Particle swarm optimizer: Economic dispatch with valve point effect using various PSO techniques
©2014
Textbook
61 Pages
Summary
Four modified versions of particle swarm optimizer (PSO) have been applied to the economic power dispatch with valvepoint effects. In order to obtain the optimal solution, traditional PSO search a new position around the current position. The proposed strategies which explore the vicinity of particle’s best position found so as far leads to a better result. In addition, to deal with the equality constraint of the economic dispatch problems, a simple mechanism is also devised that the difference of the demanded load and total generating power is evenly shared among units except the one reaching its generating limit. To show their capability, the proposed algorithms are applied to thirteen.Comparision among particle swarm optimization is given. The results show that the proposed algorithms indeed produce more optimal solutions in both cases.<br>The different PSO techniques are New PSO, SelfAdaptive PSO and Chaotic PSO Among the different PSO techniques, it is found that SelfAdaptive PSO is better than other PSO techniques in terms of better solutions, speed of convergence, time of execution and robustness but it has more premature convergence.
Excerpt
Table Of Contents
4
LIST OF TABLE
TABLE
NO.
TITLE
PAGE
NO.
5.2.(a)
Input data for IEEE 13 generator system
32
5.3.2.(a)
Result obtained by using PSO techniques
33
5.4.2.(a)
Result obtained by using APSO techniques
34
5.5.2.(a)
Result obtained by using CPSO techniques
36
5.6.2.(a)
Result obtained by using NPSO techniques
38
5.7.(a)
Comparison of all four PSO techniques
40
5
ABSTRACT
Four modified versions of particle swarm optimizer (PSO) have been applied to the economic
power dispatch with valvepoint effects. In order to obtain the optimal solution, traditional PSO
search a new position around the current position. The proposed strategies which explore the
vicinity of particle's best position found so as far leads to a better result. In addition, to deal with
the equality constraint of the economic dispatch problems, a simple mechanism is also devised that
the difference of the demanded load and total generating power is evenly shared among units except
the one reaching its generating limit. To show their capability, the proposed algorithms are applied
to thirteen.Comparision among particle swarm optimization is given. The results show that the
proposed algorithms indeed produce more optimal solutions in both cases.
The different PSO techniques are New PSO,Self Adaptive PSO and Chaotic PSO.Among the
different PSO techniques, it is found that SelfAdaptive PSO is better than other PSO techniques in
terms of better solutions, speed of convergence, time of execution and robustness but it has more
premature convergence.
7
CHAPTER1
1.1 Introduction
The main purpose of Economic Load Dispatch is to minimize the total generation cost of the plant
by considering the generator limits. In power generation fuel cost plays major role. Factors which
influence power generation at minimum cost are operating efficiencies of generator, fuel cost and
transmission losses.
Efficient generator in the system does not generate minimum cost as it may be located in an area
where fuel cost is high. If the plant is located far from the load centre, transmission losses may be
higher and the plant may be uneconomical.
The main aim is to identify the generation of different plants, such that the total operating cost is
minimum. The major component of generator operating cost is the fuel input/hour and the
maintenance cost contributes very less.
Total operating cost includes the fuel cost, cost of labour, maintenance. These costs are assumed to
be a fixed percentage of the fuel cost. After neglecting the transmission losses in economic load
dispatch we are considering only the generator units but not as the system.
We are neglecting the transmission line losses, line impedance etc., for analysis system is having
only one bus with all generations and load are connected. As there are no transmission losses, the
total load demand (Pd) is the sum of all generations.
8
1.2 Methodology in Brief
The coding of the algorithms was done on MATLAB 6.5, and the test system is the IEEE 13
generator system. Each algorithm was run for specified number of iterations and the best value
obtained was recorded, along with the graph for the average and minimum value against the number
of iterations.
The time of execution for all four algorithms were measured and recorded. Each algorithm was
executed ten times and the best and the worst value were found, the graph for these executions were
plotted.
The optimization techniques used are PSO, CPSO, NPSO and APSO, a random population is
initialized and the fitness value of each is calculated. This population is sent through a selection
process where the probability of the member of the population being selected into the matting
population is directly proportional to the previously measured fitness.
Velocity limits of the generators are initialized and it has been carried out by initializing the
generating velocities, besides those iterations are started and gbest values are found out by
continuously updating the population and also finding the fitness of the present population.
9
1.3 Organization of the Report
[1] Chapter 2 contains the basic theory on Economic Load Dispatch along with some elementary
mathematical background.
[2] In Chapter3 various available algorithms are discussed.
[3] Chapter4 explains in detail the algorithm used.
[4] Chapter5 and Chapter6 are result and conclusion.
10
CHAPTER2
2.1 Introduction to Economic Load Dispatch
Scarcity of energy resources, increasing power generation cost and evergrowing demand for
electric energy necessitates optimal economic dispatch in today's power systems. Optimal system
operation involves consideration of economy of operation, system security, emission at fossil fuel
plants, and optimal release of water at hydro generation.
Economic dispatch problem is to minimize the total cost of generating real power (production cost)
at various stations while satisfying the loads and losses in the transmission lines. In load flow
problems, two variables are specified at each bus and solutions is obtained for the variables.
In a practical power system, power plants are not loaded at the same distance from the center of
loads and their fuel cost is different. The generation capacity is more than the demand and losses.
So there is a need to schedule the generation. In an interconnected power system, the objective is to
find the real and reactive power scheduling of each power plant in such a way to minimize the
operation cost. The generators real and reactive powers are allowed to vary within certain limits to
meet a particular load demand with minimum fuel cost.
Electrical energy can not be stored, but is generated from natural sources and delivered as demand
arises. A transmission system is used for the delivery of bulk power over considerable distance and
a distribution system is used for local deliveries. An interconnected power system consists of
mainly three parts :
1. The generator, which produce electrical energy
2. The transmission line which transmit it to far away places
3. The load which use it
Such a configuration applies to all inter connected networks, where the number elements may vary.
The transmission networks are interconnected through ties so that utilities can exchange power,
share reserves and render assistance to one another in times of need. Since the sources of energy are
so diverse , the choice of one or the other is made on economic, technical or geographic basic. As
there are few facilities to store electric energy, the net production of utility must clearly track its
total load for an inter connected system, the fundamental problem is one of minimizing the source
11
expenses. The economic dispatch problem is to define the production level of each plant so that the
total cost of generation and transmission for a prescribed shecdule of loads.
· Forecasting includes determining the peak rate of supply i.e, energy demand for both long
term investment decisions and shortterm operating decisions.
· Operating applications include allocation of out put, unit startup selection, hydro thermal co
ordinations and maintenance scheduling.
· The investment planning applications cover the generation and transmission system.
2.1.1 Generator operating cost:
Factors which influence power generation at minimum cost are operating efficiencies of generator,
fuel cost and transmission losses.
Efficient generator in the system does not generate minimum cost as it may be located in an area
where fuel cost is high. If the plant is located far from the load centre, transmission losses may be
higher and the plant may be uneconomical.
The main aim is to identify the generation of different plants, such that the total operating cost is
minimum. The major component of generator operating cost is the fuel input/hour and the
maintenance cost contributes very less.
2.2 Mathematical Analysis
He we are considering only the generating units, but not as the system. We are neglecting the
transmission line losses, line impedance etc. for analysis, the system is having only one bus with all
generations and load are connected.
As there is no transmission losses, the total demand P
D
is the sum of all generation. For each plant
assume the cost function FC
i
Min FC
total
=
=
G
N
i
i
FC
1
 (1)
=
=
+
+
G
N
i
i
i
i
i
i
P
P
1
2
 (2)
12
Subject to the constraint,
D
N
i
i
P
P
G
=
=1
 (3)
The power output of any generator should not exceed the its rating nor should it be below that
necessary for stable turbine operation thus, the generations are restricted to lie within given
minimum and maximum limits. The problem is to find the real power generation for each plant such
that the objective function as defined by (2) is minimum, subject to the constrain given by (3) and
the inequality constraints given by,
max
min
i
i
i
P
P
P
where i = 1,2,3,...N
G
max
min
,
i
i
P
P
minimum and maximum generating limits
FCtotal total production cost
FCi production cost of ith plant
Pi power generation of ith plant
PD total load demand
NG total number of generating unit
Using Lagrange Multipliers,
)
(
1
=

+
=
G
N
i
i
D
total
P
P
FC
L
The minimum value will be obtained at the poing where the partials of the function to its variables
are zero.
0
=
i
P
L
0
=
L
0
)
1
0
(
=

+
=
i
total
i
P
FC
P
L
0
)
1
0
(
=

+
i
total
P
FC
G
N
total
FC
FC
FC
FC
...
...
2
1
+
+
=
i
i
i
total
dP
dFC
P
FC
=
13
Therefore optimal dispatch condition is,
=
i
i
dP
dFC
i
i
i
i
i
P
dP
dFC
2
+
=
=
i
i
i
P
2
+
0
1
=

=
=
G
N
i
i
D
P
P
L
D
N
i
i
P
P
G
=
=1
When losses are neglected, for most economic operation all plants must operate at equal
incremental production cost.
i
i
i
P
2

=
This is the coordination equation which is a function of
.So,
D
N
i
i
i
P
G
=

=1
2
=
=
+
=
G
G
N
i
i
N
i
i
i
D
P
1
1
2
1
2
The value of
has to be substituted in
i
i
i
P
2

=
,To obtain the optimal scheduling of generation.
To get the economical values of Pi, it has to undergo iterative process. Using gradient method, we
get the solutions quickly.
f
D
P
=
)
(
Expanding the left hand side in Taylors Series above an operating point
D
k
k
P
d
df
f
=
+
)
(
)
(
)
)
(
(
)
(
)
(
)
(
)
(
)
)
(
(
k
k
k
d
df
P
=
14

=
)
(
)
(
k
i
D
k
P
P
P
=
)
(
)
(
)
(
k
i
k
d
dP
P
=
=
G
N
i
i
k
k
P
1
)
(
)
(
2
1
)
(
)
(
)
1
(
k
k
k
+
=
+
Economic load dispatch problems can be solved theoretically by the following two methods they
are as follows:
1. Analytical method
2. Gradient method
2.2.1 Analytical method
In this method the is determined by solving the given parameters
Where,
=
=
+
=
G
G
N
i
i
N
i
i
i
D
P
1
1
2
1
2
, , Cost coefficients
i Index of the generator
P
D
total load demand
N
G
total number of generating units
2.2.2 Gradient method
In this method value is assumed (= 0 to 1)
2.3 Valve Point Loading
When the load demand increases the speed of the generator will decrease automatically. We know
that generator is coupled with prime mover (turbine), so to increase the speed of the generator the
15
valve point connected in the turbine is opened gradually, then the turbine starts rotating faster hence
generator gets back almost to the its original speed.
Valve point effect is added in the economic load dispatch problem to increase the accuracy of the
total fuel cost however the cost function of a generator is not always differentiable due to the valve
point effects and/or change of fuels. The valvepoint effects introduce ripples in the heatrate curve.
The fuel cost function with valvepoint loadings of the generators is usually modeled as
))
(
sin(
(min)
1
2
i
i
i
i
N
i
i
i
i
i
i
i
P
P
f
e
P
c
P
b
a
FC
G

+
+
+
=
=
Where, a
i
, b
i
, c
i
are fuel cost coefficients of generator i
e
i
and f
i
are fuel cost coefficients of generator i with valve point effect.
2.4 Problem Formulation
The economic dispatch problem is formulated as,
Min FC
total
=
=
G
N
i
i
FC
1
Subject to,
D
N
i
i
P
P
G
=
=1
max
min
i
i
i
P
P
P
Where i = 1, 2,3,...N
G
16
CHAPTER3
3.1 Evolutionary Algorithm
An evolutionary algorithm (EA) is the subset of evolutionary computation, a generic population
based metaheuristic optimization algorithm. An EA uses some mechanism inspired by biological
evolution: reproduction, mutation, recombination, natural selection and survival of the fittest.
Candidate solutions to the optimization problem play the role of individuals in a population, and the
cost function determines the environment within which the solution "live (see also fitness
function)". Evolution of the population then takes place after the repeated application of the above
operators. Artificial evolution (AE) describes a process involving individual evolutionary
algorithm; EAs are individual components that participate in artificial evolutions.
EAs consistently perform well approximating solutions to all types of problems because they don't
make any assumption about the underlying fitness landscape; this generality is show by successes in
fields as diverse as engineering, art, biology, economics, genetic, operations research, robotics,
social sciences, physics and chemistry. However, evolutionary algorithms can nonetheless the
outperformed by more field specific algorithm.
Apart from their use as mathematical optimizers, evolutionary computation and algorithms and
been used as an experimental frame work within which to validate theories about biological
evolution and natural selection, particularly through work in field of artificial life. Techniques from
evolutionary algorithm applied to the modeling of biological evolution are generally limited to
explorations of micro evolutionary processes. A limitation of evolutionary algorithms is their lack
of clear genotype phenotype distinction. In nature, the fertilized egg cell undergoes a complex
process know as embryogenesis to become a mature phenotype. This indirect encoding is believed
to make genetic search more robust (i.e. reduce the probability of fatal mutations), and also may
improve the resolvability of the organism. Recent work in the field of artificial embryogeny, or
artificial developmental systems, seeks to address these concerns.
Four evolutionary methods are used in this project they are as follows PSO, APSO, CPSO and
NPSO, these algorithms are discussed in detail.