Increasing Distributed Generation Hosting Capacity Based on a Sequential Optimization Approach Using an Improved Salp Swarm Algorithm

Abstract

In recent years, a pronounced transition to the exploitation of renewable energy sources has be observed worldwide, driven by current climate concerns and the scarcity of conventional fuels. However, this paradigm shift is accompanied by new challenges for existing power systems. Therefore, the hosting capacity must be exhaustively assessed in order to maximize the penetration of distributed generation while mitigating any adverse impact on the electrical grid in terms of voltage and the operational boundaries of the equipment. In this regard, multiple aspects must be addressed in order to maintain the proper functioning of the system following the new installations’ capacities. This paper introduces a sequential methodology designed to determine the maximum hosting capacity of a power system through the optimal allocation of both active and reactive power. To achieve this goal, an Improved Salp Swarm Algorithm is proposed, aiming to establish the appropriate operational planning of the power grid considering extensive distributed generation integration, while still ensuring a safe operation. The case study validates the relevance of the proposed model, demonstrating a successful enhancement of hosting capacity by 14.5% relative to standard models.

1. Introduction

Nowadays, the prevailing global perspective on renewable energy sources (RES) is one of growing importance and acceptance. Driven by severe climate changes and the scarcity of fossil fuels, governments, industries and communities worldwide are increasingly recognizing the need to transition from conventional thermal power plants to cleaner alternatives, such as solar, wind and hydro power. However, this transition comes with several challenges in terms of generation intermittency and grid integration issues [1]. As policy frameworks and incentives continue to evolve towards supporting the widespread deployment of distributed generation (DG), in many cases the current energy infrastructure is not prepared for the mass penetration of these new sources. In 2020, Romania reached their set target of 24% of their total energy consumption coming from RES [2]. As a European Union member, Romania must align itself with the energy policy in this partnership, resulting in a new objective of 30.7% renewable energy by 2030, achievable with the installation of 7 GW of renewable capacity. Accordingly, developers have recognized the opportunities as investment announcements, particularly in solar and wind energy, continue to accumulate. Nevertheless, the concentration of these primary energy sources in specific regions of the country (the southern territory especially) exerts high “pressure” on the existing transmission and distribution lines, reflected in overvoltage, equipment ampacity violation and excessive power losses. These issues occur as the new generation deployment exceeds the system’s hosting capacity (HC). Therefore, HC assessment and improvement are of particular interest for both grid operators and RES investors [3]. Refined as concept by Bollen et al. [4], hosting capacity refers to the amount of new generation that can be installed without affecting the power system’s performance. Given the growing concerns regarding extensive distributed generation integration, the hosting capacity problem has garnered increasing attention within the academic community in all sectors of energy systems (low, medium and high voltage networks). According to [5], four main methodologies stand out for calculating hosting capacity with respect to voltage and overloading boundaries. The first approach, namely the deterministic method, implies successively running the load flow analysis by gradually increasing the power installed in RES until the operational limit of the network has been exceeded. Despite being a suitable solution for large-scale installations [6], the main disadvantage of this method consists of its inability to model the output uncertainties in RES. To eliminate this shortcoming, the second method, known as the stochastic method, involves a probabilistic approach that analyzes multiple random operating scenarios for the RES, by using the Probabilistic Power Flow (PPF) in order to verify the system’s limits violation. However, the primary downside of the method is the computational effort it takes, as well as the loss of variables relevance when a vast number of scenarios are applied [7]. The third methodology regards the optimization-based approaches that address the hosting capacity as a maximization optimization problem, where the objective function evaluates the active power injected by the distributed generation, while the constraints model the network’s operation boundaries. Unlike the previous methods, the fourth approach, called the streamlined method, relies on multiple sensitivity analyses instead of load flow computation. Following the calculations, it creates three hosting capacity scenarios, namely the realistic, the optimistic and the most conservative one, leaving the final decision to user’s choice [8]. The methods mentioned above have been widely used in the recent literature as hosting capacity became a prevailing issue in both transmission and distribution systems. In [9], the authors propose a hybridization of the particle swarm optimization (PSO) and the gradient descent algorithm (GD) to efficiently estimate the hosting capacity of photovoltaics in distribution networks. In [10], the Grey Wolf Optimizer is applied for maximizing the hosting capacity by optimally coordinating the transformers, VAr sources and EVs, while a hybrid sine cosine artificial rabbits algorithm is employed in [11] to maximize hosting capacity by considering energy storage devices and the expansion of transmission lines. Battery energy storage systems and capacitor banks placement are used to increase PV hosting capacity in [12], using the classical Salp Swarm Algorithm. A stochastic analysis for assessing the PV hosting capacity based on Particle Swarm Optimization is conducted in [13]. The optimal allocation of Static VAr Compensators for enhancing photovoltaic hosting capacity, based on the Marine Predators Algorithm is presented in [14], while [15] applies Particle Swarm Optimization for calculating the PV hosting capacity. A Monte Carlo simulation is performed in [16] to compute the maximum hosting capacity in the standard distribution test system of South Korea based on the fast voltage stability index, while a deterministic approach is applied in [17] for a simple and quick assessment of HC in a radial grid within the University of São Paulo campus. On the other hand, a new risk and reliability perspective for HC analysis is proposed in [18] by combining users’ voltage sensitivity and reliability analysis in low-voltage networks. To address the HC of the transmission grid for distributed energy sources, the authors of [19] consider transient stability evaluation instead of steady-state analysis, while a market-based approach is proposed in [20], in which a bi-level optimization model is formulated, where both RES recovery investment and market clearing process are considered in order to maximize the HC.

In this paper, the HC problem in transmission systems is treated as an optimization problem as well. In this regard, an improved metaheuristic methodology is proposed, as the model’s complexity limits the applicability of the deterministic optimization approach. Hence, the proposed method, based on an Improved Salp Swarm Algorithm (ISSA), is structured as follows: firstly, the optimization problem determines the maximum active power installed in new production capacities, with respect to the network’s operational constraints (bus voltages and branches ampacity), while the second stage focuses on the optimal distribution of the active power to be installed in new capacities previously computed by minimizing the total active power losses. In order to maximize the active power capacity in the system, ISSA optimally applies additional control strategies including the reactive power output of existing generators, the slack bus specified voltage, transformers tap position and capacitor banks operational step deployed throughout the power grid. The main contributions of this research are summarized as follows:

• Assessment of the HC in transmission systems using an optimal reactive power management strategy;
• Development of an improved version of the conventional Salp Swarm Algorithm by introducing two new social categories within the salp population, alongside the existing leader salp and follower salps. The proposed pioneer salps move independently from the salp chain aiming to discover new promising areas within the search space, in order to compensate for the deficiencies in the exploration phase in the original SSA, while the rogue salps move around the food source, ignoring the leader salp guidance in order to improve performance during the exploitation phase.

The rest of the paper is organized as follows: Section 2 offers a brief description of the optimization problem. Section 3 presents the conventional SSA model along with the proposed enhancements. The simulation results are provided in Section 4. Finally, Section 5 concludes the study.

2. Optimization Problem Formulation

The optimization problem formulated in this paper aims to maximize the hosting capacity of a transmission network by applying an optimal reactive power management strategy (HC-ORPM). For this purpose, the optimization problem is divided into two stages, as follows: the first stage determines the maximum active power installed in new production capacities without violating the network’s operational constraints. Then, the total active power installed in the proposed capacities is redistributed throughout the network during the second stage with the objective of minimizing the active power losses.

• Objective functions

The objective function considered in the first stage, f_I(x_I), aims to maximize the total active power installed in the transmission network P_I,tot, by:

$$\begin{array}{l}\max f_I\left(x_I\right)=P_{I,tot}={\displaystyle \sum _{k=1}^{n_N-1}{P}_{IN,k}}\\ \mathrm{subject}\mathrm{to}:\begin{array}{c}{g}_I\left(x_I\right)=0\\ h_I\left(x_I\right)\le 0\end{array}\end{array}$$

(1)

where n_N is the total number of transmission network buses, P_IN,k is the active power installed at bus k, while g_I(x_I) and h_I(x_I) are the equality and inequality constraints.

For the second stage, the objective function f_II(x) is the minimization of total active power losses ΔP_T, determined based on the active power balance written for the transmission network:

$$\begin{array}{l}\min f_{II}\left(x_{II}\right)=\Delta P_T=P_{SL}+{\displaystyle \sum _{k=1}^{n_G}{P}_{gen,k}}+{\displaystyle \sum _{k=1}^{n_N-1}{P}_{IN,k}}-{\displaystyle \sum _{k=1}^{n_N}{P}_{L,k}}\\ \mathrm{subject}\mathrm{to}:\begin{array}{c}{g}_{II}\left(x_{II}\right)=0\\ h_{II}\left(x_{II}\right)\le 0\end{array}\end{array}$$

(2)

where P_SL is the active power supplied by the slack bus, P_gen,k is the active power supplied by each of the n_G generators, P_L,k is the active power demanded by the loads at bus k, g_II(x_II) and h_II(x_II) are the equality and inequality constraints for the second stage.

The control variables’ vectors for stage I, x_I, and stage II, x_II, are identical and consist of the active P_IN,k and reactive Q_IN,k powers generated at each bus k by the proposed production capacities, the specified voltage at the slack bus V_sp,SL, the reactive power generated by the existing generators Q_gen, the transformer tap position N_T for each of the n_T transformers and the operational step N_CB for each of the n_CB capacitor banks.

$$\left[x_I\right]=\left[x_{II}\right]=\left[P_{IN,1}\dots P_{IN,n_N-1},Q_{IN,1}\dots Q_{IN,n_N-1},V_{sp,SL},Q_{gen,1}\dots Q_{gen,n_G},N_{T,1}\dots N_{T,n_T},N_{CB,1}\dots N_{CB,n_{CB}}\right]$$

(3)

b.

Equality constraints

The equality constraints for both stages g_I(x_I) and g_II(x_II) are introduced in order to ensure that the load flow calculation is correctly determined for each candidate solution. For this purpose, the nodal active and reactive power balance equations are written for each of the n_N buses:

$$\begin{array}{cc}\begin{array}{l}{P}_{G,i}-P_{L,i}-V_i\cdot {\displaystyle \sum _{k=1}^{n_N}{V}_k\cdot \left[G_{ik}\cos \left({\theta }_i-{\theta }_k\right)+B_{ik}\sin \left({\theta }_i-{\theta }_k\right)\right]}=0\\ Q_{G,i}-Q_{L,i}-V_i\cdot {\displaystyle \sum _{k=1}^{n_N}{V}_k\cdot \left[G_{ik}\sin \left({\theta }_i-{\theta }_k\right)-B_{ik}\cos \left({\theta }_i-{\theta }_k\right)\right]}=0\end{array}& \mathrm{for}i=1\dots n_N\end{array}$$

(4)

where P_G,i and Q_G,i are the total active and reactive powers generated at bus i; V_i and V_k denote the voltage magnitude at buses i and k; G_ik and B_ik are the conductance and susceptance corresponding to the ik term within the admittance matrix; and θ_i and θ_k define the voltage angles at buses i and k.

c.

Inequality constraints

The inequality constraints h_I(x_I) and h_II(x_II) are integrated in the hosting capacity optimization problem in order to enforce the operational limits of the controlled devices such as the active and reactive power of the proposed generation capacities P_IN and Q_IN, the the slack bus voltage V_sp,SL, reactive power of existing generators Q_gen, the operational tap positions for the transformers N_T and operational steps for the capacitor banks N_CB, alongside the electrical network’s operational limits for bus voltages V_k and branch currents I_ik. In the following equations, the operational constraints are defined.

$$P_{IN,k}^{min}\le P_{IN,k}\le P_{IN,k}^{max},k=1\dots n_N-1$$

(5)

$$Q_{IN,k}^{min}\le Q_{IN,k}\le Q_{IN,k}^{max},k=1\dots n_N-1$$

(6)

$$V_{sp,SL}^{min}\le V_{sp,SL}\le V_{sp,SL}^{max}$$

(7)

$$Q_{gen,k}^{min}\le Q_{gen,k}\le Q_{gen,k}^{max},k=1\dots n_G$$

(8)

$$N_{T,k}^{min}\le N_{T,k}\le N_{T,k}^{max},k=1\dots n_T$$

(9)

$$N_{CB,k}^{min}\le N_{CB,k}\le N_{CB,k}^{max},k=1\dots n_{CB}$$

(10)

$$V_k^{min}\le V_k\le V_k^{max},k=1\dots n_N$$

(11)

$$\left|I_{ik}\right|\le I_{ik}^{max},\forall i,j\in N$$

(12)

Constraints (5) and (6) limit the injected active and reactive power at each bus, except the slack bus, the specified voltage at the slack bus is restricted based on Equation (7), while the existing generators’ reactive output, transformers’ plot and capacitor banks’ operation step boundaries are defined by (8), (9) and (10), respectively. The power grid operation constraints, namely the bus voltage level and the branches’ current ampacity are denoted by (11) and (12).

The inequality constraints in the first stage h_I(x_I) consist of the controlled devices limits (5)–(10) and the network’s operational limits (11)–(12). The second stage inequality constraints h_II(x_II) also integrate Equations (5)–(12), while additionally enforcing the requirement that the active power installed in the new capacities during the second stage P_II,tot be greater or equal to the active power installed after the first stage P_I,tot.

$$P_{II,tot}={\displaystyle \sum _{k=1}^{n_N-1}{P}_{IN,k}\ge P_{I,tot}}$$

(13)

d.

Adaptations for metaheuristic solvers

The hosting capacity optimization problem is solved within this study by using metaheuristic algorithms, therefore, several adaptations are introduced in the mathematical model. Firstly, the equality constraints h_I(x_I) and h_II(x_II) are enforced by performing the load flow calculation within the objective function. In this manner, all the unknown bus voltage angles θ and magnitudes U from Equation (4) are determined by applying the Newton–Raphson method for each candidate solution.

The inequality constraints (5)–(10) regarding the controlled devices for both stages, namely the proposed production capacities, existing generators, slack bus, transformers and capacitor banks, are integrated into the metaheuristic algorithms as the lower and upper bounds for the controlled variables P_IN, Q_IN, V_sp,SL, Q_gen, N_T and N_CB.

In order to integrate the other inequality constraints, the penalized objective functions F_I(x_I) and F_II(x_II) will be introduced for both stages. The majority of metaheuristic algorithms are defined to solve minimization problems, while the first stage of the HC-ORPM problem aims at maximizing the active power installed in new production capacities. Therefore, in order to avoid the necessity of implementing adjustments in the algorithms for solving maximization problems, the first stage objective function will be reformulated as a minimization problem:

$$\min F_I\left(x_I\right)=p_{LF}\cdot \left[\frac{M}{1+f_I\left(x_I\right)}+{\displaystyle \sum _{k=1}^{n_N}{p}_{V,k}+{\displaystyle \sum _{ik\in N_B}^{}{p}_{I,ik}}}\right]$$

(14)

where M is a numeric coefficient for scaling the objective function values, p_LF is the penalty coefficient for non-convergent load flow solutions, while p_V,k and p_I,ik are penalty coefficients for each bus voltage k and each branch ik.

During the optimization process, unfeasible solutions may occur, especially in the early exploration phase. The load flow calculation, performed in this case by using the Newton–Raphson method, will not converge for unfeasible solutions, therefore, the penalty coefficient p_LF is introduced with a relatively large numerical value such as 10⁵. In this manner, the unfeasible individuals are eliminated from the population, while for the feasible solutions, the p_LF values are equal to one.

The bus voltage inequality constraints, expressed in (11), are enforced by applying the p_V,k penalty voltage coefficient at each bus k. Consequently, the p_V,_k value is zero if the bus voltage value V_k is within the limits, while its value increases proportionally with the difference between the violated limit and the bus voltage value multiplied by the c_V numerical coefficient:

$$p_{V,k}=\left\{\begin{array}{c}{c}_V\cdot \left(V_k^{min}-V_{k,}\right),\mathrm{if}{V}_k<V_k^{min}\\ 0,\mathrm{if}{V}_{min}\le V_k\le V_{max}\\ c_V\cdot \left(V_k-V_k^{max}\right),\mathrm{if}{V}_k>V_k^{max}\end{array}\right.$$

(15)

The p_I,ik penalty coefficient is applied to enforce the branch currents’ constraints (12). For this purpose, for the overloaded branches, the p_I,ik coefficient is determined as the difference between the conductor’s ampacity and the current through the branch, which is scaled up using the c_I coefficient, while for the other branches, p_I,ik = 0.

$$p_{I,ik}=\left\{\begin{array}{c}\begin{array}{l}{c}_I\cdot \left(I_{ik}-I_{ik}^{max}\right),\mathrm{if}{I}_{ik}>I_{ik}^{max}\\ 0,\mathrm{if}{I}_{ik}\le I_{ik}^{max}\end{array}\end{array}\right.$$

(16)

The penalized objective function for the second stage F_II(x_II) is defined based on the f_II(x_II) objective function by integrating the penalty coefficients p_LF, p_V,k, p_I,k and p_P. The first three coefficients are also integrated in the first stage’s penalized function F_I(x_I), with the purpose of eliminating unfeasible solutions and maintaining bus voltages and branch currents within the limits. The p_P penalty coefficient is additionally introduced to enforce the inequality constraint defined in (13), which requires the total active power installed in new capacities during the second stage P_II,tot to be greater or equal to the total active power determined in stage I, P_I,tot.

$$\min F_{II}\left(x_{II}\right)=p_{LF}\cdot \left[f_{II}\left(x_{II}\right)+{\displaystyle \sum _{k=1}^{n_N}{p}_{V,k}+{\displaystyle \sum _{ik\in N_B}^{}{p}_{I,ik}}+p_P}\right]$$

(17)

However, metaheuristic solvers will encounter significant difficulties in determining new relevant values for the proposed capacities’ installed active powers that will comply with this constraint. For this reason, the total active power obtained during stage I is multiplied by α_P with values within the [0.99 … 0.999] interval in order to provide a wider search space for minimizing the active power losses. The p_P values are determined as the difference between P_I,tot and P_II,tot multiplied by the c_P coefficient when P_II,tot is less than α_P · P_II,tot, and zero otherwise.

$$p_P=\left\{\begin{array}{c}\begin{array}{l}0,\mathrm{if}{P}_{II,tot}\ge {\alpha }_P\cdot P_{I,tot}\\ c_P\cdot \left(P_{I,tot}-P_{II,tot}\right),\mathrm{otherwise}\end{array}\end{array}\right.$$

(18)

3. Improved Salp Swarm Algorithm

3.1. Salp Swarm Algorithm

Salps are marine invertebrates that share a visual resemblance to jellyfish and display swarm behavior when seeking food, forming what is known as a salp chain. This behavior inspired the development of the Salp Swarm Algorithm (SSA) proposed in [21]. The SSA has notable advantages, including its rapid convergence to the optimal value and minimal requirements in terms of adjusting its parameters [22]. These assets make this metaheuristic algorithm suitable for a multitude of optimization problems specific to engineering fields, including power systems. In [23], the SSA is applied to efficiently distribute single-phase loads amongst three phase of a distribution network, while the authors of [24] propose an SSA-based methodology for maximum power point tracking (MPPT) in photovoltaic systems. A smart and robust controller for islanded microgrids is introduced in [25], using an SSA to obtain the optimal transient response during different operating conditions. A multi-objective SSA is applied in [26] to estimate the number of PV modules and the storage battery capacity in standalone photovoltaic systems, while the authors of [27] also propose a multi-objective approach to solve transmission congestion problems.

Within a salp chain population, there is a designated leader and a group of followers. The leader is responsible for foraging for a food source, while the followers dynamically adjust their positions in relation to the salp preceding them, and consequently, in proximity to the leader. In the context of an optimization problem, at each iteration p, the leader’s position within the salp-chain $X_1^{\left(p\right)}$ is updated considering the position of the Food Source $F^{\left(p\right)}$ using the equation:

$$X_1^{\left(p\right)}=\left\{\begin{array}{l}{F}^{\left(p\right)}+c_1\left(\left(X_{max}-X_{min}\right)c_2+X_{min}\right),c_3>0.5\\ F^{\left(p\right)}-c_1\left(\left(X_{max}-X_{min}\right)c_2+X_{min}\right),c_3\le 0.5\end{array}\right.$$

(19)

where X_min and X_max are the upper and lower control variables’ limits and parameters, while c₂ and c₃ are randomly generated numbers in the [0, 1] interval. In order to compute parameter c₁, the following equation is applied:

$$c_1=2e^{-{\left(\frac{4t}{t_{MAX}}\right)}^2}$$

(20)

where t denotes the current iteration step, while t_MAX represents the total number of iterations considered.

As the leader’s position is updated, each follower adjusts its position $X_k^{\left(p\right)}$ based on the previous salp within the chain $X_{k-1}^{\left(p\right)}$:

$$X_k^{\left(p\right)}=\frac{1}{2}\left(X_k^{\left(p\right)}+X_{k-1}^{\left(p\right)}\right)$$

(21)

3.2. Improved Salp Swarm Optimization

According to [22], in light of the numerous advantages offered by the conventional SSA, an opportunity exists for modifications to enhance convergence speed while maintaining a balance between the exploration and exploitation phases. Consequently, numerous improved versions of the SSA have been proposed in the research literature. A hybrid between an SSA and SCA (Sine Cosine algorithm) is proposed in [28], while in [29], the authors introduce a sine–cosine operator and a Levy flight for improving the SSA’s performance. In [30], a Salp Swarm Algorithm hybrid is proposed by integrating the principles of simulated annealing. A chaotic map is introduced for improving the SSA’s performances in [31], and [32] proposes a mutation operator for enhancing the SSA’s exploration capabilities.

The social structure within the classic SSA consists of one Leader Salp, that moves around the Food Source and the Follower Salps which trail the leader by forming a spiral-shape chain around the food source. For the Improved Salp Swarm Algorithm, we continue the previous work [33] and we propose two new social categories, namely the Pioneer Salp and the Rogue Salp, respectively.

The Pioneer Salps are introduced to improve performance of the exploration phase as their movement is not subordinated to the Leader Salp. For this purpose, each Pioneer Salp will randomly choose, with equal probability, one of the four different position update mechanisms. In the first case, described by (22), the Pioneer Salp’s position is determined by adding a random variation to a randomly selected a salp X_R. The second Pioneer Salp position update mechanism consists of a random linear combination between two randomly selected salps X_R,1 and X_R,2, while the third option represents generating a new random position, as described in (23) and (24), respectively. The fourth alternative for determining the Pioneer Salp position is based on the opposition-based learning principle. Thus, the current position of the Pioneer Salp is determined as being “opposite” to its previous position, as given in (25). It should be mentioned that r₁ and r₂ from (22)–(25) are random numbers within the [0, 1] interval, while X_min and X_max are the lower and upper bounds for the control variables.

$$X_k^{\left(p\right)}=X_R^{\left(p-1\right)}+r_1\cdot \left[r_2\cdot \left(X_{max}-X_{min}\right)+X_{min}\right]$$

(22)

$$X_k^{\left(p\right)}=r_1\cdot X_{R,1}^{\left(p-1\right)}+\left(1-r_1\right)\cdot X_{R,2}^{\left(p-1\right)}$$

(23)

$$X_k^{\left(p\right)}=r_1\cdot \left(X_{max}-X_{min}\right)+X_{min}$$

(24)

$$X_k^{\left(p\right)}=X_{max}+X_{min}-X_k^{\left(p-1\right)}$$

(25)

On the other hand Rogue Salps are introduced to improve the ISSA’s performance in the exploitation phase. The Rogue Salps also move around the Food Source, but they disobey the Leader Salp’s guidance and determine their future position based on the Food Source and their actual position.

$$X_k^{\left(p\right)}=F_k^{\left(p-1\right)}+r_1\cdot \left[r_2\cdot F_k^{\left(p-1\right)}-\left(1-r_2\right)\cdot X_k^{\left(p-1\right)}\right]$$

(26)

where r₁ is a random number within the [−0.5, 0.5] interval, and r₂ is a random number within the [0, 1] interval. The [−0.5, 0.5] range is chosen for the r₁ values in order to assure variations in both increasing (r₁ > 0) and decreasing (r₁ < 0) the salp position values.

The classical SSA position update mechanism for the Follower Salps, given in (21), is also modified in the improved version, by generating a random linear combination between the two consecutive salps.

$$X_k^{\left(p\right)}=r_1\cdot X_{k-1}^{\left(p-1\right)}+\left(1-r_1\right)\cdot X_k^{\left(p-1\right)}$$

(27)

For each salp, except the Leader, a random number r ∈ [0, 1] is generated, determining its type. Firstly, there is a p_foll, or probability to become a Follower Salp (if r < p_foll). If the current salp is not a Follower, a new random number r ∈ [0, 1] is generated. The salp has a p_pion probability of being a Pioneer Salp (if r < p_pion), otherwise (if r ≥ p_pion) it will become a Rogue Salp. The Follower Salp probability p_foll increases linearly from $p_{foll}^{min}$ to $p_{foll}^{max}$ during the iterations, while the p_pion probability decreases linearly from $p_{pion}^{max}$ to $p_{pion}^{min}$ as iterations advance:

$$p_{foll}=\frac{p}{p_{\max }}\cdot \left(p_{foll}^{max}-p_{foll}^{min}\right)+p_{pion}^{min}$$

(28)

$$p_{pion}=\left(1-\frac{p}{p_{max}}\right)\cdot \left(p_{pion}^{max}-p_{pion}^{min}\right)+p_{pion}^{min}$$

(29)

A flowchart for the Improved Salp Swarm Algorithm is presented in Figure 1, alongside the classical SSA flowchart, where green lines indicate the steps performed by the ISSA, while the classical SSA is represented using orange lines. The source code of the proposed ISSA is publicly available at https://www.mathworks.com/matlabcentral/fileexchange/155984-improved-salp-swarm-algorithm, accessed on 4 December 2023.

4. Case Study

4.1. Network under Study

The case study presented in this paper is conducted on the IEEE-30 bus test system [34], depicted in Figure 2, consisting of nine high voltage buses with a rated voltage of 132 kV and 21 medium voltage buses with a rated voltage of 32 kV. The IEEE 30 test systems buses are connected by 41 branches represented by high and medium voltage electrical lines and transformers. The existing production capacities consist of six generators, out of which one is modeled as the slack bus. Also, two capacitor banks are connected to the IEEE 30 test system in order to provide additional reactive power support.

4.2. ISSA Performance Analysis

The proposed ISSA’s performance is analyzed within this section, firstly on the commonly used benchmark functions and then on the first stage of the proposed hosting capacity optimization problem.

4.2.1. Benchmark Functions

This section presents a comparison between the performance of the proposed ISSA and the classical SSA in solving the 23 well-known benchmark functions applied recurrently in the literature, according to [21]. The first seven functions are unimodal, while the other functions are multimodal with variable dimensions (F8–F13) and fixed dimensions (F14–F23). Firstly, a parameter analysis is conducted in order to determine the most suitable values for the four parameters introduced in the proposed ISSA: the minimum and maximum values for the p_foll and p_pion probabilities. In this regard, the probabilities are considered to be between 0.3 and 0.7 with a granulation of 0.1, resulting in ten pairs of $p_{foll}^{min}$ − $p_{foll}^{max}$ and $p_{pion}^{max}$ − $p_{pion}^{min}$. For each range of p_Foll values considered, all ten values for the p_Pion are simulated. Additionally, for each established combination of p_foll and p_pion values, ten simulations are conducted for the Rastrigin test function, considering 100 individuals and 100 iterations. The average objective function values for all the 100 simulation sets are presented in

Table 1. Parameter analysis results for ISSA.

pfoll Range	ppion Range
	0.3–0.4	0.3–0.5	0.3–0.6	0.3–0.7	0.4–0.5	0.4–0.6	0.4–0.7	0.5–0.6	0.5–0.7	0.6–0.7
0.3–0.4	15.135	12.812	11.319	10.807	11.727	9.301	10.803	11.529	13.562	14.081
0.3–0.5	11.539	11.757	11.511	13.424	12.135	11.317	11.954	10.241	10.264	9.554
0.3–0.6	13.699	12.298	11.647	9.763	13.066	13.073	11.996	9.647	9.986	12.886
0.3–0.7	11.445	12.727	11.211	12.241	11.781	12.120	8.595	12.253	13.650	10.634
0.4–0.5	13.092	13.147	12.489	10.563	13.088	11.644	9.861	10.763	11.886	10.498
0.4–0.6	12.662	12.681	7.858	9.366	11.747	11.948	10.007	10.613	8.348	8.266
0.4–0.7	10.977	11.733	10.406	10.391	12.863	11.790	8.131	8.429	9.418	10.448
0.5–0.6	11.946	10.748	8.801	7.718	10.065	9.798	9.645	12.067	10.538	10.538
0.5–0.7	12.248	10.462	8.419	8.811	11.561	9.598	7.626	8.911	8.334	10.569
0.6–0.7	11.767	8.444	7.519	7.438	8.700	9.090	7.885	9.099	7.630	7.838

.

The results reveal that the ISSA obtains its peak performance when the probability for the Follower Salps ranges between 0.6 and 0.7, and between 0.3 and 0.7 for the Pioneer Salps. In this case, the population structure during the first iterations is likely to be 70% Followers, 21% Pioneer Salps and 9% Rogue Salps, and during the final iterations, it is likely to be 60% Followers, 12% Pioneers and 28% Rogues.

The best, average and worst values obtained by the ISSA and SSA in solving the 23 popular benchmark functions during 30 consecutive iterations with 100 individuals and 100 iterations are presented in

Table 2. Results of SSA and ISSA on the benchmark test functions.

Benchmark Function	Best		Average		Worst		Std. Dev.
	ISSA	SSA	ISSA	SSA	ISSA	SSA	ISSA	SSA
F1	2.40 × 10−9	154.7873	1.93 × 10−6	527.423	9.57 × 10−5	957.2717	9.84 × 10−6	153.3441
F2	1.22 × 10−7	0.57535	1.34 × 10−5	1.800001	8.10 × 10−5	3.579042	1.62 × 10−5	0.650522
F3	4.28 × 10−10	10.76084	1.36 × 10−6	81.17091	3.61 × 10−5	202.7179	4.12 × 10−6	47.51527
F4	4.55 × 10−6	1.414139	1.09 × 10−4	4.209265	6.84 × 10−4	7.065108	1.17 × 10−4	1.363963
F5	5.880441	23.07654	7.117046	605.2254	8.570467	3795.997	0.37483	688.2878
F6	1.63 × 10−6	5.278046	8.00 × 10−6	44.6723	3.10 × 10−5	146.5104	5.72 × 10−6	31.26406
F7	1.18 × 10−4	0.000668	0.000881	0.004582	0.005056	0.021737	0.000699	0.004169
F8	−3972.39	−3049.62	−2967.1	−2042.15	−2175.19	−1379.48	376.1043	312.5094
F9	3.37 × 10−11	3.225794	8.822098	17.00788	33.85414	38.96025	7.709303	7.779694
F10	1.06 × 10−6	2.366373	2.40 × 10−5	4.20851	1.59 × 10−4	7.838593	2.51 × 10−5	0.914411
F11	5.02 × 10−11	0.520659	0.087861	1.23175	0.442987	2.443362	0.091332	0.309467
F12	2.00 × 10−7	0.176262	1.75 × 10−6	1.538313	6.10 × 10−6	4.516636	1.09 × 10−6	0.961499
F13	1.40 × 10−6	0.430243	0.000481	2.568554	0.010995	7.754975	0.002171	1.420221
F14	0.998004	0.998004	1.503895	4.671659	5.928845	12.67051	0.883455	2.769594
F15	0.000307	0.000308	0.000511	0.001446	0.001407	0.007671	0.000207	0.001435
F16	−1.03163	−1.03163	−1.03163	−1.03162	−1.03163	−1.03152	6.01 × 10−9	1.91 × 10−5
F17	0.397887	0.397887	0.397888	0.400199	0.39789	0.546348	4.41 × 10−7	0.015181
F18	3	3	3	3.002473	3.000003	3.113857	4.48 × 10−7	0.012518
F19	−3.86278	−3.86274	−3.86278	−3.83965	−3.86277	−3.70485	1.69 × 10−6	0.027524
F20	−3.32199	−3.32112	−3.27439	−3.11051	−3.20252	−2.38433	0.058589	0.147967
F21	−10.1532	−10.1526	−9.59521	−6.70491	−5.05518	−2.2562	1.594012	3.354896
F22	−10.4029	−10.4025	−9.84836	−8.04177	−3.72426	−2.44818	1.694632	2.879728
F23	−10.5364	−10.536	−9.94575	−7.53868	−3.83527	−2.39173	1.889872	3.1686

Bold values indicate the algorithm which obtains the better performance.

, alongside the standard deviation. The results demonstrate that the ISSA delivers better average and maximum objective function values on all 23 functions, while obtaining better minimum values for 20 out of 23 functions. In conclusion, the proposed ISSA delivers superior performance relative to the classical Salp Swarm Algorithm.

A convergence study is conducted for the F1, F10 and F14 benchmark functions. For these functions, the search history, the first variable trajectory of a randomly chosen salp, the average fitness functions value for the entire salp population and the convergence curve are presented in Figure 3. The search history reveals that the salps cover a significant part of the search space during the exploration phase and they concentrate around the optimal solution during the exploitation phase. In addition, the trajectory demonstrates that, during the first iterations, the first variable shows important variations, while a fine-tuning process is observed during the final stages. The convergence curves demonstrate that the salps are efficient in detecting the approximative position of the optimal solution after less than 30 iterations.

4.2.2. Hosting Capacity

This section presents a comparison of performance between the proposed ISSA and five state-of-the-art metaheuristic algorithms: Ant Lion Optimization (ALO) [35], Grey Wolf Optimization (GWO) [36], Sine Cosine Algorithm (SCA) [37], Salp Swarm Algorithm (SSA) [21] and the Whale Optimization Algorithm (WOA) [38] in solving the first stage of the hosting capacity maximization problem. In this regard, four simulation sets are considered by employing each algorithm in solving the optimization problem for ten consecutive times with different settings for population size × maximum iterations number: 50 × 50, 100 × 100, 200 × 200 and 400 × 400. It should be mentioned that 50 × 50 and 100 × 100 are common settings for medium-difficulty optimization problems. Therefore, these values were the starting point for this analysis. Given the hosting capacity problem’s increased complexity, with 70 control variables, the 200 × 200 and 400 × 400 cases were also analyzed.

The first stage of the optimization problem aims at maximizing the total active power installed in the electrical network by optimally coordinating the voltage setpoints and reactive power support provided by the generators, the transformer taps and the operational step of capacitor banks, while assuring that all line loadings and bus voltages are within the limits.

An analysis regarding the total active power installed in the transmission network (P_I,tot) determined by each algorithm under each simulation set is presented in Figure 4.

Figure 4a presents the minimum P_I,tot values obtained by the algorithm for the simulations conducted in each set. As can be observed, among the 10 consecutive simulations ISSA determines the best minimum values of 301.7 MW, 322.1 MW, 370.4 MW and 377.3 MW, respectively. The second-best performance regarding the minimum P_I,tot values is obtained by the GWO for the 50 × 50 and 200 × 200 simulation sets with 288.9 MW and 336.6 MW. The WOA qualifies second on the 100 × 100 simulation with 288.4 MW. For the 400 × 400 set, the ALO obtains the second place with a minimum P_I,tot value of 325.8 MW, with 15.8% lower relative to the result obtained by the ISSA.

The average P_I,tot values, shown in Figure 4b, also demonstrate that the ISSA delivers the best performance in the 100 × 100, 200 × 200, 400 × 400 simulation sets with 372.1 MW 382.3 MW and 383.6 MW, followed by the GWO. For the first simulation set (50 × 50), GWO occupies the first place with the average P_I,tot value of 335.8 MW while the ISSA delivers the second-best performance of 326.7 MW.

The maximum P_I,tot value, determined by each algorithm in the ten consecutive simulations conducted for each pair values for population size and iterations number, is presented in Figure 4c. The results for the 50 × 50 and 100 × 100 simulation sets reveal that GWO obtains the highest maximum P_I,tot values of 378.0 MW and 386.3 MW, while the second place is occupied by the SCA with 376.9 MW and by the ISSA with 383.3 MW. For the 200 × 200 and 400 × 400 simulations, the ISSA provides the best performance by determining maximum P_I,tot values of 386.5 MW and 386.6 MW. However, GWO provides an approximately equal performance with 386.43 MW and 386.44 MW, representing a 0.02% and 0.03% variation relative to the values determined by the ISSA. Therefore, when considering low computational efforts, GWO outperforms the ISSA, while the proposed algorithm’s superior exploration and exploitation succeeds in finding better solutions when the number of individuals and iterations are increased.

As the ISSA obtains the most consistent P_I,tot values throughout the considered simulations, a detailed analysis of the algorithms’ consistency is conducted. Figure 5 presents the standard deviation of the P_I,tot values determined by each algorithm in each simulation set. The best standard deviation values for the 50 × 50 set were 7.2 MW, obtained by ALO, and 6.6 MW, obtained by WOA for the 100 × 100 set. For the 200 × 200 and 400 × 400 simulation sets the best standard deviation values of 4.9 MW and 2.9 MW were obtained by the ISSA, which additionally obtained the best performances for minimum, average and maximum P_I,tot values. Although GWO obtained similar average and maximum P_I,tot values for the aforementioned simulation sets, the higher standard deviation values of 15.2 MW and 31.3 MW demonstrates the ISSA’s superior performance and robustness.

The standard version of the Salp Swarm Algorithm obtained the worst performance on the 400 individuals × 400 iterations simulation set, achieving a maximum P_I,tot value of 316.1 MW. On the other hand, the improved version proposed by the authors achieved 383.6 MW, the best performance on this simulation set, with P_I,tot improved by over 20% relative to the original SSA.

The hosting capacity optimization problem is characterized by an increased complexity, given the relatively large number of 70 control variables for the network under study comprised of:

• active power and corresponding power factors installed at 29 buses;
• slack bus voltage setpoint;
• reactive power outputs for five existing generators;
• tap positions for the four transformers;
• operational steps for the two capacitor banks.

Consequently, all the selected algorithms delivered poor performances for the 50 individuals × 50 iterations simulations, as the computational effort is not sufficient to provide accurate results. By increasing the population size and maximum iterations to 100, the performance is improved in terms of objective function values, but the results are characterized by a reduced consistency as the standard deviation values are relatively high.

The 200 × 200 and 400 × 400 simulation sets provide suitable computational effort, enabling both the ISSA and GWO to reach similar maximum P_I,tot values around 386.5 MW. In conclusion, the ISSA simultaneously achieves the best performance and the highest degree of robustness relative to the other five state-of-the-art metaheuristic algorithms considered in this study.

Moreover, a further increase in computational effort by considering 600 individuals and 600 iterations will only achieve a relatively reduced ISSA performance improvement, by 0.026% in terms of maximum P_I,tot values, while increasing the simulation time from 141 s to 338 s.

The average simulation time is presented in Figure 6 for each algorithm during each simulation set. The results show that the ALO is the slowest algorithm in all simulation sets, while the ISSA obtains a moderate simulation time relative to the other considered algorithms. For example, in the 400 × 400 simulation set, the fastest three algorithms are the WOA, GWO and SCA with average simulations times between 135 and 139 s; the ISSA follows very closely, with 141 s, while the SCA and ALO are the slowest with 210 s and 271 s. It should be noted that the simulations were conducted on a mid-range laptop equipped with an Intel i5, 11th generation processor (1135G7) with 2.4 GHz base frequency, SSD and 16 GB RAM.

Figure 7a shows the average convergence curves obtained by the considered algorithms during the 200 individuals × 200 iterations simulation set, while the curves for the 400 × 400 set are presented in Figure 7b,c. The results in Figure 7a clearly indicate that the ISSA obtains the best performance during the 200 × 200 simulation set, followed at considerable distance by the ALO, WOA and SCA, while the GWO and SSA are not shown, as a penalized solution increased the average objective function values during the simulations to 8949 p.u. and 2173 p.u., above the 4 p.u. threshold applied in the figure.

The convergence curves for all algorithms are presented in Figure 7b, while Figure 7c only shows the values below the 4 p.u. threshold. The GWO is the first algorithm to obtain an objective function value below 3.0 p.u., after only seven iterations, while the ISSA convergence during the first iterations is gradual and constant, requiring fourteen iterations to reach this threshold. However, after the 39th iteration, the ISSA delivers the lowest objective function values, while GWO is the second-best performant, and ALO the third. By comparison, the original SSA is the least efficient algorithm as it determines the first improvement of the initial best objective function value after 86 iterations and the average best objective function value is 3173 p.u. after 400 iterations, demonstrating the effectiveness of the proposed Rogue and Pioneer Salps.

4.3. Hosting Capacity Optimization Results

4.3.1. First Stage Results

The first stage of the proposed hosting capacity maximation problem aims at maximizing the injected active power while maintaining the line loadings and bus voltages within the allowed limits. For this purpose, the optimal reactive power management is simultaneously determined in order to maximize the hosting capacity by including all controllable devices within the transmission network: such as generators, transformers and capacitor banks. In order to analyze the performances achieved by the first stage of the HC-ORPM problem, two additional scenarios are considered:

• The first scenario consists of successive random allocations of generation capacities within the transmission network until the limits are violated. This random scenario is inspired by the current Romanian regulations for obtaining grid-connection permits, which allows all investors with the desired installed capacity and connection solution to apply. The TSO and/or DSOs will issue the grid connection permit, with the necessary network reinforcement measures if required, that will be supported by the investors;
• In the second scenario, the hosting capacity is maximized without applying the optimal reactive power management strategy, while the third scenario consists of the first stage from the proposed HC-ORPM strategy. It should be mentioned that the ISSA is applied in the second and third scenarios with a population size of 400 and maximum iterations of 400.

Figure 8 presents a boxplot of the maximum active power that can be installed within the transmission network P_I,tot, determined for 100 simulations in the random scenario, and for 10 simulations for each of the HC and HC-ORPM scenarios. The results reveal that the random scenario obtains a median value of 92.7 MW and a maximum value of 262.3 MW for the additionally installed capacity P_I,tot. In the second scenario, by applying the hosting capacity maximization considering only the active power of the new production capacities, the maximum P_I,tot value increases to 337.6 MW, representing a 28.7% improvement relative to the random scenario. The third scenario represents the proposed optimization model, where hosting capacity is determined by considering the optimal reactive power management strategy, resulting in a further increase in the maximum P_I,tot values, up to 386.6 MW, which represents a 14.5% improvement relative to the second scenario.

The active power installed at each node is shown in Figure 9 for each of the ten simulations conducted in the first stage of the proposed model. The results are presented as stacked bars, where the nodes are depicted consecutively with different colors.

The results reveal that the P_I,tot values are consistent, as the difference between the maximum value of 386.6 MW and the minimum of 377.3 is 9.3 MW, representing 2.4%, while the standard deviation of the P_I,tot values is 2.9 MW. However, the total value is obtained by different combinations of the active power values installed at each bus. As the results in Figure 9 reveal, for the best three simulations the largest difference between the P_I,tot is 0.27%, as the values are within 386.6 MW and 385.5 MW. Nevertheless, the relatively constant total active power is distributed differently across the network. For example, the active power installed at buses 1 and 2 is 0.8 MW and 0.1 MW in the first simulation, 11.9 MW and 1.9 MW during the second simulation and 8.2 MW and 5.2 MW during the third. The active power installed at node 3 is 270.5 MW for the first two simulations and 282.1 MW for the third, while 14.4 MW, 0.0 MW and 3.0 MW are connected at the fourth bus during the first three simulations. Furthermore, during the ten simulations, all active power values, except bus 3, are characterized by an important variation, as the minimum value is below 0.1 MW while the maximum values are as high as 34.7 MW at bus 6, 30.0 MW at bus 8 and 28.9 MW at buses 9, 20 and 24. These results reveal the complexity of the HC problem, as there are multiple solutions leading to similar performances.

The highest loaded branches and the bus voltage profiles obtained during the ten simulations are presented in Figure 10 and Figure 11, respectively. The results clearly reveal that the two electrical lines connected to the slack bus, 2–30 and 3–30, are loaded close to their maximum capacity, with average loadings of 99.96% and 99.93%. The hosting capacity is therefore limited by the total capacity of transmission lines directly connected to the slack bus, as this bus absorbs all the excess active power generated across the electrical network. The 3–4 transmission line reaches a maximum loading of 95.7%, qualifying as the third highest loaded branch.

The results also reveal that branch loadings present considerable variations between simulations, as the values for line 6–8 are comprised between 32.2% and 94.6%, being in turn, the 4th, 5th, 7th or 8th highest loaded branch. The highest branch loadings are observed on lines 2–30 and 3–30, while the third and fourth positions are occupied by lines 3–4, 2–4 or 6–8.

The HC-ORPM also controls the reactive power, by determining different settings for the slack bus voltage, existing and proposed generators’ reactive power output, transformer taps and operational step for the capacitor banks. Given that the solutions determined after the first stage have different active power distributions throughout the network, the reactive power management strategy is also characterized by significant variations between solutions, resulting in different bus voltages profiles, as shown in Figure 11.

Based on the results, it can be concluded that the ISSA provides a good performance in solving the hosting capacity optimization problem as it increases the total active power installed in the network, aiming to operate the most sensitive lines close to their capacity. However, as the majority of branches are not heavily loaded, there is a seemingly infinite number of combinations available for distributing the total active power to the network buses.

4.3.2. Second Stage Results

The proposed methodology for hosting capacity maximization implies the application of an optimal reactive power management strategy (HC-ORPM) consisting of the following two stages: (1) maximize the total active power installed in the power system P_I,tot and (2) minimize the active power losses by optimally redistributing the total active power P_I,tot throughout the network.

Similar to the results obtained in the first stage, the active power distribution across the network shows significant variation for different simulations, although similar P_I,tot values are achieved consistently. For this reason, identifying the optimal solution based on the first stage results is relatively complicated. Therefore, the second stage is introduced in order to provide an additional criterion for identifying the optimal solution.

The results presented in this section are determined by applying the ISSA in solving the second stage of the HC-ORPM problem for ten consecutive simulations with a population size of 400 individuals and a maximum iterations number of 400. Figure 12 presents a boxplot analysis of the active power losses obtained after the first and second stages. The results reveal that the second stage reduces the active power losses to ΔP_S2 = 17.65 MW, representing a 4.5% improvement relative to the best solution obtained in the first stage, when ΔP_S1 = 18.44 MW. Furthermore, after the second stage, the active power losses are more consistent with significantly lower interquartile range values of 0.12 MW and 0.18 MW in comparison with the 0.33 MW and 0.56 MW values achieved after the first stage.

The control variables’ values for the optimal solution determined by the ISSA after the second stage of the HC-ORPM problem are presented in

Table 3. The control variables values for the optimal solution.

Pgen,i (MW)/cosφi at Bus i
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
5.15	1.21	273.23	0.01	0.90	6.57	1.29	0.54	0.26	0.55	8.94	4.41	0.10	7.01	0.38
0.93	0.93	0.997	0.91	0.92	0.94	0.94	0.98	0.94	0.93	0.96	0.96	1.00	0.99	1.00
16	17	18	19	20	21	22	23	24	25	26	27	28	29
1.54	0.20	4.15	7.23	6.13	25.78	4.73	0.76	0.07	0.02	0.87	0.04	13.12	8.94
0.94	0.98	0.90	0.94	0.90	1.00	1.00	0.95	0.97	0.93	0.96	1.00	0.98	0.98
Qgen (MVAr)					Usp,SL (pu)	Transformer Tap				CB Step
Qgen2	Qgen5	Qgen8	Qgen11	Qgen14	Usp,30	NT9-6	NT10-6	NT12-4	NT27-28	NCB10	NCB24
67.09	21.56	30.24	−2.70	−0.90	1.0714	0	7	−3	1	16	0

. Based on the results, it can be observed that the algorithm reduces the active power losses by installing 77.3% of the total active power P_I,tot at buses with 132 kV rated voltage in order to reduce the current flow through branches. On the other hand, 71.1% of P_I,tot is installed to bus 3, which is directly connected to the slack bus, in order to reduce power losses by minimizing the distance between the injection and absorption buses.

The branch loadings and bus voltages, determined by applying control variables’ values determined in the optimal solution provided in the second stage of the HC-ORPM, are shown in Figure 13 and Figure 14. The highest loadings, of 99.96% and 99.14%, are observed on lines 2–30 and 3–30, these are the only connections to the slack bus 30, which absorbs the excess active power generated throughout the electrical network. Lines 3–4, 2–4 and 4–6 achieve loadings of 91.1%, 85.4% and 67.6%, and are also transmitting the active power toward the slack bus via buses 2 and 3. The loading for the other branches, except the electrical lines 2–6 and 22–24, are below 50%, out of which 29 branches are loaded less than 30%, and 17 are below 20%. Although the HC-ORPM aims to maximize the active power installed in the network, the objective function for the second stage aims to maximize the active power losses by installing 77.3% of the total active power P_I,tot at the eight 132 kV buses, as close as possible to the slack bus. The remaining 22.7% of P_I,tot is connected to the distribution network to supply the 94.1 MW demanded by loads connected within the medium voltage network. The active power installed in the 32 kV area is distributed as close as possible to the loads in order to contribute to losses reduction by minimizing the current through branches. For example, the largest load demand of 17.5 MW is connected to bus 21, where the largest P_IN of 25.8 MW is also installed.

The results presented in Figure 14 show that the voltage values for all 132 kV buses, namely 2–8, 28 and 30, are comprised between 1.071 p.u., at bus 30, and 1.099 p.u., at bus 3. The optimal solution aims to minimize the active power losses, therefore, the voltage values are increased, close to the upper 1.10 p.u. limit, in the 132 kV area of the network. However, the bus voltages within the 32 kV area are comprised between 1.07 p.u., at bus 12, and 1.020 p.u., at bus 1. Increased active or reactive power injections in the 32 kV area would be required in order to increase the bus voltages. However, partially redistributing the installed active power from the 132 kV area to 32 kV area will result in increased active power losses. Consequently, the transformer taps, the generators’ reactive power output and the capacitor banks’ steps are configured in order to increase the bus voltages without increasing the reactive power flow through the MV branches.

5. Conclusions

As current worldwide policies promote the mass integration of new sources of energy production (especially renewable energy sources), the hosting capacity assessment must be regardfully carried out, as the extensive connection of new generation units can lead to great disruptions in the existing power systems. In this paper, the hosting capacity problem is addressed as an optimization problem, with the aim of obtaining the maximum active power injection from distributed generation with respect to the operational boundaries of the system’s components. Due to the model’s complexity, an improved Salp Swarm Algorithm is developed in order to solve the HC problem in a two-stage approach. In addition, to investigate the efficiency of the proposed ISSA, a performance analysis alongside five state-of-the-art metaheuristic algorithms (ALO, GWO, SSA, SCA and WOA) has been conducted. In this regard, multiple simulation sets defined by different settings for population size and maximum iterations number have been used. Following the simulations, the ISSA proved to be the most robust algorithm as it achieves the best performance indices.

Regarding the hosting capacity optimization problem, the first stage results revealed that the proposed model determines a significant improvement of 47.4% and 14.5% in total active power installed in new capacities compared to two other methodologies. During the second stage, the active power losses are minimized by redistributing the total active power previously determined throughout the transmission network, while also implementing the optimal reactive power management. The obtained results show a reduction in active power losses by at least 4.5% based on the proposed approach.