Optimal power flow using archimedes optimizer algorithm

International Journal of Power Electronics and Drive Systems (IJPEDS)
Vol. 13, No. 3, September 2022, pp. 1390~1405
ISSN: 2088-8694, DOI: 10.11591/ijpeds.v13.i3.pp1390-1405  1390
Journal homepage: http://ijpeds.iaescore.com
Optimal power flow using archimedes optimizer algorithm
Mohammed Hamouda Ali1
, Ahmed Mohammed Attiya Soliman1
, Salah K. Elsayed2
1
Department of Electrical Engineering, Faculty of Engineering, Al-Azhar University, Cairo, Egypt
2
Department of Electrical Engineering, College of Engineering, Taif University, Taif, Saudi Arabia
Article Info ABSTRACT
Article history:
Received Apr 6, 2022
Revised May 23, 2022
Accepted June 11, 2022
This article proposes a new metaheuristic algorithm called Archimedes
optimization algorithm (AOA) for solving optimization problems of optimal
power flow (OPF) utilizing the renewable energy sources (RES) for
minimizing different single-objective and multi-objective functions based on
minimization of fuel cost, power losses of transmission lines, emission and
voltage profile improvement. Also, mathematical formulation of (OPF) is
introduced by converting the function with multiple objectives based on
price and weighting parameters into a single objective function. Also, the
effect of optimal RES is merged into the OPF problem. Notably, optimal
RES placement yields even more effective solution. AOA was inspired by
an intriguing physical law known as Archimedes' Principle. To prove the
effectiveness of the AOA proposed algorithm, it compared with different
recent algorithms for solving the optimal power flow problems and testing
them to one standard system of the IEEE30-bus test system. The superiority
of the proposed AOA algorithm is proven also by applying them on the
IEEE30-bus modified system with optimal allocation of renewable energy
source (RES). The results demonstrate that the proposed algorithm is more
successful and efficient than the other optimization methods in the title of
resolving OPF problems.
Keywords:
Archimedes optimization
algorithm
Metaheuristic algorithms
Multi-objective functions
Optimal power flow
Renewable energy source
This is an open access article under the CC BY-SA license.
Corresponding Author:
Mohammed Hamouda Ali
Department of Electrical Engineering, Faculty of Engineering, Al-Azhar University
P.O. Box 11751, El Nasr St, Nasr City, Cairo, Egypt
Email: Eng_MohammedHamouda@azhar.edu.eg, MohammedHamouda62.el@azhar.edu.eg
1. INTRODUCTION
The term "optimal power flow" (OPF) refers to the operation of a power system in an economical
and stable manner, which is achieved by properly setting the system's control variables, where (OPF) is a
critical and nonlinear complex optimization problem for assessing security and dependability of power
systems, whose primary goal is to select the optimal network or grid control variable solution that fulfils the
minimal objective function value while taking system constraints into consideration. where OPF aims to
optimize generator dispatch based on their limits, expected operating conditions, voltage constraints on the
bus, as well as safety margins [1], [2]. Many control variables, including generator voltage, generator actual
output power, transformer tap settings, and reactive power compensation devices, can be used in this
situation. Renewable energy sources (RES), specifically wind turbines and solar generators, have recently
been recommended for due to clean energy production and reducing operating costs. The allocation and
technical characteristics of renewable energy generators have a significant impact on the system's techno-
economic performance [1]–[5]. As a result, control variables, generator behaviour, and the establishment of
an accurate planning tool for optimal power flow in the integrated electric system [1]–[7] must be considered.

Int J Pow Elec & Dri Syst ISSN: 2088-8694 
Optimal power flow using archimedes optimizer algorithm (Mohammed Hamouda Ali)
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An objective function definition should be used to pick the optimal solution as the desired solution.
Different objectives for the OPF are considered in the electrical system. As a result, the optimal power flow
takes system constraints into account and determines the most optimal operating conditions in terms of both
system control variables,and objectives of the problem. OFP's optimal solution has been linked with technical
and economical benefits, which are typically regarded to be OPF objectives. Generally, the objective
functions of OPF may be divided into single objective functions that achieve a single goal and multi-
objective functions that achieve many objectives at the same time. These objectives might include the
generators fuel cost, emission rate of the generator, the power losses in an electric network, and security
index of the voltage [1]–[7]. As presented in [8]–[10], many optimization methods have been devised for
solve the OPF issues. These methods may be divided into two categories: conventional methods and
metaheuristics methods as presented in [9]. To address the OPF problems, several traditional approaches were
being used, including linear programming [11], nonlinear programming [12], quadratic programming [13],
newton method network flow programming [14], as well as the interior point technique [15]. The primary
drawbacks of traditional approaches are that they are unsuitable for large and complex OPF problems, which are
nonlinear and multi-modality optimization issues, as a result of the significant sophistication and nonlinear
effects of the restricted OPF issue, it has been revealed that conventional techniques may not even be capable of
handling the OPF problem solutions correctly, resulting in poor results [3], [9].
According to the literature survey in [3], [9] various metaheuristic optimization approaches
including evolutionary-inspired, bio-inspired, human-inspired, physics-inspired, hybrid, swarm and artificial
neural networks-fuzzy logic approaches, these approaches are invented and proposed to fill the gap formed
by the use of conventional methods and getting the best optimum solutions when dealing with OPF difficult
issues. Furthermore, the incorporation of new renewable sources, particularly WT and PV, into the power
system adds complexity of the OPF problem due to their intermittent power generation characteristics. As a
result, to fill the gap left by the use of conventional methods, a comprehensive overview of various
metaheuristic optimization approaches for the optimal solution of power flow issues has been invented and
proposed [3], [9]. Finally, when compared to traditional techniques, the advantages of these metaheuristic
techniques include high dependability, guaranteed best optimized solution, rapid convergence, and a low
likelihood of errors and being trapped in local minima. Because of the optimal outcomes, most researchers in
recent study work considered ametaheuristic population-based approach to resolving the OPF issue.
Several research articles used nature-inspired techniques for solving the OPF problems. Jadhav and
Bamane [16] solve the problem of OPF with a single objective function and employed the best-guided
algorithm called artificial bee colony to optimize the fuel cost. Glow-worm swarm optimization algorithm is
used to optimize emission and fuel cost as in [17]. Tan et al. [18] the fuel cost is optimized with only valve
effect by using the improved group search optimization algorithm. Power losses, fuel cost, fuel cost with valve
effect, emission are optimized in [19] employed the oppositional krill herd algorithm. Also, algorithm known as
chaotic artificial bee colony is used to optimize transient stability and fuel cost as in [20]. Also, Mukherjee and
Mukherjee [21] solve the OPF problem and employed the chaotic krill herd algorithm to optimize fuel cost, fuel
cost with valve effect, emission, power losses, and voltage deviation. Mohamed et al. [22] solving the OPF
problems with multi-objective function and employed algorithm called moth swarm to optimize emission,fuel
cost, fuel cost with only valve effect, L-index, power losses, piecewise cost, and voltage deviation.
Several research articles used the algorithms inspired by humans that were used for solving the OPF
problems.Where improved harmony search algorithm is employed in [23] to solve OPF problem using only
single objective function and optimize the fuel cost only with valve effect. Also, fuel cost, fuel cost considering
the banned regions,and fuel cost with valve effect have been optimised in [24] using algorithm called symbiotic
organisms search. Adaryani and Karami [25] fuel cost with only valve effect and emission based on using the
modified teaching-learning algorithm are optimized. Ghasemi et al. [26] solve the OPF problem and employed
algorithm known as the improved teaching-learning to optimize fuel cost, fuel cost with only valve effect,
emission, piecewise cost, and voltage deviation. Mandal and Roy [27] employed optimization algorithm known
as quasi-oppositional teaching learning algorithm to optimize emission, fuel costs with valve effect, L-index,
power losses, and L-index.
Many research papers applied evolutionary-based optimization techniques for solving the OPF
problem. Somasundaram et al. [28] authors are solving the optimal power flow problem with a single objective
function and employed the evolutionary programming algorithm to optimize the fuel cost. A faster algorithm
called evolutionary is applied in [29] to optimize fuel cost and fuel cost with only valve effect. Optimizing of
fuel cost and emission is presented in [30] by using the improved evolutionary algorithm. Power losses, fuel
cost, emission, and L-index are optimized in [31] employed enhanced self- adaptive differential evolution.
Reddy and Bijwe [32], differential evolution algorithm is applied to optimize power losses, L-index, fuel cost,
and fuel cost with only valve effect. Chaib et al. [33] solve the OPF problem and employed the backtracking
search method to optimize emission, L-index, fuel cost, fuel cost with only valve effect, piecewise cost, and
voltage deviation.

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Several research articles present the applicable physics-inspired techniques for solving the OPF issues.
Bouchekara et al. [34], the author solved the OPF problem with multi-objective function and employed
optimization method called an improved colliding bodies to optimize emission, fuel cost, fuel cost with only
valve effect, L-index, power losses, voltage deviation, and piecewise cost. Many research papers discuss
composite optimization techniques that have been used for OPF. Gacem and Benattous [35] used particle swarm
and genetic algorithm to optimize fuel cost, and cost of fuel with valve effect. Fuel cost, cost of fuel with valve
effect, emission, L-index, piecewise cost, and voltage deviation is optimized in [36] by employing Nelder-Mead
and fuzzy particle swarm optimization algorithms. Also, Singh et al. [37] with an aging leader and challengers
employed particle swarm optimization algorithm to optimize fuel cost, fuel cost with only valve effect, voltage
deviation, and power losses. The optimizations techniques based on ANNs and fuzzy logic approachs are
presented through several research articles as in references [38]-[40].
To summarise, this article presents a new population-based algorithm called archimedes optimization
algorithm (AOA) based on the physics law known as Archimedes' principle to compete with state-of-the-art and
recent optimization algorithms, including other physics-inspired methods. It is important to note that the
proposed technique strikes a balance between exploration and exploitation. Because AOA keeps a population of
solutions and investigates a large area to find the best global solution, it is well suited for solving complex
optimization problems with many locally optimal solutions.In conclusion, the following are the main
contributions of this research:
− Archimedes optimization algorithm (AOA) has been proposed as a new population-based algorithm that
mimics Archimedes' principle.
− Introduce the OPF problem formulation with different four objective-functions.
− Applying the proposed AOA for solving the optimization problems by converting the multi-objective
function (fuel cost, power losses, voltage deviation, and emission) into a single-objective function using the
price and weighting factors.
− The IEEE30-bus testing system is used in this study to assess the effect of the proposed algorithm on a
difficult test suite in metaheuristic literary works.
− The search efficiency of AOA is validated against well-established algorithms dragonfly algorithm (DA),
particle swarm optimization (PSO), sparrow search algorithm (SSA), future search algorithm (FSA).
− The AOA algorithm is also proposed for deciding the best and optimal allocation of RES.
− Finally, the modified IEEE30-bus testing system integrated with the optimal RES allocation is introduced to
test the AOA suggested algorithm's supremacy over other metaheuristic algorithms.
This paper organization is as follows: section 2 introduces the mathematical formulation model of the
OPF. The AOA proposed algorithm is discussed in section 3. Also, section 4 contains the AOA simulation
results. In this section, a thorough analysis and comparison are performed against the selected metaheuristic
algorithms. Section 5 presents the final discussion and conclusion.
2. MATHEMATICAL FORMULATION FOR OPF
The optimum power flow issue seeks to maximize an objective function by making optimal
modifications to the control variables of power system while adhering to various equality constraints and
inequality constraints. In general, the optimization issue may be mathematically stated as:
𝑚𝑖𝑛 𝐹(𝑥, 𝑢) (1)
Conditional on:
𝑔𝑗(𝑥, 𝑢) = 0 𝑗 = 1,2, … , 𝑚
ℎ𝑗(𝑥, 𝑢) ≤ 0 𝑗 = 1,2, … , 𝑝
Where function F represents the objective function, 𝑥 is a vector containing the state variables (dependent
variables), 𝑢 is a vector containing the control variables (independent variables), and 𝑔𝑗 and ℎ𝑗 are
respectively the equality and inequality requirements. The variables 𝑚 and 𝑝 represent respectively the
equality and inequality constraints numbers. In a power system, the state variables (𝑥) are as:
𝑥 = [𝑃𝐺1 , 𝑉𝐿1 … 𝑉𝐿,𝑁𝑃𝑄 , 𝑄𝐺,1 … 𝑄𝐺,𝑁𝐺 , 𝑆𝑇𝐿,1 … 𝑆𝑇𝐿,𝑁𝑇𝐿 ] (2)
where 𝑃𝐺1 denotes power of slack bus, 𝑉𝐿 denotes load bus voltage, 𝑄𝐺 denotes reactive output
power for generator, 𝑆𝑇𝐿 denotes the transmission line's apparent power flow, 𝑁𝑃𝑄 denotes the
load buses number, 𝑁𝐺 denotes the generation buses number, and 𝑁𝑇𝐿 denotes the transmission
lines number. In a power system, the control variables (u) are as:

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𝑢 = [𝑃𝐺,2 … 𝑃𝐺,𝑁𝐺, 𝑉𝐺,1 … 𝑉𝐺,𝑁𝐺 , 𝑄𝐶,1 … 𝑄𝐶,𝑁𝐶 , 𝑇1 … 𝑇𝑁𝑇] (3)
where 𝑃𝐺 is the active output power for generator, 𝑉𝐺 is the generation bus voltage, 𝑄𝐶 is the shunt
compensator reactive power injected, T is the tap setting for transformer, NC is the shunt compensator units
number, and NT is the transformers number.
2.1. Objective functions
An objective function definition should be used to pick the optimal solution as the desired solution.
In addition to the problem objectives, different objectives are assessed for the OPF, going to result in an
optimized power flow that considers system constraints and determines the finest conditions in terms of
system control variables, in furthermore to the problem objectives. OFP's best solution has been linked to
techno-economic advantages, which are commonly referred to as OPF objectives. Reduced fuel costs in
terms of annual savings are among the economic benefits, while the technical benefits are listed [3]:
− Minimization of active power losses
− Minimization of reactive power losses
− System reliability, and power quality enhancement
− Deviation of the voltage
− Stability of the voltage
2.1.1. Single objective functions
A most common objective-functions can be performed as follows [41]-[45]:
− Basic fuel costs minimization objective
This objective function is the primary aim of the OPF issue and seeks to minimize overall
fuel cost. For every generator, it may be demonstrated as a quadratic polynomial function as:
𝐹1 = ∑ 𝐹𝑖
𝑁𝐺
𝑖=1 (𝑃𝐺𝑖) = ∑ (𝑎𝑖 + 𝑏𝑖𝑃𝐺𝑖 + 𝑐𝑖𝑃2
𝐺𝑖)
𝑁𝑃𝑉
𝑖=1
$
ℎ
(4)
where, 𝐹𝑖 is the 𝑖th generator fuel cost. 𝑎𝑖, 𝑏𝑖, and 𝑐𝑖 are the cost coefficients for 𝑖th generator.
− Generation emission minimization objective
Reducing the amount of gas emitted by thermal power plants can help to reduce
pollution. The objective function for the emission gases is as:
𝐹2 = ∑ (𝛾𝑖𝑃2
𝐺𝑖 + 𝛽𝑖𝑃𝐺𝑖 + 𝛼𝑖+ζ𝑖 𝑒𝑥𝑝(𝜆𝑖𝑃𝐺𝑖)
𝑁𝐺
𝑖=1 (5)
where, 𝛾𝑖, 𝛽𝑖, 𝛼𝑖, ζ𝑖, and 𝜆𝑖 are the 𝑖th generator's emission coefficients.
− Active power losses minimization objective
The desired objective-function is to minimize real power loss, which can be presented as:
𝐹3 = ∑ 𝐺𝑖𝑗(𝑉2
𝑖 + 𝑉2
𝑗 − 2 𝑉𝑖𝑉
𝑗 cos 𝛿𝑖𝑗)
𝑁𝑇𝐿
𝑖=1 MW (6)
where, 𝐺𝑖𝑗 the transmission conductance, NTL is the transmission lines number, and 𝛿𝑖𝑗 is the
voltages phase difference.
− Voltage profile improvement
The deviations of load buses voltage from a predetermined voltage are minimized by this
objective function, it may be expressed as:
𝐹4 = 𝑉𝐷 = ∑ |𝑉𝑖 − 1|
𝑁𝑃𝑄
𝑖=1 (7)
2.1.2. Multi-objective functions
The primary goal of resolving a multi-objective problem is to optimize multiple
independent objective functions simultaneously and its definition is represented as:
𝑀𝑖𝑛 𝐹(𝑥, 𝑢) = [𝐹1(𝑥, 𝑢), 𝐹2(𝑥, 𝑢), … , 𝐹𝑖(𝑥, 𝑢)] (8)
where 𝑖 is the objective functions number, the optimization with Pareto approach or weight factors
as follows can be used to solve multi objective functions:
𝑀𝑖𝑛 𝐹5 = ∑ 𝑤𝑖
4
𝑖=1 𝐹𝑖(𝑥, 𝑢)
𝐹(𝑥, 𝑢) = 𝑤1𝐹1 + 𝑤2𝐹2 + 𝑤3𝐹3 + 𝑤4𝐹4 (9)

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𝐹(𝑥, 𝑢) = 𝑤1 ∑ (𝑎𝑖 + 𝑏𝑖𝑃𝐺𝑖 + 𝑐𝑖𝑃2
𝐺𝑖)
𝑁𝐺
𝑖=1 + 𝑤2 ∑ (𝛾𝑖𝑃2
𝐺𝑖 + 𝛽𝑖𝑃𝐺𝑖 + 𝛼𝑖+ζ𝑖 𝑒𝑥𝑝(𝜆𝑖𝑃𝐺𝑖)
𝑁𝐺
𝑖=1 +
𝑤3 ∑ 𝐺𝑖𝑗(𝑉2
𝑖 + 𝑉2
𝑗 − 2 𝑉𝑖𝑉
𝑗 cos 𝛿𝑖𝑗)
𝑁𝑇𝐿
𝑖=1 + 𝑤4 ∑ |𝑉𝑖 − 1|
𝑁𝑃𝑄
𝑖=1 (10)
where 𝑤11, 𝑤2 , 𝑤3 and𝑤4 are weight factors chosen based on the relative importance of one goal
to another. Typically, the values of the weight factors are chosen as:
∑ 𝑤𝑖
𝑛
𝑖=1 = 1 (11)
2.2. System constraints
There are already many constraints in the system that can be classified as:
2.2.1. The Equality constraints
The equality constraints for the balanced load flow equations are as:
𝑃𝐺𝑖 − 𝑃𝐷𝑖 = |𝑉𝑖| ∑ |𝑉
𝑗|(𝐺𝑖𝑗 cos 𝛿𝑖𝑗 + 𝐵𝑖𝑗𝑠𝑖𝑛𝛿𝑖𝑗)
𝑁𝐵
𝑗=1 (12)
𝑄𝐺𝑖 − 𝑄𝐷𝑖 = |𝑉𝑖| ∑ |𝑉
𝑗|(𝐺𝑖𝑗 cos 𝛿𝑖𝑗 + 𝐵𝑖𝑗𝑠𝑖𝑛𝛿𝑖𝑗)
𝑁𝐵
𝑗=1 (13)
where 𝑃𝐺𝑖 and 𝑄𝐺𝑖 are the active power and reactive power generated respectively at bus 𝑖, the
active and reactive demand of the load at bus 𝑖 are represented by 𝑃𝐷𝑖 and 𝑄𝐷𝑖, respectively. 𝐺𝑖𝑗
and 𝐵𝑖𝑗 represent conductance and susceptibility among buses 𝑖 and 𝑗 , respectively.
2.2.2. Inequality constraints
The Inequality constraints is categorized as:
Active output power of generators 𝑃𝐺𝑖
𝑚𝑖𝑛
≤ 𝑃𝐺𝑖 ≤ 𝑃𝐺𝑖
𝑚𝑎𝑥
𝑖 = 1,2, … , 𝑁𝐺 (14)
Voltages at generators buses 𝑉𝐺𝑖
𝑚𝑖𝑛
≤ 𝑉𝐺𝑖 ≤ 𝑉𝐺𝑖
𝑚𝑎𝑥
𝑖 = 1,2, … , 𝑁𝐺 (15)
Reactive output power of generators 𝑄𝐺𝑖
𝑚𝑖𝑛
≤ 𝑄𝐺𝑖 ≤ 𝑄𝐺𝑖
𝑚𝑎𝑥
𝑖 = 1,2, … , 𝑁𝐺 (16)
Tap settings of transformer 𝑇𝑖
𝑚𝑖𝑛
≤ 𝑇𝑖 ≤ 𝑇𝑖
𝑚𝑎𝑥
𝑖 = 1,2, … , 𝑁𝑇 (17)
Shunt VAR compensator 𝑄𝐶𝑖
𝑚𝑖𝑛
≤ 𝑄𝐶𝑖 ≤ 𝑄𝐶𝑖
𝑚𝑎𝑥
𝑖 = 1,2, … , 𝑁𝐶 (18)
Apparent power flows in transmission lines 𝑆𝐿𝑖 ≤ 𝑆𝐿𝑖
𝑚𝑖𝑛
𝑖 = 1,2, … , 𝑁𝑇𝐿 (19)
Magnitude of load buses voltage 𝑉𝐿𝑖
𝑚𝑖𝑛
≤ 𝑉𝐿𝑖 ≤ 𝑉𝐿𝑖
𝑚𝑎𝑥
𝑖 = 1,2, … , 𝑁𝑃𝑄 (20)
The dependent control variables can be easily incorporated into an optimization solution by using
the quadratic penalties formulation of the objective-function, which is stated:
𝐹
𝑔(𝑥, 𝑢) = 𝐹𝑖(𝑥, 𝑢) + 𝐾𝐺(∆𝑃𝐺1)2
+ 𝐾𝑄 ∑ (∆𝑄𝐺𝑖)2
+ 𝐾𝑉 ∑ (∆𝑉𝐿𝑖)2
𝑁𝑃𝑄
𝑖=1
𝑁𝑃𝑉
𝑖=1 + 𝐾𝑆 ∑ (∆𝑆𝐿𝑖)2
𝑁𝑇𝐿
𝑖=1 (21)
where 𝐾𝐺, 𝐾𝑄, 𝐾𝑉, and 𝐾𝑆 are penalty factors with large positive values, also ∆𝑃𝐺1, ∆𝑄𝐺𝑖, ∆𝑉𝐿𝑖,
and ∆𝑆𝐿𝑖 are penalty conditions that can be stated as:
∆𝑃𝐺1 = {
(𝑃𝐺1 − 𝑃𝐺1
𝑚𝑎𝑥
) 𝑃𝐺1 > 𝑃𝐺1
𝑚𝑎𝑥
(𝑃𝐺1 − 𝑃𝐺1
𝑚𝑖𝑛
) 𝑃𝐺1 < 𝑃𝐺1
𝑚𝑖𝑛
0 𝑃𝐺1
𝑚𝑖𝑛
< 𝑃𝐺1 < 𝑃𝐺1
𝑚𝑎𝑥
(22)
∆𝑄𝐺𝑖 = {
(𝑄𝐺𝑖 − 𝑄𝐺𝑖
𝑚𝑎𝑥
) 𝑄𝐺𝑖 > 𝑄𝐺𝑖
𝑚𝑎𝑥
(𝑄𝐺𝑖 − 𝑄𝐺𝑖
𝑚𝑖𝑛
) 𝑄𝐺𝑖 < 𝑄𝐺𝑖
𝑚𝑖𝑛
0 𝑄𝐺𝑖
𝑚𝑖𝑛
< 𝑄𝐺𝑖 < 𝑄𝐺𝑖
𝑚𝑎𝑥
(23)

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∆𝑉𝐿𝑖 = {
(𝑉𝐿𝑖 − 𝑉𝐿𝑖
𝑚𝑎𝑥
) 𝑉𝐿𝑖 > 𝑉𝐿𝑖
𝑚𝑎𝑥
(𝑉𝐿𝑖 − 𝑉𝐿𝑖
𝑚𝑖𝑛
) 𝑉𝐿𝑖 < 𝑉𝐿𝑖
𝑚𝑖𝑛
0 𝑉𝐿𝑖
𝑚𝑖𝑛
< 𝑉𝐿𝑖 < 𝑉𝐿𝑖
𝑚𝑎𝑥
(24)
∆𝑆𝐿𝑖 = {
(𝑆𝐿𝑖 − 𝑆𝐿𝑖
𝑚𝑎𝑥
) 𝑆𝐿𝑖 > 𝑆𝐿𝑖
𝑚𝑎𝑥
(𝑆𝐿𝑖 − 𝑆𝐿𝑖
𝑚𝑖𝑛
) 𝑆𝐿𝑖 < 𝑆𝐿𝑖
𝑚𝑖𝑛
0 𝑆𝐿𝑖
𝑚𝑖𝑛
< 𝑆𝐿𝑖 < 𝑆𝐿𝑖
𝑚𝑎𝑥
(25)
3. ARCHIMEDES OPTIMIZATION ALGORITHM OVERVIEW
AOA mimics the concept of the force of buoyancy which imposed upwards to an object partially or
completely immersed in fluid, proportional to the weight of the displaced fluid. AOA is a population-based
algorithm, and the individuals in the population are the immersed objects in the proposed approach. AOA,
like other population-based metaheuristic algorithms, begins the search process by populating objects
(candidate solutions) with random volumes, densities, and accelerations. At this point, each object is also
given a random position in fluid. AOA works in iterations until the termination condition is met after
evaluating the fitness of the initial population. AOA updates the density and volume of each object in each
iteration. The object's acceleration is updated based on the condition of its collision with any other
neighboring object. The new position of an object is determined by the updated density, volume, and
acceleration. The detailed mathematical formulation of AOA steps is given.
3.1. Algorithmic steps
The mathematical formulation of the AOA algorithm is introduced in this section. AOA, in theory,
can be thought of as a global optimization algorithm because it encompasses both exploration and
exploitation processes. Algorithm 2 shows the proposed algorithm's pseudo-code, which includes population
initialization, population evaluation, and parameter updating.The steps of the AOA are detailed
mathematically as follows.
- Step 1: Initialization
Initialize the positions of all objects using (26):
𝑂𝑖 = 𝑙𝑏𝑖 + 𝑟𝑎𝑛𝑑 × (𝑢𝑏𝑖 − 𝑙𝑏𝑖); 𝑖 = 1, 2, … , 𝑁 (26)
where 𝑂𝑖 is the object 𝑖𝑡ℎ in a population of 𝑁 objects, 𝑙𝑏𝑖and 𝑢𝑏𝑖 are respectively the lower and the upper
bounds of the search-space. Density (𝑑𝑒𝑛) and initial volume (𝑣𝑜𝑙) and for each object 𝑖𝑡ℎ using (27) and (28):
𝑑𝑒𝑛𝑖 = 𝑟𝑎𝑛𝑑 (27)
𝑣𝑜𝑙𝑖 = 𝑟𝑎𝑛𝑑 (28)
where rand is a D-dimensional vector that generates a number at random between [0, 1]. Finally, initialize
acceleration (𝑎𝑐𝑐) of object 𝑖𝑡ℎ using (29):
𝑎𝑐𝑐𝑖 = 𝑙𝑏𝑖 + 𝑟𝑎𝑛𝑑 × (𝑢𝑏𝑖 − 𝑙𝑏𝑖) (29)
inside this step, assess the initial population and choose the object with the highest fitness value. Assign
𝑥𝑏𝑒𝑠𝑡 , 𝑑𝑒𝑛𝑏𝑒𝑠𝑡 , 𝑣𝑜𝑙𝑏𝑒𝑠𝑡, and 𝑎𝑐𝑐𝑏𝑒𝑠𝑡 .
- Step 2: Update densities, volumes
The density and volume of 𝑖 object for the iteration t + 1 is updated using (30) and (31):
𝑑𝑒𝑛𝑖
𝑡+1
= 𝑑𝑒𝑛𝑖
𝑡+1
+ 𝑟𝑎𝑛𝑑 × (𝑑𝑒𝑛𝑏𝑒𝑠𝑡 − 𝑑𝑒𝑛𝑖
𝑡
) (30)
𝑣𝑜𝑙𝑖
𝑡+1
= 𝑣𝑜𝑙𝑖
𝑡+1
+ 𝑟𝑎𝑛𝑑 × (𝑣𝑜𝑙𝑏𝑒𝑠𝑡 − 𝑣𝑜𝑙𝑖
𝑡
) (31)
where 𝑣𝑜𝑙𝑏𝑒𝑠𝑡 and 𝑑𝑒𝑛𝑏𝑒𝑠𝑡 are the volume and density affiliated with the finest object discovered thus far,
and rand is an uniform random number.
- Step 3: Transfer operator and density factor
Initially, objects collide, and after a period, the objects attempt to reach an equilibrium state. This is
accomplished in AOA using the transfer factor TF, which transforms search from exploration to exploitation,
as defined by (32).

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𝑇𝐹 = exp (
𝑡−𝑡𝑚𝑎𝑥
𝑡𝑚𝑎𝑥
) (32)
Where the transfer TF gradually increases over time until it reaching1. Here 𝑡 and 𝑡𝑚𝑎𝑥are iteration number
and maximum number of iterations, respectively. Similarly, the density decreasing factor 𝑑 also aids in its
global to local search. It decreases over time when using (33).
𝑑𝑡+1
= exp (
𝑡−𝑡𝑚𝑎𝑥
𝑡𝑚𝑎𝑥
) − (
𝑡
𝑡𝑚𝑎𝑥
) =𝑇𝐹 − (
𝑡
𝑡𝑚𝑎𝑥
) (33)
Where 𝑑𝑡+1
decreases over time, allowing convergence in a previously identified outstanding region. It
should be noted that proper handling of this variable will ensure AOA's balance of exploration and
exploitation.
- Step 4.1: Exploration phase (collision between objects occurs)
If TF ≤ 0.5, an object collides, choose a random material (𝑚𝑟) and update acceleration of the object
for iteration t + 1 using (34):
𝑎𝑐𝑐𝑡+1
=
𝑑𝑒𝑛𝑚𝑟+𝑣𝑜𝑙𝑚𝑟×𝑎𝑐𝑐𝑚𝑟
𝑑𝑒𝑛𝑖
𝑡+1
×𝑣𝑜𝑙𝑖
𝑡+1 (34)
where 𝑑𝑒𝑛𝑖, 𝑣𝑜𝑙𝑖, and 𝑎𝑐𝑐𝑖 are density, volume, and acceleration of 𝑖 object. Besides that, 𝑎𝑐𝑐𝑚𝑟, 𝑑𝑒𝑛𝑚𝑟and
𝑣𝑜𝑙𝑚𝑟are the acceleration, density, and volume of random material. It is worth noting that TF ≤ 0.5
guarantees exploration during one-third of iterations. Changing the value from 0.5 to something else will
alter the exploration-exploitation behavior.
- Step 4.2: Exploitation phase (no collision between objects)
If TF > 0.5, Objects do not collide, update acceleration of the object for iteration t + 1 using (35):
𝑎𝑐𝑐𝑡+1
=
𝑑𝑒𝑛𝑏𝑒𝑠𝑡+𝑣𝑜𝑙𝑏𝑒𝑠𝑡×𝑎𝑐𝑐𝑏𝑒𝑠𝑡
𝑑𝑒𝑛𝑖
𝑡+1
×𝑣𝑜𝑙𝑖
𝑡+1 (35)
Where 𝑎𝑐𝑐𝑏𝑒𝑠𝑡 is the acceleration of the best object.
- Step 4.3: Normalize acceleration
Normalize acceleration to calculate the percentage of change using (36).
𝑎𝑐𝑐𝑖−𝑛𝑜𝑟𝑚
𝑡+1
= 𝑢 ×
𝑎𝑐𝑐𝑖
𝑡+1
−min (𝑎𝑐𝑐)
max(𝑎𝑐𝑐)−min (𝑎𝑐𝑐)
+ 𝑙 (36)
Where 𝑢 and 𝑙 are the range of normalization and set to 0.9 and 0.1, respectively. The 𝑎𝑐𝑐𝑖−𝑛𝑜𝑟𝑚
𝑡+1
determines
how much each agent will change in one step. If the object I is very far from the global optimum, the
acceleration value will be high, indicating that it is in the exploration phase; or else, it is in the exploitation
phase. This diagram depicts how the search progresses from the exploratory to the exploitation phase. In
most cases, the acceleration factor starts out high and gradually decreases. These assists search agents in
moving away from local solutions and toward the best solution globally. However, it is worth noting that
some search agents may require more time to remain in the exploration phase than usual. As a result, AOA
achieves the desired balance of exploration and exploitation.
- Step 5: Update position
If (TF ≤ 0.5) means less than 0.5 (exploration phase), the 𝑖𝑡ℎ position of the object for next iteration
t + 1 using (37):
𝑥𝑖
𝑡+1
= 𝑥𝑖
𝑡
+ 𝐶1 × 𝑟𝑎𝑛𝑑 × 𝑎𝑐𝑐𝑖−𝑛𝑜𝑟𝑚
𝑡+1
× 𝑑 × (𝑥𝑟𝑎𝑛𝑑 − 𝑥𝑖
𝑡
) (37)
where 𝐶1 is a constant equals to 2. Otherwise, if (TF > 0.5) means greater than 0.5 (exploitation phase), the
objects' positions are updated using (38):
𝑥𝑖
𝑡+1
= 𝑥𝑏𝑒𝑠𝑡
𝑡
+ 𝐹 × 𝐶2 × 𝑟𝑎𝑛𝑑 × 𝑎𝑐𝑐𝑖−𝑛𝑜𝑟𝑚
𝑡+1
× 𝑑 × (𝑇 × 𝑥𝑏𝑒𝑠𝑡 − 𝑥𝑖
𝑡
) (38)
where 𝐶2 a fixed value of 6. 𝑇 grows with time, is proportional to the transfer operator, and is defined using
𝑇 = 𝐶3 × 𝑇𝐹. 𝑇 increases with time in range [𝐶3 ×0.3,1] and initially deducts a certain percentage from the
best position. It begins with a low percentage because this results in a large difference between the best and
current positions; as a result, the step-size of the random walk will be large. As the search progresses, this

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percentage gradually increases to reduce the gap between the best and current positions. This results in an
appropriate balance of exploration and exploitation. 𝐹 is the flag to change the direction of motion using (39):
𝐹 = {
+1 𝑖𝑓 𝑃 ≤ 0.5
−1 𝑖𝑓 𝑃 > 0.5
(39)
where, 𝑃 = 2 × 𝑟𝑎𝑛𝑑 − 𝐶4.
- Step 6: Evaluation
Evaluate every object using the objective function f, and keep the best solution found so far in mind.
Assign 𝑥𝑏𝑒𝑠𝑡 , 𝑑𝑒𝑛𝑏𝑒𝑠𝑡 , 𝑣𝑜𝑙𝑏𝑒𝑠𝑡, and 𝑎𝑐𝑐𝑏𝑒𝑠𝑡.
3.2. AOA-based optimization process
This paper's holistic optimization model includes multi-dimensional parameters. The main AOA
encoding is no longer applied. The code vector for comprehensive OPF optimization is as:
[𝑃𝐺2, 𝑃𝐺5, 𝑃𝐺8, 𝑃𝐺11, 𝑃𝐺13, 𝑉1, 𝑉2, 𝑉5, 𝑉8, 𝑉11, 𝑉13, 𝑇11, 𝑇12, 𝑇15, 𝑇36, 𝑄10, 𝑄12, 𝑄15, 𝑄17, 𝑄20, 𝑄21, 𝑄23, 𝑄24, 𝑄29]
4. RESULTS OF SIMULATION
To investigate the efficacy of using AOA to resolve the OPF issue, it is investigated using one
standard test system of IEEE-30 bus test system. In this section, the simulation results of solving OPF using
AOA are compared to those obtained by other recent metaheuristic algorithms. The potential of AOA to
minimize the fuel cost, active power loss, total deviation in the voltage, and emission as a single-objective
problem for each objective and as a multi objective problem using weight factors which evaluated based on
the following cases presented below. Also, he proposed AOA algorithms' efficiency is also tested against
other algorithms through the modified IEEE30-bus test system to introduce the optimal allocation for RES
and prove its validity with minimizing of the fuel cost. The appropriate parameters of the AOA and other
methods are chosen based on empirical tests through running these algorithms considerable many times for
the test system with combination of different parameters. The application of AOA and other compared
techniques to solve OPF problem have been run on, a I7-8700 CPU, 16 GB RAM PC 2.8GHz, and
MATLAB 2018a.
4.1. Testing system description.
The standard IEEE 30-bus test system includes 6 generation power units, 41 lines and 24 load buses.
Bus no. 1 is selected as slack bus. The active and reactive power values of the total connected load are 2.834 pu
and 1.262 pu, respectively. The voltage magnitude of the power generating buses is limited between 0.95 pu and
1.1 pu, while the voltage magnitude of the remaining load buses is limited between 0.95 pu and 1.05 pu.
Furthermore, the tap changing transformers are adjustable between 0.9 and 1.1 pu. Furthermore, the VAR
compensator limit is set to fluctuate between 0 and 0.05 pu. Finally, more information about all of the buses and
lines data of the IEEE 30-bus testing system can be found and described in [46]-[48].
4.1.1. Case1: Minimization of fuel cost
The proposed AOA in this case is implemented on the IEEE 30-bus test system to reduce fuel costs.
Table 1 shows the best results obtained by the AOA as well as those obtained by other reported algorithms in
the literature. such as FSA, SSA, PSO [49] and DA [50]. According to the simulated results, the better
(minimum) fuel costs offered by AOA algorithm is 799.1543 $/hr which is better than that determined by the
other compared algorithms. Furthermore, Figure 1 shows the voltage profile of the AOA which guarantees
that the magnitudes of all voltages for all buses are within acceptable limits. Figure 2 depicts the convergence
characteristics of minimizing the fuel cost (more than 200 iterations) produced by the standard AOA and
other algorithms compared. It is observed that from this figure the AOA yields better convergence
characteristics than other compared algorithms.
4.1.2. Case2: Minimization of active power losses
For this case, the minimization of the real power loss is considered here as a single objective function.
The best simulation results yielded based on the AOA are presented in Table 2 together with the obtained results of
the other compared techniques, where AOA yielded power losses value of 2.980374 MW compared to the results
of 4.417859, 3.414704, 3.774816 and 3.8095 MW achieved by FSA , SSA , PSO and DA respectively. As in
case 1, the voltages profile for all buses are within their boundaries as shown in Figure 3. The minimizing
real power loss convergence characteristics obtained by AOA and other compared techniques is illustrated in
Figure 4, it is concluded that the AOA's convergence characteristics of minimizing real power loss
outperform with the other algorithms that were compared.

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Table 1. Optimal control variables for IEEE30-bus test system for minimizing fuel cost
FSA SSA PSO DA AOA
𝑃𝐺2 (MW) 46.28349 80 48.87814 48.93911 48.25165
𝑃𝐺5 (MW) 21.34807 15 21.47237 21.32534 21.40734
𝑃𝐺8 (MW) 23.26794 35 21.68903 21.33967 21.2471
𝑃𝐺11(MW) 14.56352 30 10 10 12.40777
𝑃𝐺13(MW) 16.68156 24.4285 12 12 11.11124
𝑉1(pu) 1.088163 0.95 1.1 1.1 1.099999
𝑉2(pu) 1.078415 1.1 1.086457 1.075001 1.086588
𝑉5(pu) 1.035299 1.070375 1.058621 1.034858 1.059408
𝑉8(pu) 1.049452 1.073196 1.066039 1.048098 1.068567
𝑉11(pu) 1.083961 1.1 1.08413 1.1 1.099741
𝑉13(pu) 1.094221 1.042241 1.1 1.1 1.099967
𝑇11 (6-9) 1.029402 0.9 0.9 0.995808 0.997018
𝑇12 (6-10) 1.078941 0.981035 1.1 1.008401 0.987031
𝑇15 (4-12) 1.062136 1.08085 1.030829 1.010048 1.005459
𝑇36 (2827) 0.968044 0.9 0.980481 0.96689 0.981402
𝑄10(MVR) 0.335860 0 0 2.614276 2.898496
𝑄12(MVR) 0.335860 0.378800 4.999604 2.327550 2.511993
𝑄15(MVR) 0.335860 0.098791 5 1.707832 4.557046
𝑄17(MVR) 0.335860 5 5 1.936540 4.807264
𝑄20(MVR) 0.335860 0 5 5 4.578255
𝑄21(MVR) 0.335860 0.205703 5 4.697618 4.954453
𝑄23(MVR) 0.335860 2.563488 0 2.858147 2.354573
𝑄24(MVR) 0.335860 3.784069 5 2.801551 4.362404
𝑄29(MVR) 0.335860 5 3.426363 5 3.406998
Fuel Cost ($/h) 802.7119 817.6356 799.5118 800.1055 799.1543
Power Losses (MW) 8.711178 25.18706 8.804382 9.023054 8.663665
Voltage Deviations(pu) 0.506489 1.438624 1.472048 1.283183 1.583447
Figure 1. The voltage profile of the AOA and other
compared algorithms for case 1
Figure 2. The convergence characteristics of AOA
and other compared algorithms for case 1
AOA compared algorithms for case 2

1399
Table 2. Optimal control variables for IEEE30-bus test system for minimizing real power loss
FSA SSA PSO DA AOA
𝑃𝐺2 (MW) 73.00357 80 80 76.21227 79.9728
𝑃𝐺5 (MW) 43.56395 50 50 50 49.99981
𝑃𝐺8 (MW) 34.90729 35 35 26.88553 34.7412
𝑃𝐺11 (MW) 30 30 10 21.95056 29.92457
𝑃𝐺13 (MW) 37.92178 40 40 40 39.91039
𝑉1(pu) 0.97198 1.038716 1.1 1.1 1.099507
𝑉2(pu) 0.964838 1.038704 1.097106 1.096669 1.099963
𝑉5(pu) 0.959585 1.038594 1.079159 1.1 1.088233
𝑉8(pu) 0.967453 1.038695 1.084461 1.089537 1.093303
𝑉11(pu) 0.951831 1.038707 1.049517 1.077027 1.098702
𝑉13(pu) 0.984971 1.038704 1.1 1.1 1.063046
𝑇11 (6-9) 0.979798 1.038544 1.1 1.002769 1.079032
𝑇12 (6-10) 0.900415 1.038584 0.9 1.024754 1.010058
𝑇15 (4-12) 0.984489 1.038668 1.1 1.1 1.019467
𝑇36 (28-27) 0.947636 1.038656 1.015115 1.007676 1.021381
𝑄10 (MVAR) 5 5 5 5 4.952362
𝑄12 (MVAR) 5 5 5 1.823587 4.656925
𝑄15 (MVAR) 5 5 5 5 4.600773
𝑄17 (MVAR) 5 5 5 1.878693 4.872392
𝑄20 (MVAR) 5 5 0 5 4.133455
𝑄21 (MVAR) 5 5 5 4.996018 4.086306
𝑄23 (MVAR) 5 5 3.740602 0 4.792458
𝑄24 (MVAR) 5 5 5 0 3.561832
𝑄29 (MVAR) 5 5 5 5 3.707280
Fuel Cost ($/h) 923.4265 968.4138 938.6007 936.42 966.5503
Power Losses (MW) 4.417859 3.414704 3.774816 3.8095 2.980374
4.1.3. Case3: Minimization of total voltage deviation
The proposed AOA is employed in this case, for minimizing the total Voltage deviation discussed in
section 2 as single objective function. The Table 3 shows the optimal variables resulting by AOA alongside
with the other compared algorithms, where the best and minimum voltage deviation value is 0.120906 pu
which observed with AOA compared to 0.138711 pu, 0.306075 pu,0.181846 pu and 0.291642 pu with FSA,
SSA, PSO and DA respectively. According to Figure 5, it is seen that the AOA also offer the best voltage
profile than the other compared algorithms. Also, Figure 6 proven that the convergence characteristic
obtained by the AOA outperforms those by the other compared algorithm.
Table 3. Optimal control variables for IEEE 30-bus test system for minimizing voltage deviation
FSA SSA PSO DA AOA
𝑃𝐺2 (MW) 46.20759 79.72873 80 43.42174 9.7903
𝑃𝐺5 (MW) 30.90417 50 15.79909 29.08509 45.8976
𝑃𝐺8 (MW) 23.94749 35 34.98085 31.68695 21.7849
𝑃𝐺11 (MW) 20.39471 30 13.34155 25.99727 28.3488
𝑃𝐺13 (MW) 25.34568 40 12.28699 24.50418 18.0528
𝑉1(pu) 1.015291 1.023154 1.046164 1.088389 1.012223
𝑉2(pu) 1.005648 1.023334 1.02455 1.04562 0.997005
𝑉5(pu) 1.019072 1.023358 1.021993 1.009017 1.019623
𝑉8(pu) 1.007426 1.023358 0.99251 0.990572 1.007383
𝑉11(pu) 1.023104 1.023177 1.043296 1.085419 1.039686
𝑉13(pu) 0.992756 1.023193 1.061513 1.02474 1.036563
𝑇11 (6-9) 0.939334 1.023104 0.902244 0.950507 0.991073
𝑇12 (6-10) 1.01692 1.023379 1.1 0.943128 0.934161
𝑇15 (4-12) 0.976926 1.023246 1.1 1.1 1.008232
𝑇36 (28-27) 0.963545 1.023257 0.938548 0.96876 0.956226
𝑄10 (MVAR) 5 5 4.992005 4.522767 3.992631
𝑄12 (MVAR) 5 5 5 2.504991 1.905802
𝑄15 (MVAR) 4.849158 5 5 5 4.122284
𝑄17 (MVAR) 5 5 0.347174 2.894850 2.425013
𝑄20 (MVAR) 4.910406 5 5 2.732917 4.994577
𝑄21 (MVAR) 5 5 0 5 4.847301
𝑄23 (MVAR) 5 5 5 2.416678 4.212442
𝑄24 (MVAR) 4.922147 5 5 0.582945 4.3825691
𝑄29 (MVAR) 5 5 0 2.658813 1.3320663
Fuel Cost ($/h). 822.525 968.0146 832.2631 828.1918 860.1368
Power Losses (MW). 7.934595 3.492302 8.511614 7.666995 10.44553

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compared algorithms for case3
and other compared algorithms for case3
4.1.4. Case4: Minimization of multi objective function without emission
For optimizing more than single objective function, simultaneously, the multi-objective function
using weighting factors as discussed in section 2 is proposed here for obtaining the maximum benefits of the
proposed test system. Table 4 shows how the AOA and other compared algorithms solved the multi-objective
OPF problem without considering emission in the IEEE-30 bus system. These findings suggest that using
AOA to solve the multi-objective OF problem is more effective than using other compared algorithms.
Where, the total objective function with the value of 836.3664 $/hr is better than all other algorithms with the
results 847.2615 $/hr, 926.823 $/hr, 844.1233$/hr and 845.088 $/hr achieved by FSA, SSA, PSO and DA
respectively without violating the consider constraints. As in previous cases, the voltage profiles of all buses
are within the specified limits, as shown in Figure 7, for all compared algorithms. Furthermore, as shown in
Figure 8, the AOA still has fast and smooth convergence characteristics when compared to other algorithms.
Table 4. Optimal control variables for IEEE30-bus test system for minimizing multi-objective function
without emission
FSA SSA PSO DA AOA
𝑃𝐺2 (MW) 55.78434 46.83946 48.60174 47.89617 49.16384
𝑃𝐺5 (MW) 24.03921 25.81327 22.56217 23.61245 22.79406
𝑃𝐺8 (MW) 19.01405 35 23.80653 19.69738 26.21244
𝑃𝐺11 (MW) 15.74422 27.06992 13.42591 20.53271 15.30012
𝑃𝐺13 (MW) 24.79774 35.95446 12 12.0331 11.4444
𝑉1(pu) 1.024729 1.037957 1.1 1.043497 1.052391
𝑉2(pu) 1.014104 1.012106 1.060285 1.028974 1.033897
𝑉5(pu) 1.002711 0.962564 1.008819 1.036575 1.004587
𝑉8(pu) 1.0187 1.018687 1.000206 0.999347 1.000761
𝑉11(pu) 1.03556 0.96675 1.053441 1.029754 1.017336
𝑉13(pu) 1.010232 1.050623 0.991688 1.016487 1.036494
𝑇11 (6-9) 1.00081 0.997347 0.928706 0.925325 0.978556
𝑇12 (6-10) 1.030657 0.957128 1.098984 1.050108 0.959816
𝑇15 (4-12) 0.99651 1.030978 0.944668 0.945414 1.041585
𝑇36 (28-27) 0.989278 0.907406 0.949064 0.953255 0.957001
𝑄10 (MVAR) 5 5 4.945975 4.941679 3.181699
𝑄12 (MVAR) 5 5 0 3.204882 3.6623086
𝑄15 (MVAR) 5 5 4.708812 1.703036 4.6046683
𝑄17 (MVAR) 5 5 0 1.821205 0.2866086
𝑄20 (MVAR) 5 5 4.999873 5 4.6719239
𝑄21 (MVAR) 5 5 5 1.237059 4.8645306
𝑄23 (MVAR) 5 5 0 0.703911 4.6938668
𝑄24 (MVAR) 5 5 5 2.438956 4.2484880
𝑄29 (MVAR) 5 5 0 2.399590 1.6325226
𝑄29 (MVAR) 5 5 0 2.399590 1.6325226
Objective Functions 847.2615 926.823 844.1233 845.088 836.3664
Fuel Cost ($/h) 812.9506 843.7939 804.9762 807.5542 803.6294
Power Losses (MW) 8.636995 6.688798 9.842343 9.652608 8.871927
4.1.5. Case5: Minimization of multi objective function with emission
The best results of solving a multi-objective OPF problem with considering emission for IEEE 30-
bus testing system attained by the AOA algorithm is shown in Table 5. From this table, it can be observed

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that the AOA outperforms other compared algorithms with it. As well as the AOA provides a best value of
865.9021 $/hr towards 878.1909 $/hr, 902.4330 $/hr, 877.0695 $/hr, and 888.3333 $/hr with the FSA, SSA,
PSO and DA respectively. The voltages profile of all buses in this case is given in Figure 9, it is recognized
that all voltages within specified limits for all compared algorithms. Moreover, the convergence
characteristics for this case obtained by AOA and other algorithms is shown in Figure 10, where AOA
convergence characteristic has fast and speed convergence, so it outperforms all other algorithms.
and other compared algorithms for case4
Table 5. Optimal control variables for IEEE30-bus test system for minimizing multi-objective function with
emission
FSA SSA PSO DA AOA
𝑃𝐺2 (MW) 61.9489 27.3040 48.2291 49.3112 52.22681
𝑃𝐺5 (MW) 23.1307 28.2910 22.1052 15.0505 22.71533
𝑃𝐺8 (MW) 21.6630 31.6452 35 25.9489 21.38361
𝑃𝐺11 (MW) 21.2544 26.3380 10 19.9303 14.97433
𝑃𝐺13 (MW) 21.3232 21.9211 12 15.1212 12.98854
𝑉1(pu) 1.0311 1.0762 1.1000 1.0391 1.0445
𝑉2(pu) 1.0233 1.0307 1.0560 1.0263 1.0267
𝑉5(pu) 0.9937 0.9553 1.0071 0.9940 1.0084
𝑉8(pu) 1.0058 0.9785 0.9961 1.0192 0.9999
𝑉11(pu) 1.0026 1.0047 1.1000 1.0378 1.0328
𝑉13(pu) 1.0248 1.0198 0.9879 1.1000 1.0077
𝑇11 (6-9) 0.9562 0.9501 0.9497 1.0116 1.0177
𝑇12 (6-10) 1.0206 0.9501 1.1000 1.0387 0.9255
𝑇15 (4-12) 1.0051 0.9501 0.9645 1.0472 0.9851
𝑇36 (28-27) 0.9879 0.9053 0.9626 0.9639 0.9648
𝑄10 (MVAR) 5 0.5307 5 0 3.8646
𝑄12 (MVAR) 5 4.0843 5 2.39735 2.8986
𝑄15 (MVAR) 5 4.7024 3.9236 1.89627 4.9887
𝑄17 (MVAR) 5 4.6202 0 2.10729 0.5656
𝑄20 (MVAR) 5 2.8588 5 3.97871 4.9662
𝑄21 (MVAR) 5 3.7336 0.01161 1.34798 4.8238
𝑄23 (MVAR) 5 3.00002 5 1.64823 4.9771
𝑄24 (MVAR) 5 1.05593 5 1.92027 4.3078
𝑄29 (MVAR) 5 0.53077 1.400453 5 2.6792
Objective Function 878.1909 902.4330 877.0695 888.3333 865.9021
Fuel Cost ($/h) 816.1303 829.7243 807.4802 809.0444 804.0073
Power Losses (MW) 7.8104 8.7264 9.418377 809.0444 9.210662
Voltage Deviations (pu) 0.1578 0.2474 0.195761 0.2906 0.120500
4.1.6. Case6: Optimal allocation for renewable energy sources for minimizing fuel cost
Where the integration of various renewable sources in the electrical power system increases the
degree of sophistication of the OPF problem as discussed in section 1, therefore, to show and confirm the
efficacy of the AOA proposed and implemented in this case to find the optimal allocation of renewable
energy sources and applied on the IEEE-30 bus testing system for minimizing the fuel costs. Table 6
illustrates the best AOA results as well as those obtained by other algorithms. According to the simulated
results, AOA algorithm introduces better (minimum) fuel cost with the optimal location at bus 25 with
766.0242 $/hr which is better than that determined at bus 30 by the other compared algorithms with values of
782.489 $/hr, 917.122 $/hr, and 857.0542 $/hr with the FSA, SSA and PSO respectively. Furthermore,
Figure 11 show the voltage profile of the AOA that guarantees that all voltage magnitudes for all buses are

 ISSN: 2088-8694
1402
within acceptable limits. The convergence characteristics of minimizing the fuel cost yielded by AOA and
other compared algorithms are shown in Figure 12. According to this figure, the AOA produces better
convergence characteristics than the other compared algorithms.
Table 6. Optimal RES allocation for IEEE 30-bus testing system for minimizing the fuel costs
DG Location
DG Size
𝐹𝑐𝑜𝑠𝑡 𝑃𝑙𝑜𝑠𝑠 VD
MW MVAr
Base Case - - - 11214.41 5.82226 1.14965
FSA 30 0.456018 0.2212981 782.489 6.35690 0.86232
SSA 30 0.253229 0.1579159 917.122 4.83847 0.78772
PSO 30 0.194454 0.1643651 857.0542 4.83432 0.76938
AOA 25 0.484643 0.2443312 776.0242 5.09091 0.63354
4.1.7. Minimization of the fuel cost with the penetration of RES
For the present case, to prove the efficiency of the proposed AOA algorithm, it compared also with
different recent algorithms to minimize and solve the OPF problem with a single objective function
represented in the reduction of fuel cost only and testing them on a modified IEEE 30-bus system that
included RES integrated with optimal allocation as present in case 6. Table 7 illustrates the results for this
case, where AOA yielded the best (minimum) fuel cost of 635.8983 $/hr, compared with 646.264547 $/hr,
688.92437 $/hr, 639.26731 $/hr, 637.9108 $/hr, achieved by FSA, SSA, PSO and DA respectively. In
addition, comparing with the first case the superiority of the proposed AOA algorithm is proven, where in
case 1, AOA introduce minimization of fuel cost with value of 799.1543 $/hr which is higher than that
determined by introduce proposed AOA with the integration of renewable energy source which adding
complexity of the optimal power flow problem and achieve fuel cost minimization with value of 635.8983
$/hr which is less than that determined in case1. Figure 13 illustrates the voltage profiles for all buses that are
within their boundaries. also, Figure 14 depicts the convergence characteristics of fuel cost by AOA and
other compared techniques, demonstrating that the AOA's convergence characteristics outperform those of
the other algorithms compared with it.

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Table 7. Optimal control variables for modified IEEE30-bus test system for minimizing fuel cost
FSA SSA PSO DA AOA
𝑃𝐺1 (MW) 134.328203 87.663917 153.90287 146.5862 157.6299
𝑃𝐺2 (MW) 50.1318584 64.489854 43.164128 41.74496 42.98818
𝑃𝐺5 (MW) 16.6277722 15.333046 15 19.46930 19.78803
𝑃𝐺8 (MW) 11.8628598 28.127361 10 10.91869 7.860474
𝑃𝐺11 (MW) 16.7113467 22.939775 10 12.48998 7.891434
𝑃𝐺13 (MW) 13.7670858 24.179165 12 12 7.627846
𝑉1(pu) 1.07213437 0.9875585 1.1 1.1 1.098665
𝑉2(pu) 1.06370895 0.9714789 1.0891724 1.091534 1.084298
𝑉5(pu) 1.07218239 0.9669598 1.0634733 1.075556 1.056581
𝑉8(pu) 1.04296082 0.9543305 1.0744262 1.073636 1.065559
𝑉11(pu) 1.07222602 0.9778671 1.1 1.016967 1.048832
𝑉13(pu) 1.04296081 0.9528972 0.95 1.055537 1.047008
𝑇11 (6-9) 1.03729456 0.9 1.1 1.026075 0.977883
𝑇12 (6-10) 1.07213628 0.9184583 1.1 1.022914 1.030724
𝑇15 (4-12) 1.07214024 0.9686594 1.1 1.022754 0.998454
𝑇36 (28-27) 1.07226459 0.9106857 1.1 1.1 1.077131
𝑄10 (MVAR) 2.67969912 0.4441304 0 3.149833 2.603767
𝑄12 (MVAR) 2.67969912 1.4258255 5 2.155722 1.15499
𝑄15 (MVAR) 2.67969912 3.0397021 5 3.474743 1.84191
𝑄17 (MVAR) 2.67969912 0.8107773 0 3.012264 2.213119
𝑄20 (MVAR) 2.67969912 4.4382199 5 0 3.070424
𝑄21 (MVAR) 2.67969912 0.5554283 5 0 3.407764
𝑄23 (MVAR) 2.67969912 0.4320592 0 1.879185 2.980572
𝑄24 (MVAR) 2.67969912 1.0959013 0 0.982048 2.07048
𝑄29 (MVAR) 2.67969912 1.2720052 5 1.774889 1.299162
Fuel Cost ($/h) 646.264547 688.92437 639.26731 637.9108 635.8983
Power Losses (MW) 8.49342960 7.7974323 9.1313042 8.273496 8.850231
5. CONCLUSION
In order to solve the OPF problem considering the fuel cost, power loss, voltage profile
improvement and emissions, a new metaheuristic algorithm has been investigated in this paper. The e
efficacy and supremacy of AOA have been evaluated based on standards for solving and optimizing the
single-objective and multi-objective function of OPF problems and modified testing system of IEEE-30 bus
with the presence of RES to prove its efficiency in finding the optimal allocation with minimization of fuel
cost. According to the results, the AOA provided a better mitigation of the objective functions in all cases
than other recently compared algorithms. The comparison results clearly show that the AOA outperformed
these recent algorithms regardless of the type of objective function, indicating the AOA's ability to solve
other real-life applications.
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BIOGRAPHIES OF AUTHORS
Mohammed Hamouda Ali is a lecturer in Electrical Engineering Department at Al-
Azhar University, Cairo, Egypt. He received his B.Eng., M.Eng. and Ph.D. degrees in Electrical
Engineering from Al-Azhar University, Cairo, Egypt, in 2011, 2016, and 2021, respectively. His
research interests are in power electronics, power system planning, optimization, operation,
power system control, power quality, reliability, and renewable energy technology. He can be
contacted at email: Eng_MohammedHamouda@azhar.edu.eg.
Ahmed Mohammed Attiya Soliman has been a lecturer in Electrical Engineering
Department at Al-Azhar University, Cairo, Egypt since 2018. He received his B.Sc., and M.Sc.
Degrees from Al-Azhar University in 2008, and 2015 respectively; and a Ph.D. degree in
Electrical Power Engineering from Al-Azhar University in 2018. He is interested in different
fields like power electronics applications, high voltage direct current (HVDC) systems, electrical
power quality, integration of renewable energy sources in electrical distribution networks, smart
grids, and optimization techniques applications in electrical power system networks. He can be
contacted at email: eng_ahmed1020@azhar.edu.eg.
Salah K. Elsayed is an Associate professor at the Electrical Engineering
Department- Faculty of Engineering- Al-Azhar University, Cairo–Egypt. He received his B.Sc.,
M.Sc. and Ph.D. Degrees from Al-Azhar University in 2005, 2009, and 2012 respectively. He is
an Associate Prof. at the Electrical Engineering Department, College of Engineering, Taif
University, Saudi Arabia. His areas of interest include Intelligent Systems Applications for
Power Systems Stability and control. He can be contacted at email:
salah_kamal1982@yahoo.com.

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