E.F.El-Saadany M. M .A. Sa I a m a A.Y. Chi khani

Indexing terms: Distribution systems, Harmonic distortion levels, Load voltage, Current harmonics, Hybrid passive filters

Abstract: The widespread application of power electronics is increasing the number of electrical loads that distort the current and voltage waveforms in electrical distribution systems. Analysis of voltage and current harmonic distortion in a realistic distribution system was carried out. The dependence of the distortion levels on the interaction between load voltage and current harmonics, loading conditions, source impedance XIR ratio, and changing the load position, has been investigated. An iterative approach is implemented in order to study the effect of voltage and current harmonic interactions on the net system distortion. Calculation of the nonlinear load susceptances at different frequencies is presented. A hybrid reactance one-port compensator is designed to reduce the harmonic distortion in practical single- phase and balanced three-phase distribution systems having different types of nonlinear loads. The harmonic analysis is carried out in the time domain to accurately represent the system nonlinearity and voltage dependency.

1 Introduction

Conventional AC electrical distribution systems are designed to operate at single and constant frequency and at specified voltage levels of constant magnitudes. However, the increasing use of non-linear and electron- ically switched loads has increased the presence of non- s;nusoidal currents and voltages in distribution systems.

Different methods have been presented in the litera- ture to describe the behaviour of distribution systems under nonsinusoidal conditions [I-31. A variety of tech- niques have been implemented in order to reduce both voltage and current harmonics. These techniques vary

0 IEE, 1998 IEE Proceedings online no. 19981888 Paper first received 15th July and in revised form 12th January 1998 E.F.El-Saadany and M.M.A. Salama are with the Department of Electri- cal and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada A.Y.Chikhani is with the Department of Electrical and Computer Engi- neering, Royal Military College, Kingston, Ontario, Canada

from passive circuits to active circuits connected in par- allel at the point of common coupling (PCC).

Passive circuit compensators can be divided into three types: capacitive compensators [4] that may gen- erate unexpected harmonic voltages owing to series res- onance, series LC compensators [ 5 , 61 that provide higher power factors than capacitive compensators, but reduce only the fundamental harmonic at the expense of increasing the remaining harmonics, and the reac- tance one-port compensator [7] that has been employed successfully with linear loads supplied from a nonsinu- soidal supply, but never applied to nonlinear loads.

Active circuits such as electronic voltage regulators, adaptive VAR compensators and active filters [8, 91 can be used in order to reduce the system distortion. These circuits can cause further distortion to the distribution system owing to the inherent nonlinearity associated with their switching operation. In order to apply any of the reduction techniques effectively, the load voltage and current waveforms should be correctly calculated.

The situation described by Fig. 1 shows a combina- tion of linear and nonlinear loads fed from a sinusoidal supply and sharing the same supply impedance. If the supply impedance is neglected, the voltage at bus 2 will be sinusoidal, resulting in certain current waveforms for each load. However, if the supply impedance is present, the distorted current (produced by the nonlin- ear loads) passing through this impedance will result in a harmonic voltage drop on it. This will be translated into a nonsinusoidal voltage at bus 2. The presence of the nonsinusoidal voltage at bus 2 will create different current waveforms for both loads.

' 1 --------L

load 1

Load 2

Fig.1 impedance

Different linear and nonlinear loads sharing the same supply

In spite of the importance of this phenomenon, very few studies [lo-121 have considered the effect of both the attenuation, that refers to the interaction between the load voltage and current harmonics, and the diver-

IEE Pvoc -Gene7 Tvansm Dcstrib , Vol 145, No 3, May 1998 320

sity, that refers to the partial cancellation of harmonic currents owing to the dispersion in harmonic current phase angles. However, these studies have dealt only with simple single-phase circuits and did not address the case of practical distribution systems.

This paper addresses two fundamental issues. The first is to investigate the effect of different system parameters on the distribution system distortion levels. The second goal of this paper is to design a hybrid pas- sive filter capable of reducing the harmonic distortion in practical single-phase and balanced three-phase dis- tribution systems loaded with nonlinear loads to acceptable levels.

2

In order to examine the effect of different system parameters on the harmonic distortion levels of electri- cal distribution systems, detailed models for the har- monic producing loads under study have been developed in the time domain. These models have been examined experimentally to verify that the harmonic contamination of both models and actual loads are in acceptable agreement [ 131. The time-domain method has the advantage of considering the phase-angle dis- persion of the harmonic currents produced by different types of non-linear loads. Thus, this method is more accurate than the current-injection method. Also, the time-domain method greatly accounts for the system non-linearity and voltage dependency. The electromag- netic transient program [14] (EMTP) is used in devel- oping the nonlinear models as well as the overall distribution system model. Since we are dealing with either balanced three-phase or single-phase systems, then the single-phase representation will be adequate for this study. Also, it is worth noting that these load models can also be used in an unbalanced three-phase study, only the system representation will be changed. The nonlinear loads used in this study are divided into three groups depending on the shape of their wave- forms.

Non-linear loads and system modelling

n

-1 t[ 0 current, A

-100

-150 Fig. 2 Voltage-current characteristics of CFL load

2. I The main loads that exist in this group are the mag- netic ballast compact fluorescent lamps (CFL) and other gas discharge lamps. Each fluorescent lamp con- sumes 40 watts when supplied from a 120 volt (RMS) power supply. In this paper, the V-I characteristics of the CFL (result of experiment) that are given in Fig. 2 will be utilised in order to obtain a representative

switched non-linear resistors as given in Fig. 3. Each resistor will have the V-I characteristics of one half of the CFL load characteristics. This model represents 25 CFL lamps connected in parallel. The current drawn

Group 1 load model

model. The CFL model is formed from two parallel

IEE Proc -Geneu Transm Distrib , Vol 145. No 3, May 1998

by this load under a sinusoidal voltage supply condi- tion is given in Fig. 4.

"s

Fig. 3 Single-phase magnetic-ballast CFL load model

200r

-200 L time, s Fig. 4 CFL load-current waveform

~ voltage, V current, A

2.2 Group 2 load model The second load group contains those loads that employ the single-phase capacitor filtered diode bridge rectifier (DBR) as their power supply. This single-phase static power converter is commonly found in electronic equipment (computers and TV sets), small adjustable speed drives and battery chargers. The circuit model for the DBR load is given in Fig. 5 and the pulsed current waveform generated by this load class is shown in Fig. 6.

Fig. 5 Capacitor-jZtered DBR load model

200r

;- 100 0, L L 7

0 " 0

$ -100

2. (51 0 c

time,s -2ooL

Fig. 6 DBR load-current waveform ~ voltage, V . . . . . . . . current, A

321

One DBR unit consumes 0.9kW when supplied from a 120 volt (RMS) power supply. The load resistance on the DC side of the converter equals 20Q for full load representation. The smoothing capacitor that is inserted in parallel with the load resistance and pro- vides a smooth output voltage has a value of 10OOpF. The snubber circuit consists of a series R-C branch connected in parallel with each diode, where R = 1Q and C = 33pF.

2.3 Group 3 load model The third load group includes those loads utilising the phase control of thyristors to control both the input voltage and the input power to the load. The major loads that appear in this class are light dimmers, heat- ing loads and controls of single-phase induction motors.

The phase angle AC voltage controller (PAVC) unit has a 0.9kW and 120 volt rating. The PAVC unit con- sists of two thyristors connected back to back with a snubber circuit connected in parallel with each thyris- tor. The snubber circuit has R = 1Q and C = 20pF. The thyristor firing angle is adjusted to 45 degrees. The circuit diagram of this load is given in Fig. 7 and the current waveform is shown in Fig. 8.

vsx ioad

Fig.7 Single-phase PAVC loud model

* O O r

k -200L t ime,s

Fig. 8 PAVC loud-currenf waveform __ voltage, V

current. A . . . . . . . . .

2.4 Distribution system configuration The realistic secondary distribution system shown in Fig. 9 is adapted for use with the harmonic analysis. This system supplies balanced three-phase and single- phase residential and commercial customers. The pri- mary and secondary distribution transformers data [ 151 are given in Fig. 9. The distribution-system feeder sec- tions data [I61 is listed in Table 1. In addition to the non-linear loads previously described, the distribution system is loaded with balanced three-phase star-con- nected linear loads with a power factor equal to 90% lagging to simulate both the induction motors and resistive heating loads.

322

1500 KVA

transformer

t t t 75 KVA other secondary Lines

with different load combinations

50m 1 t DBRI

:om R I CFLlPAVC,

1,

R 3 DBR3 PAVC3

Fig. 9 The distribution-system model

Table 1: Feeder sections data

Section Type Length, Crosssec., R, X, m mm2 Wkm R/km

1 3-phase 100.0 16.0 1.420 0.106

2 3-phase 100.0 95.0 0.239 0.081

3 single core, PVC 50.0 50.0 0.464 0.112

4 2-cores, PVC 30.0 16.0 1.380 0.080

5 2-cores, PVC 20.0 16.0 1.380 0.080

3 Study methodology

The aim of this Section is to investigate the important factors that implicitly affect the distribution system dis- tortion levels. The knowledge of these factors will allow us to determine the distribution system behaviour under different modes of operation. This will facilitate achieving the main goals of this study, that is improv- ing the distribution system performance and minimis- ing the harmonic distortion injected by different types on nonlinear loads. To achieve this result, a hybrid reactance one-port compensator will be utilised.

3.7 Voltage-current harmonic interactions The first factor that affects the harmonic distortion lev- els in practical distribution systems, is the source inter- nal impedance. This impedance refers to the secondary distribution transformer internal impedance along with the line impedance from the transformer to the distri- bution panel. The harmonic voltage drop created on this impedance due to the current harmonics injected by different types of nonlinear loads will distort the load voltage at different busses. In order to account for the interactions between the changes of current har- monics injected by the nonlinear loads and the result- ing variations in the load voltage, an iterative technique is employed. This technique can be described according to the flow chart given in Fig. 10 and by referring to the system circuit diagram shown in Fig. 11, where V,, V,,, and V,,, are the voltage drops on the supply impedance, old-load voltage and new-load voltage, respectively. The convergence of this method is highly dependent on the system's degree of distortion. More distorted systems will require more iterations. Reliable

IEE Pvoc -Gener. Tvansm. Distrib., Vol. I45, No. 3, May 1998

convergence has been reported in many case studies, although difficulties occur near sharply tuned reso- nances [17]

assumed initial sinusoidal

1 calculate the supply

supply current

1 calculate the harmonic

get the new load voltage

n = l vLnnZ 1 vLn0Z 1 vnd

I 1

draw the voltage

current waveforms

Fig. 10 Iterative-technique flow chart

1 z s 2 Z f l 3 Z f 2 1 Z f 3 5

I I I

Fig. 1 1 Distribution-system circuit diagram

3.2 Parametric investigation Different system parameters can affect the distortion levels in realistic distribution systems. The dependence of the harmonics on the system loading level, the source impedance XIR ratio and the non-linear loads positions are the main factors that might affect the sys- tem distortion level. The study of these parameters will be carried out with the assumption that the source internal impedance is present to account for the attenu- ation while performing these tests.

3.3 Harmonic reactive power compensation The analysis carried out in Sections 3.1 and 3.2 investi- gates the different operating modes of the distribution system and shows how this will affect the distortion levels. Utilising these findings, approximate, but practi- cal shapes, for both load voltage and current wave- forms have been obtained under different practical situations. Considering these realistic waveforms, a reactance one-port compensator will be designed and inserted at the point of common coupling (PCC) in order to improve the overall system performance. This filter has to limit the total harmonic distortion (THD) of both voltage and current to the acceptable limits [18]. If one filter stage does not succeed in limiting the

IEE Proc-Gener. Trunsm. Distrib., Vol. 145. No. 3, May 1998

THD, additional stages will be added until the THD levels meet the harmonic standards.

4 Computer simulation results

In this Section, the results of the performed harmonic studies described in the previous Section are presented. These results give an insight on the variation of the harmonic distortion levels as affected by the following:

the interactions between both load voltage and current harmonics the change of the supply internal impedance XIR ratio the change of the system loading conditions the interchange in the load positions.

time,s -2ooL Fig. 12 __ voltage with . , . . . . . . . . voltage without

Load voltage with and without the interactions eflect

6 c c ? 3 U

-6OOL time,s Fig. 13 ___ current with . . . . . . . . , . current without

Supply current with and without the interactions effect

4. I The interaction effect For this study, the electric distribution system will be loaded at full load by different nonlinear and linear loads. The sharing of each load type will be given as a percentage of the distribution feeder full load capacity. The load sharing is as follows: 40% DBR loads, 20% CFL loads, 20% PAVC loads and the remaining 20% represent the linear loads. The simulation will run once considering the source impedance and again with the assumption that this impedance is neglected. The results are given as follows: Fig. 12 shows the variation

323

Table 2: Effect of attenuation on the supply current harmonic levels

Percentage of fundamental current

13 15 17 19 111 113 115 TH D

Without source impedance 38.35 19.37 6.75 1.6 2.5 1.31 0.9 43.65

With source impedance 21.62 4.24 1.55 1.51 0.88 0.34 0.59 22.18

of the load voltage waveform due to the attenuation effect, the supply current waveform is given in Fig. 13, and change of the level of each of the supply current harmonic components due to the attenuation effect is reported in Table 2.

The results clearly indicate that the source impedance is an important factor that can not be neglected, since it has a great impact on the system harmonic distortion levels. These interactions take the form of harmonic cancellation, and neglecting them will result in overesti- mating the distortion levels in practical distribution systems. Therefore these interactions will be considered in the rest of this study.

4.2 Effect of changing the source impedance X/R ratio In order to investigate the effect of changing the XIR ratio of the source internal impedance, the magnitude of the source impedance for different XIR ratios should be kept constant. The distribution system will still be loaded with the same loads that consume the same load percentage. The results obtained are given in Table 3. These results show that increasing the source imped- ance XIR ratio may lead to more harmonic cancella- tion. Since the phase angle of the source impedance is directly proportional to the magnitude and phase of the load voltage, changing the XIR ratio will result in different load voltages at different buses. This will take the form of different generated harmonic currents with different phase angles, resulting in more harmonic can- cellation among the generated harmonic currents. Also, it must be maintained that for some harmonic orders, increasing the XIR ratio might result in increasing the distortion percentage, since the cancellation depends on the relative phase angle of the perspective harmonic order for different load types.

Table 3: Effect of changing the source impedance X / R ratio on the harmonic levels

XIR ratio

0.14

1.13

2.1

3.65

6

12

Percentage of fundamental current

13 15 17 19 111 113

29.63 7.55 2.67 1.33 0.4 0.5

23.38 3.82 2.1 1.65 0.36 0.76

22.1 4.01 1.71 1.6 0.72 0.5

21.62 4.2 1.55 1.51 0.88 0.34

21.5 4.39 1.5 1.47 0.9 0.26 21.47 4.54 1.48 1.44 0.99 0.21

TH D 115

0.41 30.75

0.42 23.87

0.58 22.59

0.59 22.18

0.57 22.08

0.58 22.07

4.3 Effect of loading level For the sake of this study, the loading level will be var- ied from 29% to 100% of the feeder capacity and the following assumptions should be fulfilled: a The source internal impedance XIR ratio will be

adjusted to 3.65 and kept constant. 0 The percentage sharing of each load type will remain

constant (DBR = 400/, CFL = 20%, PAVC = 20% and linear = 20%).

324

* The positions of all loads will remain unchanged. The results of this study are given in Table 4, where the supply current harmonic contents are given as percent- ages of the fundamental current for different loading percentages. The results show that as the distribution system load increases, the percentage harmonic distor- tion decreases. This inverse proportionality can be related to both the attenuation and diversity effects.

Table 4: Effect of changing the system loading level on the distortion levels

Percentage of fundamental current TH D %

loading 13 15 17 19 111 113 115

100 21.62 4.2 1.55 1.51 0.88 0.34 0.59 22.18

83 23.76 5.29 1.79 1.98 0.68 0.75 0.3 24.53

59 26.92 7.68 2.6 2.6 0.9 1.13 0.61 28.28

44 29.36 10.32 3.75 2.7 1.23 1.25 0.7 31.54

29 31.44 13.55 6.01 2.8 2.53 1.3 1.4 34.97

4.4 Effect of changing the load position The purpose of this study is to show the effect of changing the load position on the system’s total har- monic distortion. The distribution system will have the same line parameters and loads. The loads will be at their rated values and remain loaded at the same level throughout the study. The sharing of each load type will remain constant as in Section 4.3. The source inter- nal impedance will remain constant with the XIR ratio equal to 3.65. The locations of the loads will be inter- changed according to the different configurations given in Figs. 14-17. The results of this study are given in Table 5, where clearly interchanging some load posi- tions can affect the system THD. This result is attrib- uted to the interactions between the harmonic producing loads. These interactions, referring to the

t t t R3 DBR3 PAVC2

Fig. 14 Configuration I for the study of load-position interchanging

IEE Proc.-Gener. Transm Distrib., Vol. 145, No. 3, May 1998

attenuation and diversity, may result in either decreas- ing or increasing the harmonic distortion levels depend- ing on the magnitude and phase of each harmonic component.

I I t V

-

20 m

50 m

I DBR 1

t R3

50m I 4

30 m

R3 I

I l l RI C F L l R2

PAVCl CFL2 DER2

20 m

DBR3 PAVC2 Fig. 16 Conjgurution 3 for the study of loud-position interchanging

V i

t t DBRl DBR3 DBR2

Fig. 17 ConJguration 4for the study of loud-position interchanging

IEE Proc.-Gener. Tvansm. Distrib., Vol. 145, No. 3, May 1998

Table 5: Effect of changing the load positions on the dis- tribution system distortion levels

Percentage of fundamental current

13 15 17 19 111 113 115 Configuration THD

1 21.18 4.2 1.55 1.51 0.88 0.34 0.59 22.18

2 20.0 3.62 1.48 1.45 0.87 0.3 0.58 20.51

3 19.8 3.5 1.43 1.42 0.85 0.29 0.57 20.25

4 19.49 3.31 1.42 1.39 0.85 0.27 0.57 19.9

5 Hybrid filter

The main goal of the parametric investigation carried out in the previous Section is to gain better under- standing of the harmonic current behaviour under dif- ferent practical circumstances. The most valuable outcome from this study is the calculation of the load voltage and current waveforms under practical situa- tions. These waveforms are the foundations of any fil- ter design and the more accurate the waveforms are, the more efficient the filter is. In a previous study [6], the reactance one-port compensator was implemented successfully to minimise the voltage and current distor- tion in linear circuits supplied from a nonsinusoidal voltage source. In this paper, a novel approach is implemented in order to improve the distribution sys- tem performance and minimise both voltage and cur- rent distortion levels. This approach depends on calculating the equivalent-load non-linear susceptance at different harmonic frequencies, and then extends the reactance one-port compensator design to utilise it with nonlinear systems. The filter susceptance should be equal in magnitude and opposite in sign to the equiva- lent load susceptance such as

where Bc, = - B L ~ (1)

Bcn = compensator susceptance at harmonic n B,, = load susceptance at harmonic n.

In order to achieve this goal, the equivalent load sus- ceptance at different harmonic frequencies has to be calculated, and this will be the aim of the next Subsec- tion.

5. I Equivalent-load susceptance calculation The susceptance calculation in circuits having linear loads is straightforward; dividing the harmonic current by the corresponding harmonic voltage. However, in circuits with nonlinear loads, the situation is completely different. In [I91 it is assumed that if, under no load conditions, the busbar voltage is purely sinusoidal, then the AC system harmonic impedance can be evaluated by the ratio between the harmonic voltage and the har- monic current. However, this method is not accurate since it assumes that the nonlinear load is represented by constant magnitude harmonic current injection sources. This assumption implies that the voltage and current harmonics interactions are neglected.

In order to account for the load voltage and current harmonics interactions when calculating the load sus- ceptance, the procedure described in the flow chart given in Fig. 18 will be utilised.

5.2 Filter design The system configuration given in Fig. 9 will be utilised during the design stage of the reactance one-port compensator. The system will be fully loaded with the

325

1 no is the current at the desired harmonic =min?

1 recordCt!; ;,he of

1

no a l l susceptance recorded?

\TJ Fig. 18 Flow chart describing the nonlinear loud susceptance calculation

source impedance considered and the XIR ratio equal to 3.65. The equivalent load susceptance at different harmonics will be calculated using the procedure described in Section 5.1. The next step is to identily the compensator complexity, that is equal to M = N(2N - 1) for N harmonics [20]. However, according to [6], it was shown that the required complexity is much lower and not higher than N S- M s 2N. The compensator complex admittance Y,(s) or complex impedance Zc(s>, where s is the complex variable, is determined by the number of parameters equal to its complexity, that is 2N. According to eqn. 1 we have only N equations. The remaining N parameters can be chosen arbitrary by assuming the values of the compensator poles. The synthesis procedure of a reactance one-port compensa- tor [20] can be found in the Appendix (Section 8). Using this procedure, it was found that the reactance one-port filter has the structure given in Fig. 19.

26.37mH 1.095 mH 2.17 m H

zc, ( s ) 185.2j~F 401.37sF 50.6 j~F

U, =377rad/s

Fig. 19 First-stage one-port compensator

Inserting this filter in parallel at the point of com- mon coupling (PCC) succeeded in improving the sup- ply current total harmonic distortion THDI from 22.18% to 14.54% and the voltage total harmonic dis- tortion THDV from 19.9% to 10.19%. However, these distortion levels violate the recommended values listed in [17], thus another filter stage has to be designed to further improve both the voltage and current distortion levels.

5.3 Multi-stage filter design In order to enhance the distortion levels, another filter stage is to be designed. The design procedure will be

326

the same as that utilised for the first stage and shown in the Appendix (Section 8). The susceptance of the equivalent load and the first filter stage combination will be calculated using the technique described in Sec- tion 5.1. It is found that these susceptances for n E 1, 3, 5 are equal to B1 = 0.279S, B3 = 88.493 and B5 = 0.8 8 53.

The second filter stage will have the transfer function given by

- ,~ 1.385s 0.1s 11.215s (s2 + 1.22) -t (s2 + 4.02) + ( s2 + 8.02) ~ c 2 ( s i =

(2) and the compensator will have the structure given in Fig. 20.

2.551 mH 0.0165 mH 0.465mH

m m m __9, 1915uF 26.525 mF 236.5wF w , = 377 rad/s

Fig.20 Second one-port compensator stage

Applying this filter in parallel with the first stage will result in great improvement for both the distribution system THDI and THDV. The THDI will be reduced to 6.52% and the THDV will be reduced to 6.93%. This result indicates that the nonlinear susceptance calcula- tions were not too accurate leading to inaccurate filter design. This result can be attributed to the system com- plexity, since the voltages across different loads con- nected at different nodes are different owing to the presence of the feeder section impedances.

The two reactance one-port compensator stages can be combined together leading to the compensator transfer function given in eqn. 3.

1.283s 0.582s 4.805s (s2 + 1.22) -k (s2 + 4.02) + ( s2 + 8.02) Z C 2 ( S ) =

( 3 ) The structure of the two compensator stages combined together is give in Fig. 21 and the voltage and current waveforms after inserting the combined filters are given in Figs. 22 and 23, respectively. It is observed that the combined reactance one-port compensator results in decreasing the THDI to 2.26% and the THDV to 3.87%, that is within acceptable levels.

0.0965 mH 0.199 mH 2.363mH

z c (SI ___* 2 0 6 7 ~ F 1557 u F 552 UF w , ~ 3 7 7 rad/s

Fig. 21 Combined reactance one-port compensator

6 Conclusions

In this paper, a comprehensive investigation was car- ried out to evaluate the harmonic performance in a realistic electrical distribution system loaded with dif- ferent types of nonlinear loads. The parameters that affect the harmonic distortion levels have been identi- fied. The paper illustrates that the presence of the sup- ply internal impedance leads to a great reduction in the

IEE Proc -Gener Transm Disrrib , Val 145, No 3, May 1998

time, s -200L

Fig. 22 sator __ with filter , . . . . . . . . . without filter

Load voltage with and without the reactance one-port compen-

LOOr.

I t lme,s Fig.23 sator -with filter , . , . . . . . . . without filter

Supply current with and without the reactance one-port compen-

supply current distortion levels, and neglecting this effect will cause the harmonic levels to be overesti- mated. Moreover, the increase in the impedance XIR ratio will inhibit the harmonic reduction effect.

The results have also shown that increasing the sys- tem loading level is inversely proportional to the distor- tion level and has a significant effect in improving the system performance. The analysis provides clear evi- dence that changing the load position will result in changing the system harmonic distortion levels.

A new technique for calculating the nonlinear load susceptance has been presented. A hybrid passive filter based on the reactance one-port configuration has been utilised to bring the distribution system distortion to acceptable levels. The proposed filter displays the abil- ity of drastically improving the system performance by decreasing the system voltage and current harmonic distortion. The number of filter stages is highly depend- ent on the system complexity.

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monic impedance measurements’, IEEE Trans. Power Deliv.,

20 EMANUEL, A.E.: ‘Suggested definition of reactive power in nonsinusoidal systems’, Proc. Inst. Electr. Eng., 1974, 121, (7), pp.

1991, 6, (4), pp. 1721-1726

7n5-7nh 21 TUTTLE, D.F.: ‘Network synthesis’ (New York, McGraw-Hill,

1969)

8 Appendix

For the secondary distribution system that is repre- sented by the circuit diagram given in Fig. 11, the fol- lowing design procedure is followed. For this load combination, (capacitive load), and using the procedure described in Section 5.1, the load susceptances were

B,(W) ,compensator I I!

Fig.24 Plot of the compensator susceptance

327

found to be equal to E L = O.O28S, E , = 125.0s and E, = 75.75s. The plot of the required susceptance function Y,(co) is given in Fig. 24. The compensator will have such susceptance if its impedance is in the form:

(4) (s2 + 2?)(s2 + 2;)

(s2 + P?)(s2 + P $ ) ( S 2 + P,”) Z,(S) = AS

with A 2 0 and 15 PI 5 2 1 2 3 3 5 P2 522 5 5

5 I p3 (5) Assuming, for example, PI = 1.2, P, = 4.0 and P3 = 8.0, the numerator of eqn. 4 can be expressed as fol- lows:

As(s’ + 2?)(s2 + 2,”) = ass5 + + a i s (6) The coefficients a5, a3 and al can be calculated from

eqn. 1. For s = j n and n E {l , 3, 5},

This will result in

The solution for these equations’ results in the compen- sator impedance is given as

(9) 73.35S5 + 2492S3 + 16479s

(S2 + 1.2’)(S + 4.02)(S2 + 8.02) ZC(S) =

This equation can be further simplified using decompo- sition to give the following equation:

14.32s 6.61s 52.42s + (s2 + 1.22) + (s2 + 4.02) (s2 + 8.0s) 2 4 s ) =

(10)

328 IEE Proc.-Gene?. Tuansm. Distrib , Vol. 145, No. 3, May 1998