{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T09:54:23Z","timestamp":1778234063938,"version":"3.51.4"},"reference-count":27,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T00:00:00Z","timestamp":1578009600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Aiming at the fact that the independent component analysis algorithm requires more measurement points and cannot solve the problem of harmonic source location under underdetermined conditions, a new method based on sparse component analysis and minimum conditional entropy for identifying multiple harmonic source locations in a distribution system is proposed. Under the condition that the network impedance is unknown and the number of harmonic sources is undetermined, the measurement node configuration algorithm selects the node position to make the separated harmonic current more accurate. Then, using the harmonic voltage data of the selected node as the input, the sparse component analysis is used to solve the harmonic current waveform under underdetermination. Finally, the conditional entropy between the harmonic current and the system node is calculated, and the node corresponding to the minimum condition entropy is the location of the harmonic source. In order to verify the effectiveness and accuracy of the proposed method, the simulation was performed in an IEEE 14-node system. Moreover, compared with the results of independent component analysis algorithms. Simulation results verify the correctness and effectiveness of the proposed algorithm.<\/jats:p>","DOI":"10.3390\/e22010065","type":"journal-article","created":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T11:55:07Z","timestamp":1578052507000},"page":"65","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Multi-Harmonic Source Localization Based on Sparse Component Analysis and Minimum Conditional Entropy"],"prefix":"10.3390","volume":"22","author":[{"given":"Yongzhen","family":"Du","sequence":"first","affiliation":[{"name":"College of Electrical Engineering, Sichuan University, Chengdu 610065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Honggeng","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering, Sichuan University, Chengdu 610065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyang","family":"Ma","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering, Sichuan University, Chengdu 610065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2483","DOI":"10.1109\/TIE.2013.2270218","article-title":"Identification of Multiple Harmonic Sources in Power System Using Optimally Placed Voltage Measurement Devices","volume":"61","author":"Saxena","year":"2014","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.epsr.2019.01.033","article-title":"Harmonic state estimation for distribution systems based on optimization models considering daily load profiles","volume":"170","author":"Melo","year":"2019","journal-title":"Electr. Power Syst. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1145\/3308897.3308936","article-title":"Leveraging Sparsity in Distribution Grids: System Identification and Harmonic State Estimation","volume":"46","author":"Ardakanian","year":"2019","journal-title":"ACM SIGMETRICS Perform. Eval. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1701","DOI":"10.1109\/61.311191","article-title":"Power system harmonic source estimation","volume":"9","author":"Meliopoulos","year":"1994","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1109\/61.19248","article-title":"Identification of harmonic sources by a state estimation technique","volume":"4","author":"Heydt","year":"1989","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2166","DOI":"10.1109\/TPWRD.2004.843462","article-title":"Multiple harmonic source detection and equipment identification with cascade correlation network","volume":"20","author":"Lin","year":"2005","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1109\/TPWRD.2008.917656","article-title":"Optimal Harmonic Estimation Using a Particle Swarm Optimizer","volume":"23","author":"Lu","year":"2008","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1462","DOI":"10.1109\/TIM.2009.2014504","article-title":"State estimation for the localization of harmonic sources in electric distribution systems","volume":"58","author":"Muscas","year":"2009","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zang, T., He, Z., Fu, L., Chen, J., and Qian, Q. (2016). Harmonic Source Localization Approach Based on Fast Kernel Entropy Optimization ICA and Minimum Conditional Entropy. Entropy, 18.","DOI":"10.3390\/e18060214"},{"key":"ref_10","unstructured":"Farhoodnea, M., Mohamed, A., and Shareef, H. (December, January 29). A new method for determining multiple harmonic source locations in a power distribution system. Proceedings of the IEEE International Conference on Power & Energy, Kuala Lumpur, Malaysia."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2353","DOI":"10.1016\/S0165-1684(01)00120-7","article-title":"Underdetermined blind source separation using sparse representations","volume":"81","author":"Bofill","year":"2001","journal-title":"Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1016\/j.sigpro.2012.08.010","article-title":"A new blind image source separation algorithm based on feedback sparse component analysis","volume":"93","author":"Yu","year":"2013","journal-title":"Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1109\/TSP.2005.861743","article-title":"Underdetermined blind source separation based on sparse representation","volume":"54","author":"Li","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1762","DOI":"10.1016\/j.sigpro.2009.03.017","article-title":"An algorithm for mixing matrix estimation in instantaneous blind source separation","volume":"89","author":"Reju","year":"2009","journal-title":"Signal Process."},{"key":"ref_15","first-page":"499","article-title":"Generic vibration-based faults identification approach for identical rotating machines installed on different foundations, VIRM 11-Vibrations in Rotating","volume":"11","author":"Sinha","year":"2016","journal-title":"Machinery"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1177\/1475921716651394","article-title":"Sensitivity analysis of higher order coherent spectra in machine faults diagnosis","volume":"15","author":"Sinha","year":"2016","journal-title":"Struct. Health Monit."},{"key":"ref_17","unstructured":"Jolliffe, I.T. (2002). Principal Component Analysis, Springer. [2nd ed.]. Springer-Verlag Series in Statistics."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"992","DOI":"10.1109\/TNN.2005.849840","article-title":"Sparse component analysis and blind source separation of underdetermined mixtures","volume":"16","author":"Georgiev","year":"2005","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4741","DOI":"10.1016\/j.jsv.2013.04.004","article-title":"Output-only modal identification with limited sensors using sparse component analysis","volume":"332","author":"Yang","year":"2013","journal-title":"J. Sound Vib."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/j.ymssp.2018.07.026","article-title":"Enhanced sparse component analysis for operational modal identification of real-life bridge structures","volume":"116","author":"Xu","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1109\/72.761722","article-title":"Fast and robust fixed-point algorithms for independent component analysis","volume":"10","author":"Hyvarinen","year":"1999","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1129","DOI":"10.1162\/neco.1995.7.6.1129","article-title":"An information-maximization approach to blind separation and blind deconvolution","volume":"7","author":"Bell","year":"1995","journal-title":"Neural Comput."},{"key":"ref_23","unstructured":"Amari, S.I., Cichocki, A., and Yang, H.H. (1996). A new learning algorithm for blind signal separation. Advances in Neural Information Processing Systems, MIT Press."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"31","DOI":"10.20982\/tqmp.06.1.p031","article-title":"An introduction to independent component analysis: InfoMax and FastICA algorithms","volume":"6","author":"Langlois","year":"2010","journal-title":"Tutor. Quant. Methods Psychol."},{"key":"ref_25","unstructured":"Behera, S.K. (2009). Fast ICA for Blind Source Separation and Its Implementation. [Ph.D. Dissertation, National Institutes of Technology]."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1007\/BF01025996","article-title":"Mutual information functions versus correlation functions","volume":"60","author":"Li","year":"1990","journal-title":"J. Stat. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1109\/TPWRD.2008.2002968","article-title":"Harmonic load identification using complex independent component analysis","volume":"24","author":"Gursoy","year":"2008","journal-title":"IEEE Trans. Power Deliv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/65\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:42:15Z","timestamp":1760362935000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/65"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,3]]},"references-count":27,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["e22010065"],"URL":"https:\/\/doi.org\/10.3390\/e22010065","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,3]]}}}