International Journal of Eletrial Engineering. ISSN 0974-58 Volume 4, Number (0), pp. 6-8 International Reearh Publiation Houe http://www.irphoue.om Modeling, Analyi and Simulation of Robut Control Strategy on a 0kVA STATCOM for Reative Power Compenation J.K. Moharana *, M. Sengupta and A. Sengupta Department, of Eletrial Engineering, Bengal Engineering and Siene Univerity, Shibpur, Howrah-03, Wet Bengal, India. *Email: jkruhna@rediffmail.om, jkruhna@gmail.om Abtrat The STATCOM (STATi ynhronou COMpenator) i being inreaingly popular in power ytem appliation. In general, reative power ompenation for power fator and tability of the grid ytem an be improved. A imple d q tranformation and teady tate and tranient analyi are ahieved to haraterize the open loop ytem. The mall ignal heme in H ontroller ontrol the phae angle a well a modulation index of the withing pattern and with mall perturbation of referene urrent (reative urrent of load), the DC voltage nearly remain ontant. The H - ontroller deign and implementation of the STATCOM i propoed without uing independent ontrol on DC-link voltage of the STATCOM. The H - ontroller provide fat repone of the STATCOM ompenation of the reative power and beoming the table of the ytem. The pre-harge voltage of 600V on DC-link i hoen for tolerable tranient variation of the relevant repone of the STATCOM. All repone are obtained trough MATLAB imulink tool box. Index Term: STATCOM, mall ignal model, H ontroller Introdution In reent year power ytem have beome very omplex with interonneted long ditane tranmiion line. The interonneted grid tend to beome untable a the heavy load vary dynamially in their magnitude and phae angle and hene power fator. Commiioning new tranmiion ytem are extremely expenive and take
6 J.K. Moharana et al oniderable amount of time to build up. Therefore, in order to meet inreaing power demand, utilitie mut rely on power export/import arrangement through the exiting tranmiion ytem. Power eletroni devie are gaining popularity for appliation in the field of power tranmiion and ditribution ytem. The reative power (VAR) ompenation and ontrol have been reognized [] a an effiient & eonomi mean of inreaing power ytem tranmiion apability and tability. The FACTS (Flexible AC Tranmiion Sytem) devie, uh a STATCOM ha been introdued more reently whih employ a VSI with a fixed d link apaitor a a tati replaement of the ynhronou ondener. In a traditional ynhronou ondener, the field urrent of the ynhronou motor ontrol the amount of VAR aborbed/injeted and in a imilar way; the firing intant of the 3-phae inverter ontrol the VAR flow into or out of the STATCOM. Large number of apaitor bank or any other paive element are no more required. Only a fixed et of apaitor provide the required VAR ontrol, with a rapid ontrol of bu voltage and improvement of utility power fator. It offer everal advantage over onventional thyritoried onverter [] in term of peed of repone. The penalty paid for thi improvement i in term of introdution of ome harmoni, whih require eparate handling uing ative filtration tehnique. Moran et al [3] have hown in detail how the utilization of SPWM tehnique redue harmoni ditortion. It ha alo been hown that an inreae of modulation index redue the ize of the link reator and tre on withe whih are ignifiant iue in pratial implementation. The modeling and analyi of STATCOM teady tate and dynami performane with onventional ontrol method have been tudied by Shauder and Mehta [4]. In [5] the dynami repone and teady tate behavior of STATCOM with SVPWM ha been tudied and the advantage of introduing SVPWM inverter with higher value of MI are highlighted. It i alo hown that a PI Controller deigned on the bai of a linearized model [6-8] of the STATCOM ahieve very good tranient repone. The large interonneted power ytem i alled upon to work with different value of 'α ' the angle between the grid voltage and the fundamental omponent of STATCOM output voltage. Alo the modulation index (M.I) m of the inverter i to be varied over a wide range. The inherently nonlinear ytem alo experiene plant parameter variation over a wide range. In thi work it i hown that the PI Controller deigned on the bai of the plant model, linearized about a nominal point annot handle the plant parameter variation over the entire range. A a remedy the plant parameter variation i inluded a plant unertainty and a ontroller baed on H -optimization i ued. Thi give ignifiant improvement in plant repone on variation of m and α. Starting with an etablihed teady tate open loop model of the STATCOM, the preent paper goe on to develop loed loop model for invetigating tranient performane of the STATCOM. Firt, in Setion II, general modeling of the STATCOM and tudy it open-loop repone. A mall ignal model i developed to etablih the effet of the variation of the modulation index 'm' and the phae angleα. Thereafter, a loed loop ontrol heme ha been propoed with two type of ontroller deign, o that d link voltage may be kept unhanged. Thi heme i both an extenion and a ignifiant improvement of the heme uggeted by Cho et al [6].
Modeling, Analyi and Simulation 63 Modeling of the STATCOM and Analyi Operating priniple A i well known, the STATCOM i, in priniple, a tati (power eletroni) replaement of the age-old ynhronou ondener. Fig. (a) how the hemati diagram of the STATCOM at PCC through oupling indutor. The fundamental phaor diagram of the STATCOM terminal voltage with the voltage at PCC for an indutive load a well a apaitive load in operation, negleting the harmoni ontent in the STATCOM terminal voltage, are hown in Fig.(b) and (). Ideally, inreaing the amplitude of the STATCOM terminal voltage V r oa above the amplitude of the utility voltagev r a aue leading (apaitive) urrent I r a to be injeted into the r ytem at PCC. I a _ a, the real omponent of I r a, aount for the loe in the reitane of the indutor oil and the power eletroni onverter. Ideally, if the r ytem loe an be minimized to zero, I a _ a would beome zero, and I r a would be leading at perfet quadrature. Then, V r oa, whih i lagging and greater thanv r a, would alo be in phae with V r a. The STATCOM in uh a ae operate in apaitive mode (when the load i indutive)(a hown in Fig.(b)). Similarly, the operation of the STATCOM an be explained to be indutive if the load i apaitive (a given in Fig.()). (a) (b) () Figure : (a) Shemati diagram of STATCOM, (b) phaor diagram for indutive load operation and () phaor diagram for apaitive load operation. Modeling The modeling of the STATCOM, though well known, i reviewed in the line below, for the ake of onveniene. The modeling i arried out with the following aumption:. All withe are ideal. The oure voltage are balaned 3. R repreent the onverter loe and the loe of the oupling indutor 4. The harmoni ontent aued by withing ation are negligible The 3-phae tationary ab oordinate vetor with 0 0 apart from eah other
64 J.K. Moharana et al are onverted into αβ -phae tationary oordinate (whih are in quadrature). The α axi i aligned with a axi and leading β axi and both onverted into dq twophae rotating oordinate. The Park ab to dq tranformation matrix i given in () Co( ϖt ) Co( ϖt π /3) Co( ϖt + π /3) K Sin( ϖt ) Sin( ϖt π /3) Sin( ϖt + π /3) () 3 / / / V a V V, ab b V Sin( ωt α) π V Sin( ωt α ) 3 3 π Sin( ωt α + 3 The atual propoed iruit i too omplex to analyze a a whole, o that it i partitioned into everal bai ub-iruit, a hown in Fig. (a). The 3-phae ytem voltage V, ab lagging with the phae angle α to the STATCOM output voltage V o, ab and differential form of the STATCOM urrent are defined in () and (3). where, V,ϖ, R and L have their uual onnotation. The above voltage and urrent are tranformed into dq frame and given in (4) and (5) d L dt ( ) i ab R i ab + V ab V o ab, (3) [ Sinα Coα ] T V, qdo V 0 d L dt d L i dt,,,. (4) ( iq ) Riq wlid + Vq Voq (5a) ( d ) wliq Rid + Vd Vod (5b) The withing funtion S of the STATCOM an be defined a given in (6) and the modulation index, being ontant for a programmed PWM, i given by (7). m i the modulation onverion index and i the multiplying fator for tranformation of 3 three phae tationary quantitie to two phae dq rotating frame. S MI S S S a b V o, peak m v d 3 m m 3 Sin ( wt Sin ( wt Sin ( wt + ) π ) 3 π ) 3 (7) () (6)
Modeling, Analyi and Simulation 65 The STATCOM output voltage and dq tranformation are given by, V o. ab Sv d (8) V T o, qdo m [ 0 ] v d (9) The d ide urrent in the apaitor and it dq tranformation may be written a, T S i (0) i d, ab i d T o [ 0 ] i i i () m 0 q d The voltage and urrent related in the d ide i given by dvd id C () dt Now replaing () in () yield dvd m id (3) dt C The omplete mathematial model of the STATCOM in dq frame i obtained a given in (4) R w 0 i L q iq Sinα d R m V + i w i Coα (4) d d dt L L L v v 0 d m d 0 0 C Steady State and tranient Analyi The teady tate equation of I q, I d, V d, P, Q are given in (5-6) and their repone with the parameter given in Table, are hown in Fig.3 and Fig.4. V Sin α I q, I d 0 R V wl V α d Co M R Sin α P V R [ Co ( α )] Q V R Sin ( α ) (5) (6)
66 J.K. Moharana et al Table Sl.No Meaning Symbol Value Fundamental Frequeny f 50 Hz Fundamental Angular Frequeny ω 34 rad/e 3 RMS line-to-line Voltage V 45V 4 Effetive oupling Reitane R.0 Ω 5 Coupling Indutane L 5.44mH 6 DC link apaitor C d 680 μ F 7 Modulation Converion Index m 0.98 to. 8 Reitive load R l 3 Ω 9 Indutive load L l 60 mh,.06ω The open loop teady tate repone to a tep hange in phae angle α of the reative omponent of STATCOM urrent, I q, and the reative VAR, Q, of the STATCOM in apaitive mode of operation are hown in Fig..(a) and (b) 0 repetively. The orreponding tranient repone at α 0 are given in Fig.3 (a) and (b) repetively. (a) (b) Figure : Steady tate repone due to variation inα of STATCOM in apaitive mode: (a) Reative omponent of STATCOM urrent and (b) reative power generated by STATCOM The teady tate value of I q and Q vary with variation in phae angleα. Thee value (a hown in Fig. (a) and (b) repetively) are 7A and 9.45kVAR at 0 α 0 in apaitive mode of operation of STATCOM. The orreponding tranient repone (Fig.3 (a) and (b)) in generating (apaitive) mode of operation of STATCOM have high tranient eem to generate 7 A and 9.45kVAR repetively after two and half power yle. The high tranient magnitude of i q of
Modeling, Analyi and Simulation 67 the STATCOM unneearily inreae the rating of the STATCOM devie. The above alulation are on the bai of the mathematial model of the STATCOM. The pratial STATCOM rating will be limited by the hoie of the withing devie (IGBT of SEMIKRON make of part no.skm75gb3d) and the ripple urrent, the average urrent and the voltage rating of the DC link apaitor a alo the urrent rating of the oupling indutor. The above value of the tranient peak are atually out of range for the preent et-up. Realling the introdution in etion, the STATCOM i a voltage oure behind a reatane onneted in hunt to the PCC. The ytem demand fat injetion of urrent, both at fundamental and harmoni frequenie. The STATCOM voltage hould be apable of fat variation in order to ater to ytem need. State differently, the STATCOM voltage hould be a funtion of the urrent ( * * i q ) or reative power ( q ) demanded by the power ytem. The mot rudimentary ontrol ytem truture required for thi purpoe i hown in Fig.4 (a). The ontrol voltage, v ontrol (t), i generated from a prior information of the erie indutane, L, and the phae and magnitude of the PCC voltage, v (t). If thee are upplied to the referene generator, f (.), only one at the beginning of the operation, it ontitute an open-loop ontrol ytem. Although the erie indutane may not vary ignifiantly, the PCC voltage, v (t), will definitely under go oniderable variation in phae and magnitude. Some of the internal variable of the STATCOM, e.g. the DC-link voltage may alo vary whih aue a variation in the gain K hown in Fig.4 (a). An open-loop ontrol ytem will be unable to maintain the STATCOM urrent againt the diturbane. Thi all for loed-loop ontrol of the STATCOM urrent, whih i hematially hown in Fig.4 (b). The urrent or reative power ontrol f (.), i deigned to obtain the deired loed- loop bandwidth. (a) (b) 0 Figure 3: Tranient repone to a tep hange of 0 in α of STATCOM in apaitive mode: (a) Reative omponent of STATCOM urrent and (b) reative power generated by STATCOM
68 J.K. Moharana et al (a) (b) Figure 4: Shemati of STATCOM: (a) With open-loop ontrol and (b) with loedloop ontrol. Modeling of Load The R L load are onneted at the PCC of the utility oure in tar onnetion. The a, b and three phae tationary oordinate of the load urrent are onverted into α β of two phae tationary oordinate urrent and then d q of two phae rotating o-ordinate urrent. The imulation of the phae a load urrent with voltage at PCC, ative and reative omponent of the load urrent and ative and reative power are hown in Fig.5. Fig.5(a) i hown the lagging power fator of 0.77 of a phae load urrent with repet to PPC a phae voltage. Thi lagging power fator an be ompenated to unity power fator.fig.5(b) i hown the d and q axi urrent of.5a and 7.A of the above load repetively whih an be ued a referene value at the time of ontrolling d and q axi urrent of STATCOM. Fig.5() i hown the ative and reative power of 6 KW and 3 KVAR of the above load repetively. (a) (b) () Figure 5: (a) Grid phae a voltage and load phae a urrent, (b) ative and reative urrent of the load and () ative and reative power of the load Small Signal Sheme Modeling To know the dynami harateriti of STATCOM ytem, the mall ignal analyi i to be done. For a given operating point, mall ignal equivalent iruit i derived baed on the following aumption:
Modeling, Analyi and Simulation 69 i. The diturbane i mall, ii. Hene, the eond order term (produt of variation) are negligible, iii. The phae nominal value of angle α i mall. With the above aumption the equation (9) and () an be rewritten a, + vˆ ( t) M V + M vˆ + mˆ V + mˆ vˆ (7) V od od d iˆ d M I d + M iˆ d d d I d + (8) Sin ˆ α ˆ α, o α and α 0 (9) 0 By uing (5), (7), (8) and (9), and applying Laplae Tranformation, we have, L + ˆ R wl 0 iq V ˆ( α ) wl L + R M i ˆ V mˆ ( ) (0) d d 0 M C vˆ 0 d The important tranfer funtion of the tate of the STATCOM in mall ignal model an be derived a iˆ q ( ) V d wl C mˆ ( ) ) i ˆ ( ) V ( L C + R C + M ) q ˆ( α ) ) i ˆ d ( ) V d C( L + R ) mˆ ( ) ) iˆ d ( ) V d wl C ˆ α ( ) ) v ˆ d ( ) M V d ( L + R ) mˆ ( ) ) vˆ d ( ) M V ϖl ˆ α ( ) ) ) 3 CL + L CR + [ C{ R + ( L ) } + M L ] + M R (a) d (b) (a) (b) (3a) (3b) ϖ (4) Open loop repone The repone of the tate î q, î d and vˆ d (a in (-3)) are imulated in MATLAB uing the parameter given in Table-, with a variation of phae angle ˆ α 5u( t) (in degree) and a hange of modulation onverion index mˆ 0.u ( t). Fig.6 and Fig.7 how repone for a tep hange in modulation onverion index mˆ and phae angle
70 J.K. Moharana et al αˆ repetively in apaitive mode. It may be noted from the reult that a hange in mˆ ha no teady tate effet on the reative omponent of STATCOM urrent (Fig.6 (a)) in apaitive mode. It however aue v d to redue by 35V in the teady tate a hown in Fig.6 (b). However, due to a hange inαˆ by an amount given above, the STATCOM draw additional teady tate reative urrent of 37 A (Fig.7 (a)) at teady tate. Neither doe, v d, the DC-link voltage remain at it previou teady tate value. It inreae in magnitude (by50 V ) for a hange in αˆ (Fig.7 (b)). Hene the open loop dynami ugget that it i required to have loed loop ontrol of reative omponent of STATCOM urrent a well a DC-link voltage of STATCOM in order to redue tranient urge and to keep V d ontant. (a) (b) Figure 6: Tranient repone for hange in vˆ d mˆ in apaitive mode: (a) In î q and (b) (a) (b) Figure 7: Tranient repone for hange in αˆ in apaitive mode: (a) In î q and (b) vˆ d Deign of robut ontroller The tate pae model of STATCOM (4) i nonlinear in phae angleα. The model of the STATCOM in linear model are valid for mall hange in α and m about the nominal value. In any pratial et-up the variable do vary beyond mall range. Then the mall ignal model may be in adequate to repreent the ytem behaviour. The ontroller deigned on the bai of thee model may not give deired performane when applied to the original ytem. A way to overome thi problem i
Modeling, Analyi and Simulation 7 to deign robut ontroller. To the bet knowledge of the author no work i reported in the literature on robut ontrol of the nonlinear STATCOM ytem atering to linear load. The problem of ontrolling d and q axe omponent of the urrent and DC link voltage in ae of the atual nonlinear ytem may be tated a follow: given the et of equation (4), one ha to deign a ontroller() whoe output u (pleae ee Fig.8(a)) will tabilize the ytem and yield deired performane in the fae of unertainty() and perturbation(). There i no tandard method of deigning a tabilizing ontroller for a nonlinear ytem. One way that i uually followed i to perform the deign for a linearized ytem about nominal operating ondition a done in previou etion. The deign objetive are met if the ontroller i robut enough to perform well for the other operating ondition too. In the preent work perturbation to the plant are modeled a multipliative unertaintie and a robut ontroller i deigned uing the theory of H optimization [9]. Thi etion give a brief deription of the theory of the unertainty modeling, the robut tability riterion and deign proedure of the robut ontroller for the mixed enitive problem a well a a graphial deign tehnique termed loop haping. Modeling Unertainty model A robut ontroller i to be deigned for the nominal plant baed on the Mixed Senitivity problem in H -optimal Control Theory [9] a explained in brief below. Let u onider the blok-diagram in Fig. 8(a). A nominal tranfer funtion P () belong to a bounded et of tranfer funtion, K () i the H -ontroller, W ( ) i the enitivity weighting funtion and W ( ) i the omplementary enitivity weighting funtion, y (t) i the meaured output, u (t) the ontrolled input, w (t) the exogenou input (whih may inlude the diturbane) and z (t) i the ontrolled output. We an write, Figure 8: (a) Mixed Senitivity Problem and (b) modifiation of Mixed Senitivity Problem z z (5) z
7 J.K. Moharana et al The augmented plant i: whih relate: z z y w G u G W 0 I W P W P P (6) (7) The blok diagram of Fig.8(a) of the mixed enitivity problem i modified to the general ontrol problem a hown in Fig.8(b). The loed loop tranfer funtion (Fig.8(a)) from w to z i denoted by T (). The ontrol problem i to find a ontroller K () uh that The loed-loop ytem i internally table The -norm of the loed loop tranfer funtion T () atifie: T < γ for ome mall γ > 0 The perturbed tranfer funtion, reulting from variation in operating ondition, an be expreed in the form of the unertainty et a hown in Fig.9 and given in P ~ ( ) P ( ) + ΔW ( ) (8) ( ) Here, Δ i a variable tranfer funtion atifying <. It may be realled that the infinity norm ( norm) of a SISO tranfer funtion i the leat upper bound of it abolute value. Thi i alo written a Δ ω upω Δ( jω) and i the larget value of the gain on a Bode magnitude plot. The unertaintie, whih are the variation of ytem operating ondition, are thu modeled through P ~ in equation (.46). In the multipliative unertainty model in equation (8), ΔW i the normalized plant perturbation away from. If <, then Δ Δ Figure 9: Multipliative perturbation model. P ~ ( jω) W ( jω) P( jω) for all ω (9)
Modeling, Analyi and Simulation 73 So, W ( jω) provide the unertainty profile. In the frequeny plane it i the upper bound of all the normalized perturbed plant tranfer funtion away from. For the optimum ae the weighting funtion an be determined a ~ P( jω) P( jω) W ( jω) (30) P( jω) Robut tability and performane Let u onider K () ome ontroller whih i aumed to provide overall ytem tability if it provide internal tability for every plant in the unertainty et. If L denote the loop tranfer funtion ( L PK ), then the enitivity funtion S i written a S (3) The omplimentary enitivity funtion or the input-output tranfer + L PK funtion i T S (3) + PK For a multipliative perturbation model, the robut tability ondition i met if and W ( jω) L( jω) only if W T <. Thi implie that <, for all ω + L( jω) (33) or, Δ ( j ω) L( jω) < + L( jω), for allω, Δ < (34) The nominal performane ondition for an internally table ytem i given a W S <, where W i a real-rational, table, minimum phae tranfer funtion (alo alled a weighting funtion). If P() i perturbed to P ~ ( ) P ( )( + ΔW ( )) and ~ S S i perturbed to S + ( + ΔW L) + ΔW T (35) then the robut performane ondition hould therefore be [9] WS W T < and <, for all Δ < (36) + ΔW T Combining all the above, it an be hown that a neeary and uffiient ondition for robut tability and performane i W S + W T < (37) Thi i the Mixed Senitivity problem for ontroller deign. Propoed ontrol heme Fig.0 i hown the propoed ontrol trategy of mall ignal model of the STATCOM. H -mixed enitivity redued order ontroller ( H MSROC ) (.37) i applied at the reative omponent of the STATCOM urrent with referene reative omponent of the load urrent. The output of the ontroller generate a hange in phae angle αˆ and it an be onverted to a hange of modulation onverion index
74 J.K. Moharana et al mˆ by the tranfer funtion G a [(39)]. Three phae of the STATCOM output voltage 0 are generated by 0 apart from eah other with the angle umming of a hange in phae angle αˆ, initial value of phae angle α 0 and ynhronizing θ from PLL []. Then three-phae voltage of STATCOM are multiplied with the peak value of the grid ide voltage and the um of a hange in modulation onverion index mˆ a well a the initial value of the modulation onverion index M. The gate pule for the onverter are generated with the 3-phae STATCOM terminal referene voltage and SVPWM priniple. The mathematial modeling of the ytem for deigning the ontroller i given in next etion. Figure 0: Propoed ontrol heme of mall ignal model of the STATCOM. Deign of ontroller If the DC-link voltage of the STATCOM i to be kept ontant, then the variation Vˆ d hould be zero. Hene, from equation (0) ˆ M ( ) ˆ Vd L + R m ( ) MVω L ˆ( ) V ( ) α VωL d mˆ ( ) ˆ( α ) (38) ) V ( L + R ) In the preent work the author have ued the above relation (equation (38)), to ugget an important modifiation to the heme uggeted by Chou.et al [6]. Thi i a ignifiant ontribution of the preent work. Fig. how the propoed blok diagram of the loed-loop ontrol ytem with a H MSROC in order to ontrol the reative omponent of the STATCOM urrent with independent ontrol on DC-link voltage. d
Modeling, Analyi and Simulation 75 Figure : Blok diagram of loed loop ytem with H MSROC. Gm a The forward path tranfer funtion i VωL VdωLC G a Vd ( L + R ) d (39) In thi etion Iˆ q ( ) ha been modeled a from () i ( ) ˆ V LCd + RCd + M Vdω L C ( ) ˆ( ) d I q α + mˆ ( ) (40) ) ) To ahieve fat dynami repone without ontrolling the DC link voltage, then the variation of Vˆ d ( ) hould be zero. By uing equation (39), equation (40) may be modified to: ˆ V ( LCd + RCd + M ) Vω L C d I q ( ) ˆ( α ) ) )( L + R ) So that the nominal plat tranfer funtion i, ˆ I q( ) V LCd + RC P( ) ˆ( α ) ) ( + M ) d + V ω L C )( L d + R ) (4) The mixed enitivity robut ontroller [9] i now deigned for the nominal plant baed on Mixed Senitivity problem. The enitivity weighting funtion W ( ) i following bai: If the plant i P() and the loop tranfer funtion (a mentioned earlier) i ω L( )( P( ) K( )), and the enitivity funtion i weighting funtion W ( ) for enitivity S() i W S( ) then the + ω + ω ( ),where ξ i the + ξω
76 J.K. Moharana et al damping ratio. 0 Here the nominal plant, P () i taken at (i) m 0. 98 at α 0 and the perturbed ~ plant, P ( ) i taken at (ii) m. 0 and α 0. The weighting funtion W ( ) an be determined a: W ( ) ~ P( ) P( ) ~ P( ) (4) The weighting funtion W ( ) an be determined after hooing 5 % over hoot, ξ 0.69 and ω 409. 8 rad/e and teady tate error of 0. [9]. So W ( ) will be () ( 0. + 409.85) ( + 40.98) W (43).However, both weighting funtion an be finalized in uh a way that both weighting funtion obey W ( ) + W ( ) for all frequenie. Hene, by trial and error adjutment of pole and zero of the tranfer funtion of the weighting funtion, W ( ) and W ( ) will be 0 ( ) W ( ) (44) and W ( ) (45) ( + 0.4 ) + 0 The Bode plot of ( ) ( ) + W ( ) W and ( ) W hown in Fig. indiate W at all frequenie. The weighting funtion together with the nominal plant P () are ued in the Robut Control Toolbox in MATLAB to yield the mixed enitivity H -Controller K h given in (46). The ontroller i of 5 th order tranfer funtion. Figure : Bode plot of W ( ) and W ( ) for mall ignal model. 7 4 0 3 3 5 7 8.6 0 4.3 0 3.7 0 7.8 0 4.4 0 K h ( ) (46) 5 0 4 3 3 6 8 7 + 4.6 0 +.3 0 +. 0 + 9.9 0 + 4 0 It may be noted that the mall ignal model of the STATCOM ued here inlude an improvement over exiting work of Cho et al [6]. The deign of the ontroller i another ontribution of the author. The tep repone of the loed loop ytem i hown in Fig.3 (a) and the output
Modeling, Analyi and Simulation 77 ettle at 0.007 eond. The ytem an ompenate m 0A quadrature urrent a hown in Fig.3 (b). It i a good agreement with the referene quadrature urrent. (a) (b) Figure 3: Repone uing H MSROC repone of i q with referene reative urrent. : (a) Step repone of i q and (b) tep Robut ontrol tehnique generally yield high-order ontroller. The ontroller K h (46) of 5 th order i redued by Hankel norm method []. The redued-order ontroller of nd order i given in (47). Fig.4 how the omparion between the dynami of the redued-order ontroller with referene to the full order one. The loed loop tep repone of the full order ontroller and redued-order ontroller are found to be in good agreement. The ame i the ae for frequeny repone in Fig.4 (b) repetively. 7 0 8.68 0.06 0 K hr ( ) (47) 0 0 + 4.6 0 +.88 0 (a) (b) Figure 4: Repone of i q for full order and redued-order H MSROC : (a) Step repone of i q and (b) Bode plot. Simulation The H MSROC in (47) with the propoed heme i imulated uing MATLAB SIMULINK environment on variation of initial DC-link (pre-harge) voltage. An initial DC link voltage of00 V, the STATCOM with thi ontroller operate well with larger urrent pike of 0 A a hown in Fig.5 (a) of grid phae a
78 J.K. Moharana et al voltage with urrent.the power fator of the grid oure with R L load improve from 0. 77 to unity in three power yle over the open loop ituation in Fig.5(a). The ame dynami of the STATCOM phae a urrent i given in Fig.5 (b) with a urrent pike of 00 A. The urrent urge in all ae are more than the aeptable limit a per the peifiation of the STATCOM. (a) (b) Figure 5: Repone of mall ignal model of STATCOM with H MSROC at initial DC link voltage of00 V : (a) Grid phae a voltage and urrent (b) Grid phae a voltage and STATCOM phae a urrent The ame model i imulated with hanging the initial DC link voltage of550 V. It i een that with R-L load the power fator improve from 0. 77 to unity after two yle with a urrent pike of 45 A a given in Fig.6 (a) (for grid phae a ) on ompared to Fig.5(a). Fig.6 (b) how the dynami of the STATCOM urrent with tranient urge of 30 A. Fig.6 () and (d) how the dynami of d and q axe urrent of the STATCOM and their tranient ( 4 A of ative and 50A of reative urrent) die out after two power yle repetively.the ative with reative power generated by the STATCOM i hown in Fig.6 (e) and the tranient ( 8kW of ative power and 0 kvar of reative power) die out in two power yle a hown in Fig.6 (f). It i noted that the dynami of the ative urrent of STATCOM i imilar to the ative power dynami. Inae of reative urrent the dynami i imilar in nature but in phae oppoition. The prototype STATCOM an not be deigned taking the above tranient of the relevant waveform, ine that will mean exeive urge urrent rating of the olid-tate devie (whih are otherwie unneeary with repet to teady tate rating) No eparate voltage ontroller i needed for thi model. The beauty of the model i that the voltage get ontrolled indiretly and ome to teady tate at 640V after urge of 970 V repetively a hown in Fig.6(g). Thi i a ontribution of the preent work. It i noted that the ytem in ae of thi model i not o luggih but the tranient urge are not within reaonable limit.
Modeling, Analyi and Simulation 79 (a) (b) () (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) Figure 6: Repone of mall ignal model of STATCOM with H MSROC at initial DC link voltage of 550 V : (a) Grid phae a voltage and urrent, (b) Grid
80 J.K. Moharana et al phae a voltage and STATCOM phae a urrent, () Ative and reative omponent of STATCOM urrent, (d) Ative and reative omponent of STATCOM urrent (in expanion), (e) Ative and reative power generated by STATCOM, (f) Ative and reative power generated by STATCOM (in expanion), (g) DC link voltage of STATCOM, (h) DC link voltage of STATCOM (in expanion), (i) Variation of inremental of phae angle αˆ, (j) Variation of inremental of modulation onverion index mˆ, (k) Grid phae a voltage and STATCOM output phae a voltage, (l) Grid phae a voltage and STATCOM output phae a voltage (in expanion), (m) Change of STATCOM reative urrent due to hange of referene urrent (load reative urrent), (n) DC link voltage of STATCOM due to hange of referene urrent (load reative urrent) Fig.6 (i) and (j) how the orreponding dynami of inremental phae angleαˆ and modulation index mˆ repetively. Fig.6 (k) and (l) how the dynami of the grid phae a voltage with STATCOM output phae a voltage and latter i greater with lagging very mall angle than the former repetively. Appliation of mall perturbation of referene reative urrent a hown in Fig.6 (m) (reative urrent of load), a very mall hange in DC link voltage i obtained. While the referene urrent return to it original value the DC link voltage alo return to it teady tate value a hown in Fig.6 (n). Again the ame propoed model i imulated with an initial DC link voltage of 600 V.The relevant repone i obtained with a leer urrent pike of 0A and ompenate after one power yle a hown Fig.3.3 of the grid phae a voltage and urrent. The peak and rm value of the grid urrent are the bai of the deign and eletion of the devie and the peripheral of STATCOM ytem. Figure 7: Grid phae a voltage and urrent of mall ignal model of STATCOM with H MSROC at initial DC link voltage of 600 V Conluion The STATCOM (STATi ynhronou COMpenator) ha been modeled in a imple
Modeling, Analyi and Simulation 8 d q tranformation. It teady tate and tranient analyi have been ahieved to haraterize the open loop repone of the STATCOM. A mall ignal ontrol heme of the STATCOM ha been propoed without uing diret DC-link apaitor voltage ontrol. The ontroller ha been deigned on bai of mixed enitivity method. Thi heme ontrol reative urrent of the STATCOM and the DC-link voltage get ontrolled indiretly. Thi i an ahievement of the preent work. The phae angle a well a modulation index of the withing pattern i ontrolled indiretly. Different pre-harge voltage are applied on the DC-link apaitor to get proper reaonable tranient variation and ettling time of the relevant repone of the STATCOM. Thee reult give the proper idea for deigning the devie and peripheral omponent of the STATCOM. The pre-harge voltage of 600V i hoen on bai of the tolerable and aeptable tranient variation and ettling time of the relevant repone of the STATCOM. With mall perturbation of referene urrent (reative urrent of load), the DC voltage nearly remain teady tate. Thi heme alo ontrol ative and reative power indiretly. All repone are obtained trough MATLAB SIMULINK tool box. Aknowledgement The author would like to aknowledge the upport reeived from the All India Counil of Tehnial Eduation a thi work i an outome of an AICTE-RPS Projet. The author are alo thankful to Vie-Chanellor and Head, Dept. of Eletrial Engineering, Bengal Engineering and Siene Univerity, Shibpur for all upport reeived. Referene [] A.T. John, A.Ter-Gazarian and D.F.Wame, Flexible a tranmiion ytem (FACTS), IEE Power and Energy Serie, London, U.K. [] R.M.Mathur and R.K. Varma, Thyritor-baed FACTS Controller for Eletrial Tranmiion Sytem, IEEE Pre, Wiley-Interiene Publiation. [3] L.T. Moran, P.D.Zioga and G.Joo,989 Analyi and Deign of a Three- Phae Synhronou Solid- State Var Companator, IEEE Tran. Indutry Appliation, Vol. 5, No. 4, pp. 598-608 [4] C.Shauder and H.Mehta,993 Vetor analyi and ontrol of advaned tati VAR ompenator, IEE Pro, 40, No. 4, [5] M.Sengupta,J.K Moharana and A.Sengupta,003 Study on an Advaned Stati VAR Compenator withed from a Spae Vetor PWM inverter Analyi, imulation and omparion with the onventional inuoidal PWM, NPEC 003, IIT Bombay, pp 7-78,6-7 Oober [6] G.C.Cho, N.S.hoi, C.T.Rim, and G.H.Cho,99 Modeling, Analyi and Control of STATIC VAR Compenator uing three level inverter, IEEE, Ind.Soity, Annual Meet, page837-843
8 J.K. Moharana et al [7] G.C.Cho,N.S.hoi,C.T.Rim,and G.H.Cho,996 Analyi and Controller Deign of STATIC VAR Compenator Uing Three-Level GTO Inverter,IEEE,Tranation on Power Eletroni,Vol.,No.,page57-65,January [8] A. Draou, M. Benghanem and A. Tahiri, 00 Multilevel Converter and VAR Compenation, Power Eletroni Handbook, pp.599-6, Aademi Pre [9] K.Zhou and J.C.Doyle, 997 Eential of Robut Control Prentie Hall [0] M.Sengupta,J.K Moharana and A.Sengupta,003 A Nonlinear Control Modeling of A STATCOM for Reative Power Compenation NPSC 004, IIT Madra, pp 7-78,6-7 Oober [] V.Kaura and V.Blako, 997 Operation of a Phae Loked Loop Sytem under Ditorted Utility Condition,IEEE Tranation on Indutry Appliation, Vol.33, No., pp58-63 [] Safonov, M.G., R.Y. Chiang, and D.J.N. Limebeer, 990"Optimal Hankel Model Redution for Nonminimal Sytem," IEEE Tran. on Automat. Control, Vol. 35, No. 4, pp. 496-50.