Battery packs in electric vehicles and grid storage systems consist of hundreds or thousands of interconnected cells. Ensuring reliability requires rapid fault detection and localization to prevent cascading failures. Graph theory provides a mathematical framework for modeling cell interconnectivity, identifying anomalies, and isolating faulty modules without disrupting the entire system. This article explores the application of adjacency matrices, graph cuts, and Laplacian eigenmaps for fault localization in battery packs.
Adjacency matrices serve as the foundation for modeling battery pack topology. In a series-parallel configuration, each cell or module is represented as a node, while electrical connections between them form edges. The adjacency matrix A is a square matrix where A_ij = 1 if nodes i and j are connected, and A_ij = 0 otherwise. For a battery pack with n cells, A has dimensions n×n. Weighted adjacency matrices can incorporate resistance or impedance values for more precise modeling. The degree matrix D, a diagonal matrix where D_ii equals the sum of weights connected to node i, helps derive the graph Laplacian L = D - A.
Fault detection begins with real-time voltage, current, and temperature measurements. Deviations from expected behavior are mapped onto the graph as node or edge anomalies. A sudden voltage drop in a cell appears as an outlier in the corresponding node's data. Graph-based algorithms then localize the fault by analyzing neighborhood relationships. For instance, if cell k exhibits abnormal behavior, its directly connected neighbors in the adjacency matrix will show correlated deviations due to electrical coupling. This neighborhood analysis distinguishes localized faults from system-wide issues.
Graph cuts partition the battery pack into healthy and faulty regions. The normalized cut criterion minimizes edge weights between partitions while maximizing weights within partitions. For a fault localized to a subset of cells S, the cut value is the sum of edge weights between S and the remaining graph. Spectral clustering leverages the Laplacian matrix's eigenvectors to approximate optimal cuts. The Fiedler vector, the second smallest eigenvector of L, provides a bipartitioning solution. Cells are clustered based on sign changes in this eigenvector, isolating the faulty module with minimal impact on overall pack functionality.
Laplacian eigenmaps reduce dimensionality while preserving topological relationships. High-dimensional sensor data (voltage, temperature, impedance) is projected onto a lower-dimensional space using the Laplacian's eigenvectors. The first few non-trivial eigenvectors capture the most significant variance in the data, enabling visualization of fault propagation paths. For a battery pack with 100 cells, instead of monitoring 100-dimensional data, the system can track deviations in a 3D space spanned by the top three eigenvectors. This reduction simplifies real-time analysis without losing critical fault signatures.
An example implementation involves a 96-cell battery pack arranged in 12 series-connected modules of 8 parallel cells each. The adjacency matrix has a block structure reflecting this topology. When a single cell fails, voltage imbalances propagate to adjacent parallel cells but remain confined within the module due to series resistance. Graph cuts applied to the Laplacian matrix correctly isolate the affected module with 92% accuracy in experimental validations. Eigenmap projections show clear separation between the faulty module and healthy cells in the reduced space.
Performance metrics for graph-based methods include localization accuracy, computational latency, and false positive rates. Studies on 1 kWh lithium-ion packs demonstrate 85-95% fault isolation accuracy within 10 milliseconds using optimized Laplacian solvers. The method outperforms threshold-based detection by incorporating structural information through the adjacency matrix. False positives remain below 2% when accounting for sensor noise in the graph weighting scheme.
Challenges persist in dynamic reconfiguration scenarios where pack topology changes during operation. Adaptive adjacency matrices that update connection weights based on switching events maintain accuracy in such cases. Future work may integrate graph neural networks for learning fault patterns across diverse battery chemistries and configurations. The mathematical rigor of graph theory ensures these methods remain scalable for next-generation megawatt-scale storage systems.
In summary, graph-based fault localization leverages the inherent interconnectivity of battery packs to provide fast, accurate diagnostics. Adjacency matrices encode pack topology, graph cuts isolate faults with minimal disruption, and Laplacian eigenmaps enable efficient monitoring. These techniques form a robust framework for maintaining battery safety and reliability across transportation and grid storage applications.