A Fast Multi-Switched Inductor Balancing System Based on a Fuzzy Logic Controller for Lithium-Ion Battery Packs in Electric Vehicles

Based on a low cost multi-switched inductor balancing circuit (MSIBC), a fuzzy logic (FL) controller is proposed to improve the balancing performances of lithium-ion battery packs instead of an existing proportional-integral (PI) controller. In the proposed FL controller, a cell’s open circuit voltages (OCVs) and their differences in the pack are used as the inputs, and the output of the FL controller is the balancing current. The FL controller for the MSIBC has the advantage of maintaining high balancing currents over the existing PI controller in almost the entire balancing process for different lithium battery types. As a result, the proposed FL controller takes a much shorter time to achieve battery pack balancing, and thus more pack capacity can be recovered. This will help to improve the pack performance in electric vehicles and extend the serving time of the battery pack.


Introduction
The wide usage of gasoline vehicles leads to the massive emission of greenhouse gases such as carbon dioxide.Electric vehicles (EVs) are promising to reduce this emission.In EVs, lithium based batteries have become the dominant energy source due to their high-energy density, long cycle life, and low self-discharge rate [1,2].The voltage of a single cell is typically less than 4.2 V.They are usually connected in series to achieve high voltage output (around 300 V) and meet the power requirements of EVs.Such series connection may lead to a voltage or state of charge (SOC) imbalance over long periods of charging and discharging operation since the cells in the pack have different self-discharge rates and charging and discharging efficiencies.In such an imbalanced battery pack, the weakest cell determines the pack capacity.A balancing system is required to improve the pack capacity.
Battery balancing systems are generally categorized into two groups; passive and active [3].Passive balancing systems turn extra energy from the cells with higher voltage or SOC into heat through shunt resistance at a low implementation cost.Owing to thermal concerns, the balancing current has to be small, resulting in low balancing speed.Active balancing systems transfer charge or energy within a pack through balancing circuits to maximize the pack capacity.These active balancing systems can potentially have high balancing currents and high balancing speeds.
In IBCs, two families of BCs are widely used [32]; Cuk converter-based BCs (CukBCs) [18][19][20][21][22][23] and buck-boost converter-based BCs (Buck-BoostBCs) [24][25][26][27][28]. Generally, there are more components in the CukBCs than the Buck-BoostBCs.Among the Buck-BoostBCs, the multi-switched inductor BC (MSIBC) is the simplest to date, requiring only one MOSFET and one inductor for each cell [24].This paper focuses on the cost, the speed, and the final recovered pack capacity.From the cost aspect, the MSIBC was chosen.A FL controller designed for the MSIBC in this study has the merits of a faster balancing speed and the ability to recover more pack capacity compared to the PI controller in the literature [25].In the PI controller based balancing system, the balancing current is adjusted to the highest amplitude because of the largest voltage difference of the cells, which occurs at the beginning of the balancing process.The balancing current declines quickly when the voltage differences between the cells become smaller in the process of balancing.The PI controller with constant gain cannot adjust its balancing current adaptively, which severely affects its balancing performance.
In this paper, a FL controller is proposed to control a MSIBC balancing system instead of the existing PI controller.The FL controller significantly improves the balancing speed and increases the pack capacity.As a result, the FL controller for the low cost MSIBC balancing system can obtain more pack capacity and thus extend the serving time of the battery pack in an EV.The FL controller was applied to the CukBC [18,20,22,23] and the Buck-BoostBCs [33] that have more components and higher implementation costs.In the MSIBC, a proposed FL controller was designed based on the characteristics of this balancing circuit and the batteries.The FL controller changes the off-state durations of MOSFETs to adjust the amplitudes and directions of balancing currents and can maintain high balancing currents during most of the balancing process and achieve a high balancing speed.
The following parts of this paper are arranged as follows.In Section 2, the MSIBC and the operational principle of the MSIBC, with the open circuit voltage (OCV) as a balancing criterion, are explained.The design of the FL controller is explained in Section 3. Section 4 describes the experimental results with the FL and PI controllers.These are discussed in Section 5.The conclusions and suggestions for future work are provided in Section 6.

Multi-Switched Indictor Balancing Circuit
Figure 1 shows a MSIBC-based balancing system [25].It mainly consists of a MSIBC and a FL controller.The MISBC is operated in different modes.In each mode, one cell is disconnected from the BC.The mode duration, namely, the MOSFET off-state duration, affects the direction and amplitude of the balancing currents.The FL controller is implemented in the National Instruments (NI) data acquisition platform and the FPGA module.The membership functions of the FL controller are designed based on the characteristics of both the balancing circuit and the battery cell.
In the following, a four-cell battery pack is taken as an example to explain the MSIBC, which is comprised of three inductors and four power MOSFETs.There are four control signals for the MOSFETs, as shown in Figure 2, resulting in four working modes, as shown in Figure 3.A short dead-time (DT) is inserted in each mode to avoid short circuits.
Assuming that the MSIBC operates in the steady state, according to Kirchhoff's current law, the relationship between the cell balancing current and the mode duration can be expressed as: Energies 2017, 10, 1034 3 of 16 where d x represents the mode duration as percentage of one period for mode x (x = 1, 2, 3, 4) and T is the switching period.It can be seen from Equation (1) that the cell balancing current is decided by both the inductor current and the mode duration.
According to the volt-second balance, the relationship between the cell voltage and the mode duration can be expressed as [34]: In relation to the inductor resistance (R L ) and the MOSFET conduction resistance (R on ) in the MSIBC, the relationship between the inductor balancing current and the MOSFET mode duration is given by [24]: where, V p is the pack voltage.
Energies 2017, 10, 1034 3 of 18     Assuming that the MSIBC operates in the steady state, according to Kirchhoff's current law, the relationship between the cell balancing current and the mode duration can be expressed as: where x d represents the mode duration as percentage of one period for mode x (x = 1, 2, 3, 4) and T is the switching period.It can be seen from Equation (1) that the cell balancing current is decided by both the inductor current and the mode duration.
According to the volt-second balance, the relationship between the cell voltage and the mode duration can be expressed as [34]:

Balancing Principle
The MSIBC uses an inductor to transfer energy in a cell by switching MOSFETs in sequence.The direction of energy transfer is decided by both the cell voltage and the mode duration.By changing the ratio of mode-duration to cell-voltage (RMV), the cell balancing current direction can be controlled.If the RMVs of all the cells are the same, the average cell balancing current is zero.The voltages of all the cells remain the same; thus no balancing action on the cells takes place.If the RMV of one cell is higher than that of the others, this cell is charged.If the RMV of one cell is lower than that of others in the pack, this cell is discharged.If cell one in the pack is taken as an example, it can be mathematically expressed as: No Cell is balanced Cell one is charged Cell one is discharged (4) Based on this principle, the MSIBC can be controlled by constant equal mode duration, and the simulation results are shown in Figure 4.The balancing current rises to the peak, which may be too high for the battery, and then drops quickly when the voltage difference decreases.To improve the balancing performance, the FL controller is proposed to control the MSIBC by changing mode durations, leading to the possibility of regulating the amplitude and the direction of the balancing current across the whole balancing process.

Balancing Criterion oc V Estimation
The goal of the balancing operation is to equalize the cells' SOCs.The measured battery terminal voltage is only a rough indicator of the SOC as the terminal voltage consists of three components; namely, open circuit voltage (OCV), polarization voltage, and voltage drop across internal resistances [9,25,35].Of these, only the OCV directly reflects the SOC.To find the OCV, a commonly-used equivalent circuit model (ECM) of a battery is adopted, as shown in can be ignored [9].Therefore, the OCV can be approximately calculated by: ( ) where oc V represents the OCV.diff in R R + is obtained from the experiment and is taken as a constant during the balancing process.

Balancing Criterion V oc Estimation
The goal of the balancing operation is to equalize the cells' SOCs.The measured battery terminal voltage is only a rough indicator of the SOC as the terminal voltage consists of three components; namely, open circuit voltage (OCV), polarization voltage, and voltage drop across internal resistances [9,25,35].Of these, only the OCV directly reflects the SOC.To find the OCV, a commonly-used equivalent circuit model (ECM) of a battery is adopted, as shown in Figure 5.In this ECM, V b and I represent the battery terminal voltage and current, respectively.The resistance R in describes the instantaneous voltage drop.A parallel branch with a resistance R di f f and a capacitance C di f f represents the dynamics of diffusion.When the sample rate of the voltage is high, the diffusion effect of C di f f can be ignored [9].Therefore, the OCV can be approximately calculated by: where V oc represents the OCV.R di f f + R in is obtained from the experiment and is taken as a constant during the balancing process.
Energies 2017, 10, 1034 6 of 16 by: ( ) where oc V represents the OCV.diff in R R + is obtained from the experiment and is taken as a constant during the balancing process.

Fuzzy Logic Controller Design
The fuzzy theory can process imprecise information by degree of membership and is a universal approximator for non-linear mapping between input vectors and scalar outputs in terms of firing fuzzy rules to some degree [36,37].A fuzzy logic (FL) controller is the application of the fuzzy theory.Its basic structure consists of three parts; fuzzification, fuzzy inference, and defuzzification (see Figure 6).In fuzzification, crisp inputs are fuzzified into linguistic variables using membership functions (MFs).In fuzzy inference, there are two parts, an inference engine and a rule base.The inference engine decides the fuzzy logic operations, and the rule base stores the control rules based on expert knowledge.The linguistic control outputs are generated by the inference engine.In defuzzification, the linguistic outputs are converted back to the crisp output using the centre of gravity method.
The FL controller of the MISBC is designed to control the inductor currents (see Figure 6).The inductor currents are then converted into the mode duration (d) through the balancing circuit model of the pack to turn MOSFETs on and off in sequence.The mode duration controls the amplitude and direction of the balancing currents.

Fuzzy Logic Controller Design
The fuzzy theory can process imprecise information by degree of membership and is a universal approximator for non-linear mapping between input vectors and scalar outputs in terms of firing fuzzy rules to some degree [36,37].A fuzzy logic (FL) controller is the application of the fuzzy theory.Its basic structure consists of three parts; fuzzification, fuzzy inference, and defuzzification (see Figure 6).In fuzzification, crisp inputs are fuzzified into linguistic variables using membership functions (MFs).In fuzzy inference, there are two parts, an inference engine and a rule base.The inference engine decides the fuzzy logic operations, and the rule base stores the control rules based on expert knowledge.The linguistic control outputs are generated by the inference engine.In defuzzification, the linguistic outputs are converted back to the crisp output using the centre of gravity method.
The FL controller of the MISBC is designed to control the inductor currents (see Figure 6).The inductor currents are then converted into the mode duration (d) through the balancing circuit model of the pack to turn MOSFETs on and off in sequence.The mode duration controls the amplitude and direction of the balancing currents.

Membership Functions
A membership function (MF) defines how each point of the variables in the input space is mapped to a degree of membership between 0 and 1.There are two fuzzy input variables, cell OCV ( oc V ) and cell OCV difference ( d V ), and one fuzzy output variable, inductor current ( L I ).The OCV difference of cell i is defined as: The membership selection is decided by the characteristics of the battery pack and the requirement of the balancing system.A LiFePO4 battery is used as an example to explain the

Membership Functions
A membership function (MF) defines how each point of the variables in the input space is mapped to a degree of membership between 0 and 1.There are two fuzzy input variables, cell OCV (V oc ) and Energies 2017, 10, 1034 7 of 16 cell OCV difference (V d ), and one fuzzy output variable, inductor current (I L ).The OCV difference of cell i is defined as: The membership selection is decided by the characteristics of the battery pack and the requirement of the balancing system.A LiFePO 4 battery is used as an example to explain the membership design.The OCV of the LiFePO 4 battery is described by five linguistic variables, i.e., VS (very small), S (small), M (medium), L (large), and VL (very large), in triangular and trapezoidal forms, as shown in Figure 7. Since the LiFePO 4 battery has a very flat voltage around 3.3 V, the MFs concentrate on 3.3 V.The OCV difference V d and the inductor current I L are described by the same five linguistic variables as the OCV, i.e., NV (negative large), N (negative), Z (around zero), P (positive), and PV (positive large), as shown in Figures 8 and 9, respectively.The V d range covers all the possible cell voltage differences in the pack.However, in most cases, these difference values are between −0.1 V and 0.1 V, which is shown in the enlarged figure.In the control process, the output current should be near zero when the OCV difference is near zero to improve the efficiency and stability, and the output current should increase quickly when the OCV difference is above the threshold to obtain a fast balancing speed.Since the sum of the MFs at one point is not required to be one in the FL system [38], the use of Gaussian MF can achieve better performance than the use of the triangle MF; thus Gaussian MF is applied to the linguistic variable Z for the MFs of V d and I L .It can make the output inductor current drop quickly to near zero when V d is less than 20 mV, reducing the balancing loss and the chance of divergence.
Energies 2017, 10, 1034 8 of 18 all the possible cell voltage differences in the pack.However, in most cases, these difference values are between −0.1 V and 0.1 V, which is shown in the enlarged figure.In the control process, the output current should be near zero when the OCV difference is near zero to improve the efficiency and stability, and the output current should increase quickly when the OCV difference is above the threshold to obtain a fast balancing speed.Since the sum of the MFs at one point is not required to be one in the FL system [38], the use of Gaussian MF can achieve better performance than the use of the    all the possible cell voltage differences in the pack.However, in most cases, these difference values are between −0.1 V and 0.1 V, which is shown in the enlarged figure.In the control process, the output current should be near zero when the OCV difference is near zero to improve the efficiency and stability, and the output current should increase quickly when the OCV difference is above the threshold to obtain a fast balancing speed.Since the sum of the MFs at one point is not required to be one in the FL system [38], the use of Gaussian MF can achieve better performance than the use of the triangle MF; thus Gaussian MF is applied to the linguistic variable Z for the MFs of d

Fuzzy Logic Controller
The design of the proposed FL controller is explained and the procedure is summarized as follows [39,40].
Step 1: Determination of fuzzy rules.The rules describe the knowledge about the behaviour of a complex system.These rules with two inputs can be generally expressed as: where n is the total number of the fuzzy rules and k R denotes the k th rule.Step 2: Fuzzification of the input variables.Fuzzy sets for the variables are determined by the MFs.The crisp inputs are converted into a degree of membership between 0 to 1 using the following equation:

Fuzzy Logic Controller
The design of the proposed FL controller is explained and the procedure is summarized as follows [39,40].
Step 1: Determination of fuzzy rules.The rules describe the knowledge about the behaviour of a complex system.These rules with two inputs can be generally expressed as: where n is the total number of the fuzzy rules and R k denotes the kth rule.A k 1 , A k 2 (k = 1, 2, . . ., n), and B k (k = 1, 2, . . ., n) are the fuzzy sub-sets of X 1 , X 2 , and Y, respectively.In this study, X 1 and X 2 are the input linguistic variables V d and V oc , respectively, and Y is the output linguistic variable I L .The detailed fuzzy rules are given in Table 1.These rules describe the relation between the input and output of the FL controller based on the expert knowledge of LiFePO 4 battery balancing.There are a total of 25 rules.The following are two of the rules.Step 2: Fuzzification of the input variables.Fuzzy sets for the variables are determined by the MFs.The crisp inputs are converted into a degree of membership between 0 to 1 using the following equation: where x i is the input value of ith input variable, and µ A k j (x) is the fuzzy MF of the input linguistic variables V d and V oc .
Energies 2017, 10, 1034 9 of 16 Step 3: Fuzzy inference.This step applies fuzzy rules to map the given inputs to an output fuzzy set with the FL operations.Maximum-minimum composition is utilized.First, the output fuzzy set of each rule is computed with an implication operation, and then all the output sets are combined into a single fuzzy set with an aggregation operation.Mathematically, the process can be written as: where u B (I L ) denotes the aggregated output fuzzy set.
Step 4: Defuzzification of the output.This step converts the inference fuzzy output set to the crisp inductor current (I L ).The centre of gravity method is used for defuzzification.It is given by: Step 5: Obtain the mode durations for each cell.With the inductor currents, the desired mode durations can be calculated by:

Adaptivity
The FL controller has good adaptivity due to a membership design that combines triangle MF and Gaussian MF.The inductor current can be regulated based on the feedback of V oc and V d .For example, if V d is large, a high inductor current is desired; if V d is small, a low inductor current is desired.The FL controller can adaptively change the inductor current for different situations.
Considering only V d as the input with the constant OCV at 3.3 V, the equivalent gain and the corresponding inductor current are shown in Figures 10 and 11, respectively.When V d is positively or negatively large, the equivalent gain is small to keep the inductor current under the limit.As V d becomes small, the equivalent gain increases.When V d is about 25 mV, the equivalent gain reaches its peak value of 70, corresponding to the peak inductor current of 3 A. When V d continues decreasing to 15 mV, the equivalent gain starts to decrease sharply to less than 20.At the same time, the inductor current declines to near zero.This prevents the divergence of the already-balanced cells.For comparison, the gain of the PI controller is shown by the dotted line in Figure 10, which has the same peak inductor current.
Considering both V oc and V d as inputs, the inductor currents are shown in Figure 12.When V oc is higher than 3.5 V, representing a SOC higher than 90%, the inductor current is reduced and less balancing current is charged to this cell.When V oc is lower than 3 V (about 20% of the SOC), the inductor current is also reduced and less balancing current is discharged from this cell.This mechanism helps to protect the battery cells and potentially to extend their service life.
its peak value of 70, corresponding to the peak inductor current of 3 A. When d V continues decreasing to 15 mV, the equivalent gain starts to decrease sharply to less than 20.At the same time, the inductor current declines to near zero.This prevents the divergence of the already-balanced cells.For comparison, the gain of the PI controller is shown by the dotted line in Figure 10, which has the same peak inductor current.Considering both oc V and d V as inputs, the inductor currents are shown in Figure 12.When oc V is higher than 3.5 V, representing a SOC higher than 90%, the inductor current is reduced and less balancing current is charged to this cell.When oc V is lower than 3 V (about 20% of the SOC), the inductor current is also reduced and less balancing current is discharged from this cell.This mechanism helps to protect the battery cells and potentially to extend their service life.decreasing to 15 mV, the equivalent gain starts to decrease sharply to less than 20.At the same time, the inductor current declines to near zero.This prevents the divergence of the already-balanced cells.For comparison, the gain of the PI controller is shown by the dotted line in Figure 10, which has the same peak inductor current.Considering both oc V and d V as inputs, the inductor currents are shown in Figure 12.When oc V is higher than 3.5 V, representing a SOC higher than 90%, the inductor current is reduced and less balancing current is charged to this cell.When oc V is lower than 3 V (about 20% of the SOC), the inductor current is also reduced and less balancing current is discharged from this cell.This mechanism helps to protect the battery cells and potentially to extend their service life.

Experimental Results
To evaluate its performance, a prototype of the MSIBC was built to balance a four-cell battery pack.A NI FPGA module was used to implement the FL controller for the generation of the pulse width modulation (PWM) to control the MOSFETs in the MSIBC with a frequency of 5 kHz.Initially, all four cells were fully charged using an Arbin BT2000, then one cell in the pack (e.g., cell four in this test) was discharged by 20% of the tested capacity to create an imbalance scenario.

Experimental Results
To evaluate its performance, a prototype of the MSIBC was built to balance a four-cell battery pack.A NI FPGA module was used to implement the FL controller for the generation of the pulse width modulation (PWM) to control the MOSFETs in the MSIBC with a frequency of 5 kHz.Initially, all four cells were fully charged using an Arbin BT2000, then one cell in the pack (e.g., cell four in this test) was discharged by 20% of the tested capacity to create an imbalance scenario.
Once the balancing operation was completed, all the four cells were fully charged again with the Arbin BT2000 to evaluate the balancing effects.Figure 13 shows the performance evaluation platform for the MSIBC.In this platform, the PI controller is also implemented.The experimental results of the FL controller were compared with those of the PI controller.

Experimental Results
To evaluate its performance, a prototype of the MSIBC was built to balance a four-cell battery pack.A NI FPGA module was used to implement the FL controller for the generation of the pulse width modulation (PWM) to control the MOSFETs in the MSIBC with a frequency of 5 kHz.Initially, all four cells were fully charged using an Arbin BT2000, then one cell in the pack (e.g., cell four in this test) was discharged by 20% of the tested capacity to create an imbalance scenario.
Once the balancing operation was completed, all the four cells were fully charged again with the Arbin BT2000 to evaluate the balancing effects.Figure 13 shows the performance evaluation platform for the MSIBC.In this platform, the PI controller is also implemented.The experimental results of the FL controller were compared with those of the PI controller.LiFePO4 battery cells from the A123 Company and lithium nickel cobalt aluminum oxide (NCA) battery cells from Samsung were selected in this study.After a screening process, the cells with almost the same internal resistances and capacities in each type were connected in series to make the battery pack.The specifications of these two types of battery cells are listed in Table 2. LiFePO 4 battery cells from the A123 Company and lithium nickel cobalt aluminum oxide (NCA) battery cells from Samsung were selected in this study.After a screening process, the cells with almost the same internal resistances and capacities in each type were connected in series to make the battery pack.The specifications of these two types of battery cells are listed in Table 2.The inductor current and the mode duration decide the cell balancing currents based on Equation (1).The highest inductor current was set at 3 A, and the peak battery balancing current was 4 A. The balancing stopped when the OCV difference was less than 10 mV or the balancing time was longer than 3000 s.

Experimental Results for LiFePO 4 Battery Pack
The LiFePO 4 battery has flat voltage characteristics.Figures 14 and 15 show the OCVs and the average balancing currents with the FL and the PI controllers for the LiFePO 4 battery cells during the balancing process, respectively.than the PI controller, which can be seen in Figures 14b and 15b, respectively.Furthermore, with the FL controller, the three cells with the same initial OCVs keep converging as the gains among them are very small, as shown in Figure 14a.With the PI controller, until the balancing operation runs out of time, there are still significant OCV differences, as shown in Figure 15a, due to the slow balancing speed, which is a result of the small average balancing currents in the balancing process.

Experimental Results for NCA Battery Pack
In contrast with the LiFePO4 battery, the NCA battery has steep voltage characteristics.The OCVs and the average balancing currents of the NCA batteries with the FL controller and the PI controller are shown in Figures 16 and 17, respectively.With the FL controller, the balancing ends at 1500 s, when the maximum OCV difference reaches 10 mV.The balancing time for the NCA battery is slightly longer than that for the LiFePO4 battery, since the OCVs of the NCA battery drop slowly compared with those of the LiFePO4 battery.The recovered pack capacity is the highest in all cases, and this is discussed in the next section.With the PI controller, the maximum OCV difference still cannot reach 10 mV within 3000 s.The OCVs drop slowly, and the average balancing currents for the NCA battery are slightly higher than those for the LiFePO4 battery.The balancing operations with the FL controller stopped at around 1000 s when the maximum OCV difference reached 10 mV.In contrast, the balancing operation with the PI controller stopped at about 3000 s.This is because the FL controller maintains a higher balancing current for a longer time than the PI controller, which can be seen in Figures 14b and 15b, respectively.Furthermore, with the FL controller, the three cells with the same initial OCVs keep converging as the gains among them are very small, as shown in Figure 14a.With the PI controller, until the balancing operation runs out of time, there are still significant OCV differences, as shown in Figure 15a, due to the slow balancing speed, which is a result of the small average balancing currents in the balancing process.

Experimental Results for NCA Battery Pack
In contrast with the LiFePO 4 battery, the NCA battery has steep voltage characteristics.The OCVs and the average balancing currents of the NCA batteries with the FL controller and the PI controller are shown in Figures 16 and 17, respectively.With the FL controller, the balancing ends at 1500 s, when the maximum OCV difference reaches 10 mV.The balancing time for the NCA battery is slightly longer than that for the LiFePO 4 battery, since the OCVs of the NCA battery drop slowly compared with those of the LiFePO 4 battery.The recovered pack capacity is the highest in all cases, and this is discussed in the next section.With the PI controller, the maximum OCV difference still cannot reach 10 mV within 3000 s.The OCVs drop slowly, and the average balancing currents for the NCA battery are slightly higher than those for the LiFePO 4 battery.

Experimental Results for NCA Battery Pack
In contrast with the LiFePO4 battery, the NCA battery has steep voltage characteristics.The OCVs and the average balancing currents of the NCA batteries with the FL controller and the PI controller are shown in Figures 16 and 17, respectively.With the FL controller, the balancing ends at 1500 s, when the maximum OCV difference reaches 10 mV.The balancing time for the NCA battery is slightly longer than that for the LiFePO4 battery, since the OCVs of the NCA battery drop slowly compared with those of the LiFePO4 battery.The recovered pack capacity is the highest in all cases, and this is discussed in the next section.With the PI controller, the maximum OCV difference still cannot reach 10 mV within 3000 s.The OCVs drop slowly, and the average balancing currents for the NCA battery are slightly higher than those for the LiFePO4 battery.

Discussions
In the experimental results, accurate battery SOC values and the recovered pack capacities are calculated.In a battery pack connected in series, the weakest cell decides the pack capacity.Therefore, the recovered pack capacity (energy) is the ratio of the increased capacity of the weakest cell to the tested capacity (energy).The balancing results for the A123 LiFePO4 battery pack and the Samsung NCA battery pack are shown in Tables 3 and 4, respectively.
As shown in Table 3, the FL controller has better balancing performance than the existing PI controller in the LiFePO4 battery pack.With the PI controller, the final SOC difference is still 10% and 0.1299Ah is charged to the weakest cell as the recovered pack capacity.With the FL controller, the final SOC difference narrows to 7.5% and 0.17577Ah is charged to the weakest cell as the recovered pack capacity.

Discussions
In the experimental results, accurate battery SOC values and the recovered pack capacities are calculated.In a battery pack connected in series, the weakest cell decides the pack capacity.Therefore, the recovered pack capacity (energy) is the ratio of the increased capacity of the weakest cell to the tested capacity (energy).The balancing results for the A123 LiFePO 4 battery pack and the Samsung NCA battery pack are shown in Tables 3 and 4, respectively.

Figure 3 .
Figure 3. Operation modes of MSIBC in one cycle.

Figure 4 .
Figure 4. Simulation results for MSIBC with constant equal mode duration.

Figure 5 .
In this ECM, b V and I represent the battery terminal voltage and current, respectively.The resistance in R describes the instantaneous voltage drop.A parallel branch with a resistance of diffusion.When the sample rate of the voltage is high, the diffusion effect of diff C

Figure 4 .
Figure 4. Simulation results for MSIBC with constant equal mode duration.

Figure 6 .
Figure 6.Block diagram of an FL controller of MISBC.

Figure 6 .
Figure 6.Block diagram of an FL controller of MISBC.
triangle MF; thus Gaussian MF is applied to the linguistic variable Z for the MFs of d V and L I .It can make the output inductor current drop quickly to near zero when d V is less than 20 mV, reducing the balancing loss and the chance of divergence.
V and L I .It can make the output inductor current drop quickly to near zero when d V is less than 20 mV, reducing the balancing loss and the chance of divergence.
Rule 01: IF V d = NV and V oc = VS THEN I L = NV; Rule 25: IF V d = PV and V oc = VL THEN I L = PV.

Figure 10 .
Figure 10.Equivalent gain for desired inductor current with input d V at an OCV of 3.3 V.

Figure 11 .
Figure 11.Inductor current with input d V at an OCV of 3.3 V.

Figure 10 .
Figure 10.Equivalent gain for desired inductor current with input V d at an OCV of 3.3 V.

Figure 10 .
Figure 10.Equivalent gain for desired inductor current with input d V at an OCV of 3.3 V.

Figure 11 .
Figure 11.Inductor current with input d V at an OCV of 3.3 V.

Figure 12 .
Figure 12.Inductor currents with two inputs of V d and V oc .

Figure 12 .
Figure 12.Inductor currents with two inputs of d V and oc V .

Figure 14 .
Figure 14.Experimental results for the LiFePO4 battery pack with an FL controller.(a) OCV for each cell; (b) average balancing current for each cell.

Figure 14 . 18 Figure 15 .
Figure 14.Experimental results for the LiFePO 4 battery pack with an FL controller.(a) OCV for each cell; (b) average balancing current for each cell.Energies 2017, 10, 1034 14 of 18

Figure 15 .
Figure 15.Experimental results for the LiFePO 4 battery pack with a proportional-integral (PI) controller.(a) OCV for each cell; (b) Average balancing current for each cell.

Figure 15 .
Figure 15.Experimental results for the LiFePO4 battery pack with a proportional-integral (PI) controller.(a) OCV for each cell; (b) Average balancing current for each cell.

Figure 16 .
Figure 16.Experimental results for the NCA battery pack with an FL controller.(a) OCV for each cell; (b) Average balancing current for each cell.

Figure 16 .
Figure 16.Experimental results for the NCA battery pack with an FL controller.(a) OCV for each cell; (b) Average balancing current for each cell.

Figure 17 .
Figure 17.Experimental results for the NCA battery pack with PI controller.(a) OCV for each cell; (b) Average balancing current for each cell.

Figure 17 .
Figure 17.Experimental results for the NCA battery pack with PI controller.(a) OCV for each cell; (b) Average balancing current for each cell.

Table 1 .
Rule Base of an FL controller for Linguistic Variables.
and k B (k = 1, 2, …, n) are the fuzzy sub-sets of 1I .The detailed fuzzy rules are given in Table1.These rules describe the relation between the input and output of the FL controller based on the expert knowledge of LiFePO4 battery balancing.There are a total of 25 rules.The following are two of the rules.

Table 1 .
Rule Base of an FL controller for Linguistic Variables.

Table 2 .
Specifications of the two types of Battery cells.