Research Article  Open Access
Binbin Zhang, Jun Liu, "Wind Turbine Clustering Algorithm of Large Offshore Wind Farms considering Wake Effects", Mathematical Problems in Engineering, vol. 2019, Article ID 6874693, 7 pages, 2019. https://doi.org/10.1155/2019/6874693
Wind Turbine Clustering Algorithm of Large Offshore Wind Farms considering Wake Effects
Abstract
This paper proposed the SVD (singular value decomposition) clustering algorithm to cluster wind turbines into some group for a large offshore wind farm, in order to reduce the highdimensional problem in wind farm power control and numerical simulation. Firstly, wind farm wake relationship matrixes are established considering the wake effect in an offshore wind farm, and the SVD of wake relationship matrixes is used to cluster wind turbines into some groups by the fuzzy clustering algorithm. At last, the Horns Rev offshore wind farm is analyzed to test the clustering algorithm, and the clustering result and the power simulation show the effectiveness and feasibility of the proposed clustering strategy.
1. Introduction
Wind energy is renewable energy, and it can solve a shortage of fossil fuel and an environmental pollution problem. All wind turbines that will be installed by the end of 2020 can cover close to 9% of the global electricity demand [1]. Offshore wind farm is a new trend because of less planning restriction and better wind condition. Compared with the onshore wind farm, the electrical power production of offshore wind farms is higher and more stable.
There are tens or even hundreds of wind turbines in an offshore wind farm, and they bring a “dimension cruise” challenge [2] for a wind farm control [3–5], numerical simulation [6], and so on. In order to reduce the computation complexity, the common method is to establish an equivalent model for wind farm model reduction [7], and it is a key to cluster the samefeature wind turbines into a group and an equivalent single machine. In recent years, several wind turbine clustering algorithms have been proposed [8–14]. A model reduction method is proposed by a set of orthogonal modes from CFD (computational fluid dynamics) simulation [8]; however, the simulation time is too long for several wind turbines. An aggregated wind farm model is proposed by the average wind speed [9, 10]. A wind turbine clustering algorithm is considered by Hankel singular values [11] or selective modal analysis [12]. However, the wind speed at the downstream wind turbines is smaller than that at the upstream wind turbines in wind farms; this phenomenon is defined as wake effects, and these wind turbine clustering algorithms [9–12] are not considered wake effects of an offshore wind farm.
Coordinates of wind turbines are very regular in an offshore wind farm, and the wind speed and direction are stable, so wake effects of every wind turbine are very regular. Based on the wind farm wake model, wind turbines can be clustered into several groups [13, 14]. The support vector clustering technique is used to cluster wind turbines based on the wind farm layout and incoming wind direction [13]. The kmeans clustering algorithm divides wind turbines into several groups [14]. However, the wind farm wake model is a highdimensional mathematical model, and the kmeans clustering and the support vector clustering algorithms are inefficient and easily converted to a local minimum with more dimensions; at the same time, the results of two clustering algorithms are poor robustness [15]. To solve the highdimensional problem of wind turbine clustering, SVD (singular value decomposition) is an effective clustering algorithm for large datasets [15].
In this paper, the SVD clustering algorithm is proposed for an offshore wind farm to overcome the highdimensional problem. A wind farm model is firstly established based on the Jensen wake model, layout of wind farm, and incoming wind speed, a wake combination matrix of every wind turbine is built from a wind farm wake model, and wind turbines are clustered into some groups by an SVD of the wake relationship matrix. At last, an order reduction wind farm model is obtained for power maximizing, power balance control, and so on.
This paper is organized as follows: Section 2 introduces the wind turbine model and the wake model of an offshore wind farm. Then, the SVD clustering algorithm is discussed for the wake model in Section 3. The Horns Rev offshore wind farm is tested in Section 4. Finally, conclusions are drawn in Section 5.
2. Wind Farm Wake Model
There are many wakeeffect models, such as the Frandsen analytical model [16], Jensen model [17], Larsen model, and CFD (computational fluid dynamics) model [18]. In this paper, the Jensen wake model [18] is adopted because it is simple and suitable for engineering applications [18].
The Jensen wake model is based on the global momentum conservation and assumption of a linear expansion of the wake. Figure 1 shows the basic Jensen model, the radius of the wind turbine is r_{0}, the ambient wind speed is , and the wake decay constant is k. If a wind turbine is not affected by any upstream wind turbine, k = 0.04; otherwise, k = 0.08 [19]. r is the radius of the expanding wake, and it can be calculated by (1). And the wind speed inside the wake area at a distance x from the single wind turbine can be calculated by (2), where C_{T} is the wind turbine thrust coefficient:
In an offshore wind farm, a downstream wind turbine is affected by multiple wind turbines, and multiple wake effects can be combined into a single wake effect. And the combining multiple wake effects consider the shadowed areas of the upstream wind turbines. The shadow condition, between an upstream wind turbine and a downstream wind turbine, is complete shadowing, quasicomplete shadowing, partial shadowing, and no shadowing. The partial shadowing is shown in Figure 2, the wind turbines’ radius r_{0} is the same, and the swept area of the wind turbine is A_{0}. Then, the shadow area between the two wind turbines can be calculated bywhere is the distance between the upstream wind turbine i and the downstream wind turbine j along the wind direction and is the wake stream radius, which can be calculated by (1).
Based on the law of momentum conservation, the combining multiple wake model [19] of the jth wind turbine is calculated bywhere .
3. A Wind Turbine Clustering Algorithm via SVD
The layout of an offshore wind farm is regular, the distance between turbines is the same, the wake effects of some downstream wind turbines are the same, so the samewakeeffect wind turbines can be clustered as a group and equate a rescaled single wind turbine. From (4), the wind speed of downstream wind turbines is determined by the geographical location and the work condition of upstream wind turbines, and the C_{T} can be regulated by a wind turbine. Hence, the geographical location is selected as a clustering index [13, 14]. However, the clustering index is 1D data in [13, 14], and the dimension is high as the number of wind turbines increases. A 2D wake relationship matrix can be established from 1D data by analyzing (2), and it is more suitable than 1D data for an offshore wind farm and contains the relative location of wind turbines [20] The 2D wake relationship matrix is a sparse matrix. And the SVD clustering method is effective to solve the highdimensional sparse matrix clustering problem [21].
3.1. Estimation of the Wake Relationship Matrix of Every Wind Turbine
An offshore wind farm has m rows with n wind turbines, and the distance of wind turbines is regular. A wake relationship matrix of the ith row and jth column wind turbine is defined aswhere is the element of a wake relationship.
From the wind direction and the wind turbine geographical location, the wake effect between two wind turbines can be obtained. If there is a wake effect between the ijth wind turbine and the pqth wind turbine, an element of a wake relationship is β_{ij}; otherwise, the element is 0, if there is not a wake effect, or itself. So the a_{pq} of a wake relationship matrix is defined as
Generally, the shadowing condition of wind turbines can be judged using the basic geometrical relationship.
3.2. A SVD Clustering Algorithm of Offshore Wind Farm
The SVD is an orthogonal matrix reduction, the nonzero singular values contain the most information of the matrix, and it has the advantages of dimension reduction, insensitivity to matrix perturbation, scale invariance of singular values, rotation invariance of singular values, ability to solving the best approximation matrix, and so on [19]. And the proposed wind turbine clustering algorithm flow chart is shown in Figure 3 and is implemented as follows: Step 1: every wind turbine coordinate, wind direction, and wind turbine parameters, such as the radius of the wind turbine and distance between wind turbines, are obtained. Step 2: an original coordinate (X, Y) of every wind turbine is transformed into another coordinate system (x, y) in the wind direction as (7), where is the wind direction with the positive Xaxis: Step 3: the wake stream radius and shadow area of the wind farm are calculated based on Section 2. Step 4: the wake relationship matrix is established from (5) and (6). Step 5: the singular value decomposition of is calculated as follows: where U and V are the left and right singular orthogonal vectors, respectively, and , where [18]. Step 6: the values are clustered by the fuzzy clustering method [15], and these wind turbines can be clustered into k groups {, , …, }. And other parameters of the wind turbine are aggregated by a mechanical torque compensation factor method [9]. Finally, the simplified wind farm model is built.
4. Case Study
The Horns Rev offshore wind farm in Denmark [22] is used to test this clustering algorithm. It consists of eighty 2 MW wind turbines, and every wind turbine has a hub height H = 70 m and a rotor diameter D = 80 m. And the wind farm layout is parallelogram columns, the distance between two columns is 7D, the distance between turbines is 7D, 9.4D, and 10.4D for 0°, 48°, and 312°, respectively, and the angle between the first column and yaxis is approximately 7°. Its shape is shown in Figure 4, and it has 8 rows and 11 columns. The wake model of the wind farm is established under eight wind directions which are 270°, 246°, 222°, 201°, 180°, 173°, 138°, and 90° based on the wind farm layout. The clustering results are shown in Figure 5. When the wind direction is 270°, the firstcolumn wind turbines are not affected by other wind turbines, their wind speeds are the ambient wind speed, and wind speeds of othercolumn wind turbines decrease in turn. And when wind directions are 222° and 312°, the clustering results are similar to the layout of the wind farm. With the wind direction increases, the clustering results are very regular, so a wind farm clustering lookup table can be built for wind farm control and numerical simulation.
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In order to verify the clustering results, suppose that the C_{T} of all wind turbines is the same and C_{T} = 0.865 and the ambient wind speed is 12 m/s. The wind speed of each wind turbine is shown in Figure 6. The wind speed of wind turbines is the same if they are in the same group. From Figure 6, it can be seen that the clustering results are effective and feasible.
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The Horns Rev offshore wind farm power simulation is tested by the SVD clustering algorithm and detailed model in MATLAB, which considers every wind turbine powerout. And the power simulations are run on a 3.6 GHz Core i74790 CPU with 8 GB RAM using MATLAB version R2014a.
Suppose that the wind speed is 12 m/s and all wind turbines are maximizing power point tracking. And the detailed and equivalent wind farm power curves are shown in Figure 7 at the wind direction range of 180°∼270°. From Figure 7, it can be seen that the error between the equivalent model and the detailed model is negligible, and the maximum error is 0.108 MW.
However, when the wind speed of wind farms is over the rated speed, the results of the proposed clustering algorithm may be imprecise. When the ambient wind speed is 17 m/s, it is over the rated wind speed, some wind turbines are power limit controllers, and the C_{T} of them is different with the MPPT wind turbines. And the detailed and equivalent wind farm power curves are shown in Figure 8. From Figure 8, it can be seen that the maximum error is 9.98 MW, and the error may be large in some wind farm power simulation. So the proposed algorithm can be used when the ambient wind speed is less than the rated speed and the C_{T} of the samegroup wind turbines is the same.
And the computational cost of two wind farm models is shown in Table 1, and the computational efficiency of the proposed wind farm model is higher than that of the detailed model. Moreover, the SVD clustering algorithm is also used for the wind farm power control and power grid simulation considering wind farm, wind farm powermaximizing control, etc.

5. Conclusion
The main contribution of this paper is the proposed SVDbased clustering method for largescale offshore wind farms to solve the highdimensional problem. Wind turbines can be clustered into several groups based on the location of each wind turbine and wind direction, and the samegroup wind turbines, whose C_{T} is the same, can be equivalent to a single wind turbine, in order to solve the highdimensional problem in the wind farm control algorithm and numerical simulation.
Based on the layout of wind farm and wind direction, a wind farm wake model is established, a wake relationship matrix is based on the wake model, a singular matrix is calculated by SVD, and finally, wind turbines can be clustered into groups by the fuzzymeans method from singular values. SVD can reduce the high dimension of the wind farm wake model, and the clustering results are relative wind direction and are very regular. Moreover, the large wind farm power control or power grid power simulation with wind farms can reduce the computation time by clustering wind turbines into some groups using this clustering algorithm.
Data Availability
Previously reported wind turbine coordinates and the Horn Rev wind farm parameters data were used to support this study and are available at DOI: https://doi.org/10.1016/j.renene.2014.06.019. These prior studies and datasets are cited at relevant places within the text as reference [22].
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the Key R&D Project, Shaanxi, China, under Grant no. 2017GY061.
References
 Wind Power Capacity Worldwide Reaches 600 GW, 53.9 GW Added in 2018, https://wwindea.org/.
 T. R. Ayodele, A. Jimoh, J. L. Munda, and A. J. Tehile, “Challenges of grid integration of wind power on power system grid integrity: a review,” International Journal of Renewable Energy Research, vol. 2, no. 4, pp. 618–626, 2012. View at: Google Scholar
 B. Zhang, M. Soltani, W. Hu, P. Hou, Q. Huang, and Z. Chen, “Optimized power dispatch in wind farms for power maximizing considering fatigue loads,” IEEE Transactions on Sustainable Energy, vol. 9, no. 2, pp. 862–871, 2018. View at: Publisher Site  Google Scholar
 H. Zhao, Q. Wu, J. Wang, Z. Liu, M. Shahidehpour, and Y. Xue, “Combined active and reactive power control of wind farms based on model predictive control,” IEEE Transactions on Energy Conversion, vol. 32, no. 3, pp. 1177–1187, 2017. View at: Publisher Site  Google Scholar
 D.Y. Li, P. Li, W.C. Cai, Y.D. Song, and H.J. Chen, “Adaptive faulttolerant control of wind turbines with guaranteed transient performance considering active power control of wind farms,” IEEE Transactions on Industrial Electronics, vol. 65, no. 4, pp. 3275–3285, 2018. View at: Publisher Site  Google Scholar
 D. C. C. Crisostomo, A. A. F. Moura, E. P. Rocha, F. M. Cruz, and A. P. Moura, “Educational software for simulation of power and voltage control in power systems connected with wind farms,” IEEE Latin America Transactions, vol. 16, no. 6, pp. 1603–1609, 2018. View at: Publisher Site  Google Scholar
 Y. Ni, C. Li, Z. Du, and G. Zhang, “Model order reduction based dynamic equivalence of a wind farm,” International Journal of Electrical Power & Energy Systems, vol. 83, pp. 96–103, 2016. View at: Publisher Site  Google Scholar
 Y. Heggelund, C. Jarvis, and M. Khalil, “A fast reduced order method for assessment of wind farm layouts,” Energy Procedia, vol. 80, pp. 30–37, 2015. View at: Publisher Site  Google Scholar
 M. A. Chowdhury, W. X. Shen, N. Hosseinzadeh, and H. R. Pota, “A novel aggregated DFIG wind farm model using mechanical torque compensating factor,” Energy Conversion and Management, vol. 67, pp. 265–274, 2013. View at: Publisher Site  Google Scholar
 X. Zha, S. Liao, M. Huang, Z. Yang, and J. Sun, “Dynamic aggregation modeling of gridconnected inverters using Hamilton’sactionbased coherent equivalence,” IEEE Transactions on Industrial Electronics, vol. 66, no. 8, pp. 6437–6448, 2019. View at: Publisher Site  Google Scholar
 S. Ghosh and N. Senroy, “Balanced truncation based reduced order modeling of wind farm,” International Journal of Electrical Power & Energy Systems, vol. 53, pp. 649–655, 2013. View at: Publisher Site  Google Scholar
 H. A. PulgarPainemal and P. W. Sauer, “Towards a wind farm reducedorder model,” Electric Power Systems Research, vol. 81, no. 8, pp. 1688–1695, 2011. View at: Publisher Site  Google Scholar
 M. Ali, I.S. Ilie, J. V. Milanovic, and G. Chicco, “Wind farm model aggregation using probabilistic clustering,” IEEE Transactions on Power Systems, vol. 28, no. 1, pp. 309–316, 2013. View at: Publisher Site  Google Scholar
 S. Ma, H. Geng, G. Yang, and B. C. Pal, “Clusteringbased coordinated control of largescale wind farm for power system frequency support,” IEEE Transactions on Sustainable Energy, vol. 9, no. 4, pp. 1555–1564, 2018. View at: Publisher Site  Google Scholar
 S. Wierzchoń and M. Kłopotek, Modern Algorithms of Cluster Analysis, Springer, Berlin, Germany, 2018.
 S. Frandsen, R. Barthelmie, S. Pryor et al., “Analytical modelling of wind speed deficit in large offshore wind farms,” Wind Energy, vol. 9, no. 12, pp. 39–53, 2006. View at: Publisher Site  Google Scholar
 N. O. Jensen, A Note on Wind Generator Interaction, Risø National Laboratory, Roskilde, Denmark, 1983.
 R. Shakoor, M. Y. Hassan, A. Raheem, and Y.K. Wu, “Wake effect modeling: a review of wind farm layout optimization using Jensen’s model,” Renewable and Sustainable Energy Reviews, vol. 58, pp. 1048–1059, 2016. View at: Publisher Site  Google Scholar
 I. Katic, J. Højstrup, and N. O. Jensen, “A simple model for cluster efficiency,” in Proceedings of the European Wind Energy Association Conference and Exhibition, A. Raguzzi, Rome, Italy, October 1987. View at: Google Scholar
 P. Drineas, A. Frieze, R. Kannan, S. Vempala, and V. Vinay, “Clustering large graphs via the singular value decomposition,” Machine Learning, vol. 56, no. 1–3, pp. 9–33, 2004. View at: Publisher Site  Google Scholar
 F. GonzálezLongatt, P. Wall, and V. Terzija, “Wake effect in wind farm performance: steadystate and dynamic behavior,” Renewable Energy, vol. 39, no. 1, pp. 329–338, 2012. View at: Publisher Site  Google Scholar
 Y.T. Wu and F. PortéAgel, “Modeling turbine wakes and power losses within a wind farm using LES: an application to the horns rev offshore wind farm,” Renewable Energy, vol. 75, pp. 945–955, 2015. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2019 Binbin Zhang and Jun Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.