Efficient wind farm layout optimization with the FLOWERS AEP model and analytic gradients

Wind farm layout optimization (WFLO) is a set of engineering design problems wherein the positions of turbines in a wind farm are the design variables in an optimization problem. The majority of WFLO studies in the literature optimize for annual energy production (AEP) or cost of energy (COE),¹ both of which require a model to predict wind farm AEP for a given layout. AEP is a measure of the wind farm's performance in converting the wind's kinetic energy flux to electrical power; setting aside questions about the aerodynamic or generator efficiency of individual turbines, the main sources of inefficiency in power production at the wind farm level are wake interactions between turbines. In worst-case wind directions (i.e., aligned wind farm rows), wake losses can account for power deficits of up to 40% in utility-scale wind farms,^2,3 but the optimal placement of turbines can mitigate the worst of these performance impacts. Various socioeconomic or engineering restrictions on the placement of turbines may also be included in the WFLO problem, with the most fundamental being boundary constraints that enforce regions where turbines can and cannot be placed (i.e., inclusion and exclusion zones, respectively).

There are three key challenges to solving the WFLO problem: (1) multimodality, (2) dimensionality, and (3) cost. First, the solution space is highly multimodal due to the nonlinear relationship between turbine positions and wind farm AEP;^4,5 numerous locally optimal solutions exist in the design space, requiring more careful analysis and selection of optimization algorithms than for simple linear problems.⁶ Second, the design space typically spans a large number of dimensions. Assuming that the turbine positions are parameterized independently,^7–9 a wind farm of N turbines yields a WFLO problem with 2 N design variables. As the number of turbines grows, the high-dimensional design space can pose challenges for optimization solvers.⁶ Third, the computational cost of these studies can be high and can require high-performance computing resources. Even with simple AEP models with computation times on the order of a second or faster,¹⁰ repeated objective and constraint function evaluations can drive the total computation time of these studies to the order of hours or more.¹¹ The proper choice of objective function AEP model and optimization algorithm addresses some of these challenges.

Physics-based AEP estimates typically rely on the integration of low-fidelity engineering wake models across a range of expected inflow conditions.¹² Standard, “conventional” AEP models are based on numerical integration, wherein the power production is simulated in numerous independent flow scenarios (e.g., combinations of freestream wind speed, wind direction, and turbulence intensity), and AEP is obtained through rectangular quadrature across each discrete wind condition, weighted by its expected frequency of occurrence. While parallelization of these simulations can alleviate some computational cost,¹¹ there exists a fundamental cost scaling relationship with the number of discrete wind conditions. Several recent studies have investigated alternative techniques to avoid this scaling issue, including polynomial chaos expansion,^13,14 Bayesian quadrature,¹⁵ and Monte Carlo integration,¹⁶ each of which were able to reduce the cost of AEP evaluations by about an order of magnitude. However, these statistical approaches are still fundamentally based on numerical integration of an underlying wind farm model and are not expected to significantly change the AEP design space. FLOWERS is an alternative AEP model developed specifically for WFLO problems that leverages an analytical integral of the wind farm flow model across wind conditions instead of using the typical numerical integration approach. The result is a closed-form, analytical expression for wind farm AEP and its derivatives with respect to turbine position, which has been shown to reduce the cost of AEP predictions by an order of magnitude and yield a design space for the WFLO problem that is smoother and less multimodal.^17,18

The increase in smoothness and the reduction in multimodality are both important, particularly when solving the WFLO problem with a gradient-based algorithm. The most fundamental difference between gradient-based and gradient-free methods is that gradient-free algorithms rely solely on objective and constraint function evaluations to search for optimal solutions, whereas gradient-based algorithms make use of derivative information of the objective and constraint functions in addition to their values.¹⁹ Gradient-free methods have historically been a large subset of the WFLO literature,^20–29 but they scale poorly with the number of design variables⁶ and, therefore, are less appropriate for high-dimensional design spaces.³⁰ Because gradient-based methods scale better with the number of design variables, thereby addressing the challenge of dimensionality, the WFLO literature has gained increased interest in these methods,^31–39 especially for larger wind farms. The drawbacks of gradient-based methods are their sensitivity to initial conditions and an inability to guarantee global optimality, both of which can be problematic for multimodal problems.⁶ Consequently, gradient-based WFLO studies typically rely on multi-start approaches,^35,40–43 in which a large set of optimization problems is run with randomized initial conditions and analyzed in aggregate.

Gradient-based methods require gradients of the objective and constraint functions. Ideally, gradients are available in functional form as analytic derivatives (AD). Guirguis et al.⁸ first derived the analytic derivatives for a simple AEP model to use in a WFLO study, and analytic derivatives of the FLOWERS AEP model were derived in LoCascio et al.¹⁸ Another option for exact gradients is the use of automatic or algorithmic differentiation approaches, which numerically collate function gradients using a chain rule through each operation of an algorithm.¹⁹ Automatic differentiation has been used in several recent WFLO studies in the literature,^{5,9,11,16,30,40,41,44} but it can place requirements on the implementation of the wake modeling and optimization code to be compatible with off-the-shelf algorithmic differentiation software packages. (Examples of open-source wind farm modeling tools that are built around automatic differentiation include TOPFARM⁴⁵ and FLOWFarm.⁴⁶) If exact gradients are not available, finite difference (FD) methods can often provide sufficient approximations. However, FD approximations require multiple function evaluations per gradient estimate and are prone to numerical error.¹⁹ 

Our objective in this paper is to analyze the improved performance of wind farm layout optimization using a combination of the FLOWERS AEP model and a gradient-based optimization algorithm with analytic gradients. Using a multi-start approach in three case studies with different wind roses, wind farm sizes, and realistic wind farm boundaries, we illustrate the impact of the type of gradient information (AD vs FD) and the choice of AEP model (FLOWERS vs Conventional) on four quantities of interest for WFLO studies: optimal wind farm AEP, computation time of the optimization solver, number of iterations for optimization convergence, and the average time per iteration. As an addendum, we investigate the sensitivity of the optimization results to the FLOWERS model parameter tuning, which is an otherwise unknown user input to the optimization framework.

This paper is organized as follows. In Sec. II, we define the wind farm layout optimization problem, including the AEP models for the objective function and the boundary constraint function. In Sec. III, we define our case studies and discuss how we compare the performance of different combinations of flow models and derivative calculation methods used in the layout optimization problem. Our optimization results from these case studies are discussed in Sec. IV. In Sec. V, we explore the sensitivity of the FLOWERS optimization results to the parameter tuning. Finally, we summarize our results and key takeaways in Sec. VI.

We neglect a minimum spacing constraint between turbines in this formulation. We acknowledge that an explicit spacing constraint would be necessary in a realistic WFLO setting, but our simplified problem definition allows us to focus on our stated objectives of analyzing different objective functions and gradient calculation methods while avoiding the unnecessary computational cost associated with spacing constraints between every pair of turbines.¹¹ A basic feature of AEP models is that power production increases as turbine separation increases, meaning that the AEP model implicitly drives turbines to separate from one another. However, this separation is not guaranteed to meet a reasonable threshold for safety [e.g., 2–3 rotor diameters (D)]. We verify in our optimization results that the minimum spacing between all turbines is at least 1D, meaning that no turbines are colliding with one another.

To support our objective to benchmark the performance of different approaches in formulating the WFLO problem, we must distinguish between two different features of the optimization problem. First, we have two choices for the AEP model in the objective function: FLOWERS or Conventional. Second, we have two possible sources of gradient information for the gradient-based algorithm: exact analytic derivatives (AD) or finite-difference estimates (FD). As discussed previously, one of the contributions of the FLOWERS model is that AD gradients are available in the form of analytical functions. Meanwhile, AD gradient information is only available for the Conventional model via automatic differentiation; due to the details of our Conventional model's implementation (which will be discussed in Sec. III B), automatic differentiation is not feasible, and therefore, we assume that FD gradients are the only option. Consequently, we can construct three different combinations from these options: FLOWERS-AD, FLOWERS-FD, and Conventional-FD. Note that both FLOWERS-FD and Conventional-FD use FD estimates for all gradients, including for the boundary constraint function.

Pairwise comparisons of these approaches distinguish the effects of the objective function model or gradient source. By comparing FLOWERS-AD and FLOWERS-FD directly, we can investigate the impact of analytic derivatives compared with finite-difference estimates for the same objective function. Meanwhile, the comparison of FLOWERS-FD and Conventional-FD isolates the effect of the AEP model from the gradient calculation method. Finally, the FLOWERS-AD vs Conventional-FD comparison highlights the overall advantages of using the FLOWERS-based approach with analytic derivatives over a standard approach to solve the WFLO problem. Because these three pairwise comparisons are sufficient to assess the combined impact of the FLOWERS AEP model and the gradient calculation method on WFLO, the Conventional-AD approach is not included in our analysis (for other studies comparing the performance of Conventional-AD and Conventional-FD approaches, see Thomas et al.,⁵³ and Baker et al.³⁵).

The FLOWERS model is implemented in Python in an open-source repository.⁵⁴ Due to the Fourier transform of the wind rose [Eqs. (8) and (9)], the concept of wind direction discretization and resolution in the Conventional model is replaced in the FLOWERS model by a continuous Fourier series with a finite number of modes to represent the wind direction probability distribution function. We use $M=10$ Fourier modes in the FLOWERS solution [Eq. (10)]; previous work has shown that this resolution is a satisfactory balance between computational cost and precision.¹⁸ We select a parameter value of the wake expansion rate k = 0.05 corresponding to offshore conditions.⁵⁵ 

The Conventional model is implemented via FLORIS, a controls-oriented wake modeling library developed by the National Renewable Energy Laboratory (NREL).⁴⁸ The Gaussian wake model parameter $k*$ is tuned based on an ambient turbulence intensity of $7%$⁠, which is roughly representative of offshore conditions; all other model parameter values are standard in FLORIS. We use 72 wind direction bins, corresponding to $5°$ resolution, based on previous analysis of the design space of the Conventional AEP model with a Gaussian wake model.¹⁸ 

SNOPT is accessed through the pyOptSparse wrapper in Python.⁵⁶ The problem functions are scaled such that the magnitudes of all gradient elements are on the order of 1 or less, as suggested by Martins and Ning;⁶ this scaling is performed independently for each approach (e.g., FLOWERS-FD vs FLOWERS-AD) in each case study and held constant across each randomized multi-start sample. The optimality and feasibility tolerances are set at $εo$ =  $10−3$ and $εf$ =  $10−4$⁠, respectively, for all cases based on convergence studies. All optimization cases are run on individual nodes on a high-performance computing cluster, each with 18 3.0 GHz dual-core processors, with a time limit of 96 h to conserve computational resources.

We consider three different case studies in Sec. IV, as summarized in Fig. 1. These cases span different sized wind farms with qualitative differences in the wind farm boundary and wind rose, although the particular definitions are arbitrary. The small case (SM) denotes a wind farm with 10 turbines enclosed by a square boundary with a wind rose characterized by a single dominant direction and high wind speeds. The medium case (MD) defines a wind farm with 50 turbines in an irregular pentagon with a bidirectional wind rose. The large case (LG) consists of a wind farm with 250 turbines in a more complex boundary with a more uniform wind rose.

We assume that these wind farms represent offshore conditions where the mean wind flow and turbulence characteristics are approximately homogeneous throughout the domain and across different wind directions.¹⁸ The turbines are modeled as NREL 5-MW reference turbines⁵⁷ with rotor diameter $D=126$ m, hub height of 90 m, and tabulated thrust and power curves. In the objective functions, the turbines are modeled as single points at the rotor center. However, in our post-processing routine, we instead model the turbines as a three-by-three array of points over which the incident wind speed is evaluated to better capture partial wake effects.

The wind roses are sampled from the Wind Integration National Dataset (WIND) Toolkit.⁵⁸ The highest available resolution is $1°$ for the wind direction and 1 m/s for the wind speed (in the range $u∞∈[1,25]$ m/s), corresponding to 360 wind direction bins and 25 wind speed bins. This high-resolution wind rose is used in the post-processing routine for the AEP of initial and optimal layouts. For the objective functions, we evaluate AEP using a wind rose with a single average wind speed per wind direction to decrease computational cost.

Our comparison of the performance of the three WFLO approaches consists of three parts. First, to elucidate the inner workings of the SNOPT solver and the success of the optimization runs, we discuss the optimizer's exit conditions. Second, to illustrate the qualitative differences between the output solutions of different approaches, we visually compare the optimal layouts. Third, to detail differences in performance beyond the layouts themselves, we present a quantitative comparison in terms of optimal AEP, solver time, solver iterations, and solver time per iteration.

To compare optimal AEP, we delineate between the “inner-loop” AEP model in the objective function and the “outer-loop” AEP model we use in our post-processing and analysis; although the inner-loop models may change between approaches (FLOWERS or Conventional), we standardize the outer-loop model (Conventional) to provide a fair comparison of the outputs. Thus, we use the Conventional AEP model with 360 wind direction bins and 25 wind speed bins to estimate the AEP of the initial and optimal layouts in our analysis. Solver time is the wall time required for an individual optimization study with given initial conditions to be completed on a single processor. Solver iterations are the number of major iterations completed before the SNOPT algorithm exits. Ideally, the SNOPT solve exits successfully and the specified convergence criteria have been met. However, imprecise gradient estimates can cause SNOPT to terminate prematurely at a suboptimal point.⁵⁹ Finally, from total solver time and iterations we can calculate average solver time per iteration, which normalizes the effect of the convergence length on the solver time to better compare performance between approaches.

As mentioned previously, we take a multi-start approach to solve the WFLO problem with our gradient-based algorithm. For each case study defined in Fig. 1, we randomly generate 100 feasible wind farm layouts (i.e., all turbines are within the boundary and spaced at least 2D apart). The three WFLO approaches in question are run from the same 100 initial design points and converge to different optimal design points in different amounts of time and iterations. We compare the approaches in aggregate (e.g., total computation time for all 100 randomized cases in cpu-hours), in the median (e.g., median solver time or median optimal AEP across 100 randomized cases), in their best case (e.g., best optimal AEP out of 100 randomized cases), or in their variability (e.g., standard deviation of optimal AEP). We also verify that the optimal solutions are feasible, meaning that all turbines are placed within the wind farm boundary and all turbines are spaced at least 1D apart. It is possible for an optimization to be terminated prematurely (e.g., due to numerical issues or a time limit), which can result in an infeasible solution; these cases are filtered out of the multi-start distributions for all quantities of interest except aggregate computation time.

One method to decide on the number of multi-start studies is to run enough cases until the distribution of a quantity of interest (e.g., optimal AEP) has converged. Assuming that there is a finite number of local optima in the design space, it is expected that a large number of randomized initial starting points should eventually locate all of these locally optimal solutions. We expect that with more randomized starting points, the variability of the resulting solutions should reach an asymptotic limit. Thus, the convergence criterion for the multi-start study could be a sufficiently low variance of optimal AEP. In this study, we select a fixed number of multi-start studies for the sake of an even comparison between different approaches. However, we analyze the variability in optimal AEP across the multi-start study as a proxy for convergence: an approach with lower variance in optimal AEP for a fixed number of randomized studies could achieve a target variance in optimal AEP in a lower number of randomized studies. A reduction in the number of randomized multi-start studies is a linear reduction in the aggregate time. Consequently, reduced variability in optimal AEP ultimately represents a potential reduction in aggregate cost in a practical setting.

We first review the output information from the SNOPT solver to assess the behavior of the approaches. The exit conditions for each case are summarized in Table I. We see that 100 of the FLOWERS-FD cases and 99 of the FLOWERS-AD cases finished successfully; the one problematic FLOWERS-AD case will be discussed in more detail below. However, over 70% of the Conventional-FD cases terminated after numerical difficulties, which is most likely caused by inaccurate or imprecise gradient information yielding a suboptimal point that cannot be improved. Two of these Conventional-FD cases resulted in infeasible solutions, which are filtered out of our results in the remainder of this section. Based on these findings, our initial estimate of FLOWERS-AD and FLOWERS-FD is that their convergence is more well-behaved than Conventional-FD.

We highlight the output of an individual study from each approach for visualization. Figure 2 shows the optimal layout of the best individual study from the three approaches, where “best” refers to the highest optimal AEP out of the 100 randomized starts. The two FLOWERS layouts are virtually identical, with differences in AEP of less than 0.05%. The solver places seven of the ten turbines on the west boundary, which corresponds to the predominant leading edge of the farm based on this wind rose [Fig. 1(a)]. The other three turbines are placed on the east boundary to maximize spacing with the other turbines. However, the Conventional-FD layout is noticeably different, with six turbines placed on the west boundary instead of seven. While there is only a marginal discrepancy in the optimal AEP of 0.3%, this qualitative difference in layout is indicative of underlying differences in the design space of Conventional-FD vs FLOWERS-AD and FLOWERS-FD due to the choice of AEP model.

We can visualize all of the multi-start layouts by superimposing them, as shown in Fig. 3. The initial layouts show no pattern due to their randomized nature. On the other hand, the FLOWERS-AD and FLOWERS-FD layouts show a consistent placement of turbines across the 100 multi-start studies. Similar to Fig. 2, turbines are placed primarily on the west and east boundaries of the farm. The small differences between local solutions are also visible. For example, a subset of layouts have nine turbines on the east and west boundaries and single turbine placed on either the north or south boundary. Another small subset of solutions places a turbine in the interior of the wind farm. However, there are no qualitative differences between FLOWERS-AD and FLOWERS-FD. We expect the design space for these two approaches to be identical since they share the same objective function model; assuming that the effect of FD errors on the gradient descent is small, there should be little difference between the distributions of their optimal solutions. On the other hand, the Conventional-FD optimal layouts do not illustrate the same agreement with the FLOWERS layouts or the same degree of consistency. There is still a higher density of turbines along the boundary of the wind farm than in the interior, but it is evident that more turbines are placed in the interior of the wind farm than for the FLOWERS layouts. Furthermore, there is less of a distinction between the leading and trailing edges (west and east, respectively) of the farm vs the lateral edges (north and south) compared to the FLOWERS layouts. The Conventional AEP model's design space is more multimodal than the FLOWERS AEP model,¹⁸ which is reflected in the wider range of local solutions that appear as outputs of the Conventional-FD approach.

For a more quantitative analysis, we compare plots of our quantities of interest. The optimal AEP and solver time for these studies are shown in Fig. 4(a). In terms of optimal AEP, the distribution of the FLOWERS-AD and FLOWERS-FD cases is extremely similar, with medians within 0.01% of each other; this finding highlights our claim that the design spaces of these approaches are practically identical. The Conventional-FD cases display noticeably lower optimal AEP, with a median that is about 1.4% lower than the FLOWERS distributions. Another way to analyze this trend is through the lens of AEP gain per study (i.e., the relative increase in the optimal AEP against the initial AEP), which gauges the marginal improvement in the objective from the optimization process; FLOWERS-AD and FLOWERS-FD both achieve a median gain of 4.5%, while Conventional-FD yields a lower median gain of 2.9%. The Conventional-FD distribution of optimal AEP is also more variable than the FLOWERS distributions: the standard deviation for Conventional-FD is about 1.7 GWh vs 0.8 GWh for FLOWERS-AD and FLOWERS-FD. This higher variability of the Conventional-FD solutions directly follows from the higher variability of the optimal layouts themselves (Fig. 3). One might hypothesize that initial conditions with higher AEP would lead to better optimal conditions. However, our results show no pattern between better starting points and better optimal solutions; the correlation coefficient of optimal AEP vs initial AEP is less than 0.15 for these three distributions.

The computing time for these approaches is substantially different. The median time per study for the models are as follows: FLOWERS-AD, 0.9 s; FLOWERS-FD, 2.4 s; Conventional-FD 55 s. The aggregate time of the three approaches across the 100 cases show similar trends: FLOWERS-AD, 0.03 cpu-hours; FLOWERS-FD, 0.07 cpu-hours; Conventional-FD, 1.84 cpu-hours. In other words, the effect of the AD vs FD gradients on the total time is about a factor of 2.5, and the effect of the FLOWERS model vs the Conventional model is about a factor of 27. Altogether, FLOWERS-AD is about 64 times faster than Conventional-FD for the SM case.

The optimal AEP is plotted against the solver iterations in Fig. 4(b). All of the distributions display a similar shape with a skew toward higher iterations. It appears that for some of these cases, the convergence of the optimizer can hang while the step size of the gradient descent search becomes smaller, leading to high iteration counts with little marginal change in optimal AEP. The one FLOWERS-AD case that exited with numerical difficulties is visible here, with over twice the number of iterations as the next highest case. However, the FLOWERS-AD cases overall tend to converge in about 20% fewer iterations than the FLOWERS-FD and Conventional-FD cases. It follows that the AD gradients are responsible for faster convergence of FLOWERS-AD in this study, and the FD gradients appear to make convergence slightly more difficult due to numerical error.

The ratio of solver time and iterations is plotted in Fig. 4(c). The Conventional-FD solver time per iteration is about 24 times greater than for FLOWERS-FD, which in turn is about 2 times greater than the time per iteration of FLOWERS-AD. So, we observe that the Conventional AEP model and its FD gradient estimate is an order of magnitude more costly than the FLOWERS AEP model. The effect of the gradients on the solver time is less severe, but the efficiency of AD gradients is expected to be more pronounced as the dimensionality of the problem increases (e.g., the MD and LG cases).

Overall, the difference in computational cost between the approaches is a combination of convergence length, AEP model cost, and gradient calculation cost. The FLOWERS AEP model is about an order of magnitude less costly to evaluate than the Conventional AEP model, and the AD gradients are about a factor of 2 less costly than the FD gradients in this case. Furthermore, the FLOWERS model yields a smoother design space with fewer local optima than the Conventional model, and the AD gradients enable faster convergence than their FD counterparts, both of which cause the FLOWERS-AD approach to achieve convergence in fewer iterations. Consequently, the reduced convergence length and time per iteration of FLOWERS-AD result in a decrease in 2 orders of magnitude in overall solver time compared to Conventional-FD.

We again start our analysis with the SNOPT exit conditions, as shown in Table II. Similar to the trend from the SM case, we see that 100% of the FLOWERS-AD and FLOWERS-FD runs and only 20% of the Conventional-FD runs finished successfully, with the rest terminating after numerical difficulties (i.e., inaccurate or imprecise gradients). One of the Conventional-FD cases resulted in an infeasible solution, which is filtered out of the following results.

The layouts from the best studies are shown in Fig. 5. The FLOWERS layouts are again almost identical, with optimal AEP within 0.01%, and markedly different from the Conventional-FD layout, with a difference in AEP of about 0.3%. The FLOWERS layouts possess a clear row structure on the interior of the farm and regular placement of turbines along the boundary. This structured arrangement of turbines is a more natural and intuitive layout of turbines in real utility-scale wind farms so the fact that the optimizer finds mathematical optimality in this type of organization provides confidence in our solution. We note that these rows tend to be aligned with the predominant southwesterly wind in this case [Fig. 1(b)], a somewhat unexpected result, suggesting that the geometry of this particular wind farm area is more dominant than the wind direction distribution in this optimization problem. Meanwhile, the Conventional-FD layout shows less clear rows and organization, and the discrepancy with the FLOWERS layouts reinforces the divergence of the two models' design spaces.

The superimposed layouts across the multi-start studies, as shown in Fig. 6, follow the trend from the SM case. There is a similarity between the FLOWERS-AD and FLOWERS-FD optimal layouts and a contrast with the Conventional-FD layouts. The FLOWERS layouts show more consistency compared to the Conventional-FD approach, as indicated by the clustering of turbines across the boundary and at certain points within the interior. It is also interesting to note the distinct margin between the boundary and the interior region where turbines are placed by FLOWERS-AD and FLOWERS-FD, which represent a consistent spacing of about 3–4D between turbines within the farm and along its boundary. There is more variability in the Conventional-FD layouts, with less clustering of turbines in the superimposed layouts and a less distinct margin on the interior of the farm boundary.

The quantities of interest from the three approaches are plotted in Fig. 7. In terms of optimal AEP, the results are similar to the SM case: the FLOWERS-AD and FLOWERS-FD distributions are almost identical, with a difference in the median AEP of 0.01%, and both improve upon the Conventional-FD output by about 0.5%. The AEP gain reflects this trend, with a median gain for FLOWERS-AD and FLOWERS-FD of 4.9% per study and 4.4% for Conventional-FD. The standard deviation of AEP for the FLOWERS-AD and FLOWERS-FD distributions are similar (0.97 and 1.11 GWh) and are less than half the variability of the Conventional-FD distribution (2.33 GWh). This result suggests that the FLOWERS-AD and FLOWERS-FD multi-start studies converge with fewer randomized starts than Conventional-FD, which could potentially reduce the overall cost of the multi-start study. So, we can conclude that the FLOWERS-AD and FLOWERS-FD optimal layouts and their respective AEP again perform slightly better than the Conventional-FD layouts and are more consistent. We also note that there again is no substantial correlation between initial and optimal AEP for the three approaches, with correlation coefficients less than 0.12 for the three distributions.

The difference in computation time [Fig. 7(a)] between the three approaches is greater for this MD case than for the SM case. The median time per study between FLOWERS-AD (1.9 s), FLOWERS-FD (33 s), and Conventional-FD (34 min) span 3 orders of magnitude. The aggregate times are as follows: FLOWERS-AD, 0.07 cpu-hours; FLOWERS-FD, 0.92 cpu-hours; and Conventional-FD, 64.92 cpu-hours. In other words, FLOWERS-AD reduces computation time by roughly a factor of 15 relative to FLOWERS-FD and roughly a factor of 1000 relative to Conventional-FD for the MD case.

The trends from the iteration count per multi-start run [Fig. 7(b)] differ from the SM case. The FLOWERS-FD distribution now has the lowest median number of iterations (58) compared with FLOWERS-AD (86) and Conventional-FD (94). We also observe that the shape of these distributions is noticeably different. The FLOWERS-AD iteration distribution is highly skewed toward higher iteration counts, and the points on the upper end of the distribution tend to yield optimal AEP above the median. The FLOWERS-FD distribution is less skewed, but there still appears to be a positive correlation between convergence length and solution quality. Finally, the Conventional-FD distribution does not exhibit a substantial correlation between iteration count and optimal AEP.

We can draw two conclusions from these results. First, the AD gradients in FLOWERS-AD appear to enable more precise convergence to the specified optimality conditions where the FD gradients would cause the optimization to terminate. It is arguable that the marginal gain in AEP of this fine-tuned convergence is not worth the marginal cost, which could be addressed by modifying the scaling of the optimization problem as discussed in Sec. III B or by adjusting the convergence tolerance. Second, the convergence of Conventional-FD appears to be less efficient in terms of improving optimal AEP with increased iterations. The Conventional-FD approach also appears to be exiting at a suboptimal point due to numerical issues before it navigates to a locally optimal solution.

In terms of solver time per iteration [Fig. 7(c)], we again see the drastic improvement of the FLOWERS-AD and FLOWERS-FD approaches over Conventional-FD. The FLOWERS-AD median time per iteration (0.02 s) is about 29 times faster than for FLOWERS-FD (0.57 s), which is itself about 38 times faster than Conventional-FD (22.02 s). The order of magnitude difference between FLOWERS-AD and FLOWERS-FD is an increase from the SM case because of the effect of the problem's dimensionality. The number of design variables (2 N) has grown from 20 to 100, meaning that the number of AEP gradient elements has grown by a factor of 5, each of which requires two function evaluations to estimate with finite-difference methods. The AD gradient only requires one evaluation of the analytic gradient function irrespective of the number of design variables, so its cost is expected to scale much better with the number of design variables. The Conventional-FD approach again takes an order of magnitude longer than FLOWERS-FD at each iteration. Overall, FLOWERS-AD reduces the time per iteration by a factor of 1110 relative to Conventional-FD for the MD case.

We return to the idea that the total solver time is the product of the number of iterations and time per iteration, which itself is a function of the objective and gradient evaluation costs. The faster FLOWERS AEP model reduces the time per iteration by a factor of 38 relative to the Conventional model, as expected. Furthermore, the AD gradients account for a factor of 29 improvement in efficiency over FLOWERS-FD. The convergence length does not support the same pattern between the three approaches due to the complex interaction between the AEP models' design space, the gradient source, and the optimization algorithm tuning. However, the FLOWERS-AD and FLOWERS-FD approaches tend to converge in fewer iterations than Conventional-FD, and the overall computation time of the FLOWERS-AD multi-start study is about 990 times faster than the Conventional-FD study in this case.

The SNOPT exit conditions for the LG case are displayed in Table III, where we confirm that all of the FLOWERS-AD and FLOWERS-FD runs finished successfully. Only 17 of the Conventional-FD cases exited successfully, but now 62 of the runs were terminated at our imposed time limit of 96 h. Two of the Conventional-FD cases also resulted in infeasible solutions, which we have filtered out of the remainder of the results in this section.

We visualize the layouts from the best studies in Fig. 8. The FLOWERS-AD and FLOWERS-FD layouts are again qualitatively and quantitatively similar, with optimal AEP within 0.01% and clearly defined rows and columns of turbines within the interior of the farm. The Conventional-FD layout is not as uniform as the FLOWERS layouts and achieves an optimal AEP that is about 0.3% lower. The superimposed layouts across the multi-start studies are not shown for brevity, but they follow the trends of the MD and SM results in that the FLOWERS-AD and FLOWERS-FD layouts are more consistent and structured than the Conventional-FD layouts.

The quantities of interest are displayed in Fig. 9. The most glaring result from Fig. 9(a) is that this large wind farm of $N=250$ turbines exceeds the computational limits of the Conventional-FD approach. The median time per study for the models is as follows: FLOWERS-AD, 81 s; FLOWERS-FD, 88 min; and Conventional-FD, 96 h. The aggregate times of the three approaches follow a similar pattern: FLOWERS-AD, 3.9 cpu-hours; FLOWERS-FD, 164.7 cpu-hours; and Conventional-FD, 8271.6 cpu-hours. The total improvement in time between FLOWERS-AD and Conventional-FD is a factor of about 4200 in the median and 2100 in aggregate; if we allowed the Conventional-FD studies to run without a time limit, this difference in time would be even larger.

Regarding AEP, FLOWERS-AD and FLOWERS-FD achieve optimal AEP within 0.01% of each other. The quality of the Conventional-FD layouts is noticeably lower than the FLOWERS solutions, with about a 0.6% reduction in optimal AEP. In terms of AEP gain, FLOWERS-AD and FLOWERS-FD achieve a median gain of 3.6% per study compared with 3.0% for Conventional-FD. We again observe reduced variability of the FLOWERS-AD and FLOWERS-FD distributions in terms of standard deviation (3.7 and 4.7 GWh, respectively) compared with the Conventional-FD distribution (10.1 GWh); however, we acknowledge that since the majority of the Conventional-FD runs did not exit naturally, this distribution cannot be considered converged.

In terms of the convergence behavior [Fig. 9(b)], the trends from the SM and MD cases are reflected in this LG case. The FLOWERS-AD distribution possesses the most distinct tail toward high iteration counts, elucidating the ability of the AD gradients to pursue precise convergence. The shape of the Conventional-FD distribution is stunted by the artificial cutoff of our imposed time limit, meaning that the tail toward longer convergence lengths is not present. In this case, FLOWERS-AD achieves a lower median iteration count (176) than FLOWERS-FD (190) and Conventional-FD (194), which indicates that while some runs experienced prolonged convergence, a significant portion of the FLOWERS-AD runs converge in a relatively low number of iterations.

Finally, the solver time per iteration [Fig. 9(c)] follows the same hierarchy between approaches that we observed in the SM and MD cases: FLOWERS-AD is the fastest, followed by FLOWERS-FD and Conventional-FD. Quantitatively, the time per iteration is as follows: FLOWERS-AD, 0.4 s; FLOWERS-FD, 30.5 s; and Conventional-FD, 28.1 min. The memory requirements of evaluating the Conventional AEP model for this large wind farm, combined with the cost of the FD gradient estimates, result in significantly slower time per iteration than the previous case studies. Even between FLOWERS-AD and FLOWERS-FD, we observe how the FD gradient estimates are about 83 times slower than the AD gradient calculations in this case.

Overall, FLOWERS-AD is three orders of magnitude faster than Conventional-FD for this case study. The iteration counts of the three approaches are within 10% of one another, meaning that the majority of this difference in total time is due to the solver time per iteration, not the number of iterations. The FLOWERS-FD approach is about 55 times faster than Conventional-FD, illustrating the effect of the faster FLOWERS AEP model. Between FLOWERS-AD and FLOWERS-FD, the reduction in time of the AD approach by two orders of magnitude highlights the efficiency of analytic gradients for high-dimension optimization problems.

A summary of the quantitative results across the three case studies is shown in Table IV. We observe that the FLOWERS-AD and FLOWERS-FD approaches achieve similar optimal AEP, within 0.03% of each other, and improve on the Conventional-FD approach's best AEP by about 0.3% for all three case studies. We also see about 55%–64% lower standard deviation in AEP for FLOWERS-AD and FLOWERS-FD relative to Conventional-FD across the three case studies. It is important to note that while the FLOWERS-based approaches do show a consistent improvement in this quantitative benchmark, it is not guaranteed that these marginal improvements in AEP will be actualizable in real wind farms whose AEP yields are subject to a large degree of uncertainty.

The efficiency of the FLOWERS-AD approach is clear from these results, with a speed-up of two orders of magnitude in the SM case and 3 orders of magnitude in the MD and LG cases relative to Conventional-FD. While we cannot directly analyze trends in these statistics as a function of the number of turbines—due to the change in wind farm shape, sparsity, and expected wind conditions that result in different convergence behavior—this information does give us a general sense of the impact of the number of design variables on the problem's cost. The AD gradients' cost scales much more efficiently with the number of turbines, which is supported by the order-of-magnitude increase in the ratio of the FLOWERS-AD and FLOWERS-FD approaches' times between the SM and MD cases. The effect of the AEP model is more consistent across the three case studies, with a factor of 20–50 reduction in the computation time per iteration between FLOWERS-FD and Conventional-FD. The trend of the convergence behavior of the approaches is less obvious across the three case studies, but we find that FLOWERS-AD and FLOWERS-FD converge in 10%–20% fewer iterations than Conventional-FD despite the fact that the majority of the Conventional-FD runs terminated after numerical difficulties.

The WFLO problem, as defined in Eq. (1), requires as inputs a probability density function of the freestream wind conditions, an initial wind farm layout, a set of points defining the wind farm boundary, some tunable model parameters, and other known inputs. We assume that the wind farm boundary, turbine thrust and power curves, and wind rose are well-characterized, known inputs. Therefore, for layout optimization with FLOWERS, there is a single unknown tuning parameter that must be prescribed by the user: the wake expansion rate k, which is assumed to be constant for all wind turbines and across all wind conditions. The recommended values for this parameter are 0.05 in offshore conditions and 0.075 in onshore conditions.¹⁸ 

In our simple WFLO problem definition with homogeneous inflow conditions, the freestream wind does not vary throughout the domain. Consequently, optimal layouts that maximize power production are those which mitigate power losses due to wake interactions. The FLOWERS AEP model we use to predict these wake losses is physics-based, and the tuning of the wake model parameter using information about the physical characteristics of the system is important for improving the accuracy of AEP predictions. However, the optimal layouts tend to follow geometric heuristics: increase the spacing between turbines and avoid aligning turbines with the predominant wind directions. Our hypothesis is that the precise value of the wake model parameter is not critical to obtain optimal layouts from a WFLO study.

In this section, we conduct a simple parameter sensitivity study to test this hypothesis. We sweep the value of the wake expansion rate over a range $k∈[0.01,0.10]$⁠, which spans the order of magnitude of the recommended tuning, and perform the multi-start study with the FLOWERS-AD approach over 100 randomized initial conditions for each parameter value. Then, we compare the best optimal layouts from the multi-starts and the distributions of our performance metrics (optimal AEP, solver time, solver iterations, and solver time per iteration) across the parameter sweep.

We present the results from the MD case only, which are representative of the results from the SM and LG cases. The best optimal layouts from the 100 randomized starts for each parameter value are displayed in Fig. 10. Similar to our discussion of the superimposed layouts from the main multi-start study in Sec. IV B (Fig. 6), we observe clusters of turbines that indicate consistent placement of turbines across the parameter sweep. Qualitatively, we can conclude that the solutions of FLOWERS-AD are extremely similar regardless of the choice of parameter value; in other words, a user could expect to find roughly the same best layout from their multi-start study, irrespective of the value of the wake expansion rate that they choose in this prescribed range. The quantitative information supports this assessment: the AEP computed from these layouts is $1037.9±0.4$ GWh, meaning that the optimal AEP from these 10 layouts varies by less than 0.1%. We note that this variability in optimal AEP is not sensitive to the tuning of our post-processing Conventional AEP model parameters either, which are distinct from the FLOWERS model parameter.

In terms of our performance metrics in Fig. 11, we see that the parameter selection has a slight effect on convergence. In Fig. 11(b), we note a trend of higher median iteration counts with lower values of k, which coincides with a marginally higher optimal AEP for these parameter values. Smaller k translates to narrower wakes that do not dissipate as quickly with distance from the turbine, which produces a design space with sharper local optima for AEP. Meanwhile, larger values of k result in smoother, less pronounced wake regions in the AEP design space that yield smoother local optima. Our results indicate that these wider wakes lead to faster convergence, which is consistent with the results of the wake expansion continuation method developed by Thomas et al.⁵.

Peculiarly, there is also a correlation between the value of k and the solver time per iteration [Fig. 11(c)]. It is unlikely that the wake expansion parameter value would influence the run time of the AEP model and its gradient calculation. However, it is possible that the difference in time is due to internal adjustments by SNOPT to its solution approach while encountering numerical difficulties for lower values of k. Regardless, the variability of solver time per iteration is within a range of roughly 20%, while the convergence lengths span half an order of magnitude. Thus, the trend in solver time [Fig. 11(a)] is mainly explained by the differences in convergence behavior.

Our objective in this work was to benchmark and analyze the performance of wind farm layout optimization using the FLOWERS AEP model and analytic gradients compared with current reference practices. We considered three pairwise comparisons of problem formulations: FLOWERS-AD vs FLOWERS-FD to isolate the effect of analytical functions for the problem gradients compared with finite-difference estimates, FLOWERS-FD vs Conventional-FD to analyze the effect of the physics-based AEP model in the objective function, and finally FLOWERS-AD vs Conventional-FD to discuss the holistic advantages of the FLOWERS-based layout optimization formulation over a baseline approach. We tested these three combinations in three case studies, with different numbers of turbines, wind farm boundaries, and wind roses, and analyzed the optimization results from 100 randomized initial conditions for each case.

Our analysis of the quantitative performance metrics across these case studies demonstrated that the major advantage of FLOWERS-AD is in terms of computational cost. The median time per optimization for FLOWERS-AD was two orders of magnitude faster in the SM case study and three orders of magnitude faster in the MD and LG case studies than the Conventional-FD approach. We attribute this boost in efficiency to both the lower cost of evaluating the FLOWERS AEP model and the lower cost of obtaining gradient information from analytical functions (especially for problems involving larger wind farms), which results in a lower time per iteration for the solver. We also found that the FLOWERS-based approaches achieve convergence in roughly 10%–20% fewer iterations than the Conventional-FD approach, which we attribute to the well-conditioned design space of the FLOWERS AEP model.

Our qualitative analysis of the optimal layouts showed that the solutions of the FLOWERS-AD and FLOWERS-FD approaches were very similar, while the optimal layouts from the Conventional-FD approach were visually less similar and less consistent. Quantitatively, the FLOWERS AEP model led to an increase in optimal AEP of about 0.3% across all three case studies compared to the Conventional AEP model. We also found less variability in the optimal layouts and associated optimal AEP of the FLOWERS-based formulations relative to the Conventional-FD approach across the randomized multi-start cases. This finding suggests that multi-start studies with the FLOWERS objective function could likely converge with fewer randomized starts, thereby reducing aggregate costs.

To test the robustness of the approach to uncertain model parameters, we also analyzed the sensitivity of the FLOWERS optimization results to the tuning of the wake model parameter value. Across a sweep of values for the unknown wake expansion rate, we found that the best optimal layouts were almost identical and resulted in AEP predictions within 0.1% of one another. These results demonstrate negligible sensitivity of layout optimization to the tunable input parameter of FLOWERS. As wake model parameter tuning is a notoriously challenging process, this insensitivity in the optimization results makes FLOWERS-based layout optimization more reliable and user-friendly.

Overall, these results demonstrate the potential of the FLOWERS-AD approach to significantly streamline the solution of the layout optimization problem. This study was limited to the consideration of wind farms in simple, homogeneous inflow conditions (i.e., wind farms in flat terrain), and future work will focus on developing the FLOWERS model to include heterogeneous inflow effects caused by complex terrain or neighboring wind farms. We also encourage the application of the FLOWERS AEP model with analytic gradients to more realistic layout optimization problems, including different objective functions (e.g., cost of energy), different constraints (e.g., wind farm exclusion areas), or different design variable definitions (e.g., boundary-grid parameterizations).

A portion of the research was performed using computational resources sponsored by the U.S. 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 U.S. Department of Energy (DOE) (Contract No. DE-AC36-08GO28308). Funding provided by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. 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 U.S. Government purposes.

The authors have no conflicts to disclose.

Michael J. LoCascio: Conceptualization (equal); Investigation (lead); Methodology (equal); Visualization (lead); Writing – original draft (lead). Christopher J. Bay: Conceptualization (equal); Methodology (equal); Supervision (equal); Visualization (supporting); Writing – review & editing (equal). Luis A. Martínez-Tossas: Conceptualization (equal); Methodology (equal); Supervision (equal); Visualization (supporting); Writing – review & editing (equal). Jared J. Thomas: Methodology (equal); Writing – review & editing (equal). Catherine Gorlé: Conceptualization (supporting); Methodology (equal); Supervision (equal); Visualization (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.