The equalization process of the proposed equalizer consists of two stages. In the first stage, the energy is transferred among adjacent cell substrings and between two cells in a substring by corresponding outer-layer and inner-layer equalizer units, respectively. It is noted that, when N is odd, the energy can also be transferred between cells BN and BN-1, which are adjacent cells but belong to different substrings. In the second stage, the energy is transferred from the maximum voltage cell to the minimum voltage cell, which is implemented by multiple inner-layer and outer-layer equalizer units. By the first stage equalization process, adjacent cell substrings and two cells in a substring are equalized, but the imbalance among cells in the whole battery string is still large. The second stage equalization can realize the equalization of the battery string with small maximum voltage gap among cells, but its equalization speed is slower. Thus, the proposed equalization control strategy combines the first with second stage equalization, which has advantages of both the first and second stage equalization.
3.1. First Stage Equalization
In the first stage equalization, the inner-layer equalizer and outer-layer equalizer work independently. The operation of the equalizer is based on the voltage gap between two adjacent cells or cell substrings. The equalizer has two predefined equalization thresholds; one predefined threshold ∆
V for the inner-layer equalizer unit and one predefined threshold 2∆
V for out-layer equalizer unit. If the voltage gap is less than the predefined equalization threshold, the corresponding equalizer unit stops working. The working condition of inner-layer equalizer unit can be expressed as
where
VBi is the voltage of cell B
i. The working condition of outer-layer equalizer unit can be expressed as
When both the inner-equalizer unit and outer-layer equalizer unit stop working, the equalization process ends. During this stage, the operation principles of inner-layer and outer-layer equalizer units are similar. For the sake of simplification, take inner-equalizer unit as an example to analyze the working process. Assuming that cell voltage
VBi is higher than cell voltage
VBi+1. The equalization process of this stage includes two modes, i.e., Mode 1 and Mode 2. The current paths of the first stage equalization during different modes are shown in
Figure 4.
Mode 1 [t0–t1]: Bi discharges.
Figure 4a presents the inductor current path of Mode 1. Mode 1 starts when the switch S
i is turned on and the switch S
i+1 is turned off. Then,
VBi is directly applied to the terminal of the inductor
Li, and the inductor current
iLi is built up. During this mode,
Li is charged by cell B
i and
iLi increases linearly. Therefore,
iLi can be expressed as
The energy
Wi is transferred from cell B
i to
Li, and it can be expressed as
In the Formula (6), Di is the duty cycle of the switch Si, and Ts is the switching period.
Mode 2 [t1–t2]: Bi+1 charges.
Figure 4b presents the inductor current path of Mode 2. Mode 2 begins when the switches S
i and S
i+1 are turned off. The inductor current is commutated from S
i to the parasitic diode of S
i+1, and it decreases linearly due to the cell voltage is applied to the inductor in the opposite direction. During this mode, the energy is transferred into cell B
i+1 and
VBi+1 increases slowly.
iLi can be expressed as
When
iL reduces to zero, the energy transportation will be stopped by the parasitic diode of S
i+1. The freewheeling time
Td of
iL can be expressed as
In order to avoid magnetic saturation of inductors, the duty cycle of all switches should be designed to satisfy the following expression
The normal voltage range of lithium-ion battery is 2.8–4.2 V, so the duty cycle of all switches must be less than 0.4 to avoid magnetic saturation.
3.2. Second Stage Equalization
The second stage equalization forms an equivalent equalization path by controlling the conduction of the switches and regulating specific duty cycle to achieve equalization between the maximum voltage cell and minimum voltage cell. The second stage equalization includes three modes: Mode 1, the energy transfers from the maximum voltage cell to outer-layer inductor; Mode 2, the energy flows among outer-layer inductors; Mode 3, the energy transfers from outer-layer inductor to the minimum voltage cell. The paper takes even structure of proposed equalizer as an example to elaborate the operation process of the second stage equalization. It is supposed that the voltage of cell Bi is maximum and the voltage of cell Bj is minimum.
Mode 1: the energy transfers from the maximum voltage cell to outer-layer inductor.
Whether switches work or not in equalization process is related to the position of maximum voltage cell B
i in Mode 1, and the corresponding relationship is shown in
Table 1. The energy transfer process consists of two parts in Mode 1: (a) the maximum voltage cell B
i discharges; (b) the outer-layer inductor
Li+2 charges. The two parts are presented in
Figure 5.
As shown in
Figure 5a, when the inner-layer switch S
i is turned on, the analysis process is the same as the first stage equalization, the energy that stored in
Li can be obtained from Formula (6). When S
i is turned off, the inductor current
iLi flows from the parasitic diode of S
i+1 to the adjacent cell B
i+1. The equalizer unit transfers the excess energy from the highest voltage cell B
i to the adjacent cell B
i+1. On the other hand, as shown in
Figure 5b, when the outer-layer switches S
N+i is turned on, the cell B
i+1 charges the outer-layer inductor
Li+2. The energy
Wri+1 transfers from cell B
i+1 to
Li+2 can be expressed as
If the energy of cell B
i+1 maintains dynamic balance in the process of charging and discharging, the process transfers excess energy from cell B
i to
Li+2, and the energy of cell B
i+1 is unchanged. The corresponding energy relationship can be expressed as
From Formula (11), the duty cycle
DN+i of S
N+i can be obtained as
In addition, the energy
WLi+2 transferred from the cell substring (B
i, B
i+1) to
Li+2 during this process can be expressed as
Mode 2: the energy flows among outer-layer inductors.
Whether switches work or not in equalization process is related to the position of maximum voltage cell B
i and minimum voltage cell B
j in Mode 2, and the corresponding relationship is shown in
Table 2. In addition, the energy transfer process consists of three parts in Mode 2: (a) the outer-layer inductor
Li+2 discharges; (b) the outer-layer inductor
Li+3 charges; (c) the outer-layer inductor
Lj+1 charges. The three parts are presented in
Figure 6.
As shown in
Figure 6a, when the switch S
N+i is turned off,
Li+2 charges the adjacent cell substrings through the parasitic diode of switch S
N+i+1 to realize the energy transfer from
Li+2 to the cell substring (B
i+2, B
i+3). Then, as shown in
Figure 6b, when the outer-layer switch S
N+i+2 is turned on, the cell substring (B
i+2, B
i+3) charges the outer-layer inductor
Li+3. In this process, the energy
WLi+3 transfers from the cell substring (B
i+2, B
i+3) to
Li+3, which can be expressed as
If the energy of cell substring (B
i+2, B
i+3) maintains dynamic balance in the process of charging and discharging, the mode transfers excess energy from cell B
i to
Li+3, and the energy of cell substring (B
i+2, B
i+3) is unchanged in this process. The corresponding energy relationship can be expressed as
From Formula (15), the duty cycle
DN+i+2 of S
N+i+2 can be expressed as
Similarly, owing to the unchanged energy of intermediate cell substrings in this mode, it means that the duty cycle of the corresponding switches can be solved. Meanwhile, the energy that stored in the inductor
Lj-2 can be obtained from Formula (6). When the outer-layer switch S
N+j-4 is turned off, the inductor
Lj-2 charges the cell substring (B
j-2, B
j-1) through the parasitic diode of switch S
N+j-3. Then, as shown in
Figure 6c, when the switch S
N+j-2 is turned on, the cell substring (B
j-2, B
j-1) charges the outer-layer inductor
Lj+1. Similar to the above Formulas (13), (14), (15), duty cycle
DN+j-2 of S
N+j-2 can be expressed as
Mode 3: the energy transfers from outer-layer inductor to the minimum voltage cell.
Whether switches work or not in equalization process is related to the position of maximum voltage cell B
i in Mode 3, and the corresponding relationship is shown in
Table 3. In addition, the energy transfer process consists of two parts in Mode 3: (a) the outer-layer inductor
Lj+1 discharges; (b) the minimum voltage cell B
j charges. The two parts are presented in
Figure 7.
To simplify the analysis, it is assuming that all components is ideal, then the energy loss of the whole equalization process can be ignored. The energy transfers from the cell B
i to the inductor
Lj+1 can be obtained from the Formula (13). As shown in
Figure 7a, when S
N+j-2 is turned off, the inductor current
iLj+1 commutated from S
N+j-2 to parasitic diode of outer-layer switch S
N+j-1, and the outer-layer equalizer unit transfers energy from
Lj+1 to the cell substring (B
j, B
j+1). According to voltage-divider theorem, the energy
Wsj+1 transferred from
Lj+1 to the cell B
j+1 can be expressed as
On the other hand, when the inner-layer switch S
j+1 is turned on, cell B
j+1 charges the inner-layer inductor
Lj. The energy
Wrj+1 transferred from the cell B
j+1 to
Lj can be expressed as
If the energy of cell B
j+1 maintains dynamic balance in the process of charging and discharging, the mode transfers excess energy from the maximum voltage cell B
i to
Lj, and the energy of cell B
j+1 is unchanged in this process. The corresponding energy relationship can be expressed as
From Formulas (18), (19), and (20), the duty cycle
Dj+1 of S
j+1 can be expressed as
As shown in
Figure 7b, when S
j+1 is turned off,
Lj charges the cell B
j through the parasitic diode of inner-layer switch S
j to realize the energy transfer from
Lj to cell B
j.
In fact, the energy transfers from the maximum voltage cell Bi to the minimum voltage cell Bj by the second stage equalization, and the energy of other cells are unchanged. Thus, the second stage equalization decreases the maximum voltage gap between maximum voltage and minimum voltage in the battery string.