Frequency-Adaptive Virtual Flux Estimation for Grid Synchronization under Unbalanced Conditions

Transcription

1 Frequency-Adaptie Virtual Flux Etimation or Grid Synchronization under Unbalanced Condition Jon Are Suul*, Alaro Luna**, Pedro Rodriguez**, Tore Undeland* *Department o Electric Power Engineering Norwegian Unierity o Science and Technology, NTNU Trondheim, Norway jon.are.uul@elkrat.ntnu.no Abtract-Thi paper propoe a new and explicitly requencyadaptie method or Virtual Flux etimation and oltage enorle grid ynchronization under unbalanced condition. The propoed ytem i baed on uing Second Order Generalized Integrator, arranged to imultaneouly ulill the purpoe o requency-adaptie band-pa ilterin integration and quadrature ignal generation. Thi reult in a imple and eicient tructure or combined Virtual Flux etimation and eparation into poitie and negatie equence component. The propertie o the propoed Virtual Flux model i analyzed theoretically, irt a an integrator or implementing generic Virtual Flux etimation, and then with repect to equence eparation. The dynamic perormance o the propoed etimation method i teted by imulation or the cae o an unbalanced oltage drop in the grid and or a tep in grid requency. The imulation eriy the perormance to be a expected, with imilar dynamic a ynchronization baed on oltage meaurement. I. INTRODUCTION A the ue o power electronic conerter in power ytem i increain the three-phae Voltage Source Conerter (VSC) i emerging a the main topology or a wide range o application []. A large ariety o trategie or control and grid ynchronization o VSC ha thereore been uggeted and analyzed in the cientiic literature []-[4]. Operation o grid connected VSC without AC-oltage enor ha been conidered intereting or cot reduction, modularity and poible improement o reliability [5], [6]. Another intereting apect with oltage-enor-le control algorithm i that etimation method ued to replace oltage meaurement can alo be ued to etimate oltage that are not eaily aailable or real-time meaurement [7]. Voltage enor-le operation ha been inetigated and implemented on bai o eeral dierent approache or grid ynchronization. The concept o Virtual Flux, interpreting a oltage integral a a grid lux, i a common and eaily applicable method that ha been utilized or ynchronization o both traditional and enor-le control ytem [8]-[]. The introduction o the term Virtual Flux and the irt thorough analyi o application to oltage-enor-le operation o VSC wa howeer preented by Malinowki in The work o UPC in thi paper wa partially upported by the project ENE8-684-C-/ALT inanced by the Spanih Science and Innoation Minitry. **Department o Electrical Engineerin Technical Unierity o Catalonia, UPC Terraa, Spain luna@ee.upc.edu []- [4]. One o the main adantage o uing thi approach or enor-le operation i that the alue o lux or oltage behind an inductance can be eaily etimated without depending on dierentiation o the current. To aoid problem with drit and aturation o the Virtual Flux integral, many propoed implementation are baed on imple iltering tructure, although more adanced adaptie method or correction o pure integrator hae alo been uggeted [], [5], [6]. Virtual Flux etimation baed on imple iltering trategie will howeer be enitie to grid requency ariation. Een i the concept o Virtual Flux i becoming well etablihed or control o VSC, thi i till a relatiely new topic in the literature. Thereore, only a limited number o tudie hae until now conidered operation o Virtual Fluxbaed control ytem under unbalanced grid condition [6]- []. Among them, only [6] ha until now preented a generic Virtual Flux model deigned peciically or eparating poitie and negatie equence lux component in the tationary reerence rame. Conidering the drawback o exiting Virtual Flux model with repect to requency ariation and operation under unbalanced condition, thi paper propoe a new method or Virtual Flux etimation that i inherently capable o handling thee problem. The uggeted method i baed on utilizing the Second Order Generalized Integrator (SOGI) rom [], [3] a a combined requency-adaptie band-pa ilter, integrator and quadrature-ignal generator. It will be hown in the paper how the reult will be a general purpoe requency adaptie poitie and negatie equence Virtual Flux model that can be ued together with any control trategy or the conerter. Veriication and illutration o the propoed concept i proided by time-domain imulation. II. VOLTAGE SENSOR-LESS GRID SYNCHRONIZATION BASED ON VIRTUAL FLUX The lux Ψ calculated a the integral o the oltage V a gien by () i the baic tarting point o the Virtual Flux concept. For oltage-enor-le operation, thi oltage mut be calculated rom the witching tate and the DC-link oltage o the conerter []-[4], [6], [4]. Vdt () //$6. IEEE 48

2 Vg R g L g PCC Z T V R Filter inductor A. Ideal Virtual Flux Etimation or Grid Synchronization An ideal model or Virtual Flux etimation in the circuit o Fig. i hown in Fig.. The igure how the principle o the Virtual Flux etimation, a the reitie oltage drop (R i c ) i ubtracted rom the output oltage o the conerter, while the inductie lux drop (L i c ) i ubtracted rom the integrated oltage. To improe the Virtual Flux etimation, the conerter oltage can alo be compenated or the deadtime o the conerter and the oltage drop o the emiconductor deice. The reulting calculation o Virtual Flux at the grid ide o the ilter inductor i then gien by () [6]., re, VDC Vcorr, R dtl I () c, Since the Virtual Flux i the integral o the oltage, it intantaneou phae angle γ i lagging the oltage by 9º or undamental requency ignal. Thereore, the oltage phae angle θ can be eaily etimated a gien by (3). Since the integration o the Virtual Flux etimation ha a iltering eect, the intantaneou phae angle or ynchronization can uually be calculated directly rom (3), but it can alo be tracked by Phae Locked Loop [5]., arctan 9 (3), B. State-o-the-Art or Virtual Flux Etimation The implet and mot common approach or implementing the oltage integral o the Virtual Flux etimation, i to ue a irt order low-pa ilter with a crooer requency in the range around one decade below the undamental requency o the ytem [5], [6]. Thi will howeer reult in igniicant amplitude attenuation and inaccurate phae etimation. Bandpa ilter with more reedom to hape the requency repone hae alo been applied [], [5]. Another imple method or Virtual Flux calculation wa introduced by [6] and [6], baed on two cacaded low-pa ilter with crooer requency equal to the undamental requency o the grid. With thi implementation, a relatiely at tranient repone i achieed and the amplitude and phae characteritic are analytically correponding to unity gain and 9º phae hit at the undamental requency. V DC, abc re, PWM L I cab, Vc Fig.. Baic ytem under conideration Vc Vc, R R L Production/Load Reitance Virtual Flux Model Compenation Fig.. Baic concept o an ideal model or Virtual Flux etimation dt dt L C DC,, Depite o drawback with repect to accuracy and enitiity to grid requency ariation, imple implementation till make ilter-baed trategie or Virtual Flux etimation attractie compared to more adanced method baed on compenated or adaptie integration tructure like the trategy dicued in [5]. C. Modiied Per Unit Deinition o Virtual Flux Control ytem are oten analyzed and implemented in per unit alue, due to the beneit o calability and imple interpretation. Conidering the Virtual Flux integral in (), tranormation into per unit alue can be obtained by diiding by the bae alue o the lux, a gien by (4), where V b i the peak alue o the rated phae oltage and ω b i the nominal angular requency o the ytem. The bae alue or the DC-link oltage i elected to be two time V b. Neglecting the correction term, the per unit Virtual Flux model can then be expreed by (5). Vb b (4) b, b re, DC r dtl i (5) c, For grid connected conerter, there i not necearily a trong relation between the oltage and the requency a or electrical machine, ince their control i uually independent. At the ame time, ilter-baed trategie or Virtual Flux etimation are not explicitly preering the amplitude inormation, and are uually caled to obtain unity amplitude at rated requency. I deiation in requency are conidered, it can thereore be releant to deelop a Virtual Flux model that preere the ame per unit amplitude a the oltage intead o expreing the magnetic lux. Such a modiied Virtual Flux deinition i here labeled by the letter χ a deined in (6). The reulting caled Virtual Flux model can be eaily obtained by multiplying the phyical lux model with the per unit requency o the ytem a gien by (7). pub dt (6), pub re, DC r dtl pu i (7) c, D. New Method For Virtual Flux Etimation The Virtual Flux etimation uggeted in thi paper i baed on the Second Order Generalized Integrator (SOGI) dicued in [], [3]. The SOGI work a an explicitly requency-adaptie integrator or inuoidal ignal, and can be conigured a a requency-adaptie band-pa ilter by uing a eedback rom the output ignal. The reulting tructure i hown in Fig. 3 or a generic oltage, where it i clearly een that an etimate ω o the undamental requency i an explicit input to the ilter tructure. The traner unction o the ilter rom the input ariable to the iltered ariable i gien by (8), and it can be een that the requency repone can be haped by electing the gain k. In [], it i hown that a alue o k equal to reult in a good compromie between oerhoot and tabilization time. 48

3 k k Conidering the block diagram o Fig. 3 it i clearly een that the output q i the integral o the iltered output, multiplied with the reonance requency ω a gien by (9). From the preiou dicuion o Virtual Flux model, it hould be clear that the SOGI-baed ilter will be uitable or implementing a imple and explicitly requency adaptie Virtual Flux model that i caled to the ame amplitude a the grid oltage. For an ideal cae without reitance, the etimation o the lux at the grid ide o the ilter terminal i gien by (), while () gie the expreion i the per unit alue o the real lux i deired a an output. q dt (9) c dtpu l i () c c dtl i () c pu The traner unction rom the input ignal to the etimated Virtual Flux i eaily ound a a econd-order lowpa ilter gien by (). The total per unit Virtual Flux model or etimating the caled lux at the grid ide o the ilter inductor L i then gien in the Laplace-domain by (3). k q () k k,,, r i l i, k c L c (3) pu c It hould be noted that the grid angular requency ω that mut be proided to the propoed Virtual Flux model can be tracked by a traditional Phae Locked Loop (PLL), or by a Frequency Locked Loop (FLL) operating on the internal error ignal ε and the output ignal o the SOGI a dicued in [], [3]. I can urther be remarked that although Virtual Flux-baed control ytem are uually implemented or three-phae conerter, the propoed Virtual Flux model i independent o the application and could alo be ued or ingle-phae ytem. III. k (8) POSITIVE AND NEGATIVE SEQUENCE SEPARATION IN STATIONARY FRAME FOR UNBALANCED CONDITIONS For operation o conerter under unbalanced condition, detection o the poitie equence component o oltage or lux will be neceary. In [7]-[] thi ha been obtained by dierent iltering technique, or by uing a Double Reerence Frame PLL, applied on the etimated Virtual Flux SOGI q Fig. 3. Explicitly requency adaptie econd order band-pa ilter baed on a SOGI with the integral o the iltered alue a econdary output repreenting the Virtual Flux ariable. In [6] it wa howeer hown how the equence eparation can be implemented in the tationary reerence rame beore carrying out the Virtual Flux etimation. The new approach or Virtual Flux etimation propoed in thi paper will alo utilize equence eparation in the tationary reerence rame, baed on the concept o Symmetrical Component in the time-domain. The equence eparation algorithm i implemented in the two-phae tationary αβ-reerence rame by introducing a phae-hit operator q that repreent a 9º phae lag. Applying thi trategy to the generic ariable x, the calculation o poitie and negatie equence component are gien by (4) [3], [7]. x x q x q x (4) x q x x q x Implementation o the q-operator by uing the SOGI a a requency-adaptie quadrature-ignal generator () ha been thoroughly analyzed or oltage baed ynchronization method in [], [3] and will alo ere a bai or the urther inetigation in thi paper. IV. FREQUENCY ADAPTIVE VIRTUAL FLUX MODEL FOR UNBALANCED CONDITIONS Conidering the preented requency-adaptie characteritic o the SOGI-baed Virtual Flux etimation and quadrature ignal generation, it i poible to ugget a new approach where requency-adaptie ilterin Virtual Flux integration and equence eparation i merged into one operation. The bai or the propoed model will be the dicuion in ection II.D, conidering the tructure rom Fig. 3 a general building block labeled a a. Thi building block i proiding two iltered output ariable where one i the in-phae iltered image o the input ignal while the econd output i the caled integral o the irt output, i.e., the in-quradrature erion o the input ignal. A. Structure o the New Model or Virtual Flux Etimation The tarting point o the propoed Virtual Flux model i to conider the output q rom the a a caled Virtual Flux,, a deined by (6). The econd et o ignal needed or the equence eparation, which mut be phaehited by 9º with repect to the Virtual Flux ignal, are proided by inerting the ign o the in-phae output ignal. In thi way, poitie and negatie equence component o the caled Virtual Flux can be ound by (4), utilizing a minimum number o integrator or ilter. A Virtual Flux model baed on thee imple conideration i hown in Fig. 4. A een, both the Virtual Flux integral and the equence eparation are baed on the ame or each axi. When the poitie and negatie equence component o the Virtual Flux at the conerter terminal i etimated in thi way, the current induced luxe hae to be ubtracted 48

4 DC re, PWM, c, re, PWM, c, Reitance Compenation r tot r tot FLL q e e q e p q e q q e q e q q SOGI-baed Poitie and Negatie Sequence Virtual Flux model eparately or the poitie and negatie equence. Thereore, equence eparation o the current meaurement i needed a well, a hown in the igure. Howeer, till only 4 ilter are needed or the implementation, and there are no ilter connected in cacade in the lux etimation. Thereore, thi model i expected to hae ater dynamic repone than the trategy propoed in [6]. It can be een rom Fig. 4 that the preented model ha the requency a an explicit input or each operation. In thi cae, the FLL rom [8] i ued to track the ytem requency. From the point o the Virtual Flux model, the ource o the requency inormation i howeer not important, and a PLL could alo be ued or the ame purpoe. By uing or the implementation o the Virtual Flux etimation, the requency caled Virtual Flux deinition preented in ection II.C and equation (7) hould be applied, or the equence eparated lux ignal hae to be diided by the per unit grid requency a gien by (). In Fig. 4, the caled Virtual Flux model i ued, and it can be een that the current meaurement are multiplied with the per unit requency beore calculating the grid lux at the point o ynchronization. Since the lux etimation and the equence eparation i baed on utilizing the two in a imilar way a preented or equence eparation o meaured oltage in [], [3] the propoed approach can be labeled a a Dual SOGI-baed Virtual Flux (DSOGI-VF) model. B. Propertie o the Propoed Model The propertie o the a a Virtual Flux integrator hae already been dicued in ection II.D. It i howeer important to characterize the perormance o the propoed approach or eparating poitie and negatie equence component. Thi can be inetigated by tudying the traner unction rom the oltage reerence input to the Sequence eparation o Current Fig. 4. Frequency Adaptie, Dual SOGI-baed Virtual Flux model with Sequence Separation o luxe and current b X X equence eparated lux component. In balanced condition, the α- and β-component o the oltage hae equal amplitude and the β-component i lagging the α-component by 9º in time. For the teady tate requency repone, thi phae diplacement can be decribed a gien by (5) [3]. (5) re, re, From Fig. 4, the poitie equence lux component can be decribed by (6). Subtituting the expreion rom (5) into thi equation reult in (7), that can be urther expanded, by uing equation (8), into the ull expreion gien by (8). q (6) (7) k k (8) re, k k Analyzing only the teady tate requency repone, the reulting traner unction rom the α-axi oltage reerence input to the α-component o the poitie equence Virtual Flux i gien by (9). j k (9) re, j jk The requency repone o thi traner unction can alo be inetigated or negatie equence requency component by uing negatie alue or the requency ω when plotting the requency repone. A Bode-diagram howing the requency repone or both poitie and negatie requency component i gien in Fig. 5. Thi igure alo how the requency repone o the equence eparation trategy or oltage rom [3]. A can be een rom the igure, the requency repone o the amplitude i exactly equal or the propoed DSOGI-VF model and or equence eparation o meaured grid oltage. The dierence can howeer be een in the phae-repone where the poitie and negatie equence Virtual Flux component a expected are lagging the correponding oltage component by 9º. Amplitude [db] Phae angle [deg] Voltage equence eparation - Poitie equence Voltage equence eparation - Negatie equence Flux equence eparation - Poitie equence Flux equence eparation - Negatie equence Frequency [rad/] Fig. 5. Frequency repone o trategy or eparating poitie and negatie equence component o oltage and Virtual Flux 483

5 Conidering the reult in Fig. 5, it i clear that the propoed method will ulill it purpoe o achieing both Virtual Flux etimation and equence eparation in teady tate a long a the reonance requency ω i kept equal to the undamental requency o the grid. V. INVESTIGATION OF PROPOSED CONCEPT BY SIMULATION STUDIES To inetigate the dynamic perormance o the propoed Virtual Flux model, a imulation tudy i carried out with the PSCAD/EMTDC imulation otware. The imulation i baed on an intantaneou aerage model, neglecting the witching operation, o a conerter operating in a imple ytem imilar to the one hown in Fig.. A. Simulated Sytem and Control Strategy The imulation model i baed on a ector oriented current control tructure with Dual Frame PI-current controller implemented in the poitie and negatie equence reerence rame a dicued in [9]. The phae angle or ynchronization and tranormation into the poitie and negatie equence ynchronouly rotating reerence rame are obtained by the propoed Virtual Flux model rom Fig. 4 in the tationary reerence rame. The Dual Frame PI-controller are ued ince thi tructure i well etablihed and i inherently requency adaptie i the real per unit requency o the ytem i ued to calculate the decoupling term o the current controller. The oltage eedorward term o the current controller are not ued a dicued in [6], although there are poible way to ubtitute alue obtained rom oltage meaurement by alue deried rom the Virtual Flux model [3]. For implicity, only the poitie equence d-axi current reerence i ued to control the actie power balance o the conerter while all the other current reerence are et to zero. Thi imple approach reult in balanced inuoidal current, alo under unbalanced grid oltage. The DC-link oltage i iltered by a requency adaptie notch-ilter, baed on the ame requency adaptie tructure a preented or the Virtual Flux model. Thi remoe econd harmonic ocillation during unbalanced condition, o only the aerage DC-link oltage i ued to control the power low o the conerter by a PI-controller giing the poitie equence d-axi current reerence. B. Operating Condition and Simulated Cae For all imulation, a contant actie power o about.5 pu i being ed to the DC-link capacitance o the conerter. The conerter i connected to the grid through a ilter inductor with 5% inductance and a tranormer with a leakage inductance o 7 % and conduction loe o.5%. Under the initial operating condition, the grid oltage at the high oltage ide o the tranormer i contant with. pu poitie equence component. A negatie equence component o. pu i alo added to the oltage, to eriy the capability or tracking low alue o unbalance. For thee imulation, the Virtual Flux model i ynchronized to the high oltage ide o the tranormer a dicued in [7]. The oltage at the ame point i controlled directly by an ideal, controllable oltage ource that can be ued to impoe diturbance to the ytem. By thi imple approach, the etimated ariable can be directly compared to alue that are explicitly deined in the imulation model.. Tranient Repone to Unbalanced Voltage Sag To tet the tranient repone o the propoed trategy or Virtual Flux etimation and equence eparation, an unbalanced grid ault i impoed to the ytem at the high oltage ide o the tranormer. When the ault occur, the poitie equence oltage i tepped down to.733 pu and the phae angle i hited by 5º. At the ame time the negatie equence component o the oltage i et to. pu while the phae angle i hited by 5.4º. Thi correpond to the inluence rom a ditant ingle phae ault in the grid, and i the ame ault a applied in [3]. The main reult o the imulation are hown in Fig. 6, where the ault i applied at t=.4 while the ytem i V re,, [pu] g +,, [pu] ḡ,, [pu] +, - [pu] ,, [rad] -,, [rad] Time [] Fig. 6. Simulation reult howing the repone o the propoed Virtual Flux etimation when an unbalanced oltage ag occur 484

6 operating under teady tate condition. From the igure, it can be een how the oltage reerence or generation o PWM gate ignal to the conerter are becoming unbalanced when the ault occur and the control ytem i continuing to inject balanced current into the grid. The unbalanced oltage reerence are then integrated into Virtual Flux ignal and eparated into poitie and negatie equence component by the propoed etimation trategy. The amplitude o the poitie and negatie equence lux component are calculated and compared to the amplitude o the grid oltage impoed to the ytem. A can be een in the ourth plot in the igure, the detection i relatiely at and accurate with about 5 m rie-time and ery little oerhoot. The phae angle o the poitie and negatie equence lux component are hown in the two lowet plot o the igure, phae hited by 9º to correpond to the phae angle o the oltage component. The phae angle rom the oltage meaurement are hown with thin black line in the igure, and it can be een that the etimated alue are correponding ery well. Thi eriie that the propoed Virtual Flux model with equence eparation perorm a expected and doe not introduce igniicant additional delay in the grid ynchronization compared to trategie baed on equence eparation o meaured oltage.. Tranient Repone to Change in Frequency To demontrate the requency adaptiity o the propoed model by an extreme cae, a tep in requency rom 5Hz to 6Hz i impoed during the unbalanced condition. Although not a realitic cae, thi ituation i ued to demontrate how the propoed method i capable o reponding to requency change in the grid and maintain it perormance. The change in requency i applied ater the ytem ha reached teady tate in the unbalanced condition, and the main reult are hown in Fig. 7, where the time-axi i hited o the requency tep occur at t=.4. Since the change o requency i not introducing any change o oltage amplitude or power low, the inluence on the ytem i not well noticed in the plot o the αβcomponent. Howeer, in can be noticed that there i a diturbance in the amplitude o the etimated poitie and negatie equence component. The etimated phae angle are alo inluenced, and the relatie inluence i larger in the negatie equence component ince the amplitude i maller. The grid requency etimated rom the Virtual Flux model i hown in the lat plot o Fig. 7, and here it can be een that the Frequency Locked Loop ued in thee imulation i tracking the requency within a time-period in the range o m. When the requency i correctly tracked by the FLL, o that the propoed Virtual Flux model i proided with the correct alue o the grid requency, the ytem reache a new teady tate condition where the diturbance in the etimation are eliminated. Thi eriie how the propoed ytem i inherently requency adaptie. For realitic requency ariation in the grid, the repone time o the V re,, [pu] g +,, [pu] ḡ,, [pu] +, - [pu] ,, [rad] -,, [rad] [Hz] Time [] Fig. 7. Simulation reult howing the repone o the propoed Virtual Flux etimation when a large tep in grid requency occur requency tracking i expected to hae negligible inluence on the perormance. VI. CONCLUSION Thi paper ha preented a new method or Virtual Flux etimation and oltage-enor-le ynchronization during unbalanced condition. It ha irt been hown how a Second Order Generalized Integrator (SOGI) can be ued a a bai or inherently requency-adaptie Virtual Flux integration. Further on, thi paper ha preented a new coniguration o a Dual SOGI-baed Virtual Flux (DSOGI-VF) model where the unction o band-pa ilterin Virtual Flux integration and equence eparation are merged together in a imple implementation baed on the SOGI a a baic building block. The reulting tructure or Virtual Flux etimation oercome the drawback o traditional Virtual Flux model with repect to requency ariation and operation under unbalanced condition. The perormance ha been inetigated by timedomain imulation, eriying that expected reult are obtained with repect to teady-tate and tranient perormance. 485

7 ACKNOWLEDGMENT Proeor Marta Molina at NTNU i acknowledged or initiating and acilitating the cooperation between NTNU and UPC. Ignacio Candela and Juan Rocabert at UPC are alo acknowledged or proiding equipment and acilitating the work with the preented concept during a one month reearch tay o J. A. Suul in Terraa in March. REFERENCES [] S. Chakraborty, B. Kramer, B. Kropoki, A reiew o power electronic interace or ditributed energy ytem toward achieing low-cot modular deign, in Renewable and Sutainable Energy Reiew, Vol. 3, No. 9, December 9, pp [] F. Blaabjer R. Teodorecu, M. Lierre, A. V. Timbu, Oeriew o Control and Grid Synchronization or Ditributed Power Generation Sytem, in IEEE Tranaction on Indutrial Electronic, Vol. 53, No. 5, Oct. 6, pp [3] M. P. Kazmierkowki, L. Maleani, Current Control Technique or Three-Phae Voltage Source PWM Conerter: A Surey, in IEEE Tranaction on Indutrial Electronic, Vol. 45, No. 5, October 998, pp [4] M. Malinowki, M. P. Kazmierkowki, A. 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