pso based fractional order automatic generation controller for two area power system
TRANSCRIPT
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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 6545(Print),
ISSN 0976 6553(Online) Volume 5, Issue 9, September (2014), pp. 17-28 IAEME
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PSO BASED FRACTIONAL ORDER AUTOMATIC GENERATION
CONTROLLER FOR TWO AREA POWER SYSTEM
CH. Ravi Kumar Dr. P.V.Ramana Rao
Assistant Professor/E.E.E, Professor & H.O.D/E.E.E,
University College of Engg & Tech. University College of Engg & Tech.
Acharya Nagarjuna University Acharya Nagarjuna University
ABSTRACT
This paper presents the development and application of Fractional order PID controllers
based on particle swarm optimization (PSO) for load frequency control of two-area inter connected
system. The dynamic response of the system has been studied for 1% and 10% step loadperturbations in area2. The performance of the proposed FOPID controller is compared against the
traditional PID controllers based on PSO and ANFIS based intelligent controller. Comparative
analysis demonstrates that proposed FOPID controllers based on PSO reduces the settling time and
overshoot effectively, against small step load disturbances. Simulations have been performed using
MATLAB / Simulink.
Keywords:Adaptive Neuro Fuzzy Inference System (ANFIS), Fractional PID Controller, Particle
Swarm Optimization (PSO), Automatic Generation Control (AGC).
I. INTRODUCTION
In an interconnected power system Automatic Generation control or Load frequency controlis important in Electrical Power System design and operation. Large scale power system comprises
of interconnected subsystems (control areas) forming coherent groups of generators, where as
connection between the areas is made using tie-lines [1-2]. Each control area has its own generation
and is responsible for its own load and scheduled interchanges with neighbouring areas. The load in
a given power system is continuously changing and consequently system frequency deviates from
the desired normal values. Therefore to ensure the quality of power supply, a load frequency
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I A E M E
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Fig.1:Block diagram of AGC for Two area system with secondary loop
ii
j
i fBACE +P=n
1=
ijtie, (1)
Where ACEi is area control error of the ith
area
if = Frequency error of itharea
ijtie,P = Tie-line power flow error between ithand jtharea
Bi = frequency bias coefficient of ith
area
III. INTEGER ORDER PID CONTROLLER
The PID control is a widely used approach for designing a simple feedback control system
where in three constants are used to weigh the effect of the error (the P term), the integral of the error
(I term) and the derivative of the error(the D term). A typical structure of classical IOPID controlled
system [15] is shown in Fig.2.
Fig.2:Structure of PID control
P
+
sG2+1
1
2
1R
sT2+1
1
2D+s22
1
H
B2
w
ACE2
+
+
+
ACE1
+
+sG1+1
1
1D+s12
1
H
sT1+1
1
1
1
R
B1
S
T12
2
w
+
+
+
+
System
KP
Kd
Ki
d/dt
+
r(t)e(t) u(t) y(t)
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To implement a PID controller that meets the design specifications of the system under
control, the parameters [Kp, Ki, Kd, B1, B2] must be determined for the given system. An IOPID
controller is designed for frequency control in power system in this paper, whose parameters were
optimized using Particle swarm optimization.
IV. FRACTIONAL CALCULUS
Fractional calculus can have different definitions in different perspectives [15]. There are two
commonly used definitions for fractional calculus so far, that is Grunwald-Letnikov definition,
Reiman-Liouville definition.
]/)[(
0=0h
)()1+(
)+(
)h(
1=)( lim
hat
K
ta KhtfK
KtfD
(2)
t
nn
n
ta dt
f
dt
dtfD
)(1)(
)(
)-(n
1=)( (3)
Grunwald-Letnikov definition is perhaps the best known one due to its most suitability for the
realization of discrete control algorithms. The m order fractional derivative of continuous function
f(t) is given by [10]
( ) [ ]
( )
( )
dtm
tfdm
jhtfj
mx
j
jh
mLt
htfD
m
=
=
=
0)1(
0 (4)
Where [x] is a truncation andh
mtx )(= ;
jm is binomial coefficients
,!
)1+()1(=
j
jmmm
j
m
K0),=(1,= j
j
m
it can be Replaced by Gamma function,
1)+(j!
1)+(=
jm
m
j
m
. The general calculus operator including fractional order and integer order
is defined as [10]
t
a
ta
Rd
R
Rdtd
D
0)(/
>=
(5)
Where a and t are the limits related to operation of fractional differentiation, is the calculus order.
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The Laplace transform of the fractional derivative off(t)is given by
0=
1 )]([)(=)}({t
tfDsFstfDL
(6)
Where F(S)is the Laplace transform off(t). The Laplace transform of the fractional integral off(t)is
given as follows.
)(=)}({ sFstfDL (7)
V. FRACTIONAL PID (PID) CONTROLLER
The FOPID (or PID) controller involves an integrator of fractional order and a
differentiator of order , which has the following fractional order transfer function.
SK
S
KKsG
d
i
pc++=)( (8)
Where is the fractional order of the integrator and is the fractional order of the
differentiator, which both can take any value of complex numbers. The classical
controllers are particular cases of the FOPID controller. If ==1, the classical IOPID
controller is obtained. For ==0, the P controller is obtained, for =0, =1 the PD controller is
obtained. Illustration of different types of integer and fractional order controllers as and vary as
shown [15] in Fig3.
Fig.3:Illustration of IO and FO controllers
As the FOPID has two more extra tuning knobs than the classical integer-order PID
controller, the use of fractional controller (and are non-integers) gives more flexibility for the
design of a control system and gives better opportunity to adjust system dynamics if the originalsystem to be controlled is fractional system. In time domain input to the system to be controlled takes
the following form.
)(+)(+)(=)( teDKteDKteKtu dip (9)
Where is the integral order, is the differential orderKp, Ki, Kdare the parameters of PID
controller.
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A. Optimization of PID
Controllers parameters
Fractional controller has five parameters and currently there are no practical engineering
applications in frequency control of power system. Determine its parameters based on experience
like traditional controllers are not possible and so a fast and efficient way to optimize its parameters
must be found.
Particle swam optimization (PSO) is a kind of swarm intelligence algorithm, simulatingbiological predation phenomenon in nature. PSO employs swarm intelligence, which comes from
cooperation and competition between particles of a group, to guide the optimization search, with
strong convergence, global optimization and computation-efficiency [10, 15].
B. Calculation of fitness functionThe design of PI
D
controller is actually a multi dimensional function optimization problem.
The objective of controller parameters optimization is to make the control error tend to zero and
there is smaller over shoot and faster response. In order to obtain satisfactory transition of the
dynamic characteristic, the paper has used Integral squared error (ISE) performance index for the
parameters minimum objective function. ISE can be expressed as follows [15].
t
dtteISE0
2)(= (10)
C. Particle swarm optimization
Particle swarm optimization is a new population based evolutionary computation. The PSO
algorithm attempts to mimic the natural process of group communication of individual knowledge,
which occurs when such swarms flock, migrate; forage etc in order to achieve some optimum
property such as configuration or location.
In PSO the swarm is initialized with a population of random solutions. Each particle in the
swarm is a different possible set of the unknown parameters to be optimized. Representing a point in
the solution space, each particle adjusts its flying experience and shares social information among
particles. The goal is to efficiently search the solution space by swarming the particle towards thebest fitting solution encountered in previous iterations with the intent of encountering better solutions
through the course of process and eventually converging on a single minimum error solution [10].
In PSO, a swarm consists of N particles moving around in a D-dimensional search space. The
random velocity is assigned to each particle. Each particle modifies its flying based on its own and
companions experience at every iteration.
The formulas (11) and (12) are the particles velocity and position update formulas [11].
)(c+)(c+= 112211
11
1
t
id
t
gd
t
id
t
id
t
id
t
idxprxprvv (11)
+= 1tidt
id xxt
idv (12)
The ith
particle is denoted as ),(= 2,1 idiii xxxX K whose best previous solution Pbest is
represented as is ),(= 2,1 idiii pppP K current velocity (position change rate) is described
by ),(= 2,1 idiii vvvV K . Finally, the best solution achieved so far by the whole swarm is represented
as ),(= 2,1 gdiig pppP K .
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At each time step, each particle moves towards Pbestand gbestlocations. The fitness function
evaluates the performance of particle to determine whether the best fitting solution is achieved. is
the inertia weight factor, c1and c2are acceleration constant. 1rand 2r are random numbers between
zero and one. can be adjusted by the following formula (13),
tT
.=max
minmaxmax
(13)
max and min are maximum and minimum values of inertial weight coefficient, Tmaxis the maximum
of iterations, t is the current number of iterations.
D. Design of fractional order PID controller using psoWhen PI
D
controllers parameters are optimized using particle swarm optimization, the
five parameters of the fractional controller and B1,B2frequency bias coefficients of area1 and area2
can be viewed as a particle, that is K= [Kp, Ki, Kd,, , B1,B2].The seven members are assignedas real values. If there are n individual in a population, then the dimension of that population is n*7.
In this paper n is set as 10. In order to limit the evaluation value of each individual of the population,
feasible range must be set for each parameter as follows
Kp1max= Kp2max=1.5; Kp1min= Kp2min= 0; Ki1max= Ki2max= 1.5; Ki1min= Ki2min= 1;
Kd1max= Kd2max=1.5; Kd1min= Kd2min=0;
Kp1vmax=Kp1max/10; Ki1vmax=Ki1max/10; Kd1vmax=Kd1max/10;
Kp1vmin=Kp1min; Ki1vmin=Ki1min; Kd1vmin=Kd1min;
1max= 2max=2; 1min= 2min=0; 1max= 2max=2; 1min= 2min=0;
1vmax= 1max/10; 1vmin= 1min; 1vmax= 1max/10; 1vmin= 1min;
Kp2vmax=Kp2max/10; Ki2vmax=Ki2max/10; Kd2vmax=Kd2max/10;
Kp2vmin=Kp2min; Ki2vmin=Ki2min; Kd2vmin=Kd2min;
B1max= B2max=35; B1min= B2min=15; B1vmax=B1max/35; B1vmin=B1min;
B2vmax=B2max/25; B2vmin=B2min; C1=2, C2=2.Now the design steps are as follows [11-14]:
1. Randomly initialize the individuals of the population including position and velocities in the
feasible range.
2. For each individual of the population, calculate the values of the performance criterion in
(10).
3. Compare each individuals evaluation value with its personal best P id. The best evaluation
value among all Pidis denoted as Pg.
4. Modify the member velocity of each individual according to (11) where the value of is setby equation (13).
5. Modify the member position of each individual according to (12).
6.
If the number of iterations reaches the maximum, then go to step 7 otherwise go to step2.7. The latest Pg is the optimal controllers parameters.
In this study optimal parameters of fractional controller for 1% and 10% step load perturbations are:
B1=17.4191,B2=25, Kp1=0.2819, Ki1=1.5, Kd1=1.2618,1=1.1869,1=0.5524.
Kp2=1.3213, Ki2=1.5, Kd2=1.1813,2=0.8336,2=1.0770.
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VI. RESULTS AND DISCUSSIONS
In the present work Automatic Generation Control of two area interconnected power system
has been developed using PSO based IOPID Control, ANFIS controller and PSO based FOPID
controller using Matlab/Simulink package. Figs 4 to 9 respectively represent the plots of change in
system frequency for 1% and 10% step load variations in area1. The results obtained are also givenin Tables 3 and 4.
Case I: For 1% Step load Perturbation
0 2 4 6 8 10 12 14 16 18 20-3
-2.5
-2
-1.5
-1
-0.5
0
0.5x 10
-4
Time in Seconds
Deviationinfrequency(p.u.)
Change in frequency w ith PSO based IOPID control for 1% step load perturbation
Del f1
Del f2
Fig 4: Frequency deviations f1, f2withPSObased IOPID control
0 2 4 6 8 10 12 14 16 18 20-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1x 10
-3 Change in frequency with ANFIS Control for 1% step load perturbation
Time in Seconds
Dev
iationinfrequency(p.u.)
Del f1
Del f2
Fig 5:Frequency deviations f1, f2withANFIScontrol
0 2 4 6 8 10 12 14 16 18 20-5
-4
-3
-2
-1
0
1
2x 10
-4 Change in f requency w ith PSO based FOPID Controller f or 1% s tep load perturbation
Time in seconds
Deviationinfrequency(p.u.)
Del f1
Del f2
Fig 6:Frequency deviations f1, f2withPSObased FOPID controller
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Case II: For 10% Step load Perturbation
0 2 4 6 8 10 12 14 16 18 20-3
-2.5
-2
-1.5
-1
-0.5
0
0.5x 10
-3
Time in Seconds
Deviaioninfrequency(p.u.)
Change in f requency w ith PSO based IOPID Controller for 10% step load per turbation
Del f1
Del f2
Fig 7: Frequency deviations f1, f2withPSObased IOPID Control
0 2 4 6 8 10 12 14 16 18 20-7
-6
-5
-4
-3
-2
-1
0
1x 10
-3 Change in frequency with ANFIS Controller for 10% step load perturbation
Time in Seconds
Deviationinfrequency(p.u.)
Del f1
Del f2
Fig 8:Frequency deviations f1, f2withANFIScontrol
0 2 4 6 8 10 12 14 16 18 20-5
-4
-3
-2
-1
0
1x 10
-3
Time in Seconds
Deviationinfrequency(p.u.)
Change in f requency w ith PSO based FOPID controller for 10% step load perturbation
Del f1
Del f2
Fig 9:Frequency deviations f1, f2withPSObased FOPID controller
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than the classical IOPID controller, gives more flexibility for the design of a control system and
gives better opportunity to adjust system dynamics. Simulation results shows that the proposed
Fractional controller has better dynamic performance than the Integer order controller and ANFIS
controller with faster response and smaller overshoot.
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BIOGRAPHY
Ch.Ravi Kumar was born in India in 1981; He received the B.Tech degree in
Electrical and Electronics Engineering from A.S.R.College of Engineering and
Technology, Tanuku in 2003 and M.Tech degree from JNTU Anantapur, A.P.-India
in 2005. Currently he is pursuing Ph.D in Electrical Engineering and working asAsst.Professor in University college of Engineering and Technology, Acharya
Nagarjuna University, Andhra Pradesh India. His areas of Interest are Power system
operation and control, Application of Intelligent control techniques to Power systems.
P.V.Ramana Rao was born in India in 1946; He received the B.Tech degree in
Electrical and Electronics Engineering from IIT Madras, India in 1967 and M.Tech
degree from IIT Kharagpur, India in 1969. He received Ph.D from R.E.C Warangal
in 1980. Total teaching experience 41 years at NIT Warangal out of which 12 years
as Professor of Electrical Department. Currently Professor of Electrical Department
in University college of Engineering and Technology, Acharya Nagarjuna
University, Andhra Pradesh, India. His fields of interests are Power system operationand control, Power System Stability, HVDC and FACTS, Power System Protection, Application of
DSP techniques and Application of Intelligent control techniques to Power systems.