ISSN (Print) : 2320 3765
ISSN (Online): 2278 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 3, Issue 7, July 2014
DOI: 10.15662/ijareeie.2014.0307061
Economic Dispatch of Power System
Optimization with Power Generation Schedule
Using Evolutionary Technique
Girish Kumar1 , Rameshwar singh2
PG Student [Control system] , Dept. of EE, NITM. Engineering College, Gwalior, M.P, India 1
Assistant professor, Dept. of EE, NITM Engineering College, Gwalior, M.P, India 2
ABSTRACT: This paper presents an algorithm for solving optimal power flow problem through the application of
Differential Evolution (DE). Reduction in fuel costs done of power generation by proper load dispatch schedule. So the
overall costing of operation of power system can be reduced. The differential evolution algorithm is an evolutionary
algorithm that uses a rather greedy and less stochastic method than do classical evolutionary algorithms such as particle
swarm optimization (PSO) .Differential Evolution also incorporates an efficient way of self-adapting mutation using
small population. The effectiveness of algorithm has been tested for a practical economic dispatch problem on a test
system having three generating units. Thermal power generating firms needs to minimize fuel cost, for better profit at
the same time it should satisfy system load demand, real, reactive power limit, voltage limit, power transmission limit
and other limitations. For generation cost minimization Economic Load Dispatch (ELD) was developed. When cost is
the single objective, the power generation may pollute the environment. Thermal electric power could not be generated
without pollution but this pollution can be reduced for the sake of good and healthy atmospheric condition. To reduce
emission the firm has to spent money, which is burden and reduces the power generating firms profit. These two
opposite objectives should be compromised to find optimal operating condition of thermal power plants which may
yield minimum fuel cost. Differential Evolution (DE) algorithm is used to solve this complex problem.
KEYWORDS: Optimization, Economic load dispatch, Differential Evolution, PSO.
I.INTRODUCTION
Power utilities try to achieve high operating efficiency to produce cheap electricity. Competition exists in the electricity
supply industry in generation and in the marketing of electricity. The operating cost of a power pool can be reduced if
the areas with more economic units generate larger power than their load, and export the surplus power to other areas
with more expensive units. Benefits thus gained will depend on several factors like the characteristics of a pool, the
policies adopted by utilities, types of interconnections, tie-line limits and load distribution in different areas. Therefore,
transmission capacity constraints in production cost analysis are important issues in the operation and planning of
electric power systems [1]. The Differential Evolution is a parallel direct search method proposed by Storm and Price
in 1995.[2][3] DE is a population based stochastic function maximize relating to evolutionary computation, whose
simple yet powerful and straightforward features make it very attractive for numerical optimization. DE uses a rather
greedy and less stochastic approach to problem solving than do evolutionary algorithms (EA). DE combines simple
arithmetical operators with the classical operators of recombination, mutation, and selection to evolve from a randomly
generated starting population to a final solution. DE differs from conventional genetic algorithms in its use of
perturbing vectors, which are the difference between two randomly chosen parameters vectors. This method is a
heuristic, population-based optimization method that uses a population of points to search for global minimum of a
function over continuous search space. DE generates new vectors of parameters by adding the weighted difference
between two vectors to a third one. If the result provides a smaller objective function than a predetermined individual,
new result will replace the one with which it is compared. DE has been used to solve different optimization problems
including power system planning, however, the main difficulty with DE techniques, appears to be ensuring the search
diversity of population when the algorithm is approaching the region of local and global optimum. The improvement is
Copyright to IJAREEIE
www.ijareeie.com
10715
ISSN (Print) : 2320 3765
ISSN (Online): 2278 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 3, Issue 7, July 2014
mainly achieved through several mutation methods by combining the fitness sharing scheme into the solution process.
The findings are supported by detailed simulations based on mathematical function optimization before solving a
power system planning problem. The objective of the economic dispatch problem (EDP) of electric power generation,
whose characteristics are complex and highly nonlinear, is to schedule the committed generating unit outputs so as to
meet the required load demand at minimum operating cost while satisfying all units and system equality and inequality
constraints. Chaos describes the complex behaviour of a nonlinear deterministic system. The application of chaotic
sequences instead of random sequences in DE is a powerful strategy to diversify the DE population and improve the
DEs performance in preventing premature convergence to local minima.
II.
ECONOMIC LOAD DISPATCH (ELD)
The objective of an ELD problem is to find the optimal combination of power generations that minimizes the total
generation cost while satisfying equality and inequality constraints [5,6]. The fuel cost curve for any unit is assumed to
be approximated by segments of quadratic functions of the active power output of the generator. For a given power
system network, the problem may be described as optimization (minimization) of total fuel cost as defined by (1) under
a set of operating constraints.
(a) ELD Problem without Valve-Point Effects
= =1 () = =1 2 + + .(1)
Where is FT total fuel cost of generation in the system ($/hr), ai, bi, and ci are the cost coefficient of the i th generator,
Pi is the power generated by the i th unit and n is the number of generators.
The objective of ELD is to minimize the overall cost of generation. Power balance constraint The total generation cost
is minimized subjected to the following constraints Power balance constraint
power balance constraint
, , for i=1,2,n
(2)
Generation capacity constraint
(3)
=1
where Pi, min and Pi, max are the minimum and maximum power output of the i th unit, respectively. PD is the total
load demand and PLoss is total transmission losses. The transmission losses PLoss can be calculated by using B matrix
technique and is defined by (4) as,
where Bij is coefficient of transmission losses.
(b) ELD Problem Considering Valve-Point Effects
The generating units with multi-valve steam turbines exhibit a greater variation in the fuel-cost functions, The valve
opening process of multi-valve steam turbines produces a ripple-like effect in the heat rate curve of the generators.
These valve-point effects are illustrated in Fig. 1. The significance of this effect is that the actual cost curve function
of a large steam plant is not continuous but more important it is non-linear.
Fig 1 show the valve-point effects
Copyright to IJAREEIE
www.ijareeie.com
10716
ISSN (Print) : 2320 3765
ISSN (Online): 2278 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 3, Issue 7, July 2014
The valve-point effects are taken into consideration in the ELD problem by superimposing the basic quadratic fuel-cost
characteristics with the rectified sinusoid component as follows:
Pi = min
N a P 2 + b P + c + e Sin f P min P
i i
i
i
i
i
i=1 i i
i
(5)
where Fp(i) is total fuel cost of generation in ($/hr) including valve point loading, ei, fi are fuel cost coefficients of the i
th generating unit reflecting valve-point effects. Minimizing the total fuel cost of the thermal power plants while
satisfying the demand power and power system constraints is aimed at to overcome the ED problem [7].
III. DIFFERENTIAL EVOLUTION ALGORITHM
(a) Evaluation
Differential Evolution is one of the most recent population based stochastic evolutionary optimization techniques. Storn
and Price first proposed DE in 1995 [2, 3] as a heuristic method for minimizing non-linear and non-differentiable
continuous space functions. DE algorithm is a simple population based evolutionary computational algorithm for global
optimization .It is one of the accurate and fast meta-heuristic optimization algorithms that was introduced in 1995s by
Price and Storn.This evolutionary algorithm begins the search process by initial random population. DE includes three
main operators, namely, mutation, crossover, and selection. Also, it has three control parameter, namely, population
size (np), scaling coefficient (F), and crossover probability (CR) [8, 9, 10]. Differential Evolution includes Evolution
Strategies (ES) and conventional Genetic Algorithms (GA). Differential Evolution is a population based search
algorithm, which is an improved version of Genetic Algorithm. One extremely powerful algorithm from Evolutionary
Computation due to convergence characteristics and few control parameters is differential evolution. Like other
evolutionary algorithms, the first generation is initialized randomly and further generations evolve through the
application of certain evolutionary operator until a stopping criterion is reached. The optimization process in DE is
carried with four basic operations namely, Initialization, Mutation, Crossover and Selection.
(b) Initialization
The algorithm starts by creating a population vector of size P N given by equation (5) composed of individuals that
evolve over G generation. From the equation (6) each individual (G) i X, is a vector that contains as many elements as
the problem decision variable. The population size P N is an algorithm control parameter selected by the user. Each
individual or candidate solution is a vector that contains as many parameters as the problem decision variables D. In
Differential Evolution the population size P N, remains constant throughout the optimization process.
(5)
(6)
i=1,2.NP
The first step in the DE optimization process is to create an initial population of candidate solutions by assigning
random values to each decision parameter of each individual of the population. The initial population is chosen
randomly in order to cover the entire searching region uniformly. A uniform probability distribution for all random
variables is assumed as in the following equation
(7)
Where
and
are respectively, the lower and upper bound of the decision parameter
uniformly distributed random number within [0, 1] generated anew for each value of j.
and is a
(8)
i=1,2.NP
Copyright to IJAREEIE
www.ijareeie.com
10717
ISSN (Print) : 2320 3765
ISSN (Online): 2278 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 3, Issue 7, July 2014
Where Xa, Xb and Xc, are randomly chosen vectors
and a b c i. Xa, Xb and Xc, are
generated anew for each parent vector. The mutation factor F is a user chosen parameter used to control the
perturbation size in the mutation operator and to avoid search stagnation.
3.2. Crossover Operation
The crossover operation generates trial vectors by mixing the parameter of the mutant vectors with the target vectors.
For each parameter, a random value based on binomial distribution is generated in the range [0, 1] and compared
against a user defined constant referred as crossover constant. If the random number is less than the crossover constant
the parameter will come from the mutant vector, otherwise the parameter comes from parent vector as in equation (9).
The crossover operation maintains diversity in the population, preventing local minima convergence. The crossover
constant (CR) must be in the range of [0, 1]. A crossover constant of one means the trial vector will be composed
entirely of mutant vector parameters. A crossover constant near zero results in more probability of having parameters
from target vector in trial vector. A randomly chosen parameter from mutant vector is always selected to ensure that the
trial vector gets at least one parameter from mutant vector even if the crossover constant is zero. Trial vectors are
generated according to
(9)
i=1,2.NP, , j=1,2.D
Where, q, is randomly chosen index
that guarantees the trial vector gets at least one parameter
from mutant vector., is a uniformly distributed random number within [0, 1] generated a new for each value of j.
is parent vector;
is the mutant vector;
is a trial vector. possible if the global optimum of
the problem is available.
The optimization process is repeated for several generations. This allows individuals to improve
3.4. Selection Operation
Selection is the operation through which better offspring are generated. The evaluation (fitness) function of an
offspring is compared to that of its parent. The parent is replaced by its offspring if the fitness of the offspring is better
than that of its parent, while the parent is retained in the next generation if the fitness of the offspring is worse than that
of its parent. The selection operator chooses the vector that is going to compose the population in the next generation.
The selection is repeated for each pair of target / trial vector until the population for the next generation is complete.
Thus, if f denotes the cost (fitness) function under optimization (minimization), then
i=1,2.NP,
(10)
their fitness while exploring the solution space for optimal values. The iterative process of mutation, crossover and
selection on the population will continue until a user-specified stopping criterion, normally, the maximum number of
generations allowed, is met. The other type of stopping criterion, i.e. convergence to the global optimum result.
VI. RESULT AND DISCUSSION
Proposed DE Algorithm has been applied to ED problems in test cases for verifying its feasibility. These are a three
units system show in the table 1 and power Demand 850 MW [4]. The software has been written in MATLAB-7
language.
Copyright to IJAREEIE
www.ijareeie.com
10718
ISSN (Print) : 2320 3765
ISSN (Online): 2278 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 3, Issue 7, July 2014
Table 1 three unit system with VPL and without VPL ( PD 850 MW)
Case 1 Population Size Effect in 3-Generating Units without VPL
In this paper, tests were carried out for different population sizes, the various ELD problems, larger the dimension,
larger is the population size required to achieve good results, so increase the population sizes ,Achieved the minimum
global results. The DE algorithm converges to the global best solution for the 3-unit system for a population size of
10.20,30 and 50 shows in the table 3 and fig 2, fig 3, fig 4 and figures 5 in this case use C.R 0.8 or F 0.2 with 100 trails.
DE achieved quite effective result.
Table 2 population size Effect without VPL in DE with 100 trails (F=0.2 CR=0.8)
Fig 2 results of DE, 3-unit system population size 10 and trials 100
Fig 3 results of DE, 3-unit system population size 20 and trials 100
Copyright to IJAREEIE
www.ijareeie.com
10719
ISSN (Print) : 2320 3765
ISSN (Online): 2278 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 3, Issue 7, July 2014
Fig 4 results of DE, 3-unit system population size 30 and trials 100
Fig 5 results of DE, 3-unit system population size 50 and trials 100
Case ii Population Size Effect in 3-Generating Units with VPL
The DE algorithm converges to the global best solution for the 3-unit system for a population size of 10.20,30 and 50
show in the table 4 and fig 6, fig 7, fig 8 and figures 9 in this case use C.R 0.9 or F 0.3 with 100 trails.
Table 3 population size Effect with VPL in DE with 100 trails (F=0.3 CR=0.9)
Fig 6 results of DE, 3-unit system population size 10 and trials 100
Copyright to IJAREEIE
www.ijareeie.com
10720
ISSN (Print) : 2320 3765
ISSN (Online): 2278 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 3, Issue 7, July 2014
Fig 7 results of DE, 3-unit system population size 20 and trials 100
Fig 8 results of DE, 3-unit system population size 30 and trials 100
Fig 9 results of DE, 3-unit system population size 50 and trials 100
Case iii Comparative analysis
The Comparative analysis DE with achieved quite effective result. A parameter tuning was done to find optimal values
of C.R 0.8 or F 0.2 with Population sizes 50 out of 100 trails in without valve Point Loading, Comparative result show
in the table 4 and C.R. 0.9 or F 0.3 with Population sizes 50 out of 100 trials for with valve Point Effects practical
Comparative result show in the table 5, Results obtained DE algorithm have been compared then vest result with
PSO,CPSO,WIPSO and MRPSO [4], DE achieved quite effective result and global best solutions.
Copyright to IJAREEIE
www.ijareeie.com
10721
ISSN (Print) : 2320 3765
ISSN (Online): 2278 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 3, Issue 7, July 2014
Table 4 comparison result with valve point effect population size 50, 100 trails (F=0.2 CR=0.8)
Table 4 comparison result with valve point effect population size 50, 100 trails (F=0.3 CR=0.9)
V.CONCLUSION
The differential evolution algorithm has been successfully implemented to solve ED problems with the generator
constraints as linear equality and inequality constraints the algorithm is implemented for three units system. From the
result, it is clear that the proposed algorithm has the ability to find the better quality solution, computational efficiency,
less CPU time and minimum S.D. per iteration when compared to other methods such as PSO and PSO variant.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
Manish Sharma, Manjaree Pandit*, Laxmi Srivastava, Multi-area economic dispatch with tie-line constraints employing evolutionary
approach, International Journal of Engineering, Science and Technology Vol. 2, No. 3, 2010, pp. 132-149.
R. Storn and K. Price, Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces,
Technical Report TR-95-012, Berkeley, USA: International Computer Science Institute, 1995.
K. Vaisakh and L. R. Srinivas, Differential Evolution Approach for Optimal Power Flow Solution, Journal of Theoretical and Applied
Information Technology
Nagendra Singh1, Yogendra Kumar2, Constrained Economic Load Dispatch Using Evolutionary Technique Asian Journal of Technology &
Management Research [ISSN: 2249 0892] Vol. 03 Issue: 02 (Jul - Dec 2013).
Hardiansyah, A Modified Particle Swarm Optimization Technique for Economic Load Dispatch with Valve-Point Effect , I.J. Intelligent
Systems and Applications, 2013, 07, 32-41 Published Online June 2013 in MECS DOI:10.5815/ijisa.2013.07.05.
Jyoti Jain1, Rameshwar Singh2, Biogeography-Based Optimization Algorithm for Load Dispatch in Power System, International Journal of
Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3,
Issue 7, July 2013).
Neetu Agrawal, K.K.Swarnkar, Dr. S.Wadhwani, Dr. A. K. Wadhwani, Economic Load Dispatch Problem of Thermal Generators with Ramp
Rate Limit Using Biogeography - Based Optimization ,International Journal of Engineering and Innovative Technology (IJEIT) Volume 1,
Issue 3, March 2012.
Arsalan Najafi *, Hamid Falaghi, Maryam Ramezani, Combined Heat and Power Economic Dispatch Using Improved Differential Evolution
Algorithm, International Journal of Advanced Research in Computer Science and Software Engineering, Volume 2, Issue 8, August 2012
ISSN: 2277 128X.
C.N. Ravi1, Dr. C. Christober Asir Rajan2, Differential Evolution technique to solve Combined Economic Emission Dispatch 3rd
International Conference on Electronics, Biomedical Engineering and its Applications (ICEBEA'2013) January 26-27, 2013 Hong Kong
(China)
1
K.Vaisakh, 2 L.R.Srinivas, Differential Evolution Approach For Optimal Power Flow Solution Journal of Theoretical and Applied
Information Technology, 2005 - 2008 JATIT.
Copyright to IJAREEIE
www.ijareeie.com
10722
