GAETS: A Graph Autoencoder Time Series Approach Towards Battery Parameter Estimation

1 INTRODUCTION AND LITERATURE REVIEW . Energy dense Lithium-ion batteries ( LiB ) are driving the current electrification revolution in the transportation sector Balasingam & Pattipati ( 2021 ) . Rapid advancements in battery material science have enabled inexpensive and long-running LiBs . However , capacity degradation leading to poor performance are some of the drawbacks of these batteries Zhou et al . ( 2021 ) . Development of accurate State-of-Charge estimation algorithms have garnered attention over the past decade as these metrics translate to available range in an Electric Vehicle Herle et al . ( 2020 ) . Battery degradation or capacity fade occurs due to repetitive charge/discharge cycles . Decreased battery capacity over its usage is an indicator of poor health of the battery Mohtat et al . ( 2021 ) . Present battery models available in literature suffer from reliable battery capacity fade predictions Hinz ( 2019 ) . This uncertainty in prediction is chiefly due to the non-linear behaviour of battery discharge characteristics with increased battery age . Battery Ageing has also been a topic of considerable interest in the energy storage domain Ning & Popov ( 2004 ) ; Ning et al . ( 2006 ) ; Peterson et al . ( 2010 ) ; Barré et al . ( 2013 ) . Empirical battery models have been used for Remaining Useful Life prediction of batteries Saha & Goebel ( 2009 ) ; Daigle & Kulkarni ( 2016 ) ; Nuhic et al . ( 2018 ) ; Andre et al . ( 2013 ) . Simple estimators from Kalman Filter Andre et al . ( 2013 ) ; Bole et al . ( 2014 ) to Reinforcement Learning techniques Unagar et al . ( 2020 ) have been employed for battery parameter estimation . All the current battery parameter estimation techniques available in literature can be broadly classified as follows : 1 . Physics-based modeling which employs Partial Differential Equations ( PDEs ) to compute equivalent battery parameters Wang et al . ( 2011 ) 2 . Phenomenological battery modeling , achieved by focused computation of battery capacity changes corresponding to specific charge/discharge cycles Goebel et al . ( 2008 ) 3 . Data-driven estimation models , which provide meaningful insights from historical battery data and do not involve computation of complex electrochemical models Severson et al . ( 2019 ) In the present work we use a Graph-based data-driven technique to gain meaningful insights from battery datasets to improve estimate accuracy . The paper is organised as follows , a brief review of graph based forecasting techniques is covered in Section 2 , the architecture of the proposed Graphbased estimation technique is presented in Section 3 , details about experimentation are covered in Section 4 followed by conclusion in Section 5 . 2 GRAPH-BASED TIME SERIES ESTIMATION : RELATED WORK . Battery datasets typically consist of time series Voltage , Current , Temperature and State-of-Charge values as shown in Fig . 1 . Statistical Time series based estimation techniques have been popular Bhatnagar et al . ( 2021 ) for typical time series datasets . In Multivariate time series scenarios variable dependencies have been investigated to improve estimations Yu et al . ( 2018 ) . Traffic datasets are one of the prime examples wherein variation in speed and traffic density insights are useful to accurately predict vehicular traffic in other nodes of the network . There are numerous Graph Neural Network ( GNN ) based techniques which exploit the graph structure of datasets to improve prediction accuracy Seo et al . ( 2016 ) , Li et al . ( 2018 ) and Zhao et al . ( 2020 ) . All Graph-based forecasting and estimation approaches available in literature have been validated using traffic datasets . A more recent variant of Graph Neural Network dubbed GTS ( Graph for Time Series ) has been successfully employed to learn specific graph structures to improve prediction accuracy Shang et al . ( 2021b ) . In this paper we build upon the GTS architecture specifically for battery datasets . Following section elucidates the proposed architecture in greater detail . 3 GAETS ARCHITECTURE 3.1 THE METHOD . The architecture of GAETS is as depicted in 2 . The objective of the method is to forecast X̂t+T+1 : t+T+τ , given the observations Xt+1 : t+T , where X denotes training data and consists of multiple charge/discharge cycles of battery measurements . The model is denoted as X̂t+T+1 : t+T+τ = f ( A , w , Xt+1 : t+T ) , where A is the graph structure and is parameterized by w , T is the input window duration , our goal is to predict future data from time t + T + 1 to t + T + τ , here τ is the output prediction window length . The graph structure represented by adjacency matrix A ∈ { 0 , 1 } n×n is to be parameterized as it is exclusively not available to the forecasting model . GTS Shang et al . ( 2021b ) parameterizes the graph structure by considering the adjacency matrix A to be random variable on matrix Bernoulli distribution and is learned as part of the task . We use Gumbel reparameterization trick as it is used in Shang et al . ( 2021b ) because the gradients fail to flow through A in Bernoulli sampling . 3.1.1 ENCODER-DECODER . The encoder-decoder architecture is used in order to reconstruct the original time series data and to learn the intermediate represetations . The model assumes an inherent nonlinear Structural Equation Model ( SEM ) that connects the node features to the adjacency matrix . The SEM was first proposed in NOTEARS Zheng et al . ( 2018 ) . X ( j ) = f ( X ( j ) , A ) = g2 ( A T g1 ( X ( j ) ) ) ( 1 ) where g1 and g2 are neural networks , encoder and decoder respectively while j represents the node id . This can be seen as a graph autoencoder . The encoder extracts the features from the original time series data , by applying a fully connected layer on the convolved original features . hi = FC ( vec ( Conv ( Xi ) ) ) where hi is the features extracted in the lower dimensional space . Xi is the ith series of features . The matrix H of representations his generated by the encoders are used in the graph autoencoder as well as in the link predictor . In this scenario the vector representations are fed as input to the link predictor which spits out a scalar . The output scalar decides the existance of a link . These are represented as hi and hj , the vectors are concatenated and passed through two fully connected layers to finally get the scalar . The extracted features along with the sampled graph adjacency matrix is passed through the decoder . We make use of an MLP as the decoder . The loss incurred by the autoencoder also adds up to the final loss to be optimized . 3.1.2 RECURRENT NETWORKS . In order to estimate the time series predictions based on graphs , we make use of diffusion convolutional GRU as in DCRNN Li et al . ( 2018 ) and GTS Shang et al . ( 2021b ) . Rt′ = σ ( WR ? A [ Xt′ ‖Ht′−1 ] + bR ) , Ct′ = tanh ( WC ? A [ Xt′ ‖ ( Rt′ Ht′−1 ) ] + bC ) , Ut′ = σ ( WU ? A [ Xt′ ‖Ht′−1 ] + bU ) , H = Ut′ Ht′−1 + ( 1− Ut′ ) Ct′ , where WQ ? AY = K∑ k=0 ( θQk,1 ( D−1O A ) k + θQk,2 ( D−1I A ) k ) Y , where all W s are corresponding weight matrices , DI and DO are in-degree and out-degree matrix respectively . θQk,1 and θk,2 and bQ are the model parameters and K is the hyperparameter corresponding to the degree of diffusion . 3.2 TRAINING . The time series forecasting loss is the base loss which is the loss between the values forecasted by the models and the ground truth . Hence the first objective is given in 2. min ltbase ( X̂t+T+1 : t+T+τ ) = 1 τ t+T+τ∑ t′=t+T+1 ∥∥∥X̂ ′t −Xt′∥∥∥ ( 2 ) The autoencoder compares the reconstructed data against the input data . The autoencoder loss , the second objective is given in 3. min lautoencoder = 1 2n n∑ j=1 ∥∥∥X ( j ) − g2 ( AT g1 ( X ( j ) ) ) ∥∥∥ ( 3 ) where , n is the number of samples , and min lautoencoder acts as a regularizer to the original GTS ( Shang et al . ( 2021a ) ) loss . The original GTS loss is given in 2 . The final loss comprises of the loss incurred by the autoencoder and the base loss . It is given in 4. loss = ltbase ( X̂t+T+1 : t+T+τ ) + lautoencoder ( 4 ) Finally , we arrive at the expression 5 that we need to minimize . loss = 1 τ t+T+τ∑ t′=t+T+1 ∥∥∥X̂ ′t −Xt′∥∥∥+ 12n n∑ j=1 ∥∥∥X ( j ) − g2 ( AT g1 ( X ( j ) ) ) ∥∥∥ ( 5 ) 4 EXPERIMENTS . To validate the GAETS architecture we used Lithium-ion Battery Data ( 2018 ) . All the batteries listed in the dataset had multiple charge/discharge cycles . We trained the model on two sets of measurement values logged from two separate channels . The data consists of 6 measurements namely : Voltage , Current , Charge Capacity , Discharge Capacity , Charge Energy and Discharge Energy . Each measurement is treated as a node and the task is to learn the structure that represents relationship between variables as well as forecasting the battery parameter of interest . We fix the input horizon to be 80 and test the data on three different forecasting window horizons of 40 , 80 and 120 . In terms of tensor size , the training data is made of ( Xtrain , Ytrain ) where Xtrain is 1497 signals with a dimension of 6 ( for each measurement ) and horizon as 80 , while Ytrain is 1497 signals with a dimension of 6 and 40/80/120 horizon values . Validation is done on 213 signals . Meanwhile , testing data is made of ( Xtest , Ytest ) where Xtest is 1497 signals with a dimension of 6 ( for each measurement ) and 80 window ticks , while Ytest is 1497 signals with a dimension of 6 and 40/80/120 horizon values . These datasets are obtained by continuous moving window average across the original long time series data . We follow the code implementation in GTS Shang et al . ( 2021a ) , using the same setting for the graph neural network architecture and optimiser . We train each model for 200 epochs , obtain the best model for validation dataset and test it . Each model is trained from 5 different random initialisations . The performance result and confidence interval is shown in the chart . Performance is measured through MAE , RMSE and MAPE values as shown in Fig . 6 following GTS Shang et al . ( 2021a ) paper . Visual comparison of Battery Voltage , Current and Charge Energy is presented in Fig . 3,4 and 5 . 5 CONCLUSION . Battery parameter estimation remains to be a challenging and topic of relevance due to the burgeoning number of Electric Vehicles in the past decade . Accurate battery capacity estimation also helps to predict remaining usage time in battery powered consumer electronics . In this paper we introduce a Graph AutoEncoder Time Series ( GAETS ) architecture which exploits the dependencies between battery parameters ( charge capacity , voltage , current etc ) to provide precise estimates . We compare our architecture with the existing Graph Time Series architecture to highlight the effectiveness of our approach .

Summary: The paper proposes a graph neural network based forecasting algorithm for battery state of charge. Building on prior work, the paper proposes a regularization based on reconstruction loss. Evaluation on a battery dataset shows improvements.