{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T14:43:59Z","timestamp":1775832239271,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,1]],"date-time":"2023-01-01T00:00:00Z","timestamp":1672531200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Research Foundation of Education Bureau of Zhejiang Province","award":["Y202249835"],"award-info":[{"award-number":["Y202249835"]}]},{"name":"Research Foundation of Education Bureau of Zhejiang Province","award":["U1909217"],"award-info":[{"award-number":["U1909217"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["Y202249835"],"award-info":[{"award-number":["Y202249835"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U1909217"],"award-info":[{"award-number":["U1909217"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The state of charge (SOC) for a lithium-ion battery is a key index closely related to battery performance and safety with respect to the power supply system of electric vehicles. The Kalman filter (KF) or extended KF (EKF) is normally employed to estimate SOC in association with the relatively simple and fast second-order resistor-capacitor (RC) equivalent circuit model for SOC estimations. To improve the stability of SOC estimation, a two-stage method is developed by combining the second-order RC equivalent circuit model and the eXogenous Kalman filter (XKF) to estimate the SOC of a lithium-ion battery. First, approximate SOC estimation values are observed with relatively poor accuracy by a stable observer without considering parameter uncertainty. Second, the poor accuracy SOC results are further fed into XKF to obtain relative stable and accurate SOC estimation values. Experiments demonstrate that the SOC estimation results of the present method are superior to those of the commonly used EKF method. It is expected that the present two-stage XKF method will be useful for the stable and accurate estimation of SOC in the power supply system of electric vehicles.<\/jats:p>","DOI":"10.3390\/s23010467","type":"journal-article","created":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T03:50:32Z","timestamp":1672631432000},"page":"467","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":37,"title":["Stable and Accurate Estimation of SOC Using eXogenous Kalman Filter for Lithium-Ion Batteries"],"prefix":"10.3390","volume":"23","author":[{"given":"Qizhe","family":"Lin","sequence":"first","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou 325035, China"}]},{"given":"Xiaoqi","family":"Li","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou 325035, China"}]},{"given":"Bicheng","family":"Tu","sequence":"additional","affiliation":[{"name":"Ebara Great Pumps Co., Ltd., Wenzhou 325200, China"}]},{"given":"Junwei","family":"Cao","sequence":"additional","affiliation":[{"name":"Ebara Great Pumps Co., Ltd., Wenzhou 325200, China"}]},{"given":"Ming","family":"Zhang","sequence":"additional","affiliation":[{"name":"Ebara Great Pumps Co., Ltd., Wenzhou 325200, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4028-985X","authenticated-orcid":false,"given":"Jiawei","family":"Xiang","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou 325035, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Romero-Ternero, M., Oviedo-Olmedo, D., Carrasco, A., and Luque, J. (2019). A distributed approach for estimating battery state-of-charge in solar farms. Sensors, 19.","DOI":"10.3390\/s19224998"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Liu, T., Zhao, J.D., Xiang, C.Q., and Cheng, S. (2022). Research on minimization of data set for state of charge prediction. Sensors, 22.","DOI":"10.3390\/s22031101"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wang, Q., Jiang, J.Y., Gao, T., and Ren, S.R. (2022). State of charge estimation of li-ion battery based on adaptive sliding mode observer. Sensors, 22.","DOI":"10.3390\/s22197678"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Koshkouei, M.J., Kampert, E., Moore, A.D., and Higgins, M.D. (2022). Impact of lithium-ion battery state of charge on in situ QAM-based power line communication. Sensors, 22.","DOI":"10.3390\/s22166144"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Lee, J.H., and Lee, I.S. (2022). Estimation of online state of charge and state of health based on neural network model banks using lithium batteries. Sensors, 22.","DOI":"10.3390\/s22155536"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.energy.2019.03.059","article-title":"State-of-charge estimation of lithium-ion batteries based on gated recurrent neural network","volume":"175","author":"Yang","year":"2019","journal-title":"Energy"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/j.jpowsour.2004.02.033","article-title":"Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs\u2014Part 3 State and parameter estimation","volume":"134","author":"Plett","year":"2019","journal-title":"J. Power Sources"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"35957","DOI":"10.1109\/ACCESS.2018.2850743","article-title":"Unscented kalman filter-based battery soc estimation and peak power prediction method for power distribution of hybrid electric vehicles","volume":"6","author":"Wang","year":"2018","journal-title":"IEEE Access"},{"key":"ref_9","first-page":"779","article-title":"Railway sleeper crack recognition based on edge detection and CNN","volume":"28","author":"Wang","year":"2021","journal-title":"Smart Struct. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3514709","DOI":"10.1109\/TIM.2022.3180416","article-title":"Machinery fault diagnosis based on domain adaptation to bridge the gap between simulation and measured signals","volume":"71","author":"Lou","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3798","DOI":"10.1109\/TMECH.2021.3132459","article-title":"Fault detection in gears by combination of numerical simulation and generative adversarial networks","volume":"27","author":"Gao","year":"2022","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Li, C.R., Xiao, F., and Fan, Y.X. (2019). An approach to state of charge estimation of lithium-ion batteries based on recurrent neural networks with gated recurrent unit. Energies, 12.","DOI":"10.3390\/en12091592"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zhang, X.H., Huang, Y., Zhang, Z.W., Lin, H.P., Zeng, Y., and Gao, M.Y. (2022). A hybrid method for state-of-charge estimation for lithium-ion batteries using a long short-term memory network combined with attention and a Kalman filter. Energies, 15.","DOI":"10.3390\/en15186745"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Qian, C., Xu, B.H., Xia, Q., Ren, Y., Yang, D.Z., and Wang, Z.L. (2022). A Dual-input neural network for online state-of-charge estimation of the lithium-ion battery throughout its lifetime. Materials, 15.","DOI":"10.3390\/ma15175933"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Wang, Y.T., Wang, S.L., Fan, Y.C., Xie, Y.X., and Fernandez, C. (2022). A novel adaptive back propagation neural network-unscented Kalman filtering algorithm for accurate lithium-ion battery state of charge estimation. Metals, 12.","DOI":"10.3390\/met12081369"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Wang, Y.C., Shao, N.C., Chen, G.W., Hsu, W.S., and Wu, S.C. (2022). State-of-charge estimation for lithium-ion batteries using residual convolutional neural networks. Sensors, 22.","DOI":"10.3390\/s22166303"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Terala, P.K., Ogundana, A.S., Foo, S.Y., Amarasinghe, M.Y., and Zang, H.Y. (2022). State of charge estimation of lithium-ion batteries using stacked encoder\u2013decoder bi-directional LSTM for EV and HEV applications. Micromachines, 13.","DOI":"10.3390\/mi13091397"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Speyer, J.L., and Chung, W.H. (2008). Stochastic Processes, Estimation, and Control, SIAM Press.","DOI":"10.1137\/1.9780898718591"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ma, X., Qiu, D.F., Tao, Q., and Zhu, D.Y. (2019). State of charge estimation of a lithium-ion battery based on adaptive Kalman filter method for an equivalent circuit model. Appl. Sci., 9.","DOI":"10.3390\/app9132765"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3458","DOI":"10.1109\/TAC.2021.3106861","article-title":"Robust kalman filtering under model uncertainty: The case of degenerate densities","volume":"67","author":"Yi","year":"2022","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"S\u00e4rkk\u00e4, S. (2013). Bayesian Filtering and Smoothing, Cambridge University Press.","DOI":"10.1017\/CBO9781139344203"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"191","DOI":"10.2166\/hydro.2016.063","article-title":"Modeling of temporal groundwater level variations based on a Kalman filter adaptation algorithm with exogenous inputs","volume":"19","author":"Varouchakis","year":"2017","journal-title":"J. Hydroinformatics"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Liu, S.L., Cui, N.X., and Zhang, C.H. (2017). An adaptive square root unscented kalman filter approach for state of charge estimation of lithium-ion batteries. Energies, 10.","DOI":"10.3390\/en10091345"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Li, M., Zhang, Y.J., Hu, Z.L., Zhang, Y., and Zhang, J. (2021). A battery SOC estimation method based on AFFRLS-EKF. Sensors, 21.","DOI":"10.3390\/s21175698"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1080\/00207179.2016.1172390","article-title":"The eXogeneous Kalman Filter (XKF)","volume":"90","author":"Johansen","year":"2017","journal-title":"Int. J. Control."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/j.automatica.2018.05.038","article-title":"Attitude estimation by multiplicative exogenous Kalman filter","volume":"95","author":"Stovner","year":"2018","journal-title":"Automatica"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"233","DOI":"10.4173\/mic.2018.4.1","article-title":"eXogenous Kalman filter (XKF) for visualization and motion prediction of ships using live automatic identification system (AIS) data","volume":"39","author":"Fossen","year":"2019","journal-title":"Model. Identif. Control."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2354","DOI":"10.1016\/j.jfranklin.2019.11.078","article-title":"Two-stage exogenous Kalman filter for time-varying fault estimation of satellite attitude control system","volume":"357","author":"Chen","year":"2020","journal-title":"J. Frankl. Inst."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Ma, L.L., Wang, F.X., Shen, W., and Wang, J.Z. (J. Electr. Eng. Technol., 2022). Fault-tolerant control based on modified eXogenous Kalman filter for PMSM, J. Electr. Eng. Technol., early access.","DOI":"10.1007\/s42835-022-01223-y"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1016\/j.jpowsour.2014.01.057","article-title":"A survey of mathematics-based equivalent-circuit and electrochemical battery models for hybrid and electric vehicle simulation","volume":"256","author":"Seaman","year":"2014","journal-title":"J. Power Sources"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2606","DOI":"10.1007\/s11771-020-4485-9","article-title":"Simulation of second-order RC equivalent circuit model of lithium battery based on variable resistance and capacitance","volume":"27","author":"Ji","year":"2020","journal-title":"J. Cent. South Univ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"5916","DOI":"10.3390\/en8065916","article-title":"State of charge estimation of lithium-ion batteries using an adaptive cubature Kalman filter","volume":"8","author":"Xia","year":"2015","journal-title":"Energies"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Xia, B.Z., Zheng, W.H., Zhang, R.F., Lao, Z.Z., and Sun, Z. (2017). A novel observer for lithium-ion battery state of charge estimation in electric vehicles based on a second-order equivalent circuit model. Energies, 10.","DOI":"10.3390\/en10081150"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Xia, B.Z., Guo, S.K., Wang, W., Lai, Y.Z., Wang, H.W., Wang, M.W., and Zheng, W.W. (2018). A state of charge estimation method based on adaptive extended Kalman-particle filtering for lithium-ion batteries. Energies, 11.","DOI":"10.3390\/en11102755"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Guo, X.W., Kang, L.Y., Yao, Y., Huang, Z.Z., and Li, W.B. (2016). Joint estimation of the electric vehicle power battery state of charge based on the least squares method and the Kalman filter algorithm. Energies, 9.","DOI":"10.3390\/en9020100"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Falai, A., Giuliacci, T.A., Misul, D., Paolieri, G., and Anselma, P.G. (2022). Modeling and on-road testing of an electric two-wheeler towards range prediction and BMS integration. Energies, 15.","DOI":"10.3390\/en15072431"},{"key":"ref_37","first-page":"92","article-title":"The co-estimation of state of charge, state of health, and state of function for lithium-ion batteries in electric vehicles","volume":"67","author":"Ping","year":"2017","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.jpowsour.2012.02.024","article-title":"Modelling of VRLA batteries over operational temperature range using pseudo random binary sequences","volume":"207","author":"Fairweather","year":"2012","journal-title":"J. Power Sources"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1586","DOI":"10.1109\/TVT.2004.832408","article-title":"High-performance battery-pack power estimation using a dynamic cell model","volume":"53","author":"Plett","year":"2004","journal-title":"IEEE Trans. Veh. Technol."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/1\/467\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:55:21Z","timestamp":1760118921000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/1\/467"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,1]]},"references-count":39,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["s23010467"],"URL":"https:\/\/doi.org\/10.3390\/s23010467","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,1]]}}}