Article Rapid Prediction of the Open-Circuit-Voltage of Lithium Ion Batteries Based on an Effective Voltage Relaxation Model Jie Yang 1, Chunyu Du 1, Ting Wang 2, Yunzhi Gao 1, Xinqun Cheng 1, Pengjian Zuo 1, Yulin Ma 1, Jiajun Wang 1, Geping Yin 1,*, Jingying Xie 2 and Bo Lei 3 1 MIIT Key Laboratory of Critical Materials Technology for New Energy Conversion and Storage, School of Chemistry and Chemical Engineering, Harbin Institute of Technology, Harbin 150001, China; [email protected] (J.Y.); [email protected] (C.D.); [email protected] (Y.G.); [email protected] (X.C.); [email protected] (P.Z.); [email protected] (Y.M.); [email protected] (J.W.) 2 Shanghai Institute of Space Power-Sources, Shanghai 200240, China; [email protected] (T.W.); [email protected] (J.X.) 3 China Southern Power Grid Co. Ltd., Guangzhou 510623, China; [email protected] * Correspondence: [email protected]; Tel.: +86-45186403807 Received: 6 November 2018; Accepted: 4 December 2018; Published: 9 December 2018 Abstract: The open circuit voltage of lithium ion batteries in equilibrium state, as a vital thermodynamic characteristic parameter, is extensively studied for battery state estimation and management. However, the time-consuming relaxation process, usually for several hours or more, seriously hinders the widespread application of open circuit voltage. In this paper, a novel voltage relaxation model is proposed to predict the final open circuit voltage when the lithium ion batteries are in equilibrium state with a small amount of sample data in the first few minutes, based on the concentration polarization theory. The Nernst equation is introduced to describe the evolution of relaxation voltage. The accuracy and effectiveness of the model are verified using experimental data on lithium ion batteries with different kinds of electrodes (LiCoO2/mesocarbon-microbead and LiFePO4/graphite) under different working conditions. The validation results show that the presented model can fit the experimental results very well and the predicted values are quite accurate by taking only 5 min or less. The satisfying results suggest that the introduction of concentration polarization theory might provide researchers an alternative model form to establish voltage relaxation models. Keywords: lithium ion batteries; open circuit voltage; concentration polarization theory; universal relaxation model; rapid prediction 1. Introduction Due to their unique advantages in energy/power density and cycle life, rechargeable lithium-ion batteries (LIBs) are preferred in energy storage devices for most applications in portable electronics, electric vehicles (EV) and grid energy storage [1–3]. In order to ensure the reliable and safe operation of LIBs, accurate online estimation of battery states, including state of charge (SOC) [4–6], state of health (SOH) [7–9] and state of energy (SOE) [10–12], is the most crucial task for a battery management system (BMS). As a vital thermodynamic characteristic parameter reflecting the internal state of LIBs, open circuit voltage (OCV) of LIBs in equilibrium state (e-OCV) plays an important role in battery state estimation and BMS since e-OCV depends on the essential properties of electrode material and electrolyte as well as the concentration of lithium intercalated in active electrode material [13–15]. Energies 2018, 11, 3444; doi:10.3390/en11123444 www.mdpi.com/journal/energies Energies 2018, 11, 3444 2 of 15 Thus, e-OCV can be employed to estimate the SOC of LIBs because the relationship between e-OCV and SOC is very apparent and nearly monotonous for most of LIBs [16,17]. For example, the e-OCV - SOC curve or the 3-dimonsional e- OCV-SOC-T surface are obtained in the lab beforehand as a look- up table for SOC estimation [18]. The e-OCV can be predicted using models or other techniques and then the SOC can be estimated based on the look-up table [19]. This method is the most commonly used correction method for accurate SOC estimation. e-OCV can also be used to estimate the SOH of LIBs because the aging of LIBs is often associated with loss of lithium inventory, degradation of active material and increase of resistance, which would inevitably affect e-OCV value [20]. Therefore, accurate and rapid estimation of e-OCV is of importance for optimum operation of LIBs. However, it may take a quite long time (generally > 3 h) for LIBs to reach a complete equilibrium state after the current interruption, as shown in Figure 1, especially when they are operated at high charge/discharge rates and in low temperature with large SOC shifts [21,22]. This shortcoming has seriously limited the application of e-OCV. To address this problem, a few approaches including semi- empirical models, pure empirical models and equivalent circuit models were presented to predict e- OCV within a relatively short time. For example, Bergveld et al. [23] proposed a pure empirical voltage relaxation fitting-model for LIBs. Unfortunately, this model can only fit a certain type of LIBs, and its accuracy is far from satisfying when being applied to other LIBs. Unterrieder et al. [24] developed a mathematical model based on nonlinear least square estimation to describe the OCV transients. The fitted results of the model for the primary phase of the voltage relaxation curves are far from satisfying. More recently, equivalent circuit models have attracted much attention for e-OCV estimation. Waag et al. [25] presented a voltage relaxation fitting method based on a fifth-order equivalent circuit model to estimate e-OCV by taking the theoretical diffusion process into consideration, and successfully applied it to fast estimation of e-OCV. Nevertheless, it is much more sophisticated and did not always fit the measured data well. Pei [13] developed a voltage relaxation model based on a parameters-varying second-order resistance-capacitance model. It takes 20 min at least to predict e-OCV, which is time consuming for real application. Figure 1. A schematic diagram of the voltage relaxation process of a LiCoO2/graphite battery after current interruption. The basic idea of the above studies is to build relaxation process models, whose parameters are determined by a presupposed algorithm using the data measured in the first few minutes. Then, e- OCV can be obtained by extrapolation. However, most of these models cannot describe the transients of the OCV relaxation behavior very well and they are either non-universal or still time-consuming which further makes the application confining. Therefore, developing a more effective and accurate e-OCV estimation model is still urgent and challenging. This work aims at developing a semi-empirical and general model that can be put into practical application to predict the e-OCV value within a much shorter time. Previous studies [21,22] show that Energies 2018, 11, 3444 3 of 15 the time-consuming voltage relaxation process is mainly governed by temperature, charge/discharge rate, aging degree and ultimately the diffusion of lithium ion. Thus, we adopt a physics-based electrochemical model to illustrate LIBs, as shown in Figure 2 [26]. Then we deduce the OCV relaxation model by incorporating the diffusion theory and electrochemical reaction kinetics. The introduction of physicochemical principle provides us an approximate and general format of the mathematical model that ensures the generality and estimation accuracy of the OCV prediction model. In Section 4, the generality and accuracy of this model are verified on LIBs for different active materials, temperatures, depth of discharge (DOD) and charge/discharge rates. Figure 2. Schematic illustration of a lithium ion battery model. The work is organized as follows: Section 1 presents a brief review of the researches which are intended to develop effective approaches to obtain the e-OCV rapidly and analyzes the urgency of establishing an alternative voltage prediction model. Section 2 describes in details the mathematical model of the voltage relaxation. The experimental setup is specified in Section 3 and the validation results and discuss are shown in Section 4. The conclusions are given in Section 5. 2. Mathematical Model of OCV Relaxation Process 2.1. Model Description It is generally known that the electric field and concentration differences promote the diffusion of lithium ion in LIBs. This paper concentrates on the voltage relaxation process after the charging/discharging of LIBs is interrupted, which means that only the concentration difference of lithium ion is taken into consideration. For our modelling, we just focus on the simulation of the voltage relaxation behavior after a few seconds of the current interruption. The voltage transients during this period are mainly dominated by concentration polarization voltage since the time constants of the ohmic voltage drop and electrochemical polarization is so small that they can be out of consideration in the following relaxation process. So we firstly divide the diffusion path of lithium ion into infinite elements. Obviously, the concentration of lithium ion between each segment and adjacent one are different, as shown in Figure 2. Based on the concentration potential theory [27], we could adopt the Nernst equation to calculate the concentration polarization voltage between two adjacent segments. Energies 2018, 11, 3444 4 of 15 c Δ=RT i = ⋅⋅⋅ Vimi ln , 1, 2 , (1) nF ci +1 where R is the universal gas constant, T is temperature, n is the number of electrons involved in the virtual electrode reaction and as for lithium ion, we take n = 1. F is Faraday’s constant and ci is the concentration of diffusible lithium ion in corresponding segments. According to the Fick’s second law, ci is a function of time t and distance x from the electrode. We here transform Equation (1) to another form with time t as the only independent variable since the concentration of the diffusible lithium ion is time-dependent before LIBs reach equilibrium state, as shown by Equation (2): RT ct() Δ=Vt() lni , i =⋅⋅⋅ 1,2 , m i () (2) Fcti+1 The segmentation treatment makes it rational that treating the LIB as battery unit which is series connected.