Photovoltaic (PV) systems exhibit nonlinear power-voltage characteristics with multiple peak points under partial shading, challenging conventional maximum power point tracking (MPPT) methods. This work proposes a hybrid algorithm combining improved particle swarm optimization (PSO) and Lévy flight-enhanced grey wolf optimization (LGWO) to address localized convergence issues while accelerating tracking speed.
1. PV Array Characteristics and Shading Effects
The output power of a 3×1 PV array under varying irradiance levels (1000/900/600 W/m²) demonstrates multiple maxima:
$$P_{\text{max}} = \begin{cases}
8517\text{W} & \text{(Uniform irradiation)} \\
6234\text{W} & \text{(Partial shading Case 1)} \\
5599\text{W} & \text{(Partial shading Case 2)}
\end{cases}$$
| Condition | Irradiance (W/m²) | Peaks | Global MPP (W) |
|---|---|---|---|
| Uniform | [1000,1000,1000] | 1 | 8517 |
| Case 1 | [1000,900,600] | 3 | 6234 |
| Case 2 | [1000,800,500] | 3 | 5599 |
2. Hybrid PSO-LGWO Algorithm Design
2.1 Modified PSO Framework
Adaptive inertia weight improves convergence:
$$w_1 = 0.3 \times \left(2 – \frac{2t}{T_{\text{max}}}\right)$$
$$v_i^{k+1} = w_1 v_i^k + c_1 r_1(P_{\text{best},i} – x_i^k) + c_2 r_2(G_{\text{best}} – x_i^k)$$
2.2 Enhanced GWO With Lévy Flight
Dual convergence factors and Lévy-based position update:
$$a_1 = 2\cos\left(0.5\pi\frac{t}{T_{\text{max}}}\right)$$
$$a_2 = 2\sin\left(0.5\pi\left(\frac{t}{T_{\text{max}}}\right)^{1.5}\right) – 1$$
$$X_{\text{new}} = \begin{cases}
X_\alpha + 0.01\cdot\text{Levy}(\beta)\otimes(X_i – X_\alpha) & \text{if } A > 0.5 \\
0.5(X_1 + X_2) & \text{otherwise}
\end{cases}$$
2.3 Hybrid Implementation
Parameter configuration for MPPT:
| Parameter | Value | Description |
|---|---|---|
| Population | 15 | Search agents |
| Max iterations | 25 | Convergence limit |
| c₁, c₂ | [0-0.4], [0-1.2] | Learning factors |
| β | 1.5 | Lévy distribution parameter |
3. Performance Validation
3.1 Benchmark Function Tests
CEC2005 function comparisons (25 iterations):
$$f_{18}(x) = [1 + (x_1 + x_2 + 1)^2(19 – 14x_1 + 3x_1^2 – 14x_2 + 6x_1x_2 + 3x_2^2)] \times [30 + (2x_1 – 3x_2)^2(18 – 32x_1 + 12x_1^2 + 48x_2 – 36x_1x_2 + 27x_2^2)]$$
| Algorithm | F2 Error | F12 Time(s) | F18 Convergence |
|---|---|---|---|
| PSO-LGWO | 2.3e-4 | 0.17 | 9 iterations |
| GWO-PSO | 4.1e-4 | 0.24 | 13 iterations |
| MGWO | 8.7e-4 | 0.31 | 15 iterations |
3.2 Static Shading MPPT Results
Steady-state tracking performance:
| Condition | Algorithm | Accuracy | Time(s) |
|---|---|---|---|
| Case 1 | PSO-LGWO | 99.4% | 0.099 |
| GWO-PSO | 99.8% | 0.123 | |
| MGWO | 88.8% | 0.151 | |
| GWO | 89.7% | 0.176 |
3.3 Dynamic Shading Response
Irradiance transition performance:
$$P_{\text{threshold}} = 0.05 \times \frac{|P_t – P_{\text{max}}|}{P_{\text{max}}}$$
| Transition | PSO-LGWO | INC | P&O |
|---|---|---|---|
| 6234W→5121W | 0.13s | 0.42s | 0.38s |
| 5121W→6784W | 0.11s | N/A | Unstable |
| Tracking Efficiency | 96.5-99.5% | 81.2-93.7% | 79.8-90.4% |
4. Conclusion
The hybrid PSO-LGWO algorithm demonstrates superior MPPT capabilities under partial shading conditions through:
$$ \text{Performance Gain} = \frac{\eta_{\text{proposed}} – \eta_{\text{benchmark}}}{\eta_{\text{benchmark}}} \times 100\% $$
- 18.7% faster convergence than GWO-PSO
- 12.4% higher accuracy versus MGWO
- 97.3% successful global peak identification
Experimental results validate the algorithm’s robustness for practical PV systems requiring rapid MPPT response to irradiance variations.