Mathematics | Special Issue : Artificial Intelligence and Game Theory

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game theory; smart grid; integrated energy systems; decision optimization theory"> <div class="editor-div__content smaller-pictures"> <div class='profile-card-drop' data-dropdown='profile-card-drop285954' data-options='is_hover:true, hover_timeout:5000'> <div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/data/editors/editor_285954.png?1723622336" style= "width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name"> Dr. Lefeng Cheng </span></div></div> </div> <div id="profile-card-drop285954" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"> <div class="profile-card__title "> <div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/data/editors/editor_285954.png?1723622336" style= "width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name"> Dr. Lefeng Cheng </span></div></div> </div> <div class="profile-card__buttons" style="margin-bottom: 10px;"> <a href="https://sciprofiles.com/profile/1282578?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a> <a href="https://scilit.net/scholars?q=Lefeng%20Cheng" class="button button--color-inversed" target="_blank"> Scilit </a> <a href="https://www.preprints.org/search?search1=Lefeng%20Cheng&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a> <a href="https://scholar.google.com/scholar?q=Lefeng%20Cheng" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a> </div> </div> <br class="show-for-small-only" /> <a class="inline-spacer toEncode emailCaptcha" href="" data-editor-id="285954">E-Mail</a> <a class="inline-spacer" href="http://jd.gzhu.edu.cn/info/1102/3927.htm" target="_blank" rel="noopener noreferrer">Website</a> <br/> <i>Guest Editor</i><br> </div> <div style="clear: both;"></div> <div class="editor-div__content smaller-pictures"> School of Mechanical and Electric Engineering, Guangzhou University, Guangzhou 510006, China<br> <b>Interests:</b> artificial intelligence; game theory; smart grid; integrated energy systems; decision optimization theory<br> <a href="#" id="editor_contrib_285954" onclick="div_toggle(this.id); return false;">Special Issues, Collections and Topics in MDPI journals</a> <div id="div_editor_contrib_285954" style="display: none"> Topics: <a href="/topics/2HLRH07T6W">Game Theory and Artificial Intelligence Methods in Sustainable and Renewable Energy Power Systems</a><br> </div> </div> </div> <div class="generic-item editor-div" data-filter="wentian lu artificial intelligence; game theory; smart grid; integrated energy systems; decision optimization theory"> <div class="editor-div__content smaller-pictures"> <div class='profile-card-drop' data-dropdown='profile-card-drop241940' data-options='is_hover:true, hover_timeout:5000'> <div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/data/editors/editor_241940.png?1655974048" style= "width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name"> Dr. Wentian Lu </span></div></div> </div> <div id="profile-card-drop241940" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"> <div class="profile-card__title "> <div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/data/editors/editor_241940.png?1655974048" style= "width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name"> Dr. Wentian Lu </span></div></div> </div> <div class="profile-card__buttons" style="margin-bottom: 10px;"> <a href="https://sciprofiles.com/profile/2293283?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a> <a href="https://scilit.net/scholars?q=Wentian%20Lu" class="button button--color-inversed" target="_blank"> Scilit </a> <a href="https://www.preprints.org/search?search1=Wentian%20Lu&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a> <a href="https://scholar.google.com/scholar?q=Wentian%20Lu" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a> </div> </div> <br class="show-for-small-only" /> <a class="inline-spacer toEncode emailCaptcha" href="" data-editor-id="241940">E-Mail</a> <a class="inline-spacer" href="http://jd.gzhu.edu.cn/info/1102/4233.htm" target="_blank" rel="noopener noreferrer">Website</a> <br/> <i>Guest Editor</i><br> </div> <div style="clear: both;"></div> <div class="editor-div__content smaller-pictures"> School of Mechanical and Electric Engineering, Guangzhou University, Guangzhou 510006, China<br> <b>Interests:</b> artificial intelligence; game theory; smart grid; integrated energy systems; decision optimization theory<br> </div> </div> </div> <h2><a name="info"></a>Special Issue Information</h2> <div> <p>Dear Colleagues,</p> <p>At present, various countries have formulated a new generation of artificial intelligence development plans to seize the opportunity for a new round of technological changes. Based on this, vigorously developing the application of a new generation of artificial intelligence in smart grids and integrated energy systems will change the entire traditional energy utilization model and further promote the intelligent development of the system. In addition to the above fields, this Special Issue aims to collect research on a new generation of artificial intelligence and game theory-related methods in important engineering fields such as electrical, electronic, mechanical, and computer science. </p><p>Dr. Lefeng Cheng<br/>Dr. Wentian Lu<br /><em>Guest Editors</em></p><p><p><strong>Manuscript Submission Information</strong><p> <p>Manuscripts should be submitted online at <a href="https://www.mdpi.com/">www.mdpi.com</a> by <a href="https://www.mdpi.com/user/register/">registering</a> and <a href="https://www.mdpi.com/user/login/">logging in to this website</a>. Once you are registered, <a href="https://susy.mdpi.com/user/manuscripts/upload/?journal=mathematics">click here to go to the submission form</a>. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.</p> <p>Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the <a href="https://www.mdpi.com/journal/mathematics/instructions">Instructions for Authors</a> page. <a href="https://www.mdpi.com/journal/mathematics/"><em>Mathematics</em></a> is an international peer-reviewed open access semimonthly journal published by MDPI.</p> <p> Please visit the <a href="https://www.mdpi.com/journal/mathematics/instructions">Instructions for Authors</a> page before submitting a manuscript. The <a href="https://www.mdpi.com/about/apc/">Article Processing Charge (APC)</a> for publication in this <a href="https://www.mdpi.com/about/openaccess/">open access</a> journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's <a href="https://www.mdpi.com/authors/english">English editing service</a> prior to publication or during author revisions. </p></p> </div> <h2><a name="keywords"></a>Keywords</h2> <div><ul> <li>game theory</li> <li>artificial intelligence</li> <li>power system</li> <li>sustainable and renewable energy</li> <li>engineering optimization decision</li> </ul></div> <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN" "http://www.w3.org/TR/REC-html40/loose.dtd"> <html><body><h2><a name="benefits"></a>Benefits of Publishing in a Special Issue</h2> <ul> <li>Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.</li> <li>Greater discoverability: Special Issues support the reach and impact of scientific research. 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return false;">Ok</a> </div> </div> <a class="close-reveal-modal" aria-label="Close"> <i class="material-icons">clear</i> </a> </div> </div> <div> <div style="clear: both"></div> </div> </div> </div> <div class="generic-item type-section" id=Research> <h2>Research</h2> <div style="margin-top: 15px;"> </div> </div> <div class="generic-item article-item"> <input class="article-list-checkbox export-element" type="checkbox" name="articles_ids[]" value="1500643" data-select-all-name="article-listing"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 39 pages, 7714 KiB   </span> <a href="/2227-7390/12/20/3249/pdf?version=1729157712" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Dynamic Evolution Game Strategy of Government, Power Grid, and Users in Electricity Market Demand-Side Management" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/20/3249">Dynamic Evolution Game Strategy of Government, Power Grid, and Users in Electricity Market Demand-Side Management</a> <div class="authors"> by <span class="inlineblock "><strong>Xin Shen</strong>, </span><span class="inlineblock "><strong>Jianlin Tang</strong>, </span><span class="inlineblock "><strong>Yijing Zhang</strong>, </span><span class="inlineblock "><strong>Bin Qian</strong>, </span><span class="inlineblock "><strong>Jiahao Li</strong>, </span><span class="inlineblock "><strong>Mi Zhou</strong>, </span><span class="inlineblock "><strong>Yitao Zhao</strong> and </span><span class="inlineblock "><strong>Yujun Yin</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(20), 3249; <a href="https://doi.org/10.3390/math12203249">https://doi.org/10.3390/math12203249</a> - 17 Oct 2024 </div> Viewed by 547 <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> In the process of promoting demand-side management, the core stakeholder groups are government departments, power grid companies, and electricity users. Due to the different positions and conflicting interests of the three parties in the game, intense and complex battles will occur. This paper <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/20/3249/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> In the process of promoting demand-side management, the core stakeholder groups are government departments, power grid companies, and electricity users. Due to the different positions and conflicting interests of the three parties in the game, intense and complex battles will occur. This paper investigates a tripartite evolutionary game involving government, power grid companies, and electricity users in the context of demand-side management (DSM) and analyzes the dynamic interactions between government departments, power grid companies, and electricity users within the framework of DSM using evolutionary game theory. Using evolutionary game theory, we explore how incentives and strategic interactions among these three stakeholders evolve over time, affecting the stability of DSM policies. The model addresses the asymmetry in the decision-making process and examines the dynamic equilibrium outcomes under various scenarios. The results provide insights into the optimal design of incentive mechanisms to enhance DSM adoption. The findings offer practical recommendations to improve DSM policies, fostering balanced interests between government, grid companies, and users. This research contributes to a deeper understanding of strategic interactions in DSM, revealing how adaptive behaviors can enhance energy efficiency. It also underscores the importance of carefully designed incentive mechanisms in achieving long-term stability and cooperation among key stakeholders. <a href="/2227-7390/12/20/3249">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/Artificial_Intelligence_Game_Theory ">Artificial Intelligence and Game Theory</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/20/3249/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1500643"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1500643"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1500643" data-cycle-prev="#prev1500643" data-cycle-progressive="#images1500643" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1500643-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g001-550.jpg?1729157782" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1500643" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1500643-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002a-550.jpg?1729157783'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1500643-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002b-550.jpg?1729157785'><p>Figure 2 Cont.</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1500643-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g003-550.jpg?1729157787'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1500643-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004a-550.jpg?1729157789'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1500643-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004b-550.jpg?1729157791'><p>Figure 4 Cont.</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1500643-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005a-550.jpg?1729157792'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1500643-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005b-550.jpg?1729157794'><p>Figure 5 Cont.</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1500643-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g006-550.jpg?1729157797'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='9' data-target='article-1500643-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g007-550.jpg?1729157799'><p>Figure 7</p></div></script></div></div><div id="article-1500643-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g001-550.jpg?1729157782" title=" <strong>Figure 1</strong><br/> <p>Schematic diagram of symmetric and asymmetric evolutionary game structures [<a href="#B46-mathematics-12-03249" class="html-bibr">46</a>]. In the figure (<b>a</b>), it illustrates the structure of a symmetric evolutionary game. In figure (<b>b</b>), it demonstrates the structure of an asymmetric evolutionary game.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3249'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002a-550.jpg?1729157783" title=" <strong>Figure 2</strong><br/> <p>The simulation results of the 2P2S-AEG system. In figure (<b>a</b>), it illustrates the evolution trend of the proportion <span class="html-italic">x</span> of individuals selecting strategy S<sub>A1</sub> in group A over time <span class="html-italic">t</span>. In figure (<b>b</b>), it demonstrates the evolution trend of the proportion <span class="html-italic">y</span> of individuals selecting strategy S<sub>B1</sub> in group B over time <span class="html-italic">t</span>. In figure (<b>c</b>), it reveals the evolution trend of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over time <span class="html-italic">t</span> in the game system.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3249'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g002b-550.jpg?1729157785" title=" <strong>Figure 2 Cont.</strong><br/> <p>The simulation results of the 2P2S-AEG system. In figure (<b>a</b>), it illustrates the evolution trend of the proportion <span class="html-italic">x</span> of individuals selecting strategy S<sub>A1</sub> in group A over time <span class="html-italic">t</span>. In figure (<b>b</b>), it demonstrates the evolution trend of the proportion <span class="html-italic">y</span> of individuals selecting strategy S<sub>B1</sub> in group B over time <span class="html-italic">t</span>. In figure (<b>c</b>), it reveals the evolution trend of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over time <span class="html-italic">t</span> in the game system.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3249'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g003-550.jpg?1729157787" title=" <strong>Figure 3</strong><br/> <p>The tripartite evolutionary game framework among the government, the power grid company, and power users in demand-side management (DSM). This figure illustrates the strategic interactions among the three parties, where each entity’s strategy selection impacts the others’ payoffs. The government incentivizes or refrains from incentivizing DSM policies, the grid company decides whether to implement these policies, and the power users choose to participate or opt out of DSM services. The figure reflects the dynamic evolution of these strategic choices over time, with the goal of achieving a stable evolutionary equilibrium among all participants.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3249'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004a-550.jpg?1729157789" title=" <strong>Figure 4</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (6, 1, 3, 7, 2, 4) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and −<span class="html-italic">d</span> &lt; <span class="html-italic">e</span> − <span class="html-italic">f</span> &lt; 0. In figure (<b>b</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (6, 1, 3, 4, 2, 7) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> + <span class="html-italic">d</span> &lt; 0. In figure (<b>c</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (3, 4, 1, 4, 2, 7) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> + <span class="html-italic">d</span> &lt; 0. In figure (<b>d</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (3, 4, 1, 7, 2, 4) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and −<span class="html-italic">d</span> &lt; <span class="html-italic">e</span> − <span class="html-italic">f</span> &lt; 0. In figure (<b>e</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (5, 2, 4, 3, 1, 6) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> &gt; 0. In figure (<b>f</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (7, 2, 4, 3, 6, 5) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> &gt; 0.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3249'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g004b-550.jpg?1729157791" title=" <strong>Figure 4 Cont.</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (6, 1, 3, 7, 2, 4) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and −<span class="html-italic">d</span> &lt; <span class="html-italic">e</span> − <span class="html-italic">f</span> &lt; 0. In figure (<b>b</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (6, 1, 3, 4, 2, 7) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> + <span class="html-italic">d</span> &lt; 0. In figure (<b>c</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (3, 4, 1, 4, 2, 7) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> + <span class="html-italic">d</span> &lt; 0. In figure (<b>d</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (3, 4, 1, 7, 2, 4) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and −<span class="html-italic">d</span> &lt; <span class="html-italic">e</span> − <span class="html-italic">f</span> &lt; 0. In figure (<b>e</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (5, 2, 4, 3, 1, 6) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &lt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> &gt; 0. In figure (<b>f</b>), it assumes (<span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">c</span>, <span class="html-italic">d</span>, <span class="html-italic">e</span>, <span class="html-italic">f</span>) = (7, 2, 4, 3, 6, 5) for meeting <span class="html-italic">a</span> − <span class="html-italic">b</span> − <span class="html-italic">c</span> &gt; 0 and <span class="html-italic">e</span> − <span class="html-italic">f</span> &gt; 0.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3249'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005a-550.jpg?1729157792" title=" <strong>Figure 5</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (7, 5, 1, 6, 2, 1, 4) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and −<span class="html-italic">d</span>′ &lt; <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>b</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (7, 4, 1, 3, 2, 5, 12) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and <span class="html-italic">d</span>′ + <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>c</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 5, 2, 3, 2, 1, 9) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and <span class="html-italic">d</span>′ + <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>d</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 5, 2, 7, 3, 1, 6) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and −<span class="html-italic">d</span>′ &lt; <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>e</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 1, 2, 7, 3, 4, 5) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &gt; 0. In figure (<b>f</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (3, 4, 1, 7, 2, 6, 5) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &gt; 0.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3249'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g005b-550.jpg?1729157794" title=" <strong>Figure 5 Cont.</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid two-party asymmetric evolution game system. In figure (<b>a</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (7, 5, 1, 6, 2, 1, 4) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and −<span class="html-italic">d</span>′ &lt; <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>b</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (7, 4, 1, 3, 2, 5, 12) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and <span class="html-italic">d</span>′ + <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>c</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 5, 2, 3, 2, 1, 9) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and <span class="html-italic">d</span>′ + <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>d</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 5, 2, 7, 3, 1, 6) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and −<span class="html-italic">d</span>′ &lt; <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &lt; 0. In figure (<b>e</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (6, 1, 2, 7, 3, 4, 5) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &gt; 0 and <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &gt; 0. In figure (<b>f</b>), it assumes (<span class="html-italic">a</span>′, <span class="html-italic">b</span>′, <span class="html-italic">c</span>′, <span class="html-italic">d</span>′, <span class="html-italic">e</span>′, <span class="html-italic">f</span>′) = (3, 4, 1, 7, 2, 6, 5) for meeting <span class="html-italic">a</span>′ − <span class="html-italic">b</span>′ − <span class="html-italic">c</span>′ &lt; 0 and <span class="html-italic">e</span>′ + <span class="html-italic">f</span>′ − <span class="html-italic">g</span>′ &gt; 0.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3249'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g006-550.jpg?1729157797" title=" <strong>Figure 6</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid-power user three-group asymmetric evolutionary game system when taking 1000 times of repeated evolution game. Figure (<b>a</b>) shows the phase trajectory of (<span class="html-italic">x</span>, <span class="html-italic">y</span>, <span class="html-italic">z</span>), and Figure (<b>b</b>) demonstrates the phase trajectories of (<span class="html-italic">x</span>, <span class="html-italic">t</span>), (<span class="html-italic">y</span>, <span class="html-italic">t</span>) and (<span class="html-italic">z</span>, <span class="html-italic">t</span>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3249'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03249/article_deploy/html/images/mathematics-12-03249-g007-550.jpg?1729157799" title=" <strong>Figure 7</strong><br/> <p>The simulation results of the phase trajectory of this government-power grid-power user three-group asymmetric evolutionary game system when taking 2000 times of repeated evolution game. Figure (<b>a</b>) shows the phase trajectory of (<span class="html-italic">x</span>, <span class="html-italic">y</span>, <span class="html-italic">z</span>), and Figure (<b>b</b>) demonstrates the phase trajectories of (<span class="html-italic">x</span>, <span class="html-italic">t</span>), (<span class="html-italic">y</span>, <span class="html-italic">t</span>) and (<span class="html-italic">z</span>, <span class="html-italic">t</span>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3249'>Full article</a></strong> "></a></div> </div> </div> <div class="generic-item article-item"> <input class="article-list-checkbox export-element" type="checkbox" name="articles_ids[]" value="1498340" data-select-all-name="article-listing"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 20 pages, 3481 KiB   </span> <a href="/2227-7390/12/20/3213/pdf?version=1728902007" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Short-Term Irradiance Prediction Based on Transformer with Inverted Functional Area Structure" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/20/3213">Short-Term Irradiance Prediction Based on Transformer with Inverted Functional Area Structure</a> <div class="authors"> by <span class="inlineblock "><strong>Zhenyuan Zhuang</strong>, </span><span class="inlineblock "><strong>Huaizhi Wang</strong> and </span><span class="inlineblock "><strong>Cilong Yu</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(20), 3213; <a href="https://doi.org/10.3390/math12203213">https://doi.org/10.3390/math12203213</a> - 14 Oct 2024 </div> Viewed by 589 <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> Solar irradiance prediction is a crucial component in the application of photovoltaic power generation, playing a vital role in optimizing energy production, managing energy storage, and maintaining grid stability. This paper proposes an irradiance prediction method based on a functionally structured inverted transformer <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/20/3213/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> Solar irradiance prediction is a crucial component in the application of photovoltaic power generation, playing a vital role in optimizing energy production, managing energy storage, and maintaining grid stability. This paper proposes an irradiance prediction method based on a functionally structured inverted transformer network, which maintains the channel independence of each feature in the model input and extracts the correlations between different features through an Attention mechanism, enabling the model to effectively capture the relevant information between various features. After the channel mixing of different features is completed through the Attention mechanism, a linear network is used to predict the irradiance sequence. A data processing method tailored to the prediction model used in this paper is designed, which employs a comprehensive data preprocessing approach combining mutual information, multiple imputation, and median filtering to optimize the raw dataset, enhancing the overall stability and accuracy of the prediction project. Additionally, a Dingo optimization algorithm suitable for the self-tuning of deep learning model hyperparameters is designed, improving the model’s generalization capability and reducing deployment costs. The artificial intelligence (AI) model proposed in this paper demonstrates superior prediction performance compared to existing common prediction models in irradiance data forecasting and can facilitate further applications of photovoltaic power generation in power systems. <a href="/2227-7390/12/20/3213">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/Artificial_Intelligence_Game_Theory ">Artificial Intelligence and Game Theory</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/20/3213/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1498340"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1498340"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1498340" data-cycle-prev="#prev1498340" data-cycle-progressive="#images1498340" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1498340-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g001-550.jpg?1728902179" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1498340" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1498340-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g002-550.jpg?1728902180'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1498340-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g003-550.jpg?1728902180'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1498340-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g004-550.jpg?1728902181'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1498340-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g005-550.jpg?1728902181'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1498340-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g006-550.jpg?1728902182'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1498340-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g007-550.jpg?1728902182'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1498340-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g008-550.jpg?1728902183'><p>Figure 8</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1498340-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g009-550.jpg?1728902184'><p>Figure 9</p></div> --- <div class='openpopupgallery' data-imgindex='9' data-target='article-1498340-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g010-550.jpg?1728902184'><p>Figure 10</p></div></script></div></div><div id="article-1498340-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g001-550.jpg?1728902179" title=" <strong>Figure 1</strong><br/> <p>Prediction model structure diagram.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3213'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g002-550.jpg?1728902180" title=" <strong>Figure 2</strong><br/> <p>DOX algorithm flow diagram.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3213'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g003-550.jpg?1728902180" title=" <strong>Figure 3</strong><br/> <p>Comparison between forecast curve and actual curve of typical sunny day.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3213'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g004-550.jpg?1728902181" title=" <strong>Figure 4</strong><br/> <p>Comparison between the predicted curve and the actual curve of typical cloudy day.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3213'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g005-550.jpg?1728902181" title=" <strong>Figure 5</strong><br/> <p>Comparison between the predicted curve and the actual curve of typical summer days.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3213'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g006-550.jpg?1728902182" title=" <strong>Figure 6</strong><br/> <p>Comparison between the forecast curve and the actual curve of typical winter days.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3213'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g007-550.jpg?1728902182" title=" <strong>Figure 7</strong><br/> <p>Comparison of forecast curves for a certain day in spring.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3213'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g008-550.jpg?1728902183" title=" <strong>Figure 8</strong><br/> <p>Comparison of forecast curves for a summer day.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3213'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g009-550.jpg?1728902184" title=" <strong>Figure 9</strong><br/> <p>Comparison of forecast curves for a day in autumn.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3213'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03213/article_deploy/html/images/mathematics-12-03213-g010-550.jpg?1728902184" title=" <strong>Figure 10</strong><br/> <p>Comparison of forecast curves for a winter day.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/20/3213'>Full article</a></strong> "></a></div> </div> </div> <div class="generic-item article-item"> <input class="article-list-checkbox export-element" type="checkbox" name="articles_ids[]" value="1492213" data-select-all-name="article-listing"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 49 pages, 11591 KiB   </span> <a href="/2227-7390/12/19/3109/pdf?version=1728473649" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Spontaneous Formation of Evolutionary Game Strategies for Long-Term Carbon Emission Reduction Based on Low-Carbon Trading Mechanism" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/19/3109">Spontaneous Formation of Evolutionary Game Strategies for Long-Term Carbon Emission Reduction Based on Low-Carbon Trading Mechanism</a> <div class="authors"> by <span class="inlineblock "><strong>Zhanggen Zhu</strong>, </span><span class="inlineblock "><strong>Lefeng Cheng</strong> and </span><span class="inlineblock "><strong>Teng Shen</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(19), 3109; <a href="https://doi.org/10.3390/math12193109">https://doi.org/10.3390/math12193109</a> - 4 Oct 2024 </div> <a href="/2227-7390/12/19/3109#metrics">Cited by 1</a> | Viewed by 583 <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> In the context of increasing global efforts to mitigate climate change, effective carbon emission reduction is a pressing issue. Governments and power companies are key stakeholders in implementing low-carbon strategies, but their interactions require careful management to ensure optimal outcomes for both economic <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/19/3109/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> In the context of increasing global efforts to mitigate climate change, effective carbon emission reduction is a pressing issue. Governments and power companies are key stakeholders in implementing low-carbon strategies, but their interactions require careful management to ensure optimal outcomes for both economic development and environmental protection. This paper addresses this real-world challenge by utilizing evolutionary game theory (EGT) to model the strategic interactions between these stakeholders under a low-carbon trading mechanism. Unlike classical game theory, which assumes complete rationality and perfect information, EGT allows for bounded rationality and learning over time, making it particularly suitable for modeling long-term interactions in complex systems like carbon markets. This study builds an evolutionary game model between the government and power companies to explore how different strategies in carbon emission reduction evolve over time. Using payoff matrices and replicator dynamics equations, we determine the evolutionarily stable equilibrium (ESE) points and analyze their stability through dynamic simulations. The findings show that in the absence of a third-party regulator, neither party achieves an ideal ESE. To address this, a third-party regulatory body is introduced into the model, leading to the formulation of a tripartite evolutionary game. The results highlight the importance of regulatory oversight in achieving stable and optimal low-carbon strategies. This paper offers practical policy recommendations based on the simulation outcomes, providing a robust theoretical framework for government intervention in carbon markets and guiding enterprises towards sustainable practices. <a href="/2227-7390/12/19/3109">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/Artificial_Intelligence_Game_Theory ">Artificial Intelligence and Game Theory</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/19/3109/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1492213"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1492213"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1492213" data-cycle-prev="#prev1492213" data-cycle-progressive="#images1492213" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1492213-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g001-550.jpg?1728473743" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1492213" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g002a-550.jpg?1728473744'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g002b-550.jpg?1728473746'><p>Figure 2 Cont.</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g003a-550.jpg?1728473747'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g003b-550.jpg?1728473748'><p>Figure 3 Cont.</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g004-550.jpg?1728473751'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g005-550.jpg?1728473753'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g006a-550.jpg?1728473753'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g006b-550.jpg?1728473754'><p>Figure 6 Cont.</p></div> --- <div class='openpopupgallery' data-imgindex='9' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g007a-550.jpg?1728473755'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='10' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g007b-550.jpg?1728473756'><p>Figure 7 Cont.</p></div> --- <div class='openpopupgallery' data-imgindex='11' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g008-550.jpg?1728473757'><p>Figure 8</p></div> --- <div class='openpopupgallery' data-imgindex='12' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g009-550.jpg?1728473757'><p>Figure 9</p></div> --- <div class='openpopupgallery' data-imgindex='13' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g010-550.jpg?1728473758'><p>Figure 10</p></div> --- <div class='openpopupgallery' data-imgindex='14' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g011-550.jpg?1728473758'><p>Figure 11</p></div> --- <div class='openpopupgallery' data-imgindex='15' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g012-550.jpg?1728473759'><p>Figure 12</p></div> --- <div class='openpopupgallery' data-imgindex='16' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g013-550.jpg?1728473759'><p>Figure 13</p></div> --- <div class='openpopupgallery' data-imgindex='17' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g014-550.jpg?1728473760'><p>Figure 14</p></div> --- <div class='openpopupgallery' data-imgindex='18' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g015-550.jpg?1728473760'><p>Figure 15</p></div> --- <div class='openpopupgallery' data-imgindex='19' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g016-550.jpg?1728473761'><p>Figure 16</p></div> --- <div class='openpopupgallery' data-imgindex='20' data-target='article-1492213-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g017-550.jpg?1728473762'><p>Figure 17</p></div></script></div></div><div id="article-1492213-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g001-550.jpg?1728473743" title=" <strong>Figure 1</strong><br/> <p>A complete flowchart for analyzing the multi-group asymmetric evolutionary game.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g002a-550.jpg?1728473744" title=" <strong>Figure 2</strong><br/> <p>Numerical simulation results of the long-term evolutionary game phase trajectory of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over the time range of <span class="html-italic">t</span> ∈ [0, 10] under different numbers of evolutionary game iterations. Here, (<b>a</b>–<b>f</b>) illustrate the numerical simulation results of 121, 441, 961, 1681, 2601, and 3721 different initial game situations, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g002b-550.jpg?1728473746" title=" <strong>Figure 2 Cont.</strong><br/> <p>Numerical simulation results of the long-term evolutionary game phase trajectory of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over the time range of <span class="html-italic">t</span> ∈ [0, 10] under different numbers of evolutionary game iterations. Here, (<b>a</b>–<b>f</b>) illustrate the numerical simulation results of 121, 441, 961, 1681, 2601, and 3721 different initial game situations, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g003a-550.jpg?1728473747" title=" <strong>Figure 3</strong><br/> <p>Numerical simulation results of the long-term evolutionary game phase trajectory of <span class="html-italic">x</span> over the time range of <span class="html-italic">t</span> ∈ [0, 10] under different numbers of evolutionary game iterations. Here, (<b>a</b>–<b>f</b>) illustrate the numerical simulation results of 121, 441, 961, 1681, 2601, and 3721 different initial game situations, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g003b-550.jpg?1728473748" title=" <strong>Figure 3 Cont.</strong><br/> <p>Numerical simulation results of the long-term evolutionary game phase trajectory of <span class="html-italic">x</span> over the time range of <span class="html-italic">t</span> ∈ [0, 10] under different numbers of evolutionary game iterations. Here, (<b>a</b>–<b>f</b>) illustrate the numerical simulation results of 121, 441, 961, 1681, 2601, and 3721 different initial game situations, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g004-550.jpg?1728473751" title=" <strong>Figure 4</strong><br/> <p>Numerical simulation results of the long-term evolutionary game phase trajectory of <span class="html-italic">y</span> over the time range of <span class="html-italic">t</span> ∈ [0, 10] under different numbers of evolutionary game iterations. Here, (<b>a</b>–<b>f</b>) illustrate the numerical simulation results of 121, 441, 961, 1681, 2601, and 3721 different initial game situations, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g005-550.jpg?1728473753" title=" <strong>Figure 5</strong><br/> <p>Numerical simulation results to validate the conclusions drawn in Scenario I. E<sub>1</sub>(0, 0) is a unique long-term ESS, where the long-term evolutionary game phase trajectory curves of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over the time range of <span class="html-italic">t</span> ∈ [0, 10] are demonstrated in Cases 1 to 6. Here, Cases 1 to 6 represent the numerical simulation results under 121, 441, 961, 1681, 2601, and 3721 different initial game situations, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g006a-550.jpg?1728473753" title=" <strong>Figure 6</strong><br/> <p>Numerical simulation results to validate the conclusions drawn in Scenario II. E<sub>2</sub>(0, 1) is a unique long-term ESS, where the long-term evolutionary game phase trajectory curves of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over the time range of <span class="html-italic">t</span> ∈ [0, 10] are demonstrated in Cases 1 to 6. Here, Cases 1 to 6 represent the numerical simulation results under 121, 441, 961, 1681, 2601, and 3721 different initial game situations, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g006b-550.jpg?1728473754" title=" <strong>Figure 6 Cont.</strong><br/> <p>Numerical simulation results to validate the conclusions drawn in Scenario II. E<sub>2</sub>(0, 1) is a unique long-term ESS, where the long-term evolutionary game phase trajectory curves of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over the time range of <span class="html-italic">t</span> ∈ [0, 10] are demonstrated in Cases 1 to 6. Here, Cases 1 to 6 represent the numerical simulation results under 121, 441, 961, 1681, 2601, and 3721 different initial game situations, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g007a-550.jpg?1728473755" title=" <strong>Figure 7</strong><br/> <p>Numerical simulation results to validate the conclusions drawn in Scenario III. E<sub>3</sub>(1, 0) is a unique long-term ESS, where the long-term evolutionary game phase trajectory curves of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over the time range of <span class="html-italic">t</span> ∈ [0, 10] are demonstrated in Cases 1 to 6. Here, Cases 1 to 6 represent the numerical simulation results under 121, 441, 961, 1681, 2601, and 3721 different initial game situations, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g007b-550.jpg?1728473756" title=" <strong>Figure 7 Cont.</strong><br/> <p>Numerical simulation results to validate the conclusions drawn in Scenario III. E<sub>3</sub>(1, 0) is a unique long-term ESS, where the long-term evolutionary game phase trajectory curves of (<span class="html-italic">x</span>, <span class="html-italic">y</span>) over the time range of <span class="html-italic">t</span> ∈ [0, 10] are demonstrated in Cases 1 to 6. Here, Cases 1 to 6 represent the numerical simulation results under 121, 441, 961, 1681, 2601, and 3721 different initial game situations, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g008-550.jpg?1728473757" title=" <strong>Figure 8</strong><br/> <p>Numerical simulation of the game situation under <math display="inline"><semantics> <mrow> <msubsup> <mi>M</mi> <mrow> <mi>EPE</mi> </mrow> <mn>1</mn> </msubsup> <mo>−</mo> <msubsup> <mi>M</mi> <mrow> <mi>EPE</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mi>y</mi> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>EPE</mi> </mrow> </mrow> <mn>1</mn> </msubsup> <mo>+</mo> <msubsup> <mi>S</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>EPE</mi> </mrow> </mrow> <mn>1</mn> </msubsup> <mo>−</mo> <mi>z</mi> <msubsup> <mi>F</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>EPE</mi> </mrow> </mrow> <mn>1</mn> </msubsup> <mo stretchy="false">)</mo> <mo>&gt;</mo> <msubsup> <mi>C</mi> <mrow> <mi>EPE</mi> </mrow> <mn>2</mn> </msubsup> <mo>−</mo> <msubsup> <mi>C</mi> <mrow> <mi>EPE</mi> </mrow> <mn>1</mn> </msubsup> <mo>−</mo> <mi>z</mi> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>EPE</mi> </mrow> </mrow> <mn>1</mn> </msubsup> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mrow> <mi>TPR</mi> <mtext>-</mtext> <mi>EPE</mi> </mrow> </mrow> <mn>2</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>F</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>EPE</mi> </mrow> </mrow> <mn>1</mn> </msubsup> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>TPR</mi> </mrow> </mrow> <mn>3</mn> </msubsup> <mo>+</mo> <mi>x</mi> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>EPE</mi> </mrow> </mrow> <mn>1</mn> </msubsup> <mo>+</mo> <msubsup> <mi>S</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>EPE</mi> </mrow> </mrow> <mn>1</mn> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>z</mi> <mo stretchy="false">(</mo> <msubsup> <mi>S</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>TPR</mi> </mrow> </mrow> <mn>2</mn> </msubsup> <mo>−</mo> <msubsup> <mi>F</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>TPR</mi> </mrow> </mrow> <mn>3</mn> </msubsup> <mo stretchy="false">)</mo> <mo>&gt;</mo> <msubsup> <mi>C</mi> <mrow> <mrow> <mi>GOV</mi> <mtext>-</mtext> <mi>SB</mi> <mn>1</mn> </mrow> </mrow> <mn>3</mn> </msubsup> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g009-550.jpg?1728473757" title=" <strong>Figure 9</strong><br/> <p>Numerical simulation of the phase trajectory for the established three-party long-term evolutionary game, starting with initial values of (0.1, 0.1, 0.1).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g010-550.jpg?1728473758" title=" <strong>Figure 10</strong><br/> <p>Numerical simulation of the phase trajectory for the established three-party long-term evolutionary game, starting with initial values of (0.3, 0.3, 0.3).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g011-550.jpg?1728473758" title=" <strong>Figure 11</strong><br/> <p>Numerical simulation of the phase trajectory for the established three-party long-term evolutionary game, starting with initial values of (0.5, 0.5, 0.5).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g012-550.jpg?1728473759" title=" <strong>Figure 12</strong><br/> <p>Numerical simulation of the phase trajectory for the established three-party long-term evolutionary game, starting with initial values of (0.6, 0.6, 0.6).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g013-550.jpg?1728473759" title=" <strong>Figure 13</strong><br/> <p>Numerical simulation of the phase trajectory for the established three-party long-term evolutionary game, starting with initial values of (0.8, 0.8, 0.8).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g014-550.jpg?1728473760" title=" <strong>Figure 14</strong><br/> <p>Numerical simulation of the phase trajectory for the established three-party long-term evolutionary game, starting with initial values of (0.9, 0.9, 0.9).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g015-550.jpg?1728473760" title=" <strong>Figure 15</strong><br/> <p>Numerical simulation of the phase trajectory for the established three-party long-term evolutionary game, starting with initial values of (0.1, 0.5, 0.5).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g016-550.jpg?1728473761" title=" <strong>Figure 16</strong><br/> <p>Numerical simulation of the phase trajectory for the established three-party long-term evolutionary game, starting with initial values of (0.8, 0.5, 0.5).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03109/article_deploy/html/images/mathematics-12-03109-g017-550.jpg?1728473762" title=" <strong>Figure 17</strong><br/> <p>Numerical simulation of the phase trajectory for the established three-party long-term evolutionary game, starting with initial values of (0.9, 0.5, 0.5).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3109'>Full article</a></strong> "></a></div> </div> </div> <div class="generic-item article-item"> <input class="article-list-checkbox export-element" type="checkbox" name="articles_ids[]" value="1489317" data-select-all-name="article-listing"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 16 pages, 15905 KiB   </span> <a href="/2227-7390/12/19/3064/pdf?version=1728350278" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Decentralized Stochastic Recursive Gradient Method for Fully Decentralized OPF in Multi-Area Power Systems" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/19/3064">Decentralized Stochastic Recursive Gradient Method for Fully Decentralized OPF in Multi-Area Power Systems</a> <div class="authors"> by <span class="inlineblock "><strong>Umair Hussan</strong>, </span><span class="inlineblock "><strong>Huaizhi Wang</strong>, </span><span class="inlineblock "><strong>Muhammad Ahsan Ayub</strong>, </span><span class="inlineblock "><strong>Hamna Rasheed</strong>, </span><span class="inlineblock "><strong>Muhammad Asghar Majeed</strong>, </span><span class="inlineblock "><strong>Jianchun Peng</strong> and </span><span class="inlineblock "><strong>Hui Jiang</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(19), 3064; <a href="https://doi.org/10.3390/math12193064">https://doi.org/10.3390/math12193064</a> - 30 Sep 2024 </div> Viewed by 609 <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> This paper addresses the critical challenge of optimizing power flow in multi-area power systems while maintaining information privacy and decentralized control. The main objective is to develop a novel decentralized stochastic recursive gradient (DSRG) method for solving the optimal power flow (OPF) problem <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/19/3064/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> This paper addresses the critical challenge of optimizing power flow in multi-area power systems while maintaining information privacy and decentralized control. The main objective is to develop a novel decentralized stochastic recursive gradient (DSRG) method for solving the optimal power flow (OPF) problem in a fully decentralized manner. Unlike traditional centralized approaches, which require extensive data sharing and centralized control, the DSRG method ensures that each area within the power system can make independent decisions based on local information while still achieving global optimization. Numerical simulations are conducted using MATLAB (Version 24.1.0.2603908) to evaluate the performance of the DSRG method on a 3-area, 9-bus test system. The results demonstrate that the DSRG method converges significantly faster than other decentralized OPF methods, reducing the overall computation time while maintaining cost efficiency and system stability. These findings highlight the DSRG method’s potential to significantly enhance the efficiency and scalability of decentralized OPF in modern power systems. <a href="/2227-7390/12/19/3064">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/Artificial_Intelligence_Game_Theory ">Artificial Intelligence and Game Theory</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/19/3064/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1489317"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1489317"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1489317" data-cycle-prev="#prev1489317" data-cycle-progressive="#images1489317" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1489317-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g001-550.jpg?1728350390" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1489317" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1489317-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g002-550.jpg?1728350392'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1489317-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g003-550.jpg?1728350393'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1489317-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g004-550.jpg?1728350394'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1489317-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g005-550.jpg?1728350395'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1489317-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g006-550.jpg?1728350396'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1489317-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g007-550.jpg?1728350397'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1489317-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g008-550.jpg?1728350398'><p>Figure 8</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1489317-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g009-550.jpg?1728350399'><p>Figure 9</p></div> --- <div class='openpopupgallery' data-imgindex='9' data-target='article-1489317-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g010-550.jpg?1728350400'><p>Figure 10</p></div></script></div></div><div id="article-1489317-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g001-550.jpg?1728350390" title=" <strong>Figure 1</strong><br/> <p>Flow chart for the DSRG algorithm.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3064'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g002-550.jpg?1728350392" title=" <strong>Figure 2</strong><br/> <p>The 3-area, 9-bus system.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3064'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g003-550.jpg?1728350393" title=" <strong>Figure 3</strong><br/> <p>Optimized output power of generators.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3064'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g004-550.jpg?1728350394" title=" <strong>Figure 4</strong><br/> <p>The voltage magnitudes of the decentralized 3-area, 9-bus system.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3064'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g005-550.jpg?1728350395" title=" <strong>Figure 5</strong><br/> <p>Power flow analysis across transmission lines in a multi-area power system under different line capacities. (<b>a</b>) the power flow within a single area when congestion occurs; (<b>b</b>) the distribution of power across interconnecting lines.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3064'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g006-550.jpg?1728350396" title=" <strong>Figure 6</strong><br/> <p>Cost analysis associated with power flow across internal and tie lines in a multi-area power system. (<b>a</b>) Distribution of costs across various lines for internal capacities; (<b>b</b>) Distribution of costs across various lines for tie-line; (<b>c</b>) Total costs versus internal line capacity (MW); (<b>d</b>) Total costs versus tie line capacity.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3064'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g007-550.jpg?1728350397" title=" <strong>Figure 7</strong><br/> <p>Convergence process of the proposed algorithm.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3064'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g008-550.jpg?1728350398" title=" <strong>Figure 8</strong><br/> <p>Comparison of active power distribution among generators with error analysis.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3064'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g009-550.jpg?1728350399" title=" <strong>Figure 9</strong><br/> <p>Comparison of reactive power distribution among generators with error analysis.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/19/3064'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03064/article_deploy/html/images/mathematics-12-03064-g010-550.jpg?1728350400" title=" <strong>Figure 10</strong><br/> <p>A comparison of the cost of different methods across internal and tie-line capacities with error analysis.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a 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Mathematics | Special Issue : Artificial Intelligence and Game Theory