An efficient approach to compute the near-field distribution around and within a wind farm under plane wave excitation is proposed. To make the problem computationally tractable, several simplifying assumptions are made based on the geometry problem. By comparing the approximations against full-wave simulations at 500 MHz, it is shown that the assumptions do not introduce significant errors into the resulting near-field distribution. The near fields around a

Due to the rapid development of wind energy around the globe, the adverse effect of large wind turbines on radar and communication systems is raising serious concerns [

The issue of electromagnetic transmission interference, on the other hand, is more subtle and therefore more difficult to assess. There exists a shadow region directly behind a turbine structure where the scattered electromagnetic field cancels destructively with the incident electromagnetic field. Outside the shadow region, the scattered field and incident field interfere due to their phase difference, forming a multipath region characterized by a rapid spatial oscillation pattern [

Both theoretical and experimental works have been conducted on the characterization of the transmission interference problem. A simple, approximate geometrical blockage estimate can be derived based on the Fresnel zone argument [

In this paper, we set out to develop an efficient but approximate electromagnetic approach to compute the received field strength within and around a wind farm. Our approach is based on several simplifying assumptions. First, the turbine scattering is assumed to be dominated by the tower structure of the turbine, while additional scattering from the blades and nacelle structures are assumed to be of secondary importance. Second, due to the large length-to-cross section ratio of typical tower structures, the scattering process is assumed to be predominately two-dimensional (

This paper is organized as follows. In Section

The problem of interest is illustrated in Figure

Illustration of the problem statement.

While a turbine consists of multiple components including the tower, the blades, and the nacelle, it is believed that the tower gives rise to the strongest scattering and shadowing effects. Angulo et al. simulated the RCS of individual components of a turbine using physical optics (PO) and showed that the tower gives the strongest contribution [

Results of the single-turbine simulation at 500 MHz. (a) Near-field distribution around a 3D turbine model including the tower, nacelle, and blades. (b) Near-field distribution around the cone-shaped tower.

3D turbine model

3D finite tower

To test the modeling fidelity required of the turbine structure, we simulate only the cone-shaped tower using MLFMM and plot the near-field distribution in Figure

Next, we simulate the total field around a 64.5 m tall perfect conducting cylinder with the mean diameter of the cone-shaped tower, 3.3 m, at 500 MHz using FEKO. The field in the same observation plane is computed and plotted in Figure

Results of the single-turbine simulation at 500 MHz. (a) Near-field distribution around a 3D finite cylinder. (b) Near-field distribution around an infinite 2D cylinder.

3D finite cylinder

infinite 2D cylinder

Since a simple cylinder can model the tower structure with acceptable error and since the cylinder is very long compared to its cross section, we next investigate the use of a 2D simulation to model the same problem. This amounts to using an infinitely long cylinder to model the finite one, neglecting the effect of the end truncation, and in the process turning the problem into a 2D one. To carry out the 2D simulation in FEKO, we use a short (a half wavelength in height) cylinder with a 3.3 m diameter bounded by periodic boundary conditions (PBCs) on the top and the bottom. The full-wave simulation is done using the MoM solver. This setup is equivalent to an infinitely long cylinder excited by a plane wave. The observation plane is at the center of the short cylinder, and the result is plotted in Figure

The previously mentioned approximation is valid because the finite cylinder is very long compared to its cross section, and we only observe the field at a very close distance. To see the maximum range for the validity of the 2D approximation, the following investigation is carried out. We study the radiation from a uniform, finite line current of length 64.5 m. The result from numerical integration is shown as the solid blue curve in Figure

Electric field strength versus distance from a finite 64.5 m line current at 500 MHz. The field strengths of an infinite line current and an infinitesimal point current are overlaid, respectively, in red dotted and black dashed lines for comparison.

So far, the observation height has been restricted at the middle of the tower. Next, we simulate the same 3D finite cylinder using MLFMM and observe the near-field distribution in the vertical cut plane along the incident field direction. The result is shown in Figure

Near-field distribution around a 3D finite cylinder in the vertical cut plane along the incident field direction computed using MLFMM. The field inside the shadow region is relatively uniform for different observation heights.

As discussed previously, the near field of a turbine can be readily modeled by a 2D cylinder. However, computing the near field at many sampling points at 1 GHz and above is still computationally demanded since the computation time scales as the product of the number of observation positions and the number of current basis functions. To further reduce the computation time for the near field, we use the 2D far-field approximation to compute the 2D scattered field. Shown in Figure

Magnitude plot of the complex echo width (EW) of an infinite 2D cylinder.

Computed near-field distribution around an infinite 2D cylinder at 500 MHz. (a) Exact numerical integration. (b) Complex echo width (EW) approximation. (c) Fresnel approximation.

Infinite 2D cylinder

Complex EW approx

Fresnel approx

For comparison, we also compute the forward shadow using the Fresnel zone blockage formula described in [

In the previous section, we established an approximate approach to efficiently compute the field distribution near a single-turbine structure. To extend the approach to a wind farm consisting of tens or hundreds of turbines, we apply the Born approximation and assume each turbine is fully illuminated by the incident field. This approximation is expected to be the least accurate when the turbines are lined up, so that one turbine casts a shadow over subsequent turbines. However, this scenario exists only at very few incident angles, considering the slenderness of the turbine structure and the large spacing between turbines found in offshore wind farms (600 m–1000 m).

To test whether the Born approximation is reasonable, two 64.5 m long, finite cylinders, each with a 3.3 m diameter, and spaced 600 m apart, are simulated rigorously using FEKO’s MLFMM solver at 500 MHz. The near-field result is plotted in Figure

(a) Near-field distribution around two finite 3D cylinders at 500 MHz computed using the full-wave solver. (b) Near-field distribution around two infinite cylinders generated using the complex EW and the Born approximations.

Two finite cylinders

Born approx. + complex EW approx

Based on the proposed methodology, the near field around a

Near-field distribution around a 3

Near-field distribution around a 3

Near-field distribution around a 3

Finally, we discuss the computation time savings from the proposed methodology. The computation time to generate Figures

Computation time summary for Figures

3D finite cylinder − MLFMM | 2D cylinder − MoM + PBC | 2D cylinder + complex EW | |
---|---|---|---|

Matrix elements | 276 sec | 81 sec | 81 sec |

Precondition | 171 sec | 0.063 sec | 0.063 sec |

Solving | 779 sec (79 iterations) | 1 sec | 1 sec |

Near field | 692 sec | 67 sec | 0.027 sec |

| |||

Total time | 31 min | 2.5 min | 1.4 min |

Computation time summary for Figures

Two 3D finite cylinders − MLFMM | Two cylinders − complex EW + Born | |
---|---|---|

Matrix elements | 1588 sec | 81 sec |

Precondition | 185 sec | 0.063 sec |

Solving | 5905 sec (100 iterations) | 1 sec |

Near field | 17466 sec | 0.132 sec |

| ||

Total time | 7 hrs | 1.4 min |

In this paper, we have studied the near-field distribution around and within a wind farm under plane wave incidence. To make the problem computationally tractable, the following four assumptions were made. First, we assumed that the electromagnetic scattering is dominated by the tower structure and that the effect of the nacelle and blades is of secondary importance. Second, we assumed that the scattering from the tower can be regarded as a 2D problem when the observation distance is sufficiently close. Third, the 2D echo width concept was further applied to simplify the near-field computation. Lastly, we assumed that the interactions between turbines can be neglected and that the Born approximation can be applied when considering a whole wind farm. It was shown through a series of numerical tests that these assumptions can be made without introducing significant errors into the resulting near-field distribution. The near-field distributions of a

It should be noted that the plane wave incidence considered in this paper does not include any phase variations across the scatterers or any realistic decay as a function of distance from the transmitter. Thus, the distance from the transmitter to the wind farm must be sufficiently large to ensure the validity of this assumption. An appropriate modeling methodology for the close-in transmitter case is still needed. It is also important to mention that, while we are initially motivated by offshore wind farms and their impacts on marine radar/communication systems, where both the transmitter and receiver are located along the same plane, changing the elevation angle of the incident field will also be a very interesting study. However, additional formulation will be needed to account for such scenarios. Lastly, when the turbines on a farm are lined up with respect to the incident direction, the Born approximation must be modified to account for the mutual occlusion among the turbines. These will be topics of further study.

This work is supported by the Department of Energy under Grant DE-EE0005380, by the National Science Foundation under Grant ECCS-1232152, and in part by the Texas Norman Hackerman Advanced Research Program under Grant no. 003658-0065-2009.