VARYING DIMENSIONAL PARTICLE SWARM OPTIMIZERS
FOR DESIGN OF SWITCHING SIGNALS
Toshimichi Saito
1
and Kengo Kawamurae
2
1
Faculty of Science and Engineering, Hosei University, 184-8584 Tokyo, Japan
2
Hitachi Appliances, Inc., 424-0926 Shizuoka, Japan
Keywords:
Particle swarm optimizers, Swarm intelligence, Power electronics.
Abstract:
This paper presents varying dimensional particle swarm optimizers for design of switching signals in circuits
and systems. The particle position and dimension correspond to the switching phase and the number of
switches, respectively. The dimension can vary depending on an objective function in the search process, i.e.,
the number of switches is adjustable automatically. The algorithm is defined for a multi-objective problem
described by the hybrid fitness consisting of analog functions and digital logic. The algorithm is defined in a
general form and the performance is investigated in an example: design of a switching signal for single phase
inverters with a two-objective problem corresponding to total harmonic distortion and power sufficiency.
1 INTRODUCTION
The particle swarm optimizer (PSO) is a populatiob-
based optimization algorithm (Engelbrecht, 2005).
The PSO is simple in concept, is easy to implement
and is applicable to a variety of systems: image/signal
processing (Wachowiak et al., 2004), artificial neural
networks (Garro et al., 2009), power electronics (Ono
and Saito, 2009), etc. The PSO has been improved in
order to challenge various problems: multi-objective
problems, multiple solutions, escape from a trap of
local optima, variable swarm topology, etc. (En-
gelbrecht, 2005) (Parsopoulos and Vrahatis, 2004)
(Miyagawa and Saito, 2009).
This paper presents a varying dimensional parti-
cle swarm optimizer (VDPSO) for application to de-
sign of switching signals in circuits and systems. The
switching signals, which determine the on/off timing,
are characterized by the switching phases. Design of
such switching signals is a key in a variety of circuits
and systems: digital communications (Maggio et al.,
2001), switching power converters (Giral et al., 1999)
(Sundareswaran et al., 2007), etc. In the VDPSO, the
particle positions and their dimension correspond to
the switching phases and the number of switches, re-
spectively. The goal of the VDPSO is optimizing the
objective function of the switching phases to realize
a desired circuit operation. The particle dimension
is adjustable automatically in the search process fol-
lowing the objective function: two switching phases
can equalize (dimension reduction) and can separate
(dimension recovery) if some conditions are fulfilled.
Such dimension control is important because of sev-
eral reasons including 1) The number of switchings
affects performance of circuits and systems, however,
the optimal number is unknown in many cases; 2) In
optimization problems, the objective solution is of-
ten restricted in a lower dimensional subspace in the
whole search space, however, automatic identification
of the target subspace is hard. The VDPSO is defined
in multi-objective problems (MOP) described by the
hybrid fitness consisting of analog objective functions
with criterion and digital logic (Ono and Saito, 2009).
Some fitness component can increase below the crite-
rion and this increase can help escape from a trapping
solution.
After definition of the general form, the VDPSO
performance is investigated in a two-objective prob-
lem of basic dc/ac inverters whose solution is known.
The two-objective hybrid fitness function represents
the total harmonic distortion and power sufficiency
of the output waveform. Performing numerical ex-
periments, we can confirm that the dimension varies
suitably, the flexible search is realized and the parti-
cles approach to the solution. Note that we have used
the basic problem whose solution is known because
such a problem is convenient to investigate/evaluate
the algorithm performance: this paper aims at pro-
posal of a prototype of the VDPSO and investigation
of the algorithm performance. The results provide ba-
259
Saito T. and Kawamurae K..
VARYING DIMENSIONAL PARTICLE SWARM OPTIMIZERS FOR DESIGN OF SWITCHING SIGNALS.
DOI: 10.5220/0003621502590262
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 259-262
ISBN: 978-989-8425-83-6
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

sic information to improve/establish the VDPSO and
to realize effective applications in various circuits and
systems. Preliminary results along these lines can be
found in (Kawamura and Saito, 2010).
2 VARYING DIMENSIONAL PSO
Here we define the VDPSO for an optimization prob-
lem of switching signals y(t) with period T:
y(t) =

0 for a
l−1
≤ t < a
l
1 for a
l
≤ t < a
l+1
y(t+T) = y(t) (1)
where l ∈ {1,3,·· · ,N}, a
0
≡ 0 < a
1
≤ · ·· ≤a
N
<
a
N+1
≡ T and a
i
denotes the switching phase as
shown in Fig. 1 (a). y = 1 and y = 0 correspond
to switch-on and -off, respectively. Let an objective
problem be described by a set of functions of the
switching phases a
i
:
F
j
(~a) ≥ 0, ~a ≡ (a
1
,·· · , a
N
), j = 1 ∼ N
f
(2)
where the minimum values are normalized as zero.
Let the desired circuit operation corresponds to the
minimum value of F
i
for all i. The cases N
f
= 1 and
N
f
≥ 2 correspond to the uni–objective problems and
MOP, respectively. Since it is hard to find the exact
minimum value of F
i
for all i, we try to find an ap-
proximate solution~a
s
satisfying
0 ≤ F
i
(~a
s
) < C
i
, i = 1 ∼ N
f
(3)
where C
i
is the criterion of the i-th component. Note
that the problem has a margin [0,C
1
).
Let a swarm contain N
p
pieces of N-dimensional
particles. The i-th particle at a discrete time n is char-
acterized by its position~a
i
(n) ≡ (a
i1
,·· · , a
iN
) and ve-
locity ~v
i
(n) ≡ (v
i1
,·· · , v
iN
) where i = 1 ∼ N
p
is the
index of the particles. The positions correspond to
the switching phases ~a ≡ (a
1
,·· · , a
N
) and the veloc-
ity controls their movement. Each particle tries to ap-
proach a solution using two key pieces of information:
the personal best particle~a
p
i
(pbest
i
) that has the best
evaluation in the past history, and the global best par-
ticle ~a
g
(gbest) that is the best of the pbest
i
for all i.
The~a
g
is the occasional solution at time n. The search
space is the one period of time axis and is divided into
N subintervals as illustrated in Fig. 1 (b):
I
k
≡ [(k − 1)d,(k + 1)d), d = T/N, k = 1 ∼ N (4)
Note that two successive subintervals overlap I
k
∪
I
k+1
= [kd,(k + 1)d]. This overlapping plays an im-
portant role in the dimension control in the algorithm.
The algorithm is defined in the following 6 steps.
Step 1. Let n = 0. As illustrated in Fig. 1: the k-th el-
ement of the ith particle position is assigned randomly
)||(
21
δ<−
ii
aa
)(
34 ii
aa <
)(ty
t
⋅⋅⋅⋅⋅⋅
1
a
N
a
0
1
2
a
3
a
4
a
δ
T
d
(a)
0
1
t
)(ty
21 ii
aa =
43 ii
aa =
T
(c)
0
1
t
1i
a
2i
a
4i
a
3i
a
1
I
2
I
)(ty
T
(b)
Figure 1: Particles assignment and dimension control.
in I
k
: a
ik
∈ I
k
, k = 1 ∼ N. Note that if I
k
consists of
N
l
lattice points then the possible number of~a
i
is N
N
l
.
The brute force search becomes hard as N increases.
Other variables are also initialized: velocity~v
i
(n) =
~
0,
personal best~a
p
i
= ~a
i
(n) and global best~a
g
=~a
1
(n).
Step 2. If Eq. (5) is satisfied for all i then the algo-
rithm is terminated, otherwise go to Step 3.
0 ≤ F
i
(~a
g
) < C
i
, i = 1 ∼ N
f
(5)
Step 3 (Renewal of velocity and position).
~v
i
(n+ 1) = w
i
~v
i
(n) + ρ
1
(~a
p
i
−~a
i
(n)) + ρ
2
(~a
g
−~a
i
(n))
~a
i
(n+ 1) =~a
i
(n) +~v
i
(n+ 1), i = 1 ∼ N
(6)
where w, ρ
1
and ρ
2
are deterministic parameters, not
random parameters as standard PSOs. In order to
avoid overflow, speeding and stagnation; we apply
If a
ik
(n) /∈ I
k
then a
ik
(n) = RND(I
k
)
If v
i
(n) /∈ [−V
L
,V
L
] then v
ik
(n) = RND([−V
L
,V
L
])
If |v
i
(n)| < ε then v
ik
(n) = qv
ik
(n)
where q, ε and V
L
are control parameters. RND(I
k
)
means a random number on the k-th subinterval I
k
.
Step 4 (Dimension Control). As illustrated in Fig.
1 (b) and (c), the particle position is equalized if two
successive particles are sufficiently close or the order
of particles are inversed:
a
ik
= a
i(k+1)
=
a
ik
+ a
i(k+1)
2
if
|a
ik
− a
i(k+1)
| ≤ δ or a
i(k+1)
< a
ik
(7)
where k = 1 ∼ M − 1. This equalization a
ik
= a
i(k+1)
means dimension reduction of ~a in principle. If Eq.
(7) is not satisfied in the future, then a
ik
can be sep-
arated from a
i(k+1)
and the dimension can be recov-
ered in Steps 3 and 4. If ”the maximum dimension” N
is large then the particle can express very wide-band
switching signals.
Step 5 (Hybrid Fitness). The ith personal best,
i = 1 ∼ N
p
, is renewed if some fitness component(s)
ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications
260

is improved and other component(s) satisfies the cri-
terion:
~a
p
i
= ~a
i
(n+ 1) if either (A) or (B) is satisfied
(A) C
j
< F
j
(~a
i
(n+ 1)) < F
j
(~a
p
i
) for all j = 1 ∼ N
f
(B) F
j
(~a
i
(n+ 1)) < C
j
for some j and
C
k
< F
k
(~a
i
(n+ 1)) < F
k
(~a
p
i
) for k 6= j
(8)
Since Eq. (8) is constructed by analog function F
i
and
digital logic, we refer it to as hybrid fitness. The gbest
is renewed as the best of all the personal bests.
Step 6. n = n + 1, go to Step 2 and repeat until the
maximum time limit n
max
.
Note that the dimension control in Step 4 can find the
suitble number (dimension) of switchings automati-
cally. Note also that some fitness F
j
can increase be-
low the criterion C
j
before Eq. (5) is fulfilled. This
flexibility can help to search for a suitable solution.
Existing algorithms often use a weighted sum of the
objectives for the MOPs, however, it is difficult to get
the best weight values (Engelbrecht, 2005).
3 NUMERICAL EXPERIMENTS
In order to investigate the basic performance, we con-
sider the two-objective problem for switching signal
of the dc/ac inverter. Figure 2 shows an output wave-
form y(t). Since this is odd-symmetric, it is sufficient
to consider in the first quarter:
y(t) =

0 for a
l−1
≤ t < a
l
1 for a
l
≤ t < a
l+1
(9)
where the period is normalized as 2π, a
0
≡ 0 < a
1
≤
··· ≤ a
N
s
< a
N
s
+1
≡ π/2, l ∈ {1,3, · ·· , N
s
} and N
s
is
the number of the switches. Let~a ≡ (a
1
,··· ,a
N
s
). Let
us define two positive definite objective functions.
F
1
(~a) = 1−
b
2
1
2P(~a)
, F
2
(~a) =




1−
P(~a)
P
d




(10)
y(t) =
∞
∑
m
b
m
sinmt, P(~a) =
1
2π
Z
2π
0
y(t)
2
dt.
where b
m
is the Fourier sine coefficient, 0 < P(~a) < 1
is the normalized average power of y(t) and 0 < P
d
<
1 is a desired average power. F
1
relates to the total har-
monic distortion (THD) and F
1
= 0 means pure sinu-
soidal waveform. F
2
describes the power sufficiency
and F
2
(~a) = 0 gives the desired power. If both F
1
and
F
2
are minimized, we can obtain the output having de-
sired power with low distortion. It is suitable for con-
tinuous control of the ac power. Substituting N = N
s
and N
f
= 2 into the algorithm, we can implement the
)(ty
1
a
2
a
3
a
t
0
1
π
2
π
π2
)(ty
1
a
2
a
3
a
t
0
1
π
2
π
π2
Figure 2: PWM control signal of dc/ac inverters.
2π
4π
)(ty
t
0
1
)(ty
t
0
1
100
50
0
n
4π
2π
(a)
(b)
(c)
2π
4π
)(ty
t
0
1
)(ty
t
0
1
100
50
0
n
4π
2π
(a)
(b)
(c)
Figure 3: Search process for P
d
= 0.7, C
1
= 8 × 10
−2
and
C
2
= 8 × 10
−3
. (a) initial waveform at n = 0, (b) posi-
tion transition of the gbest particle, (c) waveform of the
criterion attainment at n = 112: (F
1
, F
2
) = (7.99 × 10
−2
,
7.40× 10
−3
).
VDPSO for this problem. The goal is described by
F
1
(~a
g
) ≤ C
1
and F
2
(~a
g
) ≤ C
2
. If P(~a) in F
2
is given,
the optimal solution is a
1
= ··· = a
N
s
= π(1− P(~a)/2.
That is, the optimal dimension is one. We investigate
how the algorithm reduces the dimension automati-
cally. We note again that our purpose is investigation
of basic performance of the algorithm and the search
process. For the numerical experiments, we select
C
1
, C
2
and P
d
as control parameters and the other 10
parameters are fixed after trial-and-errors: N
p
= 20,
N
s
= 17, w = 0.8, ρ
1
= ρ
2
= 2, V
L
= 0.2, ε = 10
−15
,
q = 0.1, δ = 0.01 and n
max
= 400. Fig. 3 shows a typ-
ical result: as n increases, the number of the switching
phases decreases by the dimension control in Step 4
and the criterion is attained. Fig. 3 (b) shows posi-
tion transition of the gbest particle where the number
of the switches changes from 20 to 1 by repeating di-
mension control. Fig. 4 shows the search process. As
n increases, F
2
decreases and reaches the criterion C
2
rapidly. Below the criterion C
2
, the F
2
can increase
and this increase helps decrease of F
1
and attainment
of the criterion C
1
. That is, the VDPSO can adjust
the number of switches automatically and can give the
desired solution. Table 1 summarizes results of 50 tri-
als for various parameter values and initial conditions.
We have used the three measures: SR (The successful
VARYING DIMENSIONAL PARTICLE SWARM OPTIMIZERS FOR DESIGN OF SWITCHING SIGNALS
261

)(a
0
0.06
0.12
0.18
0
0.16
0.32
0.48
0.64
0
50
100
150
1
F
2
F
1
C
2
C
n
1
F
2
F
6.0
2
C
1
C
2
F
1
F
2.0
2
10
−
1
0=n
6.0
2
C
1
C
2
F
1
F
2.0
2
10
−
1
10=n
)(b
6.0
2
C
1
C
2
F
1
F
2.0
2
10
−
1
60=n
6.0
2
C
1
C
2
F
1
F
2.0
2
10
−
1
112=n
Figure 4: Search process for C
1
= 8 × 10
−2
and C
2
= 8 ×
10
−3
. (a) Time evolution of the two fitness functions. (b)
Behavior of particles. The red point is the global best.
rate of the criterion attainment within the time limit),
#ITL (The average number of iterations to attain the
criterion for successful run) and #SW (The average
number of switches at the criterion attainment in suc-
cessful runs).
The SR is good around P
d
= 0.7 and decreases as
P
d
is apart from it. Note that #SW=2.2 for p
d
= 0.7
and C
1
= 0.08 means that the criterion can be at-
tained even in the case of 3 switchings. As C
1
de-
creases for P
d
= 0.7, #SW approaches 1. As C
1
in-
creases, the #ITE and #SW tend to increase: the cri-
teria are attained before the #SW is optimized. These
results suggest that the VDPSO is efficient if the de-
sired power P
d
is in some range, however, several
improvements are required for finding solution in a
wider range of P
d
. We have also confirmed that the
desired operation is hard to be given if the dimension
control or/and hybrid fitness is not used.
4 CONCLUSIONS
The VDPSO has been studied and its performances
have been investigated in a simple example. It is
confirmed that the dimension control can work effec-
Table 1: Basic performances (N
s
= 17, C
2
= 8× 10
−3
).
P
d
C
1
SR #ITE #SW
0.9 0.08 n/a n/a n/a
0.12 74 157 1.0
0.15 100 46 4.6
0.19 100 37 5.9
0.7 0.08 100 97 2.2
0.12 100 65 5.4
0.15 100 52 7.4
0.19 100 43 9.6
0.5 0.15 n/a n/a n/a
0.19 92 175 1.0
tively together with the hybrid fitness function. How-
ever, this paper provides a first step to develop an
efficient optimization algorithm with dimension con-
trol. Future problems are many, including analysis
of search process, role of key algorithm parameters,
comparison with other kinds of PSOs and application
to switching signals in various circuits and systems.
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