Mathematical Notation

Naming Conventions

flixOpt uses the following naming conventions:

• All optimization variables are denoted by italic letters (e.g., \(x\), \(y\), \(z\))
• All parameters and constants are denoted by non italic small letters (e.g., \(\text{a}\), \(\text{b}\), \(\text{c}\))
• All Sets are denoted by greek capital letters (e.g., \(\mathcal{F}\), \(\mathcal{E}\))
• All units of a set are denoted by greek small letters (e.g., \(\mathcal{f}\), \(\mathcal{e}\))
• The letter \(i\) is used to denote an index (e.g., \(i=1,\dots,\text n\))
• All time steps are denoted by the letter \(\text{t}\) (e.g., \(\text{t}_0\), \(\text{t}_1\), \(\text{t}_i\))

Buses

The balance equation for a bus is:

\[ \label{eq:bus_balance} \sum_{f_\text{in} \in \mathcal{F}_\text{in}} p_{f_\text{in}}(\text{t}_i) = \sum_{f_\text{out} \in \mathcal{F}_\text{out}} p_{f_\text{out}}(\text{t}_i) \]

Optionally, a Bus can have a excess_penalty_per_flow_hour parameter, which allows to penalize 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)

Flows

The flow-rate is the main optimization variable of the Flow. It's limited by the size of the Flow and relative bounds \eqref{eq:flow_rate}.

\[ \label{eq:flow_rate} \text P \cdot \text p^{\text{L}}_{\text{rel}}(\text{t}_{i}) \leq p(\text{t}_{i}) \leq \text P \cdot \text p^{\text{U}}_{\text{rel}}(\text{t}_{i}) \]

With:

• \(\text P\) being the size of the Flow
• \(p(\text{t}_{i})\) being the flow-rate at time \(\text{t}_{i}\)
• \(\text p^{\text{L}}_{\text{rel}}(\text{t}_{i})\) being the relative lower bound (typically 0)
• \(\text p^{\text{U}}_{\text{rel}}(\text{t}_{i})\) being the relative upper bound (typically 1)

With \(\text p^{\text{L}}_{\text{rel}}(\text{t}_{i}) = 0\) and \(\text p^{\text{U}}_{\text{rel}}(\text{t}_{i}) = 1\), equation \eqref{eq:flow_rate} simplifies to

\[ 0 \leq p(\text{t}_{i}) \leq \text P \]

This mathematical Formulation can be extended or changed when using OnOffParameters to define the On/Off state of the Flow, or InvestParameters, which changes the size of the Flow from a constant to an optimization variable.

LinearConverters

LinearConverters define a ratio between incoming and outgoing Flows.

\[ \label{eq:Linear-Transformer-Ratio} \sum_{f_{\text{in}} \in \mathcal F_{in}} \text a_{f_{\text{in}}}(\text{t}_i) \cdot p_{f_\text{in}}(\text{t}_i) = \sum_{f_{\text{out}} \in \mathcal F_{out}} \text b_{f_\text{out}}(\text{t}_i) \cdot p_{f_\text{out}}(\text{t}_i) \]

With:

• \(\mathcal F_{in}\) and \(\mathcal F_{out}\) 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
• \(\text a_{f_\text{in}}(\text{t}_i)\) and \(\text b_{f_\text{out}}(\text{t}_i)\) being the ratio of the flow-rate at time \(\text{t}_i\) for flow \(f_\text{in}\) and \(f_\text{out}\), respectively

With one incoming Flow and one outgoing Flow, this can be simplified to:

\[ \label{eq:Linear-Transformer-Ratio-simple} \text a(\text{t}_i) \cdot p_{f_\text{in}}(\text{t}_i) = p_{f_\text{out}}(\text{t}_i) \]

where \(\text a\) can be interpreted as the conversion efficiency of the LinearTransformer.

Piecewise Concersion factors

The conversion efficiency can be defined as a piecewise function.

Effects

Features

InvestParameters

OnOffParameters

Calculation Modes