Performance of a Cable-Driven Robot Used for Cyber–Physical Testing of Floating Wind Turbines

Abstract

Cyber–physical testing has been applied for a decade in hydrodynamic laboratories to assess the dynamic performance of floating wind turbines (FWTs) in realistic wind and wave conditions. Aerodynamic loads, computed by a numerical simulator fed with model test measurements, are applied in real time on the physical model using actuators. The present paper proposes a set of short and targeted benchmark tests that aim to quantify the performance of actuators used in cyber–physical FWT testing. They aim at ensuring good load tracking over all frequencies of interest and satisfactory disturbance rejection for large motions to provide a realistic test setup. These benchmark tests are exemplified on two radically different 15 MW FWT models tested at SINTEF Ocean using a cable-driven robot.

1. Introduction

Performing hydrodynamic model tests is standard practice for offshore structures and is often recommended or required by codes and classification societies [1,2]. As we venture into the world of sustainable energy, floating wind turbines (FWTs) play a crucial role in capturing wind power in deeper waters. However, performing hydrodynamic model testing of FWTs poses several challenges, the major one being the Froude–Reynolds scaling conflict, which prevents accurate modelling of rotor loads in hydrodynamic laboratories [3] due to the significant difference in Reynolds number. The lift and drag of a wind turbine rotor and tower follow the Reynolds scaling law, which is difficult to achieve in hydrodynamic laboratories. Several techniques have been developed to address this issue, including performance-matched rotors [4] and cyber–physical approaches [5,6]. The present paper focuses on the latter approach.

Generally speaking, cyber–physical testing is a method that combines numerical simulations with experimental testing. The goal of the method is to obtain more accurate results than what could be achieved with each method separately. In cyber–physical testing, a physical and numerical substructure are coupled in real time using a control system. This technique has been applied to FWT testing for about a decade at SINTEF Ocean under the name ReaTHM^® testing1 [6,7]. OpenFAST2 has been used to compute the rotor loads. These are applied on the physical substructure (the floater in the hydrodynamic laboratory) using a Cable-Driven Parallel Robot (CDPR) [8]. A CDPR consists of a set of winches, pulling each on a given point of the model. CDPRs are used in several laboratories [9,10]. The current alternatives to CDPRs are single fans [5,11] or multi-rotors [12,13,14,15].

There are significant differences between the actuators used in cyber–physical hydrodynamic testing: their intrusiveness in terms of weight and space, how many of the six load components they apply, how fast they respond to a commanded change in load, and how they cope with floater motions. The fidelity of cyber–physical tests relies on the performance of the control system as a whole [16,17] and of the actuators in particular. As of today, there is no standard way of assessing this performance.

The present paper proposes a set of short and targeted benchmark tests that aim at quantifying the performance of actuators used in cyber–physical FWT testing. They aim at verifying the performance of the actuator prior to running tests in a combined environment and can be repeated on a regular basis to ensure a functional setup over long test programs. The procedure is exemplified by laboratory data acquired using SINTEF Ocean’s CDPR as a particular case. Note, however, that they are relevant for any actuator solutions and for other applications of cyber–physical testing beyond FWTs.

This paper is structured as follows. First, a general overview of the control loop used in cyber–physical hydrodynamic testing is presented, with a particular focus on the CDPR actuator. Then, two case studies from model tests conducted at SINTEF Ocean are introduced involving wind/wave/current tests of 15 MW FWTs, illustrating the performance of the CDPR. In Section 4, targeted benchmark tests for the CDPR are introduced, and it is shown how the performance of the CDPR obtained from these benchmark tests can be related to the tests performed in more complex wind/wave/current conditions. The conclusions are given in Section 5.

2. Control Loop of a Cyber–Physical Model Test Using a Cable-Driven Parallel Robot (CDPR)

The typical control loop of a cyber–physical model test is illustrated in Figure 1. When using a CDPR, the actuators consist of several force-controlled winches located around the basin, which are attached to the model (referred to as the physical substructure in Figure 1). Sensors mounted on the model typically include an accelerometer, a gyrometer, and an optical motion capture system that measures the floater’s position in six degrees of freedom (6DOF). The observer block in the control loop comprises a kinematic observer that estimates the position and velocities of the model, which are then used to calculate aerodynamic loads in the numerical substructure. The resulting desired loads are fed into the allocation module, which calculates the target tension in each of the lines that yield the target load vector (commanded by the numerical substructure).

This control loop is rather generic and can be instantiated for various applications beyond FWT testing, such as the testing of wind-assisted ships [18] and the estimation of nonlinear hydrodynamic loads on moored structures [19]. In the following, we will present in more detail the working principle of the actuator (the CDPR) and the allocation module, which are of particular importance for the present paper. A thorough description of the force controller for each individual actuator has been given in [20].

2.1. Coordinate System and Kinematic Transformations

We follow the notations of [21] (Chapter 2) and define two frames of references, shown in Figure 2: The first one is Earth-fixed and denoted $\left\{n\right\}$. The second one is fixed to the FWT and denoted $\left\{b\right\}$. Two direct coordinates systems $(O,n_1,n_2,n_3)$ and $(B,b_1,b_2,b_3)$ are associated with $\left\{n\right\}$ and $\left\{b\right\}$, respectively, with reference points and orientations that depend on the application at hand. Typically, for FWTs, point B is located at the tower top, and $b_3$ is along the tower’s axis, pointing from the tower top to the tower base. The Earth-fixed $n_1$ and $n_2$ point towards the North and the East, and $\left\{n\right\}$ and $\left\{b\right\}$ coincide when the FWT is at rest.

The position of $\left\{b\right\}$ relative to $\left\{n\right\}$ is described by the position of B in $\left\{n\right\}$

$$p:={(x,y,z)}^{\top }$$

(1)

The orientation, also known as attitude, of $\left\{b\right\}$ relative to $\left\{n\right\}$ is described by the Euler angles

$$\mathsf{\Theta }:={(\varphi ,\theta ,\psi )}^{\top }$$

(2)

which denote the roll, pitch, and yaw angles, respectively. The body pose vector is

$$\eta :=(p,\mathsf{\Theta })$$

(3)

The rotation matrix $R\left(\mathsf{\Theta }\right)\in SO\left(3\right)$ maps vectors from $\left\{b\right\}$ to $\left\{n\right\}$:

$$R\left(\mathsf{\Theta }\right)=R_z\left(\psi \right)R_y\left(\theta \right)R_x\left(\varphi \right),$$

(4)

with

$$R_x\left(\varphi \right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & -\sin \varphi \\ 0& \sin \varphi & \cos \varphi \end{array}\right),R_y\left(\theta \right)=\left(\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right),R_z\left(\psi \right)=\left(\begin{array}{ccc}\cos \psi & -\sin \psi & 0\\ \sin \psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right),$$

(5)

leading to

$$R\left(\mathsf{\Theta }\right)=\left(\begin{array}{ccc}\cos \psi \cos \theta & \cos \psi \sin \theta \sin \varphi -\sin \psi \cos \varphi & \sin \psi \sin \varphi +\cos \psi \cos \varphi \sin \theta \\ \sin \psi \cos \theta & \cos \psi \cos \varphi +\sin \varphi \sin \theta \sin \psi & \sin \theta \sin \psi \cos \varphi -\cos \psi \sin \varphi \\ -\sin \theta & \cos \theta \sin \varphi & \cos \theta \cos \varphi \end{array}\right)$$

(6)

2.2. Load $\tau$ Applied by the CDPR from the Tension T in the Cables

The CDPR consists of a set of $n_a$ actuators (typically 6 or 7 for FWT testing) mounted around the hydrodynamic laboratory, each connected with a cable to a frame located on the top of the FWT. See, for example, [8] for alternative CDPR configurations. For each actuator $i\in \{1,\dots ,n_a\}$, let $p_{a,i}^n$ be the fixed position of the ith actuator $A_i$ around the basin. Similarly, let $E_i$ be the ith attachment point of the corresponding line on the FWT (often called the “end-effector” in the CDPR literature). The constant vector $r_i^b$ in $\left\{b\right\}$ describes the position of $E_i$ with respect to B. It follows that the absolute position of $E_i$ in $\left\{n\right\}$ is

$$p_{e,i}:=p+R\left(\mathsf{\Theta }\right)r_i^b$$

(7)

For each actuator $i\in \{1,\dots ,n_a\}$, let now $f_i\in {\mathbb{R}}^3$ be the force acting on the FWT at $E_i$, equal to

$$f_i=T_i{u}_i$$

(8)

where the cable tensions are gathered in a vector $T\in {\mathbb{R}}^{n_a}$, and $u_i\in {\mathbb{R}}^3$ is a unit vector given by

$$u_i:={\displaystyle \frac{p_{a,i}^n-p_{e,i}^n}{|p_{a,i}^n-p_{e,i}^n|}}\in {\mathbb{R}}^3$$

(9)

We then define the load vector $\tau \in {\mathbb{R}}^6$ applied by the CDPR. Note that $\tau$ is expressed in the local coordinate system $\left\{b\right\}$ such that ${\tau }_1$ would, for example, refer to a “surge force” and not a “North force”.

The so-called Jacobian matrix (or configuration matrix) $J\in {\mathbb{R}}^{6\times n_a}$ links the tension in the cables to the load vector $\tau$ by

$$\tau =JT$$

(10)

where

$$J=\left[q_1,q_2,\cdots ,q_{n_a}\right],\mathrm{with}{q}_i=\left[\begin{array}{c}R{\left(\theta \right)}^{\top }{u}_i\\ r_i^b\times R{\left(\theta \right)}^{\top }{u}_i\end{array}\right],i\in \{1,2,\dots ,n_a\}.$$

(11)

where the transformation $R{\left(\theta \right)}^{\top }{u}_i$ expresses line i’s direction in $\left\{b\right\}$, and the cross-product computes the moment of the force $f_i$ at the point B.

Note that J varies in time due to variations in the attitude $\mathsf{\Theta }$ of the FWT and to the variation in the position p of the FWT involved in the expression of $u_i$. To compute the load $\tau$ applied by the CDPR, it is therefore important to keep precise track of the tension in the cables and on the pose of the FWT.

2.3. Tension Allocation

In the previous section, Equation (10) linked the tension T in the cables to the load $\tau$ applied on the FWT. During cyber–physical tests, tension allocation consists of choosing T such that the resulting load $\tau =JT$ matches a desired reference load ${\tau }_{\mathrm{ref}}\in {\mathbb{R}}^{n_c}$. The integer $n_c\le 6$ is the number of components of the rotor loads that one wishes to apply. Typically, in the context of FWT testing, the heave load is not of importance and can be neglected [22], while all other load components are included, so $n_c=5$.

The allocation problem is cast as the following optimisation problem:

$$T*=\arg \underset{T}{\min }\sum _{i=1}^{n_a}{(T-T_0)}_i^2\mathrm{subject}\mathrm{to}\tilde{J}T={\tau }_{\mathrm{ref}}$$

(12)

where $T_0\in {\mathbb{R}}^{n_a}$ is a preferred set of tensions (which we will come back to), and $\tilde{J}\in {\mathbb{R}}^{n_c\times n_a}$ is the Jacobian matrix expressed in (11) but in which lines corresponding to components that are not actuated have been removed. If heave is neglected, for example, $\tilde{J}$ is obtained by removing the third line of J.

Equation (12) translates the fact that we wish to minimise deviations from a “preferred” tension while ensuring that the $n_c$ actuated components of the desired load are as requested. The preferred tension $T_0$ can be selected in different ways. If each winch is designed to apply a tension close to 20 N, $T_0$ can be chosen as ${[20\mathrm{N},\dots ,20\mathrm{N}]}^{\top }$. Note that, in general, $J{T}_0\ne 0$, meaning that applying the preferred set of tensions does not necessarily lead to zero load. If this is desirable, then $T_0$ should be chosen in the kernel (or nullspace) of $\tilde{J}$.

A direct solution to (12) can be obtained by

$$T=T_0+J^{†}({\tau }_{\mathrm{ref}}-J{T}_0)$$

(13)

where $J^{†}$ is the Moore–Penrose pseudo-inverse of $\tilde{J}$. In practice, ${\tau }_{\mathrm{ref}}$ and $\tilde{J}$ are provided at each time step, meaning that Equation (12) will be solved at every time step. In that context, it is possible to compute the pseudo-inverse $J^{†}$ efficiently in an iterative way [23]:

$$J_{i+1}^{†}=2J_i^{†}-J_i^{†}J{J}_i^{†}$$

(14)

where $J_0^{†}$ is the estimate of $J^{†}$ from the previous time step, and iterations on i are performed until convergence (typically after a few iterations).

An important remark is that the solution (13) based on pseudo-inverse does not guarantee the positivity of T. It means that the tension commanded to the winches might be negative, which is not feasible. If the CDPR is well configured, meaning that the actuators and attachment points have been wisely selected, such a situation should not happen. Optimal configuration of the CDPR has been discussed in [24], for instance. Another aspect that is not guaranteed by the present approach is the smoothness of the solutions $T\left(t\right)$, assuming that $\tau \left(t\right)$ is smooth. The interested reader is referred to [25] for details about these issues and possible solutions.

Note that even if, e.g., the heave force applied by the CDPR on the FWT is not controlled, it is possible to compute the force that was actually applied using (10). For practical cases in FWTs, it typically amounts to a couple of Newtons directed along the tower axis, which is insignificant compared to FWT buoyancy and inertia heave forces.

2.4. Adapting the Preferred Tension

In some situations, for example, when large heading variations occur, some cables might become slack or overloaded. To alleviate this, the pretension $T_0$ can be adapted [18]. One can, for example, select $T_0$ in the kernel of $\tilde{J}$ and scale it by a factor $\lambda$ following the following adaptation law:

$$\dot{\lambda }\left(t\right)=K_{\lambda }[\mathbb{I}(T_{\min }-T)-\mathbb{I}(T-T_{\max })]$$

(15)

where

$$\mathbb{I}\left(x\right)=\left\{\begin{array}{cc}1& \mathrm{if}\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{component}\mathrm{of}x\mathrm{is}\mathrm{positive}\hfill \\ 0& \mathrm{else}\hfill \end{array}\right.$$

(16)

This moves the envelope of the tensions away from bounds defined by the $[T_{\min },T_{\max }]$ tension values. The gain $K_{\lambda }$ is used to tune the speed of the adaptation.

3. Performance of the Actuator, Observed under Wind/Wave/Current Testing

This section will present the test cases, and show the performance of the CDPR when applying time-varying loads, during testing with wind, waves, and current. This will motivate the definition of the benchmarks in the next section.

3.1. Description of the Test Cases

The control system and CDPR presented in the previous section have been used to perform cyber–physical testing of several 15 MW state-of-the-art FWT designs at SINTEF Ocean. Two of these have been selected for the present study, as they differ significantly in terms of floater design, motion properties, and tower modelling strategies. The tower of FWT1 is stiffened compared to reality, while the one of FWT2 is designed to represent the first tower fore–aft and side–side tower frequency. Furthermore, the chosen FWTs support two different 15 MW generators.

The natural frequencies for the rigid body motions and tower deformations for the two FWT models are given at model scale in

Table 1. The frequencies of interest [Hz, model scale] for the tested FWT.

	FWT1	FWT2
Natural frequency—surge	0.148	0.105
Natural frequency—sway	0.150	0.121
Natural frequency—heave	0.469	0.472
Natural frequency—roll	0.247	0.243
Natural frequency—pitch	0.242	0.229
Natural frequency—yaw	0.088	0.079
Wave peak frequency ($f_p$)	0.543	0.552
Rotor frequency $\left(1p\right)$	0.80	0.90
Blade passing frequency $\left(3p\right)$	2.39	2.71
Double passing frequency $\left(6p\right)$	4.79	5.41
Natural frequency tower (wet)—fore–aft	5.65	2.15
Natural frequency tower (wet)—side–side	4.11	2.15

. The key excitation frequencies from the waves and rotor loads are also indicated.

The wind and wave tests were conducted in the Ocean Basin at SINTEF Ocean (Trondheim, Norway). The mean of the turbulent wind velocity was 11 m/s, which is close to the rated velocity for both turbines, generating a maximum mean thrust. A nominal turbulence model was used. For the considered cases, the waves were aligned with the wind and followed a JONSWAP spectrum with a significant wave height of 0.18 m (model scale) and the peak frequencies indicated in Table 1. While FWT1 was tested without a current, FWT2 was tested in a current and exhibited larger dynamic yaw angles. The rotor loads were simulated with OpenFAST, including the gyroscopic effect, tower shadowing, and blade flexibility.

In both cases, the CDPR included seven actuators placed around the basin boundaries. From each actuator, there was a thin line attached to a frame on top of the physical substructure placed in the basin. The tension in each line was measured and sampled at a frequency of 200 Hz (model scale), using a strain-type transducer with an accuracy of ±2% of measured value for significant response levels. These force transducers were used to control the CDPR and to compute the total load applied on the structure using Equation (10).

3.2. Description of the Observed Quantities

For both test cases, the commanded and measured surge and sway forces are compared (denoted ${\tau }_1$ and ${\tau }_2$, respectively) and the roll, pitch, and yaw moments at the tower top (denoted ${\tau }_4,{\tau }_5,{\tau }_6$, respectively). These loads are in a body-fixed coordinate system.

In Figure 3, Figure 4, Figure 5 and Figure 6, the power spectral density of the commanded and measured loads are plotted at the various frequency ranges, listed in

Table 2. Frequency ranges and reference to figures. F-A stands for fore–aft, and S-S stands for side–side.

Frequency Range	Min. Freq. [Hz]	Max. Freq. [Hz]	Figure FWT1	Figure FWT2
Low-frequency (LF) range	0	0.3	Figure 3a	Figure 3b
Wave-frequency (WF) range	0.3	0.8	Figure 4a	Figure 4b
Near $3p$ and FWT2 F-A S-S eigenfrequencies	2	3	Figure 5a	Figure 5b
Near $6p$ and FWT1 F-A S-S eigenfrequencies	3	6	Figure 6a	Figure 6b

. Separate plots enable the use of linear scales on the y-axis (rather than logarithmic), which provides a clearer picture of possible discrepancies. Another remark is that all the controlled components are presented, not only the dominating ones. This is important because it enables checking whether the actuator manages to apply near-zero loads when such loads are commanded.

Bode plots that contain the RAO, including the phase information between the commanded and measured signals, are presented in Figure 7 and will be commented on in detail in the next section. In the present section, we will only consider the data represented by dots in Figure 7. The solid lines correspond to the benchmark tests, which will be introduced later.

3.3. Low- and Wave-Frequency Range

The power spectra in the low-frequency range are shown in Figure 3a,b for FWT1 and FWT2, respectively. This is where the energy of the wind spectrum is concentrated, and we observe a perfect match for the low-frequency variations for both FWTs. For FWT1, loads in surge, pitch, and yaw are clearly dominating and roll is observable, while sway is insignificant. For FWT2, which exhibits larger yaw motions, all the load components are significant in this frequency range.

Figure 4a,b show the commanded and measured load spectra in the wave-frequency range. The load has a peak at the wave peak frequency $f_p$ for both FWTs. This is because the wave-induced motions of the floater, fed into the rotor simulator (numerical substructure), generate significant aerodynamic loads (damping, for example). Thus, the wave-frequency oscillations of the floater are mirrored in the commanded and measured loads. For FWT1, the surge and pitch loads are clearly dominating, while all the controlled load components except sway are important for FWT2.

The agreement between the commanded and measured loads at wave frequencies is very good for all the components of importance in the wave-frequency range. This is confirmed by the Bode plots in Figure 7, which indicate a perfect match in terms of amplitude and at zero phase lag in the LF and WF ranges.

3.4. High-Frequency Range

Figure 5a,b show the commanded and measured load spectra in the 3p region (blade passing frequency). The peak in the spectrum corresponds to the 3p frequencies of 2.39 Hz for FWT1 and 2.71 Hz for FWT2, which are visible and well captured in all five DOFs. For FWT1, surge clearly dominates the sway force, pitch, and yaw moments of the same order of magnitude, and roll is insignificant in this frequency range. For FWT2, the surge and sway forces are of the same order of magnitude, the pitch and yaw moments are of the same order of magnitude, and the roll moment is insignificant. For the non-insignificant components of the load, the CDPR tends to apply larger loads than commanded, with an error on the power spectrum of about 20%, which represents about 10% in terms of amplitude. This amplification near 3p is also visible in the Bode plots (Figure 7), which indicates a phase lag of about 20° in this frequency range. The tower fore–aft and side–side natural frequencies for FWT2 are 2.15Hz. Possible resonant vibrations that could occur at this frequency do not have any particular effect on the CDPR performance.

Figure 6a,b show the high-frequency range near $6p$ (4.79 Hz for FWT1 and 5.41 Hz for FWT2), which is also near the fore–aft and side–side natural frequencies of FWT1’s tower (5.65 Hz and 4.11 Hz, respectively). The $6p$ excitation frequency is clearly visible on the plots, while the presence of the tower’s natural frequency for FWT1 has no visible effect on the performance of the CDPR. For the previous frequency ranges, surge dominates sway for FWT1. The pitch and yaw moments have similar magnitudes, while roll is much smaller. The situation is similar for FWT2, except that the surge and sway loads are comparable. In this frequency range, discrepancies between commanded and applied loads can be large, up to 40% in energy, meaning 20% in amplitude. This is confirmed by the Bode plots in Figure 7, which in addition indicate a phase lag of over 40° between the commanded and applied load near 6p.

3.5. Summary

This section illustrated the typical performance of the actuator (here a CDPR) to apply commanded loads during realistic wind/wave/current scenarios over a wide frequency range. However, in such complex tests with combined load sources and complex floater motions, possible issues related to the actuator are difficult to detect, understand, and solve. This is why, in the next section, a set of short and targeted benchmark tests are presented. They aim at verifying the performance of the actuator prior to running tests in a combined environment and can be repeated on a regular basis to ensure a functional setup over long test programs.

4. Benchmark Tests

When investigating the performance of a control system in cyber–physical testing, it is crucial to define the purpose of the tests and hence the Quantities of Interest (QoI) [16,22,26]. When applied to an FWT, QoI are the ones used in the verification of the FWT design, or in the calibration of numerical models: motions of the floater, accelerations at the nacelle, tensions in the mooring lines, and internal loads (shear forces and bending moments) at the tower base. Frequencies of Interest (FoI) are also to be defined, which for an FWT are the ones listed in Table 2.

To achieve high fidelity, the control system orchestrating cyber–physical testing shall be “transparent” from a QoI point of view. Formulated in another way, the interconnection between the numerical and physical substructure should behave as if the two were ideally connected, i.e., the presence of the control system should not affect the QoI [16]. As the scope of the present paper is the actuator (and not the whole control system depicted in Figure 1), we transpose this requirement to the actuator as follows (Box 1).

Box 1. Requirements to the actuator.

(R) The actuator should accurately apply loads over the FoI, and this is in spite of the motions/vibrations of the floater. This accuracy is to be assessed based on the QoI for the cyber–physical tests. A practical way to decompose (R) is to require (R1) that the actuator is able to apply the loads accurately on a nearly fixed structure over the FoI. This is referred to as load tracking. And (R2) that the actuator is able to apply a zero net load, and thus not influence the dynamic behaviour of the floater, in spite of the floater motions occurring at the frequencies of interest. This is called disturbance rejection. Once (R1) and (R2) are both fulfilled, one can then verify that the combination of both (R) is fulfilled by performing combined wave–wind tests as presented in the previous section.

Note that [20] presents a controller for the CDPR in which separate components are in charge of fulfilling (R1) and (R2), which facilitates the design of the controller, validation of its performance, and troubleshooting.

4.1. Description of Benchmark Tests

For an FWT, the main challenges for actuators are (1) good load tracking for all FoI, particularly at high frequencies, and (2) disturbance rejection for large motions, which can occur at wave frequencies and natural frequencies. Therefore, three types of benchmark tests are defined to cover such situations (Box 2).

Box 2. Benchmark tests.

(B1) Chirp tests that verify load tracking performance. (B2) Free decay tests during which the actuator must apply zero load. (B3) Wave tests during which the actuator must apply zero load. (R4) Finally, wave tests during which the actuator applies wind loads.

As shown in Figure 8, benchmark test (B1) aims at verifying requirement (R1), while benchmark tests (B2) and (B3) enable verifying requirement (R2) with the structure moving at its natural frequencies and at the wave frequencies, respectively. (B4) entails a final check of the actuator’s performance in realistic conditions. Note that for some actuator types, such as fan-based solutions, it is easier to verify (R2), as the actuator applying zero load is simply deactivated. But (B2) and (B3) should still be performed to verify that the mass, the aerodynamic damping, and the power cord are not disturbing the system.

The execution of these benchmark tests will be explained and exemplified in the following.

4.1.1. Chirp Tests (B1)

Chirp tests consist of applying a constant amplitude load centred on zero at an increasing frequency, such that the whole domain of the FoI is covered. Several load components should be actuated: a surge chirp force at the tower top and then a sway chirp force. They will induce a pitch and roll moment at the tower base, respectively. An example of a chirp time series in surge and sway force is shown in Figure 9. The actual applied load should be measured and compared to the desired load. If couplings are expected between the actuators (such as possible wake/rotor interactions in multi-rotors), these can be checked using (B1).

4.1.2. Decay Tests—Zero Load (B2)

This benchmark consists of free decay tests, where the actuator shall apply zero net load on the floater. The initial conditions for the decays should be chosen as relevant for the test campaign. For an FWT, the initial condition in position/attitude can typically be the one obtained when applying the rated thrust of the turbine at the tower top. This force is suddenly removed, triggering a free decay. Note that during the free decay, the CDPR remains connected to the structure with its lines taut but applies zero load (see Section 2 on CDPR for details). A similar test should be performed with the actuator disconnected, and it is checked that the QoI are similar between the two tests. The natural periods and damping ratios for the excited degrees of freedom should match too. As in (B1), the direction of the initial force can be adjusted consistently with the test program, which can, for example, contain several wind incidence angles.

4.1.3. Wave Tests—Zero Load (B3)

In this benchmark test, the actuator is again bound to apply zero load, but this time the motions of the floater are triggered by waves. The motions will be rather energetic in the wave-frequency range, as well as in the low-frequency range due to nonlinear hydrodynamic effects. This complements (B2), in which motions occur at the natural frequency of the floater. The type of waves selected for this test should be based on the plan of the test campaign. The QoI measured during these tests should be compared to those measured during a test involving the same wave but with the CDPR disconnected.

4.1.4. Wave Tests—Wind Loads (B4)

This has been exemplified in Section 3.1.

4.2. Example of Results

Benchmark tests (B1)–(B4) were executed for both FWT1 and FWT2, as described previously. The following sections show examples of the results and their interpretation.

4.2.1. Bode Plots from (B1)

The (B1) chirp tests cover a frequency range of [0, 6.5 Hz] for FWT1 and [0, 5 Hz] for FWT2 at model scale. Based on (B1), Bode plots can be drawn to illustrate the frequency-dependent relationship between the commanded and measured loads on the floater from the CDPR. Bode plots present the amplitude ratio and the phase lag between these two quantities. Ideally, the former should be constant and equal to 1, and the latter to 0°, over the whole frequency range. Technically, the amplitude ratio in a Bode plot should be presented with a logarithmic scale, but this has not been done here as it makes deviations from 1.0 less visible.

The Bode plots for both FWT1 and FWT2 are shown in Figure 7. On each plot, data from the actual tests in the wind, wave, and current (discussed in the previous section) and from the benchmark test (B1) are superimposed. For the former, the five controlled load components are shown. Note, however, that there are some frequency ranges where the data are absent. This is because, in realistic wind/wave tests, there is very little energy in the commanded load in these frequency ranges, meaning that computing an RAO is not meaningful. In the benchmark tests, only two load components (surge and sway) were tested, but the data are available over the whole frequency range.

The Bode plot obtained from the benchmark tests compares well to the one from normal operating tests. They are also very similar for the two different FWTs tested here. The general trend is that the amplitude error increases linearly from null at low frequency to +25% at 5 Hz, while the phase lag increases linearly from 0° at low frequency to 40° at 5 Hz.

Putting this in the context of cyber–physical testing, it is important to note that this phase lag is the one related to the CDPR only. Additional time delays, not related to the actuator (and thus out of scope of the present paper), occur due to the calculation time in the numerical substructure (here, the rotor simulation) and the exchange of data between different systems. Assuming that such a time delay $t_d$ exists, it will cause an additional phase lag of $2\pi t_{d}f$, which increases linearly with the frequency f. It should be verified that such a phase lag does not compromise fidelity [16], or it should be compensated for [27] (Section 8.3.2).

On top of these linear trends, dips in the curves are observed at 4.11 Hz and 5.65 Hz for FWT1 and 2.1 Hz for FWT2, which corresponds to the towers’ natural frequencies. These larger load tracking errors are due to the large vibrations of the tower that build up when the CDPR load approaches these frequencies. It is however interesting that such force tracking errors are not visible during the wind/wave tests of FWT2, even if there was energy in the commanded load at these frequencies. This is likely because the resonant vibrations did not build up as intensely during the wind/wave tests as they do during the chirp test.

4.2.2. Comparison of Motions during Decay Tests, from (B2)

Benchmark (B2) investigates the ability to apply zero loads by conducting a free decay test, where the actuator shall apply zero net loads on the floater. Two pitch decay tests were conducted with and without the CDPR connected using FWT2. Figure 10 and

Table 3. Decay analysis results. $T_d$ is the natural period of the damped oscillation, $T_n$ is the natural period of the free undamped oscillation, P (mean) is the total linearised damping coefficient per unit mass, $P_1$ is the linear damping coefficient per unit mass, and $P_2$ is the quadratic damping coefficient per unit mass.

	CDPR Connected	CDPR Disconnected
Number of crests	16	11
Number of troughs	15	11
$T_d$ [s]	4.342	4.427
$T_n$ [s]	4.343	4.432
${\displaystyle \frac{P}{P_{crit}}}$ (mean) [–]	$2.129·{10}^{-2}$	$2.789·{10}^{-2}$
P (mean) $\left[s^{-1}\right]$	$6.163·{10}^{-2}$	$7.916·{10}^{-2}$
$P_1$ $\left[s^{-1}\right]$	$4.919·{10}^{-2}$	$6.099·{10}^{-2}$
$P_2$ $\left[de{g}^{-1}\right]$	$7.134·{10}^{-2}$	$8.199·{10}^{-2}$

show a decay analysis from both tests, where the natural period and damping coefficients are calculated.

The decay tests reveal a good match between the damping coefficients and natural periods with and without the CDPR connected.

4.2.3. Comparison of Motions—CDPR Connected and Disconnected—from (B3)

Benchmark (B3) assesses the disturbance rejection capability of the CDPR for frequency ranges covering the low frequency to the wave frequency. In the presence of waves, the CDPR is requested to apply zero load, and the Quantities of Interest are compared with their version when the CDPR is disconnected. The comparison is carried out in terms of time series and power spectra, which are shown here for FWT1. The QoI considered are the motions of the tower top, particularly the surge, pitch, and yaw, which are the ones responding significantly in these conditions3, see Figure 11.

The tests reveal a good match between the results obtained with and without the CDPR connected. The left-most peak in the power spectra corresponds to the structure’s natural frequency, while the second peak at about 0.54 Hz corresponds to the wave frequency. The location of the natural frequencies matches very well, and so does the time series of motions between the two tests. While the match is nearly perfect for the surge motion, some minor differences are observed for the pitch and yaw time series. However, it should be noted that these are small (typical amplitude of 0.5 degrees).

4.2.4. Comparison of Structural Quantities of Interest—CDPR Connected and Disconnected—from (B3)

Figure 12 compares the frequency spectra of the tensions in the three mooring lines. Similar frequency patterns to the surge motion spectrum are observed. Both the spectrum and time series demonstrate good consistency between the tests conducted with and without the CDPR connected.

This is also the case for the fore–aft bending moment measured at the tower base, see Figure 13. Figure 14 shows a close-up of the time series measurement of the internal bending moment during a slamming event, triggering high-frequency vibrations of the tower. It can be observed that the measured bending moments are the same with and without the CPDR connected.

4.2.5. Accelerations at Nacelle—CDPR Connected and Disconnected—from (B3)

Figure 15 shows the spectrum and time series of the accelerations at the nacelle level. On the lower part of the figure, the power spectrum is divided into low-frequency, wave-frequency, and 3p- and 6p-frequency ranges to better observe the frequency spectrum magnitude. Except for a discrepancy in the accelerations at 6p (but with very low energy), the accelerations compare satisfactorily with the CDPR connected and disconnected.

4.3. Summary

The defined benchmark tests enabled the assessment of SINTEF Ocean’s CDPR performance using FWTs with different dynamic behaviours. It is found that the CDPR has good disturbance rejection capabilities and very small load tracking errors in the low- and wave-frequency ranges. At frequency ranges close to 3p, a phase lag of about 20 degrees and an amplitude error of 15% are observed. Both errors keep on increasing approximately linearly with increasing frequency.

5. Conclusions

Actuators play a key role in achieving high-fidelity cyber–physical testing of floating wind turbines. There are significant differences in the the type of actuators used across hydrodynamic testing facilities, as they presently take the form of multi-rotors, single fans, single winches, and cable-driven parallel robots (CDPRs). While it is trivial to compare their intrusiveness (in terms of weight and space required on the model), or how many of the six load components they are able to apply, it has so far not been clear how to characterise their dynamic performance.

The present paper motivated and introduced (1) a set of requirements and sub-requirements to the actuator and (2) a set of simple benchmark tests that aim at checking that these requirements are fulfilled and (3) exemplified these benchmark tests using SINTEF Ocean’s CDPR and two different 15MW FWT concepts. The benchmark tests are meant to be performed prior to the initiation of the tests and repeated on a regular basis during longer test campaigns to ensure their quality. It was seen that actuator properties obtained during the benchmark and the tests under complex wind/wave/current loading were consistent.

The need to identify quantities and frequencies of interest as a basis for the assessment of the results was emphasised. It was also recommended to employ linear scales (rather than logarithmic) and splitting plots over the frequency ranges of interest (rather than showing only a single spectrum covering all frequencies) to avoid missing some important benchmark results.

The future trend for FWT concepts is to accommodate rotors with rated power beyond 15 MW. By comparing the IEA 15 MW and 22 MW reference turbines, it is seen that the range of frequencies over which the actuator should perform will not increase. However, static and dynamic rotor loads will increase significantly. To achieve this, the actuator must be upscaled. A CDPR can conveniently cope with this by adding more actuators, without increasing the weight of the reduced-scale model. A short discussion about the consequence of such a reconfiguration of the CDPR on its performance is discussed in the Appendix A.