2.2 Solution Characteristics
US-REGEN's electric sector model solution characterizes a profile of the electricity sector over time. By default, outputs include the model years 2015 to 2050, and the key solution variables of the model are
- capacity levels, which reflect new and retrofit investments and retirements;
- generation (i.e. dispatch) by technology and segment;
- inter-regional power flows by segment; and
- the price of electricity.
From those, other outputs can be derived, including fuel consumption and emissions of CO2 and other pollutants. The solution variables satisfy the optimality conditions; that is, they represent the values that minimize net present value of costs subject to constraints.
2.2.1 Optimality Conditions
The optimization formulation ensures that the key outcomes in any optimal solution will satisfy certain conditions. Equivalently, these optimality conditions describe a long-run competitive market equilibrium.
Dispatch Order: Within each segment, units will be dispatched in increasing order of marginal generation cost. Otherwise, producers' cost could be reduced (total surplus increased) by replacing the highest cost dispatched unit (i.e. marginal unit) with a less costly unit that had been bypassed in the dispatch order.
Complementary Slackness of Trade: Within each segment, if the marginal unit in one region has a higher dispatch cost than the marginal unit in an adjacent region, transmission from the adjacent region must be at its upper bound (i.e. transmission capacity must be fully utilized).[1] Otherwise, costs could be reduced by replacing the marginal unit with electricity imported from the adjacent region. Similarly, whenever the marginal generation cost is equal in adjacent regions (or more precisely, whenever the difference is less than the loss adjustment), transmission between regions during that segment must be strictly less than the upper bound.
Profitability of Investment: For any investment in capacity, the marginal unit added will have the present value of its costs (initial capital and operating costs over the lifetime) less than or equal to the present value of revenues (quantity generated in each segment multiplied by the segment price). This condition applies to both new additions and retrofit investments, and an analogous condition applies to new additions of transmission capacity (where the definition of revenues is related to the marginal value of transmission rather than the electricity price itself).[2] Otherwise, total surplus could be increased by dropping the investment in the marginal unit and its operation and foregoing the consumer benefit associated with the energy it produced, which at the margin is equal to its revenues. Further, if the present value of revenues is strictly greater than the present value of costs for the marginal investment in some technology, then there must be a constraint on that technology and investment must be equal to the upper bound. Otherwise, surplus could be increased by substituting investment of the positive-profit technology for some other investment with marginal revenue equal to marginal cost (i.e. zero net-profit).
2.2.2 Price of Electricity
The electric model reports the price of electricity for each region and time step at both the wholesale and the retail level. The wholesale price is related to the generation component of the price, which reflects energy and capacity costs of providing wholesale electricity subject to policy constraints imposed on the generation mix. This price is an output of the electric sector model's cost optimization. The remainder of the retail price reflects average transmission and distribution (T&D) costs and is estimated ex post of the model solve. Note that the wholesale or generation price also includes the market for inter-regional transmission capacity additions, which are endogenous to the electric model's optimization. However, a significant share of transmission costs is associated with intra-region capacity additions and maintenance, which are not explicitly captured within the optimization. These costs are included in the exogenously calculated T&D component of the retail price. The retail price, constructed as the sum of the generation and T&D components, is sent to the end-use model to evaluate consumer end-use energy decisions; the resulting electric load is fed back to the electric model in the next iteration.
Wholesale Price
The wholesale price is a marginal price that corresponds to the dual variable associated with the market clearance condition (i.e. supply = demand) that is enforced in every segment, region, and time period. At optimality, a dual variable (or shadow price) is equal to the amount by which the objective function could be increased (resp. decreased) if the associated constraint were relaxed (resp. tightened) by one unit. That is, the model's reported price in each segment corresponds to the marginal cost of supplying an additional MWh at that time in that region. Note that the objective function also includes costs for new inter-regional transmission as well as imputed rents accruing to existing inter-regional transmission, hence these components are included in the wholesale price. For many infra-marginal (i.e. non-peak) segments, the shadow price corresponds to the "dispatch cost" (fuel and variable O&M cost) of the marginal generating unit in the region (or in neighboring region plus the imputed transmission cost). However, in certain segments, supplying the last MWh actually requires the addition (or retention) of a unit of capacity (either in the form of new investment or deferred retirement).[3] For these segments, the shadow price will include all or a portion of the cost of that marginal capacity addition/retention. Additionally, the supplemental constraint requiring sufficient local firm capacity plus a reserve margin will bind in at least one such segment. The shadow price of this constraint is added to the shadow price on the market clearance condition to form the full wholesale price in peak segments. The result is that a small number of segments across regions and time periods will have very high prices, several orders of magnitude higher than the dispatch cost, and thus the annual average price across all segments reflects the full long-run marginal cost (including both fixed and variable components) of wholesale electricity supply (including inter-regional transmission).
Retail Price
The generation component of the retail price can be calculated in two ways. First, it can be calculated directly as the annual average wholesale price, weighted across segments by delivered load. This is analogous to the average price a de-regulated load-serving entity (LSE) would pay to provide power to its retail customers. Alternatively, the generation component can be calculated as an average cost, that is, dividing total cost (variable costs plus depreciation and rate of return on the rate base) by total end-use sales. This calculation is conceptually analogous to the process by which the generation component of the retail price is set in cost-of-service regulated regions. However, in the idealized setting of the model's intertemporal optimization, there is little practical or numerical difference between these two approaches. The distinction is primarily relevant for measuring the impacts of certain policy instruments affecting the residual value of existing generation assets. Unless the model is being applied to analyze such an instrument, we use the more straightforward approach of directly connecting the realized wholesale price to the retail price. In terms of calibration, it may not be the case that the average wholesale price calculated by the model for the base year coincides with the observed generation component of the retail rate (as reported by EIA). Such a gap may exist because of legacy investments or out-of-the-money contracts. We include an adjustment factor for observed base year discrepancies but assume that this factor is reduced to zero over time.
The T&D component of the retail price is estimated based on observed T&D costs in the base year, adjusted in future years by projected changes in the load shape calculated in the enduse model. Base year costs are derived from EIA's reported price by component from the electricity market module of NEMS, as well as SEDS reported retail price by enduse-sector and state. For model projection years, the average T&D cost per delivered MWh in each sector and region is scaled from the base year level by the change in the ratio of peak to average load within each model region. That is, total T&D costs are assumed to scale with peak load, so that average T&D costs (i.e. the T&D component of the retail rate) scale with the peak to average ratio. While a more detailed assessment would consider a range of other factors relevant for T&D expenditures, this approximation is intended to roughly capture the additional costs of T&D infrastructure upgrades that may accompany increases in system peaks driven by increased electrification. We continue to refine this calculation to reflect potential impacts of changing patterns of load and generation resources on T&D system requirements and costs.
When there is a loss factor on inter-region transmission, this condition holds when the marginal dispatch cost exceeds the adjacent region's marginal dispatch cost plus a loss adjustment. In some cases we also include a $4/MWh charge on inter-region transmission flows, as described below in Section 2.3.6. ↩︎
A similar condition exists for non-retirement decisions. All capacity has a fixed lifetime at which it must retire, but because there is also a fixed O&M charge per unit of capacity installed (regardless of dispatch), it may be optimal to retire early capacity that does not justify this cost. ↩︎
Usually this will be the segment corresponding to peak load. However, when there is a large share of output from variable resources with zero dispatch cost, the relevant peak can shift to the segment with the highest load net of variable output. Additionally, the shadow price on some infra-marginal segments may include a smaller capacity component, since the model effectively clears a capacity market in each segment, rather than clearing a single annual market for the peak, Finally, marginal capacity costs may be allocated across time periods due to subtle inflections in the dynamics of load growth and capacity evolution. ↩︎