In practical applications, the complexity of environmental conditions often leads to multi-peak phenomena in the output characteristics of photovoltaic (PV) arrays. Traditional maximum power point tracking (MPPT) methods struggle to efficiently and consistently track the global maximum power point under such conditions, which significantly impacts the performance of solar power systems. To address this challenge, I propose an improved cat group algorithm (IMCGA) that integrates an optimized multi-peak support vector machine function with the fast convergence properties of the cat swarm optimization technique. This algorithm is designed to solve multi-peak optimization problems in partially shaded solar power systems, thereby enhancing the overall efficiency of MPPT. In this article, I will explore the mathematical modeling of PV cells, analyze their multi-peak output characteristics under partial shading, and detail the construction of adaptive weights and mutation operators within the IMCGA framework. Furthermore, I will present simulation results comparing the proposed method with traditional algorithms to demonstrate its superiority.
The foundation of MPPT technology in solar power systems lies in controlling the output voltage of PV cells to match the load voltage, ensuring that the system operates at its maximum power point. This maximizes power output and improves the utilization of solar energy. However, factors like haze, building shadows, and other obstructions can cause the power-voltage (P-V) curve of PV arrays to exhibit multiple peaks and nonlinear behavior. Consequently, developing control algorithms that can rapidly adjust load characteristics under complex shading conditions is crucial for maintaining optimal performance in solar power systems.
To understand the behavior of PV cells, I begin with their fundamental principles. The photovoltaic effect in solar cells can be modeled using an equivalent circuit, as shown in Figure 1. The current-voltage (I-V) relationship is derived from this circuit and is expressed by the following equation:
$$I = I_{ph} – I_D – I_{sh}$$
where \(I_{ph}\) is the photocurrent, \(I_D\) is the diode current, and \(I_{sh}\) is the shunt current. The diode current is given by:
$$I_D = I_0 \left[ \exp\left(\frac{q(U + IR_s)}{AKT}\right) – 1 \right]$$
and the shunt current is:
$$I_{sh} = \frac{U + IR_s}{R_{sh}}$$
Combining these, the I-V characteristic equation becomes:
$$I = I_{ph} – I_0 \left[ \exp\left(\frac{q(U + IR_s)}{AKT}\right) – 1 \right] – \frac{U + IR_s}{R_{sh}}$$
Here, \(I_0\) is the reverse saturation current, \(R_s\) and \(R_{sh}\) are the series and shunt resistances, respectively, \(U\) is the output voltage, \(I\) is the output current, \(q\) is the electron charge, \(K\) is Boltzmann’s constant, \(T\) is the absolute temperature, and \(A\) is the ideality factor. For simplicity, a practical model is often used:
$$I = I_{sc} \left[ 1 – C_1 \left( \exp\left(\frac{U}{C_2 U_{oc}}\right) – 1 \right) \right]$$
where \(I_{sc}\) is the short-circuit current, \(U_{oc}\) is the open-circuit voltage, and \(C_1\) and \(C_2\) are constants defined as:
$$C_1 = \left(1 – \frac{I_m}{I_{sc}}\right) \exp\left(-\frac{U_m}{C_2 U_{oc}}\right)$$
$$C_2 = \left(\frac{U_m}{U_{oc}} – 1\right) \left[ \ln\left(1 – \frac{I_m}{I_{sc}}\right) \right]^{-1}$$
In these equations, \(I_m\) and \(U_m\) represent the current and voltage at the maximum power point. The output characteristics of solar power systems are influenced by environmental factors such as temperature and irradiance. For instance, under constant irradiance, varying temperatures alter the P-V and I-V curves, as illustrated in Table 1, which summarizes key parameters under different conditions.
| Condition | Temperature (°C) | Irradiance (W/m²) | Max Power (W) | Voltage at MPP (V) |
|---|---|---|---|---|
| Standard | 25 | 1000 | 38.5 | 17.5 |
| Low Temp | 10 | 1000 | 40.2 | 18.0 |
| High Temp | 55 | 1000 | 35.8 | 16.5 |
| Low Irradiance | 25 | 600 | 22.5 | 16.0 |
Partial shading in solar power systems exacerbates these issues, leading to multi-peak P-V curves. For example, when 50% of a PV array is shaded with an irradiance of 500 W/m² while the rest receives 1000 W/m², the P-V curve displays multiple peaks. This complexity renders traditional MPPT methods, such as perturb and observe (P&O) or incremental conductance (INC), ineffective, as they often converge to local maxima. Thus, advanced algorithms are essential for reliable operation in distributed solar power systems.
The core of MPPT technology involves adjusting the load characteristics to match the internal resistance of the PV array for maximum power transfer. In a simplified equivalent circuit, the power delivered to the load \(P_L\) is given by:
$$P_L = I^2 R_L = \left( \frac{E}{R_i + R_L} \right)^2 R_L$$
where \(E\) is the equivalent voltage source, \(R_i\) is the internal resistance, and \(R_L\) is the load resistance. Differentiating with respect to \(R_L\) yields:
$$\frac{dP_L}{dR_L} = E \cdot \frac{R_i – R_L}{(R_i + R_L)^3}$$
Setting this to zero shows that \(P_L\) is maximized when \(R_L = R_i\). Power converters with appropriate control algorithms modulate the duty cycle to achieve this match, ensuring the solar power system operates at its peak efficiency.
To tackle multi-peak MPPT, I employ swarm intelligence algorithms, particularly the cat swarm optimization (CSO) method. The basic CSO algorithm mimics the behavior of cats in searching and tracking modes. However, it tends to converge prematurely to local optima. My improved version, IMCGA, incorporates adaptive weights and mutation operators to enhance global search capabilities. The velocity update in CSO is defined as:
$$v_{k,d}(t+1) = v_{k,d}(t) + c \cdot r \cdot (x_{best,d}(t) – x_{k,d}(t))$$
where \(v_{k,d}\) is the velocity of cat \(k\) in dimension \(d\), \(x_{k,d}\) is the position, \(c\) is a constant, \(r\) is a random number, and \(x_{best,d}\) is the best position found. In IMCGA, I introduce an adaptive weight \(w\) that adjusts based on iteration progress and population diversity. The weight update formula is:
$$w = w_{min} + (w_{max} – w_{min}) \cdot \lambda \cdot \gamma$$
Here, \(w_{max} = 0.9\) and \(w_{min} = 0.3\) are the maximum and minimum weights, \(\lambda\) is an iteration factor, and \(\gamma\) is a convergence factor. The iteration factor \(\lambda\) is computed as:
$$\lambda = \sqrt{\exp(-20 \cdot t / Q)^s}$$
where \(t\) is the current iteration, \(Q\) is the maximum iterations, and \(s > 1\) is a constant. The convergence factor \(\gamma\) is derived from the fitness variance:
$$f_{avg} = \frac{\sum_{k=1}^N f(P_k)}{N}$$
$$E = \sum_{k=1}^N (f(P_k) – f_{avg})^2$$
$$\gamma = \sin(\arctan(E))$$
These adjustments ensure that the algorithm maintains high exploration in early stages and fine-tunes in later stages. Additionally, I incorporate a mutation operator to prevent stagnation. The position update with mutation is:
$$x_{k,d}(t+1) =
\begin{cases}
x_{k,d}(t+1) + (-1)^n A & \text{if } t = nT, n=1,2,\ldots \\
x_{k,d}(t+1) & \text{otherwise}
\end{cases}$$
where \(A = 0.5\) is the mutation amplitude and \(T = 50\) is the mutation period. The velocity update in IMCGA becomes:
$$v_{k,d}(t+1) = w \cdot v_{k,d}(t) + c \cdot r \cdot (x_{best,d}(t) – x_{k,d}(t))$$
and the position update is:
$$x_{k,d}(t+1) = x_{k,d}(t) + v_{k,d}(t+1)$$
For MPPT application in solar power systems, I define the output current \(I\) as the cat’s position \(x_{k,d}\) and the output power \(P\) as the fitness function. The fitness function for a partially shaded system is:
$$fit =
\begin{cases}
U(S_1, I) \cdot I & \text{if } I_2 < I \leq I_1 \\
[U(S_1, I) + U(S_2, I)] \cdot I & \text{if } I \leq I_2
\end{cases}$$
where \(U(S_1, I)\) and \(U(S_2, I)\) are the voltages under different irradiances, and \(I_1\) and \(I_2\) are current thresholds. The implementation process of IMCGA for MPPT involves initializing parameters, grouping cats based on a mixing ratio, executing search and tracking modes, and iterating until convergence. Key parameters include a population size \(N = 20\), maximum iterations \(Q = 500\), mixing ratio \(MR = 0.2\), and others as listed in Table 2.
| Parameter | Symbol | Value |
|---|---|---|
| Population Size | N | 20 |
| Max Iterations | Q | 500 |
| Mixing Ratio | MR | 0.2 |
| Inertia Weight Max | w_max | 0.9 |
| Inertia Weight Min | w_min | 0.3 |
| Mutation Amplitude | A | 0.5 |
| Mutation Period | T | 50 |
To validate the IMCGA approach, I conducted simulations in MATLAB/Simulink, comparing it with P&O, INC, and fuzzy logic methods under standard and partial shading conditions. The solar power system model consisted of PV arrays with varying irradiance levels. Under standard conditions (irradiance 1000 W/m², temperature 25°C), all methods tracked the MPP, but IMCGA demonstrated faster convergence and higher accuracy. For instance, the steady-state power output with IMCGA reached 39.0 W, compared to 38.5 W for P&O, as shown in dynamic response curves.
Under partial shading, the multi-peak P-V curve posed challenges for traditional methods. IMCGA efficiently identified the global maximum, achieving a power output of 58.0 W, while P&O and INC converged to local peaks around 55.5 W. The dynamic boost power curves further highlighted IMCGA’s superiority, with reduced oscillations and quicker stabilization. These results underscore the algorithm’s robustness in complex solar power systems.
In conclusion, the IMCGA algorithm effectively addresses multi-peak MPPT challenges in distributed solar power systems. By integrating adaptive weights and mutation operators, it enhances global search capabilities and avoids local optima. Simulation results confirm its advantages over traditional methods in terms of convergence speed and tracking accuracy. Future work could focus on real-time implementation and scalability for larger solar power systems. This research contributes to improving the efficiency and reliability of photovoltaic energy generation, supporting the broader adoption of renewable energy sources.
The mathematical models and algorithms discussed here provide a foundation for optimizing solar power systems under varying environmental conditions. As the demand for clean energy grows, advanced MPPT techniques like IMCGA will play a pivotal role in maximizing the output of photovoltaic installations. I encourage further exploration of hybrid approaches that combine swarm intelligence with other machine learning methods to achieve even greater performance in solar power system applications.