This work focuses on the optimal operation of the integrated gas and electrical power system with bi-directional energy conversion. Considering the different response times of the gas and power systems, the transient gas flow and steady- state power flow are combined to formulate the dynamic optimal energy flow in the integrated gas and power systems. With proper assumptions and simplifications, the problem is transformed into a single stage linear programming. And only a single stage linear programming is needed to obtain the optimal operation strategy for both gas and power systems. Simulation on the test case illustrates the success of the modelling and the beneficial roles of the power-to-gas are analyzed. The proposed model can be used in the decision support for both planning and operation of the coordinated natural gas and electrical power systems.