Optimal allocation of flexible AC transmission system (FACTS) for wind turbines integrated power system

The importance of flexible alternating current transmission system (FACTS) devices is increasing day by day as they provide compensation for power systems. The installation of such devices solves many issues regarding the stability and power transferability of power systems. However, for optimizing the performance of FACTS devices, the location and sizes of FACTS devices must be selected very carefully. On the other side, the utilization of optimization techniques is increasing daily, as it is capable of solving nonlinear, random problems by its stochastic nature. As a result, it became a reliable tactic for solving stochastic problems almost in every aspect of our lives. Therefore, it is adopted for solving power system problems, like, FACTS devices allocation, distributed generators allocation, determining component's parameters, and more. In this paper, the optimal size and location of several FACTS devices are selected to achieve the following single objective functions separately: fuel cost minimization and power losses minimization, then combining the mentioned objective function into one multiobjective function for gross cost minimization. The power system optimized is the Institute of Electrical and Electronics Engineers 30‐bus standard system, and the FACTS devices installed are static VAR compensator, thyristor control series compensator, and thyristor‐controlled phase shifter, while the optimization is performed using different conventional and recent optimization algorithms, including a newly developed algorithm, typically Leader Walrus Optimization Algorithm “LWaOA,” making a comparison among such algorithms to investigate the algorithm that possesses the best performance. The results of the optimization processes show the superiority of LWaOA.

As it is necessary to supply electric power to the customers with sufficient quality, power systems operators are doing their best to achieve this goal, besides accommodating new customers to the grid.For this reason, flexible alternating current transmission system (FACTS) devices are widely used in power systems, as they have a variety of capabilities that enable the power system to control the parameters of transmission lines.][3] However, the benefits of FACTS device installations on a power system mainly depend on the right sizing and placement.Consequentially, it is very important to determine the optimal sizing and placement of the desired FACTS device by which the power system could gain the maximum utilization from these installations.These techniques are successfully adopted to find the optimal sizing and location of FACTS installations in the power system, which seems to be a solo-objective function optimization process, However, this leads to enhance many parameters of the power system (i.e., voltage stability improvement, power losses reduction, power flow regulation, etc.).

| Motivation
Many contributions were presented using several optimization techniques to find the optimal sizing and location of FACTS devices.A comprehensive analysis presented in Sulaiman and Mustaffa, 4 investigates the optimal location and sizing of multiple FACTS devices in an Institute of Electrical and Electronics Engineers (IEEE) 14-bus system to minimize generation costs and power losses.In George et al., 5 optimal size and placement of thyristor control series compensator (TCSC) and DG are investigated to reduce power losses in radial power systems, typically IEEE 33-bus systems, using Ant Line Optimization, comparing the results with other techniques.Also, * in 6 , static VAR compensator (SVC) size and placement are investigated for minimizing power losses and voltage deviation in both IEEE 33-bus and IEEE 69-bus systems, using the Pareto Envelope-based Selection Algorithm II.In Sulaiman and Mustaffa, 4 the study explores the optimal allocation of FACTS devices for operation cost minimization in both IEEE 14-bus and IEEE 30-bus systems using a whale optimization algorithm, comparing the results with other techniques.In Nadeem et al., 7 the study focuses on the optimal allocation of FACTS devices and renewable energy resources in an IEEE 30-bus system, aiming to improve voltage stability index, reduce power losses, and minimize generation costs.Various optimization techniques are employed and their results are compared.In Baziar et al., 8 shunt FACTS and renewable energy system optimal sizing and placement are addressed in both IEEE 6-bus and IEEE 118-bus systems to solve the AC securityconstrained unit commitment problem, employing a hybridized algorithm of teaching learning-based optimization and grey wolf optimizer and comparing the results obtained by other optimization algorithms.In Baziar et al., 8 the optimal allocation of SVC and TCSC was investigated in IEEE 30-bus and IEEE 57-bus systems to solve optimal reactive power dispatch problem, employing a modified algorithm, known as modified lightning attachment procedure, comparing its results with other algorithms.Also, in Khan et al., 9 multiobjective multiverse optimization algorithm was developed to explore the optimal allocation of SVC and TCSC in an IEEE 57-bus system for voltage profile enhancement and power losses minimization.Furthermore, many studies [10][11][12][13][14][15][16] introduce other optimization problems and problemsolving approaches for various systems.

| Paper contributions and organization
In this paper, optimal sizing and placement of several FACTS devices are investigated in an IEEE 30-bus system, using a new modified optimization technique, where this new modified technique is considered as an enhanced version of the original technique, "Walrus Optimization Algorithm (WaOA)."This enhanced version is named as "Leader Walrus Optimization Algorithm (LWaOA)."The system itself is modified so that two wind turbines are integrated into it.Many objective functions are presented in this paper, including two separate single objective functions: generation cost minimization and power losses reduction, and combining both of them as a multiobjective function.These objective functions are executed by Gradient-Based Optimization (GBO), 17 WaOA 18 as recent optimization techniques, beside Particle Swarm Optimization (PSO) 19 and Genetic Algorithm (GA) 20 as conventional *Probabilistic load flow solution considering the optimal allocation of SVC in the radial distribution system.optimization techniques, in addition to the modified LWaOA algorithm.The outcomes of such techniques are compared, showing the performance of each technique.The objectives of the paper can be stated as: ▪ Optimal power flow evaluation of the IEEE 30-bus system with wind turbines and several FACTS devices integration.▪ Several FACTS devices placement and sizing optimization to achieve different objective functions.▪ Investigating the new modified optimization technique "LWaOA."▪ Optimization techniques outcomes comparison.
The paper is organized as follows: Section 2 presents a literature review of the FACTS devices involved, Section 3 presents such FACTS devices' modeling, Section 4 introduces the problem formulation, Section 5 presents a description of the proposed algorithm, Section 6 presents the results and discussion, and finally, Section 7 depicts the conclusion.

| SHORT NOTES ON FACTS
A FACTS device consists of a collection of conventional power system components (transformers, resistors, capacitors, and inductors) and numerous power electronics components transistors and thyristors.Due to the dependence of power flow through a transmission line on voltage magnitude, transmission line impedance, and phase angle between buses, FACTS devices have a great influence on controlling one or more of these parameters so that it can improve the power flow of the transmission lines and the power system at all. 21ncorporation of FACTS devices into a power system leads to controlling active and reactive power flow smoothly and rapidly up to a certain level, undesired reactive power flow reduction and thereby network losses, loading capability maximization, steady-state and transient stability, voltage controlling and power quality improvement, power system security and reliability improvement, power loss reduction, voltage profile enhancement, and more.In this paper, three FACTS devices are incorporated, as follows:

| TCSC
TCSC is classified as a series FACTS device, and its structure consists of a series capacitor connected in parallel with a thyristor-controlled reactor (TCR).The whole combination is connected in series to the selected transmission line, where the capacitor-TCR combination donates a smoothly controlled capacitive reactance to the system, as shown in Figure 1.
Due to the dependence of power flow in a transmission line on the transmission line's inductive reactance, the capacitive reactance delivered by TCSC enables the system operator to eliminate a portion of the inductive reactance of the transmission line.Hence, the transmission line's total impedance becomes less than its actual value that leads the transmission line to gain more power transferability.The total reactance of the line can be formulated as follows: 1 where X TL is the reactance of the transmission line, X TCSC is the reactance delivered to the transmission line by TCSC, X C TCSC is the capacitive reactance of TCSC, X L TCSC is inductive reactance of TCSC, α is the firing angle of the thyristors, and K is the compensation coefficient.
From (1) to (3), the operator can control the reactance of the TCSC delivered to a transmission line through the firing angle θ, and the compensation coefficient K expresses the effectiveness of a TCSC.
However, the TCSC has three modes of operation that can be summarized as: 1 ▪ Blocked mode: In this mode, the thyristors are not triggered, the firing angle α = 180°and no pulses activate the thyristor's gate, hence, the thyristors are not conducting, and the reactance of TCSC is fullcapacitive (i.e., X TCSC = X CTCSC ).In this mode, the TCSC acts as a fixed-series capacitor.▪ Bypass mode: In this mode, the thyristors are fully triggered, and the current flow over the thyristors is continuous and sinusoidal (i.e., α = 90°), and the reactor is completely operated with the capacitor.In this mode, the TCSC acts as a series capacitor shunted with a reactor, where the total reactance of the TCSC is slightly inductive, as the reactor's susceptance is larger than the capacitor's one.The main goal of that mode is to protect the capacitor against transient overcurrents of the line.▪ Control mode: In this mode, the thyristors are triggered so that a controlled amount of inductive current is delivered with the capacitive current of the capacitor, by which the TCSC delivers a controlled inductive or capacitive reactance to the transmission line, depending on the firing angle (α), considering the risk of subsynchronous resonance.

| Thyristor-controlled phase shifter (TCPS)
A TCPS is a series FACTS device that depends on phase shifting to control power flow in a transmission line.It changes the voltage phase of bus m so that the delivered phase of bus n is different, resulting the active power flow control to improve the stability of a power system and reduce the frequency oscillations. 22Figure 2 shows a schematic diagram for TCPS.
As shown in Figure 2, TCPS can act as an essential link for interconnecting two sections of a power system, where and where θ represents the phase shift that occurs due to the characteristics of the transmission line, and φ represents the phase shift delivered by TCPS.

| SVC
SVC is considered as the most popular shunt FACTS device, and it is able to control the voltage of a certain point injecting current into the system at the point of connection.It is adopted in voltage regulation, voltage stability restoration, transient stability maintaining, and power losses reduction.
A shunt FACTS device could be a variable impedance VAR compensator or a converter-based VAR compensator.For an SVC device, it is a variable impedance VAR compensator.It consists of one TCR shunted with several Thyristor-Switched Capacitors (TSCs), where the TCR rating is slightly greater than the rating of a single TSC, and the TCR-TSC combination allows a smoothly controlled reactive power injection by which the voltage magnitude is controlled. 23Figure 3 shows the schematic diagram for SVC that contains three TSCs and a TCR.
The effectiveness of SVC is determined by its total capacitive rating, where where Q cmax is the total capacitive reactive power of SVC, Q c is the capacitive reactive power delivered by a single TSC, and n is the number of TSCs included in SVC.
For the operation of SVC in capacitive mode, a single TSC is switched on, where the desired amount of capacitive reactive power is adjusted by the TCR (i.e., the TCR absorbs the surplus reactive power produced by the TSC so that the desired amount of capacitive reactive power is precisely delivered to the system).If the required capacitive reactive power is greater than the TSC rating, then another TSC is switched on, and the amount of capacitive reactive power is adjusted by the TCR.In other words, TSCs are switched on sequentially according to the desired amount of capacitive reactive power, and the TCR controls the value of delivered capacitive reactive power.This can be formulated as where Q creq is the required capacitive reactive power, m is the number of operating TSCs, Q c is the capacitive reactive power delivered by a single TSC, Q L is the inductive reactive power produced by the TCR, and θ is the firing angle of the thyristors.
For the operation of SVC in inductive mode, all TSCs are switched off, while the TCR is operating, and the firing angles of thyristors are adjusted so that the TCR delivers the required inductive reactive power to the system.

| TCSC
The reactance of a transmission line can be controlled through the compensating reactance of TCSC added to it, where the desired amount of the reactance can be determined by the firing angle (α), and the total reactance of the transmission line can be expressed with respect to the compensation coefficient K, as explained in Section 2.1.Figure 4 shows the equivalent model of TCSC.
Power flow formulation of a TCSC-included transmission line can be as follows: 24,25 θ θ where where V a and V b are voltage magnitudes of sending and receiving buses, respectively, α a and α b are phase angles of sending and receiving buses, respectively, G ab and B ab are the conductance and the susceptance of the transmission line matching buses a and b, respectively.

| TCPS
A TCPS is a device that influences the phase angle of a transmission line, while the voltage magnitude is unchanged, that is, the ratio is 1:1, as explained in Section 2.2.According to the equivalent model of TCPS shown in Figure 5, and considering the phase shift And, the injected active and reactive power by the TCPS to the transmission line can be formulated as 24,25

| SVC
SVC can deliver a controlled amount of reactive power by controlling the firing angle (θ) as explained in Section 2.3.Therefore, SVC can be modeled as a variable susceptance, as shown in Figure 6.
The susceptance of the SVC can be calculated as follows: 26 And, the delivered reactive power to the point of connection can be expressed as

| PROBLEM FORMULATION
As explained in Section 1, two wind turbines are integrated into the tested system, where the modified IEEE 30-bus system has four thermal generators and two wind turbines (typically generators "5" and "11"), and six FACTS devices distributed along the system, typically two TCSCs, two TCPSs, and two SVCs.The formulation of objective functions can be as follows.
4.1 | Cost of generation and power losses calculation

| Cost of generation
Regarding the modified system, the cost of generation will be the sum of the cost of all generation units, including the integrated wind turbines.Due to the uncertainty of the generated power of the wind turbines, the cost of generation can be expressed as 27 where G P ( ) is the generation cost of thermal generators, G cost j wt is the direct generation cost of the wind turbine, G cost j Rwt is the reserve cost of the wind turbine, G cost j Pwt is the underestimation's penalty cost of the wind turbine, n = {1, 2, 8, 13} where K pwtj is the penalty cost coefficient of the jth wind turbine.

| Power losses
Power losses can be formulated as follows: where G i (ab) indicates the transmission line's conductance connecting buses a and b; V a and V b indicate the voltage magnitude of buses a and b, respectively; and α a and α b are phase angles of buses a and b, respectively.

| Equality constraints
The optimization process shall submit the following equality constraints for the equilibrium of the process:  where P a gen.and P a Dem. are active generated power and load demand at bus a, respectively, Q a gen.and Q a Dem. are reactive generated power and load demand at bus a, respectively.Y ab is the admittance of the line connecting buses a and b, and θ ab is the phase angle of the line.Integrating FACTS devices in Equations ( 30) and (31):  are delivered active and reactive power by TCPS to bus a, respectively, and Q a SVC. is the delivered reactive power by SVC to bus a, where SVC cannot inject active power.

| Inequality constraints
The optimization process shall submit the following inequality constraints: Equations ( 34)-(36) refer to generators' constraints, where P i gen.and Q i gen.are the active and reactive power generated by generator i, V i gen. is the voltage of the generator i bus.These equations influence both thermal generators and wind turbines.
Equations ( 37) and (38) refer to security constraints for load buses and branches, where V j LB. is the voltage of load bus j, and S k TL. is the total power transferred by transmission line k.
Equations ( 40)-( 42) refer to the limits of TCSC, TCPS, and SVC, respectively, where k x TCSC. is the firing angle of TCSC x, φ y TCPS. is the phase shift delivered by TCPS y, and Q z SVC. is the reactive power injected by SVC z.

| The original algorithm
Inspired by nature, WaOA is an algorithm that simulates walruses' behavior during migration.Such animals live in icy places, typically near the northern pole.As a herd, the male with the longest tusks and the biggest muscles is considered to be the dominant individual who leads the group.With the annual start of the summer, the ice starts to break and melt, and the walruses prefer to migrate to rocky beaches or outcrops.Such migration is dangerous although they migrate in big numbers, due to their natural predators: the killer whale "the orca" and the polar bear.The algorithm mimics three elegant walruses' behavior: This algorithm can be mathematically modeled as (i) Initialization: Where the population is plotted as where X is the population of WaOA, n is the number of individuals, m is the number of decisions, and F is the desired objective functions.
(ii) Feeding strategy: In this step, the individual who owns the best solution (the tallest tusks) directs the other individuals to the best feeding area.The strongest individual finds the best solution.The update of the positions takes place under the guidance of the dominant (strongest) walrus, and can be mathematically modeled as where X i L1 is the updated position found for walrus i according to the feeding strategy, X i j L , 1 is its jth dimension, F i L1 is its objective function value, rnd i,j are random numbers from the interval [0, 1], SW is the best candidate solution which is considered as the strongest walrus, and I i,j are integers selected randomly between 1 and 2. This integer helps enhance the ability of exploration of the algorithm.
(iii) Migration: Simulating the walrus behavior of migration to rocky places or outcrops in this step, each Individual selects a location of another individual randomly in another area in the search space.If the newly proposed position gives better results for the objective function, it replaces the current location.This can be mathematically expressed as where X i L2 is the new generated position for the ith walrus based on the second phase, X i j L , 2 is its jth dimension, F i L2 is its objective function value, X k , k ∈ {1, 2, …, N} and k ≠ i, is the location of selected walrus to migrate the ith walrus towards it, x k,j is its jth dimension, and F k is its objective function value.
(iv) Predator escape or fight: Due to the probability of being attacked, they fight and escape away from their predators in the near around of their current locations.Simulating such behavior improves the exploitation of the algorithm for local search.This can be mathematically expressed as where X i L3 is the newly generated position for the ith walrus based on the third phase, X i j L , 3 is its jth dimension,

| The modified LWaOA algorithm
The proposed technique, known as the Leader-Based WaOA Optimization Algorithm (LWaOA), builds upon the WaOA algorithm and introduces Leader-Based Mutation-Selection 28 to enhance its performance.LWaOA addresses the limitations of the original WaOA algorithm, which includes slow convergence and susceptibility to local optima.This improvement enables the technique to achieve optimal fitness function values.The modification is grounded in the evaluation of the best location vector X t best , the second-best location vector X t best−1 , and the third-best location vector Then, the next location is updated using the following equation: Finally, the best solution is updated as below: )) < ( ).  proposed LWaOA algorithm based on the simultaneous crossover and mutation using the three best leaders.

| Benchmark functions test
To ensure an unbiased assessment, we conducted rigorous testing under uniform conditions to create a level playing field for comparison.Our evaluation involved the deployment of 50 search agents with a maximum iteration limit of 200.Additionally, we independently executed each algorithm 20 times, acknowledging the inherent stochastic nature of these algorithms.This approach allowed us to conduct a comprehensive assessment of their performance.The algorithms were implemented using MATLAB R2016a software on a high-performance computer running Windows 10 64-bit Professional, equipped with 8 GB of random-access memory.This ensured a reliable and consistent computing environment for our experiments.One crucial aspect we examined in this evaluation was the impact of parameter configurations on algorithm performance.To ensure a fair comparison, we adopted parameter values directly from the original articles authored by the developers of each respective algorithm.This method maintained consistency and eliminated the possibility of bias resulting from arbitrary parameter choices.Notably, among the algorithms we compared, including the artificial hummingbird algorithm, 30 hunter-prey optimization, 31 artificial ecosystem-based optimization, 32 and the original WaOA.The LWaOA technique stood out with exceptional performance across the seven benchmark functions. 33As demonstrated in Table 1, the results consistently outperformed those of various contemporary algorithms.LWaOA consistently achieved top rankings in terms of best, average, median, and worst results across a wide range of benchmark functions.The strength of the LWaOA algorithm lies in its remarkable consistency in uncovering optimal solutions, as evidenced by its sustained top-tier performance across diverse benchmark functions.Its outstanding performance, in contrast to other contemporary algorithms, underscores its effectiveness in tackling complex optimization problems.These findings highlight the LWaOA technique as a robust and dependable approach for addressing a broad spectrum of optimization challenges.
The convergence curves of all algorithms across the seven benchmark functions (F1, F2, F3, F4, F5, F6, and F7) are visually displayed in Figure 8.This visual representation provides valuable insights into the progression of each algorithm over iterations when applied to specific benchmark functions.Furthermore, Figure 9 showcases a series of boxplots that succinctly summarize the performance of all algorithms across the same set of benchmark functions.These boxplots offer a comprehensive overview of each algorithm's performance, encompassing their best, average, and worst outcomes.Remarkably, as illustrated in Figure 8, the LWaOA technique demonstrates exceptional convergence behavior, consistently converging towards optimal solutions across all benchmark functions.Additionally, in Figure 9, the boxplots consistently position LWaOA as a topperforming algorithm, further emphasizing its competitive advantage compared to other algorithms.These convergence curves and boxplot results strongly emphasize the robustness and effectiveness of the LWaOA algorithm, highlighting its proficiency in achieving convergence and delivering favorable results across a diverse range of benchmark functions.

| CASE STUDY TEST AND RESULTS
The required objective functions are applied to the IEEE 30-bus system with integrating wind turbines and FACTS devices, as mentioned before, where the configuration of the modified IEEE 30-bus system contains: ▪ Four thermal generators in buses 1, 2, 8, and 13. ▪ Two wind turbines in buses 5 and 11. ▪ Six FACTS devices, typically TCSC, TCPS, and SVC, two of each, where their sizes and placements are investigated.
The modified IEEE 30-bus system.IEEE, Institute of Electrical and Electronics Engineers; SVC, static VAR compensator; TCPS, thyristor controlled phase shifter; TCSC, thyristor control series compensator.
The system's schematic diagram is shown in Figure 10.Therefore, this system has 27 parameters required to be optimized, as follows: ▪ The voltages of generation buses.▪ The active power is generated by all generators excluding generator 1 due to the rule of bus 1 as a slack bus.▪ The transformation ratio for each transformer.
▪ The placement location for each FACTS device.▪ The settings of each FACTS device. 6nd the optimization processes applied to this system include: ▪ Generation cost reduction.▪ Power losses reduction.▪ Generation cost and power losses reduction, combined as a multiobjective function.Such optimization processes were carried out using five conventional and recent optimization techniques: WaOA, GBO, PSO, GA, and the new proposed technique: LWaOA.The number of populations is 200, and the number of iterations is 100, where each process was repeated 20 times.

| Optimization process 1: Generation cost reduction
In this process, the optimization took place, where the best result was obtained by the LWaOA technique.Table 2 shows the settings obtained by LWaOA, and Figure 11 shows the convergence curves for all optimization techniques.The graph in Figure 11 shows the superiority of LWaOA against the other algorithms.We can find also that PSO, WaOA, and GBO are giving promising outcomes, respectively, however, WaOA, during the last 50 iterations gave slight variations and was almost constant compared with the other algorithms, while GA gave the worst outcomes against the other algorithms.

| Optimization process 2: Power losses reduction
Also in this process, the optimization took place, and the best result was obtained by the LWaOA technique.Table 3 shows the settings obtained by LWaOA, and Figure 12 shows the convergence curves for all optimization techniques.The graph in Figure 12 again proves the superiority of LWaOA against the other algorithms.Also we can observe that PSO, WaOA, and GBO are giving promising outcomes, respectively, while GA gave the worst outcomes against the other algorithms.

| Optimization process 3: Generation cost and power losses reduction
In this process, the optimization took place, where the best result was obtained by the LWaOA technique.Table 4 shows the settings obtained by LWaOA, and Figure 13 shows the convergence curves for all optimization techniques.
The graph proves again the superiority of LWaOA against the other algorithms, and it is observed that GBO, PSO, and WaOA are giving promising outcomes, respectively, while GA gave the worst outcomes against the other algorithms.Table 5 summarizes the outcomes of all algorithms applied in the optimization processes.

| CONCLUSIONS
This paper presented power system optimization for an IEEE 30-bus system with wind turbines integration, incorporating FACTS devices, and proposed a new optimization technique for executing the optimization process.Three objective functions were executed to find the optimal placement and sizing of FACTS devices regarding achieving the objective functions, where the objective functions were: (1) generation cost reduction, (2) power losses reduction, and (3) generation cost and power losses reduction combined together.Such optimization processes were executed using different optimization techniques, the conventional PSO and GA techniques, the recent GBO and WaOA techniques, beside the proposed LWaOA technique, to evaluate the results generated by each technique.The results show the superiority of the proposed technique, as it found the best solution for optimizing the system regarding the three objective functions.Also, PSO, GBO, and WaOA generated promising results for optimizing the system according to the required objective functions.In the future, more sophisticated systems can be optimized using the proposed technique with the same FACTS devices incorporated in this paper, like, IEEE 57-bus and IEEE 118-bus systems, or incorporating other FACTS devices, like, STATCOM and SSSC.Also, the proposed technique can be compared with other recent techniques, and another recently modified techniques can be developed and tested.

F I G U R E 4
Equivalent model of TCSC.TCSC, thyristor control series compensator; TL, transmission line.F I G U R E 5 Equivalent model of thyristor control series compensator.

6
Equivalent model of SVC.SVC, static VAR compensator.

( i )
Guiding the members to feed: In this stage, the best individual's position (the dominant walrus) is tracked, leading the other individuals to promising areas, that results in big changes in individuals' positions.This step maximizes the capability of the algorithm regarding exploration and global search.(ii) Herd migration: This step makes another big change in individuals' positions.The positions of the individuals are considered as a probable destination for migration.One of these positions is randomly selected, where the other individuals move towards it.The update of the population is avoided in this stage from depending on a certain member, even if it owns the best position to avoid early convergence and trapping in local optima.(iii) Predator fight or escape: In this step, walruses are facing their predators in small areas around the positions of the individuals, causing fine changes in the positions of the individuals.The algorithm's capability for local search and exploitation is increased, and leads to better convergence.

Figure 7
Figure 7 illustrates the flowchart of the proposed LWaOA technique, which highlights the incorporation of the Leader-based mutation selection algorithm.This modification leads to enhance the exploration of the

F I G U R E 11
Convergence curve of optimization techniques for optimization process 1. GA, Genetic Algorithm; GBO, Gradient-Based Optimization; LWaOA, Leader Walrus Optimization Algorithm; PSO, Particle Swarm Optimization; WaOA, Walrus Optimization Algorithm.

F I G U R E 13
Convergence curve of optimization techniques for optimization process 3. GA, Genetic Algorithm; GBO, Gradient-Based Optimization; LWaOA, Leader Walrus Optimization Algorithm; PSO, Particle Swarm Optimization; WaOA, Walrus Optimization Algorithm.
Rwtj is the reservation cost of jth wind turbine, P wtavj is the available power from jth wind turbine, f wtj (P wt ) refers to the wind turbine's probability density function.Also, the penalty cost of a wind turbine can be expressed as wt(26)where g wtj is the coefficient of direct generation cost of the jth wind turbine, P wtsj is the scheduled power of the its objective function value, t is the iteration contour, LL j and UL j are the lower and upper bounds of the jth variable, respectively, LL j Convergence curves of all algorithms for seven benchmark functions.AEO, artificial ecosystem-based optimization; AHA, artificial hummingbird algorithm; HPO, hunter-prey optimization; LWaOA, Leader Walrus Optimization Algorithm; WaOA, Walrus Optimization Algorithm.Boxplots of all algorithms for seven benchmark functions.AEO, artificial ecosystem-based optimization; AHA, artificial hummingbird algorithm; HPO, hunter-prey optimization; LWaOA, Leader Walrus Optimization Algorithm; WaOA, Walrus Optimization Algorithm.
F I G U R E 8 LWaOA optimized settings for optimization process 1.
T A B L E 2Note: The best values obtained are in bold.Abbreviations: LWaOA, Leader Walrus Optimization Algorithm; SVC, static VAR compensator; TCPS, thyristor controlled phase shifter; TCSC, thyristor control series compensator; TTC, transformer tap changing.
T A B L E 3 LWaOA optimized settings for optimization process 2.
Note:The best values obtained are in bold.Abbreviations: LWaOA, Leader Walrus Optimization Algorithm; SVC, static VAR compensator; TCPS, thyristor controlled phase shifter; TCSC, thyristor control series compensator; TTC, transformer tap changing.F I G U R E 12Convergence curve of optimization techniques for optimization process 2. GA, Genetic Algorithm; GBO, Gradient-Based Optimization; LWaOA, Leader Walrus Optimization Algorithm; PSO, Particle Swarm Optimization; WaOA, Walrus Optimization Algorithm.
T A B L E 4 LWaOA optimized settings for optimization process 3.Note:The best values obtained are in bold.