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В1СНИК ПРИАЗОВСЬКОГО ДЕРЖАВНОГО ТЕХН1ЧНОГО УН1ВЕРСИТЕТУ
Вип. №16
2006 р.
УДК 621.311
Zhezhelenko I.1, Sayenko Y.2, Baranenko Т.3, Arndt В.4
INTERHARMONICS IN ELECTRICAL SYSTEMS
Рассмотрены методы оценки уровней интергармоник путем применения быстрого преобразования Фурье как непосредственно к случайному графику нагрузки источника интергармоник, так и к его корреляционной функции. Произведена оценка энергии дискретного спектра интергармоник тока от таких резкопеременных нагрузок, как электродуговые сталеплавильные печи и сварочные установки. Рассмотрены характерные спектры интергармоник сетевого тока непосредственных преобразователей частоты с однофазным выходом при линейном и синусоидальном законах управления. Рассмотрены методы оценки интергармоник напряжения, а также методы снижения уровней интергармоник.
Introduction
The electric power quality parameters specified by EN 50160 European Standard include a recently introduced new parameter of interharmonics, the normalizing principles being developed now.
Interharmonic currents present the same problems with heating and inductive interference as do harmonic currents. It is recommended that interharmonic currents be limited in the same manner as harmonic currents in IEEE 519-1992 [1]. The IEC limits interharmonic voltage distortion are based on the curve near the fundamental [2]. Elsewhere the limits are similar to the interharmonic levels.
The purpose of the paper is studying the reasons why interharmonics occur, developing methods forecasting their values, explaining approaches to normalizing the permissible levels and selection of methods of reducing the interharmonics levels.
Interharmonics sources in industrial power supply networks Interharmonics are harmonic fluctuations with frequencies not multiple of the supply network frequencies [3]. Interharmonics are generated by the loads which operate continuously or during short periods in transient mode. This operation mode results from the variations in the load as part of the technological process or from peculiar features of the electromagnetic processes going on during the work of these devices, e.g. alternating work of the converter valves, etc. Transient processes which cause interharmonics are, as a rule, of a random character and can be referred to the category of stationary random processes.
The main sources of interharmonics are electric arc furnaces (AF), spark welding installations, valve converters of the rolling mills and other quick-variable non-linear loads [4].
Interharmonics may appear due to subharmonic fluctuations which occur in series or parallel connections of capacitors or transformers at their considerable saturation [5].
A specific category of the loads generating significant interharmonics into supply networks are frequency converters; they are used for regulating rotation speed of electric engines in electric drive systems [4].
1 Pryazovskyi State Technical University, Prof., Doctor of Science, Eng.
2 Pryazovskyi State Technical University, Prof., Doctor of Science, Eng.
3 Pryazovskyi State Technical University, Candidate of Science, Eng.
4 University of Applied Sciences (Wurzburg Schweinfurt), Prof., Dr.-Ing.
Methods for estimating of interharmonics levels In the most simple case interharmonics levels can be defined by means of harmonic analysis of
random process I(t) or U (I), where /(/) and U (I) are enveloping curves of the rms values of current and voltage: in this case it is convenient to apply Fast Fourier Transform (FFT). It is obvious that this expansion characterizes harmonic content of a specific implementation of a random process at the interval of stationarity.
In a general case the spectral-correlation theory of random processes is used to analyze the above processes. The theory is based on FFT which interrelates the correlating function of the process p(r) and its energy spectrum (spectral density) G(a>). Energy spectrum which describes distribution
of process energy by frequencies includes a continuous Gc(a>) and a discrete G0 (co) components [6]. Thus
G(©) = G» + G». (1)
Transient processes in industrial electric supply systems refer to the processes with a mixed spectrum, i.e. they are characterized with the presence of both components of energy spectrum.
Discrete component defines energy of harmonic spectrum which includes canonical and non-canonical harmonics as well as interharmonics
co
Gd(co) = Y,Dk5((o-(oK), (2)
k=0
where Dk - dispersion of ^-harmonic; S(a>-a>K) - delta function.
Continuous spectrum is not harmonical. This type of electromagnetic disturbances, along with harmonic spectrum, produces a considerable impact on the quality of the automated process control system of such devices as frequency converters, etc.
It is known that in some cases correlation function of the random process of load variation of such consumers as welding or arc furnaces is non-attenuating which testifies to non-ergodic property of the random process. One of the most characteristic reasons of non-ergodicity of a stationary random process is the presence of periodic components. Non-attenuating component of the correlation function (the so-called "tail" of the correlation function) contains the same frequencies as the proper random process [7]. In connection with this it is recommended to make the analysis of spectral composition of low frequency periodic components of load current applying FFT not to the proper random process but to the "tail" of the correlation function thus providing for the separation of the periodic components from the random process described by one of the types of correlation functions [8]
p(T) = De-a^; (3)
p(r) = D^"'1' cos (o0z ; (4)
p(r) = De
r a , ^ cos®0T h--sin ffljr
V ®o y
(5)
where I) is process dispersion; a is attenuation coefficient; a>0 is proper frequency of correlation function.
It should be noted that the above equations characterize smoothed curves p(z).
Current interharmonics in electrical engineering installations Transient processes inside AF are most intensive during the melting period. Random process of current fluctuation during this period is not stationary as it is characterized with changes in the condition of the melted blend and the burning conditions of arcs, variations of voltage on the electrodes, other commutation operations; some more circumstances also influence in their own way.
However, this process can be decomposed into individual stationary parts with duration up to 2 min (rarely up to 5 min); the number of these parts may reach 10-12. As it was proved by numerous experiments, correlation functions of AF currents are, as a rule, represented by attenuating curves, which is a sufficient condition for ergodicity.
The described conditions allow to consider the analysis of the results of one of the implementations of the stationary part to be sufficiently representative.
Fig. 1, a shows the curve of current variations at phase A in AF-100 at the stationary part with the duration of 60 sec and Fig. 1, b - the spectrum of rms values of the current of the given graph. Discrete spectrum has been obtained by means of harmonic analysis with applying FFT.
Analysis of the discrete spectrum shows that current interharmonics occur within the range of 0-2,5 Hz, their levels may reach 10-15 % of the basic frequency current. To estimate the interharmonics energy the squared rms values should be summarized; for the graph in Fig. 1, b this value is 10,1 103 A2. The energy of continuous spectrum is determined by the area below the curve of energy spectrum Gc(a>) and is equal in numerical value to the process dispersion. Fig. 1, с shows a theoretical curve of energy spectrum of the load process under consideration, which is relevant to exponential-cosine correlation function [9]. In this case the values of the random process parameters in phase A [10] are: a = 1,47 c"1; co0 = 2,69 с:'; D = 9331,2 A2.
I. A
U, V L A
ДП r 80
- 60
- 40
- 20 - 0
■lfttJtt.f.>
0.5 1 1.5 2 2.5 3
/Hz
b)
GOO. A--S
2500
Fig. 1 - Load current (a), amplitude spectrum (b) and energy spectrum (c) of AF-100 current (phase A)
Fairly close values of energies (dispersions) corresponding with the amplitude spectrum of the load process and the energy spectrum as well as the form of the curves (Fig. 1, b and Fig. 1, c) single out FFT as the tool for analyzing interharmonics spectrum.
As it has already been noted, another approach to estimate the rms values (or amplitudes) and interharmonics frequencies is expansion into Fourier series of the non-attenuating part (the so-called ""tail") of the correlation function; the "tail" contains interharmonics of the same frequencies as the random process 7(0- To analyze the spectral composition of low frequency components of the load current in this case it is also highly recommended to use FFT [8].
In way of illustrating the above processes Fig. 2, a shows the load process of AF-100 in phase B, Fig. 2, b - correlation function for this process and Fig. 2, c - the spectrum of the squared interharmonics obtained in the result of harmonic analysis of the correlation function ""tail".
Interharmonics spectrums for various implementations of the random process i(t) during the melting period appear to be non-identical, whereas the energy values of discrete spectrums are close enough in values.
Portion of 8 0 in periodic components of load current, i.e. discrete spectrum of interharmonics is:
Sd =
YA
2 D
(6)
where AK stands for amplitudes of periodic components of load current, I) - full dispersion. For phases A, B and C of AF-100 we obtained: SoA = 21 %, 8dB = 18,8 %, 8dC = 17,4 %. Thus, the energy of discrete interharmonics spectrum for AF-100 makes 17-21 % of the total energy of the mixed spectrum.
L A
p(r). A--10
1000 800 600 400 200 0
0
15
t, sec
A 1000 800 600 400 200 0
/Hz
c)
Fig. 2 - AF-100 load current, phase B (a), correlation function of the load process (b ). and spectrum of the squared interharmonics (c)
Similar relations are true for 200 t AF, which gives grounds to regard the given results applicable to more general cases.
The portion of interharmonics for AF of lower volumes appears to be somewhat greater.
When analyzing the interharmonics of electric welding machines let us limit ourselves with single-spot machines only. Fig. 7 shows the curve of current variation during spot welding in a certain process; Fig. 8 gives the amplitude spectrum of the current calculated with the help of FFT. Noticeable interharmonics appear in rather low frequency ranges of 35-75 Hz and 140-160 Hz. Their amplitudes do not mainly exceed 20 % of the basic welding current interharmonic.
IHi
0 0.06 0.12 0.18 0.24 0.3 0.36 0.42 0.48
t, sec
.......ill
harmonic order
a)
b)
Fig. 3 - Current curve (a) and amplitude spectrum of the current curve (b) of spot welding
As the results of extensive investigations show, a common characteristic for the discrete spec-trums generated by various types of electric welding machines is a higher "density" of harmonics distribution along the frequency axis, the energy of discrete spectrums reaching 12-20 % of the total energy of the mixed spectrum.
Interharmonics of frequency converters network current Regulating of rotation speed of electric engines by controlled thyristor frequency converters is accompanied with appearance of considerable interharmonics in the curves of the network currents. Almost in all cases the network current spectrum consists of discrete and continuous parts.
Direct frequency converters (DFC), cycloconverters, are most widely spread within the electric drive systems that involve regulating of rotation speed of electric engines.
Fig. 4 shows a simplified circuit of DFC with a single-phase outlet [12]. The circuit includes two identical sets of valves VI and V2, which conduct the load current of different polarity. The f0
Fig. 4 - DFC with a single-phase outlet
frequency of the basic harmonic of the load current is determined by the Impulse-Phase Control System (IPCS). The frequencies and the rms values (amplitudes) of the network current interharmonics depend on a lot of reasons: circuit (type) of converter, operation mode, outlet frequency of DFC, type of electric engine and speed regulation function, supply voltage fluctuations, inlet electromagnetic disturbances and other factors.
However, in all cases amplitude modulation of network currents curves can be observed and specific conditions for formation of mixed, in particular, discrete harmonics spectrum depend on the above factors. Keeping in mind a wide variety of electric drive circuits based on applying DFC [13] and the absence of common function of formation of inlet/outlet current spectrums, we shall restrict our consideration to a number of specific cases; this will allow us to present the levels of the spectrums being generated.
Fig. 5 and Fig. 6 show the network currents curves of three-phase/single-phase six-impulse DFC at linear and sinusoidal control function, direct order of alternation of circuit phases and frequencies f2 = 7 Hz and /2 = 20 Hz. Fig. 7 and Fig. 8 give relevant discrete spectrums (in relative values).
Distribution of interharmonics by frequencies depends on the control function and the value of the converter outlet frequency. Calculations showed that at linear control function the interharmonics amplitudes do not basically change their values when interharmonics shift along the frequency axis in a wide range of variations of the outlet frequency f2. With this the energy of the interharmonics discrete spectrum can reach as much as 55 % of the basic harmonic energy of the DFC inlet current.
WTith sinusoidal control function amplitude values of both interharmonics and of the basic current harmonic depend on the amplitude of the modulating signal (modulation depth) and phase angle of the current at the converter outlet. In this case the energy of the interharmonics discrete spectrum can vary within a wide range, not exceeding 50 % of the energy of the basic frequency current.
wMMjy
0 0.1 0.2 0.3 0.4 0.5 0.6
/. sec
a)
Fig. 5 - Curve of DFC network current at linear control function and load frequency /2 = 7 Hz (a)
and/2 = 20 Hz (b)
i(t)
0 0.1 0.2 0.3 0.4 0.5 0.6
t, sec
a)
b)
Fig. 6 - Curve of DFC network current at sinusoidal control function and load frequency f2 = 7 Hz (a)
and f2 = 20 Hz (,b)
I!h HI1
0.8 '
0.6
Cl .4 --0.2 -■
0.6 -0.4 ■
H-1-TJ I ' T-1-1— 1 11 I
H-1-
I ' I ' 'I ' I—i1
_!_i_L
4-1—1—4-
M-M-M-
10 11 12 13 14 13
10 11 12 13 14 15
harmonic order
harmonie order
a)
//ii i --
o.s --
0.6 --
0.4 --
0.2
Fig. 7 - Amplitude spectrum of DFC network current at linear control function and load frequency f2 = 7 Hz (a) and f2 = 20 Hz (b)
///,
o.s
o.s j-
0.4 0.2
-1-
111111 il I .III. ■ 11 ■ 111
rlur"'T' I ■■ ■|'-'-"i"-L"lllJ-r ■ ■■!■'
10 11 12 13 14 15
J_L
I 1 i1 ' I -4M-
harmonic order
harmonic order
a)
Fig. 8 - Amplitude spectrum of DFC network current at sinusoidal control function and load frequency f2 = 7 Hz (a) and f2 = 20 Hz (b)
b)
'i 11 1 ■ i i
10 11 12 13 14 15
b)
It should be noted that even slight variation (± 1 %) in the network and control voltage frequencies is enough to cause a considerable shift of interharmonics spectrum along the axis of frequencies, as well as narrowing of widening of the spectrum.
The network current curves of the electric drives of the rolling mills with thyristor converters are in many cases amplitude-modulated oscillations with the modulation frequency of 5-10 Hz. Interharmonics are grouped within the range of ± (7-10) Hz of the basic or high harmonics, their relative levels very rarely exceeding 10 % in operating modes. When the electric drives of the rolling mills are idling the interharmonic level is negligibly low.
Voltage interharmonics of the supply network Voltage interharmonics can be obtained by means of harmonic analysis of the enveloping curve of the rms voltage (amplitude) values or in other words of the voltage fluctuation curve.
To illustrate this Fig. 9, a shows oscillogram of the linear voltage fluctuations on 35 kV supply buses (AF-100) and on Fig. 9, b you can see the discrete spectrum of this curve.
/. sec
a)
U, kV
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
ni
Jjlt,
/Hz
b)
Fig. 9 - Linear voltage fluctuations (a) and discrete analysis of linear voltage fluctuations (b)
on 35 kV supply buses, AF-100 Another approach is based on the similarity of random processes of current I(t) and voltage U(t)
variations connected with each other by Ohm Law. Phase voltage is
(/,,,(/) = ZJ(t). (7)
where Zc- impedance of network in the PCC of the voltage fluctuation source.
For line voltages assuming the symmetry of the processes in three phases,
U(i)= fiZJ(t). (8)
Dispersion of voltage interharmonics Dv, i.e. dispersion of voltage fluctuations, is connected with the dispersion of current fluctuations / ), by a known relation
Du = 3Z2cDJ. (9)
This also suggests the symmetry of the processes in all three phases. The Zc value depending on the power of the short-circuit in the PCC of non-linear load can be found as follows:
, u2
xc*Zc=—. (10)
K
For example, for AF depending on the power of SK on 35 kV buses, it appears that Zc« 0,5 -r- 1,7
Ohm.
Fig. 1, b gives the voltage interharmonics scale for Zc = 0,5 Ohm.
Relation between the energy of discrete and continuous spectrum are within the same limits as for current interharmonics.
Reducing voltage fluctuations with the help of correcting devices and circuit solutions results in considerable reduction of voltage interharmonics levels and their spectrum width; the same phenomenon is observed when improving the voltage curve form with the help of active or hybrid filters.
Selection of methods of reducing the interharmonics levels
Reducing the interharmonics levels in electric networks is an integral part of the task of increasing the quality of electric energy, along with reducing the levels of higher harmonics. The way the interharmonics effect the operation of the electric energy consumers is similar to the way the higher harmonics effect it. That is why the approach to minimizing the interharmonics is similar to that of minimizing the higher harmonics.
When solving the task of choosing the means for minimizing the interharmonics and higher harmonics, it is necessary to consider both the conditions of providing the required levels of non-sinusoidality of voltage and reactive power compensation and the optimal choice of the number and the point at which the filter is positioned This task does not always have a single solution and demands for performing a series of technical and economic calculations..
When selecting the interharmonics filters, a coordinated solution for a whole range of problems is needed. The main problems in question are:
1. Reducing non-sinusoidality of voltage to a permissible level.
2. Ensuring the required level of reactive power compensation.
3. Ensuring reliable operation of the interharmonics filters at deviation of the operation parameters of both the filters and the mains supply, interharmonics sources, etc. from their nominal values.
4. Absence of resonance phenomena at both interharmonics and higher harmonics frequencies.
To solve the above problems let us consider potential application areas of the filters of different
orders.
Rejector filters (filters of the first order) have limited application at minimizing the interharmonics as they are narrow-band devices.
Damping filters (filters of the second order) consist of a capacitor and a reactor with shunt active resistance (Fig. 10, a).
Fig. 10, b shows frequency characteristics of the module of full resistance of the damping filter at various values of active resistance. It is obvious that the damping filter allows to expand the pass-band. Due to this a permissible level of non-sinusoidality of voltage can be reached through installing less number of filters as compared with the filters of the first order.
Fig. 11, a shows the diagram of a complex filter (filters of the third order).
The complex filter has two resonance frequencies, one of them corresponds to a serial resonance, and the other to the parallel resonance, which allows to perform compensation of the interharmonics levels in a wide range of frequencies that are either more or less as related to the frequency of
the main harmonic. Fig. 11, b gives the example of a frequency characteristic of the module of full resistance of the complex filter which illustrates the efficiency of its application.
a)
.1 R - oo^ ■*'
R= 1 -
v.
ft¿1 R = 0,2
i | — |
b)
Fig. 10 - Diagram (a) and frequency characteristic of the module of full resistance (b)
of the damping filter
:CI
i
C2
R
III
V
a)
b)
Fig. 11- Diagram (a) and frequency characteristic of the module of full resistance (b)
of the complex filter
Conclusions
1. Amplitudes and frequencies of current and voltage mterharmonics are random values and their occurrence is determined by a combination of various parameters of electromagnetic transient processes.
2. Estimation of interharmonies amplitudes and frequencies is, as a rule, feasible for a specific implementation of a random process.
3. Relation between the energy of continuous and discrete spectrum varies within wide ranges with possible predominance of this or that component.
4. Connection between the voltage fluctuation spectrum parameters and voltage fluctuations characteristics adopted by European standards (short-time and long-time flicker "dose") is not distinct. However, reduction of voltage fluctuation levels in the network results in considerable reduction of interharmonies levels. Hence one of the approaches to normalizing the allowed values of voltage (or current) interharmonies. It is highly recommended to limit the allowed values of the equivalent energy spectrum, i.e. of the process dispersion. The required instrumentation can be created on the basis of the existing devices for measuring the correlation function and energy spectrum parameters which are widely used in radio and communication engineering as well as on the basis of computer measuring systems.
5. At calculating the spectral composition of the DFC input current, solution of the problems of electromagnetic compatibility demands for taking into account the actual wave of the load current.
6. In a common case, because of a complex nature of the amplitude spectrum of the DFC input current at any law, the numeric methods of the interharmonies calculation and analysis should be preferred.
7. Using the damping filters (filters of the second order) and complex filters (filters of the third order) for minimizing the interharmonies levels allows to decrease the number of parallel filter connections and, apart from this, reduce their specified power as compared with using the filters of the first order.
References
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Статья поступила 20.03.2006.
