Storage

Two components are available for modelling storage: Store and StorageUnit. See the Store Component and Storage Unit Component descriptions for an overview of the conceptual differences.

Stores¶

Stores have two time-dependent variables, the store dispatch \(h_{n,s,t}\) in MW and the store energy level \(e_{n,s,t}\) in MWh.

The store dispatch \(h_{n,s,t}\) is unconstrained, i.e. it can be positive (discharging) or negative (storing):

\[-\infty \leq h_{n,s,t} \leq +\infty\]

The store energy level \(e_{n,s,t}\) is constrained by maximum and minimum energy levels.

For non-extendable stores (e_nom_extendable=False), the energy level is constrained by:

Constraint	Dual Variable	Name
\(e_{n,s,t} \geq \underline{e}_{n,s,t} \hat{e}_{n,s}\)	`n.stores_t.mu_lower`	`Store-fix-e-lower`
\(e_{n,s,t} \leq \bar{e}_{n,s,t} \hat{e}_{n,s}\)	`n.stores_t.mu_upper`	`Store-fix-e-upper`

where \(\hat{e}_{n,s}\) is the nominal energy capacity, \(\underline{e}_{n,s,t}\) and \(\bar{e}_{n,s,t}\) are time-dependent restrictions on the energy level given per unit of nominal capacity.

These constraints are set in the function define_operational_constraints_for_non_extendables().

For extendable stores (e_nom_extendable=True), the energy level is constrained by:

Constraint	Dual Variable	Name
\(e_{n,s,t} \geq \underline{e}_{n,s,t} E_{n,s}\)	`n.stores_t.mu_lower`	`Store-ext-e-lower`
\(e_{n,s,t} \leq \bar{e}_{n,s,t} E_{n,s}\)	`n.stores_t.mu_upper`	`Store-ext-e-upper`

where \(E_{n,s}\) is the energy capacity to be optimised.

These constraints are set in the function [define_operational_constraints_for_extendables.

Capacity limits of Store components

For handling of capacity limits, see the Capacity Limits section.

The store energy level can also be fixed to a certain value \(\tilde{e}_{n,s,t}\):

Constraint	Dual Variable	Name
\(e_{n,s,t} = \tilde{e}_{n,s,t}\)	only in `n.model`	`Store-e_set`

These constraints are set in the function define_fixed_operation_constraints().

The power and energy variables are related by the storage consistency equation (Store-energy_balance):

\[e_{n,s,t} = \eta_{\textrm{stand},n,s}^{w_t^s} e_{n,s,t-1} - w_t^s h_{n,s,t} \quad \leftrightarrow \quad \lambda_{n,s,t}^\textrm{MSV}\]

where \(\eta_{\textrm{stand},n,s}\) represents the storage efficiency after accounting for standing losses (e.g. thermal losses in thermal storage) per hour and \(w_t^s\) is the snapshot weighting (defaulting to 1 hour). The dual variable \(\lambda_{n,s,t}^\textrm{MSV}\) represents the marginal storage value (also known as water value in the hydro-electricity literature).

These constraints are set in the function define_store_constraints().

Furthermore, there are two options for specifying the initial energy level \(e_{n,s,t=-1}\):

Set e_cyclic=False (default) and the value of e_initial in MWh.

\[e_{n,s,t=-1} = e_{n,s,\textrm{initial}}\]

Set e_cyclic=True and the optimisation sets the initial energy to be equal to the final energy level.

\[e_{n,s,t=-1} = e_{n,s,t=|T|-1}\]

Store cyclicity constraints with multiple investment periods

For how storage cyclicity constraints are handled with multiple investment periods, see the Pathway Planning section.

Mapping of symbols to component attributes

Symbol	Attribute	Type
\(h_{n,s,t}\)	`n.stores_t.p`	Decision Variable
\(e_{n,s,t}\)	`n.stores_t.e`	Decision Variable
\(E_{n,s}\)	`n.stores.e_nom_opt`	Decision Variable
\(\lambda_{n,s,t}^\textrm{MSV}\)	`n.stores_t.mu_energy_balance`	Dual Variable
\(\hat{e}_{n,s}\)	`n.stores.e_nom`	Parameter
\(\underline{e}_{n,s,t}\)	`n.stores_t.e_min_pu`	Parameter
\(\bar{e}_{n,s,t}\)	`n.stores_t.e_max_pu`	Parameter
\(\tilde{e}_{n,s,t}\)	`n.stores_t.e_set`	Parameter
\(\eta_{\textrm{stand},n,s}\)	`n.stores.standing_loss`	Parameter
\(w_t^s\)	`n.snapshot_weightings.stores`	Parameter

Storage Units¶

Storage units have three time-dependent variables, the discharge \(h_{n,s,t}^-\) in MW, the charge \(h_{n,s,t}^+\) in MW and the state of charge \(soc_{n,s,t}\) in MWh.

With a storage unit the nominal state of charge may not be independently optimised from the nominal power output (they are linked by the maximum hours parameter max_hours) and the nominal charging power is linked to the maximum discharging power.

For non-extendable storage units (p_nom_extendable=False), the charge (\(h_{n,s,t}^+\)), discharge (\(h_{n,s,t}^-\)), and state of charge (\(soc_{n,s,t}\)) variables are constrained by:

Constraint	Dual Variable	Name
\(h_{n,s,t}^- \geq 0\)	only in `n.model`	`StorageUnit-p_dispatch-lower`
\(h_{n,s,t}^- \leq \bar{h}_{n,s,t} \hat{h}_{n,s}\)	only in `n.model`	`StorageUnit-p_dispatch-upper`
\(h_{n,s,t}^+ \geq 0\)	only in `n.model`	`StorageUnit-p_store-lower`
\(h_{n,s,t}^+ \leq - \underline{h}_{n,s,t} \hat{h}_{n,s}\)	only in `n.model`	`StorageUnit-p_store-upper`
\(soc_{n,s,t} \geq 0\)	only in `n.model`	`StorageUnit-state_of_charge-lower`
\(soc_{n,s,t} \leq r_{n,s} \hat{h}_{n,s}\)	only in `n.model`	`StorageUnit-state_of_charge-upper`

where \(\hat{h}_{n,s}\) is the nominal power capacity, \(\bar{h}_{n,s,t}\) is a time-dependent restriction on the discharge per unit of nominal capacity, \(\underline{h}_{n,s,t}\) is a time-dependent restriction on the maximum charge per unit of nominal capacity (usually negative, -1 by default), and \(r_{n,s}\) is the number of hours at nominal power that fill the state of charge.

These constraints are set in the function define_operational_constraints_for_non_extendables().

For extendable storage units (p_nom_extendable=True), the charge and discharge variables are constrained by:

Constraint	Dual Variable	Name
\(h_{n,s,t}^- \geq 0\)	only in `n.model`	`StorageUnit-ext-p_dispatch-lower`
\(h_{n,s,t}^- \leq \bar{h}_{n,s,t}H_{n,s}\)	only in `n.model`	`StorageUnit-ext-p_dispatch-upper`
\(h_{n,s,t}^+ \geq 0\)	only in `n.model`	`StorageUnit-ext-p_store-lower`
\(h_{n,s,t}^+ \leq - \underline{h}_{n,s,t} H_{n,s}\)	only in `n.model`	`StorageUnit-ext-p_store-upper`

where \(H_{n,s}\) is the power capacity to be optimised.

These constraints are set in the function define_operational_constraints_for_extendables().

Capacity limits of StorageUnit components

For handling of capacity limits, see the Capacity Limits section.

All three variables can also be fixed to certain values \(\tilde{h}_{n,s,t}^-\), \(\tilde{h}_{n,s,t}^+\), and \(\tilde{soc}_{n,s,t}\):

Constraint	Dual Variable	Name
\(h_{n,s,t}^- = \tilde{h}_{n,s,t}^-\)	only in `n.model`	`StorageUnit-p_dispatch_set`
\(h_{n,s,t}^+ = \tilde{h}_{n,s,t}^+\)	only in `n.model`	`StorageUnit-p_store_set`
\(soc_{n,s,t} = \tilde{soc}_{n,s,t}\)	`n.storage_units_t.mu_state_of_charge_set`	`StorageUnit-state_of_charge_set`

An example use case would be if a storage unit must be empty and/or full every day.

These constraints are set in the function define_fixed_operation_constraints().

The charging, discharging and state of charge variables are related by (StorageUnit-energy_balance):

\[\begin{gather*}soc_{n,s,t} = \eta_{\textrm{stand};n,s}^{w_t^s} soc_{n,s,t-1}\\ + \eta_{\textrm{store};n,s} w_t^s h_{n,s,t}^+ - \eta^{-1}_{\textrm{dispatch};n,s} w_t^s h_{n,s,t}^- \\ + w_t^s\textrm{inflow}_{n,s,t} - w_t^s\textrm{spillage}_{n,s,t} \quad \leftrightarrow \quad \lambda_{n,s,t}^\textrm{MSV} \end{gather*}\]

\(\eta_{\textrm{stand};n,s}\) is the standing efficiency (e.g. due to thermal losses for thermal storage). \(\eta_{\textrm{store};n,s}\) and \(\eta_{\textrm{dispatch};n,s}\) are the efficiencies for power going into and out of the storage unit, and \(w_t^s\) is the snapshot weighting for stores (e.g. 1 hour). The dual variable \(\lambda_{n,s,t}^\textrm{MSV}\) represents the marginal storage value (also known as water value in the hydro-electricity literature).

These constraints are set in the function define_storage_unit_constraints().

Furthermore, there are two options for specifying the initial state of charge \(soc_{n,s,t=-1}\):

Set cyclic_state_of_charge=False (default) and the value of state_of_charge_initial in MWh.
Set cyclic_state_of_charge=True and the optimisation set the initial state of charge to the final state of charge.

\[soc_{n,s,t=-1} = soc_{n,s,t=|T|-1}\]

StorageUnit cyclicity constraints with multiple investment periods

For how store cyclicity constraints are handled with multiple investment periods, see the Pathway Planning section.

Mapping of symbols to component attributes

Symbol	Attribute	Type
\(h_{n,s,t}^+\)	`n.stores_t.p_dispatch`	Decision Variable
\(h_{n,s,t}^-\)	`n.stores_t.p_store`	Decision Variable
\(soc_{n,s,t}\)	`n.storage_units_t.state_of_charge`	Decision Variable
\(H_{n,s}\)	`n.storage_units.p_nom_opt`	Decision Variable
\(\textrm{inflow}_{n,s,t}\)	`n.storage_units_t.inflow`	Decision Variable
\(\textrm{spillage}_{n,s,t}\)	`n.storage_units_t.spillage`	Decision Variable
\(\lambda_{n,s,t}^\textrm{MSV}\)	`n.storage_units_t.mu_energy_balance`	Dual Variable
\(\bar{h}_{n,s}\)	`n.storage_units.p_nom`	Parameter
\(\underline{h}_{n,s,t}\)	`n.storage_units_t.p_min_pu`	Parameter
\(\bar{h}_{n,s,t}\)	`n.storage_units_t.p_max_pu`	Parameter
\(r_{n,s}\)	`n.storage_units.max_hours`	Parameter
\(\tilde{h}_{n,s,t}^-\)	`n.storage_units_t.p_dispatch_set`	Parameter
\(\tilde{h}_{n,s,t}^+\)	`n.storage_units_t.p_store_set`	Parameter
\(\tilde{soc}_{n,s,t}\)	`n.storage_units_t.state_of_charge_set`	Parameter
\(\eta_{\textrm{stand};n,s}\)	`n.storage_units.standing_loss`	Parameter
\(\eta_{\textrm{store};n,s}\)	`n.storage_units.store_efficiency`	Parameter
\(\eta_{\textrm{dispatch};n,s}\)	`n.storage_units.dispatch_efficiency`	Parameter
\(w_t^s\)	`n.snapshot_weightings.stores`	Parameter

Examples¶

Storage Units as Links & Stores

Shows how storage units can be replaced by more fundamental links and stores.

Go to example