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Reactive Power and Harmonic Compensation Based on the Generalized Instantaneous Reactive Power Theory for Three-phase Power Systems Fang Zheng Peng, Senior Member, IEEE University of Tennessee Oak Ridge National Laboratory Oak Ridge, TN 3783 1-8058 Phone: (423)576-7261,Fax: (423)24 1-6124 P.O. BOX 2009, Bldg. 9104-2 Abstract-A generalized theory of instantaneous reactive power for three-phase power systems is proposed in this paper. This theory gives a generalized definition of instantaneous reactive power, which is valid for sinusoidal or nonsinusoidal, balanced or unbal- anced, three-phase power systems with or without zero-sequence currents and/or voltages. The properties and physical meanings of the newly defined instantaneous reactive power are discussed in detail. With this new reactive power theory, it is very easy to cal- culate and decompose all components, such as fundamental ac- tiveheactive power and current, harmonic current, etc. Reactive power and/or harmonic compensation systems for a three-phase distorted power system with and without zero-sequence compo- nents in the source voltage and/or load current are then used as examples to demonstrate the measurement, decomposition, and compensation of reactive power and harmonics. I. INTRODUCTION The traditional definitions of active power, reactive power, active current, reactive current, power factor, etc., are based on the average concept for both single-phase and three-phase power systems with sinusoidal voltages and sinusoidal currents. Many contributors have attempted to redefine these quantities to deal with three-phase systems with unbalanced and distorted currents and voltages [l-51. t Prepared by the Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-7258, managed by Lockheed Martin Energy Research Corp. for the U. S. Department of Energy under contract DE-AC05-960R22464. The submitted manuscript has been authored by a contractor of the U. S. Government under contract No. DE-AC05-960R22464. Accordingly, the U. S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U. S. Government purposes. Jih-Sheng Lai, Senior Member, IEEE Oak Ridge National Laboratory? Engineering Technology Division P.O. Box 2009, Bldg. 9104-2 Oak Ridge, TN 37831-8058 Phone: (423)576-6223, Fax: (423)241-6124 Among them, Akagi and Nabae [ 1,7] have introduced and established an interesting concept of instantaneous reactive power. This concept gives an effective method to compensate for the instantaneous components of reactive power for three- phase systems without energy storage. However, this instanta- neous reactive power theory still has a conceptual limitation as pointed out in [2] that the theory is only complete for three- phase systems without zero-sequence current and voltage. To resolve this limitation and other associated problems, Willems and Nabae proposed some attractive approaches to define in- stantaneous active and reactive currents [2, 61. Their ap- proaches, however, are to deal with the decomposition of cur- rents into orthogonal components, rather than with power com- ponents, thus having some difficulties when it is necessary to separate different power components for various compensation aims. In this paper, a generalized theory of instantaneous reac- tive power for three-phase power systems is proposed. The generalized theory is valid for sinusoidal or non-sinusoidal, balanced or unbalanced three-phase systems, with or without zero-sequence current and/or voltage. Using this new reactive theory, this paper shows how to separate all different compo- nents, such as fundamental reactive power, fundamental reac- tive current, harmonic current, etc. Some interesting properties of the theory and some examples of reactive powerharmonic compensation are presented. II. DEFINITIONS, PROPERTIES AND PHYSICAL MEANINGS A. Definitions For a three-phase power system shown in Fig. 1, instanta- neous voltages, v, , vb , vc , and instantaneous currents, i, , ib , i , are qpressed as instantaneous space vectors, v and i , i.e., \*,,&It .ib 1% 1 lic 1 Fig. 2 shows the three-phase coordinates which are mutually orthogonal, representing phase ‘a‘, phase ‘b, and phase IC‘, re- spectively. The instantaneous active power of a three-phase circuit, p, can be given by OF THIS ~~~~~~~~T tS ~~~1~~~
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Page 1: Reactive Power and Harmonic Compensation Based on the .../67531/metadc... · Reactive Power and Harmonic Compensation Based on the Generalized Instantaneous Reactive Power Theory

Reactive Power and Harmonic Compensation Based on the Generalized Instantaneous Reactive Power Theory for Three-phase Power Systems

Fang Zheng Peng, Senior Member, IEEE University of Tennessee

Oak Ridge National Laboratory

Oak Ridge, TN 3783 1-8058 Phone: (423)576-7261, Fax: (423)24 1-6 124

P.O. BOX 2009, Bldg. 9104-2

Abstract-A generalized theory of instantaneous reactive power for three-phase power systems is proposed in this paper. This theory gives a generalized definition of instantaneous reactive power, which is valid for sinusoidal or nonsinusoidal, balanced or unbal- anced, three-phase power systems with or without zero-sequence currents and/or voltages. The properties and physical meanings of the newly defined instantaneous reactive power are discussed in detail. With this new reactive power theory, it is very easy to cal- culate and decompose all components, such as fundamental ac- tiveheactive power and current, harmonic current, etc. Reactive power and/or harmonic compensation systems for a three-phase distorted power system with and without zero-sequence compo- nents in the source voltage and/or load current are then used as examples to demonstrate the measurement, decomposition, and compensation of reactive power and harmonics.

I. INTRODUCTION The traditional definitions of active power, reactive power,

active current, reactive current, power factor, etc., are based on the average concept for both single-phase and three-phase power systems with sinusoidal voltages and sinusoidal currents. Many contributors have attempted to redefine these quantities to deal with three-phase systems with unbalanced and distorted currents and voltages [l-51.

t Prepared by the Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-7258, managed by Lockheed Martin Energy Research Corp. for the U. S. Department of Energy under contract DE-AC05-960R22464. The submitted manuscript has been authored by a contractor of the U. S. Government under contract No. DE-AC05-960R22464. Accordingly, the U. S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U. S. Government purposes.

Jih-Sheng Lai, Senior Member, IEEE Oak Ridge National Laboratory? Engineering Technology Division

P.O. Box 2009, Bldg. 9104-2 Oak Ridge, TN 37831-8058

Phone: (423)576-6223, Fax: (423)241-6124

Among them, Akagi and Nabae [ 1,7] have introduced and established an interesting concept of instantaneous reactive power. This concept gives an effective method to compensate for the instantaneous components of reactive power for three- phase systems without energy storage. However, this instanta- neous reactive power theory still has a conceptual limitation as pointed out in [2] that the theory is only complete for three- phase systems without zero-sequence current and voltage. To resolve this limitation and other associated problems, Willems and Nabae proposed some attractive approaches to define in- stantaneous active and reactive currents [2, 61. Their ap- proaches, however, are to deal with the decomposition of cur- rents into orthogonal components, rather than with power com- ponents, thus having some difficulties when it is necessary to separate different power components for various compensation aims.

In this paper, a generalized theory of instantaneous reac- tive power for three-phase power systems is proposed. The generalized theory is valid for sinusoidal or non-sinusoidal, balanced or unbalanced three-phase systems, with or without zero-sequence current and/or voltage. Using this new reactive theory, this paper shows how to separate all different compo- nents, such as fundamental reactive power, fundamental reac- tive current, harmonic current, etc. Some interesting properties of the theory and some examples of reactive powerharmonic compensation are presented.

II. DEFINITIONS, PROPERTIES AND PHYSICAL MEANINGS

A. Definitions

For a three-phase power system shown in Fig. 1, instanta- neous voltages, v, , vb , vc , and instantaneous currents, i, , ib , i , are qpressed as instantaneous space vectors, v and i , i.e.,

\ * , , & I t . i b

1% 1 l i c 1 Fig. 2 shows the three-phase coordinates which are mutually orthogonal, representing phase ‘a‘, phase ‘b, and phase IC‘, re- spectively. The instantaneous active power of a three-phase circuit, p , can be given by

OF THIS ~~~~~~~~T tS ~~~1~~~

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DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

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DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spe- cific commercial product, process, or service by trade name, trademark, manufac- turer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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Fig. 1. Three-phase circuit structure.

b' i

Fig. 2. Three-phase coordinates.

p = v - i , or p=v , i ,+vb ib+v , i , , (2)

where "." denotes the dot (internal) product, or scalar product of vectors. Here, we define a new instantaneous space vector q as

def q = v x i (3)

where "x" denotes the cross (exterior) product of vectors or vector product. Vector q is designated as the instantaneous reactive (or nonactive) power vector of the three-phase circuit, and the magnitude or the length of q, 4, is designated as the instantaneous reactive power, that is,

(4)

where " 11 " denotes the magnitude or the length of a vector. Equations (3) and (4) can be rewritten as

4 = IMI = llv x ill 9

1 = P

I = Y

def def - s=vi , and A=--, - P S

where v = llvll= 4- and i = llill= 4- are the instantaneous magnitudes or norms of the three-phase volt- age and current, respectively.

B. Properties

The new reactive components defined above have the follow- ing interesting properties [8].

[Property 11 A three-phase current vector, i , is always equal to the sum of i,, and iy, i.e., i i i, + iq . [Property21 iy is orthogonal to v , and ip is parallel to v, namely, v - i q = 0 and v x i , = 0 . [Property 31 All properties of the conventional reactive power theory still hold true for the new theory, such as:

2 2 - '2' , where ip = i II PI1 i 2 - . 2 = I + i q , 2 s 2 = p 2 + q 2 , a n d i =-

V P

and iy = lliqll . [Property 41 If iq = 0 , then the norm //ill or i becomes minimal for transmitting the same instantaneous active power, and the maximal instantaneous power factor is achieved, namely h = 1.

[Property51 For a three-phase system without zero se- quence voltage and current, i.e., v,+vb+v,=o and i,+ib+iPO, it is true that:

Property 1 shows that any three-phase current vector, i , can be always decomposed into the instantaneous active cur- rent vector, ip, and the instantaneous reactive current vector, & property 2 that the component, iq, indeed is the instants- neOuS reactive power because it does not contribute to any real power transmission. Property 5 makes calculation of q and p much simpler for three-phase three-wire systems. We only

respectively. In turn, we define the instantaneous active cur- rent vector, ip, the instantaneous reactive current vector, iy, the instantaneous apparent power, S, and the instantaneous power factor, h, as

2

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need to sense two phase voltages and two currents for p and q calculation without using any coordinate transformation. Fur- thermore, one can use two line-to-line voltages to calculate p and q instead of phase voltages.

C. Physical Meanings

As one can see so far that q and iq indeed represent the instantaneous reactive power and current. Actually, this can be observed more clearly from the following explanation. The instantaneous active power of each phase can be split up into two parts:

(9)

Since p = v - i = v - i p , v - i q = 0 , and q = v x i , , wecansee that pap, pbp and pcp contribute to the total power, p , and they sum up to p , i.e., pap+pbp+pcp=p. Power components pa,, pbg and pcq contribute to q and sum up to zero, i.e., puq+pbq+pcp=O. Therefore, pa,, pbq and pcq correspond to those powers that transfer or circulate between the three phases. And q represents their magnitudes and signs. It is clear from (5 ) that q. gives the amount of the power that circulates between phase "b" and phase "c". The sign of qu represents leading (for "+") or lag- ging (for "-") current from the voltages. Similarly, qb and qc represent the reactive power circulating between phases "c" and "a" and between phases "u" and "b", respectively. There- fore, the instantaneous reactive current, iq, does not convey any instantaneous active power from the source to the load (see Fig. 1), but indeed it increases the line losses and the norm (or magnitude) of the three-phase current. If q or iq is eliminated by a shunt compensator, then the norm of the source current will become minimum.

From the above definitions, properties, and discussion, we can get the following conclusions:

1) The current vector, ip, is indispensable for the instantane- ous active power @) transmission, whereas i , does not contribute to it, because p = v - i = v - ip and v - iq = 0 .

2) An inverter based compensator does not require energy storage to eliminate the instantaneous reactive power.

3) Using compensators without energy storage, the instanta- neous active power cannot be changed, and hence the minimum line losses are obtained for zero instantaneous reactive power, i.e., q=O.

111. ALTERNATIVE EXPRESSIONS In the previous section, the definitions of the instantaneous

reactive components are all based on the direct quantities of three-phase voltages and currents: v,, Vb, vc, and ia, ibr i,. If necessary, these newly defined quantities can be expressed in any other coordinates. e.g., ap0 coordinates. Here, let us ex- press the defined quantities, p , q, ip, i, etc., in aP0 coordi- nates.

For three-phase voltages and currents, v',, lib, vcr and &,, ih, i,, the a , /3 , and 0 components are expressed as

r

where, suffixes "(abc)" and ''( aP0 )" denote the corresponding coordinates, i.e.,

"(ubc) =['a 'b 1 ' 3 "(ap0) =['a 'p 'O]T

T q(abc) = [ q a q b q c ] = "(ubc) $abc) 9 and

q(ap0) =[qa qp qo]T =v(apo) xi(ap0). From (13) and 11 [C] 11 = 1 , we have

Therefore, q(apo) is identical to qcabc). Similarly, we can de- fine the instantaneous active and reactive components in ap0 coordinates as

The properties and physical meanings mentioned in Section I1 are valid and independent of coordinates. For three-phase systems without zero-sequence components, i.e., vo and io are

3

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equal to zero, the instantaneous active and reactive powers can be simplified as

p = vago. iago = vai, + vpip , (17)

0 (18)

Equations (17) and (19) are identical to the definitions de- scribed in [I]. Therefore, the pq theory described in [l] is a special case of the generalized p q theory described in this pa- per.

Using the aforementioned properties, we can simplify the calculation of p , q, i,, and i, as follows:

(20) p = v - i and ip=Tv ,and P V

i 4 =i-i,, and q = v i , = v i I/ 4ll .

In this way, one can expand the instantaneous reactive power theory to a multi-phase system [2, 81. In (21) q is no longer a vector. With the expressions of (21) one can get q directly from iq; however, one cannot express i, directly from q like (7). This makes decomposition of active/reactive/harmonic. power and current difficult, thus limiting these expressions' applica- tions to various reactive and harmonic compensation. Section IV will show the detailed explanation. To overcome this limi- tation, we introduce one auxiliary vector as follows:

(22) def v . vi =- zq Iq'

where v, is a voltage vector that is orthogonal to the voltage vector v. Fig. 3 shows the relations of these vectors. With this orthogonal voltage vector, we can re-express the instantaneous reactive current as

With the help of (22) and (23), one can decompose p and q into various components and get their corresponding current com- ponents.

t"

Iv. REACTIVE POWER AND HARMONIC COMPENSATION

Here, consider a reactive power and/or harmonic compen- sation system to see how the proposed theory can be applied for calculating and compensating for the instantaneous reactive power and harmonic current of a three-phase system (but not limited to a three-phase one). Fig. 4 shows the system configu- ration of reactive power and harmonic compensation. The compensator is connected in parallel with the load. The con- trol circuit of the compensator is also shown in Fig. 4, which includes computational circuits for the instantaneous active and reactive power of the load, p L and qL, extraction circuit of compensation power references, p ; and qg, calculation circuit of compensation current reference, i;, and current control cir- cuit. Their relations can be expressed as

(24) p , = vs . iL , qL = vs xi,, and

* * PCVS q c x v s l C = - + - , V S ' V S V S ' V S

where, pcf and q; can be assigned or extracted from p L and q L according to one's compensation requirement. p L and qL may be respectively split into two parts (dc values and ac values) as

where,

FL and qL are the instantaneous active and reactive power (dc values) originating from the symmetrical fundamental (positive-sequence) component of the load current,

are the instantaneous active and reactive power (ac values) originating from harmonic and asymmetrical fundamental (negative-sequence) component of the load current.

Furthermore, p L and iL can be respectively split into two parts (20 components and harmonic components) as

17, and cL

F L = P L Z ~ + P L ~ 9 42 = q n w +qui 1 (27)

AC Source

. p and q p L Reference& inverse

Circuit d Eq. (25) Circuit -L Calculation Extraction

V r ' 4 QL 4c

Fig. 4. System configuration of reactive power and harmonic compensator.

4

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where, PL20 and qL20

PLh and qLh

are the instantaneous active and reactive power (201 components) originating from asymmetrical fundamental (negative- sequence) component of the load current,

are the instantaneous active and reactive power (harmonic components) originating from harmonic component of load current.

To extract F L , ijL1 and j f ~ , i~~ or P L ~ ~ 4 ~ 2 ~ and p L h , qLh one can use low-pass filters or band-pass filters. Table 1 summarizes some examples of different compensation aims and their corresponding references of the Compensator. The same table and conclusions can be obtained for multi- phase systems by using (20) - (23). For example, decompos- ing the load active and reactive power, p~ and q L , into p L , ?jL, pL2" , qLzo and pLh , qLh , we can get the harmonic current from the following equation:

. PLhvS qLhvSL 2 LU =- 2 +-

VS "S

Therefore, Table 1 is also valid to (20)-(23) for a system with any number of phases.

V. SIMULATION STUDY

A. Compensation for a three-phase thyristor rectifier

First, we consider a three-phase nonlinear system without zero sequence components. Fig. 5 shows the configuration of the compensation system for a three-phase thyristor rectifier. The compensator is composed of a three-phase voltage-source pulse width modulation (PWM) inverter. Fig. 6 shows simu- lated waveforms of harmonic current compensation. To extract harmonic components, two high-pass filters are used, that is, in the reference extraction circuit, we have p i = EL = G(s)PL

and 42. = cL = G(s)qL , where G(s)=s/(s+o,) is a first-order

high-pass filter with cutoff frequency o p 2 ~ 3 0 H z . In this case, the compensator acts as an active harmonic filter. It is seen from Fig. 6 that the source current becomes sinusoidal after the compensator is started.

Fig. 7 shows simulated waveforms of reactive and har- monic current compensation. In this case, we have p i = j f L = G(s)pL and q i = qL . Obviously, the source cur- rent becomes sinusoidal and unity power factor as well without any time delay after the compensator is started.

Therefore, one can easily implement various compensation at will by just changing p and q references of the compensator. In this way, a universal control circuit can be implemented.

Fig. 5. System configuration of reactive and harmonic current compensation for thyristor rectifier.

B. Compensation for a three-phase four-wire system Here, we consider a three-phase nonlinear system contain-

ing zero sequence. Fig. 8 shows the configuration of a three- phase four-wire system, in which three single-phase diode rec- tifiers are connected in phase a, b, and c, respectively. The compensator consisting of a four-leg PWM inverter is con- nected in parallel with the loads. The control circuit of the compensator is the same as the previous systems, which in- cludes computational circuits for the instantaneous active and reactive power of the loads, p L and qL, extraction circuit of

Table 1 Different Compensation Aims and Their Corresponding References

I Source current will become sinusoidal Harmonic Current PLh 4Lh I

5

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compensation power references, p: and q i , calculation cir-

cuit of compensation current reference, il, and PWM current control circuit for the inverter. Aiming at instantaneous reac- tive power compensation only, we set pg=O and qg=qL.

.................................................................................................... ........................................................................... icu- .....................

started Fig. 6. Simulated waveforms of harmonic current compensation.

-~ ~ . -~ .............i....................................

ico ................................................

started Fig. 7. Simulated waveforms of reactive and harmonic current compensation.

Compensator

Fig. 9 shows waveforms of the system before and after reactive power compensation, where the source voltage is bal- anced and has no zero-sequence components, and the load cur- rent contains zero-sequence component. That is, v s ~ v s o + VS~+VSFO, and iLo=iLa+ia+iLc f 0. Before the compensator was started, i& and iF0. After the compensator was started, is became in phase with the source voltage immediately, and is, became zero without any time delay. This indicates that the zero-sequence current of the loads, i,,, only contributes to the instantaneous reactive power, qL.

Fig. 10 shows waveforms of the system before and after reactive power compensation, where the source voltage is un- balanced (the magnitude of vs0 is 20% smaller than vsb and vsC), thus both the source voltage and load current have zero- sequence components. In this case, the zero-sequence current, iu, contributes both to the instantaneous active power and to the instantaneous reactive power. That is, ir0 includes both instantaneous active current and reactive current components, iLop and i-, which cannot be instantaneously separated by the traditional methods. Therefore, the proposed instantaneous reactive power theory for a three-phase power system can deal with the following cases: (1) sinusoidal and nonsinusoidal waves, (2) balanced and unbalanced systems, and (3) with or without zero-sequence components. The p q theory of 111, however, is limited to a system without zero-sequence compo- nents only and cannot deal with the above examples.

VI. CONCLUSIONS In this paper, a generalized instantaneous reactive power

theory has been proposed for reactive and harmonic current compensation. Clear definitions for the instantaneous active and reactive components such as active power, reactive power, active current, reactive current, power factor, etc., have been given, and their interesting properties, relationships, and physi- cal meanings of these instantaneous quantities have been de- scribed in detail. The proposed theory is valid for sinusoidal or nonsinusoidal, balanced or unbalanced three-phase power sys- tems with or without zero-sequence components. Some appli- cation examples for reactive power and harmonic compensa- tion have been studied. This generalized reactive power theory discloses an important algorithm for instantaneous reactive power and harmonic current measurement and compensation applications.

REERENCES [ I ] H. Akagi, Y. Kanazawa, and A. Nab=, "Instantaneous Reactive Power

Compensators Comprising Switching Devices Without Energy Storage Components," IEEE Trans. ind. Appl., v01.20, pp.625-630, May/June 1984.

Jacques L. Willems, "A New Interpretation of the Akagi-Nabae Power Components of Nonsinusoidal Three-phase Situations,'' IEEE Trans. Instrum. Mens., vo1.41, no.4, August 1992.

[2]

Fig. 8. System configuration of instantaneous reactive power compensation for a three-phase four-wire load.

6

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[3] A. Ferrero and G. Superti-Furga, “A New Approach to the Definition of Power Components in Three-Phase Systems Under Nonsinusoidal Conditions,” IEEE Truns. Insfrum Meus., ~01.40, no.3, June 1991.

[7] A. Nabae and T. Tanka, “A New Definition of Instantaneous Active- Reactive Current and Power Based on Instantaneous Space Vectors on Polar Coordinates in Three-phase Circuits,” IEEWES Winter Meeting,

[4] L. S. Czamecki, ”Orthogonal Decomposition of the Currents in a 3- Paper No. 96 WM 227-9 PWRD, 19%.

phase Nonlinear Asymm~trical Circuit with a Nonsinusoidal Voltage [8] L. Rossetto and P. Tenti, “Evaluation of Instantaneous Power Terms in Source,” IEEE Trum. Insfrum. Meus., ~01.37, no.1, March 1988. Multi-Phase Systems: Techniques and Application to Power-

Conditioning Equipment,” ETEP Vol. 4, No. 6, NovemberDecember 1994. -, ”Scattered and Reactive Current, Voltage, and Power in Circuit

with Nonsinusoidal Waveforms and Their Compensation,” IEEE Trans. Instrum. Meas., ~01.40, no.3, June 1991. F. Z. Peng and J. S. Lai, ‘‘Generalized Instantaneous Reactive Power

Theory for Three-phase Power Systems.” IEEE Truns. Instrum. Meus., vo1.45, no.1, pp.293-297, February 19%. A. Nabae, et al, “Reactive Power Compensation Based on Neutral Line

Current Separating and Combining Method,” Transactions of IEE of Ja- pan, Vol. I14-D, No.7/8, July/August 1994. pp. 800-801.(in Japanese)

[5 ]

[9]

[6]

...............................................................................................

............ L .......... ........................................................................

...............................................................................................

............ L ....................................................................................

...............................................................................................

............ L ........... C..................................,...........,...........,...........

..............................................

...... ........................................

........................ ...........................................................................

icll ............ L ..........................................................,...........,...........

1 ........... ; ................................................. i ........... ; ........... i ........... 1

k b .

.................................................................................................... icc ............. L .......__.......................................,....................................

J I

1 ................................................. ................................................

ico ........... L .................................. ........... \....... ........................... started started

Fig. 9. Waveforms of instantaneous reactive power compensation with zero-sequence load current.

Fig. 10. Waveforms of instantaneous reactive power compensation with zero-sequence source voltage and load current.

7


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