【Intensive reading of nature papers】Impedance-based forecasting of lithium-ion battery performance amid uneven usage

[Intensive reading of nature papers] Impedance-based forecasting of lithium-ion battery performance amid uneven usage

Paper link: https://www.nature.com/articles/s41467-022-32422-w.pdf

Accurate forecasting of lithium-ion battery performance is essential for easing consumer concerns about the safety and reliability of electric vehicles. Most research on battery health prognostics focuses on the research and development setting where cells are subjected to the same usage patterns. However, in practical operation, there is great variability in use across cells and cycles, thus making forecasting challenging. To address this challenge, here we propose a combination of electrochemical impedance spectroscopy measurements with probabilistic machine learning methods. Making use of a dataset of 88 commercial lithium-ion coin cells generated via multistage charging and discharging (with currents randomly changed between cycles), we show that future discharge capacities can be predicted with calibrated uncertainties, given the future cycling protocol and a single electrochemical impedance spectroscopy measurement made immediately before charging, and without any knowledge of usage history. The results are robust to cell manufacturer, the distribution of cycling protocols, and temperature. The research outcome also suggests that battery health is better quantified by a multidimensional vector rather than a scalar state of health.

Accurate predictions of lithium-ion battery performance are critical to assuaging consumer concerns about the safety and reliability of electric vehicles. Most studies on battery health prediction focus on R&D settings where batteries are exposed to the same usage patterns. In practice, however, there is large variability in usage across cells and cycles, making prediction challenging. To address this challenge, we propose hereCombination of Electrochemical Impedance Spectroscopy and Probabilistic Machine Learning Methods. useA dataset of 88 commercial Li-ion coin cells generated by multi-stage charging and discharging (current varies randomly between cycles), we show that considering the future cycling protocol and charging immediately before theSingle electrochemical impedance spectroscopy measurement, and without any knowledge of usage history, one canPredicting Future Discharge Capacity Using Calibrated Uncertainty. Results are robust to cell manufacturer, distribution of cycling protocols, and temperature. The findings also show that battery health can be better quantified by multidimensional vectors rather than scalar states.

Electrification of the transportation industry is now taking place at an increasingly rapid pace, enabling significant strides towards a carbon neutral future. Fundamental to this transition has been the development of the lithium-ion battery, which powers the majority of electric vehicles (EVs) on the road today. Notwithstanding the environmental benefits of this transition, reliance on the lithium-ion battery poses significant challenges, with consumer concerns including range anxiety, fear of battery failure and charging time. Easing these concerns demands the ability to accurately forecast battery performance, and specifically when usage conditions are variable.

The electrification of transportation is happening at an increasing pace, allowing us to make significant progress towards a carbon-neutral future. Fundamental to this shift is the development of lithium-ion batteries, which power most electric vehicles on the road today. While this shift is good for the environment, the reliance on lithium-ion batteries poses significant challenges, with consumer concerns including range anxiety, battery failure and recharging times. Alleviating these concerns requires the ability to accurately predict battery performance, especially as usage conditions change.

The key challenge is the heterogeneity of the battery. Each user uses their car differently, and even across a single battery pack not all cells are necessarily charged or discharged with identical current.

The key challenge isCell heterogeneity. Every user uses a car differently, and even within a battery pack, not all cells have to be charged or discharged at the same current.

These differences mean that each cell’s internal state, including the extent of lithium plating or electrode cracking, can vary significantly both at an intra-pack and inter-pack level.

These differences mean that the internal state of each cell, including the extent of lithium plating or electrode cracking, can vary significantly at both the intra-pack and inter-pack levels.

To quantify the extent of degradation within cells, and to identify cells that have reached their End of Life (in EVs, this is typically defined as the point at which the discharge capacity has reduced to 80% of the nominal capacity), the scalar State of Health (SOH) metric is typically adopted, measured using previous cycle discharge capacity or internal resistance8–13. The problem with this approach is that batteries with the same numerical SOH do not necessarily exhibit identical levels of each degradation process (for example, lithium plating or electrode cracking), yet the impact of future cell usage on the cell’s future performance and degradation pathway depends significantly on the type of degradation that has already occurred14–16. Accurate forecasting of battery performance demands a non-invasive approach to acquire information about the cell state at a microscopic level.

To quantify the degree of degradation within a Lithium battery cell and to identify Lithium battery cells that have reached end-of-life (in EVs, this is typically defined as the point at which discharge capacity drops to 80% of nominal capacity), a scalar state of health is typically used (SOH) metric, using previous cycle discharge capacity or internal resistance measurements. The problem with this approach is that,Batteries with the same numerical SOH do not necessarily exhibit the same level of degradation in each degradation process (e.g., lithium plating or electrode cracking), but the impact of future battery use on the battery's performance and degradation pathways depends on the degradation that has already occurred. Accurate prediction of battery performance requires a non-invasive approach to obtain information on the state of the battery at the microscopic level.

Both short and long timescale forecasting of battery performance are of interest in battery prognostics. Over a short timescale, predicting how the battery would respond to a particular charging and discharging protocol can be used to develop optimal charging protocols. Short-term forecasting also encompasses SOH estimation: here, the aim is to predict the battery’s discharge capacity or internal resistance under a specific, standardised cycling protocol. Over a long timescale, the focus is on predicting the remaining useful life, end of life, or the ‘knee-point’ in the battery’s life trajectory at which degradation accelerates.

Both short- and long-scale predictions of battery performance are of interest in battery forecasting. In a short time, it is predicted that the battery willHow to respond to a specific charging and discharging protocol can be used to develop an optimal charging protocol. Short-term forecasts also includeSOH Estimation: Here, the goal is to predict the DC resistance capacity or internal resistance of the battery under a specific standardized cycling protocol. For a long time, research has focused onPredict remaining useful life, the end of life, or the "knee point" of accelerated degradation in the battery life trajectory。

Approaches to both types of forecasting can be subdivided into empirical, physics-based, and data-driven models, with some models being a hybrid of these.
Empirical approaches have been used to model long-term capacity fade with power laws but assume fixed operation over battery life and do not account for intrinsic differences in cell state at start of life. These approaches assume that all cells of the same chemistry will fade in the same way if operated in the same way, which is not observed in practice.
In physics-based approaches, the battery is either modelled mechanistically using first principles analysis of internal physical and electrochemical processes, or using equivalent circuit modelling, which models the cell as a circuit comprising resistors and capacitors that are representative of the underlying electrochemical processes. Mechanistic models aim to capture how the battery voltage responds to an externally applied current (or vice versa), which can be used to predict optimal charging protocols.However, the parameters of such models need to be updated for each individual cell and typically suffer from non-identifiability – several sets of model parameters could explain the observed data equally well, but would make drastically different predictions on test cells or on the same cell later in its life.

These two types of forecasting methods can be subdivided intoEmpirical, physics-based and data-driven models, and some models are hybrids of these models.
Empirical approaches have been used forSimulating long-term capacity decay with power law, but assumes constant operation during battery life, regardless of intrinsic differences in state at the beginning of battery life. These methods assume that all lithium battery cells of the same chemistry will fade in the same way if handled in the same way, which is not observed in practice.
In physics-based approaches , batteries are eitherMechanical modeling using first-principles analysis of the internal physical and electrochemical processes, or equivalent circuit modeling, where the battery is modeled as a circuit containing resistance and capacitance representing the underlying electrochemical process.The mechanical model is designed toCaptures how battery voltage responds to externally applied current (and vice versa), which can be used to predict optimal charging protocols. However, the parameters of this modelNeeds to be updated for each individual Li-ion cell and is often not identifiable - several sets of model parameters can explain the observed data equally well, but will have a negative effect on the test Li-ion cell or later on the same Li-ion cell. make very different predictions about the lifecycle of。

For circuit-based models, the parameters of the circuit can be fitted to either current-voltage data, or to electrochemical impedance spectra. The circuit parameters can then be used to forecast capacity degradation under standardised use conditions or to simulate the effect of different usage conditions on battery pack performance30. However, it is challenging to capture every degradation mode in an analytical model. Further, a new set of model parameters must be learnt for each cell from cycle to cycle, making it challenging to infer a general cell-to-cell model.

For circuit-based models , parameters of the circuit can be fitted to current-voltage data, or electrochemical impedance spectroscopy. Then, the circuit parameters can be used forPredict capacity degradation under standardized usage conditions or simulate the effect of different usage conditions on battery pack performance. However,Capturing each degradation mode in an analysis model is challenging. Furthermore, each cell must learn a new set of model parameters from cycle to cycle, making it challenging to infer a general cell-to-cell model。

Purely data-driven approaches to forecasting use raw data as input to a machine learning algorithm to forecast long term capacity fade, resistance increase and remaining useful life. Feature - based data-driven approaches applied machine learning on features extracted from the charging or discharging curve to predict discharge capacity, remaining useful life, and abrupt capacity decays.
Innovations in extracting features from charge/discharge curves and machine learning approaches for modelling time-series data have enabled significant improvements in the accuracy of predictions.
Further studies showed that using features of the discharge curve across a small number of initial cycles, it is possible to train machine learning models that can generalise to different cell chemistries.
Going beyond charging and discharging curves, approaches such as electrochemical impedance spectroscopy (EIS), early cycle Coulombic efficiency, current interruption and acoustic time-of-flight analysis have been used for degradation forecasting. These approaches provide a fuller description of battery state – for example, EIS captures the response of the cell over a broad frequency range, with different frequencies correlating to distinct physical, chemical and mechanical changes in the active material. Data - driven methods typically utilise data generated in the laboratory setting, where cells are charged and discharged in the same way over the entirety of their lifetimes, thus the impact of variable cell usage on future performance can be ignored (see Fig. 1). However, extrapolating the models developed for laboratory setting to field data or other realistic usage profiles such as the Worldwide Harmonized Light Vehicles Test Cycles (WLTC), where cells are cycled in vastly different ways over their lifetimes, has proved a major challenge.

A purely data-driven approach to forecastingUse raw data as input to machine learning algorithms to predict long-term capacity decline, drag increase, and remaining useful life. A data-driven approach applies machine learning toFeatures extracted from charge or discharge curves to predict discharge capacity, remaining useful life, and sudden capacity fading。

fromA Machine Learning Approach for Feature Extraction and Time Series Data Modeling in Charge-Discharge CurvesThe innovation has significantly improved the accuracy of forecasting.
Further studies showed that using the discharge profile features of a small number of initial cycles, it was possible to train machine learning models that generalized to different cell chemistries.

In addition to charge and discharge curves, methods such as electrochemical impedance spectroscopy (EIS), early-cycle Coulombic efficiency, current interruption, and acoustic time-of-flight analysis have been used for degradation prediction.
These methods provide a more comprehensive picture of the state of the battery—for example, EIS captures the battery's response over a wide range of frequencies, with different frequencies associated with different physical, chemical, and mechanical changes in the active material.
A data-driven approach usuallyUtilize data generated in a lab environment, where the batteryCharges and discharges the same way throughout its lifetime, so it can beIgnoring the effect of variable battery usage on future performance (see Figure 1). However, for extrapolation of models developed for laboratory environments to field data or other real-world usage profiles such as the Worldwide Harmonized Light Vehicle Test Cycle (WLTC), lithium battery cells cycle in vastly different ways during their lifetime, which has been That proved to be a major challenge.

Fig. 1 | Schematic comparison of the proposed approach to previous research works. Feature-based methodologies for degradation prediction have focused on constant charging protocols (the blue/red curve denotes the charge/discharge phase), using features from capacity–voltage curves as input. This necessitates knowledge of historic charging data. Our approach considers variable charging protocols (the shaded blue/red region denotes the range of currents that the charge/discharge protocols are drawn from), which is more comparable to the EV setting. Further, we employ the electrochemical impedance spectrum measured just before charging as input, without any knowledge of historic data, and predict the impact of different future usage protocols on the discharge capacity.

Figure 1 | Schematic diagram comparing the proposed method with previous research work.
The feature-based degradation prediction method focuses on the constant charging protocol (blue/red curves represent charge/discharge phases), using features of the capacity-voltage curve as input. This emphasizes the importance of knowledge of historical charging data.
Our approach considers variable charging protocols (blue/red shaded areas indicate current ranges for charging/discharging protocols), which is more similar to an EV setup. Furthermore, we use electrochemical impedance spectroscopy measured before charging as input, without knowledge of any historical data, and predict the effect of different usage protocols on the discharge capacity in the future.

In this work, we seek to identify whether there exists a sufficiently informative marker of cell health that can be used to forecast shortterm and longer term future performance, amid uneven historical and future cell usage. Figure 1 provides an illustration of our approach, and how it differs from previous approaches. We find that upon acquisition of an EIS spectrum just before charging, both next cycle and longer term cell capacity can be predicted with a test error of less than 10%.

When testing on cells subjected to similar cycling conditions to those used to train the model, our model achieves comparable accuracy to state-of-the-art forecasting models (8.2% test error versus 8.8% test error), except that our model enables forecasting with no access to any historical data, whereas previous state-of-the-art models require historical data from the cell’s cycling trajectory. In addition, when extrapolating to different operating temperatures, our model significantly outperforms the state-of-the-art model, achieving a 57% reduction in test error (from 34.2% to 14.6%).

In this work, we try toDetermine if there is a sufficiently informative lithium battery cell health signature that can be used to predict short- and long-term future performance, under unbalanced historical and future lithium battery cell usage. Figure 1 demonstrates our approach, and how it differs from previous approaches. We found that acquiring the EIS spectrum before charging can predict the next cycle and longer-term battery capacity with a test error of less than 10%.

When tested on lithium battery cells similar to the cycle conditions used to train the model, our model achieves comparable accuracy to state-of-the-art predictive models (8.2% test error vs. 8.8% test error), except Our model can make predictions without access to any historical data, whereas previous state-of-the-art models require historical data from cell cycle trajectories. Furthermore, our model significantly outperforms the state-of-the-art model when extrapolated to different operating temperatures, reducing the test error by 57% (from 34.2% to 14.6%).

We observe that our model is data-efficient, requiring just eight cells to attain a test error of less than 10%. Crucially, our approach is robust to dataset shift, attaining a test error of less than 7% on a dataset with a different distribution of cycling patterns to the training set. This is important for deployment in the field where driving patterns may be different from those used to train the model. We additionally demonstrate that, if available, using additional features based on historical capacity–voltage data can serve to augment the state representation and reduce average test error by up to 25%. Our approach is robust with respect to cell manufacturer, average usage pattern and operating temperature.

We observe that our model isData efficient data-efficientYes, only 8 units are needed to obtain a test error of less than 10%.
Crucially, our methodRobust to dataset shift, achieving a test error of less than 7% on datasets with different cyclic mode distributions. This is important for field deployments where the driving patterns may differ from those used to train the model.
We also prove that, if available,State representation can be enhanced using additional features based on historical capacity-voltage data, and reduce the average test error by up to 25%. Our method is robust across battery manufacturers, average usage patterns, and operating temperatures.

Further, our work fills a gap in publicly available data by contributing a large corpus of cycling data on cells under dynamic working conditions. Our work focuses on a set of idealised usage distributions rather than realistic driving profile in order to demonstrate the extent of generalisability of the model. Our work departs from the NASA randomised usage dataset, which randomly cycles cells for 50 cycles before measuring the next cycle discharge capacity after charging via a ‘reference’ protocol. Although several models for forecasting degradation under randomised conditions have been built based on this data, the effect of a single protocol on next cycle discharge capacity cannot be disentangled, and there is a need for a reference charge/discharge protocol every few cycles which does not concord with typical field usage.

Furthermore, we work throughProvides a large corpus of lithium battery cell cycle data under dynamic operating conditions, filling the gaps in the public data. Our work focuses on an idealized set of usage distributions rather than realistic driving profiles to demonstrate how well the model generalizes. Our work was drawn from random use data from NASA, which randomly cycled the battery for 50 cycles, then measured the discharge capacity for the next cycle after charging through a "reference" protocol. Although several models have been developed based on these data to predict degradation under stochastic conditions, the effect of a single protocol on the discharge capacity for the next cycle cannot be decoupled and requires a reference charge/discharge protocol every few cycles, which inconsistent with typical field usage.

Result

Data generation

For this study, we generate two separate datasets corresponding to commercial LiR coin cells 锂离子可充电钮扣电池 purchased from two different manufacturers, which allows us to test whether our approach is robust with respect to cell manufacturer.

In this study, we generated two independent datasets corresponding to commercial Li-ion rechargeable coin cells purchased from two different manufacturers, which allowed us to test whether our method is robust to battery manufacturers .

The first dataset corresponds to 40 Powerstream LiR 2032 coin cells (nominal capacity 1C = 35 mAh). We subject 24 cells to a sequence of randomly selected charge and discharge currents at 23 ± 2 °C for 110–120 full charge/discharge cycles. Each cycle consists of an initial diagnosis of battery state, involving acquisition of the galvanostatic EIS spectrum, followed by usage, involving a charging and discharging stage. Charging and discharging consist of a two stage and one stage Constant Current (CC) protocol, respectively; the currents are randomly selected at each cycle in the ranges 70–140 mA (2–4 C ) , 35–105 mA (1–3 C), and 35–140 mA (1–4 C) respectively. To test the model’s robustness to domain shift, we additionally cycle the remaining 16 cells under the same conditions as above, except now fixing the discharge current for all cells and cycles at 52.5mA (1.5 C) instead of randomly changing the discharge current at each cycle. The space of protocols considered is illustrated in Fig. 2 and an example of the capacity trajectories of three cells is provided in Supplementary Fig. 1 for illustration of the difference from typical monotonic capacity fade experiments. A complete description of cycling protocols is provided in the Methods and the full set of operating conditions that each cell is subjected to is detailed in Supplementary Table 1.Having used the first dataset to confirm the approach can successfully forecast discharge capacity several cycles ahead, we later significantly expand our analysis to explore the model’s robustness to cell manufacturer, changes to usage pattern and operating temperature. To achieve this, we cycle an additional 48 cells from a second manufacturer, RS Pro (nominal capacity 40 mAh), under a much wider range of usage patterns. In this case, each cell is again subjected to 100 cycles of two-stage CC charging, and one-stage CC discharging, with the three rates randomly selected at the start of each cycle. However, we now make the problem more challenging by having a different distribution of currents for each cell, to replicate the scenario in which different battery users have different average usage patterns to each other, but still exhibit random cycle-to-cycle behaviour. Of these cells, sixteen are also cycled at a higher operating temperature of 35 °C.

The first data set corresponds to 40 Powerstream LiR 2032 lithium-ion rechargeable coin cells (nominal capacity 1C = 35 mAh). We subjected 24 cells to a randomly selected charge-discharge current sequence at 23 ± 2°C for 110–120 complete charge-discharge cycles. Each cycle consists of an initial diagnosis of the battery state, including the acquisition of a constant-current EIS spectrum, followed by use, including charging and discharging phases. Charging and discharging consisted of two-level and one-level constant-current (CC) protocols, respectively; 4 C) randomly selected within the range.

To test the robustness of the model to domain shift, we cycled the remaining 16 cells under the same conditions described above, except now that the discharge current of all cells was fixed at 52.5mA (1.5 C) instead of at each cycle Randomly vary the discharge current.

The considered protocol space is shown in Fig. 2, and examples of capacity trajectories for three cells are provided in Supplementary Fig. 1 to illustrate the differences from typical monotonic capacity decay experiments. A full description of the cycling scheme is provided in the Methods, and Supplementary Table 1 details the full set of operating conditions to which each unit was subjected. After confirming that the method can successfully predict discharge capacity several cycles ago using the first dataset, we then significantly expanded our analysis to explore the relevance of the model to battery manufacturers, usage pattern changes, and operating temperature.

To achieve this, we cycled an additional 48 cells from another manufacturer, RS Pro (nominal capacity 40mAh), in wider usage mode. In this case, each cell was again subjected to 100 cycles of two-stage CC charging and one-stage CC discharging, with three rates randomly selected at the beginning of each cycle. However, we now make the problem more challenging by providing each battery with a different current distribution, to replicate scenarios where different battery users have different average usage patterns from each other, but still exhibit random cycle-to-cycle behavior. Of these cells, 16 were also cycled at a higher operating temperature of 35°C.

Fig. 2 | Proposed charge-discharge protocol. We generate battery cycling data by subjecting cells to a sequence of random charge and discharge currents. We apply two stages of constant current (CC) charging for up to 15 min each, with currents drawn from the ranges 70–140 mA (2–4 C ) a n d 3 5–105 mA (1–3 C), respectively (the blue shaded region). If the safety threshold voltage of 4.3 V is reached before the time limit, then charging is stopped. During discharging, a single constant discharge current, randomly selected in the range 35–140 mA (1–4 C), is applied (the red shaded region), until the voltage drops to 3.0 V.

Figure 2 | The proposed charging and discharging protocol. We generate battery cycling data by subjecting the battery to a series of random charge and discharge currents. We use two stages of constant current (CC) charging, each stage charges for up to 15 minutes, and the current is drawn from the range of 70-140 mA (2-4 C) and 3 5-105 mA (1-3 C) respectively (blue shaded area). Charging is stopped if the safe threshold voltage of 4.3 V is reached before the time limit. During the discharge, randomly select a constant discharge current (red shaded area) within the range of 35-140 mA (1-4 C) until the voltage drops to 3.0 V.

Capacity forecasting using EIS

We first consider the setting in which we want to predict the next cycle discharge capacity, for a cell whose usage history (including for example, cycle or calendar age, or historical capacity–voltage data) is completely unknown, if we apply a particular charging and discharging profile. We frame the problem as a regression task, and train a probabilistic machine learning model to learn the mapping $Q_n=f(s_n,a_n)$, with uncertainty estimates, where $s_n$ is the battery state at the start of the nth cycle, an is the future action (the nth cycle charge/ discharge protocol), and Qn is the discharge capacity measured at the end of the cycle. The battery state vector sn is formed from the concatenation of the real （$Z_{re}$）and imaginary（$Z_{im}$ ) components of the impedance measured at 57 frequencies, ${oh}_1，\dots {oh}_{57}$, in the range 0.02Hz-20kHz; $s_n=[Z_{re}({oh}_1),Z_{im}({oh}_1),...,Z_{re}({oh}_{57}),Z_{im}({oh}_{57})]$. The action vector is formed from the concatenation of the nth cycle charge and discharge currents.

If we apply a specific charge-discharge profile, we first consider the setting in which we want to predict the next cycle discharge capacity, for a battery whose usage history (e.g., cycle or calendar age, or historical capacity-voltage data) is completely unknown. We frame the problem as a regression task and train a probabilistic machine learning model to learn the mapping $Q_n=f(s_n,a_n)$ , with an uncertainty estimate, where$s_n$is the $The\; state\; of\; the\; battery\; at\; the\; beginning\; of\; the$ nth cycle, an is the future action (the charge/discharge protocol of the nth cycle),$Q_n$is the discharge capacity measured at the end of the cycle. Battery status vector $s_n$The real ( Z re Z_{re} of the measured impedance at 57 frequencies$Z_{re}$) and imaginary ( $Z_{im}$) components formed in series, ${oh}_1，\dots {oh}_{57}$，in the range of 0.02Hz-20kHz? $s_n=[Z_{re}({oh}_1),Z_{im}({oh}_1),...,Z_{re}({oh}_{57}),Z_{im}({oh}_{57})]$ . The motion vector is formed by the series connection of the nth cycle charge and discharge currents.

Figure 3 illustrates the accuracy of our model. Using both state and action as input, the next cycle discharge capacity is predicted with an average error of 8.2%. Importantly, both state and action (Fig. 3a) are found to be necessary to predict future cell performance: if state (Fig. 3b) or action (Fig. 3c) alone are used as inputs, the test error approximately doubles to 20.7% and 15.4% respectively. This demonstrates the importance of both the cell’s internal health and the externally selected usage in determining realised cell performance.

Figure 3 illustrates the accuracy of our model. Taking the state and action as input, the next cycle discharge capacity is predicted with an average error of 8.2%. Importantly, both state and action (Fig. 3a) were found to be necessary for predicting future cell performance: if only states (Fig. 3b) or actions (Fig. 3c) were used as inputs, the test error approximately doubled to 20.7% respectively and 15.4%. This demonstrates the importance of the internal health of the battery and the use of external options in determining the achieved battery performance.

For applications such as optimised charging and repurposing triaging, it is important that a model of battery life trajectory can forecast not only the immediate next cycle discharge capacity, but also capacity several cycles into the future49,50. With this in mind , we next investigate how the predictive accuracy of the model changes as we push the model to predict capacity further into the future. In each case, the input comprises the concatenation of the state representation at the start of the nth cycle, sn, with the ‘action’ vector an…n+j comprising all charging and discharging currents that will be applied between cycle n and cycle n + j

For applications such as optimized charging and reuse sorting, it is important that battery life trajectory models can predict not only the discharge capacity for the next cycle, but also the capacity for several cycles into the future. With this in mind, we next examine how the model's predictive accuracy changes as we push the model to predict future capacity. In each case, the input consists of the state representation sn at the beginning of the nth cycle $s_n$with the "action" vector $a_{n\dots N+}$included in loop $n$ and cycle$n+$All charge and discharge currents applied between$j$

Figure 4 shows how the coefficient of determination $R^2$changes with j. As expected, the accuracy of the model generally decreases as the forecasting interval increases. However, the model still attains $R^2$= 0.75 when projecting 40 cycles into the future.

Figure 4 shows the coefficient of determination $R^2$ withjjThe change of$j\; .$As expected, the accuracy of the model generally decreases as the prediction interval increases. However, when forecasting 40 periods into the future, the model still gets$R^2$ = 0.75。

Fig. 3 | Predicting next cycle discharge capacity. a Given knowledge of the state (the battery’s internal state, as characterised by the EIS spectrum) and the action (the next cycle charge/discharge protocol), our model predicts the next cycle discharge capacity with an error of 8.2%. Both state and action are needed to accurately forecast performance; using (b) state or ( c ) action alone is insufficient.

Figure 3 | Predicting the next cycle discharge capacity. a Given knowledge of the state (the internal state of the battery, characterized by the EIS spectrum) and the action (the next-cycle charge-discharge protocol), our model predicts the next-cycle discharge capacity with an error of 8.2%. Accurately predicting performance requires both states and actions; using (b) behavior or (c) behavior alone is not enough.

Fig. 4 | Multi-step forecasting. Our model can also forecast longer term battery performance, as quantified by (a) % test error, and (b) R2 value. Given the EIS spectrum and knowledge of the next protocols that will be applied to the cell, the discharge capacity is predicted with a test error of less than 10% up to 32 cycles in advance.

Figure 4 | Multi-step forecasting. Our model also predicts the long-term performance of the battery, quantified by (a) % test error and (b) R2 value. Considering the EIS spectra and the knowledge of the next protocol that will be applied to the battery, the test error of the discharge capacity prediction is less than 10%, up to 32 cycles in advance.

Data efficiency and robustness to domain shift

Fig. 5 | Data efficiency. The model performance, as quantified by (a) %test error and (b) R2, improves as the number of cells used to train increases. The model is dataefficient, achieving a test error of less than 10% with just eight cells in the training set.

Figure 5 | Data efficiency. Model performance quantified by (a) % test error and (b) R2 improves with increasing number of cells used for training. The model is data efficient, with a test error of less than 10% with only 8 units in the training set.

We next test the robustness of our method by investigating data efficiency and model generalisability. To test data efficiency, we measure how performance changes as the number of cells used to train the model increases. As seen in Fig. 5, there is a marked reduction in test error from 23.8% to 8.2% as the number of cells increases from two to 22. Nevertheless, the model is demonstrably data-efficient, with just eight cells needed to obtain a test error of less than 10%.

Next, we test the robustness of our method by investigating data efficiency and model generalizability. To test for data efficiency, we measure how performance changes as the number of units used to train the model increases. As shown in Figure 5, as the number of units increases from 2 to 22, the test error decreases significantly from 23.8% to 8.2%. Nonetheless, the model proved to be data efficient, requiring only 8 units to achieve a test error of less than 10%.

An important test of model generalisability is to study model accuracy when the domain distribution changes, i.e. when the model is being deployed in settings that are different from the training data.

An important test of model generalization is model accuracy when the distribution of the study domain changes, that is, when the model is deployed in a setting different from the training data.

This is important for deployment in the field as the approach needs to be robust to driving patterns that might be different from the training data8. We test model robustness by cycling an additional 16 cells from the same manufacturer, but now adjusting the cycling protocol by fixing the discharge current to 1.5C for each cell throughout its life. We use a model trained using only cells that were subjected to random discharge currents over their lifetime, to predict next-cycle discharge capacity of cells subjected to fixed discharging. To illustrate the difference in training and test datasets, the distribution of discharge capacities is shown for each in Fig. 6a.
The predictive accuracy of the model on the fixed discharge dataset is illustrated in Fig. 6b. Promisingly, the model attains a test error of just 6.3% on this domain-shifted dataset, which corresponds to R2 = 0. 76 .

This is important for field deployment, as the method needs to be robust to driving patterns that may differ from the training data. We tested the robustness of the model by cycling another 16 cells from the same manufacturer, but now adjusted the cycling protocol by fixing the discharge current of each cell at 1.5C throughout its lifetime. We use a model trained only on batteries subjected to random discharge currents during their lifetime to predict the next-cycle discharge capacity of batteries subjected to a fixed discharge. To illustrate the difference between the training dataset and the test dataset, the distribution of discharge ability is shown in Fig. 6a.
The prediction accuracy of the model on the fixed flow dataset is shown in Fig. 6b. Promisingly, the model has a test error of only 6.3% on this domain shift dataset, corresponding to $R^2$ = 0.76

Our model also outputs predictive uncertainty, which indicates how certain the model is about the quality of its predictions. It is especially important in the domain-shifted setting that the model ‘knows what it does not know’ and estimates high predictive uncertainty about data points that it is likely to obtain a high error on. We can test the model’s ability to estimate its uncertainty by observing how the average test error changes as the number of data points is reduced to include only the data points that the model is most confident about. If a model can successfully estimate its level of certainty, the average test error should reduce as the proportion of data is reduced to include only the most confidently predicted points. Figure 6c shows a 3 2 % reduction in root-mean-squared error (RMSE) as the proportion of data is reduced from 100% to the most confident 25%, demonstrating that our model has learnt which predictions it should be confident about.

Our model also outputs prediction uncertainty, which indicates how certain the model is about the quality of its predictions. In the case of domain shift, it is especially important that the model "knows what it doesn't know" and estimates high prediction uncertainty for data points on which it may obtain high errors. We can test the model's ability to estimate its uncertainty by observing how the mean test error changes as the number of data points is reduced to include only the data points for which the model is most confident. If a model is able to successfully estimate its level of certainty, then the average test error should decrease as the proportion of the data decreases to include only the most confident predicted points. Figure 6c shows that the root mean square error (RMSE) is reduced by 3.2% when the proportion of data is reduced from 100% to the most plausible 25%, suggesting that our model has learned which predictions it should be confident about.

Fig. 6 | Robustness to domain shift. a The distribution of discharge capacity is different for cells cycled under variable discharge rates (blue) compared to a fixed discharge rate (red); the overlap region of the two distributions appears darker in colour. b Our model, trained on the variable discharge rate cells, accurately predicts the discharge capacities of cells cycled under a fixed discharge rate. The colour of the plotted points denotes predicted uncertainty (see colour bar). c The model `knows what it does not know’: when we restrict the test data used to calculate the root-mean squared error (RMSE) by including only the predictions that the model is most confident about (i.e. with lowest predictive uncertainty), the RMSE reduces.

Figure 6 | Robustness to domain shift.
a The discharge capacity distribution is different for cells cycled at a variable firing rate (blue) compared to a fixed firing rate (red); the overlapping region of the two distributions is darker.
b Our model, trained on variable discharge rate cells, accurately predicts the discharge capacity of cells cycled at a fixed discharge rate. The color of the plotted points indicates the uncertainty of the forecast (see color bar).
c The model "knows what it doesn't know": When we limit the test data used to calculate the root mean square error (RMSE) to only include the predictions the model is most confident in (i.e. have the lowest prediction uncertainty), the RMSE decreases .

Comparison of state representation

Having demonstrated the ability of the EIS spectrum to capture battery state, we now benchmark this representation of battery health against other approaches utilised in the literature, including the state-of-the-art feature-based method22,51, and consider whether there are additional features to the EIS spectrum that can serve to augment battery state.

Having demonstrated the ability of EIS spectra to capture battery state, we now benchmark this representation of battery health against other methods used in the literature, including state-of-the-art feature-based methods, and consider whether EIS spectra have additional features Battery status can be enhanced.

Simple measures that have been used to forecast or estimate battery SOH include using the previous cycle discharge capacity, or the capacity throughput since cycling commenced. More advanced approaches include extracting features of the historical capacity–voltage discharge curves, as shown in Fig. 1. The state-of-the-art approach to extracting such features was implemented by Severson et al22 and inspired the approaches to feature extraction used recently by Attia et al and Paulson et al37,51. We benchmark how our EIS-based approach performs relative to those state-of-the-art features.

Simple measures for predicting or estimating battery SOH include cycle discharge capacity before use, or capacity throughput after cycle initiation. More advanced methods include extracting features of historical capacity-voltage discharge curves, as shown in Figure 1. State-of-the-art methods for extracting these features are implemented by Severson et al., rather than the more recent feature extraction methods used by Attia et al. and Paulson et al. We benchmark the performance of AIS-based methods against these state-of-the-art features.

Further, we assess whether incorporation of physical interpretations, in the form of equivalent circuit models (ECM), improves predictions. We use the widely implemented Randles circuit model, comprising a series resistance, connected with a resistance in parallel with a capacitance and a Warburg impedance element, as well as the more complex Extended Randles circuit, which adds an additional resistor-capacitor parallel combination in series to the Randles circuit.
The ECM is fitted to the spectrum (at an associated computational cost) and we use the extracted parameters as the state representation instead of raw EIS data.
In total, we consider the following features in our benchmark:

• Previous cycle discharge capacity $Q_{n-1}$.
• Capacity throughput (CT) since cycling commenced, as defined by the sum of cell charge and discharge capacities from cycles 0 to $n-1$.
• State of Health (SOH), as defined by $Q_{n-1}/Q_0$.
• State-of-the-art features of the capacity–voltage discharge curve (CVF): Following Severson et al, we form a state representation at the start of cycle n by extracting features from the capacity–voltage discharge curve after cycle n − 1. We fit e a c h curve to a spline function, linearly interpolating to measure capacity at 1000 evenly spaced voltages from $V_{m}in$ to $V_{m}ax$. T h i s 1000-dimensional capacity vector $Q_{n-1}$ is normalised by subtracting the equivalent vector from cycle 0, $Q_0$. The following features are then used as inputs: $V_{max}$, $V_{min}$, $log(var(Q_{n-1}-Q_0))$, $log(IQR(Q_{n-1}-Q_0))$. Additionally, we fit the capacity to a sigmoid$Q\left(\tilde{V}\right)=\frac{p_0}{1.0+\exp \left(p_1\left(V-p_2\right)\right)}$ where $\tilde{V}$ is the normalised voltage and use the parameters p0, p1, p2 as features.
• Equivalent circuit model parameters (ECM-R and ECM-ER): We fit equivalent circuit models using the Randles circuit (ECM-R) and Extended Randles circuit (ECM-ER) to the EIS spectra and concatenate the obtained parameters together.

In addition, we assessed whether incorporating a physical interpretation in the form of an equivalent circuit model (ECM) would improve predictions. We use the widely used Randall circuit model, which consists of a series resistor connected to a resistor in parallel with a capacitor and a Warburg impedance element, and the more complex extended Randall circuit, which adds an additional resistor-capacitor parallel combination.
ECMs are fitted to the spectrum (at the associated computational cost), and we use the extracted parameters as state representations instead of the raw EIS data.
Overall, we considered the following features in our benchmarks:

• Discharge capacity of the previous cycle $Q_{n-1}$。
• Capacity throughput (CT) since the start of the cycle, defined by the sum of the battery charge and discharge capacities from cycle 0 to n−1.
• State of health (SOH), defined as $Q_{n-1}/Q_0$。
• State-of-the-Art Characterization of the Capacity-Voltage Discharge Profile (CVF): Following Severson et al., we form a state representation at the beginning of cycle n by extracting features from the capacity-voltage discharge curve after cycle n−1. We fit a $c_h$Curve to a spline function, linearly interpolated to measure 1000 evenly spaced voltages from $V_{min}$to $V_{max}$capacity. The 1000-dimensional capacity vector Qn−1 of T hi is normalized by subtracting the equivalent vector from period 0,Q0. Then use the following features as input: $V_{max}$, $V_{min}$, $log(var(Q_{n-1}-Q_0))$, $log(IQR(Q_{n-1}-Q_0))$ .. Furthermore, we fit the capacity as a$Q\left(\tilde{V}\right)=\frac{p_0}{1.0+\exp \left(p_1\left(V-p_2\right)\right)}$, where $\tilde{V}$ is the normalized voltage and uses the parameters p0, p1, p2 as features.
• Equivalent circuit model parameters (ECM-R and ECM-ER): We fit the equivalent circuit model to the EIS spectrum using the Randles circuit (ECM-R) and the Extended Randles circuit (ECM-ER), and concatenate the obtained parameters in Together.

We note that in contrast to EIS features, the formation of a state representation using the first four aforementioned features demands access to historical current-voltage data, over at least the entirety of the previous discharge and for some features, over the entire cell lifetime. However, they benefit from the advantage of not requiring equipment to measure the EIS spectrum, which comes with an associated financial and temporal cost. Forming a state representation using the ECM parameters (extracted from the EIS spectrum) has an associated computational cost and can be considered a form of dimensionality reduction of the raw EIS data. An additional problem faced by ECMs in general is non-uniqueness, in that multiple different combinations of ECM parameters can generally explain a particular EIS spectrum equally well52.

We note that in contrast to the EIS signature, the state representation formed using the first four aforementioned features requires access to historical current-voltage data, at least throughout the previous discharge and, for some signatures, throughout the life of the battery . However, they benefit fromAdvantages of not requiring equipment to measure EIS spectra, with associated financial and time costs. Forming a state representation using ECM parameters (extracted from EIS spectra) has an associated computational cost and can be considered a form of dimensionality reduction for raw EIS data. Another problem that ECMs often face is non-uniqueness, as multiple different combinations of ECM parameters can often explain a particular EIS spectrum equally well.

Table 1 shows how the state representation impacts test error and model goodness of fit. In all cases, the model is trained to predict the next cycle discharge capacity, given the next cycle protocol and the chosen state representation. Interrogating the relative importance of features, we first consider the baseline of using EIS only (without including the protocol) and using the protocol only (without including EIS). Perhaps unsurprisingly, battery degradation is a function of both the current state and future charge/discharge protocol. As such, using both EIS and the protocol significantly outperforms using EIS only or using the protocol only.

Table 1 shows the effect of state representation on test error and model fit. In all cases, the trained model predicts the next cycle discharge capacity, given the next cycle protocol and the chosen state representation. When asking about the relative importance of features, we first consider baselines using only EIS (excluding protocol) and using only protocol (excluding EIS). Perhaps unsurprisingly, battery degradation is a function of current state and future charge and discharge protocols. Therefore, the performance of using both EIS and protocol is significantly better than that of using only EIS or only protocol.

We then explore the impact of physics-based representation of the EIS spectrum, using the Randles (ECM-R) and extended Randles (ECM-ER) equivalent circuit models. Comparing EIS + Protocol with ECM-R + Protocol and ECM-ER + Protocol reveals that these physicsbased models lose information, and using a machine learning approach to directly learn from raw data might be advantageous.

We then explored the impact of physics-based spectral representations of EIS using the Randall (ECM-R) and Extended Randall (ECM-ER) equivalent circuit models. A comparison of EIS +Protocol with ECM-R +Protocol and ECM-ER +Protocol shows that thesePhysics-based models lose information,useMachine learning methods that learn directly from raw data may be advantageousof.

We next consider the different approaches that have been reported in the literature, Qn−1, SOH, CT, and CVF, with CVF being the state-of-the-art in the battery informatics literature. In all cases, EIS + Protocol outperforms those other features with Protocol, although CVF is competitive.

Next, we consider different methods reported in the literature, Qn−1, SOH, CT and CVF, where CVF is the state-of-the-art method in the battery informatics literature. In all cases, EIS+Protocol outperforms other features of Protocol, although CVF is competitive.

Interestingly, information from capacity–voltage curve data (CVF) is complementary to EIS - combining EIS with these features leads to a significant increase in accuracy (EIS + CVF + Protocol). This is perhaps unsurprising, as EIS probes the impedance of the single ‘static’ cell discharged state (with high information content per instant state), whilst capacity–voltage curves probe how the cell state evolves continuously over the path from charged to discharged (with low information content per instant state).

Finally, the best model performance is attained by combining all of the above features to form the state representation. In this case the average test error is just 6.2%.

Interestingly, information from volume-voltage curve data (CVF) is complementary to EIS - combining EIS with these features significantly improves accuracy (EIS + CVF + Protocol). This is perhaps not surprising since EIS probes the impedance of a single "static" battery discharge state (each transient state has a high information content), whereas capacity-voltage curves probe how the battery state evolves continuously along the path from charge to discharge (Each momentary state has low information content).

Finally, the state representation is formed by combining all the above features, which leads to the best model performance. In this case, the average test error was only 6.2%.

Robustness to different cell manufacturers

We now extend our analysis to explore how robust our approach is to changing the cell manufacturer, adjusting the operating temperature and adjusting the average use pattern. We repeat our experiment on a new batch of 32 commercial LiR coin cells (of nominal capacity 1 C = 40 mAh) from RS Pro, a different manufacturer, except we now make the problem significantly more challenging by subjecting different subgroups of cells to one of four different usage distributions. These usage distributions are shown in Supplementary Table 1.

We now extend our analysis to explore the robustness of our approach to changing battery manufacturers, adjusting for operating temperature, and adjusting for average usage patterns. We repeated our experiments on a new batch of 32 commercial Li-ion rechargeable button cells (nominal capacity 1 C = 40 mAh) from another manufacturer, RS Pro, only we now made the problem more challenging , allowing different battery subgroupsEach accepts one of four different usage distributions. The distribution of these usage cases is shown in Supplementary Table 1.

We measure the accuracy of the model in two ways: firstly, we consider the case where the model is exposed to cells that have been subjected to the same distribution of protocols as the test set (random splitting), and second, the more challenging case where the model is only trained on the cells which are subjected to three of the cycling protocol distributions and tested on the remaining eight cells subjected to a different cycling protocol. This is a much harder task as the average usage on the test cells is very different to the average usage on the training cells—it is a test of whether the model can extrapolate to different average use not just different cycle-to-cycle use.

We measure the accuracy of the model in two ways:
First, we consider the modelExposure to lithium battery cells subjected to the same protocol distribution (random split) as the test setcase,
and the second, more challenging case, the modelOnly on Li-ion cells subject to distribution of the three cycling protocolstrain, andTested on remaining eight Li-ion cells subjected to different cycling protocols.
This is a more difficult task becauseThe average usage on the test cells is very different from the average usage on the training cells - this is a test to see if the model can extrapolate to different average usages, not just different cycle-to-cycle usage。

The results for different state representations are shown in Table 2 for both the case where the train/test split is random, and where the split is stratified into different usage patterns. Comparable observations are made for cells purchased from the second manufacturer: namely, the most accurate predictions are made when the state representation is formed using features of the EIS spectrum alongside those formed from the discharge curve (CVF). As expected, the model performs significantly better when it has been trained on data from some cells that have been exposed to a similar distribution of cycling patterns as those that the model is tested on. However, the model remains performant in the scaffold split scenario, and in this setting the test error reduces by 30% when the state representation is formed using the EIS spectrum alongside the features of the discharge curve, instead of solely using features of the discharge curve.

Table 2 shows the results for different state representations, where the train/test split is randomized, and where the splits are stratified into different usage patterns. Comparative observations were made on batteries purchased from a second manufacturer: namely,The most accurate predictions were made when the state representation was formed using the features of the EIS spectrum and the characteristic formed by the discharge curve (CVF).
As expected, the model performed significantly better when it was trained using data from a number of lithium battery cells exposed to a similar distribution of cycling patterns as when the model was tested.
However, the model was still performant in the scaffold splitting scenario, and in this setup, the test error was reduced by 30% when the state representation was formed using features from the EIS spectrum and the discharge curve rather than from the discharge curve alone.

These additional results further demonstrate that if available, both the EIS spectrum and discharge curve can act as informative markers of the battery’s internal state, but that they are complementary to each other.

We next verify that the model is robust with respect to changing external operating temperature. We cycle an additional 16 cells at 35 °C and test the model trained on data from cells cycled at room temperature. Table 3 shows that our model can extrapolate to cells operated at these higher temperatures, but that the EIS spectrum plays a particularly important role in characterising the battery state when the cell is not operated at the same temperature. The model obtains a test error of 34.2% when only the discharge curve features are used to characterise state, which reduces to 14.0% when both the EIS spectrum and discharge curve features are used. This further demonstrates the additional information that EIS signals contain relative to chargingdischarging curves, and supports the hypothesis that EIS implicitly tracks temperature53.

These additional results further demonstrate that, if available, both EIS spectra and discharge curves can serve as informative markers of the internal state of the battery, but they are complementary to each other.
Next, weVerify that the model is robust to changes in external operating temperature. usCycle an additional 16 Li-ion cells at 35°C and test the trained model on data from Li-ion cells cycled at room temperature.
Table 3 shows that our model can be extrapolated to batteries operating at these higher temperatures, but EIS spectra play a particularly important role in characterizing the state of the battery when the battery is not operating at the same temperature. The test error of the model is 34.2% when only the characteristics of the discharge curve are used to represent the state, and the test error of the model is 14.0% when the EIS spectrum and the characteristics of the discharge curve are used at the same time. This further demonstrates the additional information contained in the EIS signal relative to the charge-discharge profile and supports the hypothesis that EIS implicitly tracks temperature.

We make qualitatively similar observations when we test our approach on cells manufactured by RS Pro (rather than Powerstream), with EIS found to be a slightly better state representation than state-of-the-art capacity-voltage features (CVF). The best results are obtained when the two representations are combined. We test how the model performs when we split the training and testing sets randomly, and when we instead stratify the training and testing sets such that the model is tested on cells with a different usage distribution to the cells it was trained on. Usage conditions and an extended comparison of different state representations are provided in Supplementary Tables 1, 2 and 3.

We obtain qualitatively similar observations when we test our method on batteries manufactured by RS Pro (instead of Powerstream), finding that EIS is slightly better than state-of-the-art representations of the capacity-voltage signature (CVF).
Combining the two representations gives the best results. We test the model's performance when we randomly split the training and test sets when we stratify the training and test sets so that the model performs on cells that have a different usage distribution than the cells it was trained on test.
An extended comparison of the usage conditions and different state representations is provided in Supplementary Tables 1, 2 and 3.

Discussion

In this paper, we showed that the electrochemical impedance spectrum accurately characterises the internal state of a cell, and a machine learning model can be trained to accurately forecast both immediate and longer term cell performance with predictive uncertainty, even amid uneven and unknown historical cell usage. Our model achieves comparable accuracy (8.2% test error) to the state-of-the-art forecasting approach (8.8% test error) when testing on cells subjected to the same distribution of operating conditions as the cells used to train the model. However, as outlined in Fig. 1, the state-of-the-art approach demands access to historical cycling data whereas our model enables forecasting with no historical data. Additionally, our model significantly outperforms the state-of-the-art model when extrapolating to a higher operating temperature, with a 57% reduction in test error (from 34.2% to 14.6%).

In this paper, we show that electrochemical impedance spectroscopy accurately characterizes the internal state of batteries and that machine learning models can be trained to accurately predict short- and long-term battery performance even under uneven and unknown historical battery usage . Our model achieves comparable accuracy (8.8% test error) to state-of-the-art prediction methods (8.2% test error) when the test units are subjected to the same distribution of operating conditions as the units used to train the model.

However, as shown in Figure 1, state-of-the-art methods requireAccess historical cycle data, and our modelPredictions can be made without historical data. In addition, whenextrapolated to higher operating temperatures when, our model significantly outperforms the state-of-the-art, reducing test error by 57% (from 34.2% to 14.6%).

Our method is data-efficient, achieving a next-cycle test error of 9.9% with training data from just eight cells, and is robust to shifts in dataset distributions. Additionally, we find that there is scope to boost model performance by 25% if historical cycling data is available; such data can be used to derive features that augment the cell state representation. We demonstrate that our approach can be utilised across different cell chemistries, and the model is robust to different operating temperatures.

Our method is data efficient, achieving a next-epoch test error of 9.9% using only eight units of training data, and is robust to changes in the dataset distribution.
Furthermore, we find that there is room to improve model performance by 25% if historical cycle data is available; such data can be used to derive features that enhance cell state representation. We demonstrate that our method can be used for different battery chemistries and that the model is robust to different operating temperatures.

Our approach differentiates from the prior art in two important ways : First, we employ an information-rich electrical signal—EIS— which captures the response of the cell across different timescales without any knowledge of the cycling history. This is in contrast to most existing methods which employ features from the charging–discharging curve—a s i g n ificantly more coarse-grained signal—as input to machine learning models. Our results suggest significant improvements in battery management systems abound by incorporating circuitries that measure electrochemical impedance, albeit at a financial and temporal cost.

Our approach differs from the state of the art in two important ways:
First, we employ an informative electrical signal - eis - which captures the response of a lithium battery cell over different timescales without knowledge of the cycle history.
This contrasts with most existing methods that use features from charge-discharge curves—a significantly coarser-grained signal—as input to machine learning models.
The results of our study showed that byIntegrating a circuit for measuring electrochemical impedance, a significant improvement in battery management systems, albeit at a financial and time cost.

Second, we focus on uneven cycling, where the charging and discharging rates vary from cycle to cycle. This departs from previous studies on machine learning for battery degradation which focused on constant charge/discharge conditions, which are typical in battery testing. Our results problematise the concept of a single scalar State of Health, as the state of the battery is dependent on the extent of the myriad different degradation mechanisms, which in turn depends on the sequence of historic charge/discharge protocols. Rather, we suggest that a cell can be described by a multidimensional state vector, captured using informative high-dimensional measurements like EIS, and a machine learning approach can be used to predict future capacities given the state vector and future charge/discharge protocols.

Second, we focus on inhomogeneous cycling, where the charge-discharge rate varies with cycling. This differs from previous research on machine learning of battery degradation, which focused on constant charge and discharge conditions, which is typical in battery testing. Our results call into question the notion of a single scalar state of health, since the state of a battery depends on the extent of a myriad of different degradation mechanisms, which in turn depend on the sequence of historical charge-discharge protocols. Instead, we propose that batteries can be described by multidimensional state vectors, captured using informative high-dimensional measurements such as EIS, and that machine learning methods can be used to predict future capacity given state vectors and future charging/discharging protocols.

Furthermore, although in this work we only consider forecasting starting from an initially discharged state, we hypothesise that it should be possible in future work to forecast discharge capacity starting from any state of charge based on the EIS measurement, since EIS spectrum implicitly tracks state of charge.

Furthermore, although in this work we only consider predictions from the initial discharge state, we hypothesize that, in future work, it should be possible to predict discharge capacity from any charge state from EIS measurements, since EIS spectra implicitly Track state of charge.

We note that the general framework that we have laid out for predicting future battery performance given current cell state and future actions has scope to be applied in a broad range of battery diagnostic and control settings. For example, predicting the effect of a proposed charging protocol on next cycle discharge capacity as well as long term degradation is important for optimising rapid charging applications51, where a balance must be achieved between charging time and rate of cell degradation. Our work can additionally be extended to consider more complicated dynamic usage protocols, such as WLTC

We note that our general framework for predicting future battery performance, given current battery state and future actions, has applicability in a wide range of battery diagnostic and control settings. For example, predicting the impact of a proposed charging protocol on the next-cycle discharge capacity as well as long-term degradation is important for optimizing fast-charging applications, where a balance must be achieved between charging time and battery degradation rate. Our work can also be extended to consider more complex dynamic usage protocols such as WLTC

Methods

Battery cycling

For this study we cycle 88 commercial LiR coin cells purchased from two different manufacturers, Powerstream and RS Pro, in a temperature regulated laboratory at 23 ± 2 °C. A Biologic BCS-805 potentiostat is used for cycling, and photographs of the experimental setup are provided in Supplementary Fig. 2.

Across all datasets, cells are subjected to a sequence of randomly selected charge and discharge currents for 110–120 full charge/discharge cycles. Cycling commences when the cell is in the fully discharged state, and each cycle comprises the following steps: (a) resting for 20 min at the open circuit voltage, (b) acquisition of the galvanostatic EIS spectrum in the fully discharged state, © two stage CC charging, (d) resting for 20 min at the open circuit voltage, (e) acquisition of the galvanostatic EIS spectrum in the fully charged state, (f) one stage CC discharging. The galvanostatic EIS spectrum is always measured by collecting impedance measurements at 57 frequencies uniformly distributed in the log domain in the range 0.02Hz-20kHz using a sinusoidal current with amplitude of 5 mA. Cells are cycled in a temperature-controlled lab room at 23 ± 2 °C.

To generate the first dataset, we cycle 24 Powerstream LiR 2032 coin cells (nominal capacity 1 C = 35 mAh). For these cells, charging consists of a two-stage CC protocol; currents are randomly selected in the ranges 70–140mA (2C–4C) and 35mA-105mA (1C-3C) in stages 1 and 2 respectively. A time limit of 15 min is set for each charging stage such that the total charging time is constrained to be 30 min or less.

Charging will stop before the 30 min time limit if the safety threshold voltage of 4.3 V is reached. During discharging, a single constant discharge current, randomly selected in the range 35mA-140mA (1C–4C), is applied, until the voltage drops to 3.0 V.

An additional 16 cells (also manufactured by Powerstream and of nominal capacity 35 mAh) are cycled under the same conditions, except now we fix the discharge current at 52.5 mA (1.5C) for all cells and cycles, instead of randomly changing the discharge current at each cycle.

We then generate a second dataset that enables exploration of the model’s robustness to cell manufacturer, changes to usage pattern and operating temperature. We cycle 48 cells from a second manufacturer, RS Pro (nominal capacity 40 mAh), under a much wider range of usage patterns. The general six-step cycling protocol remains the same as described above, with each cell again being subjected to 100 cycles of two-stage CC charging, and one-stage CC discharging, with the three rates randomly selected at the start of each cycle. However, the distribution of currents now changes for each cell. Of these cells, sixteen are also cycled at a higher operating temperature of 35 ± 2 °C, in a temperature-controlled heating chamber. A description of the full set of operating conditions that each cell is subjected to is detailed in Supplementary Table 1.

In this study, we purchased 88 commercial LiR coin cells from two different manufacturers, Powerstream and RS Pro, and cycled them in a temperature-conditioned laboratory at 23±2°C. Cycling was performed using a Biologic BCS-805 potentiostat, a photo of the experimental setup is shown in Supplementary Fig. 2.

In all datasets, batteries were subjected to a series of randomly selected charge and discharge currents for 110-120 complete charge and discharge cycles. Cycling started when the battery was in a fully discharged state, and each cycle consisted of the following steps: (a) standing at open circuit voltage for 20 min, (b) obtaining a constant current EIS spectrum in a fully discharged state, © two-stage CC charging, ( d) Standing at open-circuit voltage for 20 min, (e) galvanostatic EIS spectrum obtained at fully charged state, (f) one-stage CC discharge. Constant current EIS spectra were always measured using a sinusoidal current with an amplitude of 5 mA by collecting impedance measurements at 57 frequencies in the logarithmic domain uniformly distributed in the range 0.02 Hz–20 kHz. Lithium battery cells are cycled in a laboratory temperature controlled at 23±2°C.

To generate the first dataset, we cycled 24 Powerstream LiR 2032 coin cells (nominal capacity 1 C = 35 mAh). For these batteries, charging consisted of a two-stage CC protocol; phase 1 and phase 2 with currents randomly selected in the range of 70-140mA (2C-4C) and 35mA-105mA (1C-3C), respectively. Set a 15-minute time limit for each charging phase, limiting total charging time to 30 minutes or less.

If the safe threshold voltage of 4.3 V is reached, charging will stop before the 30 minute time limit. During the discharge process, randomly select a constant discharge current in the range of 35mA-140mA (1C-4C) until the voltage drops to 3.0 V.

Another 16 cells (also made by Powerstream, with a nominal capacity of 35 mAh) were cycled under the same conditions, except now we fixed the discharge current at 52.5 mA (1.5C) for all cells and cycles, rather than at each Randomly change the discharge current for each cycle.

We then generate a second dataset to explore the robustness of the model to battery manufacturers, changes in usage patterns, and operating temperature. We cycled 48 cells from a second manufacturer, RS Pro (nominal capacity 40mAh), in wider usage mode. The general six-step cycling scheme was the same as above, and each cell was again subjected to 100 cycles of two-stage CC charging and one-stage CC discharging, with three rates randomly selected at the beginning of each cycle. However, the distribution of current is now changed per cell. Of these cells, 16 were also cycled at a higher operating temperature of 35±2°C in a temperature-controlled heated chamber. Supplementary Table 1 details the full set of operating conditions to which each unit was subjected.

Machine learning model

All problems in this study are framed as regression tasks. We train a probabilistic machine learning model to learn the mapping $Q_j=f(s_n,a_{n\dots })$, with uncertainty estimates, where sn is the battery state at the start of the nth cycle, an is the set of future cycling protocols applied over cycles n to j, a n d Qj is the discharge capacity at the end of the jth cycle. The battery state vector sn is formed from the concatenation of the real （$Z_{re}$）and imaginary（$Z_{im}$ ) components of the impedance measured at 57 frequencies, ${oh}_1，\dots {oh}_{57}$, in the range 0.02Hz-20kHz; $s_n=[Z_{re}({oh}_1),Z_{im}({oh}_1),...,Z_{re}({oh}_{57}),Z_{im}({oh}_{57})]$… For the task of predicting next cycle discharge capacity, the action vector an is formed from the concatenation of the nth cycle charge and discharge currents. When predicting discharge capacity several cycles, j, ahead of time, the future protocol is now formed from all charging and discharging currents that will be applied between cycle n and cycle n + j.

For the machine learning model, we use an ensemble of 10 XGBoost models58, each with 500 estimators and a maximum depth of 100. The mean and standard deviation of the predictions made by each model in the ensemble are used to quantify the predicted output and the predictive uncertainty. To test model performance we use the median R2 score and median percentage error. To obtain test metrics from a dataset comprising N cells, we randomly leave two test cells out, train on the remaining N−2 cells and repeat this process N/2 times, leaving different cells out each time.

All problems in this study are framed as regression tasks. We train a probabilistic machine learning model to learn the mapping $Q_j=f(s_n,a_{n\dots })$ , with an uncertainty estimate, where$s_n$is the state of the battery at the beginning of the nth cycle, $a_n$is the set of future cycle protocols applied to the nth to jth cycles, $Q_j$is the discharge capacity at the end of the jth cycle. Battery status vector $s_n$The real ( Z re Z_{re} of the measured impedance at 57 frequencies$Z_{re}$) and imaginary ( $Z_{im}$) components formed in series, ${oh}_1，\dots {oh}_{57}$，in the range of 0.02Hz-20kHz? $s_n=[Z_{re}({oh}_1),Z_{im}({oh}_1),...,Z_{re}({oh}_{57}),Z_{im}({oh}_{57})]$ . The motion vector is formed by the series connection of the nth cycle charge and discharge currents. When predicting several periods$At\; a\; discharge\; capacity\; of\; j$ , the future protocol is now formed by all charge and discharge currents applied between cycle n and cycle n+j.

For the machine learning model, we use an ensemble of 10 XGBoost models with 500 estimators each and a maximum depth of 100. The mean and standard deviation of the forecasts made by each model in the ensemble are used to quantify the forecast output and forecast uncertainty. To test model performance, we use the median $R^2$ Score and median percent error. In order to obtain test indicators from a dataset containing N units, we randomly set aside two test units, and among the remaining$N-Train\; on\; 2$ units and repeat this process$N/2$ times, leaving a different unit each time.

Data availability The data generated in this study are provided in the Zenobo database at https://doi.org/10.5281/zenodo.6645536.

Code availability The code required to reproduce this manuscript is available at https://github.com/PenelopeJones/battery-forecasting.