Background of Model

Linear program formulation

We qualitatively describe here the basic LP formulation of WinDS, followed by additional qualitative detail on transmission and intermittency. The Detailed Model Description section contains the actual equations/constraints used in the linear program.

The objective function in the WinDS 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, plus
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, for each power control area (PCA). 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 WinDS is subject to more than 70 different types of constraints, which result in tens of thousands of equations in the model (due primarily to the large number of regions). These constraints fall into several main categories, including:

Wind resources: 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.
Transmission constraints: In WinDS, there are several forms of constraints on transmission of both wind generation and conventional generation.

General transmission in any given time slice is constrained by the capacity of all transmission lines between any two PCAs
General transmission capacity must also be available to accommodate the transfer of firm power between PCAs (these are transfers to ensure adequate capacity is available to meet reserve margin requirements).
Wind transmission on the existing grid is constrained by:

the cost to build transmission from the wind site to the nearest existing transmission line with adequate capacity to carry the wind generation.
the total available capacity of all existing transmission lines out of the wind supply region and into a demand region.
the transmission capacity between PCAs available for generation from wind or other sources.

Wind can also be transmitted on new transmission lines constructed specifically to carry wind. Although these lines are not constrained in WinDS, 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. As can be seen from Figure 4, in later years, new transmission lines are frequently selected within WinDS, even though those lines are used only when the wind is blowing. New transmission built for wind can be constructed between wind supply/demand regions and/or within a wind supply region.

Load constraints: The primary load constraint is that the electric load in each power control area (there are 136 power control areas in WinDS) must be met in each time slice (of which there are 16) throughout a year. While the load in 2000 is based on actual loads in each PCA, the annual rate of load growth must be input and is assumed to be uniform over time and within each NERC region.
Reserve margin constraint: There are two types of reserves constraints—planning reserve margin and operating reserve. For the planning reserve margin constraint, WinDS updates its estimate (for each period) of the marginal capacity value of the next wind farm 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—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 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.
Wind Surplus: WinDS also accounts for surplus wind-generated electricity that is lost at the interconnection region level. In reality, when demand is low and the wind is blowing, there can be instances where the wind generation cannot all be used. WinDS 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 stochastically compute the expected amount of wind that cannot be used. This loss in useful wind output is taken into account when WinDS expands capacity by choosing between different generation technologies
Emissions constraints: At the national level, WinDS caps the air emissions from fossil-fueled generators for sulfur dioxide, nitrogen oxides, mercury, and carbon. The annual national emission caps and the emissions per MWH by fuel and plant type are input over time by the user.
RPS constraints: WinDS allows the user to input Renewable Portfolio Standard (RPS) constraints at either the national or state level. However, because the only nonhydroelectric renewable energy technology included at this time in WinDS is wind, the RPS constraint is essentially a constraint that requires that a certain fraction of the generation be provided by wind (both old and new). The fraction increases linearly over time from a "start year" value to a maximum value. A penalty can also be imposed for each MWh shortfall in the nation or state.

This section includes:
Qualitative model description

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