*2.1. Energy Efficiency Estimation*

Conventional DEA models, such as those proposed by Charnes et al. [18] and Banker et al. [19] (the CCR (Charnes, Cooper and Rhodes) and BCC (Banker, Charnes and Cooper), respectively) and subsequent developments, have been employed widely to evaluate the efficiency of water treatment facilities, such as WWTPs and DWTPs. DEA models can be used with a constant returns to scale (CRS) or variable returns to scale (VRS) technique. With the CRS approach, outputs increase in proportion to inputs, and producers (DWTPs in this case) are assumed to be able to linearly scale inputs and outputs without changing efficiency. By contrast, with the VRS approach, an increase or decrease in input or output does not result in a proportional change in outputs or inputs, respectively. Previous studies [1,5] have shown that the energy consumed by DWTPs to produce drinking water does not depend linearly on the pollutants removed from raw water. Hence, in this study, and as in Molinos-Senante and Sala-Garrido's studies [9], a VRS DEA model was employed to evaluate DWTP energy efficiency.

Moreover, DEA models can have an input or output orientation, depending on whether the aim of the units analyzed (DWTPs) is to minimize the use of inputs or to maximize the production of outputs. In this case study, an input orientation was adopted because the main objective of DWTPs is to produce drinking water that complies with the legal quality standards using minimum energy. The quantity of pollutants to be removed depends on the quality of the raw water and the thresholds defined by drinking water standards, which are external to the water utilities.

Let us assume that we have a set of DWTPs, *j* = 1, 2 ... , *N*, each using a vector of *M* inputs, *xj* = - *<sup>x</sup>*1*j*, *<sup>x</sup>*2*j*, ..., *xMj* , to produce a vector of *S* outputs, *yj* = - *y*1*j*, *y*2*j*, ..., *ySj* . Assuming VRS technology, the input-oriented DEA model is denoted as follows:

$$\begin{array}{c} \min \theta\_{j} \\ \text{s.t.} \\ \sum\_{k=1}^{N} \lambda\_{k} \mathbf{x}\_{ik} \le \theta\_{j} \mathbf{x}\_{ij} & 1 \le i \le M \\ \sum\_{k=1}^{N} \lambda\_{k} y\_{rk} \ge y\_{rj} & 1 \le r \le S \\ \sum\_{k=1}^{N} \lambda\_{k} = 1 \\ \lambda\_{k} \ge 0 & 1 \le k \le N\_{\prime} \end{array} \tag{1}$$

where *θj* is the energy efficiency score of the DWTP evaluated, *M* is the number of inputs considered, *S* is the number of outputs produced, *N* is the number of DWTPs evaluated, and *λk* is a set of intensity variables representing the weighting of each DWTP evaluated, *k*, in the composition of the efficient frontier.

The energy efficiency score (*θj*) ranges from 0 to 1. A DWTP is energy efficient when *θj* = 1 and inefficient when *θj* < 1. For an energy-inefficient DWTP, the value of 1 − *θj* informs us about the potential for energy savings, as it is the proportional input (energy consumption in this study) that can be achieved by DWTP *j*, given the level of outputs produced.
