A New Solution for Dynamic Economic Dispatch With Valve Point Loading Using Modified Particle Swarm Optimization

A New Solution for Dynamic Economic Dispatch With Valve Point Loading Using Modified Particle Swarm Optimization

1 D. P Dash, 2 C. Chaudhury,3 K. C. Meher,

1 Professor, Electrical Engg. Dept., Orissa engineering College, Bhubaneswar

2, 3 Asst. professor Electrical Engg. Dept., Orissa engineering College, Bhubaneswar

Abstract:- This article presents a novel optimization approach to constrained dynamic economic dispatch (DED) problems using the modified particle swarm optimization (MPSO) technique. The proposed methodology easily takes care of different constraints like transmission losses, ramp rate limits and generating limit and power balance limit. To illustrate its efficiency and effectiveness, the developed MPSO approach is tested with different number of generating units and comparisons are performed with other approaches under consideration.

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

   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 modified particle swarm optimization (MPSO) 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.

2. DED PROBLEM FORMULATION

   The objective of the DED is to schedule the outputs economically over a certain period of time under various system and operational constraints. The conventional DED problem

   Minimizes the following incremental cost function associated to dispatchable units.

   T N

   M in F Fit Pit $ (1)

   t 1i1

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

   contribute non convexity in the fuel cost function of the

   generating units [3]. Accurate modeling of the DED

   is the no. of generating units and

   Fit

   Pit

   is the fuel cost

   problem will be improved when the valve point loadings in

   in terms of its real power output

   Pit

   at timet. Taking into

   the generating units are taken into account. Previous efforts

   valve-point effects, the fuel cost of the

   i th

   thermal

   on solving DED problem have employed various mathematical programming methods and optimization techniques. Conventional method like Lagrangian

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

   relaxation [1], gradient projection method [2] and dynamic

   F P a P2 b P C e sinf P

   P

   $ / h

   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

   it it

   i it

   i it i i

   i i,min it

   (2)

   methods often use approximations to limit complexity.

   Where ai , bi , ci are cost coefficients and ei , fi

   Recently, stochastic optimization techniques such as Genetic algorithm (GA) [4-5], evolutionary

   are constants from the valve point effect of the

   i th

   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.

   generating unit, subject to the following equality and

   inequality constraints.

   1. Real power balance

      inertia factor , n is the population size ,

      r1 and

      r2 are two

      N

      Pit _ PDt _ PLt 0

      t 1

      (3)

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

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

      and

      PLt is the transmission power loss at

      i th

      interval in

      xij

      t 1 xij

      t vij

      t 1

      MW.

   2. Real power operating limits

      Pt min Pit Pt max

      (4)

      i 1,2,…….,n and j=1,2,.,d (9)

      1. MODIFIED PSO ALGORITHM

         In the conventional PSO method, the inertia weight is made constant for all the particles in a single generation,

         Where

         Pt min

         and

         Pt max

         are respectively the minimum

         but the most important parameter that moves the current

         and maximum real power output of

   3. Generating unit ramp rate limits

   i th

   generator in MW.

   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

   Pit Pi t 1 URi ,

   i 1, 2, 3,….,N

   (5)

   according to their rank, between min and max as in the

   following form:

   Pi t 1 Pit DRi,

   i 1,2,3,….,N

   (6)

   • max min Ranki

     (10)

     Where

     URi and

     DRi are the ramp-up and ramp- down

     i max

     Total Population

     limits of

     i th

     unit in MW. So the constraint given by Eq.

     (5) is modified as follows:

     max P , P DR min P

     , P UR

     So, from Eq. (9), shows that the best particle takes first rank, and the inertia weight for that particle is set to

     i min

     i t 1 i

     i max

     i t 1

     i

     (7)

     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.

     3. OVERVIEW OF PSO

     (15), and if updated velocity goes beyond maximum

     The particle swarm optimization method conducts its

     velocityVmax , than it is limited to

     Vmax

     search using a population of particles, corresponding to

     vij t 1 i vij t c1r1 pij t xij t

     individuals. It starts with a random initialization of a population of individuals in the search space and works on the social behavior of the particles in the swarm, like birds

   • c2 r2 pgi

   t xij

   t

   (11)

   flocking, fish schooling and the swarm theory. Therefore, it finds the global optimum by simply adjusting the trajectory of each individual towards its own best location and towards the best particle of the swarm at each generation of

   vij t 1 signvij t 1 minvij t 1,

   j 1,2,……,d and i 1,2,……,n

   V j max

   (12)

   volution. The position and the velocity of the i th particle

   in the d dimensional search space can be represented as

   X i xi1, xi2 ,……..x, id T and Vi vi1 , vi2 ,……..v, id T .

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

   is limited to its boundary using Eq. (18),

   Each particle has its own best position (Pbest)

   Pi t pi1 t, pi2 t,………., pid tT corresponding to the

   xij

   t 1 xij

   t vij

   t 1

   (13)

   personal best objective value obtained so far at generation

   t . The global best particle (Gbest) is denoted by

   j 1,2,……,d;

   i 1,2,…….n.

   Pg t pg1 t, pg 2 t,………., pgd tT . The new velocity

   xij

   t 1 minxij

   t 1,

   range

   j max ,

   of each particle is calculated as follows:

   vIJ t 1 vij t c1r1 pij t xij t

   xij t 1 maxxij t 1, range j min

   (14)

   c2 r2 pgi t xij t j 1,2,……..d.

   (8)

   The concept of re-initialization is introduced in the proposed MPSO method after a specific

   Where

   c1 and

   c2 are constants of acceleration coefficients

   corresponding to cognitive and social behavior, is the

   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.

   1. MPSO ALGORITHM

      1. Initialize number of population of particles dimension d with random position velocities and get the input parameters such as range [min, max] for each variable, c1, c2 and iteration counter. Set iteration counter = 0.

      2. Increment iteration counter by one.

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

      4. If stopping criterion is reached than go to step (5.9).

         Otherwise continue.

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

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

      7. 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).

      8. 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).

      9. Output the Gbest particle and its objective value.

   2. SIMULATION RESULTS

      The five unit system with non-smooth fuel cost function is used to demonstrate the performance of the proposed MPSO. 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 $.

      No. of hours	Load demand	PG1 (MW)	PG2 (MW)	PG3 (MW)	PG4 (MW)	PG5 (MW)
      1	410	12.3675	104.4735	108.9301	38.4012	140.3918
      2	435	42.4708	95.9732	113.6381	40.1022	138.6778
      3	475	72.0578	96.6296	121.2753	43.9813	139.7721
      4	530	45.0234	97.9622	116.7643	75.0224	179.8301
      5	558	19.7435	105.2582	115.7942	89.9066	197.7502
      6	608	41.9471	103.3492	116.6698	96.8961	215.1322
      7	626	11.9462	89.4341	116.7647	171.8153	221.9615
      8	654	23.6745	85.3441	117.9742	210.0113	228.9501
      9	690	47.4359	98.0986	117.7644	208.0518	231.5196
      10	704	64.1105	99.5385	116.6747	209.1853	229.5385
      11	720	43.0118	101.5421	142.6332	210.1692	230.1596
      12	740	39.7598	97.3598	164.9799	207.5818	229.3214
      13	704	42.6758	96.5389	143.9599	208.9857	228.3597
      14	690	48.6036	96.7388	118.7045	208.9947	220.6617
      15	654	19.6033	95.3773	110.7656	200.6448	201.9233
      16	580	11.1709	87.4059	112.8548	206.2245	191.5503
      17	558	11.1801	97.6698	97.4321	207.5764	178.4851
      18	608	23.5582	99.5398	113.6849	209.8158	156.1376
      19	654	21.0434	100.5196	114.6753	211.1986	188.1579
      20	704	49.4497	105.3416	114.4284	210.6318	196.1342
      21	680	34,4159	103.6783	116.0614	210.8963	213.9615
      22	605	11.7202	90.5406	108.5995	198.7053	215.0352
      23	527	10.0035	62.1432	92.0095	160.9033	223.2762
      24	463	10.0205	39.6943	83.0064	135.9715	225.6296

      Table 1: Result for five unit system with 24 h load demand

   3. CONCLUSIONS

The paper has employed the MPSO 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 conclued that the MPSO 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 problem and the large size problems will be solved by the proposed methodology.

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