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Energy inconsistency among Li-ion battery cells widely exists in energy storage systems, which contributes to the continuous deterioration of the system durability and overall performance. Researchers have proposed various kinds of battery energy equalizers to reduce such inconsistency. Among them, the inductor equalizer is a predominant type in fast equalization applications. However, it requires relatively more complex control than other types of equalizers. In order to reduce the control complexity of inductor equalizers, a bidirectional multi-input and multi-output energy equalization circuit based on the game theory is proposed in the present work. The proposed equalizer has the modularized circuit topology and the mutually independent working principle. A static game model is developed and exploited for the mathematical description and control analysis of an energy equalization circuit comprised of these equalizers. The feasible control of each equalizer was obtained by solving a series of linear equations for the Nash Equilibrium of the model among the states of charge of the battery cells. The complexity of equations grows linearly with the cell number. The equivalent simulation model for the four-cell equalization is established in the PISM software, where the operational data and simulation results justify the static game model and verify the control validation, respectively. It is concluded that the proposed inductor equalizer is suitable for large-scale battery strings in energy storage systems, electrical vehicles, and new energy power generation applications.

Li-ion batteries have been extensively used for energy storage systems in electrical vehicles, new energy power generation, and military applications because of their superior performance [

On the one hand, the specific structure of a certain type of equalizer determines its potential performance. Depending on the equalization components and topology, there are generally four types of equalizers: resistor, capacitor, transformer, and inductor equalizers [

Inductor equalizer is a predominant type in fast equalization, and its design is not as expensive or complex as the transformer equalizer [

On the other hand, the purpose of equalizer control is to realize the full potential performance of the equalizers. Therefore, working out the control method is essentially a decision problem. Game theory (GT) is the study of utilizing relevant parties in the game of multiple individuals or teams under the constraints of specific conditions and implementing corresponding strategies. It has been studied predominantly as a modeling paradigm in the mathematical social sciences especially in economics [

To simplify the control of inductor equalizer, bidirectional multi-input and multi-output energy equalization circuit (BMMEEC) is proposed. The proposed circuit has three characteristics as follows: first, in terms of the circuit topology, each equalizer consists of a dual switch group, connected in a parallel manner with the entire battery string, and an inductor is connected between every two adjacent cells. It should be indicated that no shared switches exist among equalizers; second, in terms of the controllability, each equalizer is controlled independently regardless of the coupling effect; third, in terms of the circuit operation, each equalizer works synchronously; hence the equalization time decreases. Moreover, compared with other types of equalizers, inductor equalizers have a larger current capacity than that from capacitor equalizers. Furthermore, they have lower winding precision requirements than those from transformer equalizers. In the perspective of the circuit modeling, the barrier for the mathematical description is eliminated; equalizers are independent and rational; the mathematical description is feasible and the operational data is measurable. Therefore, the CISGM can be developed and exploited for control analysis, where the equalizers are treated as independent game participants, the battery energy is regarded as the capital of the participants, and the Nash Equilibrium (NE) of the battery energy is set to be the termination of the game. As for the battery energy, it is described by the battery state of charge (SOC) in quantity. SOC is one of the most important parameters in a Li-ion battery, which is usually utilized to reflect the energy state of the battery. Reviewing the literature shows that researchers have proposed variety of accurate methods to investigate the Li-ion batteries [

This paper is organized as follows. In Section

Figure

Topology of the BMMEEC.

For a battery string that contains

Figure

The principle of the equalizer.

Figure

Consequently, in a battery string, energy redistribution for multiple cells is realized as is shown in Figure

Energy transfer from Part 1 to Part 2, caused by

In order to fully utilize the flexible topology of BMMEEC to achieve energy redistribution, it is necessary to mathematically describe and analyze its energy equalization process. Considering the following reasons, it is concluded that the game theory (GT) has high relevance and correspondence to such flexibility. (1) Independent players: in a game, players independently choose their own strategies to maximize their own benefits. (2) The comprehensive effect in benefit distribution: since all players are involved in the game, any behavior of a certain player affects the balance of the benefit. Moreover, the goal of the players is to maximize the benefit. Therefore, once they cannot get any more benefits, the game comes to the end. According to such relevance and correspondence, it is reasonable to establish a GM for the BMMEEC.

In order to simplify the model, four assumptions are made as follows.

(1) The design parameters of each cell are identical, such as the value of discharging current, discharging efficiency, battery capacity, and, most importantly, the unique correlation between the SOC and the energy. Therefore, the energy state of each cell is presented by its SOC value.

(2) The initial SOC of each cell is instantly available.

(3) Inductors have enough large inductance, i.e., enough capacity to store the energy from cells.

(4) Energy loss in the equalization process is negligible and all power components work ideally.

Each equalizer is treated as an independent player. For equalizer

In a game, the strategy is the combination of all behaviors conducted by players. In this GM, the “behavior” of each equalizer is presented as the implementation of its control plan, which is the conducting time of the actuated switch. For the equalizer

The control strategy of the game is the combination of control plans for all equalizers:

In a static game, once the behavior of each player is determined, the game result is determined. In this GM, each equalizer operates with a fixed control plan during the whole game.

In a complete information game, each player has accurate information about other players’ characteristics, strategy, and benefits. According to the second assumption of this GM, the initial SOC of each cell is known as the given condition.

In this game, benefit evaluates the effect of a player’s behavior. In other words, proper behavior can maximize a player’s benefit. In this GM, the goal of the whole equalization process is to equalize the SOC (

In (

NE is a strategy with which all players get their maximum benefit. This situation is equivalent to the SOC equality of the battery string. According to the definition of the NE, when it is applied, each player gets the maximum benefit “0” so that (

where

NE is the solution of the GM and, in order to figure it out, the equivalent equation group is derived from the two inequalities:

Substituting

In (

Applying the amperometric method, the SOC difference of a single cell can be expressed as the integral of its discharging time [

where

where

Figure

Energy transfer caused by the equalizer

On the other hand, when the switch is off, the energy stored in the inductor is released to all cells in the charging loop, and

Figure

Energy transfer caused by the equalizer

Similarity, when the switch is off, the energy stored in the inductor is released to cells that are all included in selected

In summary, when

When

Energy transfer caused by the equalizer

When the switch is off, the stored energy in the inductor is released to all cells in the charging loop but they are excluded from selected

Figure

Energy transfer caused by the equalizer

When the switch is off, the stored energy in the inductor is released to the cells in the charging loop and

In summary, when

According to the analysis above, the SOC divergence caused by one certain equalizer

The final expression of

Equation (

Elements in the coefficient matrix K.

| | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|

| -1 | -1 | -1 | | -1 | | | | | |

| | -2 | -2 | | -2 | | | | | |

| | | -3 | | -3 | | | | | |

| | | | | -4 | | | | | |

| | | | | | | | | | |

| | | | | | | | | | 1 |

For the sake of expression convenience, the SOC divergence in (

In (

Although the solution is infinite, only one feasible solution has the corresponding control strategy. Feasible requirements are as follows.

(1) Switch conducting time cannot be negative.

(2) Only one of two switches in an equalizer can be actuated. Otherwise, part of batteries in the battery string may be short-circuited. In other words, the product of

With these two requirements, the feasible solution can be uniquely determined. According to the feasible solution, the control strategy can be realized by setting the proper duty cycle of three corresponding switches:

In the BMMEEC, each equalizer exchanges the energy between upper and lower parts of cells, independently. It should be indicated that the energy equality of each cell in the battery string is the main goal of each equalizer. The control algorithm of the BMMEEC is illustrated in Figure

Flowchart of equalizer control algorithm.

The system detects the basic input parameters of the CISGM, the scale of the battery string (

The processor calculates the coefficient matrix (

If the SOC variance exceeds 5%, the system conducts the equalization procedure.

The processor solves linear equations to work out the NE of the static game model (

The controller outputs the signal to equalizers to actuate the MOSFETs.

The system redetects

The processor calculates

System conducts the termination judgment. If the SOC variance decreases after the first round of equalization, the process will return to Step

The termination criterion is met and the equalizers stop.

Taking a battery string that contains four cells, for example, the initial SOC of four batteries are set to be 0.86, 0.91, 0.93, and 0.90. Substitute the number of cells

The general solution of (

According to the first feasible requirement,

Therefore, the control can be implemented by setting the ratio among the duty cycles of the switches,

In order to verify the feasibility of the obtained control strategy, a BMMEEC for the battery string that contains four cells is established in the PISM simulation software. Figure

Simulation parameters of four-cell battery string.

Parameters | Value |
---|---|

Gating signal frequency | 4000Hz |

Battery rated capacity | 5.4 Ah |

Battery discharging current | 1.7A |

Battery rated voltage | 3.7V |

The inductance of each equalizer | 500uH |

Initial SOC of Cell_{1} | 0.86 |

Initial SOC of Cell_{2} | 0.91 |

Initial SOC of Cell_{3} | 0.93 |

Initial SOC of Cell_{4} | 0.90 |

Schematic diagram of the energy equalization circuit.

Figure _{1}. The positive value indicates that the current flows into the battery B_{1} and the battery is charging, while the negative value indicates that the current flows away from the cell C_{1} and the battery is discharging. Ammeters I2, I3, and I4 measure the discharging currents of cells C_{2}, C_{3}, and C_{4}, respectively. The positive value indicates that the current flows away from the battery and the battery is discharging, while the negative value indicates that the current flows into the battery and the battery is charging. Since only

Balancing current in one operation cycle for the ideal gating signal.

Current(A) | Time(s) | |||||||
---|---|---|---|---|---|---|---|---|

3.97998 | 3.98004 | 3.98006 | 3.98008 | 3.98010 | 3.98017 | 3.98024 | | |

I1 | 0.381 | -0.8764 | 7.3586 | 2.3505 | 1.6895 | 1.0261 | 0.3758 | -1.63e-4 |

ΔI1 | 0 | -1.2574 | 8.235 | -5.0081 | -0.6110 | -0.6634 | -1.9747 | |

| ||||||||

I2 | 3.61e-6 | 2.0959 | 1.0438 | -0.471 | -4.58e-2 | 3.64e-6 | 3.61e-6 | 7.49e-5 |

ΔI2 | 0 | 2.0959 | -1.0521 | -1.5148 | 0.4252 | 0.0458 | 0.471 | |

| ||||||||

I3 | 3.51e-6 | 2.6398 | 1.7257 | 0.0004 | 4.56e-6 | 3.54e-6 | 3.51e-6 | 5.28e-5 |

ΔI3 | 0 | 2.6398 | -0.9141 | -1.7253 | -4.00e-4 | -1.02e-6 | -3.96e-4 | |

| ||||||||

I4 | 2.68e-6 | 1.7566 | 0.8996 | -0.7335 | -0.5326 | -6.42e-4 | 2.67e-6 | 8.65e-5 |

ΔI4 | 0 | 1.7566 | -0.8570 | -1.6331 | 0.2009 | 0.5262 | 0.734 |

Waveform of the current measured by I1, I2, I3, and I4.

The waveforms of current measured by ammeters in one operation cycle reflect SOC variations of four cells. Figure

In the period from 3.97998s to 3.98004s (_{1}, C_{2}, C_{3,} and C_{4} are discharged to the inductor and the discharging speed can be determined by the slope of the current waveform.

In the period from 3.98004s to 3.98006s, two discharging loops are formed by switches _{1} gradually transformed from the discharging mode to the charging mode.

In the period from 3.98006s to 3.98008s, only _{2}, C_{3}, and C_{4} cells further decreases. Moreover, C_{2} and C_{4} cells gradually enter the charging state. The charging speed of the C_{1} cell further increases.

In the period from 3.98008 to 3.90024s, all switches are turned off and there is only one discharging loop in the circuit. The amount of electricity stored in the inductor is released to the battery through the discharging loop. Since this process has nothing to do with the control, so no specific analysis is made for this part. The current integrals are shown in the last column of Table

After several operation cycles, the four cells power consumption gradually conforms to each other. Figure

SOC waveform for the ideal gating signal.

In order to investigate the advantages of the BMMEEC, results are compared with those from the inductor-based bidirectional lossless equalization circuit [

Figure

Balancing process comparison between BMMEEC and the circuit in [

Equalization process of BMMEEC

Equalization process of the circuit in [

In Figure _{2} is discharged in Stage 1 and is charged then in Stage 2, which results in unnecessary energy transfer. In contrast, in the equalization process of the BMMEEC, only one termination criterion is needed and no unnecessary energy transfer exists. This simplifies the control and avoids extra equalization process.

All the results above are obtained when switches are actuated by ideal gating signals. However, in practical applications, control signals usually cannot reach such a precise level. Consequently, a 40KHz clock signal is applied for switches control and every 20 sequent clock cycles are regarded as one operation cycle. Consequently, the original 4000 Hz gating signal is replaced by a 1000 Hz square wave signal and a 40 kHz clock signal is used as a counting signal. The former is used for switch control as the gating signal and the latter is used for counting. The clock signal divides one square wave cycle into 10 parts. The duty cycle of each switch is changed by adjusting the number of conduction cycles of each switch every 20 clock cycles. Switches are actuated for 18, 24, and 30 clock cycles in every operation cycle until the variance of the SOC reaches the minimum value. Set the currents measured by the four ammeters to be I1′, I2′, I3^{′,} and I4′. Figure

Correlation between current waveforms for two driven signals.

Correlation between I1 and I1′

Correlation between I2 and I2′

Correlation between I3 and I3′

Correlation between I4 and I4′

Figures

Comparison of equalization results in three states.

SOC | Initial state (%) | Idle state (%) | Charging state (%) | Discharging state (%) |
---|---|---|---|---|

C_{1} | 86.00 | 88.91 | 91.91 | 85.71 |

C_{2} | 91.00 | 88.30 | 91.16 | 85.14 |

C_{3} | 93.00 | 89.73 | 92.55 | 86.60 |

C_{4} | 90.00 | 88.68 | 91.62 | 85.48 |

Variance | 8.67 | 0.36 | 0.34 | 0.39 |

SOC waveform for clock signals.

SOC waveform in idle state

SOC waveform in charging state

SOC waveform in discharging state

Figure

Based on the inductor equalizer, an equalization circuit for the battery energy, called the BMMEEC, is proposed in the present study. Then, the corresponding mathematical model deduction and simulation verifications are presented. Moreover, the CISGM model is developed for the mathematical description and control analysis of the BMMEEC and the feasible control is obtained by solving the model’s Nash Equilibrium. An equivalent simulation model of the four-cell equalization is established in the PISM. In order to justify the CISGM during the simulation, the variation of the operational data, balancing current and battery SOC, tallied with the mathematical descriptions of the BMMEEC. It is found that four cells’ SOCs are nearly identical, which verifies the control validation. Moreover, the simulation results demonstrate that the application of the BMMEEC prevents the long-period charging-discharging cycles for involved cells so that the SOC of each cell moves directly forward in harmony. It is found that the BMMEEC has the modularized circuit topology and the mutually independent working principle, compared with other inductor equalizers. Furthermore, it is observed that the control complexity of the BMMEEC has a linear correlation with the cell number. However, further studies are required. The simulations results have a certain deviation from the ideal equalization results, and the scale of the simulation equalization circuit was limited to four cells.

It is intended to expand the research in the following parts, in the near future: (1) investigate the relationship between the equalization deviation and parameters of the BMMEEC and describe the influence of the main factors on the quantity of the deviation. (2) Expand the string equalization of the simulation battery to larger scales, while controlling the voltage stress on each equalizer. (3) Perform the practical experiments to test the actual performance of the BMMEEC.

The PSIM11 simulation data used to support findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

Jiayu Wang developed and exploited the mathematical model for the multi-inductor battery energy equalization circuit and carried out simulations. Shuailong Dai designed the topology of the proposed circuit and polished the manuscript. Xi Chen, Xiang Zhang, and Zhifei Shan finalized and polished the manuscript.

This work was partially supported by the National Innovation and Entrepreneurship Training Program for College Students 201711075009.