## 1 Introduction
It is worth mentioning that an integrated approach can be one of the means to address the sustainability of atmospheric water harvesting, as well as the employment of renewable energies to run the devices [16]. Nevertheless, conducting an efficiency assessment of integrated systems can be somewhat challenging, in particular when a choice among different complex machines should be carried out in an early stage of the plant design. To address this issue, the authors, in 2022, proposed the Global Evaluation Index (GEI) formulation [21]. The GEI is a new simplified metric to evaluate, by means of a single increasing number, the energy efficiency of a multi-purpose machine, taking into account the integration degree. The need for such a metric was evident when an evaluation of integrated AWGs was required, because the existing evaluation tools concerning atmospheric water capturing techniques were not suitable, as they are focused only on water collection [21]. Instead, the GEI was designed for the assessment of multipurpose machines, encompassing HVAC functionalities as well as atmospheric water extraction. The index permits comparing similar integrated machines or comparing integrated and traditional systems (single-purpose machines providing only one useful effect). Nevertheless, the efficiency improvement process in the HVAC field can lead to hybrid solutions that combine existing machines with advanced ones, the latter used as a booster for the former plant design. For example, it is common to find retrofitted plants comprising electrical heat pumps combined with gas-boilers, sharing the heat load [22]. Multipurpose machines can even be used as a retrofit for the existing plants, covering totally, or partially, the energy load, adding the benefit of providing more than one useful effect with the same energy input of a single piece of equipment. In particular, integrated AWGs can be employed for such a scope, as described in [23], where such a system was installed in a worker village in Dubai to provide drinking water and also partially cover the cooling and heating energy required. These plant configurations, which include both integrated and non-integrated machines and can be referred to as \"hybrid\", currently lack a suitable evaluation tool. The Global Evaluation Index (GEI), as noted earlier, was designed for comparing homogenous systems and is not fully applicable to hybrid configurations. Furthermore, a review of the current literature on AWGs reveals criticisms of existing evaluation tools, particularly regarding their definitions and applicability domains [24]. This research seeks to address the knowledge gap in evaluating hybrid solutions by proposing a new and more versatile evaluation framework: the Advanced Global Evaluation Index (AGEI). The AGEI, which is a generalisation of the GEI, is specifically designed to assess both hybrid and homogenous system configurations. Additionally, the study provides a critical review of the most commonly used AWG evaluation tools to clarify their definitions and practical applicability. This study presents three key contributions:  
* A concise and reasoned review of the most commonly used tools for evaluating Atmospheric Water Generators (AWGs). * The development of the AGEI formulation, tailored for hybrid systems. * The first application of the AGEI to assess complex plant configurations, including a hybrid system combining a heat pump with a traditional drinking water supply system. The latter involves desalinated seawater that is bottled and transported to consumers. It is worth mentioning that the current research presents for the first time a comparison between atmospheric water and bottled desalinated water. This current research revisits the case study analysed in [23], which analytically explored the benefits of hybrid retrofitting. The current work expands the scope of the previous one by examining additional plant configurations and applying the AGEI for evaluation.