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Accurate modeling of the nonlinear relationship between the open circuit voltage (OCV) and the state of charge (SOC) is required for adaptive SOC estimation during the lithium-ion (Li-ion) battery operation. Online SOC estimation should meet several constraints, such as the computational cost, the number of parameters, as well as the accuracy of th...

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... extraction procedure is then repeated for each pulse response to extract the first-order model parameters as demonstrated in Figure 3. However, the detected OCV points in the charge and discharge phases, as illustrated in Figure 4, are not exactly superimposed. Indeed, the OCV curve presents a hysteresis that is well reported in the literature [30,[46][47][48]. ...

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## Citations

... As the discharge pulses are carried out at 0.2 C and the time that each pulse lasts is known, the percentage of charge remaining after each pulse can be estimated. The points marked with a star indicate the v batt value, which better approximates the corresponding v OC located in the voltage stabilization point [33]. The data obtained allow the approximation of the relationship between the v SOC and the v OC , as shown in Table 2 and displayed in Figure 3 in triangles. ...

The knowledge of battery aging is an indicator that allows controlling the performance of large battery banks. State of Health (SOH) is typically the metric used, encompassing all possible mechanisms in a percentage indicator, with the Coulomb Counting as the most common method. Hence, an in-depth study of aging based on known models provides proper information for correctly managing batteries. This article proposes an aging-sensitive 3-RC-array-equivalent electrical circuit model to characterize the behavior of batteries throughout their useful life, identifying parametric changes as complementary information to the state of health. This model was validated based on experimental tests with 2 V and 6 Ah VRLA batteries aged according to the manufacturer’s recommended use. The results reveal a proportionality through capacity degradation. Then, a control group of batteries was subjected to overcharge and over-discharge conditions. The information given by Coulomb Counting SOH and the proposed method were evaluated. The proposed method provides additional information to the SOH, enhancing the distinguishing capability between typical aging performance and misused aging performance, resulting in a useful tool capable of identifying the aging associated with parametric changes in a time-invariant system where aging is treated as an imminent multiplicative fault.

... This approach deviates from employing the hybrid pulse power characterization (HPPC) test, as the study exclusively focuses on the standard C-rate. Baccouche et al.'s research [14] emphasizes accurate modeling of the nonlinear OCV-SoC relationship crucial for adaptive Li-ion battery operation. This model, employing five parameters within double exponential and quadratic functions, closely aligns with experimental curves, boasting a mere 1 mV fitting error. ...

... Rigorous experiments validate the model's reliability and precision across diverse loads and temperatures. 759 Figure 2 illustrates the experimental OCV discharge test [14]. The test involves initially fully charging the battery, followed by discharging current pulses equivalent to 5% of the SoC step. ...

... Incorporating the nonlinear state transition function ( , ) and measurement function ( , ), where x_k represents state variables and u_k denotes input variables, the state equation can be formulated. Additionally, considering measurement noises, we can further express in (14). ...

This paper delves into the critical aspect of managing energy consumption in drone operations to achieve the utmost range and ensure accurate state of charge (SoC) estimation. Effective energy management is pivotal in determining the operational range of drones, allowing for longer distances and heavier payloads. The integration of precise energy estimation algorithms into operational planning extends the range of drones, facilitating swift, environmentally-conscious missions for sustainable and efficient logistics solutions. The paper introduces a mathematical model to understand energy consumption and battery behavior in drones, utilizing the hybrid pulse power characterization test and recursive least square with forgetting factor for parameter identification. To overcome the limitations of linear filters, the paper employs the accurate extended Kalman filter (EKF) in the nonlinear filter section. The EKF significantly enhances the battery management system by furnishing precise SoC data. The study evaluates two SoC estimation techniques: SoC-AH (ampere-hours) and SoC_EKF, using root mean square error for comparison. The SoC_EKF technique demonstrates higher accuracy, boasting a lower errors value of 0.78%, thus making it superior for precise drone battery SoC estimation. These findings contribute to the improved performance, reliability, and overall safety of drones.

... Therefore, the second-order Thevenin model is selected in this paper, and the model is shown in Figure 1. Based on the definition of SoC, the SoC variation of the battery can be expressed in Equation (1) [22]: and defines the discharge current as negative and the charge current as positive. Based on the definition of SoC, the SoC variation of the battery can be expressed in Equation (1) [22]: where SoC t represents the SoC value of the battery at the moment t, SoC 0 represents the value of SoC in the initial state, η is the coulombic efficiency, Q n represents the single rated capacity of the battery [23], I L (t) represents the current value at the moment t, and defines the discharge current as negative and the charge current as positive. ...

... Based on the definition of SoC, the SoC variation of the battery can be expressed in Equation (1) [22]: and defines the discharge current as negative and the charge current as positive. Based on the definition of SoC, the SoC variation of the battery can be expressed in Equation (1) [22]: where SoC t represents the SoC value of the battery at the moment t, SoC 0 represents the value of SoC in the initial state, η is the coulombic efficiency, Q n represents the single rated capacity of the battery [23], I L (t) represents the current value at the moment t, and defines the discharge current as negative and the charge current as positive. Based on Kirchhoff's law, and incorporating the change in battery SoC versus time, the dynamic equation of the equivalent circuit can be expressed as follows [24]: ...

With the application of lithium batteries more and more widely, in order to accurately estimate the state of charge (SoC) of the battery, this paper uses the iterated extended Kalman filter (IEKF) algorithm to estimate the SoC. The Levenberg–Marquardt (LM) method is used to optimize the error covariance matrix of IKEF. Based on the hybrid pulse power characteristics experiment, a second-order Thevenin model with variable parameters is established on the MATLAB platform. The experimental results show that the proposed model is effective under the constant current discharge condition, the Federal Urban Driving Schedule (FUDS) condition, and the Beijing dynamic stress test (BJDST) condition. The results show that the simulation error of the improved LM-IEKF algorithm is less than 2% under different working conditions, which is lower than that of the IKEF algorithm. The improved algorithm has a fast convergence speed to the true value, and it has a good estimation accuracy in the case of large changes in external input current. Additionally, the fluctuation of error is relatively stable, which proves the reliability of the algorithm.

... The higher the degree, the greater the accuracy, but it also leads to an increase in the number of parameters that need to be tuned. Logarithmic functions [34][35][36] and exponential functions [37] are other widely employed functions. The former have only three parameters to be found and exhibit good accuracy, but they cannot approach the limit values of the SOC, i.e., zero and one. ...

... They obtained an accurate model but with twelve parameters. A solution that reconciles good accuracy and low complexity is proposed in [37]. In this case, the OCV-SOC curve was modeled through two exponential terms and one square root term with only five parameters. ...

Currently, the urgent needs of sustainable mobility and green energy generation are driving governments and researchers to explore innovative energy storage systems. Concurrently, lithium-ion batteries are one of the most extensively employed technologies. The challenges of battery modeling and parameter estimation are crucial for building reliable battery management systems that ensure optimal battery performance. State of charge (SOC) estimation is particularly critical for predicting the available capacity in the battery. Many methods for SOC estimation rely on the knowledge of the open-circuit voltage (OCV) curve. Another significant consideration is understanding how these curves evolve with battery degradation. In the literature, the effect of cycle aging on the OCV is primarily addressed through the look-up tables and correction factors applied to the OCV curve for fresh cells. However, the variation law of the OCV curve as a function of the battery cycling is not well-characterized. Building upon a simple analytical function with five parameters proposed in the prior research to model the OCV as a function of the absolute state of discharge, this study investigates the dependency of these parameters on the moved charge, serving as an indicator of the cycling level. Specifically, the analysis focuses on the impact of cycle aging in the low-, medium-, and high-SOC regions. Three different cycle aging tests were conducted in these SOC intervals, followed by the extensive experimental verification of the proposed model. The results were promising, with mean relative errors lower than 0.2% for the low- and high-SOC cycling regions and 0.34% for the medium-SOC cycling region. Finally, capacity estimation was enabled by the model, achieving relative error values lower than 1% for all the tests.

... Open-circuit voltage techniques necessitate maintaining a static state for an extended period to obtain the corresponding SOC curve [6], while Coulomb counting methods tend to accumulate errors in the initial SOC estimation [7]. More sophisticated approaches, including adaptive filtering algorithms such as the Kalman Filtering algorithm [8], Particle Filter algorithm [9]- [10], and equivalent circuit models, Electrochemical Impedance Spectroscopy, EMF Method and Model Based methods [11], have been utilized to estimate SOC based on battery voltage and current data. However, these methods often require additional parameters or a reliable battery model to enhance estimation accuracy. ...

The interest to electrify all modes of transportation has increased over these past years. Their source of power is primarily a systems of energy storage like Battery packs. The SOC of a battery indicates the amount of charge stored in the battery and is a critical parameter for its safe and efficient operation of the electric vehicles. However, it's not possible to be measured directly, but can only be estimated from related measurable variables. Data-driven methods have gained significant attention in recent years due to their ability to estimate SOC accurately, even under dynamic operating conditions. This paper provides a data-driven method for SOC estimation based on the Temporal Convolutional neural network (TCN) approach. Index Terms-Electric Vehicle (EV), State of charge (SOC) estimation, Temporal Convolution Network (TCN).

... Extended Kalman filter [16,17,[119][120][121][122][123][124][125]: This represents a classical approach to state observation for dynamic systems that effectively converts (locally) a nonlinear system into a linear one. This transformation is achieved by computing the first-order Taylor series expansion, specifically the Jacobian matrix, around the estimated operating point at each time step. ...

Lithium-ion batteries are widely recognised as the leading technology for electrochemical energy storage. Their applications in the automotive industry and integration with renewable energy grids highlight their current significance and anticipate their substantial future impact. However, battery management systems, which are in charge of the monitoring and control of batteries, need to consider several states, like the state of charge and the state of health, which cannot be directly measured. To estimate these indicators, algorithms utilising mathematical models of the battery and basic measurements like voltage, current or temperature are employed. This review focuses on a comprehensive examination of various models, from complex but close to the physicochemical phenomena to computationally simpler but ignorant of the physics; the estimation problem and a formal basis for the development of algorithms; and algorithms used in Li-ion battery monitoring. The objective is to provide a practical guide that elucidates the different models and helps to navigate the different existing estimation techniques, simplifying the process for the development of new Li-ion battery applications.

... A measurement-based OCV characteristic is too irregular to be directly applied in the cell mathematical model and must be approximated [11,70,[72][73][74][75][76]. Choice of the appropriate approximating function is a further problem. ...

... The SOC value corresponding to the voltage obtained this way is calculated in reference to the total charge of all HPPC tests, that is, the Q value from the last row of Table 3. A measurement-based OCV characteristic is too irregular to be directly applied in the cell mathematical model and must be approximated [11,70,[72][73][74][75][76]. Choice of the appropriate approximating function is a further problem. ...

... Choice of the appropriate approximating function is a further problem. Several types of functions were tested, but a log-linear exponential (LLE) function [11,75] gave the best result [63]. The LEE function has the following form: ...

The aim of this research was to create an accurate simulation model of a lithium-ion battery cell, which will be used in the design process of the traction battery of a fully electric load-hull-dump vehicle. Discharge characteristics tests were used to estimate the actual cell capacity, and hybrid pulse power characterization (HPPC) tests were used to identify the Thevenin equivalent circuit parameters. A detailed description is provided of the methods used to develop the HPPC test results. Particular emphasis was placed on the applied filtration and optimization techniques as well as the assessment of the quality and the applicability of the acquired measurement data. As a result, a simulation model of the battery cell was created. The article gives the full set of parameter values needed to build a fully functional simulation model. Finally, a charge-depleting cycle test was performed to verify the created simulation model.

... Şekil 4 ve 5 oluşturulurken hata kareleri ortalamasının karekökü (RMSE) değeri kullanılmıştır (Denk. (8)). ...

Küresel ısınma ve iklim değişkenliğinin başat aktörlerinden birisi fosil yakıtların tüketilmesidir. Özellikle ulaşım endüstrisi yüksek miktarda petrol ürün gereksinimlerinden dolayı bu duruma önemli ölçüde sebep olmaktadır. Bu noktada elektrikli araçlar bir alternatif olarak düşünülmektedir. Gelişen batarya teknolojisi, özellikle de Lityum iyon (Li-ion) bataryaların geliştirilmesi ile fosil yakıtlı araçlardan elektrikli araçlara geçiş mümkün olabilmektedir. Li-ion bataryaların yüksek enerji yoğunluğu, düşükboşta deşarj etkinliği, hafıza etkisinin olmaması, hızlı şarja imkan vermeleri, uzun çevrim ömrü gibi özellikleri sayesinde elektrikli araç teknolojisinde tercih sebebi olmaktadır. Bu avantajlarının yanı sıra elektrikli araç endüstrisinde bu türde bataryaların kullanılmasıyla ortaya çıkan bazı problemlerde bulunmaktadır. Bunların en başında kalan enerji miktarını gösteren şarj durum gösterge (SoC) değerinin bulunmasıdır. Bu değerin doğru bir şekilde elde edilmesi hem batarya yönetim sistemi (BMS) için önemli bir koşul hem de kullanıcının sürüş konforu açısından vazgeçilmez bir unsurdur. SoC değerinin doğru bir şekilde elde edilmesi için kimyasal, analitik, algoritmik birçok yöntem kullanılmaktadır. Kalman filtresi tabanlı yöntemler ise SoC değeri hesabında akla gelen ilk metotlardandır. Yinelemeli yapıları sayesinde SoC değerinin gerçek değere yakın bir şekilde tahmini mümkün olmaktadır. Ancak bu filtrelerde gürültü değerlerinin SoC değişimi üzerine etkileri olabilmektedir. Bu çalışmada bu türde bir etkinin nasıl olacağı irdelenmiştir. Gürültü değerinin pozitif veya negatif yönde belirli aralıklarda değişimi için SoC ve terminal gerilim değerlerinin değişimi ele alınarak sonuçlar Turnigy Graphene 5Ah bataryası için değerlendirilmiştir.

... However, they require many parameters to be fitted and may exhibit incorrect trends outside the range or between the experimental points. Other analytical models, which are possible to find in the literature, are based on logarithmic functions, also called Nernst models [15,21,22] or exponential functions [23]. Logarithmic functions offer a good accuracy with only three parameters to be fitted but cannot be defined for an SOC equal to 0 or 1. ...

... Moreover, there are a lot of different combinations of the aforementioned functions [24-28], some of which yield higher accuracy than others. Among them, the model proposed in [23] demonstrates high accuracy and low complexity. The latter is composed of two exponentials and a quadratic term, with a total of five parameters. ...

In recent years, lithium-ion batteries (LiBs) have gained a lot of importance due to the increasing use of renewable energy sources and electric vehicles. To ensure that batteries work properly and limit their degradation, the battery management system needs accurate battery models capable of precisely predicting their parameters. Among them, the state of charge (SOC) estimation is one of the most important, as it enables the prediction of the battery’s available energy and prevents it from operating beyond its safety limits. A common method for SOC estimation involves utilizing the relationship between the state of charge and the open circuit voltage (OCV). On the other hand, the latter changes with battery aging. In a previous work, the authors studied a simple function to model the OCV curve, which was expressed as a function of the absolute state of discharge, q, instead of SOC. They also analyzed how the parameters of such a curve changed with the cycle aging. In the present work, a similar analysis was carried out considering the calendar aging effect. Three different LiB cells were stored at three different SOC levels (low, medium, and high levels) for around 1000 days, and an analysis of the change in the OCV-q curve model parameters with the calendar aging was performed.

... Independently to the method of how they have been acquired, OCV values obtained directly from measurements contain irregularities, which makes them not suitable for creating simulation models directly, so these require approximation. The approximation can be carried out using functions of various forms [3,[22][23][24][25][26][27]. Some of them are reviewed and compared in this article. ...

... A single cycle reduces the battery charge by 0.033 Q n ; therefore, a full discharge requires about 30 cycles. Other similar cycle-based test profiles used for battery model verification are DST (dynamic stress test) [1,30,34,56], ARTEMIS [38,40], and others [2,9,26,39,[57][58][59]. ...

... OCV was calculated as a mean value of recorded U batt voltage over 10 s. period before each HPPC impulse. A similar strategy of OCV measurement, but based on pulse charge and discharge tests, is presented in [15,26]. This set of data was used for OCV vs. SOC characteristic approximation function parameter identification. ...

The article describes the results of research aimed at identifying the parameters of the equivalent circuit of a lithium-ion battery cell, based on the results of HPPC (hybrid pulse power characterization) tests. The OCV (open circuit voltage) characteristic was determined, which was approximated using functions of various types, while making their comparison. The internal impedance of the cell was also identified in the form of a Thevenin RC circuit with one or two time constants. For this purpose, the HPPC pulse transients were approximated with a multi-exponential function. All of the mentioned approximations were carried out using an original method developed for this purpose, based on the PSO (particle swarm optimization) algorithm. As a result of the optimization experiments, the optimal configuration of the PSO algorithm was found. Three different cognition methods have been analyzed here: GB (global best), LB (local best), and FIPS (fully informed particle swarm). Three different swarm topologies were used: ring lattice, von Neumann, and FDR (fitness distance ratio). The choice of the cognition factor value was also analyzed, in order to provide a proper PSO convergence. The identified parameters of the cell model were used to build simulation models. Finally, the simulation results were compared with the results of the laboratory CDC (charge depleting cycle) test.