In the context of growing global environmental concerns and energy crises, the development of electric vehicles has become a primary focus of societal research. Lithium-ion batteries are widely used in fields such as electric vehicles and energy storage stations due to their compact size and high energy density. However, the voltage and capacity of a single lithium-ion battery are often low, necessitating the configuration of battery packs through series and parallel connections to meet the demands of high-voltage or high-power energy storage systems. Due to factors like manufacturing processes and usage environments, inconsistencies in parameters such as voltage and state of charge (SOC) exist among individual battery cells. As the number of charge-discharge cycles increases, these inconsistencies accumulate, leading to accelerated aging of the batteries and degradation of the overall power performance of the battery pack. Therefore, battery equalization technology is essential to reduce inconsistencies, improve the capacity utilization of the battery pack, and extend its operational lifespan.
Battery equalization is categorized into passive equalization and active equalization based on the type of energy transfer. Passive equalization, also known as energy-dissipative equalization, involves connecting individual battery cells in parallel with resistors to dissipate excess energy as heat. This method is simple in structure but results in significant energy loss. Active equalization, or energy-transfer equalization, utilizes energy storage components such as transformers, inductors, or capacitors to transfer excess energy from high-energy battery cells to low-energy battery cells. For instance, circuits based on single transformers and flyback converters have been proposed, but they suffer from large transformer size, high cost, and design challenges. Other approaches combine Buck-Boost circuits with switched inductors to reduce the number of switches and minimize energy transfer losses, yet they are not suitable for applications with a large number of battery cells and involve complex control strategies. Capacitor-based equalization circuits transfer energy between adjacent cells based on voltage differences, but their equalization capability is limited by the voltage differential between cells. Inductor-based equalization circuits offer higher energy transfer capacity compared to capacitor-based ones, but they are only suitable for energy transfer between adjacent cells, resulting in limited equalization speed and low efficiency. Topologies based on Cuk chopper circuits are applicable only to battery packs with a small number of series-connected cells. When the number of series-connected cells is large, energy must be transferred multiple times to reach low-energy cells, leading to prolonged equalization times. To address these issues, we designed a double-layer equalization scheme for lithium-ion battery packs to effectively improve the equalization speed for battery packs of a certain scale.
The double-layer equalization topology divides n battery cells into m battery groups, with the equalization circuit consisting of a bottom-layer equalization circuit and a top-layer equalization circuit. The bottom-layer equalization circuit comprises m bottom-layer equalization modules, each employing a distributed inductor Buck-Boost circuit. The Buck-Boost circuit is a non-isolated distributed structure with bidirectional current transfer paths, enabling rapid energy transfer between adjacent battery cells through inductive elements. This circuit is easy to implement, has low energy loss, and high equalization efficiency. When the number of battery cells changes, the circuit requires minimal modifications. The top-layer equalization circuit adopts a centralized inductor equalization topology, which achieves rapid energy transfer between any battery groups by controlling the on-off states of double-layer selection switches, shortening the energy transfer path and improving equalization speed. Based on the traditional centralized inductor circuit, double-layer selection switches are used instead of MOSFETs to effectively reduce circuit losses and costs. The entire battery pack shares a single equalizer, simplifying the structure. When the number of battery groups increases, only the number of double-layer selection switches needs to be increased, without altering the complexity of the circuit.
The working principle of the bottom-layer equalization circuit involves direct interactive energy transfer between two adjacent battery cells using a Buck-Boost circuit. In one cycle, the Buck-Boost circuit includes charging and discharging states. Taking the example where battery cell B1 has higher energy than battery cell B2, the equalization process consists of two states: discharging of B1 and charging of B2. During the discharging state, MOSFET Q1 is turned on, forming a loop with battery B1, MOSFET Q1, and inductor L1. Battery B1 charges inductor L1, and the current increases slowly due to the inductive reactance. The zero-state response equation is:
$$ V_1 = R_{on} i_{L1} + L_1 \frac{di}{dt}, \quad t = 0 \rightarrow t_{on} $$
where \( R_{on} \) is the total resistance of the discharge loop, \( i_{L1} \) is the current through inductor L1 (i.e., the equalization current), and \( t_{on} \) is the conduction time of MOSFET Q1. Solving this equation gives:
$$ i_{L1} = \frac{V_1}{R_{on}} (1 – e^{-t \frac{R_{on}}{L_1}}), \quad t = 0 \rightarrow t_{on} $$
When \( t = t_{on} \), MOSFET Q1 turns off, and the inductor current reaches its peak:
$$ i_{peak} = \frac{V_1}{R_{on}} (1 – e^{-t_{on} \frac{R_{on}}{L_1}}) $$
During the charging state, MOSFET Q1 is off, and the freewheeling diode D2 conducts, forming a loop with inductor L1, battery B2, and diode D2. Inductor L1 discharges to battery B2, transferring the stored energy. The inductor current is:
$$ i_{L1} = i_{max} e^{-(t – t_{on}) \frac{R_{off}}{L_1}} – \frac{V_2}{R_{off}} [1 – e^{-(t – t_{on}) \frac{R_{off}}{L_1}}], \quad t = t_{on} \rightarrow t_{off} $$
where \( i_{max} \) is the peak inductor current, and \( R_{off} \) is the total resistance of the charging loop.
The top-layer equalization circuit uses a centralized inductor topology. Assuming battery groups have achieved internal equalization and battery group P1 has higher energy than battery group P2, the equalization process involves two states. First, double-layer switch K1 is closed, and MOSFETs S1 and S3 are turned on. Battery group P1 transfers energy to inductor L, and the inductor current gradually increases from zero. The inductor current is:
$$ i_L = \frac{U_1 t}{L}, \quad 0 < t < DT $$
where \( U_1 \) is the terminal voltage of battery group P1, \( D \) is the duty cycle of the PWM control signal, and \( T \) is the period of the PWM control signal. When \( t = DT \), the inductor current reaches its maximum:
$$ i_{Lmax} = \frac{U_1}{L} DT $$
Second, double-layer switch K1 and MOSFETs S1, S3 are turned off, double-layer switch K2 is closed, and MOSFETs S2, S4 are turned on. Inductor L transfers energy to battery group P2, and the inductor current gradually decreases from its maximum. The inductor current expression is:
$$ i_L = \frac{U_1 DT}{L} – \frac{U_2 (t – DT)}{L}, \quad t = DT < t < T $$
where \( U_2 \) is the terminal voltage of battery group P2. Finally, both double-layer switches K1 and K2 are opened, and the inductor current drops to zero.
To further explore the advantage of the double-layer equalization topology in improving equalization speed, we analyzed it using a graph theory model. Since the double-layer equalization circuit consists of both top-layer and bottom-layer circuits, we introduce the graph theory models for both. The top-layer centralized inductor equalization topology is modeled as shown in the graph, where black circles represent battery groups, white circles represent energy storage elements, and arrows indicate energy transfer between battery groups through energy storage elements. A complete equalization process involves transferring energy from one battery group to another. We calculate the average number of equalization steps required to achieve overall equalization to analyze the system’s overall equalization speed. If the average number of steps is smaller, the equalization speed is faster. The formula is:
$$ T_{avg} = \frac{\sum T_{ab}}{z} $$
where \( T_{ab} \) is the cumulative sum of equalization steps between battery groups, and \( z \) is the total number of permutations between battery groups, given by \( n(n-1) \). For the centralized equalization structure, the required equalization steps are shown in Table 1.
| B1 | B2 | B3 | B4 | B5 | B6 | … | Bn |
|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 | 1 | … | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 | … | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 | … | 1 |
| 1 | 1 | 1 | 0 | 1 | 1 | … | 1 |
| 1 | 1 | 1 | 1 | 0 | 1 | … | 1 |
| 1 | 1 | 1 | 1 | 1 | 0 | … | 1 |
| … | … | … | … | … | … | … | … |
| 1 | 1 | 1 | 1 | 1 | 1 | … | 0 |
The average number of equalization steps for the centralized equalization structure is:
$$ T_{avg} = \frac{\sum T_{ab}}{z} = \frac{n(n-1)}{n(n-1)} = 1 $$
The bottom-layer equalization module uses a Buck-Boost circuit, and its graph theory model is shown in the corresponding figure. The required equalization steps are presented in Table 2.
| B1 | B2 | B3 | B4 | B5 | B6 | … | Bn |
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | … | n-1 | |
| 1 | 1 | 2 | 3 | 4 | … | n-2 | |
| 2 | 1 | 1 | 2 | 3 | … | n-3 | |
| 3 | 2 | 1 | 1 | 2 | … | n-4 | |
| 4 | 3 | 2 | 1 | 1 | … | n-5 | |
| 5 | 4 | 3 | 2 | 1 | … | n-6 | |
| … | … | … | … | … | … | … | … |
| n-1 | n-2 | n-3 | n-4 | n-5 | n-6 | … | … |
The average number of equalization steps for the bottom-layer Buck-Boost circuit is:
$$ T_{avg}’ = \frac{\sum T_{ab}}{z} = \frac{n(n-1)(n+1)/3}{n(n-1)} = \frac{n+1}{3} $$
where n is the total number of battery cells in the equalization system. It can be observed that when \( n \geq 2 \), as the number of battery cells increases, the centralized equalization circuit requires fewer equalization steps than the bottom-layer Buck-Boost circuit, meaning the centralized equalization circuit has a faster equalization speed. For the bottom-layer circuit, when \( n = 2 \), the centralized equalization structure requires the same number of equalization steps as the Buck-Boost circuit. However, considering that a centralized equalization circuit for two battery cells would require more switches than a Buck-Boost circuit for two battery cells, the bottom-layer circuit uses the Buck-Boost circuit for direct interactive energy transfer between two battery cells, ensuring equalization speed while reducing circuit losses and cost.
To effectively improve the equalization speed of lithium-ion battery packs, we designed a hierarchical fuzzy logic control strategy. The entire equalization process for the batteries is as follows. First, bottom-layer intra-group equalization is performed. The SOC values of individual battery cells within each group are compared. The inputs to the fuzzy logic controller are the average SOC value of two battery cells and the SOC difference between them. This allows high-energy battery cells to discharge and low-energy battery cells to charge. When the SOC difference \( \Delta SOC \) reaches the equalization start threshold \( \Delta SOC_{setB} \), the bottom-layer equalization circuit is activated. When \( \Delta SOC \) falls below the equalization end threshold \( \Delta SOC_{offB} \), the equalization circuit stops. To prevent frequent switching actions during equalization, the equalization start threshold \( \Delta SOC_{setB} \) is set to 2%, and the equalization end threshold \( \Delta SOC_{offB} \) is set to 0.1%. Second, top-layer inter-group equalization is conducted. The average SOC of the entire battery pack is calculated. For n battery groups, denoted as p1, p2, etc., the SOC of each group is taken as the average SOC of the individual cells within the group, denoted as \( SOC_{p1} \), \( SOC_{p2} \), etc. The average SOC is:
$$ SOC_{av} = \frac{SOC_{p1} + SOC_{p2} + SOC_{p3} + \ldots + SOC_{pn}}{n} $$
Using the average \( SOC_{av} \) as a criterion, the battery groups are divided into two intervals: Q1 and Q2. Q1 represents the charging battery group interval, where each element’s SOC value is less than \( SOC_{av} \); Q2 represents the discharging battery group interval, where each element’s SOC value is greater than \( SOC_{av} \). Thus, the entire battery group interval is:
$$ Q_{all} = Q_1 + Q_2 = \{a_1, a_2, \ldots, a_n\} + \{b_1, b_2, \ldots, b_n\} $$
where the entire battery group interval includes the charging and discharging battery group intervals. Here, a denotes the SOC values of the charging battery groups, and b denotes the SOC values of the discharging battery groups. The charging battery group interval Q1 is sorted in ascending order based on SOC values to identify the battery group with the minimum SOC value (\( SOC_{min} \)). The discharging battery group interval Q2 is sorted in descending order based on SOC values to identify the battery group with the maximum SOC value (\( SOC_{max} \)). The discharging battery group with \( SOC_{max} \) and the charging battery group with \( SOC_{min} \) are paired as input variables for fuzzy logic control. If \( SOC_{max} – SOC_{min} > \Delta SOC_{setP} \), equalization is initiated; if \( SOC_{max} – SOC_{min} \leq \Delta SOC_{offP} \), inter-group equalization ends. Since the consistency requirement for battery groups typically sets \( \Delta SOC \) at 7%, we have \( 2\Delta SOC_{setB} + 2\Delta SOC_{setP} \leq \Delta SOC \). The inter-group equalization threshold \( \Delta SOC_{setP} \) is set to 1.5%, and the inter-group equalization end threshold \( \Delta SOC_{offP} \) is set to 0.1%.
After identifying the batteries to be equalized, a fuzzy logic control algorithm is used to dynamically adjust the equalization current, effectively shortening the equalization time and enabling rapid battery equalization. The fuzzy logic controller consists of a fuzzifier, a control rule base, an inference engine, and a defuzzifier. First, input variables are converted into fuzzy variables via the fuzzifier. The input variables are \( \Delta SOC \) and \( SOC_{ave} \), calculated as:
$$ \Delta SOC = |SOC_i – SOC_j| $$
$$ SOC_{ave} = \frac{SOC_i + SOC_j}{2} $$
where \( SOC_i \) and \( SOC_j \) represent the SOC values of the discharging and charging batteries or battery groups, respectively. Based on simulation experiments where the maximum SOC is 79% and the minimum SOC is 66%, the range for \( \Delta SOC \) is set to (0, 15%), and for \( SOC_{ave} \) to (0, 80%). After fuzzification, \( \Delta SOC \) and \( SOC_{ave} \) become fuzzy variables u(x) and u(y), each divided into five fuzzy intervals: Very Small (VS), Small (S), Medium (M), Large (L), and Very Large (VL). The output variable \( I_{equ} \) has a range of (0, 3), within the safe operating range of lithium-ion batteries, and its fuzzy variable u(z) is also divided into VS, S, M, L, VL.
Second, the fuzzy variables are sent to the inference engine, which processes them based on the control rule base derived from knowledge and practical experience. In the inference engine, membership degrees for each variable in the five fuzzy intervals are determined using triangular membership functions, chosen for faster computation and more intuitive results. The FLC membership functions for the three variables are as shown in the corresponding figure. The control rule base is established as shown in Table 3.
| \( I_{equ} \) | \( \Delta SOC \) | |||||
|---|---|---|---|---|---|---|
| VS | S | M | L | VL | ||
| VS | VS | S | M | L | L | |
| S | S | M | L | L | VL | |
| \( SOC_{ave} \) | M | M | L | L | VL | VL |
| L | VS | M | L | L | VL | |
| VL | VS | S | M | L | L | |
Third, the fuzzy logic control process is designed using the fuzzy logic design tool in MATLAB software, and the relationship between input and output variables is analyzed to ensure it meets the fuzzy logic rules. The correspondence between the output variable and the two input variables is as shown in the corresponding figure. It can be observed that when \( \Delta SOC \) and \( SOC_{ave} \) are small, the equalization current is also small, preventing over-discharge of the batteries and reducing deviation from the actual equalization current. When \( \Delta SOC \) and \( SOC_{ave} \) are large, the equalization current is larger, effectively shortening equalization time and minimizing battery energy loss, consistent with the rules in the fuzzy logic rule base.
Fourth, the output from the inference engine is a fuzzy variable, not a numerical equalization current value, so defuzzification is performed using the centroid method, which provides smoother and more reasonable results. The formula is:
$$ I_{equ} = \frac{\int z_I \phi(z_I) dz}{\int \phi(z_I) dz} $$
where \( \phi(z_I) \) is the fuzzy variable output from the inference engine. Fifth, the difference between the defuzzified equalization current value and the actual equalization current value is used as the input to a PID controller, and the output PWM control signal controls the switching of the transistors, as shown in the corresponding figure.
To validate the effectiveness of the designed double-layer equalization scheme, we built a corresponding circuit simulation model on the MATLAB/Simulink software platform. We selected lithium-ion batteries of type 18650 with parameters of 3.7 V/3.2 Ah. The initial SOC values of the individual battery cells were set to 79%, 76%, 75%, 73%, 72%, 69%, 68%, and 66%. To further verify the performance of the equalization topology and control strategy, we conducted experiments on both aspects.
For the equalization topology experiment, we compared the designed double-layer equalization topology with a traditional double-layer equalization topology using the maximum value method for equalization control. The SOC value changes are shown in the corresponding figures. In the static equalization experiment, the traditional double-layer equalization topology required 1,737.61 s to complete equalization, while the designed equalization topology required 1,452.06 s, reducing the equalization time by 285.55 s. This indicates that the designed equalization topology improves equalization speed by 16.43% compared to the traditional double-layer equalization topology. Therefore, the designed equalization topology achieves faster equalization.
For the hierarchical fuzzy logic control strategy experiment, we compared it with the maximum value method under three conditions: static, charging, and discharging. In the static equalization experiment, as shown in the corresponding figure, the maximum value method required 1,452.06 s to achieve equalization, while the hierarchical fuzzy logic control strategy required 1,083.75 s, reducing the equalization time by 368.31 s. This corresponds to a 25.36% improvement in equalization speed with the fuzzy logic strategy compared to the maximum value method, and the energy transfer efficiency also increased by 7%. The comparison shows that the hierarchical fuzzy logic equalization strategy performs better.
In the charging equalization experiment, as shown in the corresponding figure, the maximum value method required 1,363.49 s to achieve equalization, while the proposed strategy required 1,080.51 s, reducing the equalization time by 282.98 s. This results in a 20.75% improvement in equalization speed with the proposed strategy compared to the maximum value method. In the discharging experiment, as shown in the corresponding figure, the maximum value method required 1,423.03 s to achieve equalization, while the proposed strategy required 1,106.14 s, reducing the equalization time by 316.89 s, corresponding to a 22.27% improvement in equalization speed. Based on the results from the static, charging, and discharging equalization experiments, it is verified that the hierarchical fuzzy logic control equalization strategy algorithm offers superior equalization speed.
In conclusion, to address the equalization speed issue for series lithium-ion battery packs of a certain scale, we designed a double-layer equalization scheme for lithium-ion battery packs. First, we proposed a double-layer equalization topology and used a graph theory model to verify its superiority in equalization speed. Second, we designed a hierarchical fuzzy logic control strategy to dynamically adjust the equalization current, thereby improving equalization speed. Finally, we built a circuit simulation model and compared it with traditional double-layer equalization topologies. The results show that the designed equalization topology improves equalization speed by 16.43%. Under static, charging, and discharging conditions, compared to the maximum value method, the fuzzy logic control strategy improves equalization speed by 25.36%, 20.75%, and 22.27%, respectively. This validates the effectiveness of the proposed equalization scheme and demonstrates enhanced equalization speed for lithium-ion battery packs. The research contributes to the advancement of battery management systems for electric vehicles and energy storage applications, ensuring longer lifespan and better performance of lithium-ion battery packs. Future work may focus on optimizing the control parameters and extending the scheme to larger-scale battery systems.