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Harmonic Analysis of Static var Compensator Operating Under Distorted Voltages E. V. Liberado and J. A. Pomilio F. P. Marafão School of Electrical and Computer Engineering University of Campinas Campinas, Brazil, 13083-852 Emails: [email protected], [email protected] Group of Automation and Integrated Systems Unesp - Univ Estadual Paulista Sorocaba, Brazil, 18087-180 Email: [email protected] Abstract—The static var compensators (SVC) are still in use in both transmission and distribution grids, due to their reduced costs and high reliability. Especially in distribution grids, on which the voltage harmonic distortion limits are less restrictive, it may cause a considerable harmonic distortion impact when operating under non sinusoidal voltage conditions. Therefore, this paper describes a harmonic analysis methodology for the SVC, considering non-ideal source voltages, in order to determine harmonic power terms that may be used to design and also control harmonic power filters. Application examples are presented and the operation of an experimental SVC prototype is also analysed. I. I NTRODUCTION Static var compensator (SVC) is a well-known technology that has been studied and used for years in voltage control applications [1]–[3]. Many SVC units are indeed being repaired in order to extent their lifetime [4], which shows that it is still a cost-effective and reliable solution. Due to its capacity of absorbing a variable capacitive or inductive reactive power, voltage and load balancing applications has been also proposed for the SVC, considering its installation in distribution grids [5]–[10]. However, a well-know drawback of the SVC is the harmonic distortion caused by its operation. On the SVC traditional application scenario, i.e., the tranmission and medium-voltage distribution grids [11], the voltage harmonic distortion is highly limited, thus the harmonic distortion caused by the SVC is mainly due to the thyristor valves’s switching, for which passive filters are applied. Nevertheless, in low-voltage-level distribution grids the limits for voltage harmonic distortion (HDu ) are higher than the ones from medium- and high-voltage- levels. In this scenario, the SVC capacitors may cause more harmonic distortion in the currents than the thyristors’ switching. Moreover, the passive filters may be useless in grids with high short-circuit level [12] and, unlike the active power filter (APF), their current cannot be adjusted to achieve determined goals. Therefore, this paper presents a methodology of SVC harmonic analysis in order to estimate the harmonic power demand to be addressed for compensation, either by a passive filter or an APF to be installed closer to the SVC [13]. Recommended harmonic distortion limits are used to determine the harmonic source voltages [14], [15], in order to evaluate the harmonic current drained by the SVC capacitors, as well as the influence of harmonic voltages in the operation of the thyristor-controlled reactor (TCR). II. SVC H ARMONIC L INE C URRENTS Consider the SVC topology represented in Fig. 1, composed by three delta-connected fixed capacitors (CSV C ) and three delta-connected TCR branches (LT CR ). This is the simplest SVC topology and it is suitable for distribution grid applications, due to their reduced number of switching components. The SVC is fed by a three-phase three-wire voltage source through three balanced line impedances Zs = Rs + jωLs . As the voltages in distribution grids may be distorted, unbalanced and asymmetrical, the source voltages are generically defined from the fundamental frequency to a determined harmonic order Kh as: um = Kh √ k 2 · Um sin kωt + φkm m = 1, 2, 3. (1) k=1 On the SVC side, the line voltages can be written as: uSV C,1 = u1 − Rs · iSV C,1 − Ls · ı̆SV C,1 uSV C,2 = u2 − Rs · iSV C,2 − Ls · ı̆SV C,2 uSV C,3 = u3 − Rs · iSV C,3 − Ls · ı̆SV C,3 (2) on which ı̆SV C,m is the time-derivative of iSV C,m , m = 1, 2, 3. Considering that the sum of the line voltages and currents is zero at any instant (which is also valid for their time derivatives and integrals) on the proposed circuit, the line currents can be written in terms of the currents of each SVC branch as: c 978-1-5090-3792-6/16/$31.00 2016 IEEE 299 iSV C,1 = iSV C,12 − iSV C,31 iSV C,2 = iSV C,23 − iSV C,12 iSV C,3 = iSV C,31 − iSV C,23 , (3) u3 u2 Rs Rs uZs,3 Ls u1 uZs,2 Ls uZs,1 iTSVCR C,mn = 2 on which iSVC,1 LTCR uSVC,12 φkT CR,mn CSVC CSVC CSVC iSVC,31 iSVC,12 LTCR Fig. 1. SVC circuit for harmonic current analysis. uSV C,12 = uSV C,1 − uSV C,2 = u1 − u2 − Rs (iSV C,1 − iSV C,2 ) − Ls (ı̆SV C,1 − ı̆SV C,2 ) = u12 − (3 · Rs · iSV C,12 + 3 · Ls · ı̆SV C,12 ) = u23 − (3 · Rs · iSV C,23 + 3 · Ls · ı̆SV C,23 ) uSV C,23 uSV C,31 = u31 − (3 · Rs · iSV C,31 + 3 · Ls · ı̆SV C,31 ) (4) From (4), one may observe that in case of unbalanced line impedances, their delta-equivalent can be obtained by using a wye-delta transformation and then used in (4) and in the following formulation. By using the superposition theorem, the current of the SVC branches mn = 12, 23, 31 can be split into iCap SV C,mn and iTSVCR C,mn in order to determine each of these components separately. Therefore, the current of each SVC capacitor branch can be expressed by: 2 αmn αmn − π/2 , tb,mn = ti0,mn − , ω ω on which αmn is the conduction angle of the TCR branch mn and ti0,mn may be determined iteratively for each semicycle of iTSVCR C,mn by incrementing t until reach the following conditions: T CR T CR • iSV C,mn = 0 and ı̆SV C,mn < 0 for the positive semicycle and T CR T CR • iSV C,mn = 0 and ı̆SV C,mn > 0 for the negative semicycle. Additionally, iTSVCR C,mn = 0 for t < ta,mn and t > tb,mn . Term Cmn is the integration constant and may be determined for each ta as: Cmn = Kh √ 2 · ITk CR,mn sin kωta + φkT CR,mn k=1 Finally, the current of SVC branches are expressed by: T CR iSV C,12 = iCap SV C,12 + iSV C,12 T CR iSV C,23 = iCap SV C,23 + iSV C,23 iSV C,31 = k 2 · ICap,mn sin kωt + φkCap,mn , iCap SV C,31 + (6) iTSVCR C,31 and the SVC line currents can be determined by using (3). k=1 on which III. SVC H ARMONIC C OMPLEX P OWER k ICap,mn = k Umn In the sequel, the fundamental and harmonic positive- and negative-sequence components of the SVC line currents can be calculated using a Fast Fourier Transform algorithm and the Fortescue’s symmetric components matrix. Assuming that the harmonic voltages will not be compensated by the shunt harmonic filters, only the fundamental positive-sequence voltages are used in the complex power calculation and it is expressed as: 2 , 1 (3 · Rs ) + 3 · kωLs − kωCSV C ⎞ ⎛ 1 3 · kωLs − ⎜ kωCSV C ⎟ = φkmn − arctan ⎝ ⎠. 3 · Rs 2 φkCap,mn 2 (3 · Rs ) + (kω (LT CR + 3 · Ls )) kω (LT CR + 3 · Ls ) = φkmn − arctan . 3 · Rs ta,mn = Thus, the SVC line-to-line voltages can be expressed as: √ k Umn Terms ta and tb are the instants that delimit the conduction time interval of each TCR thyristor. If a phase reference signal synchronized with the fundamental-frequency line-to-line voltages is used to control the thyristors switching, the TCR current semi-cycles are symmetrical. Thus, the conduction time interval of each TCR thyristor is the same and 1 uSVC,31 iCap SV C,mn = 2ITk CR,mn sin kωt + φkT CR,mn − Cmn , ITk CR,mn = LTCR Kh Kh √ k=1 iSVC, 2 iSVC,23 3 Rs Ls iSVC,3 uSVC,23 Correspondingly, the current of each TCR branch valid in the interval ta,mn ≤ t ≤ tb,mn is expressed by: From the SVC capacitor’s equivalent impedance, the resonant frequency between CSV C and Ls is calculated by: fo = 1 . 2π 3 · Ls CSV C √ (5) f,p f,p f,p − Żsf,p · I˙SV U̇SV C = U̇ C (7) on which U̇ f,p and Żsf,p are the fundamental-positive sequence components of the source voltages and line impedances. Then, 300 SV C1 SV C2 QCap SV C CR QT SV C CSV C LT CR 57.6 kvar 67.6 kvar 81.9 kvar 98.8 kvar 1052 μF 1235 μF 4.7 mH 3.9 mH the complex positive- and negative-sequence power components for each harmonic k are calculated by: k,p k,p k,p f,p ˙k,p ṠSV C = PSV C + jQSV C = U̇SV C · ISV C (8) Ṡ k,n = P k,n + jQk,n = U̇ f,p · I˙k,n SV C SV C SV C SV C SV C Therefore, the SVC harmonic complex power to be addressed for compensation can be defined for a determined set of harmonic orders Ks ≤ k ≤ Kh as [16]: c c c c c ṠSV C = SSV C φSV C = PSV C + jQSV C , (9) on which QcSV C 2 2 c c c (PSV , C ) + (QSV C ) , φSV C = arctan c PSV C Kh Kh 2 2 k,p k,n c PSV PSV + , PSV C = 3.0 · C C c SSV C = QcSV C k=Ks k=Ks k=Ks k=Ks harmonic current [A], frequency [Hz] TABLE I E VALUATED SVC S . Kh Kh 2 2 Qk,p Qk,n = 3.0 · + . SV C SV C Eqs. (8) allow the selective compensation of unbalanced harmonic currents [17], which may be useful in applications on which the SVC operation may be unbalanced (i.e., TCR branches with different conduction angles). Thus it is interc esting to calculate ṠSV C considering a large set of harmonic orders, in order to evaluate which are the critical ones. Also, for applications on which the SVC operation may be c unbalanced, ṠSV C should be calculated for a set of combinations of TCR angles, then only its highest value is conduction c,max considered ṠSV . This value, as well as the conduction C angle combination that results in it, may vary according to the number of harmonic orders considered in the calculation of c ṠSV C. IV. A PPLICATION EXAMPLES Table I presents two SVCs designed for load balancing in a 220 V/60 Hz/75 kVA distribution grid. The line impedances were defined in terms of the low-voltage winding impedances of a three-phase-75 kVAdistribution transformer, thus Rs = 6.5mΩ and Ls = 100.0μH. In this case, the resonant frequency of the SVC capacitors is 283.3 Hz (CSV C1 ) and 261.5 Hz (CSV C2 ). If the cable impedances are considered, the resonant frequencies can be reduced. Fig 2 presents the resonant frequency and harmonic current magnitudes of one capacitor branch of SV C1 when considering the transformer impedances plus a variablelength cable impedance equal to 0.34 + j0.56Ω/km. The 300 5 ICap1 7 ICap1 11 ICap1 fo 250 200 150 100 50 0 0 25 50 75 100 125 150 175 line length [m] 200 225 250 275 300 Fig. 2. Resonant frequency and harmonic currents of one SV C1 capacitor branch in terms of a variable-length line impedance. f k line voltages were Um = 127V and Um = 6.35V, k = 5, 7, 11; m = 1, 2, 3. One may observe that the longer are the cables, the lower is the resonant frequency and the capacitor harmonic currents are considerably reduced. Alternatively, a small series inductance may be inserted between the delta-connected SVC capacitors and the line impedances in order to atenuate the harmonic currents that are close to the resonant frequency [18]. However, it is important to observe that the power losses increase proportionally to the line impedance. Therefore, by adopting the transformer impedance plus a 25 m cable impedance, Rs = 15.0mΩ and Ls = 137.1μH and the resonant frequencies of SV C1 and SV C2 capacitors are 241.9 Hz and 223.3 Hz in the next two application examples. A. Example 1: IEEE recommended limits In the first application example the recommended limits from IEEE Std-519-2014 [14] were adopted to define the harmonic source voltages, expressed in terms of the individual harmonic distortion factor HDu and the fundamental k f = HDu · Um , k = 5, 7, 11. As the frequency voltage as Um nominal voltage was 220 V, the limit of individual harmonic voltages is 5% [14], thus HDu = 0 · · · 5%. The harmonic currents and complex power to be compensated were calculated for SV C1 and SV C2 considering all the odd harmonic orders between the 3rd and 51st . Also, these calculations were repeated for a set of 51 values of TCR conduction angles into the [π/2, π] interval, resulting in the evaluation of 513 = 132, 651 combinations of SVC operation. Fig. 3 presents the odd harmonic (3rd -13th ) current magnitudes of one capacitor branch of SV C1 in function of the source voltages T HDu , which varied from 0 to 8.66%. Note that the capacitor harmonic currents that correspond to the harmonic source voltages dramatically increases in function of T HDu . The fundamental-frequency capacitor current remained fixed and equal to 92.94 A. Fig. 4 presents the harmonic currents of one TCR branch of SV C1 . Every curve of each harmonic order is plotted in function of the conduction angles for one value of (T HDu ). In 301 40 35 100 3 5 7 9 11 13 h= h= h= h= h= h= 80 25 h ISV C,1[A] ICh ap,12[A] 30 h= h= h= h= h= h= 20 15 10 3 5 7 9 11 13 60 40 20 5 0 0 1 2 3 4 T HDu % 5 6 7 0 1.5708 8 Fig. 3. Harmonic currents of one capacitor of SV C1 in function of the source voltage’s T HDu . 16 h= h= h= h= h= h= 14 ITh C R,12[A] 12 3 5 7 9 11 13 2.1991 2.5133 αSV C,12[rad] 2.8274 3.1416 Fig. 5. SV C1 harmonic line currents in function of the TCR conduction angles and the source voltage’s T HDu . c,max ṠSV C TABLE II OF SV C1 AND SV C2 AND C ORRESPONDING TCR C ONDUCTION A NGLES FOR E ACH C ASE OF S OURCE VOLTAGE T HDu . T HDu % 10 c,max ṠSV C [kV A] α12 [rad] α23 [rad] α31 [rad] 1.9478 1.9164 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.9792 1.9164 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.9478 1.9164 1.9164 1.8221 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.9478 1.9478 1.9164 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 SV C1 8 6 4 2 0 1.5708 1.885 1.885 2.1991 2.5133 αSV C,12[rad] 2.8274 3.1416 Fig. 4. Harmonic currents of one TCR branch of SV C1 in function of the conduction angles and the source voltage’s T HDu . relation to the capacitors behavior, it is possible to observe that the TCR currents do not vary significantly with the increasing of T HDu . Fig. 5 presents the harmonic components in the line current of SV C1 , calculated only for the SVC balanced operation cases, i.e., considering the same conduction angle for the three TCR branches. Also in this Figure every curve of each harmonic component is function of one value of T HDu and the TCR conduction angles. The triplen harmonics are absent in the line currents, thus the most relevant harmonics in the line currents are the ones that are also present in the source voltages (5th , 7th and 11th ), due to the currents drained by the SVC capacitors. Table II presents the maximum values of harmonic complex power for each source voltage T HDu among all combinations of SV C1 and SV C2 operation. The corresponding TCR conduction angles indicate that in most cases the maximum harmonic complex power occurs for unbalanced operation of the TCR, on which the triplen harmonics caused by the TCR’s switching are present in the line currents. B. Example 2: PRODIST limits In the second application example, the HDu limits proposed in [15] were adopted to the source voltages. The HDu for the 0.00 0.87 1.73 2.60 3.46 4.33 5.20 6.06 6.93 7.79 8.66 5.34 50.26◦ 6.92 35.31◦ 9.27 9.09◦ 12.07 9.16◦ 14.93 9.26◦ 17.81 9.28◦ 20.71 9.29◦ 23.61 9.31◦ 26.52 9.31◦ 29.44 9.32◦ 32.36 9.32◦ 0.00 0.87 1.73 2.60 3.46 4.33 5.20 6.06 6.93 7.79 8.66 6.30 50.52◦ 7.42 41.44◦ 9.09 31.17◦ 11.10 8.58◦ 13.39 8.40◦ 15.72 8.25◦ 18.07 8.13◦ 20.44 8.04◦ 22.81 8.03◦ 25.20 7.98◦ 27.59 7.93◦ 3.1102 1.5708 1.8535 1.8535 1.8850 1.8850 1.8850 1.8850 1.8850 1.8850 1.8850 SV C2 3.1102 3.1102 1.5708 1.8535 1.8535 1.8535 1.8535 1.8535 1.8850 1.8850 1.8850 5th harmonic varied from 5% to 7.5%, while the HDu for the 7th varied from 5% to 6.5% and the 11th HDu remained constant and equal to 4.5%. Table III presents the maximum harmonic power of SV C1 and SV C2 and the corresponding TCR conduction angles for this application example. As expected, the harmonic complex power continued to increase in relation to the previous example, as the HDu limits of 5th and 7th harmonics are higher in these cases than the ones adopted before. 302 TABLE III c,max ṠSV C OF SV C1 AND SV C2 AND C ORRESPONDING TCR C ONDUCTION A NGLES C ONSIDERING THE T HDu L IMITS P ROPOSED IN [15]. T HDu % c,max ṠSV C [kV A] α12 [rad] α23 [rad] α31 [rad] 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8850 1.8850 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8535 1.8850 1.8535 1.8535 SV C1 32.28 35.14 38.00 40.86 43.42 45.98 8.38 8.99 9.60 10.23 10.56 10.90 27.51 29.83 32.15 34.48 36.49 38.52 9.34◦ 9.36◦ 9.42◦ 9.43◦ 9.45◦ 9.51◦ 8.38 8.99 9.60 10.23 10.56 10.90 1.8850 1.8850 1.8850 1.8850 1.8850 1.8850 SV C2 7.95◦ 7.93◦ 7.91◦ 7.94◦ 7.93◦ 7.96◦ 1.8850 1.8850 1.8850 1.8850 1.8850 1.8850 Fig. 6. Experimental setup: left: SVC front panel and capacitor banks (on top), right: TCR and linear loads. TABLE IV E XPERIMENTAL R ESULTS FOR THE SVC O PERATION . V. E XPERIMENTAL R ESULTS An experimental SVC prototype with QCap SV C = 7.5 kvar, QTSVCR C = 10 kvar, CSV C = 137μF and LT CR = 40mH was set to compensate reactive power and load imbalance of a 6.2 kVA three-wire linear unbalanced RL load. Fig. 6 presents the experimental SVC and load, which were fed by a three-phase programmable voltage source. Two source voltage cases were tested: A) 220 V/60 Hz and B) 220 V/60 Hz + HDu = 2% for 5th harmonic. The line impedances were neglected. Table IV presents the measured RMS line voltages and currents, line T HDu and T HDi (current total harmonic distortion) factors and the magnitude c of ṠSV C , which were calculated in these cases considering all harmonic components present in the line currents, by extracting the fundamental-frequency current from the measured ones. In the first case (A), both load and SVC were connected to the voltage source. As the load is purely resistive/inductive, all the harmonic distortion observed in the line currents may be accounted to the SVC, particularly the TCR. Fig. 7 presents the measured line currents in this case. When the source voltages were distorted (cases B), the line currents were not much distorted when only the load was connected to the source (case B.1). However, the currents became highly distorted when the SVC was turned on (case B.2), mainly due to the SVC capacitors, as shown in the results of case B.3, on which both load and TCR were turned off. Fig. 8 presents the waveforms of the line currents in these cases. Regarding the harmonic power, Table V presents a comc,max parison between the magnitude of ṠSV C , calculated using (9) and considering the odd harmonic orders from the 3rd to c the 51st for T HDu = 0 and 2%, and SSV C of cases A and B.2. As the conduction angles that compensate the load in c,max that cases were different than the ones that result in ṠSV C , c the experimental SSV C resulted in lower values in relation to the maximum expected ones even in case B.2, on which the A B.1 B.2 B.3 U1 [V ] U2 [V ] U3 [V ] 126.62 126.75 126.53 126.80 126.52 126.41 126.49 126.69 126.54 127.28 127.28 127.10 I1 [A] I2 [A] I3 [A] 11.77 11.72 11.28 13.84 15.17 19.19 12.11 11.55 11.50 19.56 19.63 19.80 T HDu,1 % T HDu,2 % T HDu,3 % 0.52 0.60 0.48 2.09 2.14 2.18 2.24 2.11 2.07 2.17 2.23 2.22 T HDi,1 % T HDi,2 % T HDi,3 % 11.13 10.35 12.33 1.55 2.04 1.51 16.44 10.76 23.18 10.83 10.47 10.98 c SSV C [VA] 493.20 104.13 763.36 802.92 TABLE V c,max C OMPARISON B ETWEEN T HEORETICAL SSV C c SSV . C T HDu = 0 case A T HDu = 2% case B.2 AND THE E XPERIMENTAL c SSV C [V A] α12 [rad] α23 [rad] α31 [rad] 685.64 493.20 1.5708 1.9208 1.9792 2.1288 2.0106 2.5408 1036.76 763.36 2.5133 1.9288 2.4819 2.1168 2.5133 2.5368 measured line currents included also the harmonic currents drained by the load. VI. C ONCLUSION A methodology to estimate the harmonic power absorbed by the SVC was presented. It is a relevant parameter to design SVC harmonic filters, especially in distribution applications, on which even when the voltage harmonic distortion is under recommended limits, the SVC capacitors may cause more current harmonic distortion than the TCR. Even though the source voltages in the presented examples and results were balanced and symmetrical, the methodology allows the using of any sort of voltage conditions, including 303 definition of a selective harmonic compensation for active filters and can be also addressed to distributed compensators. ACKNOWLEDGMENT The authors would like to thank the Research National Council for Scientific and Technological Development - CNPq, (grants 302257/2015-2 and 487471/2012-1) and Varixx Co. for provide the thyristor’s control modules. R EFERENCES Fig. 7. Line currents with SVC and load fed by sinusoidal voltages. Fig. 8. Line currents for cases B.1 (top), B.2 (middle) and B.3 (bottom). unbalanced and asymmetrical cases. Some alternatives to reduce the SVC harmonic power include: to adopt an optimum placement strategy for the SVC, on which the minimization of the power losses should be considered; the insertion of a small-rated power series inductance between the grid and the SVC capacitors; or the adoption of a complete electronic compensator composed by TCR and active filter. Moreover, the harmonic complex power terms allows the [1] R. Gutman, J. J. Keane, M. E. Rahman, and O. 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