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

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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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