2.1.1. Weighted Proportion (Wepro) Model

The Wepro model is a simplified approach to model residential electricity load profiles in cities by adjusting and matching the proportion of city's weighted profiles with the households' profiles through the existing household profile generators. First, it is necessary to collect information on the citizens' age groups (AG), gender (GD) and labour force (LF). In this case, a figure for annual electricity consumption is not required, since we only focus on providing the share of hourly electricity load profiles. Second, we coupled the share of age groups and labour force and applied this share to proportionally fit the total population. The population is categorised into three groups by age: 0–17 years old, 18–64 years old and over 64 years old. Thus, the sum of the composition of these age groups represents the city's population by age group is expressed in Equation (1):

$$\text{Tag} = A\text{G}\_1\text{°}\_0 + A\text{G}\_2\text{°}\_0 + A\text{G}\_3\text{°}\_0 \tag{1}$$

where Tag is the total share of the age groups' share in the city. *AG*<sup>1</sup> is the age group for people aged 0 to 17, and *AG*<sup>2</sup> for people aged 18 to 64 and *AG*<sup>3</sup> for people over the age of 64. In more detail, each age group has gender information, although we can also identify gender information at the higher level of the age groups, giving totals for each gender in the city. In this model, more details on the gender composition of each age group is required as expressed in Equation (2):

$$Tmf = Ml\% + Fm\%\tag{2}$$

where *Tmf* is the total share of male's share and female's share in the city. *Ml* is Male and *Fm* is Female. We also need to identify the city's labour force composition. The shares of employment and unemployment represent the city's labour force is formed in Equation (3):

$$T I f = E m^{\diamondsuit} + \mathcal{L} I n^{\diamondsuit} \tag{3}$$

where *Tlf* is the total share of employment's share and unemployment's share in the city. *Em* is Employment and *Un* = Unemployment. The labour force data are measured on the basis of the labour force population, which is only derived from one of the age groups. In this case, the labour force is included in AG2 = 18–64 years old. Here the labour force is the proper set of age groups, labour force being an aspect of the age groups but not equal to age groups as shown in Equation (4):

$$AG = \langle \text{AG}\_{1,} \text{ AG}\_{2,} \text{ AG}\_{3} \rangle \text{ and } LF = \langle \text{AG}\_{2} \rangle$$

$$LF \subset AG \tag{4}$$

As mentioned, we employ the household profile generators in this case LPG and ALPG to generate the household load profiles. The first step is to select the household profiles to be modelled by the profile generators. The fundamental consideration is that the selected household profiles in the profile generators should represent the city's characteristics in term of age groups, gender structure and labour force, this being the focus of our study. This means that the selected household profiles should represent the city's profiles proportionally as depicted in Figure 1.

**Figure 1.** Weighted proportion structure of the city's main parameters: age group, labour force composition and gender structure.

• Capacity, fairness of allocation and rounding number

We apply the capacity model based on the amsterdam's age groups share in Figure 1 for selecting which household profiles to be modelled. The main goal is to determine the number of the occupants's profiles to be modelled as shown in the following expression of Equation (5):

$$Tamt = AG1wt + AG2wt + AG3wt \tag{5}$$

where *Tamt* is the total number of the occupants' profiles. *AG*1*wt* is the number in age group 1 based on it's weight. *AG*2*wt* is the number in age group 2 based on it's weight and *AG*3*wt* is the number in age group 3 based on it's weight. The share of the occupants for each group are converted to decimal form to provide the results of the total number of occupant-profiles from each age group.

Furthermore, the capacity model can also be extended to determine the gender of the selected profiles as expressed in Equation (6) if it is supported by the profile generators. In this case, it is applicable to LPG, since LPG provides detail characters of the occupants' gender information:

$$\text{Tg} = \begin{aligned} \text{(AG1m} \ast \text{AG1wt)} + \text{(AG1f} \ast \text{AG1wt)} + \text{(AG2m} \ast \text{AG2wt)} + \text{(AG2f} \ast \text{AG2wt)} + \text{(AG3m} \ast \text{AG3wt)}\\ \text{AG3wt)} + \text{(AG3f} \ast \text{AG3wt)} \end{aligned} \tag{6}$$

Here *Tg* is the total number of combinations of the occupants' gender. AG1m is the share of males in age group 1. AG1f is the share of females in age group 1. AG2m is the share of males in age group 2. AG2f is the share of females in age group 2. AG3m is the share of males in age group 3 and AG3f is the share of females in age group 3. In this case a widely used fairness sharing technique called max-min fairness can be applied in sharing the allocations if it is required.

Therefore, the application of the Wepro model to the case-study city namely amsterdam is as follows: First, the city's population is represented by the sum of the composition of age groups in amsterdam. We grouped the city's age groups into three categories: 0–17 years old = 17.5%; 18–64 years old = 70.3%; and above 64 years old = 12.2% [57,58] using the formula in Equation (1):

$$\text{Tag} = 17.5\% + 70.3\% + 12.2\%$$

*Tag* = 100%

In more detail, the gender structure is classified into three age groups. For the age group of 0 to 17-year-olds, 51.58% are male and 48.42% female. In the age group of 18- to 64-year-olds, 50.24% are male and 49.75% female. Finally, for the age group above 65, we identified 46.24% male and 53.75% female [57,58]. Therefore, Equation (2) is presented to identify the gender at the city level:

$$Tmf = 49.5\% + 50.5\%$$

$$Tmf = 100\%$$

Furthermore, the labour force data are measured on the basis of the labour force population, which is only derived from age group among 18- to 64-year-olds. The unemployment rate is recorded as 6.7% [56]. In this case, Equation (3) is used to identify the employment and unemployment shares.

$$Tlf = 93.3\% + 6.7\%$$

$$Tlf = 100\%$$

Here, Equation (4) is used where the labour force is the proper set of age groups, labour force being an aspect of the age groups but not equal to age groups:

*AG* = *{*0–17 years old, 18–64 years old, 64+*} and LF* = *{*15–64 years old*}*

$$170.3\% \text{ aged } 18-64 \subset 100\% \text{ aged } 0-17, \text{ 18-64, over } 64$$

We coupled the share of age groups and labour force and applied the Proportional matched profile to the total population as the city's main characteristics. Therefore, as displayed in Figure 2, the Amsterdam's main profile should reflect: The age groups, labour force and gender classes.

This means that from the age group percentage: The aged 0–17 group is nearly 20%, aged 18–64 is 70% and the rest 10% is for aged over 64. From this 70% where the aged 18–64, there is about 93% of this age group are people with work and the rest are not working. Furthermore, each age group illustrates a slight difference in the share of gender information, except for the aged over 64, where the female populations are slightly more dominant than the male populations.

**Figure 2.** The application of the Weighted proportion (Wepro) model's structure to the case-study city, namely amsterdam, The Netherlands. It consists of the amsterdam's age group share, labour force composition share and gender share of each age group.

• Capacity, fairness of allocation and rounding number

Furthermore, Equation (5) is presented, where the weighted city's age group values are applied into a simple capacity model, in order to determine the capacity of the allocation. Therefore, based on the weighted values, we have ten capacity of the households profiles. It means, we can only select maximum ten occupants from the household profiles generators:

$$Tamt = 17.5\% + 70.3\% + 12.2\%$$

$$Tamt = 1.75 + 7.03 + 1.2$$

$$Tamt = 2 + 7 + 1$$

$$Tamt = 10$$

Furthermore, if it is supported by the profile generators, the capacity model can also be extended to determine the gender of the selected profiles as expressed in Equation (6). In this case, it is applicable to LPG, since LPG provides detail characters of the occupants gender information:

$$T\mathcal{g} = (1.03 + 0.96) + (3.51 + 3.48) + (0.46 + 0.54)$$

$$T\mathcal{g} = (1\mathbf{m} + 1\mathbf{f}) + (4\mathbf{m} + 3\mathbf{f}) + 1\mathbf{f}$$

As shown in the Equation (5), age group 1 has two allocations, age group 2 has seven allocations and age group 3 has one allocation. Thus, there are currently two resources for two allocations, which after the division between them, resulting in 1. Furthermore, AG1m has an excess of 0.03, where the excess can be taken and divided among the remaining demands, which is only AG1f. Therefore, AG1f = 1. As a result of the capacity and fairness of allocation model depicted in Figure 3, there will be two occupants: one male and one female in age group 0–17. Furthermore for the case Age group 2 and Age group 3, we cannot fully apply the max-min fairness. We simply apply rounding number because we have only two resources per age group. For instance, for age group 3, there are two resources for only one allocation. Therefore, rounding number is applied to the highest weight between the resources.

**Figure 3.** The application of the Wepro model's structure to the case-study city, namely amsterdam, The Netherlands. It consists of the amsterdam age group share, labour force composition share and gender share of each age group and their capacity of the occupants to be modelled.

As a result, the age group 18–64 should consist of seven adult occupants with six of them working people and one person not working. Considering the gender share is quite balance in this age group, then it is either four females and three males, or four males and three females in the occupants' list. Lastly, for the aged over 64, which has only one allocation, based on the results of the capacity and fairness of allocation model, we apply rounding value to the one which has the highest share to represent the senior age group. Therefore, we selected a female senior to represent this age group.
