【nature論文精讀】 Impedance-based forecasting of lithium-ion battery performance amid uneven usage

文章目錄

• 【nature論文精讀】 Impedance-based forecasting of lithium-ion battery performance amid uneven usage
• • Result
  • • Data generation
    • Capacity forecasting using EIS
• Data efficiency and robustness to domain shift
• • • Comparison of state representation
    • Robustness to different cell manufacturers
  • Discussion
  • Methods
  • • Battery cycling
    • Machine learning model

論文連結：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.

锂離子電池性能的準确預測對于緩解消費者對電動汽車安全性和可靠性的擔憂至關重要。大多數關于電池健康預測的研究都集中在電池處于相同使用模式的研發環境中。然而，在實際操作中，跨細胞和周期的使用存在很大的可變性，是以預測具有挑戰性。為了解決這一挑戰，我們在這裡提出了電化學阻抗譜測量與機率機器學習方法的結合。利用88個通過多級充放電産生的商業锂離子硬币電池的資料集(在周期之間電流随機變化)，我們表明，考慮到未來的循環協定和充電前立即進行的單一電化學阻抗譜測量，并且不需要任何使用曆史知識，可以用校準的不确定性預測未來的放電容量。結果對電池制造商、循環協定的分布和溫度具有魯棒性。研究結果還表明，電池健康狀況可以通過多元向量而不是标量狀态來更好地量化。

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.

交通運輸業的電氣化正在以越來越快的速度進行，使我們能夠在實作碳中和的未來方面取得重大進展。這一轉變的基礎是锂離子電池的發展，目前在路上行駛的大多數電動汽車都是由锂離子電池驅動的。盡管這種轉變對環境有利，但對锂離子電池的依賴帶來了重大挑戰，消費者擔心的問題包括裡程焦慮、電池故障和充電時間。緩解這些擔憂需要準确預測電池性能的能力，特别是當使用條件變化時。

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.

關鍵的挑戰是電池的異質性heterogeneity。每個使用者使用汽車的方式都不同，即使是在一個電池組中，也不是所有的電池都必須以相同的電流充電或放電。

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.

這些差異意味着每個電池的内部狀态，包括锂電鍍或電極開裂的程度，在電池組内和電池組間的水準上都可能存在顯著差異。

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.

為了量化锂電池單體内的退化程度，并識别已達到生命終點的锂電池單體(在電動汽車中，這通常定義為放電容量降低到标稱容量的80%的點)，通常采用标量健康狀态(SOH)度量，使用以前的循環放電容量或内阻測量。這種方法的問題是，具有相同數值SOH的電池在每個降解過程中不一定表現出相同的水準(例如，锂電鍍或電極開裂)，但未來電池使用對電池的性能和降解途徑的影響取決于已經發生的降解。電池性能的準确預測需要一種非侵入性的方法來擷取微觀水準上電池狀态的資訊。

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.

電池性能的短和長時間尺度預測在電池預測中都是感興趣的。在短時間内，預測電池将如何響應特定的充放電協定，可用于開發最佳充電協定。短期預測還包括SOH估計:在這裡，目标是在特定的标準化循環協定下預測電池的直流電阻容量或内阻。在很長一段時間内，研究的重點是預測剩餘的使用壽命，即壽命的結束，或者電池壽命軌迹中加速退化的“膝點”。

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.

這兩種類型的預測方法可以細分為經驗模型、基于實體的模型和資料驅動模型，有些模型是這些模型的混合。

經驗方法Empirical approaches已被用于模拟長期容量随功率定律衰減，但假設在電池壽命期間固定運作，不考慮電池壽命開始時狀态的内在差異。這些方法假設所有具有相同化學性質的锂電池單體如果以相同的方式操作，将以相同的方式褪色，這在實踐中沒有觀察到。

在基于實體的方法physics-based approaches中，電池要麼使用内部實體和電化學過程的第一性原理分析進行機械模組化，要麼使用等效電路模組化，将電池模組化為包含代表底層電化學過程的電阻和電容的電路。 機械模型旨在捕捉電池電壓如何響應外部施加的電流(反之亦然)，可用于預測最佳充電協定。然而，這種模型的參數需要為每個單獨的锂電池單體更新，并且通常存在不可識别性——幾組模型參數可以同樣很好地解釋觀察到的資料，但會對測試锂電池單體或同一锂電池單體後期的生命周期做出截然不同的預測。

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.

對于基于電路的模型，電路的參數可以拟合到電流-電壓資料，或電化學阻抗譜。然後，電路參數可用于預測标準化使用條件下的容量退化或模拟不同使用條件對電池組性能的影響。然而，在分析模型中捕獲每種退化模式.是具有挑戰性的。此外，從一個周期到另一個周期，每個細胞都必須學習一組新的模型參數，這使得推斷一個一般的細胞到細胞模型具有挑戰性。

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.

純資料驅動的預測方法使用原始資料作為機器學習算法的輸入，來預測長期容量衰退、阻力增加和剩餘使用壽命。資料驅動方法将機器學習應用于從充電或放電曲線中提取的特征，以預測放電容量、剩餘使用壽命和容量突然衰減。

從充放電曲線中提取特征和時間序列資料模組化的機器學習方法的創新，使得預測的準确性有了顯著提高。

進一步的研究表明，利用少量初始周期的放電曲線特征，可以訓練出可以推廣到不同細胞化學性質的機器學習模型。

除了充電和放電曲線之外，電化學阻抗譜(EIS)、早周期庫侖效率、電流中斷和聲學飛行時間分析等方法已用于降解預測。

這些方法提供了對電池狀态更全面的描述——例如，EIS捕獲了電池在寬頻率範圍内的響應，不同的頻率與活性材料中不同的實體、化學和機械變化相關。

資料驅動方法方法通常利用在實驗室環境中産生的資料，其中電池在其整個生命周期内以相同的方式充電和放電，是以可以忽略可變電池使用對未來性能的影響(見圖1)。然而，将為實驗室環境開發的模型外推算到現場資料或其他實際使用概況，如全球協調輕型車輛測試周期(WLTC)，锂電池單體在其一生中以截然不同的方式循環，這已被證明是一個重大挑戰。

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.

圖1 |所提方法與以往研究工作的對比示意圖。

基于特征的退化預測方法專注于恒定充電協定(藍色/紅色曲線表示充電/放電階段)，使用容量-電壓曲線的特征作為輸入。這強調了曆史收費資料的知識的重要性。

我們的方法考慮了可變充電協定(藍色/紅色陰影區域表示充電/放電協定的電流範圍)，這更類似于電動汽車設定。此外，我們使用充電前測量的電化學阻抗譜作為輸入，不了解任何曆史資料，并預測未來不同使用協定對放電容量的影響。

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%).

在這項工作中，我們試圖确定是否存在一個足夠資訊的锂電池單體健康标記，可以用來預測短期和長期的未來性能，在不平衡的曆史和未來锂電池單體使用。圖1展示了我們的方法，以及它與以前的方法的不同之處。我們發現，在充電前擷取EIS頻譜，可以預測下一個周期和更長期的電池容量，測試誤差小于10%。

當在與用于訓練模型的周期條件相似的锂電池單體上進行測試時，我們的模型達到了與最先進的預測模型相當的精度(8.2%的測試誤差對8.8%的測試誤差)，除了我們的模型可以在不通路任何曆史資料的情況下進行預測，而以前的最先進的模型需要來自細胞周期軌迹的曆史資料。此外，當外推到不同的工作溫度時，我們的模型顯著優于最先進的模型，測試誤差降低了57%(從34.2%降至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.

我們觀察到，我們的模型是資料高效 data-efficient的，隻需要8個單元就可以獲得小于10%的測試誤差。

至關重要的是，我們的方法對資料集偏移dataset shift具有魯棒性，在具有不同循環模式分布的資料集上獲得小于7%的測試誤差。這對于在駕駛模式可能不同于用于訓練模型的駕駛模式的現場部署非常重要。

我們還證明，如果可用，使用基于曆史容量-電壓資料的附加特征可以增強狀态表示，并将平均測試誤差降低高達25%。我們的方法在電池制造商、平均使用模式和操作溫度方面是穩健的。

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.

此外，我們的工作通過提供動态工作條件下锂電池單體循環資料的大型語料庫，填補了公開資料的空白。我們的工作側重于一組理想化的使用分布，而不是現實的駕駛剖面，以證明模型的泛化程度。我們的工作脫離了NASA的随機使用資料，該資料随機循環電池50個周期，然後通過“參考”協定充電後測量下一個周期的放電容量。盡管基于這些資料已經建立了在随機條件下預測退化的幾個模型，但單個協定對下一個周期放電容量的影響無法解耦，并且需要每隔幾個周期就有一個參考充電/放電協定，這與典型的現場使用情況不一緻。

Result

Data generation

For this study, we generate two separate datasets corresponding to commercial LiR coin cells 锂離子可充電鈕扣電池 purchased from two different manufacturers, which allows us to test whether our approach is robust with respect to cell manufacturer.

在這項研究中，我們生成了兩個獨立的資料集，對應于從兩個不同的制造商購買的商業锂離子可充電鈕扣電池，這使我們能夠測試我們的方法對于電池制造商是否穩健。

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.

第一個資料集對應于40個Powerstream LiR 2032锂離子可充電鈕扣電池(标稱容量1C = 35 mAh)。我們将24個電池置于随機選擇的23±2°C充放電電流序列中，進行110-120個完整的充放電循環。每個周期包括電池狀态的初始診斷，包括恒流EIS譜的擷取，然後是使用，包括充電和放電階段。充電和放電分别由兩級和一級恒流(CC)協定組成;電流分别在70-140 mA (2-4 C)、35-105 mA (1-3 C)和35-140 mA (1-4 C)的範圍内随機選擇。

為了測試模型對域移位的魯棒性，我們在上述相同的條件下對其餘16個單元進行循環，除了現在将所有單元的放電電流固定在52.5mA (1.5 C)，而不是在每個周期随機改變放電電流。

所考慮的協定空間如圖2所示，補充圖1中提供了三個單元的容量軌迹示例，以說明與典型單調容量衰減實驗的差異。方法中提供了對循環方案的完整描述，補充表1較長的描述了每個單元所承受的全套操作條件。在使用第一個資料集确認該方法可以成功預測幾個周期前的放電容量後，我們随後顯著擴充了我們的分析，以探索模型對電池制造商、使用模式變化和工作溫度的相關性。

為了實作這一目标，我們在更廣泛的使用模式下，從另一家制造商RS Pro(标稱容量40毫安時)額外循環48個電池。在這種情況下，每個電池再次受到100個兩級CC充電和一級CC放電的循環，在每個循環開始時随機選擇三個速率。然而，我們現在通過為每個電池提供不同的電流分布使問題更具挑戰性，以複制不同的電池使用者彼此具有不同的平均使用模式，但仍然表現出随機的周期到周期行為的場景。在這些電池中，有16個也在35°C的較高工作溫度下循環。

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.

圖2 |提出的充放電協定。我們通過将電池置于一系列随機充放電電流中來生成電池循環資料。我們采用兩級恒流(CC)充電，每級充電長達15分鐘，電流分别從70-140 mA (2-4 C)和3 5-105 mA (1-3 C)的範圍内抽出(藍色陰影區域)。如果在時間限制之前達到4.3 V的安全門檻值電壓，則停止充電。在放電過程中，随機在35-140 mA (1-4 C)範圍内選擇一個恒定的放電電流(紅色陰影區域)，直到電壓降至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 ) Q_n = f(s_n, a_n) Qn​=f(sn​,an​), with uncertainty estimates, where s n s_n sn​ 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 r e Z_{re} Zre​）and imaginary（ Z i m Z_{im} Zim​ ) components of the impedance measured at 57 frequencies, ω 1 ， … ω 57 ω_1，…ω_{57} ω1​，…ω57​, in the range 0.02Hz-20kHz; s n = [ Z r e ( ω 1 ) , Z i m ( ω 1 ) , . . . , Z r e ( ω 57 ) , Z i m ( ω 57 ) ] s_n =[Z_{re}(ω_1),Z_{im}(ω_1),...,Z_{re}(ω_{57}),Z_{im}(ω_{57})] sn​=[Zre​(ω1​),Zim​(ω1​),...,Zre​(ω57​),Zim​(ω57​)]. The action vector is formed from the concatenation of the nth cycle charge and discharge currents.

如果我們應用特定的充放電剖面，我們首先考慮我們想要預測下一個周期放電容量的設定，對于一個使用曆史(例如，周期或月曆年齡，或曆史容量-電壓資料)完全未知的電池。我們将問題架構為一個回歸任務，并訓練一個機率機器學習模型來學習映射 Q n = f ( s n , a n ) Q_n = f(s_n, a_n) Qn​=f(sn​,an​)，具有不确定性估計，其中 s n s_n sn​是第 n n n個循環開始時的電池狀态，an是未來的動作(第n個循環充電/放電協定)， Q n Q_n Qn​是循環結束時測量的放電容量。電池狀态矢量 s n s_n sn​在57個頻率下測量的阻抗的實（ Z r e Z_{re} Zre​）和虛（ Z i m Z_{im} Zim​）分量的串聯形成， ω 1 ， … ω 57 ω_1，…ω_{57} ω1​，…ω57​，在0.02Hz-20kHz範圍内; s n = [ Z r e ( ω 1 ) , Z i m ( ω 1 ) , . . . , Z r e ( ω 57 ) , Z i m ( ω 57 ) ] s_n =[Z_{re}(ω_1),Z_{im}(ω_1),...,Z_{re}(ω_{57}),Z_{im}(ω_{57})] sn​=[Zre​(ω1​),Zim​(ω1​),...,Zre​(ω57​),Zim​(ω57​)]。動作矢量由第n個循環充電和放電電流的串聯形成。

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.

圖3說明了我們模型的準确性。以狀态和動作作為輸入，預測下一循環放電容量，平均誤差為8.2%。重要的是，狀态和動作(圖3a)都被發現是預測未來單元性能所必需的:如果僅使用狀态(圖3b)或動作(圖3c)作為輸入，測試誤差大約翻倍，分别為20.7%和15.4%。這證明了電池的内部健康和外部選擇的使用在決定實作電池性能方面的重要性。

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

對于優化充電和再利用分類等應用，重要的是電池壽命軌迹模型不僅可以預測下一個循環的放電容量，而且可以預測未來幾個循環的容量。考慮到這一點，我們接下來将研究模型的預測準确性如何變化，因為我們推動模型預測未來的容量。在每種情況下，輸入包括第n個周期開始時的狀态表示 s n s_n sn​與“動作”向量 a n … N + j a_{n…N +j} an…N+j​包括在循環 n n n和循環 n + j n +j n+j之間應用的所有充電和放電電流

Figure 4 shows how the coefficient of determination R 2 R^2 R2changes with j. As expected, the accuracy of the model generally decreases as the forecasting interval increases. However, the model still attains R 2 R^2 R2= 0.75 when projecting 40 cycles into the future.

圖4顯示了決定系數 R 2 R^2 R2随 j j j的變化情況。正如預期的那樣，随着預測區間的增加，模型的精度一般會下降。然而，當預測未來40個周期時，模型仍然得到 R 2 R^2 R2 = 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.

圖3 |預測下一循環放電容量。a給定狀态(電池的内部狀态，由EIS譜表征)和動作(下一個循環充放電協定)的知識，我們的模型預測下一個循環放電容量的誤差為8.2%。要準确預測業績，既需要狀态，也需要行動;僅使用(b)行為或( c )行為是不夠的。

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.

圖4 |多步預測。我們的模型還可以預測電池的長期性能，通過(a) %測試誤差和(b) R2值進行量化。考慮到EIS光譜和将應用于電池的下一個協定的知識，放電容量預測的測試誤差小于10%，最多可提前32個周期。

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.

圖5 |資料效率。由(a) %測試誤差和(b) R2量化的模型性能随着用于訓練的細胞數量的增加而提高。該模型具有資料效率，在訓練集中隻有8個單元的情況下，測試誤差小于10%。

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%.

接下來，我們通過調查資料效率和模型的可泛化性來測試我們方法的穩健性。為了測試資料效率，我們測量性能如何随着用于訓練模型的單元數的增加而變化。如圖5所示，随着單元數從2個增加到22個，測試誤差從23.8%顯著降低到8.2%。盡管如此，該模型證明是資料高效的，隻需要8個單元就可以獲得小于10%的測試誤差。

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.

模型泛化性的一個重要測試是研究領域分布變化時的模型精度，即當模型部署在與訓練資料不同的設定中時。

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 .

這對于現場部署非常重要，因為該方法需要對可能與訓練資料不同的駕駛模式具有魯棒性。我們通過循環來自同一制造商的另外16個電池來測試模型的穩健性，但現在通過在整個生命周期中将每個電池的放電電流固定在1.5C來調整循環協定。我們使用一個僅使用在其生命周期内受到随機放電電流的電池訓練的模型，來預測受到固定放電的電池的下一個周期放電容量。為了說明訓練資料集和測試資料集的差異，放電能力的分布如圖6a所示。

模型在固定流量資料集上的預測精度如圖6b所示。有希望的是，該模型在這個域移位資料集上的測試誤差僅為6.3%，對應于 R 2 R^2 R2 = 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.

我們的模型還輸出預測不确定性，這表明模型對其預測品質的确定程度。在域移位的情況下，模型“知道它不知道的東西”并估計資料點的高度預測不确定性是特别重要的，它可能會在這些資料點上獲得很高的誤差。我們可以通過觀察平均測試誤差如何随着資料點數量的減少而變化來測試模型估計其不确定性的能力，以隻包括模型最有信心的資料點。如果一個模型能夠成功地估計其确定性水準，那麼平均測試誤差應該随着資料比例的減少而減少，隻包括最自信的預測點。圖6c顯示，當資料比例從100%降低到最可信的25%時，均方根誤差(RMSE)減少了3.2%，這表明我們的模型已經學會了它應該對哪些預測有信心。

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.

圖6 |對域移位的魯棒性。

a與固定放電率(紅色)相比，在可變放電率(藍色)下循環的細胞的放電容量分布是不同的;兩個分布的重疊區域顔色較深。

b我們的模型在可變放電率單元上訓練，準确地預測了在固定放電率下循環的單元的放電能力。标繪點的顔色表示預測的不确定性(見顔色條)。

c模型“知道它不知道的東西”:當我們限制用于計算均方根誤差(RMSE)的測試資料，隻包括模型最有信心的預測(即具有最低的預測不确定性)，RMSE會降低。

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.

在示範了EIS光譜捕捉電池狀态的能力後，我們現在将這種電池健康狀況的表示與文獻中使用的其他方法進行基準測試，包括最先進的基于特征的方法，并考慮EIS光譜是否有其他特征可以增強電池狀态。

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.

用于預測或估計電池SOH的簡單措施包括使用之前的循環放電容量，或循環開始後的容量吞吐量。更先進的方法包括提取曆史容量-電壓放電曲線的特征，如圖1所示。提取這些特征的最先進的方法是由Severson等人實作的，而不是最近由Attia等人和Paulson等人使用的特征提取方法。我們對基于ais的方法相對于這些最先進的功能的性能進行基準測試。

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 Q_{n−1} Qn−1​.
• Capacity throughput (CT) since cycling commenced, as defined by the sum of cell charge and discharge capacities from cycles 0 to n − 1 n−1 n−1.
• State of Health (SOH), as defined by Q n − 1 / Q 0 Q_{n−1}/Q_0 Qn−1​/Q0​.
• 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 i n V_min Vm​in to V m a x V_max Vm​ax. T h i s 1000-dimensional capacity vector Q n − 1 Q_{n−1} Qn−1​ is normalised by subtracting the equivalent vector from cycle 0, Q 0 Q_{0} Q0​. The following features are then used as inputs: V m a x V_{max} Vmax​, V m i n V_{min} Vmin​, l o g ( v a r ( Q n − 1 − Q 0 ) ) log(var(Q_{n−1} - Q_0)) log(var(Qn−1​−Q0​)), l o g ( I Q R ( Q n − 1 − Q 0 ) ) log(IQR(Q_{n−1} - Q_0)) log(IQR(Qn−1​−Q0​)). Additionally, we fit the capacity to a sigmoid Q ( V ~ ) = p 0 1.0 + exp ⁡ ( p 1 ( V − p 2 ) ) Q\left( \tilde{V} \right) =\frac{p_0}{1.0+\exp \left( p_1\left( V-p_2 \right) \right)} Q(V~)=1.0+exp(p1​(V−p2​))p0​​ where V ~ \tilde{V} 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.

此外，我們評估了以等效電路模型(ECM)的形式納入實體解釋是否能改善預測。我們使用廣泛應用的蘭德爾電路模型，包括一個串聯電阻，與一個并聯電容和一個Warburg阻抗元件的電阻連接配接，以及更複雜的擴充蘭德爾電路，它在蘭德爾電路中增加了一個額外的電阻-電容并聯組合。

ECM拟合到頻譜(在相關的計算成本下)，我們使用提取的參數作為狀态表示，而不是原始EIS資料。

總的來說，我們在基準測試中考慮了以下特性:

• 前一循環放電容量 Q n − 1 Q_{n−1} Qn−1​。
• 自循環開始以來的容量吞吐量(CT)，由周期0到n−1的電池充放電容量之和定義。
• 運作狀況(SOH)，定義為 Q n − 1 / Q 0 Q_{n−1}/Q_0 Qn−1​/Q0​。
• 容量-電壓放電曲線(CVF)的最新特征:繼Severson等人之後，我們通過從周期n−1後的容量-電壓放電曲線中提取特征，在周期n開始時形成一個狀态表示。我們拟合一個 c h c_h ch​曲線到一個樣條函數，線性插值來測量1000個均勻間隔電壓從 V m i n V_{min} Vmin​到 V m a x V_{max} Vmax​的容量。T h i的1000維容量向量Qn−1通過從周期0,Q0中減去等效向量進行歸一化。然後使用以下特征作為輸入: V m a x V_{max} Vmax​, V m i n V_{min} Vmin​, l o g ( v a r ( Q n − 1 − Q 0 ) ) log(var(Q_{n−1} - Q_0)) log(var(Qn−1​−Q0​)), l o g ( I Q R ( Q n − 1 − Q 0 ) ) log(IQR(Q_{n−1} - Q_0)) log(IQR(Qn−1​−Q0​)).。此外，我們将容量拟合為一個 Q ( V ~ ) = p 0 1.0 + exp ⁡ ( p 1 ( V − p 2 ) ) Q\left( \tilde{V} \right) =\frac{p_0}{1.0+\exp \left( p_1\left( V-p_2 \right) \right)} Q(V~)=1.0+exp(p1​(V−p2​))p0​​ ，其中 V ~ \tilde{V} V~為歸一化電壓，并使用參數p0, p1, p2作為特征。
• 等效電路模型參數(ECM-R和ECM-ER):我們使用Randles電路(ECM-R)和Extended Randles電路(ECM-ER)拟合等效電路模型到EIS譜，并将得到的參數連接配接在一起。

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.

我們注意到，與EIS特征相比，使用前面提到的前四個特征形成的狀态表示需要通路曆史電流-電壓資料，至少在之前的整個放電過程中，對于某些特征，在整個電池壽命中。然而，他們受益于不需要裝置來測量EIS光譜的優勢，這帶來了相關的财務和時間成本。使用ECM參數(從EIS譜中提取)形成狀态表示具有相關的計算成本，可以認為是原始EIS資料降維的一種形式。ECM通常面臨的另一個問題是非唯一性，因為ECM參數的多個不同組合通常可以同樣好地解釋特定的EIS譜。

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.

表1顯示了狀态表示對測試誤差和模型拟合優度的影響。在所有情況下，訓練模型預測下一個周期放電容量，給定下一個周期協定和選擇的狀态表示。在詢問特性的相對重要性時，我們首先考慮隻使用EIS(不包括協定)和隻使用協定(不包括EIS)的基線。也許不出意外，電池的退化是目前狀态和未來充放電協定的一個函數。是以，同時使用EIS和協定的性能明顯優于僅使用EIS或僅使用協定的性能。

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.

然後，我們使用蘭德爾(ECM-R)和擴充蘭德爾(ECM-ER)等效電路模型，探讨了基于實體的EIS頻譜表示的影響。EIS +Protocol與ECM-R +Protocol和ECM-ER +Protocol的比較表明，這些基于實體的模型會丢失資訊，使用機器學習方法直接從原始資料中學習可能是有利的。

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.

接下來，我們考慮文獻中報道的不同方法，Qn−1,SOH, CT和CVF，其中CVF是電池資訊學文獻中最先進的方法。在所有情況下，EIS +Protocol優于Protocol的其他特性，盡管CVF具有競争力。

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%.

有趣的是，來自容量-電壓曲線資料(CVF)的資訊是EIS的補充-将EIS與這些特征結合起來會顯著提高準确性(EIS + CVF +Protocol)。這也許并不令人驚訝，因為EIS探測單個“靜态”電池放電狀态的阻抗(每個瞬間狀态具有高資訊含量)，而容量-電壓曲線探測電池狀态如何在從充電到放電的路徑上連續演變(每個瞬間狀态具有低資訊含量)。

最後，通過結合上述所有特征來形成狀态表示，進而獲得最佳的模型性能。在這種情況下，平均測試誤差僅為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.

我們現在擴充我們的分析，以探索我們的方法在改變電池制造商、調整操作溫度和調整平均使用模式方面的魯棒性。我們在另一家制造商RS Pro的新一批32塊商業锂離子可充電紐扣電池(标稱容量為1 C = 40 mAh)上重複了我們的實驗，隻是我們現在讓問題變得更具挑戰性，讓不同的電池亞組分别接受四種不同的使用分布中的一種。這些使用情況的分布見補充表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.

我們通過兩種方式來衡量模型的準确性:

首先，我們考慮模型暴露于與測試集受到相同協定分布(随機分裂)的锂電池單體的情況，

其次，更具有挑戰性的情況，即模型隻在受到三種循環協定分布的锂電池單體上訓練，并在受到不同循環協定的其餘八個锂電池單體上測試。

這是一項更加艱巨的任務，因為測試單元格上的平均使用量與訓練單元格上的平均使用量非常不同——這是一個測試模型是否可以外推到不同的平均使用量，而不僅僅是不同的周期到周期的使用。

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.

表2顯示了不同狀态表示的結果，其中訓練/測試分割是随機的，以及分割分層為不同的使用模式。對從第二個制造商購買的電池進行了比較觀察:即，當使用EIS譜的特征與放電曲線(CVF)形成的特征形成狀态表示時，進行了最準确的預測。

正如預期的那樣，當它使用來自一些锂電池單體的資料進行訓練時，模型的表現明顯更好，這些锂電池單體暴露在與模型測試時相似的循環模式分布中。

然而，該模型在支架分裂場景中仍然具有性能，在這種設定下，當使用EIS譜和放電曲線的特征而不是單獨使用放電曲線的特征來形成狀态表示時，測試誤差降低了30%。

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.

這些額外的結果進一步證明，如果可用，EIS譜和放電曲線都可以作為電池内部狀态的資訊标記，但它們是互相補充的。

接下來，我們驗證該模型對于外部操作溫度的變化是穩健的。我們在35°C下循環另外16個锂電池單體，并在室溫下循環的锂電池單體資料上測試訓練的模型。

表3顯示，我們的模型可以外推到在這些更高溫度下工作的電池，但當電池不在相同溫度下工作時，EIS譜在表征電池狀态方面起着特别重要的作用。僅使用放電曲線特征表征狀态時，模型的測試誤差為34.2%，同時使用EIS譜和放電曲線特征時，模型的測試誤差為14.0%。這進一步證明了EIS信号相對于充放電曲線所包含的額外資訊，并支援了EIS隐式跟蹤溫度的假設。

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.

當我們在RS Pro(而不是Powerstream)制造的電池上測試我們的方法時，我們得到了質的相似的觀察結果，發現EIS比最先進的容量電壓特征(CVF)的狀态表示略好。

将兩種表示法結合起來可以得到最好的結果。當我們随機分割訓練集和測試集時，當我們對訓練集和測試集進行分層時，我們會測試模型的表現，這樣模型就會在與它所訓練的細胞具有不同使用分布的細胞上進行測試。

在補充表1、2和3中提供了使用條件和不同狀态表示的擴充比較。

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%).

在本文中，我們展示了電化學阻抗譜準确地表征了電池的内部狀态，并且可以訓練機器學習模型，即使在不均勻和未知的曆史電池使用情況下，也可以準确地預測短期和長期電池性能。我們的模型達到了與最先進的預測方法(8.2%的測試誤差)相當的準确性(8.8%的測試誤差)，當測試單元受到相同的操作條件分布時，用于訓練模型的單元。

然而，如圖1所示，最先進的方法需要通路曆史循環資料，而我們的模型可以在沒有曆史資料的情況下進行預測。此外，當外推到更高的工作溫度時，我們的模型顯著優于最先進的模型，測試誤差降低了57%(從34.2%降至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.

我們的方法是資料高效的，僅使用八個單元的訓練資料就實作了9.9%的下一個周期測試誤差，并且對資料集分布的變化具有魯棒性。

此外，我們發現，如果曆史循環資料可用，則有空間将模型性能提高25%; 這樣的資料可用于派生增強單元狀态表示的特征。我們證明了我們的方法可以用于不同的電池化學成分，并且該模型對不同的操作溫度具有魯棒性。

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.

我們的方法在兩個重要方面與現有技術不同:

首先，我們采用了資訊豐富的電信号- eis -它在不了解周期曆史的情況下捕捉锂電池單體在不同時間尺度上的響應。

這與大多數現有的方法形成了對比，這些方法使用來自充放電曲線的特征——一種明顯更粗粒度的信号——作為機器學習模型的輸入。

我們的研究結果表明，通過整合測量電化學阻抗的電路，電池管理系統有了顯著的改進，盡管在财務和時間上都有成本。

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.

其次，我們關注不均勻循環，其中充放電速率随循環而變化。這與之前關于電池退化的機器學習的研究不同，以前的研究集中在恒定的充放電條件下，這是電池測試中的典型情況。我們的結果對單一标量健康狀态的概念提出了質疑，因為電池的狀态取決于無數不同退化機制的程度，而這又取決于曆史充放電協定的順序。相反，我們建議電池可以用多元狀态向量來描述，使用資訊性高維測量(如EIS)來捕獲，并且可以使用機器學習方法來預測給定狀态向量和未來充電/放電協定的未來容量。

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.

此外，盡管在這項工作中，我們隻考慮從初始放電狀态開始預測，但我們假設，在未來的工作中，應該有可能根據EIS測量從任何電荷狀态開始預測放電容量，因為EIS譜隐含地跟蹤電荷狀态。

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

我們注意到，鑒于目前電池狀态和未來行動，我們為預測未來電池性能所制定的總體架構，在廣泛的電池診斷和控制設定中具有應用範圍。例如，預測所提議的充電協定對下一個周期放電容量以及長期退化的影響對于優化快速充電應用是重要的，其中必須在充電時間和電池退化率之間實作平衡。我們的工作還可以擴充到考慮更複雜的動态使用協定，比如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.

在這項研究中，我們從兩家不同的制造商Powerstream和RS Pro購買了88個商業LiR紐扣電池，在23±2°C的溫度調節實驗室中循環。使用Biologic BCS-805恒電位器進行循環，實驗裝置的照片見補充圖2。

在所有資料集中，電池都受到一系列随機選擇的充放電電流，進行110-120個完整的充放電周期。當電池處于完全放電狀态時開始循環，每個循環包括以下步驟:(a)在開路電壓下靜置20分鐘，(b)在完全放電狀态下獲得恒流EIS譜，©兩級CC充電，(d)在開路電壓下靜置20分鐘，(e)在完全充電狀态下獲得恒流EIS譜，(f)一級CC放電。恒流EIS譜總是通過收集均勻分布在0.02Hz-20kHz範圍内的對數域57個頻率的阻抗測量值，使用振幅為5 mA的正弦電流來測量。锂電池單體在溫度控制在23±2°C的實驗室中循環。

為了生成第一個資料集，我們循環24個Powerstream LiR 2032硬币電池(标稱容量1 C = 35 mAh)。對于這些電池，充電包括兩個階段的CC協定;電流分别随機選擇在70-140mA (2C-4C)和35mA-105mA (1C-3C)範圍内的階段1和階段2。每個充電階段設定15分鐘的時間限制，使總充電時間限制在30分鐘或更短。

如果達到4.3 V的安全門檻值電壓，将在30分鐘時限前停止充電。在放電過程中，在35mA-140mA (1C-4C)範圍内随機選擇一個恒定的放電電流，直到電壓降至3.0 V。

另外16個電池(也由Powerstream制造，标稱容量為35毫安時)在相同的條件下循環，除了現在我們将所有電池和周期的放電電流固定在52.5 mA (1.5C)，而不是在每個周期随機改變放電電流。

然後，我們生成第二個資料集，以探索模型對電池制造商的魯棒性、使用模式的變化和操作溫度。我們從第二個制造商RS Pro(标稱容量40毫安時)循環48個電池，在更廣泛的使用模式下。一般的六步循環方案與上述相同，每個電池再次受到100個兩階段CC充電和一階段CC放電的循環，在每個周期開始時随機選擇三個速率。然而，電流的分布現在改變了每個單元。在這些電池中，有16個也在溫度控制的加熱室中，在35±2°C的較高操作溫度下循環。補充表1較長的描述了每個單元所承受的全套操作條件。

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 … j ) Q_j = f(s_n, a_{n…j}) Qj​=f(sn​,an…j​), 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 r e Z_{re} Zre​）and imaginary（ Z i m Z_{im} Zim​ ) components of the impedance measured at 57 frequencies, ω 1 ， … ω 57 ω_1，…ω_{57} ω1​，…ω57​, in the range 0.02Hz-20kHz; s n = [ Z r e ( ω 1 ) , Z i m ( ω 1 ) , . . . , Z r e ( ω 57 ) , Z i m ( ω 57 ) ] s_n =[Z_{re}(ω_1),Z_{im}(ω_1),...,Z_{re}(ω_{57}),Z_{im}(ω_{57})] sn​=[Zre​(ω1​),Zim​(ω1​),...,Zre​(ω57​),Zim​(ω57​)]… 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.

本研究中的所有問題都被框定為回歸任務。我們訓練一個機率機器學習模型來學習映射 Q j = f ( s n , a n … j ) Q_j = f(s_n, a_{n…j}) Qj​=f(sn​,an…j​)，具有不确定性估計，其中 s n s_n sn​是第n個循環開始時的電池狀态， a n a_n an​是應用于第n到第j個循環的未來循環協定集， Q j Q_j Qj​是第j個循環結束時的放電容量。電池狀态矢量 s n s_n sn​在57個頻率下測量的阻抗的實（ Z r e Z_{re} Zre​）和虛（ Z i m Z_{im} Zim​）分量的串聯形成， ω 1 ， … ω 57 ω_1，…ω_{57} ω1​，…ω57​，在0.02Hz-20kHz範圍内; s n = [ Z r e ( ω 1 ) , Z i m ( ω 1 ) , . . . , Z r e ( ω 57 ) , Z i m ( ω 57 ) ] s_n =[Z_{re}(ω_1),Z_{im}(ω_1),...,Z_{re}(ω_{57}),Z_{im}(ω_{57})] sn​=[Zre​(ω1​),Zim​(ω1​),...,Zre​(ω57​),Zim​(ω57​)]。動作矢量由第n個循環充電和放電電流的串聯形成。當預測幾個周期 j j j的放電容量時，未來的協定現在由在周期n和周期n + j之間應用的所有充電和放電電流形成。

對于機器學習模型，我們使用10個XGBoost模型的內建，每個模型有500個估計器，最大深度為100。集合中每個模型所作預測的平均值和标準差用于量化預測輸出和預測不确定性。為了測試模型性能，我們使用中位數 R 2 R^2 R2分數和中位數百分比誤差。為了從包含N個單元的資料集中獲得測試名額，我們随機留出兩個測試單元，在剩餘的 N − 2 N-2 N−2個單元上進行訓練，并重複此過程 N / 2 N/2 N/2次，每次留出不同的單元。

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.