Remarks on Computational Efficiency

In (6.18), we described the applied reduction of work load during the sample evalu-ation. “Early Rejection” had a very high impact on the parallel chains without noise that started from a distant point in parameter space. Due to the nature of the model, it is possible to stop many evaluations at a very early stage because of extreme deviations in the output or numerical issues that may arise, since not every possible parameter combination makes sense. The effective work load could be reduced by 67.7 %. That is, by using “Early Rejection”, three times as many samples could be evaluated than using the regular approach.

6.6 Discussion and Conclusion

This chapter shows the applicability of parameter estimation and uncertainty quan-tification of lithium-ion cells by Bayesian model inversion using the Markov Chain Monte Carlo sampling approach.

We started with an introduction to parameter estimation in general, and then focussed on estimating dynamic parameters and their uncertainties in a computational model of a lithium-ion cell. We gave some insight into the modelling of the prior and the set-up of the algorithm. Due to the complexity of the model, we parallelized the approach and implemented “Early Stopping” as an additional means of reducing computing times. We compared the results of synthetic measurements and presented the statistical efficiency by investigating the integrated autocorrelation time.

The analysis of statistics in Table6.3 and IACT in Table6.4 indicate a sharp distribution and a higher statistical efficiency for individual chains. Only the scatter plots shown in Fig.6.5reveal the inferior sample coverage of the posterior in the individual chain case. This necessitates the use of parallel chains.

The proposed approach has been shown to be appropriate for investigating the dynamic properties of lithium-ion cells in the presence of noise. However, addi-tional work must be done to incorporate stationary and quasi-stationary effects and influences, such as the open-circuit voltage and geometric quantities.

Acknowledgments The authors would like to acknowledge the financial support of the “COMET K2—Competence Centres for Excellent Technologies Programme” of the Austrian Federal Ministry for Transport, Innovation and Technology (BMVIT), the Austrian Federal Ministry of Economy, Family and Youth (BMWFJ), the Austrian Research Promotion Agency (FFG), the Province of Styria and the Styrian Business Promotion Agency (SFG).

References

1. Beskos A, Papaspiliopoulos O, Roberts GO (2006) Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12(6):1077–1098

2. Doyle M, Fuller TF, Newman J (1993) Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell. J Electrochem Soc 140(6):1526–1533

3. Geyer CJ (1992) Practical markov chain monte carlo. Stat Sci 7(4):473–483

4. Morris M (1991) Factorial sampling plans for preliminary computational experiments. Tech-nometrics 33(2):161–174

5. Newman J, Thomas-Alyea KE (2004) Electrochemical systems, 3rd edn., John Wiley & Sons, New York. ISBN: 978-0-471-47756-3, http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471477567.html

6. Papaspiliopoulos O, Roberts GO (2008) Retrospective markov chain monte carlo methods for dirichlet process hierarchical models. Biometrika 95(1):169–186

7. Saltelli A, Tarantola S, Campolongo F, Ratto M (2004) Sensitivity analysis in practice: a guide to assessing scientific models. Halsted Press, USA

8. Solonen A, Ollinaho P, Laine M, Haario H, Tamminen J, Järvinen H (2012) Efficient mcmc for climate model parameter estimation: Parallel adaptive chains and early rejection. Bayesian Anal 7(3):715–736

9. Speltino C, Domenico DD, Fiengo G, Stefanopoulou A (2009) Experimental identification and validation of an electrochemical model of a lithium-ion battery. In: American control conference

Abstract This chapter presents a data-based approach to nonlinear observer design for battery state of charge (SoC) estimation. The SoC observer is based on a purely data-driven model in order to allow for the application of the proposed concepts to any type of battery chemistry, especially when conventional physical modelling is not eas-ily possible. In order to cope with the complex nonlinear dynamics of the battery, an integrated workflow for experiment design, model creation and automated observer design is proposed. The nonlinear battery model is constructed using a proven training algorithm based on the architecture of local model networks (LMNs). One impor-tant advantage of LMNs is that they offer local interpretability, which enables the extraction of local linear battery impedance models for automated nonlinear observer design. The proposed concepts are validated experimentally using real measurement data from a lithium-ion power cell.

7.1 Introduction

In the automotive industry, data-driven methodologies are becoming more and more important due to the constantly increasing demands. Such methods create models based on measured input and output data from the process and require little or no phys-ical or formal information, see e.g. [29]. Especially in engine calibration, data-based approaches have been established as an important tool for systematically dealing

C. Hametner(

B

Christian Doppler Laboratory for Model Based Calibration Methodologies, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Vienna, Austria

e-mail: [email protected] S. Jakubek

Institute of Mechanics and Mechatronics, Vienna University of Technology, Wiedner Hauptstr.

8-10, 1040 Vienna, Austria e-mail: [email protected]

A. Thaler and D. Watzenig (eds.), Automotive Battery Technology, 111 Automotive Engineering: Simulation and Validation Methods,

DOI: 10.1007/978-3-319-02523-0_7, © The Author(s) 2014

with the growing complexity of automotive systems, see e.g. [10, 12]. In this con-text, the optimisation of combustion engines and hybrid electrical vehicles comprises the calibration of various controller parameters for both feedforward and feedback controllers in engine and hybrid control units. Thereby calibration is understood as the optimisation of vehicles and their subsystems through proper parametrisation of various controller parameters.

One important requirement for an integrated methodology for the complete cali-bration workflow is that both the experiment on the testbed and the model architec-ture must be designed such that the model is able to cover all relevant effects, and all parameter interactions can be taken into account in the optimisation procedure.

Such a model-based calibration workflow consists of the following steps: Experiment design, nonlinear system identification and controller/observer analysis and design.

Besides engine calibration, the optimisation of hybrid components has become an important issue in recent years. Hybrid electrical vehicles require an accurate online observation of the electric power supply. In this context, the development of the battery management system (BMS) and the energy management system (EMS) is thus a challenging task. One essential part of the BMS is a battery model, which must be accurate under the specific loads and environmental conditions. One of the most important functions of the BMS is determining the state of charge (SoC) of the battery, as well as the charge and discharge control. Knowledge about the state of charge, which cannot be measured directly, is thus essential in order to extend battery life and preserve the usable capacity.

The present work describes the model-based calibration workflow using data-driven models in general and presents the adaptation/extension of the proposed con-cepts to battery modelling and the design of the associated nonlinear observer (see also [11]). The remainder of this chapter is structured as follows: Sect.7.2 pro-vides an overview on the three major steps of the data-driven calibration workflow.

Section7.3presents the application of the proposed concepts for SoC observer design and demonstrates the performance using real measurement data from a lithium-ion cell.