A Graph-informed Kriging model for sample-efficient Bayesian optimization of wind farm layouts

Zhen-Fan Wang; Kai Zhang; Xianping Du; Zhaolong Han; Dai Zhou; Onur Bilgen

cs.LGOct 26, 2026

A Graph-informed Kriging model for sample-efficient Bayesian optimization of wind farm layouts

Authors: Zhen-Fan Wang, Kai Zhang, Xianping Du, Zhaolong Han, Dai Zhou, Onur Bilgen

Organizations: State Key Laboratory of Ocean Engineering, School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China · School of Ocean Engineering and Technology, Sun Yat-Sen University & Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, 519000, China · Shenzhen Research Institute of Shanghai Jiao Tong University, Shenzhen, 518063, China · Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, New Jersey 08854, USA

Abstract

Wind farm layout optimization becomes increasingly difficult as turbine count grows, especially when high-fidelity aerodynamic models are used. Conventional Kriging-based Bayesian optimization often suffers from low sample efficiency in high-dimensional layout spaces because sparse training data provide limited domain coverage. We develop a Graph-informed Kriging framework that embeds reusable aerodynamic interaction information into the surrogate mean function. The wind farm is represented as a directed aerodynamic graph, where pairwise turbine interactions defined in a low-dimensional relative-coordinate space are assembled into a farm-scale mean. This structured mean captures the dominant variation in annual energy production (AEP), allowing the Gaussian process to model only the residual structure and uncertainty. Numerical experiments show that Graph-informed Kriging consistently improves AEP prediction accuracy over ordinary and generic-trend Kriging under sparse-data conditions, with increasing benefits as layout dimension and wake-interaction strength grow. Embedded in Bayesian optimization, it substantially improves sample efficiency and achieves competitive solutions for the IEA Wind Task 37 benchmark with about 280 AEP evaluations. In a controlled 16-turbine case, it also outperforms ordinary and multi-fidelity Kriging under the same high-fidelity evaluation budget. The framework is further extended to a 48-turbine wind farm, demonstrating applicability to large-scale CFD-based layout optimization. These results show that Graph-informed surrogate modeling can enable more efficient use of high-fidelity aerodynamic models for reliable large-scale wind farm layout design.

Figures & tables

Study	Flow/turbine model	Optimization strategy	Turbine count	Main feature
King et al. (2017)	RANS+ADM	Discrete-adjoint gradient optimization	16	Adjoint sensitivities are obtained at a cost largely independent of the number of design variables, but the resulting gradient-based optimization remains local and can be sensitive to the initial layout.
Antonini et al. (2020)	RANS+ADM	Continuous-adjoint gradient optimization	14	Extends efficient adjoint-based CFD optimization to complex terrain, while retaining the initialization sensitivity associated with local gradient-based search.
Cruz and Carmo (2020)	RANS+ADM	Genetic algorithm	20	Direct CFD-in-the-loop evolutionary optimization; demonstrates feasibility but also the high evaluation demand of population-based search.
Wang et al. (2024b)	RANS+ADM	Kriging-based Bayesian optimization	8	Reduces the number of CFD evaluations through Kriging-based BO, but surrogate construction becomes increasingly data-demanding as the layout dimension grows.
Bempedelis et al. (2024)	LES+ADM	Kriging-based Bayesian optimization	16	Direct LES-in-the-loop BO demonstrates feasibility of high-fidelity surrogate optimization, while also illustrating its substantial sampling cost.
Shao et al. (2025)	RANS+ADM	Kriging-based Bayesian optimization	8	Reduces both the cost of each AEP evaluation and the number of expensive optimization evaluations through PCE and surrogate modeling.

Table 1: Representative CFD-based WFLO studies

Figure 1: Overview of the GIK-driven Bayesian optimization framework. The framework leverages a graph-theoretic mean function to capture global aerodynamic trends, guiding the efficient exploration of the design space while minimizing AEP evaluations.

Figure 2: Spatial relations of a turbine pair used in the Graph-informed model.

Figure 3: The wind rose and wind farm boundary for case studies: $(a)$ Wind 1, characterized by a unimodal directional distribution; $(b)$ Wind 2, characterized by a multidirectional distribution; $(c)$ square wind farm boundary (Boundary 1); $(d)$ circular wind farm boundary (Boundary 2); and $(e)$ irregular non-convex wind farm boundary (Boundary 3).

Figure 4: Predictive performance of the $g_{\mathrm{GIK}}(\boldsymbol{z})$ model under varying coupling orders $m$ . Top row ( $a-d$ ) and bottom row ( $e-h$ ) correspond to wind scenarios 1 and 2, respectively. Scatter plots show predicted versus true AEP, where the dashed diagonal line denotes perfect agreement. The solid red line indicates the linear fit obtained using Orthogonal Distance Regression. Color intensity represents point density estimated via a Gaussian kernel.

Figure 5: MAPE comparison of GIK and conventional Kriging with constant, linear, and polynomial mean functions for Wind 1 and Wind 2 shown in Figs. 3 ( $a$ ) and ( $b$ ). $(a,b)$ Effect of training sample size from $1N$ to $10N$ with $N=50$ . $(c,d)$ Effect of turbine number from 25 to 100 at fixed wind farm density. $(e,f)$ Effect of wind farm density with $N=50$ , where the sparse, moderate, and dense cases correspond to $50D\times 50D$ , $30D\times 30D$ , and $20D\times 20D$ domains, respectively.

Figure 6: Evolutions of optimized AEPs obtained from BO-GIK and BO-OK. $(a)$ – $(c)$ show the results for Wind 1 with Boundaries 1–3, respectively, while $(d)$ – $(f)$ show those for Wind 2 with Boundaries 1–3, respectively. The lines represent the mean best-so-far AEP over 10 independent runs, and the bars shown at 50-iteration intervals indicate the corresponding minimum-to-maximum range across the 10 runs.

Figure 7: Computational time profiles of BO-GIK and BO-OK across the six optimization cases. ( $a$ )–( $c$ ) show the results for Wind 1 with Boundaries 1–3, respectively, while ( $d$ )–( $f$ ) show those for Wind 2 with Boundaries 1–3, respectively. The lines represent the mean computational time per optimization iteration over 10 independent runs, and the bars shown at 50-iteration intervals indicate the corresponding minimum-to-maximum range across the 10 runs.

		BO-GIK ( $m=2$ , 200 iter.)		BO-GIK ( $m=1$ , 200 iter.)		BO-OK (500 iter.)
Wind	Boundary	AEP (GWh)	Time (min)	AEP (GWh)	Time (min)	AEP (GWh)	Time (min)
1	1	466.5	52.3	465.7	38.5	436.4	127.1
	2	458.0	52.9	457.7	42.5	426.6	128.4
	3	729.8	65.4	729.2	52.9	695.9	181.3
2	1	365.5	54.3	365.0	49.3	354.7	117.9
	2	356.7	59.8	356.4	49.1	345.8	128.1

Table 2: Optimization performance and computational cost of BO-GIK and the extended BO-OK runs across the six analytical-wake-model WFLO cases. BO-GIK results are reported after 200 iterations, whereas BO-OK results are reported after 500 iterations. The optimized AEP and cumulative computation time are averaged over 10 independent runs.

Figure 8: Prediction errors of selected candidate samples during the optimization process. The prediction error is defined as $\left|\widehat{\mathrm{AEP}}-\mathrm{AEP}\right|/\mathrm{AEP}\times 100\%$ , where $\widehat{\mathrm{AEP}}$ is the surrogate-predicted AEP of the selected candidate before objective evaluation, and $\mathrm{AEP}$ is the corresponding actual value obtained from the wake model.

Figure 9: The ( $a$ ) wind farm boundary and ( $b$ ) wind rose for the IEA Wind Task 37 considered by Thomas et al. (2023) .

Method	Mean AEP (GWh)	Min. AEP (GWh)	Max. AEP (GWh)	Runtime (min)	AEP evaluations
SNOPT+WEC	2906.3	2898.8	2910.1	16.5	17,626
DEBO	–	–	2913.2	$\sim$ 111	106,065
CMA-ES	–	–	2906.6	$\sim$ 60	$\sim$ 95,000
BO-GIK	$\boldsymbol{2908.6}$	$\boldsymbol{2907.0}$	$\boldsymbol{2911.2}$	$\boldsymbol{\sim 60}$	$\boldsymbol{81+200}$

Table 3: Optimization performance and computational cost of GIK and selected direct optimization methods. Results for SNOPT + WEC, DEBO and CMA-ES are taken from Thomas et al. (2023) . The ’-’ indicates that the corresponding quantity is not reported in the original study. Runtime denotes the average time per optimization run where multiple runs are performed. For BO-GIK, the total number of AEP evaluations is decomposed into initial evaluations (81 evaluations) and subsequent infilling evaluations performed during the optimization iterations.

Figure 10: Optimization performance comparison among BO-GIK, BO-OK, and BO-MFK for the 16-turbine ADM-RANS-based WFLO case. ( $a$ ) Evolution of optimized AEP during the optimization process. ( $b$ ) Prediction errors of the surrogate models at the selected candidate layouts during sequential optimization. The prediction error is defined as $\left|\widehat{\mathrm{AEP}}-\mathrm{AEP}\right|/\mathrm{AEP}\times 100\%$ , where $\widehat{\mathrm{AEP}}$ is the surrogate-predicted AEP of the selected candidate before objective evaluation, and $\mathrm{AEP}$ is the corresponding actual value obtained from ADM-RANS. BO-MFK results are obtained with different low-to-high fidelity sample ratios ( $Q=10$ , $20$ , and $40$ ).

Figure 11: Estimated computational cost of surrogate modeling and candidate search in BO, relative to one CFD-based AEP evaluation, for the present implementations of OK, GIK and MFK, with $n_{\mathrm{HF}}=16$ , 66, and 116 on the ( $a$ ) 32- and ( $b$ ) 64-core workstations.

Figure 12: ( $a$ ) The wind turbine layout and boundary in Lillgrund wind farm. ( $b$ ) The wind rose for the CFD-based 48-turbine WFLO studies. The turbine positions and original wind data are obtained from Sebastiani et al. (2021) , and the wind data are then grouped into six directional sectors, with the occurrence frequency and mean wind speed of each sector used to define the wind rose.

Figure 13: The CFD-based WFLO results obtained using the GIK-based BO framework. ( $a$ ) Directional AEP distributions under different wind conditions. ( $b$ ) Comparison of the baseline and optimized layouts, with turbine-level AEP distributions. ( $c$ ) Velocity contour at the hub height of the optimized layout.

Two-turbine database preparation	Initial CFD evaluations	BO infill CFD evaluations	Total
$\sim 12$ h	$\sim 2.3$ h/eval $\times$ 48 evals	$\sim 2.3$ h/eval $\times$ 80 evals	$\sim 306.4$ h

Table 4: Computational wall-clock time breakdown for the 48-turbine CFD-based WFLO case.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 14: ( $a$ ) Setups for wind farm CFD simulations. ( $b$ ) Mesh dependency test. A $4\times 4$ wind farm, characterized by streamwise and spanwise turbine spacings of $5D$ and $3D$ , respectively, is used for the test. ( $c$ ) Validation of the predicted row-averaged power output of ADM-RANS against the reference ALM-LES data from Cao et al. (2025) .

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Optimal Transport-based Permutation-Invariant Bayesian Optimization of Offshore Wind Farm Layouts

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Bayesian Optimization

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Bayesian Optimization (BO) is widely and successfully adopted for solving optimization problems having an expensive-to-evaluate, black-box, and non-convex objective function. However, the vanilla BO algorithm is not able to exploit possible symmetries characterizing the target problem. An intuitive case is given by optimal location problems, whose decision variables refer to a finite set of points within a continuous space, with the order of points not affecting the value of the objective function. We refer to this setting as optimization over layouts to distinguish from optimization over point-clouds where, instead, the order of points counts. As an instance of optimization over layouts we consider a real-life industrial-relevant application, that is the optimization of the layout of an offshore wind farm: given identical wind turbines, switching any pair of them has not any effect on the annual energy production. Based on Optimal Transport theory, we propose a Permutation-Invariant BO approach, namely PIBO, proved to provide better wind farm layouts when compared to the vanilla BO approach while cutting computation time roughly in half.

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