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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content=" MMC-HVDC based power system, offshore wind farms, fault ride-through, asymmetrical AC faults, negative sequence control. " lang="en" name="keywords"/> <base href="/html/2409.16743v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S1" title="In Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S2" title="In Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">SYSTEM DESCRIPTION</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S3" title="In Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">IMPLEMENTATION OF NEGATIVE SEQUENCE CONTROL</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S4" title="In Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">SIMULATION STUDIES</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S5" title="In Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span><span class="ltx_text ltx_font_smallcaps">CONCLUSION</span></span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.1">\AtBeginShipoutNext</span><span class="ltx_ERROR undefined" id="p1.2">\AtBeginShipoutDiscard</span> </div> <span class="ltx_note ltx_role_thanks" id="id1"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">thanks: </span>N. Cherat, V. Nougain, P. Palensky, and A. Lekić are with the Department of Electrical Sustainable Energy, Delft University of Technology, Delft, The Netherlands. (e-mail: N.Cherat@student.tudelft.nl, {V.Nougain, P.Palensky, A.Lekic}@tudelft.nl). M. Majstorović is with the University of Belgrade, Belgrade, Serbia. (e-mail: majstorovic@etf.bg.ac.rs) This project has received funding from the European Union’s HORIZON-WIDERA-2021-ACCESS-03 under grant agreement No. 101079200. </span></span></span> <h1 class="ltx_title ltx_title_document">Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Naajein Cherat, Vaibhav Nougain, Milovan Majstorović, Peter Palensky, <span class="ltx_text ltx_font_italic" id="id3.1.id1">Senior Member, IEEE</span>, and Aleksandra Lekić, <span class="ltx_text ltx_font_italic" id="id4.2.id2">Senior Member, IEEE</span> </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.1">Fault ride-through capability studies of MMC-HVDC connected wind power plants have focused primarily on the DC link and onshore AC grid faults. Offshore AC faults, mainly asymmetrical faults have not gained much attention in the literature despite being included in the future development at national levels in the ENTSO-E HVDC code. The proposed work gives an event-triggered control to stabilize the system once the offshore AC fault has occurred, identified, and isolated. Different types of control actions such as proportional-integral (PI) controller and super-twisted sliding mode control (STSMC) are used to smoothly transition the post-fault system to a new steady state operating point by suppressing the negative sequence control. Initially, the effect of a negative sequence current control scheme on the transient behavior of the power system with a PI controller is discussed in this paper. Further, a non-linear control strategy (STSMC) is proposed which gives quicker convergence of the system post-fault in comparison to PI control action. These post-fault control operations are only triggered in the presence of a fault in the system, i.e., they are event-triggered. The validity of the proposed strategy is demonstrated by simulation on a <math alttext="\pm" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mo id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><csymbol cd="latexml" id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">±</annotation></semantics></math>525 kV, three-terminal meshed MMC-HVDC system model in Real Time Digital Simulator (RTDS).</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Index Terms: </h6> MMC-HVDC based power system, offshore wind farms, fault ride-through, asymmetrical AC faults, negative sequence control. </div> <span class="ltx_note ltx_note_frontmatter ltx_role_publicationid" id="id2"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">publicationid: </span>pubid: <span class="ltx_text ltx_inline-block" id="id2.1" style="width:433.6pt;">979-8-3503-9042-1/24/<math alttext="\$" class="ltx_Math" display="inline" id="id2.1.m1.1"><semantics id="id2.1.m1.1b"><mo id="id2.1.m1.1.1" xref="id2.1.m1.1.1.cmml">$</mo><annotation-xml encoding="MathML-Content" id="id2.1.m1.1c"><csymbol cd="latexml" id="id2.1.m1.1.1.cmml" xref="id2.1.m1.1.1">currency-dollar</csymbol></annotation-xml><annotation encoding="application/x-tex" id="id2.1.m1.1d">\$</annotation><annotation encoding="application/x-llamapun" id="id2.1.m1.1e">$</annotation></semantics></math>31.00 ©2024 European Union </span> <span class="ltx_text ltx_inline-block" id="id2.2" style="width:433.6pt;"> </span></span></span></span> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Offshore wind farms are experiencing rapid growth as a sustainable energy solution, attributed to their advantages of reduced wind variability and space constraints <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib1" title="">1</a>]</cite>. High voltage direct current (HVDC) is a proven technology for the grid integration of offshore wind farms. Modular multilevel converters (MMC) are used in HVDC transmission systems due to their distinctive features such as modular structure, high reliability, effective redundancy, simple fault identification, and clearance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib2" title="">2</a>]</cite>. Owing to its grid-forming ability and control capability, the MMC-HVDC-based power system is considered a cost-effective option for the integration of offshore wind farms <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib3" title="">3</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">One of the critical contingencies affecting the stability of MMC-based HVDC systems is the short circuit fault <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib4" title="">4</a>]</cite>. This can result in power system instability. An effective fault ride-through strategy is critical to avoid HVDC converter station disconnection from the AC grid. The literature for research on short-circuit faults in HVDC systems has primarily focused on the DC link and onshore AC grid faults <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib5" title="">5</a>]</cite>, neglecting the offshore AC fault contingency. Asymmetrical AC faults (single-phase-to-ground, phase-to-phase, and phase-to-phase-to-ground) represent the most probable fault contingencies in the AC power system <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib6" title="">6</a>]</cite>. On the occurrence of an asymmetrical fault, large negative sequence currents will be generated in the system. Due to the over-current limitations imposed by the converters, the control strategy of negative sequence current under asymmetric AC faults is of great significance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib6" title="">6</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">During an unbalanced onshore grid fault, the impact of negative sequence current control schemes on an onshore AC transmission line is analyzed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib7" title="">7</a>]</cite>. Two scenarios are considered, the suppression of negative sequence current and injection of negative sequence current proportionally to the negative sequence voltage. It is shown that injecting negative sequence current enhances fault detection capabilities and improves the performance of protection schemes in comparison to suppression of negative sequence current.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Work <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib8" title="">8</a>]</cite> proposes a control strategy that utilizes negative sequence voltages to facilitate controlled injection of negative sequence currents during offshore asymmetric AC faults. It is shown that by actively managing negative sequence currents and voltages, the strategy ensures a controlled level of fault current and prevents overvoltage conditions in healthy phases post-fault. This approach not only enhances the stability and reliability of the offshore wind power transmission system but also minimizes the risk of protection mal-operation. An enhancement to the control strategy proposed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib8" title="">8</a>]</cite> is done in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib3" title="">3</a>]</cite>. The per unit value of the MMC valve-side voltage positive sequence component is used as the droop coefficient, aiming to reduce over-adjustment issues in AC voltage during fault periods. The over-adjustment of AC voltage can lead to a significant reduction in the active power transmission of the HVDC system in case of a large voltage drop.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">Paper <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib9" title="">9</a>]</cite> proposes a high-performance fault ride-through method for an MMC-integrated offshore wind farm system, showcasing responses to symmetric and asymmetric faults. A method for continuous power transmission under faults and effective suppression of overcurrent and modulation is explained in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib9" title="">9</a>]</cite>. Negative sequence current coordinated control strategy for severe asymmetric offshore AC faults is discussed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib6" title="">6</a>]</cite>. When there is a severe asymmetric AC fault, the control strategy ensures that the negative sequence current of the offshore MMC is suppressed to zero, while the GSC (grid-side converters) cooperates to reduce the positive sequence current. The strategy involves limiting the positive sequence current of the GSC and reducing the positive sequence voltage of the offshore MMC to reserve margins for the modulation of the negative sequence voltage.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">This paper focuses on an event-triggered suppression of negative sequence current to improve the overall transient behavior of the power system during an unbalanced offshore grid fault, restoring system stability after the fault is cleared. Linear (PI) and non-linear (STSMC) controller are tested for their effectiveness in restoring the system. Since non-linear controllers have a faster convergence compared to their linear PI counterparts, the proposed non-linear controller is well suited to counter the non-linear dynamics of MMC converters. The proposed control strategy is tested on a <math alttext="\pm" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><mo id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><csymbol cd="latexml" id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">±</annotation></semantics></math>525 kV three-terminal meshed MMC-HVDC system modeled in Real Time Digital Simulator (RTDS). The rest of the paper is organized as follows. Section <span class="ltx_ERROR undefined" id="S1.p6.1.1">\@slowromancap</span>ii@ outlines the system description and the main controllers used. Section <span class="ltx_ERROR undefined" id="S1.p6.1.2">\@slowromancap</span>iii@ explains the implementation of negative sequence control. Section <span class="ltx_ERROR undefined" id="S1.p6.1.3">\@slowromancap</span>iv@ validates the proposed controller for various asymmetrical AC faults. Finally, Section <span class="ltx_ERROR undefined" id="S1.p6.1.4">\@slowromancap</span>v@ provides the conclusion and scope for future work.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">SYSTEM DESCRIPTION</span> </h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">Figure <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S2.F1" title="Figure 1 ‣ II SYSTEM DESCRIPTION ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">1</span></a> illustrates the configuration of the three-terminal meshed MMC-HVDC system used in this study. The system consists of an offshore wind plant, an offshore converter, and two onshore converters. The rated power of the MMC-HVDC system is 2GW and the rated voltage is <math alttext="\pm" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mo id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><csymbol cd="latexml" id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">±</annotation></semantics></math>525 kV. A bipolar configuration is used for HVDC in this study. Converters CSA1 and CSA3 are connected to strong AC grids. The offshore wind farms are connected to the DC grid through the converter CSA2. A cable of 300 km in length and 400 km in length connect the CSA2 with CSA1 and CSA3 respectively. The onshore converters CSA1 and CSA3 are also connected via cables.</p> </div> <figure class="ltx_figure" id="S2.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="347" id="S2.F1.g1" src="extracted/5878475/3T.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S2.F1.4.2.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S2.F1.2.1" style="font-size:90%;">Configuration of the <math alttext="\pm" class="ltx_Math" display="inline" id="S2.F1.2.1.m1.1"><semantics id="S2.F1.2.1.m1.1b"><mo id="S2.F1.2.1.m1.1.1" xref="S2.F1.2.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S2.F1.2.1.m1.1c"><csymbol cd="latexml" id="S2.F1.2.1.m1.1.1.cmml" xref="S2.F1.2.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.2.1.m1.1d">\pm</annotation><annotation encoding="application/x-llamapun" id="S2.F1.2.1.m1.1e">±</annotation></semantics></math>525 kV, 2GW, three-terminal MMC-HVDC system</span></figcaption> </figure> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.2">Two-level control typical for the Voltage Source converter (VSC) is used for the MMC converters. The upper level of control generates the reference voltage wave. The lower level control provides valve firing signals and ensures that cell capacitor voltage in each submodule (SM) remains constant according to a predetermined value. The reference is generated by using vector control philosophy where the 3-<math alttext="\phi" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mi id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">italic_ϕ</annotation></semantics></math> ABC system quantities are transformed into DQ system quantities by Park’s transformation. This is done to have a flexible and independent control of active and reactive power using simple DC parameters (in the DQ system) of their equivalent 3-<math alttext="\phi" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><mi id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><ci id="S2.p2.2.m2.1.1.cmml" xref="S2.p2.2.m2.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">italic_ϕ</annotation></semantics></math> parameters (in the ABC system). The upper-level control has a decoupled inner current controller whose references are based on user-defined MMC operating mode.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.6">Onshore converter, CSA1 acts as the slack converter controlling the DC voltage of the whole system. The control mode used is <math alttext="V_{dc}/V_{ac}" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mrow id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml"><msub id="S2.p3.1.m1.1.1.2" xref="S2.p3.1.m1.1.1.2.cmml"><mi id="S2.p3.1.m1.1.1.2.2" xref="S2.p3.1.m1.1.1.2.2.cmml">V</mi><mrow id="S2.p3.1.m1.1.1.2.3" xref="S2.p3.1.m1.1.1.2.3.cmml"><mi id="S2.p3.1.m1.1.1.2.3.2" xref="S2.p3.1.m1.1.1.2.3.2.cmml">d</mi><mo id="S2.p3.1.m1.1.1.2.3.1" xref="S2.p3.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.p3.1.m1.1.1.2.3.3" xref="S2.p3.1.m1.1.1.2.3.3.cmml">c</mi></mrow></msub><mo id="S2.p3.1.m1.1.1.1" xref="S2.p3.1.m1.1.1.1.cmml">/</mo><msub id="S2.p3.1.m1.1.1.3" xref="S2.p3.1.m1.1.1.3.cmml"><mi id="S2.p3.1.m1.1.1.3.2" xref="S2.p3.1.m1.1.1.3.2.cmml">V</mi><mrow id="S2.p3.1.m1.1.1.3.3" xref="S2.p3.1.m1.1.1.3.3.cmml"><mi id="S2.p3.1.m1.1.1.3.3.2" xref="S2.p3.1.m1.1.1.3.3.2.cmml">a</mi><mo id="S2.p3.1.m1.1.1.3.3.1" xref="S2.p3.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S2.p3.1.m1.1.1.3.3.3" xref="S2.p3.1.m1.1.1.3.3.3.cmml">c</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1"><divide id="S2.p3.1.m1.1.1.1.cmml" xref="S2.p3.1.m1.1.1.1"></divide><apply id="S2.p3.1.m1.1.1.2.cmml" xref="S2.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.2.1.cmml" xref="S2.p3.1.m1.1.1.2">subscript</csymbol><ci id="S2.p3.1.m1.1.1.2.2.cmml" xref="S2.p3.1.m1.1.1.2.2">𝑉</ci><apply id="S2.p3.1.m1.1.1.2.3.cmml" xref="S2.p3.1.m1.1.1.2.3"><times id="S2.p3.1.m1.1.1.2.3.1.cmml" xref="S2.p3.1.m1.1.1.2.3.1"></times><ci id="S2.p3.1.m1.1.1.2.3.2.cmml" xref="S2.p3.1.m1.1.1.2.3.2">𝑑</ci><ci id="S2.p3.1.m1.1.1.2.3.3.cmml" xref="S2.p3.1.m1.1.1.2.3.3">𝑐</ci></apply></apply><apply id="S2.p3.1.m1.1.1.3.cmml" xref="S2.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.3.1.cmml" xref="S2.p3.1.m1.1.1.3">subscript</csymbol><ci id="S2.p3.1.m1.1.1.3.2.cmml" xref="S2.p3.1.m1.1.1.3.2">𝑉</ci><apply id="S2.p3.1.m1.1.1.3.3.cmml" xref="S2.p3.1.m1.1.1.3.3"><times id="S2.p3.1.m1.1.1.3.3.1.cmml" xref="S2.p3.1.m1.1.1.3.3.1"></times><ci id="S2.p3.1.m1.1.1.3.3.2.cmml" xref="S2.p3.1.m1.1.1.3.3.2">𝑎</ci><ci id="S2.p3.1.m1.1.1.3.3.3.cmml" xref="S2.p3.1.m1.1.1.3.3.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">V_{dc}/V_{ac}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">italic_V start_POSTSUBSCRIPT italic_d italic_c end_POSTSUBSCRIPT / italic_V start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT</annotation></semantics></math>, where the d-axis current reference <math alttext="i_{dref}" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><msub id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml"><mi id="S2.p3.2.m2.1.1.2" xref="S2.p3.2.m2.1.1.2.cmml">i</mi><mrow id="S2.p3.2.m2.1.1.3" xref="S2.p3.2.m2.1.1.3.cmml"><mi id="S2.p3.2.m2.1.1.3.2" xref="S2.p3.2.m2.1.1.3.2.cmml">d</mi><mo id="S2.p3.2.m2.1.1.3.1" xref="S2.p3.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.2.m2.1.1.3.3" xref="S2.p3.2.m2.1.1.3.3.cmml">r</mi><mo id="S2.p3.2.m2.1.1.3.1a" xref="S2.p3.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.2.m2.1.1.3.4" xref="S2.p3.2.m2.1.1.3.4.cmml">e</mi><mo id="S2.p3.2.m2.1.1.3.1b" xref="S2.p3.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.2.m2.1.1.3.5" xref="S2.p3.2.m2.1.1.3.5.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><apply id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p3.2.m2.1.1.1.cmml" xref="S2.p3.2.m2.1.1">subscript</csymbol><ci id="S2.p3.2.m2.1.1.2.cmml" xref="S2.p3.2.m2.1.1.2">𝑖</ci><apply id="S2.p3.2.m2.1.1.3.cmml" xref="S2.p3.2.m2.1.1.3"><times id="S2.p3.2.m2.1.1.3.1.cmml" xref="S2.p3.2.m2.1.1.3.1"></times><ci id="S2.p3.2.m2.1.1.3.2.cmml" xref="S2.p3.2.m2.1.1.3.2">𝑑</ci><ci id="S2.p3.2.m2.1.1.3.3.cmml" xref="S2.p3.2.m2.1.1.3.3">𝑟</ci><ci id="S2.p3.2.m2.1.1.3.4.cmml" xref="S2.p3.2.m2.1.1.3.4">𝑒</ci><ci id="S2.p3.2.m2.1.1.3.5.cmml" xref="S2.p3.2.m2.1.1.3.5">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">i_{dref}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">italic_i start_POSTSUBSCRIPT italic_d italic_r italic_e italic_f end_POSTSUBSCRIPT</annotation></semantics></math> is controlled by the DC voltage loop and q-axis current reference <math alttext="i_{qref}" class="ltx_Math" display="inline" id="S2.p3.3.m3.1"><semantics id="S2.p3.3.m3.1a"><msub id="S2.p3.3.m3.1.1" xref="S2.p3.3.m3.1.1.cmml"><mi id="S2.p3.3.m3.1.1.2" xref="S2.p3.3.m3.1.1.2.cmml">i</mi><mrow id="S2.p3.3.m3.1.1.3" xref="S2.p3.3.m3.1.1.3.cmml"><mi id="S2.p3.3.m3.1.1.3.2" xref="S2.p3.3.m3.1.1.3.2.cmml">q</mi><mo id="S2.p3.3.m3.1.1.3.1" xref="S2.p3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.3.m3.1.1.3.3" xref="S2.p3.3.m3.1.1.3.3.cmml">r</mi><mo id="S2.p3.3.m3.1.1.3.1a" xref="S2.p3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.3.m3.1.1.3.4" xref="S2.p3.3.m3.1.1.3.4.cmml">e</mi><mo id="S2.p3.3.m3.1.1.3.1b" xref="S2.p3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.3.m3.1.1.3.5" xref="S2.p3.3.m3.1.1.3.5.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p3.3.m3.1b"><apply id="S2.p3.3.m3.1.1.cmml" xref="S2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p3.3.m3.1.1.1.cmml" xref="S2.p3.3.m3.1.1">subscript</csymbol><ci id="S2.p3.3.m3.1.1.2.cmml" xref="S2.p3.3.m3.1.1.2">𝑖</ci><apply id="S2.p3.3.m3.1.1.3.cmml" xref="S2.p3.3.m3.1.1.3"><times id="S2.p3.3.m3.1.1.3.1.cmml" xref="S2.p3.3.m3.1.1.3.1"></times><ci id="S2.p3.3.m3.1.1.3.2.cmml" xref="S2.p3.3.m3.1.1.3.2">𝑞</ci><ci id="S2.p3.3.m3.1.1.3.3.cmml" xref="S2.p3.3.m3.1.1.3.3">𝑟</ci><ci id="S2.p3.3.m3.1.1.3.4.cmml" xref="S2.p3.3.m3.1.1.3.4">𝑒</ci><ci id="S2.p3.3.m3.1.1.3.5.cmml" xref="S2.p3.3.m3.1.1.3.5">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m3.1c">i_{qref}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m3.1d">italic_i start_POSTSUBSCRIPT italic_q italic_r italic_e italic_f end_POSTSUBSCRIPT</annotation></semantics></math> is controlled by AC voltage loop. Onshore converter, CSA3 uses control mode <math alttext="P_{ac}/Q_{ac}" class="ltx_Math" display="inline" id="S2.p3.4.m4.1"><semantics id="S2.p3.4.m4.1a"><mrow id="S2.p3.4.m4.1.1" xref="S2.p3.4.m4.1.1.cmml"><msub id="S2.p3.4.m4.1.1.2" xref="S2.p3.4.m4.1.1.2.cmml"><mi id="S2.p3.4.m4.1.1.2.2" xref="S2.p3.4.m4.1.1.2.2.cmml">P</mi><mrow id="S2.p3.4.m4.1.1.2.3" xref="S2.p3.4.m4.1.1.2.3.cmml"><mi id="S2.p3.4.m4.1.1.2.3.2" xref="S2.p3.4.m4.1.1.2.3.2.cmml">a</mi><mo id="S2.p3.4.m4.1.1.2.3.1" xref="S2.p3.4.m4.1.1.2.3.1.cmml">⁢</mo><mi id="S2.p3.4.m4.1.1.2.3.3" xref="S2.p3.4.m4.1.1.2.3.3.cmml">c</mi></mrow></msub><mo id="S2.p3.4.m4.1.1.1" xref="S2.p3.4.m4.1.1.1.cmml">/</mo><msub id="S2.p3.4.m4.1.1.3" xref="S2.p3.4.m4.1.1.3.cmml"><mi id="S2.p3.4.m4.1.1.3.2" xref="S2.p3.4.m4.1.1.3.2.cmml">Q</mi><mrow id="S2.p3.4.m4.1.1.3.3" xref="S2.p3.4.m4.1.1.3.3.cmml"><mi id="S2.p3.4.m4.1.1.3.3.2" xref="S2.p3.4.m4.1.1.3.3.2.cmml">a</mi><mo id="S2.p3.4.m4.1.1.3.3.1" xref="S2.p3.4.m4.1.1.3.3.1.cmml">⁢</mo><mi id="S2.p3.4.m4.1.1.3.3.3" xref="S2.p3.4.m4.1.1.3.3.3.cmml">c</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.4.m4.1b"><apply id="S2.p3.4.m4.1.1.cmml" xref="S2.p3.4.m4.1.1"><divide id="S2.p3.4.m4.1.1.1.cmml" xref="S2.p3.4.m4.1.1.1"></divide><apply id="S2.p3.4.m4.1.1.2.cmml" xref="S2.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.p3.4.m4.1.1.2.1.cmml" xref="S2.p3.4.m4.1.1.2">subscript</csymbol><ci id="S2.p3.4.m4.1.1.2.2.cmml" xref="S2.p3.4.m4.1.1.2.2">𝑃</ci><apply id="S2.p3.4.m4.1.1.2.3.cmml" xref="S2.p3.4.m4.1.1.2.3"><times id="S2.p3.4.m4.1.1.2.3.1.cmml" xref="S2.p3.4.m4.1.1.2.3.1"></times><ci id="S2.p3.4.m4.1.1.2.3.2.cmml" xref="S2.p3.4.m4.1.1.2.3.2">𝑎</ci><ci id="S2.p3.4.m4.1.1.2.3.3.cmml" xref="S2.p3.4.m4.1.1.2.3.3">𝑐</ci></apply></apply><apply id="S2.p3.4.m4.1.1.3.cmml" xref="S2.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.p3.4.m4.1.1.3.1.cmml" xref="S2.p3.4.m4.1.1.3">subscript</csymbol><ci id="S2.p3.4.m4.1.1.3.2.cmml" xref="S2.p3.4.m4.1.1.3.2">𝑄</ci><apply id="S2.p3.4.m4.1.1.3.3.cmml" xref="S2.p3.4.m4.1.1.3.3"><times id="S2.p3.4.m4.1.1.3.3.1.cmml" xref="S2.p3.4.m4.1.1.3.3.1"></times><ci id="S2.p3.4.m4.1.1.3.3.2.cmml" xref="S2.p3.4.m4.1.1.3.3.2">𝑎</ci><ci id="S2.p3.4.m4.1.1.3.3.3.cmml" xref="S2.p3.4.m4.1.1.3.3.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m4.1c">P_{ac}/Q_{ac}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m4.1d">italic_P start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT / italic_Q start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT</annotation></semantics></math> where the d-axis current reference, <math alttext="i_{dref}" class="ltx_Math" display="inline" id="S2.p3.5.m5.1"><semantics id="S2.p3.5.m5.1a"><msub id="S2.p3.5.m5.1.1" xref="S2.p3.5.m5.1.1.cmml"><mi id="S2.p3.5.m5.1.1.2" xref="S2.p3.5.m5.1.1.2.cmml">i</mi><mrow id="S2.p3.5.m5.1.1.3" xref="S2.p3.5.m5.1.1.3.cmml"><mi id="S2.p3.5.m5.1.1.3.2" xref="S2.p3.5.m5.1.1.3.2.cmml">d</mi><mo id="S2.p3.5.m5.1.1.3.1" xref="S2.p3.5.m5.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.5.m5.1.1.3.3" xref="S2.p3.5.m5.1.1.3.3.cmml">r</mi><mo id="S2.p3.5.m5.1.1.3.1a" xref="S2.p3.5.m5.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.5.m5.1.1.3.4" xref="S2.p3.5.m5.1.1.3.4.cmml">e</mi><mo id="S2.p3.5.m5.1.1.3.1b" xref="S2.p3.5.m5.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.5.m5.1.1.3.5" xref="S2.p3.5.m5.1.1.3.5.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p3.5.m5.1b"><apply id="S2.p3.5.m5.1.1.cmml" xref="S2.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S2.p3.5.m5.1.1.1.cmml" xref="S2.p3.5.m5.1.1">subscript</csymbol><ci id="S2.p3.5.m5.1.1.2.cmml" xref="S2.p3.5.m5.1.1.2">𝑖</ci><apply id="S2.p3.5.m5.1.1.3.cmml" xref="S2.p3.5.m5.1.1.3"><times id="S2.p3.5.m5.1.1.3.1.cmml" xref="S2.p3.5.m5.1.1.3.1"></times><ci id="S2.p3.5.m5.1.1.3.2.cmml" xref="S2.p3.5.m5.1.1.3.2">𝑑</ci><ci id="S2.p3.5.m5.1.1.3.3.cmml" xref="S2.p3.5.m5.1.1.3.3">𝑟</ci><ci id="S2.p3.5.m5.1.1.3.4.cmml" xref="S2.p3.5.m5.1.1.3.4">𝑒</ci><ci id="S2.p3.5.m5.1.1.3.5.cmml" xref="S2.p3.5.m5.1.1.3.5">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.5.m5.1c">i_{dref}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.5.m5.1d">italic_i start_POSTSUBSCRIPT italic_d italic_r italic_e italic_f end_POSTSUBSCRIPT</annotation></semantics></math> is controlled by the active power loop and q-axis current reference, <math alttext="i_{qref}" class="ltx_Math" display="inline" id="S2.p3.6.m6.1"><semantics id="S2.p3.6.m6.1a"><msub id="S2.p3.6.m6.1.1" xref="S2.p3.6.m6.1.1.cmml"><mi id="S2.p3.6.m6.1.1.2" xref="S2.p3.6.m6.1.1.2.cmml">i</mi><mrow id="S2.p3.6.m6.1.1.3" xref="S2.p3.6.m6.1.1.3.cmml"><mi id="S2.p3.6.m6.1.1.3.2" xref="S2.p3.6.m6.1.1.3.2.cmml">q</mi><mo id="S2.p3.6.m6.1.1.3.1" xref="S2.p3.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.6.m6.1.1.3.3" xref="S2.p3.6.m6.1.1.3.3.cmml">r</mi><mo id="S2.p3.6.m6.1.1.3.1a" xref="S2.p3.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.6.m6.1.1.3.4" xref="S2.p3.6.m6.1.1.3.4.cmml">e</mi><mo id="S2.p3.6.m6.1.1.3.1b" xref="S2.p3.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.6.m6.1.1.3.5" xref="S2.p3.6.m6.1.1.3.5.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p3.6.m6.1b"><apply id="S2.p3.6.m6.1.1.cmml" xref="S2.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S2.p3.6.m6.1.1.1.cmml" xref="S2.p3.6.m6.1.1">subscript</csymbol><ci id="S2.p3.6.m6.1.1.2.cmml" xref="S2.p3.6.m6.1.1.2">𝑖</ci><apply id="S2.p3.6.m6.1.1.3.cmml" xref="S2.p3.6.m6.1.1.3"><times id="S2.p3.6.m6.1.1.3.1.cmml" xref="S2.p3.6.m6.1.1.3.1"></times><ci id="S2.p3.6.m6.1.1.3.2.cmml" xref="S2.p3.6.m6.1.1.3.2">𝑞</ci><ci id="S2.p3.6.m6.1.1.3.3.cmml" xref="S2.p3.6.m6.1.1.3.3">𝑟</ci><ci id="S2.p3.6.m6.1.1.3.4.cmml" xref="S2.p3.6.m6.1.1.3.4">𝑒</ci><ci id="S2.p3.6.m6.1.1.3.5.cmml" xref="S2.p3.6.m6.1.1.3.5">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.6.m6.1c">i_{qref}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.6.m6.1d">italic_i start_POSTSUBSCRIPT italic_q italic_r italic_e italic_f end_POSTSUBSCRIPT</annotation></semantics></math> is controlled by the reactive power loop. The control loops for CSA1 and CSA3 come under the class of grid-following (GFL) control as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S2.F2" title="Figure 2 ‣ II SYSTEM DESCRIPTION ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">2</span></a>. GFL converter requires a dedicated unit to identify the grid voltage angle and calculate the proper phase shift. This is done using phase-locked loops (PLLs).</p> </div> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="398" id="S2.F2.g1" src="extracted/5878475/GFL.png" width="538"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S2.F2.2.1.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S2.F2.3.2" style="font-size:90%;">MMC control for grid-following converters</span></figcaption> </figure> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1">Lower-level control keeps the capacitor voltages across all submodules (SMs) within an acceptable range. This is achieved by selectively switching SMs based on the direction of arm currents. However, this regulation of capacitor voltages results in circulating currents among the three-phase units. This current does not influence the currents in the AC and DC sides, but they distort the arm current and increase the rated current of the submodules. A control loop to suppress the circulating current is included to reduce the effects of the circulating current.</p> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.1">The MMC model used in the present study assumes that the capacitor voltages of each submodule (SM) are internally balanced. Hence, there’s no need to specify the specific SMs for insertion; only the total number of SMs needs to be specified. The control input is thus simplified to an overall deblock integer signal and the number of SMs to be inserted.</p> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.1">The offshore converter, CSA2 uses a grid-forming (GFM) control. GFM converters are responsible for establishing and regulating grid voltages at the point of common coupling (PCC), especially in islanded operation mode. The control objective is to stabilize grid frequency and regulate voltage amplitude. GFM converters can self-synchronize to the grid without the need for a dedicated unit. An overview of the different control algorithms for GFM converters is provided in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib10" title="">10</a>]</cite>. Out of the variety of voltage control schemes developed, the simplest approach is to directly feed the voltage magnitude and phase angle which are generated from outer control loops (Fig. <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S2.F3" title="Figure 3 ‣ II SYSTEM DESCRIPTION ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">3</span></a>) to the modulation . Lack of current controllability may cause overcurrent tripping of the converter during grid faults. During the fault, control loop is switched to a dual loop consisting of an inner current loop which will be explained further in section <span class="ltx_ERROR undefined" id="S2.p6.1.1">\@slowromancap</span>iii@.</p> </div> <figure class="ltx_figure" id="S2.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="170" id="S2.F3.g1" src="extracted/5878475/grid_forming.png" width="299"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S2.F3.2.1.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S2.F3.3.2" style="font-size:90%;">MMC control for offshore converter</span></figcaption> </figure> <div class="ltx_para" id="S2.p7"> <p class="ltx_p" id="S2.p7.1">In this study, a Type 4 wind turbine is used, connecting the stator of the permanent magnet synchronous machine (PMSM) to the grid via two full-scale back-to-back converters. VSCs linked to the grid regulate DC voltage and reactive power, while those connected to the PMSM reduce reactive power and enhance generator efficiency.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">IMPLEMENTATION OF NEGATIVE SEQUENCE CONTROL</span> </h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">In the event of unbalanced faults, conventional generation units typically act as voltage sources in the positive sequence circuit, leading to high fault currents. However, power converter-interfaced generation units function as controlled current sources in both the positive and negative sequence circuits, while the zero sequence circuit remains mostly open due to transformer configuration <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib7" title="">7</a>]</cite>. When there is an asymmetrical fault, large negative sequence currents are generated in the system. A common strategy during unbalanced fault conditions is suppressing the negative sequence current to prevent the switching valves from uncontrolled fault currents.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">When a fault is detected, the sequence components are used to control the offshore converter CSA2. An event-triggered control strategy is activated to regulate the sequence components of voltage and current signals during a fault. The sudden increase in negative sequence current serves as a fault detection mechanism, triggering the proposed control strategy. The presence of a fault is detected by comparing the negative sequence current across two consecutive time steps; if the difference exceeds a set limit, a fault is indicated. The value considered in this study is 0.45 p.u. The positive and negative sequence control for the offshore MMC is given in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S3.F5" title="Figure 5 ‣ III IMPLEMENTATION OF NEGATIVE SEQUENCE CONTROL ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">5</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S3.F5" title="Figure 5 ‣ III IMPLEMENTATION OF NEGATIVE SEQUENCE CONTROL ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">5</span></a> respectively.</p> </div> <figure class="ltx_figure" id="S3.F5"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="290" id="S3.F5.g1" src="extracted/5878475/Positive_seq.png" width="538"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F5.2.1.1" style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text" id="S3.F5.3.2" style="font-size:90%;">MMC offshore converter control - Positive sequence</span></figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="231" id="S3.F5.g2" src="extracted/5878475/Neg-Seq.png" width="359"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F5.4.1.1" style="font-size:90%;">Figure 5</span>: </span><span class="ltx_text" id="S3.F5.5.2" style="font-size:90%;">MMC offshore converter control - Negative sequence</span></figcaption> </figure> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">Conventional PI controllers are implemented in all control loops of the model. Numerous studies have underscored the advantages of employing non-linear controllers over conventional PI controllers to enhance power system performance. The authors in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib12" title="">12</a>]</cite> explain the different non-linear controllers such as Model Predictive Control (MPC), Back-Stepping Control (BSC), and Sliding Mode Control (SMC). Out of the three, SMC has the benefit of high robustness against system uncertainties and fast transient response. However, the chattering problem of SMC can lead to system oscillations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib13" title="">13</a>]</cite>. A second-order SMC, super twisting sliding mode controller (STSMC), has become popular recently, which not only preserves the robustness of the classical SMC but also overcomes the chattering issue.</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.1">In the SMC controller, a sliding surface is defined that guides the system states towards desired values. This sliding surface is typically based on tracking error, with the goal of minimizing this error to zero. When the states reach this surface, the controller uses a sign function to maintain the condition, which can lead to chattering phenomena due to the discontinuous nature of the sign function. Higher-order SMC such as STMSCs, are used to address the chattering issues and improve the performance of the controller.</p> </div> <div class="ltx_para ltx_noindent" id="S3.p5"> <p class="ltx_p" id="S3.p5.4">The sliding surface <math alttext="\dot{S}" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><mover accent="true" id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml"><mi id="S3.p5.1.m1.1.1.2" xref="S3.p5.1.m1.1.1.2.cmml">S</mi><mo id="S3.p5.1.m1.1.1.1" xref="S3.p5.1.m1.1.1.1.cmml">˙</mo></mover><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><apply id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1"><ci id="S3.p5.1.m1.1.1.1.cmml" xref="S3.p5.1.m1.1.1.1">˙</ci><ci id="S3.p5.1.m1.1.1.2.cmml" xref="S3.p5.1.m1.1.1.2">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">\dot{S}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">over˙ start_ARG italic_S end_ARG</annotation></semantics></math> for STSMC is proposed as outlined in Eq. <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S3.E1" title="In III IMPLEMENTATION OF NEGATIVE SEQUENCE CONTROL ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">1</span></a> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#bib.bib11" title="">11</a>]</cite>. The value of <math alttext="x" class="ltx_Math" display="inline" id="S3.p5.2.m2.1"><semantics id="S3.p5.2.m2.1a"><mi id="S3.p5.2.m2.1.1" xref="S3.p5.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><ci id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">italic_x</annotation></semantics></math> is determined using the error signal obtained from comparing the measured actual value <math alttext="y(t)" class="ltx_Math" display="inline" id="S3.p5.3.m3.1"><semantics id="S3.p5.3.m3.1a"><mrow id="S3.p5.3.m3.1.2" xref="S3.p5.3.m3.1.2.cmml"><mi id="S3.p5.3.m3.1.2.2" xref="S3.p5.3.m3.1.2.2.cmml">y</mi><mo id="S3.p5.3.m3.1.2.1" xref="S3.p5.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.p5.3.m3.1.2.3.2" xref="S3.p5.3.m3.1.2.cmml"><mo id="S3.p5.3.m3.1.2.3.2.1" stretchy="false" xref="S3.p5.3.m3.1.2.cmml">(</mo><mi id="S3.p5.3.m3.1.1" xref="S3.p5.3.m3.1.1.cmml">t</mi><mo id="S3.p5.3.m3.1.2.3.2.2" stretchy="false" xref="S3.p5.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.3.m3.1b"><apply id="S3.p5.3.m3.1.2.cmml" xref="S3.p5.3.m3.1.2"><times id="S3.p5.3.m3.1.2.1.cmml" xref="S3.p5.3.m3.1.2.1"></times><ci id="S3.p5.3.m3.1.2.2.cmml" xref="S3.p5.3.m3.1.2.2">𝑦</ci><ci id="S3.p5.3.m3.1.1.cmml" xref="S3.p5.3.m3.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m3.1c">y(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m3.1d">italic_y ( italic_t )</annotation></semantics></math> to the reference value <math alttext="y^{*}(t)" class="ltx_Math" display="inline" id="S3.p5.4.m4.1"><semantics id="S3.p5.4.m4.1a"><mrow id="S3.p5.4.m4.1.2" xref="S3.p5.4.m4.1.2.cmml"><msup id="S3.p5.4.m4.1.2.2" xref="S3.p5.4.m4.1.2.2.cmml"><mi id="S3.p5.4.m4.1.2.2.2" xref="S3.p5.4.m4.1.2.2.2.cmml">y</mi><mo id="S3.p5.4.m4.1.2.2.3" xref="S3.p5.4.m4.1.2.2.3.cmml">∗</mo></msup><mo id="S3.p5.4.m4.1.2.1" xref="S3.p5.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.p5.4.m4.1.2.3.2" xref="S3.p5.4.m4.1.2.cmml"><mo id="S3.p5.4.m4.1.2.3.2.1" stretchy="false" xref="S3.p5.4.m4.1.2.cmml">(</mo><mi id="S3.p5.4.m4.1.1" xref="S3.p5.4.m4.1.1.cmml">t</mi><mo id="S3.p5.4.m4.1.2.3.2.2" stretchy="false" xref="S3.p5.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.4.m4.1b"><apply id="S3.p5.4.m4.1.2.cmml" xref="S3.p5.4.m4.1.2"><times id="S3.p5.4.m4.1.2.1.cmml" xref="S3.p5.4.m4.1.2.1"></times><apply id="S3.p5.4.m4.1.2.2.cmml" xref="S3.p5.4.m4.1.2.2"><csymbol cd="ambiguous" id="S3.p5.4.m4.1.2.2.1.cmml" xref="S3.p5.4.m4.1.2.2">superscript</csymbol><ci id="S3.p5.4.m4.1.2.2.2.cmml" xref="S3.p5.4.m4.1.2.2.2">𝑦</ci><times id="S3.p5.4.m4.1.2.2.3.cmml" xref="S3.p5.4.m4.1.2.2.3"></times></apply><ci id="S3.p5.4.m4.1.1.cmml" xref="S3.p5.4.m4.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m4.1c">y^{*}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m4.1d">italic_y start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t )</annotation></semantics></math> as given in Eq. <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S3.E2" title="In III IMPLEMENTATION OF NEGATIVE SEQUENCE CONTROL ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">2</span></a>. Since both the terms of the sliding surface are continuous, the chattering is reduced.</p> <table class="ltx_equation ltx_eqn_table" id="S3.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\dot{S}(y)=-\alpha\sqrt{(}x)\operatorname{sgn}(x)-\beta\int(\operatorname{sgn}% (x)),\alpha,\beta>0" class="ltx_math_unparsed" display="block" id="S3.E1.m1.1"><semantics id="S3.E1.m1.1a"><mrow id="S3.E1.m1.1b"><mover accent="true" id="S3.E1.m1.1.2"><mi id="S3.E1.m1.1.2.2">S</mi><mo id="S3.E1.m1.1.2.1">˙</mo></mover><mrow id="S3.E1.m1.1.3"><mo id="S3.E1.m1.1.3.1" stretchy="false">(</mo><mi id="S3.E1.m1.1.1">y</mi><mo id="S3.E1.m1.1.3.2" stretchy="false">)</mo></mrow><mo id="S3.E1.m1.1.4" rspace="0em">=</mo><mo id="S3.E1.m1.1.5" lspace="0em">−</mo><mi id="S3.E1.m1.1.6">α</mi><msqrt id="S3.E1.m1.1.7"><mo id="S3.E1.m1.1.7.2" stretchy="false">(</mo></msqrt><mi id="S3.E1.m1.1.8">x</mi><mo id="S3.E1.m1.1.9" rspace="0.167em" stretchy="false">)</mo><mi id="S3.E1.m1.1.10">sgn</mi><mo id="S3.E1.m1.1.11" stretchy="false">(</mo><mi id="S3.E1.m1.1.12">x</mi><mo id="S3.E1.m1.1.13" stretchy="false">)</mo><mo id="S3.E1.m1.1.14">−</mo><mi id="S3.E1.m1.1.15">β</mi><mo id="S3.E1.m1.1.16" rspace="0em">∫</mo><mo id="S3.E1.m1.1.17" stretchy="false">(</mo><mi id="S3.E1.m1.1.18">sgn</mi><mrow id="S3.E1.m1.1.19"><mo id="S3.E1.m1.1.19.1" stretchy="false">(</mo><mi id="S3.E1.m1.1.19.2">x</mi><mo id="S3.E1.m1.1.19.3" stretchy="false">)</mo></mrow><mo id="S3.E1.m1.1.20" stretchy="false">)</mo><mo id="S3.E1.m1.1.21">,</mo><mi id="S3.E1.m1.1.22">α</mi><mo id="S3.E1.m1.1.23">,</mo><mi id="S3.E1.m1.1.24">β</mi><mo id="S3.E1.m1.1.25">></mo><mn id="S3.E1.m1.1.26">0</mn></mrow><annotation encoding="application/x-tex" id="S3.E1.m1.1c">\dot{S}(y)=-\alpha\sqrt{(}x)\operatorname{sgn}(x)-\beta\int(\operatorname{sgn}% (x)),\alpha,\beta>0</annotation><annotation encoding="application/x-llamapun" id="S3.E1.m1.1d">over˙ start_ARG italic_S end_ARG ( italic_y ) = - italic_α square-root start_ARG ( end_ARG italic_x ) roman_sgn ( italic_x ) - italic_β ∫ ( roman_sgn ( italic_x ) ) , italic_α , italic_β > 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="S3.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="e=y^{*}(t)-y(t),x=K_{i}e+K_{P}e," class="ltx_Math" display="block" id="S3.E2.m1.3"><semantics id="S3.E2.m1.3a"><mrow id="S3.E2.m1.3.3.1"><mrow id="S3.E2.m1.3.3.1.1.2" xref="S3.E2.m1.3.3.1.1.3.cmml"><mrow id="S3.E2.m1.3.3.1.1.1.1" 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<td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.6">where <math alttext="\alpha=1.5\sqrt{H}" class="ltx_Math" display="inline" id="S3.p5.5.m1.1"><semantics id="S3.p5.5.m1.1a"><mrow id="S3.p5.5.m1.1.1" xref="S3.p5.5.m1.1.1.cmml"><mi id="S3.p5.5.m1.1.1.2" xref="S3.p5.5.m1.1.1.2.cmml">α</mi><mo id="S3.p5.5.m1.1.1.1" xref="S3.p5.5.m1.1.1.1.cmml">=</mo><mrow id="S3.p5.5.m1.1.1.3" xref="S3.p5.5.m1.1.1.3.cmml"><mn id="S3.p5.5.m1.1.1.3.2" xref="S3.p5.5.m1.1.1.3.2.cmml">1.5</mn><mo id="S3.p5.5.m1.1.1.3.1" xref="S3.p5.5.m1.1.1.3.1.cmml">⁢</mo><msqrt id="S3.p5.5.m1.1.1.3.3" xref="S3.p5.5.m1.1.1.3.3.cmml"><mi id="S3.p5.5.m1.1.1.3.3.2" xref="S3.p5.5.m1.1.1.3.3.2.cmml">H</mi></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.5.m1.1b"><apply id="S3.p5.5.m1.1.1.cmml" xref="S3.p5.5.m1.1.1"><eq id="S3.p5.5.m1.1.1.1.cmml" 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For this study, the value of H is set to 10. <br class="ltx_break"/>The traditional PI controller within the inner current control loop has been substituted with STSMC for regulating sequence components in the offshore MMC, CSA2. The remaining control loops, including outer control loops, circulating current suppression control, and onshore converter control loops, continue to utilize state-of-the-art PI controllers.</p> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">SIMULATION STUDIES</span> </h2> <figure class="ltx_table" id="S4.T1"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T1.7.1.1" style="font-size:90%;">TABLE I</span>: </span><span class="ltx_text" id="S4.T1.8.2" style="font-size:90%;">Control mode and parameters of MMC converters</span></figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T1.5" style="width:433.6pt;height:291pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(69.3pt,-46.5pt) scale(1.46960327508904,1.46960327508904) ;"> <table class="ltx_tabular ltx_align_middle" id="S4.T1.5.5"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T1.5.5.6.1"> <td class="ltx_td ltx_align_left ltx_border_t" id="S4.T1.5.5.6.1.1" rowspan="2"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.6.1.1.1">Parameter</span></td> <td class="ltx_td ltx_align_center ltx_border_t" colspan="3" id="S4.T1.5.5.6.1.2"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.6.1.2.1">Converters</span></td> </tr> <tr class="ltx_tr" id="S4.T1.5.5.7.2"> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.7.2.1"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.7.2.1.1">CSA 1</span></td> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.7.2.2"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.7.2.2.1">CSA2</span></td> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.7.2.3"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.7.2.3.1">CSA3</span></td> </tr> <tr class="ltx_tr" id="S4.T1.3.3.3"> <td class="ltx_td 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xref="S4.T1.3.3.3.3.m1.1.1.2">subscript</csymbol><ci id="S4.T1.3.3.3.3.m1.1.1.2.2.cmml" xref="S4.T1.3.3.3.3.m1.1.1.2.2">𝑃</ci><apply id="S4.T1.3.3.3.3.m1.1.1.2.3.cmml" xref="S4.T1.3.3.3.3.m1.1.1.2.3"><times id="S4.T1.3.3.3.3.m1.1.1.2.3.1.cmml" xref="S4.T1.3.3.3.3.m1.1.1.2.3.1"></times><ci id="S4.T1.3.3.3.3.m1.1.1.2.3.2.cmml" xref="S4.T1.3.3.3.3.m1.1.1.2.3.2">𝑎</ci><ci id="S4.T1.3.3.3.3.m1.1.1.2.3.3.cmml" xref="S4.T1.3.3.3.3.m1.1.1.2.3.3">𝑐</ci></apply></apply><apply id="S4.T1.3.3.3.3.m1.1.1.3.cmml" xref="S4.T1.3.3.3.3.m1.1.1.3"><csymbol cd="ambiguous" id="S4.T1.3.3.3.3.m1.1.1.3.1.cmml" xref="S4.T1.3.3.3.3.m1.1.1.3">subscript</csymbol><ci id="S4.T1.3.3.3.3.m1.1.1.3.2.cmml" xref="S4.T1.3.3.3.3.m1.1.1.3.2">𝑄</ci><apply id="S4.T1.3.3.3.3.m1.1.1.3.3.cmml" xref="S4.T1.3.3.3.3.m1.1.1.3.3"><times id="S4.T1.3.3.3.3.m1.1.1.3.3.1.cmml" xref="S4.T1.3.3.3.3.m1.1.1.3.3.1"></times><ci id="S4.T1.3.3.3.3.m1.1.1.3.3.2.cmml" xref="S4.T1.3.3.3.3.m1.1.1.3.3.2">𝑎</ci><ci id="S4.T1.3.3.3.3.m1.1.1.3.3.3.cmml" xref="S4.T1.3.3.3.3.m1.1.1.3.3.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.3.3.3.3.m1.1c">P_{ac}/Q_{ac}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.3.3.3.3.m1.1d">italic_P start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT / italic_Q start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T1.5.5.8.3"> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.8.3.1"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.8.3.1.1">Rated active power [MW]</span></td> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.8.3.2">2000</td> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.8.3.3">2000</td> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.8.3.4">2000</td> </tr> <tr class="ltx_tr" id="S4.T1.5.5.9.4"> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.9.4.1"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.9.4.1.1">AC Grid Voltage [kV]</span></td> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.9.4.2">400</td> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.9.4.3">220</td> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.9.4.4">400</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.4"> <td class="ltx_td ltx_align_left ltx_border_t" id="S4.T1.4.4.4.2"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.4.2.1">DC link Voltage [kV]</span></td> <td class="ltx_td ltx_align_center ltx_border_t" colspan="3" id="S4.T1.4.4.4.1"> <math alttext="\pm" class="ltx_Math" display="inline" id="S4.T1.4.4.4.1.m1.1"><semantics id="S4.T1.4.4.4.1.m1.1a"><mo id="S4.T1.4.4.4.1.m1.1.1" xref="S4.T1.4.4.4.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.T1.4.4.4.1.m1.1b"><csymbol cd="latexml" id="S4.T1.4.4.4.1.m1.1.1.cmml" xref="S4.T1.4.4.4.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.4.4.4.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S4.T1.4.4.4.1.m1.1d">±</annotation></semantics></math>525</td> </tr> <tr class="ltx_tr" id="S4.T1.5.5.10.5"> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.10.5.1"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.10.5.1.1">Number of Submodules per arm</span></td> <td class="ltx_td ltx_align_center" colspan="3" id="S4.T1.5.5.10.5.2">200</td> </tr> <tr class="ltx_tr" id="S4.T1.5.5.11.6"> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.11.6.1"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.11.6.1.1">MMC arm inductance [mH]</span></td> <td class="ltx_td ltx_align_center" colspan="3" id="S4.T1.5.5.11.6.2">39.7</td> </tr> <tr class="ltx_tr" id="S4.T1.5.5.5"> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.5.1"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.5.1.1">MMC arm capacitance [<math alttext="\mu" class="ltx_Math" display="inline" id="S4.T1.5.5.5.1.1.m1.1"><semantics id="S4.T1.5.5.5.1.1.m1.1a"><mi id="S4.T1.5.5.5.1.1.m1.1.1" xref="S4.T1.5.5.5.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.T1.5.5.5.1.1.m1.1b"><ci id="S4.T1.5.5.5.1.1.m1.1.1.cmml" xref="S4.T1.5.5.5.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.5.5.5.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.T1.5.5.5.1.1.m1.1d">italic_μ</annotation></semantics></math>F]</span></td> <td class="ltx_td ltx_align_center" colspan="3" id="S4.T1.5.5.5.2">15000</td> </tr> <tr class="ltx_tr" id="S4.T1.5.5.12.7"> <td class="ltx_td ltx_align_left" id="S4.T1.5.5.12.7.1"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.12.7.1.1">Transformer leakage reactance [pu]</span></td> <td class="ltx_td ltx_align_center" colspan="3" id="S4.T1.5.5.12.7.2">0.18</td> </tr> <tr class="ltx_tr" id="S4.T1.5.5.13.8"> <td class="ltx_td ltx_align_left ltx_border_b" id="S4.T1.5.5.13.8.1"><span class="ltx_text ltx_font_bold" id="S4.T1.5.5.13.8.1.1">AC converter bus voltage [kV]</span></td> <td class="ltx_td ltx_align_center ltx_border_b" colspan="3" id="S4.T1.5.5.13.8.2">275</td> </tr> </tbody> </table> </span></div> </figure> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">This section presents an in-depth analysis of the transient response of a 3-terminal meshed MMC-HVDC system under various asymmetrical faults (L-G, L-L, and L-L-G). Table <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S4.T1" title="TABLE I ‣ IV SIMULATION STUDIES ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">I</span></a> provides details of the control mode and parameters of the MMC converters. As illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S2.F1" title="Figure 1 ‣ II SYSTEM DESCRIPTION ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">1</span></a>, the fault occurs on the line connecting the offshore wind plant and the offshore MMC (CSA2). It is assumed that the AC fault will clear on its own after 300 milliseconds. A fault resistance of 0.001 <math alttext="\Omega" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mi id="S4.p1.1.m1.1.1" mathvariant="normal" xref="S4.p1.1.m1.1.1.cmml">Ω</mi><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><ci id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1">Ω</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">\Omega</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">roman_Ω</annotation></semantics></math> is considered.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.1">Initially, the fault is introduced to the system without applying the proposed sequence current control. The most severe fault scenario, L-L-G, is chosen to evaluate the system’s response. The DC link voltage and active power signal for the different converters are given in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S4.F6" title="Figure 6 ‣ IV SIMULATION STUDIES ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">6</span></a>. It is observed that once the fault is cleared, the system remains unstable and the voltage magnitude and phase angle control is not able to stabilize the system. The active power control of 1 GW of the onshore converter CSA3 makes the power signal for CSA3 remain constant.</p> </div> <figure class="ltx_figure" id="S4.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="326" id="S4.F6.g1" src="extracted/5878475/Default.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F6.2.1.1" style="font-size:90%;">Figure 6</span>: </span><span class="ltx_text" id="S4.F6.3.2" style="font-size:90%;">DC link voltage and active power plots for different MMC’s</span></figcaption> </figure> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.1">Subsequently, the sequence current control scheme is activated upon fault occurrence. The DC link voltage and active power signals for different faults (L-G, L-L, and L-L-G) using the conventional PI controller are given in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S4.F9" title="Figure 9 ‣ IV SIMULATION STUDIES ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">9</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S4.F9" title="Figure 9 ‣ IV SIMULATION STUDIES ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">9</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S4.F9" title="Figure 9 ‣ IV SIMULATION STUDIES ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">9</span></a>. It is evident that with the introduction of sequence control, the system reverts back to a stable state post fault clearance. Due to its severity, the L-L-G fault necessitates a longer period for the system to stabilize compared to less severe faults.</p> </div> <figure class="ltx_figure" id="S4.F9"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="326" id="S4.F9.g1" src="extracted/5878475/LLG-PI.png" width="598"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F9.2.1.1" style="font-size:90%;">Figure 7</span>: </span><span class="ltx_text" id="S4.F9.3.2" style="font-size:90%;">DC link voltage and active power plots for different MMC’s during L-L-G fault using PI controller</span></figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="326" id="S4.F9.g2" src="extracted/5878475/LL_PI.png" width="598"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F9.4.1.1" style="font-size:90%;">Figure 8</span>: </span><span class="ltx_text" id="S4.F9.5.2" style="font-size:90%;">DC link voltage and active power plots for different MMC’s during L-L fault using PI controller</span></figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="326" id="S4.F9.g3" src="extracted/5878475/LG_PI.png" width="598"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F9.6.1.1" style="font-size:90%;">Figure 9</span>: </span><span class="ltx_text" id="S4.F9.7.2" style="font-size:90%;">DC link voltage and active power plots for different MMC’s during L-G fault using PI controller</span></figcaption> </figure> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.1">Further, the DC link voltage and active power signals for different faults using STSMC for the inner current loop are given in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S4.F13" title="Figure 13 ‣ IV SIMULATION STUDIES ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">13</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S4.F13" title="Figure 13 ‣ IV SIMULATION STUDIES ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">13</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S4.F13" title="Figure 13 ‣ IV SIMULATION STUDIES ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">13</span></a>. Non-linear controller (STSMC) helps to restore the system back to its stable state quickly irrespective of the type of fault. This is a result of quicker convergence by the sliding surface to the post-fault operating point. A comparison between the PI and STSMC controller when the system is subjected to L-L-G fault is given in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2409.16743v1#S4.F13" title="Figure 13 ‣ IV SIMULATION STUDIES ‣ Event-Triggered Non-Linear Control of Offshore MMC Grids for Asymmetrical AC Faults"><span class="ltx_text ltx_ref_tag">13</span></a>. As evident, STSMC finds faster settling to the new operating point both for the DC voltage and active power plots, indicating the effectiveness of STSMC controller compared to the state-of-the-art PI controller.</p> </div> <figure class="ltx_figure" id="S4.F13"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="326" id="S4.F13.g1" src="extracted/5878475/LLG-STSMC.png" width="598"/></div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F13.2.1.1" style="font-size:90%;">Figure 10</span>: </span><span class="ltx_text" id="S4.F13.3.2" style="font-size:90%;">DC link voltage and active power plots for different MMC’s during L-L-G fault using STSMC controller</span></figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="326" id="S4.F13.g2" src="extracted/5878475/LL_STSMC.png" width="598"/></div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F13.4.1.1" style="font-size:90%;">Figure 11</span>: </span><span class="ltx_text" id="S4.F13.5.2" style="font-size:90%;">DC link voltage and active power plots for different MMC’s during L-L fault using STSMC controller</span></figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="326" id="S4.F13.g3" src="extracted/5878475/LG_STSMC.png" width="598"/></div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F13.6.1.1" style="font-size:90%;">Figure 12</span>: </span><span class="ltx_text" id="S4.F13.7.2" style="font-size:90%;">DC link voltage and active power plots for different MMC’s during L-G fault using STSMC controller</span></figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="326" id="S4.F13.g4" src="extracted/5878475/LLG_PI_STSMC.png" width="598"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F13.8.1.1" style="font-size:90%;">Figure 13</span>: </span><span class="ltx_text" id="S4.F13.9.2" style="font-size:90%;">DC link voltage and active power plots for CSA2 converter using PI and STSMC controller during L-L-G faults</span></figcaption> </figure> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span><span class="ltx_text ltx_font_smallcaps" id="S5.1.1">CONCLUSION</span> </h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">This paper explores the impact of negative current suppression control during an unbalanced offshore AC fault. The control strategy is applied to a three-terminal meshed MMC-HVDC system that links an offshore wind plant to two onshore AC grids. Various fault scenarios such as line-to-ground, line-to-line, and line-to-line-to-ground faults are analyzed using this control method. The study demonstrates that controlling sequence current components is more effective than voltage amplitude and phase angle control of GFM converters in restoring system stability after fault clearance. The paper also discusses the advantages of using a non-linear controller, particularly focusing on the STSMC controller, over a linear one. The findings indicate that the STSMC controller achieves faster restoration of system stability than the conventional PI controller. Finally for future research, it would be beneficial to identify the optimal level of negative sequence current that ensures system stability while also improving the capabilities of the protection system.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> Y. Zhang, A. Shotorbani, L. Wang, and W. Li, “Distributed voltage regulation and automatic power sharing in multi-terminal hvdc grids,” <em class="ltx_emph ltx_font_italic" id="bib.bib1.1.1">IEEE Transactions on Power Systems</em>, vol. 35, no. 5, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> E. Mehrasa, M.and Pouresmaeil, S. Zabihi, and J. P. S. 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