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Article

Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method

1
Department of Electrical Engineering, College of Engineering, Northern Border University, Arar 73222, Saudi Arabia
2
Electrical Engineering Department, College of Engineering, Jouf University, Sakaka 72388, Saudi Arabia
3
Department of Electrical and Computer Engineering, College of Engineering and Information Technology, Ajman University, Ajman P.O. Box 346, United Arab Emirates
4
Institute of Information Engineering, Automation and Mathematics, Slovak University of Technology in Bratislava, 81107 Bratislava, Slovakia
5
John von Neumann Faculty of Informatics, Obuda University, 1034 Budapest, Hungary
*
Author to whom correspondence should be addressed.
Appl. Sci. 2022, 12(16), 7959; https://doi.org/10.3390/app12167959
Submission received: 8 July 2022 / Revised: 1 August 2022 / Accepted: 4 August 2022 / Published: 9 August 2022
(This article belongs to the Special Issue Advances in Power Flow Analysis of Power System)

Abstract

:
A key component of the design and operation of power transmission systems is the optimal power flow (OPF) problem. To solve this problem, several optimization algorithms have been developed. The primary objectives of the program are to minimize fuel costs, reduce emissions, improve voltage profiles, and reduce power losses. OPF is considered one of the most challenging optimization problems due to its nonconvexity and significant computational difficulty. Teaching–learning-based optimization (TLBO) is an optimization algorithm that can be used to solve engineering problems. Although the method has certain advantages, it does have one significant disadvantage: after several iterations, it becomes stuck in the local optimum. The purpose of this paper is to present a novel adaptive Gaussian TLBO (AGTLBO) that solves the problem and improves the performance of conventional TLBO. Validating the performance of the proposed algorithm is undertaken using test systems for IEEE standards 30-bus, 57-bus, and 118-bus. Twelve different scenarios have been tested to evaluate the algorithm. The results show that the proposed AGTLBO is evidently more efficient and effective when compared to other optimization algorithms published in the literature.

1. Introduction

Performing power networks efficiently is one of the essential goals in the electricity sector. One of the most crucial tools for accomplishing this task is to use the optimal power flow (OPF) method to optimize the control variables of the power systems. Under various operating conditions in the power systems, it is crucial to find the optimal values for these variables to maintain secure and cost-effective operations. Meanwhile, OPF problems are complex, nonlinear, and nonconvex, with uneven and non-derivative components. As a result, traditional approaches cannot find suitable solutions. This, therefore, has always required a robust optimization approach, which has been realized and investigated [1]. The complexity of optimization problems across multiple disciplines cannot be solved using a universally applicable solution method in today’s engineering environment. In recent years, hundreds of novel approaches have been proposed, and many have proven successful in resolving certain optimization challenges [2,3,4]. Over the past half-century, electrical engineering experts have focused their attention on the classic OPF problem. Its goals have evolved significantly over the years. By analyzing this problem and considering nonlinear constraints, the power of generation units and settings of controllable devices can be controlled. Traditionally, linear, and nonlinear programming have been used to solve this problem. The problem’s nonlinear, convex, and non-derivative nature has recently resulted in a trend towards new optimization methods. This study improves teaching–learning-based optimization (TLBO) to be more effective for a wide range of different OPF problems.
Numerous constrained optimization issues have recently been addressed using population-based metaheuristic algorithms such as engineering design and operation. In power system operation, a significant concern is the optimal flow of power, which is one of the main issues addressed by many implementations of these algorithms.
Metaheuristic optimization approaches have been proposed to address a variety of OPF problems, including tabu search (TS) [5], a new self-adaptive TLBO with various distributions [6], Harris Hawks optimization (HHO) [7], a grasshopper optimization algorithm (GOA) for a grid-connected microgrid OPF controller [8], multi-objective differential evolution (DE) algorithm [9], teaching–learning–studying-based optimizer (TLSBO) [10], an improved colliding bodies optimization (ICBO) [11], artificial bee colony (ABC) [12], chaotic ABC algorithm [13], adaptive group search optimization (AGSO) [14], modified honey bee mating optimization (MHBMO) [15], moth swarm algorithm (MSA) [16], a hybrid particle swarm optimization (PSO) based on DE (DEEPSO) in the controllable wind-photovoltaic energy systems [17], an enhanced self-adaptive DE [18], slime mold algorithm (SMO) [19], resolving the OPF issue with increased renewable energy penetration via equilibrium optimizer algorithm (EOA) [20], a hybrid PSO and SFLA algorithm [21], differential search algorithm (DSA) [22], a hybrid HHO based on DE (HHODE) [23], improved artificial bee colony (IABC) [24], grey wolf optimizer (GWO) [25], chaotic invasive weed optimization (CIWO) algorithms [26], a Sine-Cosine algorithm [27], TLBO algorithm enhanced with Lévy mutation (LTLBO) [28], an improved strength Pareto evolutionary algorithm [29], ant lion algorithm (ALA) [30], interior search algorithm (ISA) [31], Jaya and modified Jaya algorithm [32,33], a gaussian bare-bones levy-flight firefly algorithm (GBLFA) [34], an new improved adaptive DE [35], a new modified interior search algorithm (MISA) [36], social spider optimization (SSO) [37], glow-worm swarm optimization (GSO) algorithm [38], an adaptable online primary regulation is formulated in [39] in order to optimize operation for microgrid to minimizing power losses, white shark optimizer (WSO) for OPF of hybrid systems with renewable energy sources [40], an enhanced equilibrium optimizer (EEO) [41], a developed turbulent flow of a water-based optimizer (TFWO) [42] to OPF with single and multi-objective functions applied on the IEEE 30-bus and 57-bus systems, teaching learning-based algorithm (TLBA) for losses and fuel costs optimization of hybrid alternating current (AC) and multi-terminal direct current (DC) power grids [43], etc., which verified the effectiveness of these methods in solving the OPF problems.
Studies in this field show that each article has tried to provide an algorithm that can work better than previous methods. This shows the importance of the OPF problem and the need for new methods that can perform better in avoiding local-optimal solutions while converging rapidly. As a result of the students’ learning and the teacher’s instruction in a typical classroom, Rao et al. proposed a new efficient algorithm in 2012 [44]. Several implementations of this fundamental algorithm have been proposed in several disciplines over the past few decades, and many updated versions have been offered. Still, there is evidence from the literature that TLBO can converge to the optimal local solution [10,28]. Alternatively, the metaheuristic algorithms can be modified to avoid local optima, although generally, these algorithms have the issue of a relatively slow convergence rate. Therefore, this paper develops a new version of the TLBO algorithm, which combines TLBO with adaptive Gaussian mutation to enhance local optima avoidance ability and fast convergence features.
The following are the most significant contributions of the proposed AGTLBO algorithm:
  • A new adaptive Gaussian mutation is integrated into the conventional TLBOs to enhance its exploration and exploration capabilities.
  • The proposed algorithm can resolve a wide range of OPF problems for IEEE 30-bus, 57-bus, and 118-bus standard power systems with better outcomes than the state-of-the-art metaheuristic algorithms.
  • Many-objective OPF problems to reduce the loss, emission, fuel cost, and voltage deviation in 12 cases are considered and solved by the proposed method to confirm its performance and applicability for various kinds of OPF problems.
The present paper is organized into four sections: Section 2 presents formulations of the OPF problems, Section 3 discusses the principles and functions of TLBO, and Section 4 presents simulation results and discusses the conclusions. In Section 5, the conclusions of the paper are given.

2. OPF Problem Formulation

According to [14,15,16], the non-linear OPF optimization aims at objectives tradeoff in the power network subject to various inequality and equality constraints expressed as follows.
Min   J ( x ,   u )
Subject   to :   g ( x ,   u ) = 0
h ( x ,   u )     0
It is assumed that u and x are the system’s decision and state variables, respectively. J ( x , u ) denotes the objective function of the problem, g ( x , u ) , the equality constraints of the load flow equation, and h ( x , u ) , the inequality constraints of the transmission network’s physical limits on the decision and state variables.
The variables controlling the OPF problems include the active power outputs from the generation systems, the voltage magnitudes from the generator buses, changes in the transformer tap points, and the reactive power compensation. Additionally, voltage magnitudes in load buses, maximum transmission capacities of branches, active power outputs of reference buses, and reactive generation of generators are state variables in this context.

2.1. Problem Constraints

The OPF is subjected to equality and inequality constraints related to systems and units. These constraints are outlined below.

2.1.1. Equality Constraints

There are two types of power supply constraints applicable to system buses—real and reactive [14]:
P G i P D i = V i j = 1 N B V j ( G i j cos δ i j + B i j sin δ i j )  
Q G i Q D i = V i j = 1 N B V j ( G i j sin δ i j B i j cos δ i j )
Vi and Vj represent the magnitudes of the voltages, while i and j are the indices of the bus. PGi and QGi are the generated units’ real and reactive power outputs, respectively. The real load demand is represented by PDi and the reactive load demand by QDi. δij is the voltage phase angle difference between nodes i and j. Gij and Bij are the conductance and susceptance of line ij, respectively. NB is the number of buses.

2.1.2. Inequality Constraints of the OPF Problem

The vector h ( x , u ) includes the following groups of inequality constraints.
  • Constraints relating to the generation units
Active and reactive output generation and bus voltage magnitudes are limited in their respective bounds at each generation bus i (i = 1, 2, …, NG) as follows [14]:
P G i min P G i P G i max Q G i min Q G i Q G i max V G i min V G i V G i max
where V G i min and V G i max express the voltage magnitude range of ith generator; P G i min and P G i max and the power output allowed ith generator. Furthermore, Q G i min and Q G i max present the range of the power output of ith generator. Finally, NG indicates the number of the generators in the network.
  • According to [14], the transformers’ taps can be adjusted as follows.
    T i min T i T i max ;   i = 1 , ,   N T
    where T i min and T i max present the upper and lower values for ith.
  • As stated in [14], the generation of volt ampere reactive (VAR) compensating units are constrained as follows.
    Q C i min Q C i Q C i max ;   i = 1 , ,   N C
    where Q C i min and Q C i max are the allowable range of VAR injection of ith compensating unit and NC is the number of the VAR compensating units in the network. The highest apparent power transfer across branches and the acceptable range of load bus voltage magnitudes are known as security constraints [14]:
    V L i min V L i V L i max
    S l i S l i max
The magnitudes boundary of voltage ith load bus are identified by V L i min and V L i max . Additionally, the apparent power via ith transmission line and its higher bound represent by S l i and S l i max , respectively.

2.2. Constraint Handling

It is worth noting that the optimization process considers the decision variables’ allowable ranges. In order to tackle constrained optimization problems, one of the most common approaches is to apply penalized values to violations of constraints on the objective function. Therefore, the problem is converted from a constrained to an unconstrained optimization problem [45]. The fitness function of the OPF problem, including the penalty costs, is expressed as follows [26]:
J = i = 1 N G F i ( P G i ) + λ P ( P G 1 P G 1 lim ) 2 + λ Q i = 1 N G ( Q G i Q G i lim ) 2 + λ V i = 1 N P Q ( V L i V L i lim ) 2 + λ S i = 1 N T L ( S l i S l i lim ) 2
where λP, λV, λQ and λS are the penalty factors, and NPQ and NTL are the number of the load buses and transmission lines in the network, respectively; the supplementary variable ulim is given as [26]:
u lim = { u ;   u min u u max u min ;                 u u min u max ;                 u u max

3. Proposed Algorithm

3.1. TLBO

The TLBO algorithm was inspired by the classroom teaching and learning method [46]. Teachers use education in the classroom to enhance student awareness. By sharing their experiences, students often seek to expand their knowledge. The TLBO algorithm simulates both processes, which are described below.
As a first step, a set of initial points is created randomly within the feasible space. The point with the highest awareness is the teacher, while the others are the students. Until the stopping criteria are met, both phases of teaching and learning continue. These two phases are described below.

3.1.1. Teaching Phase

As mentioned, the teaching by a teacher is aimed at increasing the average knowledge of class members (Xmean). Therefore, this process is modeled by the following relation [46]:
X n e w = X i + r a n d ( X t e a c h e r ( T F X m e a n ) )
where X i and X n e w denote the status of the ith student before and after the teaching, Xteacher is the status of the teacher, Xmean is the average status of the class and r a n d represents a uniformly distributed random number in the range [0, 1]. Additionally, TF is a random number that can be 1 or 2 and indicates the student’s learning rate. T F can be obtained by the following equation [46]:
T F = round ( r a n d ) + 1
where r a n d denotes a random number in the range [0, 1].

3.1.2. Learning Phase

Students choose one student at random to share their knowledge during the learning process. The student with less knowledge learns from the other. Mathematically, this phase is defined as follows [46]:
X n e w = { X i + r a n d ( X i i X i ) ; if C o s t ( X i i ) < C o s t ( X i ) X i + r a n d ( X i X i i ) ;   else .
The index ii in (15) describes the haphazardly selected student, and r a n d is a random number in the range [0, 1]. For a minimization problem, the smaller a student’s cost function, i.e., C o s t ( X i ) , the more information they have. If the solution is better, a new point is agreed upon in the teaching and learning processes.

3.2. GTLBO Using Gaussian Mutation Strategy

3.2.1. Gaussian Distribution

There are many fields in which the Gaussian distribution is widely used. In this case, we have a simple distribution with a mean (μ), variance (σ2), and standard deviation. Using the following formula, we can determine the probability density function for this distribution [47]:
f μ , σ 2 ( x ) = 1 σ 2 π e ( x μ ) 2 2 σ 2
Below is a demonstration of the normal distribution of a real-valued random variable Y:
Y N ( μ , σ 2 )
The Gaussian mutation may therefore be characterized by a Gaussian distribution with μ = 0 and σ = 1 as follows:
X n e w ( j ) = X i ( j ) + N j ( 0 , 1 )
where Nj (0,1) specifies that a random number is created for each value of j. The mutation shifts a small amount of the local optimum to new solutions in order to prevent conventional TLBO from sticking in the local optima. As a result, the power of the algorithm can be effectively increased. In fact, in the GTLBO algorithm, the phases of learning and teaching are calculated as the following equations.
X n e w = X i + r a n d ( X t e a c h e r ( T F X m e a n ) ) + N ( 0 , 1 )
X n e w = { X i + r a n d ( X i i X i ) + N ( 0 , 1 ) ; if C o s t ( X i i ) < C o s t ( X i ) X i + r a n d ( X i X i i ) ; else .

3.2.2. Adaptive GTLBO

In GTLBO, as the number of iterations increases, the mutation rate for the population needs to be reduced. Random values N (0,1) significantly affect the quality of the result. Therefore, if the random value is slightly larger at first, the algorithm can observe a suitable jump in the solutions to move faster towards the global optimization and escape from the local optimization. On the other hand, by increasing the number of iterations of the algorithm, the obtained optimal value becomes very close to the optimal global solution. Hence, the value N (0,1) needs to be more limited. Hence, we have made it dependent on the number of iterations, and the final formula can be given as follows:
X n e w = X i + r a n d ( X t e a c h e r ( T F X m e a n ) ) + ( N ( 0 , ϕ I t e r ) ) D
X n e w = { X i + r a n d ( X i i X i ) + ( N ( 0 , ϕ I t e r ) ) D if   C o s t ( X i i ) < C o s t ( X i ) X i + r a n d ( X i X i i ) + ( N ( 0 , ϕ I t e r ) ) D ;   else .
ϕ I t e r = ϕ S t a r t + ( ( I t e r I t e r m a x ) 3 ( ϕ S t o p ϕ S t a r t ) )
where this paper assumes ϕ S t a r t = 10 9 , ϕ S t o p = 10 4 , and D is the number of the dimension in problem.
TLBO performance is evaluated using the modified learning phase [28] based on the following modifications to (22):
X n e w = { X i i + r a n d ( X i i X i ) + ( N ( 0 , ϕ I t e r ) ) D if C o s t ( X i i ) < C o s t ( X i ) X i + r a n d ( X i X i i ) + ( N ( 0 , ϕ I t e r ) ) D ;   else .
The flowchart of the proposed algorithm is shown in Figure 1 which is adapted from [6,28].

3.3. Solution Method

In order to solve OPF problems, the AGTLBO algorithm is used. The following sections detail the execution of the proposed AGTLBO for OPF problems:
  • Step 1: Initialize AGTLBO using the parameters from the under-study test system. The maximum number of iterations and population (Npop) for the 30-bus system have been chosen 500 and 25, respectively, 1000 and 40 for the 57-bus system, and 5000 and 60 for the 118-bus system.
  • Step 2: Using Equations (21)–(24), generate the starting population or students depending on the number of populations   X i = X j min + r a n d ( X j max ( X j min ) = [ P G 2 : P G N G ,   V G 1 : V G N G , T 1 : T N T ,   Q C i : Q C N C ] for i = 1: Npop, the lowest and maximum values of control variables   X j min and X j max   ( j = 1 : D ) , system constraints, and OPF constraint constraints using Equations (4)–(10).
  • Step 3: Using the system and OPF problem constraints, determine the under-study objective function of the initially generated solutions based on (11) and (12) and save them as initial answers of the initial population.
  • Step 4 (Adaptive Gaussian Teaching phase): Enhance the answers from the previous step based on (21), apply the system and OPF problem constraints, and calculate their under-study objective function based on (11) and (12). Then, during the subsequent iteration, choose the best answer (new student) from the current and prior solutions provided.
  • Step 5: Compare the best student solution (new student) to the previous step’s greatest global solution (teacher). If the new student solution is superior to the best global (teacher) solution, the former takes precedence; otherwise, the old (teacher) solution is retained in the algorithm memory.
  • Step 6 (Adaptive Gaussian Learning phase): Based on (24) and the system and optimal OPF problem constraints, enhance the previous step solutions and construct their under-study objective function based on (11) and (12). Then, during the subsequent iteration, choose the student’s answer from among the current and previously generated solutions.
  • Step 7: Compare the student solution to the previous step’s teacher solution. If the student solution is superior to the teacher solution, the student solution takes precedence over the teacher solution; otherwise, the prior student solution is stored in the algorithm memory.
  • Step 8: Has the algorithm termination requirement been met? Proceed to Step 4 and continue the optimization process if yes; if not, proceed to Step 3.

4. Numerical Results

The proposed method is used to solve OPF problems in IEEE 30- and 57-bus power systems [16] to verify the efficiency and effectiveness of the AGTLBO algorithm. Both simulations were run on a Pentium IV E5200 PC with 2 GB RAM and MATLAB 7.6. The values of penalty factors in (11) are nominated as λ P = 5,000,000, λ V = λ Q = 5,000,000 and λ S = 1,000,000. On each event, each algorithm was run 25 times. The maximum number of iterations and population for the 30-bus system was 500 and 25, respectively, 1000 and 40 for the 57-bus system, and 5000 and 60 for the 118-bus system.

4.1. Cases Definition

The following 12 cases were examined to determine the efficacy of AGTLBO.

4.1.1. The Studied Cases for the IEEE 30-Bus System

  • Minimizing the fuel cost (Case 1).
  • Minimizing piecewise quadratic fuel cost functions (Case 2).
  • Minimizing the emission (Case 3).
  • Minimizing the actual power loss (Case 4).
  • Minimizing the fuel cost considering valve point effect (VPE) (Case 5).
  • Minimizing the fuel cost and actual power loss (Case 6).
  • Minimizing the fuel cost and voltage deviation (Case 7).
  • Minimizing the fuel cost, emissions, voltage deviation and losses (Case 8).

4.1.2. The Studied Cases for the IEEE 57-Bus System

  • Minimizing the fuel cost (Case 9).
  • Minimizing the fuel cost while improving voltage profile (Case 10).
  • Minimizing the fuel cost, emissions, voltage deviation and losses (Case 11).

4.1.3. The Studied Case for the IEEE 118-Bus System

  • Minimizing the fuel cost (Case 12).

4.2. EEE 30-Bus Test System

The generator data, bus information, and the lower and upper bounds for the decision variables of the IEEE 30-bus test system (Figure 2) can be found in [48,49,50]. On buses 1, 2, 5, 8, 11, and 13, there are six generators, and on lines 6–9, 6–10, 4–12, and 28–27, there are four transformers with off-nominal tap ratios [49]. The entire device demand is 2.834 p.u. at a 100 MVA base. All load buses’ upper and lower voltage ranges are fixed to 1.05–0.95 in p.u., respectively.
Table 1 summarizes all AGTLBO’s optimal options for the 30-bus power grid.

4.2.1. Case 1: Minimizing the Fuel Cost

In natural power systems, generation units use fossil fuel resources such as oil and natural gas, and their fuel costs can be formulated as follows [16]:
O F 1 = i = 1 N G F i ( P G i ) = i = 1 N G ( α i + b i P G i + c i P G i 2 ) + P C o s t
P C o s t = λ P ( P G 1 P G 1 l i m ) 2 + λ Q i = 1 N G ( Q G i Q G i l i m ) 2 + λ V i = 1 N P Q ( V L i V L i l i m ) 2 + λ S i = 1 N T L ( S l i S l i l i m ) 2  
A fuel cost is given by Fi, and an output is given by PGi, respectively, for the ith generator. There are three cost coefficients called αi, bi and ci, as well as NG, which indicates the number of generators overall. Detailed information about the cost coefficients can be found in [51]. P C o s t is the penalty cost of constraints violations. In Table 2, TLBO and AGTLBO algorithms are compared in terms of fuel costs (USD/h), emissions (t/h), power losses (MW), and V.D. (p.u.). A comparison is made between these results and those reported in recent literature such as TS (Tabu Search) [5], ABC (Artificial Bee Colony) [12], AGSO (Adaptive Group Search Optimization) [14], MHBMO (Modified Honey Bee Mating Optimisation) [15], MSA (Moth Swarm Algorithm) [16], FPA (Flower Pollination Algorithm) [16], MFO (Moth-Flame Optimization) [16], MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21], GWO (Grey Wolf Optimizer) [25], MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45], DE (Differential Evolution) [49], SKH (Stud Krill Herd Algorithm) [52], SFLA-SA (Shuffle Frog Leaping Algorithm and Simulated Annealing) [53], ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51], MGBICA (Modified Gaussian Bare-Bones Imperialist Competitive Algorithm) [54], IEP (Improved Evolutionary Programming) [50], PSOGSA (Particle Swarm Optimization and Gravitational Search Algorithm) [55], and EP (Evolutionary Programming) [56]. As Table 2 shows, the proposed algorithm (Case 1) is suitable and effective for optimizing this objective function compared to other algorithms. According to the obtained simulation data, the minimum cost obtained from AGTLBO is 800.4811 USD/h, which is less comparing to result of other algorithms. By employing the AGTLBO, the objective function of TLBO, TS, ABC, AGSO, MHBMO, MSA, FPA, MFO, MPSO-SFLA, GWO, MICA-TLA, DE, SKH, SFLA-SA, ARCBBO, MGBICA, IEP, PSOGSA, and EP are decreased by 0.1930 USD/h, 1.8090 USD/h, 0.1790 USD/h, 1.2690 USD/h, 1.5040 USD/h, 0.0290 USD/h, 2.3170 USD/h, 0.2050 USD/h, 1.2690 USD/h, 0.9290 USD/h, 0.5680 USD/h, 1.9090 USD/h, 0.0330 USD/h, 1.3090 USD/h, 0.0350 USD/h, 0.6600 USD/h, 1.9790 USD/h, 0.0180 USD/h, and 3.0890 USD/h, respectively, which demonstrates the effectiveness of AGTLBO.

4.2.2. Case 2: Minimizing the Total Fuel Cost Considering Multi-Fuel Sources

A few generators in existing power systems are driven by several fuels, resulting in a cost function with piecewise quadratic characteristics. As a result, the OPF problem’s objective function for these structures can be defined as follows [16]:
F ( P G i ) = { α i 1 + b i 1 P G i + c i 1 P G i 2 ; P G i min P G i P G i 1 α i 2 + b i 2 P G i + c i 2 P G i 2 ; P G i 1 P G i P G i 2 α i k + b i k P G i + c i k P G i 2 ; P G i k 1 P G i P G i max
where cost coefficients for the ith generator are represented by αik, bik and cik for fuel type k.
These circumstances are commonly considered just for the two first generators in several earlier papers, as described below [16]:
O F 2 = ( i = 1 2 α i k + b i k P G i + c i k P G i 2 ) + ( i = 3 N G α i + b i P G i + c i P G i 2 ) + P C o s t
Reference [49] provided the data for the units with several fuel types. Table 3 summarizes the optimization results of the TLBO and AGTLBO algorithms, and these results are compared to those published in the previous works such as LTLBO (Lévy Teaching–Learning-Based Optimization) [28], MDE (Modified Differential Evolution) [49], MFO (Moth-Flame Optimization) [16], IEP (Improved Evolutionary Programming) [50], MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45], FPA (Flower Pollination Algorithm) [16], SSO (Social Spider Optimization) [37], GABC (Gbest Guided Artificial Bee Colony Algorithm) [57], MSA (Moth Swarm Algorithm) [16], and MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21]. The table shows that AGTLBO has adequate optimization ability compared to other approaches. According to the obtained simulation data, the minimum cost obtained from AGTLBO is 646.4511 USD/h, which is less comparing to result of other algorithms. By employing the AGTLBO, the objective function of TLBO, LTLBO, MDE, MFO, IEP, MICA-TLA, FPA, SSO, GABC, MSA, and MPSO-SFLA are decreased by 0.6833 USD/h, 0.9804 USD/h, 1.3949 USD/h, 2.8216 USD/h, 2.8609 USD/h, 0.6491 USD/h, 4.9257 USD/h, 16.9007 USD/h, 0.5789 USD/h, 0.3853 USD/h, and 1.0989 USD/h, respectively, which demonstrates the effectiveness of AGTLBO.

4.2.3. Case 3: Minimizing the Emission

One of the most pressing concerns around fossil-fuel-based power production today is the emissions of different contaminants caused by the combustion of these fuels by generation units. As a result, the sum of pollution is known as a single component (as in the current research) or conjunction with other functions in the latest optimization articles. In this work, the emission function is given as [16]:
O F 3 = i = 1 N G F E i ( P G i ) = i = 1 N G ( α i + β i P G i + γ i P G i 2 + ξ i exp ( λ i P G i ) ) + P C o s t
where FEi signifies the amount of emission caused by the ith thermal generator. γ i , β i , ξ i and λ i are the emission coefficients of the ith generator, while α i (ton/h), γ i (ton/h MW2), and β i (ton/h MW) are related to SOX. Additionally, λ i (1/MW) and ξ i (ton/h) are related to NOX.
Table 4 compares the findings of AGTLBO, the original TLBO and several state-of-the-art methods such as ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51], ABC (Artificial Bee Colony) [12], GBICA (Gaussian Bare-Bones Imperialist Competitive Algorithm) [54], FPA (Flower Pollination Algorithm) [16], AGSO (Adaptive Group Search Optimization) [14], DSA (Differential Search Algorithm) [22], MFO (Moth-Flame Optimization) [16], MSA (Moth Swarm Algorithm) [16], MPSO-SFLA (Modified Shuffle Frog Leaping Algorithm) [53], and MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21]. It can be shown that AGTLBO’s solution has a lesser cost function value than those achieved from other approaches. According to the obtained simulation data, the minimum cost obtained from AGTLBO is 0.2048 t/h, which is less comparing to result of other algorithms. By employing the AGTLBO, the objective function of TLBO, ARCBBO, ABC, GBICA, FPA, AGSO, DSA, MFO, MSA, MSFLA, and MPSO-SFLA are decreased by 0.0 t/h, 0.0 t/h, 0.0 t/h, 0.0001 t/h, 0.0004 t/h, 0.0011 t/h, 0.0010 t/h, 0.0001 t/h, 0.0 t/h, 0.0008 t/h, and 0.0004 t/h, respectively, which demonstrates the effectiveness of AGTLBO.
There is no difference between the proposed algorithm and the TLBO, ARCBBO, ABC, or MSA algorithms, even though it has the same solution. The solution of 0.2048 is the global optimal solution to this problem, which can be described as the simplest objective function of load distribution. However, this problem involves other complicated functions and constraints to judge the performance of these optimization algorithm.

4.2.4. Case 4: Minimizing the Real Power Loss

One of the most critical priorities of a power grid is to distribute electrical power so that the transmitting network’s power losses are negligible. As a result, it has been recognized as a single function in this part, and optimization has been performed. The following is a summary of the objective function [16]:
O F 4 = P L o s s = k = 1 k = ( i , j ) N T L g k ( V i 2 + V j 2 2 V i V j cos δ i j ) + P C o s t
where PLoss is the total active power losses of the transmission network. gk is the conductance of branch k. Vi and Vj are the voltages of ith and jth bus, respectively. NTL represents the number of transmission lines, δij indicates the phase difference of voltages between bus i and bus j. Table 5 demonstrates the results of this case for the proposed algorithm in comparison with ABC (Artificial Bee Colony) [12], FPA (Flower Pollination Algorithm) [16], MFO (Moth-Flame Optimization) [16], MSA (Moth Swarm Algorithm) [16], DSA (Differential Search Algorithm) [22], GWO (Grey Wolf Optimizer) [25], Jaya [32], ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51], EEA (Efficient Evolutionary Algorithm) [58], EGA (Enhanced Genetic Algorithm) [58], EGA-DQLF (Enhanced Genetic Algorithm to Decoupled Quadratic Load Flow) [59], and ALC-PSO (Particle Swarm Optimization with an Aging Leader and Challengers) [60]. According to the findings, the proposed improved algorithm was able to find a satisfactory solution in the required number of iterations. According to the obtained simulation data, the minimum cost obtained from AGTLBO is 3.0906 MW, which is less comparing to result of other algorithms. By employing the AGTLBO, the objective function of TLBO, ABC, FPA, MFO, MSA, DSA, GWO, Jaya, ARCBBO, EEA, EGA, EGA-DQLF, and ALC-PSO are decreased by 0.0198 MW, 0.0172 MW, 0.4755 MW, 0.0205 MW, 0.0099 MW, 0.0039 MW, 0.3194 MW, 0.0129 MW, 0.0103 MW, 0.1917 MW, 0.1534 MW, 0.1102 MW, and 0.0794 MW, respectively, which demonstrates the effectiveness of AGTLBO.

4.2.5. Case 5: Minimizing the Fuel Cost Considering Valve-Point Effects (VPE)

The generators have typically functioned with fossil fuels in power systems. The fuel consumption can be controlled by adjusting the valves of fuel supply reservoirs. Therefore, by altering the fuel cost function of Case 1, VPE can be modeled [53]:
O F 5 = O F 1 + i = 1 N G | d i sin [ e i ( P G i min P G i ) ] |
where the cost coefficients for the ith generating unit are di and ei.
Table 6 shows the results attained by the AGTLBO algorithm and the solutions obtained in previous publications such as PSO (Particle Swarm Optimization) [11], and SP-DE (Differential Evolution Integrated Self-Adaptive Penalty) [61], demonstrating the efficacy of the proposed AGTLBO. According to the obtained simulation data, the minimum cost obtained from AGTLBO is 832.1624 USD/h, which is less comparing to result of other algorithms. By employing the AGTLBO, the objective function of TLBO, PSO, and SP-DE are decreased by 0.3701 USD/h, 0.5247 USD/h, and 0.3189 USD/h, respectively, which demonstrates the effectiveness of AGTLBO.
To compare and better test the performance of the AGTLBO, the problem is considered a multiobjective optimization issue by solving more than one objective function at the same time for Cases 6, 7, and 8.

4.2.6. Case 6: Minimizing the Fuel Cost and Real Power Loss

The network power losses and the fuel cost function are considered a starting point of multiobjective optimization. The coefficient is calculated in the same way as in the previous papers. This cost function of this Case 6 is given as follows [61]:
O F 6 = O F 1 + λ p P L o s s
where the factor 𝜆p is selected as 40, similar to [61].
According to the results shown in Table 7, it can be concluded that the proposed AGTLBO outperforms the original TLBO algorithm and the two other approaches such as MSA (Moth Swarm Algorithm) [16] and SF-DE (Differential Evolution Integrated Superiority of Feasible Solutions) [61].

4.2.7. Case 7: Minimizing the Fuel Cost and Voltage Deviation

The results cannot achieve the required network voltage profile if only the simple fuel cost function is considered. As a result, the goal function can be divided into two parts. As in Case 1, the first part is the fuel cost function. The second part is to reduce load bus voltage differences from the reference value of 1 p.u. to increase the network voltage profile. Generally, this optimization’s objective function can be described as below [61]:
O F 7 = O F 1 + λ v i = 1 N P Q | V i 1.0 |
where the value of factor 𝜆v is set to 100 [61].
In Table 8, the minimum fuel cost (USD/h) and V.D. (p.u.) of the TLBO and AGTLBO algorithms are compared, and the findings are linked to those published in the recent literature such as MPSO (Modified Particle Swarm Optimization) [16], MFO (Moth-Flame Optimization) [16], MOICA (Multi-Objective Imperialist Competitive Algorithm) [62], NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62], BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62], MNSGA-II (Modified Non-dominated Sorting GA-II) [62], and MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62]. The table’s findings demonstrate AGTLBO’s ability, as it was able to find appropriate solutions for this situation.

4.2.8. Case 8: Minimizing the Fuel Cost, Emission, Voltage Deviation and Losses

We found all four objective functions in the last case of OPF problem in the 30-bus scheme, similar to MSA [16] and MOMICA [62], which can be considered a systematic, objective function for OPF problems. The weight factors are nominated as in [61] with 𝜆v = 21, 𝜆p = 22 and 𝜆e = 19 for the steadiness of problem different objectives. It can be observed that the proposed algorithm is a suitable and accurate algorithm for the problem of optimum control flow in power systems, based on the results compared with the other algorithms such as MFO (Moth-Flame Optimization) [16], MSA (Moth Swarm Algorithm) [16], MOICA (Multi-Objective Imperialist Competitive Algorithm) [62], NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62], BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62], MNSGA-II (Modified Non-dominated Sorting GA-II) [62], and MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62] in Table 9.
The bus voltage profile for the best solution obtained in a 30-bus system is shown in Figure 3. It can be seen from the figure that the voltages in the bus are often within the specified range.

4.3. IEEE 57-Bus Test System

A standard IEEE 57-bus test system (Figure 4) is used to evaluate the performance of the AGTLBO algorithm in greater-scale power systems. The system comprises seven generating units, 80 branches, 15 tap shifting transformers, three shunt reactive compensators, and a combined demand of 1250.8 MW. Reference [63] was used to collect the data for this evaluation device. A general range of voltages for generator buses and the decision variables for tap shifting transformers is set to 1.06–0.94 p.u. All load bus voltages are set to 1.06 and 0.94 p.u., respectively. We selected 0.0 and 0.3 in p.u. as the minimum and maximum ranges for the shunt reactive power compensation.

4.3.1. Case 9: Minimizing the Fuel Cost

To show the efficiency of the proposed algorithm in the first case of power flow optimization for the 57-bus scheme, we looked at the same simple basic objective function that we observed in Case 1. The objective function is designated as defined in (25). Table 10 represents the ultimate results of the proposed algorithm with many solutions published in prior studies. Table 11 also gives a comparison study between the outcomes achieved by proposed methods and other recent works such as ICBO (Improved Colliding Bodies Optimization) [11], MPSO (Modified Particle Swarm Optimization) [16], MFO (Moth-Flame Optimization) [16], DSA (Differential Search Algorithm) [22], LTLBO (Lévy Teaching–Learning-Based Optimization) [28], MICA (Modified Imperialist Competitive Algorithm) [45], ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51], and EADDE (Evolving Ant Direction Differential Evolution) [63], in terms of cost function. It is obvious that the improved technique offered in this work can attain superior solutions. According to the obtained simulation data, the minimum cost obtained from AGTLBO is 41,678.310 USD/h, which is less comparing to result of other algorithms. By employing the AGTLBO, the objective function of TLBO, ICBO, MPSO, MFO, DSA, LTLBO, MICA, ARCBBO, and EADDE are decreased by 1.5810 USD/h, 19.0200 USD/h, 0.3660 USD/h, 8.1020 USD/h, 8.5100 USD/h, 1.2350 USD/h, 4.7380 USD/h, 7.6900 USD/h, and 35.3100 USD/h, respectively, which demonstrates the effectiveness of AGTLBO.

4.3.2. Case 10: Minimizing the Fuel Cost While Improving Voltage Profile

The key protection and operation indexes are bus voltages. In the OPF dilemma, taking only a cost-based target will result in a viable solution that delivers an unacceptable voltage profile. To account for voltage profile enhancement, the objective function may be thought of as defined in (33).
Based on the findings in Table 12, the voltage profile of the 57-bus configuration in this case is clearly superior to that of Case 9. In addition, as compared to originally approved approaches such as MPSO (Modified Particle Swarm Optimization) [16], MFO (Moth-Flame Optimization) [16], MSA (Moth Swarm Algorithm) [16], EADDE (Evolving Ant Direction Differential Evolution) [63], MICA (Modified Imperialist Competitive Algorithm) [45], and MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45], the proposed algorithm was able to reach suitable and effective solutions. It can be seen from Table 12 that AGTLBO had a low value in emissions, fuel cost, and power losses, but the MSA algorithm was close. The worst in terms of power loss was the MICA algorithm, and in terms of fuel cost was the EADDE algorithm. The best algorithm in terms of voltage deviation is MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm).
Table 13 shows the optimal decision variables achieved by various algorithms for further study.

4.3.3. Case 11: Minimizing the Fuel Cost, Emission, Voltage Deviation and Losses

We found all four objective functions in the ultimate case of power flow optimization, as in [16,62]:
O F 9 = i = 1 N G ( α i + b i P G i + c i P G i 2 ) + O F 3 + λ v i = 1 N P Q | V i 1.0 | + λ p P L o s s
The weight elements are designated as 𝜆v = 1000, 𝜆p = 100 and 𝜆e = 1000 to stabilize the problem dissimilar goals.
The findings in Table 14 show that AGTLBO is a suitable and accurate algorithm for solving the OPF problem in power systems in comparison with MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62], MOICA (Multi-Objective Imperialist Competitive Algorithm) [62], MNSGA-II (Modified Non-dominated Sorting GA-II) [62], BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62], and NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62]. Table 15 also contains the final solution for this scenario.
In addition, the voltage profile is given for all three cases of the 57-bus system in Figure 5. It is clear from the figure that the allowable limits of the bus voltage have been well-observed for the best solutions.

4.4. Case 12: IEEE 118-Bus System

A large-scale IEEE 118-bus system [16] is being investigated for additional verification of the proposed AGTLBO algorithm’s effectiveness and scalability. The number of control variables increases from 24 to 130 compared to the IEEE 30-bus system. Increasing the number of generators complicates the OPF problem. Active and reactive power requirements for the system are 4242 MW and 1439 MVAR, respectively. To be comparable to other state-of-the-art heuristic algorithms described in the literature, the objective function in examples 1 and 9 is to minimize the fuel cost. Table 16 summarizes the best solutions achieved using the suggested AGTLBO algorithm. In Table 17, this result is compared to the results of other algorithms under investigation and some other techniques reported in the literature, including CS-GWO [65]; MSA [16], FPA [16], MFO [16], PSOGSA [44], IABC [66], MCSA [67], MRao-2 and Rao algorithms [68], SSO [32], FHSA [57], ICBO [11], GWO [25], and EWOA [69]. According to this table, the AGTLBO outperforms various optimization techniques used to solve the large-scale OPF. The CS-GWO achieves praiseworthy results while working with such a large-scale system. According to the obtained simulation data, the minimum cost obtained from AGTLBO is 129,543.60 USD/h, which is less than the results of other algorithms. By employing the AGTLBO, the objective function of CS-GWO, MSA, FPA, MFO, PSOGSA, IABC, MCSA, MRao-2, Rao-2, Rao-3, Rao-1, SSO, FHSA, ICBO, GWO, and EWOA are decreased by 0.40 USD/h, 97.10 USD/h, 145.10 USD/h, 164.50 USD/h, 190.0 USD/h, 318.40 USD/h, 330.0 USD/h, 1914.20 USD/h, 1947.1 USD/h, 2249.50 USD/h, 2274.30 USD/h, 2536.8 USD/h, 2594.70 USD/h, 5578.0 USD/h, 10,405.0 USD/h, and 10,632.0 USD/h, respectively, which demonstrates the effectiveness of AGTLBO.
Based on the results obtained in this study, it is evident from comparing the effects that the CS-GWO and AGTLBO algorithms are very close in their solutions. In addition, the new, improved CS-GWO algorithm is modern and robust, while the investigated system’s dimensions are considerable. As a result of this minor difference, the proposed algorithm can be further improved in the future. In contrast, the proposed algorithm has an Std It that is much lower, thereby making it more reliable. Moreover, the algorithm’s execution time is more reasonable and acceptable than those of other algorithms.
In Figure 6, the bus voltage profile is given for this system, which indicates that the voltage constraints are satisfied for the best solution.

4.5. Discussion on Results

In this section, for a suitable statistical description of the proposed method, general results of all cases from different systems in comparison with the ABC [70], BBO [71], and PSO [72] algorithms are presented. We have selected the following three basic algorithms. The population of these algorithms has been designated twice as TLBO algorithms for each case. The optimization conditions are the same and fair for all algorithms. In addition, the value of the control parameters of each algorithm is based on the given references [70,71,72].
The results for all single-objective cases are given in Table 18. The proposed AGTLBO in all cases has a good advantage over other algorithms. On the other hand, the optimization process is simple. Hence, no additional time is required to perform the optimization process compared to the original TLBO. AGTLBO is also the most robust algorithm among these algorithms. The important point is that as the dimension increases, the issue of AGTLBO superiority over other algorithms becomes more prominent, which was evidenced in the case of OPF for the 118-bus system.
It appears from these tables that the AGTLBO outperforms many optimization approaches used to solve OPF. Furthermore, the AGTLBO obtains appreciated results when dealing with such a large-scale system, which may explain the growing interest in including the Gaussian mutation strategy in the OPF solution methods.
The convergence curves of the algorithms for all cases are given in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, from which the proposed algorithm has a suitable convergence to obtain the optimal solution. The proposed algorithm’s convergence characteristic in case 12 compared to the original algorithm is given in Figure 13. The proposed algorithm has a suitable and effective convergence speed.
In addition, the detailed results for the different test cases on the IEEE-30 bus test system, including the reactive power generation in the value of the generator, the voltage magnitude at all the buses and the transmission power via lines, are shown in Table 19.

5. Conclusions

This paper develops a new effective improved teaching–learning-based optimization (TLBO) algorithm called adaptive Gaussian TLBO (AGTLBO) to solve the optimum power flow (OPF) problem. This problem entails choosing decision variables such as transformer pulses, generator outputs, reactive power sources in the most efficient way possible and generator voltages. The adaptive Gaussian mechanism was incorporated into the teaching and learning phases and its mathematical relations to increase the exploration and exploitation capabilities of the conventional TLBO. Then, the solution method was expressed step by step using the proposed AGTLBO for solving the OPF problem. The AGTLBO was applied to three test networks, i.e., standard IEEE 30-, 57- and 118-bus systems, in 12 cases with a combination of different objective functions such as fuel cost, losses, voltage deviation, and pollution. Using the proposed method on these 12 cases was compared with other state-of-the-art optimization methods. It was shown that AGTLBO has a significant advantage in finding the optimal solutions compared to other algorithms. It has been shown from statistical and comparative analyses that AGTLBO produces excellent performance and outperforms both TLBO and other algorithms for most applications. Despite the longer computation time, AGTLBO outperforms all the different algorithms due to its superior results. In summary, it can be concluded that AGTLBO can be an effective alternative to solving OPF. Further reducing the computational burden with parallel computing will be a future challenge and opportunity. The multi-objective AGTLBO in a multi-objective OPF problem is under development and will be proposed soon. In addition, statistical results showed that this algorithm can be useful for practical applications and can also be used to solve other engineering problems well. In future work, the multi-objective model of OPF problem using an efficient method can be investigated, and the integration of intermittent renewable generations can also be considered in the optimization model. This approach provides robust and high-quality OPF solutions for single and multi-objective optimization problems, such as optimizing total fuel costs, minimizing active power losses, and minimizing pollution and voltage deviations. Compared with the original TLBO, three original algorithms, and many other well-known metaheuristic techniques reported in the literature, the proposed AGTLBO achieves acceptable and effective OPF solutions and converges to the global optimum with fewer iterations.

Author Contributions

Conceptualization and methodology, A.A. and M.A.; software, Z.A.M. and A.M.; validation, M.A. and Z.A.M.; formal analysis and investigation, A.A.; writing—original draft preparation, A.A.; writing—review and editing, M.A. and A.M.; visualization, A.M.; supervision. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Flowchart of the proposed AGTLBO algorithm.
Figure 1. Flowchart of the proposed AGTLBO algorithm.
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Figure 2. Single line diagram of IEEE 30-bus test system.
Figure 2. Single line diagram of IEEE 30-bus test system.
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Figure 3. The voltage profile of the Cases 1 to 8.
Figure 3. The voltage profile of the Cases 1 to 8.
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Figure 4. Single line diagram of IEEE 57-bus test system.
Figure 4. Single line diagram of IEEE 57-bus test system.
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Figure 5. The voltage profile of the Cases 9 to 11.
Figure 5. The voltage profile of the Cases 9 to 11.
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Figure 6. The voltage profile of Case 12.
Figure 6. The voltage profile of Case 12.
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Figure 7. Convergence of algorithms for Case 1.
Figure 7. Convergence of algorithms for Case 1.
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Figure 8. Convergence of algorithms for Case 2.
Figure 8. Convergence of algorithms for Case 2.
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Figure 9. Convergence of algorithms for Case 3.
Figure 9. Convergence of algorithms for Case 3.
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Figure 10. Convergence characteristics for Case 4.
Figure 10. Convergence characteristics for Case 4.
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Figure 11. Convergence of algorithms for Case 5.
Figure 11. Convergence of algorithms for Case 5.
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Figure 12. Convergence of algorithms for Case 9.
Figure 12. Convergence of algorithms for Case 9.
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Figure 13. Convergence of algorithms for Case 12.
Figure 13. Convergence of algorithms for Case 12.
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Table 1. Optimal control variables at the lowest cost (best solution) for each test case.
Table 1. Optimal control variables at the lowest cost (best solution) for each test case.
Control Variables SettingsLimitsCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8
LowerUpper
PG1 (MW) 150250177.1160139.999664.064051.4932198.7677102.6327176.3251122.2170
PG2 (MW)208048.744554.999967.559379.998344.754755.536249.093052.5169
PG5 (MW)155021.322524.181050.000049.999718.561738.112521.558731.4731
PG8 (MW)103521.293334.998735.000034.999710.000035.000022.146735.0000
PG11 (MW)103011.944218.376130.000029.999910.001030.000012.146726.7432
PG13 (MW)124012.001717.574340.000039.999812.001226.648612.001621.0381
VG1 (p.u.)0.951.11.08351.07561.06281.06181.08101.06971.04271.0729
VG2 (p.u.)0.951.11.06041.05831.05621.05711.05771.05751.02301.0573
VG5 (p.u.)0.951.11.03341.03281.03751.03821.03021.03571.01511.0326
VG8 (p.u.)0.951.11.03831.04141.04391.04451.03681.04371.00651.0411
VG11 (p.u.)0.951.11.09231.09371.08061.08461.09681.09821.05951.0401
VG13 (p.u.)0.951.11.05021.05631.05421.05421.06061.05760.98791.0235
T6–9 (p.u.)0.91.11.06691.08861.04571.02371.08101.03471.08301.0999
T6–10 (p.u.)0.91.10.92420.90000.94030.97340.92950.94980.90000.9533
T4–12 (p.u.)0.91.10.97830.98530.99850.99750.99800.99110.93971.0322
T28–27 (p.u.)0.91.10.97370.97760.97600.97660.97700.97490.97081.0050
QC10 (MVAR)0.05.04.24043.61024.32784.54475.00000.74855.00003.5672
QC12 (MVAR)0.05.02.80950.08955.00004.56684.61040.36590.27880.3126
QC15 (MVAR)0.05.04.32754.81974.62814.62103.84744.48304.99293.8845
QC17 (MVAR)0.05.05.00005.00004.98984.99744.86065.00000.00504.9997
QC20 (MVAR)0.05.05.00004.07094.12754.21754.82194.26134.99884.9991
QC21 (MVAR)0.05.04.99724.95955.00004.99954.97615.00004.94075.0000
QC23 (MVAR)0.05.03.15382.96993.22533.24103.57003.25034.99954.3039
QC24 (MVAR)0.05.05.00005.00004.99834.99984.99735.00004.99935.0000
QC29 (MVAR)0.05.02.76962.60552.51832.54352.47182.55202.63072.6200
Cost ($/h)--800.4811646.4511944.3385967.6336832.1624858.9928803.7385830.1559
Emission (t/h)--0.36620.28350.20480.20730.43790.22900.36390.2529
Power losses (MW)--9.02226.72963.22333.090610.68634.53009.87185.5823
V.D. (p.u.)--0.91090.90460.90170.90860.85920.94280.09470.2975
1PG1 is not a control variable; it is the slack generator in the slack bus.
Table 2. Optimal results by different algorithms for Case 1.
Table 2. Optimal results by different algorithms for Case 1.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO0.3662800.4810.91099.0222
TLBO0.3668800.6740.91209.0198
TS (Tabu Search) [5]-802.290--
ABC (Artificial Bee Colony) [12]0.3651800.6600.92099.0328
AGSO (Adaptive Group Search Optimization) [14]0.3703801.750--
MHBMO (Modified Honey Bee Mating Optimization) [15]-801.985-9.4900
MSA (Moth Swarm Algorithm) [16]0.3665800.5100.90369.0345
FPA (Flower Pollination Algorithm) [16]0.3596802.7980.36799.5406
MFO (Moth-Flame Optimization) [16]0.3685800.6860.75779.1492
MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21]-801.750-9.5400
GWO (Grey Wolf Optimizer) [25]-801.410-9.3000
MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45]-801.049-9.1895
DE (Differential Evolution) [49]-802.390-9.4660
SKH (Stud Krill Herd Algorithm) [52]0.3662800.514-9.0282
SFLA-SA (Shuffle Frog Leaping Algorithm and Simulated Annealing) [53]-801.790--
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51]0.3663800.5160.88679.0255
MGBICA (Modified Gaussian Bare-Bones Imperialist Competitive Algorithm) [54]0.3296801.141--
IEP (Improved Evolutionary Programming) [50]-802.460--
PSOGSA (Particle Swarm Optimization and Gravitational Search Algorithm) [55]-800.4990.12679.0339
EP (Evolutionary Programming) [56]-803.570--
Table 3. Optimal results by different algorithms for Case 2.
Table 3. Optimal results by different algorithms for Case 2.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO0.2835646.45110.90466.7296
TLBO0.2835647.13440.89276.8409
LTLBO (Lévy Teaching–Learning-Based Optimization) [28]0.2835647.43150.88966.9347
MDE (Modified Differential Evolution) [49]-647.8460-7.095
MFO (Moth-Flame Optimization) [16]0.28336649.27270.470247.2293
IEP (Improved Evolutionary Programming) [50]-649.3120--
MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45]-647.1002-6.8945
FPA (Flower Pollination Algorithm) [16]0.28083651.37680.312597.2355
SSO (Social Spider Optimization) [37]-663.3518--
GABC (Gbest Guided Artificial Bee Colony Algorithm) [57]-647.03000.80106.8160
MSA (Moth Swarm Algorithm) [16]0.28352646.83640.844796.8001
MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21]-647.5500--
Table 4. Optimal results by different algorithms for Case 3.
Table 4. Optimal results by different algorithms for Case 3.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO0.2048944.33850.90173.2233
TLBO0.2048944.54460.79793.2944
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51]0.2048945.15970.86473.2624
ABC (Artificial Bee Colony) [12]0.2048944.43910.84633.2470
GBICA (Gaussian Bare-Bones Imperialist Competitive Algorithm) [54]0.2049944.6516--
FPA (Flower Pollination Algorithm) [16]0.2052948.94900.42764.4920
AGSO (Adaptive Group Search Optimization) [14]0.2059953.6290--
DSA (Differential Search Algorithm) [22]0.2058944.4086-3.2437
MFO (Moth-Flame Optimization) [16]0.2049945.45530.70973.4295
MSA (Moth Swarm Algorithm) [16]0.2048944.50030.87393.2358
MSFLA (Modified Shuffle Frog Leaping Algorithm) [53]0.2056---
MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21]0.2052---
Table 5. Optimal results by different algorithms for Case 4.
Table 5. Optimal results by different algorithms for Case 4.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO0.2073967.63360.90863.0906
TLBO0.2072967.71950.88213.1104
ABC (Artificial Bee Colony) [12]0.2073967.68100.90083.1078
FPA (Flower Pollination Algorithm) [16]0.2076967.11380.38933.5661
MFO (Moth-Flame Optimization) [16]0.2073967.67850.91563.1111
MSA (Moth Swarm Algorithm) [16]0.2073967.66360.88873.1005
DSA (Differential Search Algorithm) [22]0.2083967.6493-3.0945
GWO (Grey Wolf Optimizer) [25]-968.3800-3.4100
Jaya [32]-967.68270.12723.1035
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51]0.2073967.66050.89133.1009
EEA (Efficient Evolutionary Algorithm) [58]-952.3785-3.2823
EGA (Enhanced Genetic Algorithm) [58]-967.9300-3.2440
EGA-DQLF (Enhanced Genetic Algorithm to Decoupled Quadratic Load Flow) [59]-967.86000.12183.2008
ALC-PSO (Particle Swarm Optimization with an Aging Leader and Challengers) [60]-967.7683-3.1700
Table 6. Optimal results by different algorithms for Case 5.
Table 6. Optimal results by different algorithms for Case 5.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO0.4379832.16240.859210.6863
TLBO0.4382832.53250.865410.6870
PSO (Particle Swarm Optimization) [11]-832.6871--
SP-DE (Differential Evolution Integrated Self-Adaptive Penalty) [61]0.43651832.48130.750410.6762
Table 7. Optimal results by different algorithms for Case 6.
Table 7. Optimal results by different algorithms for Case 6.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO0.2290858.99280.94284.5300
TLBO0.2307859.01910.91524.5687
MSA (Moth Swarm Algorithm) [16]0.2289859.19150.928524.5404
SF-DE (Differential Evolution Integrated Superiority of Feasible Solutions) [61]0.2289859.14580.927314.5245
Table 8. Optimal results by different algorithms for Case 7.
Table 8. Optimal results by different algorithms for Case 7.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO0.3639803.73850.09479.8718
TLBO0.3565804.23910.10569.9565
MPSO (Modified Particle Swarm Optimization) [16]0.3636803.97870.12029.9242
MFO (Moth-Flame Optimization) [16]0.36355803.79110.105639.8685
MOICA (Multi-Objective Imperialist Competitive Algorithm) [62]-805.03450.1004-
NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62]-804.96120.099-
BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62]-804.96390.1021-
MNSGA-II (Modified Non-dominated Sorting GA-II) [62]-805.00760.0989-
MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62]0.3552804.96110.09529.8212
Table 9. Optimal results by different algorithms for Case 8.
Table 9. Optimal results by different algorithms for Case 8.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO0.2529830.15590.29755.5823
TLBO0.2538831.31860.29825.5847
MFO (Moth-Flame Optimization) [16]0.25231830.91350.331645.5971
MSA (Moth Swarm Algorithm) [16]0.25258830.6390.293855.6219
MOICA (Multi-Objective Imperialist Competitive Algorithm) [62]0.267831.22510.40466.0223
NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62]0.2491834.64330.44485.8935
BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62]0.2479833.03450.39455.6504
MNSGA-II (Modified Non-dominated Sorting GA-II) [62]0.2527834.56160.43085.6606
MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62]0.2523830.18840.29785.5851
Table 10. Optimal control variables of IEEE 57-bus test system for Case 9.
Table 10. Optimal control variables of IEEE 57-bus test system for Case 9.
Control Variables SettingsAlgorithms
EADDE (Evolving Ant Direction Differential Evolution) [63]GSA (Gravitational Search Algorithm) [64]ABC (Artificial Bee Colony) [12]MICA (Modified Imperialist Competitive Algorithm) [45]LTLBO (Lévy Teaching–Learning-Based Optimization) [28]AGTLBO
PG1 (MW)143.150142.369142.811142.520142.870143.910
PG2 (MW)95.29092.63090.03388.00188.95392.021
PG3 (MW)45.32045.31844.51544.59644.78845.324
PG6(MW)73.60072.35574.20072.08672.57770.000
PG8 (MW)464.850464.743454.848460.325460.906458.585
PG9 (MW)83.44084.99996.88598.35494.17996.025
PG12 (MW)361.240363.951362.772360.179361.687360.000
VG1 (p.u.)1.0501.0591.0421.0331.0381.060
VG2(p.u.)1.0481.0581.0411.0371.0421.059
VG3 (p.u.)1.0411.0601.0391.0281.0311.055
VG6 (p.u.)1.0491.0601.0551.0441.0471.060
VG8 (p.u.)1.0561.0601.0641.0601.0601.060
VG9 (p.u.)1.0341.0601.0371.0271.0281.039
VG12 (p.u.)1.0411.0461.0411.0191.0221.044
T4–18 (p.u.)1.0510.9000.9380.9610.9011.012
T4–18 (p.u.)0.9070.9001.0501.0041.0800.966
T21–20 (p.u.)1.0380.9090.9751.0211.0191.011
T24–25 (p.u.)1.0041.0590.9501.0381.0341.100
T24–25 (p.u.)0.9620.9991.0130.9721.0250.970
T24–26 (p.u.)0.9860.9221.0001.0291.0261.026
T7–29 (p.u.)0.9840.9321.0130.9830.9840.990
T34–32 (p.u.)0.9081.0880.9130.9630.9710.964
T11–41 (p.u.)0.9231.0390.9000.9330.9000.906
T15–45 (p.u.)0.9911.0431.0130.9510.9540.976
T14–46 (p.u.)0.9821.0250.9880.9350.9400.966
T10–51 (p.u.)0.9890.9541.0000.9490.9500.967
T13–49 (p.u.)0.9660.9290.9630.9040.9120.933
T11–43 (p.u.)0.9721.0990.9630.9410.9480.965
T40–56 (p.u.)0.9970.9690.9631.0170.9930.990
T39–57 (p.u.)1.0021.0620.9250.9850.9700.947
T9–55 (p.u.)1.0451.0940.9880.9740.9720.996
QC18 (MVAR)9.03015.24316.00018.66017.5908.950
QC25 (MVAR)8.17014.40315.00013.62017.41018.110
QC53 (MVAR)20.13015.10214.00014.31015.08015.410
Cost ($/h)41,713.62041,695.87241,693.95941,683.04841,679.54541,678.310
Emission (t/h)-----1.901
V.D. (p.u.)1.098--1.449-1.618
Power losses (MW)16.090--15.26615.15915.064
Table 11. Optimal results by different algorithms for Case 9.
Table 11. Optimal results by different algorithms for Case 9.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO1.90141,678.3101.61815.064
TLBO1.92841,679.8911.65115.153
ICBO (Improved Colliding Bodies Optimization) [11]-41,697.3301.31715.547
MPSO (Modified Particle Swarm Optimization) [16]1.94441,678.6761.34015.127
MFO (Moth-Flame Optimization) [16]2.00441,686.4121.29415.611
DSA (Differential Search Algorithm) [22]-41,686.8201.083-
LTLBO (Lévy Teaching–Learning-Based Optimization) [28]-41,679.545-15.159
MICA (Modified Imperialist Competitive Algorithm) [45]-41,683.0481.44915.266
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51]-41,686.000-15.377
EADDE (Evolving Ant Direction Differential Evolution) [63]-41,713.6201.09816.090
Table 12. Optimal results by different algorithms for Case 10.
Table 12. Optimal results by different algorithms for Case 10.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO1.9241,707.970.7115.79
TLBO2.0041,715.200.7116.17
MPSO (Modified Particle Swarm Optimization) [16]2.0141,721.610.6816.25
MFO (Moth-Flame Optimization) [16]2.0141,718.870.6816.22
MSA (Moth Swarm Algorithm) [16]1.9641,714.980.6815.92
EADDE (Evolving Ant Direction Differential Evolution) [63]-42,051.440.7919.32
MICA (Modified Imperialist Competitive Algorithm) [45]-41,974.430.5420.30
MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45]-41,959.180.5419.91
Table 13. Optimal control variables of IEEE 57-bus test system for Case 10.
Table 13. Optimal control variables of IEEE 57-bus test system for Case 10.
Control Variables SettingsAlgorithms
EADDE (Evolving Ant Direction Differential Evolution) [63]MICA (Modified Imperialist Competitive Algorithm) [45]MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45]AGTLBO
PG1 (MW)153.820163.830152.900142.857
PG2 (MW)83.67099.99399.80588.694
PG3 (MW)71.56042.11142.71845.094
PG6(MW)54.79029.67137.04671.816
PG8 (MW)506.750467.689474.497459.858
PG9 (MW)79.930100.00077.78796.653
PG12 (MW)319.600367.810385.955361.617
VG1 (p.u.)1.00580.99230.99081.0224
VG2(p.u.)1.00260.99971.00201.0202
VG3 (p.u.)1.01081.02321.01911.0140
VG6 (p.u.)1.03700.99981.00001.0282
VG8 (p.u.)1.06191.00381.00241.0477
VG9 (p.u.)1.02481.01391.01651.0152
VG12 (p.u.)1.02081.03991.04211.0093
T4–18 (p.u.)0.98401.10000.95701.0141
T4–18 (p.u.)1.01670.92701.00830.9424
T21–20 (p.u.)1.00470.97540.97250.9949
T24–25 (p.u.)1.02051.03551.07000.9914
T24–25 (p.u.)1.02191.08571.06181.0112
T24–26 (p.u.)1.00810.99580.99571.0256
T7–29 (p.u.)1.00770.99550.99921.0016
T34–32 (p.u.)0.92760.91210.91940.9397
T11–41 (p.u.)0.90070.90000.90000.9000
T15–45 (p.u.)0.93320.90730.91180.9553
T14–46 (p.u.)0.95400.99240.99580.9566
T10–51 (p.u.)1.00141.02231.02590.9714
T13–49 (p.u.)0.94990.90000.90000.9275
T11–43 (p.u.)0.98930.99120.95950.9506
T40–56 (p.u.)0.90020.96481.02720.9934
T39–57 (p.u.)1.02520.90000.90000.9427
T9–55 (p.u.)1.00621.01621.01850.9975
QC18 (MVAR)19.62004.70000.00005.0604
QC25 (MVAR)19.150019.770020.780016.7240
QC53 (MVAR)11.090029.110030.000015.2132
Cost ($/h)42,051.4441,974.434641,959.177441,707.9679
Emission (t/h)---1.9194
V.D. (p.u.)0.78820.54160.53915.7901
Power losses (MW)19.3220.30319.9090.7099
Table 14. Optimal results by different algorithms for Case 11.
Table 14. Optimal results by different algorithms for Case 11.
AlgorithmEmission (t/h)Fuel Cost (USD/h)V.D. (p.u.)Power Losses (MW)
AGTLBO1.432841,929.3870.952613.2563
TLBO1.435741,932.0130.952813.6420
MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62]1.49641,983.0590.797013.6969
MOICA (Multi-Objective Imperialist Competitive Algorithm) [62]1.760541,998.5660.874813.3353
MNSGA-II (Modified Non-dominated Sorting GA-II) [62]1.496542,070.8250.889614.4557
BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62]1.533641,994.0191.074212.6090
NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62]1.517442,065.9961.042013.9764
Table 15. Optimal decision variables settings for Case 11.
Table 15. Optimal decision variables settings for Case 11.
Control Variables AGTLBOControl Variables AGTLBO
PG1 (MW)149.2946T24–25 (p.u.)0.9861
PG3 (MW)51.8000T24–26 (p.u.)1.0194
PG2 (MW)100.0000T24–25 (p.u.)1.0999
PG6(MW)100.0000T7–29 (p.u.)1.0262
PG9 (MW)100.0000T11–41 (p.u.)1.0999
PG8 (MW)378.9279T34–32 (p.u.)0.9220
PG12 (MW)384.0338T15–45 (p.u.)0.9771
VG2(p.u.)1.0559T10–51 (p.u.)1.0018
VG1 (p.u.)1.0599T14–46 (p.u.)0.9731
VG3 (p.u.)1.0420T13–49 (p.u.)0.9265
VG6 (p.u.)1.0487T11–43 (p.u.)0.9330
VG9 (p.u.)1.0321T39–57 (p.u.)0.9085
VG8 (p.u.)1.0600T40–56 (p.u.)0.9700
VG12 (p.u.)1.0237T9–55 (p.u.)1.0140
T4–18 (p.u.)1.0298QC18 (MVAR)3.2131
T4–18 (p.u.)0.9801QC25 (MVAR)19.4187
T21–20 (p.u.)0.9831QC53 (MVAR)18.2685
Table 16. Optimal decision variables settings for Case 12.
Table 16. Optimal decision variables settings for Case 12.
Actual power output of generators (MW)
PG1~PG924.2000.0000.0100.000403.00085.73020.00011.00020.100
PG10~PG180.030196.000281.00610.9107.15016.0000.0605.00048.410
PG19~PG2741.90019.000194.02049.30530.90032.400149.991148.2910.000
PG28~PG36354.488350.901458.2000.0000.0100.00015.87319.6170.000
PG3~PG45431.9860.0003.599507.0010.0000.0000.0000.000233.400
PG46~PG5438.0900.0104.10129.0086.00535.10036.4000.0100.000
Voltage magnitude of generators (p.u.)
VG1~VG91.0171.0451.0371.0801.1001.0331.0321.0361.030
VG10~VG181.0671.0931.0991.0561.0441.0511.0341.0281.017
VG19~VG271.0251.0441.0581.0331.0321.0331.0521.0601.060
VG28~VG361.0681.0731.0811.0531.0571.0511.0431.0281.060
VG3~VG451.0711.0771.0741.0951.0721.0711.0751.0591.062
VG46~VG541.0491.0361.0321.0261.0311.0381.0231.0471.063
Transformers’ tap (p.u.)
T1~T91.0401.0500.9710.9771.0001.0100.9760.9700.980
VAR compensating units (MVAR)
QC1~QC929.9000.0000.0002.00019.6696.0009.00027.99129.900
QC10~QC1429.99959.000129.99961.00011.0050Cost (USD/h)129,543.56PLoss (MW)76.2098
Table 17. Optimal results for Case 12.
Table 17. Optimal results for Case 12.
OptimizerMinMeanMaxStd.Time (s)
AGTLBO129,543.6129,551.9129,562.48.5737.2
CS-GWO (Crisscross Search Based Grey Wolf Optimizer) [65]129,544.0129,558.9129,568.810.74252.5
MSA (Moth Swarm Algorithm) [16]129,640.7----
FPA (Flower Pollination Algorithm) [16]129,688.7----
MFO (Moth-Flame Optimization) [16]129,708.1----
PSOGSA (Particle Swarm Optimization and Gravitational Search Algorithm) [44]129,733.6----
IABC (Improved Artificial Bee Colony Optimization) [66]129,862.0129,895.0-40.84157.8
MCSA (Modified Crow Search Optimizer) [67]129,873.6----
MRao-2 (Modified Rao-2 Algorithm) [68]131,457.8---1160.3
Rao-2 [68]131,490.7---804.6
Rao-3 [68]131,793.1---806.7
Rao-1 [68]131,817.9---808.0
SSO (Social Spider Optimization) [68]132,080.4----
FHSA (Fuzzy Harmony Search Algorithm) [001]132,138.3132,138.3132,138.30.0-
ICBO (Improved Colliding Bodies Optimization) [11]135,121.6----
GWO (Grey Wolf Optimizer) [25]139,948.1142,989.3145,484.6797.81766.2
EWOA (Effective Whale Optimization Algorithm) [69]140,175.8----
Table 18. Statistical results to show the performance of proposed AGTLBO and other algorithms.
Table 18. Statistical results to show the performance of proposed AGTLBO and other algorithms.
OptimizerMinMeanMaxStd.Time (s)
Case 1
AGTLBO800.4811800.5316800.55870.07626.7
TLBO800.6735800.8705801.13240.54226.6
ABC800.6802800.9642802.15061.81925.1
BBO 800.9527802.3304805.39193.99428.9
PSO 800.7016801.4378803.88262.07527.0
OptimizerMinMeanMaxStd.Time (s)
Case 2
AGTLBO646.4511646.6973646.90050.09426.8
TLBO647.1344647.8260648.43260.92726.7
ABC647.4256648.2013648.85911.74826.4
BBO647.7739649.2835651.70723.69029.0
PSO647.6495648.9424650.66252.51427.3
OptimizerMinMeanMaxStd.Time (s)
Case 3
AGTLBO0.204820.204830.204840.00027.0
TLBO0.204840.204900.204950.00627.0
ABC0.204840.205050.205570.00926.2
BBO0.204900.205320.206030.01029.1
PSO0.204840.204960.205510.00927.5
OptimizerMinMeanMaxStd.Time (s)
Case 4
AGTLBO3.09063.09113.09200.00027.5
TLBO3.11043.11333.12750.06227.3
ABC3.13143.16823.20740.10326.8
BBO3.17253.24903.31780.19929.5
PSO3.17093.17703.22350.07127.7
OptimizerMinMeanMaxStd.Time (s)
Case 5
AGTLBO832.1624832.2364832.30470.40227.4
TLBO832.5325832.8761833.44220.95327.3
ABC832.5860833.0923833.49861.04626.9
BBO832.9234834.7060836.60112.81429.4
PSO832.5966832.9648834.57461.98328.0
OptimizerMinMeanMaxStd.Time (s)
Case 9
AGTLBO41,678.31041,684.74441,692.1924.3671.3
TLBO41,679.89041,690.37641,710.3838.1572.5
ABC41,681.70441,695.05841,712.7445.9083.8
BBO41,697.55241,728.91841,765.75133.4107.6
PSO41,683.95141,697.12041,714.5475.6288.9
OptimizerMinMeanMaxStd.Time (s)
Case 12
AGTLBO129,545.56129,552.92129,562.38.53740.2
TLBO129,550.49129,607.63129,691.072.14741.8
ABC129,677.83129,732.46130,896.6446.81284
BBO129,734.06129,985.25132,268.8995.21595
PSO129,703.48129,867.01131,573.7610.71359
Table 19. The state variables for the best solutions for different test cases in the IEEE 30-bus system.
Table 19. The state variables for the best solutions for different test cases in the IEEE 30-bus system.
VariablesLimitsCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8
MinMax
Reactive power generation in the generators
QG1 (MVar)−2020013.04356.2865−1.0687−1.41836.46582.39136.14825.8087
QG2 (MVar)−20109.98509.99809.99539.98709.998310.00009.67829.9999
QG5 (MVar)−158030.039828.529524.188924.115329.311525.890249.882826.4629
QG8 (MVar)−156029.783528.760429.137829.238429.644529.404841.530929.2938
QG11 (MVar)−105027.944730.913221.115219.852732.300626.296830.447620.7654
QG13 (MVar)−15600.22294.91664.17804.17468.13196.036614.56938.7635
Voltage magnitude at all the buses
V1 (p.u.)0.951.11.08351.07561.06281.06181.08101.06971.04271.0729
V2 (p.u.)0.951.11.06041.05831.05621.05711.05771.05751.02301.0573
V3 (p.u.)0.951.051.05001.05001.05001.05001.05001.05001.00811.0500
V4 (p.u.)0.951.051.04221.04391.04681.04711.04311.04601.00011.0447
V5 (p.u.)0.951.11.03341.03281.03751.03821.03021.03571.01511.0326
V6 (p.u.)0.951.051.03941.04151.04391.04451.03911.04361.00341.0411
V7 (p.u.)0.951.051.02901.03001.03321.03391.02751.03240.99991.0296
V8 (p.u.)0.951.11.03831.04141.04391.04451.03681.04371.00651.0411
V9 (p.u.)0.951.051.03931.03551.04161.04811.03571.04991.00001.0000
V10 (p.u.)0.951.051.04671.04741.04531.04501.04371.04781.00841.0087
V11 (p.u.)0.951.11.09231.09371.08061.08461.09681.09821.05951.0401
V12 (p.u.)0.951.051.05001.05001.05001.05001.05001.05001.00871.0119
V13 (p.u.)0.951.11.05021.05631.05421.05421.06061.05760.98791.0235
V14 (p.u.)0.951.051.04011.04021.04021.04011.03961.04040.99961.0018
V15 (p.u.)0.951.051.03991.04001.03931.03921.03891.04021.00051.0018
V16 (p.u.)0.951.051.04261.04291.04151.04131.04121.04311.00051.0044
V17 (p.u.)0.951.051.04331.04391.04201.04181.04071.04421.00081.0051
V18 (p.u.)0.951.051.03371.03341.03211.03211.03191.03380.99440.9953
V19 (p.u.)0.951.051.03321.03261.03111.03101.03091.03310.99400.9947
V20 (p.u.)0.951.051.03831.03761.03591.03581.03571.03810.99941.0000
V21 (p.u.)0.951.051.03811.03871.03671.03641.03521.03920.99971.0000
V22 (p.u.)0.951.051.03871.03921.03731.03701.03581.03971.00041.0007
V23 (p.u.)0.951.051.03651.03621.03551.03551.03521.03701.00000.9999
V24 (p.u.)0.951.051.03201.03181.03061.03051.02941.03260.99450.9945
V25 (p.u.)0.951.051.03661.03511.03441.03431.03301.03671.00001.0000
V26 (p.u.)0.951.051.01931.01781.01701.01691.01561.01930.98200.9820
V27 (p.u.)0.951.051.04791.04571.04531.04531.04361.04771.01221.0122
V28 (p.u.)0.951.051.03531.03771.03991.04051.03481.03970.99991.0369
V29 (p.u.)0.951.051.03651.03391.03331.03331.03141.03581.00001.0000
V30 (p.u.)0.951.051.02201.01951.01901.01901.01711.02150.98510.9851
The transmission power via i-th line to j-th (MW)
i-thj-thminmaxCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8
12−130130115.30389.887536.766626.3472129.99864.6271114.88478.3493
13−13013062.331250.276427.318025.184168.890338.048961.803844.0164
24−656532.664727.418918.814819.827835.342021.851732.465624.1042
34−13013058.139146.774524.618622.529864.600635.042757.475340.7758
25−13013063.124257.993838.929839.268566.448347.105464.386852.0719
26−656544.234236.467324.714625.469948.731228.874343.785631.9784
46−909051.432040.390126.431725.366859.188831.382454.311235.0825
57−707014.595115.90788.93538.679013.317512.427820.729714.2529
67−13013034.695936.808329.234728.908934.433233.249534.582635.1701
68−323211.19071.03981.26981.225720.92361.014414.84670.9188
69−656537.385041.517521.848514.327041.866521.537139.614926.9585
610−323221.534526.597315.38149.585921.063513.952126.133419.9261
911−656530.390335.962636.685835.973933.813539.893932.781133.8585
910−656530.715533.637032.965933.100329.843335.341730.535535.6623
412−656532.197226.31979.14459.114532.404517.808839.805621.6933
1213−656512.003818.249140.217640.217014.496827.323819.273322.7903
1214−32327.46447.48878.07498.07627.60367.58467.39377.4230
1215−323217.659617.798220.329920.326718.045218.248017.953617.5549
1216−32327.04597.03739.55989.55917.51587.29167.20736.6202
1415−16161.67971.72782.35602.35201.68521.85182.00941.6884
1617−16163.70793.76966.28366.26273.95494.06383.78913.2547
1518−16165.69895.68156.94446.94865.97045.75745.72535.4246
1819−16162.58792.50203.82083.82202.77002.61632.66372.2690
1920−32328.23778.14997.31477.30487.88398.14748.30128.4000
1020−32329.33659.39998.16348.14589.07659.31159.32229.6304
1017−32325.93645.99264.13994.09225.42675.81378.86036.3076
1021−323216.426616.729216.695016.672116.315416.974416.319816.7705
1022−32327.79137.98667.97277.95837.71818.15747.72328.0305
2122−32322.47052.21732.38522.41692.59102.23172.72442.5827
1523−16164.77324.94816.80646.79875.03155.45865.19825.0669
2224−16165.63526.10306.18716.16195.46376.67415.56576.4935
2324−16161.80131.91223.44053.43882.23182.23292.00052.0263
2425−16161.67911.07551.88031.87561.52551.30151.80401.4386
2526−16164.25934.25954.25964.25964.25984.25934.26484.2648
2527−16165.95955.38454.89984.92495.74195.10916.12515.6398
2827−656519.034018.421317.013117.037118.952217.659119.244818.4261
2729−16166.20476.19876.19626.19696.19566.19666.20806.2077
2730−16167.11777.12527.12907.12797.13167.12677.13607.1364
2930−16163.96523.94893.94033.94283.93603.94333.95653.9554
828−32323.09874.54964.34634.35752.09754.47184.40754.6084
628−323216.234713.919512.713812.729017.704313.243815.978213.8691
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Alanazi, A.; Alanazi, M.; Memon, Z.A.; Mosavi, A. Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method. Appl. Sci. 2022, 12, 7959. https://doi.org/10.3390/app12167959

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Alanazi A, Alanazi M, Memon ZA, Mosavi A. Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method. Applied Sciences. 2022; 12(16):7959. https://doi.org/10.3390/app12167959

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Alanazi, Abdulaziz, Mohana Alanazi, Zulfiqar Ali Memon, and Amir Mosavi. 2022. "Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method" Applied Sciences 12, no. 16: 7959. https://doi.org/10.3390/app12167959

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