Instantaneous Inactive Power

A New Approach to Compensation of Instantaneous Inactive Power

A.F.

Zakeril,

K. Kanz

il and

M. Tavakoli Bina2 'The ACECR Research Institute, Tehran,Iran

2K.

N. Toosi University of Technology, Tehran,Iran

E-mails: Zakeril4ggmail.com, khkanziggmail.com, tavakolig kntu.ac.ir

Abstract-The theory of the instantaneous three-phase powers have been developing in different ways. The zero sequence components of the load affect the source during the process of instantaneous inactive or reactive power compensation. Different approaches have been used to overcome this issue, such as Clark transformation to separate zero sequence components from the phase quantities. This paper introduces a new approach to a previously proposed theory of the instantaneous reactive power compensation, minimization method. The proposed method is applicable to all balanced three-phase systems as well as unbalanced system without zero sequence components. A separate paper discusses a new way ofovercoming the effects of the zero sequence components on the sourcecurrents, extending the new approach to all unbalanced systems.

Keywords-Instantaneousinactive power, zero sequence components, unbalanced networks, reactive powercompensation.

I. Introduction voltage. The phase voltages and line currents are transformed tothe

a,/o

frames as follows:

The instantaneous active power in a three-phase system could be describedas the instantaneous energy flow between

two subsystems. The instantaneous three-phase reactive ^{a a a a}

power, however, is concerned with the energy exchanged

i|= [C] 'b

=

[C]

V^b between

phases,

which do not contribute to any kind of . .

energy transfer between the source and the load. This is

0lo

LCj L0 different from that ofasingle-phase circuit. ^In asingle-phase

circuit, the instantaneous reactive power corresponds to an 1 L_{2 2} oscillatingenergytransfer between thesourceand the load.

[C

^]

VT VX3X

Some researchers have been working on finding the \13 2 12

instantaneous powersby direct use of the load terminalphase X X

/-

voltages (va(t), vb(t), vc(t)) and linecurrents(ia(t), ib(t), ic(t)) (1)

intheir analysis [1], [2]. The obtained powers, however, are The instantaneous reactive power q(t), for a three phase greatly influenced by zero sequence voltage and current. As system is

expressed by:

these components are part of the line currents and phase voltages, hence, usages of these kinds of methods aremostly

restricted to three-phase systems without zero sequence q(t) q

Vaa60

⁼

va

(t)

i8 (t)- v8

(t)

'a

^(t) (2) components (ZSC). Additionally, the product of zero

sequence voltage and current contributes ^{to a} pure

instantaneous active power, whichconverterswith

significant

^{1 A}

energy storage elements or a separate power source are Vsa Zs Vv

needed to supply this power. The line currents are divided i3

into the sum of their active and reactive parts, and using a Vb

minimization method

they

^are obtained. This method is bT

simpleto implement, but the drawn currents from the source V

might contain zero sequence current depending on the load Vsc Z

zerosequencevoltage. C Lo

Akagi-Nabae presented the concept of instantaneous + L m

reactive power in [3] and [4], which gives an effective way of - compensation without any energy storage element. The

method is not affected by the ZSC, neither current nor Figure 1: Three-phase systems including ZSC for an unbalanced non-linear load.

Based on the transformation principles, both the positive Assume both the load phase voltages and line currents and negative sequence components contribute to the contain no ZSC (sum of the phase voltages and line currents instantaneous reactive power expressed by (2) in the steady are zero), then using (4), we have:

state, whereas ZSC do not. Assume an unbalanced three-

phase system which its sum of the load phase voltages is

qx(t) Vb (t) 'c(t)

^-

Vc (t)ib (t)

nonzero (see Fig. 1). Then, the results from minimization method show that the sum of the minimized source current

(i,o

(t)) is also nonzero. Consequently, the minimized source (-va(t)-

vc

(t))

ic

(t)-

vc

(t) (_

ia

(t)-

ic

(t)) currents are different from those of Akagi-Nabae method.

Also, their compensated reactive powers are different. =

(t) ia (t)

^{- v (t)}

ic (t)= qy

^{(t) (6)}

Therefore, these practical issues make the minimization

method

inappropriate

to treat ZSC. This research work aims ^a

X5

topropose newdefinitions toremedy these issues. Later on, a ^lb

generalized instantaneous reactive power theory for Va Load-a

balanced/unbalanced systems introduced in [5].The ib Vb

instantaneous active and reactive powers are defined as - -

below: Vb Load-b

p(t)=v.i q(t)=vxi (3)

ic

42

Where v is the phase voltages vector, and i is the line vc

Load-c

currents vector. The active power ]9 (t) and the reactive

power

q

^(t)arepresented byinternal andexternal productsof (a)

v and, irespectively. Using these definition definitions, the 1 1

equations described by

(1)

and

(2)

are a special case of

(3) ka

[5]. Here we present the first part of a research to introduce a

new approach to both reactive and inactive power +

ib_

_

compensation. The decomposition of inactive power as well

as therequired samplingmethod are suggested inthe second Vb

part of the research ina separate paper. In otherwords, this ic paper proposes the framework of the ^new

approach

for all

balanced networks as well as unbalanced networks without

vc

zsC.

II. Inactive and reactive power (b)

Figure 2: Three-phase systems excluding ZSC, (a) generic balanced

The instantaneous reactivepowerdefined

by (3)

is loads;(b)unbalanced resistive load without ZSC.

expandedinthreeorthogonalaxesbasedonthe instantaneous In^asimilarmanner, it^canbe shown that values ofthe

phase

voltagesand linecurrents as:

q () = q (t) q (t) Q (t) for all balanced networks q(t)

=qx

^{(t) x}±

qy

^(t)

y~

±

qz (t)

2 aswellasunbalanced networks withoutZSC,resultingin

(Vb (t) ic (t) - Vc (t)ib (t))

x

(4) q(t) 3 Q(t)

³

q (t) + + q (t)

+

(Vc (t) 'a (t) -Va (t)

ic

(t)) y^

± (va(t)ib (t) -vb (t) ia (t)) zConsidering Fig. 2(a), for the balanced resistive load,R,

v

a

(t) 'b (t) V

^b

(t) 'a (0)

^z we get Q⁼ Va(t)'b(t)^-Vb(t) ia(t)⁼0,and thus using (7) the Themagnitudeof q(t)isdesignatedasthe instantaneous reactive power iszeroand active power is

equal

to R

(i2 (t)

+ inactive power, thatis,

i2

(t)+

i72

(t)). Meanwhile, if the resistive load is replaced by q(t) =

qr (t)

+

qy (t)

+

q2 (1)

(5) a balanced inductive one, L, the reactive power equals 3 L

(ba (t)

dt

- b (t) a ) and the active power is zero.

625

Forall balanced

three-phase loads, llq(t)II presents

^{the pure}

ia(t) ia(t)

^{h (t)}

(t)

reactive power.However,for the unbalanced resistive load Fi

(t)

^{Va(t) Va(t)}

without ZSC shown inFig. 2(b), the active power equals | (t)

(t)

a b and the reactive power

q(t)

^{is nonzero}and ^!i (t) a b (

b(t) vb(t)

⁽

R Vb(t) V a i_Vb(t)

can be obtained from(7):

Va(t) v ic(t)

q(t)=

\3Q(t)

=V3

^Va

2()vb()

⁽⁸⁾

v_(t)

R These vectors generate noinactive power, as using (3) WhereRis the resistance connected betweenphasesaand their cross products withthe phase voltage vector v are zero:

b. This nonzero power is neither active nor reactive as no reactive generator or absorber element is present on Fig. 2.

This is actually a kind of inactive power due to vx i 0 v i =

unbalanced load which contains no ZSC. We call it Non-zero 0 v x iz 0

(10)

Inactive Component - NIC. In a general case, where both

reactive power and NIC are present, generating

llq

^(t) Now, we consider the sum of the entries in each column from

(7)

would

compensate

both of them. In a

general

of

(9)

^to

form

^abasicsourcecurrent vector

iB

^{threecurrents}

unbalanced three-phase four-wire network in which ZSC are in total as follows:

present, there are some other forms of inactive power in addition to the instantaneous reactive power and NIC. These

are detailed in another paper by introducing two other kinds

Va

(t) Va(t)

of inactivepower. at I +vb

it (I)

II

Vb(t)~~~~~ bVb(t)

III. Compensation approach

iBbI ia (t)

^{+ b}

+(ic)

^{+ (t) (11)}

Considering (5), the instantaneous inactive power is iBc va (t)

v(

C ⁽

always zero, if and only if all the three inactive power

B(c)

a

vc

^{+ b(I)}

Vc

^{+ i ()} components

(qx

(t),

qy

(t) and

qz

(t)) are zero for all ^t. The

va(t) Vb(t)

new approach starts from this concept, and focuses on the

generalized definition of the instantaneous inactive power. These basic source currents

produce

^no inactive power.

Hence,using (4), providing This canbe shown

by expressing

inactive power in

(3) by

its determinant format and

expanding

it into three different

Vb(t)

*b

determinants.

b_

J = h t- Jresultsin

qx

(t) 0.Inthe same manner,

vc(t) icj(t)

vc(I) -=~c(I) and a )=^-a() result in

qy

(t)= 0and Va

(t) Vb

()

Vc (t)

va

(t)

^ia

(t)

^vb

(t) 'b (t)

ⁱ_Ba ⁱ_Bb ⁱ_Bc

qz (t)

⁼0,

respectively.

In our proposed compensation approach three +

instantaneous currents are generated from each line

current, Va(t) Vb(t)

vC(t) based on the corresponding phase voltages ratios (PVR ia(t)

ia(t) vb()

v (t) method). Thus, nine current components are present in the

Va(t)

a

Va(t)

analysis. Inotherwords,three current vectors areproducedas

follows:

Va(t)

VbJ(t)

VC()

(t)

Va

'b(t) ib(t)

Vb (t) vb (t)

Both (16) and (17) represent the proposed compensation

approach. However,

ifa

phase voltage

^iszeroat^a

given time,

thenall entries on the row and column correspondingto that

Va (t) vb (t) vc(t)

(12) phase in(9) are zero.

va Vb(t)

ic

^{( V}

vcvt) (t) 1c (t) vc(t)

^{1t A Simulationresults}

Supposing the phase voltages and line currents of a Where all the determinants in therighthand sidearezero, three-phase system correspond to those are shown in Fig. 3.

as their second row is a multiplicand of the third one. Also, the compensation currents, using (17), is supplied by a Therefore, ifthecompensatedcurrents in(12)are drawn from PWM-converter as shown in Fig. 1. The PVR method is the source, then the compensatorsuppliesthe needed inactive simulated with MATLAB, which Fig. 4 illustrates the source powerof the load. However, the active powersuppliedbythe currents given by (16). The following points can be observed sourcehas tobe equaltothe load active power. So,using (3), fromsimulation results:

the instantaneous active power delivered byiBa'

iBb

and

iBc

is: * Fig. 4 shows that the compensated source currents

isa, isb

^and

isc

areneither sinusoidal norproportionaltothe

PL

⁼ Va 'Ba+ Vb

'Bb

+ VC 'Bc phase voltages.

i

i i

=

(V2

^{+ v}

2±V

^{( a + b + c (13) * The rms values of}the

compensated

^source

currents,

Va Vb

vC

^{shown in}

Fig. 4,

^are

equal

^to 1.0219 and 0.9210 and 1.2716

P.U.,respectively.

Whereas the load instantaneous active power is:

* The rms values of the load currents, shown in fig 3b,

PL ⁼ Val'a^{±Vb ±b+} Vc 'c ^{(14) are}

equal

^to

1.1481,

0.9533 and 1.2604

P.U., respectively,

PL=Val +

b

lb+V C(14)

Tomatch the active powers obtainedby (13) and(14), an I _ instantaneous divisor y is defined:

Y B (15) ~

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I|

^II

IlE Il1 PX.(15

^{I. I I}

PL

Now,

^thecurrentsIBa,

iBb' andIBC

aredivided

by

y . - ;-- Hence,the active power drawn from thesourceisequalto

the load active power, while the inactive power still remains

zero. Thesenew sourcecurrentsare: 2

---a ---

sa lBa(t)

a5g

^10.1

R001 0M2 MOM2

⁰⁰³

0.035'

⁰⁰

isb il3B()b (16)

I

SC _ _IBc Figure3a: Ahypotheticalthree-phase non-sinusoidal loadvoltages.

The difference between the load and the source currents whichare

bigger

than those of thecompensatedsource is the compensation current, which should be supplied by a currents.

PWM-converter. Thiscanbe obtainedas:

* The rmsvalues of the compensator currents, shown in

_ca _t) _a _t) _a _Fig.

^{5 are}

equal

^to

0.1441,

^{0.1861 and}

0.500P.U.,

|tca(I) la(l)

a

's| la respectively.

. . . *~~~~~~~~Theaverage active power for both the load active

LlC(t) Llc(l ) Ll c

culrrents

and the compensated source currents are the

samne

and equal to3.2892

P.U.

627

*Fig

6 shows that the ^source current

produces oniy m~comnaff

ROR

curret

active power as same asload active power. I I I

*Fig

7 the

compensator

current

produces oniy

reactive -- ^----

poweras same asload^one.

51811 (3115MWDIM 0103 U UU5B01 UI0U2 iU5 0030T04Q

The Threes~~~~~~~~~~~igr 5 Tecopestin uretssppiety hePM-covrtr

-2 O Da 1815 010 0I9 111 MM 0 UO 01 00 02 03 3

Figure3b: Ahypothetical three-phasenon-sinusoidal loadcurrents./~A~~~i

The ouc cv ent 0~~~~~~~~~~~~~~^{MODS Sol} KIM5 0C02 01025 113 OMNI 0204

2 iI I Te ur fhr

haThe

^I

4j

.0-- -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~---, O 0005 001 0~~~~U15 0>02 0025 003 0035 004 Cl~~~~~~ UGU5a QOl B0015 W02 f025 RO03 0U35 004

I

I I I~~~~~77~ Figure6:thesourcecurrentproducesactive power identicaltothe load active

I ~~~power.

a Q005 QOl A00151 002 32 03 03 0.04 ME hcr saopwrbkjcA~nsb~

Figure4:Thecompensatedsourcecurrentsobtainedby simulating (16).

B. TheZSC issue U 005 TeIa ecieQ he hs a,~j

Considering (16),

the ^sumof the ^sourcecurrents^canbe ~

obtainedasfollows: G

a >0 0 >01(2( 09915 002 0>0 0103 0>3 0

so sa sb SC -

(v

^+v V a

+v

b'V

++v

cbb cC lO

Thus, the source zero sequence current is nonzero if the simple to implement, applicable to balanced three-phase sum of the phase voltages is nonzero. In other words, load systems along with unbalanced systems without ZSC. The terminal zero sequence voltage causes the source zero proposal has also been extended tothose unbalanced systems

sequence currents. including ZSC, which will beappeared as a separate paper in

Considering Fig. 4, the sum of the source currents is zero. nearfuture.

Otherwise, the necessity of a ground path at the source-end is References unavoidable for zero sequence current to flow. The phase

voltages and line currents in Fig. 3 werealso applied toboth [1] M. Depenbrock, "A Generally Applicable Tool for Analyzing

Akagi-Nabae

and minimizationmethods,which outcomes are Power

Relations",

IEEE

Transactions

on PowerSystems, 8(2):381-387, the same as those of the PVR method.

[2] L. S.,Czarnecki, "Physical ReasonsofCurrent RMSValueIncrease in Power Systems with Non-sinusoidal Voltage", IEEE Transactions on

IV.Conclusion Power Delivery,8(1):437-443, January 1993.

[3] H. Akagi, "Trends in Active Power Line Conditioners", IEEE

The generalized theory of instantaneous reactive power Transactions on Power Electronics, 9(3):263- 268, May 1994.

is used to develop a new way of compensation, together with

presenting inactive and reactive power concepts. Somepresentinginactive and reactive power concepts. Some Power[4] H. Akagi, Y. Kanazawa, and A. Nabae, "Instantaneous ReactiveCompensator Comprising Switching Devices without Energy

examples are given which show the presence of inactive Storage Elements", IEEE Transactions on Industrial Applications, IA- power rather than reactive power in unbalanced three-phase 20(3):625-630, May 1984.

systems. This paper proposes a new approach (PVR method)

to compensate instantaneous reactive power as well as non-_tocompensateinstantaneou reactive power ^as

wellaPower

[5] F. Z. Peng and L-S. Lai,Theory for Three-Phase Power"Generalized InstantaneousSystems", IEEE Transactions onReactive zeroinactivepower

(NIC).

Thecompensation approachis Instrumentation, 1998.

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