The abundance of natural gas in the United States and the need for cleaner electric power have prompted widespread installation of gas-fired power plants and caused electric power systems to depend heavily on reliable gas supplies. The use of gas generators for peak load and reserve generation causes high intra-day variability in withdrawals from high pressure gas transmission systems, which leads to gas price fluctuations and supply disruptions that affect electric generator dispatch and threaten the security of both power and gas systems. In this manuscript, we investigate different gas compressor operation policies and their influence on the affected power system. Specifically, we consider constant pressure boost ratios and dynamic adjustment of these ratios to track pressure set-points. We also implement a joint optimization of generator dispatch schedules and gas compressor protocols using a dynamic gas flow model. We develop tractable, physically accurate implementations that are compared using an integrated model of test networks for power and gas systems with 24 and 25 nodes, which are coupled through gas-fired generators. This demonstrates the benefits that can be achieved with globally optimized gas system operations and increased gas-electric coordination.