Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method

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.

$$\mathrm{Min}J\left(x,u\right)$$

(1)

$$\mathrm{Subject}\mathrm{to}:g\left(x,u\right)=0$$

(2)

$$h\left(x,u\right)\le 0$$

(3)

It is assumed that u and x are the system’s decision and state variables, respectively. $J\left(x,u\right)$ denotes the objective function of the problem, $g\left(x,u\right)$, the equality constraints of the load flow equation, and $h\left(x,u\right)$, 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]:

$$P_{Gi}-P_{Di}=V_i{\displaystyle \sum }_{j=1}^{NB}{V}_j\left(G_{ij}\cos {\delta }_{ij}+B_{ij}\sin {\delta }_{ij}\right)$$

(4)

$$Q_{Gi}-Q_{Di}=V_i{\displaystyle \sum }_{j=1}^{NB}{V}_j\left(G_{ij}\sin {\delta }_{ij}-B_{ij}\cos {\delta }_{ij}\right)$$

(5)

V_i and V_j represent the magnitudes of the voltages, while i and j are the indices of the bus. PG_i and QG_i are the generated units’ real and reactive power outputs, respectively. The real load demand is represented by PD_i and the reactive load demand by QD_i. δ_ij is the voltage phase angle difference between nodes i and j. G_ij and B_ij 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 $h\left(x,u\right)$ 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]:

$$\begin{array}{c}{P}_{Gi}^{\min }\le P_{Gi}^{}\le P_{Gi}^{\max }\\ Q_{Gi}^{\min }\le Q_{Gi}^{}\le Q_{Gi}^{\max }\\ V_{Gi}^{\min }\le V_{Gi}^{}\le V_{Gi}^{\max }\end{array}$$

(6)

where $V_{Gi}^{\min }$ and $V_{Gi}^{\max }$ express the voltage magnitude range of ith generator; $P_{Gi}^{\min }$ and $P_{Gi}^{\max }$ and the power output allowed ith generator. Furthermore, $Q_{Gi}^{\min }$ and $Q_{Gi}^{\max }$ 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.

  $$T_i^{\min }\le T_i\le T_i^{\max };\forall i=1,\dots ,NT$$

  (7)

  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.

  $$Q_{Ci}^{\min }\le Q_{Ci}\le Q_{Ci}^{\max };\forall i=1,\dots ,NC$$

  (8)

  where $Q_{Ci}^{\min }$ and $Q_{Ci}^{\max }$ 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]:

  $$V_{Li}^{\min }\le V_{Li}\le V_{Li}^{\max }$$

  (9)

  $$S_{li}\le S_{li}^{\max }$$

  (10)

The magnitudes boundary of voltage ith load bus are identified by $V_{Li}^{\min }$ and $V_{Li}^{\max }$. Additionally, the apparent power via ith transmission line and its higher bound represent by $S_{li}$ and $S_{li}^{\max }$, 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]:

$$J={{\displaystyle \sum }}_{i=1}^{NG}{F}_i\left(P_{Gi}\right)+{\lambda }_P{\left(P_{G1}-P_{G1}^{\lim }\right)}^2+{\lambda }_Q{{\displaystyle \sum }}_{i=1}^{NG}{\left(Q_{Gi}-Q_{Gi}^{\lim }\right)}^2+{\lambda }_V{{\displaystyle \sum }}_{i=1}^{NPQ}{\left(V_{Li}-V_{Li}^{\lim }\right)}^2+{\lambda }_S{{\displaystyle \sum }}_{i=1}^{NTL}{\left(S_{li}-S_{li}^{\lim }\right)}^2$$

(11)

where λ_P, λ_V, λ_Q and λ_S 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 u^lim is given as [26]:

$$u^{\lim }=\{\begin{array}{l}u; u^{\min }\le u\le u^{\max }\\ u^{\min }; u\le u^{\min }\\ u^{\max }; u\ge u^{\max }\end{array}$$

(12)

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 (X_mean). Therefore, this process is modeled by the following relation [46]:

$$X_{new}=X_i+rand\ast \left(X_{teacher}-\left(T_F\ast X_{mean}\right)\right)$$

(13)

where $X_i$ and $X_{new}$ denote the status of the ith student before and after the teaching, X_teacher is the status of the teacher, X_mean is the average status of the class and $rand$ represents a uniformly distributed random number in the range [0, 1]. Additionally, T_F 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]:

$$T_F=\mathrm{round}\left(rand\right)+1$$

(14)

where $rand$ 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]:

$$X_{new}=\{\begin{array}{l}{X}_i+rand\ast \left(X_{ii}-X_i\right);\mathrm{if}Cost(X_{ii})<Cost(X_i)\\ X_i+rand\ast \left(X_i-X_{ii}\right); \mathrm{else}.\end{array}$$

(15)

The index ii in (15) describes the haphazardly selected student, and $rand$ is a random number in the range [0, 1]. For a minimization problem, the smaller a student’s cost function, i.e., $Cost(X_i)$, 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]:

$$f_{\mu ,{\sigma }^2}\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{(x-\mu )}^2}{2{\sigma }^2}}$$

(16)

Below is a demonstration of the normal distribution of a real-valued random variable Y:

$$Y\sim N\left(\mu ,{\sigma }^2\right)$$

(17)

The Gaussian mutation may therefore be characterized by a Gaussian distribution with μ = 0 and σ = 1 as follows:

$$X_{new}\left(j\right)=X_i\left(j\right)+N_j\left(0,1\right)$$

(18)

where N_j (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.

$$X_{new}=X_i+rand\ast \left(X_{teacher}-\left(T_F{X}_{mean}\right)\right)+N\left(0,1\right)$$

(19)

$$X_{new}=\{\begin{array}{l}{X}_i+rand\ast \left(X_{ii}-X_i\right)+N\left(0,1\right);\mathrm{if}Cost(X_{ii})<Cost(X_i)\\ X_i+rand\ast \left(X_i-X_{ii}\right);\mathrm{else}.\end{array}$$

(20)

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:

$$X_{new}=X_i+rand\ast \left(X_{teacher}-\left(T_F{X}_{mean}\right)\right)+{\left(N\left(0,{\varphi }_{Iter}\right)\right)}^D$$

(21)

$$X_{new}=\{\begin{array}{l}{X}_i+rand\ast \left(X_{ii}-X_i\right)+{\left(N\left(0,{\varphi }_{Iter}\right)\right)}^D\mathrm{if} Cost(X_{ii})<Cost(X_i)\\ X_i+rand\ast \left(X_i-X_{ii}\right)+{\left(N\left(0,{\varphi }_{Iter}\right)\right)}^D; \mathrm{else}.\end{array}$$

(22)

$${\varphi }_{Iter}={\varphi }_{Start}+\left({\left(\frac{Iter}{Ite{r}_{max}}\right)}^3\left({\varphi }_{Stop}-{\varphi }_{Start}\right)\right)$$

(23)

where this paper assumes ${\varphi }_{Start}={10}^{-9}$, ${\varphi }_{Stop}={10}^{-4}$, 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):

$$X_{new}=\{\begin{array}{l}{X}_{ii}+rand\ast \left(X_{ii}-X_i\right)+{\left(N\left(0,{\varphi }_{Iter}\right)\right)}^D\mathrm{if}Cost(X_{ii})<Cost(X_i)\\ X_i+rand\ast \left(X_i-X_{ii}\right)+{\left(N\left(0,{\varphi }_{Iter}\right)\right)}^D; \mathrm{else}.\end{array}$$

(24)

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 $X_i=X_j^{\min }+rand\ast (X_j^{\max }-(X_j^{\min })=\left[P_{G2}^{}:P_{GNG}^{}, V_{G1}^{}:V_{GNG}^{},T_1:T_{NT}, Q_{Ci}:Q_{CNC}\right]$ for i = 1: Npop, the lowest and maximum values of control variables $X_j^{\min }$ and $X_j^{\max } \left(j=1:D\right)$, 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 ${\lambda }_P$ = 5,000,000, ${\lambda }_V$ = ${\lambda }_Q$ = 5,000,000 and ${\lambda }_S$ = 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. Optimal control variables at the lowest cost (best solution) for each test case.

Control Variables Settings	Limits		Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8
	Lower	Upper
PG1 (MW) 1	50	250	177.1160	139.9996	64.0640	51.4932	198.7677	102.6327	176.3251	122.2170
PG2 (MW)	20	80	48.7445	54.9999	67.5593	79.9983	44.7547	55.5362	49.0930	52.5169
PG5 (MW)	15	50	21.3225	24.1810	50.0000	49.9997	18.5617	38.1125	21.5587	31.4731
PG8 (MW)	10	35	21.2933	34.9987	35.0000	34.9997	10.0000	35.0000	22.1467	35.0000
PG11 (MW)	10	30	11.9442	18.3761	30.0000	29.9999	10.0010	30.0000	12.1467	26.7432
PG13 (MW)	12	40	12.0017	17.5743	40.0000	39.9998	12.0012	26.6486	12.0016	21.0381
VG1 (p.u.)	0.95	1.1	1.0835	1.0756	1.0628	1.0618	1.0810	1.0697	1.0427	1.0729
VG2 (p.u.)	0.95	1.1	1.0604	1.0583	1.0562	1.0571	1.0577	1.0575	1.0230	1.0573
VG5 (p.u.)	0.95	1.1	1.0334	1.0328	1.0375	1.0382	1.0302	1.0357	1.0151	1.0326
VG8 (p.u.)	0.95	1.1	1.0383	1.0414	1.0439	1.0445	1.0368	1.0437	1.0065	1.0411
VG11 (p.u.)	0.95	1.1	1.0923	1.0937	1.0806	1.0846	1.0968	1.0982	1.0595	1.0401
VG13 (p.u.)	0.95	1.1	1.0502	1.0563	1.0542	1.0542	1.0606	1.0576	0.9879	1.0235
T6–9 (p.u.)	0.9	1.1	1.0669	1.0886	1.0457	1.0237	1.0810	1.0347	1.0830	1.0999
T6–10 (p.u.)	0.9	1.1	0.9242	0.9000	0.9403	0.9734	0.9295	0.9498	0.9000	0.9533
T4–12 (p.u.)	0.9	1.1	0.9783	0.9853	0.9985	0.9975	0.9980	0.9911	0.9397	1.0322
T28–27 (p.u.)	0.9	1.1	0.9737	0.9776	0.9760	0.9766	0.9770	0.9749	0.9708	1.0050
QC10 (MVAR)	0.0	5.0	4.2404	3.6102	4.3278	4.5447	5.0000	0.7485	5.0000	3.5672
QC12 (MVAR)	0.0	5.0	2.8095	0.0895	5.0000	4.5668	4.6104	0.3659	0.2788	0.3126
QC15 (MVAR)	0.0	5.0	4.3275	4.8197	4.6281	4.6210	3.8474	4.4830	4.9929	3.8845
QC17 (MVAR)	0.0	5.0	5.0000	5.0000	4.9898	4.9974	4.8606	5.0000	0.0050	4.9997
QC20 (MVAR)	0.0	5.0	5.0000	4.0709	4.1275	4.2175	4.8219	4.2613	4.9988	4.9991
QC21 (MVAR)	0.0	5.0	4.9972	4.9595	5.0000	4.9995	4.9761	5.0000	4.9407	5.0000
QC23 (MVAR)	0.0	5.0	3.1538	2.9699	3.2253	3.2410	3.5700	3.2503	4.9995	4.3039
QC24 (MVAR)	0.0	5.0	5.0000	5.0000	4.9983	4.9998	4.9973	5.0000	4.9993	5.0000
QC29 (MVAR)	0.0	5.0	2.7696	2.6055	2.5183	2.5435	2.4718	2.5520	2.6307	2.6200
Cost ($/h)	-	-	800.4811	646.4511	944.3385	967.6336	832.1624	858.9928	803.7385	830.1559
Emission (t/h)	-	-	0.3662	0.2835	0.2048	0.2073	0.4379	0.2290	0.3639	0.2529
Power losses (MW)	-	-	9.0222	6.7296	3.2233	3.0906	10.6863	4.5300	9.8718	5.5823
V.D. (p.u.)	-	-	0.9109	0.9046	0.9017	0.9086	0.8592	0.9428	0.0947	0.2975

¹P_G1 is not a control variable; it is the slack generator in the slack bus.

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]:

$$O{F}_1={\displaystyle \sum }_{i=1}^{NG}{F}_i\left(P_{Gi}\right)={\displaystyle \sum }_{i=1}^{NG}\left({\alpha }_i+b_i{P}_{Gi}+c_i{P}_{Gi}^2\right)+P_{Cost}$$

(25)

$$P_{Cost}={\lambda }_P{\left(P_{G1}-P_{G1}^{lim}\right)}^2+{\lambda }_Q{\displaystyle \sum }_{i=1}^{NG}{\left(Q_{Gi}-Q_{Gi}^{lim}\right)}^2+{\lambda }_V{\displaystyle \sum }_{i=1}^{NPQ}{\left(V_{Li}-V_{Li}^{lim}\right)}^2+{\lambda }_S{\displaystyle \sum }_{i=1}^{NTL}{\left(S_{li}-S_{li}^{lim}\right)}^2$$

(26)

A fuel cost is given by F_i, and an output is given by PG_i, respectively, for the ith generator. There are three cost coefficients called α_i, b_i and c_i, as well as NG, which indicates the number of generators overall. Detailed information about the cost coefficients can be found in [51]. $P_{Cost}$ is the penalty cost of constraints violations. In

Table 2. Optimal results by different algorithms for Case 1.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	0.3662	800.481	0.9109	9.0222
TLBO	0.3668	800.674	0.9120	9.0198
TS (Tabu Search) [5]	-	802.290	-	-
ABC (Artificial Bee Colony) [12]	0.3651	800.660	0.9209	9.0328
AGSO (Adaptive Group Search Optimization) [14]	0.3703	801.750	-	-
MHBMO (Modified Honey Bee Mating Optimization) [15]	-	801.985	-	9.4900
MSA (Moth Swarm Algorithm) [16]	0.3665	800.510	0.9036	9.0345
FPA (Flower Pollination Algorithm) [16]	0.3596	802.798	0.3679	9.5406
MFO (Moth-Flame Optimization) [16]	0.3685	800.686	0.7577	9.1492
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]	0.3662	800.514	-	9.0282
SFLA-SA (Shuffle Frog Leaping Algorithm and Simulated Annealing) [53]	-	801.790	-	-
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51]	0.3663	800.516	0.8867	9.0255
MGBICA (Modified Gaussian Bare-Bones Imperialist Competitive Algorithm) [54]	0.3296	801.141	-	-
IEP (Improved Evolutionary Programming) [50]	-	802.460	-	-
PSOGSA (Particle Swarm Optimization and Gravitational Search Algorithm) [55]	-	800.499	0.1267	9.0339
EP (Evolutionary Programming) [56]	-	803.570	-	-

, 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]:

$$F\left(P_{Gi}\right)=\{\begin{array}{l}{\alpha }_{i1}+b_{i1}{P}_{Gi}+c_{i1}{P}_{Gi}^2;P_{Gi}^{\min }\le P_{Gi}\le P_{Gi1}\\ {\alpha }_{i2}+b_{i2}{P}_{Gi}+c_{i2}{P}_{Gi}^2;P_{Gi1}\le P_{Gi}\le P_{Gi2}\\ \dots \\ {\alpha }_{ik}+b_{ik}{P}_{Gi}+c_{ik}{P}_{Gi}^2;P_{Gik-1}\le P_{Gi}\le P_{Gi}^{\max }\end{array}$$

(27)

where cost coefficients for the ith generator are represented by α_ik, b_ik and c_ik for fuel type k.

These circumstances are commonly considered just for the two first generators in several earlier papers, as described below [16]:

$$O{F}_2=\left({\displaystyle \sum }_{i=1}^2{\alpha }_{ik}+b_{ik}{P}_{Gi}+c_{ik}{P}_{Gi}^2\right)+\left({\displaystyle \sum }_{i=3}^{NG}{\alpha }_i+b_i{P}_{Gi}+c_i{P}_{Gi}^2\right)+P_{Cost}$$

(28)

Reference [49] provided the data for the units with several fuel types.

Table 3. Optimal results by different algorithms for Case 2.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	0.2835	646.4511	0.9046	6.7296
TLBO	0.2835	647.1344	0.8927	6.8409
LTLBO (Lévy Teaching–Learning-Based Optimization) [28]	0.2835	647.4315	0.8896	6.9347
MDE (Modified Differential Evolution) [49]	-	647.8460	-	7.095
MFO (Moth-Flame Optimization) [16]	0.28336	649.2727	0.47024	7.2293
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]	0.28083	651.3768	0.31259	7.2355
SSO (Social Spider Optimization) [37]	-	663.3518	-	-
GABC (Gbest Guided Artificial Bee Colony Algorithm) [57]	-	647.0300	0.8010	6.8160
MSA (Moth Swarm Algorithm) [16]	0.28352	646.8364	0.84479	6.8001
MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21]	-	647.5500	-	-

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]:

$$O{F}_3={\displaystyle \sum }_{i=1}^{NG}{F}_{Ei}\left(P_{Gi}\right)={\displaystyle \sum }_{i=1}^{NG}\left({\alpha }_i+{\beta }_i{P}_{Gi}+{\gamma }_i{P}_{Gi}^2+{\xi }_i\exp ({\lambda }_i{P}_{Gi})\right)+P_{Cost}$$

(29)

where F_Ei signifies the amount of emission caused by the ith thermal generator. ${\gamma }_i$, ${\beta }_i$, ${\xi }_i$ and ${\lambda }_i$ are the emission coefficients of the ith generator, while ${\alpha }_i$ (ton/h), ${\gamma }_i$ (ton/h MW²), and ${\beta }_i$ (ton/h MW) are related to SOX. Additionally, ${\lambda }_i$ (1/MW) and ${\xi }_i$ (ton/h) are related to NOX.

Table 4. Optimal results by different algorithms for Case 3.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	0.2048	944.3385	0.9017	3.2233
TLBO	0.2048	944.5446	0.7979	3.2944
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51]	0.2048	945.1597	0.8647	3.2624
ABC (Artificial Bee Colony) [12]	0.2048	944.4391	0.8463	3.2470
GBICA (Gaussian Bare-Bones Imperialist Competitive Algorithm) [54]	0.2049	944.6516	-	-
FPA (Flower Pollination Algorithm) [16]	0.2052	948.9490	0.4276	4.4920
AGSO (Adaptive Group Search Optimization) [14]	0.2059	953.6290	-	-
DSA (Differential Search Algorithm) [22]	0.2058	944.4086	-	3.2437
MFO (Moth-Flame Optimization) [16]	0.2049	945.4553	0.7097	3.4295
MSA (Moth Swarm Algorithm) [16]	0.2048	944.5003	0.8739	3.2358
MSFLA (Modified Shuffle Frog Leaping Algorithm) [53]	0.2056	-	-	-
MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21]	0.2052	-	-	-

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]:

$$O{F}_4=P_{Loss}={\displaystyle \sum }_{\begin{array}{l}k=1\\ k=\left(i,j\right)\end{array}}^{NTL}{g}_k(V_i^2+V_j^2-2V_i{V}_j\cos {\delta }_{ij})+P_{Cost}$$

(30)

where P_Loss is the total active power losses of the transmission network. g_k is the conductance of branch k. V_i and V_j are the voltages of ith and jth bus, respectively. NTL represents the number of transmission lines, δ_ij indicates the phase difference of voltages between bus i and bus j.

Table 5. Optimal results by different algorithms for Case 4.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	0.2073	967.6336	0.9086	3.0906
TLBO	0.2072	967.7195	0.8821	3.1104
ABC (Artificial Bee Colony) [12]	0.2073	967.6810	0.9008	3.1078
FPA (Flower Pollination Algorithm) [16]	0.2076	967.1138	0.3893	3.5661
MFO (Moth-Flame Optimization) [16]	0.2073	967.6785	0.9156	3.1111
MSA (Moth Swarm Algorithm) [16]	0.2073	967.6636	0.8887	3.1005
DSA (Differential Search Algorithm) [22]	0.2083	967.6493	-	3.0945
GWO (Grey Wolf Optimizer) [25]	-	968.3800	-	3.4100
Jaya [32]	-	967.6827	0.1272	3.1035
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51]	0.2073	967.6605	0.8913	3.1009
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]	-	967.8600	0.1218	3.2008
ALC-PSO (Particle Swarm Optimization with an Aging Leader and Challengers) [60]	-	967.7683	-	3.1700

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]:

$$O{F}_5=O{F}_1+{\displaystyle \sum }_{i=1}^{NG}\left|d_i\sin \left[e_i\left(P_{Gi}^{\min }-P_{Gi}\right)\right]\right|$$

(31)

where the cost coefficients for the ith generating unit are d_i and e_i.

Table 6. Optimal results by different algorithms for Case 5.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	0.4379	832.1624	0.8592	10.6863
TLBO	0.4382	832.5325	0.8654	10.6870
PSO (Particle Swarm Optimization) [11]	-	832.6871	-	-
SP-DE (Differential Evolution Integrated Self-Adaptive Penalty) [61]	0.43651	832.4813	0.7504	10.6762

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]:

$$O{F}_6=O{F}_1+\lambda p\ast P_{Loss}$$

(32)

where the factor 𝜆p is selected as 40, similar to [61].

According to the results shown in

Table 7. Optimal results by different algorithms for Case 6.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	0.2290	858.9928	0.9428	4.5300
TLBO	0.2307	859.0191	0.9152	4.5687
MSA (Moth Swarm Algorithm) [16]	0.2289	859.1915	0.92852	4.5404
SF-DE (Differential Evolution Integrated Superiority of Feasible Solutions) [61]	0.2289	859.1458	0.92731	4.5245

, 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]:

$$O{F}_7=O{F}_1+\lambda v\ast {\displaystyle \sum }_{i=1}^{NPQ}\left|V_i-1.0\right|$$

(33)

where the value of factor 𝜆v is set to 100 [61].

In

Table 8. Optimal results by different algorithms for Case 7.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	0.3639	803.7385	0.0947	9.8718
TLBO	0.3565	804.2391	0.1056	9.9565
MPSO (Modified Particle Swarm Optimization) [16]	0.3636	803.9787	0.1202	9.9242
MFO (Moth-Flame Optimization) [16]	0.36355	803.7911	0.10563	9.8685
MOICA (Multi-Objective Imperialist Competitive Algorithm) [62]	-	805.0345	0.1004	-
NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62]	-	804.9612	0.099	-
BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62]	-	804.9639	0.1021	-
MNSGA-II (Modified Non-dominated Sorting GA-II) [62]	-	805.0076	0.0989	-
MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62]	0.3552	804.9611	0.0952	9.8212

, 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. Optimal results by different algorithms for Case 8.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	0.2529	830.1559	0.2975	5.5823
TLBO	0.2538	831.3186	0.2982	5.5847
MFO (Moth-Flame Optimization) [16]	0.25231	830.9135	0.33164	5.5971
MSA (Moth Swarm Algorithm) [16]	0.25258	830.639	0.29385	5.6219
MOICA (Multi-Objective Imperialist Competitive Algorithm) [62]	0.267	831.2251	0.4046	6.0223
NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62]	0.2491	834.6433	0.4448	5.8935
BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62]	0.2479	833.0345	0.3945	5.6504
MNSGA-II (Modified Non-dominated Sorting GA-II) [62]	0.2527	834.5616	0.4308	5.6606
MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62]	0.2523	830.1884	0.2978	5.5851

.

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. Optimal control variables of IEEE 57-bus test system for Case 9.

Control Variables Settings	Algorithms
	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)	143.150	142.369	142.811	142.520	142.870	143.910
PG2 (MW)	95.290	92.630	90.033	88.001	88.953	92.021
PG3 (MW)	45.320	45.318	44.515	44.596	44.788	45.324
PG6(MW)	73.600	72.355	74.200	72.086	72.577	70.000
PG8 (MW)	464.850	464.743	454.848	460.325	460.906	458.585
PG9 (MW)	83.440	84.999	96.885	98.354	94.179	96.025
PG12 (MW)	361.240	363.951	362.772	360.179	361.687	360.000
VG1 (p.u.)	1.050	1.059	1.042	1.033	1.038	1.060
VG2(p.u.)	1.048	1.058	1.041	1.037	1.042	1.059
VG3 (p.u.)	1.041	1.060	1.039	1.028	1.031	1.055
VG6 (p.u.)	1.049	1.060	1.055	1.044	1.047	1.060
VG8 (p.u.)	1.056	1.060	1.064	1.060	1.060	1.060
VG9 (p.u.)	1.034	1.060	1.037	1.027	1.028	1.039
VG12 (p.u.)	1.041	1.046	1.041	1.019	1.022	1.044
T4–18 (p.u.)	1.051	0.900	0.938	0.961	0.901	1.012
T4–18 (p.u.)	0.907	0.900	1.050	1.004	1.080	0.966
T21–20 (p.u.)	1.038	0.909	0.975	1.021	1.019	1.011
T24–25 (p.u.)	1.004	1.059	0.950	1.038	1.034	1.100
T24–25 (p.u.)	0.962	0.999	1.013	0.972	1.025	0.970
T24–26 (p.u.)	0.986	0.922	1.000	1.029	1.026	1.026
T7–29 (p.u.)	0.984	0.932	1.013	0.983	0.984	0.990
T34–32 (p.u.)	0.908	1.088	0.913	0.963	0.971	0.964
T11–41 (p.u.)	0.923	1.039	0.900	0.933	0.900	0.906
T15–45 (p.u.)	0.991	1.043	1.013	0.951	0.954	0.976
T14–46 (p.u.)	0.982	1.025	0.988	0.935	0.940	0.966
T10–51 (p.u.)	0.989	0.954	1.000	0.949	0.950	0.967
T13–49 (p.u.)	0.966	0.929	0.963	0.904	0.912	0.933
T11–43 (p.u.)	0.972	1.099	0.963	0.941	0.948	0.965
T40–56 (p.u.)	0.997	0.969	0.963	1.017	0.993	0.990
T39–57 (p.u.)	1.002	1.062	0.925	0.985	0.970	0.947
T9–55 (p.u.)	1.045	1.094	0.988	0.974	0.972	0.996
QC18 (MVAR)	9.030	15.243	16.000	18.660	17.590	8.950
QC25 (MVAR)	8.170	14.403	15.000	13.620	17.410	18.110
QC53 (MVAR)	20.130	15.102	14.000	14.310	15.080	15.410
Cost ($/h)	41,713.620	41,695.872	41,693.959	41,683.048	41,679.545	41,678.310
Emission (t/h)	-	-	-	-	-	1.901
V.D. (p.u.)	1.098	-	-	1.449	-	1.618
Power losses (MW)	16.090	-	-	15.266	15.159	15.064

represents the ultimate results of the proposed algorithm with many solutions published in prior studies.

Table 11. Optimal results by different algorithms for Case 9.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	1.901	41,678.310	1.618	15.064
TLBO	1.928	41,679.891	1.651	15.153
ICBO (Improved Colliding Bodies Optimization) [11]	-	41,697.330	1.317	15.547
MPSO (Modified Particle Swarm Optimization) [16]	1.944	41,678.676	1.340	15.127
MFO (Moth-Flame Optimization) [16]	2.004	41,686.412	1.294	15.611
DSA (Differential Search Algorithm) [22]	-	41,686.820	1.083	-
LTLBO (Lévy Teaching–Learning-Based Optimization) [28]	-	41,679.545	-	15.159
MICA (Modified Imperialist Competitive Algorithm) [45]	-	41,683.048	1.449	15.266
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51]	-	41,686.000	-	15.377
EADDE (Evolving Ant Direction Differential Evolution) [63]	-	41,713.620	1.098	16.090

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. Optimal results by different algorithms for Case 10.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	1.92	41,707.97	0.71	15.79
TLBO	2.00	41,715.20	0.71	16.17
MPSO (Modified Particle Swarm Optimization) [16]	2.01	41,721.61	0.68	16.25
MFO (Moth-Flame Optimization) [16]	2.01	41,718.87	0.68	16.22
MSA (Moth Swarm Algorithm) [16]	1.96	41,714.98	0.68	15.92
EADDE (Evolving Ant Direction Differential Evolution) [63]	-	42,051.44	0.79	19.32
MICA (Modified Imperialist Competitive Algorithm) [45]	-	41,974.43	0.54	20.30
MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45]	-	41,959.18	0.54	19.91

, 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. Optimal control variables of IEEE 57-bus test system for Case 10.

Control Variables Settings	Algorithms
	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)	153.820	163.830	152.900	142.857
PG2 (MW)	83.670	99.993	99.805	88.694
PG3 (MW)	71.560	42.111	42.718	45.094
PG6(MW)	54.790	29.671	37.046	71.816
PG8 (MW)	506.750	467.689	474.497	459.858
PG9 (MW)	79.930	100.000	77.787	96.653
PG12 (MW)	319.600	367.810	385.955	361.617
VG1 (p.u.)	1.0058	0.9923	0.9908	1.0224
VG2(p.u.)	1.0026	0.9997	1.0020	1.0202
VG3 (p.u.)	1.0108	1.0232	1.0191	1.0140
VG6 (p.u.)	1.0370	0.9998	1.0000	1.0282
VG8 (p.u.)	1.0619	1.0038	1.0024	1.0477
VG9 (p.u.)	1.0248	1.0139	1.0165	1.0152
VG12 (p.u.)	1.0208	1.0399	1.0421	1.0093
T4–18 (p.u.)	0.9840	1.1000	0.9570	1.0141
T4–18 (p.u.)	1.0167	0.9270	1.0083	0.9424
T21–20 (p.u.)	1.0047	0.9754	0.9725	0.9949
T24–25 (p.u.)	1.0205	1.0355	1.0700	0.9914
T24–25 (p.u.)	1.0219	1.0857	1.0618	1.0112
T24–26 (p.u.)	1.0081	0.9958	0.9957	1.0256
T7–29 (p.u.)	1.0077	0.9955	0.9992	1.0016
T34–32 (p.u.)	0.9276	0.9121	0.9194	0.9397
T11–41 (p.u.)	0.9007	0.9000	0.9000	0.9000
T15–45 (p.u.)	0.9332	0.9073	0.9118	0.9553
T14–46 (p.u.)	0.9540	0.9924	0.9958	0.9566
T10–51 (p.u.)	1.0014	1.0223	1.0259	0.9714
T13–49 (p.u.)	0.9499	0.9000	0.9000	0.9275
T11–43 (p.u.)	0.9893	0.9912	0.9595	0.9506
T40–56 (p.u.)	0.9002	0.9648	1.0272	0.9934
T39–57 (p.u.)	1.0252	0.9000	0.9000	0.9427
T9–55 (p.u.)	1.0062	1.0162	1.0185	0.9975
QC18 (MVAR)	19.6200	4.7000	0.0000	5.0604
QC25 (MVAR)	19.1500	19.7700	20.7800	16.7240
QC53 (MVAR)	11.0900	29.1100	30.0000	15.2132
Cost ($/h)	42,051.44	41,974.4346	41,959.1774	41,707.9679
Emission (t/h)	-	-	-	1.9194
V.D. (p.u.)	0.7882	0.5416	0.539	15.7901
Power losses (MW)	19.32	20.303	19.909	0.7099

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]:

$$O{F}_9={\displaystyle \sum }_{i=1}^{NG}\left({\alpha }_i+b_i{P}_{Gi}+c_i{P}_{Gi}^2\right)+O{F}_3+\lambda v\ast {\displaystyle \sum }_{i=1}^{NPQ}\left|V_i-1.0\right|+\lambda p\ast P_{Loss}$$

(34)

The weight elements are designated as 𝜆v = 1000, 𝜆p = 100 and 𝜆e = 1000 to stabilize the problem dissimilar goals.

The findings in

Table 14. Optimal results by different algorithms for Case 11.

Algorithm	Emission (t/h)	Fuel Cost (USD/h)	V.D. (p.u.)	Power Losses (MW)
AGTLBO	1.4328	41,929.387	0.9526	13.2563
TLBO	1.4357	41,932.013	0.9528	13.6420
MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62]	1.496	41,983.059	0.7970	13.6969
MOICA (Multi-Objective Imperialist Competitive Algorithm) [62]	1.7605	41,998.566	0.8748	13.3353
MNSGA-II (Modified Non-dominated Sorting GA-II) [62]	1.4965	42,070.825	0.8896	14.4557
BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62]	1.5336	41,994.019	1.0742	12.6090
NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62]	1.5174	42,065.996	1.0420	13.9764

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. Optimal decision variables settings for Case 11.

Control Variables	AGTLBO	Control Variables	AGTLBO
PG1 (MW)	149.2946	T24–25 (p.u.)	0.9861
PG3 (MW)	51.8000	T24–26 (p.u.)	1.0194
PG2 (MW)	100.0000	T24–25 (p.u.)	1.0999
PG6(MW)	100.0000	T7–29 (p.u.)	1.0262
PG9 (MW)	100.0000	T11–41 (p.u.)	1.0999
PG8 (MW)	378.9279	T34–32 (p.u.)	0.9220
PG12 (MW)	384.0338	T15–45 (p.u.)	0.9771
VG2(p.u.)	1.0559	T10–51 (p.u.)	1.0018
VG1 (p.u.)	1.0599	T14–46 (p.u.)	0.9731
VG3 (p.u.)	1.0420	T13–49 (p.u.)	0.9265
VG6 (p.u.)	1.0487	T11–43 (p.u.)	0.9330
VG9 (p.u.)	1.0321	T39–57 (p.u.)	0.9085
VG8 (p.u.)	1.0600	T40–56 (p.u.)	0.9700
VG12 (p.u.)	1.0237	T9–55 (p.u.)	1.0140
T4–18 (p.u.)	1.0298	QC18 (MVAR)	3.2131
T4–18 (p.u.)	0.9801	QC25 (MVAR)	19.4187
T21–20 (p.u.)	0.9831	QC53 (MVAR)	18.2685

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. Optimal decision variables settings for Case 12.

Actual power output of generators (MW)
PG1~PG9	24.200	0.000	0.010	0.000	403.000	85.730	20.000	11.000	20.100
PG10~PG18	0.030	196.000	281.006	10.910	7.150	16.000	0.060	5.000	48.410
PG19~PG27	41.900	19.000	194.020	49.305	30.900	32.400	149.991	148.291	0.000
PG28~PG36	354.488	350.901	458.200	0.000	0.010	0.000	15.873	19.617	0.000
PG3~PG45	431.986	0.000	3.599	507.001	0.000	0.000	0.000	0.000	233.400
PG46~PG54	38.090	0.010	4.101	29.008	6.005	35.100	36.400	0.010	0.000
Voltage magnitude of generators (p.u.)
VG1~VG9	1.017	1.045	1.037	1.080	1.100	1.033	1.032	1.036	1.030
VG10~VG18	1.067	1.093	1.099	1.056	1.044	1.051	1.034	1.028	1.017
VG19~VG27	1.025	1.044	1.058	1.033	1.032	1.033	1.052	1.060	1.060
VG28~VG36	1.068	1.073	1.081	1.053	1.057	1.051	1.043	1.028	1.060
VG3~VG45	1.071	1.077	1.074	1.095	1.072	1.071	1.075	1.059	1.062
VG46~VG54	1.049	1.036	1.032	1.026	1.031	1.038	1.023	1.047	1.063
Transformers’ tap (p.u.)
T1~T9	1.040	1.050	0.971	0.977	1.000	1.010	0.976	0.970	0.980
VAR compensating units (MVAR)
QC1~QC9	29.900	0.000	0.000	2.000	19.669	6.000	9.000	27.991	29.900
QC10~QC14	29.9995	9.0001	29.9996	1.000	11.0050	Cost (USD/h)	129,543.56	PLoss (MW)	76.2098

summarizes the best solutions achieved using the suggested AGTLBO algorithm. In

Table 17. Optimal results for Case 12.

Optimizer	Min	Mean	Max	Std.	Time (s)
AGTLBO	129,543.6	129,551.9	129,562.4	8.5	737.2
CS-GWO (Crisscross Search Based Grey Wolf Optimizer) [65]	129,544.0	129,558.9	129,568.8	10.7	4252.5
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]	129,862.0	129,895.0	-	40.8	4157.8
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]	132,138.3	132,138.3	132,138.3	0.0	-
ICBO (Improved Colliding Bodies Optimization) [11]	135,121.6	-	-	-	-
GWO (Grey Wolf Optimizer) [25]	139,948.1	142,989.3	145,484.6	797.8	1766.2
EWOA (Effective Whale Optimization Algorithm) [69]	140,175.8	-	-	-	-

, 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. Statistical results to show the performance of proposed AGTLBO and other algorithms.

Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 1
AGTLBO	800.4811	800.5316	800.5587	0.076	26.7
TLBO	800.6735	800.8705	801.1324	0.542	26.6
ABC	800.6802	800.9642	802.1506	1.819	25.1
BBO	800.9527	802.3304	805.3919	3.994	28.9
PSO	800.7016	801.4378	803.8826	2.075	27.0
Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 2
AGTLBO	646.4511	646.6973	646.9005	0.094	26.8
TLBO	647.1344	647.8260	648.4326	0.927	26.7
ABC	647.4256	648.2013	648.8591	1.748	26.4
BBO	647.7739	649.2835	651.7072	3.690	29.0
PSO	647.6495	648.9424	650.6625	2.514	27.3
Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 3
AGTLBO	0.20482	0.20483	0.20484	0.000	27.0
TLBO	0.20484	0.20490	0.20495	0.006	27.0
ABC	0.20484	0.20505	0.20557	0.009	26.2
BBO	0.20490	0.20532	0.20603	0.010	29.1
PSO	0.20484	0.20496	0.20551	0.009	27.5
Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 4
AGTLBO	3.0906	3.0911	3.0920	0.000	27.5
TLBO	3.1104	3.1133	3.1275	0.062	27.3
ABC	3.1314	3.1682	3.2074	0.103	26.8
BBO	3.1725	3.2490	3.3178	0.199	29.5
PSO	3.1709	3.1770	3.2235	0.071	27.7
Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 5
AGTLBO	832.1624	832.2364	832.3047	0.402	27.4
TLBO	832.5325	832.8761	833.4422	0.953	27.3
ABC	832.5860	833.0923	833.4986	1.046	26.9
BBO	832.9234	834.7060	836.6011	2.814	29.4
PSO	832.5966	832.9648	834.5746	1.983	28.0
Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 9
AGTLBO	41,678.310	41,684.744	41,692.19	24.36	71.3
TLBO	41,679.890	41,690.376	41,710.38	38.15	72.5
ABC	41,681.704	41,695.058	41,712.74	45.90	83.8
BBO	41,697.552	41,728.918	41,765.75	133.4	107.6
PSO	41,683.951	41,697.120	41,714.54	75.62	88.9
Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 12
AGTLBO	129,545.56	129,552.92	129,562.3	8.53	740.2
TLBO	129,550.49	129,607.63	129,691.0	72.14	741.8
ABC	129,677.83	129,732.46	130,896.6	446.8	1284
BBO	129,734.06	129,985.25	132,268.8	995.2	1595
PSO	129,703.48	129,867.01	131,573.7	610.7	1359

. 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. The state variables for the best solutions for different test cases in the IEEE 30-bus system.

Variables		Limits		Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8
		Min	Max
Reactive power generation in the generators
QG1 (MVar)		−20	200	13.0435	6.2865	−1.0687	−1.4183	6.4658	2.3913	6.1482	5.8087
QG2 (MVar)		−20	10	9.9850	9.9980	9.9953	9.9870	9.9983	10.0000	9.6782	9.9999
QG5 (MVar)		−15	80	30.0398	28.5295	24.1889	24.1153	29.3115	25.8902	49.8828	26.4629
QG8 (MVar)		−15	60	29.7835	28.7604	29.1378	29.2384	29.6445	29.4048	41.5309	29.2938
QG11 (MVar)		−10	50	27.9447	30.9132	21.1152	19.8527	32.3006	26.2968	30.4476	20.7654
QG13 (MVar)		−15	60	0.2229	4.9166	4.1780	4.1746	8.1319	6.0366	14.5693	8.7635
Voltage magnitude at all the buses
V1 (p.u.)		0.95	1.1	1.0835	1.0756	1.0628	1.0618	1.0810	1.0697	1.0427	1.0729
V2 (p.u.)		0.95	1.1	1.0604	1.0583	1.0562	1.0571	1.0577	1.0575	1.0230	1.0573
V3 (p.u.)		0.95	1.05	1.0500	1.0500	1.0500	1.0500	1.0500	1.0500	1.0081	1.0500
V4 (p.u.)		0.95	1.05	1.0422	1.0439	1.0468	1.0471	1.0431	1.0460	1.0001	1.0447
V5 (p.u.)		0.95	1.1	1.0334	1.0328	1.0375	1.0382	1.0302	1.0357	1.0151	1.0326
V6 (p.u.)		0.95	1.05	1.0394	1.0415	1.0439	1.0445	1.0391	1.0436	1.0034	1.0411
V7 (p.u.)		0.95	1.05	1.0290	1.0300	1.0332	1.0339	1.0275	1.0324	0.9999	1.0296
V8 (p.u.)		0.95	1.1	1.0383	1.0414	1.0439	1.0445	1.0368	1.0437	1.0065	1.0411
V9 (p.u.)		0.95	1.05	1.0393	1.0355	1.0416	1.0481	1.0357	1.0499	1.0000	1.0000
V10 (p.u.)		0.95	1.05	1.0467	1.0474	1.0453	1.0450	1.0437	1.0478	1.0084	1.0087
V11 (p.u.)		0.95	1.1	1.0923	1.0937	1.0806	1.0846	1.0968	1.0982	1.0595	1.0401
V12 (p.u.)		0.95	1.05	1.0500	1.0500	1.0500	1.0500	1.0500	1.0500	1.0087	1.0119
V13 (p.u.)		0.95	1.1	1.0502	1.0563	1.0542	1.0542	1.0606	1.0576	0.9879	1.0235
V14 (p.u.)		0.95	1.05	1.0401	1.0402	1.0402	1.0401	1.0396	1.0404	0.9996	1.0018
V15 (p.u.)		0.95	1.05	1.0399	1.0400	1.0393	1.0392	1.0389	1.0402	1.0005	1.0018
V16 (p.u.)		0.95	1.05	1.0426	1.0429	1.0415	1.0413	1.0412	1.0431	1.0005	1.0044
V17 (p.u.)		0.95	1.05	1.0433	1.0439	1.0420	1.0418	1.0407	1.0442	1.0008	1.0051
V18 (p.u.)		0.95	1.05	1.0337	1.0334	1.0321	1.0321	1.0319	1.0338	0.9944	0.9953
V19 (p.u.)		0.95	1.05	1.0332	1.0326	1.0311	1.0310	1.0309	1.0331	0.9940	0.9947
V20 (p.u.)		0.95	1.05	1.0383	1.0376	1.0359	1.0358	1.0357	1.0381	0.9994	1.0000
V21 (p.u.)		0.95	1.05	1.0381	1.0387	1.0367	1.0364	1.0352	1.0392	0.9997	1.0000
V22 (p.u.)		0.95	1.05	1.0387	1.0392	1.0373	1.0370	1.0358	1.0397	1.0004	1.0007
V23 (p.u.)		0.95	1.05	1.0365	1.0362	1.0355	1.0355	1.0352	1.0370	1.0000	0.9999
V24 (p.u.)		0.95	1.05	1.0320	1.0318	1.0306	1.0305	1.0294	1.0326	0.9945	0.9945
V25 (p.u.)		0.95	1.05	1.0366	1.0351	1.0344	1.0343	1.0330	1.0367	1.0000	1.0000
V26 (p.u.)		0.95	1.05	1.0193	1.0178	1.0170	1.0169	1.0156	1.0193	0.9820	0.9820
V27 (p.u.)		0.95	1.05	1.0479	1.0457	1.0453	1.0453	1.0436	1.0477	1.0122	1.0122
V28 (p.u.)		0.95	1.05	1.0353	1.0377	1.0399	1.0405	1.0348	1.0397	0.9999	1.0369
V29 (p.u.)		0.95	1.05	1.0365	1.0339	1.0333	1.0333	1.0314	1.0358	1.0000	1.0000
V30 (p.u.)		0.95	1.05	1.0220	1.0195	1.0190	1.0190	1.0171	1.0215	0.9851	0.9851
The transmission power via i-th line to j-th (MW)
i-th	j-th	min	max	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8
1	2	−130	130	115.303	89.8875	36.7666	26.3472	129.998	64.6271	114.884	78.3493
1	3	−130	130	62.3312	50.2764	27.3180	25.1841	68.8903	38.0489	61.8038	44.0164
2	4	−65	65	32.6647	27.4189	18.8148	19.8278	35.3420	21.8517	32.4656	24.1042
3	4	−130	130	58.1391	46.7745	24.6186	22.5298	64.6006	35.0427	57.4753	40.7758
2	5	−130	130	63.1242	57.9938	38.9298	39.2685	66.4483	47.1054	64.3868	52.0719
2	6	−65	65	44.2342	36.4673	24.7146	25.4699	48.7312	28.8743	43.7856	31.9784
4	6	−90	90	51.4320	40.3901	26.4317	25.3668	59.1888	31.3824	54.3112	35.0825
5	7	−70	70	14.5951	15.9078	8.9353	8.6790	13.3175	12.4278	20.7297	14.2529
6	7	−130	130	34.6959	36.8083	29.2347	28.9089	34.4332	33.2495	34.5826	35.1701
6	8	−32	32	11.1907	1.0398	1.2698	1.2257	20.9236	1.0144	14.8467	0.9188
6	9	−65	65	37.3850	41.5175	21.8485	14.3270	41.8665	21.5371	39.6149	26.9585
6	10	−32	32	21.5345	26.5973	15.3814	9.5859	21.0635	13.9521	26.1334	19.9261
9	11	−65	65	30.3903	35.9626	36.6858	35.9739	33.8135	39.8939	32.7811	33.8585
9	10	−65	65	30.7155	33.6370	32.9659	33.1003	29.8433	35.3417	30.5355	35.6623
4	12	−65	65	32.1972	26.3197	9.1445	9.1145	32.4045	17.8088	39.8056	21.6933
12	13	−65	65	12.0038	18.2491	40.2176	40.2170	14.4968	27.3238	19.2733	22.7903
12	14	−32	32	7.4644	7.4887	8.0749	8.0762	7.6036	7.5846	7.3937	7.4230
12	15	−32	32	17.6596	17.7982	20.3299	20.3267	18.0452	18.2480	17.9536	17.5549
12	16	−32	32	7.0459	7.0373	9.5598	9.5591	7.5158	7.2916	7.2073	6.6202
14	15	−16	16	1.6797	1.7278	2.3560	2.3520	1.6852	1.8518	2.0094	1.6884
16	17	−16	16	3.7079	3.7696	6.2836	6.2627	3.9549	4.0638	3.7891	3.2547
15	18	−16	16	5.6989	5.6815	6.9444	6.9486	5.9704	5.7574	5.7253	5.4246
18	19	−16	16	2.5879	2.5020	3.8208	3.8220	2.7700	2.6163	2.6637	2.2690
19	20	−32	32	8.2377	8.1499	7.3147	7.3048	7.8839	8.1474	8.3012	8.4000
10	20	−32	32	9.3365	9.3999	8.1634	8.1458	9.0765	9.3115	9.3222	9.6304
10	17	−32	32	5.9364	5.9926	4.1399	4.0922	5.4267	5.8137	8.8603	6.3076
10	21	−32	32	16.4266	16.7292	16.6950	16.6721	16.3154	16.9744	16.3198	16.7705
10	22	−32	32	7.7913	7.9866	7.9727	7.9583	7.7181	8.1574	7.7232	8.0305
21	22	−32	32	2.4705	2.2173	2.3852	2.4169	2.5910	2.2317	2.7244	2.5827
15	23	−16	16	4.7732	4.9481	6.8064	6.7987	5.0315	5.4586	5.1982	5.0669
22	24	−16	16	5.6352	6.1030	6.1871	6.1619	5.4637	6.6741	5.5657	6.4935
23	24	−16	16	1.8013	1.9122	3.4405	3.4388	2.2318	2.2329	2.0005	2.0263
24	25	−16	16	1.6791	1.0755	1.8803	1.8756	1.5255	1.3015	1.8040	1.4386
25	26	−16	16	4.2593	4.2595	4.2596	4.2596	4.2598	4.2593	4.2648	4.2648
25	27	−16	16	5.9595	5.3845	4.8998	4.9249	5.7419	5.1091	6.1251	5.6398
28	27	−65	65	19.0340	18.4213	17.0131	17.0371	18.9522	17.6591	19.2448	18.4261
27	29	−16	16	6.2047	6.1987	6.1962	6.1969	6.1956	6.1966	6.2080	6.2077
27	30	−16	16	7.1177	7.1252	7.1290	7.1279	7.1316	7.1267	7.1360	7.1364
29	30	−16	16	3.9652	3.9489	3.9403	3.9428	3.9360	3.9433	3.9565	3.9554
8	28	−32	32	3.0987	4.5496	4.3463	4.3575	2.0975	4.4718	4.4075	4.6084
6	28	−32	32	16.2347	13.9195	12.7138	12.7290	17.7043	13.2438	15.9782	13.8691

.

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.