Trajectory investment problem

General description

Recall the annual investment problem in Xpansion :

$$ \min_{x \in \mathcal{X}} \quad C^T x + \text{ANTARES}(x) $$ over a set of investment variables specified by the user, where :

• \(x\) is the vector of the capacities installed for each candidate
• \(C\) contains the fixed cost annuities of those candidates
• \(\text{ANTARES}(x)\) is the operating cost of the system for a given investment level.

Switching to a pluriannual vision

We want to switch to a pluriannual vision and optimise the investments over several possible trajectories described on a diverging tree of scenarios

Figure 1 - Trajectory tree made up of annual Xpansion studies

The optimisation problem we now want to solve is, denoting by \(n \in \mathcal{T}\) a node in the tree:

\[ \begin{aligned} \min_{x, dx^+, dx^-} \quad & \sum_{n \in \mathcal{T}} wIC_n^T dx_n^+ + wDC_n^T dx_n^- + wOC_n^Tx_{n} + w_{n} \times ANTARES_n(x_n)\\\\ \text{s.t.} \quad & \forall i,n \quad x_{i,n} = x_{i, \text{parent}(n)} + dx^+_{i,n} - dx^-_{i,n} \\\\ & \forall i,n \quad dx^+_{i,n} \geq 0, \quad dx^-_{i,n} \geq 0 \\\\ & \forall i,n \quad 0 \leq x_{i,n} \leq X_{i,n}^{max} \\ \end{aligned} \]

• \(wIC_n\) contains the already weighted (for probability of the node) and discounted one-time payment investment costs per MW.
• \(wDC_n\) contains the already weighted (for probability of the node) and discounted one-time payment retirement costs per MW.
• \(wOC_n\) contains the already weighted (for represented duration of the node & probability) and discounted operation and maintenance fixed costs.
• \(w(n) = P(n) \times \sum_{y = y_n}^{y = y_n + d_n - 1} \frac{1}{(1+r)^{y - y_0}}\).
• \(P(n)\) is the probability of realisation of node \(n\) : \(P(n) = P_{\text{parent}(n)}(n) \times P(\text{parent}(n))\).
• \(P(root) = 1\).

On the \(dx^{+/-}\) variables

Note : In our model, \(x_{i,n}\) is the capacity available during the period represented by \(n\), and this means the decisions \(dx_{i,n}^{+/-}\) represent the variation of capacity during the period between \(\text{parent}(n)\) and \(n\) (i.e. the capacity being built or decommisionned during the period represented by \(\text{parent}(n)\), with effective entry into service at the beginning of \(n\)).

We can impose the decisions to be the same in all children of a given node (see trajectory constraints) if we want the investment decision of a given period to be independent of what scenario will materialize in the next period when the new capacities enter into service.

• In the example from Figure 1, this would mean that the capacity we install in the period [2040, 2050] is independent of wether 2050_A or 2050_B will be realised.