Sizing (Investment Optimization)

Overview

Sizing introduces a capacity decision variable \(S\) that replaces the fixed nominal capacity \(\bar{P}_f\). The solver optimizes both the capacity and the dispatch simultaneously.

Variables

Symbol	Code	Domain	Description
\(S_f\)	flow_size[flow]	\(\geq 0\)	Invested flow capacity
\(y_f\)	flow_size_indicator[flow]	\(\{0, 1\}\)	Binary: invest yes/no (optional only)
\(S_s\)	storage_capacity[storage]	\(\geq 0\)	Invested storage capacity
\(y_s\)	storage_size_indicator[storage]	\(\{0, 1\}\)	Binary: invest yes/no (optional only)

Mandatory Sizing

When mandatory=True, the component must be built. The capacity is continuous:

\[ S^- \leq S_f \leq S^+ \quad \text{(mandatory)} \]

where \(S^-\) = min_size and \(S^+\) = max_size.

Optional Sizing

When mandatory=False, a binary indicator \(y_f\) gates the capacity:

\[ S^- \cdot y_f \leq S_f \leq S^+ \cdot y_f \]

When \(y_f = 0\): \(S_f = 0\) (not built). When \(y_f = 1\): \(S_f \in [S^-, S^+]\).

Binary Invest

When \(S^- = S^+\), the sizing reduces to a binary yes/no decision at exactly that capacity.

Flow Rate Bounds with Sizing

With sizing, the fixed capacity \(\bar{P}_f\) is replaced by the variable \(S_f\). The relative bounds scale by the invested size:

\[ S_f \cdot \underline{p}_{f,t} \leq P_{f,t} \leq S_f \cdot \bar{p}_{f,t} \quad \forall \, t \]

Similarly for fixed profiles:

\[ P_{f,t} = S_f \cdot \pi_{f,t} \quad \forall \, t \]

Storage Sizing

The same pattern applies to storage capacity. The charge state bounds become:

\[ S_s \cdot \underline{e}_s \leq E_{s,t} \leq S_s \cdot \bar{e}_s \quad \forall \, t \]

Investment Effects

Investment costs contribute to effect totals. For each effect \(k\):

Per-Size

\[ \Phi_k^{\text{invest,per\_size}} = \sum_{f} \gamma_{f,k} \cdot S_f + \sum_{s} \gamma_{s,k} \cdot S_s \]

where \(\gamma_{f,k}\) is Sizing.effects_per_size[k].

Fixed

\[ \Phi_k^{\text{invest,fixed}} = \sum_{f} \phi_{f,k} \cdot y_f + \sum_{s} \phi_{s,k} \cdot y_s \]

where \(\phi_{f,k}\) is Sizing.effects_fixed[k]. Only applies when mandatory=False (binary indicator exists).

Total

The direct investment contribution to effect \(k\) is:

\[ \Phi_k^{\text{invest}} = \Phi_k^{\text{invest,per\_size}} + \Phi_k^{\text{invest,fixed}} \]

This feeds into the effect total equation and can be further weighted by cross-effect contributions.

Parameters

Symbol	Description	Reference
\(S_f\)	Flow capacity variable	flow_size[flow]
\(S_s\)	Storage capacity variable	storage_capacity[storage]
\(y_f\), \(y_s\)	Binary invest indicator	flow_size_indicator, storage_size_indicator
\(S^-\)	Minimum size	Sizing.min_size
\(S^+\)	Maximum size	Sizing.max_size
\(\gamma_{f,k}\)	Per-size investment cost	Sizing.effects_per_size
\(\phi_{f,k}\)	Fixed investment cost	Sizing.effects_fixed

See Notation for the full symbol table.