{"id":"https://openalex.org/W7157682008","doi":"https://doi.org/10.48550/arxiv.2604.25473","title":"Complex-Vector Power and Cross-Phase Unbalance in Three-Phase Systems","display_name":"Complex-Vector Power and Cross-Phase Unbalance in Three-Phase Systems","publication_year":2026,"publication_date":"2026-04-28","ids":{"openalex":"https://openalex.org/W7157682008","doi":"https://doi.org/10.48550/arxiv.2604.25473"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.25473","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.25473","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.25473","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5087995465","display_name":"Juan Carlos Bravo Rodr\u00edguez","orcid":"https://orcid.org/0000-0002-4896-0903"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bravo-Rodr\u00edguez, Juan Carlos","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5004480043","display_name":"Juan Carlos Del-Pino-L\u00f3pez","orcid":"https://orcid.org/0000-0001-9953-1167"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"del-Pino-L\u00f3pez, Juan Carlos","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5007731285","display_name":"Francisco Casado-Machado","orcid":"https://orcid.org/0000-0001-6278-8508"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Casado-Machado, Francisco","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10573","display_name":"Power Quality and Harmonics","score":0.644599974155426,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10573","display_name":"Power Quality and Harmonics","score":0.644599974155426,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10305","display_name":"Power System Optimization and Stability","score":0.07010000199079514,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11102","display_name":"HVDC Systems and Fault Protection","score":0.059700001031160355,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/ac-power","display_name":"AC power","score":0.6351000070571899},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.6026999950408936},{"id":"https://openalex.org/keywords/antisymmetric-relation","display_name":"Antisymmetric relation","score":0.5694000124931335},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5245000123977661},{"id":"https://openalex.org/keywords/electric-power-system","display_name":"Electric power system","score":0.47999998927116394},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.46399998664855957},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.4603999853134155},{"id":"https://openalex.org/keywords/power","display_name":"Power (physics)","score":0.4447999894618988},{"id":"https://openalex.org/keywords/voltage","display_name":"Voltage","score":0.4424000084400177}],"concepts":[{"id":"https://openalex.org/C108755667","wikidata":"https://www.wikidata.org/wiki/Q1930258","display_name":"AC power","level":3,"score":0.6351000070571899},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.6026999950408936},{"id":"https://openalex.org/C152401794","wikidata":"https://www.wikidata.org/wiki/Q583760","display_name":"Antisymmetric relation","level":2,"score":0.5694000124931335},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5267000198364258},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5245000123977661},{"id":"https://openalex.org/C89227174","wikidata":"https://www.wikidata.org/wiki/Q2388981","display_name":"Electric power system","level":3,"score":0.47999998927116394},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.46399998664855957},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.4603999853134155},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.4447999894618988},{"id":"https://openalex.org/C165801399","wikidata":"https://www.wikidata.org/wiki/Q25428","display_name":"Voltage","level":2,"score":0.4424000084400177},{"id":"https://openalex.org/C173871940","wikidata":"https://www.wikidata.org/wiki/Q471846","display_name":"Three-phase","level":3,"score":0.35010001063346863},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3416000008583069},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.3393999934196472},{"id":"https://openalex.org/C168167062","wikidata":"https://www.wikidata.org/wiki/Q1117970","display_name":"Component (thermodynamics)","level":2,"score":0.32710000872612},{"id":"https://openalex.org/C59407501","wikidata":"https://www.wikidata.org/wiki/Q550341","display_name":"Volt-ampere reactive","level":5,"score":0.32409998774528503},{"id":"https://openalex.org/C148043351","wikidata":"https://www.wikidata.org/wiki/Q4456944","display_name":"Current (fluid)","level":2,"score":0.30219998955726624},{"id":"https://openalex.org/C165296897","wikidata":"https://www.wikidata.org/wiki/Q552380","display_name":"Symmetrical components","level":4,"score":0.2992999851703644},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.2906000018119812},{"id":"https://openalex.org/C196796808","wikidata":"https://www.wikidata.org/wiki/Q132629","display_name":"Electrical network","level":2,"score":0.2874000072479248},{"id":"https://openalex.org/C22762622","wikidata":"https://www.wikidata.org/wiki/Q628904","display_name":"Operating point","level":2,"score":0.2800000011920929},{"id":"https://openalex.org/C64424096","wikidata":"https://www.wikidata.org/wiki/Q750454","display_name":"Power factor","level":3,"score":0.2669000029563904},{"id":"https://openalex.org/C122770356","wikidata":"https://www.wikidata.org/wiki/Q1656753","display_name":"Identifiability","level":2,"score":0.26330000162124634},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2605000138282776}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.25473","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.25473","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.25473","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.25473","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Unbalanced":[0],"three-phase":[1,104],"systems":[2,106],"still":[3],"lack":[4],"a":[5,29,68,137,164,178],"compact":[6,165],"phasor-domain":[7],"representation":[8,39],"of":[9,45,52,140,168],"power":[10,31,47,81,157],"that":[11,26,72,111,170,180],"makes":[12],"phase":[13],"asymmetry":[14],"explicit":[15,67,177],"while":[16,65],"remaining":[17],"consistent":[18],"with":[19,48,146],"established":[20,172],"apparent-power":[21,115,142,173,186],"definitions.":[22],"This":[23],"paper":[24],"addresses":[25],"point":[27],"through":[28],"complex-vector":[30],"formulation":[32,100,161],"for":[33,117],"sinusoidal":[34],"steady-state":[35],"operation.":[36],"The":[37,99,159],"proposed":[38,160],"supplements":[40],"the":[41,49,59,95,113,118,141],"classical":[42],"dot-product":[43],"expression":[44],"complex":[46,124],"cross":[50],"product":[51],"voltage":[53,120],"and":[54,62,86,89,126,149,155],"current":[55],"phasors,":[56],"thereby":[57],"retaining":[58],"usual":[60],"active":[61,154],"reactive":[63,156],"terms":[64],"making":[66,176],"cross-phase":[69,87,147],"unbalance":[70,148,169],"vector":[71],"captures":[73],"antisymmetric":[74],"interphase":[75],"relations.":[76],"In":[77],"this":[78],"way,":[79],"apparent":[80],"is":[82,92,101,144,181],"separated":[83],"into":[84],"intraphase":[85],"contributions,":[88],"its":[90],"norm":[91,116],"preserved":[93],"under":[94],"power-invariant":[96],"Fortescue":[97],"transformation.":[98],"extended":[102],"to":[103],"four-wire":[105],"by":[107,175],"introducing":[108],"equivalent":[109],"coordinates":[110],"preserve":[112],"effective":[114],"chosen":[119],"reference.":[121],"Only":[122],"standard":[123],"numbers":[125],"matrices":[127],"are":[128],"required.":[129],"Numerical":[130],"examples":[131],"show":[132],"operating":[133],"conditions":[134],"in":[135],"which":[136],"non-negligible":[138],"part":[139],"structure":[143],"associated":[145],"cannot":[150],"be":[151],"inferred":[152],"from":[153,184],"alone.":[158],"thus":[162],"provides":[163],"phasor-based":[166],"descriptor":[167],"complements":[171],"theories":[174],"component":[179],"not":[182],"accessible":[183],"scalar":[185],"representations.":[187]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-04-30T00:00:00"}