# A Thermal Discomfort Index for Demand Response Control in Residential Water Heaters
## 1 Introduction  
Energy demand management is a crucial idea for the modern paradigm of smart cities. The concept of smart electricity networks, or _smart grids_, refers to electrical grids enhanced by including operation and management features to improve the controlling of production and distribution of energy [1]. Smart grids are mainly oriented to maintain a reliable and secure infrastructure to allow properly satisfying the demand growth, the integration of distributed energy resources, smart storage, and other features related to smart devices and real-time information provided to clients [2]. Information and Communication Technologies (ICT) are very closely connected to smart grids, as they provide the basis for communicating and processing information that is very useful at different levels to implement the aforementioned services [3]. The daily consumption pattern of electricity involves periods with higher-than-average electricity consumption (_peaks_) and other periods with lower-than-average electricity consumption (_valleys_). A common situation in the daily operation of electric grids is that electricity generation and transmission systems may not always meet peak demandrequirements. In these situations, power demand management strategies are helpful tools for operation of electric grids. Power demand management refers to the proper administration of power consumption for end consumers in a smart grid in order to promote better energy utilization. Two of the most widely applied actions for power demand management are load management, whose main goal is modifying, reducing, or shifting the demand, and energy conservation, which is mainly focused on reducing the demand, e.g., via technological improvements. In turn, several other actions have been applied for demand management, including fuel substitution and load building [4]. Among load management techniques, the most used are peak reduction, oriented to reduce power consumption in periods of maximum demand; valley filling, whose main goal is to promote energy utilization in off-peak periods,; and load shifting from peak to off-peak periods. Several demand response and demand management tools can be applied to mitigate overloads in the electrical system. One of the simplest yet most effective methods for direct load control is allowing the electric company to remotely control user devices. This method is properly applied to control those devices with a thermostat, especially those that have important thermal inertia. On the one hand, remote control is a very effective technique to achieve peak reduction and load shifting at critical periods. On the other hand, the benefits of the achieved reduction of energy consumption in the overall operating cost of the electrical system must be weighted against the loss of comfort that the users of the controlled devices may have. In order to define an economic value to the loss of comfort associated with an intervention, to be taken into account in the business model of the electric company, the discomfort of users must be properly evaluated with quantifiable metrics in advance. In this line of work, this article proposes and evaluates a methodology to calculate a Thermal Discomfort Index (TDI) associated with a remote intervention for load management from the electric company. The computed index evaluates the discomfort for users, generated by the intervention of electric water heater appliances. The main goal of the index is to be a valuable tool for defining a method to manage a set of water heaters to defer electrical demand by properly identifying those appliances that have the lower impact on user comfort to be interrupted first. This way, the values of the computed TDI make it possible to decide in which order water heaters should be interrupted to minimize the overall total discomfort of a set of users. The key aspect of the proposed strategy is to know in advance the value of TDI in order to decide if it is economically profitable to carry out an intervention.