The Online Journal on Electronics and Electrical Engineering (OJEEE)
Vol. (3) – No. (1)
Particle Swarm Algorithm-Based Self-Excited
Induction Generator Steady State Analysis
Ahmed. E. Kalas*, Medhat. H. Elfar*, Soliman. M. Sharaf **
*Electrical Engineering Dept., Faculty of Engineering, Port-Said, Suez Canal University, Egypt
**Electrical Engineering Dept., Faculty of Engineering, Helwan University, Egypt
Abstract-In this paper, Novel method, Particle Swarm
Optimization (PSO) algorithm, based technique is proposed to
estimate and analyze the steady state performance of self-excited
induction generator (SEIG). The steady state analysis of SEIG is
formulated as a multidimensional optimization problem. In this
novel method the tedious job of deriving the complex coefficients
of a polynomial equation and solving it, as in previous methods,
is not required. The steady state open loop behavior of SEIG
driven by a regulated prime mover is formulated in a simple and
straightforward way without mathematical manipulation. By
comparing the simulation results obtained by the proposed
method with those obtained by the conventional methods, a good
agreement between these results is obtained. The comparison
validates the effectiveness of this proposed technique.
Key Words- SEIG, PSO, steady state analysis
I.
INTRODUCTION
The self-excited induction generators (SEIG) have been
found suitable for energy conversion for remote locations.
Self-excited induction generators (SEIG) are frequently
considered as the most economical solution for powering
costumers isolated from the utility grid. SEIG has many
advantages such as simple construction, absence of DC power
supply for excitation, reduced maintenance cost, good over
speed capability, and self short-circuit protection capability.
Unlike induction generators connected to the power utility
grid, both frequency and voltage are not fixed but depend on
many factors, such as generator parameters, excitation
capacitor, speed, and load. This makes the SEIG steady state
analysis is more difficult. Major drawbacks of SEIG are
reactive power consumption, its relatively poor voltage and
frequency regulation under varying prime mover speed,
excitation capacitor and load characteristics [1], [2].
Over the past 25 years, many researchers have analyzed the
steady state performance of the three-phase SEIG on the basis
of its approximate equivalent circuit. Two different solution
methods have been used to determine the steady-state
analysis of the three-phase SEIG driven by constant-speed
prime mover. The loop impedance approach is the first
method [3]-[9]. Using loop impedance approach, for a given
load and speed, two simultaneous nonlinear equation are
derived in terms of frequency ( F ) and magnetizing reactance
( X m ) of the generator by separating the real and imaginary
parts of the loop impedance of the equivalent circuit. The
equations are then solved for F and X m . Usually, the
Newton-Raphson method is used to solve the simultaneous
nonlinear equations, which requires much mathematical effort
to derive the elements of the Jacobean matrix. The second
method is the nodal admittance approach [10]-[14], in this
Reference Number: W10-0070
method the equivalent admittance of the circuit at the air gap
point is separated into real and imaginary parts. The real part
of the admittance is independent of X m and usually expressed
by a high order polynomial of F [12]. However, the
imaginary part of the admittance is a nonlinear function of
both F and X m . This method requires the combined
admittance of the load, capacitive reactance and stator
leakage reactance to be evaluated, which will result in lengthy
algebraic manipulations. For complex R-L loads, the degree
of the polynomial is seven.
Both methods have a common disadvantage that detailed
algebraic derivations for the coefficients of the equations are
required. In addition, the earlier approaches are not flexible as
the coefficients are valid only for a given circuit
configuration. Inclusion of core loss resistance or load
inductance will increase the order of the equations. Chan [10]
simplified the algebraic derivations in the nodal admittance
approach by introducing an iterative technique to find the
value of F from real part of admittance. Chan [14] employed
a symbolic programming technique to reduce the manual
derivations of the equations. However these techniques still
require some degree of manual manipulation of the
mathematical equations before these techniques are applied.
Recently, many researchers have proposed different
techniques to tackle these problems based on user-friendly
computer software that have been developed for numericalbased solution of a set of nonlinear equations[15-19].
Different from conventional optimization method, particle
swarm optimization (PSO), introduced by Kennedy and
Eberhart in 1995 [20-22], uses a simple mechanism that
mimics swarm behaviour in birds flocking and fish schooling
to guide the particles to search for globally optimal solutions.
This paper proposes a novel method for evaluating the
SEIG steady state characteristics using PSO algorithm that
does not require the detailed derivation of nonlinear
equations.
II. STEADY STATE MODELING OF SEIG
Most of the steady state models of SEIG developed by
different researchers are based on per phase equivalent circuit.
These models use the following two basic methods; i) Loop
impedance method and ii) Nodal admittance method. The
steady state model based on nodal admittance method is
presented here. In the analysis that follows, the following
assumptions are made:
-
The core loss in the machine is neglected.
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The Online Journal on Electronics and Electrical Engineering (OJEEE)
-
All the machine parameters in the equivalent circuit are
assumed to be constant except the magnetizing reactance
which is assumed to be affected by the magnetic
saturation.
The per-phase equivalent circuit of a three-phase SEIG
with a R-L load and excitation capacitor is shown in Fig. 1,
where R1 , X 1 , R2 , X 2 and X m represent the stator resistance,
stator leakage reactance, rotor resistance, rotor leakage
reactance, and magnetizing reactance respectively, RL , X L ,
and X c represent the load resistance, load reactance, and
excitation capacitor reactance, respectively and F and
v represent the per unit(p.u.) frequency and speed,
respectively. The reactances are specified at a base or rated
frequency. The circuit is normalized to the p.u. frequency by
dividing all parameters and voltages by the p.u. frequency F .
Using nodal analysis, the circuit can be represented by three
parallel admittances Y1 , Ym and Yr , where
Y1 
(Yc  YL )Ys
Yc  YL  Ys
Yc 
Ys 
2
jF
Xc
1
R1 / F  jX1
Ym 
YL 
1
jX m
Yr 
1
RL / F  jX L
1
( R2 / F  v)  jX 2
optimization technique has been successfully used in many
research area such as function optimization, fuzzy system
control, ANN training etc and has become a new and hot spot
of research in the world. The following is a brief introduction
to the operation of the PSO algorithm. Consider a swarm of
particles. Each particle represents a potential solution and has
a position in the problem space represented by a position
vector xi. A swarm of particles moves through the problem
space with the moving velocity of each particle represented
by a velocity vector vi. At each time step, a fitness function f
representing a quality measure is calculated by using xi as
input. Each particle keeps track of its individual best position,
xpbest, which is associated with the best fitness it has
achieved so far. Furthermore, the best position among all the
particles obtained so far in the swarm is kept track of as
xgbest. This information is shared by all particles. The PSO
algorithm is implemented in the following iterative procedure
to search for the optimal solution.
(i)
(ii)
(iii)
(iv)
The nodal equation for node “a” is found to be
E1 (Y1  Ym  Yr )  0
(1)
Under normal operating condition, the self-excitation, E1  0 .
Thus
(2)
(Y1  Ym  Yr )  0
R1
F
IL
VL
F
RL
F
jX
 jX
F
L
jX 1
jX
a
2
jX
E
F
m
(v)
Initialize a population of particles with random positions
and velocities of N dimensions in the problem space.
Define a fitness measure function to evaluate the
performance of each particle.
Compare each particle’s present position xi with its
xpbest based on the fitness evaluation. If the current
position xi is better than xpbest, then set xpbest = xi.
If xpbest is updated, then compare each particle’s xpbest
with the swarm best position xgbest based on the fitness
evaluation. If xpbest is better than xgbest, then set xgbest
= xpbest.
At iteration k, a new velocity for each particle is updated
by
vi(k+1)=wvi(k)+c1r1(xpbest(k)-xi(k))+c2r2(xgbest(k)-xi(k))
(3)
(vi) For each particle, change its position according to the
following equation.
2
I2
I1
c
Vol. (3) – No. (1)
R2
F v
xi(k+1) = xi(k) + vi(k+1)
(4)
(vii) Repeat steps (iii)-(vi) until a criterion, usually a
sufficiently good fitness or a maximum number of
Yc
Ym
YL
Yr
iterations is achieved. The final value of xgbest is
Figure (1): Per-phase equivalent circuit of a three-phase SEIG.
regarded as the optimal solution of the problem.




III. OVERVIEW OF PSO
Particle swarm optimization technique was first proposed
by Kennedy and Eberhart in 1995 [20]. PSO is motivated
from the simulation of the behavior of social systems such as
fish schooling and birds flocking. The PSO has been found to
be fast in solving nonlinear, non-differentiable, multimodal
optimization problems. Comparing with genetic algorithm,
PSO’s advantages lie on its easy implementation and few
parameters to adjust, also the PSO algorithm requires less
computation time and less memory. Particle swarm
Reference Number: W10-0070
In (3), c1 and c2 are positive constants representing the
weighting of the acceleration terms that guide each particle
toward the individual best and the swarm best positions
xpbest and xgbest, respectively, r1 and r2 are uniformly
distributed random numbers in [0, 1]; w is a positive inertia
weight developed to provide better control between
exploration and exploitation; N is the number of particles in
the swarm. The velocity vi is limited to the range [-vmax,
vmax]. If the velocity violates this limit, it is set to the
relevant upper or low-bound value.
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The Online Journal on Electronics and Electrical Engineering (OJEEE)
Constant speed operation
If the induction generator is driven by constant speed prime
mover to supply constant load impedance Z L with fixed value
of X c , the equation(2) is solved to find the values of
unknown values F and X m . Once the values of F and
X m are known, the performance of the generator (voltage,
current, and power at various points of the circuit) can easily
be determined. In most of the previous methods of analysis,
the determination of F and X m is tedious and timeconsuming task. In this study, equation(2) is solved as
optimization problem by using PSO technique. The value of
total admittance is considered as fitness function and the
problem space is two dimension. The various characterisitic
of the geneerator can be obtained from its equivalent circuit
but that requires to run PSO for different possible values of a
particular parameter. The no-load charactersitics is the
variation of no-load voltage against the excitation capacior C
for a constant speed v , this characteristic can easily be
generated from run PSO for various values of X c at noload( Z L   ).
Knowing F and X m is the first step to get these
characteristic, hence we will focus on the comparison
between the values calculated using PSO and those calculated
by the previous methods. The machine parameters used are
indicated in appendix I [19]. The results found for excitation
capacitor varies from 22 to 44 microfarad at constant speed of
1500 r.p.m are shown in Fig. 2. The solid line represents PSO
results and dashed line represnets the previous methods
results, the figure indicates that PSO results are very close to
the corresponding previous methods values. Fig. 2(a) shows
the variations of magnetizing reactance against the excitation
Reference Number: W10-0070
140
130
Magnetizing reactance,Xm
120
110
100
90
80
70
60
20
25
30
35
Excitation capacitor,micro-F
40
45
40
45
(a)
50
49.95
49.9
Frequency,F
To ensure the phenomenon of self-excitation, equation (2)
must be satisfied. By investigating equation(2), the all
parameters of induction generator, except the magnetizing
reactance, are considered as constant. The magnetizing
reactance X m is assumed to be a variable and depends on
magnetic saturation. The other parameters as X c , F , v and
Z L are considered as adjustable parameters in the circuit, thus
equation (2) is nonlinear equation having five variables
( X m , X c , F , v , Z L ). However, based on the generator
operation condition (Constant speed-constant frequencyvariable speed), some of these variable parameters can be
considered as fixed parameters. To determine the unknown
variables, the previous methods separate the complex
equation(2) to two nonlinear scalar equations and these two
equations are rearranged to be expressed in unknown terms,
this requires step-by-step algebraic manipulations to do this
task. The procedure of obtaining the unknown parameters for
constant speed operation by using PSO is described in the
next section.
capacitor. The variations of the frequency against excitation
capacitor is shown in Fig. 2(b). Moreover, the no-load
terminale voltage of the generator for various values of
excitation capacitor is shown in Fig. 2(c). The percentage
errors of X m , F and VL calculated using PSO and the
previous methods is shown in Fig. 3, the error introduced by
using the PSO with respect to the previous methods reaches
up 4% which is acceptable value.
49.85
49.8
49.75
49.7
20
25
30
35
Excitation capacitor,micro-F
(b)
260
240
220
No-load terminal voltage,V
IV. PROBLEM FORMULATION
Vol. (3) – No. (1)
200
180
160
140
120
100
80
60
20
25
30
35
Excitation capacitor,micro-F
40
45
(c)
Figure (2): 2 No-load characteristics of IG. (-.) PSO results;
(--) previous methods results
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The Online Journal on Electronics and Electrical Engineering (OJEEE)
5
Vol. (3) – No. (1)
50
4
49.5
3
49
Frequency,F
2
Error-%
1
0
48.5
48
-1
47.5
-2
47
-3
-4
46.5
-5
20
25
30
35
Excitation capacitor,micro-F
40
0
200
400
45
600
Load power-W
800
1000
1200
(b)
120
Figure (3): The relative error (--)Voltage; (-.)Frequency;
(-*) Magnetizing reactance
115
magnetizing reactance,Xm
110
The load characteristic is the variation of terminal voltage
against the generator output power and such a characteristic
can again generated from applying PSO for various values of
Z L . In this case, X c and v are considered as fixed parameters.
Fig. 4 shows the comprison of PSO and previous methods
results for excitation capacitor (40μF) and speed of 1pu. The
figure indicates that PSO results are very close to the
corresponding previous methods values. Fig. 4(a) shows the
variations of the terminal voltage versus the load power, it
can be observed from this figure that this characteristic is
divided to two regions, the stable region and unstable region.
As Z L is decreased the load power increases while the load
voltage decreases, until the load power reaches its maximum
value that region represents the stable region. If Z L is
decreased more, both load power and load voltage decrease
that represents the unstable region. The variations of the
frequency against the load power is shown in Fig. 4(b), where
the magnetizing reactance variation versus the load power is
shown in Fig. 4(c).
105
100
95
90
85
80
75
0
200
400
600
Load power
800
1000
1200
(c)
Figure (4): The load characteristics of IG. (-.) PSO results;
(--) previous methods results
It can be observed from Fig. 5 that the maximum errors of
X m and F and VL are within the tolerance level(±5%).
5
4
260
3
2
240
Error-%
Terminal voltage,Vt
1
220
0
-1
200
-2
180
-3
-4
160
-5
140
0
200
400
600
Load power-W
(a)
Reference Number: W10-0070
800
1000
1200
0
100
200
300
400
500 600
Load power
700
800
900
1000 1100
Figure (5): The relative error. (--)Voltage; (-.)Frequency;
(-*) Magnetizing reactance
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The Online Journal on Electronics and Electrical Engineering (OJEEE)
V. CONCLUSION
In this paper, novel method (PSO) is proposed to evaluate
the performance characteristics of a SEIG as optimization
problem, instead of step by step analytical derivation of
several equations as in the previous methods. PSO, a modern
and efficient optimization technique, is used to obtain the
steady state characteristics of SEIG. The PSO simulation
results are compared with the previous methods results,
closeness between the results prove the validity of PSO
technique. The maximum relative error between PSO and
previous method is 4% for no-load characteristics and 2% for
the load characteristics that is acceptable relative error.
[9]
[10]
[11]
Appendix I
[12]
Induction machine data [19]
3-phase, 4-pole, 50Hz,delta connected, squirrel cage
induction motor, 1.5-kW, 220V, the parameters: R1  5.033 ,
X 1  5.605 , R2  4.667 , and X 2  5.605 .
Air gap voltage: the variation of air gap voltage with
magnetizing reactance at rated frequency for the induction
machine is as given:
[13]
[14]
E
 596.03  12.035 X m  0.1374 X m2  5.636  10  4 X m3
F
[15]
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