[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83981-en":3,"doc-seo-83981-105":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},83981,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Reachability Analysis for Power Systems with Heterogeneous Resources via Jordan Transformation","A computationally efficient framework bounds the evolution of transmission-level power system dynamics with synchronous generators, grid-forming and grid-following inverters, and uncertain power injections or withdrawals. Reduced-order device models and a frequency-divider network representation yield a linear ODE model for reachable-set computation under bounded disturbances across buses. Real Jordan transformation separates non-oscillatory from oscillatory modes, using interval reachability with contraction-based ball bounds for certified over-approximations. Experiments on a modified IEEE 39-bus system validate reachable tubes against EMT simulations and enable multi-second reachable sets computed in under a second.","Reachability Analysis for Power Systems with Heterogeneous Resources via Jordan Transformation  \nDamola Ajeyemi, Antonin Colot, Sairaj Dhople, Emiliano Dall’Anese, and Saber Jafarpour  \narXiv :2607 .05599v1 [ ee ss . SY] 6 Jul 2026  \nAbstract—This paper develops a computationally efficient framework for reachability analysis of transmission-level power system dynamics with synchronous generators, grid-forming and grid-following inverters, and uncertain power injections/withdrawals. Starting from reduced-order device models and a frequency-divider representation, we derive a linear ordinary-differential-equation model suitable for efficient reachable-set computation under bounded disturbances across network buses. The proposed reachability method combines interval reachability and contraction-based bounds to construct certified over-approximations for the linear ordinary-differentialequation model. A real Jordan transformation separates nonoscillatory modes, handled through a linear embedding system, from oscillatory modes, enclosed using contraction-based ball bounds. Numerical experiments on a modified IEEE 39-bus system validate the reachable tubes against high-fidelity electromagnetic-transient (EMT) simulations, and demonstrate multi-second reachable sets computed in sub-second time.  \nI. INTRODUCTION  \nPower systems are experiencing increasing variability due to the rapid growth of inverter-based resources and large-scale electrified loads such as data centers [1]–[3] . System operators can benefit significantly from tools that, in near real time, reliably estimate the range of possible operating conditions and transient responses following combinations of disturbancesand events [4], [5] . Traditional tools for small-signal analysis characterize local modal properties around an operating point, but are not designed to certify behavior under large disturbances. Conversely, transient stability analysis relies on timedomain simulation, which produces individual trajectories but cannot characterize worst-case behavior over uncertainty sets [6],[7] . As modern grids experience increasing variability, operators must reason about sets of possible frequencies, powers, and flows emerging from variations of loads across the system, rather than individual simulated scenarios.  \nReachability analysis provides a model-based framework for bounding the evolution of uncertain dynamical systems [8],[9] . Instead of simulating a single trajectory, sets of states are propagated through the dynamics to obtain a reachable tube containing all trajectories induced by admissible bounded disturbances. In power systems, these tubes can estimate, e.g., worst-case frequency deviations, rate of change of frequency (RoCoF), and settling time following disturbances;  \nD. Ajeyemi and E. Dall’Anese are with the Division of Systems Engineering, Boston University. Antonin Colot is with Elia Grid International. Sairaj Dhople is with the Department of Electrical and Computer Engineering, University of Minnesota. Saber Jafarpour is with the Department of Computer Science, University of Colorado Boulder. Corresponding author: D. Ajeyemi, email: [dajeyemi@bu.edu](dajeyemi@bu.edu).  \nsuch elements are key to certifying performance and ensuring secure operation. Because exact reachable-set computation for power transmission systems is generally intractable, existing methods either consider short horizons or employ computationally expensive set representations [10]–[17] . The former may miss the full post-disturbance evolution from transient response to settling, while the latter can scale poorly with system dimension. Zonotope-based methods, differentialinclusion methods, and Hamilton–Jacobi techniques [18] can therefore be computationally prohibitive for large networks.  \nThis paper develops a computationally efficient reachability framework for transmission-level power systems with heterogeneous devices, including synchronous generators and grid-forming and gri","cbCaivAETlxDTcm5","https://ap.wps.com/l/cbCaivAETlxDTcm5","pdf",5824555,3,1,19,"English","en",105,"# Introduction\n## Motivation for set-based certification\n## Limitations of traditional analysis methods\n# Proposed Reachability Framework\n## Reduced linear model construction\n## Hybrid interval–contraction via real Jordan decomposition\n## Mode-decoupled reachable-tube computation\n# Validation and Experiments\n## IEEE 39-bus case study\n## Comparison with high-fidelity EMT simulations","[{\"question\":\"What systems and uncertainties does the paper target for reachability analysis?\",\"answer\":\"It considers transmission-level dynamics including synchronous generators, grid-forming inverters, grid-following inverters, and uncertain power injections/withdrawals across network buses under bounded disturbances.\"},{\"question\":\"How does the proposed method make reachable-set computation efficient?\",\"answer\":\"It derives a reduced linear ODE model using reduced-order device/network representations and then computes reachable tubes with a hybrid interval–contraction approach that is mode-decoupled after a real Jordan decomposition.\"},{\"question\":\"Why is Jordan transformation important in the proposed framework?\",\"answer\":\"The real Jordan transformation separates non-oscillatory modes from oscillatory modes, enabling interval bounds for non-oscillatory dynamics and contraction-based ball bounds for oscillatory dynamics.\"}]",1784191830,48,{"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},"reachability-analysis-for-power-systems-with-heterogeneous-resources-via-jordan-transformation","",{"@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/reachability-analysis-for-power-systems-with-heterogeneous-resources-via-jordan-transformation/83981/",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","2026-07-26","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},"What systems and uncertainties does the paper target for reachability analysis?","Question",{"text":75,"@type":76},"It considers transmission-level dynamics including synchronous generators, grid-forming inverters, grid-following inverters, and uncertain power injections/withdrawals across network buses under bounded disturbances.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method make reachable-set computation efficient?",{"text":80,"@type":76},"It derives a reduced linear ODE model using reduced-order device/network representations and then computes reachable tubes with a hybrid interval–contraction approach that is mode-decoupled after a real Jordan decomposition.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is Jordan transformation important in the proposed framework?",{"text":84,"@type":76},"The real Jordan transformation separates non-oscillatory modes from oscillatory modes, enabling interval bounds for non-oscillatory dynamics and 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