In recent years, the rapid growth of the electric vehicle industry, particularly in China EV markets, has highlighted the critical need for enhancing the safety and efficiency of battery systems. The battery shell, as a primary protective component, plays a vital role in safeguarding the battery cells from mechanical impacts, thereby preventing potential hazards such as short circuits, fires, or explosions. However, the substantial weight of the battery shell, which can account for up to half of the total battery module mass, poses challenges for achieving overall vehicle lightweighting without compromising safety. To address this, I propose a novel optimization method that integrates an improved Firefly Algorithm (FA) with a Backpropagation Neural Network (BPNN) for the structural optimization of electric vehicle battery shells. This approach aims to achieve reliable 3D topology optimization while minimizing mass and maximizing performance, contributing to the advancement of China EV technologies. The method leverages the nonlinear mapping capabilities of BPNN and enhances FA to avoid common pitfalls like local optima and overfitting, ensuring robust parameter fitting and structural design.
The core of this research lies in the development of an improved FA for parameter fitting of the electric vehicle battery shell. Traditional FA simulates the flashing behavior of fireflies, where brightness represents the fitness value of a solution, and positions denote potential solutions. Fireflies move toward brighter individuals based on relative distance, as defined by the Euclidean distance formula: $$D_{ij} = \|X_i – X_j\| = \sqrt{\sum_{k=1}^{n} (X_{i,k} – X_{j,k})^2}$$ where \(D_{ij}\) is the distance between individuals \(X_i\) and \(X_j\), and \(n\) is the population size. However, standard FA suffers from real-time position updates and rapid positioning issues, which can lead to inefficient searches. To mitigate this, I introduce a random vector and apply stochastic and micro-element processing to the proportional light intensity attraction. The updated position formula for the improved FA is: $$X_i^{t+1} = X_i^t + \beta_{i,j}^t (X_j^t – X_i^t) + \lambda \cdot \delta$$ where \(X_i^{t+1}\) is the position of individual \(X_i\) at time \(t+1\), \(\beta_{i,j}^t\) is the proportional light intensity attraction between \(X_i\) and \(X_j\) at time \(t\), \(\lambda\) is the step factor, and \(\delta\) is a random vector. The micro-element and stochastic processing of the attraction is given by: $$dx_\beta = \beta_{i,j}^t (X_j^t – X_i^t)$$ $$\lambda(t) = \lambda_0 \cdot e^{-c \frac{t}{T}}$$ $$dx_\lambda = \lambda \cdot \delta$$ Here, \(\lambda(t)\) is the step factor at iteration \(t\), \(\lambda_0\) is the initial step factor, \(c\) is a reduction constant, and \(T\) is the maximum number of iterations. This improvement enhances the algorithm’s ability to explore the solution space efficiently, which is crucial for optimizing the complex parameters of electric vehicle battery shells.
To achieve precise fitting of the shell structure parameters, I employ a response surface optimization method, which refines the fitting function by analyzing the sensitivity of independent variables. Given the multitude of variables in electric vehicle battery shells, a Monte Carlo approach combined with response surface methodology is used for sensitivity analysis. The relationship between the shell thickness variables and the output mass target is expressed as: $$\hat{M} = c_0 + \sum_{i=1}^{n} c_i V_i + \sum_{i=1}^{n} \sum_{j=1}^{n} c_{i,j} V_i V_j$$ where \(\hat{M}\) is the predicted mass, \(c_0\), \(c_i\), and \(c_{i,j}\) are constants and coefficients, and \(V_i\) and \(V_j\) represent different shell variables. Since the battery shell can be treated as a black-box system in the context of China EV applications, the mathematical model decomposes as: $$y = f_0 + \sum_{i=1}^{12} f_i(V_i) + \sum_{i<j} $$\int_0^1="" +="" \,="" \ldots,="" a="" analysis,="" assessment="" battery="" condition:="" construction="" decomposition="" designs.
While the improved FA effectively optimizes parameters, it falls short in performing 3D topology optimization for the battery shell. To overcome this, I integrate a BPNN, which excels in nonlinear mapping, self-learning, and generalization capabilities. However, standard BPNN training methods, such as gradient descent or Gauss-Newton algorithms, are prone to gradient vanishing or explosion and overfitting due to the full-rank Jacobian matrix. Therefore, I adopt a Levenberg-Marquardt gradient descent combined with Gauss-Newton algorithm for training. The weight error correction formula is: $$w_n = w_{n-1} – (J^T J + \alpha S)^{-1} J e_{n-1}$$ where \(w_n\) is the weight偏差 at iteration \(n\), \(J\) is the Jacobian matrix, \(\alpha\) is a correlation coefficient, \(S\) is the identity matrix, and \(e\) is the error. For 3D topology optimization, the goal is to minimize both mass and flexibility, formulated as an interpolation model: $$\text{minimize} \quad f(d); \quad \text{minimize} \quad M$$ $$\text{subject to} \quad y(d) = v_k – v_0 \leq 0$$ $$\text{find} \quad D_k = [d_1, d_2, \ldots, d_k]^T$$ $$d_1 \leq d_i \leq d_n, \quad n=1,2,3,\ldots,k$$ where \(y(d)\) is the constraint function, \(v_k\) is the optimized shell volume, \(v_0\) is the initial volume, \(D_k\) is the design variable vector, \(f(d)\) is the flexibility, and \(M\) is the mass. The flexibility is computed as: $$f(d) = D^T K D = \sum_{E=1}^{K} (d_i^E)^p d_E^K d_E$$ where \(K\) is the stiffness matrix, and \(E\) represents its elements. Despite BPNN’s strengths, it is susceptible to local optima and slow training. Thus, I combine it with the improved FA to adaptively adjust weights and thresholds, leading to the improved FA-BPNN approach.
The workflow of the improved FA-BPNN method begins with data initialization, where BPNN weights, thresholds, and FA population parameters are set. The process involves minimizing the mean squared error of the regression loss function, with stop conditions and maximum iterations defined. The algorithm outputs the current best individual position and light intensity, updates positions, and iteratively searches for optimal solutions. If a better individual is found, coordinates and intensity are updated; otherwise, the search continues. The fitness function for the improved FA-BPNN is: $$\text{fitness} = \frac{1}{h} \sum_{i=1}^{h} (x_h – \hat{x}_h)^2$$ where \(h\) is the number of individuals, \(x_h\) is the actual value, and \(\hat{x}_h\) is the predicted value. The objective function is: $$R = \frac{N \cdot \sum_{i=1}^{37} O_{s,i} \cdot O_{t,i} – \left( \sum_{i=1}^{37} O_{s,i} \cdot O_{t,i} \right)^2}{\sqrt{N \cdot \sum_{i=1}^{37} O_{s,i}^2 – \left( \sum_{i=1}^{37} O_{s,i} \right)^2} \cdot \sqrt{N \cdot \sum_{i=1}^{37} O_{t,i}^2 – \left( \sum_{i=1}^{37} O_{t,i} \right)^2}}$$ where \(R\) is the objective function, \(N\) is the sample size, \(O_{s,i}\) is the predicted output, and \(O_{t,i}\) is the actual output of the test set. This integrated approach ensures efficient optimization for electric vehicle battery shells, enhancing the safety and performance of China EV models.
To evaluate the performance of the improved FA-BPNN, I compare it with other optimization algorithms, such as Sparrow Search Algorithm-BPNN (SSA-BPNN) and Whale Optimization Algorithm-BPNN (WOA-BPNN). The experimental setup includes an input layer with 7 nodes and an output layer with 2 nodes, initial weights of 0.01, allowable error of 0.01, and a maximum of 1000 iterations. The residual sum of squares (RSS) and mean absolute percentage error (MAPE) are used as metrics to assess predictive accuracy. The results, summarized in the table below, demonstrate the superiority of the improved FA-BPNN in optimizing electric vehicle battery shell structures.
| Algorithm | Minimum RSS | Average RSS | Minimum MAPE (%) | Average MAPE (%) |
|---|---|---|---|---|
| SSA-BPNN | 0.08 | 0.11 | 0.61 | 0.63 |
| WOA-BPNN | 0.04 | 0.06 | 0.59 | 0.61 |
| Improved FA-BPNN | 0.03 | 0.03 | 0.45 | 0.45 |
As shown in the table, the improved FA-BPNN achieves the lowest RSS and MAPE values, with an average RSS of 0.03 and an average MAPE of 0.45%, significantly outperforming SSA-BPNN and WOA-BPNN. This indicates that the proposed method offers superior prediction capabilities, which is essential for reliable optimization in electric vehicle applications. Furthermore, to validate the practical effectiveness of the optimized battery shell structure, I conduct simulation tests on a selected optimization scheme. The results, presented in the following table, compare predicted and simulated values for key parameters, such as frequency and mass, with relative errors calculated to assess accuracy.
| Parameter | Predicted Value | Simulated Value | Relative Error (%) |
|---|---|---|---|
| Frequency (Hz) | 103.3 | 102.8 | 0.48 |
| Mass (kg) | 190.1 | 193.5 | 1.76 |
| Upper Shell Thickness (mm) | 4.5 | 4.5 | 0.00 |
| Lower Shell Base Thickness (mm) | 1.0 | 1.0 | 0.00 |
| Rear Plate Thickness (mm) | 1.5 | 1.5 | 0.00 |
| Lug 1 Thickness (mm) | 7.5 | 7.5 | 0.00 |
| Lug 2 Thickness (mm) | 8.0 | 8.0 | 0.00 |
| Lug 3 Thickness (mm) | 7.0 | 7.0 | 0.00 |
| Lug 4 Thickness (mm) | 8.0 | 8.0 | 0.00 |
The simulation results reveal that the relative errors for frequency and mass are 0.48% and 1.76%, respectively, both within the acceptable range of less than ±3%. This confirms that the improved FA-BPNN method provides reliable optimization solutions for electric vehicle battery shells, enhancing structural integrity and contributing to the safety goals of China EV development. The minimal errors in thickness parameters further validate the method’s precision in handling complex design variables. Additionally, the optimization process effectively reduces mass while maintaining performance, aligning with the lightweighting objectives of modern electric vehicle manufacturing.
In conclusion, the integration of an improved Firefly Algorithm with a Backpropagation Neural Network presents a robust approach for optimizing electric vehicle battery shell structures. This method addresses limitations of traditional algorithms by incorporating stochastic elements and adaptive parameter tuning, resulting in enhanced prediction accuracy and structural reliability. The experimental outcomes demonstrate significant improvements in RSS and MAPE compared to alternative methods, and simulation tests confirm the practical viability of the optimized designs with errors within acceptable limits. This research contributes to the advancement of China EV technologies by providing a scalable solution for battery safety and efficiency. Future work will focus on refining the prediction accuracy through the incorporation of radial basis function neural networks and exploring real-time applications in electric vehicle production lines. As the demand for safer and more efficient electric vehicles grows, such optimization methods will play a pivotal role in shaping the future of sustainable transportation.