## 5 Case Study
### Feature Extraction
#### 5.3.2 Inertia Tank Tendency Temperature  
The inertia tank operates based on the same principles as the laboratory but with different values. It has a set point, meaning that if the tank's temperature falls below this value, the heat pump is activated. The tank also has upper and lower thresholds. The tank heats to the set point plus the upper threshold, and once this value is reached, the heat pump turns off until the temperature drops below the set point minus the lower threshold. This behavior can be observed in Figure 9 from 09:00 to 17:00. In this system, the upper threshold is set to 0.7 \({}^{\circ}\)C and the lower threshold is set to 1 \({}^{\circ}\)C.  
Figure 8: Temperature, set point, and thresholds of the laboratory. This plot visualizes the oscillation of the laboratory’s temperature between the upper and lower thresholds. Figure 9: Temperature, set point, and thresholds of the inertia tank. This plot visualizes the oscillation of the inertia tank’s temperature between the upper and lower thresholds. The laboratory only has two modes, either becoming hotter by the use of the fan coil or colder by the difference with the exterior temperature. For the inertia tank, the tendency function has to differentiate if the fan coil is ON or OFF. Four distinct modes are distinguished based on the status of the fan coil (ON/OFF) and the status of the heat pump (ON/OFF). If the fan coil is OFF, the behavior of the inertia tank is straightforward, resembling that of the laboratory. If the temperature is below the set point minus the lower threshold, it tends to be heated to the set point plus the upper threshold, if the temperature is above the set point, it tends to be cooled tending toward the temperature of the machine room, which is the environment with which the inertia tank is in contact. If the fan coil is ON, it is observed that the temperature of the inertia tank reaches a certain equilibrium temperature. This is due to a balance between the capacity of the fan coil to extract heat and the capacity of the heat pump to introduce heat. Figure 10a,b show an example where the laboratory's temperature is below the set point for long enough to observe how the inertia tank reaches a equilibrium temperature when the set point is at 40 \({}^{\circ}\)C from 08:00 to 17:00. Finding this equilibrium temperature is essential to define the tendency function of the tank. To identify the equilibrium temperature of the inertia tank, an alternative variable was chosen that expresses the same idea: the difference between the set point and the actual temperature. This approach offers greater utility by providing insights into temperature differences at low and high set points without directly involving the tank temperatures. The procedure was as follows: from the dataset, cases were initially selected where both the fan coil and the heat pump were activated; next, the temperatures of the inertia tank were classified according to their set points; following that, the difference between the set point and the tank temperature was computed, and only the positive values were stored. Because only temperatures below the set point are of interest, only positive differences were stored. The data obtained are shown in Table 3.  
The set points of 39.7 \({}^{\circ}\)C and 47.0 \({}^{\circ}\)C are identified as outliers that can be dismissed due to their low number of samples, which are considered non-representative. The remaining points in Figure 11 follow an exponential function with the general expression \(f(x)=Ae^{bx+c}\) with \(A,b,c\in\mathbb{R}\). Using the curve_fit function from the scipy.optimize package [27], an exponential curve was successfully fitted to the data. The parameters \(A\), \(b\)  
Figure 10: Temperatures on Sunday 4 February 2024. (**a**) Temperature and set point of the laboratory. (**b**) Temperature and set point of the inertia tank.