Global Constraints¶

Global constraints apply to more than one component at once and are stored under a unique name in n.global_constraints. Several pre-defined types of global constraints are available (type). For instance, for defining emission limits, transmission expansion limits, or technology capacity expansion limits using the sense and constant attributes.

By default, global constraints span across all investment periods. For models with multiple investment periods, global constraints can be limited to affect only single investment period (e.g. an emission limit for a specific year) by specifying the in the attribute investment_period. The shadow price of each global constraint is stored in the attribute mu.

Naming Convention

Global constraints carry the name scheme GlobalConstraint-{name} in the Linopy model.

Primary Energy¶

The primary energy constraints (type="primary_energy") are used to limit byproducts of primary energy consumption of a carrier (e.g. CO₂ emissions) in generators, storage units and stores. They depend on the generator efficiency and carrier-specific attributes (e.g. co2_emissions).

Suppose there is a global constraint defined for CO₂ emissions (carrier_attribute) with sense "<=" (sense) and a limit \(\Gamma\) (constant). Emissions can come from generators whose energy carriers have CO₂ emissions, and from stores and storage units whose storage medium releases or absorbs CO₂ when it is converted. Only stores and storage units with non-cyclic state of charge that is different at the start (\(t=-1\)) and end (\(t=|T|-1\)) of the optimisation period contribute.

If the specific emissions of carrier \(s\) is \(\rho_s\) (n.carriers.co2_emissions) and the generator with carrier \(s\) at node \(n\) and snapshot \(t\) has efficiency \(\eta_{n,s,t}\) then the CO₂ constraint is

\[\begin{gather*}\sum_{n,s,t} w_t^g \cdot \eta_{n,s,t}^{-1} \cdot g_{n,s,t}\cdot \rho_s + \sum_{n,s}\left(e_{n,s,t=-1} - e_{n,s,t=|T|-1}\right) \cdot \rho_s\\ + \sum_{n,s}\left(soc_{n,s,t=-1} - soc_{n,s,t=|T|-1}\right) \cdot \rho_s \leq \Gamma \quad \leftrightarrow \quad \mu\end{gather*}\]

The first sum is over generators; the second sum is over stores; the third over storage units. The shadow price \(\mu\) would represent the CO₂ price in this case.

This global constraint is added in the function define_primary_energy_limit().

Mapping symbols to component attributes

Symbol	Attribute	Type
\(g_{n,s,t}\)	`n.generators_t.p`	Decision Variable
\(e_{n,s,t}\)	`n.stores_t.e`	Decision Variable
\(soc_{n,s,t}\)	`n.storage_units_t.soc`	Decision Variable
\(\mu\)	`n.global_constraints.mu`	Dual Variable
\(w_t^g\)	`n.snapshot_weightings.generators`	Parameter
\(\eta_{n,s,t}\)	`n.generators.efficiency`	Parameter
\(\rho_s\)	`n.carriers.co2_emissions`	Parameter
\(\Gamma\)	`n.global_constraints.constant`	Parameter

Operational Limit¶

The operational constraints can limit the net production of a carrier taking into account generator, storage units and stores (type="operational_limit"). For example, this can be used to limit the usage of gas in the system to a certain amount \(\Gamma\) (constant in MWh). With sense "<=" (sense), the constraint would be given by

\[\begin{gather*}\sum_{n,s,t} w_t^g \cdot g_{n,s,t}+ \sum_{n,s}\left(e_{n,s,t=-1} - e_{n,s,t=|T|-1}\right) \\ + \sum_{n,s}\left(soc_{n,s,t=-1} - soc_{n,s,t=|T|-1}\right) \leq \Gamma \quad \leftrightarrow \quad \mu\end{gather*}\]

The first sum is over generators; the second sum is over stores; the third over storage units. Structurally, it is similar to the primary energy limit, but without the consideration of specific emissions and efficiencies. The shadow price \(\mu\) (in currency/MWh) would represent the reduction in system costs if the operational limit were relaxed by one unit.

This global constraint is added in the function define_operational_limit().

Mapping symbols to component attributes

Symbol	Attribute	Type
\(g_{n,s,t}\)	`n.generators_t.p`	Decision Variable
\(e_{n,s,t}\)	`n.stores_t.e`	Decision Variable
\(soc_{n,s,t}\)	`n.storage_units_t.soc`	Decision Variable
\(\mu\)	`n.global_constraints.mu`	Dual Variable
\(w_t^g\)	`n.snapshot_weightings.generators`	Parameter
\(\Gamma\)	`n.global_constraints.constant`	Parameter

Volume Limit on Transmission Expansion¶

This global constraint can be used to limit the expansion volume in MWkm of transmission lines and links (type="transmission_volume_expansion_limit"). The carrier_attribute specifies the subset of carriers to consider. These can be individual carriers or concatenated by commas, e.g. "AC", "DC", "AC,DC", or a Link carrier such as "H2 pipeline". With sense="<=", the constraint is defined as

\[\sum_{l\in L_{\textrm{carriers}}} d_{l} F_{l} \leq \Gamma \quad \leftrightarrow \quad \mu\]

where \(L_{\textrm{carriers}}\) is the set of lines and links with the specified carriers, \(\Gamma\) is the maximum allowed volume expansion in MWkm, \(d_{l}\) is the distance of line or link \(l\) in km and \(F_{l}\) is the capacity of line or link \(l\) in MW. The shadow price \(\mu\) represents the marginal benefit of expanding the transmission capacity in currency/MWkm/a.

This global constraint is added in the function define_transmission_volume_expansion_limit().

Mapping symbols to component attributes

Symbol	Attribute	Type
\(F_l\)	`n.{lines,links}.p_nom_opt`	Decision Variable
\(\mu\)	`n.global_constraints.mu`	Dual Variable
\(d_l\)	`n.{lines,links}.length`	Parameter
\(\Gamma\)	`n.global_constraints.constant`	Parameter

Cost Limit on Transmission Expansion¶

This global constraint can be used to limit the total investment cost in currency/a of transmission lines and links (type="transmission_expansion_cost_limit"). The carrier_attribute specifies the subset of carriers to consider. These can be individual carriers or concatenated by commas, e.g. "AC", "DC", "AC,DC", or a Link carrier such as "H2 pipeline". With sense="<=", the constraint is defined as

\[\sum_{l\in L_{\textrm{carriers}}} c_{l} F_{l} \leq \Gamma \quad \leftrightarrow \quad \mu\]

where \(L_{\textrm{carriers}}\) is the set of lines and links with the specified carriers, \(c_{l}\) is the capital cost of line or link \(l\) in currency/MW/a, \(F_{l}\) is the capacity of line or link \(l\) in MW and \(\Gamma\) is the maximum allowed cost of line expansion in currency/a. The shadow price \(mu\) represents how much the total system cost could be reduced if the spending limit was increased by one currency/a.

This global constraint is added in the function define_transmission_expansion_cost_limit().

Mapping symbols to component attributes

Symbol	Attribute	Type
\(F_l\)	`n.{lines,links}.p_nom_opt`	Decision Variable
\(\mu\)	`n.global_constraints.mu`	Dual Variable
\(c_l\)	`n.{lines,links}.capital_cost`	Parameter
\(\Gamma\)	`n.global_constraints.constant`	Parameter

Expansion Limit¶

This global constraint can be used to limit the total capacity of components of a carrier (type=tech_capacity_expansion_limit). This global constraint can be specific to an investment period by setting the investment_period attribute and specific to a bus by setting the bus attribute. This constraint is mainly used for networks with multiple investment periods, where land usage and building rate restrictions need to be applied for a range of active components of a particular carrier (carrier_attribute) in a certain region.

Warning

Currently, only the capacities of extendable components are considered, i.e. generators, storage units and stores with extendable=True. The capacities of non-extendable components are not considered in this constraint.

For example, the capacities of all onshore wind generators (carrier_attribute="onshore wind") at a certain bus (bus="DE") should be smaller (sense="<=") than a hypothetical technical potential of 200 GW for onshore wind in the specific region (constant=200e3). Then the technology capacity expansion constraint across all active components is given by

\[\sum_{s | b_s<=a<b_s+L_s} G_{n,s} \leq \Gamma \quad a \in A \quad \leftrightarrow \quad \mu\]

where \(A\) are the investment periods, \(s\) are all extendable generators of the specified carrier, \(b_s\) is the build year of an asset \(s\) with lifetime \(L_s\). In this example, the shadow price \(\mu\) would represent the marginal benefit of expanding the capacity of onshore wind in currency/MW/a.

In general, the constraint would iterate over all investment variables for generators \(G_{n,s}\), lines and transformers \(P_{l}\), links \(F_{l}\), stores \(E_{n,s}\) and storage units \(H_{n,s}\) for the specified carrier and bus. For components connecting two buses, the bus selection is done by bus0.

This global constraint is added in the function define_tech_capacity_expansion_limit().

Mapping symbols to component attributes

Symbol	Attribute	Type
\(G_{n,s}\)	`n.generators.p_nom_opt`	Decision Variable
\(H_{n,s}\)	`n.storage_units.p_nom_opt`	Decision Variable
\(E_{n,s}\)	`n.stores.e_nom_opt`	Decision Variable
\(F_l\)	`n.links.p_nom_opt`	Decision Variable
\(P_l\)	`n.{lines,transformers}.s_nom_opt`	Decision Variable
\(\mu\)	`n.global_constraints.mu`	Dual Variable
\(b_s\)	`n.{<component>}.build_year`	Parameter
\(L_s\)	`n.{<component>}.lifetime`	Parameter
\(a\), \(A\)	`n.investment_periods`	Parameter
\(\Gamma\)	`n.global_constraints.constant`	Parameter

Growth Limit per Carrier¶

This carrier-specific constraint type implements absolute and relative growth limits per carrier which constrains new installed capacities for each investment period. It can be defined by providing the attributes n.carriers.max_growth and n.carriers.max_relative_growth.

Suppose the absolute growth limit for a specific carrier \(s\) is \(\Gamma_s\) (max_growth) and the relative growth limit is \(\gamma_s\) (max_relative_growth). With the growth limit constraint, for each investment period \(a\), the new installed capacity \(G_{a,s}\) of all components with carrier \(s\) is limited by:

\[G_{a,s} \leq \gamma_s \cdot G_{a-1,s} + \Gamma_s \quad \forall a, s \quad \leftrightarrow \quad \mu_{a,s}\]

where \(G_{a,s}\) represents the sum of all newly built nominal capacities of extendable components with carrier \(s\) in period \(a\). The relative growth limit \(\gamma_s\) allows the new capacity in period \(a\) to be proportional to the capacity added in the previous period \(a-1\).

This constraint only applies to networks with multiple investment periods and only considers components that are newly activated in each period. For the constraint to take effect, at least one of the attributes max_growth or max_relative_growth must be set to a finite value for the carrier.

Warning

The relative and abolute growth limits are additive, i.e. the absolute growth limit applies in addition to the relative growth limit.

In general, the constraint would iterate over all investment variables for generators \(G_{n,s}\), lines and transformers \(P_{l}\), links \(F_{l}\), stores \(E_{n,s}\) and storage units \(H_{n,s}\) for the specified carrier. For components connecting two buses, the bus selection is done by bus0.

This global constraint is added in the function define_growth_limit() and carries the name Carrier-growth_limit.

Mapping symbols to component attributes

Symbol	Attribute	Type
\(G_{n,s}\)	`n.generators.p_nom_opt`	Decision Variable
\(H_{n,s}\)	`n.storage_units.p_nom_opt`	Decision Variable
\(E_{n,s}\)	`n.stores.e_nom_opt`	Decision Variable
\(F_l\)	`n.links.p_nom_opt`	Decision Variable
\(P_l\)	`n.{lines,transformers}.s_nom_opt`	Decision Variable
\(\gamma_s\)	`n.carriers.max_relative_growth`	Parameter
\(\Gamma_s\)	`n.carriers.max_growth`	Parameter