Modelling of random variation of three-phase voltage unbalance in electric distribution systems using the trivariate Gaussian distribution

Y J.Wang

Abstract: This paper deals with probabilistic modelling of the random variation of three-phase voltage unbalance. With field measurement and an analysis of three-phase voltage unbalance, trivariate Gaussian distribution has been employed to model the random variation of three-phase voltages, and a simulation method based on the Monte Carlo technique has been developed to generate voltage data, the statistical properties of which are close to those of the measured voltages. Comparison of the probability distribution functions of the complex voltage unbalance factor between Monte Carlo simulation and the measurement is then carried out for validation.

1 Introduction

Three-phase voltage unbalance is frequently encountered in commercial and residential distribution networks, in partic- ular near the further ends of electric feeders. It is also a power system imperfection often found in electric railways, steel rolling mills and steel refineries. Voltage unbalance not only increases power losses, but also has a detrimental effect on electric rotating machines. A typical example is the three-phase induction motor. Its negative-sequence impedance is much lower than its positive-sequence imped- ance. A slight level of voltage unbalance may lead to a significant increase in its negative-sequence current, result- ing in overheating and possible damage to stator windings.

The causes of voltage unbalance can be categorised into structural and functional factors. The former refers to asymmetry of three-phase impedances of overhead lines and cables, while the latter refers to uneven distribution of single-phase loads over three phases. In commercial and residential distribution systems, the proportion of single- phase loads is rather high. Unbalance of loads in three phases always exists. In industrial distribution systems, elec- tric railway locomotives are large single-phase loads that often cause voltage unbalance. In addition, steel refineries using AC arc furnaces absorb randomly-varying three- phase currents, producing both harmonics and voltage unbalance.

As three-phase power flowing in three phases varies with random switching on/off of loads, the level of voltage unbalance also fluctuates at random. Consideration of the random variation of voltage unbalance is of importance for the establishment of the limits of the level of voltage unbal- ance. An appropriate voltage unbalance limit is a key

0 IEE, 2001 ZEE Proceedings onlie no. 20010372 DOL 10.1049/ipgtd:20010372 Paper fmt received 8th November 2000 and in revised form 21sl February 2001 The author is with the Department of Electrical Engineering, National Yunlin University of Science & Technology, 123 Section 3 University Road, Tou-Liu, Yunlin 640, Taiwan, Republic of China

factor of good quality electricity for the utilities, electric appliance manufacture and customers.

The issue of voltage unbalance in Taiwan has drawn much attention since the government started the electrifica- tion of the west coast railway lines in the 1970s. However, the voltage unbalance limit was not studied and established until the construction of the high-speed railway system in recent years. The voltage unbalance problem was studied in Europe much earlier. This was attributed mainly to fast development of the European railway networks after World War 11. The railway power supply caused considera- ble voltage unbalance, but also stimulated European elec- trical engineers to study the problem. The channel tunnel railway system that links the UK and France again enabled a more profound problem to be ascertained: the time- varying and probabilistic characteristics of the level of volt- age unbalance [ 1, 21.

This paper studies the probability features of voltage unbalance. Modelling of random variations of three-phase line-to-line voltages is carried out using trivariate Gaussian distribution. This is followed by a Monte Carlo simulation technique developed for simulating the voltage variations. Field measurement has also been carried out to collect volt- age data at a substation. The results of Monte Carlo simu- lation are compared with field recorded data to validate the proposed probabilistic model.

2 Literature review

The majority of technical documents dealing with voltage unbalance are based on deterministic models that evaluate the effect of voltage unbalance upon electric rotating machines [3] , semiconductor rectifiers [4], and electric rail- ways [I , 2, 51. Relatively few papers have discussed proba- bilistic models of randomly-varying voltage unbalance.

Pierrat and Morrison [6] published an analytical model, capable of expressing the probability distribution of the voltage unbalance factor (VUF) analytically. However, the model involved modified Bessel functions that are compli- cated and hard to calculate. Moreover, the model assumed a phase displacement of 120" between the three-phase volt- ages, which is somewhat restricted since in a practical

219 IEE Proc-Gener. Transm. Distrib., Vol. 148, No. 4, J d y 2001

power system the phase displacements between three-phase voltages may differ from the ideal value of 120".

The Monte Carlo method [7] is a powerful tool for simu- lating voltage unbalance caused by randomly- varying loads. Wang and Pierrat [8] simulated random variation of three-phase active and reactive powers using six correlated Gaussian random variables. The Monte Carlo technique was applied to generate active and reactive power data from a six-dimensional Gaussian distribution, and then the probability distribution of the W F at the point of common coupling (PCC) could be calculated numerically. Another simulation combining the Monte Carlo method, the three-phase load flow calculation and the network reduction method was carried out in Wang's work [9] in which a stationary stochastic process was introduced to consider both the statistical and the time-varying character- istics of the VUF at the PCC. The methods used by [8, 91, though capable of finding the probability distribution of the W F , are computationally intensive, and can apply only to power systems of limited scale.

Unlike the methods proposed by [8, 91 that simulate random variations of three-phase active and reactive powers, this paper deals with probabilistic modelling of voltage unbalance with a novel approach that directly simulates three-phase line-to-line voltages using a trivariate Gaussian distribution model. Most studies concerning volt- age unbalance used the W F as an index that measures the level of voltage unbalance. This paper uses the complex voltage unbalance factor (CVUF ) to indicate voltage unbalance. The probability distributions of both the magni- tude and the angle of the C W F are to be investigated.

3 Complex voltage unbalance factor

The level of three-phase voltage unbalance is often meas- ured by the VUF. The International Electrotechnical Commission (IEC) defines the VUF as the ratio of the negative-sequence voltage V, to the positive-sequence volt- age vp

V U F = IVn/Vpl x 100% (1) The CVUF is an extension of the VUF, and is defined by

CVUF = (Vn/Vp) x 100% = r e e3' ( 2 )

Obviously, the magnitude z of the CVUF is equal to the VUF while the angle 8 is the phase angle by which the negative-sequence voltage leads the positive-sequence volt- age.

4 Calculation of the CVUF

Normally, calculation of the CVUF requires the three- phase line-to-neutral voltages (including magnitudes and phase angles) to be known, which allows the positive- and negative-sequence voltage phasors to be obtained by the symmetrical component operation. Nevertheless, the volt- age phase angle is more difficult to measure than the volt- age magnitude. And in case the system neutral is not accessible, or in case the three-phase voltages come from a A-connected transformer bank where the system neutral does not exist, only line-to-line voltages are measurable. A practical method for calculating the CVUF has been devel- oped that requires only the magnitudes of three-phase line- to-line voltages [lo]

r = ( A - d m ) / d m (3)

with

( 5 ) A = V,Z, + v2b + Vd", B v:a + v$, + vb",

where vu&, Vb, and V, are the magnitudes of three-phase line-to-line voltages. The value of 0 in eqn. 4 must satisfy

sign(sin8) = sign('& - v,",) (6) to avoid undeterinined values related to the arc-tangent function. Eqns. 3 6 allow the CVUF to be calculated from the magnitudes of the line-to-line voltages without the need of symmetrical components operation and measurement of phase angles.

Fig. 1 shows the relation between line voltages and the parameters z and 0 of the CVUF. The contours of constant z and constant 8 are drawn in a plane, the abscissa and the ordinate of which are line voltage ratios VadVb, and VcalVbc, respectively. Fig. 1 also allows the values of z and 8 to be looked up quickly as soon as the line voltage ratios VadVb, and VcaIVbc are available.

1.10

1.05

x P ?m 1.00

0.95

0.90 0.

Fi 1 Curves showing contours of constant zand 0 in a plane whose abscirsa a J m ordmate ' are ratios of line voltages V,dv,,

"abNbc

K . f l h c , respectively

5

The measurement was carried out at the building of the Department of Electrical Engineering, National Yunlin University of Science and Technology, using a digital power analyser capable of recording three-phase voltages, currents, active and reactive powers. The time interval between two successive measurements is programmable. The digital power analyser was connected to the secondary of a 11.4kVI220-380V, 1 000kVA distribution transformer that supplied electric power to the building. The electric loads in the building included lots of single-phase appli- ances such as fluorescent lights, personal computers, air conditioners, etc. and a few three-phase loads, such as water pumping motors. Two sessions of measurement were performed. The first session lasted 24 hours with measure- ment interval of 30 seconds. The second session lasted seven days with measurement interval of five minutes.

5. I Measured data during 24 hours The first session of measurement was performed in July lasting for 24 hours. The time interval between two succes- sive automatic measurements was set to 30 seconds. The

Presentation of the measured data

280 IEE Proc-Gener. Transm. Dislrib.. Vol. 148. No. 4, July 2001

378 1 374

370

> 366

362

? Q

I' I

0 480 960 1440 1920 2400 2880 358 I

time, multiple of 30 s 378 r

358A 480 960 1440 I i 2 0 2400 2d80 time, multiple of 30 s

378r 374

370

a 366 m

362 I' I

358i 480 960 1440 19'20 $00 2d80 time, multiple of 30 s

Fig.2 Variutions of line voltuges vab, vbc and V, during 24h

I I 1

0 1 0 480 960 1440 I920 2400 2880

time step, multiple of 30 s 250 r a

0 23

200 3 > 0 m 5

E I : 150 vl

1001 I I

0 480 960 1440 1920 2400 2880 time step, multiple of 30 s

b Fig.3 during 24 h

Vurhtions of (a) the magnitude z und (b) the ungle 0 ofthe CVUF

results of the measurement are shown in Fig. 2, in which the variations of three-phase line voltages with time are dis- played. The magnitude z and the angle 6 of the corre- sponding CVUF are calculated using eqns. 3-6, and are depicted in Fig. 3. It is seen that most W F values vary between 0.14.3%, and that most values of the angle are withm the interval (1200, 240").

5.2 Measured data during one week The second session of measurement was carried out at the same place but the period lasted for seven days. Because of the memory limit of the digital power analyser, the time interval between two successive measurements was extended to five minutes. Fig. 4 shows the measured line voltage Vub against the time steps. The waveforms of Vb, and V, are quite similar to that of VOb and are not shown due to the space limit. The corresponding magnitude and angle of the C W F are calculated using eqns. 3-6 and are depicted in Fig. 5. It is seen that the magnitude and the angle of the CVUF measured during a week are quite different from those measured during a day, showing significant variation of voltage unbalance.

378 1 374 t I

vu-

0 288 576 864 1152 1440 1728 2016 time, multiple of 5 min

Memured line voltuge V, during one week Fig. 4

0.32 r

8 6 0.24 U

2 f 0.16 0 m s c .- 5 0.08 E

I

0 288 576 864 1152 1440 1728 2016 time, multiple of 5 min

a 260 r

0 ;d 220 V

U 3 > 0 m 5 c

180 - 0,

m

1

0 288 576 864 1152 1440 1728 2016 time, multiple of 5 min

b Variations of (U) the magnitude utut (h) the mgle ofthe VUF during Fig. 5

one week

IEE Proc.-Gene,. Trunsm. Dislvib.. Vol. 148, No. 4, July 2001 28 1

6 Trivariate Gaussian distribution

6.1 Theoretical background The random variations of three-phase line voltages can be modelled using an appropriate multivariate probability distribution. This paper attempts to model the three-phase line voltages using trivariate Gaussian distribution, i.e. three-dimensional normal distribution. According to Kirch- hoff s voltage law, the phasor sum of three-phase line volt- ages must be zero. That is, if expressed graphically, the phasors of three-phase line voltages must form a triangle.

Let the three sides of the triangle be denoted by Wl, W2 and W3. Then W,, W2 and W3 can be treated as three cor- related Gaussian random variables. Any pair of the ran- dom variables are (statistically) correlated. The correlation is measured by the correlation coefficients pI2, ~ 2 3 , ~ 3 1 . Fig. 6 illustrates this relation conceptually. The correlated random variables W,, W2 and W3 then constitute a trivari- ate Gaussian distribution that has nine parameters: the expected values W , , W 2 and W3, the standard deviations q, 02, o3 and the correlation coefficients pI2, ~ 2 3 , ~ 3 1 .

Fig. 6 the three sides of LI trumgle Their correlation coefficients are p,*, p23 and

Correkated Gaus,stun random vciriuhles W,, W, m ~ d W, repwentitis

The joint probability density function of W,, W2 and W3 can be written as

where

(8) [w - W]"[c,]-l[w - W]

2 Q =

w1 -W1

w3 - w3

[w -W] = [ w2 -.I (9)

I 02 pl2fl102 p13al03

[cw] == p125152 02" P23f l203 (10) [ P130103 P 2 3 0 2 0 3 03"

and l[Cw]-ll means the determnant of the inverse of [C,]. The matrix [C,] in eqn. 10 is the covariance matrix of random variables W,, W2 and W3.

6.2 Computer generation of Gaussian random variables Before generating random variables Wl , W2 and W, whose joint distribution is described by eqn. 7 , we need to gener- ate three statistically independent Gaussian random varia- bles Yl, Y2 and Y3, each with zero mean and unit variance,

282

by Box and Muller's method [I 11

Y1 = ~ ~ C O S ( 2 7 r X 2 ) (11)

Y, = JZL(FJsin(27r~2) (12)

y3 = 2 / -21nos in (27r~4) (13) where XI, X2, X3 and X, are uniformly distributed random variables within the interval [0, I]. Most programming lan- guage compilers and program libraries provide high-per- formance random number generators that generate uniform random variables with satisfactory statistical prop- erties, and can be used to generate XI, X2, X3 and X4. We now take a linear transformation of Yl, Y2 and Y3

where

and

[WI = [TI[Yl (14)

(15)

(16)

[q = T21 T22 (17)

[W] = [W w2 W3It

[y] = [yi y 2 Y3It [ TI1 T12 Tl;]

T31 T32 T33

A linear transformation of Yl, Y2 and Y3 yields W,, W2 and W3 that are also Gaussian, and have also zero means. The covariance matrix [C,] of the new Gaussian random variables W,, W2 and W3 is given by [12]

[ C W I = [ 5 w Y 1 [ 2 1 t (18) Since Yl, Y2 and Y3 are independent, and all have zero means and unit variances, the covariance matrix [C,] of Y,, Y2 and Y3 is a unitary matrix. Eqn. 18 can then be written as

Let the transformation matrix [a be expressed as a lower triangular matrix

[ C W I = [q . [V (19)

0 [q = [ T21 T 2 2 : ] (20)

T31 T32 T33

Solving eqns. 10, 19 and 20 allows the elements of [a to be expressed in terms of ol, 02, a,, p12, f i 3 and p31

Ti2 = T13 = T23 = 0 (21)

T11 = 51 (22)

T21 = P l 2 0 2 (23)

T31 = P1353 (24)

T32 = 0 3 4 1 - p:3 - (p23 - p12p13)2/(1 - p:,) (27)

To consider the means of W,, W2 and W3, it suffices to add W , , W2 and W 3 to the right-hand side of eqn. 14, which yields

Tl1 0

IEE Proc.-Cener. Transm. Distrih., Vol. 148, No. 4, July 2001

The procedure of generating correlated Gaussian random variables is briefly summarised as follows. Eqns. 11-13 allow us to generate independent Gaussian random varia- bles Y,, Y2 and Y3 from random numbers XI, X2, X3 and X,. We then use eqn. 28 to transform Y,, Y2 and Y3 to the desired correlated Gaussian random variables W, , W2 and W3 that simulate random variations of three-phase line voltages.

7 simulation

Comparison of measurement and Monte Carlo

The method of generating correlated Gaussian random variables developed in the previous Section is used to simu- late the three-phase line voltages. Before the simulation can be carried out, the nine parameters of the trivariate Gaus- sian distribution described by eqn. 7 must be identified from the measured data. A simulation program in FOR- TRAN has been developed to perform the Monte Carlo simulation. Each simulation run generates 20000 sets of three-phase voltages. The statistical analysis of the simu- lated voltages then gives the probability density functions (PDFs) of the voltages, the magnitude and the angle of the CVUF. The PDFs obtained from the measured data and from the simulation are depicted for comparison.

7.1 Comparison with data measured in session one Tables 1 and 2 list statistical properties of the voltage data measured in session one (a period of 24h), including the means, the standard deviations and the correlation coeff- cients of the voltages. It is noted that the correlation between three-phase voltages is very high.

Table 1: Means and standard deviations of line voltages measured during 24 hours

va b vbc Vca

mean (V) 367.9287 368.6985 367.8775

standard deviation (V) 2.810266 2.789217 2.833485

Table 2: Correlation coefficients of line voltages measured during 24 hours

va b vbc Vca

Vab 0.993085 0.984217

Vbc 0.993085 1 0.994703

Vca 0.984217 0.994703 1

355 365 375 385 vabr "

Fig. 7 Comparison o the &nsityJilnctwns of line voltage Vc,b obtained rom the Monte Carlo s h u h o n (solid line) und jkom the meuswed datu &to- gram) during 24h

IEE Psoc.-Geney. Tsansrn. Distrib.. Vol. 148, No. 4, July 2001

Fig. 7 shows the PDFs of vab. The histogram represents the PDF of the measured data while the solid line refers to that of the Monte Carlo simulation. The PDFs of meas- ured and simulated data for vbc and V, are similar to that of Klb and are not shown. Although the PDFs of the simu- lated voltages do not agree exactly with those of the meas- ured data, the Gaussian distribution provides a good fit for the distribution of the voltages. Comparison of the PDFs of the CVUF is shown in Figs. Sa and b. It can be observed in the figures that for the magnitude of the CVUF, the PDFs agree well with each other, but not that well for the angle. Nevertheless, the mean and the variance of the angle from the simulation are very close to those of the measured data.

magnitude of the CVUF a, % a

0 0 0

0

0.01 8

angle of the CVUF 8, deg b

Com arisan ofthe PDFs of the CVUF obtained tom Monte Curlo Fig.8 skulation (soh lines) andpom the meusurement (scatterechots) durhg 24 h U the magnitude b the angle

7.2 Comparison with data measured in session two Tables 3 and 4 list statistical properties of the voltages measured in session two (a period of one week). The corre- lation coefficients listed in Table 4 show that the three-

Table 3: Means and standard deviations of line voltages measured during one week

Vab vbc Vca

mean (VI 366.9032 367.9995 367.4633

standard deviation (V) 2.136667 2.022527 2.028547

Table 4 Correlation coefficients of line voltages measured during one week

Vab vbc VC,

Vab 0.994629 0.992072

vbc 0.994629 1 0.995335

V,, 0.992072 0.995335 1

283

phase voltages are strongly correlated. Fig. 9 compares the PDFs of Vab between measurement (histogram) and simu- lation (solid line). The comparisons for Vh, and V, are quite similar and are not shown. The normality assumption for the voltages is acceptable. Fig. 10a shows the PDFs of

vabi Fig.9 Comparison of the denrity fmctions of line voltage V obtained rom Monte Carlo shulatwn (solid I&) undfvom measured data (h$ogram) &.ing one week

.- E

e

- p P

a

O‘03,r

magnitude of the CVUF Z, % a

0

0 I 140 180 220 260

b angle of the CVUF 0, deg

Fig. 10 Com arison of the PDFs of the CVUF obtainedfvom Monte Carlo sunulation (soldines) d j ? o m mewrernent (scattered dots) during one week a the magnitude h the angle

the magnitude of the CVUF calculated from the measured (scattered dots) and from the simulated (solid lines) voltage data. Fig. 10b is the comparison for the angle of the CVUF. Both the magnitude and the angle of the CVUF calculated from the simulated data agree well with the measurement.

8 Conclusions

This paper has proposed a method to simulate voltage unbalance resulting from random variation of three-phase voltages. The proposed method is working on line-to-line voltages. It needs only the magnitudes of the voltages to be known. The voltage phase angles are not required, which largely simplifies the calculation and measurement.

The trivariate Gaussian distribution model has been use to model random variations of three-phase voltages. A method based on the Monte Carlo technique has also been developed to generated random variables of the trivariate Gaussian distribution, and to simulate three-phase line voltages. The probability distributions of the CVUF (including the magnitude and the angle) can then be obtained using the Monte Carlo simulation.

The results of the proposed simulation method have been compared with data from field measurement and good agreement has been obtained.

9 References

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284 IEE Proc.-Gener. Trunsm. Distrih.. Vol. 148. No. 4, July 2001