National-scale example model math¶
Complete mathematical formulation for a Calliope pathway optimisation model (based on the national-scale example model). The downloadable math YAML file only includes additions to the pre-defined base calliope YAML
A guide to math documentation¶
If a math component's initial conditions are met (the first if statement), it will be applied to a model.
For each objective, constraint and global expression, a number of sub-conditions then apply (the subsequent, indented if statements) to decide on the specific expression to apply at a given iteration of the component dimensions.
In the expressions, terms in bold font are decision variables and terms in italic font are parameters.
The decision variables and parameters are listed at the end of the page; they also refer back to the global expressions / constraints in which they are used.
Those parameters which are defined over time (timesteps) in the expressions can be defined by a user as a single, time invariant value, or as a timeseries that is loaded from file or dataframe.
Note
For every math component in the documentation, we include the YAML snippet that was used to generate the math in a separate tab.
Download the national-scale example model math formulation as a YAML file
Objective¶
min_cost_optimisation¶
Minimise the total cost of installing and operating all technologies in the system. If multiple cost classes are present (e.g., monetary and co2 emissions), the weighted sum of total costs is minimised. Cost class weights can be defined in the indexed parameter objective_cost_weights.
\[
\begin{array}{l}
\min{}\!\!:\\[2em]
\quad \text{if } (\bigvee\limits_{\substack{\text{node} \in \text{nodes} \\ \text{tech} \in \text{techs} \\ \text{cost} \in \text{costs} \\ \text{investstep} \in \text{investsteps}}} (cost))\land{}(\text{config.ensure\_feasibility}\mathord{=}\text{true})\!\!:\\
\qquad \sum\limits_{\text{cost} \in \text{costs}} (\sum\limits_{\text{investstep} \in \text{investsteps}} (\sum\limits_{\text{[nodes,techs]} \in \text{[nodes,techs]}} (\textbf{cost}_\text{node,tech,cost,investstep}) \times \textit{investstep\_resolution}_\text{investstep}) \times \textit{objective\_cost\_weights}_\text{cost}) + \sum\limits_{\text{investstep} \in \text{investsteps}} (\sum\limits_{\text{timestep} \in \text{timesteps}} (\sum\limits_{\text{[carriers,nodes]} \in \text{[carriers,nodes]}} (\textbf{unmet\_demand}_\text{node,carrier,timestep,investstep} - \textbf{unused\_supply}_\text{node,carrier,timestep,investstep}) \times \textit{timestep\_weights}_\text{timestep}) \times \textit{investstep\_resolution}_\text{investstep}) \times \textit{bigM}\\[2em]
\quad \text{if } (\bigvee\limits_{\substack{\text{node} \in \text{nodes} \\ \text{tech} \in \text{techs} \\ \text{cost} \in \text{costs} \\ \text{investstep} \in \text{investsteps}}} (cost))\land{}(\neg (\text{config.ensure\_feasibility}\mathord{=}\text{true}))\!\!:\\
\qquad \sum\limits_{\text{cost} \in \text{costs}} (\sum\limits_{\text{investstep} \in \text{investsteps}} (\sum\limits_{\text{[nodes,techs]} \in \text{[nodes,techs]}} (\textbf{cost}_\text{node,tech,cost,investstep}) \times \textit{investstep\_resolution}_\text{investstep}) \times \textit{objective\_cost\_weights}_\text{cost})\\[2em]
\quad \text{if } (\neg (\bigvee\limits_{\substack{\text{node} \in \text{nodes} \\ \text{tech} \in \text{techs} \\ \text{cost} \in \text{costs} \\ \text{investstep} \in \text{investsteps}}} (cost)))\land{}(\text{config.ensure\_feasibility}\mathord{=}\text{true})\!\!:\\
\qquad \sum\limits_{\text{investstep} \in \text{investsteps}} (\sum\limits_{\text{timestep} \in \text{timesteps}} (\sum\limits_{\text{[carriers,nodes]} \in \text{[carriers,nodes]}} (\textbf{unmet\_demand}_\text{node,carrier,timestep,investstep} - \textbf{unused\_supply}_\text{node,carrier,timestep,investstep}) \times \textit{timestep\_weights}_\text{timestep}) \times \textit{investstep\_resolution}_\text{investstep}) \times \textit{bigM}\\[2em]
\quad \text{if } (\neg (\bigvee\limits_{\substack{\text{node} \in \text{nodes} \\ \text{tech} \in \text{techs} \\ \text{cost} \in \text{costs} \\ \text{investstep} \in \text{investsteps}}} (cost)))\land{}(\neg (\text{config.ensure\_feasibility}\mathord{=}\text{true}))\!\!:\\
\qquad 0\\[2em]
\end{array}
\]
equations:
- where: any(cost, over=[nodes, techs, costs, investsteps])
expression: "sum(\n sum(\n sum(cost, over=[nodes, techs])\n * investstep_resolution,\n\
\ over=investsteps\n )\n * objective_cost_weights,\n over=costs\n) + $unmet_demand"
- where: NOT any(cost, over=[nodes, techs, costs, investsteps])
expression: $unmet_demand
sub_expressions:
unmet_demand:
- where: config.ensure_feasibility=True
expression: "sum(\n sum(\n sum(unmet_demand - unused_supply, over=[carriers,
nodes]) * timestep_weights,\n over=timesteps\n )\n * investstep_resolution,\n\
\ over=investsteps\n) * bigM"
- where: NOT config.ensure_feasibility=True
expression: '0'
sense: minimise
active: true
Subject to¶
flow_capacity_per_storage_capacity_min¶
Set the lower bound of storage flow capacity relative to its storage capacity.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{storage\_cap}_\text{node,tech,investstep}) \land \exists (\textit{flow\_cap\_per\_storage\_cap\_min}))\!\!:\\[2em]
\quad \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \geq \textbf{storage\_cap}_\text{node,tech,investstep} \times \textit{flow\_cap\_per\_storage\_cap\_min}\\
\end{array}
\]
flow_capacity_per_storage_capacity_max¶
Set the upper bound of storage flow capacity relative to its storage capacity.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{storage\_cap}_\text{node,tech,investstep}) \land \exists (\textit{flow\_cap\_per\_storage\_cap\_max}_\text{tech}))\!\!:\\[2em]
\quad \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \leq \textbf{storage\_cap}_\text{node,tech,investstep} \times \textit{flow\_cap\_per\_storage\_cap\_max}_\text{tech}\\
\end{array}
\]
source_capacity_equals_flow_capacity¶
Set a supply technology's flow capacity to equal its source capacity.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{source\_cap}_\text{node,tech,investstep}) \land \textit{source\_cap\_equals\_flow\_cap}\mathord{=}\text{true})\!\!:\\[2em]
\quad \textbf{source\_cap}_\text{node,tech,investstep} = \textbf{flow\_cap}_\text{node,tech,carrier,investstep}\\
\end{array}
\]
force_zero_area_use¶
Set a technology's area use to zero if its flow capacity upper bound is zero.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{area\_use}_\text{node,tech,investstep}) \land \textit{flow\_cap\_max}_\text{node,tech}\mathord{=}\text{0})\!\!:\\[2em]
\quad \textbf{area\_use}_\text{node,tech,investstep} = 0\\
\end{array}
\]
area_use_per_flow_capacity¶
Set a fixed relationship between a technology's flow capacity and its area use.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{area\_use}_\text{node,tech,investstep}) \land \exists (\textit{area\_use\_per\_flow\_cap}))\!\!:\\[2em]
\quad \textbf{area\_use}_\text{node,tech,investstep} = \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \times \textit{area\_use\_per\_flow\_cap}\\
\end{array}
\]
area_use_capacity_per_loc¶
Set an upper bound on the total area that all technologies with area_use can occupy at a given node.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{area\_use}_\text{node,tech,investstep}) \land \exists (\textit{available\_area}))\!\!:\\[2em]
\quad \sum\limits_{\text{tech} \in \text{techs}} (\textbf{area\_use}_\text{node,tech,investstep}) \leq \textit{available\_area}\\
\end{array}
\]
flow_capacity_systemwide_max¶
Set an upper bound on flow capacity of a technology across all nodes in which the technology exists.
\[
\begin{array}{l}
\forall{}
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{flow\_cap\_max\_systemwide}_\text{tech,investstep}))\!\!:\\[2em]
\quad \sum\limits_{\text{node} \in \text{nodes}} (\textbf{flow\_cap}_\text{node,tech,carrier,investstep}) \leq \textit{flow\_cap\_max\_systemwide}_\text{tech,investstep}\\
\end{array}
\]
flow_capacity_systemwide_min¶
Set a lower bound on flow capacity of a technology across all nodes in which the technology exists.
\[
\begin{array}{l}
\forall{}
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{flow\_cap\_min\_systemwide}))\!\!:\\[2em]
\quad \sum\limits_{\text{node} \in \text{nodes}} (\textbf{flow\_cap}_\text{node,tech,carrier,investstep}) \geq \textit{flow\_cap\_min\_systemwide}\\
\end{array}
\]
balance_conversion¶
Fix the relationship between a conversion technology's outflow and consumption.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{conversion} \land \neg (\textit{include\_storage}_\text{tech}\mathord{=}\text{true}))\!\!:\\[2em]
\quad \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_out\_inc\_eff}_\text{node,tech,carrier,timestep,investstep}) = \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep})\\
\end{array}
\]
flow_out_max¶
Set the upper bound of a technology's outflow.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{carrier\_out}_\text{node,tech,carrier}) \land \neg (\exists (\textbf{operating\_units}_\text{node,tech,timestep,investstep})))\!\!:\\[2em]
\quad \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} \leq \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \times \textit{timestep\_resolution}_\text{timestep} \times \textit{flow\_out\_parasitic\_eff}_\text{tech}\\
\end{array}
\]
flow_out_min¶
Set the lower bound of a technology's outflow.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{flow\_out\_min\_relative}) \land \neg (\exists (\textbf{operating\_units}_\text{node,tech,timestep,investstep})))\!\!:\\[2em]
\quad \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} \geq \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \times \textit{timestep\_resolution}_\text{timestep} \times \textit{flow\_out\_min\_relative}\\
\end{array}
\]
flow_in_max¶
Set the upper bound of a technology's inflow.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{carrier\_in}_\text{node,tech,carrier}) \land \neg (\exists (\textbf{operating\_units}_\text{node,tech,timestep,investstep})))\!\!:\\[2em]
\quad \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep} \leq \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \times \textit{timestep\_resolution}_\text{timestep}\\
\end{array}
\]
source_max¶
Set the upper bound of a supply technology's source consumption.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{source\_cap}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad \textbf{source\_use}_\text{node,tech,timestep,investstep} \leq \textit{timestep\_resolution}_\text{timestep} \times \textbf{source\_cap}_\text{node,tech,investstep}\\
\end{array}
\]
storage_max¶
Set the upper bound of the amount of carrier a technology can store.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{storage}_\text{node,tech,timestep,investstep}))\!\!:\\[2em]
\quad \textbf{storage}_\text{node,tech,timestep,investstep} \leq \textbf{storage\_cap}_\text{node,tech,investstep}\\
\end{array}
\]
storage_discharge_depth_limit¶
Set the lower bound of the stored carrier a technology must keep in reserve at all times.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{storage}_\text{node,tech,timestep,investstep}) \land \exists (\textit{storage\_discharge\_depth}))\!\!:\\[2em]
\quad \textbf{storage}_\text{node,tech,timestep,investstep} - (\textit{storage\_discharge\_depth} \times \textbf{storage\_cap}_\text{node,tech,investstep}) \geq 0\\
\end{array}
\]
system_balance¶
Set the global carrier balance of the optimisation problem by fixing the total production of a given carrier to equal the total consumption of that carrier at every node in every timestep.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!:\\[2em]
\quad \text{if } (\text{config.ensure\_feasibility}\mathord{=}\text{true})\land{}(\bigvee\limits_{\text{tech} \in \text{techs}} (carrier_export))\!\!:\\
\qquad \sum\limits_{\text{tech} \in \text{techs}} (\textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) - \sum\limits_{\text{tech} \in \text{techs}} (\textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}) - \sum\limits_{\text{tech} \in \text{techs}} (\textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep}) + \textbf{unmet\_demand}_\text{node,carrier,timestep,investstep} + \textbf{unused\_supply}_\text{node,carrier,timestep,investstep} = 0\\[2em]
\quad \text{if } (\text{config.ensure\_feasibility}\mathord{=}\text{true})\land{}(\neg (\bigvee\limits_{\text{tech} \in \text{techs}} (carrier_export)))\!\!:\\
\qquad \sum\limits_{\text{tech} \in \text{techs}} (\textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) - \sum\limits_{\text{tech} \in \text{techs}} (\textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}) + \textbf{unmet\_demand}_\text{node,carrier,timestep,investstep} + \textbf{unused\_supply}_\text{node,carrier,timestep,investstep} = 0\\[2em]
\quad \text{if } (\neg (\text{config.ensure\_feasibility}\mathord{=}\text{true}))\land{}(\bigvee\limits_{\text{tech} \in \text{techs}} (carrier_export))\!\!:\\
\qquad \sum\limits_{\text{tech} \in \text{techs}} (\textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) - \sum\limits_{\text{tech} \in \text{techs}} (\textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}) - \sum\limits_{\text{tech} \in \text{techs}} (\textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep}) = 0\\[2em]
\quad \text{if } (\neg (\text{config.ensure\_feasibility}\mathord{=}\text{true}))\land{}(\neg (\bigvee\limits_{\text{tech} \in \text{techs}} (carrier_export)))\!\!:\\
\qquad \sum\limits_{\text{tech} \in \text{techs}} (\textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) - \sum\limits_{\text{tech} \in \text{techs}} (\textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}) = 0\\[2em]
\end{array}
\]
foreach:
- nodes
- carriers
- timesteps
- investsteps
equations:
- expression: sum(flow_out, over=techs) - sum(flow_in, over=techs) - $flow_export
+ $unmet_demand_and_unused_supply == 0
sub_expressions:
flow_export:
- where: any(carrier_export, over=techs)
expression: sum(flow_export, over=techs)
- where: NOT any(carrier_export, over=techs)
expression: '0'
unmet_demand_and_unused_supply:
- where: config.ensure_feasibility=True
expression: unmet_demand + unused_supply
- where: NOT config.ensure_feasibility=True
expression: '0'
balance_demand¶
Set the upper bound on, or a fixed total of, that a demand technology must dump to its sink in each timestep.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{demand})\!\!:\\[2em]
\quad \text{if } (\exists (\textit{sink\_use\_equals}_\text{timestep,node,tech,investstep}))\land{}(\textit{sink\_unit}\mathord{=}\text{per\_area})\!\!:\\
\qquad \textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep} = \textit{sink\_use\_equals}_\text{timestep,node,tech,investstep} \times \textbf{area\_use}_\text{node,tech,investstep}\\[2em]
\quad \text{if } (\exists (\textit{sink\_use\_equals}_\text{timestep,node,tech,investstep}))\land{}(\textit{sink\_unit}\mathord{=}\text{per\_cap})\!\!:\\
\qquad \textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep} = \textit{sink\_use\_equals}_\text{timestep,node,tech,investstep} \times \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_cap}_\text{node,tech,carrier,investstep})\\[2em]
\quad \text{if } (\exists (\textit{sink\_use\_equals}_\text{timestep,node,tech,investstep}))\land{}(\textit{sink\_unit}\mathord{=}\text{absolute})\!\!:\\
\qquad \textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep} = \textit{sink\_use\_equals}_\text{timestep,node,tech,investstep} \times 1\\[2em]
\quad \text{if } (\neg (\exists (\textit{sink\_use\_equals}_\text{timestep,node,tech,investstep})) \land \exists (\textit{sink\_use\_max}))\land{}(\textit{sink\_unit}\mathord{=}\text{per\_area})\!\!:\\
\qquad \textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep} \leq \textit{sink\_use\_max} \times \textbf{area\_use}_\text{node,tech,investstep}\\[2em]
\quad \text{if } (\neg (\exists (\textit{sink\_use\_equals}_\text{timestep,node,tech,investstep})) \land \exists (\textit{sink\_use\_max}))\land{}(\textit{sink\_unit}\mathord{=}\text{per\_cap})\!\!:\\
\qquad \textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep} \leq \textit{sink\_use\_max} \times \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_cap}_\text{node,tech,carrier,investstep})\\[2em]
\quad \text{if } (\neg (\exists (\textit{sink\_use\_equals}_\text{timestep,node,tech,investstep})) \land \exists (\textit{sink\_use\_max}))\land{}(\textit{sink\_unit}\mathord{=}\text{absolute})\!\!:\\
\qquad \textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep} \leq \textit{sink\_use\_max} \times 1\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- carriers
- timesteps
- investsteps
where: base_tech=demand
equations:
- where: sink_use_equals
expression: flow_in_inc_eff == sink_use_equals * $sink_scaler
- where: NOT sink_use_equals AND sink_use_max
expression: flow_in_inc_eff <= sink_use_max * $sink_scaler
sub_expressions:
sink_scaler:
- where: sink_unit=per_area
expression: area_use
- where: sink_unit=per_cap
expression: sum(flow_cap, over=carriers)
- where: sink_unit=absolute
expression: '1'
balance_demand_min_use¶
Set the lower bound on the quantity of flow a demand technology must dump to its sink in each timestep.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{sink\_use\_min}) \land \neg (\exists (\textit{sink\_use\_equals}_\text{timestep,node,tech,investstep})) \land \textit{base\_tech}_\text{tech}\mathord{=}\text{demand})\!\!:\\[2em]
\quad \text{if } (\textit{sink\_unit}\mathord{=}\text{per\_area})\!\!:\\
\qquad \textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep} \geq \textit{sink\_use\_min} \times \textbf{area\_use}_\text{node,tech,investstep}\\[2em]
\quad \text{if } (\textit{sink\_unit}\mathord{=}\text{per\_cap})\!\!:\\
\qquad \textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep} \geq \textit{sink\_use\_min} \times \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_cap}_\text{node,tech,carrier,investstep})\\[2em]
\quad \text{if } (\textit{sink\_unit}\mathord{=}\text{absolute})\!\!:\\
\qquad \textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep} \geq \textit{sink\_use\_min} \times 1\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- carriers
- timesteps
- investsteps
where: sink_use_min AND NOT sink_use_equals AND base_tech=demand
equations:
- expression: flow_in_inc_eff >= sink_use_min * $sink_scaler
sub_expressions:
sink_scaler:
- where: sink_unit=per_area
expression: area_use
- where: sink_unit=per_cap
expression: sum(flow_cap, over=carriers)
- where: sink_unit=absolute
expression: '1'
balance_supply_no_storage¶
Fix the outflow of a supply technology to its consumption of the available source.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{supply} \land \neg (\textit{include\_storage}_\text{tech}\mathord{=}\text{true}))\!\!:\\[2em]
\quad \textbf{flow\_out\_inc\_eff}_\text{node,tech,carrier,timestep,investstep} = \textbf{source\_use}_\text{node,tech,timestep,investstep} \times \textit{source\_eff}\\
\end{array}
\]
balance_supply_with_storage¶
Fix the outflow of a supply technology to its consumption of the available source, with a storage buffer to temporally offset the outflow from source consumption.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{storage}_\text{node,tech,timestep,investstep}) \land \textit{base\_tech}_\text{tech}\mathord{=}\text{supply})\!\!:\\[2em]
\quad \text{if } (\textit{timesteps}_\text{timestep}\mathord{=}\text{timesteps[0]} \land \neg (\textit{cyclic\_storage}\mathord{=}\text{true}))\!\!:\\
\qquad \textbf{storage}_\text{node,tech,timestep,investstep} = \textit{storage\_initial} \times \textbf{storage\_cap}_\text{node,tech,investstep} + (\textbf{source\_use}_\text{node,tech,timestep,investstep} \times \textit{source\_eff}) - \textbf{flow\_out\_inc\_eff}_\text{node,tech,carrier,timestep,investstep}\\[2em]
\quad \text{if } (((\textit{timesteps}_\text{timestep}\mathord{=}\text{timesteps[0]} \land \textit{cyclic\_storage}\mathord{=}\text{true}) \lor \neg (\textit{timesteps}_\text{timestep}\mathord{=}\text{timesteps[0]})) \land \neg (\textit{lookup\_cluster\_first\_timestep}\mathord{=}\text{true}))\!\!:\\
\qquad \textbf{storage}_\text{node,tech,timestep,investstep} = ((1 - \textit{storage\_loss}_\text{tech})^{\textit{timestep\_resolution}_\text{timestep-1}}) \times \textbf{storage}_\text{node,tech,timestep-1,investstep} + (\textbf{source\_use}_\text{node,tech,timestep,investstep} \times \textit{source\_eff}) - \textbf{flow\_out\_inc\_eff}_\text{node,tech,carrier,timestep,investstep}\\[2em]
\quad \text{if } (\textit{lookup\_cluster\_first\_timestep}\mathord{=}\text{true} \land \neg (\textit{timesteps}_\text{timestep}\mathord{=}\text{timesteps[0]} \land \neg (\textit{cyclic\_storage}\mathord{=}\text{true})))\!\!:\\
\qquad \textbf{storage}_\text{node,tech,timestep,investstep} = ((1 - \textit{storage\_loss}_\text{tech})^{\textit{timestep\_resolution}_\text{timestep=lookup\_cluster\_last\_timestep[timestep]}}) \times \textbf{storage}_\text{node,tech,timestep=lookup\_cluster\_last\_timestep[timestep],investstep} + (\textbf{source\_use}_\text{node,tech,timestep,investstep} \times \textit{source\_eff}) - \textbf{flow\_out\_inc\_eff}_\text{node,tech,carrier,timestep,investstep}\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- carriers
- timesteps
- investsteps
where: storage AND base_tech=supply
equations:
- expression: storage == $storage_previous_step + source_use * source_eff - flow_out_inc_eff
sub_expressions:
storage_previous_step:
- where: timesteps=get_val_at_index(timesteps=0) AND NOT cyclic_storage=True
expression: storage_initial * storage_cap
- where: "(\n (timesteps=get_val_at_index(timesteps=0) AND cyclic_storage=True)\n\
\ OR NOT timesteps=get_val_at_index(timesteps=0)\n) AND NOT lookup_cluster_first_timestep=True"
expression: (1 - storage_loss) ** roll(timestep_resolution, timesteps=1) * roll(storage,
timesteps=1)
- where: lookup_cluster_first_timestep=True AND NOT (timesteps=get_val_at_index(timesteps=0)
AND NOT cyclic_storage=True)
expression: (1 - storage_loss) ** select_from_lookup_arrays(timestep_resolution,
timesteps=lookup_cluster_last_timestep) * select_from_lookup_arrays(storage,
timesteps=lookup_cluster_last_timestep)
source_availability_supply¶
Set the upper bound on, or a fixed total of, a supply technology's ability to consume its available resource.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{source\_use}_\text{node,tech,timestep,investstep}) \land (\exists (\textit{source\_use\_equals}) \lor \exists (\textit{source\_use\_max}_\text{timestep,node,tech})))\!\!:\\[2em]
\quad \text{if } (\exists (\textit{source\_use\_equals}))\land{}(\textit{source\_unit}_\text{tech}\mathord{=}\text{per\_area})\!\!:\\
\qquad \textbf{source\_use}_\text{node,tech,timestep,investstep} = \textit{source\_use\_equals} \times \textbf{area\_use}_\text{node,tech,investstep}\\[2em]
\quad \text{if } (\exists (\textit{source\_use\_equals}))\land{}(\textit{source\_unit}_\text{tech}\mathord{=}\text{per\_cap})\!\!:\\
\qquad \textbf{source\_use}_\text{node,tech,timestep,investstep} = \textit{source\_use\_equals} \times \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_cap}_\text{node,tech,carrier,investstep})\\[2em]
\quad \text{if } (\exists (\textit{source\_use\_equals}))\land{}(\textit{source\_unit}_\text{tech}\mathord{=}\text{absolute})\!\!:\\
\qquad \textbf{source\_use}_\text{node,tech,timestep,investstep} = \textit{source\_use\_equals} \times 1\\[2em]
\quad \text{if } (\neg (\exists (\textit{source\_use\_equals})) \land \exists (\textit{source\_use\_max}_\text{timestep,node,tech}))\land{}(\textit{source\_unit}_\text{tech}\mathord{=}\text{per\_area})\!\!:\\
\qquad \textbf{source\_use}_\text{node,tech,timestep,investstep} \leq \textit{source\_use\_max}_\text{timestep,node,tech} \times \textbf{area\_use}_\text{node,tech,investstep}\\[2em]
\quad \text{if } (\neg (\exists (\textit{source\_use\_equals})) \land \exists (\textit{source\_use\_max}_\text{timestep,node,tech}))\land{}(\textit{source\_unit}_\text{tech}\mathord{=}\text{per\_cap})\!\!:\\
\qquad \textbf{source\_use}_\text{node,tech,timestep,investstep} \leq \textit{source\_use\_max}_\text{timestep,node,tech} \times \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_cap}_\text{node,tech,carrier,investstep})\\[2em]
\quad \text{if } (\neg (\exists (\textit{source\_use\_equals})) \land \exists (\textit{source\_use\_max}_\text{timestep,node,tech}))\land{}(\textit{source\_unit}_\text{tech}\mathord{=}\text{absolute})\!\!:\\
\qquad \textbf{source\_use}_\text{node,tech,timestep,investstep} \leq \textit{source\_use\_max}_\text{timestep,node,tech} \times 1\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- timesteps
- investsteps
where: source_use AND (source_use_equals OR source_use_max)
equations:
- where: source_use_equals
expression: source_use == source_use_equals * $source_scaler
- where: NOT source_use_equals AND source_use_max
expression: source_use <= source_use_max * $source_scaler
sub_expressions:
source_scaler:
- where: source_unit=per_area
expression: area_use
- where: source_unit=per_cap
expression: sum(flow_cap, over=carriers)
- where: source_unit=absolute
expression: '1'
balance_supply_min_use¶
Set the lower bound on the quantity of its source a supply technology must use in each timestep.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{source\_use\_min}) \land \neg (\exists (\textit{source\_use\_equals})) \land \textit{base\_tech}_\text{tech}\mathord{=}\text{supply})\!\!:\\[2em]
\quad \text{if } (\textit{source\_unit}_\text{tech}\mathord{=}\text{per\_area})\!\!:\\
\qquad \textbf{source\_use}_\text{node,tech,timestep,investstep} \geq \textit{source\_use\_min} \times \textbf{area\_use}_\text{node,tech,investstep}\\[2em]
\quad \text{if } (\textit{source\_unit}_\text{tech}\mathord{=}\text{per\_cap})\!\!:\\
\qquad \textbf{source\_use}_\text{node,tech,timestep,investstep} \geq \textit{source\_use\_min} \times \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_cap}_\text{node,tech,carrier,investstep})\\[2em]
\quad \text{if } (\textit{source\_unit}_\text{tech}\mathord{=}\text{absolute})\!\!:\\
\qquad \textbf{source\_use}_\text{node,tech,timestep,investstep} \geq \textit{source\_use\_min} \times 1\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- timesteps
- investsteps
where: source_use_min AND NOT source_use_equals AND base_tech=supply
equations:
- expression: source_use >= source_use_min * $source_scaler
sub_expressions:
source_scaler:
- where: source_unit=per_area
expression: area_use
- where: source_unit=per_cap
expression: sum(flow_cap, over=carriers)
- where: source_unit=absolute
expression: '1'
balance_storage¶
Fix the quantity of carrier stored in a storage technology at the end of each timestep based on the net flow of carrier charged and discharged and the quantity of carrier stored at the start of the timestep.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } ((\textit{include\_storage}_\text{tech}\mathord{=}\text{true} \lor \textit{base\_tech}_\text{tech}\mathord{=}\text{storage}) \land \neg (\textit{base\_tech}_\text{tech}\mathord{=}\text{supply} \lor \textit{base\_tech}_\text{tech}\mathord{=}\text{demand}))\!\!:\\[2em]
\quad \text{if } (\textit{timesteps}_\text{timestep}\mathord{=}\text{timesteps[0]} \land \neg (\textit{cyclic\_storage}\mathord{=}\text{true}))\!\!:\\
\qquad \textbf{storage}_\text{node,tech,timestep,investstep} = \textit{storage\_initial} \times \textbf{storage\_cap}_\text{node,tech,investstep} - \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_out\_inc\_eff}_\text{node,tech,carrier,timestep,investstep}) + \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep})\\[2em]
\quad \text{if } (((\textit{timesteps}_\text{timestep}\mathord{=}\text{timesteps[0]} \land \textit{cyclic\_storage}\mathord{=}\text{true}) \lor \neg (\textit{timesteps}_\text{timestep}\mathord{=}\text{timesteps[0]})) \land \neg (\textit{lookup\_cluster\_first\_timestep}\mathord{=}\text{true}))\!\!:\\
\qquad \textbf{storage}_\text{node,tech,timestep,investstep} = ((1 - \textit{storage\_loss}_\text{tech})^{\textit{timestep\_resolution}_\text{timestep-1}}) \times \textbf{storage}_\text{node,tech,timestep-1,investstep} - \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_out\_inc\_eff}_\text{node,tech,carrier,timestep,investstep}) + \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep})\\[2em]
\quad \text{if } (\textit{lookup\_cluster\_first\_timestep}\mathord{=}\text{true} \land \neg (\textit{timesteps}_\text{timestep}\mathord{=}\text{timesteps[0]} \land \neg (\textit{cyclic\_storage}\mathord{=}\text{true})))\!\!:\\
\qquad \textbf{storage}_\text{node,tech,timestep,investstep} = ((1 - \textit{storage\_loss}_\text{tech})^{\textit{timestep\_resolution}_\text{timestep=lookup\_cluster\_last\_timestep[timestep]}}) \times \textbf{storage}_\text{node,tech,timestep=lookup\_cluster\_last\_timestep[timestep],investstep} - \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_out\_inc\_eff}_\text{node,tech,carrier,timestep,investstep}) + \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep})\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- timesteps
- investsteps
where: (include_storage=true or base_tech=storage) AND NOT (base_tech=supply OR base_tech=demand)
equations:
- expression: "storage == $storage_previous_step -\n sum(flow_out_inc_eff, over=carriers)
+ sum(flow_in_inc_eff, over=carriers)"
sub_expressions:
storage_previous_step:
- where: timesteps=get_val_at_index(timesteps=0) AND NOT cyclic_storage=True
expression: storage_initial * storage_cap
- where: "(\n (timesteps=get_val_at_index(timesteps=0) AND cyclic_storage=True)\n\
\ OR NOT timesteps=get_val_at_index(timesteps=0)\n) AND NOT lookup_cluster_first_timestep=True"
expression: (1 - storage_loss) ** roll(timestep_resolution, timesteps=1) * roll(storage,
timesteps=1)
- where: lookup_cluster_first_timestep=True AND NOT (timesteps=get_val_at_index(timesteps=0)
AND NOT cyclic_storage=True)
expression: (1 - storage_loss) ** select_from_lookup_arrays(timestep_resolution,
timesteps=lookup_cluster_last_timestep) * select_from_lookup_arrays(storage,
timesteps=lookup_cluster_last_timestep)
set_storage_initial¶
Fix the relationship between carrier stored in a storage technology at the start and end of the whole model period.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{storage}_\text{node,tech,timestep,investstep}) \land \exists (\textit{storage\_initial}) \land \textit{cyclic\_storage}\mathord{=}\text{true})\!\!:\\[2em]
\quad \textbf{storage}_\text{node,tech,timestep=timesteps[-1],investstep} \times ((1 - \textit{storage\_loss}_\text{tech})^{\textit{timestep\_resolution}_\text{timestep=timesteps[-1]}}) = \textit{storage\_initial} \times \textbf{storage\_cap}_\text{node,tech,investstep}\\
\end{array}
\]
foreach:
- nodes
- techs
- investsteps
where: storage AND storage_initial AND cyclic_storage=True
equations:
- expression: "storage[timesteps=$final_step] * (\n (1 - storage_loss) ** timestep_resolution[timesteps=$final_step]\n
) == storage_initial * storage_cap"
slices:
final_step:
- expression: get_val_at_index(timesteps=-1)
active: true
balance_transmission¶
Fix the relationship between between carrier flowing into and out of a transmission link in each timestep.
\[
\begin{array}{l}
\forall{}
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission})\!\!:\\[2em]
\quad \sum\limits_{\text{[nodes,carriers]} \in \text{[nodes,carriers]}} (\textbf{flow\_out\_inc\_eff}_\text{node,tech,carrier,timestep,investstep}) = \sum\limits_{\text{[nodes,carriers]} \in \text{[nodes,carriers]}} (\textbf{flow\_in\_inc\_eff}_\text{node,tech,carrier,timestep,investstep})\\
\end{array}
\]
symmetric_transmission¶
Fix the flow capacity of two transmission technologies representing the same link in the system.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission})\!\!:\\[2em]
\quad \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_cap}_\text{node,tech,carrier,investstep}) = \textbf{link\_flow\_cap}_\text{tech,investstep}\\
\end{array}
\]
export_balance¶
Set the lower bound of a technology's outflow to a technology's carrier export, for any technologies that can export carriers out of the system.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep}))\!\!:\\[2em]
\quad \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} \geq \textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep}\\
\end{array}
\]
flow_export_max¶
Set the upper bound of a technology's carrier export, for any technologies that can export carriers out of the system.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep}) \land \exists (\textit{export\_max}))\!\!:\\[2em]
\quad \text{if } (\exists (\textbf{operating\_units}_\text{node,tech,timestep,investstep}))\!\!:\\
\qquad \textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep} \leq \textit{export\_max} \times \textbf{operating\_units}_\text{node,tech,timestep,investstep}\\[2em]
\quad \text{if } (\neg (\exists (\textbf{operating\_units}_\text{node,tech,timestep,investstep})))\!\!:\\
\qquad \textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep} \leq \textit{export\_max}\\[2em]
\end{array}
\]
unit_commitment_milp¶
Set the upper bound of the number of integer units of technology that can exist, for any technology using integer units to define its capacity.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{operating\_units}_\text{node,tech,timestep,investstep}) \land \exists (\textbf{purchased\_units}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad \textbf{operating\_units}_\text{node,tech,timestep,investstep} \leq \textbf{purchased\_units}_\text{node,tech,investstep}\\
\end{array}
\]
available_flow_cap_binary¶
Limit flow capacity to zero if the technology is not operating in a given timestep.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{available\_flow\_cap}_\text{node,tech,carrier,timestep,investstep}))\!\!:\\[2em]
\quad \textbf{available\_flow\_cap}_\text{node,tech,carrier,timestep,investstep} \leq \textit{flow\_cap\_max}_\text{node,tech} \times \textbf{operating\_units}_\text{node,tech,timestep,investstep}\\
\end{array}
\]
available_flow_cap_continuous¶
Limit flow capacity to the value of the flow_cap decision variable when the technology is operating in a given timestep.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{available\_flow\_cap}_\text{node,tech,carrier,timestep,investstep}))\!\!:\\[2em]
\quad \textbf{available\_flow\_cap}_\text{node,tech,carrier,timestep,investstep} \leq \textbf{flow\_cap}_\text{node,tech,carrier,investstep}\\
\end{array}
\]
available_flow_cap_max_binary_continuous_switch¶
Force flow capacity to equal the value of the flow_cap decision variable if the technology is operating in a given timestep, zero otherwise.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{available\_flow\_cap}_\text{node,tech,carrier,timestep,investstep}))\!\!:\\[2em]
\quad \textbf{available\_flow\_cap}_\text{node,tech,carrier,timestep,investstep} \geq \textbf{flow\_cap}_\text{node,tech,carrier,investstep} + ((\textbf{operating\_units}_\text{node,tech,timestep,investstep} - \textbf{purchased\_units}_\text{node,tech,investstep}) \times \textit{flow\_cap\_max}_\text{node,tech})\\
\end{array}
\]
flow_out_max_milp¶
Set the upper bound of a technology's ability to produce carriers, for any technology using integer units to define its capacity.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) \land \exists (\textbf{operating\_units}_\text{node,tech,timestep,investstep}) \land \exists (\textit{flow\_cap\_per\_unit}))\!\!:\\[2em]
\quad \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} \leq \textbf{operating\_units}_\text{node,tech,timestep,investstep} \times \textit{timestep\_resolution}_\text{timestep} \times \textit{flow\_cap\_per\_unit} \times \textit{flow\_out\_parasitic\_eff}_\text{tech}\\
\end{array}
\]
flow_in_max_milp¶
Set the upper bound of a technology's ability to consume carriers, for any technology using integer units to define its capacity.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}) \land \exists (\textbf{operating\_units}_\text{node,tech,timestep,investstep}) \land \exists (\textit{flow\_cap\_per\_unit}))\!\!:\\[2em]
\quad \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep} \leq \textbf{operating\_units}_\text{node,tech,timestep,investstep} \times \textit{timestep\_resolution}_\text{timestep} \times \textit{flow\_cap\_per\_unit}\\
\end{array}
\]
flow_out_min_milp¶
Set the lower bound of a technology's ability to produce carriers, for any technology using integer units to define its capacity.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) \land \exists (\textbf{operating\_units}_\text{node,tech,timestep,investstep}) \land \exists (\textit{flow\_out\_min\_relative}))\!\!:\\[2em]
\quad \text{if } (\exists (\textit{flow\_cap\_per\_unit}))\!\!:\\
\qquad \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} \geq \textbf{operating\_units}_\text{node,tech,timestep,investstep} \times \textit{timestep\_resolution}_\text{timestep} \times \textit{flow\_cap\_per\_unit} \times \textit{flow\_out\_min\_relative}\\[2em]
\quad \text{if } (\exists (\textbf{available\_flow\_cap}_\text{node,tech,carrier,timestep,investstep}))\!\!:\\
\qquad \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} \geq \textbf{available\_flow\_cap}_\text{node,tech,carrier,timestep,investstep} \times \textit{timestep\_resolution}_\text{timestep} \times \textit{flow\_out\_min\_relative}\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- carriers
- timesteps
- investsteps
where: flow_out AND operating_units AND flow_out_min_relative
equations:
- where: flow_cap_per_unit
expression: flow_out >= operating_units * timestep_resolution * flow_cap_per_unit
* flow_out_min_relative
- where: available_flow_cap
expression: flow_out >= available_flow_cap * timestep_resolution * flow_out_min_relative
storage_capacity_units_milp¶
Fix the storage capacity of any technology using integer units to define its capacity.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{storage}_\text{node,tech,timestep,investstep}) \land \exists (\textbf{purchased\_units}_\text{node,tech,investstep}) \land \exists (\textit{storage\_cap\_per\_unit}))\!\!:\\[2em]
\quad \textbf{storage\_cap}_\text{node,tech,investstep} = \textbf{purchased\_units}_\text{node,tech,investstep} \times \textit{storage\_cap\_per\_unit}\\
\end{array}
\]
flow_capacity_units_milp¶
Fix the flow capacity of any technology using integer units to define its capacity.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{operating\_units}_\text{node,tech,timestep,investstep}) \land \exists (\textit{flow\_cap\_per\_unit}))\!\!:\\[2em]
\quad \textbf{flow\_cap}_\text{node,tech,carrier,investstep} = \textbf{purchased\_units}_\text{node,tech,investstep} \times \textit{flow\_cap\_per\_unit}\\
\end{array}
\]
flow_capacity_max_purchase_milp¶
Set the upper bound on a technology's flow capacity, for any technology with integer capacity purchasing.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{purchased\_units}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad \text{if } (\exists (\textit{flow\_cap\_max}_\text{node,tech}))\!\!:\\
\qquad \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \leq \textit{flow\_cap\_max}_\text{node,tech} \times \textbf{purchased\_units}_\text{node,tech,investstep}\\[2em]
\quad \text{if } (\neg (\exists (\textit{flow\_cap\_max}_\text{node,tech})))\!\!:\\
\qquad \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \leq \textit{bigM} \times \textbf{purchased\_units}_\text{node,tech,investstep}\\[2em]
\end{array}
\]
flow_capacity_min_purchase_milp¶
Set the lower bound on a technology's flow capacity, for any technology with integer capacity purchasing.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{purchased\_units}_\text{node,tech,investstep}) \land \exists (\textit{flow\_cap\_min}))\!\!:\\[2em]
\quad \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \geq \textit{flow\_cap\_min} \times \textbf{purchased\_units}_\text{node,tech,investstep}\\
\end{array}
\]
storage_capacity_max_purchase_milp¶
Set the upper bound on a technology's storage capacity, for any technology with integer capacity purchasing.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{purchased\_units}_\text{node,tech,investstep}) \land \exists (\textit{storage\_cap\_max}_\text{tech}))\!\!:\\[2em]
\quad \textbf{storage\_cap}_\text{node,tech,investstep} \leq \textit{storage\_cap\_max}_\text{tech} \times \textbf{purchased\_units}_\text{node,tech,investstep}\\
\end{array}
\]
storage_capacity_min_purchase_milp¶
Set the lower bound on a technology's storage capacity, for any technology with integer capacity purchasing.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{purchased\_units}_\text{node,tech,investstep}) \land \exists (\textit{storage\_cap\_min}))\!\!:\\[2em]
\quad \textbf{storage\_cap}_\text{node,tech,investstep} \geq \textit{storage\_cap\_min} \times \textbf{purchased\_units}_\text{node,tech,investstep}\\
\end{array}
\]
unit_capacity_max_systemwide_milp¶
Set the upper bound on the total number of units of a technology that can be purchased across all nodes where the technology can exist, for any technology using integer units to define its capacity.
\[
\begin{array}{l}
\forall{}
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{purchased\_units}_\text{node,tech,investstep}) \land \exists (\textit{purchased\_units\_max\_systemwide}))\!\!:\\[2em]
\quad \sum\limits_{\text{node} \in \text{nodes}} (\textbf{purchased\_units}_\text{node,tech,investstep}) \leq \textit{purchased\_units\_max\_systemwide}\\
\end{array}
\]
unit_capacity_min_systemwide_milp¶
Set the lower bound on the total number of units of a technology that can be purchased across all nodes where the technology can exist, for any technology using integer units to define its capacity.
\[
\begin{array}{l}
\forall{}
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{purchased\_units}_\text{node,tech,investstep}) \land \exists (\textit{purchased\_units\_max\_systemwide}))\!\!:\\[2em]
\quad \sum\limits_{\text{node} \in \text{nodes}} (\textbf{purchased\_units}_\text{node,tech,investstep}) \geq \textit{purchased\_units\_min\_systemwide}\\
\end{array}
\]
async_flow_in_milp¶
Set a technology's ability to have inflow in the same timestep that it has outflow, for any technology using the asynchronous flow binary switch.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{async\_flow\_switch}_\text{node,tech,timestep,investstep}))\!\!:\\[2em]
\quad \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}) \leq (1 - \textbf{async\_flow\_switch}_\text{node,tech,timestep,investstep}) \times \textit{bigM}\\
\end{array}
\]
async_flow_out_milp¶
Set a technology's ability to have outflow in the same timestep that it has inflow, for any technology using the asynchronous flow binary switch.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{async\_flow\_switch}_\text{node,tech,timestep,investstep}))\!\!:\\[2em]
\quad \sum\limits_{\text{carrier} \in \text{carriers}} (\textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) \leq \textbf{async\_flow\_switch}_\text{node,tech,timestep,investstep} \times \textit{bigM}\\
\end{array}
\]
ramping_up¶
Set the upper bound on a technology's ability to ramp outflow up beyond a certain percentage compared to the previous timestep.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{flow\_ramping}_\text{tech}) \land \neg (\textit{timesteps}_\text{timestep}\mathord{=}\text{timesteps[0]}))\!\!:\\[2em]
\quad \text{if } (\exists (\textit{carrier\_out}_\text{node,tech,carrier}) \land \neg (\exists (\textit{carrier\_in}_\text{node,tech,carrier})))\!\!:\\
\qquad \frac{ \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} }{ \textit{timestep\_resolution}_\text{timestep} } - \frac{ \textbf{flow\_out}_\text{node,tech,carrier,timestep-1,investstep} }{ \textit{timestep\_resolution}_\text{timestep-1} } \leq \textit{flow\_ramping}_\text{tech} \times \textbf{flow\_cap}_\text{node,tech,carrier,investstep}\\[2em]
\quad \text{if } (\exists (\textit{carrier\_in}_\text{node,tech,carrier}) \land \neg (\exists (\textit{carrier\_out}_\text{node,tech,carrier})))\!\!:\\
\qquad \frac{ \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep} }{ \textit{timestep\_resolution}_\text{timestep} } - \frac{ \textbf{flow\_in}_\text{node,tech,carrier,timestep-1,investstep} }{ \textit{timestep\_resolution}_\text{timestep-1} } \leq \textit{flow\_ramping}_\text{tech} \times \textbf{flow\_cap}_\text{node,tech,carrier,investstep}\\[2em]
\quad \text{if } (\exists (\textit{carrier\_in}_\text{node,tech,carrier}) \land \exists (\textit{carrier\_out}_\text{node,tech,carrier}))\!\!:\\
\qquad \frac{ (\textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} - \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}) }{ \textit{timestep\_resolution}_\text{timestep} } - \frac{ (\textbf{flow\_out}_\text{node,tech,carrier,timestep-1,investstep} - \textbf{flow\_in}_\text{node,tech,carrier,timestep-1,investstep}) }{ \textit{timestep\_resolution}_\text{timestep-1} } \leq \textit{flow\_ramping}_\text{tech} \times \textbf{flow\_cap}_\text{node,tech,carrier,investstep}\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- carriers
- timesteps
- investsteps
where: flow_ramping AND NOT timesteps=get_val_at_index(timesteps=0)
equations:
- expression: $flow - roll($flow, timesteps=1) <= flow_ramping * flow_cap
sub_expressions:
flow:
- where: carrier_out AND NOT carrier_in
expression: flow_out / timestep_resolution
- where: carrier_in AND NOT carrier_out
expression: flow_in / timestep_resolution
- where: carrier_in AND carrier_out
expression: (flow_out - flow_in) / timestep_resolution
ramping_down¶
Set the upper bound on a technology's ability to ramp outflow down beyond a certain percentage compared to the previous timestep.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{flow\_ramping}_\text{tech}) \land \neg (\textit{timesteps}_\text{timestep}\mathord{=}\text{timesteps[0]}))\!\!:\\[2em]
\quad \text{if } (\exists (\textit{carrier\_out}_\text{node,tech,carrier}) \land \neg (\exists (\textit{carrier\_in}_\text{node,tech,carrier})))\!\!:\\
\qquad -1 \times \textit{flow\_ramping}_\text{tech} \times \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \leq \frac{ \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} }{ \textit{timestep\_resolution}_\text{timestep} } - \frac{ \textbf{flow\_out}_\text{node,tech,carrier,timestep-1,investstep} }{ \textit{timestep\_resolution}_\text{timestep-1} }\\[2em]
\quad \text{if } (\exists (\textit{carrier\_in}_\text{node,tech,carrier}) \land \neg (\exists (\textit{carrier\_out}_\text{node,tech,carrier})))\!\!:\\
\qquad -1 \times \textit{flow\_ramping}_\text{tech} \times \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \leq \frac{ \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep} }{ \textit{timestep\_resolution}_\text{timestep} } - \frac{ \textbf{flow\_in}_\text{node,tech,carrier,timestep-1,investstep} }{ \textit{timestep\_resolution}_\text{timestep-1} }\\[2em]
\quad \text{if } (\exists (\textit{carrier\_in}_\text{node,tech,carrier}) \land \exists (\textit{carrier\_out}_\text{node,tech,carrier}))\!\!:\\
\qquad -1 \times \textit{flow\_ramping}_\text{tech} \times \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \leq \frac{ (\textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} - \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}) }{ \textit{timestep\_resolution}_\text{timestep} } - \frac{ (\textbf{flow\_out}_\text{node,tech,carrier,timestep-1,investstep} - \textbf{flow\_in}_\text{node,tech,carrier,timestep-1,investstep}) }{ \textit{timestep\_resolution}_\text{timestep-1} }\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- carriers
- timesteps
- investsteps
where: flow_ramping AND NOT timesteps=get_val_at_index(timesteps=0)
equations:
- expression: -1 * flow_ramping * flow_cap <= $flow - roll($flow, timesteps=1)
sub_expressions:
flow:
- where: carrier_out AND NOT carrier_in
expression: flow_out / timestep_resolution
- where: carrier_in AND NOT carrier_out
expression: flow_in / timestep_resolution
- where: carrier_in AND carrier_out
expression: (flow_out - flow_in) / timestep_resolution
area_use_bounding¶
Area use in a given investment period is the area use in the previous period plus the new area used minus old used area that has been decommissioned between the previous period and this one.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{area\_use}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad \textbf{area\_use}_\text{node,tech,investstep} = \sum\limits_{\text{vintagestep} \in \text{vintagesteps}} (\textbf{area\_use\_new}_\text{node,tech,vintagestep} \times \textit{available\_vintages}_\text{tech,vintagestep,investstep}) + (\textit{area\_use\_initial}_\text{node,tech} \times \textit{available\_initial\_cap}_\text{tech,investstep})\\
\end{array}
\]
flow_cap_bounding¶
Flow capacity in a given investment period is the capacity in the previous period plus the new capacity installed minus old capacity that has been decommissioned between the previous period and this one.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{flow\_cap}_\text{node,tech,carrier,investstep}))\!\!:\\[2em]
\quad \textbf{flow\_cap}_\text{node,tech,carrier,investstep} = \sum\limits_{\text{vintagestep} \in \text{vintagesteps}} (\textbf{flow\_cap\_new}_\text{node,tech,carrier,vintagestep} \times \textit{available\_vintages}_\text{tech,vintagestep,investstep}) + (\textit{flow\_cap\_initial}_\text{tech,node} \times \textit{available\_initial\_cap}_\text{tech,investstep})\\
\end{array}
\]
limit_flow_cap_new_max_rate¶
Limit maximum rate of new capacity build-out
\[
\begin{array}{l}
\forall{}
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{flow\_cap}_\text{node,tech,carrier,investstep}) \land \exists (\textit{flow\_cap\_new\_max\_rate}))\!\!:\\[2em]
\quad \text{if } (\neg (\textit{investsteps}_\text{investstep}\mathord{=}\text{investsteps[0]}))\!\!:\\
\qquad \sum\limits_{\text{[vintagesteps,nodes]} \in \text{[vintagesteps,nodes]}} ((\textbf{flow\_cap\_new}_\text{node,tech,carrier,vintagestep}\vee{}0) \times \textit{available\_vintages}_\text{tech,vintagestep,investstep}) \leq \sum\limits_{\text{node} \in \text{nodes}} (\textbf{flow\_cap}_\text{node,tech,carrier,investstep-1}) \times \textit{flow\_cap\_new\_max\_rate}\\[2em]
\quad \text{if } (\textit{investsteps}_\text{investstep}\mathord{=}\text{investsteps[0]} \land \textit{flow\_cap\_initial}_\text{tech,node}\mathord{>}\text{0})\!\!:\\
\qquad \sum\limits_{\text{[vintagesteps,nodes]} \in \text{[vintagesteps,nodes]}} ((\textbf{flow\_cap\_new}_\text{node,tech,carrier,vintagestep}\vee{}0) \times \textit{available\_vintages}_\text{tech,vintagestep,investstep}) \leq \sum\limits_{\text{node} \in \text{nodes}} (\textit{flow\_cap\_initial}_\text{tech,node}) \times \textit{flow\_cap\_new\_max\_rate}\\[2em]
\end{array}
\]
equations:
- expression: sum(default_if_empty(flow_cap_new, 0) * available_vintages, over=[vintagesteps,
nodes]) <= sum($prev_flow_cap, over=nodes) * flow_cap_new_max_rate
foreach:
- techs
- carriers
- investsteps
sub_expressions:
prev_flow_cap:
- where: NOT investsteps=get_val_at_index(investsteps=0)
expression: roll(flow_cap, investsteps=1)
- where: investsteps=get_val_at_index(investsteps=0) AND flow_cap_initial > 0
expression: flow_cap_initial
where: flow_cap AND flow_cap_new_max_rate
link_storage_level¶
Link the storage level at the end of one investmentstep to the start of the next.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ cost }\negthickspace \in \negthickspace\text{ costs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{storage}_\text{node,tech,timestep,investstep}) \land \neg (\textit{investsteps}_\text{investstep}\mathord{=}\text{investsteps[0]}))\!\!:\\[2em]
\quad \textbf{storage}_\text{node,tech,timestep=timesteps[0],investstep} = \textbf{storage}_\text{node,tech,timestep=timesteps[-1],investstep-1} \times ((1 - \textit{storage\_loss}_\text{tech})^{\textit{timestep\_resolution}_\text{timestep=timesteps[-1]}})\\
\end{array}
\]
equations:
- expression: "storage[timesteps=$initial_step] == roll(\n storage[timesteps=$final_step]
* (\n (1 - storage_loss) ** timestep_resolution[timesteps=$final_step]\n ),\n\
\ investsteps=1\n)"
foreach:
- nodes
- techs
- costs
- investsteps
slices:
final_step:
- expression: get_val_at_index(timesteps=-1)
initial_step:
- expression: get_val_at_index(timesteps=0)
where: storage AND NOT investsteps=get_val_at_index(investsteps=0)
source_cap_bounding¶
Storage capacity in a given investment period is the capacity in the previous period plus the new capacity installed minus old capacity that has been decommissioned between the previous period and this one.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{source\_cap}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad \textbf{source\_cap}_\text{node,tech,investstep} = \sum\limits_{\text{vintagestep} \in \text{vintagesteps}} (\textbf{source\_cap\_new}_\text{node,tech,vintagestep} \times \textit{available\_vintages}_\text{tech,vintagestep,investstep}) + (\textit{source\_cap\_initial}_\text{node,tech} \times \textit{available\_initial\_cap}_\text{tech,investstep})\\
\end{array}
\]
storage_cap_bounding¶
Storage capacity in a given investment period is the capacity in the previous period plus the new capacity installed minus old capacity that has been decommissioned between the previous period and this one.
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{storage\_cap}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad \textbf{storage\_cap}_\text{node,tech,investstep} = \sum\limits_{\text{vintagestep} \in \text{vintagesteps}} (\textbf{storage\_cap\_new}_\text{node,tech,vintagestep} \times \textit{available\_vintages}_\text{tech,vintagestep,investstep}) + (\textit{storage\_cap\_initial}_\text{node,tech} \times \textit{available\_initial\_cap}_\text{tech,investstep})\\
\end{array}
\]
Where¶
flow_out_inc_eff¶
Outflows after taking efficiency losses into account.
Used in:
- balance_storage
- balance_supply_no_storage
- balance_conversion
- balance_transmission
- balance_supply_with_storage
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}))\!\!:\\[2em]
\quad \text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission})\!\!:\\
\qquad \frac{ \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} }{ (\textit{flow\_out\_eff}_\text{tech} \times \textit{flow\_out\_parasitic\_eff}_\text{tech} \times (\textit{flow\_out\_eff\_per\_distance}^{\textit{distance}_\text{tech}})) }\\[2em]
\quad \text{if } (\neg (\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission}))\!\!:\\
\qquad \frac{ \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} }{ (\textit{flow\_out\_eff}_\text{tech} \times \textit{flow\_out\_parasitic\_eff}_\text{tech}) }\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- carriers
- timesteps
- investsteps
where: flow_out
equations:
- where: base_tech=transmission
expression: "flow_out / (\n flow_out_eff * flow_out_parasitic_eff *\n flow_out_eff_per_distance
** distance\n)"
- where: NOT base_tech=transmission
expression: flow_out / (flow_out_eff * flow_out_parasitic_eff)
flow_in_inc_eff¶
Inflows after taking efficiency losses into account.
Used in:
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}))\!\!:\\[2em]
\quad \text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission})\!\!:\\
\qquad \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep} \times \textit{flow\_in\_eff}_\text{tech} \times (\textit{flow\_in\_eff\_per\_distance}^{\textit{distance}_\text{tech}})\\[2em]
\quad \text{if } (\neg (\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission}))\!\!:\\
\qquad \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep} \times \textit{flow\_in\_eff}_\text{tech}\\[2em]
\end{array}
\]
cost_var¶
The operating costs per timestep of a technology.
Used in:
Unit: cost_per_time
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ cost }\negthickspace \in \negthickspace\text{ costs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{cost\_export}) \lor \exists (\textit{cost\_flow\_in}_\text{tech,cost,investstep}) \lor \exists (\textit{cost\_flow\_out}_\text{tech,cost}))\!\!:\\[2em]
\quad \text{if } (\exists (\textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep}))\land{}(\textit{base\_tech}_\text{tech}\mathord{=}\text{supply})\!\!:\\
\qquad \textit{timestep\_weights}_\text{timestep} \times (\sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_export} \times \textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep}) + \sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_flow\_out}_\text{tech,cost} \times \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) + \textit{cost\_flow\_in}_\text{tech,cost,investstep} \times \textbf{source\_use}_\text{node,tech,timestep,investstep})\\[2em]
\quad \text{if } (\exists (\textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep}))\land{}(\neg (\textit{base\_tech}_\text{tech}\mathord{=}\text{supply}))\!\!:\\
\qquad \textit{timestep\_weights}_\text{timestep} \times (\sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_export} \times \textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep}) + \sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_flow\_out}_\text{tech,cost} \times \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) + \sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_flow\_in}_\text{tech,cost,investstep} \times \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}))\\[2em]
\quad \text{if } (\neg (\exists (\textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep})))\land{}(\textit{base\_tech}_\text{tech}\mathord{=}\text{supply})\!\!:\\
\qquad \textit{timestep\_weights}_\text{timestep} \times (\sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_flow\_out}_\text{tech,cost} \times \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) + \textit{cost\_flow\_in}_\text{tech,cost,investstep} \times \textbf{source\_use}_\text{node,tech,timestep,investstep})\\[2em]
\quad \text{if } (\neg (\exists (\textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep})))\land{}(\neg (\textit{base\_tech}_\text{tech}\mathord{=}\text{supply}))\!\!:\\
\qquad \textit{timestep\_weights}_\text{timestep} \times (\sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_flow\_out}_\text{tech,cost} \times \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}) + \sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_flow\_in}_\text{tech,cost,investstep} \times \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}))\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- costs
- timesteps
- investsteps
where: cost_export OR cost_flow_in OR cost_flow_out
equations:
- expression: timestep_weights * ($cost_export + $cost_flow_out + $cost_flow_in)
sub_expressions:
cost_export:
- where: flow_export
expression: sum(cost_export * flow_export, over=carriers)
- where: NOT flow_export
expression: '0'
cost_flow_in:
- where: base_tech=supply
expression: cost_flow_in * source_use
- where: NOT base_tech=supply
expression: sum(cost_flow_in * flow_in, over=carriers)
cost_flow_out:
- expression: sum(cost_flow_out * flow_out, over=carriers)
cost_investment_flow_cap¶
The investment costs associated with the nominal/rated capacity of a technology.
Used in:
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ cost }\negthickspace \in \negthickspace\text{ costs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{flow\_cap}_\text{node,tech,carrier,investstep}) \land (\exists (\textit{cost\_flow\_cap}_\text{cost,tech,vintagestep}) \lor \exists (\textit{cost\_flow\_cap\_per\_distance})))\!\!:\\[2em]
\quad \text{if } (\exists (\textbf{flow\_cap\_new}_\text{node,tech,carrier,vintagestep}))\land{}(\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission})\!\!:\\
\qquad \sum\limits_{\text{vintagestep} \in \text{vintagesteps}} ((\textit{cost\_flow\_cap}_\text{cost,tech,vintagestep} + (\textit{cost\_flow\_cap\_per\_distance} \times \textit{distance}_\text{tech})) \times 0.5 \times \textbf{flow\_cap\_new}_\text{node,tech,carrier,vintagestep})\\[2em]
\quad \text{if } (\exists (\textbf{flow\_cap\_new}_\text{node,tech,carrier,vintagestep}))\land{}(\neg (\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission}))\!\!:\\
\qquad \sum\limits_{\text{vintagestep} \in \text{vintagesteps}} (\textit{cost\_flow\_cap}_\text{cost,tech,vintagestep} \times \textbf{flow\_cap\_new}_\text{node,tech,carrier,vintagestep})\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- carriers
- costs
- investsteps
where: flow_cap AND (cost_flow_cap OR cost_flow_cap_per_distance)
equations:
- where: flow_cap_new
expression: sum($cost_sum * flow_cap_new, over=vintagesteps)
sub_expressions:
cost_sum:
- where: base_tech=transmission
expression: (cost_flow_cap + cost_flow_cap_per_distance * distance) * 0.5
- where: NOT base_tech=transmission
expression: cost_flow_cap
cost_investment_storage_cap¶
The investment costs associated with the storage capacity of a technology.
Used in:
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ cost }\negthickspace \in \negthickspace\text{ costs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{cost\_storage\_cap}_\text{cost,tech,vintagestep}) \land \exists (\textbf{storage\_cap}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad \text{if } (\exists (\textbf{storage\_cap\_new}_\text{node,tech,vintagestep}))\!\!:\\
\qquad \sum\limits_{\text{vintagestep} \in \text{vintagesteps}} (\textit{cost\_storage\_cap}_\text{cost,tech,vintagestep} \times \textbf{storage\_cap\_new}_\text{node,tech,vintagestep})\\[2em]
\end{array}
\]
cost_investment_source_cap¶
The investment costs associated with the source consumption capacity of a technology.
Used in:
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ cost }\negthickspace \in \negthickspace\text{ costs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{cost\_source\_cap}_\text{cost,tech,vintagestep}) \land \exists (\textbf{source\_cap}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad \text{if } (\exists (\textbf{source\_cap\_new}_\text{node,tech,vintagestep}))\!\!:\\
\qquad \sum\limits_{\text{vintagestep} \in \text{vintagesteps}} (\textit{cost\_source\_cap}_\text{cost,tech,vintagestep} \times \textbf{source\_cap\_new}_\text{node,tech,vintagestep})\\[2em]
\end{array}
\]
cost_investment_area_use¶
The investment costs associated with the area used by a technology.
Used in:
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ cost }\negthickspace \in \negthickspace\text{ costs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{cost\_area\_use}) \land \exists (\textbf{area\_use}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad \text{if } (\exists (\textbf{area\_use\_new}_\text{node,tech,vintagestep}))\!\!:\\
\qquad \sum\limits_{\text{vintagestep} \in \text{vintagesteps}} (\textit{cost\_area\_use} \times \textbf{area\_use\_new}_\text{node,tech,vintagestep})\\[2em]
\end{array}
\]
cost_investment_purchase¶
The investment costs associated with the binary purchase of a technology.
Used in:
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ cost }\negthickspace \in \negthickspace\text{ costs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textit{cost\_purchase}) \land \exists (\textbf{purchased\_units}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad \text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission})\!\!:\\
\qquad ((\textit{cost\_purchase} + (\textit{cost\_purchase\_per\_distance} \times \textit{distance}_\text{tech})) \times \textbf{purchased\_units}_\text{node,tech,investstep}) \times 0.5\\[2em]
\quad \text{if } (\neg (\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission}))\!\!:\\
\qquad \textit{cost\_purchase} \times \textbf{purchased\_units}_\text{node,tech,investstep}\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- costs
- investsteps
where: cost_purchase AND purchased_units
equations:
- where: base_tech=transmission
expression: (cost_purchase + cost_purchase_per_distance * distance) * purchased_units
* 0.5
- where: NOT base_tech=transmission
expression: cost_purchase * purchased_units
cost_investment¶
The installation costs of a technology, including annualised investment costs and annual maintenance costs.
Used in:
Unit: cost
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ cost }\negthickspace \in \negthickspace\text{ costs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{cost\_investment\_flow\_cap}_\text{node,tech,carrier,cost,investstep}) \lor \exists (\textbf{cost\_investment\_storage\_cap}_\text{node,tech,cost,investstep}) \lor \exists (\textbf{cost\_investment\_source\_cap}_\text{node,tech,cost,investstep}) \lor \exists (\textbf{cost\_investment\_area\_use}_\text{node,tech,cost,investstep}) \lor \exists (\textbf{cost\_investment\_purchase}_\text{node,tech,cost,investstep}))\!\!:\\[2em]
\quad \text{if } (\exists (\textit{cost\_depreciation\_rate}))\!\!:\\
\qquad \frac{ \sum\limits_{\text{timestep} \in \text{timesteps}} (\textit{timestep\_resolution}_\text{timestep} \times \textit{timestep\_weights}_\text{timestep}) }{ 8760 } \times ((\textit{cost\_depreciation\_rate} \times (\sum\limits_{\text{carrier} \in \text{carriers}} ((\textbf{cost\_investment\_flow\_cap}_\text{node,tech,carrier,cost,investstep}\vee{}0)) + (\textbf{cost\_investment\_storage\_cap}_\text{node,tech,cost,investstep}\vee{}0) + (\textbf{cost\_investment\_source\_cap}_\text{node,tech,cost,investstep}\vee{}0) + (\textbf{cost\_investment\_area\_use}_\text{node,tech,cost,investstep}\vee{}0) + (\textbf{cost\_investment\_purchase}_\text{node,tech,cost,investstep}\vee{}0)) \times (1 + \textit{cost\_om\_annual\_investment\_fraction})) + \sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_om\_annual} \times \textbf{flow\_cap}_\text{node,tech,carrier,investstep}))\\[2em]
\quad \text{if } (\neg (\exists (\textit{cost\_depreciation\_rate})) \land \textit{cost\_interest\_rate}_\text{tech,cost}\mathord{=}\text{0})\!\!:\\
\qquad \frac{ \sum\limits_{\text{timestep} \in \text{timesteps}} (\textit{timestep\_resolution}_\text{timestep} \times \textit{timestep\_weights}_\text{timestep}) }{ 8760 } \times ((\frac{ 1 }{ \textit{lifetime}_\text{tech} } \times (\sum\limits_{\text{carrier} \in \text{carriers}} ((\textbf{cost\_investment\_flow\_cap}_\text{node,tech,carrier,cost,investstep}\vee{}0)) + (\textbf{cost\_investment\_storage\_cap}_\text{node,tech,cost,investstep}\vee{}0) + (\textbf{cost\_investment\_source\_cap}_\text{node,tech,cost,investstep}\vee{}0) + (\textbf{cost\_investment\_area\_use}_\text{node,tech,cost,investstep}\vee{}0) + (\textbf{cost\_investment\_purchase}_\text{node,tech,cost,investstep}\vee{}0)) \times (1 + \textit{cost\_om\_annual\_investment\_fraction})) + \sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_om\_annual} \times \textbf{flow\_cap}_\text{node,tech,carrier,investstep}))\\[2em]
\quad \text{if } (\neg (\exists (\textit{cost\_depreciation\_rate})) \land \textit{cost\_interest\_rate}_\text{tech,cost}\mathord{>}\text{0})\!\!:\\
\qquad \frac{ \sum\limits_{\text{timestep} \in \text{timesteps}} (\textit{timestep\_resolution}_\text{timestep} \times \textit{timestep\_weights}_\text{timestep}) }{ 8760 } \times ((\frac{ (\textit{cost\_interest\_rate}_\text{tech,cost} \times ((1 + \textit{cost\_interest\_rate}_\text{tech,cost})^{\textit{lifetime}_\text{tech}})) }{ (((1 + \textit{cost\_interest\_rate}_\text{tech,cost})^{\textit{lifetime}_\text{tech}}) - 1) } \times (\sum\limits_{\text{carrier} \in \text{carriers}} ((\textbf{cost\_investment\_flow\_cap}_\text{node,tech,carrier,cost,investstep}\vee{}0)) + (\textbf{cost\_investment\_storage\_cap}_\text{node,tech,cost,investstep}\vee{}0) + (\textbf{cost\_investment\_source\_cap}_\text{node,tech,cost,investstep}\vee{}0) + (\textbf{cost\_investment\_area\_use}_\text{node,tech,cost,investstep}\vee{}0) + (\textbf{cost\_investment\_purchase}_\text{node,tech,cost,investstep}\vee{}0)) \times (1 + \textit{cost\_om\_annual\_investment\_fraction})) + \sum\limits_{\text{carrier} \in \text{carriers}} (\textit{cost\_om\_annual} \times \textbf{flow\_cap}_\text{node,tech,carrier,investstep}))\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- costs
- investsteps
where: cost_investment_flow_cap OR cost_investment_storage_cap OR cost_investment_source_cap
OR cost_investment_area_use OR cost_investment_purchase
equations:
- expression: "$annualisation_weight * (\n $depreciation_rate * (\n sum(default_if_empty(cost_investment_flow_cap,
0), over=carriers) +\n default_if_empty(cost_investment_storage_cap, 0) +\n\
\ default_if_empty(cost_investment_source_cap, 0) +\n default_if_empty(cost_investment_area_use,
0) +\n default_if_empty(cost_investment_purchase, 0)\n ) * (1 + cost_om_annual_investment_fraction)\n\
\ + sum(cost_om_annual * flow_cap, over=carriers)\n)\n"
sub_expressions:
annualisation_weight:
- expression: sum(timestep_resolution * timestep_weights, over=timesteps) / 8760
depreciation_rate:
- where: cost_depreciation_rate
expression: cost_depreciation_rate
- where: NOT cost_depreciation_rate AND cost_interest_rate=0
expression: 1 / lifetime
- where: NOT cost_depreciation_rate AND cost_interest_rate>0
expression: (cost_interest_rate * ((1 + cost_interest_rate) ** lifetime)) / (((1
+ cost_interest_rate) ** lifetime) - 1)
cost¶
The total annualised costs of a technology, including installation and operation costs.
Used in:
Unit: cost
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ cost }\negthickspace \in \negthickspace\text{ costs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\text{if } (\exists (\textbf{cost\_investment}_\text{node,tech,cost,investstep}) \lor \exists (\textbf{cost\_var}_\text{node,tech,cost,timestep,investstep}))\!\!:\\[2em]
\quad \text{if } (\exists (\textbf{cost\_var}_\text{node,tech,cost,timestep,investstep}))\land{}(\exists (\textbf{cost\_investment}_\text{node,tech,cost,investstep}))\!\!:\\
\qquad \textbf{cost\_investment}_\text{node,tech,cost,investstep} + \sum\limits_{\text{timestep} \in \text{timesteps}} (\textbf{cost\_var}_\text{node,tech,cost,timestep,investstep})\\[2em]
\quad \text{if } (\exists (\textbf{cost\_var}_\text{node,tech,cost,timestep,investstep}))\land{}(\neg (\exists (\textbf{cost\_investment}_\text{node,tech,cost,investstep})))\!\!:\\
\qquad \sum\limits_{\text{timestep} \in \text{timesteps}} (\textbf{cost\_var}_\text{node,tech,cost,timestep,investstep})\\[2em]
\quad \text{if } (\neg (\exists (\textbf{cost\_var}_\text{node,tech,cost,timestep,investstep})))\land{}(\exists (\textbf{cost\_investment}_\text{node,tech,cost,investstep}))\!\!:\\
\qquad \textbf{cost\_investment}_\text{node,tech,cost,investstep}\\[2em]
\quad \text{if } (\neg (\exists (\textbf{cost\_var}_\text{node,tech,cost,timestep,investstep})))\land{}(\neg (\exists (\textbf{cost\_investment}_\text{node,tech,cost,investstep})))\!\!:\\
\qquad 0\\[2em]
\end{array}
\]
foreach:
- nodes
- techs
- costs
- investsteps
where: cost_investment OR cost_var
equations:
- expression: $cost_investment + $cost_var_sum
sub_expressions:
cost_investment:
- where: cost_investment
expression: cost_investment
- where: NOT cost_investment
expression: '0'
cost_var_sum:
- where: cost_var
expression: sum(cost_var, over=timesteps)
- where: NOT cost_var
expression: '0'
active: true
Decision Variables¶
flow_cap¶
A technology's flow capacity, also known as its nominal or nameplate capacity.
Used in:
- balance_demand_min_use
- flow_cap_new
- flow_capacity_systemwide_min
- cost_investment
- flow_capacity_min_purchase_milp
- flow_capacity_units_milp
- cost_investment_flow_cap
- source_availability_supply
- balance_supply_min_use
- flow_capacity_systemwide_max
- flow_in_max
- flow_out_max
- flow_capacity_per_storage_capacity_min
- ramping_up
- limit_flow_cap_new_max_rate
- balance_demand
- available_flow_cap_continuous
- available_flow_cap_max_binary_continuous_switch
- flow_capacity_per_storage_capacity_max
- flow_out_min
- ramping_down
- flow_capacity_max_purchase_milp
- symmetric_transmission
- flow_cap_bounding
- area_use_per_flow_capacity
- source_capacity_equals_flow_capacity
Unit: power
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\quad \textit{flow\_cap\_min} \leq \textbf{flow\_cap}_\text{node,tech,carrier,investstep}\\
\quad \textbf{flow\_cap}_\text{node,tech,carrier,investstep} \leq \textit{flow\_cap\_max}_\text{node,tech}\\
\end{array}
\]
link_flow_cap¶
A transmission technology's flow capacity, also known as its nominal or nameplate capacity.
Used in:
Unit: power
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{transmission})\!\!:\\[2em]
\quad 0 \leq \textbf{link\_flow\_cap}_\text{tech,investstep}\\
\quad \textbf{link\_flow\_cap}_\text{tech,investstep} \leq inf\\
\end{array}
\]
flow_out¶
The outflow of a technology per timestep, also known as the flow discharged (from storage technologies) or the flow received (by transmission technologies) on a link.
Used in:
- async_flow_out_milp
- flow_out_min_milp
- flow_out_inc_eff
- flow_out_max_milp
- system_balance
- flow_out_max
- export_balance
- cost_var
- flow_out_min
- ramping_up
- ramping_down
Unit: energy
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\exists (\textit{carrier\_out}_\text{node,tech,carrier}))\!\!:\\[2em]
\quad 0 \leq \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep}\\
\quad \textbf{flow\_out}_\text{node,tech,carrier,timestep,investstep} \leq inf\\
\end{array}
\]
flow_in¶
The inflow to a technology per timestep, also known as the flow consumed (by storage technologies) or the flow sent (by transmission technologies) on a link.
Used in:
- async_flow_in_milp
- flow_in_max_milp
- system_balance
- flow_in_max
- cost_var
- flow_in_inc_eff
- ramping_up
- ramping_down
Unit: energy
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\exists (\textit{carrier\_in}_\text{node,tech,carrier}))\!\!:\\[2em]
\quad 0 \leq \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep}\\
\quad \textbf{flow\_in}_\text{node,tech,carrier,timestep,investstep} \leq inf\\
\end{array}
\]
flow_export¶
The flow of a carrier exported outside the system boundaries by a technology per timestep.
Used in:
Unit: energy
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\exists (\textit{carrier\_export}))\!\!:\\[2em]
\quad 0 \leq \textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep}\\
\quad \textbf{flow\_export}_\text{node,tech,carrier,timestep,investstep} \leq inf\\
\end{array}
\]
area_use¶
The area in space utilised directly (e.g., solar PV panels) or indirectly (e.g., biofuel crops) by a technology.
Used in:
- balance_demand
- balance_demand_min_use
- cost_investment_area_use
- force_zero_area_use
- balance_supply_min_use
- source_availability_supply
- area_use_bounding
- area_use_new
- area_use_per_flow_capacity
- area_use_capacity_per_loc
Unit: area
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\exists (\textit{area\_use\_min}) \lor \exists (\textit{area\_use\_max}_\text{tech}) \lor \exists (\textit{area\_use\_per\_flow\_cap}) \lor \textit{sink\_unit}\mathord{=}\text{per\_area} \lor \textit{source\_unit}_\text{tech}\mathord{=}\text{per\_area})\!\!:\\[2em]
\quad \textit{area\_use\_min} \leq \textbf{area\_use}_\text{node,tech,investstep}\\
\quad \textbf{area\_use}_\text{node,tech,investstep} \leq \textit{area\_use\_max}_\text{tech}\\
\end{array}
\]
source_use¶
The carrier flow consumed from outside the system boundaries by a supply technology.
Used in:
- source_max
- balance_supply_no_storage
- balance_supply_min_use
- source_availability_supply
- cost_var
- balance_supply_with_storage
Unit: energy
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{supply})\!\!:\\[2em]
\quad 0 \leq \textbf{source\_use}_\text{node,tech,timestep,investstep}\\
\quad \textbf{source\_use}_\text{node,tech,timestep,investstep} \leq inf\\
\end{array}
\]
source_cap¶
The upper limit on a flow that can be consumed from outside the system boundaries by a supply technology in each timestep.
Used in:
- source_max
- cost_investment_source_cap
- source_cap_new
- source_cap_bounding
- source_capacity_equals_flow_capacity
Unit: power
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\textit{base\_tech}_\text{tech}\mathord{=}\text{supply})\!\!:\\[2em]
\quad \textit{source\_cap\_min} \leq \textbf{source\_cap}_\text{node,tech,investstep}\\
\quad \textbf{source\_cap}_\text{node,tech,investstep} \leq \textit{source\_cap\_max}\\
\end{array}
\]
storage_cap¶
The upper limit on a carrier that can be stored by a technology in any timestep.
Used in:
- storage_max
- balance_supply_with_storage
- storage_capacity_min_purchase_milp
- balance_storage
- storage_capacity_max_purchase_milp
- storage_discharge_depth_limit
- set_storage_initial
- storage_cap_bounding
- flow_capacity_per_storage_capacity_min
- storage_capacity_units_milp
- cost_investment_storage_cap
- storage_cap_new
- flow_capacity_per_storage_capacity_max
Unit: energy
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\textit{include\_storage}_\text{tech}\mathord{=}\text{true} \lor \textit{base\_tech}_\text{tech}\mathord{=}\text{storage})\!\!:\\[2em]
\quad \textit{storage\_cap\_min} \leq \textbf{storage\_cap}_\text{node,tech,investstep}\\
\quad \textbf{storage\_cap}_\text{node,tech,investstep} \leq \textit{storage\_cap\_max}_\text{tech}\\
\end{array}
\]
storage¶
The carrier stored by a storage technology in each timestep.
Used in:
- storage_max
- link_storage_level
- balance_storage
- storage_discharge_depth_limit
- set_storage_initial
- storage_capacity_units_milp
- balance_supply_with_storage
Unit: energy
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\textit{include\_storage}_\text{tech}\mathord{=}\text{true} \lor \textit{base\_tech}_\text{tech}\mathord{=}\text{storage})\!\!:\\[2em]
\quad 0 \leq \textbf{storage}_\text{node,tech,timestep,investstep}\\
\quad \textbf{storage}_\text{node,tech,timestep,investstep} \leq inf\\
\end{array}
\]
purchased_units¶
Integer number of a technology that has been purchased,
for any technology set to require integer capacity purchasing.
This is used to allow installation of fixed capacity units of technologies (
if flow_cap_max == flow_cap_min) and/or to set a fixed cost for a technology,
irrespective of its installed capacity.
On top of a fixed technology cost,
a continuous cost for the quantity of installed capacity can still be applied.
Since technology capacity is no longer a continuous decision variable, it is possible for these technologies to have a lower bound set on outflow/consumption which will only be enforced in those timesteps that the technology is operating. Otherwise, the same lower bound forces the technology to produce/consume that minimum amount of carrier in every timestep.
Used in:
- unit_capacity_max_systemwide_milp
- unit_capacity_min_systemwide_milp
- flow_capacity_max_purchase_milp
- flow_capacity_units_milp
- storage_capacity_min_purchase_milp
- available_flow_cap_max_binary_continuous_switch
- storage_capacity_max_purchase_milp
- unit_commitment_milp
- storage_capacity_units_milp
- flow_capacity_min_purchase_milp
- cost_investment_purchase
Unit: integer
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{Z}\;\!\!:\\[2em]
\text{if } (\textit{cap\_method}\mathord{=}\text{integer})\!\!:\\[2em]
\quad \textit{purchased\_units\_min} \leq \textbf{purchased\_units}_\text{node,tech,investstep}\\
\quad \textbf{purchased\_units}_\text{node,tech,investstep} \leq \textit{purchased\_units\_max}\\
\end{array}
\]
operating_units¶
Integer number of a technology that is operating in each timestep, for any technology set to require integer capacity purchasing.
Used in:
- flow_out_min_milp
- flow_export_max
- available_flow_cap_binary
- flow_capacity_units_milp
- available_flow_cap_max_binary_continuous_switch
- flow_in_max
- flow_out_max_milp
- flow_in_max_milp
- unit_commitment_milp
- flow_out_max
- flow_out_min
Unit: integer
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{Z}\;\!\!:\\[2em]
\text{if } (\textit{integer\_dispatch}\mathord{=}\text{true} \land \textit{cap\_method}\mathord{=}\text{integer})\!\!:\\[2em]
\quad 0 \leq \textbf{operating\_units}_\text{node,tech,timestep,investstep}\\
\quad \textbf{operating\_units}_\text{node,tech,timestep,investstep} \leq inf\\
\end{array}
\]
available_flow_cap¶
Flow capacity that will be set to zero if the technology is not operating in a given timestep and will be set to the value of the decision variable flow_cap otherwise.
Used in:
- available_flow_cap_binary
- flow_out_min_milp
- available_flow_cap_continuous
- available_flow_cap_max_binary_continuous_switch
Unit: power
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\textit{integer\_dispatch}\mathord{=}\text{true} \land \exists (\textit{flow\_cap\_max}_\text{node,tech}) \land \neg (\exists (\textit{flow\_cap\_per\_unit})))\!\!:\\[2em]
\quad 0 \leq \textbf{available\_flow\_cap}_\text{node,tech,carrier,timestep,investstep}\\
\quad \textbf{available\_flow\_cap}_\text{node,tech,carrier,timestep,investstep} \leq inf\\
\end{array}
\]
async_flow_switch¶
Binary switch to force asynchronous outflow/consumption of technologies with both flow_in and flow_out defined. This ensures that a technology with carrier flow efficiencies < 100% cannot produce and consume a flow simultaneously to remove unwanted carrier from the system.
Used in:
Unit: integer
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{Z}\;\!\!:\\[2em]
\text{if } (\textit{force\_async\_flow}\mathord{=}\text{true})\!\!:\\[2em]
\quad 0 \leq \textbf{async\_flow\_switch}_\text{node,tech,timestep,investstep}\\
\quad \textbf{async\_flow\_switch}_\text{node,tech,timestep,investstep} \leq 1\\
\end{array}
\]
unmet_demand¶
Virtual source of carrier flow to ensure model feasibility. This should only be considered a debugging rather than a modelling tool as it may distort the model in other ways due to the large impact it has on the objective function value. When present in a model in which it has been requested, it indicates an inability for technologies in the model to reach a sufficient combined supply capacity to meet demand.
Used in:
Unit: energy
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\text{config.ensure\_feasibility}\mathord{=}\text{true})\!\!:\\[2em]
\quad 0 \leq \textbf{unmet\_demand}_\text{node,carrier,timestep,investstep}\\
\quad \textbf{unmet\_demand}_\text{node,carrier,timestep,investstep} \leq inf\\
\end{array}
\]
unused_supply¶
Virtual sink of carrier flow to ensure model feasibility. This should only be considered a debugging rather than a modelling tool as it may distort the model in other ways due to the large impact it has on the objective function value. In model results, the negation of this variable is combined with unmet_demand and presented as only one variable: unmet_demand. When present in a model in which it has been requested, it indicates an inability for technologies in the model to reach a sufficient combined consumption capacity to meet required outflow (e.g. from renewables without the possibility of curtailment).
Used in:
Unit: energy
Default: 0
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ timestep }\negthickspace \in \negthickspace\text{ timesteps, }
\text{ investstep }\negthickspace \in \negthickspace\text{ investsteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\text{config.ensure\_feasibility}\mathord{=}\text{true})\!\!:\\[2em]
\quad -inf \leq \textbf{unused\_supply}_\text{node,carrier,timestep,investstep}\\
\quad \textbf{unused\_supply}_\text{node,carrier,timestep,investstep} \leq 0\\
\end{array}
\]
area_use_new¶
Additional area use commissioned in an investstep.
Used in:
Unit: area
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ vintagestep }\negthickspace \in \negthickspace\text{ vintagesteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\exists (\textbf{area\_use}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad 0 \leq \textbf{area\_use\_new}_\text{node,tech,vintagestep}\\
\quad \textbf{area\_use\_new}_\text{node,tech,vintagestep} \leq \textit{area\_use\_new\_max}\\
\end{array}
\]
flow_cap_new¶
Additional flow capacity commissioned in an investstep.
Used in:
Unit: power
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ carrier }\negthickspace \in \negthickspace\text{ carriers, }
\text{ vintagestep }\negthickspace \in \negthickspace\text{ vintagesteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\exists (\textbf{flow\_cap}_\text{node,tech,carrier,investstep}))\!\!:\\[2em]
\quad 0 \leq \textbf{flow\_cap\_new}_\text{node,tech,carrier,vintagestep}\\
\quad \textbf{flow\_cap\_new}_\text{node,tech,carrier,vintagestep} \leq \textit{flow\_cap\_new\_max}_\text{tech,vintagestep}\\
\end{array}
\]
source_cap_new¶
Additional source capacity commissioned in an investstep.
Used in:
Unit: power
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ vintagestep }\negthickspace \in \negthickspace\text{ vintagesteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\exists (\textbf{source\_cap}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad 0 \leq \textbf{source\_cap\_new}_\text{node,tech,vintagestep}\\
\quad \textbf{source\_cap\_new}_\text{node,tech,vintagestep} \leq \textit{source\_cap\_new\_max}\\
\end{array}
\]
storage_cap_new¶
Additional storage capacity commissioned in an investstep.
Used in:
Unit: energy
\[
\begin{array}{l}
\forall{}
\text{ node }\negthickspace \in \negthickspace\text{ nodes, }
\text{ tech }\negthickspace \in \negthickspace\text{ techs, }
\text{ vintagestep }\negthickspace \in \negthickspace\text{ vintagesteps }
\!\!,\\
\forall\mathbb{R}\;\!\!:\\[2em]
\text{if } (\exists (\textbf{storage\_cap}_\text{node,tech,investstep}))\!\!:\\[2em]
\quad 0 \leq \textbf{storage\_cap\_new}_\text{node,tech,vintagestep}\\
\quad \textbf{storage\_cap\_new}_\text{node,tech,vintagestep} \leq \textit{storage\_cap\_new\_max}\\
\end{array}
\]
Parameters¶
objective_cost_weights¶
Weightings for cost classes to apply in the objective function.
Used in:
Default: 1
investstep_resolution¶
Used in:
Default: 1
bigM¶
BigM is a large value used to define certain optimisation problems. See https://en.wikipedia.org/wiki/Big_M_method for more information. This value should be larger than the largest values that any decision variables can take, but should not be too large (i.e., do not set it greater than 3 orders of magnitude above the numeric range of the model). If too large, numerical problems may arise in the optimisation.
Used in:
Default: 1000000000.0
available_initial_cap¶
Used in:
Default: 0
base_tech¶
Should be the name of one of the abstract base classes, from which some initial parameter defaults will be derived and with which certain base math will be triggered.
Used in:
- balance_demand_min_use
- source_use
- flow_out_inc_eff
- cost_investment_flow_cap
- balance_supply_min_use
- link_flow_cap
- cost_var
- balance_demand
- source_cap
- balance_supply_no_storage
- balance_conversion
- balance_transmission
- storage
- balance_supply_with_storage
- storage_cap
- balance_storage
- symmetric_transmission
- flow_in_inc_eff
- cost_investment_purchase
carrier_out¶
Carrier(s) produced by this technology. Only transmission, conversion, storage, and supply technologies can define this parameter
Used in:
cost_flow_in¶
Cost per unit of flow_in in each timestep.
Used in:
Unit: \(\text{energy}^{-1}\).
Default: 0
cost_flow_out¶
Cost per unit of flow_in in each timestep.
Used in:
Unit: \(\text{energy}^{-1}\).
Default: 0
cost_interest_rate¶
Used when computing levelized costs and technology depreciation_rate (relative to lifetime).
Used in:
Unit: fraction
Default: 0
flow_cap_initial¶
Used in:
Default: 0
flow_cap_max¶
Limits flow_cap to a maximum.
Used in:
- available_flow_cap_binary
- force_zero_area_use
- flow_capacity_max_purchase_milp
- available_flow_cap_max_binary_continuous_switch
- available_flow_cap
Unit: power.
Default: inf
flow_cap_max_systemwide¶
Limits the sum of flow_cap over all nodes in the model to a maximum. If cap_method=integer, this will be scaled by the number of integer units of a technology purchased.
Used in:
Unit: power or \(\frac{\text{power}}{\text{unit}}\).
Default: inf
flow_out_eff¶
Conversion efficiency from the technology to sink/flow_out (tech dependent). Set as value between 1 (no loss) and 0 (all lost).
Used in:
Unit: fraction.
Default: 1.0
flow_ramping¶
limit maximum outflow / inflow / outflow - inflow (technology base class dependent) to a fraction of maximum capacity, which increases by that fraction at each timestep.
Used in:
Unit: \(\frac{\text{fraction}}{\text{hour}}\).
Default: 1.0
lifetime¶
Must be defined if fixed capital costs are defined. A reasonable value for many technologies is around 20-25 years.
Used in:
Unit: years.
Default: inf
carrier_in¶
Carrier(s) consumed by this technology. Only transmission, conversion, storage, and demand technologies can define this parameter
Used in:
area_use_initial¶
Used in:
Default: 0
area_use_max¶
If set to a finite value, limits the upper bound of the area_use decision variable to this value.
Used in:
Unit: \(\text{area}^{2}\).
Default: inf
flow_cap_per_storage_cap_max¶
ratio of maximum charge/discharge (kW) for a given storage capacity (kWh).
Used in:
Unit: \(\text{hour}^{-1}\)
Default: inf
flow_out_parasitic_eff¶
Additional losses as flow gets transferred from the plant to the carrier, e.g. due to plant parasitic consumption. Set as value between 1 (no loss) and 0 (all lost).
Used in:
Unit: fraction.
Default: 1.0
include_storage¶
When true, math will be triggered to allow discontinuous carrier inflow and outflows across timesteps.
Used in:
Default: False
source_cap_initial¶
Used in:
Default: 0
source_unit¶
Sets the unit of Source to either absolute (e.g. kWh), per_area (e.g. kWh/m2), or per_cap (e.g. kWh/kW). per_area uses the area_use decision variable to scale the source while per_cap uses the flow_cap decision variable.
Used in:
Default: absolute
storage_cap_initial¶
Used in:
Default: 0
storage_cap_max¶
Limit upper bound of storage_cap decision variable.
Used in:
Unit: energy.
Default: inf
storage_loss¶
Rate of storage loss per hour, used to calculate lost stored flow as (1 - storage_loss)^hours_per_timestep.
Used in:
Unit: \(\frac{\text{fraction}}{\text{hour}}\).
Default: 0
flow_in_eff¶
Conversion efficiency from source/flow_in (tech dependent) into the technology. Set as value between 1 (no loss) and 0 (all lost).
Used in:
Unit: fraction.
Default: 1.0
available_vintages¶
Used in:
- area_use_bounding
- flow_cap_bounding
- storage_cap_bounding
- source_cap_bounding
- limit_flow_cap_new_max_rate
Default: 0
cost_flow_cap¶
Cost per unit of the decision variable flow_cap.
Used in:
Unit: \(\text{power}^{-1}\).
Default: 0
cost_source_cap¶
Cost per unit source_cap.
Used in:
Unit: \(\text{power}^{-1}\).
Default: 0
cost_storage_cap¶
Cost per unit storage_cap, i.e., the maximum available capacity of the storage technology's "reservoir".
Used in:
Unit: \(\text{energy}^{-1}\).
Default: 0
distance¶
Used for ..._per_distance constraints. If not defined, it will be automatically derived from latitude/longitude of nodes in a link.
Used in:
Default: 1.0
sink_use_equals¶
Required amount of carrier removal from the system (e.g., electricity demand, transport distance). Unit dictated by source_unit.
Used in:
Default: nan
source_use_max¶
Maximum sink use to remove a carrier from the system (e.g., biofuel, coal, rainfall, wind flow). Unit dictated by source_unit.
Used in:
Default: inf
timestep_resolution¶
Used in:
- flow_out_min_milp
- source_max
- link_storage_level
- balance_storage
- flow_in_max
- flow_out_max_milp
- flow_in_max_milp
- ramping_up
- cost_investment
- set_storage_initial
- flow_out_max
- flow_out_min
- balance_supply_with_storage
- ramping_down
timestep_weights¶
Used in:
available_area¶
Used in:
area_use_min¶
Limits the lower bound of the area_use decision variable to this value.
Used in:
Unit: \(\text{area}^{2}\).
area_use_per_flow_cap¶
If set, forces area_use to follow flow_cap with the given numerical ratio (e.g. setting to 1.5 means that area_use == 1.5 * flow_cap).
Used in:
Unit: \(\frac{\text{area}^{2}}{\text{power}}\).
cap_method¶
One of 'continuous' (LP model) or 'integer' (integer/binary unit capacity).
Used in:
cost_area_use¶
Cost per unit area_use.
Used in:
Unit: \(\text{area}^{-2}\).
cost_depreciation_rate¶
Applied to "annualise" investment costs so they are comparable to variable costs. If not provided, this will be calculated using technology lifetime and cost_interest_rate.
Used in:
Unit: fraction.
cost_export¶
Cost per unit of flow_export in each timestep. Usually used in the negative sense, as a subsidy.
Used in:
Unit: \(\text{energy}^{-1}\).
cost_flow_cap_per_distance¶
Cost per unit of the decision variable flow_cap and per unit distance of a transmission link. Applied to transmission links only.
Used in:
Unit: \((\text{power}\times\text{distance})^{-1}\)
cost_om_annual¶
Annual costs applied per unit flow_cap. These costs are not subject to being recalculated relative to technology lifetime, only scaled to reflect the fraction of one year that the model represents (e.g., 7 days ~= 0.02 of a year).
Used in:
Unit: \(\text{power}^{-1}\).
cost_om_annual_investment_fraction¶
Add an additional cost to total investment costs (except cost_om_annual) that is a fraction of that total.
Used in:
Unit: fraction / total investment.
cost_purchase¶
Cost applied to the variable purchased_units. Requires the parameter cap_method to be integer.
Used in:
Unit: \(\text{purchased\_unit}^{-1}\)
cost_purchase_per_distance¶
Cost applied if the binary variable purchased is 1 or per unit of the integer variable units. Requires the parameter cap_method to be integer.
Used in:
Unit: \((\text{purchased\_units}\times\text{distance})^{-1}\)
cyclic_storage¶
If true, link storage levels in the last model timestep with the first model timestep. inter_cluster_storage custom math must be included if using time clustering and setting this to true. This must be set to false if using operate mode.
Used in:
Unit: boolean.
export_max¶
If carrier_export is defined, limit the allowed export of produced carrier for a technology.
Used in:
Unit: power.
flow_cap_min¶
Limits flow_cap to a minimum. NOTE: this will force flow_cap to a minimum value unless cap_method is set to integer. If cap_method=integer, this will be scaled by the number of integer units of a technology purchased.
Used in:
Unit: power or \(\frac{\text{power}}{\text{unit}}\).
flow_cap_min_systemwide¶
Limits the sum of flow_cap over all nodes in the model to a minimum. NOTE: this will force the sum of flow_cap to a minimum value unless cap_method is set to integer.
Used in:
Unit: power.
flow_cap_new_max_rate¶
Used in:
flow_cap_per_storage_cap_min¶
ratio of minimum charge/discharge (kW) for a given storage capacity (kWh).
Used in:
Unit: \(\text{hour}^{-1}\)
flow_cap_per_unit¶
Set the capacity of each integer unit of a technology purchased, if cap_method is integer.
Used in:
Unit: \(\frac{\text{power}}{\text{unit}}\).
flow_in_eff_per_distance¶
Total link efficiency will be calculated as \(\text{flow\_in\_eff}\times{}\text{flow\_in\_eff\_per\_distance}^\text{distance}\). Set as value between 1 (no loss) and 0 (all lost).
Used in:
flow_out_eff_per_distance¶
Total link efficiency will be calculated as \(\text{flow\_out\_eff}\times{}\text{flow\_out\_eff\_per\_distance}^\text{distance}\). Set as value between 1 (no loss) and 0 (all lost).
Used in:
flow_out_min_relative¶
Set to a value between 0 and 1 to force minimum flow_out as a fraction of the technology rated capacity. If non-zero and cap_method is continuous, this will force the technology to operate above its minimum value at every timestep.
Used in:
Unit: fraction.
force_async_flow¶
If True, non-zero flow_out and flow_in cannot both occur in the same timestep.
Used in:
integer_dispatch¶
When true, will limit per-timestep out/inflows relative to the number of units of a technology that are in operation.
Requires cap_method=integer.
Used in:
purchased_units_max_systemwide¶
sets the upper bound of the sum across all nodes of the decision variable units for a particular technology.
Used in:
Unit: integer.
purchased_units_min_systemwide¶
sets the lower bound of the sum across all nodes of the decision variable units for a particular technology.
Used in:
Unit: integer.
sink_unit¶
Sets the unit of Sink to either absolute x-unit: energy), per_area x-unit: energy/area), or per_cap x-unit: energy/power). per_area uses the area_use decision variable to scale the sink while per_cap uses the flow_cap decision variable.
Used in:
sink_use_max¶
Maximum sink use to remove a carrier from the system (e.g., electricity demand, transport distance). Unit dictated by source_unit.
Used in:
sink_use_min¶
Minimum sink use to remove a carrier from the system (e.g., electricity demand, transport distance). Unit dictated by source_unit.
Used in:
source_cap_equals_flow_cap¶
If true, the decision variables source_cap and flow_cap are forced to equal one another.
Used in:
source_eff¶
Conversion efficiency from the technology from source. Set as value between 1 (no loss) and 0 (all lost).
Used in:
Unit: fraction.
source_use_equals¶
Required amount of carrier removal from the system (e.g., biofuel, coal, rainfall, wind flow). Unit dictated by source_unit.
Used in:
source_use_min¶
Minimum source use to add a carrier from the system (e.g., biofuel, coal, rainfall, wind flow). Unit dictated by source_unit.
Used in:
storage_cap_min¶
Limit lower bound of storage_cap decision variable.
Used in:
Unit: energy.
storage_cap_per_unit¶
Set the storage capacity of each integer unit of a technology purchased.
Used in:
Unit: \(\frac{\text{energy}}{\text{unit}}\).
storage_discharge_depth¶
Defines the minimum level of storage state of charge, as a fraction of total storage capacity.
Used in:
Unit: fraction.
storage_initial¶
Set stored flow in device at the first timestep, as a fraction of total storage capacity.
Used in:
Unit: fraction.