Calculations of diversified harmonic currents in electric distribution systems

Y.G. Hegazy and M.M.A. Salama

Abstract: A comprehensive analysis of the characteristics of distribution-system harmonics is provided. This analysis leads to the formulation of harmonic diversity factor tables. These tables give a reliable estimate of harmonic currents in distribution feeders and enable the engineers to determine the amount of background distortion of the voltage waveform. The use of the proposed tables will ease the computational difficulties in performing distribution system harmonic studies. Both deterministic and probabilistic methods were integrated in the conducted analysis in this paper to generate the diversity factor tables. The proposed method and the analysis results are presented in detail.

1 Introduction

The proliferation of electronic switching into a wide variety of power electronics devices has raised serious concems over the adequacy of the available harmonic, analysis methods in handling the different issues associated with the use of these devices. One of the issues, which still need more investiga- tion, is the determination of the diversified harmonic currents produced by nonlinear loads with harmonic phase angle dispersion taken into consideration. Diversified harmonic currents are the vector sum of individual harmonic currents produced by a large number of loads. Phase angle dispersion is a result of one of the following reasons: change in operating conditions of the device, variations of circuit parameters, line impedance and transformer connection [I]. Although the importance of this subject has been addressed in the IEEE standard-519, not much work has been done to develop a rigorous method for the addition of harmonic vectors. Mansoor et al. [2-31 attempted to study this point by performing Monte- Carlo simulations to compute the cumulative harmonic currents produced by a large number of single-phase capacitor-filtered diode bridge rectifier loads. The results of this study indicated that, owing to the phase angle dispersion, a significant amount of harmonic cancellation takes place. However, the use of random vectors does not necessarily reflect the true performance of the system. On the other hand, the effect of the phase angle dispersion has been included implicitly in both power-system harmonic studies conducted by using an iterative harmonic load-flow program and the reports from field measurements [&5]. Neither the Monte-Carlo method nor the iterative load-flow programs succeeded in offering a reliable approach to handle the underlying problem.

0 IEE, 2003 IEE Proceedhgs online no. ZW30818 doi: lO.lM9/ip-~:ZW30818 Paper fin1 m i v d 10th May 2W2 and in revised form 1st May ZW3 Y.G. Hegay is with the Electrical Power and Machines Department, Ain Shams University, Facnliy of Engimcfing. I Sacayat Sr., Abbasia, Cam, Egypt M.M.A. Salama is with the Department of Electrical &Computer En?jineering, University of Waterloo, Waterloo. Ontario, Canada N2L 3G1

IEE Proc-Gpner. Tronrm DiFrrib. Vol. I50, No. 6, Nouonber 2”

This paper provides a comprehensive analysis of the characteristics of distribution-systnn harmonics. This ana- lysis leads to the formulation of harmonic diversity factor tables. The use of the proposed tables will ease the computational difficulties in performing distribution-system harmonic studies. The formulation of the problem in hand, along with the proposed approach to solve it, is discussed in Section 2.

2 Problem formulation

Distribution-system harmonic studies may involve the analysis of feeders supplying more than one nonlinear load. In this case, the harmonic components of the feeder current would be a result of a vector summation of all the harmonics produced by individual loads. The calculation of the net harmonic current of such a feeder requires knowledge of the terminal voltage phase angle of each nonlinear load, a clear relationship identifying the depen- dence of the load current on its terminal voltage and the number of energised loads, which is normally a random variable [6]. In addition to the required amount of data for this analysis, the whole process has to be repeated whenever the feeder loading condition is changed, which might be a result of turning on or off a nonlinear load. On the other hand. there are many restrictions on the sue of the systems and the number of the loads, which can be handled using the available computer software. Hence, the problem of estimating the net harmonic current injected by a large number of nonlinear loads needs to be thoroughly investigated. In an engineering form the underlying problem can be outlined as follows:

2.1 Given A three-phase distribution feeder supplies a large number of different types of both single-phase and three-phase non- linear loads. The load on this feeder varies according to certain daily demand characteristics. The share of each load in the total demand is known. The line parameters are given as well as the number of nonlinear loads from each type.

2.2 Required An index is required, which reflects the realistic character- istics of distribution-system harmonics. This index shall

651

include the effect of the different parameters involved in the generation and propagation of harmonics. It shall also &ve an indication of the amount of cancellation and attenuation experienced by harmonic currents generated by grouped loads, for different modes of operation of the system.

2.3 Approach If the total number of the harmonic current vectors to be summed is Nand these currents are generated by K, J , L numbers of harmonic sources types X, Y and Z, respectively, then the following argument is valid for each harmonic order, regardless of the distribution of the harmonic producing devices along the feeder [6]:

where I, is the harmonic current vector injected by the ith device type X, i= 1,2, ... K; I y I is the harmonic current vector injected by the ith device type Y, i= I ,2,. . .A Izj is the harmonic current vector injected by the ith device type Z, i= 1,2,. . .L; I, is the net harmonic current of the considered feeder, i= 1,2 ,_.. and N = K + J + L .

Hence, the harmonic producing devices can be treated as clusters of different devices. The contribution of each cluster of devices in the total feeder current can be studied separately and the sum of all the contributions from the different clusters will result in the net harmonic current of the feeder. In addition, the net harmonic current produced by a certain cluster can be expressed in terms of the sum of the magnitudes of the harmonic currents in this cluster as:

M

XIj,, = DF,O. M . I h (2) i= I

where M is the number of similar devices in the same cluster; h is the harmonic order; I,, is the magnitude of the hth harmonic current; lIzi is the magnitude of the hth harmonic current phasor of device number i. D E is the diversity factor of the hth harmonic current of cluster type 0.

The harmonic diversity factor is defined as the ratio between the vector sum of certain harmonic currents to the algebraic sum of the same harmonic currents. The diversity factor is a function of several parameters, such as the harmonic order, the type of the harmonic producing device, the number of devices, the feeder parameters and the percentage loading of the nonlinear load. This index reflects the amount of cancellation in the total harmonic current and the attenuation in the net harmonic injection. The use of such an index allows comparison between different feeders and gives a realistic indication of the severity of the existing harmonics. Hence, the work in tlus paper will be devoted towards the calculations of typical values for diversity factors of the harmonic currents produced by the common distribution system nonlinear loads.

3 Working strategy

The proposed working strategy in this paper is divided into the following three phases:

3. I Phase 1: Load identification The criteria considered for choosing the distribution-system nonlinear loads for the harmonic study in this paper are as follows:

652

(i) The load has been witnessed as a harmonic-producing load and the harmonics injected by this load have negative effects on the performance of the system. (U) The load exists in large numbers in the eledtric power distribution networks. (ui) The load is proliferating and expected to be continually used in the future.

The chosen loads are then grouped into four groups based on the common features in the waveforms of their currents. The grouped loads include:

Croup A : The main elements of this group are the three- phase power converters, to represent the industrial loads. The most common form of static power converter is the six- pulse bridge rectifier type. It is widely used in distribution systems as the front-end for DC drives and adjustable speed drives (ASD). Hence, the six-pulse bridge rectifier as a DC drive is considered to be an example of this group. The circuit model and the harmonic spectra of the current d r a m by this load are found in Figs. 1 and 2, respectively.

Goup B: Group B loads are different versions of the single-phase capacitor filtered bridge rectifiers. Typical examples of this group are television sets, personal computers and fluorescent lights, which utilise electronic ballasts. The circuit model for this group and the harmonic spectra associated with its current are shown in Figs. 1 and 2, respectively.

A , ;$$+Ro

e B C

D2 D4 D6 I 0 2 D4 T group A

- - group C

Fig. 1 Circuit model of load groups

90

group B

DII.’ group D

3 5 7 9 11 13 15 17

harmonic order

Fig. 2 Harmonic spectra of load group currents

Croup C The phasecontrolled single-phase loads represent the main element of this group. Heating loads and light dimmers are the most widely used loads, which

I€€ Proc . -Ge~r T r m . Drnri!.. Vol. ISO, No. 6, Notionher 2W3

employ phase control of thyristors. The equivalent circuit of this load and the harmonic spectra of the current waveform of this load are depicted in Figs. 1 and 2, respectively.

Goup D: The loads of group D are mainly iron-core reactors. These types of loads include transformers, fluorescent and other gas discharge lighting. A nonlinear inductor is used as a circuit representation of these loads. The equivalent circuit of such a load and the associa- ted harmonic currents of its waveform are presented in Figs. 1 and 2.

3.2 Phase 2: parameters investigation In this phase, the parameters which have the highest impact on the harmonic performance of the underlying loads, are investigated. These parameters are classified into explicit and implicit parameters:

Explicit parumtevs are the parameters which control the generation of the harmonic currents from individual nonlinear loads. These parameters can be extracted from a Fourier transform of the distorted current waveforms.

Impkitparameters are the parameters that arise from the interaction between harmonic producing loads when they are working in groups.

In this phase, both the implicit and explicit parameters are investigated. Among the examined explicit parameters is the percentage loading of the harmonic producing device and the firing angles of thyristors in power-electronics-based loads. On the other hand, the number of working devices in one cluster and the parameters of the feeder supplying this cluster are considered as the implicit parameters.

The time domain analysis method is adopted in this phase as a modelling environment for the loads under study. First, a detailed time domain model of each load is developed, then the developed model for each load is solved under the following modes of operation:

(i) The percentage loading of each harmonic source is varied from 20% to 100% of its full load value. (ii) For groups A and C loads, the thyristors are symmetrically fired and the firing angle of each thyristor vanes in the range Oiart90" under full load conditions of the rectifier. (iii) The percentage loading of each harmonic source is varying from 20% to 100% of its full load value for different thyristor firing angles.

Then load current is analysed for each case study to determine its harmonic contents and to examine the effect of the parameter variation on the harmonic behaviour of this load. Finally, the harmonic interaction between the different loads is studied by analysing the harmonic contents of the considered loads, when they are working in groups. For this purpose, the distorted current wave- forms of four different distribution feeders are analysed under different modes of operations. The equivalent circuits of the feeders under study are shown in Fig. 3. Each of these feeders is rated 0.2kV and supplies a certain number of identical harmonic producing loads from the same

vb

b c d

Fig. 3 Radial distribution feeder undrr study

IEE P r o c - h e r . T r m . Dirt,$, VoL 150, No. 6, Nowmbrr 2003

group. The impedance of each line segment is Z , = 0.33 + j0.66p.u.

The harmonic spectra of the input current to each of these four feeders are calculated after solving their detailed time domain models under the following conditions:

(i) The studied harmonic sources are fully loaded. (ii) The number of harmonic sources supplied by each feeder is incremented by one source and the maximum number of sources per feeder is limited by the feeder capacity. (iii) The X/R ratio of the feeder impedance is changing in a range defined by the feeder parameters. (iv) The firing angles of the thyristors are varying in the range Oia<90" for each of these cases.

The computer software packages used to perform this analysis are: altemative transient program (ATP), to develop time domain models of the loads; electrical distribution system analysis program (EDSA) to perform energy flow analysis and harmonic currents injection; MATLAB in order to perform frequency domain analy- ses. The results obtained for this phase are presented in Section 5 .

3.3 Phase 3: Calculations of harmonic diversity factors The parameters affecting the harmonic performance of each load can vary deterministically or probahilistically, depend- ing on the nature of these parameters and the working conditions. In phase 2 the underlined parameters are examined deterministically. However, power-electronics- based loads normally undergo random variations of their operating conditions, due to the random fluctuations of their firing angles. Hence, to obtain correct estimates for the diversity factors of their harmonic currents, both probabil- istic analysis and deterministic analysis have to be performed. The diversity factors calculated from the two models are combined in one harmonic diversity factor defined by

DF = d D m (3) where DF, is the deterministic diversity factor and DF, is the probabilistic diversity factor. The model used to perform probabilistic harmonic analysis is developed according to the following proposed approach. The combined results of both probabilistic and deterministic models are presented in Section 6.

4 Probabilistic modelling

In order to include the random behaviour of the line current harmonics of the power electronics loads under study, the fuing angles of each converter are assumed to be varying randomly and independently. The following assumptions are made during the calculations of the probability distribution of the random harmonic currents produced by the following converters: [I I . Balanced three-phase supply with zero inductance. 2. Flat DC current.

A probabilistic model for loads of both groups A and B are developed using the detailed method presented by the authors in [8]. However, in this Section, the steps used to develop the probabilistic model for group A loads is presented as an example for the utilised approach.

The amplitude of the hth harmonic current of the six-pulse power converter (load type A) can be

653

written by

(4)

where V, is the maximum line-to-line voltage and R,, is the load resistance, h=6k+ 1, k= 1,2 ,.....

The phase angle of the hth harmonic current can be expressed as

( 5 ) x

$h = -ha f- [ I ~ (-I)'] 2

In phasor notation the real and imaginary components of the hth harmonic current can be formulated as

cos a. cosh a X ( h ) = (-1)' [- 6&V' n2R,h

6 8 V m Y ( h ) = (-1)'" [- n2R,h cosa.sinha

This result implies that the real and imaginary components X ands Y are determined by the random variable a, therefore X and Y become random variables and the parameters of their probability density functions (p.d.f.) can be determined accordingly. To determine the p.d.f. of the summation of the harmonic currents let S be the sum of X components of N loads and W be the sum of their Y components, where

N N S = and W = K

i= I i= I

According to the central limit theorem [7], the sum of the X and Y components are normal distribution functions, regardless of the nature of the probability distribution function of the components. To determine the parameters of the p.d.f. of both S and W, the random variable a is assumed to follow a uniform distribution from a , to a*. Accordingly, the mean pxj and variance a,' of each random variable X j are calculated by

P . ~ = 1; W ) . f

a.:. = 1: [ X ( h ) - pJ2f(a)da

Pyi = 1; Y ( h ) ' f ( W

a:i = 1; [Y(h) - ~ ~ c , l ~ f ( a ) d a

(9)

(10)

Similarly, the mean pyj and variance 0:. for the random variable Yj are defined by

(11)

(12)

The sum of the X component S has a mean and variance given by

Similarly, the sum of Y component W has a mean and variance given by

With the virtue of the central limit theorem, the random variables S and Ware normally distributed with marginal

654

probability density functionsfJS) and h( !+), where

Finally, with the assumption of having the real and imaginary components of harmonic currents represented by their expected values, the amplitude of the hth harmonic current component is calculated hy

I rh = Jm (16)

where I',h is the expected value of the hth harmonic current component, phr, and phw are the mean for the hth components of S and W, respectively.

5 Results

The results obtained for the previously described phases are presented accordingly.

5. I Results of phase 2 In phase 2 parameters are investigated which affect the generation and propagation of harmonics from the under- lined loads. These parameters are classified to explicit and implicit parameters. Investigation revealed the following results:

5.7. 7 Explicit parameters: Examination of the ex- plicit parameters, which determine the characteristics of the harmonic currents of the loads under study, produced a tremendous amount of results. However, owing to the limited space in this paper, only a small portion of these results will be presented, as the authors have provided more results in [6]. The following are these results sorted by group order:

Group A loads: The variations of harmonic currents of each converter with its percentage loading are calculated for different firing angles. Figure 4 shows the amplitude of the 5th harmonic current under different loading conditions. The associated phase angles of this harmonic current and other harmonic currents are presented in Fig. 5.

E 25,

20 40 60 80 100 percenfage loading

Fig. 4 Power converter 5th harmonic current agaim percentage loading for different f ir ing angles

8 180

$ 6 0 - 5

P O -60 A M

6 120

A 411 : -120 -180

a

0 20 40 60 80 100 firing angles, degrees

Fig. 5 Bring angles

Power contierrer harmonic currents p lum angles against

Group B: Figure 6 shows the harmonic spectra of group B loads for different loading conditions. The dispersion of

IEE Proc.-Gemr. Trmm. Dlstrib.. Vol 150, No. 6, Nmembpr 2W3

90

2 70 60

2 50 P 40

30

= 10

E 80

5 20 "

20 40 60 80 100 loading, %

Fig. 6 luudmg

Grmtp B loud7 harmonic currentsfor difwent percentage

Y)

% 150.

loading, % E 'I"

Fig. 1 percentage loading

the harmonic currents phase angles is depicted in Fig. 5 . (Figure 7)

Group C loads: The variations of the harmonic currents of each phase-controlled resistive load are illustrated in Fig. 8. The tiring angles of thyristors are varied in the range from 30 to 120 to cover the range of the expected operating conditions.

Group B lo& harmonic rurremphose anglesfor dflerenr

0.30

Z 0.25 - 6

L

g 0.20 - 2 015 - p 0' 0.10 -

30 60 90 120 firing angles, degrees

Fig. 8 Group C harmonic airrentsfor diferentfiring angles

Group D loads: In practice, these loads have constant operating conditions, therefore their harmonic content will be the same as that shown in Fig. 2.

It is clear from the results obtained in this phase that the harmonic currents, produced by each of the studied loads, experienced changes with the increase in the percentage loading of each load. These changes can take the form of harmonic reduction or harmonic addition, depending on the type of load. In addition, the thyristor firing angles play an important role in determining both the magnitude and the phase angle of the produced harmonic current.

5.7.2 /mp/icit parameters: This Section presents some of the results obtained for the examined implicit parameters. The effect of increasing the number of the supplied loads on the harmonic contents of the feeder current is presented first, followed by the effect of valying the feeder line parameters.

Harmonic currents agaimt number of loads: Figures W 1 show the characteristics of the summation of harmonic

IEE Proc.-Ce~r. T r m . Dirlrrb. Vol 150, No. 6, Nowmber 2W3

1 2 3 4 5 6 7 6 9 1 0 number of convener^

Fig. 9 loads for diferentfiring angles

Feeder 5rh harmonic w e n t agoimt nmber of group A

01 I 0 2 4 6 8 10 12 14 16

numberof B loads

Fig. 10 Feeder harmonic curre,zt against nunlber of group B loah

30 - D l I

I 0 5 10 15 20

number of group C loads

Fig. 11 of group C for dffermrfring anyla

Feeder percentage third harmonic current againct number

currents produced by loads A-C, respectively. The feeder supplying these loads has X / R ratio equal to 2 for the analyses leading to these results.

Figure 9 shows the relationship between the harmonic current component of group A loads against the number of on converters for constant fuing angle. Similar results were obtained for other harmonic components, but are not shown due to the limited space allowed for this paper.

Figure IO illustrates the percentage harmonic currents against the number of loads for load type B. Figure 11 shows the relationship between the third harmonic current component of group C loads against the number of on loads for constant firing angle.

Effect of feeder parameters: Table 1 presents the harmo- nic current components for load groups A-D, for different X / R ratios. The loads are considered fully loaded in this study, and the analyses were performed for a constant 30" f i n g angle for both loads A and C . N is the number of loads and h is the order of the harmonic current component. More results are found in [q. The results in this Section reveal that the phase angles dispersion of the injected harmonic currents in distribution systems reduce the levels of the net harmonic currents in these systems. In addition, the magnitudes of the harmonic current components of a radial distribution feeder-supplying harmonic producing

655

loads are inversely proportional to the X/R ratio of this feeder. The levels of the collective harmonic currents of radial distribution feeders decrease with the increase of the number of harmonic sources supplied by these feeders. The injected harmonic currents from group B loads and group D loads reach constant levels, as the number of the loads from these groups supplied by one feeder exceeds a certain figure.

5.2 Results of phase 3 In phase 3 the harmonic diversity factors are cdlcuhted. As illustrated before, two diversity factors have to be combined together to obtain the harmonic diversity factor, i.e the deterministic diversity factor and the probabilistic diversity factor. In the following, tables for each diversity factor are calculated and presented.

5.2. 1 Deterministic modelling results: The follow- ing tables present the calculated harmonic diversity factors

Table 1: Harmonic currents of loads A-C for different XIR ratios

using the deterministic model. In these tables, OF3 denotes the diversity factor of the third harmonic current and same notation goes for all other harmonics. The calculated diversity factors are for full-load condition with the !iring angles ranging from 30 to 50". Table2 shows the deterministic diversity factors for group A loads. Tables 3-5 present the deterministic diversity factors for loads C-D, respectively.

These results show that the calculated deterministic diversity factors decrease with increase in both the number of working loads and the line reactance to resistance ratio. The pattern of the diversity factor of the same harmonic current component differs from one load to another.

5.2.2 Probabilistic modelling results: Here, the parameters of the p.d.f. of the summation of harmonic currents are calculated, then the expected harmonic currents are formubated using the equations in Section 4. Tables 6

Group A

xm h = 5 h = 7 h = l l 5 7 11 5 7 11

N= 4 N= 8 N= 10

2 15.4 11.6 5.9 12.7 8.8 4.3 11.8 7.2 3.5

5 15.7 10.8 . 5.7 11.5 8.3 4.1 10.8 6.9 3.0

10 15.3 10.4 5.5 11.1 . 8.1 3.8 10.8 6.5 2.9

Group B N= 5 N= 10 N= 15

xm h = 3 h = 5 h = 7 3 5 7 3 5 7

2 78.2 47.1 18.0 75.1 36.2 12.2 72 29 11.1

5 75.1 45.3 16.1 72.3 34.5 10.1 68.2 28.4 8.3

10 72.0 42.0 11.1 69.1 32.1 9.1 62.2 25.2 5.2

Group C

WR h = 3 h = 5 h = 7 3 5 7 3 5 7

N= 10 N= 15 N=20

2 21.1 3.2 2.7 20.1 2.2 2.5 19.9 2.5 1.71

5 2.4 1.6 2.3 1.56 1.11 2.1 1.07 1.0 1.5

10 2.01 1.3 2.1 1.42 1.04 1.89 0.65 0.78 1.24

Group D

WR h = 3 h = 5 h = 7 3 5 7 3 5 7

N= 20 N= 40 N= 60

2 20.1 10.3 3.57 19.8 9.9 3.38 19.7 9.86 3.17

5 19.9 10.2 3.5 19.6 9.87 3.31 19.6 9.85 3.11

10 19.8 10.1 3.37 19.5 9.85 3.22 19.5 9.83 3.08

Table 2: Deterministic diversity factors of group A loads

N= 4 N= 8 N= 10

XIR OF, OF; DFq, OF, DFT DFn DF; DFq,

2 0.96 0.82 0.76 0.18 0.69 0.7 0.75 0.61 0.51

5 0.91 0.9 0.76 0.71 0.73 0.58 0.66 0.61 0.43

10 0.68 0.82 0.7 0.77 0.65 0.44 0.64 0.54 0.36

656 IEE Proc.-Gener T r m m D i ~ f r i h Vol. 150, No. 6, Noaember 2W3

Table 3: Deterministic diversity factors of group B loads

2 1 .oo 0.83 0.8 0.99 0.65 0.46 0.96 0.53 0.44 5 0.99 0.79 0.64 0.95 0.6% 0.55 0.90 0.54 0.41 10 0.98 0.77 0.6 0.96 0.59 0.41 0.9 0.48 0.41

Table 4 Deterministic diversity factors of group C loads

N=5 N= 10 N=20

WR OF, DF, 06 OF, 06 OF, 06 OF,

2 0.9 0.85 0.81 0.75 0.6 0.64 0.6 0.45 0.53 5 0.86 0.87 0.86 0.72 0.6 0.63 0.6 0.41 0.51 10 0.81 0.83 0.87 0.69 0.56 0.61 0.57 0.44 0.44

Table 5: Deterministic diversity factors of group D loads

N=20 N=40 N=60

WR OF3 0.5 06 OF, OF, 06 OF, OF, 06

2 0.95 0.67 0.94 0.91 0.92 0.92 0.89 0.91 0.88 5 0.94 0.96 0.94 0.93 0.92 0.89 0.93 0.92 0.84 10 0.93 0.94 0.90 0.91 0.91 0.86 0.91 0.91 0.82

Table 6 Expected 5th and 7th harmonic currents of group A

Table 7: Expected 3rd and 5th harmonic currents of group C

2 0.319 -0.102 0.3349 0.08 -0.008 0.1174 4 OD823 -0.231 0.383 0.019 -0.034 0.039 6 0.252 -0.361 0.44 -0.07 -0.016 0.071 8 0.846 -0.212 0.871 0.21 -0.177 0.261

10 1.09 -0.573 1.238 0.131 0.193 0.233

and 7 show the calculated p.d.f. parameters and the expected harmonic currents for loads type A and C , respectively. Probabilistic diversity factors are then derived and presented in Table 8.

The calculated diversity factors in Table 8 show that the random variation of the firing angle a in the power- electronic-based loads results in random changes in the collective harmonic currents. The trend of these changes is to reduce the level of each harmonic current rather than to increase it.

6 Diversity factors

The combined diversity factors defined by (3) are presented in this Section. The diversity factors of loads B and D are those presented in Table2, because there is no random change in their operating conditions. Table 9 presents the combined harmonic diversity factors for load groups A and C . The firing angles range from 30" to 50". The results for all ranges of variations of firing angles are found in [a. IEE Proc-Gener. T r m . Dirtrib, Vol. ISO, No, 6, Nor;emher 2W3

N P3X P3y le3 P S X 115" . c s

2 -0.052 -0.019 0.02 0.148 -0.018 0.149 4 -0.285 -0.244 0.375 -0.175 0.248 0.303 6 -0.358 -0.36 0.508 -0.28 0.17 0.327 8 -0.978 -0.575 1.135 -0.751 -0.314 0.816 10 -1.972 -0.566 2.05 -1.06 -0.342 1.121

Table 8: Probabilistic diversity factors of load groups A and C

Group A Group C N OF, OF, N OF, OF, OF,

2 0.87 0.44 2 0.81 0.93 0.90 4 0.75 0.34 4 0.78 0.91 0.88 6 0.6 0.31 6 0.75 0.81 0.84 8 0.8 0.28 8 0.73 0.70 0.81 10 0.63 0.18 10 0.72 0.59 0.79

7 Conclusions

In this paper, the diversity factors of harmonic currents produced by common distribution-system nonlinear loads are calculated. The parameters involved in the generation and propagation of these harmonic currents are studied in details. In addition, the random nature of these harmonic

657

Table 9: Combined probabilistic and deterministic diversity factors for load groups A and C

Group A

N = 4 N= 8 N= 10

x/R OF, OF, OF, 7 OF, OF, OF, 1 6 OF, OF, 1

2 0.84 0.53 0.72 0.79 0.44 0.59 0.6% 0.33 0.51 5 0.82 0.55 0.70 0.75 0.45 0.55 0.64 0.33 0.50

10 0.81 0.53 0.67 0.78 0.43 0.55 0.63 0.31 0.48

Group C

XJR OF, O h OF, OF, OF, DFz OF, D h

2 0.83 0.87 0.84 0.76 0.70 0.77 0.73 0.59 0.71

5 0.81 0.88 0.86 0.75 0.71 0.76 0.72 0.59 0.70

10 0.79 0.87 0.87 0.73 0.70 0.75 0.70 0.57 0.69

N= 4 N = 8 N= 10

currents is incorporated into the performed analyses. The obtained results showed that diversity factors tend to decrease with the increase in both the number of the supplied loads and the line reactance to resistance ratio. In addition, the value for the diversity factor of each individual harmonic tends to reach a constant level for a large number of supplied loads. The phase angle dispersion and the random nature of harmonic generation are the key factors leading to the diversity of the harmonic currents. Finally, harmonic diversity factors (OF, are a valuable tool in obtaining a reliable and timely assessment of the harmonic distortion in a given distribution system.

8 References

1

2

'IEEE guide for hamonic control and reactive compensation of static power converters' (TEEE Swndard 519-1981, New York, 1981) Mansoor, A., Grady, W.M., and Chowdhury, A.H.: 'An investigation of harmonic attenuation and diversity among distnhuted single-phase

power electronic loads', IEEE T r m . POWW Deli"., 1995, 10, (I), pp. 461413 Gmdy, W.M., Mansoor, A., and Fuchs, E.: 'Net harmonic currents praduced by distributed single-phase power electronic loads' (PAQ, NY, 1995) Hammam, S.A., Ortmeyer, T.H., Katimoto, N., and Hiyama, T.: 'Harmonic performance of individual and grouped loads'. Proc. 3rd Tnt. Cod . on Power system harmonin, Athens, Greece. 1989, pp. 211- 283

5 Emanuel, A.E., Orr, J.A., and Gulachenski, E.M.: 'A survey of h m o l u c voltages and mrrenll at distribution suhstationr'. IEEE

3

4

Tram Power D&, 1991, 6, (4), pp. 188H890 Hegazy, Y.G.: 'Identification of hamonic current characteristics in electric distribution svstcms'. Ph.D. Thesis. UniVeniN of Waterloo.

6

Waterloo. Ontano, &ada 1996

, ~ ~ , , r r ~~~~ ~ ~~ ~ ~~

8 Hegary, T.G., ahd %lama, M.M.A.: 'Prbhab&stic represenlation of harmonic aments produced by AC/DC static power converters'. hoc. IEEE Industry Applications Swiety, Orlando, FL, October 1995, pp. 168W695

658 I€€ Proc-Gemr. T r ~ m r DVtrib.. Vol. 150, No. 6, Nouemher 2W3