{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T09:13:16Z","timestamp":1762161196024,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2015,9,1]],"date-time":"2015-09-01T00:00:00Z","timestamp":1441065600000},"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>A six-dimensional nonlinear hydropower system controlled by a nonlinear predictive control method is presented in this paper. In terms of the nonlinear predictive control method; the performance index with terminal penalty function is selected. A simple method to find an appropriate terminal penalty function is introduced and its effectiveness is proved. The input-to-state-stability of the controlled system is proved by using the Lyapunov function. Subsequently a six-dimensional model of the hydropower system is presented in the paper. Different with other hydropower system models; the above model includes the hydro-turbine system; the penstock system; the generator system; and the hydraulic servo system accurately describing the operational process of a hydropower plant. Furthermore, the numerical experiments show that the six-dimensional nonlinear hydropower system controlled by the method is stable. In addition, the numerical experiment also illustrates that the nonlinear predictive control method enjoys great advantages over a traditional control method in nonlinear systems. Finally, a strategy to combine the nonlinear predictive control method with other methods is proposed to further facilitate the application of the nonlinear predictive control method into practice.<\/jats:p>","DOI":"10.3390\/e17096129","type":"journal-article","created":{"date-parts":[[2015,9,1]],"date-time":"2015-09-01T10:55:58Z","timestamp":1441104958000},"page":"6129-6149","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Nonlinear Predictive Control of a Hydropower System Model"],"prefix":"10.3390","volume":"17","author":[{"given":"Runfan","family":"Zhang","sequence":"first","affiliation":[{"name":"Institute of Water Resources and Hydropower Research, Northwest A&F University, Yangling 712100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Diyi","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute of Water Resources and Hydropower Research, Northwest A&F University, Yangling 712100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyi","family":"Ma","sequence":"additional","affiliation":[{"name":"Institute of Water Resources and Hydropower Research, Northwest A&F University, Yangling 712100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,9,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1126\/science.1200990","article-title":"Mekong hydropower development","volume":"332","author":"Edward","year":"2011","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.epsr.2004.05.006","article-title":"Design and simulation of a nonlinear fuzzy controller for a hydropower plant","volume":"73","author":"Mahmoud","year":"2005","journal-title":"Electr. 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