Elsevier

Applied Energy

Volume 161, 1 January 2016, Pages 330-336
Applied Energy

State-space model with non-integer order derivatives for lithium-ion battery

https://doi.org/10.1016/j.apenergy.2015.10.025Get rights and content

Highlights

  • We use fractional order model to describe lithium-ion battery behavior.

  • We use Oustaloup approximation to obtain the state-space equation.

  • Particle swarm optimization is employed to identify model parameters.

  • Good accuracy and robustness is obtained when cycles, aging levels and cells vary.

Abstract

Lithium ion batteries have attracted wide attention due to their high energy density, long cycle life, and environmental friendliness and are widely used in electrical vehicles. An accurate and reliable battery model is needed in battery management systems (BMS) to monitor battery operating conditions, including state of charge (SOC), state of health (SOH), etc. This paper presents a state-space model with non-integer order derivatives for electrochemical batteries with a constant phase element (CPE) in order to accurately describe battery dynamics. The proposed model is a combination of electrochemical impedance spectroscopy and the 1-RC model. The Oustaloup recursive approximation was selected for model parametric identification and potential implementation. A particle-swarm optimization (PSO) algorithm was used to identify three model parameters by using time-domain test data. The model accuracy and robustness were validated by using datasets from different driving cycles, aging levels and cells of the same chemistry. The proposed FOM showed good accuracy and robustness. It is suitable for research on battery reliability, including issues like SOC estimation, SOH prediction, and charging control.

Introduction

Chemistry-based energy storage systems are critical for sustainable road transportation. Today, lithium ion batteries have attracted wide attention due to their high energy density, long cycle life, and environmental friendliness [1]. To ensure safety and extend battery life, accurate and reliable battery models should be used in battery management systems (BMS) to monitor conditions such as state of charge (SOC) [2], and state of health (SOH) [3].

Electrochemical models are generally more accurate than equivalent circuit-based models in describing battery dynamics. However, electrochemical models are less practical in applications because of the large number of model parameters needed, highly complex structure, and common over-fitting issue [4]. Moreover, partial differential equations (PDEs) associated with electrochemical models may require heavy storage and computation resources [4], [5], [6], [7]. As an alternative, equivalent circuit models (ECMs) are gaining popularity in BMS design because of their relatively simple structure and model parameters. ECMs are composed of electric components such as resistors, inductors, and capacitors [8], [9]. Plett et al. provided a detailed introduction to various types of ECMs evolved from a primitive model, including a simple model with only SOC as a state, and a one-state hysteresis model adding an additional “hysteresis state” to the model state equation [10]. Resistance–capacitance (RC) networks have been widely used to construct ECMs, which leads to many commonly used models, e.g., 1-RC model [11], 2-RC model [12], and 1-RC with hysteresis [13], etc. In these models, the variation of dynamics caused by temperature and SOC are modeled as parametric functions [14], [15]. Recently ECMs have been widely used in battery management, including SOC estimation [16], SOH prediction [17], and charging control [15].

One major shortcoming of ECMs is their structural mismatch with the electrochemical impedance spectroscopy (EIS) measurement. The EIS measured in small-voltage excitation can provide important details about battery internal states and degradation. For example, in a lithium ion battery the EIS is often composed of two sections in the operating frequency range: (1) a straight line in low frequency and (2) a semi-ellipse in middle frequency. The low-frequency straight tail is generally believed to be caused by mass diffusion of lithium ions, while the middle frequency semi-ellipse is caused by dynamics such as charge-transfer reaction and double-layer effect [18]. An ECM with 1-RC network has an ideal semi-circle in the EIS diagram, which obviously does not agree with measured data. High-order RC networks can improve EIS accuracy, but this inevitably results in a large number of model parameters. Fractional-order models (FOM) can address the discrepancy between the complexity and accuracy in ECMs, which uses non-integer derivatives instead of integer derivatives for states. Existing studies have demonstrated that batteries inherently possess fractional-order properties because of mass transport, diffusion dynamics, and memory hysteresis [19], [20]. With better accuracy and fewer parameters, FOMs are attracting the attention of battery researchers. For example, Xu et al. proposed a fractional Kalman filter for SOC estimation [20], where the model was structured with a Warburg element and a parallel circuit consisting of a constant phase element (CPE) with resistance. The fractional order of the Warburg element was fixed at 0.5 and the other model parameters were identified using the least-squares method in the time domain. Sabatier et al. proposed a fractional-order model using a Warburg element to estimate the crankability of lead–acid batteries with a fractional order fixed at 0.5 [21]. However, some researchers have demonstrated that optimal fractional order is not 0.5 and that fixing the fractional order may decrease model accuracy [22].

The main contribution of this paper is to take advantage of fractional order modeling method to describe an electrochemistry-based system and to derive a state-space model with non-integer derivative using a structure similar to the commonly used resistor–capacitor model for batteries. This modeling method has the potential to mimic the input–output behavior of complex electrochemical dynamics inside a battery with fewer model parameters than a physics-based model. The proposed FOM is structured with an ohmic resistor and a parallel circuit consisting of a CPE and a resistor. The ohmic resistor is used to depict the resistive impedance, and the parallel circuit is mainly used to describe the diffusion dynamics. The high complexity of electrochemical models and the structural mismatch of ECMs could be mitigated. The fractional order is optimized as the variant value with the respect to the different SOC ranges to maintain the model accuracy. The performance of the FOM has been validated by the battery testing data at the different aging levels. The remainder of the paper is organized as follows: Section 2 reviews the fractional-order calculus; Section 3 derives the fractional-order model for lithium ion batteries and introduces the Oustaloup recursive approximation process of fractional-order operator; Section 4 identifies the model parameters using time-domain test data; and Section 5 discusses the model accuracy and robustness, followed by the conclusion in Section 6.

Section snippets

Fractional-order calculus

The introduction of fractional-order calculus (FOC) is usually associated with Leibniz and L’Hospital in the late of the 17th century [23]. Many real-world physical systems, such as electrochemical, biological, material, and viscoelastic systems display fractional-order dynamics, because their behavior is governed by fractional-order differential equations. For example, in electricity, if at least one of the electrodes has a rough surface, the current through a capacitor is proportional to the

Fractional-order model for a lithium ion battery

The electrode process of a lithium ion battery is not a simple chemical reaction, but rather a series of processes with different phenomenon. Typical battery dynamics include charge-transfer reaction on electrode/electrolyte interface, lithium ion diffusion in electrodes, double-layer effects, and resistance/capacitance growth of the anode-insulating film, etc. [22], [18]. Therefore, acquiring an accurate mathematical model to describe the working behaviors of the battery is of critical

Model parameter identification and discussion

This section discusses the battery model parameters based on the testing data and discusses the results.

Conclusion

A fractional-order model structured with an ohmic resistor and a parallel circuit consisting of a CPE and a resistor was proposed to describe battery dynamics. The derived model with non-integer derivative was a combination of the electrochemical impedance spectroscopy and 1-RC model and adopts similar a structure as the commonly used resistor–capacitor model. The Oustaloup recursive approximation characterized by its easy implementation was introduced to model the fractional differentiation

Acknowledgments

This research work is supported by NSF China with Grants 51375044 and 51205228 and University Talent Introduction 111 Project B12022.

References (39)

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