[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83239-en":3,"doc-seo-83239-105":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},83239,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Tight Formulations for Unit Commitment with Different Levels of Details – Part I: Models and Theoretical Insights","Unit commitment (UC) is central to optimal power system operation but becomes computationally challenging in large-scale investment and stochastic settings due to UC’s binary decisions. This work studies how formulation quality and model size jointly affect performance and addresses the lack of a clear guidance across generator types. It introduces UC models with multiple levels of detail, each derived from the convex hull, and provides new tightness proofs for ramping as well as start-up and shut-down costs and capabilities.","Tight Formulations for Unit Commitment with Different Levels of Details – Part I: Models and  \nTheoretical Insights  \nMaaike B. Elgersma, Karen I. Aardal, Mathijs M. de Weerdt, and Germán Morales-España  \narXiv :2607 .07421v1 [math .OC] 8 Jul 2026  \nAbstract—The unit commitment (UC) problem is paramount for optimal operation of power systems, but it faces computational limitations in large-scale settings, especially in investment or stochastic models, because of the binary variables that it contains. A lot of research has attempted to improve the computational performance of UC models, either by reducing model size, resulting in lower fidelity and accuracy, or by improving the tightness of the formulation. Tightness and model size are the best a priori indicators of the computational performance of UC models, but there is no clear overview of what the best formulation is for different generators. In this research, we define models with different levels of detail, and present a formulation for each level that is based on the convex hull. We show new proofs on the tightness of well-known formulations for ramping, and for start up and shut down costs and capabil ties. These models, with a different level of detail, can be incorporated into large scale problems to reduce the computational burden, as demonstrated in Part II.  \nIndex Terms—Unit commitment (UC), mixed-integer linear programming (MILP), linear programming (LP), convex hull, tight formulation, optimal investments, ramping, minimum up and down times, start-up and shut-down costs/capabilities/trajectories.  \nNOMENCLATURE  \nAn overview of the notation used throughout this paper is given below.  \nSets and indices:  \ng ∈ G units (generators)  \nG 1 units (generators) with a minimum up time of 1 (⊆ G) t ∈ T time periods, where t0 is the first  \nT0 = T \\ {t0 }  \nTgup = T \\ {t0 , . . . , t0 + Tupg}  \nTgup = T \\ {t0 , . . . , t0 + Tdng} Parameters:  \nCnlg  \nCpg  \nCsug  \nCsdg  \nDt  \nPsgui  \nPsgdi  \nPg  \nPg  \nPsu  \ng  \nPsd  \ng  \ncost of non-load of unit g  \ncost of energy production of unit g cost of starting up unit g once cost of shutting down unit g once demand in time period t  \noutput of unit g in time period i of the start-up trajectory  \noutput of unit g in time period i of the shut-down trajectory  \nminimum output of unit g in one time period  \nmaximum output of unit g in one time period maximum output of unit g when it starts up  \nmaximum output of unit g when it shuts down  \n~~ ~~ up  \nRg  \n~~ ~~ dn  \nRg  \nTupg  \nTdng  \nTsug  \nTsdg  \nmaximum amount that unit g can ramp-up its output in one time period  \nmaximum amount that unit g can ramp-down its output in one time period  \nminimum up time of unit g minimum down time of unit g duration of start-up trajectory of unit g  \nduration of start-up trajectory of unit g  \nVariables:  \ncsgut  \ncsgdt  \npgt  \nptgrtaj  \nugt  \nuinvg  \nvgt  \nwgt  \ncosts associated with starting up unit g in time period t costs associated with shutting down unit g in time period t  \namount of energy that unit g produces in time period t  \namount of energy that unit g produces in its start-up or shut-down trajectory in time period t  \nindicates whether unit g is turned on or off in time period t  \nindicates the number of investments in unit g indicates whether unit g is starts up in time period t or not  \nindicates whether unit g is shuts down in time period tor not  \nI. INTRODUCTION THE unit commitment (UC) problem is one of the most  \nimportant problems for power system management [1],[2] . It is a traditional optimization problem that obtains the best operational schedule for a group of units, such as thermal generators, nuclear power plants, and renewable generators, but it can also be used for other types of units, such as electrolyzers [3] and block bids [4] . It is typ cally formulated as a Mixed Integer Linear Program (MILP) that minimizes system-wide operational costs of power generators for 24- 48 hours, while making sure that the demand is me","cbCaiasEj1tXD6ds","https://ap.wps.com/l/cbCaiasEj1tXD6ds","pdf",965046,3,1,13,"English","en",105,"# Introduction\n## Motivation and computational challenges\n## Modeling UC constraints and need for different levels of detail\n# Nomenclature\n## Sets and indices\n## Parameters\n## Variables\n# Abstract and index terms","[{\"question\":\"Why does the unit commitment (UC) problem become hard in large-scale investment or stochastic models?\",\"answer\":\"UC becomes computationally limited because it includes binary variables, which increase solving difficulty. The challenge is amplified in large-scale investment or uncertainty-aware formulations.\"},{\"question\":\"How does the paper handle varying modeling needs across different generator contexts?\",\"answer\":\"It defines UC models with different levels of detail and proposes a formulation for each level based on the convex hull. This enables selecting the most suitable detail level for the given context while keeping LP relaxations strong.\"},{\"question\":\"What tightness properties does the paper prove for common UC formulations?\",\"answer\":\"It provides new proofs on the tightness of well-known formulations for ramping and for start-up and shut-down costs and capabilities. These results support stronger formulations without substantially increasing model size.\"}]",1784186159,33,{"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},"tight-formulations-for-unit-commitment-with-different-levels-of-details-part-i-models-and-theoretical-insights","",{"@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/tight-formulations-for-unit-commitment-with-different-levels-of-details-part-i-models-and-theoretical-insights/83239/",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","2026-07-25","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 does the unit commitment (UC) problem become hard in large-scale investment or stochastic models?","Question",{"text":75,"@type":76},"UC becomes computationally limited because it includes binary variables, which increase solving difficulty. The challenge is amplified in large-scale investment or uncertainty-aware formulations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper handle varying modeling needs across different generator contexts?",{"text":80,"@type":76},"It defines UC models with different levels of detail and proposes a formulation for each level based on the convex hull. This enables selecting the most suitable detail level for the given context while keeping LP relaxations strong.",{"name":82,"@type":73,"acceptedAnswer":83},"What tightness properties does the paper prove for common UC formulations?",{"text":84,"@type":76},"It provides new proofs on the tightness of well-known formulations for ramping and for start-up and shut-down costs and capabilities. 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