  Lithium-ion battery packs demand effective active equalization systems to
enhance their usable capacity and lifetime. Despite numerous topologies and
control schemes proposed in the literature, conducting quantitative analyses,
comprehensive comparisons, and systematic optimization of their performance
remains challenging due to the absence of a unified mathematical model at the
pack level. To address this gap, we introduce a novel, hypergraph-based
approach to establish the first unified model for various active battery
equalization systems. This model reveals the intrinsic relationship between
battery cells and equalizers by representing them as the vertices and
hyperedges of hypergraphs, respectively. With the developed model, we identify
the necessary condition for all equalization systems to achieve balance through
controllability analysis, offering valuable insights for selecting the number
of equalizers. Moreover, we prove that the battery equalization time is
inversely correlated with the second smallest eigenvalue of the hypergraph's
Laplacian matrix of each equalization system. This significantly simplifies the
selection and optimized design of equalization systems, obviating the need for
extensive experiments or simulations to derive the equalization time.
Illustrative results demonstrate the efficiency of the proposed model and
validate our findings.
