A methodology for design and fatigue analysis of power cables for wave energy converters

International Journal of Fatigue 122 (2019) 61–71

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

A methodology for design and fatigue analysis of power cables for wave energy converters

T

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Artjoms Kuznecovsa, Jonas W. Ringsberga, , Shun-Han Yanga, Erland Johnsonb, Andreas Andersonc a

Chalmers University of Technology, Department of Mechanics and Maritime Sciences, SE-412 96 Gothenburg, Sweden RISE Research Institutes of Sweden, Division of Safety and Transport, Mechanics, Box 857, SE-501 15, Borås, Sweden c RISE Research Institutes of Sweden, Division of Safety and Transport, Electronics, Box 857, SE-501 15, Borås, Sweden b

A R T I C LE I N FO

A B S T R A C T

Keywords: Cable design Fatigue Dynamic cables Power cables Wave energy converter

The recent development of subsea power cables for various offshore marine renewable energy technologies has identified the need for new cables that have low structural stiffness properties. This type of cable is referred to as dynamic cable because of its high bending flexibility compared to static cables. The current study presents a cable design model and simulation models that were developed for the design and fatigue analysis of dynamic cables. These models were applied on a subsea dynamic power cable with a design that is suitable for a floating point-absorbing wave energy converter (WEC), where the cable must withstand cyclic loads imposed by the motions of the WEC, the waves and the ocean currents. The cable design model is presented with its detailed design and dimensioning methodology for cables with multiorder helical structures, with respect to desired (target) mechanical properties. The cable design model is verified against a verification study in the literature. A simulation model of a fatigue test rig for accelerated rotational bending is presented. The results from the numerical simulations and the subsequent fatigue analyses are compared against results from experiments using the test rig. The influence of the dynamic effects and mechanical properties on the fatigue life of the cable is discussed. This study contributes to a better understanding of the fatigue failure mechanisms of the cable, and it also highlights the importance of further development of numerical models.

1. Introduction Subsea power cables and their derivatives are broadly utilized. Examples of energy production applications are various types of floating wave energy converters, floating wind energy platforms and innovative hydrokites [1]. Moreover, umbilicals and power cables are broadly used in the offshore oil and gas industry for controlling and powering subsea equipment such as pumps or manifolds. Wave energy is currently regarded as a promising source of renewable energy, and the market for wave energy production has a potential for growth [2]. Recently, advances have been made in the technological enhancement and economic viability of several wave energy converter (WEC) concepts. To increase the economic potential and competitiveness of WECs, a high technical performance level is required. Reducing the cost of WECs and thus lowering the levelized cost of energy (LCOE), is the main challenge that interested parties are facing today [2]. In the case of floating WECs, the cost issues can be

overcome by increasing the service life of the WEC systems, the moorings and, in particular, their power cables [3]. Hence, mechanical fatigue and loss of functionality of subsea power cables are important aspects for successful WEC installations. A cable is a structure that is flexible in both bending and torsion, due to low bending and torsional stiffness properties, while maintaining a high tensile strength due to relatively large axial stiffness. Typically, cables consist of helical (laid around a central core in a helix) and cylindrical components. Thus, cable mechanics are similar to that of wire ropes and flexible pipes, and the same theories regarding structural responses can be applied. Most of the fundamental research on helical structures is based on Love’s [4] general thin rod theory, which describes the equilibrium of a single helical rod due to general forces and moments. Phillips, Costello and LeClair used Love’s formulation to create analytical models for twisted wire ropes [5–7]. Lutchansky [8] investigated the effect of internal friction during bending, and Vinogradov and Atatekin [9] showed that the responses inside a bent cable

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Corresponding author. E-mail addresses: [email protected] (A. Kuznecovs), [email protected] (J.W. Ringsberg), [email protected] (S.-H. Yang), [email protected] (E. Johnson), [email protected] (A. Anderson). https://doi.org/10.1016/j.ijfatigue.2019.01.011 Received 22 October 2018; Received in revised form 16 January 2019; Accepted 19 January 2019 Available online 22 January 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

Llay lay length [m] LR one fourth of a circumference length in the cable rig [m] LS, LL short and long cables lengths respectively [m] m slope in the S-N curve [–] N number of cycles [–] R bending radius [m] Rlay layer radius [m] RX, RY, RZ Euler angles around the x, y and z axes respectively [rad]

α lay angle [deg] A cross-section area [m2] C S-N curve constant [–] E, G the elastic and shear moduli [Pa] EA, EI, GKV axial (in tension) [N], bending [N m2] and torsion [N m2/rad] stiffness I moment of inertia [m4] KV torsion constant [m4/rad]

were hysteretic. These analytical models and findings were incorporated into a general multilayer model by Witz and Tan for axialtorsional loading [10] and for bending [11], and the model showed very good agreement with the experiments. To simplify the rather advanced thin rod model, the orthotropic model was introduced by Hobbs and Raoof [12], in which every helical layer was represented by an orthotropic cylinder. Jolicoeur [13] showed that this homogenization of the cross-section of the cable is appropriate to accurately capture responses due to axial and torsional loads. In the work by Velinsky [14], Costello’s model [5] was linearized, and the theory was extended to wire ropes with complex crosssections including multistrand geometries. Phillips and Costello [15] generalized Velinsky’s approach, which was later summarized by Costello [16]. Velinsky [17] extended the linearized theory presented by Phillips and Costello [15], and proposed a design procedure for multilay wire strands that allowed for different configurations. The effect of multiple design parameters was investigated, and a methodology for sizing the individual wires in the strands was developed, but with respect to only one or two coupled stiffness properties at a time. There are examples of studies in the literature that present multistrand models without homogenization for better prediction of stresses in the individual wires of multiorder helical structures. Inagaki et al. [18] presented an analytical model for the mechanical analysis of second-order helical structures (helix-in-helix) based on the hierarchical approach introduced by Leech [19] with a friction model from Papailiou [20]. Another analytical study on high-order helical structures is the work by Xiang et al. [21] that considers a second-order helix configuration in multistrand wires under tension and torsion without friction. Some of the models and approaches described above were implemented in the Helica software [22] that is used in this study for the analysis of a dynamic cable. The software incorporates a multilayer model that includes the homogenization concept and the friction between layers. The methodology presented in this paper is further extended by applying the hierarchical approach, i.e., the submodeling technique, to make the multiorder modeling feasible [23]. A specific focus of this study is on the WEC application of dynamic cables. Floating point-absorbing WECs are intentionally placed in offshore locations where the scatter diagram shows high wave energy densities. WECs are designed to operate with large responses (e.g., heave motions) to maximize the power take-off. Thus, a submerged power cable connected to a WEC should be designed for a large number of cyclic fatigue loads caused by the motions of the WEC, the waves and the ocean currents [24]. Reliable power transmission from a WEC requires a durable marine power cable design with a mechanical strength that is adequate to withstand the environmental conditions for the typical installation lifetime of 25 years. Several investigations in the literature have contributed to develop the theory and the models required for the analysis of cable structures connected to WECs. Thies et al. [24] assessed the mechanical loading conditions and failure modes of a specific dynamic power cable connected to floating marine energy converters, and fatigue failure of the cable components was identified as the governing failure mode. Yang

et al. [3] presented a study on the numerical fatigue assessment of a flexible and dynamic power cable connected to a floating point-absorbing WEC. They adopted a first-principle design approach where the bending of the cable was identified as the major source of fatigue stresses and where the curvature responses of the cable depended on the environment conditions and loads the cable was subjected to. The necessity of developing a more detailed numerical cable model that can capture intrinsic failure mechanisms was discussed. Nasution et al. [25,26] carried out experiments and finite element (FE) simulations to investigate the fatigue life of copper conductors in power cables for different loading conditions. Different fatigue failure mechanisms were analyzed, and the fatigue properties were summarized in S-N curves. In a similar study by Buitrago et al. [27], an experimental methodology was developed for the fatigue assessment of subsea power cables. It confirmed that the fatigue performance of the cable that they studied was governed by the fatigue failure of the conductor. Identification of the correct failure mode and the failure mechanisms of power cables is important for the cable design and the design of fatigue experiments. Karlsen [28] found in an experimental investigation of the failure mechanisms of power cables that rotatingbending tests are more conservative in comparison to cantilever beam and three-point bending tests [29]. Some studies have shown that the fatigue damage of cables used for dynamical applications can be significantly reduced by an appropriate selection of its mechanical properties [3,29]. Therefore, an appropriate first design strategy would be to estimate the required mechanical characteristics of the cable based on a global load analysis followed by a detailed cable design. Research on marine power cables is scarce in public literature, especially in regard to cable designs (geometry, material and mechanical properties) for floating offshore WECs. One reason for this may be because few commercially available WEC concepts exist, thus, the design of cables for this application is still under development. Recently, some advances have been made in the design methodologies regarding the cross-sectional arrangement of flexible cables. Miller and Albertani [31] proposed a design procedure for single-point power-mooring cables for WECs. The procedure was based on the required axial stiffness of the cables, and they evaluated the fatigue life of structural members made of synthetic, composite and metal alloy materials. Yang et al. [32] presented a design procedure for the section layout of umbilical cables with respect to the geometry, the mechanics and the thermal distribution with a particle swarm optimization algorithm. Despite the evolving research within the topic, there is a gap in the design methods regarding how to obtain marine dynamic cables with the required mechanical properties in tension, torsion and bending, simultaneously. The main objective of this study was to develop an efficient methodology for the dimensioning of complex cables with multiorder helix configurations, which is a typical structure for subsea applications in the marine environment. This study introduces a new practical design methodology and numerical fatigue analysis of flexible power cables that facilitate the development of the cables and increase the understanding of the intrinsic cable failure mechanisms. The cable and simulation models developed for the design and fatigue analysis of dynamic cables are presented. They are applied on a 62

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subsea dynamic power cable with a design suitable for a floating pointabsorbing WEC. Section 2 presents the methodology, which includes the cable design model, three different cable models, and a fatigue analysis procedure. In Section 3, the simulation model of a fatigue test rig for the accelerated testing of the cables is presented. The results from a verification study of the methodology is presented in Section 4 together with a comparison of the results from the experiments and simulations of the accelerated fatigue tests. The conclusions of the study are presented in Section 5.

carry out the fatigue analysis of the cable and its components. It should be highlighted that a fatigue analysis can be carried out using the DeepC results without using Helica; see Yang et al. [3,36,37]. The simulation software package SESAM was used to simulate and analyze the motions and responses of the cable. This software has a number of modules that are used to simulate the environmental loads (DeepC [35]) and to simulate the structural response of a cable subjected to motions, e.g., motions caused by environmental loads from a marine environment (Helica [22]). The developed cable models were used to simulate and analyze the cable in the fatigue test rig. It became necessary to design detailed models to determine the degree of complexity that is required to mimic the observations and results from the experiments and full-scale measurements.

2. Methodology The cable analyzed in this study was a cable that was specifically designed for and mounted on Waves4Power’s WEC concept WaveEL 3.0 [33]. The WaveEL 3.0 is a floating wave energy point-absorber that extracts wave energy from the heaving motions of the WEC and converts them into electrical energy. This WEC was installed together with the cable in full-scale at a test site in Runde, Norway. Fig. 1 shows the principal layout of the WEC system. It consists of a floating buoy moored to the seabed with taut moorings, which allows the buoy to heave according to the motions of the waves and the power take-off system of the WEC. The power produced by the WEC is transmitted to a power-collecting hub via a submerged free-hanging dynamic cable; another power cable goes from the hub to the shore along the seabed. Dynamic cables for WECs are catenary (free-hanging) to allow them to move with the buoy, and they are subjected to both static and dynamic loads. The configuration of the cable system and the static gravity force determine the magnitude and distribution of the global tensile, bending and torque loads [25]. The motions of the buoy that the cable is connected to, as well as the motions of the cable that are caused by the waves and the ocean currents, result in cyclic dynamic loads that lead to fatigue, fretting and wear damages in the cable. The structure of dynamic cables is highly complex. It consists of cylindrical and helical members with different material and mechanical properties, and there is a strong interaction between the members. It is important to understand these interactions and to be able to represent them correctly in a simulation model to make reliable predictions of the fatigue performance and failure modes of the cable. Thus, simulation models for dynamic cables are developed with different degrees of complexity (i.e., how detailed the components of the cable are modeled) by making use of submodeling techniques, which is also known as the hierarchical approach. It is important to outline for which purpose a specific simulation model can be used, as well as its delimitations and its ability to capture information, e.g., modes of failure or other response modes. To depict various levels of complexity, three cable models are presented in Fig. 2. The “simple” model is a model with an axisymmetric solid tube geometry. The “intermediate” model includes the main components that a cable consists of and the cable mechanics theory needed to describe their characteristics on this level of detail. Finally, the “complex” model is a shared name for the models that include subcomponents, where each one of them has a high-degree resolution of their subcomponents, e.g., the copper threads, and cable mechanics interaction models among all the components. Fig. 3 shows an overview of the procedure developed for a cable design, via analysis of the motions and responses in a test rig/in-field installation, to analyze fatigue and assess mechanical degradation; see Sections 2.1–2.3 for details. It starts with a cable design tool in MATLAB [34] called the “Chalmers Cable Design” (CCD) tool. The geometric properties, material data and other relevant properties for a cross-section of a dynamic cable and its characteristics are given as inputs. The cross-sectional properties of the cable are used as inputs to the cable motions analysis in the SESAM DeepC software [35], where the simple cable model in Fig. 2(a) is used. The responses from the DeepC simulation, and an intermediate cable model in Fig. 2(b) defined by the CCD tool, form a model which is used in the SESAM Helica software [22] to

2.1. Cable model for cross-sectional analysis and optimization Most cables consist of several conductor wires that are braided together with some lay angle α —the angle that the component forms with the axis of a cable, see Fig. 4. The distance along the center axis that a helical component needs to make one complete revolution with some layer radius Rlay is defined as the lay length Llay . The inner layer radius Rin is the radius of a central core around which the helical component is laid around. The relation between the lay angle α and the lay length Llay is given by Eq. (1),

Llay =

2πRlay tan(α )

(1)

Every conductor has a conducting metallic core. If the cable is intended for fixed installation, i.e., if it is subjected only to static loads, the core of the conductor is usually either solid or consists of a few wires. However, if a high degree of flexibility (i.e., low bending stiffness) of the cable is desired, the core is often constructed with thin wires, which are assembled in strands that are helically laid together around a center, see Fig. 5. The Helica code—the kernel of the CCD tool—has restrictions on the modeling complexity of helical structures. In the current version (version 2.5), it does not allow for the direct modeling of helical components that in turn are composed of helical components. The conductors in a dynamic power cable are an example of such a structure. A submodeling procedure for the “complex” cable model, see Fig. 2(c), was developed to overcome the restrictions in Helica. Fig. 5 outlines how different components of a cable were designed as submodels and assembled together; these assemblies were used to estimate the mechanical properties of complex dynamic cables. The obtained crosssectional properties of the submodel are assigned to the general helical components of a higher-level model to prescribe the tangential, radial and torsional moments of inertia together with its elastic and shear moduli. Therefore, the submodel with the lowest level of complexity consists only of a number of solid bodies, such as the conductor strands. Materials in the models are assumed to be linear elastic. Contact between components within one helical layer is neglected during analysis in Helica. This limitation required additional modeling techniques for cross-sections without a center core. An absence of the

Fig. 1. Principal layout of the WaveEL 3.0 system: (1) gravity anchor, (2) mooring line, (3) floater, (4) buoy, (5) power cable and (6) hub. 63

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Fig. 2. Cable models with different levels of complexity: (a) simple, (b) intermediate, and (c) complex.

(b). By applying the submodeling approach, multiorder helical cable geometries and designs can be handled using the CCD tool and Helica together. 2.2. Chalmers cable design tool The Chalmers Cable Design (CCD) tool, see Fig. 3, was developed to enable cable modeling and to optimize cable dimensions with respect to the desired mechanical properties such as the axial, bending and torsional stiffnesses. The CCD tool also provides a possibility to carry out detailed parametric studies of cable parameters to determine how these influence specific cable properties and characteristics. The CCD tool can be used during the final design step for cable dimensioning after the required characteristics and a suitable cross-sectional layout are determined. The CCD tool was developed on the MATLAB virtual platform [34] with the SESAM Helica software version 2.5 [22] as a kernel for the cross-sectional analysis of the cable. A thorough description of the models and algorithms that Helica is based on are presented in the literature [38]. The MATLAB Global Optimization Toolbox functionality is used in the optimization part of the CCD tool. The CCD tool has a large flexibility in cable design, but the user has to give initial inputs for the basic and conceptual cable design. The data about components, their arrangement in the cable and material properties have to be provided. From the initial cable design model, it is possible to optimize the model and its characteristics by defining the constraints, design variables and desired mechanical properties in the optimization algorithm. A constrained optimization of a cable built in the CCD tool is possible using a genetic algorithm (GA), which is based on evolutionary selection of solutions [39]. The GA has proven to be a better algorithm in identifying global minima of objective functions in comparison with traditional optimization methods, which are often sensitive to initial values of the design variables. In the GA optimization, the design variables are treated as independent parameters whose values are allowed to vary. Examples of typical design variables for a cable are the nominal diameter of a conductor wire, the lay angles, the insulation thickness and the number of helical components. Some of the cable properties that can be defined in the CCD tool are the axial, bending and torsional stiffnesses and the mass per unit length of the cable. In

Fig. 3. Outline of the fatigue analysis procedure developed for dynamic cables, including its steps and connected simulation models.

Fig. 4. Definition of the basic parameters of the helical components.

center core in a Helica model leads to unrestricted deformation of the innermost helical layer towards the center, which results in underestimated values of the stiffness properties of the cable. The solution to this limitation was to introduce a “dummy” core component in the CCD tool; see the cable model in Fig. 9. The dummy component is assigned only a radial stiffness in order to simulate “ring contact” of the components within a layer. To conclude, the Helica code in the CCD tool is called for crosssectional analyses as many times as required and as many times as there are different levels (helical orders) in the “complex” model. The firstorder helical model is therefore the “intermediate” model, see Fig. 2

Fig. 5. Principal design of a (left) dynamic cable with all its components, (middle) one of its conductors, and (right) strands composed by copper wires. 64

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which results in lower fatigue life bounds [44]. The full-stick tube model is based on the Euler-Bernoulli beam theory, and it assumes that the plane sections remain plane during bending and that the helix element is placed in its actual off-center position. This approach yields conservative fatigue life results from a design perspective. The cable installed in WaveEL 3.0 is an example of a structure with a Seale arrangement of conductors and synthetic fiber ropes (see the far-left image in Fig. 9). Seale is a construction where two neighboring layers have the same amount of helical components and where the outer elements rest in the valleys between the inner elements. In Helica version 2.5, all components in the Seale arrangements have to be modeled in one layer with the same inner layer radius Rin (see Fig. 4) during fatigue analysis for correct redistribution of frictional forces, and the intermediate cable model has to be modified accordingly. It was found in this study that this technique has to be used with caution. This is a suitable approach for fatigue analysis, however, it should not be used in cross-sectional analysis and optimization.

this study, the objective function was defined to satisfy a condition where an optimized numerical model of the cable fits the target values of axial, bending and torsional testing of a real cable; see Section 4.2 for details. Note that the goal parameters in the optimization can be given different weight coefficients in order to enable rank priorities of their fulfillment. As a result of the limitations of the Helica code, the contact and friction between components within one helical layer are omitted, e.g., the locking behavior cannot be predicted. This assumption is valid if the components within one layer are arranged with some tangential space [18]. Therefore, an analytical procedure for obtaining the appropriate layer inner radius during optimization, which prevents contact within one layer in a nondeformed condition, was adopted in the CCD tool using an assumption for thin wires in accordance with Costello [16]. The minimum inner layer radius just before contact is defined as 99% of the design radius. This is a common practice in cable design: to leave some space between the components of the cable in favor of decreasing the friction. The CCD tool has a built-in feature that checks the compliance with IEC 60228 [40] and IEC 60502 [41] regulations for the conductors of the insulated cables and the power cables with extruded insulation, respectively. The IEC 60228 and IEC 60502 are part of the ISO standard 13628-5 – Subsea umbilicals [42]. These can be used as additional constraints in the GA optimization module of the CCD tool. It should be noted that only the geometric dimensions, material properties and mechanical properties can be optimized in the CCD tool, while the electromechanical and thermomechanical properties of the cable are used only as constraints.

3. Fatigue test rig 3.1. Fatigue test rig overview One of the main causes of fatigue damage in free-hanging or floating marine power cables is the cyclic variation of axial and bending stresses from the motions of the WEC, the waves and the ocean currents. According to Yang et al. [3], the stresses in the cable are dominated by the bending normal stress, but the superposition from the axial normal stress can be important in some locations along the length of the cable and should therefore be part of the stress analysis. Other studies reaffirm that bending is the most critical loading regime in terms of fatigue damage [30]. A test rig was designed for accelerated fatigue testing of full-scale cables in a laboratory environment. The purpose of the experiments was to examine the fatigue characteristics of a power cable, to follow its mechanical degradation as a function of the number of load cycles and to examine its intrinsic failure mechanisms. An illustration of the test rig is shown in Fig. 7. The test rig can be used for accelerated rotatingbending fatigue tests of cables, and it offers a possibility to carry out fatigue testing at loading frequencies of 1.5–2 Hz. The upper end B of the cable is fixed to the rig in all 6 degrees of freedom, while the end A is allowed to rotate around a vertical axis through point B with a pre-described rotational frequency. For every revolution around the vertical axis, to avoid accumulated twisting of the end points A and B, the end point A is rotated back around a horizontal axis that goes through point A with the same frequency as the rotational frequency around the vertical axis; point A is fixed in the other degrees of freedom. This rotation is achieved by transmitting motions from the driving axis of the rail through the bevel gear and the belting. Further, the bending radius R is set by adjusting the bracings of the cable in height and along the rail. The bending radius is defined as the distance from the connection points to the intersection of the local longitudinal cable axes. The cable in the test rig is, during one revolution, evenly bent in all in-plane directions at the radius R. During a test, the conductors of the cable were connected to measurement equipment that measured the electrical conductivity and the temperature on the surface of the cable

2.3. Fatigue analysis The mechanical degradation due to the accumulation of fatigue damage in the cable was assumed to follow the stress-based (S-N) phenomenological fatigue approach. Other degradation mechanisms such as fretting, wear between individual wires and the bird-caging phenomenon were disregarded. Several studies pointed out that the strain-based approach has to be applied for copper material due to its nonlinear stress-strain relationship and poor creep properties [7,8]. In the present study, however, simplifications of the material properties were unavoidable due to the restrictions of the Helica solver. Thus, the S-N approach was adopted in the fatigue assessment of the load responses from the numerical analyses using the simulation model of the cable fatigue test rig model (see Section 3). In the fatigue analysis procedure of the test rig results, the time series of the curvatures and axial forces of the cable were used as inputs to the intermediate cable model created in the CCD tool and the Helica code. The fatigue evaluation was carried out in 16 helix positions covering one pitch of the helix element. At each helix position, the fatigue damage was calculated in 8 circular hotspots, see Fig. 6 and [43] for details. This procedure was used to identify the most fatiguecritical location along the length of the cable and in its cross-section. The rainflow counting method was used to calculate the fatigue stress cycles from the DeepC and Helica calculations. The PalmgrenMiner rule was utilized to calculate the accumulated fatigue damage from the fatigue stress history. The partial fatigue damage for a fatigue stress cycle was calculated using the Basquin equation in Eq. (2), where the material parameters for a copper material that are commonly used in power conductors, as presented by Nasution et al. [25], were used: m = 6.238 and C = 6.098×1019.

N (Δσ )m = C

(2)

The friction forces in the subcomponents of the cable (second order helix), such as the strands, cannot be handled in Helica in consistency with the general stick/slip behavior applied to the first-order helix components. To get around the problem, the forces in the subcomponents are calculated by applying the full-stick tube assumption,

Fig. 6. Illustration of the positions where the fatigue assessments are carried out in a cable model [43]. 65

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The cross-section of the cable in DeepC was defined with the “simple” model with global stiffness properties obtained by the CCD tool. The constant full-slip bending stiffness was assigned at all sublevels of the model because of small critical curvature values found during the cross-sectional analysis using the CCD tool. Consequently, the full-slip condition in the cable was assumed to initiate directly after the load was applied. This assumption is made on a global level as if the components in the “intermediate” model were not restricted from relative motions; note that this is not to be confused with the full-stick assumption for the subcomponents of the conductors during fatigue analysis, see Section 2.3. 4. Results and discussion The low-voltage dynamic power cable was tested in the cable fatigue test rig for different bending curvatures, i.e., bending radii R. The numerical models of that cable were developed and designed using the CCD tool due to uncertainties in the dimensions and material properties. These cable models were used in the simulations of the cable fatigue test rig, and the results were compared against the results from the experiments. Moreover, a case study of a flexible pipe was performed for comparison. Fig. 7. Illustration of the cable fatigue test rig.

4.1. Benchmark study in three points in the middle region of the cable. After each test, the cable was dissected for detailed analysis of its mechanical degradation. Each test was run until a predefined interruption criterion was reached—in this case, a 15% drop in electrical conductivity—or after a given number of load cycles when the test was judged as a run-out test. The initial results from the fatigue tests in the cable rig showed long fatigue lives with several run-outs. It motivated the development of the numerical simulation model of the test rig. Since the predefined interruption criteria cannot be considered in the cable models as failure criteria, the fatigue stresses in the simulated cables in the rig tests were used to predict their fatigue lives according to the procedure presented in Section 2.3. These values were then compared against the results from the experiments, at least for the test objects that were not deemed as run-outs. Another important argument for the development of the test rig simulation model was to enable detailed analysis and evaluation of the curvature and axial force distributions along the cable in consideration of the dynamic effects caused by the rotational speed.

The Helica software was previously validated, and it demonstrated good agreement with experimental results [46]. The CCD tool cable modeling and cross-sectional analysis were validated against a case study of flexible pipes that was presented by Witz [47]. This case study enabled a comparison against ten different cross-sectional analysis algorithms with mechanical tests of the same flexible riser, see Fig. 8. Since both risers and cables are composed of helical structures, the same analysis method can be applied for both cases. Table 1 presents a comparison of the results in terms of the axial, torsional and bending stiffnesses of the riser. The mean value and standard deviation results from [47] were calculated after excluding the highest and the lowest values; the results from CCD-Helica were in good agreement with these mean values (within 4% difference). The largest difference is noticed for the bending stiffness, which had a large scatter in the results among the participants; moreover, not many results were reported. This benchmark study shows that the CCD-Helica methodology compares well with other tools used for calculating mechanical properties of flexible cables and risers.

3.2. Simulation model of the rig

4.2. Cable design using the CCD tool

The simulation model was developed in the software SESAM DeepC [35], which is a coupled floater motion analysis software that uses the two subprograms SIMO and RIFLEX for finite element analyses. This software enabled the possibility to connect the results to the Helica software. The cable is represented in DeepC as an axisymmetric line with outer geometric dimensions and mechanical properties that can be calculated using the CCD tool. The cable end B is fixed in all 6 degrees of freedom, while the end A has pre-described motions that mimic the exact test conditions in the test rig with translations and rotating Euler angles for every time increment. In the DeepC subprogram RIFLEX, the cable was modeled with nonlinear beam elements with cubic interpolation [45]. A convergence analysis of the length of the beam elements resulted in a 10 mm mesh density. The time step increments were set to 0.005 s to achieve a solution that was numerically stable. The simulation requires between 5 and 10 full revolutions/load cycles due to the transient responses in the start and interruption of the simulations. After completing a simulation, the curvatures and axial forces along the cable for the “stable” load cycles were exported to Helica for further post-processing and analysis; see Fig. 3. To exclude the undesired end effects, the results used for the fatigue analyses were taken at some distance from the boundaries.

The numerical model of the cable tested was named the Numerical Dynamic Power Cable (NDPC) to distinguish it from the real cable. Fig. 9 presents sketches of the cable and its components, which are modeled in detail in the NDPC model.

Fig. 8. Schematic of the case study flexible riser in Witz [47]. 66

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Table 1 Results from a verification exercise of the CCD-Helica methodology compared against the data in Witz [47]. ITCON and IACON indicate if the riser is torsionally or axially constrained, respectively. CW and CCW are the clockwise and counter-clockwise applied torsional load, respectively. GKV [kNm2/rad]

EA [MN] ITCON = 0

Mean value Standard deviation CCD-Helica

127 12 126

ITCON = 1

129 13 127

EI [Nm2]

IACON = 0

IACON = 1

Full slip

CW

CCW

CW

CCW

4.1 1.2 4.3

175 10 180

87 7 79

209 15 215

The NDPC model was developed and optimized to fit target values that were based on the axial, bending and torsional testing of the cable. This was the chosen procedure, which aimed to achieve a numerical model that was as close as possible to the “real” cable despite the uncertainties in dimensions and material properties. The initial design of the NDPC cable prior to optimization had a Seale construction consisting of three conductors that were laid together with synthetic ropes. The function of the synthetic ropes was to take up axial loads, and they were only given an axial stiffness in the numerical model since they are very flexible in all other directions. Polyurethane (PUR) was chosen as the material for the outer protective sheathing. The conductors had a polyethylene (PE) insulation and a conducting core made of copper with a 50 mm2 nominal cross-sectional area. The copper conductors were composed of two wire strand layers, which were concentrically laid around a strand core. The strands were also concentric with three layers of copper wires laid around the center wire. The torsional stiffness for the “intermediate” model can be obtained for both the clockwise and counter-clockwise directions. However, the torsional moment of inertia for a general helical component in the “complex” model can only be specified in one direction in Helica. This complicates the correct modeling of regular and lang lays. In a regular lay, the wires in the strands are laid in the opposite direction to the lay direction of the strands, while for a lang lay, the wires and the strands are laid in the same direction [16]. The conceptual design of the cable was assumed to follow the lang lay configuration (both for the strands and the conductors) due to its better fatigue and abrasion resistance compared to the regular configuration [48]. It has to be noted that with such a design approach, it is difficult to obtain a torsion-free configuration of the cable, which is an important design criterion for some applications [32]. However, it is impossible to achieve a torque-balanced design with the initial concept of the cable studied, see Fig. 9; hence, consideration of this aspect was omitted. The outer cable diameter and the cross-sectional areas of the conductors were important constraints. The dimensions of the cable design should be in accordance with the IEC regulations; see IEC (1993) and IEC (2004). The lay angle is the dominating geometrical parameter that affects the axial, bending and torsional stiffnesses in helical structures [49]. Thus, the lay lengths of the helical components within every subcomponent were chosen as independent design variables together with the conductor insulation thickness. The axial and bending

2564 2535 1618

stiffnesses were prioritized goal parameters in the optimization process, since they have the largest influence on the fatigue life of the cables studied in this project, for the loads they were subjected to. The provided data for the physical cable stiffness properties were based on few experiments, which showed large scatter. It was therefore decided to define target design values for the NDPC cable that were close to these data; these values were used in the CCD tool for optimization. Table 2 presents the resulting mechanical properties of the optimized NDPC cable. In addition, the outputs from the optimization process are solutions for the design variables and mechanical properties of the subcomponents of the cable. The CCD tool found a perfect match for the axial and bending stiffnesses of the NDPC cable, but there was some deviation in the torsional stiffness. Since torsional stiffness has a negligible influence on the fatigue life (i.e., fatigue stresses) of the cable studied, considering the loads it is subjected to, the difference was accepted. Additionally, the increase in mass per unit length was considered marginal.

4.3. Numerical simulations The cable mechanical properties presented in Table 2 were used in the simulation model of the fatigue test rig to study, among others, the sensitivity in the results with regard to some specific parameters. The “simple” model in Fig. 2(a) was used as the cable model with different bending radii, R, and cable lengths, LR ; see Table 3 and Fig. 7. The bending radii were chosen to match the bending radii used in the experiments. Some simulations were also carried out for different cable lengths, where at least one of them was from the experiments, to evaluate the dependence of the installed cable length on the fatigue life of the cable. The letters S and L denote the short and long cable lengths, respectively. If LR is defined as one fourth of a circumference with radius R , the short cable length was defined as LS = 1.01·LR and the long cable length as LL = 1.05·LR ; the value of the short cable length was used the most in the simulations. Fig. 10 presents an example of the axial force and curvature results for the cases labeled R4S and R8S. The axial forces are rather small, and they act in tension for the larger bending radius, R8S, and in compression for the smaller bending radius, R4S. As expected, the R4S has a larger curvature than the R8S, where both maxima are close to half of the respective length of the cables. Fig. 11 presents screenshots from the motions of the cables from a

Fig. 9. Sketches for every submodel of the NDPC cable with (left) dummy core in the center, (middle) a conductor, and (right) the conductor strands. 67

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Table 2 Mechanical properties of the studied cable and its components. Source

Component

Axial stiffness, EA [MN]

Bending stiffness, EI [Nm2]

Torsional stiffness, GKv [Nm2/rad]

Mass per unit length [kg/m]

Target design values Calculated using the CCD tool

NDPC NDPC NDPC NDPC

4.00 0.077 1.38 4.00

4.00 0.002 0.77 4.00

3.00 0.003 0.62 3.70

2.00 0.024 0.52 2.13

cable strand conductor cable

calculation. Furthermore, the conductor in the intermediate model (used for fatigue analysis) is assigned a lower stiffness than the stiffness of its subcomponents. This results in lower fatigue stresses compared to the outer and the inner strands. Thus, the fatigue stresses at the hotspots are reduced, which gives a longer fatigue life. Hence, the curve represents the conductor component in a component fatigue test; cf. [44]. A parametric study of the sensitivity in fatigue life due to variations in the axial stiffness in tension (EA), the bending stiffness (EI) and the torsional stiffness (GKv) was carried out. The simple cable model was used, with different values of these stiffnesses, in the fatigue test rig simulation model. The fatigue life of the conductor for each case is presented in Fig. 13. The torsional stiffness was also varied as 1.5, 3 and 4.5 Nm2/rad but it did not affect the results at all; hence, those results are not reported here. The fatigue life results show that, for the current loading and boundary conditions, the value of the bending stiffness affects the fatigue life of the cable the most. Fig. 14 presents the relationship between the installed bending radius and the minimum curvature, as well as the relationship between the bending radius and the fatigue life of the cable. It is also shown that the minimum curvature radius is smaller compared to the bending radius installed in the test rig; this is an effect from the dynamic response and gravity (see Fig. 11). The results also show that the longer cables have shorter fatigue lives than the shorter cables.

Table 3 Summary of simulation cases using the fatigue test rig simulation model. Simulation

Bending radius [mm]

Cable length [mm]

R4S R4 L R5S R6S R8S R8L

400 400 500 600 800 800

635 660 793 952 1269 1330

side and top view, respectively. The rotational speed and inertial effects gave rise to sideways motions. The middle point of the cable is indicated. The results from these dynamic simulations show that the bending curvatures were larger due to dynamic effects than those observed in the static solution. It was concluded that for a correct estimation of the fatigue life in the fatigue test rig, a static solution of the bending of the cable is insufficient and that a dynamic simulation model is needed in the design of such an experiment. Furthermore, the results also show the importance of well-controlled installed cable lengths to achieve the curvature response that is expected in the design of the experiment. Mechanical failure of the conductor was identified as the fatigue performance criteria of the cable [28]. Thus, the fatigue lives of the copper conductor and its components were calculated with respect to the cable length by using the “intermediate” cable model shown in Fig. 2(b) with Helica. The results in Fig. 12 show an example from the simulation case R8S; the other simulations showed the same trends. The outer strands within the conductor have the lowest fatigue life, followed by the inner and the core strands. This is consistent with the assumption of plane bending, where normal stresses at a large distance from the neutral axis are larger compared to the normal stresses close to the neutral axis. If the “intermediate” cable model is used, see Section 2.3 for its assumptions and limitations, the results will give shorter fatigue lives. A validation by experiments is recommended to define which of the boundaries represent the fatigue failure criterion the best. In addition, the curve denoted as “Conductor” corresponds to the fatigue life based on the loads in the cable that have been distributed on the nominal conductor core cross-sectional area in the stress

4.4. Comparison of results from experiments and numerical simulations The results presented in Sections 4.1 and 4.2 show that the numerical simulation models are valuable for sensitivity analyses and parametric studies. These models can also be employed to clarify the part or component of a cable that is expected to fail first due to fatigue. The results from the fatigue test rig experiments and the numerical simulations using the NDPC cable model are presented in Fig. 15. Note that the dissection of the cables from the experiments could not confirm which component (e.g., wires) that lead to the first mechanical failure (e.g., fracture). It only confirmed that the largest damage (partly broken wires and wear) occurred in the middle of each cable where the curvature was the largest; the parts of the cable near the end points were unaffected. Thus, the fatigue results for all strands and the conductor from the numerical simulations are presented for comparison together

Fig. 10. Axial force (left) and bending curvature (right) distributions along the cable length for the simulations R4S and R8S. The normalized cable length coordinate denotes 0 – the moving cable end (A) and 1 – the rigidly fixed (B) cable end. 68

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Fig. 11. Cable profiles from a side view (left) and from a top view (right) from four simulations.

with the results from the experiments. Future work should develop a better procedure for experimental testing that can define the degradation or failure of a property, which can be quantified/measured and used in numerical simulation models as a corresponding fatigue failure criterion. The number of experiments was too low to be able to draw solid conclusions. However, despite the large scatter in the results from the experiments, it seems that the conductor curve is closer to the experimental results than the strand results. It has also been discussed in Section 4.3 that a fatigue test in the rig is sensitive to the installed cable length, i.e., LS and LR , as this will influence the curvature levels. Moreover, the fatigue lives of the strands are obtained in a conservative way, as elaborated in Section 2.3. The uncertainties in the parameters included in the simulation models, which were measured in the laboratory experiments (such as the axial stiffness in tension, EA, the bending stiffness, EI, and the torsional stiffness, GKv) must be better understood and quantified. The simulation models can be used for parametric and parameter sensitivity analyses, but these need to be compared against measured data. The thermomechanical influence from electric current, and whether it acts together with mechanical loads to shorten mechanical life, was not studied either. The effect is known if the electric current gives rise to certain phenomenon, e.g., high temperatures inside the cable, that influence the creep properties of copper [28]. For the NDPC cable, the majority of the fatigue results from the experiments are shifted towards longer fatigue lives compared to the predictions made by the simulation model. There could be several reasons for this, and some of them are mentioned hereafter.

Fig. 13. Sensitivity analysis of the influence of bending and axial stiffnesses on the fatigue life of a conductor in the NDPC cable.

assumptions were made, and data from the literature were used.

• The mechanical properties such as the stiffness properties have a •

• A cable simulation model is only a model that has been developed to try to mimic the real cable as realistically as possible. In this project, several material properties of the cable were not available. Hence,

large influence on the fatigue results, as shown in Fig. 13. More thorough measurements of these values, including their scatter, could have assisted in the understanding of deviations between the experiments and the results from the numerical simulations. Stress-strain relations for soft annealed and hard drawn copper wires are different [50], and it is not fully clear when the simulation models are developed which copper material is actually used in the cables. This should be investigated in more detail in future works.

Fig. 12. Fatigue life of components in the NDPC cable from the R8S simulation. 69

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Fig. 14. Relationships between the installed bending radius, the actual minimum curvature radius, and the fatigue life of the components of the NDPC cable.

5. Conclusions

responses and, consequently, the fatigue results. The outermost strands within the conductors were predicted to have the shortest fatigue lives due to rotational bending. It was shown that the simulation model of the fatigue test rig can serve as an important tool in the design and analysis of forthcoming experiments before they are started. The simulation models and numerical procedures developed can also be applied in other applications, presuming that there is sufficient information available to create the hydrodynamic and structural models of the objects and their components. Further development of the methodology for the numerical simulation and prediction of the mechanical degradation of dynamic cables and their components is needed. This study did not include wear or fretting damages to the conductor wires, despite knowing that these accelerate wire breaks and shorten the mechanical life. In turn, to incorporate these phenomena, it is believed that direct multiorder helical models are required to estimate the stresses and relative motions between the subcomponents. However, the load analysis methodology with the simple model and the fatigue analysis corresponding to a component test (i.e., the conductor curves in Fig. 15) show that this level of model detail is sufficient, at least in an early design stage when the different cables and their properties should be compared with regard to their fatigue lives.

This study contributed a new numerical modeling method of complex multiorder helical structures based on a hierarchical approach. Design and optimization procedures with respect to the mechanical properties of the cable were established. The developed CCD tool, including the cable mechanics that are inherent to the software used, were verified against the results from the literature. The tool proved its applicability for the generic design of geometric and mechanical properties of helical structures, which can be an asset for future works and to improve the design of flexible pipes, umbilicals and power cables. The fatigue performance of a dynamic subsea power cable for wave energy applications was examined. The cable fatigue life due to rotational bending was investigated both numerically and experimentally. For these purposes a new accelerated fatigue test rig was devised and built for the study of a cable that is identical to the one used for WECs. The numerical simulation model of the test rig was utilized to estimate the responses of the cable due to imposed deformations. A fatigue analysis of the numerical cable was conducted, and the simulation results were compared with the tests; the simulations and the tests demonstrated a general agreement. The importance of dynamic effects due to rotational motions and the accuracy in the installed cable length in the test rig was revealed. Additionally, it was found that mechanical properties and, in particular, the bending stiffness of the cable had a large influence on the motion

Fig. 15. Comparison of the fatigue lives (N) from the experiments and the numerical simulations using the NDPC cable. The arrows mark the run outs in the experiments. 70

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Acknowledgment

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