Enhancing Power System Performance via TCSC Technology Allocation With Enhanced Gradient-Based Optimization Algorithm

This paper addresses the critical challenge of optimal allocation of Thyristor-Controlled Series Compensator (TCSC) devices in transmission power systems through an innovative optimization framework. Leveraging an Enhanced Gradient-Based Algorithm (EGBA) augmented with a crossover operator, the proposed methodology seeks to promote diversity in the solutions generated in each iteration, aiming to maximize the efficiency of power transmission networks. The algorithm incorporates key components such as the Gradient Search Process (GSP) and Local Escaping Process (LEP) to guide the exploration process and prevent premature convergence to suboptimal solutions. Additionally, the crossover operator, a novel addition in the EGBA, facilitates the exchange of TCSC configurations between solutions, contributing to solution diversity and potentially revealing novel optimal allocations. Initially, the EGBA and GBA performances are estimated using the CEC 2017 benchmarks. Moreover, to assess the practical applicability of the suggested EGBA, it is specifically tailored and implemented to enhance the operation of transmission power systems. The primary objective is to minimize technical power losses, considering varying numbers of TCSC devices with experimentation on two distinct IEEE power systems, one with 30 buses and another with 57 buses. The results are analyzed to validate the ability of the EGBA method in optimizing power systems and addressing technical losses. The novel proposed EGBA method significantly reduces power losses compared to the original GBA method in both tested power systems. In the first system, the EGBA achieved 0.85%, 2.99%, and 1.32% lower losses than the GBA when optimizing for one, two, and three TCSC devices, respectively. In addition, the objective of enhancing the security margin of the transmission lines is involved to optimize the power flow besides the minimization function of power losses. Similarly, in the second system, the EGBA outperformed the original GBA by 5.19%, 6.32%, and 5.12% for the same TCSC configurations. The simulation results demonstrate that the proposed EGBA is not only more effective but also more efficient than the original GBA and other recent approaches.

SECTION I. Introduction

A static non-linear programming issue that takes into consideration the electrical elements of massive transmission power grids is called the optimal power flow (OPF). While optimizing vital objectives, the primary purpose of the challenge is to identify the steady-state functioning points of all electric elements accessible to the power systems [1], [2]. The OPF problem takes into account several individual goals, including entire power losses, the fuel costs for power generation electricity, voltage deviations, polluted emissions, and voltage stability index [3]. Furthermore, the issue of OPF requires that a set of operational and physical constraints be accurately met. These constraints include those enforced by devices and network limitations, such as switchable capacitor banks, transmission line capacity limits, bus voltages, transformer taps, active and reactive generators’ power, and transformer taps [4], [5]. Essentially, in order to obtain other dependent variables such as the voltage magnitude at other buses, and the reactive power of the generators, the control variables such as voltage magnitude at generation buses, the active power of the generators, transformer tap settings, and injected reactive power at capacitor buses of the OPF problem must first be determined [6], [7], [8].

Real-world electrical networks frequently use TCSC technology due to the fact that is a robust and reasonably priced series FACTs device with exceptional performance that enables precise, dependable power flow regulation of power lines [9]. The use of TCSC devices, which provide series compensating characteristics, is one of the most affordable approaches to increase the transmission network’s real electrical carrying capacity [10], [11]. In order to reduce area frequency fluctuations and tie-line power, three series FACTs apparatuses, which are the thyristor controlled phase shifter (TCPS), the TCSC, and the static synchronous series compensator (SSSC), are being considered and emulated in Automatic Generation Control (AGC) assessments pertaining to multi-area associated electrical networks [12]. The Improved Particle Swarm Optimization Technique (IPSOT) has been employed as a solution tool in conjunction with the Integral of Time multi-plied Squared Error (ITSE) as the goal of reduction to construct the damper controllers. In terms of vibration dampening at area frequencies and tie-line transmission powers, the TCSC-AGC has demonstrated better performance than the TCPS and SSSC. Sensitivity testing has also been conducted to demonstrate the TCSC-AGC’s resilience through the practical deployment of the TCSC in transmission networks that highlighted its benefits over SSSC [13]. A variety of conventional and metaheuristic methods have recently been developed to address OPF such as sequential unconstrained methodology [14], interior point method [15], fuzzy linear framework [16], linear and nonlinear programming [17], [18], and Newton-based method [19] are examples of conventional methodologies. It should be highlighted, therefore, that these techniques are not helpful for significant electrical networks and do not result in globally optimum solutions. Thus, scientists have worked to develop metaheuristic techniques to overcome the shortcomings of earlier approaches [20].

Consequently, developing metaheuristic techniques is essential to overcoming the above-described limitations. Using a range of heuristic (population-based) approaches to tackle different OPF challenges has been more common in the last 20 years [21], [22]. Various solutions for augmenting the procedures have been characterized to reduce the power losses. An improved social spider optimizer has been outlined in [23] that balances the movement patterns of male and female spiders to minimize power losses. An alteration was made to the JAYA algorithm, enhancing its capability to refine generated solutions by incorporating adjustments based on the worst and best solutions regarding voltage profile and losses [24]. Gorilla troops optimizer was developed for designing fractional order controller integrating tilt integral derivative for stabilizing a three-area hybrid power system in [25] while a grasshopper optimization was hybridized with bald eagle searching algorithm [26] to address the unit commitment. In [27], graphical processing units (GPU-native sparse direct solver) have been employed to improve the overall performance alternating current OPF analysis. An invasive weed optimizer (IWO) was presented in [28] to combinational approach for OPF research with the inclusion of FACTs while an Emended Crow Search Algorithm (ECSA) was utilized on the OPF with adjustments involving a novel bat technique [29]. A placement methodology for TCSC in power systems that takes into account both line interruptions and normal operation was developed in [30]. In order to minimize the power system’s voltage stability index, voltage deviation and power loss, the stochastic OPF problem has been handled in [31] with the use of an adaptive Lightning Attachment Procedure Technique (ALAPT) with the inclusion of renewable sources. In [32], the impact of hydro constraints has been investigated in the operation of hydrothermal systems via a nonlinear multi-period hydrothermal OPF model. While the study resulted in a notable reduction in transmission line overloads during the IEEE 5 bus and 14 bus networks, it failed to determine the appropriate TCSC sizing in the investigated networks, which has a significant impact on these applications. In [33], the OPF has been solved by the error bound which could be represented by greatest discrepancy between the best possible outcomes of OPF models. In [34], Social Network Search Approach (SNSA) has been employed on a multi- OPF and applied on practical electrical system. Gorilla Troops Technique (GTT) was carried out on the IEEE 30 bus grid in [35] and addition of TCSC modules to the IEEE 30 bus system in [36]. However, the allocation and size of the TCSC devices have not been studied. To increase the voltage stability and available transfer capability, a multi-objective PSO for the multi-objective optimal allocation model for TCSC has been performed in [37]. This study presented a chaos initialization approach and then set up a variable inertia weight configuration for the IEEE-30 bus system, which can only be used to one transmission network. To find the best position and compensation level of TCSC devices, an enhanced version of GA was presented in [38]. To improve the transfer capacity that is accessible in power systems, the proposed GA was integrated with twofold mutation probabilities. A modified version of Subtraction-Average-Based Optimizer (SAO) for TCSC allocation, which lessens losses in electric power grids, is provided in [39]. This study modifies the usual SAO by incorporating a cooperative learning strategy powered by the leader solution.

To obtain OPF solutions, the authors employed an improved particle swarm optimization technique. The OPF of IEEE 30- and 57-bus networks provided with TCSC and TCPS was carried out in [40]. The analysis considered the positions of the FACTs devices to be fixed. Improved particle swarm optimisation was used in [41] to solve the OPF created with serial FACTs devices, where it was capable of handling challenging OPF situations. Using a generalized interline power flow controller (GIPFC), the OPF problem has been tackled in [42] and improved the efficiency of the current transmission lines. Employing the ameliorated ant lion optimization technique, the authors were able to find the solutions for the OPF modeled with FACTs. To generate high-quality OPF solutions, the success history-based adaptive differential evolution (SHADE) technique has been combined with the overpowering of feasible solutions constraints addressing strategy. Using the adaptive parallel seeker optimization technique, OPF solutions have been achieved in [43] while taking into account the best possible allocation of TCSC. Case studies with OPF were conducted under both nominal and contingency settings. Numerous methods exhibit remarkable convergence characteristics and proficiently establish inequality bounds. Nonetheless, these conventional strategies are susceptible to getting trapped in local minima due to their reliance on the initial setup, rendering them incapable of attaining the true optimal outcome. Furthermore, while dealing with discrete and integer variables, each technique struggles and necessitates modelling specific versions of the OPF. Therefore, creating metaheuristic methods is crucial to getting over the restrictions mentioned above [44].

In order to enhance the power system available transfer capability, optimizing the TCSC parameters and places was addressed for mitigating the network congestion [45]. The system’s sensitivity to parameter changes was estimated in this study using sensitivity factor-based approaches. These approaches, nevertheless, are limited to linear approximations of system behaviour and may occasionally fail to identify the globally best solution, which could end up to inadequate outcomes in nonlinear systems. Sensitivity-based approaches are therefore less effective when many TCSCs are needed to relieve electrical grid congestion. Several heuristic/metaheuristic techniques can be used to concurrently locate and optimise TCSCs in order to overcome this problem. In [46], the integration of FACTs devices into the system has been coded using Newton Raphson load flow equations to determine their optimal locations while a heuristic optimization algorithms in [47] has been adopted to identify suitable locations for FACTs devices and optimize their parameters. In [48], a GA has been designed to optimize a nonlinear objective function, focusing on the practical incorporation of FACTs devices within a congested network. In [49], Teaching-Learning-Based Optimizer (TLBO), Gray Wolf Optimizer (GWO) and Particle Swarm Optimizer (PSO) have been applied and compared for optimizing the TCSC reactance for stable system operation and congestion mitigation. In this study, the suitable locations were determined in a separate pre-stage employing the line utilization factors. Thereafter, TLBO, GWO and PSO have been contrasted with the application on the standard IEEE-30 bus system where the findings indicated that TLBO provided better performance than PSO and GWO. Unfortunately, because the pre-specified TCSC placement using the line utilisation factor was predicated on a single static operating situation, it might not be able to adjust efficiently to changes in real time brought about by fluctuating loads and generation patterns. Under the changed circumstances in the power system, this static placement might not be beneficial. Additionally, under certain operating conditions, the pre-specified placements might not always find the best places to mitigate congestion, which could result in suboptimal performance in congestion management and voltage stability.

This study demonstrates an Enhanced Gradient-Based Algorithm (EGBA) incorporating the Gradient Search Process (GSP) and Local Escaping Process (LEP) for handling different benchmark functions and the TCSC optimal allocation issue. The exceptional indicated solution contains GSP and LEP to guide the exploration process and prevent premature convergence to suboptimal solutions. Initially, the effectiveness of the proposed EGBA and GBA is evaluated using the CEC 2017. Besides, to assess the practical applicability of the proposed EGBA is specifically implemented for the optimal allocation of TCSC devices to minimize technical power losses in transmission power systems. Furthermore, the accuracy and superiority of the proposed EGBA over the others can be observed while considering a range of TCSC devices.

• A new metaheuristic approach called EGBA is developed.

• The EGBA algorithm is utilized to determine the best placement and rating for TCSC devices that are integrated with the IEEE 30-bus and IEEE 57-bus power network.

• It is successfully implemented for enhancing power system performance via TCSC allocation under varying numbers of TCSC devices.

• Additionally, the goal of improving the security margin of transmission lines is incorporated to optimize power flow, alongside the objective of minimizing power losses.

• The voltage profile is further improved for all buses based on the proposed EGBA with average improvement of 7.22% and 8.08% for both systems.

• The results of the simulation reveal that the proposed EGBA outperforms many other modern alternatives as well as the original GBA in terms of effectiveness and efficacy. Also, the proposed method is proven to be superior by the statistics and convergence analysis results considering CEC 2017 benchmark functions.

The following is the structure of the remaining portions of the paper: The approach to use for minimizing power losses when TCSC devices are present is discussed in Section II. The optimization frameworks for the original GBA and its improved version (EGBA) are provided in Section III. The simulation findings, which are divided into two subsections, are presented in Section IV. The experimental research conducted to identify the optimal EGBA version is covered in the first subsection. The allocation and size problem for TCSC devices to lower the power network losses is provided in the second subsection. This solution is derived using the suggested EGBA and additional competitive metaheuristic algorithms. The findings of the current study are discussed in Section V.

SECTION II. Problem Formulation: TCSC Allocation in Transmission Networks

A. TCSC Modeling

Among the various FACTs devices, the TCSC stands out as a popular choice due to its numerous benefits. These benefits include its effectiveness, rapid response time, and cost-efficiency [50]. The TCSCs can operate in two modes: inductive and capacitive, allowing them to either increase or decrease the reactance of a transmission line. Figure 1(a) illustrates how a TCSC is connected in series with a transmission line within a power network. Its internal structure consists of a capacitor (C) in parallel with an inductor (L), with their combined behavior controlled by a thyristor-based valve. The valve’s operation is determined by the extinction angle ($\alpha$ ), which can be adjusted within a range of 90° to 180° [51]. The TCSC compensator acts as a variable capacitor, changing the reactance (resistance to AC current) of the transmission line as depicted in Fig. 1(b) [52]. Essentially, the TCSC’s reactance replaces the original reactance of the transmission line ($X_{Line}$ ). To avoid overcompensating the line, the necessary $X_{TCSC}$ value can be calculated using a specific equation [53].

$$\begin{align*} X_{TCSC} (\alpha)& =\frac {X_{L} (\alpha)\times X_{C}}{X_{L} (\alpha)+X_{C} } \tag {1}\\ X_{L} (\alpha)& =\left ({{\frac {\pi }{\pi -sin (2\alpha)-2\alpha }}}\right)X_{L,max} \tag {2}\\ X_{L\,,max} & =\left ({{2\pi f}}\right)L,\,\,X_{C} =\frac {-1}{j\left ({{2\pi f}}\right)C} \tag {3}\end{align*}$$

FIGURE 1.

(a) TCSC Circuit model (b) Configuration of transmission line with TCSC [54].

By replacing the terms $X_{L}(\alpha)$ and $X_{C}$ , the formulation of Eq. (1) becomes as follows:

$$\begin{equation*} X_{TCSC} (\alpha)=\frac {\left ({{\frac {\pi }{-2\alpha -sin (2\alpha)+\pi }}}\right)X_{L,max} \times X_{C}}{X_{C} +\left ({{\frac {\pi }{-2\alpha -sin (2\alpha)+\pi }}}\right)X_{L,max}} \tag {4}\end{equation*}$$

B. Losses Minimization and Constraints

Reducing total network losses is the main objective since it improves voltage profile and electrical system performance. For computing purposes, this objective function (OF) can be mathematically described as follows [55]:

$$\begin{align*} {P}_{Loss} \!=\!\sum \limits _{m=1}^{Nbus} \left ({{ \sum \limits _{\substack { {n=1} \\ {m}\ne n \\ }}^{Nbus} {G}_{mn} \left ({{V_{m}^{2} -2\!\times \! \left ({{ {V}_{m} {V}_{n} {cos} \left ({{\theta _{mn}}}\right)\!+\! {V}_{n}^{2}}}\right) }}\right.}}\right) \tag {5}\end{align*}$$ where the variables $\theta _{mn}$ and $V_{mn}$ indicate the phase angle and voltage difference, respectively, between buses m and n, while $N_{bus}$ denotes the number of buses. Moreover, $G_{mn}$ indicates the conductance of the transmission line connecting buses m and n.

Lots of equality and inequalities constraints pertaining to both dependent and independent variables must be satisfied in order to address the TCSC allocation problem.

The following are the control variables for optimum TCSC allocation challenges:

1. Reactance compensation for each TCSC device that needs to be deployed.

2. Each TCSC device that is installed will require the selection of potential transmission lines.

3. Injecting reactive power into the transmission system with the use of current Var sources.

4. The voltage of the generator

5. The output powers of the generator.

6. Transformer tap configurations.

Therefore, Equations. (6) and (7) indicate that the specifications for reactance compensation, independent variables, and TCSC places are required to be satisfied.

$$\begin{align*} -50\% X_{Line_{TCSC,k}} & \geq X_{TCSC} (\alpha)_{k} \geq +50\% X_{Line_{TCSC,p}}, \\ & \quad {k}=1,2,\ldots N_{TCSC} \tag {6}\\ N_{lines} & \geq Line_{TCSC,k} \geq {1, k}=1,2,\ldots N_{TCSC} \tag {7}\end{align*}$$ where Line$_{TCSC,k}$ refers to the potential lines for fixing TCSC devices; $N_{lines}$ expresses the entire number of transmission lines; $N_{TCSC}$ signifies the entire number of allocated TCSC devices; $X_{Line_{TCSC,k}}$ designates the reactance of the corresponding lines which are designated for fixing TCSC device.

Regarding independent variables, Equations (8)–(11) control the limitations for tap settings, generator output powers, generator voltage, and reactive power injection from Var sources, respectively [56].

$$\begin{align*} {Tp}_{k}^{min} & \le {Tp}_{k} \le {Tp}_{k}^{max}, k=1,2,\ldots Nt \tag {8}\\ {Pgn}_{m}^{min} & \le {Pgn}_{m} \le {Pgn}_{m}^{max},m=1,2,\ldots Ngn \tag {9}\\ {Vgn}_{m}^{min} & \le {Vgn}_{m} \le {Vgn}_{m}^{max},m=1,2,\ldots Ngn \tag {10}\\ {QI}_{Vr}^{min} & \le {QI}_{Vr} \le {QI}_{Vr}^{max}, Vr = 1,2,\ldots Nq \tag {11}\end{align*}$$ where Nq designates the entire number of VAr sources, Ngn indicates the entire number of generation plants, and Nt stands for the entire number of transformers. Pgn describes the real generators’ power output; Tp reveals the tap values that describe the tap transformers. Besides, the two symbols Vgn and QI illustrate the generators’ voltages and the reactive power that are injected by VAr sources, respectively.

Furthermore, with regards to dependent variables, the restrictions pertaining to apparent power flow across the transmission lines, bus voltage, and generators’ reactive power output are handled by Eqs. (12)–(14).

$$\begin{align*} \left |{{ {SF}_{L}}}\right | & \le {SF}_{L}^{max}, L=1,2,\ldots N_{lines} \tag {12}\\ {V}_{m}^{min} & \le {V}_{m} \le {V}_{m}^{max}, m=1,2,\ldots Nbus \tag {13}\\ {Qg}_{m}^{min} & \le {Qg}_{m} \le {Qg}_{m}^{max},m=1,2,\ldots Ng \tag {14}\end{align*}$$ where SF signifies transmission flow constraints and Qg indicates the generators’ reactive power.

While minimizing network losses, it’s crucial to maintain the balance between active and reactive power at each bus in the system. This balance is ensured by fulfilling specific equality constraints, which are achieved through the execution of a load flow analysis.

SECTION III. Proposed EGBA for Optimal TCSC Allocation in Power Systems

The GBA serves as a powerful metaheuristic approach adept at addressing intricate optimization problems through the amalgamation of population-based and gradient-based methodologies [57]. Within this framework, the navigation of search agents in the problem space is orchestrated by Newton’s method, a technique intricately woven into the structure of the GBA [58]. In an effort to refine and amplify this methodology, an innovative adaptation EGBA is introduced. The proposed EGBA distinguishes itself by seamlessly integrating a crossover strategy into the foundational GBA structure, thereby enriching the diversity exhibited by the generated search agents. This advanced design incorporates a crossover strategy, contributing to the creation of a more diverse and randomly configured population in subsequent iterations. Crucially, the essential elements of the GSP and LEP, inherent in the GBA, remain integral to the proposed EGBA. The deliberate retention of these core mechanisms ensures that the modified version upholds the fundamental principles of directing the search toward promising areas and circumventing entrapment in local optima.

A. Initialization

The commencement of the EGBA involves the initiation of a set of initial search solutions, each evolving with respect to its position along a path determined by gradients. The maximum number of iterations is denoted by $t_{Mx}$ . This process is articulated through the following expression:

$$\begin{equation*} X_{_{k}} =X_{_{Mn}} +rd_{0} \times \left ({{X_{_{Mx}} -X_{_{Mn}}}}\right); k=1:N_{x} \tag {15}\end{equation*}$$

Here, the notation $X_{k}$ represents each searching individual within the population, and “$X_{Mn}$ ” and “$X_{Mx}$ ” denote the lower and upper boundaries of the dimensions (Dim), respectively. rd₀ represents a randomly generated values within the boundary [0,1]. $N_{x}$ denotes the count of search individuals within the population. This initialization stage marks the inception of the EGBA algorithm, setting the groundwork for subsequent gradient-guided movements of the search individuals within the defined solution space.

B. GSP Exploration and Convergence Enhancement

The GSP is harnessed within the optimization framework to augment the exploration of the scanning universe and expedite the convergence towards the optimal solution. This method leverages gradient-based techniques to guide the search process effectively. The iterative refinement of findings in each cycle is accomplished through the application of the following mathematical expression:

$$\begin{align*} X_{k}^{\ast } =rd_{1} \left ({{(1-rd_{2})A_{k} +rd_{2} B_{k}}}\right)+(1-rd_{1})C_{k};k=1:N_{x} \tag {16}\end{align*}$$

In this equation:

$X^{\ast }_{k}$ and $X_{k}$ correspond to the updated and previous solution vectors associated with the solution position.

rd₁ and rd₂ represent two randomly generated values within the boundary [0,1].

A, B and C denote three newly assessed solutions, calculated as follows:

$$\begin{align*} A_{k} & \!=\!X_{Best} +rand\times \sigma _{1} \times \left ({{X_{R1} +X_{R2}}}\right)-GSP;k=1:N_{x} \tag {17}\\ B_{k} & =X_{k} +rand\times \sigma _{1} \times \left ({{X_{Best} +B_{k}}}\right)-GSP;k=1:N_{x} \tag {18}\\ C_{k} & =B_{k} +\sigma _{2} \times \left ({{A_{k} -B_{k}}}\right); k=1:N_{x} \tag {19}\\ GSP& =\sigma _{1} \times randn\left ({{\frac {2\times X_{k} \times \Delta X}{\varepsilon +yp_{j} -yq_{j}}}}\right) \tag {20}\end{align*}$$ Here:

$\sigma _{1}$ is a pivotal parameter subject to variations based on the sine function.

$\sigma _{2}$ represents a randomized parameter.

randn and rand denote a generated integer number and a uniformly distributed generated number within the boundary [0,1].

$X_{Best}$ symbolizes the optimal searching solution yielding the minimum objective score.

$X_{R1}$ and ${X} _{R2}$ illustrate two randomly chosen and distinct solutions.

This intricate process, involving the exploration and convergence-enhancing mechanisms of the GSP, demonstrates the sophistication and adaptability embedded within the EGBA method for optimization endeavors.

C. LEP for Avoiding Local Optima

The LEP is a crucial component employed within the optimization framework to steer the program away from local optima, thereby enhancing the algorithm’s adaptability. Following each iteration, the EGBA method refines its findings through the utilization of the ensuing mathematical formulation:

$$\begin{align*} X_{k}^{\ast } & \!\!=\!\!\left.{{\begin{cases} {X_{k}^{\ast } +D_{k} +L_{2} \left ({{X_{R1} -X_{R2}}}\right)} & {if ~rd_{3} \lt 0.5} \\ {X_{k}^{\ast } +D_{k} +\frac {L_{2}}{2}\left ({{X_{R1} -X_{R2}}}\right)} & {Else} \\ \end{cases}}}\right \} \!~if~rd_{4} \!\lt \!\Psi \tag {21}\\ D_{k} & =\phi _{1} \left ({{L_{1} X_{Best} -L_{2} X_{k}}}\right)+\sigma _{1} \phi _{2} \left ({{L_{3} A_{k} -B_{k}}}\right) \tag {22}\end{align*}$$

In this expression:

$\Psi$ signifies the likelihood of activating the LEM step.

rd₃ and rd₄ are random values within the range [0,1].

$\phi _{1}$ and $\phi _{2}$ represent two randomly generated values using a uniform distribution function within the interval [-1, 1].

$L_{1}$ , $L_{2}$ , and $L_{3}$ are three random numbers produced through the following equations:

$$\begin{align*} L_{1} & =\left ({{2\times \zeta \times rd_{5}}}\right)-\left ({{\zeta -1}}\right) \tag {23}\\ L_{2} & =\left ({{rd_{5} \times \zeta }}\right)-\left ({{\zeta -1}}\right) \tag {24}\\ L_{3} & =\left ({{rd_{5} \times \zeta }}\right)-\left ({{\zeta -1}}\right) \tag {25}\\ \zeta & =\begin{cases} \displaystyle 0 & {M_{1} \gt 0.5} \\ \displaystyle 1 & {Else} \end{cases} \tag {26}\end{align*}$$ Here, $M_{1}$ indicates a randomly generated number within the set [0,1]. $$\begin{align*} X_{_{k}} =\begin{cases} \displaystyle {X_{R3}} & if~M_{2} \lt 0.5 \\ \displaystyle {X_{Mn} +rd_{0} \times \left ({{X_{_{Mx}} -X_{_{Mn}}}}\right)} & {Else} \end{cases} \tag {27}\end{align*}$$

D. Crossover Strategy Incorporation

In this research endeavor, an advanced EGBA is introduced, featuring an augmented crossover operator seamlessly integrated with the original EGBA. This augmentation is designed to significantly enhance the diversity of the solutions generated by the algorithm. The application of the crossover operator is strategically orchestrated for each solution in every iteration, contingent upon a predefined crossover probability. The operational paradigm of the crossover operation unfolds as follows:

$$\begin{align*} X\_new_{k,j} & =\begin{cases} \displaystyle {X_{SR,j}} & if ~rd_{6} \lt CR_{p} \\ \displaystyle {X_{k,j}^{\ast }} & {Else} \end{cases} \\ k& =1:N_{x};j=1:Dim \tag {28}\end{align*}$$ where, X_new corresponds to the new generated solution position; $X^{\ast }_{k}$ stands for the upgraded one after either the GSP as activated in Eq. (15) or the LEP as activated in Eq. (21) while $X_{k}$ is the previous solution position. CR_p indicates the crossover probability while rd₆ is a randomly generated values within the boundary [0,1].

Based on that model, a new solution vector is synthesized by exchanging components between the current upgraded solution vector and a randomly selected one from the population. The activation of this crossover operation is governed by a condition based on the crossover probability (CR_p) which is set in this study to 25%. Consequently, a random value (rd₆) within the range [0,1] is generated where the crossover is employed when it is less than crossover probability, showcasing a judicious selection criterion. On the other side, any dimension of the new generated solution position (X_new) may exceed the permissible limit. Therefore, each dimension should be preserved if it is exceeded. The mathematical representation of the preservation mechanism is encapsulated in the following expression:

$$\begin{align*} & X\_new_{k,j} \!=\!\begin{cases} \displaystyle X_{Mn,j} & if~X\_new_{k,j} \lt X_{Mn,j} \\ \displaystyle X_{Mx,j} & if~X\_new_{k,j} \gt X_{Mx,j} \\ \displaystyle X\_new_{k,j}& Else \end{cases} \\ & \quad \qquad \qquad \qquad \qquad k\!=\!1:N_{x};j\!=\!1:Dim \tag {29}\end{align*}$$

E. Objective Evaluation and Constraints Handling of TCSC Allocation in Power Systems

In addressing the TCSC allocation problem, due consideration is accorded to both equality and inequality constraints. To satisfy the equality criteria inherent in load flow balancing equations, the Newton-Raphson (NR) technique is employed. This technique is particularly significant in power network engineering as it upholds power balancing requisites, portraying the system’s steady state. Consequently, the NR approach serves as a crucial tool for illustrating three-phase circuits, prominently utilized by the MATPOWER [59] framework.

Within the scope of operational constraints, two distinct categories emerge, namely decision variables and dependent variable constraints. Decision variables persist in adhering to their defined limits, with any overruns triggering a random regeneration within the specified bounds, thereby ensuring compliance with the constraints articulated in Eqs. (6)–​(11). This preservation mechanism, delineated in Eq. (29), plays a pivotal role in maintaining the integrity of the decision variables.

Moreover, the objective function, encompassing the second category of constraints related to dependent variables, is designed to extend and penalize violations. Consequently, if a solution violates any of the corresponding constraints, it faces rejection in the subsequent iteration. The objective function (OF_t), along with the overall network losses (OJ) defined in Eq. (5), can be expressed as follows:

$$\begin{align*} OF_{_{t}} = OJ+\lambda _{1} \sum \limits _{L=1}^{NLd} {\Delta V_{L} ^{2}} +\lambda _{2} \sum \limits _{g=1}^{Ng} {\Delta Q_{g}^{2}} +\lambda _{3} \sum \limits _{l=1}^{Nlines} {\Delta SF_{l}^{2}} \tag {30}\end{align*}$$ where, $\Delta V_{L}$ , $\Delta Q_{g}$ , and $\Delta$ SF_l are characterized by: $$\begin{align*} \Delta V_{L} & =\begin{cases} \displaystyle {V}_{L}^{min} -{V}_{L} & if~V_{L} \lt {V}_{L}^{min} \\ \displaystyle {V}_{L}^{max} -{V}_{L}& if~V_{L} \gt {V}_{L}^{max} \end{cases} \tag {31}\\ \Delta Q_{g} & =\begin{cases} \displaystyle {Q}_{g}^{min} -{Q}_{g} & if ~Q_{g} \lt {Q}_{g}^{min} \\ \displaystyle {Q}_{g}^{max} -{Q}_{g} & if~Q_{g} \gt {Q}_{g}^{max} \end{cases} \tag {32}\\ \Delta SF_{l}& ={SF}_{l}^{max} -{SF}_{l} \quad if ~SF_{l} \gt {SF}_{l}^{max} \tag {33}\end{align*}$$

Additionally, penalty factors, denoted as $\lambda _{1}$ , $\lambda _{2}$ , and $\lambda _{3}$ , are introduced to penalize violations in load voltages, reactive outputs from generators, and line power flows, respectively. The depicted graphical representation in Figure 2 illustrates the principal stages comprising the proposed EGBA, shedding light on the key components and sequential progression integral to the methodology.

FIGURE 2.

EGBO flowchart.

SECTION IV. Simulation Results

In the following paragraphs, we investigate the use of the novel EGBA in two different cases. Simulations based on test functions are firstly engaged, focusing on the CEC 2017 evaluations in particular. Such simulated efforts involve a thorough contrast, comparing the EGBA’s performance versus several recently developed metaheuristic methods. Afterward, we extend the simulations to tackle the complexities of TCSC allocating problems in the context of power systems. the applications are implemented with focus on two IEEE standard electrical networks, with 30 and 57 buses, respectively, and assess the effectiveness of the EGBA in terms of optimizing the distribution of TCSC devices.

A. Examination of Application Performance Utilizing CEC 2017 Benchmarks

The assessment of the effectiveness of optimization strategies necessitates a rigorous validation of their performance, and benchmark functions serve as integral tools for this purpose. This section delves into the comprehensive evaluation of the proposed EGBA and GBA by leveraging the CEC 2017 competition as a standardized benchmark [60]. This competition features an array of routines specifically crafted to appraise various attributes, encompassing diverse functions with unique characteristics. Across a spectrum of 28 benchmarking functions, our evaluation employs a dimensionality of 30 control variables, each bounded within the range of [-100, 100]. The scrutiny of the proposed EGBA extends to a comparative analysis with the conventional GBA, specifically considering the CEC 2017 benchmarks. The visual representation of the convergence characteristics of both EGBA and GBA is elucidated in figure 3. To establish a comprehensive benchmark, we extend our analysis to include a variety of contemporary optimization techniques as manifested in table 1. These encompass the Aquilla Optimization Technique (AOT) [61], GTT [62], Red Kite Algorithm (RKA) [63], and SAO [64], Slime Mould Optimizer (SMO) [65] and Dwarf mongoose optimization (DMO) [66]. The specific configurations for each method under comparison are meticulously outlined in Table 3. In order to ensure a robust and comprehensive evaluation, a total of fifty independent operations are executed for each technique across diverse benchmarks, thereby mitigating the impact of inherent randomness. The tabulated data in Table 3 provides a comprehensive array of statistical metrics, encompassing the best, mean, worst, and standard deviation (Std) outcomes for the various techniques being compared. Notably, upon careful examination of Table 3, it becomes evident that the proposed EGBA technique demonstrates a notable superiority in efficacy, consistently registering the most favorable statistical indices across a significant majority of benchmark functions. This is highlighted by the fact that the proposed EGBA exhibits an impressive improvement percentage of 90.17% when compared to AOT. Similarly, in contrast to SAO, the EGBA method displays a substantial improvement percentage of 89.29%. Furthermore, it outperforms RKA with a noteworthy improvement percentage of 83.04%, while showcasing a commendable improvement percentage of 80.36% when compared to GBA and GTT. Finally, it demonstrates a substantial improvement percentage of 62.5% when compared to the original GBA. These results underscore the robustness and efficacy of the proposed EGBA technique, positioning it as a highly competitive and effective optimization approach when compared to a diverse set of contemporary methods across varied benchmark functions.

TABLE 1 Parameters of the Compared Algorithms

TABLE 2 Statistical Indices of EGBA, GBA, GTT, DMO, SAO, RKA and AOT for CEC 2017 Problems

TABLE 3 Outcomes of the Proposed EGBA With Respect to the Original GBA and Recent Approaches for TCSC Device Allocations With Respect to Scenario 1

FIGURE 3.

Converging trends of EGBA and GBA for CEC 2017 benchmarks.

B. Proposed EGBA for Fixing TCSC Devices in IEEE 30-Bus Transmission Network

The IEEE standard 30-bus system illustrated in Fig. 4 [68] is employed in this section to manage the best TCSC allocations. This system has four parts which are 30 nodes, 41 lines, 9 compensators and 4 transformers [69]. In this system, the tap positions are 0.90 p.u. and the maximum generating voltage is 1.10 p.u. Besides, the generator bus has limits of 1.10 and 0.90 p.u, whilst the voltage limitations for the load buses are 1.05 and 0.95 p.u. The original GBA and several more modern algorithms, such as AEO, SAO, GWO, AOT and DMO, are compared with the proposed EGBA. Both implemented algorithms are run 20 times independently, with 300 iterations and 50 searching individuals for each algorithm. Three distinct situations are examined, taking into consideration one, two, and three TCSC devices, depending on the number of candidate devices provided.

FIGURE 4.

Schematic representation of the IEEE 30-bus system [70].

1) Scenario 1

The proposed EGBA is used to optimize the allocation of a single TCSC device in order to achieve lowest possible power losses. The acquired outcomes are contrasted with the original GBA, AEO, SAO, GWO, AOT and DMO. In addition to this, Table 3 displays the best control variables, which are the Var source’s injection power, the output power and generators’ voltage, and the tap value as well as the location and size of the TCSC device. As illustrated from this table, the proposed EGBA generates the least amount of power loss of 2.8067 MW. Additionally, with a -49.98 % size reduction from the installed line reactance, the transmission line (28–27) is determined to be the optimal placement for the TCSC in the proposed EGBA.

When comparing the proposed EGBA to the initial scenario, power losses were reduced by 51.88%. Moreover, the proposed EGBA achieves a noteworthy decrease percentage of 0.32% in the power losses when comparing its results with those of the original GBA. In addition, the proposed EGBA achieves a decrease percentage of over 1.33% and 0.53% in comparison to the findings achieved by the AEO and SAO. In comparison to the GWO, AOT and DMO, the proposed EGBA reduces power losses by 8.13%, 6.53%. and 7.56%, respectively.

Besides, Figure 5 displays the convergence curves for the proposed EGBA and the original GBA. The numerical findings unequivocally demonstrate that the proposed EGBA yields considerable economic advantages and outperforms the original GBA in searching. The proposed EGBA converged in less iterations, according to the convergence curves manifested in Figure 5. In order to statistically assess the methodologies that were compared, figure 6 displays the outcomes associated with the proposed EGBA and the original GBA for scenario 1. The corresponding statistical results of the calculated Losses (MW) for this scenario are shown in Table 4. Upon aggregating the lowest indexes from the acquired target values, it is evident that the suggested EGBA offers excellent performance. The original GBA, AEO, SAO, GWO, AOT and DMO get the mean of acquired losses of 2.8331, 3.007, 2.930, 3.472, 3.080, and 3.065 MW, respectively, whereas the mean losses, found in the proposed EGBA, is 2.8092 MW which is lower than the mentioned algorithms. Compared to the outcomes attained by the original GBA, AEO, SAO, GWO, AOT and DMO, the proposed EGBA attains reductions in improvement of the acquired mean of 0.85%, 7.04%, 4.30%, 23.59%, and 9.11%, respectively. According to the worst achieved losses, the proposed EGBA records the lowest losses of 3.038 MW; in contrast, the losses received by GBA, AEO, SAO, GWO, AOT and DMO are 2.8452, 3.180, 3.188, 3.849, 3.172, and 3.109 MW, respectively. The proposed EGBA provides improvement reductions of 1.04%, 12.93%, 13.21%, 36.69%, 12.65% and 10.41%, respectively, compared to the findings obtained by the original GBA, AEO, SAO, GWO, AOT and DMO.

TABLE 4 Statistical Analysis of the Proposed EGBA With Regard to the Original GBA and Recent Approaches for Scenario 1

FIGURE 5.

Convergence curves for the proposed EGBA versus the original GBA with respect to scenario 1.

FIGURE 6.

Twenty runs for outcomes of proposed EGBA with regard to the original GBA for scenario 1.

2) Scenario 2

The proposed EGBA is used to optimize the allocation of a single TCSC device in order to achieve lowest possible power losses. The acquired outcomes are contrasted with the original GBA, AEO, SAO, GWO, AOT and DMO. In addition to this, Table 5 displays the best control variables, which are the Var source’s injection power, the output power and generators’ voltage, and the tap value as well as the location and size of the TCSC devices. From this table, the proposed EGBA generates the least amount of power loss of 2.7799 MW. Additionally, the transmission lines (28–27) and (2–5) are chosen by the planned EGBA with compensation values of 49.99% and 25.55 percent subtraction from the installed line reactance.

TABLE 5 Outcomes of the EGBA Versus the Original GBA and Recent Approaches for TCSC Device Allocations With Respect to Scenario 2

When comparing the proposed EGBA to the initial scenario, power losses were reduced by 47.66 %. Moreover, the proposed EGBA achieves a noteworthy decrease percentage of 2.12% in the power losses when comparing its results with those of the original GBA. Also, EGBA achieves a decrease percentage of over 3.13% and 1.44 % in comparison to AEO and SAO. In comparison to the GWO, AOT and DMO, the proposed EGBA reduces power losses by 16.08 %, 7.74 %. and 8.14 %, respectively. Besides, Figure 7 displays the convergence curves for the proposed EGBA and the original GBA. The numerical findings unequivocally demonstrate that the proposed EGBA yields considerable economic advantages and outperforms the original GBA in searching. The proposed EGBA converged in less iterations, according to the convergence curves manifested in Figure 7.

FIGURE 7.

Convergence curves for the proposed EGBA versus the original GBA with respect to scenario 2.

In this regard, figure 8 displays the outcomes associated with the proposed EGBA and the original GBA for scenario 2. The corresponding statistical results of the calculated Losses (MW) for this scenario are shown in Table 6. It is clear that the proposed EGBA provides high performance when it aggregates the smallest objective indices. The original GBA, AEO, SAO, GWO, AOT and DMO get the mean of acquired losses of 2.8813, 2.988, 2.938, 3.501, 3.105, and 3.063 MW, respectively, whereas the mean losses, found in the proposed EGBA, is 2.7974 MW which is lower than the mentioned algorithms. Compared to the outcomes attained by the original GBA, AEO, SAO, GWO, AOT and DMO, the proposed EGBA attains reductions in improvement of the acquired mean of 3.00%, 6.81%, 5.03%, 25.15%, 11.00%, and 9.49%, respectively. According to the worst achieved losses, the proposed EGBA finds the lowest losses of 2.8234 MW; in contrast, the losses received by GBA, AEO, SAO, GWO, AOT and DMO are 3.1302, 3.214, 3.168, 4.18, 3.181 and 3.119MW, respectively. The proposed EGBA provides improvement reductions of 10.87%, 13.83%, 12.20%, 48.05%, 12.66% and 10.44%, respectively, compared to GBA, AEO, SAO, GWO, AOT and DMO.

TABLE 6 Statistical Analysis of the Proposed EGBA With Regard to the Original GBA and Recent Approaches for Scenario 2

FIGURE 8.

Twenty runs for outcomes of proposed EGBA with regard to the original GBA for scenario 2.

3) Scenario 3

In this scenario, the proposed EGBA is used to optimize the allocation of a single TCSC device in order to achieve lowest possible power losses. The acquired outcomes are contrasted with the original GBA, AEO, SAO, GWO, AOT and DMO where Table 7 displays the regarding control variables. As illustrated from this table, the proposed EGBA generates the least amount of power loss of 2.7596 MW. Additionally, the transmission lines (4-12) and (28-27), and (2–5) are chosen by the planned EGBA with compensation values of 49.8449.99% and -49.9425.55 and -25.20 percent from the installed line reactance, respectively. When comparing the proposed EGBA to the initial scenario, power losses were reduced by 47.32 %. Moreover, the proposed EGBA achieves a noteworthy decrease percentage of 1.45 % in the power losses when comparing its results with those of the original GBA. In addition, the proposed EGBA achieves a decrease percentage of over 4.36% and 2.22% in comparison to the findings achieved by the AEO and SAO. In comparison to the GWO, AOT and DMO, the proposed EGBA reduces power losses by 15.49%, 7.59%. and 9.33%, respectively. Besides, Figure 9 displays the convergence curves for the proposed EGBA and the original GBA. The numerical findings unequivocally demonstrate that the proposed EGBA yields considerable economic advantages and outperforms the original GBA in searching. The proposed EGBA converged in less iterations, according to the convergence curves manifested in Figure 9.

TABLE 7 Outcomes of the Proposed EGBA With Respect to the Original GBA and Recent Approaches for TCSC Device Allocations With Respect to Scenario 3

FIGURE 9.

Convergence curves for the proposed EGBA versus the original GBA with respect to scenario 3.

Moreover, figure 10 displays the outcomes associated with the proposed EGBA and the original GBA for scenario 3. The corresponding statistical results of the calculated Losses (MW) for this scenario are shown in Table 8. It is clear that the proposed EGBA provides high performance when it aggregates the fewest objective indices. The original GBA, AEO, SAO, GWO, AOT and DMO get the mean of acquired losses of 2.8194, 3.010, 2.918, 3.468, 3.079, and 3.071 MW, respectively, whereas the mean losses, found in the proposed EGBA, is 2.7826 MW which is lower than the mentioned algorithms. Compared to the outcomes attained by the original GBA, AEO, SAO, GWO, AOT and DMO, the proposed EGBA attains reductions in improvement of the acquired mean of 1.32%, 8.17%, 4.86%, 24.63%, 10.65% and 10.36%, respectively. According to the worst achieved losses, the proposed EGBA finds the lowest losses of 2.8166 MW; in contrast, the losses received by GBA, AEO, SAO, GWO, AOT and DMO are 2.8559, 3.536, 3.189, 3.856, 3.198 and 3.143 MW, respectively. The proposed EGBA provides improvement reductions of 1.39%, 25.54%, 13.22%, 36.90%, 13.54% and 11.59%, respectively, compared to GBA, AEO, SAO, GWO, AOT and DMO.

TABLE 8 Statistical Analysis of the Proposed EGBA With Regard to the Original GBA and Recent Approaches for Scenario 3

FIGURE 10.

Twenty runs for outcomes of proposed EGBA with regard to the original GBA for scenario 3.

The secure operation of the IEEE 30 power networks depends on the entire load buses’ voltage profile in comparison to the first scenario, based on the candidate TCSC devices. In this regard, the proposed EGBA is used to compute the voltage magnitude of the buses, and the results are displayed in Figure 11 in comparison to the initial case. Also, the regarding improvement is drawn in the secondary vertical axis.

FIGURE 11.

Voltage profile and improvements regarding the three TCSC installations (Case 3) against the initial case.

Looking at the numerical values in this figure, the initial cases range from 0.9012 to 1.05, with an average initial value of approximately 1.00. After the additions of the three TCSC devices, represented by “Case 3,” the values generally increase, with the mean value rising to approximately 1.088. This indicates an average improvement of around 8.8% across all cases. While some individual cases show smaller improvements around 4.8%, others demonstrate more substantial enhancements, reaching up to 19.0%. However, the mean improvement of 8.8% suggests that, on average, the additions of the three TCSC devices are effective in positively impacting the outcomes. This analysis underscores the overall effectiveness of the TCSC devices installations based on the proposed EGBA in improving the voltage profile in the system.

C. Enhancing Security Margin Alongside Power Loss Minimization for the IEEE 30-Bus Transmission Network

Power system congestion has a substantial impact on the fluctuation of voltage and current, which can cause unintentional variations in the distribution of power throughout the network. A security margin enhancement is required to be included as a critical aim due to this issue. The security margin is a measure of the system’s capacity to accept variations in power flow while maintaining safety thresholds [49]. It basically gauges the power system’s ability to endure the loss of any part, such a generator or a transmission line, without compromising overall system security. The Security Margin Index (SMI$_{\mathrm {L}}$ ) for each individual transmission line (L) can be mathematically modelled as follows [49]:

$$\begin{equation*} SMI_{L} =\frac {{SF}_{L}^{max} -\left |{{ {SF}_{L}}}\right |}{{SF}_{L}^{max} }, L=1,2,\ldots. N_{lines} \tag {34}\end{equation*}$$ where SF_L signifies actual transmission flow and ${SF}_{L}^{max}$ is the rated transmission flow constraint in the line (L).

Therefore, the Overall Security Margin (OSM) can be formulated as the summation of the Security Margin Index (SMI$_{\mathrm {L}}$ ) of all transmission lines as follows:

$$\begin{equation*} OSM=\sum \limits _{L=1}^{N_{lines}} {SMI_{L}} \tag {35}\end{equation*}$$

Based on that, a higher security margin indicates a greater ability of the system to tolerate disturbances and component failures without violating safety constraints. The objective to enhance the security margin involves optimizing the power flow such that the system remains robust against potential disruptions. To optimize the minimization function of power losses and enhance the security margin, the following function is considered:

$$\begin{equation*} OF=\omega _{1} \times \frac {P_{Loss}}{P_{Loss,max}}+\omega _{2} \times \frac {OSM_{max}}{OSM} \tag {36}\end{equation*}$$ where, $\omega _{1}$ and $\omega _{2}$ are the weighting factors while $P_{Loss,max}$ and OSM_max are two set values regarding the losses and the overall security margin objectives. This function seeks to minimize the combined objective of the normalized losses and the reciprocal of the security margin. Thus, it maximizes the security margin by minimizing the potential deviations and ensuring the system can handle component losses effectively. Also, the choice of weights reflects the relative significance assigned to each objective, taking into account the trade-offs and priorities among them. Proper weight selection involves understanding the problem domain, stakeholder preferences, and system requirements. To implement this, two different scenarios are addressed. At first (Scenario 4), only the minimization of the normalized reciprocal of the OSM is considered by setting $\omega _{1} = 0$ and $\omega _{2}=1$ . Second (Scenario 5), both normalized functions are considered of equal importance by setting $\omega _{1} = 1$ and $\omega _{2}=1$ .

In both scenarios, the proposed EGBA and the original GBA are implemented. Table 9 tabulates their obtained outcomes for TCSC device allocations regarding scenarios 4 and 5 where Figs. 12 and 13 display their convergence properties. As shown, the results demonstrate that the proposed EGBA derives better performance than the GBA. In Scenario 4, the proposed EGBA successfully maximizes the OSM from 26.63 to 29.91 with 10.96% improvement while the GBA increases it to 29.85. Similar results are attained in Scenario 5, the proposed EGBA simultaneously maximizes the OSM and minimizes the power losses to 29.55 and 2.817 MW while the GBA achieves lower OSM of 29.48 and higher losses of 2.84 MW. Alternatively, Fig. 14 illustrates the values of the SMI of each line regarding Scenarios 4 and 5 utilizing the proposed EGBA versus the initial scenario.

TABLE 9 Outcomes of the Proposed EGBA With Respect to the Original GBA for TCSC Device Allocations Regarding Scenarios 4 and 5

FIGURE 12.

Convergence curves for the proposed EGBA versus the original GBA with respect to scenario 4.

FIGURE 13.

Convergence curves for the proposed EGBA versus the original GBA with respect to scenario 5.

FIGURE 14.

SMI regarding Scenarios 4 and 5 utilizing the proposed EGBA versus the initial scenario.

As shown, the EGBA provides substantial improvements over the initial scenario. EGBA in both scenarios 4 and 5 demonstrates enhanced performance in the security margin for the IEEE 30-bus transmission network. EGBA-Scenario 4 and EGBA-Scenario 5 show significant improvements over the initial scenario in most instances. For example, the SMI for the first line improves from 55.21% in the initial scenario to 65.52% in EGBA-Scenario 4 and further to 78.43% in EGBA-Scenario 5. This trend is consistent across many lines, indicating that the EGBA algorithm enhancements are effective. These improvements highlight the efficacy of the proposed modifications in optimizing the power system performance.

D. Proposed EGBA for Fixing TCSC Devices in IEEE 57-Bus Transmission Network

This part uses the standard IEEE 57-bus transmission network that is shown in Figure 15. There are 57 nodes, 7 generators, 80 lines, 3 capacitive sources on buses, and 17 on-load tap changing transformers in the aforementioned system. The system description is taken from [71]. In order to lower the power losses, the three situations under study are examined with consideration for one, two, and three TCSC devices. Where Table 10 displays their determined control variables, the proposed EGBA and the original GBA are implemented. The results demonstrate that the proposed EGBA reduces power losses by 9.2209663, 9.0879067, and 9.0300855 MW for the scenarios 6-8, while the original GBA reduces power losses by 9.3262325, 11.277533, and 9.1171028 MW, respectively. Alternatively, Fig. 16 indicates the converging features, where iterations to identify and create the best individual are utilized. The numerical findings unequivocally demonstrate that the proposed EGBA yields considerable economic advantages and outperforms the original GBA in searching. The proposed EGBA converged in less iterations, according to the convergence curves manifested in Fig. 16.

TABLE 10 Outcomes of the Proposed EGBA and the Original GBA for TCSC Device Allocations With Respect to Scenarios 6-8

FIGURE 15.

Graphic representation for IEEE 57-bus power system [72].

FIGURE 16.

Convergence curves for the proposed EGBA versus the original GBA with respect to scenarios 6-8.

To evaluate the overall efficiency of the proposed EGBA in addressing the optimal allocation of TCSC devices in the IEEE 57-bus transmission network, the distribution of the objective function across 20 runs is visually depicted in Figure 17 and summarized in Table 11 for scenarios 6-8. From this data, the following conclusions can be inferred:

• In the sixth scenario, the suggested EGBA yields significant improvements over GBA, with a reduction in the best outcome from 9.326 MW to 9.221 MW, a decrease in mean losses from 9.910 MW to 9.421 MW, and a drop in worst losses from 10.408 MW to 9.754 MW. This indicates improvements of approximately 0.1129%, 5.89%, and 6.68%, respectively, showcasing the EGBA’s effectiveness in minimizing losses.

• In the seventh scenario, EGBA demonstrates superiority over GBA across all metrics. It achieves reductions in best, mean, and worst outcomes from 11.278 MW to 9.088 MW, 10.019 MW to 9.423 MW, and 10.494 MW to 10.277 MW, respectively. These improvements correspond to approximately 19.63%, 6.15%, and 2.07%, highlighting the EGBA’s consistent ability to optimize TCSC device allocation and reduce losses.

• In the eighth scenario, the proposed EGBA finds the least losses of 9.6106 MW according to the mean acquired losses, whereas the original GBA gets losses of 10.1027 MW.

TABLE 11 Statistical Analysis of the Proposed EGBA With Regard to the Original GBA for Scenarios 6-8

FIGURE 17.

Twenty runs for outcomes of proposed EGBA with regard to the original GBA for scenarios 6-8.

Overall, the numerical results demonstrate the significant improvements achieved by EGBA over GBA in minimizing losses across different scenarios in the IEEE 57-bus transmission network. The improvements range from approximately 0.11% to 19.63%, underscoring the effectiveness of EGBA in enhancing the allocation of TCSC devices and optimizing power grid performance.

The IEEE 57 power networks’ secure operation relies on the voltage profile of all load buses, with the proposed EGBA used to compute bus voltage magnitudes, as shown in Figure 18, illustrating improvements compared to the initial scenario. The initial voltage values range from 0.9359 to 1.0598, averaging around 1.00, while after adding three TCSC devices (Case 3), values generally increase, with the mean rising to approximately 1.0651, indicating an average improvement of about 8.8%. Although some cases show smaller enhancements around 1.1%, others demonstrate more substantial improvements of up to 11.7%. Overall, the mean improvement of 7.22% suggests the effectiveness of TCSC device installations in enhancing the system’s voltage profile, as determined by the proposed EGBA.

FIGURE 18.

Voltage profile and improvements regarding scenario 8 against the initial case.

In order to address the maximization of the security margin for the IEEE 57-bus system, two different scenarios are addressed. Only the minimization of the normalized reciprocal of the OSM is considered in Scenario 9 while both normalized functions in Eq. (36) are considered of equal importance in Scenario 10.

In both scenarios, the proposed EGBA and the original GBA are implemented. Table 12 tabulates their obtained outcomes for TCSC device allocations regarding scenarios 9 and 10 where Figs. 19 and 20 display their convergence properties. As shown, the results demonstrate that the proposed EGBA derives better performance than the GBA. In Scenario 9, the proposed EGBA successfully maximizes the OSM from 32.15 to 37.4 with 16.33% improvement while the GBA increases it to 35.51 with only 10.45% improvement. Similar results are attained in Scenario 10, the proposed EGBA simultaneously maximizes the OSM and minimizes the power losses to 35.5 and 12.17 MW while the GBA achieves lower OSM of 35 and higher losses of 15.81 MW. Fig. 21 illustrates the thermal limit (MVA) and the power flows of each line regarding Scenarios 9 and 10 utilizing the proposed EGBA versus the initial scenario. As shown, both EGBA-Scenario 9 and Scenario 10 exhibit significant overall improvements in the security margins compared to the initial scenario, indicating the effectiveness of the EGBA in enhancing system security. The EGBA provides significant ability in healing several over loadings in lines numbers (1, 2, 3, 6, 14, 15, 16, 17, 23, 25, 26, 27, 28, 57, 58 and 72) in the initial scenario. EGBA in both scenarios 9 and 10 demonstrates enhanced performance in the security margin for the IEEE 57-bus transmission network.

TABLE 12 Outcomes of the Proposed EGBA With Respect to the Original GBA for TCSC Device Allocations Regarding Scenarios 9 and 10

FIGURE 19.

Convergence curves for the proposed EGBA versus the original GBA with respect to scenario 9.

FIGURE 20.

Convergence curves for the proposed EGBA versus the original GBA with respect to scenario 10.

FIGURE 21.

MVA thermal limit and flows of each line regarding Scenarios 9 and 10 utilizing the proposed EGBA versus the initial scenario.

SECTION V. Conclusion

This work proposes a novel EGBA to handle two distinct IEEE power systems, with 30 and 57 buses, respectively, and a variety of TCSC devices. These systems are considered to be challenging engineering issues with limited optimal solutions. Important elements of the algorithm, such as the Local Escaping Process (LEP) and Gradient Search Process (GSP), direct the exploration phase and avoid an early convergence to less-than-ideal solutions. Furthermore, a novel feature to the EGBA is the crossover operator which allows TCSC configurations to be exchanged between solutions, and accordingly increasing solution diversity and possibly disclosing new optimal allocations. The applicability of the proposed method is demonstrated by applying it to the CEC 2017 single objective optimisation functions, and its robustness is examined through statistical evaluation and convergence results. Additionally, the proposed EGBA is contrasted with the approaches of AEA, AOT, GTT, RKA, SAO, and SMO. It has been discovered that the proposed EGBA has an exceptional ability to avoid becoming stuck in local optima once it has been implemented on unimodal and multimodal functions. The proposed EGBA is more reliable in reaching the optimum values over several runs, according to the statistical analysis of the benchmark functions. The proposed EGBA effectively reduces power losses compared to several techniques. It also converges faster than the original GBA in reducing power losses. Besides minimizing losses, the EGBA aims to enhance the security margin of transmission lines to optimize power flow. For the IEEE 30 bus system, the EGBA increased the OSM with a 10.96% improvement while it raised the OSM with a 16.33% improvement for the IEEE 57 bus system. Additionally, the EGBA significantly mitigates several line overloads.

ACKNOWLEDGMENT

The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2024/01/29312).