I Introduction
Lithiumion batteries (LiBs) have found fastgrowing use in many industrial sectors, such as electric vehicles, smart grid, renewable energy and consumer electronics. This trend has driven a surge of research on dynamic modeling for LiBs, which plays a foundational role in advanced management of LiB systems [2]. Various practical applications strongly demand LiB models that can offer both high predictive accuracy and computational efficiency with a broad range of Crates. This need, however, has not been fully met yet.
The equivalent circuit models (ECMs) are widely used in the battery engineering sector. They leverage electrical circuits to simulate a LiB’s current/voltage dynamics. Their structural and mathematical simplicity make them well conducive to control and estimation design. However, with the same reason, ECMs offer relatively limited accuracy, even though recent studies have led to some more accurate designs
[26]. Electrochemical modeling is another important means to describe the dynamics of LiBs, which uses electrochemcial principles to comprehensively characterize the electrochemical reactions, Liion diffusion and concentration changes in the electrode/electrolyte as well as various associated processes during charging/discharging of LiBs. Compared to ECMs, they can reproduce the current/voltage responses with much better accuracy. A wellknown electrochemical model is the socalled DoyleFullerNewman (DFN) model, which is broadly considered reliable and precise enough for almost all LiB management scenarios [4]. The accuracy yet comes with enormous computational complexity. This hence has motivated an incessant search for streamlined electrochemical models. The single particle model (SPM) is the most parsimonious one, which represents each electrode as a spherical particle and delineates Liion intercalation and diffusion in the particles [22]. The SPM has spurred a wide range of improved models for higher accuracy under different conditions. They usually supplement it with characterization of thermal behavior [8, 25], electrolyte dynamics [12, 16, 9, 13], and stress buildup [13]. Another important line of research lies in applying numerical model order reduction methods to the DFN or other electrochemical models, with the aim of enabling computationally fast implementation [27, 24, 23, 14, 11].Both ECMs and electrochemical models can be viewed as physicsbased models as they build upon physical principles. Meanwhile, the growing data abundance for today’s LiB systems due to the ubiquitous onboard sensing makes it appealing to identify models simply from measurement data. Machine learning (ML) holds much promise here, with its success in various datadriven modeling tasks. A close inspection shows that these two ways of modeling are constructively complementary. Physicsbased models can offer physical interpretations of LiBs’ dynamic behaviors and extrapolate to any charging/discharging scenarios meeting the model assumptions. However, they either require much computation, as in the case of the DFN, or have inadequate accuracy when the model assumptions are not satisfied—for instance, ECMs and the SPM are designed for low/medium Crates and can poorly predict LiBs’ dynamics at high Crates. MLbased modeling extracts blackbox approximations from data in a convenient and efficient manner and requires only fixed computational costs once a model is trained. Yet, in practical applications, its performance can be constrained by the data informativeness and may face the pitfall of producing physically inconsistent results. Therefore, hybrid physicsML modeling has emerged recently, with the aim of combining the respective merits and overcoming the challenges of each way. The study in [19] couples a onedimensional electrochemical model with different kinds of neural networks (NNs). In [18], recurrent NNs are used to learn the residuals between a LiB’s terminal voltage and the SPM’s output voltage. In [5], a simplified SPM and a lumped thermal model are combined with a neural network in series to predict the terminal voltage. While these studies offer promising results, the hybrid modeling of LiBs is still underexplored, without living up to its potential. Note that a related line of research is to apply ML, e.g., NNs and Gaussian processes, to estimate the stateofcharge, stateofhealth and temperature of LiBs [3, 21, 20], which yet is beyond the scope of this paper.
In the hybrid models surveyed above, the input of an NN includes the current and output voltage of the physical model, and the output of the NN is the residual or actual terminal voltage. However, the mapping from the NN’s input to its output would not be oneonone physically, due to the dynamics inside LiBs. The NN, and consequently the hybrid model, will hence be limited in the predictive capability, even if achieving satisfactory training accuracy. To overcome this key limitation, this work contributes a new perspective into hybrid modeling: the NN should be additionally informed of the internal state of the physical model. This will allow the NN to be aware of the ongoing dynamics of the physical model and thus learn more effectively what is missed from the physical model in comparison to the measurement data. This perspective leads us to develop the following specific contributions.

We develop two new hybrid physicsML models for LiBs which integrate the SPMT with the FNNs. The first model, named HYBRIDI, leverages an NN to capture the residuals of SPMT, and the second, named HYBRIDII, uses an NN to predict the voltage based on the SPMT. Different from the literature, both of them feed the stateofcharge (SoC) information of the SPMT to the NN as an additional critical input.

We evaluate the two hybrid models by extensive simulations. They both demonstrate high predictive accuracy when applied to testing datasets and are found applicable to a broad range of Crates as high as 10 C. The use of FNNs also make the architectures of the two models relatively parsimonious.
With the exceptional accuracy, the proposed hybrid models can find prospective use in various energy storage applications, especially those that require highpower charging/discharging.
This paper is organized as follows. Section II discusses the construction of the proposed hybrid models. Section III presents an overview of the SPMT and FNN models. Section IV contains evaluation settings and simulation results. Finally, section V concludes our work with remarks and future research.
Ii Hybrid PhysicsML Modeling for LiBs
This section constructs hybrid models that blend the SPMT with an FNN to describe the current/voltage dynamics of LiBs. The aim is leveraging the SPMT to create a physicsinformed FNN, thus endowing the hybrid models with high predictive accuracy and ability across a broad Crate range. Next, we begin with a brief introduction to the SPMT and FNNs and then present the hybrid model design. A further description of the SPMT and FNN is offered in Section III.
The SPMT is an improved version of the SPM, which contains an extra bulk thermal model to describe a LiB’s thermal behavior [1]. Its dynamics hence mainly includes the lithiumion diffusion in the electrodes, electrochemical reaction kinetics on the surface of the electrodes, and lumped thermodynamics. When a charging/discharging current is applied to the SPMT, the lithiumion concentrations in the electrodes, , will change due to the reaction and transport, and so will the lumped temperature ; consequently, the SPMT’s output voltage, , will evolve through time. The SPMT is more accurate than the SPM but still valid for only low to medium Crates (up to 1C) [7].
FNNs are an important class of ML methods designed to approximate complex functions. An FNN is composed of multiple layers of different interconnected functions, analogous to a brain comprising many interconnected neurons. Its network structure contains no cycle or feedback connections, making FNNs the simplest type of NNs and easier to train and implement. The theoretical performance of FNNs is guaranteed by the universal approximation theorem, which generally states that a continuous vectorvalued function in the real space can be approximated with arbitrary accuracy by an FNN
[10].We develop our first hybrid model, HYBRIDI, based on residual modeling. Its structure is depicted in Fig. 1. As aforementioned, the SPMT as a reduced electrochemical model is limited to low to medium Crates and would give poor prediction at high Crates. An FNN is thus used to learn the residual errors of the SPMT relative to the measurements. The FNN’s output then is , where is the actual voltage. The input selection is critical, as it decides the information that the FNN can exploit. A primary criterion is to enable the FNN to utilize the SPMTbased physics both sufficiently and economically.
Here, we designate the input to include the current , temperature , initial SoC , and bulk and surface SoC in the negative electrode, denoted as and , respectively. Note that and offer an aggregated yet efficient representation of the SPMT’s state, so feeding them into the FNN will provide an awareness of the SPMT’s overall internal condition, in the benefit of residual learning and prediction.
The second hybrid model, HYBRIDII, is shown in Fig. 2. It connects the SPMT and an FNN in cascade to learn the terminal voltage directly. The FNN uses the same set of variables from the SPMT as in HYBRIDI, but its output tries to emulate . By design, this structure also uses the SPMTbased physics to ensure the FNN’s effectiveness of learning.
Remark 1
The pivotal difference of the above hybrid modeling design from the literature is that information about the physical model’s state is fed as part of the input to the ML model. This integration provides a closer physicsML integration than the means described in the literature. It particularly allows the ML to associate the physical model’s inherent internal dynamics with the measurement data, thus greatly enhancing the predictive accuracy. This design is the first of its kind to our knowledge.
Remark 2
The choice is nonunique for the variables used to represent the SPMT’s state and feed to the FNN. For instance, an expedient way is to just use the full state of the SPMT. This, however, will cause extremely high training and computational costs. We find out that just some simple, aggregated state variables will suffice. After much trialanderror search, we identify that and offer a favorable choice in terms of both computational efficiency and prediction performance.
Iii Overview of SPMT and FNNs
Following Section II, this section provides an overview of the SPMT and FNNs.
Iiia The SPMT model
Developed in [7], the SPMT model couples the SPM model with a bulk thermal model to predict the electrochemical and thermal behaviors simultaneously.
The SPM simplifies each electrode of a LiB cell as a spherical particle and neglects the electrolyte dynamics. The transport of the lithium ions inside a particle is governed by the Fick’s diffusion law in the spherical coordinates:
where is the solidphase lithiumion concentration of positive () or negative () electrode, and is solidphase diffusion coefficient. The diffusion is subject to the following boundary conditions:
where is the particle radius and is the molar flux at the particle surface. Here, is given by
where is the specific interfacial area, is the Faraday’s constant, is an electrode’s surface area, and is an electrode’s thickness. Further, results from the electrochemical kinetics and depends on the overpotential of the electrodes . The relation is characterized by the ButlerVolmer equation:
where the exchange current density is a function of kinetic reaction rate , constant electrolytephase lithiumion concentration , solidphase lithiumion concentration at the particle surface and maximum concentration in the solid phase
By assuming anodic and cathodic charge transfer coefficient , the above indicates that can be expressed as
The terminal voltage is
where , are equilibrium potential and , are solidelectrolyte interphase film resistance.
The charging/discharging of LiBs is accompanied by temperature buildup. The change in temperature can be intense at large currents and notably affects the lithiumion diffusion and electrochemical kinetics. Characterizing and incorporating this effect will help improve the accuracy of the SPM. Here, the temperature dependence of and is the most important, which is governed by the Arrhenius law:
where is or , is the lumped temperature, is universal gas constant and is activation energy. Based on the energy balance principle, the change of is assumed to follow
where is cell bulk density, is lumped specific heat capacity, denotes the heat generation rate which is contributed by ohmic heat and entropic heat and is the convective heat removal rate with the ambience. Further, and are given by
where is the ambient temperature, is the convective heat transfer coefficient and the anodic/cathodic bulk concentration is given by:
We define the anodic surface SoC and bulk SoC as:
Summarizing the foregoing equations will lead to a complete formulation of the SPMT model. It offers good accuracy when low to medium currents are applied. Yet, its prediction performance at high Crates is harmed by the idealistic simplification of an electrode as a particle and absence of the electrolyte dynamics.
IiiB The FNN Model
FNNs are a basic yet extremely important type of NNs, with numerous successful applications. We offer an overview of them here, which is mainly based on [6, 15].
Consider an unknown function , which is a mapping from a dimensional input to a dimensional output . An FNN aims to approximate it by defining a parameterized mapping and learning the collection of parameters from the data . The structure of an FNN includes three parts interconnected in series, namely, the input layer, hidden layers, and output layer. The input layer takes the input and passes it to the first hidden layer. A hidden layer makes a nonlinear transformation of its input. For example, the first hidden layer will transform into , where
is a chosen nonlinear mapping often called as activation function,
is the weight matrix, and is a correction term. The following hidden layers then run the same nonlinear transformation sequentially. The final layer is the output layer, which generates an output value to match . Considering an layer FNN, it can be described in a general form:where and are the input and output of the th layer, respectively. For the FNN, is the collection of and for . Note that the information flows only in the forward direction from to in the above network model, which is why the model is called as feedforward NN.
The training of the FNN is to identify from the measurement data . A common approach is based on maximum likelihood, which minimizes the following cost function:
where is the databased empirical distribution of and , and
is the probability distribution of
over the parameter space based on the FNN model. Under some assumptions, can be reduced to a mean squared error cost:The minimization is usually achieved using the stochastic gradient descent algorithm. The computation of the gradient can be complicated, especially for multilayer FNNs, but it can still be done efficiently and exactly by the backpropagation algorithm and its generalizations.
Iv Hybrid Model Validation
In this section, we perform simulation to validate the effectiveness of the proposed HYBRIDI and HYBRIDII models.
Iva Evaluation Setting
The evaluation setting is summarized as follows:

The DFN model, which is acknowledged as a generic and reliable electrochemical model, is used as the benchmark to assess the HYBRIDI and HYBRIDII.

We use the DUALFOIL simulation package [17] to run the DFN model representing an LCO/graphite battery that operates between 3.1 and 4.1 V to generate synthetic data as the ground truth.

The synthetic data are divided into the training and test datasets. The training datasets are produced by applying constant discharging currents at 0.1/0.2/1/2/4/6/8/10 C and variable currents created based on the Urban Dynamometer Driving Schedule (UDDS) and US06 at . The test datasets are obtained by applying constant discharging currents at 0.5/1/3/5/7/10 C and variable currents created based on the UDDS and US06 at . Here, the UDDSbased training and testing current profiles are generated by modifying the standard UDDS profile differently so that they differ from each other notably. They are labeled as UDDSA and UDDSB, respectively. Besides, all variable current profiles are scaled to a maximum current of around 10 C. In all cases, the initial temperature C.

Both the HYBRIDI and HYBRIDII employ a fourlayer FNN. The FNN has two hidden layers, each consisting of 32 nodes. The input and output of the FNN are as specified in Section II
. The rectified linear unit (ReLu) function is chose as the activation function for the two hidden layers, and the linear activation function chosen for the output layer. Keras, a Pythonbased deep learning library, is used to create, train and implement the FNN. Because the quantities of the input variables vary across different orders of magnitude, the input data are preprocessed by normalization, as often recommended in the practice of NNs.

We utilize the rootmeansquare error as a metric to evaluate a model’s performance:
where is the true voltage at time , is the modelbased voltage prediction, and is the total number of data values. Furthermore, a relative error reduction () is introduced to quantify the improvement of the HYBRIDI and HYBRIDII over the SPMT, which is defined as
IvB Validation of the HYBRIDI and HYBRIDII Models
We begin with validating the HYBRIDI model. Table I summarizes its performance over all the training datasets and compares it with the baseline SPMT. Further, Figs. 45 offers a visual evaluation of the training results under constant and variable (UDDSA) current profiles. We observe that the HYBRIDI offers remarkable accuracy in all training scenarios. It consistently outperforms the SPMT with a considerable margin, especially at medium to very high currents. Further, we apply the trained HYBRIDI model to the test datasets to appraise its prediction performance. Table II and Figs. 67 demonstrate the results. As is seen, the HYBRIDI can still retain the high accuracy in the testing phase, proving its strong predictive ability.
Input profile  RMSE (SPMT)  RMSE (HYBRIDi@)  RER (%) 

CC0.1C  2.31 mV  1.58 mV  31.60 
CC0.2C  4.37 mV  1.44 mV  44.95 
CC1C  21.59 mV  2.87 mV  86.71 
CC2C  56.40 mV  4.72 mV  91.63 
CC4C  94.28 mV  8.88 mV  90.58 
CC6C  118.86 mV  7.40 mV  93.77 
CC8C  150.06 mV  7.83 mV  94.78 
CC10C  191.05 mV  10.87 mV  94.31 
UDDSA  23.32 mV  8.60 mV  63.12 
US06  33.30 mV  9.65 mV  71.02 
Input profile  RMSE (SPMT)  RMSE (HYBRIDi@)  RER (%) 

CC0.5C  10.92 mV  7.82 mV  28.39 
CC1C  19.15 mV  5.05 mV  73.63 
CC3C  76.37 mV  9.19 mV  87.97 
CC5C  101.85 mV  7.19 mV  92.94 
CC7C  137.85 mV  8.77 mV  93.64 
CC10C  207.29 mV  10.67 mV  94.85 
UDDSB  17.59 mV  6.75 mV  61.63 
US06  35.05 mV  9.12 mV  73.98 
For the HYBRIDII model, its training performance is illustrated in Table III. Table IV further shows its performance across the different test datasets, and Fig. 8 displays its prediction under the US06based test dataset. These results show that the HYBRIDII is also greatly effective in grasping and forecasting the dynamics of LiBs.
Input profile  RMSE (SPMT)  RMSE (HYBRIDii@)  RER (%) 

CC0.1C  2.31 mV  4.24 mV  83.55 
CC0.2C  4.37 mV  3.43 mV  21.51 
CC1C  21.59 mV  4.65 mV  78.46 
CC2C  56.40 mV  4.57 mV  91.90 
CC4C  94.28 mV  8.90 mV  90.56 
CC6C  118.86 mV  8.75 mV  92.63 
CC8C  150.06 mV  8.89 mV  94.07 
CC10C  191.05 mV  13.03 mV  93.18 
UDDSA  23.32 mV  8.48 mV  63.64 
US06  33.30 mV  10.69 mV  67.90 
Input profile  RMSE (SPMT)  RMSE (HYBRIDii@)  RER (%) 

CC0.5C  10.92 mV  15.35 mV  40.57 
CC1C  19.15 mV  3.56 mV  81.41 
CC3C  76.37 mV  12.18 mV  84.05 
CC5C  101.85 mV  7.80 mV  92.34 
CC7C  137.85 mV  9.95 mV  92.78 
CC10C  207.29 mV  11.98 mV  94.22 
UDDSB  17.59 mV  8.37 mV  52.42 
US06  35.05 mV  10.36 mV  70.44 
Finally, we highlight that the HYBRIDI and HYBRIDII models provide higher testing accuracy and better predictive performance than the existing hybrid models for LiBs, as extensive simulation reveals. This underscores the efficacy of the proposed design that feeds a physicsbased model’s state information into the ML model.
V Conclusions
The everincreasing adoption of LiBs across various sectors presents a pressing demand for accurate and computationally efficient models. In this paper, we proposed two new hybrid models that integrate physicsbased electrochemical modeling with datadriven ML to meet this need. The model development was driven by the novel perspective that informing the ML model of the electrochemical model’s state information will bring significant improvements to the prediction performance. We conducted extensive simulations and illustrated that the proposed hybrid models can offer exceptionally high predictive accuracy for LiBs operating at a wide range of Crates. This suggests their strong potential for enhancing various LiB systems, especially when they run at high power levels. Our future work will include experimental validation and the application of the proposed models to different battery management tasks.
References
 [1] (1985) A general energy balance for battery systems. Journal of The Electrochemical Society 132 (1), pp. 5–12. External Links: Document, Link Cited by: §II.
 [2] (2010) Algorithms for advanced batterymanagement systems. IEEE Control Systems Magazine 30 (3), pp. 49–68. Cited by: §I.
 [3] (2018) Neural networkbased state of charge observer design for lithiumion batteries. IEEE Transactions on Control Systems Technology 26 (1), pp. 313–320. Cited by: §I.
 [4] (1993) Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell. Journal of The Electrochemical Society 140 (6), pp. 1526–1533. External Links: Document, Link Cited by: §I.
 [5] (2020) Coestimation of lithiumion battery state of charge and state of temperature based on a hybrid electrochemicalthermalneuralnetwork model. Journal of Power Sources 455, pp. 227935. External Links: ISSN 03787753, Document, Link Cited by: §I.
 [6] (2016) Deep learning. MIT Press. Note: http://www.deeplearningbook.org Cited by: §IIIB.
 [7] (2011) Singleparticle model for a lithiumion cell: thermal behavior. Journal of The Electrochemical Society 158 (2), pp. A122. External Links: Document, Link Cited by: §II, §IIIA.
 [8] (2011) Singleparticle model for a lithiumion cell: thermal behavior. Journal of The Electrochemical Society 158, pp. A122. External Links: Document Cited by: §I.
 [9] (2015) Simplification of physicsbased electrochemical model for lithium ion battery on electric vehicle. part i: diffusion simplification and single particle model. Journal of Power Sources 278, pp. 802 – 813. External Links: ISSN 03787753, Document, Link Cited by: §I.
 [10] (1989) Multilayer feedforward networks are universal approximators. Neural Networks 2 (5), pp. 359 – 366. External Links: ISSN 08936080, Document, Link Cited by: §II.
 [11] (2014) Computational efficiency of solving the dfn battery model using descriptor form with legendre polynomials and galerkin projections. In 2014 American Control Conference, Vol. , pp. 260–267. Cited by: §I.
 [12] (2013) Extension of physicsbased single particle model for higher charge–discharge rates. Journal of Power Sources 224, pp. 180 – 194. External Links: ISSN 03787753, Document, Link Cited by: §I.
 [13] (2017) A single particle model for lithiumion batteries with electrolyte and stressenhanced diffusion physics. Journal of The Electrochemical Society 164 (4), pp. A874–A883. External Links: Document, Link Cited by: §I.
 [14] (2018) A reducedorder model of a lithiumion cell using the absolute nodal coordinate formulation approach. IEEE Transactions on Control Systems Technology 26 (3), pp. 1001–1014. Cited by: §I.

[15]
(2020)
Asymptotic prediction error variance for feedforward neural networks
. IFAC. Note: In press Cited by: §IIIB.  [16] (2017) Battery state estimation for a single particle model with electrolyte dynamics. IEEE Transactions on Control Systems Technology 25 (2), pp. 453–468. Cited by: §I.

[17]
(2008) Fortran programs for the simulation of electrochemical systems. [online]..
Note: Available:
http://www.cchem.berkeley.edu/jsngrp/fortran.html Cited by: 2nd item. 
[18]
(2017)
Hybrid electrochemical modeling with recurrent neural networks for liion batteries
. In 2017 American Control Conference (ACC), Vol. , pp. 3777–3782. Cited by: §I.  [19] (2011) Hybrid neural net model of a lithium ion battery. In ASME 2011 Dynamic Systems and Control Conference and Bath/ASME Symposium on Fluid Power and Motion Control, Vol. 2, pp. 239–246. External Links: Document Cited by: §I.
 [20] (2019) Battery health prediction under generalized conditions using a gaussian process transition model. Journal of Energy Storage 23, pp. 320 – 328. External Links: ISSN 2352152X, Document, Link Cited by: §I.
 [21] (2018) Battery stateofcharge estimation based on regular/recurrent gaussian process regression. IEEE Transactions on Industrial Electronics 65 (5), pp. 4311–4321. Cited by: §I.
 [22] (2006) Online estimation of the state of charge of a lithium ion cell. Journal of Power Sources 161 (2), pp. 1346 – 1355. External Links: ISSN 03787753, Document, Link Cited by: §I.
 [23] (2007) Control oriented 1d electrochemical model of lithium ion battery. Energy Conversion and Management 48 (9), pp. 2565 – 2578. External Links: ISSN 01968904, Document, Link Cited by: §I.
 [24] (2009) Comparison of reduced order lithiumion battery models for control applications. In Proceedings of the 48th IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, Vol. , pp. 3276–3281. Cited by: §I.
 [25] (2015) State of charge estimation of a lithium ion cell based on a temperature dependent and electrolyte enhanced single particle model. Energy 80, pp. 731 – 739. External Links: ISSN 03605442, Document, Link Cited by: §I.
 [26] (2021) Nonlinear doublecapacitor model for rechargeable batteries: modeling, identification, and validation. IEEE Transactions on Control Systems Technology 29 (1), pp. 370–384. External Links: Document Cited by: §I.
 [27] (2017) A computationally efficient implementation of an electrochemistrybased model for lithiumion batteries. IFACPapersOnLine 50 (1), pp. 2169 – 2174. Note: 20th IFAC World Congress External Links: ISSN 24058963, Document, Link Cited by: §I.
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