NREL: Regional Energy Deployment System (ReEDS): Linear program formulation

Linear program formulation

This section qualitatively describes the basic LP formulation of ReEDS, followed by additional qualitative detail on transmission and variability. Simplified Model Description and Detailed Model Description contain the actual equations/constraints used in the linear program.

The objective function in the ReEDS linear program is a minimization of all the costs of the U.S. electric sector including:

the present value of the cost for both generation and transmission capacity installed in each period

the present value of the cost for operating that capacity during the next 20 years to meet load, i.e., fixed and variable operation and maintenance (O&M) and fuel costs

the cost of several categories of ancillary services and storage.

By minimizing these costs while meeting the system constraints (discussed below), the linear program determines which types of new capacity are the most economical to add in each period, in each balancing authority. Simultaneously, the linear program determines what capacity should be dispatched to provide the necessary energy in each of the 16 annual time-slices. Therefore, the capacity factor for each dispatchable technology in each region is an output of the model, not an input.

The cost minimization that occurs within ReEDS is subject to more than 70 different types of constraints, which result in hundreds of thousands of equations in the model (due primarily to the large number of regions). These constraints fall into several main categories, including:

Resource constraints: The total amount of wind capacity of each type (onshore, offshore shallow, offshore deep) installed in each region, in each wind class must be less than the wind resource potentially available.

Similarly, the total amount of CSP capacity installed in each region, in each insolation class must be less than the solar resource potentially available; geothermal capacity installed in each balancing authority in each price bin must be less than the recoverable geothermal resource in the area; and annual generation from biofuels—whether in dedicated biomass plants or cofired in coal plants—is constrained by the amount of biomass produced in each balancing authority.

Transmission constraints: In ReEDS, there are several forms of constraints on transmission of both renewable and conventional generation:

General transmission in any given time-slice is constrained by the capacity of all transmission lines between any two balancing authorities.
General transmission capacity must also be available to accommodate the transfer of firm power between balancing authorities (these are transfers to ensure adequate capacity is available to meet reserve margin requirements).
Wind and CSP transmission on the existing grid is constrained by:
- The cost to build transmission from the wind/CSP site to the nearest existing transmission line with adequate capacity to carry the expected generation.
- The total available capacity of all existing transmission lines out of the supply region and into a demand region.
- The transmission capacity between balancing authorities available for generation from renewable or conventional sources.
Wind and CSP can also be transmitted on new transmission lines constructed specifically to carry them. Although these lines are not constrained in ReEDS, the model does include a cost for their construction that varies with the length and capacity of the line, as well as the slope of the terrain in the origination and destination regions, and the population density of those regions. New transmission built for wind and CSP can be constructed between supply/demand regions and/or within a supply region.

Load constraints: The primary load constraint is that the electric load in each balancing authoritiy (there are 134 of them in ReEDS) must be met in each time-slice (of which there are 16) throughout a year. While the load in 2006 is based on actual loads in each balancing authority, the annual rate of load growth must be input. The load growth varies with each investment period and varies by NERC region.

There is an option in ReEDS to subdivide certain time-slices in certain regions if there are substantial amounts of both wind and baseload capacity compared to load. The mini-slices are a 20%-60%-20% hourly division and the wind capacity factor is adjusted for the 20% segments to represent those hours when the wind blows most and least. This allows ReEDS to more finely capture the variations in wind resource and their impact on base, intermediate, and peaking generation.

Reserve margin constraint: There are two types of reserve constraints: planning reserve margin and operating reserve. For the planning reserve margin constraint, each period ReEDS updates its estimate of the marginal capacity value of the next wind farm or CSP plant built in each region, using a detailed statistical approach. The capacity value is set equal to the amount of load that could be added—along with the wind or CSP—without changing the risk of a shortage in generation capacity at peak load times (Effective Load Carrying Capability or ELCC). The approach accounts for the dispersion of the wind and CSP sites contributing to the load and the correlation in the output of those sites.

Operating reserve constraint: The operating reserve requirement induced by each new wind farm is also modified each period for each region. It is assumed that the operating reserve requirement induced by wind is statistically independent from the normal operating reserve requirement induced by load variability and forced outages. Thus, the additional operating reserve requirements due to wind are not proportional to the amount of wind, but rather to the variance in the sum of the normal operating reserve and that due to only the wind generation. This means that the operating reserves induced by wind are generally low per unit of wind capacity initially, but can grow quickly if the wind capacity becomes a significant part of system capacity—especially if the output of the new wind capacity is highly correlated with that of existing wind capacity.

CSP facilities, as presently modeled, are assumed to have six hours of thermal storage, so CSP capacity does not increase the operating reserve requirement the way wind does. ReEDS is currently being modified to include CSP facilities without storage. Consequently, the operating reserve requirements will be modified to account for this.

Wind Surplus: ReEDS also accounts for surplus wind—generated electricity that is curtailed if wind plus must—run conventional output exceeds the load. In reality, when demand is low and the wind is blowing, there can be instances where the wind generation can not all be used. ReEDS uses the variance of the sum of all wind generation in the interconnect—together with a load duration curve and the forced outage rates of conventional technologies—to statistically compute the expected amount of wind that can not be used. This loss in useful wind output is taken into account when ReEDS expands capacity by choosing between different generation technologies.

The six-hour thermal storage assumed for CSP capacity also means that CSP does not have an issue with surplus.

Emissions constraints: At the national level, ReEDS caps the emissions from fossil-fueled generators for sulfur dioxide, nitrogen oxides, mercury, and carbon dioxide. The annual national emission caps and the emissions per MWh by fuel and plant type are inputs to the model.

In carbon-constrained scenarios, CO₂ can be either capped or taxed, and either a cap or tax can be finely adjusted to match proposed legislation.

RPS constraints: ReEDS allows the user to input Renewable Portfolio Standard (RPS) constraints at either the national or state level. All renewable generation counts toward the national RPS requirement. The renewable generation sources include wind, CSP, geothermal, and biopower (including the biomass fraction of cofiring plants). State RPS requirements do not include hydroelectric power generation. The RPS can ramp in either linearly over time or according to an externally defined profile. A penalty can also be imposed for each MWh shortfall in the nation or state.

Background of Model

Linear program formulation