Power-to-gas technology provides an emerging pathway for promoting green and low-carbon transformation of energy systems. Through the processes of electrolyzing water and the methanation reaction, it converts surplus renewable energy into hydrogen and natural gas, offering an effective approach for large-scale integration of renewable energy sources. However, the optimization of existing integrated energy systems has yet to finely model the operational characteristics of power-to-gas technology, severely limiting the energy conversion efficiency of systems. To address this issue, this paper proposes an integrated energy system operation strategy considering the slow dynamic response characteristics of power-to-gas. Firstly, based on the technical features of power-to-gas, an operational model for electrolyzing water to produce hydrogen is constructed, considering the transition relationships among cold start-up, hot start-up, and production states of a methanation reaction, thereby building a power-to-gas operation model considering slow dynamic response characteristics. This model finely reflects the impact of power-to-gas operational states on methanation, facilitating accurate representation of the operational states of methanation. Then, considering the energy conversion constraints and power balance of various coupled devices within integrated energy systems, an optimization model for the operation of the integrated energy system is constructed with the total daily operation cost of the system as the optimization objective. Finally, simulation comparisons are conducted to demonstrate the necessity of considering the slow dynamic response characteristics of power-to-gas technology for integrated energy system operation. The case study results indicate that the proposed power-to-gas operation model can accurately simulate the methanation process, facilitating the rational conversion of surplus renewable energy into natural gas energy and avoiding misjudgments in system operation costs and energy utilization efficiency.