Multinode optimization model

The introduction of multiple spatial nodes into the model is the second big extension of the minimal model that is possible. Similar to the intertemporal model expansion it also adds an index level to all variables and parameters. This addition is perpendicular to the intertemporal modeling and both extensions do not interact in any complex way with each other. Here, the multinode model extension will be shown without the intertemporal extension, i.e., it is shown as an extension to the minimal model. In this case the variable vector of the optimization problem reads:

\[x^{\text{T}}=(\zeta, \underbrace{\rho_{vct}}_{\text{commodity variables}}, \underbrace{\kappa_{vp}, \widehat{\kappa}_{vp}, \tau_{vpt}, \epsilon^{\text{in}}_{vcpt}, \epsilon^{\text{out}}_{vcpt}}_{\text{process variables}}, \underbrace{\kappa_{af}, \widehat{\kappa}_{af}, \pi^{\text{in}}_{aft}, \pi^{\text{out}}_{aft}}_{\text{transmission variables}}).\]

Here, \(\zeta\) represents the total annualized system cost across all modeled vertices \(v\in V\), \(\rho_vct\) the amount of commodities \(c\) taken from a virtual, infinite stock at vertex \(v\) and time \(t\), \(\kappa_{vp}\) and \(\widehat{\kappa}_{vp}\) the total and the newly installed process capacities of processes \(p\) at vertex \(v\), \(\tau_{vpt}\) the operational state of processes \(p\) at vertex \(v\) and time \(t\), \(\epsilon^{\text{in}}_{vcpt}\) and \(\epsilon^{\text{out}}_{vcpt}\) the total inputs and outputs of commodities \(c\) to and from process \(p\) at vertex \(v\) and time \(t\), \(\kappa_{af}\) and \(\widehat{\kappa}_{af}\) the installed and new capacities of a transmission line \(f\) linking two vertices with the arc \(a\) and \(\pi^{\text{in}}_{aft}\) and \(\pi^{\text{out}}_{aft}\) the in- and outflows into arc \(a\) via transmission line \(f\) at time \(t\).

There are no qualitative changes to the cost function only the sum of all units now extends over processes and transmission lines.

Transmission capacity constraints

Transmission lines are modeled as unidirectional arcs in urbs. This means that they have a input site and an output site. Furthermore, an arc already specifies a commodity that can travel across it. However, from the modelers point of view the transmissions rather behave like non-directional edges, linking both sites with the identical capacity in both directions. This behavior is then ensured by the transmission symmetry rule, which sets the capacity of both unidirectional arcs to be identical:

\[\begin{split}&\forall a\in V\times V\times C,~f\in F:\\ &\kappa_{af}=\kappa_{a^{\prime}f},\end{split}\]

where \(a^{\prime}\) is the inverse arc of \(a\). The transmission capacity is then calculated similar to process capacities in the minimal model:

\[\begin{split}&\forall a\in V\times V\times C,~f\in F:\\ &\kappa_{af}=K_{af}+\widehat{\kappa}_{af},\end{split}\]

where \(K_{af}\) represents the already installed and \(\widehat{\kappa}_{af}\) the new capacity of transmission \(f\) installed in arc \(a\).

Transmission capacity limit rule

Completely analogous to processes also transmission line capacities are limited by a maximal and minimal allowed capacity \(\overline{K}_{af}\) and \(\underline{K}_{af}\) via:

\[\begin{split}&\forall a\in V\times V\times C,~f\in F:\\ &\underline{K}_{af}\leq \kappa_{af}\leq \overline{K}_{af}\end{split}\]

Commodity dispatch constraints

Apart from these time independent rules, the time dependent rules governing the unit utilization are amended with respect to the minimal model by the introduction of transmission lines.

Amendments to the Vertex rule

The vertex rule is changed, since additional commodity flows through the transmission lines occur in each vertex. The commodity balance function is thus changed to:

\[\begin{split}&\forall c \in C,~t\in T_m:\\\\ &\text{CB}(c,t)= \sum_{(c,p)\in C^{\mathrm{in}}_p}\epsilon^{\text{in}}_{vcpt}+ \sum_{(a,f)\in A^{\mathrm{in}}_{v}}\pi^{\text{in}}_{aft}- \sum_{(c,p)\in C^{\mathrm{out}}_p}\epsilon^{\text{out}}_{vcpt}- \sum_{(a,f)\in A^{\mathrm{out}}_{v}}\pi^{\text{out}}_{aft}.\end{split}\]

Here, the new tuple sets \(A^{\mathrm{in,out}}_v\) represent all input and output arcs \(a\) connecting vertex \(v\), respectively. The commodity balance is thereby allowing for commodities to leave the system at vertex \(v\) via arcs as well as processes. Apart from this change to the commodity balance the vertex rule and the other rules restricting commodity flows remain unchanged with respect to the minimal model.

Global CO2 limit

In addition to the general vertex specific constraint for the emissions of environmental commodities as discussed in the minimal model, there is a hard coded possibility to limit the CO2 emissions across all modeled sites:

\[\begin{split}-w\sum_{v\in V\\t\in T_{m}}\text{CB}(v,\text{CO}_2,t)\leq \overline{L}_{\text{CO}_2,y}\end{split}\]

Transmission dispatch constraints

There are two main constraints for the commodity flows to and from transmission lines. The first restricts the total amount of commodity \(c\) flowing in arc \(a\) on transmission line \(f\) to the total capacity of the line:

\[\begin{split}&\forall a\in V\times V\times C,~f\in F,~t\in T_m:\\ & \pi^{\text{in}}_{aft}\leq \kappa_{af}.\end{split}\]

Here, the input into the arc \(\pi^{\text{in}}_{aft}\) is taken as a reference for the total capacity. The output of the arc in the target site is then linked to the input with the transmission efficiency \(e_{af}\)

\[\begin{split}&\forall a\in V\times V\times C,~f\in F,~t\in T_m:\\ & \pi^{\text{out}}_{aft}= e_{af}\cdot \pi^{\text{in}}_{aft}.\end{split}\]

These constraints finalize the multinode feature.