1. Introduction
To achieve the goal of limiting global warming to 1.5 °C set by the United Nations in the Paris Climate Agreement, renewable energy sources, including wind energy, must be extended in the future [
1]. Since offshore locations have vast wind potential and are not as constrained by populated or protected areas as onshore locations, particularly offshore wind farms, they are expected to grow substantially [
2]. While most offshore wind farms are bottom-fixed platforms installed in shallow waters, a few projects are already implemented in deeper waters, such as the floating wind farms Hywind Scotland, WindFloat Atlantic, and Hywind Tampen [
3,
4,
5]. In all implemented projects, the power cables connecting the FOWTs are laid on the seabed [
3,
4,
5]. However, the power cables can also be suspended, which means they are submerged but do not touch the seabed. These suspended configurations can reduce the length compared to cables laying on the seabed. This length reduction has the advantages of reduced inter-array energy loss and lower investment costs for the power cables [
6]. This work analyzes the fatigue damage of suspended inter-array power cables between two floating offshore wind turbines using a stress factor-based fatigue analysis approach.
Ikhennicheu et al. [
7] published various offshore dynamic power cable configurations connecting a floating platform and the seabed. The most common dynamic cable configurations described in the literature are the catenary and lazy-wave shapes. The lazy-wave shape distinguishes itself from the catenary shape through buoyancy modules attached to the hanging power cable. Thies et al. [
8] analyzed a dynamic umbilical’s tension and fatigue life in lazy-wave and catenary configurations connecting a wave energy converter with a static power cable at the seabed. They showed that the lazy-wave configuration reduces the maximum tension and the number of fatigue cycles in the dynamic umbilical compared with the catenary configuration. A reduction in the maximum tension resulted in a significant extension of the fatigue life of the dynamic umbilical. Thies et al. [
9] obtained the tensions and fatigue of a 66 kV cable connecting a FOWT to the seabed in a lazy-wave shape. The platform’s motion was the main contributor to the loads on the umbilical. Rentschler et al. [
6] presented a design optimization method for inter-array cable configurations for FOWTs. Their study included an estimation of the fatigue life, the performance in extreme weather conditions, and selected economic parameters for lazy-wave power umbilical configurations. They applied the method to dynamic umbilical configurations at different water depths, resulting in reduced buoyancy sections and a lower position of the wave shape than in the original design. Bakken [
10] calculated the fatigue life of a dynamic power cable connecting a FOWT to the seabed using the numerical programs SIMA and Bflex. The lowest fatigue life occurred at the locations with the maximum tension range, and the main contributors to the fatigue of the copper conductor were local friction effects. Hu et al. [
11] investigated the bending behavior of a copper conductor inside a dynamic power cable experimentally and numerically. They stated that the nonlinear bending of the power cable is the main factor leading to fatigue failures. Prediction models for the fatigue of dynamic power cables were developed by Svensson [
12] using experiments and finite element simulations. Interlayer friction forces were an important factor in the calculation of the fatigue life of the power cable. Zhao et al. [
13] studied the behavior and fatigue of power cables in lazy-wave and double-wave configurations connected to a FOWT in shallow water. They detected the hang-off point as the most critical point regarding the tension and fatigue of the power cable for both shapes. The second buoyancy section of the double wave configuration was critical for fatigue life. Ballard et al. [
14] estimated the fatigue life of a lazy-wave-shaped umbilical attached to a wave energy converter. Curvature-induced fatigue was identified as the critical parameter in their setup, causing fatigue damage.
The general fatigue properties of a copper conductor inside a dynamic subsea power cable were determined by Karlsen et al. [
15]. They suggested using strain-cycle curves for copper conductors due to the limited applicability of traditional S–N curves for this material. Marta et al. [
16] assessed a dynamic power cable’s stresses and fatigue life at floating offshore renewable energy installations. Fretting was identified as a major crack initiation mechanism in copper conductors, causing fatigue failure. The governing failure mode in all their cases was fatigue. Nasution et al. [
17] established the S–N curve of a copper conductor of power cables used in offshore wind farms through experimental data and finite element simulations. Nasution et al. [
18] assessed the effects of tension and bending loads on copper power conductors. They identified inter-layer friction caused by curvature as the main reason for fatigue failures.
Yang et al. [
19] presented the dynamic motions and fatigue damage of a power cable connecting a wave energy converter to a static hub in a freely hanging catenary configuration. They calculated a long fatigue life for a simplified cable, disregarding internal effects such as wear or fretting. Suspended power cables between FOWTs were described by Rapha and Domínguez [
20]. Schnepf et al. [
21] performed feasibility studies for different suspended power cable configurations. They identified configurations with buoys as more suitable than configurations with buoyancy modules. The behavior of a suspended power cable connecting a FOWT with a floating production storage and offloading unit (FPSO) was analyzed by Schnepf et al. [
22]. The results showed that suspended configurations with buoys attached evenly over the power cable length could reduce its tension compared to a freely hanging configuration and a configuration with buoys attached to its middle section. Ahmad et al. [
23] proposed a design method for suspended inter-array power cable configurations between two FOWTs. They concluded that attaching several buoys leads to lower maximum tensions in the power cable and that copper conductors might be more suitable for suspended power cables than aluminum conductors. To the authors’ knowledge, no work has yet been performed assessing the fatigue of a suspended power cable. Analyzing the fatigue life of a suspended power cable due to dynamic loading is important for identifying possible design optimizations and ensuring the reliability and cost-effectiveness of the proposed setup. Additionally, no study has yet been published on a simplified method to obtain stress factors for calculating the fatigue of power cables.
This work is organized as follows:
Section 2 describes the software used, the setup of the FOWTs and the connecting suspended power cable in deep water, the environmental conditions applied, and the fatigue assessment.
Section 3 presents and discusses the results of the stress actor calculations and the fatigue analyses of the case study. Conclusions are drawn in
Section 4.
2. Methodology and Numerical Setup
This section describes the applied software for the present study, the numerical setup of the suspended power cable configuration between two FOWTs, the environmental conditions, and the applied fatigue calculation method.
2.1. Numerical Tools
The present study uses the numerical software OrcaFlex version 11.2d [
24] in combination with Python v3.10 [
25]. This software can perform global static and dynamic analyses of marine systems, including analyses of the behavior of wind turbines. A time-domain solution procedure determines the interactions between floating structures and the environmental loads (including wave, wind, and current loads). Moreover, fatigue calculations can be carried out based on the results of dynamic simulations. Line structures such as power cables can be modeled with finite elements.
The finite element software UFLEX version 2.8 [
26] is used to determine cable properties. The software assumes that the problem is two-dimensional with respect to tension and torsion, while bending loads, including their three-dimensional extent, are considered. Filled bodies can be modeled as beam elements and tubular bodies as shell elements. UFLEX provides modeling capabilities for complex cross-section geometries, contact and friction stresses, nonlinear relationships between curvature and bending moment, cross-section geometry ovalization, and nonlinear material models. This software has been validated by other finite element software as well as experiments described by Sævik and Bruaseth [
27], Dai [
28], Sævik and Gjøsteen [
29], and SINTEF [
30].
2.2. Numerical Models
This section describes the FOWT system, the properties of the dynamic power cable, and the setup of the suspended power cable configuration.
2.2.1. Floating Offshore Wind Turbine System
The reference wind turbine in the present study is the 5MW OC3-Hywind FOWT based on Jonkman et al. [
31] and Jonkman [
32]. This FOWT uses a spar concept, and the stability of the platform is achieved through the large draft of the cylindrical buoy.
Figure 1 shows the geometry of the FOWT, and
Table 1 lists its general parameters. The setup of the FOWT in the dynamic analysis software is based on Schnepf et al. [
21].
2.2.2. Power Cable Properties
This study uses a 66 kV dynamic power cable designed by Nexans [
33].
Table 2 describes the general properties of the power cable, and
Figure 2 shows its cross-section. The cable contains three copper conductors consisting of 19 copper wires, accumulating to a total copper cross-sectional area of 285 mm
2. Two layers of armored steel wires with a diameter of 4.2 mm are used to protect and stabilize the cable. The helically laid armor wires have a lay angle of 10° and −10°, respectively. A lay angle of 10° is used for the entire copper conductor, while the second and third copper wire layers have lay angles of 2° and 4°, respectively. The remaining insulation and binding components are made of polymer materials, such as cross-linked polyethylene (XLPE).
Table 3 describes the material properties of the different cable components. The fiber-optic wire shown in the cable cross-section is neglected in the present study based on the procedures of previous works such as Thies et al. [
8], Rentschler et al. [
6], and Svensson [
12].
Figure 3 shows the copper wires’ stress–cycle curve (S–N curve) in the dynamic umbilical. The curve is similar to the one obtained by Nasution et al. [
17]. The S–N curve for the armor steel wires is shown in
Figure 3. It is similar to the S–N curves for steel with cathodic protection provided by DNV [
34]. This study uses S–N curves and not strain-cycle curves as proposed by Karlsen et al. [
15], based on the procedures of the majority of similar studies, such as Marta et al. [
16], Ballard et al. [
14], and Nasution et al. [
18]. Power cable manufacturers, such as Nexans, usually only provide S–N curves for their products under the current standard engineering approach [
33].
The power cable’s diameter and weight increase during its lifetime because marine growth is dependent on the water depth. The calculation of the marine growth effects in this study is based on the information given by NORSOK [
35] and DNV [
36]. The start-of-life (SOL) state and the end-of-life (EOL) state power cable properties are specified in
Table 4.
2.2.3. Suspended Power Cable Setup
The inter-array power cable is suspended between the two FOWTs without touching the seabed or sea surface.
Figure 4 shows an example of a suspended power cable setup. The distance between the wind turbines is 1134 m, equivalent to 9 times the rotor diameter of the FOWTs. The power cable is 1260 m long and divided into sections of different discretization sizes. A section of 10 m from the hang-off location has a segment length of 0.1 m, and the free-hanging cable sections have a segment length of 1.0 m. Attached to the power cable are three buoys with a distance of 300 m between them. The properties of the buoys are described in
Table 5. The buoys have an anti-marine growth coating. Bend stiffeners to prevent excessive cable bending are attached to both ends of each buoy. The cable is discretized with a segment length of 0.31 m at the subsea buoys and 0.12 m at the bend stiffeners. The power cable hang-off consists of an I-tube, as shown in
Figure 5. The center of its bell mouth is 70 m below the still-sea water level (SWL) at an 8 m radius from the spar center axis. It has a length of 7 m and is placed at an angle of 25° from the spar. The bell mouth opening has a diameter of 2.3 m and a minimum radius of 1.9 m. The mooring systems of the two FOWTs are mirrored, as shown in
Figure 6. The nacelles of the wind turbines are always rotated towards the incoming wind direction.
2.3. Environmental Conditions
The wave, wind, and current conditions are based on a location in the North Sea on a latitude of 61° N with a water depth of 320 m from Spyrou et al. [
37], Komen et al. [
38], WAMDI [
39], Papadopoulos et al. [
40], and Asplin et al. [
41].
Table 6 presents all environmental parameters for the 30 load cases used in the present study, along with their occurrence probabilities. Load case 30 represents extreme weather when the wind turbine is idling. The FOWT is in operation in all other load cases. The Torsethaugen spectrum is applied to model irregular waves, and the NPD spectrum is used to simulate the wind [
24]. The current velocity at a specific height z is estimated through the following formula given by DNV [
36]:
where
vc,SWL is the current velocity at the still water level, and d is the water depth. The six directions of the environmental loads toward the power cable and their appearance probabilities are shown in
Figure 7. Wind and waves are aligned in all load cases to obtain conservative results.
2.4. Fatigue Assessment
The fatigue assessment in this study is carried out in four steps. First, the nonlinear bending behavior of the power cable is determined. Second, dynamic analyses of the entire setup are performed to obtain the axial tension and the bending curvatures of the power cable. Third, the stresses of the cable components are calculated through stress factors. This work proposes a simplified and effective method to estimate these stress factors for dynamic power cables. Fourth, fatigue damage is determined by applying the rainflow counting method and the Miner–Palmgren rule.
2.4.1. Nonlinear Bending Behavior Estimation
Dynamic power cables show nonlinear bending behavior due to friction, contact, and slip effects between the wires in the same layer and between wires of different layers [
11]. The nonlinear bending behavior of the power cable is determined through finite-element analysis in UFLEX. A stepwise curvature is applied to the cable, and each step’s corresponding bending moment is calculated. The obtained relationship between the bending curvature and the bending moment is applied to the power cable in the dynamic analyses.
2.4.2. Calculation of Stresses Using Stress Factors
The stresses to determine damage are obtained by the following formula [
24]:
where
S is the stress and
Kt and
Kc are the tension and curvature stress factors.
T is the effective tension,
Cx and
Cy are the curvature components, and
ϴ is the circumferential location of the fatigue point.
In the following part, this study proposes a simplified method for estimating the stress factors in a dynamic power cable in the early design stages when results from validated software or experiments are unavailable. If the power cable has a relatively low axial tension but a high bending utilization during its operation, the tension stress factors can be derived using the equations for the simple area stress of composite beams. When axially loaded, the strain is continuous across the cross-section of a composite beam, but the stress is discontinuous [
42]. From Hook’s law, the following can be deduced for a beam with two components when an axial load is applied:
where
ε is the strain and
σ1 and
σ2 are the axial stresses of cable components 1 and 2.
E1 and
E2 are Young’s moduli of the respective cable components. The tension must equal the stresses times their respective areas in the longitudinal direction:
Combining both equations results in the following:
where
σa1 and
σa2 are the axial stresses of cable components 1 and 2, and
A1 and
A2 are the areas of cable components 1 and 2. The formulas for the tension and stress factors become:
An assumption is made to derive the equation for the curvature stress factor for copper and armor wires. A total of 100% utilization of the cable capacity with zero tension is assumed to be equal to reaching the limit of any of the materials in the cable due to curvature. From the capacity curve, the maximum curvature of the cable can be determined. The following formulas are established for estimating a conservative curvature stress factor for two different cable components:
where
Y1 and
Y2 are the yield strengths of components 1 and 2, and
Cmax is the maximum allowable curvature of the cable.
The results obtained from the stress factor method are compared with those from the validated software, UFLEX. The stress factors are calculated through a detailed analysis of the stresses of the different cable components. Several axial tension and curvature loads are applied to analyze the stress distribution among the cable components.
2.4.3. Dynamic and Fatigue Damage Analysis
Dynamic analyses are performed for all load cases in each loading direction in OrcaFlex. All load-case simulations have a duration of one hour and a time step of 0.1 s. Each simulation uses a different random seed to enable independent wave and wind conditions. All load cases are considered for the fatigue analysis with their occurrence probabilities. Furthermore, all cases of one loading direction are scaled to the appearance probability of each loading direction. The simulations determine the axial tensions and the curvatures for each node along the power cable. Marine growth is considered by running two different fatigue analyses with the power cable in the SOL and EOL states.
The fatigue analysis is performed using the rainflow counting method [
43]. This method uses the stress time history resulting from the dynamic analyses to calculate the fatigue damage. The stress reversal points are grouped into different stress ranges. In the rainflow counting method, four reversal points are considered to detect stress cycles. If the outer points bind the inner points, a cycle is counted, and the difference between the stress ranges of the inner points is assigned as the amplitude of this cycle. This method assumes that the fatigue damage resulting from a closed cycle in the irregular amplitude loading equals the fatigue damage caused by one cycle in a constant amplitude test. The damage caused by half-cycles is neglected in this procedure because there are few of them in every realistic case.
The cumulative damage is calculated according to the Miner–Palmgren rule [
44]. This theory assumes constant damage during a given stress range. The following formula expresses the Miner–Palmgren rule:
where
D is the accumulated damage,
i is a specific stress range, and
k is the total number of stress ranges. Additionally,
n is the number of cycles counted in a specific stress range, and
N is the number of cycles until damage occurs in a specific stress range. The values for the different power cable components for
N are obtained from the S–N curves. Based on these formulas, the fatigue damage and the fatigue life in years of the dynamic power cable can be determined.