[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83383-en":3,"doc-seo-83383-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83383,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Preconditioner Based Acceleration Method for Solving EMTP Linear Equations","Electromagnetic transient program (EMTP) computation is constrained by ill-conditioned linear systems whose instability and complexity significantly slow iterative solvers. Existing EMTP acceleration work often targets implementation or modeling, while the ill-conditioning of the EMTP admittance matrix is insufficiently addressed. This letter systematically links the matrix’s physical origins to mathematical pathologies induced by network topology. A preconditioner-based strategy then accelerates computation while maintaining numerical accuracy, validated by simulation results showing efficiency and robustness.","Preconditioner-Based Acceleration Method for Solving EMTP Linear Equations  \nQi Lou, Student Member, Yijun Xu, Senior Member, Yang Cao, Student Member, Wei Gu, Senior Member,  \nFei Zhang, Member,  \narXiv :2607 .08442v1 [ ee ss . SY] 9 Jul 2026  \nAbstract—The computational speed of electromagnetic transient programs (EMTP) is severely limited by both the curse of dimensionality and the ill-conditioned system matrix, which collectively degrade solver performance. However, existing research on EMTP acceleration has largely overlooked the issue of ill-conditioning. This letter presents a first systematic, EMT-oriented investigation of the ill-conditioning of the EMTP admittance matrix by establishing a link between its physical origins and mathematical pathologies, thereby revealing the underlying mechanism by which network topology induces ill-conditioning. Building upon these structural insights, a preconditioner-based strategy is developed that significantly accelerates computation while preserving numerical accuracy. Simulation results demonstrate the outstanding efficiency and robustness of the proposed approach.  \nIndex Terms—Electromagnetic transients program(EMTP), preconditioner, power system dynamics, ill-conditioning.  \nI. Introduction  \nCOmputational efficiency remains the central challenge  \nin electromagnetic transient (EMT) simulations [1] . While significant efforts have been devoted to developing improved linear solvers, including the Conjugate Gradient (CG) [2] and Minimal Residual (MR) methods [3], solver-level acceleration alone has not fully resolved this challenge. The fundamental computational requirementsO (n) for CG and O(nnz·m)1 for MR-continue to present critical limitations that hinder complete solution of this persistent challenge [4] .  \nFrom a broader perspective, EMT acceleration has been pursued through multiple complementary directions. Most existing efforts focus on physical modeling or implementation aspects, such as network decoupling [5], model simplification [6], and high-performance computing [7],[8] . Solver-oriented acceleration from the mathematical perspective, which can complement these approaches, has received limited attention in EMT simulations.  \nTo further enhance EMTP computing efficiency from this solver-oriented perspective, this letter investigates the ill-conditioning of EMTP nodal equations. This focus is motivated by the fact that the condition number κ directly governs numerical stability and computational complexity: for example, the Krylov subspace dimension in MR methods grows linearly with κ, while CG exhibits a complexity of O ( √κ) [9] .  \nQ. Lou, Y. Xu, Y Cao, F. Zhang and W. Gu are with the Electrical Engineering Department, Southeast University, Nanjing, Jiangsu 210096, China (e-mail: {louqi, yijunxu, yangcao, zhangfei, [wgu}@seu.edu.cn](wgu}@seu.edu.cn)).  \n(Corresponding author: Wei Gu.)  \n1 Here, nnz is the number of non-zero matrix entries, and m is the dimension of the Krylov subspace.  \nMotivated by this, this letter tends to accelerate the EMTP by pioneering an investigation into the illconditioning characteristics of EMT simulations through the development of tailored preconditioning techniques. Although preconditioning has been reported in long-term time-domain simulations [10], existing applications are mainly restricted to systems without power electronic devices and rely on empirical performance comparisons, lacking a systematic physical interpretation for preconditioner design in EMTP nodal equations. By explicitly exploiting the structural and physical characteristics of EMT modeling, our method ensures numerically stable transformations while maintaining solution accuracy, enabling efficient resolution of EMTP.  \nII. PRELIMINARIES of EMTP  \nEMTP analyzes power system transients using discretization methods and nodal analysis [11] . Its matrix form is  \nG⃗v(t) = ⃗iinj (t) + ⃗Ihistory ≡ ⃗i(t) . (1)  \nHere, ⃗v(t) denotes the nodal voltages, ⃗i(t) denotes th","cbCaif8Qh4Yis4AF","https://ap.wps.com/l/cbCaif8Qh4Yis4AF","pdf",4131217,3,1,6,"English","en",105,"# Abstract\n# Introduction\n# Preliminaries of EMTP\n# Analysis of the Causes of Ill-Conditioning","[{\"question\":\"Why do ill-conditioned matrices limit EMTP computation speed?\",\"answer\":\"EMTP solver performance degrades when the system matrix is ill-conditioned, which makes numerical stability worse and increases iterative solver cost (e.g., Krylov subspace growth with the condition number).\"},{\"question\":\"How does the document explain the physical causes of EMTP ill-conditioning?\",\"answer\":\"It analyzes how grounded admittances (large equivalent admittances tied to power sources or capacitors) and switch devices (strong coupling of voltages and currents when switches are closed) generate extreme sensitivity in the admittance matrix.\"},{\"question\":\"What is the proposed acceleration approach and what does it preserve?\",\"answer\":\"The letter develops a preconditioner-based strategy derived from structural and physical insights into EMT modeling, accelerating computation significantly while preserving numerical accuracy and robustness, as confirmed by simulations.\"}]",1784187124,15,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"preconditioner-based-acceleration-method-for-solving-emtp-linear-equations","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/preconditioner-based-acceleration-method-for-solving-emtp-linear-equations/83383/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why do ill-conditioned matrices limit EMTP computation speed?","Question",{"text":75,"@type":76},"EMTP solver performance degrades when the system matrix is ill-conditioned, which makes numerical stability worse and increases iterative solver cost (e.g., Krylov subspace growth with the condition number).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the document explain the physical causes of EMTP ill-conditioning?",{"text":80,"@type":76},"It analyzes how grounded admittances (large equivalent admittances tied to power sources or capacitors) and switch devices (strong coupling of voltages and currents when switches are closed) generate extreme sensitivity in the admittance matrix.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the proposed acceleration approach and what does it preserve?",{"text":84,"@type":76},"The letter develops a preconditioner-based strategy derived from structural and physical insights into EMT modeling, accelerating computation significantly 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