an evolutionary programming based neuro-fuzzy technique ... · iii. evolutionary programming...

ISSN (Print) : 2278-8948, Volume-2, Issue-1, 2013

134

An Evolutionary Programming based Neuro-Fuzzy Technique for Multiobjective Generation Dispatch with Nonsmooth Fuel Cost and Emission Level Functions

S.K. Dash

Department of Electrical Engineering, Gandhi Institute for Technological

Advancement,Badaraghunathpur,Madanpur, Bhubaneswar,752054, Orissa, India

Email: [email protected]

Abstract – A combined approach involving an EP based

fuzzy coordination and an ANN(artificial neural network)

methods along with a heuristic rule based search algorithm

has been propounded in this paper in order to obtain the

best fit optimal generation schedules for multiobjective

generation dispatch problem with non-smooth

characteristic functions satisfying various practical

constraints. Initially, the economy objective function is

minimized, followed by minimization of emission level

objective function. Then, both the objectives are mixed

through a fuzzy coordination method to form a fuzzy

decision making (FDM) function. Maximizing the FDM

function then solves the original two-objective problem.

The minimization and maximization tasks of this

optimization problem are solved by the evolutionary

programming technique and the results are trained

through a radial basis function ANN to reach a

preliminary generation schedule. Since, some practical

constraints may be violated in the preliminary stage, a

heuristic rule based search algorithm is developed to reach

a feasible best compromising generation schedule which

satisfies all practical constraints in the final stage. The

proposed EP based neuro-fuzzy technique has been

applied to standard IEEE-30 bus test system and the

results are presented. Simulation results indicate that the

accuracy and the capability of very fast computation of

generation schedule by this technique seem to be very

promising for its suitability for on-line multiobjective

generation dispatching with any kind of characteristic

functions. This technique can be extended to other higher

test case systems as well with suitable assumptions.

Index Terms— Multiobjective generation dispatch, Fuzzy

coordination method, Evolutionary, Programming (EP),

Radial basis function ANN(Artificial Neural Network),

Neuro-fuzzy technique, Heuristic Rule.

I. INTRODUCTION

The purpose of the multiobjective generation

dispatch is to generate the optimal amount of the

generated power for the fossil fuel based generating

units in the system by minimizing the fuel cost and

emission level simultaneously subject to various system

constraints. In this procedure both the objectives conflict

each other. Therefore, it is difficult to handle them by

conventional approaches, which can optimize a single

objective function. Some of the optimization techniques

for multiobjective generation dispatch problem such as

goal programming [1], goal-attainment technique [2],

classical technique based on coordination equation [3],

etc. have been proposed with varying degree of success.

These classical approaches need to introduce a

compromising factor in order to decide the optimal

solution and these results a complicated problem

formulation. Further, these methods are not fast enough

in terms of execution time. Moreover, they do not have

a mechanism to show the vague or „fuzzy‟ preference of

the human decision-maker in obtaining a compromising

solution in presence of such conflicting objectives.

Fuzzy systems provide tools for representing and

manipulating inexact concepts and the ambiguity

prevalent in human interpretation and thought processes.

Further, fuzzy sets [4] can be applied for decision

making in multiple objectives involving various

constraints. Amongst various applications of fuzzy

systems, Srinivasan, Chang and Liew [5] have proposed

a fuzzy optimal search technique for a multiobjective

generation scheduling problem. Many interesting

applications of fuzzy sets in the power field have been

reported in the literature during last decade. Hota et al.

[6] have described a simple and efficient technique

based on fuzzy set theory for the economic emission

load dispatch problem. Further, they have developed an

interactive fuzzy satisfying method [7] to solve

multiobjective generation dispatch problem. The major

advantage of fuzzy technique applied to this kind of

problems lies in having a mechanism to show the

International Journal of Advanced Electrical and Electronics Engineering (IJAEEE)


135

vague/fuzzy preference of the human decision-maker in

obtaining a compromising solution in the presence of

conflicting objectives. However, the execution time of

these fuzzy techniques appears not to be very promising

for real time operation where the execution time is

crucial.

A very fast solution method for multiobjective

generation dispatch problem is only obtained by using

artificial neural network (ANN). Such an approach has

been described by Hota et al. [8]. They have developed

an ANN based method to obtain well-coordinated

economic emission load dispatch solutions suitable in

terms of accuracy and speed. However, this method

lacks the vague/fuzzy preference of the human decision-

maker in obtaining the compromising solutions.

Therefore, an integrated approach combining a fuzzy

coordination and an artificial neural network methods

along has been developed recently [9]. In this work the

authors have proposed a neuro-fuzzy approach, where

the conventional quadratic fuel cost and emission level

functions of the generators are considered, for which

conventional optimization technique has been used. The

major disadvantage of this approach is its incapability of

handling non smooth fuel cost and emission level

functions. On the other hand, if the non-linear fuel cost

and emission level functions are considered, then the use

of global optimization techniques such as genetic

algorithm, simulated annealing and evolutionary

programming may be undertaken. Wong et al [10], have

successfully applied simulated annealing to economic

dispatch problems. Wong and Fung [11], have

demonstrated a method to solve short-term

hydrothermal scheduling problem by using simulated

annealing optimizaton technique. Hota et al [12], have

described a novel method for economic emission load

dispatch with nonsmooth fuel cost and emission level

functions using a simulated annealing based goal-

attainment method. Wong et al [13], have proposed a

combined genetic algorithm/ simulated annealing/ fuzzy

set approach to short-term generation scheduling with

take-or-pay fuel contract. In their work, they have

considered nonsmooth cost functions of the generating

units owing to the effects of valve-point loading. In

recent years, another powerful optimization technique

called as evolutionary programming (EP) is being

continuously applied to various power system

optimization problems due to its more powerful ability

in finding the global optimum solutions as compared to

genetic algorithm or simulated annealing technique.

Yang et al [14], have developed an efficient general

economic dispatch algorithm for units with nonsmooth

fuel cost functions based on EP technique. In this work

the authors have compared the results of ED problems

when solved by genetic algorithm, simulated annealing

and EP. They have shown that the EP method is able to

give a cheaper schedule at a less computation time.

Hanzheng et al [15], described a solution method for

unit commitment using Lagrangian relaxation combined

with evolutionary programming. Hota et al [16], have

developed an evolutionary programming based

algorithm for solution of short-term hydrothermal

scheduling problem. They have also shown that when

compared to simulated annealing based algorithm for

short-term hydrothermal scheduling, EP based algorithm

is able to obtain a cheaper hydrothermal schedule at

reduced execution time. In this paper, an evolutionary

programming (EP) based neuro-fuzzy technique is

proposed to solve the multiobjective generation dispatch

problem with nonsmooth characteristic functions i.e.,

fuel cost and emission level functions. The developed

EP based algorithm is tested on IEEE 30-bus test system

and the results are presented. In order to verify the

successful working of the proposed EP based algorithm,

numerical results obtained from EP based algorithm are

compared with those obtained from previous

conventional optimization algorithm when the quadratic

fuel cost and emission level functions are considered.

Simulation results show that the proposed EP based

neuro-fuzzy approach is capable of not only solving the

multiobjective generation dispatch problem with any

type of fuel cost and emission level functions, analytical

or empirical curves, but also obtaining the very fast

global or near global compromising solution.

II. PROBLEM FORMULATION

The present formulation in this dissertation work

treats the multiobjective generation dispatch problem as

a minimization problem which is concerned with the

attempt to minimize each objective simultaneously. The

equality and inequality constraints of the system must

meanwhile be satisfied. The multiobjective generation

dispatch problem has been formulated as the following

two-objective optimization problem that deals with the

cost of generation and emission level as objective

functions. The generating units involved are all thermal

units and assumed operating on-line throughout.

Equality and inequality system constraints as well as

transmission loss have also been included in the

problem formulation for completeness of the problem

under study.

A . Cost of generation

Considering a system having N buses and NL lines

let the first NG buses have sources for power generation.

Taking into account the valve-point effects, the fuel cost

function of each generating unit is expressed as the sum

of a quadratic and a sinusoidal function Therefore, the



136

total cost of generation C in terms of control variable

PG‟s can be expressed as:

hrRCNG

iiiiiiiiii PGPGedcPGbPGaf

1

min2

1))(sin(5.0

where, PGi is the real power output of an ith

generator, NG denotes the number of generators

and edcba iiiii,,,, are the fuel cost curve

coefficients of an ith generator.

B. Emission level

The combustion of fuel used in fossil based generating

units gives rise to four basic forms of emission. Those

are oxides of sulphur (SOx), oxides of nitrogen (NOx),

carbon dioxides (CO2), and particulates. In the present

work, however, all the four forms of emission are

treated together as a single emission criterion. The

amount of emission from a fossil-based generating unit

depends upon the amount of power generated by that

unit which is the sum of a quadratic and an exponential

function in the present work [13]. Therefore, the total

emission level E from all the generating units in the

system then can be expressed as:

hrlbENG

iiiiiiiii PGPGPGf

1

2

2)exp(5.0

where, iiiii,,,, are the emission curve

coefficients of the ith generating unit.

C. Equality and inequality constraints

The following equations and inequalities are satisfied in

the present formulation of multiobjective generation

dispatch problem.

Generator load balance

The real power balance between generation and the load

must be maintained at all time while assuming the load

at any time as constant.

NG

iLDi PPPG

1

(3)

where, PD is the estimated real power demand and PL is

the total transmission system loss of the real power. The

total system real power transmission loss is represented

as:

NG

iiiPGAPL

1

2

(4)

where, Ai are the loss coefficients and are evaluated

from base load flow solutions. Evaluation of these

coefficients is very fast and simple unlike the evaluation

of conventional B-coefficients which is more involved

and time consuming. The effectiveness and validity of

this loss coefficient formulation for generation dispatch

problem has been well established. Moreover, these loss

coefficients can also be updated on a real-time basis

with the change in the system operating condition.

Lower and upper limits of generator output

Each generating unit is constrained by its lower and

upper limits of real power output as shown below to

ensure stable operation.

PGPGPG iii

maxmin (5)

where, PGi

min and PGi

max are the minimum and

maximum real power output of ith unit, respectively.

III. EVOLUTIONARY PROGRAMMING

TECHNIQUE

Evolutionary programming (EP) is a powerful

general-purpose technique for solving complex

real-world optimization problems. It is also a

stochastic optimization technique and can search

for global optimum solution. Like genetic

algorithm (GA), this technique works on

population of trial solutions, imposes random

changes to those solutions to create offsprings, and

incorporates the use of selection to determine

which solutions to maintain into future generations

and which are to be removed from the pool of trials

[17]. But in contrast to GA, the individual

component of a trial solution in EP technique is

viewed as a behavioral trait, not as a gene. In other

words, EP technique emphasizes the behavioral

link between parents and offsprings rather than the

genetic link. It is assumed that whatever genetic

transformation occurs, the resulting change in each

behavioral trait will follow a Gaussian distribution

with zero mean difference and some standard

deviation. The key feature of EP is in its

probabilistic nature of selection by conducting a

stochastic tournament for survival at each

generation. The probability that a particular trial

solution will be maintained is made a function of

its rank in the population. The production of an

offspring population is called a generation. Many



137

such generations are required for the population to

converge to an optimum solution, the number

increasing according to the problem difficulty. In

the EP algorithm the maximum number of

generations, i.e., maximum number of iteration is

defined.

A. Implementation of the EP Algorithm

The implementation of the algorithm is done for

following three cases of optimization.

a) Minimization of total generation cost, i.e., f1

in (1).

b) Minimization of total emission level, i.e., f2 in

(2).

c) Maximization of fuzzy decision making

function, FDM in (18) to be discussed in the

following section.

a) First select arbitrarily a dependent generating

unit from among the committed NG units. Let the

unknown generation PGd be the dependent

generation. The PGd can be calculated by

assuming that the non-dependent generations, i.e.,

the PG j for j = 1, 2, …, NG but , are known.

Further, since the power loss is a function of the

generation outputs and system topology, to

determine the output of dependent unit PGd ,

A-loss coefficients are also required as shown in

(4). Therefore, PGd is calculated as:

NG

djj

jLDd PGPPPG1

(6)

In determining the optimal generation schedule for

the ELD problem according to the above

mentioned problem solving formulation, the main

objective is to determine the non-dependent

generations which have been assumed to be known

by some method. In this work, evolutionary

programming algorithm has been applied to

determine the non-dependent generations and

hence, the global optimal generation schedule with

the minimization of total generation cost has been

obtained. The EP technique implemented to solve

the economic load dispatch problem is stated in the

following subsections.

(i) Representation of trial solution vector

According to the formulation for solving the

problem, a dependent generation PGd from

committed generator, is randomly selected. The

generations from non-dependent generators, i.e.,

PGj for j = 1, 2, …, NG, j d are together taken as a

(NG-1)-dimensional trial vector.

Let ],...,,,...,,[)1()1(21 PPGPGPGPGP NGddi

be the trial vector designating the ith individual of

a population to be evolved.

(ii) Initialization of a population of trial vectors

(Parents)

Taking the population size to be NP, each initial

parent trial vector Pi, i = 1, 2, … , NP, is selected at

random from a feasible range in each dimension.

This is done by setting the jth component of each

parent as:

],[maxmin

PGPGPG jjjrand for j = 1, 2, …,

(d-1),(d+1), …, NG (7)

where, ],[maxmin

PGPG jjrand denotes a

uniform random variable ranging over

],[maxmin

PGPG jj.

(iii) Generation of offspring population

An offspring Pi

' is generated according to the

relative value of the objective function f(Pi)

associated with the trial vector Pi. If f(Pi) is

relatively low, the offspring trial solution is

generated near the current parent solution Pi. On

the other hand, if the f(Pi) is relatively high, the

Pi

' is will be searched within a wider range. To

generate an offspring Pi

' from each parent Pi, a

Gaussian random variable with zero mean and

standard deviation proportional to the scaled cost

values of the parent trial solution is added to the

each component of Pi as given by the following



138

expressions:

],...,,,...,,[''

)1(

'

)1(

'

2

'

1

'

PGPGPGPGPGP NGddi

(8)

and ),0(2'

jjjMPGPG for j = 1, 2, … ,

NG, and j d (9)

where, represents a Gaussian random variable

with mean zero and standard deviation . The

standard deviation indicates the range of the

offspring generated around the parent trial solution

and is given by:

)()

minmax

min

(PGPG

f

Pjj

i

j

f (10)

where, fmin is the minimum cost value among the

NP trial solutions and is a scaling factor.

(iv) Competition and selection

After generation of offspring population,

competition and selection procedure is

implemented to determine which solutions are to

be maintained into the next generation and which

are to be removed from the competing pool of

trials. The NP parent trial vectors and their

corresponding NP offsprings compete with each

other in the competing pool for survival. To do this

a competitor Pr is selected at random from among

the 2NP trial solutions, where „r‟ is an integer as

given by:

]1]1,0[2[1

randN Pr (11)

In the above equation, rand1[0,1] is an random

number ranging over [0,1] and value of r is taken

to be the greatest integer less than or equal to the

value of the expression in the right hand side. After

a stochastic competition, the score for each trail

vector is calculated as:

N P

mmPi ww

1

(12)

and wm = 1, if rand2[0,1] > )()(

)(

PPP

ir

i

ff

f

= 0, otherwise.

where, rand2[0,1] is another uniform random

number generated between 0 and 1.

After the competition is over, the 2NP trial

solutions in the competing pool are sorted

according to their obtained scores from highest to

the lowest. Thereafter, the first NP trial solutions

from the sorted pool are selected as the new parent

vectors for the next generation.

(v) Stopping rule

The iterative procedure of generating new trials by

selecting those with minimum function values from

the competing pool consists of equal number of

parents and offsprings is terminated when there is

no significant improvement in the solution. It can

also be terminated when a preset number of

iteration is reached. In the present work, the latter

method is employed.

The initial values of all components (non-

dependent generations) of each parent are specified

or generated at random before starting the process

of evolution. Consequently, the dependent

generation is calculated. Thereafter, all the

generation levels are checked against their

corresponding limits and the generator-load

balance is checked. If all the constraints are

satisfied, then the current non-dependent

generations are taken as the components of the

final feasible parent. If otherwise then, using the

current value of the dependent generation, all the

generation levels are again calculated and all the

constraints are checked till all of them are satisfied.

The process of generating feasible parent vectors

continues till the iteration count equals NP. Similar

checking of constraints is performed for generation

of each feasible offspring. At the end of the

solution process the trial vector with minimum

function value among the NP trial vectors will be

the global optimum solution.

(b) In this case same EP procedure is adopted for

the emission level function minimization as shown

in (2).



139

(c) In many problems, the objective is more

naturally stated as the minimization of some cost

function g(x) rather than the maximization of some

utility or profit function u(x). Even if the problem

is naturally stated in maximization form, this alone

does not guarantee that the utility function will be

nonnegative for all x as we require in fitness

function. As a result, it is often necessary to map

the underlying natural objective function to a

fitness function form through one or more

mappings. The duality of cost minimization and

profit maximization is well known. In normal

operational research work, to transform a

minimization problem to a maximization problem

we simply multiply the cost function by a minus

one. In evolutionary programming work, this

operation alone is insufficient because the measure

thus obtained is not guaranteed to be nonnegative

in all instances. With evolutionary programming

algorithm, the following cost-to-fitness

transformation is commonly used:

)()(max

xgxf C when g(x) < Cmax, (13)

= 0 otherwise.

There are a variety of ways to choose the

coefficient Cmax. It may be taken as an input

coefficient, as the largest g value observed thus far,

as the largest g value in the current population, or

the largest of the last k generations. Perhaps more

appropriately, Cmax should vary depending on the

population variance. The fuzzy decision making

maximization problem is transformed into a

general minimization problem by using Equation -

13 which is shown below:

f(PG) = Cmax – FDM(PG) when FDM < Cmax,

(14)

= 0 otherwise.

where, Cmax is taken as an input coefficient for

simplicity.

The EP procedure as described for case (a) remains

same except the objective function which is

replaced by the above function as described in

Equation – 14.

IV. THE PROPOSED EP - BASED NEURO-

FUZZY TECHNIQUE

The basic block diagram of the proposed EP-based

neuro-fuzzy technique for multiobjective

generation dispatch with nonsmooth characteristic

functions has been shown in Fig. 1. Initially, the

economy objective function, i.e., the cost of

generation of the multiobjective generation

dispatch problem is minimized, followed by

minimization of emission level objective function

using the global optimization technique namely

evolutionary programming. Then, both the

objectives are combined through a fuzzy

coordination method to form a fuzzy decision

making (FDM) function. The original two-

objective problem is then solved by maximizing

the FDM function by using evolutionary

programming technique. After this optimization,

the results are trained by a radial basis function

ANN to reach a preliminary generation schedule.

Since, some practical constraints may be violated

in the preliminary schedule, a heuristic rule based

search algorithm is developed to reach a feasible

best compromising generation schedule which

satisfies all practical constraints in the final stage.

Figure 1: Basic block diagram of proposed EP based

neuro-fuzzy technique for multiobjective generation

dispatch with nonsmooth characteristic functions

A. Fuzzy coordination method

In this method, the fuzzy decision making function [4] is

represented by introducing the membership function in

the fuzzy set theory. The idea of the membership

function is to replace the concept that each variable has

a precise value. Rather, each variable is assigned a

degree of membership for each possible value of the

variable. Fig. 2 depicts the membership function *c for

the fuzzy variable signifying total fuel cost fc. This

function describes numerically how the decision-maker

is satisfied by which level of the index chosen. The

decision-maker is fully satisfied with the cost if *c = 1

and not satisfied at all if *c = 0. Therefore, the value of

the membership function indicates the adaptability of

the economy index.



140

c

0

1

fc

f fcm cd

Figure 2: The membership function for fuzzy fuel

cost fc

The membership function for an ith objective

function fi(PG) is defined as:

iid i

id im

PGf f PG

f f( )

( )

(15)

where, fim is the minimum permissible value of the

objective function assumed to be known previously

and the parameter fid is the least permissible

desired value beyond which the objective is

unsatisfactory for the decision-maker. The fuzzy

decision making function (FDM) for ith objective

is defined as below.

0, i(PG) 0

FDMi(PG)= i(PG), 0 i(PG) 1 (16)

1, i(PG) 1

Consequently, FDMi becomes 1 when the ith

objective value is most desirable, and it is 0 (zero)

when the objective value is most undesirable. The

combined fuzzy decision making function (FDM)

is obtained as:

FDM

FDMi

i

PG( )

1

2

(17)

The optimal (best compromising) solution of the

multiobjective generation dispatch problem is

obtained by solving the following optimization:

Maximize FDM (18)

subject to PG

where, PG: NG-dimensional vector of decision

variables, : the set of feasible solutions

B. Procedure of neuro-fuzzy approach

As shown in Fig. 3, the design procedure of the

proposed integrated approach consisting of EP-

based fuzzy coordination and ANN methods along

with a heuristic rule based search algorithm for

optimal solutions involves four major steps, viz.

training set creation, training, testing and heuristic

search. In the proposed approach, the minimum

cost of generation and minimum emission level are

calculated by evolutionary programming

technique. Then the optimization of economy-

emission is done by evolutionary programming

based fuzzy coordination method. For

determination of generation dispatches of thermal

units, neural networks of supervised learning are

needed. This is because, the optimal generation

schedule of the thermal units (outputs) for each

total system load demand (input) in the training set

are required to be known in advance by some

suitable method. A radial basis function ANN

called as RBANN is employed in the present work

for training and testing due to its auto configuring

architecture and faster learning ability. The EP-

based fuzzy coordination method as described

earlier has been applied to create the necessary

training set. In the training process, the RBANN is

presented with a series of pattern pairs; each pair

consists of an input pattern and a target output

pattern. The training pattern 'p' is described by:

t(p) = {( input (p) ), ( output (p) )}

= {( PD (p)), ( PG1(p) , PG2 (p) , ... , PGNG

(p) )} (19)

The sum of the squared errors (SSE) between the

actual and the desired (target) outputs over the

entire training sets is used as the measure to find

out the convergence of the network. The RBANN



141

used is trained by the orthogonal least squares

learning algorithm. Training is continued until the

given error-goal in terms of SSE is reached. Once

the RBANN is trained, there after only the Steps 3

and 4 are used to obtain the optimal solutions of

multiobjective generation dispatch for any given

load PD. In the Step-3 only a preliminary

generation schedule is obtained since, the practical

constraints of lower and upper limits of real power

generation outputs of the generators may be

violated in the preliminary schedule. Therefore, a

heuristic rule based search algorithm is developed

in the Step-4 to reach a feasible best compromising

generation schedule, which satisfies all the

practical constraints.

C. Heuristic rule based search algorithm

for determination of final schedule

In this work the following heuristic rules are

applied to refine the preliminary schedule and to

reach the final best compromising generation

schedule.

i) Heuristic rule on lower limits of generators

Let PGPG ii

min , if PGPG ii

min

for i =1,2,…,NG (20)

ii) Heuristic rule on upper limits of generators

Let PGPG ii

max , if PGPG ii

max

for i =1,2,…,NG (21)

When the ANN output for a particular generator

either crosses lower limit or upper limit, the

generation is fixed at its corresponding limit. The

removed generation (in case of exceeding upper

limit) or added generation (in case of exceeding

lower limit) is so small that even if it is neglected,

then also the percentage error is very much well

within the acceptable limits for all practical

purposes. However, in this work, the removed

generation or added generation of a generator is

equally shared by remaining generators

accordingly as described below. If more than one

generator exceeds their limits, then also the

heuristic rule based search algorithm may be

extended in similar ways.

i) Removed generation case:

For an n-generator case, let the generation level of

jth generator is PGj such that PGj > PGjmax

. So, the

removed generation is PGj - PGjmax

and

accordingly, an amount (PGj - PGjmax

)/(n-1) is

added to each generation level of remaining n-1

generators i.e., PGi for i = 1 to n and i ≠ j.

ii) Added generation case:

For an n-generator case, let the generation level of

jth generator is PGj such that PGj < PGjmin

. So, the

added generation is PGjmin

- PGj and accordingly,

an amount (PGjmin

- PGj)/(n-1) is subtracted from

each generation level of remaining n-1 generators

i.e., PGi for i = 1 to n and i ≠ j.

Step 1. Training set creation

Figure 3: Design procedure of the proposed EP-

based neuro-fuzzy technique for multiobjective

generation dispatch problem



142

V. SYSTEM STUDIES

The proposed EP based neuro-fuzzy technique has

been applied to the IEEE-30 bus test system. The 30-bus

test system consists of three generators in the first three

bus and 40 transmission lines. Table 1 summarizes the

operating limits of the three generators. The fuel cost

function and the emission level function data are given

in Table 2 and Table 3, respectively. A base case load of

240 MW has been considered for 30-bus system. The

loss coefficients are evaluated from the base case load

flows.

Table 2 Fuel cost function coefficients

A. Simulation studies on 30-bus test system

Prior to applying the EP based fuzzy coordination

method to the multiobjective generation dispatch

problem with nonsmooth fuel cost curves of the

generators, it has been initially applied to the problem

with conventional quadratic fuel cost and emission level

functions of the generators to prove its satisfactory

working. The control parameters of the EP algorithm are

maximum iteration number, population size and scaling

factor and the most appropriate values of these

parameters are set to 500, 50 and 0.001, respectively.

These values are obtained after testing and evaluating

different combinations. For a load demand of 240 MW

the most economical, minimum emission and best

compromising solution using fuzzy coordination method

were computed by sequential quadratic programming

approach as described in Reference-9 and the proposed

EP approach, and the comparison is shown in Table 4.

From this table it is observed that results obtained from

both the methods closely match with each other. This

indicates that the generation dispatch results can be

accurately obtained by the proposed EP approach. In

order to explore the converging characteristics of the

EP, different random initial solutions were given to the

proposed EP algorithm along with the above mentioned

control parameters for the most economical solution

corresponding to a load demand of 240 MW. The

optimal solutions corresponding to each random initial

solution (trial) are observed. The total cost variation of

the most economical generation schedule obtained from

proposed EP approach when executed 10 times with

different random initial solutions were observed. It was

found that about 90% of the solutions after execution of

each trial were converged approximately at the global

optimum solution. This further indicates that the EP

technique has more powerful ability to achieve the

global optimum solution.

In this work, all the minimization and maximization

tasks are performed by using the evolutionary

programming approach that represents the most

powerful tool for global optimization. The objectives

f¬1(PG) and f¬2(PG) are minimized separately from

80% to 120% in steps of 5% of base case load of 240

MW to obtain the most economical and minimum

emission solutions, respectively, considering loss and

the results are presented in Tables 5 and 6, respectively.

It is observed from Tables 5 and 6 that the generator

allocations are not coincident. This may be accounted

from the fact that they are optimized based on different

performance indices. At base load the total fuel cost is

1559.3 R/h from most economical solution but increases

to 1575.8 R/h corresponding to minimum emission

solution. The total emission level from most economical

solution is found to be 622.0449 lb/h but it decreases to

612.6412 lb/h corresponding to the minimum emission

solution. Thus, Tables 5 and 6 clearly demonstrate the

conflicting nature of the two objective functions. The

total fuel cost found from minimum emission solution,

i.e., 1575.8 R/h is set as the maximum of the desired

value f1dmax of the objective f1(PG). The total fuel cost

found from most economical solution, i.e., 1559.3 R/h is

set as the minimum permissible value f1m of the same

objective. Similarly, f2dmax and f2m correspond to the

total emission level from the most economical and

minimum emission solutions, respectively. These

solutions are 622.0449 lb/h and 612.6412 lb/h,

respectively, for the present case load demand. In this

method the range of the objective values are fixed in this

way to guarantee the generation of noninferior solutions.

However, after setting the fim as described above, the

decision-maker may set the value of fid as per his/her

satisfaction. It is to be noted here that, fid must be less



143

than or equal to fidmax and certainly more than the fim.

The following test case is computed to demonstrate the

applicability of the developed algorithm as given in

previous section in obtaining the best compromising

solutions. In this work no priority of the objectives is

assumed, and the desired values of the objectives are set

as f1d = f1dmax and f2d = f2dmax. The computations

were carried out according to the procedure given in

section-4.1 for the load values from 80% to 120% of

base load in steps of 5%. The best compromising

solutions consisting of optimum generations using fuzzy

coordination method are presented in Table 7.

Performance indices obtained from best compromising

solutions by fuzzy coordination method were compared

with those obtained from most economical and the

minimum emission solutions in Tables 5 and 6. It is

clear from these tables that the best compromising

solutions force both the performance indices to remain

in between those obtained from most economical and

the minimum emission solution procedures as expected.

Table 4 Comparison of generation dispatch solutions

obtained by Sequential Quadratic Programming and

Evolutionary Programming Techniques

Table 5 Most economical solutions for 30-bus system

Table 6 Minimum Emission solutions for 30-bus

system

Table 7 Best compromising solutions using fuzzy

coordination method for 30-bus system

A radial basis function ANN model, namely RBANN is

designed for the 30-bus test system. There is only 1

input node (load demand) for the model. The optimal

loads of the thermal units in the system, i.e., ,..., are the

output nodes. Therefore, there are 3 output nodes for the

RBANN. The number of neurons in the single hidden

layer is equal to the number of iterations required for

training and is set adaptively for RBANN. It is not

unusual to get good performance on training data

followed by much worse performance on test data. This

can be guarded against by ensuring that the training data

are uniformly distributed. The cases used to train the

networks are as follows: PD is taken as base load i.e.,

240MW. The range of load demand is chosen between

80% to 120% of base load in steps of 5% for RBANN.

Therefore, 9 different training patterns were generated

covering the system load from 192 MW to 288 MW.

The training patterns are already given Table 7. The

RBANN was trained with its corresponding 9 patterns to

reach the error-goal (convergence target) which was

SSE = 0.001. RBANN required only 9 iterations in

reaching the convergence target. To achieve the best

performance on the test data and good generalization an

appropriate value of spread factor (SF) is set.

Computations were carried out for different values of SF

to find the best value as per the guideline given in

Reference-18. For a given set of test patterns the

percentage mean absolute error (% MAE) is recorded

for each value of SF. Then the value of SF



144

corresponding to the minimum of the % MAE is taken

as the best value of SF. The best SF is found to be 18 for

RBANN. For the performance evaluation of the

proposed neuro-fuzzy technique, 4 numbers of test cases

(load levels other than those in training sets but within

80% to 120% of the base load) are considered. These

test cases were generated by fuzzy coordination method.

The test cases were computed by the RBANN, which

was trained earlier taking the best value of SF i.e., 18.

The final optimal generation schedule obtained from the

RBANN along with heuristic rule based search

algorithm was compared with those obtained from fuzzy

coordination method and the %Error=

3

1

3

1

3

1 i

F

ii i

NF

i

F

i PGPGPG 100 was

also computed where PGF

iand PG

NF

iare

generation schedule obtained from fuzzy coordination

method and proposed neuro-fuzzy technique,

respectively. The comparison of best compromising

generation dispatch solutions between fuzzy

coordination method and the neuro-fuzzy technique are

shown in Table 8. From this table it is observed that the

generation schedule obtained from neuro-fuzzy

technique closely matches to that of the fuzzy

coordination method.

Table 8 Comparison of best compromising generation

dispatch solutions between the fuzzy coordination

method and EP based neuro-fuzzy technique for 30-bus

system

The most significant advantage of the proposed neuro-

fuzzy technique is that once the RBANN is trained for a

given range of load levels of a multiobjective generation

dispatch problem then, the computation of best

compromising generation schedule corresponding to a

new load demand only requires Steps 3 and 4. It may be

noted that both Step-1 and Step-2 require relatively

lengthy computational effort while that of Steps 3 and 4

require only fraction of a second. However, it is

significant to note that the first two steps are simulated

off-line only. Present case studies on the test system

demonstrates that the absolute % error in scheduling

found to be much less than even 1% and when

computed on a 2.4 GHz P-IV machine. All the computer

programs were implemented using MATLAB 6.1 and

run on a Pentium-IV PC with Windows 98 operating

system. The proposed EP based fuzzy coordination

method required about an average time of 9 minutes of

total computer time to obtain the best (global optimum)

multiobjective generation schedule with non-smooth

characteristic functions for each load level. But, the

average execution time (Steps 3 and 4 of Fig. 3) neuro-

fuzzy method for a given load demand is found to be

only 0.1 sec. Therefore, the accuracy and the capability

of very fast computation of generation schedule of the

proposed EP based neuro-fuzzy technique seem to be

very promising for its suitability for on-line

multiobjective generation dispatching with non-smooth

characteristic functions.

VI. CONCLUSION

An integrated approach combining an evolutionary

programming based fuzzy coordination and an artificial

neural network methods along with a heuristic rule

based search algorithm has been developed in this paper

to obtain the best compromising generation schedules

for multiobjective generation dispatch problem with

nonsmooth characteristic functions, satisfying various

practical constraints that are suitable both in terms of

speed and accuracy, while allowing more flexibility in

operation. Initially, the economy objective function is

minimized, followed by minimization of emission level

objective function. Then, both the objectives are

combined through a fuzzy coordination method to form

a fuzzy decision making (FDM) function. Maximizing

the FDM function then solves the original two-objective

problem. After this optimization, the results are trained

by a radial basis function ANN to reach a preliminary

generation schedule. Since, some practical constraints

may be violated in the preliminary schedule, a heuristic

rule based search algorithm is developed to reach a

feasible best compromising generation schedule which

satisfies all practical constraints in the final stage.

Simulation results indicate that the accuracy and the

capability of very fast computation of generation

schedule of the proposed EP based neuro-fuzzy

technique seem to be very promising for its suitability

for on-line multiobjective generation dispatching with

any kind of characteristic functions. The minimization

and maximization tasks of the optimization problem

considered are solved by the evolutionary programming

technique. There are two advantages to use EP: first, the

output can be represented exactly, secondly, comparing

to genetic algorithms the time-consuming encoding-

decoding manipulations are avoided. Though applied to

a moderate size test system, this technique may be

applied to large size systems effectively. Further, in the

present case studies only two objective functions such as

fuel cost and emission level functions are taken. In



145

future works objective functions like security,

reliability, etc., may be addressed while solving

multiobjective generation dispatch problems through

neuro-fuzzy technique.

VII. REFERENCES

[1] J.Nanda, D.P.Kothari and K.S.Lingamurthy,

'Economic Emission Load Dispatch through Goal

Programming Technique', IEEE Trans. on Energy

Conversion, Vol.3, 1988, pp.26-32.

[2] P.K.Hota, R.Chakrabarti & P.K.Chattopadhyay,

„Multiobjective Generation Dispatch using Goal-

attainment Method‟, Journal of Institution of

Engineers(India), pt. EL, Vol.82, September-2001,

pp.122-127.

[3] J.Nanda, H.Lakshman and M.L.Kothari,

'Economic Emission Load Dispatch with Line Flow

Constraints using a Classical Technique', IEE Proc.-

Generation Transmission and Distribution, Vol.141,

No.1, 1994, pp.1-10.

[4] H. J. Zimmermann, „Fuzzy set theory and its

applications‟, Kluwer-Nijhoff publishing, 1985.

[5] D.Srinivasan, C.S.Chang & A.C.Liew,

‟Multiobjective Generation Scheduling using Fuzzy

Optimal Search Technique‟, IEE Proc.- Generation

Transmission and Distribution,1 Vol.141, No.3, May-

1994, pp.233-242.


„A Fuzzy-Set Based Optimization Technique for

Economic Emission Load Dispatch‟, Journal of

Institution of Engineers(India), pt. EL, Vol.80,

November-1999, pp.99-103.


„Economic Emission Load Dispatch through an

Interactive Fuzzy Satisfying Method‟, Electric Power

Systems Research, Vol.54(3), 2000, pp.151-157.


„An Integrated Approach to Economic Emission Load

Dispatching using Neural Network and Goal-attainment

Methods‟, Electric Machines and Power Systems,

Vol.27, No.10, 1999, pp.1085-1096.


„Multiobjective Generation Dispatch through a Neuro-

Fuzzy technique”, Electric Power Components and

Systems, Vol. 32, No. 11, 2004, pp.1191-1206.

[10] K.P.Wong and C.C.Fung, “Simulated annealing

based economic dispatch algorithm”, IEE Proc.-

Generation Transmission and Distribution, Vol.140,

No.6, November-1993, pp. 509-515.

[11] K.P.Wong and Y.W.Wong, “Short-term

hydrothermal scheduling Part-I: Simulated annealing

approach”, IEE Proc.- Generation Transmission and

Distribution, Vol.141, 1994, pp. 497-501.

[12] P.K.Hota, R.Chakrabarti and P.K.Chattopadhyay,

“A simulated annealing-based goal-attainment method

for economic emission load dispatch with nonsmooth

fuel cost and emission level functions”, Electric

Machines and Power Systems, Vol.28, No.11, 2000, pp.

1037-1051.

[13] K.P.Wong and Suzannah, Y.W.Wong,“Combined

genetic algorithms/ simulated annealing/ fuzzy set

approach to short-term generation scheduling with take-

or-pay fuel contract”, IEEE Trans. on Power Systems,

Vol.11, No.1, 1996, pp. 128-136.

[14] H.Yang, P.Yang and C.Huang, “Evolutionary

programming based economic dispatch for units with

non-smooth fuel cost functions”, IEEE Trans. on Power

Systems, Vol.11, No.1, 1996, pp. 112-118.

[15] H.Duo, H.Sasaki, T.Nagata and H.Fujita,“A

solution for unit commitment using Lagrangian

relaxation combined with evolutionary programming ”,

Electric Power Systems Research, Vol.51, 1999, pp. 71-

77.

[16] P.K.Hota, R.Chakrabarti and P.K.Chattopadhyay,

“Short-term hydrothermal scheduling through

evolutionary programming technique”, Electric Power

Systems Research, Vol.52, 1999, pp. 189-196.

[17] D.B.Fogel, “An introduction to simulated

evolutionary optimization”, IEEE Trans., Neural

Networks, Vol.5, No.1, 1994, pp. 3-14.

[18] H.Desmuth and M.Beale, “Neural Network Tool

Box User‟s Guide”, Math Works, Natick, MA, 1994.

an evolutionary programming based neuro-fuzzy technique ... · iii. evolutionary programming...

Documents