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\begin{document}
\title{Welcome to Authorea!}
\author[ ]{DAMANJEET KAUR}
\affil[ ]{}
\vspace{-1em}
\date{}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\textbf{A Hybrid Approach to Find Maximum Loadability Limits of a Power
System}\\
Damanjeet kaur, Assistant Professor, UIET, Panjab University Chandigarh,
India
\href{mailto:djkb14@rediffmail.com}{\nolinkurl{djkb14@rediffmail.com}}
\href{mailto:damaneee@pu.ac.in}{\nolinkurl{damaneee@pu.ac.in}}\\
\textbf{~}\\
\textbf{Nomenclature}\\\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
FA\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Firefly Algorithm\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{N\textsubscript{B}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Total number of buses\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
LIA\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Load Incremental Algorithm\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{N\textsubscript{o}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Total number of buses excluding slack buses\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
DE\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Differential Evolution\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{N\textsubscript{G}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Number of \emph{PV} buses\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
DEPSO\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
~Differential Evolution Particle swarm optimization\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{N\textsubscript{PQ}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Number of \emph{PQ} buses\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
PSO\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Particle swarm optimization\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\selectlanguage{greek}\emph{θ\textsubscript{ij}}\selectlanguage{english}\textsubscript{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
voltage angle difference between buses \emph{i} and j in
radians\emph{}\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
NP\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Non-linear Programming\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{B\textsubscript{ij}}\textsubscript{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Transfer susceptance between buses \emph{i} and \emph{j} in p.u\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{P\textsubscript{Gi}}\textsubscript{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Real Power generation at \emph{i\textsuperscript{th}} bus\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{G\textsubscript{ij}}\textsubscript{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Transfer conductance between buses \emph{i} and \emph{j} in p.u\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{Q\textsubscript{Gi}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Reactive power generation at \emph{i\textsuperscript{th}} bus\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Minimum power output in MW of unit \emph{j}\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{Q\textsubscript{Di}}\textsubscript{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Reactive power demand at \emph{i\textsuperscript{th}} bus\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Maximum power output in MW of unit \emph{j}\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{P\textsubscript{Di}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Real power demand at \emph{i\textsuperscript{th}} bus\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{V\textsubscript{i}\textsuperscript{min}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{~}Minimum voltage at \emph{i\textsuperscript{th}} bus in p.u\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\textbf{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Increment load at bus \emph{i} in addition to base load\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{V\textsubscript{i}\textsuperscript{max}}\textbf{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Maximum voltage at \emph{i\textsuperscript{th}} bus in p.u.\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{Itmax}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Maximum number of iterations\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{V\textsubscript{i}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Voltage at \emph{i\textsuperscript{th}} bus in p.u.\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{n}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Total number of fireflies i.e. size of population\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{Is}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Light intensity of a firefly at the source\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
~\\
~\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Minimum value of incremental load for \emph{u\textsuperscript{th}} load
bus of firefly \emph{f}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\selectlanguage{greek}γ\selectlanguage{english}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Light absorption coefficient of a firefly\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Maximum value of incremental load for \emph{u\textsuperscript{th}} load
bus of firefly \emph{f}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{r\textsubscript{fh}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Cartesian distance between two fireflies f and h\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{x\textsubscript{f,k}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{k\textsuperscript{th}} component of the spatial coordinate
\emph{x\textsubscript{f}} of \emph{f\textsuperscript{th}} firefly\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\selectlanguage{greek}β\selectlanguage{english}\emph{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Attractiveness of a firefly with respect to other fireflies\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{x\textsubscript{h,k}}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{k\textsuperscript{th}} component of the spatial coordinate
\emph{x\textsubscript{h}} of \emph{h\textsuperscript{th}} firefly\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{I}\textsubscript{0}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Original light intensity.\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
APSO\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Adaptive PSO\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\emph{N\textsubscript{c}}~\\
\emph{~}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Number of buses violating minimum voltage constraint.\\
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
CPSO\emph{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Chaotic PSO\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
\selectlanguage{greek}δ\selectlanguage{english}\emph{}\\
\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
A small step of incremental load\\
\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}
~\\
~\\
\textbf{1.} \textbf{Introduction}\\
It is always important to find maximum loadability limits of power
system as it decides the secure operation of power system in safe
boundary conditions. This problem became significant with exponential
increase in load demand. The increasing load demand has put burden on
the power system. To meet ever increasing load demand there is need of
power system expansion/upgradation. In order to postpone the
upgardation/expansion, power engineers load the power system fully while
all technical constraints are met and system is in safe limits. For safe
operation of power system it is important to find its capacity of being
loaded while operating point is within boundaries. To know the safe
margin between the operating point and maximum loadability it is
important to find maximum loadability limits. The major factor of
concern in finding loadability limits is voltage profile of the system.
The drop in voltage profile causes voltage instability. ~\\
To solve this problem, authors solved mainly loadability maximization
problem while technical constraints are not violated {[}1-18{]}. Few
authors considered problem of loadability maximization and load
curtailment simultaneously {[}4-8{]}. While few others considered cost
of generator {[}9{]}, steady and dynamic security as additional factors
in loadability maximization {[}10-11{]}.\\
To solve maximum loadability limits problem, various techniques are
available in the literature In earlier years conventional methods based
on continuation power flow {[}1{]} and Interior Point method for
non-linear and sequential problem were developed {[}2,4-7{]}. Few
authors modified the interior point algorithm to solve the problem when
zone-based constraints are imposed on the loads {[}8{]}. While few
authors calculated the nearest bifurcation point from an interior or an
exterior point of the feasible region {[}12-13{]}. Some authors solved
the problem by optimal adjustment of generation and load under
constraints and treated the problem as an optimal power flow problem
{[}14-15{]}.\\
But in these methods {[}1-10, 12-15{]}, the selection of parameters
directly affects the performance of the algorithm. To overcome this,
later on population based evolutionary algorithms became common due to
their simplicity, ease of application and ability to provide optimal or
near optimal solution. For last decade, authors proposed PSO based
approaches to find maximum loadability limits {[}9, 16-18{]} because of
its simplicity. But it is found that sometimes PSO goes into local
minima. To overcome this problem authors implemented other methods in
addition with PSO. Authors implemented genetic algorithm in addition
with PSO to solve maximum loadability limit problem for small size
network {[}16{]}. Later on {[}9{]} solved the problem using Multi agent
based PSO. Authors implemented particle swarm optimization and breeding
feature of Genetic algorithm to find maximum loadability limits and cost
of generation for the same {[}9{]}.\\
Authors implemented various PSO based approaches as simple PSO, APSO and
CPSO to find maximum loadability limits considering different
limitations {[}17{]}. Afterwards {[}18{]} proposed PSO and DE to find
loadability limits because of excellent convergence characteristics of
DE and implemented it successfully on 118-bus and 30-bus system
{[}9{]}.\\
In this paper, another population based firefly algorithm in combination
with load incremental algorithm is proposed to find maximum loadability
limits of power system. Firefly algorithm is a nature inspired algorithm
proposed by X.S.Yang {[}19-21{]}. The algorithm is based on the
behaviour of fireflies. The algorithm is sufficiently random in nature
and produces optimal solution in less computational time. It has been
applied successfully on NP hard problems {[}22-32{]}. It has been found
that firefly has ability to provide better solution than other existing
methods in various engineering applications viz. economic dispatch {[}
22-23{]}, travelling salesman problem {[}24{]}, job scheduling {[}25{]},
automatic generation control {[}26{]}, image compression {[}27{]},
highly nonlinear multimodal design problems {[}28{]} and multiobjective
problems {[}29{]}. Authors found that FA is consistent and has better
performance than existing evolutionary algorithms {[}30{]}.\\
~Keeping this in view, in this paper problem of finding maximum
loadability limits has been solved using firefly algorithm. The results
obtained using firefly is updated using LIA by increasing load in small
steps on \emph{PQ} buses while voltage constraint is satisfied. LIA
gives quality solution because of its ability of modifying solution
precisely. Due to presence of LIA optimal solution is obtained for small
population size and less number of iterations in FA which otherwise
directly affects the quality of solution. The applicability and
effectiveness of the proposed approach is tested on 30-bus, 118 bus and
300-bus test systems. From literature survey it is found that approach
proposed by {[}18{]} has better results than other existing evolutionary
algorithms {[}9, 16-17{]}. Therefore in this paper results are compared
with results obtained in {[}18{]}. On comparison it is found that the
results obtained using proposed approach are better than {[}18{]}.\\
\textbf{2.} \textbf{Problem Formulation}\\
To find maximum loadability limits of power system, the objective is to
increase the active power loading on load buses of transmission network
in such a manner that load is maximized while voltage constraint is not
violated. Mathematically it can be given as:\\
(1)~\\
Where \textbf{}is increment load at bus \emph{i} in addition to base
load while voltage constraint is not violated.\\
Subject to\\
(2)\\
(3)\\
(4)\\
(5)\\
(6)\\
\textbf{3. Solution Approach}\\
\textbf{~}\\
In this section, firefly algorithm is introduced and then its
implementation in combination with load incremental algorithm to solve
maximum loadability problem is discussed.\\
~\\
\textbf{3.1} \textbf{Overview of Firefly Algorithm}\\
The firefly algorithm is a recent nature inspired technique which mimics
the behaviour of fireflies. Fireflies flash light in the sky in summer
in tropical and temperate regions during night. This flash of light is
the mode of communication between fireflies. Firefly with less light
intensity flashes are attracted towards high intensity flashes of
fireflies. Light intensity changes with distance from other fireflies
and some intensity is lost in medium. As algorithm mimics the behavior
of fireflies, the solution of any problem is evaluated in terms of light
intensity in the presence of some variables for light intensity lost and
distance between fireflies. For simplicity in describing FA, the
following three idealized rules are framed {[}19-21{]}:\\
(i). All fireflies are unisex so that one firefly will be attracted to
other fireflies regardless of their sex.\\
(ii). ~Attractiveness is proportional to their brightness, thus for any
two flashing fireflies, the less brighter one will move towards the
brighter one. The attractiveness is proportional to the brightness and
they both decrease as their distance increases. If there is no brighter
one than a particular firefly, it will move randomly.\\
(iii). The brightness of a firefly is affected or determined by the
landscape of the objective function.\\
~\\
The steps of firefly algorithm are as mentioned below:\\
\emph{(a)} \emph{Random Generation of Fireflies}\\
Initially a random population of fireflies is generated within the
\emph{d}-dimensional search space of the given problem. Each firefly
represents the solution of the problem. In this paper \emph{d} is
\emph{N\textsubscript{PQ}}.\\
\emph{(b)} \emph{Light Intensity or Brightness}\\
In firefly algorithm, light intensity or brightness of a firefly is
calculated for objective function \emph{f(x)} to be optimized. For a
maximization problem, the brightness is directly proportional to the
value of the objective function evaluated for a problem. In the simplest
case for maximum optimization problems, the brightness \emph{I} of a
firefly at a particular location \emph{x} can be chosen as \emph{f(x)}
at that point.\\
\emph{(c ) Attractiveness towards Brightness}\\
As fireflies with less light intensity move towards high intensity
fireflies which depends on attractiveness and light absorption. The
attractiveness is relative; it is seen in the eyes of the beholder or
judged by the other fireflies. Light intensity decreases with the
distance from its source, and it is also absorbed in the media, so we
should allow the attractiveness to vary with the degree of absorption.
In the simplest form, the light intensity \emph{I}(\emph{r}) varies
according to the inverse square law \emph{I}(\emph{r}) =
\emph{Is/r}\textsuperscript{2}. For a given medium with a fixed \selectlanguage{greek}γ,
\selectlanguage{english}\emph{I} between two fireflies f \emph{} and \emph{h} \emph{} varies
with \emph{r\textsubscript{fh}} between them \emph{i.e.}\\
~\\
~(7)\\
Where\\
The firefly's attractiveness is proportional to the light intensity as
seen by adjacent fireflies. It can be defined as . Initially ~at
\emph{r} = 0.In the implementation, the actual form of attractiveness
function can be any monotonically decreasing function such as the
following generalized form :\\
(8)\\
Generally \emph{m} is considered as 2.\\
\emph{(d)} \emph{~Movement}\\
The movement of a firefly \emph{h} is attracted to another more
attractive (brighter) firefly \emph{f} is determined as:\\
(9)\\
In the above equation, the second term is due to the attraction while
the third term is randomization introduced in the algorithm. Generally
the value of \selectlanguage{greek}α \selectlanguage{english}lies between 0 and 1.\\
~\\
\textbf{3.2} \textbf{Algorithm}\\
To solve maximum loadability limits problem, FA and LIA are used in
combination. The pseudo codes and flowchart for the same is as given
below:\\
\textbf{FA( ): To find maximum loadability limits}\\
\textbf{~}\\
Input: Read linedata, busdata, number of iterations, size of population,
number of PQ\\
~\\
buses, ~~\\
Output: Maximum loadability limits\\
~\\
Step1. Generate a population of fireflies of size . Each firefly
represents incremental load ~for given number of buses. For firefly
\emph{f}, incremental load for \emph{u\textsuperscript{th}} load bus can
be given as:\\
~\\
(10)\\
Step2. Set Counter=1, b=1.\\
Step3. While (Counter\textless{} Number of iterations)\\
for (b=1: n) /* for loop*/\\
a. Add b\textsuperscript{th} firefly with original load of PQ buses of
test system.\\
b. Run load flow analysis of given test system in the presence of this
additional load with base load on PQ buses.\\
c. Check for voltage constraint as in (4). Add penalty
(\emph{k\textsubscript{i}}) for constraint violation on buses and
evaluate its fitness consisting of total load increment and penalty for
constraint violation as given below:\\
(11)\\
d. b=b+1;\\
End /*end of for loop*/\\
~\\
Step4. Sort fireflies corresponding to their fitness (light
intensity).\\
Step5. Update fireflies with less light intensity by moving towards more
light intensity fireflies as in (9).\\
Step5. Counter= Counter+1.\\
If counter\textless{}= Number of iterations\\
Go to Step3\\
else\\
Exit ()\\
End/* end of if\\
End /*end of while loop*/\\
~\\
Step6. Return best solution\\
Pseudo code of Firefly Algorithm\\
\textbf{~}\\
\textbf{Load Incremental Algorithm ( ): To improve the solution of FA}\\
~\\
Input\textbf{:} Number of PQ buses, Incremental load \selectlanguage{greek}δ, \selectlanguage{english}best solution of
FA\\
Output: Modified solution is obtained\\
Step1. Consider FA best solution as input for load modification.\\
Step2. s=1\\
Step3. While (s\textless{}=N\textsubscript{PQ})\\
a. Load of s\textsuperscript{th} bus= Load of s\textsuperscript{th} bus+
\selectlanguage{greek}δ /\selectlanguage{english}*Increase load of s\textsuperscript{th} bus*/\\
b. Perform load flow analysis\\
c. If voltage constraint is not satisfied\\
then\\
(i). Load of s\textsuperscript{th} bus= Load of s\textsuperscript{th}
bus- \selectlanguage{greek}δ\selectlanguage{english}\\
(ii). s=s+1\\
(iii). ~Go to Step 3\\
~\\
Else\\
Go to Step 3.a\\
~\\
End /*end of while loop*/\\
~\\
Step4. Return best solution\\
~\\
Pseudo code of Load Incremental Algorithm\\
\textbf{Insert Fig 1(a) here}\\
\textbf{Fig 1(a)-Flowchart of the FA Algorithm}\\
\textbf{~}\\
\textbf{Insert Fig 1(b) here}\\
\textbf{Fig 1(b)-Flowchart of the LIA Algorithm}\\
\textbf{4. Results and Conclusions}\\
The proposed approach is tested on 30 bus, 118-bus and 300-bus test
systems {[}31{]}. The program for the problem is written in MATLAB
software. The linedata and busdata for test systems are as in
{[}31{]}.\\
\emph{4.1 Selection of Parameters}\\
The performance of firefly algorithm is affected by the values of \selectlanguage{greek}α, γ
\selectlanguage{english}and \selectlanguage{greek}β \selectlanguage{english}parameters. The value of \selectlanguage{greek}α \selectlanguage{english}varies from 0 to 1. Its value is
generated randomly between 0 and 1 which adds randomness to the
algorithm. The value of \selectlanguage{greek}γ \selectlanguage{english}can vary from {[}0 . If \selectlanguage{greek}γ \selectlanguage{english}is considered 0 that
means the attractiveness is constant between fireflies and there is no
movement of fireflies which may give single optimum local maxima. If \selectlanguage{greek}γ\selectlanguage{english}=[?]
then there is no attraction i.e. sight of other fireflies is
short-sighted with respect to other fireflies which leads each firefly
to roam in a completely random way {[}19-21{]}. To avoid this, value of
\selectlanguage{greek}γ \selectlanguage{english}is varied between 0.01 and 100 in an iterative manner which finally
results \selectlanguage{greek}γ\selectlanguage{english}=1 while \selectlanguage{greek}β\selectlanguage{english}\textsubscript{0} is taken as 1. In this paper, \selectlanguage{greek}α \selectlanguage{english}is
varied as number of iterations increases as in PSO {[}32-33{]}.\\
\emph{4.2 Selection of Population Size and Number of Iterations}\\
The quality of solution obtained using FA depends upon number of
iterations and population size. Hence first step to solve loadability
problem is to select a suitable population size and number of iterations
for 30 bus, 118 bus and 300 bus test systems. As in AI based techniques
it is difficult to predict a single value obtained as a best or worst
solution. Therefore in this paper, 50 trials are run for each population
size for given number of iterations for all test cases.\\
In this paper population size is varied from 20 -50 with step size of 10
i.e. 20, 30, 40 and 50 respectively for all test cases while iterations
are changed from 20 to 100 with step size of 10. The steps of population
size and number of iterations selection is discussed in detail for
30-bus test system.\\
On varying population size and number of iterations for 30-bus system,
the results obtained are shown in Fig 2. It is clear from Fig 2 that a
small population size of 20 and \emph{itmax} 20 gives optimal solution
while there is no variation in solution quality as number of iterations
is increased for population size 20. It is also evident from Fig 2 that
as size of population increases from 30-40 and onwards, it takes more
number of iterations for optimal solution. Similar pattern is observed
for increase in number of iterations for a given population size as
solution is not improving while it increases computational burden
Therefore a small population size is considered as good option for
quality solution for 30-bus test system. In this paper, a small
population size and less number of iterations converge to a quality
solution because of LIA. In LIA the solution obtained after given number
of iteration from FA is considered as input and is incremented in small
steps and the solution is modified precisely. Therefore for 30-bus test
system, population size of 20 and \emph{itmax} of 20 is considered for
comparison and validity of suggested algorithm. ~It is also evident from
Fig 2 that loadability lie in a narrow range of 277.79 MW to 280.34 MW
for various population size and number of iterations because of LIA
which modifies load on \emph{PQ} buses one by one till there is scope of
load addition.\\
\textbf{~}\\
\textbf{Insert Fig 2 here}\\
\textbf{Fig 2-Variation of Population Size and Iterations for 30-bus
Test System}\\
Similarly, an iterative process is carried for 118 bus and 300 bus test
systems which results population size of 30 and \emph{itmax} 30 for
118-bus test system. While for 300-bus test system, it is found that a
population size of 30 and \emph{itmax} 40 gives promising results. With
increase in population size beyond 30 and \emph{itmax} 40 in 300-bus
test system does not improve the solution rather computational time
increases.\\
\selectlanguage{greek}\emph{4.3 Step size of δ}\selectlanguage{english}\\
In LIA, the best solution obtained using FA is modified by increasing
load in small steps on \emph{PQ} buses. The step size of incremental
load affects the computational time and quality of solution. If very
small step size is considered, time for solution increases. On other
hand if step size is high, the solution obtained using FA is not
modified significantly. In this paper, load incremental step is varied
from 0.01 to 0.00001 which results \selectlanguage{greek}δ\selectlanguage{english}=0.001 as most suitable step size
for load increment in LIA algorithm.\\
From above discussion it is clear that a small population size and less
number of iterations for a given test system can give an optimal
solution because of LIA which increase the loadability precisely with a
small incremental step.\\
\emph{4.4 30-bus Test System {[}31{]}}\\
The validity and effectiveness of proposed approach is tested on 30-bus
test system which consists of 20 \emph{PQ} buses, 6 generators and 41
branches. The minimum and maximum voltage limits are considered as 0.95
p.u and 1.0 p.u as given in {[}31{]}..\\
As discussed in Section 4.2 for 30-bus test system, the size of firefly
and number of iterations are considered as 20. While other parameters
considered are as \selectlanguage{greek}γ\selectlanguage{english}=1, \selectlanguage{greek}β\selectlanguage{english}\textsubscript{0}=1 and \selectlanguage{greek}δ\selectlanguage{english}=0.001. To find initial
loadability on 30-bus system, load flow analysis is performed on base
case which results total generation and loading as given below:\\
To solve maximum loadability problem, a population of fireflies of size
corresponding to \emph{PQ} buses is generated. i.e 20 *20. Fitness
function for each firefly is evaluated as given in (11) in Section 3.2.
The program is run for given number of iterations i.e. 20. The result
obtained using firefly algorithm after 20\textsuperscript{th} iteration
is considered as input to LIA. A small step size of load is incremented
on load buses as explained in Section 3.2 which yields total load of
2.8438 p.u while voltage at all buses is within limits as given below:\\
The final load in MW and voltage profile obtained using FAand LIA is
tabulated in Table 1. It is clear from 3\textsuperscript{rd} colum of
Table 1 that for this load increase there is no voltage constraint
volation.\\
\textbf{~}\\
\textbf{Insert Table 1 here}\\
\textbf{Table 1-Load and Voltage Profile of 30-bus test system with
proposed approach}\\
~\\
To check the performance of suggested approach, results are compared
with base case and DEPSO based method. As in loadability problem,
voltage constraint is the main constraint which should be satisfied. To
study voltage profile in detail for updated load, the voltage profile
along with voltage angles for base case and DEPSO are studied and
mentioned in Table 2. On looking at 2\textsuperscript{nd} column of
Table 2 that minimum voltage of 0.961 p.u. occurs at node 8 in base
case. After solving loadability problem using DEPSO and proposed
approach, minimum voltage of magnitude 0.95 (approx) occurs at node 8
(as tabulated in 4\textsuperscript{th} and 6\textsuperscript{th} column
of Table 2) which is in prescribed limits while minimum voltage angle
-3.96 degree occurs at node 19 in base case which further decreases to
-10.91 and -9.67 degrees in DEPSO and proposed approach respectively.\\
~\\
\textbf{Table 2-Complete Voltage Profile along with Voltage Angles for
Base case, DEPSO and Proposed Approach for 30-bus system}\\
\textbf{Insert Table 2 here}\\
~\\
To analyze the results in detailed manner, proposed approach results are
compared for \emph{P\textsubscript{G,} Q\textsubscript{G},
P\textsubscript{D,} Q\textsubscript{D}}, voltage magnitude, voltage
angle and losses with base case and DEPSO. It is clear from last column
of Table 3 that the proposed approach increases loadability to 50.32\%
from base case while it is 42.32 \% in DEPSO. ~On comparing the real and
reactive power losses with DEPSO, it is found that there is no
significant difference between two methods while the proposed approach
yields nearly 8\% more load increase than DEPSO.\\
\textbf{Table 3-Comparison of Base case, DEPSO and Proposed Approach for
30-bus system}\\
\textbf{Insert Table 3 here}\\
To check the validity of the suggested approach. The results obtained
from ~suggested approach are compared with existing methods. The
statistical analysis for 50 trials for populations size 20 and
\emph{itmax} 20 is carried out. The results obtained ~are compared with
existing DE and DEPSO based approaches. The best, worst and mean
solutions for the same are tabulated and compared in Table 4. On looking
at Table 4 it is clear that the worst solution obtained using proposed
approach is even better than the other existing methods. The best
solution yields maximum line loadability of 2.8438 pu while it is 2.6709
p.u and 2.6974 p.u in DE and DEPSO respectively. The average value of
loadability of suggested approach is even better than best solution
obtained in DE and DEPSO while total computataional time is slightly
more than other methods due to LIA.\\
~\\
\textbf{Table 4-Statstical Analysis of Proposed Method for 30-bus System
for 50 trials}\\
\textbf{Insert Table 4 Here}\\
\emph{4.5 118-Bus Test System{[}31{]}}\\
The busdata and linedata for 118 bus test system is as given in
{[}31{]}. The busdata consists of 54 generators, 64 load buses, 14
shunts, 186 branches and 9 transformers. The minimum and maximum
voltages are 0.94 p.u and 1.0 p.u respectively as given in {[}31{]}.\\
First of all load flow analysis is performed for the test system for
base case which results total load on the system as mentioned below:\\
To increase loadability for the selected population size and
\emph{itmax} as in Section 4.2\emph{,} a random population of fireflies
of size 30*64 is generated. ~Each firefly represents a small incremental
load which can be added to the base load of \emph{P-Q} buses. Consider
each firefly and perform load flow analysis for load increase
(incremental load +original load on \emph{PQ} buses) as explained in
Section 3.2. After given number of iterations of FA, the load is further
increased using LIA as in Section 3.2. Finally total load obtained using
FA and LIA in combination is as given below:\\
The total load on system is 65.63 p.u. ~The load on buses after
modification is tabulated in Table 5 while voltage on all buses is
within limits as shown in Fig 3. ~The voltage profile for base case and
proposed approach is shown in Fig 3. On looking at Fig 3, it is clear
that with increase in load (FA+LIA) on system, minimum bus voltage of
0.94 p.u. occurs at bus118. The results for voltage magnitude and
voltage angles for base case and suggested approach are compared as in
Table 6. It is found that for base case minimum voltage 0.943 p.u.
occurs at bus 76. For base case, minimum voltage angle of 7.05 deg at
bus 41 while it is -54.28 degree at bus 1 for suggested approach.\\
\textbf{Table5-Final Load on all Buses in MW}\\
\textbf{Insert Table 5 Here}\\
~\\
\textbf{Table 6-Comparison of Base Case and Proposed Approach}\\
\textbf{Insert Table 6 Here}\\
~\\
\textbf{Insert Fig 3 Here}\\
\textbf{Fig 3-Voltage Profile of 118-bus system}\\
~\\
The effectiveness of the suggested approach is tested by comparing its
results with existing DE and DEPSO based methods as in Table 7. It is
clear from 3\textsuperscript{rd} column of Table 7 that the total load
using proposed approach is more than the other existing DE and DEPSO
methods. On comparing the results it is found that that the proposed
approach increases loadability to 54.73 \% from base load while it is
33.47\% and 34.39\% in DE and DEPSO methods respectively. The suggested
approach yields nearly 9\% more loadability than DEPSO.\\
To validate the proposed approach, the statistical analysis of method is
done for 50 trials for population size 30 and 40 iterations. The best,
average and worst solution obtained during these trials are tabulated in
Table 8. Looking at Table 8, it is found that best, average and mean
solution of firefly and LIA based approach is better than existing
methods. The total computation time of the proposed approach is 234.08
seconds while it is less in DE and DEPSO methods. The computational time
is affected by the step size of incremental load which is added in LIA.
But solution obtained is better than other existing methods.\\
\textbf{Table 7-Comparison with Other Existing Methods}\\
\textbf{Insert Table 7 Here}\\
\textbf{Table 8-Statstical Analysis of Proposed Method for 118-bus
System}\\
\textbf{Insert Table 8 Here}\\
\emph{4.6 300-bus system{[}31{]}}\\
To check the applicability of proposed approach on a large test system,
it is tested on 300-bus system which consists of 69-generators,
201-\emph{PQ} buses, 29-shunts, 411- Branches and 107-Transformers. The
busdata and linedata is as given in {[}31{]}. Generally voltage
constraint for transmission system is \selectlanguage{ngerman}± 5\% while for study purposes, in
this paper voltage constraint is relaxed and considered from 0.9 p.u to
1.0 pu. The coding of buses is considered as in {[}31{]}.\\
Load flow analysis is done on 300-bus test system for base case which
results total generation and load as given below:\\
To solve the maximum loadability problem, a population of size 30* 201
is generated. Each firefly presents a small incremental load on
\emph{PQ} buses in addition to base load. Fitness of each firefly is
evaluated as in (11) and modified for next iteration corresponding to
the brightness of the firefly. After 40 iterations, the solution
obtained using FA is modified with LIA which gives the solution. The
results are compared with base case for various parameters viz. loading,
voltage magnitude and voltage angle. The results are tabulated in Table
9. It is clear from 4\textsuperscript{th} column of Table 9 that the
base case has minimum voltage of 0.9287p.u. at bus numbers 282 while it
decreases to 0.9 p.u at node 282 using FA+LIA algorithm. The total load
on the system is increased to 23.99 pu which is nearly 24 \% more than
the base case which causes voltage angle to decrease from -37.54 degrees
to -57.14 degrees at bus 528.\\
The competence of the proposed approach in providing quality solution
for a small population size and less number of iterations, in this work,
statistical analysis for two cases is carried out. Case 1 consists of
population size 20 and \emph{itmax} 20 while Case 2 has selected
population size of 30 and \emph{itmax} 40 as in Section 4.2. ~For both
cases 50 trials are run and results obtained are shown in Fig 4. It is
clear from Fig 4 that Case 1 provides a solution which is comparable
with Case 2. Case 2 has bigger population size and more iterations but
solution is not significantly better.\\
Looking at Fig 4 it is found that the results for both cases lies in a
narrow range from 23988 MW (worst) to 23994 MW (best) due to LIA which
increases load precisely on \emph{PQ} buses till voltage constraint is
met. The statistical analysis is done for both cases and presented in
detail in Table 10. Best, mean and worst solution for both cases is
compared. It is clear from 3\textsuperscript{rd} column of Table 10 that
the best solution in both cases is almost same while mean and worst
solution are also comparable. But there is difference in computational
time for both cases per trial. It is 420.4 sec more in case 2 per
trial.\\
\textbf{Table 9-Comparison of Base Case and Proposed Approach}\\
\textbf{Insert Table 9 Here}\\
\textbf{~~Table 10-Comparison of ~Case 1 and Case2 ~}\\
\textbf{Insert Table 10 Here}\\
~\\
\textbf{Insert Fig 4 Here}\\
~\\
Fig 4-comparison of Case 1 and Case 2 for 50 trials\\
From above discussion it is clear that application of LIA in addition to
FA improves the quality of solution and solution lie in very narrow
range. Although it increases the computational time to some extent yet
it improves the results significantly.\\
\textbf{5. Conclusions:}\\
~The proposed approach to find maximum loadability limits of the power
system has ability to provide an optimal solution. The proposed approach
is tested on 30-bus, 118 bus and 300-bus systems. The results are
compared on various parameters with existing methods and base case. The
quality of solution obtained for test systems is better than other
existing techniques. On comparison it is found the maximum loading limit
using proposed approach is more than other existing methods while
voltage constraint is within its limits. From statistical analysis it is
found that the mean and worst results obtained using proposed approach
are better than other methods for small size of population and less
number of iterations. On comparing the computational time it is found
that the proposed approach takes more time because of step increase of
load in LIA. But it does not limit the effectiveness and applicability
of proposed approach to big size network. The effectiveness of suggested
approach on large network of 300-bus system is analyzed which provides
quality solution.\\
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\selectlanguage{english}
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