EQUIVALENT EQUATION OF EARTH RESISTANCE FOR RING ELECTRODE OF WIND TURBINE

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EQUIVALENT EQUATION OF EARTH RESISTANCE FOR RING ELECTRODE OF WIND TURBINE

YASUDA Yoh FUJII Toshiaki

Kansai University Otowa Electric Co., Ltd. [email protected] [email protected]

ABSTRACT Earth resistance is a one of the most important issues for wind turbine lightning protection. Although various equations have been proposed to simulate earth resistance, it is still difficult to accurately estimate an actual wind turbine earth impedance because of its complex shape combined with a concrete foundation, auxiliary vertical and ring electrodes. This report clarified how the shape and size of the foundation and the auxiliary electrodes affects to the earth resistance using FDTD (Finite Difference Time Domain) calculation. An equivalent equation to practically simulate wind turbine earthing can be proposed with some correction parameters and improved conventional equations.

1 INTRODUCTION

Lightning protection of wind power generation is becoming an important public issue as installations of wind turbines have greatly increased worldwide and its commutative capacity has exceeded at 150 GW [1]. Wind turbines are often struck by lightning because of their open-air locations, special shape and very-high construction. As well as seriously damaging blades, accidents where low-voltage and control circuits breakdown frequently occur in many wind farms. Although some reports, including IEC TR 61400-24: 2002 [2], have indicated a methodology for protection against such accidents, a standardized solution remains to be established. This problem has recently surfaced as an important issue. Among several elements of lightning protection, an earthing (grounding) system is one of the most important points to be considered. However, few investigations and reports have been published as far as wind turbine earthing except of a few papers [3]-[6].

Although various equations including the famous Sunde's equation have been proposed to simulate earth resistance of various shape of electrode, it is still difficult to accurately estimate the earth resistance of wind turbine. One of the reasons of this is because an earthing system of a wind turbine is too complex to simply express by conventional equations. A general turbine earthing includes a block of concrete foundations with reinforcement bars and combination of auxiliary vertical

rods and ring earth electrodes as recommended in IEC 61400-24. For this reason, lack of a universal method to estimate earthing system tends to inhibit cost-effective foundation design in the practical field.

Therefore, the aim of this report is to establish a universal and practical method with simple equivalent equations to simulate various shape, size, type of combination of turbine foundation and electrodes. At the first step of our investigation, a numerical calculation using Finite Difference Time Domain (FDTD) method is employed to clarify how the shape and size of the foundation affects to the earth resistance.

2 SUNDE’S EQUATION FOR RING ELECTRODE

To estimate a steady resistance of a ring electrode, there exist a famous and authorized equivalent equation proposed by Sunde [7];

(1)

where a is a radius of a cross-section of the ring, d is a burying depth of the ring, and r is an outer radius of the ring.

Although the Sunde’s equivalent equation describing a ring electrode is authorized and its accuracy is already confirmed experimentally, it is doubtful that the equation would be suitable to describe a ring earth electrode of actual wind turbine. This is because an earthing system of a wind turbine is a combination of a ring earth electrode and a turbine foundation itself, which is a relatively huge block of reinforcing concrete and its presence is expected to give a significant effect to a combined resistance. Since it is difficult to algebraically describe the combined resistance of the turbine foundation and the ring electrode, it is not recommended to employ the Sunde’s equation on its own to estimate earth resistance of an actual wind turbine. Therefore it would be helpful to estimate the combined resistance using a numerical calculation such as FDTD method.

30th International Conference on Lightning Protection - ICLP 2010(Cagliari, Italy - September 13th -17th, 2010)

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3 MODEL OF WIND TURBINE FOUNDATION FOR FDTD ANALYSIS

A Finite Difference Time Domain (FDTD) method is a computing calculation algorithm in which Maxwell’s electromagnetic equations are computationally treated as difference equations in both time and space domains [8]. While the FDTD methods was initially applied for an antenna analysis, with increased CPU power in PC machines, various investigations into high voltage engineering including lightning surge and earth system analysis have employed the algorithm [9].

For the present modeling, we assume a simplified ideal foundation with the following assumptions;

(i) a tower is not considered and the lightning current is assumed to be strike on the top surface of the foundation,

(ii) the foundation is made of a polygonal solid concrete block, and

(iii) a reinforced bar in the foundation is simulated by copper frame surrounding the foundation.

Other parameter conditions in the present FDTD calculations are shown in Table 1.

Figure 1 shows a model of a typical wind turbine earthing system with an octagonal foundation. The inner radius a is fixed as 0.50 m, and buried depth d of the ring earth electrode is supposed to be 3.50 m in the present calculation.

4 RESULT OF FDTD CALCULATION

Figure 2 shows a result of the FDTD calculation using the octagonal turbine foundation model. In the right graph, a simultaneous resistance (namely, impedance) R(t) at each time is given by R(t) = V(t) / I(t), where V(t) is a simultaneous potential on the earth surface above the centre of the turbine foundation and I(t) is a simultaneous current flowing to the foundation. Here, we define a “steady resistance” in the present FDTD calculation as the simultaneous resistance at t = 10.0 µs because the resistance is considered to be enough steady at the moment. As shown in Fig.4, it is clear that the large the ring earth electrode becomes, the lower the steady the resistance suppressed.

Figure 3 is the graph to clarify the tendency. Dots in the figure denote the result of each FDTD calculation varying the size of the octagonal ring electrode. The figure also illustrates an approximate line of the FDTD results and a line from Sunde’s formula from Eq.(1). In the following discussion, to compare with the Sunde’s formula Eq.(1) describing in the previous chapter, the equivalent radius of the octagonal (ring) electrode with a x-meter side is defined as re;

500 2,500

1,000

1,000 air

soil

impulse current source concrete

bonding bar (copper)

reinfocing bar (copper)

outer ring earth electrode (copper)

700 tan 22.5°

14,000 tan 22.5° 15,000 tan 22.5°

(16,000 + x) tan 22.5° (15,000 + x) tan 22.5°

x x

x

Figure 1. Model of wind turbine earthing system

Table 1. Parameters for FDTD calculation

domain of space

100 m × 100 m × 50 m

step size of space

0.50 m (200 splits for x and y axes, 100 splits for z axis)

step size of time

4.5×10–10 s (satisfying Courant’s stable condition)

air 1 soil 10

concrete 6 relative

permittivity conductor (copper) 1

air 0 soil 5.0 ×10–4 S/m

concrete 58×10-4 S/m conductivity

conductor (copper) 58×106 S/m permeability air 4π×10–7 H/m

soil 1 conductor (copper) 1

relative permeability

cast iron 60 resistivity soil 2000 Ωm velocity of electromagnetic wave 3×108 m/s

crest width 1 µs wave tail 70 µs

lightning (lump wave)

crest peak 30 kA

boundary condition

the second-order Liao’s a boundary bsorbing condition [10]


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2 1+ 2( )x2 = π re2

∴re =2 1+ 2( )

πx

(2)

From the calculation result, it became clear that the tendency of earth resistance against the outer octagonal ring electrode is quite similar to the Sunde’s conventional equivalent equation. However, the significant difference is clearly shown between the calculation and the conventional equation. The reason for the difference between the two lines is consider to existence of a huge concrete foundation. According to another calculation by the authors, it is confirmed that a calculation result by a simple ring electrode model without a concrete foundation agrees well with that of Sunde’s formula. It is also confirmed that few difference cannot be seen between the result by octagonal ring and that by circular ring. Since it is very difficult to obtain an algebraic formula of a wind turbine’s complex earthing system as shown in Fig.1 with a foundation block and ring earth electrode (in some cases, with a group of long vertical rods), a FDTD calculation can be realize to solve it by a

numerical calculation where the Maxwell’s equations are solved directly.

To estimate tendency of a wind turbine ring earth electrode against an equivalent radius, the authors propose a novel equivalent equation here. To simplify a proposed equivalent equation, suppose two improvement constants, i.e. α and β, according to Sunde’s formula in Eq.(1). Thus, the proposed equivalent equation describing the wind turbine earthing system with the ring earth electrode can be defined as following;

€

ʹ′ R =ρ

α ⋅2π 2re

ln β ⋅8re

2ad (3)

According to a least square method to approximate the dots given by FDTD calculation, the improvement constants α and β are determined as α = 1.85 and β = 3.82.

5 RESULT OF FDTD CALCULATION AND COMPARISON WITH SUNDE’S FORMULA

It is very important to estimate the maximum potential rise of an earthing system of a electric power equipment as well as a wind turbine from the viewpoint of human safety and device protection against lightning. In general, it is almost impossible to estimate the maximum potential rise of an earthing system without understanding its impedance character or frequency character (i.e. inductive or capacitive) in advance. If the earthing system consists of somewhat inductive or capacitive characteristic, the moment when the maximum potential rise is measured is not same as the moment when an applied impulse current reaches the maximum point at all. It is, therefore, very difficult to estimate a function between the maximum potential rise Vmax and the maximum current of the impulse Imax. The FDTD calculation enables it to obtain the maximum potential rise without any background information because of the calculation method bases on a direct numerical computing using different equations of Maxwell’s equations.

Figure 2. Waveforms of potential rise and simultaneous resistance (impedance)

Figure 3. Comparison of Sunde's equation and proposed equation from FDTD results.

equivalent radius: re [m]

stea

dy re

sist

ance

: R [Ω

]

FDTD results

proposed equation; Eq.(3)

Sunde’s formula


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From a careful investigation on the results from the present FDTD calculation as shown in Fig.4, it becomes clear that the group of given dots of the maximum potential rise from FDTD calculation can be approximate in same manner of Eq.(4). Thus,

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Vmax =ρImaxʹ′ α ⋅2π 2re

ln ʹ′ β ⋅ 8re2ad

(4)

where, parameters αʹ′ and βʹ′ are improvement constants as same as those in Eq.(3). After the result of the present FDTD calculation, the improvement constants αʹ′ and βʹ′ can be determined as αʹ′ = 1.50 and βʹ′ = 3.77, respectively.

The reason why there exist a slight difference between α or β given from Fig.3 and αʹ′ or βʹ′ given from Fig.4 is estimated that the impedance of the turbine earthing system is not completely resistive but is slightly inductive as shown in the left graph in Fig.2. Figure 4 also illustrates a reference curve (a bleu line) which is given by a product of the maximum applied current Imax and the estimated steady resistance Rʹ′ given in the previous chapter. As mentioned above, the maximum potential of the earthing system Vmax cannot be given simply by a product of Imax and R, it is natural that there exists an evident difference between two curves. In the present result, as the FDTD results (red-coloured dots) are distributed just above the reference curve (a bleu line), it is clearly understood that the reason of the difference is caused by the inductive characteristic of the whole earthing system of the present model.

As mentioned above, the estimate approximation equation given in Eq.(4) does not have any substantial physical reason. However, the proposed equation is considered to be useful to expect the maximum (i.e. worst) potential rise because it is designed simply in similar manner of the well-known Sunde’s formula.

6 CONCLUSIONS

Improving the conventional Sunde's equation with some correction parameters according to the size and shape of earthing system, it is possible to propose a novel equivalent equation;

€

ʹ′ R =ρ

α ⋅2π 2re

ln β ⋅8re

2ad (5)

in order to estimate practical wind turbine earth resistance. The present proposed equation is not only covers to a steady earth resistance, but also helps to estimate a transient potential rise of earthing system to consider accurate lightning protection of wind turbine.

7 REFERENCES

[1] Global Wind Energy Council: “Global wind power boom continues despite economic woes”, 2010. http://www.gwec.net/fileadmin/documents/PressReleases/PR_2010/Annex%20stats%20PR%202009.pdf

[2] International Electrotechnical Commission, Final Drought for International Standard, “Wind Turbine Generation System - 24: Lightning Protection”, IEC 61400-24, 2010 (88/186/FDIS).

[3] I. Cotton: “Windfarm Earthing”, Proc. of 11th International Symposium on High Voltage Engineering (ISH 99), 1999; 288-291.

[4] M. Lorentzou, N. Hatziargyriou, B. Papadias: “Analysis of Wind Turbine Grounding Systems”, Proc. of 10th Mediterranean Electrotechnical Conference (MELECON2000), 2000; 936- 939.

[5] Y. Yasuda, T. Ueda: “FDTD Transient Analysis of Ring Earth Electrode”, Proc. of 41st International Universities Power Engineering Conference (UPEC2006), 2006; 133-136.

[6] Y. Yasuda, T. Fuji, T. Ueda: “Transient Analysis of Ring Earth Electrode for Wind Turbine”, European Wind Energy Conference (EWEC2007), 2007; BL3.212.

[7] E. D. Sunde: “EARTH CONDUCTION EFFECTS IN TRANSMISSION SYSTEMS”, McGraw-Hill Book Company, Inc., 1968.

[8] K. S. Yee: “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media”, IEEE Transaction on Antennas and Propagation, 1996; 14(3): pp.302-307.

[9] K. Tanabe, A. Asakawa: “Computer Analysis of Transient Performance of Grounding Grid Element Based on the Finite-Difference Time-Domain Method”, IEEE Transaction on Power Energy, 2003; 120-B(8/9): pp.209-212.

[10] Z. P. Liao, H. L. Wong, B.-P. Yang, Y.-F. Yuan, “A transmitting boundary for transient wave analysis,” Science Sinica, 1984; 27(10): 1063-1076.

Figure 4. Estimated approximation equation for the maximum potential rise against radius.

equivalent radius: re [m]

max

. pot

entia

l ris

e: V

max

[V]

FDTD results

proposed equation; Eq.(4)

R′ × Imax


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EQUIVALENT EQUATION OF EARTH RESISTANCE FOR RING ELECTRODE OF WIND TURBINE