Comparison of Linear MPC and Explicit MPC for Battery Cell Balancing Control

Abstract

This paper presents and compares two model predictive control (MPC) approaches for battery cell state-of-charge (SOC) balancing. In both approaches, a linearized discrete-time model that takes into account individual cell capacities is used. The first approach is a linear MPC controller that effectively regulates multiple cells to track a target SOC level while satisfying physical constraints. The second approach is based on explicit MPC implementation to reduce online computation while achieving a comparable performance. The simulation results suggest that explicit MPC can deliver the same balancing performance as linear MPC, while achieving faster online execution. Specifically, explicit MPC reduces the computation time by $37.3\%$ in a five-cell battery example, with the cost of higher offline computation. However, simulation results also reveal a significant limitation for explicit MPC for battery systems with a larger number of cells. As the number of cells increases and/or the prediction horizon increases, the computational requirements grow exponentially, making its application to SOC balancing for large battery systems impractical. To the best of the authors’ knowledge, this is the first study that compares MPC and explicit MPC algorithms in the context of battery cell balancing.

1. Introduction

The electric vehicle industry is regarded as one of the key measures to combat global warming [1]. However, the widespread adoption of lithium-Ion batteries, which is one of the main power sources for EVs, is still held back by issues such as state-of-charge (SOC) and voltage imbalance that can compromise driving range and slow down further development of the industry [2,3,4]. One of the primary causes of these imbalances lies in the cell-level characteristics within battery packs. Packs typically consist of multiple cells connected in series, but due to differences in manufacturing, temperature, internal resistance, and self-discharge rates, imbalance between cells gradually occurs [5,6]. This imbalance reduces the capacity of the entire pack, as the weakest cell limits the overall charge and discharge range. It also decreases efficiency, shortens battery lifespan, and may cause safety risks such as overvoltage or undervoltage in individual cells [7,8]. Therefore, to improve EV driving range, battery cell balancing control has been proposed to reduce differences among cells [7,8,9,10].

To overcome cell imbalance, two main balancing methods are commonly used: dissipative and redistributive. Dissipative balancing is simple but inefficient—it equalizes cell voltages by discharging excess energy from higher-charged cells as heat through resistors, leading to energy loss [11]. In contrast, redistributive (or active) balancing uses power electronic circuits to transfer energy directly from cells with a higher SOC to those with a lower SOC, improving the energy efficiency and preserving the overall pack performance [12,13,14], using balancing circuits [15] such as flyback DC/DC converters [12] or half-bridge converters [16]. However, the redistributive balancing method requires advanced battery management systems to track and regulate cell SOC and voltage. In this paper, the redistributive method is primarily adopted.

The aim of active balancing is to prevent cell undervoltage by maintaining safe voltage levels, with many control methods explored in the literature. For example, ref. [17] discussed simple feedback control and refs. [18,19] implemented rule-based control to monitor and track SOC. Although these studies have shown encouraging results, they usually adopt simple control strategies that fall short in maximizing the utilization of energy stored in battery cells. The authors in [20,21] used heuristic control and the results appear to have been effective. In this paper, model predictive control (MPC), a real-time receding horizon control approach, is investigated as a more effective alternative. Specifically, we apply both linear MPC and explicit MPC for battery cell SOC balancing. The linear MPC relies on online computation to find the optimal balancing currents and requires only minimal memory, whereas the explicit MPC performs offline computation at the expense of significantly higher memory usage. Compared to existing MPC-based balancing [7,8,10,12], we focus on linear MPC to simplify the required onboard computation. Moreover, we provide a numerical analysis on the real-time implementability and scalability for large-scale battery packs. To the best of the authors’ knowledge, this work is the first to focus on a computational comparison between MPC and explicit MPC in the context of battery SOC balancing. We firstly confirm that both linear MPC and explicit MPC can generate the same control inputs for battery balancing. Based on this, we investigate three factors that may affect the number of regions in explicit MPC: cell number, prediction horizon, and control horizon. The results provide numerical insights for future researchers on how to select appropriate cell numbers, prediction horizons, and control horizons when deploying explicit MPC for battery balancing. The results also offer practical insights into the selection of linear versus explicit MPC depending on whether the battery management system (BMS) hardware has more memory or a higher CPU performance.

Our contributions are summarized below.

• A linear MPC controller is designed to regulate the SOC of battery cells to track a target SOC level.
• An explicit MPC controller is also designed to perform the same balancing task, whose performance and computation requirements are compared with the linear MPC.
• To provide insights on the scalability and real-time implementability of explicit MPC, an analysis is conducted on how the offline computation effort for the explicit MPC approach changes as the prediction horizon increases.

The remainder of this paper is organized as follows. Section 2 introduces system model and two MPC formulations, while the system settings and simulation results are presented in Section 3. Section 4 concludes the paper with future work directions.

2. Formulation

2.1. System Model

The serial-connected battery considered here is shown in Figure 1, where N cells are connected to provide current i to the load, e.g., an EV. We consider the conventional hierarchical topology, namely from cells to modules and from modules to pack, as the basis for our simulations. We introduce the state-of-charge (SOC) variable $s_n\left(t\right)\in [0,1]$ for the n-th cell. Its continuous-time dynamics under balancing current $i_n\left(t\right)$ (A) are

$${\dot{s}}_n\left(t\right)=-{\displaystyle \frac{1}{3600C_n}}{i}_n\left(t\right),$$

(1)

where $C_n$ is the cell capacity in Ah; the factor 3600 converts the capacity from Amp-Hour to Amp-Second. Because manufacturing tolerances vary, each $C_n$ may differ.

Stacking all cell SOCs and currents gives

$$\begin{array}{cc}\hfill x\left(s\right)=& \left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ ⋮\\ s_N\left(t\right)\end{array}\right],\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill i\left(t\right)=& \left[\begin{array}{c}{i}_1\left(t\right)\\ i_2\left(t\right)\\ ⋮\\ i_N\left(t\right)\end{array}\right].\hfill \end{array}$$

(3)

With the diagonal matrix

$$M_c=-3600\left[\begin{array}{cccc}{C}_1& 0& \dots & 0\\ 0& C_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & C_N\end{array}\right],$$

(4)

the vector form of (1) is

$$\dot{x}\left(t\right)=M_c^{-1}i\left(t\right).$$

(5)

Using a sampling time $T_s$ and forward Euler, we have

$$x_{k+1}=x_k+T_s{M}_c^{-1}{i}_k,$$

(6)

where $i_k$ is the balancing current vector at time step k. It should be noted that this is different from the main discharge current i shown in Figure 1. Define

$$\begin{array}{}\mathrm{(7a)}& A=I,\hfill \mathrm{(7b)}& B=T_s{M}_c^{-1},\hfill \end{array}$$

where I is the identity matrix, which we define as A to enhance readability and generalization of the model. So that

$$x_{k+1}=A{x}_k+B{i}_k.$$

(8)

For convenience in controller design, we shift SOC by $\overline{s}$, and with a slight abuse of notation:

$$x_k=\left[\begin{array}{c}{s}_1\left(k\right)-\overline{s}\\ s_2\left(k\right)-\overline{s}\\ ⋮\\ s_N\left(k\right)-\overline{s}\end{array}\right],$$

(9)

where $\overline{s}$ denotes the target SOC level for all cells to track. Note that by using $s_n\left(k\right)-\overline{s}$ instead of $s_n\left(k\right)$ as state variables, we essentially reformulate the tracking control problem into a regulation problem, which can be easier for implementation.

With sampling time $T_s$ and number of cells N, the matrices in (7) reduce to the following explicit forms

$$A=\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& 1& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 1\end{array}\right],B=-T_s·\left[\begin{array}{cccc}{\displaystyle \frac{1}{3600C_1}}& 0& \dots & 0\\ 0& {\displaystyle \frac{1}{3600C_2}}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\displaystyle \frac{1}{3600C_N}}\end{array}\right]$$

(10)

Balancing currents are subject to magnitude limits and charge-conservation:

$$\begin{array}{ccc}\hfill i_{min}\le i_n\left(k\right)\le i_{max},& \hfill & \hfill n=1,\dots ,N,\end{array}$$

(11)

$$\begin{array}{c}\hfill \sum _{n=1}^N{i}_n\left(k\right)=0.\end{array}$$

(12)

Constraint Equation (12) enforces overall charge conservation within the pack, meaning the sum of the balancing currents is zero. Note that this is simplified from the power conservation constraint and has been widely used in battery cell balancing literature [8].

Note that [22] implemented multiple estimation methods for lithium-ion batteries, and their results show that, after noise removal, the SOC dynamics are highly linear. Furthermore, since the battery balancing has been studied in literature [13,23,24,25] solely on simulation, our model is sufficient to meet the research objectives. Therefore, to better highlight the computational comparison between linear MPC and explicit MPC, we adopt a simplified battery model in this work.

2.2. Linear MPC

The linear MPC controller aims to regulate the SOC of all cells to a target value $\overline{s}$. At each time step, the controller solves a finite-horizon optimization problem to determine the optimal balancing current sequence. The optimization problem can be formulated as follows.

$$\begin{array}{}\mathrm{(13a)}& {\displaystyle \underset{u}{min}}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \sum _{k=1}^p}\parallel x_k{\parallel }_{Q_x}^2+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{k=0}^{p-1}}{\parallel u_k\parallel }_{Q_u}^2\hfill \mathrm{(13b)}& \mathrm{s}.\mathrm{t}.& \mathrm{system}\mathrm{dynamics}\left(8\right),k=0,1,\dots ,p-1\hfill \mathrm{(13c)}& \hfill & \mathrm{input}\mathrm{constraints}\left(11\right) \mathrm{and} \left(12\right),k=0,1,\dots ,p-1\hfill \mathrm{(13d)}& \hfill & x_0=\mathrm{initial}\mathrm{state},\hfill \end{array}$$

where p is the prediction horizon, $Q_x$ and $Q_u$ are weight matrices for the states and inputs, respectively. The prediction horizon p is how many future steps the controller considers when optimizing. $Q_x$ and $Q_u$ are symmetric weight matrices that penalize state deviations and control effort; they are chosen to be diagonal in our simulations. For more mathematical definitions and derivations of MPC, readers are referred to [26].$u_k$ is the balancing current and $i_k$ introduced in Section 2.1. Based on references [26,27,28] in the MPC field, we present the following derivation: The optimization problem (13) can be formulated as a quadratic programming (QP) problem,

$$\begin{array}{cc}\hfill \underset{u}{min}& {\displaystyle \frac{1}{2}}{u}^{\top }(H_x+M_u)u+f_x^{\top }u\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left\{\begin{array}{c}{M}_{AB}u\le x_{max}-M_{Ak}{x}_0\hfill \\ -M_{AB}u\le -x_{min}+M_{Ak}{x}_0\hfill \\ u\le u_{max}\hfill \\ -u\le -u_{min}\hfill \\ Du\le {\Delta }_{max}-r_u\hfill \\ -Du\le -{\Delta }_{min}+r_u\hfill \end{array}\right.\hfill \end{array},$$

(14)

where

$$\begin{array}{cc}\hfill M_{AB}& =\left[\begin{array}{cccc}B& 0& \dots & 0\\ AB& B& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ A^{p-1}B& A^{p-2}B& \dots & B\end{array}\right],\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill M_{Ak}& =\left[\begin{array}{c}A\\ A^2\\ ⋮\\ A^p\end{array}\right].\hfill \end{array}$$

(16)

In the QP problem (13), $x_{max}$ and $x_{min}$ are the upper and lower bounds of the system states, while $u_{max}$ and $u_{min}$ are the upper and lower bounds of the control inputs, respectively. ${\Delta }_{max}$ and ${\Delta }_{min}$ represent the maximum and minimum changes in control inputs between two time steps. This setting is introduced to ensure more stable control inputs for the system. We denote

$$M_x=\left[\begin{array}{ccccc}{Q}_x& 0& 0& \dots & 0\\ 0& Q_x& 0& \dots & 0\\ 0& 0& Q_x& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & Q_x\end{array}\right].$$

(17)

Moreover,

$$f_x=x_0^{\top }{M}_{Ak}^{\top }{M}_x{M}_{AB},$$

(18)

$$M_u=\left[\begin{array}{ccccc}{Q}_u& 0& 0& \dots & 0\\ 0& Q_u& 0& \dots & 0\\ 0& 0& Q_u& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & Q_u\end{array}\right],$$

(19)

$$r_u=\left[\begin{array}{c}-u_{-1}\\ 0\\ 0\\ ⋮\\ 0\end{array}\right].$$

(20)

2.3. Explicit MPC

For embedded applications with limited computational resources, explicit MPC provides an alternative approach by pre-computing the control law offline [29]. The explicit MPC approach for the same optimization problem transforms the online optimization into a function evaluation problem. The following mathematical derivation is also mainly inspired by [29]. We have

$$f_x=x_0^{\top }{M}_{Ak}^{\top }{M}_x{M}_{AB}=x_0^{\top }F,$$

(21)

It is worth noting that $f_x$ and F are terms that are independent of u. They are intermediate variables introduced to simplify the mathematical derivation and improve readability. The above equation can be rewritten as a finite-horizon problem:

$$\begin{array}{cc}\hfill \underset{u}{min}& {\displaystyle \frac{1}{2}}{u}^{\top }(H_x+M_u)u+x_0^{\top }Fu,\hfill \\ \hfill \mathrm{subject}\mathrm{to}& \left\{\begin{array}{c}{M}_{AB}u\le x_{max}-M_{Ak}{x}_0,\hfill \\ -M_{AB}u\le -x_{min}+M_{Ak}{x}_0\hfill \\ u\le u_{max}\hfill \\ -u\le -u_{min}\hfill \end{array}\right.\hfill \end{array}$$

(22)

Furthermore, denote

$$\begin{array}{cc}\hfill H& =H_x+M_u\hfill \end{array}$$

(23a)

$$\begin{array}{cc}\hfill G& =\left[\begin{array}{c}{M}_{AB}\\ -M_{AB}\\ I\\ -I\end{array}\right]\hfill \end{array}$$

(23b)

$$\begin{array}{cc}\hfill W& =\left[\begin{array}{c}{x}_{max}\\ -x_{min}\\ u_{max}\\ -u_{min}\end{array}\right]\hfill \end{array}$$

(23c)

$$\begin{array}{cc}\hfill S& =\left[\begin{array}{c}-M_{Ak}\\ M_{Ak}\\ 0\\ 0\end{array}\right].\hfill \end{array}$$

(23d)

where

$$H_x=M_{AB}^T{M}_x{M}_{AB}.$$

(24)

Equation (22) can be written as

$$\begin{array}{cc}\hfill \underset{u}{min}& {\displaystyle \frac{1}{2}}{u}^{\top }Hu+x_0^{\top }Fu\hfill \\ \hfill \mathrm{subject}\mathrm{to}& Gu\le W+S{x}_0\hfill \end{array}$$

(25)

the solution of this optimize problem is

$$\begin{array}{}\mathrm{(26a)}& \hfill H{u}^{*}+F^{\top }{x}_0+G^{\top }{\lambda }^{*}& =0\hfill \mathrm{(26b)}& \hfill {\lambda }_i^{*}(G^{i}u-W^i-S^i{x}_0)& =0,i\in \mathcal{A}\hfill \mathrm{(26c)}& \hfill {\lambda }^{*}& \ge 0\hfill \mathrm{(26d)}& \hfill Gu& \le W+S{x}_0\hfill \end{array}$$

where $\mathcal{A}$ is the set of active constraints, i.e., constraints that the equality holds, corresponding to the optimal solution ${\mathbf{u}}^{*}$. Then, the KKT condition above gives rise to the following linear system:

$$\left[\begin{array}{cc}H& {\left(G^{\mathcal{A}}\right)}^{\top }\\ G^{\mathcal{A}}& 0\end{array}\right]\left[\begin{array}{c}{\mathbf{u}}^{*}\\ {\lambda }^{*}\end{array}\right]=\left[\begin{array}{c}-F^{\top }{x}_0\\ W^{\mathcal{A}}+S^{\mathcal{A}}{x}_0\end{array}\right]$$

(27)

Since H is positive definite, we have

$${\mathbf{u}}^{*}=-H^{-1}\left({\left(G^{\mathcal{A}}\right)}^{\top }{\lambda }^{*}+F^{\top }{x}_0\right).$$

(28)

Therefore,

$${\mathbf{u}}^{*}=K\left(\mathcal{A}\right)x_0+g\left(\mathcal{A}\right),$$

(29)

where

$$\begin{array}{c}\hfill K\left(\mathcal{A}\right)=H^{-1}{\left(G^{\mathcal{A}}\right)}^{\top }{\left(G^{\mathcal{A}}{H}^{-1}{\left(G^{\mathcal{A}}\right)}^{\top }\right)}^{-1}\\ \hfill ·\left(S^{\mathcal{A}}+G^{\mathcal{A}}{H}^{-1}{F}^{\top }\right)-F^{\top }\end{array}$$

(30)

$$g\left(\mathcal{A}\right)=H^{-1}{\left(G^{\mathcal{A}}\right)}^{\top }{\left(G^{\mathcal{A}}{H}^{-1}{\left(G^{\mathcal{A}}\right)}^{\top }\right)}^{-1}{W}^{\mathcal{A}}$$

(31)

The optimal control to be implemented is given by

$$\begin{array}{cc}\hfill u^{*}& =[I0\cdots 0]{\mathbf{u}}^{*}\hfill \\ \hfill & =[I0\cdots 0]K\left(\mathcal{A}\right)x_0+[I0\cdots 0]g\left(\mathcal{A}\right)\hfill \end{array}$$

(32)

where $K\left(\mathcal{A}\right)$ is a matrix whose first $n_u$ rows define the gain $K_A$, and $g\left(\mathcal{A}\right)$ is a vector whose first $n_u$ entries define $g_A$. Then, we have the following affine feedback control law:

$$u^{*}=K_A{x}_0+g_A.$$

(33)

The main advantage of explicit MPC is the reduction in online computational requirements, making it suitable for real-time implementation on embedded systems. However, as will be seen shortly, the offline computation increases exponentially with the state dimension and prediction horizon length, which is a significant limitation for high-dimensional systems. In terms of memory requirements, linear MPC computes optimal control online, so the information it stores grows polynomially with respect to the prediction horizon, whereas explicit MPC must store the control law and inequalities for many regions. As the number of regions grows exponentially with respect to the prediction horizon, explicit MPC often needs much more memory.

3. Simulation Setup and Results

In this section, the numerical settings of the system along with the simulation results are presented. Both MPC controllers are applied to the same battery system.

3.1. System Settings

Both the linear and explicit MPC controllers are configured using the same system setup to ensure consistency and simplify comparison. The battery system consists of five cells, each with a capacity of $4.1$ Ah. In many battery balancing studies, the initial imbalance deviations are larger. For example, ref. [30] performs balancing during charging, where the initial SOC difference between the maximum and minimum can reach 0.3. References [23,31] conduct balancing during discharging, with selected initial SOC deviations of about 0.1. Authors in [32] also provide an example under worst case conditions, where the initial SOC deviation reaches 0.45. In our simulation, initial SOC deviation is set to 0.2. The initial state $x_0$ is given by:

$$x_0={\left[\begin{array}{ccccc}-0.1& -0.05& 0.1& 0.05& 0\end{array}\right]}^T,$$

(34)

with the target SOC level given by $\overline{s}=0.5$. In other words, tracking x to 0 actually corresponds to tracking the SOC to 0.5, and in this case, the range of x is between $-0.5$ and $0.5$. The sampling time $T_s$ is set to 20 s, and the total simulation duration is 500 min. For the weights, we test several different values in the linear MPC and select the set that yields a relatively smooth balancing current. The MPC weights are chosen as 10 for output tracking, $0.1$ for input magnitude, and 0 for input rate, emphasizing accurate SOC regulation while minimizing control efforts. The values of $i_{max}$ and $i_{min}$ are $0.3$ and $-0.3$, respectively.

3.2. Simulation Results

Figure 2 and Figure 3 present cell SOC deviations and balancing currents for the linear MPC controller. Specifically, Figure 2 presents results with a prediction horizon $P=10$ and control horizon $m=10$, while Figure 3 has a prediction horizon $P=4$ and control horizon $m=3$. As can be seen, it takes approximately 200 min to complete the battery SOC balancing. We can also observe that the system reaches balancing faster when the prediction horizon is longer. It should be noted that once the system finishes balancing, the power converter stops operating. This can also be observed from the balancing current trajectory in Figure 2, Figure 3 and Figure 4; after the cell SOCs return to 0.5, the balancing current drops to 0.

Figure 4 presents the cell SOC deviations and balancing current for an explicit controller with a prediction horizon $P=4$ and control horizon $m=3$. Note that the weights for the cost function is same as for the linear MPC discussed above. It can be observed that the trend and converge time are the same as Figure 3 for the linear MPC, demonstrating that the explicit MPC can deliver the same performance, even though a large amount of computation is moved to offline.

3.3. Computation Analysis

The offline computation for explicit MPC increases exponentially with the prediction horizon due to the rapidly growing number of regions in the state-space partition. The processor of our computation platform is 13th Gen Intel(R) Core(TM) i9-13900F and for our 5-cell battery balancing problem with prediction horizon $p=4$ and control horizon $m=3$, the number of regions is 16,807 and the offline computation time is 502 s. For online computation, the explicit MPC requires 0.1637 s, while linear MPC (with the same p and m) requires 0.2610 s. It can then be observed that the online computation time decreases $37.3\%$.

When $p=5$ and $m=4$, the number of regions is 59,049, which is a huge computation burden. In fact, when $m=p$, the complexity follows approximately $O\left(2^{2(n_x+n_u)p}\right)$, where $n_x$ is the state dimension, $n_u$ is the control input dimension, and p is the prediction horizon. In the cell balancing problem studied in this paper, $n_x=n_u=N$, where N is the number of cells in a battery pack. Then, we have that the complexity follows approximately $O\left(2^{4Np}\right)$.

Table 1. Performance metrics for different number of cells ($p=m=3$).

N	2	3	4	5	6
Regions	49	343	2401	16,807	117,649
Offline Time	0.079 s	1.747 s	30.927 s	479.361 s	6892.412 s
Online Time	0.109 s	0.114 s	0.115 s	0.116 s	0.637 s

shows the computation performance when $p=m=3$ with the number of battery cells N ranging from 2 to 5, Figure 5 plots the number of regions and offline computation time with respect to N. In a lower dimension system where N equals to 2, 3, or 4, the offline computation time is negligible. When the N is equal to 5, the offline computation time rises to 8 min. When the N increases to 6, the offline computation time rises abruptly to about 115 min, which is $14.3$ times larger than the case when $N=5$.

Table 2. Performance metrics for varying prediction horizons ($p=m$) with $N=5$.

$\mathit{p}=\mathit{m}$	2	3	4	5	6
Region	3125	16,807	59,049	161,051	371,293
Offline Time	47.036 s	479.361 s	3122.997 s	15,869.483 s	60,357.946 s
Online Time	0.118 s	0.116 s	0.140 s	0.147 s	0.162 s

and Figure 6 present the computation performance results when the system dimension remains unchanged and the prediction and control horizon increase gradually, which also confirm the exponential increase, but is slightly manageable compared with the number of battery cells. Since all systems use $N=5$, each case is the same MPC problem with different prediction and control horizons. When $p=m=2$ or 3, the online computation times are almost the same. As n and m increase, the online time also increases to some extent.

In Table 1, the memory usage of the five battery packs with different dimensions N is 9 KB, 37 KB, 287 KB, 2862 KB, and 26,528 KB. We can see that as the number of cells increases, the memory required for the explicit approach grows exponentially, indicating that deploying explicit MPC in high-dimension battery systems is not feasible. Finally, it is worth noting that the online computation time required by explicit MPC stays relatively stable when the system dimension changes. Therefore, if the offline computation is affordable with computing devices, such as high-performance computing clusters, then the real-time implementation should not be an issue for production-grade microcontrollers.

4. Conclusions

This paper explores two model predictive controla (MPCa), i.e., linear MPC and explicit MPC, for redistributive battery cell balancing in a serial-connected battery pack. A linearized discrete-time model is derived, taking into account individual cell capacities and enforcing charge conservation. Both controllers are implemented under identical system settings to ensure a fair comparison. Simulation results show that both MPC controllers successfully achieve SOC balancing. The linear MPC, although requiring online optimization at each time step, provides consistent performance with a moderate computational cost. In contrast, the explicit MPC achieves a faster online execution, reducing the onboard computation time by $37.3\%$, making it suitable for battery systems with a small number of cells. This set of experiments also highlights that, even though explicit MPC shifts most computations offline, the control performance remains comparable to that of linear MPC. However, explicit MPC incur a substantial offline computational burden as the prediction horizon and/or number of cells increase. In fact, the exponential growth in offline efforts may limit the scalability of explicit MPC in battery systems with a large number of cells. The computation analysis notes that although both exhibit exponential growth, the increase in system dimension has a greater influence on the offline computation time compared to the increase in prediction and control horizons. Future work will scale up the simulation with more cells in the battery pack and explore methods to reduce the offline complexity of explicit MPCs, such as region merging or approximate controller synthesis. Moreover, we plan to improve the control strategy with further simulation validation and extend the model to a more realistic system by employing voltage-based SOC estimation. An experimental analysis will also be conducted to assess the performance of the current control design. Another direction is to develop a hybrid control framework that switches between explicit MPC and linear MPC to balance offline and online computation. In addition, we will investigate reduced-order MPC for this system and compare its performance against conventional explicit MPCs. Finally, building on these algorithm developments, we aim to deploy the controllers onto real hardware platforms for experimental validation.