GenX Model Formulation

GenX is a complex model comprised of numerous equations. Briefly described below is the objective function formulation and the representation of storage used in GenX. For greater detail on other aspects of the model, see the working paper published by Sepulveda and Jenkins [70].

Objective Function

The GenX optimization is broken down to a cost-minimizing objective function comprised of four pieces.

They are the combined sums of:

1. the annualized amortization of investments for new plants and fixed operation and maintenance (O&M) costs of total capacity;

2. the variable O&M costs of generation and storage which includes the fuel consumption costs of plants;

3. the cost of unserved energy; and

4. the startup costs for thermal generation plants.

In plain language, the objective function is the underlying method by which GenX determines how to invest in generation resources, how to dispatch, cycle, and commit them, and who to serve. The full objective function for GenX considers several other additional components, transmission and distribu-tion network reinforcement costs for example. These are not used in this research however, and are not included here.

GenX takes the four pieces described above and, subject to input assumptions (Section 3.1.2), con-figuration settings for GenX (Section 3.2.2), and policy choices of the user(Section 3.1.1), will deploy generation resources and build out a least-cost power system.

Energy Storage Representation

GenX provides two ways to represent energy storage technologies. The first type of storage, where energy and power capacity is linked by a fixed energy to power ratio, is already described in the GenX working paper [70], and will be only very briefly described here. The second type, where energy and power capacity are independently determined, is not included in the working paper, and is detailed in greater depth below.

All storage technologies in GenX are characterized by their charging efficiency, discharging efficiency, and self-discharge rate. As noted, the first type of storage is modeled using a fixed energy to power ratio. Using this ratio is a common way to model energy storage (see Sepulveda et al. [69], Schill and Zerrahn [68], Sisternes, Jenkins, and Botterud [71], Denholm and Hand [26]). The energy to power ratio shows the power capacity (MW) per unit of energy storage energy capacity (MWh). Lithium-ion batteries are included as a technology option in this research, and are modeled in this manner

The first energy storage technology type is represented by equations 3.2 - 3.8 below.

Γt = Γt −1− Θt

η_discharge + (ηcharge×Π) (3.2)

Γt = µ_storage¹ ×Installed Storage Capacity (3.3)

Πt =η_charge¹ ×Installed Storage Capacity (3.4)

Πt = µ_storage¹ ×Installed Storage Capacity−Γt (3.5)

Θt = ηdischarge×Installed Storage Capacity (3.6)

Θt = Γt (3.7)

Θt

η_discharge + (ηcharge×Π) ≤Installed Storage Capacity (3.8)

WhereΓis the storage state of charge,Θis its discharge rate,ηis its efficiency,Πis its charge rate, and µis the energy to power ratio.

Equation 3.2 tracks the storage energy level (or state of charge) throughout the year. Equation 3.3 sets the maximum energy storage level at any given time as based on the capacity built. Equations 3.4 and 3.5 constrain the charge rate, which must be less than either the net installed power capacity or the remaining energy capacity in each time step. Equations 3.6 and 3.7 constrain the discharge rate, which must be less than either the net installed capacity or the current energy storage level. Finally, Equation 3.8 constrains the combined charging and discharging rates to be less than the total installed capacity.

Long duration energy storage technologies are modeled very similarly to the method described above, but with no fixed energy to power ratio. Thus, the only two equations from above that are affected are equations 3.3 and 3.5, which utilize energy to power ratios.

To constrain the maximum energy storage, minimum and maximum input durations are included as model inputs. Equation 3.3 is modified by constraining the maximum energy storage to be between these minimum and maximum input durations multiplied by power capacity. This change is reflected in Equation 3.10. Equation 3.5 is modified by constraining the charge rate to be less than equal to the remaining available storage capacity, as shown in Equation 3.12. Thus, the full set of equations for energy storage with decoupled energy and power capacities is below.

Γt = Γt −₁− Θt

η_discharge + (ηcharge×Π) (3.9)

Duration_min≤ Energy Storage Capacity

Installed Storage Capacity ≤Duration_max (3.10)

Π_t =η_charge¹ ×Installed Storage Capacity (3.11)

Πt =Energy Storage Capacity−Γt (3.12)

Θt = ηdischarge×Installed Storage Capacity (3.13)

Θt = Γt (3.14)

Θt

η_discharge + (ηcharge×Π) ≤Installed Storage Capacity (3.15)

WhereΓis the storage state of charge,Θis its discharge rate,ηis its efficiency,Πis its charge rate, and µis the energy to power ratio.

It is appropriate to model these energy storage devices differently due to the physical differences between lithium-ion and LDES technologies. The amount of power lithium-ion batteries can produce is physically linked to the amount of energy the battery may store. This is due to the technology design, where the same device that is delivering electrical current (the power) is also storing energy (in the form of lithium ions stored by the anode in the battery). While this ratio may change from battery to battery, depending on how the lithium-ion battery was designed and constructed, the ratio will remain fixed.

LDES technologies, on the other hand, can be sized more independently. In most LDES technologies the energy storage is separate from the power component. Take power to hydrogen for example. In this LDES technology, electricity is used to create hydrogen through the use of an electrolyzer. The hydrogen is then stored in an entirely separate environment, for example in a pressurized cannister or perhaps a salt cavern. To turn hydrogen back into power, it is then run through a fuel cell. Again, this is separate from the energy storage mechanism. Thus, separating the power and energy capacity components allows the model an extra degree of freedom to determine an optimal solution, and provides greater insight into optimal LDES technology options.