A New Approach For Dynamic Economic Dispatch Using Hybrid Modified Particle Swarm Optimization

A New Approach For Dynamic Economic Dispatch Using Hybrid Modified Particle Swarm Optimization

1 D. P DASH, 2 M. BASU, 3 J. PATTANAIK

1 Assoc.Professor, Electrical Engg. Dept O.E.C. Bhubaneswar

2 Professor, Power Engg. Dept., Jadavpur University

3 Research Scholar, Electrical Engg.Dept., Jadavpur University

ABSTRACT

This article presents a novel optimization approach to constrained dynamic economic dispatch (DED) problems using the hybrid particle swarm optimization (HPSO) technique. The proposed methodology easily takes care of different constraints like transmission losses, ramp rate limits and also uses for non-smooth cost functions. To illustrate its efficiency and effectiveness, the developed HPSO approach is tested with different number of generating units and comparisons are performed with other approaches under consideration.

Convexity in the fuel cost function [3]. Accurate modeling of the DED problem will be improved when the valve point loadings in the generating units are taken into account. Previous efforts on solving DED problem have employed various mathematical programming methods and optimization techniques. Conventional method like Lagrangian relaxation [1], gradient projection method [2] and dynamic programming etc, when used to solve DED problem suffer from myopia for non-linear, discontinuous search space, leading them to a less desirable performance and these methods often use approximations to limit complexity.

1. INTRODUCTION

   The dynamic economic dispatch (DED) is an extension of the traditional economic dispatch problem used to determine the schedule of real- time control of power system operation so as to meet the load demand at the minimum operating cost under various system and operational constraints. DED procedure follows the dynamic connection by handling the ramp rate limits of generating units and by modifying the steady state cost to include the extra fuel consumption. The DED problem is not only the most accurate formulation of the economic dispatch problem (EDP).

   Most of the literature addresses DED problem with convex cost function [1-2]. However, in reality, large steam turbines have steam admission valves, which contribute non

   Recently, stochastic optimization techniques such as Genetic algorithm (GA) [4-5], evolutionary programming (EP) [6-7], simulated annealing (SA) [8-9] and particle swarm optimization (PSO) [10-12] have been given much attention by many researches due to their ability to seek for the near global optimal solution. However, all the previous work mentioned above neglected the non-smooth characteristic of generator, which actually exist in the real power system.

   This paper presents a novel optimization method based on hybrid particle swarm optimization (HPSO) algorithm applied to dynamic economic dispatch in a practical power system while considering some nonlinear characteristics of a generator such as ramp rate limits, generators constraints, power loss and non-smooth cost function. The proposed

   methodology emerges as a robust optimization technique for solving the DED problem for different size power system.

   c. Generating unit ramp rate limits

   Pit

   Pi t 1

   URi ,

   i 1, 2, 3,….,N

   (5)

2. DED PROBLEM FORMULATION

   The objective of the DED is to schedule the

   Pi t 1

   Pit

   DRi,

   i 1,2,3,….,N

   (6)

   outputs economically over a certain period of

   time under various system and operational constraints. The conventional DED problem

   Where URi and

   ramp- down limits of

   DRi

   i th

   are the ramp-up and

   unit in MW. So the

   minimizes the following incremental cost function associated to dispatchable units.

   constraint given by Eq. (5) is modified as

   follows:

   M in F

   T N

   Fit

   Pit

   $ (1)

   max Pi min , Pi t 1

   DRi

   min Pi max , Pi t 1

   URi

   (7)

   t 1i 1

   Where F is the total operating cost over the whole dispatch period, T is the no. of intervals in the scheduled horizon, N is the no. of generating

3. OVERVIEW OF PSO

   The particle swarm optimization method

   units and

   Fit Pit

   is the fuel cost in terms of its

   conducts its search using a population of

   real power output Pit at timet. Taking into

   particles, corresponding to individuals. It starts with a random initialization of a population of

   valve-point effects, the fuel cost of the

   i th

   individuals in the search space and works on the

   thermal generating unit is expressed as the sum of a quadratic and a sinusoidal function in the following form is given by

   social behavior of the particles in the swarm, like birds flocking, fish schooling and the swarm theory. Therefore, it finds the global optimum by simply adjusting the trajectory of each individual

   Fit Pit

   a P 2

   bi Pit

   Ci ei sin fi Pi,min _ Pit

   $ / h

   towards its own best location and towards the

   i it

   (2)

   best particle of the swarm at each generation of evolution. The position and the velocity of the

   i th particle in the d dimensional search space can

   Where

   ai ,

   bi ,

   ci are cost coefficients and

   be represented as X

   , x ,……..,x

   T and

   ei , fi are constants from the valve point effect

   i i1 i2 id

   of the

   i th

   generating unit, subject to the

   Vi i1 , vi2 ,……..,vid

   T . Each particle has its

   following equality and inequality constraints.

   own best position (Pbest)

   Pi t pi1 t , pi2 t ,……… pid t T

   1. Real power balance

      N

      P _ P _ P 0

      (3)

      corresponding to the personal best objective value obtained so far at generation t . The global best particle (Gbest) is denoted by

      it Dt Lt

      t 1

      P t p t , p t ,……… p t T . The new

      Where t = 1, 2 T, is the total power demand at

      g g1 g 2 gd

      time t and

      PLt

      is the transmission power loss

      velocity of each particle is calculated as follows:

      at i th

      interval in MW.

      vIJ t 1

      vij t

      c1r1

      pij t

      xij t

   2. Real power operating limits

   c2 r2

   p gi t

   xij t

   j 1,2,……..d.

   (8)

   Pt min

   Pit

   Pt max

   (4)

   Where

   c1 and

   c2 are constants of acceleration

   Where

   Pt min

   and

   Pt max

   are respectively the

   coefficients corresponding to cognitive and social behavior, is the inertia factor , n is the

   minimum and maximum real power output of

   population size ,

   r1 and

   r2 are two independent

   i th

   generator in MW.

   random numbers. Thus, the position of each particle at each generation is updated according to the following equation:

   xij t 1 xij t vij t 1

   i 1,2,…….,n and j=1,2,.,d (9)

4. MODIFIED PSO :

   In the conventional PSO method, the inertia weight is made constant for all the particles in a single generation, but the most important parameter that moves the current position towards the optimum position is the inertia weight . In modified PSO, the particle position is adjusted such that the highly fitted particle (best particle) moves slowly when compared to the lowly fitted particle. This can be achieved by selecting different values for each particle according to their rank, between min and max as in the following form:

   The concept of re-initialization is introduced in the proposed HPSO methodafter a specific number of generations if there is no improvement in the convergence of the algorithm. At the end of the method the specific generation is re-initialized with new randomly generated individuals. The number of new individuals is selected from k least individuals of the original population, where k is the percentage of total population to be changed. This re-initialization of population is performed after checking the change in the Fbest value in each and every specific generation.

5. SEQUENTIAL QUADRATIC PROGRAMMING (SQP):

   Sequential quadratic programming (SQP) [13] is widely used to solve practical optimization

   max

   min

   Ranki

   (10)

   problems. It outperforms every other nonlinear

   i max

   Total Population

   programming method in terms of efficiency, accuracy and percentage of successful solutions.

   So, from Eq. (9), shows that the best particle takes first rank, and the inertia weight for that particle is set to minimum value while for the lowest particle takes the maximum inertia weight, which particle move a high velocity. The velocity of each particle is updated using Eq. (15), and if updated velocity goes beyond maximum velocity Vmax , than it is limited to

   Vmax

   The method closely mimics Newtons method for constrained optimization just as is done for unconstrained optimization. At each major iteration, an approximation is made of the Hessian of the Lagrange function using Broyden- Fletcher-Goldfarb-Shanno (BFGS) quasi- Newton updating method. This is then used to generate a quadratic programming subproblem whose solution is used to form a search direction for a line search procedure.

   As the objective function to be minimized is

   vij t 1

   i vij t

   c1r1 pij t

   xij t

   (11)

   nonconvex, SQP requires a local minimum for

   c2 r2

   p gi t

   xij t

   an initial solution. In this paper, SQP is used as a local optimizer for fine-tunning the better region explored by AIS. Here, the formulation of SQP

   vij t 1

   sign vij t 1

   min vij t 1 ,

   V j max

   (12)

   subroutine is taken from [15].

   For each iteration, a QP is solved to obtain the search direction which is used to update the

   j 1,2,……,d

   and i

   1,2,……,n

   control variables. QP problem can be described as follows

   The new particle position is obtained by using Eq. (17), and if any particle position beyond the

   Minimize the following

   F d

   1 d d

   rang e specified, it is limited to its boundary using Eq. (18),

   subject to

   k k 2

   k K k

   xij t 1

   xij t

   vij t 1

   gi k

   g k d k 0

   j 1,2,……,d;

   i 1,2,…….n.

   (13)

   i 1,…., me

   g

   g d 0

   xij t 1

   min xij t 1 ,

   rangej max ,

   i k k k

   xij t 1

   max xij t 1 ,

   range

   j min

   (14)

   i me

   1,…, m

   where

   k the Hessian matrix of the Lagrangian function at the k th iteration

   d k the search direction at the k th iteration

   k the real power vector at the k th iteration

   g k constraints from (3) to (4)

   me number of equality constraints

6. m number of constraints

   L , F g

   where is the vector of Lagrangian multiplier.

   k is calculated using quasi-Newton formula given by,

   [min, max] for each variable, c1, c2 and iteration counter. Set iteration counter = 0.

   1. Increment iteration counter by one.

   2. Find out the fitness function of all particles in the population and update the objective function.

   3. If stopping criterion is reached than go to step (5.9). Otherwise continue.

   4. Evaluate the inertia factor according to Eq. (10).

   5. Update the velocity given in Eq. (11) and correct it using Eq. (12).

   6. Update the position of each particle using Eq. (13) and if the new position goes out of range, set it to boundary value using Eq. (14).

   7. For every 5 generations, the {Fbest, new value} is compared with the {Fbest, old value}. If there is no change, then use the re- initialization concept and go to step (5.3).

   8. Output the Gbest particle and its objective value.

      qk qk

      k Sk Sk k

   9. solve the DED problem using the SQP

   k 1

   where

   qk Sk Sk

   k

   Sk k 1 k

   k Sk

   method with the selected solution obtained from PSO.

7. SIMULATION RESULTS

   qk L k 1 , k 1 L k , k 1

   For each iteration of the QP sub-problem the direction d k is calculated using the objective function. The solution obtained forms a new iterate given by,

   k 1 k k dk

   The step length value k is determined to produce a considerable reduction in an augmented Lagrangian merit function as follows

   L , , F g g g

   2

   where is a nonnegative scalar. The procedure

   The five unit system with non-smooth fuel cost function is used to demonstrate the performance of the proposed HPSO. We have used the same system data as done by Panigrahi et al. [8]. The load demand of the system is taken over 24 hour. The result of the proposed method is given in Table 1. The earlier reported result for the cost is 47356 $. For the present simulation, the cost is found to be 44568 $.

8. CONCLUSIONS

The paper has employed the HPSO algorithm on constrained of dynamic economic dispatch problem. The proposed approach has produced comparable to or better than those generated by other algorithms, and the solution has superior quality and good convergence characteristics.

from this limited comparative study, it can be

is repeated until the value of some tolerance value.

6. HPSO ALGORITHM

S k has reached

concluded that the HPSO can be effectively used to solve non-smooth as well as smooth constrained economic load dispatch problems. In the future, the work will can be made to incorporate more realistic constraints to the

6.1) Initialize number of population of particles dimension d with random position velocities and get the input parameters such as range

problem and the large size problems will be solved by the proposed methodology.

Table1: Result for five unit system with 24 h load demand

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