5.2.3. Results

Figure 3 depicts the amount of energy used by each of the four devices under the proposed SV allocation algorithm and the actual energy usage according to the REDD database [31]. The proposed optimization problem has been constrained. Thus, showing how with the same amount or less energy than in the actual REDD scenario. ES (see Figure 4) is higher at each time slot.

**Figure 3.** Power usage with the proposed Shapley Value (SV) load allocation algorithm vs. real usage according with the REDD [31] database.

**Figure 4.** Energy satisfaction (ES) with the proposed SV load allocation using the same or less energy than real REDD [31] database vs. ES satisfaction using REDD [31] database.




**Table 4.** Device-based satisfaction.



**Table 5.** Load description.

Figure 5 depicts how power is used in both scenarios. Figure 6 shows the resulting ES from that energy usage. Lighting is suggested to be on early in the morning and in the afternoon when the satisfaction derived from them is the highest. Therefore, the refrigerator needs to be off for one hour because it is a priority (see Table 4 to have lighting on since its contribution to the total ES is higher. However, analyzing ES brought by refrigerator in Figure 6 is slightly lower for the proposed algorithm since it was turned off. According to the REDD database [31], the stove is in on status during one hour but it is on at a time that is not bringing any satisfaction, hence the proposed algorithm recommends to not turn it on under the energy restrictions. For the microwave, in the REDD database [31], it is on at different times of the day however it is not on when the satisfaction is the highest. For example, in the morning it is preferable to have lighting on instead of the microwave, since the energy usage of lighting is less than the energy usage of the microwave.

**Figure 5.** Power usage by each device with the proposed SV load allocations vs. real usage according with the REDD [31] database.


**Table 6.** Time-based satisfaction.


**Table 7.** Device-based satisfaction.


For the calibration part, we made sure the algorithm was suggesting those profiles where the Energy Satisfaction was highest at a minimum energy usage. In the following section, we will be testing the algorithm for a different set of devices.

#### *5.3. Data Characterization for the Algorithm Testing.*

Eight devices were selected according to the ones used in Ogunjuyigbe et al. [8] for testing purposes. For the algorithm simplicity, all lights were considered as a single appliance. Ogunjuyigbe et al. [8] used some loads for which there are not data available in the REDD database [31], such as TV, AC, Radio, and phone. Hence, for these ones, Unit Wattage data from [8] was used. Additionally, *ui* was randomly generated for those devices, such that a comparison can be made with the proposed algorithm's *ui* output. Table 5 describes the electrical appliances used by a user, their rating, their optimum CLoU, *tu* and their optimum LoU, *tt* for a responsible consumption.

#### 5.3.1. House Head's Satisfaction

The algorithm also needs practical input data for satisfaction to create the model. Data from Ogunjuyigbe et al. [8] (*σ<sup>t</sup>* and *σ<sup>d</sup>*) are being used for this purpose. Data from

time-based satisfaction was mapped into satisfaction levels described in Table 1 in the following fashion,

$$\Omega\_t = \begin{cases} 0, & \text{if } \sigma^t = 0.5\\ \begin{array}{ll} 6 - \frac{2\left(1 - \sigma^t[t]\right)}{0.5}, & \text{if } \sigma^t \ge 0.5\\ 6\sigma^t[t] - 3, & \text{if } \sigma^t < 0.5 \end{array} \end{cases} \tag{15}$$

Dissatisfaction and 'indifference' values (0, 1, 2 and 3) are mapped into negative values and zero (−3, −2, −1 and 0, respectively). The proposed model introduced negative values when low satisfaction, thus making it preferable to have them 'OFF', representing by itself, before the optimization problem, a more accurate satisfaction model, which allows making decisions not only for energy and economic savings but also responsibly fulfilling the customer's satisfaction. Table 6 shows complete resulting time-based satisfaction table and Table 7 shows device-based satisfaction after mapping d-domain satisfaction found in Ogunjuyigbe et al. [8] (*σ<sup>d</sup>*) by using Equation (16).

$$
\Delta \left[ t \right] = 10 \sigma^d \left[ t \right]\_\prime \tag{16}
$$

5.3.2. Results

One of the main results to report is that the implementation of the SV optimization provided a consumption pattern that represents an energy consumption less or equal than the initial actual use and a higher energy satisfaction for almost all hours in all devices. Figure 7 provides the graphical comparison of the power consumption at each hour for the actual based case and the case with the SV optimization. Figure 8 presents the energy satisfaction at each hour showing how the SV optimization outperforms the base case, particularly increasing its advantage in the early morning hours and the late evening hours.

**Figure 7.** Power usage by each device with the proposed SV load allocations vs. real usage according to the REDD [31] database.

**Figure 8.** ES Satisfaction with the proposed SV load allocation using the same or less energy than real REDD [31] database vs. real usage according to the REDD [31] database.

Figure 9 depicts a comparison of the hourly power consumption of all devices between the proposed SV allocation algorithm and the actual power consumption. In Figure 9 (left side), the algorithm attempts to meet desired time-based and device-based satisfaction tables (See Tables 6 and 7) while consuming equal or less hourly power than the one shown in 9 (right side). The energy reduction was of approximately 75%, from 32.6 KWh to 7.35 KWh. Equally important is the energy satisfaction increase of 40% with the SV algorithm, from 5500 to 7825. A consumption plan is scheduled for the user by managing devices based on the SV game theory approach. Next, a reliability signal or economic signal will be sent to a Human-Machine Interface (HMI). A reliability signal will ensure that the electric system keeps operating when a house is not connected to the grid, while an economic signal ensures the same purpose when connected to the grid. This way, the customer is aware of the situation and can make a final decision based on the available information.

**Figure 9.** Power usage by each device with the proposed SV load allocations (**left** side) vs. real usage according to the REDD (**right** side) [31] database.

Figure 10 includes the ES for each device and it seems that the better SV performance is due mainly to the microwave use in the morning and the TV at night. Since operational status vectors, *u* for TV, AC, Radio, and phone, were randomly generated, Figure 9 shows an atypical consumption pattern. Figure 10 shows a comparison between the hourly ES obtained through the SV allocation algorithm and the ES obtained in the actual case representation, for each of the devices. When attempting to meet the energy constraints imposed by the actual case scenario, with the proposed SV algorithm, the ES obtained at each hour is equal or higher for almost each of the hours for every device. This is one of the most important results of this study. Only in the case of the laptop, the ES is higher at the last hour of the day in the actual case scenario.

Similarly, Ogunjuyigbe et al. [8] presented their results in 24 h plots for three different daily budget constraints to provide a maximum satisfaction at those predefined budgets. On the other hand, the present research used energy constraints rather than budget, and thus including a key component of the research when penalizing excessive and low consumption, because of its detrimental impact in the quality of life. He compared a 'desired satisfaction' with an 'achieved satisfaction'. The 'achieved satisfaction' was the output of their load-satisfaction algorithm, which is analogous to the present 'SV load allocation' algorithm. We did not choose to compare the output results with the 'desired satisfaction' (as Seen in Tables 6 and 7). Instead, we compare it with an 'actual' scenario represented by using the REDD database. Ogunjuyigbe et al. [8] implemented a genetic algorithm (GA) approach. While the GA approach may have a good convergence speed and good efficiency, the present approach does not have to deal with convergence times and offers a more intuitive optimization framework.

**Figure 10.** ES derived from each device with the proposed SV load allocation algorithm vs. real usage according to the REDD [31] database.
