In the context of the global energy revolution and domestic carbon neutrality goals, actively exploring optimal solutions for low-carbon and clean energy generation has become an inevitable trend. Among these, solar photovoltaic power generation stands out due to its abundant energy reserves and clean, pollution-free advantages. Maximum Power Point Tracking (MPPT) is a crucial method to enhance the photoelectric conversion efficiency of solar power systems. To reduce power loss and improve energy utilization, researching MPPT technology holds significant practical importance. This paper focuses on the MPPT control of solar power systems, using the Whale Optimization Algorithm (WOA) as the foundational design, and proposes a multi-strategy fused Improved Whale Optimization Algorithm (IWOA) to address issues such as low control accuracy in conventional MPPT methods, difficulty in tracking the global MPP under partial shading conditions, poor dynamic tracking performance of typical intelligent optimization algorithms, and susceptibility to local MPP entrapment.
The solar power system consists of photovoltaic (PV) arrays, power conversion circuits, control systems, and loads. The PV array, composed of multiple solar cells, converts solar energy into electrical energy. The output characteristics of the PV array are nonlinear and influenced by external environmental factors such as irradiance and temperature. Under uniform illumination, the output power-voltage (P-U) curve exhibits a single peak, while under partial shading, it becomes multi-peaked due to mismatched outputs from shaded and unshaded cells. This multi-peak characteristic poses challenges for MPPT control, as conventional methods may fail to locate the global maximum power point (MPP).
The mathematical model of a PV cell is based on the single-diode equivalent circuit. The output current I is given by:
$$I = I_{pv} – I_0 \left[ \exp\left(\frac{q(U + IR_s)}{AkT}\right) – 1 \right] – \frac{U + IR_s}{R_p}$$
where \(I_{pv}\) is the photocurrent, \(I_0\) is the reverse saturation current, \(q\) is the electron charge, \(A\) is the diode ideality factor, \(k\) is Boltzmann’s constant, \(T\) is the temperature, \(R_s\) is the series resistance, and \(R_p\) is the parallel resistance. For practical engineering applications, a simplified model under standard test conditions (STC: 1000 W/m², 25°C) is often used, with parameters adjusted for actual conditions using compensation coefficients. The output power \(P\) of the PV array is calculated as \(P = U \times I\), and its variation with voltage under different irradiance and temperature conditions forms the basis for MPPT.
MPPT technology aims to match the load impedance with the internal impedance of the PV array dynamically, ensuring maximum power transfer. This is achieved by controlling the duty cycle of a DC-DC converter, such as a Boost circuit, which adjusts the equivalent load resistance. The relationship between the input resistance \(R_i\) and output resistance \(R_o\) of the Boost converter is:
$$R_i = \frac{R_o}{(1 – d)^2}$$
where \(d\) is the duty cycle. By regulating \(d\), the operating point on the P-U curve is shifted to the MPP. Conventional MPPT methods include the Constant Voltage Method (CVM), Perturb and Observe (P&O), and Incremental Conductance (INC). However, these methods suffer from steady-state oscillations, low accuracy, and susceptibility to local optima under partial shading. For instance, P&O involves perturbing the voltage and observing power changes, but it oscillates around the MPP and may misjudge under rapidly changing conditions. INC uses the derivative of power with respect to voltage (\(\frac{dP}{dU} = 0\)) at the MPP, but it requires high-precision sensors and may still oscillate.
To overcome these limitations, intelligent optimization algorithms have been introduced for MPPT control. These algorithms, such as Particle Swarm Optimization (PSO) and Genetic Algorithm (GA), excel at solving multi-peak optimization problems. The Whale Optimization Algorithm (WOA), inspired by the bubble-net feeding behavior of humpback whales, is a metaheuristic algorithm with simple principles, ease of implementation, and fast convergence. However, basic WOA tends to fall into local optima and has low convergence precision. Thus, an Improved Whale Optimization Algorithm (IWOA) is proposed, incorporating multiple strategies to enhance performance.
The basic WOA consists of three phases: encircling prey, bubble-net attacking, and random search. The position update equations are as follows. For encircling prey:
$$D = |C \cdot X^*(t) – X(t)|$$
$$X(t+1) = X^*(t) – A \cdot D$$
where \(X^*\) is the best candidate solution, \(X\) is the current position, \(A\) and \(C\) are coefficient vectors calculated as \(A = 2a \cdot r_1 – a\) and \(C = 2 \cdot r_2\), with \(a\) decreasing linearly from 2 to 0 over iterations, and \(r_1\), \(r_2\) are random vectors in [0,1]. For bubble-net attacking (exploitation):
$$X(t+1) = D’ \cdot e^{bl} \cdot \cos(2\pi l) + X^*(t)$$
where \(D’ = |X^*(t) – X(t)|\), \(b\) is a constant, and \(l\) is a random number in [-1,1]. The algorithm switches between encircling and attacking based on a probability \(p\). For random search (exploration) when \(|A| > 1\):
$$D = |C \cdot X_{rand} – X|$$
$$X(t+1) = X_{rand} – A \cdot D$$
where \(X_{rand}\) is a random whale position. Despite its advantages, WOA often converges prematurely to local optima. The IWOA addresses this through several improvements:
First, a nonlinear convergence factor \(a\) is introduced to balance global exploration and local exploitation. The modified \(a\) is given by:
$$a = 2 – \frac{2}{1 + e^{-t/T_{max}}}$$
where \(t\) is the current iteration and \(T_{max}\) is the maximum iterations. This nonlinear decrease allows rapid global search initially and finer local search later.
Second, an adaptive weight \(\omega(t)\) and Lévy flight strategy are incorporated to enhance local search randomness and avoid local optima. The weight update is:
$$\omega(t) = \omega_i + (\omega_f – \omega_i) \exp(-\alpha t / T_{max})$$
where \(\omega_i\) and \(\omega_f\) are initial and final weights, and \(\alpha\) is a constant. The position update in bubble-net attacking becomes:
$$X(t+1) = \omega(t) \cdot D’ \cdot e^{bl} \cdot \cos(2\pi l) + X^*(t)$$
Additionally, Lévy flight is applied to the step size \(D’\):
$$D’ = \text{Lévy}(\lambda) \otimes (X^*(t) – X(t))$$
where \(\otimes\) denotes element-wise multiplication, and Lévy(\(\lambda\)) is a random path following Lévy distribution, which occasionally generates large steps to escape local optima.
Third, Cauchy mutation and opposition-based learning are introduced with a probability \(P_s\) to perturb the population and prevent premature convergence. The opposition-based solution \(X^*_{best}(t)\) is computed as:
$$X^*_{best}(t) = ub + r \cdot (lb – X_{best}(t))$$
where \(ub\) and \(lb\) are upper and lower bounds, and \(r\) is a random vector. The new position is:
$$X_{new}(t+1) = b \cdot (X^*_{best}(t) – X_{best}(t))$$
where \(b\) is a communication coefficient decreasing over time. Cauchy mutation is applied as:
$$X_{new}(t+1) = \text{Cauchy} \otimes X_{best}(t)$$
The probability \(P_s\) for alternating these strategies is:
$$P_s = \exp\left(-\theta \left(1 – \frac{t}{T_{max}}\right)^2\right)$$
with \(\theta = 0.05\). These strategies collectively improve IWOA’s ability to navigate complex search spaces.
To validate IWOA’s performance, benchmark functions are used for testing. The table below summarizes the test functions, including unimodal (F1-F5) and multimodal (F6-F8) types:
| Function | Name | Range | Optimum |
|---|---|---|---|
| F1 | Sphere | [-100,100] | 0 |
| F2 | Schwefel2.22 | [-10,10] | 0 |
| F3 | Schwefel1.2 | [-100,100] | 0 |
| F4 | Schwefel2.21 | [-100,100] | 0 |
| F5 | Quartic | [-1.28,1.28] | 0 |
| F6 | Rastrigin | [-5.12,5.12] | 0 |
| F7 | Ackley | [-32,32] | 0 |
| F8 | Griewank | [-600,600] | 0 |
IWOA is compared with WOA, PSO, Grey Wolf Optimizer (GWO), and Firefly Algorithm (FA) over 30 independent runs. The parameters are set as: population size 50, maximum iterations 500. The results, including best value, mean, and standard deviation, demonstrate IWOA’s superiority. For example, on F1, IWOA achieves 0.00E+00 best value, while WOA reaches 6.28E-87. The convergence curves show IWOA’s faster and more precise convergence without stagnation.
For MPPT application, IWOA is integrated with a solar power system model in MATLAB/Simulink. The system includes a PV array, Boost converter, and resistive load. The duty cycle \(d_n(t)\) of the Boost converter is the optimization variable, and the objective function is maximizing output power:
$$f(d_n(t)) = \max P(d_n(t)) = \max V_{PV}(d_n(t)) \times I_{PV}(d_n(t))$$
with \(d_n(t) \in [0,1]\). The convergence condition is set as \(|d_n(t) – d_n(t-1)| / d_n(t) < 1\%\), and a restart mechanism triggers if power change exceeds 5% to handle sudden environmental changes.
Simulations are conducted under four conditions: uniform illumination, static partial shading, partial shading mutation, and continuously changing irradiance. Under uniform illumination (1000 W/m²), IWOA achieves 99.95% tracking accuracy with minimal oscillation, outperforming P&O (low accuracy due to steady-state oscillation), PSO (significant oscillations), and WOA (slower convergence). In static partial shading, such as when three series-connected PV modules receive irradiances of [1000, 800, 600] W/m², the P-U curve has multiple peaks. P&O fails to track the global MPP, while IWOA, WOA, and PSO succeed, but IWOA converges fastest with the least oscillation. The table below summarizes results for partial shading condition 2:
| Algorithm | Theoretical MPP (W) | Actual Power (W) | Convergence Time (s) |
|---|---|---|---|
| IWOA | 425.5 | 425.4 | 0.0595 |
| WOA | 425.5 | 425.4 | 0.0750 |
| PSO | 425.5 | 425.4 | 0.1229 |
| P&O | 425.5 | 201.2 | 0.2241 |
Under partial shading mutation, where irradiance changes abruptly from uniform to partial shading, IWOA quickly re-tracks the MPP with minimal fluctuation, whereas WOA may temporarily trap in local optima. For continuously changing irradiance (e.g., from 800 W/m² to 1000 W/m² linearly), IWOA maintains precise tracking without oscillations, while WOA shows slight deviations. These results validate IWOA’s robustness and dynamic performance in solar power systems.
In conclusion, the proposed IWOA effectively enhances MPPT control in solar power systems by addressing local optima entrapment and improving convergence speed and accuracy. Future work will focus on applying IWOA to larger-scale PV arrays and hardware implementation for real-world validation. The integration of advanced optimization algorithms like IWOA is pivotal for maximizing the efficiency of solar power systems, contributing to sustainable energy solutions.