The data used in this study were obtained from Chad’s National Electricity Company (SNE) and pertain to the medium-voltage distribution network of the city of N’Djamena. The observation period covers the year 2023. The collected data include the characteristics of MV feeders and MV/LV transformers, outage histories, power demand figures, the geographical locations of transformers and photovoltaic sites, and the information required to calculate reliability indices.

The studied network section consists of three medium-voltage (15 kV) feeders supplying a total of 90 MV/LV (15/0.4 kV) transformer substations, serving approximately 22,560 customers across various residential, commercial, and administrative areas of N’Djamena. The network features a conventional looped topology that allows for supply restoration operations in the event of a fault.

Reliability assessment is based on operational records from 2023. Failure parameters were derived from actual events observed on the network (outage frequency and duration), assuming a mean time to repair of 12 hours. Network reconfiguration maneuvers are carried out using a mixed switching strategy combining manual operations with existing network switching devices to isolate faulty sections and restore power to healthy sections.

The SAIFI, SAIDI, CAIDI, EENS, ECOST, AENS, and VAENS indices were calculated both before and after the integration of distributed photovoltaic generation to quantitatively assess the impact of the proposed method on distribution network reliability.

Parameters for Evaluating the Reliability of the Conventional Network

This involves evaluating the conventional distribution network by determining, for each feeder:

-Customer-based reliability indices (SAIFI, SAIDI, and CAIDI);

-And energy-based reliability indices (EENS, ECOST, AENS, and VAENS).

1)Determinationofcustomer-basedreliabilityindices

where n = total number of load points, λ_i = average failure rate of each segment i, N_i = number of customers shed at the i^th node, and N = total number of customers served.

2)Determinationofenergy-basedreliabilityindices

where U_i is the annual outage duration for load point i, La is the average load at each load point busbar, P is the received active power, C is the cost of the received active power, and H is the duration or period (in hours) of the received active power availability.

A wide range of optimization techniques has been investigated for the optimal allocation of distributed generation, including genetic algorithms, mathematical programming and particle-swarm-based methods. Recent studies have also employed improved or hybrid metaheuristic algorithms to determine the optimal location and sizing of renewable distributed generation while satisfying technical and reliability constraints [22]-[25].

In this study, an Improved Particle Swarm Optimization algorithm (IPSO) is therefore proposed. The objective is to optimize the position (x, y coordinates) of each production site relative to the position of the corresponding transformer substation while simultaneously maximizing production to meet local consumption needs as well as a significant portion of the demand in the area supplied by that same transformer. To achieve this, an improved Particle Swarm Optimization algorithm (IPSO) is used as follows:

Let X (x_i, y_i) be the coordinates of any given physical site (where x_i is the longitude and y_i is the latitude of site i); the IPSO initiates the optimization process using the initial positions of the PV production sites.

X 0 k : initial coordinate of a PV production site.

The determination of PV production site locations proceeds in two stages:

-Stage 1: Determining the production location based on energy demand and the capacities and locations of MV/LV substations in the zone. The search process yields the optimal production site location, Pbest;

-Stage 2: Determining the production location for a neighboring zone, based on the capacities and locations of MV/LV substations. The search process yields the optimal location for the neighboring production site, Lbest; this continues until the optimal site locations for the entire distribution network, Gbest, are determined.

The principles governing changes in velocity and position are defined as follows:

where ω is the weighting function; C₁ and C₂ are the weighting factors; rand₁ and rand₂ are two uniformly distributed random numbers generated independently between 0 and 1; V i k and V i k + 1 are the particle velocities at iterations k and k + 1, respectively; and ( x i k , x i k + 1 ) and ( y i k , y i k + 1 ) are the longitude and latitude of site i at iterations k and k + 1, respectively.

The weighting function is given by the equation:

where: ω_max = initial weight, ω_min = final weight, itermax = maximum number of iterations, and iter = current iteration.

The following table (Table 1) presents the principal parameters of the proposed IPSO algorithm.

Table 1. Parameters of the proposed IPSO algorithm.

The proposed IPSO algorithm was implemented using a swarm of 50 particles and a maximum of 100 iterations. A constant inertia weight of 0.7 was selected based on preliminary numerical tests to provide a suitable balance between global exploration and local exploitation during the optimization process. The cognitive and social learning coefficients were fixed at 1.5 and 2.0, respectively. Since PSO is inherently stochastic, the optimization was independently executed 15 times, and the solution with the best objective function value was retained. The search constraints were enforced using a boundary projection strategy: whenever a particle exceeded the admissible search domain, its position was projected back onto the corresponding boundary. This implementation was specifically configured for the simultaneous optimization of the geographical coordinates and capacities of the distributed photovoltaic sites.

2.3.1. Conceptual Diagram of the Improved PSO Applied to PV Placement

General principle

Each particle in the swarm represents a candidate solution defined by:

- The geographical coordinates of the PV sites (x, y);

- The power injected by each PV site;

-The distribution network transformers have fixed positions, which impose a rigid network topology.

The operating cycle of the IPSO (Improved Particle Swarm Optimization) algorithm can be represented by Figure 6 below.

Figure 6. Flowchart of the IPSO algorithm for optimizing the coordinates of PV production sites.

2.3.2. Complete Mathematical Formulation of the Problem

1)Decisionvariables

The optimization problem aims at determining the optimal geographical coordinates of the photovoltaic sites. The decision vector is defined as

where:

L a t i is the latitude of PV site i; L o n i is the longitude of PV site i; N = 25 is the number of PV sites considered in this study.

Each particle of the swarm therefore represents a complete geographical configuration of all photovoltaic sites.

2)Objectivefunctions

The objective is to determine an optimal geographical distribution of the photovoltaic sites while maintaining realistic installation locations and improving the electrical performance of the distribution network.

The optimization problem is formulated as

where:

is the Euclidean distance between the i^th photovoltaic site and its associated transformer.

The penalty function is defined as:

where M is a sufficiently large penalty coefficient

The first term of Equation (12) minimizes the distance between each photovoltaic site and its associated distribution transformer, thereby reducing feeder extension requirements and facilitating local power injection.

The second term of Equation (12) maximizes the geographical dispersion of photovoltaic sites by penalizing excessively close installations, thus preventing unrealistic clustering of generation units.

The third term of Equation (12) represents a penalty function that guarantees feasible geographical solutions.

The weighting coefficients α, β, and γ regulate the relative contribution of each objective component.

Constraints

There are at least four constraints to be met in this optimization: constraints regarding voltage limits, power limits, and transformer capacity limits, as well as constraints concerning the zones authorized for injection into a conventional grid transformer. These constraints are formulated as follows:

Voltage constraints:

PV power constraints:

Transformer capacity constraints:

Geographical constraints:

To efficiently inject power generated at a production site into the conventional distribution grid (following the assessment and optimization of the power produced and the energy demand at each PV site), an intelligent interface system is installed, comprising:

-A pure sine wave inverter sensitive to distribution grid fluctuations;

-An intelligent stabilizer capable of controlling and maintaining the frequency and voltage of both the production site and the grid within acceptable limits;

-An intelligent meter capable of real-time monitoring of energy produced, energy consumed, and energy fed into the grid;

-An intelligent residual-current circuit breaker capable of continuously monitoring the inputs and outputs of the PV injection system regarding both the site and the grid.

The following images (Figure 7) show the schematic diagram of the PV injection process:

Figure 7. Architecture of photovoltaic grid injection system.

Principle

The DG sources at the PV production site are considered to operate in MPPT mode. They are connected to the load bus via the smart interface system. In a hybrid meshed configuration, when there is insufficient power from the site (e.g., at night) to meet load demands, the smart interface system can draw energy from the Conventional Distribution Network (CDN) via the MV/LV transformer. Conversely, when surplus power is available from the site’s DG sources, it can be injected into the CDN grid via the smart interface, the line, and the MV/LV transformer.

The additional constraints to be met during injection, in this instance as well as in other cases cited in the literature [26], are as follows:

The power flow, P_ij, through each branch (i, j) must be less than or equal to the branch’s maximum load limit: P_ij ≤ P_chmax;Power balance constraint: P_total(t) = P_RDC(t) + P_Site(t);Converter rating constraint: (P_Site(t))² + (Q_Site(t))² ≤ (S_Site(t))².

Where P_total is the total active power of the CDN, and V_min and V_max are the minimum and maximum bus voltage magnitude limits set by the grid. P_Site, Q_Site, and S_Site are the active, reactive, and apparent power produced by the PV production site, respectively.

RelationshipbetweenIPSOoptimizationandreliabilityassessment

It should be emphasized that the IPSO algorithm does not directly minimize the reliability indices. The optimization stage aims at determining the spatial configuration of the distributed PV units by minimizing a weighted objective function combining the PV-to-transformer connection distance and the corresponding integration cost. The resulting optimal PV locations are subsequently introduced into the distribution-network reliability assessment model. The reliability indices, including SAIFI, SAIDI, CAIDI, EENS, AENS and ECOST, are then calculated for the optimized configuration and compared with the reference configuration under identical network, failure-rate, repair-time, load and PV-availability assumptions. Therefore, the reported improvement in reliability represents the effect of the optimized spatial allocation of PV generation, rather than a direct optimization of the reliability indices by IPSO.

In the simulations performed using ETAP, the output of the photovoltaic installations is assumed to be constant in order to evaluate the impact of photovoltaic site optimization on grid performance under steady-state conditions. Temporal variations in solar irradiance and module temperature were not taken into account in this study.

Each photovoltaic installation is equipped with a smart inverter that ensures normal grid-connected operation while enabling islanded operation when the upstream grid becomes unavailable. When a major fault, such as a short circuit, triggers the opening of protection devices, the faulty section is isolated from the rest of the grid. The inverters then maintain power supply to the local loads connected to their islanded zone, provided that the available photovoltaic power is sufficient.

Given the grid’s looped configuration and the adopted protection strategy, no energy injection into other grid sections is permitted during islanded operation. Loads located outside the zone supplied by the photovoltaic installation remain de-energized until the main power supply is restored and the grid returns to its normal configuration. The following table (Table 2) presents the key simulation assumptions adopted.

Table 2. Key simulation assumptions.

After carefully applying the optimization methods described above, we obtained optimized coordinates corresponding to the actual site locations. These new site positions were optimized relative to the fixed locations of the transformer substations within the conventional distribution network. The following table (Table 4) presents the results for each PV generation site.

Table 4. Actual and optimized coordinates.

We used Excel software to illustrate the locations of actual production sites in relation to the production sites optimized based on the fixed positions of the equivalent transformers. The following (Figure 8) illustrates these locations.

Figure 8.Overview of the coordinates of actual transformer sites and optimized sites, illustrated using Excel software.

To inject generated power effectively and efficiently, it is necessary based on the power demand at each generation site to optimize (specifically, to maximize) the power produced by that site autonomously, while adhering to the grid constraints described in paragraph 2.4 above.

The results obtained after applying the selected optimization method are presented in Table 5 below.

Table 5.Recorded power and optimized power for PV generation sites.

The following Figure 9 shows the power outputs recorded at actual sites compared to the optimized power outputs for each production site.

Figure 9.Graph of recorded power levels (red) versus optimized power levels (green).

Figure 9 above shows that the higher the actual power generated, the greater the power obtained after optimization. This is due to the demand at the sites in question, which feature high and varied load concentrations. Regarding Figure 10, a nearly steady increase is observed in the difference between the recorded power and the optimized power as the recorded power levels rise. This indicates a proportional relationship between the power differences and the actual power recorded at each PV generation site.

Figure 10. Representation of power deviations as a function of measured power levels.

After optimizing the locations and output of the photovoltaic generation sites using the Particle Swarm Optimization (PSO) algorithm in MATLAB, we proceeded to inject this power into the study area’s conventional distribution grid using ETAP simulation software. This step was taken to verify the effectiveness of the chosen method. The setup diagram for simulating these PV injections is shown in Figure 11 below.

Figure 11. Single-line diagram of the hybrid distribution network.

The results obtained from injecting the photovoltaic output of the 25 production sites are recorded in Table 6 below.

Table 6. Reliability indices.

Figure 12.System interruption frequency and customer interruption duration.

Figure 13.Expected Energy Not Supplied (EENS) and Average Energy Not Supplied per served customer (AENS).

Figure 14. Cost of Expected Energy Not Supplied (ECOST) and Average Energy Not Supplied per Served Customer (VAENS).

When optimized PV generation is injected, a remarkable improvement (reduction) is observed in the System Average Interruption Frequency Index (SAIFI), and a very sharp decrease occurs in the Customer Average Interruption Duration Index (CAIDI) (Figure 12 above). There is also a very sharp decrease in Expected Energy Not Supplied (EENS) and a remarkable decrease in Average Energy Not Supplied per customer (AENS) (Figure 13 above). Furthermore, a very sharp decrease is seen in the Cost of Expected Energy Not Supplied (ECOST), alongside a remarkable improvement in the Cost of Expected Energy Not Supplied per customer (VAENS) (Figure 14 above). The sharp reductions in CAIDI, EENS, and ECOST stem from the fact that outage durations in distribution networks are often long, unlike outage frequencies. Consequently, when injection is performed strategically—targeting residential areas prone to these issues—outage durations improve significantly, explaining these sharp decreases. This indicates that when system interruption duration decreases significantly, the impact on the values of other indices becomes substantial. Moreover, optimizing the locations and power outputs of generation sites leads to further reductions across all indices. This demonstrates that optimizing the coordinates and power outputs of PV sites has a significant impact on the reliability indices of looped hybrid distribution networks.

From the preceding Table 4, we derive the percentage improvements for each reliability index when photovoltaic generation site coordinates are optimized and their power outputs are strategically injected into the conventional network. Table 7 below presents the percentage gains for these various reliability indices.

Table 7.Percentage gains in reliability indices.

Figure 15. Percentage gain in the improvement of reliability indices for the hybrid distribution network.

It can be observed from Table 7 and Figure 15 above that optimizing the geographical coordinates of photovoltaic production sites yields significant gains across all reliability indices. This indicates that optimizing the geographical coordinates of PV sites relative to a target distribution substation (for power injection) has a positive impact on the reliability index values of an electrical distribution network.