1 Introduction and Literature Review
Energy dense Lithiumion batteries (LiB) are driving the current electrification revolution in the transportation sector Balasingam and Pattipati (2021). Rapid advancements in battery material science have enabled inexpensive and longrunning LiBs. However, capacity degradation leading to poor performance are some of the drawbacks of these batteries Zhou et al. (2021). Development of accurate StateofCharge 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 nonlinear 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 and 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 and Goebel (2009); Daigle and 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:

Physicsbased modeling which employs Partial Differential Equations (PDEs) to compute equivalent battery parameters
Wang et al. (2011) 
Phenomenological battery modeling, achieved by focused computation of battery capacity changes corresponding to specific charge/discharge cycles Goebel et al. (2008)

Datadriven 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 Graphbased datadriven 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 Graphbased Time Series Estimation: Related Work
Battery datasets typically consist of time series Voltage, Current, Temperature and StateofCharge 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 Graphbased 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 , given the observations , where denotes training data and consists of multiple charge/discharge cycles of battery measurements. The model is denoted as , where is the graph structure and is parameterized by , is the input window duration, our goal is to predict future data from time to , here is the output prediction window length. The graph structure represented by adjacency matrix 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
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 in Bernoulli sampling.3.1.1 EncoderDecoder
The encoderdecoder 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).
(1) 
where and are neural networks, encoder and decoder respectively while 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.
where
is the features extracted in the lower dimensional space.
is the th series of features. The matrix of representationss 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
and , 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).
where
where all s are corresponding weight matrices, and are indegree and outdegree matrix respectively. and and are the model parameters and
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.
(2) 
The autoencoder compares the reconstructed data against the input data. The autoencoder loss, the second objective is given in 3.
(3) 
where, is the number of samples, and 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.
(4) 
Finally, we arrive at the expression 5 that we need to minimize.
(5) 
4 Experiments
To validate the GAETS architecture we used Lithiumion 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 (
, ) where is 1497 signals with a dimension of 6 (for each measurement) and horizon as 80, while 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 (, ) where is 1497 signals with a dimension of 6 (for each measurement) and 80 window ticks, while 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.
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