Simulating Annual Variation in Load, Wind, and Solar by Representative Hour Selection

The spatial and temporal variability of renewable generation has important economic implications for electric sector investments and system operations. This study describes a method for selecting representative hours to preserve key distributional requirements for regional load, wind, and solar time series with a two-orders-of-magnitude reduction in dimensionality. We describe the implementation of this procedure in the US-REGEN model and compare impacts on energy system decisions with more common approaches. The results demonstrate how power sector modeling and capacity planning decisions are sensitive to the representation of intra-annual variation and how our proposed approach outperforms simple heuristic selection procedures with lower resolution. The representative hour approach preserves key properties of the joint underlying hourly distributions, whereas seasonal average approaches over-value wind and solar at higher penetration levels and under-value investment in dispatchable capacity by inaccurately capturing the corresponding residual load duration curves.


INTRODUCTION
A key research question for energy system modeling is what role wind and solar could play in the transition to a low-carbon energy system.To explore this question effectively, a model must capture the strong effect of temporal and spatial variability on the fundamental economics of intermittent renewable energy.Ideally, a model would maintain full hourly resolution to represent this variability.In practice, computational constraints require more compact alternatives for use in large-scale energy system or integrated assessment models with national or global coverage and long (multidecadal) timeframes.This issue is common to any model seeking to understand the implications of electric sector investments, including utility-scale resource planning models, regional or national power sector models used for policy-making, and global integrated assessment models used to inform issues like climate change mitigation where the power sector plays an important role (IPCC, 2014).In this paper, we propose a "representative hours" method for capturing the essential economic implications of intra-annual variability in electric sector capacity planning models in a computationally efficient manner, namely with resolution two orders of magnitude lower than hourly.
a Electric Power Research Institute,3420 Hillview Avenue,Palo Alto,CA 94304. b Prior to the emergence of wind and solar power as major potential resources of electricity supply, the variability of electricity demand was the only significant source of intra-annual variability that an energy system model needed to account for.The commonly employed "seasonal average" approach emerged as a compact alternative to full hourly resolution that captured key characteristics of load variability.This approach uses a limited number of segments to capture the load curve (peak, shoulder, and base in summer, winter, and fall/spring).While simple approaches of this type can be effective at reproducing a load duration curve, they poorly represent the distribution and co-variation with load of renewable resources, as well as the co-variation among regions needed to effectively model power transmission.The seasonal average approach assigns wind and solar coefficients to each segment based on average resource availability during the corresponding load period.This replicates average wind and solar capacity factors, but completely misses key intra-annual extremes, such as periods where load is high, but wind and solar are low.Existing studies show the relevance of increased temporal resolution to model outputs when wind and solar power are options (Ludig et al., 2011;Nicolosi et al., 2011;Pina et al., 2011) and provide overviews of approaches to incorporate this increased source of variability in models (Merrick, 2016;Nahmmacher et al., 2016;Sullivan et al., 2014).Examples of more advanced approaches to capturing intra-annual temporal variability are varied (Munoz and Mills, 2015;Ueckerdt et al., 2015;Wogrin et al., 2014;Van der Weijde and Hobbs, 2012;Swider and Weber, 2007). 1  Bridging the gap between short-run variations in the power system (including higher levels of temporal, spatial, and technical detail) and long-run investment decisions has recently become an active area of research.Collins et al. (2017) survey the strengths, shortcomings, and applicability of different methods for integrating these shorter-term power system dynamics into longer-term optimization models.Two primary categories of approaches are either to soft-link a capacity planning model with a more detailed operational model or to employ a direct integration approach by improving temporal, spatial, and technical features of a single model.The suitability of these two approaches varies by application, but methods like the representative hours procedure detailed in this paper are most relevant to the direct integration approach. 2 The generalized objective of any such approach is to find the unique hours of the year, not only in terms of electricity demand, but in terms of joint demand and wind and solar availability (Merrick, 2016).This framing suggests the use of clustering methods, which are employed elsewhere to find the number of unique hours (Mantzos andWiesenthal, 2016, Nahmmacher et al., 2016).The operations research literature on aggregation of linear programs also points to clustering methods and associated guarantees on model accuracy (Rogers et al., 1991). 3For a sample dataset, Merrick (2016) shows how a clustering method that reduces the number of hours from 8,760 to the order of 1,000 guarantees no aggregation error is introduced in any model output relative to the original problem.However, this order of magnitude is still intractably high for a detailed electricity model.Currently, most numerical applications in a dynamic investment (i.e., capacity planning) context are only able to reflect intra-annual variability with resolution on the order of 100 hours due to computational limitations.
Despite these emerging innovations, many prominent models nevertheless mimic variability in stylized ways using simple approaches, prioritizing increased resolution in other modeling dimensions over intra-annual temporal detail.Integrated assessment models may even use restrictive approaches to representing renewables like upper bounds on wind and solar shares, substitution functions with limited flexibility, fixed backup requirements, and explicit integration cost markups that do not change in penetration levels (Collins et al., 2017). 4The seasonal average approach remains common in capacity planning models for U.S. policy-making, including the National Energy Modeling System (NEMS) created by the U.S. Department of Energy's Energy Information Administration (EIA, 2014), and the Integrated Planning Model (IPM) used by the U.S. Environmental Protection Agency (EPA, 2010).It is also the default method for capturing variability used by the MARKet Allocation (MARKAL) and TIMES family of energy models.MARKAL/TIMES is used by 79 institutions in 38 countries,5 and is thus one of the most widely applied and emulated platforms for energy modeling.Although the platforms allow user-defined temporal resolution, MARKAL/TIMES models typically adopt load-duration-targeted seasonal average approaches.In addition to their influence on policy-makers and decision-makers, these models have propagated their hour-selection methodologies to other modeling frameworks like the OSeMOSYS open-source family of models (Howells et al., 2011).
In this paper, we propose a representative hours method to bridge this gap between the recognized shortcomings of existing approaches to representing power system variability in capacity planning models and need for computationally tractable solutions.Our novel representative hours method, used by the US-REGEN model (EPRI, 2017;Blanford et al., 2014), consists of strategic selection of particular hours during a calibration year that satisfy simultaneously key distributional requirements for load, wind, and solar time series across multiple inter-connected model regions.In particular, to reduce the resolution from thousands to hundreds of hours, we use a priori information about the relevance of different hours to the model solution.A key principle of the representative hours method is that the extreme points of the annual load, wind, and solar distributions must be captured.These hours are important for representing potential capacity shortfalls as well as the potential extent of surplus renewable energy production.The relevant extremes include not only the peaks and minimums of the individual series, but crucially also the joint extremes, for example the moments when load is high and both wind and solar are low, represented for each region.However, while extreme hours are essential, by themselves they do not suitably reflect the distribution across the entire year.To supplement the selected extreme hours, we also include several hours identified by a standard clustering algorithm to represent the interior of the distribution.The selected hours are then weighted to sum to a full year while minimizing the sum of errors between the approximated and hourly duration curves for each regional series.The resulting set of "representative hours" is used as the domain for dispatch of electric generation and transmission capacity in the model, where load, wind, solar availability factors are equal to their levels in the actual underlying hour and the duration is equal to the weight.This novel augmented clustering approach recognizes the importance of boundary events in system operations, price extremes, and consequently the profitability of new capacity investments.
We provide a range of diagnostic tests demonstrating the method's performance, as well as a comparison to a more typical approach of choosing a small number of points calculated as seasonal averages.Although the shortcomings of these seasonal average approaches are well known (as described earlier in the literature review), we provide new explanatory evidence comparing these simplified approaches to our proposed representative hours method and to the underlying hourly data.This comparison underscores that substantial differences in capacity planning decision arise not simply due to a small number of chosen intra-annual segments but due to the manner in which these segments are selected.We also demonstrate that a clustering-only approach to hour selection yields a significant improvement over the seasonal average approach but that some information about the capacity value of resources is lost, suggesting that the extreme hour selection proposed in this paper is a useful complement to clustering in certain contexts.
In the next section, we provide a description of the underlying comprehensive dataset and US-REGEN modeling context and an exposition of the three components of the "representative hours" method: (a) selection of "extreme" hours; (b) selection of "cluster" hours; and (c) weighting of hours.We also briefly describe our implementation of the more typical approach based on seasonal averages.The most meaningful metric presented here is an ex-post calculation of marginal value curves for wind and solar using a static model with full hourly resolution as well as approximations based on our method and the more typical approach.In Section 3.1, we show that the "representative hours" method reproduces far more closely the value of wind and solar as measured in the hourly model.These considerations are shown to be most important in investment environments where widespread deployment of renewable generation is possible.In Section 3.3, we perform sensitivity analysis to demonstrate that our method improves on either a clustering-only approach, or a seasonal average approach with more hours.A presentation and discussion of our results follow.

METHODS
The representative hours method has been implemented in the US-REGEN model, a detailed equilibrium model of electricity investment and dispatch with customizable sub-regions of the continental U.S., as shown in Figure 1.Parameterization is based on synchronized hourly data series for each region for load, as provided by FERC at the service territory level (FERC, 2010), and for renewable resource availability.Profiles for potential wind and solar output were developed for the model by AWS Truepower based on detailed simulations at potential sites in each region, as described in EPRI (2017).Although the simulations produced a range of profiles in each region corresponding to differentiated quality classes, which are represented in the model, for the purposes of representative hour selection we consider only a single profile for each region averaged over all classes.It is also possible to include multiple classes in certain regions, as discussed in the Appendix.In this analysis, all profiles are based on data from 2010.The model can be solved as a dynamic optimization through 2050 with five-year time steps, or alternatively as a static equilibrium with dispatch and capacity rental for a single year using full hourly resolution as well as more aggregate configurations.

Representative Hours Method: "Extreme" Hour Selection
The first phase of the hour selection algorithm is to identify a minimal set of hours that adequately covers each relevant extreme in each region.The focus on the role of extreme hours is supported by numerical experiments conducted by (Merrick, 2016) that identify these hours as the source of aggregation error in clustering methods.To begin, the algorithm identifies the hours with minimum and maximum values of load, wind, and solar individually, that is, six hours per region.Next, the algorithm identifies the hours at each vertex of the three two-dimensional planes (four each), and of the three-dimensional load-wind-solar space, that is, eight hours corresponding to each possible combination of maximum and minimum in each dimension, again in each region.If there were no overlap among these selected extremes, this would result in 26 points in 15 regions, or 390 candidate hours.However, the two-and three-dimensional extremes often coincide, and one extreme is sometimes represented by the same hour in multiple regions.For our 2010 profiles, overlap between dimensions and regions reduces this total to 223 unique extreme hours.
The simplest approach might be simply to proceed using all 223 extreme hours.However, though the extremes themselves are unique, other hours may be quite similar in terms of their joint values in the three dimensions of load, wind, and solar, and moreover may be near extremes in multiple (usually neighboring) regions.Thus, it is possible to reduce substantially the number of hours needed to adequately represent the extremes by allowing other "qualifying" hours within a certain radius (based on a Euclidean norm) from the true extreme.Geometrically, one may imagine a "bubble" around each extreme point in each region (in one, two, or three dimensions respectively), as shown in Figure 2 for the three-dimensional space in Texas.The selection algorithm is designed to find the minimum number of hours such that all 390 bubbles are populated with at least one hour.The algorithm is implemented as a straightforward integer programming problem in GAMS/ CPLEX with a binary decision variable for each hour corresponding to whether it is selected, a constraint that the sum of selected qualifying hours for each extreme is greater than or equal to one, and a minimand equal to the sum of selected hours.If the radius for each bubble is set to zero, the algorithm must choose the extremes themselves, that is, the 223 unique extreme hours.With a nonzero bubble radius, the algorithm can take advantage of hours that are near-extreme in multiple regions or dimensions and choose fewer total hours.Some bubbles may be more important than others and thus may be assigned a smaller radius or tolerance.There is naturally a trade-off here between accuracy and computational tractability of the model, whose solution time is convex in the number of representative hours.Our goal is to arrive at a configuration on the order of 100 hours, including the cluster hours described in the next section.Table 1 summarizes the relationship between radii of the various bubbles and the minimum number of spanning hours.The current analysis uses the configuration shown in bold with 83 extreme hours.

Representative Hours Method: "Cluster" Hour Selection
By design, the first phase has focused only on characterizing the convex hull of the three-dimensional space in each region.While extreme hours in one region may in fact be interior points that are more centrally located in another region (as illustrated in Figure 2), it is clear that the interior of the load-wind-solar space remains under-sampled by the algorithm described above.Thus, we add a second phase in which a second set of hours is selected based on a standard clustering algorithm.In contrast to the first phase, such an algorithm will by design identify hours near the center of the joint distribution.Whereas in the first phase, the number of selected hours was an outcome of the algorithm (and the choice of qualifying radii), in this clustering algorithm the number of clusters is an input.With one cluster, the algorithm will attempt to find the centroid of the entire distribution.
With multiple clusters, the algorithm will first partition the space into clusters and then identify the centroid of each cluster, with the objective of minimizing the sum of distances from each point to its corresponding centroid.By definition, the hours nearest the selected cluster centroids will have virtually no overlap with the hours identified as extremes, thus providing a complementary subset.Again, there is a trade-off between sampling accuracy and computational tractability.In this analysis, we applied the cluster algorithm phase with a target of 20 hours, for a total of 103 representative hours.

Representative Hours Method: Hour Weighting
Finally, the representative hour method must calculate weights for the hours selected by the first two phases such that the sum equals 8,760, or a full year.The weights are chosen to minimize the error not only with respect to the annual average, but also with respect to the shape of the cumulative distribution function (i.e., sorted duration curve) of each series in each region.The objective of the error minimization is the sum of squared differences between each representative hour's sort position in the full hourly curve and its sort position in the representative weighted curve.An additional subtlety is that errors are more heavily weighted at points where the sorted duration curve is steeper, that is, where errors in the sort position will lead to a greater distortion of the shape of the curve.We note that the residual error after optimal weights are chosen is an indicator of whether the number of selected hours by the first two phases has been sufficient.

Alternative Method ("Seasonal Average" Approach)
Traditionally, modeling the electric sector in a reduced-form context required only a relatively simple representation of the load duration curve, with a small number of segments, typically less than ten or even five, capturing the peak, shoulder, and base load periods (Santen et al., 2017;Merrick, 2016).Many models were developed with this structure, which allows a reasonably ef-Table 1: Relationship between bubble radius (% difference from true extreme allowed in each dimension) and number of selected extreme hours.
fective approximation of trade-offs between high fixed-cost / low variable-cost base load generators and low fixed-cost / high variable-cost peaking generators in a conventional power system.However, with the growing importance of intermittent renewable energy, models with this type of structure face challenges in adding variation in wind and solar to the framework.A typical response to these challenges is illustrated by the National Energy Modeling System (NEMS) used and made publicly available by the U.S. Energy Information Administration (EIA).The NEMS model, as described recently in EIA ( 2014), uses nine segments traditionally defined to capture the load curve (peak, shoulder, and base in summer, winter, and fall/spring), and assigns wind and solar coefficients to each segment based on average resource availability during the corresponding load period.For this analysis, we have reproduced a similar set of coefficients based on the same 2010 hourly data used in our representative hours method. 6Section 3.3.1 performs a robustness test of the paper's conclusions by comparing the representative hours approach (with 103 total segments) with a seasonal average approach that yields a comparable number of segments ( 108) to demonstrate how shortcomings of the seasonal average approach are not necessarily due to the small number of segments but to the selection procedure itself.
The difficulty with this type of approach is that it insufficiently describes both the individual distributions of wind and solar resource availability and the joint distribution of wind and solar with load.It also makes no attempt to capture regional correlation, meaning that transmission between model regions during a given segment occurs with non-simultaneous conditions on the two ends of the transaction.Although the NEMS model also employs other ad hoc constraints to account for potential moments of both low and high renewable output, its underlying approach of using a small number of load-based segments with averaged wind and solar coefficients is quite common and can lead to a substantial misrepresentation of the value of renewable technologies.

Diagnostic Results
We begin with an assessment of the representative hour (103 segments) and the seasonal average (9 segments) methods with respect to preserving distributional characteristics of the underlying hourly time series for load, wind, and solar.While the peak, minimum, and average of each series are preserved to a high degree of accuracy by construction in the representative hour method, the seasonal average method only ensures accurate capture of the averages and the peak load (see Table A1 in the Appendix).
Figure 3 shows duration curves (values sorted in descending order and weighted by segment length) for both the hourly and the approximated series for Texas.Results are qualitatively similar across other model regions, as shown in the Appendix.The representative hour method successfully captures the shape of the annual distribution for all three individually, but the seasonal average method underestimates variation in wind and solar.Most importantly, we assess the performance of each method with respect to correlation among the series by examining residual load duration curves.Residual load is calculated as demand less the available generation from intermittent renewable resources, which represents load that must be met through dispatchable assets.We illus-6.This procedure and approximate number of segments are similar to other models mentioned in the literature review in Section 1 like IPM (EPA, 2010).The choice of NEMS for this comparison was made because the code was publicly available, and thus we could reproduce the NEMS methodology with a high degree of confidence.trate residual load in Texas with a hypothetical introduction of 80 GW of wind and solar respectively (roughly equivalent to peak load in 2010), re-sorted to form a duration curve.
As shown in Figure 4, the hourly residual load duration curves indicate two important properties of wind and solar: (i) they alone contribute little to capacity needs, as peak residual load is unaffected by large renewable capacity additions; and (ii) they provide energy disproportionately at hours with low residual load.These two properties drive decreasing returns to renewable energy over the long term, as discussed below and elsewhere in the literature (Grubb, 1991;Fripp and Wiser, 2008;Edenhofer et al., 2013).The representative hour method preserves both properties with a limited downward shift in residual on the left side of the curve and a much larger downward shift on the right side.By contrast, the seasonal average method results in a larger contribution to the residual peak and a more limited contribution at the low end of the residual load curve.These results indicate that the representative hour method is much more likely to preserve key economic properties of intermittent renewable technologies in a reduced-form model than the more typical seasonal average approach.Note how the seasonal average method does a comparable job to the representative hour approach in capturing the residual load duration curve with no renewable deployment.As such, differences between approximation methods are smaller at lower wind and solar penetration levels.
We have also examined correlations between multiple time series (e.g., between load and resources, between different resource types, and across regions).In particular, models should reflect the joint distribution of load, wind, and solar characteristics to reflect the economics of intermittent renewable technologies.As shown in Figure 5, the representative hour method captures correlation coefficients between load and renewable output well, but the simplified seasonal average approach does not sufficiently represent these characteristics.In addition to the interdependence of load and renewables, correlations between different regions can be an important dynamic in trade outcomes and large-scale system balancing.Figure 5 shows how the representative hours capture the cross-correlations in the underlying data better than the heuristic approach, which understates the heterogeneity across regions.
We now turn to a comparison of model outputs.We begin with an illustrative analysis using the static version of the model, described above, in which the full hourly resolution can be used (as well as the two approximation methods).The experiment, based on Blanford (2015), consists of a series of static model simulations where the cost of wind (resp.solar) is systematically varied from current levels down to zero.As the cost decreases (and all other system parameters remain unchanged), total wind (resp.solar) deployed in the U.S. increases, thus revealing a marginal value curve for each resource.This measurement of value is a strong indicator for the role of renewable technologies in a dynamic simulation where costs and other components of the system evolve over time.
Figure 6 shows the results of this analysis using hourly resolution as well as the reduced-form methods.Again, the representative hour method closely matches the results from the hourly simulation, while the seasonal average method, which fails to account for key distributional impacts on the value of renewable energy profiles, significantly overestimates the marginal value curve for this dataset.The representative hour value curve tracks the hourly simulation results closely but tends to fall beneath these curves, especially for higher levels of solar deployment.

Dynamic Model Results
To characterize the electric sector transition to a long-run equilibrium and to evaluate the impacts of model representations of annual variation on these outcomes, we conduct a dynamic analysis in US-REGEN through 2050 using the representative hour and seasonal average approaches described above.These two variability specifications are run under three policy scenarios: 1. Reference (no additional climate policies); 2. Carbon tax of $25/t-CO 2 beginning in 2025; 3. Carbon tax of $50/t-CO 2 .Scenario assumptions are detailed in the Appendix.
These experiments demonstrate how model results are sensitive to the segment selection procedure and its accuracy in approximating temporal and spatial distributions of load, wind, and solar.Model recommendations for capacity investments (Figure 7) as well as other dispatch, emissions, and cost metrics are sensitive to these specifications across a range of market settings but are most responsive under scenarios that incentivize low-carbon technologies.The seasonal average approach described above does not account for the correlations between intermittent resources and load, which ceteris paribus incorrectly values these resources or dispatchable assets.
Cumulative investments through 2050 in solar (wind) are 113 GW (35 GW) larger with the seasonal average approach under baseline policy conditions and 217 GW (156 GW) larger under a $50/t-CO 2 tax compared with the representative hour approach.Moreover, investments in conventional capacity, in particular gas combined cycle and gas turbines, are considerably lower in the seasonal average approach.An assessment of capacity adequacy of the dynamic model solution against the true underlying hourly distribution shows that the seasonal average approach leads to a capacity shortfall of as much as 200 GW nationally, while the representative hour approach, which explicitly accounts for extreme hours driving capacity needs, falls short by a maximum of only 34 GW.
Although the specifications above lead the seasonal average approach to overestimate the value of renewables in these scenarios, the direction of the bias depends strongly on the implementation details and data of different approaches (Merrick, 2016).

Number of Segments in the Seasonal Average Approach
Previous sections compare the representative hour approach with a seasonal average method that replicates the procedure and approximate number of segments used in models like NEMS and IPM.The sensitivity in this section illustrates how the shortcomings of the seasonal average approach are not due to the small number of segments but to the selection procedure itself.In this sensitivity, a total of 108 segments results from using 12 months (instead of 3 seasons) and 9 output blocks for each region (instead of 3 time periods).
As shown in Figure 8, the higher-resolution seasonal average method more accurately replicates the load distribution compared with the nine-segment case; however, this method still does not adequately capture the respective duration curves for wind and solar resources.The seasonal average method insufficiently describes both the individual distributions of wind and solar resource availability and joint distributions with load and across regions.Such features are prominent shortcomings regardless of the number of selected segments (within a computationally feasible range).
As a result, Figure 9 shows how the 108-and 9-segment seasonal average cases produce similar recommendations for capacity mixes.Thus, the underlying approach of targeting load-based segments with averaged wind and solar coefficients can lead to misrepresentations of the value of renewable technologies, and increasing the number of segments alone will not remedy this shortcoming.

Selection Procedure with Clustering Only
Beyond the seasonal average approach, a comparatively more sophisticated selection method involves a clustering algorithm only (Merrick, 2016;Mantzos and Wiesenthal, 2016;Nahmmacher et al., 2016).This sensitivity compares the representative hour method with a clustering-only approach (i.e., without the modifications described in Section 2) to understand how the inclusion of additional extremes impacts the accuracy of results.The clustering approach uses an equivalent number of segments (103) to the representative hour method.
Figure 10 demonstrates how outputs from the clustering-only approach is more similar to the representative hour method than to the seasonal average one.Clustering with a sufficiently large number of segments represents the interior of the joint distribution while suitably capturing some of the extreme points.However, the wind and solar capacity additions suggested by the clustering approach are still considerably higher than the representative hour approach.Joint extremes in load, wind, and solar distributions are important to capture in energy models owing to their ability to drive decisions about system investments, operations, and renewable curtailment.Conversely, these results indicate that, if modelers are resource constrained in their hour selection procedure, adopting a standard clustering approach is preferable to a seasonal average method and can achieve a significant percentage of the gains described in this paper.An important caveat is that the representative hour methodology is likely more robust for any given dataset, especially if a more limited number of segments is chosen and the coverage from a standard clustering algorithm is less likely to capture extreme values (Merrick, 2016).

DISCUSSION AND CONCLUSION
The results described here demonstrate how power sector modeling and capacity planning decisions are sensitive to the representation of intra-annual variation and how our proposed ap- proach significantly outperforms simple heuristic selection procedures while maintaining computational tractability.In particular, the value of intermittent renewable energy from wind and solar is inextricably linked with the timing of their production relative to load.Clustering methods have been shown to guarantee that a model with aggregated temporal resolution will reproduce the outputs of a model with full hourly resolution (Merrick, 2016).Since such a resolution is still too great for a dynamic, national-level model, we have reduced the resolution further by drawing upon knowledge of the influence of extreme hours on electric sector investments.The goal of our representative hour approach is to model dynamic investment decisions that reflect as accurately as possible (with reasonable computational tractability) the true economic implications of intermittency.
We first demonstrate that key properties of the joint underlying hourly distributions are well-preserved by the representative hour approach for our comprehensive U.S. dataset.The changing shape of the residual load duration curve with increasing penetration of wind and solar capacity is an essential attribute of renewable value, and the approximated curve with weighted representative hours closely matches that based on hourly data.By contrast, the seasonal average approach yields a much poorer fit for residual load.We note that Ueckerdt et al. (2015) has focused on parameterizing directly the relationship between wind and solar capacity and the shape of the residual load duration curve, which is a compact and useful approach for modeling a single region.However, because our method is based on a sampling of simultaneous hourly data across multiple interconnected regions, we are able to reflect spatial variation of resources, including spatial variation in temporal correlations, as well as transmission capacity requirements.In this regard, the representative hour approach is a significantly more powerful (though computationally intensive) aggregation method, and one more suitable for detailed regional electricity dispatch and investment models.Moreover, in the conventional seasonal average approach, each region's averages are derived separately, so that only seasonal correlation between regions is retained, making it very difficult to relate transmission flows in a given model segment to actual capacity requirements.
We next demonstrate that model results using the representative hour approach for aggregation align closely with results from an otherwise identical hourly model.The marginal value curve, which describes how the value of wind and solar additions at the margin falls with cumulative national capacity, can be calculated in a rental context with a single year at hourly resolution, making it an ideal point of comparison for alternative aggregation approaches.Again, the seasonal average approach misses the mark, over-valuing wind and solar at higher levels of penetration as a result of not accurately capturing the corresponding residual load duration curves.
Finally, we show results for a dynamic simulation with alternative carbon policies.Consistent with the marginal value curve results, the seasonal average approach leads to a greater deployment of wind and solar and lower investment in dispatchable capacity than the representative hour approach, particularly at higher carbon prices.Although the hourly benchmark is not available in this setting, the capacity results from the aggregate model can be compared to the hourly data, which reveals much larger shortfalls in the seasonal average approach due to its lack of attention to extreme moments.In Section 3.3.1,we demonstrate how shortcomings of the seasonal average approach are not necessarily due to the small number of segments but to the selection procedure itself.We also demonstrate that a clustering-only approach to hour selection yields a significant improvement over the seasonal average approach but that some information about the capacity value of resources is lost, suggesting that the extreme hour selection proposed in this paper is a useful complement to clustering in certain contexts.Overall, these experiments illustrate how a representative hour approach can provide more reliable energy modeling insights and accurate asset valuation relative to simplified approaches that appear in many similarly oriented models.
One important limitation of the representative hour approach, and indeed of the simpler seasonal average and any similar approach, is that the chronology of hours is not preserved.Thus, it is not possible to explicitly model electricity storage (unless one makes an unrealistic assumption of an unlimited reservoir size) or operational constraints on ramp rates and start-up/shut-down cycles.While it is possible to study these issues with full hourly resolution, and in the case of operational constraints with a unit commitment formulation, it is not currently possible to conduct national-level dynamic investment analysis in such a context.One idea is to represent the year with a small number of full weeks or days, possibly at slightly less than hourly resolution, and examine the deployment of electricity storage, subject to reservoir constraints, within each of these "representative weeks" or "representative days" (Nahmmacher et al., 2016;Poncelet et al., 2016;De Sisternes et al., 2016;Nelson et al., 2012).However, it is unclear whether such an approach sufficiently captures annual variation at the hourly level, nor whether a weekly horizon is sufficient to capture the potential value of storage.A more conservative approach is to complement dynamic investment modeling using representative hours with separate but harmonized supplementary analysis using hourly resolution for a single year.For example, Blanford (2015) shows that unless costs of electricity storage become far lower than they are today, bulk storage does not significantly change the optimal mix of generation technology in a static equilibrium for a range of policy scenarios.Other novel approaches propose formulations to retain some chronological information across non-adjacent intra-annual segments to allow for storage modeling (Wogrin et al., 2016).Future research should investigate how novel modeling approaches could be applied to long-time-horizon capacity planning models to capture chronology in a computationally efficient manner (Train, 2009).A key research question for future work is to what extent opportunities for electricity storage and operational constraints fundamentally change the value of generation investments and the trade-offs between intermittent renewable and dispatchable technologies during the transformation to a low-carbon energy system (Bistline, 2017).

APPENDIX A. Model and Diagnostic Results
The U.S. Regional Economy, Greenhouse Gas, and Energy (US-REGEN) model and analysis in this paper is organized into 15 state-based regions.For two regions, South-Atlantic (VA, NC, and SC) and Florida, a wind series was not simulated by AWS Truepower because of the low quality of wind resources in those regions.Otherwise, a single series for wind and for solar is assigned in each region based on a weighted average of identified potential sites across a range of quality classes.These classes are represented individually in the model, with the same hours and weights derived in the methodology described in this paper based on the weighted average regional series.There is typically a strong correlation within a region between different quality classes, and we have not observed large errors with this approach.However, it is relatively straightforward to explicitly include multiple quality classes within a region, for example a high-quality wind class in a region that is expected to be important in the model solution.Rather than add an additional wind dimension for this class, we have found it more practical to add a duplicate region with a new wind series and unchanged values for the load and solar series.
Figure 11 compares summary statistics for the load, wind, and solar time-series data across the three approaches on a regional basis.The underlying hourly data (8,760) is compared with the results from the representative hour approach (103) and seasonal average method (9) for the average, maximum, and minimum values in each time series.The peaks, minimums, and averages are preserved to a high degree of accuracy by construction in the representative hour method; however, We have also examined correlations between multiple time series (e.g., between load and resources, between different resource types, and across regions).In particular, models should reflect the joint distribution of load, wind, and solar characteristics to reflect the economics of intermittent renewable technologies.Lamont (2008) shows how the marginal value of a generating technology depends both on a generator's average capacity factor and also on the covariance between marginal system costs and output from that generator.Thus, this theoretical insight motivates the importance of examining the correlation coefficients between load and renewable availability as metrics for systematically testing the quality of variability approximations.
As shown in Figure 5, the simplified load-targeting approach captures load crudely but does not sufficiently represent the full variability spectrum for wind and solar availability let alone its interdependence.Extreme hours in wind and solar availability are poorly captured with the load-targeting approach due to its focus on averaging renewable characteristics during applicable seasons and load segments, which dampens the resource fluctuations.Demand, wind speeds, and solar radiation not only vary within a region but also between model regions.Correlations between different regions can be an important dynamic in trade outcomes and large-scale system balancing.Figure 14 shows the interregional correlation coefficients across US-REGEN model regions for the actual 2010 data (left panel) and two approximations (middle and right panels). 7The representative-hour (103-segment) data capture the cross-correlations in the complete data better than the seasonal average approach, which understates the heterogeneity across regions.

B. Scenario Details for the Static Analysis
In the static model, annualized investment (rental) costs in Table 2 are equal to total investment cost multiplied by a capital charge rate.The charge rate c is based on the discount rate r and technology lifetime n through the equation c = r / (1 -(1 + r) -n .Storage assumptions, which are varied in the analysis, include a charge capacity cost of $800/kW, storage capacity cost of $120/kWh, and charge penalty of 25%.Transmission capacity costs are $3.85 million per mile for a 6.4 GW line.

C. Scenario Details for the Dynamic Analysis
The scenario specification for the dynamic analysis uses many common assumptions across the reference (i.e., no policy) case and the two carbon tax cases.For this study, we use the electric sector version of the US-REGEN model.Additional documentation about the model structure and assumptions can be found in EPRI (2017).For each of the three scenarios in these numerical experiments, all model features are held constant save for the representation of temporal variability.
All scenarios use fuel prices from the 2015 Annual Energy Outlook (EIA, 2015).Technology cost and performance assumptions come from the most recent EPRI Integrated Generation Technology Options report.In line with AEO 2015 assumptions, there are no forced retirements for existing coal units in the reference case, though retirements for economic reasons are possible in any period.Limitations on new transmission and nuclear capacity additions are based on EPRI expertise and historical experience.
Policies in all scenarios include existing state RPS requirements (as of January 2013), MATS, CWA §316(b), RCRA CCR, and CAA §111(b) performance standards for fossil units.Performance standards under CAA §111(d) for existing sources (i.e., the Clean Power Plan) are not included.Other state and regional policies include RGGI and California's AB32.Extensions of the ITC and PTC are not assumed.Rooftop solar is modeled as a "behind-the-meter" technology and, as per current regulatory practice, receives the retail rate for generated electricity instead of the wholesale price.The carbon tax scenarios apply rates of the $25/t-CO 2 and $50/t-CO 2 to power sector CO 2 emissions beginning in 2025.
Figure 15 provides an additional comparison between the representative hour and seasonal average approaches for regional trade flows.

Figure 1 :
Figure 1: Regional structure of the US-REGEN model.

Figure 2 :
Figure 2: Normalized hourly load, wind, and solar data for the Texas region (red), bubbles around corner points (black), and chosen segments from the full hour selection procedure (blue).

Figure 3 :
Figure 3: Duration curves for load (top panel), wind (middle panel), and solar (bottom panel) in Texas.The hourly duration curve (black) is approximated with 103 segments (red) and 9 segments (blue).

Figure 4 :
Figure 4: Residual Load Duration curves with 80 GW wind (top panel) and 80 GW solar (bottom panel).The hourly duration curve (black) is approximated with 103 segments (red) and 9 segments (blue).Solid lines at the top of each panel show curves without renewable deployment, and dotted lines show wind and solar penetration scenarios.

Figure 5 :
Figure 5: Correlation coefficient comparison for load and wind (left panel) and load and solar (right panel) across all 15 model regions for all 8,760 hours (blue), the 103-segment approximation (green), and 9-segment approximation (yellow).

Figure 6 :
Figure 6: Marginal value curves for wind (left panel) and solar (right panel) using the full hourly data (black), representative hour approach (red), and seasonal average approach (blue).

Figure 7 :
Figure 7: Cumulative electric sector capacity investments through 2050 (GW) for three carbon policy scenarios under the representative hour approach (left) and seasonal average approach (right).

Figure 8 :
Figure 8: Duration curves for load (top panel), wind (middle panel), and solar (bottom panel) in Texas.The hourly duration curve (black) is approximated with 103 segments using the representative hour approach (red) and with 108 segments using the seasonal average approach (blue).

Figure 9 :
Figure 9: Cumulative electric sector capacity additions through 2050 (GW) for the reference (left three columns) and $50 CO 2 tax (right three columns) scenarios.Values shown for the representative hour (RH) approach, seasonal average approach with 9 segments (SA 9), and seasonal average approach with 108 segments (SA 108).

Figure 10 :
Figure 10: Cumulative electric sector capacity additions through 2050 (GW) for the reference (left three columns) and $50 CO 2 tax (right three columns) scenarios.Values shown for the representative hour (RH) approach, cluster-only approach with 103 segments, and seasonal average approach with 9 segments (SA 9).

Figure 11 :
Figure 11: Comparison of the average, maximum, and minimum hourly values for the annual time-series data for load, wind, and solar.Each section compares the underlying hourly data (8,760), the representative hour approach (103), and the seasonal average method (9).Columns represent the 15 model regions.

Figure 12 :
Figure 12: Duration curves for load (left panel), wind (middle panel), and solar (right panel) in the Northwest Central region.The hourly duration curve (black) is approximated with 103 segments (red) using the representative hour approach and 9 segments (blue) using the seasonal average method.
7. Data were not available for the South Atlantic and Florida model regions, since no wind capacity was installed in 2010.

Figure 14 :
Figure 14: Interregional correlation coefficients for existing wind resources.

Figure 15 :
Figure 15: Annual regional trade (TWh) under the $50/t-CO 2 carbon tax scenario under the representative hour (top) and seasonal average (bottom) approaches.Negative values indicate that flows move in the opposite direction of the arrow in a given period.