{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T17:12:28Z","timestamp":1776100348804,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,3]],"date-time":"2023-02-03T00:00:00Z","timestamp":1675382400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U21A20146"],"award-info":[{"award-number":["U21A20146"]}],"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":["1908085MF215"],"award-info":[{"award-number":["1908085MF215"]}],"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":["201904a05020007"],"award-info":[{"award-number":["201904a05020007"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003995","name":"Natural Science Foundation of AnHui Province","doi-asserted-by":"publisher","award":["U21A20146"],"award-info":[{"award-number":["U21A20146"]}],"id":[{"id":"10.13039\/501100003995","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003995","name":"Natural Science Foundation of AnHui Province","doi-asserted-by":"publisher","award":["1908085MF215"],"award-info":[{"award-number":["1908085MF215"]}],"id":[{"id":"10.13039\/501100003995","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003995","name":"Natural Science Foundation of AnHui Province","doi-asserted-by":"publisher","award":["201904a05020007"],"award-info":[{"award-number":["201904a05020007"]}],"id":[{"id":"10.13039\/501100003995","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100017668","name":"Key Research and Development Project of Anhui Province","doi-asserted-by":"publisher","award":["U21A20146"],"award-info":[{"award-number":["U21A20146"]}],"id":[{"id":"10.13039\/501100017668","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100017668","name":"Key Research and Development Project of Anhui Province","doi-asserted-by":"publisher","award":["1908085MF215"],"award-info":[{"award-number":["1908085MF215"]}],"id":[{"id":"10.13039\/501100017668","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100017668","name":"Key Research and Development Project of Anhui Province","doi-asserted-by":"publisher","award":["201904a05020007"],"award-info":[{"award-number":["201904a05020007"]}],"id":[{"id":"10.13039\/501100017668","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The secure operation of smart grids is closely linked to state estimates that accurately reflect the physical characteristics of the grid. However, well-designed false data injection attacks (FDIAs) can manipulate the process of state estimation by injecting malicious data into the measurement data while bypassing the detection of the security system, ultimately causing the results of state estimation to deviate from secure values. Since FDIAs tampering with the measurement data of some buses will lead to error offset, this paper proposes an attack-detection algorithm based on statistical learning according to the different characteristic parameters of measurement error before and after tampering. In order to detect and classify false data from the measurement data, in this paper, we report the model establishment and estimation of error parameters for the tampered measurement data by combining the the k-means++ algorithm with the expectation maximization (EM) algorithm. At the same time, we located and recorded the bus that the attacker attempted to tamper with. In order to verify the feasibility of the algorithm proposed in this paper, the IEEE 5-bus standard test system and the IEEE 14-bus standard test system were used for simulation analysis. Numerical examples demonstrate that the combined use of the two algorithms can decrease the detection time to less than 0.011883 s and correctly locate the false data with a probability of more than 95%.<\/jats:p>","DOI":"10.3390\/s23031683","type":"journal-article","created":{"date-parts":[[2023,2,3]],"date-time":"2023-02-03T01:40:25Z","timestamp":1675388425000},"page":"1683","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["Detection of False Data Injection Attacks in Smart Grids Based on Expectation Maximization"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7332-7096","authenticated-orcid":false,"given":"Pengfei","family":"Hu","sequence":"first","affiliation":[{"name":"School of Electrical Engineering, Anhui Polytechnic University, Wuhu 241000, China"},{"name":"Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Chinese Ministry of Education, Wuhu 241000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2679-4641","authenticated-orcid":false,"given":"Wengen","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Anhui Polytechnic University, Wuhu 241000, China"},{"name":"Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Chinese Ministry of Education, Wuhu 241000, China"}]},{"given":"Yunfei","family":"Li","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Anhui Polytechnic University, Wuhu 241000, China"},{"name":"Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Chinese Ministry of Education, Wuhu 241000, China"}]},{"given":"Minghui","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Anhui Polytechnic University, Wuhu 241000, China"},{"name":"Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Chinese Ministry of Education, Wuhu 241000, China"}]},{"given":"Feng","family":"Hua","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Anhui Polytechnic University, Wuhu 241000, China"},{"name":"Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Chinese Ministry of Education, Wuhu 241000, China"}]},{"given":"Lina","family":"Qiao","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Anhui Polytechnic University, Wuhu 241000, China"},{"name":"Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Chinese Ministry of Education, Wuhu 241000, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Abur, A., and Exposito, A.G. (2004). Power System State Estimation: Theory and Implementation, CRC Press.","DOI":"10.1201\/9780203913673"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1109\/TPWRD.1986.4308016","article-title":"Mutiple bad data identwication for state estimation by combinatorial oftimization","volume":"1","author":"Monticelli","year":"1986","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"806","DOI":"10.1016\/j.epsr.2007.05.021","article-title":"Identification of interacting bad data in the framework of the weighted least square method","volume":"78","author":"Granelli","year":"2008","journal-title":"Electr. Power Syst. Res."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Harvey, M., Long, D., and Reinhard, K. (March, January 28). Visualizing nistir 7628, guidelines for smart grid cyber security. Proceedings of the 2014 Power and Energy Conference at Illinois (PECI), Champaign, IL, USA.","DOI":"10.1109\/PECI.2014.6804566"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Zanero, S. (2018, January 28\u201331). When cyber got real: Challenges in securing cyber-physical systems. Proceedings of the 2018 IEEE Sensors, New Delhi, India.","DOI":"10.1109\/ICSENS.2018.8589798"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1836","DOI":"10.1109\/TPWRS.2008.2002298","article-title":"Vulnerability assessment of cybersecurity for SCADA systems","volume":"23","author":"Ten","year":"2008","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1109\/MSP.2010.49","article-title":"Smart-grid security issues","volume":"8","author":"Khurana","year":"2010","journal-title":"IEEE Secur. Priv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1109\/JPROC.2011.2161428","article-title":"Cyber\u2013physical security of a smart grid infrastructure","volume":"100","author":"Mo","year":"2012","journal-title":"Proc. IEEE"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Teixeira, A., Amin, S., Sandberg, H., Johansson, K.H., and Sastry, S.S. (2010, January 15\u201317). Cyber security analysis of state estimators in electric power systems. Proceedings of the 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, USA.","DOI":"10.1109\/CDC.2010.5717318"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Metke, A.R., and Ekl, R.L. (2010, January 19\u201321). Smart grid security technology. Proceedings of the 2010 Innovative Smart Grid Technologies (ISGT), Gaithersburg, MD, USA.","DOI":"10.1109\/ISGT.2010.5434760"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Liu, Y., Reiter, M.K., and Ning, P. (2009, January 9\u201313). False data injection attacks against state estimation in electric power grids. Proceedings of the 2009 ACM Conference on Computer and Communications Security (CCS), Chicago, IL, USA.","DOI":"10.1145\/1653662.1653666"},{"key":"ref_12","first-page":"157","article-title":"Power system state estimation based on network attack node credibility","volume":"39","author":"Xie","year":"2018","journal-title":"Chin. J. Sci. Instrum."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1922","DOI":"10.1109\/TPWRS.2020.3026951","article-title":"Power systems decomposition for robustifying state estimation under cyber attacks","volume":"36","author":"Ahmadi","year":"2021","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Jia, L., Thomas, R.J., and Tong, L. (2012, January 4\u20137). Impacts of malicious data on real-time price of electricity market operations. Proceedings of the Hawaii International Conference on System Sciences, Maui, HI, USA.","DOI":"10.1109\/HICSS.2012.313"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1109\/TSG.2011.2161892","article-title":"Integrity data attacks in power market operations","volume":"2","author":"Xie","year":"2011","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_16","unstructured":"Choi, D.H., and Xie, L. (2012, January 5\u20138). Malicious ramp-induced temporal data attack in power market with look-ahead dispatch. Proceedings of the 2012 IEEE Third International Conference on Smart Grid Communications (SmartGridComm), Tainan, Taiwan."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1109\/TSG.2011.2123925","article-title":"Modeling load redistribution attacks in power systems","volume":"2","author":"Yuan","year":"2011","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5349","DOI":"10.1109\/TSG.2021.3106246","article-title":"Targeted false data injection attacks against AC state estimation without network parameters","volume":"12","author":"Du","year":"2021","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1626","DOI":"10.1109\/TSG.2020.3033520","article-title":"Network parameter coordinated false data injection attacks against power system AC state estimation","volume":"12","author":"Liu","year":"2021","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1109\/TII.2022.3172688","article-title":"False data injection enabled network parameter modifications in power systems: Attack and detection","volume":"19","author":"Liu","year":"2022","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3919","DOI":"10.1109\/TSG.2018.2843721","article-title":"Detection and characterization of intrusions to network parameter data in electric power systems","volume":"10","author":"Molzahn","year":"2019","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2476","DOI":"10.1109\/TSG.2015.2388545","article-title":"Detecting false data injection attacks in AC state estimation","volume":"6","author":"Chaojun","year":"2015","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1109\/TII.2017.2720726","article-title":"Joint-transformation-based detection of false data injection attacks in smart grid","volume":"14","author":"Singh","year":"2018","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2892","DOI":"10.1109\/TII.2018.2875529","article-title":"Detecting false data injection attacks against power system state estimation with fast go-decomposition approach","volume":"15","author":"Li","year":"2019","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2318","DOI":"10.1109\/TSG.2022.3141803","article-title":"A highly discriminative detector against false data injection attacks in AC state estimation","volume":"13","author":"Cheng","year":"2022","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Chen, Y., Hayawi, K., Zhao, Q., Mou, J., Yang, L., Tang, J., Li, Q., and Wen, H. (2022). Vector auto-regression-based false data injection attack detection method in edge computing environment. Sensors, 22.","DOI":"10.3390\/s22186789"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Almasabi, S., Alsuwian, T., Javed, E., Irfan, M., Jalalah, M., Aljafari, B., and Harraz, F.A. (2021). A novel technique to detect false data injection attacks on phasor measurement units. Sensors, 21.","DOI":"10.3390\/s21175791"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3271","DOI":"10.1109\/TII.2018.2825243","article-title":"Online False Data Injection Attack Detection with Wavelet Transform and Deep Neural Networks","volume":"14","author":"Yu","year":"2018","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"31762","DOI":"10.1109\/ACCESS.2019.2902910","article-title":"Detection of False Data Injection Attacks in Smart Grid Utilizing ELM-Based OCON Framework","volume":"7","author":"Xue","year":"2019","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Almasabi, S., Alsuwian, T., Awais, M., Irfan, M., Jalalah, M., Aljafari, B., and Harraz, F.A. (2022). False Data Injection Detection for Phasor Measurement Units. Sensors, 22.","DOI":"10.3390\/s22093146"},{"key":"ref_31","first-page":"102844","article-title":"Ensemble unsupervised autoencoders and Gaussian mixture model for cyberattack detection","volume":"59","author":"An","year":"2022","journal-title":"Inf. Process. Manag. Libr. Inf. Retr. Syst. Commun. Netw. Int. J."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Sheng, T., Wu, W., Sun, H., Wang, Z., Sun, Q., and Ma, J. (2018, January 22\u201325). A fully distributed topology identification approach for active distribution network based on multi-agent framework. Proceedings of the 2018 IEEE Innovative Smart Grid Technologies-Asia (ISGT Asia), Singapore.","DOI":"10.1109\/ISGT-Asia.2018.8467917"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"22713","DOI":"10.1109\/ACCESS.2017.2756844","article-title":"State estimation in smart distribution system with low-precision measurements","volume":"5","author":"Chen","year":"2017","journal-title":"IEEE Access"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1109\/MCOM.2017.1700180","article-title":"Defense mechanisms against data injection attacks in smart grid networks","volume":"55","author":"Jiang","year":"2017","journal-title":"IEEE Commun. Mag."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1109\/TNS.2003.823015","article-title":"An improved maximum likelihood approach to image reconstruction using ordered subsets and data subdivisions","volume":"51","author":"Sheng","year":"2004","journal-title":"IEEE Trans. Nucl. Sci."},{"key":"ref_36","unstructured":"Duan, X., Sun, G., and Tao, Y. (2011, January 25\u201328). Moving target detection based on genetic k-means algorithm. Proceedings of the 2011 IEEE 13th International Conference on Communication Technology, Jinan, China."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Watanabe, M., and Yamaguchi, K. (2003). The EM Algorithm and Related Statistical Models, CRC Press.","DOI":"10.1201\/9780203913055"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Boyd, S., and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press.","DOI":"10.1017\/CBO9780511804441"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/3\/1683\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:23:02Z","timestamp":1760120582000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/3\/1683"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,3]]},"references-count":38,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["s23031683"],"URL":"https:\/\/doi.org\/10.3390\/s23031683","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,3]]}}}