International Journal of Power Electronics and Energy
ISSN: ABCD-WXYZ, Vol: 1, No: 2
CASER
2012
www.caser.in
Selective Harmonic Elimination Using Pattern
Search Method in Single phase Voltage - Source
Inverters
a
M. Narayanana , K. Keerthivasanb
Karpagam University, Coimbatore – 641021, b Karpagam University, Coimbatore – 641021
Email: a [email protected], [email protected]
Abstract — Variable speed drives are very commonly used for many industrial applications in the industrial
sectors, commercial building and municipal water system for achieving energy savings. The problem with these
drives is even though it improves efficiency, it pollutes the quality of supply by introducing harmonics.
Therefore to reduce the effect of harmonics in the system, appropriate techniques are needed to eliminate
dominant harmonics. An Voltage Source inverter output contains both the fundamental and the harmonics.
Suitable technique has to be used to solve the transcendental equations representing inverters output. So in this
research work, Pattern Search (PS) method has been employed to eliminate the low order harmonics. Using
pattern search technique, the transcendental equations are solved and switching times (angles) that produce
only the fundamental without generating any specific harmonic order (3 rd, 5 th, 7 th and 11 th) is found. This is
referred to as harmonic elimination or programmed harmonic elimination as the switching angles are chosen
(programmed) to eliminate specific harmonics.
Keywords – Harmonic Elimination, Unipolar Pulse Width Modulation, Pattern Search Method, Total Harmonic
Distortion.
1. INTRODUCTION
In the past few decades the power electronics
field witnessed a rapid growth in power
semiconductors and digital techniques, which makes it
possible to build systems with high efficiency, reliable
and low cost. One of these systems is the Voltage
Source Inverter (VSI), which is used in different fields
of power converters such as UPS, SMPS, drives,
tractions and HVDC. VSI is used to convert DC
supply to AC supply at required number of phases and
values of voltage and frequency. The problem with
power converters is it affects the quality of supply. In
the case of inverter, the output contains frequencies
other than fundamental [4].
waveform without any kind of distortion. If the
current or voltage waveforms are distorted from its
ideal form it is termed as harmonic distortion. This
harmonic distortion could result because of many
reasons. In today’s world, prime importance is given
by the engineers to derive a method to reduce the
harmonic distortion. Harmonic distortion was very
less in the past when the designs of power systems
were very simple and conservative. But, nowadays
with the use of complex designs in the industry
harmonic distortion has increased as well. Also due to
the wide use of adjustable AC & DC drives in
industrial applications, analysis of power quality is
very important [15].
Power quality is very important to commercial
and industrial power system designs. Ideally, the
electrical supply should be a perfect sinusoidal
So to reduce the harmonic content in the
inverters output Selective Harmonic Elimination Pulse
Width Modulation is used. The output voltage
waveform of an inverter is given by Fourier series
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CASER
International Journal of Power Electronics and Energy
ISSN: ABCD-WXYZ, Vol: 1, No: 2
2012
www.caser.in
expansion which contains both the fundamental and
the harmonics. These transcendental equations are
solved using techniques such as iterative numerical
techniques [1], Selective harmonic elimination [6],
Programmed PWM technique [7], elimination using
resultants [16], etc., to compute the switching angles.
However, the drawback of these methods is heavy
computation burden and complicated hardware [8].
Here a method is presented to solve the transcendental
equations whose solutions are highly reliable with fast
converging characteristics.
Direct Search (DS) methods are evolutionary
algorithms used to solve constrained optimization
problems. DS method does not require any information
about the gradient of the objective function at hand,
while searching for an optimum solution. This family
includes PS algorithms, Simplex Methods (SM)
(different from the simplex method used in linear
programming), Powell Optimization (PO) and others.
The PS method is a technique that is suitable to solve a
variety of optimization problems that lie outside the
scope of the standard optimization methods. Generally,
PS has the advantage of being very simple in concept,
easy to implement and computationally efficient.
Hence it can be used to solve the transcendental
equations involving harmonics.
Figure 1: Power circuit of a single – phase VSI
2. PROBLEM FORMULATION
The power circuit of a basic single - phase
voltage source inverter is shown in Fig. 1.
Figure 2: Unipolar PWM output waveform
The Fourier series expansion of unipolar waveform
[13] is given by:
Fig 2. shows the inverter output voltage
waveform under unipolar switching scheme. If the
waveform contains (K+1) switching angles per quarter
v(ωt) = (4Vdc /π) {
+cycle, (K) harmonics will be eliminated. In
this paper, K is equal to 4.
sin(n t ) / n
[(cos(nθ1 ) –
n 1, 3, 5
cos(nθ2 ) + cos(nθ3 ) - cos(nθ4 ) + cos(nθ5 ) ]}
(1)
The problem is to find the unknown angles with
transcendental equations as follows:
cosθ1 - cosθ2 + cosθ3 - cosθ4 + cosθ5
=M
cos3θ1 - cos3θ2 + cos3θ3 - cos3θ4 +cos3θ5 = 0
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(2)
(3)
CASER
International Journal of Power Electronics and Energy
ISSN: ABCD-WXYZ, Vol: 1, No: 2
cos5θ1 - cos5θ2 + cos5θ3 - cos5θ4 +cos5θ5 = 0
(4)
cos7θ1 - cos7θ2 + cos7θ3 - cos7θ4 + cos7θ5 = 0
(5)
First Step: The Pattern search begins at the initial
point X0 that is given as a starting point by the user. At
the first iteration, with a scalar of magnitude 1 called
mesh size, the pattern vectors are constructed as [0 1],
[1 0], [−1 0],and [0 −1], they may be called direction
vectors. Then the PS algorithm adds the direction
vectors to the initial point X 0 to compute the following
mesh points:
cos11θ1 - cos11θ2 + cos11θ3 - cos11θ +cos11θ5 = 0 (6)
Where M = 4Vdc/ π V1
(7)
This formulated problem will be solved using
PS method whose objective function aims to minimize
the harmonic equations subject to the constraints [1113],
0o < θ1 < θ2 < θ3 < θ4 < θ5 < 90o
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X 0 + [1 0]
X 0 + [0 1]
(8)
X 0 + [−1 0]
Therefore the aim is to control the magnitude
st
of the 1 harmonic (M Є [0, 0.9]) and to eliminate the
undesired low-order selected harmonics given by the
equation (3) to (6).
X 0 + [0 −1]
Figure: 3 illustrates the formation of the mesh and
pattern vectors. The algorithm computes the objective
function at the mesh points in the order shown.
3. PATTERN SEARCH METHOD
The PS optimization routine is a derivative
free evolutionary technique that is suitable to solve a
variety of optimization problems that lie outside the
scope of the standard optimization methods. Generally,
PS has the advantage of being very simple in concept,
and easy to implement and computationally efficient
algorithm [11]. Unlike other heuristic algorithms, such
as GA, PS possesses a flexible and well-balanced
operator to enhance and adapt the global and fine tune
local search.
The PS algorithm proceeds by computing a
sequence of points that may or may not approach the
optimal point. The algorithm starts by establishing a
set of points called mesh, around the given point. This
current point could be the initial starting point supplied
by the user or it could be computed from the previous
step of the algorithm. The mesh is formed by adding
the current point to a scalar multiple of a set of vectors
called a pattern. If a point in the mesh is found to
improve the objective function at the current point, the
new point becomes the current point at the next
iteration.
Figure 3: PS Mesh points and Pattern
The algorithm polls the mesh points by
computing their objective function values until it finds
one whose value is smaller than the objective function
value of X0 . If there is such point, then the poll is
successful and the algorithm sets this point equal to
X 1.
After a successful poll, the algorithm steps to
iteration 2 and multiplies the current mesh size by 2,
This may be better explained by the following:
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International Journal of Power Electronics and Energy
ISSN: ABCD-WXYZ, Vol: 1, No: 2
(this is called the expansion factor and has a default
value of 2). The mesh at iteration 2 contains the
following points: X1 + 2*[1 0], X1 + 2*[0 1], X1 +2*[-1
0] and X1 + 2*[0 -1] . The algorithm polls the mesh
points until it finds one whose value is smaller than the
objective function value of X 1 . The first such point it
finds is called X 2 , and the poll is successful. As the
poll is successful, the algorithm multiplies the current
mesh size by 2 to get a mesh size of 4 at the third
iteration.
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All the parameters involved in the PS optimization
algorithm can be pre-defined subject to the nature of
the problem being solved.
4. SIMULATION RESULTS
The proposed PS method is used to eliminate
the lower order harmonics, 3rd , 5th , 7th and 11th from the
inverter voltage waveform for any desired value of
modulation index M. In this section, the results are
registered for M=0.8.Table 1 summaries the
parameters used in the PS method to solve the SHE
problem.
Second Step: Now if in iteration 3, none of the mesh
points has a smaller objective function value than the
value at X2 , the poll is called an unsuccessful poll. In
this case, the algorithm does not change the current
point at the next iteration, i.e., X3 = X2 . At the next
iteration, the algorithm multiplies the current mesh
size by 0.5, a contraction factor, so that the mesh size
at the next iteration is smaller. The algorithm then
polls with a smaller mesh size. The PS optimization
algorithm will repeat the illustrated steps until it finds
the optimal solution for the minimization of the
objective function. The PS algorithm stops when any
of the following conditions occurs:
Table 1:Parameters of PS method
PS parameter
Value
Mesh tolerance
10e-006
Maximum iterations
100 * no. of variables
Maximum
evaluations
2000 * no. of variables
function
Table 2: PS based computed switching angles
to eliminate 3, 5, 7 and 11 harmonics at M = 0.8
The mesh size is less than the mesh tolerance.
The number of iterations performed by the
algorithm reaches the value of maximum iteration
number.
θ1
θ2
θ3
θ4
θ5
objfun
18.04
26.584
37.157
53.983
57.68
9.4e007
The total number of objective function evaluations
performed by the algorithm reaches the value of
maximum function evaluations.
Table 3: Corresponding magnitudes of the
Fourier coefficients.
The distance between the point found at one
successful poll and the point found at the next
successful poll is less than the specified tolerance.
The change in the objective function from one
successful poll to the next successful poll is less
than the objective function tolerance.
22
Co-eff
B1
B3
B5
B7
B11
Desired
0.8
0
0
0
0
Actual
0.8
4.44e016
0
2.22e016
9.4032e007
International Journal of Power Electronics and Energy
ISSN: ABCD-WXYZ, Vol: 1, No: 2
CASER
2012
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Table 3. illustrates the obtained set of
solution for M= 0.8. Moreover, the corresponding
amplitudes of the selected harmonics to be eliminated
are computed and the results are presented in Table 4.
The results shows the low-order harmonics are well
attenuated, and the amplitude of the 1st harmonic is set
to the desired value at the same time.
After obtaining the switching angles
through the MATLAB using PS method, the single
phase inverter circuit is fired with the obtained
solution. Simulations were carried out on a Intel Core
2 Duo 1.8GHz, 512 MB RAM processor. The rating of
the proposed AC drive system is given in Table 4.
Figure 4: Simulation Diagram for unipolar case
Table 4: Simulation parameters
PS parameter
Value
Input DC voltage
230 V
Fundamental frequency
50 Hz
Rload
10 ohms
Lload
6.5 mH
Figure 5: Switching pattern for IGBT 1 and IGBT 2
Let us consider an voltage source inverter
connected to an RL load as shown below.
Figure 6: Switching pattern for IGBT 3 and IGBT 4
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International Journal of Power Electronics and Energy
ISSN: ABCD-WXYZ, Vol: 1, No: 2
CASER
2012
www.caser.in
method to determine efficiently the required switching
angles to implement the SHE strategy. The Total
Harmonic Distortion (THD) of the inverters output
voltage waveform is 37.84% which is due to the
higher harmonic content. Further reduction in THD is
possible when the number of eliminated harmonics is
increased. This, will in turn increases the number of
the estimated switching angles.
Figure 7: Input DC voltage, Output voltage and Load
current waveforms for M=0.8
Figure 9: Output voltage harmonic spectrum of
unipolar PWM (M=0.5, Elimination of the 3rd , 5th , 7th
and 11th harmonics)
Figure 8: Output voltage harmonic spectrum of
unipolar PWM (M=0.8, Elimination of the 3rd , 5th , 7th
and 11th harmonics)
Figure 5 and Figure 6 shows the control signal
of IGBT switches. The resultant inverters output
voltage waveform, load current waveform and DC
input voltage waveform are shown in Fig. 7. To
evaluate the performance of the proposed method, the
harmonic spectrum of the output voltage is
investigated. The results are plotted in Fig. 8.
According to the obtained results, the selected loworder harmonics (3rd , 5th , 7th and 11th ) are well
attenuated. The nearest component to the fundamental
is the 13th harmonic, which is not eliminated. The
harmonic spectrum verifies the capability of the PS
Figure 10: Output voltage harmonic spectrum of
unipolar PWM (M=0.6, Elimination of the 3rd , 5th , 7th
and 11th harmonics)
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International Journal of Power Electronics and Energy
ISSN: ABCD-WXYZ, Vol: 1, No: 2
CASER
2012
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Figure 13: Load current harmonic spectrum (M=0.5)
Figure 11: Output voltage harmonic spectrum of
unipolar PWM (M=0.7, Elimination of the 3rd , 5th , 7th
and 11th harmonics)
Figure 14: Load current harmonic spectrum (M=0.7)
Figure 12: Output voltage harmonic spectrum of
unipolar PWM (M=0.75, Elimination of the 3rd , 5th , 7th
and 11th harmonics)
Figure 9 to Figure 12 shows the output voltage
harmonic spectrum for different values of modulation
index eliminating 3rd , 5th , 7th and 11th harmonics.
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International Journal of Power Electronics and Energy
ISSN: ABCD-WXYZ, Vol: 1, No: 2
Figure 15: Load current harmonic spectrum (M=0.75)
method and the corresponding output voltage and load
current THDs are shown in Table 5.
The waveform for the load current with RL load
and harmonic spectrum for load current are shown in
Figure 13 to Figure 15.
6. CONCLUSION
The research work has successfully
implemented selective harmonic elimination. The
similar work using Pattern Search methodology can be
carried out for further cases and best results can be
obtained. The estimation of 5 switching angles per
quarter cycle has been performed, while minimizing a
pre-selected number of harmonics and at the same
time, controlling the fundamental component. The
validity of the estimated angles has been verified from
the obtained harmonic spectrum and THD, where a
complete elimination of the selected harmonics is
achieved. When the low order harmonics are
eliminated, only higher order harmonics will appear at
the output and need to be attenuated by designing a
suitable low pass filter to get nearly sinusoidal output.
Further work should focus on real time implementation
of SHE-PWM inverters.
Table 5: Fourier coefficients of the harmonics
based on the PS Method
Modulat
ion
Index
M = M=0.
0.5
6
M=0.
7
M=0.7
5
M=0.8
Desired
B1
0.5
0.6
0.7
0.75
0.8
Actual
B1
0.5
0.6
0.7
0.75
0.8
B3
0
2.22e
-016
0
0
4.44e016
B5
2.22e016
0
2.22e
-016
4.44e016
0
B7
3.33e016
5.55e
-016
3.33e
-016
1.665e
-016
2.22e016
7.77e016
9.403
2e007
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