a comprehensive study of novel metaheuristic …...da dragonfly algorithm de differential evolution...
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A Comprehensive Study of Novel Metaheuristic Techniques for Optimal Power
Flow Solution
A PhD Synopsis
submitted to Gujarat Technological University for the Degree of
the Doctorate in Philosophy
in
Electrical Engineering
by
Hitarth Buch
149997109002
under supervision of
Prof. (Dr.) Indrajit N Trivedi
Professor – Power Electronics
Vishwakarma Government Engineering College, Chandkheda, Ahmedabad
Gujarat Technological University, Ahmedabad
June - 2020
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Contents
List of Abbreviations ............................................................................................................... 2
1) Abstract ............................................................................................................................. 5
2) State of the art of the research topic ............................................................................... 6
3) Definition of the problem ............................................................................................... 10
4) Objective and Scope of work ......................................................................................... 12
5) Original contribution by the thesis ............................................................................... 14
6) The methodology of research, results/comparisons..................................................... 14
7) Results/comparisons ....................................................................................................... 17
8) Assessment of Techno-Economic-Environmental Benefits ......................................... 26
9) Achievements concerning objectives ............................................................................. 27
10) Conclusion ....................................................................................................................... 29
11) Publications ..................................................................................................................... 30
12) References ........................................................................................................................ 32
2
List of Abbreviations
ABC Artificial Bee Colony Algorithm
ACO Ant Colony Optimization
ALO Ant Lion Optimizer
AGSA Adaptive Group Search Algorithm
AMFO Adaptive Moth Flame Optimization
ANN Artificial Neural Network
APFPA Adaptive Flower Pollination Algorithm
ARCBBO Adaptive Real Coded Biogeography Based Optimization
AST Average Simulation Time
Avg Average
BBO Biogeography Based Optimization Algorithm
BCS Best Compromised Solution
BLDC Brushless DC Motor Design
B-MMOFPA Bio-inspired Modified Multi-Objective Flower Pollination Algorithm
BSA Backtracking Search Optimization Algorithm
CABC Chaotic Artificial Bee Colony Algorithm
CKHA Chaotic Krill Herd Algorithm
CMICA Combined Modified Imperialist Competitive Algorithm
CSDHA Chaotic Self-Adaptive Differential Harmony Search Algorithm
DA Dragonfly Algorithm
DE Differential Evolution
DSA Differential Search Algorithm
DV Voltage Deviation
E Emission
EA Evolutionary Algorithm
EADDE Evolutionary Ant Direction Differential Evolution
EP Evolutionary Programming
ES Evolutionary Strategy
ESDE-MC Enhanced Self-adaptive Differential Evolution with Mixed Crossover
FCMF Fuel Cost with Multifuel
FCPOZ Fuel Cost with Prohibited Operating Zone
FCVPL Fuel Cost with Valve-Point Loading
FHSA Fuzzy Harmony Search Algorithm
FPA Flower Pollination Algorithm
GA Genetic Algorithm
GD Generational Distance
GEM Grenade Explosion Method
GOA Grasshopper Optimization Algorithm
GS Gradient Search
GSA Group Search Algorithm
GSO Glowworm Swarm Optimization
GWO Grey Wolf Optimizer
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HC Hillside Climbing
HIPSO Hierarchical Interactive Particle Swarm Optimization
HSA Harmony Search Algorithm
HV Hypervolume
ICA Imperialist Competitive Algorithm
ICBO Improved Colliding Bodies Optimization
ICLS Iterated Citizen Look Search Algorithm
IEEE Institute of Electrical and Electronics Engineers
IGD Inverse Generational Distance
IMO Ions Motion Optimization
IP Interior Point
ISPEA Improved Strength Pareto Evolutionary Algorithm
KHA Krill Herd Algorithm
LCA League Championship Algorithm
LDI-PSO Linearly Decreasing Inertia Weight Particle Swarm Optimization
LP Linear Programming
LTLBO Levy Mutation Strategy for Teaching Learning Based Optimization
MDE Multi-objective Differential Evolution
MF Multifuel/Multi-fuel
MFO Moth-Flame Optimization
MGBICA Modified Gaussian Bare-bones Imperialist Competitive Algorithm
MICA Modified Imperialist Competitive Algorithm
MOALO Multi-Objective Ant Lion Optimizer Algorithm
MOCSO Multi-Objective Cat Swarm Optimization
MODA Multi-Objective Dragonfly Algorithm
MOEA/D Multi-Objective Evolutionary Algorithm Based on Decomposition
MOFA-CPA Multi-Objective Firefly Algorithm with a Constraints-Prior Pareto-
Domination Approach
MOGWO Multi-Objective Grey Wolf Optimizer Algorithm
MOICA Multi-Objective Imperialist Competitive Algorithm
MOIMO Multi-Objective Ions Motion Optimization Algorithm
MOMFO Multi-Objective Moth Flame Optimization Algorithm
MOMICA Multi-Objective Modified Imperialist Competitive Algorithm
MOMVO Multi-Objective Multi-Verse Optimization Algorithm
MOO Multi-Objective Optimization
MOOPF Multi-Objective Optimal Power Flow
MOPSO Multi-Objective Particle Swarm Optimization
MoS Metric of Spacing
MOSSA Multi-Objective Salp Swarm Algorithm
MSA Moth Swarm Algorithm
MSFLA Modified Shuffled Frog Leap Algorithm
MTLBO Modified Teaching Learning Based Optimization Algorithm
MVO Multi-Verse Optimizer Algorithm
NFL No Free Lunch
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NLP Non-Linear Programming
NR Newton Raphson
NS Non-dominated Sorting
NSGA Non-dominated Sorting Genetic Algorithm
NSGWO Non-dominated Sorting Grey Wolf Optimizer
NSIMO Non-dominated Sorting Ions Motion Algorithm
NSMFO Non-dominated Sorting Moth Flame Optimization Algorithm
NSMOOGSA Non-dominated Sorting Multi-Objective Opposition-based Gravitation
Search Algorithm
NSSCA Non-dominated Sorting Sine Cosine Algorithm
OPF Optimal Power Flow
PAES Pareto Archived Evolution Strategy
PL Active Power Loss
POZ Prohibited Operating Zone
PSO Particle Swarm Optimization
PV Pressure Vessel
QFC Quadratic Fuel Cost
QL Reactive Power Loss
QP Quadratic Programming
SA Simulated Annealing
SCA Sine Cosine Algorithm
SD Standard Deviation
SKHA Stud Krill Herd Algorithm
SP Satellite Pipe
SSA Symbiotic Search Algorithm
TFC Total Fuel Cost
TLBO Teaching Learning Based Optimization
TS Tabu Search
TSA Tree-Seed Algorithm
VPL Valve-Point Loading
VSI Voltage Stability Index
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1) Abstract
Optimal economic planning and operation of power production have always remained a point
of great significance in the power industry. When power is delivered from several producing
units to meet a continually fluctuating load, economic competence is desired, and the output of
each generator is set to maintain the fuel cost to the bare minimum. The optimal power flow
(OPF) problem encompasses economic dispatch and the power flow. Hence, in the operation
of the power system, OPF is an essential everyday optimization task.
Researchers have developed many classical and heuristic methods to solve the OPF problem.
Because of the valve-point loading effect, use of multiple fuels, transformer tapping, and shunt
compensation, the OPF is a non-linear, non-convex, and intermittent problem. Classical
methods often fail to determine global optima for complex OPF problems as these methods
need differentiable and continuous objective functions and also usually tend to get caught in
the local optima. These limitations can be overcome if the OPF problem is solved using
derivative-free approaches. Metaheuristic methods directly use "fitness" information instead of
function derivatives or other linked information. It follows stochastic rather than deterministic
transition rules. Thus, metaheuristic techniques can be considered as an excellent alternative to
classical methods for solving the non-linear and non-convex OPF problem. Hence,
performance evaluation of metaheuristics for solving the OPF problem is an exciting but
challenging task. Therefore, a thorough performance assessment of metaheuristics has been
carried out in this research.
As the first step towards OPF optimization using different metaheuristics, 8 different
algorithms were applied to optimize 22 single-objective optimal power flow objective
functions on IEEE 57-bus and IEEE 118-bus test systems. The results were compared,
considering different performance indicators. Various statistical techniques were also applied
to check the performance of the approaches. In this case, the MFO provided the best results
amongst all algorithms for majority performance indicators.
The above results motivated to develop a new variant of the MFO, i.e., the Adaptive MFO
(AMFO), to solve complex OPF problems. The AMFO uses adaptive step size to update moth
position. Fourteen unimodal and multimodal single-objective benchmark functions were used
to assess the performance of the AMFO and MFO. After confirming the efficacy of AMFO on
benchmark functions, the AMFO was tested on the IEEE 118-bus test system for optimizing
13 different complex objective functions. The performance of AMFO is evaluated in terms of
various performance criteria. For the majority of these criteria, the AMFO provided better
results as compared to the rest of the approaches demonstrating its efficiency to deal with the
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OPF problem. Three different statistical tests were also employed to confirm that AMFO
results are not obtained by chance.
For real-world problems, no single solution exists that concurrently optimizes every objective.
Such problems are said to have conflicting objectives, and there is possibly an infinite number
of best solutions. The OPF also is one such problem. Multi-objective optimization algorithms
are required to be applied for solving such problems. The concept mooted in the NFL algorithm
applies to multi-objective problems as well.
In this work, two multi-objective versions of existing single-objective IMO were proposed.
The first version is based on the Non-dominated Sorting Ions Motion Algorithm, i.e., NSIMO.
The NSIMO uses elitist non-dominated sorting and crowding distance approaches to attain
different non-domination levels and to retain diversity between the optimum set of solutions.
The second version is achieved by integrating single-objective IMO with external storage and
leader selection strategy. This storage maintains the best non-dominated solutions obtained so
far during optimization, and the leader selection strategy helps the search process towards the
least crowded region of the Pareto front. The suggested methods were applied to several multi-
objective benchmark functions having distinct characteristics. The outcomes were compared
based on various performance metrics. The results obtained were analyzed with recently
proposed algorithms. Both proposed approaches provided competitive, if not better, results for
majority benchmark functions.
After confirming its efficacy on standard benchmark functions, the NSIMO and MOIMO were
also applied for solving the multi-objective OPF problem on different test systems. The results
were assessed in terms of different performance standards over 30 independent runs. The
algorithms were also ranked for HV values based on the statistical tests. Both approaches stood
as strong contenders for solving the MOOPF problem.
2) State of the art of the research topic
After the introduction of the idea of the Optimal Power Flow (OPF) by Carpentier [1] in 1962,
the OPF importance is increasingly recognized. In the present era, it has been developed into
an important and essential tool to determine the most economical and secure state of power
system planning and operation. Numerous classical methods like Newton Raphson (NR)
Method, Linear Programming (LP) Method, Nonlinear Programming (NLP) method, Quadratic
Programming (QP) method, and the Interior Point (IP) method have been applied to solve the
OPF problem [8], [13], [26]–[33], [34]–[42].
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For example, in [8], Rosehart et al. proposed the combination of interior-point method and goal
programming method to optimize active and reactive power dispatch while maximizing voltage
security in IEEE 57-bus and IEEE 118-bus test systems. In [4], Mangoli et al. employed the
LP method to minimize the quadratic fuel cost for 6-bus and modified IEEE 30-bus test
systems. Bottero et al. [7] proposed a reduced Hessian method for optimizing the fuel cost. The
P-Q decomposition method was employed in [6] for three different bus systems of 5-bus, 30-
bus, and 962-bus test systems. This work optimized quadratic fuel cost and active power loss.
Momoh optimized quadratic fuel cost, active power loss, and voltage deviation using the LP
method in [21] on the 39-bus test system. Sun et al. [11] employed a quasi-newton based
approach to reduce active power loss on a portion of the North-Eastern United States. The fuel
cost and power loss were minimized in [5] using the IP method on the well-known IEEE Test
Systems with 30, 57, 118, and 300 buses.
The main disadvantage of a classical numerical optimization method is that it often converges
to local optima due to the gradient-based search method it employs [22]. Handling inequality
constraints becomes difficult with gradient-based and newton methods. OPF relationships are
simplified to ensure convexity while implementing Non-linear Programming (NLP), and
Quadratic Programming (QP) as these methods rely on convexity to obtain the optimal
solution. Linear Programming (LP) is meant to handle linear functions for the input-output
relationship. Interior Point (IP) method is computationally fast and efficient; however,
improper step size may lead to an infeasible solution for the sub-linear problem in the original
non-linear domain. Moreover, the IP method suffers from a lousy initial, optimality, and
termination criteria. Classical numerical methods also become sluggish with growing system
size. These drawbacks make classical approaches unsuitable for non-linear, multimodal, large,
and complex OPF problems. A detailed review of these methods can be found in [23], [24].
The stochastic search method of the Evolutionary Algorithm (EA) can efficiently explore the
search space in quest of global optima. Some of the earliest works that applied stochastic
population-based methods to OPF were the Genetic Algorithm [25], Evolutionary
Programming [26], etc. Numerous EAs were applied to solve the OPF problem. A few standard
OPF objectives were optimized in [27] for different IEEE systems using Differential Search
Algorithm (DSA). Daryani et al. [28] proposed improvement on the standard Group Search
Algorithm (GSA) algorithm with adaptation to develop the Adaptive Group Search Algorithm
(AGSA) to perform a study on OPF. Chaib et al. [29] applied the Backtracking Search
Optimization Algorithm (BSA) to optimize the OPF problem with consideration of valve-point
loading effects and multi-fuel in fuel cost. Improved Colliding Bodies Optimization (ICBO)
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algorithm is applied to the OPF problem in reference [30]. ICBO increases the number of
colliding bodies during each iteration to improve algorithm performance. Ref. [31] proposed
and applied Moth Swarm Optimization Algorithm (MSA) on numerous objectives of OPF for
various bus systems. Chaotic Artificial Bee Colony (CABC) is formulated by incorporating the
Chaos theory in the Artificial Bee Colony in [32], and security-constrained OPF was solved.
Adaptive Real Coded BBO (ARCBBO) was suggested in [33] to augment exploration
capability, and population diversity of the Biogeography Based Optimization Algorithm
(BBO) and applied to the OPF problem. Dynamic adjustment of control parameters in the
Adaptive Partitioning Flower Pollination Algorithm (APFPA) [34] revealed better
convergence speed and improved accuracy than FPA for OPF solutions. In the Fuzzy Harmony
Search Algorithm (FHSA) [35], the impact of fuzzy-based automatic adjustment of pitch
adjustment rate and bandwidth of HSA was examined in solving the problem of OPF. In
solving OPF, the Levy Mutation Strategy for Teaching Learning Based Optimization (LTLBO)
[36] proposed a Levy mutation operator to improve exploration at initial search stages. Krill
Herd Algorithm (KHA) [37] and its modified version - Stud Krill Herd Algorithm (SKHA)
[38], [39] were also popular in solving OPF problem. Grenade Explosion Method (GEM) [40]
and Glowworm Swarm Optimization (GSO) [41] also exhibited encouraging outcomes for
OPF, as reported in the literature. Modified Imperialist Competitive Algorithm (MICA) and
TLBO were crossbred in [42] to enhance local search and convergence of the original
Imperialist Competitive Algorithm (ICA) algorithm in obtaining OPF solutions.
OPF problem is solved for many different objectives, which are often contradicting. So, multi-
objective formulations and solutions of OPF are also available in the literature. A common
approach to deal with multiple objectives in OPF is the weighted sum in which the
predetermined weight factor is assigned to each objective. The weight factor is finalized after
a few tests with a judgment on the desired outcome of specific objectives. While the references
listed above for OPF dealt mostly with single-objective cases, weighted sum approach was
presented using Differential Search Algorithm (DSA) [27], Backtracking Search Optimization
Algorithm (BSA) [29], Improved Colliding Bodies Optimization (ICBO) [30], Moth Swarm
Algorithm (MSA) [31] and Adaptive Partitioning Flower Pollination Algorithm (APFPA) [34].
However, the weighted sum approach provides only one Pareto solution with a set of weight
factors during a complete run of the algorithm. For obvious reasons, a set of solutions with
multi-objective algorithms is preferred to a single solution with a weighted sum approach. A
limited number of publications reported a multi-objective approach to the problem of OPF.
Multi-objective Evolutionary Algorithm Based on Decomposition (MOEA/D) was adopted to
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solve MOOPF with 2 and 3-objectives in [43], where constraints were dealt with a static
penalty function approach. MOOPF witnessed the implementation of Multi-Objective Particle
Swarm Optimization (MOPSO) in [43] and the Non-dominated Sorting Genetic Algorithm-II
(NSGA-II) in [43][44]. An Enhanced Self-adaptive Differential Evolution with Mixed
Crossover (ESDE-MC) strategy was proposed in [45] to perform MOOPF on IEEE 30-bus and
IEEE 57-bus systems for mostly 2-objective cases together with a couple of 4-objective cases.
Niknam et al. [46] proposed a mutation operator to come up with a Modified Shuffled Frog
Leap Algorithm (MSFLA) and solved a 2-objective case (cost and emission) on the IEEE 30-
bus system. Shabanpour-Haghighi et al. in [47] studied the 2-objective (cost and emission)
optimization case for the two standard bus systems using Modified Teaching Learning Based
Optimization Algorithm (MTLBO). The study cases of ref. [47] were reperformed in [48]
using Modified Gaussian Bare-bones Imperialist Competitive Algorithm (MGBICA) and in
[49] using Improved Strength Pareto Evolutionary Algorithm (ISPEA). Non-dominated
Sorting Multi-Objective Opposition-based Gravitation Search Algorithm (NSMOOGSA) [50]
was applied to several 2-objective cases of the IEEE 30-bus system. Multi-Objective
Differential Evolution (MDE) based solution methodology was investigated in [51] on several
2-objective, a few 3 and 4-objective cases of OPF for IEEE 57-bus system. In studying the
MOOPF problem, the Imperialist Competitive Algorithm was modified to propose a Combined
Modified Imperialist Competitive Algorithm (CMICA) [52], Multi-Objective Modified
Imperialist Competitive Algorithm (MOMICA) [53] and Multi-Objective Imperialist
Competitive Algorithm (MOICA) [54]. Bio-inspired modified multi-objective flower
pollination algorithm (B-MMOFPA) [55] exhibited competitive performance on several 2 and
3-objective cases of OPF. In summary, the reference papers performed the optimization task
applying different algorithms with two or more objectives on single or multiple test systems.
As a next step, the existing literature is analyzed in terms of the optimized objective functions.
The following important observations were made:
1. Majority contributors [47], [50], [53], [61], [63], [65], [66], [69], [72], [73], [75]–[84],
[85]–[89] have considered QFC as objective functions. Very few contributors have
considered valve-point loading [51], [76], [90]–[97], [98]–[105] POZ [74], [76], [86]–
[89], and multifuel effects [90][90] while optimizing the fuel cost. In [42], [90], authors
have considered the simultaneous effect of valve-point loading effect and POZ to
minimize the fuel cost. In [91], authors have considered the simultaneous effect of
valve-point loading and multifuel to optimize the OPF problem. It is important to note
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that both these case studies are conducted on the IEEE 30-bus test system. Moreover, a
handful of contributors have analyzed the IEEE 30-bus test system with practical
generator consideration, and much less attention is paid to analyzing the practical
generator model for IEEE 57-bus or IEEE 118-bus test system. Thus, simultaneous
consideration of valve-point loading effect, multifuel, and POZ in quadratic fuel cost
on a larger system like IEEE 57-bus and IEEE 118-bus test system is an unexplored
research area.
2. Another important aspect is system size. Majority researchers have considered IEEE
30-bus test system for OPF optimization like in references [50], [51], [52], [53], [56],
[58], [61], [62], [66], [67], [70], [73], [75], [76], [78]–[83], [84], [90]–[92], [94], [95],
[97], [98], [100], [101], [106], [107], [110]–[116]. Very few researchers have
considered the IEEE 57-bus test system as presented in references [47], [49], [50], [51],
[53]–[56], [58], [62], [65], [69], [72], [84], [94], [95], [98], [110], [112], [114], [116]
and IEEE 118-bus test system in references [46], [49], [51], [55], [76], [81], [90], [91],
[97], [98], [100], [105], [108], [110], [113]–[115], [117]–[119]. Fig. 1 graphically
represents number of publications for solving the OPF problem considering the size of
systems. Clearly, single-objective, and multi-objective OPF optimization on a larger
sized system needs much more attention.
3. Many optimization approaches have been applied to solving the OPF problem. As
suggested in NFL [100] theorem, no algorithm can solve all the problems with the same
efficiency; thus, there is always room for new or enhanced algorithms to solve the OPF
problem more efficiently. The application of recent algorithms or their enhanced
version to solve the OPF problem can be a promising task.
3) Definition of the problem
As discussed previously, the OPF problem has become a vital part of the power system
economics in the last few decades. The OPF is a non-linear, non-convex, large-scale, and static
programming problem that enhances objective functions while meeting a set of equality and
inequality limitations. The active and reactive power equilibrium equations are the equality
constraints, and the limits on the state and control variables are the inequality constraints of the
OPF problem. The state variables contain load bus voltages, active and reactive power
generation at the slack bus, and apparent power flow. The control variables include active
power generation at PQ buses except at slack bus, transformer tap settings, the reactive power
output of shunt compensation capacitors, and the generator bus voltages.
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Fig. 1 Classification of publications based on the size of the test system
The shortcomings of classical approaches require the introduction of new procedures or
enhancement of present methods [22], [101]. Different population-based optimization
algorithms have been used to solve a complex constrained optimization problem in the field of
power systems, including the OPF problem. Some of these techniques include PSO [57], GWO
[95], MFO [59], ABC [84], TS [102], DE [65], GA [25], ACO [58], DA [103], etc.
Thus, in recent times, different well-proposed metaheuristic algorithms have been successfully
applied for the solution of the OPF problem though most of them have not extensively
examined the OPF problem. For example, most researchers have not considered a realistic
generator fuel cost model for the IEEE 118-bus test system. More often, the comparison is
based on the best values and average simulation time rather than statistical tests conducted over
independent runs. Besides, enhancement of the search performance of the available
metaheuristic techniques for solving the OPF problem is also an equally essential but mainly
untouched aspect.
Even with the efficient single-objective optimization, optimizing a single-objective function is
not enough in the power system since there are various objective functions to be minimized.
Improving the value of one objective often deteriorates the value of other objectives. Also, as
already discussed, the OPF comprises multiple objective functions like voltage deviation,
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active power loss, emissions, etc. and simultaneously optimizing all these objective functions
is extremely important. Hence, the OPF is a multi-objective optimization problem wherein
concurrent optimization of multiple objective functions is required.
As suggested by the No Free Lunch algorithm [100], no algorithm can be considered as the
best algorithm for solving all the problems. Thus, there is always a necessity for a new or
enhanced version of an existing algorithm that can efficiently solve the considered real-world
problem. Therefore, the identification of an algorithm that can efficiently solve the practical
multi-objective OPF problem is of prime importance.
In this work, the following problem is addressed: Out of all contemporary algorithms, is it
possible to identify an algorithm that provides better results for the practical OPF problem on
a medium-sized power system network? This problem can be divided into two sub-problems:
(i) How efficient is it to modify the search process of an existing algorithm to optimize real-
world issues? (ii) How to design and develop novel multi-objective metaheuristic algorithms
and validate their performance against the contemporary multi-objective optimization
algorithms on standard benchmark functions and practical OPF problem?
4) Objective and Scope of work
Optimizing various economic and technical objectives in an Optimal Power Flow problem is
essential to maximize system stability, profitability, and minimize carbon emissions. Different
metaheuristic algorithms were applied to solve different OPF optimization problems. This
research work also focuses on the performance enhancement of existing optimization
algorithms and the development of multi-objective optimization algorithms from existing
single-objective algorithms. The application of these modified algorithms for solving single-
objective and multi-objective OPF problems was studied.
The scope of this research is:
• to evaluate the most recently introduced metaheuristic approaches for optimizing
the OPF problem on IEEE 57-bus, and 118-bus test systems
A large number of algorithms have been proposed in the recent past. The NFL theorem
[100] suggests that the performance of an algorithm heavily relies on the type of the
problem under consideration. In this study, eight contemporary algorithms [104]–[111]
for single-objective OPF optimization for the realistic generator model were applied to
the IEEE 57-bus and IEEE 118-bus test systems. A total of 22 case studies were
considered including fuel cost for realistic generator model, carbon emission, voltage
13
stability index, voltage deviation, active and reactive power loss. A rigorous
performance analysis was performed considering the minimum and average values of
objective functions, and computational time. Different statistical tests were performed
to rank the algorithms based on their robustness. In the end, the best performing
metaheuristic technique for solving the considered OPF problem was identified.
• to propose and assess the Adaptive version of Moth-Flame Optimization (AMFO)
for solving the optimal power flow problem
Moth-Flame Optimization (MFO) was identified as the best performing algorithm for
optimizing the OPF problem. The promising results of MFO led to the development of
an enhanced version of MFO, i.e., Adaptive MFO (AMFO), to obtain better
searchability in an existing algorithm. The performance of AMFO was compared with
basic MFO on different standard single-objective benchmark functions. After
confirming the efficacy of AMFO on different benchmark functions, the AMFO was
also employed to optimize the single-objective OPF problem on the IEEE 118-bus test
system, and results were compared with contemporary algorithms for various criteria
including non-parametric statistical tests.
• to design and develop a non-dominated sorting-based and archive-based Ions
Motion Algorithm (IMO) and their applications on the above test systems
Over the years, many archive-based multi-objective optimization algorithms have been
proposed. Although existing multi-objective algorithms can approximate the true
Pareto optimal front of a given problem, they are not able to solve all optimization
problems according to the NFL theorem [100]. Therefore, it is likely that a new
algorithm can solve a problem more efficiently than the existing techniques in the
literature. This fact has motivated us to propose the multi-objective version of existing
single-objective IMO. The two proposed versions, i.e., Non-dominated Sorting based
Ions Motion Algorithm (NSIMO) and Archive-based Ions Motion Algorithm
(MOIMO), were tested on standard multi-objective benchmark functions. After
confirming their efficacy on standard benchmark functions, these approaches were
applied on a large-scale, highly constrained non-linear MOOPF problem. 10 test cases
of IEEE 30-bus, IEEE 57-bus, and IEEE 118-bus test systems were taken into
14
consideration. The findings were compared in terms of various multi-objective
performance indices apart from BCS and computational time.
5) Original contribution by the thesis
This research addressed different single and multi-objective OPF problems with a quadratic
and realistic cost function. The contributions of the research work are listed as follows:
1. Comprehensive analysis of 8 contemporary algorithms for 22 single-objective OPF
objectives including for the realistic generator model
2. Introduction of an enhanced version of MFO, i.e., AMFO and assessment of its viability
to optimize the challenging single-objective OPF problems
3. Design, development, and evaluation of non-dominated sorting and archive-based
approaches for optimizing the conflicting multi-objective problems
6) The methodology of research, results/comparisons
This research work can be divided into the following stages:
1. A literature study was conducted on the history of the OPF problem. This study has
presented an overall contextual on OPF formulation, past/current advancements,
and solution approaches. The research has also encompassed general terms of
metaheuristic optimization methods, single/multi-objective(s) and metaheuristic
approaches, and their advantages over the conventional techniques.
2. A practical single and multi-objective OPF problem was developed with chosen
objectives and constraints.
3. The results of a literature study on the single and multi-objective evolutionary
algorithms were further evaluated, given preceding inferences and findings.
4. Eight different contemporary algorithms were selected, and their performance
assessment is carried out based on various parameters for solving the OPF problem
for IEEE 57-bus and IEEE 118-bus test systems. The critical performance analysis
is performed on multiple parameters, and MFO is identified as the best performing
metaheuristic technique for solving the OPF problem.
5. In step 4, the best performing algorithm is identified, and the MATLAB code for
the enhanced version of the defined algorithm is developed to obtain better
searchability. The standard MFO algorithm updates the moth position based on the
distance of moth to the flame. Instead of updating the moth position on the above
concept, here, the moth position is adjusted adaptively based on the best, worst, and
15
current moth position. The step size determines how far the new moth position is
from the current moth position. As shown in Eq. (1), the step size varies inversely
with iteration. With increasing iteration count, the step size reduces.
𝑋𝑖𝑡+1 = (
1
𝑡)
|𝑏𝑒𝑠𝑡 𝑓(𝑡)−𝑓𝑖(𝑡)
𝑏𝑒𝑠𝑡 𝑓(𝑡)−𝑤𝑜𝑟𝑠𝑡 𝑓(𝑡)|
(1)
The adaptive step size is added to the current moth position to obtain a new moth
position.
𝑀𝑜𝑡ℎ𝑝𝑜𝑠(𝑡+1) = 𝑀𝑜𝑡ℎ𝑝𝑜𝑠(𝑡) + 𝑝 ∗ 𝑋𝑖𝑡+1 (2)
Fig. 2 presents a detailed flowchart of AMFO. The performance of AMFO is
compared with basic MFO on different standard single-objective benchmark
functions. AMFO is also employed to optimize the OPF problem on IEEE 118-bus
test system, and results are compared with contemporary algorithms for various
criteria.
6. In a paper on NSGA-II [112], [113], elitist non-dominated sorting and diversity
preserving crowding distance approaches were introduced. The same procedure can
be integrated into the single-objective approach for categorization of the population
in different non-domination stages with calculated crowded distance. An elitist non-
dominated sorting for finding distinct non-domination phases is defined first, and
then crowding distance method to maintain the variety amongst the optimum set of
solutions has been elucidated. Fig. 3 presents a generic concept of a non-dominated
sorting approach. In this work, the above approach is integrated into single-
objective IMO resulting in NSIMO. The NSIMO is tested for various multi-
objective benchmark functions. After thoroughly analyzing its performance with
other recently introduced multi-objective optimization techniques for optimizing
benchmark functions, NSIMO is applied to simultaneously optimize 10 conflicting
multi-objectives for IEEE 30-bus, 57-bus, and 118-bus test systems.
16
Fig. 2 Flowchart of AMFO
7. Finally, as the fourth contribution, another multi-objective version of the Ions
Motion Algorithm (IMO), i.e., archive-based MOIMO, is proposed. MOIMO
approach first generates the population of ions within the upper and lower bounds
of variables. This approach then computes the objective values of each ion and
determines the non-dominated ones. If the storage has a vacancy, then the non-
dominated solutions are added into it. If the storage is completely occupied, the
archive maintenance is run to remove the solution with a crowded neighborhood.
In this case, the solutions are ranked and chosen through a roulette wheel. After
eliminating the required number of solutions from the storage, the new solutions
are added to the storage. After updating the storage, an ion is selected from the non-
dominated solutions having the least crowded neighborhood. The next step is to
update the positions. All the above steps are repeated (except initialization), till the
termination criteria are met. Fig. 4 presents a schematic representation of an
archive-based multi-objective approach. The performance of MOIMO is compared
with different contemporary methods for different benchmark functions and OPF
problems for IEEE 30-bus, 57-bus, and 118-bus test system.
17
7) Results/comparisons
We have implemented eight different metaheuristic techniques [104]–[109], [114], [115] to
solve the OPF problem on standard IEEE 57-bus and IEEE 118-bus test systems. Total 22
different objective functions test cases are considered. The population size is selected to be 25,
and each approach is examined for 30 independent runs with 500 iterations per run. Fig. 5
presents the comparison of all algorithms for the best solution, the average solution, and the
average solution time over 30 separate runs. MFO attains the best performance for all
performance metrics.
Three statistical tests, Quade test [116], Friedman test [117], and Friedman aligned test [118],
are performed to rank the algorithms. For all three statistical analyses, as shown in Table 1,
MFO performed the best.
After confirming the efficacy of MFO [108] to deal with the OPF problem, an enhanced version
of MFO, i.e., adaptive MFO, is proposed on similar lines of adaptive cuckoo search algorithm
[119], [120]. Adaptive MFO is compared with basic MFO for 14 benchmark functions for
average value, the best value, standard deviation, and the average computational time. Both
algorithms are run for 30 independent runs. As presented in Fig. 6, for most benchmark
functions, AMFO provided better results as compared to MFO. Thirteen different objective
functions test cases of IEEE 118-bus test systems are considered. Algorithms are evaluated for
the best value, average simulation time, and standard deviation. As presented in Fig. 7, AMFO
provided the best results of minimum values. However, AMFO produced inferior results for
simulation speed. Table 2 compares different algorithms based on various statistical tests. It is
visible that AMFO stands first in all statistical analyses demonstrating its effectiveness to solve
the OPF problem.
After assessing the performance of metaheuristic approaches for single-objective OPF, the next
step is the simultaneous optimization of conflicting multiple OPF objectives. In this work,
existing single-objective IMO is extended to non-dominated sorting based multi-objective
IMO. Initially, NSIMO is analyzed for various standard multi-objective benchmark functions.
The results are compared based on GD, IGD, spread, HV, and average computation time. Fig.
8 presents the rank obtained by NSIMO for these performance metrics. NSIMO achieved the
top two positions for all performance parameters for most instances except computational time.
18
Fig. 3 Schematic representation of a non-dominated sorting-based algorithm
Fig. 4 Schematic representation of an archive-based multi-objective algorithm
19
Fig. 5 Performance comparison of 8 algorithms for solving single-objective OPF
problem
After confirming the performance of NSIMO on different benchmark functions, NSIMO, along
with three other non-dominated sorting multi-objective metaheuristics [121]–[123] are applied
to optimize 10 OPF problems. The results are compared in terms of the median value of HV,
the standard deviation in HV, computational time, best-compromised solution, and obtained
the best Pareto front.
Table 1 Composite ranking of different algorithms based on statistical tests
Position Algorithm Friedman Algorithm Quade Algorithm Friedman
Aligned
1 MFO 2.046154 MFO 2.039154 MFO 16.34615
2 GWO 2.692308 GWO 2.691969 GWO 22.76538
3 DA 3.980769 ALO 4.003485 MVO 34.25769
4 IMO 4.707692 GOA 4.717485 GOA 42.89231
5 MVO 4.853846 DA 4.789862 SCA 43.67692
6 ALO 4.884615 MVO 4.896846 ALO 44.67308
7 SCA 4.942308 IMO 4.981462 DA 45.06538
8 GOA 7.892308 SCA 7.879723 IMO 74.32308
MFO GWO DA SCA ALO MVO GOA IMO
0
5
10
15
20
25
21
0
2
0 0
2
1
0
11
1
0 0
6
4
0
2
5 5
0
4
0
5
0
3
Nu
mb
er o
f in
stan
ces
Algorithm
Best Solution Average Solution Average Simulation Time
20
Fig. 6 Comparison of AMFO and MFO for standard benchmark functions
Fig. 9 ranks algorithms in terms of minimum computational time. NSSCA obtains 1st and 2nd
rank for five occasions. The newly proposed NSIMO ranks 1st for three instances while 2nd
for two times. NSMFO does not get rank on the first two places and secures only 3rd and 4th
rank. Fig. 10 ranks algrithms in terms of median values of hypervolume. Higher the
hypervolume value, the better is the performance. In this case, NSSCA and NSGWO obtain
1st rank for four instances. In terms of median values of HV, NSIMO performs worst as it fails
to secure the 1st position. It hardly secures 2nd position that too only once. The algorithm
which has a minimum standard deviation determined over several independent runs can be
considered as a robust algorithm. Fig. 11 ranks algorithms in terms of minimum standard
deviation in HV values over 30 separate runs. In this case, NSIMO and NSMFO obtain 1st
position for four instances, while NSIMO attains 2nd position for three cases. Thus, NSIMO
can be considered as a consistent performer than the rest of the algorithms. Apart from the
above performance metrics, algorithms are also analyzed based on statistical tests, as shown in
Table 3. These tests are performed on variation in median hypervolume values over 30
individual runs. As can be seen, NSIMO ranks on top while NSGWO ranks last for both the
tests.
AMFO MFO
0
2
4
6
8
10
12
11
5
9
5
9
5
11
3
Num
ber
of
inst
ance
s
Algorithm
Average value Best value Standard deviation Average time taken
21
Fig. 7 Performance assessment of single-objective algorithms for optimizing the OPF
problem
Finally, the archive-based multi-objective IMO algorithm is devised and applied on 18 standard
multi-objective benchmark functions. The MOIMO algorithm is compared with other
contemporary archive-based multi-objective algorithms [104], [124]–[128] for GD [129], IGD
[130], Spread [131], and MoS value [132], HV [133] and resulting SD (standard deviation) in
these values for 30 independent runs as shown in Fig. 12. MOIMO provides better results for
almost all performance metrics presenting its further worthiness for solving real-world multi-
objective problems. Fig. 12 also compares algorithms in terms of the average value of
Computational Time (CT). MOMVO achieved minimum CT for 13 instances, followed by
MOIMO for five instances.
After confirming the competitive performance of MOIMO on standard benchmark functions,
MOIMO, along with seven algorithms, is also applied to solve ten multi-objective OPF
problems on the IEEE 30-bus, 57-bus, and 118-bus test system. The performance of algorithms
is compared using median HV values, simulation time, the standard deviation in HV values,
and ranking of statistical tests. Fig. 13 compares algorithms in terms of average simulation
time. MOGWO, MOMVO, and MOSSA obtain 1st rank for three instances, followed by
MOIMO for a single instance.
12
2
1
0 0 0
1 1
0
12
0
1
0 0
1 1
0 00
2
4
0
4
0 0 0
3
AMFO MFO GWO DA SCA ALO MVO GOA IMO
0
2
4
6
8
10
12
Num
ber
of
inst
ance
s
Algorithm
Best value Average value Avg. simulation time
22
Table 2 Comparison of various algorithms based on a statistical test on the IEEE 118-
bus test system
Position Algorithm Friedman Algorithm Quade Algorithm Friedman
Aligned
1 AMFO 1.6077 AMFO 1.3467 AMFO 240.0038
2 MFO 2.8462 MFO 2.5318 MFO 287.4577
3 GWO 3.5462 GWO 3.1504 GWO 362.9231
4 ALO 4.9269 ALO 4.8824 ALO 513.8731
5 GOA 5.6308 GOA 5.6459 GOA 666.4308
6 MVO 5.7923 DA 5.9544 DA 683.8538
7 DA 5.8538 MVO 6.1984 MVO 748.0885
8 IMO 5.9038 IMO 6.3157 IMO 765.7385
9 SCA 8.8923 SCA 8.9743 SCA 1,001.13
Fig. 8 Best performance of NSIMO for various performance metrics
Fig. 14 compares archive-based algorithms for better median HV value. In this case, MOMFO
achieved the best position for six instances. The proposed MOIMO algorithm attained 1st and
2nd position for two and five cases presenting its competitive performance.
All archive-based algorithms are compared for the minimum standard deviation in HV values.
Fig. 15 presents the ranking of algorithms for minimum standard deviation values. For this
case, MOIMO ranks 1st for maximum, i.e., four instances. To confirm that HV values are not
obtained by chance, the algorithms are also compared based on statistical test ranking. As
shown in Table 4, MOMVO ranked 1st for all tests while MOIMO and MOMFO presented
poor performance obtaining the last two positions.
1 2 3 4
-2
0
2
4
6
8
10
12
14
Nu
mb
er o
f in
stan
ces
Rank
GD IGD Spread HV Computational time
23
Fig. 9 Ranking of non-dominated sorting (NS) based algorithms for computational time
Fig. 10 Ranking of non-dominated sorting (NS) based algorithms for median HV values
Table 3 Ranking of NS algorithms based on statistical analyses
Rank Algorithm Friedman test Algorithm Quade test
1 NSIMO 3.96 NSIMO 3.55333
2 NSMFO 4.733333333 NSSCA 5.28663
3 NSSCA 5.693333333 NSMFO 5.66667
4 NSGWO 5.933333333 NSGWO 6.013
24
Fig. 11 Ranking of NS algorithms for standard deviation in HV
Table 4 Ranking of algorithms based on statistical tests
Rank Algorithm Friedman
test
Algorithm Quade
test
Algorithm Friedman
aligned
test
1 MOMVO 16.6 MOMVO 3.29331 MOMVO 13.82
2 MOGWO 18.066666 MOALO 3.31331 MODA 14.44
3 MOALO 20.466666 MODA 3.46003 MOALO 16
4 MOSSA 20.866666 MOSSA 3.953333 MOSSA 18.04
4 MODA 20.866666 MOGWO 4.04001 MOGWO 18.6
6 MOIMO 24.4 MOIMO 4.86003 MOIMO 22.42
7 MOMFO 27.933333 MOMFO 5.08 MOMFO 22.68
NSSCA NSIMO NSGWO NSMFO
0
1
2
3
4
5
0
4
2
44
3
1
2
5
2 2
11 1
5
3R
ank
Algorithm
1st 2nd 3rd 4th
25
Fig. 12 Comparison of different archive-based algorithms for optimizing standard
benchmark functions
Fig. 13 Ranking of algorithms in terms of computational time
GD SD in GD IGD SD in IGD Spread SD in Spread MoS SD in MoS CT
-2
0
2
4
6
8
10
12
14
16
18
Nu
mb
er o
f in
stan
ces
Comparison criteria
MOMVO MOALO MODA MOIMO
26
Fig. 14 Ranking in terms of hypervolume values
Fig. 15 Ranking in terms of standard deviation (HV)
8) Assessment of Techno-Economic-Environmental Benefits
The best-compromised solutions obtained using MOIMO and NSIMO were compared with the
reported methods in the literature [134]. MOIMO was applied to optimize QFC and Emission
in Case 2 concurrently. As shown in Table 5, MOIMO provided an annual fuel cost saving of
117988 $, along with a reduction in emission of 298.716 tons. Quadratic fuel cost and active
power loss were minimized for the IEEE 30-bus test system in Case 11. As shown in Table 5,
NSIMO achieved a daily saving of 0.631 $/h resulting in an annual saving of 5527.56 $. Table
6 illustrated that NSIMO achieved a power loss reduction of 4839.9 MW. In case 12, the
27
quadratic fuel cost ($/h) and emission (ton/h) were simultaneously optimized. In this case,
NSIMO achieved savings of 12.253 $/h resulting in a yearly saving of 107336 $. At the same
time, it achieved a yearly carbon emission reduction of 350.4 tons. In case 15, quadratic fuel
cost, emission, and active power loss were simultaneously optimized. In this case, NSIMO
provided yearly fuel cost saving of 112828 $, yearly emission reduction of 297.64 tons, and
yearly active power loss reduction of 737.592 MW.
Table 5 Ranking of algorithms based on statistical tests
Economic benefits Environmental benefits
Competitive
methods
Savings
($/h)
Annual
savings
($)
Competitive
methods
Reduction
in tons/hr
Annual
savings
in tons
MOIMO MOFA-
CPA
[134]
13.469 117988 MOIMO MOFA-
CPA
[134]
0.0341 298.716
838.551 852.02 0.2449 0.279
NSIMO MOFA-
CPA
0.631 5527.56
-------
833.309 833.94
NSIMO MOFA-
CPA
12.253 107336 NSIMO MOFA-
CPA
0.04 350.4
839.767 852.02 0.239 0.279
NSIMO MOFA-
CPA
12.88 112828 NSIMO MOFA-
CPA
0.034 297.64
855.027 867.907 0.23 0.264
9) Achievements concerning objectives
This research work has led to the following accomplishments summarized below:
1. A literature survey revealed that the researchers had considered a limited number of
algorithms and OPF test cases for comparing the metaheuristic techniques. Also,
algorithms are generally compared in terms of best value, average value, standard
deviation, and average simulation time. Judging algorithms based on a limited number
28
of test studies may turn into skewed results. For a fair comparison, in this work,
probably for the first time, eight contemporary metaheuristic approaches were applied
to optimize the 22 single- objective OPF objectives having varying complexity,
including a realistic generator model. The performance evaluation was done on the
average simulation time, the best solution, the average solution, and the standard
deviation. Moreover, algorithms were also ranked based on three statistical tests. MFO
performed the best irrespective of the size of the network and the complexity of the
objective function. [Publication 1-2]
Table 6 Ranking of algorithms based on statistical tests
Technical benefits
Competitive methods Savings (MW/h) Annual savings (MW)
NSIMO MOFA-CPA 0.5525 4839.9
4.455 5.0075
NSIMO MOFA-CPA 0.0842 737.592
4.45 4.5342
2. An improved variant of the MFO, i.e., AMFO, was proposed. Performance evaluation
of AMFO was carried out for standard single-objective benchmark functions. The
AMFO provided competitive results than its basic version proving its candidature to
solve real-world problems. Hence, 13 different OPF test cases were optimized using
AMFO. The results were compared with other contemporary algorithms in terms of the
best solution, average simulation time, and average values. Three statistical tests were
employed to validate the results by each algorithm. The AMFO achieved the first rank
for all the statistical tests. [Publication 3]
3. Multi-objective versions, i.e., archive-based and non-dominated sorting-based variants
of the existing single-objective algorithm, were proposed. These proposed approaches,
i.e., NSIMO and MOIMO, were tested on standard multi-objective benchmark
functions. Results thus obtained were compared with other contemporary multi-
objective algorithms. Newly introduced algorithms provided competitive outcomes for
all benchmark functions, if not better. Thus, the NSIMO and MOIMO emerged as
strong contenders for the optimization of any real-world multi-objective optimization
problem. [Publication 4]
29
4. The NSIMO, along with the other three algorithms, i.e., NSGWO, NSSCA, and
NSMFO, were applied to optimize 10 OPF objective functions for IEEE 30-bus, 57-
bus, and 118-bus test system. The performance assessment was done for different
performance parameters wherein the NSIMO provided competitive results amongst all
algorithms. [Publication 5]
5. The MOIMO, along with seven other algorithms, were applied to solve 10 OPF
objective functions. The performance was examined based on the compromised
solutions, median HV value, computational time, the standard deviation in HV value.
Algorithms were also examined in terms of statistical tests. The MOMVO ranked 1st
for all three statistical analyses.
10) Conclusion
This work presented a comprehensive analysis of the novel metaheuristic techniques for
optimizing the OPF problem. As an initial step, the existing single-objective metaheuristic
approaches were applied to optimize the OPF problem on different test systems. After various
experimental analysis, the MFO presented its best capability to deal with complex OPF
problem.
To enhance the searchability, the existing single-objective MFO was equipped with an adaptive
step size mechanism. The performance of the AMFO was validated for different single-
objective benchmark functions in terms of the average value, the best value, average simulation
time, and standard deviation. After confirming the performance of the AMFO for benchmark
functions, 13 single-objective OPF test cases were optimized bearing various complex and
practical restraints. The evaluation was done based on the best solution, average simulation
time, and standard deviation. Besides, the performance of algorithms was tested using three
statistical tests for best values obtained over 30 independent runs. The AMFO achieved the
first rank for all statistical tests.
This work also proposed the multi-objective editions of the newly introduced IMO algorithm,
i.e., NSIMO and MOIMO. The NSIMO and MOIMO were employed on various standard
multi-objective unconstrained, constrained, and engineering design benchmark functions. The
outcome of these algorithms was compared with different non-dominated sorting and archive-
based approaches. Both approaches have shown enough virtues amongst the contemporary
multi-objective algorithms and projected themselves as a substitute for solving multi-objective
optimization problems.
30
The non-dominated sorting based and archive-based approaches were also applied to optimize
different conflicting multi-objective OPF problems. The algorithms were also compared in
terms of different numerical results, and the best Pareto fronts were obtained. To confirm that
results were not found by chance, statistical tests on the results of HV was also performed, and
ranks were decided. For non-dominated approaches, Quade and Friedman’s analysis was
performed. For both these tests, the NSIMO ranked first.
In nutshell, following outcomes are achieved from this research work:
▪ There is always a quest for a new or enhanced version of the existing algorithm.
▪ The OPF is a non-linear, non-convex, discontinuous objective function which cannot
be solved efficiently using conventional approaches.
▪ The modern metaheuristic techniques can be an excellent alternative to conventional
methods as metaheuristic techniques are derivative-free and treat every problem as a
black box.
▪ MFO is found to be a promising tool for solving the single-objective OPF problem.
▪ An enhanced version of MFO provides better results than original MFO but at the cost
of higher computational efforts.
▪ Two versions of existing single-objective IMO are proposed. But for the multi-
objective OPF problem, no algorithm can provide the best-compromised solution
majority cases. BCS cannot be the only yardstick to compare the performance of
algorithms.
▪ The proposed multi-objective algorithms are also compared in terms of statistical test
ranking. The newly proposed algorithms provide competitive results as compared to
those already available in the literature.
11) Publications
Published
• Buch, Hitarth, Indrajit N. Trivedi, and Pradeep Jangir. "Moth flame optimization to
solve optimal power flow with non-parametric statistical evaluation validation." Cogent
Engineering 4.1 (2017): 1286731.
• Buch, Hitarth, and Indrajit N. Trivedi. "On the efficiency of metaheuristics for solving
the optimal power flow." Neural Computing and Applications 31.9 (2019): 5609-5627.
31
• Buch, Hitarth, and Indrajit N. Trivedi. "An Efficient Adaptive Moth Flame
Optimization Algorithm for Solving Large-Scale Optimal Power Flow Problem with
POZ, Multifuel and Valve-Point Loading Effect." Iranian Journal of Science and
Technology, Transactions of Electrical Engineering 43.4 (2019): 1031-1051.
• Buch, Hitarth., and Indrajit N. Trivedi. "A new non-dominated sorting ions motion
algorithm: Development and applications." Decision Science Letters 9.1 (2020): 59-76.
Accepted
• Buch, Hitarth, Indrajit N. Trivedi, Hitesh Karkar, and Prasanta Ghosh. " A Non-
dominated Sorting Ions Motion Algorithm for solving the Multi-objective Optimal
Power Flow Problem." 2020 the 8th International Conference on Smart Energy Grid
Engineering between 12-14 August 2020 in Toronto, Canada.
Under review
• Buch, Hitarth., and Indrajit N. Trivedi. " Ions Motion Optimization algorithm for multi-
objective optimization problem." in Decision Science Letters
32
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