multi objective economic load dispatch problem using a ... · 4 [email protected] may...

1

Multi Objective Economic Load Dispatch problem using

A-Loss Coefficients

D.Poornima1, Sishaj P. Simon2

B.Sonia3 ,T.Sunita4

1,3,4Vignan’s Institute of Information

Technology, Duvvada.

2NIT, Tiruchirapalli, [email protected] [email protected] [email protected] [email protected]

May17,2017

Abstract

Multi Objective Economic Load Dispatch (MOELD) is

one of the main objectives of power system operation

while dispatching the output power of various

generating units. The main objective of this problem is

to minimize the Generation Cost, Emissions of fossil

fuel plants and Transmission losses in the network.

This problem is an extension of Economic-Emission

dispatch problem which also includes the minimization

of transmission losses. In this paper, the transmission

losses are evaluated using nominal A-loss coefficients

which can be derived for any transmission network

from the knowledge of load flow analysis at few

operating conditions using perturbation method. As

the evaluation of these loss coefficients involves more

than one operating conditions of the network in

contrast to that for conventional B-loss coefficients,

International Journal of Pure and Applied MathematicsVolume 114 No. 8 2017, 143-153ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

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these are proven to be accurate in calculating

transmission losses. So, these A-loss coefficients are

used in this paper for solving MOELD problem.

Conventional Weighed Sum (WS) approach and

Strength Pareto Genetic Algorithms (SPGA) are used

to solve the problem of MOELD and the effectiveness

of the algorithms are compared based on the results

obtained for IEEE 30 bus system with 6 generating

units.

Key Words : Multi Objective Economic Load

Dispatch, A-Loss coefficients, Economic-Emission

dispatch, Weighed sum approach, Strength Pareto

Genetic Algorithm

1 Introduction:

Economic Load dispatch problem plays a key role in load dispatch

process which minimize the cost of generation by suitable

scheduling of committed generating units, while satisfying different

operational constraints. Due to the U.S, Clean Air Act amendments

1990, the power generation companies using fossil fuels are enforced

to revise their strategies such that atmospheric emissions are

reduced. So, the emissions are included as one of the objectives to be

minimized. As the average transmission losses can also be

minimized by properly distributing the generation among various

power plants, losses are also considered as one of the objectives to be

minimized in this paper. By considering these three objectives at

a time, the problem is converted into a Multi Objective Optimization

Problem (MOOP).

The transmission losses of power system network are conventionally

calculated using B-loss coefficients which are derived at a particular

operating condition of the network by making some assumptions. So,

the calculation of transmission losses is not so accurate with these

loss coefficients. In literature, A-loss coefficients have been proposed

[1], [2] to be effective in calculating transmission losses. These

coefficients are evaluated using perturbation method by considering

more than one operating conditions of the power system network

which makes it effective and accurate in calculating losses.

So, A-loss coefficients are used in solving MOELD problem in this

paper, which are proven to be very effective. The Multi Objective

Optimization Problems (MOOP)have been solved using different

techniques in literature. Blaze Gjorgiev& Marko Cepin.,[3] proposed

Weighted Sum approach to solve the Economic-Environmental

power dispatch and the constraints are handled using a penalty

function. M.S. Osmana, M.A. Abo-Sinnab, A.A. Mousab., [4]

International Journal of Pure and Applied Mathematics Special Issue

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presented a novel Multi Objective Genetic Algorithm for solving

Economic-Emission load dispatch problem in which the non-

dominated solutions iteratively updated based on the concept of ε-

dominance. M. A. Abido, [5]presented a new methodology based on

Strength Pareto Evolutionary Algorithm (SPEA) to solve the

Economic -Environmental power dispatch problem which uses a

clustering algorithm to bring about Pareto-optimal solution. A best

compromise non dominated solution is extracted using Fuzzy set

theory.

In this paper, the three objective MOELD problems is solved by

using A-loss coefficients with Weighed Sum approach and Strength

Pareto Genetic Algorithms which are proven to be effective in

literature and results are compared to justify the effective method.

The IEEE 30 bus system with 6 generating units is chosen as the

test system to carry out the simulations

2 Formulation of Multi Objective Economic

Load dispatch (MOELD)

The MOELD problem minimizes three competitive objectives

while satisfying different operational constraints and is

formulated as below.

A. Objective functions:

a) Cost of generation:

The cost of generation of thermal plants can be expressed as a

quadratic function of its real power output (Pgi) and is given by

Eq (1)

.

where, nt is number of generating units, ai, bi, ci are cost

coefficients of ith generating unit.

b) Emissions of pollutants:

The amount of pollutants released into atmosphere can be

expressed as a function of the output power of the plant (Pgi) with

the help of emission coefficients (α, β, ) as given by Eq (2).

2

21

( ) ( ) ( ) / ..............(2)

nt

gi i i gi i gii

F P P P ton h

2

11

( ) ( ) ( ) $ / ..............(1)

nt

gi i gi i gi ii

F P a P b P c h


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c)Transmission losses:

The transmission losses can be expressed as a function of

output power(Pgi) of generating units with help of A-loss

coefficients as given in Eq (3).

2

31

( ) ( ) ............(3)

nt

gi i gii

F P AP MW

B. Constraints:

a) Equality constraint:

This is also calledPower balance constraint which sets the total

power generation to be equal to t sum of load demand and

transmission losses in the network. It is expressed as in Eq (4).

1

..................(4)

nt

gi load lossi

P P P

where,Pload is the total demand and Ploss is the transmission

losses calculated using Eq (3).

3 Solution Methodologies

The Multi Objective Optimization Problem can be solved using

various methods like conventional weighed sum approach in

which a single objective function is defined which is a weighed

combination of the objectives of the problem and evolutionary

algorithms with pareto set approach. These methodologies are

explained below in brief.

A. Weighed Sum Approach:

In this approach, the MOOP problem can be solved by

converting it to a one objective optimization problem by using the

linear combination of all objectives as a weighed sum such that

𝑤𝑖𝑚𝑖=1 = 1 and wi 0 (i=1,2,…,m) where m is the number of

objective functions and wi is the weighing coefficient of ith

objective function. This technique requires well known domain

knowledge to assign appropriate weighing coefficients to each

objective function. So this problem is solved using different


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weightage combinations as shown in Table1 to locate non-inferior

solution set. The best compromise weightage combination is

determined with a fuzzy mechanism known as membership

function. In this paper, Newton-Raphson method and Real coded

Genetic algorithms are chosen to solve the problem using this

approach.

a) Newton - Raphson method:

In this method, the constrained MOELD problem which is

formed as a single objective function F is altered into

unconstrained scalar optimization problem using Lagrangian

multiplier function as shown in Eq (6). The optimality conditions

are derived by taking the partial derivatives of this augmented

objective function with respect to Pgi, λ.

The augmented objective function is given by,

𝑳 = 𝑤𝑖𝐹𝑖 + 𝜆(𝑃𝑙𝑜𝑎𝑑 + 𝑃𝑙𝑜𝑠𝑠 −𝑚𝑖=1

𝑃𝑔𝑖𝑛𝑔𝑗= ) ………… . . (6)

b) Real coded Genetic Algorithm (GA):

It is a randomised search algorithm which is guided by the

principle of natural genetic systems. This algorithm is robust and

requires no auxiliary information and can offer significant

advantages in solution methodologies. The process of GA is

explained as follows to solve this MOELD problem.

1) Generate a random feasible solution set which is known as

population

2) Assign fitness value to each member of the population based on

its evaluation.

3) Select solutions with lowest fitness value (value of F which is the

weighed combination of three objective functions) to be parent the

new solutions during reproduction process.

4) The new solution set replaces the less fitted old solutions based

on selection rate.


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5) Continue the process from step 2 till the convergence criterion is

satisfied.

Non Pareto solution set is formed for different combinations of

weightages among which the optimum solution is extracted based

on the membership value.

c) Membership function:

This is one of the effective method in Fuzzy logic which

derives Pareto-optimal solution from a group of non-inferior

solutions. The membership function of ith objective function of a

solution is given by Eq (7),

𝜇 𝐹𝑖 =

1 ∶ 𝐹𝑖 ≤ 𝐹𝑖 ,𝑚𝑖𝑛𝐹𝑖 ,𝑚𝑎𝑥 −𝐹𝑖

𝐹𝑖𝑖 ,𝑚𝑎𝑥 −𝐹𝑖 ,𝑚𝑖𝑛 ∶ 𝐹𝑖 ,𝑚𝑖𝑛 < 𝐹𝑖 < 𝐹𝑖 ,𝑚𝑎𝑥

0 ∶ 𝐹𝑖 ≥ 𝐹𝑖 ,𝑚𝑎𝑥

……………(7)

Where 𝐹𝑖 ,𝑚𝑎𝑥 and 𝐹𝑖 ,𝑚𝑖𝑛 are the maximum and minimum

values of ith objective function. The normalized membership

function corresponds to kth non-dominated solution is given by

∑∑

∑K

1k

m

1i

ki

m

1i

ki

kD

)F(μ

)F(μ

μ

= =

==

……………(8)

The solution with maximum kDμ value is chosen as the Pareto

optimal solution.

B) Strength Pareto Genetic Algorithm (SPGA):

It is one of the potential algorithm for Multi Objective

Optimization Problems which works based on the Pareto set

approach. In this approach, non-dominated solution set is

determined using Pareto dominance principle which is defined for

a minimization problem as,

∀i∈{1,2,….m} : fi(x1) ≤ fi(x2)

∃j∈{1,2,….m}: fi(x1)<fi(x2)

Here x1 is known as non-dominated solution within the set


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{x1, x2}.The non dominatedsolution within the entire search space

is known as Pareto-optimalsolutions. A value known as strength

is assigned to each solution within the range [0,1) for evaluation.

This strength is defined to be proportional to the number of

solutions covered by it. The fitness of an individual is calculated

asthe sum of the strengths of all external Pareto solutions by

which it is covered. The steps to be followed to solve MOELD

problem using SPGA method are explained below.

1) An initial population of random feasible solutions is generated.

2) The non-dominated solutions are identified using dominance

principle to update the archive set.

3) Assign fitness value to each member and sort out the population

from maximum to minimum fitness value.

4) The generation values correspond to maximum fitness value is

considered as pareto

optimal solution.

C) Steps to form initial feasible solution set using A-loss coefficients:

1. Create the population of real power outputs of generators except

for slack bus power (Pg1)

2. Assume Pg1=0.

3. Calculate losses by using Eq (3).

4. For each combination of chromosomes, calculate the generation

of the slack bus (Pg1) by

using the Eq (10).

1

2

...............(10)

nt

g load losses gii

P P P P

5. Calculate the losses by using Eq (3).

6. Calculate the difference in losses evaluated in steps 3 and 5. If

it is more than 0.0001, go to

step 3 and repeat the steps. Otherwise stop the process.

4. Simulation Results:

The Multi Objective Economic Load Dispatch problem is solved


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for IEEE-30 bus system at a total load of 283.4MW, using

SCILAB-5.4.1 software. The transmission losses are calculated

using nominal A loss coefficients. Newton-Raphson method (NR

method), Genetic algorithms are used to solve the problem with

Weighed Sum approach and Strength Pareto Genetic Algorithm

is also used to solve the problem and results are compared with

these algorithms.

Table 1

Different weightage combinations

Combination

number w1 w2 w3

1 1 0 0

2 0 1 0

3 0 0 1

4 0.85 0.15 0

5 0.7 0.3 0

6 0.55 0.45 0

7 0.4 0.6 0

8 0.25 0.75 0

9 0.1 0.9 0

10 0.85 0 0.15

By applying membership technique, the most optimal solution

obtained by these three methods is determined. The respective

normalized membership values.

0246810

0200400600800

1000

Cost of generation

Emissions Losses

Lo

sses

(M

W)

Co

st (

$/h

), E

mis

sio

n

s(to

n/h

)

Objective considered to be minimized

Non dominated solutions


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Fig 1: Non-inferior solutions with NR method and WS approach

From the membership values, we can justify the effectiveness of

Strength Pareto Genetic algorithm in solving Multi Objective

Optimization problems.

5. Conclusion:

In this paper, the formulation and implementation of Multi

objective optimization problems have been explained. The Multi

Objective Economic Load Dispatch problem is solved in this paper

which optimizes three objectives, Cost of generation, Emissions

and the Transmission losses of power system operation. The

losses are calculated using nominal A loss coefficients. Both

conventional and evolutionary algorithms are used to solve the

problem with Weighed Sum approach and Strength Pareto

Genetic Algorithm. The use of A loss coefficients in calculating

transmission losses, while solving Multi objective ELD is

attempted successfully. The feasibility of the above algorithms

has been checked by validating them in IEEE 30 bus system. It is

observed that Strength Pareto Genetic algorithm gives better

result when compared with Weighed Sum approach.

6. References:

[01] J.Nanda., L.L. Lai, “A novel approach to computationally efficient

algorithms for transmission loss and line flow formulations,”

Elsevier, Electrical Power and Energy Systems 21 (1999) 555–

560.

[02] C. H. Ram Jethmalani, PoornimaDumpa, Sishaj P. Simon and K.

Sundareswaran, “Transmission Loss Calculation using A and B

Loss Coefficients in Dynamic Economic Dispatch Problem,”Int. J.

Emerg. Electr. Power Syst. DOI 10.1515/ijeeps-2015-01812016

0

5

10

15

0

200

400

600

800

1000

1 4 7 10 13 16 19 22 25 28

Lo

sses

, M

W

Co

st

($/h

), E

mis

sio

ns(

ton/h

s

Weightage combination number

NR method


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[03] Blaze Gjorgiev& Marko Cepin., “A multi-objective optimization

based solution for the combined economic-environmental power

dispatch problem”, Engineering Applications of Artificial

Intelligence 26 (2013) 417–429.

[04] M.S. Osmana, M.A. Abo-Sinnab, A.A. Mousab,“An ε-dominance-

based multi objective genetic algorithm for economic emission

load dispatch optimization problem”, Electric Power Systems

Research 79 (2009) 1561–1567.

[05] M. A. Abido., “Environmental/Economic power dispatch using

multi objective evolutionary algorithms”, IEEE transactions on

power systems, Vol. 18, No. 4, November 2003

[06] M. A. Abido., “Multi objective evolutionary algorithms for Electric

power dispatch problem”, IEEE transactions on evolutionary

computation,Vol. 10, No. 3, June 2006

[07] Bin Shi a, Lie-Xiang Yan a, WeiWub, “Multi-objective

optimization for combined heat and power economic dispatch

with power transmission loss and emission reduction”, Energy 56

(2013) 135e143

[08] B.Venkateswara Rao , G.V.Nagesh Kumar , M.Ramya Priya ,

and P.V.S.Sobhan, "“Implementation of Static VAR Compensator

for Improvement of Power System Stability”", International

Conference on Advances in Computing, Control, and

Telecommunication Technologies, ACT 2009 organized by

ACEEE and CPS, Trivandrum, Kerala, India, , 28-29 December,

2009, Pages: 453-457.

[09] M. A. Abido, “A new multi objective evolutionary algorithm for

environmental/economic power dispatch” IEEE Power Eng. Soc.

Summer Meeting, Vancouver, BC, Canada, Jul. 15–19, 2001.,

1263–1268.


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