Economic Load Dispatch Using

PSO

ContentsIntroductionProblem formulationConventional ApproachPractical swarm optimizationSteps of ImplementationCase StudiesResult analysisConclusionReferences

Introduction The main objective of ELD is to distribute

the total power demand among the generating units by minimizing the selected cost criteria, subject to load and operation constraints.

Traditional methods for solving ELD problem Lambda iteration methodGradient based methodDynamic programming method

Practical swarm optimization(PSO) is suggested by Kennedy & Eberhart based on analogy of Swarm of birds and school of fish

PSO mimics the behavior of individuals in swarm to maximize the survival of species. This method is based on Metaphor of social interaction

Main advantages of PSO algorithm are,Easy implementationRobustness to control parametersCompetition efficiencyApplicable to non-linear & non-continuous

PROBLEM FORMULATIONThe objective of economical load dispatch

problem is to minimize the total fuel cost by satisfying the system constraints.

i.e. Min of subject to

An objective function is defined by augmenting with equality constraints and Lagrangain multiplier(λ)---Lambda Iteration method

Neglecting Transmission line Losses The objective function for the above

economic load dispatch problem can be written as

The condition for optimal system operation

= = ….= λ

Including Transmission Line Losses the Objective function for optimal

operation of the system including the transmission line losses is

In this case the condition for optimal operation of the system is modified as

= λ where,

= = Incremental Transmission line losses

PARTICLE SWARM OPTIMIZATION Conventional methods have essential assumption that is

incremental cost curves of the units are monotonically increasing piecewise-linear functions.

PSO uses a number of solutions (particles) that constitute a swarm and looks for best solution.

Particle (X): It is a candidate solution represented by an m-dimensional vector

Population, Pop (t): It is a set of n particle at time t, i.e.

i i1 i2 inX (t) [X (t),X (t), X (t)]

1 2 nPop(t) X (t),X (t), X (t)

Continued…Each particle keeps track of its best position that it has

so far achieved. 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 particle in the neighborhood of that particle. This value is called Gbest

After finding the two best values, the particle updates its velocity and positions

1.

2.

(u 1) (u) (u)i i 1 i i

(u)2 i i

v w * v C * rand()*(pbest p )C * rand()*(gbest p )

(u 1) (u) (u 1)i i ip p v

Continued… The inertia weight ‘W’ is set according to the

following equation

Inertia weight W is introduced to enable the swarm to fly in the larger search space. W often decreases linearly from about 0.9 to 0.4 during a run.

max minmax

max

w ww w *ITERITER

Steps of Implementation1. Initialize the fitness function i.e.., total cost

function from the individual cost function of the various generating stations

2. Initialize the PSO parameters population size c1,c2 , error gradient etc.,

3. Input the fuel cost functions MW limits of generating stations along with b-coefficient matrix and the total power demand

4. At the first step of execution problem a large no of vectors of active power satisfying the MW limits are randomly allocated

5. For each vector of active power the value of fitness function is calculated all values obtain in an iteration are compared to obtain p-best . At each iteration all values of the whole population till then are compared to obtain the g-best . At each step these values are updated

6. At each step error gradient is checked and the value of g-best is plotted till it comes with in the pre specified range

7. This final value of g-best is the ,minimum cost and active power vector represents the economic load dispatch solution

Flow Chart

Case studies

Unit a b c

1 600 100 0.00156 7.95 561

2 400 100 0.00194 7.85 310

3 200 550 0.00482 7.97 78

 

PSO ParametersFor each load, 20 trials were performed to

observe the evolutionary process and to compare their solution quality.

PARAMETERS:Population Size: 100Maximum No Of Iteration: 20Inertia Weight Factor (W): Wmax=0.9 & Wmin=0.4Acceleration Constant: C1=2 & C2=2Error Gradient: 1e-06

Result Analysis of 3 Unit SystemTransmission Line Losses Neglected case

Transmission Line Losses Included case

S.No Load(MW) (Rs/hr)

PSO Method(Rs/hr)

1 585 5821.44 5821.4395

2 700 6838.41 6838.404351

3 800 7738.50 7738.4946

S.No Load(MW)

λ-Method(Rs/Hr)

PSO Method(Rs/hr)

1 585 5886.94 5886.911604

2 700 6934.79 6934.78119

3 800 7867.23 7867.202213

Unit Pmax (MW)

Pmin (MW)

a ($/MW2)

b ($/MW)

c (MW)

1 125 10 0.15240 38.53973 756.79886

2 150 10 0.10587 46.15916 451.32513

3 225 35 0.02803 40.39655 1049.9977

4 210 35 0.0354 38.30553 1243.5311

5 325 130 0.02111 36.32782 1658.5596

6 315 125 0.01799 38.27041 1356.6592

ij

0.000140 0.000017 0.000015 0.000019 0.000026 0.000022 0.000017 0.000060 0.000013 0.000016 0.000015 0.000020 0.000015 0.000013 0.000065 0.000017 0.000024 0.000019 B 0.000019 0.000016 0.000017 0.000071 0.000030 0.000025 0.000026 0.000015 0.000024 0.000030 0.000069 0.000032 0.000022 0.000020 0.000019 0.000025 0.000032 0.000085

Case study-2: 6 units Thermal power plant The Characteristics of the plant and b-coefficients

Result Analysis of 6 Unit System

S.No Load(MW) Lambda method(Rs/hr) PSO method(Rs/hr)

1 800 40675.97 40675.9682

2 900 45464.09 45464.08097

3 1000 50363.71 50363.69128

• Transmission Line Losses neglected case

S.No Load(MW) Lambda method(Rs/hr) PSO method(Rs/hr)

1 800 41896.63 41896.62871

2 900 47045.16 47045.15634

3 1000 57871.60 57870.36512

• Transmission Line Losses included case

Reliability & ConvergenceReliability of 6 unit system Reliability of 3 unit system

Convergence of 6 unit system Convergence of 3 unit system

ConclusionIn PSO method the convergence tend to be

improving as the complexity of the system increases.

The solution for the higher order systems can be obtained in much less time duration than the conventional Lambda iteration method

Irrespective of the run of the program it is capable of obtaining same result for the problem(Reliability)

References Swarup K.S. , Rohit Kumar P., “A new

evolutionary computation technique for economic dispatch with security constraints” ;Electrical Power and Energy Systems 28 (2006) pp 273–283

Wadhwa, C.L (2009). Electrical Power System. New Age publishers.

D. N. Jeyakumar, T. Jayabarathi, and T. Raghunathan, “Particle swarm optimization for various types of economic dispatch problems,” Int. J. Elect. Power Energy Syst., vol. 28, no. 1, pp. 36–42, 2006.