With　increasing　uncertainties　in　the　source　and　load,　uncertain　energy　flow　calculation　is　important　for　optimal　energy　scheduling　of　the　integrated　electricity　and　natural　gas　system.　A　holomorphic　embedding　affine　energy　flow　method　for　natural　gas　network　is　proposed　to　address　the　issues　of　the　complexity　of　inverting　the　affine　matrix　and　difficulties　in　applying　it　to　the　looped　network　of　the　affine　power　flow　method　for　natural　gas　network.　On　this　basis,　the　energy　flow　equations　of　the　electricity　and　gas　subsystems　are　restructured　according　to　the　energy　flow　at　the　coupling　devices,　and　a　holomorphic　embedding　electricity-gas　affine　multi-energy　flow　model　is　constructed.　An　affine　energy　flow　alternating　recursion　calculation　method　is　proposed　to　reduce　the　dependency　on　initial　values　and　improve　the　efficiency　of　multi-energy　flow　calculation.　The　germ　solution　of　the　status　variables　of　the　no-load　conditions　of　the　power　grid　is　used　as　the　starting　point　for　multi-energy　flow　calculation.　Based　on　the　recursion　relationships　of　the　power　series　coefficients　of　affine　status　variables,　the　power　series　coefficients　of　affine　status　variables　of　the　power　grid,　gas　network,　and　coupling　devices　are　calculated　alternately　and　recursively,　and　the　energy　flow　distribution　interval　of　integrated　electricity　and　natural　gas　system　is　solved.　Simulation　results　verify　the　advantages　of　the　proposed　method　in　terms　of　conservatism,　convergence,　and　computational　efficiency.　©　2024　Power　System　Technology　Press.　All　rights　reserved.