In this modeling approach, the single-diode equivalent photovoltaic model is adopted. This model offers the best trade-off between accuracy and computational complexity [11][12]. The generalized I-V relationship for a module with Ns series-connected cells is:

Where I p h is the photogenerated current [A], I 0 is the diode saturation current [A], n is the ideality factor, V t = k T q the thermal voltage [V] with k the Boltzmann constant, q the electron charge and T the absolute temperature [K], and R s , R s h are the series and shunt resistances [Ω].

The photogenerated current I p h varies linearly with solar irradiance G [W/m²] and depends on temperature T according to [13][14]:

The diode saturation current (I₀) is strongly temperature-dependent:

where E g ≈ 1.121 eV is the silicon bandgap energy. The open-circuit voltage Voc decreases with temperature at the rate K_V ≈ −0.34 %/˚C, while the short-circuit current I s c increases slightly at the rate K_I ≈ +0.065 %/˚C.

Table 1 presents the electrical parameters of the 250 W PV module at Standard Test Conditions (STC: G = 1000 W/m², T = 25˚C, AM = 1.5).

The internal single-diode parameters used in the simulation are: photogenerated current I p h = 8.9397 A, diode saturation current I 0 = 4855 × 10⁻¹⁰ A, ideality factor n = 1, series resistance R s = 0.31 Ω, and shunt resistance R s h = 188.26 Ω. These values were extracted from the module datasheet (Voc, Isc, Vmp, Imp, KV, KI). The simulated STC operating points (Voc, Isc, Vmp, and Imp) coincide with the datasheet values, resulting in negligible errors and confirming our simulated parameters.

Table 1. Electrical parameters of the 250 W PV module (STC).

The corresponding I-V and P-V curves are presented in Figures 1-2.

Figure 1 and Figure 2 illustrate the simulated curves of the 250 W PV module. The I-V nonlinearity and the uniqueness of the MPP at each irradiance level are clearly visible. The short-circuit current I s c evolves proportionally to G, while Voc varies logarithmically. With temperature, Voc decreases significantly (−0.34%/˚C) and I s c increases slightly (+0.065%/˚C), resulting in a net reduction of maximum power of approximately 0.40%/˚C.

The validation results indicate perfect agreement between the simulated and manufacturer-reported STC operating points. This outcome is expected because the single-diode parameters were extracted directly from the datasheet through an iterative fitting procedure. Therefore, the obtained model accurately reproduces the reference electrical characteristics before being used for MPPT performance evaluation.

Figure 1. I-V and P-V curves at temperature T = 25˚C.

Figure 2. I-V and P-V curves at irradiance G = 1000 W/m².

The dynamic modeling of the Boost converter is based on the averaged state-space method, introduced by Middlebrook and Ćuk [15][16]. This approach consists in deriving the differential equations of the circuit for each of the two switching phases and then weighting them by the duty cycle D. This method provides a linearized model that is valid in Continuous Conduction Mode (CCM) and can be used directly for analysis and sizing [17][18].

Applying the averaged state-space method, the equations governing the dynamic behavior of the Boost converter are:

In steady state, the derivatives vanish and we obtain the characteristic conversion ratio of the Boost converter:

The minimum sizing of inductance L and capacitance C is performed to meet the specified ripple criteria: inductor current ripple I L ≤ 5 % of the average inductor current I_L and output voltage ripple Δ V o u t ≤ 1 % of V p v .

The design parameters selected for the boost converter are summarized in Table 2. The values adopted in the simulation (L = 2 mH and C = 470 µF) well exceed these minimum constraints and guarantee robust CCM operation over the entire studied temperature range (25 - 70˚C).

Table 2. Boost converter design parameters.

The complete PV-Boost-MPPT system was implemented in MATLAB/Simulink using the averaged state-space model of the Boost converter operating in CCM. The initial duty cycle was set to D₀ = 0.50. The load resistance was fixed at RL = 9.2 Ω, corresponding to a nominal output voltage of 48 V. The PWM switching frequency was fs = 20 kHz, while the MPPT sampling period was TPO = 0.05 s. Each simulation was carried out over a duration of 4 s.

5.1.1. Tracking Efficiency

Tracking efficiency ( η M P P T ) is the main performance criterion. It represents the ratio between the energy actually extracted by the chopper (or inverter) and the maximum theoretical energy available over the simulation duration [19]:

Where T s i m represents the total simulation time. The reference power P M P P ( t ) was generated from the PV model by directly computing the theoretical maximum power point corresponding to the instantaneous irradiance and temperature conditions. At each simulation instant, P M P P ( t ) represents the global maximum of the P-V characteristic, whereas P M P P T ( t ) corresponds to the power extracted by the simulation.

5.1.2. Convergence Dynamics and Thermal Behavior

Over a 4 s simulation window, the system reaches steady state after approximately 24 iterations (1.2 s × 1/0.05 s) where 0.05 s corresponds to the sampling period T P O .

The importance of MPPT is amplified by thermal variations. When the temperature increases from 25˚C to 70˚C, the voltage V_MPP decreases, shifting the optimal operating point according to:

Without MPPT (fixed D = 0.5), the system operates at a fixed point that significantly deviates from the actual MPP, producing only 179.4 W at 25˚C (183.65 W on average over the 25˚C - 70˚C range).

With active P & O MPPT, the controller dynamically adjusts the duty cycle, reaching a steady-state power of 249.9 W at 25˚C corresponding to an average tracking efficiency of 96.80%.

5.1.3. Sensitivity Analysis of Step ΔD

The choice of the perturbation step ΔD is the most critical tuning parameter of the P & O algorithm. A systematic sensitivity analysis (Figure 6) was carried out by varying ΔD from 0.001 to 0.030 under fixed STC conditions (G = 1000 W/m², T = 25˚C).

Figure 6.Impact of the perturbation step ΔD on the tracking efficiency η_MPPT (left axis, blue) and the convergence time t_conv (right axis, red). The dashed green line indicates the optimal value ΔD = 0.005 at (T = 25˚C, G = 1000 W/m²).

The convergence time t_conv is defined as the first instant at which the extracted power enters and remains within ±2% of the theoretical maximum power P_MPPP. Similarly, the retracking time t_retrack is defined as the time required to return and remain within the same tolerance band after a temperature step change.

For each value, the tracking efficiency η_MPPT and the convergence time t_con were recorded at steady state (Table 4).

Table 4. Impact of step ΔD on system performance.

1) Speed vs. Accuracy: When the perturbation step is too small (ΔD < 0.002), oscillations are minimized, but convergence is considerably slowed, with a response time exceeding 2.5 s.

2) Stability vs. Efficiency: Conversely, a too-large step (ΔD > 0.015) speeds up convergence but reduces efficiency below 92% due to excessive residual oscillations around the MPP.

3) Optimal Value: The optimal perturbation step ΔD = 0.005 provides the best compromise, yielding a tracking efficiency of 96.8% and a convergence time of 1.2 s under fixed STC conditions (G = 1000 W/m², T = 25˚C). This finding aligns with the recommendations of Femia et al. [20], who advise setting the step size to approximately 1/200 of the duty cycle’s dynamic range to minimize steady-state oscillations while ensuring sufficient responsiveness to environmental changes.

To assess the robustness of the optimal perturbation step (ΔD = 0.005), a stepwise temperature profile consisting of four levels (25˚C, 40˚C, 55˚C, and 70˚C) was applied during a 4 s simulation. Figure 7 illustrates the dynamic response of the PV power, duty cycle, and PV voltage.

1) Retracking Performance: The P & O algorithm effectively retracks the new maximum power point following each temperature change, with a re-stabilization time of 0.04 to 0.08 s per step.

2) Duty Cycle Adaptation: The controller dynamically adapts the duty cycle from D = 0.375 at 25˚C to approximately D = 0.49 at 70˚C to compensate for the reduction in MPP voltage (V_mpp).

In open-loop control with fixed D = 0.50, the extracted power increases slightly with temperature, from 179.4 W at 25˚C to 186.2 W at 70˚C. This phenomenon is explained by the slight increase in short-circuit current I_sc with T (KI = +0.065%/˚C) which shifts the intersection of the I-V curve with the fixed load line toward slightly higher voltage and current values. However, this power remains far below the theoretical MPP, which decreases from 250 W to 203.7 W over the same temperature range, fully justifying the use of P & O MPPT.

Figure 7. Dynamic response of the PV-Converter-MPPT P & O system under successive temperature steps (25˚C → 40˚C → 55˚C and 70˚C).

Table 5 provides a detailed quantitative comparison of the system performance under the two control configurations. Four independent simulations were performed at constant temperatures of 25˚C, 40˚C, 55˚C, and 70˚C, under constant irradiance of 1000 W/m² (STC conditions).

Table 5. Performance comparison between optimized P & O MPPT and open-loop control under thermal constraints.

These results show that the gain provided by P & O MPPT is greater at lower temperatures.

1) Power Gain: The MPPT system delivers an average gain of +25.6% compared to a fixed duty cycle (D = 0.50).

2) Temperature Impact: At 25˚C, the gain reaches its maximum of +39.3%, underscoring the critical importance of active tracking even when available power is at its maximum.

3) Overall Efficiency: The average tracking efficiency remains at 96.80% across the entire temperature range, confirming the validity of the Boost converter component sizing (L = 2 mH, C = 470 µF) in Continuous Conduction Mode (CCM).

Figure 8 shows a histogram comparing the power gains and the overall system efficiency under different operating conditions.

Figure 8. Comparative histogram of PV power extracted using open-loop control (fixed D = 0.50), Optimized P & O MPPT, and the theoretical maximum power point (ideal MPP) at different temperatures (G = 1000 W/m²).

The results confirm that the P & O algorithm delivers near-optimal power extraction in steady state (average η_MPPT = 96.80%, ranging from 97.09% at 25˚C to 96.46% at 70˚C), which is consistent with the performances reported in the literature for similar boost converter configurations [21][22]. The average gain of +25.6% (reaching +39.3% at 25˚C) compared to a fixed duty cycle underscores the critical importance of active tracking to maximize the energy efficiency of the photovoltaic generator [23].

The main limitation of fixed-step P & O observed in this study is the persistence of a low-amplitude oscillation around the maximum power point (P_MPP) in steady state. Adaptive step variants, proportional to |ΔP/ΔV|, would drastically reduce these oscillations without sacrificing convergence speed [24][25]. Although P & O is effective under thermal variations, its main limitation arises during rapid irradiance changes (G > 500 W/m²), where the power variation due to the perturbation may be dominated by the environmental variation, leading to mistracking [26]. To overcome these shortcomings, artificial intelligence-based approaches, such as artificial neural networks (ANN) or fuzzy logic, offer superior dynamic performance and tracking accuracy by eliminating residual oscillations [27][28].