Techno-Economic Assessment of Fixed and Variable Reactive Power Injection Using Thyristor-Switched Capacitors in Distribution Networks

Abstract

This paper presents a hybrid optimization framework for solving the optimal reactive power compensation problem in medium-voltage smart distribution networks. Leveraging Julia’s computational environment, the proposed method combines the global search capabilities of the Chu & Beasley genetic algorithm (CBGA) with the local refinement efficiency of the interior-point optimizer (IPOPT). The objective is to minimize the annualized operating costs by reducing active power losses while considering the investment and operating costs associated with thyristor-switched capacitors (TSCs). A key contribution of this work is the comparative assessment of fixed and time-varying reactive power injection strategies. Simulation results on the IEEE 33- and 69-bus test feeders demonstrate that the proposed CBGA-IPOPT framework achieves annualized cost reductions of up to 11.22% and 12.58% (respectively) under fixed injection conditions. With variable injection, cost savings increase to 12.43% and 14.08%. A time-domain analysis confirms improved voltage regulation, substation reactive demand reductions exceeding 500 kvar, and peak loss reductions of up to 32% compared to the uncompensated case. Benchmarking shows that the hybrid framework not only consistently outperforms state-of-the-art metaheuristics (the sine-cosine algorithm, the particle swarm optimizer, the black widow optimizer, and the artificial hummingbird algorithm) in terms of solution quality but also demonstrates high solution repeatability across multiple runs, underscoring its robustness. The proposed method is directly applicable to real-world distribution systems, offering a scalable and cost-effective solution for reactive power planning in smart grids.

1. Introduction

1.1. Background and Motivation

Reactive power compensation in medium-voltage distribution networks (MVDNs) is a well-established and essential strategy employed by electric utilities to enhance the operational efficiency and reliability of power distribution systems [1]. Its primary objectives are to minimize technical losses and maintain voltage levels within permissible limits, particularly in feeders with resistive–inductive load profiles [2]. This is typically achieved through the strategic deployment of reactive power sources—primarily capacitor-based devices—that inject reactive power into the network, thereby improving the power factor and voltage regulation [3].

Historically, fixed or stepwise-switched capacitor banks have been deployed along distribution feeders to meet reactive power demands [4]. Although effective under certain conditions, these static compensation schemes often fall short in addressing the temporal variability of active and reactive loads throughout the day. This is because they rely on pre-determined capacitor switching schedules or fixed reactive power injections that do not adapt to real-time load fluctuations. As a result, they inject a constant amount of reactive power regardless of the actual load level [5]. This lack of adaptability can lead to undercompensation during peak demand periods and overcompensation during low-load periods, causing voltage deviations and inefficient reactive power utilization [6].

To address these challenges, modern distribution networks are increasingly adopting flexible and adaptive reactive compensation technologies. Flexible AC transmission system (FACTS) devices—such as static VAR compensators (SVCs), static synchronous compensators (STATCOMs), and thyristor-switched capacitors (TSCs)—offer enhanced dynamic reactive power control as well as voltage support [7,8]. Among these alternatives, TSCs are gaining traction as a practical and cost-effective solution for medium-voltage applications, given their simplified structure, rapid switching capabilities, and modular scalability [2,9].

In comparison with SVCs and STATCOMs, TSCs provide several operational and economic benefits. They eliminate the need for sophisticated power electronics and continuous modulation, which leads to reduced capital costs and lower maintenance requirements [10]. TSCs also offer higher reliability and energy efficiency, especially in applications where stepwise compensation is sufficient to respond to system dynamics. Moreover, their robust construction allows for seamless integration with existing fixed-capacitor banks through controlled switching [11].

1.2. Operating Principles and Configuration of TSCs

Figure 1 illustrates the single-phase equivalent circuit of a TSC, a well-established technology for stepwise reactive power compensation in MVDNs. This configuration includes a capacitor bank connected in series with a pair of thyristors ($T_1$ and $T_2$) arranged in anti-parallel, thus allowing for controlled switching [12]. A protection inductor is also integrated to limit the inrush current and suppress transients during switching events, thereby safeguarding the power electronic components.

This arrangement provides a practical and reliable solution for reactive power regulation in response to time-varying load conditions [13]. Compared to continuously controllable devices such as STATCOMs, TSCs offer a simpler structure and incur lower capital costs, which makes them particularly attractive for stepped compensation schemes in radial distribution feeders. The use of thyristor-based switching eliminates the need for mechanical contactors, thereby improving the device’s switching response and extending its operational lifespan [14]. These characteristics align with smart grid objectives, providing digitally controllable and modular reactive power support for voltage stability and power factor correction. In the context of modern smart distribution networks, TSCs serve as an intermediate solution between fixed capacitors and fully dynamic FACTS [15]. Their simple control logic and scalability make them well-suited for hybrid optimization methodologies—such as the one proposed in this work—that seek to minimize operating costs while ensuring compliance with technical constraints under realistic and time-varying loading conditions.

1.3. Problem Statement and Research Gap

While fixed-capacitor banks offer simplicity and low costs, TSCs represent a more flexible and technically advanced alternative for reactive power compensation in modern MVDNs [2]. Despite their advantages over traditional solutions such as SVCs and D-STATCOMs, the optimized integration of TSCs into MVDNs remains underexplored in the specialized literature [9].

Several recent studies have addressed the optimal placement and sizing of FACTS in both transmission and distribution systems. For example, the work by [16] examined FACTS deployment in an IEEE 39-bus system with electric vehicle penetration, using particle swarm optimization (PSO) to improve voltage profiles and reduce operating costs. Similarly, ref. [17] applied the seeker optimization algorithm for the placement of thyristor-controlled series capacitors (TCSCs), achieving enhanced voltage regulation.

In the context of distribution networks, hybrid strategies combining fuzzy logic and ant colony algorithms were proposed by [18], while heuristic approaches based on voltage and loss indices were explored in [19]. Analytical–heuristic hybrid frameworks have also been introduced in [20,21] to address multi-objective FACTS allocation. However, many of these models are limited to maximum demand scenarios, which restricts their adaptability to real-world conditions with variable load profiles.

To address these limitations, the authors of [2] proposed a master–slave framework for optimal FACTS allocation in MVDNs, which includes unified power flow controllers (UPFCs), TCSCs, and SVCs. The methodology integrates the black widow optimizer (BWO) with a hybrid discrete–continuous encoding and a successive approximations power flow model. Simulations on IEEE 33-, 69-, and 85-bus systems revealed that SVCs offer the best trade-off between performance and cost.

Expanding on the role of TSCs, the study by [9] introduced an approach based on the artificial hummingbird algorithm (AHA) for TSC siting and sizing, targeting annualized operating costs minimization. The method, validated on 33- and 69-bus systems, outperformed several metaheuristics including, the sine-cosine algorithm (SCA), the Chu & Beasley genetic algorithm (CBGA), PSO, and BWO.

Additionally, the authors of [22] developed a hybrid methodology that combines the SCA for identifying candidate TSC locations and sizes with the interior-point optimizer (IPOPT) for solving the optimal power flow. Applied to the IEEE 33-bus test system, the proposed SCA-IPOPT approach achieved a 12.43% reduction in operating costs under variable reactive power injection conditions, outperforming benchmark metaheuristic methods.

1.4. Contributions and Scope

In contrast to prior studies, this paper introduces a novel and comprehensive methodology for the optimal siting and sizing of TSCs in MVDNs. The primary objective is to minimize the annualized operating costs associated with energy losses while considering the capital investments required for TSC deployment. This work also provides a comparative analysis of two compensation strategies (i.e., fixed and variable reactive power injection) to evaluate the impact of daily reactive demand fluctuations on both technical and economic performance.

The proposed methodology leverages a hybrid optimization approach that combines the global search efficiency of the CBGA with the local refinement capabilities of the IPOPT. This hybrid scheme ensures robust exploration and rapid convergence, allowing for the precise handling of the mixed-integer nonlinear programming (MINLP) characteristics inherent in the problem [22].

The proposed combination of the CBGA and the IPOPT is specifically designed to exploit the complementary strengths of global metaheuristics and deterministic local solvers [23,24]. The CBGA is particularly effective for exploring high-dimensional discrete search spaces, making it well-suited for determining the optimal siting of TSCs in distribution networks. However, metaheuristic algorithms often lack the precision required to accurately refine continuous decision variables, such as device size, especially in the presence of complex nonlinear constraints [22].

To address this limitation, the IPOPT was integrated into the framework in order to handle the nonlinear OPF subproblem in the slave stage. The IPOPT leverages second-order derivative information and an interior-point methodology to efficiently solve large-scale, sparse, and non-convex nonlinear programming problems [25,26]. This ensures that each candidate solution provided by the CBGA is accurately evaluated in terms of to both feasibility and cost optimality.

Thus, the hybrid CBGA-IPOPT framework enables a clear decomposition of the original MINLP problem: the CBGA performs global exploration over binary placement variables, while the IPOPT conducts local refinement over continuous sizing variables. This master–slave coordination improves convergence speed, enhances numerical stability, and reduces variability across optimization runs. Comparative simulations with other standalone metaheuristics and classical solvers such as PSO, BWO, and BONMIN demonstrate that this hybrid approach yields superior performance in terms of solution quality, robustness, and computational efficiency. These findings support the use of the CBGA-IPOPT combination as a highly effective methodology for reactive power compensation planning in MVDNs.

While the proposed hybrid optimization framework yields promising results, it is important to acknowledge its limitations. On the one hand, the effectiveness of a variable reactive power injection strategy relies on the availability of accurate short-term load and generation forecasts. Inaccuracies in these predictions may affect the optimality of the compensation schedule. On the other hand, the computational cost of the CBGA–IPOPT approach may increase with network size and complexity, which could limit its scalability in real-time applications. Future work will focus on integrating robust optimization techniques to handle forecast uncertainties, as well as on exploring parallel computing strategies to improve scalability and computational efficiency.

1.5. Document Structure

The remainder of this paper is organized as follows. Section 2 presents the mathematical formulation of the optimization problem, including the objective function, the power flow constraints, and the TSC siting and sizing model for both fixed and variable reactive power injection scenarios. Section 3 describes the proposed hybrid solution methodology, which combines the CBGA for discrete TSC placement with the IPOPT for continuous sizing optimization within a master–slave strategy. Section 4 presents the test systems used, i.e., the 33- and 69-bus radial feeders, along with their electrical characteristics, time-varying load profiles, and economic parameters. Section 5 outlines and analyzes the simulation results, including a comparative assessment of the proposed method against state-of-the-art approaches under fixed and variable compensation scenarios, as well as a detailed techno-economic evaluation. Finally, Section 6 summarizes the main findings and contributions of this work, and it outlines potential future research directions such as multi-objective optimization and uncertainty modeling for advanced reactive power planning.

2. Mathematical Formulation of the Optimization Problem

This section presents the mathematical formulation of the optimal siting and sizing of TSCs in MVDNs. The problem is structured as an MINLP model, incorporating both discrete (binary) and continuous decision variables [2]. This formulation includes the nonlinear power flow equations, investment cost considerations, and operational constraints associated with distribution networks. The objective is to minimize the annualized operating costs, which include both energy losses and the investment made in TSC integration [9].

2.1. Objective Function

The overall objective function is designed to minimize the total annualized cost, denoted by f, which comprises two components [22]:

$$minf=f_1+f_2,$$

(1)

This objective function combines the annualized cost of technical losses and the annualized TSC investment cost for the techno-economic optimization of reactive power planning. This dual-component formulation ensures a balanced compromise between improving energy efficiency and limiting capital expenditure, enabling the economically sustainable operation of the distribution network [9].

Note that, in (1), the term $f_1$ represents the annualized cost associated with the distribution network’s technical energy losses, calculated based on the power flow over a typical year of operation. On the other hand, $f_2$ accounts for the annualized investment cost related to the deployment of TSCs, considering their size and associated capital expenditure [2]. Together, these two components quantify the total cost to be minimized in the optimal planning and operation of reactive power compensation devices in MVDNs.

The cost of energy losses is expressed as follows [27]:

$$f_1=C_{\mathrm{kWh}}T\sum _{h\in \mathcal{H}}\sum _{k\in \mathcal{N}}\sum _{m\in \mathcal{N}}{v}_{kh}{v}_{mh}{Y}_{km}cos({\theta }_{kh}-{\theta }_{mh}-{\varphi }_{km}){\Delta }_h,$$

(2)

where $C_{\mathrm{kWh}}$ is the unit energy cost, T is the number of days in a year, $\mathcal{H}$ is the set of time intervals (e.g., hours), $\mathcal{N}$ is the set of buses, $Y_{km}$ and ${\varphi }_{km}$ are the magnitude and angle of the admittance between buses k and m, and $v_{kh}, {\theta }_{kh}$ are the nodal voltage magnitude and angle at bus k in period h.

The annualized TSC investment cost is defined by a cubic cost model [2].

$$f_2=T\left({\displaystyle \frac{k_1}{k_2}}\right)\sum _{k\in \mathcal{N}}{q}_k^{\mathrm{TSC}}\left({\omega }_1{\left(q_k^{\mathrm{TSC}}\right)}^2+{\omega }_2{q}_k^{\mathrm{TSC}}+{\omega }_3\right),$$

(3)

where $q_k^{\mathrm{TSC}}$ is the TSC capacity at bus k, ${\omega }_1,{\omega }_2,{\omega }_3$ are the cost model coefficients, and $k_1/k_2$ is the annuitization factor over the planning horizon.

2.2. Power Flow Balance Constraints

To ensure a proper energy balance in the system, the active and reactive power injected at each bus must satisfy the network’s steady-state constraints. The power balance equations are presented below [28].

$$\begin{array}{cccc}\hfill p_{kh}^g-p_{kh}^d& =\sum _{m\in \mathcal{N}}{v}_{kh}{v}_{mh}{Y}_{km}cos({\theta }_{kh}-{\theta }_{mh}-{\varphi }_{km}){\Delta }_h,\hfill & \hfill & \forall k\in \mathcal{N},\forall h\in \mathcal{H},\hfill \end{array}$$

(4)

$$\begin{array}{cccc}\hfill q_{kh}^g-q_{kh}^d+q_{kh}^{\mathrm{TSC}}& =\sum _{m\in \mathcal{N}}{v}_{kh}{v}_{mh}{Y}_{km}sin({\theta }_{kh}-{\theta }_{mh}-{\varphi }_{km}){\Delta }_h,\hfill & \hfill & \forall k\in \mathcal{N},\forall h\in \mathcal{H}.\hfill \end{array}$$

(5)

Here, $p_{kh}^g$ and $q_{kh}^g$ denote the generated active and reactive powers at bus k in period h, $p_{kh}^d$ and $q_{kh}^d$ are the load demands, and $q_{kh}^{\mathrm{TSC}}$ represents the reactive power injected by the TSC.

2.3. Branch Current Flow Limits

To ensure a safe operation, the current flowing through each distribution line must not exceed its thermal limit. The real and imaginary current components from bus k to bus m in period h are as follows [9]:

$$\begin{array}{cc}\hfill i_{kmh}^r& =y_{km}\left(v_{kh}cos({\theta }_{kh}-{\varphi }_{km})-v_{mh}cos({\theta }_{mh}-{\varphi }_{km})\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill i_{kmh}^i& =y_{km}\left(v_{kh}sin({\theta }_{kh}-{\varphi }_{km})-v_{mh}sin({\theta }_{mh}-{\varphi }_{km})\right).\hfill \end{array}$$

(7)

The current magnitude must satisfy the following [29]:

$$\sqrt{{\left(i_{kmh}^r\right)}^2+{\left(i_{kmh}^i\right)}^2}\le I_{km}^{max},\forall (k,m)\in \mathcal{L},\forall h\in \mathcal{H},$$

(8)

where $I_{km}^{max}$ is the maximum permissible current.

2.4. Voltage Regulation Constraints

The voltage magnitude at each bus must be maintained within operational limits, which are typically set by regulatory standards [30].

$$\begin{array}{cccc}\hfill v_{sh}& =V_{\mathrm{nom}},{\theta }_{sh}=0,\hfill & \hfill & \forall h\in \mathcal{H},\hfill \end{array}$$

(9)

$$\begin{array}{cccc}\hfill V_{min}& \le v_{kh}\le V_{max},\hfill & \hfill & \forall k\in \mathcal{N},\forall h\in \mathcal{H}.\hfill \end{array}$$

(10)

Equation (10) provides a general formulation of the voltage regulation constraints, where the lower and upper bounds can be flexibly defined based on the voltage level of the system and the applicable regulatory standards [31]. In this study, the voltage limits were set to $0.90$ p.u. and $1.10$ p.u. to ensure consistency with typical utility practices and to prevent violations of the constraints in the base case (i.e., with no TSCs installed) [9]. This particular choice of bounds also ensures that all evaluated compensation strategies are benchmarked against a common, realistic, and technically feasible operating condition. By doing so, it guarantees a fair comparison, as each strategy is assessed under the same voltage constraints that reflect practical system limits and regulatory requirements, thereby facilitating an equitable and meaningful evaluation of their performance.

2.5. TSC Siting and Sizing Constraints

The binary variable $x_k\in \{0,1\}$ indicates whether a TSC is installed at bus k. The TSC sizing and operating profile are constrained as follows [9]:

$$\begin{array}{cccc}\hfill 0\le q_k^{\mathrm{TSC}}& \le x_k{q}_{max}^{\mathrm{TSC}},\hfill & \hfill & \forall k\in \mathcal{N},\hfill \end{array}$$

(11)

$$\begin{array}{cccc}\hfill -q_k^{\mathrm{TSC}}\le q_{kh}^{\mathrm{TSC}}& \le q_k^{\mathrm{TSC}},\hfill & \hfill & \forall k\in \mathcal{N},\forall h\in \mathcal{H},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \sum _{k\in \mathcal{N}}{x}_k& \le x_{max}^{\mathrm{TSC}},\hfill \end{array}$$

(13)

where $q_{max}^{\mathrm{TSC}}$ is the maximum capacity of an individual TSC unit, and $x_{max}^{\mathrm{TSC}}$ defines the maximum number of devices allowed in the network.

2.6. Fixed Reactive Power Scenario

In the case of constant (fixed) reactive power injection, Equation (12) is replaced by [5]

$$q_{kh}^{\mathrm{TSC}}=q_k^{\mathrm{TSC}},\forall k\in \mathcal{N},\forall h\in \mathcal{H},$$

(14)

implying a time-invariant reactive power injection profile from the TSCs.

2.7. Problem Classification and Solving Strategy

The overall model described by Equations (1)–(14) is an MINLP problem, which poses significant challenges due to its non-convexity, its nonlinearities, and the presence of both binary and continuous variables. Recent advances in computational optimization have shown that hybrid methods combining metaheuristic global search with deterministic local solvers can effectively handle such problems [2]. This research adopted the CBGA for the siting decision and the IPOPT for solving the continuous subproblem.

3. Solution Methodology

To address the problem regarding the optimal siting and sizing of TSCs in MVDNs under both fixed and variable reactive power injection scenarios, this work employs a hybrid master–slave optimization strategy. This framework effectively combines discrete and continuous optimization by coordinating two stages: the CBGA for determining the optimal installation sites and an IPOPT to compute the corresponding TSC sizes through an optimal power flow (OPF) formulation. The detailed structure of this solution methodology is presented below.

3.1. Master Stage: CBGA for Optimal TSC Location

The CBGA is a robust evolutionary optimization method well-suited for solving combinatorial problems that involve binary or integer decision variables [32]. In this context, the algorithm searches for the optimal locations of TSCs across the distribution network while aiming to minimize the combined cost of energy losses and capital investment.

Each candidate solution, or chromosome, is encoded as a binary vector $\mathbf{x}={[x_1,x_2,\dots ,x_n]}^{\top }$$\in {\{0,1\}}^n$, where each element $x_k$ indicates the presence ($x_k=1$) or absence ($x_k=0$) of a TSC at bus k. The initial population is randomly generated and evaluated using a fitness function derived from solving a nonlinear programming subproblem (handled by the IPOPT), which determines the optimal reactive power injection levels $q_k^{\mathrm{TSC}}$.

The CBGA then performs some genetic operations like selection (e.g., tournament or roulette wheel), crossover (uniform or single-point), and mutation (bit flipping with low probability) in order to evolve the population towards better solutions [33]. An elitist strategy ensures that the best-performing solutions are retained across generations. This iterative process continues until a termination criterion is satisfied, typically a maximum number of generations or convergence of the best fitness value.

3.2. Slave Stage: IPOPT for TSC Sizing

Once the CBGA has defined a binary configuration $\mathbf{x}$ for TSC installation, the corresponding sizing problem is formulated as a nonlinear OPF subproblem [34]. This subproblem is governed by the nonlinear dynamics of power systems, including nodal power balances, voltage constraints, line thermal limits, and device capacity limits.

The IPOPT solves this continuous subproblem efficiently via a barrier-function approach, transforming inequality constraints into a series of unconstrained minimization problems [22]. It computes a search direction based on Newton’s method and performs a line search to ensure feasibility and descent in the objective function.

For each installation vector $\mathbf{x}$ provided by the CBGA, the IPOPT returns an optimal vector of TSC sizes ${\mathbf{q}}^{\mathrm{TSC}}={[q_1^{\mathrm{TSC}},\dots ,q_n^{\mathrm{TSC}}]}^{\top }$, ensuring operating compliance and cost-effectiveness. The resulting fitness value, composed of operating losses and investment costs, is used to guide the CBGA’s evolutionary search.

3.3. Summary of the Optimization Framework

Our hybrid master–slave structure enables the effective decomposition of the complex MINLP problem by decoupling the combinatorial siting decision (master stage) from the continuous sizing optimization (slave stage) [22]. This modular approach leverages the global exploration capabilities of metaheuristics and the high-precision convergence of local solvers, thereby enhancing solution quality, computational efficiency, and scalability.

The overall procedure is summarized in the following pseudocode (Algorithm 1):

Algorithm 1 Hybrid CBGA-IPOPT for optimal TSC siting and sizing.

1: Initialize population of binary vectors $\mathbf{x}$ (TSC locations) [CBGA] 2: while termination criterion not met do 3:     for each individual in population do 4:         Fix TSC installation pattern $\mathbf{x}$ 5:         Solve continuous OPF problem for sizing ${\mathbf{q}}^{\mathrm{TSC}}$ [IPOPT] 6:         Evaluate fitness using objective function 7:     end for 8:     Apply genetic operators: selection, crossover, mutation 9:     Update population (elitism strategy) 10: end while 11: Return best $\mathbf{x}$ and ${\mathbf{q}}^{\mathrm{TSC}}$

This hybrid methodology ensures that both the location and size of the TSCs are jointly optimized while observing technical and economic constraints, providing a powerful decision-making tool for reactive power compensation in MVDNs.

4. Test System Information

Two well-established radial distribution test systems consisting of 33 and 69 nodes were employed to evaluate the effectiveness of the proposed CBGA-IPOPT approach in addressing the optimization problem under study [2]. The topological configuration of each network is depicted in Figure 2, while detailed electrical parameters, including line impedances and peak load demands, are provided in

Table 1. Main parameters for the 33-node test system.

Node i-j	${\mathit{R}}_{\mathit{ij}}$ ($\mathbf{\Omega }$)	${\mathit{X}}_{\mathit{ij}}$ ($\mathbf{\Omega }$)	${\mathit{P}}_{\mathit{j}}$ (kW)	${\mathit{Q}}_{\mathit{j}}$ (kvar)	Node i-j	${\mathit{R}}_{\mathit{ij}}$ ($\mathbf{\Omega }$)	${\mathit{X}}_{\mathit{ij}}$ ($\mathbf{\Omega }$)	${\mathit{P}}_{\mathit{j}}$ (kW)	${\mathit{Q}}_{\mathit{j}}$ (kvar)
1–2	0.0922	0.0477	100	60	17–8	0.7320	0.5740	90	40
2–3	0.4930	0.2511	90	40	2–19	0.1640	0.1565	90	40
3–4	0.3660	0.1864	120	80	19–20	1.5042	1.3554	90	40
4–5	0.3811	0.1941	60	30	20–21	0.4095	0.4784	90	40
5–6	0.8190	0.7070	60	20	21–22	0.7089	0.9373	90	40
6–7	0.1872	0.6188	200	100	3–23	0.4512	0.3083	90	50
7–8	1.7114	1.2351	200	100	23–24	0.8980	0.7091	420	200
8–9	1.0300	0.7400	60	20	24–25	0.8960	0.7011	420	200
9–10	1.0400	0.7400	60	20	6–26	0.2030	0.1034	60	25
10–11	0.1966	0.0650	45	30	26–27	0.2842	0.1447	60	25
11–12	0.3744	0.1238	60	35	27–28	1.0590	0.9337	60	20
12–3	1.4680	1.1550	60	35	28–29	0.8042	0.7006	120	70
13–14	0.5416	0.7129	120	80	29–30	0.5075	0.2585	200	600
14–15	0.5910	0.5260	60	10	30–31	0.9744	0.9630	150	70
15–16	0.7463	0.5450	60	20	31–32	0.3105	0.3619	210	100
16–17	1.2860	1.7210	60	20	32–33	0.3410	0.5302	60	40

and 

Table 2. Main parameters for the 69-node test system.

Node i-j	${\mathit{R}}_{\mathit{ij}}$ ($\mathbf{\Omega }$)	${\mathit{X}}_{\mathit{ij}}$ ($\mathbf{\Omega }$)	${\mathit{P}}_{\mathit{j}}$ (kW)	${\mathit{Q}}_{\mathit{j}}$ (kvar)	Node i-j	${\mathit{R}}_{\mathit{ij}}$ ($\mathbf{\Omega }$)	${\mathit{X}}_{\mathit{ij}}$ ($\mathbf{\Omega }$)	${\mathit{P}}_{\mathit{j}}$ (kW)	${\mathit{Q}}_{\mathit{j}}$ (kvar)
1–2	0.0005	0.0012	0.00	0.00	3–36	0.0044	0.0108	26.00	18.55
2–3	0.0005	0.0012	0.00	0.00	36–37	0.0640	0.1565	26.00	18.55
3–4	0.0015	0.0036	0.00	0.00	37–38	0.1053	0.1230	0.00	0.00
4–5	0.0251	0.0294	0.00	0.00	38–39	0.0304	0.0355	24.00	17.00
5–6	0.3660	0.1864	2.60	2.20	39–40	0.0018	0.0021	24.00	17.00
6–7	0.3810	0.1941	40.40	30.00	40–41	0.7283	0.8509	1.20	1.00
7–8	0.0922	0.0470	75.00	54.00	41–42	0.3100	0.3623	0.00	0.00
8–9	0.0493	0.0251	30.00	22.00	42–43	0.0410	0.0478	6.00	4.30
9–10	0.8190	0.2707	28.00	19.00	43–44	0.0092	0.0116	0.00	0.00
10–11	0.1872	0.0619	145.00	104.00	44–45	0.1089	0.1373	39.22	26.30
11–12	0.7114	0.2351	145.00	104.00	45–46	0.0009	0.0012	29.22	26.30
12–13	1.0300	0.3400	8.00	5.00	4–47	0.0034	0.0084	0.00	0.00
13–14	1.0440	0.3450	8.00	5.50	47–48	0.0851	0.2083	79.00	56.40
14–15	1.0580	0.3496	0.00	0.00	48–49	0.2898	0.7091	384.70	274.50
15–16	0.1966	0.0650	45.50	30.00	49–50	0.0822	0.2011	384.70	274.50
16–17	0.3744	0.1238	60.00	35.00	8–51	0.0928	0.0473	40.50	28.30
17–18	0.0047	0.0016	60.00	35.00	51–52	0.3319	0.1114	3.60	2.70
18–19	0.3276	0.1083	0.00	0.00	9–53	0.1740	0.0886	4.35	3.50
19–20	0.2106	0.0690	1.00	0.60	53–54	0.2030	0.1034	26.40	19.00
20–21	0.3416	0.1129	114.00	81.00	54–55	0.2842	0.1447	24.00	17.20
21–22	0.0140	0.0046	5.00	3.50	55–56	0.2813	0.1433	0.00	0.00
22–23	0.1591	0.0526	0.00	0.00	56–57	1.5900	0.5337	0.00	0.00
23–24	0.3463	0.1145	28.00	20.00	57–58	0.7837	0.2630	0.00	0.00
24–25	0.7488	0.2475	0.00	0.00	58–59	0.3042	0.1006	100.00	72.00
25–26	0.3089	0.1021	14.00	10.00	59–60	0.3861	0.1172	0.00	0.00
26–27	0.1732	0.0572	14.00	10.00	60–61	0.5075	0.2585	1244.00	888.00
3–28	0.0044	0.0108	26.00	18.60	61–62	0.0974	0.0496	32.00	23.00
28–29	0.0640	0.1565	26.00	18.60	62–63	0.1450	0.0738	0.00	0.00
29–30	0.3978	0.1315	0.00	0.00	63–64	0.7105	0.3619	227.00	162.00
30–31	0.0702	0.0232	0.00	0.00	64–65	1.0410	0.5302	59.00	42.00
31–32	0.3510	0.1160	0.00	0.00	11–66	0.2012	0.0611	18.00	13.00
32–33	0.8390	0.2816	14.00	10.00	66–67	0.0470	0.0140	18.00	13.00
33–34	1.7080	0.5646	19.50	14.00	12–68	0.7394	0.2444	28.00	20.00
34–35	1.4740	0.4873	6.00	4.00	68–69	0.0047	0.0016	28.00	20.00

. Both feeders operate at a nominal line-to-ground substation voltage of 12.66 kV, with voltage limits defined between 0.90 p.u. and 1.10 p.u., in accordance with standard utility constraints [35]. These test systems were selected due to their widespread use in the literature as benchmarking platforms for the validation of power flow and optimization techniques in distribution networks, enabling a consistent and rigorous comparison with prior studies such as that by [9].

To assess the operating performance of the TSCs in the 33- and 69-bus test feeders, a time-varying load profile was applied to simulate realistic demand conditions. Specifically, the peak power consumption values per node, as provided in Table 1 and Table 2, served as the baseline demand for each bus. These baseline values were then scaled for each time period using the percent load factors presented in Figure 3, which represent the relative variation in power demand throughout the day. This approach enables a more accurate and dynamic evaluation of the TSCs’ behavior under changing load conditions, reflecting the challenges and performance requirements associated with the operation of modern distribution networks.

Additionally, the parameters used for estimating the expected operating costs associated with the TSCs are presented in

Table 3. Parameterization of the $f_2$ objective function (investments in TSCs).

Parameter	Value	Unit	Parameter	Value	Unit
${\omega }_1$	1.50	USD/Mvar3	${\omega }_2$	−713.00	USD/Mvar2
${\omega }_3$	153,750	USD/Mvar	T	365	days
$k_1$	${\displaystyle \frac{6}{2190}}$	1/day	$k_2$	10	years
${\Delta }_h$	${\displaystyle \frac{1}{2}}$	hour	$C_{\mathrm{kWh}}$	0.1390	USD/kWh

. These parameters include the relevant technical and economic data required for cost-based performance evaluation and were adapted from the work presented by [2]. This information served as the basis for quantifying the economic impact of TSC deployment on the distribution networks under study.

5. Simulation Results

The computational implementation of the proposed methodology was carried out using the Julia programming language, version 1.9.2 [36], executed on a personal computer equipped with an AMD Ryzen 7 3700 processor sourced from AMD, Santa Clara, USA, operating at 2.3 GHz, 16.0 GB of RAM, and a 64-bit version of Microsoft Windows 10 Single Language. Julia was selected due to its high-performance computing capabilities, which combine the execution speed of low-level languages with the syntactic simplicity of their high-level counterparts, an essential feature for handling computationally demanding optimization tasks.

For mathematical modeling and optimization, the JuMP package was employed due to its expressive syntax and compatibility with a wide range of solvers. The nonlinear programming subproblems were solved using the Ipopt solver [36], which is well-suited for large-scale, sparse, and non-convex optimization problems. Additionally, the CBGA was implemented in the Julia environment to determine the optimal placement of TSCs. The CBGA interacted with Ipopt within a master–slave framework, where the CBGA handled the binary decision variables (location) and the Ipopt optimized the continuous sizing variables.

The analysis comprised the following:

• A comparative assessment against existing literature-reported methodologies where TSCs are located and sized while assuming a fixed reactive power injection profile throughout the day.
• The evaluation of a variable reactive power injection strategy, where the TSC output is adjusted on an hourly basis to better match the dynamic load conditions and assess its effectiveness in improving network performance.

5.1. Fixed-Step Operation Scenario

To assess the performance of the proposed CBGA-IPOPT framework, a comparative analysis was conducted against four state-of-the-art metaheuristic algorithms: SCA, PSO, BWO, and AHA [9]. These algorithms were selected based on their relevance and reported success in solving nonlinear, mixed-integer optimization problems for power systems. PSO is one of the most widely adopted swarm-based techniques in reactive power compensation studies, offering fast convergence and ease of implementation [16]. Moreover, SCA is a math-inspired algorithm known for its balance between exploration and exploitation, and it has been effectively applied to the optimal placement of FACTS [22]. BWO and AHA, on the other hand, are recent bio-inspired methods that have demonstrated competitive performance in solving complex engineering problems involving multiple decision variables and constraints. This selection ensured algorithmic paradigm diversity—from classical to emerging strategies—and provided a robust baseline to validate the superiority of the proposed hybrid framework in terms of technical performance, economic efficiency, and computational robustness.

5.1.1. Comparative Analysis for the 33-Bus Grid

Numerical simulations applying the proposed master–slave metaheuristic optimization methodology to the 33-bus test system, along with a comparative analysis against existing methods from the literature, are presented in

Table 4. Comparative results for the 33-bus grid.

Method	Location (Node)	Size (Mvar)	Objective Function (USD/Year)	Expected Reduction (%)
BONMIN	[6, 18, 30]	[0.0000, 0.1138, 0.4551]	100,221.38	11.10
CBGA	[13, 30, 31]	[0.1528, 0.3227, 0.1157]	100,139.21	11.18
PSO	[14, 30, 31]	[0.1486, 0.3244, 0.1157]	100,107.24	11.21
BWO	[14, 30, 32]	[0.1486, 0.3337, 0.1064]	100,093.29	11.22
AHA	[14, 30, 32]	[0.1486, 0.3337, 0.1064]	100,093.29	11.22
CBGA-IPOPT	[14, 30, 32]	[0.1486, 0.3337, 0.1064]	100,093.29	11.22

.

The results presented in this table demonstrate the effectiveness of CBGA-IPOPT in determining the optimal location and size of TSCs in a MVDN. The primary performance metric under consideration was the annualized objective function value, which combines the costs of energy losses and the investment in TSCs, measured in USD per year.

Among the evaluated methodologies, the BONMIN solver, a benchmark MINLP approach implemented in GAMS, achieved a total annual cost of USD 100,221.38, with a corresponding expected reduction of 11.10% (with respect to the absence of TSCs, i.e., USD 112,740.90). However, its solution involved a zero reactive power injection at one of the selected nodes (bus 6), which may indicate either an unnecessary installation or convergence limitations due to the inherent complexity of the non-convex problem space.

The metaheuristic approaches (CBGA, PSO, and BWO) provided more consistent and efficient solutions, with annual costs of USD 100,139.21, USD 100,107.24, and USD 100,093.29, respectively. These values correspond to operating cost reductions of approximately 11.18%, 11.21%, and 11.22%. Notably, all these methods identified bus 30 as a critical location for TSC installation, highlighting its electrical significance in minimizing network losses.

The AHA produced identical results to BWO, reflecting its comparable ability to effectively navigate the search space. However, the proposed CBGA-IPOPT stood out due to its superior convergence behavior. By integrating the CBGA for discrete location selection and the IPOPT for continuous sizing optimization, this method achieved the best-known cost of USD 100,093.29, with the added benefit of reduced solution variability across different runs and improved numerical stability.

Additionally, CBGA-IPOPT consistently identified the same three optimal nodes (14, 30, 32) and associated TSC sizes (0.1486, 0.3337, 0.1064 Mvar), which indicates a high degree of repeatability and robustness. These results affirm the advantage of combining the global exploration capabilities of the CBGA with the IPOPT’s precise local search, especially for MINLP problems involving complex network constraints.

In summary, while all optimization techniques demonstrated improvements over the baseline solution, CBGA-IPOPT proved to be the most reliable and effective, offering the highest cost reduction and the most consistent results. This validates its suitability for reactive power compensation planning in radial distribution systems under real-world operational constraints.

5.1.2. Results for the 69-Bus Grid

Table 5. Comparative results for the 69-bus grid.

Method	Location (Node)	Size (Mvar)	Objective Function (USD/year)	Expected Reduction (%)
BONMIN	Does not converge
CBGA	[21, 61, 65]	[0.0662, 0.4840, 0.0635]	104,695.69	12.55
PSO	[23, 61, 64]	[0.0526, 0.4009, 0.1504]	104,679.61	12.56
BWO	[21, 61, 64]	[0.0647, 0.4363, 0.1125]	104,658.03	12.58
AHA	[21, 61, 64]	[0.0647, 0.4363, 0.1125]	104,658.03	12.58
CBGA-IPOPT	[21, 61, 64]	[0.0647, 0.4363, 0.1125]	104,658.03	12.58

presents the numerical results obtained for the 69-bus test system using different optimization methods, including the proposed CBGA-IPOPT. The benchmark case, i.e., system operation without reactive power compensation, corresponds to an annualized cost of approximately USD 119,715.63. This cost served as the baseline for evaluating the performance of the studied methods in terms of costs reduction.

As observed in the table, the BONMIN solver was unable to converge in this larger and more complex distribution feeder, reinforcing the challenges faced by exact solvers in high-dimensional, nonlinear, mixed-integer formulations. In contrast, all metaheuristic strategies yielded feasible and high-quality solutions.

Among the standalone metaheuristic methods, BWO, AHA, and CBGA exhibited a notable performance, achieving cost reductions ranging from approximately 12.55% to 12.58% when compared to the benchmark case. Although PSO also yielded competitive results, its reduction was slightly lower in comparison. These findings highlight the effectiveness of evolutionary and bio-inspired optimization techniques in addressing the complexity of reactive power compensation in large-scale distribution networks, particularly under nonlinear and mixed-integer constraints.

The best-performing strategy, CBGA-IPOPT, achieved the lowest annualized cost, i.e., USD 104,658.03, corresponding to a 12.58% reduction relative to the benchmark. This configuration, consistently identified across BWO, AHA, and CBGA-IPOPT, includes nodes 21, 61, and 64, with reactive capacities of 0.0647, 0.4363, and 0.1125 Mvar. The consistency in both location and size across multiple algorithms indicates the robustness and optimality of the identified solution.

Furthermore, CBGA-IPOPT combines the global exploration capabilities of CBGA for placement decisions with the precision of IPOPT in solving the nonlinear continuous sizing subproblem. This synergy allows our methodology to efficiently and reliably handle the combinatorial complexity and nonlinear dynamics of the power flow model.

In summary, the results confirm that the proposed CBGA-IPOPT not only surpasses classical solvers like BONMIN in terms of feasibility and robustness but also provides superior economic performance compared to other state-of-the-art metaheuristics. This makes it a strong candidate for real-world applications involving optimal reactive power planning for large distribution networks.

5.2. Variable-Step Operation Scenario

A significant advantage of CBGA-IPOPT is its ability to handle flexible reactive power injection over time, allowing for a more efficient and adaptive use of TSCs in distribution networks. For the 33-bus system, this time-dependent operation leads to finely tuned TSC sizes (0.1786 Mvar, 0.4022 Mvar, and 0.1365 Mvar) installed at the same nodes identified with the fixed-injection strategy. This dynamic approach reduces the annual operating cost to 98,729.21 USD/year, achieving a 12.43% cost reduction relative to the benchmark scenario. This enhanced capacity to modulate reactive power in response to hourly demand variations contributes to more effective voltage regulation and lower system losses when compared to traditional fixed injection schemes.

A similar improvement is observed in the 69-bus distribution system. By enabling variable reactive power injection, the hybrid strategy optimally allocates TSC capacities of 0.0683 Mvar, 0.5380 Mvar, and 0.1450 Mvar at nodes 21, 61, and 64. This is consistent with the locations found under fixed-injection conditions but yields superior performance due to the temporal flexibility introduced. In this case, the annual operating cost is reduced to 102,861.83 USD/year, corresponding to a 14.08% reduction with respect to the uncompensated baseline. This confirms the method’s adaptability to complex load dynamics and validates its efficacy in reducing long-term operational expenditure.

In both test feeders under variable reactive power injection conditions, CBGA-IPOPT consistently outperforms its fixed counterpart, as well as other conventional metaheuristic approaches like BONMIN, PSO, and BWO. These results emphasize the value of master–slave coordination between discrete location decisions and continuous sizing optimization, and they position the proposed method as a robust and economically efficient tool for planning dynamic reactive power compensation in modern radial distribution networks.

5.3. Techno-Economic Assessment

Table 6. Economic analysis regarding TSC installation in MVDNs.

Case	${\mathit{f}}_1$ (USD)	${\mathit{f}}_2$ (USD)	f (USD)	Net Profit (USD)
The 33-bus grid
Without TSCs	112,740.88	0	112,740.87	0
Fixed	91,052.01	9041.29	100,093.30	12,647.56
Variable	87,713.87	11,015.33	98,729.21	14,011.66
The 69-bus grid
Without TSCs	119,715.63	0	119,715.63	0
Fixed	95,240.70	9417.33	104,658.03	15,057.60
Variable	91,331.77	11,530.06	102,861.83	16,853.80

presents a techno-economic evaluation of the proposed TSC-based compensation strategy under three operation scenarios for the 33-bus and 69-bus test systems: (i) no compensation, (ii) fixed reactive power injection, and (iii) variable reactive power injection. For each scenario, the annual cost of energy losses ($f_1$), the annualized cost of investment in TSCs ($f_2$), and the total cost ($f=f_1+f_2$) are reported. Additionally, the net profit is defined as the difference between the base-case cost and the total cost with compensation.

For the 33-bus system, the base case yields an annual energy losses cost of USD 112,740.88, which serves as the reference for comparison. Implementing fixed-step compensation reduces this cost to USD 91,052.01 and requires an annualized investment of USD 9,041.29, for a total cost of USD 100,093.30 and a net profit of USD 12,647.56. When the system operates under a variable reactive power injection strategy, the cost of energy losses further decreases to USD 87,713.87, with a slightly higher investment of USD 11,015.33. The resulting total cost of USD 98,729.21 leads to a greater net profit of USD 14,011.66, confirming that variable compensation provides enhanced economic benefits, given its better alignment with time-varying load demands.

In the case of the 69-bus system, the base-case annual losses cost is USD 119,715.63. The fixed-step scenario reduces this value to USD 95,240.70 and requires an investment of USD 9417.33, resulting in a total cost of USD 104,658.03 and a profit of USD 15,057.60. With variable compensation, the cost of energy losses drops to USD 91,331.77, and the investment cost increases to USD 11,530.06, leading to a minimized total cost of USD 102,861.83 and a net profit of USD 16,853.80. These results show that the flexibility of time-varying compensation enables a better system performance, even though slightly higher investments are required.

In both test feeders, the variable compensation strategy offers the highest net profit, highlighting the economic advantages of flexible TSC operation. Although the variable approach incurs slightly higher investment costs, the greater reduction in operational losses leads to superior total savings. These findings confirm that incorporating temporal flexibility into reactive power planning can significantly improve the overall efficiency and cost-effectiveness of distribution systems. The proposed CBGA-IPOPT successfully identifies optimal configurations that balance technical performance with economic feasibility, making it a robust solution for smart grid planning in the presence of dynamic demand profiles.

5.4. General Performance Analysis

The superior performance of the CBGA–IPOPT framework can be attributed to its hybrid structure, which effectively combines the strengths of global and local optimization techniques. CBGA is particularly adept at exploring the high-dimensional binary search space related to TSC placement, maintaining genetic diversity to avoid premature convergence and ensure broad exploration of potential solutions. Once promising candidate configurations are identified, IPOPT acts as a high-precision local optimizer, solving the nonlinear OPF subproblem for each candidate to refine solutions, satisfy system constraints, and minimize operational costs.

This master–slave synergy functions as a coordinated search process: CBGA (the master) explores and generates candidate solutions, while IPOPT (the slave) provides precise local refinement and validation of these solutions. This collaborative dynamic allows the framework to consistently identify high-quality solutions with low variance across simulations, as reflected in the stable voltage profiles, significant loss reductions, and cost savings observed in both test systems. Essentially, the master–slave relationship ensures a thorough exploration of the solution space combined with meticulous local optimization, leading to robust and reliable outcomes.

5.5. Complementary Analysis

To assess the impact of the proposed compensation strategies on operating performance, Figure 4 and Figure 5 present the time-domain evolution of three key metrics over a 24 h period for the 33- and 69-bus grids: (a) reactive power injection at the substation, (b) minimum nodal voltage profile, and (c) total active power losses. These results enable a comparative evaluation of how each compensation strategy influences the dynamic behavior of the networks.

5.5.1. Reactive Power Injection at the Substation Terminals

• 33-bus system. The benchmark case demands considerable reactive support from the upstream grid, with peak injections exceeding 2000 kvar during high-load periods. The fixed compensation approach reduces this demand but still exceeds 1400 kvar. In contrast, the proposed variable compensation strategy effectively tracks the reactive power requirements, dynamically adjusting its output and peaking around 1300 kvar. This reflects a reduction of approximately 700 kvar compared to the benchmark, demonstrating a more efficient and adaptive compensation behavior.
• For the 69-bus system, a similar trend is observed, with higher magnitudes due to its larger size. The benchmark scenario shows peak substation injections near 2300 kvar. Fixed compensation improves this performance but still requires more than 1700 kvar. The variable compensation strategy adapts more precisely to the network’s needs, maintaining peak injections near 1600 kvar and achieving reductions above 1000 kvar during the early morning and evening periods. These results highlight the scalability and responsiveness of the proposed approach.

5.5.2. Minimum Voltage Profile Performance

• 33-bus system. Under benchmark conditions, the minimum nodal voltage drops to 0.91 p.u. during peak demand, potentially violating standard voltage limits. The fixed compensation strategy improves the profile to approximately 0.93 p.u., while the variable strategy consistently maintains the minimum voltage above 0.95 p.u. This indicates enhanced compliance with voltage regulation standards and improved supply reliability.
• 69-bus system. The extended topology and higher demand accentuate voltage regulation challenges. The benchmark case exhibits minimum voltages close to 0.91 p.u., while fixed compensation improves these values to about 0.93 p.u. Once again, the variable strategy outperforms both alternatives, keeping the voltages above 0.94 p.u. across the 24 h horizon. This confirms our control strategy’s robustness in maintaining voltage levels within acceptable bounds across more complex networks.

5.5.3. Total Active Power Losses

• 33-bus system. Power losses in the benchmark scenario peak at approximately 186 kW during periods of high load. The fixed compensation reduces these values to around 151 kW, while the variable compensation strategy limits the maximum losses to 127 kW. This represents a reduction of nearly 32% compared to the benchmark and about 16% compared to the fixed approach, underscoring the energy management efficiency of this approach.
• In the 69-bus system, the losses reach 198 kW in the benchmark case. Fixed compensation reduces this value to approximately 159 kW, and the variable approach further reduces the peak to 154 kW. Although the relative improvement is slightly smaller than in the 33-bus system, the absolute reduction remains significant, offering operating cost savings and mitigating thermal stress on the distribution infrastructure.

5.5.4. Overall Assessment

This comparative analysis across both test systems reveals some consistent trends:

• The benchmark scenario exposes critical performance limitations, including an excessive reactive power demand, voltage deviations, and elevated power losses.
• The fixed compensation strategy achieves moderate improvements but lacks flexibility to effectively accommodate daily load variations.
• The proposed variable compensation strategy demonstrates the most favorable performance in both networks, achieving

  –

  Significantly lower substation reactive power injections;

  –

  Consistent and improved voltage regulation across all time intervals;

  –

  Noticeable reductions in active power losses throughout the day.

These findings validate the proposed strategy’s scalability and efficiency, confirming its ability to enhance power quality and system reliability in medium-voltage distribution systems of different sizes and load profiles.

6. Conclusions and Future Work

This paper presented a hybrid master–slave optimization framework for the optimal siting and sizing of TSCs in MVDNs under both fixed and variable reactive power injection scenarios. The proposed methodology incorporates the CBGA to solve the discrete placement subproblem and the IPOPT for continuous sizing, providing an efficient solution of the underlying MINLP problem. The optimization model integrates nonlinear power flow equations, investment cost considerations, and operational constraints regarding voltage limits and compensation capacity.

Simulation studies conducted on standard 33- and 69-bus radial distribution feeders demonstrated the robustness and performance of the proposed strategy. In the fixed reactive injection scenario, the hybrid CBGA-IPOPT achieved annualized operating cost reductions of approximately 11.22% and 12.58% for the 33- and 69-bus systems, respectively. These results outperform conventional solvers such as BONMIN and other metaheuristic approaches including PSO and BWO. In the variable reactive injection case, which leverages temporal flexibility in compensation, the economic benefits were even more significant, with savings of 12.43% and 14.08%.

A complementary time-domain analysis further validated the advantages of the proposed strategy. In both test feeders, the variable reactive power compensation approach consistently reduced the reactive power demand at the substation, achieving smoother and lower daily injection profiles when compared to fixed compensation. The minimum nodal voltages were maintained above 0.95 p.u. throughout the 24 h period, ensuring regulatory compliance and improved reliability. Additionally, the total active power losses were minimized, with the variable approach reducing peak losses by up to 32% compared to the benchmark. These findings confirm the strong correlation between flexible compensation planning and improved network performance in terms of energy efficiency, voltage stability, and cost-effectiveness.

Another key contribution lies in the demonstrated repeatability of the identified TSC configurations across different simulation runs, highlighting the robustness of the proposed hybrid optimization framework. The coordinated use of global exploration (CBGA) and local exploitation (IPOPT) proved highly effective for addressing high-dimensional, non-convex MINLP problems under realistic distribution network conditions.

Future work will aim to enhance both the robustness and practical applicability of the proposed framework. In particular, the authors plan to integrate uncertainty modeling—via robust or stochastic optimization—to address forecast errors in load demand, distributed generation, and energy pricing. To improve scalability and reduce computation times in large-scale systems, parallel implementations of the CBGA–IPOPT approach and surrogate modeling techniques will also be investigated. Additional research directions include the following:

• Extending the current methodology to a multi-objective framework that concurrently minimizes energy losses, enhances voltage profiles, and reduces greenhouse gas emissions, offering a holistic decision-making tool for sustainable planning.
• Incorporating stochastic or robust optimization techniques to account for uncertainties in load demand, distributed generation output, and energy pricing, thereby increasing the resilience of the proposed planning strategy.
• Exploring the joint deployment and control of diverse reactive power devices such as SVCs, STATCOMs, and PV-STATCOMs within a coordinated planning framework in order to maximize system flexibility and dynamic response.
• Implementing hardware-in-the-loop (HIL) or real-time digital simulator (RTDS) platforms to experimentally validate the proposed methodology, facilitating its translation from theoretical design to practical deployment in modern smart distribution grids.

In summary, the proposed hybrid optimization framework, supported by detailed simulation results and time-domain analysis, offers a powerful and scalable tool for enhancing the planning and operational efficiency of distribution networks through optimized reactive power compensation.