Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method

: 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.


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 2. 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-ofthe-art metaheuristic algorithms.3. 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.

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 (2) 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.

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

Equality Constraints
There are two types of power supply constraints applicable to system buses-real and reactive [14]: 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.

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]: where V min Gi and V max Gi express the voltage magnitude range of ith generator; P min Gi and P max Gi and the power output allowed ith generator.Furthermore, Q min Gi and Q max Gi 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.
where T min i and T max i 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.
where Q min Ci and Q max Ci 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]: S li ≤ S max li (10) The magnitudes boundary of voltage ith load bus are identified by V min Li and V max  Li .Additionally, the apparent power via ith transmission line and its higher bound represent by S li and S max li , respectively.

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]: (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]:

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.

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]: 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]: where rand denotes a random number in the range [0, 1].

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]: 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.

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 (σ 2 ), and standard deviation.Using the following formula, we can determine the probability density function for this distribution [47]: Below is a demonstration of the normal distribution of a real-valued random variable Y: The Gaussian mutation may therefore be characterized by a Gaussian distribution with µ = 0 and σ = 1 as follows: 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.

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: where this paper assumes φ Start = 10 −9 , φ 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): The flowchart of the proposed algorithm is shown in Figure 1 which is adapted from [6,28].

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 Npop, the lowest and maximum values of control variables X min j and X max j (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.

Solution Method
In order to solve OPF problems, the AGTLBO algorithm is used.The follow tions detail the execution of the proposed AGTLBO for OPF problems:

•
Step 1: Initialize AGTLBO using the parameters from the under-study test s The maximum number of iterations and population (Npop) for the 30-bus sy have been chosen 500 and 25, respectively, 1000 and 40 for the 57-bus syste 5000 and 60 for the 118-bus system.

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.

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

•
Minimizing the fuel cost and actual power loss (Case 6).

•
Minimizing the fuel cost and voltage deviation (Case 7).

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.

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]: 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, 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 800.4811USD/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.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]: 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]: 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 646.4511USD/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.

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]: where F Ei 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 MW 2 ), and β i (ton/h MW) are related to SOX.Additionally, λ i (1/MW) and ξ i (ton/h) are related to NOX.
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.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]: 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 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 (En-hanced 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  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]: where the cost coefficients for the ith generating unit are d i and e i .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 832.1624USD/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]: 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].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]: where the value of factor λv is set to 100 [61].

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. 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.

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

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 1250.8MW.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 1.06-0.94p.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.

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 41,678.310USD/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 35.3100USD/h, respectively, which demonstrates the effectiveness of AGTLBO.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.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).We found all four objective functions in the ultimate case of power flow optimization, as in [16,62]: 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.

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 com-

Number load bus
Voltage magnitude in p.u.

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 129,543.60USD/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.   [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.[67] 129,873.6 ----MRao-2 (Modified Rao-2 Algorithm) [68] 131,457.8---1160.3Rao-2 [68] 131,490.7 ---804.6Rao-3 [68] 131,793.1 ---806.7 Rao-1 [68] 131,817.9---808.0SSO (Social Spider Optimization) [68] 132,080.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.

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 AG-TLBO 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.

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 Figures 7-13, 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.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.

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.

Figure 2 .
Figure 2. Single line diagram of IEEE 30-bus test system.

Figure 3 .
Figure 3.The voltage profile of the Cases 1 to 8.

Figure 3 .
Figure 3.The voltage profile of the Cases 1 to 8.

Figure 4 .
Figure 4. Single line diagram of IEEE 57-bus test system.

Figure
Figure Single line diagram of IEEE 57-bus test system.

33 Figure 5 .
Figure 5.The voltage profile of the Cases 9 to 11.

Figure 5 .
Figure 5.The voltage profile of the Cases 9 to 11.

Figure 6 .
Figure 6.The voltage profile of Case 12.

Table 1 .
Optimal control variables at the lowest cost (best solution) for each test case.

Control Variables Limits Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Figure 2. Single
line diagram of IEEE 30-bus test system.

Table 1
summarizes all AGTLBO's optimal options for the 30-bus power grid.

Table 1 .
Optimal control variables at the lowest cost (best solution) for each test case.

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

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

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

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

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

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

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

Table 9 .
Optimal results by different algorithms for Case 8.

Table 9 .
Optimal results by different algorithms for Case 8.

Table 10 .
Optimal control variables of IEEE 57-bus test system for Case 9.

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

Table 12 .
Optimal results by different algorithms for Case 10.

Table 13 .
Optimal control variables of IEEE 57-bus test system for Case 10.

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

Table 15 .
Optimal decision variables settings for Case 11.

Table 16 .
Optimal decision variables settings for Case 12.

Table 17 .
Optimal results for Case 12.

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.

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.