A state-of-charge (SOC) versus open-circuit-voltage (OCV) model developed for batteries should preferably be simple, especially for real-time SOC estimation. It should also be capable of representing different types of lithium-ion batteries (LIBs), regardless of temperature change and battery degradation. It must therefore be generic, robust and adaptive, in addition to being accurate. These challenges have now been addressed by proposing a generalized SOC-OCV model for representing a few most

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... idely used LIBs. The model is developed from analyzing electrochemical processes of the LIBs, before arriving at the sum of a logarithmic, a linear and an exponential function with six parameters. Values for these parameters are determined by a nonlinear estimation algorithm, which progressively shows that only four parameters need to be updated in real time. The remaining two parameters can be kept constant, regardless of temperature change and aging. Fitting errors demonstrated with different types of LIBs have been found to be within 0.5%. The proposed model is thus accurate, and can be flexibly applied to different LIBs, as verified by hardware-in-the-loop simulation designed for real-time SOC estimation. Energies 2016, 9, 900 2 of 16 state. It effectively reflects the concentration ratio of resultants to reactants during battery charging and discharging, and is therefore determined by the inherent electrochemical properties of the battery. This EMF is subsequently used as the approximate OCV. The SOC-OCV function is therefore representative for a particular battery, and is generally a nonlinear monotone function between SOC and OCV for all LIBs. It is hence widely used in battery management systems (BMS) for correcting SOC calculation. Specific cases can be found in [6] [7] [8] [9] , where model based estimation of battery SOC and capacity has been developed using the SOC-OCV relationship. It has also been revealed in [10] that the accuracy of the SOC-OCV curve has great influence on the SOC value estimated. The same applies to battery capacity estimation, which has commonly been relied on for state-of-health (SOH) determination. It is consequently important to determine the SOC-OCV relationship precisely, if an accurate estimation of the battery state is necessary. For this, some studies have proposed diversified methods for OCV modeling with each having distinctive pros and cons [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] . Xiong in [11] proposed a novel systematic state-of-charge estimation framework for accurately estimating SOC of the battery, where the relationship between battery SOC and OCV is highly employed. With the accurate battery model and adaptive filter based battery SOC estimator, the SOC of the battery pack can be accurately estimated. Reference [12] developed an EMF model as a function of the battery temperature, terminal voltage under open-circuit condition (not steady-state) and its slope. Its model parameters were determined from experimental data, but its accuracy gradually reduces as the battery ages. References [13, 14] next use exponential and logarithmic functions for describing the relationship among OCV, EMF and time. However, like in [12], they result in battery models with high complexity, and are therefore difficult for usage in real time. Reference [15] proposed an alternative adaptive OCV estimation method based on battery diffusion principles. This method demonstrates high accuracy with its estimated SOC and capacity, but it is complex and has difficulty in online estimation because of its many coupled and non-coupled parameters. Reference [16] then employed a dynamic hysteresis model for predicting the OCV, where a hysteresis voltage has been included in the function for SOC. This model demonstrates high accuracy with Li 4 Ti 5 O 12 (LTO), LiFePO 4 (LFP), LiMn 2 O 4 (LMO) and LiNi 1/3 Mn 1/3 Co 1/3 O 2 (LNMCO) batteries, but its OCV hysteresis is generally not suitable for real-time model updating. Reference [17] proposed an OCV model structure in which simplified hyperbolic and exponential functions are used to represent phenomenological characteristics associated with the lithium-ion intercalation/deintercalation process. The developed SOC-OCV model applying to LiFePO 4 battery demonstrated higher accuracy compared to five OCV models summarized in [18] . However, its adaptability to other types of lithium-ion batteries needs to be further investigated. Reference [19] developed another type of OCV model that generates OCV vs. SOC curves based on the electrode half-cell data, which is able to be used for battery diagnostics and prognostics, and is an effective method especially for determining the degree of battery degradation in a quantitative manner. This approach requires half-cell data and thus opening the cells to reach high accuracy, which has difficulty in real-time SOC estimate applications. This paper expands the main ideas in [10] and introduces a new model structure for the SOC-OCV relationship with some distinctive features that are especially important for model updating in real time: (1) the model uses four base functions that capture the fundamental electrochemical foundations over low, middle, and high SOC ranges; (2) it fits the experimental data for a large class of batteries of different types well, with very high accuracy; (3) it is simple and contains much fewer numbers of parameters than common existing models such as piece-wise interpolation types; (4) due to its simplicity, it becomes uniquely suitable for real-time updating on the parameter values. In other words, it is desirable for data-driven model identification, which is essential for adaptive battery management systems that can accommodate aging, environment variations, fault diagnosis, SOC estimation, and SOH monitoring. Parameters of the generalized model must next be optimized for mapping out the SOC-OCV characteristics of different LIBs. For this, a nonlinear iterative algorithm has been developed, which is beyond the concepts presented in [10] . A real-time SOC estimation algorithm is then presented Energies 2016, 9, 900 3 of 16