DIFFERENTIAL EVOLUTION BASED TUNING OF PID CONTROLLER FOR AN AUTOMATIC VOLTAGE REGULATOR SYSTEM

Transcription

1 DIFFERENTIAL EVOLUTION BASED TUNIN OF PID CONTROLLER FOR AN AUTOMATIC VOLTAE REULATOR SYSTEM 1 M. SIVA RAMAKRISHNA, L. RAVI SRINIVAS 1 P. Student, Professor, udlavalleru Engneerng College, jntuk, autonomous E-mal: 1 msrk.projects@gmal.com, lravsrnvas@gmal.com Abstract Ths paper presents the mplementaton of DE algorthm for PID controller tunng for automatc voltage regulator systems so that more expedtous settlng to rated voltage s ascertaned and AVR s closed loop system compensated wth a PID controller. In ths paper two AVR examples are consdered. In one plant saturaton non-lnearty s neglected and second plant saturaton non-lnearty s consdered. Dfferental evolutons are habtuated to tune the parameters of PID controller, to procure optmal soluton. Optmal control parameters (proportonal gan, ntegral gan, dervatve gan) are obtaned by mnmzng the objectve functon ITAE (ntegral tme absolute error). Smulatons are done to show the performance of PID controlled AVR system tuned utlzng z-n method and dfferental evolutons are utlzed. Index Terms PID, saturaton non-lnearty, dfferental evoluton. I. INTRODUCTION The AVR system s a closed loop control system compensated wth a PID or PSS controller. Power system stablzer (PSS) has two drawbacks one s sx tunng parameters and hgh gan s requred, so PID controller utlzed because three tunng parameters and low gan s requred. The AVR s a component of synchronous machne. The AVR mantans the generator termnal voltage level under mundane operatng condtons at dfferent load levels. Automatc voltage regulator s manly used where the supply voltage flutng and supply voltage not stable. The engenderer exctaton system mantans engenderer voltage as requred and controls the reactve power flow utlzng AVR. AVR conssts of amplfer, sensor, excter, and engenderer. Amplfer may be magnetc, rotatng, or electronc and electronc amplfer s characterzed by a gan may wthal nclude a tme constant as shown n fg1. The engenderer exaltaton system mantans engenderer voltage constant and regulates supply lne voltage. Sensor senses the output voltage and feedback to nput. Any ac component present output voltage s rectfed by sensor. Among conventonal PID tunng methods Zegler-Nchols[] s most popular method however, ths may engender mmensely colossal overshoot. Soft computng technque has been proposed to fnd the optmal control parameters. Rudmentary dfferental evoluton optmzaton Technque proposed by storn and prnce. DE s a pussant soft computng technque smple evolutonary algorthm for optmzaton of authentc valued, mult modal functons. DE s generally consdered as relable, precse and robust optmzaton technque. In general the advantages of de are 1. Able to locate the precse ecumencal optmum rrespectve of the ntal parameter values. Has rapd convergence 3. Utlze few control parameters thus facle and smple to utlze. 4. Effcacous n non lnear constrant optmzaton quandares wth penalty functons. 5. Effcent algorthm wthout matrx multplcaton. 6. Parallel operatng nature. 7. Faclty to fnd the non dfferental able, streptous and tme dependent functons. Now a day s computng technques play paramount role for tunng PID controller. For tunng PID controller there are so many technques example fuzzy logc[6],[17], chaotc ant swarm (cas), chaotc optmzaton [1], artfcal bee colony (abc)[10],[11], frut fly optmzaton algorthm[15] genetc algorthms[16],[0], Partcle swam optmzaton (PSO)[5],[8],[19], dfferental evolutons[3] and other methods[9],[18]. Other then soft computng technques one heurstc method s Zegler-Nchols method. In PSO we engender the partcles and veloctes, veloctes are ntegrated to partcles to we get ncpent poston of partcles. In dfferental evoluton we engender the three desultory vectors, and weghted dfferences of two vectors are ntegrated to the thrd vector. Dfferental evoluton s teraton senstve. The DE algorthm wthal utlzes a unform crossover that can take chld vector parameter from one parent more often than t does from others. It have feature of parallel operatng nature. In PID controller ncrementng proportonal gan rse tme and study state error decreases, peak overshoot and settlng tme ncrease and stablty degraded. Incrementng ntegral gan rse tme and study state error decrease, overshoot and settlng tme ncrease, stablty degraded. Incrementng dervatve gan rse tme, overshoot and settlng tme decreases and stablty amelorate. A PID controller s very low cost compared to other controllers. It has the feature of robust nature. Facle to control, PID controller always avalable n market, n PID controller, the proportonal controller s drectly proportonal to error, the proportonal controller doesn t perform the 40

2 ntegral operaton, so that ntegral operaton have more expedtous settlng feature, but the ntegral operaton gves peak overshoot, ths drawbacks overcome by dervatve acton. PID controller s better than ndvdual of PID controllers lke P, PI, PD controllers. Because of PID controller have better performance than ndvdual of PID controllers. After the ntroducton, modelng of AVR system n secton II, plant specfcatons n secton III, controller desgn and, Zegler-Nchols method dfferental evoluton are explaned n secton IV, smulaton results n secton V, concluson n secton VI. II. MODELIN OF AN AVR SYSTEM [13], [14] Usng above table mentoned nomnal parameters of plants. The open loop transfer functon for automatc voltage regulaton system s shown below. V t ( s ) V ( s ) s s 0.1s s 1.51 s 11 Where (s) s change the termnal voltage. V t V (s) Is change erence voltage. I. CONTROL STRATEY A smple AVR system contans four man components, namely amplfer, excter, generator [7], and sensor. For mathematcal modelng and transfer functon of the four components, these components must be lnearzed, whch takes nto account the major tme constant and gnores (the saturaton n plant1 and consdered n plant or) other nonlneartes. The transfer functon model of each component conssts of a gan and a tme constant and s gven as PLANTS SPECIFICATIONS [1],[4] Table.1 Fg 1.block dagram of an AVR system wthout PID controller Fg.block dagram of an AVR system wth PID controller Fg.3 Block dagram wth saturaton and PID controller The transfer functon of AVR system wth PID controller s shown n below. 3 ds Vt ( s) 0.1k (0.1 kp kd ) s (0.1 k 10k p) s 10k V ( s) s s 0.555s ( k ) s (1 10k ) s10k d p The block dagram of AVR wth PID has shown n fg. AVR conssts of amplfer, excter, generator, and sensor. Amplfer may be magnetc, rotatng, or electronc and electronc amplfer s characterzed by a gan may also nclude a tme constant as shown n fg.1. Tunng of PID controller s usng Zegler-Nchols method and dfferental evoluton (DE). 41

3 1) PID CONTROLLER DESIN [1] A PID controller calculates the error value, the dfference between a measured process varable and a desred set pont. The controller parameters adjusted mnmze error those are proportonal gan kp, ntegral gan k, dervatve gan kd. The transfer functon of PID controller s shown below. u( k kp kds e( s u( 1 kp1 TdS e( TS ) ZIULAR-NICHOLAS METHOD [] For determnng PID parameters The Zegler-Nchols method s a heurstc method. It was ntroduced by John. Zegler and Nathanel B. Nchol. Followng steps Step1: Frst we set the K and Kd to zero. Step: And then the Kp gan s ncreased (from zero) untl t reaches the crtcal ultmate gan Ku, at that pont the output of the loop begns to oscllate wth a constant ampltude. Step3: The three gans of PID are set by usng only ultmate gan Ku and oscllaton of perod Tu. Calculate the Kp, K, Kd depends on the type of controller used as shown n the Table.. Table. 3) DIFFERENTIAL EVOLUTION [3] Dfferental evoluton s a stochastc, populaton-predcated optmzaton algorthm for solvng nonlnear optmzaton quandares the algorthm was developed by Storn and Prce n In dfferental evoluton four man operatons there are ntalzaton, mutaton, crossover and cull. Mutaton, crossover, cull perpetuated untl ceasng crteron s reached as shown n Fg.4. Step 1- Intalzaton Here we ntalze upper and lower bounds of each control varables [Xmn, Xmax] and sze of the populaton denoted wth N. The ntal populaton s chosen randomly n the nterval [Xmn, Xmax] by unform probablty dstrbuton. Ftness value or cost value s calculated for each set of control varables. By usng below formula generate the control varables as shown below. X X ( L) rand [ 0,1].( X ( H ) X ( L) ), 1,,,... N rand s a random number les between 0 and 1. The ntal process produces populaton Np ndvduals of X randomly. Ftness Functon: T ITAE t e( dt 0 Where e( V ( V ( V ( Reference voltage V t ( Termnal voltage The performance ndex ITAE tunng gets systems whch settle much more quckly. Tme s multpled wth absolute error and ntegrates wth n the lmts 0 to T. We get ntegral tme absolute error (ITAE). Step- Mutaton Operaton Mutaton operaton ncreases the search space. A new vector s generated based on the present ndvdual X as fallows 1 r Y X F (( X r1 X r ) ( X r 3 X 4 )) The mutaton factor was selected as F [0,1.], and the upper lmt of 1. for F was selected dependng on convenent; r1, r, r3 and r4 are selected randomly and those are should not be same. Step 3- Crossover Operaton Mutant vector and a present ndvdual are culled by a bnomal dstrbuton to progress the crossover operaton to engender a progeny. The dversty of populaton has been ncreased. Each gene of the t th j ndvdual s reproduced from the new vector Y ( Y, Y,..., Y ) and the present 1 K ( X 1, X K ndvdual X,..., X ). That s Fg.4 flow chart of dfferental evoluton 1 X h, f a random number Cr Yh 1 Yh, otherwse Where =1,,..., Np; h=1,,..., nc; nc s the decson parameters of dmenson; K s the no.of 4

4 genes; and the crossover factor s set to be C [0,1 ]. Step 4- Estmaton and Selecton: The parent vector s superseded by ts scon f the ftness or cost of the scon s better than that of the parent. Smlarly, f the ftness of the scon s cost than that of the parent then parent s retaned for the next generaton. Two forms are presented as follows 1 X arg mn{ F( X ), F( Y 1 )} X h 1 argmn{ ( 1 b F X )} Where arg mn means the argument of the mnmum 1 and X s the best ndvdual. r COMPARISON WITH OTHER METHODS FOR PLANT1 Table.4 Step5- Repeat step to step 4 untl desred ITAE s reached as shown n Fg.. IV. SIMULATION RESULTS COMPARISON WITH OTHER METHODS FOR PLANT Table.5 COMPARISON OF TUNIN VALUES TABLE 3 CONCLUSION Fg5.Responses of plant 1 For plant 1 and plant are controlled utlzng PID controller. The PID controller s tuned utlzng DE algorthm s desgned for an Automatc Voltage Regulator system to proved the tunng preponderatng of the DE algorthm, compared wth other methods. Dfferental evoluton gves the better results. DE algorthm shows the better results for the performance measure ntegral tme absolute error (ITAE). In plant 1 the partcle sze s les between 0 to 1, mutaton factors s 0.5, crossover rato 0.7. Populaton sze 100. In plant the partcle sze s les between 0 to 1.5, mutaton factors s 0.5, crossover rato 0.8. Populaton sze 100. REFERENCES Fg6.Responses of plant [1] Adaptve PSO based tunng of PID controller for an Automatc Voltage Regulator system Femmy Nrmal, J.; Jeraldn Auxlla, D. Crcuts, Power and Computng Technologes (ICCPCT), 013 Internatonal Conference on Year:013 Pages: ,DOI: /ICCPCT IEEE Conference Publcatons. [] J.. Zegler, N. B. Nchols (194), "Optmum settngs for automatc controllers,", Transactons on ASME, 64(8), [3] Dfferental Evoluton, an Alternatve Approach to Evolutonary Algorthm Lee, K.; El-Sharkaw, M. Modern Heurstc Optmzaton Technques: Theory and Applcatons 43

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