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Koopman-Based Model Predictive Control for Simultaneous State of Charge and Temperature Balancing of Lithium-Ion Cells
Jihoon Moon
TL;DR
Cell-to-cell electrical and thermal variations produce SoC and temperature imbalances, while nonlinear electro-thermal dynamics complicate control with efficient optimization. The paper combines cell-specific Koopman predictors with regularized feedforward compensation and constrained MPC. In simulations, KMPC achieves balancing comparable to NMPC with substantially lower optimization time.
Problem
Cell-to-cell electrical and thermal variation creates coupled SoC and temperature imbalances, while nonlinear electro-thermal fidelity complicates efficient control synthesis.
Source: §1 · p. 1Method
The framework combines cell-specific EDMD Koopman predictors with Tikhonov-regularized feedforward compensation and constrained KMPC for residual-error regulation.
Source: §1 · p. 2, §6.1 · p. 6Results
KMPC achieves balancing performance comparable to NMPC, with optimization times of 0.0028 s for KMPC and 0.0259 s for NMPC.
Source: §7.3 · pp. 8–9Takeaways & Limitations
The reported comparison supports KMPC as a lower-computation alternative to NMPC for the simulated balancing setting.
Source: §7.3 · pp. 8–9The study assumes electro-thermal states are directly available, although practical battery-management systems must estimate states such as SoC and core temperature.
Source: §8 · p. 9
Abstract
from arXiv · showhide
Cell-to-cell electrical and thermal variations produce nonuniform state of charge (SoC) and temperature distributions in lithium-ion battery packs. This paper proposes a Koopman-operator-based framework for simultaneous SoC and temperature balancing of series-connected cells. Cell-specific Koopman predictors identified using extended dynamic mode decomposition are assembled into neighboring-cell error dynamics that retain heterogeneity and thermal-disturbance effects. A Tikhonov-regularized feedforward controller provides numerically robust compensation without excessive current amplification, while constrained Koopman-based model predictive control (KMPC) regulates the residual errors through a convex quadratic program. Three-cell simulations validate the controller under two consecutive Urban Dynamometer Driving Schedule cycles. KMPC achieves balancing performance comparable to nonlinear model predictive control (NMPC). Relative to the uncontrolled case, KMPC reduces the RMSEs of the neighboring-cell SoC and core temperature differences by 66-81% and 57-76%, respectively, while reducing the computation time per optimization from 0.0259s for NMPC to 0.0028s for KMPC.
1 Introduction
Electrical and thermal heterogeneity jointly create SoC and temperature imbalances, motivating simultaneous balancing. The paper addresses the need for a nonlinear, heterogeneity-aware representation that still supports efficient control optimization.
1 Introduction
- Cell-to-cell parameter variation creates nonuniform SoC and temperature, which can compromise safety and performance, accelerate degradation, and contribute to thermal-runaway risk.Even cells from the same production batch may differ in key parameters by more than 1%.Source: §1 · p. 1
- Electrical and thermal imbalances reinforce one another, and a 3–5°C cell temperature difference can induce current nonuniformity and shorten cycle life.Under a shared current input, thermal and SoC balancing may conflict, motivating simultaneous regulation.Source: §1 · p. 1
- Existing balancing work spans active current allocation, distributed objectives, heterogeneity-aware models, nonlinear optimization, and learning-based control.Prior work also shows that balancing one state can worsen another under some operating conditions.Source: §1 · p. 2
- A control-oriented model is sought that captures nonlinear coupling and heterogeneity while preserving the linear structure needed for efficient optimization.Simplified or locally linearized models may lose accuracy across broad operating ranges, while nonlinear MPC may require online nonconvex optimization.Source: §1 · p. 2
- The paper develops cell-specific Koopman predictors and combines them with regularized feedforward compensation and constrained MPC for simultaneous SoC and temperature balancing.The framework retains heterogeneity and thermal-disturbance effects while using a convex quadratic program for residual-error regulation.Source: §1 · p. 2
2 Preliminaries
The preliminaries describe active balancing hardware and the Koopman framework used to turn nonlinear cell dynamics into predictors suited to control. EDMD supplies finite-dimensional lifted models for systems with control inputs.
2.1 Active-Balancing Hardware for Series-Connected Cells
- Bidirectional active-balancing DC/DC converters allow independent regulation of each series cell’s balancing current and can accelerate balancing relative to passive methods.The hardware supports larger balancing currents between cells.Source: §2.1 · p. 2
2.2 Koopman Operator Theory [23]
- The Koopman operator evolves observable functions linearly even when the underlying state-transition map is nonlinear; practical EDMD models use a finite-dimensional approximation [23, 24].The exact representation is generally infinite-dimensional.Source: §2.2 · p. 3
2.3 Koopman Operator with Control
- For controlled systems, EDMD approximates the Koopman operator in a finite-dimensional observable space and identifies lifted predictor matrices from measured state–input data.The lifted state is a vector of selected observables, and the predictor maps it forward before recovering original state coordinates.Source: §2.3 · p. 3
3 Electro-Thermal Model
The electro-thermal model represents cell-specific electrical and thermal behavior while assuming known, fixed parameters over the study horizon. It couples a Thevenin electrical model with a two-state lumped thermal model.
3 Electro-Thermal Model
- The heterogeneous model is motivated by cell-to-cell differences in electrical and thermal parameters.The stated assumption separates the balancing-control problem from parameter-estimation uncertainty.Source: §3 · p. 3
- The study assumes cell-specific electrical and thermal parameters are known and constant over the operating horizon, excluding parameter-estimation uncertainty from the balancing design.Robustness to parameter uncertainty is outside the study’s scope.Source: §3 · p. 3
- Each cell’s electrical subsystem uses a first-order Thevenin equivalent circuit with SoC and polarization voltage as states and terminal voltage as output.SoC is updated by Coulomb counting, and current is positive during discharge.Source: §3 · pp. 3–4
- A two-state lumped thermal model tracks core and surface temperatures, with heat transfer through conductive and convective resistances and transient response set by thermal capacitances.Heat generation is computed from current times the difference between open-circuit and terminal voltage.Source: §3 · p. 4
4 Koopman operator for Electro-Thermal Model
The paper constructs cell-specific finite-dimensional Koopman predictors from nonlinear electro-thermal data, including ambient temperature as a disturbance. These predictors provide a tractable linear representation for controller synthesis.
4 Koopman operator for Electro-Thermal Model
- The nonlinear OCV relationship couples electrical and thermal dynamics through heat generation, motivating finite-dimensional Koopman predictors for tractable controller synthesis.Nonlinear control may increase online computational requirements and complicate closed-loop stability analysis.Source: §4 · p. 4
- Each cell’s state, current input, and ambient-temperature disturbance are included in the heterogeneous dynamics used to construct the Koopman approximation.A spatially uniform ambient condition is represented by the same disturbance for all cells.Source: §4 · p. 4
- EDMD identification uses measured state, input, and disturbance transitions with lifted-state snapshots to estimate each cell’s predictor matrices by least squares.The Moore–Penrose pseudoinverse is used in the stated least-squares estimates.Source: §4 · pp. 4–5
5 Error Dynamics
The balancing objective is consensus among neighboring cells, represented through lifted edge errors. The global error dynamics retain cell-specific heterogeneity and thermal disturbances.
5 Error Dynamics
- The balancing objective is to make neighboring cells’ physical states converge to consensus.The stated objective is that the norm of each neighboring-cell state difference tends to zero.Source: §5 · p. 5
- Each lifted edge error is the difference between the lifted states of adjacent cells.For the path spanning tree, edges connect cells (i, i + 1); the formulation extends to a general spanning tree through its incidence matrix.Source: §5 · p. 5
- The global lifted error dynamics include cell-specific dynamics, inputs, and thermal disturbances.The model includes A∆Z(k) and E∆D(k) terms alongside the error and control terms.Source: §5 · p. 5
- The reconstructed physical-state error is obtained by mapping the lifted edge error through the reconstruction matrix.When the original electro-thermal states are included in the lifting dictionary, the reconstruction matrices can be chosen as a common state-selection matrix.Source: §5 · p. 5
6 Controller Design
The controller combines feedforward compensation for heterogeneous dynamics and thermal disturbances with feedback balancing. Regularized compensation limits current amplification, and constrained KMPC regulates the residual through a convex quadratic program.
6.1 Composite Controller Design
- The composite input combines nominal current, balancing feedback, and feedforward compensation for heterogeneous dynamics and thermal disturbances.The feedforward term compensates the combined disturbance while feedback retains the cell-balancing role.Source: §6.1 · p. 6
- The Tikhonov-regularized feedforward compensator addresses ill-conditioned inversion that can make exact cancellation infeasible or produce excessive currents.The stated issue arises because ambient temperature acts only through surface-temperature dynamics, so the required range condition is not guaranteed.Source: §6.1 · p. 6
- The regularization parameter trades cancellation accuracy against feedforward-current magnitude.When H(k) is in Range(B∆), the residual approaches zero as λ approaches zero from above; controller synthesis assumes a negligible residual over the study’s operating region.Source: §6.1 · p. 6
6.2 Cell Balancing Controller Design
- KMPC determines balancing feedback from the compensated error dynamics while enforcing balancing-current bounds.Only the first optimal control action is applied before the optimization is repeated at the next sampling instant.Source: §6.2 · pp. 6–7
- The quadratic objective, linear prediction dynamics, and affine input constraints make the KMPC problem a convex quadratic program.With a positive-definite current-effort penalty, every feasible instance has a unique global optimizer.Source: §6.2 · pp. 6–7
7 Simulation Results and Comparison
The simulations validate the Koopman predictors on a held-out WLTP profile and evaluate balancing for three series-connected cells over two UDDS cycles. KMPC reduces SoC imbalance and optimization time relative to the uncontrolled case and NMPC, respectively.
7.1 Koopman Operator Model Validation
- The predictors were trained on nine NEDC, UDDS, and HWFET excitation cases, formed by scaling each profile to 1C, 2C, and 3C.Each cell had 900 training rollouts from 100 initial conditions per excitation case; ambient temperature was held at each rollout’s initial surface temperature.Source: §7.1 · p. 7, Figure 2 · p. 8
- On a WLTP current profile excluded from training, predicted SoC and Vc closely matched simulation, while temperature predictions had modest discrepancies.Core- and surface-temperature RMSEs were 1.01 °C and 0.61 °C, respectively.Source: §7.1 · p. 7, Figure 3 · p. 9
7.2 Simulation setup
- The balancing controllers were evaluated over two consecutive UDDS cycles for three cells, and KMPC and NMPC used identical physical cell-to-cell error penalties and balancing-current constraints.Both used a prediction horizon of 10 and componentwise balancing-current limits of −20 A and 20 A.Source: §7.3 · p. 8, §7.2 · p. 8
7.3 Balancing results
- KMPC reduced neighboring-cell SoC-difference RMSEs by 66–81% versus uncontrolled operation and cut solve time from 0.0259 s for NMPC to 0.0028 s.Both controllers substantially reduced temperature differences; KMPC temperature-pair RMSEs were 0.46 °C and 0.45 °C, versus 0.34 °C and 0.50 °C with NMPC.Source: §7.3 · pp. 8–9
8 Conclusion
The paper presents a Koopman-based framework for balancing SoC and temperature in heterogeneous series-connected cells. Simulations show balancing comparable to NMPC with shorter optimization times, while future work identifies state estimation and broader operating conditions as open needs.
8 Conclusion
- Under two consecutive UDDS cycles, KMPC achieved SoC and core-temperature balancing comparable to NMPC, with optimization time reduced from 0.0259 s to 0.0028 s.The study reports that its identified predictors reproduced electrical and thermal responses under an unseen current profile with sufficient accuracy for control.Source: §8 · p. 9
- Cell-specific Koopman predictors were combined into neighboring-cell error dynamics, with Tikhonov-regularized feedforward compensation and constrained KMPC regulating residual errors.The compensator avoids direct use of a potentially ill-conditioned pseudoinverse and accounts for nominal current, parameter mismatch, and thermal disturbances.Source: §8 · p. 9
- The study assumes electro-thermal states are directly available; future work calls for estimation-error-aware control, degradation modeling, and evaluation across broader conditions.The stated broader conditions include ambient temperatures, cycling profiles, and high and low SoC operation; coordination with pack heating, ventilation, and air conditioning is also future work.Source: §8 · pp. 9–10