A Particle-Swarm-Confinement-based Zonotopic Space Filtering Algorithm and Its Application to State of Charge Estimation for Lithium-Ion Batteries
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摘要: 荷电状态是衡量锂离子电池剩余电量的关键指标，其准确估计对电池管理系统至关重要。该文提出了一种粒子群约束下的多胞空间滤波算法，用于解决系统存在未知但有界噪声时的状态估计问题。该算法能够准确检测并重新映射异常粒子，从而确保搜索过程的稳定性。通过采用法向量缩放的方法调整超平面位置，将粒子群限制在多胞搜索空间区域内，以优化状态估计的效率。该粒子群优化算法具备良好的适应性，能够有效减少估计冗余并增强鲁棒性，尤其适用于高维系统。将该算法应用于锂离子电池荷电状态分析的实验结果表明，该算法能够对锂离子电池荷电状态变化情况进行有效估计。Abstract:
Objective The State of Charge (SOC) is a critical indicator for evaluating the remaining capacity and health status of lithium-ion batteries, which are widely deployed in electric vehicles, portable electronics, and energy storage systems. Accurate SOC estimation is essential for maintaining safe operation, extending battery life, and optimizing energy utilization. However, practical SOC estimation is complicated by measurement uncertainties and disturbances, particularly Unknown But Bounded (UBB) noise arising from sensor errors, environmental fluctuations, and battery aging. Conventional filtering algorithms, such as Kalman filters, often depend on probabilistic noise assumptions and tend to perform poorly when actual noise characteristics deviate from Gaussian distributions. This study addresses these limitations by proposing a Particle-Swarm-Confinement-based Zonotopic Space Filtering (PSC-ZSF) algorithm to enhance estimation robustness and reduce conservatism, with specific emphasis on high-dimensional dynamic systems such as lithium-ion battery SOC estimation. Methods The PSC-ZSF algorithm combines the robustness of set-membership filtering with the global optimization capabilities of Particle Swarm Optimization (PSO), integrating geometric uncertainty representation with heuristic search strategies. A zonotopic feasible state set is first constructed by propagating system model predictions and refining them with measurement updates, thereby representing the bounded uncertainty in system states. A swarm of particles is then randomly initialized within this zonotopic space to explore potential state estimates. Particle movement follows PSO-based velocity and position updates, leveraging both individual experience and swarm intelligence to identify optimal state estimates. Fitness functions quantify the consistency between candidate states and observed measurements, guiding particle convergence toward more plausible regions. To maintain algorithm stability, a boundary detection mechanism identifies particles that exceed the zonotopic feasible region. Out-of-bound particles are projected back into the feasible set by solving a quadratic programming problem that minimizes positional distortion while preserving spatial characteristics. Additionally, a dynamic contraction strategy adaptively tightens the zonotopic boundaries by scaling the normal vectors of the defining hyperplanes, effectively shrinking the search space as the particle swarm converges. This contraction improves estimation precision and reduces conservatism without incurring excessive computational overhead. The approach utilizes Minkowski sum properties intrinsic to zonotopes and utilizes efficient geometric computations to balance accuracy and efficiency. For experimental validation, the PSC-ZSF algorithm is applied to SOC estimation of lithium-ion batteries modeled by a discrete-time equivalent circuit that incorporates polarization resistance and capacitance effects. Real-world data are collected from a 18650 lithium-ion battery undergoing constant current discharge at room temperature. The system model considers UBB process and measurement noise, with parameters calibrated through empirical measurements. The performance of the proposed method is benchmarked against Ellipsoidal Set-Membership Filtering (ESMF) and Zonotopic Set-Membership Filtering (ZSMF) methods by comparing feasible state set volumes and the tightness of estimated boundaries.Results and Discussions The proposed PSC-ZSF algorithm demonstrates reliable confinement of particle swarms within the zonotopic feasible region throughout iterative optimization, effectively preventing particle divergence and improving estimation stability and reliability ( Fig. 1 ). Comparative analysis indicates that PSC-ZSF consistently achieves significantly smaller feasible state set volumes at each time step compared to ESMF and ZSMF methods, reflecting reduced estimation redundancy and improved compactness (Fig. 3 ). The ESMF method guarantees that the true state remains enclosed; however, it produces overly conservative ellipsoidal bounds, especially under conditions of rapid system dynamics, which compromises estimation informativeness and responsiveness. The ZSMF method improves upon this by employing zonotopic bounds but still yields relatively broad estimation regions due to fixed zonotope geometries and cautious boundary updates. In contrast, PSC-ZSF adaptively refines the zonotopic boundaries based on real-time particle swarm distributions, leading to consistently tighter, more accurate boundaries that closely track the true SOC and polarization voltage trajectories (Figs. 4 and5 ). This adaptive boundary contraction strategy enhances estimation precision while preserving robustness. Moreover, computational complexity analysis shows that although particle projection and boundary scaling introduce additional per-iteration operations, the accelerated convergence of PSC-ZSF reduces overall iteration requirements. This trade-off ensures computational feasibility for real-time SOC estimation in battery management systems.Conclusions This study proposes a Particle-Swarm-Confinement-Based Zonotopic Space Filtering (PSC-ZSF) algorithm that integrates set-membership filtering with PSO to address state estimation under unknown but bounded noise. The PSC-ZSF algorithm ensures that particle swarms remain confined within a zonotopic feasible region through optimal projection and dynamically contracts the zonotope boundaries via hyperplane scaling. This approach improves estimation accuracy and reduces conservatism. Application to lithium-ion battery SOC estimation confirms the approach’s superiority over conventional approaches, providing more precise and stable state boundaries while maintaining computational efficiency suitable for real-time applications. Future work will focus on extending the PSC-ZSF algorithm to complex dynamic systems such as autonomous vehicle navigation and smart grid state estimation to further assess scalability and practical applicability. -
表 1 开路电压与荷电状态的对应关系
SOC
(%)0 10 20 30 40 50 60 70 80 90 100 UOC
(V)2.782 3.629 3.711 3.758 3.792 3.826 3.873 3.938 4.012 4.088 4.193 
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表 2 等效电路的各参数值
R0 (Ω) Rp (Ω) I (A) Cp (F) $ \Delta t $ (s) $ {{{Q}}_{{\text{re}}}} $ (Ah) $ \eta $ 0.0421 0.0311 1 2369.8 5 1200 0.9
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