Mathematical Modeling of Battery Degradation Based on Direct Measurements and Signal Processing Methods

Abstract

This paper proposes and evaluates the behavior of a new health indicator to estimate the capacity fade of lithium-ion batteries and their state of health (SOH). This health indicator is advantageous because it does not require the acquisition of data from full charge–discharge cycles, since it is calculated within a narrow SOC interval where the voltage vs. SOC relationship is very linear and that is within the usual transit range for most practical charge and discharge cycles. As a result, only a small fraction of the data points of a full charge–discharge cycle are required, reducing storage and computational resources while providing accurate results. Finally, by using the battery model defined by the Nernst equation, the behavior of future charge–discharge cycles can be accurately predicted, as shown by the results presented in this paper. The proposed approach requires the application of appropriate signal processing techniques, from discrete wavelet filtering to prediction methods based on linear fitting and autoregressive integrated moving average algorithms.

1. Introduction

The use of lithium-ion (Li-ion) batteries in energy storage systems (ESS) and electric mobility applications is increasing due to their attractive characteristics such as high specific energy and power density, high charge and discharge efficiency, long life, and low cost. [1]. However, the impedance rise and capacity fade over the cycle life are still the focus of concern. By comparing the impedance or the capacity of the pristine cell with that of the aged or degraded cell, it is possible to quantify the degradation mechanisms since the capacity fades due to the change of some identified kinetic parameters [2]. Today, there is a growing interest in reusing electric mobility batteries for second-life applications such as ESS. Although they have reached the end of their life, second-life batteries from electric vehicle applications still have a residual capacity of about 70–80% [3,4]. However, the performance and efficiency of the reused battery system is highly dependent on the use of these batteries, which require a similar state of health (SOH), capacity, and remaining useful life (RUL). Therefore, it is of utmost importance to establish battery grading techniques, making it necessary to develop fast methods to evaluate the degradation of the batteries in terms of capacity, SOH, and RUL. Such techniques play a key role in the application of battery management strategies, while allowing to extend the battery lifetime and increase the reliability of the battery system. For example, an accurate estimation of the state of charge (SOC) allows avoiding overcharging and over-discharging of the battery, thus prolonging its lifetime [5]. However, electrochemical batteries exhibit a highly nonlinear behavior [5], making accurate estimation of SOC, SOH, and RUL a complex task.

Due to the importance of the battery in electric mobility and energy storage applications, the analysis of the SOH is a priority for researchers. Therefore, numerous methods have been developed over the years to evaluate and estimate the SOH. They can be broadly classified into model-based, data-driven, direct [6], and indirect measurement methods. Model-based methods typically use the battery equivalent circuit or mathematical models such as Nernst or Shepherd to calculate the SOC. However, these models generally do not allow the capacity fade within cycles to be described. Data-driven methods handle large databases and usually predict the SOC using various mathematical techniques such as fuzzy logic [7,8] or neural networks [9]. Data-driven methods treat the battery system as a black box and still provide an accurate response but at the cost of high computational requirements that hinder on-line predictions. Direct methods are typically based on electrochemical impedance spectroscopy (EIS) since the impedance of the cell typically increases gradually with the number of charge–discharge cycles [10]. Due to the wide range of frequencies analyzed in EIS measurements, this technique can characterize internal reaction mechanisms associated with battery aging processes, such as ion concentrations, particle size, dendrite growth, or electrolyte and electrode aging [11,12]. As a result, EIS is more accurate than the conventional method based on temperature, current, and voltage readings from the battery management system (BMS) [10]. EIS provides detailed information about the SOH, and requires long rest periods to reduce the impact of the battery relaxation effect on EIS measurement [13]. Therefore, EIS is difficult to implement in real-world applications due to the lack of practical access to the battery chemistry [14] and the need for a variable frequency generator, making it more suitable for laboratory testing than for direct use in practical applications [6,15,16]. Finally, Coulomb counting is a very common and simple method to determine the capacity of the battery but its accuracy is highly dependent on the precision of the measurements, while presenting a cumulative error [17].

It is very attractive to develop battery degradation models based on the evolution of health indicators to determine the battery SOH and RUL. Common indicators of SOH are battery capacity and internal impedance [18]. Therefore, techniques based on the battery capacity and electrochemical impedance spectroscopy (EIS) are commonly used for this purpose [19]. However, capacity is difficult to measure in on-line processes [18], while impedance spectroscopy is a complex and costly technique [18] that usually requires disconnecting the battery from the load, and interrupting the normal operation of the system, with the associated downtime implications and costs. Therefore, indirect methods for an on-line evaluation of the capacity fade, SOH, and RUL without the need to disconnect the battery from the load require further research efforts.

Since the open-circuit voltage (OCV) at the battery terminal is directly related to the SOC, it can be used to determine the beginning/end of charge/discharge when using the ampere-hour integration method. However, the battery must be allowed to rest for some time before the OCV is measured to ensure voltage stability. This makes it difficult for real-time applications [5]. In addition, the OCV–SOC curve has a certain hysteresis, i.e., for the same OCV, different SOC values are possible when charging and discharging, which affects the accuracy of the SOC calculation [20].

Researchers often use easily measured variables such as temperature, current, and voltage during charge–discharge cycles to determine health indicators (HIs) [18]. The choice of an appropriate HI greatly affects the accuracy of the predictive battery aging model and, thus, the accuracy of RUL and SOH estimation. Battery HIs are typically calculated by processing the battery charge–discharge data using signal processing and/or artificial intelligence algorithms. Battery health indicators can be broadly classified into book-keeping, direct, and indirect HIs. The decrease in the usable capacity of the battery and the increase in its internal resistance are key characteristics of battery aging; so, capacity and internal resistance are the most commonly used HIs to characterize the aging process. While the capacity fade is a book-keeping HI, the internal resistance is a direct HI. Indirect HIs are related to the extraction of indirect parameters from battery charge–discharge data that are strongly correlated with battery aging [21]. Indirect HIs found in the literature include average discharge voltage, maximum discharge current, constant current charge time, charge terminal voltage change, or charge temperature change rate [21].

For an accurate real-time evaluation, depending on the estimation method used, the system will need to store a large amount of data, requiring high storage and processing capabilities. In addition, a complete discharge of the charge stored in the battery or an expensive device capable of measuring other parameters such as internal resistance may also be required. Signal processing plays a leading role in the calculation of HIs for battery diagnosis and prognosis. Raw acquired battery data are usually denoised and filtered; subsequently, linear interpolation, extrapolation, look-up tables, or other processing methods are typically applied to determine the electromotive force. For example, time-domain and frequency-domain signal processing methods are used in EIS approaches because EIS is based on wide frequency-band measurements [15]. In [22], empirical mode decomposition and particle filtering methods are applied to predict the end-of-life of Li-ion batteries, while other authors [23,24] have applied neural networks for this purpose. In this paper, signal processing methods such as discrete wavelet filtering and prediction methods based on linear fitting and autoregressive integrated moving average algorithms are applied.

This paper proposes a voltage-based battery health indicator. It is an indirect HI that, as shown in the paper, is highly correlated with the degradation of battery capacity, since for the analyzed batteries the Pearson correlation coefficient R is around 0.99. The main advantage of calculating the proposed HI is that it is not necessary to collect data of the full cycle, i.e., from a fully charged state to a fully discharged state or vice versa, since it is calculated within a narrow SOC interval where the voltage–SOC relationship is very linear. This interval is found within the usual range of most practical charge and discharge cycles, despite their profiles. Many approaches found in the technical literature require the information of the complete charge–discharge cycles [25,26,27,28]; however, these approaches are not realistic for practical applications, thus this paper contributes in this area. Since the method proposed in this paper requires only the information of a narrow SOC interval, only a small fraction of the data points of a complete charge–discharge cycle are required, thus reducing the storage and computational resources while also providing similar results to other methods. Finally, by using the battery model defined by the Nernst equation, the behavior of future charge–discharge cycles can be accurately predicted using experimental data, as shown by the results presented in this paper.

The rest of the paper is organized as follows. Section 2 describes the methodology proposed in this paper to determine the HI. Section 3 describes the battery dataset used in this paper. Section 4 presents and discusses the experimental results. Finally, Section 5 concludes this paper.

2. Methodology

Several mechanisms are known to affect the lifetime of Li-ion cells. These include an increase in internal impedance, loss of active material due to mechanical stress caused by structural changes in the anode and cathode, or loss of cyclable lithium caused by the growth of solid electrolyte interphase at the anode due to lithium consumption and electrolyte decomposition [29]. According to [30], degradation mechanisms of Li-ion batteries can be divided into loss of active materials in the electrodes and loss of lithium inventory (LLI), although it is not easy to distinguish the aging mechanism. Therefore, it is necessary to quantify the degradation level of the battery in order to assess the health status of the battery.

2.1. Battery Degradation Quantification

Probably the most common way to quantify the state of health of a cell is its capacity, expressed in Ah, which is directly related to the amount of charge it can hold. It is defined as

$$C={\displaystyle {\int }_{t_0}^{t_f}i(t)dt} [\mathrm{Ah}]$$

(1)

Note that (1) corresponds to the Coulomb counting method, which integrates the current over time over one charge cycle, where the cell is completely discharged at t₀ and fully charged at t_f. The degradation or fading of the cell capacity in cycle n, ΔC_n, can be calculated as

$$\Delta C_n=C_{rated}-C_n\hspace{1em}[\mathrm{Ah}]$$

(2)

where n = 1, 2, 3, …, n_max is the number of cycles and n = 1 corresponds to the initial cycle of the battery where C_{n = 1} = C_rated.

The SOC of a battery is related to the remaining charge Q_remaining [C] in the battery; so, it is often defined as the ratio of Q_remaining to the rated charge Q_rated as [5]

$$SOC(t)=\frac{Q_{remaining}(t)}{Q_{rated}(t)} [-]$$

(3)

The SOC can be calculated using the Coulomb counting method, which integrates the cell current over time during the usage period. The SOC at any time t of the discharging phase can be calculated as

$$SOC(t)=SO{C}_0-\frac{1}{C_{rated}}{\displaystyle {\int }_{t_0}^{t}i(t)dt} [-]$$

(4)

where SOC₀ is the state of charge at the initial time of t₀; C_rated [Ah] is the rated capacity of the cell, which is obtained from the cell manufacturer; and i(t) is the instantaneous cell current. The current i(t) in (4) is positive when discharging and negative when charging.

Although the SOH has different definitions, it is related to the ratio of the effective cell capacity to the initial capacity. One possible definition is based on the ratio of the difference between the actual cell capacity C_actual and the end-of-life capacity C_EOL (70% of C_initial) [31] and the difference between the initial capacity C_initial and the end-of-life capacity C_EOL [10], which can be expressed as

$$SOH=\frac{C_{actual}-C_{EOL}}{C_{initial}-C_{EOL}} [-]$$

(5)

However, in practical applications, it is difficult to measure the actual capacity value of the cell because it is necessary to integrate (1) from a fully discharged state to a fully charged state. In this paper, it is proposed to use an indirect method to estimate the current capacity of the battery. This method is based on the calculation of the root mean square (RMS) of the difference ΔV_RMS between the first discharge cycle of the cell, which is taken as the reference cycle, and the current cycle. Note that the ΔV_RMS can be used as an indicator of the state of health of the cell. The ΔV_RMS,n [V] indicator is calculated at a generic cycle n in the interval [SOC_min, SOC_max] as follows:

$$\Delta V_{\mathrm{RMS},n}=\sqrt{\frac{1}{SO{C}_{\max }-SO{C}_{\min }}{\displaystyle {\int }_{SO{C}_{\min }}^{SO{C}_{\max }}{\left[V_{ref}(SOC)-V_n(SOC)\right]}^{2}dSOC}} [\mathrm{V}]$$

(6)

where n = 1 corresponds to the initial or reference cycle and n is a generic cycle.

Figure 1 explains the ΔV_RMS indicator. It shows the evolution of the cell terminal voltage versus the SOC value calculated from (4) during the first discharge cycle and during the n-th cycle.

In this paper, the SOC interval [SOC_min, SOC_max] = [0.55, 0.75] is chosen to calculate the ΔV_RMS indicator. Since the proposed ΔV_RMS indicator is calculated in a narrow SOC interval, this HI is quite flexible because it does not require the cell to be fully charged and discharged. As seen in Figure 1, the linear region of the voltage vs. SOC curve falls within this interval. This interval is typically within the normal transit area for most charge and discharge cycles, regardless of their profile.

2.2. Correlation between ΔV_RMS and the Cell Capacity Fade ΔC

As explained, due to the impossibility of a direct measurement of the capacity fade ΔC in practical charge–discharge cycles, this paper proposes an indirect measurement based on the indicator ΔV_RMS.

Since the values of ΔC and ΔV_RMS are on different scales, they have to be normalized for a fair comparison. The min–max normalization is used for this purpose because it is one of the most commonly used methods to normalize data, which can be calculated as [32],

$$x_{norm}=\frac{x-x_{\min }}{x_{\max }-x_{\min }}$$

(7)

where x is ΔC or ΔV_RMS, and x_min and x_max are the minimum and maximum values of x, respectively.

$$\Delta C_{norm}=\frac{\Delta C_n-0}{0.3C_{rated}-0}$$

(8)

$$\Delta V_{RMS,norm}=\frac{\Delta V_{RMS,}{}_n-0}{(\Delta V_{RMS,}{}_n-\Delta V_{RMS,}{}_{cut-off})-0}$$

(9)

In this paper, the cell is considered to reach its end of life when C = 0.7 C_rated; so, ΔC_end-of-life = 0.3 C_rated [31]. ΔV_RMS,cut-off is obtained from the manufacturer datasheet.

However, the behavior and accuracy of ΔV_RMS in quantifying the capacity fade needs to be evaluated. The Pearson correlation R is a useful statistic for this purpose. Figure 2 compares the behavior of the two indicators ΔC and ΔV_RMS for one of the cells analyzed, where a strong correlation is seen with R = 0.99. More details can be found in Section 4.

2.3. Forecasting Methods

As explained, to predict the future evolution of the cell capacity, it is necessary to apply suitable mathematical methods. In this section, it is proposed to apply two approaches, namely, linear fitting model (LF) and autoregressive integrated moving average (ARIMA).

As shown in Figure 3, the forecasting algorithm predicts the value of ΔC or ΔV_RMS of the cell at the current cycle according to the degradation model built from the evolution of ΔC or ΔV_RMS. Positioned at the current cycle, the forecasting algorithm takes the past values of ΔC or ΔV_RMS in the training interval (see the gray area in Figure 3) and fits these measured values to a straight line in the case of the LF method or uses lagged values of ΔC or ΔV_RMS and lagged errors in the case of ARIMA, as explained in the following paragraphs.

ARIMA was developed in the 1970s [33] and has since been widely used for forecasting and time series analysis [34,35]. ARIMA builds a time domain forecasting model based on the most recent measurements combined with the long-term historical trend [36]. ARIMA models have been used to forecast time series of economic variables [37], pandemic data [38], climate-related variables [26], the remaining useful life of fuel cells [39], and the behavior of electrochemical batteries, among other applications. Nonseasonal (p, d, q) ARIMA models are based on p autoregressive terms, d nonseasonal differences, and q lagged forecast errors. A (p, d, q) ARIMA model can be expressed as

$${\widehat{y}}_j={\alpha }_1{\Delta }^d{y}_{j-1}+\dots +{\alpha }_p{\Delta }^d{y}_{j-p}+{\epsilon }_j+{\gamma }_1{\epsilon }_{j-1}+\dots +{\gamma }_q{\epsilon }_{j-q}$$

(10)

where y_j and ${\widehat{y}}_j$ denote the measured and predicted values of ΔC and ΔV_RMS, respectively, of the cell at the time instant jΔT, where ΔT is the sampling time; ${\epsilon }_j=y_j-{\widehat{y}}_j$ is an error term; ${\Delta }^d{y}_j={(1-L)}^d{y}_j$; and L is the lag operator, i.e., Ly_j = y_j−1 and L^dy_j = y_j−q. Thus, y_j is obtained by applying d successive integrations to ${\Delta }^d{y}_j$.

2.4. Nernst Equation

In this work, the Nernst model is used to predict the pattern of a future discharge cycle using data from the reference cycle and the estimated degradation data. If the SOC is known, the terminal cell voltage can be determined by applying the Nernst equation [40,41] as

$$v_n(t)=E_0-R{i}_n(t)+{\mu }_1\ln [SOC(t)]+{\mu }_2\ln [1-SOC(t)]$$

(11)

where v_n(t) and i_n(t) are the terminal cell voltage and current (which is positive during the discharge stage and negative during the charge stage) at the n-th cycle, E₀ is the standard cell potential, R is the internal resistance of the cell, and μ₁ and μ₂ are constant parameters. For a set of m measurements at different times jΔT in the same cycle, where j = 1, 2, 3, …, m, the set of parameters in (11) can be determined as

$$\left[\begin{array}{c}{E}_0\\ R\\ {\mu }_1\\ {\mu }_2\end{array}\right]={\left[\begin{array}{cccc}1& i_{n,1}& \ln (SO{C}_{n,1})& \ln (1-SO{C}_{n,1})\\ 1& i_{n,2}& \ln (SO{C}_{n,2})& ln(1-SO{C}_{n,2})\\ \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots \\ 1& i_{n,m}& \ln (SO{C}_{n,m})& \ln (1-SO{C}_{n,m})\end{array}\right]}^{-1}\left[\begin{array}{c}{v}_{n,1}\\ v_{n,2}\\ \dots \\ \dots \\ v_{n,m}\end{array}\right]$$

(12)

where v_i,j is the terminal cell voltage at cycle number i, with i = 1, 2, …, n, and at the j-th sample of the cycle, where j = 1, 2, …, m.

To predict the behavior of a future cycle (terminal voltage vs. SOC) at different stages of cell aging, the following steps are proposed:

▪

Reference cycle. The instantaneous values of current and battery terminal voltage are measured during the full discharge cycle and the instantaneous values of the SOC are calculated. From these, the four parameters of the Nernst equation are determined.

▪

Future cycle n. By applying the LF or ARIMA forecasting methods, the estimated values of the ΔV_RMS,n and the capacity fade ΔC_n of cycle n are determined. If the rated capacity C_rated of the cell and the capacity fade at the n-th cycle ΔC_n are known, the state of charge SOC_n at the n-th cycle can be calculated as

$$SO{C}_n(t)=\frac{1}{C_{rated}-\Delta C_n}{\displaystyle {\int }_0^{t}i(t)dt}$$

(13)

Finally, once the SOC_n of the n-th cycle is known, the terminal cell voltage v_n for this cycle is obtained from ΔV_RMS,n and (11) so that the terminal voltage is predicted vs. the SOC characteristic curve of the n-th cycle.

3. Data Description

The data for the cells studied in this paper were obtained from the NASA Ames Prognostic Center of Excellence (PCoE) public database [42]. This database contains the terminal voltage v(t) and current i(t) measurements of the charge and discharge cycles and impedance of several commercial 18650 Li-ion cells under various conditions with a non-uniform sampling rate. Cell voltage, current, and temperature measurements were performed using a PXI-chassis-based DAQ. The measurements are grouped into six different datasets. NASA Ames researchers recommend dataset number 6 for predicting cell remaining charge and remaining useful life (RUL).

Table 1. Cell degradation test parameters.

Cell Name	Number of Cycles	Discharged Voltage	Discharge Current	Charged Voltage	Charge Current	Ambient Temperature	End of Life	Rated Capacity
B0005	168	2.7 V	2.0 A	4.2 V	1.5 A	24 °C	30% fade in rated capacity(2–1.4 Ah)	2 Ah
B0006	168	2.5 V
B0007	168	2.2 V
B0018	132	2.5 V

summarizes the key parameters during the cell degradation tests performed at a constant discharge current.

The algorithms and methods used in this paper were programmed by the authors of this work in the Matlab^® R2022b environment.

4. Results

This section details the tests performed in this paper, and explains and discusses the results obtained. As explained, the method proposed in this paper takes the first discharge cycle as the reference cycle. Based on the measured values of the terminal cell voltage and current, the capacity of the cell and the parameters of the Nernst model are determined in the first cycle. According to the process shown in Figure 1, the ΔV_RMS,n health indicator is calculated in each cycle for all analyzed cells.

4.1. Cell Capacity Fade Measurement and Estimation

Figure 4 shows the evolution of the capacity degradation ΔC of cell B0005 vs. the number of cycles as well as the evolution of the predictor ΔV_RMS. Note that ΔC is calculated directly from the experimental data by applying (2), whereas ΔV_RMS is calculated by applying (6) from the RMS value of the difference between the cell terminal voltage of the initial or reference cycle until the last cycle.

The results presented in Figure 4 show a similar trend of ΔC and ΔV_RMS. To quantify the degree of correlation,

Table 2. Parameters of the analyzed cells when discharged at a C-rate of 1.

Dataset	Cell Name	Cycles Considered	Correlation Coefficient R of ΔVRMS vs. ΔC
6	B0005	168	0.991
	B0006	168	0.990
	B0007	168	0.993
	B0018	132	0.982

shows the values of the Pearson correlation coefficient R between the capacity degradation ΔC and the proposed estimator ΔV_RMS for the four analyzed Li-ion cells calculated over the whole lifetime of the cells.

The results presented in Table 2 show a high value of the correlation coefficient R for all analyzed cells throughout their entire lifetime, thus validating the usefulness and accuracy of the ΔV_RMS capacity degradation estimator. Since it is demonstrated that ΔV_RMS is a good estimator of the capacity fade, it is used to evaluate the SOH of the cell.

4.2. Data Filtering

As observed in the previous subsection (see Figure 4), raw ΔC values and calculated ΔV_RMS values show a significant ripple due to internal chemical changes of the cell [43], mainly due to the recovery effect. The recovery effect is due to an overhang of the negative electrode over the positive electrode due to a design feature of the cells. This is a common effect found in aging tests, although it complicates the design of RUL forecasting tools [44,45]. In order to improve the modeling accuracy and to reduce the dependence of the results on the internal chemical changes, these data are filtered by discrete wavelet transform (DWT). Although other filtering approaches are possible [22], in this paper, a three-stage DWT was applied to the calculated ΔV_RMS values because it offers several advantages, such as low computational complexity, a simultaneous localization in the time and frequency domains, and the ability to separate the fine details of the signal. The applied process is summarized in Figure 5, which shows the original ΔV_RMS signal, the approximated or filtered signal, and the three details (high-pass filters) removed from the signal corresponding to the successive application of low-pass and high-pass filters made by the three-level DWT. It is noted that the discrete Meyer mother wavelet has been applied because it is commonly used in fault diagnosis approaches [46].

Figure 6 shows the results of the three-level DWT filter. It compares the original values of the ΔV_RMS with the filtered ones vs. the number of charge and discharge cycles of the cells.

As can be seen in Figure 6, the DWT process applied to the ΔV_RMS signal can significantly reduce the noise of the time series while preserving the overall trend, thus also increasing the quality of the data used during the subsequent forecasting stage.

4.3. Capacity Fade Forecasting Results

This section analyzes the evolution of the capacity fade of the four analyzed cells with the number of charge–discharge cycles.

Figure 7 shows three ARIMA predictions at different cycles for each analyzed cell. The training range varies between 30% (left), 50% (middle), and 80% (right) of the cell lifetime. Each plot shows the behavior of the measured capacity fade ΔC (black line, which corresponds to the objective time series) and the forecasting results using the proposed health indicator ΔV_RMS to estimate ΔC (blue line) and the forecast directly using ΔC (red line). Note that the solid lines of ΔV_RMS and ΔC refer to known values while the dashed lines refer to predicted values.

From Figure 7, it can be seen that both forecasting methods improve their accuracy as more data points are available. It can also be seen that both methods give similar results for the same initial conditions.

To measure the performance of the methods,

Table 3. Comparative RMSE results between the proposed health indicator ΔV_RMS and the capacity fade ΔC with respect to the measured capacity fade ΔC (objective time series) for each studied cell with a discharge current of 2 A (C-rate of 1).

Cell		B0005		B0006		B0007		B0018
Forecast Start Point (% of Total Useful Life)	Method	ΔVRMS [Ah]	ΔC [Ah]	ΔVRMS [Ah]	ΔC [Ah]	ΔVRMS [Ah]	ΔC [Ah]	ΔVRMS [Ah]	ΔC [Ah]
30%	LF	0.375	0.220	0.275	0.067	0.267	0.156	0.226	0.149
	ARIMA	0.235	0.164	0.230	0.054	0.168	0.067	0.172	0.049
40%	LF	0.303	0.136	0.134	0.128	0.181	0.071	0.219	0.050
	ARIMA	0.068	0.026	0.083	0.191	0.161	0.124	0.263	0.056
50%	LF	0.204	0.052	0.070	0.182	0.091	0.024	0.167	0.046
	ARIMA	0.059	0.024	0.080	0.089	0.060	0.026	0.077	0.073
60%	LF	0.154	0.027	0.065	0.137	0.066	0.029	0.089	0.064
	ARIMA	0.054	0.030	0.033	0.064	0.036	0.023	0.044	0.056
70%	LF	0.110	0.029	0.070	0.113	0.045	0.036	0.046	0.078
	ARIMA	0.060	0.025	0.020	0.033	0.040	0.015	0.040	0.087
80%	LF	0.075	0.037	0.064	0.081	0.030	0.036	0.056	0.092
	ARIMA	0.038	0.034	0.024	0.033	0.021	0.022	0.023	0.037

shows the root-mean-square error (RMSE) of the blue (ΔV_RMS prediction) and red lines (ΔC prediction) of Figure 7, which is calculated against the black line (objective data series) at different cycles.

The results presented in Table 3 show a trend such that the prediction accuracy improves with the amount of data considered (number of cycles in the training set). The mean values of the RMSE for the ARIMA forecasting are 0.064 and 0.046 for ΔV_RMS and ΔC, respectively. In the case of LF, the mean values are 0.112 and 0.069 for ΔV_RMS and ΔC, respectively. Since the mean RMSE values of ΔV_RMS and ΔC are similar, it can be concluded that ΔV_RMS can be used as an estimator of the capacity fade ΔC, while requiring less data (see Figure 1). It is also shown that ARIMA generally provides better forecasting results than the LF method.

Figure 8 shows the experimental values of the cell terminal voltage vs. the SOC at the reference cycle (blue line, using the Nernst fitting from experimental data) and at different cycles (black dashed lines), and the predictions of future discharge cycles (solid magenta line) using the Nernst model described by (11) and ΔV_RMS,n and ΔC_n. The SOC is calculated by applying (13) once the estimated value of ΔC_n is known, while the cell terminal voltage in Figure 8 is the voltage obtained from (11) minus ΔV_RMS,n. The magenta line (predicted future discharge cycles) should be very close to the black dashed line corresponding to the objective data.

The results presented in Figure 8 show that the discharge behavior depends on the predicted values of ΔV_RMS,n and ΔC_n at cycle n. However, it accurately approximates the discharging behavior when the same discharge current is applied.

Table 4. Comparative results between the experimental measurements of the cell terminal voltage vs. SOC and the Nernst equation adjusted with the proposed degradation model with a discharge current of 2 A (C-rate of 1).

Cell	B0005	B0006	B0007	B0018
Forecast Start Point (% of Total Useful Life)	RMSE	RMSE	RMSE	RMSE
25%	0.221	0.172	0.101	0.134
50%	0.125	0.145	0.075	0.077
80%	0.135	0.616	0.068	0.307

shows the RMSE values between the measured data (dashed black lines) and predictions made by the proposed degradation model and the Nernst equation (solid magenta lines).

From the results presented in Figure 8 and Table 4, it can be concluded that at a given cycle n, if some known discharge conditions are expected, the discharge behavior of the system can be approximated using the proposed ΔV_RMS health indicator.

4.4. Capacity Fade Forecasting at Different Discharge Conditions

This section analyzes the evolution of the capacity fade of a fifth cell that was discharged at a discharge C-rate of 2, i.e., at a discharge current of 4 A instead of 2 A, as was performed in the previous section.

Figure 9 shows three ARIMA predictions at different cycles for each analyzed cell. The training range varies between 30% (left), 50% (middle), and 80% (right) of the cell lifetime. Each plot shows the behavior of the measured capacity fade ΔC (black line, which corresponds to the objective time series) and the forecasting results using the proposed health indicator ΔV_RMS to estimate ΔC (blue line) and the forecast directly using ΔC (red line). The solid lines of ΔV_RMS and ΔC refer to known values, while the dashed lines refer to predicted values.

Table 5. Comparative RMSE results between the proposed health indicator ΔV_RMS and the capacity fade ΔC with respect to the measured capacity fade ΔC (objective time series) for cell B0034 with a discharge current of 4 A (C-rate of 2).

Forecast Start Point (% of Total Useful Life)	Method	ΔVRMS [Ah]	ΔC [Ah]
30%	LF	0.149	0.052
	ARIMA	0.161	0.034
40%	LF	0.101	0.058
	ARIMA	0.094	0.046
50%	LF	0.072	0.095
	ARIMA	0.070	0.043
60%	LF	0.061	0.093
	ARIMA	0.057	0.044
70%	LF	0.047	0.083
	ARIMA	0.050	0.032
80%	LF	0.023	0.065
	ARIMA	0.051	0.058

shows the root-mean square error (RMSE) of the blue (ΔV_RMS prediction) and red lines (ΔC prediction) of Figure 9, which is calculated against the black line (objective data series) at different cycles.

The trend in the results presented in Table 5 is that the prediction accuracy improves with the number of cycles in the training set. The mean values of the RMSE for the LF forecasting are 0.076 and 0.074 for ΔV_RMS and ΔC, respectively. In the case of ARIMA the mean values are 0.080 and 0.043 for ΔV_RMS and ΔC, respectively. Since the mean RMSE values of ΔV_RMS and ΔC are similar, it can be concluded that ΔV_RMS, can be used as an estimator of the capacity fade ΔC, even with a C-rate of 2.

Figure 10 shows the experimental values of the terminal voltage of cell B0034 vs. the SOC at the reference cycle (blue line, using the Nernst fitting from experimental data) and at different cycles (black dashed lines) and the predictions of future discharge cycles (solid magenta line) using the Nernst model described by (11) and ΔV_RMS,n and ΔC_n. The SOC is calculated by applying (13) once the estimated value of ΔC_n is known, while the cell terminal voltage in Figure 8 is the voltage obtained from (11) minus ΔV_RMS,n. The magenta line (predicted future discharge cycles) should be very close to the black dashed line corresponding to the objective data.

The results presented in Figure 10 show that the discharge behavior based on the predicted values of ΔV_RMS,n and ΔC_n at cycle n approximates the measured discharging behavior, thus validating the usefulness of the proposed approach.

Table 6. Comparative RMSE results between the experimental measurements of the terminal voltage for cell B0034 vs. SOC and the Nernst equation adjusted with the proposed degradation model with a discharge current of 4 A (C-rate of 2).

Cell	B0034
Forecast Start Point (% of Total Useful Life)	RMSE
25%	0.105
50%	0.072
80%	0.070

shows the RMSE values between the measured data (dashed black lines) and predictions made by the proposed degradation model and the Nernst equation (solid magenta lines).

The results presented in Figure 10 and Table 6 exhibit that, at a given cycle n, the discharge behavior of the system can be approximated using the proposed ΔV_RMS health indicator at different discharge currents.

5. Conclusions

The efficiency and performance of a battery is highly dependent on its past and present use. To ensure optimal performance, it is of utmost importance to develop battery grading techniques based on fast methods to determine battery degradation in terms of capacity fade, health status, and remaining useful life. Signal processing plays a leading role in the development of predictive and prognostic methods to determine the state of health and degradation status. This paper has proposed and evaluated the behavior of a new health indicator to estimate the capacity fade of lithium-ion batteries, which is directly related to the state of health. It has been shown that the proposed estimator is highly correlated with the capacity fade since the Pearson correlation coefficient R between the two indicators is around 0.99 for the analyzed batteries. The proposed health indicator is calculated within a narrow SOC interval where the voltage–SOC has a very linear relationship. This narrow SOC interval is within the normal range of most practical charge–discharge cycles, so it is not necessary to collect data from full charge and discharge cycles. Thus, only a small fraction of the data points of a full charge–discharge cycle are required, minimizing computational and storage resources while providing accurate results. The results presented in this paper have shown that by using the Nernst battery model, the behavior of future charge–discharge cycles can be accurately predicted. The approach proposed in this paper could be applied in many applications where the batteries are not fully charged or discharged, such as in electric mobility or smartphone applications. Finally, the method proposed in this paper is based on various signal processing methods such as discrete wavelet filtering and prediction methods based on linear fitting and autoregressive integrated moving average algorithms.