Natural Gas Flow Equations: Uniqueness and an MI-SOCP Solver

Abstract

The critical role of gas fired-plants to compensate renewable generation has increased the operational variability in natural gas networks (GN). Towards developing more reliable and efficient computational tools for GN monitoring, control, and planning, this work considers the task of solving the nonlinear equations governing steady-state flows and pressures in GNs. It is first shown that if the gas flow equations are feasible, they enjoy a unique solution. To the best of our knowledge, this is the first result proving uniqueness of the steady-state gas flow solution over the entire feasible domain of gas injections. To find this solution, we put forth a mixedinteger second-order cone program (MI-SOCP)-based solver relying on a relaxation of the gas flow equations. This relaxation is provably exact under specific network topologies. Unlike existing alternatives, the devised solver does not need proper initialization or knowing the gas flow directions beforehand, and can handle gas networks with compressors. Numerical tests on tree and meshed networks with random gas injections indicate that the relaxation is exact even when the derived conditions are not met.

Natural Gas Flow Solvers Using Convex Relaxation

“…The relaxation is provably exact in NGNs with non-overlapping cycles and a single fixed-pressure node. This significantly extends the claim of [8], where exactness was proved for non-overlapping cycles and a single fixed-pressure node, but did not allow for compressors in cycles. Having compressors in cycles is a typical arrangement, e.g., when two compressors are connected in parallel.…”

“…Therefore, the (un)-constrained energy function minimization approaches of (7)-(8) apply to the gas flow and water flow problems alike. For water networks, the decomposition or 'stitching' technique of [20] extends (7)- (8) to water network setups with |N ψ | = 1 and pumps, but not lying on cycles. A similar technique can be used to solve the GF problem with compressors not on cycles and |N ψ | = 1.…”

“…Under Condition 1, the nodal injections are fixed a priori and the GF task aims at finding the associated (ψ, φ). Albeit Definition 1 considered the GF task with multiple fixedpressure nodes, the setup of a single fixed-pressure node is commonly met; see [10], [8], [11], [25]. Regarding Condition 2, although it may seem restrictive at the outset, it is satisfied by several practical gas networks [26].…”

“…The contribution of this work is on four fronts: First, Section III establishes that the nonlinear steady-state GF equations enjoy a unique solution even with multiplicative compressors. Building on [8] where uniqueness was shown for GF setups with a single fixed-pressure node, here uniqueness is non-trivially generalized to setups with multiple fixedpressure nodes. Second, Section IV reformulates the GF task as a convex minimization.…”

“…The obtained solver can handle GF setups with a single fixed-pressure node and compressors not on cycles. Third, Section V expands the analytical arXiv:1906.01711v1 [math.OC] 4 Jun 2019 claims for the MI-QCQP gas flow solver of [8]. Different from the convex minimization approach, this solver applies to any GF setup and any network.…”