*3.3. Power Satisfaction*

The power satisfaction will not only depend on the user input satisfaction set by the household's head. PS also depends on the energy consumption patterns at each hour of the day. Thus, PS depends on the number of hours of continuous usage (CLoU) and the total length of use in a day (LoU). To find the *PS* at time *t*, we need to analyze the previous 24 h, i.e., from time *t* − 24 until time *t* − 1. It is suggested that *PS* cannot be tested based on how satisfied a person is at *t* but it will be affected by its perception of the last 24 h.

#### *3.4. Equation of Power Satisfaction*

Power satisfaction at a given time/hour can be expressed as the function in Equation (2) below:

$$PS\_{i}[t] = \left(\boldsymbol{u}\_{i}\left[t\right] - \beta\_{i}[t]\right)\hat{\boldsymbol{t}}\_{ui}\ \left[t\right] - \boldsymbol{t}\_{u\_{i}}\left[-\gamma\left[t\right]\right]\hat{\boldsymbol{t}}\_{ti}\left[t\right] - \boldsymbol{t}\_{t\_{i}}\left[\right]\right)\boldsymbol{u}\_{i}[t] \tag{2}$$

where,

$$
\beta\_i[t] = \gamma\_i[t] = \frac{\kappa\_i[t] + 3}{48 - t\_{u\_i} - t\_{t\_i}} \tag{3}
$$

where, *α* is related to the customer's answer, *β* is related to CLoU and *γ* is the related to LoU. The main idea is to penalize the initial satisfaction, *α* according to an excess or poor consumption. Variables *β* and *γ* depends on *α* and convert *t u* and *t t*, respectively, into a value that can be deducted from *α* and,

$$\alpha\_i \begin{bmatrix} \mathbf{t} \end{bmatrix} = \frac{\Omega\_i \begin{bmatrix} \mathbf{t} \end{bmatrix}}{\mathbf{t}\_{t\_i}} \,, \tag{4}$$

where Ω*i* is the time-based input satisfaction for device *i* and it is divided by the responsible LoU, *tti* , hence *α* will reach its maximum if it used the expected time *tti* hours during a 24-h period. To find PS, 24 h of experiment is needed. For each device *i*, the array *ui* = {*un* : *n* = 1, 2, . . . 24} where *un* is 0 or 1. It is the input vector for each device with the operational status of the devices, it will be one (1) when 'ON' and (0) if it is 'OFF'. For further details, see previous work reference [19].
