## 8. Transmission interfaces

In some applications, such as energy exchange studies, it is important to enforce flow limits not only on individual lines, but also on groups of transmission lines. These groups are known as interfaces. More precisely, an interface is composed by two sets of lines: the inbound and the outbound lines. The flow across the interface is defined as the sum of the flow in all inbound lines minus the sum of the flow in all outbound lines. An upper and a lower limit may be imposed on the flow across the interface, and a penalty is imposed if the limit is exceeded.

### Sets and constants

Symbol | Unit | Description
--- | --- | ---
$L^\text{inbound}_{si}$ |  | Set of inbound lines for interface $i$ in scenario $s$.
$L^\text{outbound}_{si}$ |  | Set of outbound lines for interface $i$ in scenario $s$.
$M^\text{limit-down}_{sit}$ | MW | Lower flow limit for interface $i$ at time at time $t$ and scenario $s$ (negative number).
$M^\text{limit-up}_{sit}$ | MW | Upper flow limit for interface $i$ at time at time $t$ and scenario $s$ (positive number).
$Z^\text{overflow}_{sit}$ | $/MW | Overflow penalty for interface $l$ at time $t$ and scenario $s$.
$\text{IF}$ |  | Set of transmission interfaces.

### Decision variables

Symbol | JuMP name | Unit | Description | Stage
--- | --- | --- | --- | ---
$y^\text{i-flow}_{sit}$ | `interface_flow[s,i,t]` | MW | Flow across interface $i$ at time $t$ and scenario $s$. | 2
$y^\text{i-overflow}_{sit}$ | `interface_overflow[s,i,t]` | MW | Flow above limit for interface $i$ at time $t$ and scenario $s$. | 2

### Objective function terms

- Penalty for exceeding interface limits:

\[ \sum_{s \in S} p(s) \left[ \sum_{i \in \text{IF}} \sum_{t \in T} y^\text{i-overflow}_{sit} Z^\text{overflow}_{sit} \right]\]

### Constraints

- Definition of interface flow (`eq_if_flow`):

\[y^\text{i-flow}_{sit} = \sum_{b \in B} y^\text{inj}_{sbt} \left[ \sum_{l \in L^\text{outbound}_{si}} \delta_{sbl} - \sum_{l \in L^\text{inbound}_{si}} \delta_{sbl} \right]\]

- Interface flow limits (`eq_if_limit_up` and `eq_if_limit_up`)

\[\begin{align*} y^\text{i-flow}_{sit} & \leq M^\text{limit-up}_{sit} + y^\text{i-overflow}_{sit} \\ -y^\text{i-flow}_{sit} & \leq -M^\text{limit-down}_{sit} + y^\text{i-overflow}_{sit} \end{align*}\]

## 9. Contingencies

## 10. Reserves

Theme

This document was generated with Documenter.jl version 1.15.0 on Wednesday 5 November 2025. Using Julia version 1.12.1.