A Bayesian framework for on-line degradation assessment and residual life prediction of secondary batteries inspacecraft

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Abstract

The paper presents a Bayesian framework consisting of off-line population degradation modeling and on-line degradation assessment and residual life prediction for secondary batteries in the field. We use a Wiener process with random drift, diffusion coefficient and measurement error to characterize the off-line population degradation of secondary battery capacity, thereby capturing several sources of uncertainty including unit-to-unit variation, time uncertainty and stochastic correlation. Via maximum likelihood, and using observed capacity data with unknown measurement error, we estimate the parameters in this off-line population model. To achieve the requirements for on-line degradation assessment and residual life prediction, we exploit a particle filter-based state and static parameter joint estimation method, by which the posterior degradation model is updated iteratively and the degradation state of an individual battery is estimated at the sametime.

A case study of some Li-ion type secondary batteries not only shows the effectiveness of our method, but also provides some useful insights regarding the necessity of on-line updating and the apparent differences between the population and individual unit degradation modeling and assessment problems.

Highlights

► A Bayesian framework for secondary battery degradation modeling. ► A random-effect Wiener process with drift for modeling capacity degradation. ► Off-line parameter estimation using MLE for population degradation. ► On-line state and parameters joint estimation using PF for individual degradation. ► A thorough case study and several significative conclusions.

Introduction

A secondary or rechargeable battery is a critical component in any photovoltaic power system. Moreover, it constitutes the most common source of electrical power for long life spacecraft. In such settings, the consequences of secondary battery degradation or failure can range from reduced performance to operational impairment or even catastrophic failure[1]. Since the degradation induced by solid electrolyte interface/interphase (SEI) growth and dissolution, space radiation damage, etc. [2], [3] is inevitable while the power requirement is increasing from several watts to several kilowatts over a life of 15–20 years [4], assessing how much degradation has occurred, as well as predicting the residual life of a battery has become increasingly important to ensure that the battery operates normally and within the desired limits.

Most current research concerning secondary battery management involves modeling the state-of-charge (SOC) in each charge–discharge cycle. From this basis, the state-of-life (SOL) and the health state of the battery can then be evaluated. These characteristics are important aspects of any battery management system (BMS), and help prevent the battery from over-charging or excessive discharge. Such models vary from a collection of basis functions [5] to detailed formulations derived from physical analysis of the cell [6], [7]. Dynamic models for lithium-ion batteries that take into consideration nonlinear equilibrium potentials, rate and temperature dependencies, thermal effects and transient power response have also been developed [8], [9]. Least squares estimation, state estimation techniques [6], [10] and machine learning methods [11], [12], [13] have all been used to estimate model parameters and solve the degradation prediction problem, based on various sources of physical test and telemetry data including cyclic charge–discharge current and voltage, body and ambient temperature, and EISpulse.

Due to the development of prognostics and health management techniques, assessing the amount of long-term degradation, including feature extraction, process modeling, state estimation and prediction, and then predicting battery residual life has become increasingly important. Refs. [6], [5] have used a charge–discharge model and Kalman Filter or Extended Kalman Filter (EKF)-based methods to extract degradation features such as capacity and resistance in each cycle and hence characterize the aging properties of a battery. Ref. [14] uses a first-principles aging model based on the Maximum Charge Voltage (MCV) and the cell life friction (LF) to analyze the aging process. To monitor the remaining life of a spacecraft battery, [11] employs the electrolyte resistance and SOC for a NiCd battery in a neural network to derive a fuzzy logic SOC prediction model. Because the aging mechanism of a secondary battery is complex, data-driven methods usually receive the most attention in efforts to predict long-term degradation. Based on physical tests and mathematical modeling, several different degradation models describing feature evolution with time have been presented. Ref. [13] uses a linear degradation model with a Coulombic efficiency factor to characterize capacity degradation. Exponential models for the same test data are presented in [15], [16]. Ref. [5] relies on a low-pass filter to model capacity and resistance fading. Ref. [11] investigated a decision-level fusion of data-driven algorithms, like Autoregressive Integrated Moving Average (ARIMA) and neural networks, for both battery diagnostics and prognostics.

However, there is still a large gap between these recent successes and engineering requirements with respect to long-term degradation assessment and residual life prediction. A more flexible framework is still needed that combines the sensor data from battery monitors, the models developed, appropriate state estimation and prediction algorithms, and yet provides reasonable predictions in the face of uncertainty.

  • (1)

    Aging and degradation studies that concentrate on the internal physics, and are based on tightly controlled, destructive testing, are impractical for field applications because of their off-line, invasive nature. On the other hand, most of the empirical or semi-empirical models have limited predictive capability given the strong dependence of the behavior of such special batteries on factors such as temperature and charge/discharge cycling protocols. Appropriate statistical modeling techniques that combine a degradation mechanism, observed history data and on-line monitoring data should be devised to solve this problem.

  • (2)

    Uncertainty management is the most challenging aspect of long-term performance prediction. The sources include unit-to-unit variation, stochastic load and environmental stress, disturbances and unknown or vague behaviors in the system, errors in measurement or degradation feature extraction procedure (e.g., capacity calculation through amp-hour integration). Current models and algorithms generally focus on only a few of these uncertainties, and therefore cannot hope to furnish satisfactory results. Further considerations are still necessary to cope with these factors appropriately and thus avoid the divergence of the model predictions from actual battery performance.

  • (3)

    For degradation level assessment and residual life prediction, filter-based methods, such as the Kalman filter or particle filter, are widely recognized as powerful, and potentially useful for jointly estimating both the state and parameter values. But recent efforts, such as the diffusion process model of [13], [16] for the unknown parameters, appear impractical due to their excessive uncertainty.

In this paper, we present a method for state space modeling and prediction that combines both population and individual information, and can appropriately manage several kinds of uncertainty. This new approach represents a hybrid of two more traditional methods of estimation: off-line population modeling and online recursive updating of the individual unit observations.

Section snippets

A framework for degradation-based life prediction

Our new approach has two primary components; see the schematic displayed in Fig. 1. The first component is an off-line model of population degradation and corresponding reliability. The second aspect is an online model of individual unit degradation and residual life prediction. The two parts are tightly linked, with aspects of the off-line model providing a starting point for calculations that update the online model for a specific individual unit as relevant observed data become available.

The

Degradation feature determination

Although degradation feature extraction is important for reasonably modeling and precisely predicting the degradation process of a secondary battery, it has not been stated explicitly in current research, especially for spacecraft applications. In general, several performance measures including voltage, current, temperature, etc. can be measured during the battery operation. Which measure, or set of measures, is the most appropriate for characterizing battery degradation, i.e., relates closely

Population degradation modeling and reliability assessment

In this section, we develop a population degradation model for secondary battery capacity. Fitting this model will require observed data concerning capacity and the use of suitable methods of parameter estimation. Suppose there are n observed realizations of the capacity degradation process Xi(t), i=1,,n. Then each observed sample of degradation data has the following form:

  • From each realization, we observe mi measurements Yi(tij) at times tij, j=1,,mi, i=1,,n.

  • These observations have i.i.d.

Individual degradation modeling and residual life prediction

In current engineering practice, and especially in prognostics and health management (PHM) and condition-based maintenance (CBM), the assessment of operating product degradation and the prediction of its residual life is becoming increasingly important. Much recent research has focused on these problems with respect to an individual product, where the degradation process for the unit is based on unit-specific observations, and the unit's degradation level is derived using state

A case study

In this section, we describe the results of degradation modeling and residual life prediction for some Li-ion secondary batteries, using the methods we have outlined. We have used data from the NASA Ames Prognostics Center of Excellence (PCoE), where commercially available Li-ion 18,650-sized rechargeable batteries are tested. Sample Li-ion batteries were run through different operational profiles including charge, discharge, rest and EIS at room temperature. All charging was carried out in a

Conclusions

In this article we have described a Bayesian framework for solving an important engineering problem about online degradation assessment and residual life prediction. Our proposed solution combines several statistical tools, including maximum likelihood estimation, particle filtering, Bayesian inference and degradation modeling. A case study using NASA data from Li-ion battery test results has demonstrated that this framework and our proposed solution is able to furnish useful predictions with

Acknowledgment

This research was supported by the National Natural Science Foundation of China under Grant Nos. 71071158 and 70901024 and the Chinese Postdoctoral Science Foundation. The authors would like to thank the editor, Carlos Guedes Soares, and two referees for providing valuable comments that improved the original manuscript.

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