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Abstract—Lithium-ion batteries have gained rapid popularity as an important means for the power sector. It is essential to model the battery in order to predict its ...
Battery Equivalent Circuits and Brief Summary of Components Value Determination of Lithium Ion A Review Johnny Wehbe*, Nabil Karami, Member,IEEE Faculty of Electrical Engineering, University Of Balamand Lebanon [email protected], [email protected] Abstract—Lithium-ion batteries have gained rapid popularity as an important means for the power sector. It is essential to model the battery in order to predict its behavior under various operating conditions to avoid any wrong operation and to safely manage to increase its life time. This paper introduces the most commonly used battery modelling techniques and their respective equivalent circuit models. IndexTerms—Lithium ion,battery,modeling,equivalent circuit.

management system development. Several models have been introduced in literature, but the most common ones will be highlighted on. The rest of this paper is organized as follows. Section II presents the most common Li-ion battery models with their equivalent circuit models. A brief summary of some methods for parameters identification is presented in section III. Section IV discusses further aspects that might affect battery performance and models, and section V concludes this paper.

I. INTRODUCTION In earlier years, lithium batteries started being used as lithium metal. Since lithium metal has instability properties, designing a lithium metal battery had failed attempts. Therefore, scientists began using lithium ions instead of metal, because they provided a safer environment despite having a slightly lower density than the lithium metal. Lithium ion (Li-ion) batteries gained wide popularity over the nickel-cadmium due to their high energy density, typically twice. Li-ion also have high power density, and low selfdischarge, which made it suitable for modern fuel gauge applications. On the other hand, Li-ion batteries have their drawbacks. They require protection circuit to keep safe operation. The cell is also fragile to extreme temperatures, thus its temperature should be always monitored. In addition, it is noticed that the capacity of the Li-ion battery decreases after a year or two, even if unused. Therefore, to ensure better operations of these batteries, and to study their behavior concerning their charging/discharging lifecycle and other parameters, it is essential to model these batteries. In this paper, the most common battery equivalent circuit models will be reviewed. There are other battery models such as the electrochemical models, but these result in partial differential equations with large number of unknown parameters. This complexity often leads to significant requirement of time and computation. Battery equivalent models have been studied especially for the purpose of vehicle power management control and battery

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II. MODELING Before introducing the various models, it is necessary to introduce some definitions, which will be grasped later on: 1) A battery is said to be fully charged if its voltage, after being charged for a period of time, reaches v=vh. 2) A battery is said to be fully discharged when its voltage, after being drained for a period of time, reaches v=vl. 3) The capacity of a cell C is the highest amount of ampere-hours that can be used from a cell before it is fully discharged, starting with the cell fully charged. 4) The nominal capacity Cn is the amount of Ah that can be used from a cell, starting at full charge. 5) The State of Charge (SOC) of the cell is the ratio of C and Cn, as stated in definitions 3 and 4. Now that the above terms are familiar, some mathematical models will be given, involving SOC. 0

(1)

where, is the cell SOC, is instantaneous cell is the cell nominal capacity. is the Cell current, is equal to 1. For Columbic efficiency. For discharge, charge, is less than or equal to 1.

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The above relation can be sampled using rectangular approximation, yielding: (2) Equation (2) is the state vector equation for SOC, as an input. It will be used in the upcoming including models. A. Models with only SOC as a state Figure 1: The simple model

The first three models that are going to be introduced have only the SOC as a state. Having this, they are considered limited in estimating the cell terminal voltage, which can be improved by adding more states. 1) The combined model Having SOC, the terminal voltage can be estimated using several methods, such as Shepherd model, Unnewehr universal model, and Nernst model [1]. The combined model will be: Δ

ln

There is a minor issue in the last two models presented. This may be seen during the rest periods of the discharge phase as presented in Fig. 2. After the discharge is finished, it is noticed from Fig. 2 (a) and (b) that the voltage relaxes, taking a value less than the true open circuit voltage. This is solved by the hysteresis effect. One more state should be added to the model to take into account the slow degeneration in voltage. This model adjusts the output equation:

ln 1

where, is the terminal voltage, is the internal resistance, is the polarization resistance. The unknowns , , , , may be found using least-squares estimation. A simple summary is found in [2]. 2) The simple model Using the parameters of the combined model, the output equation can be furtherly modeled into more parts: ln

3) The zero-state hysteresis model

is the where, signifies the sign of the current, subtraction of the charge/discharge curve divided by two. is subtracted too. Then the loss More models exist in which more states are added. For instance, the one-state hysteresis model [2] estimates hysteresis oscillating between maximum and minimum values as a result of changes in the sign of the current. Other model is the Enhanced Self-Correcting (ESC) model [2], where the relaxation effect is modeled as a low pass filter.

ln 1

An easier model than the combined model is derived as: Δ

(3) where is the Open Circuit Voltage of the cell. Eq. (3) can be modeled as shown in Fig. 1. R1 and R2 in Fig. 1 are the charge and discharge resistances, respectively. The diodes present in the simple model equivalent circuit are assumed ideal. Eq. (2) and (3) are what comprise the simple model.

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battery model can be modeled using Randle’s equivalent circuit [3] as shown in Fig. 3.

Figure 3: Randle's EECM model

2) V-I based EECMs The previous method is tedious because measuring the impedance requires a signal generator, therefore not practical. This V-I based method takes advantage of available measurements, like load current, terminal voltage, and temperature. It also uses linear structure to avoid complex nonlinear elements. The model has an ideal voltage source, a resistor for internal resistance, and some RC networks to model the dynamic behavior of the battery. Known models are the first order [5] and second order [6]. Figure 4 shows the general order of such EECM model.

Figure 4: m-th order EECM Figure 2: Modeling discharge [2]. (a): Combined model discharge, (b): Simple model discharge, (c): Zero-state hysteresis model discharge

C. Other electrical models

Results of Fig. 2 show that the performance of the zerostate hysteresis model is better than the simple model. Fig. 2 (c) compares the cell’s true voltage with the zero-state hysteresis model voltage estimation over discharge pulses.

Other electrical models exist like the Thevenin model [7] and its upgraded version, the Dual Polarization (DP) model [8].

B. Equivalent Electric Circuit Models (EECMs)

The Thevenin battery model consists of an ideal battery voltage, internal resistance, capacitance, and overvoltage resistance. This model is commonly used for Li-ion battery modeling. In this model, the voltage response to current excitation is used to determine the model parameters. The model is shown in Fig. 7.

EECMs in general use a mixture of basic electric elements to model batteries. They are divided into two parts, the impedance based EECMs and the V-I based EECMs.

1) Thevenin model

1) Impedance based EECMs: The concept of this model is to make use of measuring the battery’s impedance by generating an AC current, and recording the AC voltage response. After that, the complex impedance can be calculated using an FFT analyzer. This

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III. PARAMETER IDENTIFICATION METHODS A. Step-current charging and discharging test The model parameters of the battery can be repeatedly discharged and rested regularly at constant duty cycle from full state of charge until depleted [9]. The voltages of each RC circuit can be expressed as: where Figure 1: Thevenin circuit model

Depending on the accuracy level needed, the RC branches are placed. More branches are needed for detailed simulation. 2) Dual Polarization (DP) model This is an improved electric circuit model. The previous electric models states are still not accurate enough since their circuit elements can change, depending on certain conditions of the battery. For improving this, the DP model [8] considers the polarization characteristics of the Li-ion battery. The modified Thevenin model has an additional RC branch. The equivalent capacitances include electrochemical polarization capacitances, and are used to describe the transient response during the cycles.

While discharging, by a constant dc current, the capacitors can be modeled by an open circuit, and the resistances can be modeled by the following equation: / is the discharging dc current. where The capacitors are modeled by: / B. Weighted recursive least square (WRLS) • • •

Makes use of the linear structure of EECMs. Only requires current and voltage as inputs. Can be used to estimate battery SOC and state of health (SOH) [10]. IV. DISCUSSION

Figure 2: DP circuit model

The following is the electrical behavior as found in [8]:

where, and respectively.

/

/

/

/

are the voltages on

and

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The previous models presented differ by their level of accurateness. Each can be chosen according to the application needed. These models can be furtherly upgraded to take into account some real life factors, such as temperature effect and ageing factor. A battery model might change over time due to these factors. Temperature for instance, not only has influence on OCV, but also on battery capacity. Battery capacity increases as temperature increases, and decreases in colder weathers. Therefore, including the temperature effect in battery models can improve the battery’s performance. In [11], a test was done on a commercial li-ion battery to analyze the effect of temperature on the parameters of high power li-ion battery. The results have shown some variation with temperature of the main impedances of the li-ion battery. On the other hand, ageing effect will also affect the battery model. In fact, battery ageing can cause the battery to fail, due to the capacity fading, or increasing of the internal resistance. There are two types of ageing; calendar ageing due to storage, and cycling due to charge/discharge usage.

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V. CONCLUSION In this paper, the most common Li-ion battery models are presented, and their different types of equivalent circuits. For instance, the combined model was first discussed, and then added to it a zero state model, and then a one state model, and comparisons were shown on the discharge cycle of the Li-ion battery. Afterwards, common EECM models were presented, including 1st order RC network and the 2nd order RC network. RC networks can be increased depending on the accuracy needed in the application. Also, in this paper, a brief summary of parameters identification methods is presented. More methods exist in literature. REFERENCES [1] Hussein, H. H., & Batarseh, I. (2011, July). An overview of generic battery models. In Power and Energy Society General Meeting, 2011 IEEE (pp. 1-6). [2] Plett, G. L. (2004). Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification. Journal of power sources, 134(2), 262-276. [3] Mauracher, P., &Karden, E. (1997). Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification. Journal of power sources, 67(1), 69-84. [4] Zhang, C., Li, K., Mcloone, S., & Yang, Z. (2014, June). Battery modelling methods for electric vehicles-A review. In Control Conference (ECC), 2014 European (pp. 2673-2678). IEEE.

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[5] Chiang, Y. H., Sean, W. Y., &Ke, J. C. (2011). Online estimation of internal resistance and open-circuit voltage of lithium-ion batteries in electric vehicles. Journal of Power Sources, 196(8), 3921-3932. [6] Chen, M., & Rincon-Mora, G. A. (2006). Accurate electrical battery model capable of predicting runtime and IV performance. IEEE transaction on Energy conversion, 21(2), 504-511. [7] Hentunen, A., Lehmuspelto, T., &Suomela, J (2014). TimeDomain Parameter Extraction Method for Thevenin-Equivalent Circuit Battery Models. IEEE transaction on Energy conversion, 29(3). [8] He, H., Xiong, R., Zhang, X., Sun, F., & Fan, J. (2011). Stateof-charge estimation of the lithium-ion battery using an adaptive extended Kalman filter based on an improved Thevenin model. IEEE transaction on Vehicular Technology, 60(4), 1461-1469. [9] Hsieh, Y. C., Lin, T. D., & Chen, R. J. (2012, November). Liion battery model exploring by intermittent discharging. IEEE in Renewable Energy Research and Applications (ICRERA), 2012 International Conference on (pp. 1-5). [10] Verbrugge, M., & Koch, B. (2006). Generalized Recursive Algorithm for Adaptive Multiparameter Regression Application to Lead Acid, Nickel Metal Hydride, and Lithium-Ion Batteries. Journal of The Electrochemical Society,153(1), A187A201. [11] Gomez, J., Nelson, R., Kalu, E. E., Weatherspoon, M. H., & Zheng, J. P. (2011). Equivalent circuit model parameters of a high-power Li-ion battery: Thermal and state of charge effects. Journal of Power Sources, 196(10), 4826-4831.

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