Research paperInitialization of a fractional order identification algorithm applied for Lithium-ion battery modeling in time domain
Introduction
Electrochemical cells are one of the most important devices used for generating energy from chemical reactions in many electrical systems. Batteries are composed of one or more electrochemical cells. Several kinds of batteries are distinguished by their internal chemical reactions such as Alkaline, Lead-acid, Lithium-ion and others. Lithium-ion batteries have many important properties which include high density, durability and high specific power [1]. For these reasons, they are nowadays integrated into several applications such as electric vehicles [2], spacecrafts [3] and electronic devices [4]. Accuracy of battery model is requested for these applications, hence some aspects such as chemical characteristics, thermodynamic properties and diffusion process should be considered.
Several models that differ among communities, have been developed to ensure accurate description of the internal battery behavior. Electrochemists proposed models that takes into consideration the chemical properties of the batteries. The main drawback is that it takes long characterization time which complicates their use in real time applications such as in the battery management systems [5]. Mathematicians developed some useful stochastic models although they generally lack the physical meaning [6], [7], [8]. Electrical engineers aim to describe the internal behavior of batteries using equivalent electric circuits composed of resistors, capacitors and voltage sources [9], [10]. The main advantage of these methods lays in the simplicity of their simulation in different applications and their ability to reflect all internal battery phenomena.
The equivalent electric circuits made of three components in a series connection are mainly based on spectroscopic experiments. The first component is the resistance of electrolyte and connection within the cell. The second component is the charge transfer impedance that corresponds to dynamics within the medium to high frequencies. The third component corresponds to the diffusion impedance whose dynamics are covering the low up to medium frequencies. Diffusion phenomenon is the most difficult process to account for considering the internal battery behavior modeling. The main three diffusion models are: Nernst, Restrictive and Semi-infinite [11]. In this paper, only Nernst diffusive model will be considered.
The main disadvantage of the spectroscopic experiments is that they are very restrictive in terms of time. Indeed, characterizing the low frequency behavior of a battery could lead to very long experiments. In order to get over this problem, links existing between frequency and time domains could be relevantly used to identify models derived from equivalent electric circuits without spectroscopy. In this paper, the proposed method is based on time domain fractional modeling. Identification results applied to thermal systems [12], [13], [14] and batteries [15], [16] have shown that fractional models are highly suitable to describe the diffusion behavior. Diffusion impedance, also known as Warburg impedance, is directly simulated using a fractional impedance with only one half order integrator which is simulated using a fractional operator. The fractional model parameters are linked to the corresponding physical parameters.
Once the battery model is determined based on the fractional modeling, a gray-box model identification is required to retrieve physical battery components. Non linear identification algorithm, such as Levenberg–Marquardt could be used for this purpose. As it is an iterative algorithm, an initialization process is required to ensure the convergence of the algorithm to the real system parameters. Therefore, the more this initialization is closer to the real values, the more the algorithm will be able to converge quickly and accurately.
The objective of this paper is to present an initialization method that insures accurate convergence of the Levenberg–Marquardt algorithm. The presented method is based on a very poor a priori information about the impedance characteristics, and allows a convenient initialization of the model parameters which leads to a proper convergence of the identification algorithm. Simulations using noiseless data are performed in order to check the ability of the initialization procedure combined with Marquardt algorithm to obtain the impedance physical parameters. Noisy simulations are also provided in order to confirm this ability.
This paper is divided into six sections. The equivalent electric circuit battery modeling is firstly introduced in section two, then the fractional modeling and its application to the battery behavior are presented in section three. The identification algorithm and its initialization method are given in section four. In section five, the identification results are displayed and they are compared to a time domain simulator derived from this electric circuit that is described also in this section. Finally, the conclusion and perspectives of this work are presented in section six.
Section snippets
Battery electrochemical impedance model
Several models have been developed to describe the internal behavior of the electrochemical cell [17], [18]. These models are used to predict the state of charge, the state of health and the global performance of the battery [19], [20].
Storage and generation of electrical energy involved in batteries are governed by electrochemical processes whose dynamics can be clearly distinguished by:
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charge transfer phenomenon which corresponds to the flux of oxidizing and reducing species from one
Battery fractional modeling
Fractional calculus has been a widely used topic in many applications such as heat transfer [12], [13] and batteries [15], [16]. Fractional field which is also known as non integer order presents an accurate solution for simulating diffusion behavior. In this section, Eq. (1) given in Laplace domain is approximated by a state space model in order to provide time domain data for the required identification process.
Identification results and discussions
In this section, simulation results of fractional modeling in time domain are presented. Initialization method and final identification results are presented first using noiseless voltage signal δV(t) then using noisy voltage signal δV*(t) with a Signal to Noise Ratio (SNR) of 20 dB. Results indicate the advantage of using the proposed initialization method to obtain parameters values that converge to the requested physical parameters.
Conclusion
An initialization method for identification algorithm is proposed in this paper in order to accurately calculate the physical parameters of Lithium-ion battery. Parameters are calculated from voltage response in time domain. Modified Randles circuit is used to model the internal impedance behavior of the battery. The diffusion impedance is characterized using fractional order model.
The ability of the fractional model to properly estimate the impedance physical parameters has been tested using
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