**1. Introduction**

In the context of climate change, energy demand for urban water supplies has emerged as a relevant issue [1]. In the future, more energy is expected to be required to treat and supply drinking water to citizens due to the adaption of water systems to the effects of climate change and to new regulatory requirements. Goal 6 of the Sustainable Development Goals adopted by the United Nations (UN) in 2015 involves the achievement of universal and equitable access to safe and affordable drinking water for all by 2030. The achievement of this goal will require the construction of many more drinking water treatment plants (DWTPs), which will increase the amount of energy required for drinking water supplies worldwide.

Urban water utilities use energy to extract, convey, treat, and distribute drinking water. Previous studies have focused on the evaluation of energy requirements for one or several activities in the urban water cycle. Several studies have focused on the quantification and evaluation of the economic and environmental effects of energy use to supply drinking water to major cities and on the comparison of energy use among cities and countries [2,3]. Other studies have involved a more detailed analysis of individual drinking-water supply stages and quantification of the energy required to treat raw water, i.e., the energy used by DWTPs [4]. The aim of these studies was to quantify and compare the energy intensity of DWTPs, that is, the energy consumed (kWh) per unit volume (m3) of drinking water produced (kWh/m3) [5]. However, as Santana et al. [6] and Molinos-Senante and Sala-Garrido [7] determined, the energy required to treat raw water depends on several factors, including the quality of the raw water and of drinking water standards, as well as the water treatment technology used. Sowby and Burian [8] also emphasized this issue; after analyzing the energy requirements for drinking water supplies in 109 cities in the United States, they concluded that energy intensity is an overly simplistic metric that is not adequate for the comparison of DWTP performance.

To overcome the comparability limitation and to facilitate benchmarking between DWTPs, a few studies have focused on evaluating DWTP energy efficiency. Molinos-Senante and Sala-Garrido [9] defined energy efficiency as a "synthetic index that incorporates both the quality of the raw water being processed and the energy required to treat it." Although several methods can be used to estimate the energy efficiency of DWTPs, data envelopment analysis (DEA) has been used in the few studies conducted to date. DEA is a non-parametric method based on mathematical programming techniques that integrates multiple inputs (energy use) and outputs (concentrations of several pollutants removed from raw water and the volume of treated water) into a synthetic index; namely, the energy efficiency score. Molinos-Senante and Guzman [10] computed the energy efficiency of a sample of DWTPs using the DEA approach, investigating the presence of economies of scale in these facilities. Applying the more advanced metafrontier DEA model, Molinos-Senante and Sala-Garrido [9] compared energy efficiency among DWTPs using different treatment technologies. Recently, Ananda [11,12] computed the environmental efficiency and productivity change of a sample of 49 Australian urban water utilities using DEA, with a focus on economic issues, but integration of greenhouse gas emissions as undesirable outputs.

These studies [9–12] contributed to the literature by providing estimated energy efficiency scores for DWTPs derived from the application of a holistic and integrated approach. However, they have ignored the deterministic nature of the DEA methodology; as statistical inferences cannot be drawn from conventional DEA (energy) efficiency scores [11] and regression analysis cannot be conducted to explore the determinants of previously estimated scores [12]. Moreover, conventional DEA models do not integrate data variability into the (energy) efficiency assessment, which negatively impacts the robustness and reliability of the results.

In the framework of efficiency assessment, two main alternative methodological approaches have been proposed to explore the causality between factors and efficiency scores. Cazals et al. [13] proposed the *order* − *m* method, in which a fraction of the sample is used to estimate efficiency scores. However, selection of the *m* value is challenging, and it affects efficiency scores [14]. Alternatively, Simar and Wilson [15] proposed a double-bootstrap DEA procedure for the estimation of efficiency scores that overcomes the two main limitations of conventional DEA models; i.e., it allows exploration of the determinants or factors affecting efficiency scores [16], and it permits bias correction and the calculation of confidence intervals for the scores. Despite the relevance of this type of analysis, the bootstrap approach has not been used to evaluate the energy efficiency of DWTPs.

Against this background, the objectives of this study were twofold. The first objective was to assess the energy efficiency of a sample of DWTPs with consideration of data variability, i.e., to estimate bias-corrected energy efficiency scores and their confidence intervals. The second objective was to explore the determinants DWTP energy efficiency. To do so, we employed the double-bootstrap DEA approach proposed by Simar and Wilson [17]. Empirical application was performed with a large sample (*n* = 146) of Chilean DWTPs. Although many scholars have examined the energy intensity of urban water cycle activities in recent times, few studies have assessed the energy efficiency of water treatment plants and none has involved the application of a robust methodological approach such as double-bootstrap DEA. This paper contributes to the current body of literature in the water–energy nexus field by presenting for the first time bias-corrected energy efficiency scores and discussing factors affecting the energy efficiency of a sample of DWTPs.
