Download Optimal Switching in a Three-Level Inverter: an

2016 ICSEE International Conference on the Science of Electrical Engineering
Optimal Switching in a Three-Level Inverter: an
Analytical Approach
Almog Salman, Yair Cohen, Lior Simchon, Tirza Routtenberg, Raul Rabinovici
Department of Electrical and Computer Engineering
Ben-Gurion University of the Negev
Beer-Sheva 84105, Israel
Abstract—This paper addresses the problem of optimal pulse
width modulation (PWM) voltage switching signals that minimizes the total harmonic distortion (THD) for a three-level
inverter. Previous works have focused on the derivation of ad-hoc
methods that achieve low voltage THD values by using frequency
domain analysis and brute-force search. The use of optimization
techniques in power systems, especially in THD minimization,
can improve system performance and reduce computational
complexity, thus providing a manageable method with claims to
optimality. In this paper, an optimization method, the Lagrange
multipliers technique, is used to solve the THD minimization
problem. For a given modulation index it is shown that the
minimization of the voltage THD results in a single-switch
solution. Finally, the performance of the proposed optimal singleswitch solution is examined in simulations. The proposed method
reduces computational complexity, compared to the brute-force
methods, and increases solution accuracy by using analytical
THD time domain expressions.
Index Terms—Multilevel inverter, pulse width modulation
(PWM), total harmonic distortion (THD), harmonics, optimization, Lagrange multipliers.
and analyzed for low switching frequencies, i.e. for each
quarter of the square wave there are three switches. Then,
the THD expression is formulated in the time domain and
the problem is described in the context of an optimization
framework. Estimation and detection theory have been shown
to be powerful tools in smart grid analysis (see e.g. [7], [8],
[9], and [10]). Similarly, this work demonstrates the use of
modern optimization tools in classical power system problems.
The optimization is conducted by using Lagrange multipliers
and results in a counter-intuitive solution; in this case, less
is more. That is, single switching is the optimal THD minimization solution. Finally, the performance of the proposed
optimal single-switch solution is examined in simulations. It
is shown that the proposed method reduces the computational
complexity, compared to brute-force methods, and increases
solution accuracy.
I. I NTRODUCTION
Multilevel inverters (MLI) play an important role in industrial power applications. In recent decades, MLI applications
have grown rapidly and modulation methods have gained
prominence in MLI research [1]. The main advantage of MLIs
is that their output voltage can be generated with low harmonics. Thus, it is found that the harmonics decrease proportionately to the inverter level. Modulation patterns of inverters
are generally characterized by performance measures, such
as the commonly used total harmonic distortion (THD) [2]
[3], the peak common-mode voltage (CMV) [4], the harmonic
distortion factor (HDF) [5], and specific harmonic-amplitude
restrictions [6]. MLI power efficiency is affected by switching
losses, which are directly related to the switching frequency.
High efficiency can be obtained by using a low switching
frequency. However, reducing the switching frequency affects
the harmonic content and increases THD values. Therefore,
both the inverter voltage level and the switching frequency
should be selected carefully in order to achieve low THD and
high efficiency. Optimal selection requires an accurate THD
analysis of inverter pulse width modulation (PWM) output.
THD minimization techniques for MLI have been discussed
extensively in the literature and are usually based on off-line
algorithms and the use of lookup tables. The well known
Selective Harmonic Elimination (SHE) method [11] reduces
the THD by removing low-order harmonics. The SHE method
requires the solution of a set of nonlinear equations. Therefore,
SHE solutions are generally obtained by using numerical
approaches. The number of the canceled harmonics is a
function of the number of switching angles. This technique
demands offline calculations that are stored in a lookup table.
The synchronous optimal pulse width modulation (SOPWM)
technique [12] operates with low switching frequencies and
is usually used in motor systems. Similar to the approach
outlined in this paper, the SOPWM method is obtained by
using optimization methods. However, the SOPWM method
approximates the objective function as a finite harmonic sum,
which results in a less accurate solution that requires higher
computational complexity. Another method is the Criteria
Based Modulation (CBM) method [13] that offers a direct
search for any given objective function. The CBM method
performs an offline search by mapping all possible switching
permutations. In each of these methods, a switching set is
stored in a lookup table for each modulation index.
A. Summary of results
In this paper we consider the problem of THD minimization
in a three-level inverter. First, the THD problem is described
c
978-1-5090-2152-9/16/$31.00 2016
IEEE
B. Related works
The rest of the paper is organized as follows. In Section 2
2016 ICSEE International Conference on the Science of Electrical Engineering
the THD problem is formulated in the time domain, followed
by an optimization solution, which proposed in Section 3.
Simulations are conducted in the MATLAB environment and
discussed in Section 4. Conclusions are presented in Section
5.
II. THD F ORMULATION
In this section we present a mathematical analysis of an
MLI output voltage. The output voltage and the related THD
expression are formulated in the time domain as a function of
the switching angles. A general MLI output voltage signal is
represented by a stepped waveform with N voltage levels and
M switches. In the following we discuss a three-level inverter,
i.e. N = 3, with three switching angles per quarter wave, i.e.
M = 12. Therefore, the MLI output voltage is given by:

θ ∈ R1
 VDC
−VDC
θ ∈ R2 ,
(1)
f (θ, α) =

0
otherwise
where
R1 = [α1 , α2 ] ∪ [α3 , π − α3 ] ∪ [π − α2 , π − α1 ],
R2 = [π + α1 , π + α2 ] ∪ [π + α3 , 2π − α3 ] ∪ [2π − α2 , 2π − α3 ],
and α = [α1 , α2 , α3 ]T , in which α1 , α2 , α3 are the switching
angles. The inverter input DC voltage level is denoted by
VDC . It should be noted that according to these definitions
the switching angles should satisfy:
π
0 ≤ α1 ≤ α2 ≤ α3 ≤ .
(2)
2
where the associated Fourier coefficients [15] are given by:

P3
4
VDC m=1 (−1)m+1 cos(nαm )
odd n
 πn
(5)
Vn =

0
even n
for any n ∈ N. The output voltage THD describes the ratio
between the total harmonic components and the fundamental
desired harmonic, V1 . The THD is defined by [16]:
T HD(α) =
sP
∞
2
n=2 Vn
2
V1
=
sP
∞
2
n=1 Vn
2
V1
−1 .
In practice, evaluating the THD values from (6) is performed
by using a truncated THD expression with a limited number
of harmonics. Accurate THD values can be obtained by using
the time domain THD expression. In particular, the Parseval
identity implies:
∞
X
Vn2
+
Wn2
n=1
1
=
π
Z
2π
|f (θ, α)|2 dθ
.
The output voltage, f (θ, α), from (1) is presented in Fig. 1.
This signal is 2π periodic, and it can be seen that the signal is
an odd function that has a quarter-wave symmetry. This signal
can be expanded by Fourier series [14] as follows:
f (θ, α) =
∞
X
Wn cos(nt) + Vn sin(nt) .
(3)
n=1
The cosine coefficients, Wn , n = 1, . . . , ∞, are eliminated
due to the odd function characteristics and the even sine
coefficients, Vn , are eliminated due to quarter-wave symmetry
[11]. Therefore, the Fourier series representation of the output
voltage signal is given by:
f (θ, α) =
∞
X
n=1,3,5,...
Vn sin(nt) ,
(4)
(7)
0
By substituting (4) into (7), it can be verified that [17]
∞
X
Vn2 =
n=1,3...
1
2π
Z
2π
|f (θ, α)|2 dθ
0
(8)
4 2
= VDC
(α2 − α1 − α3 + π) .
π
Substitution of (8) into (6) results in
s
4 2
π VDC (α2 − α1 − α3 + π)
T HD(α) =
−1 .
V12
Fig. 1. Typical output voltage for a three-level inverter, as a function of
α1 , α2 , α3 .
(6)
(9)
The modulation index describes the ratio between the inverter
input DC voltage and the fundamental voltage amplitude, and
is defined as
m(α) =
V1
.
VDC
(10)
By substituting (10) in (9), one obtains the THD analytical
expression
T HD(α) =
s
4(α2 − α1 − α3 ) + 2π
−1 .
πm2 (α)
(11)
Usually, the fundamental harmonic amplitude, V1 , is based on
the system requirements, i.e. a given value of the modulation
index, m(α). By calculating V1 using (5) with n = 1 and
substituting into (10), the modulation index result is
m(α) =
4
(cos(α1 ) − cos(α2 ) + cos(α3 )) .
π
(12)
2016 ICSEE International Conference on the Science of Electrical Engineering
III. THD O PTIMIZATION
for any n = 1, 2, 3. Differentiating (12) and (16) we get:
The THD factor from (11) is a nonlinear function of the
switching angles, where the switching angles should satisfy
the condition in (2) for a given modulation index m∗ . Under
this condition, this minimization problem can be formulated
into a nonlinear optimization problem, where the objective
function is the THD, as follows:
∂g(α)
= 2(−1)n ,
and
∂αn
∂m(α)
4
= (−1)n sin(αn ) .
∂αn
π
min T HD(α)
α

 0 ≤ α1 ≤ α2 ≤ α3 ≤
such that

m(α) = m∗
π
2
.
(13)
According to (11), the THD expression, which is the objective
function in (13), is a positive and monotonic function. Therefore, the optimization problem from (13) can be replaced by
the following equivalent optimization problem:
min FT HD (α) = T HD2 (α) + 1
α

 0 ≤ α1 ≤ α2 ≤ α3 ≤ π2
,
such that

m(α) = m∗
(14)
where FT HD (α) is the modified objective function and m∗
is the desired modulation index. Constrained optimization
problems can be solved by using the Lagrange multipliers
method [18]. For the optimization problem from (14) the
Lagrangian is defined as
−L(α, λ) = FT HD (α) − λ(m(α) − m∗ )
2g(α)
− λ(m(α) − m∗ ) ,
=
πm2 (α)
(15)
where
g(α) = 2(α2 − α1 − α3 ) + π .
(16)
In order to minimize the Lagrangian, the partial derivatives
of L(α, λ) w.r.t. α1 , α2 , α3 , and λ should be equated to zero.
That is,
(20)
By using the constraint m(α) = m∗ and substituting (20) in
(19) one obtains
4
3
n+1
sin(αn )
g(α) + λm∗ − m∗ = 0 . (21)
2(−1)
π
It can be seen that (21) implies that
πm∗
sin(αn ) =
n = 1, 2, 3.
4g(α) + λm3∗
(22)
Since the sine function is an injective function within [0, π2 ],
(22) implies that the optimal solution is obtained where all
three switching angles are equal, i.e. α1 = α2 = α3 . That is,
a single switching angle provides the minimal THD value.
Since all switching angles are equal, using α = αn for
any n=1,2,3 will provide a single switch expression. Finally,
the THD expression is obtained for a single switch from (11):
r
π 2 − 2πα
T HD(α) =
−1 .
(23)
8 cos2 α
Similarly to (23), the modulation index expression is obtained
for a single switch from (12):
4
cos α .
(24)
π
The single switch α is given for any fixed modulation index
m(α) =
4
(25)
α = cos−1 ( m∗ ) .
π
This section implies that for any given modulation index the
switching angles for which the voltage output THD is minimal
are all equal and can be directly calculated based on (25).
IV. S IMULATION
In this section we examine and discuss the performance of
the proposed analytical method solution. All simulations were
performed by using the MATLAB software with a Lenovo
∂L(α, λ)
(17) G50 Intel i5 CPU with 4GB RAM. The SHE and CBM
= m(α) − m∗ = 0
∂λ
methods were implemented with three switching angles per
and
quarter wave, where the SHE solution was obtained using
the fmincon.m MATLAB function. The CBM method was
∂L(α, λ)
=
implemented for low and high resolution of the switching
∂αn
angle steps. In Fig. 2, the THD of the different methods is
∂g(α)
∂m(α)
(18) presented versus the modulation index. It can be seen that the
∂m(α)
2 ∂αn m(α) − 2g(α) ∂αn
−λ
=
π
m3 (α)
∂αn
SHE method provides the highest THD values; the behavior
of the CBM method can be attributed to the discrete search
=0,
approach, with its inherent resolution errors.
for n = 1, 2, 3. It can be verified that Equation (18) can be Simulation runtime was analyzed for each method and is
rewritten as
presented in Table 1. In order to achieve lower THD via the
SHE or CBM methods, higher resolution is required. How∂m(α)
π ∂m(α) 3
∂g(α)
m(α) − 2g(α)
−λ
m (α) = 0 , (19) ever, this increases problem complexity, resulting in longer
∂α
∂α
2 ∂α
n
n
n
2016 ICSEE International Conference on the Science of Electrical Engineering
runtime, which precludes real-time applications or requires
larger lookup table sizes. Since the proposed optimal switch
solution has a significantly lower runtime it does not require
any lookup table. Thus, the proposed approach is suitable for
real-time applications.
Fig. 2. The THD for the proposed optimal solution, CBM with low/high
resolution and SHE method versus modulation index.
Method
Grid res.[rad]
Runtime[sec]
No. of values in table
SHE
0.01
0.001
5.1
41.7
92
902
3
1
0.2
0.011
0.28
27.4
4960
125580
15288900
CBM
proposed solution has been developed for theoretical models
and is valid for pure resistor loads. One of the known uses of
inverters is the electrical motor drive, which is characterized
by an RL circuit. Therefore, future work to minimize THD
for RL loads is required and can be done as an extension to
the presented method.
ACKNOWLEDGMENT
TABLE I
RUNTIME TABLE FOR THE
Fig. 3. THD mapping with fixed modulation index m=0.9 and feasible
α1 , α2 .α3 is obtained by (10)
SHE AND CBM.
In Fig. 3. the following scenario was implemented: the modulation index was predefined to a specific value, α1 and α2 were
mapped, and α3 was evaluated from (12). Valid switching sets
were selected from (2). The obtained THD value is presented
versus the mapped switching angles, for each value of α1 and
α2 , and presented in Fig. 3. It can be seen that the minimal
THD values are obtained for α1 = α2 , which is equivalent to
the proposed solution of a single switch α3 . This simulation
validates the proposed analytical solution presented in this
paper, and shows that the optimal solution can be calculated
analytically for any given modulation index.
V. C ONCLUSION
In this paper, we discuss the THD problem for MLI and develop an analytical approach based on optimization methods.
This method has proven more efficient for the THD problem
since it provides a global minimum THD value and since
switching angles are calculated online and no lookup tables
are used. Optimized analytical THD has two advantages: first,
it is a simple analytical continuous expression that has realtime applications; second, a single switching signal provides
the lowest switching losses in a simple hardware design. The
This research was partially supported by THE ISRAEL
SCIENCE FOUNDATION (grant No. 1173/16)
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