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| Fig. 1: Commonly used terms to describe wind turbine specifications. (Image source: Wikimedia Commons, as modified by P. Carpenter.) |
When comparing the economics of a wind farm to other sources of power generation - such as gas-turbines, coal power plants, or solar energy - a commonly utilized metric is the levelized cost of energy, or LCOE. In its simplest form, the LCOE is a ratio of the net present value of the average annual costs over the lifetime of a plant (capital expenditure, operating expenditure and fuel costs included) divided by the average annual power produced over the lifetime of the plant. [1] This metric allows for comparison between types of power generation that often have very different cost structures. Gas-fired power plants, for example, often have relatively low installment and construction costs but maintain high continued operating costs due to the price of natural gas and wear of turbines at high temperatures and pressures. In contrast, renewable energy power plants often require high upfront capital expenditure, but have low operating costs because their source of energy (wind or solar, commonly) is free, abundant, and renews itself. Thus, examining the LCOE for a given project allows developers and investors to normalize disparities in CAPEX and OPEX to understand the average price of electricity produced over the lifetime of the project.
Wind farms are particularly expensive to build. In 2022, the average cost to install a single wind turbine in the United States (with a generator nameplate capacity of 3.2 megawatts) was $4.52 million dollars. [2] For a utility-scale wind farm, which may contain dozens or even hundreds of wind turbines, the total project capital expenditure can easily reach hundreds of millions to billions of dollars. However, because wind is free, and wind turbines have relatively low maintenance costs, the continued operating expenditure remains low. Thus, a simple expression for the LCOE of wind farms can be described as:
| LCOE | = | CAPEX + OPEX Average Annual Energy Production (AEP) |
For wind farm developers, minimizing the LCOE of a given project involves an optimization of both minimizing the CAPEX required to build the farm and maximizing the total energy outputted. While there are many factors that contribute to the capital and operating expenditures of the LCOE, this report seeks to understand the metrics that are used to understand and evaluate the average annual energy production of a wind farm, as well as examine production trends among US onshore wind farms.
Before understanding how the energy output of a turbine is described, it is important to understand the power that can be produced from wind in the first place. [5] So, we will first consider some potential wind site in the absence of a wind turbine . The energy in the wind, described as the energy flux φ , is given by
where ρ is the mass density of the air measured in kg/m3 and V represents the speed of the wind at a given site and time, measured in m/s. φ then has units of W/m2. Now, suppose some turbine with blades of length L is introduced to the site. Because the hub (the central spinning component at the center of a turbine) is small in diameter compared to the length of the blade (See Fig. 1), we will omit its diameter in this discussion. With this idealized turbine, we can then say that the total area swept out by the blades as they rotate is given by:
where r, the radius of the turbine swept area, is approximated by L. With these two equations, we can now describe the energy flux for the given cross-sectional area of our wind turbine, placed perpendicular to the direction of wind speed. The total available wind power that can be captured by a given turbine as a function of the wind speed is given as:
Here, it is important to note that this relation describes the total power that can be produced from the wind at a given site. Due to other thermodynamic and engineering factors - which will not be considered for the sake of simplicity - most wind turbines convert only ~50% of this available power into electricity. [3]
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| Fig. 2: Histogram of real hourly windspeed data for Stanford, CA at 80m above ground level. [6-8] Note the significant distribution of wind speeds away from the average. (Image source: P. Carpenter) |
The cubic relationship between power and wind speed has significant implications on the performance of a wind turbine. For a given location, the wind speed often varies significantly, even on an hourly basis. The intermittency of the wind resource can be well-approximated by the probabilistic Weibull distribution, which shows the probability that the wind moves at a given speed for any chosen time. [5] Fig. 2, which shows annual hourly wind data for Stanford University, demonstrates the broad distribution of wind speeds - from 0 m/s to 19 m/s - around the average wind speed (5.8 m/s).
When the wind speed changes by even a small factor, the effect of this variability will be drastically magnified in the wind power. To quantify this effect, consider some initial wind speed, Vref, that produces power output, Pref, for a given turbine. If the wind speed decreases by some factor, say 10%, we can analyze the new power output relative to the reference power. All else equal, our analysis yields
| P Pref |
= | V3 Vref3 |
= | (0.9 Vref)3 Vref3 |
= | (0.9)3 | = | 0.73 |
This value tells us that when the wind speed decreases by just 10% from some reference value, the total potential power decreases by 27%. When the wind speed decreases by 50% from the reference, the effect is further magnified, leading to a total wind power that is just 12.5% of the initial value. Because the power output of a turbine is highly sensitive variations in windspeed at a given location and time, no turbine will ever produce at 100% of the manufacturer-specified generator capacity (also called the turbine's nameplate capacity) all the time. To describe the intermittency of this power output, the capacity factor of a wind turbine is defined as:
| Capacity Factor (CF) | = | Actual AEP Nameplate Capacity * 8760 hours/year |
× 100% |
where, as a ratio of watt-hours to watt-hours, the capacity factor is a dimensionless quantity. The output of a capacity factor calculation tells us how much time a turbine spends producing at its maximum rated capacity per year. So, a turbine that produces at an average of 50% of its rated capacity for an entire year will have a capacity factor of 50%. Note that this equation explicitly links the annual energy production and the capacity factor, such that our LCOE for a wind farm can be rewritten as
| LCOE | = | CAPEX + OPEX 8760 hours/year × CF |
Thus, maximizing the AEP of a wind farm to minimize the LCOE requires, by definition, maximizing the capacity factor. If we return to our initial equation for the available power that can be generated by wind, we find that there are two key parameters that affect the available power to be captured by wind turbines: the turbine blade length, and the wind speed at a given site.
Though there is little to be done about the variability of wind speeds for a given site, there is one way that access to wind resource can be improved. For much of the world, increasing in height above ground level will lead to an increase in the average wind speed. Thus, to the limit that our technology can provide, increasing the height of a wind turbine can help improve the average wind speed, and thus average power production for a given site. As wind turbine technology has continued to mature over the past two decades, examining the average turbine hub height (the height at which the central turbine hub is above the ground) for US land-based turbines depicts a similar conclusion: the average hub height of a wind turbine installed in 2022 was 98.1 meters above the ground, representing a 73% increase from just two decades earlier. [2]
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| Fig. 3: Graph representing the downward trend in turbine specific power for turbines installed in a given year. [2] (Courtesy of the DOE) |
The second parameter that can drastically affect the available power from the wind is the swept area of a turbine, which for our purposes, has been approximated by the blade length, L. However, because many different turbine models operate in a range of nameplate capacities and at different heights, this metric is more difficult to compare on an equal basis. Thus, the specific power of a turbine is used, which is defined as a ratio of the turbine's nameplate capacity (in W) to the swept area (in m2). [2] Qualitatively, this value represents the fraction of swept area that is devoted to each watt of power produced. For regions with high average wind speeds and less variation, where turbines spend more time producing power at their nameplate capacity, there is little difference in performance between turbines with high or low specific power. However, in regions with low wind speeds, lower specific power turbines will outperform a turbine with a high specific power, because there is more area available to capture wind energy per unit watt of power generated. Though the mathematical relationship is complex, we can qualitatively conclude that the specific power shares an inverse relationship with the capacity factor of a turbine. With all else equal, decreasing the specific power of a turbine will increase the capacity factor at a given site.
Fig. 3 demonstrates how this conclusion has manifested in US turbines over the past two decades. Originally designed for sites with lower wind speeds, turbines with lower specific power have dominated the US market, such that the average specific power of a new turbine in 2022 was 233 W/m2, down from 393 W/m2 for turbines installed from 1998-1999. [2]
Ultimately, as these two trends - increasing hub height and decreasing specific power - have evolved in the past two decades, the capacity factor of wind farms has significantly increased. In 1998-99, the average capacity factor of a wind turbine facility entering service was 20%. In 2018, the capacity factor of new wind plants entering service had more than doubled, reaching over 40%. [2] Continuing to improve the capacity factor of modern wind turbines will be an important factor in enabling the continued development of wind energy in the US, as it will help improve the LCOE of wind relative to other sources of power.
Finally, as wind continues to grow in the US, it will also be of increasing importance to effectively integrate wind power with the rest of the electric grid. Historically, the electric grid lacked the capacity to store energy, meaning that the demand for electricity must be matched exactly by the total power generated at every hour of the day, every day of the year. For renewable power technologies like wind, the intermittency of the wind resource means that power generation for a wind farm is both unreliable and difficult to predict. As such, it has historically been difficult to integrate large- scale turbine facilities into the electric grid, with grid operators often needing to rely on 'on-demand' gas power plants in the event of no wind. [4] However, improving the capacity factor of wind turbines somewhat decreases the intermittency of power production, making it more reliable as a power source. Thus, even beyond the LCOE, it is thought that improving the capacity factor - and thus increasing the reliability - of wind turbines provides additional benefits in reducing the costs of integrating such a variable source of power with the rest of our energy system. [4]
© Peter Carpenter. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] K. Hansen, "Decision-Making Based on Energy Costs: Comparing Levelized Cost of Energy and Energy System Costs," Energy Strategy Rev. 24, 68 (2019).
[2] R. Wiser et al., "Land-Based Wind Market Report: 2023 Edition," U.S. Office of Energy Efficiency and Renewable Energy, DOE/GO-102023-6055, August 2023.
[3] "Wind Energy," University of Michigan, CSS07-09, August 2023.
[4] S. D. Ahmed et al., "Grid Integration Challenges of Wind Energy: A Review," IEEE Access 8, 10857 (2020).
[5] O. Friedman, "Wind Turbines," Physics 240, Stanford University, Fall 2021.
[6] Pfenninger and I. Staffell, "Long-Term Patterns of European PV Output Using 30 Years of Validated Hourly Reanalysis and Satellite Data," Energy 114, 1251 (2016).
[7] I. Staffell and S. Pfenninger, "Using Bias-Corrected Reanalysis to Simulate Current and Future Wind Power Output," Energy 114, 1224 (2016).
[8] M. M. Rienecker et al., "MERRA: NASA's Modern-Era Retrospective Analysis for Research and Applications," J. Clim. 24, 3624 (2011).
