*3.1. Stock and Di*ff*usion Rate*

Since the overall consumption of electricity is impacted by the total quantity of equipment, it is crucial to calculate the adoption rates for the population as well as the total sales numbers of end-use equipment. The sales are the sum of initial purchases of equipment and the replacement purchases, which includes replacements-on-burnout and early retirements. The calculations for replacements

involve the age of the equipment within the stock and a retirement function, which represents the percentage of failed equipment in a vintage stock:

$$\text{Stock}(y) = \text{Sales}(y) + \text{Stock}(y-1), \tag{3}$$

$$\text{Sales}(y) = \text{First\\_purclasses}(y) + \text{Replacements}(y),\tag{4}$$

First purchases, shown in Equation (4), represent an increase in the stock quantity that can be due to new construction projects such as housing subsidies by the Public Authority of Housing Welfare (PAHW) or an increased rate of equipment diffusion per household, as shown in Equation (5):

$$\text{First purchases}(y) = H(y)D(y),\tag{5}$$

where *H*(*y*) represents the number of new households based on [22]. *D*(*y*) is the equipment diffusion rate per household. Equipment diffusion rates are not available as input data, but are projected according to a macroeconomic model using a logistic function [25,26]:

$$D(y) = \frac{\alpha}{1 + \gamma + e^{-(\beta\_1 I(y) + \beta\_2 E(y) + \beta\_3 II(y))}}\tag{6}$$

where *I*(*y*) denotes the average annual income per household (*y*), whereas *E*(*y*) is the electrification rate, *U*(*y*) is the urbanization rate, and γ and β are the parameters for scale. For the case of Kuwait, since the income, electrification, and urbanization rates are relatively high, diffusion rates for equipment are reflective of this phenomenon in the analysis. The logistic function, by definition, has a maximum value of one at which the saturation level is reached. However, some households have more than one appliance or equipment of the same type. Therefore, the logistic function is scaled by the parameter α, as seen in Equation (6), which is the saturation level [25]. As the climate conditions directly impact the air conditioner ownership rates, cooling degree days (CDD) were used instead of an urbanization rate in the equation above to calculate the diffusion rates of AC units. For some appliances, the sale price affects the diffusion rate as purchases depend on affordability. Therefore, a price variable was added for some appliances based on [27]. Replacement stock are attained from previous sales as in Equation (7):

$$\text{Replacements}(y) = \sum\_{i=1}^{L} \text{Sales}(y-i) \times \text{Retricements}(i) \tag{7}$$

In Equation (7), *Retirements* (*i*) represents the probability of the equipment retiring at a given lifetime for each year up to its entire lifetime (*L*), and is modeled using a Weibull distribution [16,28]:

$$Rettiments(i) = 1 - e^{-(i/\lambda)^k} \tag{8}$$

where *i* is the number of years after the equipment is purchased; λ is a scale parameter; and *k* is a shape parameter, which determines the way the failure rate changes through time. These parameters were estimated for each equipment based on [29].

#### *3.2. Unit Energy Consumption*

The next section describes the methods and assumptions for determining the average unit energy consumption (*UEC*) for each piece of equipment. *UEC* depends on the typical product used (size and rated power), the use patterns, and equipment efficiency. Therefore, the *UEC* model includes information on equipment usage and lifetime profiles as well as stock energy efficiencies by vintage and efficiency improvement profiles [5,30]. The assumption of the efficiency improvement of the appliances over time was made based on [31,32], and the likely improvement was 1–5%, depending on the equipment, considering the technical limitation of the technology.
