In power networks with increasing shares of sustainable energy resources such as wind and solar radiation, the supply of energy is subject to natural fluctuations. Since the power demand itself is also a varying quantity, there is a need for large-scale storage methods.

Gas networks as storage agents in the power system have been subject to recent investigations, since they can generate electrical power to compensate for a lack in available power, and vice versa convert electrical energy into gas through processes such as hydrolysis when there is excessive supply and low demand.

A mathematical model of the connection between power and gas networks must take into account the different natures of the corresponding infrastructures. Power supply and demand change over time and due to various factors and are hence inherently stochastic. But while the transport of power can be modelled as instantaneous, the control of gas reserves as a stabilization must consider the temporal and spatial component of the gas flow in the network. This makes controlling the system complex and underlines the need for large-scale simulation tools.

In the article “Efficient simulation of coupled gas and power networks under certain demands” (April 2022, European Journal of Applied Mathematics) [1], the authors Eike Fokken, Simone Göttlich and Michael Herty provide such a software tool and carry out simulation experiments in this direction.

The mathematical model in the paper and the corresponding open-source software package called ‘grazer’ includes models of a power network, a gas network, and interconnections between the two, which symbolize physical gas power plants that transform either of the quantities gas and power into the other at given efficiency rates.

The networks are modelled as one graph, with different properties of the nodes and arcs in the power and the gas network.

In the former, the physical supply and demand of power needs to be modelled. This reduces to the time-dependent quantities active and reactive power at each node. The connections between nodes carry information in form of the circuit admittance. Some of the nodes in the power network have a stochastic nature, modelling the fluctuating demand in the network. The authors model this by sampling from a cut-off Ornstein-Uhlenbeck process since this type of process can represent a realistic power demand through its time-dynamic mean.

The gas network, on the other hand, has a temporal and a spatial aspect. Here, arcs are either controlled units like valves or compressors, or pipelines with a physical extension. The modelled quantities in here are gas pressure and flux and the two must satisfy a balance law that describes the macroscopic movement of the gas through pipelines. Boundary conditions are imposed to ensure pressure equality at nodes, and controlled arcs are realistically modelled to allow pressure changes, either an increase in pressure through compressors or a decrease through valves.

Numerically, solutions are computed at fixed time increments using Newton’s method. The equations imposed to the quantities in the network can mostly be easily discretised. A difficulty in the discretisation arises from the solution of the balance laws that must hold for the gas, which the authors overcome by employing a suitable implicit box scheme. Further care must be taken when drawing the discrete samples from the Ornstein-Uhlenbeck process, this is done by discretising the corresponding SDE at finer time-steps.

A strength of the numerical experiments carried out in the study is the usage of existing open-source data for both the power and the gas network. An existing power network dataset is slightly altered to include the stochastic nature from the model and the two networks are connected through simulated gas power plants. The authors then simulate their model and give exemplary results for gas pressure deviations that result from the stochastic power demand over the course of 24 hours. They further employ an optimal control framework to solve for compressor and valve states in a setting where operating network agents relates to costs.

[1] FOKKEN, E., GÖTTLICH, S., & HERTY, M. (2022). Efficient simulation of coupled gas and power networks under uncertain demands. _European Journal of Applied Mathematics, 1-27. doi:10.1017/S0956792522000079