Assessing the Validity of Analytical Equations for Offshore Power Cable Bending with Fixed and Loose Tube Fiber Strain Sensors

Abstract

Background

Subsea power cable failures in offshore wind farms result in significant financial losses. One common failure mode is submarine power cable bending.

Objective

The primary objective of this study is to validate two analytical models using strain readings obtained from a novel 3-point bending setup designed for power cable specimens. The setup incorporates two types of optical fiber sensors for simultaneous strain measurement.

Methods

A 3-point bending setup is constructed, integrating optical fiber sensors installed on the embedded fiber optic cable within the submarine power cable. One set of sensors is fixed to the fiber optic cable sheath, while a second set consists of loose tube fibers that are inside the fiber optic cable. The strain readings of the fixed sensors are compared to two analytical models. The first analytical model assumes a constant power cable curvature, while the second model considers variable curvature.

Results

The analytical models both predict nearly flat strain profiles and are in line with each other. The strain data, however, approaches zero strain away from the cable center. Model assumptions such as perfect sensor positioning and zero slip of the fiber optic cable cause this discrepancy. The results of the constant curvature model agree well with strain averages of the fixed sensors around the central region of the power cable, and both scale linearly with amplitude. Finally, the strain readings from the loose tube fibers demonstrate high reproducibility, facilitating the development of a calibration curve for estimating power cable curvature.

Conclusions

The analytical models surpass existing models by providing good agreement with the measured strain around the cable center. Moreover, the highly reproducible strain readings from the loose tube fibers allow estimating power cable curvature.

Appendices

Appendix 1 Analytical Model Details

General

Parametrization

A 3D parametrization of a helical path based on a path length parameter s is found in literature [29]. This parametrization $\vec {r}(s)$ describes a helical path of pitch radius R and pitch length L and includes the effects of bending and torsion, with bending occurring in the xz plane. It therefore can serve as a model for the fiber optic cable path. The generalized parametrization $\vec {r}(s)$ is:

$$\begin{aligned} x(s) =\ {}&x_{c}(s) + R\ cos\left( \left[ \frac{2\pi }{L}+\varphi \right] \cdot s + \beta \right) \cdot cos(\kappa s) \\ y(s) =\ {}&y_{c}(s) - R\ sin\left( \left[ \frac{2\pi }{L}+\varphi \right] \cdot s + \beta \right) \\ z(s) =\ {}&z_{c}(s) + R\ cos\left( \left[ \frac{2\pi }{L}+\varphi \right] \cdot s + \beta \right) \cdot sin(\kappa s) \end{aligned}$$

in which $\vec {r}_{c}(s) =$($x_{c}$, $y_{c}$, $z_{c}$) are the coordinates of the central axis of the power cable, $\beta$ the helix orientation at $s=0$, $\varphi$ the cable twist, and $\kappa$ the power cable curvature. The sign of the y coordinate is reversed to make the helix left-handed. The following parameters are chosen for the bending setup: $R=$ 56 mm, $L=$ 3200 mm, $\beta = 3\pi /2$ and $\varphi =0$. Finally, the path length parameter s runs from $-L/2$ to L/2.

Calculation of curvature

The 3D curvature of the fiber optic cable $k_{FO}(s)$ can be calculated as follows:

$$\begin{aligned} k_{FO}(s)&= \frac{\Vert \vec {r'}(s) \times \vec {r''}(s)\Vert }{\Vert \vec {r'}(s)\Vert ^3}\\&= \frac{\sqrt{\left( z''y'-y''z'\right) ^2 + \left( x''z'-z''x'\right) ^2 + \left( y''x'-x''y'\right) ^2}}{\left( {x'}^2+{y'}^2+{z'} ^2\right) ^{3/2}} \end{aligned}$$

with the accents denoting the derivatives w.r.t. path length s. The curvature depends in other words on the first and second derivatives of each of the components. Since $\vec {r}(s) = \vec {r}_{c}(s) + \vec {r}_{FO}(s)$, the (second) derivative of the total parametrization is the sum of the (second) derivatives of $\vec {r}_{c}(s)$ and $\vec {r}_{FO}(s)$, e.g.:

$$\begin{aligned} \vec {r'}(s) = \vec {r'}_c(s) + \vec {r'}_{FO}(s) \Rightarrow x'(s) = x_c'(s) + x_{FO}'(s). \end{aligned}$$

Writing $D(s) = \kappa s$ and $E(s)=\left( \frac{2\pi s}{L}+\beta \right)$, the first and second derivatives of $\vec {r}_{FO}(s)$ become:

$$\begin{aligned}x_{FO}'(s) =& -\frac{2\pi R}{L}\ sin(E) \cdot cos(D) \\ &- R\ cos(E) \cdot sin(D) \cdot \left[ \kappa + s\cdot \kappa '\right] \\x_{FO}''(s) =& - \left( \frac{2\pi }{L}\right) ^2 R\ cos(E) \cdot cos(D) \\&+ \frac{4\pi R}{L}\ sin(E) \cdot sin(D) \cdot \left[ \kappa + s\cdot \kappa '\right] \\& - R\ cos(E) \cdot cos(D) \cdot \left[ \kappa + s\cdot \kappa '\right] ^2 \\& - R\ cos(E) \cdot sin(D) \cdot \left[ 2\kappa ' + s\cdot \kappa ''\right] \\y_{FO}'(s) =& -\frac{2\pi R}{L} \cdot cos(E) \\y_{FO}''(s) =& \left( \frac{2\pi }{L}\right) ^2 R\ sin(E) \\z_{FO}'(s) = &-\frac{2\pi R}{L}\ sin(E) \cdot sin(D) \\&+ R\ cos(E) \cdot cos(D) \cdot \left[ \kappa + s\cdot \kappa '\right] \\z_{FO}''(s) =& - \left( \frac{2\pi }{L}\right) ^2 R\ cos(E) \cdot sin(D) \\&- \frac{4\pi R}{L}\ sin(E) \cdot cos(D) \cdot \left[ \kappa + s\cdot \kappa '\right] \\&- R\ cos(E) \cdot sin(D) \cdot \left[ \kappa + s\cdot \kappa '\right] ^2 \\& + R\ cos(E) \cdot cos(D) \cdot \left[ 2\kappa ' + s\cdot \kappa ''\right] \end{aligned}$$

In what follows, two models are discussed: the first model assumes a constant power cable curvature $\kappa$. This is the model that [29] uses. The second model is based on a fit to the 3D camera data. The power cable has in this case a variable curvature.

Constant Curvature Model

The parametrization of the central axis of the power cable is as follows:

$$\begin{aligned} x_{c}(s) =&\ \frac{1}{\kappa } \left[ (cos(\kappa s)-1) + \left| cos\left( \kappa L_{wheel}\right) -1 \right| \right] , \\ y_{c}(s) =&\ 0,\\ z_{c}(s) =&\ \frac{1}{\kappa } \cdot sin(\kappa s). \end{aligned}$$

The parameter $\kappa$ is here determined by assuring that $x_{c}(0) > \Delta x$ for $L_{wheel} = 1800$ mm, the distance between the cable eye and a wheel support. Table 2 shows which power cable curvatures $\kappa$ are used for all calculations. The third column shows the bending radius $R_0$ of the power cable.

No explicit expressions of the first and second derivatives of $\vec {r}_{c}(s)$ are given here. Note however that the above formulas for $x'_{FO}(s)$ and $x''_{FO}(s)$ simplify for this model since the derivatives of $\kappa$ vanish.

Variable Curvature Model

General

This model is based on data from the 3D camera setup. First, we assume the central axis of the power cable symmetric around $s=0$ for $x_c(s)$ and hence select an even function. A fourth degree polynomial is chosen for $x_c(s)$, because it captures the slack that the power cable has in the vertical direction. For $y_c(s)$ a first degree polynomial is fitted to capture the coordinate data optimally. The following information regarding the power cable setup is used to constrain $x_c(s)$: $x_{c}(0) = \Delta x$ and $x_{c}(\pm L_{wheel}) = 0$. Thus the following parametrization is used:

$$\begin{aligned} x_{c}(s) &= A\left( s^2-L_{wheel}^2\right) s^2\\&\quad-\Delta x \left( \frac{s}{L_{wheel}}\right) ^2 + \Delta x, \\ y_{c}(s) &=Bs+C\\ z_{c}(s) &= s, \end{aligned}$$

with fit parameters A, B and C dependent on the displacement $\Delta x$. Table 3 lists all fit parameters used in this work.

Table 3 Variable curvature model: fit parameters A, B, and C

Power cable curvature

Again the first and second derivative of $\vec {r}_{c}(s)$ are not given here. The power cable curvature is variable, and is described by

$$\begin{aligned} \kappa (s) &= sgn\left[ x_c''(s)\right] \cdot \frac{\Vert \vec {r}'_{c}(s) \times \vec {r''}_{c}(s)\Vert }{\Vert \vec {r'}_{c}(s)\Vert ^3} \\&\equiv sgn\left[ x_c''(s)\right] \cdot \frac{\sqrt{F(s)}}{\left[ G(s)\right] ^{3/2}}, \end{aligned}$$

with the first factor representing the sign function of $x_c''(s)$ to assure the curvature being signed. If this factor is neglected, numerical errors appear at the inflection points of $\kappa (s)$. These errors are occurring because of the strict positive definition of 3D curvature. The $sgn\left[ x_c''(s)\right]$ corrects for this and propagates also to the first and second derivatives of $\kappa (s)$. Furthermore, the explicit expression for F(s) and G(s) are the following:

$$\begin{aligned} F(s)&= \sum _{i \ne j} (x_i'' x_j'-x_j'' x_i')^2 \\&= \left( z_c''y_c'-y_c''z_c'\right) ^2 + \left( x_c''z_c'-z_c''x_c'\right) ^2 \\&\quad+ \left( y_c''x_c'-x_c''y_c'\right) ^2\\ G(s)&= \sum _{i} x_i'^2 = {x_c'}^2+{y_c'}^2+{z_c'}^2 \end{aligned}$$

The vector notation $x_i$ is here used for either components $x_{c}$, $y_{c}$, or $z_{c}$.

Derivatives of the power cable curvature

To be able to calculate $k_{FO}(s)$, we need to calculate $x''(s)$ and in turn $x_{FO}''(s)$. Hence it is necessary to derive first and second derivative of $\kappa$ too (see explicit expressions of $x'_{FO}(s)$ and $x''_{FO}(s)$). Because of above relationship between $\kappa (s)$ and F(s) or G(s), the derivatives of $\kappa (s)$ are expressed in derivatives of F(s) and G(s). First, the explicit expressions of both derivatives of the curvature are given, after which the compact versions of the derivatives of F(s) and G(s) will be given.

Derivatives of power cable curvature

$$\begin{aligned} \kappa '(s) = sgn\left[ x_c''(s)\right] \cdot \frac{F'G-3FG'}{2\sqrt{F}\left( G\right) ^{5/2}} \end{aligned}$$

$$\begin{aligned} \kappa ''(s) =&\ sgn\left[ x_c''(s)\right] \\&\cdot \frac{\left( F''G-2F'G'-3FG''\right) \cdot 2FG - \left( F'G-3FG'\right) \left( GF'+5FG'\right) }{4\left( F\right) ^{3/2}\left( G\right) ^{7/2}} \end{aligned}$$

Derivatives of F(s) and G(s)

$$\begin{aligned}F'(s)& = 2\cdot \sum _{i\ne j} (x_i'' x_j'-x_j'' x_i')(x_i''' x_j' - x_j''' x_i') \\F''(s)& = 2\cdot \sum _{i\ne j} \Big [ (x_i''' x_j' - x_j''' x_i')^2 \\&\quad+ (x_i''x_j'-x_j''x_i')(x_i^{(4)}x_j' \\&\quad+ x_i'''x_j'' - x_j^{(4)}x_i - x_j'''x_i'') \Big ] \\G'(s)& = 2\cdot \sum _i x_i' x_i'' \\G''(s) &= 2\cdot \sum _i \left[ (x_i'')^2 + x_i'x_i'''\right] \end{aligned}$$

Appendix 2 Projection on the FBG Sensor Plane

Assume $\vec {r}(s)=\left( x(s), y(s), z(s)\right)$ to be a 3D parametrization of a helix with s the path length parameter. At each point along the helix, we can define an orthonormal local coordinate system that consists of a tangential vector $\vec {e}_T$ and normal vector $\vec {e}_N$:

$$\begin{aligned} \vec {e}_T(s)&=\frac{\vec {T}(s)}{||\vec {T}(s)||},\\ \vec {e}_N(s)&= \frac{\vec {T'}(s)}{||\vec {T'}(s)||}. \end{aligned}$$

These two vectors span the osculating plane for each s. The local radius of curvature $\rho (s)$ of $\vec {r}(s)$ lies inside the osculating plane. It is the radius of the osculating circle with center M. The normal vector $\vec {e}_N(s)$ points to M. The local curvature $k_{FO}(s)$ is furthermore the inverse of $\rho (s)$.

The FBG sensors are mounted in the plane that is spanned by unit vectors $\vec {e}_T(s)$ and $\vec {e}_x$. The relative orientation of the sensor w.r.t. to the osculating circle will influence how much bending strain is measured. Figure 11 shows a cross section of the fiber optic cable in the $(\vec {e}_N(s), \vec {e}_x)$-plane (bottom right). The local radius of curvature $\rho (s)=((k_{FO}(s))^{-1}$ connects M with the center of the fiber optic cable of radius r. The sensors are indicated in red, the in-plane unit vectors are shown in green and the tangential unit vector is in orange.

When $\vec {e}_N$ and $\vec {e}_x$ are aligned, the bending strain captured by the FBG sensors is maximal. When $\vec {e}_N \perp \vec {e}_x$ however, no bending strain is detected. The bending strain is thus dependent on the angle between the osculating plane and the sensor plane, or equivalently: the angle between their normals, $(\vec {e}_T \times \vec {e}_x)$ and $(\vec {e}_T \times \vec {e}_N)$:

$$\begin{aligned} cos(\theta ) &= (\vec {e}_T \times \vec {e}_x) \cdot (\vec {e}_T \times \vec {e}_N) \\&= (\vec {e}_T \cdot \vec {e}_T)(\vec {e}_x \cdot \vec {e}_N)\\&\quad - (\vec {e}_T \cdot \vec {e}_N)(\vec {e}_x \cdot \vec {e}_T) = \vec {e}_x \cdot \vec {e}_N \end{aligned}$$

The projection onto the FBG sensor plane in equation (1) is in other words equivalent with multiplying $k_{FO}$ with the x component of the unit normal vector:

$$\begin{aligned} cos(\theta ) = e_{N_x}(s) = \frac{x''(s)}{\sqrt{x''(s)^2+y''(s)^2+z''(s)^2}}. \end{aligned}$$

Fig. 11

Cross section of the fiber optic cable in the $(\vec {e}_N, \vec {e}_x)$-plane. The relative orientation of $\vec {e}_N$ and $\vec {e}_x$ influences the amount of bending strain that is detected by the FBG sensors. The local radius of curvature is shown in blue and unit vectors are indicated in green or orange