Fundamental Techniques for Optimal Control of Reconfigurable Battery Systems: System Modeling and Feasible Search Space Construction

Technical Summary: Fundamental Techniques for Optimal Control of Reconfigurable Battery Systems

Problem Statement
Reconfigurable Battery Systems (RBSs) offer a promising solution for improving fault tolerance, charge/thermal balance, and energy delivery by dynamically altering battery connections via switches. However, optimizing RBS performance requires solving high-dimensional, nonlinear integer programming problems. The authors identify two critical challenges hindering this process:

1. Dynamic System Modeling: Existing literature lacks a unified method to model RBSs under dynamic reconfiguration. As switch states change, battery cells may join or quit operations, altering system dimensions and requiring the manual reconstruction of Kirchhoff's Current Law (KCL) and Voltage Law (KVL) equations for every configuration. This process is computationally inefficient and difficult to automate.
2. Search Space Complexity: The number of possible switch state vectors (SSVs) grows exponentially with the number of switches and decision steps. For large-scale RBSs, searching the complete solution space ($B^{N_S}$) is time-consuming and often futile, as the majority of configurations result in infeasible states such as short circuits, open circuits, or undesired system voltages.

Methodology
The paper proposes a unified framework addressing both modeling and search efficiency:

1. Unified Modeling via Complete System Configuration (CSC):

   • Instead of removing branches corresponding to "off" switches, the authors propose retaining all physical branches (batteries, switches, wires) to maintain a stable mesh structure known as the Complete System Configuration (CSC).
   • Switch states are incorporated into the model by defining a variable switch resistance ($R_{S_n}^m$). When a switch is "on," it exhibits low on-resistance; when "off," it exhibits high off-resistance (effectively an open circuit).
   • This approach allows the formulation of a single, unified state-space model ($\dot{X} = AX + B i_t$) where the system matrices depend on the SSV but the model structure remains invariant. This eliminates the need for frequent model reconstruction during iterative optimization.
   • The model is discretized for optimal control formulation, with parameters updated based on State of Charge (SoC) look-up tables at each sampling interval.

2. Feasible Search Space Construction:

   • To mitigate the "curse of dimensionality," the authors develop an algorithm to actively construct a feasible search space ($B_f^{N_S}$) rather than searching the complete space.
   • The method involves mapping system configurations to SSVs based on hierarchical module structures (Series-then-Parallel, $C_{sp}$, and Parallel-then-Series, $C_{ps}$).
   • Algorithm 1 utilizes Dynamic Programming (DP) to enumerate all feasible Parallel-then-Series configurations. It iteratively builds valid cell connection patterns (flags for series, parallel, or bypassed cells) while enforcing constraints on normalized system voltage and avoiding short/open circuits.
   • Only SSVs corresponding to these valid configurations are retained, drastically reducing the search space size.

Key Results

• Model Accuracy: The proposed unified model was validated against experimental data from a 3-cell RBS testbed.
  • System-level simulation showed Mean Absolute Errors (MAE) of 52 mV for terminal voltage and 46 mA for input current.
  • Cell-level simulations demonstrated significantly higher accuracy than a "simple model" (which assumes equal current distribution). The proposed model achieved cell current MAEs $\le$ 20 mA and voltage MAEs $\le$ 7 mV, compared to much higher errors in the simple model, particularly during transient parallel connections.
• Search Efficiency:
  • For a 10-cell RBS, the ratio of feasible SSVs to the total possible SSVs was found to be less than $2 \times 10^{-10}$.
  • In a case study minimizing SoC imbalance over 20 decision steps using a Genetic Algorithm (GA):
    • Searching the complete space failed to converge; the algorithm could not find feasible initial solutions, and the objective function remained stagnant.
    • Searching the narrowed feasible space successfully reduced the total SoC imbalance from 0.30 to 0.031 within 220 generations.

Significance and Claims
The paper claims to lay a solid foundation for the effective formulation and efficient solution of RBS optimal control problems. Its primary contributions are:

1. Unified Modeling: Providing a single mathematical framework that accommodates various RBS designs and their hybrid combinations without requiring model reconstruction, thereby facilitating topology design and control formulation.
2. Search Space Reduction: Demonstrating that narrowing the search space to only feasible solutions is not merely an optimization heuristic but a necessity for solving large-scale RBS control problems, as it ensures safe operation and enables convergence where standard methods fail.

The authors conclude that these techniques enable the formulation of diverse optimal control problems to enhance system performance, with future work planned to explore specific optimization algorithms and applications.