State-space model with non-integer order derivatives for lithium-ion battery
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)
- et al.
A data-driven based adaptive state of charge estimator of lithium-ion polymer battery used in electric vehicles
Appl Energy
(2014) - et al.
Method for estimating capacity and predicting remaining useful life of lithium-ion battery
Appl Energy
(2014) - et al.
A novel modeling methodology of open circuit voltage hysteresis for LiFePO4 batteries based on an adaptive discrete Preisach model
Appl Energy
(2015) - et al.
Cell state-of-charge inconsistency estimation for LiFePO4 battery pack in hybrid electric vehicles using mean-difference model
Appl Energy
(2013) - et al.
Experimental investigation of the lithium-ion battery impedance characteristic at various conditions and aging states and its influence on the application
Appl Energy
(2013) - et al.
A data-driven multi-scale extended Kalman filtering based parameter and state estimation approach of lithium-ion polymer battery in electric vehicles
Appl Energy
(2014) - et al.
State-of-charge and capacity estimation of lithium-ion battery using a new open-circuit voltage versus state-of-charge
J Power Sour
(2008) Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification
J Power Sour
(2004)- et al.
Online estimation of model parameters and state-of-charge of LiFePO4 batteries in electric vehicles
Appl Energy
(2012) - et al.
Linear parameter varying battery model identification using subspace methods
J Power Sour
(2011)
A comparative study of equivalent circuit models for li-ion batteries
J Power Sour
Charging time and loss optimization for LiNMC and LiFePO4 batteries based on equivalent circuit models
J Power Sour
Fund Battery Dyn
J Power Sour
Fractional system identification for lead acid battery state of charge estimation
Signal Process
A new method to estimate the state of charge of lithium-ion batteries based on the battery impedance model
J Power Sour
A fractional order model for lead–acid battery crankability estimation
Commun Nonlinear Sci Numer Simulat
An electrochemical-based impedance model for lithium-ion batteries
J Power Sour
An approximate method for numerically solving fractional order optimal control problems of general form
Comput Math Appl
Two direct Tustin discretization methods for fractional-order differentiation/integrator
J Franklin Inst
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2023, Communications in Nonlinear Science and Numerical SimulationCitation Excerpt :As for LIBs, the FO models were proven to capture the system characteristics at high fidelity, while keeping reasonable calculation requirements [17–19]. In recent years, FO models have been widely used in battery state estimation and reliability analysis [20–25]. It should be pointed out that all the FO observers mentioned previously are independent of the order of the system.