Simple statistical models and sequential deep learning for Lithium-ion batteries degradation under dynamic conditions: Fractional Polynomials vs Neural Networks
Clara B. Salucci, Azzeddine Bakdi, Ingrid K. Glad, Erik Vanem, Riccardo De Bin
SSimple statistical models and sequential deep learning forLithium-ion batteries degradation under dynamic conditions:Fractional Polynomials vs Neural Networks
Clara B. Salucci , Azzeddine Bakdi ,Ingrid K. Glad , Erik Vanem , Riccardo De Bin February 12, 2021
Abstract
Longevity and safety of Lithium-ion batteries are facilitated by efficient monitoring and adjustmentof the battery operating conditions: hence, it is crucial to implement fast and accurate algorithmsfor State of Health (SoH) monitoring on the Battery Management System. The task is challengingdue to the complexity and multitude of the factors contributing to the battery capacity degrada-tion, especially because the different degradation processes occur at various timescales and theirinteractions play an important role. This paper proposes and compares two data-driven approaches:a Long Short-Term Memory neural network, from the field of deep learning, and a MultivariableFractional Polynomial regression, from classical statistics. Models from both classes are trainedfrom historical data of one exhausted cell and used to predict the SoH of other cells. This work Corresponding author.Department of Mathematics, University of Oslo, 0851 Oslo, Norway, [email protected] Department of Mathematics, University of Oslo, 0851 Oslo, Norway, [email protected] Department of Mathematics, University of Oslo, 0851 Oslo, Norway, [email protected] Department of Mathematics, University of Oslo, 0851 Oslo, Norway;DNV GL Group Technology and Research, 1322 Høvik, Norway, [email protected] Department of Mathematics, University of Oslo, 0851 Oslo, Norway, [email protected] a r X i v : . [ s t a t . A P ] F e b ses data provided by the NASA Ames Prognostics Center of Excellence, characterised by varyingloads which simulate dynamic operating conditions. Two hypothetical scenarios are considered:one assumes that a recent true capacity measurement is known, the other relies solely on the cellnominal capacity. Both methods are effective, with low prediction errors, and the advantages ofone over the other in terms of interpretability and complexity are discussed in a critical way. Keywords
Lithium-ion batteries; State of Health prediction; Long Short-Term Memory; Multivariable Frac-tional Polynomial; Linear regression; NASA Randomized Battery Usage Data Set
Transports account for the largest share of greenhouse gas emissions, hence it is fundamental tooptimise sustainable and low-emission solutions such as electric batteries. Lithium-ion batteries(LIBs) are the most popular battery technology, as they offer important advantages compared toother battery types such as lead-acid, nickel-cadmium or nickel-metal-hydride [1, 2]. The perfor-mance of LIBs is nevertheless destined to deteriorate over time (calendar ageing) and usage (cycleageing), as they are complex electrochemical systems sensitive to operating conditions and theirnonlinear characteristics are time-varying due to ageing. To cope with the increasing demand forhigh-performance and durability of rechargeable batteries, Prognostics and Health Management(PHM) received tremendous attention over the recent years. PHM plays a crucial role: it en-ables the operators to monitor the State of Health (SoH) of the battery, defined in Section 2.2,and take actions to maintain availability and reliability. The LIB prognostics phase includes fivesteps [3]: measurement, feature extraction, SoH estimation, SoH prediction, and Remaining UsefulLife (RUL) estimation. Various approaches are proposed in the literature for SoH estimation ofLIBs, which may be categorised as: (i) experimental approaches, including direct SoH testing andexperiment-based static models; and (ii) adaptive data-based methods, including filters, observers,expert systems, statistical and machine learning (ML) methods. This article focuses on statistical2nd ML methods.Experimental direct testing methods [4] include internal resistance and impedance measurement,battery energy level, and incremental capacity analysis methods [5]. Common measurement-basedmodels for SoH estimation include coulomb counting, destructive tests [3], and other data-fittingmodels obtained only from test measurements. The experimental methods can only be conductedoffline and are highly time-consuming: hence they are inappropriate for the Battery ManagementSystem (BMS). Adaptive models are based on online parameter estimation using: physical modelssuch as equivalent circuits [6], and electrochemical models [7]; purely data-driven models; domainknowledge; hybrid models. Among the approximate physical models, the widest classes of SoHestimation methods are filters and state observers: especially, Kalman filter and its unscented andextended versions are widely adopted for the estimation of State of Charge (SoC), defined e.g. in [8],and have their applications successfully extended to SoH estimation [9] and sliding-mode observers[10]. Pure knowledge-based SoH estimation approaches comprise expert systems such as fuzzy logic[11] and Bayesian networks [12] with structures designed by experts: these are however limited,though widely integrated with the other approaches [13].Data-driven SoH estimation models are abundant in the literature: they are usually based onpartial segments of charging/discharging curves, and mainly rely on ML methods. For example,[14] developed an energy-based segmentation called ”energy of equal distance voltage difference”to estimate SoH using deep neural network (NN). A gate recurrent unit-convolutional NN wasdeveloped in [15], and deep convolutional NN (CNN) in [16] to estimate SoH from full trajectoriesof fixed charge curves; whereas [17] extended the idea to multiple cells SoH estimation from their(voltage, current, temperature) trajectories over a sliding window using Long Short-Term Memory(LSTM). A more comprehensive overview on data-driven models can be found in [8].While exhibiting good prediction abilities, ML approaches lack interpretability and require largeamounts of data: the complexity of ML models, which are difficult to be handled and checked, risksreplacing the complexity of the physical problem that was to be avoided by considering a data-drivenapproach. For instance, a NN acts as a black box: it provides results based on transformationsunavailable to the user. The example of [18], in which a transformation of the response is amongthe inputs, shows that the NN may be forced to find spurious relationships: in fact, the prediction3rror is not 0 as it should have been, had the right relationship been identified. Furthermore, thisexample points out that recognizing potential issues can be difficult when dealing with black-boxmethods: in contrast, a linear regression with the same response transformation among the inputswould have provided a clear indication of the one-to-one relationship between the input and theoutcome. Motivated by this case, we aim at promoting the use of simple statistical methods tomodel LIB degradation, recognising that complex models have the additional downside of beingnot implementable in the BMS. We first design a ML approach based on a Deep LSTM RegressionNetwork (D–LSTM–RN), then present an alternative perspective in which a statistical model isidentified via Multivariable Fractional Polynomial (MFP) approach. We discuss the advantagesin terms of interpretability (e.g. explanation of the individual feature contribution to ageing ac-celeration), generalisability (ML methods are generally prone to overfitting, which is more easilyhandled for simpler models), and portability (e.g. computational efficiency and implementabilityon the BMS), without substantial loss of performance in terms of prediction ability. Both our MLand statistical approaches are non-destructive; they make use of measured variables only, withoutneed of computing SoC, open circuit voltage (OCV), or any derived quantity; and they are basedon LIB historical data, without using inputs from specific reference cycles or particular segments.Other works considering linear regression in connection to LIB capacity estimation or predictionare [19, 20, 21, 22, 23]. Different statistical methods have also been proposed, for example: [24]used support vector machines to develop a curve-similarity factor to estimate SoH from chargingvoltage segments; [25] estimate SoH using Gaussian process regression based on another form ofpartial segments designed as n equispaced voltage points. Further methods require the segment tobe a full cycle, such as approximate weighted total least squares [26] to estimate the rated dischargecapacity from arbitrary total capacity. Due to their common idea of partial segments, the accuracyof these methods is subject to the availability of long deep monotonic segments, and they cannotbe applied to SoH estimation for LIBs under dynamic conditions and calendar ageing.Conversely, history-based SoH prediction approaches can explain the degradation of batteryhealth as influenced by the whole LIB history, and they are crucial in PHM for their simultaneousadvantages of predicting the future performance and optimising the present operating conditions.Different operating conditions have different effects on the LIB ageing behaviour; a SoH prediction4odel with capabilities of predictive and prescriptive analytics, such as those considered in thisarticle, enables the BMS to adjust temperature and charge/discharge currents to increase longevityand facilitate safe, high-performance operation.The LIB degradation emerges from a complex interplay between many influencing elements,degradation mechanisms (internal side reactions), degradation modes, and observed degradationeffects [27]. The influencing elements include: cell and pack design factors; production factors;and application (stress) factors; they influence internal side reactions through complex irreversiblephysical and chemical processes, which in turn lead to the various degradation modes of Lithiumdepletion, active material loss, electrolyte decomposition, and increase of internal resistance [27]. Asa result, the observed LIB degradation effects are capacity fade and power fade. While the designand production factors of influence are fixed and depend on the monitored LIB, the stress factors aredynamic features that may accelerate the LIB degradation behaviour. They can be extracted frommeasured variables or estimated states and they include: exposure to elevated and low voltages;Depth of Discharge (DoD); cycle bandwidth; cycling frequency; high and low temperatures; highdischarge rates (Section 2.1).A critical review [3] emphasises that the degradation effects originate from various processes andtheir interactions: studying the ageing mechanism is challenging as these processes occur simulta-neously, they have different time scales, and it should be avoided to analyse them independently.However, very few multi-factor SoH degradation analyses are reported in the literature: [28] stud-ied the effects of current, cycling limits, and temperature on battery ageing using four dependentmodels of these factors; [29] conducted a weight analysis to study the influence of voltage, capacityand internal resistance inconsistency on module capacity; [30] conducted orthogonal experiments tostudy the impact degree of single and multiple stress factors on capacity loss. Unfortunately, thesefactors are considered independently or in subgroups, as pointed out in Section 2.1. Thus, thispaper also aims at exploring how various combinations of stress factors affect the LIB degradation,and the degree of impact of different factors to ageing acceleration.The remainder of this article is organised as follows: Section 2 overviews the stress factorsfor LIBs, introduces the problem of SoH degradation in relation to the battery capacity fade,and presents the experimental data used for the analysis; Section 3 introduces the LSTM neural5etwork, and presents our deep LSTM regression network model and its results; Section 4 describesthe MFP algorithm in the context of linear regression, and presents the MFP models and theirresults; Section 5 compares the two methodologies with each other and with recently publishedmethodologies applied on the same dataset; Section 6 summarises the main results and providesconcluding remarks. LIB capacity decrease is attributed to multiple factors and processes and their interactions overvarious timescales, including: exposure to elevated voltages; Depth of Discharge (DoD); cycle band-width; cycling frequency; elevated temperatures; and high discharge rates [31, 32].Temperature has a significant impact on performance, safety, and ageing of LIBs. The temper-ature range ( − ◦ C , +60 ◦ C) is considered acceptable [33], whereas the desired temperature rangeis (+15 ◦ C , +35 ◦ C) [34]. The effects of temperature on LIBs can be classified into low-temperatureand high-temperature effects: in both cases, extreme temperatures affect both calendar and cycleageing. Low temperatures may cause slow chemical reactions and charge transfer, decreased ionicconductivity, and Lithium plating [35]. Discharging at low temperatures results in power limitation,while low-temperature charge produces a reduced power capability and cold cranking [36]. Hightemperatures may cause loss of Lithium and increase of internal resistance, which in turn produceloss of capacity and of power, respectively. Furthermore, the operating temperature affects theState of Charge (SoC) of the battery, and extreme temperatures may accelerate Lithium platingon the anode. Temperature-based features will be designed to account for the influence of tem-perature on the battery ageing behaviour. Fast charging is a desired aspect in batteries; however,high charge-discharge rates may cause mechanical-induced damage of active particles in LIBs andaccelerate the capacity fade [37]. Lithium plating is also associated with fast charging, particularlyin combination with low temperatures. The effects of different current rates on coulomb efficiencyand capacity loss are studied in [38], and [39] explores the impact on capacity estimation, confirmingthat the C-rate (or, equivalently, the current intensity) is an important feature for capacity fade6rediction. In this work, we consider the discharge current I .The charge–discharge cycling frequency is related to mechanical stress on LIBs: it affects thedegradation behaviour, especially when it is extreme [40], but high frequency may also come asa consequence of the battery ageing. The data at hand are characterised by a varying cyclingfrequency, which we consider in the analysis through the cycle duration, the proportion of cycleslasting less than the default duration, and the rest time. The rest time affects the recovery effects,too, and the charge balancing of the battery, and influences its lifetime [41]. Besides temperature,both cycle current and rest time affect the SoC estimation, a complex process which often leads touncertain results: this work advantageously avoids relying on SoC estimation.High voltages and overcharge contribute to Lithium plating and electrolyte decomposition, whichaccelerate the battery ageing [42]. The Depth of Discharge DoD = SoC − SoC is also usuallyconsidered as a stress factor; however, [43] showed that the adopted SoC range (SoC , SoC ) playsa bigger role: in fact, though batteries cycled with range (100%, 25%) degraded faster than thosewith (100%, 40%), it was also observed that (100%, 40%) degraded much faster than (85%, 25%),despite the same DoD. Furthermore, (100%, 50%) showed a faster degradation than (85%, 25%),despite the lower DoD. In our work, the initial voltage and the voltage difference of the cycle accountfor the stress factors of high voltages, DoD, and SoC range.All these stress factors are accounted for, directly or indirectly, in our model. However, thereare redundancies in the input feature space, to account for the presence of nonlinear relationshipsbetween the features. These factors are analysed in previous works separately; however, they shouldnot be analysed independently as their interactions play an important role in LIBs ageing. Thisanalysis models the battery capacity loss as a process of all factors simultaneously, and researcheswhich combinations of features contribute mostly in explaining the capacity fade, measuring theirdegree of importance. The dataset we use is part of the “Randomized Battery Usage Data Set” by the NASA AmesPrognostics Center of Excellence (PCoE) [44], which comprises ageing data for 18650 LIBs cycledunder randomly generated current profiles. The aim of the experiment was to mimic the dynamic7perating condition of batteries used in real-life. In fact, though laboratory data have enabledimportant progress in the study of LIB deterioration, they are typically gathered under particularand unrealistic conditions, for example with small temperature variation and constant current. Thisrandomised dataset constitutes an effort towards a better approximation of real-life conditions; it isa well-known benchmark that is widely used for training, testing, and comparing various methods inthe literature, hence we adopt it to validate our work. The data considered in this study pertain tofour battery cells, RW9, RW10, RW11 and RW12, that were operated in a controlled environmentunder different modes; the two modes relevant for this analysis are: • Reference discharge: a controlled full discharge cycle, occurring immediately after a con-trolled full charge cycle, allows to compute the cell capacity periodically. Thus, the capacitypredicted by the models can be compared to the true capacity. During a reference chargecycle, the cell is initially charged at a constant current of 2 A, until the battery reaches themaximum voltage of 4.2 V; then, the voltage is kept constant until the charging current dropsto 0.01 A. During the subsequent reference discharge cycle, the cell is discharged at 1 A untilthe voltage reaches the threshold of 3.2 V. • Random walk (RW) steps: the current load is selected at random from the set: { -4.5 A, -3.75 A, -3 A, -2.25 A, -1.5 A, -0.75 A, 0.75 A, 1.5 A, 2.25 A, 3 A, 3.75 A, 4.5 A } with negative values implying charging and positive values implying discharging. The ran-domly selected charge or discharge lasts for 5 minutes, unless the voltage reaches the rangeboundaries [3.2 V, 4.2 V]: in this case, the RW step is immediately stopped and a new currentis selected from the set to proceed with another step. Between every two charge/dischargesteps, a short resting time ( < C d = (cid:90) t cutoff I d dt ; (1)in particular, since the discharging current for the reference cycles throughout the experiment isalways I d = 1 A, the magnitude of the discharging capacity corresponds to the discharging time,expressed in hours. It should be noted, however, that the batteries are not likely to be at equilibriumat the beginning of each reference cycle, since there were too short resting periods (or no restingperiods at all) to allow reaching the steady state. Consequently, the initial voltage of the cycles isuncertain, and generally different from the desired value of 4.2 V. This translates into an uncertaintyon the benchmarked capacity [47], the determination of which would involve an accurate study of thebattery transient dynamics, possibly complemented by gathering of data from cycles interspersedwith longer resting periods of the cell. An adjustment for the transient effects is introduced inSection 3.2.2; the adjusted capacities for the four cells considered in the study are shown in Figure1.Figure 1: True capacity values for cells RW9 (blue), RW10 (green), RW11 (red), RW12 (brown)adjusted as explained in Section 4.2.2 9igure 2 shows the RW9 data for voltage, current intensity and temperature for the first andthe last 50 RW steps respectively: it can be noticed that there is a consistent difference betweenthe two voltage curves: in the first steps, the voltage rarely hits the range boundaries of 3.2 and 4.2V; while in the last steps, this happens more frequently. As a result, the last steps are also shorter,as many of them last for less than 5 minutes: therefore, the total time is only 1 hour comparedto the 3.5 hours of the first RW steps. There is also a significant difference in the temperature,which is higher in the second sub-figure. All these effects are a clear indication of the cell healthdegradation.Figure 2: Measurements of voltage (red), current intensity (blue) and temperature (green) duringthe first 50 RW steps (left sub-figure) and the last 50 RW steps (right sub-figure) of cell RW9. Long short-term memory (LSTM) is a type of Recurrent Neural Network (RNN), i.e. a multi-layerNN. The LSTM architecture was originally introduced by Hochreiter and Schmidhuber [48] withthe purpose of overcoming the vanishing or exploding gradients problem [49], by allowing constanterror flow through self-connected units embedded in the LMST cell. This key feature of LSTMmakes it capable of learning long-term dependencies, as opposed to vanilla-RNN.The computational unit of a NN is the neuron, often called node or cell. The LSTM-NN has aparticular neuron, called LSTM cell or memory cell, which will be explained in the following basedon the description in [50] and [51]. The state of the network at the k -th LSTM cell, ( c k , H k ), is10omposed of: the cell state c k , which encloses the learnt information up to step k ; and the hiddenstate H k , which is the output of the cell. The network state ( c k , H k ) is fed back as input by the cellat the next step, k + 1, which can optionally modify the state by adding or removing information.The learnable weights of an LSTM layer are: the input weights W I , the recurrent weights
W R ,and the bias B . Inside the LSTM cell, the network state is modified through three main steps thatare controlled by gates: • Forget : a “forget gate” decides how much information to keep from the previous cell state,through a sigmoid activation function σ ( x ) = [1 + exp( − x )] − , f k +1 = σ ( W I f x k +1 + W R f H k + B f ) . (2) • Update : an “input gate” decides which are the values that shall be updated, again through asigmoid function, i k +1 = σ ( W I i x k +1 + W R i H k + B i ); (3)in addition, another sigmoid or hyperbolic tangent layer produces new candidate values ˜ c t +1 that could be used to update the cell state,˜ c k +1 = σ ( W I ˜ c x k +1 + W R ˜ c H k + B ˜ c ) or ˜ c k +1 = tanh( W I ˜ c x k +1 + W R ˜ c H k + B ˜ c ) . (4)The combination of these two procedures, added up to the product of the previous cell stateby the forget gate, creates an update to the cell state, c k +1 = f k +1 (cid:12) c k + i k +1 (cid:12) ˜ c k +1 , (5)where (cid:12) indicates the element-wise multiplication of vectors.11 Output : Lastly, the sigmoidal “output gate” o k +1 = σ ( W I o x k +1 + W R o H k + B o ) (6)decides how much of the information carried by the newly updated cell state ought to beadded to the hidden state, H k +1 = o k +1 (cid:12) σ ( c k +1 ) or H k +1 = o k +1 (cid:12) tanh( c k +1 ) , (7)where a sigmoid or tanh function has been applied to the updated cell state c k +1 .At the end of the process, the weights W I , W R and the biases B of a LSTM cell are concatenationsof each gate’s weights and biases: W I = W I i W I f W I ˜ c W I o , W R = W R i W R f W R ˜ c W R o , B = B i B f B ˜ c B o . (8) Using exclusively sensor data, a minimum and sufficient set of input features for the Deep LSTMregression network (D–LSTM–RN) is designed to account for all the degradation acceleration fac-tors, without redundancy and without the need for State of Charge (SoC) estimation. Stress factorsduring the j -th RW step are therefore represented by six features: · V j , initial voltage at the beginning of the cycle; · ∆ V j , cycle signed difference between initial and final voltage values; · ∆ t j , duration of the cycle; 12 T min,j , minimum temperature during the cycle; · T max,j , maximum temperature during the cycle; · ¯ I j , mean current during the cycle.Temperature and current features are directly associated to the temperature- and high-discharge-rate-related battery degradation factors mentioned in Section 2.1; ∆ t j accounts for the cyclingfrequency in these experiments; while voltage features are sufficient to express the exposure toelevated voltages, DoD and cycle bandwidth. The need for manual feature engineering is thusremoved, as this process is achieved in the deep hidden layers of the D–LSTM–RN structure as partof the learning problem. The capacity-fade-prediction problem is here considered as a multivariate-sequence-to-one-scalarregression problem using D–LSTM–RN. The target variable may be the capacity drop between anypair of reference discharge cycles p and n , occurring at time points t p < t n . The capacity fade inthe period [ t p , t n ] is modelled as the cumulative effect of all RW charge and discharge steps that thecell experienced between t p +1 and t n − (excluding reference charge/discharge cycles); the sequencelength varies and it is in the order of S ≈ · Short-term prediction: we assume that the capacity C ( t p ) at cycle p is known, and the targetvariable is ∆ C ( t p , t n ) = C ( t p ) − C ( t n ) , (9)where p and n are two consecutive reference cycles, and C ( t p ) is the true capacity at t p ; · Long-term prediction: we assume that the only known capacity is the cell nominal capacity,and the target variable is 13 C ( t p , t n ) = ˆ C ( t p ) − C ( t n ) , (10)where p and n are two consecutive reference cycles, and ˆ C ( t p ) is the capacity estimated bythe D-LSTM-RN at cycle p .The true capacity used for short-term prediction is not directly computed from Equation 1. Infact, in order to account for the batteries not having reached equilibrium (Section 2.2), a correctionhas been introduced: the voltage curves of each reference discharge have been interpolated witha monotone Hermite spline, which allowed to extrapolate how much longer it would have takenif the cycles had started from the threshold voltage value of 4.2 V, instead of the observed ones.This is but a first step towards a more accurate capacity estimation, but it allows to introducea small correction (Figure S2 in the Supplementary Material). Note that, in the long-term case,data pertaining to the reference cycles are not used at all, making this approach more useful inpractice; however, due to the use of the estimated capacity at cycle p , the long-term model suffersfrom cumulative prediction error.Given the aforementioned sufficient set of features, the battery degradation model can be ap-proximated effectively by D–LSTM–RN in spite of the non-linearities, dynamics, and time-variantcharacteristics of the true ageing process. The structure of the deep learning model is illustrated inFigure 3 with one training example. The model consists of: an input layer through which the sixfeatures are normalised and the multivariate sequence of S ≈ L of N = 200 LSTM cells that outputs the whole sequence of hidden states H ( t ) ( t = 1 , . . . , S ); a second LSTM layer L of N = 200 LSTM cells that outputs the last hiddenstate H ( S ); a fully connected layer L ; and a regression layer that predicts the capacity fade.Training the network consists of optimizing the network parameters to minimise the loss func-tion, here the mean squared error between the true and predicted target values. The networkparameters are the input weights W I ∈ R N × , W I ∈ R N × N , W I ∈ R × N and biases B ∈ R N × , B ∈ R N × , B ∈ R for L , L , and L , respectively; and the recurrent weights W R ∈ R N × N and W R ∈ R N × N of LSTM layers L , and L , respectively. The model istrained using back propagation through time and stochastic gradient descent method with batch14 … … … … … … 𝐿𝑆𝑇𝑀 𝐿𝑆𝑇𝑀 𝑗2 𝐿𝑆𝑇𝑀 𝑁 𝐿𝑆𝑇𝑀 𝐿𝑆𝑇𝑀 𝑖1 𝐿𝑆𝑇𝑀 𝑁 𝑉 𝑡 ∆𝑉 𝑡 ∆𝑡 𝑡 𝑇𝑚𝑖𝑛 𝑡 𝑇𝑚𝑎𝑥 𝑡 𝐼ҧ 𝑡 𝐶 Given
𝐶ሺ𝑇𝑝ሻ ? 𝐶ሺ𝑇𝑛ሻ 𝑇0 𝑇𝑝 +1 𝑇𝑛 -1 Time
𝑆 ≈ 1500 cycles 𝑡 = 1, … , 𝑆 𝑘 𝑡ℎ RW: 𝑋 𝑘 ∈ ℝ Sequence input layer 𝐿 : hidden LSTM layer 1 𝐿 : LSTM layer 2 𝐻 ሺ 𝑡 ሻ ∈ ℝ 𝑁 × 𝑡 = , … , 𝑆 ℎ 𝑖 ሺ𝑡ሻ ℎ 𝑗 ሺ𝑡ሻ 𝐻 ሺ𝑡 − 1ሻ 𝐻 ሺ 𝑆 ሻ ∈ ℝ 𝑁 × 𝐻 ሺ𝑡 − 1ሻ 𝐿 : F u ll y c onn ec t e d l a y e r R e g r e ss i on ou t pu t l a y e r 𝑦 ∈ ℝ C a p a c i t y f a d e ∆ 𝐶 ሺ 𝑇 𝑝 , 𝑇 𝑛 ሻ Figure 3: Structure of the deep LSTM regression network with one RW example sequence of cycles.size b = 10, gradient threshold τ = 10, and constant learning rate γ = 0 .
03. Cell RW9 is used fortraining the D–LSTM–RN model; RW10, RW11, RW12 are used for testing. The computationaltime in the training phase is in the range of 1 hour for 600 iterations, and it is negligible in thetesting phase.
To evaluate accuracy in prediction, we consider the Root Mean Squared Error:RMSE( ˆ
C, C ) = (cid:118)(cid:117)(cid:117)(cid:116) n n (cid:88) i =1 (cid:0) C i − ˆ C i (cid:1) . (11)In particular, note that we compare RMSEs on the capacity scale, despite the target variable beingthe capacity drop between two consecutive reference cycles. This is done to ease the comparisonbetween the different methods we used, and with other works in the literature using the same data.A normalised version where the deviation is divided by the true capacity is also provided:RMSE norm ( ˆ C, C ) = (cid:118)(cid:117)(cid:117)(cid:116) n n (cid:88) i =1 (cid:18) C i − ˆ C i C i (cid:19) . (12)15he data-gathering for the cells under study extended well beyond their End of Life (EoL): this, infact, is defined to occur when the present capacity of the battery reaches 70% or 80% of its nominalcapacity [52, 53]. SoH prediction after EoL is practically less important and it is generally ignoredin some works in the literature: for this reason, along with the two metrics RMSE and RMSE norm computed on the whole test set, we report their RMSE EoL and RMSE
EoLnorm counterparts computedon data up to the cell EoL, defined to occur when the capacity is 80% of the nominal capacity.Figure 4 depicts capacity fade prediction results using the D-LSTM-RN models described inSection 3.2.2. Both short-term and long-term predictions for the four batteries RW9 (training),RW10, RW11, and RW12 are shown. Predictions are compared to the true capacity values, and thenormalised error metrics are displayed for each cell. It emerges that the short-term predictions arein all cases very close to the true values, with RMSE norm ranging from 1.8% to 7.25% and RMSE
EoLnorm from 1.15% to 1.80%. The predictions for the long-term case suffer, as expected, from cumulativeerror which is reflected in higher RMSE norm values (from 6.8% to 13.45%). However, when consid-ering the capacity fade only up to the cell EoL we get much smaller errors: the RMSE
EoLnorm is lessthan 1% for cell RW10 and about 2% for cell RW11, while it increases to almost 9% for RW12.16 hort-term Long-term R W RMSE norm : normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : normEoL : Time C apa c i t y ( A h ) PredictedTrue R W EoL RMSE normEoL : norm : Time C apa c i t y ( A h ) PredictedTrue
EoL RMSE normEoL : norm : Time C apa c i t y ( A h ) PredictedTrue R W RMSE norm : normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : normEoL : Time C apa c i t y ( A h ) PredictedTrue R W RMSE norm : normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : normEoL : Time C apa c i t y ( A h ) PredictedTrue
Figure 4: Short-term (left column) and long-term (right column) capacity fade prediction resultsusing D-LSTM-RN, together with the considered error metrics. Predictions are shown as red crosses;true capacity values are shown as green stars. The first row (cell RW9) reports the training errorof the models. 17
Multivariable Fractional Polynomials
The Multivariable Fractional Polynomial (MFP) approach of Sauerbrei and Royston [54] consistsin using an algorithm to find the best input transformations in a multivariable linear regression.A multivariable linear regression assumes a linear relationship between the response, or targetvariable, y and a set of inputs (also called features or covariates) x , ..., x p , y = E [ y | x , ..., x p ] + ε = β + β x + ... + β p x p + ε (13)where β , ..., β p are the regression coefficients, ε is an error term, and E [ y | x , ...x p ] = β + β x + ... + β p x p is the expected value of y conditioned on x , ..., x p . Note that linearity is assumed withrespect to the regression coefficients β , ..., β p , not necessarily to x , ..., x p : on the contrary, it isimportant to consider possible nonlinear contributions from the inputs, which if not accounted formay lead to misspecified final models [54]. The MFP method, implemented in R with the mfp package [55], is chosen to this end, as the inputs are transformed by using the most suitable FPfunctions.Given a unidimensional input x , a FP function of first degree is defined as x l , where the power l can be either integer or fraction, positive or negative, from the predefined set A = { -2,-1, -0.5, 0, 0.5, 1, 2, 3 } where x ≡ log x . The best l in the set A is considered to be that yielding the lowest deviance d = − (cid:96) ( ˆ β ML ), where (cid:96) ( ˆ β ML ) is the maximised log-likelihood function. In a multivariate settingwith p > mfp package allows for FPs of degree m >
1: the second-degree polynomial is preferred over the first-degree on the basis of a χ test,and so on. Further details about the MFP method with FPs of a generic degree m can be found in1854, 56, 57]. For the analysis presented in this article, FPs with maximum permitted degree m = 2were considered, but only first-degree models were eventually selected by the algorithm.Considering a training set of n labelled pairs ( y , x l ), where x l is the q -dimensional vector ofinputs x , ..., x p transformed according to the MFP algorithm and l = ( l , . . . , l q ) the set of selectedpowers, the linear model can be written as y = E [ y | ˜ X ] + ε = ˜ X β + ε , (14)where: y is a vector of n observations of the target variable; ˜ X is a n × ( q + 1) matrix assumed tohave full rank q + 1, containing a column of 1’s in the first position to account for the intercept term β , and the transformed inputs in the remaining columns; and ε is a vector of n independent errorterms that are here assumed to be N (0 , σ )-distributed. In general, q (cid:54) = p due to the possibility ofintroducing an m -degree polynomial for each input x , ..., x p ; however, q = p in our case where onlyfirst-degree polynomials were selected. Given the training set, one can learn the linear relationshipthrough the least squares criterion, i.e. by solvingˆ β = argmin β ∈ R q (cid:107) y − ˜ X β (cid:107) . (15)Despite constituting such a simple setup, linear regression proved to work extremely well inmany non-trivial contexts and different applications, e.g. [58, 59, 60]. An attractive characteristicof multiple linear regression is the interpretation of the regression coefficients β j for j = 1 , ..., q ,which may be read as the change in E [ y | x l , ...x l q q ] for an increase of one unit in x l j j , holding allother features constant. Importantly, this enables immediate identification of the most relevantfeatures, and allows to study the effect of one single feature while adjusting for the effects of theothers. In addition, the coefficient of determination R can be interpreted, in the context of linearregression, as the proportion of total variability in the outcomes that is explained by the model:19 = 1 − RSSTSS with RSS = (cid:80) ni =1 ( y i − ˆ y i ) residual sum of squaresˆ y i = ˆ β + (cid:80) qj =1 ˆ β j x lq TSS = (cid:80) ni =1 ( y i − ¯ y ) total sum of squares¯ y = n (cid:80) ni =1 y i (16)with 0 ≤ R ≤
1. For this reason, R provides an interesting diagnostic measure. However, since R increases every time new features are added to the regression equation, one should preferablyconsider its penalised version R adj = 1 − RSS n − p − n − , (17)called adjusted coefficient of determination , that increases only when newly added features increasethe variability explained by the model.It is to avoid inclusion of overabundant features which do not add valuable information, thatthe regression models considered in this work have undergone a variable selection mechanism: astepback procedure based on the AIC criterion. As a consequence, only the subset of the mostsignificant variables to predict the outcome is kept into the model. Variable selection is importantfor both theoretical and practical reasons, such as: achieving a reduction of the model variance,thus improving the prediction accuracy; facilitating the model interpretation and providing a cleanerview of the data-generating process; and reducing the computational and usage time of the model,which makes it more portable. Selecting variables is an important and delicate point in particularfor explanatory models such as linear regression, since including or excluding highly correlatedfeatures may lead to significantly different interpretations of their effects [61].20 .2 Implementation The structure of the dataset is such that capacity measurements are only available at the n referencecycles performed throughout the cell life: as a consequence, inputs from the RW steps occurringbefore each reference cycle need to be summarised, to construct a n × ( p + 1) input matrix. Regard-ing the number and choice of features, quantities reflecting the well-known stress factors describedin Section 2.1 were chosen, along with covariates describing characteristics of this particular exper-iment. The following inputs have been considered for each reference cycle i occurring after m RWdischarge steps and l RW charge steps: · C prev = C i − , true capacity at the previous reference discharge cycle; · ∆ t = (cid:80) m + lk =1 ( t endik − t startik ), total duration of the cycle; · s V in = m (cid:80) mk =1 v ink , average of the initial voltages of each k -th RW discharge step; · Ě ∆ V = m (cid:80) mk =1 ( v endik − v startik ), average of the voltage drops of the RW discharge steps; · ¯ I = m (cid:80) mk =1 ( I ik ), average of the current intensities of the RW discharge steps; · s T min = m (cid:80) mk =1 ( T minik ), average of the minimum temperatures of each RW discharge step; · s T max = m (cid:80) mk =1 ( T maxik ), average of the maximum temperatures of each RW discharge step; · λ = short steps m , proportion of steps in which the voltage reaches the boundaries before thedefault duration (300 s); · ∆ t rest , a nonlinear smooth saturation function of the time elapsed between the last RW stepof the cycle and the beginning of the reference cycle:∆ t rest = 11 + exp (cid:20) − (cid:0) ˜ tσ t< (cid:1)(cid:21) with ˜ t = t start,refi − t end, ( m + l ) i σ t< is the variance of the observations such that ˜ t <
20 h; · C approx = m (cid:80) mk =1 ( ˆ C ik ), average of rough capacity estimates at each RW discharge step,obtained through a preliminary linear regression model described in the Supplementary Ma-terial. The purpose of C approx is to compensate for the potential information loss that comesas a consequence of summarising a large number of RW steps in each cycle. Three regression models were considered:a) a model including all features presented in Section 4.2.1, but C prev and C approx ;b) a model including all features presented in Section 4.2.1, but C prev ;c) a model including all features presented in Section 4.2.1, with no exceptions.Model c assumes that C prev , the true capacity measured at the most recent reference cycle, isknown and can be used to add important information to the regression model. Model b reflects themore realistic scenario in which C prev is undetermined, but it includes C approx as an approximatesurrogate with the purpose of maximising the information extracted from the data sensed duringthe RW discharges; model a relies uniquely on the data directly measured by sensors, withoutadditional true or estimated capacity values. In each case, the target variable for the analysis is thechange in the battery capacity at time t compared to its nominal capacity, where capacity valueshave been adjusted as described in Section 3.2.2: y = ∆ C ( t ) = C ( t ) − C ( t ) . (18)After having undergone the MFP and variable selection procedures, the three final models are: MFPa ∆ C a,i = α + α · s T min,i + α · s T max,i + α · λ i + ε i MFPb ∆ C b,i = β + β · C approx,i + β · ∆ t rest,i + β · s T min,i + β · λ i + β · s V in,i + ε i ; MFPc ∆ C c,i = γ + γ · C prev,i + γ · ∆ t rest,i + γ · C approx,i + γ · s T min,i + ε i .Cell RW9 is used for training each model, while RW10, RW11 and RW12 are used for testing. The trained models are reported in Table 1: the estimated regression coefficients are shown togetherwith their standard errors, the corresponding p-values, and the R and R adj coefficients.23ovariate Est Std err p-value R R adj MFPa intercept 0.43 0.17 0.0142 0.985 0.984 s T min s T max -0.53 0.08 9.13e-08 λ C approx -3.99 0.40 1.83e-11∆ t rest -0.26 0.07 0.60e-03 s T min -0.06 0.01 8.39e-09 λ s V in -0.10 0.04 0.0230MFPc intercept 4.63 0.21 < C prev -1.36 0.13 4.87e-12∆ t rest -0.34 0.04 5.23e-10 C approx -0.87 0.13 7.83e-08 s T min -0.03 0.005 1.05e-05Table 1: Results for the three models MFPa, MFPb and MFPc, reporting the estimated regressioncoefficients (Est), standard errors (Std err) and corresponding p -values, together with the R andthe adjusted R coefficients.MFPa includes only s T min and s T max , together with the proportion λ of steps interrupted due tothe voltage reaching the boundaries before the default duration. The temperatures are the mostsignificant variables. Increasing s T max by one degree while holding all other features constant impliesan increase in the capacity drop of 0.54; this effect is counterbalanced by s T min having an oppositecoefficient. When they have close values, the effect of the two variables is close to 0, meaning thata narrow temperature range, hence a controlled temperature variation, does not affect the batteryhealth seriously. Concerning λ , it also appears strongly significant and it is included as a cubiceffect with a positive regression coefficient.With MFPb we included the average of the approximate capacity for each RW step, C approx .This led to increased values of both R and R adj : the square root of C approx is in fact the mostsignificant feature of the model, correctly associated with a negative coefficient: a higher value ofthe square root of C approx involves a smaller capacity gap. A mild beneficial effect is attributed alsoto s V in and s T min : the initial voltage is not extremely significant; while the minimum temperaturecontinues to have a very low p-value, which makes it the second most important feature in themodel. s T min is the only temperature variable selected for MFPb, and its effect is smaller than and24pposite to that of MFPa: without s T max in the model, s T min is left alone to account for the effect ofextreme temperatures. ∆ t rest is the third covariate for importance and it produces a reduction inthe capacity variation: in fact, as confirmed from studies such as [62, 63], an apparent increase inthe capacity of LIBs can be achieved by allowing the battery to rest for some time. The resting timewas not selected in MFPa: this might be ascribed to the correlation that exists between ∆ t rest and λ , as λ has a reduced effect in MFPb: its p-value changed from 0.0007 to 0.0146 and the feature isnow included as a linear effect with a smaller coefficient estimate. Concurrently, since λ is both acause and a consequence of capacity fade in the experimental settings of the NASA datasets, it islikely that the reduction in its significance compared to MFPa is strongly connected to the presenceof C approx .MFPc is the result of variable selection starting from the full set of inputs described in Section4.2.1, including the most recent true capacity value, C prev . The inclusion of C prev adds a greatdeal of exact information to the model, which unsurprisingly results in a further increase of both R and R adj , almost reaching their maximum value of 1. The results of MFPc seem consistentwith those of MFPb: the most important feature is now the square root of C prev , which also hasthe larger (in magnitude) estimated coefficient. Interestingly, the second most significant covariateis now ∆ t rest , with a stronger effect also in its coefficient: this could again be explained with itsrelation to λ , not present in this model. However, once again, the absence of λ should be alsorelated to its connection with C prev and C approx : this emerges clearly as λ was strongly significantin MFPa, where neither C prev nor C approx were considered; less important in MFPb where C approx was included; not present at all in MFPc where both the capacity measures are part of the model. C approx continues to be extremely significant, but it is now included linearly and it is less importantthan in MFPb. Finally, s T min has an even smaller effect than in MFPb, but persists in being animportant input to predict the change in capacity.For the sake of interpretability and explanation, MFPc seems the best choice: it has the highestcoefficients of determination, it is reasonably sparse and all the included variables are extremelysignificant. However, it assumes that the true capacity is known at every previous cycle, which ishardly the case: then, MFPb constitutes a valid alternative as it takes advantage of approximatecapacity estimates derived directly from the steps sensor data. However, it is noticeable that MFPa25lso reaches high R and R adj while comprising only three features which can be very easily obtainedin practice.When it comes to accuracy in prediction, the normalised RMSEs are presented in Figure 5, wherethe predicted capacity fade according to each model is compared to the true values. The grey arearepresents the 90% prediction interval, which has been computed using basic asymptotic results atalmost no additional computational cost. The first row (cell RW9) reports the training error of eachmodel. The results show that the difference in the performances of the three considered modelsis not huge: all of them have good predictive accuracy with RMSE norm and RMSE EoLnorm rangingrespectively from 2.22% to 11.69% and from 3.21% to 7.18%. The errors reflect the similarities anddissimilarities in the production phase and operational history of the four cells, which also emergesin Figure 1 in Section 2.2. Considering RMSE norm , there is a consistent improvement going frommodel MFPa to MFPb and MFPc for batteries RW10 and RW11, while the minimum RMSE
EoLnorm is obtained in model MFPa; for cell RW12, remarkably, we obtain better results with model MFPaaccording to both the error metrics. 26
FPa MFPb MFPc R W RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue R W RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue R W RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue R W RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue
RMSE norm : EoL
RMSE normEoL : Time C apa c i t y ( A h ) PredictedTrue
Figure 5: Capacity fade prediction results using the three models MFPa (left column), MFPb(middle column), and MFPc (right column), together with the considered error metrics. Predictionsare shown as red crosses; true capacity values are shown as green stars; the grey area is the 90%prediction interval. The first row (cell RW9) reports the training error of the models.
Both D-LSTM-RN and the MFP models have good prediction abilities. A comparison betweenthe two approaches can be done, provided that the distinctions characterising the analyses areconsidered, above all: the difference in the target variable, which is the capacity fade between the27urrent cycle and the most recently occurred reference cycle for the D-LSTM-RN, and that betweenthe current cycle and the nominal capacity for the MFP models; and the possibility for the LSTM-NN to handle a multivariate sequence of sensor data gathered during the RW phases, while thedata need to be summarised to a single observation vector for the MFP. Nevertheless, the closenessof the prediction errors obtained with the two approaches implies that both their performancesare good, and there is not a significant improvement in choosing the deep learning architecture,especially when the true capacity at the most recent reference cycle is not known. In fact, ontop of having a much higher computational efficiency (for this study, the computational time wasabout 10 seconds for training MFP against 1 hour for D-LSTM-RN), the statistical approach offersmany key advantages: it gave the chance to interpret the results and discuss the effect of eachfeature, while allowing to rank the most important ones; it easily provided prediction intervals byusing fundamental results from the classic asymptotic theory; it allowed feature selection, to thebenefit of prediction accuracy, interpretability and portability; it does not require the tuning ofmany tuning parameters, as opposed not only to D-LSTM-RN, but to most of the ML methodscommonly used for capacity prediction of LIBs; and its much smaller number of parameters (4-6for MFP against > × for D-LSTM-RN) grants it a crucial practical advantage for PHM, as itcan be easily implemented on the BMS to monitor and optimise the battery performance.Table 2 lists the results of the most recently published methodologies chosen to model thecapacity curve for the NASA Randomized dataset. A one-to-one comparison would not take intoconsideration several differences across the studies, including the data-split into training and testand, more importantly, fundamental dissimilarities in the aim and methodology (estimation vsprediction, using particular cycles or segments vs the whole historical sequence): concerning thepredictive objective of our study, in fact, the only work close enough is [64] (though different inmethodology), that reports much higher errors than ours; while for [18], it should be remindedthat the results were achieved having included a transformation of the response among the inputs.Compared to our results, Table 2 gives an overview of general performances in contemporary works:an indirect comparison shows the inline performance of our models, which are competitive to mostrecent studies. A detailed report of the most commonly considered error metrics obtained in ouranalysis can be found in Table S1 in the Supplementary Material.28ethod Dataset Train - test Evaluationmetric RangeDCNNDCNN-TL [16]DCNN-ETL 10-year sourcedataset +NASAfirst 20 cells Train: 16 cellsTest: 4 cells RMSE norm MaxE norm ∼ > . < < norm norm MAE norm
MaxE norm
RMSE
EoLnorm
MAE
EoLnorm
MaxE
EoLnorm
This paper has proposed two novel approaches, drawn on both machine learning and classicalstatistics, to predict the SoH of Lithium-ion batteries under randomised load conditions recreated inthe NASA Ames PCoE. The SoH capacity-related definition has been considered, and an adjustmentto the capacity values obtained by integration of the discharge current have been introduced toaccount for the battery transient effects. Both the D-LSTM and MFP regressions are based onhistorical data from LIBs and rely on sensor data solely. It was shown that the degradationbehaviour of the batteries under examination is influenced by their historical data, as supported bythe low prediction errors achieved with both methods, in a multi-factor perspective which allows tostudy the impact of different factor combinations. We have shown that the use of simple statisticalmodels which linearly include suitably-transformed inputs has competitive performance, besides29mplementability, while offering additional advantages such as: the opportunity to easily select themost significant inputs, and the interpretability of their coefficients; the capability of producingprediction intervals around the outcome; the absence of large numbers of hyperparameters; thegreat computational efficiency and consequent implementability on the BMS, which enables theadjustment of operating conditions for potential increase of longevity and safety.A possible shortcoming of the study is that the four considered cells, despite showing differentdegradation patterns, are all characterised by the same chemical composition and cycled underidentical laboratory conditions: this could result in poor generalisation properties, therefore itcould be interesting to extend the study to different cells or cells cycled under different conditions.However, in real applications, it is often easy to find cells treated under similar conditions: hence,training the model on one exhausted cell to predict the SoH evolution of the similar ones remainsa valuable possibility. Different sets of inputs could also have been selected, where additional carecould have been posed on the correlations existing between the various stress factors. Besides,more efficient methods could have been adopted for the capacity adjustment, as opposed to splineswhich can be poor in extrapolating. The next sensible step would be to extend the methods,developed on laboratory data (albeit randomised), to the more complex and challenging situationsthat characterise real applications.
Acknowledgments
This research is funded by the Norwegian Research Council research-based innovation centerBigInsight, project no 237718. 30 upplementary Material
Preliminary linear model
The models MFPb and MFPc described in Section 4.2.2 include the feature C approx = 1 m m (cid:88) k =1 ( ˆ C k ) (S1)where ˆ C k = I k · ˆ t dis,k are rough capacity estimates at each RW step, obtained multiplying thecurrent intensity I k by the discharge time t dis,k estimated from a linear regression model:ˆ t dis,k = ˆ γ + ˆ γ s T k + ˆ γ I k + ˆ γ s V k + ˆ γ ∆ V k + ˆ γ ∆ T k + ˆ γ t triangle,k + two-way interactions + three-way interactions. (S2)In particular, ˆ t dis,k is an estimate of the time that we would have observed if the corresponding k -th RW step had been a complete reference discharge, rather than a small portion of dischargeoccurring under a randomly selected current load. To train the model, we used piecewise linearinterpolation between each pair of capacity measurements, considering it an approximate responseat each RW step, since we lack capacity measurements during the RW phase of the cycles. Theinputs included in model (S2) are, for each RW discharge step or reference discharge profile k : · s T k , the average capacity of the discharge process; · I k , the current intensity. It constant and equal to 1 A during the reference cycles; for the RWsteps, it is a constant and randomly selected value in the set { } . · s V k , the average voltage of the discharge process · ∆ V k , the signed difference between the beginning and final voltage values of the dischargeprocess 31 ∆ T k = T max − T min , the temperature range during the discharge process · t triangle,k = ( t k · A ) /b k , where t k is the duration of the discharge, A is the voltage dropcharacterising the previous reference cycle, and b is the voltage drop of the current discharge.The input t triangle,k is motivated by a simple geometrical consideration: looking at the voltage-vs-time curve of a reference cycle (Figure S1), we interpret a RW step as a portion of such a curve.Approximating the curve with a straight line, we get two similar rectangular triangles having,respectively, the duration of the reference cycle T and the duration of the RW step t as longercathetus. The proportion T : t = A : b (S3)is then valid under the approximation, and it implies t triangle = ( t · A ) /b . Note that, for this inputto be defined and positive, the dataset was filtered in order to exclude all the RW steps having b = ∆ V ≤ t triangle . Black curve: voltage curveof a reference discharge profile; light blue line: linear approximation of the voltage curve; T : totalduration of the reference discharge; t : duration of a discharge RW step; A : voltage drop of thereference discharge; b : voltage drop of the RW discharge.32 etailed prediction error results Model Test cell RMSE RMSE norm
MAE MAE norm
MaxE norm
RMSE
EoLnorm
MAE
EoLnorm
MaxE
EoLnorm
MFPa RW9 0.0417 3.88% 0.0339 2.93% 12.14% 1.71% 1.61% 2.43%RW10 0.0816 6.49% 0.0672 5.20% 16.31% 4.02% 3.19% 7.97%RW11 0.1208 11.69% 0.0913 7.65% 49.25% 3.22% 2.57% 5.75%RW12 0.0661 4.38% 0.0549 3.58% 10.24% 3.84% 3.15% 7.00%MFPb RW9 0.0303 2.62% 0.0241 2.06% 6.41% 1.43% 0.97% 3.47%RW10 0.0614 4.50% 0.0487 3.67% 9.62% 4.39% 3.52% 9.13%RW11 0.1163 11.23% 0.0975 8.23% 41.88% 4.73% 3.66% 9.61%RW12 0.1392 9.59% 0.1269 8.41% 20.45% 7.18% 6.10% 11.95%MFPc RW9 0.0183 1.49% 0.0147 1.19% 3.52% 0.96% 0.79% 1.72%RW10 0.0352 2.22% 0.0222 1.60% 7.38% 3.37% 2.38% 7.38%RW11 0.0907 8.52% 0.0795 6.46% 36.71% 3.96% 3.26% 7.37%RW12 0.1383 9.35% 0.1281 8.49% 15.15% 6.94% 5.84% 12.24%D-LSTM-RN RW9 0.0225 1.96% 0.0172 1.47% 4.56% 0.85% 0.54% 2.03%short-term RW10 0.0264 2.12% 0.0202 1.61% 6.16% 1.24% 0.88% 2.82%RW11 0.0625 7.25% 0.0408 3.74% 36.01% 1.81% 1.27% 4.22%RW12 0.0257 1.80% 0.0181 1.17% 7.17% 1.15% 1.00% 1.74%D-LSTM-RN RW9 0.0603 5.22% 0.0477 4.02% 11.70% 0.92% 0.66% 1.71%long-term RW10 0.0797 6.80% 0.0612 5.09% 16.36% 0.91% 0.77% 1.62%RW11 0.1303 10.98% 0.1087 8.82% 27.04% 2.04% 1.82% 2.93%RW12 0.2052 13.45% 0.1884 12.18% 22.14% 9.00% 7.93% 14.41%
Table S1: Capacity fade prediction errors for each model considered in the machine learning ap-proach and in the classical statistics approach for the batteries RW9 (training), RW10, RW11, andRW12. 33 igure: difference between original and adjusted capacity values
Figure S2: Relative difference between capacity values before and after introducing a correctionaccounting for the uncertain initial voltage of the reference cycle.34 eferences [1] B. Saha and K. Goebel, “Modeling Li-ion battery capacity depletion in a particle filteringframework,”
Proceedings of the Annual Conference of the Prognostics and Health Mngt Society ,pp. 1–10, 2009.[2] C. Sbarufatti, M. Corbetta, M. Giglio, and F. Cadini, “Adaptive prognosis of Lithium-ionbatteries based on the combination of particle filters and radial basis function neural networks,”
Journal of Power Sources , vol. 344, pp. 128–140, 2017.[3] M. Berecibar, I. Gandiaga, I. Villarreal, N. Omar, J. Van Mierlo, and P. Van den Bossche,“Critical review of state of health estimation methods of Li-ion batteries for real applications,”
Renewable and Sustainable Energy Reviews , vol. 56, pp. 572 – 587, 2016.[4] N. Noura, L. Boulon, and S. Jemei, “A review of battery state of health estimation methods:Hybrid electric vehicle challenges,”
World Electric Vehicle Journal , vol. 11, p. 66, 2020.[5] D. Stroe and E. Schaltz, “Lithium-ion battery state-of-health estimation using the incrementalcapacity analysis technique,”
IEEE Transactions on Industry Applications , vol. 56, no. 1,pp. 678–685, 2020.[6] H. Meng and Y.-F. Li, “A review on prognostics and health management (PHM) methods ofLithium-ion batteries,”
Renewable and Sustainable Energy Reviews , vol. 116, p. 109405, 2019.[7] B. Bole, C. Kulkarni, and M. Daigle, “Adaptation of an electrochemistry-based Li-ion bat-tery model to account for deterioration observed under randomized use,” 2014. PHM 2014 -
Proceedings of the Annual Conference of the Prognostics and Health Management Society 2014 .[8] M. Ng, J. Zhao, Q. Yan, G. Conduit, and Z. W. Seh, “Predicting the state of charge andhealth of batteries using data-driven machine learning,”
Nature Machine Intelligence , vol. 2,pp. 161–170, 2020.[9] N. Wassiliadis, J. Adermann, A. Frericks, M. Pak, C. Reiter, B. Lohmann, and M. Lienkamp,“Revisiting the dual extended kalman filter for battery state-of-charge and state-of-health35stimation: A use-case life cycle analysis,”
Journal of Energy Storage , vol. 19, pp. 73 – 87,2018.[10] Y. Feng, C. Xue, Q. Han, F. Han, and J. Du, “Robust estimation for state-of-charge andstate-of-health of Lithium-ion batteries using integral-type terminal sliding-mode observers,”
IEEE Transactions on Industrial Electronics , vol. 67, pp. 4013–4023, 2020.[11] P. Singh and D. Reisner, “Fuzzy logic-based state-of-health determination of lead acid bat-teries,” in , pp. 583–590,2002.[12] Z. He, M. Gao, G. Ma, Y. Liu, and S. Chen, “Online state-of-health estimation of Lithium-ionbatteries using dynamic bayesian networks,”
Journal of Power Sources , vol. 267, pp. 576 – 583,2014.[13] M. Zeng, P. Zhang, Y. Yang, C. Xie, and Y. Shi, “SOC and SOH joint estimation of the powerbatteries based on fuzzy unscented kalman filtering algorithm,”
Energies , vol. 12, p. 3122, 2019.[14] W. Liu and Y. Xu, “Data-driven online health estimation of Li-ion batteries using a novelenergy-based health indicator,”
IEEE Transactions on Energy Conversion , vol. 35, pp. 1715–1718, 2020.[15] Y. Fan, F. Xiao, C. Li, G. Yang, and X. Tang, “A novel deep learning framework for state ofhealth estimation of Lithium-ion battery,”
Journal of Energy Storage , vol. 32, p. 101741, 2020.[16] S. Shen, M. Sadoughi, M. Li, Z. Wang, and C. Hu, “Deep convolutional neural networkswith ensemble learning and transfer learning for capacity estimation of Lithium-ion batteries,”
Applied Energy , vol. 260, p. 114296, 2020.[17] P. Li, Z. Zhang, Q. Xiong, B. Ding, J. Hou, D. Luo, Y. Rong, and S. Li, “State-of-healthestimation and remaining useful life prediction for the Lithium-ion battery based on a variantlong short term memory neural network,”
Journal of Power Sources , vol. 459, p. 228069, 2020.[18] P. Venugopal and T. Vigneswaran, “State-of-Health estimation of Li-ion batteries in electricvehicle using IndRNN under variable load condition,”
Energies , vol. 12, pp. 1–29, 2019.3619] M. Berecibar, F. Devriendt, M. Dubarry, I. Villarreal, N. Omar, W. Verbeke, and J. Mierlo,“Online state of health estimation on nmc cells based on predictive analytics,”
Journal ofPower Sources , vol. 320, pp. 239–250, 2016.[20] X. Tang, Z. Changfu, K. Yao, C. Guohua, L. Boyang, H. Zhenwei, and G. Furong, “A fast esti-mation algorithm for Lithium-ion battery state of health,”
Journal of Power Sources , vol. 396,2018.[21] J. Kong, F. Yang, Y. Zhao, and K.-L. Tsui, “Battery prognostics at different operating condi-tions,”
Measurement , vol. 151, 2020.[22] P. Khumprom and N. Yodo, “A data-driven predictive prognostic model for Lithium-ion bat-teries based on a deep learning algorithm,”
Energies , vol. 12, p. 660, 2019.[23] K. Tseng, J.-W. Liang, W. Chang, and S.-C. Huang, “Regression models using fully dischargedvoltage and internal resistance for state of health estimation of Lithium-ion batteries,”
Ener-gies , vol. 8, pp. 2889–2907, 04 2015.[24] X. Feng, C. Weng, X. He, X. Han, L. Lu, D. Ren, and M. Ouyang, “Online state-of-healthestimation for Li-ion battery using partial charging segment based on support vector machine,”
IEEE Transactions on Vehicular Technology , vol. 68, pp. 8583–8592, 2019.[25] R. R. Richardson, C. R. Birkl, M. A. Osborne, and D. A. Howey, “Gaussian process regressionfor in situ capacity estimation of Lithium-ion batteries,”
IEEE Transactions on IndustrialInformatics , vol. 15, p. 127–138, 2019.[26] G. L. Plett, “Recursive approximate weighted total least squares estimation of battery celltotal capacity,”
Journal of Power Sources , vol. 196, pp. 2319 – 2331, 2011.[27] X. Han, L. Lu, Y. Zheng, X. Feng, Z. Li, L. Jianqiu, and M. Ouyang, “A review on the keyissues of the Lithium ion battery degradation among the whole life cycle,” eTransportation ,vol. 1, p. 100005, 2019.[28] V. Muenzel, J. de Hoog, M. Brazil, A. Vishwanath, and S. Kalyanaraman, “A multi-factorbattery cycle life prediction methodology for optimal battery management,” in
Proceedings of he 2015 ACM Sixth International Conference on Future Energy Systems , e-Energy ’15, (NewYork, NY, USA), p. 57–66, Association for Computing Machinery, 2015.[29] H. Zou, H. Zhan, and Z. Zheng, “A multi - factor weight analysis method of Lithium-ionbatteries based on module topology,” in , pp. 61–66, 2018.[30] Y. Cui, C. Du, G. Yin, Y. Gao, L. Zhang, T. Guan, L. Yang, and F. Wang, “Multi-stress factormodel for cycle lifetime prediction of Lithium ion batteries with shallow-depth discharge,” Journal of Power Sources , vol. 279, pp. 123 – 132, 2015. 9th International Conference onLead-Acid Batteries – LABAT 2014.[31] K. Smith, M. Earleywine, E. Wood, J. Neubauer, and A. Pesaran, “Comparison of plug-inhybrid electric vehicle battery life across geographies and drive cycles,” in
SAE 2012 WorldCongress & Exhibition , SAE International, 2012.[32] K. Uddin, S. Perera, W. Widanage, L. Somerville, and J. Marco, “Characterising Lithium-ionbattery degradation through the identification and tracking of electrochemical battery modelparameters,”
Batteries , vol. 2, p. 13, 2016.[33] Y. Ji, Y. Zhang, and C.-Y. Wang, “Li-ion cell operation at low temperatures,”
Journal of TheElectrochemical Society , vol. 160, no. 4, pp. A636–A649, 2013.[34] A. Pesaran, S. Santhanagopalan, and G. H. Kim, “Addressing the impact of temperatureextremes on large format Li-ion batteries for vehicle applications,” National Renewable EnergyLaboratory (U.S.), May 2013.[35] S. Ma, M. Jiang, P. Tao, C. Song, J. Wu, J. Wang, T. Deng, and W. Shang, “Temperatureeffect and thermal impact in Lithium-ion batteries: A review,”
Progress in Natural Science:Materials International , vol. 28, pp. 653 – 666, 2018.[36] K. Chen, Z. Yu, S. Deng, Q. Wu, J. Zou, and X. Zeng, “Evaluation of the low temperatureperformance of Lithium manganese oxide/Lithium titanate Lithium-ion batteries for start/stopapplications,”
Journal of Power Sources , vol. 278, pp. 411 – 419, 2015.3837] C. Snyder,
The effects of charge/discharge rate on capacity fade of Lithium ion batteries . PhDthesis, Rensselaer Polytechnic Institute, 2016.[38] S. S. Madani, E. Schaltz, and S. Kær, “Effect of current rate and prior cycling on the coulombicefficiency of a Lithium-ion battery,”
Batteries , vol. 5, p. 57, 2019.[39] C. Armenta-Deu, J. P. Carriquiry, and S. Guzm´an, “Capacity correction factor for Li-ionbatteries: Influence of the discharge rate,”
Journal of Energy Storage , vol. 25, p. 100839, 2019.[40] M. Uno and K. Tanaka, “Influence of high-frequency charge–discharge cycling induced bycell voltage equalizers on the life performance of Lithium-ion cells,”
IEEE Transactions onVehicular Technology , vol. 60, pp. 1505–1515, 2011.[41] M. Reichert, D. Andre, A. R¨osmann, P. Janssen, H.-G. Bremes, D. Sauer, S. Passerini, andM. Winter, “Influence of relaxation time on the lifetime of commercial Lithium-ion cells,”
Journal of Power Sources , vol. 239, pp. 45 – 53, 2013.[42] D. Juarez-Robles, A. A. Vyas, C. Fear, J. A. Jeevarajan, and P. P. Mukherjee, “Overchargeand aging analytics of Li-ion cells,”
Journal of The Electrochemical Society , vol. 167, p. 090547,2020.[43] B. Xu, A. Oudalov, A. Ulbig, G. Andersson, and D. S. Kirschen, “Modeling of Lithium-ion battery degradation for cell life assessment,”
IEEE Transactions on Smart Grid , vol. 9,pp. 1131–1140, 2018.[44] B. Bole, C. Kulkarni, and M. Daigle, “Randomized battery usage data set.” http://ti.arc.nasa.gov/project/prognostic-data-repository . NASA Ames Prognostics Data Reposi-tory, NASA Ames Research Center, Moffett Field, CA.[45] J. Vetter, P. Nov´ak, M. Wagner, C. Veit, K.-C. M¨oller, J. Besenhard, M. Winter, M. Wohlfahrt-Mehrens, C. Vogler, and A. Hammouche, “Ageing mechanisms in Lithium-ion batteries,”
Jour-nal of Power Sources , vol. 147, pp. 269 – 281, 2005.3946] A. Kirchev, “Battery management and battery diagnostics,” in
Electrochemical Energy Storagefor Renewable Sources and Grid Balancing (P. T. Moseley and J. Garche, eds.), ch. 20, pp. 411–435, Elsevier, 2015.[47] M. Salehabadi, M. Zandigohar, N. Lotfi, and P. Fajri, “Investigating the sources of uncer-tainty in capacity estimation of Li-ion batteries,” in , pp. 1–6, 2019.[48] S. Hochreiter and J. Schmidhuber, “Long short-term memory,”
Neural Computation , vol. 9,pp. 1735–80, 1997.[49] F. Informatik, Y. Bengio, P. Frasconi, and J. Schmidhuber, “Gradient flow in recurrent nets:the difficulty of learning long-term dependencies,”
A Field Guide to Dynamical RecurrentNeural Networks , 2003.[50] MathWorks, “Long short-term memory networks.” mathworks.com/help/deeplearning/ug/long-short-term-memory-networks.html . Accessed: 17/01/2021.[51] C. Olah, “Understanding lstm networks.” https://colah.github.io/posts/2015-08-Understanding-LSTMs/ , 2015. Accessed: 17/01/2021.[52] B. Arachchige, S. Perinpanayagam, and R. Jaras, “Enhanced prognostic model for Lithium ionbatteries based on particle filter state transition model modification,”
Applied Sciences , vol. 7,p. 1172, 2017.[53] S. Ng, Y. Xing, and K. L. Tsui, “A naive Bayes model for robust remaining useful life predictionof Lithium-ion battery,”
Applied Energy , vol. 118, pp. 114 – 123, 2014.[54] W. Sauerbrei and P. Royston, “Building multivariable prognostic and diagnostic models: trans-formation of the predictors by using fractional polynomials,”
Journal of the Royal StatisticalSociety: Series A (Statistics in Society) , vol. 162, pp. 71–94, 1999.[55] G. Ambler and A. Benner, mfp: Multivariable Fractional Polynomials , 2015. R package version1.5.2. 4056] W. Sauerbrei, C. Meier-Hirmer, A. Benner, and P. Royston, “Multivariable regression modelbuilding by using fractional polynomials: Description of SAS, STATA and R programs,”
Com-putational Statistics & Data Analysis , vol. 50, pp. 3464 – 3485, 2006.[57] P. Royston and D. G. Altman, “Regression using fractional polynomials of continuous covari-ates: Parsimonious parametric modelling,”
Journal of the Royal Statistical Society. Series C(Applied Statistics) , vol. 43, pp. 429–467, 1994.[58] T. Isobe, E. D. Feigelson, M. G.Akritas, and G. J. Babu, “Linear regression in astronomy. I.,”
Astrophysical Journal , vol. 364, p. 104, 1990.[59] R. Theobald and S. Freeman, “Is it the intervention or the students? Using linear regressionto control for student characteristics in undergraduate STEM education research,”
CBE—LifeSciences Education , vol. 13, pp. 41–48, 2014. PMID: 24591502.[60] B. Gravesteijn, D. Nieboer, A. Ercole, H. Lingsma, D. Nelson, B. Van Calster, E. Steyer-berg, C. ˚Akerlund, K. Amrein, N. Andelic, L. Andreassen, A. Anke, A. Antoni, G. Audibert,P. Azouvi, M. Azzolini, R. Bartels, P. Barzo, and R. Beauvais, “Machine learning algorithmsperformed no better than regression models for prognostication in traumatic brain injury,”
Journal of Clinical Epidemiology , vol. 122, pp. 95–107, 2020.[61] G. Heinze, C. Wallisch, and D. Dunkler, “Variable selection – a review and recommendationsfor the practicing statistician,”
Biometrical Journal , vol. 60, pp. 431–449, 2018.[62] A. Eddahech, O. Briat, and J.-M. Vinassa, “Lithium-ion battery performance improvementbased on capacity recovery exploitation,”
Electrochimica Acta , vol. 114, pp. 750–757, 2013.[63] B. Epding, B. Rumberg, H. Jahnke, I. Stradtmann, and A. Kwade, “Investigation of significantcapacity recovery effects due to long rest periods during high current cyclic aging tests inautomotive Lithium ion cells and their influence on lifetime,”
Journal of Energy Storage , vol. 22,pp. 249 – 256, 2019.[64] R. R. Richardson, M. A. Osborne, and D. A. Howey, “Battery health prediction under general-ized conditions using a Gaussian process transition model,”
Journal of Energy Storage , vol. 23,pp. 320 – 328, 2019. 4165] P. Tagade, K. S. Hariharan, S. Ramachandran, A. Khandelwal, A. Naha, S. M. Kolake, andS. H. Han, “Deep Gaussian process regression for Lithium-ion battery health prognosis anddegradation mode diagnosis,”
Journal of Power Sources , vol. 445, p. 227281, 2020.[66] R. R. Richardson, M. A. Osborne, and D. A. Howey, “Gaussian process regression for forecast-ing battery state of health,”