Power Calculation Algorithm for Single-Phase Droop-Operated Inverters Considering Nonlinear Loads

Transcription

1 Internatonal Conference on Renewable Energes and Power Qualty (ICREPQ 18) Salamanca (Span), 1 th to 3 th March, 018 eçãtuä XÇÜzç tçw céãü dâtä àç ]ÉâÜÇtÄ (RE&PQJ) ISSN X, No.16 Aprl 018 Power Calculaton Algorthm for Sngle-Phase Droop-Operated Inverters Consderng Nonlnear Loads J. El Marachet*, J. Matas **, Helena Martín, Abdullah Abusorrah** * Department of Electronc Engneerng, Unversdad Poltécnca de Cataluña (UPC), Barcelona, Span. Department of Electrc Engneerng, Unversdad Poltécnca de Cataluña (UPC), Barcelona, Span. **Renewable Energy Research group, Kng Abdulazz Unversty (KAU), Jeddah, Saud Araba. Abstract The average actve and reactve powers, P av and Q av, are crucal parameters that have to be calculated when sharng common loads between parallelzed droop-operated sngle-phase nverters. However, low-pass flters () wth very low cut-off frequency should be used to mnmze the dstorton mpact n the ampltude and frequency references proded by the droop equatons. Ths forces the control to operate at a very low dynamc velocty, degradng the stablty of the parallelzed system. For ths reason, dfferent solutons had been proposed to ncrease the droop operaton velocty n lterature, but wth the consderaton of only sharng lnear loads. The ssues derved from the sharng of nonlnear loads had not been properly consdered. Ths paper proposes a method to calculate P av and Q av usng second order generalzed ntegrators (SOGI) that ncrease the velocty of the droop control algorthm consderng nonlnear loads as the desgn worst case scenaro. Then t s employed a double SOGI (DSOGI) approach to flter the current non-snusodal waveform and prode the fundamental component, whch results n a faster transent response and mproves the system's stablty. The proposed calculaton method shows to be faster than other approaches when consderng nonlnear loads. Smulatons are proded to valdate the proposal. Keywords-component; Actve and reactve power calculaton, sngle-phase nverters, nonlnear loads, nverter parallelzaton, droop method, trade-off speed and accuracy. I. INTRODUCTION The calculaton of the averaged actve, P av, and reactve, Q av, powers s an mportant aspect n the droop based local control algorthm used to parallel sngle-phase nverters wthout ntercommuncaton, snce t has a crtcal nfluence on the transent response speed of the nverter and n the system stablty [1]-[3]. The calculaton of P av and Q av had been usually performed by the multplcatons of the nverter delvered output current, o, wth the nverter output ltage, v o, and wth ts π/ phase shfted verson, v o, for obtanng the actve and reactve nstantaneous powers, p and q, respectvely. A should be appled to acheve the averaged values of p and q and for remong the double frequency component resultng for the multplcaton of these snusodal sgnals [4], [5] and [6]-[1]. In ths operaton, v o can be obtaned by dfferent approaches such as a transport delay (TD) n [13] and [14], an etended three-phase dq SRF approach appled to sngle-phase systems n [15] and [16], and a method usng the quadrature output of a SOGI flter n [17]. In [18] a method based on SOGI for calculaton of powers and later cancellng the double frequency component smlarly to [6] had been proposed. Although the calculaton tme was reduced n one order of magntude aganst lnear loads, ths method employed also a for obtanng P av and Q av, whch slows down the transent response. Moreover, a proposal based on a dscrete Fourer transform (DFT) was presented n [19] for etractng averaged values, but t ntroduced a severe delay that makes t unsutable when abrupt changes of load occur. For ths reason, n [0] a method was presented for calculatng the actve and reactve powers through optmzng a cost functon of P and Q by means of LMS adaptve algorthm. However, appromatons acheved n the P and Q epressons are only possble n steady state and aganst lnear loads. In general, all these proposals have n common the objectve of tryng to enhance somehow the droop-operated system stablty by means of acheng a fast and accurate calculaton of the averaged powers for the droop references. However, the valdty of these approaches s only partal when sharng nonlnear loads. Ths paper proposes a modfcaton n the power evaluaton scheme proposed n [6] and [18], usng a DSOGI approach for obtanng the fundamental component of the nverter output current, of, whch s used n the power calculaton aganst a RC rectfed nonlnear load wth a 15% total harmonc dstorton (THD). The flterng capablty of the DSOGI s determned by ts dampng factor parameter, ξ, that wll be desgned for keepng the rpple n the calculated powers below a predefned desred value. The use of the DSOGI allows the remong of the from the ntal calculaton scheme and reduces the tme for obtanng the averaged powers. Ths 19 RE&PQJ, Vol.1, No.16, Aprl 018

2 DSOGI structure must be desgned wth a proper value of ξ for gng a lower settlng tme durng transents. The proposal s then compared wth the classcal droop approach and wth [6] and [18] under the assumpton of causng the same ampltude rpple at the obtaned powers for a gven RC rectfed nonlnear load. The obtaned responses show to be faster when abrupt load changes occur and determne to whch pont the speed and stablty of the system can be enhanced. In Secton II the calculaton block n a droop based local control structure s contetualzed and descrbed, for a snglephase nverter. In Secton III, an advanced method for calculatng P-Q based on [18] s shown. Secton IV proposes the novel calculaton of actve and reactve powers based on a DSOGI flter approach and also shows the smulaton results for valdatng the proposal. II. POWER CALCULATION IN SINGLE-PHASE DROOP- OPERATED INVERTERS Ths secton deals wth the power calculaton of a sngle-phase nverter when sharng a common load wth another parallelzed nverter usng the droop method. Ths secton and the rest of the paper s focused on the power calculaton dynamcs of one of the nverters n attempt to ew the problems when sharng a nonlnear load, to propose a soluton to deal wth the dstorted currents, and to see the relatonshp between the transent speed and the flterng capablty of the power calculaton, whch wll mprove the stablty of the system and the accuracy of power sharng. Fg. 1 represents a basc scheme of a sngle-phase nverter that s operated wth the droop method. In ths fgure t can be seen that the control scheme s composed by a P-Q power calculaton block, a droop method block, and nverter's control nner loops plus pulse wdth modulaton block (PWM). The P- Q block uses the nverter's output ltage and current, v o and o, to delver the averaged powers, P av and Q av, to the droop block that uses them to generate the nverter's reference ltage, v ref, to command the nverter power swtches through the nner loops plus PWM control block. Sngle-Phase Power Inverter (Inv. #1) Power Swtches PWM CTRL L vref L DROOP Method C o o v o P-Q Power CALC v o Load (Inv. #) Fg. 1. Droop-based control scheme of a sngle-phase nverter. Fg. shows the tradtonal power calculaton that obtans the averaged powers, after the multplcatons between ltages and current to produce an nstantaneous actve, p, and reactve, q, powers. Then, s are used to obtan the respectve averaged powers [17]. -π/ p q Flterng Flterng o Averaged Power products powers Fg. Block dagram of conventonal P-Q power calculaton prodng averaged powers. The droop control method determnes the operatng frequency and ampltude ltage of the nverter through the followng equatons, when lne mpedance and output nverter s mpedance are consdered to be manly nductve: = (1) = () where m and n are the droop coeffcents, ω n and V n are the nomnal frequency and ampltude and ω * and V * are the proded frequency and ampltude references. These references are used to generate the followng snusodal ltage reference for the nverter's nner control loops to follow. = ( ) (3) Assumng that the output ltage and current of the nverter are [15] ( )= ( ) (4) ( )= ( ) (5) where V and I are the ltage and current ampltudes, s the fundamental frequency and o s the phase angle between v o and o. The quadrature ltage, wth a π/ delay, s defned as ( )= sn( ) (6) So, the nstantaneous actve and reactve powers could be formulated as = ( ) ( )= ( ) = = (7) And, n a smlar way, = ( ) ( )= ( ) = = (8) where P av and Q av are the average actve and reactve powers and and are the oscllatng components at twce of the fundamental operatng frequency proded by the droop method. The s used to flter these nstantaneous powers, see Fg., should have a low cut-off frequency value, fc, n order to flter properly the double frequency components, and. Ths value s typcally of one or two order of magntude lower than the nverter's operatng frequency [1], []. The fc value fnally determnes the speed of the droop method, whch s too slow. Moreover, fc should be reduced more to be able to handle the sharng of nonlnear loads. Also, nonlnear loads may ntroduce strong dstortons n the nverter current, [3], whch are drectly conveyed to the nstantaneous powers. Therefore, calculaton of powers become much more comple and not only the double frequency power components shall be removed. In sngle phase parallelzed nverters the man concern when sharng nonlnear loads s to ad the ecessve power rpple nduced by these loads. The power rpples cause 0 RE&PQJ, Vol.1, No.16, Aprl 018

3 strong dstortng swngs n the proded droop frequency and ampltude references, ω * and V *, whch, n turn, cause dstorton n nverter's ltage reference, v ref, and prokes a bad droop operaton of the system. In fact, the value of fc should be desgned to be typcally less than 1Hz n order to ad the mpact of nonlnear loads. III. ADVANCED P-Q POWER CALCULATION METHOD A SOGI s a specal lnear flter wth one nput, v n, and two outputs, v d and v q, one n-phase and the other π/ delayed wth respect to the nput, respectvely. These outputs have the followng band-pass flter (BPF) and transfer functons relatonshps regardng to the nput n ( s) ξω s Hd ( s) = = (9) v ( s) s ξ ω s ω n v q( s) ξ Hq( s) = = (10) v ( s) s ξ ω s ω where ξ s the flter dampng factor and ω ts tunng center frequency. These two parameters determne the settlng tme of the transent response of ths flter, whch s (11) Fg. 3 shows the proposed P-Q calculaton method n [18] for acceleratng the calculaton of the actve and reactve powers. o -π/ p q SOGI1 SOGI p q Fg. 3. P-Q calculaton block scheme based on [18]. The SOGI1 and SOGI n Fg. 3 are used for etractng the pulsatng double frequency power components, and, respectvely. These SOGI are tuned both at ω o and ξ 1 =ξ =1. The s are used for a better flterng and prode the averaged powers P av.and Q av. Fg. 3 do not show the method for generatng the π/ delay snce t s not mentoned n [18]. Therefore another SOGI, SOGI0, tuned at ω o and ξ=0.707, s used for generatng ths delay as shown n Fg. 4, for adng delay ssues reported n [13], [14], [19]. ξ SOGI0 ξ o vq p q SOGI1 SOGI p q Fg. 4. P-Q calculaton block scheme of Fg. 3 usng an addtonal SOGI for generatng the -π/ delay. Fg. 5 shows the smulatons results after usng the P-Q scheme of Fg. 4 when sharng a lnear load that produces a current perturbaton from 4A to 8A at tme 1s. For sake of smplcty Fg. 5 only shows the actve power, P av. The dynamcs are compared wth the obtaned by the conventonal droop method depcted n Fg., named as P droop, usng a wth fc=1hz. The P av power of Fg. 4 s named as P adv to dfferentate t, from now on, from the other methods. The cut-off frequency for the advanced power calculator was desgned to be 10Hz. Lnear current (A) Actve Power P (W) Fg. 5. P droop and P adv transent response for a lnear load current perturbaton from 4A to 8A at 1s: up) Detal of the perturbaton; down) P droop and P adv calculated powers. As shown n Fg. 5, the P adv dynamcs remove the double frequency component and s much faster than the P droop, whch stll contans a small double frequency component. These results are compatble wth those reported n [18]. However, the perfect dynamc behaor depcted n Fg. 5 vanshes when a nonlnear load that nduces dstorton n current s shared. Fg. 6 shows the dynamcs of the system when a nonlnear load s used. In ths case, the load s a rectfer supplyng a RC load that draws a hghly dstorted current wth 4A peak and that suffers a perturbaton that pushes the peak to 8A. The smulaton parameters are shown n Table I. TABLE I. SIMULATION PARAMETERS FOR FIG. 4. V n 311V ω n π50(rad/s) R at t <1s; R at t >1s 1100Ω;37 Ω C 470µF Current THD 15% 0.7, 1 fc droop; fc adv 1Hz; 10Hz Padv (10Hz) As shown n Fg. 6, the dynamcs of the proposed method n [18] were never consdered usng a nonlnear load, smlarly to other proposals mentoned n Secton I. Thus, n the presence 1 RE&PQJ, Vol.1, No.16, Aprl 018

4 of nonlnear loads the method has ecessve steady state rpple that corrupts the calculated powers, oppostely to the stated n [18]. Fortunately, the flterng capabltes of the n Fg. 4 can be mproved by reducng fc to.hz. Therefore, the rpple of P adv s dmnshed untl the same level than the produced by the conventonal droop method. Fg. 7 shows ths stuaton and also shows how the advanced method s stll faster calculatng P adv than the conventonal droop controller. Note that both methods have been compared under the same dynamcal and dstorton-attenuaton condtons, beng better the advanced one but wth less effectty than ntally argued. Nonlnear current (A) Actve Power P (W) Actve Power P (W) Fg. 6. P droop and P adv transent response for a nonlnear rectfer-type load perturbaton n current from 4A peak to 8A peak at 1s: up) Detal of the dstorted load current perturbaton from 4A peak to 8A peak; mddle) P droop and P adv calculated powers; down) Detal of the calculated powers showng ther rpple. Actve Power P (W) Padv (10Hz) Padv (10Hz) Padv (.Hz) Actve Power P (W) Fg. 7. P droop and P adv transent response for a nonlnear rectfer-type load perturbaton from 4A peak to 8A peak at 1s: up) P droop and P adv calculated powers, for P adv wth a wth fc=.hz; down) Detal of the calculated powers showng ther rpple. Note also that there s a postve offset n the calculated actve power at steady state, snce the mean value of P adv s slghtly hgher than ths of P droop, see lower plot of Fg. 7. IV. PROPOSED P-Q DSOGI POWER CALCULATION METHOD Fg. 8 shows the proposed power calculaton method that conssts n a modfcaton of Fg. 4 to enhance the dynamcal response when sharng nonlnear loads. ξ SOGI0 ξ vq OF DSOGI SOGI3 SOGI4 o d d ξc ξ ξ OF p' q' SOGI1 SOGI p q Padv (.Hz) Fg. 8. Proposed P-Q calculaton method for dealng wth nonlnear loads and usng a DSOGI approach. When the proposed nonlnear load s connected, the output current of the nverter can be epressed as [3]: ( ) = ( ) (h ) (1) where the sub nde h represents the harmonc number, N the mamum set of harmoncs, I DC the DC component, and are the ampltudes of the fundamental and harmonc components, respectvely. The fundamental frequency s and h represents ts harmonc multples. Fnally, and are the phase-shft of the fundamental and of each harmonc component, respectvely. Then, the nstantaneous powers should be redefned as: = ( ) h h h (13) = ( ) h h h (14) Consequently (13) and (14) contan the DC, double frequency and hgher harmonc order components of the powers and the subtracton of only the double frequency component s not enough for the proper calculaton of P av and Q av. Then, t becomes necessary the flterng of the measured current, o, n order to reject ts harmoncs components preously for resultng n a smpler and faster calculaton of powers as n (7) and (8). For ths purpose, n Fg. 8, the DSOGI, formed by SOGI3 & 4, s used to flter the non-snusodal output current and to prode ts fundamental component. Then, t s obtaned the product wth the n-phase and the quadrature ltages and RE&PQJ, Vol.1, No.16, Aprl 018

5 generate nstantaneous powers wth only the double frequency components and wthout thrd or hgher order harmoncs. Later, SOGI1 and SOGI are used for only remong the double frequency components wth the help of the subtractng blocks. The resultng value s named as P DSOGI for the actve power, and Q DSOGI for the reactve one. In Fg. 8, the DSOGI BPF flterng acton s so strong that allows to remove the from the scheme, whch accelerates further the P-Q dynamc response. Also, the subharmonc components of current are rejected due to the BPF behaor of the SOGI flter. In ths case, because the DSOGI s used n the current, the transent response speed s determned by (11) and t relays manly n ξc, snce the frequency proded by the droop method, ω*, vares n a small range around the nomnal ω n. Therefore, the DSOGI dampng factor s tuned to flter the non-snusodal current to a pont n whch the produced power rpple s dentcal n ampltude as the conventonal droop controller. In ths case, ths s acheved for ξc=ξ 3 = ξ 4 =0.1. Fg. 9 shows the smulaton results, whch yeld that the proposed method s faster calculatng the actve power and acheves a lower steady state rpple. Actve Power P (W) Actve Power P (W) Padv (.Hz) P DSOGI (ch=0.1) Padv (.Hz) P (ch=0.1) DSOGI Fg. 9. Actve power transent responses for a nonlnear rectfer-type load perturbaton from 4A peak to 8A peak at 1s: up) Fundamental current of proded by the DSOGI flter; mddle) P DSOGI, Padv and Pdroop calculated powers; down) Detal of calculated powers at steady-state. Note that the calculated P DSOGI power has not postve offset error n steady state n contradstncton wth P adv. Ths means that the proposed method s also more accurate than proposed n [18]. Table II yelds the measured rse tme for the transent responses depcted n Fg. 9, whch shows that the proposed method mples a 60.00% and a 79.69% reducton n the rse tme regardng P adv and P droop, respectvely. TABLE II. RISE TIME MEASUREMENTS FROM FIG. 9. Measurements 35ms P droop P adv 165ms P DSOGI 66ms Improvements Improvement P DSOGI vs P adv 60.00% Improvement P DSOGI vs P droop 79.69% V. CONCLUSIONS In ths work a P-Q calculaton method has been proposed for sngle-phase parallelzed nverters wth the purpose of mprong the speed and accuracy of the power calculaton when they are sharng nonlnear loads. The dynamc response of the power calculaton used n the conventonal droop method and n another advanced method s analyzed frst to show ther lmtatons n speed and accuracy when sharng a RC-rectfed type nonlnear load. For ths reason a novel calculaton method has been proposed and compared wth the preous ones. The smulatons results, under the same dstortng condtons n current, show how the proposed method obtans the P-Q powers at a 76.69% faster than the proposed one for the conventonal droop controller. Also t obtans also the P-Q powers at a 60.00% faster than the advanced one based on [18]. Ths mprovement supposes an enhancement n the droop speed operaton under nonsnusodal condtons n current that may lead to a better dynamc performance of non-herarchcally controlled nverters n mcrogrds. Further works wll be carred out to determne the mprovements n load sharng dynamcs for the paralleled systems and n ts stablty. REFERENCES [1] E. A. A. Coelho, P. C. Cortzo and P. F. D. Garca, "Small sgnal stablty for sngle phase nverter connected to stff AC system," Conf. Record of the 1999 IEEE Industry Applcatons Conf. 3th IAS Annual Meetng (Cat. No.99CH36370), pp l.4, [] M. Soshnskaya, W. H. J. Graus, J. M. Guerrero, J. C. Vasquez, "Mcrogrds: eperences barrers and success factors", Renew. Sustan. Energy Rev., l. 40, pp , 014. [3] L. S. Araújo, D. I. Narváez, T. G. Squera and M. G. Vllalva, "Modfed droop control for low ltage sngle phase solated mcrogrds," 016 IEEE Int. Conf. on Automatca (ICA-ACCA), pp. 1-6, 016. [4] "IEEE Standard defntons for the measurement of electrc power quanttes under snusodal, nonsnusodal, balanced or unbalanced condtons, IEEE Standard ", Mar [5] J. Lu, Y. Wen, Yngchao Zhang and W. Wen, "A novel power calculaton method based on second order general ntegrator," 016 IEEE 8th Int. Power Electroncs and Moton Control Conf. (IPEMC- ECCE Asa), Hefe, pp , 016. [6] E. T. Andrade, P. E. M. J. Rbero, J. O. P. Pnto, C. L. Chen, J. S. La and N. Kees, "A novel power calculaton method for droop-control mcrogrd systems," 01 7 th Annual IEEE Appled Power Electroncs Conf. and Ep., (APEC), pp , 01. [7] Z. Ren, M. Gao, Q. Mo, K. Lu, W. Yao, M. Chen and Z. Qan "Power Calculaton Method Used n Wreless Parallel Inverters Under Nonlnear Load Condtons," n Proc. of APEC, pp , 010. [8] E.C. Furtado, L.A. Agurre and L.A.B. Torres, "UPS Parallel Balanced Operaton Wthout Eplct Estmaton of Reactve Power - A Smpler Scheme," IEEE Trans. on Crcuts and Systems II: Epress Brefs, l.55, no.10, pp , Oct [9] J.M. Guerrero, J. Matas, L. G. Vcuna; M. Castlla and J. Mret, "Decentralzed Control for Parallel Operaton of Dstrbuted Generaton 3 RE&PQJ, Vol.1, No.16, Aprl 018

6 Inverters Usng Resstve Output Impedance," IEEE Trans. on Industral Electroncs, l.54, no., pp , Apr [10] S. A. O. Slva, R. Nochadlo and R.A. Modesto, Sngle-phase PLL structure usng modfed p-q theory for utlty connected systems, n Proc. of IEEE PESC, pp , 008. [11] H. Wang, X. Yue, X. Pe and Y. Kang, A new method of power calculaton based on parallel nverters, n Proc. of IEEE EPE-PEMC, pp , 009. [1] Yu, D. Xu and K. Ma, A Novel Accurate Actve and Reactve Power Calculaton Method for Paralleled UPS System, n Proc. of APEC, pp , 009. [13] Zheng Ren; Mngzh Gao; Qong Mo; Kun Lu; We Yao; Mn Chen; Zhaomn Qan, "Power calculaton method used n wreless parallel nverters under nonlnear load condtons," Appled Power Electroncs Conf. and Ep., (APEC), th Annual IEEE, pp.1674,1677, 1-5 Feb [14] We Yu; Dehong Xu; Kuan Ma, "A Novel Accurate Actve and Reactve Power Calculaton Method for Paralleled UPS System," Appled Power Electroncs Conf. and Ep., 009. (APEC 009). 4 th Annual IEEE, pp.169,175, Feb. 009 [15] H. Akag, E.H. Watanabe, M. Aredes, "Instantaneous Power Theory and Applcaton to Power Condtonng," Pscataway, NJ: John Wley & Sons, Inc., 007, pp. 5-8 [16] Mngzh Gao; Shangda Yang; Cheng Jn; Zheng Ren; Mn Chen; Zhaomng Qan, "Analyss and epermental valdaton for power calculaton based on p-q theory n sngle-phase wreless-parallel nverters,"appled Power Electroncs Conference and Eposton (APEC), th Annual IEEE, pp.60,64, 6-11 March 011 [17] J. Matas, M. Castlla, L. G. d. Vcuña, J. Mret and J. C. Vasquez, "Vrtual Impedance Loop for Droop-Controlled Sngle-Phase Parallel Inverters Usng a Second-Order General-Integrator Scheme," n IEEE Trans. on Power Electroncs, l. 5, no. 1, pp , Dec [18] S. Tolan and P. Sensarma, "An mproved droop controller for parallel operaton of sngle-phase nverters usng R-C output mpedance," 01 IEEE Int. Conf. on Power Electroncs, Drves and Energy Systems (PEDES), Bengaluru, 01, pp [19] Y. Yang, F. Blaabjerg and H. Wang, "Low ltage rde-through of sngle-phase transformerless photoltac nverters," 013 IEEE Energy Converson Congress and Eposton, Denver, CO, 013, pp [0] Y. Yang and F. Blaabjerg, "A new power calculaton method for snglephase grd-connected systems," 013 IEEE Internatonal Symposum on Industral Electroncs, Tape, Tawan, 013, pp [1] J. M. Guerrero, N. Berbel, J. Matas, L. G. de Vcuna and J. Mret, "Decentralzed Control for Parallel Operaton of Dstrbuted Generaton Inverters n Mcrogrds Usng Resstve Output Impedance," IECON 006-3nd Annual Conference on IEEE Industral Electroncs, Pars, 006, pp [] H. Song and K. Nam, Dual current control scheme for PWM converter under unbalanced nput ltage condtons, IEEE Trans. Ind. Electron., l. 46, no. 5, pp , Oct [3] IEEE Standard Defntons for the Measurement of Electrc Power Quanttes Under Snusodal, Nonsnusodal, Balanced, or Unbalanced Condtons - Redlne," n IEEE Std (Reson of IEEE Std ) - Redlne, l., no., pp.1-5, March RE&PQJ, Vol.1, No.16, Aprl 018