Time domain simulation of Li-ion batteries using non-integer order equivalent electrical circuit

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Abstract

For electric vehicle (EV) or hybrid EV (HEV) development and integration of renewables in electrical networks, battery monitoring systems have to be more and more precise to take into account the state-of-charge and the dynamic behavior of the battery. Some non-integer order models of electrochemical batteries have been proposed in literacy with a good accuracy and a low number of parameters in the frequential domain. Nevertheless, time simulation of such models required to approximate this non-integer order system by an equivalent high integer order model. An adapted algorithm is then proposed in this article to simulate the non-integer order model without any approximation, thanks to the construction of a 3-order generalized state-space system. This algorithm is applied and validated on a 2.3 A.h Li-ion battery.

Highlights

► We model Li-ion battery using equivalent electrical circuit. ► These circuits include classically non-integer order systems. ► These systems cannot be included directly for numerical simulation. ► We propose a methodology for time-domain simulation. ► This methodology is validated on a commercial Li-ion cell.

Introduction

The increasing of power electricity demand and environmental concerns has spurred worldwide interest in the development of alternative resources and storage systems, whatever the application domain as distribution grids or electric and hybrid vehicles. As a consequence, researches on advanced rechargeable batteries are receiving much more attention to boost performances and lifetime while decreasing production costs [1].

To cope with the complexity of future electrical systems and to find favorable or optimal operation strategies, simulation-based design methods are becoming very important for designers. But, for these simulations, suitable and accurate models of system components are mandatory, even if such models for energy storage devices are difficult to obtain especially for different operation profiles [2], [3]. Besides, such models are also required for battery monitoring or battery management systems to improve the precision state-of-charge (SoC) and voltage estimations, especially if new sensitive battery technologies are introduced.

Several models can be proposed in the literacy, classified from the minimum to maximum degrees of knowledge. Thus, on the one hand, some black box models as transfer functions can be introduced to describe only behavior of the battery input and output variables, as current and voltage [4]. On the other hand, some physicochemical models can rebuild the full processes occurring in the battery, using spatial resolution by means of resolution of partial differential equations. In that case, models are not easily usable for a designer, especially for commercial batteries, and need a very high computational power and parameterization effort [5], [6]. By the way, other models balance these two categories, basing on an analogy between physicochemical phenomena and common electrical elements as capacitances, resistances and so on. Such models are called equivalent electrical circuits and offer a good compromise between calculation times, parameterization effort and precision of the simulation [7].

For electrical equivalent circuits, which are developed from physical considerations, several experimental tests allow to identify parameters. The first one consists in step-response with the hypothesis that all parameters are not influenced by charge or discharge currents. The non-linear and non-stationary behaviors of the battery are then ignored leading to less accurate models. In comparison, parameterization from impedance-based measurements allows taking into account the influence of operating condition on model parameters for small-signal behavior in order to avoid nonlinearities. However, there are some difficulties during the impedance spectroscopy measurements thanks to the non-stationary behavior of the battery. Then, for a given battery current, a quasi-stationary condition is restricted to the minimum frequency which corresponds to a short measurement time compared to total charge or discharge time. Hence, the low-frequency portion of battery impedance cannot be determined for high currents [8]. Anyway, some methods of parameterization for high currents are found in literacy to overcome these difficulties [9].

Besides these experimental precautions, equivalent electrical circuit simulation is not as easy as classical Thevenin ones, as passivation film and species transfer can be modeled by constant phase elements or CPE. Such nonelectrical elements are non-integer order systems and allow modeling a distributed physical phenomenon with a very low number of parameters in frequency domain. Nevertheless, for practical simulations, such elements have to be approximated by a network of electrical R-C parameters in order to implement them in numerical simulators. The compactness of such models can then disappear as the order of equivalent R-C is directly linked to the desired precision. As an example, [9] approximates a CPE element described by 2 parameters using a 7-order (i.e. 14 parameters) equivalent integer-order model. Moreover, the physical meaning of each parameter can be lost in the approximation procedure, which may prevent the designer to come back over the system parameter from simulations, e.g. to estimate the health or the crankability of the battery.

Indeed, the authors propose in this paper a new methodology for time simulation of battery, basing on the construction of a generalized state-space system from the equivalent electrical circuits but without any approximation of CPE elements. Moreover, the simulation takes into account the evolution of model parameters against charge/discharge currents and the state of charge.

Results will be illustrated on a commercial lithium iron phosphate/graphite cell (LiFePO4/C) with a nominal capacity of 2.3 A.h (ANR26650 m1, A123Systems Company Ltd).

Section snippets

Equivalent non-integer order electrical circuit

The model used in this study is based on an equivalent electrical circuit where parameters could be tabulated according to state-of-charge, current, temperature and so on. This model consists in an open-circuit voltage (denoted OCV) associated with a well-known complex impedance composed in four elements. The first one is a resistance (R1) to model the global ohmic behavior of the battery including metallic connection between poles and electrodes, conductivity of metallic contacts, intercell

Algorithm for time-domain simulation

The insertion of non-integer order impedances in the equivalent circuit model of the battery made it possible to have a reliable and compact frequency model. The non-integer approach is then at first place a frequential approach. It is nevertheless essential to transform this frequential model into time domain, which is a mandatory step for the study of the dynamic behavior of the battery itself, and the integration of a battery in an electrical system.

Aoun et al. [16] presents an overview of

Results and discussion

In this section, the dynamic response derived from the non-integer order model of the battery in Section 3 has been calculated and compared to experimental results.

Conclusion

In this article, a new methodology of time simulation of electrochemical battery is proposed to reduce the order of equivalent electrical order circuits without any physical simplification. The good agreement between measured and simulated voltages tends to prove the validity of the method, even if parameterization tests must be accurately developed using Electro Impedance Spectroscopy, as model parameters are closely linked to State of Charge of the battery and charge/discharge conditions.

Acknowledgements

The authors would gratefully thank Dr. K.T. Dong and Dr. A. Kirchev (CEA/LITEN/LSE) for their help to obtain experimental results.

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