Renewable Energy 36 (2011) 1529e1544
Contents lists available at ScienceDirect
Renewable Energy
journal homepage: www.elsevier.com/locate/renene
A practical multi-objective PSO algorithm for optimal operation management of
distribution network with regard to fuel cell power plants
Taher Niknam*, Hamed Zeinoddini Meymand, Hasan Doagou Mojarrad
Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz, P.O. 71555-313, Iran
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 9 June 2010
Accepted 23 November 2010
Available online 18 December 2010
In this paper a novel Multi-objective fuzzy self adaptive hybrid particle swarm optimization (MFSAHPSO)
evolutionary algorithm to solve the Multi-objective optimal operation management (MOOM) is presented. The purposes of the MOOM problem are to decrease the total electrical energy losses, the total
electrical energy cost and the total pollutant emission produced by fuel cells and substation bus.
Conventional algorithms used to solve the multi-objective optimization problems convert the multiple
objectives into a single objective, using a vector of the user-predefined weights. In this conversion several
deficiencies can be detected. For instance, the optimal solution of the algorithms depends greatly on the
values of the weights and also some of the information may be lost in the conversion process and so this
strategy is not expected to provide a robust solution. This paper presents a new MFSAHPSO algorithm for
the MOOM problem. The proposed algorithm maintains a finite-sized repository of non-dominated
solutions which gets iteratively updated in the presence of new solutions. Since the objective functions
are not the same, a fuzzy clustering technique is used to control the size of the repository, within the
limits. The proposed algorithm is tested on a distribution test feeder and the results demonstrate the
capabilities of the proposed approach, to generate true and well-distributed Pareto-optimal non-dominated solutions of the MOOM problem.
2010 Elsevier Ltd. All rights reserved.
Keywords:
Fuzzy self adaptive hybrid particle swarm
optimization (FSAHPSO)
Optimal operation management (OOM)
Multi-objective optimization
Fuel cell power plant (FCPP)
1. Introduction
Power deregulation and restructuring have created increasing
interest in distributed generation (DG), which is expected to play an
increasingly important role in the electric power system. DG can be
defined as a small-scale generating unit located close to the load
being served. A wide variety of DG technologies and types are
included: renewable energy sources such as wind turbines and
photovoltaic, micro-turbines, fuel cells, and energy storage devices
such as batteries [1,2].
We prefer to use fuel cells (FCs) because with these systems, low
power generation can reach a high efficiency. FCs appear as one of
the most promising due to their good efficiency even at partial load,
and especially due to their clean electric generation, with only water
and heat as by-products. Also, their low noise and static operation
makes them suitable to be used even in domestic generations [3e5].
* Corresponding author. Tel.: þ98 711 7264121; fax: þ98 711 7353502.
E-mail addresses: taher_nik@yahoo.com, niknam@sutech.ac.ir (T. Niknam),
h.zeinaddini@gmail.com (H.Z. Meymand), hasan_doagou@yahoo.com (H.D.
Mojarrad).
0960-1481/$ e see front matter 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.renene.2010.11.027
Studies carried out by researching centers have revealed that
fuel cell power plants (FCPPs) contribution in energy production
systems will be enhanced to more than 25% in near future [6].
Therefore, it is necessary to study the impact of FCPPs on the power
systems, especially on the distribution networks.
Since the X/R ratio (X and R are respectively reactance and
resistance of transmission line) of distribution lines is small and the
structure of distribution network is radial, MOOM is one of the most
important schemes in the distribution networks, which can be
affected by FCPPs. In a general view, optimal operation management
in power systems refers to the optimal use of all equipments, to
generate and control active and reactive powers with the lowest cost
and meet the physical and technical constraints.
Many researchers have investigated the optimal operation of the
distribution network and particularly the topic of Volt/Var control.
For instance, a supervisory Volt/Var control scheme, based on the
new measurements and computer resources which were available
at the substation bus was presented in [7]. A centralized Volt/Var
control algorithm for the distribution system management considering summation of power losses and power demands as the
objective function was presented in [8]. The supervisory control
systems for integrated Volt/Var control at the substation and feeders
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T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
Nomenclature
X
n
Ng
Nt
Nc
Nd
Nb
Ri
Ii
PG
Pgi
state variables vector including active power of FCPPs
number of state variables
number of FCPPs
number of transformers
number of capacitors
number of load variation steps
number of branches
resistance of ith branch (U)
current of ith branch (A)
active power of all FCPPs during the day (kW)
active power of ith FCPP during the day (kW)
t
Pgi
active power of ith FCPP for tth load level step (kW)
Tap
tap vector representing tap position of all transformers
in the next day
Tapi
tap vector including tap position of ith transformer in
the next day
current tap positions of ith transformer during time t
Tapti
QC
capacitors reactive power vector including reactive
power of all capacitors in the next day (kVar)
Qci
capacitors reactive power vector including reactive
power of ith capacitor in the next day (kVar)
reactive power of ith capacitor for tth load level step
Qcit
(kVar)
hj
electrical efficiency of jth FC
part load ratio of jth FC for tth load level step
PLRtj
power generated at substation bus of distribution
Psub
feeders for tth load level step (kWh)
cost of electrical energy generated by FCPPs for tth load
CFC
level step ($)
Csubstation cost of power generated at substation bus for tth load
level step ($)
energy price for tth load level step ($/kWh)
Pricet
t
emission of FCPP for tth load level step (kg)
EFC
t
emission of large scale sources (substation bus that
EGrid
connects to grid) for tth load level step (kg)
nitrogen oxide pollutants of FCPP for tth load level step
NOtx FC
(kg)
sulphur oxide pollutants of FCPP for tth load level step
SOt2 FC
(kg)
NOtx Grid nitrogen oxide pollutants of grid for tth load level step
(kg)
SOt2 Grid sulphur oxide pollutants of grid for tth load level step
(kg)
Pmin;FC minimum active power of ith FCPP (kW)
Pmax;FC Maximum active power of ith FCPP (kW)
jPijLine j
Absolute power flowing over distribution lines (kW)
Line
Pij;max
maximum transmission power between the nodes
i and j (kW)
minimum tap positions of ith transformer
Tapmin
i
were presented in [9]. An approach for modeling local controllers
and coordinating the local and centralized controllers at the distribution system management was presented in [10] and [11]. A
heuristic and algorithmic combined technique for reactive power
optimization with time varying load demand in distribution systems
was presented in [12]. Volt/Var control in distribution systems using
a time-interval was described in [13]. An improved evolutionary
programming and its hybrid version combined with the nonlinear
interior point technique to solve the optimal reactive power
dispatch problems was proposed in [14]. A voltage regulation
Tapmax
i
Pfmin
Pfmax
Pf t
Vit
Vmax
Vmin
fi (X)
gi (X)
hi (X)
fimin
fimax
m
maximum tap positions of ith transformer
minimum power factor at substation bus
maximum power factor at substation bus
current power factor at substation bus during time t
voltage magnitude of ith bus during time t (Volt)
maximum value of voltage magnitudes of ith bus (Volt)
minimum value of voltage magnitudes of ith bus (Volt)
the ith objective function
equality constraints
inequality constraints
lowest limit of ith objective function
highest limit of ith objective function
the number of non-dominated solutions
uk
weight of kth objective function
t
current iteration number
u
inertia weight
c1 and c2 weighting factors of the stochastic acceleration terms
(Learning factors)
rand1 () random function in the range of [0,1]
rand2 () random function in the range of [0,1]
best previous experience of ith particle that is recorded
Pbesti
best particle among the entire population
Gbest
the jth chaotic variable
cxji
number of individuals for CLS
Nchoas
0
initial population for CLS
Xcls
F(X)
objective function values of the multi-objective OOM
problem
values of the augmented Fi (X)
Fi(X)
number of equality constraints of the OOM problem
Neq
number of inequality constraints of the OOM problem
Nueq
penalty factor
k1
penalty factor
k2
NSwarm number of the swarms
velocity of ith state variable
vi
position of ith state variable
xi
normalized membership value for ith non-dominated
Nmi
solution
size of repository
m1
cumulative probability for jth individual
Cj
List of abbreviations
PSO
particle swarm optimization
FSAPSO fuzzy self adaptive PSO
HPSO1 hybrid PSO-tent equation
HPSO2 hybrid PSO-logistic equation
FSAHPSO fuzzy self adaptive hybrid PSO
OOM
optimal operation management
MOOM multi-objective OOM
FCPP
fuel cell power plant
DG
distributed generation
CLS
chaotic local search
coordination method of distributed generation system at which
distribution system voltage regulation is coordinated by controlling
its reactive power output according to its real power output was
proposed in [15]. Optimal use of voltage support distributed
generation to support voltage in distribution feeders was presented
in [16]. Voltage and reactive power control in distribution systems
and how the presence of synchronous machine-based distributed
generation would be affect the control was presented in [17]. A
practical algorithm for optimal operation management of distribution network including fuel cell power plants was presented in [18].
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
In all the above-mentioned studies, the MOOM problem is
considered as a single objective one. In this paper a multi-objective
approach is used to solve the problem.
Based on the above discussion, the optimal operation management
(OOM) is a multi-objective optimization problem whose objectives are
not the same and commensurable. Due to equipment existing in
distribution systems, such as static var compensators (SVCs), fuel cell
power plants (FCPPs), load tap changers (LTCs) and voltage regulators
(VRs), the MOOM problem is modeled as a mixed integer nonlinear
and non-differentiable optimization problem. Therefore, it is difficult
to solve the problem by conventional approaches that convert the
multiple objectives into a single objective by using a vector of userpredetermined weights [19,20]. These approaches have several
drawbacks. For example, the values of the weights have a major impact
on the final solution, some optimal solutions may not be found if the
objective functions are not convex, and they may not work successfully if objective functions have a discontinuous-variable space [19,20].
Due to the simple concept, easy implementation and quick
convergence, nowadays particle swarm optimization (PSO) has
attracted much attention and has obtained wide applications in
various kinds of nonlinear optimization problems. However, the
performance of traditional PSO greatly depends on its parameters,
which often suffers from the problem of being trapped in local optima.
In order to overcome local optima problems, we propose a chaotic
local search and adjustable parameters of PSO that greatly improve
the performance of algorithm. Therefore, in this paper a novel multiobjective fuzzy self-adaptive hybrid particle swarm optimization
(MFSAHPSO) algorithm is proposed and implemented to solve the
multi-objective optimal operation management problem. In the
proposed approach, objective functions are the total electrical energy
losses, the total cost of electrical energy generated by FCPPs and
substation bus and the total emission of FCPPs and substation bus. The
proposed algorithm maintains a finite-sized repository of nondominated solutions which gets iteratively updated in the presence of
new solutions. An external memory has been used for the storage of
non-dominated solutions found in the search process. Since the
objective functions are not similar, a fuzzy clustering algorithm is
utilized to manage the size of the external memory.
The main contribution of the paper is the presentation of the
multi-objective optimization algorithm for the MOOM problem
which utilizes the concept of Pareto optimality. In other words, the
MFSAHPSO algorithm obtains a set of various solutions demonstrating different trade-off among the objective functions.
The remainder of the paper is organized as follows. In Section 2,
the MOOM problem is formulated. FCPP is modeled in Section 3.
Section 4 describes the principles of multi-objective optimization.
A fuzzy-based clustering to control the size of repository is presented in Section 5. Section 6 deals with the proposed MFSAHPSO
algorithm. The application of the MFSAHPSO algorithm in the
MOOM problem is illustrated in Section 7. In Section 8, the feasibility of the MFSAHPSO algorithm is demonstrated by the implementation on a distribution system.
2. Optimal operation management of distribution networks
regarding FCPPs
In the multi-objective OOM problem, total electrical energy
losses, total electrical energy cost and the total emission have been
considered as the objectives which satisfies various constraints. Its
mathematical model can be described as follows.
2.1. Objective functions
With the proposed MOOM problem, the objective function
consists of three terms: (i) total active power losses; (ii) total cost of
1531
electrical energy; (iii) total emission. Objective functions can be
described as:
2.1.1. Minimization of the power losses
Minimizing the electrical energy losses of distribution network
in the presence of FCPPs is of great importance in optimal operation
problem. The minimization of the total real power losses can be
calculated as follows:
min f1 ðXÞ ¼
Nd
X
t
PLoss
¼
t¼1
Nd X
Nb
X
t ¼1 i¼1
Ri jIit j
2
(1)
where, Nd is the number of load variation steps, Nb is the number of
branches, Ri is the resistance of ith branch, Ii is the current of ith
branch and X is the state variables vector including active power of
FCPPs, Tap of transformers and capacitor reactive power that can be
described as follows:
X ¼ PG ; Tap; QC 1n
n ¼ Nd Ng þ Nt þ Nc
(2)
where, n is the number of state variables and PG is the active power
of all FCPPs during the day, Tap is the tap vector representing tap
position of all transformers in the next day, Q C is the capacitors
reactive power vector including reactive power of all capacitors in
the next day. These variables can be described as follows:
i
h
PG ¼ Pg1 ; Pg2 ; .; PgNg
i
h
1
2
Nd
; i ¼ 1; 2; 3; .; Ng
Pgi ¼ Pgi
; Pgi
; .; Pgi
(3)
where, Ng is the number of FCPPs, P gi is the active power of the ith
t is the active power of ith FCPP for tth load
FCPP during the day, Pgi
level step.
Tap ¼ Tap1 ; Tap2 ; .; TapNt
i
h
;
Tapi ¼ Tap1i ; Tap2i ; .; TapNd
i
i ¼ 1; 2; 3; .; Nt
(4)
where, Nt is the number of transformers, Tapi is the tap vector
including tap position of ith transformer in the next day, Tapti is the
tap position of ith transformer for tth load level step.
QC ¼ Qc1 ; Qc2 ; .; QcNc
i
h
Qci ¼ Qci1 ; Qci2 ; .; QciNd ;
i ¼ 1; 2; 3; .; Nc
(5)
where, Nc is the number of capacitors, Q ci is the capacitors
reactive power vector including reactive power of ith capacitor in
the next day, Qcit is the reactive power of ith capacitor for tth load
level step.
2.1.2. Minimization of the total cost of electrical energy
Minimization of the summation of costs of electrical energy
generated by FCCPs and power of substation bus can be described
as follows. In [21] the authors introduce a cost model for the FCPP
operating strategy which can be utilized as:
min f2 ðXÞ ¼
Nd
X
Costt ¼
t¼1
Nd X
t ¼1
t
t
CFC
þ Csubstation
(6)
where, CtFC is the cost of electrical energy generated by FCPPs and
Ctsubstation is the cost of power generated at substation bus for tth
load level step. These vectors can be defined as follows:
t
CFC
¼ 0:04$=KWh Ng
X
Pgt j
j¼1
hj
where, hj is electrical efficiency of jth FC.
(7)
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T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
PLRtj ¼
Pgt j
Pmaxj
For PLRj < 0:050hj ¼ 0:2716
For PLRj 0:050hj ¼ 0:9033PLR5j 2:9996PLR4j
(8)
þ3:6503PLR3j 2:0704PLR2j þ 0:3747
where, PLRtj is part load ratio of jth FC.
t
t
Csubstation
¼ pricet Psub
(9)
Ptsub
where,
is the power generated at substation bus of distribution
feeders and pricet is energy price of tth load level step.
2.1.3. Minimization of the total emissions
Minimization of summation of FCPPs and substation bus emissions is one of the major objectives of the OOM problem that can be
described as follows:
min f3 ðXÞ ¼
Nd
X
Emissiont ¼
t¼1
Nd
X
t
t
EFC þ EGrid
(10)
t¼1
t
EFC
t
where,
is the emission of FCPP, EGrid
is the emission of large
scale sources (substation bus which is connected to grid). These
variables can be defined as follows:
t
EFC
¼ NOtx FC þSOt2 FC ¼ ð0:01361þ0:00272Þkg=MWh Ng
X
Pgt j
(11)
j¼1
where,
is the nitrogen oxide pollutants of FCPP and SOt2 FC is
the sulphur oxide pollutants of FCPP for tth load level step.
t
EGrid
¼ NOtx Grid þ SOt2 Grid
t
¼ ð2:29518 þ 3:58338Þkg=MWh Psub
(12)
SOt2 Grid
where,
is the nitrogen oxide pollutants of grid and
is the sulphur oxide pollutants of grid for tth load level step.
2.2. Constraints
Constraints are defined as follows:
Active power constraints of FCPPs:
t
t
t
Pmin;FC
Pgi
Pmax;FC
t
Pmin;FC
is minimum active power of the ith FCPP and
maximum active power of the ith FCPP.
(13)
t
Pmax;FC
is
Distribution line limits:
Line t
P
< P Line
ij
ij;max
(14)
Line t
P
and P Line are the absolute power flowing in distribution
ij
ij;max
lines and maximum transmission power between the nodes i and j,
respectively.
Tap of transformers:
Tapmin
< Tapti < Tapmax
i
i
(15)
Tapmin
and Tapmax
are the minimum and maximum tap positions of
i
i
the ith transformer, respectively.
Unbalanced three-phase power flow equations.
Substation power factor.
Pfmin Pf t Pfmax
(16)
Pf t
Pfmin , Pfmax and
are the minimum, maximum and current
power factor at the substation bus during time t.
Bus voltage magnitude.
Vmin Vit Vmax
(17)
Vit ,
NOtx FC
NOtx Grid
Fig. 1. Models of FC power plants, (a) PQ Model with simultaneous three-phase control
(b). PQ Model with independent three-phase control, (c) PV Model with simultaneous
three-phase control, (d) PV Model with independent three-phase control.
Vmax and Vmin are the voltage magnitudes of the ith bus during
the time t and the maximum and minimum values of voltage
magnitudes, respectively.
3. Fuel cell power plant modeling
Fuel cell is a great development in alternate energy field. Fuel
cell, in simple word is an electrochemical energy generating device.
It has become one of the most attractive and interesting aspects of
modern technology. There are a lot of things that are yet to be
developed in this field and also fuel cell technology is vast and
involves various applications. Many experts all around the world
are researching on Fuel Cells [3e5].
Generally, FCPPs in distribution load flow can be modeled as PV
or PQ models. PQ buses are nodes (buses) where both the real
power (P) and reactive power (Q) are given. PV buses are nodes
where the real power (P) is given, but the reactive power (Q) must
be determined each iteration.
Since distribution networks are unbalanced three-phase
systems, FCPPs can be controlled and operated in two forms: (i)
Simultaneous three-phase control, and (ii) Independent threephase control or single-phase control. Regarding the control
methods, four different models can be used for simulation of these
sources as shown in Fig. 1.
It must be taken into account that when FCPPs are considered as
the PV models, they should be able to generate reactive power to
maintain their voltage magnitudes. Many researchers have presented several procedures to model generators connected to
distribution networks as the PV buses [22e24]. Fig. 1 shows
a model of the FC power plants based on the type of their control. In
this paper, the FCPPs are modeled as the PQ model with simultaneous three-phase control (Fig. 1a).
4. Principles of multi-objective optimization
Many real-world problems include simultaneous optimization
of several objective functions. Generally, these functions are noncommensurable and have often competing and conflicting objectives. Multi-objective optimization with such conflicting objective
function leads to a set of optimal solutions, instead of one optimal
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
solution. The reason why many solutions are optimal is that no one
can be considered to be better than any others regarding all
objective functions. These optimal solutions are known as Paretooptimal solutions.
A general multi-objective optimization problem consists of
a number of objectives that should be optimized simultaneously
associated with a number of equality and inequality constraints. It
can be formulated as follows [19,20].
Minimize F ¼ ½f1 ðXÞ; f2 ðXÞ; .; fn ðXÞT
gi ðXÞ < 0 i ¼ 1; 2; .; Nueq
Subject to :
hi ðXÞ ¼ 0 i ¼ 1; 2; .; Neq
1533
(18)
where, fi(X) is ith objective function, gi(X) and hi(X) are the equality
and inequality constraints, respectively, X is the vector of the
optimization variables, n is the number of objective functions.
Fig. 2. Flowchart of CLS.
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T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
Fig. 3. Membership functions of inputs and outputs.
For a multi-objective optimization problem, any of the two
solutions X1 and X2 can have one of two possibilities: one dominates the other or none dominates the other. In a minimization
problem, without loss of generality, a solution X1 dominates X2 if
the following two conditions are satisfied:
In this procedure, a fuzzy membership function is used to
recognize the best compromise solution. In other words, decision
making is done while the repository is being filled. For any individual in the repository, the membership function of each objective
function is defined as follows:
cj˛f1; 2; .; ng; fj ðX1 Þ fj ðX2 Þ
1; fi ðXÞ fimin
0; fi ðXÞ fimax
mfi ðXÞ ¼ f max f ðXÞ
>
i
i
>
; fimin fi ðXÞ fimax
>
>
: f max f min
dk˛f1; 2; .; ng; fk ðX1 Þ < fk ðX2 Þ
(19)
If any of the above condition is violated, solution X1 does not
dominate solution X2. If X1 dominates solution X2, X1 is called the
non-dominated solution. The solutions that are non-dominated
within the entire search space, are denoted as Pareto-optimal which
constitute the Pareto-optimal set or Pareto-optimal front.
Pareto-dominance conception is utilized to evaluate the suitability of each particle and in this way determine which particles
should be selected to be stored in the repository of non-dominated
solutions. The repository absorbs superior current non-dominated
solutions and eliminates inferior solutions in the repository
through interacting with the generated population in any iteration.
A candidate solution can be added to the repository if it satisfies any
of the following conditions [25]:
(i) The repository is full but the candidate solution is nondominated and it is in a less crowded region than at least one
solution.
(ii) The repository is not full and the candidate solution is not
dominated by any solution in the repository.
(iii) The candidate solution dominates any of the solutions existing
in the repository.
(iv) The repository is empty.
In addition, the repository should be maintained in such a way
that all the solutions be non-dominated. One important criterion to
measure the performance of a multi-objective optimization algorithm is to check if the solutions derived from this algorithm can
spread along the entire Pareto front in a graceful manner.
8
>
>
>
>
<
i
(20)
i
where, fmin
and fmax
are the lowest and highest limits of ith
i
i
objective function, respectively.
and fmax
are evaluated
In the proposed algorithm, the values of fmin
i
i
using the results achieved by optimizing each objective separately.
For each individual in the repository, the normalized membership value is evaluated as follows:
n
P
NmðjÞ ¼
k¼1
uk mfk Xj
m P
n
P
j¼1 k¼1
(21)
uk mfk Xj
where, m is the number of non-dominated solutions. uk is the
weight of kth objective function. This membership function shows
a type of decision making criteria that is adaptive and can change
with the available decision options. In the fuzzy-based clustering
the normalized membership values are sorted and the best individuals are selected and stored in the repository.
6. The proposed MFSAHPSO algorithm
Some studies have been recently reported to implement a multiobjective PSO (MOPSO) to solve power system problems. A fuzzified
MOPSO to solve the economic emission dispatch (EED) problem
Table 1
Fuzzy rules for inertia weight correction Du.
5. Fuzzy-based clustering for the control of the repository
size
Since the objective functions are imprecise, a fuzzy-based clustering procedure has been utilized to control the size of repository.
Various artificial intelligence techniques are described in Ref. [26].
Du
NBF
u
S
M
L
S
M
L
ZE
PE
PE
NE
ZE
ZE
NE
NE
NE
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
with heat dispatch and the development of several distribution
preservation mechanisms for dealing with multi-objective optimization case presented in [25]. A modified MOPSO to optimize an
energy management system where the problem is solved in three
phases by dividing the original optimization problem into partial
problems is proposed in [27]. A MOPSO based approach to solve the
congestion management problem where the cost and congestion are
simultaneously minimized is presented in [28].
1535
6.2.1.1. CLS 1. In the first CLS method which is based on the logistic
equation, the related equation is defined by the following equation:
i
h
Cxi ¼ cx1i ; cx2i ; .; cxNg
; i ¼ 0; 1; 2; .; Nchoas
i
1Ng
j
j
j
cx iþ1 ¼ 4 cxi 1 cxi ; j ¼ 1; 2; .; Ng
j
j
(24)
cx i ˛½0; 1; cx0 ;f0:25; 0:5; 0:75g
cx j0 ¼ rand $
j
6.1. Particle swarm optimization (PSO) algorithm
PSO method is a population-based optimization technique that
was first introduced by Kennedy and Eberhart [29] in which each
solution called ‘‘particle” flies around a multidimensional search
space. During the flight, every particle adjusts its position according
to its own experience, as well as the experience of neighboring
particles, using the best position encountered with itself and its
neighbors. The swarm direction of a particle is defined by its history
experience and the experience of its neighbors. A particle status on
the search space is described by two factors: its position and
velocity, which are updated by following equations:
ðtþ1Þ
Vi
ðtþ1Þ
Xi
ðtÞ
þ c1 $rand1 ð$Þ$ Pbesti Xi
ðtÞ
þ c2 $rand2 ð$Þ$ Gbest Xi
ðtÞ
¼ u$Vi
ðtÞ
¼ Xi
ðtþ1Þ
þ Vi
where, cxi indicates jth chaotic variable, Nchoas is the number of
individuals for CLS, Ng is the number of FCPPs and rand () is
a random number between [0,1].
At first, a particle randomly selected from the repository (Xg) is
0 ). X 0 is scaled into
considered as an initial population for CLS (Xcls
cls
[0,1] according the following equation:
i
h
0 ¼ x1
Xcls
; x2 ; .; xNg
cls;0 cls;0
cls;0 1Ng
i
h
Ng
Cx0 ¼ cx10 ; cx20 ; .; cx0
j
cx j0
¼
j
x cls;0 P min;FC
P jmax;FC P jmin;FC
;
(25)
j ¼ 1; 2; .; Ng
Then, the chaos population for CLS is generated as follows:
(22)
(23)
where, t is the current iteration number, u is the inertia weight, c1
and c2 are Weighting factors of the stochastic acceleration terms,
which pull each particle towards the Pbesti and Gbest positions,
rand1() and rand2() are two random functions in the range of [0,1],
Pbesti is the best previous experience of ith particle that is recorded
and Gbest is the best particle among the entire population.
The Eq. (22) is used for the calculation of ith particle’s velocity
considering three terms: the particle’s previous velocity, the
distance between the particle’s best previous and current positions,
and finally, the distance between the position of the best particle in
the swarm and ith particle’s current position.
i
h
Ng
i
Xcls
¼ x1cls;i ; x2cls;i ; .; xcls;i
; i ¼ 1; 2; .; Nchoas
1Ng
j
j
j
j
j
x cls;i ¼ cx i1 P max;FC P min;FC þ P min;FC ; j ¼ 1:2; .; Ng
(26)
The objective functions are evaluated for all individuals of CLS.
Non-dominated solutions should be found and stored into a separate memory subsequently. The way that one of the non-dominated
6.2. Multi-objective fuzzy self-adaptive hybrid PSO (MFSAHPSO)
algorithm
The standard PSO algorithm is not suitable to resolve multiobjective optimization problems. Thus, in order to render the PSO
algorithm capable of dealing with multi-objective problems, some
modifications seem to be necessary. In this paper, the standard PSO
algorithm is modified and improved to facilitate a multi-objective
optimization approach, i.e., Multi-objective fuzzy self adaptive hybrid
particle swarm optimization (MFSAHPSO), in which Pareto-dominance is employed to handle the problem. Through incorporation of
certain global attraction mechanisms, the repository of previously
found non-dominated solutions would make the convergence toward
globally non-dominated solutions possible. The following section
describes the proposed chaotic local search and modified parameters
of PSO, in order to improve algorithm performance.
6.2.1. Chaotic local search
Due to the properties of including all points in a given space,
inherent stochastic property and irregularity of chaos, a chaotic search
can traverse every state in a certain space, and each state can be visited
only once, thus it is helpful to avoid being trapped in local optima.
Therefore, to improve the search behavior, we propose a chaotic PSO
method that combines PSO with chaotic local search (CLS).
There are two CLS procedures which can be shown as follows:
Fig. 4. Single line diagram of distribution test system.
1536
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
Fig. 5. Daily energy price and load variations.
solutions is replaced with a randomly selected particle from the
swarm, is described in the flowchart presented in Fig. 2.
6.2.1.2. CLS 2. The second CLS is based on the Tent equation, which
generates the chaos variables randomly. It can also be introduced to
the process of the chaotic local search, which can be defined by the
following equation:
i
h
Ng
Cxi ¼ cx1i ; cx2i ; .; cxi
; i ¼ 0; 1; 2; .; Nchoas
1Ng
(
j
j
2cx i ; 0 < cx i 0:5
j
j ¼ 1:2; .; Ng
cx iþ1 ¼
2 1 cx ji ; 0:5 < cx ji 1
(27)
j
cx 0 ¼ randð$Þ
The procedure of CLS 2 is similar to CLS 1. The only difference
which can be detected in this method is that the chaotic variables
are defined in it as described in Eq. (27).
6.2.2. PSO parameters
There are three tuning parameters u, c1 and c2 as shown in
Eq. (22) that greatly affects the algorithm performance.
6.2.2.1. Learning factors c1 and c2. A self-adaptive manipulation of c1
and c2 is considered to avoid the cumbersome task of first localizing
and then fine-tuning of these parameters. In the proposed method,
two tuning parameters c1 and c2 are considered as the two new
variables that are incorporated with control variables vector X. The
new control variables vector for particles in this paper will be:
X ¼ PG ; Tap; QC ; c1 ; c2
(28)
Also, Pbesti , Gbest and Vi which represent the best previous
position of the swarm, best global position of the swarm and
velocity respectively, increase their dimension. In this case each
particle will additionally be endowed with the ability of adjusting
its parameters by aiming at the parameters it had while getting the
best position in the past and the parameters of the leader, which
was managed to bring the best particle to its privileged position. As
a consequence, particles not only use their cognition of individual
thinking and social cooperation to improve their positions, but also
improve the way they do it by accommodating themselves to the
best known conditions: namely, their conditions when get the best
so far position and the leader’s conditions.
6.2.2.2. Inertia weight u. The inertia weight u is used to control the
impact of the previous history of velocities on the current velocity.
Relatively large inertia weight has more global search ability while
a relatively small inertia weight results in a faster convergence.
Suitable selection of the inertia weight can prepare a balance
between the global and local exploration abilities, thus on average,
less iterations are required to find the optimum. It is probably
impossible to find a specific inertia weight u which can work well
in all cases but the following fuzzy adaptive PSO (FAPSO) algorithm,
based on a fuzzy system, has been found to work in practice.
Based on this kind of knowledge, in this paper a fuzzy system is
developed to adjust the best fitness (BF) and the inertia weight (u)
as the input variables, and the inertia weight correction (Du) as the
output variable.
Both the positive and negative corrections are required for the
inertia weight. Therefore, a range of 1.0 to þ1.0 has been preferred
for the inertia weight correction.
Table 2
Comparison of average and standard deviation for 20 trails (Cost objective function).
Method
Average ($)
Standard
deviation ($)
Worst
solution ($)
Best
solution ($)
PSO
FSAPSO
HPSO1
HPSO2
FSAHPSO1
FSAHPSO2
5394.38341
5316.55594
5264.64750
5253.90303
5226.80219
5225.88578
258.54169
183.91965
36.54415
30.50328
0.11858
0.05930
5903.74821
5839.64726
5339.96647
5337.90188
5227.03684
5226.01741
5253.39978
5248.02901
5236.11074
5235.73017
5226.68104
5225.81423
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
1537
Table 3
Comparison of average and standard deviation for 20 trails (Emission objective function).
Method
Average (kg)
Standard deviation (kg)
Worst solution (kg)
Best solution (kg)
PSO
FSAPSO
HPSO1
HPSO2
FSAHPSO1
FSAHPSO2
2.558672211434Eþ08
2.530555730333Eþ08
2.291333891898Eþ08
2.276178003996Eþ08
2.152619511675Eþ08
2.152455425820Eþ08
1.417747859292Eþ07
1.347361774972Eþ07
1.222370857028Eþ07
1.171336984518Eþ07
2.069856101850Eþ04
1.677316587295Eþ04
2.958487767121Eþ08
2.911045763755Eþ08
2.637268834505Eþ08
2.608028287274Eþ08
2.152950708848Eþ08
2.152850530412Eþ08
2.496487075342Eþ08
2.470151031060Eþ08
2.237074093361Eþ08
2.222851993425Eþ08
2.152392169719Eþ08
2.152254123057Eþ08
ukþ1 ¼ uk þ Du
(29)
The normalized best fitness (NBF) in this multi-objective
problem is considered as the normalized membership value corresponding with the selected global best.
Triangular membership functions are used for every input and
output as illustrated in Fig. 3 in which S (Small), M (Medium), L
(Large) are three linguistic values for inputs (NBF, u) and NE
(Negative), ZE (Zero), PE (Positive) are the linguist values for the
output of inertia weight correction (Du).
The Mamdani-type fuzzy rule is used to formulate the conditional statements. For example:
If (NBF is S) and (u is M), THEN (Du is NE).
The fuzzy rules in Table 1 are used to adjust the inertia weight
correction (Du). Each rule represents a mapping from the input
space to output space.
7. Application of the MFSAHPSO algorithm to the MOOM
problem
To apply the MFSAHPSO algorithm in the MOOM problem, the
following steps should be taken and repeated:
Step 1: Define the input data:
Input data includes network configuration, line impedance,
characteristics of FCPPs, emission functions and prices of Fuel cell
and substation bus.
Step 2: Transfer the constraint MOOM problem to an unconstraint one:
The multi-objective OOM problem should be transformed into
an unconstrained one by constructing an augmented objective
function incorporating penalty factors for any value violating the
constraints as follows.
2
6 f1 ðXÞ þ k1
6
6
F1 ðXÞ
6
6
4
5
FðXÞ ¼ F2 ðXÞ
¼ 6 f2 ðXÞ þ k1
6
F3 ðXÞ 31 6
6
4
f3 ðXÞ þ k1
2
3
2
hj ðXÞ
N
eq P
!
þ k2
j¼1
N
eq P
!
j¼1
N
eq P
!
2
hj ðXÞ
j¼1
2
hj ðXÞ
þ k2
þ k2
N
ueq P
where, F(X) is the objective function values of the multi-objective
OOM problem. F1(X), F2(X) and F3(X) are the values of the
augmented f1 (X), f2 (X) and f3(X), respectively. Neq and Nueq are the
number of equality and inequality constraints, respectively. hj(X)
and gj(X) are the equality and inequality constraints, respectively.
While k1 and k2 are the penalty factors. Since the constraints should
be met, the values of the parameters should be high. In this paper
the values have been considered to be 10,000,000. This number has
been selected based on the value of emission (one of the objective
functions) which its value is high.
In the MOOM problem, the values of the objective functions FðXÞ
are calculated as follows:
At first, the distribution load flow is run based on the state
variables (active power of FCPPs). Based on the results of distribution load flow, the objective functions values (f1 ðXÞ, f2 ðXÞ and
f3 ðXÞ) are calculated and the constraints are checked. Then the
augmented objective functions are calculated using the values of
objective functions, constraints and penalty factors.
Step 3: Generate the initial population and initial velocity:
The initial population and initial velocity for each particle are
randomly generated as follows:
3
X1
7
6 X2
7
population ¼ 6
5
4.
XN swarm
h
i
Ng
X0 ¼ x10 ; x20 ; .; x0
x j0 ¼ rand ð$Þ xmax
xmin
; j ¼ 1; 2; .; Ng
þ xmin
i
i
i
h i
j
Xi ¼ x i
; i ¼ 1; 2; 3; .; Nswarm
1n x ji ¼ 4 x ji1 1 x ji1
n ¼ Nd Ng þ Nt þ Nc
2
(31)
h
i2
Max 0; gj ðXÞ
!3
7
7
!7
h
i2 7
7
Max 0; gj ðXÞ
7
7
j¼1
!7
h
i2 7
N
ueq P
5
Max 0; gj ðXÞ
j¼1
N
ueq P
j¼1
(30)
31
2
Table 4
Comparison of average and standard deviation for 20 trails (PLoss objective function).
Method
PSO
FSAPSO
HPSO1
HPSO2
FSAHPSO1
FSAHPSO2
Average
(kWh)
Standard
deviation (kWh)
Worst
solution (kWh)
Best
solution (kWh)
864.15868
832.74085
797.02531
775.25353
669.67644
666.36276
82.47944
64.43453
59.66481
56.29530
2.43229
1.44245
1019.55105
938.70674
924.19666
902.43226
673.21316
668.98124
755.21160
735.25546
716.95827
710.39056
667.65610
662.22731
3
V1
6 V2
7
7
velocity ¼ 6
4.
5
VN swarm
Vi ¼ ½vi 1n ; i ¼ 1; 2; 3; .; Nswarm
max
vi ¼ randð$Þ
vmin
þ vmin
i
i
vi
n ¼ Nd Ng þ Nt þ Nc
(32)
where, vi and xi are the velocity and position of ith state variable,
respectively, Ng is the number of FCPPs, Nd is the number of load
1538
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
Fig. 6. Emission and Cost Pareto-optimal set of PSO algorithm.
Fig. 7. Emission and Cost Pareto-optimal set of FSAHPSO1 algorithm.
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
1539
Fig. 8. Emission and Cost Pareto-optimal set of FSAHPSO2 algorithm.
variation steps, rand () is a random function generator between
0 and 1, n is the number of state variables.
Step 4: i¼1.
Step 5: Select the ith individual.
The values of the objective functions are evaluated for ith
individual using the result of the distribution load flow.
Step 6: If the individual is a non-dominated solution, it is stored
in the repository and the fuzzy clustering is used to control the size
of repository.
Step 7: Select local best solution as follows:
At first the initial generated population was considered as the
local best solution. It is updated when one of the following
Fig. 9. PLoss and Cost Pareto-optimal set of PSO algorithm.
1540
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
Fig. 10. PLoss and Cost Pareto-optimal set of FSAHPSO1 algorithm.
conditions is satisfied, otherwise it would be the same as the
previously mentioned population:
(i) If the current population dominates the former local best, it is
considered as the local best.
(ii) If none dominates the other, the one that its normalized
membership function is greater, will be considered as the local
best.
Step 8: If all of the individuals are selected, go to step 9, otherwise set up i ¼ iþ1 and return to step 5.
Step 9: Select global best as follows:
At first the normalized membership values are calculated
(Eq. (21)) for non-dominated solutions in the repository.
N m ¼ ½Nm1 ; Nm2 ; .; N mm 1m1
(33)
where, Nmi is the normalized membership value for the ith nondominated solution and m1 is regarded as the size of repository.
Cumulative probabilities are calculated as:
Ci ¼ ½C1 ; C2 ; .; Cm 1m1
where,
C1 ¼ Nm1
C2 ¼ C1 þ Nm2
«
Cm1 ¼ Cm11 þ N mm1
(34)
In the mentioned equations, Cj is the cumulative probability for
the jth individual.
The roulette wheel is used for the stochastic selection of the best
global position as follows:
A number between 0 and 1 is randomly generated and
compared with the calculated cumulative probability. The first term
of cumulative probabilities (Cj), which is greater than the generated
number, is selected and the associated position is considered as the
best global position.
Step 10: If the algorithm of MFSAHPSO1is used go to CLS1 and if
MFSAHPSO2 is utilized go to CLS2.
Step 11: Check the termination criteria:
The values of the objective functions of each individual are
evaluated by using the results of the distribution load flow. If the
individual is non-dominated, store it in the repository and use the
fuzzy clustering to control its size, else the termination criteria
should be checked. If the termination criteria is satisfied, finish
the algorithm, otherwise the initial population should be replaced
with the new population of the swarms and then goes back to
step 4.
8. Simulation results
The proposed MFSAHPSO algorithm is tested on a distribution
test system. Fig. 4 shows the test system that is an 11-kV radial
distribution system having two substations (S/S-1, S/S-2), four
feeders (F1, F2, F3, F4) and 70 nodes (represented by numbers
1e70). The related information of this network is given in [30].
It is assumed that 12 FCPPs are located in this network at buses
4, 28, 29, 39, 41, 49, 50, 58, 59, 62, 65 and 66 and each of these
sources can generate an active power of 250 kW. In addition, 12
capacitors are placed in the network at buses 3, 9, 27, 28, 37, 40, 48,
57, 60, 61, 63 and 67 and the reactive power of each capacitor is
200 kVar.
In the daily MOOM problem, it is assumed that daily load variations and daily energy price variations can be changed as shown in
Fig. 5.
At first, the total cost of electrical energy, the total emission and
the total electrical energy losses objectives are the separate optimized to find the extreme points of the trade-off front. The best
results obtained by separately optimization of the objectives are
shown in Tables 2e4 respectively. These tables present a comparison among the results of PSO, FSAPSO (Fuzzy Self Adaptive PSO),
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
1541
Fig. 11. PLoss and Cost Pareto-optimal set of FSAHPSO2 algorithm.
HPSO1 (Hybrid PSO-Tent equation), HPSO2 (Hybrid PSO-Logistic
equation), FSAHPSO1 and FSAHPSO2 algorithms for 20 random tails
for three objective functions. PSO algorithm is explained in Section
6.1 in detail. FSAPSO algorithm includes self-adaptive method for
tuning the parameters c1 and c2 and fuzzy adaptive technique for
adjusting the inertia weight u in a suitable way. HPSO1 algorithm
takes advantages of the combination of PSO and chaotic local
search (Tent equation) to improve the search behavior and in the
Fig. 12. PLoss and Emission Pareto-optimal set of PSO algorithm.
1542
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
Fig. 13. PLoss and Emission Pareto-optimal set of FSAHPSO1 algorithm.
same way HPSO2 algorithm uses the Logistic equation of chaotic
local search. FSAHPSO1 algorithm joins the properties of FSAPSO
and HPSO1 methods and in a similar way, FSAHPSO2 utilizes the
combination of FSAPSO and HPSO2 algorithms.
As shown in the tables, the algorithm is capable of finding the
global solutions for each objective function.
The proposed approach has been implemented to optimize the
objectives simultaneously. The distribution of the Pareto-optimal
Fig. 14. PLoss and Emission Pareto-optimal set of FSAHPSO2 algorithm.
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
1543
Fig. 15. Three-dimensional Pareto-optimal set of FSAHPSO1 algorithm.
set over the trade-off surface is shown in Figs. 6e14. The Pareto
front for Emission and Cost objectives obtained with PSO, FSAHPSO1
and FSAHPSO2 algorithms are shown in Figs. 6e8 respectively.
The Pareto front for PLoss and Cost objectives are shown in
Figs. 9e11.
The Pareto front for PLoss and Emission objectives are shown in
Figs. 12e14.
The three-dimensional Pareto front for the three objectives with
FSAHPSO1 and FSAHPSO2 algorithms are shown in Figs. 15 and 16.
It is noticeable that the proposed technique preserves the
diversity of the non-dominated solutions over the Pareto-optimal
front and solves the problem effectively.
As mentioned in Section 5, a fuzzy-based clustering procedure
has been utilized to control the size of repository. In all cases we
Fig. 16. Three-dimensional Pareto-optimal set of FSAHPSO2 algorithm.
1544
T. Niknam et al. / Renewable Energy 36 (2011) 1529e1544
have obtained 100 non-dominated solutions through this technique except for PLoss and Emission objectives with PSO algorithm
(Fig. 12) in which 42 non-dominated solutions are found.
The non-dominated solutions that represent the best solutions
for the objective functions (given in Tables 2e4) are shown in Figs.
6e16 with cursor.
The close agreement of the results clearly shows the capability
of the proposed technique to handle multi-objective optimization
problems as the best solution for each objective along which
a manageable set of non-dominated solutions can be obtained.
9. Conclusion
In this paper, a novel multi-objective FSAHPSO optimization
technique has been proposed and applied to daily Multi-objective
Optimal Operation Management (MOOM) problem in distribution
system regarding Fuel Cell Power Plants (FCPPs). The proposed
algorithm utilizes the concept of Pareto optimality. One of the most
important advantages of the multi-objective formulation is that it
obtains several non-dominated solutions allowing the system
operator to use his personal preference in selecting any of the
mentioned solutions, for implementation. In order to control
the size of the repository, a fuzzy-based clustering has been used.
The results proved that the proposed approach is an efficient
technique to solve multi-objective optimization problems where
multiple Pareto-optimal solutions can be found utilizing this
method. In addition, the non-dominated solutions in the obtained
Pareto-optimal set are well distributed and reveal acceptable
diversity characteristics.
References
[1] Zangeneh A, Jadid Sh, Rahimi-Kian A. Promotion strategy of clean technologies
in distributed generation expansion planning. Renewable Energy 2009;34:
2765e73.
[2] Dicorato M, Forte G, Trovato M. Environmental-constrained energy planning
using energy-efficiency and distributed-generation facilities. Renewable
Energy 2008;33:1297e313.
[3] Soroudi A, Ehsan M, Zareipour H. A practical eco-environmental distribution
network planning model including fuel cells and non-renewable distributed
energy resources. Renewable Energy 2011;36:179e88.
[4] Yu S, Hong D, Lee Y, Lee S, Ahn K. Development of a catalytic combustor for
a stationary fuel cell power generation system. Renewable Energy 2010;35:
1083e90.
[5] Moreira MV, da Silva GE. A practical model for evaluating the performance of
proton exchange membrane fuel cells. Renewable Energy 2009;34:1734e41.
[6] Ackerman T, Anderson G, Soder L. Distributed generation: a definition. Electric
Power Syst Res 2001;57:195e204.
[7] Baron ME, Hsu MY. Volt/var control at distribution substations. IEEE Trans
Power Systems 1999;14:312e8.
[8] Roytelman I, Wee BK, Lugtu RL. Volt/var control algorithm for modern
distribution management system. IEEE Trans Power Systems 1995;10:
1454e60.
[9] Borozan V, Baran ME, Novosel D. Integrated volt/var control in distribution
systems. IEEE Power Eng Soc Winter Meet 2001;3:1485e90.
[10] Roytelman I, Ganesan V. Modeling of local controllers in distribution network
applications. IEEE Trans Power Deliv 2000;15:1232e7.
[11] Roytelman I, Ganesan V. Coordinated local and centralized control in distribution management systems. IEEE Trans Power Deliv 2000;15:718e24.
[12] Deng Y, Ren X, Zhao C, Zhao D. A heuristic and algorithmic combined
approach for reactive power optimization with time varying load demand in
distribution systems. IEEE Trans Power Systems 2002;17:1068e72.
[13] Hu Z, Wang X, Chen H, Tailor GA. Volt/var control in distribution systems
using a time-interval based approach. IEE Proc Generation Trans Distribution
2003;150:548e54.
[14] Lu WYS, Yu DC. A novel optimal reactive power dispatch method based on an
improved hybrid evolutionary programming technique. IEEE Trans Power
Syst 2004;19:913e8.
[15] Kim TE, Kim JE. Voltage regulation coordination of distributed generation
system in distribution system. Proc IEEE Power Eng Soc Summer Meet 2001;1:
480e4.
[16] Kashem MA, Ledwich G. Multiple distributed generators for distribution
feeder voltage support. IEEE Trans Energy Conversion 2005;20:676e84.
[17] Viawan FA, Karlsson D. Voltage and reactive power control in systems with
synchronous machine-based distributed generation. IEEE Trans Power Deliv
2008;23:1079e87.
[18] Niknam T, Zeinoddini Meymand H, Nayeripour M. A practical algorithm for
optimal operation management of distribution network including fuel cell
power plants. Renewable Energy 2010;35:1696e714.
[19] Lin CM, Gen M. Multi-criteria human resource allocation for solving multistage combinatorial optimization problems using multiobjective hybrid
genetic algorithm. Expert Syst Appl 2008;34:2480e90.
[20] Chang PC, Chen SH, Liu CH. Sub-population genetic algorithm with mining
gene structures for multiobjective flowshop scheduling problems. Expert Syst
Appl 2007;33:762e71.
[21] El-Sharkh MY, Tanrioven M, Rahman A, Alam MS. Cost related sensitivity
analysis for optimal operation of a grid-parallel PEM fuel cell power plant.
J Power Sources 2006;161:1198e207.
[22] Niknam T. Application of honey bee mating optimization on distribution state
estimation including distributed generators. J Zhejiang Univ Sci A 2008;9:
1753e64.
[23] Losi A, Russo M. Dispersed generation modeling for object-oriented distribution load flow. IEEE Trans Power Deliv 2005;20:1532e40.
[24] Niknam T. A new approach based on ant colony optimization for daily volt/var
control in distribution networks considering distributed generators. Energy
Convers Manag 2008;49:3417e24.
[25] Wang L, Singh Ch. Environmental/economic power dispatch using fuzzified
multi-objective particle swarm optimization algorithm. Electric Power Syst
Res 2007;77:1654e64.
[26] Kalogirou SA. Artificial intelligence for the modeling and control of combustion processes: a review. Prog Energy Combust Sci 2003;29:515e66.
[27] Kitamura S, Mori K, Shindo S, Izui Y, Ozaki Y. Multiobjective energy
management system using modified MOPSO. IEEE Int Conf Syst Man Cybern
2005;4:3497e503.
[28] Hazra J, Sinha AK. Congestion management using multi-objective particle
swarm optimization. IEEE Trans Power Syst 2007;22:1726e34.
[29] Kennedy J, Eberhart R, Particle swarm optimization, IEEE international conf.
on neural networks, Piscataway, NJ, 4 1995 1942e1948.
[30] Das D. A fuzzy multi-objective approach for network reconfiguration of
distribution systems. IEEE Trans Power Deliv 2006;21:202e9.