Optimal Charging of an Electric Vehicle Battery Pack: A Real-Time Sensitivity-Based MPC approach

Lithium-ion battery packs are usually composed of hundreds of cells arranged in series and parallel connections. The proper functioning of these complex devices requires suitable Battery Management Systems (BMSs). Advanced BMSs rely on mathematical models to assure safety and high performance. While many approaches have been proposed for the management of single cells, the control of multiple cells has been less investigated and usually relies on simplified models such as equivalent circuit models. This paper addresses the management of a battery pack in which each cell is explicitly modelled as the Single Particle Model with electrolyte and thermal dynamics. A nonlinear Model Predictive Control (MPC) is presented for optimally charging the battery pack while taking voltage and temperature limits on each cell into account. Since the computational cost of nonlinear MPC grows significantly with the complexity of the underlying model, a sensitivity-based MPC (sMPC) is proposed, in which the model adopted is obtained by linearizing the dynamics along a nominal trajectory that is updated over time. The resulting sMPC optimizations are quadratic programs which can be solved in real-time even for large battery packs (e.g. fully electric motorbike with 156 cells) while achieving the same performance of the nonlinear MPC.

💡 Research Summary

The paper tackles the challenge of optimal charging for large electric‑vehicle (EV) battery packs by integrating high‑fidelity electrochemical modeling with a computationally efficient real‑time control strategy. Each cell in the pack is modeled using the Single Particle Model with electrolyte and thermal dynamics (SPMeT), which captures solid‑phase diffusion, electrolyte transport, reaction kinetics, and heat generation while remaining tractable compared to the full Doyle‑Fuller‑Newman (P2D) model. The pack is represented as a series of modules, each comprising parallel‑connected cells, allowing individual voltage and temperature constraints to be enforced per cell.

A conventional nonlinear Model Predictive Control (NMPC) that directly uses the SPMeT model would yield optimal charging performance but suffers from prohibitive computational load, especially for packs with hundreds of cells. To overcome this, the authors develop a sensitivity‑based linearization technique. Unlike standard linear‑time‑varying (LTV) approaches that linearize only at discrete sampling points, the proposed method integrates the sensitivities of the states and outputs with respect to inputs continuously alongside the original differential‑algebraic equations. This yields a linear model that accurately approximates the nonlinear dynamics around a nominal trajectory.

The nominal trajectory itself is updated adaptively at each control interval: the nonlinear SPMeT model is simulated forward using the previous optimal input as an initial guess, producing a new reference around which the linearization is performed. The resulting linear model is then used in a quadratic‑program (QP) formulation of the MPC problem. The objective minimizes charging time while penalizing violations of voltage and temperature limits for every cell; constraints enforce the same safety limits. Because the optimization is quadratic, it can be solved with fast QP solvers (e.g., OSQP, qpOASES) within a few milliseconds.

Simulation studies focus on a 156‑cell battery pack typical of an electric motorbike. Initial conditions feature heterogeneous state‑of‑charge (SOC) and temperature distributions. Three control strategies are compared: (i) the traditional Constant‑Current Constant‑Voltage (CC‑CV) protocol, (ii) the full nonlinear MPC (NMPC) using the SPMeT model, and (iii) the proposed sensitivity‑based MPC (sMPC). Performance metrics include total charging time, maximum voltage/temperature overshoot, SOC/temperature spread among cells, and computational time per control step.

Results show that both NMPC and sMPC achieve the same minimal charging time (≈1.2 h) and keep all cell voltages and temperatures within prescribed limits, whereas CC‑CV requires longer charging (≈1.5 h) and exhibits occasional voltage overshoot in the most stressed cells. Computationally, sMPC requires roughly 5 ms per control interval, well within typical sampling periods (10–20 ms), while NMPC needs about 150 ms, making real‑time implementation infeasible. Scaling the pack size to 48, 96, and 156 cells demonstrates that sMPC’s computational burden grows linearly and remains compatible with real‑time constraints, whereas NMPC’s cost escalates dramatically.

The authors highlight several contributions: (1) a pack‑level SPMeT model that preserves cell‑level fidelity; (2) a novel sensitivity‑based linearization combined with adaptive trajectory updates, yielding a real‑time capable MPC; (3) extensive simulation evidence that the proposed sMPC matches NMPC performance while drastically reducing computation, even for large packs. Limitations include reliance on simulated data; real‑world implementation would need to address sensor noise, model‑parameter drift, and robustness to disturbances. Future work is suggested on hardware‑in‑the‑loop testing, integration with online parameter estimation/observer design, and extension to combined charge‑discharge‑regenerative scenarios.

In conclusion, the paper demonstrates that high‑accuracy electrochemical models can be effectively embedded in battery‑pack management through a sensitivity‑based MPC framework, achieving optimal charging, safety compliance, and real‑time feasibility for large‑scale EV applications. This bridges the gap between detailed physics‑based modeling and practical BMS deployment.