The physical electrical system to be sized is represented in Figure 1. It’s a standalone PV system consisting of a PV generator, a battery bank, a charge controller, and a hybrid inverter. Optimally sizing such a system involves determining the optimal sizes of its main components (PV generator and battery bank) to best meet the user's energy demand while minimizing the cost of energy.

Figure 1.

Physical system to size

The conventional intuitive method is based on simple formulas that consider only averaged values and involve intuitive choices for storage autonomy. Traditionally, the PV generator size P_P [W_p] and the battery storage capacity S_BAT [Ah] are respectively estimated using eq. (1) and eq. (2), as mentioned in [7]:.

where E_D [Wh/day] is the daily energy demand, G_R [W/m²] is the reference irradiance value (1000 W/m²), I_D [Wh/m²∕day] is the average daily solar irradiation, η_g [%] is the overall efficiency of the PV generator, N_D [days] is the number of autonomy days, U_BAT [V] is the voltage across the battery bank, η_BAT [%] is the battery efficiency and DOD_MAX [%] is the maximum depth of discharge of the batteries.

The global solar irradiance measured on an inclined plane G_IT(t) [in W/m²] is expressed as the sum of the direct or beam solar irradiance TBI(t), the diffuse component TDI(t) from the sky dome and the fraction TRI(t) reflected by the ground, as described by eq. (3). These three components are respectively given by eq. (5), eq. (4) and eq. (6).

In general, meteorological stations provide only the global irradiance measured on a horizontal plane G_HI(t) [in W/m²].

In this study, the irradiance received by photovoltaic modules tilted at an angle β is estimated from this horizontal measurement using the Erbs model presented in [27] for the direct – diffuse separation and the Hay–Davies transposition model also presented in [27] to reconstruct the global irradiance incident on the tilted surfaces.

where the hourly diffuse fraction F_d(t) of the horizontal global irradiance is given by eq. (7):

R_g(t) is a function of the atmospheric transmittance. It is also called geometric factor and can be found using eq. (8):

A(t) is the anisotropy index that is a function of the atmospheric transmittance for beam radiation and is given by eq. (9):

where G_HI(t) [W/m²] is the global horizontal irradiance, HBI(t) [W/m²] is the global horizontal direct or beam irradiance, HDI(t) [W/m²] is the global horizontal direct or beam irradiance, and ρ is the ground reflectance constant that varies based on the surroundings of the tilted surface.

According to [28], the commonly used values for ground reflectance are ρ = 0.2 for hot and humid tropical locations, ρ = 0.5 for dry tropical locations, and ρ = 0.9 for snow-covered terrain. Here, β [°] denotes the tilt angle of the PV modules, h [°] the solar altitude, i [°] the incidence angle of the solar radiation, ω [°] is the hour angle, δ [°] is the declination angle, L [°] is the latitude of the installation site, a_m [°] is the PV module azimuth angle, and a_s [°] is the sun azimuth angle.

k(t) is the clearness index and mathematically defined by eq. (10):

where G₀(t) [W/m²] is the extraterrestrial irradiance and given by eq. (11) as a function of the solar altitude h [°] and the day number of the year N. This irradiance varies by ±3.3% around the solar constant (1367 W/m²) due to the ellipticity of the Earth's orbit:

The evaluation of the tilted global irradiance model was performed using the coefficient of determination R² given by eq. (12), which indicates the proportion of the variability in measured data explained by the model, and the Root Mean Square Error (RMSE) given by eq. (13), which represents the typical average deviation between modeled and measured values in the same unit (W/m²):

where: G_TI(t)measured is the global tilted irradiance measured, G_TI(t)modeled is the global tilted irradiance modelled, and GTImeasured¯ is the annual average of the global tilted irradiance measured.

In order to simulate the system behavior under operating conditions and optimaly size it, it’s necessary to model its main components (PV generator and Battery bank). Thus, the PV generator is modeled by its instantaneous output power P_PV(t) [W] by equation eq. (14) as it’s mentioned in [29]. It’s mainly depended on the global tilted irradiance G_TI(t) [W/m²] receives on its surface inclined β [°] relative to the horizontal at each time t of the year at the installation site and the PV modules temperature T_mod(t) [°C] given by eq. (15):

where P_P [W_p] is the peak power of the PV generator, G_R and T_aR are respectively reference irradiance and temperature values (1000 W/m² and 25 °C), F [%] is the soiling factor representing the power reduction due to dust (0.85), c is the temperature coefficient (0.386%/°C), T_a(t) is the instantaneous ambient temperature [°C], G_TI(t) is the instantaneous global tilted irradiance [W/m²], T_NOCT is module temperature at 800 W/m² irradiance and 25 °C ambient temperature.

When it’s come to the Batteries Bank, it is modeled by its storage capacity S_BAT [Ah] and state of charge SOC [Wh]. To optimize the battery storage capacity which is a key design parameter, eq. (2) has been reformulated as expressed in eq. (16) by replacing the daily energy requirement _D [Wh/days] with the average hourly consumption E_MH [Wh/h] given by eq. (17) and the number of autonomy days N_D [days] with the number of autonomy hours N_H [h]:

where t_MAX [h] corresponds to the number of hours with non-zero power demand.

The state of charge SOC(t) [Wh] is given as a function of the surplus or deficit (positive for surplus and negative for deficit) of production ∆P(t) [W] and the battery efficiency η_BAT [%] by eq. (18):

where ∆t [h] is simulation time step. In this study, a time step of 1h is considered.

The production deficit or excess at each time t ∆P(t) [W] is given by eq. (19):

where η_Reg [%] and η_Ond [%] are the efficiencies of the charge controller and the inverter, respectively. P_CH(t) [W] is the load power at each time t of the year.

The state of charge is constrained to evolve within a specific range to avoid rapidly degrading the batteries due to overcharging or excessive discharging, as shown in the system of eq. (20):

where SOC_MAX [Wh] is the maximum state of charge of the battery, SOC_MIN [Wh] is the minimum state of charge of the battery, P [%] is the percentage term representing the maximum energy level not to be exceeded in the batteries during charging and DOC_MAX [%] is the maximum depth of discharge of the battery.

The technical and economic performance of the system is evaluated using respectively the Loss of Power Supply Probability (LPSP) and the Levelized Cost of Energy (LCOE). The LPSP [%] is the indicator generally used to assess or measure the technical reliability of electricity generation systems [30]. It is expressed here by eq. (21):

where LPS(t) [Wh] corresponds to the energy deficit at each hour of the year. It is expressed by eq. (22):

where ∆P [W] is previously given by eq. (19). It corresponds to the difference between the power supplied by the PV generator and the power demanded by the user's load. The LCOE [EUR/kWh] is the indicator most used to assess the cost per kWh produced by the system or to measure the economic feasibility of electricity generation systems [30]. It is expressed here by eq. (23):

where NPV [EUR] is the Net Present Value, defined by eq. (24), taking into account the total initial investment cost C_IT [EUR], the operation and maintenance costs C_OM [EUR] which correspond to a certain percentage m [%] of the investment cost and the replacement costs C_R [EUR] of certain component respectively given by eq. (25), eq. (26) and eq. (27). CRF [%] is the Capital Recovery Factor which is precisely the annualization factor used to convert a Net Present Value (NPV) into an equivalent annual cost:

where C_Smod [EUR/W_p] and C_SBAT [EUR/W_p] correspond to the specific costs of a module a battery, respectively. C_Reg [EUR] and C_Ond [EUR] correspond to the costs of the charge controller and the inverter, respectively. C_SUP–INST [EUR] represents the cost of the mounting structures and the installation of the system and is taken here as 40% of the purchase cost of the PV generator modules and batteries bank. CRF [%], F_a [%] and F a j [%] correspond to the annualization factor, the overall discount factor for maintenance costs, and the discount factor for the replacement cost of each component j that needs to be replaced in the system, respectively. They are respectively given by eq. (28), eq. (29), and eq. (30) [30]:

where D_S [years] is the lifetime of the system, a_r [%] is the real interest rate, i [%] is the inflation rate and k is the k-th replacement.

N R j is the number of replacements of each component j defined by eq. (31):

where D_j [years] is the lifetime of each component j and D_S [years] is the system lifetime.

The lifetime of the batteries is a parameter that plays a role in the economic analysis of autonomous PV systems with storage. However, designers of these systems often use the battery lifetime provided by the manufacturer for a specific use where the charge-discharge cycles are regular. In autonomous PV systems with storage, the cycling of the storage is not regular due to the fluctuation of solar resource.

To account for this aspect in the sizing of these systems, a battery aging calculation model already proposed in [31] and based on manufacturer data has been integrated into the iterative approach developed in this study to predict the storage lifetime.

Thus, to evaluate the battery lifetime, the process followed is presented in Figure 2. In this process, the state of charge profile of the battery obtained from the system simulation is first subjected to the Rainflow cycle counting algorithm, which converts it into a series of depth of discharge values with j going from 1 to n. After that, the Miner’s rule is applied initially to assess the total fraction of damage T_VA [%] expressed by eq. (32), also called the annual aging rate. The battery lifetime is finally deduced in years from the annual aging rate using eq. (33) below.

where n corresponds to the cycle number counted and N C j(DOD_j) is the number of cycles associated with depth of discharge number j. This number is obtained by using the fitting power law equation (eq. (34)) below with two coefficients (a and b). These coefficients depend on the type of battery technology meaning that this fitting model is unique for every battery and is derived based on information from the manufacturer datasheet. For the study-case battery the fitted model is shown with a red line in Figure 7a.

For example, if the aging rate is 25%, then the battery lifetime is deduced to be 4 years. Miner’s rule states that for a given depth of discharge, the damage caused to the battery by a cycle is equal to the inverse of the number of cycles associated with that depth of discharge.

Figure 2.

Batteries bank aging calculation model

Based on the mathematical models of the main components of the physical system, performance indicators, and storage aging, the optimization problem for sizing is formulated according to the system of eq. (33) below. The problem thus consists of finding a peak power P_P [W_P] of the pv generator and storage autonomy N_H [h] which are capable of covering the user's needs as widely as possible and at the lowest cost.

To solve the previously defined optimization problem, an iterative algorithm developed in the MATLAB environment was implemented. The flowchart presented in Figure 3 illustrates the overall structure of the proposed optimization methodology. The algorithm begins by importing the weather and load data, the battery aging characteristics provided by the manufacturer, the techno-economic parameters and unit costs and the optimization constraints. For each considered number of autonomy hours, N_H the peak power of the PV generator P_Pis iteratively increased from PPMIN to PPMAX with a predefined step ∆P_P.

For each configuration (N_H ,P_P), an hourly energy balance is performed over one full year (8760 hours). At each time step ∆P(t) which corresponds to de difference between the PV generator production and the load demand is evaluated. The battery state of charge (SOC) is updated depending on whether the system is in charging or discharging mode, while strictly enforcing the operational limits SOC_MIN and SOC_MAX Whenever the battery reaches its lower bound and all the load is not met, the unmet load is quantified through the instantaneous loss of power supply LPS(t). After completing the annual simulation, the Loss of Power Supply Probability (LPSP) is calculated. The algorithm then evaluates and stores the battery lifetime using the rainflow cycle counting method, the total annual energy produced and the Levelized Cost of Energy (LCOE). The process continues until the maximum autonomy hours NHMAX is reached.

Finally, only configurations whose LPSP is below to the LPSP_MAX (LPSP ≤ LPSP_MAX) are considered technically feasible. The configuration offering the best compromise between system reliability (low LPSP) and economic performance (minimum LCOE), ensuring both technical feasibility and cost-effectiveness is selected among all feasible configurations as the optimal solution.

Figure 3.

Flowchart of the Sizing Program

To apply the iterative approach developed in this study, hourly profiles of solar irradiation, ambient temperature, and household consumption from a peri-urban neighborhood of Ouagadougou (Zagtouli) illustrated in Figure 4 were used. The meteorological data come from the weather station installed in this locality of Burkina Faso. The consumption curve presented in Figure 4c shows two peaks: one at 5 AM and another at 8 PM. The solar irradiation and ambient temperature profiles fluctuate throughout the year, illustrating the variation in solar resources and their influence on temperature. The technical and financial characteristics of the system components, as well as their lifespan, are listed in Table 1. The interest and inflation rates used are the same as those in the LCOE calculation in [32]. The photovoltaic modules used are 265 W_p panels assembled by Faso Energy, a Burkinabe company specializing in photovoltaic cell assembly. These panels are oriented due south with a 12° tilt, corresponding to the latitude of Ouagadougou. Table 2 presents the number of battery cycles depending on the depth of discharge for a specific application. These data were provided by the manufacturer. Table 3 specifies the technical constraints adopted in the system design.

Figure 4.

(a) Hourly profile of irradiance on a horizontal plane; (b) Hourly profile of ambient temperature; (c) Hourly profile of household consumption

After simulating the sizing model using the input parameters previously defined in Figure 4 and Table 1, Table 2, and Table 3, the following results were obtained:

The Figure 5 presents the evolution of the measured global horizontal irradiance (GHI), the measured global tilted irradiance (measured GTI), and the modeled tilted irradiance (GTI model) over two consecutive days of the year. A typical daily solar cycle is observed, with a rapid increase in the morning, a peak of about 1000 W/m² near solar noon, and a symmetric decline in the afternoon.

The measured tilted irradiance remains higher than the horizontal irradiance during daylight hours, as expected, since tilted PV modules intercept a larger portion of the direct solar radiation.

The modeled tilted irradiance obtained using the Erbs and Hay-Davies methods (orange curve), closely matches the measurements, accurately reproducing both the amplitude and the daily pattern.

A coefficient of determination (R²) of 0.96, indicating that 96% of the variability in measured irradiance is explained by the model, and an RMSE of 65.30 W/m², representing the typical deviation between modeled and measured values, confirm the model’s reliability in estimating tilted-plane irradiance from horizontal data. Minor deviations, particularly near solar noon, mainly reflect atmospheric variability or measurement uncertainties.

Figure 5.

Measured and modeled global irradiance on a tilted surface over two consecutive days

Figure 6a illustrates the PV generator size and LPSP as a function of battery autonomy. For an autonomy of 1 to 13 hours, the PV generator size remains fixed at 25,600 Wp, while the LPSP decreases linearly from 70% to 11%. This means that with an autonomy of 13 hours or less, it is impossible to achieve an LPSP of 10% or lower, regardless of the PV generator used. In this range (1h to13 h of autonomy), the optimization algorithm progressively increases the PV generator size up to its maximum allowable value (26.5 kWp) in an attempt to satisfy the LPSP constraint. However, due to the very limited storage capacity associated with low autonomy levels, most of the daytime PV production cannot be effectively stored and is therefore curtailed. As a result, the nighttime household demand remains largely unmet, and the LPSP constraint cannot be achieved even with the maximum PV generator size. Between 13 and 16 hours, the generator size drops rapidly from 26,500 Wp to 1,060 Wp and remains constant beyond that, while the LPSP decreases from 11% to 9.35%, then declines more slowly. From 13 h of autonomy onward, the storage capacity becomes sufficiently large to cover a significant share of the household’s nighttime demand. his allo s the optimization algorithm to drastically reduce the V generator size hile still satisfying the LPSP ≤ 10% constraint. Consequently, the sharp decrease in PV generator size observed in this interval reflects a transition from a PV-dominated strategy to a storage-driven reliability improvement. An autonomy beyond 17 hours slightly reduces the LPSP but significantly increases storage size, impacting the LCOE. Thus, a minimum autonomy of 17 hours and a PV generator of 1,060 Wp are required to meet 90% of the household’s annual energy demand. This region therefore corresponds to an optimal trade-off, where a relatively moderate storage autonomy (around 16–17 h) enables high supply reliability with a minimal PV generator size, thereby defining a techno-economic “s eet spot” for system design.

Figure 6b shows the curves of LCOE and storage lifespan as a function of storage autonomy. For an autonomy of 13 hours or less, the LCOE (0.05 EUR) and storage lifespan (1.80 years) remain constant due to the stable size of the PV generator and the low storage capacity. However, this autonomy does not meet the 10% LPSP threshold. Between 13 and 16 hours, the LCOE increases rapidly (0.05 to 0.20 EUR), while the storage lifespan rises to 2 years. In this interval, the rapid increase in LCOE is mainly driven by the growth of storage capacity, which becomes the dominant cost component of the system, while the PV generator size is significantly reduced. Beyond 16 hours, the storage lifespan grows linearly, and the LCOE fluctuates, likely due to variations in the number of battery replacements required for the storage system. Beyond 16 h of autonomy, further increases in storage capacity do not reduce the PV generator size any further, but they continue to increase the investment and replacement costs of the battery system. The observed LCOE fluctuations are therefore mainly associated with variations in the predicted storage lifetime and, consequently, in the number of battery replacements over the project lifetime.

Figure 6.

(a) PV Generator and LPSP evolution as a function of storage autonomy; (b) LCOE and Battery lifetime evolution as a function of storage autonomy

In order to estimate the storage lifetime associated with the optimal configuration, a cycle-based aging methodology as adopted. his approach combines the manufacturer’s cycle-life data, the simulated annual SOC profile, and a cycle counting procedure based on the Rainflow algorithm. The overall objective is to quantify the cumulative battery degradation under real operating conditions.

Figure 7a presents the aging curve of the AGM batteries used, obtained by fitting the data from manufacturer (Table 2) using MATLAB. This curve makes it possible to determine the number of regular cycles for any given Depth of Discharge (DOD). This aging model constitutes the reference framework for evaluating battery degradation once the actual operating cycles are identified from the simulated SOC profile. For the fit power law model used, the estimated coefficients are a=3.128×10 and b=−1.519. The goodness-of-fit indicators confirm the adequacy of the selected model, with a coefficient of determination R² and a root mean square error RMSE=87.74 cycles. The high R² value indicates that more than 98% of the variability in the manufacturer’s cycle-life data is explained by the fitted function.

Figure 7b now presents the state-of-charge (SOC) profile of the storage from the optimal configuration simulation, which ranges between 90% and 27% and complies with the defined storage management strategy. This SOC profile represents the real operating conditions of the battery over one year and serves as the input signal for the cycle counting analysis.

Figure 7c classifies the cycles by amplitude or DOD using the Rainflow algorithm. Based on the SOC profile shown in Figure 7b, the algorithm identifies 378 cycles. Each identified cycle is characterized by its amplitude (DOD), which allows associating it with the corresponding allowable number of cycles derived from the aging curve shown in Figure 7a. he cumulative damage is then calculated using Miner’s rule, hich assumes that the total degradation corresponds to the linear accumulation of partial damages induced by each cycle. When the accumulated damage reaches unity, the battery is considered to have reached its end of life. Based on this approach, the storage lifetime in the optimal configuration is estimated at approximately 2 years. This relatively short lifetime highlights the significant impact of the cycling profile and storage autonomy on battery degradation, confirming the strong coupling between reliability requirements and economic performance.

Figure 7.

(a) Battery cycle life as a function of Depth of Discharge (DOD) based on manufacturer data and fitted aging model; (b) Battery Annual hourly State of Charge; (c) Distribution of cycle Depth of Discharge obtained using the Rainflow counting algorithm

In the conventional method, the PV generator size is calculated using the average daily energy demand (4500 Wh), obtained by summing the hourly load over one day, the average daily solar irradiation at the Zagtouli site (5 kWh∕m²∕day), and an overall system efficiency of 0.7. This global efficiency results from the product of the charge controller efficiency (85%), inverter efficiency (90%), and dust-related losses (85%).

This simplified sizing approach relies on averaged quantities and does not explicitly consider the hourly interaction between PV production, storage operation, and load demand during the design phase. The storage capacity is determined assuming a fixed autonomy of one day, a battery bank voltage of 48 V, a battery efficiency of 85%, and a maximum depth of discharge (DOD) of 70%, as commonly adopted in conventional rural electrification practices.

To allow a fair comparison with the proposed approach, the conventional configuration was subsequently simulated under real hourly operating conditions using the same load profile and solar resource data. This simulation shows that the conventional solution achieves a reliability level comparable to that of the optimized configuration, with an LPSP of 9.02%, while the optimized solution yields an LPSP of 9.35%.

Despite these similar reliability levels, significant differences are observed in terms of economic performance. As shown in Table 4, the optimized configuration obtained from the proposed iterative optimization procedure consists of a PV generator size of 1060 Wp and 16 hours of storage autonomy, corresponding to a storage capacity of 100.60 Ah and an LCOE of 0.20 EUR/kWh.

When comparing both approaches, a 35.92% reduction in storage capacity is achieved with the optimized configuration, leading to a corresponding 37.50% reduction in LCOE. This indicates that the proposed method identifies a more cost-effective solution for nearly identical reliability levels.

The main difference between the two approaches therefore lies not in the achieved reliability, but in the sizing philosophy. The proposed optimization explicitly accounts for hourly system dynamics, storage aging, and economic trade-offs during the design stage, whereas the conventional method relies on fixed autonomy assumptions and averaged indicators. As a result, the optimized approach ensures a better compromise between economic performance and reliability while preserving the required comfort level of electricity supply.