# Prediction of the Remaining Life of a Lithium-Ion Battery Using a Frugal Data-Based Approach

Antoine Gaignard, Christophe Farges, Franck Cazaurang

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Antoine Gaignard, Christophe Farges, Franck Cazaurang. Prediction of the Remaining Life of a Lithium-Ion Battery Using a Frugal Data-Based Approach. 2025 6th International Conference on Control and Fault-Tolerant Systems (SysTol), Oct 2025, Ayia Napa, France. pp.186-191, 10.1109/Sys-Tol66549.2025.11267345. [hal-05489704](https://hal.science/hal-05489704v1)

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# Prediction of the remaining life of a lithium-ion battery using a frugal data-based approach

Antoine Gaignard<fnref n="1" />, Christophe Farges<fnref n="1" /> and Franck Cazaurang<fnref n="1" />

*Abstract*—Predictive maintenance focuses on predicting the end-of-life of components to optimize human intervention and parts usage, though it necessitates continuous monitoring. A key aspect of predictive maintenance is calculating the Remaining Useful Life (RUL), which estimates how much longer a system can function effectively. Traditionally, there are three main approaches to calculating RUL: model-based, data-driven, and hybrid methods. Data-driven approaches, whether statistical or neural network-based, typically require large datasets. This paper proposes a model-free approach to calculate the RUL of lithium-ion batteries. The method uses a polynomial to approximate capacity over cycles, with the coefficients determined through least mean squares optimization under linear constraints. With an average prediction horizon of 30% remaining lifetime, this approach is both computationally efficient and minimizes data requirements.

## I. INTRODUCTION

Maintenance can be classified into three primary approaches: curative, preventive, and predictive [1]. Curative maintenance is performed after a failure occurs, but it carries the risk of unexpected downtime. Preventive maintenance is scheduled at fixed intervals to reduce the likelihood of incidents, though it may lead to suboptimal use of parts, causing an increase in the operating costs. Predictive maintenance forecasts the end-of-life of components, optimizing human intervention and part usage, though it requires continuous monitoring [2], [3]. A key element of predictive maintenance is calculating the Remaining Useful Life (RUL), which estimates how much longer a system will function effectively.

There are several methods to determine the RUL, including data-driven techniques such as statistical methods and neural networks. These methods rely on the existing data from previous devices, the training data, to make predictions. They range from a simple probabilistic approach, such as the average lifetime of the device and its associated standard deviation [4], to more complex trends and characteristics identification with deep learning methods. The main advantage of data-driven approaches is that the exact behaviour and the physical equations behind it are not needed. On the other hand, since these approaches require training data, scarcity of accurate data can lead to inaccurate predictions [5]. The other drawbacks include the time required by those methods, as training a network can take several hours to several days depending on the training data size and the architecture of the network.

Another possibility for RUL estimation is a model-based approach. It relies on the knowledge of the physical equations to construct a model. This is why model-based approaches are also named physics-based approaches. One of the main advantages of these approaches is that they do not need large quantities of data to accurately identify the coefficients and parameters of the equations [6]. Another advantage is that the results are easily explained since they stem from the physical equations. Lastly, unlike data-based methods that require the same experimental setup, namely the same solicitations and the same environmental conditions, model-based method can be used with different experimental setup [7]. The main drawback of physics-based methods is that they require a comprehensive knowledge of the phenomenons to be modelled [8]. Also, some equations used to create the models can be empirical relationship, for example the Arrhenius equation in chemical cinetic, blurring the line between model and data.

The third possibility is a hybrid method that combines both data-based and model-based methods [9]. They are generally helpful when a system has several physical phenomenons, but one of them is not well understood or necessitates too many equations to be accurately described. The well-known or easily described phenomenons are put in equations and the other phenomenons are replaced with a neural network [10]. These methods do not require a deep knowledge of the physics thanks to the data-driven parts and can be used with a wide range of experimental setup thanks to the model-based part.

This paper focuses on calculating the RUL for a lithium-ion battery using a model-free approach that necessitates a low amount of data. It is difficult to classify our proposed method among the aforementioned categories. Indeed, unlike classical data-based methods that require to know the evolution of several batteries' capacity to train a network or use an average lifespan value, our method only requires the battery's capacity history, up to the current cycle. Moreover, unlike model-based method that require the equations to describe the chemical reaction inside the battery and the various processes lowering the capacity over the time, our method makes use of a polynomial approximation to represent the evolution of the capacity over the cycles. As a result, our method requires a low amount of computational power and time. The main limitation of our approach is that the early RUL estimations are less precise than data-based methods.

The proposed approach is applied to the experimental data from the CALCE database [11]. This dataset needs some prefiltering for an optimal result, hence, in the rest of the paper the measured capacity refers to the capacity

<footnote n="1">¹All Authors are with Univ. Bordeaux, CNRS, Bordeaux INP, IMS, UMR 5218, F-33400, Talence, France firstname.lastname@ims-bordeaux.fr</footnote>
that has been prefiltered. Capacity values are collected at regular intervals and used to optimize the coefficients of a polynomial function that predicts future capacity values. In this paper, we propose to obtain the polynomial coefficients as the solution of an optimization problem with a least square objective under linear constraints, making it efficient in terms of computation resources and data required. The resultant polynomial is then used to predict future capacity. The RUL is then obtained as the time remaining until the battery's capacity reaches a specified threshold, typically 70% of the initial capacity [12]. Thanks to the low computational cost of this process, it is then possible to iterate it at each new battery cycle in order to update the polynomial coefficients, thereafter refining the RUL estimation.

The rest of the paper is divided in four parts. Section II is a more detailed presentation of the CALCE database and shows how a well-chosen polynomial can accurately describe the evolution of the battery capacity over the cycles. Section III covers the optimization problem, notably the linear constraints that are a major components of our approach. It also covers the results of the RUL estimation done for 6 batteries of the database. In section IV, several metrics are used to evaluate RUL performances and our results are compared to other methods from the literature, especially data-driven methods. Finally, section V is a short conclusion.

## II. EVOLUTION OF THE BATTERY CAPACITY WITH AGEING

### A. Presentation of the database

The experimental data used in this paper comes from the database of the CALCE research team at the A. James Clark School, University of Maryland, USA. Specifically, the cycling tests for the CS2-33, CS2-34, CS2-35, CS2-36, CS2-37 and CS2-38 batteries are used. They are part of a batch of CS2 type batteries, which are lithium-ion batteries with a cobalt oxide cathode (LiCoO₂). They have a nominal capacity of 1.1 Ah and are of prismatic geometry. Batteries 33 and 34 were cycled with a constant current of 0.55 A (0.5 C) and batteries 35 to 38 at 1.1 A (1 C). These batteries have already been used, for example, in the following papers, [13], [14] and [15].

The typical measure for calculating the RUL of a battery is its capacity, which can be determined by integrating the current delivered during the discharge phase (a method known as Coulomb counting). In the dataset used, each discharge and charge cycle is complete, ranging from 0% to 100% State Of Charge (SOC). By convention, a SOC of 100% represents the maximum energy the battery can store in its current condition, even if this maximum is lower than the initial capacity.

The curve showing the evolution of the capacity over the cycles are drawn in Figs. 2 and 1 by taking the capacity at each cycle. The raw measurement curves contain numerous outliers, characterized by large downward and upward spikes caused by issues during data collection. To address this issue, the data undergoes pre-filtering. The filtering criterion is set such that if the capacity decreases by more than a

threshold of 0.1 A in a single cycle (roughly 100% of the initial capacity), the data point is considered an outlier and is filtered out.

Upon reviewing the data, it is observed that there are no consecutive outliers. The removed points are thus replaced with the average capacity between the preceding and succeeding cycles. The choice of 0.1 A as the threshold is heuristic, ensuring that false points are eliminated without affecting valid data. Replacing the outliers with averages assumes that the capacity drop is locally linear, which is a reasonable approximation given the small local variations in capacity.

After filtering the outliers as described, the resulting curves, shown in Fig. 2, are used to calculate the RUL.

### B. Polynomial approximation of the capacity evolution over the cycles

Although the batteries originate from the same production batch, there is significant variation in capacity across the batteries in the final cycles, as depicted in Fig. 2. Despite this variation, all six curves display a similar pattern in their evolution. Physically, the formation of an initial Solid Electrolyte Interface (SEI) layer leads to a sharp decline in capacity during the first few cycles. After this, a moderate decrease in capacity is observed, attributed to the growth of the SEI

<figure><img src="image_2.png" /><figcaption>Fig. 1. Raw capacities measurement over the cycles for six CS2 batteries</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. filtered capacities over the cycles for six CS2 batteries</figcaption></figure>
layer [16], along with other less significant phenomena, as discussed in [17]. Towards the end of the battery's life, the lithium plating phenomenon causes a rapid drop in capacity. To model these three stages in the battery's lifespan, this paper proposes the use of a third-order polynomial.

Fig. 3 displays the actual measured capacity of the battery CS2-33 and a third-order polynomial approximation of the form:

$$C_{\text{poly}}(N) = \frac{a}{3} \cdot N^3 + \frac{b}{2} \cdot N^2 + c \cdot N + d \quad (1)$$

where N denotes the variable associated with a cycle of charge and discharge.

<figure><img src="image_4.png" /><figcaption>Fig. 3. CS2-33 battery capacity and third-order polynomial approximation</figcaption></figure>

Fig. 3 confirms that the choice of a polynomial is appropriate to describe the evolution of capacity as a function of the number of cycles. A polynomial of order 3 allows the derivative function to be calculated explicitly and simply:

$$\frac{dC_{\text{poly}}}{dN} = a \cdot N^2 + b \cdot N + c, \quad (2)$$

as well as the second derivative, which cancels out at the inflection point:

$$N_I = \frac{-b}{2a}. \quad (3)$$

Equations (2) and (3) are used to set linear constraints during the optimisation of the polynomial coefficients in the next section.

### III. BATTERY CAPACITY PREDICTION FOR REMAINING USEFUL LIFE ESTIMATION

#### A. Polynomial coefficients optimisation with least square optimization under linear constraints

The proposed methodology for calculating the RUL relies on predicting battery capacity using a polynomial approximation, with the coefficients being optimized at regular intervals. The coefficients of the polynomial, denoted as $\theta$, are determined by solving the following optimization problem, which minimizes the 2-norm between the measurements and a polynomial subject to linear constraints:

$$\min_{\theta} \frac{1}{2} \|M \cdot \theta - C_{\text{mes}}\|_{2}^{2} \quad (4)$$

subject to:

$$\begin{bmatrix} -2N_{I}^{\min} & -1 & 0 & 0 \\ 2N_{I}^{\max} & 1 & 0 & 0 \end{bmatrix} \cdot \theta \le \begin{bmatrix} 0 \\ 0 \end{bmatrix} \quad (5)$$

$$\begin{bmatrix} N_{I}^{2} & N_{I} & 1 & 0 \\ N_{I}^{2} & N_{I} & 1 & 0 \\ \vdots & \vdots & \vdots & \vdots \\ N_{\text{total}}^{2} & N_{\text{total}} & 1 & 0 \end{bmatrix} \cdot \theta \le \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \quad (6)$$

where

$$M = \begin{bmatrix} 0 & 0 & 0 & 1 \\ \frac{1}{3} & \frac{1}{2} & 1 & 1 \\ \vdots & \vdots & \vdots & \vdots \\ \frac{N^3}{3} & \frac{N^2}{2} & N & 1 \end{bmatrix}, \quad C_{\text{mes}} = \begin{bmatrix} C_{\text{mes}}(0) \\ \vdots \\ C_{\text{mes}}(N) \end{bmatrix}, \quad \theta = \begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix} \quad (7)$$

with $C_{\text{mes}}(0)$ corresponding to the measured initial capacity and $C_{\text{mes}}(N)$ corresponding to the capacity measured at cycle N, from which the RUL is estimated.

The inequality (5) ensures that the inflection point is positioned neither too early nor too late in the battery lifespan. It can translate as:

$$N_{I}^{\min} \le N_{I} \le N_{I}^{\max} \quad (8)$$

For the considered batteries, we chose $N_{I}^{\min} = 150$ and $N_{I}^{\max} = 600$.

The inequality (6) imposes that the capacity is decreasing for all cycles, ensuring that the polynomial follows the continuous degradation of the battery:

$$\frac{dC_{\text{poly}}}{dN} \le 0. \quad (9)$$

Let $N_{\text{final}}$ be the numerical solution to the equation:

$$\frac{a}{3} \cdot N^3 + \frac{b}{2} \cdot N^2 + c \cdot N + d = 0.7 \cdot C_{\text{mes}}(0) \quad (10)$$

and N the current cycle.

Then the remaining number of life cycles is:

$$RUL(N) = N_{\text{final}} - N. \quad (11)$$

This optimization problem was solved using the MATLAB function `lsqlin`, which is based on the interior point algorithm [18]. The optimization was performed in Matlab R2024b on a Windows 11 system with an Intel Core i9-12900H processor and 64 GB of available RAM. The computation times for batteries 33, 34, 35, 36, 37, and 38 were 167 s, 170 s, 164 s, 265 s, 177 s, and 212 s, respectively. The computation time is calculated between the start of the first cycle and the end of the last cycle. Since the RUL estimation is done every 5 cycles, the computation time covers between 120 and 160 optimizations, depending on the battery. As a comparison with neural network approaches the computation time for a single cycle is 10 s for the simplest network, MLP and over 60 s for a more complex network like Dual-LSTM [19]. As on average there are 140 iteration the total time required is between 1400 s and 9000 s. Table I highlights that classical data-driven methods requires a lot of time contrary to the proposed method. Our proposed method, in order of magnitude, is in minute while neural networks are in hours.
<table><caption>TABLE I
COMPUTATION TIME COMPARISON BETWEEN PROPOSED THE APPROACH AND POPULAR NEURAL NETWORKS</caption><thead><tr><th></th><th>MLP</th><th>RNN</th><th>LSTM</th><th>GRU</th><th>Dual-LSTM</th><th>Detransformer</th><th>this paper (polynomial)</th></tr></thead><tbody><tr><th>total time (s)</th><td>1400</td><td>3500</td><td>8680</td><td>6860</td><td>8960</td><td>3780</td><td><b>193</b></td></tr></tbody></table>

## B. Results and graphical representation

Figs. 4 and 5 represent the fitted polynomials for several cycles.

<figure><img src="image_5.png" /><figcaption>Fig. 4. Polynomial approximation at 3 stages in the battery life (CS2 35)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Polynomial approximation at 3 stages in the battery life (CS2 36)</figcaption></figure>

The fitted polynomial follows the data very closely up to the cycle when the optimization is done, but for the future cycle it may deviate from the real data. The threshold at which the battery is considered unusable is denoted with the dotted horizontal black line. For the measured capacities this threshold is reached at cycle 546 for Fig.4 and 503 for Fig. 5. It means that the true End of Life (EOL) for the battery of Fig. 4 is 546 cycles. At cycle 546 the remaining useful life is then 0 cycle conversely at cycle 0 the RUL is 546 cycles. Thus the true RUL over the cycles can be represented by a line of slope -1 and y-intercept 546. Please note that the threshold is here set specifically at 80% of the initial capacity in accordance with article [15]. The estimated end of life is the cycle at which the fitted polynomial reach that same dotted horizontal line, and the RUL is then the

difference between the current cycle and the estimated EOL. For example, for cycle 421 the corresponding polynomial is drawn in magenta. It reaches the EOL for cycle 570 and 503. The current cycle is 421 so the RUL is $570 - 421 = 149$ and $468 - 421 = 47$. This calculation is plotted every fifth cycle and then all the calculated RUL are presented on the same graph, see Figs. 6 and 7.

<figure><img src="image_7.png" /><figcaption>Fig. 6. RUL estimation over the cycles for battery 33</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. RUL estimation over the cycles for the 6 batteries</figcaption></figure>

Fig. 7 is a representation of the estimated RUL for the six batteries discussed earlier. The x-axis represents the current number of battery cycles, while the y-axis shows the remaining cycles, as calculated for each cycle. In these plots, the diagonal lines with a slope of -1 represent the actual RUL.

The quality of the results can be assessed using several metrics [20], with the primary one being the prediction horizon. Let $N_{tube10}$ denote the cycle at which the RUL curve first enters a 10% band around the true RUL and
remains within it thereafter, and $N_{0.7}$ represents the cycle corresponding to the true end-of-life. The prediction horizon is then defined as:

$$P_H = N_{0.7} - N_{\text{tube10}}. \qquad (12)$$

The greater this value, the greater the time available to organize maintenance. For predictive maintenance Fig. 6 is very useful as it indicates at what time the RUL estimation can be used as a trusted signal to schedule the maintenance. Indeed maintenance requires an oversight of the system as a whole, during all its life rather than just at one cycle. Such a graph of RUL over the cycle shows that the proposed RUL estimation does not work only for a certain specific cycle, and that our method has a stable performance throughout the cycles. Finally, it denotes a coherent behaviour as the estimated RUL curve tends to a slope of -1, like the true RUL.

The calculation of the prediction horizons for the six batteries is summarized in Table II, ranging from 150 to 300 cycles, or between 25% and 50% of the battery's total lifespan.

<table><caption>TABLE II
SUMMARY TABLE OF RUL CALCULATIONS FOR THE 6 BATTERIES</caption><thead><tr><th></th><th>N<sub>0.7</sub></th><th>N<sub>tube10</sub></th><th>P<sub>H</sub></th><th>RUL at N<sub>tube10</sub> (%)</th></tr></thead><tbody><tr><td>bat 33</td><td>624</td><td>331</td><td>293</td><td>46.9</td></tr><tr><td>bat 34</td><td>610</td><td>386</td><td>224</td><td>36.7</td></tr><tr><td>bat 35</td><td>661</td><td>478</td><td>183</td><td>27.7</td></tr><tr><td>bat 36</td><td>623</td><td>476</td><td>147</td><td>23.6</td></tr><tr><td>bat 37</td><td>731</td><td>501</td><td>230</td><td>31.5</td></tr><tr><td>bat 38</td><td>794</td><td>511</td><td>283</td><td>35.6</td></tr><tr><td>mean</td><td>673</td><td>447</td><td>227</td><td>33.6</td></tr></tbody></table>

Similar to capacity, which shows significant variation, both the prediction and RUL also exhibit considerable dispersion. As seen in Fig. 6, the RUL is notably underestimated in the early cycles due to the rapid capacity drop caused by the formation of the initial SEI layer. During the intermediate cycles, the RUL tends to be overestimated, as multiple values of the polynomial coefficients may correspond to the same initial part of the curve before the inflection point occurs. After the inflection, the results stabilize, aligning with the final values of the coefficients.

## IV. COMPARISON WITH OTHER METHODS

In many papers the quality of the RUL prediction is not measured with the prognostic horizon, $P_H$, but with the Relative Error (RE), Absolute Error (AE), Mean Absolute Error (MAE) and Root Mean Squared Error. They are defined as:

$$RE(N) = \frac{|N_{0.7} - RUL(N)|}{N_{0.7}} \qquad (13)$$

$$AE(N) = |N_{0.7} - RUL(N)| \qquad (14)$$

$$MAE(N) = \frac{1}{N_{0.7}} \sum_{i=1}^{N_{0.7}} |C_{\text{poly}}(i) - C_{\text{mes}}(i)| \qquad (15)$$

$$RMSE(N) = \sqrt{\frac{1}{N_{0.7}} \sum_{i=1}^{N_{0.7}} (C_{\text{poly}}(i) - C_{\text{mes}}(i))^2}. \qquad (16)$$

The prognostic horizon is a metric that focuses on the maintenance aspect of the RUL estimation. Indeed, when planning intervention it is crucial to know when a device will be offline so as not to disrupt the whole system. How the degradation evolves is not as useful to schedule maintenance. It is however useful for adapting the usage of a device to increase its lifespan. The four metrics (13) - (16) focus on measuring how close the predicted capacity is to the measured capacity, hence how the degradation evolves. As such they indicate if for a certain cycle the prediction sitcks to the measure but do not give an overview of the RUL estimation in general.

Table III presents the RE and RMSE metrics for the batteries 35, 36 and 37.

<table><caption>TABLE III
COMPARISON BETWEEN PREDICTION METHOD FROM [13] AND THIS PAPER</caption><thead><tr><th></th><th></th><th>bat 35</th><th>bat 36</th><th>bat 37</th></tr></thead><tbody><tr><td rowspan="2">RE (%)</td><td>article [13]</td><td><b>3.4</b></td><td><b>4</b></td><td><b>2.3</b></td></tr><tr><td>this paper</td><td>7.8</td><td>5.5</td><td>9</td></tr><tr><td rowspan="2">RMSE</td><td>article [13]</td><td>0.032</td><td>0.052</td><td>0.039</td></tr><tr><td>this paper</td><td><b>0.015</b></td><td><b>0.013</b></td><td><b>0.019</b></td></tr></tbody></table>

Article [13] is a data based method, using ELM network and LSTM network to estimate and predict the State Of Health (SOH) of the batteries from CALCE dataset, in the context of RUL estimation the SOH and capacity prediction are the same since SOH is the normalised capacity in [13]. As training set, the whole capacity degradation curve of a battery is used and then another battery capacity degradation curve is used as validation set. Focusing on the result from battery 37, the proposed method has an absolute error of 28 cycles which is a relative error of 5.5% whereas the two fully trained neural network in series have an absolute error of 22 cycles which is a relative error of 4%. Our proposed method performs almost as well as [13] especially given that the MAE and RMSE of the third-order polynomial are lower than those of the neural networks. It demonstrates that adequate polynomial coefficients can accurately describe the capacity fade and predict the future capacity degradation.

Table IV presents the relative error for battery CS2 35 at several cycles.

<table><caption>TABLE IV
COMPARISON BETWEEN PREDICTION METHODS FROM [15] AND THIS PAPER APPROACH FOR BATTERY CS2 35</caption><thead><tr><th>RE (%)</th><th>cycle 211</th><th>cycle 316</th><th>cycle 421</th><th>cycle 507</th></tr></thead><tbody><tr><td>model A [15]</td><td><b>13.1</b></td><td><b>12.3</b></td><td>32.9</td><td>3.9</td></tr><tr><td>model B [15]</td><td>∞</td><td>∞</td><td>12.6</td><td>∞</td></tr><tr><td>model C [15]</td><td>19.1</td><td>22.9</td><td><b>0.2</b></td><td>22.6</td></tr><tr><td>This paper</td><td>15.9</td><td>29.1</td><td>3.8</td><td><b>0.9</b></td></tr></tbody></table>

There are 3 models used in article [15], model A is a sum of 2 exponential, model B is a second-order polynomial and
<table><caption>TABLE V
EARLY RUL ESTIMATION FROM [14]</caption><thead><tr><th>Metric</th><th>MLP</th><th>RNN</th><th>GRU</th><th>DeTransformer</th><th>MMMe</th></tr></thead><tbody><tr><td>RE (%)</td><td>40</td><td>16</td><td>13</td><td>7.6</td><td>0.2</td></tr><tr><td>RMSE</td><td>0.10</td><td>0.11</td><td>0.09</td><td>0.07</td><td>0.03</td></tr></tbody></table>

model C is a model that combines both the exponential and the polynomial model. Focusing on the line of model B, it can be seen that a second order polynomial rarely converges, and so rarely is able to estimate a RUL. Then it is worth noting that on the later stages of the battery life the model C and our method are the most efficient at predicting the RUL. With less than 1% relative error, our method clearly well-estimates the RUL of battery 35.

The proposed approach is efficient when sufficient data are available, which means that early predictions lack accuracy. However, methods that use full capacity history as training data can start predicting accurately very early in the battery life. For example, in article [14] several popular neural network type are tested at cycle 65. Table ?? presents the early prediction for those networks. The comparison with our method is not relevant since cycle 65 is too early to plan maintenance since the prediction on one cycle is not enough to plot a graph similar to Fig. 6 which is the one useful to organise maintenance.

## V. CONCLUSION AND PERSPECTIVES

In this paper we propose a simple approximation of the battery capacity over the cycle using a third-order polynomial. This approximation makes possible the use of linear constraints during the optimization process. The proposed method enables accurate predictions starting from the midpoint of the battery's life, making it suitable for maintenance planning. With an average of over 30% of the remaining life when the prediction enters the 10% band around the actual RUL. This method is particularly valuable because it does not require large datasets or lengthy training times, unlike other statistical or neural network approaches. The authors intend to use these results as a benchmark for comparison with other methods, particularly model-based approaches.

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