Multi-energy systems have been developed to supply the multi-energy users economically by considering the physical limitations of different energy networks. This paper proposes a new entity called multiple energy distribution company (MEDC) to meet the electricity, gas, and heat demands of consumers in the presence of renewable energy resources (RESs) and multi-energy conversion technologies with the lowest operating cost. To achieve a more accurate scheduling model, a multi-energy flow model is used that involves practical constraints of the power distribution network, heating distribution network (HDN) and natural gas distribution network simultaneously. A variable mass flow and temperature control strategy is applied in the HDN to make a high-performance energy supply scheme. Multi-energy storage systems (MESSs) and integrated demand response (IDR) are also considered to increase the flexibility of the MEDC for serving multi-type energy demands. Moreover, a hybrid robust-stochastic optimization technique is adopted to handle the system uncertainties, where the uncertainties related to RESs and energy prices are addressed under a scenario-based stochastic programming and a robust optimization technique, respectively. The simulation results demonstrate that the efficient use of MESSs and IDR improves the performance of multi-energy generation units in the presence of multi-energy distribution network constraints and reduces the total operation cost by 15%.