Small-Signal Modeling of Power Electronic Converters with Resonant Controllers

Transcription

1 Small-Signal Moeling of Power Electronic Converter with Reonant Controller Peter W. ehn, Stephen Porucky Abtract--While time omain electromagnetic tranient imulation i an increibly powerful tool for the analyi of large ignal event in power ytem, many mall-ignal phenomena may alternatively be tuie uing eigenvalue analyi an/or tranfer function. Eigenvalue or tranfer function bae analyi require ytem linearization to be carrie out about a tatic operating point. In the cae of power ytem, where voltage an current ignal are AC, linearization i mae poible through ue of the ynchronou or q erence frame tranformation. Since machine moel an moel of mot FACTS evice an their controller are reaily available in the q-frame, the proce of linearizing uch equipment i relatively traight-forwar. Recently, reonant controller have emerge a an alternative to q-frame controller for regulation of gri connecte converter, both in FACTS evice an in interface converter for itribute reource. Although thee control ytem behave omewhat imilar to q-frame controller uner balance operating conition, their behaviour uner unbalance operation i unique. Thi paper evelop a mall ignal moel of a VSC ytem, where reonant current controller are ue for regulation of the gri current. Dynamic of a DC voltage control loop are inclue. Small ignal ynamic are valiate againt time omain imulation. eywor: reonant control, FACTS, HVDC, active rectifier, eigenvalue, mall ignal analyi, converter, VSC. I. INTRODUCTION HEN attempting to regulate gri connecte AC to DC converter, uch a FACTS evice, q-frame controller are a tanar approach []. Uner balance gri voltage operating conition the q-frame control trategy i a imple an effective olution. Recently, -frame reonant controller have emerge to erve in the place of q-frame controller [], [3]. The primary ifference between q-frame an -frame controller i that the -frame reonant controller are able to track both balance an unbalance current erence comman [4]. However, one rawback of the -frame current control i the appearance of timeinvariant moulation block in it control loop. With the appearance of uch time-invariant term, control eign an parameterization technique uch a eigenvalue analyi cannot be performe. Similar linearization technique of a time-invariant ytem for VSC bae HVDC tranmiion W Thi work wa fune by the Natural Science an Engineering Reearch Council of Canaa P.W. ehn an S. Porucky are with the Department of Electrical an Computer Engineering, Univerity of Toronto, Toronto, Ontario, Canaa, M5S 3G4 ( lehn@ecf.utoronto.ca, tephen.porucky@utoronto.ca) Paper ubmitte to the International Conference on Power Sytem Tranient (IPST9) in yoto, Japan June 3-6, 9 control i utilize in [5]. Thi paper will evelop a linearization technique applicable to VSC with -frame controller, thu making mall ignal ytem moeling poible. Valiation of the new ytem moel i carrie out by comparing the linearize mall ignal moel with large ignal imulation reult obtaine from MATAB/SIMUIN. II. TYPICA VSC CONTRO STRUCTURE The electrical iagram of a VSC bae AC to DC controller i hown in Fig.. The icuion of thi paper i with repect to thi 3-phae AC to DC converter, which may be ue a a builing block to evelop more elaborate FACTS controller, VSC bae HVDC ytem or a part of an electrical rive ytem. v G i R v t VSC Fig.. Single line iagram for erie voltage ource converter C i cap R loa Bae on time-averaging aumption, Fig. provie a large-ignal -frame moel of the converter ynamic, ueful for computationally efficient time-omain imulation. The inner -frame current control loop, given in Fig.4, contain a reonant controller for both the an -axi current. Uner unbalance operating conition, thi controller can (i) reject unbalance gri voltage itortion an (ii) track unbalance current erence, if eire. Thi eliminate not only the nee for eparate controller for poitive an negative equence component, but alo eliminate the filtering require to eparate the poitive from negative equence component, a icue in [4]. Only two block in the iagram of Fig. are not linear timeinvariant. One of thee block appear after the DC voltage controller. It i a moulator ue to create the require frame current erence. The other i the AC to DC power converion equation, which relate the converter -ie voltage an current to it ac-ie voltage an current in the frame. Embee in thi power equation i a emoulation function. In Fig.3, the emoulating function an the power equation have been eparate. The extraction of the emoulating element tranfer the -frame power converion equation into the q-frame. Neither the emoulation function, nor the power equation are linear time-invariant.

2 j t e ω p I i v t αβ 3 v tα iα v v i tβ β C j i q Fig.. Block iagram of DC voltage control moel with frame current control j t e ω j t e ω p I v tαβ v tq 3 v ti v i v tq q C j i q Fig. 3. Block iagram of DC voltage control moel with frame current control moifie v gαβ x q e jω t x αβ τ y αβ ε αβ ir ip ω Fig. 4. Inner AC current loop of Fig. v tαβ v Rαβ R To allow ue of linear analyi (e.g. eigenvalue analyi) an control eign technique (e.g. root locu, boe plot or linear tate pace control technique) the tructure of Fig. mut be linearize. Thi i achieve by manipulation of the equivalent ytem hown in Fig. 3. The proce, will create a q-frame equivalent of the -frame current controller. Unlike the original -frame moel the q-frame equivalent will: (i) contain only time-invariant block (ii) be linearizable about a nominal operating point. III. DQ-FRAME EQUIVAENT OF PHA BETA-FRAME RESONANT CURRENT CONTROER Utilizing the block iagram bae erence frame tranformation approach of [6], the e jt term can be move through the current controller. The tranformation approach epicte in Fig. 5, where the tranfer function parameter τ may, in general, be complex. ( jω )τ y q x q Fig.5. Shifting of rotating pace vector e jω t y αβ By performing thi hift through a tranfer function, each pole an zero of the tranfer function i hifte by j. The reulting hift in each -frame tranfer function can be viewe in Table I. Thee reult are in agreement with thoe of Zmoo, Holme, an Boe [7]. TABE I TRANSFORMATION OF APHA-BETA FRAME CURRENT CONTROER Frame Equivalent q-frame Tranfer Function Tranfer Function ip ip ir ω ( jω) ir ( j ω) ( ω ) ω = ir j ( 4 ω ) 4ω = C ( ) j C ( ) Re Im Uing reult from Table I, the block iagram of Fig. 6 may be contructe uing the relation: vq = ( CR ( ) jci ( ))( ε jεq ) () v = ( ε C ε C ) j( ε C ε C ) () q R q I q R I

3 where ip i the proportional current control gain, ir i the reonant current control gain, i the error of the -axi current control, q i the error of the q-axi current control. From the q-frame equivalent block iagram of Fig.6, one can note a change in tructure. While the gri voltage an the AC plant ynamic are imply replace by their q-frame equivalent, the current controller contain cro coupling tranfer function between the an q-axe. IV. SMA SIGNA MODEING AND INEARIZATION ABOUT BAANCED SOURCE OPERATING POINT Having hifte the e jω t moulator rightwar through the current loop allow the moulator/emoulator function to be cancelle. A can be oberve in Fig. 6 the reulting current loop i now in the q-frame. Combining the DC voltage control loop of Fig. 3 with the current control loop of Fig. 6 yiel a complete, large-ignal, q-frame moel, a hown in Fig. 7. Fig. an Fig. 7 are theore equivalent an may be interchangeably ue for large-ignal imulation (bae on time averaging aumption). The only non-linearity in Fig. 7 reie in the AC to DC power converion equation. Thi function can be eaily linearize uing mall-ignal converion of a non-linear ytem. Similar technique are utilize in DC to DC converter, [9]. A Taylor erie wa applie to the non-linear power equation (3) to obtain our linear moel (4). The time varying mall ignal variable in (4) are enote with an inflection above them. Capital variable tan for the teay tate operating point value of the given ytem. i 3 vti v i = v tq q 3 I [ ] i = v t I v tqiq ivt iqvtq v (4) V V where v t i the -axi VSC AC ie terminal voltage, v tq i the q-axi VSC AC ie terminal voltage, i the -axi AC line current, i q i the q-axi AC line current, i the DC link voltage of the VSC, an i the current from the VSC into the DC link. Now that each element of th-frame equivalent to the current controller i linear, controller eign technique an linear analyi technique, incluing eigenvalue analyi an ytem parameterization, can be performe. For thee technique to be eaily implemente the tate pace repreentation of the cloe loop ytem wa evelope an can be viewe in the Appenix. The mall ignal tate pace repreentation in the Appenix take on the form of (5) an (6). (3) x = Ax Bu t (5) yˆ = Cx (6) [ T x = x x x x 3 xq xq x q3 i iq v ] (7) Where x i the vector of mall ignal ytem variable, u the ytem input an y the ytem output. With repect to (7), the firt tate variable i relate to the DC voltage controller. The following two group of three are linearly inepenent tate that repreent the tate of the an q axi current controller. The next three tate are the ytem output tate. With the linearize tate pace repreentation, a time omain imulation can be eaily performe an the mall ignal moel of the control ytem can be valiate. V. MODE VAIDATION VIA TIME DOMAIN SIMUATIONS The performance of the mall ignal moel for the DC voltage controller wa tete againt time omain imulation uing MATAB/SIMUIN. The repone of the two ytem moel were compare by viewing the DC link voltage, the - axi AC line current, an the q-axi AC line current with repect to two input tep change: (i) a A (.7p.u.) tep in DC loa current (ii) a 5V (.67p.u.) tep in the DC voltage erence. The ytem parameter an teay tate operating point value can be viewe in Table II below. Parameter for a low power 4.5 kva VSC are ue to allow future laboratory valiation. TABE II PARAMETERS USED IN SIMUATION Sytem Parameter Variable/Symbol Value Gri voltage Vg 8 V ll Converter Parameter Variable/Symbol Value AC interface inuctance.5 mh AC interface reitance R.5 DC link capacitor C 7 F Converter Rating kva rating Sbae 4.5 kva AC voltage rating Vbae V ln AC current rating Ibae.5 A DC voltage rating V_rate 375 V Steay State Operating Variable/Symbol Value Conition D-axi gri voltage Vg 69.7 V Q-axi gri voltage Vgq V D-axi terminal voltage Vt 6.66 V Q-axi terminal voltage Vtq V D-axi AC line current I 5.46 A Q-axi AC line current Iq A DC link current I A DC link voltage V 375 V

4 v g ε ( ω ) ( 4 ω ) ω 4ω v t R ω v gq i q ε q ω 4ω ( ω ) ( 4 ω ) v tq ω R i q Fig.6. Dq-frame equivalent of -frame current loop of Fig.4 p I v tq 3 v ti v i v tq q C j i q Fig. 7. Dq-frame block iagram equivalent of Fig.. A. DC oa Current Step Repone The converter i firt operate a an active rectifier an a tep change in DC loa current i introuce to the ytem. At time econ the DC loa current i increae from A to A. The repone of large an mall ignal ytem are compare in Fig 8. DC ink Voltage (V) D-axi Current (A) Q-axi Current (A) DC Voltage Repone to DC oa Step Fig. 8 Repone to DC loa tep change Small Signal Moel arge Signal Moel D-axi Current Repone to DC oa Step Q-axi Current Repone to DC oa Step Time () From viewing Fig. 8 one can note very little eviation between the large an mall ignal ytem repone a a reult of a DC loa tep. B. DC ink Voltage Reference Step Repone A change in DC link voltage erence i mae to the imulate ytem at time econ, with the DC voltage erence being increae from 375 to 4 V. The repone of the large an mall ignal ytem are given in Fig. 9. From viewing Fig. 9, one can again note very little eviation between the large an mall ignal ytem repone. The imulation reult of Fig. 8 an 9 valiate the accuracy of the evelope mall ignal moel. DC ink Voltage (V) D-axi Current (A) Q-axi Current (A) DC Voltage Repone to DC Voltage Reference Step Fig. 9 Repone to DC voltage erence tep change VI. APPICATIONS Small Signal Moel arge Signal Moel D-axi Current Repone to DC Voltage Reference Step Q-axi Current Repone to DC Voltage Reference Step Time () Small ignal converter moel have many application ranging from tability analyi to control eign. In power ytem, one application i to employ the mall ignal moel to invetigate the ytem ynamic after a fault event. In contrat to large ignal imulation moel, a mall ignal moel allow

5 the ynamic to be relate back to ytem an control parameter value. In contrat, eveloper of electric rive, VSC bae HVDC ytem or win turbine ytem with back-to-back converter interface, eigner may intea be interete in the DC-ie input/output impeance of a converter, a thi impeance may be ue for invetigating DC ie interaction. Two application will be preente. The firt invetigate the mall ignal ynamic of a VSC interface energy ource, where the AC gri i ubjecte to a itant line-line fault or itant 3-phae fault. The econ application invetigate the DC-ie output impeance of the VSC when it i operate a a unity power factor active rectifier. A. Converter Dynamic ue to Ditant Gri Fault Two type of itant gri fault will be coniere, namely a 3-phae fault an a line-line fault. Gri voltage before, uring an after the fault are ummarize in the table below. DC ink Voltage (V) D-axi Current (A) Q-axi Current (A) DC Voltage Repone to Symmetrical Fault Small Signal Moel arge Signal Moel D-axi Current Repone to Symmetrical Fault Q-axi Current Repone to Symmetrical Fault Time () Fig. Small ignal converter repone to a itant 3-phae fault. TABE III OCA GRID VOTAGE VARIATION DUE TO FAUT Va Phaor Vb Phaor Vc Phaor Balance Fault Pre-fault Fault Pot-fault Unbalance Fault Pre-fault Fault Pot-fault The repone to a balance itant fault i hown in Fig.. During the 3-phae fault a ignificant ip in gri voltage introuce ome inaccuracy in the linearize moel behavior. However, when the fault i cleare the recovery ynamic of the linearize ytem are highly accurate. The repone to a line-line itant fault i hown in Fig.. Three trace are hown in Fig. ; the linearize ytem behavior of the VSC with reonant control, the large ignal behavior of the VSC with reonant control an the large ignal behavior of a claical q-frame controller. We can make three key obervation:. Depite -frame an q-frame controller being ue almot interchangeably in the literature, a large ifference exit in their repone to unbalance event.. Depite the ability of the reonant current control loop to reject any gri voltage imbalance, unbalance current till flow in the ytem, ue to control action of the DC voltage regulator. 3. The linearize moel accurately capture the ynamic aociate with the unbalance fault. B. Converter DC-Sie Output Impeance when Operate a a Unity Power Factor Active Rectifier When inter-connecting multiple converter, a one in back-to-back HVDC ytem, electric rive an cacae power upply ytem, the circle criterion i ometime ue to etermine tability of the inter-connecte ytem, []. DC ink Voltage (V) D-axi Current (A) DC Voltage Repone to Source Unbalancing D-axi Current Repone to Source Unbalancing 5 Q-axi Current (A) Q-axi Current Repone to Source Unbalancing Time () Fig. Converter repone to a itant line-line fault. Application of the circle criterion require the output (or input) impeance of each converter to be known. When operating a a unity power factor active rectifier, the tranfer function from to (ee Fig.) give the -ie output impeance of the converter. Thi may be foun from the linearize moel from: V ( ) Z( ) = = Cm ( I A) Bm ( 7) I ( ) Small Signal Moel Reonant Current Control arge Signal Moel Reonant Current Control arge Signal Moel q-frame Current Control A A where A = with ub-matrice given in the appenix, A3 A4 C m = [ ], an B m = [ -/C] T. Fig. an Fig. 3 how the magnitue an angle of the DCie output impeance. Uing ucceive time omain imulation, the mall ignal impeance i verifie at five icrete frequencie, a hown by the x marker in Fig.. The linearize reult again how excellent accuracy up to everal hunre hertz.

6 VII. CONCUSIONS A complete mall ignal moel of the VSC with αβ-frame control evelope an valiate. Since linearization of the ytem mut be carrie out aroun a inuoial operating point, the αβ-frame control an ytem moel mut be firt converte into equivalent q-frame control an ytem moel. The converion i carrie out uing a imple block iagram manipulation approach. The reulting q-frame moel i finally linearize. The reulting tate pace matrix equation i evelope parametrically o that uer may explore the effect of controller gain, parameter value an operating point on the ytem ynamic. The evelope moel ha been valiate againt time omain imulation reult. Two application of the moel, one to power ytem ynamic an the other to motor rive/hvdc ytem tability analyi give brief example of how the moel might be ue. Phae of Impeance (Degree) Magnitue of Impeance Frequency (Hz) Phae of Output DC Impeance - Fig. Magnitue an phae of output DC impeance x A A = x t A3 A 4 Magnitue of Output DC Impeance Frequency (Hz) VIII. APPENDIX I P u ip P ip 3 3 ippi ipiq V C V C C I A = A = 4ω P ip irω ir A3 = irω irω 3 3 3iR Iq 3 3 ip I irω I ω ir I irω I q V C VC V C V C VC 4ω irω R ip ip P ω A4 = ir R ip ω 3 3iR Iq 3( ) 3( Vtq ipiq ) irωi Vt ip I 3iP PI I V C V C V C V C V C yˆ = x u = v iq v g v gq i loa IX. REFERENCES T [] M. Cichowla an M.P. amierkowki, Comparion of current control technique for PWM rectifier, IEEE International Sympoium on Inutrial Electronic, Vol.4, November, pp [] D.N. Zmoo an D.G. Holme, Stationary frame current regulation of PWM inverter with zero teay-tate error, IEEE Tran. On Power Electronic, Vol 8, May 3, pp [3] J.G. Hwang, M. Winkelnkemper, an P.W. ehn, Deign of an Optimal Stationary Frame Controller for Gri Connecte AC-DC Converter, 3n Annual Conference on IEEE Inutrial Electronic 6, Nov. 6, pp [4] J.G. Hwang, an P.W. ehn, DC pace vector controller an it application to converter control, P.E.S.C. 8, June 8, pp [5] D. Jovcic,.A amont an. Xu, VSC tranmiion moel for analytical tuie, IEEE Power Engineering Society General Meeting 3, Vol. 3, July 3, pp [6] C. Sao an P.W. ehn, A block iagram approach to erence frame tranformation of converter ynamic moel, IEEE 8 th Canaian Conf. Elec. An Comp. Eng., CCECE6, May 6. [7] D.N. Zmoo, D.G. Holme, an G.H. Boe, Frequency-Domain Analyi of Three-Phae inear Current Regulator, IEEE Tran. on In. App., Vol. 37, April/March, pp [8] J.G. Hwang, P.W. ehn, an M. Winkelnkemper, Control of AC- DC-AC converter with minimize DC link capacitance uner gri itortion, 6 IEEE International Sympoium on Inutrial Electronic, Vol., July 6, pp. 7-. [9] R. W. Erickon an D. Makimovi, Funamental of Power Electronic, n e., Springer ScienceBuine Meia, C,, pp [] M.Viyaagar, Nonlinear Sytem Analyi, n e., Prentice-Hall, 993.