PRACTICAL ECONOMIC LOAD DISPATCH PROBLEM WITH
SHUFFLED DIFFERENTIAL EVOLUTION
1
1
2
S. REDDI KHASIM , 2B. RAMAKRISHNA
3
V.VEERA NAGI REDDY
(M.Tech Student, Department of Electrical & Electronics Engineering, MJR College of Engineering &
Technology, Pileru, JNTU, Ananthapur, Andhra Pradesh-India, [email protected])
(Assistant Professor,Department of Electrical & Electronics Engineering,MJR College of Engineering &
Technology, Pileru, JNTU, Ananthapur, Andhra Pradesh-India)
3
(Head of the Department, Electrical & Electronics Engineering, MJR College of Engineering &
Technology, Pileru, JNTU, Ananthapur, Andhra Pradesh-India)
Abstract— Solving Practical economic
load dispatch problem if valve point loading
effects are considered may cause a change in the
objective function formulation and adds
complexity. This paper proposes shuffled
differential evolution (SDE) algorithm, a strong
optimization technique, which reduces the cost of
generation also integrates novel differential
mutation operator to address the problem under
study. To validate proposed methodology, detailed
simulation results are obtained on standard test
systems having thirteen units and forty units are
presented and discussed. To demonstrate the
superiority of proposed method, a comparison is
made with other settled nature-inspired algorithm
Index Terms—Nonconvex economic dispatch, Valve
point loading effects, Shuffled differential evolution.
I. INTRODUCTION
The aim of power industry is to generate
electrical energy at minimum cost while satisfying all
the limits and constraints imposed on generating
units. Economic Load dispatch (ELD) is the one of
the optimization problem in power industry. ELD
determines the optimal power solution to have
minimum generation cost while meeting the load
demand. But due to environment issues which are
arising from the emissions produced by the fossil
fuels in thermal power plant, it has become
mandatory for power utility to consider the
environment constraints along with the economy. But
both of the objectives i.e. minimum fuel cost and
minimum emission are of conflicting nature. This
problem
leads
to
the
formulation
of
multiobjectiveeconomic load dispatch. Or it can also
be formulated as combined economic and emission
dispatch (CEED)problem. This paper deals with
Economic dispatch problem of all thermal units only.
The problem of economic operation of a
power system or optimal power flow can be stated as:
Allocating the load (MW) among the various units of
generating stations and among the various generating
stations that the overall cost of generation for the
given load demand is minimum.
Traditional
non-linear
programming
solutions using quadratic polynomials in iterative
search algorithms reveal some short comings mainly
from polynomial approximation fails to design nonlinear characteristics of power generation units, that
is valve point loadings and prohibited operating
zones and limits, these results in non-optimal
solutions and consumes more time for computations.
To overcome above said limitations,
sophisticated solution algorithms named differential
evolution (DE), an evolutionary computation method
for optimizing non-linear and non-differential
continuous space functions. Differential evolution
also suffers from the problem of premature
convergence, then the objective function loses its
diversity.
Shuffled frog leaping algorithm (SFLA) is
newly developed metaheuristic algorithm for
combinational optimization having few parameters,
high performance and easy programming.
The main benefits of SFLA are its fast
convergence while its main drawbacks are mainly
due to the insufficient learning mechanism for the
swarm that could lead to noncomprehensive solution
domain exploration. In order to overcome the
intrinsic limitations of DE and SFLA, emphasizing at
the same time their benefits, an innovative technique
called shuffled differential evolution (SDE)
characterized by a novel mutation operator has been
designed. The main contributions of this paper are:
(1)
Presenting a novel mutation operator to
enhance the search ability of the SDE and
the mutation operator is specific to this
work and has been never presented in the
previous search works in the area.
(2)
Applying the proposed methodology to
two benchmark ED problems with valve
point loading effects and the results are
presented.
(3)
The best results obtained from the solution
of the ED problem by adopting the SDE
algorithm are compared to those published
in the recent state of the art literatures.
II. FORMULATION OF OBJECTIVE FUNCTION
The main aim of the objective function is to
minimize the total cost of generation and at the
same time to meet the demand with minimum
amount of losses and have to satisfy both equality
and inequality constraints.
Objective function,
Where Fi is the fuel cost of ith unit.
by ripples in the multi-valve steam turbine generated
units. The Valve-point effect is illustrated below:
Fig 1. Valve Point loading effect
The Incremental fuel cost of such unit is given by
2.2 Constraints
(a) Equality Constraints
The total power generated should be the
same as the total load demand plus the total
transmission losses. In this work, transmission power
losses have not been considered and the active power
balance can be expressed as:
(i) Total generation=Total Demand(neglecting losses)
Such a resultant equation is called as Co-ordination
equation.
(ii) Total generation = Total Demand + Losses
(considering losses).
Such a resultant equation is called as ExactCoordination equation.
(b) Inequality Constraints
The inequality constraint is given by
Generally the cost function is given by
Fi = aiPGi2 + biPGi + Ci
2.1. Valve Point loading effects
The incremental fuel cost is given by
𝜕𝐹𝑖
= (𝑎𝑖 ∗ 𝑃) + (𝑏𝑖)
𝜕𝑃𝐺𝑖
Where the Incremental fuel cost is nothing but the
slope of fuel cost curve.
In complex non-linear fuel cost function, a
sinusoidal component is super imposed on heat rate
curve to take account of valve point loading caused
Pmin is the minimum operating limit of the ith unit.
Pmax is the maximum operating limit of the ith unit.
This is because of some units may not be
operated all at a same time because of Crewconstraints. The optimal load dispatch problem must
then incorporate this startup and shut down cost for
without endangering the system security. The power
generation limit of each unit is then given by the
inequality constraints. The maximum limit Pmax is
the upper limit of power generation capacity of each
unit. On the other hand, the lower limit Pmin pertains
to the thermal consideration of operating a boiler in a
thermal or nuclear generating station. An operational
unit must produce a minimum amount of power such
that the boiler thermal components are stabilized at
the minimum design operating temperature.
III PROPOSED OPTIMIZATION TECHNIQUE
(a) Shuffled differential evolution
To address nonconvex economic dispatch
problems the adoption of a hybrid solution technique
based on a combination of differential evolution (DE)
and shuffled frog leaping algorithm (SFLA) is
proposed in this paper.
To overcome the limitations of the Differential
evolution, An innovative technique called shuffled
differential evolution (SDE) characterized by a novel
mutation operator is proposed here. The proposed
algorithm is based on the shuffling property of SFLA
and DE algorithm. Similarly to other evolutionary
algoritmms, in SDE a population is initialized by
randomly generating candidate solutions. The fitness
of each candidate solution is then calculated and the
population is sorted in descending order of their
fitness and partitioned into memeplexes as shown in
the below figure.
(b) Proposed algorithm
The following is the pseudo code for
implementing the SDE optimization.
Begin;
Initialize the SDE parameters
Randomly generate a population of solutions
(frogs);
Foriis = 1 to SI (maximum no. of generations);
For each individual (frog); calculate fitness of
frogs;
Sort the population in descending order of their
fitness;
Determine the global best frog;
Divide population into m memeplexes; /*memeplex
evolution step*/
Forim = 1 to m;
Forie = 1 to IE (maximum no. of memetic
evolutions)
Determine the best frog;
For each frog
Generate new donor vector (frog) from mutation
(using DE/memeplexbest/2)
Apply crossover
Evaluate the fitness of new frog;
If new frog is better than old
Replace the old with new one
Fig 3(a). Framework of SDE
In this connection it is important to
note that, unlike classicSFLA, all the frogs
in each memeplex (i) participate in the
populationevolution; (ii) have local search
capabilities enhanced by theDE based
memetic evolution process; and (iii) can
perform a globalexploration of the solution
domain. These features give an edge tothe
proposed
algorithm
over
classic
evolutionary based solution techniques in
finding global optimal solution.
End if
End for
End for
End for
/*end of memeplex evolution step*/
Combine the evolved memeplex;
Sort the population in descending order of their
fitness;
Update the global best frog;
End for
End
(c) Control parameters
The evolutionary operators and the control
parameters characterizing the proposed SDE
algorithm are presented in the following section. In
SDE the memetic evolution step of SFLA is replaced
by a differential evolution iteration characterized by
the DE/Memeplex-best.2 mutation, the crossover
operator and the selection. The memetic evolution
step of SDE algorithm is shown in Fig below
Fig 3.(c). Memetic evolution step
IV. SIMULATION RESULTS
The proposed algorithm is implemented
using a MATLAB in order to demonstrate the
performance of the proposed SDE method, it was
tested on three benchmark functions. In the next
section, ED problem is solved with valve point
loading effects considered, 13 and 40 unit test system
and the results are compared with well settled natureinspired and bio-inspired optimization alogorithms.
4.1 Thirteen unit thermal system
A system of thirteen thermal units with the
effects of valve-point loading was studied in this
case. The expected load demand to be met by all the
three generating units is 1800MW . The system data
can be found and the convergence profile of the cost
function is depicted in fig.1. The dispatch results
using the proposed method and other algorithms are
given in Table 1. The global optimal solution for this
test system is 17963.8293 $/h. From Table 1, it is
clear that the proposed method SDE reported the
global optimum solution. The mean values also
highlighted.
The hybrid algorithm is applied on 13-unit system
with the effects of valve-point loading.
Fig 4.1 Convergence profile of the total cost for
13 units with PD=1800MW
Table 1: Comparisons of simulation results of
different methods for 13-unit case study system with
PD = 1800 MW
IGA_MU
HQPSO
Unit
SDE
[41]
[42]
1
628.3151
628.3180
628.3185
2
148.1027
149.1094
222.7493
3
224.2713
223.3236
149.5995
4
109.8617
109.8650
60.0000
5
109.8637
109.8618
109.8665
6
109.8643
109.8656
109.8665
7
109.8550
109.7912
109.8665
8
109.8662
60.0000
109.8665
9
60.0000
109.8664
109.8665
10
40.0000
40.0000
40.0000
11
40.0000
40.0000
40.0000
12
55.0000
55.0000
55.0000
13
55.0000
55.0000
55.0000
Total
power
1800.0000 1800.0000
1800.0000
in MW
Total
cost in 17963.9848
17963.9571
17963.8293
$/h
Table 1 shows the best dispatch solutions obtained
by the proposed method for the load demand of 1800
MW. The convergence profile for SDE method is
presented in Fig. 5. The results obtained by the
proposed methods are compared with those available
in the literature as given in Table 2. Though the
obtained best solution is not guaranteed to be the
global solution, the SDE has shown the superiority to
the existing methods. The minimum cost obtained by
SDE method is 17963.8293 $/h, which is the best
cost found so far and also compared the SDE method
with the IGA_MU and HQPSO methods. The
minimum cost for IGA_MU and HQPSO is
17963.9848 $/h and 17963.9571 $/h respectively.
The results demonstrate that the proposed
algorithm outperforms the other methods in terms of
better optimal solution. Fig. 5.4 shows the variations
of the fuel cost obtained by SDE for 100 different
runs and convergence results for the algorithms are
presented in Table 2 for 1800MW load.
Fig. 4.1.1 Distribution of total costs of the SDE
algorithm for a load demand of 1800 MW for 100
different trials for 13-unit case study
Fig.4.1.1 shows the variations of the fuel cost
obtained by SDE for 100 different runs and
convergence results for the algorithms are presented
in Table 2 for 1800MW load.
Table 2: Convergence results (100 trial runs) for
13-unit test system with PD = 1800 MW
Average
Minimum
Maximum
Method
cost
cost ($/h)
cost ($/h)
($/h)
IGA_
17963.9848
NA
NA
MU [41]
18273.86
HQPS
17963.9571
18633.0435
10
O [42]
17972.87
17963.8293
17975.3434
SDE
74
Table 2 shows the convergence results for 100
trials for 13-unit test system with load 1800 MW and
compared the minimum, average and maximum cost
for IGA_MU and HQPSO methods. It has been
observed that minimum, average and maximum costs
for SDE proposed method is 17963.8293 $/h,
17972.8774 $/h and 17975.3434 $/h respectively and
also observed that the proposed method minimum,
average and maximum cost values are low compared
with the IGA_MU [41] and HQPSO methods.
The proposed hybrid algorithm is applied on 13unit system with effects of valve-point loading with
Pd=2520MW.
Fig. 4.1.2 Convergence profile of the total cost
for 13-unit generating units with PD = 2520 MW
Table 3 shows the best dispatch solutions obtained
by the proposed method for the load demand of 2520
MW. The convergence profile for SDE method is
presented in Fig. 4.1.2. The results obtained by the
proposed methods are compared with those available
in the literature as given in Table 3. Though the
obtained best solution is not guaranteed to be the
global solution, the SDE has shown the superiority to
the existing methods. The minimum cost obtained by
SDE method is 24169.9177 $/h, which is the best
cost found so far and also compared the SDE method
with the GA_MU and FAPSO-NM methods.
Table 3 Comparisons of simulation results of
different methods for 13-unit case study system
with PD = 2520 MW
GA_MU
FAPSOUnit
SDE
[48]
NM [20]
1
628.3179
628.32
628.3185
2
299.1198
299.20
299.1993
3
299.1746
299.98
299.1993
4
159.7269
159.73
159.7331
5
159.7269
159.73
159.7331
6
159.7269
159.73
159.7331
7
159.7302
159.73
159.7331
8
159.7320
159.73
159.7331
9
159.7287
159.73
159.7331
10
159.7073
77.40
77.3999
11
73.2978
77.40
77.3999
12
77.2327
87.69
92.3999
13
92.2598
92.40
87.6845
Total
2520.000
2520.000
2520.000
power
0
0
0
in MW
Total
24170.75
24169.91
cost in
24169.92
50
77
$/h
considering the 4th decimal.
4.2 Forty unit thermal system
The ED problem has been solved for a 40
thermal units power system considering the effects of
valve-point loading effects. The load demand is
10500 MW. The system data can be found in and
given in table.
Fig. 4.1.3 Distribution of total costs of the SDE
algorithm for a load demand of 2520 MW for 100
different trials for 13-unit case study.
The results demonstrate that the proposed
algorithm outperforms the other methods in terms of
better optimal solution. Fig. 4.1.3 shows the
variations of the fuel cost obtained by SDE for 100
different runs and convergence results for the
algorithms are presented in Table 4 for 2520 MW
load.
Table 4 Convergence results (100 trial runs) for
13-unit test system with PD = 2520 MW
Average
Minimum
Maximum
Method
cost
cost ($/h)
cost ($/h)
($/h)
GA_M
24170.755 24429.1202 24759.312
U [48]
FAPS
24170.440
24169.920 24170.0017
O-NM
2
[20]
24169.917
24178.834
24170.0960
SDE
6
6
Table 4 shows the convergence results for 100
trials for 13-unit test system with load 1800 MW and
compared the minimum, average and maximum cost
for GA_MU and FAPSO-NM methods. It has been
observed that minimum, average and maximum costs
for SDE proposed method is 17963.8293 $/h,
17972.8774 $/h and 17975.3434 $/h respectively.
The results obtained by the proposed methods are
compared with those available in the literature, the
solution quality of SDE is better than those obtained
by other methods. Use of memeplex/best mutation
scheme often eliminate trapping of SDE algorithm
into local minimum and provides global minimum.
Analyzing the data it is worth noting as the identified
solution satisfies all the system constraints. From the
results obtained by SDE method, it is clear that the
power balance constraint is satisfied even after
Fig. 4.2 Convergence profile of the total cost for
40 generating units with PD = 10500 MW
The Fig. 4.2 demonstrates the union profile for the
aggregate expense for 40 creating units with heap of
10500 MW. The outcomes got by applying the SDE
solution algorithm are condensed in Table 4.
Dissecting the information, it can be seen as the SDE
strategy succeeds in discovering a tasteful
arrangement. The base expense got by SDE strategy
is 121412.5355 $/h, which is the best cost discovered
in this way. This announcement is additionally
affirmed by examining Table 5 which compresses the
base, normal, and greatest expense acquired by other
settled calculations. The examination of these similar
results exhibits that the proposed methodology shows
better execution thought about than other settled
systems reported in the writing. In this manner, the
proposed technique can incredibly improve the
looking capacity; guarantee nature of normal
solutions, and additionally proficiently deals with the
framework imperatives. Fig. 4.2.1 demonstrates the
varieties of the fuel expense got by SDE for 100
different runs for forty unit system.
The simulation relults for the forty-unit case
study system with the load demand of 10500 mega
watts with SDE and the convergence results of 100
trails for the same units and load is illustrated in the
below tabular columns and the respective results:
Table 5 Simulation results for 40-unit case study
system with PD = 10500 MW
Un
Unit
SDE
SDE
it
1
110.7889
21
523.2794
2
110.7998
22
523.2794
3
97.3999
23
523.2794
4
279.7331
24
523.2794
5
87.7999
25
523.2794
6
140.0000
26
523.2794
7
259.5996
27
10.0000
8
284.5996
28
10.0000
9
284.5996
29
10.0000
10
130.0000
30
87.7999
11
94.0000
31
190.0000
12
94.0000
32
190.0000
13
214.7598
33
190.0000
14
394.2794
34
164.7998
15
394.2794
35
200.0000
16
394.2794
36
194.3978
17
489.2794
37
110.0000
18
489.2794
38
110.0000
19
511.2794
39
110.0000
20
511.2794
40
511.2794
Total
power in
10500.0000
MW
Total
cost in
121412.5355
$/h
Table 5 demonstrates the meeting results for 40unit test framework with burden 10500 MW and
looked at the base, normal and most extreme expense
for BBO and ACO routines. It has been watched that
base, normal and most extreme expenses for SDE
proposed
technique
is
121412.5355$/h,
121474.0032$/h and 121521.0211$/h separately.
Table 6. Convergence results (100 trial runs) for
40-unit test system with PD = 10500 MW
Minimu
Average
Maximu
Met
m
cost cost
m
cost
hod
($/h)
($/h)
($/h)
121426.9
121508.0
121688.6
BB
325
634
O [16] 530
121811.3
121930.5
122048.0
AC
800
660
O [50] 700
121412.5
121474.0
121521.0
SDE
355
032
211
Fig. 4.2.1 Distribution of total costs of the SDE
algorithm for a load demand of 10500 MW for 100
different trials for 40-unit case study
5.3 Convergence characteristic and computational
efficiency
The merging profiles for SDE strategy for three
unit framework with thirteen unit system with 1800
MW and 2520MW heap and forty unit test
framework with 10500 MW are presented in fig. 4.1,
fig. 4.1.2 and fig. 4.2 individually. From the union
profiles, it was clear that the base expense is acquired
by SDE inside 100iterationsfor three unit framework
and thirteen unit framework for forty unit framework.
Fig. 4.1.1, fig. 4.1.3 and fig.4.2.1 demonstrates the
joining profiles for the higher number of emphases
for thirteen unit framework and forty unit framework
separately.
Table 7. CPU time comparison for 40-unit test
system.
CPU time
Method
in sec
BBO [16]
42.98
ACO [50]
92.54
SDE
27.69
Fig.4.2.3 CPU times of SDE method for different
systems
It is observed that from Table 7, the SDE
method is computationally efficient than the
mentioned methods. The AverageCPU times of SDE
method for different systems are shown inFig. 5.9.it
is observed that the minimum cost producedby SDE
is least compared with other methods emphasizingits
better solution quality. Many trials with different
initial population have been carried out to test the
consistency of the SDEalgorithm. Due to the inherent
randomness involved the performanceof heuristic
search algorithms are judged out of a numberof trials.
V. CONCLUSION
In this paper a strong optimization
technique: Shuffled differential evolution (SDE) for
non convex Economic Dispatch problems has been
presented. The performance of the proposed
technique has been evaluated for 13 and 40 unit
system in MATLAB simulation. The results shows
the effectiveness in reducing the total cost of
generating when compelled with the other settled
nature-inspired algorithms. The successful optimizing
performance on the validation data sets illustrated the
efficiency of the SDE and shows that it can be used
as a reliable tool for ED problem.
VI. REFERENCES
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[3]. Walters DC, Sheble GB. Genetic algorithm
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loading. IEEE Trans Power Syst 1993;8(3):1325–32.
[4]. Neto JXV, Bernert DLA, Coelho LS. Improved
quantum-inspired evolutionary algorithm with
diversity information applied to economic dispatch
problem with prohibited operating zones. Energy
Convers Manage 2011;52(1):8–14.
[5]. Liang ZX, Glover JD. A zoom feature for a
dynamic programming solution to economic dispatch
including transmission losses. IEEE Trans Power
Syst1992;7(2):544–50.
[6]. Yang HT, Yang PC, Huang CL. Evolutionary
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Author’s Profile:
S.REDDI KHASIM received
B.Tech degree from SVCET
CHITTOOR, JNTU, Ananthapur in
2012. Currently he is pursuing his
Master’s degree from MJRCET
PILERU, JNTU Ananthapur in the
department of Electrical and
Electronics
Engineering.
His
current interests include Electrical
machines,
Electrical
power
systems.
B.RAMA KRISHNA received
B.Tech degree from SITAMS
CHITTOOR, JNTU Hyderabad in
2007, Master’s degree from NIT
CALICUT in 2013 in the
department of Electrical and
Electronics Engineering. Currently
he is working as Assistant
professor in EEE dept. of MJRCET
PILERU, JNTU, Ananthapur. His
interests include Power Electronics
and Semiconductor drives.
Sri V.VEERA NAGI REDDY,
MJR College of Engineering and
Technology (MJRCET), Piler,
Chittoor(dt) A.P, India. He
obtained Masters degree from
JNTU,
Anantapur,
India.
Currently he is working as HOD in
Electrical
and
Electronics
Engineering
department
at
MJRCET, A.P, INDIA. His
research interests include Robotics,
smart grids, distributed generation
systems and renewable energy
sources, power system operation
and control.