Co-located VRE-Storage Module

elec_vre_stor!(EP::Model, inputs::Dict)

This function defines the decision variables, expressions, and constraints for the electrolyzer component of each co-located ELC, VRE, and storage generator.

The total electrolyzer capacity of each resource is defined as the sum of the existing electrolyzer capacity plus the newly invested electrolyzer capacity minus any retired electrolyzer capacity:

\[\begin{aligned} & \Delta^{total,elec}_{y,z} = (\overline{\Delta^{elec}_{y,z}}+\Omega^{elec}_{y,z}-\Delta^{elec}_{y,z}) \quad \forall y \in \mathcal{VS}^{elec}, z \in \mathcal{Z} \end{aligned}\]

One cannot retire more energy capacity than existing elec capacity:

\[\begin{aligned} &\Delta^{elec}_{y,z} \leq \overline{\Delta^{elec}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{elec}, z \in \mathcal{Z} \end{aligned}\]

For resources where $\overline{\Omega_{y,z}^{elec}}$ and $\underline{\Omega_{y,z}^{elec}}$ are defined, then we impose constraints on minimum and maximum energy capacity:

\[\begin{aligned} & \Delta^{total,elec}_{y,z} \leq \overline{\Omega}^{elec}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{elec}, z \in \mathcal{Z} \\ & \Delta^{total,elec}_{y,z} \geq \underline{\Omega}^{elec}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{elec}, z \in \mathcal{Z} \end{aligned}\]

Constraint 2 applies ramping constraints on electrolyzers where consumption of electricity by electrolyzer $y$ in time $t$ is denoted by $\Pi_{y,z}$ and the rampping constraints are denoated by $\kappa_{y}$.

\[\begin{aligned} \Pi_{y,t-1} - \Pi_{y,t} \leq \kappa_{y}^{down} \Delta^{\text{total}}_{y}, \hspace{1cm} \forall y \in \mathcal{EL}, \forall t \in \mathcal{T} \end{aligned}\]

\[\begin{aligned} \Pi_{y,t} - \Pi_{y,t-1} \leq \kappa_{y}^{up} \Delta^{\text{total}}_{y} \hspace{1cm} \forall y \in \mathcal{EL}, \forall t \in \mathcal{T} \end{aligned}\]

In constraint 3, electrolyzers are bound by the following limits on maximum and minimum power output. Maximum power output is 100% in this case.

\[\begin{aligned} \Pi_{y,t} \geq \rho^{min}_{y} \times \Delta^{total}_{y} \hspace{1cm} \forall y \in \mathcal{EL}, \forall t \in \mathcal{T} \end{aligned}\]

\[\begin{aligned} \Theta_{y,t} \leq \Pi^{total}_{y} \hspace{1cm} \forall y \in \mathcal{EL}, \forall t \in \mathcal{T} \end{aligned}\]

The regional demand requirement is included in electrolyzer.jl

inverter_vre_stor!(EP::Model, inputs::Dict, setup::Dict)

This function defines the decision variables, expressions, and constraints for the inverter component of each co-located VRE and storage generator.

The total inverter capacity of each resource is defined as the sum of the existing inverter capacity plus the newly invested inverter capacity minus any retired inverter capacity:

\[\begin{aligned} & \Delta^{total, inv}_{y,z} = (\overline{\Delta^{inv}_{y,z}} + \Omega^{inv}_{y,z} - \Delta^{inv}_{y,z}) \quad \forall y \in \mathcal{VS}^{inv}, z \in \mathcal{Z} \end{aligned}\]

One cannot retire more inverter capacity than existing inverter capacity:

\[\begin{aligned} & \Delta^{inv}_{y,z} \leq \overline{\Delta^{inv}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{inv}, z \in \mathcal{Z} \end{aligned}\]

For resources where $\overline{\Omega^{inv}_{y,z}}$ and $\underline{\Omega^{inv}_{y,z}}$ are defined, then we impose constraints on minimum and maximum capacity:

\[\begin{aligned} & \Delta^{total, inv}_{y,z} \leq \overline{\Omega^{inv}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{inv}, z \in \mathcal{Z} \\ & \Delta^{total, inv}_{y,z} \geq \underline{\Omega^{inv}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{inv}, z \in \mathcal{Z} \end{aligned}\]

The last constraint ensures that the maximum DC grid exports and imports must be less than the inverter capacity. Without any capacity reserve margin or operating reserves, the constraint is:

\[\begin{aligned} & \eta^{inverter}_{y,z} \times (\Theta^{pv}_{y,z,t} + \Theta^{dc}_{y,z,t}) + \frac{\Pi_{y,z,t}^{dc}}{\eta^{inverter}_{y,z}} \leq \Delta^{total, inv}_{y,z} \quad \forall y \in \mathcal{VS}^{inv}, \forall z \in \mathcal{Z}, \forall t \in \mathcal{T} \end{aligned}\]

With only capacity reserve margins, the maximum DC grid exports and imports constraint becomes:

\[\begin{aligned} & \eta^{inverter}_{y,z} \times (\Theta^{pv}_{y,z,t} + \Theta^{dc}_{y,z,t} + \Theta^{CRM,dc}_{y,z,t}) + \frac{\Pi_{y,z,t}^{dc} + \Pi_{y,z,t}^{CRM,dc}}{\eta^{inverter}_{y,z}} \\ & \leq \Delta^{total, inv}_{y,z} \quad \forall y \in \mathcal{VS}^{inv}, \forall z \in \mathcal{Z}, \forall t \in \mathcal{T} \end{aligned}\]

With only operating reserves, the maximum DC grid exports and imports constraint becomes:

\[\begin{aligned} & \eta^{inverter}_{y,z} \times (\Theta^{pv}_{y,z,t} + \Theta^{dc}_{y,z,t} + f^{pv}_{y,z,t} + r^{pv}_{y,z,t} + f^{dc,dis}_{y,z,t} + r^{dc,dis}_{y,z,t}) + \frac{\Pi_{y,z,t}^{dc} + f^{dc,cha}_{y,z,t}}{\eta^{inverter}_{y,z}} \\ & \leq \Delta^{total, inv}_{y,z} \quad \forall y \in \mathcal{VS}^{inv}, \forall z \in \mathcal{Z}, \forall t \in \mathcal{T} \end{aligned}\]

With both capacity reserve margins and operating reserves, the maximum DC grid exports and imports constraint becomes:

\[\begin{aligned} & \eta^{inverter}_{y,z} \times (\Theta^{pv}_{y,z,t} + \Theta^{dc}_{y,z,t} + \Theta^{CRM,dc}_{y,z,t} + f^{pv}_{y,z,t} + r^{pv}_{y,z,t} + f^{dc,dis}_{y,z,t} + r^{dc,dis}_{y,z,t}) \\ & + \frac{\Pi_{y,z,t}^{dc} + \Pi_{y,z,t}^{CRM,dc} + f^{dc,cha}_{y,z,t}}{\eta^{inverter}_{y,z}} \leq \Delta^{total, inv}_{y,z} \quad \forall y \in \mathcal{VS}^{inv}, \forall z \in \mathcal{Z}, \forall t \in \mathcal{T} \end{aligned}\]

In addition, this function adds investment and fixed O&M related costs related to the inverter capacity to the objective function:

\[\begin{aligned} & \sum_{y \in \mathcal{VS}^{inv}} \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST, inv}_{y,z} \times \Omega^{inv}_{y,z}) + (\pi^{FOM, inv}_{y,z} \times \Delta^{total,inv}_{y,z})\right) \end{aligned}\]

investment_charge_vre_stor!(EP::Model, inputs::Dict, setup::Dict)

This function activates the decision variables and constraints for asymmetric storage resources (independent charge and discharge power capacities (any STOR flag = 2)). For asymmetric storage resources, the function is enabled so charging and discharging can occur either through DC or AC capabilities. For example, a storage resource can be asymmetrically charged and discharged via DC capabilities or a storage resource could be charged via AC capabilities and discharged through DC capabilities. This module is configured such that both AC and DC charging (or discharging) cannot simultaneously occur.

The total charge/discharge DC and AC capacities of each resource are defined as the sum of the existing charge/discharge DC and AC capacities plus the newly invested charge/discharge DC and AC capacities minus any retired charge/discharge DC and AC capacities:

\[\begin{aligned} & \Delta^{total,dc,dis}_{y,z} =(\overline{\Delta^{dc,dis}_{y,z}}+\Omega^{dc,dis}_{y,z}-\Delta^{dc,dis}_{y,z}) \quad \forall y \in \mathcal{VS}^{asym,dc,dis}, z \in \mathcal{Z} \\ & \Delta^{total,dc,cha}_{y,z} =(\overline{\Delta^{dc,cha}_{y,z}}+\Omega^{dc,cha}_{y,z}-\Delta^{dc,cha}_{y,z}) \quad \forall y \in \mathcal{VS}^{asym,dc,cha}, z \in \mathcal{Z} \\ & \Delta^{total,ac,dis}_{y,z} =(\overline{\Delta^{ac,dis}_{y,z}}+\Omega^{ac,dis}_{y,z}-\Delta^{ac,dis}_{y,z}) \quad \forall y \in \mathcal{VS}^{asym,ac,dis}, z \in \mathcal{Z} \\ & \Delta^{total,ac,cha}_{y,z} =(\overline{\Delta^{ac,cha}_{y,z}}+\Omega^{ac,cha}_{y,z}-\Delta^{ac,cha}_{y,z}) \quad \forall y \in \mathcal{VS}^{asym,ac,cha}, z \in \mathcal{Z} \end{aligned}\]

One cannot retire more capacity than existing capacity:

\[\begin{aligned} &\Delta^{dc,dis}_{y,z} \leq \overline{\Delta^{dc,dis}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,dc,dis}, z \in \mathcal{Z} \\ &\Delta^{dc,cha}_{y,z} \leq \overline{\Delta^{dc,cha}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,dc,cha}, z \in \mathcal{Z} \\ &\Delta^{ac,dis}_{y,z} \leq \overline{\Delta^{ac,dis}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,ac,dis}, z \in \mathcal{Z} \\ &\Delta^{ac,cha}_{y,z} \leq \overline{\Delta^{ac,cha}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,ac,cha}, z \in \mathcal{Z} \end{aligned}\]

For resources where $\overline{\Omega_{y,z}^{dc,dis}}, \overline{\Omega_{y,z}^{dc,cha}}, \overline{\Omega_{y,z}^{ac,dis}}, \overline{\Omega_{y,z}^{ac,cha}}$ and $\underline{\Omega_{y,z}^{dc,dis}}, \underline{\Omega_{y,z}^{dc,cha}}, \underline{\Omega_{y,z}^{ac,dis}}, \underline{\Omega_{y,z}^{ac, cha}}$ are defined, then we impose constraints on minimum and maximum charge/discharge DC and AC power capacity:

\[\begin{aligned} & \Delta^{total,dc,dis}_{y,z} \leq \overline{\Omega}^{dc,dis}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,dc,dis}, z \in \mathcal{Z} \\ & \Delta^{total,dc,dis}_{y,z} \geq \underline{\Omega}^{dc,dis}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,dc,dis}, z \in \mathcal{Z} \\ & \Delta^{total,dc,cha}_{y,z} \leq \overline{\Omega}^{dc,cha}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,dc,cha}, z \in \mathcal{Z} \\ & \Delta^{total,dc,cha}_{y,z} \geq \underline{\Omega}^{dc,cha}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,dc,cha}, z \in \mathcal{Z} \\ & \Delta^{total,ac,dis}_{y,z} \leq \overline{\Omega}^{ac,dis}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,ac,dis}, z \in \mathcal{Z} \\ & \Delta^{total,ac,dis}_{y,z} \geq \underline{\Omega}^{ac,dis}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,ac,dis}, z \in \mathcal{Z} \\ & \Delta^{total,ac,cha}_{y,z} \leq \overline{\Omega}^{ac,cha}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,ac,cha}, z \in \mathcal{Z} \\ & \Delta^{total,ac,cha}_{y,z} \geq \underline{\Omega}^{ac,cha}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{asym,ac,cha}, z \in \mathcal{Z} \\ \end{aligned}\]

Furthermore, for storage technologies with asymmetric charge and discharge capacities (all $y \in \mathcal{VS}^{asym,dc,dis}, y \in \mathcal{VS}^{asym,dc,cha}, y \in \mathcal{VS}^{asym,ac,dis}, y \in \mathcal{VS}^{asym,ac,cha}$), the charge rate, $\Pi^{dc}_{y,z,t}, \Pi^{ac}_{y,z,t}$, is constrained by the total installed charge capacity, $\Delta^{total,dc,cha}_{y,z}, \Delta^{total,ac,cha}_{y,z}$. Similarly the discharge rate, $\Theta^{dc}_{y,z,t}, \Theta^{ac}_{y,z,t}$, is constrained by the total installed discharge capacity, $\Delta^{total,dc,dis}_{y,z}, \Delta^{total,ac,dis}_{y,z}$. Without any activated capacity reserve margin policies or operating reserves, the constraints are as follows:

\[\begin{aligned} & \Theta^{dc}_{y,z,t} \leq \Delta^{total,dc,dis}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,dc,dis}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Pi^{dc}_{y,z,t} \leq \Delta^{total,dc,cha}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,dc,cha}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Theta^{ac}_{y,z,t} \leq \Delta^{total,ac,dis}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,ac,dis}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Pi^{ac}_{y,z,t} \leq \Delta^{total,ac,cha}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,ac,cha}, z \in \mathcal{Z}, t \in \mathcal{T} \\ \end{aligned}\]

Adding only the capacity reserve margin constraints, the asymmetric charge and discharge DC and AC rates plus the 'virtual' charge and discharge DC and AC rates are constrained by the total installed charge and discharge DC and AC capacities:

\[\begin{aligned} & \Theta^{dc}_{y,z,t} + \Theta^{CRM,dc}_{y,z,t} \leq \Delta^{total,dc,dis}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,dc,dis}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Pi^{dc}_{y,z,t} + \Pi^{CRM,dc}_{y,z,t} \leq \Delta^{total,dc,cha}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,dc,cha}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Theta^{ac}_{y,z,t} + \Theta^{CRM,ac}_{y,z,t} \leq \Delta^{total,ac,dis}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,ac,dis}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Pi^{ac}_{y,z,t} + \Pi^{CRM,ac}_{y,z,t} \leq \Delta^{total,ac,cha}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,ac,cha}, z \in \mathcal{Z}, t \in \mathcal{T} \\ \end{aligned}\]

Adding only the operating reserve constraints, the asymmetric charge and discharge DC and AC rates plus the contributions to frequency regulation and operating reserves (both DC and AC) are constrained by the total installed charge and discharge DC and AC capacities:

\[\begin{aligned} & \Theta^{dc}_{y,z,t} + f^{dc,dis}_{y,z,t} + r^{dc,dis}_{y,z,t} \leq \Delta^{total,dc,dis}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,dc,dis}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Pi^{dc}_{y,z,t} + f^{dc,cha}_{y,z,t} \leq \Delta^{total,dc,cha}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,dc,cha}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Theta^{ac}_{y,z,t} + f^{ac,dis}_{y,z,t} + r^{ac,dis}_{y,z,t} \leq \Delta^{total,ac,dis}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,ac,dis}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Pi^{ac}_{y,z,t} + f^{ac,cha}_{y,z,t} \leq \Delta^{total,ac,cha}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,ac,cha}, z \in \mathcal{Z}, t \in \mathcal{T} \\ \end{aligned}\]

With both capacity reserve margin and operating reserve constraints, the asymmetric charge and discharge DC and AC rate constraints follow:

\[\begin{aligned} & \Theta^{dc}_{y,z,t} + \Theta^{CRM,dc}_{y,z,t} + f^{dc,dis}_{y,z,t} + r^{dc,dis}_{y,z,t} \leq \Delta^{total,dc,dis}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,dc,dis}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Pi^{dc}_{y,z,t} + \Pi^{CRM,dc}_{y,z,t} + f^{dc,cha}_{y,z,t} \leq \Delta^{total,dc,cha}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,dc,cha}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Theta^{ac}_{y,z,t} + \Theta^{CRM,ac}_{y,z,t} + f^{ac,dis}_{y,z,t} + r^{ac,dis}_{y,z,t} \leq \Delta^{total,ac,dis}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,ac,dis}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Pi^{ac}_{y,z,t} + \Pi^{CRM,ac}_{y,z,t} + f^{ac,cha}_{y,z,t} \leq \Delta^{total,ac,cha}_{y,z} \quad \forall y \in \mathcal{VS}^{asym,ac,cha}, z \in \mathcal{Z}, t \in \mathcal{T} \\ \end{aligned}\]

In addition, this function adds investment and fixed O&M costs related to charge/discharge AC and DC capacities to the objective function:

\[\begin{aligned} & \sum_{y \in \mathcal{VS}^{asym,dc,dis} } \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST,dc,dis}_{y,z} \times \Omega^{dc,dis}_{y,z}) + (\pi^{FOM,dc,dis}_{y,z} \times \Delta^{total,dc,dis}_{y,z})\right) \\ & + \sum_{y \in \mathcal{VS}^{asym,dc,cha} } \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST,dc,cha}_{y,z} \times \Omega^{dc,cha}_{y,z}) + (\pi^{FOM,dc,cha}_{y,z} \times \Delta^{total,dc,cha}_{y,z})\right) \\ & + \sum_{y \in \mathcal{VS}^{asym,ac,dis} } \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST,ac,dis}_{y,z} \times \Omega^{ac,dis}_{y,z}) + (\pi^{FOM,ac,dis}_{y,z} \times \Delta^{total,ac,dis}_{y,z})\right) \\ & + \sum_{y \in \mathcal{VS}^{asym,ac,cha} } \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST,ac,cha}_{y,z} \times \Omega^{ac,cha}_{y,z}) + (\pi^{FOM,ac,cha}_{y,z} \times \Delta^{total,ac,cha}_{y,z})\right) \end{aligned}\]

lds_vre_stor!(EP::Model, inputs::Dict)

This function defines the decision variables, expressions, and constraints for any long duration energy storage component of each co-located VRE and storage generator ( there is more than one representative period and LDS_VRE_STOR=1 in the Vre_and_stor_data.csv).

These constraints follow the same formulation that is outlined by the function long_duration_storage!() in the storage module. One constraint changes, which links the state of charge between the start of periods for long duration energy storage resources because there are options to charge and discharge these resources through AC and DC capabilities. The main linking constraint changes to:

\[\begin{aligned} & \Gamma_{y,z,(m-1)\times \tau^{period}+1 } =\left(1-\eta_{y,z}^{loss}\right) \times \left(\Gamma_{y,z,m\times \tau^{period}} -\Delta Q_{y,z,m}\right) \\ & - \frac{\Theta^{dc}_{y,z,(m-1) \times \tau^{period}+1}}{\eta_{y,z}^{discharge,dc}} - \frac{\Theta^{ac}_{y,z,(m-1)\times \tau^{period}+1}}{\eta_{y,z}^{discharge,ac}} \\ & + \eta_{y,z}^{charge,dc} \times \Pi^{dc}_{y,z,(m-1)\times \tau^{period}+1} + \eta_{y,z}^{charge,ac} \times \Pi^{ac}_{y,z,(m-1)\times \tau^{period}+1} \quad \forall y \in \mathcal{VS}^{LDES}, z \in \mathcal{Z}, m \in \mathcal{M} \end{aligned}\]

The rest of the long duration energy storage constraints are copied and applied to the co-located VRE and storage module for any long duration energy storage resources $y \in \mathcal{VS}^{LDES}$ from the long-duration storage module. Capacity reserve margin constraints for long duration energy storage resources are further elaborated upon in vre_stor_capres!().

solar_vre_stor!(EP::Model, inputs::Dict, setup::Dict)

This function defines the decision variables, expressions, and constraints for the solar PV component of each co-located VRE and storage generator.

The total solar PV capacity of each resource is defined as the sum of the existing solar PV capacity plus the newly invested solar PV capacity minus any retired solar PV capacity:

\[\begin{aligned} & \Delta^{total, pv}_{y,z} = (\overline{\Delta^{pv}_{y,z}} + \Omega^{pv}_{y,z} - \Delta^{pv}_{y,z}) \quad \forall y \in \mathcal{VS}^{pv}, z \in \mathcal{Z} \end{aligned}\]

One cannot retire more solar PV capacity than existing solar PV capacity:

\[\begin{aligned} & \Delta^{pv}_{y,z} \leq \overline{\Delta^{pv}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{pv}, z \in \mathcal{Z} \end{aligned}\]

For resources where $\overline{\Omega^{pv}_{y,z}}$ and $\underline{\Omega^{pv}_{y,z}}$ are defined, then we impose constraints on minimum and maximum capacity:

\[\begin{aligned} & \Delta^{total, pv}_{y,z} \leq \overline{\Omega^{pv}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{pv}, z \in \mathcal{Z} \\ & \Delta^{total, pv}_{y,z} \geq \underline{\Omega^{pv}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{pv}, z \in \mathcal{Z} \end{aligned}\]

If there is a fixed ratio for capacity (rather than co-optimizing interconnection sizing) of solar PV built to capacity of inverter built ($\eta_{y,z}^{ILR,pv}$), also known as the inverter loading ratio, then we impose the following constraint:

\[\begin{aligned} & \Delta^{total, pv}_{y, z} = \eta^{ILR,pv}_{y, z} \times \Delta^{total, inv}_{y, z} \quad \forall y \in \mathcal{VS}^{pv}, \forall z \in Z \end{aligned}\]

The last constraint defines the maximum power output in each time step from the solar PV component. Without any operating reserves, the constraint is:

\[\begin{aligned} & \Theta^{pv}_{y, z, t} \leq \rho^{max, pv}_{y, z, t} \times \Delta^{total,pv}_{y, z} \quad \forall y \in \mathcal{VS}^{pv}, \forall z \in Z, \forall t \in T \end{aligned}\]

With operating reserves, the maximum power output in each time step from the solar PV component must account for procuring some of the available capacity for frequency regulation ($f^{pv}_{y,z,t}$) and upward operating (spinning) reserves ($r^{pv}_{y,z,t}$):

\[\begin{aligned} & \Theta^{pv}_{y, z, t} + f^{pv}_{y,z,t} + r^{pv}_{y,z,t} \leq \rho^{max, pv}_{y, z, t} \times \Delta^{total,pv}_{y, z} \quad \forall y \in \mathcal{VS}^{pv}, \forall z \in Z, \forall t \in T \end{aligned}\]

In addition, this function adds investment, fixed O&M, and variable O&M costs related to the solar PV capacity to the objective function:

\[\begin{aligned} & \sum_{y \in \mathcal{VS}^{pv}} \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST, pv}_{y,z} \times \Omega^{pv}_{y,z}) + (\pi^{FOM, pv}_{y,z} \times \Delta^{total,pv}_{y,z}) \right) \\ & + \sum_{y \in \mathcal{VS}^{pv}} \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}} (\pi^{VOM, pv}_{y,z} \times \eta^{inverter}_{y,z} \times \Theta^{pv}_{y,z,t}) \end{aligned}\]

stor_vre_stor!(EP::Model, inputs::Dict, setup::Dict)

This function defines the decision variables, expressions, and constraints for the storage component of each co-located VRE and storage generator. A wide range of energy storage devices (all $y \in \mathcal{VS}^{stor}$) can be modeled in GenX, using one of two generic storage formulations: (1) storage technologies with symmetric charge and discharge capacity (all $y \in \mathcal{VS}^{sym,dc} \cup y \in \mathcal{VS}^{sym,ac}$), such as lithium-ion batteries and most other electrochemical storage devices that use the same components for both charge and discharge; and (2) storage technologies that employ distinct and potentially asymmetric charge and discharge capacities (all $y \in \mathcal{VS}^{asym,dc,dis} \cup y \in \mathcal{VS}^{asym,dc,cha} \cup y \in \mathcal{VS}^{asym,ac,dis} \cup y \in \mathcal{VS}^{asym,ac,cha}$), such as most thermal storage technologies or hydrogen electrolysis/storage/fuel cell or combustion turbine systems. The following constraints apply to all storage resources, $y \in \mathcal{VS}^{stor}$, regardless of whether or not the storage has symmetric or asymmetric charging/discharging capabilities or varying durations of discharge.

The total storage energy capacity of each resource is defined as the sum of the existing storage energy capacity plus the newly invested storage energy capacity minus any retired storage energy capacity:

\[\begin{aligned} & \Delta^{total,energy}_{y,z} = (\overline{\Delta^{energy}_{y,z}}+\Omega^{energy}_{y,z}-\Delta^{energy}_{y,z}) \quad \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z} \end{aligned}\]

One cannot retire more energy capacity than existing energy capacity:

\[\begin{aligned} &\Delta^{energy}_{y,z} \leq \overline{\Delta^{energy}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z} \end{aligned}\]

For resources where $\overline{\Omega_{y,z}^{energy}}$ and $\underline{\Omega_{y,z}^{energy}}$ are defined, then we impose constraints on minimum and maximum energy capacity:

\[\begin{aligned} & \Delta^{total,energy}_{y,z} \leq \overline{\Omega}^{energy}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z} \\ & \Delta^{total,energy}_{y,z} \geq \underline{\Omega}^{energy}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z} \end{aligned}\]

The following two constraints track the state of charge of the storage resources at the end of each time period, relating the volume of energy stored at the end of the time period, $\Gamma_{y,z,t}$, to the state of charge at the end of the prior time period, $\Gamma_{y,z,t-1}$, the DC and AC charge and discharge decisions in the current time period, $\Pi^{dc}_{y,z,t}, \Pi^{ac}_{y,z,t}, \Theta^{dc}_{y,z,t}, \Theta^{ac}_{y,z,t}$, and the self discharge rate for the storage resource (if any), $\eta_{y,z}^{loss}$. When modeling the entire year as a single chronological period with total number of time steps of $\tau^{period}$, storage inventory in the first time step is linked to storage inventory at the last time step of the period representing the year. Alternatively, when modeling the entire year with multiple representative periods, this constraint relates storage inventory in the first timestep of the representative period with the inventory at the last time step of the representative period, where each representative period is made of $\tau^{period}$ time steps. In this implementation, energy exchange between representative periods is not permitted. When modeling representative time periods, GenX enables modeling of long duration energy storage which tracks state of charge between representative periods enable energy to be moved throughout the year. If there is more than one representative period and LDS_VRE_STOR=1 has been enabled for resources in Vre_and_stor_data.csv, this function calls lds_vre_stor!() to enable this feature. The first of these two constraints enforces storage inventory balance for interior time steps $(t \in \mathcal{T}^{interior})$, while the second enforces storage balance constraint for the initial time step $(t \in \mathcal{T}^{start})$:

\[\begin{aligned} & \Gamma_{y,z,t} = \Gamma_{y,z,t-1} - \frac{\Theta^{dc}_{y,z,t}}{\eta_{y,z}^{discharge,dc}} - \frac{\Theta^{ac}_{y,z,t}}{\eta_{y,z}^{discharge,ac}} + \eta_{y,z}^{charge,dc} \times \Pi^{dc}_{y,z,t} + \eta_{y,z}^{charge,ac} \times \Pi^{ac}_{y,z,t} \\ & - \eta_{y,z}^{loss} \times \Gamma_{y,z,t-1} \quad \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z}, t \in \mathcal{T}^{interior}\\ & \Gamma_{y,z,t} = \Gamma_{y,z,t+\tau^{period}-1} - \frac{\Theta^{dc}_{y,z,t}}{\eta_{y,z}^{discharge,dc}} - \frac{\Theta^{ac}_{y,z,t}}{\eta_{y,z}^{discharge,ac}} + \eta_{y,z}^{charge,dc} \times \Pi^{dc}_{y,z,t} + \eta_{y,z}^{charge,ac} \times \Pi^{ac}_{y,z,t} \\ & - \eta_{y,z}^{loss} \times \Gamma_{y,z,t+\tau^{period}-1} \quad \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z}, t \in \mathcal{T}^{start} \end{aligned}\]

The last constraint limits the volume of energy stored at any time, $\Gamma_{y,z,t}$, to be less than the installed energy storage capacity, $\Delta^{total, energy}_{y,z}$.

\[\begin{aligned} & \Gamma_{y,z,t} \leq \Delta^{total, energy}_{y,z} & \quad \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

The next set of constraints only apply to symmetric storage resources (all $y \in \mathcal{VS}^{sym,dc} \cup y \in \mathcal{VS}^{sym,ac}$). For storage technologies with symmetric charge and discharge capacity (all $y \in \mathcal{VS}^{sym,dc} \cup y \in \mathcal{VS}^{sym,ac}$), since storage resources generally represent a 'cluster' of multiple similar storage devices of the same type/cost in the same zone, GenX permits storage resources to simultaneously charge and discharge (as some units could be charging while others discharge). The simultaneous sum of DC and AC charge, $\Pi^{dc}_{y,z,t}, \Pi^{ac}_{y,z,t}$, and discharge, $\Theta^{dc}_{y,z,t}, \Theta^{ac}_{y,z,t}$, is limited by the total installed energy capacity, $\Delta^{total, energy}_{o,z}$, multiplied by the power to energy ratio, $\mu_{y,z}^{dc,stor}, \mu_{y,z}^{ac,stor}$. Without any capacity reserve margin constraints or operating reserves, the symmetric AC and DC storage resources are constrained as:

\[\begin{aligned} & \Theta^{dc}_{y,z,t} + \Pi^{dc}_{y,z,t} \leq \mu^{dc,stor}_{y,z} \times \Delta^{total,energy}_{y,z} \quad \forall y \in \mathcal{VS}^{sym,dc}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Theta^{ac}_{y,z,t} + \Pi^{ac}_{y,z,t} \leq \mu^{ac,stor}_{y,z} \times \Delta^{total,energy}_{y,z} \quad \forall y \in \mathcal{VS}^{sym,ac}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

Symmetric storage resources with only capacity reserve margin constraints follow a similar constraint that incorporates the 'virtual' discharging and charging that occurs and limits the simultaneously charging, discharging, virtual charging, and virtual discharging of the battery resource:

\[\begin{aligned} & \Theta^{dc}_{y,z,t} + \Theta^{CRM,dc}_{y,z,t} + \Pi^{dc}_{y,z,t} + \Pi^{CRM,dc}_{y,z,t} \\ & \leq \mu^{dc,stor}_{y,z} \times \Delta^{total,energy}_{y,z} \quad \forall y \in \mathcal{VS}^{sym,dc}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Theta^{ac}_{y,z,t} + \Theta^{CRM,ac}_{y,z,t} + \Pi^{ac}_{y,z,t} + \Pi^{CRM,ac}_{y,z,t} \\ & \leq \mu^{ac,stor}_{y,z} \times \Delta^{total,energy}_{y,z} \quad \forall y \in \mathcal{VS}^{sym,ac}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

Symmetric storage resources only subject to operating reserves have additional variables to represent contributions of frequency regulation and upwards operating reserves while the storage is charging DC or AC ($f^{dc,cha}_{y,z,t}, f^{ac,cha}_{y,z,t}$) and discharging DC or AC ($f^{dc,dis}_{y,z,t}, f^{ac,dis}_{y,z,t}, r^{dc,dis}_{y,z,t}, r^{ac,dis}_{y,z,t}$). Note that as storage resources can contribute to regulation and reserves while either charging or discharging, the proxy variables $f^{dc,cha}_{y,z,t}, f^{ac,cha}_{y,z,t}, f^{dc,dis}_{y,z,t}, f^{ac,dis}_{y,z,t}, r^{dc,dis}_{y,z,t}, r^{ac,dis}_{y,z,t}$ are created for storage components.

\[\begin{aligned} & \Theta^{dc}_{y,z,t} + f^{dc,dis}_{y,z,t} + r^{dc,dis}_{y,z,t} + \Pi^{dc}_{y,z,t} + f^{dc,cha}_{y,z,t} \\ & \leq \mu^{dc,stor}_{y,z} \times \Delta^{total,energy}_{y,z} \quad \forall y \in \mathcal{VS}^{sym,dc}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Theta^{ac}_{y,z,t} + f^{ac,dis}_{y,z,t} + r^{ac,dis}_{y,z,t} + \Pi^{ac}_{y,z,t} + f^{ac,cha}_{y,z,t} \\ & \leq \mu^{ac,stor}_{y,z} \times \Delta^{total,energy}_{y,z} \quad \forall y \in \mathcal{VS}^{sym,ac}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

For symmetric storage resources with both capacity reserve margin and operating reserves, DC and AC resources are subject to the following constraints:

\[\begin{aligned} & \Theta^{dc}_{y,z,t} + \Theta^{CRM,dc}_{y,z,t} + f^{dc,dis}_{y,z,t} + r^{dc,dis}_{y,z,t} + \Pi^{dc}_{y,z,t} + \Pi^{CRM,dc}_{y,z,t} + f^{dc,cha}_{y,z,t} \\ & \leq \mu^{dc,stor}_{y,z} \times \Delta^{total,energy}_{y,z} \quad \forall y \in \mathcal{VS}^{sym,dc}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Theta^{ac}_{y,z,t} + \Theta^{CRM,ac}_{y,z,t} + f^{ac,dis}_{y,z,t} + r^{ac,dis}_{y,z,t} + \Pi^{ac}_{y,z,t} + \Pi^{CRM,ac}_{y,z,t} + f^{ac,cha}_{y,z,t} \\ & \leq \mu^{ac,stor}_{y,z} \times \Delta^{total,energy}_{y,z} \quad \forall y \in \mathcal{VS}^{sym,ac}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

Long duration energy storage constraints are activated by the function lds_vre_stor!(). Asymmetric storage resource constraints are activated by the function investment_charge_vre_stor!().

In addition, this function adds investment, fixed O&M, and variable O&M costs related to the storage capacity to the objective function:

\[\begin{aligned} & \sum_{y \in \mathcal{VS}^{stor}} \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST, energy}_{y,z} \times \Omega^{energy}_{y,z}) + (\pi^{FOM, energy}_{y,z} \times \Delta^{total,energy}_{y,z}) \right) \\ & + \sum_{y \in \mathcal{VS}^{sym,dc} \cup \mathcal{VS}^{asym,dc,dis}} \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}} (\pi^{VOM,dc,dis}_{y,z} \times \eta^{inverter}_{y,z} \times \Theta^{dc}_{y,z,t}) \\ & + \sum_{y \in \mathcal{VS}^{sym,dc} \cup \mathcal{VS}^{asym,dc,cha}} \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}} (\pi^{VOM,dc,cha}_{y,z} \times \frac{\Pi^{dc}_{y,z,t}}{\eta^{inverter}_{y,z}}) \\ & + \sum_{y \in \mathcal{VS}^{sym,ac} \cup \mathcal{VS}^{asym,ac,dis}} \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}} (\pi^{VOM,ac,dis}_{y,z} \times \Theta^{ac}_{y,z,t}) \\ & + \sum_{y \in \mathcal{VS}^{sym,ac} \cup \mathcal{VS}^{asym,ac,cha}} \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}} (\pi^{VOM,ac,cha}_{y,z} \times \Pi^{ac}_{y,z,t}) \end{aligned}\]

vre_stor!(EP::Model, inputs::Dict, setup::Dict)

This module enables the modeling of 1) co-located VRE and energy storage technologies, and 2) optimized interconnection sizing for VREs. Utility-scale solar PV and/or wind VRE technologies can be modeled at the same site with or without storage technologies. Storage resources can be charged/discharged behind the meter through the inverter (DC) and through AC charging/discharging capabilities. Each resource can be configured to have any combination of the following components: solar PV, wind, DC discharging/charging storage, and AC discharging/charging storage resources. For storage resources, both long duration energy storage and short-duration energy storage can be modeled, via asymmetric or symmetric charging and discharging options. Each resource connects to the grid via a grid connection component, which is the only required decision variable that each resource must have. If the configured resource has either solar PV and/or DC discharging/charging storage capabilities, an inverter decision variable is also created. The full module with the decision variables and interactions can be found below.

Figure. Configurable Co-located VRE and Storage Module Interactions and Decision Variables

This module is split such that functions are called for each configurable component of a co-located resource: inverter_vre_stor(), solar_vre_stor!(), wind_vre_stor!(), stor_vre_stor!(), lds_vre_stor!(), and investment_charge_vre_stor!(). The function vre_stor!() specifically ensures that all necessary functions are called to activate the appropriate constraints, creates constraints that apply to multiple components (i.e. inverter and grid connection balances and maximums), and activates all of the policies that have been created (minimum capacity requirements, maximum capacity requirements, capacity reserve margins, operating reserves, and energy share requirements can all be turned on for this module). Note that not all of these variables are indexed by each co-located VRE and storage resource (for example, some co-located resources may only have a solar PV component and battery technology or just a wind component). Thus, the function vre_stor!() ensures indexing issues do not arise across the various potential configurations of co-located VRE and storage module but showcases all constraints as if each decision variable (that may be only applicable to certain components) is indexed by each $y \in \mathcal{VS}$ for readability.

The first constraint is created with the function vre_stor!() and exists for all resources, regardless of the VRE and storage components that each resource contains and regardless of the policies invoked for the module. This constraint represents the energy balance, ensuring net DC power (discharge of battery, PV generation, and charge of battery) and net AC power (discharge of battery, wind generation, and charge of battery) are equal to the technology's total discharging to and charging from the grid:

\[\begin{aligned} & \Theta_{y,z,t} - \Pi_{y,z,t} = \Theta_{y,z,t}^{wind} + \Theta_{y,z,t}^{ac} - \Pi_{y,z,t}^{ac} + \eta^{inverter}_{y,z} \times (\Theta_{y,z,t}^{pv} + \Theta_{y,z,t}^{dc}) - \frac{\Pi^{dc}_{y,z,t}}{\eta^{inverter}_{y,z}} \\ & \forall y \in \mathcal{VS}, \forall z \in \mathcal{Z}, \forall t \in \mathcal{T} \end{aligned}\]

The second constraint is also created with the function vre_stor!() and exists for all resources, regardless of the VRE and storage components that each resource contains. However, this constraint changes when either or both capacity reserve margins and operating reserves are activated. The following constraint enforces that the maximum grid exports and imports must be less than the grid connection capacity (without any policies):

\[\begin{aligned} & \Theta_{y,z,t} + \Pi_{y,z,t} \leq \Delta^{total}_{y,z} & \quad \forall y \in \mathcal{VS}, \forall z \in \mathcal{Z}, \forall t \in \mathcal{T} \end{aligned}\]

The second constraint with only capacity reserve margins activated is:

\[\begin{aligned} & \Theta_{y,z,t} + \Pi_{y,z,t} + \Theta^{CRM,ac}_{y,z,t} + \Pi^{CRM,ac}_{y,z,t} + \eta^{inverter}_{y,z} \times \Theta^{CRM,dc}_{y,z,t} + \frac{\Pi^{CRM,dc}_{y,z,t}}{\eta^{inverter}_{y,z}} \\ & \leq \Delta^{total}_{y,z} \quad \forall y \in \mathcal{VS}, \forall z \in \mathcal{Z}, \forall t \in \mathcal{T} \end{aligned}\]

The second constraint with only operating reserves activated is:

\[\begin{aligned} & \Theta_{y,z,t} + \Pi_{y,z,t} + f^{ac,dis}_{y,z,t} + r^{ac,dis}_{y,z,t} + f^{ac,cha}_{y,z,t} + f^{wind}_{y,z,t} + r^{wind}_{y,z,t} \\ & + \eta^{inverter}_{y,z} \times (f^{pv}_{y,z,t} + r^{pv}_{y,z,t} + f^{dc,dis}_{y,z,t} + r^{dc,dis}_{y,z,t}) + \frac{f^{dc,cha}_{y,z,t}}{\eta^{inverter}_{y,z}} \leq \Delta^{total}_{y,z} \quad \forall y \in \mathcal{VS}, \forall z \in \mathcal{Z}, \forall t \in \mathcal{T} \end{aligned}\]

The second constraint with both capacity reserve margins and operating reserves activated is:

\[\begin{aligned} & \Theta_{y,z,t} + \Pi_{y,z,t} + \Theta^{CRM,ac}_{y,z,t} + \Pi^{CRM,ac}_{y,z,t} + f^{ac,dis}_{y,z,t} + r^{ac,dis}_{y,z,t} + f^{ac,cha}_{y,z,t} + f^{wind}_{y,z,t} + r^{wind}_{y,z,t} \\ & + \eta^{inverter}_{y,z} \times (\Theta^{CRM,dc}_{y,z,t} + f^{pv}_{y,z,t} + r^{pv}_{y,z,t} + f^{dc,dis}_{y,z,t} + r^{dc,dis}_{y,z,t}) + \frac{\Pi^{CRM,dc}_{y,z,t} + f^{dc,cha}_{y,z,t}}{\eta^{inverter}_{y,z}} \\ & \leq \Delta^{total}_{y,z} \quad \forall y \in \mathcal{VS}, \forall z \in \mathcal{Z}, \forall t \in \mathcal{T} \end{aligned}\]

The rest of the constraints are dependent upon specific configurable components within the module and are listed below.

vre_stor_capres!(EP::Model, inputs::Dict, setup::Dict)

This function activates capacity reserve margin constraints for co-located VRE and storage resources. The capacity reserve margin formulation for GenX is further elaborated upon in cap_reserve_margin!(). For co-located resources ($y \in \mathcal{VS}$), the available capacity to contribute to the capacity reserve margin is the net injection into the transmission network (which can come from the solar PV, wind, and/or storage component) plus the net virtual injection corresponding to charge held in reserve (which can only come from the storage component), derated by the derating factor. If a capacity reserve margin is modeled, variables for virtual charge DC, $\Pi^{CRM, dc}_{y,z,t}$, virtual charge AC, $\Pi^{CRM, ac}_{y,z,t}$, virtual discharge DC, $\Theta^{CRM, dc}_{y,z,t}$, virtual discharge AC, $\Theta^{CRM, ac}_{o,z,t}$, and virtual state of charge, $\Gamma^{CRM}_{y,z,t}$, are created to represent contributions that a storage device makes to the capacity reserve margin without actually generating power. These represent power that the storage device could have discharged or consumed if called upon to do so, based on its available state of charge. Importantly, a dedicated set of variables and constraints are created to ensure that any virtual contributions to the capacity reserve margin could be made as actual charge/discharge if necessary without affecting system operations in any other timesteps (similar to the standalone storage capacity reserve margin constraints).

If a capacity reserve margin is modeled, then the following constraints track the relationship between the virtual charge variables, $\Pi^{CRM,dc}_{y,z,t}, \Pi^{CRM,ac}_{y,z,t}$, virtual discharge variables, $\Theta^{CRM, dc}_{y,z,t}, \Theta^{CRM, ac}_{y,z,t}$, and the virtual state of charge, $\Gamma^{CRM}_{y,z,t}$, representing the amount of state of charge that must be held in reserve to enable these virtual capacity reserve margin contributions and ensuring that the storage device could deliver its pledged capacity if called upon to do so without affecting its operations in other timesteps. $\Gamma^{CRM}_{y,z,t}$ is tracked similarly to the devices' overall state of charge based on its value in the previous timestep and the virtual charge and discharge in the current timestep. Unlike the regular state of charge, virtual discharge $\Theta^{CRM,dc}_{y,z,t}, \Theta^{CRM,ac}_{y,z,t}$ increases $\Gamma^{CRM}_{y,z,t}$ (as more charge must be held in reserve to support more virtual discharge), and the virtual charge $\Pi^{CRM,dc}_{y,z,t}, \Pi^{CRM,ac}_{y,z,t}$ reduces $\Gamma^{CRM}_{y,z,t}$. Similar to the state of charge constraints in the stor_vre_stor!() function, the first of these two constraints enforces storage inventory balance for interior time steps $(t \in \mathcal{T}^{interior})$, while the second enforces storage balance constraint for the initial time step $(t \in \mathcal{T}^{start})$:

\[\begin{aligned} & \Gamma^{CRM}_{y,z,t} = \Gamma^{CRM}_{y,z,t-1} + \frac{\Theta^{CRM, dc}_{y,z,t}}{\eta_{y,z}^{discharge,dc}} + \frac{\Theta^{CRM,ac}_{y,z,t}}{\eta_{y,z}^{discharge,ac}} - \eta_{y,z}^{charge,dc} \times \Pi^{CRM,dc}_{y,z,t} - \eta_{y,z}^{charge,ac} \times \Pi^{CRM, ac}_{y,z,t} \\ & - \eta_{y,z}^{loss} \times \Gamma^{CRM}_{y,z,t-1} \quad \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z}, t \in \mathcal{T}^{interior}\\ & \Gamma^{CRM}_{y,z,t} = \Gamma^{CRM}_{y,z,t+\tau^{period}-1} + \frac{\Theta^{CRM,dc}_{y,z,t}}{\eta_{y,z}^{discharge,dc}} + \frac{\Theta^{CRM,ac}_{y,z,t}}{\eta_{y,z}^{discharge,ac}} - \eta_{y,z}^{charge,dc} \times \Pi^{CRM,dc}_{y,z,t} - \eta_{y,z}^{charge,ac} \times \Pi^{CRM,ac}_{y,z,t} \\ & - \eta_{y,z}^{loss} \times \Gamma^{CRM}_{y,z,t+\tau^{period}-1} \quad \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z}, t \in \mathcal{T}^{start} \end{aligned}\]

The energy held in reserve, $\Gamma^{CRM}_{y,z,t}$, also acts as a lower bound on the overall state of charge $\Gamma_{y,z,t}$. This ensures that the storage device cannot use state of charge that would not have been available had it been called on to actually contribute its pledged virtual discharge at some earlier timestep. This relationship is described by the following constraint (as also outlined in the storage module):

\[\begin{aligned} & \Gamma_{y,z,t} \geq \Gamma^{CRM}_{y,z,t} \quad \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z}, t \in \mathcal{T} \\ \end{aligned}\]

The overall contribution of the co-located VRE and storage resources to the system's capacity reserve margin in timestep $t$ is equal to (including both actual and virtual DC and AC charge and discharge):

\[\begin{aligned} & \sum_{y \in \mathcal{VS}^{pv}} (\epsilon_{y,z,p}^{CRM} \times \eta^{inverter}_{y,z} \times \rho^{max,pv}_{y,z,t} \times \Delta^{total,pv}_{y,z}) \\ & + \sum_{y \in \mathcal{VS}^{wind}} (\epsilon_{y,z,p}^{CRM} \times \rho^{max,wind}_{y,z,t} \times \Delta^{total,wind}_{y,z}) \\ & + \sum_{y \in \mathcal{VS}^{sym,dc} \cup \mathcal{VS}^{asym,dc,dis}} (\epsilon_{y,z,p}^{CRM} \times \eta^{inverter}_{y,z} \times (\Theta^{dc}_{y,z,t} + \Theta^{CRM,dc}_{y,z,t})) \\ & + \sum_{y \in \mathcal{VS}^{sym,ac} \cup \mathcal{VS}^{asym,ac,dis}} (\epsilon_{y,z,p}^{CRM} \times (\Theta^{ac}_{y,z,t} + \Theta^{CRM,ac}_{y,z,t})) \\ & - \sum_{y \in \mathcal{VS}^{sym,dc} \cup \mathcal{VS}^{asym,dc,cha}} (\epsilon_{y,z,p}^{CRM} \times \frac{\Pi^{dc}_{y,z,t} + \Pi^{CRM,dc}_{y,z,t}}{\eta^{inverter}_{y,z}}) \\ & - \sum_{y \in \mathcal{VS}^{sym,dc} \cup \mathcal{VS}^{asym,dc,cha}} (\epsilon_{y,z,p}^{CRM} \times (\Pi^{ac}_{y,z,t} + \Pi^{CRM,ac}_{y,z,t})) \end{aligned}\]

If long duration energy storage resources exist, a separate but similar set of variables and constraints is used to track the evolution of energy held in reserves across representative periods, which is elaborated upon in the long_duration_storage!() function. The main linking constraint follows (due to the capabilities of virtual DC and AC discharging and charging):

\[\begin{aligned} & \Gamma^{CRM}_{y,z,(m-1)\times \tau^{period}+1} = \left(1-\eta_{y,z}^{loss}\right)\times \left(\Gamma^{CRM}_{y,z,m\times \tau^{period}} -\Delta Q_{y,z,m}\right) \\ & + \frac{\Theta^{CRM,dc}_{y,z,(m-1)\times \tau^{period}+1}}{\eta_{y,z}^{discharge,dc}} + \frac{\Theta^{CRM,ac}_{y,z,(m-1)\times \tau^{period}+1}}{\eta_{y,z}^{discharge,ac}} \\ & - \eta_{y,z}^{charge,dc} \times \Pi^{CRM,dc}_{y,z,(m-1)\times \tau^{period}+1} - \eta_{y,z}^{charge,ac} \times \Pi^{CRM,ac}_{y,z,(m-1)\times \tau^{period}+1} \\ & \forall y \in \mathcal{VS}^{LDES}, z \in \mathcal{Z}, m \in \mathcal{M} \end{aligned}\]

All other constraints are identical to those used to track the actual state of charge, except with the new variables for the representation of 'virtual' state of charge, build up storage inventory and state of charge at the beginning of each period.

vre_stor_operational_reserves!(EP::Model, inputs::Dict, setup::Dict)

This function activates either or both frequency regulation and operating reserve options for co-located VRE-storage resources. Co-located VRE and storage resources ($y \in \mathcal{VS}$) have six pairs of auxilary variables to reflect contributions to regulation and reserves when generating electricity from solar PV or wind resources, DC charging and discharging from storage resources, and AC charging and discharging from storage resources. The primary variables ($f_{y,z,t}$ & $r_{y,z,t}$) becomes equal to the sum of these auxilary variables as follows:

\[\begin{aligned} & f_{y,z,t} = f^{pv}_{y,z,t} + f^{wind}_{y,z,t} + f^{dc,dis}_{y,z,t} + f^{dc,cha}_{y,z,t} + f^{ac,dis}_{y,z,t} + f^{ac,cha}_{y,z,t} & \quad \forall y \in \mathcal{VS}, z \in \mathcal{Z}, t \in \mathcal{T}\\ & r_{y,z,t} = r^{pv}_{y,z,t} + r^{wind}_{y,z,t} + r^{dc,dis}_{y,z,t} + r^{dc,cha}_{y,z,t} + r^{ac,dis}_{y,z,t} + r^{ac,cha}_{y,z,t} & \quad \forall y \in \mathcal{VS}, z \in \mathcal{Z}, t \in \mathcal{T}\\ \end{aligned}\]

Furthermore, the frequency regulation and operating reserves require the maximum contribution from the entire resource to be a specified fraction of the installed grid connection capacity:

\[\begin{aligned} f_{y,z,t} \leq \upsilon^{reg}_{y,z} \times \Delta^{total}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}, z \in \mathcal{Z}, t \in \mathcal{T} \\ r_{y,z,t} \leq \upsilon^{rsv}_{y,z}\times \Delta^{total}_{y,z} \hspace{4 cm} \forall y \in \mathcal{VS}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

The following constraints follow if the configurable co-located resource has any type of storage component. When charging, reducing the DC and AC charge rate is contributing to upwards reserve and frequency regulation as it drops net demand. As such, the sum of the DC and AC charge rate plus contribution to regulation and reserves up must be greater than zero. Additionally, the DC and AC discharge rate plus the contribution to regulation must be greater than zero:

\[\begin{aligned} & \Pi^{dc}_{y,z,t} - f^{dc,cha}_{y,z,t} - r^{dc,cha}_{y,z,t} \geq 0 & \quad \forall y \in \mathcal{VS}^{sym,dc} \cup \mathcal{VS}^{asym,dc,cha}, z \in \mathcal{Z}, t \in \mathcal{T}\\ & \Pi^{ac}_{y,z,t} - f^{ac,cha}_{y,z,t} - r^{ac,cha}_{y,z,t} \geq 0 & \quad \forall y \in \mathcal{VS}^{sym,ac} \cup \mathcal{VS}^{asym,ac,cha}, z \in \mathcal{Z}, t \in \mathcal{T}\\ & \Theta^{dc}_{y,z,t} - f^{dc,dis}_{y,z,t} \geq 0 & \quad \forall y \in \mathcal{VS}^{sym,dc} \cup \mathcal{VS}^{asym,dc,dis}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Theta^{ac}_{y,z,t} - f^{ac,dis}_{y,z,t} \geq 0 & \quad \forall y \in \mathcal{VS}^{sym,ac} \cup \mathcal{VS}^{asym,ac,dis}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

Additionally, when reserves are modeled, the maximum DC and AC charge rate and contribution to regulation while charging can be no greater than the available energy storage capacity, or the difference between the total energy storage capacity, $\Delta^{total, energy}_{y,z}$, and the state of charge at the end of the previous time period, $\Gamma_{y,z,t-1}$, while accounting for charging losses $\eta_{y,z}^{charge,dc}, \eta_{y,z}^{charge,ac}$. Note that for storage to contribute to reserves down while charging, the storage device must be capable of increasing the charge rate (which increases net load):

\[\begin{aligned} & \eta_{y,z}^{charge,dc} \times (\Pi^{dc}_{y,z,t} + f^{dc,cha}_{o,z,t}) + \eta_{y,z}^{charge,ac} \times (\Pi^{ac}_{y,z,t} + f^{ac,cha}_{o,z,t}) \\ & \leq \Delta^{energy, total}_{y,z} - \Gamma_{y,z,t-1} \quad \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

Finally, the maximum DC and AC discharge rate and contributions to the frequency regulation and operating reserves must be less than the state of charge in the previous time period, $\Gamma_{y,z,t-1}$. Without any capacity reserve margin policies activated, the constraint is as follows:

\[\begin{aligned} & \frac{\Theta^{dc}_{y,z,t}+f^{dc,dis}_{y,z,t}+r^{dc,dis}_{y,z,t}}{\eta_{y,z}^{discharge,dc}} + \frac{\Theta^{ac}_{y,z,t}+f^{ac,dis}_{y,z,t}+r^{ac,dis}_{y,z,t}}{\eta_{y,z}^{discharge,ac}} \\ & \leq \Gamma_{y,z,t-1} \quad \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

With the capacity reserve margin policies, the maximum DC and AC discharge rate accounts for both contributions to the capacity reserve margin and operating reserves as follows:

\[\begin{aligned} & \frac{\Theta^{dc}_{y,z,t}+\Theta^{CRM,dc}_{y,z,t}+f^{dc,dis}_{y,z,t}+r^{dc,dis}_{y,z,t}}{\eta_{y,z}^{discharge,dc}} + \frac{\Theta^{ac}_{y,z,t}+\Theta^{CRM,ac}_{y,z,t}+f^{ac,dis}_{y,z,t}+r^{ac,dis}_{y,z,t}}{\eta_{y,z}^{discharge,ac}} \\ & \leq \Gamma_{y,z,t-1} \quad \forall y \in \mathcal{VS}^{stor}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

Lastly, if the co-located resource has a variable renewable energy component, the solar PV and wind resource can also contribute to frequency regulation reserves and must be greater than zero:

\[\begin{aligned} & \Theta^{pv}_{y,z,t} - f^{pv}_{y,z,t} \geq 0 & \quad \forall y \in \mathcal{VS}^{pv}, z \in \mathcal{Z}, t \in \mathcal{T} \\ & \Theta^{wind}_{y,z,t} - f^{wind}_{y,z,t} \geq 0 & \quad \forall y \in \mathcal{VS}^{wind}, z \in \mathcal{Z}, t \in \mathcal{T} \end{aligned}\]

wind_vre_stor!(EP::Model, inputs::Dict, setup::Dict)

This function defines the decision variables, expressions, and constraints for the wind component of each co-located VRE and storage generator.

The total wind capacity of each resource is defined as the sum of the existing wind capacity plus the newly invested wind capacity minus any retired wind capacity:

\[\begin{aligned} & \Delta^{total, wind}_{y,z} = (\overline{\Delta^{wind}_{y,z}} + \Omega^{wind}_{y,z} - \Delta^{wind}_{y,z}) \quad \forall y \in \mathcal{VS}^{wind}, z \in \mathcal{Z} \end{aligned}\]

One cannot retire more wind capacity than existing wind capacity:

\[\begin{aligned} & \Delta^{wind}_{y,z} \leq \overline{\Delta^{wind}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{wind}, z \in \mathcal{Z} \end{aligned}\]

For resources where $\overline{\Omega^{wind}_{y,z}}$ and $\underline{\Omega^{wind}_{y,z}}$ are defined, then we impose constraints on minimum and maximum capacity:

\[\begin{aligned} & \Delta^{total, wind}_{y,z} \leq \overline{\Omega^{wind}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{wind}, z \in \mathcal{Z} \\ & \Delta^{total, wind}_{y,z} \geq \underline{\Omega^{wind}_{y,z}} \hspace{4 cm} \forall y \in \mathcal{VS}^{wind}, z \in \mathcal{Z} \end{aligned}\]

If there is a fixed ratio for capacity (rather than co-optimizing interconnection sizing) of wind built to capacity of grid connection built ($\eta_{y,z}^{ILR,wind}$), then we impose the following constraint:

\[\begin{aligned} & \Delta^{total, wind}_{y, z} = \eta^{ILR,wind}_{y, z} \times \Delta^{total}_{y, z} \quad \forall y \in \mathcal{VS}^{wind}, \forall z \in Z \end{aligned}\]

The last constraint defines the maximum power output in each time step from the wind component. Without any operating reserves, the constraint is:

\[\begin{aligned} & \Theta^{wind}_{y, z, t} \leq \rho^{max, wind}_{y, z, t} \times \Delta^{total,wind}_{y, z} \quad \forall y \in \mathcal{VS}^{wind}, \forall z \in Z, \forall t \in T \end{aligned}\]

With operating reserves, the maximum power output in each time step from the wind component must account for procuring some of the available capacity for frequency regulation ($f^{wind}_{y,z,t}$) and upward operating (spinning) reserves ($r^{wind}_{y,z,t}$):

\[\begin{aligned} & \Theta^{wind}_{y, z, t} + f^{wind}_{y,z,t} + r^{wind}_{y,z,t} \leq \rho^{max, wind}_{y, z, t} \times \Delta^{total,wind}_{y, z} \quad \forall y \in \mathcal{VS}^{wind}, \forall z \in Z, \forall t \in T \end{aligned}\]

In addition, this function adds investment, fixed O&M, and variable O&M costs related to the wind capacity to the objective function:

\[\begin{aligned} & \sum_{y \in \mathcal{VS}^{wind}} \sum_{z \in \mathcal{Z}} \left( (\pi^{INVEST, wind}_{y,z} \times \Omega^{wind}_{y,z}) + (\pi^{FOM, wind}_{y,z} \times \Delta^{total,wind}_{y,z}) \right) \\ & + \sum_{y \in \mathcal{VS}^{wind}} \sum_{z \in \mathcal{Z}} \sum_{t \in \mathcal{T}} (\pi^{VOM, wind}_{y,z} \times \Theta^{wind}_{y,z,t}) \end{aligned}\]

write_vre_stor(path::AbstractString, inputs::Dict, setup::Dict, EP::Model)

Function for writing the vre-storage specific files.

write_vre_stor_capacity(path::AbstractString, inputs::Dict, setup::Dict, EP::Model)

Function for writing the vre-storage capacities.

write_vre_stor_charge(path::AbstractString, inputs::Dict, setup::Dict, EP::Model)

Function for writing the vre-storage charging decision variables/expressions.

write_vre_stor_discharge(path::AbstractString, inputs::Dict, setup::Dict, EP::Model)

Function for writing the vre-storage discharging decision variables/expressions.