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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.

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.

With dynamic electricity pricing, the operation of water distribution systems (WDS) is expected to become more variable. The pumps moving water from reservoirs to tanks and consumers can serve as energy storage alternatives if properly operated. Nevertheless, the optimal scheduling of WDS is challenged by the hydraulic law for which the pressure along a pipe drops proportionally to its squared water flow. The optimal water flow (OWF) task is formulated here as a mixed-integer non-convex problem incorporating flow and pressure constraints, critical for the operation of fixed-speed pumps, tanks, reservoirs, and pipes. The hydraulic constraints of the OWF problem are subsequently relaxed to second-order cone constraints, and a penalty term is appended to its objective to promote solutions feasible for the water network. The modified problem can be solved as a mixed-integer second-order cone program, which is analytically shown to yield WDS-feasible minimizers under certain sufficient conditions. By weighting the penalty in the objective of the relaxed problem, its minimizers can attain arbitrarily small optimality gaps, thus providing OWF solutions. Numerical tests using real-world demands and prices on benchmark systems demonstrate the relaxation to be exact for several cases, including setups where the sufficient conditions are not met.Index Terms-Water flow equations, convex relaxation, secondorder cone constraints, optimal water flow.

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