FACTA UNIVERSITATIS 

Series: Electronics and Energetics Vol. 35, No 4, December 2022, pp. 495-512 

https://doi.org/10.2298/FUEE2204495P 

© 2022 by University of Niš, Serbia | Creative Commons License: CC BY-NC-N 

Original scientific paper  

FUZZY-BASED REAL-CODED GENETIC ALGORITHM  

FOR OPTIMIZING NON-CONVEX ENVIRONMENTAL 

ECONOMIC LOSS DISPATCH 

Shradha Singh Parihar1, Nitin Malik2 

1Gautam Buddha University, Greater Noida, India 
2The NorthCap University, Gurugram, India 

Abstract. A non-convex Environmental Economic Loss Dispatch (NCEELD) is a 

constrained multi-objective optimization problem that has been solved for assigning 

generation cost to all the generators of the power network with equality and inequality 

constraints. The objectives considered for simultaneous optimization are emission, 

economic load and network loss dispatch. The valve-point loading, prohibiting operating 

zones and ramp rate limit issues have also been taken into consideration in the generator 

fuel cost. The tri-objective problem is transformed into a single objective function via the 

price penalty factor. The NCEELD problem is simultaneously optimized using a fuzzy-

based real-coded genetic algorithm (GA). The proposed technique determines the best 

solution from a Pareto optimal solution set based on the highest rank. The efficacy of the 

projected method has been demonstrated on the IEEE 30-bus network with three and six 

generating units. The attained results are compared to existing results and found superior 

in terms of finding the best-compromise solution over other existing methods such as GA, 

particle swarm optimization, flower pollination algorithm, biogeography-based 

optimization and differential evolution. The statistical analysis has also been carried out 

for convex multi-objective problem. 

 

Key words: Multi-objective optimization, non-convex Environmental Economic Loss 

Dispatch, price penalty factor, Pareto optimality, real-coded genetic 

algorithm, valve-point loading, prohibiting operating zones, ramp rate limit 

 

 

 
Received March 2, 2022; revised June 22, 2022; accepted July 6, 2022 

Corresponding author: Nitin Malik 

The NorthCap University, Sector 23A, Gurugram, India 
E-mail: nitinmalik77@gmail.com 



496 S. S. PARIHAR, N. MALIK 

List of Abbreviations: 

CEED: Combined emission and economic dispatch   

ED: Emission dispatch  

ELD: Economic load dispatch  

FPA: Flower Pollination Algorithm 

FRCGA: Fuzzy-based real-coded genetic algorithm  

GA: Genetic algorithm  

N/w: Network 

NCEELD: Non-convex Environmental Economic Loss Dispatch  

NSGA: Non-dominated sorting genetic algorithm 

POZs: Prohibiting operating zones 

PPF: Price penalty factor  

PSO: Particle swarm optimization  

RCGA: Real-coded genetic algorithm  

RRL: Ramp rate limit 

VPL: Valve point loading 

1. INTRODUCTION 

1.1. Motivation 

The electrical power networks traditionally functioned to minimize total generation fuel 

cost and were less bothered about the harmful emissions generated in the network [1-3]. 

After the US clean air act of 1990 (amended in 2010) and similar legislation in several 

other countries, the public concern towards the pollutants like COX, SO2 and NOX produced 

from the thermal power plant has grown. This, in turn, forces the utilities to deliver the power 

to the consumers with simultaneous minimum total generator fuel cost and total emission 

level [4-22]. A high degree of non-linearity and complexity is present in the modern 

generator’s cost curve function because of the presence of valve point loading (VPL) effect 

and other effects, the resultant approximate solutions lead to a lot of revenue loss over time 

which is also affected by the network losses. To overcome this, the optimal amount of 

generated power of the thermal units are to be determined by minimizing emission, loss and 

cost simultaneously while satisfying all practical constraints, hence, generating a large-scale 

highly constrained non-linear multi-objective optimization problem.  

1.2. Literature survey 

The economic load dispatch (ELD) [1-3] is a real-world problem that, earlier, only 

considers the minimization of the generator fuel cost. Therefore, emission dispatch (ED) is 

considered in [4] for the very first time. Hence, both generator fuel cost and harmful 

environmental emissions should be treated as competing objectives. The combined 

emission and economic dispatch (CEED) minimize harmful emissions and generating unit 

cost simultaneously to obtain optimal generation for each network (N/w) unit satisfying 

various practical constraints. In [5-9], the authors presented weighted-sum or price penalty 

factor (PPF) based methods where all the considered objectives are treated as a unit function. 

Conventional genetic algorithm (GA) and differential evolution have been presented in [10] 

and [11], respectively to demonstrate the effect of VPL on the generators cost function but 



 Fuzzy-based Real-Code Genetic Algo for Optimizing Non-Convex Environment Economic Loss Dispatch 497 

GA requires large CPU time for the optimization.  A fast initialization approach has been 

presented in [12] to solve non-convex economic dispatch problem but is usually stuck in 

local minima. A new whale optimization approach has been presented in [13] and have 

high computational efficiency.  A flower pollination algorithm (FPA) is demonstrated in 

[14] for solving ELD and CEED problem in larger N/w.  

Many evolutionary algorithms such as non-dominated sorting genetic algorithm 

(NSGA) [15], squirrel search algorithm [16], evolutionary programming [17] and NSGA-

II [18] have been proposed for solving the bi-objective problem. The evolutionary programming 

has a slow convergence rate for large problem. A mine-blast algorithm has been developed in 

[19] to incorporate the valve point loading effect for solving the environmental economic load 

dispatch problem. A new global particle swarm optimization (PSO) is developed in [20] to solve 

bi-objective problem without and with transmission losses. A fuzzified PSO technique [21], 

harmony search [22] and Cuckoo search [23] is applied to optimize the solution for the CEED 

problem. The PSO approach deals with the problem of partial optimism. 

1.3. Paper contributions  

a) As most of the research has been carried out considering only two objectives (fuel cost 
and emissions), the authors have incorporated additional objective (network loss) to 

make the problem formulation more comprehensive and find better solution by merging 

two soft-computing techniques (RCGA and Fuzzy) for finding the best compromised 

solution out of the obtained Pareto solutions. Moreover, it has been found from the 

exhaustive literature review that the non-convex multi-objective optimization problem 

formulation with simultaneous minimization of three objective functions (emission, fuel 

cost and network loss) at different load demands has not been explored before.  

b) The different non-linearities like valve-point loading, prohibiting operating zones (POZs) 
and ramp rate limit (RRL) are considered in this article for three conflicting objectives.  

c) As all the considered objectives are competitive, the method generates multiple non-
dominated Pareto optimal solutions rather than a single best solution from which the best-

compromised solution is selected based on the highest fuzzy membership function value.  

d) To validate the proposed methodology, three test cases have been considered at different 
load demands and the results are compared with already published methods based on GA 

[25], PSO [25, 26], FPA [27], Biogeography-based Optimization [28] and differential 

evolution [29]. 

2. MATHEMATICAL MODELING 

The practical non-convex EELD problem has three conflicting objectives which aim to 

minimize generating cost, amount of harmful emissions and losses of the complex and non-

linear network. To formulate a non-convex EELD problem following objectives and 

operating constraints are given below: 

2.1. Non-convex economic load dispatch  

It is more practical for fossil fuel-based generators to introduce the steam valve-point 

loading effect in a turbine by adding a rectified sinusoidal term to the quadratic cost 



498 S. S. PARIHAR, N. MALIK 

equation which leads to non-smooth and non-convex function having manifold minimas 

[10]. Total generator fuel cost based on active power output can be represented as [14] 

  𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓1 =  𝐹𝑇 = ∑ (𝑎𝑖 𝑃𝑖
2 + 𝑏𝑖 𝑃𝑖 + 𝑐𝑖 )

𝑁
𝑖=1 + |𝑒𝑖 × sin (𝑓𝑖 × (𝑃𝑚𝑖𝑛 − 𝑃𝑖 ))|   (1) 

where Pi represents the output power generation of i
th unit. ai, bi, ci, ei, and fi are the generator 

fuel cost coefficients.  

2.2. Emission dispatch (ED) 

The goal of ED is to minimize the total environmental degradation due to fossil fuel 

burning to produce power. The total pollution level of the environment that needs to be 

minimized is given as [14]: 

 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓2 =  𝐸𝑇 = ∑ 10
−2 × (𝛼𝑖 + 𝛽𝑖 𝑃𝑖 + 𝛾𝑖 𝑃𝑖

2)𝑁𝑖=1 + 𝜉𝑖 exp (𝜆𝑖 𝑃𝑖 ) (2) 

where i, i, i, i,  i represents the pollution coefficients of the i
th generating unit. 

2.3. Loss dispatch 

The loss dispatch aims to minimize power loss without considering the generator cost 

and harmful emission of the network. To minimize loss [14] 

  𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓3 =  𝑃𝐿 = ∑ ∑ 𝑃𝑖 𝐵𝑖𝑗 𝑃𝑗 + ∑ 𝐵𝑖𝑜 𝑃𝑖 + 𝐵𝑜𝑜
𝑁
𝑖=1

𝑁
𝑗=1

𝑁
𝑖=1   (3) 

where Bij, Bio and Boo  represents the line loss coefficients. 

2.4. Non-convex Environmental Economic Loss Dispatch (NCEELD) 

The NCEELD problem is to be formulated having an economy, harmful emissions and 

losses of the network as competing objectives. The proposed complex problem can be 

written as 

  𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐶 = 𝑓1 + (𝑝𝑓𝑒) ∗ 𝑓2 + (𝑝𝑓𝑙) ∗ 𝑓3       (4) 

where ′𝑃𝑓𝑒′  and ′𝑃𝑓𝑙′  are the PPF for emission and loss respectively. 𝑓1   represents total 
generator fuel cost, 𝑓2   represents total emission and 𝑓3  represents total N/w loss. The ratio 
of the max value of f1 to the max value of f2 gives PPF for emission, whereas, the ratio of 

the max value of f1 to the max value of f3 of the corresponding generator gives PPF for loss. 

The procedure for finding PPF for emission and loss can be given as: 

(a) The generator fuel cost ($/hr) is calculated at its maximum output using (1) for the 

convex and non-convex problems. 

(b) The emission release from every generator (lb/hr or kg/hr) is calculated at its maximum 

output using (2). 

(c) The losses of each are calculated at its maximum output using (3). 

(d) 𝑃𝑓𝑒[𝑖], 𝑃𝑓𝑙[𝑖] (𝑖 = 1,2 . . . 𝑛) for each generator is determined as in (5) and (6). 

 𝑝𝑓𝑒[𝑖] =
∑ (𝑎𝑖+𝑏𝑖𝑃𝑖

𝑚𝑎𝑥 
+𝑐𝑖𝑃𝑖

𝑚𝑎𝑥 2
)

𝑁
𝑖=1 +|𝑒𝑖×sin {𝑓𝑖×(𝑃𝑖𝑚𝑖𝑛

𝑚𝑎𝑥 
−𝑃𝑖

𝑚𝑎𝑥 
)}|

∑ 10−2×(𝛼𝑖+𝛽𝑖𝑃𝑖
𝑚𝑎𝑥 +𝛾𝑖𝑃𝑖

𝑚𝑎𝑥 2)𝑁𝑖=1 +𝜉𝑖exp (𝜆𝑖𝑃𝑖
𝑚𝑎𝑥 )                      

 ($/𝑙𝑏)     (5) 

 𝑝𝑓𝑙[𝑖] =
∑ (𝑎𝑖+𝑏𝑖𝑃𝑖

𝑚𝑎𝑥 
+𝑐𝑖𝑃𝑖

𝑚𝑎𝑥 2
)𝑁𝑖=1 +|𝑒𝑖×sin {𝑓𝑖×(𝑃𝑖𝑚𝑖𝑛

𝑚𝑎𝑥 
−𝑃𝑖

𝑚𝑎𝑥 
)}|

∑ ∑ 𝑃𝑖
𝑚𝑎𝑥 𝐵𝑖𝑗𝑃𝑗

𝑚𝑎𝑥 +∑ 𝐵𝑖𝑜𝑃𝑖
𝑚𝑎𝑥 +𝐵𝑜𝑜

𝑁
𝑖=1

𝑁
𝑗=1

𝑁
𝑖=1                     

 ($/𝑝𝑢)  (6) 

where 𝑃𝑖
𝑚𝑎𝑥  is the maximum capacity of the unit.    



 Fuzzy-based Real-Code Genetic Algo for Optimizing Non-Convex Environment Economic Loss Dispatch 499 

(e) 𝑃𝑓𝑒[𝑖] and 𝑃𝑓𝑙[𝑖]  (i=1, 2... n) are sorted in ascending order. 
(f) 𝑃𝑖

𝑚𝑎𝑥  is added starting from the generator unit with the smallest 𝑃𝑓𝑒[𝑖] for harmful 
emissions and the generator unit with the smallest 𝑃𝑓𝑙[𝑖] for the loss until ∑ 𝑃𝑖

𝑚𝑎𝑥 ≥ 𝑃𝐷 . 
(g) The 𝑃𝑓𝑒[𝑖] and 𝑃𝑓𝑙[𝑖] linked with the last generator unit is the PPF for emission and 

loss, respectively for a given load 𝑃𝐷 . 
(h) The 𝑃𝑓𝑒[𝑖] and 𝑃𝑓𝑙[𝑖] for particular load are determined. Eq. (4) is optimized subject 

to constraints in case of the tri-objective minimization problem. 

For the convex EED problem, the ′𝑃𝑓𝑒 ′ selected is 43.55981 $/Kg and 44.07915 $/Kg 
[27] for three generator unit network at 400 MW and 500 MW respectively. For non-

convex problem considering standard IEEE 30-bus network, 𝑃𝑓𝑒 ′ and  ′𝑃𝑓𝑙′ calculated for 
load PD of 2.834 p.u is 5932.9377 $/lb & 10445.0680 $/p.u and for load PD = 4.32 p.u is 

10949.4251 $/lb & 19612.6323 $/p.u respectively using method given in reference [8].  

The optimization process is subjected to the following constraints:  

a) The active power output of a generating unit is constrained by its bounds for a stable 

operation and is given as: 

 𝑃𝑖
𝑚𝑖𝑛 ≤ 𝑃𝑖 ≤ 𝑃𝑖

𝑚𝑎𝑥        𝑖 = 1,2, … . , 𝑁 (7) 

b) The total generated power balances the sum of the active power loss (PL) and total load 

demand (PD). Therefore,  

 ∑ 𝑃𝑖 − (𝑃𝐷 + 𝑃𝐿 ) = 0
𝑁
𝑖=1   (8) 

where PL is denoted as B-coefficients. The error in loss coefficients is considered to be 

constant as in ref [14]. 

c) Generator ramp rate limits: The inclusion of ramp rate limits changes the operating 

limits of the generator as [24] 

 𝑀𝑎𝑥(𝑃𝑖
𝑚𝑖𝑛 , 𝑃𝑖

𝑜 − 𝐷𝑅𝑖 ) ≤ 𝑃𝑖 ≤ 𝑀𝑖𝑛(𝑃𝑖
𝑚𝑎𝑥 , 𝑃𝑖

𝑜 + 𝑈𝑅𝑖 ) (9) 

where, 𝑃𝑖
𝑜 is the previous operating point of ith generator and DRi & URi are the down and 

up ramp rate limits respectively.  

e) Prohibited operating zones: If any power plant works in these zones, some faults might 
occur for the machines or accessories such as pumps or boilers. Therefore, to prevent theses 

faults, the power generation limits must be changed so that they satisfy the POZ constraint. 

This feature can be included in the non-convex multi-objective problem formulation as [24] 

 

min

1

1

max

L

i i i

U L

i ik i ik

U

izi i i

P P P

P P P P

P P P

−

  


  


 

 (10)  

Here zi are the number of prohibited zones in i
th generator curve, k is the index of prohibited 

zone of ith generator, P
ik

L
 is the lower limit of kth prohibited zone, and P

ik−1

U
 is the upper limit 

of kth prohibited zone of ith generator. 



500 S. S. PARIHAR, N. MALIK 

3. SOLUTION METHODOLOGY 

The paper implemented FRCGA on three- and six generator networks, to identify the 

best-compromised solution amongst the available set of Pareto optimal solutions. The 

techniques used in the algorithm are as follows: 

3.1. Pareto optimality  

It is defined as the degree of efficacy in multi-objective and multi-criteria solutions and 

represents a condition where economic resources and its output have been assigned in such 

a manner that no objective can be made better without losing the well-being of the other. 

There is no way to improve one part of a Pareto optimal solution set without making another 

part worse. A state U will dominate state V if U is superior to V in at least one objective 

function and not worse in regard to the other objective functions. A decision vector ‘u’ will 

dominate another vector ‘v’ (as m˂n) if 

 𝑓𝑗 (𝑢) ≤  𝑓𝑗 (𝑣)          ⩝ 𝑗 = 1,2,3, , , 𝑖   (11) 

 and  𝑓𝑗 (𝑢) ˂ 𝑓𝑗 (𝑣)   for at least one j (12) 

where j shows a total number of objectives considered for simultaneous optimization. The 

reduction in fuel cost of generator increases the environmental emissions and vice-versa. 

As the considered objectives are conflicting in nature so instead of getting an optimal 

solution a set of non-dominated (Pareto-optimal) solutions have been obtained, hence, 

Pareto-optimal solution has been considered. 

3.2. Real-Coded Genetic algorithm  

In a real-coded genetic algorithm (RCGA) for optimization, the output of each 

generator in the system is illustrated as a floating point rather than a binary number 

resulting in high precision solution [30]. For discontinuous, non-differentiable and discrete 

objective functions the algorithm is proved to be effective and superior to binary coded 

genetic algorithm. The outputs of all the generating units generate a solution string known 

as chromosome. The initial population is randomly generated in a given search space. The 

RCGA loop comprises pre-processing, three genetic operations and post-processing. It 

performs a global optimization to identify the best solution to the formulated problem and 

iterates until the convergence criteria is met. To estimate the fitness value for each 

individual to optimize NCEELD problem mentioned by (4) for a given load while 

satisfying limits shown in (7) and (8): 

𝑀𝑖𝑛 𝐶 = (𝑓1 + 𝛼[∑ 𝑃𝑖
𝑁
𝑖=1 − (𝑃𝐷 + 𝑃𝐿 )])

2 + ([𝑝𝑓𝑒 ∗ (𝑓2 + 𝛼[∑ 𝑃𝑖
𝑁
𝑖=1 − (𝑃𝐷 + 𝑃𝐿 )]

2]) +
                                     ([𝑝𝑓𝑙 ∗ (𝑓3 + 𝛼[∑ 𝑃𝑖

𝑁
𝑖=1 − (𝑃𝐷 + 𝑃𝐿 )]

2])       (13) 

where α represents the penalty parameter that occurs if N/w load demand is not satisfied. 

This guarantees that a feasible solution gets higher fitness as compared to an infeasible 

solution. 



 Fuzzy-based Real-Code Genetic Algo for Optimizing Non-Convex Environment Economic Loss Dispatch 501 

3.3. Fuzzy approach based on min-max proposition 

To optimize three conflicting objectives (fuel cost, emission and N/w loss) simultaneously 

is a tedious task as there are no single criteria to finalize the merit of the available non-dominated 

solutions. Due to the conflicting nature of the objectives, it is hard to find the best solution. 

Every objective is assigned a degree of satisfaction based on the membership functions provided 

by the fuzzy method. The membership functions represent the degree of membership in fuzzy 

sets in the range [0,1].  (Fi) is monotonically decreasing function given as [9]: 

   

min

max

min max

max min

max

1;

( ) ;

0;

i i

i i

i i i i

i i

i i

F F

F F
F F F F

F F

F F






 −
=  

−
 


  (14) 

where F
i

min
 represents the expected minimum value and F

i

max
 represents the expected 

maximum value of objective function i.  

The membership function value signifies how much a solution satisfies Fi on a scale of 

0 to 1. The fuzzy min-max proposition to nominate the best solution amongst many solutions 

can be given as [9] 

  µ𝑏𝑒𝑠𝑡𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 = 𝑀𝑎𝑥{min [µ(𝐹𝑗 )]
𝑘 }  (15) 

where k is the number of Pareto-optimal solutions.  

 

Each objective is expected to attain higher satisfaction for each solution. The best-

compromised solution is identified based on the highest rank among k solutions. The 

pseudo-code to solve NCEELD problem is shown below 

Step I: Initialise the cost coefficients, generator limits, load demand and the min-max 

values of each objective. 

Step II: Create a random population to define the number of generators within specified 

limits. 

Step III: Evaluate the fitness of the constrained tri-objective problem of the network with 

prohibiting operating zones and ramp rate limits. 

Step IV: Single point crossover is used for pairing and mating of the selected chromosomes. 

Step V: Mutant is created on a random basis. 

Step VI: Create new chromosomes and offspring for convergence check. 

Step VII: Select the fittest individual for the next generation. 

Step VIII: Check the convergence criteria. If the maximum counter is reached, jump to 

Step IX. Else, Step IV. 

Step IX: Calculate the membership value of the Pareto optimal solutions using (14). The Fmin 

and Fmax value of each objective are determined by optimizing all the objectives 

independently to determine the endpoints of the obtained Pareto front. 

Step X: The degree of satisfaction attained for each objective is used to find the best-

compromise solution based on min-max proposition as given in (15).    



502 S. S. PARIHAR, N. MALIK 

4. RESULTS AND DISCUSSION  

To validate the performance, FRCGA has been employed to solve NCEELD problem 

on two networks having 3 and 6 generators satisfying all the operational network constraints at 

various power demands. The network data for 3 and 6 generating units is given in the 

appendix (Table 13, Table 14, Table 15 and Table 16). A program to imitate results for 

both the test N/w is written on MATLAB 7.10. The standard IEEE-30 bus network with 

six generator units is presented in Fig.1. 

 

Fig. 1 One-line diagram of 30-bus network 

 

To demonstrate the superiority of the FRCGA, three different test cases have been 

identified at different network complexity. The convergence test was carried out employing 

the same evaluation function for the same no. of iterations for convex case. The results for 

one trial of 250 iterations are shown in Fig. 2, Fig. 3 and Fig. 4 for optimized cost, emission 

and loss function respectively. It can be seen that FRCGA converges faster for the population 

size of 500. 



 Fuzzy-based Real-Code Genetic Algo for Optimizing Non-Convex Environment Economic Loss Dispatch 503 

 

Fig. 2 Convergence characteristic for best fuel cost solution for different pop sizes 

 

Fig. 3 Convergence characteristic for best emission solution for different pop sizes  

 

Fig. 4 Convergence characteristic for best n/w loss solution for different pop sizes   

0 50 100 150 200 250
606

608

610

612

614

616

618

620

622

no. of iteration

fu
e
l 
c
o
s
t

 

 

popsize=200

popsize=300

popsize=500

popsize=400

0 50 100 150 200 250
0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

no. of iteration

e
m

is
s
io

n

 

 

popsize=200

popsize=300

popsize=400

popsize=500

0 50 100 150 200 250
0

0.02

0.04

0.06

0.08

0.1

no. of iteration

s
y
s
te

m
 l
o
s
s

 

 

popsize=200

popsize=300

popsize=400

popsize=500



504 S. S. PARIHAR, N. MALIK 

Hence, the optimal settings for both cases are the same, with the exception of population 

size and are mentioned in Table 1 

Table 1 FRCGA parameters for different case studies 

Parameters Selected value 

population size 200 (case 1) 

500 (case 2 & 3) 

Selection rate 0.3 

Mutation rate 0.2 

Trials 60 

Iterations 250 

4.1. Environmental Economic Dispatch  

Three and six generator networks have been tested without considering the effect of 

VPL in the network. Table 2 illustrates the best cost and emission linked with the network 

at two different power demands of 400 MW and 500 MW. When cost minimization is 

performed, the generating fuel cost and N/w emissions are 20792.88 $ and 206.3426 Kg, 

respectively, but the cost of the generator increases to 20846.60 $, and the network harmful 

emission reduces to 200.1578 Kg in ED case at power demand of 400 MW. For 500 MW, 

the generator cost and N/w emissions are 25453.26 $ and 319.5089 Kg when cost 

minimization is performed, but the cost rises to 25500.40 $ and emission reduces to 

311.0776 Kg. Using min and max values of each objective function, the membership value 

of the non-dominated solutions is determined. 

Table 2 Best solution for ELD and ED of 3-unit N/w at PD=400 MW and 500 MW 

 Load demand 

400 MW 500 MW 

ELD ED ELD ED 

P1(MW) 81.4957 106.4685 103.5167 130.8372 

P2(MW) 175.8190 151.1246 217.1612 190.1187 

P3(MW) 149.8137 149.7724 190.9736 190.7181 

Fuel cost ($) 20792.88 20846.60 25453.26 25500.40 

Emission (Kg) 206.3426 200.1578 319.5089 311.0776 

Loss (MW) 7.5560 7.3865 11.9239 11.6800 

The simultaneous optimization of the environmental emission and the generator fuel 

cost is carried out to determine a best-compromise solution. In Table 3 and Table 4, five 

intermediate Pareto solutions are listed from the attained Pareto solution set using the 

presented approach with its membership values. Solution 5 is selected as the best solution 

having the highest rank of 0.1584 and 0.1110 at 400 MW and 500 MW respectively. 



 Fuzzy-based Real-Code Genetic Algo for Optimizing Non-Convex Environment Economic Loss Dispatch 505 

Table 3 Pareto optimal solutions for the convex-EED problem at PD=400 MW (3-unit N/w) 

Solution 

number 

Cost ($) Emission 

(Kg) 
µ𝟏 µ𝟐 µ𝒎𝒊𝒏 

1 20845.74 203.7849 0.0160 0.4135 0.0160 

2 20843.59 200.6626 0.0560 0.9184 0.0560 

3 20812.80 205.3911 0.6293 0.1539 0.1539 

4 20838.31 200.3850 0.1544 0.9633 0.1544 

5 20838.09 200.2123 0.1584 0.9912 0.1584 

Table 4 Pareto optimal solutions for the convex-EED problem at PD=500 MW (3-unit N/w) 

Solution 

number 

Cost ($) Emission 

(Kg) 

µ𝟏 µ𝟐 µ𝒎𝒊𝒏 

1 25497.79 312.3221 0.0553 0.8524 0.0553 

2 25497.63 311.0877 0.0586 0.9988 0.0586 

3 25497.56 312.2660 0.0602 0.8590 0.0602 

4 25496.93 311.1103 0.0737 0.9961 0.0737 

5 25495.17 311.1194 0.1110 0.9950 0.1110 

The summarized result for a best-compromised solution for three generating unit 

network is tabulated in Table 5 and is compared with the other methods such as GA [25], 

PSO [25] and FPA [27].  

Table 5 Best solution for the convex-EED problem at PD=400 MW and 500 MW (3-unit N/w) 

 
Best-Compromised solution 

400 MW 500 MW 

FRCGA GA [25] PSO [25] FPA [27] FRCGA GA [25] PSO [25] FPA [27] 

P1 (MW) 102.8514 102.617 102.612 102.4468 129.3252 128.997 128.984 128.8074 

P2 (MW) 154.0217 153.825 153.809 153.8341 192.4745 192.683 192.645 192.5906 

P3 (MW) 150.5278 151.011 150.991 151.1321 189.8764 190.11 190.063 190.2958 

Fuel cost ($) 20838.09 20840.10 20838.30 20838.10 25495.17 25499.40 25495.00 25494.70 

Emission (Kg) 200.2123 200.256 200.221 200.2238 311.1194 311.273 311.15 311.155 

Loss (MW) 7.4090 7.41324 7.41173 7.4126 11.6882 - - - 

Total cost ($) 29559.59 29563.20 29559.90 29559.81  39209.7 39220.10 39210.20 39210.15 

The comparison depicts that the total generation cost incurred in solving EED problem 

from the FRCGA approach is lower than that incurred using other optimization approaches 

in both test cases. Thus, FRCGA succeeds to obtain the global minimum solution and 

performs superior to these algorithms in respect of all parameters. The total network losses 

for the best-compromised solution are 7.4090 MW and 11.6882 MW for power demand of 

400 MW and 500 MW, respectively. For 30-bus N/w, the best-compromised solution attained 

has the value of 0.1999 lb/hr and 619.90 $/hr respectively for harmful environmental emission 

and cost, respectively at load demand of 2.834 p.u and is in close agreement with 0.1969 lb/hr 

and 623.87 $/hr as mentioned in [20]. Fig. 5 is the Pareto front drawn between the fuel cost and 

the emission points which was found to have an inverse relationship between the two objectives. 



506 S. S. PARIHAR, N. MALIK 

 
Fig. 5 Pareto front between generator fuel cost ($/hr) and emission (lb/hr) for convex EED 

4.2. Environmental Economic Loss Dispatch with valve-point loading 

The performance of the FRCGA on the NCEELD problem is examined for the first time 

on the IEEE 30-bus network at two different loading conditions. Three objectives (fuel cost, 

environmental emission and losses) are simultaneously considered and optimized to obtain 

minimum network generation cost. The total generation cost comes out to be 1810.10 $/hr at 

2.834 p.u load demand which is found to be superior to published results at 2.834 p.u.  

The minimum-maximum limits for fuel cost with VPL effect, harmful environmental 

emissions and losses for load demand of 2.834 p.u and 4.32 p.u are given in Table 6. For the 

load of 2.834 p.u, the values attained for cost and emission is 608.02 $/hr and 0.1938 lb/hr 

that is found to be less when compared to 626.96 $/hr & 0.2110 lb/hr [26], 613.342 $/hr & 

0.2028 lb/hr [28] and 613.338 $/hr & 0.1953 lb/hr [29],  respectively.  The membership values 

of all the Pareto optimal solutions for the NCEELD problem are obtained. Five intermediate 

solutions are tabulated in Table 7 and Table 8 for PD=2.834 p.u and PD=4.32 p.u respectively.  

Table 6 Min-max limit for fuel cost with VPL effect, emission and loss at 2.834 p.u and 4.32 p.u 

  Load (p.u) 

2.834 4.32 

Cost ($/hr) Minimum 608.02 965.93 

Maximum 646.19 980.67 

Emission 

(lb/hr) 

Minimum 0.1938 0.2263 

Maximum 0.2211 0.2422 

Loss (p.u) Minimum 0.0209 0.0514 

Maximum 0.0379 0.0612 

Table 7 Pareto optimal set of NCEELD problem with VPL effect for load PD=2.834 p.u 

Solution 

Number 

Cost 

($/hr) 

Emission 

(lb/hr) 

Loss 

(p.u) 
1 2 3 µ𝑚𝑖𝑛  

1 622.74 0.1973 0.0262 0.6144 0.8704 0.6894 0.6144 

2 622.62 0.2001 0.0228 0.6174 0.7697 0.8862 0.6174 

3 621.53 0.2022 0.0228 0.6461 0.6911 0.8863 0.6461 

4 619.84 0.2021 0.0268 0.6905 0.6961 0.6558 0.6558 

5 614.99 0.2027 0.0255 0.8174 0.6742 0.7284 0.6742 

615 620 625 630 635 640 645 650
0.193

0.194

0.195

0.196

0.197

0.198

0.199

0.2

0.201

cost

e
m

is
s
io

n



 Fuzzy-based Real-Code Genetic Algo for Optimizing Non-Convex Environment Economic Loss Dispatch 507 

Table 8 Pareto optimal set of NCEELD for load PD=4.32 p.u 

Solution 

Number 

Cost 

($/hr) 

Emission 

(lb/hr) 

Loss 

(p.u) 

Total cost 

($/hr) 
1 2 3 µ𝒎𝒊𝒏 

1 973.38 0.2326 0.0555 4490.2 0.4944 0.6013 0.5774 0.4944 

2 972.85 0.2329 0.0563 4515.4 0.5301 0.5831 0.4959 0.4959 

3 972.96 0.2335 0.0541 4489.12 0.5228 0.5451 0.7296 0.5228 

4 972.76 0.2332 0.0545 4475.29 0.5365 0.5666 0.6788 0.5365 

5 972.22 0.2335 0.0543 4497.8 0.5728 0.5480 0.7045 0.5480 

The results reveal that the best-compromise solution for load demand of 2.834 p.u is 

2099.20 $/hr and for load PD=4.32 p.u is found to be 4497.82 $/hr with the highest rank of 

67.42% and 54.80% respectively depending upon its membership value of each objective. 

Fig. 6 depicts the convergence criteria of 30-bus network on two different loads which 

reveal that the convergence of load PD= 2.834 p.u and PD= 4.32 p.u is attained faster even 

for the complex multi-objective minimization problem. 

 

 
Fig. 6 Convergence characteristic for total generation cost for different load conditions 

4.3. Environmental Economic Loss Dispatch with valve-point loading, POZs and RRL 

For this test case, all the mentioned practical constraints and non-linear characteristic 

of non-convex multi-objective problem are considered. Due to which this test case is more 

complex than other test cases considered above. Data for the ramp rate limits and POZs 

has been taken from appendix (Table 15 and Table 17). The generator ramp rate limit needs 

to be satisfied as generator output cannot change (increase or decrease its output) arbitrarily 

to any value, the change has to within the up/down ramp rate limits. The inclusion of ramp 

rate limits changes the operating limits of the generator. The minimum-maximum limits of 

fuel cost, emission and loss evaluated for the six-unit system with POZs and RRL are given 

in Table 9 with load demand 2.834 pu. The results presented in Table 10 provides the 

intermediate solutions obtained using RCGA. The best solution is ranked on the basis of 

its performance for all the objectives considered. Therefore, overall rank for extreme points 

is zero. The rank of best solution is found to be 0.6685 which indicated that all three 

objectives are satisfied at least 66.85 % for load of 2.834 p.u. 

0 50 100 150 200 250
0

1

2

3

4

5

6
x 10

4

iteration

fi
tn

e
s
s
 f
u
n
c
ti
o
n

 

 

load= 2.834 pu

load= 4.32 pu



508 S. S. PARIHAR, N. MALIK 

Table 9 Min-max limit for fuel cost with VPL effect, emission and loss with POZs and 

RRL at 2.834 p.u 

Cost($/h) Emission(lb/h) Loss(pu) 

minimum maximum minimum maximum minimum maximum 

611.2998 645.3562 0.1942 0.2073 0.0256 0.0358 

Table 10 Pareto optimal set of NCEELD with POZs and RRL for load PD=2.834 p.u 

 Cost 
($/h) 

Emission 
(lb/h) 

Loss 
(pu) 

µ1 µ2 µ3 µ𝑚𝑖𝑛  

SOL.1 624.7335    0.1975     0.0257 0.6055        0.7473    0.9901 0.6055        
SOL.2 623.7816    0.1989     0.0242  0.6335         0.6421    1.0000 0.6335         
SOL.3 623.3781    0.1979     0.0291 0.6453         0.7208    0.6589 0.6453         
SOL.4 621.4747    0.1987     0.0283    0.7012         0.6598    0.7379 0.6598    
SOL.5 620.3646    0.1985     0.0256 0.7338         0.6685    1.0000 0.6685    

 

The results clearly showed that all the constraints, such as VPL effect, POZs, RRL, 

generation limits and power balance constraints were fully satisfied for all considered test 

cases of tri-objective optimization problem. Due to the non-convexity constraints 

introduced in test system, the cost increases from 608.0296 $/hr to 611.2998 $/hr, emission 

increases from 0.1938 lb/hr to 0.1942 lb/hr and system loss from 0.0209 p.u to 0.0256 p.u. 

4.4. Statistical Analysis 

Table 11 lists the comparison of different approaches for cost and emission minimization 

in terms of their minimum, maximum, mean and median values, respectively, for IEEE 30-

bus N/w. The cost minimum (Cmin), cost mean (Cmean), cost median (Cmedian), emission minimum 

(Emin), emission mean (Emean) and emission median (Emedian) values obtained for the ELD and 

ED problem, respectively, are found to be lowest as compared to other published work. The 

statistical comparison of CEED problem has also been shown in Table 12 in terms of their 

mean and standard deviation. The values of Cmean and Emean obtained from solving convex 

CEED problem also demonstrates the superiority of the method. The value of cost standard 

deviation (Cstd) and emission standard deviation (Estd) attained from the proposed approach 

of FRGCA are 7.127 and 0.0057, respectively which is less than that obtained from other 

approaches. This clearly shows that the obtained results lie close to its mean value as 

compared to other published methods. 

Table 11 Statistical comparison of ELD and ED minimization for IEEE 30-bus N/w at load 

PD=2.834 p.u 

[1] Fuel cost 
minimization 

 Cmin Cmax Cmean Cmedian 

Proposed approach  601.31 610.07 603.20 602.23 
GQPSO [31] 606.38 611.86 609.49 609.66 
SAIWPSO [32] 605.99 606.00 605.99 605.99 
NGPSO [20] 605.99 605.99 605.99 605.99 

[2] Emission 
minimization 

 Emin Emax Emean Emedian 

Proposed approach 0.1938 0.2295 0.1941 0.1940 
GQPSO [31] 0.1942 0.1946 0.1944 0.1944 
SAIWPSO [32] 0.1941 0.1941 0.1941 0.1941 
NGPSO [20] 0.1941 0.1941 0.1941 0.1941 



 Fuzzy-based Real-Code Genetic Algo for Optimizing Non-Convex Environment Economic Loss Dispatch 509 

Table 12 Statistical comparison of CEED minimization for IEEE 30-bus N/w at load 

PD=2.834 p.u 

 Cmean Cstd Emean Estd 

Proposed approach 622.62 7.127 0.2012 0.0057 

GQPSO [31] 644.09 12.2       0.2109 0.0095 

SAIWPSO [32] 623.76 - 0.1970 - 

NGPSO [20] 623.86 - 0.1969 - 

5. CONCLUSION 

The fuzzy-based RCGA is demonstrated to solve multi-objective environmental economic 

loss dispatch problem considering non-convex and non-smooth fuel cost function. The multi-

objective minimization problem is transformed into the constrained single-objective problem 

by the use of price penalty factor which blends all competing objectives (generator cost, 

environmental emission and system losses). Because the objectives are inversely related, a set 

of Pareto optimal solutions are attained rather than a single optimal solution for a given 

objective. Furthermore, a fuzzy approach is exploited to extract best-compromised solution as 

per the highest rank based on their membership values. The convergence of the NCEELD 

problem at different load demand is also analyzed considering the different practical operating 

limits (POZs, RRL and VPL) of the network. The total generation cost of the network attained 

from the proposed method for different test cases has been compared to the other techniques 

which validate the solution to NCEELD problem for small and large networks. The statistical 

analysis also validates the FRGCA approach. The percentage reduction in Cstd and Estd values 

are 41.5% and 40% as compared to ref. [31]. The proposed work can further be extended for 

the study of integration of renewable energy sources and for practical transmission networks 

considering dynamic non-convex CEELD problem. 

APPENDIX  

Table 13 Generator cost, emission coefficients & generation constraints for three generating 

unit network 

Cost 

Coefficients 

 G1 G2 G3 

ai 0.03546 0.02111 0.01799 

bi 38.30553 36.32782 38.27041 

ci 1243.5311 1658.5696 1356.6592 

Emission 

Coefficients 

αi 0.00683 0.00461 0.00461 

βi -0.54551 -0.5116 -0.5116 

𝛾i 40.2669 42.89553 42.89553 

Unit limits 
Pmin (p.u)   35 130 125 

Pmax(p.u) 210 325 315 



510 S. S. PARIHAR, N. MALIK 

Table 14 B-coefficients for three generating unit network 

 

Bij * 0.0001 

 

0.71 0.3 0.25 

0.3 0.69 0.32 

0.255 0.32 0.8 

Table 15 Generator fuel cost, emission coefficients and N/w generation constraints for 30-

bus N/w 

 

Cost 

coefficients 

 G1 G2 G3 G4 G5 G6 

ai 100 120 40 60 40 100 

bi 200 150 180 100 180 150 

ci 10 10 20 10 20 10 

ei 200 200 200 200 200 200 

fi 0.0050 0.0060 0.0010 0.0009 0.0009 0.0015 

 

Emission 

coefficients 

αi 4.091 2.543 4.258 5.326 4.258 6.131 

βi -5.554 -6.047 -5.094 -3.550 -5.094 -5.555 

𝛾i 6.490 5.638 4.586 3.380 4.586 5.151 
𝜁i 0.0002 0.0005 0.00001 0.002 0.000001 0.00001 
𝜆i 2.857 3.333 8.000 2.000 8.000 6.667 

Generator unit 

constraints 

Pmin (p.u) 0.05 0.05 0.05 0.05 0.05 0.05 

Pmax (p.u) 0.5 0.6 1.0 1.2 1.00 0.60 

Ramp rate limits 
DRi(up)/h 0.08 0.11 0.15 0.18 0.15 0.18 

DRi(dn)/h 0.08 0.11 0.15 0.18 0.15 0.18 

Table 16 B-coefficients for six generating unit network 

Bij 

0.1382 -0.0299 0.0044 -0.0022 -0.0010 -0.0008 

-0.0299 0.0487 -0.0025 0.0004 0.0016 0.0041 

0.0044 -0.0025 0.0182 -0.0070 -0.0066 -0.0066 

-0.0022 0.0004 -0.0070 0.0137 0.0050 0.0033 

-0.0010 0.0016 -0.0066 0.0050 0.0109 0.0005 

-0.0008 0.0041 0.0066 0.0033 0.0005 0.0244 

Bo -0.0107 0.0060 -0.0017 0.0009 0.0002 0.0030 

Boo 0.00098573      

Table 17 POZs of units for IEEE-30 bus N/w 

Unit 1 2 5 

POZ [0.10  0.15] [0.25  0.30] [0.50  0.55] 

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