Circuit for Balancing Harmonic-Polluted Three-Phase Networks

Transcription

1 Electrical Pwer Quality and Utilisatin, Jurnal Vl. XVI, N. 1, 2013 Circuit fr Balancing Harmnic-Plluted Three-Phase Netwrks Mhamed M. Saied Electrical Engineering Department, Cllege f Engineering and Petrleum, Kuwait University; m.saied@ieee.rg Summary: This paper presents a circuit capable f balancing harmnics-plluted three-phase pwer netwrks supplying single-phase lads. It takes int accunt the negative and psitive sequences exhibited by the fifth and seventh harmnics, respectively. The cncept extends the relatins f the Steinmetz circuit, which are based exclusively n fundamental frequency cnsideratins. The Steinmetz circuit des nt, therefre, guarantee the balancing f the different harmnics in the line currents. Furthermre, it can lead t resnances between the capacitive and inductive elements in the tw added balancing branches. A prcedure fr identifying the tplgy and the circuit parameters f the prpsed cnnectin is given. The results f a detailed case study are presented in rder t demnstrate the superirity f the prpsed circuit ver the cnventinal Steinmetz cnnectin. Key wrds: harmnic analysis, pwer quality, lad balancing, frequency scan, Steinmetz circuit 1. INTRODUCTION Single-phase lads lead t the unbalanced peratin f the pwer netwrks which are usually designed based n the ideal situatin f balanced lading. Althugh a cnsiderable percentage f the lads in these netwrks are balanced three-phase nes, the adverse effects f eventually existing single-phase lads were the tpics f many investigatins under bth balanced [1 4], and unbalanced supply cnditins [5]. Fr instance, the single-phase lad f the pure real admittance (r cnductance) bc = G Siemens (r W -1 ) acrss the terminals b and c f the threephase supply shwn in Fig. 1 will lead t the extremely unbalanced currents: I a = 0, I b = GV, and I c = GV, where V is the netwrk s rated line vltage, with the vltage V bc as reference. Adding the capacitive and inductive susceptances ca = + jg / 3 and ab = jg / 3acrss the line terminals c,a and a,b, respectively, as depicted in Fig. 2, will result in the Steinmetz circuit. It can be shwn that the line currents I a, I b and I c will then cnstitute a psitive sequence system f three-phase currents having equal magnitudes f GV / 3 and 120 degrees phase shifts, [1]. Althugh mst f the studies in the literature address the peratin f the Steinmetz balancing circuit under pure sinusidal perating cnditins [1 5], several recent investigatins deal with the harmnic perfrmance and the ptential resnance prblems in pwer netwrks cmprising such balancing circuits [6 13]. Fr example, the theretical and experimental studies in refer ences [8 10] address the high frequency respnse as well as the pssible parallel and series resnances that can take place between the capacitr f the Steinmetz circuit and the pwer netwrk s inductance. References [11,13] investigate the increase in the vltage distrtin and the identificatin and mitigatin f the pssible harmnic disturbances resulting frm the backgrund vltage distrtin. Relatins were presented fr identifying the series resnance appearing, if a Steinmetz circuit is cnnected t a harmnic-plluted supply and fr cases invlving nnlinear lads injecting harmnic currents. It shuld be nted that the balancing capability f the Steinmetz circuit depends n tw imprtant cnditins: Fig. 1. A single phase lad f cnductance G cnnected t the terminals b, c f a symmetrical three-phase netwrk f line vltage V. [1] Fig. 2. The Steinmetz circuit cmprising the tw balancing reactive elements ca = + jg / 3 and / 3 ab = jg used fr balancing the single phase lad f cnductance G cnnected t the terminals b, c f a symmetrical three-phase netwrk f line vltage V. [1] 19

2 (i) The phase sequence f the supply vltage shuld be psitive (i.e. a-b-c). (ii) The supply frequency shuld be equal t that used fr the calculatin f the capacitance C frming ca cnnected between c&a and the inductance L frming ab cnnected between a&b. In ther wrds, G/ 3 = wc and G/ 3 = 1/( wl) where w 0 is the fundamental angular frequency. With regard t the harmnic respnse f the balancing circuit, it shuld be taken int cnsideratin that the harmnics f therders h = 6p + 1 (p = 1,2,3,etc.), i.e. h = 7,13,19,...,are f a psitive sequence, whereas thse having h = 6p 1 (p = 1,2,3,etc.), i.e. h = 5,11,17,..., exhibit the characteristics f a negative sequence system. Fig. 3 shws the wavefrms f the line currents I a, I b and I c fr the single phase resistive lading described by the circuit depicted in Fig. 1. Currents are given in per unit based n the rms lad current (VG). The 60-Hz supply vltage is assumed t include fifth and seventh harmnic cmpnents f 4% and 3%, respectively. As expected, I a = 0 (thinnest curve), while the tw ther line currents I b (curve f medium thickness) and I c (thickest curve) are identical but f ppsite signs. Because the lad is purely resistive, and due t the absence f any inductances r capacitances in the unbalanced circuit f Fig.1, the line currents in Fig. 3 exhibit the same percentage harmnic cntent and distrtin factr f the supply vltage. This shuld be cmpared with the three strngly distrted line currents resulting frm the use f the Steinmetz balancing circuit given in Fig. 4. The fundamental cmpnents f I a, I b and I c cnstitute a three-phase balanced system, while the much higher harmnic cntent f any f the three line currents is due t the fact that the tw abve mentined cnditins i) and ii) are nt satisfied fr the fifth and seventh harmnics. Fig. 5 illustrates the dependence f the amplitudes f the harmnic currents in the three line currents I a, I b and I c n the harmnic rder (r the per unit frequency) h. Only integer dd harmnic rders h are relevant in this plt. As expected, the fundamental cmpnents f the three line currents are equal t 1/ 3 per unit based n GV. Regarding the fifth harmnics, it can be seen that their percentage in the line currents I a, I b and I c are, respectively, 2.8, 1.2 and 3.8 times higher than thse in the supply vltage. The crrespnding magnificatin factrs fr the seventh harmnic are 4.1, 0.9 and 3.5, respectively. The fifth and seventh harmnic cmpnents in the supply vltage cnstitute symmetrical 3-phase systems f negative and psitive sequences, respectively. The per unit magnitudes f the negative sequence cmpnents in the line currents are given in Fig. 6 versus the harmnic rder. Of special interest are the values crrespnding t h = 1 (fundamental), h = 5 and h = 7. The plt shws the absence f any negative sequence cmpnent fr h = 1, since the Steinmetz circuit was riginally designed fr the fundamental frequency. The crrespnding values fr h = 5 and h = 7 are and per unit, respectively. 2. THE PROPOSED CIRCUIT The abve cmments indicate that the adverse harmnic effects f the Steinmetz cnnectin are due t the fact that the admittances f the tw added branches ca = + jg / 3 and ab = jg / 3, shwn in Fig. 2, were determined with the bjective f balancing the circuit at the fundamental frequency w 0. In the fllwing sectin a mdified balancing circuit is suggested fr balancing the circuit at the pwer frequency w 0 and, simultaneusly, capable f mitigating the adverse harmnic effects assciated with the first tw characteristic harmnics (i.e. the fifth and the seventh). The balancing admittances ca,, ab, will nw be selected such that the circuit will be balanced at the fundamental frequency w 0, the fifth as well as the seventh harmnic frequencies 5 w 0 and 7 w 0, respectively. Accrdingly, the six fllwing cnditins must be satisfied by ca,, ab,. Fig. 3. Plts f the three line currents I a, I b and I c fr the unbalanced cnditin shwn in Fig. 1 with a supply vltage plluted by 4% fifth and 3% seventh harmnics, respectively. Currents are in per unit based n the rms lad current (VG) I a = 0 (thinnest curve), I b (curve f medium thickness) and I c (thickest curve) Fig. 4. Plts f the three line currents I a, I b and I c fr the balanced cnditin shwn in Fig. 2 (using the Steinmetz circuit) with a supply vltage plluted by 4% fifth and 3% seventh harmnics, respectively. Currents are in per unit based n VG I a = 0 (thinnest curve), I b (curve f medium thickness) and I c (thickest curve) 20 Electrical Pwer Quality and Utilizatin, Jurnal Vl. XVI, N 1, 2013

3 Fig. 5. Plts f the frequency characteristics f the three line currents I a, I b and I c fr the balanced cnditin shwn in Fig. 2 (using the Steinmetz circuit). I a (thinnest curve), I b (curve f medium thickness) and I c (thickest curve) Fig. 6. Plts f the negative sequence cmpnent in the three line currents I a, I b and I c fr the circuit shwn in Fig. 2 (i.e. using the Steinmetz circuit) versus the harmnic rder (r the per unit frequency) h. Fig. 7. The mdified balancing circuit. Fig. 8. The frequency respnse f the per unit susceptances f ca,, ab,. thin curve: susceptance f ca,, thick curve: susceptance f ab, ca, ( w ) = + jg / 3 ( w) = [1/ ( jwc ) + ( jwl) / (( jw) LC+ 1)] (2) 2 1 ca, ab, ca, ab, ca, ab, ( w ) = jg / 3 (5 w ) = jg / 3 (5 w ) = + jg / 3 (7 w ) = + jg / 3 (7 w ) = jg / 3 (1) ( w) = [( jwl ) + ( jwl ) / (( jw) L C + 1)] (3) 2 1 ab, After substituting these tw relatins in the previusly listed six simultaneus equatins (1), an analytical clsed-frm slutin culd be fund using the sftware Mathematica as: G C11 = ( ) (4) w The signs f the admittances take the psitive sequence nature f bth the fundamental and the seventh harmnic as well as the negative sequence f the fifth harmnic int accunt. The availability f the abve six equatins implies that a mdified balanced circuit can be identified, including three elements within each f the admittance branches, such as the suggested circuit shwn in Fig. 7. It includes the still unknwn 6 circuit elements (C 11, L 1, C 1 ), and (L 22, L 2, C 2 ) cnstituting ca,, ab,, respectively. At any angular frequency, ω, the tw admittances are given by: G C1 = ( ) (5) w L1 = (6) Gw 21

4 L = (7) Gw L2 = (8) Gw G C2 = ( ) (9) w Where w 0 is the fundamental angular frequency in rad./s. 3. SAMPLE RESULTS The effectiveness f the abve suggested balancing circuit and the validity f the slutin technique are demnstrated by Fig. 8. It gives tw curves fr the frequency respnse f the per unit susceptances (i.e. the imaginary parts f ca,, ab, ). They are depicted by the thin and the thick curves, respectively. The base value f the susceptances is the cnductance G f the single-phase lad. The curves shw that at the fundamental, the 5 th harmnic and the 7 th harmnic frequencies, the tw admittances exhibit equal susceptances magnitudes (0.57 per unit r,equivalently, G / 3 Siemens) but f ppsite signs, as expected. The imprvement achieved thrugh the use f the suggested balancing circuit f Fig. 7 is manifested in the wavefrms f the three line currents I a, I b and I c depicted in Fig. 9. The equatins f these line currents are given in the Appendix. Cmparing these curves with thse given earlier in Fig. 4 (btained using the Steinmetz circuit), it can be clearly seen that these wavefrms are balanced and their shapes are much clser t the pure sine waves f the fundamental pwer frequency. In fact the time curves f the line currents are similar t the wavefrm f any f the supply line vltages as well as t the line currents I b and I c in Fig. 3. Fig. 10 illustrates the per unit amplitudes f the mdified harmnic cmpnents in the three line currents I a, I b and I c resulting frm the applicatin f the suggested balancing circuit f Fig. 7 versus the harmnic rder h. Only integer dd harmnic rders h are relevant in this plt. The curves shw that fr each f the fundamental (h = 1) as well as the 5 th and 7 th harmnic frequencies (i.e. h = 5 and h = 7), the per unit amplitudes f the harmnic cmpnents in the three line currents and are equal. Results f cmputing the phase angles indicated als that the phase shifts between each tw f the line currents are 2 p/3 radians in the crrespnding phase sequence (i.e. negative fr h = 5 and psitive fr h = 7), This implies that the suggested circuit is capable f balancing the 5 th and 7 th harmnic currents in additin t thse f the pwer frequency. The superirity f the mdified balancing circuit f Fig. 7 can be recgnized by cmparing the plts f Fig.10 with thse depicted in Fig. 5 fr the Steinmetz circuit The per unit magnitudes f the negative sequence cmpnents in the line currents, as functins f the harmnic rder h, are given in Fig. 11. Of special interest are the values crrespnding t h = 1 (fundamental), h = 5 and h = 7. The plt shws zer values f the negative sequence cmpnents fr all f them, since the balancing admittances ca,, ab, were selected, accrding t Eq. (1), such that the circuit will be balanced at these three frequencies. The advantage f the suggested circuit will be recgnized if this plt is cmpared with that shwn in Fig.6 fr the Steinmetz cnnectin. It is seen frm Eqs. (1) That there are tw cnditins which can be frmulated fr each additinally cnsidered harmnic. With these equatins it will be pssible t identify tw circuit elements. They must be reactive (in rder t avid hmic lsses), and they shuld be f ppsite nature (i.e. an inductance in ne f the tw branches ab r ca and a capacitance in the ther branch). The extensin f the prpsed balancing prcedure in rder t take ne mre f the backgrund harmnics f the supply vltage (h = 11 fr example) int accunt is based n the additin f ne mre reactive element t each f the tw admittances ca,, ab,. Accrdingly, tw mre equatins shuld be added t Eqs. (1), and slved simultaneusly. They are: ca, (11 w ) = jg / 3, ab, (11 w ) = + jg / 3. It is believed that it is pssible t design different LC structures. Fig. 9. Plts f the three line currents I a, I b and I c using the suggested circuit given in Fig. 7 with a supply vltage plluted by 4% fifth and 3% seventh harmnics, respectively. Currents are in per unit based n VG I a (thinnest curve), I b (curve f medium thickness) and I c (thickest curve) Fig. 10. Plts f the per unit harmnic cntent in the three line currents I a, I b and I c fr the balanced cnditin shwn in Fig.2 (using the mdified circuit shwn in Fig. 7). I a (thinnest curve), I b (curve f medium thickness) and I c (thickest curve) 22 Electrical Pwer Quality and Utilizatin, Jurnal Vl. XVI, N 1, 2013

5 harmnic rder h was investigated. Of special interest are the values crrespnding t h = 1 (fundamental), h = 5 and h = 7. Results shw a zer value f the negative sequence fr all f them. REFERENCES Fig. 11. Plts f the per unit negative sequence cmpnent in the line currents I a, I b and I c fr the circuit shwn in Fig. 7 (i.e. using the mdified circuit) versus the per unit frequency. 4. CONCLUSIONS The Steinmetz circuit des nt guarantee the balancing f the different harmnics in the line currents. Fr a single-phase resistive lad cnnected acrss the tw lines b and c, three strngly distrted line currents will result. Regarding the fifth harmnics, their percentage in the line currents I a, I b and I c are, respectively, 2.8, 1.2 and 3.8 times higher than thse in the supply vltage. The crrespnding magnificatin factrs f the seventh harmnic are 4.1, 0.9 and 3.5, respectively. Results indicate the absence f any negative sequence cmpnent fr h = 1, since the Steinmetz circuit is designed fr the fundamental frequency. The crrespnding values fr h = 5 and h = 7 are and per unit, respectively. A mdified circuit is suggested fr balancing the pwer netwrk at the pwer frequency w 0 and is, simultaneusly, capable f mitigating the adverse effects resulting frm the fifth and the seventh harmnics in the supply vltage. The balancing admittances will nw be selected such that the netwrk will be balanced at h = 1, h = 5 and h = 7. The results f a case study are presented in rder t validate the perfrmance f the prpsed circuit and t demnstrate its superirity ver the cnventinal Steinmetz cnnectin. The imprvement is reflected in the wavefrms f the three line currents. They are mre balanced and much clser t the pure sine waves f the fundamental pwer frequency. The time curves f the line currents are similar t the wave frm f the supply line vltages. Fr each f the fundamental as well as the 5 th and 7 th harmnic frequencies, the amplitudes f the harmnic cmpnents in the three line currents I a, I b and I c are equal with 2p / 3 radians phase differences. This implies that the suggested circuit is capable f balancing the 5 th and 7 th harmnic currents in additin t thse f the pwer frequency. The dependence f the magnitudes f the negative sequence cmpnents in the line currents n the 1. Miller T. J.: Reactive Pwer Cntrl in Electric Systems. Bk, Wiley-Interscience, 1982, Chapter Gyugyi L., Ott R., Putman T.: Principles and Applicatin f Static Thyristr-Cntrlled Shunt Cmpensatrs. IEEE Trans.n Pwer Apparatus & Systems, Vl. 97, Sept. /Oct. 1978, pp Chen T.: Criteria t Estimate the Vltage Unbalances due t High Speed Railway Demands. IEEE Trans. n Pwer Systems, Vl. 9, N. 3, August 1994, pp Lee S., Wu C.: On-line Reactive Pwer Cmpensatin Schemes fr Unbalanced three-phase fur Wire Distributin Feeders. IEEE Trans. n Pwer Delivery, Vl. 8, N. 4, Oct. 1993, pp Jrdi O., Sainz L., Chindris M.: Steinmetz System Design under Unbalanced Cnditins. Eurpean Transactins n Electric Pwer, Vl. 12, N. 4, 2007, pp Czarnecki L.: Reactive and Unbalanced Currents Cmpensatin in three-phase asymmetrical Circuits under nn-sinusidal Cnditins. IEEE Trans. Instrum. Meas., Vl. 38, N. 3, Jan. 1989, pp Czarnecki L. Minimizatin f Unbalanced and Reactive Currents in asymmetrical Circuits with nn-sinusidal Vltage. Prc. Inst. Elect. Eng. B, Vl. 139, N. 4, July 1992, pp Car M., Sainz L., Pedra J.: Study f the Pwer System Harmnic Respnse in the Presence f the Steinmetz Circuit. Electric Pwer Systems Research 2006, Vl. 76, N.12, pp Sainz L., Pedra J., Car M.: Backgrund Vltage Distrtin Influence n Pwer Electric Systems in the Presence f the Steinmetz Circuit. Electric Pwer Systems Research 2009, Vl. 79, N. 1, pp Sainz L., Pedra J., Car M.: Steinmetz Circuit Influence n Electric System Harmnic Respnse. IEEE Trans.n Pwer Delivery, Vl. 20(1), N. 2, 2005, pp Sainz L., Car M., Car E.: Analytical Study f the Series Resnance in Pwer Systems With the Steinmetz Circuit. IEEE Trans. n Pwer Delivery, Vl. 24, N. 4, 2009, pp Abdelaziz A.., Mekhamer S.F., Ismael S.M.: Surces and Mitigatin f Harmnics in Industrial Electrical Pwer Systems: State f the Art. Online Jurnal n Pwer and Energy Engineering (OJPEE), Vl. 3, N. 4, pp , Saied M.: Effect f the Backgrund Harmnics and the Netwrk s Impedance Characteristics n the Operatin f Shunt Capacitrs. STM Jurnal f Pwer Electrnics and Pwer Systems, Vl. 3, Issue1, pp , APPENDIX Relatins fr the Line Currents: Cnsidering the circuit in Fig. 2, the fllwing equatins can be btained using Ohm s and Kirchhff s laws: I a = V ab ab V ca ca I b = V bc G V ab ab I c = V ca ca V bc G (A-1) (A-2) (A-3) Fr the mdified balancing circuit in Fig. 7, the crrespnding relatins will be: I a = V ab ab, V ca ca, I b = V bc G V ab ab, I c = V ca ca, V bc G (A-4) (A-5) (A-6) 23

6 It shuld be nted that nce the mdified circuit in Fig. 7 is analyzed, the results crrespnding t the Steinmetz circuit in Fig. 2 can be easily btained by substituting the fllwing special admittance values: C11 = G/ ( w 3) L 1 = 0 L22 = 3/( wg) and L 2 = 0 The riginal unbalanced circuit invlving nly the single phase lad (Fig. 1) can be simulated by inputting the fllwing special values: C 11 = 0 and L 22 =. Prf. Mhamed M. Saied Received the B.Sc. degree (Hnrs) in Electrical Engineering frm Cair University, Egypt in 1965, then the Dipl.- Ing. and Dr-Ing. degrees frm RWTH Aachen, Germany, in 1970 and Frm 1974 t 1983, he was at Assiut University, Egypt. In 1983, he jined Kuwait University where he served as full prfessr and EE Department Chairman. He spent ne-year sabbatical leave (1998) as a Visiting Prfessr at Cair University. Since his retirement in 2009, Prf. Saied. Wrks as an Independent Researcher, Giza, Cair, Egypt. He is an IEEE Senir Member since m.saied@ieee.rg 24 Electrical Pwer Quality and Utilizatin, Jurnal Vl. XVI, N 1, 2013