a comparison of the algorithms used in electricity power systems
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electricityTRANSCRIPT
A comparison of the algorithms used in electricity power systems
By Zikai LUStudent ID:9644350
PS5University Language Centre
Group 3027/08/2015
Word count:2230
TABLE OF CONTENTS1. Introduction..................................................................................................2
2. The introduction of three algorithms............................................................3
2.1Genetic algorithm (GA)..................................................................3
2.2Differential algorithm (DE)..............................................................4
2.3Particle swarm optimization (PSO)................................................5
3. Formulation of the economic dispatch problem...........................................6
3.1Transmission line losses................................................................7
3.2 valve-point effects.........................................................................8
3.3 multiple fuels.................................................................................9
4. The process of solving economic dispatch problem based on difference
algorithm.........................................................................................................10
5. Case study: Comparison of the best simulation results of DE, GA, PSO. .11
6. Conclusions...............................................................................................12
1. IntroductionAlong with the development of human economic society, electricity becomes
the most essential aspect of energy consumption. In China, primary energy,
such as coal, oil, natural gas, hydro energy and nuclear energy, is often
transformed into electricity by power plants, so the non-renewable resource
consumption is huge. What’s more, according to the statistics, electricity
demand grows rapidly. ‘China ‘s generation capacity reached 847GW at the
end of 2009,and it is expected to reach 1186GW in 2020,to become the
largest power production country in the world.’[1] Therefore, It is important to
implement the load distribution and improve the efficiency of the operation to
reduce the production cost.
An electricity power system is a complex system, which is composed of a
power plant, transmission system, distribution system and power load. The
primary aim of electricity power generation is to plan the use of the existing
energy and equipment so as to meet the required load demand at a minimum
running cost while satisfying the requirements of safety and power quality. In
recent years, many experts and scholars at home and abroad have devoted
themselves to the study of economic dispatch in electricity power systems,
trying to find a better solution to obtain more economic benefits.
The economic dispatch of an electricity power system is characterized by high
dimension, nonlinear and non-differentiable,so it is complex to calculate. So
with more and more research on the algorithm, various algorithms have been
2
applied to solve the problem of economic dispatch in power systems.
This paper, first of all, will introduce three different algorithms, the genetic
algorithm, the differential algorithm and the particle swarm optimization
algorithm. Secondly, this paper will discuss the theory of the economic
dispatch and introduce the formulation of economic dispatch problem and
present factors that will affect the solving of the economic dispatch problem.
Then the paper will introduce the process of solving the economic dispatch
problem based on difference algorithm. Finally, this paper will compare the
best simulation results of the genetic algorithm, the differential algorithm and
the particle swarm optimization algorithm.
2. The introduction of three algorithms
2.1Genetic algorithm (GA)
The genetic algorithm was first proposed in 1975 by Professor Holland of the
University of Michigan in America, which is based on biological natural
selection and genetic optimization. The genetic algorithm is highly parallel
arithmetic, randomized and adaptive, which simulates the nature of life and
intelligence generation and the evolution process, according to the principle of
"natural selection and survival of the fittest". The genetic algorithm imitates
Mendel's theory of genetic variation to find a better solution to solve the
problem that a survival of the fittest chromosome during the iterative process,
including the selection, crossover and mutation operation, eventually
converge to the best adaptive value of the environment and obtain the optimal
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solution to the problem. When the genetic algorithm is applied to the
optimization system, the solution space is mapped into the genetic space and
each possible solution is called a chromosome or individual, which is
expressed by the encoding. The genetic algorithm is a general optimization
algorithm, where the encoding technology and the genetic operation is
relatively simple. In addition, the genetic algorithm is considered to be a
robust method because no restrictions on the solution space are made during
the process. [2] The genetic algorithm can search in the whole solution space
and the convergence performance is good, which can converge to the global
optimal solution. In view of the above advantages, the genetic algorithm has
been widely used in the optimization of complex systems.
2.2Differential algorithm (DE)
The differential algorithm is a kind of intelligent algorithm to solve complex
optimization problems, of which the basic idea is firstly using the disturbance
of the differences between the individuals of the current population to produce
a new intermediate population, then through restructure and selection
producing a new population, and finally after many iterations finding the best
individual. [3] The classical differential evolution algorithm uses real encoding,
so it can be more effective to solve the real problem. The differential evolution
algorithm includes mutation, crossover and selection operation, and its
biggest characteristic is mutation operation. The mutation operation is the
main evolutionary process of the differential algorithm, which uses the
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difference vector of different individuals in the population to disturb the target
vector so as to achieve individual mutation. The disturbance is random and
independent, which makes the differential algorithm have inherent parallelism.
Mutation operation more effectively uses the population distribution, which
improves the searching performance of the algorithm. But unlike the selection
operation of the genetic algorithm, the differential algorithm uses the one-to-
one selection mechanism. In a word, the differential algorithm can be easily
used to calculate the power system economic load distribution of the power
system and it is also simple in structure, easy to operate and has good
optimization ability.
2.3Particle swarm optimization (PSO)
Kennedy and Eberhart proposed the particle swarm optimization algorithm in
1995, which is based on the experience of the individual such as found in fish
schooling and bird flocking. [4] The particle imitates birds foraging process
and the particles are randomly distributed in the solution space of a problem
or a function. The particle swarm optimization algorithm uses the current
location of the particle to assess the function. In the process of iterative
optimization, the particle can determine the moving path in the search space
by four factors. These factors are: firstly, the current position of the particle,
secondly, their historical best position, thirdly, in a group of one or more of the
best particle position and the last is random disturbance. Only when all the
particles of the previous generation move their position can they complete a
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whole iteration. During the process of optimization, the whole group mutually
cooperates and moves toward the best point of the fitness function. So the
particle swarm optimization algorithm is based on the individual's cooperation
and competition to complete the search of the optimal solution in the complex
search space, which is a kind of evolutionary computation technology based
on swarm intelligence. The particle swarm optimization algorithm has the
advantages of parallel processing and good robustness. Its biggest advantage
is easy to achieve and has fast convergence. As a member of the modern
heuristic algorithm, in recent years, the application of the particle swarm
optimization algorithm has gradually shown its advantages and broad
application prospects in the power system.
3. Formulation of the economic dispatch problem
The optimization problem of economic dispatch in an electricity power system
is actually a problem of using a specific algorithm to distribute each
generating unit in the power system, and get the optimal solution to meet the
load requirements of the power system. Therefore, whether the economic
dispatch of a power system can achieve high efficiency, depends on the
selection of scheduling the algorithm. If the optimal algorithm can achieve the
efficiency of scheduling, the load distribution is the best solution to solve the
problem. The objective of the economic dispatch problem is to minimize the
total fuel cost of each generating unit subjected to the operating constraints of
a power system. In general, the formulation of the economic dispatch problem
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can be modeled as:
min F=∑i=1
N
F i ( pi ) (1)
Traditionally, the fuel cost curve is approximated using a simple smooth
quadratic function [4] given by:
F i ( pi )=ai p i2+bi p i+ci
(2)
Where F is the total generation cost ($/hr.), N is the number of dispatchable
units, pi is the generation output of dispatchable unit i (MW), and F i ( pi )
is the
fuel cost function of dispatchable unit i ($/hr.) and a i, b i and c i are the fuel-cost
coefficients of generator i.
Subjected to
∑i=1
n
pi=p l+D (3)
pimin≤ p i≤ pimax (4)
Where pl is the total active power demand, D is the total active loss of
transmission, piminis the minimum operating limit of unit i, pimaxis the maximum
operating limit of unit i.
There are many factors regarding the economic dispatch problem of power
systems, such as valve-point effects, transmission line losses, multiple fuel
options and so on. These factors will have an influence on the solving of the
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load distribution problem. The following will present these factors.
3.1Transmission line losses
The network loss is also called transmission line losses, which refers to a part
of the energy in the process of conveying electricity energy in the form of heat
emission, resulting in transmission loss in the transmission process. When the
load density is high and the transmission distance is short, the transmission
line losses can be ignored. However, when the load density is low and the
power transmission distance is long, the transmission line losses can reach
20% to 30% of the total load. This is the important factor that transmission line
losses affect the load distribution. The B matrix loss formula is usually used to
calculate transmission line losses. The mathematical model of transmission
line losses is as follows:
PL=∑i=1
N
∑j=1
N
P iBijP j+∑i=1
N
B i0 Pi+¿B00¿ (5)
PL= [Pg1 Pg2 ⋯ Pgn ] [B11 B12 ⋯ B1nB21 B22 … B2n⋮ ⋮ ⋱ ⋮Bn1 Bn2 ⋯ Bnn
][Pg1
Pg2
⋮Pgn
]+[Pg1 Pg2 … Pgn ][B10B20⋮Bn0
]+B00 (6)
Where Bij, Bi0, B00 are transmission line loss coefficients.
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3.2 valve-point effects
In practical work, the active power of the generator is gradually increased
from the smallest to the maximum value, the generating unit consumption
curve is increased by the ups and downs and is not a linear increase. The
generating unit ideal curve becomes a pulsating rose shape, as shown in
Figure 3-1. When each steam valve starts to open, the valve-point effect
results in ripples in the fuel cost curve. Neglecting the valve point effect has a
certain effect on the accuracy of the solution. So to think about the valve-point
effects, sinusoidal functions can be added to the quadratic fuel cost function
as follows:
F i ( pi )=ai p i2+bi p i+ci+|g isin (hi ( pi−pi
min ))| (7)
Where giand hi are coefficient of the generating units reflecting valve-point
loading effects.
Figure 3-1
3.3 multiple fuels
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In addition, the actual problem of economic load distribution should not only
consider the valve-point effect of optimization problem, but also consider the
different generating units supplied with multiple fuels which lead to the
different results. It requires choosing reasonable fuel for different generating
units and reducing the use of expensive and rare resources, ultimately
achieving the goal of generating minimum cost. Different from the traditional
cost function, the cost function of generating units supplied with multiple fuels
should be represented as a few piecewise functions “reflecting the effects of
fuel type changes and each segment of the hybrid cost function implies some
information about the type of fuel being burned or the operational
characteristics of the unit”[5]. The piecewise quadratic function with the valve-
point loading effect is described as.
F i ( pi )={ai1 pi12 +bi1 pi1+c i1+|gi1sin (hi1 ( p i1
min−p i1) )|, fuel1, pimin≤ pi≤ pi1
a i2 pi22 +bi2 pi2+c i2+|gi 2sin (hi2 ( pi2min−pi2) )|, fuel2 , pi1≤ pi≤ pi2
⋮
a¿ p¿2+b¿ p¿+c¿+|g¿sin (h¿ ( p¿
min−p¿) )|, fueln , p¿−1≤ pi≤ pimax
(8)
Where a¿, b¿, c¿, g¿and h¿are cost coefficient of generating unit i.
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Figure 3-2
4. The process of solving economic dispatch problem based on difference
algorithm
Step1: Input parameters of generating units and the inequality constraints set
the parameters of the differential algorithm, which include: population size,
mutation factor, crossover rate, the maximum number of iterations and
termination condition.
Step2: To initialize the population, each individual in the group is a solution to
the optimization problem of economic dispatch, and each individual is
composed of a set of decision variables, which represent the output of
generating units during the process of economic dispatch.
pij=pij,min+rand ( p ij,max−p ij,min) (1)
Step3: According to the mathematical model of economic dispatch with
multiple fuels and valve-point loading effects to evaluate the objective function
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value of each individual in the initial population, evaluate its objective function
(fitness) value and then record the optimal individual value.
Step4: mutation operation: each individual vector in the population has been
mutated, and there has been generating variation.
vik+1=xr1
k +F ( xr 2k −xr3k ) (2)
Step5: Crossover operation: crossover operation can increase the diversity of
the perturbed parameter vectors.
ui , jk +1={v i , j
k +1 , rand lij≤CR 或 j=rand (i)x i , jk , rand lij>CR且 j ≠ rand (i)
(3)
Step6: Selection operation: to satisfy the various constraints, calculating the
degree of adaptation in individuals, for each individual selection operation.
x ik+1={x ik ,∧if ƒ ( xik )≤ƒ (uik+1 )
u ik+1 , else
(4)
Step7: Determine the number of iterations to see whether the maximum
number of iterations has been reached, if not, then jump to step4. If achieved,
then stop and output the optimal solution in the population.
5. Case study: Comparison of the best simulation results of DE, GA, PSO
This case study contains 10 generating units with both multiple fuels and a
valve-point loading effect [4]. The total load demand is 2700 MW not
considering transmission line losses.
Tab.5-1 Comparison of results of DE with the GA and the PSO
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Methods GA [4] PSO [4] DE[6]
Output power (MW)
Fuel type
PG (MW) Fuel type
PG (MW) Fuel type
PG (MW)
1 2 216.2974 2 225.5729 2 220.08
2 1 211.6352 1 208.2240 1 211.66
3 1 281.2672 1 278.8078 2 382.67
4 3 241.5130 3 238.0062 3 239.28
5 1 264.4552 1 282.4136 1 279.77
6 3 242.3118 3 239.6464 3 240.33
7 1 284.1604 1 285.4269 1 287.69
8 3 240.5808 3 239.1045 3 239.28
9 3 438.6755 3 425.5856 3 426.24
10 3 280.2956 1 277.2121 1 272.48
Total power (MW)
2700
Total cost ($/h)
624.5050 624.5074 623.93
6. Conclusions
From the comparisons of the results obtained by the DE algorithm, the GA
algorithm and the PSO algorithm, the results show that the DE algorithm has
higher efficiency and robustness to solve the economic load distribution of
multi fuels and it can effectively improve the energy utilization rate and
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economic benefit of the power system.
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References
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