# A Thermal Discomfort Index for Demand Response Control in Residential Water Heaters
## 3 Related Works  
This section reviews relevant related works about load reduction techniques and thermal discomfort evaluation. Several works in the literature have described the main concepts of peak load reduction strategies and specific implementations in smart grids [9; 16]. In particular, direct load control allows utilities to remotely manage electricity demand by modifying the operation of end-use devices to perform load shifting [11; 17]. A special type of devices to perform load shifting are thermostat-controlled appliances (TCA). The main feature of TCA is that they provide flexibility to select the desired temperature, i.e., the _thermostat set point_[18; 19]. In addition, some TCAs can also store energy in the form of heat, providing a great advantage over non-storage appliances when performing load shifting strategies. This is the case of domestic electric water heaters, which are the focus of this article. [PERSON] et al. [20] proposed and analyzed an interactive demand-side management strategy for electric water heaters, but the research did not elaborate on specific aspects of thermal discomfort of users. A recent article by [PERSON] et al. [21] introduced a complex strategy to minimize thermal discomfort related with electric water heater control. The proposed strategy requires information to build a time-varied weight matrix based on detected utilization patterns of domestic electric water heaters. In turn, the weight matrix and other information are used to generate a customer satisfaction prediction index. Due to the large amount of information required to build the model, the approach is difficult to apply in practice. Demand response control strategies for TCA can be effectively implemented provided that thermal comfort is not compromised. Thus, quantifying the impact on user comfort is a crucial aspect, and it has been the focus of several works. [PERSON] et al. [22] evaluated the thermal discomfort in demand response control of several appliances used for heating, ventilation, and air conditioning in a university building. A _daily discomfort score_ was proposed for demand response events to reduce the cost of energy, but the proposed strategy requires knowing the real temperature. Regarding electric water heaters, the study of the hot water utilization profile by end-users is a key aspect for estimating discomfort. [PERSON] and [PERSON] [23] studied whether a smart heating system can benefit from good predictions of the user behavior. [PERSON] et al. [24] proposed an algorithm for estimating domestic hot water utilization, but the technique requires the installation of temperature and vibration sensors. The main factor that defines comfort is the water temperature. Thus, a model to estimate water temperature from measured information is crucial for the effectiveness of the comfort evaluation. [PERSON] et al. [25] proposed a water heater model to estimate the temperature of the water in the tank as a function of time and the related variables, including the thermal losses and the water utilization. Data from smart meters, recorded at 15 min intervals, were used for validation. The model was proposed to be applied in a multiobjective demand-side management program. The study by [PERSON] et al. [26] provided a comprehensive empirical analysis of a simplified energy consumption model for water heaters considering the variation of the temperature of the water in the tank. Results of the proposed Water Heater Analysis Model (WHAM) model were compared with data from water heaters simulation programs (TANK, WATSIM, and WATSMPL). WHAM obtained very accurate prediction results, while being significantly faster. In addition, WHAM requires less detailed engineering information about the water heaters. Finally, comfort evaluation must be considered in the problem of controlling a subset of electric water heaters, e.g., by building a ranking to sort all interruptible devices according to appropriate criteria. [PERSON] et al. [27] proposed a scheduling strategy based on a temperature state priority list.