With　the　widespread　implementation　of　distributed　generation　(DG)　and　the　integration　of　soft　open　point　(SOP)　into　the　distribution　network　(DN),　the　latter　is　steadily　transitioning　into　a　flexible　distribution　network　(FDN),　the　calculation　of　carbon　flow　distribution　in　FDN　is　more　difficult.　To　this　end,　this　study　constructs　a　model　for　low-carbon　optimal　operations　within　the　FDN　on　the　basis　of　enhanced　carbon　emission　flow　(CEF).　First,　the　carbon　emission　characteristics　of　FDNs　are　scrutinized　and　an　improved　method　for　calculating　carbon　flow　within　these　networks　is　proposed.　Subsequently,　a　model　for　optimizing　low-carbon　operations　within　FDNs　is　formulated　based　on　the　refined　CEF,　which　merges　the　specificities　of　DG　and　intelligent　SOP.　Finally,　this　model　is　scrutinized　using　an　upgraded　IEEE　33-node　distribution　system,　a　comparative　analysis　of　the　cases　reveals　that　when　DG　and　SOP　are　operated　in　a　coordinated　manner　in　the　FDN,　with　the　cost　of　electricity　generation　was　reduced　by　40.63　percent　and　the　cost　of　carbon　emissions　by　10.18　percent.　The　findings　indicate　that　the　judicious　optimization　of　areas　exhibiting　higher　carbon　flow　rates　can　effectively　enhance　the　economic　efficiency　of　DN　operations　and　curtail　the　carbon　emissions　of　the　overall　network.　©　2025　The　Authors.