Bus
A Bus is a simple nodal balance between its incoming and outgoing flow rates.
\[ \label{eq:bus_balance}
\sum_{f_\text{in} \in \mathcal{F}_\text{in}} p_{f_\text{in}}(\text{t}_i) =
\sum_{f_\text{out} \in \mathcal{F}_\text{out}} p_{f_\text{out}}(\text{t}_i)
\]
Optionally, a Bus can have a excess_penalty_per_flow_hour parameter, which allows to penaltize the balance for missing or excess flow-rates.
This is usefull as it handles a possible ifeasiblity gently.
This changes the balance to
\[ \label{eq:bus_balance-excess}
\sum_{f_\text{in} \in \mathcal{F}_\text{in}} p_{f_ \text{in}}(\text{t}_i) + \phi_\text{in}(\text{t}_i) =
\sum_{f_\text{out} \in \mathcal{F}_\text{out}} p_{f_\text{out}}(\text{t}_i) + \phi_\text{out}(\text{t}_i)
\]
The penalty term is defined as
\[ \label{eq:bus_penalty}
s_{b \rightarrow \Phi}(\text{t}_i) =
\text a_{b \rightarrow \Phi}(\text{t}_i) \cdot \Delta \text{t}_i
\cdot [ \phi_\text{in}(\text{t}_i) + \phi_\text{out}(\text{t}_i) ]
\]
With:
- \(\mathcal{F}_\text{in}\) and \(\mathcal{F}_\text{out}\) being the set of all incoming and outgoing flows
- \(p_{f_\text{in}}(\text{t}_i)\) and \(p_{f_\text{out}}(\text{t}_i)\) being the flow-rate at time \(\text{t}_i\) for flow \(f_\text{in}\) and \(f_\text{out}\), respectively
- \(\phi_\text{in}(\text{t}_i)\) and \(\phi_\text{out}(\text{t}_i)\) being the missing or excess flow-rate at time \(\text{t}_i\), respectively
- \(\text{t}_i\) being the time step
- \(s_{b \rightarrow \Phi}(\text{t}_i)\) being the penalty term
- \(\text a_{b \rightarrow \Phi}(\text{t}_i)\) being the penalty coefficient (
excess_penalty_per_flow_hour)