The　interval　power　flow　(IPF)　method　is　widely　employed　to　address　the　uncertainties　of　renewable　energy　sources　(RESs)　in　power　systems.　However,　limited　research　exists　on　the　application　of　mathematical　optimization-based　approaches　to　compute　IPF　results.　Furthermore,　a　comprehensive　framework　for　analyzing　the　derived　IPF　results　and　formulating　appropriate　countermeasures　is　still　lacking.　Therefore,　this　paper　proposes　a　novel　linear　programming-based　framework　of　IPF　analysis　for　distribution　systems,　designed　to　enhance　IPF　calculation　efficiency　and　keep　system　state　variables　within　recommended　limits　utilizing　controllable　equipment.　First,　a　linearized　IPF　model　is　proposed　to　improve　calculation　efficiency.　The　over-limit　of　system　state　variables　is　analysed　based　on　the　IPF　results.　Then,　A　countermeasure　strategy　utilizing　controllable　equipment　is　proposed　to　maintain　system　security　under　potential　extreme　scenarios.　The　output　intervals　of　the　controllable　equipment　are　determined　as　scheduling　references　ensuring　secure　operation　under　the　uncertainties.　The　numerical　results　demonstrate　that　the　linearized　formulation　computes　the　IPF　results　6.57　times　faster　than　the　non-linear　method,　with　insignificant　calculation　errors　(below　0.06　%　for　magnitudes　and　0.02　degrees　for　angles).　The　countermeasure　method　can　successfully　keep　state　variables　within　predefined　ranges　and　provide　system　operators　with　effective　scheduling　reference　intervals　of　controllable　equipment　under　uncertainties.