A Comprehensive Study of Novel Metaheuristic …...DA Dragonfly Algorithm DE Differential Evolution DSA Differential Search Algorithm DV Voltage Deviation E Emission EA Evolutionary

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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