AN ANGLE ON SOLAR AND WIND VARIABILITY

AN ANGLE ON SOLAR AND WIND VARIABILITY
Victor Diakov

It wasn’t that long ago that solar and wind energy were perceived as exotic supplements to our usual electricity consumption. Yet NREL (National Renewable Energy Lab) research shows that we can realistically generate 20 percent (U.S. DOE, 2008) to 80 percent (Hand et al., 2012) of electricity from renewable sources — mostly wind and solar. The computer models simulating solar and wind variability effects on balancing the electric load with generation are extremely complex. For example, to account for hourly variations in wind generation, literally millions of inputs are used. My colleagues at NREL and I have developed an approach that simplifies the complexity.

The concept involves vectors representing data arrays and is supported by computer model results. The quadratic1 load-matching model was proposed by W. Short (Short and Diakov, 2012). Our work focuses on generic features of load matching with wind and solar resources, and is based on the hour-to-hour variations of demand and generation at tens of thousands of potential wind and solar sites throughout the year in the United States.

As the share of wind- and solar-generated power increases, flexible generation2 will have to cover the difference between variable load and the aggregate “traditional” baseload plus wind and solar generation. We can reduce our reliance on flexible generation by taking advantage of geographic diversity for wind and solar resources3 (NERC 2009;Milligan et al. 2009).

As our calculations show, aggregating wind and photovoltaic (PV) sites reduces the resource variability from 45-50 degrees to about 20 degrees. If we represent the load, average PV generation and average wind generation as vectors, we find that based on the angle between the aggregated generation profile and the load, the required amount of dispatchable generation is no more than 20 percent of the total electric load. Considering that these numbers (20 degrees and 20 percent) hold in our study for both the Western Electricity Coordinating Council (WECC) region and the United States as a whole, it is likely that they are similar for other geographic regions as well. Taking transmission limitations into consideration (Diakov, Short, Gilchrist, 2012) doesn’t change the amount of PV and wind by much: For WECC, the existing large-scale transmission limitations reduce the amount of PV and wind from 80 percent to 70 percent of the electric load.

APPROACH
Our model (Short and Diakov, 2012) answers the following question: What is the best match to U.S. loads that could be achieved with wind power and photovoltaics? (Specifically, we examined the wind, solar resources and loads in the United States as a whole and in the area of the WECC. Our primary decision variables are where and how much resource Wi (wind and/or photovoltaic) should be built at each wind and PV site i. Over all 8,760 hours in a year, the model minimizes the sum of the squared difference between loads (lt) and the generation from the selected wind and solar sites:

The sum of all the positive ?t is the total amount of generation required from dispatchable generators. Similarly, the sum of the negative ?t is the total amount of all energy curtailed or sent to storage. The wind and solar generation are traded off solely on the basis of how well they meet load, not their relative economics.

INPUT DATA
For the wind resource, we are using NREL data developed for the Western Wind and Solar Integration Study (WWSIS) (Potter et al. 2008) and the Eastern Wind Integration and Transmission Study (EnerNex 2011). The data include three years of generation information, 2004–2006, for 32,000 potential wind sites in the western United States and about 6,000 potential wind sites in the eastern United States.

For the photovoltaic resource, we are using hourly insolation data for the same years for 949 U.S. sites found in the National Solar Radiation Data Base (Wilcox et al. 2007) and converting that to power generation from a south-oriented PV panel with a 10-degree tilt using the PVWatts calculator (nrel.gov/rredc/pvwatts). To prevent unreasonable overuse of the sites that have generation profiles best matching the load profiles, we limited the PV capacity at any single site to 1 gigawatt (GW).

We aggregated the load data from Ventyx’s Velocity Suite product, which is based on hourly historical demand for the same years from FERC Form 714 Part III Schedule 2.

It is convenient to represent hourly data (PV, wind generation or load) for the year as vectors, each vector having 8,760 components (representing hours in the year). Three vectors — load, average PV generation4 and average wind generation — are used to build a 3-D representation of input data (see figure 1, page 31). The three axes on figure 1 are i) the load, ii) the combination of average PV and load (the combination is chosen in such a way as to make the second axis perpendicular to the first one), and iii) the combination of average wind generation with the other two axes (again, the third axis is perpendicular to the other two).

MODEL RESULTS
To demonstrate the effect of scaling up wind and PV resources, we compare three loadmatching cases: A) the WECC load with only one wind and one PV generation profile, i.e., no geographic diversity; B) the WECC load with available WECC-area wind and PV resources; and C) the full U.S. load with available U.S. wind and PV resources. None of the cases demonstrates exact load matching, which tells us that in order to balance the load, the wind and PV resources have to be at least partially backed up by flexible generators and/or energy storage. As evident in figure 2a, the renewable sources overproduce significantly in early spring, before experiencing a large production shortfall during late summer months.

The table on this page compares these three cases. For case C (continental United States), the model built 950 GW of wind capacity and 580 GW of PV. We would need another 472 GW of dispatchable capacity to meet all loads. To ensure that no energy was wasted, we’d also need 461 GW of storage charging capacity. The shortfalls occur for 5,387 hrs and would need to be made up by 723 terawatt-hours (TWh) from dispatchables. Overproduction occurs in 3,372 hours and totals 324 TWh.

The data from the table show that the overall shortfall for the year consistently exceeds the total overproduction, and more so for a poor load-matching case A. This property of the quadratic load matching allows for a straightforward geometric explanation. The hourly load (or generation) profile for the year is considered as a vector in an 8,760-dimensional space (figure 2b). Expression 1 of the formula on page 31 essentially minimizes the distance between the tip of the wind-plus-PV generation vector and the tip of the given load vector.

Let’s consider first case A with just one wind and one PV site, with no constraints on how much capacity can be built on each of them. Between building wind (43.6-degree angle with load) and PV (52.2-degree angle with load) capacity, the model chooses the combination that minimizes the angle between combined (wind plus PV) production and load (37.2 degrees), figure 3a. Figure 3b schematically shows both the load and the combined production (line “one PV and one wind site”). The distance between vector tips is minimal when vector ? closing the gap is perpendicular to the generation vector. A larger angle between the generation and the load (which means a worse load match) results in a larger overall shortfall (figure 3b vs. 3c).

Based on the angle between load and the combined wind/PV generation (note that this is just one number), one can determine the length of the mismatch ? and estimate the overall shortfall as fractions of yearly load. Thus, this angle is an important characteristic of renewable generation.

As seen in the table on page 32, combining numerous wind and PV resources from the WECC area significantly decreases the variability of the aggregated generation profile. Respectively, the angle between the latter and the load (load^production) is reduced to 20.5 degrees. Broadening the area from WECC to the entire continental United States results in a slightly smaller “variability” angle, 18.9 degrees. The angle between generation and ? is exactly 90 degrees for the case A in the table. The same angle for case B only approximately equals 90 degrees, because the generation capacity of the sites involved is limited. The fact that this angle is close to 90 degrees demonstrates that the vector approach represents a valid approximation. Further, based on close load^production angle values, only marginal improvement in load matching is possible when expanding the geographic region from WECC to the entire United States.

Figure 3d helps illustrate how the variability angle characterizes the balance between load and its best matching variable generation. The length of the load vector serves as the unit length. As long as the angle between and generation vectors is close to 90 degrees, the length of is given by sin(a) (equals 0.35 for a=20 degrees), and the overall shortfall is approximately sin2(a) (equals 0.12 for a=20 degrees). Notably, the width of the distribution of is also determined by the variability angle a, as sin(a). cos(a) (equals 0.32 for a =20 degrees). Assuming a normal distribution of (the result does not change much for a bi-modal distribution) with standard deviation 0.32 and a 0.12 average, gives us a total shortfall of 19 percent of the total load and a surplus of 8 percent of the load (766 TWh shortfall and 305 TWh surplus for the continental United States — these are remarkably close to the numbers obtained from the detailed model and given above in connection with the case C in the table).