International Journal of Advances in Engineering Science and Technology
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238
ISSN: 2319-1120
Half-wave Symmetry SHE PWM Scheme for Five
level Voltage Source Inverter
Anitha.R 1,Anbarasu.P 2
Assistant Professor1,2
Department of Electrical & Electronics Engineering,
1
Maharaja Institute of Technology,Arasur,Coimbatore,
2
Sri Eshwar College of Engineering ,Kinathukadavu,Coimbatore
1
E-mail: aanithaflora@gmail.com
, anbarasu.ambi@gmail.com.2
1,2
Abstract— This paper introduces a Half-wave symmetry Selective Harmonic Elimination(SHE) Pulse Width Modulation (PWM)
scheme to obtain more solutions to SHE PWM problems. Initially various SHE PWM techniques are analyzed and the problems are
listed. Then a new proposed method and its generalized formulation are proposed. The specific features of this scheme is the format is
simple, has broad solution space and has flexibility in PWM waveforms. Implementation of this method is done in a five level cascaded
voltage source inverter. Set of solutions for a specific modulation index is obtained and illustrated. Simulations are done using Mat lab
and the performance of inverter is verified.
Index Terms- Half-wave symmetry, Multilevel inverter, Pulse Width Modulation(PWM), Selective Harmonic
Elimination(SHE)
I.
INTRODUCTION
M
ulti level inverters have greatly come in to existence in recent years and hence to overcome the problems related to it is
significant. Many technique have been developed to minimize or eliminate harmonics present. Eliminating the harmonics,
particularly the lower order harmonics can give high quality output. This in turn reduces current ripple and torque pulsation which
improves the overall system performance.
To obtain the solution of the non linear transcendal equations with trigonometric functions are complex. To solve these
equations many iteration methods and algorithms are proposed. The algorithms include (i)introducing transformation steps to
ensure convergence and finding all sets of solutions through, for instance a sequential homotopy- based computation.[1] (ii)use of
resultant theory and high order polynomials.[2-4] (iii) walsh functions.[5] (iv)optimization technique such as genetic algorithm.
[6,7](v)colony of ants.[8] (vi)optimization search.[9,10].The multilevel SHE-PWM method is investigated in comparison with
multi-carrier-based sine pulse width modulation(SPWM)[11] method. It is concluded that the SHE-PWM method offers a
significantly higher converter bandwidth than the multi-carrier-based SPWM method and also reduces the switching frequency
without changing the waveform quality over the full range of modulation index.
Much effort has been made in obtaining effective initial values to guarantee rapid convergence of Newton iteration.[12] A
generalised formulation for multilevel SHE-PWM converters with non-equal DC source was also reported[13]. A method
manipulating the well-known SHE-PWM technique to control power flow into each H-Bridge of a cascaded multilevel rectifier at
a low switching frequency while still achieving a high-quality waveform was described.[14] Instead of eliminating specific loworder harmonics, an optimization strategy considering all the harmonics and the total harmonic distortion (THD) as a global
problem in conjunction with one specific actual grid code was proposed[15]. A new active harmonic elimination technique was
recently introduced to the line frequency method aiming to eliminate higher order of harmonics by simply generating the opposite
of the harmonics to cancel them. The main disadvantage is that it uses a high switching frequency. Solutions trajectories of the
harmonic elimination problem were mathematically addressed.
A systematic design approach for applying signal processing methods as an addition to conventional control to simultaneously
achieve two control objectives, like fast transient response and efficient harmonic filtering, was reported in [16]. A realtime
algorithm for calculating switching angles that minimises THD for step modulation of multilevel inverters was proposed in [17].
This method cannot be employed for multilevel inverters in which the power devices operate at a frequency higher than the
fundamental. Generalized formulations for quarter- and half-wave symmetry SHEPWM problems for multilevel inverters were
proposed in literature [18,19]. The main advantages of these methods include high flexibility of PWM waveforms, broad solution
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Half-wave Symmetry SHE PWM Scheme for Five level Voltage Source Inverter
space and ease of generalization to voltage levels for any converter topologies.
The quarter-wave symmetry assumption guarantees that the even harmonics will be zero and that all harmonics will be either
in phase or anti-phase with the fundamental. Although this simplified the definition of the problem, the quarter-wave symmetry
constraints limit the solution space.In order that more solutions for multilevel SHE-PWM problems can be obtained, the symmetry
requirements are loosened or abolished, and more generalized techniques of SHE-PWM method can be formulated. The main
purpose of this paper is to propose a new SHEPWM scheme and try to obtain a full set of multilevel SHE-PWM schemes.
Multiple solution property of each SHE-PWM scheme is investigated and compared.
II. MULTIPLE SOLUTION PROPERTY OF DIFFERENT SHE-PWM SCHEME
In order to obtain the same number of switching angles in one output cycle, we choose N, 2N and 4N as the number of
independent switching angles for quarter-wave symmetry, half-wave symmetry and asymmetry SHE-PWM schemes. The
Fourier series expansion of the SHE-PWM waveform is given by
(1)
where ω is the radian frequency of the output voltage. Owing to the PWM waveform characteristics of half-wave symmetry,
an = bn = 0 when n is even. When n is odd, the generalized expressions of an and bn for any number of switching angles and any
number of voltage levels are given by
(2)
where E is the voltage of a level and the coefficient of pk can be expressed as follows:
(3)
The number of independent switching angles and nonlinear equations for each SHE-PWM scheme are different.
A. Quarter Wave Symmetry She-Pwm Formulation
The quarter-wave symmetry assumption guarantees that the even harmonics will be zero and that all harmonics will be either
in phase or anti phase with the fundamental voltage. Although this is convenient, the quarter-wave symmetry constraint limits the
solution space. On abolishing the symmetry requirements, more generalized techniques of the SHE-PWM method can be formulated
for traditional two-level [17] and five-level [18] inverters, which brings about infinite solutions.
Some solutions may have merit relative to others with respect to system losses, ripple characteristics, distribution of uneliminated
harmonics, THD, or some other system aspects. For example, these unique harmonic profiles can be chosen such that they place
significant energy in triplen harmonics, a benefit for applications with balanced three-phase wye-connected loads. However, four
times the number of switching anglesand harmonic elimination constraints compared with quarter symmetry method result in
asymmetry SHE-PWM equations too complicated to be practical.
Multilevel inverters are mainly employed in three-phase medium- and high-voltage systems in which all the triple harmonics
are absent in output voltage, so only the low-order non-triple harmonics in the line-to-neutral voltages are to be eliminated. For a
multilevel inverter with L output levels, given a desired fundamental V1, a corresponding modulation index M can be expressed as
(4)
The generalized formulation can be expressed as
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IJAEST, Volume 2, Number 2
Anitha.R ,and Anbarasu.P
(5)
(6)
where n = 5, 7, 11, . . . , 3N–2 when N is odd; n = 5, 7, 11, . . ., 3N–1 when N is even; N is the number of switching angles in a quarter
cycle; L is the number of levels of the output phase voltage; and ak is the kth switching angle.
Fig 1. Quarter wave symmetry SHE PWM scheme waveform
B. Half-Wave Symmetry She-Pwm Formulation
With the same harmonic elimination and fundamental control tasks, if the quarter-wave symmetry constraint is relaxed to
half-wave symmetry, half the number of switching angles and harmonic elimination constraints in contrast with asymmetry SHE-PWM
method are needed. The complexity of half-wave symmetry SHE-PWM techniques is reduced, and the property of infinite solutions
remains. Infinite solutions provide infinite opportunities for optimization design.
For half-wave symmetry SHE-PWM scheme shown in where the second half-cycle is extended according to
(7)
where f is frequency of output voltage and t is time in range of (0, 1/(2f )).
The generalized SHE-PWM equations can be expressed as
(8)
(9)
(10)
(11)
where E is the level voltage, M is the modulation index, pk is the same as in (6) and ak is the kth switching angle with the
constraint
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Half-wave Symmetry SHE PWM Scheme for Five level Voltage Source Inverter
(12)
A1 and B1 are two constants that satisfy the equation
(13)
Fig 2. Half wave symmetry SHE PWM scheme waveform
No even-order harmonics exist owing to the symmetry property. Two equations are needed for each low- and oddorder
harmonic to be eliminated. There are 2N independent switching angles and non-linear equations. Therefore the same number of loworder harmonics can be eliminated as when assuming quarter-wave symmetry.
A Novel Classification of Evolutionary Algorithms: A Critical Assessment
C. Asymmetry She-Pwm Scheme
Ignoring the symmetry constraint, an asymmetry SHE-PWM scheme can be obtained. That freedom in the initial phase angle
of fundamental voltage does not mean infinite solutions as stated above is also true in the asymmetry SHE-PWM scheme. Another
constraint that does not exist in quarter and half-wave symmetry SHE-PWM schemes is that the DC component must be zero.
Therefore the generalized formulation of asymmetry SHE-PWM scheme is as follows
(14)
(15)
(16)
(17)
(18)
Two equations for each low-order harmonic including odd and even-order harmonics are required. By ignoring the DC
component constraint, the asymmetry SHE-PWM scheme can eliminate the same harmonics as quarter- and half-wave symmetry
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IJAEST, Volume 2, Number 2
Anitha.R ,and Anbarasu.P
schemes with the same number of switching angles in one cycle. However, DC component is very harmful to both loads and power
supplies and must be removed. Therefore one more low-order harmonic is left un-eliminated compared to quarter- and half-wave
symmetry SHE-PWM schemes, although many more independent switching angles and equations are involved, leading to a higher
computation cost.
Fig 3. Asymmetry SHE PWM scheme waveform.
III.PROPOSED FORMULATION OF HALF-WAVE SYMMETRY MULTILEVEL SHE-PWM PROBLEMS
Switching angles in the first half cycle of a typical half-wave symmetry SHE-PWM waveform of a multilevel inverter with L
levels are shown in Fig. 1. The problem under consideration is to find appropriate switching angles, namely, α1, α2, α3, . . . , αN, so
that the (N − 1)/2 non-triplen odd harmonics can be eliminated and the control of the fundamental voltage is also achieved. The Fourier
series expansion of the SHE-PWM waveform is given by
(19)
where ω is the radian frequency of the output voltage. Owing to the PWM waveform characteristics of half-wave symmetry,
an = bn = 0 when n is even. When n is odd, the generalized expressions of an and bn for any number of switching angles and any
number of voltage levels (even or odd, provided that the waveform is physically correct and can be implemented) are given by
(20)
where E is the voltage of a level and the coefficient of pk can be expressed as follows:
(21)
Multilevel inverters are mainly employed in three-phase medium- and high-voltage systems, in which all the triple harmonics
are absent in output voltage, so only the low-order nontriple harmonics in the line-to-neutral voltages are to be eliminated.
IV.SOLUTION TO HALF-WAVE SYMMETRY SHE PWM PROBLEMS FOR MULTI LEVEL INVERTER
A.Methods For Obtaining Initial Values
In half-wave SHE PWM problems the phasing of fundamental voltage is free to vary. The equations and initial values are dependent on
initial phase of output voltage. Suppose reference is sinusoidal signal with initial phase angle
equal to output fundamental voltage.
Then substitute ωt with ωt+
and obtain initial value based on rules of (i) equal area (ii)super position of centre of gravity.
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Half-wave Symmetry SHE PWM Scheme for Five level Voltage Source Inverter
The alternative methods for finding the initial phase angle are using a multi carrier based SPWM, in this reference sinusoidal signal
should have same initial phase angle with output fundamental voltage. The amplitude should be close to that determined by the
modulation index M. b. trail and error method if more set of solutions needed.
B. Resolving Multi Level She Pwm Equation
The method used in this paper is not analytical but rather numerical in nature.
Step 1: The number of switching angles in half cycle, the modulation index M, and the two constraints A1 and B1 are
decided.
Step 2: The series of initial values composing of rising and falling edges are obtained.
Step 3:The judgement with numerical result in true solution
C.Solution To She Pwm Equation
Take a five level inverter and 20 switching angles over a half cycle as example. With M=0.8, the SHE-PWM non linear
equations are formulated and resolved. Lower order harmonics are to be eliminated. Twelve sets of solution Sa-Sj are obtained as
listed in table. The switching angle with positive and negative signs represent the rising edge and falling edges of SHE PWM wave,
respectively. The corresponding output waveform is drawn with respect to the set of solutions.
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IJAEST, Volume 2, Number 2
Anitha.R ,and Anbarasu.P
V.SIMULATION
To verify the validity of the solutions obtained with the new SHE-PWM scheme proposed in this paper, simulations on a
PSIM platform of a single-phase five-level cascade inverter are carried out. The parameters for simulation are as follows: voltage of
each level E = 100 V, frequency of the output voltage f = 50 Hz and resistance of the load R = 136 V. Sa and Sb, which are the first
and second group of solutions at M = 0.8, are adopted in the simulations. The simulation waveforms of the line-to-neutral voltage are
shown.
It can be observed from the simulated waveforms that all the harmonics up to 1.5 kHz, which includes all voltage harmonics
up to and including the 30th, have been eliminated and the goals of fundamental control are achieved in both cases.
VI. CONCLUSION
A novel half-wave symmetry SHE-PWM scheme for multilevel voltage source inverters is proposed in this paper. This
method is simple in formulation and flexible in SHE-PWM waveform. The proposed method brings a large number of new solutions to
SHE-PWM problems and therefore provides more opportunities for practical optimisation design. Many solutions can be obtained with
one group of equations, which makes the process of solving the equations much simpler. The advantages of the proposed method have
been proved by numerical examples.
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