Document Zbl 1478.93396

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Sci.</a> <a href="/?q=in%3A472815" title="Articles in this Issue">543, 437-453 (2021)</a>. </div> <div class="abstract">Summary: This paper investigates the observer-based predictive event-triggered load frequency control (LFC) for a multi-area power system in a smart grid incorporating electric vehicles (EVs) under denial-of-service (DoS) attacks. The DoS attacks were launched in control channels with long duration to block the data transmission process and interrupt the LFC command. This situation becomes even more severe with the integration of EVs in the multi-area power system. The control design formulates the LFC problem as a disturbance attenuation problem in the presence of both long-duration DoS attacks and external disturbances. First, an event-triggering scheme was developed based on the observer in the presence of DoS attacks. Second, a model-based predictive control approach was presented. Then, by using the Lyapunov theory, stability for the multi-area power system with EVs was derived under consideration of the long-duration DoS attacks. Moreover, two algorithms were provided to obtain the model predictive event-triggered control (MPETC) parameters, LFC gains, and control of DoS attacks simultaneously. Finally, a simulation of a two-area power system with EVs was studied to illustrate the effectiveness of the proposed method.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7436106">Cited in <strong>18</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C65" title="MSC2020">93C65</a> </td> <td class="space"> Discrete event control/observation systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B53" title="MSC2020">93B53</a> </td> <td class="space"> Observers </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B36" title="MSC2020">93B36</a> </td> <td class="space"> $H^\infty$-control </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C80" title="MSC2020">93C80</a> </td> <td class="space"> Frequency-response methods in control theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B45" title="MSC2020">93B45</a> </td> <td class="space"> Model predictive control </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aload+frequency+control">load frequency control</a>; <a href="/?q=ut%3Amodel+predictive+control">model predictive control</a>; <a href="/?q=ut%3Amulti-area+power+system">multi-area power system</a>; <a href="/?q=ut%3Adenial-of-service+%28DoS%29+attacks">denial-of-service (DoS) attacks</a>; <a href="/?q=ut%3Aevent-triggering+communication">event-triggering communication</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1478.93396" data-ciurl="/ci/07436106" data-biburl="/bibtex/07436106.bib" data-amsurl="/amsrefs/07436106.bib" data-xmlurl="/xml/07436106.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07436106.pdf" title="Zbl 1478.93396 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.ins.2020.07.042" aria-label="DOI for “Observer-based event triggering $H_\infty$ LFC for multi-area power systems under DoS attacks”" title="10.1016/j.ins.2020.07.042">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Liu, S.; Luo, W.; Wu, L., Co-design of distributed model-based control and event-triggering scheme for load frequency regulation in smart grids, IEEE Trans. Syst. Man Cybern. Syst., 1-9 (2018)</td> </tr><tr> <td>[2]</td> <td class="space">C. Peng, H.T. Sun, M.J. Yang, Y.L. Wang, A survey on security communication and control for smart grids under malicious cyber attacks, IEEE Trans. Man Cybern. Syst. 49 (2019) 1554-1569.</td> </tr><tr> <td>[3]</td> <td class="space">Singh, V. P.; Kishor, N.; Samuel, P., Distributed multi-agent system-based load frequency control for multi-area power system in smart grid, IEEE Trans. Ind. Electron., 64, 5151-5160 (2017)</td> </tr><tr> <td>[4]</td> <td class="space">Li, Y.; Zhang, P.; Ma, L., Denial of service attack and defense method on load frequency control system, J Franklin Inst., 356, 8625-8645 (2019) · <a href="/1423.93047" class="nowrap">Zbl 1423.93047</a></td> </tr><tr> <td>[5]</td> <td class="space">Mo, Y.; Kim, T. H.-J.; Brancik, K.; Dickinson, D.; Lee, H.; Perrig, A.; Sinopoli, B., Cyber-physical security of a smart grid infrastructure, Proc. IEEE., 100, 195-209 (2012)</td> </tr><tr> <td>[6]</td> <td class="space">Sridhar, S.; Hahn, A.; Govindarasu, M., Cyber-physical system security for the electric power grid, Proc. IEEE, 100, 210-224 (2012)</td> </tr><tr> <td>[7]</td> <td class="space">Pham, T. N.; Trinh, H., Load frequency control of power systems with electric vehicles and diverse transmission links using distributed functional observers, IEEE Trans. Smart Grid., 7, 238-252 (2015)</td> </tr><tr> <td>[8]</td> <td class="space">Yilmaz, M.; Krein, P. T., Review of battery charger topologies, charging power levels, and infrastructure for plug-in electric and hybrid vehicles, IEEE Trans. Power Electron., 28, 2151-2169 (2012)</td> </tr><tr> <td>[9]</td> <td class="space">Liu, H.; Hu, Z.; Song, Y.; Wang, J.; Xie, X., Vehicle-to-grid control for supplementary frequency regulation considering charging demands, IEEE Trans. Power Syst., 30, 3110-3119 (2014)</td> </tr><tr> <td>[10]</td> <td class="space">Gaur, P.; Soren, N.; Bhowmik, D., Impact assessment of vehicle-to-grid technology in LFC of multi-area solar-thermal power system, Int. J. Renew. Energy Res., 8, 1580-1590 (2018)</td> </tr><tr> <td>[11]</td> <td class="space">Padhy, S.; Panda, S.; Mahapatra, S., A modified GWO technique based cascade PI-PD controller for AGC of power systems in presence of Plug in Electric Vehicles, Eng. Sci. Technol. Int J., 20, 427-442 (2017)</td> </tr><tr> <td>[12]</td> <td class="space">Liu, H.; Huang, K.; Yang, Y.; Wei, H.; Ma, S., Real-time vehicle-to-grid control for frequency regulation with high frequency regulating signal, Prot. Control Mod. Power Syst., 3, 13 (2018)</td> </tr><tr> <td>[13]</td> <td class="space">Pham, T. N.; Trinh, H.; Wong, K. P., Integration of electric vehicles for load frequency output feedback $H_\infty$ control of smart grids, IET Gener. Transm. Dis., 10, 3341-3352 (2016)</td> </tr><tr> <td>[14]</td> <td class="space">Zhao, F.; Yuan, J.; Wang, N.; Zhang, Z.; Wen, H., Secure load frequency control of smart grids under deception attack: a piecewise delay approach, Energies, 12, 2266 (2019)</td> </tr><tr> <td>[15]</td> <td class="space">Peng, C.; Li, J.; Fei, M., Resilient event-triggering $H_\infty$ load frequency control for multi-area power systems with energy-limited dos attacks, IEEE Trans. Power Syst., 32, 4110-4118 (2017)</td> </tr><tr> <td>[16]</td> <td class="space">Liu, J.; Gu, Y.; Zha, L.; Liu, Y.; Cao, J., Event-triggered $H_i \mathit{nfty}$ load frequency control for multiarea power systems under hybrid cyber attacks, IEEE Trans. Syst. Man Cybern. Syst., 49, 1665-1678 (2019)</td> </tr><tr> <td>[17]</td> <td class="space">Z. Cheng, D. Yue, X. Lan, C. Huang, S. Hu, $ H_\infty$ prediction triggering control of multi-area power systems load frequency control under dos attacks, in: Advanced Computational Methods in Energy, Power, Electric Vehicles, and Their Integration, Springer, Singapore, 2017, pp. 477-487.</td> </tr><tr> <td>[18]</td> <td class="space">Camacho, E. F.; Alba, C. B., Model Predictive Control (2007), Springer-Verlag: Springer-Verlag London</td> </tr><tr> <td>[19]</td> <td class="space">Zeng, G. Q.; Xie, X. Q.; Chen, M. R., An adaptive model predictive load frequency control method for multi-area interconnected power systems with photovoltaic generations, Energies, 10, 1840 (2017)</td> </tr><tr> <td>[20]</td> <td class="space">Yousef, A. M., Model predictive control approach based load frequency controller, WSEAS Trans. Sys. Ctrl., 6, 265-275 (2011)</td> </tr><tr> <td>[21]</td> <td class="space">Mohamed, T. H.; Bevrani, H.; Hassan, A. A.; Hiyama, T., Decentralized model predictive based load frequency control in an interconnected power system, Energy Convers. Manag., 52, 1208-1214 (2011)</td> </tr><tr> <td>[22]</td> <td class="space">Liu, A.; Bai, L., Distributed model predictive control for wide area measurement power systems under malicious attacks, IET Cyber-Phys. Syst. Theory Appl., 3, 111-118 (2018)</td> </tr><tr> <td>[23]</td> <td class="space">Sun, Y.-C.; Yang, G.-H., Robust event-triggered model predictive control for cyber-physical systems under denial-of-service attacks, Int. J. Robust Nonlinear Control., 29, 4797-4811 (2019) · <a href="/1426.93067" class="nowrap">Zbl 1426.93067</a></td> </tr><tr> <td>[24]</td> <td class="space">Yin, X.; Yue, D.; Hu, S.; Peng, C.; Xue, Y., Model-based event-triggered predictive control for networked systems with data dropout, SIAM J. Control Optim., 54, 567-586 (2016) · <a href="/1331.93149" class="nowrap">Zbl 1331.93149</a></td> </tr><tr> <td>[25]</td> <td class="space">Yin, X.; Yue, D.; Hu, S., Model-based event-triggered predictive control for networked systems with communication delays compensation, Int. J. Robust Nonlinear Control., 25, 3572-3595 (2015) · <a href="/1336.93065" class="nowrap">Zbl 1336.93065</a></td> </tr><tr> <td>[26]</td> <td class="space">Zhang, H.; Cheng, P.; Shi, L.; Chen, J., Optimal DoS attack scheduling in wireless networked control system, IEEE Trans. Control Syst. Technol., 24, 843-852 (2016)</td> </tr><tr> <td>[27]</td> <td class="space">Guo, Y.; Zhang, L.; Zhao, J.; Wen, F.; Salam, A.; Mao, J.; Li, L., Networked control of electric vehicles for power system frequency regulation with random communication time delay, Energies, 10, 621 (2017)</td> </tr><tr> <td>[28]</td> <td class="space">De Persis, C.; Tesi, P., Input-to-state stabilizing control under denial-of-service, IEEE Trans. Autom. Control., 60, 2930-2944 (2015) · <a href="/1360.93629" class="nowrap">Zbl 1360.93629</a></td> </tr><tr> <td>[29]</td> <td class="space">Hu, S.; Yue, D.; Xie, X.; Chen, X.; Yin, X., Resilient event-triggered controller synthesis of networked control systems under periodic dos jamming attacks, IEEE Trans. Cybern., 49, 4271-4281 (2019)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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