Combined switching harmonics from multiple grid-connected single-phase inverters

D.G.lnfield

Abstract: The paper is concerned with the manner in which random phase harmonics, and in particular switching harmonics, caused by multiple devices connected to a common single phase supply, combine. Simple cancellation rules are suggested based on probabilistic integrals and Monte Carlo simulation. The analysis is general in that it applies equally to harmonics of all orders (of the fundamental) and to both current sources (generators) and sinks (loads). Formulas for both equal and unequal harmonic magnitudes are presented.

1 Introduction

Power electronic devices connected to the electricity network are now commonplace and the use of such equip- ment, connected to the electricity distribution system in particular, has a significant impact on power quality. High harmonic current components can result in increased distri- bution losses, but more seriously the resulting voltage harmonics may reach a level where they disrupt the opera- tion of other equipment supplied by the network. As such power electronic devices become widespread, and in partic- ular as inverters for a range of embedded generation sources such as photovoltaics are installed, it is imperative that the manner in which the harmonics from these multi- ple sources and sinks combine can be quantified.

Recent research [l, 21 makes clear two distinct processes at work when power electronic devices are opcrated in parallel, i.e. through connection to a common bus. These processes have been termed attenuation and cancellation (or diversity). The first of these relates to the fact that currents (absorbed or supplied at a point of common coupling to the network) are reflected into voltages depend- ent on the connection impedance, and these voltage varia- tions will affect the operation of other sources and sinks. The generally identified effect is to reduce the additional contributions to harmonic current. The second process concerns the phase relationship between the various harmonic components of the current. Clearly, if two sources or sinks of harmonic current are identical in all regards except their phase, then the possibility exists for perfect cancellation when they are in anti-phase, direct alge- braic addition of the currents when perfectly in phase, and all states between these two. This paper is concerned solely with the cancellation of random phase harmonics, and arose from a study of multiple inverter operation; it briefly presents a mathematical formulation of the problem together with a combination of exact and numerical solu-

0 IEE, 2001 IEE Proceedings online no. 20010434 DOL 10.1049/ipgtd:20010434 Paper received 25th September 2000 The author is with CREST (Centre for Renewable Energy Systems Tcchnol- ow), Department of Electronic and Electrical Engineering, Loughborough University, Loughborough LEI 1 3TU, UK

tions to the relevant probabilistic equations. It is of particu- lar relevance to the case of switching harmonics from self- commutated inverters which are random in phase, and builds on earlier work on harmonic cancellation [3, 41.

2 currents of random phase

2. I Current magnitudes are arbitrary and initially assumed equal, so we can conceive of this problem as being simply one of the evaluation of the expected value of the magni- tude of the sum of two unit vectors of random phase, as shown in Fig. 1. This case accurately represents the case of switching harmonics from two similar self-commutated inverters connected to a common supply point. Only the relative phase is relevant in this case, so the first unit vector can be selected as lying along the real axis with no loss of generality. The probability density function (PDF) for the phase of the second vector is uniform on the interval [0, 2 4 with magnitude U21-c. The expected value of the magnitude of the sum of these two vectors, denoted E[C/,], is thus given by

Probabilistic integrals for equal harmonic

The case of two harmonic currents

1/(27r) LT(2 + 2cos#l)1/2d#l = 4/7r (1)

which is in agreement with eqn. 9 of [4] for unity ampli- tude.

Fig. 1 Addition of hiu wfor . s (p1u~~or.s)

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A diversity factor is defined for the harmonics of order h injected at a point, arccording to [2, 51, as

where I i is the phasor for the harmonic current of order injected into the ith load. A diversity factor of 0 indicates perfect cancellation, whilst a value of 1 corresponds to none. For the case under consideration here the numerator is given by eqn. 1, the denominator is simply 2, and so the diversity factor takea the value of 2/n -- 0.64, indicating a significant cancellation effect even when n = 2.

2.2 The case of n > 2 identical currents of random phase The PDFs remain a!; for n = 2, and the expected value for the sum of n randonlly phased unit vectors is given by

2n2n 2n

1/(27r)% /s. . ./ [(cos $1 +cosq!12 +. . . cos & ) 2

+(sin 41 + sin $ 2 + . . . sin &)2 ] d4l,d452 . ’ . d4n

0 0 0 1/2

(3) Closed form solutions do not exist for n > 2. However, numerical evaluation of the integral has been undertaken for n = 3, 4, 5 using the proprietary package MathcadTM. The results are shown in Table 1 together with the numeri- cal value of the expression 4/11(2n)’” which is proposed here as a useful approximation to the diversity factor.

Table 1: Comparisons of proposed expression with numeri- cal integration results

Number of Diversiiy factor Value of expression 4/z unit vectors calculaied by MathcadrM (2n)’lZ for diversity factor

3 0.525 0.5198 4 0.450 5 0.402

0.4506 0.4026

The agreement with the numerical solutions is good to two significant figures and of course, the proposed expres- sion is consistent with the exact solution for n = 2 (indeed this was a key motivation for the chosen form). Support for the approximate formula as applied to larger values of n is provided by the Monte Carlo simulation results of Section 4.

For large n, it ha!; been shown [3, 41 that approximate formulas can be deduced by application of the central limit theorem [6]. In parti~:ular, for equal harmonic magnitudes, EPI,] = ( ~ d 4 ) ’ / ~ and the corresponding expression for diversity factor is, then simply (d4n)’”. Note that since (dn)/2 is very close in value to (2d2)lq 0.886 compared to 0.900, the two alternative formulas for the diversity factor for n equal magnitude harmonics are in good agreement. The derivation of the expression for large n and how it ena- bles extension of the results to cover unequal harmonic magnitudes is discussed in Section 3.

2.3 An expression for the variance Because this is a sta1:istical problem in practice, an expres- sion for the variance of the magnitude of the vector sum is useful, particularly in the context of Monte Carlo simula-

428

This can be rearranged to give 2r2,” 2 5

1

( 5 ) which can be simplified to give

(6) Finally, provided it can be assumed that Eph] = 4r~’ /~ /n 2‘/*, consistent with the proposed expression for the diversity factor, the variance is given simply by n(1 - U$). For the particular case of n = 2, the variance is unambiguously 2 - 16/d, since Eph] is then exactly equal to 4/n as given by eqn. 1. T h s is in agreement with [4], eqn. 10. A normalised standard deviation can usefully be defined as the ratio of the standard deviation to the mean. This is

which takes the constant value of 0.483.

3 Extension to unequal magnitudes for large n

It will not always be accurate to assume that the harmonic magnitudes are equal, as has been done in the preceding sections. In particular, there may well be differing harmonic impedances due to cables, and perhaps transformers, between the inverters and the point of common coupling, which will result in unequal magnitudes. This problem can be approached by considering the case of large numbers of sources/sinks. Sherman [3] and later Rowe [4] realised that for large n the central limit theorem (CLT) could be applied to provide the expected value for the sum of har- monic vectors of futed, but random, phase. The theorem states that, under general conditions, the s m of a large number of random variables with arbitrary probability dis- tributions will have a normal (or Gaussian) PDF, and if the individual distributions have zero mean and variance p? then the variance of the sum will be Cp?.

Let us assume we are summing n harmonic vectors given by the phase and magnitude pairs (ei, ai), where the magni- tudes ai are fixed and the 6, are distributed uniformly over [07 24. Choosing arbitrary axes, these vectors are resolved into the components aicosOi and aisinOi. The means of both these distributions are zero and the variances are equal and given by

IEE Proc.-Gener. Trunsm. Distrib , Vol. 148. No. 5, Septen~ber 2001

2T

E[.: cos2 01 = a:/(27r) /cosH2dH 0

27T

= nf/(47r) [B + 1 . sinEii] = a ; / 2 ( 8 ) 0

The next stage is to note that, for large n, from the CLT, the sums Ca,cose, and &,sine, are normally distributed with mean zero and variance (&?)/2. The final harmonic magnitude we are concerned to calculate is for S = C COS^,)^ + (Za,~inO,)~]'~~. This is straightforwardly done since it is well known (see, for example, [6]) that the proba- bility distribution for such a sum is a Rayleigh distribution with PDF given by

S - exp(-S2/2a2) + (9)

with mean d ( d 2 ) and variance d ( 2 - d2), where d is the variance of both of the summed distributions. In this case, as derived above, d = (Za2)/2 and so finally we have the mean value as (1/2)d(7&,2), which in the case of unit vectors reduces to (n7~/4)I/~ as already quoted in Section 2.2. The diversity factor for the general case is calculated by dividing the mean value by the algebraic sum of the magni- tudes,

This is simply (Za.i')'/2/(n'/2ki) times the corresponding factor for unit vectors. The variance is (1 - d4) Ca' and thus the normalised standard deviation is given by

As already mentioned, d d 2 is very close in value to (242)ln; 7d4 is thus similar in magnitude to 81s. Not surprisingly, then, the value for the normalised standard deviation derived assuming large n is in reasonable agree- ment with that derived in Section 2.3.

4 Monte Carlo simulation results

Monte Carlo simulations, which are a numerically feasible way of investigating larger values of n, have been carried out for up to 100 independent identical sources. With n = 50, the results of five different random test sequences, for

0'3 r

I I 1 I I I 1

o : l s l 100 200 300 400 500

number of averages Fig. 2 cmc of 50 mkpwdent sourceF

~ sene5 I ~ senes 4 ~ sencs 2 ___ series 5 ~ senes 3

increasing numbers of samples up to 500, are shown in Fig. 2. It is apparent that for averaged sample numbers above 450 the results have converged acceptably, as would be expected since the spread which decreases with (1/ds), where s the number of averages, is then only 2%.

Table 2 summarises the averaged results of 2500 Monte Carlo simulations for 10, 15, 20, 50 and 100 independent harmonic sources of equal magnitude; the values confirm the accuracy of the proposed approximate formula for the diversity factor and also the alternative expression derived from the central limit theorem.

Table 2: Harmonic reduction for larger n

Number of Monte Carlo (n /4n) sources result 4/n (217)''~ from cLT

10 0.28 0.2847 0.2802

15 0.23 0.2325 0.2288

20 0.20 0.2013 0.1982

50 0.13 0.1273 0.1253

100 0.09 0.0900 0.0886

5 measurement

Further remarks and comparisons with

5. I Summary of results The agreement between the proposed simplified formula for diversity factor (which is exact for n = 2) and the numerical calculations is excellent. What is perhaps surpris- ing is how well the alternative formula for the diversity factor based on the central limit theorem performs for small n. Fig. 3 shows a comparison between the two alter- native expressions for n unit vectors up to 20.

0.7 ,-

o.l t O I ' " " " " " " ' " ~ ~ ' 0 20

n Fig.3 upproximation

~ approx.

Comparison of result from central limit theorem with proposed

CLT _-_

It has been shown, following [4], that for large n, varia- tions in the magnitude of the vectors can be taken into account by multiplying the diversity factor by (&ai), the root mean square harmonic magnitude divided by dn times the algebraic sum of the magnitudes. Since the results derived from the CLT remain accurate for small n, it is justified to use this same scaling factor to generalise the proposed formula, 4/11(2n)'", which is exact for n = 2, to varying harmonic magnitudes. This results in the following general expression for the diversity factor:

The useful formulas that have been derived here can thus be summarised as:

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(i) DF = 4/n(2n)”’ for all harmonic vectors of unity magni- tude and uniform PDF across [O, 2 4 for the phase. (ii) DF = 2(2Cri,2)1’2/(n7-cCrif) for harmonics of magnitude U,

and uniform PDF across [O, 2 4 for phase. (iii) Normalised standard deviation = (7d2-\j2)(1 ~ 8/d)’j2 = 0.483. In all of these the term 41n(2n)”’ can be replaced with (7d4n)’”, or equivalently (242)ln replaced with dlri2, with- out significant loss of accuracy, especially for larger n.

5.2 Comparison with experiment During the multiple inverter study which prompted this work, measurements of switching harmonics were made on up to five identical inverters all connected directly to the 230V network supply at the same point. It is highly likely that groups of similar inverters will be used in this way on larger installations. Fig. 4 shows the results for the diver- sity factor for the switching harmonic calculated from the average of 20 independent experiments for each number of inverters from 1 to 5. The switching frequency for these inverters was 20 kHi:. Following the discussion of Section 2.3, the normalised error associated with these points (0.483(1/1/s), where J’ is the number of samples) is in this case -11%. It is expected that attenuation will further reduce the harmonic contribution so that the measured levels should lie below those predicted to be caused by cancellation alone. Furthermore, attenuation will be most significant when phase angles are similar, and this will be more likely statistically the larger the numbers of inverters. This explains why the case for five inverters lies further below the prediction than for the smaller groupings. The results also show that for small numbers of inverters connected directly together at the point of common coupling, cancellation effects dominate over attenuation effects. Since the actual harmonic level will be lower than the prediction basad on cancellation alone, formulas presented in this paper provide a usefully conservative esti- mate for practical application.

0.8

I I

O 1 2 3 4 5

::: ~ ,

number of inverters Fig. 4 Comjmiton ivith (ciprunent

experiment + prediction

6 Conclusions

An approximate formula for the diversity factor is presented for the case of n equal vectors with a uniform distribution of phases. This is shown to be accurate for both large and small n. Previous analyses based on large n and using the central limit theorem are found to be in close agreement with the proposed formula, even surprisingly for small n.

It has been shown that the results can be generalised to deal with nonequal magnitudes in a simple manner. The normalised variation around the mean is found to be constant, i.e. independent of n and the values of the harmonic magnitudes, a,. All the formulas are straightfor- ward to apply, and should assist system designers and engi- neers of the distribution network operators charged with assessing the implications of proposed new generation installations using inverter connected sources, and in partic- ular photovoltaics.

The implications for power systems and the possible future wide-scale deployment of grid connected inverters are encouraging in that there will be a marked tendency for cancellation of the expected random phase contributions from switching to harmonic distortion. For groupings of more than three inverters the diversity factor will be < 0.5, falling to < 0.3 for collections of 10 or more. Such signifi- cant cancellation may favour installations of multiple grid connected inverters for photovoltaic applications rather than the more conventional single inverter arrangement.

7 Acknowledgments

This work was undertaken as part of a project on multiple inverter operation supported by the Engineering and Physi- cal Sciences Research Council, EPSRC, under grant GR/ L62894. I am gratetul to Peter Onions of CREST for undertaking the Monte Carlo simulations and making the measurements on the inverters.

References

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