Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) Optimizing Linear Quadratic Regulator Controller On DC-DC Converter

Transcription

1 Internatonal Conference on Control, Automaton and Systems 28 Oct , 28 n COEX, Seoul, Korea Partcle Swarm Optmzaton (PSO) and Genetc Algorthm (GA) Optmzng Lnear Quadratc Regulator Controller On DC-DC Converter S. Eshtehardha 1, M. Bayat Poudeh 1, S. A, Emam 2 and M. Moradyan 1 1 Department of Electrcal Engneerng, Islamc Azad Unversty, Najaf abad Branch, Isfahan, Iran (Eshtehardha@gmal.com, poudeh@gmal.com) 2 Department of Electrcal Engneerng, Islamc Azad Unversty, Khomenshahr Branch, Isfahan, Iran (emam_dna@yahoo.com) Abstract: In ths paper, two dfferent control methods on DC-DC Converter are compared wth together. hese converters are used for the stablzaton or the control of DC voltage of a battery. In addton to other applcatons, DC converters feed electrc vehcles (trucs, electrc vehcles, and subway locomotves), telephone sets and cvl nverters. Lately, mprovement the performance of the DC-DC converter s one of the goals of the engneers n the ndustres. In ths way, several control methods are used to control Buc converters. In ths paper, Lnear Quadratc Regulator controller s used to optmze the DC-DC converter performance. Also other controller, by the way, Genetc Algorthm (GA) and Partcle Swarm Optmzaton (PSO) are used wth these control methods, to obtan the best coeffcents n them. he results are shown the capablty of the control methods n the mprovement of the above-mentoned converter functonng. Keywords: Genetc Algorthm, Partcle Swarm Optmzaton, Lnear Quadratc Regulator, DC-DC converter 1. INRODUCION A buc converter or step-down swtch mode power supply can also be called a swtch mode regulator. Popularty of a swtch mode regulator s due to ts farly hgh effcency and compact sze and a swtch mode regulator s used n place of a lnear voltage regulator at relatvely hgh output, because lnear voltage regulators are neffcent. he DC converter s a devce whch transforms AC to DC. hs devce s also nown as an AC to DC converter. A Chopper can be consdered as a DC equvalent of an AC transformer wth a convertble constant convertble n a contnuous form. Le a transformer, the converter can be employed for stepwse ncrease or reducton of DC source voltage. he converters are wldly used for the control of motor voltage n electrc cars, celng elevators, mne excavaton etc. her specfc features are the precse control of acceleraton wth hgh effcency and fast dynamc response [1]. Snce the power devces used n lnear regulators have to dsspate a farly large amount of power, they have to be adequately cooled, by mountng them on heatsns and the heat s transferred from the heatsns to the surroundng ar ether by natural convecton or by forced-ar coolng. Heatsns and provson for coolng maes the regulator buly and large. In applcatons where sze and effcency are crtcal, lnear voltage regulators cannot be used. Converters are also employed n DC motors to return the energy to ts source. In ths way, t results n the savng of energy n the transportaton systems n prolonged stoppage. Converters are also used n DC voltage regulators along wth an nductor to produce a DC current source especally for the current source nverters [2]. A swtch mode regulator overcomes the drawbacs of lnear regulators. Swtched power supples are more effcent and they tend to have an effcency of 8% or more. hey can be pacaged n a fracton of the sze of lnear regulators. Unle lnear regulators, swtched power supples can step up or step down the nput voltage. Some control methods have stated the ssue of control through pole placement [1]. Another method s the use of state feedbac n the control of DC-DC converters [2]. In modelng area of DC-DC converters, a varety of models are presented whch comprses desrable responses by admnstraton of control methods. Most of the artcles have concentrated on controllng desgns of PID and PI [3]. Of other control methods, the feedbac loop s another among others, [4], and []. he use of LQR method for the mprovement of Buc converter functon s the subject presented n [6]. In ths paper, Secton 2 descrbes the model of the Buc converter. he LQR control method s detaled n Secton 3, Genetc algorthm s shown n Secton 4, another optmzaton method, Partcle Swarm Optmzaton, s descrbed n Secton. he computer smulaton results are presented and dscussed n Secton 6, and Fnally, Secton 7 concludes ths paper. 2. BUCK CONVERER DC-DC converters were referred to as choppers earler, when SCRs were used. Nowadays, IGBs and MOSFEs are the devces used for dc-dc converson and these crcuts can be classfed as swtch mode power supply crcuts. he abbrevaton or acronym for swtch mode power supply s SMPS [7]. In a Buc converter the average amount of output voltage V out s less than the nput voltage V n. he regulator crcut uses the power swtch depcted n Fg. 1. he dc-dc converters can have two dstnct modes of operaton: Contnuous conducton mode (CCM) and dscontnuous conducton mode (DCM).he functon of the crcut s dvded nto two parts. he frst part starts 1669

2 when the swtch s turned on at t=. he nput current whch s rsng passes through L flter nductor, C flter capactor, and R charge resstance. he second part starts when the swtch s turned off at t=t 1. Due to the presence of stored energy n the nductor, the nductor current contnues passng from L, C, load and D. Now t s requred to show the effect of on and off duratons of swtch n equatons (1) and (2) to obtan the mean values of state equatons.. X X = AX + BV 1 dc X 1 = L, X2 = V o, X =, X2 A = da1 + (1 d) A 2, B = db 1 + (1 d)b2 (3) t on d = Fg. 1. Buc converter he nductor current declnes untl the second swtch swtchng n the next cycle. In ths model, Vo s the system output voltage and V ref s the converter voltage. he converter state equatons n low-frequency state, s presented. Swtch ON: V dc V o = ( R s + R L ) L + L( d L / dt) L = C( dv o / dt) + V o / R,, X = A1 X + B 1 V dc, ( R L + R S ) 1 1 B 1 =. A, L 1 = L L 1 1 C RC Fg. 2. Swtch on for a tme duraton D Swtch OFF: ( R d + R L ) L + L( d L / dt) + Vo =, L = C( dv o / dt) + V o / R, X = A2 X + B2 V dc, ( R L + R d ) 1 2 = L L A, B 2 = 1 1. C RC Fg. 3. Swtch off for tme duraton (1-D) (1) Fg. 4.ey waveforms 3. LQR DESIGN he LQR desgn problem has been extensvely nvestgated for the past four decades [8-12]. Consequently, the theory of lnear optmal control has been very well developed. Central to the optmal stablzer desgn problem s the formulaton of the cost functon and the choce of the state and control weghtng matrces. Although the need for the desgn of a cost functon that reflects the physcal characterstcs and practcaltes of the studed system s reasonably understood, the choce of the elements of the weghtng matrces that would translate ths understandng stll remans a dffculty. hs s manly due to the nteractve nature of the states and controls of lnear multvarable dynamc systems. In the followng we present a novel systematc approach to ths problem whch overcomes the shortcomngs of exstng approaches. In ths method, the feedbac gan matrx s determned f J energy functon s optmzed. o acheve equlbrum among range control parameters, response speed, settlng tme, and proper overshoot rate, all of whch guarantee the system stablty, the LQR s employed. 3.1 LQR ALGORIHM. For a system n the form of X = AX + BU the LQR method determnes the K matrx of the equaton U () t = - X () t to mnmze the 167

3 J = ( X QX + U RU ) dt functon. R and Q matrces express the relaton between error and energy expense rate. R and Q are also the defnte postve matrces. From the above equaton, we have []: J = ( X QX + X K RKX ) dt = ( X ( Q + K RK) X ) dt 3.2 FLOW CHAR OF LQR CONROLLER Statement of the Problem Step1 Gven the plant as x(t) = Ax(t) + Bu(t) the performance ndex as J = = (X QX + X K (X (Q + K RK)X)dt RKX)dt and the ts condtons x ( t ) = x; x( ) =, Fnd the optmal control, ndex. Step 2 Step 3 Step 4 Soluton of the Problem Solve the matrx algebrac Rccat equaton 1 A P + PA PBR B P + Q = Solve the optmal state x* (t) from * 1 * X ( t) = ( A BR B P) X ( t) wth ntal condton x(t ) = x. Obtan the optmal control u* (t) from * 1 * u ( t) = R B PX ( t) Obtan the optmal performance ndex from 1 * 2at * * J = e x ( t ) P x ( ) 2 t (4) 4. GENEIC ALGORIHM Genetc Algorthms (GA) s a technque that mmcs natural evoluton to solve problems n wde varety of domans. In 196 the frst serous nvestgaton nto Genetc Algorthms (GAs) was undertaen by John Holland. Genetc algorthms have become popular due to several factors. hey have been successfully appled to self-adaptve control systems and to functon optmzaton problems. In nature, ndvduals best suted to competton for scanty resources survve. Adaptng to changng envronment s essental for the survval of ndvduals of each speces. Whle the varous features that unquely characterze an ndvdual determne ts survval capacty, the features n turn are determned by the ndvdual s genetc content [13]. hey are a powerful search technque, yet are computatonally smple. he search method they use s robust snce t s not lmted le other search methods wth regard to assumptons about the search space. Genetc algorthms are dfferent from normal search methods encountered n engneerng optmzaton n the followng ways: 1- GA wor wth a codng of the parameter set not the parameters themselves. 2- GA search from a populaton of ponts, not a sngle pont. 3- GA use probablstc transton rules, not determnstc transton rules. he genetc algorthm s an algorthm whch s based on natural evoluton and the survval of the best chromosome. here are three basc dfferences between genetc algorthm and optmzaton classcal methods. Frstly, the genetc algorthm wors on the encoded strngs of the problem parameters. Each strng s the representatve of one answer to the problem, and the real quanttes of the parameters are obtaned from the decodng of these strngs. Secondly, the genetc algorthm s a search algorthm whch wors on a populaton of search spaces []. hs qualty causes the genetc algorthm to search dfferent response spaces smultaneously reducng the possblty of beng entrapped at local optmzed ponts. hrdly, the genetc algorthm does not need prevous data from the problem response space such as convexty and dervable. It s only necessary to calculate a response functon named ftness functon. hs functon expresses the rate of response proxmty to the goal functon of the ntended algorthm. Dfferent methods are used for encodng the parameters n genetc algorthm [14]. A GA operates through a smple cycle of stages as follows: 1- Creaton of a populaton of strngs (usng a random generator). 2- Evaluaton of each strng (usng a ftness functon). 3- Selecton of best strngs (usng a sutable selecton method). 4- Genetc manpulaton to create the new populaton of strngs. A smple genetc algorthm s composed of three operators: selecton, crossover, and mutaton. 1- Selecton s a process where an old strng s carred through nto a new populaton dependng on the value of ftness functon. Due to ths move, strngs wth better ftness values get larger numbers of copes n the next generaton. Selectng good strngs for ths operaton can be mplemented n many dfferent ways. 2- A smple crossover follows the selecton s made n three steps. Frst, the newly selected strngs are pared together at random. Second, an nteger poston "n" along every par of strngs s selected unformly at random. Fnally, based on a probablty of crossover (p c ), 1671

4 the pared strngs undergo crossng over at the nteger poston "n" along the strng. 3- Mutaton s smply an occasonal random alteraton of a strng poston based on probablty of mutaton (p m ). In a bnary code, ths nvolves changng a 1 to a and vce versa. he mutaton operator helps n avodng the possblty of mstang a local mnmum for a global mnmum. Several practcal means of decdng when to stop regeneraton are used. hese are: 1- Stop after a perod of tme (Maxmum number of generatons). 2- Stop when the average ftness s close to the mnmum (or the maxmum) of the ftness. 3- Stop when there s no mprovement n the maxmum (or mnmum) value of the ftness. 4- Stop after fndng a better soluton than the one proposed by other means or crteron. - Stop after fndng the desred maxmum (or mnmum). he genetc algorthm flowchart s shown n Fg.. Fg.. Genetc algorthm flowchart. PARICLE SWARM OPIMIZAION (PSO) he partcle swarm optmzaton s an evolutonary computaton technque developed by Eberhart and Kennedy nspred by the socal behavor of brd flocng and fsh schoolng []. PSO s a populaton based optmzaton tool. In PSOs, that are nspred by flocs of brds and shoals of fsh, a number of smple enttes (partcles) are placed n the parameter space of some problem or functon, and each evaluates the ftness at ts current locaton. he advantages of PSO compared to other evolutonary computatonal technques are [16]: 1. PSO s easy to mplement. 2. here are few parameters to be adjusted n PSO. 3. All the partcles tend to converge to the best soluton qucly. In PSO each partcle adjusts ts flght accordng to ts own and ts companon s flyng experence. Each partcle eeps trac of ts co-ordnates n the soluton Space whch are assocated wth the best soluton (ftness) that has acheved for by that partcle. hs value s called personal best, J pbest. Another best value obtaned so far by any partcle n the neghborhood of that partcle. hs value s called global best, X gbest. he basc concept of PSO les n acceleratng each partcle towards ts J pbest and the X gbest locatons, wth a random weghted acceleraton at each tme step. Each partcle tres to modfy ts poston usng the nformaton such as the current postons, the current veloctes, the dstance between the current poston and J pbest, the dstance between the current poston and the X gbest. he mathematcal equatons for the searchng process are [17]: + 1 v = wv + c1 rand1 (...)( pbest s ) + c rand (...)( gbest s ) K+1 K +1 I = XI + v () x (6) Where v Velocty of partcle at teraton W weghtng functon c 1 and c 2 weghtng factor rand unformly dstrbuted random number between and 1. s Current poston of the partcle at teraton pbest pbest of partcle gbest best value obtaned by any partcle so far he frst term n (14) s the former velocty of the partcle(s), the second s the cognton modal, whch expresses the thought of the partcle tself, and the thrd represents the socal model. he three parts together determnes the space searchng ablty. he frst part has the ablty to search for local mnmum. he second part causes the swarm to have a strong ablty to search for global mnmum and avod local mnmum. he thrd part reflects the nformaton sharng among the partcles. Under the nfluence of the three parts, the partcle can reach the best poston [18]. In the above procedure, the maxmum velocty vmax determnes the resoluton of ftness regons are searched between the present poston and target poston. If max v too hgh, partcle mght fly past good soluton. If max v s too small, the convergence could be slower. Accordng to experence of PSO, vmax taes often 1% to % of the dynamc range of the velocty. he PSO algorthm used n ths thess can be brefly dscussed by the followng steps. 1: Intalze a populaton of pop partcles wth random postons wthn the lower and upper bound of the problem space. Smlarly ntalze randomly pop veloctes assocated wth the partcles. 2: Evaluate the optmzaton ftness functons J for the ntal populaton. 3: Fnd the mnmum ftness value for ftness functons J n step 2 and call t J pbest and let the partcle assocated wth t be X pbest. 4: Intally set J gbest equal to J pbest. 1672

5 : Update the weght Q usng the followng equaton ( wmax wmn ter w = w max maxter (7) Where w max ntal weght w mn fnal weght Maxter maxmum teraton number Iter current teraton number Vo (V) LQR1 LQR2 LQR3 6: Update the velocty of each partcle usng () 7: Chec V for the range [V max, V mn ]. If not, set t to the lmtng values. 8: Update the poston of each partcle usng (6) whch gves the new populaton. 9: Repeat 7 for the new populaton. 1: Evaluate the optmzaton ftness functons J for new populaton. 11: Obtan J pbest for ftness functons J n step 1. 12: Compare the J pbest obtaned n step 11 wth J gbest. If J pbest s better than J gbest then set J gbest to J pbest. 13: Stop f convergence crtera are met, otherwse go to step. he stoppng crtera are, good ftness value, reachng maxmum number of teratons, or no further mprovement n ftness [18]. 6. RESUL OF SIMULAION Wth regard to the state equatons for the converter and tang nto consderaton n able 1, the Buc converter equatons are. Open loop response wll be n the form of Fg. 6. o optmze the V o the LQR control method wth GA and PSO s used. V o me (sec) x 1-3 Fg. 7. Response Buc converter response wth LQR controller 6.2 HE APPLICAION OF GENEIC ALGORIHM FOR ADJUSING LQR CONROLLING COEFFICIENS For ths purpose the genetc algorthm on the system, wthn several successve seasons, has mproved the convergence of the system and has brought the best chromosome of that generaton nto optmum amount from generaton to generaton. 6.3 HE APPLICAION OF PARICLE SWARM OPIMIZAION FOR ADJUSING LQR CONROLLING COEFFICIENS By consderng Eq (3), Fg. 6, the Buc converter open loop response s obtaned. By applyng LQR controllng on the system and wth regard to Fg. 7. n able. 2, the results of Fg. 8, are obtaned. By applyng Genetc Algorthms for the optmzaton of LQR controller response, the results n Fg. 9 are obtaned by applyng Partcle swarm optmzaton me (sec) x 1-3 Fg. 6. Open Loop response of Buc converter As t s evdent from the study of outputs, the open loop system cannot hold the output n constant and deal condtons. Of course, the outputs wll eventually approxmate the deal rate wth regard to the stablty of the system HE RESUL OF SIMULAION WIH LQR CONROL MEHOD By applyng LQR controllng on the system and wth regard to Fg. 7 n able 2, V o me (sec) x 1-8 Fg. 8. Buc converter response wth LQR-GA controller 1673

6 Vo (V) Symbol Value me (sec) x 1-4 Fg. 9. Buc converter response wth LQR-PSO controller ABLE 1 Buc converter parameters. R L R d R on C L R L V dc m 1.7 LQR1 LQR2 LQR3 LQR-GA LQR-PSO. 1. mf ABLE 2 LQR parameters. 1 = 1 Q, R = [ 1] 46 Q =, R = [ 966].9 12 Q = [ 8.6] R =, 3 Q =.48 mh.97, R = [ 168] 21.3 Q = , R = [ ] 7. CONCLUSION o use a converter t s mportant that the voltage rpple of converter s reduced and a completely DC voltage output s produced. he optmum desgn method for LQR controller s able to control the dynamc behavour of the converter. he use of genetc algorthm and Partcle Swarm Optmzaton for the calculaton of optmum coeffcents of the gans n the desgn of LQR controllers can brng about optmum dynamc response. Also, the results of LQR controller method are more satsfactory wth Partcle Swarm Optmzaton than the results of genetc algorthm method. REFERENCES [1] M. Bayat Poodeh, S. Eshtehardha, A. Kyoumars, M. Atae Optmzng LQR and Pole placement to Control Buc Converter by Genetc Algorthm, Internatonal Conference on Control, Automaton and System 27 (ICCAS'7), pp , Oct. 27. [2] M. Bayat Poodeh, S. Eshtehardha, M. Namnabat, Optmzed State Controller on DC-DC Converter, Internatonal Conference on Power Electroncs 27 (ICPE7), pp. 3-8, Oct. 27. V. [3] M. Namnabat, M. Bayat Poodeh, S. Eshtehardha, Comparson the Control Methods n Improvement the Performance of the DC-DC Converter, Internatonal Conference on Power Electroncs 27 (ICPE7), pp , Oct. 27. [4] S. Uran and M. Mlanovc, State Controller for buc Converter, EUROCON, 23. IEEE. 23, Vol. 1, Sept. 23, pp [] A.H. Zaer, M. Bayat Poodeh, S. Eshtehardha, Improvement of Cû Converter Performance wth Optmum LQR Controller Based on Genetc Algorthm, IEEE. 27., Internatonal Conference on Intellgent & Advanced Systems, Nov. 27. [6] Y. He, and F. L. Luo, Study of sldng mode control for DC-DC converters, PowerCon 24., Nov. 24., Vol.2, pp [7] 7s1p1.htm [8] F. H. F. Leung, P. K. S. am, and C. K L, An mproved LQR-based controller for swtchng DC-DC converters, IEEE rans., Ind Electron., Oct. 1993, pp [9] M. AHANS, and P. FALB, Optmal control: an ntroducton to the theory and ts applcaton, McGraw- Hll, [1] F. L. LEWIS, Optmal control, John Wley, [11] B. D. O. ANDERSON, and J. B. MOORE, Optmal control: lnear quadratc methods, Prentce-Hall, [12] W. S. LEVINE, and M. AHANS, On the determnaton of the optmal constant output feedbac gans for lnear multvarable systems, IEEE rans., 197, AC-, (I), pp , 197. [13] A. A. Eldamaty, Dampng Interarea And orsonal Oscllatons Usng Facts Devces, A thess submtted For the Degree of Doctor of Phlosophy n the Department of Electrcal Engneerng Unversty of Sasatchewan Sasatoon, Sasatchewan, pp ,. [14] R. L. Haupt and S. E. Haupt, Practcal Genetc Algorthms, John wley & Sons, Inc, New yor, 24. [] Kennedy J. and Eberhart R, Partcle swarm optmzaton, IEEE nternatonal conference on neural networs, vol. 4, pp , 199. [16] Basar S, Applcaton of Evolutonary Computaton to Engneerng Opmzaton Problems, Lecture Notes, Short term course on EMOA, (27). [17] M. Wlljuce Iruthayarajan, S. Basar optmzaton of PID parameters usng Genetc Algorthms and Partcle Swarm Optmzaton, Internatonal Conference on Informaton and Communcaton echnology n Electrcal Scences, Dec. 2-22, 27. pp [18] S. F. Fazullah, Dampng Enhancement of Mult machne Power System hrough SACOM, A thess submtted Unversty of Petroleum and Mnerals (KFUPM), Saud Araba, pp.6-9, March. 1674