Economic Load Dispatch for Multi-Generator Systems with Units Having Nonlinear and Discontinuous Cost Curves Using Gravity Search Algorithm

International Journal of Applied Power Engineering (IJAPE)
Vol. 3, No. 3, December 2014, pp. 166~174
ISSN: 2252-8792  166
Journal homepage: http://iaesjournal.com/online/index.php/IJAPE
Economic Load Dispatch for Multi-Generator Systems with
Units Having Nonlinear and Discontinuous Cost Curves Using
Gravity Search Algorithm
Tunir Dey
Electrical Engineering, National Institute of Technology, NIT Durgapur, India
Article Info ABSTRACT
Article history:
Received Okt 7, 2014
Revised Nov 10, 2014
Accepted Nov 23, 2014
Economic Load Dispatch aims at distributing the load demand between
various generation stations in a system such that the total cost of generation
is minimum. This is of vital importance since it not only reduces the
operation cost of the generation utility but also helps in conserving fast
dwindling energy resources.
Modern day power systems are large interconnected systems with a large
number of generator units each having its own cost curve. Ideally the cost
function of a unit is a quadratic function of the power generated by the unit
and the cost curve obtained is a smooth parabola. But in practice cost curves
deviate from the idealised one due the several reasons such as valve point
effect, multi fuel operation, existence of forbidden zones etc. and as such
may not be continuous or analytic. Also for a large interconnected system it
becomes essential to consider the effect of transmission losses.
Conventional numerical method based approaches work well with systems
without losses but for large systems with losses obtaining convergence
becomes difficult as the number of iterations required as well as the
computational time are very high. These methods fail entirely if non ideal
cost curves are considered. Hence soft computing based methods become
essential. Here Gravity Search Algorithm(GSA) has been used to for finding
economic load scheduling in a multi generator system, given a certain load
demand, and taking into consideration the effects of practical constraints on
the idealised load curve. The algorithms for finding the economic scheduling
has been written in Matlab and has provided satisfactory results based on the
given tolerance values. Also the traditional and soft computing based
approaches have been compared to demonstrate the advantages of one over
theother.
Keyword:
Economic Load Dispatch
Gravity Search Algorithm
Ideal Generator
Practical Cost curves
Total Cost-Loss with Iterations
Copyright © 2014 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Tunir Dey,
Electrical Engineering,
National Institute of Technology,
NIT Durgapur, India.
Email: t.deynitdgp@gmail.com
1. INTRODUCTION
Economic load dispatch (ELD) is an important optimization task in power system operation for
allocating generation among the committed units such that the constraints imposed are satisfied and
the energy requirements in terms of Rupee/hour or $/hr are minimized. Improvements in scheduling
the unit outputs can lead to significant cost savings. Traditional dispatch algorithms employ Lagrangian
multipliers and require monotonically increasing incremental cost curves. Unfortunately, the input–output

IJAPE ISSN: 2252-8792 
Economic Load Dispatch for Multi-Generator Systems with Units Having Nonlinear …(Tunir Day)
167
characteristics of modern units are inherently highly nonlinear because of valve-point loadings, rate limits,
etc., and furthermore they may generate multiple local minimum points in the cost function. Classical
dispatch algorithms require that these characteristics be approximated; however, such approximations are not
desirable as they may lead to suboptimal operation and hence huge revenue loss over-time.
In light of the nonlinear characteristics of the units, there is a demand for techniques that do not
have restrictions on the shape of the fuel-cost curves. Classical calculus-based techniques fail to address these
types of problems satisfactorily. In the recent years, several attempts have been made by the researchers to
solve ELD problems with intelligent techniques, such as hybrid interior point method (IPM) assisted
differential evolution (DE) (IPM-DE) ,hybrid DE with biogeography-based optimization (BBO) (DE-BBO),
BBO, combined particle swarm optimization (PSO) with real-valued mutation (CBPSO-RVM), improved
coordinated aggregation-based PSO (ICA-PSO), quantum PSO (QPSO), ant colony optimization (ACO),
hybrid genetic algorithm (GA)-pattern search (PS)-sequential quadratic programming (SQP) (GA-PS-SQP),
PSO with both chaotic sequences and crossover operation (CCPSO), new PSO (New PSO), PSO with
crazy particles (PSO-Crazy), simple PSO (SPSO), PSO with time varying acceleration coefficients (PSO-
TVAC), real coded GA (RCGA), self organizing hierarchical PSO (SOH-PSO), PSO with chaotic and
Gaussian approaches (PSOCG) , bacterial foraging with Nelder–Mead algorithm (BF-NM) , new PSO with
local random search (LRS) (NPSO-LRS), new PSO (NPSO), PSO with LRS (PSO-LRS), DE combined with
SQP (DEC-SQP), improved GA with multiplier updating (IGA-MU), improved fast evolutionary,
evolutionary programming (IFEP), Hopfield model (HM).
Gravitational search algorithm (GSA) is one of the latest heuristic optimization algorithms, which
was first introduced by Rashedi et al. based on the metaphor of gravitational interaction between masses.
GSA is inspired by the Newton theory that says: ―Every particle in the universe attracts every other particle
with a force that is directly proportional to the product of their masses and inversely proportional to the
square of the distance between them‖. Gravity is a force, pulling together all matter. Promising results
were reported in for benchmark function optimization problems by adopting GSA.
2. ECONOMIC LOAD DISPATCH
The definition of economic dispatch is ―The operation of generation facilities to produce energy at
the lowest cost to reliably serve consumers, recognizing any operational limits of generation and transmission
facilities‖. In traditional economic dispatch, the operating cost is reduced by proper allocation of the amount
of power to be generated by different generating units. Most electric power systems dispatch their own
purchased power in a way that may be said to meet this definition.
Generally there are two types of constraints:
a) Equality Constraints
b) Inequality Constraints
a) Equality constraints:
The equality constraints are the basic load flow equations of active and reactive power.
p=1,2,3,4....
where ep and fp are the real and imaginary components of voltage at the pth
node and Gpq
and Bpq
are
the nodal conductance and susceptance between the pth
and qth
nodes.
b) Inequality constraints:
The KVA loading of a generator can be represented as,

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168
The KVA loading should not exceed a pre-specified value to limit the temperature rise. The
maximum active power generated ‗P‘ from a source is also limited by the flame instability of the boiler. If the
power generated out of a generator falls below a pre-specified value Pmin, the unit is not put on the bus bar.
The maximum reactive power is limited by overheating of rotor and minimum reactive power is
limited by the stability limit of machine. Hence, the generator reactive power Q should not be outside the
range stated by inequality for it‘s stable operation.
1) Input-output curve for Ideal Generator:
The factors influencing power generation are operating efficiencies of generators, fuel cost and
transmission losses. The total cost of generation is a function of the individual generation of the sources
which can take values within certain constraints. The problem is to determine the generation of different
plants such that total operating cost is minimum.
Figure 1. Input-Output characteristics of generator units
Incremental fuel rate=dF/dP
Incremental efficiency = dP/dF
Where F is the fuel input in million Btu per hour and P is the power output in MW. Incremental fuel cost is
expressed in terms of Rs per MWhr which is obtained by multiplying the incremental fuel rate by fuel cost in
Rs per Btu.
3. PRACTICAL COST CURVES
In practically cost curve are not continuous smooth ones.
Discontinuities arise due to:
3.1. Multi Fuel Operation
Many modern units use multiple fuels depending on power demand for efficient generation. This
gives rise to a piecewise continuous cost curve with discontinuities at the points corresponding to the power
generations at which change of fuel takes place.

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169
Figure 2. Cost Curve for Multi fuel operation
3.2. Valve Point Consideration
The generating units with multi-valve steam turbines exhibit a greater variation in the fuel-cost
functions. The valve opening process of multi-valve steam turbines produces a ripple-like effect in the heat
rate curve of the generators. When the valve point effect is considered in the input-output curve, the
possibility of non-convex curves must be accounted for if extreme accuracy is desired. If non-convex input-
output curves are to be used, equal incremental cost methodology cannot be used, since there are multiple
outputs for any given value of incremental cost. Thereby the effects of valve point loading is modelled as a
recurring rectified sinusoid contribution and added to the basic quadratic cost function.
Figure 3. Cost curve with Valve point effect
3.3. Forbidden Zone
There are certain areas in power system where no generation takes place due to vibration of turbine
and rotor which is generally known as forbidden zone. This factor gives regions of discontinuity in the input-
output curve.
Figure 4. Cost curve for Forbidden zone

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These are the various factors which can hamper the linearity of the incremental fuel cost curve; hence the
theoretical results will be different from practical ones. During solution of ELD using GSA we have
considered effects of multiple fuel operation, forbidden zones and valve point effect.
4. GRAVITY SEARCH ALGORITHM
The classical optimization algorithms are incapable of providing suitable solution for
optimization problems containing high dimensional search space as well as nonlinear behaviour. This
is because the search space increases exponentially with problem size. Algorithms inspired by the
behaviours of natural phenomena are being used as effective methods to tackle such problems. It is
shown by many researchers that these algorithms are well suited to solve complex computational
problems such as optimization of objective functions, pattern recognition, control objectives, image
processing, filter modelling, etc. Various heuristic approaches have been adopted by researches so far, for
example Genetic Algorithm, Simulated Annealing, Ant Colony Search Algorithm, Particle Swarm
Optimization, etc. These algorithms are progressively analyse or powered by researchers in many different
areas. These algorithms solve different optimization problems.
a) Implementation of GSA in ELD program:
The basic principle and mathematical structure of GSA is dealt with in the preceding section. These
fundamental steps have to be modified in order to suit the problem at hand, that of Economic Load Despatch
for multi generator systems. The implementation of GSA in for the above problem comprises three functions;
namely,
 Formulation of a suitable objective function.
 Generation of Initial Sample solutions.
 Development of a GSA based computational
module for calculating the minimum generation cost for a system under given load demand condition.
1) Formulation of objective function:
An objective function forms the basis of any optimization model. It is the function whose value is to
be optimized (either maximized or minimized depending on the problem). In case of ELD the value of the
objective function is to be minimized. The problem of ELD has the factors that are to be considered for any
solution, namely,
 The cost of generation.
 Transmission loss or cost of transmission cost.
 Mismatch between sum of power demand and transmission losses and the generated power
obtained from ELD solution.
We mostly consider systems with transmission losses in our project. Also in any solution set of the power
balance equation,
∑ Pg=Pl + Pd
where Pg= power generated by each unit;
Pl= power loss in each line;
Pd= total power demand,
obtained by classical numerical methods or by meta heuristic approach would not satisfy the equation
perfectly, creating a mismatch between the two sides. The equation under such conditions can be
approximated as:
| ∑ Pg - ∑ Pl – Pd | = δ
where δ is a small deviation.
Thus we see all three factors play a role in case of a ELD problem and so the objective function in general is
taken to be a linear combination of the three factors; ie
Objective Function = X1*Cost of generation + X2*Cost of loss+ X3*mismatch

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Where, X1, X2 and X3 are weights assigned to the three factors respectively. However in order to minimise
the objective function the factor with the highest weight age is optimised more at the cost of the factors
having lesser weights.
For most of our problems we have concentrated on the cost of generation alone and taken X1, X2 and X3 to
be 1, 0 and 0 respectively.
2) Generation of initial sample solutions:
As is the case for any stochastic method GSA also needs a set for initial sample solutions in order to start
computations towards the best result. From equality constraint of ELD solutions we have,
∑ Pg - Pl – ∑ Pd = 0
where the loss of power due to transmission Pl can be expressed using transmission loss or B coefficients as:
Substituting the value of Pl in the equation 3.22 we get a second order equation involving the power
generated by the various generating units. If we assume the system comprise of ‗n‘ generating units then we
randomly allocate power generations with the respective permissible range of operation for (n-1) units
following the equation:
Pg
i
= Pi
min + rand (i) X ( Pi
max – Pi
min )
where Pg
i
denotes the generation by the ith
unit Pi
min and Pi
max denote the minimum and maximum permissible
generation of the ith
unit respectively.
rand (i) denotes a random number between 0 and 1.
The allocation of generations to (n-1) units reduces equation ∑ Pg - Pl – ∑ Pd = 0 to a quadratic equation in
Pg
n
. If the solution of this equation yields a real value for Pg
n
such that Pmin
n
< Pg
n
< Pmax
n
then the solution set
represent a valid solution and is taken to be a part of the initial population.
3) Development of a GSA based computational module for calculating the minimum generation cost for a
system under given load demand condition:
Taking into consideration the basic steps of GSA listed in the previous section, a GSA based ELD module for
a generalised system having n generators is implemented as follows:
 A ‗n‘ dimensional space is used for the problem.
 Each solution set of the population is treated as a point mass, the mass assigned to it being
inversely proportional to the cost of generation (or the value of the objective function) since it
is a minimization problem.
 The new positions of the masses are the modified power generations allocated to the ‗n‘
generating units.
 Step d is repeated till the exhaustion of allowed number of iterations in order to obtain the best
possible solution.
5. RESULTS
16 Generator lossless systems using Lagrangian multiplier:
 Power Demand(MW) =3000
 Lambda = 4.830629999976126
 Total Generation Cost ($/h) = 9.637785672117201e+003
 Total Generated Power = 2.999997340557906e+003
 Elapsed Time = 15.371956240090096 seconds

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16 Generator systems with transmission loss using GSA:
 Power Demand(MW) = 2630
 Total Power Generation = 2.657131496253033e+003
 Total Generated Cost ($/h) = 3.254862358344061e+004
 Total Loss = 27.104321170969200
 Elapsed Time = 2.837629 seconds
The following results obtained for 10 generating units by considering multiple fuel systems and valve point
effect using GSA:
 Total generation (MW) = 2700
 Power Demand (MW) = 2700
 Total generation cost ($/h) = 624.0555
 Elapsed time = 18.693614 seconds
Generation allocated to various units of 10 unit multiple fuel system:
Unit = 1 Generation=218.37 Fuel Type = 2
The Figure of variation of total cost and loss with iterations have shown in Figure 5 and Figure 6
respectively,
Figure 5. Variation of total cost with iterations Figure 6. Variation of total loss with iterations
6. DISCUSSION
From comparison of the various results obtained and observations made during the execution of the
MATLAB codes the following points were noted:
1) The Lagrangian Multiplier method used for 16 generator systems having no transmission losses
gave much slower convergence than GSA used for 16 generator system having transmission losses.

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For GSA based system the execution time did not increase linearly with the number of units. The increase of
execution time reduced with the increase in the number units. Hence it is advantageous for systems having
large number of units.
7. CONCLUSION
In this project, Gravity Search Algorithm has been used as a tool for solving ELD problem for
practical generating units having nonlinear or discontinuous cost curves in an efficient way. GSA has proved
to be quite efficient in solving the problem both in terms of execution time and the minimisation of the
objective function. The execution time depended on the number of units considered and the types of
nonlinearities or discontinuities included in the definition of the cost curve of the units. As such the execution
time increased at a rate slower than that of the increase in the number of units and thus GSA based solutions
were found to be more efficient for larger systems.
8. FUTURE WORK
This project concentrates mainly on the minimisation of generation cost and in most cases losses,
mismatch and other factors have not been weighed into the objective function. Hence there is scope for
including of these factors into the objective functions and study the effects it has on the results. The objective
functions may be further generalised to develop a optimization module for Optimal Power Flow Management
(OPFM).
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