The revolution of hydraulic fracturing 1 has dramatically increased the supply and lowered the cost of natural gas in the United States driving an expansion of natural gas-fired generation capacity in many electrical grids 2. Unrelated to the natural gas expansion, lower capital costs 3 and renewable portfolio 4 standards are driving an expansion of intermittent renewable generation capacity such as wind and photovoltaic generation. These two changes may potentially combine to create new threats to the reliability of these interdependent energy infrastructures. Natural gas-fired generators are often used to balance the fluctuating output of wind generation. However, the time-varying output of these generators results in time-varying natural gas burn rates that impact the pressure in interstate transmission pipelines. Fluctuating pressure impacts the reliability of natural gas deliveries to those same generators and the safety of pipeline operations. We adopt a partial differential equation model of natural gas pipelines and use this model to explore the effect of intermittent wind generation on the fluctuations of pressure in natural gas pipelines. The mean square pressure fluctuations are found to grow linearly in time with points of maximum deviation occurring at the locations of flow reversals.

The revolution of hydraulic fracturing [1] has dramatically increased the supply and lowered the cost of natural gas in the United States driving an expansion of natural gas-fired generation capacity in many electrical grids [2]. Unrelated to the natural gas expansion, lower capital costs [3] and renewable portfolio [4] standards are driving an expansion of intermittent renewable generation capacity such as wind and photovoltaic generation. These two changes may potentially combine to create new threats to the reliability of these interdependent energy infrastructures. Natural gasfired generators are often used to balance the fluctuating output of wind generation. However, the time-varying output of these generators results in time-varying natural gas burn rates that impact the pressure in interstate transmission pipelines. Fluctuating pressure impacts the reliability of natural gas deliveries to those same generators and the safety of pipeline operations. We adopt a partial differential equation model of natural gas pipelines and use this model to explore the effect of intermittent wind generation on the fluctuations of pressure in natural gas pipelines. The mean square pressure fluctuations are found to grow linearly in time with points of maximum deviation occurring at the locations of flow reversals. PACS numbers: 89.30.an, The ongoing evolution to intermittent wind and solar electric generation is causing many electrical grid operators to use more agile natural gas-fired electric generation to balance these new stochastic resources. This interdependence causes a cascade of the fluctuations of renewable generation into the systems that supply fuel to the gas-fired generators, i.e. natural gas pipelines. We develop a model of the coupling between electrical grid fluctuations and natural gas pipeline systems, analyze the resulting fluctuations of pipeline pressure, and draw conclusions about the impact of renewable electrical generation on the stability and security of natural gas pipelines.

We prove that if a c 1 smooth change of variable ~:R--.R generates a bounded v composition operator f--for in the space Ap(R)= L p , p#2, then ~ is linear (affine).We also prove that for a nonlinear c 1 mapping ~, the norms of exponentials e '~" as Fourier multipliers in LP(N) tend to infinity (hEN, I;q--~c). In both results the condition ~Ec' is sharp, it cannot be replaced by the Lipschitz condition.

We prove that if the indicator-function IE of a measurable set E C R is a Fourier multiplier in the space LP(R) for some p ~ 2 then E is an open set (up to a set of measure zero).

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