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					                            IPASJ International Journal of Electrical Engineering (IIJEE)
                                                                        Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm
A Publisher for Research Motivation........                                       Email: editoriijee@ipasj.org
Volume 1, Issue 6, December 2013                                                                ISSN 2321-600X




    A Thermal Economic Dispatch Problem Solving
          By Particle Swarm Optimization
                                            Manjeet Singh1, Divesh Thareja2
       1
        Department of Electrical and Electronics Engineering, Assistant Professor, HCTM Technical Campus, Kaithal, India.
                       2
                         Department of Electrical Engineering, Assistant Professor, BGIET, Sangrur, India.




                                                      ABSTRACT
The primary objective of the ELD is to minimize the total cost of generation while maintaining the operational costs of the
available generation resources. This paper consist a new approach to solve thermal units economic dispatch (ED) problems.
Economic dispatch is a highly constrained optimization problem in power system encompassing interaction among decision
variables. The particle swarm optimization (PSO) technique is demonstrated. This paper presents a Particle Swarm
Optimization (PSO) based solution for optimal flow with generating units having fuel costs curves while satisfying the
constraints.PSO has been examined and tested for standard 10 generating units.
Index Terms: Particle swarm optimization, non-smooth fuel cost functions, valve point effects.

  1. INTRODUCTION
Economic dispatch is generation allocation problem and defined as the process of calculating the generation of the
generating units so that the system load is meet while satisfying all the constraints. Historically economic dispatch is
being carried out since 1920. It was the time when engineers were concerned with the problem of economic allocation
of generation or the proper division of the load among the generating units available. The methods for solving this kind
of problem include traditional operational research algorithms (such as linear programming, quadratic programming,
dynamic programming, gradient methods and Lagrange relaxation approaches) and modern methods (such as
simulated annealing and evolutionary algorithms). Some of these methods are successful in locating the optimal
solution, but they are usually slow in convergence and require heavy computational cost [2]. Economic load dispatch
problem plays an important role in the operation of power systems. It is a method to determine the most efficient, low
cost and reliable operation of a power system by dispatching the available electricity generation resources to supply the
load on the system. The primary objective of the ELD is to minimize the total cost of generation while maintaining the
operational costs of the available generation resources. The economic load dispatch is very important optimized
problem solution in power system operations for allocating generation among the committed units such that the system
constraints imposed are satisfied and energy requirements in terms of Rupees per hour ( Rs/hr) or Dollars per
hour($/hr)are minimized.Eberhart and Kennedy suggested a particle swarm optimization (PSO) based on the analogy
of swarm of bird and school of fish [4]. The PSO mimics the behaviour of individuals in a swarm to maximize the
survival of the species. In PSO, each individual makes his decision using his own experience together with other
individuals’ experiences [3]. Partical swarm optimization (PSO) is one of the modern heuristic algorithms, which can
be used to solve non linear optimization problems [3]. It is a population-based search algorithm and searches in parallel
using a group of particles similar to Genetic Algorithms (GA). The original PSO suggested by Kennedy and Eberhart is
based on the analogy of swarm of bird and school of fish [4]. Each particle in PSO makes its decision using its own
experience and its neighbor’s experiences for evolution. That is, particles approach to the optimum through its present
velocity, previous experience, and the best experience of its neighbors [2].In this paper, an alternative approach is
proposed to the non-smooth ED problem using a PSO which focuses on equality and inequality constraints when
modifying each individual’s search. The equality constraint (i.e., the supply/demand balance) is easily satisfied by
specifying a variable (i.e., a generator output) at random in each iteration as a slag generator whose value is determined
by the difference between the total system demand and the total generation excluding the slag generator. However, the
inequality constraints in the next position of an individual produced by the PSO algorithm can violate the inequality
constraints. In this case, the position of any individual violating the constraints is set to its minimum or maximum
position depending on the velocity evaluated.



Volume 1, Issue 6, December 2013                                                                                      Page 5
                                 IPASJ International Journal of Electrical Engineering (IIJEE)
                                                                            Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm
A Publisher for Research Motivation........                                           Email: editoriijee@ipasj.org
Volume 1, Issue 6, December 2013                                                                    ISSN 2321-600X

The main objective of this study is to introduce the use of Particle Swarm Optimization (PSO) technique to the subject
of power system economic load dispatch. In this paper, the PSO method has been employed to solve economic dispatch
problem with a valve point effects.

2. PROBLEM FORMULATION
2.1Objective Function.
The objective of the economic dispatch problem is to minimize the total fuel cost of thermal power plants subjected to
the operating constraints of a power system. The simplified cost function of each generator can be represented as a
quadratic function as described in (1).

             =               )        (1)

                 =     +          +         (2)

Where,
   Total generation cost.
  Cost function of generator i.
    , ,cost coefficients of generator i.
  Power of generator i.
N number of generators.

2.2 Cost Function Considering Valve-Point Effects.
The generator with multi-valve steam turbines has very different input-output curve compared with the smooth cost
function. Typically, the valve point results in, as each steam valve starts to open, the ripples like in Fig. 1. To take
account for the valve-point effects, sinusoidal functions are added to the quadratic cost functions as follows:

                                                                                       (3)

Where     and    are the coefficients of unit i reflecting valve point effects.




                           Figure1.Fuel cost versus power output for four valve steam turbine unit.

2.3 System Constraints.
2.3.1 Equality /Active power balance equation.
  For power balance, an equality constraint should be satisfied. The total generated power should be the same as total
load demand plus the total line loss.

                     = +                          (4)
Where     the total system demand and             is the total line loss.

2.3.2 Inequality Constraints.
  Generation output of each generator should be laid between maximum and minimum limits. The corresponding
inequality constraints for each generator are:

Volume 1, Issue 6, December 2013                                                                                  Page 6
                           IPASJ International Journal of Electrical Engineering (IIJEE)
                                                                     Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm
A Publisher for Research Motivation........                                    Email: editoriijee@ipasj.org
Volume 1, Issue 6, December 2013                                                             ISSN 2321-600X

                                                                        (5)
Where        and         are the minimum and maximum output of generator i, respectively.



 3. IMPLEMENTATION OF PSO FOR ED PROBLEMS

3.1. Particle Swarm Optimization.
Particle swarm optimization (PSO) is a population-based optimization method first proposed by Kennedy and Eberhart
in 1995, inspired by social behaviour of bird flocking or fish schooling. The PSO as an optimization tool provides a
population-based search procedure in which individuals called particles change their position (state) with time. In a
PSO system, particles fly around in a multidimensional search space. During flight, each particle adjusts its position
according to its own experience (This value is called pbest), and according to the experience of a neighboring particle
(This value is called gbest), made use of the best position encountered by itself and its neighbor (Fig. 2).




                                    Figure2. Concept of a searching point by PSO.

This modification can be represented by the concept of velocity. Velocity of each agent can be modified by the
following equation:

                                                                                               (6)

Using the above equation, a certain velocity, which gradually gets close to pbest and gbest can be calculated .The
current position (searching point in the solution space) can be modified by the following equation.

                                                                                         (7)

NP is the number of particles in a group.
NG is the number of members in particles.
 is the pointer of iterations (generations).
  is the inertia weight factor
C1 and C2 are the acceleration constant.
R1 and R2 are uniform random values in the range [0, 1].
    is the velocity of    member of      particle at iteration,
  is the current position of    member of      particle at    iteration.

Suitable selection of inertia weight w provides a balance between global and local explorations,thus requiring less
iteration on average to find a sufficiently optimal solution. As originally developed,      often decreases linearly from
about 0.9 to 0.4 during a run. In general, the inertia weight is set according to the following equation [4].

                                                                                (8)

Where,

Volume 1, Issue 6, December 2013                                                                                Page 7
                                     IPASJ International Journal of Electrical Engineering (IIJEE)
                                                                                           Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm
A Publisher for Research Motivation........                                                          Email: editoriijee@ipasj.org
Volume 1, Issue 6, December 2013                                                                                   ISSN 2321-600X

       is the maximum number of iterations (generations).
  is the current number of iterations.


4. SIMULATION RESULTS AND DISCUSSION

The proposed PSO algorithm is tested on standard 10 generating units. The test system consists of 10 thermal units
without losses (Table1), 24 hour load.
The optimal setting of the PSO control parameters are: c1=2, c2=2, numbers of particles is 10 and number of iterations
is 100.
The Inertia weight was kept between 0.4 and 0.9

4.1The OPF with quadratic fuel cost functions.
The proposed PSO is applied to standard 10 generating units. The obtained results using PSO are given in Tables 1.
Fig.3. shows the cost convergence of PSO for various numbers of generations. It was clearly shown that there is no
rapid change in the fuel cost function value after 40 iterations. Hence it is clears from the Fig3.
                                       Table.1 Ten-unit generator characteristics.


                  Hours                         P(MW)      Hours     P(MW)     Hours        P(MW)        Hours   P(MW)
                  1                             1036       7         1702      13           2072         19      1776
                  2                             1110       8         1776      14           1924         20      2072
                  3                             1258       9         1924      15           1776         21      1924
                  4                             1406       10        2072      16           1554         22      1628
                  5                             1480       11        2146      17           1480         23      1332
                  6                             1628       12        2220      18           1628         24      1184
                  Total                         2114300
                  cost($/hr)
                  CPT(sec)                      7.691574



                                                 6
                                            x 10
                                     2.24


                                     2.22


                                      2.2


                                     2.18
                       Cost($/hr.)




                                     2.16


                                     2.14


                                     2.12


                                      2.1
                                            0        10    20   30    40       50     60      70    80     90    100
                                                                       No of Iterations.



                                       Figure3. Convergence characteristic of the 10 generating units

Volume 1, Issue 6, December 2013                                                                                                 Page 8
                           IPASJ International Journal of Electrical Engineering (IIJEE)
                                                                                                Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm
A Publisher for Research Motivation........                                                               Email: editoriijee@ipasj.org
Volume 1, Issue 6, December 2013                                                                                        ISSN 2321-600X

                                                    4
                                             x 10
                                        13


                                        12


                                        11


                                        10
                          Cost($/hr.)




                                         9


                                         8


                                         7


                                         6


                                         5
                                             0              5          10                  15           20        25
                                                                            No of Hours.




                                                        Figure4.Cost-Load curve for Test problem.




5. CONCLUSION
The proposed algorithm has been tested on the 10 generating units. Particle Swarm Optimization has implemented on
ten generating units having twenty four hours load with non-smooth fuel cost functions. It was clearly shown that there
is no rapid change in the fuel cost function value after 40 iterations The PSO is successfully and effectively used for
minimization the overall cost functions.

REFERENCE
[1] A. J. Wood, and B. F. Wollenbergy, “Power Generation, Operation, and Control”, New York, NY, John
    Wiley & Sons, Inc., 1984.
[2] J. Kennedy and R. C. Eberhart, “Swarm Intelligence, San Francisco”, CA, Morgan Kaufmann Publishers,
     2001.
[3] K. Y. Lee and M. A. El-Sharkawi (Editors), “Modern Heuristic Optimization Techniques with Applications
     to Power Systems”, IEEE Power Engineering Society (02TP160), 2002.
[4] J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” Proceedings of IEEE International
    Conference on Neural Networks (ICNN’95), Vol. IV, pp. 1942-1948, Perth, Australia, 1995.
[5]H. Yoshida, K. Kawata, Y. Fukuyama, S. Takayama, and Y. Nakanishi, “A particle swarm optimization for
    reactive power and voltage control considering voltage security assessment”, IEEE Trans. Power Syst., vol.
   15, pp. 1232–1239, Nov. 2000.
[6] M. Clerc and J. Kennedy, “The particle swarm-explosion, stability, and convergence in a multidimensional
    Complex space”, IEEE Trans. Evol.Comput., vol. 6, no. 1, pp. 58–73, Feb. 2002.
[7] M. A. Abido, “Optimal design of power-system stabilizers using particle swarm optimization” , IEEE Trans.
    Energy Conv., vol. 17, no. 3, pp. 406–413, Sept. 2002.
[8] S. Pothiya, I. Ngamroo, W. Kongprawechnon, "Application of multiple tabu search algorithm to solve
    dynamic economic dispatch considering generator constraints", in 1November 2007.
[9] Z. X. Liang and J. D. Glover, “A zoom feature for a dynamic programming solution to economic dispatch
     including transmission losses”, IEEE Trans. on Power Systems, Vol. 7. No. 2, pp. 544-550, May 1992.




Volume 1, Issue 6, December 2013                                                                                                      Page 9

				
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Description: IPASJ International Journal of Electrical Engineering (IIJEE) Web Site: http://www.ipasj.org/IIJEE/IIJEE.htm A Publisher for Research Motivation........ Email: editoriijee@ipasj.org Volume 1, Issue 6, December 2013 ISSN 2321-600X