# Accelerated SOH Balancing in Lithium-ion Battery Packs Using Finite Set MPC

Arman Oshnoei, Hoda Sorouri, Ashkan Safari, Pooya Davari, Martin Zacho, Anders Dahl Johnsen, Remus Teodorescu

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Arman Oshnoei, Hoda Sorouri, Ashkan Safari, Pooya Davari, Martin Zacho, et al. Accelerated SOH Balancing in Lithium-ion Battery Packs Using Finite Set MPC. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0161 . hal-05108315

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# Accelerated SOH Balancing in Lithium-ion Battery Packs Using Finite Set MPC

Arman Oshnoei, Senior Member, IEEE, Hoda Sorouri, Student Member, IEEE
Ashkan Safari, Member, IEEE, Pooya Davari, Senior Member, IEEE
Martin Zacho, Anders Dahl Johnsen, and Remus Teodorescu, Fellow, IEEE

***Abstract***—State of Health (SOH) balancing is crucial for extending lithium-ion battery life. This paper introduces a Finite Control Set Model Predictive Control (FCS-MPC) method for an 8-cell pack, achieving faster SOH uniformity than traditional methods by dynamically adjusting control actions and optimizing cell usage to reduce degradation.

***Index Terms***—Battery Management System, State of Health, Battery Aging, Active Balancing

## I. INTRODUCTION

The increasing demand for electric vehicles is driving higher deployment of lithium-ion (Li-ion) batteries. However, the manufacturing processes for these batteries—such as mixing, coating, calendaring, slitting, and formation [1]—often fail to achieve consistent quality, even within the same production line [2], [3]. These variations can lead to issues such as overcharging or over-discharging [4], [5], due to what is known as the "shortboard effect" [6], which results in energy loss and reduced capacity. Consequently, cells with lower capacity experience faster degradation, and those in poorer health degrade even more rapidly. These imbalances can lead to premature system failures, resulting in decreased capacity, lower efficiency, and shorter operational lifespans. Therefore, balancing cells with different degradation levels is crucial to maintaining battery efficiency,

This work is funded by Battery Management Systems for Next-Generation Data Centers (BMS-DC) project, granted by Energy Cluster Denmark in 2024. Also, it is funded by Smart Battery Villum Investigator Grant (222860) for Villum Fonden, Denmark.

Arman Oshnoei, Hoda Sorouri, Pooya Davari, and Remus Teodorescu are from the Department of Energy, Aalborg University, Aalborg, Denmark ({aros, hoso, pda, ret}@energy.aau.dk)

Ashkan Safari is from the Mechanical, Automotive and Materials Engineering Department, University of Windsor, Windsor, ON, Canada (ashkansafari@ieee.org, safari8@uwindsor.ca)

Martin Zacho is from the Center of Excellence, Energy Storage, Schneider Electric, Region of Southern Denmark, Denmark (martin.zacho@se.com)

Anders Dahl Johnsen is from Eluminate, Hillerød, Capital Region of Denmark, Denmark (anders.johnsen@eluminate.dk)

which is important in various applications, such as electric vehicles [7]–[14].

Most recent battery balancing approaches are based on the State of Charge (SOC) of the cells. For instance, in [15], the authors propose a cell balancing method for lithium-ion batteries using a deep reinforcement learning approach with switched shunt resistors. This method, utilizing a metaheuristic algorithm, improves battery capacity, reduces SOC variance, and minimizes switching operations. In [16], an automatic balancing strategy for islanded bipolar DC microgrids is proposed, which balances multiple battery units using an adaptive virtual resistance algorithm to ensure optimal balancing and energy support within the microgrid. Additionally, [17] presents a machine learning-driven battery pack reconfiguration technique to enhance SOC balance. Validated through an in-house simulation tool, this method integrates advanced battery models and extended Kalman filtering for accurate SOC estimation. Another approach, a hybrid duty cycle balancing technique proposed in [18], combines duty cycle and cell-to-pack balancing to improve SOC control, accelerating SOC convergence and achieving balance in approximately 400 seconds.

While these SOC-based balancing methods focus on equalizing the charge levels across all cells, ensuring that each cell reaches a similar charge state to prevent undercharging or overcharging, they primarily aim to optimize the short-term performance and immediate energy output of the battery pack. In contrast, methods that focus on the State of Health (SOH) of the cells prioritize long-term health and degradation management by balancing based on the cells' overall condition, including capacity fade and internal resistance. While SOC-based balancing helps maintain uniform energy distribution in the short term, SOH-focused balancing strategies target extending the battery pack's life by preventing weaker cells from degrading faster than healthier ones, making SOH-based balancing increasingly important.

However, most of the existing SOH-based approaches focus primarily on estimating the SOH of
the cells [19]–[25], with relatively few studies presenting effective balancing strategies for managing SOH differences in battery units. Moreover, these balancing techniques are still in the early stages of development and have not yet achieved optimal performance. For example, [22] introduces an SOH balancing control method for modular multilevel-converter-based battery energy storage systems, which analyzes the relationship between SOH, depth of discharge (DOD), and life cycles to establish criteria for power distribution among battery packs. Another study [26] develops an online SOH estimation method for second-life batteries based on the depth of charge and charge rate to balance capacity differences, using a conventional battery as the reference module. Moreover, [27] describes the use of reconfigurable battery packs to address cell imbalances that affect the performance of large battery systems, considering both system reconfigurability and the SOH of individual cells. Lastly, the different AI-based SOC estimation methods, and an intelligent cell balancing methodology are manifested in [28], and [29], respectively.

Although these studies have explored various balancing strategies, a crucial factor that affects their overall effectiveness is the speed at which battery packs reach a balanced state. Faster balancing is particularly advantageous in applications requiring high performance and reliability. Accordingly, this paper proposes a novel SOH balancing method based on Finite Control Set Model Predictive Control (FCS-MPC) for an 8-cell lithium-ion battery pack. The method leverages the FCS-MPC's capability to dynamically adjust control actions, effectively balancing the degradation rates among cells. By selectively bypassing cells that have undergone greater aging, the method reduces their usage frequency, thereby extending the overall lifespan of the battery pack. The effectiveness of the proposed FCS-MPC method is further enhanced by optimizing the prediction horizon, which allows the controller to better anticipate future states and adjust control actions more effectively. Simulation results demonstrate that the proposed FCS-MPC method achieves cell balancing significantly faster than traditional methods, reaching equilibrium in as few as 50 cycles with an appropriately chosen prediction horizon.

## II. BATTERY IMBALANCE AND PROPOSED BMS ARCHITECTURE

### A. Battery imbalance

Typically, a battery storage system is designed with several high-voltage battery packs connected in parallel to achieve a large overall capacity. Within each pack,

<figure><img src="image_2.png" /><figcaption>Fig. 1. Showing the performance of series-connected cells in a battery pack under (A) ideal conditions and (B) real conditions.</figcaption></figure>

multiple cells are connected in series to attain the required voltage. Conventional battery management strategies often treat these series-connected cells as a single “high-voltage battery”, which means the individual states of each cell are not separately managed or controlled. This can lead to significant balancing challenges.

Figure 1 provides a visual comparison of two different scenarios for a set of series-connected cells, emphasizing the effects of imbalances in SOC and SOH. In Figure 1(A), the ideal scenario is depicted. Here, three cells (A1, A2, A3) are connected in series, all having the same SOH and initial SOC. This uniformity allows the cells to reach their full charging capacity simultaneously, as represented by the blue-colored sections that indicate the remaining energy. During operation, the cells discharge evenly, and the BMS maintains a balanced condition, allowing the cells to utilize their entire capacity without exceeding the upper SOC limit (red dashed line) or dropping below the lower SOC limit (yellow dashed line). However, Figure 1(B) shows a more realistic and challenging scenario, where the cells (B1, B2, B3) have different SOHs and initial SOCs. In this case, cell B3 has the lowest SOH and initial SOC, while cell B1 has
a relatively higher SOH and initial SOC. As a result, the charging process must stop as soon as the healthiest cell (B1) reaches its upper SOC limit to avoid overcharging. Similarly, during discharging, the process must halt when the weakest cell (B3) reaches its lower SOC limit to prevent overdischarging. This situation leads to inefficient use of the battery pack's capacity, as not all cells can be fully charged or discharged, causing energy wastage. The presence of cells with varying SOH exacerbates these challenges. Cells with lower SOH tend to degrade faster, while those with higher SOH age more slowly, creating a widening imbalance over time. Consequently, achieving effective SOH balancing becomes crucial to maximizing the battery pack's performance and longevity, especially in applications where maintaining uniform cell health is essential.

## B. Proposed BMS architecture

To address the challenges associated with SOH imbalance, the proposed battery system, illustrated in Figure 2, consists of n cells connected in series. Each cell, referred to as the i-th cell, is integrated with a half-bridge circuit containing two complementary switches. During normal operation, the top MOSFETs (M1, M2, M3) are turned on, and their corresponding bottom ones (M'1, M'2, M'3) remain off. This configuration allows for the overall load current to flow through all cells in series, maintaining the required DC bus voltage. The unique feature of this design is its ability to bypass individual cells without disrupting the total load current. By selectively engaging the appropriate switch, cells with a lower SOH can be bypassed to prevent further degradation, while cells with a higher SOH can bear more of the load. This dynamic adjustment promotes balanced aging across the battery pack, optimizing the usage of each cell according to its health status. This bypass technique naturally results in a pulsed current flow through the cells, a method shown in prior research [10] to potentially enhance the lifespan of Li-ion batteries. Thus, the proposed battery system is designed not only to extend the overall cycle life of the battery pack but also to achieve effective SOH balancing under real-world conditions where cell characteristics may vary.

## III. SOH BALANCING PRINCIPLE

The SOH is commonly determined by assessing the decline in the battery's maximum capacity and the rise in its internal resistance. SOH, when characterized in terms of capacity degradation, can be considered as:

$$SOH = \frac{A_{max}}{A_{rated}} \qquad (1)$$

where $A_{max}$ represents the maximum capacity that can be released in the current state, and $A_{rated}$ denotes the nominal capacity. Another way to express battery SOH is by using the increase in internal resistance, as shown in equation (2).

$$SOH = \frac{R_E - R_C}{R_E - R_N} \qquad (2)$$

In this context, $R_E$ represents the internal resistance at the end of the battery's life, $R_C$ is the current internal resistance, and $R_N$ denotes the nominal internal resistance of a new battery. However, the former requires a lengthy full charge/discharge test and is sensitive to changes in ambient temperature. Therefore, using a more easily measured variable to define SOH could be more practical. In [13, 14], the connection between battery life cycles and aging was studied, showing a nearly linear relationship between the increase in life cycles and capacity degradation. This allows for defining SOH based on life cycles, as discussed in (3) [15, 16].

$$SOH(t) = \frac{C_R}{C_T} = SOH(t=0) - \frac{C_{acu}}{C_T} \qquad (3)$$

That, $C_R$, and $C_{acu}$ are the remaining cycle numbers from the current state to the end of use, as well as the accumulated cycle life manifested until the current status, respectively. $C_T$ is considered as the total number of cycles. The initial SOH value is represented by $SOH(t=0)$. In order to provide a correlation between $C_T$, and DOD, the following can be expressed:

$$C_T = \partial \cdot DOD^{-\epsilon} \qquad (4)$$

In which, $\partial$, and $\epsilon$ are the constants varying for different battery types. They are 694, and 0.795 for a typical lithium-ion battery [6]. Consequently, the new form of SOH can be derived as:

$$SOH = \frac{C_{left}}{C_T} = SOH(t=0) - \frac{C_{acu}}{\partial \cdot DOD^{-\epsilon}} \qquad (5)$$

It is clear that as the DOD increases, the rate of change in the SOH speeds up. Theoretically, by implementing diverse DOD levels on batteries with different levels of health, the SOH of the batteries will converge to a singular curve. This method aids in achieving battery health equilibrium, ensuring a more consistent degradation rate over time. In the next phase, the proposed battery pack is modeled, utilizing 8 individual battery cells:

$$ \left\{ \begin{array}{l} SoH_1(t) = SoH_1(t=0) - \frac{C_{acu,1}}{\partial \cdot DOD_1^{-\epsilon}} \\ \\ \vdots \\ \\ SoH_8(t) = SoH_8(t=0) - \frac{C_{acu,8}}{\partial \cdot DOD_8^{-\epsilon}} \end{array} \right. \qquad (6) $$
The next section will detail the proposed FCS-MPC approach and its application in achieving SOH uniformity among battery cells.

## IV. PROPOSED CONTROL SYSTEM

FCS-MPC is a predictive control method that directly determines the optimal switching states of power electronic devices, such as the MOSFETs used in BMS. Unlike conventional control strategies that rely on continuous control signals, FCS-MPC considers a finite set of discrete control actions, making it highly suitable for systems with binary switching requirements.

In the context of SoH balancing, the proposed FCS-MPC method dynamically adjusts the binary switching states (ON/OFF) of the MOSFETs to achieve uniform degradation across all cells in the battery pack. The main advantage of FCS-MPC lies in its ability to optimize control actions over a prediction horizon, ensuring that each cell's health is managed effectively while minimizing energy losses and thermal stress. By directly controlling the switching actions, the FCS-MPC approach enhances the overall longevity and performance of the battery pack.

The following subsections present the mathematical formulation of the proposed FCS-MPC strategy, including the state update equations, cost function, constraints, and optimization problem.

### A. Mathematical formulation of FCS-MPC

The goal of the proposed FCS-MPC method is to minimize the variance in SOH among cells while reducing the number of switching actions to lower energy losses and thermal stress on the components. The SOH of each cell at the next control cycle $k + 1$ is predicted based on its current SoH, the binary switching state of the MOSFET, and a constant degradation rate:

$$SOH_{i}(k + 1) = SOH_{i}(k) + M_{i}(k) \cdot u_{max} - d \quad (7)$$

where: $SOH_i(k)$ is the SOH of cell $i$ at cycle $k$; $M_i(k) \in \{0, 1\}$ is the binary switching state of the MOSFET controlling cell $i$ at cycle $k$, where $M_i(k) = 1$ indicates that the MOSFET is ON, allowing a balancing action (e.g., charging or bypassing current), and $M_i(k) = 0$ indicates that the MOSFET is OFF, and no balancing action is applied; $u_{max}$ represents the maximum allowable SOH adjustment magnitude per cycle; and $d$ is the degradation rate per cycle, representing the natural degradation of each cell.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Proposed battery system architecture</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Control diagram of SOH equalization for Li-ion batteries.</figcaption></figure>
## B. Cost function

The objective of the FCS-MPC is to minimize a cost function that evaluates the effectiveness of the chosen switching states in achieving SOH uniformity while penalizing excessive switching actions. The cost function is defined as:

$$J = \sum_{t=0}^{N-1} \left( \sum_{i=1}^{M} \left( SOH_i(k+t) - \overline{SOH}(k+t) \right)^2 + \lambda \sum_{i=1}^{M} s_i(k+t) \right) \quad (8)$$

where N is the prediction horizon, specifying the number of future cycles considered in the optimization; L is the number of cells in the battery pack; $SOH_i(k+t)$ is the predicted SoH of cell i at future time step $k+t$; $\overline{SOH}(k+t) = \frac{1}{L} \sum_{i=1}^{L} SOH_i(k+t)$ is the mean SOH across all cells at time step $k+t$; and $\lambda$ is a regularization parameter that penalizes switching actions to reduce energy consumption and extend the lifetime of the MOSFETs.

## C. Constraints

The FCS-MPC is subject to the following constraints to ensure safe operation and binary switching:

$$SOH_{min} \leq SOH_i(k) \leq SOH_{max}, \quad \forall i = 1, \dots, M \quad (9)$$

where $SOH_{min}$ and $SOH_{max}$ are the minimum and maximum allowable SOH values to ensure safe operation. The binary switching state $M_i(k)$ of each MOSFET must be either ON or OFF at any given control cycle $k$, ensuring discrete control actions for balancing purposes. This is expressed mathematically as:

$$M_i(k) \in \{0, 1\}, \quad \forall i = 1, \dots, L \quad (10)$$

Given that the degradation of SOH is affected by the DOD, an additional constraint is introduced to maintain safe operating limits:

$$DOD_{min} \leq DOD_i(k) \leq DOD_{max}, \quad \forall i = 1, \dots, L \quad (11)$$

where $DOD_{min}$ and $DOD_{max}$ are the minimum and maximum allowable DOD values.

At each control cycle $k$, the MPC controller solves the following optimization problem to determine the optimal set of binary switching states $M_k = [M_1(k), M_2(k), \dots, M_L(k)]^T$:

$$M_k^* = \arg \min_{M_k \in \{0,1\}^L} J \quad (12)$$

<figure><img src="image_5.png" /><figcaption>Fig. 4. SOH balancing of each cell using FCS-MPC with a prediction horizon of 1.</figcaption></figure>

where $M_k^*$ is the vector of optimal binary switching states for all cells at cycle $k$ that minimizes the cost function $J$. Once the optimal switching states $M_k^*$ are determined, the SOH for each cell is updated as:

$$SOH_i(k+1) = SOH_i(k) + M_i^*(k) \cdot u_{max} - d \quad (13)$$

where $M_i^*(k)$ is the optimal switching state for cell $i$ at cycle $k$.

## V. SIMULATION RESULTS

Simulations were conducted on a battery pack comprising 8 distinct cells to evaluate the effectiveness of the proposed FCS-MPC method for SOH balancing. The simulations were implemented using the Spyder integrated development environment. The initial SOH values for the cells were set as follows: [0.95, 0.92, 0.97, 0.90, 0.93, 0.88, 0.96, 0.94]. Additionally, the DOD for each cell was initialized to: [32%, 26%, 34%, 36%, 45%, 35%, 30%, 28%]. These values were chosen to represent a realistic scenario where cells in a battery pack may have varying levels of health and discharge, which could affect the overall balancing process. The results of the proposed FCS-MPC method were compared against a conventional mean balancing method. Additionally, further simulations were conducted to compare the performance of the FCS-MPC method with two different prediction horizons, demonstrating how varying the prediction horizon can impact the speed and effectiveness of the balancing process. Figure 4 shows the SoH balancing results using the FCS-MPC method with a prediction horizon of 1 over 300 cycles. The cells gradually converge towards uniform SoH levels, with balancing occurring within the first 100 cycles. While this result demonstrates a significant improvement over traditional methods, the speed of
<figure><img src="image_6.png" /><figcaption>Fig. 5. SOH balancing of each cell using mean balancing method.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Variance of SOH across cells over 300 cycles using FCS-MPC with a prediction horizon of 1.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Variance of SOH across cells over 300 cycles using mean balancing method.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Duty cycles for each cell using FCS-MPC with a prediction horizon of 1 over 300 cycles.</figcaption></figure>

convergence can be further enhanced by increasing the prediction horizon.

Figure 5 presents the results of the traditional mean balancing method over 300 cycles. The SOH values decline steadily, with a noticeable variance in degradation rates among the cells. The cells with initially lower SOH values degrade more quickly, resulting in increased disparity over time. This demonstrates that the mean balancing method is less effective in achieving uniform cell health, particularly compared to the FCS-MPC approach.

Figure 6 illustrates the variance of SoH across all cells using the FCS-MPC method with a prediction horizon of 1. The variance decreases sharply in the initial 50 cycles but takes nearly 100 cycles to reach near-zero levels. This shows that while the FCS-MPC method with a prediction horizon of 1 is effective, there is potential for even faster balancing with a longer prediction horizon.

Figure 7 shows the variance of SOH across cells over 150 cycles for the traditional mean balancing method. The variance decreases gradually, but the rate of reduction is significantly slower than with the FCS-MPC

method. This further emphasizes the superiority of MPC-based control for SOH balancing. Figure 8 highlights the switching actions for each cell using the FCS-MPC method with a prediction horizon of 1. The figure shows frequent switching activity, particularly within the first 100 cycles, as the controller attempts to balance the cells’ SOH. To demonstrate the benefits of increasing the prediction horizon, simulations were extended to use a prediction horizon of 3. Figure 9 shows the SOH balancing results using the FCS-MPC method with a prediction horizon of 3. With a longer prediction horizon, the cells reach a balanced state much faster, achieving significant convergence within the first 50 cycles. This rapid balancing is a result of the controller’s ability to anticipate future states more accurately and optimize the control actions more effectively.

Figure 10 depicts the variance of SOH across all cells for the FCS-MPC method with a prediction horizon of 3. The variance decreases more sharply and reaches near-zero levels within 50 cycles, demonstrating a much faster balancing process compared to a prediction horizon of
<figure><img src="image_10.png" /><figcaption>Fig. 9. SOH balancing of each cell using FCS-MPC with a prediction horizon of 3.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10. Variance of SOH across cells over 300 cycles using FCS-MPC with a prediction horizon of 3.</figcaption></figure>

1. The use of a higher prediction horizon thus provides a more effective balancing strategy.

Figure 11 presents the switching actions for the FCS-MPC method with a prediction horizon of 3. The switching activity is mainly concentrated within the

<figure><img src="image_12.png" /><figcaption>Fig. 11. Duty cycles for each cell using FCS-MPC with a prediction horizon of 3 over 300 cycles.</figcaption></figure>

initial cycles, and there is minimal switching after cycle 50. This reduced switching activity further supports the effectiveness of a longer prediction horizon in achieving faster convergence with fewer switching actions.

By comparing the two prediction horizons, it is evident that a higher prediction horizon allows for faster balancing, enhancing the overall performance of the battery management system. However, this also increases the computational burden, which may affect real-time implementation. Thus, selecting the appropriate prediction horizon requires a balance between the desired speed of SoH balancing and the available computational resources.

## VI. CONCLUSION

This paper proposed a new method for SOH (State of Health) balancing in an 8-cell lithium-ion battery pack using Finite Control Set Model Predictive Control (FCS-MPC). The method adjusts control actions to balance the aging rates among cells by selectively bypassing those with more wear, reducing their use and extending the overall lifespan of the battery pack. Simulation results show that the proposed FCS-MPC method achieves much faster SOH balancing than traditional methods, reaching balance in as few as 50 cycles. The study also examined how changing the prediction horizon affects the performance of the FCS-MPC method. It was found that increasing the prediction horizon improves the speed and effectiveness of the balancing process, with a prediction horizon of 3 leading to faster balance compared to a horizon of 1. However, this improvement comes with a trade-off of higher computational demands, highlighting the need to choose the prediction horizon carefully based on specific system needs. While the proposed method offers significant improvements in SOH balancing, future work could make it even more practical by considering real-world limitations, such as minimum voltage requirements for battery strings or modules and acceptable voltage differences between parallel strings. Taking these factors into account could make the method useful for a wider range of battery systems, ensuring both effective SOH balancing and compliance with operational voltage limits.

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