Electric Power Systems Research 76 (2006) 1055–1063

Study of the power system harmonic response in thepresence of the Steinmetz circuit

Manuel Caro a, Luis Sainz b,∗, Joaquın Pedra b

a IDOM Ingenierıa y Arquitectura, C. Barcas, 2, 46002 Valencia, Spainb Department of Electrical Engineering, ETSEIB-UPC, Av. Diagonal, 647, 08028 Barcelona, Spain

Received 22 June 2005; received in revised form 21 October 2005; accepted 17 December 2005Available online 17 February 2006

Abstract

The Steinmetz circuit is constituted by single-phase loads, e.g. traction systems, delta-connected with reactances to load the network withbalanced currents. A parallel resonance between the capacitive reactance of the Steinmetz circuit and the power system reactances can be produced,and therefore harmonic voltage distortion in the presence of non-linear loads increases. The paper studies this resonance analytically and presentsan expression to locate it. Experimental measurements have also been made to validate the obtained analytical results.©

K

1

psctanpbTcrutt

ot[c

0d

2006 Elsevier B.V. All rights reserved.

eywords: Power quality; Frequency scan; Harmonic analysis

. Introduction

A power system is expected to operate under balanced three-hase conditions; however, single-phase loads such as tractionystems can be connected. These loads consume unbalanced lineurrents and cause unequal voltage drops in distribution lines,hus resulting in unbalanced load bus voltages [1]. These unbal-nced operating conditions can produce undesirable effects inetworks [2,3]. For this reason, circuits consisting of single-hase loads, inductors and capacitors suitably connected haveeen developed in order to reduce unbalance in power systems.hese circuits, which are more commonly known as Steinmetzircuits, allow the network to be loaded with symmetrical cur-ents. There are different studies on the Steinmetz circuit designnder sinusoidal balanced or unbalanced conditions. Some ofhese studies propose analytical expressions and optimizationechniques for its characterization [4].

Moreover, in the last few years there has been an increasef non-linear devices in electric networks. This fact has ledo a growing presence of harmonic voltages in power systems5,6], and the Steinmetz circuit design must consider this cir-

a power system to balance a single-phase load, a parallel reso-nance is produced between its capacitors and the supply systeminductors. In non-sinusoidal conditions, this resonance must belocated to prevent harmonic problems when the Steinmetz circuitis connected. The resonance problem was pointed out in [11].Afterwards, in [12], this problem was studied and the resonancewas characterized numerically. In [12], several curves were fit-ted numerically from the power system harmonic impedancesto predict the resonance at the 5th, 7th and 11th harmonics only.

The present paper continues the above studies on power sys-tem harmonic behavior in the presence of the Steinmetz circuit.In particular, the parallel resonance between the Steinmetz cir-cuit and the supply system reactances is analytically studied.An expression to locate this resonance is obtained from the ana-lytical study of the power system harmonic impedances. Thisexpression allows the resonance for any value of the single-phaseload of the Steinmetz circuit and any harmonic to be predicted.Laboratory measurements have been made to validate the theo-retical expressions obtained in the analytical study.

2. Steinmetz circuit design

umstance [7–10]. Thus, if the Steinmetz circuit is connected to

∗ Corresponding author. Tel.: +34 93 4011759; fax: +34 93 4017433.E-mail address: sainz@ee.upc.edu (L. Sainz).

Fig. 1 presents a network with the supply system, the Stein-metz circuit and the non-linear load.

In the Steinmetz circuit of Fig. 1, the single-phase load ismodeled with the impedance Z = Y−1 = R + jkX , where k

378-7796/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2005.12.023

- Lk - Lk L L

1056 M. Caro et al. / Electric Power Systems Research 76 (2006) 1055–1063

is the studied harmonic, RL the load resistance and XL is the loadreactance at the fundamental frequency of the supply voltage.This single-phase load is delta-connected to a capacitor (Z- 2k =Y-

−12k = −jX2/k) and a pure inductance (Z- 1k = Y-

−11k = jkX1) to

load the network with balanced currents (i.e. I-B1 = a-2 · I-A1,

I-C1 = a- · I-A1, where a- = ej2π/3).The procedure to design the Steinmetz circuit, i.e. the deter-

mination of the capacitor X2 and the inductance X1 from thesingle-phase load impedance, is presented below.

The three-phase harmonic currents consumed by the Stein-metz circuit are determined from the three-phase harmonic volt-ages (Fig. 1):

⎡⎢⎣

I-Ak

I-Bk

I-Ck

⎤⎥⎦ =

⎡⎢⎣

1 0 −1

−1 1 0

0 −1 1

⎤⎥⎦

⎡⎢⎣

Y- 1k 0 0

0 Y- Lk 0

0 0 Y- 2k

⎤⎥⎦

⎡⎢⎣

V- ABk

V- BCk

V- CAk

⎤⎥⎦ .

(1)

The symmetrical components of the Steinmetz circuit three-phase fundamental currents (I-A1, I-B1, I-C1) are then obtainedwith the Fortescue transformation:

⎡⎢⎣

I-01

I

⎤⎥⎦ = 1

⎡⎢⎣

1 1 1

1 a a2

⎤⎥⎦

⎡⎢⎣

I-A1

I

⎤⎥⎦ (a = ej(2π/3)). (2)

f

m

coatc

| − 1/X1 + 2λmλ2/RL}2

λmλ2/RL)2},(4)

wpf

f

Fig. 1. Studied system.

Fig. 2. Harmonic behavior of the supply system and the Steinmetz circuit.

i.e. the negative sequence current to be zero:

√3

(1

X2+ 1

X1

)− 2λ2

RL= 0

1

X2− 1

X1+ 2λmλ2

RL= 0

⎫⎪⎪⎬⎪⎪⎭ ⇒ X1 =

√3RL

λ2(1 + √3λm)

;

X2 =√

3RL

λ2(1 − √3λm)

. (5)

From (5), the circuit under discussion (with capacitors andinductors) turns out to be feasible (X2 > 0) only when the fun-damental power factor of the loads satisfies the condition 1 ≥λ >

√3/2. In the paper, the studied values of the power fac-

tor are λ = (1, . . ., 0.9) as they are the usual values in powersystems.

3. Power system harmonic response

According to Fig. 1, the harmonic response of the system“

-p1

I-n13 - -

1 a-2 a-

-B1

I-C1

-

After that, the unbalance factor is determined analyticallyrom the sequence currents I-01, I-p1 and I-n1:

- i1 = I-n1

I-p1= I-A1 + a-

2I-B1 + a-I-C1

I-A1 + a-I-B1 + a-2I-C1

= V- AB1Y- 11(1−a-2)+V- BC1Y- L1(a-

2−a-) + V- CA1Y- 21(a- − 1)

V- AB1Y- 11(1−a-)+V- BC1Y- L1(a- − a-2) + V- CA1Y- 21(a-

2 − 1)

= V- AB1Y- 11 + a-2V- BC1Y- L1 + a-V- CA1Y- 21

V- AB1Y- 11 + a-V- BC1Y- L1 + a-2V- CA1Y- 21

1 − a-2

1 − a-. (3)

Thereby, the final expression of this factor depends on theonsideration of the balanced or unbalanced voltages at the pointf common coupling [4]. In this paper, the balanced voltagesre considered (V- BC1 = a-

2V- AB1 and V- CA1 = a-V- AB1) in ordero simplify the study of the circuit design under non-sinusoidalonditions. The obtained unbalance factor is

m- i1| = |I-n1||I-p1|

=

√√√√ {√3(1/X2 + 1/X1) − 2λ2/RL}2 + {1/X2

4{(λ2/RL)2 + (1/X2 − 1/X1 −

here λ = RL/ZL1 is the fundamental power factor of the single-hase load, ZL1 the magnitude of the load impedance at theundamental frequency and λm = ((1/λ)2 − 1)1/2.

Finally, the symmetrizing reactive elements are obtained byorcing the fundamental current unbalance factor (4) to be zero,

seen” from the non-linear load must be studied to determine

M. Caro et al. / Electric Power Systems Research 76 (2006) 1055–1063 1057

the power system harmonic response. This implies analyzingthe passive set formed by the supply system and the Steinmetzcircuit, which is shown in Fig. 2.

The equivalent harmonic impedance matrix of the studied setallows its harmonic behavior to be characterized. According toFig. 2, this matrix relates the kth harmonic three-phase voltagesand currents at the non-linear node as follows:⎡⎢⎣

V- Ak

V- Bk

V- Ck

⎤⎥⎦ = Z- Bk

⎡⎢⎣

I-A

I-B

I-C

⎤⎥⎦ =

⎡⎢⎣

Z- AA Z- AB Z- AC

Z- BA Z- BB Z- BC

Z- CA Z- CB Z- CC

⎤⎥⎦

⎡⎢⎣

I-Ak

I-Bk

I-Ck

⎤⎥⎦ =

⎡⎢⎣

Y- S + Y- 1 + Y- 2 −Y- 1 −Y- 2

−Y- 1 Y- S + Y- 1 + Y- L −Y- L

−Y- 2 −Y- L Y- S + Y- 2 + Y- L

⎤⎥⎦

−1

k

⎡⎢⎣

I-Ak

I-Bk

I-Ck

⎤⎥⎦ ,(6)

where Y- Sk = Z-−1Sk = (RS + jkXS)−1 corresponds to the admit-

tance of the power supply system and Y- Lk, Y- 1k and Y- 2k corre-spond to the admittances of the Steinmetz circuit components(Section 2). The voltage node method, which considers pointN in Fig. 2 as the reference bus, has been applied to obtain(6).

Finally, it can be observed that the harmonic behaviordepends not only on the harmonic impedance matrix (i.e. thepower system impedances) but also on the non-linear load cur-rent injections.

The presence of single- and three-phase non-linear loads isstudied in the paper considering that the supply voltages arebalanced and the Steinmetz circuit designed with (5) consumes

j0.0493 pu, which feeds a Steinmetz circuit, RL = 1.368 pu andλ = 1.0 (according to (5), X1 = X2 = 2.372 pu).

The magnitudes of the Z- Bk impedances (Z- AAk to Z- CCk) andthe positive- and negative-sequence driving impedances (|Z- Apk|to |Z- Cnk|) ((7) and (8)) are plotted in Fig. 3 (broken and contin-uous lines, respectively).

From Fig. 3, it can be noticed that:

• The connection of the Steinmetz circuit causes a parallel res-onance in the Z- Bk impedances and the driving impedances.In the example, this resonance is close to the fifth harmonicfor all the impedances (kp ≈ 251/50 = 5.02, where 251 Hz isthe frequency of the measured parallel resonance and 50 Hzis the fundamental frequency.)

• The asymmetrical resonant behavior of the circuit leads to anasymmetrical effect on the harmonic voltages, and the mostcritical resonance takes place in phases A and C, betweenwhich the capacitor is connected. Both phases have the high-

io

4r

4

Z

i

symmetrical currents.The harmonic currents injected by single-phase non-linear

loads in balanced conditions have all the odd harmonics and theymay not be in any sequence (I-Ak, I-Bk and I-Ck, where k = 1, 3,5, 7, . . .). Therefore, the diagonal and the non-diagonalimpedances of the matrix Z- Bk (Z- Bk impedances, i.e. Z- AAk toZ- CCk) must be studied to characterize the power system har-monic behavior. The calculation of both sets of impedances isnecessary because a resonance in any of these impedances couldcause a high level of distortion in the corresponding voltages.

The harmonic currents injected by three-phase non-linearloads in balanced conditions have only the first and the fifthorder harmonics (k = 1, 7, . . . and k = 5, 11, . . ., respectively).The first order harmonic currents are a set of positive sequence(I-Bk = a-

2I-Ak and I-Ck = a-I-Ak, where k = 1, 7, . . .) and thefifth order harmonic currents are a set of negative sequence(I-Bk = a-I-Ak and I-Ck = a-

2I-Ak, where k = 5, 11, . . .). Therefore,the matrix Z- Bk can be reduced to three driving impedances since

V- Gk = (Z- GGk + a-2Z- GHk + a-Z- GFk)I-Gk = Z- GpkI-Gk (7)

for the first order harmonic currents and

V- Gk = (Z- GGk + a-Z- GHk + a-2Z- GFk)I-Gk = Z- GnkI-Gk (8)

for the fifth order harmonic currents. In (7) and (8), GHF are thephases corresponding to the sequence ABCABC. . . (e.g. withG = B, HF = CA).

As an example, the harmonic response of the network of Fig. 2has been measured in the laboratory and is presented in Fig. 3.Considering the base values UB = 230 V and SB = 2.7 kVA,this network comprises the supply system, Z- S = 0.0219 +

est harmonic impedance, and will therefore have the highestharmonic voltages.

In the following sections, this power system harmonic behav-or is analytically studied and the above-presented parallel res-nance is analytically located.

. Analytical study of the power system harmonicesponse

.1. Power system harmonic impedances

In the study, the magnitudes of the Z- Bk impedances (Z- AAk to

- CCk) are obtained from (6) and the magnitudes of the drivingmpedances (|Z- Apk| to |Z- Cnk|) are obtained from (7) and (8).

The magnitudes of the Z- Bk impedances are

|Z- AAk| = |Y- 2S+Y- S(Y- 1+Y- 2 + 2Y- L) + Y- 1(Y- 2 + Y- L) + Y- 2Y- L|k

Y- Sk|D- k||Z- BBk| = |Y- 2

S+Y- S(Y- 1+2Y- 2 + Y- L) + Y- 1(Y- 2 + Y- L) + Y- 2Y- L|kY- Sk|D- k|

|Z- CCk| = |Y- 2S + Y- S(2Y- 1+Y- 2+Y- L) + Y- 1(Y- 2 + Y- L) + Y- 2Y- L|k

Y- Sk|D- k||Z- ABk| = |Z- BAk| = |Y- 1k(Y- S + Y- 2 + Y- L)k + Y- 2kY- Lk|

Y- Sk|D- k||Z- ACk| = |Z- CAk| = |Y- 2k(Y- S + Y- 1 + Y- L)k + Y- 1kY- Lk|

Y- Sk|D- k||Z- BCk| = |Z- CBk| = |Y- Lk(Y- S + Y- 1 + Y- 2)k + Y- 1kY- 2k|

Y- Sk|D- k|,

(9)

1058 M. Caro et al. / Electric Power Systems Research 76 (2006) 1055–1063

Fig. 3. Measured impedance/frequency matrix in the presence of the Steinmetz circuit (continuous line: driving impedances; broken line: Z- Bk impedances).

where

|D- k| = |Y- 2S + 2Y- S(Y- 1 + Y- 2 + Y- L)

+ 3(Y- 1(Y- 1 + Y- 2) + Y- 2Y- L)|k. (10)

The magnitudes of the driving impedances are

|Z- Ahk| = |2Y- Sk + Y- 1k + Y- 2k + 4Y- Lk + jδk

√3(Y- 2 − Y- 1)k|

2|D- k||Z- Bhk| = |Y- 1k − 2(Y- 2 + Y- L)k − Y- Sk − jδk

√3(Y- S + Y- 1 + 2Y- 2)k|

2|D- k||Z- Chk| = |Y- 2k − 2(Y- 1 + Y- L)k − Y- Sk + jδk

√3(Y- S + 2Y- 1 + Y- 2)k|

2|D- k|(h = p, n),

(11)

where δk = 1 for Z- Apk, Z- Bpk and Z- Cpk, i.e. for the positive-sequence harmonics k = 1, 7, . . . or δk = −1 for Z- Ank, Z- Bnk andZ- Cnk, i.e. for the negative-sequence harmonics k = 5, 11, . . ..

In the following analysis of (9) and (11), the supply systemadmittances and the Steinmetz circuit components are

Y- Sk ≈ −j1

kXS, Y- 1k = −j

λ2(1 + √3λm)

k√

3RL,

Y- 2k = jkλ2(1 − √

3λm)√3RL

, Y- Lk = 1

RL(1 + jkλm), (12)

where the supply system resistance, RS, has been neglected and(5) has been considered to obtain the admittances of the Stein-metz circuit components.

It can be noted that the magnitude of the impedances, i.e. thepower system harmonic response, depends on four variables: theharmonic order, k, the supply system fundamental reactance, XS,the single-phase load resistance, RL, and the single-phase loadfundamental power factor, λ.

M. Caro et al. / Electric Power Systems Research 76 (2006) 1055–1063 1059

4.2. Normalization of the power system harmonicimpedances

By using the single-phase load resistance, RL, as reference,the Z- Bk impedances and the driving impedances ((9) and (11))can be normalized. In this way, the magnitude of the normal-ized impedances (|Z- AAk|/RL to |Z- CCk|/RL and |Z- Apk|/RL to|Z- Cnk|/RL, respectively) depends on three variables only: theharmonic order, k, the single-phase load fundamental power fac-tor, λ, and the ratio of the supply system fundamental reactanceto the single-phase load resistance, XS/RL.

For example, the magnitude of the normalized impedance,|Z- AAk|/RL, can be expressed as

|Z- AAk|RL

= R2L|Y- 2

S+Y- S(Y- 1+Y- 2+2Y- L)+Y- 1(Y- 2+Y- L)+Y- 2Y- L|kRLY- SkR

2L|D- k|

,

(13)

where (13) has only the terms RLY- Sk, RLY- 1k, RLY- 2k and RLY- Lk,and these terms are

RLY- Sk = −j1

kXS/RL, RLY- 1k = −j

λ2(1 + √3λm)

k√

3,

RLY- 2k = jkλ2(1 − √

3λm)√3

, RLY- Lk = 1

1 + jkλm. (14)

rtprvX

4

soaomprs

4r

iwo

Fig. 4. (a and b) Location of the maximum value of the normalized impedancemagnitudes.

with respect to the harmonic k. From (15), note that the minimumvalue of the impedance magnitude denominator coincides withthe minimum value of the term |D- k| (10).

In order to work with only three variables (k, λ and XS/RL),the above conclusion can be extended to magnitude of the nor-malized impedances (|Z- AAk|/RL to |Z- CCk|/RL and |Z- Apk|/RLto |Z- Cnk|/RL) and the normalized value of the term |D- k|, i.e.R2

L · |D- k| (13).Therefore, for a given power factor λ and a ratio XS/RL,

the parallel resonance is at the same harmonic kp for all theimpedances (Z- Bk impedances and driving impedances) andcoincides with the minimum of the term |D- k| or R2

L · |D- k| (15).To illustrate this, the system of Fig. 2 with XS/RL = 0.018

and λ = 1.0 is studied. The results obtained for the impedancesZ- Apk/RL and Z- Ank/RL are shown in Fig. 4(a and b). Themagnitude of these impedances (|Z- Apk|/RL and |Z- Ank|/RL) isplotted in Fig. 4(a), and their numerator (Num{|Z- Apk|/RL} andNum{|Z- Ank|/RL}) and the term |D- k| of their denominators areplotted in Fig. 4(b). It can be noted that the maximum value of thenormalized impedance magnitudes coincides with the minimumvalue of the term |D- k|. The same is true of the other impedancemagnitudes.

According to the above study, Fig. 5 shows the harmonic ofthe parallel resonances, kp, for different values of the power fac-tor λ and the ratio XS/RL. The parallel resonance has been locatedas the minimum value of the term |D | (continuous lines). It canb

•

The same is true of the other impedance magnitudes.It can be noted that the ratio XS/RL is equal to the inverse of the

atio (1/λ) · SS/SL, i.e. λ · SL/SS = λ · XS/ZL = XS/RL, where SS ishe short-circuit power at the PCC bus and SL is the apparentower of the single-phase load. Therefore, considering the usualange of SS/SL ratio values [1], and the fundamental power factoralues, λ = (1, . . ., 0.9), the XS/RL values used in the study areS/RL = (0, . . ., 0.2).

.3. Analytical location of the parallel resonance

Considering the variables λ and XS/RL, the following three-tage study of the magnitudes of the normalized impedancesbtained from (9) and (11) allows the harmonic kp of the par-llel resonance to be analytically located. In the first and sec-nd stages, the parallel resonance of the normalized impedanceagnitudes is numerically analyzed and an approximation is

roposed to locate this resonance. In the third stage, the parallelesonance is analytically located on the basis of the previoustages.

.3.1. First stage: numerical location of the parallelesonance

The parallel resonance, i.e. the maximum value of thempedance magnitudes (|Z- AAk| to |Z- CCk| and |Z- Apk| to |Z- Cnk|)ith respect to the harmonic k coincides with the minimum valuef the impedance magnitude denominator

Den{|Z- AAk|} = · · · = Den{|Z- CCk|} = |Y- Sk||D- k|Den{|ZApk|} = · · · = Den{|ZCnk|} = 2|D- k|

(15)

- k

e noted that:

The harmonic of the parallel resonance, kp, increases withdecreasing the power factor λ or the ratio XS/RL.

1060 M. Caro et al. / Electric Power Systems Research 76 (2006) 1055–1063

Fig. 5. Location of parallel resonances (continuous line: numerical location ofthe |D- k| minimum value; broken line: analytical location of the |D- k| minimumvalue, (18)).

• If the power factor of the single-phase load is close to theunity value, the probability of having the parallel resonanceat the low harmonics (kp = 3, 5 or 7) increases.

• The measured parallel resonances of Fig. 3 (XS/RL = 0.036and λ = 1.0) are labeled with point P. The numerical studyis in agreement with the measured harmonic behavior of thesystem.

• The simulated parallel resonances of Fig. 4 (XS/RL = 0.018and λ = 1.0) are labeled with point Q.

In this section, it has been proved that the location of the par-allel resonance by means of the minimum value of the term |D- k|or R2

L · |D- k| provides results in agreement with the maximumvalue of the normalized impedance magnitudes. However, theanalytical determination of these parallel resonances from (10)is too complicated and it is not possible to obtain an expressionto locate this resonance. Therefore, an approximation, which ispresented below, must be used.

4.3.2. Second stage: approximation to analytically locatethe parallel resonance

The seven terms of |D- k| (10) are numerically studied to ana-lyze if some can be neglected.

In order to do this and according to (12), the last six termsof |D- k|, 2Y- SkY- 1k, 2Y- SkY- 2k, 2Y- SkY- Lk, 3Y- 1kY- 2k, 3Y- 1kY- Lk and3Y- 2kY- Lk, are referred to the first term, Y-

2Sk, and the following

ratios are numerically studied:

c-1k = 2Y- SkY- 1k

Y-2Sk

= −λ2(√

3λm + 1)2√3

(XS

RL

),

c-2k = 2Y- SkY- 2k

Y-2Sk

= −λ2(√

3λm − 1)2√3k2

(XS

RL

)

c-3k = 2Y- SkY- Lk

Y-2Sk

= − 1

1 + k2λ2m

2

k

XS

RL(kλm + j),

c-4k = 3Y- 1kY- 2k

Y-2Sk

= λ4(3λ2m − 1)k2

(XS

RL

)2

c-5k = 3Y- 1kY- Lk

Y-2Sk

= −λ2(√

3λm + 1)

1 + k2λ2m

k√

3

(XS

RL

)2

(kλm + j)

c-6k = 3Y- 2kY- Lk

Y-2Sk

= −λ2(√

3λm − 1)

1 + k2λ2m

k3√

3

(XS

RL

)2

(kλm + j).

(16)

To analyze the contribution of the different terms to |D- k|, thereal and imaginary parts of the complex ratios, chrk = Re{c-hk}and c = Im{c } with h = 1–6, have been calculated and plot-tpbrt

•

•

•

fc

|

4r

nk

Fig. 6. Study of the |D- k| terms contribution.

hik -hk

ed in Fig. 6 for the different values of the ratio XS/RL. Theower factor equal to 1 is considered in Fig. 6. The curves haveeen calculated for the corresponding harmonic of the parallelesonance, kp, e.g. for XS/RL = 0.036 and λ = 1 (point P in Fig. 5)he harmonic kp = 5 is used.

From Fig. 6, it can be noted that:

The imaginary parts of c-1k, c-2k and c-4k are zero and are notplotted.For λ = 1, λm = ((1/λ)2 − 1)1/2 = 0; thus, the real parts of c-3k,c-5k and c-6k are zero and are not plotted.The contribution of the ratios c-1k, c-4k and c-5k to |D- k|is lower than that of the other ratios at the harmonicof the parallel resonance, and therefore they can be neg-lected.

The same conclusion can be drawn for power factors differentrom the unity value. In these cases, the real parts c-3k, c-5k and

-6k are not zero and must also be analyzed.Therefore, the term |D- k| can be approximated as

D- k|approx ≈ |Y- 2Sk + 2Y- Sk(Y- 2k + Y- Lk) + 3Y- 2kY- Lk|. (17)

.3.3. Third stage: analytical location of the parallelesonance

To analytically locate the harmonic of the parallel reso-ance, the minimum of |D- k|approx with respect to the harmonicis analytically obtained, and the expression to locate this

M. Caro et al. / Electric Power Systems Research 76 (2006) 1055–1063 1061

resonance is

kp =√

G1λm + 4λG2 + √G3λm + G4

3xrλG5

G1 = 36λ2x2r + 14

√3λ2xr + 12(λ2 + 1)

G2 = 3√

3λ2x2r − (4λ4 + 7.5λ2 − 10.5)xr − √

3(λ2 − 1)

G3 = 216√

3λ5x4r + λ3(144λ4 − 612λ2 + 756)x3

r

+ λ3(128√

3λ4 − 534√

3λ2 + 438√

3)x2r

− λ(192λ6 − 505λ5 + 564λ2 − 252)xr

+ λ(24√

3λ4 − 48√

3λ2 + 24√

3)

G4 = λ4(−216λ2 + 324)x4r + √

3λ4(48λ4 − 540λ2 + 504)x3r

+ λ2(256λ8 − 768λ6 + 1254λ4 − 1383λ2 + 657)x2r

− √3λ2(64λ6 − 280λ4 + 384λ2 − 168)xr

− 24λ6 + 84λ4 − 96λ2 + 36

G5 = (4λ2 − 3)(9λ2x2r + 12λλmxr − 4(λ2 − 1)),

(18)

where λm = ((1/λ)2 − 1)1/2 and xr = XS/RL.This expression is plotted in Fig. 5 (broken lines). It can

be observed that the analytical expressions allow the parallelresonance to be located with an acceptable error.

For λ = 1, (18) can be rewritten as

k

√

(ie

rf.

f

w1w

wFn[

ahrdi

resonance at the fifth harmonic when the Steinmetz circuit isconnected, can be calculated as XS = 0.0397 · 1.368 = 0.0543 pu.It must be noted that this reactance is close to the reactance ofthe example, XS = 0.0493 pu.

5. Resistance influence on the power system harmonicresponse

The resistances of the supply system and the Steinmetz cir-cuit inductance have not been considered in the previous study.In general, these resistances could damp the power system har-monic response and shift the system resonance frequencies.

In this section, this influence is numerically analyzed from(9) and (11) by considering the above-mentioned resistances,i.e. Y- Sk = Z-

−1Sk = 1/(RS + jkXS) and Y- 1k = Z-

−11k = 1/(R1 +

jkX1).It has been numerically verified that the supply system

resistance, RS, damps the Z- Bk impedances and the drivingimpedances but does not modify the resonance frequencies ofthese impedances.

As an example, the network of Fig. 2, which comprises a sup-ply system (XS = 0.144 pu and RS/XS = 0, 0.1 and 0.5) feedinga Steinmetz circuit (RL = 4 pu, λ = 1.0 and X1 = X2 = 6.928 pu),is analyzed. The driving impedances of phase A are obtainedfrom (11) and are plotted in Fig. 7. It can be observed thattifXib

F

p,λ=1 =

√√√ 6(6√

3(XS/RL) − 2

+√

27(XS/RL)2 + 3√

3(XS/RL) + 4)

9(XS/RL). (19)

As an example, the harmonic resonance shown in Fig. 3XS = 0.0493 pu, RL = 1.368 pu and λ = 1), which was measuredn the laboratory, can be predicted with the obtained analyticalxpressions ((18) or (19)), and the result is kp = 4.95.

It is observed that (18) allows the harmonic kp of the parallelesonance for any value of the ratio xr = XS/RL and the poweractor λ to be predicted within the studied ranges, i.e. xr = (0,. ., 0.2) and λ = (1, . . ., 0.9). This expression has been calculatedrom the analytical study of (9) and (11).

In the previous study of [12], the expression of the ratio xr,hich provokes the parallel resonance only at the 5th, 7th and1th harmonics, was numerically obtained for any power factorithin the studied range:

xrk = (E1k + E2kλ + E3kλ2)

−1, 0.88 ≤ λ ≤ 0.965

xrk = (F2kλ eλF2k )−1

, 0.965 < λ ≤ 1,(20)

here k = 5, 7 and 11 and the parameters E1k, E2k, E3k, F1k and2k were calculated from close fitting of the curves obtainedumerically from (11). Their values were shown in Table II of12].

With (20), the harmonic resonance shown in Fig. 3 can bepproximately predicted since this resonance is close to the fiftharmonic. First, the value of the ratio xrk, which provokes aesonance at the fifth harmonic, is calculated from (20) with theata of Table II in [12], xr5 = 25.17−1 = 0.0397. Once this ratios determined, the supply system reactance, which provokes the

he supply system resistances influence the value of the driv-ng impedances but not the resonance frequency. This resonancerequency can be determined by means of (18) or (19), i.e. fromS/RL = 0.144/4 = 0.036, the harmonic of the parallel resonance

s kp = 4.95. This resonance frequency can also be determinedy means of Fig. 5 (point P).

ig. 7. Influence of the supply system resistance on the driving impedances.

1062 M. Caro et al. / Electric Power Systems Research 76 (2006) 1055–1063

Fig. 8. Location of the parallel resonances (continuous line: analytical results;broken line: measured results).

It has also been numerically verified that the resistance ofthe Steinmetz circuit inductance does not modify the obtainedpower system harmonic response (this has been verified withresistance values up to 10X1). Actually, the fact that the terms2Y- SkY- 1k, 3Y- 1kY- 2k and 3Y- 1kY- Lk can be neglected to locatethe parallel resonance with |D- k|approx instead of |D- k| has beenstudied. If a resistance is added to the Steinmetz circuit induc-tance (i.e. Y- 1k = 1/(R1 + jkX1)), its harmonic admittance, andtherefore the terms 2Y- SkY- 1k, 3Y- 1kY- 2k and 3Y- 1kY- Lk, i.e. itsinfluence on the power system harmonic response, will be evensmaller.

Thus, although the power system resistances are not con-sidered, the analytical characterization of the power systemresonance frequencies presented in Section 4 is valid.

6. Experimental measurements of the power systemharmonic response

As an example, the harmonic response of the network in Fig. 2has been measured in the laboratory and the parallel resonancehas been located to validate the study. Considering the basevalues UB = 230 V and SB = 2.7 kVA, this network comprisesthe supply system, Z- S = 0.0219 + j0.0493 pu, which feeds aSteinmetz circuit, RL = 1.368 pu and λ = 1.0 (according to (5),X1 = X2 = 2.372 pu).

apta

pd

f0X

ar

It can be noted that, although the resistances are neglected inthe study, the analytical results of the resonance frequency arequite in agreement with the experimental measurements.

7. Conclusion

The paper has studied the power system harmonic responsein the presence of the Steinmetz circuit. A parallel resonanceis produced between the capacitive reactance of the Steinmetzcircuit and the supply system reactances. This resonance is morecritical in those phases between which the capacitor is con-nected. An expression to locate the harmonic of the parallelresonance has been obtained analytically from the power sys-tem harmonic impedances. This theoretical expression allowsthe parallel resonance to be predicted if the Steinmetz circuitis connected to a power system in order to balance a single-phase load. Experimental measurements have been made in thelaboratory to validate the proposed expression.

Acknowledgment

This work was supported by grant DPI2005-03785.

References

[

[

[

MdBH

The frequency response measurements have been made with4.5 kVA AC ELGAR Smartwave Switching Amplifier as theower source, which can generate sinusoidal waveforms of arbi-rary frequencies, and a YOKOGAWA DL 708 E digital scopes the measurement device.

The measures of the power system harmonic behavior areresented in Fig. 3. The resonance shown in Fig. 3 can be pre-icted with the analytical results of Section 4, Fig. 5 or (18).

Measurements of the parallel resonance have also been madeor five different power factors, λ = 1.0, 0.975, 0.95, 0.925 and.9. They have been plotted in Fig. 8 (broken line). The ratioS/RL = 0.0493/1.368 = 0.036 must be used to obtain this plot.

In this figure, the results obtained with (18) are also plotteds a continuous line. The adequacy of the obtained analyticalesults can be verified.

[1] T.-H. Chen, Criteria to estimate the voltage unbalances due to high-speedrailway demands, IEEE Trans. Power Syst. 9 (3) (1994) 1672–1678.

[2] S.Y. Lee, C.J. Wu, On-line reactive power compensation schemes forunbalanced three-phase four wire distribution feeders, IEEE Trans.Power Deliv. 8 (4) (1993) 1958–1965.

[3] S.T. Sobral, Interference between faulted power circuits and communi-cation circuits or pipelines—simplification using the decoupled method,IEEE Trans. Power Deliv. 6 (4) (1991) 1599–1606.

[4] O. Jordi, L. Sainz, M. Chindris, Steinmetz system design under unbal-anced conditions, ETEP 12 (4) (2002) 283–290.

[5] G.T. Heydt, Electric Power Quality, Stars in a Circle Publications, WestLafayette, Indiana, USA, 1991.

[6] J. Arrillaga, C.P. Arnold, Computer Analysis of Power Systems, JohnWiley & Sons, New York, 1990.

[7] L.S. Czarnecki, Reactive and unbalanced currents compensation inthree-phase asymmetrical circuits under non-sinusoidal conditions, IEEETrans. Instrum. Meas. 38 (3) (1989) 754–759.

[8] L.S. Czarnecki, Minimization of unbalanced and reactive currents inthree-phase asymmetrical circuits with non-sinusoidal voltage, Proc.IEE-B 139 (4) (1992) 347–354.

[9] L. Sainz, O. Jordi, M. Chindris, Steinmetz system study under non-sinusoidal conditions, in: Proceedings of the IASTED InternationalConference PES, 2001, pp. 106–111.

10] M. Chindris, A. Cziker, S. Stefanescu, L. Sainz, Fuzzy logic controllerfor Steinmetz circuitry with variable reactive elements, in: Proceed-ings of the 8th International Conference OPTIM, vol. 1G.3, 2002, pp.233–238.

11] L. Sainz, M. Caro, J. Pedra, Study of electric system harmonic response,IEEE Trans. Power Deliv. 19 (2) (2004) 868–874.

12] L. Sainz, J. Pedra, M. Caro, Steinmetz circuit influence on the electricsystem harmonic response, IEEE Trans. Power Deliv. 20 (2) (2005)1143–1156.

anuel Caro was born in Manzanares, Spain, in 1979. He obtained his B.S.egree in industrial engineering from the Universitat Politecnica de Catalunya,arcelona, Spain, in 2002. He works for IDOM Ingenierıa y Arquitectura.is main field of research is power system quality.

M. Caro et al. / Electric Power Systems Research 76 (2006) 1055–1063 1063

Luis Sainz was born in Barcelona, Spain, in 1965. He received the B.S.degree in industrial engineering and the Ph.D. degree in engineering fromthe Universitat Politecnica de Catalunya (UPC), Barcelona, Spain, in 1990and 1995, respectively. Since 1991, he has been a professor with the ElectricalEngineering Department, UPC. His main field of research is power systemharmonics.

Joaquın Pedra was born in Barcelona, Spain, in 1957. He received the B.S.degree in industrial engineering and the Ph.D. degree in engineering from theUniversitat Politecnica de Catalunya (UPC), Barcelona, Spain, in 1979 and1986, respectively. Since 1985, he has been a professor with the ElectricalEngineering Department, UPC. His research interests are in the areas ofpower system quality and electric machines.