Abstract
Photovoltaic (PV) systems are pivotal in achieving carbon neutrality, yet their efficiency is highly susceptible to environmental factors. Maximum Power Point Tracking (MPPT) remains a critical challenge, as conventional algorithms often suffer from slow convergence and local optima entrapment. This paper proposes a Multi-Strategy Beluga Whale Optimization (MBWO) algorithm to address these limitations. By integrating Logistic chaotic mapping for population initialization, Cauchy distribution mutation for enhanced exploration, and adaptive weight factors for refined local exploitation, MBWO significantly improves tracking speed and accuracy. Simulations under static and dynamic shading conditions demonstrate that MBWO outperforms traditional BWO and Particle Swarm Optimization (PSO) in both convergence time and power extraction efficiency.
1. Introduction
The global shift toward renewable energy underscores the importance of solar power systems. A key challenge lies in optimizing energy harvest through MPPT, which ensures PV systems operate at their maximum power point (MPP) despite environmental fluctuations. Traditional MPPT methods, such as Perturb and Observe (P&O) or Incremental Conductance (IC), fail under partial shading conditions where power-voltage (P-V) curves exhibit multiple peaks. While metaheuristic algorithms like PSO offer global optimization capabilities, their parameter-heavy frameworks and slow convergence hinder practical applications.
The Beluga Whale Optimization (BWO) algorithm, inspired by beluga social behaviors, provides a simpler parameter structure and faster convergence. However, its vanilla form struggles with local optima and unstable outputs post-convergence. This work introduces MBWO, a multi-strategy enhanced variant tailored for MPPT control. Key innovations include:
- Logistic chaotic mapping for diversified population initialization.
- Nonlinear balance factors and cosine-optimized falling probability to refine exploration-exploitation transitions.
- Cauchy mutation and adaptive weighting to escape local optima and stabilize outputs.
2. Mathematical Model of Photovoltaic Systems
2.1 PV Cell Equivalent Circuit
The single-diode model (Fig. 1) describes a PV cell’s output characteristics. Kirchhoff’s laws yield the current-voltage relationship:I=Iph−Id[eq(V+IRs)nkT−1]−V+IRsRshI=Iph−Id[enkTq(V+IRs)−1]−RshV+IRs
where IphIph is photocurrent, IdId is diode saturation current, RsRs and RshRsh are series/shunt resistances, qq is electron charge, kk is Boltzmann’s constant, and TT is temperature.
2.2 Output Characteristics Under Partial Shading
Partial shading creates multi-peak P-V curves (Table 1), complicating MPPT. For example, a 4-series PV array under shading patterns (Combination 3: 1000, 800, 600, 400 lx) exhibits four distinct peaks (Fig. 3).
Table 1: Shading Patterns and Corresponding MPPs
| Pattern | S1 (lx) | S2 (lx) | S3 (lx) | S4 (lx) | MPP (W) |
|---|---|---|---|---|---|
| 1 | 1000 | 1000 | 1000 | 1000 | 852.6 |
| 2 | 1000 | 1000 | 800 | 400 | 551.8 |
| 3 | 1000 | 800 | 600 | 400 | 412.6 |
3. Beluga Whale Optimization (BWO) Algorithm
BWO mimics beluga swimming, foraging, and whale-fall behaviors. Its three phases are:
3.1 Exploration Phase
Belugas swim synchronously or mirror each other. Position updates follow:Xi,jt+1={Xi,pt+(Xr,pt−Xi,pt)⋅sin(2πr2),if j evenXi,pt+(Xr,pt−Xi,pt)⋅cos(2πr2),if j oddXi,jt+1={Xi,pt+(Xr,pt−Xi,pt)⋅sin(2πr2),Xi,pt+(Xr,pt−Xi,pt)⋅cos(2πr2),if j evenif j odd
where r1,r2∈(0,1)r1,r2∈(0,1) are random numbers.
3.2 Exploitation Phase
Levy flight enhances convergence:Xit+1=r3⋅Xbestt−r4⋅Xit+C1⋅L⋅(Xrt−Xit)Xit+1=r3⋅Xbestt−r4⋅Xit+C1⋅L⋅(Xrt−Xit)C1=r5⋅(1−t/Tmax),L=μ⋅0.05∣ν∣1/βC1=r5⋅(1−t/Tmax),L=∣ν∣1/βμ⋅0.05
where μ,ν∼N(0,1)μ,ν∼N(0,1), β=1.5β=1.5, and C1C1 controls Levy intensity.
3.3 Whale-Fall Phase
Weak individuals are replaced using:Xit+1=r6⋅Xit−r7⋅Xstept,Xstept=e−2t/Tmax⋅(Ub−Lb)Xit+1=r6⋅Xit−r7⋅Xstept,Xstept=e−2t/Tmax⋅(Ub−Lb)
4. Multi-Strategy Enhanced BWO (MBWO)
4.1 Chaotic Population Initialization
Logistic mapping replaces random initialization to avoid clustering:rt+1=4⋅rt⋅(1−rt),Xi,j=Lb+(Ub−Lb)⋅ri,jrt+1=4⋅rt⋅(1−rt),Xi,j=Lb+(Ub−Lb)⋅ri,j
4.2 Nonlinear Balance Factor
A decreasing exponential balance factor accelerates early-stage exploration:Bf=B0⋅2rt⋅exp(−π2⋅tTmax)Bf=B0⋅2rt⋅exp(−2π⋅Tmaxt)
4.3 Cauchy Mutation for Global Exploration
Cauchy-distributed perturbations enhance diversity:Xi′=Xbestt+Xbestt⋅Cauchy(0,1)Xi′=Xbestt+Xbestt⋅Cauchy(0,1)
4.4 Adaptive Weighting for Local Exploitation
An exponentially decaying weight prioritizes global search initially and local refinement later:ω=ωf+(ωi−ωf)⋅exp(−α⋅t/Tmax)ω=ωf+(ωi−ωf)⋅exp(−α⋅t/Tmax)
where ωi=0.9ωi=0.9, ωf=0.4ωf=0.4, and α=0.7α=0.7.
4.5 MPPT Control Framework
MBWO maps beluga positions to DC/DC converter duty cycles. Termination and restart mechanisms ensure adaptability:
- Termination: Halt if iterations exceed TmaxTmax or position changes <1%.
- Restart: Reactivate if ∣(Pt−Pm)/Pm∣>0.03∣(Pt−Pm)/Pm∣>0.03.
5. Simulation Results
5.1 Static Shading Conditions
Combination 1 (Uniform Illumination):
| Algorithm | Tracking Time (s) | Power (W) | Accuracy (%) |
|---|---|---|---|
| PSO | 0.26 | 849.0 | 99.58 |
| BWO | 0.181 | 851.1 | 99.83 |
| MBWO | 0.074 | 851.7 | 99.89 |
Combination 3 (Partial Shading):
| Algorithm | Tracking Time (s) | Power (W) | Accuracy (%) |
|---|---|---|---|
| PSO | 0.30 | 409.6 | 99.27 |
| BWO | 0.27 | 410.5 | 99.49 |
| MBWO | 0.108 | 411.7 | 99.78 |
5.2 Dynamic Shading Conditions
Switching from Combination 2 to 3 at t=1t=1 s:
| Algorithm | Tracking Time (s) | Power (W) | Accuracy (%) |
|---|---|---|---|
| BWO | 0.210 | 410.1 | 99.39 |
| MBWO | 0.103 | 411.3 | 99.68 |
6. Conclusion
The proposed MBWO algorithm significantly advances MPPT control under complex shading conditions. By integrating chaotic initialization, nonlinear balance factors, Cauchy mutation, and adaptive weights, MBWO achieves faster convergence (59% improvement over BWO) and higher accuracy (99.89% under uniform illumination). Future work will explore hardware-in-the-loop validation and scalability for larger PV arrays.