INTRODUCTION
In the context of the global transition toward renewable energy and electric mobility, Lithium-ion batteries play a pivotal role owing to their high energy density, long cycle life, and stable performance. However, charging speed remains a significant bottleneck, exacerbating range anxiety and limiting practical usability [1,3]. Increasing the C-rate (e.g., 2C, 3C, or even 12C) can shorten charging time but introduces severe electro-thermal and chemical degradation risks. The heat generation rate, proportional to I2R, causes rapid temperature rise; meanwhile, high current density accelerates lithium plating and SEI cracking, leading to increased internal resistance and capacity loss [4–7]. Experimental studies have also reported aluminum dissolution and SEI damage, which can result in up to ~25% capacity loss after 500 fast-charging cycles [6]. These risks not only compromise operational safety but also reduce the lifecycle cost advantage of energy storage systems. To reconcile the trade-offs between charging speed, durability, and safety, three main research directions have been actively pursued: (i) the development and substitution of advanced electrode materials, (ii) thermal management during charging, and (iii) charging current control strategies. Recent studies have explored electrode materials such as TiNb2O7 and Li4Ti5O12 with nano-architectures to shorten the Li⁺ diffusion path, accelerate surface kinetics, and stabilize the crystal structure under high charging currents [8–10]. Although promising, this approach entails long R&D cycles, high costs, and commercialization challenges. Thermal management solutions employing advanced heat-transfer fluids circulating between cells have been shown to reduce inter-cell temperature gradients and suppress local hot spots, thereby enhancing overall system safety and longevity [8–11]. However, these solutions require additional installation space and complex mechanical structures and, crucially, only control heat externally—offering limited capability to directly mitigate the internal electrochemical processes responsible for heat generation and abnormal temperature rise. In contrast, research on charging current control strategies represents a more practical and scalable path, leveraging software and algorithmic approaches to maximize efficiency using existing hardware while maintaining adaptability to cell states and operating conditions.
Traditional charging control strategies—such as constant-current–constant-voltage (CC–CV), multi-stage CC–CV (MCC–CV), and pulse charging (see Fig. 1)—remain the fundamental benchmarks for fast charging. However, when applied to series-connected cell strings, they exhibit inherent limitations due to their fixed parameters, lack of adaptability to individual cell SoC/SoH/temperature states, and the necessity to constrain the entire string to the charging current of the weakest cell.
Numerous studies have optimized MCC–CV charging with electro-thermal and aging constraints at the cell or module level, demonstrating reduced charging time and lower energy loss compared to conventional CC–CV. Pulse charging, on the other hand, has been systematically analyzed as a class of strategies that mitigate polarization and diffusion limitations, showing particular advantages at low temperatures. Nonetheless, it still requires state-dependent parameter tuning and does not inherently address heterogeneity among cells in series configurations [12].
The current research trend is shifting toward model-based optimal control (using equivalent circuit or electrochemical models, ECM/EChM) that explicitly incorporates electro-thermal and aging phenomena to compute adaptive charging profiles. These methods employ optimal programming or model predictive control (MPC) frameworks with voltage, temperature, power loss, and cost constraints to determine the optimal charging currents. Some works have coupled electro-thermal-aging models and solved the optimization problem using collocation or spectral methods, or adopted predictive control strategies to achieve fast charging while ensuring safety and extending cell lifetime [13].
Recently, the integration of fast-charging control with active thermal management has emerged as a promising approach that enables coordinated regulation of both current and temperature, maintaining permissible temperature rise even under harsh environments [14]. Another important direction involves embedding anti-lithium-plating constraints directly into the optimization problem—through electrode potential limits or SoC–temperature–current density proxies—to ensure safety under high C-rates [15].
At the pack or series-string level, a fundamental bottleneck arises because a common charging current must be imposed across all cells, causing overall charging progress to be limited by the weakest one. Consequently, series–parallel (S/P) reconfiguration has been proposed as a system-level enabler to overcome this limitation by enabling differentiated current allocation per cell, thereby shortening charging time without sacrificing lifespan. Research efforts—from reconfigurable circuit architectures and switch-loss modeling to experimental validation—have confirmed its potential to enhance fast-charging efficiency at the pack scale.
In parallel, a new research branch combines reinforcement learning and predictive control to jointly optimize fast-charging currents and active cell balancing across the pack, leveraging electro-thermal coupled models and aging-awareness to enforce real-time constraints under heterogeneous cell conditions. Beyond technical constraints, several recent works have also incorporated time-of-use (ToU) electricity pricing into the cost function to co-optimize charging cost, duration, and lifetime—an approach particularly suited for battery energy storage systems (BESS) and electric vehicles (EVs) operating under scheduled charging windows.
Although numerous studies on optimal charging control for Lithium-ion batteries have been conducted in recent years, most have focused solely on optimizing the charging process under homogeneous cell conditions or addressing only a single aspect—such as voltage limits, temperature constraints, or energy cost optimization. In contrast, real-world battery packs typically operate under highly heterogeneous conditions: individual cells may differ simultaneously in SoC, SoH, and initial temperature; they are also subjected to large current constraints, time-varying electricity prices, and multi-objective trade-offs among safety, performance, and cost. To date, few studies have comprehensively validated such compound scenarios experimentally— where the controller must dynamically adapt to variable operating conditions and the topological constraints of a reconfigurable architecture [16–20]. Consequently, the practical applicability of existing optimal charging strategies remains an open challenge, particularly when electro-thermal safety, energy balancing, and cost optimization must all be guaranteed at the cell level for large-capacity battery systems.
To overcome these limitations, this paper proposes an optimal cell-level fast-charging control strategy based on a coupled electro–thermal model and a flexible series–parallel (S/P) reconfiguration mechanism. The proposed method enables independent, real-time current regulation for each cell while strictly enforcing operational limits on voltage, temperature, and SoH-dependent current, and incorporates time-of-use (ToU) electricity pricing into the objective function to minimize ToU-weighted charging energy cost (battery-side), with a straightforward extension to grid-side cost by incorporating charger efficiency. By using switching variables and a projection-type topological constraint, the control structure decouples intrinsic cell dynamics (invariant with respect to interconnection topology) from inter-cell constraints (dependent on reconfiguration mode).
Each cell is assigned a dedicated fast-charging profile with SoH- and temperature-adaptive current limits, preventing overvoltage and overheating while synchronizing all cells to reach the target SoC simultaneously. The optimization framework simultaneously considers (i) voltage–temperature–aging constraints, (ii) operational disturbances (ambient temperature variations, internal resistance fluctuation), and (iii) ToU-based cost weighting. Active S/P reconfiguration throughout the charging process allows differentiated current distribution across cells, thereby eliminating dependency on the weakest cell, shortening charging time, reducing cost, and maintaining electro-thermal safety.
A comprehensive set of simulation and experimental scenarios is designed to reflect real-world operating conditions—spanning variations in SoC, SoH, temperature, maximum allowable current, electricity pricing, and objective-function weighting—to thoroughly validate the controller’s adaptability, safety, and economic efficiency. The results demonstrate that the proposed strategy not only maintains safety under all conditions but also achieves high charging speed, uniform terminal states, and significant energy-loss reduction—representing a major step toward practical deployment in next-generation Battery Management Systems (BMS). Compared with the two costly hardware-oriented approaches—material redesign and active cooling—the proposed method offers three distinct practical advantages: (1) Real-time adaptability: charging current is continuously modulated according to each cell’s current state (SoC/SoH/SoT, polarization voltage, and estimated internal resistance); (2) Hardware compatibility: the lightweight model and controller can be readily implemented on existing BMS platforms; (3) Joint techno-economic optimality: the strategy achieves the target SoC within safe operating limits while minimizing cost under ToU pricing.
These advantages make the approach particularly effective for large-scale energy storage systems (ESS) and EV battery packs comprising multiple series–parallel modules with inherently heterogeneous cells. The main contributions of this paper are summarized as follows:
· Designs an active series–parallel (S/P) reconfiguration mechanism that enables cell-wise current allocation, overcoming the “bottleneck” limitation of traditional series configurations based on cell’s coupled electro–thermal model with real-time state feedback (SoC, SoT, polarization voltage) to optimally regulate fast charging current. Note that the experimental setup used in this research is seen as a cell-level emulation platform for series-parallel reconfigurable charging instead of a complete hardware realization.
· Proposes a multi-objective optimal control framework at the cell level, integrating electro-thermal, degradation/SoH, and ToU cost constraints to shorten charging time while maintaining safety and durability. Charging time is not minimized in a strict free-terminal-time optimal control sense, but rather enforced through a predefined charging window.
· Demonstrates through validation that all cells reach the target SoC (≥ 97%) within electro-thermal safety limits while simultaneously reducing charging costs under ToU pricing.
The structure of this paper is organized as follows. Section 2 presents the cell’s coupled electro–thermal model and the fast-charging model based on series–parallel reconfiguration. A unified representation for an n-cell system is proposed using switching variables and projection-type topological constraints, effectively decoupling the “cell dynamics” from the “inter-cell coupling constraints.” Section 3 formulates the optimal control problem for the fast-charging process under practical operating conditions, including the definition of constraints, construction of the objective function, and selection of the optimization algorithm. Section 4 provides simulation and experimental results, while Section 5 concludes the paper and outlines directions for future research.
THE CELL’S COUPLED ELECTRO–THERMAL MODEL AND FAST-CHARGING MODEL BASED ON RECONFIGURATION
Assume that n cells are connected in series during discharge to supply electrical energy to the load. To accelerate the charging process—where each cell can have an optimal charging current depending on its operating condition—a series–parallel reconfigurable structure is adopted, as illustrated in Fig. 2. During discharge, the cells operate in a series configuration, whereas during charging, they are reconfigured into independent parallel branches. The switching process is governed by binary variables z∈{0,1} together with a projection-type topological constraint, which effectively decouples the cell dynamics (invariant across configurations) from the inter-cell coupling constraints (which vary according to the discharge/charge mode).
In this study, we employ a cell model (or supercell—formed by multiple cells connected in series or parallel and treated as a single equivalent cell with higher power, current, and voltage capacity), as illustrated in Fig. 3. The cell’s coupled electro-thermal model combines a temperature-dependent first-order ECM for SoC with an ARX thermal model for SoT, linked through a physics-informed heat input q ˙ c
The cell model is represented in a coupled electro–thermal form as expressed in Eq (1) below.
with
(2)
By denoting the superscript index corresponding to cell i(i = 1,2, ..., n), the electro–thermal coupled model for cell i can be expressed in the form of Eq (2)
(3)
in which
(4)
The vectors representing the n cells are defined as follows:
where In is the n×n identity matrix, and 1n is a column vector of ones with n elements. The control variables are defined as in Eq (6).
Let z denote the switching operator, the model input of model U(k)of n cells is described by Eq (7) below
with z = 1 : u i ( k ) = u d ( k ) | ∀ i ; z = 0 : u i ( k ) = u c i ( k ) = I c i ( k )
The topological constraint for single current control in discharge mode is
where Pn is projection matrix with its elements are P nij = δ ij - 1 n
(9)
with
(10)
The models (9) and (10) represent the n cells within a single framework: series topology in discharge, and parallel topology in charge to obtain n fully independent channels. All electro-thermal equations are retained per cell, denoted by the superscript (i). This is precisely the benefit of using the projection matrix P n = I n - 1 n 1 n 1 n ⊤
The coupled electro-thermal model requires the identification of the following parameters: the relationship VOC = gOCV (SoC, Tc), R1(Tc), C1(Tc), R0(Tc), a1(Ta),..., ana(Ta), b1(Ta),..., bnb(Ta). To identify these parameters, experimental data is required. The parameters of model (1) are specified in [21] which we recently published.
FORMULATION OF THE OPTIMAL CONTROL PROBLEM UNDER OPERATING CONDITIONS
Charging model
Based on (9), the one-step-ahead predictor for SoC and ToC corresponding to the charging current U(k), z = 0 is
The charging model described by Eq. (11) characterizes the electro–thermal dynamics of the cell during the charging process, enabling prediction of the evolution of state variables X(k)—including SoC, ToC, cell voltage, and the average cell temperature—as a function of the controlled charging current X(k) at each sample time step. Through the state matrices A T ^ c ( k ) , B T ^ c ( k ) Φ ( k ) ϑ ( T a ( k ) )
Safety-related constraints
High charging current is hazardous and can rapidly damage the cell; therefore, it must be kept within the manufacturer’s specified limits. The charging-current constraint is
where u dis _ max i SoH i = C i C nom i u dis _ max i = I max f ( SoH i )
Objective functions of fast charging control
Fast-charging speed is a key objective, the control strategy minimizes the SoC mismatch of the cells in the string, whose initial states are SoC(0)=[SoC1 (0), SoC2 (0),...,SoCn (0)], to a desired value SoCd after N charging control steps. Here, N reflects the desired fast-charging duration specified by the user; equivalently, the user seeks SoC (kN) ≈ SoCd* 1n with kN = NΔT. The fast-charging requirement is specified by a maximum allowable charging duration NΔT, under a given maximum charging duration, the fast-charging objective is achieved implicitly by minimizing the terminal SoC deviation and allowing the charging current to be optimally allocated so that all cells reach the desired SoC as early as possible within the predefined time window. The time-related objective for fast charging is defined as:
An equivalent formulation could treat the charging duration as a decision variable; however, in this study, we adopt a fixed-horizon formulation consistent with practical BMS operation, where maximum charging time is typically specified by the user or scheduling system.
After the interval NΔT, the cells’ SoC must reach the desired value SoCd, i.e., we minimize the following cost JSoC
To improve charging efficiency, we also minimize the energy loss during charging; the loss-related objective JE is given by (16).
with V RC ( k ) = V RC 1 ( k ) , V RC 2 ( k ) , ⋯ , V RC n ( k ) T
Here, VRC represents the polarization voltage of the cell, which captures the internal electrochemical overpotential associated with diffusion and charge-transfer dynamics. The product V RC ( k ) T U ( k )
Economic cost is another key objective that helps users reduce their electricity bill under time-of-use tariffs. In practice, electricity prices differ between peak and off-peak hours and are higher during peaks; therefore, choosing larger or smaller charging currents at different times should be decided so that the total cost is minimized. The electricity cost is computed by (17)
where p(k) is the time-of-use price at sampling instant k(in currency/kWh), and VC ( k ) = V c 1 ( k ) , V c 2 ( k ) , ⋯ , V c n ( k ) T · V c ( k ) T U ( k )
Finally, to penalize large currents when SoC is close to 1, we impose a safety-and-longevity objective that enforces smaller current as SoC→1; this is a mandatory engineering requirement. The penalty for SoC > 0.9 is defined in (18)
Formulation of the objective function
With the above analyses, the optimal fast-charging control under operating conditions can be posed as the following constrained optimization problem
(19)
in which γ1, γ2, γ3, γ4, γ5 > 0are user-tunable weights reflecting priority. When the user places greater emphasis on a particular objective term, the corresponding weight is increased.
Optimal charging control algorithm
To solve the optimal charging-current problem—minimizing a composite objective that includes charging time, the terminal SoC error relative to the desired SoC, charging losses, peak temperature rise, and electricity cost—subject to inequality/equality constraints on cell voltage, average cell temperature, and charging current, a suitable approach is Sequential Quadratic Programming (SQP). At each iteration, SQP forms a local quadratic approximation around the current iterate: the objective is expanded to second order using the Lagrangian Hessian (estimated via BFGS/DFP to remain positive definite), while the constraints are linearized via the Jacobian. The controller then solves a local QP subproblem and updates the iterate using a line-search/merit function or a trust-region step to ensure descent of the objective and progressive satisfaction of the constraints. SQP is well suited to the fast-charging problem considered here because the optimization is a smooth, constrained nonlinear program (with an coupled electro-thermal model and an ARX thermal model) and features tight time-indexed constraints. By exploiting derivatives, SQP achieves locally superlinear convergence even with active constraints, enabling rapid computation of feasible charging currents that respect cell voltage and temperature limits. Current, temperature, and voltage constraints are enforced within the QP via an active-set mechanism, which immediately identifies any violated constraints and facilitates tuning of the objective weights. SQP embeds naturally in a real-time BMS: at each sampling instant the algorithm can be warm-started from the previous solution, requiring only a few iterations, with stable compute time; for small to medium numbers of cells the QP subproblems solve very quickly. The method also accommodates time-of-use (ToU) pricing without altering the algorithmic framework, supports soft constraints for time-varying tariffs, and maintains feasibility under elevated ambient temperature or low SoH. In this study we implement the optimizer in MATLAB using fmincon with the ‘sqp’ or ‘sqp-legacy’ algorithm.
SIMULATION AND EXPERIMENTAL RESULTS
The experimental system
The experimental system is designed as a cell-level emulation platform for a series–parallel reconfigurable charging architecture, aiming to reproduce the essential current-allocation and electro–thermal constraint characteristics of S/P reconfiguration. It should be noted that the experimental setup focuses on validating the proposed cell-level optimal charging and current allocation strategy under reconfigurable constraints, rather than reproducing all hardware-level switching losses and parasitic effects of a full-scale S/P reconfigurable battery pack. It employs OPA549 power amplifiers [22] in combination with PI controllers to independently regulate the charging current of each cell, thereby optimizing the fast-charging process while ensuring cell protection. The experimental setup consists of the following components: a Dell Inspiron 15 3000 computer (Core i5, 32 MB RAM) running MATLAB R2020a, an OPA549 power amplifier circuit [22], a National Instruments (NI) cDAQ-9174 data acquisition chassis [23], three NI 9239 analog input modules (each with four ±10 V analog input channels), and one NI 9263 analog output module [25]. Each cell’s current, surface temperature, and terminal voltage are measured individually, requiring a total of 12 analog input channels—hence the use of three NI 9239 modules. Each cell also requires a dedicated analog output channel to deliver the optimal charging current (computed in MATLAB) to the OPA549 amplifier, thus necessitating one NI 9263 module (which provides four ±10 V analog output channels). The OPA549 amplifier is capable of supplying up to 8 A charging current, with an input voltage range of 0–5 V and an output voltage up to ±15 V. A photograph of the actual experimental setup is shown in Fig. 4.
In this experiment, we use four SAMSUNG 18560 cells, cells 1 through 4, connected in series to form a battery string. Each cell is instrumented with sensors for current, terminal voltage, and ambient temperature; these measurements are sent to the BMS. Series configuration is used during discharge and parallel reconfiguration during charge, both governed by controlled charge and discharge switches. For n cells, n+1 discharge switches and 2n charge switches are required. The BMS acquires per-cell current, voltage, and temperature, actuates the pack’s charge and discharge switches, and computes/estimates cell states such as SoC, SoT, and SoH to supply the optimal charging controller. It also enforces safety functions (limits on current, temperature, and voltage). The optimal charging controller receives the cell states (SoC, SoH, temperature, voltage) from the BMS, computes the optimal charging currents for each cell (I c 1 , * , I c 2 , * , I c 3 , * , I c 4 , *
Each cell is charged independently by its own charger circuit. In this study, an OPA549 power amplifier serves as the current-regulating element, with an integrated over-current protection I c i ≤ I cmax i I c i I c i , *
The per-cell charging current error is defined by (21)
The PI charging-current control law is given by the following expression (22)
where KP and KI are the controller gains.
The input–output relation between the control voltage V c i △ V i
where A is the OPA549 gain, V OC i = g OCV ( SoC i , T c i ) V RC i R 1 ∥ C 1
The transfer function for measurement noise and shunt dynamics (modeled as first order due to the RC filter and OPA bandwidth) is given by (25)
with ωp approximating the closed-loop bandwidth of the shunt path. The approximate closed-loop transfer function of the per-cell charging system (cell i, PI current controller, OPA549, shunt resistor) is shown in (26)
with parameters: ωp = 6.24 rad/s, Rs = 0.5W, Kp = 1.7513 V/A, KI = 12.9303 V/A.s.
Validation of the electro–thermal model
The electro-thermal coupled model of the cell is illustrated in Fig. 3, which is taken from our previous study in [21], where the output of the electrical model, q·cell, serves as the input to the thermal model, and the output of the thermal model, Tcell, is one of the inputs to the electrical model. The electro–thermal model parameters consist of the open-circuit voltage relationship VOC = gOCV(SoC, Tcell), the temperature-dependent electrical parameters R1(Tcell), C1(Tcell), and R0(Tcell), together with the coefficients na, nb, a1(Tamb), ..., ana(Tamb), and b1(Tamb), ..., bnb(Tamb) of the ARX-based thermal model. All model parameters were identified using experimental data obtained over a wide operating temperature range from −25°C to 45°C.
To validate the accuracy of the proposed electro–thermal model under fast-charging conditions, the predicted terminal voltage and cell temperature were systematically compared with experimental measurements. The results demonstrate a high level of agreement between simulation and experiment throughout the charging process. Specifically, the mean prediction error of the cell terminal voltage is approximately 10–15 mV, while the mean temperature prediction error of the ARX-based thermal model is approximately 0.6–0.8°C [21]. These small discrepancies confirm that the proposed electro–thermal model is capable of accurately capturing both electrical and thermal dynamics of the cell during fast charging, and is therefore suitable for real-time optimal charging control and constraint enforcement.
Simulation and experimental scenarios
In this section, four simulation and experimental scenarios are presented to validate the proposed optimal fast-charging control strategy under different operating conditions. These scenarios enable a comprehensive assessment of the influence of heterogeneous SoC/SoH, ambient temperature, current limits, and cost–lifetime trade-off priorities on the optimal charging current profiles. Fig. 5 illustrates the daily time-of-use (ToU) electricity pricing profile, which serves as the case study for this research.
We conduct simulations and experiments for four scenarios with distinct parameter settings and verification objectives, as described below:
Scenario (a): The initial SoC(0) and SoH of all cells are identical, while the ambient temperatures differ. The current limit is set to 2C, charging begins at 1000 minutes, and all weighting factors in the objective function are set to 1. This scenario is designed to verify the controller’s thermal adaptability and its ability to ensure electro–thermal safety under non-uniform initial temperature conditions. The test evaluates whether the controller can automatically adjust each cell’s charging current according to its actual temperature to maintain both safety and thermal balance. The optimal controller should automatically reduce the charging current for hotter cells and increase it for cooler ones, ensuring that the temperature of all cells remains within the safety threshold (45°C). This scenario verifies the controller’s capability for state-dependent current coordination, a key feature of the proposed strategy. Equal weighting factors eliminate prioritization among cost, loss, or thermal criteria, allowing a pure assessment of the influence of initial thermal conditions on control performance.
Scenario (b): The initial SoC(0) values of all cells are the same, but the SoH and ambient temperatures differ. This scenario validates the comprehensive adaptability of the optimal fast-charging controller under conditions where cells differ in both health and thermal environments. Variations in SoH cause differences in internal resistance, current-handling capability, and heat generation, while temperature disparities affect electro–thermal dynamics and safety limits. The controller must automatically allocate charging currents considering both effects—reducing current for degraded or hotter cells, and increasing it for healthier or cooler ones—to ensure electro–thermal safety, prevent overvoltage and overheating, and still achieve the target SoC within the specified time. Simulation and experimental results from this case demonstrate the controller’s ability for multi-criteria adaptive current coordination, the stability of the coupled electro–thermal model, and the practical feasibility of the proposed strategy for real battery packs with heterogeneous aging and operating conditions.
Scenario (c): Both SoC(0) and SoH differ among cells, and the charging current limit is increased to 3C. This scenario examines the controller’s robustness under harsher conditions, where cells vary in both energy and health states and experience strong thermal and electrical stress due to high charging currents. Increasing the current limit to 3C enables clear observation of the transition mechanism between CC and CV phases and allows evaluation of voltage and temperature control performance in high-power, high-heat regimes. Differences in SoC and SoH introduce substantial variations in electro–thermal dynamics across cells, creating a critical test for the controller’s ability to dynamically distribute charging currents under safety constraints and manage the CC→CV phase transition optimally. The results confirm the controller’s ability to maintain electro–thermal safety, limit temperature peaks, and keep voltage and SoC within acceptable bounds, demonstrating the stability and reliability of the proposed strategy under high-current and non-uniform pack conditions typical of real systems.
Scenario (d): The initial SoC(0) values of all cells are identical, while SoH and temperature differ; charging starts when the electricity price is high, and the weighting factors are increased (γ4 = 1.3, γ5 = 1.3). This scenario evaluates the controller’s capability to adapt charging behavior to dynamic ToU pricing and real operating conditions, particularly its load-shifting capability and ability to dynamically adjust charging currents to minimize energy loss and cost while maintaining electro–thermal safety. By increasing γ4 and γ5, the controller is biased toward reducing peak power and limiting heat generation during high-price periods, reflecting intelligent current scheduling aligned with the ToU tariff. Simultaneous variations in SoH and temperature create a complex multi-constraint environment, testing the controller’s stability and efficiency in optimizing across three objectives: safety, cost, and performance.Simulation and experimental results confirm the controller’s ability to flexibly adjust charging currents over time and per-cell states, and to achieve optimal charging–load-shifting behavior consistent with ToU variations and heterogeneous operating conditions in real battery systems.
Simulation results
Scenario a: At the beginning of the charging process, all cells have the same initial state of charge SoC(0) = 25%, and the desired final SoC is 97%. SoH of all cells is also identical at 97%, while the ambient temperatures differ among cells: cell 1 starts at 35°C, cell 2 at 30°C, and cells 3 and 4 at 37°C and 40°C, respectively. The maximum charging current limit is 2C, the maximum allowable cell temperature is 45°C, and the maximum cell voltage is 4.3 V. Charging begins at 1000 minutes, when the electricity price is 2200 VND/kWh, and all objective function weights are set to 1. The fast-charging process lasts 98 minutes, and the results are presented in Fig. 6.
The simulation results for scenario (a) demonstrate that the proposed optimal fast-charging controller can effectively regulate the distribution of charging current according to the initial thermal conditions of each cell while maintaining electro–thermal safety throughout the charging process. At the start of charging, all cells have identical initial states of charge, SoC(0) = 25%, and identical states of health, SoH = 97%, but significantly different initial temperatures (ranging from 30°C to 40°C). The controller automatically supplies higher current to cooler cells (e.g., those starting at 30°C) while limiting the current for hotter cells (e.g., those starting at 37–40°C) to ensure that each cell’s local temperature remains below the safety threshold of 45°C. As a result, the cell temperatures increase in a controlled manner and then gradually decline toward the safe region during the final phase, confirming that the controller successfully maintains the thermal margins within prescribed limits. Simultaneously, the cell voltages remain below the upper limit of 4.3 V throughout the entire process. When any cell voltage approaches this threshold, its charging current is sharply reduced, reflecting the controller’s real-time voltage protection mechanism at the cell level. This adaptive current regulation leads to temporary differences in SoC accumulation rates among cells during the early stage; however, all cells eventually converge to the target SoC of approximately 97% at the end of the 98-minute charging period. This indicates that the optimization successfully achieves the final energy target while strictly complying with voltage and temperature constraints. Moreover, the RC branch voltage difference (VRC) gradually decreases toward zero over time, showing that the electrochemical transients are effectively damped as the charging current tapers during the final phase. This confirms that the controller not only accelerates SoC progression but also ensures that each cell reaches a stable voltage and safe temperature before the end of the charging session. Overall, the simulation verifies that when the cells differ only in their initial thermal conditions (while SoC and SoH are uniform), the controller can autonomously prioritize thermal and voltage safety for each cell while still achieving fast-charging progress across the entire pack. This fulfills the core validation objective of scenario (a) — to demonstrate the controller’s temperature-adaptive capability in achieving the target SoC within strict voltage, temperature, and 2C current limit constraints.
Scenario b: SoC(0) for cells 1, 2, 3, and 4 are 0.3, 0.4, 0.3, and 0.4, respectively. The corresponding SoH are 0.95, 0.9, 0.8, and 0.99 for cells 1 through 4. The ambient temperatures are 40°C for cell 1, 30°C for cell 2, 37°C for cell 3, and 35°C for cell 4. The maximum charging current limit is 2C, the maximum allowable temperature is 45°C, and the maximum cell voltage is 4.3 V. The desired final SoC at the end of charging is 0.97, and all objective function weights are set to 1. Charging begins at 1000 minutes, and the process completes after 67 minutes. The corresponding results are presented in Fig. 7.
The simulation results for scenario b clearly demonstrate the multi-criteria adaptability of the proposed optimal fast-charging controller under dual heterogeneity in both cell health (SoH) and initial thermal conditions. At the beginning of charging, the cells not only differ in their SoC (ranging from 0.3 to 0.4) but also exhibit significant disparities in aging level (SoH from 0.8 to 0.99), with initial ambient temperature differences of up to 10oC. This creates a stringent and realistic operating scenario: more degraded cells (e.g., cell 3 with SoH = 0.8) typically possess higher internal resistance, are more prone to heat generation, and are more likely to reach voltage limits during fast charging, while hotter cells (e.g., cell 1 at 40°C) have a smaller remaining thermal safety margin compared to cooler ones (e.g., 30°C). The controller responded appropriately — the optimal charging currents were distributed non-uniformly across the cells rather than enforced to be identical. Healthier cells with more favorable thermal conditions (e.g., cell 4 with SoH = 0.99 and initial temperature = 35°C) received higher charging currents during the early phase, while degraded or hotter cells were assigned lower currents to ensure electro–thermal safety. Consequently, the SoC rise rates differed among cells in the initial stage; however, all cells gradually converged toward the target SoC ≈ 0.97 by the end of the 67-minute charging period, confirming that the controller still achieved the desired overall energy target at the pack level despite highly non-uniform initial conditions. Thermally, the trajectories of Tc(t) indicate that each cell’s temperature remained below the 45°C limit throughout the process, even for the hottest cell (40°C). The initial temperature rise due to large charging currents peaked and then decreased as the controller actively reduced the current over time, demonstrating the algorithm’s ability to dynamically “brake” hot or aged cells to prevent local overheating. Simultaneously, the cell voltage profiles VRC show that voltage peaks approaching the 4.3 V limit were handled by immediate current reduction, confirming that the cell-level protection mechanism functioned as intended. The RC branch voltages VRC progressively decayed toward zero for all cells, indicating that polarization and internal transient effects were gradually suppressed as charging transitioned into the tapering phase, ensuring a stable end state before termination. Overall, Scenario (b) confirms that the proposed controller can simultaneously: (i) detect and compensate for differences in SoH and temperature among cells; (ii) redistribute charging current according to each cell’s individual electro–thermal risk rather than applying a uniform current; and (iii) achieve the target SoC within the fast-charging time frame (67 minutes) without violating voltage or temperature constraints. This outcome fulfills the validation goal of Scenario (b) — to evaluate the controller’s ability for multi-criteria adaptive current coordination (balancing electro–thermal safety and cell longevity) in a battery pack with heterogeneous health states and thermal conditions, closely reflecting real-world post-aging operating scenarios.
Scenario c: The initial SoC of cells 1 and 2 are 0.3, while those of cells 3 and 4 are 0.2. The corresponding SoH are 0.95, 0.9, 0.8, and 0.99 for cells 1 through 4, respectively. The maximum charging current limit is increased to 3C, and the maximum allowable cell voltage is 4.3 V. The desired final SoC at the end of the charging process is 0.97, and the maximum allowable temperature is 45°C. All objective function weights are set to 1, and the charging process begins at 450 minutes. The charging operation completes after 64 minutes, and the corresponding results are illustrated in Fig. 8.
The simulation results of scenario c show that during the initial phase, the cells are subjected to very high instantaneous charging currents, reflecting the controller’s utilization of the full 3C limit to accelerate SoC rise. However, the current is not distributed uniformly among the cells: those with better health conditions (e.g., cell 4 with SoH = 0.99) receive larger currents and sustain them for a longer duration, whereas more degraded cells (e.g., cell 3 with SoH = 0.8) experience an earlier current reduction. This behavior confirms that the controller continues to selectively regulate the charging current according to each cell’s health condition, consistent with the principle that “healthier cells charge faster, weaker cells are protected.” The cell voltage profiles further show that whenever a cell voltage approaches the 4.3 V threshold, its charging current is sharply reduced. This represents the gradual transition from the constant-current (CC) to the constant-voltage (CV) region at the individual cell level. As a result of this strategy, no cell exceeds the maximum voltage limit, even under a high overall current constraint. The CC→CV transition does not occur simultaneously for all cells but follows different sequences depending on their initial SoC, SoH, and charging rate, confirming that the controller manages each cell’s CC/CV phase independently rather than enforcing a uniform switching point for the entire pack. Regarding temperature, the trajectories of T_c(t) show that cell temperatures increase rapidly in the early stage due to the high 3C current, reaching local peaks near 45°C, and then decrease as the charging current is reduced. This indicates that the controller actively “eases off” when a cell enters a thermally sensitive region to prevent exceeding the 45°C temperature constraint. This demonstrates that even in high-power fast-charging conditions, the algorithm prioritizes electro–thermal safety rather than merely minimizing charging time. The VRC values gradually converge toward zero, indicating that polarization and transient internal resistance effects are diminished as the current tapers in the final phase, meaning that the cells are brought to a stable voltage state before charging completion. Finally, all cells converge to the target SoC of approximately 0.97 after 64 minutes, despite non-uniform initial conditions and time-varying individual current limits. This proves that the controller successfully achieves the overall system energy target within a shortened charging duration. In summary, the simulation of Scenario (c) confirms three key points: (i) the control strategy can operate at high charging currents (3C) while respecting per-cell voltage and temperature constraints; (ii) the transition from CC to CV mode is managed independently for each cell based on its SoH and SoC, rather than using a single switching point for the entire pack; and (iii) the controller achieves the target SoC rapidly (within 64 minutes), demonstrating safe and effective fast charging under high-power, heterogeneous cell conditions. This outcome fulfills the core verification objective of scenario c.
Scenario d: The initial SoC of all cells is 0.3, while their SoH differ, with values of 0.95, 0.9, 0.85, and 0.99 for cells 1 through 4, respectively. The corresponding cell temperatures are 35°C, 30°C, 37°C, and 35°C. The weighting factors associated with safety and cost optimization are increased to γ4 = 1.3 and γ5 = 1.3. The maximum charging current limit is 3C, the maximum allowable cell voltage is γ5 = 1.3, and the maximum allowable temperature is 45°C. The desired final SoC at the end of the charging process is 0.97, and charging begins at 450 minutes. The charging process completes after 62 minutes, and the corresponding results are presented in Fig. 8.
The simulation results of scenario (d) clearly show that in the early charging phase, the cells receive high charging currents for a short duration, after which the current rapidly decreases and does not remain at the 3C level for long. This current reduction is not only intended to protect the weaker cells (for example, the one with SoH = 0.85 and a higher initial temperature) but also to limit the instantaneous power consumption, thereby reducing internal heat generation and conductive losses (I2R) during high electricity price periods. It can be observed that after the initial high-current phase, the controller quickly transitions to lower stepwise current levels instead of maintaining a prolonged high current as in the pure high-power case. This behavior aligns with the cost optimization objective: rather than “charging as fast as possible at any cost,” the controller chooses to “charge adequately, safely, and with lower losses”. From the voltage perspective, the Vc(t) trajectories show that whenever a cell’s voltage approaches the 4.3 V limit, the charging current for that cell is immediately reduced. This mechanism ensures that no cell exceeds the voltage constraint while all continue progressing toward the target SoC of 0.97. At the same time, this voltage regulation helps reduce internal resistance losses since the cells are not driven into over-polarized regions; this is evidenced by the rapid decay and early convergence of the polarization voltage VRC toward zero. The early reduction in polarization indicates that the controller avoids maintaining large currents when the cell is electrochemically stressed—meaning it actively “minimizes losses” by preventing unnecessary ohmic and transient energy dissipation. Regarding temperature, the thermal trajectories of all cells remain below the 45°C limit. The different initial temperatures (ranging from 30°C to 37°C) cause varying heating rates in the early phase, but the peak temperatures of each cell subsequently decline as the current is reduced. This confirms that the increased safety weighting factor (γ4) prompts the controller to reduce current for hotter or more degraded cells (with lower SoH), preventing extended operation in high heat generation zones, thereby mitigating thermal risks and reducing thermal losses. An important observation is that although the controller behaves more “conservatively” in terms of instantaneous power and electricity cost, all cells still converge to approximately SoC = 0.97 within 62 minutes. This means that the total fast-charging duration is only slightly longer than in the case where absolute speed is prioritized. Thus, the controller achieves a reasonable trade-off: energy cost and electro–thermal losses are reduced while still meeting the final SoC target and maintaining voltage and temperature within safety limits. In summary, the simulation results of scenario d demonstrate that the controller not only adjusts current according to cell health and temperature—as in previous scenarios—but also actively optimizes when and how much current should be supplied to minimize losses (reducing I2R and polarization) and overall energy cost throughout the charging cycle. This behavior represents the load-shifting and adaptive charging profile optimization focused on safety–cost priorities that scenario d was designed to verify.
The experimental results
The experimental results for scenarios (a), (b), (c), and (d) are presented in Fig. 10, 11, 12, and 13, respectively.
The experimental results of the four scenarios (a, b, c, and d) show a high degree of consistency between the simulation, numerical computation, and actual measurements, thereby confirming both the accuracy of the coupled electro–thermal model and the practical feasibility of the proposed optimal fast-charging controller. Across all scenarios, the measured trajectories of SoC, cell voltage Vc, charging current Ic, and cell temperature Tc closely follow the simulation results, with maximum voltage deviations below 30 mV and maximum temperature deviations below 1.5°C. In scenario (a), where SoC and SoH are uniform but the initial temperatures differ, the controller automatically allocates charging current according to each cell’s initial temperature, maintaining electro–thermal safety throughout the entire charging process. In scenario (b), where both SoH and temperature vary, the more degraded cells receive reduced current earlier, while healthier and cooler cells maintain higher current levels—clearly reflecting the algorithm’s multi-criteria adaptability. In scenario (c), with the maximum charging current limit increased to 3C, the experimental data clearly capture the phase transition from constant-current (CC) to constant-voltage (CV) charging. Despite the high current level, both voltage and temperature remain stably controlled below 4.3 V and 45°C, demonstrating the controller’s reliability under high-power operation. Meanwhile, in scenario (d), with increased safety and cost weighting factors (γ4 = γ5 = 1.3), the charging current is regulated earlier, and the peak temperature is significantly reduced—indicating load-shifting behavior and reduced energy losses, consistent with the objective of cost optimization and cell longevity preservation.
The “computed” curves (in dark blue or orange) represent results from the optimal control model, while the “experimental” data points (hollow circles) correspond to actual measurements from real cells under the same conditions. The minor deviations observed in the SoC–Vc plots during the initial phase are primarily due to measurement noise and acquisition latency; however, both curves converge to the same final state with less than 2% error. This demonstrates that the electrical model within the controller accurately replicates the cell’s charging characteristics and voltage protection mechanism. The maximum temperature deviation between simulation and experiment is less than 1.5°C, confirming that the coupled electro–thermal model is well-calibrated and capable of reliable real-time prediction. The experimentally measured charging current exhibits slightly larger fluctuations due to the response characteristics of the power supply and sensors but still closely follows the computed current profile, demonstrating the controller’s stability under real operating conditions. Overall, the four experimental scenarios validate that the proposed optimal fast-charging controller operates stably under heterogeneous SoC, SoH, and temperature conditions, ensuring electro–thermal safety, achieving high charging efficiency, and reducing both energy losses and electricity cost—thus confirming the accuracy, robustness, and practical applicability of the proposed control method.
CONCLUSIONS
This paper has proposed an optimal cell-level fast-charging control strategy based on a coupled electro–thermal model and a series–parallel reconfigurable architecture, aiming to ensure electro–thermal safety, adapt to each cell’s SoH, and optimize energy cost according to real-time ToU electricity pricing. The method is formulated as a constrained optimal control problem, in which the charging current for each individual cell is adaptively regulated according to operating conditions, including SoC, SoH, ambient temperature, and instantaneous electricity price. A key innovation of this study is the use of switching variables and projection-type topological constraints to decouple the intrinsic cell dynamics (invariant to connection configuration) from the inter-cell coupling constraints (dependent on the reconfiguration mode). This structure allows the controller to be scalable to large battery packs while maintaining independent cell-level operation. Simulation results across four distinct scenarios — ranging from homogeneous to heterogeneous conditions in SoC, SoH, and temperature, as well as cases with varying current limits and objective function weightings — have demonstrated the stability, flexibility, and safety of the proposed controller. Specifically, the strategy can adaptively allocate charging currents according to temperature and aging levels, smoothly control the CC to CV transition even at high current rates (3C), and perform load shifting to reduce energy losses and electricity costs during high-price periods, all while ensuring that each cell reaches its target SoC within the predefined charging time. Throughout the entire process, all safety constraints—voltage (≤ 4.3 V), temperature (≤ 45°C), and maximum current—are strictly satisfied. Importantly, experimental validation on a hardware setup with four independently controlled cells showed a high degree of agreement between simulation, computation, and measured data, with maximum voltage and temperature errors below 30 mV and 1.5°C, respectively. The controller automatically reduced current for cells with lower SoH or higher temperature, while assigning higher current to healthier and cooler cells, accurately reflecting the dynamic current coordination mechanism under real operating conditions. The measured charging current, voltage, and temperature closely followed the model predictions, confirming both the accuracy of the coupled electro–thermal model and the practical implementability of the proposed optimal controller in a battery management system (BMS). This research contributes a comprehensive approach to safe, efficient, and cost-effective fast charging at the cell level, with scalability to large-capacity battery systems—where heterogeneity in SoC, SoH, and temperature is inevitable. The obtained results not only demonstrate the technical feasibility of the proposed method but also highlight its practical potential for future applications in battery energy storage systems (BESS) and electric vehicles (EVs), where the demand for fast, safe, and energy-efficient charging is becoming increasingly critical. Future work will incorporate converter efficiency and switching losses to evaluate grid-side energy cost at pack scale.
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