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
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 gGIK(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×50D , 30D×30D , and 20D×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 AEP−AEP/AEP×100% , where AEP is the surrogate-predicted AEP of the selected candidate before objective evaluation, and 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
∼ 111
106,065
CMA-ES
–
–
2906.6
∼ 60
∼ 95,000
BO-GIK
2908.6
2907.0
2911.2
∼60
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 AEP−AEP/AEP×100% , where AEP is the surrogate-predicted AEP of the selected candidate before objective evaluation, and 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 nHF=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
∼12 h
∼2.3 h/eval × 48 evals
∼2.3 h/eval × 80 evals
∼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 4×4 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) .
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.
Antonio Candelieri, Laurens Bliek
Department of Economics Management and Statistics, University of Milano-Bicocca, Milan, Italy. · Department of Industrial Engineering and Innovation Sciences, Eindhoven University of Technology, Eindhoven, the Netherlands. · Eindhoven AI Systems Institute, Eindhoven, the Netherlands.
This study provides a comprehensive benchmarking of conventional machine learning (ML), ensemble learning, deep neural networks, recurrent architectures, Transformers, graph based models, and hybrid ensemble deep learning approaches under complementary renewable energy scenarios. Three datasets are considered: a large scale WEC dataset, a 16 WEC dataset, and operational 10 min SCADA measurements at the Penmanshiel wind farm. For structured WEC layout data, tree ensembles exhibited a clear advantage over conventional ML and neural predictors because randomized partitioning and boosting efficiently captured nonlinear layout power interactions without requiring explicit feature representation learning. The Extra Trees was the strongest model, achieving considerable results. Relative to the MLP baseline, this corresponds to an approximately 63.7% reduction in MAE, demonstrating the suitability of randomized tree ensembles for high dimensional structured WEC data. Also, STGCN reduced the MAE to approximately 167.0 kW and achieved R = 0.93 by explicitly learning spatial and temporal turbine interactions. The best overall forecasting accuracy was obtained by the RF BiLSTM hybrid, with an MAE=150.5 kW. Compared with standalone LSTM, this represents an approximately 75% reduction in MAE, while improving on STGCN by approximately 10.0%. Finally, the experiments reveal that no single AI architecture is universally optimal: randomized and boosted ensembles are particularly effective for structured WEC surrogate modeling, graph networks become advantageous when explicit spatial interactions dominate, and ensemble recurrent hybrids provide the strongest balance when nonlinear tabular relationships and temporal dynamics coexist.
Majid Masoumi, Asghar Dashtiy, Mohammad Dehghan +1
Department of Computer Science Yazd University Yazd · Department of Mechanical Engineering Yazd University Yazd
Accurate modeling of wind turbine power curves is crucial for optimal wind farm operation. Nearly all existing power curve models focus on temporal variables such as wind speed and temperature while overlooking the influence of terrain covariates, which governs inflow wind conditions and thus also affects wind power production. This paper proposes a nonparametric spatio-temporal Gaussian process model that integrates temporal environmental covariates with spatial terrain features. The model falls in the category of spatial-temporal Gaussian process models with data on a grid. The challenge to be addressed is that the spatio-temporal modeling require certain temporal alignment among the data, a property that the wind farm data does not have. Our solution strategy is to construct a shared representative temporal covariate set which not only aligns the temporal inputs but also has a size an order of magnitude smaller than the original data size. With this transformation, our resulting model is able to employ a separable kernel structure that captures both spatial and temporal dependencies. Empirical analysis on a real wind farm dataset shows that our method improves predictive accuracy over existing baselines and can be used to quantify the various impact of the terrain characteristics on turbine performance.
Ahmadreza Chokhachian, V. Roshan Joseph, Yu Ding
H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology