Finding a better fit for lithium ion batteries: A simple, novel, load dependent, modified equivalent circuit model and parameterization method

Highlights

• The modified ECM uses switching RC network values.

• Novel parameter identification method with load dependence.

• Enhanced accuracy by 77.4% in drive cycle and 87.6% in constant current.

• Significant usefulness for a large format lithium iron phosphate prismatic cell.

Abstract

Equivalent circuit models (ECM) of lithium ion batteries are used in many applications because of their ease of implementation and low complexity. The accuracy of an ECM is critical to the functionality and usefulness of the battery management system (BMS). The ECM accuracy depends on the parametrization method, and therefore different experimental techniques and model parameter identification methods (PIM) have been widely studied. Yet, how to account for significant changes in time constants between operation under load and during relaxation has not been resolved. In this work a novel PIM and modified ECM is presented that increases accuracy by 77.4% during drive cycle validation and 87.6% during constant current load validation for a large format lithium iron phosphate prismatic cell. The modified ECM uses switching RC network values for each phase, which is significant for this cell and particularly at low state-of-charge for all lithium ion batteries. Different characterisation tests and the corresponding experimental data have been trained together across a complete State-of-Charge (SoC) and temperature range, which enables a smooth transition between identified parameters. Ultimately, the model created using parameters captured by the proposed PIM shows an improved model accuracy in comparison with conventional PIM techniques.

Introduction

Electrification is inevitable. The transformation from internal combustion engine (ICE) transportation to electric drive is dramatically increasing in recent years, due to aggressive policies worldwide driven by air quality, greenhouse emissions and national economic competition. Between 2030–40, a significant majority of countries and/or cities will ban the sales of ICE passenger vehicles, including China, USA, Germany, UK and many more [1,2]. As a consequence the sales of EVs have been increasing dramatically. There are over 2 million electric vehicles sold in 2018, up from just a few thousand in 2010. Bloomberg NEF forecasted that the sales of annual passenger EV will rise to 10 million in 2025, 28 million in 2030 and 56 million by 2040 [3]. During this global transport industry paradigm shift, the lithium-ion battery plays a central role in the majority of EVs [3].

While researching and developing lithium-ion batteries with new materials and manufacturing, the usage of a robust battery model is pivotal at the application level. Models enable the battery management system (BMS) to improve battery performance and prolong lifetime [4]. There are predominantly three types of models in the literature, which are data-driven models, physics-based models and equivalent circuit models (ECM) [5]. Data-driven models, such as neural networks [6] and support vector machine [7], etc., do not require a physical interpretation of the battery's internal dynamics, making it suitable for simulating complex or unknown systems. However, the model training usually relies upon a large amount of data and complex optimization, and there is no guarantee models will work under different operating conditions. The physics-based models capture the physical behaviours through solving equations such as lithium diffusion equations and charge conservation equations. Newman, Doyle, Fuller et al. established the foundations for these physics-based model [8,9]. Later, other electrochemical models have been proposed to describe different battery properties, such as capacity fade and electrode particle geometry etc. [[10], [11], [12], [13], [14], [15], [16]]. The main two drawbacks of physics-based models for BMS applications are 1. High complexity of parameterization; 2. High computational power. There are more than 30 parameters needed to be fitted and parameterised including salt concentration, electrode/separator thickness, conductivity of electrolyte etc. Quite often the parameters in the physics-based model require ex-situ experimental measurements which are time/cost inefficient [17,18]. Also, the computational speed is another disadvantage of this type of models. Further, it is challenging to scale such model into multi-dimensional or pack level analysis [14,19,20]. Although these drawbacks can be mitigated through reduced order models (ROMs) ECMs are still the model of choice for many applications.

ECMs describe the battery terminal voltage-current dynamics using passive electrical components (resistors and capacitors) and measured/parameterised look-up tables or simple mathematical functions. The ease of implementation and low model complexity make ECM feasible in real-time applications such as integrated BMS, and they are regularly embedded in microprocessors and deliver precise simulation/feedback signals in real-time [21]. The reader is referred to recent reviews by He et al. [22] and Hu et al. [23] on ECMs. A typical ECM consists of the battery open circuit voltage (OCV) and a series ohmic resistor $R_0$ and several resistor-capacitor (RC) networks [24]. The battery OCV can be measured directly from experimental data using low-rate constant current (CC) (dis)charge (giving a pseudo-OCV), or using a pulsed current (dis)charge with long rest periods between pulses (true OCV) [25]. The RC values, on the other hand, need to be identified by fitting the model's voltage prediction under current load to measurements using optimization algorithms [26,27]. This represents the model parametrization procedure.

The parameter identification method (PIM) is critical for the ECM model accuracy. The model accuracy is correlated with what and how PIM is used for the certain Li-ion battery. There are various PIMs for ECM parametrization in the literature, including genetic algorithm (GA) [28], particle swarm optimization (PSO) algorithm [29], and the least-squares method [30,31]. A recent study conducted by Lai et al. [32] compared 9 different popular PIMs for 9 different ECMs in the entire SoC area and demonstrated the importance of the PIM to the model accuracy.

The ECM parameters depend on the operating conditions. The popular methods for capturing this parameter dependency include offline parameterized look-up tables [[33], [34], [35]] and the online adaptive parameter estimation algorithms such as recursive least squares methods [26,36] and the dual Kalman filter algorithms [37]. Existing PIMs mainly focus on capturing the parameter dependence on the SoC, temperature and current directions. However, another important dependency factor for ECM parameters that is generally overlooked is the type of current loads. Different current profiles have been used for ECM parameter estimation, such as the pulsed current test (different types of pulse design as in Ref. [34,35,38]), drive cycles [39,40] and constant-current charging and discharging [[41], [42], [43]]. However, few works have considered the effect of choosing different current profiles on the identified parameters and the model accuracy [39,40]. The battery is an electrochemical system with complex internal dynamics, and ECM is an approximate reduced-order model. Therefore, different current excitations will reveal different system properties. As a result, the ECM parameters will vary under different load conditions [44,45]. Waag et al. [45] analysed the dependency of the ECM parameters on the frequency characteristics of the load current, and proposed an application-specific parametrization method by taking into consideration the frequency spectrum of the load current. However, the switching scheme between different load conditions is not addressed in Ref. [45]. Further, because of the nonlinearity of the battery dynamics, it is not straightforward to translate the frequency domain analysis to time-domain implementation. The influence of the load current profile on the ECM parameters was also studied in Ref. [40]. However, the study is limited to a few standard drive cycles that are usually used for battery characterization, and the underlying mechanism is not explained.

There is one key difference in the current load that the studies in the literature were missing, which is the difference in the underload and the relaxation. Generally the model parameters were identified without distinguishing the two different working conditions and the underload and relaxation test data are used together for ECM parametrization [34,38]. However, the test data analysis on the chosen cell in this paper shows that the battery performs distinctly differently during underload and relaxation, in terms of the scale of magnitude of time constants of the RC networks. It shows that using the same parameter set cannot capture both the underload and the relaxation dynamics accurately. Therefore, this paper proposes a parameter switching scheme between underload and relaxation working conditions to address this problem.

Further, this study delivers a novel PIM that captures ECM parameter dependence on load switching, SoC (0–100%) and temperatures (10$°C$, 20$°C$, 30$°C$ and 40$°C$). The novelty of the proposed PIM is that the time constants of the RC networks are independent from SoC and temperature. The rest parameters, i.e., the resistor values, then become linear-in-the-variable which can be readily obtained using computationally efficient least squares optimization solvers. This also enables the simultaneous estimation of all the resistor values under all SoC and temperatures levels with parameter constraints to ensure a smooth transition between different temperature levels. In theory, the battery internal resistance increases while the operating temperature decreases. However, in the conventional way of parameter identification, the resistance values might zig-zag across a range of temperatures without using parameter constraints [27]. The ultimate purpose of this novelty is to provide a temperature dependent ECM model e.g. Ref. [19], in many BMS and modelling applications, which is critical for the model accuracy. This proposed PIM shows a better fit for large current/large power/large heat generation applications. Unlike smaller cells, large format cells have significant heat generation challenges, therefore training the model within the operating temperature window is important and inevitable.

The rest of the paper is structured as follows. The ECM equations are expressed in Section 2. The experimental design for battery characterization and model validation is introduced in Section 3. Then, in Section 4 there is a detailed data analysis and the introduction of the novelties of the proposed PIM. In Section 5, the modelling results are validated against the experimental data from Section 3. Also, there is one comparative study between the proposed PIM and conventional method. Please note, in the main paper there are only experimental data and its corresponding validation results under a single thermal chamber temperature. The rest results at other temperature levels can be found in the supplementary material A. Lastly, Section 6 concludes this study.