Natural Gas Flow Solvers Using Convex Relaxation

Abstract

In addition to their lower emissions and fast ramping capabilities, gas-fired electric power generation is increasing due to newly discovered supplies and declining prices. The vast infrastructure development, gas flow dynamics, and complex interdependence of gas with electric power networks call for advanced computational tools. Solving the equations relating gas injections to pressures and pipeline flows lies at the heart of natural gas network (NGN) operation, yet existing solvers require careful initialization and uniqueness has been an open question. In this context, this work considers the nonlinear steady-state version of the gas flow (GF) problem. It first establishes that the solution to the GF problem is unique under arbitrary gas network topologies, compressor types, and sets of specifications. For GF setups where pressure is specified on a single (reference) node and compressors do no appear in cycles, the GF task is posed as an convex minimization. To handle more general setups, a GF solver relying on a mixed-integer quadraticallyconstrained quadratic program (MI-QCQP) is also devised. This solver can be used for any GF setup and any network. It introduces binary variables to capture flow directions; relaxes the pressure drop equations to quadratic inequality constraints; and uses a carefully selected objective to promote the exactness of this relaxation. The relaxation is provably exact in networks with non-overlapping cycles and a single fixed-pressure node. The solver handles efficiently the involved bilinear terms through McCormick linearization. Numerical tests validate our claims, demonstrate that the MI-QCQP solver scales well, and that the relaxation is exact even when the sufficient conditions are violated, such as in networks with overlapping cycles and multiple fixed-pressure nodes.

Distributed robust synergistic scheduling of electricity, natural gas, heating and cooling systems via alternating direction method of multipliers

“…Natural gas network is a very complex nonlinear nonconvex system, which is comprised of gas sources, pipelines, and compressors. 34 In this paper, the steady-state natural gas flow based on the Weymouth equations is employed. The natural gas flow in the natural gas pipeline is modeled as (16), which ensures the gas flow in one direction along the pipeline.…”

“…In order to solve the nonconvexity caused by Weymouth equations, a binary variable-based method 34 and convex relaxation technique are adopted to transform the (16) into the MISOCP form. First, with regard to nonconvexity caused by sign function in (16), the binary variable x mn, t is introduced to replace the sign function to represent the direction of natural gas in the gas pipeline mn.…”

“…Noted that the constraints (22) are tight, and the exactness of relaxation was proved in natural gas networks. 34 2z mn,m,t −2z mn,n,t + π n,t −π m,t ≥ α mn Ágf 2 mn,t ,…”

“…Finally, the scheduling model for IES considering multiple uncertainties is turned into the robust scheduling model with objective function (35) and constraints (2) to ( 9), ( 11), ( 12), ( 14), ( 15), ( 17), ( 18), ( 20), ( 22), ( 23), (32) to (34), and (36).…”

“…Natural gas network is a very complex nonlinear nonconvex system, which is comprised of gas sources, pipelines, and compressors 34 . In this paper, the steady‐state natural gas flow based on the Weymouth equations is employed.…”