Economic Dispatch

Contents:

• Economic Dispatch definition.

• Basic Methods of Economic Dispatch.

• Economic Dispatch problem formulation.

• Particle Swarm Optimization.

• Particle Swarm Algorithm and its application to Economic Dispatch.

• Test Systems.

• Simulation Results.

• Conclusion.

• Further presentation.

• References.

Economic Dispatch:

The determination of the most optimal sharing of theload between the given generators.

The main objective of the Economic Dispatch problemis to minimize the total cost of generation whileconsidering all the units to be ON.

Basic methods of Economic Dispatch:

The Basic concept involved in Economic Dispatch problemis that the incremental costs of all the generators should beequal for a given load.

The basic methods are:

• Lambda iteration method or Lagrange Relaxation method.• Gradient method.• Newton’s double gradient method.

Economic Dispatch problem formulation:

Aim: To minimize the total cost of generation.

Subject to the constraints:

Power balance constraint:

Power generation limit constraint:

Lagrange Relaxation method:Lagrange function:

Solution:

Step 1:

This results in the following relation with (N-1) equations:

Step 2: The Nth equation will be the power balance constraint equation:

Step 3: Solve the ‘N’ equations to get the optimal solution.

Example problem:

Consider 3 generators with the following cost functions:

Solution:

Step 1: Equate the incremental costs of the generators.

Step 2:

Load Demand is 850 MW.

Step 3: Solve the equations obtained, the solution will be:

Advantages and drawbacks of these methods:

• The main advantage of these methods are that theyare accurate, fast and easy to implement for thetypical cost function characteristic (quadratic costfunction) of the generator.

• The main drawback of these methods is that theybecome very difficult when the cost function curvedeviates from quadratic nature which occurs moreoften practically.

Types of fuel cost functions used:

1). Quadratic fuel cost function:

2). Cubic fuel cost function:

3). Fuel cost function with valve point loading:

Particle Swarm Optimization (PSO):

• Proposed by James F. Kennedy and Russell .C Eberhart in March1995 [5].

• The algorithm imitates a swarm of birds searching for food.

• Similarly while performing an optimization problem (Forexample: Economic Dispatch), a set of potential solutions calledparticles are initialized in the problem space and these particlessearch for the optimal solution.

Rules for choosing Particles:

• The independent variables are taken as the dimensionsof the particle.

• The dependent variables are to be estimated fromindependent variables.

What is a particle?

A particle is a possible solution for a given optimization problemthat exists in the given problem space.

Problem space is the space in which all the constraints involvedin the problem are satisfied by the independent variables andthe dependent variables.

Particle velocity:

Particle velocity is the rate of change of position of a givenparticle. Using particle velocity, we can update the positions ofthe particles and find new possible solutions (ultimately thebest or optimal solution).

Without particle velocity, we can’t find new possible solutions.

The Particle swarm algorithm:

Step 1: Initialize different particles in the given problem space (The coordinatesof each particle are the output powers of the generators in case of EconomicDispatch Problem).

Step 2: Evaluate the objective function (In case of Economic Dispatch Problem,the total cost of generation) at each and every particle.

Step 3: Initialize every particles best value as their current position and the globalbest position as the overall best (overall least cost) among all the particles.

Step 1 and Step 2 Step 3

Step 4: If it is the first iteration, then initialize the velocities of the particles,otherwise update the velocity of the particle using the velocity update equation.

Step 5: Update the position of the particle using the position update equation.

Step 6: Go to step 2. After repeating step 2, check if the particles position isbetter than it’s previous position. If it is better than it’s previous position, thenupdate it’s best value to the current position.

Stopping criteria: The maximum number of iterations, initialized beforesimulation.

Step 4 and Step 5 Step 6 and Step 2 Step 3

Example problem:

Consider 3 generators with the following cost functions:

Solution:Load Demand is 850 MW.

Step 1: Take P1,P2 and P3 as the dimensions of the particle. Consider P3 asthe slack generator if its limits are not violated. Otherwise either P1 or P2

will be the slack generator (in order to satisfy the power balance constraint).

Step 2: Initialize various particles in the problem space and assign initialvelocities to all the particles.

Step 3: The Objective function is the total cost of generation and henceimplement the PSO algorithm.

Path of the particles:

Optimal solution of Generator output powers

Position updates of the particles:

Iter 1

Iter 2

Iter 4

Initial positionIter 3

Gbest updates of the population:

Optimal solution of Generator output powers

Gbest 3

Gbest 2

Gbest 1Initial Gbest

Advantages of Particle Swarm Optimization:

1. It is the latest, the most robust and the most efficientsearch algorithm, among all the optimization techniqueslike Genetic algorithm, linear programming, dynamicprogramming, etc.

2. It is easier to implement and can obtain better optimalsolutions for many optimization problems in ElectricalEngineering.

Application of Particle Swarm Optimization(PSO) to Economic Dispatch problem:

Aim: To minimize the total cost of generation.

Subject to the constraints:Power balance constraint:

Power generation limit constraint:

Particle: All the generators sharing the load are the members ordimensions of the particle.

P1, P2, P3,……,PN

Particle members

Velocity update equation:

Position update equation:

Where,

pbestij – The current best position of the particle ‘i’ and generator ‘j’.gbestj – The current global best position of the generator ‘j’.w – Inertia weight factor.iter - Current iteration.np – Number of particles.N – Number of generators in the given system.c1, c2 – Acceleration constants.rand1, rand2 – Random numbers generated between ‘0’ and ‘1’.

i= 1,2,….np, j=1,2,….N

Inertia Weight (w):

The concept of inertia was proposed by Russel C. Eberhart and Y. Shiin 2000 [6].

The inertia weight can be either implemented as a fixed value orcan be dynamically changing.

Essentially, this parameter controls the exploration of the searchspace, therefore an initially higher value allows the particles tomove freely in order to find the global optimum neighborhood fast.

Once the optimal region is found, the value of the inertia weightcan be decreased in order to narrow the search, shifting from anexploratory mode to an exploitative mode. Commonly, a linearlydecreasing inertia weight has produced good results in manyapplications.

Inertia weight (w):

The formula for inertia weight is given by:

Where,wmax – Initial inertia weight.wmin – Final inertia weight.iter – Current iteration.itermax – Maximum number of iterations.

Iter 1

Iter 2

Iter 4

Initial positionIter 3

With Inertia Weight:

The solution converged at iteration 84. The solution obtained is:

Without Inertia Weight:

The solution converged at iteration 158. The solution obtained is:

Flow Chart for Economic Dispatch problem using PSO:Start

Initialize a population of particles containing the power outputs of all the generators

Initialize parameters such as the size of population, initial/final inertia weights, particle velocity and acceleration constants.

Evaluate the cost function at each and every particle’s current position.

Compare each individual’s evaluation value with it’s previous best value (pbest). If it is better than it’s previous position, update pbest.

Evaluate gbest, the best among the pbests of all the particles.

Update the velocities and positions of all the particles using the velocity and position update equations.

The stopping criteria is the total number of iterations initialized

before simulation.

Stopping criteria satisfied?

Yes

No

Stop

1). Quadratic fuel cost function:

System 1:

Total load demand = 850 MW

S.No Pmin (MW) Pmax (MW) ai bi ci

1. 150 600 561 7.92 0.001562

2. 100 400 310 7.85 0.001940

3. 50 200 78 7.97 0.004820

System 2:


S.No Pmin (MW) Pmax (MW) ai bi ci

1. 150 600 459 6.48 0.00128

2. 100 400 310 7.85 0.00194

3. 50 200 78 7.97 0.00482


System 1:


S.No Pmin (MW)Pmax

(MW)ai bi ci di

1. 320 800 749.55 6.95 9.68 X 10-4 1.27 X 10-7

2. 300 1200 1285 7.051 7.38 X 10-4 6.453 X 10-8

3. 275 1100 1531 6.531 1.04 X 10-3 9.98 X 10-8


System 1:


S.No Pmin (MW) Pmax (MW) ai bi ci ei fi

1. 100 600 561 7.92 0.001562 300 0.0315

2. 100 400 310 7.85 0.001940 200 0.0420

3. 50 200 78 7.97 0.004820 150 0.0630

System 2:


S.No Pmin (MW) Pmax (MW) ai bi ci ei fi

1. 36 114 94.705 6.73 0.00690 100 0.084

2. 36 114 94.705 6.73 0.00690 100 0.084

3. 60 120 309.54 7.07 0.02028 100 0.084

4. 80 190 369.03 8.18 0.00942 150 0.063

5. 47 97 148.89 5.35 0.01140 120 0.077

6. 68 140 222.33 8.05 0.01142 100 0.084

7. 110 300 287.71 8.03 0.00357 200 0.042

8. 135 300 391.98 6.99 0.00492 200 0.042

9. 135 300 455.76 6.60 0.00573 200 0.042

10. 130 300 722.82 12.9 0.00605 200 0.042

11. 94 375 635.20 12.9 0.00515 200 0.042

12. 94 375 654.69 12.8 0.00569 200 0.042

13. 125 500 913.40 12.5 0.00421 300 0.035

14. 125 500 1760.4 8.84 0.00752 300 0.035

15. 125 500 1728.3 9.15 0.00708 300 0.035

16. 125 500 1728.3 9.15 0.00708 300 0.035

17. 220 500 647.85 7.97 0.00313 300 0.035

18. 220 500 649.69 7.95 0.00313 300 0.035

19. 242 550 647.83 7.97 0.00313 300 0.035

20. 242 550 647.81 7.97 0.00313 300 0.035

21. 254 550 785.96 6.63 0.00298 300 0.035

22. 254 550 785.96 6.63 0.00298 300 0.035

23. 254 550 794.53 6.66 0.00284 300 0.035

24. 254 550 794.53 6.66 0.00284 300 0.035

25. 254 550 801.32 7.10 0.00277 300 0.035

26. 254 550 801.32 7.10 0.00277 300 0.035

27. 10 150 1055.1 3.33 0.52124 120 0.077

28. 10 150 1055.1 3.33 0.52124 120 0.077

29. 10 150 1055.1 3.33 0.52124 120 0.077

30. 47 97 148.89 5.35 0.01140 120 0.077

31. 60 190 222.92 6.43 0.00160 150 0.063

32. 60 190 222.92 6.43 0.00160 150 0.063

33. 60 190 222.92 6.43 0.00160 150 0.063

34. 90 200 107.87 8.95 0.0001 200 0.042

35. 90 200 116.58 8.62 0.0001 200 0.042

36. 90 200 116.58 8.62 0.0001 200 0.042

37. 25 110 307.45 5.88 0.0161 80 0.098

38. 25 110 307.45 5.88 0.0161 80 0.098

39. 25 110 307.45 5.88 0.0161 80 0.098

40. 242 550 647.83 7.97 0.00313 300 0.035

System 3:


S.NoPmin

(MW)Pmax

(MW)ai bi ci ei fi

1. 0 680 550 8.1 0.00028 300 0.0352. 0 360 309 8.1 0.00056 200 0.0423. 0 360 307 8.1 0.00056 200 0.0424. 60 180 240 7.74 0.00324 150 0.0635. 60 180 240 7.74 0.00324 150 0.0636. 60 180 240 7.74 0.00324 150 0.0637. 60 180 240 7.74 0.00324 150 0.0638. 60 180 240 7.74 0.00324 150 0.0639. 60 180 240 7.74 0.00324 150 0.063

10. 40 120 126 8.6 0.00284 100 0.08411. 40 120 126 8.6 0.00284 100 0.08412. 55 120 126 8.6 0.00284 100 0.08413. 55 120 126 8.6 0.00284 100 0.084

Unit Pmin (MW) Pmax (MW)Generation

(MW)

Cost

($/hour)

1 150 600 393.169842 3916.464844

2 100 400 334.603750 3153.867432

3 50 200 122.226408 1124.023682

Total Generation & Total Cost 850.000000 8194.355469

1). Quadratic fuel cost function:System 1:

GeneratorReference results

Simulation resultsLambda iteration method [8] Modified PSO method [1]

Generation

(MW)Cost ($/hour)

Generation

(MW)

Cost

($/hour)

Generation

(MW)Cost ($/hour)

1 393.2 3916.6389 393.170 3916.3645 393.169842 3916.464844

2 334.6 3153.8069 334.604 3153.8435 334.603750 3153.867432

3 122.2 1123.9103 122.226 1124.1481 122.226408 1124.023682

Total 850.0 8194.3561 850.000 8194.3561 850.000000 8194.355469

System 2:


(MW)

Cost

($/hour)

1 150 600 600 4807.8

2 100 400 187.130170 1846.906174

3 50 200 62.869830 598.124152


Generator Lambda iteration method [8] Simulation results

Generation

(MW)Cost ($/hour)

Generation

(MW)Cost ($/hour)

1 600.0 4807.799805 600.000000 4807.799805

2 187.1 1846.647435 187.139252 1846.984009

3 62.9 598.382896 62.860718 598.046021

Total 850.0 7252.830136 850.000000 7252.829590


System 1:

Unit Pmin (MW) Pmax (MW) Generation (MW) Cost ($/hour)

1 320 800 725.142479 6346.720693

2 300 1200 909.856153 8359.945220

3 275 1100 865.001368 8023.072721


Generator Lambda Iteration method [8] Simulation results

Generation

(MW)Cost ($/hour) Generation (MW) Cost ($/hour)

1 726.9 6361.7858 725.142479 6346.720693

2 912.7 8825.8381 909.856153 8359.945220

3 860.4 7862.9324 865.001368 8023.072721

Total 2500.0 23050.5563 2500.000000 22729.738635


System 1:

Unit Pmin (MW) Pmax (MW) Generation (MW) Cost ($/hour)

1 150 600 300.266900 3087.509906

2 100 400 400.000000 3767.124609

3 50 200 149.733100 1379.437214

Total Generation & Total Cost 850 8234.071729

System 2:Unit Pmin (MW) Pmax (MW) Generation (MW) Cost ($/hour)

1 36 114 110.873094 926.316990

2 36 114 111.206585 931.873081

3 60 120 97.400045 1190.551107

4 80 190 179.733103 2143.550384

5 47 97 87.925632 708.586285

6 68 140 140.000000 1596.464320

7 110 300 259.602353 2612.933970

8 135 300 284.599877 2779.840728

9 135 300 284.600405 2798.244060

10 130 300 130.000000 2502.065000

11 94 375 168.799904 2959.460472

12 94 375 94.000000 1908.166840

13 125 500 214.759790 3792.070018

14 125 500 304.519592 5149.699318

15 125 500 394.279370 6436.586289

16 125 500 394.279370 6436.586289

17 220 500 489.279396 5296.711310

18 220 500 489.279496 5288.767855

19 242 550 511.279486 5540.931726

20 242 550 511.279396 5540.909788

21 254 550 523.279393 5071.290149

22 254 550 523.279593 5071.294204

23 254 550 523.279490 5057.225500

24 254 550 523.279389 5057.223473

25 254 550 523.279390 5275.088941

26 254 550 523.279386 5275.088854

27 10 150 10.000000 1140.524000

28 10 150 10.000000 1140.524000

29 10 150 10.000000 1140.524000

30 47 97 89.062397 727.4469524

31 60 190 190.000000 1643.991252

32 60 190 190.000000 1643.991252

33 60 190 190.000000 1643.991252

34 90 200 200.000000 2101.017035

35 90 200 172.284687 1666.484581

36 90 200 200.000000 2043.727035

37 25 110 110.000000 1220.166122

38 25 110 110.000000 1220.166122

39 25 110 110.000000 1220.166122

40 242 550 511.279380 5540.929436



(MW)

Cost

($/hour)

1 0 680 628.318531 5749.919673

2 0 360 224.399475 2154.834619

3 0 360 297.548894 2780.578148

4 60 180 159.733100 1559.001704

5 60 180 60.000000 716.064

6 60 180 60.000000 716.064

7 60 180 60.000000 716.064

8 60 180 60.000000 716.064

9 60 180 60.000000 716.064

10 40 120 40.000000 474.544

11 40 120 40.000000 474.544

12 55 120 55.000000 607.591

13 55 120 55.000000 607.591


System 3:

Comparison of simulation results:

Fuel cost function with valve point loading:

Unit GA [3] EP [4] MPSO [1]Simulation

Results

1. 300.00 300.26 300.27 300.267

2. 400.00 400.00 400.00 400.000

3. 150.00 149.74 149.73 149.733

Total Power

Generation (MW)850.00 850.00 850.00 850.00

Total Cost of

Generation ($/Hour)8237.60 8234.07 8234.07 8234.07

System 1:

System 2:Method

Total Cost of

Generation ($/Hour)

IFEP [2] 122624.35

MPSO [1] 122252.265

Simulation Result 121441.1761

MethodTotal Cost of

Generation ($/Hour)

IFEP [2] 17994.07

PPSO [8] 17971.01

Simulation Result 17988.92

System 3:

PPSO – Personal best Oriented PSO. It is an improved form of PSOfor which the position update equation will be updated withrespect to pbest.

Conclusions:

1) The Particle Swarm algorithm is a robust algorithm insolving optimization problems like Economic Dispatch.

2) It is capable of producing more optimal solutionswhen compared to Genetic Algorithm (GA), ANN,Dynamic Programming, etc.

3) It is easier to implement in optimization problemswhen compared to other artificial intelligencetechniques.

Further Presentation:

• Particle Swarm Optimization algorithm has been successfullyimplemented for Economic Dispatch problem, but thedisadvantages of Economic Dispatch problem can be overcomewith Unit Commitment problem.

• The Economic Dispatch algorithm can be implemented as auseful function while solving Unit Commitment problem afterthe decision for the ON/OFF status of all the units have beendone.

References:

[1]. Jong-Bae Park, Ki-Song Lee, Joong-Rin Shin, Kwang Y. Lee, “A ParticleSwarm Optimization for Economic Dispatch with Nonsmooth CostFunctions.”, IEEE Transactions on Power Systems, Vol. 20, No. 1, pp 34 –42, February 2005.

[2]. N. Sinha, R. Chakrabarti, and P. K. Chattopadhyay, “Evolutionaryprogramming techniques for economic load dispatch.,” IEEE Trans. Evol.Comput., vol. 7, pp. 83–94, Feb. 2003.

[3]. D. C. Walters and G. B. Sheble, “Genetic algorithm solution ofeconomic dispatch with the valve point loading,” IEEE Trans. PowerSystems, vol. 8, pp. 1325–1332, Aug. 1993.

[4]. H. T. Yang, P. C. Yang, and C. L. Huang, “Evolutionary programming based economic dispatch for units with nonsmooth fuel cost functions.”, IEEE Trans. Power Syst., vol. 11, no. 1, pp. 112–118, Feb. 1996.

[5]. Yamille del Valle, Ganesh Kumar Venayagamoorthy, SalmanMohagheghi, Jean Carlos Hernandez and Ronald G. Harley, “ParticleSwarm Optimization: Basic Concepts, Variants and Applications inPower Systems.”, IEEE Transactions on Evolutionary Computation, Vol.12, No. 2, pp 171 – 195, April 2008.

[6]. J. Kennedy and R. Eberhart, “Particle swarm optimization.”, in Proc.IEEE Int. Conf. Neural Networks. (ICNN), Nov. 1995, vol. 4, pp. 1942–1948.

[7]. R. Eberhart and Y. Shi, “Comparing inertia weights and constrictionfactors in particle swarm optimization.”, in Proc. IEEE Congress Evol.Comput, Jul. 2000, vol. 1, pp. 84–88.

[8]. C. H. Chen and S. N. Yeh, “Particle Swarm Optimization for EconomicPower Dispatch with Valve-Point Effects.”, 2006 IEEE PES Transmissionand Distribution Conference and Exposition Latin America, Venezuela.

[9]. A. J. Wood and B. F. Wollenberg, “Power Generation, Operation, andControl.”, New York: Wiley, 1984.

Acknowledgements:

• Koneru Lakshmaiah College of Engg.• Prof J.Pal.• IIT Kharagpur.

Economic Dispatch

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