Current control with asymmetrical regular sampled pulse width modulator applied in parallel active filter

Transcription

1 BLLEIN OF HE POLISH ACADEMY OF SCIENCES ECHNICAL SCIENCES, Vol. 64, No. 2, 2016 DOI: /bpasts Current control wth asymmetrcal regular sampled pulse wdth modulator appled n parallel actve flter. PŁAEK 1 and. OSYPIŃSKI 2 1 Insttute of Control and Industral Electroncs, Warsaw nversty of echnology, 75 Koszykowa St., Warsaw, Poland 2 Medcom Company, 78A Jutrzenk St., Warsaw, Poland Abstract. hs paper presents an analyss of the propertes of pulse wdth modulator wth a sngle (symmetrcal regular sampled PWM) and double (asymmetrcal regular sampled PWM) control samplng of the nput sgnal (low-frequency control nput wave) n presence of a trangular auxlary sgnal. In ths paper, a comparson of the characterstcs of these modulators used n the control system wth a lnear proportonal controller s presented. he artcle provdes the relatons derved for the maxmum amplfcaton of regulators for whch the control system operates stably. Analyss results have been confrmed by smulaton and expermental studes of a commercal actve flter nstalled n an ndustral plant. Key words: Actve Power Flter, asymmetrcal regular sampled PWM, current control. 1. Introducton he use of P-type current regulators n the Actve Power Flter (APF) not requrng ntensve calculatons, s partcularly attractve n systems contanng non-statonary loads. Other solutons, such as predctve current regulators [1], requre much more tme for calculaton, whch may be an obstacle to achevng good dynamc propertes. he repettve control [2 4] s useful f the reference sgnal s of repettve nature. In APF systems for fast Current Control, dfferent forms of hysteress controllers are appled. he most popular soluton s a modulator wth constant hysteress value. hs soluton s characterzed by a varable frequency of the current rpple n the nductance at the nverter output of APF. Solutons wth varable hysteress wdth [5] allow for achevng a constant frequency, however, they are stll not synchronous systems. hs makes t dffcult to optmze parameters of the LCL rpple flter. he use of proportonal controllers allows for obtanng possbly low-order characterstc equatons descrbng the current control system, whch s partcularly useful from the vewpont of preservng the stable operaton of the power system wth APF reactve power compensatng capactve load. In addton to the constant operatng frequency of the transstors, a PWM modulator [6] should ensure good dynamc and statc parameters of the closed loop APF output current control system. Paper [7] dscusses the control system wth a proportonal controller wth a samplng frequency, whch s double and quadruple of the swtchng frequency. Snce the APF requres 6 modulators, the proposed modulator s too complex to apply. he artcle ncludes a smulaton study of the control system wth symmetrcal regular sampled (SRS) pulse wdth modulator (PWM) wth sngle samplng frequency and wth asymmetrcal regular sampled (ARS) PWM wth double samplng frequency. e-mal: tadeusz.platek@ee.pw.edu.pl Smulaton studes show an ncrease of the crtcal proportonal controller gan wth symmetrcal regular sampled PWM wth a doubled number of samples n the perod of the trangular auxlary sgnal for whch the control system s stable. Also, the analyss ncluded n ths artcle demonstrates the possblty of ncreasng the value of the gan of the proportonal controller n a closed system wth asymmetrcal regular sampled PWM. Publcatons [8 14] devoted to applcatons n power electroncs systems wth the asymmetrcal regular sampled PWM are not related to control systems wth a proportonal controller. From the pont of vew of the topcs covered n ths artcle, the most nterestng paper s publcaton [15]. hs paper addresses the dgtal control system wth asymmetrcal regular sampled PWM wth dfferent types of controllers, but the ssues crtcal to the gan of the proportonal controller addressed n the present artcle are not dscussed. One of the objectves of the work presented s to compare the dynamc and statc propertes of the control system wth a proportonal controller and symmetrcal regular sampled or asymmetrcal regular sampled PWM. Carred out n ths artcle s an analyss takng nto account the tme between the nstant of measurement of the controlled value and the actual determnaton of the magntude of the sampled reference (PWM computaton delay [15]). Control system wth proportonal controller has non-zero value of dsturbance error. From the pont of vew of mnmzng the dsturbance errors and the reducng the tme of decay of transent component of ths error advantages of asymmetrcal regular sampled PWM compared to the symmetrcal regular sampled PWM manfest themselves only under the condton of an approprately small value of the PWM computaton delay. he artcle also descrbes a mechansm allowng for elmnaton of the mpact of ths error on the output current of the APF. hs s a feature of APF wth supervsory system of lnk voltage whch allows the use of output current proportonal controller. Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated 287

2 . Płatek and. Osypńsk 2. Descrpton of the nput crcut of actve power flter Fg. 1 shows a schematc dagram of the analyzed medum-voltage power supply system [16]. Controlled twelve-pulse converter s powered va a transformer r2 wth a vector group of Dd0y5 connectons. APF contanng two voltage nverters and medum voltage transformer r1 wth a vector group of Dy5 compensates harmonc current drawn by the non-lnear load n the form of h12 converter. he chokes at the output of the nverter wth the C capactors and the leakage nductance of the transformer r1 form LCL flter type [17, 18] n each phase. he APF has two two-level nverters (1_1 6_2) [19, 20] wth FF600R12IS4F hybrd modules. 3. Descrpton of control system he control system of one phase APF wth a voltage regulator n the crcut shown n Fg. 2 mplements the algorthm descrbed n the lterature [13, 21, 22]. Supervsory u voltage regulator stablzes the set value, proportonal to u. Sgnals u 1, u 2, u 3 are n phase wth the phase voltages u c1, u c2, u c3, whch cause the controller output sgnal u n to determne the ampltude and phase (0 or π rad) of the actve component of the nverter output currents. Such a control system provdes dschargng or chargng of C capactor n transent states that are related to the change of actve power drawn by the load. he control system of the flter uses the measurement sgnals of the load currents to determne the waveforms ( 1, 2, 3 ) that provde the set-pont values of the nverter output currents. Set-pont sgnal of current 1 contans three components: the 1,p() s a component of the set-pont sgnal of current n phase wth the voltage u c1, controllng the voltage u. he 1,h component contans the sum of selected harmoncs of load currents, the current 1,,f comprses the fundamental frequency component. Snce the APF nverters are connected to the grd through r1 transformer wth Dy5 connecton group, set-pont sgnals ( 1, 2, 3 ) should depend on the dfference of currents L1 L2, L2 L3, L3 L1 respectvely, whch ndrectly results from the formulas that relate the currents on both sdes of the transformer: ( )/ 3 1 = õz l 1 l2 (1) ( )/ 3 2 = õz l2 l3 (2) ( )/ 3 3 = õz l3 l1, (3) Fg. 1. he dagram of the analyzed crcut wth actve flter he vectoral varables: u s, u, s, L, l, c, u c, _1, _2,, u_ o1, u_ o2 are defned as follows: u s =[u s1, u s2, u s3 ] vector of the mans source voltages u =[u 1, u 2, u 3 ] vector of the phase voltages s =[ s1, s2, s3 ] vector of the lne currents L =[ L1, L2, L3 ] vector of the ac load currents l =[ l1, l2, l3 ] vector of the APF output current c =[ c1, c2, c3 ] vector of the C capactor currents u c =[u c1, u c2, u c3 ] vector of the C capactors voltages _1 =[ 1_1, 2_1, 3_1 ] vector of the INV1 output currents _2 =[ 1_2, 2_2, 3_2 ] vector of the INV2 output currents =[ 1, 2, 3 ] vector of the sum nverters currents u o_1 =[u o1_1, u o2_1, u o3_1 ] vector of the INV1 output voltages u o_2 =[u o1_2, u o2_2, u o3_2 ] vector of the INV2 output voltages. wheren υ z means transformer turns rato. Above dependences have been determned wthout takng nto account the magnetzng current of the transformer and C capactors currents. Values of k L and k coeffcents are the constants of current transducers. he purpose of band-pass flters and phase shfters, used n the load current measurement crcut, s to compensate for phase shfts of ndvdual harmonc ntroduced both by the load current measurng transducers and r1 transformer [21], [23 25]. he prncple of compensaton of ndvdual L,n harmoncs n load current nvolves forcng such APF output current, so the vector of ts nstantaneous values l,n satsfes the relatonshp: =, (4) l,n L,n where n s the harmonc order (n > 1). hs equalty s satsfed also for the ndvdual components of the vectors and thus also for ther ampltudes, whch wll be marked as I Lm,n and I lm,n omttng ndces 1, 2, 3: I = I. (5) Lm, n lm,n Snce the load s three-phase balanced wthout neutral, the ampltude dfference of the currents L1,n L2,n s 3 tmes the ampltude of the harmonc n any phase. For the summng nodes 288 Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated

3 Current control wth asymmetrcal regular sampled pulse wdth modulator appled n parallel actve flter Fg. 2. Equvalent dagram for actve flter control system for one phase APF at the nputs of proportonal controllers we can wrte equalty, whose fulfllment ensures compensaton of the n order harmonc of the load current: 3 =, (6) I Lm, nklku f kref 0. 5kIm, n wheren k uf denotes the gan of the band-pass flter, I m,n s the ampltude of the n order harmonc of current and k ref s the scale factor. he selecton of the k ref coeffcent allows for such condtonng of measurement sgnal level of the load current that leads to the rght compensaton of selected harmoncs. From equatons (1) and (3) the followng relatonshp prevals: I 3I m,n lm,n =. (7) õz After substtutng (5) and (7) n (6), we obtan the relatonshp based on whch one can determne the value of k ref assurng compensaton of ndvdual harmoncs: k k õ z ref =. (8) 6k Lku f Fg. 3. Dagram of the output current control system of voltage nverter wth nductve load smulaton tests were performed. Research was carred out for L = 80 μh, the frequency of the trangular carrer f c = 15 khz (f c = 1/ c ), k = 1, = 720 V, and for the maxmum value of the trangular carrer = 5.5 V. he varables expressed n the p.u. system shown n Fgs. 5, 6, 8 11 are defned by the followng formulas: ū c = u c /, ū L = u L /, ū = u /, ū r = u r /, ī = /I b, ū = u /I b. Base current I b defned by relaton b ( L f ) I = / 4 (9) s equal to the maxmum value of the nverter (1,2) output current shown n Fg. 3 for u = 0, u c = 0. c o mplement the chosen flter control we have used a popular MS320F28335 DSP appled n power electroncs [26]. 4. Output current control system of voltage nverter wth nductve load Comparatve analyss of the three types of modulators was performed based on the systems shown n Fgs. 3 and 4. Modulators shown n Fgs. 4b and 4c contan two systems of S & H. Pulses trg1 (error samplng) are ahead of pulses trg2 (PWM updatng) by τ w tme. he mnmum tme τ w s assocated wth the processng tme of the A/D and the tme requred for the error calculaton and ts multplcaton by the constant. o examne the propertes of the modulators shown n Fg. 4 from the pont of vew of maxmum gan,cr of the proportonal controller whch ensures stable operaton of the system, Fg. 4. Dagrams of tested systems of PWM modulators: a) double-edge naturally sampled PWM, b) symmetrcal regular sampled PWM, c) asymmetrcal regular sampled PWM Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated 289

4 . Płatek and. Osypńsk he analyss presented n ths artcle has been based on the assumpton of a very large value of L/R (R s the parastc resstance of the L nductor) characterstc for hgh power devces. 5. Current control system wth double-edge naturally sampled PWM For the control system shown n Fg. 3 wth pulse wdth modulaton, nvolvng the drect comparson of u auxlary sgnal wth the output of the current controller u r (double-edge naturally sampled PWM [27], Fg.4a), the crtcal gan,cr can be determned by comparng the slew rate of the two sgnals: du / dt du dt. (10) r / If ths condton s not satsfed, a repeated change n the output of the comparator n the perod of c leads to unstable operaton of the control system. Fg. 5 shows the border state for whch the control system s stable. he output sgnal of the controller s: u = (u k )K. (11) r If for analyss we assume constant value of the set-pont sgnal u, then we obtan r ( d dt) dur / dt = kkr /. (12) he du /dt dervatve depends on the maxmum value and the perod of the auxlary trangle wave: From (10 13) we obtan du 4 dt k K =. (13) c r d dt 4. (14) c Boundary condtons for whch the current can be formed, occur for two cases: u c = /2 or u c = /2. Maxmum slew rate of the current occurs when voltage s present across the nductance of the choke L. For ths state the followng equalty s vald: K 4 L f c r,cr =. (15) k 6. Current control system wth symmetrc and asymmetrc sampled PWM Fgs. 6a and 6b show the waveforms made for both types of modulators for gans wth values close to the crtcal values and tme τ w =0. Fgs. 6a and 6b show that twofold ncrease of the value of reduces the steady state dsturbance error twce. Smulatons were performed for u = 0, therefore the value of AV means dsturbance steady state error. Below, the dscrete sgnal of the control error s determned and formulas are derved descrbng the crtcal value of gan for the modulators shown n Fgs. 4b and 4c. he block dagram shown n Fg. 3 can be represented n the form of a pulse automatc control system (Fgs. 7a and 7b). he Modulator MOD comprses a Zero Order Holder (ZOH) as shown n Fg. 4b or Fg. 4c. he Zero Order Holder (ZOH) conssts of a pulsar of the samplng perod p and the block G p (s) wth the transfer functon: G p ( s) s e p 1 =. (16) s Replacement of two S & H systems by one of the modulators of Fgs. 4b and 4c results from the assumpton of zero lead tme τ w. Inverter (1, 2) s represented by block K INV [15, 28 30]: Fg. 5. Waveforms of voltage ū r, ū, ū L and of current ī for the modulator n Fg. 4a for u = 0, for ū c = 0.47 and for = Fg. 6. Waveforms of voltage ū r, ū, ū L and of current ī for the symmetrcal (a) and for asymmetrcal regular sampled PWM (b) for u = 0, for u c / = 0.25, for = 0.95,cr and for τ w = Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated

5 Current control wth asymmetrcal regular sampled pulse wdth modulator appled n parallel actve flter I AV 1 k = G + Z L 1 ( s) c sl y x t = m p + [ G 1] y x, (24) Fg. 7. Average value model block dagram representaton of actve flter control system (a) and ts transformed form (b) K INV =. (17) he followng magntudes u oav, u LAV and AV represent mean values of the nverter output voltage, the voltage across the nductor L and the output current of the nverter, respectvely. he mean values are calculated for one perod of the p. Fg. 6 shows that for τ w = 0, the sample values are equal to the average current values for both types of modulaton: = ( mp), (18) AV where m s the number of the next sample of the S &H2 system. herefore the adopton of the model shown n Fg. 7 s justfed. Fg. 7b shows the system n the form convenent for determnng the total error ε u of controlled varable AV. he transfer functon G yx (z) of a closed loop, wth the nput sgnal beng varable x and the output beng varable y, s defned by the dependence Y y =. (19) X G x ransfer functon G yxo (s) of contnuous porton of the open loop s s kkr 1 Gyxo( s) e p =. (20) 2 2 L s ransfer functon G yxo (s) corresponds to the pulse transfer functon G yxo (z): G yxo kkr p =. (21) 2 L z 1 he transfer functon of closed system G yx (z) has the form: where: G yx 1 b =, (22) z b kkr p b = 1. (23) L Based on the dagram shown n Fg. 7b, we can wrte: where Z s the operator of the dscrete transform, L 1 s the operator nverse Laplace transform. he overall devaton of the system shown n Fg. 7 s defned by the followng relatonshp: å ( m) = u. (25) ε u Accordng to the defnton of control devaton gven n [31], t results that u sgnal s equal to ths devaton for k =1 only. he Z transform of error ε u defned by (25) can be determned on the bass of (22, 24): where: E u AV = + + Z L z 1 z b ( s) c sl a z b t = m p z z b 1 1 ( k 1) +, (26) Kr p a =. (27) L he ε u (m) control error contans two components: the error assocated wth the reference sgnal u (trackng error) and the error assocated wth the u c dsturbance sgnal (dsturbance error): ε u (m) = ε u (m) + ε uc (m). (28) he frst component of equaton (26) s a transform of trackng error E u (z), the second s a transform of dsturbance error E uc (z). he Z transform E u (z) has the form: E u z 1 z b a z b = + By forcng u (t )= 1(t) we obtan: E u z = z b + a z ( z 1)( z b). (29). (30) he orgnal of trackng error s descrbed by the followng relatonshp: m m 1- b ε u ( m) = b 1 ( m) + a 1( m). (31) 1- b Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated 291

6 . Płatek and. Osypńsk When we nsert a, b gven by (27, 23) nto equaton (31), we obtan: m u 1 k k K L k 1 k ε ( m) = r p 1 ( m) + ( m) 1. (32) he frst component of equaton (32) descrbes the transent trackng error ε u,t, whle the second s the steady-state trackng error ε u,ss. Due to smplfcaton, the control errors can be descrbed wthout markng that they are dscrete functons. he transent trackng error ε u,t wll decrease asymptotcally to zero f the absolute of the base of power n the frst factor of equaton (32) s lower than unty. he condton for asymptotc stablty of the system s the fulflment of nequalty: 0 kkr p < 2 L <. (33) he maxmum value of, for whch the condton (33) s true s the crtcal value of gan,cr. he asymmetrcal samplng method s not equvalent to doublng the samplng frequency; however, when substtutng τ w = 0, the model shown n Fg. 7 (correct for the average current) dstngushes between symmetrcal regular sampled and asymmetrcal regular sampled based on the amount of samples. For control system wth symmetrcal regular sampled PWM ( p = c ) we obtan relatons descrbng,cr : 4 L fc Kr, cr =. (34) k,cr for control systems wth symmetrcal and wth double-edge naturally sampled PWM are descrbed by the same expressons (15, 34). For control system wth asymmetrcal regular sampled PWM ( p = c /2) we obtan: K 8 L f c r, cr =. (35) k he mnmum tme settngs of the control system can be determned from the condton: kkr L p = 1. (36) Fg. 8. Waveforms of the reference sgnal ū and of ī current for the symmetrcal (a) and asymmetrcal regular sampled PWM (b) for mnmum-tme settng of, for u c = 0 and for τ w = 1.5 μs Equaton (36) s a result of adoptng zero value of the power base of the frst part (32). Such mnmum-tme settng of ensures aperodc stablty. One can assume the mnmum tme settng of as an approxmate crteron for the selecton of gan. In the control system of such settngs, there s a theoretcal possblty of obtanng a steady state after p tme. Asymmetrcal regular sampled PWM offers the possblty of a faster decay of the ε u,t component. Smulaton studes demonstrate the advantages of asymmetrcal regular sampled PWM n terms of decay tme t d of the transent error component. Fg. 8a shows a smulaton of the relatve value of ī current at step change n setpont value of ū for a regularly sampled symmetrc PWM whle Fg. 8b for a regularly sampled asymmetrc PWM system for mnmum-tme settngs ( =,cr /2). Smulaton studes were performed for τ w = 1.5 μs. Steady-state trackng error ε u,ss descrbed by the second part of equaton (32) for both types of modulaton can be determned from the relatonshp: k 1 åε u,s =. (37) k For k = 1 s error ε u,ss = 0, whch means that the control system shown n Fgs. 7a and 7b, representng the block dagram shown n Fg. 3 wth the deal nductance L, s for u c = 0 an astatc system. From the above analyss one can conclude that for τ w = 0, both types of modulaton are equvalent n terms of steady state trackng error of the control system wth zero dsturbance sgnal (u c = 0). 7. Dsturbance error n control systems wth symmetrc and asymmetrc sampled PWM Accordng to equaton (25), dsturbance error ε uc for u = 0 s: uc ε =. (38) he second component of equaton (26) s a dscrete transform E uc (z) of the dsturbance error ε uc : E uc AV 1 c( s) z 1 = Z L sl. (39) z b t = m p By forcng u c (t) = c 1(t), we obtan the pulse transfer functon E uc (z): cp z Euc =. (40) L z 1 z b ( )( ) 292 Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated

7 Current control wth asymmetrcal regular sampled pulse wdth modulator appled n parallel actve flter Orgnal of dsturbance error s descrbed by the followng relatonshp: k K k K L ε ( m) = c r p c 1 ( m) + 1 ( m) uc 1 r m k K r r. (41) he frst component of equaton (41) descrbes the transent dsturbance error ε uc,t, whle the second s the steady-state dsturbance error ε uc,ss. Equaton (41) shows that for,cr /2, transent dsturbance error ε uc,t decays faster for asymmetrcal regular sampled PWM. he steady-state dsturbance error ε uc,ss for both types of modulaton can be determned from the relatonshp: c ε uc, s ( m) = 1( m). (42) k K r From equatons (32, 41) t follows that dsturbance errors more strongly depend on the value. Fg. 9a shows a smulaton of the relatve value of ī current at step change n setpont value of ū c for a regularly sampled symmetrc PWM, whle Fg. 9b for a regularly sampled asymmetrc PWM system for mnmum-tme settngs ( =,cr /2). Smulaton studes were performed for τ w = 1.5 μs. System wth asymmetrc sampled PWM offers the possblty of a sgnfcantly faster decay of the ε uc,t component. From the vewpont of the control tme, the snusodal course of u c voltage may be taken as constant. he analyss wll be lmted to the determnaton of steady-state dsturbance error ε uc,ss. Below we wll determne the offset current n the control system wth symmetrcal regular sampled PWM. At steady state (Fg. 10), the average voltage value across the nductance L for perod c s zero. Hence the equalty u ( τ) = 0 ôτ + u ô Lmax Lmn c (43) (where u Lmax = 0.5 u c, u Lmn = 0.5 u c ) s vald, from whch τ tme can be determned: uc τô = c. (44) he average error for the perod c resultng from the non-zero voltage u c and fnte gan value of (for τ w = 0) can be determned from the smlarty of trangles (ABC and ADE): + ôw τ kkr =. (45) c ôτ c Substtutng (44) to (45) we obtan: 2u c ôw τ =. (46) k he average value of the output current AV n steady state operaton of the nverter for the perod c s: AV,ss = τw + Δ τw, (47) where AV,ss s a mean value of the current n a steady state, Δ τw means an addtonal error due to the non-zero lead tme τ w. he error s defned by the relatonshp: uc Δ Ä τw ôw = τô L w. (48) Fg. 9. Waveforms of the dsturbance sgnal ū c and of ī current for the symmetrcal (a) and for the asymmetrcal regular sampled PWM (b) for mnmum-tme settng of, for u = 0 and for τ w = 1.5 μs 8. Steady-state dsturbance error for non-zero PWM computaton delay Below, equatons are derved descrbng the steady-state dsturbance error takng nto account the tme τ w for both types of modulaton. Fg. 10. Waveforms of voltage ū r ū,ū L and of current ī for the symmetrcal regular sampled PWM for u = 0, for u c < 0 and for,cr Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated 293

8 . Płatek and. Osypńsk Steady-state dsturbance error ε uc,ss s descrbed, accordng to (38 ), by the new relatonshp: ε uc,ss = AV,ss (49) hus, the steady-state dsturbance error value s equal to the negatve average AV,ss : 2 u + 2u åε uc,ss. (50) c c uc, s = + ôτ k 2L w For u c <0, n the range of small values of τ w, steady-state dsturbance error decreases wth the ncrease of τ w, whle for u c > 0, the absolute value of the error decreases (Fg. 12). Below, a formula s derved defnng the crtcal gan,cr n the control system wth asymmetrcal regular samplng for small values of τ w / c. he formulas (43, 44) are vald for systems rrespectve of the type of PWM modulaton. On the bass of smulaton studes for u c 0, t was observed that for small values of τ w / c > 0, the value of the sample taken before the negatve apex of the trangular auxlary voltage corresponds exactly to the mnmum value of the output current of the nverter. Smlarly, for u c 0 t has been observed that for small values of τ w / c > 0, the value of the sample taken pror to the postve apex of auxlary trangular voltage corresponds exactly to the maxmum output current of the nverter. he frst case s llustrated by the waveforms shown n Fg. 11. sng the smlarty of trangles (FGH and ADE or ABC an ADE, respectvely) we derve the followng two relatonshps: kkrcmn 4 = (51) ôτ ôτ w c w + kk c rc τw ôw =. (52) ôτ w c After tme ( c /2 τ ) the value of the current decreases from max to τw, whch s shown by equaton uc c max = τw ôw + ôτ. (53) L 2 he ncrease of the current n tme τ s descrbed by the followng relaton: 0.5 uc max mn = ôτ. (54) L From equatons (44, 51 54), we obtan a relaton that descrbes the crtcal gan for small values of τ w / c for the control system wth the sampled regular asymmetrcal PWM: 8 L fc = 1 uc τô Kr, cr + 2 w. (55) k u 2 ( ) c c Runnng a smlar analyss for u c 0 allows for generalzaton of the above relaton for postve and negatve values of u c : 8 L f = 1 uc ôτ K r, cr 2 w. (56) k c w ( 0.5 ) uc 2 c Fg. 11. Waveforms of voltage ū r, ū, ū L and of current ī for the asymmetrcal regular sampled PWM for u =0, for u c <0 and for =,cr Fg. 13a shows that the ncrease of τ w causes a decrease of,cr to values descrbed by equaton (34). Comparng the rght sdes of (34) and (56), we obtan the equaton whch can be used for determnng the lmt value of τ w for whch (56) s met. he lmt of τ w, for whch (56) s true s descrbed by the followng relaton: ô 1 uc 0 < τ w <. (57) 8 4 Formula (56) for τ w 0 takes the form: K c 8 L f c r,cr =. (58) k Equaton (58) s dentcal to equaton (35), obtaned on the assumpton that the nverter may be replaced by a lnear crcut wth an output voltage equal to the average value for the c perod, set at the tme of samplng the output current. It s clear from the above relatons that the control system of asymmetrcal regular sampled PWM allows for settng hgher values of the gan controller as compared to the value of the symmetrcal regular sampled PWM, especally for small values of τ w / c > 0, so further consderatons wll apply to ths case. he value of the AV,ss average current s equal to: max + mn AV,ss =. (59) 2 sng formulas (44, 51 53), for u c 0, we obtan: 2 u u + 2u c c c AV, ss = c. (60) kk rc 4L Havng run an dentcal analyss for u c 0 and consderng a lnear nature of the control crcut (ε uc,ss = AV,ss for u = 0), the followng formula for steady-state dsturbance error s obtaned: 294 Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated

9 Current control wth asymmetrcal regular sampled pulse wdth modulator appled n parallel actve flter u u 2u c c c åε uc,ss s = + c. (61) kk rc 4L Smulaton studes (Fg. 6b) for the case when τ w = 0 and,cr, showed that the sampled reference u rm2 s constant n tme and waveforms of u r, u L and are smlar to the waveforms for the crcut wth symmetrcal regular sampled PWM for τ w = 0. hus, the error for the system of asymmetrcal regular sampled PWM for τ w = 0 s specfed by the frst component of the formula (61): å = u c ε uc,ss. (62) k For zero value of tme τ w, the steady-state dsturbance error n the system wth asymmetrcal regular sampled PWM can be half of the steady-state dsturbance error n the system of symmetrcal regular sampled PWM. Fgs. 12 and 13 show the crtcal gans and steady-state errors δ uc,ss expressed n the p.u. system as functon of τ w / c, determned on the bass of smulatons. It s assumed that the fxed state s at the tme when the current at the begnnng and end of the perod c does not dffer by more than 1% of the base value I b, as descrbed by formula (9). Crtcal gan s referenced to the value gven by formula (34), whle the error s referenced to the base value of I b current. åε δä uc,ss = uc,ss I (63) Fg. 12. Relatve value of crtcal gan (a) and relatve value of error (b) n a system wth symmetrcal regular sampled PWM for u = const b Fg. 13. Relatve value of crtcal gan (a) and relatve value of error (b) n a system wth asymmetrcal regular sampled PWM for u = const metrcal regular sampled PWM, steady-state error ε u,ss strongly depends on the tme τ w. For asymmetrcal regular sampled PWM, ths dependence does not occur. In real systems, the average value of the output voltage of the nverter depends on the smultaneous mnmum off-tme of the two transstors of each branch of the nverter (t dead ). For contnuous output current of the nverter u oav value s descrbed by the relatonshp: uoav = Kr u tdead fcsgn( ). (64) he magntudes used n the above formula are ndcated n Fgs. 3 and 4. he second component changes replacement gan of the open loop system, for whch Δu s nput and u oav s output sgnal, respectvely. hs fact causes a certan mpact of t dead on the value of the crtcal gan,cr. In ths artcle, ths effect s not evaluated, whereas smulaton tests, the results of whch are shown below were made for the nverter models that take nto account the non-zero value of t dead. In the control crcut of the output current of the nverter shown n Fg. 3, dsturbance error ε uc,ss was descrbed n equaton (50) for symmetrcal regular sampled PWM and n equaton (61) for asymmetrcal regular sampled PWM. If the voltage u c s a snusodal functon, the dsturbance error ε uc,ss, assumng τ w = 0 for both modulators, also vares snusodally. hs means that even for a reference value of u = 0, at the output of the nverter shown n Fg. 3 the current component wll appear n phase wth the voltage u c. Indvdual characterstcs were obtaned for the followng data: waveform 1 for ū c = 0.44, waveform 2 for ū c = 0.25, waveform 3 for u c = 0, waveform 4 for ū c = 0.25, waveform 5 for ū c = he δ uc,ss varables for waveform 3 show the relatve values of steady-state error for the case u c = 0. herefore, the δ uc,ss varables for ths case can be treated as relatve values of steady-statc error assocated wth the set pont u. For sym- 9. APF output current error he control system wth a voltage regulator n the crcut n the APF enforces an addtonal component of the reference sgnal of ts output current, compensatng the dsturbance error of the proportonal current controller, ensurng also the reactve nature of the output current of the parallel actve flter n a steady state operaton. Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated 295

10 . Płatek and. Osypńsk By neglectng the r1 magnetzng currents and C capactors currents, consderatons for error n APF output current can be reduced to evaluaton of the overall error n total output current () of two nverters. For clarty of the followng consderatons, the vector magntudes were replaced by ther components wthout ndces 1, 2, 3: As shown n Fg. 1, output current of each phase has two components: = _1 + _2. (65) Fg. 2 shows a control dagram for the frst phase of the APF. For the remanng phases the scheme s the same. In order to generalze the below analyss for errors n any APF symbols of phase voltage, current, setpont sgnal and erorr ndex 1 s omtted. Setpont sgnal of current contans three components: p() + f + =. (66) he p() s a component of the setpont sgnal of current n phase wth the u c voltage, controllng the voltage u. he h component contans the sum of selected harmoncs of load currents accordng to the respectve phases as shown n Fg. 2. he f current comprses two components: f fsn fcos h = +, (67) where the fundamental frequency component of fsn s n phase wth the voltage u c, and the fundamental frequency component of fcos s orthogonal to u c voltage. In the APF system, s the setpont for total output current of the nverters and, therefore, the total error ε s descrbed by the followng relatonshp: åε = (68) he error ε s dscrete n dgtal regulaton, but for the followng analyss t wll be presented as a contnuous quantty. For the control system of the total output current of two actve flter nverters, the setpont sgnal s the sum of u _1 + u _2 hus, error ε u s descrbed by the new relatonshp: åε u = k (69) u ref In the APF system n the states of unsettled average voltage u, we can dstngush fve components of the nverters output current : the fundamental frequency components of p()sn and fsn that are n phase wth the u c voltage, the fundamental frequency component of fcos orthogonal to u c voltage, the component contanng hgher harmoncs h and the car component ncludng sdeband harmoncs of f c : = (70) p() fsn he car component of current does not depend on the coeffcent of, whle the error for the remanng components can be wrtten as the sum of the errors of the fundamental components ε p(), ε fsn, ε fcos and component contanng hgher harmoncs ε h : fcos h car ε = ε p() + ε fsn + ε fcos + ε h, (71) wheren ε, ε p(), ε fsn, ε fcos and ε h are errors correspondng to setpont sgnals, p(), fsn, fcos and h, respectvely: ε p() = p() p() (72) ε fsn = fsn fsn (73) ε fcos = fcos fcos (74) ε h = h h. (75) he u c value n the actve flters s the phase voltage u c1, u c2 or u c3. Snce n the steady state operaton of the flter, the mean (for the perod of the mans voltage) value of,av s constant, current does not contan the actve component ( p() + fsn = 0). hs condton s acheved by forcng an approprate value u that contans a component p() proportonal to u whch s a value dependent on the output sgnal of the u n voltage regulator. Reducng the mpact of steady state dsturbance error of proportonal regulator on output current of the APF by p() and fsn reference sgnal components means for τ w = 0 satsfyng the equalty: uc kref ( p() + fsn ) =. (76) k K he rght sde of equaton (76) s a dsturbance error ε uc,ss descrbed by (62) and the left sde of the equaton s assocated wth the component of the reference sgnal of nverter current ensurng no actve component n the output current n a steady operatng condton of the APF. Another concluson drawn from the foregong dscusson s that the current descrbed by formula (70) contans a non-zero sum of the components p() and fsn only n the states of non-zero error of the average voltage u,av, assocated wth the change of actve power load. In the states of a steady state operaton (wthout takng nto consderaton heat losses n the APF system), the sum of the p() + fsn components takes a value of zero, whch means that the actve flter can compensate only the reactve and deformaton power, and the accuracy of the compensaton depends on the value of ε fcos and ε h errors. he nfluence on output current of the steady state dsturbance error forced by mans voltage hgher harmoncs s not compensated. 10. Smulaton tests of the power supply system APF Smulaton tests of the power supply system shown n Fg. 1 were performed wth the followng parameters: r1 (400 V / 400 V), r2 (400 V / 400 V / 400 V), L l = 20 μh, L p = 80 μh (h1-h2 and h7-h12 rectfer sde dsperson nductance of r2), L s = 3 μh, armature nductance L a = 1 mh, armature resstance R a = 10 mω, k = 1, k uf = 1, parameters of voltage controller u : n = 1.25 s, K n = , constant of multpler r 296 Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated

11 Current control wth asymmetrcal regular sampled pulse wdth modulator appled n parallel actve flter Fg. 14. Smulaton of waveforms of reference currents 1,p(), 1,h, current 1(500 A/dv), current error ε1,h (500 A/dv) and voltage uc1 (500 V/dv) n compensaton system wth symmetrcal regular sampled PWM Fg. 16. Smulaton of waveforms of currents: L1, 1 s1, c1 (2 ka/dv) and voltages ucv, u12 (1 kv/dv) n compensaton system wth symmetrcal regular sampled PWM Fg. 15. Smulaton of waveforms of reference currents 1,p(), 1,h, current 1 (500 A/dv), current error ε1,h (500 A/dv) and voltage uc1 (500 V/dv) n compensaton system wth asymmetrcal regular sampled PWM Fg. 17. Smulaton of waveforms of currents: L1, 1 s1, c1 (2 ka/dv) and voltages ucv, u12 (1 kv/dv) n compensaton system wth asymmetrcal regular sampled PWM system km = 20, armature current Ia = 992 A, armature voltage a = 619 V, frng angle of SCR converter α = 50º, kref = 0.5, kl = 1/ 3, C = 2.7 mf, C = 50 μf, L = 80 μh, = 720 V, fc = 15 khz, τw = 1.5 μs, tdead = 2 μs and for the maxmum value of the trangular carrer = 5.5 V. Fgs. 14 and 15 show the results of smulaton research on compensaton for 11th and 13th harmonc for symmetrcal and asymmetrcal regular sampled PWM n stable operaton of the flter for whch the 1 output current has no fundamental components (1 = 1,h), respectvely. Waveforms of errors ε1,h for both types of modulators ndcate lower nstantaneous values of error for asymmetrcal regular sampled PWM. Fgs. 14 and 15 also show waveforms llustratng the compensaton forced by the set-pont component 1,p() of the error of the nverter output current 1 of APF nverters, resultng from the non-zero value of nstantaneous uc1 voltage. For a system wth symmetrcal regular sampled PWM, for whch the APF worked stably at a lower value of Kr, the maxmum value 1,p() requred to compensate the ε1,p() error must be larger than the value requred n the system of asymmetrcal regular sampled PWM. he smulaton (Fgs. 16 and 17) was carred out for the case of a closed Sw1 n Fg. 2, whch means the compensaton of reactve power and deformaton reactve power shft. 297 Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated

12 . Płatek and. Osypńsk Waveforms shown n Fg. 16 were obtaned for a system wth symmetrcal regular sampled PWM. he system worked steadly for k = and the obtaned output coeffcents were HD = 7.74% and HD u = 1.33%, respectvely. For a system wth asymmetrcal regular sampled PWM (Fg. 17), stable operaton was ensured by the followng product k = V/A, and sgnfcantly better results were acheved: HD = 4.41%, HD u = 1.1%. 11. Expermental studes In order to verfy the correctness of the analyss of the propertes of the modulator wth asymmetrc sampled PWM, studes have been conducted wth FAS-400k-400 actve flter produced n MEOM company. ests were conducted n ndustral condtons n a coal mne. APF works as a compensator of hgh harmonc currents consumed by a lftng machne n the mne (for the case of open S w1 n Fg. 2). Flter parameters, C, C, L, f c, t dead correspond wth those assumed for the smulaton studes. he remanng parameters are: k L = , k uf = 2.72, υ z = 26, k = , k ref = 0.65, r1 (6 kv / 400 V / 1.6 MVA / short-crcut voltage 5.68%). Due to the fact that the APF nverters are connected to the medum voltage network va r1 transformer, a problem appears concernng phase shfts n the general case, whch are dfferent for the dfferent harmoncs. Phase error was also ntroduced by ndustral current transducers used for measurng the load current. herefore, the control system uses ndvdual selecton of phase correcton-values for each harmonc. Waveforms shown n Fgs. 18a, 18b, and 18c were taken n the actve flter n ts normal operaton state. In practce, waveforms shown n Fg. 18b wth suffcent approxmaton satsfy the followng equaton, whch shows the small control error of flter output current: 2 1 1,h kref / k =. (77) Fg. 19. me waveforms of current s1 and frequency spectrum of ths current (values should be multpled by 1.2) n power system wth flter off Fg. 20. me waveforms of current s1 and frequency spectrum of ths current (values should be multpled by 1.2) n power system wth flter on (f c = 15 khz, k = 0.021) wth symmetrcal regular sampled PWM Equaton (77) was obtaned on the bass of the current control loop (Fg. 2) assumng full compensaton of the selected harmoncs whch means acceptance of ε 1,h = 0. (a) (b) (c) Fg. 18. Expermental results of a system wth asymmetrcal regular sampled PWM. (a) waveform of load current L1 (CH1: 180 A/dv) and current setpont sgnal 1,h (CH2: 1 A/dv) and current 1 (CH3: 500 A/dv) for compensaton of 11th and 13th; (b) waveform of setpont sgnal 1,h (CH1: 0.5 A/dv), of output flter current 1 (CH3: 500 A/dv); (c) waveform of load current L1 (CH1:90A/dv) and nput man current s1 (CH2: 120 A/dv) 298 Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated

13 Current control wth asymmetrcal regular sampled pulse wdth modulator appled n parallel actve flter Fg. 21. me waveforms of current s1 and frequency spectrum of ths current (values should be multpled by 1.2) n power system wth flter on (f c = 15 khz, k = 0.045) wth asymmetrcal regular sampled PWM Lack of components wth voltage rpple u c1 n waveforms of reference sgnal 1,h and current 1 ndcates that the dsturbance error of proportonal controller ε uc,ss forced by ths voltage does not affect the output current of the APF. he expermental results were obtaned for the flter output current control system wth symmetrcal or asymmetrcal regular sampled PWM. Fgs and able 1 present the currents measured n the 6 kv man grd together wth the nput current harmonc dstorton factor HD and the r.m.s. values of ndvdual harmoncs (the system wth a dsabled flter, and a system wth enabled flter wth a preset compensaton of 11th, 13th and 23th current harmoncs). HD factor equal to 14.8 % was reduced to 5.45% when the flter was on for k = V/A and to 4.85% for k = V/A. f c = 15kHz able 1 I I harm HD I 11h I 13h I 23h A rms A rms % A rms A rms A rms APF off APF on/sym k = APF on/asym k = able 1 also shows the mpact of the gan of on HD current drawn from the mans. Greater decrease n the value of 11th, 13th and 23th current harmoncs s vsble for the control system of greater value of. 12. Conclusons he results of smulaton and expermental tests show that the control system wth asymmetrcal regular sampled PWM allows for ncreasng the gan of the proportonal controller n relaton to the gan of the control system wth symmetrcal regular sampled PWM. hs enables reducton of the steady-state dsturbance error of the control system and reducton of the tme decay of ts transent component. Crtcal gan of the proportonal controller n a closed control system wth asymmetrcal regular sampled PWM strongly depends on the PWM computaton delay tme τ w n the range of ts small values (τ w / c. < 0.15). From the pont of vew of mnmzng the dsturbance errors the advantages of asymmetrcal regular sampled PWM compared to the symmetrcal regular sampled PWM manfest themselves only under the condton of an approprately small value of τ w tme. he use of a proportonal current controller n APF s justfed by the partcular property of an APF nvolvng the elmnaton of the nfluence of snusodal steady-state dsturbance error of the proportonal controller on the output current of the actve flter comprsng a voltage regulaton system n the crcut. he use of a proportonal controller provdes very good dynamc parameters of the APF, whch allows for achevng good compensatng results n supply systems wth nonlnear, non-statonary SCR type rectfer. References [1] Y. Zhang, Q. Zhang, Z. L, Y. Zhangt, Comparatve Study of Model Predctve Current Control and Voltage Orented Control for PWM Rectfers, 2013 Internatonal Conference on Electrcal Machnes and Systems, Busan, Korea, (2013). [2] S. Fukuda and H. Kamya, Adaptve learnng algorthm asssted current control for actve flters, n Proc. IEEE Ind. Appl. Conf., 1, (2001). [3] R. Costa Castello, R. Grno, R. Cardoner -Parpal, and E. Fossas, Hgh performance control of a sngle-phase shunt actve flter, IEEE ransactons on Control Systems echnology, 17, (2009). [4] A. Garca-Cerrada, O. Pnzon-Ardla, V. Felu-Batlle, P. Roncero-Sanchez, and P. Garca-Gonzalez, Applcaton of a Repettve Controller for a hree-phase Actve Power Flter, Power Electroncs, IEEE ransactons on, 22, (2007). [5] Y. L, X. Hao, X. Yang, R. Xe,. Lu, A varable-band hysteress modulated mult-resonant sldng-mode controller for three-phase grd-connected VSI wth an LCL-flter, ECCE Asa Downunder (ECCE Asa), 2013 IEEE, (2013). [6] D.G. Holmes,.A. Lpo, Pulse Wdth Modulaton For Power Converters, IEEE PRESS, Wlley-Interscence, [7] H. Fujta, A Sngle-Phase Actve Flter sng an H-Brdge PWM Converter Wth a Samplng Frequency Quadruple of the Swtchng Frequency, IEEE ransactons on Power Electroncs, 24(4), (2009). [8] D.G. Holmes and B. McGrath, Opportuntes for harmonc cancellaton wth carrer-based PWM for a two-level and multlevel cascaded nverters, IEEE rans. Ind. Appl., 37(2), (2001). [9] Z. Y-Feng, Z. Zheng, Y. Ha-zhu,. Ka, A new control method for neutral-pont-clamped three-level PWM rectfers, Consumer Electroncs, Communcatons and Networks (CECNE), (2011). [10] S.W Lu, F. Ln, X.J You,.Q. Zheng, Research of Mult-module PWM Rectfer for Feedng System of Hgh-speed Maglev Vehcles, IEEE rans on Industral Electroncs, , (2006). Bull. Pol. Ac.: ech. 64(2) 2016 nauthentcated 299

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