A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732615622"> <meta name="title" content="A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions"> <meta name="description" content="Due to their exceptional operational versatility, doubly fed induction machines (DFIM) are widely employed in power systems comprising variable renewable energy-based electrical generation sources, such as wind farms and pumped-storage hydropower plants. However, their starting and grid synchronization methods require numerous maneuvers or additional components, making the process challenging. In this paper, a soft start method for DFIM, inspired by the traditional synchronization method of synchronous machines, is proposed. This method involves matching the frequencies, voltages, and phase angles on both sides of the main circuit breaker, by adjusting the excitation through the controlled power converter at standstill conditions. Once synchronization is achieved, the frequency is gradually reduced to the rated operational levels. This straightforward starting method effectively suppresses large inrush currents and voltage sags. The proposed method has been validated through computer simulations and experimental tests, yielding satisfactory results." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/machines-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions"> <meta name="dc.creator" content="José M. Guerrero"> <meta name="dc.creator" content="Kumar Mahtani"> <meta name="dc.creator" content="Itxaso Aranzabal"> <meta name="dc.creator" content="Julen Gómez-Cornejo"> <meta name="dc.creator" content="José A. Sánchez"> <meta name="dc.creator" content="Carlos A. Platero"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Machines 2024, Vol. 12, Page 847"> <meta name="dc.date" content="2024-11-25"> <meta name ="dc.identifier" content="10.3390/machines12120847"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="Due to their exceptional operational versatility, doubly fed induction machines (DFIM) are widely employed in power systems comprising variable renewable energy-based electrical generation sources, such as wind farms and pumped-storage hydropower plants. However, their starting and grid synchronization methods require numerous maneuvers or additional components, making the process challenging. In this paper, a soft start method for DFIM, inspired by the traditional synchronization method of synchronous machines, is proposed. This method involves matching the frequencies, voltages, and phase angles on both sides of the main circuit breaker, by adjusting the excitation through the controlled power converter at standstill conditions. Once synchronization is achieved, the frequency is gradually reduced to the rated operational levels. This straightforward starting method effectively suppresses large inrush currents and voltage sags. The proposed method has been validated through computer simulations and experimental tests, yielding satisfactory results." > <meta name="dc.subject" content="doubly fed induction generators" > <meta name="dc.subject" content="hydroelectric power generation" > <meta name="dc.subject" content="renewable energy" > <meta name="dc.subject" content="start-up" > <meta name="dc.subject" content="synchronization" > <meta name="dc.subject" content="wind energy generation" > <meta name ="prism.issn" content="2075-1702"> <meta name ="prism.publicationName" content="Machines"> <meta name ="prism.publicationDate" content="2024-11-25"> <meta name ="prism.volume" content="12"> <meta name ="prism.number" content="12"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="847" > <meta name="citation_issn" content="2075-1702"> <meta name="citation_journal_title" content="Machines"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions"> <meta name="citation_publication_date" content="2024/12"> <meta name="citation_online_date" content="2024/11/25"> <meta name="citation_volume" content="12"> <meta name="citation_issue" content="12"> <meta name="citation_firstpage" content="847"> <meta name="citation_author" content="Guerrero, José M."> <meta name="citation_author" content="Mahtani, Kumar"> <meta name="citation_author" content="Aranzabal, Itxaso"> <meta name="citation_author" content="Gómez-Cornejo, Julen"> <meta name="citation_author" content="Sánchez, José A."> <meta name="citation_author" content="Platero, Carlos A."> <meta name="citation_doi" content="10.3390/machines12120847"> <meta name="citation_id" content="mdpi-machines12120847"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2075-1702/12/12/847"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2075-1702/12/12/847/pdf?version=1732611295"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2075-1702/12/12/847/pdf?version=1732611295"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2075-1702/12/12/847/pdf?version=1732611295"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2075-1702/12/12/847/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2075-1702/12/12/847/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2075-1702/12/12/847/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2075-1702/12/12/847/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2075-1702/12/12/847/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2075-1702/12/12/847/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/machines-logo-social.png?8600e93ff98dbf14" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2075-1702/12/12/847" /> <meta property="og:title" content="A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions" /> <meta property="og:description" content="Due to their exceptional operational versatility, doubly fed induction machines (DFIM) are widely employed in power systems comprising variable renewable energy-based electrical generation sources, such as wind farms and pumped-storage hydropower plants. However, their starting and grid synchronization methods require numerous maneuvers or additional components, making the process challenging. In this paper, a soft start method for DFIM, inspired by the traditional synchronization method of synchronous machines, is proposed. This method involves matching the frequencies, voltages, and phase angles on both sides of the main circuit breaker, by adjusting the excitation through the controlled power converter at standstill conditions. Once synchronization is achieved, the frequency is gradually reduced to the rated operational levels. This straightforward starting method effectively suppresses large inrush currents and voltage sags. The proposed method has been validated through computer simulations and experimental tests, yielding satisfactory results." /> <meta property="og:image" content="https://pub.mdpi-res.com/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g001-550.jpg?1732611374" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); 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class='html-article-content'> <span itemprop="publisher" content="Multidisciplinary Digital Publishing Institute"></span><span itemprop="url" content="https://www.mdpi.com/2075-1702/12/12/847"></span> <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> <h1 class="title hypothesis_container" itemprop="name"> A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions </h1> <div class="art-authors hypothesis_container"> by <span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13444604' data-options='is_hover:true, hover_timeout:5000'> José M. 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width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a>, </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13444605' data-options='is_hover:true, hover_timeout:5000'> Kumar Mahtani</div><div id="profile-card-drop13444605" 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="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Kumar Mahtani</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/2549838?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=Kumar%20Mahtani" class="button button--color-inversed" target="_blank"> Scilit </a><a href="https://www.preprints.org/search?search1=Kumar%20Mahtani&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a><a href="https://scholar.google.com/scholar?q=Kumar%20Mahtani" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div></div><sup> 2</sup><span style="display: inline; 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Platero</div><div id="profile-card-drop13444609" 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="/profiles/48115/thumb/Carlos_A._Platero.bmp" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Carlos A. 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width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a></span> </div> <div class="nrm"></div> <span style="display:block; height:6px;"></span> <div></div> <div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-item"><sup>1</sup></div> <div class="affiliation-name ">Department of Electrical Engineering, Escuela de Ingeniería de Bilbao, Universidad del País Vasco—Euskal Herriko Unibersitatea, 48013 Bilbao, Spain</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>2</sup></div> <div class="affiliation-name ">Department of Electrical Engineering, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, 28006 Madrid, Spain</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>3</sup></div> <div class="affiliation-name ">Department of Hydraulics, Energy and Environmental Engineering, Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 28040 Madrid, Spain</div> </div> <div class="affiliation"> <div class="affiliation-item"><sup>*</sup></div> <div class="affiliation-name ">Author to whom correspondence should be addressed. </div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Machines</em> <b>2024</b>, <em>12</em>(12), 847; <a href="https://doi.org/10.3390/machines12120847">https://doi.org/10.3390/machines12120847</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 30 October 2024</span> / <span style="display: inline-block">Revised: 16 November 2024</span> / <span style="display: inline-block">Accepted: 22 November 2024</span> / <span style="display: inline-block">Published: 25 November 2024</span> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Section <a href="/journal/machines/sections/Electrical_Machines_and_Drives">Electrical Machines and Drives</a>)<br/> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-1529093" aria-controls="drop-supplementary-1529093" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-1529093" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2075-1702/12/12/847/pdf?version=1732611295" data-name="A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions" data-journal="machines">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2075-1702/12/12/847/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2075-1702/12/12/847/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2075-1702/12/12/847/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2075-1702/12/12/847/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g001.png?1732611371" title=" <strong>Figure 1</strong><br/> <p>Schemes for DFIM start-up: (<b>a</b>) Opposite phase sequence; (<b>b</b>) Autotransformer and variable resistors; (<b>c</b>) Stator short-circuit; (<b>d</b>) Auxiliary converter.</p> "> </a> <a href="https://pub.mdpi-res.com/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g002.png?1732611374" title=" <strong>Figure 2</strong><br/> <p>Electrical scheme for the proposed start-up method [<span class="html-italic">U<sub>grid</sub></span>: grid voltage; <span class="html-italic">f<sub>grid</sub></span>: grid frequency; <span class="html-italic">U<sub>s</sub></span>: stator voltage; <span class="html-italic">f<sub>s</sub></span>: stator frequency; <span class="html-italic">U<sub>r</sub></span>: rotor voltage; <span class="html-italic">f<sub>r</sub></span>: rotor frequency; <span class="html-italic">ΔU</span> = |<span class="html-italic">U<sub>grid</sub></span>|−|<span class="html-italic">U<sub>s</sub></span>|; <span class="html-italic">Δ<sub>f</sub></span> = <span class="html-italic">f<sub>grid</sub></span>−<span class="html-italic">f<sub>s</sub></span>; <span class="html-italic">Δφ</span>: phase difference between <span class="html-italic">U<sub>grid</sub></span> and <span class="html-italic">U<sub>s</sub></span> phasors].</p> "> </a> <a href="https://pub.mdpi-res.com/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g003.png?1732611377" title=" <strong>Figure 3</strong><br/> <p>Conceptual algorithm of the proposed DFIM start-up method.</p> "> </a> <a href="https://pub.mdpi-res.com/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g004.png?1732611378" title=" <strong>Figure 4</strong><br/> <p>Simulation model for the stator synchronization-based start-up method.</p> "> </a> <a href="https://pub.mdpi-res.com/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g005.png?1732611379" title=" <strong>Figure 5</strong><br/> <p>Simulation results for a stator synchronization start-up [Mechanical rotor speed (<span class="html-italic">ω<sub>mec</sub></span>); torque (<span class="html-italic">T<sub>m</sub></span>); RMS values of rotor and stator currents (<span class="html-italic">I<sub>r</sub></span> and <span class="html-italic">I<sub>s</sub></span>, respectively); rotor voltage (<span class="html-italic">U<sub>r</sub></span>), in p.u].</p> "> </a> <a href="https://pub.mdpi-res.com/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g006.png?1732611381" title=" <strong>Figure 6</strong><br/> <p>Grid and stator voltages during synchronization using the proposed DFIM start-up method.</p> "> </a> <a href="https://pub.mdpi-res.com/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g007.png?1732611382" title=" <strong>Figure 7</strong><br/> <p>Experimental setup: electrical scheme [<span class="html-italic">U<sub>exc</sub></span> = Excitation voltage of the SG].</p> "> </a> <a href="https://pub.mdpi-res.com/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g008.png?1732611387" title=" <strong>Figure 8</strong><br/> <p>Experimental setup: overview [(1): VFD; (2): SCIM; (3): SG; (4): DFIM; (5): Autotransformer; (6): DC excitation system; (7): main CB; (8): Synchronoscope; (9) and (10): Voltmeter; (11): <span class="html-italic">I<sub>r</sub></span> measurement; (12): <span class="html-italic">I<sub>s</sub></span> measurement; (13): Voltmeter; (14): Oscilloscope].</p> "> </a> <a href="https://pub.mdpi-res.com/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g009.png?1732611388" title=" <strong>Figure 9</strong><br/> <p>Experimental results for a stator synchronization start-up [Mechanical rotor speed (<span class="html-italic">ω<sub>mec</sub></span>); torque (<span class="html-italic">T<sub>m</sub></span>); RMS values of rotor and stator currents (<span class="html-italic">I<sub>r</sub></span> and <span class="html-italic">I<sub>s</sub></span>, respectively); rotor voltage (<span class="html-italic">U<sub>r</sub></span>), in p.u].</p> "> </a> </div> <a class="button button--color-inversed" href="/2075-1702/12/12/847/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">Due to their exceptional operational versatility, doubly fed induction machines (DFIM) are widely employed in power systems comprising variable renewable energy-based electrical generation sources, such as wind farms and pumped-storage hydropower plants. However, their starting and grid synchronization methods require numerous maneuvers or additional components, making the process challenging. In this paper, a soft start method for DFIM, inspired by the traditional synchronization method of synchronous machines, is proposed. This method involves matching the frequencies, voltages, and phase angles on both sides of the main circuit breaker, by adjusting the excitation through the controlled power converter at standstill conditions. Once synchronization is achieved, the frequency is gradually reduced to the rated operational levels. This straightforward starting method effectively suppresses large inrush currents and voltage sags. The proposed method has been validated through computer simulations and experimental tests, yielding satisfactory results.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=doubly+fed+induction+generators">doubly fed induction generators</a>; <a href="/search?q=hydroelectric+power+generation">hydroelectric power generation</a>; <a href="/search?q=renewable+energy">renewable energy</a>; <a href="/search?q=start-up">start-up</a>; <a href="/search?q=synchronization">synchronization</a>; <a href="/search?q=wind+energy+generation">wind energy generation</a></div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-machines-12-00847' type='intro'><h2 data-nested='1'> 1. Introduction</h2><div class='html-p'>Addressing environmental challenges is one of the foremost issues in today’s world. They are inherently tied to the adoption of cleaner energy sources for electrical power generation [<a href="#B1-machines-12-00847" class="html-bibr">1</a>,<a href="#B2-machines-12-00847" class="html-bibr">2</a>]. In this regard, wind power and hydropower generation are among the most extensively adopted renewable energy sources worldwide, due to their abundant availability, scalability, and relatively low environmental impact.</div><div class='html-p'>In the realm of wind power generation [<a href="#B3-machines-12-00847" class="html-bibr">3</a>,<a href="#B4-machines-12-00847" class="html-bibr">4</a>], doubly fed induction machines (DFIM) technology remains the most widely implemented, as part of the notably known Type-III wind energy conversion systems. Similarly, DFIMs are also predominantly utilized in pumped storage hydropower plants [<a href="#B5-machines-12-00847" class="html-bibr">5</a>,<a href="#B6-machines-12-00847" class="html-bibr">6</a>,<a href="#B7-machines-12-00847" class="html-bibr">7</a>,<a href="#B8-machines-12-00847" class="html-bibr">8</a>,<a href="#B9-machines-12-00847" class="html-bibr">9</a>,<a href="#B10-machines-12-00847" class="html-bibr">10</a>,<a href="#B11-machines-12-00847" class="html-bibr">11</a>,<a href="#B12-machines-12-00847" class="html-bibr">12</a>], where water is pumped or turbined based on demand requirements.</div><div class='html-p'>DFIMs play a crucial role in integrating variable renewable energy into the grid [<a href="#B13-machines-12-00847" class="html-bibr">13</a>,<a href="#B14-machines-12-00847" class="html-bibr">14</a>,<a href="#B15-machines-12-00847" class="html-bibr">15</a>], ensuring voltage and frequency controls under varying primary energy inputs, according to applicable regulations. Designed for adjustable speed operation, DFIM systems offer numerous operational advantages, including high energy yield, reduced mechanical stress, low power output fluctuations, and precise control over both active and reactive power.</div><div class='html-p'>Typically, DFIM systems are directly grid-connected, featuring a reduced-power converter [<a href="#B16-machines-12-00847" class="html-bibr">16</a>] connected to the rotor that operates at variable frequency. This configuration facilitates the straightforward adjustment of the DFIM’s operating point, to accommodate varying primary energy inputs.</div><div class='html-p'>Despite the aforementioned benefits, DFIMs also present certain drawbacks [<a href="#B17-machines-12-00847" class="html-bibr">17</a>,<a href="#B18-machines-12-00847" class="html-bibr">18</a>,<a href="#B19-machines-12-00847" class="html-bibr">19</a>]: being directly grid-connected, they exhibit high sensitivity to external grid faults or disturbances. Additionally, they require regular maintenance due to the wear and tear of slip rings and brushes, and in the case of wind turbines, gearbox maintenance also becomes necessary.</div><div class='html-p'>Another significant limitation of DFIMs is their start-up process, extensively addressed in the literature [<a href="#B20-machines-12-00847" class="html-bibr">20</a>,<a href="#B21-machines-12-00847" class="html-bibr">21</a>,<a href="#B22-machines-12-00847" class="html-bibr">22</a>,<a href="#B23-machines-12-00847" class="html-bibr">23</a>,<a href="#B24-machines-12-00847" class="html-bibr">24</a>,<a href="#B25-machines-12-00847" class="html-bibr">25</a>,<a href="#B26-machines-12-00847" class="html-bibr">26</a>,<a href="#B27-machines-12-00847" class="html-bibr">27</a>,<a href="#B28-machines-12-00847" class="html-bibr">28</a>,<a href="#B29-machines-12-00847" class="html-bibr">29</a>,<a href="#B30-machines-12-00847" class="html-bibr">30</a>,<a href="#B31-machines-12-00847" class="html-bibr">31</a>,<a href="#B32-machines-12-00847" class="html-bibr">32</a>,<a href="#B33-machines-12-00847" class="html-bibr">33</a>,<a href="#B34-machines-12-00847" class="html-bibr">34</a>,<a href="#B35-machines-12-00847" class="html-bibr">35</a>,<a href="#B36-machines-12-00847" class="html-bibr">36</a>,<a href="#B37-machines-12-00847" class="html-bibr">37</a>,<a href="#B38-machines-12-00847" class="html-bibr">38</a>,<a href="#B39-machines-12-00847" class="html-bibr">39</a>]. Most of the proposed approaches involve additional elements such as auxiliary power converters and switches to adjust DFIM conditions to the grid, necessitating multiple maneuvers during the start-up process. Consequently, this complexity not only challenges the start-up procedure but also increases the overall cost of electric generation units.</div><div class='html-p'>This paper proposes a soft start-up method for DFIMs, based on the synchronization with the power system at standstill conditions. This approach simplifies DFIM starting and eliminates the need for additional components. Analogous to the paralleling of synchronous generators, this technique primarily involves aligning DFIM conditions with those of the grid, and, once achieved, synchronizing with the grid by closing the main circuit breaker (CB) at standstill conditions. Subsequently, using the rotor’s power converter for voltage and frequency control, the machine accelerates to operating speed. This procedure mitigates inrush currents and voltage sags while reducing the complexity of maneuvers and devices required during the start-up process.</div><div class='html-p'>This paper is structured as follows: <a href="#sec2-machines-12-00847" class="html-sec">Section 2</a> provides a brief overview of DFIM start-up procedures. <a href="#sec3-machines-12-00847" class="html-sec">Section 3</a> describes the proposed start-up method in detail. <a href="#sec4-machines-12-00847" class="html-sec">Section 4</a> presents computer simulation results for a DFIM start-up using the proposed method. <a href="#sec5-machines-12-00847" class="html-sec">Section 5</a> reports experimental test results under the same conditions as the simulations, with a discussion of the obtained results. Finally, <a href="#sec6-machines-12-00847" class="html-sec">Section 6</a> concludes the paper by summarizing its main ideas and contributions.</div></section><section id='sec2-machines-12-00847' type=''><h2 data-nested='1'> 2. State of the Art</h2><div class='html-p'>The approaches to DFIM start-up have been classified into four main groups. The main advantages and drawbacks of each of them are summarized in <a href="#machines-12-00847-t001" class="html-table">Table 1</a>. In the following, the categories are described based on <a href="#machines-12-00847-f001" class="html-fig">Figure 1</a>.</div><section id='sec2dot1-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 2.1. Opposite Phase Sequence-Based Start-Up Methods</h4><div class='html-p'>In these methods, the excitation power converter placed in the rotor of the DFIM and an additional switch (S1, initially closed) is utilized, as illustrated in <a href="#machines-12-00847-f001" class="html-fig">Figure 1</a>a. The start-up procedure involves feeding both the stator and the rotor of the DFIM through the power converter, with two phases swapped at the connection between the stator and the rotor using S1. Under these conditions, the machine is driven to its rated speed. Then, S1 is opened and the CB connecting the DFIM stator to the power system is closed.</div><div class='html-p'>The main advantage of this method is its short starting time [<a href="#B20-machines-12-00847" class="html-bibr">20</a>]. However, the dead time during the commutation between S1 and the CB can induce pulsed torques that may severely damage the machine shaft if synchronization is not achieved shortly after the opening of S1. In [<a href="#B20-machines-12-00847" class="html-bibr">20</a>], these pulsed torques are examined and compared with other methods.</div></section><section id='sec2dot2-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 2.2. Reduced Voltage-Based Start-Up Methods</h4><div class='html-p'>There are various methods to perform a DFIM start-up at low voltage. One approach uses an autotransformer (AT) connected to the stator terminals and a variable resistor (<span class='html-italic'>R<sub>var</sub></span>) in the rotor terminals [<a href="#B21-machines-12-00847" class="html-bibr">21</a>], as shown in <a href="#machines-12-00847-f001" class="html-fig">Figure 1</a>b. Consequently, two additional switches (S2 and S3, initially closed) are needed to connect and disconnect these elements. The variable resistor enables a progressive start-up. When the motor reaches approximately 95% of its rated speed [<a href="#B22-machines-12-00847" class="html-bibr">22</a>], S2 is disconnected. At this point, the AT functions as a current limiter. Simultaneously, S3 is opened, and S1 is closed, feeding the rotor through the power converter. Finally, the CB is connected, bypassing the AT.</div><div class='html-p'>Other techniques rely on the converter to control the variable speed drive, to achieve reduced voltage-based start-up without the need of the AT and the variable resistor [<a href="#B23-machines-12-00847" class="html-bibr">23</a>]. One of the techniques for starting a DFIM at low voltage involves short-circuiting the stator winding [<a href="#B24-machines-12-00847" class="html-bibr">24</a>,<a href="#B25-machines-12-00847" class="html-bibr">25</a>,<a href="#B26-machines-12-00847" class="html-bibr">26</a>,<a href="#B27-machines-12-00847" class="html-bibr">27</a>,<a href="#B28-machines-12-00847" class="html-bibr">28</a>,<a href="#B29-machines-12-00847" class="html-bibr">29</a>,<a href="#B30-machines-12-00847" class="html-bibr">30</a>,<a href="#B31-machines-12-00847" class="html-bibr">31</a>,<a href="#B32-machines-12-00847" class="html-bibr">32</a>,<a href="#B33-machines-12-00847" class="html-bibr">33</a>,<a href="#B34-machines-12-00847" class="html-bibr">34</a>]. This technique is most commonly used for large generator units due to its low pulsed torques and currents. <a href="#machines-12-00847-f001" class="html-fig">Figure 1</a>c shows the electrical circuit for a stator short-circuit start-up. Only one additional switch (S1, initially closed) is required to perform the short-circuit. First, the rotor converter increases the rotor frequency (<span class='html-italic'>f<sub>r</sub></span>) up to the rated frequency. Once the DFIM reaches its rated speed, S1 is opened, and after meeting the synchronization conditions, the CB is closed. With this start-up method, the power converter must supply the rotor with a voltage proportional to <span class='html-italic'>f<sub>r</sub></span> and reduce the voltage as the frequency decreases (constant V/Hz ratio) [<a href="#B24-machines-12-00847" class="html-bibr">24</a>,<a href="#B25-machines-12-00847" class="html-bibr">25</a>]. This ensures that the air gap flux remains constant [<a href="#B24-machines-12-00847" class="html-bibr">24</a>,<a href="#B25-machines-12-00847" class="html-bibr">25</a>,<a href="#B26-machines-12-00847" class="html-bibr">26</a>].</div><div class='html-p'>Additionally, the stator short-circuit approach requires fast switching between S1 and the CB, e.g., in [<a href="#B28-machines-12-00847" class="html-bibr">28</a>] this process occurs in 40 ms. Otherwise, the machine and grid conditions may differ significantly, leading to an abrupt grid connection. Several variants of stator short-circuit start-ups exist: using Field Oriented Control (FOC) to accelerate the machine [<a href="#B28-machines-12-00847" class="html-bibr">28</a>,<a href="#B29-machines-12-00847" class="html-bibr">29</a>,<a href="#B30-machines-12-00847" class="html-bibr">30</a>,<a href="#B31-machines-12-00847" class="html-bibr">31</a>,<a href="#B32-machines-12-00847" class="html-bibr">32</a>], applying maximum torque start-up [<a href="#B33-machines-12-00847" class="html-bibr">33</a>], or implementing power loss reduction techniques [<a href="#B34-machines-12-00847" class="html-bibr">34</a>,<a href="#B35-machines-12-00847" class="html-bibr">35</a>]. The dynamic behavior of this approach under converter and sensor faults has also been addressed in previous works [<a href="#B36-machines-12-00847" class="html-bibr">36</a>].</div><div class='html-p'>Similarly, the short-circuit can be applied to the rotor winding [<a href="#B23-machines-12-00847" class="html-bibr">23</a>]. In this case, the power converter feeds the stator through an additional switch. The start-up is conducted as if the machine were an ordinary squirrel-cage induction machine-based variable frequency drive. Once synchronization with the grid is achieved, the power converter is disconnected from the stator, which is then fed directly from the grid, and subsequently connected to the rotor [<a href="#B26-machines-12-00847" class="html-bibr">26</a>]. Improvements such as FOC can also be applied in this case [<a href="#B37-machines-12-00847" class="html-bibr">37</a>].</div></section><section id='sec2dot3-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 2.3. Auxiliary Converter-Based Start-Up Methods</h4><div class='html-p'>Auxiliary converter-based start-up methods are softer than the previously presented alternatives, but they require additional electronic devices [<a href="#B32-machines-12-00847" class="html-bibr">32</a>,<a href="#B38-machines-12-00847" class="html-bibr">38</a>,<a href="#B39-machines-12-00847" class="html-bibr">39</a>].</div><div class='html-p'>In [<a href="#B38-machines-12-00847" class="html-bibr">38</a>], an auxiliary power converter is installed at the stator terminals to initiate the DFIM operation. The electrical scheme is shown in <a href="#machines-12-00847-f001" class="html-fig">Figure 1</a>d. Initially, S1, S2, and S3 are closed, allowing the auxiliary converter to feed the stator at the rated frequency. The converter also supplies the rotor to achieve the required rotor speed. Once the operational point is set, the CB is closed under proper synchronization conditions of voltage, frequency, and phase, while the auxiliary power converter remains bypassed. At this point, S2 and S3 are opened, concluding the start-up process. Meanwhile, in [<a href="#B32-machines-12-00847" class="html-bibr">32</a>], the DC bus of the rotor power converter feeds with an additional inverter the machine in the stator and rotor at the same time.</div><div class='html-p'>The use of DC/DC converters in the main rotor power converter is another option for synchronizing the machine [<a href="#B39-machines-12-00847" class="html-bibr">39</a>]. When both sides of the DC bus have the same voltage, the DC bus is closed, connecting the grid side converter (GSC) with the rotor side converter (RSC).</div></section><section id='sec2dot4-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 2.4. Proposed Start-Up Methods</h4><div class='html-p'>It is worth noting that all the previously discussed methods require additional elements, such as switches, resistors, or power electronic devices, making their implementation more complex and costlier.</div><div class='html-p'>The method proposed in this work solves these issues through a simple synchronization-based approach that reduces complexity and costs. One of the main advantages of this method is that only the main CB is needed for the start-up. Additionally, it retains the benefits of other soft start-up methods, such as avoiding high inrush currents and pulsing torques.</div></section></section><section id='sec3-machines-12-00847' type=''><h2 data-nested='1'> 3. Description of the Proposed Soft Start-Up Method</h2><div class='html-p'>This section describes the theoretical background of the proposed soft start-up method and outlines the steps to implement it (<a href="#sec3dot1-machines-12-00847" class="html-sec">Section 3.1</a>). To this end, the dynamic equations of the DFIM are presented (<a href="#sec3dot2-machines-12-00847" class="html-sec">Section 3.2</a>). Since the present method is based on synchronization to the power system, the conditions on both sides of the switch/CB must be equal in terms of phase sequence, voltage amplitude, frequency, and phase angle. Initially, the DFIM does not rotate unless it is driven by an external torque, because the stator windings are open before synchronization by the time the DFIM is energized through the rotor. This synchronization at standstill conditions is analyzed (<a href="#sec3dot3-machines-12-00847" class="html-sec">Section 3.3</a>). Finally, using the rotor’s power converter voltage/frequency control, the machine is accelerated, which is addressed at the end of this section (<a href="#sec3dot4-machines-12-00847" class="html-sec">Section 3.4</a>).</div><div class='html-p'>The electrical scheme required for the implementation of the proposed method in pumped storage hydropower plants is shown in <a href="#machines-12-00847-f002" class="html-fig">Figure 2</a>.</div><div class='html-p'>As shown in <a href="#machines-12-00847-f002" class="html-fig">Figure 2</a>, the stator of the DFIM is connected to the system through the main power transformer and the main CB, which is equipped with an automatic synchronizing relay. The rotor winding is supplied via two back-to-back converters (grid side and rotor side converters, interfaced by a DC link), powered through an excitation transformer. The shaft is mechanically coupled to the hydro turbine.</div><section id='sec3dot1-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 3.1. Method Overview</h4><div class='html-p'>The algorithm that summarizes the start-up process is presented in <a href="#machines-12-00847-f003" class="html-fig">Figure 3</a>. At the initial conditions, the DFIM is at standstill conditions (zero mechanical speed, <span class='html-italic'>ω<sub>mec</sub></span> = 0), with the main CB in an open state, i.e., isolated from the grid. In this scenario, grid synchronization conditions must be met.</div><div class='html-p'>To achieve the synchronization conditions (voltage amplitude, frequency, and phase angle), the rotor’s power converter must control the voltage frequency and amplitude at its terminals. The frequency of the rotor’s currents should be defined according to Equation (1):<div class='html-disp-formula-info' id='FD1-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mi>m</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mi>p</mi> </mrow> </mfrac> </mstyle> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> where <span class='html-italic'>p</span> is the number of pole pairs of the DFIM, <span class='html-italic'>f<sub>s</sub></span> is the stator frequency and <span class='html-italic'>f<sub>r</sub></span> is the rotor frequency.</div><div class='html-p'>From Equation (1), it follows that for a start-up from <span class='html-italic'>ω<sub>mec</sub></span> = 0, the electrical frequency of the rotor (<span class='html-italic'>f<sub>r</sub></span>) imposed at the terminals of the power converter must be equal to the stator frequency (<span class='html-italic'>f<sub>s</sub></span>), which is also the grid frequency (<span class='html-italic'>f<sub>grid</sub></span>). Furthermore, for a start-up from zero <span class='html-italic'>ω<sub>mec</sub></span>, which is a common scenario for motor start-ups, the imposed voltage at the power converter terminals must be the rotor rated voltage (<span class='html-italic'>U<sub>r</sub></span>) for <span class='html-italic'>f<sub>r</sub></span> = <span class='html-italic'>f<sub>s</sub></span>. This ensures that the induced stator voltage (<span class='html-italic'>U<sub>s</sub></span>) matches the grid voltage (<span class='html-italic'>U<sub>grid</sub></span>) prior to synchronization (<span class='html-italic'>U<sub>r</sub></span> = <span class='html-italic'>U<sub>grid</sub></span>/<span class='html-italic'>r<sub>t</sub>,</span> with <span class='html-italic'>r<sub>t</sub></span> being the rotor/stator voltage ratio). The imposed voltage at the power converter terminals should be controlled under a constant flux strategy. It must be noticed that in the case of turbine start-up mode, i.e., the shaft can be already rotating, <span class='html-italic'>f<sub>r</sub></span> should be the difference between shaft frequency and <span class='html-italic'>f<sub>grid</sub></span> to impose <span class='html-italic'>f<sub>s</sub></span> = <span class='html-italic'>f<sub>grid</sub></span> and then perform the CB closure. The following explanations describe the motor start-up mode, as it is more critical than the turbine one due to the counter torque present in this mode of operation.</div><div class='html-p'>Provided that the phase sequence is identical on both sides of the CB, once the converter control has matched the voltage and frequency values with those of the grid (Δ<span class='html-italic'>U</span> = |<span class='html-italic'>U<sub>grid</sub></span>|−|<span class='html-italic'>U<sub>s</sub></span>| = 0 and Δ<span class='html-italic'>f</span> = <span class='html-italic'>f<sub>grid</sub></span>−<span class='html-italic'>f<sub>s</sub></span> = 0), the breaker (CB) is closed when the phase shift is null (Δφ = 0). The machine is thus synchronized to the grid at standstill conditions. Since the rotor’s frequency is the same as the stator’s, their fluxes are synchronized, preventing the generation of a torque. Consequently, the shaft remains at <span class='html-italic'>ω<sub>mec</sub></span> = 0 after synchronization.</div><div class='html-p'>After synchronization, the power converter gradually reduces its output frequency, causing the shaft to start rotating according to (1). The rotor voltage (<span class='html-italic'>U<sub>r</sub></span>) should also be proportionally reduced to maintain the DFIM flux (constant <span class='html-italic'>U<sub>r</sub></span>/<span class='html-italic'>f<sub>r</sub></span>). The starting process concludes once the power converter reaches the rated <span class='html-italic'>f<sub>r</sub></span>. During this process, the DFIM accelerates from a unitary slip (<span class='html-italic'>s</span>) value to the rated <span class='html-italic'>s</span> value in a downward ramp progression.</div><div class='html-p'>The main advantage of this method over the methods analyzed in <a href="#sec2-machines-12-00847" class="html-sec">Section 2</a> is that it does not require additional elements such as switches, breakers, autotransformers, variable resistances, or auxiliary converters. Only the main CB is needed. Moreover, the method is characterized by its simplicity and safety, as it avoids any short-circuit conditions. Additionally, inrush currents and voltage sags during start-up are eliminated.</div><div class='html-p'>However, despite the benefit of using only the main CB, synchronization conditions must be carefully observed in the stator. Consequently, the high-frequency pulses generated by the power converter, due to pulse width modulation (PWM), will flow through the DFIM during the start-up process. This results in increased electrical insulation stress. Finally, it is important to note that the lower the frequency variation in the power converter, the smoother the start-up will be.</div></section><section id='sec3dot2-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 3.2. Dynamic Equations of the DFIM</h4><div class='html-p'>In order to set up the dynamic equations of a DFIM [<a href="#B40-machines-12-00847" class="html-bibr">40</a>], it is assumed that there is a sinusoidal field distribution along the periphery of the airgap, no zero-sequence currents, and constant saturation throughout the process. In the per unit (p.u.) system, the phasor equations for the stator and rotor are given in the excitation or stator-flux reference frame by Equations (2) and (3), respectively:<div class='html-disp-formula-info' id='FD2-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <munder> <mrow> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow> <munder> <mrow> <mi>Ψ</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>+</mo> <mi>j</mi> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msub> <mrow> <munder> <mrow> <mi>Ψ</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div><div class='html-disp-formula-info' id='FD3-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <munder> <mrow> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow> <munder> <mrow> <mi>Ψ</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>+</mo> <mi>j</mi> <msub> <mrow> <mo>(</mo> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mi>m</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <mo>)</mo> <munder> <mrow> <mi>Ψ</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div> where <span class='html-italic'>Ψ<sub>s</sub></span> is the stator flux, <span class='html-italic'>Ψ<sub>r</sub></span> is the rotor flux, <span class='html-italic'>ω<sub>0</sub></span> is the angular speed, <span class='html-italic'>R<sub>s</sub></span> is the stator resistance, <span class='html-italic'>R<sub>r</sub></span> is the rotor resistance, and <span class='html-italic'>I<sub>s</sub></span> and <span class='html-italic'>I<sub>r</sub></span> are the stator and rotor currents, respectively.</div><div class='html-p'>The stator and rotor currents and fluxes are related by Equations (4) and (5) as follows:<div class='html-disp-formula-info' id='FD4-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msub> <mrow> <munder> <mrow> <mi>Ψ</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div><div class='html-disp-formula-info' id='FD5-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <munder> <mrow> <mi>Ψ</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div> where <span class='html-italic'>X<sub>s</sub></span> is the stator leakage reactance, <span class='html-italic'>X<sub>m</sub></span> is the magnetizing reactance and <span class='html-italic'>X<sub>r</sub></span> is the rotor leakage reactance of the DFIM.</div><div class='html-p'>Substituting Equations (4) and (5) into Equations (2) and (3), the following Equations (6) and (7) are obtained:<div class='html-disp-formula-info' id='FD6-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <munder> <mrow> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mfenced> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo> </mo> <mo>+</mo> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>+</mo> <mi>j</mi> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfenced> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div><div class='html-disp-formula-info' id='FD7-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <munder> <mrow> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mi>s</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mfenced> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>+</mo> <mfenced separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>+</mo> <mi>j</mi> <mi>s</mi> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfenced> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div></div><div class='html-p'>Additionally, the mechanical characteristics are expressed by Equations (8) and (9):<div class='html-disp-formula-info' id='FD8-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>d</mi> <mi>ω</mi> </mrow> <mrow> <mi>m</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>H</mi> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div><div class='html-disp-formula-info' id='FD9-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> <mo>·</mo> <mi>I</mi> <mi>m</mi> <mfenced open="{" close="}" separators="|"> <mrow> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>·</mo> <msubsup> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mi>*</mi> </mrow> </msubsup> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div> where <span class='html-italic'>T<sub>m</sub></span> is the mechanical torque, <span class='html-italic'>T<sub>r</sub></span> is the mechanical counter-torque in the shaft of the DFIM and <span class='html-italic'>H</span> is the rotor inertia constant.</div><div class='html-p'>These equations completely define the transient evolution of the machine state variables.</div></section><section id='sec3dot3-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 3.3. Stator Synchronization Analysis</h4><div class='html-p'>The power converter should provide the magnetizing current during the process and should impose the synchronization voltage. Consequently, <span class='html-italic'>I<sub>s</sub></span> should be zero when closing the CB. According to Equations (6) and (7), Equations (10) and (11) are derived, respectively, as the time derivatives should be null and <span class='html-italic'>f<sub>r</sub></span> = <span class='html-italic'>f<sub>s</sub></span>:<div class='html-disp-formula-info' id='FD10-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mfenced open="" close="|" separators="|"> <mrow> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mi>s</mi> <mi>y</mi> <mi>n</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <munder> <mrow> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <mi>j</mi> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div><div class='html-disp-formula-info' id='FD11-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mfenced open="" close="|" separators="|"> <mrow> <msub> <mrow> <munder> <mrow> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mi>s</mi> <mi>y</mi> <mi>n</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mfenced> </mrow> <mrow> <mi>j</mi> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <msub> <mrow> <munder> <mrow> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div></div><div class='html-p'>This implies that the converter should impose a voltage slightly larger than the rated one. As this is usually not possible, the strategy should be adjusted: <span class='html-italic'>U<sub>r</sub></span> will be set at its rated value and its phase angle (<span class='html-italic'>φ</span>) should be such that the stator voltage phase angle coincides with <span class='html-italic'>U<sub>grid</sub></span> phase angle (zero phase shift condition). Taking <span class='html-italic'>U<sub>grid</sub></span> as a reference, then, from Equation (6) just before closing the CB:<div class='html-disp-formula-info' id='FD12-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mfenced open="" close="|" separators="|"> <mrow> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mi>s</mi> <mi>y</mi> <mi>n</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>j</mi> <msub> <mrow> <mfenced open="" close="|" separators="|"> <mrow> <msub> <mrow> <mi>I</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mi>s</mi> <mi>y</mi> <mi>n</mi> <mi>c</mi> </mrow> </msub> </mrow> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div></div><div class='html-p'>Introducing Equation (12) into Equation (7), and assuming that <span class='html-italic'>U<sub>r</sub></span> = 1 p.u. (with unknown phase angle, <span class='html-italic'>φ<sub>r</sub></span>):<div class='html-disp-formula-info' id='FD13-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mi>j</mi> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>j</mi> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mfenced> <msub> <mrow> <mi>I</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math> </div> <div class='l'> <label >(13)</label> </div> </div></div><div class='html-p'>Therefore, the <span class='html-italic'>I<sub>r</sub></span> amplitude and the phasor difference between <span class='html-italic'>U<sub>grid</sub></span> and the stator voltage at the synchronization instant are given by Equations (14) and (15), respectively:<div class='html-disp-formula-info' id='FD14-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mfenced open="" close="|" separators="|"> <mrow> <msub> <mrow> <mi>I</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mi>s</mi> <mi>y</mi> <mi>n</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msqrt> <msubsup> <mrow> <mi>R</mi> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(14)</label> </div> </div><div class='html-disp-formula-info' id='FD15-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mfenced open="" close="|" separators="|"> <mrow> <msub> <mrow> <munder> <mrow> <mo>∆</mo> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mi>s</mi> <mi>y</mi> <mi>n</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <msqrt> <msubsup> <mrow> <mi>R</mi> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(15)</label> </div> </div></div><div class='html-p'>Due to the typical values of DFIM parameters, this voltage difference is very small. Thus, the closing transient <span class='html-italic'>I<sub>s</sub></span> will be also small and lower than its rated value. The initial (just after closing CB) time derivative of <span class='html-italic'>I<sub>s</sub></span> is expressed by Equation (16):<div class='html-disp-formula-info' id='FD16-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mfenced open="" close="|" separators="|"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> </mfenced> </mrow> <mrow> <mi>s</mi> <mi>y</mi> <mi>n</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>·</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <msqrt> <msubsup> <mrow> <mi>R</mi> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(16)</label> </div> </div></div></section><section id='sec3dot4-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 3.4. DFIM Acceleration</h4><div class='html-p'>Once the DFIM is synchronized with the grid, it should be accelerated by reducing the magnetizing current provided by the power converter. As a consequence, the power consumption from the grid increases to compensate for this reduction.</div><div class='html-p'>To better analyze this process, starting electromagnetic transients shall be neglected. This can be safely completed because the starting is designed to be smooth, i.e., electrical variables will change slowly, and their time (<span class='html-italic'>t</span>) decay is fast. So, time derivatives in Equations (6) and (7) will be assumed null. Additionally, to keep the magnetic flux constant, the ratio between <span class='html-italic'>U<sub>r</sub></span> and <span class='html-italic'>f<sub>r</sub></span> will also be constant (<span class='html-italic'>U<sub>r</sub></span>/<span class='html-italic'>f<sub>r</sub></span> = 1 p.u.). Under these assumptions, <span class='html-italic'>I<sub>r</sub></span> can be calculated as in Equation (17):<div class='html-disp-formula-info' id='FD17-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <munder> <mrow> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mfenced> <mo>−</mo> <mi>j</mi> <msub> <mrow> <mi>s</mi> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> <msub> <mrow> <munder> <mrow> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <mi>s</mi> <msubsup> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mi>s</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mfenced> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mfenced> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(17)</label> </div> </div></div><div class='html-p'>Therefore, <span class='html-italic'>I<sub>s</sub></span> follows Equation (18):<div class='html-disp-formula-info' id='FD18-machines-12-00847'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <munder> <mrow> <mi>U</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>−</mo> <mi>j</mi> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> <msub> <mrow> <munder> <mrow> <mi>I</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(18)</label> </div> </div></div><div class='html-p'>These equations, along with Equations (8) and (9), allow calculating the machine’s starting dynamics. The acceleration process finalizes when the power converter reaches the rated rotor’s frequency, i.e., the machine slip is reduced from the unitary value to the rated value.</div></section></section><section id='sec4-machines-12-00847' type=''><h2 data-nested='1'> 4. Computer Simulations</h2><div class='html-p'>To validate the proposed method, numerous computer simulations have been performed using Matlab-Simulink<sup>®</sup> (R2023a). For these simulations, a conventional 0.52-kW DFIM model was considered. In this section, the simulation model and the results of the proposed start-up technique are described.</div><section id='sec4dot1-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 4.1. Simulation Model</h4><div class='html-p'>In <a href="#machines-12-00847-f004" class="html-fig">Figure 4</a> the simulation model method is shown. <a href="#machines-12-00847-t002" class="html-table">Table 2</a> lists the rated values of the DFIM’s most significant parameters.</div><div class='html-p'>In this simulation model, the 0.52-kW DFIM is connected to a 400-V, 50-Hz three-phase grid through a CB equipped with a synchronizer. Additionally, high-value resistances have been connected in parallel to the DFIM grid terminals in order to avoid convergence errors during the simulation.</div><div class='html-p'>In the rotor circuit, the power converter (fed by a 120-V three-phase grid) is connected in series with a 1-mH choke inductance (equivalent to the secondary of the excitation transformer seen in <a href="#machines-12-00847-f002" class="html-fig">Figure 2</a>). The grid-side converter is controlled to maintain the voltage at the DC bus (<span class='html-italic'>U<sub>DC</sub></span>), which employs a 1600-μF capacitor. The setpoint of <span class='html-italic'>U<sub>DC</sub></span> depends on <span class='html-italic'>f<sub>r</sub></span>.</div><div class='html-p'>On the machine side of the power converter, the inverter generates a PWM signal to obtain the required voltage and frequency at each instant. When starting the machine at standstill conditions, the frequency through the windings is about 49.95 Hz, a value close to the grid’s frequency, allowing for soft synchronization. Once the machine is synchronized, assuming the conditions for voltage, frequency, and phase angle are met, this frequency decreases to about 2 Hz as a downward ramp function. The total acceleration <span class='html-italic'>t</span> has been set at 5 s to achieve a soft start-up. If the ramp is too steep, there is a risk of overcurrent due to the DFIM’s moment of inertia (<span class='html-italic'>J</span>).</div></section><section id='sec4dot2-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 4.2. Results</h4><div class='html-p'><a href="#machines-12-00847-f005" class="html-fig">Figure 5</a> shows the simulation results for the main parameters (<span class='html-italic'>ω<sub>mec</sub></span>, <span class='html-italic'>T<sub>m</sub></span>, <span class='html-italic'>I<sub>r</sub></span>, <span class='html-italic'>I<sub>s</sub></span>, and <span class='html-italic'>U<sub>r</sub></span>). All variables are in per unit (p.u.) using DFIM rated values as bases.</div><div class='html-p'>In <a href="#machines-12-00847-f005" class="html-fig">Figure 5</a>, synchronization was achieved at t = 0.242 s. Initially, while the rotor’s current maintained a constant value of 0.45 p.u. to magnetize the DFIM, the stator’s current was zero because the machine was not yet coupled to the grid. On the other hand, <span class='html-italic'>Ur</span> was set to obtain <span class='html-italic'>U<sub>s</sub></span> = <span class='html-italic'>U<sub>grid</sub></span> = 1 p.u on both sides of the CB before synchronization. Once the voltage, frequency, and phase angle conditions were optimal, synchronization was performed.</div><div class='html-p'>As shown in <a href="#machines-12-00847-f005" class="html-fig">Figure 5</a>, after closing the CB, a transient occurred in <span class='html-italic'>I<sub>s</sub></span> and <span class='html-italic'>I<sub>r,</sub></span> while the machine’s shaft remained nearly stationary with negligible torque. The frequency downward ramp was applied at t = 0.3 s. When the frequency slope started at 0.3 s, a larger transient appeared in currents, and consequently in torque, during the first 0.7 s of evolution (up to t = 1 s), as the result of overcoming DFIM inertia. After these transients, the currents and the torque returned to their soft starting state. In this case, both rotor’s and stator’s currents did not exceed 0.5 p.u., except in the initial transient where <span class='html-italic'>I<sub>r</sub></span> reached 1 p.u. Once the DFIM was accelerated for approximately 5 s under the constant flux condition, the magnetizing current was proportionally divided into the stator’s and rotor’s currents according to the machine’s impedances. Furthermore, at the beginning <span class='html-italic'>U<sub>r</sub></span> and <span class='html-italic'>U<sub>DC</sub></span> experienced a transient decrease of around 15% and the torque initially pulsed following the current behavior. The acceleration process ends when <span class='html-italic'>U<sub>r</sub></span> is brought down to its rated value. Therefore, the DFIM attains its rated speed.</div><div class='html-p'>Finally, in <a href="#machines-12-00847-f006" class="html-fig">Figure 6</a> the grid and stator voltages are shown. The voltage on both sides of the CB during the synchronization process is plotted. Before the CB was closed, the stator voltage pulsed due to the PWM converter feeding the rotor of the machine. After closing the CB, <span class='html-italic'>U<sub>grid</sub></span> was imposed.</div></section></section><section id='sec5-machines-12-00847' type=''><h2 data-nested='1'> 5. Experimental Tests</h2><div class='html-p'>Obtaining a deeper understanding of the behavior of the previous parameters involved in the start-up process requires thoroughly performed experimental tests. In this section, the experimental setup and the results are described.</div><section id='sec5dot1-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 5.1. Experimental Setup</h4><div class='html-p'>The experimental setup implemented in the laboratory is shown in <a href="#machines-12-00847-f007" class="html-fig">Figure 7</a> and <a href="#machines-12-00847-f008" class="html-fig">Figure 8</a>. It consists of a variable frequency drive (VFD) (1) with a squirrel cage induction machine (SCIM) (2) that drives a synchronous generator (SG) (3). The SG feeds the rotor of a 0.52-kW DFIM (4) with the frequency governed by the SCIM speed and the voltage level controlled by the DC excitation system (6) of the SG. The SG is used to provide constant <span class='html-italic'>U</span>/<span class='html-italic'>f</span> control for the DFIM rotor’s excitation. However, a controlled power converter could be used instead for rotor feeding in this setup. The DFIM’s nameplate parameters are listed in <a href="#machines-12-00847-t003" class="html-table">Table 3</a> and its equivalent circuit parameters correspond to the ones shown in <a href="#machines-12-00847-t002" class="html-table">Table 2</a>.</div><div class='html-p'>Therefore, the conditions of <span class='html-italic'>f<sub>s</sub></span> = <span class='html-italic'>f<sub>grid</sub></span> and <span class='html-italic'>U<sub>s</sub></span> = <span class='html-italic'>U<sub>grid</sub></span> can be easily imposed on the DFIM stator before the closure of the main CB (7). In addition, the voltages can be adjusted using the autotransformer (5) included on the grid side of the CB. With the help of a synchronoscope (8) and two voltmeters (9, 10), the CB is closed when adequate conditions are met.</div><div class='html-p'>Furthermore, the rotor and stator currents (11, 12) and voltages are recorded using a 4-channel oscilloscope (14). An additional voltmeter (13) is used to monitor the SG excitation voltage as a safety measure. Once synchronized, the variable speed drive is controlled to decrease the frequency from 50 Hz to 8 Hz and the SG operates under <span class='html-italic'>U<sub>r</sub></span>/<span class='html-italic'>f<sub>r</sub></span> = constant control.</div></section><section id='sec5dot2-machines-12-00847' type=''><h4 class='html-italic' data-nested='2'> 5.2. Results</h4><div class='html-p'><a href="#machines-12-00847-f009" class="html-fig">Figure 9</a> shows the values recorded during a start-up process, which lasts 15 s. The synchronization was performed using two sinusoidal voltage signals because the rotor was fed from an SG.</div><div class='html-p'>As shown in <a href="#machines-12-00847-f008" class="html-fig">Figure 8</a>, the magnetizing current is provided by the rotor’s feeding at an initial constant value of 0.44 p.u. Once synchronization is performed at <span class='html-italic'>t</span> = 2 s, the stator inrush current does not contribute to the DFIM’s magnetization. At <span class='html-italic'>t</span> = 3.5, the <span class='html-italic'>f<sub>r</sub></span> is made to change and the shaft begins its rotational motion. At this point, while the machine is accelerated under the constant flux condition for 14.5 s, the proportion of magnetizing current supported by the stator and rotor varies, as the <span class='html-italic'>f<sub>r</sub></span> shifts linearly downward from 50 Hz (standstill initial condition) to 8 Hz (final operating point). This causes <span class='html-italic'>I<sub>r</sub></span> to decrease and <span class='html-italic'>I<sub>s</sub></span> to increase.</div><div class='html-p'>As the simulation results have also shown, the inrush currents are negligible, and the final currents do not exceed 0.4 p.u. However, the start-up is smoother than in the computer simulations, as the initial transient observed from t = 0.3 to <span class='html-italic'>t</span> = 1 s in <a href="#machines-12-00847-f005" class="html-fig">Figure 5</a> does not appear in <a href="#machines-12-00847-f009" class="html-fig">Figure 9</a>. The results shown in <a href="#machines-12-00847-f009" class="html-fig">Figure 9</a> suggest that the transient is caused by a simulation discontinuity rather than by a physical phenomenon. Additionally, the torque depicted in <a href="#machines-12-00847-f009" class="html-fig">Figure 9</a> does not exhibit the oscillations shown in <a href="#machines-12-00847-f005" class="html-fig">Figure 5</a>. Thus, the experimental results are consistent with the simulations: before and after synchronization, and throughout the controlled acceleration.</div><div class='html-p'>If the start-up current is compared with other techniques found in the literature, it can be observed that it presents one of the lowest values. As the RSC only has to provide the magnetizing current (around 0.4 p.u.), this is the higher reachable value at the start. Meanwhile, opposite phase sequence-based techniques reach around 5–6 times the rated <span class='html-italic'>I<sub>s</sub></span> [<a href="#B20-machines-12-00847" class="html-bibr">20</a>]. Reduced voltage-based techniques, using autotransformers, reduce up to around 0.8 p.u. the inrush current [<a href="#B21-machines-12-00847" class="html-bibr">21</a>].</div><div class='html-p'>Using stator short circuits and, then, performing the synchronization allows obtaining <span class='html-italic'>I<sub>s</sub></span> = 0.6–1.5 p.u. [<a href="#B25-machines-12-00847" class="html-bibr">25</a>,<a href="#B26-machines-12-00847" class="html-bibr">26</a>,<a href="#B28-machines-12-00847" class="html-bibr">28</a>,<a href="#B29-machines-12-00847" class="html-bibr">29</a>,<a href="#B30-machines-12-00847" class="html-bibr">30</a>,<a href="#B31-machines-12-00847" class="html-bibr">31</a>,<a href="#B32-machines-12-00847" class="html-bibr">32</a>,<a href="#B33-machines-12-00847" class="html-bibr">33</a>,<a href="#B34-machines-12-00847" class="html-bibr">34</a>,<a href="#B35-machines-12-00847" class="html-bibr">35</a>,<a href="#B36-machines-12-00847" class="html-bibr">36</a>], while rotor short circuits reach <span class='html-italic'>I<sub>s</sub></span> = 1 p.u. during the DFIM start-ups [<a href="#B37-machines-12-00847" class="html-bibr">37</a>]. Finally, the use of auxiliary power converter techniques obtain <span class='html-italic'>I<sub>s</sub></span> = 0.5 p.u. [<a href="#B39-machines-12-00847" class="html-bibr">39</a>], a value that is in the actual range of the proposed method, and it is an indicator of its viability.</div></section></section><section id='sec6-machines-12-00847' type='conclusions'><h2 data-nested='1'> 6. Conclusions</h2><div class='html-p'>This paper proposes a soft start method for DFIM applications. The method feeds the rotor with a frequency equal to the grid frequency, at a standstill and isolated machine conditions, producing the required stator voltage for grid synchronization. Synchronization occurs at standstill conditions through the main CB closure, keeping the rotor still as stator and rotor fluxes are identical, hence no electromotive forces are produced. After synchronization, the rotor frequency and voltage decrease maintaining a constant DFIM flux, gradually increasing the rotor mechanical speed. This enables soft frequency reduction until the desired speed, avoiding high torque transients.</div><div class='html-p'>The proposed method stands out from the state-of-the-art because no additional elements are required (switches or breakers, autotransformers, variable resistances, or auxiliary converters, among others). The start-up is achieved using only the main CB and simple converter control. This approach ensures safety by avoiding short-circuit conditions, and it eliminates inrush currents and voltage sags during start-ups.</div><div class='html-p'>Computer simulations and experimental tests were conducted for this work. The experimental results are not only consistent with the computer simulations but also demonstrate significant improvements, as some transients observed in the simulations were not present in the experimental measurements. Furthermore, the required currents are substantially lower compared to other start-up methods.</div><div class='html-p'>The DFIM start-up method has been validated under free shaft conditions, typical for variable speed hydro reversible generators. However, the main limitation of the method is related to the excitation power converter capacity, because it must be able to introduce the magnetizing current into the rotor up to the synchronization is completed and the rotor frequency starts to decrease. Future works shall aim to adapt the method for loaded start-ups, and also, to overcome this previous limitation by studying alternative DFIM synchronization methodologies that will not utilize the rotor converter to provide the magnetizing current.</div></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>Conceptualization, J.M.G., K.M. and C.A.P.; methodology, J.M.G., I.A., J.G.-C. and C.A.P.; software, J.M.G. and K.M.; validation, J.M.G., K.M. and J.A.S.; formal analysis, K.M. and J.A.S.; investigation, K.M. and C.A.P.; resources, J.M.G. and C.A.P.; data curation, J.M.G. and K.M.; writing—original draft preparation, J.M.G. and K.M.; writing—review and editing, I.A., J.G.-C., J.A.S. and C.A.P.; visualization, J.M.G. and K.M.; supervision, C.A.P.; project administration, C.A.P.; funding acquisition, C.A.P. All authors have read and agreed to the published version of the manuscript.</div></section><section class='html-notes'><h2>Funding</h2><div class='html-p'>This research received no external funding.</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>Datasets are available on request from the authors.</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>The authors declare no conflicts of interest.</div></section><section id='html-glossary'><h2 >Nomenclature and Abbreviations</h2><table class='html-array_table'><tbody ><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>f<sub>grid</sub></span></td><td align='left' valign='middle' class='html-align-left' >Grid frequency (in Hz)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>f<sub>r</sub></span></td><td align='left' valign='middle' class='html-align-left' >Rotor frequency (in Hz)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>f<sub>s</sub></span></td><td align='left' valign='middle' class='html-align-left' >Stator frequency (in Hz)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>H</span></td><td align='left' valign='middle' class='html-align-left' >Rotor inertia constant (in s)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>I<sub>r</sub></span></td><td align='left' valign='middle' class='html-align-left' >Rotor current (in A)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>I<sub>s</sub></span></td><td align='left' valign='middle' class='html-align-left' >Stator current (in A)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>J</span></td><td align='left' valign='middle' class='html-align-left' >Moment of inertia (in kg·m<sup>2</sup>)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>p</span></td><td align='left' valign='middle' class='html-align-left' >Number of pole pairs</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>R<sub>r</sub></span></td><td align='left' valign='middle' class='html-align-left' >Rotor resistance (in Ω)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>R<sub>s</sub></span></td><td align='left' valign='middle' class='html-align-left' >Stator resistance (in Ω)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>r<sub>t</sub></span></td><td align='left' valign='middle' class='html-align-left' >Stator/Rotor voltage ratio</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>R<sub>var</sub></span></td><td align='left' valign='middle' class='html-align-left' >Variable resistance (in Ω)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>s</span></td><td align='left' valign='middle' class='html-align-left' >Slip</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>T<sub>m</sub></span></td><td align='left' valign='middle' class='html-align-left' >Mechanical torque (in N·m)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>T<sub>r</sub></span></td><td align='left' valign='middle' class='html-align-left' >Mechanical counter-torque (in N·m)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>t</span></td><td align='left' valign='middle' class='html-align-left' >Time (in s)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>U<sub>DC</sub></span></td><td align='left' valign='middle' class='html-align-left' >DC bus voltage (in V)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>U<sub>grid</sub></span></td><td align='left' valign='middle' class='html-align-left' >Grid voltage (in V)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>U<sub>r</sub></span></td><td align='left' valign='middle' class='html-align-left' >Rotor voltage (in V)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>U<sub>s</sub></span></td><td align='left' valign='middle' class='html-align-left' >Stator voltage (in V)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>X<sub>m</sub></span></td><td align='left' valign='middle' class='html-align-left' >Magnetizing reactance (in Ω)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>X<sub>r</sub></span></td><td align='left' valign='middle' class='html-align-left' >Rotor leakage reactance (in Ω)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>X<sub>s</sub></span></td><td align='left' valign='middle' class='html-align-left' >Stator leakage reactance (in Ω)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>φ</span></td><td align='left' valign='middle' class='html-align-left' >Phase angle (in rad)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>Ψ<sub>r</sub></span></td><td align='left' valign='middle' class='html-align-left' >Rotor flux (in Wb)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>Ψ<sub>s</sub></span></td><td align='left' valign='middle' class='html-align-left' >Stator flux (in Wb)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>ω<sub>0</sub></span></td><td align='left' valign='middle' class='html-align-left' >Angular speed (in rad/s)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><span class='html-italic'>ω<sub>mec</sub></span></td><td align='left' valign='middle' class='html-align-left' >Mechanical rotor speed (in rad/s)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >AC</td><td align='left' valign='middle' class='html-align-left' >Alternative Current</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >AT</td><td align='left' valign='middle' class='html-align-left' >Autotransformer </td></tr><tr ><td align='left' valign='middle' class='html-align-left' >CB</td><td align='left' valign='middle' class='html-align-left' >Circuit Breaker </td></tr><tr ><td align='left' valign='middle' class='html-align-left' >DC</td><td align='left' valign='middle' class='html-align-left' >Direct Current</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >DFIM</td><td align='left' valign='middle' class='html-align-left' >Doubly Fed Induction Machine</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >FOC</td><td align='left' valign='middle' class='html-align-left' >Field Oriented Control </td></tr><tr ><td align='left' valign='middle' class='html-align-left' >GSC</td><td align='left' valign='middle' class='html-align-left' >Grid Side Converter </td></tr><tr ><td align='left' valign='middle' class='html-align-left' >PWM</td><td align='left' valign='middle' class='html-align-left' >Pulse Width Modulation </td></tr><tr ><td align='left' valign='middle' class='html-align-left' >RMS</td><td align='left' valign='middle' class='html-align-left' >Root Mean Square</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >RSC</td><td align='left' valign='middle' class='html-align-left' >Rotor Side Converter</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >SCIM</td><td align='left' valign='middle' class='html-align-left' >Squirrel Cage Induction Machine </td></tr><tr ><td align='left' valign='middle' class='html-align-left' >SG</td><td align='left' valign='middle' class='html-align-left' >Synchronous Generator </td></tr><tr ><td align='left' valign='middle' class='html-align-left' >VFD</td><td align='left' valign='middle' class='html-align-left' >Variable Frequency Drive</td></tr></tbody></table></section><section id='html-references_list'><h2>References</h2><ol class='html-xx'><li id='B1-machines-12-00847' class='html-x' data-content='1.'>Khalid, M. 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"https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f001"></a> </div> </div> <div class="html-fig_description"> <b>Figure 1.</b> Schemes for DFIM start-up: (<b>a</b>) Opposite phase sequence; (<b>b</b>) Autotransformer and variable resistors; (<b>c</b>) Stator short-circuit; (<b>d</b>) Auxiliary converter.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_machines-12-00847-f001"> <div class="html-caption"> <b>Figure 1.</b> Schemes for DFIM start-up: (<b>a</b>) Opposite phase sequence; (<b>b</b>) Autotransformer and variable resistors; (<b>c</b>) Stator short-circuit; (<b>d</b>) Auxiliary converter.</div> <div class="html-img"><img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g001.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g001.png" alt="Machines 12 00847 g001" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g001.png" /></div> </div> <div class="html-fig-wrap" id="machines-12-00847-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f002"> <img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g002.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g002.png" alt="Machines 12 00847 g002" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> Electrical scheme for the proposed start-up method [<span class='html-italic'>U<sub>grid</sub></span>: grid voltage; <span class='html-italic'>f<sub>grid</sub></span>: grid frequency; <span class='html-italic'>U<sub>s</sub></span>: stator voltage; <span class='html-italic'>f<sub>s</sub></span>: stator frequency; <span class='html-italic'>U<sub>r</sub></span>: rotor voltage; <span class='html-italic'>f<sub>r</sub></span>: rotor frequency; <span class='html-italic'>ΔU</span> = |<span class='html-italic'>U<sub>grid</sub></span>|−|<span class='html-italic'>U<sub>s</sub></span>|; <span class='html-italic'>Δ<sub>f</sub></span> = <span class='html-italic'>f<sub>grid</sub></span>−<span class='html-italic'>f<sub>s</sub></span>; <span class='html-italic'>Δφ</span>: phase difference between <span class='html-italic'>U<sub>grid</sub></span> and <span class='html-italic'>U<sub>s</sub></span> phasors].  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_machines-12-00847-f002"> <div class="html-caption"> <b>Figure 2.</b> Electrical scheme for the proposed start-up method [<span class='html-italic'>U<sub>grid</sub></span>: grid voltage; <span class='html-italic'>f<sub>grid</sub></span>: grid frequency; <span class='html-italic'>U<sub>s</sub></span>: stator voltage; <span class='html-italic'>f<sub>s</sub></span>: stator frequency; <span class='html-italic'>U<sub>r</sub></span>: rotor voltage; <span class='html-italic'>f<sub>r</sub></span>: rotor frequency; <span class='html-italic'>ΔU</span> = |<span class='html-italic'>U<sub>grid</sub></span>|−|<span class='html-italic'>U<sub>s</sub></span>|; <span class='html-italic'>Δ<sub>f</sub></span> = <span class='html-italic'>f<sub>grid</sub></span>−<span class='html-italic'>f<sub>s</sub></span>; <span class='html-italic'>Δφ</span>: phase difference between <span class='html-italic'>U<sub>grid</sub></span> and <span class='html-italic'>U<sub>s</sub></span> phasors].</div> <div class="html-img"><img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g002.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g002.png" alt="Machines 12 00847 g002" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g002.png" /></div> </div> <div class="html-fig-wrap" id="machines-12-00847-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f003"> <img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g003.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g003.png" alt="Machines 12 00847 g003" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> Conceptual algorithm of the proposed DFIM start-up method.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_machines-12-00847-f003"> <div class="html-caption"> <b>Figure 3.</b> Conceptual algorithm of the proposed DFIM start-up method.</div> <div class="html-img"><img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g003.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g003.png" alt="Machines 12 00847 g003" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g003.png" /></div> </div> <div class="html-fig-wrap" id="machines-12-00847-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f004"> <img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g004.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g004.png" alt="Machines 12 00847 g004" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g004-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> Simulation model for the stator synchronization-based start-up method.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_machines-12-00847-f004"> <div class="html-caption"> <b>Figure 4.</b> Simulation model for the stator synchronization-based start-up method.</div> <div class="html-img"><img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g004.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g004.png" alt="Machines 12 00847 g004" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g004.png" /></div> </div> <div class="html-fig-wrap" id="machines-12-00847-f005"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f005"> <img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g005.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g005.png" alt="Machines 12 00847 g005" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g005-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f005"></a> </div> </div> <div class="html-fig_description"> <b>Figure 5.</b> Simulation results for a stator synchronization start-up [Mechanical rotor speed (<span class='html-italic'>ω<sub>mec</sub></span>); torque (<span class='html-italic'>T<sub>m</sub></span>); RMS values of rotor and stator currents (<span class='html-italic'>I<sub>r</sub></span> and <span class='html-italic'>I<sub>s</sub></span>, respectively); rotor voltage (<span class='html-italic'>U<sub>r</sub></span>), in p.u].  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_machines-12-00847-f005"> <div class="html-caption"> <b>Figure 5.</b> Simulation results for a stator synchronization start-up [Mechanical rotor speed (<span class='html-italic'>ω<sub>mec</sub></span>); torque (<span class='html-italic'>T<sub>m</sub></span>); RMS values of rotor and stator currents (<span class='html-italic'>I<sub>r</sub></span> and <span class='html-italic'>I<sub>s</sub></span>, respectively); rotor voltage (<span class='html-italic'>U<sub>r</sub></span>), in p.u].</div> <div class="html-img"><img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g005.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g005.png" alt="Machines 12 00847 g005" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g005.png" /></div> </div> <div class="html-fig-wrap" id="machines-12-00847-f006"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f006"> <img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g006.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g006.png" alt="Machines 12 00847 g006" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g006-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f006"></a> </div> </div> <div class="html-fig_description"> <b>Figure 6.</b> Grid and stator voltages during synchronization using the proposed DFIM start-up method.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_machines-12-00847-f006"> <div class="html-caption"> <b>Figure 6.</b> Grid and stator voltages during synchronization using the proposed DFIM start-up method.</div> <div class="html-img"><img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g006.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g006.png" alt="Machines 12 00847 g006" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g006.png" /></div> </div> <div class="html-fig-wrap" id="machines-12-00847-f007"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f007"> <img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g007.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g007.png" alt="Machines 12 00847 g007" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g007-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f007"></a> </div> </div> <div class="html-fig_description"> <b>Figure 7.</b> Experimental setup: electrical scheme [<span class='html-italic'>U<sub>exc</sub></span> = Excitation voltage of the SG].  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_machines-12-00847-f007"> <div class="html-caption"> <b>Figure 7.</b> Experimental setup: electrical scheme [<span class='html-italic'>U<sub>exc</sub></span> = Excitation voltage of the SG].</div> <div class="html-img"><img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g007.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g007.png" alt="Machines 12 00847 g007" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g007.png" /></div> </div> <div class="html-fig-wrap" id="machines-12-00847-f008"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f008"> <img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g008.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g008.png" alt="Machines 12 00847 g008" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g008-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f008"></a> </div> </div> <div class="html-fig_description"> <b>Figure 8.</b> Experimental setup: overview [(1): VFD; (2): SCIM; (3): SG; (4): DFIM; (5): Autotransformer; (6): DC excitation system; (7): main CB; (8): Synchronoscope; (9) and (10): Voltmeter; (11): <span class='html-italic'>I<sub>r</sub></span> measurement; (12): <span class='html-italic'>I<sub>s</sub></span> measurement; (13): Voltmeter; (14): Oscilloscope].  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_machines-12-00847-f008"> <div class="html-caption"> <b>Figure 8.</b> Experimental setup: overview [(1): VFD; (2): SCIM; (3): SG; (4): DFIM; (5): Autotransformer; (6): DC excitation system; (7): main CB; (8): Synchronoscope; (9) and (10): Voltmeter; (11): <span class='html-italic'>I<sub>r</sub></span> measurement; (12): <span class='html-italic'>I<sub>s</sub></span> measurement; (13): Voltmeter; (14): Oscilloscope].</div> <div class="html-img"><img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g008.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g008.png" alt="Machines 12 00847 g008" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g008.png" /></div> </div> <div class="html-fig-wrap" id="machines-12-00847-f009"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f009"> <img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g009.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g009.png" alt="Machines 12 00847 g009" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g009-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#fig_body_display_machines-12-00847-f009"></a> </div> </div> <div class="html-fig_description"> <b>Figure 9.</b> Experimental results for a stator synchronization start-up [Mechanical rotor speed (<span class='html-italic'>ω<sub>mec</sub></span>); torque (<span class='html-italic'>T<sub>m</sub></span>); RMS values of rotor and stator currents (<span class='html-italic'>I<sub>r</sub></span> and <span class='html-italic'>I<sub>s</sub></span>, respectively); rotor voltage (<span class='html-italic'>U<sub>r</sub></span>), in p.u].  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_machines-12-00847-f009"> <div class="html-caption"> <b>Figure 9.</b> Experimental results for a stator synchronization start-up [Mechanical rotor speed (<span class='html-italic'>ω<sub>mec</sub></span>); torque (<span class='html-italic'>T<sub>m</sub></span>); RMS values of rotor and stator currents (<span class='html-italic'>I<sub>r</sub></span> and <span class='html-italic'>I<sub>s</sub></span>, respectively); rotor voltage (<span class='html-italic'>U<sub>r</sub></span>), in p.u].</div> <div class="html-img"><img data-large="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g009.png" data-original="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g009.png" alt="Machines 12 00847 g009" data-lsrc="/machines/machines-12-00847/article_deploy/html/images/machines-12-00847-g009.png" /></div> </div> <div class="html-table-wrap" id="machines-12-00847-t001"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href='#table_body_display_machines-12-00847-t001'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#table_body_display_machines-12-00847-t001"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 1.</b> DFIM start-up methods: state of the art sum-up. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_machines-12-00847-t001"> <div class="html-caption"><b>Table 1.</b> DFIM start-up methods: state of the art sum-up.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Start-Up Type</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Method</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >References</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Advantages</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Disadvantages</th></tr></thead><tbody ><tr ><td colspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Opposite phase sequence-based</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[<a href="#B20-machines-12-00847" class="html-bibr">20</a>]</td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>✓</dt><dd><div class='html-p'>Fast start-up</div></dd></dl></td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>×</dt><dd><div class='html-p'>High oscillation torque and high torsional forces in the shaft</div></dd></dl></td></tr><tr ><td rowspan='3' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Reduced voltage-based</td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' >With autotransformer </td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[<a href="#B21-machines-12-00847" class="html-bibr">21</a>,<a href="#B22-machines-12-00847" class="html-bibr">22</a>] </td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>✓</dt><dd><div class='html-p'>Partial winding start-up and smoother than other methods</div></dd></dl></td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>×</dt><dd><div class='html-p'>Slow start-up</div></dd><dt id=''>×</dt><dd><div class='html-p'>Additional autoransformer and rotor resistances to start the machine are required</div></dd></dl></td></tr><tr ><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' >Stator short-circuited </td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[<a href="#B23-machines-12-00847" class="html-bibr">23</a>,<a href="#B24-machines-12-00847" class="html-bibr">24</a>,<a href="#B25-machines-12-00847" class="html-bibr">25</a>,<a href="#B26-machines-12-00847" class="html-bibr">26</a>,<a href="#B27-machines-12-00847" class="html-bibr">27</a>,<a href="#B28-machines-12-00847" class="html-bibr">28</a>,<a href="#B29-machines-12-00847" class="html-bibr">29</a>,<a href="#B30-machines-12-00847" class="html-bibr">30</a>,<a href="#B31-machines-12-00847" class="html-bibr">31</a>,<a href="#B32-machines-12-00847" class="html-bibr">32</a>,<a href="#B33-machines-12-00847" class="html-bibr">33</a>,<a href="#B34-machines-12-00847" class="html-bibr">34</a>,<a href="#B35-machines-12-00847" class="html-bibr">35</a>,<a href="#B36-machines-12-00847" class="html-bibr">36</a>]</td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>✓</dt><dd><div class='html-p'>Lower power losses in the start-up than other methods</div></dd><dt id=''>✓</dt><dd><div class='html-p'>Acceptable soft start-up </div></dd></dl></td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>×</dt><dd><div class='html-p'>High power commutation switch to support the stator short-circuit currents is needed</div></dd><dt id=''>×</dt><dd><div class='html-p'>Reduced starting torque</div></dd></dl></td></tr><tr ><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' >Rotor short-circuited</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[<a href="#B23-machines-12-00847" class="html-bibr">23</a>,<a href="#B26-machines-12-00847" class="html-bibr">26</a>,<a href="#B37-machines-12-00847" class="html-bibr">37</a>] </td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>✓</dt><dd><div class='html-p'>Fast DFIM start-up compared to other methods</div></dd></dl></td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>×</dt><dd><div class='html-p'>The power converter must be able to carry the DFIM up to its rated speed by feeding the stator (it can increase the converter rated power)</div></dd><dt id=''>×</dt><dd><div class='html-p'>High power commutation switch to support the rotor short-circuit currents is needed</div></dd></dl></td></tr><tr ><td colspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Auxiliary converter-based </td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >[<a href="#B32-machines-12-00847" class="html-bibr">32</a>,<a href="#B38-machines-12-00847" class="html-bibr">38</a>,<a href="#B39-machines-12-00847" class="html-bibr">39</a>]</td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>✓</dt><dd><div class='html-p'>Soft start-up of the machine</div></dd><dt id=''>✓</dt><dd><div class='html-p'>Auxiliary converters can be used for power enhancement in case of low load</div></dd><dt id=''>✓</dt><dd><div class='html-p'>Synchronization in DC is simpler than in 3-phase AC</div></dd></dl></td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>×</dt><dd><div class='html-p'>Additional power electronic devices and switches required</div></dd></dl></td></tr><tr ><td colspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Proposed start-up</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' > </td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>✓</dt><dd><div class='html-p'>Simple to execute</div></dd><dt id=''>✓</dt><dd><div class='html-p'>No additional devices needed</div></dd><dt id=''>✓</dt><dd><div class='html-p'>Soft start with no torque pulses and low inrush currents</div></dd></dl></td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><dl class='html-bullet'><dt id=''>×</dt><dd><div class='html-p'>Slow start-up as a consequence of the soft start-up</div></dd></dl></td></tr></tbody> </table> </div> <div class="html-table-wrap" id="machines-12-00847-t002"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href='#table_body_display_machines-12-00847-t002'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#table_body_display_machines-12-00847-t002"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 2.</b> Simulated DFIM’s main rated parameters. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_machines-12-00847-t002"> <div class="html-caption"><b>Table 2.</b> Simulated DFIM’s main rated parameters.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Parameter</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Magnitude</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Units</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >Real power (<span class='html-italic'>P<sub>n</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >0.52</td><td align='center' valign='middle' class='html-align-center' >kW</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Stator voltage (<span class='html-italic'>U<sub>n</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >400</td><td align='center' valign='middle' class='html-align-center' >V</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Frequency (<span class='html-italic'>f<sub>n</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >50</td><td align='center' valign='middle' class='html-align-center' >Hz</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Stator/rotor voltage ratio (<span class='html-italic'>r<sub>t</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >10/1</td><td align='center' valign='middle' class='html-align-center' >V/V</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Stator resistance (<span class='html-italic'>R<sub>s</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >30.0</td><td align='center' valign='middle' class='html-align-center' >Ω</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Stator inductance (<span class='html-italic'>L<sub>s</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >0.120</td><td align='center' valign='middle' class='html-align-center' >H</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Equivalent rotor resistance (<span class='html-italic'>R<sub>r</sub>’</span>)</td><td align='center' valign='middle' class='html-align-center' >30.0</td><td align='center' valign='middle' class='html-align-center' >Ω</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Equivalent rotor inductance (<span class='html-italic'>L<sub>r</sub>’</span>)</td><td align='center' valign='middle' class='html-align-center' >0.120</td><td align='center' valign='middle' class='html-align-center' >H</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Mutual inductance (<span class='html-italic'>L<sub>m</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >2.432</td><td align='center' valign='middle' class='html-align-center' >H</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Moment of inertia (<span class='html-italic'>J</span>)</td><td align='center' valign='middle' class='html-align-center' >0.0015</td><td align='center' valign='middle' class='html-align-center' >kg·m<sup>2</sup></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Number of pole pairs (<span class='html-italic'>p</span>)</td><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' > </td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Speed (<span class='html-italic'>n<sub>n</sub></span>)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1400</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >rpm</td></tr></tbody> </table> </div> <div class="html-table-wrap" id="machines-12-00847-t003"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href='#table_body_display_machines-12-00847-t003'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-1702/12/12/847/display" href="#table_body_display_machines-12-00847-t003"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 3.</b> Tested DFIM’s nameplate parameters. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_machines-12-00847-t003"> <div class="html-caption"><b>Table 3.</b> Tested DFIM’s nameplate parameters.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Rated Parameters</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Magnitude</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Units</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >Real power (<span class='html-italic'>P<sub>n</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >0.52</td><td align='center' valign='middle' class='html-align-center' >kW</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Frequency (<span class='html-italic'>f<sub>n</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >50</td><td align='center' valign='middle' class='html-align-center' >Hz</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Stator voltage (<span class='html-italic'>U<sub>n</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >380/660</td><td align='center' valign='middle' class='html-align-center' >V</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Stator current (<span class='html-italic'>I<sub>n</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >1.6/0.92</td><td align='center' valign='middle' class='html-align-center' >A</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Rotor voltage (<span class='html-italic'>U<sub>n,r</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >65 (Y)</td><td align='center' valign='middle' class='html-align-center' >V</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Rotor current (<span class='html-italic'>I<sub>n,r</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >6 (Y)</td><td align='center' valign='middle' class='html-align-center' >A</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >Speed (<span class='html-italic'>n<sub>n</sub></span>)</td><td align='center' valign='middle' class='html-align-center' >1400</td><td align='center' valign='middle' class='html-align-center' >rpm</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Power factor (<span class='html-italic'>cosφ<sub>n</sub></span>)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.72</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' > </td></tr></tbody> </table> </div> </section><section class='html-fn_group'><table><tr id=''><td></td><td><div class='html-p'><b>Disclaimer/Publisher’s Note:</b> The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). 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A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions. <em>Machines</em> <b>2024</b>, <em>12</em>, 847. https://doi.org/10.3390/machines12120847 </p> <div style="display: block"> <b>AMA Style</b><br> <p> Guerrero JM, Mahtani K, Aranzabal I, Gómez-Cornejo J, Sánchez JA, Platero CA. A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions. <em>Machines</em>. 2024; 12(12):847. https://doi.org/10.3390/machines12120847 </p> <b>Chicago/Turabian Style</b><br> <p> Guerrero, José M., Kumar Mahtani, Itxaso Aranzabal, Julen Gómez-Cornejo, José A. Sánchez, and Carlos A. Platero. 2024. "A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions" <em>Machines</em> 12, no. 12: 847. https://doi.org/10.3390/machines12120847 </p> <b>APA Style</b><br> <p> Guerrero, J. 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A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions. <em>Machines</em> <b>2024</b>, <em>12</em>, 847. https://doi.org/10.3390/machines12120847 </p> <div style="display: block"> <b>AMA Style</b><br> <p> Guerrero JM, Mahtani K, Aranzabal I, Gómez-Cornejo J, Sánchez JA, Platero CA. A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions. <em>Machines</em>. 2024; 12(12):847. https://doi.org/10.3390/machines12120847 </p> <b>Chicago/Turabian Style</b><br> <p> Guerrero, José M., Kumar Mahtani, Itxaso Aranzabal, Julen Gómez-Cornejo, José A. Sánchez, and Carlos A. Platero. 2024. "A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions" <em>Machines</em> 12, no. 12: 847. https://doi.org/10.3390/machines12120847 </p> <b>APA Style</b><br> <p> Guerrero, J. M., Mahtani, K., Aranzabal, I., Gómez-Cornejo, J., Sánchez, J. A., & Platero, C. A. (2024). A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions. <em>Machines</em>, <em>12</em>(12), 847. https://doi.org/10.3390/machines12120847 </p> </div> </div> <div class="info-box no-margin"> Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. 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A Soft Start Method for Doubly Fed Induction Machines Based on Synchronization with the Power System at Standstill Conditions