# State-of-the-Art Economic Load Dispatch of Power Systems Using Particle
Swarm Optimization

Mahamad Nabab Alam The author is with the Department of Electrical
Engineering, Indian Institute of Technology, Roorkee, India (e-mail:
<EMAIL_ADDRESS>
###### Abstract

Metaheuristic particle swarm optimization (PSO) algorithm has emerged as one
of the most promising optimization techniques in solving highly constrained
non-linear and non-convex optimization problems in different areas of
electrical engineering. Economic operation of the power system is one of the
most important areas of electrical engineering where PSO has been used
efficiently in solving various issues of practical systems. In this paper, a
comprehensive survey of research works in solving various aspects of economic
load dispatch (ELD) problems of power system engineering using different types
of PSO algorithms is presented. Five important areas of ELD problems have been
identified, and the papers published in the general area of ELD using PSO have
been classified into these five sections. These five areas are (i) single
objective economic load dispatch, (ii) dynamic economic load dispatch, (iii)
economic load dispatch with non-conventional sources, (iv) multi-objective
environmental/economic dispatch, and (v) economic load dispatch of microgrids.
At the end of each category, a table is provided which describes the main
features of the papers in brief. The promising future works are given at the
conclusion of the review.

###### Index Terms:

Dynamic economic dispatch, economic load dispatch, environmental/emission
dispatch, particle swarm optimization, valve-point loading effect.

##  I Introduction

Economic operation of an electric power system involves unit commitment (UC)
and economic load dispatch (ELD). The first one is related to the optimum
selection of generating units from available options to supply a particular
load demand economically, whereas the second one is related to the optimum
power generation from each of the committed (selected) generating units to
supply dynamically varying load demand economically [1]. Proper handling of
these two issues not only reduces fuel consumption costs significantly but
also reduces transmission losses as well as environmental emission
considerably. The issues related to the economic operation of power systems
have been widely studied in various books [2, 3, 4, 5, 6].

Normally, the ELD problem is formulated as an optimization problem where the
objective is to minimize the total cost of fuel consumption while supplying
the given load demand successfully and maintaining system operation within the
specified limits [5]. Commonly, the fuel consumption cost is represented as a
simple quadratic function of power generation of the committed generating
units along with many non-linear characteristics of that unit. Further, a set
of equality and inequality constraints are considered in this minimization
problem. Also, some additional non-linear features like valve-point loading
effects and multi-fuel input options are considered in the objective function
which makes the optimization problem non-convex. Furthermore, generators
prohibited operating zones and ramp rate limits make the overall optimization
problem exceedingly complex [6].

Mathematical techniques like Gradient method, Basepoint and participation
factor method, Newton method, and Lambda-iteration method have already been
found to be ineffective in solving ELD problems of modern power systems. Also,
dynamic programming, non-linear programming, and their modified versions
suffer from dimensionality issues in solving ELD problems of modern power
systems which are having a large number of generating units. Recently,
different metaheuristic optimization approaches have proven to be very
effective with promising results in solving ELD problems, such as, simulated
annealing (SA) [7], tabu search (TS) [8], artificial neural network (ANN) [9],
pattern search (PS) [10], evolutionary programming (EP) [11], genetic
algorithm (GA) [12], differential evolution (DE) [13], and particle swarm
optimization (PSO) [14, 15]. Metaheuristic algorithms provide high-quality
solutions in relatively less time in solving highly constrained problems [16].
Among these algorithms, PSO has shown great potential in solving ELD problems
efficiently and effectively [17]. The simple concept, fast computation, and
robust search ability are considered to be the most attractive features of
PSO.

Although PSO is a very efficient algorithm in solving ELD problems, however,
it may suffer from trapping into local minimums during the search process. To
handle such trapping into local minima, many modified and hybrid versions of
PSO algorithm have been developed for solving ELD problems. Valley et al.
[18], have presented PSO, its variants and their applications in solving
various issues of power systems in a very comprehensive way. AlRashidi et al.
[19], have presented another comprehensive survey considering the application
of different PSO algorithms in solving ELD problems. Lee et al. [20], have
discussed the merits and demerits of PSO in solving ELD problems of power
system operations. As a large number of publications involving the solution of
ELD problems using various PSO algorithms are available in the literature, so
a new literature review is needed to obtain a broad idea about the ability of
PSO in solving ELD problems in modern power systems prospectives.

This paper presents a comprehensive survey of the application of PSO in
solving ELD problems in electric power systems. Initially, ELD problem
formulation and the concept of the PSO algorithm have been discussed. After
that, this survey paper covers 14 years of publications from 2003 to 2016 and
discusses some of the important contributions available by reputed publishers.
The published papers have been classified into five different categories and
discussion related to their problem formulation, PSO methodology used, testing
of the technique for the formulated model, the output results, and its
effectiveness have been analyzed. These five categories are (i) single
objective economic dispatch, (ii) dynamic economic load dispatch, (iii)
economic dispatch with non-conventional sources, (iv) multi-objective
environmental/economic dispatch, and (v) economic dispatch of microgrids.

The remainder of this paper is organized as follows. In Section 2, details of
ELD problem formulation is described. In Section 3, the concept of the PSO
algorithm is discussed. Review of the application of PSO for solving ELD
problems is discussed in Section 4. Finally, conclusions are presented in
Section 5.

##  II Review of problem formulation of economic load dispatch problems

The ELD problem is formulated as an optimization problem of minimization of
total cost of power generation to meet a particular load demand subjected to
the constraints related to the generator’s power output.

###  II-A Conventional problem formulation

Mathematically, ELD problem can be formulated as [15]:

$FT=\min\sum_{i=1}^{n}FC_{i}(PG_{i})$ (1)

In eqn. (1), $FT$ is the total cost of generation, $PG_{i}$ is the output
power $i^{th}$ generator, $FC_{i}(\cdot)$ is the cost function of $i^{th}$
generator and $n$ is the number of generating unit in the system. The cost
function of a $i^{th}$ generator is expressed as [18];

$FC_{i}(PG_{i})=a_{i}+b_{i}PG_{i}+c_{i}PG_{i}^{2}$ (2)

In eqn. (2), $a_{i}$, $b_{i}$ and $c_{i}$ are coefficients of the cost
function of $i^{th}$ generator.

The objective function of ELD problem defined in eqn. (1) is subjected to
various constraints which are defined as follows [15, 18, 20];

####  II-A1 Requirement related to power balance

Mathematically, the requirement related to power balance of any power system
area is defined as follows [15];

$\sum_{i=1}^{n}PG_{i}=PD+PN_{loss}+SR$ (3)

In eqn. (3), $PN_{loss}$ is the total electric power loss in the transmission
network, $PD$ is the sum of total power demand of the area and $SR$ is some
excess power requirement which is known as spinning reserve. The total
electric power loss in the transmission network is defined as follows [15];

$PN_{loss}=\sum_{i=1}^{n}\sum_{j=1}^{n}PG_{i}B_{i,j}PG_{j}+\sum_{j=1}^{n}B_{01}PG_{j}+B_{00}$
(4)

In eqn. (4), $B_{00}$, $BB_{01}$ and $BB_{i,j}$ are the coefficients of power
loss in the transmission networks.

####  II-A2 Requirements related to power generation capacity of generating
unit

The power generation from any generator must be within their minimum and
maximum generation capacity. Boundary limits on each generator is expressed as
follows [15, 20];

$\textit{PG}_{i,min}\leq\textit{PG}_{i}\leq\textit{PG}_{i,max}$ (5)

In eqn. (5), $PG_{i,min}$ is the minimum power generation and $PG_{i,max}$ is
the maximum power generation limits of $i^{th}$ generator.

####  II-A3 Requirements related to generator ramp rate limits

The power output of generating units in certain interval are subjected to the
following set of constraints known as ramp rate limits and are expressed as
follows [15, 18, 20];

$\displaystyle PG_{i}-PG_{i,0}\leq UR_{i}$ (6) $\displaystyle
PG_{i,0}-PG_{i}\leq DR_{i}$ (7) $\max(PG_{i,min},PG_{i,0}-DR_{i})\leq
PG_{i}\leq\min(P_{i,max},PG_{i,0}+UR_{i})$ (8)

In eqn. (8), $UR_{i}$ is ramp-up, $DR_{i}$ is ramp-down and $PG_{i,0}$ is the
previous generator output of $i^{th}$ generator.

####  II-A4 Requirements related to prohibited operating zones of generator

The smooth power output within the minimum and maximum generation limits of
the generators is not possible. There are always some intervals in which power
generation are not available. This unavailable power intervals are known as
prohibited operating zones of generator and are expressed as follows [15, 20];

$\displaystyle\textit{PG}_{i,min}\leq\textit{PG}_{i}\leq\textit{PG}_{i,1}^{lower}$
(9)
$\displaystyle\textit{PG}_{i,j-1}^{upper}\leq\textit{PG}_{i}\leq\textit{PG}_{i,j}^{lower}$
(10)
$\displaystyle\textit{PG}_{i,PZ_{i}}^{upper}\leq\textit{PG}_{i}\leq\textit{PG}_{i,max}$
(11) $\forall j=2,3,...,PZ_{i}\ \forall i=1,2,...,n_{PZ}$

In eqns. (9)-(11), $PG_{i,j}^{lower}$ is the lower and $PG_{i,j}^{upper}$ is
the upper prohibited operating zone for $\forall i=1,2,...,n_{PZ}$ and
$\forall j=1,2,...,PZ_{i}$ for $i^{th}$ generator. Further, $PZ_{i}$ is the
number of the prohibited zones of $i^{th}$ generator and $n_{PZ}$ is the
number of generators with such zones.

####  II-A5 Requirements related to the spinning reserve of the area

Any power generating station must have a certain excess generation to supply
during emergency or peak loading conditions. A constraint is imposed to handle
the spinning reserve requirement of the system as follows [5];

$\sum_{i=1}^{n}SPG_{i}\geq SR$ (12)

In eqn. (12), $SPG_{i}=\max(PG_{i,max}-PG_{i},SPG_{i,max})$ is the reserve
contribution of $i^{th}$ generator, $SPG_{i,max}$ is the maximum reserve
contribution of $i^{th}$ generator and $SR$ is the surplus spinning reserve
capacity after load demand is met.

###  II-B Some additional cost functions

Many cost functions are available in the literature to express practical
effects affecting cost functions of generating units committed to supplying a
given load demand. In this work, some commonly used cost functions in the
literature are considered.

####  II-B1 Cost function with valve-point effects

Cost of generation of electricity varies with multiple valve opening and
closing. The effects of the valve-point of the cost function is expressed as
follows [15, 18, 20]:

$\displaystyle FC_{i}(PG_{i})=a_{i}+b_{i}PG_{i}+c_{i}PG_{i}^{2}+|e_{i}\times
sin(f_{i}$ (13) $\displaystyle\times(PG_{i,min}-PG_{i}))|$

In eqn. (13), $e_{i}$ and $f_{i}$ are coefficients related to the valve-point
effects of $i^{th}$ generator. Fig. 1 shows the characteristic of the cost
function of a generator with the valve-point effects.

Figure 1: A typical characteristic of cost function of generator with valve-
point effects.

####  II-B2 Cost function with multiple fuel input options

Modern generators are equipped to use multiple types of fuel options for
generating electric power. The cost function with multiple fuel input options
of generator is expressed as follows [18, 20];

$F_{i}(PG_{i})=\left\\{\begin{array}[]{ll}a_{i1}+b_{i1}PG_{i}+c_{i1}PG_{i}^{2},\\\
\qquad\qquad\ fuel\ 1,\ PG_{i,min}\leq PG_{i}\leq PG_{i,1}\\\
a_{i2}+b_{i2}PG_{i}+c_{i2}PG_{i}^{2},\\\ \qquad\qquad\ fuel\ 2,\ PG_{i,1}\leq
PG_{i}\leq PG_{i,2}\\\ ...\\\ a_{ik}+b_{ik}PG_{i}+c_{ik}PG_{i}^{2},\\\
\qquad\qquad\ fuel\ k,\ PG_{i,k-1}\leq PG_{i}\leq PG_{i,max}\\\
\end{array}\right.$

In the above equations, $a_{ik}$, $b_{ik}$ and $c_{ik}$ are the cost
coefficients of $i^{th}$ generator for $k^{th}$ type of fuel. A typical
characteristic of cost function with multiple fuel input options is shown in
Fig. 2.

Figure 2: A typical characteristic of cost function of generator with multiple
fuel input options.

####  II-B3 Cost function with valve-point effects and multiple fuel input
options

The electricity generation cost function with valve-point effects and multiple
fuel input options is expressed as follows [18, 20];

$F_{i}(PG_{i})=\left\\{\begin{array}[]{ll}a_{i1}+b_{i1}PG_{i}+c_{i1}PG_{i}^{2}+|e_{i1}\times
sin(f_{i1}\\\ \times(PG_{i,min}-PG_{i}))|,\\\ \qquad\qquad\ fuel\ 1,\
PG_{i,min}\leq PG_{i}\leq PG_{i,1}\\\
a_{i2}+b_{i2}PG_{i}+c_{i2}PG_{i}^{2}+|e_{i2}\times sin(f_{i2}\\\
\times(PG_{i,min}-PG_{i}))|,\\\ \qquad\qquad\ fuel\ 2,\ PG_{i,1}\leq
PG_{i}\leq PG_{i,2}\\\ ...\\\
a_{ik}+b_{ik}PG_{i}+c_{ik}PG_{i}^{2}+|e_{ik}\times sin(f_{ik}\\\
\times(PG_{i,min}-PG_{i}))|,\\\ \qquad\qquad\ fuel\ k,\ PG_{i,k-1}\leq
PG_{i}\leq PG_{i,max}\\\ \end{array}\right.$

In the above eqns., $e_{ik}$ and $f_{ik}$ are the cost coefficients
corresponding to valve-point effects and multiple fuel input options of
$i^{th}$ generator for $k^{th}$ type of fuel.

####  II-B4 Cost function with emissions of harmful gases in the environment

Thermal power plants emit harmful gases in the environment constantly [21,
22]. A penalty is imposed by the authorities to limit emission to the minimum
level. The cost function representing this effect is expressed as follows;

$\displaystyle
E_{i}(PG_{i})=\alpha_{i}+\beta_{i}PG_{i}+\gamma_{i}PG_{i}^{2}+\zeta_{1i}\times
exp(\lambda_{1}\times PG_{i})+$ (14) $\displaystyle\zeta_{2i}\times
exp(\lambda_{2}\times PG_{i})$

In eqn. (14), $\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$, $\zeta_{1i}$,
$\zeta_{2i}$, $\lambda_{1}$ and $\lambda_{2}$ are the coefficients of emission
function.

####  II-B5 Cost function in the presence of wind turbines

The objective cost function in the presence of wind turbines is normally
expressed by the following equation as expressed in [23];

$FC_{W}=\sum_{t=1}^{T}\sum_{i=1}^{nw}fw_{i}(PG_{i}^{t})S_{i}^{t}+C_{i}S_{i}^{t}(1-S_{i}^{t-1})$
(15)

In eqn. (15), $fw_{i}(\cdot)$ is the cost function of $i^{th}$ wind turbine,
$S_{i}^{t}$ is state of $i^{th}$ generator having value either 0 or 1 (0 is
OFF 1 is ON) at time step $t$, $T$ is the maximum time step, $nw$ is the
number of wind turbines and $C_{i}$ is cold start cost of $i^{th}$ generator.

####  II-B6 Cost function of hydroelectric power generation

The cost function of hydroelectric power generation can be expressed as
follows [5];

$FC_{H}=\gamma_{h}\sum_{t=1}^{T}fh_{i}(PG_{i}^{t})$ (16)

In eqn. (16), $FC_{H}$ is the total cost of hydroelectric power generation,
$fh_{i}(\cdot)$ is a function representing water discharge during
hydroelectric power generation and $\gamma_{h}$ is a factor which converts the
rate of water discharge term in equivalent cost. Hydroelectric power
generation is subjected to two main constraints as follows;

a) Reservoir water head limits:

$\textit{H}_{min}\leq\textit{H}\leq\textit{H}_{max}$ (17)

In eqn. (17), $H_{min}$ is the minimum and $H_{max}$ is the maximum water head
$H$ of the reservoir.

b) Water discharge limits:

$\textit{Q}_{min}\leq\textit{Q}\leq\textit{Q}_{max}$ (18)

In eqn. (18), $Q_{min}$ is the minimum and $Q_{max}$ is the maximum possible
water discharge $Q$ of the hydroelectric turbine.

##  III Review of particle swarm optimization

Particle swarm optimization (PSO) is a swarm intelligence based nature
inspired meta-heuristic optimization technique developed by James Kennedy and
Russell Eberhart [24] in 1995. It is inspired by the social behavior of birds
flocking. The swarm is made of potential solution known as particles. Each
particle flies in the search space with the certain velocity and keeps a
memory of their best position held so far, and the swarm keeps a memory of the
overall best position of the swarm obtained by any particle held during the
fly. The next position of each particle in the search space is decided by the
present movement, best individual position and the best position of the swarm
obtained so far. The present movement of the particle is scaled by a factor
called inertial weight $w$ whereas individual best, and the overall best
experiences are scaled by acceleration factors $c_{1}$ and $c_{2}$
respectively. Also, these experiences are perturbed by multiplying with two
randomly generated numbers $r_{1}$ and $r_{2}$ between [0,1] respectively. The
classical PSO algorithm is represented by two mathematical equation described
below.

Let us assume that the initial population (swarm) of size $N$ and dimension
$D$ is denoted as $\bf{X}$ = [$\bf{X}$1,$\bf{X}$2,…,$\bf{X}$N]T, where
${}^{\prime}T^{\prime}$ denotes the transpose operator. Each individual
(particle) $\bf{X}$i $(i=1,2,...,N)$ is given as
$\bf{X}$i=$[X_{i,1},X_{i,2},...,X_{i,D}]$. Further, the initial velocity held
by the swarm is denoted as $\bf{V}$ = [$\bf{V}$1,$\bf{V}$2,…,$\bf{V}$N]T.
Thus, the velocity particle $\bf{X}$i $(i=1,2,...,N)$ is given as
$\bf{V}$i=$[V_{i,1},V_{i,2},...,V_{i,D}]$. Here, the index index $j$ varies
from 1 to $D$ and the index $i$ varies from 1 to $N$. The detailed algorithms
of various methods are described below for the purpose of completeness [25].

${V}_{i,j}^{k+1}=w\times{V}_{i,j}^{k}+c_{1}r_{1}\times({Pbest}_{i,j}^{k}-{X}_{i,j}^{k})+c_{2}r_{2}\times({Gbest}_{j}^{k}-X_{i,j}^{k})$
(19) $X_{i,j}^{k+1}=X_{i,j}^{k}+V_{i,j}^{k+1}$ (20)

In eqn. (19), $\textit{Gbest}_{j}^{k}$ represents $j^{th}$ component of the
best particle of the population and $\textit{Pbest}_{i,j}^{k}$ represents
personal best $j^{th}$ component of $i^{th}$ particle up to iteration $k$.
Fig. 3 shows the graphical representation of the PSO search mechanism in
multidimensional space.

Figure 3: Graphical representation of PSO search mechanism in the search
space.

Later, many PSO variants have been developed to improved results. Some of them
are discussed below.

###  III-A Time varying inertial weight of PSO

Earlier, inertia weight $w$ of the PSO was considered a fixed value between
[0.4, 0.9]. Later, it was found that varying inertia with time (iteration)
provides faster convergence. Normally, varying inertia with iteration is
expressed as follows;

$w=w_{max}-k\times(w_{max}-w_{min})/\textit{Maxite}$ (21)

In eqn. (21), $k$ is the current iteration count whereas $Maxite$ is the
maximum iteration count set. The value of inertial factor $w$ is used to
decrease linearly from $w_{max}$ to $w_{min}$ as the iteration increases.

###  III-B Time varying acceleration factors of PSO

Earlier, the values of acceleration factors $c_{1}$ and $c_{2}$ of PSO were
considered to be equal to 2, but later, it was observed that varying
acceleration factors with time (iteration) provide a better solution. Time
varying acceleration coefficients (TVAC) of PSO are expressed as follows [26];

$\displaystyle c_{1}=c_{1,max}-k\times(c_{1,max}-c_{1,min})/\textit{Maxite}$
(22) $\displaystyle
c_{2}=c_{2,min}+k\times(c_{2,max}-c_{2,min})/\textit{Maxite}$ (23)

In eqn. (22), $c_{1,min}$ is the minimum and $c_{1,max}$ is the maximum limits
of acceleration factor $c_{1}$ whereas in eqn. (23), $c_{2,min}$ is the
minimum and $c_{2,max}$ is the maximum limits of acceleration factor $c_{2}$.

###  III-C Constriction factor PSO

In [27], Maurice Clerc and James Kennedy introduced the concept of
constriction factor to PSO to solve problems of multidimensional search space
efficiently. PSO with constriction factor shows great potential in solving
very complex problems effectively. Mathematically, velocity equation of PSO
with constriction factor is represented as follows;

${V}_{i,j}^{k+1}=K[{V}_{i,j}^{k}+c_{1}r_{1}\times({Pbest}_{i,j}^{k}-{X}_{i,j}^{k})+c_{2}r_{2}\times({Gbest}_{j}^{k}-X_{i,j}^{k})]$
(24)

In eqn. (24) $K$ is known as the constriction factor of PSO, and is defined as
follows;

$K=2\kappa/|2-\phi-\sqrt{\phi^{2}-4\phi}|$ (25)

where $\phi=c_{1}+c_{2}>4$ and $\kappa\in[0,1]$ [28, 29].

####  III-C1 PSO algorithm

A typical PSO algorithm is given for completeness as discussed in [25] as
follows

  1. 1.

Set $w_{min}$, $w_{max}$, $c_{1}$ and $c_{2}$ parameters

  2. 2.

Initialize positions $\bf{X}$ and velocities $\bf{V}$ of each particle of the
population

  3. 3.

Evaluate particles fitness i.e.,
${F}$${}_{i}^{k}=f(\bf{X}$${}_{i}^{k}),\forall i$ and find the index $b$ of
the best particle

  4. 4.

Select $\bf{Pbest}$${}_{i}^{k}=\bf{X}$${}_{i}^{k},\forall i$ and
$\bf{Gbest}$${}^{k}=\bf{X}$${}_{b}^{k}$

  5. 5.

Set iteration count $k=1$

  6. 6.

$w=w_{max}-k\times(w_{max}-w_{min})/\textit{Maxite}$

  7. 7.

Update velocity and position of particles
${V}$${}_{i,j}^{k+1}=w\times{V}$${}_{i,j}^{k}+c_{1}r_{1}({Pbest}$${}_{i,j}^{k}-{X}$${}_{i,j}^{k})+c_{2}r_{2}({Gbest}$${}_{j}^{k}-X_{i,j}^{k});\;\forall
j$ and $\forall i$ $X_{i,j}^{k+1}=X_{i,j}^{k}+V_{i,j}^{k+1};\;\forall j$ and
$\forall i$

  8. 8.

Evaluate the fitness of updated particles i.e.,
$F_{i}^{k+1}=f(\bf{X}$${}_{i}^{k+1}),\forall i$ and find the index $b1$ of the
best particle at this iteration

  9. 9.

Update Pbest of each particle of the population $\forall i$ If
$F_{i}^{k+1}<F_{i}^{k}$ then $\bf{Pbest}$${}_{i}^{k+1}=\bf{X}$${}_{i}^{k+1}$
else $\bf{Pbest}$${}_{i}^{k+1}=\bf{Pbest}$${}_{i}^{k}$

  10. 10.

Update Gbest of the population If $F_{b1}^{k+1}<F_{b}^{k}$ then
$\bf{Gbest}$${}^{k+1}=\bf{Pbest}$${}_{b1}^{k+1}$ and set $b=b1$ else
$\bf{Gbest}$${}^{k+1}=\bf{Gbest}$k

  11. 11.

If $k>\textit{Maxite}$ then go to step 12 else $k=k+1$ and go to step 6

  12. 12.

Optimum solution is obtained as $\bf{Gbest}$k

Fig. 4 shows the flowchart of the PSO algorithm discussed above.

Figure 4: Flowchart of the PSO algorithm.

####  III-C2 Parameter selection of PSO

Parameter selection of PSO is of extreme importance. Many researchers have
given various sets of parameters of the algorithm in the literature. The
following parameters of the PSO algorithms are used commonly for solving ELD
problems in power systems [30, 25]:

  * •

Population size: 10 to 50

  * •

Initial velocity: 10 % of position

  * •

Inertial weight: 0.9 to 0.4

  * •

Acceleration factors ($c_{1}$ and $c_{2}$): 2 to 2.05

  * •

For constriction factors $c_{1}$ and $c_{2}$: 2.025 to 2.1

  * •

Maximum iteration (Maxite): 500 to 10000

##  IV Review of the application of PSO for solving ELD problems

The most relevant research papers, in solving ELD problems using PSO,
published in years 2002 through 2016, are considered in this article for
presentation and discussion. It has been identified five important and related
areas of ELD, and the relevant papers published by well-known publishers in
the general area of economic dispatch using PSO are classified under one of
these five categories. The identified categories are as follows:

  * •

Single-objective economic load dispatch (SOELD)

  * •

Dynamic economic load dispatch (DELD)

  * •

Economic load dispatch with non-conventional sources (ELDNCS)

  * •

Multi-objective environmental/economic load dispatch (MOELED)

  * •

Economic load dispatch of micro-grids (ELDMG)

At the end of each category, a table is given which gives a brief idea of each
of the research papers discussed in detail.

Fig. 5 shows a pie distribution of various publications considered in each of
the category discussed above. This pie chart has six division (five for the
categories discussed and one for the rest of the discussion including the
formulation ELD problem and developed of the classical PSO algorithm). There
are 41, 8, 6, 16 and 17 papers under SOELD, DELD, ELDNCS, MOELD, and ELDMG
respectively whereas 30 publications are under general discussion about ELD
problems and PSO algorithm.

Figure 5: Distribution of the publications in the different areas of economic
dispatch using PSO.

Fig. 6 shows the year-wise development of all the publications considered in
this work. From this figure, it can be observed that there are three types of
publication considered which are a) books, b) conference papers and c) journal
papers. Most of the paper considered are from various journals.

Figure 6: Considered Publications for ELD problems using PSO yearly.

Fig. 7 shows classification of all the publication considered in this paper.
In Fig. 7(a), classification of the entire publications have been shown
concerning their type as shown in Fig. 6 whereas in Fig. 7(b), classification
of all the journal papers have been shown as per the database from where the
papers have been considered. Under this aegis, 92 journal and 21 conference
papers and five books have been considered in this work. Further, out of 92
journal papers, 53 are from $Elsevier$, 27 are from $IEEE$, nine are from
$IET$, two are from $Taylor\&Francis$, and one is from $Springer^{\prime}s$
database.

Figure 7: Classifications of all the publications considered in this paper

###  IV-A Single objective economic load dispatch

The single objective economic load dispatch (ELD) deals with only the cost
minimization of the generating system while satisfying all the constraints to
dispatch power economically. In this section, research papers which possess
objective functions of non-smooth, and non-convex, with or without valve-point
loading effects, with constraints of generation limits of individual units,
multi-fuel input options, ramp rate limits and prohibited operating zones have
been analyzed. The problem formulation includes one or more or a combination
of two or more practical constraints mentioned above. The different types of
PSO algorithms have been used in solving the formulated problems. These PSO
algorithms can be categorized in the following four categories:

  * •

ELD using classical PSO

  * •

ELD using a modified/improved/enhanced version of PSO

  * •

Fuzzy controlled PSO

  * •

ELD using a hybrid version of PSO.

Now, research findings of various research papers of each class are explained
in detail in the following subsections.

####  IV-A1 ELD using classical PSO

Park et al. [31], introduced a new approach to the PSO algorithm for solving
ELD problems considering non-smooth objective functions. Further, a new
constraint handling (CH) approach has been introduced. The suggested CH
approach can satisfy the constraints within a reasonable computation time to
provide faster convergence.

Gaing [32], used a basic version of PSO algorithm with a new solution process
for solving ELD problems considering generator constraints. The following
generator constraints have been considered: a) ramp-rate limits and b)
prohibited operating zones. Also, the transmission network losses have also
been considered.

####  IV-A2 ELD using modified/improved/enhanced version of PSO

Park et al. [33], proposed a modified PSO (MPSO) algorithm for solving various
types of ELD problems. The cost functions of the ELD problems have been
formulated in three ways, a) smooth cost function, b) non-smooth cost function
having valve-paint effects and c) non-smooth cost function having multiple
fuel input options. In the MPSO algorithm, a strategy has been adopted to
enhance the convergence rate and to reduce the search space dynamically.

Selvakumar and Thanushkodi [34], suggested a new PSO algorithm named NPSO-LRS
based on a Local Random Search (LRS) approach for solving the ELD problem.
Also, the NPSO-LRS algorithm bifurcates the cognitive behavior into a bad and
good experience components. This modification significantly improves the
exploration ability of the particles to obtain the global optimum solution.
The non-convex ELD problems considered include: a) valve-point loading effects
(EDVL), b) prohibited operating zone (EDPOZ), and c) valve-point loading
effects and multiple fuel (EDVLMF) options.

Cai et al. [35], described a chaotic PSO algorithm (CPSO) by using chaotic
local search and an adaptive inertia weight for solving ELD problems
considering generator constraints. Also, two CPSO algorithms were presented
based on the Tent and logistic equations.

Selvakumar and Thanushkodi [36], proposed a new anti-predatory PSO (APSO)
algorithm for solving the EDVL and EDVLMF problems. In PSO, the foraging
activity includes the social and cognitive behaviors of the swarm of birds. In
the proposed approach, the anti-predatory activity and the foraging activity
have been used to help the particles to escape from the predators in the
classical PSO. This improves the ability of the particle to escape from local
minima and to explore the entire search space efficiently.

Saber et al. [37], introduced a modified PSO (MPSO) algorithm for solving ELD
problems of a higher order cost function. In the MPSO algorithm, a new
velocity vector has been considered. To analyze the importance of the higher
order cost function, sensitivity studies of higher order cost polynomials have
been performed for ELD problems.

PSO with crazy particles (PSO-crazy) was used for solving non-convex ED (NCED)
problems by Chaturvedi et al. [38]. In the PSO-crazy algorithm, the velocities
of crazy particles are randomly adjusted to maintain the momentum of the
particles to avoid saturation in the feasible reason.

Chaturvedi et al. [39], developed time-varying acceleration coefficients PSO
(PSO_TVAC) algorithm for solving NCED problems of power systems.

Park et al. [40], introduced an improved PSO (IPSO) algorithm for solving non-
convex ELD problems. To escape from a local minimum and to enhance the global
search ability of particles, chaotic sequence based inertia-weight and new
crossover operation have been proposed in the IPSO algorithm. Further,
efficient equality and inequality constraint handling approach has been
introduced.

Meng et al. [41], discussed a quantum-inspired PSO (QPSO) algorithm for
solving various types of ELD problems of power systems. The proposed QPSO
employs self-adaptive probability selection and chaotic sequence mutation.

Zhisheng [42], developed a quantum-behaved PSO, namely QPSO algorithm for
solving ELD problems of power systems.

Neyestani et al. [43], introduced a modified PSO (MPSO) algorithm for solving
various types of ELD problems. In this algorithm, an attempt has been made to
control the diversity of small a population to avoid premature convergence.

Subbaraj et al. [44], proposed modified stochastic acceleration factors based
PSO (PSO-MSAF) algorithm for solving large-scale ELD problems with various
generator constraints and transmission losses.

Safari et al. [45], proposed a new iteration PSO (IPSO) algorithm for solving
various types of ELD problems.

Saber et al. [46], proposed a new hybrid PSO and DE optimization (PSDEO)
algorithm to solve ELD problems of higher order non-smooth cost functions with
various practical constraints. The modified PSDEO combines the advantages and
disadvantages of both algorithms. In the algorithm, PSO exploits and DE
explores the search space to obtain the global optimum solution efficiently.

Dieu et al. [47], introduced a newly improved PSO (NIPSO) algorithm for
solving non-smooth ELD problems. The proposed NIPSO considers time-varying
acceleration coefficients (TVAC), a variation of inertia weight with a sigmoid
function, particle guidance by pseudo-gradient and quadratic programming (QP)
to obtain the initial condition.

Chalermchaiarbha and Ongsakul [48], proposed a new PSO algorithm called
stochastic weight trade-off particle swarm optimization (SWT_PSO) for solving
the ELD problems of power systems. In this algorithm, stochastic inertial
weight and acceleration factors are adjusted at each iteration to increase
diversity.

Hosseinnezhad and Babaei [49], proposed $\theta$-PSO algorithm for solving
various types of ELD problems. In the proposed algorithm, the velocity
component of the conventional PSO has been replaced by a phase angle vector.
This approach also takes care of various constraints related to transmission
loss and generator operational limitations.

Hosseinnezhad et al. [50], proposed species based quantum PSO (SQPSO)
algorithm for solving different types of ELD problems of power systems. In
this SQPSO algorithm, particles are treated as a group at each iteration in
addition to the QPSO algorithm. This approach improves the exploration ability
of the particles which leads to achieving the global best optimum solution.

Sun et al. [51], discussed random drift PSO (RDPSO) for solving various types
of ELD problems of power systems. The concept of RDPSO is inspired by the free
electron model of conductors and uses some special evolution equation which
leads the particles to reach the optimum global point in the search space.

Basu [52], proposed a modified PSO (MPSO) algorithm for solving different
kinds of ELD problems of power systems. In this MPSO, a Gaussian random number
generator has been used in calculating the new velocity vector of each
particle. This approach improves the exploration ability of the particles and
provides much better results.

Hsieh and Su [53], proposed a new PSO algorithm based on Q-learning for
solving the ELD problems. In this algorithm, Q-learning and PSO approaches
have been integrated to form the QSO algorithm. In QSO algorithm, the best
particle is considered to be the one whose cumulative value of the objective
function is the best which is unlike the conventional PSO where the best
particle is the one whose current value of the objective function is the best
at that particular iteration.

Jadoun et al. [54], proposed a modified dynamically controlled PSO (DCPSO) for
solving multiple-area ELD problems of power systems. In DCPSO algorithm,
velocity vectors of the particles are controlled by introducing exponential
constriction functions. This approach helps the particles to explore the
entire search space rapidly. In multiple-area ELD modeling, tie-lines have
been proposed to exchange power as per the requirement.

####  IV-A3 Fuzzy controlled PSO

Niknam [55], developed a fuzzy adaptive PSO utilizing the Nelder-Mead (FAPSO-
NM) algorithm for solving ELD problems. Here at each iteration, the NM
algorithm served the objective of a local search for the FAPSO algorithm.
Thus, FAPSO-NM improves FAPSO performance significantly.

Niknam et al. [56], introduced a novel method, named Fuzzy Adaptive Modified
PSO (FAMPSO), for solving non-convex ED problems. A new mutation operator has
been introduced to take care of premature convergence. Further, inertia weight
and acceleration factors of PSO are tuned using the fuzzy system.

Mahdad et al. [57], proposed a fuzzy controlled parallel PSO (FCP-PSO)
algorithm for solving large-scale non-convex ED problems. In the proposed
algorithm, PSO parameters are adjusted dynamically using fuzzy rules. The
parallel execution of PSO executed in a decomposed network procedure is found
to explore the local search space effectively.

Niknam et al. [58], described a new adaptive PSO (NAPSO) algorithm suitable
for solving ELD problems with multi-fuel input options and prohibited
operation zones of generators. Further, a new mutation approach has been used
in adaptive PSO (APSO) to escape particles from local minima and search the
global optimum. In the APSO algorithm, fuzzy rules have been used to tune the
inertial weight of PSO whereas a self-adaptive adjustment approach has been
used to adjust other parameters of PSO, such as the cognitive and social
parameters.

Niknam et al. [59], developed a new hybrid algorithm using variable DE (VDE)
and fuzzy adaptive PSO (FAPSO) for solving the ELD problems. The proposed
algorithm named FAPSO-VDE considers, the fuzzy rules to adjust parameters
adaptively, the PSO to maintain population diversity of the population and the
DE to optimize the problem.

####  IV-A4 ELD Using Hybrid Version of PSO

Chen et al. [60], proposed a new hybrid PSO-RLD algorithm by combining PSO
with recombination and dynamic linkage discovery (RLD). The RLD employ a
selection operator to adapts the linkage configuration for any type of
objective function of the optimization problem. Further, the recombination
operator cooperates with PSO by using the configurations which are discovered.

Coelho and Lee [61], suggested a combination of chaotic sequences and Gaussian
probability distribution functions with PSO in solving ELD problems. The
chaotic sequence with logistic map helps the particles to escape from local.
The chaotic variables can travel over the whole search space to explore the
possibility of the global optimum solution.

Chaturvedi at el. [62], proposed time-varying acceleration coefficients (TVAC)
to PSO to develop a self-organizing hierarchical PSO (SOH_PSO) algorithm. In
this algorithm, the problems of stagnation and premature convergence have been
addressed by utilizing reinitialization of velocity vectors and the TVAC
respectively. These strategies provide a high-quality, robust solution
efficiently even for non-smooth and discontinuous cost functions.

Kuo [63], proposed a hybrid optimization algorithm using simulated annealing
and PSO called SA-PSO. The stochastic search ability of this algorithm pushes
the particles to be in the feasible reason for the search space and thus the
solution time reduces drastically with a high-quality solution.

Coelho et al. [64], proposed a new hybrid chaotic PSO with implicit filtering
(HPSO-IF) algorithm for solving ELD problems. In the proposed algorithm, the
chaotic sequence provides a high exploration ability in the whole search
space, whereas, the fine-tuning of the final results are obtained by the IF in
the PSO.

Vlachogiannis et al. [65], introduced a new hybrid optimization algorithm
called improved coordinated aggregation-based PSO (ICA-PSO). In this
algorithm, all the particles can be attracted by the other particles having
better fitnesses in the population than its own except the particle having the
best fitness. Further, in this algorithm, the size of the population has been
adjusted adaptively.

Victoire and Jeyakumar [66], presented a hybrid algorithm for solving ELD
problems of power systems. The proposed algorithm integrates sequential
quadratic programming (SQP) and PSO, named PSO-SQP. In this algorithm, the SQP
has been used as fine-tuning of each improved result in the PSO run.

Sun et al. [67], proposed a modified QPSO with differential mutation (DM)
method, named QPSO-DM, for solving ELD problems.

Kumar et al. [68], presented a hybrid multi-agent based PSO (HMAPSO) algorithm
for solving ELD problems considering valve point effect. The proposed
technique integrates a) Multi-agent system (MAS), b) deterministic search, c)
PSO and d) bee decision-making process.

Chakraborty et al. [69], proposed a new hybrid PSO algorithm, which is
inspired by quantum mechanics for solving various types of ELD problems. The
developed algorithm is named hybrid quantum inspired PSO (HQPSO). In HQPSO
algorithm, velocity and position vectors of particles are adjusted in a more
diverse manner to explore the entire search space to find the global best
solution. Also, a special feature has been introduced which increases particle
size from single to multiple.

Sayah and Hamouda [70] proposed two new hybrid methods, a) by combining
evolutionary programming (EP) and efficient PSO (EPSO) termed EP-PSO and b) by
combining neural network (NN) and efficient PSO (EPSO) termed as NN-EPSO, for
solving ELD problems considering valve-point loading effects.

Abarghooee et al. [71], proposed a hybrid algorithm using an enhanced
gradient-based optimization method and a simplified PSO for solving ELD
problems of power systems. An attempt has been made to obtain the global or
near-global, fast and robust solution in highly constrained ELD problems.

Table I gives various details like type of algorithm, modeling of ELD problem,
the size of the test system, etc., for each of the papers reviewed above in
this subsection.

TABLE I: Single Objective Economic Load Dispatch

Type of PSO | Type of the ELD problem | Test system | References
---|---|---|---
PSO with dynamic process | SCF, NSCF | 3-unit | [31]
PSO | Non-linear with RRL, POZ, NSCF | 6, 15, 40-unit | [32]
MPSO | SCF, NSCF with VL effects, NSCF with VL effects and MF input | 3, 40-unit | [33]
NPSO-LRS | Non-convex with EDPO, EDVL, EDVLMF | 6, 10, 40-unit | [34]
Chaotic PSO (CPSO) | Non-linear Characteristic of Generators, VL, POZ, RRL, NSCF | 6, 15-unit | [35]
APSO | EDVL, EDVLMF | 10, 40-unit | [36]
MPSO | ELD with Higher Order Cost Function | 6, 15-unit | [37]
PSO_crazy | NCED, NCCF | 3, 6-unit | [38]
PSO_TVAC | NCED, NCCF | 13, 15, 38-unit | [39]
IPSO | NCCF, POZ, RRL | 140-unit Korean System | [40]
Quantum-inspired PSO (QPSO) | NSCF | 3, 13, 40-unit | [41]
Quantum-behaved PSO (QPSO) | ELD | 3, 13-unit | [42]
MPSO | NSCF with RRLs and POZs, EDVLMF | 6-unit | [43]
PSO-MSAF | Large Scale ED Problems with POZ, RRL, Transmission Losses | 15,40-unit | [44]
Iteration PSO (IPSO) | Non-continuous, Non-smooth with POZ, RRL | 6, 15-unit | [45]
PSDEO | Higher Order NSCF | 6, 15-unit | [46]
NIPSO Based on PSO-TVAC | Non-convex | 13, 40-unit | [47]
SWT_PSO | ELD | 10, 15, 40 | [48]
$\theta$-PSO | ELD with generator constraints | 6, 13, 15, 40 gen. | [49]
SQPSO | ELD with various other constraints | 6, 15, 40 gen. | [50]
RDPSO | ELD with generator constraints | 6, 15, 40 gen. | [51]
MPSO | ELD with generator constrants | 3, 6, 40, 140 gen. | [52]
QSO | ELD problems | 3, 40 gen. | [53]
DCPSO | Multiple area ELD | 4, 40, 140 gen. | [54]
FAPSO-NM | Non-linear, Non-smooth, and Non-convex with VL Effects | 13, 40-unit | [55]
FAMPSO | Non-convex Economic Dispatch (NCED) | 13, 40-unit | [56]
FCP-PSO | Large Scale Non-convex ED with POZs | 40-unit | [57]
NAPSO, Fuzzy | EDLV, EDVLMF and POZs | 6, 10, 15, 40, 80-unit | [58]
FAPSO-VDE | Non-convex ED with Valve-point Loading Effects | 13, 40-unit | [59]
PSO_RLD | ELD | 3, 40-unit | [60]
PSO, Gaussian and Chaotic Signals | Non-linear Generating Characteristics, RRL, POZ | 15, 20-unit | [61]
SOH_PSO | Non-convex | 6, 15, 40-unit | [62]
SA-PSO | Non-linear Cost Function | 6, 13, 15, 40-unit | [63]
HPSO-IF | Valve-point Loading Effects | 13-unit | [64]
ICA-PSO | ELD | 6, 13, 15, 40-unit | [65]
PSO-SQP | Valve-point Loading Effects | 3, 13, 40-unit | [66]
QPSO-DM | NSCF, Non-linear Characteristics of Generators, RRL, POZ | 6, 15-unit | [67]
HMAPSO | ED with Valve-point Loading Effects | 13, 40-unit | [68]
HQPSO | ELD with various constraints | 6, 10, 15, 40 gen. system | [69]
NN-EPSO | EDVL | 13, 40-unit | [70]
EGSSOA | POZ, RRL, EDVL, EDVLMF and Transmission Network Losses | 10, 15, 40, 80-unit | [71]
ED: economic dispatch; VL: valve-point loading; POZ: prohibited operating
zones; RRL: ramp rate limits
NSCF: non-smooth cost function; NCED: non-convex economic dispatch; NCCF: non-
convex cost function
NCCF: non-convex cost function; EDVL: ED with valve-point loading; EDVLMF:
EDVL and multiple fuels
EDVLMF: EDVL and multiple fuels; MF: multiple fuels; MF: multiple fuels; SCF:
smooth cost function
EDPO: ED with prohibited operation; ELD: economic load dispatch; ELDVL: ELD
with valve-point loading.

###  IV-B Dynamic economic load dispatch

The dynamic economic dispatch problem includes dynamic characteristics or
parameters such as spinning reserve constraints, etc., while solving the
problem. The dynamic economic dispatch (DED) is the real-time problem in any
power system [72]. PSO based DED tasks are analyzed to attract the attention
of researchers in this particular area.

Victoire et al. [73], proposed a hybrid optimization algorithm considering PSO
and SQP for solving reserve constrained DED problems of generator considering
valve-point effects. In the proposed hybrid algorithm, the SQP is used as a
local optimizer to fine-tune the reason for the solution for the PSO run.
Thus, the PSO works as the main optimizer whereas the SQP guides the PSO to
obtain better results for solving very DED problems.

Panigrahi et al. [74], proposed a novel adaptive variable population PSO
approach to DED problem of power systems. In the proposed DED model, various
system constraints have been considered such as transmission losses, ramp rate
limits, prohibited operating zones, etc.

Baskar and Mohan [75], suggested an improved PSO (IPSO) algorithm suitable to
solve security constraints ELD problems. In the proposed IPSO algorithm
constriction factor approach (CFA) has been considered to update the velocity
equation of PSO. Security constraints in the paper include bus voltage and
line flow limits.

Baskar and Mohan [76], proposed an improved PSO (IPSO) algorithm suitable to
solve contingency constrained ELD problems. In the proposed IPSO algorithm
constriction factor approach (CFA) with eigenvalue analysis has been
considered to update the velocity equation of PSO. In the proposed problem
formulation, a twin objective a) minimization of severity index and b)
minimization of fuel cost have been considered.

Wang et al. [77], introduced an improved chaotic PSO (ICPSO) algorithm
suitable for solving DED problems considering valve-point effects of the
generators. In ICPSO algorithm, premature convergence has been controlled
using chaotic mutation to improve PSO results. Further, effective constraints
handling strategies have been proposed.

Wang et al. [78], presented a chaotic self-adaptive PSO (CSAPSO) algorithm
suitable to solve dynamic DED problems considering valve-point effects of the
generators. In the presented algorithm, an approach has been used to adjust
velocity dynamically. Further, a chaotic local search has been used to
overcome premature convergence of the algorithm. Also, a random adjustment
strategy has been incorporated to handle constraint violations effectively.

Niknam and Golestaneh [79], proposed an enhanced adaptive PSO (EAPSO)
algorithm suitable to solve DED problems. In DED modeling, transmission
network losses, ramp-rate limits of generating units and valve-point effects
have been considered. In the proposed algorithm, tuning of social and
cognitive terns have been proposed to be accomplished dynamically and
adaptively. Also, linearly varying inertial weight has been considered.

Table II gives various details like type of algorithm, modeling of ELD
problem, the size of the test system, etc., about each paper reviewed above in
this subsection.

TABLE II: Dynamic Economic Load Dispatch

Type of PSO | Type of the ELD problem | Test system | References
---|---|---|---
Hybrid PSO with SQP | Reserve Constrained DED | 10-unit | [73]
Adaptive PSO (APSO) | DED with Transmission Losses, RRLs, POZs, NSCF | 3, 6, 15-unit | [74]
IPSO, CFA | Security Constraints DED | IEEE 14-bus, 66-bus Indian Utility | [75]
IPSO, CFA | Contingency Constraints ELD (CCELD) | IEEE 30-bus, IEEE 118-bus | [76]
ICPSO | DED with Valve-point Effects | 10-unit, 30-unit | [77]
CSAPSO | DED with Valve-point Effects | 10-unit, 30-unit | [78]
EAPSO | DED with RRLs and with Valve-point Effects and | 10-unit, 30-unit | [79]
| Transmission Losses |  |
DED: dynamic economic dispatch; TL: transmission line;
CCELD: contingency constrained economic load dispatch.

###  IV-C Economic load dispatch with con-conventional sources

Non-conventional resources like wind energy, solar energy, tidal energy, etc.,
considered while solving ELD problems in power systems. These energy sources
have almost no fuel cost but may suffer power quality problems. A large number
of researchers are working on this issue to make these sources reliable to
supply electric energy and synchronized to the grid so that fuel consumption
of conventional sources can be reduced considerably [80]. In this section, the
research papers are analyzed considering non-conventional sources along with
conventional sources, to dispatch electric power economically using various
PSO algorithms.

Wang and Singh [81], proposed a modified multi-objective PSO (MPSO) algorithm
suitable for solving bi-objective ED problems with wind penetration in the
system. Fuzzy rules have been applied to control wind penetration into the
system which creates a security problem in the system. A compromise between
economic and security requirements has been considered in this paper to
achieve both the objectives (economic and security).

Li and Jiang [82], proposed PSO algorithm-based model to evaluate and lower
the risk arises due to high wind power penetration and its variability into
the system. In the proposed model, integrated risk management (IRM) and value
at risk (VaR) have been used to access the risk and to establish an optimum
trade-off between the risk and the profit in the system operations.

Wu et al. [83], proposed PSO algorithm based ED problems considering combined
heat and power (CHP) system consisting thermal, waste heat boiler, gas boiler,
fuel cell, PV, wind turbine, battery, and electric load. The random features
of PV power, wind power, thermal and electrical load have been handled using
chance-constrained programming (CCP).

Firouzi et al. [84], proposed a fuzzy self-adaptive learning particle swarm
optimization (FSALPSO) algorithm and dynamic economic emission dispatch with
wind power and load uncertainties. A roulette wheel technique has been used to
model wind power and load uncertainties to generate various scenarios. Also, a
fuzzy adaptive approach has been considered to tune algorithm parameters.

Table III gives various details like type of algorithm, modeling of ELD
problem, the size of the test system, etc., about each of the papers reviewed
above in this subsection.

TABLE III: Economic Load Dispatch with Non-Conventional Sources

Type of PSO | Type of the ELD problem | Test system | References
---|---|---|---
MPSO, Fuzzy | ED and Security Impact (Wind and Thermal) | IEEE 30-bus 6-generator | [81]
MOPSO, Fuzzy | ELD and Security Impact (Wind and Thermal) | IEEE 30-bus 6-generator | [85]
PSO, CCP | Combined Heat and Power Dispatch of micro-grid | A typical 6-generator System | [83]
PSO | Lowering the risk (with Wind Power) | IEEE 30-bus 6-generator, Shanghai Network | [82]
FSALPSO, PSO Variants | DEED in presence of wind generation | A typical 10-generator system | [84]

###  IV-D Multi-objective environmental/economic load dispatch

Multi-objective environmental/economic dispatch (EED) problems include not
only fuel consumption but also emission of SOx, NOx, etc. while solving
economic load dispatch. The EED is a multi-objective problem with objectives
of minimizing the emissions as well as the cost of generation [86]. There has
been a diversity of research about EED problems. Here, it has been analyzed
research works dealing with EED problems using various PSO algorithms.

Jeyakumar et al. [87], described the multi-objective problem by means of four
different models, namely, a) multi-area ED as MAED, b) piecewise quadratic
cost function considering multi-fuel option as PQCF, c) cost as well as
emission minimization as CEED and d) ED with the prohibited operating zone as
ED with POZ.

Huang and Wang [88], proposed a novel hybrid optimization technique to
considers the network of radial basis function (RBF) for real-time power
dispatch (RTPD) by combining enhanced PSO (EPSO) algorithms and orthogonal
least-square (OLS) method. The OLS algorithm gives the number of centers in
the hidden layer, and the EPSO algorithm gives fine-tuned parameters in the
network.

Wang and Singh [89], proposed fuzzified multi-objective PSO (FMOPSO) algorithm
suitable to solve multi-objective EED problems of power systems. Here,
minimization of fuel cost and emissions has been considered as objectives to
meet.

AlRashidi and El-Hawary [90], presented a new hybrid optimization algorithm by
combining Newton-Raphson and PSO suitable to solve multi-objective EED
problems of power system operation. Further, a new inequality constraint
handling mechanism has been incorporated into the proposed optimization
approach.

Wang and Singh [91], proposed an improved PSO algorithm considering a combined
deterministic and stochastic model of ELD problems and simultaneously
considering environmental impact.

Agrawal et al. [92], proposed a fuzzy clustering-based PSO (FCPSO) algorithm
for solving highly constrained multi-objective EED problem involving
conflicting objectives. The niching and fuzzy clustering technique has been
used to direct the particles towards lesser-explored regions of the Pareto
front. Further, an adaptive mutation operator has been used to prevent
premature convergence. Also, a fuzzy-based approach has been used to make a
compromise with objectives.

Wang and Singh [93], discussed the solution of multi-area environment/economic
dispatch (MAEED) problems using an improved multi-objective PSO (MOPSO). The
objectives are to obtain optimum ELD and to minimize pollutant emissions. In
the proposed model, tie-line transfer limits and a reserve-sharing scheme have
been used to ensure the ability of each area to fulfill its reserve
requirement.

Cai et al. [94], introduced a multi-objective chaotic PSO (MOCPSO) algorithm
suitable for solving EED problems. The comparison of performances of the
proposed MOCPSO and the conventional MOPSO algorithm has been performed.

Zhang et al. [95], proposed a bare-bones multi-objective PSO (BB-MOPSO)
algorithm suitable for solving EED problems of power systems operations. In
this algorithm, constraint handling strategy, mutation operator, crowding
distance, fuzzy membership functions and an external repository of elite
particles have been used to make BB-MOPSO algorithm much more efficient for
multiple objectives optimization problems.

Chalermchaiarbha and Ongsakul [96], proposed a new elitist multi-objective PSO
(EMPSO) algorithm for solving multiple objectives ELD problems of power
systems. In the proposed algorithm, fuzzy multi-attribute decision making is
utilized to obtain a good compromise among the conflicting objectives.

Zeng and Sun [97], proposed an improved PSO algorithm for solving the CHP-DED
problems with various systems constraints of power systems. In the proposed
algorithm, chaotic mechanism, TVAC, and self-adaptive mutation scheme have
been considered. Also, various constraints handing approaches have been
utilized.

Jadoun et al. [98], proposed a new modified modulated PSO (MPSO) algorithm for
solving various types of economic emission dispatch (EED) problems of power
systems. In this algorithm, the velocity vector of the conventional PSO has
been modified by the truncated sinusoidal constriction function in the
velocity equation. Further, the conflicting objectives of the EED problem,
which is compromised of economic dispatch and emission minimization are
combined in a fuzzy framework by suggesting adjusted fuzzy membership
functions which are then optimized using the proposed MPSO.

Jiang et al. [99], proposed a newly modified gravitational acceleration
enhanced PSO (GAEPSO) algorithm for solving multiple objectives wind-thermal
economic and emission dispatch problems of the power system. In this
algorithm, the velocity of each particle is simultaneously updated using PSO
and gravitational search algorithm (GSA). The concepts of updating the
velocity vector using PSO provides enough exploration whereas the GSA provides
enough exploitation to each particle. These features of the proposed GAEPSO
make it a faster and more efficient algorithm in solving ELD problems.

Mandal et al. [100], proposed a newly modified self-adaptive PSO algorithm for
solving emission constrained economic dispatch problems of power systems. The
proposed algorithm is a self-organizing hierarchical PSO with time-varying
acceleration coefficients (SOHPSO_TVAC).

Liu et al. [101], proposed cultural multi-objective QPSO (CMOQPSO) algorithm
for solving the EED problem of power systems. In this algorithm, population
diversity is maintained by introducing a cultural evolution mechanism in the
QPSO algorithm. Believe space, available in the cultural evolution mechanism
is utilized to avoid premature convergence. This feature leads to explore the
entire search space effectively and gives much better results.

Table IV provides various details, such as the type of algorithm used,
modeling of the ELD problem, size of the test system, etc., about each of the
papers reviewed above in this subsection.

TABLE IV: Multi-Objective Environmental/Economic Dispatch

Type of PSO | Type of the ELD problem | Test system | References
---|---|---|---
PSO, CEP | Multi-fuel Option, Combined EED with RRL | 6, 10, 15, 16-gen. systems | [87]
EPSO, OLS | Real-time Power Dispatch | IEEE 30-bus six-gen. | [88]
FMOPSO | Bi-objective Cost as well as Pollutant Emission Minimization | 14-generator system | [89]
PSO, Newton-Raphson | Minimization of Real Power Loss, Fuel Cost, and Gaseous emission | IEEE 30-bus six-gen. | [90]
Improved PSO | Deterministic and Stochastic Model of ELD with Environmental Impact | IEEE 30-bus six-gen. | [91]
FCPSO, Niching | Highly Constrained Multi-objective EED | IEEE 30-bus six-gen. | [92]
MOPSO | Reserve-constrained Multi-are EED (MAEED) | Typical 7-gen. system | [93]
MOCPSO | Fuel Cost and Emission Minimization | IEEE 30-bus six-gen. | [94]
BB-MOPSO, Fuzzy | Multi-objective EED | IEEE 30-bus six-gen. | [95]
EM PSO | Multi-objective ELD problems with various systems constraints | 6 and 18 gen. | [96]
Improved PSO | CHP based DED problems with various systems constraints | 10 gen. | [97]
MPSO | EED | 6, 10 and 40-gen. systems | [98]
GAEPSO | wind-thermal economic and emission dispatch | 6 and 40-gen. systems | [99]
SOHPSO_TVAC | emission constrained EED | two typical test systems | [100]
CMQPSO | EED | 6 and 40-gen. systems | [101]

###  IV-E Economic load dispatch of micro-grids

The sustainable development goal of countries can be achieved through a
provision of access to clean, secure, reliable and affordable energy. This can
only be achieved by renewable power generation. To access such electric power
we need excellent micro-grids technologies. Various researchers are now
focusing on the technical and economical suitability of micro-grids [102, 103,
104, 105, 106].

Moghaddama et al. [107], presented a comprehensive literature review on ELD
problems related to micro-grids. In this work, the primary focus has been
given to the application of PSO algorithms in solving issues related to the
economic operations of micro-grids. Basu et al. [108], proposed a CHP-based
micro-grids economic scheduling considering network losses. Nikmehr and
Ravadanegh [109], proposed the optimum power dispatch of micro-grids
considering probabilistic model using PSO. Also, [110], introduced the
economic scheduling of multi-micro-grids using PSO. Wu et al. [111], proposed
the economic operation of CHP based micro-grid system considering photovoltaic
arrays (PV), wind turbines (WT), diesel engines (DE), fuel cells (FC), micro-
turbines (MT) and battery system (BS). The proposed model has been solved
using an improved PSO.

Yao et al. [112], proposed a quantum-inspired PSO (QPSO) algorithm for solving
the green energy based ELD problems of smart grids. In the proposed QPSO, a
quantum-inspired evolutionary algorithm (QEA) which is based on quantum
computing has been utilized to obtain better and faster global optimum
results. In the ELD problem, wind power uncertainty and carbon tax have been
considered while formulating the problem.

Faria et al. [113], discussed a modified PSO for solving the ELD problems
including the demand response and distributed generation (DG) resources of
modern smart grids systems. Hu et al. [114], discussed fuzzy-adaptive PSO
(FAPSO) algorithm for minimizing distribution network loss using optimum load
response to the consumers.

Wu et al. [115], proposed multi-objective PSO (MPSO) for solving the ELD
problems of the risk-based wind-integrated power system. The proposed modeling
considers wind power uncertainty and optimum power dispatch simultaneously. A
probabilistic model has been considered to predict wind availability and risk
related to supply the load demand.

Cheng et al. [116], proposed the PSO algorithm for solving energy management
of a hybrid generation system (HGS). The proposed HGS consists of power
generation from the photovoltaic array, wind turbine, micro-turbine, battery
banks, and the utility grid.

Elsied et al. [117], proposed energy management of microgrids considering
minimization of the energy cost, carbon dioxide, and pollutant emissions while
maximizing the power of the available renewable energy resources using binary
PSO (BPSO) algorithm.

Li et al. [118], proposed chaotic binary PSO (CBPSO) for solving the ELD
problems of micro-grids. A new fuzzy-based modeling has been developed to
minimize systems losses, pollutant emission and the cost of supplying the
power demand.

Table V gives various details such as the type of algorithm used, modeling of
the ELD problem, size of the test system, etc., about each of the papers
reviewed above in this subsection.

TABLE V: Economic Dispatch in Micrgrids

Type of PSO | Type of the ELD problem | Test system | References
---|---|---|---
FSAPSO | A survey paper in micro-grids power dispatch | A typical micro-grid test system | [107]
PSO | CHP-dispatch with losses and emission in micro-grids | IEEE 14-bus five-gen. | [108]
PSO, ICA | Micro-grids with WT and PV power dispatch | A typical three-gen. micro-grid test system | [109]
PSO | Micro-grids with WT and PV with losses power dispatch | A typical three-gen. micro-grid test system | [110]
PSO | Micro-grids with PV, WT, DE, FC, MT, BT with losses | A typical seven-gen. micro-grid test system | [111]
| using actual mathematical modelling of the resources |  |
QPSO | ELD of smart grids with uncertainty and carbon tax | Modified IEEE 30-bus six-gen. | [112]
Modeified PSO | ELD with demand response and presence of DG | A real test system | [113]
FAPSO | ELD with demand response from consumers | A typical 18-bus system with 3-WT | [114]
MPSO | ELD with risk based WT | IEEE 30-bus six-gen. | [115]
PSO | ELD of HGS with PV, WT, MT, BT and utility system | IEEE 30-bus six-gen. | [116]
BPSO | EED of micro-grids with various renewable resources | A typical micro-grid system | [117]
CBPSO, fuzzy | Multi objective EED | A typical distribution network | [118]
CHP: Combined heat and power;
HGS: Hybrid generation system;
ICA: Imperialist competitive algorithm;
FSAPSO: Fuzzy self adaptive particle swarm optimization;

Table VI gives more details about the various database and their journals from
which the papers have been considered in this study. Further, the serial
numbers of all those references taken from a particular journal have also been
included in this table.

TABLE VI: Various database and the journals considered in this study

Publishers | Name of journals | References | No. of papers
---|---|---|---
Elsevier | Applied Energy | [55, 105] | 2
Applied Soft Computing | [38, 58, 70, 100, 101, 104] | 6
Chaos, Solitons & Fractals | [64] | 1
Electric Power Systems Research | [25, 36, 37, 66, 76, 85, 89, 91] | 8
Energy | [56, 84] | 2
Energy Conversion and Management | [35, 48, 59, 67, 74, 77, 94, 117] | 8
Engineering Applications of Artificial Intelligence | [43, 72, 93] | 3
Expert Systems with Applications | [42, 45, 78] | 3
Information Sciences | [95] | 1
International Journal of Electrical Power & Energy Systems | [22, 39, 44, 49, 50, 52, 54, 61, 68] | 15
| [75, 80, 87, 98, 99, 111] |
Renewable and Sustainable Energy Reviews | [17, 23, 107] | 3
| Renewable Energy | [116] | 1
IEEE | IEEE Transactions on Evolutionary Computation | [18, 19, 27, 92] | 4
IEEE Transactions on Industrial Informatics | [51, 112] | 2
IEEE Transactions on Power Systems | [1, 9, 11, 32, 33, 34, 40, 41, 62, 63] | 16
| [65, 73, 82, 88, 90, 108] |
IEEE Transactions on Smart Grid | [106, 110, 113, 118] | 4
| IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics | [60] | 1
IET | IET Generation, Transmission & Distribution | [12, 13, 69, 71, 79, 86, 114, 115] | 8
IET Renewable Power Generation | [102] | 1
Taylor & Francis | Electric Power Components and Systems | [96, 97] | 2
Springer’s | Neural Computing and Applications | [53] | 1

##  V Conclusion

This paper presents a literature review of the application of particle swarm
optimization (PSO) algorithm for solving various types of economic load
dispatch (ELD) problems in power systems. A survey of papers and reports of
the period 2003-2016 addressing various aspects of economic load dispatch
using the PSO algorithm have been presented in this paper along with a brief
discussion of a simple ELD model and the classical PSO algorithm. ELD problems
have been identified and classified into five important groups. These groups
are (i) single objective economic dispatch, (ii) dynamic economic load
dispatch, (iii) economic dispatch with non-conventional sources, (iv) multi-
objective environmental/economic dispatch and (v) economic dispatch of micro-
grids. An attempt has been made to include more and more descriptions to point
out the unique and important aspects of each paper considered. In summary,
there are several promising approaches with the help of PSO in smart grid
technologies for further progress. Some of the areas are as follows:

  1. (a)

Electric Vehicle Charging in Smart Grids: PSO can be applied to charging
optimization of electric vehicle (charging plan of each vehicle while
satisfying the requirements of the individual vehicle owners without
distribution network congestion) and its coordination to minimize power losses
and improve the voltage profile of the smart grid.

  2. (b)

Protection of Smart Grid: Since smart grids are more flexible, fault current
levels in the grid are variable. The conventional protection system may not be
effective in such a situation. Research can be focused to apply improved PSO
algorithms to limit the fault currents in smart grids by the size of
thyristor-controlled impedance

  3. (c)

Resource Scheduling: Coordinated scheduling of distributed energy resources
(including residential energy sources) and balancing of supply and demand
across time with the help of optimization algorithms.

## References

  * [1] B. Chowdhury and S. Rahman, “A review of recent advances in economic dispatch,” _IEEE Transactions on Power Systems,_ , vol. 5, no. 4, pp. 1248–1259, Nov 1990.
  * [2] A. Wood, B. Wollenberg, and G. Sheblé, _Power Generation, Operation and Control_. Wiley, 2013.
  * [3] J. Catalão, _Electric Power Systems: Advanced Forecasting Techniques and Optimal Generation Scheduling_. CRC Press, 2012.
  * [4] K. Lee and M. El-Sharkawi, _Modern Heuristic Optimization Techniques: Theory and Applications to Power Systems_ , ser. IEEE Press Series on Power Engineering. Wiley, 2008.
  * [5] D. Kothari, D.thari and J. hillon, _Power System Optimization_. PHI Learning, 2010.
  * [6] J. Zhu, _Optimization of Power System Operation_ , ser. IEEE Press Series on Power Engineering. Wiley, 2015.
  * [7] H. Singh, A. Isaacs, T. Ray, and W. Smith, “A simulated annealing algorithm for constrained multi-objective optimization,” in _IEEE Congress on Evolutionary Computation_ , June 2008, pp. 1655–1662.
  * [8] J. Tippayachai, W. Ongsakul, and I. Ngamroo, “Nonconvex economic dispatch by enhanced tabu search algorithm,” in _IEEE Power Engineering Society General Meeting_ , vol. 2, July 2003, pp. 908–913.
  * [9] J. Kumar and G. Sheble, “Clamped state solution of artificial neural network for real-time economic dispatch,” _IEEE Transactions on Power Systems,_ , vol. 10, no. 2, pp. 925–931, May 1995.
  * [10] J. Dhillon, J. Dhillon, and D. Kothari, “Generation pattern search for different kinds of economic load dispatch,” in _IEEE Large Engineering Systems Conference on Power Engineering_ , Oct 2007, pp. 250–255.
  * [11] H.-T. Yang, P.-C. Yang, and C.-L. Huang, “Evolutionary programming based economic dispatch for units with non-smooth fuel cost functions,” _IEEE Transactions on Power Systems,_ , vol. 11, no. 1, pp. 112–118, Feb 1996.
  * [12] C.-L. Chiang, “Genetic-based algorithm for power economic load dispatch,” _IET Generation, Transmission Distribution,_ , vol. 1, no. 2, pp. 261–269, March 2007.
  * [13] S.-K. Wang, J.-P. Chiou, and C. Liu, “Non-smooth/non-convex economic dispatch by a novel hybrid differential evolution algorithm,” _IET Generation, Transmission Distribution,_ , vol. 1, no. 5, pp. 793–803, September 2007.
  * [14] S. Prakash, N. Gopinath, and O. Prakash, “A new approach for real power optimization of generation with nonsmooth cost function using particle swarm optimization,” in _IEEE National Conference on Emerging Trends In New Renewable Energy Sources And Energy Management (NCET NRES EM)_ , Dec 2014, pp. 59–62.
  * [15] M. N. Alam, A. Mathur, and K. Kumar, “Economic load dispatch using a differential particle swarm optimization,” in _In Proc. IEEE 1 st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES)_, Jul. 2016, pp. 1–5.
  * [16] S. Sahoo, K. Dash, and A. Barisal, “Solution of economic load dispatch by evolutionary optimization algorithms: A comparative study,” in _IEEE International Conference on Control, Instrumentation, Energy and Communication (CIEC)_ , Jan 2014, pp. 259–263.
  * [17] A. Mahor, V. Prasad, and S. Rangnekar, “Economic dispatch using particle swarm optimization: A review,” _Renewable and Sustainable Energy Reviews_ , vol. 13, no. 8, pp. 2134 – 2141, 2009.
  * [18] Y. del Valle, G. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, and R. Harley, “Particle swarm optimization: Basic concepts, variants and applications in power systems,” _IEEE Transactions on Evolutionary Computation,_ , vol. 12, no. 2, pp. 171–195, April 2008.
  * [19] M. R. AlRashidi and M. E. El-Hawary, “A survey of particle swarm optimization applications in electric power systems,” _IEEE Transactions on Evolutionary Computation,_ , vol. 13, no. 4, pp. 913–918, Aug 2009.
  * [20] K. Lee and J.-B. Park, “Application of particle swarm optimization to economic dispatch problem: Advantages and disadvantages,” in _IEEE PES Power Systems Conference and Exposition_ , Oct 2006, pp. 188–192.
  * [21] M. Abdullah, A. Bakar, N. Rahim, H. Mokhlis, and C. Tan, “Implementation of hybrid particle swarm optimization for combined economic-emission load dispatch problem,” in _IEEE 8th International Power Engineering and Optimization Conference (PEOCO)_ , March 2014, pp. 402–407.
  * [22] P. K. Roy and S. Bhui, “Multi-objective quasi-oppositional teaching learning based optimization for economic emission load dispatch problem,” _International Journal of Electrical Power & Energy Systems_, vol. 53, pp. 937 – 948, 2013.
  * [23] R. Boqiang and J. Chuanwen, “A review on the economic dispatch and risk management considering wind power in the power market,” _Renewable and Sustainable Energy Reviews_ , vol. 13, no. 8, pp. 2169 – 2174, 2009.
  * [24] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in _IEEE International Conference on Neural Networks_ , vol. 4, Nov 1995, pp. 1942–1948.
  * [25] M. N. Alam, B. Das, and V. Pant, “A comparative study of metaheuristic optimization approaches for directional overcurrent relays coordination,” _Electric Power Systems Research_ , vol. 128, pp. 39–52, 2015.
  * [26] T. Subhani, C. Babu, and A. Reddy, “Particle swarm optimization with time varying acceleration coefficents for economic dispatch considering valve point loading effects,” in _IEEE Third International Conference on Computing Communication Networking Technologies (ICCCNT)_ , July 2012, pp. 1–8.
  * [27] M. Clerc and J. Kennedy, “The particle swarm - explosion, stability, and convergence in a multidimensional complex space,” _IEEE Transactions on Evolutionary Computation,_ , vol. 6, no. 1, pp. 58–73, Feb 2002.
  * [28] N. Kumar, U. Nangia, and K. Sahay, “Economic load dispatch using improved particle swarm optimization algorithms,” in _IEEE Power India International Conference (PIICON)_ , Dec 2014, pp. 1–6.
  * [29] M. N. Alam, K. Kumar, and A. Mathur, “Economic load dispatch considering valve-point effects using time varying constriction factor based particle swarm optimization,” in _2015 IEEE UP Section Conference on Electrical Computer and Electronics (UPCON)_ , Dec. 2015, pp. 1–6.
  * [30] Y. Shi and R. Eberhart, “A modified particle swarm optimizer,” in _IEEE World Congress on Computational Intelligence_ , May 1998, pp. 69–73.
  * [31] J.-B. Park, K.-S. Lee, J.-R. Shin, and K. Lee, “Economic load dispatch for nonsmooth cost functions using particle swarm optimization,” in _IEEE Power Engineering Society General Meeting_ , vol. 2, July 2003, pp. 938–943.
  * [32] Z.-L. Gaing, “Particle swarm optimization to solving the economic dispatch considering the generator constraints,” _IEEE Transactions on Power Systems,_ , vol. 18, no. 3, pp. 1187–1195, Aug 2003.
  * [33] J.-B. Park, K.-S. Lee, J.-R. Shin, and K. Lee, “A particle swarm optimization for economic dispatch with nonsmooth cost functions,” _IEEE Transactions on Power Systems,_ , vol. 20, no. 1, pp. 34–42, Feb 2005.
  * [34] A. Selvakumar and K. Thanushkodi, “A new particle swarm optimization solution to nonconvex economic dispatch problems,” _IEEE Transactions on Power Systems,_ , vol. 22, no. 1, pp. 42–51, Feb 2007.
  * [35] J. Cai, X. Ma, L. Li, and P. Haipeng, “Chaotic particle swarm optimization for economic dispatch considering the generator constraints,” _Energy Conversion and Management_ , vol. 48, no. 2, pp. 645 – 653, 2007.
  * [36] A. I. Selvakumar and K. Thanushkodi, “Anti-predatory particle swarm optimization: Solution to nonconvex economic dispatch problems,” _Electric Power Systems Research_ , vol. 78, no. 1, pp. 2 – 10, 2008.
  * [37] A. Y. Saber, S. Chakraborty, S. A. Razzak, and T. Senjyu, “Optimization of economic load dispatch of higher order general cost polynomials and its sensitivity using modified particle swarm optimization,” _Electric Power Systems Research_ , vol. 79, no. 1, pp. 98 – 106, 2009.
  * [38] K. T. Chaturvedi, M. Pandit, and L. Srivastava, “Particle swarm optimization with crazy particles for nonconvex economic dispatch,” _Applied Soft Computing_ , vol. 9, no. 3, pp. 962 – 969, 2009.
  * [39] ——, “Particle swarm optimization with time varying acceleration coefficients for non-convex economic power dispatch,” _International Journal of Electrical Power & Energy Systems_, vol. 31, no. 6, pp. 249 – 257, 2009.
  * [40] J.-B. Park, Y.-W. Jeong, J.-R. Shin, and K. Lee, “An improved particle swarm optimization for nonconvex economic dispatch problems,” _IEEE Transactions on Power Systems,_ , vol. 25, no. 1, pp. 156–166, Feb 2010.
  * [41] K. Meng, H. G. Wang, Z. Dong, and K. P. Wong, “Quantum-inspired particle swarm optimization for valve-point economic load dispatch,” _IEEE Transactions on Power Systems,_ , vol. 25, no. 1, pp. 215–222, Feb 2010.
  * [42] Z. Zhisheng, “Quantum-behaved particle swarm optimization algorithm for economic load dispatch of power system,” _Expert Systems with Applications_ , vol. 37, no. 2, pp. 1800 – 1803, 2010.
  * [43] M. Neyestani, M. M. Farsangi, and H. Nezamabadi-pour, “A modified particle swarm optimization for economic dispatch with non-smooth cost functions,” _Engineering Applications of Artificial Intelligence_ , vol. 23, no. 7, pp. 1121 – 1126, 2010.
  * [44] P. Subbaraj, R. Rengaraj, S. Salivahanan, and T. Senthilkumar, “Parallel particle swarm optimization with modified stochastic acceleration factors for solving large scale economic dispatch problem,” _International Journal of Electrical Power & Energy Systems_, vol. 32, no. 9, pp. 1014 – 1023, 2010\.
  * [45] A. Safari and H. Shayeghi, “Iteration particle swarm optimization procedure for economic load dispatch with generator constraints,” _Expert Systems with Applications_ , vol. 38, no. 5, pp. 6043 – 6048, 2011.
  * [46] A. Saber and D. Rahman, “Economic load dispatch using particle swarm differential evolution optimization,” in _IEEE Power and Energy Society General Meeting_ , July 2011, pp. 1–8.
  * [47] V. N. Dieu, P. Schegner, and W. Ongsakul, “A newly improved particle swarm optimization for economic dispatch with valve point loading effects,” in _IEEE Power and Energy Society General Meeting_ , July 2011, pp. 1–8.
  * [48] S. Chalermchaiarbha and W. Ongsakul, “Stochastic weight trade-off particle swarm optimization for nonconvex economic dispatch,” _Energy Conversion and Management_ , vol. 70, pp. 66–75, Jun 2013.
  * [49] V. Hosseinnezhad and E. Babaei, “Economic load dispatch using $\theta$-PSO,” _International Journal of Electrical Power & Energy Systems_, vol. 49, pp. 160–169, Jul 2013.
  * [50] V. Hosseinnezhad, M. Rafiee, M. Ahmadian, and M. T. Ameli, “Species-based quantum particle swarm optimization for economic load dispatch,” _International Journal of Electrical Power & Energy Systems_, vol. 63, pp. 311–322, Dec 2014.
  * [51] J. Sun, V. Palade, X.-J. Wu, W. Fang, and Z. Wang, “Solving the power economic dispatch problem with generator constraints by random drift particle swarm optimization,” _IEEE Transactions on Industrial Informatics_ , vol. 10, no. 1, pp. 222–232, Feb 2014.
  * [52] M. Basu, “Modified particle swarm optimization for nonconvex economic dispatch problems,” _International Journal of Electrical Power & Energy Systems_, vol. 69, pp. 304–312, Jul 2015.
  * [53] Y.-Z. Hsieh and M.-C. Su, “A q-learning-based swarm optimization algorithm for economic dispatch problem,” _Neural Computing and Applications_ , vol. 27, no. 8, pp. 2333–2350, Oct 2015.
  * [54] V. K. Jadoun, N. Gupta, K. Niazi, and A. Swarnkar, “Multi-area economic dispatch with reserve sharing using dynamically controlled particle swarm optimization,” _International Journal of Electrical Power & Energy Systems_, vol. 73, pp. 743–756, Dec 2015.
  * [55] T. Niknam, “A new fuzzy adaptive hybrid particle swarm optimization algorithm for non-linear, non-smooth and non-convex economic dispatch problem,” _Applied Energy_ , vol. 87, no. 1, pp. 327 – 339, 2010.
  * [56] T. Niknam, H. D. Mojarrad, and M. Nayeripour, “A new fuzzy adaptive particle swarm optimization for non-smooth economic dispatch,” _Energy_ , vol. 35, no. 4, pp. 1764 – 1778, 2010.
  * [57] B. Mahdad, K. Srairi, T. Bouktir, and M. El Benbouzid, “Fuzzy controlled parallel pso to solving large practical economic dispatch,” in _IEEE International Energy Conference and Exhibition (EnergyCon)_ , Dec 2010, pp. 34–40.
  * [58] T. Niknam, H. D. Mojarrad, and H. Z. Meymand, “Non-smooth economic dispatch computation by fuzzy and self adaptive particle swarm optimization,” _Applied Soft Computing_ , vol. 11, no. 2, pp. 2805 – 2817, 2011.
  * [59] ——, “A novel hybrid particle swarm optimization for economic dispatch with valve-point loading effects,” _Energy Conversion and Management_ , vol. 52, no. 4, pp. 1800 – 1809, 2011.
  * [60] Y. ping Chen, W.-C. Peng, and M.-C. Jian, “Particle swarm optimization with recombination and dynamic linkage discovery,” _IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics,_ , vol. 37, no. 6, pp. 1460–1470, Dec 2007.
  * [61] L. dos Santos Coelho and C.-S. Lee, “Solving economic load dispatch problems in power systems using chaotic and gaussian particle swarm optimization approaches,” _International Journal of Electrical Power & Energy Systems_, vol. 30, no. 5, pp. 297 – 307, 2008.
  * [62] K. Chaturvedi, M. Pandit, and L. Srivastava, “Self-organizing hierarchical particle swarm optimization for nonconvex economic dispatch,” _IEEE Transactions on Power Systems,_ , vol. 23, no. 3, pp. 1079–1087, Aug 2008.
  * [63] C.-C. Kuo, “A novel coding scheme for practical economic dispatch by modified particle swarm approach,” _IEEE Transactions on Power Systems,_ , vol. 23, no. 4, pp. 1825–1835, Nov 2008.
  * [64] L. dos Santos Coelho and V. C. Mariani, “A novel chaotic particle swarm optimization approach using hnon map and implicit filtering local search for economic load dispatch,” _Chaos, Solitons & Fractals_, vol. 39, no. 2, pp. 510 – 518, 2009.
  * [65] J. G. Vlachogiannis and K. Y. Lee, “Economic load dispatch a comparative study on heuristic optimization techniques with an improved coordinated aggregation-based pso,” _IEEE Transactions on Power Systems,_ , vol. 24, no. 2, pp. 991–1001, 2009.
  * [66] T. A. Victoire and A. Jeyakumar, “Hybrid pso-sqp for economic dispatch with valve-point effect,” _Electric Power Systems Research_ , vol. 71, no. 1, pp. 51 – 59, 2004.
  * [67] J. Sun, W. Fang, D. Wang, and W. Xu, “Solving the economic dispatch problem with a modified quantum-behaved particle swarm optimization method,” _Energy Conversion and Management_ , vol. 50, no. 12, pp. 2967 – 2975, 2009\.
  * [68] R. Kumar, D. Sharma, and A. Sadu, “A hybrid multi-agent based particle swarm optimization algorithm for economic power dispatch,” _International Journal of Electrical Power & Energy Systems_, vol. 33, no. 1, pp. 115 – 123, 2011.
  * [69] S. Chakraborty, T. Senjyu, A. Yona, A. Saber, and T. Funabashi, “Solving economic load dispatch problem with valve-point effects using a hybrid quantum mechanics inspired particle swarm optimisation,” _IET Generation, Transmission & Distribution_, vol. 5, no. 10, p. 1042, 2011.
  * [70] S. Sayah and A. Hamouda, “A hybrid differential evolution algorithm based on particle swarm optimization for nonconvex economic dispatch problems,” _Applied Soft Computing_ , vol. 13, no. 4, pp. 1608 – 1619, 2013.
  * [71] R. Azizipanah-Abarghooee, T. Niknam, M. Gharibzadeh, and F. Golestaneh, “Robust, fast and optimal solution of practical economic dispatch by a new enhanced gradient-based simplified swarm optimisation algorithm,” _IET Generation, Transmission Distribution,_ , vol. 7, no. 6, pp. 620–635, June 2013\.
  * [72] F. Zaman, S. M. Elsayed, T. Ray, and R. A. Sarker, “Configuring two-algorithm-based evolutionary approach for solving dynamic economic dispatch problems,” _Engineering Applications of Artificial Intelligence_ , vol. 53, pp. 105–125, Aug 2016.
  * [73] T. Victoire and A. Jeyakumar, “Reserve constrained dynamic dispatch of units with valve-point effects,” _IEEE Transactions on Power Systems,_ , vol. 20, no. 3, pp. 1273–1282, Aug 2005.
  * [74] B. Panigrahi, V. R. Pandi, and S. Das, “Adaptive particle swarm optimization approach for static and dynamic economic load dispatch,” _Energy Conversion and Management_ , vol. 49, no. 6, pp. 1407 – 1415, 2008.
  * [75] G. Baskar and M. Mohan, “Security constrained economic load dispatch using improved particle swarm optimization suitable for utility system,” _International Journal of Electrical Power & Energy Systems_, vol. 30, no. 10, pp. 609 – 613, 2008.
  * [76] ——, “Contingency constrained economic load dispatch using improved particle swarm optimization for security enhancement,” _Electric Power Systems Research_ , vol. 79, no. 4, pp. 615 – 621, 2009.
  * [77] Y. Wang, J. Zhou, H. Qin, and Y. Lu, “Improved chaotic particle swarm optimization algorithm for dynamic economic dispatch problem with valve-point effects,” _Energy Conversion and Management_ , vol. 51, no. 12, pp. 2893 – 2900, 2010.
  * [78] Y. Wang, J. Zhou, Y. Lu, H. Qin, and Y. Wang, “Chaotic self-adaptive particle swarm optimization algorithm for dynamic economic dispatch problem with valve-point effects,” _Expert Systems with Applications_ , vol. 38, no. 11, pp. 14 231 – 14 237, 2011.
  * [79] T. Niknam and F. Golestaneh, “Enhanced adaptive particle swarm optimisation algorithm for dynamic economic dispatch of units considering valve-point effects and ramp rates,” _IET Generation, Transmission Distribution,_ , vol. 6, no. 5, pp. 424–435, May 2012.
  * [80] S. Banerjee, K. Dasgupta, and C. K. Chanda, “Short term hydro-wind-thermal scheduling based on particle swarm optimization technique,” _International Journal of Electrical Power & Energy Systems_, vol. 81, pp. 275–288, Oct 2016.
  * [81] L. Wang and C. Singh, “Pso-based multi-criteria economic dispatch considering wind power penetration subject to dispatcher’s attitude,” in _IEEE 38th North American Power Symposium_ , Sept 2006, pp. 269–276.
  * [82] X. Li and C. Jiang, “Short-term operation model and risk management for wind power penetrated system in electricity market,” _IEEE Transactions on Power Systems,_ , vol. 26, no. 2, pp. 932–939, May 2011.
  * [83] Z. Wu, W. Gu, R. Wang, X. Yuan, and W. Liu, “Economic optimal schedule of chp microgrid system using chance constrained programming and particle swarm optimization,” in _IEEE Power and Energy Society General Meeting_ , July 2011, pp. 1–11.
  * [84] B. Bahmani-Firouzi, E. Farjah, and R. Azizipanah-Abarghooee, “An efficient scenario-based and fuzzy self-adaptive learning particle swarm optimization approach for dynamic economic emission dispatch considering load and wind power uncertainties,” _Energy_ , vol. 50, pp. 232 – 244, 2013.
  * [85] L. Wang and C. Singh, “Balancing risk and cost in fuzzy economic dispatch including wind power penetration based on particle swarm optimization,” _Electric Power Systems Research_ , vol. 78, no. 8, pp. 1361 – 1368, 2008\.
  * [86] A. Gholami, J. Ansari, A. Kazemi, and M. Jamei, “Environmental/economic dispatch incorporating renewable energy sources and plug-in vehicles,” _IET Generation, Transmission & Distribution_, vol. 8, no. 12, pp. 2183–2198, Dec 2014.
  * [87] D. Jeyakumar, T. Jayabarathi, and T. Raghunathan, “Particle swarm optimization for various types of economic dispatch problems,” _International Journal of Electrical Power & Energy Systems_, vol. 28, no. 1, pp. 36 – 42, 2006\.
  * [88] C.-M. Huang and F.-L. Wang, “An rbf network with ols and epso algorithms for real-time power dispatch,” _IEEE Transactions on Power Systems,_ , vol. 22, no. 1, pp. 96–104, Feb 2007.
  * [89] L. Wang and C. Singh, “Environmental/economic power dispatch using a fuzzified multi-objective particle swarm optimization algorithm,” _Electric Power Systems Research_ , vol. 77, no. 12, pp. 1654 – 1664, 2007.
  * [90] M. R. AlRashidi and M. E. El-Hawary, “Hybrid particle swarm optimization approach for solving the discrete opf problem considering the valve loading effects,” _IEEE Transactions on Power Systems,_ , vol. 22, no. 4, pp. 2030–2038, Nov 2007.
  * [91] L. Wang and C. Singh, “Stochastic economic emission load dispatch through a modified particle swarm optimization algorithm,” _Electric Power Systems Research_ , vol. 78, no. 8, pp. 1466 – 1476, 2008.
  * [92] S. Agrawal, K. Panigrahi, and M. Tiwari, “Multiobjective particle swarm algorithm with fuzzy clustering for electrical power dispatch,” _IEEE Transactions on Evolutionary Computation,_ , vol. 12, no. 5, pp. 529–541, Oct 2008\.
  * [93] L. Wang and C. Singh, “Reserve-constrained multiarea environmental/economic dispatch based on particle swarm optimization with local search,” _Engineering Applications of Artificial Intelligence_ , vol. 22, no. 2, pp. 298 – 307, 2009.
  * [94] J. Cai, X. Ma, Q. Li, L. Li, and H. Peng, “A multi-objective chaotic particle swarm optimization for environmental/economic dispatch,” _Energy Conversion and Management_ , vol. 50, no. 5, pp. 1318 – 1325, 2009.
  * [95] Y. Zhang, D.-W. Gong, and Z. Ding, “A bare-bones multi-objective particle swarm optimization algorithm for environmental/economic dispatch,” _Information Sciences_ , vol. 192, pp. 213 – 227, 2012.
  * [96] S. Chalermchaiarbha and W. Ongsakul, “Elitist multi-objective particle swarm optimization with fuzzy multi-attribute decision making for power dispatch,” _Electric Power Components and Systems_ , vol. 40, no. 14, pp. 1562–1585, Oct. 2012.
  * [97] Y. Zeng and Y. Sun, “An improved particle swarm optimization for the combined heat and power dynamic economic dispatch problem,” _Electric Power Components and Systems_ , vol. 42, no. 15, pp. 1700–1716, Oct 2014.
  * [98] V. K. Jadoun, N. Gupta, K. Niazi, and A. Swarnkar, “Modulated particle swarm optimization for economic emission dispatch,” _International Journal of Electrical Power & Energy Systems_, vol. 73, pp. 80–88, Dec 2015.
  * [99] S. Jiang, Z. Ji, and Y. Wang, “A novel gravitational acceleration enhanced particle swarm optimization algorithm for wind-thermal economic emission dispatch problem considering wind power availability,” _International Journal of Electrical Power & Energy Systems_, vol. 73, pp. 1035–1050, Dec 2015\.
  * [100] K. Mandal, S. Mandal, B. Bhattacharya, and N. Chakraborty, “Non-convex emission constrained economic dispatch using a new self-adaptive particle swarm optimization technique,” _Applied Soft Computing_ , vol. 28, pp. 188–195, Mar 2015.
  * [101] T. Liu, L. Jiao, W. Ma, J. Ma, and R. Shang, “Cultural quantum-behaved particle swarm optimization for environmental/economic dispatch,” _Applied Soft Computing_ , vol. 48, pp. 597–611, Nov 2016.
  * [102] C. Chen, S. Duan, T. Cai, B. Liu, and G. Hu, “Smart energy management system for optimal microgrid economic operation,” _IET Renewable Power Generation,_ , vol. 5, no. 3, pp. 258–267, May 2011.
  * [103] N. Augustine, S. Suresh, P. Moghe, and K. Sheikh, “Economic dispatch for a microgrid considering renewable energy cost functions,” in _IEEE PES Innovative Smart Grid Technologies (ISGT)_ , Jan 2012, pp. 1–7.
  * [104] A. Khare and S. Rangnekar, “A review of particle swarm optimization and its applications in solar photovoltaic system,” _Applied Soft Computing_ , vol. 13, no. 5, pp. 2997–3006, May 2013.
  * [105] J. Soares, M. A. Fotouhi Ghazvini, Z. Vale, and P. de Moura Oliveira, “A multi-objective model for the day-ahead energy resource scheduling of a smart grid with high penetration of sensitive loads,” _Applied Energy_ , vol. 162, pp. 1074–1088, Jan 2016.
  * [106] H. Abdi, E. Dehnavi, and F. Mohammadi, “Dynamic economic dispatch problem integrated with demand response (deddr) considering non-linear responsive load models,” _IEEE Transactions on Smart Grid_ , vol. 7, no. 6, pp. 2586–2595, Nov 2016.
  * [107] A. A. Moghaddam, A. Seifi, and T. Niknam, “Multi-operation management of a typical micro-grids using particle swarm optimization: A comparative study,” _Renewable and Sustainable Energy Reviews_ , vol. 16, no. 2, pp. 1268 – 1281, 2012.
  * [108] A. Basu, A. Bhattacharya, S. Chowdhury, and S. Chowdhury, “Planned scheduling for economic power sharing in a chp-based micro-grid,” _IEEE Transactions on Power Systems,_ , vol. 27, no. 1, pp. 30–38, Feb 2012.
  * [109] N. Nikmehr and S. Najafi-Ravadanegh, “Probabilistic optimal power dispatch in multi-microgrids using heuristic algorithms,” in _IEEE Smart Grid Conference (SGC)_ , Dec 2014, pp. 1–6.
  * [110] N. Nikmehr and S. Ravadanegh, “Optimal power dispatch of multi-microgrids at future smart distribution grids,” _IEEE Transactions on Smart Grid,_ , vol. 6, no. 4, pp. 1648–1657, July 2015.
  * [111] H. Wu, X. Liu, and M. Ding, “Dynamic economic dispatch of a microgrid: Mathematical models and solution algorithm,” _International Journal of Electrical Power & Energy Systems_, vol. 63, pp. 336 – 346, 2014.
  * [112] F. Yao, Z. Y. Dong, K. Meng, Z. Xu, H. H.-C. Iu, and K. P. Wong, “Quantum-inspired particle swarm optimization for power system operations considering wind power uncertainty and carbon tax in australia,” _IEEE Transactions on Industrial Informatics_ , vol. 8, no. 4, pp. 880–888, Nov 2012\.
  * [113] P. Faria, J. Soares, Z. Vale, H. Morais, and T. Sousa, “Modified particle swarm optimization applied to integrated demand response and dg resources scheduling,” _IEEE Transactions on Smart Grid_ , vol. 4, no. 1, pp. 606–616, Mar 2013.
  * [114] W. Hu, Y. Hu, Z. Chen, and B. Bak-Jensen, “Fuzzy adaptive particle swarm optimisation for power loss minimisation in distribution systems using optimal load response,” _IET Generation, Transmission & Distribution_, vol. 8, no. 1, pp. 1–10, Jan 2014.
  * [115] J. Wu, K. Wang, B. Zhang, J. Shao, J. Yao, T. Ge, and D. Zeng, “Optimal economic dispatch model based on risk management for wind-integrated power system,” _IET Generation, Transmission & Distribution_, vol. 9, no. 15, pp. 2152–2158, Nov 2015.
  * [116] Y.-S. Cheng, M.-T. Chuang, Y.-H. Liu, S.-C. Wang, and Z.-Z. Yang, “A particle swarm optimization based power dispatch algorithm with roulette wheel redistribution mechanism for equality constraint,” _Renewable Energy_ , vol. 88, pp. 58–72, Apr 2016.
  * [117] M. Elsied, A. Oukaour, H. Gualous, and O. A. Lo Brutto, “Optimal economic and environment operation of micro-grid power systems,” _Energy Conversion and Management_ , vol. 122, pp. 182–194, Aug 2016.
  * [118] P. Li, D. Xu, Z. Zhou, W.-J. Lee, and B. Zhao, “Stochastic optimal operation of microgrid based on chaotic binary particle swarm optimization,” _IEEE Transactions on Smart Grid_ , vol. 7, no. 1, pp. 66–73, Jan 2016.