## 3 Literature Review for Water Heating Modelling  
Several works dealing with EWH modelling can be found in the literature. First papers dealing with the use of ERWH for DR introduced the so-called one-zone/mass model (i.e., one capacitor to model the storage capacity of water, or \"1C\") to simulate the dynamic of water temperature inside the tank/reservoir using a first-order differential equation [14] or a stochastic first order-partial differential equation (SPDE) [15]. Ref. [14] develops the aggregation of ERWH using Monte Carlo methodologies, whereas [15] applies probability density functions obtained through the resolution of SPDE. This last approach considerably simplifies the aggregation process and reduces the simulation time. Models reflect the dynamics of water temperature assuming a uniform distribution of temperature into the reservoir, but this is not true because laboratory measurements (see Section 4) clearly state that some significant variations of the temperature appear through the tank (stratification) after most of the water draws (e.g., water draws around 10-20% of tank capacity) due to cooking, shower, or bath requirements. Consequently, the estimation of temperature and thermostat control are wrong. In [16], a two-node/masses ERWH model (i.e., the use of two capacitors to model the storage capacity, or 2C) is developed. The model accounts for both thermal dynamics and users' water consumption. The model assumes fixed volumes for the cold zone (bottom of the tank) and warm zone (top of the tank), and the temperature in each node is assumed to be uniform. Unfortunately, there is some lack of validation of the stratification with real data. Water draw patterns were estimated using average residential demand profiles. Ref. [17] presents a hybrid model that changes between one and two masses/zones (1C to 2C) depending on the service of WHs. The authors use a first-order model to model  
Figure 2: Example of individual and aggregated appliance demand curves: (**a**) Recordings for 80 L ERWH and 80 L HPWH; (**b**) Simulation of the average aggregated demand with DR control (40 WH loads with a standard water service profile in winter, granularity of the simulation output 15 min). WH dynamics when the WH is charged or fully discharged (after a major water draw). The simulator remains with this model until a draw occurs, and it uses a model very close to [16]. The height (capacity) of the lower (cold) zone goes down when the appliance demands power to get the thermostat setpoint and remains with this model until the WH is fully charged. Then, it changes to a one-zone model. The author's simulation results show the effectiveness of WH responses and their dependence on hot water usage with this hybrid model. The main problem is that the model is not tested against measured data to make a comparison of its performance possible. Through a partial differential equation, a physical-based model is developed in [18] to describe the temperature profiles at different WH tank locations. The WH has two resistors that operate independently. Sensors are located at different positions, and the model allows us to know the temperatures at each site and evaluate the ON/OFF state of each resistor. The model was validated against field test data for more than 10 days. The validation scenario is quite limited because only four water draws were tested. Another main drawback is that the computational complexity is like [15] but significantly higher than in the cases of the one- or two-node models [14; 16; 17]. In DR scenarios, the simulation time is very relevant because WHs need a further aggregation process, and such a model requires an excessive amount of time to obtain results to be useful for aggregators in some services or markets--for instance, in Ancillary Services (AS). The physical model presented in [19] divides a large WH tank into eight zones/masses (8C) to develop a more complex model to account for water stratification. The simulated response of the joint model represents the measurement data quite accurately.