# Applications of Probabilistic Forecasting in Demand Response
## 2 Materials and Methods
### Demand Response and Physically Based Load Models
#### 2.5.1 Elemental Models  
To manage the computational complexity of the aggregation processes through the consideration of several hundreds of elemental/individual models, physical processes (in our case heat conversion and transport) must be simplified to some extent to only account for main-state variables which can be of interest for evaluating and performing load management and demand response. Some examples of these basic models are depicted in Figure 1. Firstly, Figure 1a illustrates a basic third-order model for Electric Heating (EH) and the connection with each electrical component of the model (notice that it is a thermal-electrical equivalent), which is the physical appliance and its \"environment\". The picture shows various heat sources, some indicated with yellow arrows (like solar radiation) and others in magenta, representing internal heat generation loads (such as other appliances, inhabitants, or ventilation leaks in the dwelling). Heat sources are modeled through current sources. In the equivalent, voltage nodes (in green) represent temperatures (state variables in PBLMs) inside the dwelling, walls, ceilings, or ground (according to the relevance of heat exchanges through other dwellings or flats downstairs and upstairs (Figure 1 on the left)). Capacitances (red/yellow triangle) represent the specific heat capacities of envelope building materials, and air and furniture to a minor extent. Resistors (arrow in brown) represent the heat conduction/convection coefficients that model heat fluxes (losses or gains according to seasons) through walls and windows. Heat exchange through solar radiation (current sources H\({}_{\text{SW}}\) and H\({}_{\text{W}}\) on the top) can be considered or simplified. It depends on the relevance of these inputs for each specific dwelling (for instance, north or south orientation) and season (winter or summer). Figure 1b,c presents the physical device and the elemental equivalent RC network for Water Heaters (WHs). Heat Pump (HPWHs) and Electrical Resistor Water Heaters (ERWHs) can be described by the same PBLM network. In the elemental HPWH model, heating is driven by two independent heat sources: a main source (the compressor of the Heat Pump) and an ancillary source, the heating resistor (Figure 1b). Operation settings between the resistor and Heat Pump can be considered through the \"MODE\" switch (Figure 1c). Obviously, the model can be adapted for ERWH loads by setting H\({}_{\text{HP}}\) = 0. Heat exchanges in the water tank are modeled in two ways: conventional current sources for energy conversion (in red in Figure 1c), and two dependent current sources representing the water inlet (\"cold\" input to tank) and outlet (\"heat\" output, the load service). In the equivalent, node voltages are again temperatures (state variables) in the different zones of the water tank (WH1-2, Figure 1c), justified by water stratification inside the tank (hot water at the top and cold water at the bottom of reservoir). Finally, electrical capacitances C\({}_{1}\) and C\({}_{2}\) represent the heat storage capacity (red/yellow triangles) of each tank zone (i.e., WH1 and WH-2), while resistors model heat losses through conduction/convection between the boundaries of the tank zones (thermocline) and between the tank and its surroundings (pipelines, dwelling). It is important to underline that, in some cases, there is potential feedback between the operation of a class of load (end-use) and others in the same dwelling, and it can be considered with this PBLM approach. For instance, HPWHs cool the surrounding of the load (Figure 1b, heat exchange) and this can affect EH performance because the internal  
Figure 1: Elemental PBLMs and their environment: (**a**) Electric Heating (EH) in flats; (**b**) water heating (WH) appliance; (**c**) WH Electrical–Thermal equivalent. load changes (Figure 1a, at the bottom): it can support cooling in summer, or it can increase heat \"losses\" in winter.