre loa angle γ of x 3 Varable Structure Control The funamental theory of varable structure systems can be foun n [4] The control law of VSC s a lnear state feeback whose coeffcents are pecewse constant functons Conser the lnear tme-nvarant controllable system gven by: &X = AX + BU (7) Where, X s n-mensonal state vector, U s m- mensonal control force vector, A s a n n system matrx, an B s n m nput matrx The VSC control laws for the system of (7) are gven by: u = ψ T X = n j= ψ x ; =,,, m j j (8) Where the feeback gans are gven as α j, f x ; =,, m ψ j = α j, f x ; j =,, n T σ ( X ) = C X =, =,, m C are the swtchng vectors Some conventonal esgn proceures reporte n the lterature for selectng the elements of the swtchng vectors C can be foun n [5] Fg Nonlnear moel of SMIB power system The state space moel s gven as follows: x x = x k 3 x4 k x 3 sn( x ) + k x x 3 αx 4 + k4 Dx + 3 k5x4 k u u Where, k = R c, k = ω B EE, k 3 = ω B R c k 4 =, k5 = ω B L HX H L H The parameters of the system shown n Fgure use n ths stuy are [3]: DC converter rate at 8 MW, 3KV system, Machne ratng 8 MVA On a 8 MVA base: X = pu, R c = 3 pu, L = 5 pu, H = 7 s, D = 5 s -, an α = - s - (6) 4 Partcle Swarm Optmzaton The Partcle swarm optmzaton (PSO) s an evolutonary computaton technque evelope by Eberhart an Kenney [6] nspre by socal behavour of br flockng or fsh schoolng PSO algorthm apple n ths stuy can be escrbe as follows: Step : Intalze a populaton (array) of partcles wth ranom postons an veloctes v on menson n the problem space The partcles are generate by ranomly selectng a value wth unform probablty over the th optmze search mn max space [ x, x ] Step : For each partcle x, evaluate the esre optmzaton ftness functon,, n varables Step 3: Compare partcles ftness evaluaton wth x pbest, whch s the partcle wth best local ftness value If the current value s better than that of x pbest,

3 then set x pbest equal to the current value an x pbest locatons equal to the current locatons n - mensonal space Step 4: Compare ftness evaluaton wth populaton overall prevous best If current value s better than x gbest, the global best ftness value then reset x gbest to the current partcle s array nex an value Step 5: Upate the velocty v There are two ways of upatng the veloctes an are gven below: a) Inerta weght (PSO-w): v ( k ) = w( k )v ( k ) + ran ( x pbest ( k ) x ( k )) + ran ( xgbest ( k ) x ( k )) Where, k s the number of teraton, s the number of the partcles that goes from to n, s the menson of the varables, an ran, s a unformly strbute ranom number n (, ),, are acceleraton constants an are set, as recommene by nvestgators [7], equal to The weght w s often ecrease lnearly from about 9 to 4 urng the search process b) Constrcton Factor (PSO-cf): v ( k ) = K [ v ( k ) + ran ( xpbest ( k ) x ran ( xgbest ( k ) x, K = 4 ( k )) + ( k ))] where = +, > 4 k,,, ran,, are smlar to Inerta weght metho For both methos the partcle s velocty n the th menson s lmte by max some maxmum value v Ths lmt further mproves the exploraton of the problem space In max ths stuy, v s propose as: max max v = η x where η s a small constant value chosen by the user, usually between - of [7] For ths stuy t x max was foun emprcally that a value of η proves satsfactory results Step 6: Upate poston of the partcles, x ( t) = v ( t) + x ( t ) Step 7: Loop to, untl a crteron s met, usually a goo ftness value or a maxmum number of teratons (generatons) m s reache 5 Propose Desgn of VS Stablzer usng PSO In the present work, teratve heurstc PS optmzaton algorthm s apple n the followng way: Step : The control sgnals, u an u of state space equaton (), are of VSC type an are gven as follows: The states of the system are: X = [ω I P me δ e ] where, P me = P m P mesre (9) δe = δ γ () u = ψ () x α f σ x > ψ = α f σ x < () σ = C x = (3) u = ψ T X (4) α j f x > ψ j = α j f x < j =,,4 (5) σ = C T X (6) = Step : The optmum values of the VS stablzer, whch nclues C anα j, are foun by the PSO algorthm n the followng way: ) Generate ranom values for feeback gans an swtchng vector values ) Evaluate a performance nex that reflects the objectve of the esgn In ths stuy the followng objectve functons are use: ISE = e P me δ + + ω + I ITAE = t t ( δ + P + ω + I )t e me (7) (8) Where, ISE : Integral of square error objectve functon an ITAE : Integral of tme multple by absolute value of error crteron By mnmzng such objectve functons the control objectves wll be satsfe ) Use PSO to generate new feeback gans an swtchng vector values as escrbe n Secton 4 v) Evaluate the objectve functons n Step for the new feeback gans an swtchng vector Stop f the maxmum number of teratons s reache; otherwse go to Step

4 6 Smulaton Results The system escrbe by equaton (6) was smulate wth the followng ntal contons [3]: x = 5, x =, x 3 =, x 4 =66sn(x ()) The two versons of PSO were use wth the followng settngs: k max = 5 teratons,n =, PSO-w: w mn = 4, w max = 9, =, =, PSO-cf: =, = The stoppng crtera use s to termnate the search process f there s no more mprovement n ftness value for the last teratons or f the maxmum number of teratons, 5, s reache The algorthm has been run for trals snce the PSO starts wth ntal ranom values Tables an present the optmal control sgnals u an u (n terms of swtchng vector an feeback gan) when usng fferent objectve functons The summary of the performance nces (objectve functon an computatonal tme) of the runs for the two objectve functons an usng the two PSO algorthms when apple to the nonlnear SMIB system s gven n Table 3 Table : VSC optmal settngs: u Objectve functon C α PSO-w ISE ITAE PSO-cf ISE ITAE The ynamcal behavour of the SMIB system wth the propose PSO esgn s shown n Fgures, 3, an 4 Spee Devaton (Hz) Table 3: Summary of trals of runnng PSO for fferent objectve functons Objectve functon Average value of Best value of PSO-w:ISE&ITAE PSO-cf:ITAE PSO-cf:ISE Tme (s) Fg Frequency evaton Average Computato n tme (Mns) PSO ISE w ITAE PSO ISE cf ITAE Table : VSC optmal settngs: u T C PSO-w ISE [ ] ITAE [ ] PSO-cf ISE [ ] ITAE [ ] α T PSO-w ISE [ ] ITAE [ ] PSO-cf ISE [ ] ITAE [ ] Machne Angle Devaton PSO-cf & PSO-w: ITAE PSO-cf & PSO-w: ISE Tme (s) Fg 3: Devaton of Machne s angle

5 Mechancal Power PSO-cf an PSO-w: ISE PSO-cf an PSO-w: ITAE No Control Tme (s) Fg 4: Mechancal Power From the obtane results t can be conclue that the propose esgn metho of VSC can be apple successfully to nonlnear systems The propose metho requres no nonlnear transformaton or lnearzaton of the moel Thus, a systematc smple way of esgnng VSC controller s acheve The control objectves of mnmzng the frequency evaton an followng a esre angle were satsfe To emonstrate the effectveness of the present esgn algorthm, comparson wth the results of the esgn metho propose n [3] has been mae It s qute clear that the ynamcal behavour of the present metho s very smlar to that of [3] n terms of overshoot an s better n terms of settlng tme, Tables 4 an 5 On the other han, the propose metho s much smpler an oes not requre any nonlnear transformatons to arrve at the optmal values for both swtchng vector an feeback gans Table 4: Comparson of Settlng tme (secons) PSO: w & cf Parameter [3] ITAE ISE P m 6 9 ω Angle Table 5: Comparson of Overshoot (%) PSO: w & cf Parameter [3] ITAE ISE P m ω Angle 7 Conclusons ) The control objectves of mnmzng frequency evaton an followng a esre angle, state n secton, are all satsfe by the propose VS-PSO stablzer, Fgures an 3 ) The two versons of PSO showe very smlar results wth slght out performance of PSO-w n terms of achevng smaller objectve functon values, Table 3) The ISE objectve functon reuces slghtly the ntal overshoot n control terms, Fgure 4) From the reporte results t can be conclue that the propose esgn of VS stablzer can be apple successfully to nonlnear systems Acknowlegment The frst author woul lke to acknowlege the support of the Petroleum Insttute The secon an thr authors woul lke to thank Kng Fah Unversty of Petroleum & Mnerals for the support they receve References [] Y Y NU, Ttle of the Paper, Internatonal ournal of Scence an Technology, VolX, NoX, 9XX, pp XX-XX [] Y N YU, Electrc power system ynamcs, Acaemc press, 983 [] Y H Song, Emergng technques for power system ynamcs stablsaton, IEE colloquum on power system ynamcs stablsaton, Lonon, 998, pp /-/5 [3] Q Lu an Y Sun, Nonlnear Stablzng control of multmachne systems, IEEE Transactons on Power Systems, vol 4, no, 989, pp 36-4 [4] W Chapman, M D Ilc, C C Kng, L Eng, H Kanfman, Stablzng a multmachne power system va ecentralze feeback lnearzng exctaton control, IEEE Trans Power Syst, vol 8, no 3, 993, pp [5] X Y L, Y H Song, X C Lu, Y Lu, A nonlnear optmal-varable-am strategy for mprovng multmachne power system transent stablty, IEE Proceengs- Generaton, Transmsson, an Dstrbuton, vol 43, no 3, 996, pp 49-5 [6] Y Z Sun, X L, S Yan, Y H Song, M M Farsang, Novel Decentralze Robust Exctaton control for Power System Stablty Improvement, IEEE Internatonal Conference on Electrc Utlty Deregulaton an Restructng an Power Technologes,, pp

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