Continuous-inflow and outflow LDF in DSM

[JakobBD]

Currently, The lifetime distribution function assumes that all input to the DSM happens on 1 January at once.
This is irrelevant for longer $lifetimes >> 1 year$. But for plastics packaging we have much shorter lifetimes.
If instead we take time averages of the lifetime distribution functions, we would get

$\lim_{lifetime\rightarrow 0} stock = annual\_inflow * lifetime\_in\_years$,

which is the actual limit, instead of some other, incorrect value of the LDF.

However, we should make this behaviour optional.

[JakobBD]

Proposed methodology:

• 2 "modes":
  • One with "time instant" like now, which receives a vector of points in time, such that the survival function at input year = output year is always zero.
  • One with "time average", where each time step represents the average stock and flow in that period.

Time instant implementation:

• first get a difference matrix (previously called "remaining ages") $D_{o,i} = \max(0, t_o - t_i)$, where $i$ and $o$ are input year and output year indices and $t$ is the time vector.
• get a matrix "end of life share by year" $E_{o,i} = \rho_i(D_{o,i})$ with the lifetime distribution function $\rho$ (e.g. log-normal; NOT the survival function), the parameters of which may depend on the input year.
• Get the survival function $S_{o,i} = 1 - \sum_{\tilde{o}=1,...,o} (E_{\tilde{o},i})$, i.e. one minus the cumsum over $o$.

Changes for average formulation

• instead of a vector of time instants, this method needs a vector of time interval lengths $l$ and a vector $s$ with the starting points of the time intervals. Since we can't assume that such a vector is passed, I'd add a method to create it from a vector of time instances, where each interval bound is in the midpoint between two time instances. The starting point vector is one element longer than $t$, as $s_{n+1}$ is needed as the end point of the interval $n$.
  I assume that $t$ indices run from 1 to $n$. Then $s_i=(t_i + t_{i-1})/2$ for $i=2,...,n$, and extrapolation yields $s_1=s_2 - (t_2 - t_1)$ as well as $s_{n+1}=s_n + (t_n - t_{n-1})$. . The vector of interval lengths is then $l_i = (s_{i+1}-s_{i})$ for $i=1,...n$.
• $\tilde{D}_{o,i} = \max(0, s_o - s_i)$ is also required in the adapted/extended version.
• instead of getting $E_{o,i} = \rho_i(R_{o,i})$, we now get it via the integral average formulation
  $E_{o,i} = [P_i (\tilde{D} _{o,i})-P_i(\tilde{D} _{o+1,i})]/l_i$ . Here, $P$ is the scipy survival function belonging to each ldf. Its parameters (mean, std) can be dependent on $t_i$.
• The calculation of $S_{o,i}$ remains the same as for the time instant implementation