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Understanding the connection between water supply and energy consumption in a circular economy is crucial for managing the demands of these resources. Sankey diagrams have the potential to enhance the visualisation of resource flows. However, there is a knowledge gap regarding the effective use of Sankey diagrams to capture the water-energy nexus and circularity of water. This study focused on effectively utilizing Sankey diagrams to model the nexus of water and energy flows and to assess the circularity of water. A comprehensive literature review was conducted on major studies on water material flows and modelling. This was supported by a review of approaches for modelling circularity, which have evolved from the material circularity index. The newly developed framework in this paper models water resource flows at all stages of the water supply chain, embedding the nexus as an interaction and considering resource intensity and material circularity index at each node. A city-level case study in South Africa was then used to apply the recommendations, model circularity, and the water-energy nexus using Sankey diagrams. In terms of circularity, the results indicate that water loss in the study area was very high, accounting for up to 40% of the total water supply and the water recycle rate was still very low, ranging from 3% to 7%. In terms of the nexus, the residential sector was found to have the highest energy intensity for water supply, primarily due to energy demand for water heating. Exploring the use of renewable or solar energy for water heating will be of interest and value. The new approach presented in this study allows for the simultaneous consideration of objectives, enabling a more robust analysis of energy-effective water use and water circularity along the water supply chain. [Display omitted] • A new framework for modelling the water energy nexus enriched by circularity options, energy intensity factors and confidence modelling. • Options for embedded circularity through the water supply chain. • The urgent challenge to address massive water loses in water supply systems for water scarce regions. • New Sankey diagrams showcasing how to model nexus and circularity. [ABSTRACT FROM AUTHOR]

This paper addresses the critical challenge of sizing hybrid renewable energy systems over their lifespan while accounting for uncertainties in energy sources and load demands. By leveraging stochastic programming, we introduce a novel modeling technique that ensures robust optimization without compromising numerical tractability. Unlike conventional methods, which determine system parameters for the entire project duration upfront, our approach enables annual adjustments to the renewable components' sizing. This dynamic strategy alleviates early-stage under-utilization, aligning system capacity with evolving demand patterns. Central to our approach is the reformulation of the two-stage stochastic program as a quasi-optimal control. This reduces the number of optimization variables significantly, simplifying the solution process. Moreover, thousands of constraints are replaced with a system of differential equations, enhancing computational efficiency. By minimizing capital costs and dynamic operating expenses, we achieve an optimal system size. Through a real-world application in a rural area of South Africa, we demonstrate the effectiveness of our approach. The results showed a remarkable improvement in problem-solving efficiency, reduced under-utilization, and heightened adaptability to unpredictable weather conditions. This innovative framework represents a substantial leap forward in hybrid renewable energy systems design, offering a more agile and efficient solution for sustainable energy generation. [ABSTRACT FROM AUTHOR]