Allow increase in space heating adoption

[robinhasse]

Problem

We currently assume a time-invariant factor between the product of HDD, U-values and floor space on the one side and the UE demand for space heating of the other hand. This implies that the extend to which theoretical useful energy demand is served does not change over time.
While this is a reasonable assumption in medium to high income countries, we see esp. in AFR, IND and BRA historic increases in this factor.

As we keep it constant in projection, we underestimate future growth and get trend breaks that don't look good.

Solution

I'd suggest the following solution:

• linear regression of the historic $\delta$ over time yields $\dot{\delta}_0$
• continue all region with negative slopes $\dot{\delta}_0$ constantly
• project $\delta$ in regions with a positiv historic slope with a asymptotic function that starts with the historic slope $\dot{\delta}_0$ from the linear regression and approaches a constant until a given mid-term period (e.g. 2040)
  • Let's say, we want the function to start with a value of $\delta_0$ and a slope of $\dot{\delta}_0$ in the last historic period $t_0$ and approach a fraction $\epsilon$ of this slope until $t_1$ with $\Delta t = t_1 - t_0$:

$\delta (t) = \delta_0 + \dfrac{\dot{\delta}_0}{c} \cdot \left[ 1 - \exp \left( - c \cdot (t - t_0) \right) \right]$
with the rate $c = -\dfrac{\ln \epsilon}{\Delta t}$