Modified Takagi-Sugeno Fuzzy Logic Based Controllers for a Static Compensator in a Multimachine Power System

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1 Mofe Takag-Sugeno Fuzzy Logc Base Controllers for a Statc Compensator n a Multmachne Power System Salman Mohaghegh, Ronal G. Harley* School of Electrcal an Computer Engneerng Georga Insttute of Technology Atlanta GA USA rharley@ece.gatech.eu Ganesh K. Venayagamoorthy Department of Electrcal an Computer Engneerng Unversty of Mssour-Rolla MO USA gkumar@eee.org * Emertus Professor, Unversty of Kwa-Zulu Natal, Durban, South Afrca Abstract- Takag-Sugeno (TS) base fuzzy logc controllers have been esgne for controllng a STATCOM n a multmachne power system. Such controllers o not nee any pror knowlege of the plant to be controlle an can effcently control a STATCOM urng fferent sturbances n the network. Two fferent approaches for the TS fuzzy logc controller are propose: a conventonal TS fuzzy logc esgn an a mofe TS fuzzy logc esgn base on shrnkng span membershp functons. Smulaton results, along wth a comparson of the conventonal TS fuzzy logc controller performance wth that of the propose controller are presente. Keywors- Takag-Sugeno fuzzy logc controller; shrnkng span fuzzy membershp functons; Statc Compensator; multmachne power system. I. INTRODUCTION Statc Compensators (STATCOMs) are power electronc base shunt Flexble AC Transmsson System (FACTS) evces whch can control the lne voltage at the pont of connecton to the electrc power network. Regulatng the reactve power njecte to the network an the actve power rawn from t by ths evce, proves control over the ac lne voltage an the DC bus voltage nse the evce respectvely [1]. A power system contanng generators an FACTS evces s a nonlnear system. It s also a non-statonary system snce the power network confguraton changes contnuously as lnes an loas are swtche on an off. In recent years most of the papers have suggeste methos for esgnng STATCOM controllers usng lnear control technques, n whch the system equatons are lnearze at a specfc operatng pont an base on the lnearze moel, PI controllers are tune at that pont n orer to have the best possble performance [2]-[4]. The rawback of such PI controllers s that ther parameters are mostly tune base on a tral an error approach. Moreover, ther performance egraes as the system operatng contons change. Nonlnear aaptve controllers on the other han can gve goo control capablty over a we range of operatng contons, but they have a more sophstcate structure an are more ffcult to mplement compare to lnear controllers. In aton, they nee a mathematcal moel of the system to be controlle [5]-[7]. Fuzzy logc controllers offer solutons to ths problem. They are nonlnear controllers an nee no pror plant nformaton. Moreover, they can prove effcent control over a we range of system operatng contons. Conventonal fuzzy logc controllers have been wely apple n power systems [8]-[11]. Ths paper esgns a conventonal an two mofe Takag-Sugeno fuzzy logc base controllers for a STATCOM connecte to a multmachne power system, usng Shrnkng Span Membershp Functons (SSMF) [12] an backpropagaton (steepest escent) tranng metho [18],[13]. Smulaton results are prove to compare the performance of both the conventonal an the mofe TS fuzzy controllers wth that of the conventonal PI controller. II. STATCOM IN A MULTIMACHINE POWER SYSTEM Fgure 1 shows a STATCOM connecte to a multmachne power system. The system s a 10 bus, 500 kv, 5000 MVA power network an s smulate n the PSCAD envronment. The generators are moele together wth ther automatc voltage regulator (AVR), excter, governor an turbne ynamcs taken nto account. Detale parameters of the network can be foun n [14]. 2637

2 Fg 1. STATCOM n the multmachne system The STATCOM s frst controlle usng a conventonal PI controller as escrbe n [2]. D-axs an Q-axs voltage evatons are erve from the fference between actual an reference values of the power network lne voltage an the DC bus voltage (nse the STATCOM) respectvely, an are then passe through two PI controllers (Fg. 2). Those values n turn etermne the moulaton nex an nverter output phase shft apple to the PWM moule. Controllng the voltage V at the pont of connecton to the network s the man objectve of the STATCOM consere n ths paper. system, or approxmatng a low orer moel for a hgher orer system. Even n such contons the soluton s not necessarly trval, an sometmes uncertantes assocate wth real lfe problems further exacerbate the relablty of such approaches. Fuzzy logc s a tool that can compensate for the above problems, snce t s the only technque that can eal wth mprecse, vague or fuzzy nformaton [15]. Fuzzy logc controllers consst of a set of lngustc control rules base on fuzzy mplcatons an the rule of nference. By provng an algorthm, they convert the lngustc control strategy base on expert knowlege nto an automatc control strategy [16]. In contrast to the mathematcal moels or other expert systems, fuzzy logc controllers allow the representaton of mprecse human knowlege n a logcal way, wth approxmate terms an values, rather than forcng the use of precse statements an exact values; thus makng them more robust, more compact an smpler [17]. Also, as oppose to most neural network base controllers, n most of the cases fuzzy logc controllers o not nee a moel of the plant to be controlle. Fg 2. STATCOM nternal control structure Parameters of the STATCOM PI controllers are tune at one specfc operatng pont, so that the controller proves satsfactory an stable performance when the system s expose to small changes n reference values as well as large sturbances such as a three phase short crcut on the power network. III. FUZZY LOGIC CONTROLLERS Analytcal approaches have tratonally been use for moelng an control of power networks. However, these mathematcal moels/equatons are acheve uner certan restrctve assumptons, such as lnearzng a nonlnear Fuzzy logc systems prove a nonlnear mappng from a set of crsp nputs to a set of crsp outputs, usng both ntuton an mathematcs. In orer to o that, each fuzzy logc system s assocate wth a set of rules, whch heurstcally efne the ynamcs of the plant to be controlle. For nstance n a mult-nput sngle output fuzzy system: j Rule j: If u 1 s F 1,..., an If u n s j F n, Then y s j G. Usng fferent fuzzfers such as Sngleton, Gaussan an Trangular fuzzfers, any set of crsp nputs s mappe to a fuzzy set. Varous rules n the rule base are apple to the fuzzy nput ata, n orer to create a fuzzy output. Ths 2638

3 output s n turn efuzzfe to generate a crsp output value. A efuzzfer s a mappng that, gven a fuzzy set, etermnes the best crsp representatve of that set. Maxmum, Center of Gravty an Centro are the most popular efuzzfers apple to fuzzy logc systems. Trangular fuzzfers an the centro efuzzfer are use n ths paper as the mappng technques. Two stnct fuzzy logc methos for esgnng controllers are the Maman metho an the Takag- Sugeno metho [16],[18]. Some of the avantages of both methos are as follows: Maman Metho: Is ntutve, Has wesprea acceptance, Is well sute to human nput. Rule 1: If Rule 2: If Rule 3: If Rule 4: If Rule 5: If V s negatve bg, Then e s f1( V ), V s negatve small, Then e s f 2 ( V ), V s zero, Then e s f 3 ( V ), V s postve small, Then e s f 4 ( V ), V s postve bg, Then e s f ( ). 5 V Takag-Sugeno Metho: Is computatonally effcent, Works well wth lnear, aaptve an optmzaton technques, Has guarantee contnuty of the output surface, Is well sute to mathematcal analyss. In short, the Maman metho s the better opton for statc systems (wth slow changng ynamcs) an the Takag-Sugeno metho s more effcent for ynamc systems (wth fast changng ynamcs). Due to the above mentone ssues an the fact that a STATCOM (or any other FACTS evce) n a power system goes through fast changes n terms of system parameters an ynamcs, the latter metho s selecte for esgnng the STATCOM fuzzy logc controller n ths paper. IV. STATCOM FUZZY LOGIC CONTROLLER A. Conventonal Takag-Sugeno Controller Fuzzy varables are as V, V, e, e, an fuzzy c q sets wth lngustc characterstcs of negatve bg, negatve small, zero, postve small an postve bg are assgne to each varable an equal-span trangular functons have been selecte as the fuzzy membershp functons (Fg. 3). Both nputs of the fuzzy controllers have the same membershp functon escrpton as n Fg. 3, however the subntervals x 's are heurstcally selecte base on the characterstcs of each control loop n orer to prove the best ampng/stablzaton performance. A Takag-Sugeno (TS) type fuzzy rule base s been assgne for each combnaton of nput/output varable [16]. As an example for the lne voltage loop: Fg 3. Membershp functons of the nput fuzzy sets Where the f ' s are typcal lnear functons whose coeffcents are selecte n a way that proves optmal performance. Usng the most popular centro efuzzfer, the fnal control output s gven by (1): e = 5 = 1 w. f ( V ), 5 w = 1 where the a certan value of the nput sgnal w ' s are the membershp values of each rule for V. The same rule base s use for the secon controller (.e., DC lnk voltage control). B. Shrnkng Span Membershp Functon (SSMF) Takag- Sugeno Controller There are several fferent esgn parameters assocate wth a fuzzy logc controller, each of whch play an mportant role n the effcency of the controller. Selectng the proper fuzzy membershp functons, subntervals an the efuzzfcaton metho, are some of them. Due to smplcty, most researchers ten to esgn the nput/output fuzzy membershp functons usng the equalspan trangular functons. However, these functons o not (1) 2639

4 necessarly prove the optmum soluton for all problems. Instea a pror knowlege of the plant to be controlle, an ts ynamcs, mght lea to fferent stanar or nonstanar fuzzy membershp functons wth varous physcal shapes n orer to esgn a more effcent fuzzy logc controller. Moreover, when the control response s closer to the system set pont, t can be ntutvely seen that the fuzzy membershp functons for that specfc lngustc term shoul have narrower spans [19], n orer to be prove smoother result wth less oscllatons. Shrnkng span membershp functons (SSMFs) are propose n orer to compensate for the above problems[12]. Ths metho creates trangular membershp functons wth shrnkng spans (Fg. 4), n a way that the controller generates large an fast control actons when the system output s far from the set pont an makes moerate an slow changes when t s near the set pont. [12] The functon s a trangular functon efne as n equaton (3): x a f x < a c a x ( x; a, b, c) = f a x < c c 0 otherwse an the subntervals x 's are erve as followng: x = s m m where s [0,1 ] s the shrnkng factor for the nput varable x. By applyng fferent shrnkng spans to an nput varable, fferent results are acheve. A typcal shrnkng span of 0.7 s selecte for ths work., (3) (4) It shoul be note that wth the mathematcal efnton of the membershp functons n (2) an (4), the unverse of scourse s lmte to [-1,1], however ths can easly be change by ntroucng a scalng factor. Usng a smlar approach, membershp functons can be efne for the output varable [12], although n the Takag-Sugeno metho, lnear functons are apple as a mappng tool between the nput an the output. Fg 4. Shrnkng span membershp functons The etals of esgnng a SSMF fuzzy controller n a general case (multple nput multple output systems) s rgorously escrbe n [12]. Nevertheless, t s brefly revste here for ths specfc problem (sngle nput sngle output system). Dfferent trangular functons for the nput varable can be expresse as n (2): F = for ( x; x 1, x, x + 1), = m,..., m where m s the nex for the nput set, resultng n 2m+1 lngustc terms for that nput varable x. In ths work, the parameter m s selecte to be 5, therefore 11 shrnkng span membershp functons are assgne to the nput varable. (2) C. Aaptve Takag-Sugeno Controller Both the conventonal an SSMF fuzzy logc controllers propose above, have tme nvarant membershp functons an output mappngs between nput an output, n other wors all the esgn parameters, reman constant throughout the control performance. However n a more realstc approach, especally n a control process wth fast changng ynamcs, an aaptve esgn wll result n a more effcent an a more robust performance. In an aaptve fuzzy logc controller, the membershp functons an the lnear mappngs can be a functon of tme. Several tranng technques have been propose n the lterature n orer to upate the fuzzy controller parameters [18],[13]. The backpropagaton metho s selecte for ths specfc control problem. In ths approach, an error functon s efne an the fuzzy controller parameters are trane n a way that the error functon s mnmze. Lne voltage evaton at the 2640

5 mle of the transmsson lne,.e., V, s assume as the error functon. A conventonal Takag-Sugeno fuzzy controller s selecte as explane n secton A. The coeffcents of the nput-output mappng lnear functons (equaton (1)) are assume to be the tme varant parameters of the controller. It shoul be note that the subntervals of the fuzzy membershp functons (Fg. 3) can be trane an upate as well. However ue to the smple structure of the controller n ths work, these parameters are consere to be constant an only the mappng functons coeffcents unergo tranng. All the coeffcents are upate n a recton that mnmzes V. Even changng the number of membershp functons or the shrnkng factor oes not make a sgnfcant fference n the performance of the SSMF fuzzy controller. Fgures 6 an 7 show some of the typcal waveforms of the generators. V. SIMULATION RESULTS In orer to evaluate the performance of fferent controllers, a 150 ms three phase short crcut s apple to the system at the mle of the transmsson lne, where the STATCOM s connecte. Smulaton results for the fferent controllers appear n Fg. 5. Fg 6. Generator 3 termnal voltage urng the fault Fgure 5 shows that the fuzzy controllers are more successful than the PI controller n ampng the lne voltage swngs at the pont of connectng the STATCOM. Ths s because the PI controller has been fne tune at only one operatng pont, whle a severe fault lke a three phase short crcut changes the operatng conton of the network rastcally. Fg 7. Generator 3 spee evatons urng the fault Fg 5. Lne voltage urng a 150 ms three phase short crcut at the mle of the transmsson lne. Although the mofe Takag-Sugeno metho works more effcently, ue to the rather smple structure of the controller an straghtforwar rule base, the conventonal Takag-Sugeno controller works almost equally effectve. Fg 8. Inverter moulaton nex urng the fault 2641

6 The controllers can also be compare n terms of the control acton generate by each one. The moulaton nex apple to the STATCOM nverter s a measure of control acton an t s clear from Fg. 8 that the fuzzy controllers are faster n senng the approprate control sgnal, whch n turn means less control acton prove. VI. CONCLUSION Dfferent Takag-Sugeno base fuzzy logc esgns are propose for a STATCOM connecte to a multmachne power system: a conventonal esgn, a SSMF esgn an an aaptve controller usng the backpropagaton (steepest escent) metho. The frst two controllers are smulate an the results show better an faster ampng compare to that of the conventonal PI controller. SSMF fuzzy controllers can aapt to almost all problems wth fferent natures an prove to be more effectve than the conventonal approach, especally when there s not enough nformaton avalable on the ynamcs an behavor of the plant to be controlle. Even though both conventonal an SSMF fuzzy controllers esgne an smulate n ths work, ten to rely on the fuzzy nference an reasonng, they stll slghtly epen on the nature of the plant. In other wors, a better knowlege of the ynamcs of the STATCOM n ths specfc power system wll lea to a better tunng of the controller, whch n turn prouces better results at fferent operatng contons an uner varous faults apple to the system. A soluton to ths problem can be an aaptve fuzzy controller whose esgn parameters start from an ntal set, but can be upate an trane base on the real tme performance of the controller. Such a controller wll further elmnate the epenency of the esgn on the expert knowlege on the system an ts ynamcs. Smulatons are carre out by the authors n orer to mplement the aaptve Takag-Sugeno fuzzy logc controller usng the backpropagaton metho. Further results wll be prove n the follow up papers. ACKNOWLEDGEMENTS The authors gratefully acknowlege fnancal support from the Duke Power Company, Charlotte, North Carolna, an from the Natonal Scence Founaton Grant ECS # REFERENCES [1] N.G. Hngoran an L. Gyugy, Unerstanng FACTS, Concepts an Technology of Flexble AC Transmsson Systems, IEEE, New York 1999, ISBN [2] L.Y. Dong, L. Zhang an M.L. Crow, A New Control Strategy for the Unfe Power Flow Controller, IEEE Power Engneerng Socety Wnter Meetng, New York, NY, USA, January 27-31, 2002, Vol.1, pp [3] D. Shen an P.W. Lehn, Moelng, Analyss an Control of a Current Source Inverter-Base STSTCOM, IEEE Transactons on Power Delvery, Vol.17, No.1, Jan 2002, pp [4] P. Rao, M.L. Crow an Z. Yang, STATCOM Control for Power System Voltage Control Applcatons, IEEE Transactons on Power Delvery, Vol. 15, No. 4, October 2000, pp [5] F. Lu et al, The Nonlnear Internal Control of STATCOM: Theory an Applcaton, Internatonal Journal of Electrcal Power & Energy Systems, Vol. 25, Issue 6, 2003, pp [6] Q. 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