R.Kameswara Rao. et al. / International Journal of Engineering Science and Technology (IJEST)
EFFECTIVE HARMONIC MITIGATION TECHNIQUES USING WAVELETS BASED ANALYSIS R.KAMESWARA RAO1 Electrical and Electronics Engineering Department, Jntu College Of Engineering, Kakinada, Andhra Pradesh, India
[email protected] http://jntuk.edu.in
G.RAVI KUMAR2 Electrical and Electronics Engineering Department, Bapatla Engineering College, Bapatla, Andhra Pradesh, India
[email protected] http://www.becbapatla.ac.in
S.S.TULASI RAM3 Electrical and Electronics Engineering Department, Jntu College Of Engineering, Kakinada, Andhra Pradesh, India
[email protected] http://jntuk.edu.in Abstract - The harmonic currents are generated by single phase nonlinear loads. The mirror surplus harmonic elimination method is used mostly because of its less computation. The Selective Harmonic Elimination Pulse Width Modulation (SHEPWM) based methods can provide theoretically the highest quality output among all the PWM methods. In this paper SHEPWM model of a multilevel series connected voltage source inverter is developed which can be used for an arbitrary number of levels and switching angles. This paper describes the analysis of harmonic elimination methods with wavelet transform method. For the analysis of harmonic components Wavelet bior6.8 version is used. The harmonics must be reduced in order to reduce the size of the filters. Selective harmonic elimination method totally eliminates a particularly selected harmonic either of lower order or higher order, there by Total Current Harmonic Distortion (THD) can be reduced to a large extent. Keywords: THD; SHEPWM; PWM; OMTHD; OHSW 1. Introduction Distributed single phase power electronic loads are very significant source of harmonics in electric power distribution system. The harmonic currents generated by single phase nonlinear loads such as desk top computers and florescent lamps cause appreciable distortion in distribution feeder when consider more in number. The mirror surplus harmonic elimination method is used mostly because of its less computation. The SHEPWM based methods can provide theoretically the highest quality output among all the PWM methods. In this paper SHEPWM [1] model of a multilevel series connected voltage source inverter is developed which can be used for an arbitrary number of levels and switching angles. 1.1 Programmed PWM technique In 1973, the selected harmonic elimination method for PWM inverters was introduced for single-cell (two and three-level) inverters. This method is sometimes called as programmed PWM technique. It illustrates the general quarter-wave symmetric triple-level programmed PWM switching pattern. The square wave is chopped m times per half cycle. Owing to the symmetries in the PWM waveform, only odd harmonics exist. The Fourier
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coefficients of odd harmonics in triple level programmed PWM inverters with odd switching angles are given by 4E bn [cos n 1 cos n 2 .. (1) j 1 cos n j (1) n ... cos n m ] Where n is harmonic order. Any m harmonics can be eliminated by solving m equations obtained from setting above equation equal to zero. Usually Newton iteration method [2] is used to solve these non-linear equations, the condition [3] must be satisfied is
0 1 2 ............ m 2
(2)
Each cell of such a single-phase inverter switches and produces a three-level {-1, 0, 1} PWM waveform. This results in a five-level -2, -1, 0, 1, 2 inverter output. Theoretically, odd harmonics can be eliminated from the inverter’s spectrum while keeping the fundamental components of both cells equal to each other. The switching angles must be obtained from the following system of nonlinear transcendental equations:
m
(1)
i 1
i 1 m
(1) i 1 m
(1)
i 1
i 1
i 1 m
(1)
cos i 4 M
cos i 4 M m
cos 3 i ( 1) i 1 cos 3 i 0 i 1
i 1
m
cos(4m 3) i (1) i 1 cos (4m - 3) i 0
(3) Where αi is the switching angle of the first cell, βi is the switching angles of the other cell, and M is the modulation index. This paper explicitly requires an even fundamental power sharing among cells. The Convergence of numerical procedures used to solve (3) is sensitive to the starting values of switching angles and requires considerable computation i 1
i 1
1.2 Mirror surplus harmonic PWM technique Elimination of low-order harmonics from only one cell, which will be called a general SHEPWM method, can be obtained by solving a system of m equations. m
(1)
i 1
i 1
m
(1)
cos i 4 M
i 1
cos 3 i 0
i 1 m
(1)
i 1
i 1
cos( 2m 1) i 0
(4)
The first significant surplus harmonic from this cell has amplitude A2m+1. If it is desired to eliminate A2m+1 from the output spectrum of the single-phase inverter, the other cell must produce the 2m+1 harmonic of amplitude -A2m+1. To preserve the elimination 2m-1 of the low-order odd harmonics and to set the amplitude 2m+1 of the harmonic to -A2m+1, the number of switching angles in the second cell must be increased by one to m+1. The switching angles of the second cell fulfill the following system of m+1 equation:
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(1)
i 1
i 1
m 1
(1)
i 1
i 1
cos i 4 M
cos 3 i 0
... m 1
(1)
i 1
i 1 m 1
(1)
cos( 2m 1) i 0
i 1
i 1
cos( 2m 1) i
( 2m 1) A2 m 1 4
(5)
An unexpected benefit of such a 2m+1 harmonic cancellation is that the whole first cluster of significant harmonics from the second cell becomes nearly a mirror image of the first cluster of significant harmonics from the first cell. Thus, the solution of (4) and (5) approximates very closely the solution of (3). Similar models can be developed for three-phase systems. The difference between single- and three-phase calculations is that for a three-phase system triplen harmonics need not be included in the set of harmonics selected for elimination. The cancellation of harmonics using (4) and (5) will be called a mirror surplus harmonic PWM technique. This cancellation has been checked for several values of m and for a wide range of the modulation index M. Since the system of equations as in (4) do not have analytical solutions, it is difficult to find a theoretical explanation for the proposed method. It will be an important topic for future research. Nevertheless, the proposed approach is a practical way of finding an approximate solution to (3) and hence the harmonic suppression in double-cell series-connected inverters. The main advantage of this method is that the proposed approach is a practical way of finding an approximate solution of programmed PWM technique equations and hence harmonic suppression is achieved in double cell series connected inverters. 1.3 Optimal minimization of total harmonic distortion technique The basic idea for this method is to adjust switching angles in order to minimize the output voltage THD. To minimize the THD, it is necessary for its partial derivative to be zero with respect to each switching angle. This implies that the partial derivative of its square is also zero because the value of THD is always positive. After development and some mathematical simplifications, the square THD of the chosen multi-level generalized waveform (periodic with odd quarter-wave symmetric characteristic) is given by: 2 2 THD 8
2 S 2 Vk 1V12 k 1
S
j 2
j V j2 2V j
S Vk cos k k 1
2
Vi i 1 1 j 1
(6)
It is assumed that the dc sources were all equal. It is also noticed that there is no control on the fundamental component of the output voltage. This means that Optimized Method of THD (OMTHD) technique is applied only for THD minimization with no constraint on the value of fundamental component; where as, the primary objective in any method of inverter control is the adjustment of the fundamental component to the desired value. OMTHD technique is applied to the cascaded multi-level inverter with unequal dc sources to minimize THD while producing the desired fundamental component at the output. On the other hand, the fundamental component must have the desired value Vf. This implies that switching angles must also satisfy the following equation:
V1dc cos ( 1 ) V2dc cos ( 2 ) ...... VSdc cos ( S ) ma
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2. Harmonics A harmonic is a signal or wave whose frequency is an integral (whole-number) multiple of the frequency of some reference signal or wave. For instance, with a fundamental frequency of 60Hz, the 3rd harmonic frequency is 180Hz (3 x 60Hz).
Fig.1. Harmonic waveform
The current drawn by non-linear loads is periodic but not sinusoidal. Periodic waveforms are described mathematically as a series of sinusoidal waveforms that are summed together. Sinusoidal components are integer multiples of the fundamental frequency which is 60 Hz in the United States. Harmonics are multiples of the fundamental frequency, as shown in Figure 1. Total harmonic distortion is the contribution of all the harmonic frequency currents to the fundamental. The total harmonic distortion, or THD, of a signal is a measurement of the harmonic distortion present and is defined as the ratio of the sum of the powers of all harmonic components to the power of the Fundamental frequency. 2.1 Selective Harmonic Elimination Method This method totally eliminates a particularly selected harmonic either of lower order or of higher order; thereby THD can be reduced to a large extent [4]. This provides the highest quality output to the system when compared to the other methods and a pure sinusoidal wave can be obtained. This method can be applied to different inverters of multi levels. This paper focuses on series-connected voltage-source PWM inverters. The Selective Harmonic Elimination pulse width modulation (SHEPWM) based methods [5] can theoretically provide the highest quality output among all the PWM methods. The SHEPWM method presented in this paper offers the same number of control variables as the number of inverter levels. The results given in are only for a five-level inverter allowing up to seven switching angles without taking into account that inverter cells should equally share the output power. The SHEPWM model of a multilevel series-connected voltagesource inverter is developed which can be used for an arbitrary number of levels and switching angles. 3. Multi Level Inverter MULTI-LEVEL inverter [6] and [7] is recently used in many industrial applications such as ac power supplies, static VAR compensators, drive systems etc. One of the significant advantages of multi-level structure is the harmonic reduction in the output waveform without increasing switching frequency or decreasing the inverter output power. The output voltage waveform of a multi-level inverter is composed of a number of levels of voltages, typically obtained from capacitor voltage sources. The so-called multi-level starts from three levels [8]. As the number of levels increases, the output THD approaches zero. The number of achievable voltage levels, however, is limited by voltage unbalance problems, voltage clamping requirement, circuit layout, and packaging constraints. Therefore an important key in designing an effective and efficient multi-level inverter is to ensure that the total harmonic distortion (THD) in the output voltage waveform is small enough. The total harmonic distortion of the output current decreases as the number of levels of the generated output voltage increases. The multi level starts from 3 levels. The main advantage of multi level structure is the harmonic reduction in the output waveform without increasing switching frequency or decreasing the inverter output power. As the number of levels increases, the output THD approaches zero. 4.
Wavelet Analysis
In mathematics, wavelets, wavelet analysis and the wavelet transform refers to the representation of a signal in terms of a finite length or fast decaying oscillating waveform known as mother wavelet.
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This waveform is scaled and translated to match the input signal. In formal terms, this representation is wavelet series, which is the coordinate representation of square integrable function with respect to a complete, orthonormal set of basis function for the Hilbert space of square integral functions. The word wavelet is due to Morlet and Grossman in the early 1980s. They used the French word ondelettemeaning “small wave”. A little later it was transformed into English by translating “ondo” into “wave”-giving wavelet. Wavelet analysis [9] represents a next logic step describing a windowing technique with variable sized regions. The Wavelet analysis allows the use of longtime intervals when more precise low-frequency information is needed and shorter regions when high frequency information is needed.
Wavelet analysis does not use a time-frequency region, but rather a time-scale region. Wavelet theory is applicable to several other subjects. All wavelet transforms may be considered to be forms of time-frequency representation and therefore are, related to the subject of harmonic analysis. Almost all practically useful discrete wavelet transforms make use of filter banks containing finite impulse response filters. The wavelets forming a Continuous Wave Transform (CWT) are subject to Heisenberg’s uncertainty principle and equivalently, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle The principal difference between the CWT and the Discrete Wave Transform (DWT) is that CWT operates over every possible scale and translation whereas DWT uses a specific subset of all scale and translation values. There are number of ways of defining a wavelet (or a wavelet family). 4.1 Scaling filter: The wavelet is entirely defined by the scaling filter g – a low pass impulse response (FIR) filter of length of length 2N and sum1. In Biorthogonal wavelets, separate decomposition and reconstruction filters the time reverse of the decomposition. Daubechies and Symlet wavelets can be defined by the scaling filter. 4.2 Scaling function: Wavelets are defined by the wavelet function Ψ (t) (i.e; the mother wavelet) and scaling function Φ (t) (also called father wavelet) in the time domain. The scaling function filters the lowest level of the transform and ensures that the spectrum is covered. For a wavelet with compact support, Φ (t) can be considered finite in length and is equivalent to the scaling filter g. Meyer wavelets can be defined by scaling functions. 4.3 Wavelet function: The wavelet only has a time domain representation as the wavelet function Ψ (t). Mexican hat wavelets can be defined by a wavelet function. The continuous wavelet transform was developed as an alternative approach to the short time Fourier transforms to overcome the resolution problem.
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5.
Simulation Results
The figures.4, 8 and 12 shows the simulation diagrams,fig.5, 9 and 13 show the comparative analysis of THD using FFT Fig. 6,10,11,14,15 show the harmonics analysis using wavelet.
5.1 Simulation Of First Cell Of Single Phase Double Cell Series Connected PWM Inverter
Fig: 4 Schematic diagram of first cell for single phase double cell series connected PWM inverter.
Fig 5: single phase spectra and wave forms for first cell (a) output voltage waveform (b) frequency spectra
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Fig 6: Wavelet Analysis of single phase spectra and wave forms for first cell output voltage waveform detailed coefficient spectra.
Fig 7: Wavelet Analysis of single phase spectra THD wave forms for first cell output voltage waveform detailed coefficient spectra.
5.2 Simulation Result For Second Cell Of Single Phase Double Cell Series Connected PWM Inverter
Fig 8: Schematic diagram of a single phase double cell series connected PWM inverter.
5.3 Simulation Result For Single Phase Double Cell Series Connected PWM Inverter
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Fig 9. (a) Inverter output voltage (b) Frequency spectrum
Fig 10: Wavelet analysis of a single phase double cell series connected PWM inverter.
Fig 11: Wavelet analysis of a single phase double cell series connected PWM inverter THD Analysis.
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5.4 Simulation Result For Optimized Harmonic Stepped Waveform (OHSW)
Fig 12: Schematic diagram of OHSW Technique.
Fig 13: Harmonic spectra of OHSW
Fig 14: Wavelet analysis of OHSW Technique.
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Fig 15: Wavelet analysis of OHSW Technique THD analysis.
6.
Conclusion
A more effective method of harmonic suppression is described. Mirror surplus harmonic elimination PWM technique is the effective method to eliminate selected lower order harmonics compared to other pulse width modulation techniques with added advantage of eliminating the selected harmonics completely. In the selected two harmonic elimination method angles were found to generate pulses to the inverter and observed that THD reduced and also noticed that selected lower order harmonics were eliminated effectively, at output stage in mirror surplus technique and it required only two cells to eliminate any number of harmonics. This is an added advantage for this method where as in OHSW technique the THD was reduced effectively. As seen from the above observations Wavelets will support in a more effective way to analyze harmonics in current and voltage waveforms. References [1]
P. M. Bhagwat and V. R. Stefanovic. (1983). “Generalized structure of a Multilevel PWM inverter,” IEEE Trans. Ind. Applicant., vol. IA-19, pp.1057–1069. [2] Tjalling J. Ypma, (2004). “Historical development of the Newton-Raphson method, SIAM Review 37 (4), 531–551, 1995. Doi: 10.1137/1037125.P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics, Vol. 35. Springer, Berlin, ISBN 3-540-21099-7. [3] Carrara, D. Casini, S. Gardella, and R. Salutari. (1993). “Optimal PWM for the control of multilevel voltage source inverter,” in Proc. 5th European Conf. Power Electronics and Applications, vol. 4, Brighton, U.K., pp. 255–259, Sept.13–16. [4] “Multilevel Selective Harmonic Elimination PWM Technique in Series-Connected Voltage Inverters”, (2000). IEEE Transactions on Industry Applications, vol. 32, no. 1, pp. 1454-1461, [5] Ajay Maheswari, Khai D. T. Ngo, (1993). “Synthesis of 6-step PWM waveforms with Selective Harmonic Elimination”, IEEE Transactions on Power electronics, Vol.8, No.4, pp. 554-561. [6] M. Marchesoni and M. Mazzucchelli. (1993). “Multilevel converters for high power ac drives: A review,” in Proc. IEEE Int. Symp. Industrial Electronics, pp. 38–43. [7] Prasad N. Enjeti, Member, IEEE, Phoivos D. Ziogas, Senior Member, IEEE, and James F. Lindsay, Senior Member, IEEE,” A Current Source PWM Inverter with Instantaneous Current Control Capability”, Industry Apllications Society Annual Meeting, Conference record of the 1988 IEEE. [8] J. S. Lai and F. Z. Peng. (1996). “Multilevel converter-A new breed of power converters” IEEE Trans. Ind. Applicat., vol. 32, pp. 2348-2357. [9] Amara Graps, “An Introduction toWavelets, (1995).” IEEE Computational Science and Engineering, Vol.2, No.2, also available on the Web at http://www.amara.com/current/wavelet.html with a variety of otherwaveletsresources.
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Biography R. Kameswara Rao received B. Tech and M.Tech degrees in Electrical Engineering from JNTU College of Engineering, Kakinada. He is presently working as Associate professor in Electrical and Electronics Engineering department in the same college. His areas of interest include Power quality and Harmonics analysis. G.RaviKumar graduated from Andhra University College of Engineering, Visakhapatnam, India and received M.Tech from JNTU College of Engineering, Kakinada. He is currently working as Associate professor in EEE Department at Bapatla Engineering College, Bapatla, India. His areas of interest are Power System operation and Control, Power System Protection. Dr.S.S.Tulasi Ram received B. Tech, M.Tech and Ph.D degrees in Electrical Engineering from JNTU College of Engineering, Kakinada. He is currently working as professor of Electrical and Electronics Engineering in the same college. His areas of interest include high voltage engineering, Power system analysis and control.
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