Orthogonal components of the three-phase current at asymmetrical active-reactive load in 4-wire circuit Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

UDC 621.31

doi: 10.20998/2074-272X.2016.3.10

Iu.A. Sirotin

ORTHOGONAL COMPONENTS OF THE THREE-PHASE CURRENT AT ASYMMETRICAL ACTIVE-REACTIVE LOAD IN 4-WIRE CIRCUIT

Purpose. For the unbalanced sinusoidal mode with asymmetric voltage in 3-phase 4-wire to receive the orthogonal 4-component decomposition of 3-phase current, are classified symmetry/asymmetry of active and reactive load elements separately. Methodology. The methodology is based on the vector approach, which with one voice allows to analyze the energy characteristics of a 4-wire and 3-wire circuits as balanced and unbalanced modes. At asymmetrical voltage the matrix representation methodology of the equivalent conductivities is used. Results. For 3-phase 4-wire network with a sinusoidal unbalanced mode with asymmetric voltage obtained 4-component orthogonal decomposition of the 3-phase current. The components have a clear electro-energetic sense and are classified irrespective by the load condition. Originality. The resulting decomposition current develops the theory Currents' Physical Components (CPC) for 4-wire circuit with asymmetric voltage. For the first time the unbalanced current is classified by activity and reactivity of asymmetry load elements. Practical value. Practical value of the obtained orthogonal decomposition current and the power equation is a possibility of their utilization for the increase both quality of delivery and quality of consumption of electrical energy. References 8, figures 1.

Key words: three-phase circuit, active and reactive power, power shift, power equation, unbalanced current and mode, active-reactive asymmetrical load, asymmetrical voltage, currents' physical components (CPC).

Для 3- фазной схемы электроснабжения рассмотрен синусоидальный несимметричный режим. При несимметричном напряжении и ассиметричной активно-реактивной нагрузке для 4- проводной сети получено ортогональное разложение трехфазного тока. Четыре составляющие разложения классифицированы активностью/реактивностью и симметрией/асимметрией нагрузки и имеют однозначный электроэнергетический смысл. Для 4- проводной цепи с несимметричной нагрузкой при несимметричном напряжении полученное уравнение мощности развивает теорию токовых физических составляющих (Currents' Physical Components - CPC). Библ. 8, рис. 1.

Ключевые слова: трехфазная цепь, активная и реактивная мощность, мощность сдвига, уравнение мощности, несбалансированный ток и режим, активно-реактивная несимметричная нагрузка, несимметричное напряжение, Currents' Physical Components (CPC).

Introduction. Active-reactive unbalanced load not only consumes electrical energy (EE) of active power, but also the EE of inactive components of total power (TP) which leads to additional losses. An effective solution to the problem of reducing losses and increasing the accuracy of accounting EE is the combined use of compensating devices and differential accounting means consumption EE. However, even in a sinusoidal mode, taking into account existing means of measuring energy efficiency they measure EE due to only the symmetry of the load active and reactive elements (active power and reactive power of shift). In real conditions of asymmetry voltage components of TP due to the asymmetry of the active-reactive load elements lead to additional losses, however, are not measured are not counted and will not be compensated.

Problem definition. Compensation, measuring and accounting for components of TP are related, complementary objectives of effective EE consumption. These tasks are the same positions should be solved within the framework of the general theory of power using orthogonal decomposition of 3-phase current [1-6] The mutual orthogonal component decomposition can uniquely estimate the losses caused by them independently. Widely used power theory Currents' Physical Components (CPC) [2, 4-6] uses a methodology of orthogonal decomposition. At the sinusoidal unbalanced mode, 3-phase current comprises two orthogonal components: balanced and unbalanced. Balanced component (due to the symmetry of the active-reactive load elements) comprises orthogonally reactive current and active for three, and for a 4-wire circuit. The

asymmetry of the active-reactive load elements, both in symmetric and asymmetric under voltage leads to unbalance current.

Unfortunately, even in a sinusoidal mode CPC theory developed either for 3-wire or 4-wire circuits with a symmetrical voltage [2, 4-6]. Thus, for 3-wire circuit is decomposed into two components with unbalanced voltage in the CPC power theory of unbalance current, using the method of symmetrical components [6], which is not shared by the asymmetry of the load active and reactive elements explicitly.

The goal of the work is for unbalanced mode with asymmetric voltage in 3-phase 4-wire to obtain orthogonal 4-component decomposition of the 3-phase current, are classified by symmetry/asymmetry of separate active and reactive load elements separately.

Periodic power processes. When considering a 3-phase 4-wire circuit we assume that the voltage in the phases are measured relative to the neutral (Fig. 1).

Ia

U a

B

Ib 1

Ub

—» C

I с V. \u с

Fig. 1. 3-phase 4-wire power supply with unbalanced load - sinusoidal mode

© Iu.A. Sirotin

A

n

At any time, instantaneous values (i.v.) of voltages (relative to a «neutral» conductor) and i.v. of currents in the phases are considered as 3-dimensional vectors (matrix columns) of an arithmetic 3-dimensional space R(3)

u(t) = [(t) ub(t) uc(t)]r , i(t) = [(t) ib(t) ic(t)]r,(1)

hereinafter ris the transposition sign.

Steady power mode in a 3-phase section <A, B, C> is determined by 3-D ^-periodic curves of current and voltage:

u(t) = u(t + T ^ i(t ) = i(t + T ).

An ensemble of 3-D (3-phase) 7-periodic vector curves

x(t) = [Xa (t) Xb (t) Xc(t)]r, t e (v, v + T) (2) with finite root mean square (rms) value

llx ||=

v+T

- Jx(t)rx(t)dt

< œ

(3)

(4)

form a Hilbert space

L(23)(T) = {x(t), t e (v,v + T):||x ||< <»}.

For vector curves x(t), y(t) e L(23)(T) a scalar product (SP) is determined

v+T v+T

< x, y >= T Jx(ty y(t)dt = T J(x(t), y(t))dt (5)

vv

as integral averaged of scalar products of i.v. in a 3-D space R(3).

In particular, for active power

v+T v+T

< i,u >= T Ji{t)Tu(t)dt = T J(i(t), u(t))dt = P . (6)

v p(t) v

Instantaneous power p(t) = iTu = ia (t)ua (t) + ib (t)ub (t) + ic (t)uc (t) (7) equals to the electricity rate through the section <A, B, C>. In the space (4) the inequality of Cauchy-Schwarz is correct

< x,y > < || x || • || y ||. (8)

In particular, active power does not exceed the apparent (total) power

P = < i,u > < || u || • || i ||.

Sinusoidal mode and 3-complexes.

3-D curves of i.v. sinusoidal processes of voltage and current

u(t) = 42^.e[UeJwt], i(t) = Ie]Wt]. (9)

are T-periodic (To = 2n) and fully determined by 3-complexes of voltage and current

An ensemble of 3-form a 3-D complex space C(3) with a complex SP

(X,Z) = XrZ* = XX + XbZ*b + XCZ* . (12) Hereinafter * is a sign of complex conjugation. Thus, for rms

il x ii2==2 Xmxmxm =ix i2=x2.

^^m ^^m

In particular,

ii u ii=i U i = U , iiiii=iIi = I. (13)

For a couple of sinusoidal processes x(t),

z(t) e Z(23)(T) the equality is correct

< x,z >= Re[XrZ*] = Re[ZrX*]. (14) So, if 3-complexes are orthogonal then corresponding 3-D curves are orthogonal, too. The converse is not true.

From (14) it follows that at the sinusoidal mode active power is adequately represented in terms of 3-complexes of voltage and current

P = < i,u > = Re[IrU*] = Re[UrI*]. (15) The temporal shift of 3-D curve of i.v. of sinusoidal voltage u± (t) = u(t - T/4) is equivalent to a rotation of

the 3-complex of voltage in the space C(3) to 90°

u±(t) = 4l^e[ULe]at] = jUe]Wt]. (16)

Here i| u± ii = iiu i| . Because of < uL,u > = me[-jUrU*] = me[-j i U|2] = 0 , 3-D curves of voltages are orthogonal ( u 1 u± ).

Integral determination of reactive power (known as power of shift) is represented in terms of 3-complexes of voltage and current

Q =< u > = fte[-jUr I * ] = Jm[UrI * ]. (17) Powers (15) and (17) are connected by complex power - SP of 3-complexes of voltage and current

S = UrI* = Re[UrI*] + j Jm[UrI*] = P + jQ . (18) At the sinusoidal mode at symmetrical load the equation of powers is correct

P2 + Q2 = ii i ii - ii u ii. (19)

Equivalent conductivities of load current. At the

sinusoidal mode 3-complexes of current and voltage permit to determine equivalent conductivities of current in the section <A, B, C>

Ym = Gm - jBm = U^ ' m e>a ^ C}

(20)

~Ua ' Ue*- ' ' Ia ' ' J ejVa ' a UaYa Ya 0 0 Ua

U = Ub = Ubej , I = Ib = Ibein , (10) I = UbYb = 0 Yb 0 Ub = YU

Uc Ucej Ic Ij UcYc 0 0 y- _ Uc

- vectors of complex rms of voltage and current.

3-complexes (10) are calculated by 3-D curves of i.v. of sinusoidal processes of voltage and current

r—v+T r—v+T

U Ju(t)e"otdt, I Ji(t)e-j0tdt. (11)

and represent a 3-complex of 3-phase current in the matrix form

(21)

by using a diagonal matrix

Y = diag{Ya ,Yb ,Yc}. (22)

For a 4-wire circuit with a star-type load equivalent conductivities (20) in the section <A, B, C> equal to conductivities of phase loads.

Active power and power of shift are adequately represented by quadratic forms of the 3-complex of voltage

P = Re[UAY*U*], Q = Jm[UAY V]. (23) Active power (power of shift) depends only on conductivities of active (reactive) load elements

P =yZGm\UUm\2, Q =YJBm\(jm P. (24)

Losses of total 3-phase current per one Q

\| i \\2 = Re[ IA I* ] = (G2 + B2m)\Um \2 . (25)

Active and reactive current. For 3-D curve of sinusoidal current (9) it is correct

i(t) = ^¡m^.e[YUeJWt], I = YU . (26)

Algebraic form of complex equivalent conductivities (20) permits to resolute the diagonal matrix (22)

Y = G - jB,

(27)

< iR,u± > = Re[-jUTIR ] = Jm[UrBU ] =

= V Bm\Ûm\2 =< i,u±> = Q .

^^m

(38)

In the resolution (30) active (reactive) current is caused by summarily symmetry and asymmetry of active (reactive) load elements.

Balanced current component. A sinusoidal mode is balanced if 3-complexes of current and voltage (10) are collinear (parallel I \\U) [7, 8]

I\\U » I = /U ( / = /' + JP", p* 0). (39) A mode is really balanced [7, 8] if Jm[/] = /" = 0 . If the load is symmetrical then the mode is balanced at any unbalanced voltage.

For an unbalanced mode the 3-complex of components of current balanced with 3-phase voltage equals to projection of 3-complex of voltage in the space C(3)

(ru )u

IS = ( v v)v =- 2

\U\2

(40)

G = diag{Ga, Gb, Gc } , B = diag{Ba, Bb, Bc } (28) and divide 3-complex of current into two components associated with active and reactive load elements

I = IA + Ir , IA = Gu , IR =-jBU = Bul . (29) Resolution of 3-D curve of current (26)

i(t) = Ia (t) + Ir (t) (30) into active and reactive current

iA (t) = Sfte[G UeJ'wt ], iR (t) = V2^e[B U1ejat ] (31) is orthogonal in the space of 3-D curves (4). Because of the quantity

ITa IR = (G U)'(- jBU )* = j^Ul GmBm (32)

is pure imaginary then 3-D curves (31) are orthogonal

< iA,iR >= Re[IaIR] = 0 ^ iA 1 iR . (33)

Because of orthogonality of the resolution (29) for losses per one Q the Pythagoras equality is correct

l|i||2 = II ia II2 + II ir II2. (34)

Losses of active and reactive current

I\iA, II2 = Re[ IA IA ] = ^ GHUm I2, (35)

m

I\ir, 112 = Re[ IA IR ] = 2 BlIUm I2 (36) determine losses of total current (25). Here

I\ ia II2 ^ I\ i IP , I\ iR IP ^ \i IP.

Active current guarantees EE supply with active power of total current (24)

< u,iA >= Re[UAI*A] = Re[UAGU*] =

= Y Gm I Um I2 =< i,U >= P . (37)

Reactive current guarantees EE transmission of power of shift of total current (24)

Hereinafter:

V = [da ib »c]A, \ U \2 =U2a +vb +u2 = 1 (41) is the ort of the 3-complex of voltage

U = I U\ v , Um = UUm (m e{a, b, c}). (42) In terms of conductivities of the 3-complex of balance current (40)

A *

(IU )U = (S*/u2)■ U = ySU . (43)

Is =-

\U\2

Hereinafter:

ys = S 7 U2 = fad + Ybo2b + YcDI (44) is the equivalent complex conductivity of the balanced current component;

S * = (UAI * )* = IAU * = UAYU * is the complex conjugate power.

In terms of the ort of the 3-complex of voltage (42) the active and reactive power have equivalent form of representation:

P = Re[S] = UAGU* = U2 Y Gmuin ; (45)

^^m

Q = Jm[S?] = UABU* = U2 Ym BmU2m . (46)

Equivalent complex conductivity (44) of 3-D curve of balance current

is (t) = Sne[ y Ue1^ ], Is = y s U (47)

in all phases is the same and equal to weighted average sum of equivalent complex phase conductivities (20). Weighting factors are determined by the ort of the 3-complex of voltage (42).

If voltage is symmetric with direct sequence (DS)

then

v = (1/V3)[1 a* a]A, o2a = ol = ic2 = 1/3 , (48)

where a = el120° = -\J2 + jV^/2 . If the load is unbalanced

Vs * Ya, ys ^ Yb, ys ^ Yb, (49)

then the mode is unbalanced at any voltage

The complex conductivity of the balance current (47)

ys , (50)

determines conductivities associated with symmetry of active and reactive load elements

2 (51)

9s = GaUl + GbUb + GcU1 ;

BaOl + Bbul + B^l.

(52)

These conductivities equal to weighted average sums of phase conductivities. If the load is unbalanced then

9s * Ga , 9s * Gb, gs * Gc ;

(53)

bs * Ba , bs * Bb , bs * Bc. (54)

Conductivities (51, 52) characterize the symmetry of active and reactive load elements by phases for the 3-phase voltage.

3-complex (43) of the balanced component has two components: active and reactive

IsA =9 sU, IsR =- jb sU = b U (56)

and guarantees resolution of the balanced current

is (t) = lsA (t) + isR (t) (57)

into components associated with active and reactive load elements:

isA (t) = V2we[ Isa^0 ] = V2Re[g Ue0 ], (58)

isR(t) = V2we[IsReiot] = V2Re[bUie0]. (59) 3-D curves (58) h (59) are orthogonal because

< isA,isR > = Re[j= 0 . (60)

Because of orthogonality of the resolution (57) for current component the Pythagoras equality is correct

isA (t) 1 isR (t) ^ I]A + IR = Is2; (61)

I2a =9U2, isR =bU2.

From (58) it follows

< u,iSA >= Re[UrI*A] = Re[UrgSU*] = P (62)

Because of balanced active current is really parallel with voltage (IsA ||U ^ IsA = gSU) it guarantees EE supply of active power (62) with minimal losses [7]

11 isA || < || iA 11< || i || . (63)

Here P = < u,isA > = || u || • || isA ||. Balanced reactive current guarantees EE supply of power of shift

< u1,isR > = Re[-J'uTKr ] = < u1 J >= Q . (64) Because of balanced reactive current is really

parallel with voltage IsR ||U1 ^ it guarantees EE supply of reactive power of shift with minimal losses

|| isR || < || iR 11< || i ||. Here

|Q| = |<u1,isR >|=||«||-||iäR||

(65)

ID = I — IS , ( ID 1 IS ).

(66)

Unbalanced component (66) can be represented by using vector product in the space of 3-complexes C(3) [7, 8].

From (21) and (47) it follows

ID = I - Is = #U-ysU = ^-^7 = yDU. (67)

Yd

The matrix form of the 3-complex of unbalance current

D 0 0 Ua Id = YdU = 0 ydb 0 Ub . (68)

0 0 yJU

Uses a complex diagonal matrix

yD = diag{yD a ,yD b ,ydc } (69)

of equivalent conductivities of the unbalance current

D = Y m ~ Vs, m e {a, b, c}. (70)

If the voltage is symmetric to DS then (48) and

D = (2Ym Jk V3, m e {a, b, c}. (71)

Complex conductivities (70) characterize dissipation by phases of load conductivities regarding balance conductivity. Unbalance (asymmetry) determines the unbalanced current

iu (t) = V2^e[ IDe}Wt ], (72)

which is orthogonal to voltage

<u,iu > = Re[U( ydU)*] = Re[U2V D2m(Ym-is)] =

m

(73)

= Re[U2(y o2mYm) — ys)] = 0.

Unbalanced current and asymmetry of load conductivities. At the unbalanced mode the unbalanced component of the 3-complex of current (unbalance current) is determined as an orthogonal complement to the balanced component (40)

Here from (67) the resolution follows

I = Is + ID , i(t) = is (t) + iu (t). (74)

Dissipation (unbalance) by phases of separately active and reactive load elements

9Dm = Gm — 9s , &Dm = Bm - bs, m £ {a,b,c} (75) is represented by diagonal matrices of conductivities of active and reactive load elements:

9d = diag{gD a ,gD b ,gD c},

bD = diag{bD a ,bD b ,bD c}. Unbalance by phases (asymmetry of phase conductivities) separately of active and reactive load elements determines resolution of 3-complex of unbalance current into two components

Id = IDA + hn , (76)

Ida= 9d U = [gD aUa Q^Ub g^Uc ]Y, (77) IDR=-j bD U = -j[b D aU a b^ b^ ]Y. (78) The resolution of the unbalanced current is correct

iu (t) = iuA (t) + iuR (t), (79)

where iuA (t) = V2Re[ IDAejat ] = V2Re[gD Uejat ], (80)

iuR (t) = V25Re[ IDRe]at ] = V^Re^ U1ej}t ] (81) are the components determined by asymmetry of active and reactive load elements.

3-D curves (80) and (81) are orthogonal because

< iuA,iuR > = Re[IYAIDr ] = Re[ j Y ^9dm&dm ] = 0 •

m

Because of orthogonality of the resolution (79) the Pythagoras equality is correct

(t ) 1

iuR (t) ^ ID

+12 = i2 I 1 DÎ? — 1 D

iDA = Ua + obvlb + t&Dc) .

IDr = U 2(^b2 a +^b2 b +^b2 c ).

(82)

(83)

(84)

So, for the resolution of current two dichotomous factors are used:

• the first factor is determined by activity and reactivity of load elements;

• the second factor is determined by symmetry and asymmetry of load elements by phases.

Resolution of 3-phase current and power equation of unbalanced mode. Combination of values of two factors classifying the load:

• «activity/reactivity» - the first factor)

l = l a + ir ;

• («symmetry/asymmetry» - the second factor)

to use them not only for separated measurement and recording of inactive components of TP but also to solve the compensation problem at sinusoidal unbalanced mode.

Conclusions. For a 3-phase 4-wire network with a sinusoidal unbalanced mode at asymmetric voltage a 4-component orthogonal decomposition of the 3-phase current is obtained. Components having a clear power sense independently classify the load condition. The resulting decomposition expands the CPC theory to 4-wire circuits with unbalanced voltage imbalance by resolution of the unbalance current into two components determined by the asymmetry of the active and reactive load elements.

permitted to obtain four mutually orthogonal components of 3-phase current

isA, iuA, lsR, iuR ,

which guarantee resolution into four mutually orthogonal 3-phase current components

i = iA + iR = ((a + Ra ) + (R + RA) . (85)

iA lR

Because of current resolution (85) is orthogonal then identity (equation of losses per one Q) is correct

llill2 =11 isA l2 + IliuA l2 + IlisR l2 + IliuR l2 . (86) Multiplication of equation (86) by square of rms of voltage II u II2 gives the equation for powers of sinusoidal unbalanced mode

ST = P2 + Q2 + DG + DB . (87)

Here:

St =\\i\\-\\u\\ (88)

is the total power;

P = \\ isA \\ • \\ U \\ = < i,U > (89)

is the active balance power determined by symmetry of active load elements;

\Q\=\\isR\\-\\uJ\= \< i,UL >\ (90)

is the reactive balance power determined by symmetry of active load elements;

Dg = \\iuA \\*\\U\\ (91)

is the unbalance power determined by asymmetry of active load elements;

Dg = \\ iuR \\ * \\ u \\ (92)

is the unbalance power determined by asymmetry of reactive load elements;

Power equation (87) generalizes the equation for the sinusoidal unbalanced mode [7]

(93)

12 • U2 = P2 + Q2 + Du2,

because of Du2 = DG2 + DB2 .

Practical value of the received orthogonal decomposition of current and power equations is the ability

REFERENCES

1. 1459-2010 IEEE Standard definitions for the measurement of electric power quantities under sinusoidal, nonsinusoidal, balanced, or unbalanced conditions. doi: 10.1109/ieeestd.2010.5439063.

2. Czarnecki L.S. Orthogonal decomposition of the currents in a 3-phase nonlinear asymmetrical circuit with a nonsinusoidal voltage source. IEEE Transactions on Instrumentation and Measurement, 1988, vol.37, no.1, pp. 30-34. doi: 10.1109/19.2658.

3. Ferrero A., Superti-Furga G. A new approach to the definition of power components in three-phase systems under nonsinusoidal conditions. IEEE Transactions on Instrumentation and Measurement, 1991, vol.40, no.3, pp. 568577. doi: 10.1109/19.87021.

4. Lev-Ari H., Stankovic A.M. A decomposition of apparent power in polyphase unbalanced networks in nonsinusoidal operation. IEEE Transactions on Power Systems, 2006, vol.21, no.1, pp. 438-440. doi: 10.1109/tpwrs.2005.860903.

5. Czarnecki L.S., Haley P.M. Unbalanced power in four-wire systems and its reactive compensation. IEEE Transactions on Power Delivery, 2015, vol.30, no.1, pp. 53-63. doi: 10.1109/tpwrd.2014.2314599.

6. Czarnecki L.S., Bhattarai P.D. Currents' physical components (CPC) in three-phase systems with asymmetrical voltage. Przeglqd Elektrotechniczny, 2015, no.6, pp. 40-47. doi: 10.15199/48.2015.06.06.

ZSirotin Iu.A. Vectorial instantaneous power and energy modes in three-phase circuits. Tekhnichna elektrodynamika -Technical electrodynamics, 2013, no.6, pp. 57-65. (Rus). 8. Sirotin Iu.A. Non-pulsed mode of supply in a three-phase system at asymmetrical voltage. Przeglad Elektrotechniczny, 2013, no.7, pp. 54-58.

Received 29.02.2016

Iu.A. Sirotin, Doctor of Technical Science, Professor, National Technical University «Kharkiv Polytechnic Institute», 21, Frunze Str., Kharkiv, 61002, Ukraine, e-mail: yuri_sirotin@ukr.net

i = i +1

s u

How to cite this article:

Sirotin Iu.A. Orthogonal components of the three-phase current at asymmetrical active-reactive load in 4-wire circuit.

Electrical engineering & electromechanics, 2016, no.3, pp. 62-66. doi: 10.20998/2074-272X.2016.3.10.