Abstract
Photovoltaic (PV) systems are highly sensitive to environmental factors, which directly impact their energy conversion efficiency. Traditional maximum power point tracking (MPPT) algorithms often suffer from low accuracy and susceptibility to local optima. To address these challenges, this study proposes a multi-strategy enhanced Beluga Whale Optimization (MBWO) algorithm. Compared to the original BWO, MBWO incorporates Logistic chaotic mapping for population initialization, improves balance factors and predation probability, introduces Cauchy mutation to enhance global exploration, and integrates adaptive weighting for refined local search. Simulation results on MATLAB/Simulink demonstrate that MBWO achieves superior tracking speed, accuracy, and stability under complex shading conditions, making it a robust solution for real-world MPPT applications.
1. Introduction
The global transition toward carbon neutrality has intensified the demand for renewable energy systems, particularly solar PV systems. As a core technology in PV systems, MPPT ensures optimal energy harvest by dynamically adjusting the operating point to the maximum power point (MPP). However, partial shading or rapid environmental changes often create multi-peak power-voltage (P-U) curves, rendering conventional MPPT methods (e.g., perturb-and-observe, incremental conductance) ineffective. While intelligent algorithms like Particle Swarm Optimization (PSO) offer global optimization capabilities, their slow convergence and parameter sensitivity limit practicality.
The Beluga Whale Optimization (BWO) algorithm, inspired by beluga behaviors, provides a parameter-efficient framework for MPPT. This work enhances BWO through multi-strategy fusion, focusing on faster convergence, improved accuracy, and adaptability to dynamic conditions.
2. Photovoltaic System Modeling
2.1 PV Cell Mathematical Model
A PV cell’s output characteristics are governed by the single-diode equivalent circuit (Fig. 1). The current-voltage relationship is derived as:I=Iph−Id(eq(V+IRs)nkT−1)−V+IRsRshI=Iph−Id(enkTq(V+IRs)−1)−RshV+IRs
where IphIph is the photocurrent, IdId is the diode saturation current, RsRs and RshRsh are series and shunt resistances, nn is the ideality factor, and TT is the cell temperature.
2.2 Multi-Peak Characteristics Under Partial Shading
Partial shading creates multiple local MPPs, complicating MPPT. Table 1 summarizes the P-U characteristics under three shading patterns (uniform, static shading, dynamic shading).
Table 1: P-U Characteristics Under Different Shading Conditions
| Shading Pattern | Irradiance (S1–S4, W/m²) | MPP Power (W) |
|---|---|---|
| Uniform | 1000, 1000, 1000, 1000 | 852.6 |
| Static Shading | 1000, 1000, 800, 400 | 551.8 |
| Dynamic Shading | 1000→800, 800→600 | 412.6→411.3 |
3. Beluga Whale Optimization (BWO) Algorithm
BWO mimics beluga swimming, hunting, and whale-fall behaviors. Its three phases are:
3.1 Exploration Phase
Belugas update positions synchronously or mirroringly:Xi,jt+1={Xi,pt+r1sin(2πr2)⋅(Xr,pt−Xi,pt),if j evenXi,pt+r1cos(2πr2)⋅(Xr,pt−Xi,pt),if j oddXi,jt+1={Xi,pt+r1sin(2πr2)⋅(Xr,pt−Xi,pt),Xi,pt+r1cos(2πr2)⋅(Xr,pt−Xi,pt),if j evenif j odd
where r1,r2∈[0,1]r1,r2∈[0,1] are random numbers, and Xr,ptXr,pt is a randomly selected individual.
3.2 Exploitation Phase
Belugas converge toward the best solution using Lévy flight:Xit+1=r3Xbestt−r4Xit+C1⋅LF⋅(Xrt−Xit)Xit+1=r3Xbestt−r4Xit+C1⋅LF⋅(Xrt−Xit)
where C1=r5(2−t/Tmax)C1=r5(2−t/Tmax), and LFLF follows a Lévy distribution.
3.3 Whale-Fall Phase
Weak individuals are replaced to maintain population diversity:Xit+1=r6Xit−r7XsteptXit+1=r6Xit−r7XsteptXstept=e−2t/Tmax⋅(Ub−Lb)Xstept=e−2t/Tmax⋅(Ub−Lb)
4. Multi-Strategy Enhanced BWO (MBWO)
4.1 Chaotic Initialization via Logistic Mapping
Logistic chaotic sequences replace random initialization to improve population diversity:rt+1=4⋅rt(1−rt)rt+1=4⋅rt(1−rt)Xi,j=Lb+(Ub−Lb)⋅ri,jXi,j=Lb+(Ub−Lb)⋅ri,j
4.2 Nonlinear Balance Factor
A nonlinear balance factor BfBf accelerates early-stage exploration:Bf=B0⋅2r8⋅exp(−π2⋅tTmax)Bf=B0⋅2r8⋅exp(−2π⋅Tmaxt)
4.3 Cauchy Mutation for Global Exploration
Cauchy-distributed perturbations enhance escape from local optima:Xi′=Xbestt+Xbestt⋅Cauchy(0,1)Xi′=Xbestt+Xbestt⋅Cauchy(0,1)
4.4 Adaptive Weighting for Local Refinement
An exponentially decaying weight prioritizes global search initially and local refinement later:ω=0.4+0.5⋅exp(−0.7tTmax)ω=0.4+0.5⋅exp(−Tmax0.7t)Xit+1=ω⋅(r9Xbestt−r10Xit)+C1LF(Xrt−Xit)Xit+1=ω⋅(r9Xbestt−r10Xit)+C1LF(Xrt−Xit)
5. MPPT Control Using MBWO
5.1 Algorithm Workflow
- Position Representation: Each beluga represents a duty cycle for the DC/DC converter.
- Termination Criteria:
- Maximum iterations reached.
- Relative position difference < 1%.
- Restart Mechanism: Triggers if power deviation exceeds 3%:
∣Pt−Pm∣Pm>0.03Pm∣Pt−Pm∣>0.03
5.2 Simulation Results
Table 2: Performance Comparison Under Uniform Irradiance
| Algorithm | Tracking Time (s) | Power (W) | Accuracy (%) |
|---|---|---|---|
| PSO | 0.26 | 849.0 | 99.53 |
| BWO | 0.18 | 851.1 | 99.82 |
| MBWO | 0.07 | 851.7 | 99.89 |
Table 3: Performance Under Dynamic Shading
| Algorithm | Tracking Time (s) | Power (W) | Accuracy (%) |
|---|---|---|---|
| PSO | 0.30 | 409.6 | 99.27 |
| BWO | 0.27 | 410.5 | 99.46 |
| MBWO | 0.10 | 411.3 | 99.68 |
6. Conclusion
The proposed MBWO algorithm significantly enhances MPPT performance under partial shading and dynamic conditions. Key innovations include chaotic initialization, Cauchy mutation, and adaptive weighting, which collectively improve convergence speed by 59% and accuracy by 0.3–0.5% compared to BWO. Future work will focus on hardware-in-the-loop validation and scalability for large PV arrays.