Annual energy production (AEP) is often the objective function in wind plant layout optimization studies. The conventional method to compute AEP for a wind farm is to first evaluate power production for each discrete wind direction and speed using either computational fluid dynamics simulations or engineering wake models. The AEP is then calculated by weighted-averaging (based on the wind rose at the wind farm site) the power produced across all wind directions and speeds. We propose a novel formulation for time-averaged wake velocity that incorporates an analytical integral of a wake deficit model across every wind direction. This approach computes the average flow field more efficiently, and layout optimization is an obvious application to exploit this benefit. The clear advantage of this new approach is that the layout optimization produces solutions with comparable AEP performance yet is completed 2 orders of magnitude faster. The analytical integral and the use of a Fourier expansion to express the wind speed and wind direction frequency create a relatively smooth solution space for the gradient-based optimizer to excel in comparison to the existing weighted-averaging power calculation.
The layout of a wind plant is a primary design element that influences its performance. Optimizing the layout can be thought of as a wake avoidance problem, wherein turbines are placed such that they avoid the wakes from other turbines as much as possible. Power losses from wake interactions can be on the order of 10 %–20 % in wind farms
In controls and optimization applications, the wake velocity deficit is approximated with low-fidelity analytical models. The classical top-hat model parameterizes the wake expansion rate and computes the wake deficit as a function of downstream position
Layout optimization studies leverage these low-fidelity models to approximate the wake velocity within the wind farm. Turbines are placed to minimize wake interactions and thereby maximize annual energy production (AEP) of the plant. Gradient-based optimization algorithms leverage the derivative of the objective function to choose search directions for optimal solutions, while gradient-free optimization only evaluates the objective function (thereby avoiding its derivatives) and is useful for discontinuous and noisy functions. Gradient-free algorithms are common practice in industry for small wind farms
AEP is an integral quantity. The total power production of a wind plant is calculated based on the wind speed flowing through each turbine. For a single wind speed and direction, this procedure is straightforward. Figure
The inspiration for the FLOW Estimation and Rose Superposition (FLOWERS) flow field model is to analytically compute the average wake velocity given the frequency and magnitude of the wind speed for every direction. Since the average wake velocity is conceptualized similarly to AEP, extending the FLOWERS approach to calculating AEP would be straightforward. We hypothesize that the analytical integration will considerably reduce the computational cost of average wake velocity and AEP calculations compared to the numerical integration.
In this paper, we first derive the equations for the time-averaged wake velocity and a new formulation for AEP in Sect.
To derive a mathematical formulation for the time-averaged flow distribution, we use the classical Jensen (top-hat) wake deficit model
We transform Eq. (
The power
We apply this novel formulation of time-averaged wake velocity and AEP to the wind plant layout optimization problem. For
The primary benefits of FLOWERS lie in its suitability to drive layout optimization as a wake avoidance problem, despite the fact that simplifications made to develop FLOWERS might induce some errors in the predicted magnitude of AEP. The optimizer relies on the objective function to provide a quantitative metric to compare possible solutions; in this case with gradient-based optimization, the ratio of the objective function evaluated for two different solutions is of importance. In other words, the objective function's output itself is not necessarily critical as long as the mapping between inputs and outputs in the function remains consistent. If we think of the layout optimization problem as a wake avoidance problem, then the objective function must be able to approximate wake magnitudes and downstream influence to minimize their interactions. Turbines aligned with predominant wind directions and turbines with close spacing will reduce AEP in the FLOWERS optimization, just as it will in the numerical integration-based optimization. Wake avoidance can be achieved despite a less accurate estimate of AEP because the factors that cause positive or negative changes to AEP are still present. The gradient throughout the optimization space will be different because the objective functions are not identical. However, with a sufficiently strict convergence criterion, we predict that the FLOWERS optimization will still find a similar quality solution to the numerical integration technique. A more accurate AEP estimate can be added as a final post-processing step once the layout optimization is complete.
We start by comparing the AEP estimates for an illustrative test case of three turbines aligned with a predominant wind direction. The AEP for the numerical integration approach is computed using the Jensen wake deficit model with the same nominal (i.e., based on ambient turbulence intensity and excluding wake-added turbulence) wake expansion coefficient as in FLOWERS
(
Annually averaged flow field comparison for three turbines aligned with the predominant wind direction with 6
Flow field comparison for a 60-turbine wind farm with 6
The FLOWERS AEP is 2.9 % lower than the result from the numerical integration approach. Substantial wakes only exist for three discrete wind directions in this example, so the profile of
For a more realistic wind rose, the discrepancies in AEP between FLOWERS and the Jensen numerical integration approach are more substantial. We consider a larger wind farm of 60 turbines with 6
One important feature of the FLOWERS solution is its smoothness. Despite using the discrete top-hat model, the flow field in Fig.
We now examine the differences in AEP between FLOWERS and the Jensen numerical integration more broadly. A total of 40 randomized test cases were generated. A random number of turbines between 4 and 50 was chosen for each. The layout of the turbines is randomized within a square boundary of side length 25
A total of 40 test cases with random layouts and random wind roses to compare AEP more generally.
We compare the computation time and percent difference in AEP between the two methods in Fig.
Comparison of computational cost and relative difference in AEP between FLOWERS and Jensen for the randomized cases.
We should note that the implementation of the Jensen numerical integration in FLORIS Version 2 is non-vectorized (i.e., calculations are mostly performed through for-loops instead of vector operations). Comparisons in computation time with a vectorized code such as PyWake would likely show a smaller discrepancy
The discrepancy in AEP between the two methods is more pronounced in these randomized cases. The free-stream wind speed is not held constant here as it was in the first example. More variations in
This difference in AEP between the two methods is not necessarily a fatal flaw. FLOWERS is likely not a reliable prediction of AEP for a wind farm, but it is difficult to expect a highly accurate and precise estimate of AEP from a low-fidelity wake model anyway. However, as we will illustrate shortly, it is still possible to use the FLOWERS AEP in the layout optimization problem. In fact, the FLOWERS AEP calculation is better suited for layout optimization problems than the numerical integration method. More precise AEP estimates can always be generated as a final post-processing step after the layout optimization is complete.
Before addressing the utility of the FLOWERS formulation in the layout optimization problem, we can explore how to further improve the computational time. For the Jensen numerical integration method, a common approach to reduce the cost of calculating AEP is to reduce the number of wind speed–direction bins, thereby reducing the number of simulations that must be run. Figure
The effect of the resolution of wind direction bins in the Jensen model
The trade-off of sparse sampling of the wind rose is that the AEP from numerical integration is highly sensitive to the number of bins chosen. The AEP varies by as much as 40 % as we reduce the number of wind direction bins from 72 to 9. To reduce the computational cost by a factor of 2, AEP fluctuates by about 2 %; to reduce the cost by a factor of 5, AEP changes up to 10 %. The sensitivity manifests as both overestimates and underestimates of AEP, so it is not possible to assume a conservative underestimate of AEP, for example.
The equivalent idea in the FLOWERS formulation is to reduce the number of Fourier series modes. Each term in the discrete Fourier series is a single arithmetic expression, so the cost should also scale linearly with the number of terms. Figure
With better understanding of the low sensitivity of AEP accuracy to the number of Fourier terms in FLOWERS, we have the opportunity to further reduce the computational cost. To hone in on the appropriate number of Fourier terms to use, we return to the 40 randomized test cases from Sect.
Influence of number of Fourier modes in the FLOWERS solution for the 40 randomized AEP test cases.
We therefore recommend using this truncated FLOWERS solution. By using only one-eighth of the Fourier terms, there is a reduction in cost of roughly a factor of 6 with virtually no trade-off in accuracy. There is no reason to use the extended Fourier series if it increases the computational cost of the FLOWERS solution with no benefit to accuracy.
The wind rose used for the layout optimization studies performed in Sect.
Consider nine turbines placed within a square boundary of side length 12
The AEP that drives the gradient-based optimization is different between both optimizers. However, we wish to compare the quality of the optimal solutions for both without confounding the differences in AEP discussed in Sect.
The first comparison is against the Jensen numerical integration model. The 5
Figure
The optimal layouts for the FLOWERS and Jensen optimizers (
A comparison of cost and performance for 10 cases with randomized initial conditions. Five Fourier modes are used for the FLOWERS solution, and 72 wind direction bins are used for the Jensen numerical integration. FLOWERS is on average about 48 times faster than Jensen and achieves an AEP gain that is 1.5 % higher.
To investigate this result more generally, we consider nine additional multistart cases with randomized initial conditions (10 in total). Figure
The superior performance of FLOWERS compared to the Jensen optimizer connects back to the smooth nature of the formulation. The FLOWERS optimization space is smooth and continuous because of the Fourier transform and analytical integration. On the other hand, the Jensen optimization space is coarse because of the discrete model and numerical integration. The gradient-based optimizer thrives in the smoother optimization space of FLOWERS. More refined adjustments of the turbine positions are possible, and the optimizer is less likely to become stuck in local optimal solutions in the smooth landscape. In the discrete space of the Jensen optimization, it is more difficult for the optimizer to explore the optimization space with equivalent precision and efficiency.
To test the effect of wind rose resolution on the optimization performance, we use
A multistart study, now for the Jensen model with
The optimal layouts for the FLOWERS and Gauss optimizers (with
We should note that the 10 randomized initial layouts tested here represent a limited sample size for a multistart study. The results still allow for a comprehensive discussion of the differences between the FLOWERS and Jensen models. Future work could expand on the scope of the multistart experiments to investigate whether FLOWERS converges to more consistent solutions than Jensen and also whether FLOWERS converges to a solution in fewer iterations than Jensen.
Comparing FLOWERS to a smoother wake model, in this case the Gaussian wake model
Multistart experiment for FLOWERS against the Gauss model with
The results of one of the optimization studies are shown in Fig.
The trade-off for improved performance by using the Gauss numerical integration is in computational cost. The Gaussian optimization took 10 800 s (3 h), while the FLOWERS optimization only required 50.2 s. This is an improvement by a factor of 216 for FLOWERS.
Figure
This experiment suggests that the smoothness of the Gaussian model compared with the Jensen model is the most likely explanation for the improved performance. While the number of wind direction bins is unchanged, the flow field for each simulation is smoother with the Gaussian model. When a turbine's position is adjusted, there is no binary switch between being within the wake or outside of it; this discrete change in wake velocity would cause the AEP to be sensitive to slight perturbations in the turbine positions. On the other hand, in the Gaussian model, a small change in turbine position results in a similarly small change in the wake deficit because of the smooth profile. This continuity produces a smoother solution space, which enables the optimizer to move along more subtle gradients and achieve more optimal solutions than the Jensen optimizer.
While the quality of solutions between FLOWERS and the Gauss optimizer is comparable, there is no contest in terms of cost. The FLOWERS optimization is 2 orders of magnitude faster than the Gauss optimizer and can produce optimal solutions with equivalent performance. Moreover, we are only comparing the results for a wind plant with nine turbines. As illustrated in Fig.
We have demonstrated that the FLOWERS AEP is insensitive to the number of Fourier series terms and have used the truncated series to achieve similar performance to the Gauss optimization. We also previously showed that the AEP calculated from numerical integration is extremely sensitive to the resolution of the wind direction bins. For a fair and comprehensive comparison, the Gauss optimization should be performed with a limited number of wind direction bins to mimic the reduction in cost that was implemented in FLOWERS. To match
The 10 randomized cases for FLOWERS against the Gauss model with
As we have demonstrated, FLOWERS is able to match the performance of conventional layout optimization methods despite simplifications in its formulation. The discrepancies in AEP estimates between FLOWERS and the Jensen and Gauss numerical integrations did not inhibit its application to the optimization problem. However, we could enhance the accuracy of FLOWERS by improving the following.
The objective of this paper was to develop a novel analytical formulation of annually averaged wake velocity to use in a layout optimization problem and demonstrate its effect on reducing the computational cost of these studies. We derived the equations for the analytical integration of the top-hat wake deficit model. The wind speed and wind direction frequency distributions were expressed as a Fourier series to facilitate the integration.
The annually averaged wake velocity was used to compute AEP. We approximated the average power by using the average wake speed cubed rather than an average of the cube of the wake speed, which introduces error when there are pronounced wakes or the wind speed varies significantly across different wind directions. Also, the local wind speed's effect on turbine power production and thrust was not accounted for. These simplifications introduced error that led to the AEP computed in FLOWERS differing from the Jensen numerical integration approach by about 13 %.
Fortunately, these limitations in the accuracy of the FLOWERS AEP do not preclude its use in the optimization problem. The FLOWERS optimizer built around the Jensen wake model finds optimal wind plant layouts with AEP comparable to an optimizer that numerically integrates a Gaussian wake model. This finding is unexpected but promising because it implies that the mathematical formulation behind FLOWERS compensates for a more simplistic wake model to achieve similar results to a more sophisticated wake model. The clear advantage of FLOWERS, then, is the robust layout optimization performance while achieving a reduction in computational cost of 2 orders of magnitude. We believe that this improvement in computation time will scale better with wind farms containing more than the nine turbines studied here.
This achievement could translate to the difference between running an optimization study in 10 min versus 5 d, or between running the study on a personal laptop versus a high-performance computer cluster. This technique could open the door for other areas of research in layout optimization, including optimization under uncertainty, by making these studies more accessible and less costly. Moreover, the new conceptualization of the wake velocity deficit could inspire brand new areas of research in wake modeling and wind plant control and optimization.
This paper serves as a foundation for future work on the FLOWERS formulation. Since the motivation of this approach was to improve computational cost, one avenue to explore is further optimization of the FLOWERS code. Wind plant layout and yaw steering co-design is a popular area of research, and another potential application for FLOWERS if yaw deflection models could be included in the formulation. Future work will also focus on studying the effects of superposition methods and model uncertainty in the FLOWERS formulation. We also plan to validate the performance of the FLOWERS optimal solutions with high-fidelity simulations.
This code is currently under development and not publicly available yet. Information can be obtained from the corresponding authors in the meantime.
The data can be obtained from the corresponding authors.
MJL developed the software, conducted the investigation, and wrote the manuscript. CJB conceptualized the study, developed the software, and supervised the study. MB performed formal analysis and edited the manuscript. GEB acquired funding and edited the manuscript. PF acquired funding. LAMT conceptualized the study, performed formal analysis, and supervised the study.
The contact author has declared that neither they nor their co-authors have any competing interests.
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The authors would like to thank Bart Doekemeijer, Nicholas Hamilton, Jennifer King, Patrick Moriarty, Rafael Mudafort, Eric Simley, and Andrew P. J. Stanley for their feedback and support. The views expressed in the article do not necessarily represent the views of the DOE or the US Government. The US Government retains and the publisher, by accepting the article for publication, acknowledges that the US Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for US Government purposes.
A portion of the research was performed using computational resources sponsored by the US Department of Energy's Office of Energy Efficiency and Renewable Energy and located at the National Renewable Energy Laboratory. This work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the US Department of Energy (DOE) under contract no. DE-AC36-08GO28308. Funding was provided by the US Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office and the US Department of Energy Office of Science Office of Workforce Development for Teachers and Scientists under the Science Undergraduate Laboratory Internships Program. Majid Bastankhah acknowledges funding from Innovate UK (grant no. 89640).
This paper was edited by Jens Nørkær Sørensen and reviewed by Paul van der Laan and one anonymous referee.