Energy and Power En gi neering, 2011, 3, 113-119
doi:10.4236/epe.2011.32015 Published Online May 2011 (http://www.SciRP.org/journal/epe)
Economic Dispatch with Multiple Fuel Options Using CCF
Radhakrishnan Anandhakumar, Srikrishna Subramanian
Department of Electrical Engineering, FEAT, Annamalai University,
Annamalainagar, India
Received February 6, 2011; revised March 24, 20 1 1; accepted April 2, 2011
Abstract
This paper presents an efficient analytical approach using Composite Cost Function (CCF) for solving the
Economic Dispatch problem with Multiple Fuel Options (EDMFO). The solution methodology comprises
two stages. Firstly, the CCF of the plant is developed and the most economical fuel of each set can be easily
identified for any load demand. In the next stage, for the selected fuels, CCF is evaluated and the optimal
scheduling is obtained. The Proposed Method (PM) has been tested on the standard ten-generation set system;
each set consists of two or three fuel options. The total fuel cost obtained by the PM is compared with earlier
reports in order to validate its effectiveness. The comparison clears that this approach is a promising alterna-
tive for solving EDMFO problems in practical power system.
Keywords: Economic Load Dispatch, Composite Cost Function, Multiple Fuel Options, Piecewise Quadratic
Function, Mathematical Model
1. Introduction
The economic dispatch problem in a power system is to
determine the optimal combination of power outputs for
all generating units which will minimize the total cost
while satisfying the load and operational constraints [1].
The economic dispatch problem is very complex to solve
because of its colossal dimension, a non-linear objective
function, and a large number of constraints. Conven-
tional techniques lambda iteration method and quadratic
programming offer good results, but when the search
space is nonlinear and has discontinuities, they become
very complicated with a slow convergence ratio and do
not always seek the optimal solution. New numerical
methods are needed to cope with these difficulties, espe-
cially those with high speed search for the optimal and
not being trapped in local minima [2].
The stochastic search algorithms such as Simulated
Annealing (SA) and Genetic Algorithm (GA) have been
applied to determine the optimal generation schedule for
economic dispatch problem in a power system [3]. SA is
applied in many power system problems, but setting the
control parameters of the SA algorithm is a difficult task,
and the convergence speed is slow when applied to a real
system. The GA methods have been employed to suc-
cessfully to solve complex optimization problem, recent
research has identified some deficiencies in GA per-
formance. Particle swarm optimization method (PSO)
has been applied for solving economic dispatch problems
with various operating constraints [4].
A novel optimization approach, Artificial Immune
System (AIS) has been applied to solve constrained eco-
nomic load dispatch problem [5]. This approach utilizes
the clonal selection principle and evolutionary approach
wherein cloning of antibodies is performed followed by
hyper mutation. A novel coding scheme for practical
economic dispatch by modified particle swarm optimiza-
tion approach has also been proposed to solve economic
dispatch problem [6]. The heuristic search technique,
Differential Evolution has been suggested for solving
economic dispatch problems [7]. Bacterial Foraging-
Nelder Mead method has been applied for the solution of
economic dispatch problems [8].
In certain fossil fire systems, the generation cost func-
tion is represented as a segmented piecewise quadratic
function. The generating unit, supplied with multi-fuel
sources like coal, natural gas or oil suffers with the
problem of determining the most economic fuel to burn.
Such a problem has been solved using the Hierarchical
Method (HM) of Lagrangian multipliers method to find
the incremental fuel cost for subsystems comprising sets
of units [9]. The solution searches for the optimal for
various choices of fuel and generation range of the units
iteratively. A Hopfield neural network approaches to

114 R. ANANDHAKUMAR ET AL.
economic dispatch problems has been proposed [10]. An
improved adaptive Hopfield Neural Network (HNN)
approach has been proposed for finding the solution for
economic dispatch with multiple fuel options [11]. The
HNN suffers with slow convergence rate and normally
takes a large number of iterations. A hybrid real coded
GA method has been presented for solving the economic
dispatch problem with multip le fuel options [12 ].
An enhanced Lagrangian neural network has been ap-
plied to solve the economic load dispatch problems with
piecewise quadratic cost functions [13]. In this method
the convergence speeds are enhanced by employing by
momentum technique and providing criteria for choosing
the learning rate. Economic dispatch solutions with
piecewise quadratic cost functions has been solved by
using improved genetic algorithm [14]. In order to im-
prove the effectiveness of genetic algorithm multi-stage
algorithm and directional crossover methods are pro-
posed and projection method is introduced to satisfy a
linear equality constraint from power balance. The heu-
ristic search techniques such as PSO [15] ,Taguchi
method (TM) [16], Evolutionary Programming (EP) [17]
and its improved version are also been applied to solve
the economic dispatch problems with multiple fuel op-
tions [18,1 9] .
The CCF is a non-iterative direct method, gives the
most economic dispatches of the online units with less
computation time. In th is paper, the CCF is used to solve
the economic dispatch problem with multiple fuel op-
tions.
2. Problem Formulation
2.1. Nomenclature
ai, bi, ci Fuel cost coefficients of the units
asj, bsj, csj Fuel cost coefficients of the set S with k
fuel options
A, B, C Composite cost coefficients
AS, BS, CS Composite cost coefficients of the set S
AP, BP, CP Composite cost coefficients of the plant
FCSj Fuel cost function of the set S with k fuel
options, in $/h
FCS Fuel cost function of the set S, in $/h
FCP Fuel cost function of the plant, in $/h
k Number of fuel options in a plant
N Number of generation units
min
S
P Minimum power generation of set S, in
MW
max
S
P Maximum power generation of set S, in
MW
PS Economic dispatch of the set S, in MW
PG Power generation of the plant, in MW
PD Power demand, in MW
S Set of generating units in a plant
λ Incremental production cost, in $/MWh
2.2. Economic Dispatch Problem with Multiple
Fuels
The main objective of economic dispatch is to find the
optimal combination of power generation that minimizes
the total generation cost while satisfying equality and
inequality constraints. A piecewise quadratic function is
used to represent the input-output curve of a generator
with multiple fuel options. For a generator with k fuel
options, the cost curve is divided into k discrete regions
between lower and upper boun d s. The economic dispatch
problem with piecewise quadratic function is defined as
Minimize 
1
N
ii
i
F
P



2m
111
2
2221 2
2m
1
,1,
,2,
,,
iiiiiiii
iiii iiii
ii
ik iik iikikii
aPbP cfuelPPP
aPbPcfuelPPP
FP
aPbP cfuelkPPP

 
 



 


in
1
ax
(1)
where Fi(Pi) is the fuel cost function of the ith unit, Pi is
the power output of the ith unit, N is the number of gen-
erating units in the system and aik, b
ik and cik are cost
coefficients of the ith unit using fuel type k.
Minimization of the generation cost is subjected to the
following constraints:
1) The power balance constraints
1
N
iD
i
PP


 (2)
where
D
P is the total system demand in MW.
2) Generating capacity constraints
min max
iii
PPP (3)
where, and are the minimum and maxi-
mum power outputs of the ith unit.
min
i
Pmax
i
P
3. Composite Cost Function (CCF)
The composite cost coefficients were reported in the lit-
erature [9].


12
11 11
N
A
aa a (4)


112 2NN
Bbaba baA (5)
2;
G
AP B


 where
GD
PP
2
i
Gi
i
b 
Pa

N where (6) 1, 2,,i

R. ANANDHAKUMAR ET AL.
115


12
222 2
1122
44 4
N
NN
Ccc c
bababa BA

 

4
(7)
2
P
PGPG P
F
CAPBPC (8)
The A, B and C are the composite cost coefficients and
can be easily calculated by using Equations (4), (5) and
(7) respectively. For a particular load demand, the opti-
mal generation of units is directly computed using Equa-
tion (6).
4. Proposed Approach for Economic
Dispatch with MFO
The proposed methodology consist two stages of the
most economic fuel identification and economic sched-
uling. In the first stage, the composite cost function of
the plant is developed as de tailed in the previous section.
The incremental cost of the plant for a particular load
demand is calculated and the generation dispatch for
each unit is determined. The dispatch of each set directly
indicates the most economical fuel and feasible operating
region. In the second stage, the dispatch of the generating
units is refined within the feasible operating region. The
composite cost function is developed with the selected
fuels and is solved to obtain the most economic dispatch
of generating units.
4.1. Identification of the Most Economic Fuel
Consider a plant consists of ‘S’ set of generation, each
set consists of ‘k’ fuel options.
The fuel cost function of the set ‘S’ with ‘k’ fuel op-
tions is,
2
min max
,
;
1, 2,,;1,2,,
SjSj SSj SSj
SSS
FCaPb Pc
PPP
SNj


k
(9)
The composite cost function of set ‘S’ is calculated by
using Equa tions (4), (5) and (7).
2
SSSSS S
F
CAPBPC (10)

12
11 11
Sk
A
aa a (11)

112 2Sk
Bbaba baAkS
(12)


12
222 2
1122
444 4
Sk
kk S
Cccc
bababa BA

 

S
(13)
In this manner, the CCF for the plant is calculated and
is given as,
2
P
PGPG P
F
CAPBPC (14)
The composite coefficients are constant for any load
demands.
The incremental production cost of the plant for the
demand is calculated by
2;where
P
GP GD
A
PBP P


 (15)
The economic dispatch of the set ‘S’ is calculated as,
2
S
S
S
B
P
A
 
 (16)
The above equation provides the dispatch of each set
to meet the load demand. Based on this dispatch, the
most economical fuel and the feasible operating region
of each set can be easily identified.
4.2. Economic Dispatch of the Selected Fuels
The composite cost function is developed using the most
economic fuels. The generation dispatch is refined within
the feasible limits as detailed in Section 3.
The computational flow of the proposed methodology
for solving economic dispatch problem with multiple
fuel options is presented as a flow chart in Figure 1.
Start
Read the cost coefficients, power
Ge neration limits and load
Compute composite cost coefficients A, B, and C for each set and
evaluate the composite cost function for the plant
Compute optimum power generation required for each set to
meet the load demand
Ident ify t he most econ omical f uel of each set
Compute composite coefficients A, B, C for the selected fuels
Obtain the economic schedule and ca lculate total fu el c ost
Stop
Figure 1. Flow chart of the proposed method.

R. ANANDHAKUMAR ET AL.
116

5. Numerical Simulation Results and
Discussion
The proposed technique has been implemented in
MATLAB on a 2.10 GHz core2Duo processor PC. The
simulation studies have been carried out on ten-generat-
ing unit system with multiple fuel options [9]. This
problem includes one objective function with ten vari-
able parameters (P1, P2,, P10), one equality and
twenty inequality constraints, i.e. power balance con-
straint and maximum and minimum limits of each gen-
erating unit.

By the proposed strategy, the economic dispatch solu-
tion for the given system can be obtained in two stages: 1)
evaluate the composite fuel cost function for the plant
and is solved to identify the most economic fuel and fea-
sible operating region of each set, and 2) the optimal
dispatches are calculated by solving composite fuel cost
function with selected fuels.
The implementation of the proposed strategy for the
given system is detailed as follows. The equivalent cost
function of set ‘S’ is calculated using the Equations (11),
(12) and (13). In the selected sample system the number
of sets in the plant is ten and the number of units is
twenty nine.
For example, the equivalent cost function of the set 5
is,
2
555
0.0000448070.0493 91.5841FCP P (17)
Similarly, for set 8, the equivalent cost function is,
2
888
0.0000673690.04404 126.6098FCPP (18)
In this manner, the equivalent cost function of each set
is determined. By combining these cost coefficients, the
equivalent cost function of the plant is calculated.
The equivalent cost function of the plant is,
06 2
7.32870.3933 1518
PGG
FCe PP

 (19)
The incremental cost of the plant is obtained by using
Equation (15) .
For a load demand of 2400 MW, the λ is 0.4285 and
the dispatch of each set is determined by using Equation
(16). Then, the dispatches of the set 5 and 8 are 278 MW
and 265 MW respectively. In this manner, the dispatch of
each set is identified and it indicates the most economical
fuel and the feasible operating region. For the set 5, the
dispatch is 278 MW, then the most economical fuel is 1
and the feasible operating region is 190 MW to 338 MW.
For the set 8, the dispatch is 265 MW, then the most
economical fuel is 3 and the feasible operating region is
200 MW to 265 MW.
Similarly, the economic dispatch is performed to de-
termine the optimal dispatches within the feasible oper-
ating region by using the composite cost function of the
selected fuels. During the calculation of dispatch, if the
generation of any unit violates the effective limits, its
generation are fixed at the violated limit. Then that unit
is eliminated from the dispatch procedure. The genera-
tion of all the units except the violated unit is recalcu-
lated using the above procedure with total generation
equal to the load demand minus the generation of the
limit violated unit.
The simulation is performed for various load demands
of 2400 MW, 2500 MW, 2600 MW and 2700 MW. The
optimal dispatches obtained by the proposed methodol-
ogy and HM [9] for the above mentioned load demands
are compared and the comparison is presented in Table 1.
These results signify that the proposed CCF always pro-
vides better solution than HM [9]. Though the HM
method is an iterative mathematical approach, suffers
with the assumption of initial la mbda values. In addition ,
two operating points having same incremental cost also
exist and it requires valid assumption to choose the op-
timal fuel for a particular demand, improper selection
will lead infeasible solution. Additionally, the proposed
CCF is a direct or non iterative method, it does not de-
mand any initial guess values for economic dispatch of
units for the given loa d demand.
The comparison of total fuel cost obtained by pro-
posed methodology, HM [9], HNN [10], AHNN [11],
HGA [12], Modified PSO (MPSO) [15], TM [16], Im-
proved Fast EP (IFEP) [17], Fast EP (FEP) [17], Classi-
cal EP (CEP) [17], PSO [18], and Improved EP (IEP) [19]
is presented in Table 2. As seen the comparison, the gen-
eration costs obtained by CCF are lowest among the re-
sults. Moreover, the optimal fuel cost obtained through
the CCF method is exactly same as the HGA [12], except
the load demand of 2600 MW. However, the proposed
method directly provides the optimal schedule and it
utilizes the CCF for fuel selection and economic sched-
uling. These comparison results confirm that the CCF
provides better solution quality.
The salient features of proposed approach over exist-
ing methods are:
 The approach is conceptually simple.
 It is a non iterative method.
 The simplified generalized expression directly gives
the most economical fuel and the feasible operating
region for a particular load demand.
 It also provides the most economic schedule of gen-
eration with less computational effort.
 It requires negligible computational time, hence it is
suitable for on line applications.
 The performance of the proposed is independent of
system size; hence it is suitable for system of any
size.

R. ANANDHAKUMAR ET AL.
117
Table 1. Comparison of simulation results between proposed method and HM.
LOAD DEMAND = 2400 MW LOAD DEMAND = 2500 MW
PM HM [9] PM HM [9]
UNIT
FTGEN (MW) FT GEN (MW)FT GEN (MW)FTGEN (MW)
1 1 189.7405 1 193.2 2 206.5190 2 206.6
2 1 202.3427 1 204.1 1 206.4573 1 206.5
3 1 253.8953 1 259.1 1 265.7391 1 265.9
4 3 233.0456 3 234.3 3 235.9531 3 236.0
5 1 241.8297 1 249.0 1 258.0177 1 258.2
6 3 233.0456 1 195.5 3 235.9531 3 236.0
7 1 253.2750 1 260.1 1 268.8635 1 269.0
8 3 233.0456 3 234.3 3 235.9531 3 236.0
9 1 320.3832 1 325.3 1 331.4877 1 331.6
10 1 239.3969 1 246.3 1 255.0562 1 255.2
LOAD DEMAND = 2600 MW LOAD DEMAND = 2700 MW
PM HM [9] PM HM [9]
UNIT
FTGEN (MW) FT GEN (MW)FT GEN (MW)FTGEN (MW)
1 2 216.5442 2 216.4 2 218.2499 2 218.4
2 1 210.6058 1 210.9 1 211.6626 1 211.8
3 1 278.1441 1 278.5 1 280.7228 1 281.0
4 3 239.0967 3 239.1 3 239.6315 3 239.7
5 1 275.5154 1 275.4 1 278.4973 1 279.0
6 3 239.0967 3 239.1 3 239.6315 3 239.7
7 1 285.7585 1 285.6 1 288.5845 1 289.0
8 3 239.0967 3 239.1 3 239.6315 3 239.7
9 1 343.8134 1 343.3 3 428.5216 3 429.2
10 1 271.5861 1 271.9 1 274.9667 1 275.2
Table 2. Comparison of the total cost with different techniques.
TOTAL FUEL COST ($/h)
TECHNIQUES 2400 MW 2500 MW 2600 MW 2700 MW
HM [9] 488.500 526.700 574.030 625.180
HNN [10] 487.870 526.130 574.260 626.120
AHNN [11] 481.700 526.230 574.370 626.240
HGA [12] 481.7226 526.2388 574.3808 623.8092
MPSO [15] 481.723 526.239 574.381 623.809
TM [16] 481.6 ----- ----- 623.7
IFEP [17] ----- 526.25 ----- -----
FEP [17] ----- 526.26 ----- -----
CEP [17] ----- 526.25 ----- -----
PSO [18] ----- ----- ----- 623.88
IEP [19] 481.779 526.304 574.473 623.851
PROPOSED
METHOD 481.7226 526.2338 574.0105 623.8092

R. ANANDHAKUMAR ET AL.
118
6. Conclusions
The economic dispatch problem with multiple fuel op-
tions is a complex optimization problem whose impor-
tance may increase as competition in power generation
intensifies. This paper presents economic dispatch prob-
lem with multiple fuel options using composite cost
function. The proposed CCF based solution for economic
dispatch with MFO offers a best contribution in the area
of economic dispatch. In contrast to the HM [9], this
approach fully explores the cost coefficients and gives a
promising value of power for providing improved eco-
nomic dispatch. The HM method requires the valid as-
sumptions such as initial value of lambda and it itera-
tively solves the problem. The proposed methodology is
a non-iterative method that directly gives the optimal
generation schedule of the generating set and it does not
require any assumptions. The numerical results demon-
strate that the proposed approach offers a better conver-
gence rate, minimum cost to be achieved and better solu-
tion than the existing methods. As power systems are
usually large scale systems, the proposed meth od may be
suggested for the solution of economic load dispatch
problems and it is also suitable for online applicatio ns.
7. Acknowledgements
The authors gratefully acknowledge the authorities of
Annamalai University, Annamalainagar, Tamilnadu,
India, for their continued support, encouragement, and
the extensive facilities prov ided to carry out this research
work.
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