Optimization and Feasibility Analysis of Grid-Connected Photovoltaic Systems for Zero-Energy Housing in Algeria

2.1 Methodology

This paper evaluates the impact of electric vehicle (EV) integration on a residential load profile in northern Algeria. A novel approach is suggested to optimize the grid-connected PV system in a ZEH using the WCAMFO algorithm. The proposed approach relies on the standard electric load profile of an Algerian household with and without EV integration. Considering two energy scenarios, an optimization analysis is first performed to select the best design of the grid-connected PV system components based on the FIT schemes implemented in Algeria. Figure 1 illustrates the fundamental schematic of the system.

Figure 1. Fundamental schematic of a photovoltaic (PV) grid-connected system

The optimization process prioritizes minimizing NPC and cost of electricity (COE) while adhering to constraints such as the RF, ZEH requirement, and available roof space. The performance of the proposed WCAMFO-based optimization is compared to that of HOMER software to assess their relative effectiveness. Additionally, a sensitivity analysis explores the influence of load consumption, RF, available roof area, ZEH constraints, and PV capacity on the configuration of the optimal hybrid energy system.

2.2 Proposed approach

This study uses a hybrid metaheuristic method called WCAMFO. Introduced by Khalilpourazari and Khalilpourazary [76], the WCAMFO algorithm combined the strengths of two existing algorithms: the water cycle algorithm (WCA) and moth flame optimization (MFO). By merging these two algorithms, the WCAMFO algorithm enhanced performance in the exploration and exploitation phases.

2.2.1 Water cycle algorithm

The WCA represents a nature-inspired optimization method introduced by Eskandar et al. [77] in 2012. Taking inspiration from the natural movement of water in ecosystems, this metaheuristic approach models how streams and rivers converge toward the sea to address constrained optimization problems [78, 79]. The optimization process of the WCA is illustrated in Figure 2.

Figure 2. Water cycle algorithm (WCA) optimization process

An initial stream matrix is generated with dimensions $\mathrm{N}_{\mathrm{p}} \times \mathrm{D}$, where $\mathrm{N}_{\mathrm{p}}$ represents the population size and D denotes the total number of variables in the system [77].

Total population $=\left[\begin{array}{c}\text { Sea } \\ \text { River }_1 \\ \text { River }_2 \\ \text { River }_3 \\ \vdots \\ \text { Stream }_{\mathrm{N}_{\mathrm{sr}}+1} \\ \text { Stream }_{\mathrm{N}_{\mathrm{sr}}+2} \\ \text { Stream }_{\mathrm{N}_{\mathrm{sr}}+3} \\ \vdots \\ \text { Stream }_{\mathrm{N}_{\mathrm{p}}}\end{array}\right]=\left[\begin{array}{ccccc}\mathrm{x}_1^1 & \mathrm{x}_2^1 & \mathrm{x}_3^1 & \ldots & \mathrm{x}_{\mathrm{D}}^1 \\ \mathrm{x}_1^2 & \mathrm{x}_2^2 & \mathrm{x}_3^2 & \cdots & \mathrm{x}_{\mathrm{D}}^2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \mathrm{x}_1^{N_p} & \mathrm{x}_2^{N_p} & \mathrm{x}_3^{N_p} & \cdots & \mathrm{x}_{\mathrm{D}}^{N_p} \end{array}\right]$  (1)

where, $\mathrm{N}_{\mathrm{s}}$ represents the total count of rivers and a single sea:

$N_s=$ Number of rivers $+\underbrace{1}_{\text {sea}}$   (2)

The remaining population $\mathrm{N}_{\text {streams}}$ is given in Eq. (3):

$\mathrm{N}_{\text {streams }}=\mathrm{N}_{\mathrm{p}}-\mathrm{N}_{\mathrm{s}}$   (3)

The quantity of water conveyed to a river or the sea is determined by the volume of water in flow. An estimation of the allocated streams for rivers and the sea is computed in the following manner:

$NS_n=\operatorname{round}\left\{\left|\frac{\operatorname{Cos} t_n-\operatorname{Cost}_{N_s+1}}{\sum_{n=1}^{N_s} C_n}\right| \times N_{\text {streams }}\right\}$   (4)

where, $NS_n$ is the count of streams that contribute to individual rivers and the sea.

During the exploitation phase, the updated positions of variables are acquired using the following equations:

$\vec{X}_{\text {stream }}(t+1)=\vec{X}_{\text {stream }}(t)+$ rand $\times C \times\left(\vec{X}_{\text {sea }}(t)-\vec{X}_{\text {stream }}(t)\right)$   (5)

$\vec{X}_{\text {stream }}(t+1)=\vec{X}_{\text {stream }}(t)+$ rand $\times C \times\left(\vec{X}_{\text {river }}(t)-\vec{X}_{\text {stream }}(t)\right)$   (6)

$\vec{X}_{\text {river }}(t+1)=\vec{X}_{\text {river }}(t)+$ rand $\times C \times\left(\vec{X}_{\text {sea }}(t)-\vec{X}_{\text {river }}(t)\right)$   (7)

Eqs. (5) and (6) are used for streams that merge into the sea and their respective rivers.

The iteration index is denoted by t. 1 < C < 2 denotes a constant, and rand signifies a random number uniformly distributed between 0 and 1. When a stream shows a lower cost function value than its associated river, their positions are interchanged. Similarly, in cases where a river exhibits a higher cost function value than the sea, the river and sea are interchanged.

The evaporation mechanism serves to prevent local optima and occurs when a river or stream is close to the sea, as determined by the following criterion:

$\left\{\begin{array}{c}\text { if }\left\|\overrightarrow{\mathrm{X}}_{\text {sea }}^{\mathrm{t}}-\overrightarrow{\mathrm{X}}_{\text {river }_{\mathrm{j}}}^{\mathrm{t}}\right\|<\mathrm{d}_{\max } \text { or rand }<0.1 \mathrm{j}=1,2,3, \ldots, \mathrm{~N}_{\mathrm{sr}}-1 \\ \text { perform raining process } \\ \text { end }\end{array}\right.$   (8)

where, $\mathrm{d}_{\max }$ is a small positive value that approaches zero over time and diminishes according to the specified formula:

$d_{\max ^{i+1}}=d_{\max ^i}-\frac{d_{\max ^i}}{\text {max iteration}}$   (9)

After fulfilling the evaporation process, the algorithm proceeds with the raining step. During this process, a new population of streams is generated at various locations. Calculation of the positions for the new streams is performed using the following formula:

$X_{\text {stream }}^{\text {new}}=LB+$ rand $\times(UB-LB)$   (10)

In the context of LB (lower bound) and UB (upper bound), which define the optimization constraints, we can implement a mechanism for constrained problems to generate streams flowing directly to the sea. This approach enables a deeper exploration of the solution space in proximity to the sea, potentially leading to optimal solutions:

$X_{\text {stream}}^{\text {new}}=X_{\text {sea}}+\sqrt{\mu} \times \operatorname{randn}(1, D)$   (11)

where, μ is a coefficient associated with proximity to the sea.

2.2.2 Moth flame optimization algorithm

Developed by Mirjalili [80], the Moth-Flame Optimization (MFO) algorithm is a nature-inspired metaheuristic method that replicates the navigation behavior of moths, using moonlight as a reference for orientation to solve optimization problems, as illustrated in Figure 3.

Figure 3. Conceptual model of the fly behavior of moths

The MFO algorithm treats candidate solutions as moths, using their positions in the search space to represent the problem variables. This enables the exploration of different dimensional spaces by adjusting the positional vectors of the moths. The algorithm employs a population-based approach, which represents moths as a matrix for efficient manipulation:

$M=\left[\begin{array}{ccccc}m_{1,1} & m_{1,2} & \ldots & \ldots & m_{1, d} \\ m_{2,1} & m_{2,2} & \ldots & \ldots & m_{2, d} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ m_{n, 1} & m_{n, 2} & \ldots & \ldots & m_{n, d}\end{array}\right]$   (12)

where, n represents the total moth population and d indicates the dimensionality of the problem space. The fitness value for each moth is stored in an array using the following representation:

$O M=\left[\begin{array}{c}\mathrm{OM}_1 \\ \mathrm{OM}_2 \\ \vdots \\ \mathrm{OM}_n\end{array}\right]$   (13)

Each moth's fitness value is computed by evaluating the fitness function using the moth's position vector (stored in the first row of matrix M). The resulting value from the fitness function is then allocated to the respective moth as its fitness score, such as OM1 in OM. In addition to moths, another important component of the algorithm is flames. A matrix akin to the one used for moths is employed to represent flames and is structured as follows:

$F=\left[\begin{array}{ccccc}F_{1,1} & F_{1,2} & \ldots & \ldots & F_{1, d} \\ F_{2,1} & F_{2,2} & \ldots & \ldots & F_{2, d} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ F_{n, 1} & F_{n, 2} & \ldots & \ldots & F_{n, d}\end{array}\right]$   (14)

where, n denotes the moth count and d represents the dimensionality. The matrices M and F share identical dimensions in Eq. (14). An array stores the fitness values of the flames, structured as:

$OF=\left[\begin{array}{c}OF_1 \\ OF_2 \\ \vdots \\ OF_n\end{array}\right]$   (15)

The MFO algorithm employs a three-component system to find the global optimum in optimization problems, structured as:

$MFO=(I, P, T)$   (16)

where, I create a population of moths along with their corresponding fitness values:

$I: \varnothing \rightarrow M, OM$   (17)

In addition, the P function moves the moths:

$P: M \rightarrow M$   (18)

The function T returns a Boolean value: true when termination conditions are met, false otherwise:

$T: M \rightarrow\{$true, false$\}$   (19)

To update the moth position, we use the following equation:

$M_i=S\left(M_i, F_j\right)$   (20)

where, $\mathrm{D}_{\mathrm{i}}$ indicates i-th moth and j-th flame, respectively, and S is the spiral function as follows:

$S\left(M_i, F_j\right)=D_i \cdot e^{b t} \cdot \cos (2 \pi t)+F_j$   (21)

where, $D_i$ is the distance between $M_i$ and $F_j$, $b$ is constant, $t$ is randomly chosen in [-1, 1], and $D_i$ is calculated using the following equation:

$D_i=\left|F_j-M_i\right|$   (22)

To enhance the exploitation phase, the number of flames is gradually decreased in relation to the number of iterations, following the equation shown below:

$N_{\text {flames}}=\operatorname{round}\left(N-I \times \frac{N-1}{T}\right)$   (23)

During iteration I, N indicates the maximum flame count and T denotes the total iteration limit. The number of flames decreases as iterations progress.

2.2.3 Hybrid Water Cycle–Moth Flame Optimization algorithm

The WCAMFO hybrid algorithm aims to combine the strengths of the WCA and MFO algorithms, harnessing the benefits of each. While the WCA excels in the exploration phase, it lacks an efficient exploitation operator. On the other hand, the MFO algorithm, which emulates moths spiralling around artificial lights, demonstrates proficiency in exploiting the solution space. However, it is prone to occasionally becoming trapped in local optima [81].

In the WCAMFO algorithm, the foundational algorithm is the WCA. The initial enhancement in the WCA involves employing the spiral motion of moths to update the positions of streams and rivers. In the standard adjustment process of the WCA, the primary focus is on the region between the stream and the river. When determining the next position, the new position of the stream depends on how far it is from the river into which it flows [76]. Conversely, the MFO algorithm facilitates positional adjustments by moths freely in the vicinity of their associated flame. This flexibility enables the streams and rivers to use the spiral movement of moths to update their positions. By incorporating this capability, the hybrid WCAMFO algorithm significantly enhances its exploitation ability, leading to improved optimization performance [76].

Two processes are introduced in the WCAMFO algorithm to improve the randomization, first the raining process: This process is triggered when the gap between a river or stream and the sea drops below a predefined threshold value dmax. In the WCAMFO algorithm, the raining process is employed to generate new solutions. The other process is the random walk, which allows the random movement of streams within the solution space via a random walk, specifically using a Levy flight.

The WCAMFO algorithm allows streams to move to new positions using the following equation, which helps to make the algorithm more random [76]:

$\mathrm{x}_{\mathrm{i}+1}=\mathrm{x}_{\mathrm{i}}+\operatorname{Levy}(\operatorname{dim}) \otimes \mathrm{x}_{\mathrm{i}}$   (24)

where, $x_{i+1}$ represents the next position, $x_i$ denotes the immediate location, and dim indicates the number of decision variables in the problem space. Levy flight is determined through the application of the following formula:

$\operatorname{Levy}(x)=\frac{0.01 \times \sigma \times r_1}{\left|r_2\right|^{1 / \beta}}$   (25)

where, r₁ and r₂ are randomly generated numbers between 0 and 1, and σ is calculated as follows:

$\sigma=\left(\frac{\Gamma(1+\beta) \times \sin \left(\frac{\pi \beta}{2}\right)}{\Gamma\left(\frac{1+\beta}{2}\right) \times \beta \times 2^{\left(\frac{\beta-1}{2}\right)}}\right)^{\frac{1}{\beta}}$   (26)

2.3 Mathematical formulation of hybrid system components

2.3.1 Photovoltaic model

The impact of temperature on the PV generator’s hourly output power P_PV is considered. The calculation of P_PV incorporates the PV generator’s area A_PV (m²) and the solar radiation on the tilted plane module G(t) (kW/m²) using Eq. (27). Of note, the PV module’s efficiency declines with rising temperature.

$P_{PV}=\eta_{PV} \cdot \eta_{inv} \cdot A_{PV} \cdot \gamma \cdot \frac{G}{G_{STC}}\left[1+\gamma_T\left(T-T_{STC}\right)\right]$   (27)

Solar PV module performance incorporates multiple efficiency factors: $\eta_{\mathrm{PV}}$ is the PV module efficiency, $\eta_{\text {inv }}$ denotes the inverter efficiency; $\mathrm{G}_{\text {STC}}$ is the irradiance under standard test conditions (STC), which is $1 \mathrm{~kW} / \mathrm{m}^2$; $\mathrm{T}$ and $\mathrm{T}_{\mathrm{STC}}$ indicate the PV cell's operating temperature and the temperature at STC (25 ℃), respectively. $\gamma_{\mathrm{T}}$ characterizes the module's maximum power variation per degree Celsius relative to the STC temperature of 25 ℃. For crystalline silicon, this coefficient typically ranges from -0.5 to -0.3%/℃ [82]. The derating factor $\gamma$ accounts for the system performance losses, assumed to be 0.9 throughout this study [83]. The temperature at which a PV cell is operating can be estimated using a linear approximation using Eq. (28) [84].

$T=T_a+\frac{G}{0.8}\left(T_{NOCT}-20\right)$   (28)

where, $T$ (℃) is the PV cell operating temperature, $T_a$ (℃) is the ambient air temperature, $G$ (kW/m²) is the incident solar irradiance on the PV module surface, and $T_{NOCT}$ (℃) is the Nominal Operating Cell Temperature (typically between 45 and 48 ℃), which represents the cell temperature under standard test conditions of 800 W/m² irradiance, 25 ℃ ambient temperature, and wind speed of 1 m/s. This formulation reflects the increase in cell temperature with increasing solar irradiance and ambient temperature.

2.3.2 Inverter

The hourly AC power output of the inverter is calculated as:

$E_{inv, out}(t)=\eta_{inv} \cdot E_{PV}(t)$   (29)

where, $E_{\text {inv, out }}(t)(\mathrm{W})$ is the inverter output power at hour $t$, $E_{P V}(t)(\mathrm{W})$ is the DC power generated by the PV array at the same time, and $\eta_{\text {inv}}$ denotes the inverter efficiency. This equation represents the conversion of DC power produced by the PV array into usable AC power, accounting for inverter losses.

2.3.3 Objective function

The NPC serves as the objective function in the optimization problem, which can be calculated as follows [85]:

$NPC=\cos t-C_{ex} \times \frac{1}{CRF}$   (30)

where, $C_{ex}$ represents the net grid interaction cost, calculated as the difference between the energy purchased and the energy sold, the overall cost of the grid-connected system comprises the initial expense, replacement costs, and the ongoing operation and maintenance expenses for the components throughout the life span of the system, as given in Eq. (31) [85]:

$\operatorname{cost}=\sum_{\mathrm{k}}\left[\left(\mathrm{C}_{\mathrm{I}, \mathrm{k}}+\mathrm{C}_{\mathrm{O}, \mathrm{k}} \times \frac{1}{\mathrm{CRF}}+\mathrm{C}_{\mathrm{R}, \mathrm{k}} \times \mathrm{R}_{\mathrm{k}}\right) \times \mathrm{P}_{\mathrm{k}}\right]$   (31)

The component indicator k delineates cost elements where C_I, C_R, and C_O represent initial, replacement, and operational costs per kilowatt ($/kW), respectively. R_k indicates the single payment present worth, while CRF designates the capital recovery factor, both calculated through specified equations [18, 86]:

$R_k=\sum_{n=1}^N \frac{1}{(1+j)^{L \times n}}$   (32)

$CRF=\frac{j(j+1)^{T_p}}{(j+1)^{T_p}-1}$   (33)

In this equation, j represents the real interest rate while N denotes the number of replacements for component k. L indicates the component's lifetime, and Tp represents the project duration (25 years), which aligns with the PV panels' expected lifespan. The real interest rate calculation employs Fisher's formula as follows [26]:

$j=\frac{j'-f}{1+f}$   (34)

where, j represents the real interest rate, j' denotes the nominal interest rate, and f indicates the annual inflation rate.

2.3.4 Renewable fraction and zero-energy constraints

The RF represents the proportion of the total electrical energy supplied to the residential load that originates from renewable sources, namely PV generation in this study. It is calculated as the ratio between the annual PV energy production and the total annual energy supplied to the system, which includes both PV energy and energy purchased from the grid. An RF value close to unity indicates a high level of renewable penetration and reduced dependency on grid electricity. The calculation of the RF is determined as follows [87]:

$E_{t t}=E_{PV}+E_{purch}$   (35)

$RF=\frac{E_{PV}}{E_{\text {tot}}}$   (36)

where, $\mathrm{E}_{\text {purch}}$ represents the annual energy purchased from the grid (kWh/yr) is the annual PV energy (kWh/yr), and $\mathrm{E}_{\text {tot}}$ is the annual energy production (kWh/yr).

It should be noted that the RF is an annual, energy-based indicator and does not explicitly account for temporal mismatches between PV generation and load demand. In practical grid-connected applications, periods of surplus PV generation and grid electricity imports may still occur, even when high RF values are achieved. Consequently, while RF is a useful metric for evaluating the long-term contribution of renewable energy, it does not fully capture short-term operational constraints or grid interaction dynamics. For this reason, a zero-energy balance is introduced as a constraint to ensure that the overall annual energy consumption of the system is balanced by renewable energy generation. To achieve a ZEH, the total energy produced by the PV system must be equal to or greater than the household energy demand:

$E_{PV} \geq E_{\text {load}}$   (37)

where, $\mathrm{E}_{\mathrm{PV}}$ refers to the total electrical energy output of the PV system and $\mathrm{E}_{\text {load }}$ is the amount of energy consumed by the house.

2.4 Electric vehicle charging characteristics

Several factors influence the process of EV charging, including the type of vehicle, battery technology, battery capacity, battery state of charge (SOC), charging power level, and charging behavior [88]. The following mathematical model simulates the typical charging profile of lithium-ion EV batteries [89]:

$\mathrm{P}(\mathrm{t})= \begin{cases}\mathrm{P}_{\mathrm{r}} & 0<\mathrm{t} \leq \mathrm{t}_1 \\ \mathrm{P}_{\mathrm{r}} \frac{\mathrm{t}_2-\mathrm{t}}{\mathrm{t}_2-\mathrm{t}_1} & \mathrm{t}_1<\mathrm{t} \leq \mathrm{t}_2\end{cases}$   (38)

where, $\mathrm{P}(\mathrm{t})$ denotes the charging power at time $\mathrm{t}$, $\mathrm{P}_{\mathrm{r}}$ represents the rated charging power determined by the selected charging mode, and $t_1, t_2$ are the respective start and end times that define the variation in charging power as the battery nears full capacity.

This study only uses slow charging power (Mode 1) among the three charging classifications presented in Table 1 and Figure 4. The EV characteristics are summarized in Table 2.

Table 1. Parameters of the electric vehicle (EV) battery charging profile

Table 2. Characteristics of the electric vehicle (EV) chosen

Figure 4. Charging profile of lithium-ion electric vehicle (EV) batteries [90]

2.5 Designed system specification

The technical and economic specifications are detailed in Table 3. The PV system features the CEM240P-60, a multi-crystalline module produced by the Algerian Manufacturer Condor. This module delivers 240 Wp with 14.7% efficiency at an upfront cost of 1312 \$/kWp. DC to AC conversion is handled by an inverter operating at 95% efficiency, requiring an initial investment of 600 \$/kW and replacement cost of 300 \$/kW, with a ten-year service life.

Table 3. Technical characteristics and cost-related attributes of the system

Sonelgaz, Algeria's national utility company, manages the power grid, while the Regulatory Commission for Electricity and Gas oversees electricity pricing. Residential customers typically use the 54 M tariff rate. Table 4 provides comprehensive details about Algeria's residential feed-in tariff structure, which consists of four pricing tiers, the first tier applies to consumption between 0 and 125 kWh at a rate of 1.778 DZD/kWh, the second tier covers consumption between 126 and 250 kWh at a rate of 4.178 DZD/kWh, the third tier applies to consumption between 251 and 1000 kWh at a rate of 4.182 DZD/kWh, and the fourth tier applies to consumption exceeding 1,000 kWh at a rate of 5.479 DZD/kWh.

Table 4. Residential feed-in tariff rate

The modeling and optimization of the system in this study were carried out using MATLAB software. The code was developed to incorporate the WCAMFO hybrid algorithm, which is employed to identify the optimal configuration of a grid-connected PV system under two distinct scenarios.

Scenario 1: The grid-connected PV system is configured such that all the produced PV energy is sold to the grid, while all the energy consumed by the household is purchased from the grid. In other words, any energy generated by the PV system is fed back into the grid, and households rely entirely on the grid for their energy needs. Eq. (39) shows the hourly profile of PV energy transactions with the grid, encompassing both energy purchases and injections.

$\mathrm{E}_{\text {purch}}(t)=E_{\text {load}}(t)$ and $E_{\text {sold}}(t)=E_{PV}(t)$   (39)

Scenario 2: The home purchases only the electricity needed from the grid. Any surplus energy generated by the PV system is fed into the grid and sold to the electricity provider. This setup allows the home to offset its electricity consumption by using PV-generated electricity and potentially earn credits or financial compensation for the excess energy supplied to the grid.

$\begin{gathered}\mathrm{E}_{\text {purch }}(\mathrm{t}) \\ =\mathrm{E}_{\text {load }}(\mathrm{t})-\mathrm{E}_{\mathrm{PV}}(t), \mathrm{E}_{\text {sold }}(\mathrm{t})=0, \text { if } \mathrm{E}_{\text {load }}(\mathrm{t})>\mathrm{E}_{\mathrm{PV}}(t) \\ =\mathrm{E}_{\mathrm{PV}}(\mathrm{t})-\mathrm{E}_{\text {load }}(\mathrm{t}), \mathrm{E}_{\text {purch }}(\mathrm{t})=0, \text { if } \mathrm{E}_{\text {load }}(\mathrm{t})<\mathrm{E}_{\mathrm{PV}}(t)\end{gathered}$   (40)

Both energy scenarios' optimization problems are formulated using Eq. (41):

$\begin{array}{r}\operatorname{Min}\{\operatorname{NPC}(\mathrm{x})\} \mathrm{x} \in \mathrm{ss} \quad \mathrm{x}=\left(\mathrm{P}_{\mathrm{PV}}\right) \\ \text { Subject to }\left\{\begin{array}{c}\mathrm{S} \leq 16 \mathrm{~m}^2 \\ \mathrm{RF} \geq 30 \% \\ \mathrm{E}_{\mathrm{PV}} \geq \mathrm{E}_{\mathrm{Load}}\end{array}\right.\end{array}$   (41)

NPC serves as the objective function. The constraints include the RF, available roof area, and zero energy consumption requirements. The analysis evaluates the hybrid grid-PV system over a 25-year lifecycle to assess its technical, economic, and environmental performance under two different energy scenarios. These scenarios are constructed based on previously established input parameters. The population size is set to 50, with the algorithm executed over 100 iterations. First, the WCAMFO algorithm is compared with PSO. Figure 9 presents the convergence behavior of both algorithms in minimizing the NPC under the basic load scenario. WCAMFO converges to the optimal NPC value of −223.269 $ within fewer than five iterations and exhibits stable convergence without oscillatory behavior. In contrast, PSO requires approximately 10–12 iterations to reach a comparable solution. These results demonstrate the faster convergence rate of WCAMFO. The improved performance of WCAMFO is attributed to its hybrid search mechanism, which provides an effective balance between global exploration and local exploitation. Conversely, PSO is more susceptible to premature convergence and oscillations due to its velocity update formulation and sensitivity to parameter settings. Consequently, WCAMFO achieves reduced computational effort and improved stability, making it suitable for complex nonlinear optimization problems such as energy management systems with EV integration.

Figure 9. Net present cost (NPC) convergence comparison between Water Cycle–Moth Flame Optimization (WCAMFO) and particle swarm optimization algorithm (PSO) for basic load

Figures 10 and 11 illustrate the convergence curves for the different load profile scenarios. WCAMFO achieves the optimal solution within three iterations for both scenarios. However, the first scenario achieves a superior minimum value (NPC), whereas the second energy scenario yields a higher RF.

Figure 10. Net present cost (NPC) convergence curve for basic load using the Water Cycle–Moth Flame Optimization (WCAMFO) algorithm

Figure 11. Net present cost (NPC) convergence curve for load & electric vehicle (EV) using the Water Cycle–Moth Flame Optimization (WCAMFO) algorithm

Initially, the system is simulated using the basic load profile for both energy scenarios to identify the optimal solution that meets the criteria of an RF exceeding 30%, a roof area of less than 16 m², and zero energy. Subsequently, the simulation was repeated for the load profile incorporating the integrated EV, employing identical configurations, and the outcomes were compared and analyzed.

Beyond the convergence performance, the obtained optimization results provide important insights into the practical integration of PV systems in residential buildings in Algeria. The rapid convergence of the WCAMFO algorithm within a few iterations indicates its suitability for real-world applications where computational efficiency is essential for system sizing and economic assessment. The lower NPC observed under the basic load scenario reflects the reduced system stress and more stable demand pattern, whereas the load + EV scenario introduces higher energy demand and variability, resulting in increased NPC despite improved RF.

Furthermore, for the purpose of validating the outcomes generated through the WCAMFO algorithm, a comparative analysis was conducted with the HOMER software. The comparison focused on key metrics such as NPC, COE, and RF. The results are summarized in Tables 6 and 7.

Table 6. Simulation results for the basic load

Note: NPC = net present cost; COE = cost of electricity; PV = photovoltaic; RF = renewable fraction; WCAMFO = Water Cycle–Moth Flame Optimization; Sc1 = scenario1; Sc2 = scenario2; HOMER = Hybrid Optimization Model for Electric Renewables

Table 7. Simulation results for Load + EV

Note: NPC = net present cost; COE = cost of electricity; PV = photovoltaic; RF = renewable fraction; WCAMFO = Water Cycle–Moth Flame Optimization; HOMER = Hybrid Optimization Model for Electric Renewables

For the basic load, the optimal solution consists of a 2.28 kWp PV system and a 2.5 kW converter. In y scenario 1, the COE for the system is -0.023 \$/kWh, while in Energy scenario 2, it is -0.0032 \$/kWh. The NPC is -1941.88\$ for scenario 1 and -223.4\$.

For scenario 2. The optimal system demonstrates significant performance, achieving a RF of 69% in scenario 1 and 82.6% in scenario 2. Furthermore, both energy scenarios result in an annual surplus of electricity sold to the grid, with 3693.7 kWh/year for Energy scenario 1 and 2815.11 kWh/year for Energy scenario 2, surpassing the energy purchased from the grid, which amounts to 1658.31 kWh/year for scenario 1 and 779.73 kWh/year for scenario 2.

Upon connecting the EV, the energy obtained from the grid increases significantly, reaching 7356.48 kWh/year for scenario 1 (compared to the initial 1658.31 kWh/year) and 6326.03 kWh/year for scenario 2 (compared to the initial 779.73 kWh/year). This increase in grid energy consumption directly correlates with a notable decrease in the RF, dropping from 69% to 33.4% for scenario 1 and 82.6% to 36.8% for scenario 2. Consequently, the expenses related to grid purchases also escalate, rising from 44.88\$/year to 237.06\$/year for scenario 1 and from 20.95\$/year to 203.79\$/year for scenario 2.

Moreover, the NPC of the system increased from -1941.88\$ (basic load) to 1093.8\$ (load + EV) for scenario 1 and from -223.4\$ (basic load) to 2988.76\$ (load + EV) for scenario 2, which can be attributed to the heightened grid purchases. Despite this, the PV system successfully met the energy requirements of the load + EV in both energy scenarios and generated a total income of 316.99\$ and 195.7\$, respectively. The HOMER simulations closely aligned with our WCAMFO outcomes, thus validating the reliability of our optimization approach. For the Basic Load, WCAMFO algorithm outperformed HOMER in economic terms, showing negative NPC values. WCAMFO showed an NPC of -223.4\$, compared to HOMER's positive 31.44\$, reflecting substantial economic benefits. The COE was also negative in WCAMFO energy scenarios, implying earnings from energy produced. Additionally, the RF was higher in WCAMFO (82.6%) than in HOMER (81.3%). For Load + EV, both algorithms indicated increased costs. WCAMFO scenario 1 had a positive NPC (1093.85\$), and scenario 2 showed even higher costs (2988.76\$), similar to HOMER (3048\$). Despite this, WCAMFO demonstrated lower COE values, indicating more cost-effective solutions. RF values were lower due to the additional load from the EV, but WCAMFO still showed slightly better economic performance compared to HOMER.

The results unequivocally demonstrate that the WCAMFO algorithm performs comparably to HOMER, with slight variations in the results. This validation establishes the reliability and accuracy of the WCAMFO algorithm and further validates its versatility. Figures 12 and 13 present the hourly energy balance simulations for the basic load in scenarios 1 and 2, respectively.

Figure 13. Hourly energy balance for the basic load (Scenario 2): (a) winter day and (b) summer day

Figures 14 and 15 display the hourly energy balance simulations for the load + EV in both energy scenarios. Figures 12(a)-15(a) for Energy scenarios 1 and 2 show that during winter days, the available PV power is insufficient for fulfilling the energy needs of the household. Conversely, Figures 12(b) and 13(b) demonstrate that on summer days, the energy generated by the PV system exceeds the load in both energy scenarios. This indicates that the PV system adequately fulfills the load demand. In scenario 1, excess power is purchased from the grid, and PV energy is sold back to the grid. In Energy scenario 2, surplus energy is consumed internally, and any remaining surplus is injected into the grid. Furthermore, Figures 14(b) and 15(b) illustrate that from 07:00 to 22:00, the PV-generated energy surpasses the load demand. Nevertheless, between midnight and 06:00, the load experiences an increase caused by the charging of EV batteries, predominantly occurring during the nighttime from 01:00 to 06:00.

Figure 14. Hourly energy balance for load + EV (scenario 1): (a) winter day and (b) summer day

Figure 15. Hourly energy balance for load+EV (scenario 2): (a) winter day and (b) summer day

4.1 Monthly variation analysis of energy generation in the designed system

Figure 16 shows that the monthly purchased energy from the grid is consistently lower than the energy sold to the grid in the basic load case. This leads to positive energy balances annually for all months.

Figure 16. Average monthly energy output of the designed system for the basic load (scenario 1, scenario 2)

Conversely, Figure 17 illustrates that in both load + EV energy scenarios, monthly grid electricity purchases exceed the amount of energy exported back to the grid. This outcome leads to negative energy balances throughout the months, indicating an overall energy deficit.

Figure 17. Average monthly energy output of the designed system for the load + EV (scenario 1, scenario 2)

4.2 Average monthly net grid interaction cost

Figures 18 and 19 show the variation of the net grid interaction $\operatorname{cost}\left(\mathrm{C}_{\mathrm{ex}}\right)$, which is found to be more favorable in scenario 1 than in scenario 2 for both profiles. In the basic load profile, $\mathrm{C}_{\mathrm{ex}}$ shows an increasing trend during the summer months, peaking in July at 70.57 \$/month for scenario 1 and 57.2 \$/month for scenario 2.

Figure 18. Average monthly net grid interaction cost ($C_{ex}$) for the basic load (scenario 1, scenario 2)

Figure 19. Average monthly net grid interaction cost $C_{ex}$ for the load + EV (scenario 1, scenario 2)

Conversely, a decline is observed throughout the winter, culminating in a minimum level in December at 16.04 \$/month for scenario 1 and 10.77 \$/month for scenario 2. For the load + EV case, $\mathrm{C}_{\mathrm{ex}}$ follows a similar trend, rising during the summer months, reaching its maximum value in July (54.25 \$/month for scenario 1 and 39.67 \$/month for scenario 2), and decreasing during winter, culminating in a minimum in the month of December (0.21 \$/month for scenario 1 and −5.86 \$/month for scenario 2). Overall, these findings underscore the advantageous impact on the system’s profitability, especially during the summer season.

4.3 Monthly variation of the carbon mitigation factor CO₂

Based on estimated carbon emissions from the grid in Algeria, which amount to 517 g/kWh [85], Figures 20 and 21 show the monthly variation of the carbon mitigation for the basic load and load + EV, respectively. The mitigation of CO₂ emissions is considerably greater during the summer months than during winter in both considered energy scenarios. Furthermore, Energy scenario 1 demonstrates a higher level of CO₂ gas attenuation for both load profiles. In fact, it reached 86.22 kg in August for the basic load profile and 335.48 kg for the load + EV profile.

Figure 20. Monthly variation of the carbon mitigation factor CO₂ for the basic load in the two energy scenarios

Figure 21. Monthly variation of the carbon mitigation factor CO₂ for (load + EV) in both energy scenarios

4.4 Impact of RF on net present cost, cost of electricity, photovoltaic production, and grid sales profiles

Figures 22 and 23 illustrate the strong impact of the RF on various parameters, including NPC, COE, PV production, and grid sales, for the (load + EV) profile.

Figure 22. Effect of renewable fraction (RF) on net present cost (NPC), cost of electricity (COE), photovoltaic (PV) production, and grid sales (load + EV in scenario 1)

Figure 23. Effect of renewable fraction (RF) on net present cost (NPC), cost of electricity (COE), photovoltaic (PV) production, grid sales, and grid purchases (load + EV in scenario 2)

The COE experiences a decline, reducing from 0.009 \$/kWh to -0.0375 \$/kWh in energy scenario 1and from 0.0254 \$/kWh to -0.346 \$/kWh in scenario 2, as the RF increases from 40% to 90%. The NPC exhibits a similar trend, decreasing from 1475.99 \$ to -43,071.5 \$ in scenario 1 and from 3601.82 \$ to -33,394.6 \$ in scenario 2, with an increase in RF from 30% to 90%. Moreover, PV production and grid sales demonstrate an upward trend in both energy scenarios as the RF increases from 30% to 90%. In scenario 1, PV production and grid sales rise from 3152.6 to 66,207.93 kWh/yr. Similarly, in scenario 2, PV production increases from 2727.34 to 55,583.2 kWh/year, whereas grid sales increase from 1734.67 to 54,403.2 kWh/year.

4.5 Effect of photovoltaic-rated power on net present cost and grid sales

Figures 24 and 25 showcase the significant impact of PV rated capacity on NPC, grid sales and purchases, and roof space for the load + EV profile. In scenario 1, the NPC decreases from 1475.99\$ to -43,071.5\$ when the PV-rated capacity increases from 2 to 41 kWp. Similarly, in Energy scenario 2, the NPC decreases from 3601.82\$ to -33,394.6\$ with the same increase in PV-rated capacity.

Figure 24. Effect of photovoltaic (PV)-rated power on net present cost (NPC) and grid sales (load + EV in scenario 1)

Figure 25. Effect of photovoltaic (PV)-rated power on net present cost (NPC), and grid sales (load + EV in scenario 2)

Consequently, the increase in PV-rated capacity leads to a substantial increase in the quantity of surplus energy exported to the grid in both energy scenarios. For scenario 1, grid sales increase from 3152.6 to 66,207.93 kWh/year with grid purchases of 7356.48 kWh. In scenario 2, grid sales increase from 1734.67 to 54,403.2 kWh/year, which leads to a decrease in grid purchases from 6363.81 to 6176.02 kWh/year. A notable positive correlation exists between PV capacity and the amount of roof space needed. This relationship emphasizes the importance of considering available space when decisive optimal PV capacity for a given installation.

From a practical perspective, these findings are particularly relevant to Algerian residential buildings, where the deployment of PV systems is often constrained by limited roof area, variability in household electricity consumption, and economic considerations. In northern coastal regions, characterized by moderate solar irradiance, the optimization results indicate that cost-effective PV integration requires careful system sizing and effective load management strategies. Conversely, in the high plateau and southern regions, higher solar irradiance levels substantially enhance PV energy production, leading to reduced grid dependency and lower NPC under comparable system configurations.

In this context, although the proposed WCAMFO-based optimization framework is validated through comparison with HOMER, its generalization capability is further supported by its evaluation under multiple load profile scenarios, including EV integration and different seasonal operating conditions. These scenarios represent a wide range of realistic operating environments and demonstrate the robustness of the algorithm in responding to variations in energy demand and solar resource availability. Furthermore, the framework is data-driven and parameterized, which facilitates its adaptation to other geographical regions by incorporating region-specific solar irradiance, temperature, and load characteristics.

The authors extend their prayers to the late Professor Abdel-Rahman Hamidat, their cherished mentor, who regrettably succumbed to Covid-19 on August 16, 2021. They beseech God's grace, forgiveness, and blessings to encompass him.