Active Load Balancing in a Three-Phase Network by Reactive Power Compensation

Transcription

1 Ative Load Balaning in a hree-phase Network by eative Power Compensation Adrian Pană Politehnia University of imisoara omania. ntrodution. Brief overview of the auses, effets and methods to redue voltage unbalanes in three-phase networks During normal operating ondition, a first ause of voltage unbalane in three-phase networks omes from the asymmetrial struture of network elements (eletrial lines, transformers et.. Best known example is the asymmetrial struture of an overhead line, as a result of asymmetrial spatial positioning of the ondutors. Also the total length of the ondutors on the phases of a network may be different. his asymmetry of the network element is refleted in the asymmetry of the phase equivalent impedanes (self and mutual, longitudinal and transversal. he impedane asymmetry auses then different voltage drop on the phases and therefore the voltage unbalane in the network nodes. As an example of orretion method for this asymmetry is the well-known method of transposition of ondutors for an overhead eletrial line, whih allows reduing the voltage unbalane under the admissible level, of ourse, with the ondition of a balaned transfer on the phases. But the main reason of the voltage unbalane is the s supply, many of whih are unbalaned, single-phase - onneted between two phases or between one phase and neutral. Many unbalaned s, having small power values (a few tens of watts up to 5- kw, are onneted to low voltage networks. But the most important unbalane is produed by high power single-phase industrial s, with the order MW power unit, that are onneted to high or medium voltage eletrial networks, suh as welding equipment, indution furnaes, eletri rail tration et. Current and voltage unbalanes aused by these s are most often aompanied by other forms of disturbane: harmonis, voltage sags, voltage flutuations et. (Czarneki, 995. Current unbalane, whih an be assoiated with negative and zero sequene omponents flow, lead to inreased longitudinal losses of ative power and energy in eletrial networks, and therefore lower effiieny. Voltage unbalane auses first negative effets on the rotating eletrial mahines. t is assoiated with inreased heating additional losses in the windings, whose size depends on amount of negative sequene voltage omponent. t also produes parasiti ouples, whih is manifested by harmful vibrations. Both effets an redue the useful life of eletrial mahines and therefore signifiant material damage. ransformers, apaitor banks, some protetion systems (e.g. distane protetion, threephase onverters (three-phase retifiers, AC-DC onverters et. are also affeted by a threephase unbalaned system supply voltages.

2 Power Quality Monitoring, Analysis and Enhanement egarding limiting voltage unbalanes, as they primarily due to unbalaned s, the main methods and means used are aimed at preventing respetively limiting the unbalanes. From the measures intended to prevent unbalanes, are those who realize a natural balane. t may mention here two main methods: balaned repartition of single-phase s between the phases of the three-phase network. his is partiularly the ase of single-phase s supplied at low voltage; onneting unbalaned s to a higher voltage level, whih generally orresponds to the solution of short-iruit power level inreasing at their terminals. s the ase of industrial s, high power (several MVA or tens of MVA to whih power is supplied by its own transformer, other than those of other s supplied at the same node. Under these onditions the voltage unbalane fator will derease proportionally with inreasing the short-iruit power level. From the ategory of measures to limit unbalaned onditions are: balaning iruits with single-phase transformers (ott and V iruit (UE, 998; balaning iruits through reative power ompensation (teinmetz iruit, single and three phase, whih may be applied in the form of dynami ompensators type VC (tati Var Compensator (Gyugyi et al., 98; Gueth et al., 987; an et al., 99; Czarneki et al., 994; Mayordomo et al., ; Grünbaum et al., ; aid et al., 9. high performane power systems ontrollers - based on self-ommutated onverters tehnology (e.g. type ACOM - tati Compensator (Dixon, 5. his hapter is basially a theoretial development of the mathematial model assoiated to the iruit proposed by Charles Proteus teinmetz, whih is founded now in major industrial appliations.. Load balaning mehanism in the teinmetz iruit As is known, teinmetz showed that the voltage unbalane aused by unbalaned urrents produed in a three-phase network by onneting a resistive (with the equivalent ondutane G between two phases, an be eliminated by installing two reative s, an indutane (a oil, having equivalent suseptane BL G/ and a apaitane (a apaitor with equivalent suseptane BC G/. he ensemble of the three reeivers, forming a delta onnetion (, an be equalized to a perfetly balaned three-phase s, in star onnetion (, having on eah branh an equivalent admittane purely resistive (ondutane with the value G (Fig.. a b Fig.. teinmetz montage and its equivalene with a balaned, purely resistive

3 Ative Load Balaning in a hree-phase Network by eative Power Compensation o explain how to ahieve balaning by reative power ompensation of a three-phase network supplying a resistive, it will onsider suessively the three reeivers, supplied individually. For eah reeiver will determine the phase urrents, whih are then deomposed by referene to the orresponding phase to neutral voltages, to find ative and reative omponents of urrents, whih are used then to determine the ative and reative powers flow on the phases of the network. t is assumed that the phase-to-neutral voltages and phase-to-phase voltages at soure forms perfetly symmetrial three-phase sets. Also ondutor s impedanes are negleted. herefore, for the ase of supplying the resistive having equivalent ondutane G between and phases (Fig.., a urrent in phase with the applied voltage is formed on the phase ondutor: a Fig.. esistive supplied between and phases b he equation to alulate the rms value is: U G ( U G, ( where U is the rms value of phase-to-neutral voltage, onsidered the same on the three phases. On the phase ondutor, an equal but opposite urrent like the one on the phase is formed: ( he two urrents are now reported eah to the orresponding phase-to-neutral voltage, in order to find the ative respetively reative omponents of eah other. For this, the omplex plane is assoiated to the phase-to-neutral voltage; its phasor is positioned in the real axis,

4 Power Quality Monitoring, Analysis and Enhanement positive diretion. t is noted that the urrent phasor on the phase, (, is leading the orresponding phase-to-neutral voltage phasor, U, with an phase-shift equal to π /6 rad, whih means that the reative omponent has apaitive harater: π π ( ( a j( r ( os j( sin U G j U G (4 6 6 t an now dedue the equations for the ative and reative power flow on phase: * P jq U ( U G j U G (5 On the phase, the urrent phasor is lagging the voltage U with a phase-shift equal to π /6 rad, whih means that the reative omponent has indutive harater. By a similar alulation with the above, ative and reative powers flow on the phase are obtained: * P jq U ( U G j U G (6 t may be noted that the resistive supplied between two phases, absorbs ative power equal on the two phases. But it ours also on the reative power flow on the network phases, absorbing reative power on the phase, but returning it to the soure on the phase, without modifying the reative power flow on all three phases. On this ensemble, result: ( şi Q Q Q (7 P P P U G U G A similar demonstration an be done for supplying the apaitive having the equivalent suseptane BC G/, between phases and (Fig.. he urrent formed on the phase ondutor is leading the voltage with a phase-shift equal to π / rad (the omplex plane assoiated to the phase-to-phase voltageu : he rms value an be determined by the equation: j U BC (8 U B U G (9 he urrent formed on the phase, have the same rms value and is opposite to that on the phase: C ( Now reporting the two urrents to the orresponding phase-to-neutral voltages, it an be determined the ative and reative omponents of this, and then the ative and reative powers on the two phases: * P jq U ( U G j U G (

5 Ative Load Balaning in a hree-phase Network by eative Power Compensation * P jq U ( U G j U G ( t is noted that the apaitive absorbs the same reative power on the two phases at whih it is onneted. t ours also on the ative powers flow, absorbs ative power on phase, but returns it to the soure on the phase. On all three phases of the network, results: P P P şi Q Q Q ( U BC U G ( a Fig.. he apaitive supplied between phases and b he same method applies now to the ase of indutive, having equivalent suseptane BL G/ supplied between and phases (Fig. 4. he urrent formed on the phase ondutor is lagging the supplying voltage with an phase-shift equal to π / rad: he rms value an be determined using the equation: j U BL (4 U B U G (5 he urrent formed on the phase, have the same rms value and is opposite to that on the phase: L (6

6 4 Power Quality Monitoring, Analysis and Enhanement a Fig. 4. ndutive supplied between and phases Now reporting the two urrents to the orresponding phase-to-neutral voltages, it an be determined the ative and reative omponents of this, and then the ative and reative powers on the two phases: * P jq U ( U G j U G (7 * P jq U ( U G j U G (8 t is noted that the indutive absorbs the same reative power on the two phases at whih it is onneted. t ours also on the ative powers flow, absorbs ative power on phase, but returns it to the soure on the phase. On all three phases of the network, results: P P P and b Q Q Q ( U B U G (9 L o dedue the power flow on the network phases that supply simultaneously the three s previously onsidered, we an apply the superposition theorem (Fig. 5. Ative and reative powers flow run on the network phases that supply the ensemble of all three s are obtained by algebrai addition of ative and reative power previously deduted for the individual supplying iruits. t obtains:, Q Q Q ( P P P U G P P P U G Q Q Q ( P P P U G Q Q Q (

7 Ative Load Balaning in a hree-phase Network by eative Power Compensation 5 Q Q Q Q ( P P P P U G a Fig. 5. he ensemble of the three s t notes that after the ompensation, in the supplying network of the ensemble of the three reeivers, only ative power flows, the same on the three phases. he ompensation ondues to maximize the power fator ( Q and to ative balaning on the three phases: P Pphase U G (4 he ensemble of the three s, in onnetion an be equated with three equal ative s, single-phase, eah having equivalent ondutane with value G, in onnetion (Fig.. he three urrents have the same phase-shift with the orresponding phase-to-neutral voltages and have the same rms value (Fig. 5: U G (5 b. Most ommon appliations of teinmetz iruit teinmetz iruit is usually applied to balane large s, with values of order MW of power, whose ontribution to the voltage unbalane in the onneting node is very high. Figure 6.a presents simplified supply iruit diagram of a three-phase network, of a singlephase indution furnae. Furnae oil is onneted seondary of the single-phase transformer. ts primary is onneted between two phases of a medium voltage network. Capaitane C onneted in parallel with the, is to ompensate its reative omponent and therefore to improve the power fator. Capaitane C and indutane L are sized to ahieve balaning ative omponent, as shown above. But the ative power of the is variable, depending on the speifi tehnologial proess. o ensure adaptive single-phase balaning, apaitor banks having the apaitanes C and C and indutane L have to allow the ontrol of these values aording to variation.

8 6 Power Quality Monitoring, Analysis and Enhanement a b Fig. 6. implified iruit of teinmetz installation for -balaning appliations in the ase: a indution furnae; b railway eletri tration Another important appliation of the teinmetz iruit is balaning in three-phase networks whih supplies eletri tration railway, equivalent to a single-phase. Figure 6.b shows the simplified iruit diagram of a substation supply of setion from an eletri railway line. ine the is hanging rapidly and within large limits, the ompensator elements must satisfy the same requirements. s needed a dynami balaning (Grünbaum et al.,. he solution applied use a VC (tati Var Compensator realized by a C (hyristor Controlled eator. Controlling the thyristors (onneted bak-to-bak whih are in series with the indutanes L, L and L allow a dynami ontrol of indutive and apaitive urrents on the branhes of the ompensator. hus is performed a dynami ompensation of reative (inreasing the power fator and dynami balaning of ative in the supply network. Appliation of the teinmetz iruit is a simple solution, relatively inexpensive and effiient, whih an be applied to any voltage level at any value of power, a perfet balaning on the three phases is obtained. However, it presents the following inonvenienes: - the frequeny of 5 Hz, the ensemble - ompensator an be equated with impedane perfetly balaned, but on other frequenies, namely the higher harmonis that an be generated by the same or lose s, it auses a strong unbalane; - dimensioning the ompensator should be made taking into aount the restrition to avoid parallel resonane between the equivalent apaitane of the ompensator and the network equivalent indutane seen at the point of onnetion to the network, for harmoni frequenies present in the steady operation onditions, the apaitors will be inluded in passive filters for harmoni urrents. 4. he generalized teinmetz iruit he iruit available for a single phase ative onneted between two phases an be extended to a ertain unbalaned three-phase with ative and reative omponents (indutive and / or apaitive. he mathematial model for sizing the ompensator elements and determining the urrents flow and powers flow for the purpose of understanding the ompensation mehanism and for oneption of ontrol algorithms

9 Ative Load Balaning in a hree-phase Network by eative Power Compensation 7 depend on the presene or not of neutral ondutor and so of the presene or not of zero sequene omponents of urrents. n the following, the two situations are analyzed. 4. he generalized teinmetz iruit for three-phase three-wire networks A ertain three-phase eletri is onsidered, onneted to a three-wire network supplied by a balaned phase to phase voltages set. n suh situations usually an only know the values of the phase urrents and phase to phase voltages, network neutral don t exist or is not aessible. he set of phase to phase voltages is onsidered symmetrial (Fig. 7b, and the equivalent iruit of the is taken in onnetion, whose elements, for pratial reasons, are onsidered like admittanes (Fig. 7a. a b Fig. 7. he equivalent onnetion with admittanes for a ertain three-phase : a - definition of eletrial quantities, b - phasor diagram of voltages For the network in figure 7 we therefore have the following sets of equations: G jb G jb U U G jb (6 U (7 U U U U U U ( a U a U a U U a U a U (8 (a a U U U U (a U U U U (9 Using the equations (6 ( it obtains: ( U G B G B j G B G B U G B B j G B G U B G B j G G B (

10 8 Power Quality Monitoring, Analysis and Enhanement Neessary and suffiient ondition for the three phase urrents to form a balaned set is the anellation of the negative sequene urrent omponent: i ( a a ( Putting the anellation onditions for the real and imaginary parts of i obtained by substituting equations ( in ( we obtain the onditions: ( G G G B B ( G G B B B ( his system of equations define the relationship that should exist between the six elements of the equivalent onnetion of a, so that, from the point of view of the network it appear as a perfetly balaned ( i. hese onditions an be obtained by hanging (ompensating the equivalent parameters using a parallel ompensation iruit, also in onnetion, so that equations ( (Gyugyi et al., 98 to be satisfied for the ensemble - ompensator (Fig. 8. Fig. 8. he ensemble unbalaned - shunt ompensator he problem lies in determining the elements of the ompensator, so that, knowing the elements of the equivalent iruit of the, to obtain an ensemble whih is perfetly balaned from the point of view of the network, as it means that after the ompensation, the urrents on the phases satisfy the ondition: a a (4 he ompensator will not produe hanges in the total ative power absorbed from the network (whih would mean further losses and hene will ontain only reative elements ( G G G. n equations ( will be replaed so:

11 Ative Load Balaning in a hree-phase Network by eative Power Compensation 9 G G G G ; G G ( ; ( ; ( B B B B B B B B B From the equations ( resulting the equation system: B B A B B B B (5 (6 Where: A G B G G B B G B B G B (7 Unknowns are therefore: B, B and B. With two equations and three unknowns, we are dealing with indeterminay. A third equation, independent of the first two, whih expresses a relationship between the three unknowns, will result by imposing any of the following onditions: a. full ompensation of reative power demand from network; b. partial ompensation of reative power demand (up to a required level of power fator;. voltage ontrol on the bus bars trough the ontrol of reative power demand; d. install a minimum reative power for the ompensator; e. minimize ative power losses in the supply network of the. n this hapter we will onsider only the operation of the ompensator sized aording to the a riterion, other riteria an be treated similarly. 4.. izing the ompensator elements based on the riterion of total ompensation of reative power demand from the network Aording to a riterion, in addition to balaning, ompensation should also lead to anellation of the reative power absorbed from the network on the positive sequene (osϕ. his is equivalent to the additional ondition: ( m (8 is the positive sequene omponent orresponding to the urrent of the ensemble - ompensator. But for this it an write the ondition: ( a a, (9 beause a a, where, and are the urrents absorbed by the network after the ompensation. As the supplementary ondition will be: ( mean: G G ( B B m (4 (4

12 Power Quality Monitoring, Analysis and Enhanement Assoiating now the equations ( and (4, where the equations (5 are replaed, the system of three equations with three unknowns is obtained: B A B B B C (4 where: C ( G B G B olving the system (4 leads to the following solutions: mean: (4 B ( A C B ( B C B A C ( ( B B G G B B G G B B G G ( ( Using now the equations of transformation of a delta onnetion iruit in a equivalent onnetion iruit, is ahieved: (44 (45 G G G G G G G B B B hese equivalenes are illustrated in Figure 9. (46 Fig. 9. Equivalene of the ensemble ompensator with a balaned ative

13 Ative Load Balaning in a hree-phase Network by eative Power Compensation 4.. he ompensation iruit elements expressed by using the sequene omponents of the urrents Expressing the ompensation iruit elements by using the sequene omponents of the urrents will allow a full interpretation of the mehanism of ompensation. For this, lets onsider again the general three-phase unbalaned, supplied from a balaned three-phase soure, without neutral, represented by the equivalent iruit as shown in Fig. 7. he three absorbed urrents will be: U ( a (a U (a a ( a U (a (a a We apply the known equations for the sequene omponents: (47, ( a a ( a a ( where (orresponding to the referene phase,. eplaing the equations (47 in equations (48 the symmetrial omponents depending on the admittanes are obtained: (48 and are the positive, negative and zero sequene omponents U ( (a a U ymmetrial omponents of urrents on the ompensator phases are obtained by the same way: ( j B B B U j B B B U U B B j U B B B ( a a ( ( Writing symmetrial omponents of the phase urrents absorbed from the network by the ensemble - ompensator: (49 (5 (5

14 Power Quality Monitoring, Analysis and Enhanement he sizing onditions reeive the form: m( e( m( ( m( U B B B e( U ( B B m( U ( B B B olving this system of equations give the following solutions: B m( e( m( U B m( e( U B m( e( m( U (5 (5 ine the positive sequene and negative sequene urrents flow an be onsidered independent, ompensator also an be deomposed into two independent ompensators, fititious or real. herefore one ompensator will be symmetrial ( and produe ompensation (anellation of the reative omponent of the positive sequene urrents and the other will be unbalaned (-, and will ompensate the negative sequene urrent. he mehanism of the ompensation of the sequene urrents omponents is illustrated in Figure. Fig.. Compensator representation by two independent ompensators: one for the positive sequene ompensation and other for the negative sequene ompensation of the urrent

15 Ative Load Balaning in a hree-phase Network by eative Power Compensation he elements of the two ompensators will be: B B B m( U B e( m( U U B m( U B e( m( U U Expressing then real and imaginary parts of symmetrial omponents depending on the elements of the equivalent iruit of the, i.e.: it obtain: ( m( B B B U e( G B G G B U m( G B B G B U, (54 (55 ( B B B B B B ( B B B B G G ( B B B B G G B B B B ( G G t is noted that the sum of - ompensator elements values is zero. (56 B B B (57 nstead, the sum of the ompensator elements will be equal and opposite to the sum of the reative elements of the. 4.. Currents flow in the ensemble - ompensator expressed by symmetrial omponents On the basis of equations (5, the urrents on the branhes of the two and - fititious ompensators an be determined, using real and imaginary parts of the sequene urrents of :

16 4 Power Quality Monitoring, Analysis and Enhanement m ( ( m( e m ( ( m( e With these equations we an determine the urrents on the phases of both fititious ompensators, and then the urrents flow in symmetrial omponents, into the ensemble - ompensator. ( j,,,,, ( j ( j,,,,,,,, ompensator produes a three-phase set of positive sequene urrents, whih ompensate the reative omponent of the positive sequene urrent on eah phase, and - ompensator produes a three-phase set of negative sequene urrents, whih ompensate the negative sequene urrent on eah phase (both ative and reative omponent: jm( a jm( a m j ( (58 (59 (6 a ( a ( (6 he urrents on the three phases, after ompensation, represent a balaned set, positive sequene and ontain only the ative omponent (they have zero phase-shift relative to the orresponding phase-to-neutral voltage, equal to the ative omponent of positive sequene urrent: e ( e( ( a a e Figure shows the ompensation mehanism of urrents symmetrial omponents using phasor diagram. tarting from the three phasors of unbalaned urrents, onsidered arbitrary, but hek the ondition (whih means they have a ommon origin in the enter of gravity of the triangle formed by the peaks of the phasors, were determined the referene (6

17 Ative Load Balaning in a hree-phase Network by eative Power Compensation 5 symmetrial omponents phasors (orresponding to phase, Figure.a. hey will allow for the determination of the phasors and beause: m( (6 a b Fig.. Phasor diagram illustrating the ompensation mehanism of the urrent symmetrial omponents: a - determination of symmetrial omponents of referene (phase, b - ompensation of the negative sequene omponent, - ompensation of the imaginary part of the positive sequene omponent Currents on the phases of the ensemble - ompensator are then obtained, first by ompensating the negative sequene (Figure.b and then by ompensating the positive sequene (Figure. realized on the basis of equations:

18 6 Power Quality Monitoring, Analysis and Enhanement s d i s d i a a s d i a a Obviously that in pratial appliations is not eonomially to use two ompensators. A single ompensator, having variables suseptanes will be suffiient to produe both positive sequene ompensation (inreasing power fator and the negative sequene urrents ompensation ( balaning. ( Currents and ompensation iruit elements expressed by urrents Using the urrents equations on the phases with ative and reative omponents and supposing their indutive harater, mean: j a r j a ( a r a ( j, and replaing into the sequene urrents equations (48 results: a r (65 ( a a a j ( r r r (66 a a r a r j r r a r a By developing the equations (58, we obtain relations between urrents respetively suseptanes on the ompensator branhes respetively ative and reative omponents of urrents on the phases: ( a a ( r r r ( a a ( r r r ( a a ( r r r B ( a a ( r r r U B ( a a ( r r r U B ( a a ( r r r U (67 (68

19 Ative Load Balaning in a hree-phase Network by eative Power Compensation 7 And for the fititious and - ompensators, taking into aount the equations (58, are obtained: ( r r r ( a a ( r r r ( a a ( r r r ( a a ( r r r B B B ( r r r 9 U B a a ( r r r U B a a ( r r r U B a a ( r r r U (69 ( he urrents and powers flow into the ensemble -ompensator expressed in phase omponents Analytial determination of the urrents and powers flow into the ensemble - ompensator is useful for performing alulations for sizing or for heking the auray of ompensation in real installations. Using equations (59, written to the omplex plane assoiated with phase, is determined, the urrent equations,, respetively written in omplex plans assoiated to orresding phase-to-neutral voltages (noted respetively, making the,, operations: esult:,, a,, a (7,,,,, ( j (,,,,, ( j (,,,,, ( j ( Combining now the equations (69 with (7 the equations for the urrents on the and - phases, relatively to the orresponding phase-to-neutral voltages are obtained: (7

20 8 Power Quality Monitoring, Analysis and Enhanement j ( r r r a a a j r r r j j [ ( ] [ ( ] a a a r r r [ ( ] a a a r r r Using now the equations (69, (7, and (7, for the urrents on the ompensator phases are obtained the equations: (7 ( j ( a a a jr ( a a a jr a a a r t an be seen lear that the ompensator provides on eah phase a reative urrent omponent whih is equal and opposite to the reative omponent of urrent. As the ative omponents of the ompensating urrents, it is positive or negative, being equal to the differene between the ative omponent of positive sequene urrent (whih is the network on eah phase after ompensation and the ative omponent of eah urrent. Can be now alulated the urrents absorbed on eah phase by the ensemble - ompensator: (74 t s obtained: (75 (76 ( a a a e( he ative and reative powers on the three phases of the ompensator an now be easily alulated: P U a U a a a ( r ( 6 r U a a a P U a U a a a ( r r U ( a a a 6 P U a U a a a ( r r U ( a a 6 a (77

21 Ative Load Balaning in a hree-phase Network by eative Power Compensation 9 P P P P P, P P, P P (78 o, the positive sequene ompensator intervenes only on the reative power flow, and the negative sequene ompensator, although it hange the ative power on eah phase of the ompensated network, on the ensemble of the three phases don t hange the ative power balane beause: P P P (79 his means that on some phase(s the - ompensator absorbs ative power, and on the other(s debits ative power as otherwise noted above. He thus produes a redistribution of ative power between the phases, balaning them. t an be said that it made ative power ompensation. After ompensation: and so: P P P U ( a a a (8 P P U ( (8 ph ph a a a ph,, ph,, For the reative powers are obtained the equations: Q U 4 6 r r r ( a a Q U r 4 r r 6 ( a a Q U r r 4 r 6 ( a a (8 Q Q Q U ( r r r (8 Q U r U ( r r r Q U r U ( r r r Q U r U ( r r r For the ensemble -ompensator: Q Q Q Q Q Q Q Q Q Q Q Q (84 (85

22 4 Power Quality Monitoring, Analysis and Enhanement On eah phase the ompensator ompensates the reative power orresponding to the positive sequene omponent of urrents and - ompensator the reative power orresponding to the negative sequene omponent of urrents. On eah phase the ompensator produes a reative urrent equal and opposite sign to the reative urrent. he ompensator is symmetrial and makes the total ompensation for the reative power of the. Qph Qph (86 ph,, ph,, he - ompensator is unbalaned, the unbalane depends on the unbalane. On eah phase - intervenes with different reative powers absorbed or debited, but on the ensemble of the three phases it don t affets the reative power flow beause: (87 Qph ph,, t an be said therefore that the -ompensator performs a redistribution of reative power between phases of the ompensated network. 4. teinmetz generalized iruit for three-phase four-wire networks For the ase of three-phase four-wire network, displaying the mathematial model will be brief, mathematial development method is the same as in the previous ase. Consider an unbalaned, whih may be an individual reeiver or an equivalent redued at the interest setion of the network (e.g. bars of a low voltage transformer, supplied from a three-phase four-wire network. he neutral ondutor indiates the presene of single-phase reeivers, typial soures of urrent unbalane. Equivalent iruit of suh a will always be n onnetion. Artifiial balaning on the network phases that supply suh a an be done, as in the ase of three-wire networks, by stati reative power soures, whih makes shunt ompensation. But this time will be used simultaneously two suh three-phase ompensators, ontaining only reative iruit elements, whih need will be justified during the exposure of the mathematial model: one with onnetion and another with n onnetion. Figure shows the simplified eletrial iruit for an unbalaned, using the two ompensators mentioned, where were speified in some of the notations used in the mathematial model. he mathematial model used the following hypotheses: - was onsidered only steady operating onditions; - three - phase sets of phase-to-phase and phase-to-neutral voltages in the interest setion are perfetly balaned; - do not onsider non-sinusoidal regime: supplying voltages waves are perfetly sinusoidal and the elements and the network are onsidered linear.

23 Ative Load Balaning in a hree-phase Network by eative Power Compensation 4 Fig.. eative power shunt ompensators installed to balane a supplied from a three-phase four-wire network - equivalent eletrial iruit 4.. Criteria for sizing of the ompensators elements etting values of the two ompensators suseptanes will be done on the basis of some riterions regarding both unbalaned regime and power fator improvement, interdependent ations for unbalaned power distribution networks. Moreover, a simultaneous approah of those two questions must be made neessarily beause, as has been noted in the previous ase, they are always interrelated. o ompensate will use a single ompensator or both, as the aim is only balaning, only power fator improvement or a simultaneous ation. But the reative power flow hange produed by the ompensators requires intervention to hek the voltage level in the network buses, whih is an element of restrition. On the other hand, reative power flow ontrol an be a method of voltage ontrol. We annot forget the tehno-eonomi effiieny of the ompensation, whih an be maximized by minimization the investment osts for failities that maximize the benefits offset by inreasing power quality and effiieny of eletriity use. herefore, to sizing the elements of the two ompensators an be applied one of the following riteria, or sub-riteria: a. power fator improvement without taking into aount the unbalaned regime; a. power fator improvement in the supply network by the full ompensation of the reative urrent omponent of positive sequene, using a symmetrial ompensation; a. minimize total ative power losses in the supply network; b. balaning without taking into aount the improvement of power fator;. power fator improvement and unbalane dereasing;. minimize unbalane by anellation the zero sequene urrent by ompensation and power fator inrease by full ompensation of positive sequene reative power;. redue the unbalane by full ompensation of the negative sequene urrent and power fator inrease by full ompensation of the positive sequene reative power;

24 4 Power Quality Monitoring, Analysis and Enhanement. full balaning and power fator maximization; -. intervention of ompensator only for the negative sequene urrent flow; -. minimization of the ative power losses on the ompensators; -. minimization of the installed reative power for the ompensators; 4. balaning and reative power ompensation to a required power fator; d. Minimization of the total ative power losses in the supply network. As in the ase of the three-wire network are interest the omponents of the ompensator and the urrents and powers flow on the ensemble -ompensator. hese objetives are neessary to both sizing the ompensation equipment and proess ontrol, when the ompensation is subjet of dynami ontrol. For the present study was hosen to present the ase of dimensioning the two ompensators elements by applying the riterion C (full balaning and power fator maximization. his riterion orresponds to a regime that an be onsidered the most advantageous from the tehnial point of view of network operating onditions. As before, the symmetrial omponents method is applied, based on symmetrial omponents of urrents equations orresponding on the phases and on the two ompensators, based on the phase omponents: ( a a a j ( r r r a ( a a ( r r j ( a a r ( r r a ( a a ( r r j ( a a r ( r r j ( 6 j 6 j ( ( ( ( (88 (89 j ( j ( ( he meaning of the notations in the above equations is no longer needed to be explained. 4.. Determination of ompensation urrents and suseptanes Maximizing the power fator requires the full ompensation of positive sequene reative power, so the anellation of the imaginary omponent of positive sequene urrent. otal balaning requires, as in the ase of three-wire network, the negative sequene urrent (9

25 Ative Load Balaning in a hree-phase Network by eative Power Compensation 4 anellation of urrents, by the anellation of its real and imaginary parts. But this time, in the network is also present the zero sequene omponent of urrent; on the neutral ondutor of the flow a urrent three times greater than this. he reative shunt ompensation produed by the two ompensators will have to anel that omponent, also by the anellation of its real and imaginary parts. From the analytial point of view results, for the sequene omponents of urrents on the phases of the ensemble - ompensator, the following five onditions: where: ( ( ( ( ( m, e, m, e, m (9 o determine the six urrents on the branhes of the two ompensators, a sixth ondition is needed. n this paper we onsider the ondition -, as shown above, results from the ondition: (9 (9 Equation (9 refers to the reative omponents of urrents on the branhes of the ompensator. Applying equations (88 (89 (9 in (9 and by joining the onditions (9 and (9 form a system of six equations with six unknowns, whih, expressed in matrix, has the form: A B C D E n (94, A, B, C, D, E are known quantities, with expressions that an be written depending on ative and reative omponents of urrents on the phases, respetively on the sequene omponents of these: ( r r r m( A B ( a a a r r e( ( a a r r r ( ( a a a r r e( ( a a r r r ( C 6m D E 6m (94 (95

26 44 Power Quality Monitoring, Analysis and Enhanement olving the system of equations (94, we obtain the urrents equations on the branhes of two ompensators: A E A DE ADE,, B C DE C E B C DE,, 6 6 ( ( ( a a r a a r a a r ( ( ( a a a a a a (96 (97 mmediately results the equations for the six suseptanes: ( ( B B B B B B U a a r U a a ( ( U a a r U a a ( ( U a a r U a a ( ( ( ( ( ( B G G B B G G s s s s s B G G B B G G s s s s s B G G B B G G s s s s s (98 ( he urrent flow on the ensemble - ompensator, in phase omponents he urrents equations on the phases of ompensator respetively, written for the omplex plans reported to phase-to-neutral voltages involved (notation "*", are: ( ( ( ( ( ( * j * j * j ( * * * j j j ( Using equations for alulating the six ompensation urrents resulted from the riterion C- (97 urrents on the phases of two ompensators an be dedued: ( ( * a a a a a j ( ( * a a a a a j ( ( * a a a a a j (

27 Ative Load Balaning in a hree-phase Network by eative Power Compensation 45 ( * j * j ( * j ( a a r a a r a a r ( As a useful observation, the rms values of the three phase urrents on the phases of ompensator are equal, as expeted, the three urrents forming a symmetrial and balaned set, of negative sequene: ( / (4 * * * a a a a a a a a a Calulating now the urrents absorbed on the phases of the ensemble - ompensator, with the equations: t obtains: * * * * * * * * * * * * (5, ( ph phase,, (6 * * * * ph r ph r ph r ph r ph r ph r ( a a a ( ( * * * e e (7 t finds that the sizing onditions (9 are satisfied: the negative and zero sequene urrents, and the reative omponent of the positive sequene urrent were aneled by ompensation. n other words, balaning and full ompensation are obtained (osϕ osϕ. After the ompensation the phase urrents beome purely ative, balaned with the rms value equal to the arithmeti average of the three ative urrents of the. But let's not forget the urrents flow on neutral ondutors. o determine the urrent equation on the neutral ondutor, using the formula: N (8 Using the urrents equations written for the omplex plane assoiated with phase : j a r ( a r a r ( r a ( ( a r a r r a a j j a j j (9 result: ( ( ( ( N a a a r r j r r r a a (

28 46 Power Quality Monitoring, Analysis and Enhanement he urrent on the neutral ondutor of ompensator: where: N, ( j ( ( a j j a j j ( Using equations (97 for the urrents on the phases of ompensator, for the urrent N results: so that: N N, ( N N N (4 t an be onluded that: N (5 o the ompensator injet on the neutral ondutor a urrent equal and opposite to that on the neutral ondutor of the, aneling it he powers flow into the ensemble - ompensator he ative powers equations on the phases are: P U U a a P U U a a P U U a a (6 he ative powers on the phases of the ompensator are: ( ( ( P U U * a a a a P U U * a a a a P U U * a a a a (7 t may be noted that the ompensator modifies on eah phase the ative power flow: on some phase debits while on the others absorbs ative power. But on the ensemble of the three phases, the ative power flow is not affeted beause: (8 P ph ph,,

29 Ative Load Balaning in a hree-phase Network by eative Power Compensation 47 nstead, the ompensator does not intervene at all on the ative power flow: P P P (9 After ompensation: ( ( ( P U U a a a a P U U a a a a P U U a a a a ( t is noted one again that the presene of the ompensator does not affet the ative power absorbed from the network (the same value before and after ompensation. Pph Pph U ( a a a ( ph,, ph,, For the reative powers on the three phases into the ensemble - ompensator it obtains: Q U r Q U r Q U r ( ( ( Q U U * r a a Q U U * r a a Q U U * r a a ( ( ( Q U U Q U U ( Q U U ( * r a a r * r a a r * r a a r (4 t is noted one again the full ompensation of reative power. he two ompensators determine together on the eah phase a reative urrent flow, equal and opposite to the reative urrent. t an be observed that: (5 Q ph ph,, o, the ompensator absorbs reative power on some phase and debits reative power on the others, but it does not affet the reative power flow on the ensemble of the three phases. herefore it an be said that the ompensator realizes a redistribution of reative power between the three phases. t an be observed also: Qph Qph (6 ph,, ph,,

30 48 Power Quality Monitoring, Analysis and Enhanement n fat, the ompensator is that one whih effetively realizes the reative power ompensation of the he urrents flow into the ensemble - ompensator expressed by symmetrial omponents of urrents Expressing the ompensation urrent depending on symmetrial omponents of the urrents allows a omplete interpretation of the mehanism of the reative power ompensation and balaning. Considering the same sizing riterion (-, the sizing onditions (9 will be written as: ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( m m m e e e m m m e e e m m m (7 Expressing the sequene omponents of the urrents on the phases of the two ompensators expressed by the ompensation urrents, it s obtained the equation system: ( ( ( ( ( ( ( ( 6 ( ( ( ( 6 ( m e m e m Adding the additional ondition (9, results the matrix form: m( s e( m( 6 6 e( m( 6 6 he solutions have the equations: (8 (9

31 Ative Load Balaning in a hree-phase Network by eative Power Compensation 49 ( ( ( ( m( m ( ( ( ( ( e m e m e m e m ( m( ( ( ( ( ( ( m m e m m e m ( ( Now it an dedue the equations for the urrents on the phases of the: ompensator, ompensator, and the ensemble of these, expressed with symmetrial omponents of urrents on the phases: ( ( ( j e ( a a a e( j ( ( a a a e( j ( ( j jm e ( ( ( a a m a a e j j ( ( ( ( a j a jm a a e h a a (4 a a e( ( a e (5 a e( t an be observed that on the phases of the ensemble -ompensator a three phase positive sequene urrent it s formed, orresponding to the ative omponents of the positive sequene urrents. o explain the ompensation mehanism, the equations of the urrents on the phases of the two ompensators are written so as to highlight the omponents of positive, negative and zero sequene sets:

32 5 Power Quality Monitoring, Analysis and Enhanement ( m( e( ( ( ( ( ( ( j a a jm e a a jm e (6 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( jm jm e e jm a m a m e e m j j j a jm a jm e e jm (7 t s found that the urrents on the phases of ompensator an be deomposed in two threephase sets, both of negative sequene: one equal and opposite two the negative sequene of the urrents on the phases (noted, whih anels, one whih depends on the ative and reative omponents of the zero sequene urrent on the phases (. he urrents on the phases of the ompensator an be deomposed in three three-phase sets: one of positive sequene (, equal and opposite to the set formed by the reative omponents of positive sequene urrents of the, whih anels, one of negative sequene (, equal and opposite to the set of the urrents on the phases of the ompensator (, the two sets aneling eah other, the third one, of zero sequene (, equal and opposite to the zero sequene set of urrents on the phases, whih anels. t also observes that the urrent on the neutral ondutor of the ompensator ompensates the urrent on the neutral ondutor, beause: N ( he ompensation suseptanes expressed depending on the sequene omponents of the urrent Based on the equations ( and (, an be established equations for the alulation of the sixths ompensation suseptanes expressed depending on the ative and reative omponents of the sequene urrents (available for the riterion -: ( ( ( ( B e m e m U B m ( m ( U B e m e m U ( ( ( ( (9

33 Ative Load Balaning in a hree-phase Network by eative Power Compensation 5 ( ( ( ( ( ( ( ( B m m U B m U e m B m U e m (4 ine the urrents flow on the three symmetrial sequenes are independent, these suseptanes an be deomposed to form five fititious ompensators:,,,, B B B B B B B B B B B B B B B B B B B B B (4 in whih: ( m( B U B m ( U B U e ( m( e ( ( m B U B m ( U B U e ( ( m e m ( (4 (4 B B B (44 U ( e( m( e( m( B B m U B B U B B U (45 t an observe that the suseptanes onstituted in an be determined with the same equations as in the ase of the three-wire network (equations 54. o ompensate the imaginary parts of the positive sequene urrents, this time it s used a symmetrial ompensator in star onnetion (, whih is not neessary the onnetion with the neutral ondutor of the network. he zero sequene omponents of the urrents are ompensated by a mix ompensator. he and ompensators an be grouped into a single reeiver, onnetion (, whih must allowed the losure of zero sequene ompensation urrents, whih is possible only in an n onnetion.

34 5 Power Quality Monitoring, Analysis and Enhanement Fig.. he deomposition of the ompensators in three fititious ompensators (,,, and the ompensation mehanism illustrated by symmetrial omponents a b d Fig. 4. he phasor diagrams illustrate the ompensation mehanism for unbalaned in three-phase four-wire network

35 Ative Load Balaning in a hree-phase Network by eative Power Compensation 5 he five ompensators an be redued to three, eah one realizing the ompensation on one of the sequenes. Figure presents the three fititious ompensators and the sequene urrents flow into the ensemble - ompensator and Figure 4 presents the phasor diagram and the ompensation mehanism. n the last one is first illustrated the three phases urrents of the onstituted in a threephase unbalaned set, by whih were determined the referene sequene urrents (fig. 4.a. t an be observed the suessive intervention of the fititious ompensators whih anels the zero sequene omponents (fig. 4.b, the negative sequene omponent (fig. 4. and the reative omponent of the positive sequene urrent (fig. 4.d. After the ompensation remains a three-phase urrents set perfetly balaned and purely ative (in phase with orresponding simple voltages. 5. Conlusions Unbalane of phase equivalent impedane and/or unbalane on a three-phase eletrial network, determine unequal voltage drops on the three phases and hene the voltage unbalane. he main negative effets of urrent and voltage unbalanes in an eletrial network onsist of yields redution of proesses and dysfuntions at a wide range of equipment, effets whih an be equated with damages aused by additional energy losses, deterioration of quality of proesses and life shortening of the equipments. One of the most popular means of mitigating the unbalane in three-phase networks is the teinmetz iruit. t may be used to onvert a purely ative, onneted between two phases, in an equivalent three-phase ative, perfetly balaned, using the shunt reative ompensation. Ative balaning method in a three-phase network by shunt reative ompensation an be generalized to any three-phase s, supplied by three or four wire networks. his hapter presents the developed mathematial model for this method, inluding the sizing of reative ompensation elements, the urrents flow into the ompensator, respetively the urrents and powers flow into the -ompensator ensemble. For this, it has been used as tools both the symmetrial omponents method and the phase omponents method. Analytial relations for power flow into the -ompensator ensemble established in the mathematial model, helps to explain the mehanism of balaning by reative ompensation, showing how the ompensators determine the ative and reative power redistribution between the phases of the network, thus ahieving both reative power ompensation on positive sequene (for inrease the power fator to the required level or for network voltage ontrol and balaning (anellation of urrents on negative and zero sequene. hunt reative ompensation ahieved with passive reative elements (reators and apaitors, is an effiient solution for optimization the operating onditions of networks, by inreasing the power fator, voltage ontrol and balaning. Balaning ompensators remain effiient due to relatively low ost ompared to equipment type ACOM, for the ase of large s having relatively slow variation. hey will ontains reators and apaitors fixed sized so that it an ompensate the higher voltage unbalanes. he ontrol of ompensation urrents depending on the urrents will be made by stati swithing equipment, with an individual ommand on the branhes of the ompensator. uh equipment is therefore VC type and it will have like ontrol elements reators or apaitors

36 54 Power Quality Monitoring, Analysis and Enhanement ontrolled by thyristors (Cs hyristor - Controlled eators respetively CCs hyristor - Controlled Capaitors. One of the disadvantages of suh a ompensator is given by the high value of reative power that must be installed in passive reative elements, whih must be at least equal to the reative. Another disadvantage is given by the need to adopt effiient measures to mitigate the non-sinusoidal regime resulted from the operation of the VC and to avoid the parallel resonanes that may be produe between ompensator and network and whih an amplify the non-sinusoidal onditions. 6. eferenes Czarneki, L.,. & Hsu,., M. (994. hyristor ontrolled suseptanes for balaning ompensators operated under nonsinusoidal onditions, EE Proeedings on Eletri Power Appliations, Vol. 4, No. 4, 994, pp Czarneki, L.,. (995. Power related phenomena in three-phase unbalaned systems. EEE ransations on Power Delivery, Vol., No., 995, pp Dixon, J., Morán, L., odríguez, J., Domke,. (5. eative Power Compensation ehnologies, tate of-the-art eview, Proeedings of the EEE, Vol. 9, No., Deember 5, pp Grünbaum, L., Petersson, A., horvaldsson, B. (. FAC improving the performane of eletrial grids, ABB eview (peial eport on Power ehnologies, ear, pp. -8 Gueth, G., Enstedt, P., ey, A., Menzies,., W. (987. ndividual phase ontrol of a stati ompensator for ompensation and voltage balaning, EEE ransations on Power ystems, Vol., 987, pp Gyugyi, L., Otto,., Putman,. (98, Priniples and Appliations of tati hyristor - Controlled hunt Compensators, EEE ransations on PA, Vol. PA-97, Otober 98, No. 5, pp Mayordomo, J., G., zzeddine, M., Asensi,. (. Load and Voltage Balaning in Harmoni Power Flows by Means of tati VAr Compensators, EEE ransations on Power Delivery, Vol. 7, No., July, pp aid,., K. & Pirouti, M. (9. Neural network-based Load Balaning and eative Power Control by tati VA Compensator, nternational Journal of Computer and Eletrial Engineering, Vol., No., April 9, pp. 5-. an,., L. & Chi, J., W. (99. On-line reative power ompensation shemes for unbalaned three phase four wire distribution feeders, EEE ransation on Power Delivery, Vol. 8, No. 4, 99, pp UE [nternational Union for Eletroheat] (998. Power Quality Working group WG, Guide to quality of eletrial supply for industrial installations, Part 4: Voltage unbalane, January 998

37 Power Quality â Â Monitoring, Analysis and Enhanement Edited by Dr. Ahmed Zobaa BN Hard over, 64 pages Publisher neh Published online, eptember, Published in print edition eptember, his book on power quality written by experts from industries and aademis from various ounties will be of great benefit to professionals, engineers and researhers. his book overs various aspets of power quality monitoring, analysis and power quality enhanement in transmission and distribution systems. ome of the key features of books are as follows: Wavelet and PCA to Power Quality Disturbane Classifiation applying a BF Network; Power Quality Monitoring in a ystem with Distributed and enewable Energy oures; ignal Proessing Appliation of Power Quality Monitoring; Pre-proessing ools and ntelligent ehniques for Power Quality Analysis; ingle-point Methods for Loation of Distortion, Unbalane, Voltage Flutuation and Dips oures in a Power ystem; -transform Based Novel ndies for Power Quality Disturbanes; Load Balaning in a hree-phase Network by eative Power Compensation; Compensation of eative Power and ag Voltage using uperonduting Magneti Energy torage; Optimal Loation and Control of Flexible hree Phase hunt FAC to Enhane Power Quality in Unbalaned Eletrial Network; Performane of Modifiation of a hree Phase Dynami Voltage estorer (DV for Voltage Quality mprovement in Distribution ystem; Voltage ag Mitigation by Network eonfiguration; ntelligent ehniques for Power Quality Enhanement in Distribution ystems. How to referene n order to orretly referene this sholarly work, feel free to opy and paste the following: Adrian Pana (. Ative Load Balaning in a hree-phase Network by eative Power Compensation, Power Quality â Â Monitoring, Analysis and Enhanement, Dr. Ahmed Zobaa (Ed., BN: , neh, Available from: neh Europe University Campus ep i lavka Krautzeka 8/A 5 ijeka, Croatia Phone: 85 ( Fax: 85 ( neh China Unit 45, Offie Blok, Hotel Equatorial hanghai No.65, an An oad (West, hanghai, 4, China Phone: Fax: