Electricity and natural gas networks, as two major energy transport infrastructure, have traditionally been planned and operated independently from each other. Electricity generation was dominated by coal, oil and nuclear power stations prior to 1990, when the "dash for gas" brought a significant number of gas-fired power stations into the generation mix. These geographically dispersed large power stations have created a loose link between natural gas and electricity networks at the energy source. As the pace of decarbonising our electricity section accelerates, the two energy networks will progressively become more closely linked by end users, driven by the electrification of heat and major efficiency improvements. This presents critical new challenges to the traditional network modelling, operation and optimisations, in particular as they were developed independently for natural gas and electricity networks. The traditional methods do not take into account of the substantial rise in the interaction between the two networks, i.e. how a change in gas demand/resource might impact the demand/generation of the electrical system and vice versa. The vision of this research is to develop a statistical model for combined gas and electricity systems at the distribution level that can efficiently simulate the interactions across the energy vector under severe uncertainties. The developed model will then be fed to the novel optimal operation strategies to manage the two systems for encouraging increased use of renewables and infrastructure and promoting customer interaction with the systems. This fellowship will address this vision by developing highly efficient network sampling methodologies, multi-vector probabilistic energy flow and optimisation tools that will transform the modelling and analysis of highly integrated systems. These new developments will: i) enable detailed real-time analyses of energy flows and capacity bottlenecks of the highly integrated energy systems with high accuracy and in reasonable time scale, ii) assist network operators to optimise the performance of the existing energy systems to minimise the cost of integrating low carbon generation and demand, and iii) assist policy makers to design effective policies and regulations for economic and sustainable energy network development.