**1. Introduction**

A Li-ion battery is a type of secondary battery in which lithium ions move from a negative electrode to a positive electrode through an electrolyte during a discharge process. Lithium is the lightest solid element and has the lowest standard reduction potential. Therefore, when applied as an electrode material, the battery cell can obtain an electromotive force higher than 3 V, and a high energy density per weight and a high energy density per volume can be obtained. Li-ion batteries have less memory effect and less self-discharge as well. In addition to these advantages, due to their reliable performance, long life cycle, and advantages such as reduced pollution, Li-ion batteries are one of the most promising power sources for portable electronics, electric vehicles, renewable energy storage devices, et cetera [1,2]. However, high energy density batteries with low thermal stability electrode materials may have low safety performance [3–5]. In particular, battery failure caused by extreme conditions such as excessive external force, high temperature, low temperature, overcharge, and over-discharge becomes a serious problem [6–8].

Overcharging the Li-ion battery can be one of the most important safety issues. Overcharging occurs when a charging current is forced in after the battery reaches its upper cutoff voltage or state of charge (SoC) limit. It is usually caused by a malfunction of the battery charger or by an inaccurate estimate of the condition of the battery in the battery management system (BMS). When the Li-ion battery is overcharged, in addition to the increase in internal resistance, decomposition of the binder and electrolyte, formation of insoluble products, blocked electrode pores, and gas generation may occur sequentially [9]. Especially, severe expansion of the battery may occur due to gas accumulation [10], and when the internal pressure exceeds the limit, structural deformation, rupture, and an internal short circuit may occur [11,12]. Lithium metal and moisture in the air may react after the battery ruptures, and even flammable gases may ignite [8,13]. Heat generated by side reactions and internal short circuits can accelerate battery failure mechanisms through natural positive feedback, leading to thermal runaway, eventually causing battery ignition or explosion [8,14,15]. In the case of over-discharge, a significant loss of active lithium and positive electrode material occurs [16], and a decrease in battery capacity occurs [17,18].

**Citation:** Kim, J.; Kowal, J. A Method for Monitoring State-of-Charge of Lithium-Ion Cells Using Multi-Sine Signal Excitation. *Batteries* **2021**, *7*, 76. https://doi.org/10.3390/batteries 7040076

Academic Editors: Duygu Kaus and Kai Peter Birke

Received: 30 July 2021 Accepted: 3 November 2021 Published: 9 November 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Battery over-discharging also results in a change in the solid electrolyte interphase (SEI) on the anode surface and an increase in impedance [18]. Gas is also generated from the decomposition of the SEI, which can cause battery swelling [19].

Accurate SoC estimation has the following advantages:


Nonetheless, there is still no way to directly measure the SoC of a battery cell. Therefore, it must be estimated or inferred indirectly from the measured current, voltage, and temperature.

Li-ion cell SoC estimation methods can be listed as follows [20–22]:

• SoC estimation method based on open circuit voltage (OCV)

In order to represent the thermodynamic state of the energy of the cell electrode, the OCV according to the chemical composition of the electrode is used. OCV is the voltage of the electrode when it has been stable for a sufficient time without flowing current. Cell voltage is related to temperature and electrode particle surface concentration, whereas cell SoC is related to particle average concentration. In other words, OCV is measured when the electrodes of an electrochemical cell reach equilibrium and there is no voltage deviation depending on the position inside the electrode, which reflects the Gibbs free energy at thermodynamic equilibrium. The OCV represents a strong dependence on SoC in most batteries. Nevertheless, it is impractical for real-time or continuous state estimation since a relaxation time of several hours is generally required to reach electrochemical equilibrium. This is especially problematic for battery applications where resting time can never exist. Furthermore, Li-ion batteries have a flatter OCV compared to lead-acid batteries, making SoC estimation difficult. In addition, cell temperature and cell state of health (SoH) can also lead to SoC estimation errors based on OCV.

• SoC estimation method based on ampere counting

In the SoC estimation method based on ampere counting, the accumulated amount of charging and discharging current relative to the discharging capacity is defined as ΔSoC and calculated with the set initial SoC. Since only cumulative current information is used for SoC estimation, it has the advantage of requiring relatively low performance for the hardware and software of the BMS. However, this method also has weaknesses in estimating cell SoC. It becomes a problem if the SoC of the battery cell is entirely dependent on the initial SoC. Since only ΔSoC is calculated, it is unavoidable to misestimate the SoC if the initial SOC setting is incorrect. In addition, it is impossible to know exactly the total capacity and coulombic efficiency of a cell, which should be approximated, and both of these approximations contribute to the error in cell SoC estimation. Self-discharge currents and leakage currents from electronic circuits for measuring cell performance increase errors as well. Furthermore, errors in voltage, current, and temperature measurements contribute to increasing the estimation error. These errors are more integrated and intensified as battery cell operating time increases. As a result, the uncertainty in the SOC estimate is exacerbated by the accumulated measurement errors. Therefore, this method can show reliable estimation results for a short period of time only if the initial conditions are well known, unless there is a feedback mechanism for error correction.

• SoC estimation method based on heuristic data

The heuristic based SoC estimation method is a method based on experimental data. Statistical rules or patterns found from data obtained through various cell charge and discharge experiments are used. These methods include fuzzy logic, neural networks, and support vector machines (SVM). Reliable estimation results can be obtained when the learning technique is implemented with a large amount of experimental data under different conditions. However, it takes a lot of time for the necessary experimental data to be properly secured.

• SoC estimation method based on adaptive control

An adaptive control based SoC estimation method such as a Kalman filter (KF) [23] or a sliding mode observer [24,25] compares the actual SoC measurement result with the estimated value and gradually reduces the difference according to the feedback principle. Although this method has a high estimation performance, it is relatively complicated to implement and thus has a disadvantage of high cost.

• SoC estimation method based on equivalent circuit model

The electrochemical impedance spectroscopy (EIS) method is a well-established technique for determining the dynamic behavior of electrochemical systems [26,27]. It is used to characterize battery impedance behavior over a wide frequency range [28,29]. Using EIS, the measured spectrum in an electrochemical system can be interpreted as an impedance spectrum of a lumped element model consisting of resistors, inductors, and capacitors. It can be used to implement dynamic simulation models [30]. The SoC [21,22,31] and SoH [28,32,33] of the battery cell can be estimated through the equivalent circuit model obtained through EIS. Moreover, the commonly used definition for battery end of life (EoL) is the predefined battery impedance increase at nominal conditions [34–36]. Nevertheless, EIS equipment is generally used in laboratories for general propose impedance measurements [3], which are not suitable for battery monitoring purposes. For EIS measurements, battery cells need to be detached from the operating load, and it is usually time consuming for impedance measurements in a wide frequency range. In addition, EIS equipment can be an excessive investment for battery monitoring systems, especially heavy and bulky for portable devices. Moreover, without algorithms for cell state estimation, EIS measurements alone cannot estimate cell SoH and SoC.

Unlike the method using EIS, the proposed method can be used to estimate the SoC of the cell during operation. There is a growing interest in the use of cell impedance to monitor the condition of batteries. The papers by Qahouq [37] and Waag et al. [38] deal with the impedance measurement of the operating cell but do not cover cell SoC and SoH estimation using the measured impedance. In the paper by Do et al. [39], the measured impedance is not adjusted according to the temperature change, and the computational complexity increases by using the extended KF. In addition, this paper does not deal with the method of estimating the cell SoH and SoC from the measured impedance. In the papers by Huang et al. [40] and Howey et al. [41], the cell SoC is estimated by measuring the cell impedance, but the effect of temperature on the cell impedance is not considered. The paper by Fleischer et al. [33] shows a good result of estimating the state of a cell in operation. However, a physical model and an equivalent circuit model are used, and so-called mutation algorithms are used. Since nonlinear differential equations and matrix operations are required, the complexity of the operation is increased. On the other hand, the proposed method is relatively simple but accurate. Using the cell temperature and impedance at two frequencies, SoH and SoC of the cells in operation can be estimated.
