Uncertainty, Geometry, and Phase Transitions in Energy Networks: A Unified Framework for Cascades and Optimal Control

Uncertainty, Geometry, and Phase Transitions in Energy Networks: A Unified Framework for Cascades and Optimal Control | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (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);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Uncertainty, Geometry, and Phase Transitions in Energy Networks: A Unified Framework for Cascades and Optimal Control Nam Anh Quach, Dung Phuong Thi Bui This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9188413/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Emerging economies increasingly face critical energy bottlenecks as highly stochastic industrial demand interacts with grid infrastructures designed under deterministic assumptions. This paper develops a unified mathematical framework that integrates robust optimization, network dynamics, and stochastic optimal control to characterize and manage this structural vulnerability. First, we introduce the concept of robust capacity geometry , showing that demand uncertainty induces an anisotropic contraction of the feasible operating region, governed by the topology-dependent covariance of network flows. Second, we establish an endogenous stabilization bound , proving the existence of a spectral phase transition: when demand variance exceeds a critical threshold relative to the network’s algebraic connectivity, decentralized stabilization mechanisms fail and cascading failures emerge endogenously. Third, we formulate a stochastic control problem and solve the associated Hamilton–Jacobi–Bellman equation, deriving a dynamically optimal load-shedding policy that allocates deficits according to marginal system value. Numerical simulations on a stylized regional grid validate the theoretical results, demonstrating (i) a substantial contraction of the feasible region under stochastic demand, (ii) a spectral collapse of network connectivity during cascades, and (iii) a reduction of aggregate economic loss exceeding 50% under the optimal policy relative to conventional rolling blackout strategies. Collectively, the results provide a rigorous operational framework for managing energy systems under uncertainty, with direct implications for infrastructure planning and real-time control in rapidly industrializing economies. Applied Mathematics Operations Research Network Dynamics Phase Transitions Cascading Failures Spectral Graph Theory Energy Networks Stochastic Processes Complex Networks Hamilton–Jacobi–Bellman Equation Energy Systems under Uncertainty Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9188413","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":610005391,"identity":"d88117fd-9d38-4933-a363-a10b49fdba13","order_by":0,"name":"Nam Anh Quach","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9UlEQVRIiWNgGAWjYLACxgYQycP4+E8FkGZmbiBaC7MBzxmQFkbitbBJ8LYhuDiBOfvpxM+FO6wT+9vPHpCQnFcbzd8O1PKjYhtOLZY9uZulZ55JT5xxJi/BwHDb8dwZhxkbGHvO3MapxeBA7gZp3rbDiQ03eAwSErcdy20AamFmbMOj5fzbzb9BWuYDtRw4OOdY7nyCWm7kbgPbsuEGj2FjY0NN7gbCWt5us+ZtSzfeeCbHmJnh2IHcjUAtB/H65Xzu5tu8bday846fMf/NUFOXO+/84YMPflTg1gIFzDDGYTB5gJB6ZC11RCgeBaNgFIyCkQYAJ8Fh8iX1/zgAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0000-7962-6695","institution":"Hong Bang International University","correspondingAuthor":true,"prefix":"","firstName":"Nam","middleName":"Anh","lastName":"Quach","suffix":""},{"id":610005523,"identity":"fbc90d33-ab60-40c6-8c98-a3d2288c89fc","order_by":1,"name":"Dung Phuong Thi Bui","email":"","orcid":"","institution":"Bank for Investment and Development of Vietnam","correspondingAuthor":false,"prefix":"","firstName":"Dung","middleName":"Phuong Thi","lastName":"Bui","suffix":""}],"badges":[],"createdAt":"2026-03-22 00:48:17","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9188413/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9188413/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105564650,"identity":"a10391f9-5ac4-4a50-baf1-7e9b4ce62dc9","added_by":"auto","created_at":"2026-03-27 12:50:20","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":528581,"visible":true,"origin":"","legend":"","description":"","filename":"OptimalEnergyDistribution6update.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9188413/v1_covered_7d47de8f-da33-4114-9284-c726e8b326f7.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eUncertainty, Geometry, and Phase Transitions in Energy Networks: A Unified Framework for Cascades and Optimal Control\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Network Dynamics, Phase Transitions, Cascading Failures, Spectral Graph Theory, Energy Networks, Stochastic Processes, Complex Networks, Hamilton–Jacobi–Bellman Equation, Energy Systems under Uncertainty","lastPublishedDoi":"10.21203/rs.3.rs-9188413/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9188413/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eEmerging economies increasingly face critical energy bottlenecks as highly stochastic industrial demand interacts with grid infrastructures designed under deterministic assumptions. This paper develops a unified mathematical framework that integrates robust optimization, network dynamics, and stochastic optimal control to characterize and manage this structural vulnerability.\u003c/p\u003e \u003cp\u003eFirst, we introduce the concept of \u003cem\u003erobust capacity geometry\u003c/em\u003e, showing that demand uncertainty induces an anisotropic contraction of the feasible operating region, governed by the topology-dependent covariance of network flows. Second, we establish an \u003cem\u003eendogenous stabilization bound\u003c/em\u003e, proving the existence of a spectral phase transition: when demand variance exceeds a critical threshold relative to the network\u0026rsquo;s algebraic connectivity, decentralized stabilization mechanisms fail and cascading failures emerge endogenously. Third, we formulate a stochastic control problem and solve the associated Hamilton\u0026ndash;Jacobi\u0026ndash;Bellman equation, deriving a dynamically optimal load-shedding policy that allocates deficits according to marginal system value.\u003c/p\u003e \u003cp\u003eNumerical simulations on a stylized regional grid validate the theoretical results, demonstrating (i) a substantial contraction of the feasible region under stochastic demand, (ii) a spectral collapse of network connectivity during cascades, and (iii) a reduction of aggregate economic loss exceeding 50% under the optimal policy relative to conventional rolling blackout strategies.\u003c/p\u003e \u003cp\u003eCollectively, the results provide a rigorous operational framework for managing energy systems under uncertainty, with direct implications for infrastructure planning and real-time control in rapidly industrializing economies.\u003c/p\u003e","manuscriptTitle":"Uncertainty, Geometry, and Phase Transitions in Energy Networks: A Unified Framework for Cascades and Optimal Control","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-24 05:57:16","doi":"10.21203/rs.3.rs-9188413/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"bd2cceed-7ea5-4675-a506-75c0281f3ee9","owner":[],"postedDate":"March 24th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":64909141,"name":"Applied Mathematics"},{"id":64909142,"name":"Operations Research"}],"tags":[],"updatedAt":"2026-03-24T05:57:16+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-24 05:57:16","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9188413","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9188413","identity":"rs-9188413","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}