*4.2. Information Analysis*

The model relies on consumption during standby mode in order to determine whether the appliance is operative or not. Standby energy (*U*) is calculated by the following equations,

$$\hat{F}\_n(t) = \frac{1}{n} \sum\_{i=1}^n E\_{i\epsilon\_i't} \tag{1}$$

$$F = \nabla \widehat{F}\_{\mathfrak{n}}(t) \tag{2}$$

$$
\mathcal{U} = \max(F),
\tag{3}
$$

where *F*\$*n*(*t*)represents the empirical cumulative distribution of energy consumption *Ei<sup>t</sup>* function over time, and *F* represents the gradient of *F*\$*n*(*t*). The algorithm assumes that switching from standby to normal operation corresponds to the maximum energy gradient; therefore, the standby consumption is calculated as the energy that determines the maximum gradient of the cumulative distribution of measured consumption.

In actual operation, the appliance may perform several cycles in close succession (e.g., two washes, one immediately after the other for the washing machine). The model can separate cycles in close succession based on their energy-consumption profile. The disaggregation of cycles is computed with the following equations,

$$
\mathcal{Y} = \nabla \theta\_A \tag{4}
$$

$$
\Delta \mathcal{Y} = ZCI(\mathcal{Y}),
\tag{5}
$$

where *θ<sup>A</sup>* is the energy consumption of aggregated cycles, *Y* is the gradient of consumption and Δ*Y* is the position over time of the relative maxima of energy consumption in aggregated cycles. The *ZCI*(*Y*) function interpolates from the points it identifies near the zero crossings to linearly approximate the actual zero crossings. Maxima characterized by energy consumption above the value that corresponds to the percentile set as a parameter of the model (e.g., 85% of the consumption of the aggregated cycle) identify separate cycles and are used by the algorithm to disaggregate nearby cycles. Typically, energy consumption is higher during the first stage of the cycle (i.e., water heating) for washing machines and dishwashers.

As anticipated, the model neglects operations that last less than or that consume less than the thresholds set as input parameters in order to exclude those situations that may not represent real operating cycles. The corresponding data (spurious data) are stored separately in the results.

Then the model allocates energy consumption in time slots. In particular, hours between 8 a.m. and 8 p.m. can be grouped in a customized number of time slots provided as input. Moreover, the algorithm accounts for the standard time slots defined by the Italian Energy Authority (ARERA) as follows [46]:


For each individual period and cycle of operation, the following quantities are calculated: total energy consumption, average 15-min energy consumption, maximum and minimum consumption, and maximum/minimum ratio as an indicator of the variability of the consumption during operation.

The code then performs a monthly analysis, and calculates the following quantities:


Similar quantities are calculated with reference to the monthly energy consumption. For household appliances with cyclic operation, the following additional quantities are calculated by differentiating cycles according to energy consumption and duration:


Moreover, for single cycles and for aggregated cycles (i.e., cycles before the application of the algorithm that distinguish cycles in close sequence) the following monthly quantities are calculated:


Finally, the code performs a series of energy balance checks and then results are printed and plotted in graphs, and the relevant variables are saved in .xlsx and .mat files.

#### **5. Results and Discussion**

In this section, the analysis on the operation of a washing machine is used as an example to describe the results and the logic of the model. Indeed, this electrical appliance has many features that can be representative of other appliances, i.e., it is cyclic, and it can be used to illustrate the main features of the model. Similar remarks apply to other appliances.

#### *5.1. Calibration of Parameters*

As the algorithm can analyse all domestic appliances, the control parameters need to be tuned before deploying the model to the analysis of real data. Hence, Dataset A has been used for calibration of these parameters, which are then are applied to Dataset B in order to validate the model. Table 3 shows the range of parameters and the calibrated values obtained for washing machines. In detail, parameter st\_by\_prc is the reference percentile of energy used to evaluate the usability of standby energy. Parameter min\_dur represents the minimum length of useful operative cycles, and the calibrated value (30 min) has been obtained from all cyclic appliances. Parameter min\_Wh represents the minimum energy of useful operative cycles (i.e., 100 Wh for washing machines). All the other parameters are equal for cyclic appliances.


**Table 3.** Parameter tuning for washing machines.

#### *5.2. Algorithm Results*

The energy analysis starts with the calculation of the standby consumption of the appliance. Figure 5 depicts the cumulative distribution of the energy consumption for a washing machine from which the standby energy is equal to 0.15 Wh. The model uses this value to distinguish among periods of operation and inactivity for the specific appliance.

**Figure 5.** Cumulative distribution of the energy consumption.

The energy consumption profile in a typical cycle for washing machines is shown in Figure 6: an initial peak of consumption corresponding to water heating is followed by a longer phase (depending on the type of cycle set by the user as well as the type, model, and age of the appliance) with lower consumption. The algorithm calculates the gradient of energy demand and finds consumption peaks, which correspond to relative maxima (i.e., zero gradient), for each aggregate cycle (multiple nearby cycles). The consumption of the peaks that separate individual nearby cycles (i.e., the first phase of operation) exceeds a specific percentile of the cycles' consumption, the value of which is selected among the parameters of the model. Once the various cycles have been recognized according to this methodology, the model determines the start and the end time (which corresponds to the lowest consumption before the peak of consumption of the next cycle), and duration of the single cycles. Tests have demonstrated that the algorithm is reliable in distinguishing nearby cycles, as shown in Figure 7. A finer sample period could produce better accuracy, but for the purposes of this investigation, the quarter-hour interval is appropriate.

**Figure 6.** Automatic identification of the phases in the consumption profile of a single washing cycle of a washing machine.

**Figure 7.** Automatic detection of two consecutive washing cycles.

Figure 8 illustrates a bubble chart with relevant information on cycle length and consumption of the same washing machine. The start hour during the day is displayed (abscissa) versus duration (ordinate), and the consumption is represented by the coloured scale of bubbles, whose size is proportional to the ratio between the highest and lowest quarter-hour consumption in the cycle. Larger bubble sizes in washing machines (and dishwashers) indicate washing cycles at a medium-high temperature.

Figure 9 depicts the subdivision of records (upper chart) and consumption (lower chart) based on the type of operation: appliance operative, standby mode, off, spurious data, and sensor not active (i.e., n/a label). Washing cycles correspond to a small fraction of the time but are associated with most of the consumption, unlike standby mode. Moreover, Figure 10 shows the subdivision of the records on a monthly basis, whereas Figure 11 illustrates the monthly (histogram on the left) and overall subdivision of consumption (pie chart on the right) for ARERA time slots. The upper graphs in Figure 12 show the monthly (histogram on the left) and overall (pie chart on the right) allocation of consumption according to the customized daytime slots, in the specific case three time slots (i.e., 8 a.m. to 12 p.m., 12 p.m. to 4 p.m., 4 p.m. to 8 p.m.), whereas the bottom graphs represent the

number of operating hours (histogram on the left) and the overall operating hours (pie chart on the right).

**Figure 8.** Duration, start time, and energy consumption of a washing machine's cycles.

**Figure 9.** Subdivision of records (upper graph) and consumption (lower graph) based on the type of operation for a washing machine.

**Figure 10.** Monthly subdivision of records based on the type of operation for a washing machine.

**Figure 11.** Monthly allocation of consumption in the three time slots defined by ARERA.

**Figure 12.** Monthly allocation of consumption according to the user-defined daytime slots.
