Multi objective economic load dispatch using hybrid fuzzy, bacterial

International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME
43
MULTI OBJECTIVE ECONOMIC LOAD DISPATCH USING HYBRID
FUZZY, BACTERIAL FORAGING-NELDER–MEAD ALGORITHM
Bharathkumar S1
, Arul Vineeth A D2
, Ashokkumar K3
, Vijay Anand K4
1,4
II Year ME Power System, EEE Department, Anna University Regional centre ,
Coimbatore
2
II Year ME Information Technology, IT Department, Anna University Regional centre,
Coimbatore,Tamilnadu,India
3
II Year ME Control and Instrumentation, EEE Department, Anna University Regional
centre, Coimbatore
ABSTRACT
In this paper, a new approach is proposed to solve the economic load dispatch (ELD)
problem. Power generation, spinning reserve and emission costs are simultaneously
considered in the objective function of the proposed ELD problem. In this condition, if the
valve-point effects of thermal units are considered in the proposed emission, reserve and
economic load dispatch (ERELD) problem, a non-smooth and non-convex cost function will
be obtained. Frequency deviation, minimum frequency limits and other practical constraints
are also considered in this problem. For this purpose, ramp rate limit, transmission line losses,
maximum emission limit for specific power plants or total power system, prohibited
operating zones and frequency constraints are considered in the optimization problem. A
hybrid method that combines the bacterial foraging (BF) algorithm with the Nelder–Mead
(NM) method (called BF–NM algorithm) is used to solve the problem. In this paper, the
performance of the proposed BF–NM algorithm is compared with the performance of other
classic (non-linear programming) and intelligent algorithms such as particle swarm
optimization (PSO) as well as genetic algorithm (GA), differential evolution (DE) and BF
algorithms. The simulation results show the advantages of the proposed method for reducing
the total cost of the system.
Index Terms- Economic Dispatch, Differential Evolution, Evolutionary Algorithms, Valve
Point Loading Effects, Prohibited Operating Zones, Piecewise Quadratic Cost Functions.
INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING
& TECHNOLOGY (IJEET)
ISSN 0976 – 6545(Print)
ISSN 0976 – 6553(Online)
Volume 4, Issue 3, May - June (2013), pp. 43-52
© IAEME: www.iaeme.com/ijeet.asp
Journal Impact Factor (2013): 5.5028 (Calculated by GISI)
www.jifactor.com
IJEET
© I A E M E

44
I. INTRODUCTION
The economic load dispatch of power plants is one of the most important problems in
power system operations. In this regard, the ELD minimizes the generation cost of power
plants so that the generated power satisfies the load demand by considering practical system
constraints [1–3]. This is an extremely important problem in restructured power systems.
Due to increasing sensitivity regarding power plant emissions, the ELD must be
performed such that the environmental emissions of power plants are minimized [4].
Furthermore, during a specific period of time, the emission constraint is considered in [5–7]
to solve the ELD problem. The prohibited power generation zone is another constraint that
can be considered in the ELD problem [8–10]. In addition, economic dispatch can be solved
by considering frequency constraints [1].
To develop a complete model of the ELD problem, the effect of the spinning reserve
constraint [11–13] as well as the valve-point effect [14–21] and transmission line losses
[22,24] can also be taken into account. In most studies, the generation cost function is
considered to be quadratic function, but a cubic cost function more closely conforms to the
generation cost [14]. Therefore, the use of a cubic cost function leads to more accurate
modelling of power plant costs.
The ELD problem is an optimization problem; thus, a large number of methods are
available to solve this problem. Recently, stochastic search algorithms such as PSO, GA,
direct search, and DE algorithms [17] have been successfully used to solve the ELD problem.
Each of these algorithms has its own advantages and disadvantages. For example, the direct
search method and GA have slow execution speeds, and the PSO algorithm requires the
execution of many repeated stages. The above-mentioned search methods determine the local
optimal point but cannot find that optimal solution.
The BF algorithm is a new optimization algorithm that has recently been considered
to solve the real world optimization problem. It covers a wide search region but has low
convergence speed. In this respect, a hybrid method combines BF algorithm and NM method
(BF–NM algorithm) with the combination of the fuzzy logic is used. By combining these
three methods, the search power of intelligent methods and the precision of conventional
methods are simultaneously employed [27]. Addition, the transmission losses, maximum
emission limit, and practical constraints of the power plants are considered in the problem.
The frequency deviation, minimum frequency limit and maximum permissible environmental
emission constraints are also used in the problem to assure the power system security. The
simulation results validate the performance and accuracy of the proposed method for solving
the ERELD problem by placing practical constraints in power system.
II. ECONOMIC LOAD DISPATCH
A. Problem formulation
The proposed ERELD problem consists of an objective function and practical constraints.
The objective function and constraints are introduced in following subsections.
Objective Functions: The objective of the classical economic dispatch is to minimize the total
system cost (1) by adjusting the power output of each of the generators connected to the grid.
The total system cost is modelled as the sum of the cost function of each generator.

45
(1))(....
1
321 i
N
i
inCost PFFFFFF ∑=
=+++=
where Pi and Fi are the output power and the generation cost of ith
generating units and N
is the number of power plants.
The cost function of each generator establishes the relationship between the power
injected to the system by the generator and the cost incurred to load the machine to that
capacity. Generators are typically modelled by smooth quadratic functions such as (2), in
order to simplify the corresponding optimization problem, as well as to facilitate the
application of proposed technique. The cost function is generally considered to be a square
cost function [29,30]. However, a cubic cost function is more appropriate and accurate. So,
the proposed total generation cost can be expressed as follows:
(2))PPP(min 3
i
2
i
1
iCost i
N
i
iii dcbaF +++= ∑=
Valve-Point Effect: If the power output of a generator with multi-valve steam turbines is
increased to meet the increased demand, various steam valves should to be opened in
sequence. As shown in Fig. 1, the valve-point effect can be considered by adding the absolute
value.
Fig. 1. A cost function of a unit with valve-point effect and prohibited operating zones.
of a sinusoidal function with a cubic cost function [14–18]. Thus, the cost function is
modified as follows:
(3)))]P-(Psin(f.(e
)PPP[(min
iminiii
3
i
2
i
1
iCost ++++= ∑=
i
N
i
iii dcbaF
Power Plant Spinning Reserve Cost Function: Plants should have enough spinning reserve to
provide energy without interruption for customers. This reserve provides cost for the system
[12]. Thus,
(4))(
....
1
321
i
N
i
i
nCost
RFR
FRFRFRFRF
∑=
=
+++=

46
where FRcost is the total reserve cost of the whole system and Ri is the reserve for the ith
unit.
The determination of spinning reserve values to minimize the FRcost function is one of the
main objectives in power system operations. Therefore,
(5))RR(min 2
i
1
iCost ∑=
++=
N
i
ribiri cbaFR
where ari, bri and cri are the coefficients of the reserve cost of the ith
generator.
Multiple Fuel types: Some generating units are capable of operating using different types of
fuels. The use of multiple fuel types may result in multiple cost curves that are not
necessarily parallel or continuous. The lower region of the resulting cost curve determines
which fuel type is most economical to burn.
Fig. 2 Fuel cost function of a thermal generation unit
supplied with multiple fuel types
This cost function can be represented by a piecewise curve (see Fig. 2), and the segments
are defined by the range in which each fuel is used (6). The ED problem with piecewise
quadratic cost curves is very difficult to solve by standard techniques. Piecewise quadratic
cost functions have as many segments as fuel types.
F୧൫Fୋ౟
൯ ൌ
‫ە‬
ۖ
‫۔‬
ۖ
‫ۓ‬a୧,ଵ ൅ b୧,ଵPୋ౟
൅ c୧,ଵPୋ౟
ଶ
, Pୋ౟
ଵ
൏ Pୋ౟
൏ Pୋഠ
ଵതതതത
a୧,ଶ ൅ b୧,ଶPୋ౟
൅ c୧,ଶPୋ౟
ଶ
, Pୋ౟
ଶ
൏ Pୋ౟
൏ Pୋഠ
ଶതതതത
‫ڭ‬
a୧,୩ ൅ b୧,୩Pୋ౟
൅ c୧,୩Pୋ౟
ଶ
‫ڭ‬
Pୋ౟
୩
൏ Pୋ౟
൏ Pୋഠ
୩തതതത
ሺ6ሻ
where Pୋ౟
୩
and Pୋഠ
ଵതതതത are the lower and upper bound respectively of the kth
fuel of unit i, and ai,k
bi,k ci,k are the kth
fuel cost coefficients of unit i.
Prohibited Operating Zones: Generating units may have certain regions where operation is
either undesired or impossible due to physical limitations of

47
the machine components or issues related to instability. These regions produce discontinuities
in the cost curve since the unit must operate under or over certain specified limits. This type
of cost functions results in non-convex sets of feasible solution points, which are modelled as
follows:
൞
ܲ௜,௠௜௡ ൑ ܲ௜ ൑ ܲ௜,ଵ
௟
ܲ௜,௞ିଵ
௨
൑ ܲ௜ ൑ ܲ௜,௞
௟
ܲ௜,௭
௨
൑ ܲ௜ ൑ ܲ௜,௠௔௫
௟
(7)
where Pl
i,k and Pu
i,k are the lower and upper limits of the kth
of the prohibited zone of ith
generating unit and z is the number of the prohibited zones of the ith
unit, respectively.
The Proposed Objective Function: Fuel, spinning reserve and emission costs are in conflict
with each other. In other words, as the minimum generation and reserve costs and minimum
emission do not occur at a single point, it is necessary to optimize them, simultaneously.
Multi-objective optimization methods can be used to solve this optimization problem. To
generate the non-inferior solutions of a multi-objective optimization problem, the weighting
method can be used [7,31]. This approach aggregates all objective functions in a weighted
combination, producing a single one [31]. Therefore, the ERELD problem can be converted
into a scalar optimization problem as follows:
min ߶ ൌ ෍ ቂ‫ݓ‬ଵ ቀ‫ܨ‬௖௢௦௧,௜ ൅ ‫ܴܨ‬௖௢௦௧,௜ ൅ ቚ݁௜. sin ቀ݂௜൫ܲ௜,௠௜௡ െ ܲ௜൯ቁቚቁ
ே
௜ୀଵ
൅ ‫ݓ‬ଶ൫ܳ௥,௜. ‫ܨ‬ா௠௜௦௦௜௢௡,௜൯ቃ ሺ8ሻ
where w1 and w2 are non-negative weights, such that w1 + w2 = 1. w1 and w2 are used to
make a trade-off between emission and total cost (energy and reserve costs). So these
weighting factors vary between w1 = 1.0, w2 = 0.0 and w1 = 0.0, w2 = 1.0. It means that, if w1
= 1.0, w2 = 0.0, the economic and reserve dispatch will be performed instead of ERELD.
Also, if w1 = 0.0, w2 = 1.0, the emission dispatch will be performed instead of ERELD. If w1
= w2, emission and total cost (energy and reserve cost) have similar importance. Many studies
use this weight setting to convert a multi-objective problem into a single-objective.
III. THE HYBRID BF–NM ALGORITHM WITH FUZZY LOGIC
In order to solve the proposed OPF problem, the hybrid bacterial foraging (BF)
algorithm and the Nelder–Mead (NM) method are employed to minimize the cost function of
the problem. The BF algorithm is a stochastic optimization algorithm. It covers a wide search
region, but it has low convergence speed. In this respect, the BF algorithm and the NM
method can be combined [27]. By combining these two methods, the search power of the
intelligent methods and the precision of conventional methods are simultaneously exploited.
Therefore, in this section, the BF algorithm and the NM method are first introduced. The BF–
NM combinational algorithm is then presented to solve the proposed ERELD problem.

48
Algorithm
Step 1. Form the objective function for the ERELD problem.
Step 2. Set the initial count of the bacteria.
Step 3. Determine the P0 for tth
hour.
Step 4. Check the constrains are satisfied the Update limits for tth
hour.
Step 5. Random selection of each bacteria.
Step 6. Find the localized optimized value for the each bacteria by Nelder Mead
algorithm.
Step 7. Find the localized optimized value satisfy the objective function if not the
bacteria are been places locally but the fuzzy logic according to the membership
function and rule based.
Step 8. Check for the Optimized feasible solution for the problem.
Fig. 3. Convergence properties of different optimization algorithms
The bacteria movement is been given by
Swimming: (9)
(i)(i)
(i)
)(l)k,j,(),,1(
T
∆∆
∆
+=+ iClkj ii
θθ
Tumble: (10)
(i)(i)
(i)
)(l)k,1,j(),,1(
T
∆∆
∆
++=+ iClkj ii
θθ
Where θi
(j,k,l) represents the position of ith
bacterium at jth
chemotaxis, kth
reproduction, and lth
elimination and dispersal, respectively. Also, C(i) and ∆(i) are the
movement length and direction random vector, respectively. If the value of the cost function
in ith
chemotactic step is smaller than the value of the cost function in i-1th
chemotactic step,
the moving direction will be correct and the bacterium swims in the same direction.
Otherwise, the moving direction will be incorrect. In this case, a new random direction
(tumble) is set for this bacterium.

49
In this A simple method has been developed for finding a local minimum point from a
function of several variables by Nelder and Mead [28]. The Nelder–Mead method is used to
compare the objective function values in n + 1 of the vertex for the solution of the n
dimensional optimization problem. At each stage, one new vertex is generated.
Then, if the new vertex has less objective function value relative to the previous
vertices, this new vertex is replaced with the worst vertex of the previous level. As an
example, the NM method, which is a pattern search for a problem with 2 variables, compares
the values of the objective function at the three vertices of a triangle. The Nelder–Mead direct
search method is shown in Fig. 4 for the minimization of a non-linear function in a two-
dimensional space is shown in the diagram. By moving toward the minimum point in this
method, the size of the triangle becomes increasingly small.
Fig. 4. The Nelder–Mead algorithm.
So that the bacteria need not search for food in every area the Nelder Mead simplex
algorithm would search locally for the food so that the speed is enhanced if the values are not
satisfied the constrain then the fuzzy logic is used to relocate the bacterial search in the new
area which are been modelled mathematically according to the ERELD problem.
The Convergence of the optimal value is very fast and accuracy more than any other
model which would lead to solve the problem more effective than any other optimization
technique for any real world problem.
IV. CONCLUSION
In this paper, by considering spinning reserve, emission, and the valve-point effects, a
new ELD problem was presented and solved using the BF–NM algorithm. In this problem,
the frequency constraints the practical constraints of power plants and the maximum emission
limit were also considered. By investigating the simulation results, it was found that if the
frequency constraints are inserted in the proposed problem, it can be solved by controlling the
frequency within the permissible limit. Thus, the frequency-constrained ERELD effectively
increases social welfare for consumers and GENCOs. The simulation results confirm the
validity of proposed FC-ERELD problem solved by BF–NM algorithm with fuzzy in
comparison with conventional method and other optimization algorithms.

50
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52
AUTHOR’S PROFILE
S.Bharathkumar received the B.E. degree from the Department of Electrical
and Electronics Engineering, Anna University Tirunelveli from Vins
Christian College of Engineering Nagercoil, in 2011, and is currently
pursuing the M.E degree in the Department of Electrical and Electronics
Engineering in Power Systems Engineering, Anna University Regional
Centre Coimbatore. His research interests include Real time Optimization,
Power Systems Optimization, Non linear Controlling Techniques, Fuzzy improvising
Systems and Digital Image Processing.
A.D.Arul Vineeth received the B.E. degree from the Department of
Computer Science Engineering, Anna University Tirunelveli from Sun
Engineering Nagercoil, in 2011, and is currently pursuing the M.Tech
degree in the Department of Information Technology, Anna University
Regional Centre Coimbatore. His research interests include Real time
Optimization, Cloud Computing, Soft Data Computing, and Network
Security and in Fuzzy improvising Systems.
K.Ashokkumar received the B.E. degree from the Department of Electrical
and Electronics Engineering, Anna University Chennai from Odaiyappa
College of Engineering and Technology Theni, in 2009, and is currently
pursuing the M.E degree in the Department of Electrical and Electronics
Engineering Control and Instrumentation, Anna University Regional Centre
Coimbatore. His research interests include Real time Control Systems, Non
linear Controlling Techniques.
K. Vijay Anand received the B.E. degree from the Department of Electrical
and Electronics Engineering, Madras University from IFET Engineering
College Villuppuram, in 2004, and completed MBA (HRM), Annamalai
University Chidambaram in 2011 and is currently pursuing the M.E degree
in the Department of Electrical and Electronics Engineering in Power
Systems Engineering, Anna University Regional Centre Coimbatore. His
research interests include Real time Optimization and in Power Systems Optimization.

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