Mission-Specific Prognosis of Li-ion Batteries using Hybrid Physics-Informed Neural Networks

New transportation modalities such as electric powered verticaltakeoff and landing aircraft and logistic applicationslike delivery of packages with drones require highly reliableand powerful electric batteries for operation. A challengingbut very important task hereby is the precise forecasting ofthe degradation of battery state-of-health (SOH) and stateof-charge (SOC). While high-fidelity electrochemistry basedmodels can provide precise predictions of the SOC, they canbe computationally expensive. On the other hand, purely datadrivenapproaches require a large amount of training data inorder to learn the input to output relation. In this research animproved hybrid physics-informed machine learning modelis introduced, that conserves the electrochemistry based lawsand is implemented with data-driven layers to compensate forunknown portions of internal voltage drop during discharge.Preliminary results indicate that the model can predict dischargefor a large variety of loads, accurately predicts capacitydegradation over age and can be enhanced through extractinginformation from cell temperature data as surrogate for aging.


PROBLEM STATEMENT
State-of-the-art Li-ion battery cells are a crucial part of the electric powered vehicles and contribute up to 40% of vehicle cost (Lutsey & Nicholas, 2019).Hence, there is a demand in models that can predict SOH and SOC depending on wide load level variations taking into account the underlying electrochemistry of Li-ion battery cells.The existing modeling approaches that have been trying to address these challenging predictions have found the following roadblocks: • Purely model based approaches have difficulties to replicate the complex non-linear behavior of battery aging processes without using experimental validation data.• Data-driven approaches require a large amount of data that come along extensive efforts to build test beds for Kajetan Fricke et al.This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 United States License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
battery aging under real-world conditions.• Time constraints associated with experimental testing of battery life make it impractical to gather data from real life operation for prognosis of battery health degradation.
In this PhD research we aim to address these roadblocks by (i) developing a physics-informed machine learning model, which uses previously gathered data on battery degradation and aims to forecast the SOH and SOC degradation on new developed battery cells with only some available data history; (ii) validating the trained model against gathered data from a accelerated battery life aging test conducted on a self-designed battery cycling test bed; and (iii) contribute to the state-of-the art literature regarding Li-ion battery degradation through implementation of a new model using additional available sensor data (e.g.cell temperature) for SOH prediction.

CONTRIBUTIONS TO THE STATE-OF-THE ART
The main output of this research is a physics-informed machine learning model for battery degradation prognosis that: • Estimate SOC during constant and variable current discharge depending on load levels • Track and forecast capacity degradation over lifetime as function of operating conditions • Predict Battery failure taking into account current levels and temperature build-up during discharge The resulting hybrid physics-informed machine learning model is designed to accurately estimate remaining useful life of Liion batteries and detect battery cells that are close to failure and require replacement.The model is also designed to be deployable for different Li-ion battery cell chemistries by utilizing readily available data from fleet operators and minor adjustments of model parameters.

RESEARCH PLAN
The specific objectives of this research include these four primary tasks: Objective 1: Develop a test bed for accelerated lifing of Li-ion batteries-Design an accelerated battery lifecycle test bed that utilizes integrated self-designed printed circuit boards to autonomously cycle Li-ion battery cells.The goal is to gather a rich dataset of battery aging depending on load level variations as training data for hybrid machine learning models.
Objective 2: Implement an algorithm for solving ordinary differential equations using hybrid physics-informed neural networks-In this task we develop a new machine learning approach that implements hybrid models combining physics-informed and data-driven kernels, where data-driven kernels are used to reduce the gap between predictions and observations of time series forecasting.
Objective 3: Create a framework for pro-active monitoring of new components using Bayesian transfer learning-The goal here is to build a transfer learning model that uses a combined electrochemistry based and machine learning approach to predict SOC and SOH of Li-ion batteries taking into account battery fleets with large load level ranges.
Objective 4: Develop a method for battery degradation prognosis as function of cell temperature and load levels-Using temperature build-up and current data from discharge cycles we aim to build a Bayesian model for remaining useful life and failure prediction.

Progress to date
In the first step of this PhD research, we designed and built a test bed for accelerated battery life testing that integrates selfdesigned printed circuit boards, which, in combination with a microcontroller, perform continuous full charge and discharge cycles of battery packs at controlled current levels.Here, we contribute to the literature by generating a life cycle dataset for batteries subjected to a wide range of load levels with either constant or variable loading conditions.This rich dataset allows to study battery aging under real world operating conditions, as they are expected in battery powered vehicles.With the goal of building a hybrid battery degradation model, we estimate the battery capacity at each 20th cycle through reference discharges at constant current levels of 1C.Fig. 1 (left) shows the self-developed testbed in operation, where 6 battery packs placed on separate levels are connected to one charger board each and a load board, used for discharge, is placed on the top level.In Fig. 1 (right) the capacity degradation over cumulative energy, generated from our life cycle study, is shown for each battery until reaching end of life (marked by X).By collecting this dataset we already contribute to the literature for battery aging research, covering a wide battery aging range, which is used in this research as foundation for battery degradation prognosis with hybrid machine learning.
In the second part of this research, we implemented a Python framework for the solution of ordinary differential equations through recurrent neural networks (RNN) as further detailed in (Nascimento, Fricke, & Viana, 2020).This approach is used to build hybrid models, where ODEs governing reduced order  To demonstrate the capabilities of this approach, in one case study we implement fatigue crack growth integration through Euler's forward method combining a data-driven stress intensity range model with a physics-based crack length increment model in order to reduce the gap between predictions and observations.In a second case study, we implemented a two degree of freedom mass-spring-damper system within an RNN cell, where parameter identification of the damping coefficients could be successfully demonstrated.
Using this RNN integration framework we extend the capabilities of the RNN cell by implementing a hybrid electrochemistry and data-driven battery SOC model to predict the voltage curves during discharge depending on current and battery cell temperature inputs (Fig. 2).The cell predicts the voltage output at each time step, where the state, similar to a state space representation, is integrated over the entire time series.The general voltage drop is captured through the Nernst and Butler-Volmer equations (blue blocks), as introduced in (Karthikeyan, Sikha, & White, 2008) and (Daigle & Kulkarni, 2013), while the remaining discrepancy between actual and predicted voltage is captured through the non-ideal voltage multi-layer perceptron (green block), as it was presented by (Nascimento, Corbetta, Kulkarni, & Viana, 2021).
Our contribution here, further enhances the hybrid approach by introducing the effects of large load variations and cell temperature build-up during discharge to the prediction model.The proposed model is trained, on constant loading discharge curves detailed in Fig. 3, whereas the variable loading curves are used for validation.Fig. 4 illustrates the discharge prediction results on early life variable loading (top) and predictions on the same battery later in life (bottom), where our findings validate that the model predictions significantly improve through aging parameter updates.The aging parameters in form of available Li-ions (q max ) and internal resistance (R b ) are updated using battery discharge curves throughout cycle life.Fig. 5 shows our Gaussian process model (GP) predicting the degradation of q max and the increase of the internal resistance R b as function of cumulative energy and loading levels, where our model captures the influence of different load levels on both aging parameters.Using the predicted aging parameters we established predic-Figure 5. Gaussian process model for q max and R b over lifetime with mean function and 95% CI.
Figure 6.Forecast of q max and R o as function of cumulative energy for constant current at 9.3A.Predictions are done with and without q max and R b update using early life data.
tions of discharge curves on aged batteries.Fig. 7 shows the discharge curve prediction for 9.3A load level at a battery age of 4kWh and 6kWh, respectively.Without early life data, the model prediction deviates from the discharge curve, whereas our investigation shows, adding early life data helps to improve the discharge curve prediction significantly.
Figure 7. Voltage curves for 9.3A constant discharge at 4kWh (left) and 6kWh (right) using predicted q max and R b .

Remaining work
We will further contribute by enhancing the model to predict cell failure and using temperature build-up data during discharge cycles as surrogate for age.Fig. 8 shows the failure classification including 95% confidence intervals (CI) for different load levels as function of cumulative energy.The presented GP model already establishes a prediction method for battery failure as function of age on different load levels and will be fused with the battery aging model in the next step.In Fig. 9 one can see a clear trend between the within-mission temperature build-up and battery age (Fig. 9, left).Here, we will contribute by extracting the slope of the temperature curves to build a GP capacity prediction model with load level and temperature slope as input (Fig. 9, right).In this research we also propose a model to capture loading variations over life time.Fig. 10 shows the capacity degradation curve (full circles) of a battery that was initially aged at 16A until reaching 2.15Ah, moved to a mild load of 7.5A until 2.0Ah and aged at a more aggressive load of 17.5A until failure at 1.92Ah.Here, we will establish a model that can predict battery degradation depending on load level changes, where the model is used to move the battery to an equivalent age at the current load level (empty circles).Additionally, we contribute through re-using previously aged battery cells.For this purpose, previously deployed cells with similar aging history were assembled to new battery packs.Those second life battery packs are then subjected to milder load levels compared to the initial deployment.Fig. 11 shows the capacity degradation curves of four second life batteries, where significant life extension of up to 4kWh has been observed.With this approach we will introduce a model that can be beneficial for operators of battery fleets that intend to re-purpose used battery cells.

Conclusion
In this research work an enhanced hybrid physics-informed machine learning model is proposed to predict the SOH and SOC of Li-ion batteries.Through a self-designed test bed

Figure 2 .
Figure 2. Hybrid physics-informed battery discharge model

Figure 3 .
Figure 3. Current and voltage discharge curves

Figure 4 .
Figure 4. Random loading discharge prediction on early life (top plot) and aged battery (bottom plot) with updated and model parameters

Figure 8 .
Figure 8. Failure classification: current levels vs age

Figure 9 .
Figure 9. Capacity as function of cell temperature.

Figure 11 .
Figure 11.Aged batteries with 2nd life at low current levels.