This paper proposes a composite optimization algorithm combining an enhanced cuckoo search strategy with variable-step perturb and observe (P&O) method to address the multi-peak power-voltage (P-U) characteristics of photovoltaic (PV) arrays under partial shading conditions. The algorithm integrates population initialization optimization and adaptive switching mechanisms to improve tracking speed and accuracy while minimizing power oscillation.
1. Photovoltaic Array Output Characteristics
1.1 Mathematical Model of PV Cell
The single-diode equivalent circuit model describes PV cell behavior:
$$I = I_{ph} – I_o\left[\exp\left(\frac{q(V + IR_s)}{nKT_j}\right) – 1\right] – \frac{V + IR_s}{R_{sh}}$$
Simplified for practical applications:
$$I \approx I_{ph} – I_o\left[\exp\left(\frac{q(V + IR_s)}{nKT_j}\right) – 1\right]$$
Where temperature-dependent parameters are defined as:
$$I_{ph} = I_{sc}\left[1 + \alpha(T – T_{ref})\right]\frac{G}{G_{ref}}$$
$$I_o = \beta T^3 \exp\left(-\frac{E_g}{kT}\right)$$
| Parameter | Description | Value |
|---|---|---|
| \(I_{sc}\) | Short-circuit current | 7.84A |
| \(V_{oc}\) | Open-circuit voltage | 36.3V |
| \(P_{max}\) | Maximum power | 213W |
| \(n\) | Ideality factor | 1.3-1.5 |
1.2 Partial Shading Characteristics
Under partial shading conditions, the P-U curve exhibits multiple local maxima:
$$P_{array} = \sum_{i=1}^N V_i \cdot \min(I_{1}, I_{2}, …, I_{N})$$
| Module | Uniform (W/m²) | Pattern 1 (W/m²) | Pattern 2 (W/m²) |
|---|---|---|---|
| PV1 | 1000 | 1000 | 800 |
| PV2 | 1000 | 900 | 700 |
| PV3 | 1000 | 800 | 600 |
| PV4 | 1000 | 700 | 500 |
2. Hybrid MPPT Algorithm Design
2.1 Enhanced Cuckoo Search Algorithm
Improved initialization strategy for population distribution:
$$x_i^{init} = V_{min} + \frac{i}{N}(V_{max} – V_{min}) \quad i=1,2,…,N$$
Adaptive discovery probability:
$$P_a = 0.25 + 0.1 \cdot \frac{t}{t_{max}}$$
Modified Lévy flight step size:
$$L(\lambda) = \frac{\phi \cdot \mu}{|v|^{1/\beta}} \cdot (x_i – x_{best})$$
$$\phi = \left[\frac{\Gamma(1+\beta) \cdot \sin(\pi\beta/2)}{\Gamma((1+\beta)/2) \cdot \beta \cdot 2^{(\beta-1)/2}}\right]^{1/\beta}$$
2.2 Variable-step P&O Integration
Switching condition:
$$\Delta P/P_{max} < 1\% \quad \text{and} \quad \Delta V < 0.5V$$
Adaptive step size calculation:
$$\delta V = \delta_{min} + k \cdot \left|\frac{dP}{dV}\right|$$
$$k = 0.05 \cdot \frac{V_{oc}}{P_{max}}$$
3. Simulation Results and Analysis
MATLAB/Simulink simulation parameters:
| Parameter | Value |
|---|---|
| Array Configuration | 5S4P |
| Simulation Time | 2s |
| Sampling Frequency | 10kHz |
| DC Bus Voltage | 200V |
3.1 Uniform Irradiation Performance
Comparison of tracking characteristics:
$$t_{settle} = \frac{1}{\omega_n \zeta} \sqrt{1 – \zeta^2}$$
| Algorithm | Settling Time (s) | Overshoot (%) | Efficiency (%) |
|---|---|---|---|
| Standard CS | 0.56 | 4.2 | 98.1 |
| Proposed Method | 0.16 | 0.8 | 99.4 |
3.2 Partial Shading Performance
Multi-peak tracking results:
$$MPP_{detection} = \arg\max_{V \in [V_{min}, V_{max}]} P(V)$$
| Shading Pattern | Global MPP (W) | Tracking Time (s) | Accuracy (%) |
|---|---|---|---|
| Pattern 1 | 3360 | 0.17 | 99.2 |
| Pattern 2 | 2540 | 0.21 | 98.7 |
4. Conclusion
The hybrid MPPT algorithm demonstrates superior performance in both uniform and partial shading conditions. Key improvements include:
- 63% faster convergence compared to standard cuckoo search
- Power oscillation reduction from ±15W to ±1W
- 98.5% average tracking accuracy under dynamic shading
This research provides an effective solution for maximizing energy harvest in partially shaded PV systems through intelligent algorithm integration and adaptive control strategies.