Comparitive Analysis of Lambda Iteration Method and Particle Swarm Optimization for Economic Emission Dispatch Problem

Comparitive Analysis of Lambda Iteration Method and Particle Swarm Optimization for Economic Emission Dispatch Problem

Vipandeep Kour

Assistant Professor, Electrical Engineering Department,

IET Bhaddal, Ropar, India

Lakhwinder Singh

Professor,

Electrical Engineering Department, BBSBEC, Fatehgarh Sahib, India

Abstract – With the increase in energy demands, it becomes necessary to operate the thermal power plants most economically, which gives rise to Economic Load Dispatch (ELD) problem. The problem of ELD in power system is to plan the power output for each committed generating unit in such a way that operating cost is minimized while meeting load demand, power operating limits. In present scenario, expansion in power generation has resulted to increase in emissions in the environment which is the prime concern for power system planners. In order to get clean energy, emission control has become an important operational objective in addition to minimization of operating cost. The combination of both the objectives i.e. economic dispatch and emission dispatch have resulted in multi-dimensional power system optimization problem called Economic Emission Dispatch (EED) problem. In this paper, EED problem has been solved using Lambda Iteration Method and Particle Swarm Optimization (PSO) technique. Both the solution methodologies have been validated on three generator system and five generator system. Comparative analysis of the results obtained by both the methods has been performed.

Keywords Economic Load Dispatch, Economic Emission Dispatch, Lambda Iteration Method, Particle Swarm Optimization.

1. INTRODUCTION

   The modern power system is a vast interconnected system in which the main task is to allocate the load demand among participating generators at minimum possible cost. Planning the output of each generator in a power system while minimizing the fuel cost and satisfying various system constrains is termed as economic dispatch. The system constraints include matching the power generation with the load, operating the generators within permissible limits and maintaining system stability. When fossil fuels are burnt, toxic gases are released, such as oxides of carbon, oxides of sulphur and oxides of nitrogen [1]. These gases cause pollution in atmosphere and hence disturb the ecological balance leading to global warming. With the increasing energy production to meet the increasing demands, the emission of pollutants has also increased, thus making the environment unfit for the survival of living beings. To meet clean energy requirement, there is need to minimize the emissions along with the fuel cost which is termed as emission dispatch. While minimizing the emissions, there is need to satisfy the system constraints. When economic dispatch and emission dispatch problems are combined together becomes as Economic Emission Dispatch (EED) problem [2].

   Different techniques have been reported in the literature pertaining to solution of economic emission dispatch problem. Senthil et al. gave an improved Tabu search algorithm of three generator system, six generator system with emission constraints and thirteen generator system with valve point effect loading [3]. Abido presented a multi-objective evolutionary algorithm for environmental/economic power dispatch problem as non-linear constrained multi-objective optimization problem and used Strength Pareto Evolutionary Algorithm (SPEA) to solve the formed multi-objective problem [4]. In another attempt, Abido also presented a Multi-Objective Particle Swarm Optimization (MOPSO) for environmental/economic dispatch problem [5].

   Thakur et al. used PSO algorithm to solve the problem of Combined Economic and Emission Dispatch (CEED) with use of penalty factors. The authors focused on reducing NO2 and SO2 emissions [6]. Valle et al. provided a detailed literature on PSO, its concepts, variants and application in the field of power systems. The authors have performed a vast study on this optimization technique [7]. Arunachalam et al. presented a new approach to solve CEED problem using a Hybrid Particle Swarm Optimization (HPSO) and Firefly algorithm where a multi-objective optimization problem with the valve point effect using a price based penalty factor was solved [8]. Das et al. used PSO and Teaching Learning Based Optimization (TLBO) for solution of ELD problem used to find the optimum solution with lowest fuel cost for four different network consisting of three, six, fifteen, and twenty generating units respectively for different load demand. Results obtained using PSO and TLBO are compared with the results obtained using lambda iteration method [9].

   Particle swarm optimization is a population based stochastic optimization technique developed by Eberhart and Kennedy in 1995 to optimize nonlinear functions [10]. It is inspired by the social behavior of bird flocking or fish schooling while searching for food. PSO optimizes problem by having a population of particles, moving these particles around in the search space according to simple mathematical formula over the particle's position and velocity. Each particle's movement is influenced by its local best known position and is also guided toward the best known positions in the search space, which are updated as better positions are found by other particles. This moves the swarm toward the best solutions.

   Each particle keeps track of its coordinates in the solution space, which are associated with the best solution (fitness) that has been achieved so far by that particle. This value is called personal best, Pbest. Another best value that is tracked by the PSO is the best value obtained so far by any other particle in the neighborhood of that particle. This value is called Gbest. The basic concept of PSO lies in accelerating each particle toward

   The objective function (FT) is equal to the total cost for supplying the indicated load (PLOAD). The problem is to minimize function (FT) subject to the constraint that the sum of the power generated must equal the load. When transmission losses are neglected, the total fuel cost (FT), is stated as

   = 1 + 2 + + (1)

   Pbest and the Gbest locations, with a random weighted

   = ( )

   (2)

   acceleration at each time. Each particle tries to modify its

   =1

   position using the following information: current positions, current velocities, distance between the current position and Pbest, and distance between the current position and the Gbest.

   =1

   PSO has a flexible and well balanced method to adapt the global exploration abilities with faster convergence rates [10]. The efficiency of PSO is due to the fact that each particle uses the information of the best particle and improves itself accordingly. PSO has been successfully implemented to solve power system problems including EED and many other multi- objective functions [11-14]. The key objectives of PSO over

   The operating cost of plant can be represented as shown in fig. 2. This cost is usually approximated by one or more quadratic segments. So, the fuel cost curve is a quadratic curve in active power generation. The fuel cost function without valve-point loading of the generating unit is given by

   () = 2 + + Rs/hr (3)

   The economic load dispatch problem can be described as an optimization (minimization) process with the following objective function

   other optimization techniques can be listed as [15]:

   • Unlike other algorithms, it is derivative free algorithm.

   • It does not depends on the ature of objective function, i.e. continuity and convexity.

   • The solution is not trapped in the local minima.

   • It does not require an initial solution to start the algorithm.

   In PSO, the individuals of the swarm are not created or destroyed, that is, the population remains stable and the particles follow the path of cooperation over competition. The particles while moving within the search space retains a memory of its best point ever attained. This best position is communicated to all other particles.

   The intent of the paper is to implement PSO to optimize

   (4)

   EED problem and to compare traditional lambda iteration method with PSO in order to prove the better computational efficiency of later over the former.

2. PROBLEM FORMULATION

   1. Economic Dispatch

      Consider a system of N thermal-generating units connected to a single bus-bar serving the electrical load is shown in fig. 1. The input to each unit is Fi. The output of each unit is Pgi. The total cost of this system is the sum of the costs of each of the individual units. The essential constraint on the operation of this system is that the sum of the output powers must equal the load demand [16].

      Fig. 2. Operating cost characteristics of fossil fuel fired generator

   2. Emission Dispatch

   Fossil fuels constitute a significant repository of carbon. Burning such fuels, result in the conversion of carbon to carbon dioxide, which is then released into the atmosphere causing an increase in the earth's levels of atmospheric carbon dioxide, which enhances the greenhouse effect and contributes to global warming. Other emissions produced from a generating station are oxides of sulphur and oxides of nitrogen. Sulphur and nitrogen oxides cause smog and acid rain [17-18]. Very tall flue-gas stacks can be built on plants, so that pollutants would get diluted when they are put in the atmosphere. While this helps in reducing local contamination, it does not help in solving the global issues. Total emissions of the system shown in fig. 1 are given by ET, such that

   = 1 + 2 + + (5)

   = ( )

   (6)

   =1

   Fig. 1. N thermal units committed to serve a load PLOAD

   The emission of the thermal power plant can be formulated as a second order polynomial function as

   () = 2 + + Kg/hr (7)

   The emission dispatch problem can be described as an optimization (minimization) process with the following objective function

   B. Particle Swarm Optimization

   In the PSO algorithm, a random population of particles is created and the optimization is achieved by the movement of

   =1

   (8)

   particles towards the global best position. The particles update their position and velocity according to their self-experience as

3. SYSTEM CONSTRAINTS

   Optimization of a given function is done when certain constraints are being satisfied. The constraints that are considered during the optimization of cost function and emission function formulated in eq. 4 and eq. 8 are:

   1. Equality Constraints

      id

      1 1

      best id

      id

      2 2

      best d

      id

      The equality constraints are represented by the power

      well as the social interaction between other particles. Economic emission dispatch problem is solved by implementing PSO.

      The modified velocity and position of each particle can be derived using the present velocity and the distance from Pbest id to Gbest d by the velocity and position equations as given in the following equations:

      balance constraint, that is, the power balance equation, where

      Vid(t+1) = w.v t + c r (P

      x t) + c r (G x t) (11)

      the total power generation must cover the total power demand and the power loss.

      xid(t+1) = xidt +v1d

      (t+1) (12)

      =1

      = +

      (9)

      Where, x represents the position of particle, v represents velocity of the particle, and the ith particle is denoted as xi =

   2. Inequality Constraints

   The inequality constraints are power generation limits of thermal power generators. Upper and lower bounds on the generation of each generator are to be fulfilled and can be expressed as

   (xi1, xi2,…, xid) in the dth dimensional search space. The previous best position of the ith particle is stored in memory and represented by Pbest i = Pbest i1, Pbest i2 ,.., Pbest id. Also, for a particle, the description of rate of velocity is denoted as vi = vi1, vi2, …, vid. The best position among the Pbest is represented as Gbest d. Fig. 4 shows the flow chart of PSO algorithm.

   Pmin P Pmax

   (i = 1, 2,, n) (10)

   gi gi gi

4. SOLUTION METHODOLOGY

   The purpose is to solve economic emission dispatch problem using the conventional optimization technique, lambda iteration method, and compare the results obtained by solving the same using PSO algorithm.

   A. Lambda Iteration Method

   Lambda iteration method is an iterative type of computational technique shown in fig. 3. The optimum operating point of any generator set, within a specified limits, is found using this method.

   Fig. 1. Lambda iteration method

   Fig. 2. Basic PSO algorithm

5. RESULTS AND DISCUSSION

   Economic emission dispatch problem has been solved using lambda iteration method and particle swarm optimization technique. The validity of the proposed techniques has been verified on two generator sets: 3 generator set for a demand of 350, 400 and 450 MW and 5 generator set for a demand of 400, 500 and 600 MW [3][19].

   Equations (4) and (8) were solved using the proposed optimization techniques to get desired results for the two generator sets. All the PSO based optimization was done in MATLAB R2015a on 32 bit Intel Core i3 Computer with 2GB RAM with Windows 7 operating system.

   1. Results for 3 Generator Set

      Table 1 shows the values of powers generated and fuel cost in three generator set calculated from Lambda iteration method for economic dispatch. Table II shows the power output and total emissions for emission dispatch calculated from lambda iteration method.

      TABLE I

      POWER OUTPUT & FUEL COST FOR A 3 GENERATOR SET FOR ECONOMIC DISPATCH CALCULATED FROM LAMBDA ITERATION METHOD

      Demand (MW)	P1	P2	P3	Fuel cost (Rs/hr)
      350	200.52	94.00	56.32	1487.34
      400	200	119.32	81.00	1716.09
      450	200	144	106	1957.34

      TABLE III

      POWER OUTPUT & FUEL COST FOR A 3 GENERATOR SET FOR EMISSION DISPATCH CALCULATED FROM LAMBDA ITERATION METHOD

      Demand (MW)	P1	P2	P3	Total Emissions (kg/hr)
      350	109.52	96.51	144.4	694.93
      400	123.41	109.23	167.32	655.83
      450	142.89	127.07	180	753.98

      Table III shows the values of powers generated by each of the generator in three generator set calculated from PSO for economic dispatch. Table IV shows the power output of each unit for emission dispatch calculated from PSO.

      TABLE IIIII

      POWER OUTPUT & FUEL COST FOR A 3 GENERATOR SET FOR ECONOMIC DISPATCH CALCULATED FROM PSO

      Demand (MW)	P1	P2	p>P3	Fuel cost (Rs/hr)
      350	182.185	107.258	60.082	1406.273
      400	192.716	122.60	84.148	1715.771
      450	195.891	142.719	110.775	1956.557

      TABLE IVV

      POWER OUTPUT & FUEL COST FOR A 3 GENERATOR SET FOR EMISSION DISPATCH CALCULATED FROM PSO

      Demand (MW)	P1	P2	P3	Total Emissions (kg/hr)
      350	185.636	82.935	81.427	682.891
      400	124.3	114	160.751	654.3982
      450	143	126.651	179.518	752.412

   2. Results for 5 Generator Set

      Table V shows the values of powers generated and fuel cost in five generator set calculated from Lambda iteration method for economic dispatch. Table VI shows the power

      output and total emissions for emission dispatch calculated from lambda iteration method.

      TABLE V

      POWER OUTPUT & FUEL COST FOR A 5 GENERATOR SET FOR ECONOMIC DISPATCH CALCULATED FROM LAMBDA ITERATION METHOD

      Demand (MW)	P1	P2	P3	P4	P5	Fuel cost (Rs/hr)
      400	105.45	69.98	85	30	110.0	1169.19
      500	153.99	82.98	85	30	149.23	1418.58
      600	187.06	93.48	108.73	30.22	180.6	1632.54

      TABLE VI

      POWER OUTPUT & FUEL COST FOR A 5 GENERATOR SET FOR EMISSION DISPATCH CALCULATED FROM LAMBDA ITERATION METHOD

      Demand (MW)	P1	P2	P3	P4	P5	Total Emissions (kg/hr)
      400	68.78	55.02	116.95	52.93	106.31	318.99
      500	88.40	70.72	139.38	66.01	135.75	451.89
      600	118.67	40	173.95	86.17	181.12	670.01

      Table VII shows the values of powers generated by each of the generator in five generator set calculated from PSO for economic dispatch. Table VIII shows the power output of each unit for emission dispatch calculated from PSO.

      TABLE VII

      POWER OUTPUT & FUEL COST FOR A 5 GENERATOR SET FOR ECONOMIC DISPATCH CALCULATED FROM PSO

      Demand (MW)	P1	P2	P3	P4	P5	Fuel cost (Rs/hr)
      400	123.30	61.20	87.23	32.32	91.74	1168.33
      500	189.33	70.47	84.78	75.33	80.00	1430.66
      600	185.38	93.15	99.69	44.68	175.28	1631.85

      TABLE VIII

      POWER OUTPUT & FUEL COST FOR A 5 GENERATOR SET FOR EMISSION DISPATCH CALCULATED FROM PSO

      Dema nd (MW)	P1	P2	P3	P4	P5	Total Emissi ons (kg/hr)
      400	70.35	53.52	115.34	54.45	105.53	317.32
      500	89.58	135.64	135.42	68.53	69.76	450.28
      600	116.45	55.64	170.53	88.53	170.45	652.91

   3. Comparison of Results obtained from Lambda Iteration Method and PSO technique

   The results from both the techniques were compared and the comparison results are presented in Table IX for three generator set and in Table X for five generator set.

   TABLE IX

   COMPARISON BETWEEN LAMBDA ITERATION METHOD & PSO FOR 3 GENERATOR SET

   Demand (MW)	Fuel Cost (Rs/hr)		Total Emissions (Kg/hr)
   	Lambda Iteration Method	PSO	Lambda Iteration Method	PSO
   200	858.43	858.38	446.08	443.33
   250	1060.27	1058.22	472.26	471.70
   300	1273.27	1273.21	515.74	510.07

   TABLE X

   COMPARISON BETWEEN LAMBDA ITERATION METHOD & PSO FOR 5 GENERATOR SET

   Demand (MW)	Fuel Cost (Rs/hr)		Total Emissions (Kg/hr)
   	Lambda Iteration Method	PSO	Lambda Iteration Method	PSO
   400	1169.19	1168.34	318.99	317.33
   500	1418.58	1403.66	451.89	450.28
   600	1632.54	1631.85	670.02	652.91

6. CONCLUSION

In this paper, two optimization techniques (lambda iteration and PSO) have been implemented to solve economic emission dispatch problem and the obtained results have been compared. Lambda iteration method is a conventional method but PSO is a new optimization technique which is a population based search algorithm. PSO shows better results along with fast convergence characteristics hence the optimized results of PSO are better than lambda iteration method. As far as fuel cost is concerned, it is small for three generators set but it is reasonably good for five generators set.

APPENDIX

The Particle Swarm Optimization parameters used are: Population size 100

No. of iterations 80

Cognitive coefficient, C1 2

Social coefficient, C2 2

Inertia weight, w : 1 for 3 gen. set, 1.5 for 5 gen. set

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