Marine Structures 79 (2021) 103019

Contents lists available at ScienceDirect

Marine Structures
journal homepage: www.elsevier.com/locate/marstruc

Feasibility of modal expansion for virtual sensing in offshore wind

jacket substructures
Dawid Augustyn a, b, *, Ronnie R. Pedersen b, Ulf T. Tygesen b, Martin D. Ulriksen c,
John D. Sørensen a
a
Department of the Built Environment, Aalborg University, Denmark
b
Ramboll Energy, Esbjerg, Denmark
c
Department of Energy Technology, Aalborg University, Denmark

A R T I C L E I N F O A B S T R A C T

Keywords: The present paper investigates the feasibility of modal expansion-based virtual sensing in the
Virtual sensing context of offshore wind jacket substructures. For this specific application, issues have been re­
Modal expansion ported when expanding wind-driven brace vibrations and wave-driven vibrations in the splash-
Ritz vectors
zone based on a sensor network placed solely above the sea level. These limitations are
Wind turbines
Jacket substructures
addressed in this paper by extending the sensor network with sub-sea vibration sensors and a
wave radar sensor, which allow for capturing local brace vibration modes and the wave-driven
vibration response. The brace expansion is thus improved by including the local brace vibra­
tion modes in the expansion basis, while the representation of wave-driven vibrations is improved
by including load-dependent Ritz vectors computed based on input from the wave radar sensor.
The merit of the proposed extension is explored using a numerical model of an offshore wind
turbine supported by a jacket substructure in a simulation setting with different operational and
environmental conditions. It is documented that the extended setup provides an improvement in
the expansion-based estimation of both wind- and wave-driven vibrations. The former
improvement is particularly relevant for operational cases, while the latter is relevant for idling
cases. Despite the documented improvements, a systematic reduction in the expansion quality is
observed for higher wind speeds in operational cases for both the basic and the extended setup. It
is contended that this phenomenon is due to the operational variability of the controller, which
violates the fundamental assumption of the structural system being linear and time-invariant.

1. Introduction

Structural vibration response composes a pivotal part in many civil and mechanical engineering applications; i.a., structural
monitoring (including fatigue estimation) [1,2] and control [3,4]. In practice, direct vibration measurements can be obtained from an
installed sensor network, which covers a limited number of locations. The response at the remaining locations can subsequently be
estimated using, e.g., virtual sensing methods [5–9], where the response at the unmeasured (virtual) locations is predicted based on the
available measurements.
Virtual sensing has been widely adopted for vibration estimation in offshore structures. Successful applications have been

* Corresponding author. Ramboll Energy, Esbjerg, Denmark.

E-mail address: dawa@ramboll.com (D. Augustyn).

https://doi.org/10.1016/j.marstruc.2021.103019
Received 15 June 2020; Received in revised form 1 February 2021; Accepted 2 May 2021
Available online 1 June 2021
0951-8339/© 2021 Elsevier Ltd. All rights reserved.
D. Augustyn et al. Marine Structures 79 (2021) 103019

demonstrated for offshore (oil and gas) jacket structures [10,11] and monopile substructures of offshore wind turbines [12,13]. A
recent publication by Henkel et al. [14] investigates the feasibility of a particular virtual sensing method, namely, modal expansion, for
application to offshore wind jacket substructures. These structures are different from the previous applications due to their lattice
topology and the coupled wind-wave loading. Henkel et al. [14] report high accuracy of the expansion in the leg elements, while low
expansion quality is obtained for the brace elements due to the inherent limitations of the employed setup. In particular, the findings
suggest that the implemented sensor network, which only includes vibration sensors above the sea level, does not allow for an adequate
representation of the local brace modes or the quasi-static contribution from the wave loading. In the literature, the latter issue has
been addressed in the context of system reduction [15–17] and earthquake engineering [18–20] by applying load-dependent Ritz
vectors. A similar approach is implemented in this paper and its feasibility in the context of virtual sensing of offshore wind jacket
substructures is investigated.
The present paper addresses the limitations that Henkel et al. [14] report on modal expansion for virtual sensing of wind turbine
jacket substructures. More specifically, we examine an extended setup in which sub-sea vibration sensors and a wave radar sensor are
added to capture both local brace modes and wave-governed modes, which are then included in the modal expansion basis. Two
expansion scenarios—the basic one employed in Ref. [14] and the proposed extension incorporating sub-sea and wave sensors—are
tested. The study is conducted using a numerical model of a 5 MW wind turbine with a jacket substructure in a simulation setting with
different operational and environmental conditions.
The remainder of the paper is organized as follows. In Sec. 2, we outline the modal expansion theory used for virtual sensing, Sec. 3
establishes the setup of the numerical wind turbine case study, and Sec. 4 presents the appertaining virtual sensing results. The results
are summarized in Sec. 4.3 and further discussed in Sec. 5, while the paper closes with some concluding remarks in Sec. 6.

2. Modal expansion theory

Modal expansion requires the structural system in question to be linear and time-invariant (LTI). Obviously, wind turbines violate
this due to environmental and operational variability [21,22]. Yet, previous modal expansion studies concerning offshore wind tur­
bines have operated on the premise of LTI conditions and resulted in, to some extent, adequate results [12–14]. The premise of the
present study is to apply the well-established modal expansion method and focus on improving the expansion quality in cases where
low quality has been reported. Consequently, this study also adapts the LTI assumption, which implies that the structural system can be
described by
Mü(t) + Cu̇(t) + Ku(t) = f(t), (1)

where M, C, K ∈ Rna ×na are the mass, damping, and stiffness matrices, na is the number of degrees of freedom (dof), ü(t), u̇(t), u(t) ∈ Rna
are the acceleration, velocity, and displacement vectors, and f(t) ∈ Rna is the load vector. It is assumed that the system matrices are
positive definite, M, C, K ≻ 0, and that the damping in system (1) is classically distributed. The latter implies, as specified by Caughey
[23], that M− 1 K and M− 1 C commute such the eigenvectors of system (1) equal the undamped ones.

2.1. Modal expansion

Let the output—here taken as displacements, but the same relations hold for velocities and accelerations—be partitioned into nm
measured outputs, um (t) ∈ Rnm , and ne = na − nm virtual, expanded outputs, ue (t) ∈ Rne . Then,
[ ]
um (t)
u(t) = , (2)
ue (t)

and the aim of modal expansion is to estimate ue (t) based on um (t). If u(t) is governed by nq modes, a modally truncated approximation
writes
[ ]
Φm (t)
u(t) ≈ Φ(t)q(t) = q(t), (3)
Φe (t)

where q(t) ∈ Rnq contains the modal displacements associated with the nq governing modes and Φ(t) ∈ Rna ×nq is the expansion matrix,
which is partitioned into Φm (t) ∈ Rnm ×nq and Φe (t) ∈ Rne ×nq . Φ will be specified in Subsec. 2.2.
Assuming nm ≥ nq and rank(Φm (t)) = nq , an estimate on q(t) that minimizes ||Φm (t)q(t) − um (t)||2 is given by
( )− 1
q(t) = Φm (t)T Φm (t) Φm (t)T um (t) = Φm (t)† um (t),
̃ (4)

with superscript † and overhead ∼ denoting, respectively, pseudo-inversion and an estimate. Thus, an estimate of the virtual part of
the displacement output can be computed as
ue (t) = Φe (t)̃
̃ q(t). (5)

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2.2. The expansion matrix

The expansion matrix, which includes a mixture of dynamic and static modes, is given by
[ ]
Φ(t) = Φ(d) Φ(s) Φ(R) (t) , (6)

where Φ(d) contains eigenvectors of system (1), while Φ(s) and Φ(R) (t) capture the static response due to, respectively, wind and wave
loading. The modes are extracted for the dof corresponding to the measured, Φm (t), and virtual, Φe (t), locations. It is assumed that the
static modes are calculated individually for specific sea states. The static wave modes, Φ(R) (t), are constructed based on input from a
wave radar sensor.

2.2.1. Dynamic modes

The dynamic modes, Φ(d) , contain a subset of the eigenvectors of system (1). The particular eigenvectors are selected such that the
governing dynamics of the wind turbine system are adequately described. Compared to previous studies [12–14], where only the first
few dynamic modes of the jacket/monopile substructure were considered, we include higher modes with significant local brace
participation.
In this study, both the measured and the virtual partition of Φ(d) are obtained from the numerical model of system (1). Alterna­
tively, the measured partition can be taken directly as the experimental mode shapes or as a combination of these and the model
predictions using, e.g, the SEREP method [24] or the local correspondence principle [10,25].

2.2.2. Static wind modes

The static displacement response to wind loading can be computed as

Φ(s) = K− 1 Fs , (7)

in which Fs ∈ Rna ×6 collects 6 linearly independent unit loads (3 translational and 3 rotational) that are applied to the top of the wind
turbine tower. This procedure follows the approach suggested by Iliopoulos et al. [12] who, however, restrict it to lateral translation.

2.2.3. Static wave modes

The static displacement response to wave loading is—as done by Skafte et al. [11] for oil and gas structures—estimated as Ritz
vectors [26], thus

Φ(R) (t) = K− 1 f R (t), (8)

where f R (t) ∈ Rna is the wave load estimated for a given sea state. The wave loading can be reconstructed by using information from, e.
g., a wave radar. The wave radar captures a time history of the wave surface elevation, which is subsequently used to estimate wave
kinematics, using appropriate wave theory, e.g., Stokes fifth-order wave theory [27]. The wave kinematics are used to estimate the
wave forces acting on individual structural members using the Morison equation [28]. A further discussion on estimating wave
loading, with special focus on practical issues, is provided in Subsec. 5.4.

2.3. Validity of LTI model for modal expansion

In practice, the LTI assumption is always violated, as each system changes its properties with respect to both loading conditions
(non-linear system) and time (time-variant system). In the context of an offshore wind jacket substructure, the system is non-linear and
time-variant due to, i.a., non-linear wind loads, controller variability, non-linear wave forces, and soil. The impact of each violation on
modal expansion is briefly discussed in this subsection.
The non-linearities introduced due to wind forces stem from aero-dynamic coupling between the blades and air particles. Assuming
no large deformation and/or plasticity are present in the substructure, and this assumption is valid for normal production cases [29],
the non-linearities introduced from wind forces do not affect modal expansion in the substructure.
The controller variability is known to alter the modal parameters (including mode shapes) of wind turbine structures [21,22]. Since
each wind turbine structure is coupled to a substructure, any alternation of the mode shapes of the turbine affects the substructure
dynamics as well. The effect of neglecting the controller variability, which has been reported as one of the sources of reduced
expansion quality for higher wind speeds in Ref. [14], is further investigated in Subsec. 5.1 in the present paper.
The wave forces are non-linear as they include the effect of the relative velocity between the wave and the structure, as typically
modelled by use of the relative-velocity Morison equation [28]. For bottom-fixed offshore structures, the effect of relative velocity is
small [28], and therefore the error introduced by neglecting it in this study with a bottom-fixed jacket substructure (to comply with the
LTI assumption) is not critical.
The soil stiffness is non-linear with respect to the applied load. Variable soil stiffness affects the mode shapes used to expand both
the static and the dynamic part of the response. In the LTI model, the soil stiffness has to be linearized for representative conditions.
Consequently, any variation in soil stiffness is neglected. For a typical offshore wind application, the soil stiffness is linearized ac­
cording to the initial stiffness [30], which is representative for power production cases where the majority of fatigue damage is
generated. For extreme cases, a different linearization point could be selected to account for the modified soil stiffness. Consequently,

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by deriving a number of modal expansion sets, representative for each loading condition and hence soil stiffness configuration, the soil
stiffness variation can, indirectly, be accounted for.

3. Case study setup

The case study is based on synthetic displacement data obtained from a numerical model of a 5 MW wind turbine exposed to
different load cases. The model is described in Subsec. 3.1. The displacement data are extracted from both physical output locations
and virtual output locations. The former is used as input to the modal expansion, while the latter is used to validate the expanded,
virtual results. The particular locations are specified in Subsec. 3.2 for the basic and extended sensor configurations. The load cases
considered in this study are described in Subsec. 3.3.

3.1. Modelling

The jacket substructure and its appertaining wave loading are modelled using ROSAP (Ramboll Offshore Structural Analysis
Package), version 53 [31]. The jacket substructure considered in this study, which is depicted in Fig. 1a, has a total height of
approximately 50 m. The substructure comprises four legs, each with a diameter of approximately 2 m, and three brace bays, each with
a diameter of approximately 0.5 m. The substructure model includes, i.a., soil-pile interaction, local joint flexibility, scour, marine
growth, and appurtenance masses. The water depth is 40 m and the soil conditions are characterized as clay. The substructure includes
30 m grouted piles. The soil-structure interaction is modelled by use of soil curves linearized according to the API method [30]. The
substructure carries a representative 5 MW turbine modelled in LACflex aero-elastic code [32]. The turbine includes a 70 m tubular
tower with a diameter ranging between 4 m and 6 m. Along the tower, three concentrated masses are assumed to emulate the effect of
secondary-structures. The aero-elastic code employs a modal-based representation of the turbine (including the tower, rotor, and
blades), while the substructure is represented as a Craig-Bampton superelement [33]. The structural damping is modelled according to
the Rayleigh model [34] assuming 0.5% and 1% modal damping in the first and second bending modes, respectively.
The aero-hydro-servo-elastic simulation is performed in a sequentially coupled manner as described by Nielsen et al. [35]. The key
steps of the procedure are as follows. 1) the substructure model and corresponding wave loading are reduced to a superelement with 30
internal modes accounting for internal substructure dynamics. A convergence study has been performed to ascertain that the reduced
model (including 30 modes) adequately captures the relevant modal parameters of the non-reduced system. Subsequently, 2) the wind
loading is computed through aero-elastic analyses, in which the synchronized wave loading and the substructure superelement are
included. Finally, 3) the force-controlled recovery run outlined by Nielsen et al. [17] is performed, where the response of the sub­
structure is recovered and relevant measurements are extracted.

3.2. Sensor layout

The two employed sensor configurations are depicted in Fig. 1c and d. Each configuration consists of physical sensors, which are
assumed installed on the structure to deliver the displacement measurements constituting um (t) in (2), and virtual sensors, which are
placed at locations where the displacement response is estimated through modal expansion. In this study, the displacement

Fig. 1. Instrumentation of the jacket substructure. a) sensor levels, b) substructure direction, c) basic sensor layout, and d) extended sensor layout.

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measurements are obtained directly from the numerical model. In practice, the vibration sensors would be composed of accelerom­
eters, so the displacements would be obtained through double integration of the temporal acceleration signals and/or a linear
transformation of the temporal strain signals. The basic configuration described in Subsec. 3.2.1 corresponds to the setup utilized by
Henkel et al. [14], while the extended setup described in Subsec. 3.2.2 contains sub-sea vibration sensors and a wave radar sensor.

3.2.1. Basic sensor configuration

The basic sensor setup, which is depicted in Fig. 1c, contains 7 bi-axial vibration sensors (measuring displacements). The sensors
are located in the nacelle, on the tower (at the bottom and at approximately 2/3 of the tower height), and at the top of each jacket leg.
The measurements are expanded to the virtual locations indicated in Fig. 1c. These virtual locations include nodes in the K-joints and
X-joints. The joints are named after the level at which they are positioned. The level numbering is indicated in Fig. 1a.

3.2.2. Extended sensor configuration

The extended sensor setup includes bi-axial sub-sea vibration sensors in the lower X-joints at level 15. The approximate locations of
the sensors are indicated in Fig. 1d. In addition to the displacement sensors, a wave radar sensor is included to capture the wave surface
elevation. Subsequently, the wave surface elevation is used to compute the static wave modes, as elaborated in Subsec. 2.2.3.

3.3. Load cases

Two turbine states are considered in the simulations; 1) operational where the control system is active and 2) idling where the
control system is inactive and the blades are pitched 90◦ to minimize the wind loading. The load cases are defined in accordance with
the IEC standard [36] from which load cases 1.2 and 6.4 are considered. These two load cases—which are further described in Subsecs.
3.3.1 and 3.3.2 and summarized in Table 1—are assumed to be representative for the operational and idling turbine states.

3.3.1. Operational
The vibration response of the turbine and its jacket substructure is simulated for a wind speed range of U ∈ [4, 29] m/s, which is
where the turbine is assumed to generate power. The lower and upper limits of U = 4 m/s and U = 29 m/s are denoted the cut-in and
cut-out wind speeds. While operating within these limits, the control system is activated to maximize the energy yield by altering the
rotor’s angular velocity and the blades’ pitch angle up to the rated wind speed, which is approximately 12 m/s for this particular
turbine. Above the rated wind speed and up to the cut-out wind speed, the control system stabilizes the rotor’s angular velocity and
alters the blades’ pitch angle to produce the rated power and minimize the wind loads. A varying turbulence intensity is applied
according to the IEC normal turbulence model [36]. Consequently, stochastic wind speed time-series (turbulent wind) are generated
for each load case. Each time series has a duration of 800 s of which the first 200 s is neglected to avoid an initiation disturbance.
In addition to the wind loads, wave loading is incorporated by applying irregular sea states characterized by the wave height, Hs ,
and the wave period, Tp . For these, we assume Hs ∈ [0.5, 5.5] m and Tp ∈ [0.4, 6.4] s, which is a representative set for the substructure
considered in this study. The irregular waves are generated based on the JONSWAP spectrum [37] with the peak-enhanced factor γ =
3.3, while the drag and mass coefficients are set to 0.65 and 2, respectively. The wind and wave loading is assumed to be fully aligned
and approaching the structure from direction N, see Fig. 1b. Stationary conditions are assumed within each sea state.

3.3.2. Idling
The vibration response of the turbine and jacket substructure is simulated below the cut-in wind speed and above the cut-out wind
speed. Under these conditions, the blades are pitched 90◦ to minimize the wind-induced loading, and the angular velocity of the rotor is
negligible. As a result, the structural response is predominantly governed by the wave loading.
The response of the structure is simulated for two wind speeds; U = 3 m/s, which is below the cut-in wind speed, and U = 35 m/s,
which corresponds to the representative extreme wind speed [36]. The wave loading is, in analogy to the procedure for the operational
cases, generated by applying irregular waves characterized by wave height and period. The wave parameters used in the two simu­
lations are Hs ∈ {0.4, 6.4} m and Tp ∈ {3.7, 6.7} s, corresponding to a representative set for the substructure considered in this study.
The same loading direction and alignment as selected in the operational cases are used in the idling cases.

3.4. Modal expansion performance indicators

The quality of the conducted modal expansion is assessed based on four performance indicators. Let ui , u ̃i ∈ RN denote the
measured and estimated displacement signals at sensor i, then the first performance indicator is the time response assurance criterion
(TRAC) [38] defined by

Table 1
Load case definitions according to IEC [36] and representative site-specific parameters.
Turbine state Wind speed, U [m/s] Turbulence, TI [− ] Wave height, Hs [m] Wave period, Tp [s] Direction

Operational 4–29 0.21–0.09 0.5–5.5 3.8–6.6 N

Idling 3 and 35 0.23 and 0.08 0.4 and 6.4 3.7 and 6.7 N

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( )2
uTi ̃
ui
T = ( ) ∈ [0, 1], (9)
(uTi ui ) ̃uTi u
̃i

hence yielding a measure of the temporal correlation between the measured and estimated signals. T = 0 indicates no correlation,
while T = 1 indicates full temporal correlation.
Since the TRAC does not account for the amplitudes of the signals, the coefficient of determination (CoD),
[( ) ] 2
E ui − u
̃i
R2 = 1 − ∈ ( − ∞, 1), (10)
Var[ui ]

is introduced to capture potential amplitude errors. Here, E[.] and Var[.] denote the expectation and variance operators.
Two metrics related to estimation of the amplitude range uncertainty are employed, namely, bias and coefficient of variation (CoV).
The bias is defined as the expected value of the cumulative amplitude range ratios of the displacement signal, thus
⎡ ⎤
Δui ⎦
b=E ⎣ , (11)
Δ̃ui

where Δui ∈ Nm is a cumulative rainflow count of the measured displacements over one time-series, Δ̃
ui ∈ Nm is a cumulative rainflow
count of the predicted displacements, and m is the number of rainflow count bins. The CoV is defined as
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
⎡ ⎤
√
√
√
√Var⎣Δui ⎦
Δ̃ ui

cv = , (12)
b

which is the standard deviation of the cumulative amplitude range ratios for all amplitude bins normalized to the bias.

4. Case study results

Modal expansion results are presented for the two sensor configurations described in Subsec. 3.2 and the load cases described in
Subsec. 3.3. The expansion setups are summarized in Table 2, and a subset of the modes constituting the expansion matrix, Φ, is
illustrated in Fig. 2.

4.1. Basic expansion setup

The basic setup includes sensors located above the seawater level with simple access. As a result, the captured dynamic response is
dominated by the first 2 global bending modes and the first torsion mode, so 5 dynamic modes in total are included in Φ(d) . Even
though 14 signals are obtained in this setup (allowing to include up to 14 modes in the expansion basis), the frequency content of the
signals is dominated by the first 5 modes. Consequently, there is no benefit of including more modes. With this basic setup, the local
brace modes cannot be captured, hence local brace vibrations are neglected in this setup. The static wind modes included in Φ(s) are
established in accordance with (7). The 6 static wind modes represent wind load applied at the top of the tower, i.e., 3 translations and
3 rotations. Since no sub-sea sensors are available in the basic setup, the static wave modes are not included in Φ. The expansion
matrix, Φ, contains 11 modes in total in the basic setup.

4.1.1. Operational results

Displacement time-series obtained at leg level 50 and brace level 15 for wind speed U = 6 m/s are presented in Fig. 3. Here, both
the measured signals and the expansion-based estimates are shown, and it can be seen that the displacements in the leg element are
expanded well, while the expansion for the brace element yields an underestimation of the amplitudes.

Table 2
Overview of sensors and modes included in the expansion setups.
Setup Sensor location Expansion basis

Above water Below water Dynamic Static

Global Local Wind Wave

Basic Yes No Yes No Yes No

Extended Yes Yes Yes Yes Yes Yes

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Fig. 2. A subset of modes included in the expansion matrix, Φ. a) 2nd global bending mode included in Φ(d) , b) static wind mode due to unit
translation included in Φ(s) , c) local brace mode included in Φ(d) , and d) static wave (Ritz) mode included in Φ(R) (t).

Fig. 3. Measured (blue) and expanded (red) displacements using the basic setup. u23 , leg level 50 (top) and u47 , brace level 15 (bottom). (For
interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

The TRAC and CoV results averaged for all operational load cases are presented in Fig. 4. Evidently, high TRAC values are obtained
for the leg elements; with the highest value being 1 (observed at level 50) and the lowest being 0.90 (observed at level 10, which is
close to the mudline). A general trend of reduced TRAC is observed for the leg elements in the lower part of the substructure.
A large variation in TRAC values is observed for the brace elements. The brace element at level 55—which is the one closest to the
transition piece—obtains a TRAC value of 1, the brace element at the intermediate level 25 a TRAC value of 0.75, and the brace
element at the lowest level (i.e., number 15) a TRAC value of 0.50. In accordance with the leg element observations, the brace element
expansion decreases in quality for the lower part of the substructure. The CoV ranges between 0.05 and 0.15 for the leg elements and
0.05 and 0.50 for the brace elements. Generally, the lowest uncertainty is observed in the top part of the jacket, while the largest
uncertainty is observed in the lowest brace level 15.
To investigate the effect of the operational variability on the expansion quality, the TRAC value as a function of wind speed is
presented in Fig. 5 for the different leg and brace elements. The highest TRAC values are generally obtained for the lowest wind speed,

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Fig. 4. Averaged expansion results for the operational load cases using the basic setup.

while a gradual quality reduction is observed between U = 4 m/s and U = 12 m/s. The expansion quality stabilizes for U > 12 m/s,
which is the rated wind speed. The impact of the operational variability is further discussed in Sec. 4.3.

4.1.2. Idling results

The average TRAC and CoV values for the idling cases are presented in Fig. 6. Evidently, we obtain TRAC values between 0.4 and
0.8 and CoV values between 0.2 and 0.7, which are lower than the corresponding results for the operational cases. The reduced
expansion quality is governed by the increased wave contribution, which is captured poorly in the basic expansion setup.

4.2. Extended expansion setup

The extended setup includes additional displacement sensors located sub-sea and a wave radar sensor. As a result, Φ can, compared
to the configuration used for the basic setup, be extended with 7 local brace modes included in Φ(d) and 1 static wave Ritz mode
included in Φ(R) (t). Thus, the expansion matrix, Φ, contains 19 modes in the extended setup.

Fig. 5. TRAC values as functions of the wind speed obtained using the basic setup; a) leg levels and b) braces levels.

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Fig. 6. Averaged expansion results for the idling load cases using the basic setup.

4.2.1. Operational load case

The measured and expanded displacements at leg level 50 and brace level 15 for U = 6 m/s are presented in Fig. 7. The average
TRAC and CoV values for all the leg and brace levels are presented in Fig. 8. TRAC values close to 1 are observed across all the leg levels
and at brace levels 55 and 15. The lowest TRAC value of 0.85 is observed at brace level 25. The CoV values for all the leg elements and
the brace elements at levels 55 and 15 are below 0.05, while the CoV value is 0.15 for the brace element at level 25.
The TRAC value as a function of wind speed is presented in Fig. 9 for the different leg and brace elements. Evidently, TRAC values of
1 are obtained for all wind speeds at every level except brace level 25. Here, we observe an average TRAC value of 0.95 below rated
wind speed and an average TRAC value of 0.80 above rated wind speed. This variation is further discussed in Subsec. 5.1.

4.2.2. Idling load cases

The average TRAC and CoV values for the idling cases are presented in Fig. 10. TRAC values close to 1 are observed across all the leg
and brace levels, and the CoV values are below 0.05 for the leg elements and 0.10 for the brace elements.

Fig. 7. Measured (blue) and expanded (red) displacements using the extended setup. u23 , leg level 50 (top) and u47 , brace level 15 (bottom). (For
interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

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Fig. 8. Averaged expansion results for the operational load cases using the extended setup.

Fig. 9. TRAC values as functions of the wind speed obtained using the extended setup; a) leg levels and b) braces levels.

4.3. Summary of the results

The expansion quality gained by including vibration sensors below the water level and a wave radar sensor is summarized in this
subsection. The operational and idling results are discussed in Subsec. 4.3.1 and Subsec. 4.3.2, respectively.
The averaged expansion quality indicators for the basic and extended setups are presented in Table 3. The expansion quality across
the considered load cases and virtual sensor locations is increased after implementing the extended setup. In particular, the average
TRAC value is increased from 0.69 to 0.99, while the average CoV value is reduced from 0.26 to 0.05. The CoD is increased from − 3.70
to 0.99, while the bias is reduced from 1.15 to 1.03.

4.3.1. Operational results

The expansion results, in terms of TRAC and CoV values, for the operational cases are presented in Figs. 11 and 12 for comparison
purposes. Implementation of the extended expansion setup yields an increase in average TRAC value for all leg elements from 0.95 to 1,
while the corresponding CoV value is reduced from 0.10 to 0.05. For the brace elements, the average TRAC value is increased from 0.70

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Fig. 10. Averaged expansion results for the idling load cases using the extended setup.

Table 3
Averaged quality indicators for the basic and extended expansion setups.
Setup Load case TRAC (T ) CoD (R2 ) Bias (b) CoV (cv )

Legs Braces Legs Braces Legs Braces Legs Braces

Basic Operational 0.95 0.70 0.90 0.75 1.05 2.00 0.10 0.25
Idling 0.50 0.60 − 9.50 − 7.00 0.80 0.75 0.30 0.40
Extended Operational 1.00 0.95 1.00 0.95 1.05 1.05 0.05 0 05
Idling 1.00 1.00 1.00 1.00 1.00 1.00 0.05 0.05

Fig. 11. TRAC values obtained using the basic (blue) and extended (orange) setup for the operational load cases. Legs (left) and braces (right). (For
interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

to 0.95, and the CoV value is reduced from 0.25 to 0.05.

4.3.2. Idling results

The expansion improvement obtained by implementing the extended setup in the idling cases is indicated in Figs. 13 and 14. We
observe an increase in average TRAC value for all leg elements from 0.50 to 1, while the corresponding CoV value is reduced from 0.30

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D. Augustyn et al. Marine Structures 79 (2021) 103019

Fig. 12. CoV values obtained using the basic (blue) and extended (orange) setup for the operational load cases. Legs (left) and braces (right). (For
interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 13. TRAC values obtained using basic (blue) and extended (orange) setup for the idling load cases. Legs (left) and braces (right). (For
interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

to 0.05. For the brace elements, the average TRAC value is increased from 0.60 to 1, and the CoV value is reduced from 0.40 to 0.05.

5. Discussion

This section offers a discussion on the presented results and some practical aspects of the implemented modal expansion method.
The effect of operational variability on the expansion quality is discussed in Subsec. 5.1, followed by a discussion in Subsec. 5.2 on the
importance of including higher-order dynamics and wave loading information in the expansion. An optimal sensor placement strategy
is discussed in Subsec. 5.3, and the section closes by addressing the limitations and practical feasibility of modal expansion for offshore
wind application in Subsec. 5.4.

5.1. Effect of operational variability on the expansion quality

The time-variant and non-linear effects in the substructure are negligible when the full system (wind turbine and substructure)
operates under operational conditions. However, the wind turbine system exhibits time-variant and non-linear behaviour, which
affects its dynamic properties. The non-linearities are promoted by, among other factors, large deflections of the blades and potential
contact problems in the rotor, while the time-variant effects stem from the controller and environmental (temperature, humidity, etc.)

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D. Augustyn et al. Marine Structures 79 (2021) 103019

Fig. 14. Uncertainties obtained using the basic (blue) and extended (orange) setup for the idling load cases. Legs (left) and braces (right). levels.
(For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

variation. As a result, the modal properties of the combined system (including the substructure mode shapes used in the expansion,
Φ(d) ), are time-variant, thus a reduced expansion quality is expected when, as done in the present study, operating with an LTI basis.
The observations reported below support this proposition.
A systematic reduction in the expansion quality is observed in Figs. 5 and 9 for higher wind speeds in operational cases. To
investigate this effect, expansion results in the brace element at level 25 for two wind-only cases are presented in Fig. 15. The first case
considers the turbine during operation with the control system activated and the second case an idling setting with the control system
being inactive. Evidently, when the control system is active, the TRAC value decreases monotonically as the wind speed increases up to
the rated wind speed, while the TRAC value remains more or less constant when the control system is inactive. As the controller
parameters alter to optimize the power output, the modal parameters become time-variant. This time-variance cannot be captured in
the LTI model, hence resulting in the noted expansion quality reduction for the case with an active controller.

5.2. On the higher-order dynamics and wave loading

The extended setup adds two additional sensor types (sub-sea vibration sensors and a wave sensor). The added value of each sensor
type varies for different structures and operational conditions. In this subsection, the value of including the two sensor types separately
is discussed in the context of typical offshore wind substructures and operational conditions.
A wave sensor improves the expansion quality for structures whose response is driven by wave loading. In the context of offshore
wind applications, such structures are 1) the ones carrying old turbines generating low wind loading compared to wave loading, 2) all
turbines in idling cases, where wind forces are significantly reduced compared to operational cases, and 3) monopile structures, which
attract more wave loading compared to, e.g., jacket substructures. To quantify the improvement in the expansion quality due to wave
radars only, two additional sensor setups are defined. The first one corresponds to the basic setup with a wave radar, while the second
one is the extended setup with the wave radar removed. For the sake of brevity, we present the results for the idling cases as these are
wave-driven, and hence the effect of the wave radar is the most profound, although the same (qualitative-wise) conclusion holds for the
operational cases. Results attained using the four setups (including the basic and the extended one) are presented in Figs. 16 and 17 for
the TRAC and CoV values, respectively. The TRAC values are improved for both the leg and brace elements when the wave radar is

Fig. 15. TRAC value as a function of wind speed obtained for an inactive and active control system.

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D. Augustyn et al. Marine Structures 79 (2021) 103019

Fig. 16. TRAC values obtained using modified (with/without Ritz modes) basic and extended setups for the idling load cases.

Fig. 17. CoV values obtained using modified (with/without Ritz modes) basic and extended setups for the idling load cases.

included, as indicated in Fig. 16. The largest improvement is observed for the basic setup for wind speed U = 3 m/s, where the TRAC
value is increased from 0.3 to 0.95 for the brace elements by including only the wave radar. The improvement in the extended setup is
less profound; with the TRAC value increasing from 0.65 to 1. A similar trend—namely, improvement of the expansion quality when
the wave radar is included—is displayed by the CoV values presented in Fig. 17.
The sub-sea vibration sensors improve the expansion quality for structures whose response is driven by high-frequency dynamic
response. For a typical offshore wind turbine under power production conditions, the turbulent wind loading results in dynamic
response of the blades, which is further transferred to the substructure due to blade-brace coupling [39]. Consequently, local brace
modes are activated and therefore the expansion requires sub-sea sensors to accurately capture these vibrations.

5.3. Optimal sensor placement

The location and number of physical sensors affect the expansion quality. In general, increasing the number of installed sensors
improves the expansion quality, as exemplified for the basic and extended sensor setup in Subsec. 4.3. Due to practical and economic
constraints, a limited number of sensors can be installed in real-life applications, which necessitates a careful selection of the sensor
locations. The modal expansion method delivers optimal (in an ℓ2 -norm sense) modal displacements, which are subsequently used to
expand for virtual sensing. The precision of the virtual sensing estimate depends on the linear dependence of the mode shapes in the

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expansion basis matrix [40]. The linear dependence can be reduced to increase the expansion quality by placing sensors according to,
e.g., the effective independence (EI) method [41]. In this method, the optimal sensor placement is derived by maximizing some
suitable norm of the Fisher information matrix, effectively minimizing linear dependence of the available mode shapes. The merit of
placing sensors using the EI method in the context of modal expansion-based virtual sensing has been verified by Andersen et al. [40].
In this study, the sensor distribution is selected based on the EI method under the assumptions that; 1) all locations are accessible, 2)
the cost of installation is immaterial, and 3) one level of bracing can be monitored as a sub-sea location. These engineering judgement
criteria are chosen to increase the robustness of the sensor setup with respect to potential sensor failure during operation. Conse­
quently, brace level 15 is chosen, as the braces at this level are the longest and hence the most prone to excessive vibrations. Given
other brace levels should be monitored instead, the expansion quality would be reduced. Recall that the chosen sensor setup and mode
shapes in the expansion basis yield a high linear correlation between the dynamic modes (first bending) and the static wind modes
(lateral deflection). Despite the large linear correlation between the two mode shapes, their strain energy differs [12]. Moreover, a
large correlation is not problematic, as a full linear correlation resulting in a singular expansion matrix, and hence poor expansion, is
not possible.

5.4. Limitations and practical feasibility

This paper investigates the theoretical feasibility of modal expansion based on data generated from a numerical model, without
taking into account any of the associated practical challenges and issues. In this subsection, a brief discussion on the limitations and a
few practical challenges related to modal expansion is provided.
In this study, virtual sensing is applied based on displacements, according to Eqs. (4) and (5). In practice, displacements are rarely
measured directly. For a typical offshore wind application, accelerations are preferred due to a better signal-to-noise ratio [12]. The
accelerations are then double-integrated with respect to time to obtain a high-frequency part of the displacement response. The issue
with this approach is that the low-frequency (quasi-static) content of the response is lost. Hence, it has to be augmented with infor­
mation from an additional sensor type, which is able to capture the quasi-static response, e.g., a strain gauge, inclinometer, or GPS
sensor. Consequently, sensor fusion techniques [42] have to be applied to combine different types of sensors to reconstruct the
displacement signals. Alternatively, the displacements can be obtained directly from acceleration signals by applying the Walsh
Transform as proven in the context of a seismic application [43]. In the present study, the displacements captured in the physical
sensors are directly obtained from the numerical model, hence circumventing the above-mentioned issues.
The wave-induced displacements are expanded based on the Ritz vectors, as described in Subsec. 2.2.3. The Ritz vectors are derived
based on the stiffness matrix and wave forces, see Eq. (8). In practice, the wave forces have to be estimated based on information from
in-situ sensors, e.g., wave radars [44] or pressure transducers [45]. These sensors can be quite expensive and instrumenting each wind
turbine in a park would require a substantial investment. Some researchers suggest instrumenting only a small number of structures,
the so-called fleet leaders, and expanding/correlating this information to the remaining, lightly instrumented structures [46].
Moreover, even when a wave radar is directly installed on a structure, reconstructing wave forces is challenging because of the waves’
irregular nature, their high frequency content [47], and the wave-structure interaction [48]. In the present study, the wave forces are
obtained directly from the numerical model using a generic theoretical framework as described in Subsec. 2.2.3, hence neglecting a
number of potential practical issues. Consequently, further research is required to investigate the practical application of wave
reconstruction in the context of modal expansion.

6. Conclusions

This paper investigates the feasibility of modal expansion-based virtual sensing in the context of offshore wind jacket substructures.
Two different expansion setups, namely, a basic and an extended one, are employed. It is evidenced how the basic setup, which only
includes sensors above the seawater level, fails to deliver high-quality expansion results during idling conditions, where the vibrations
are governed solely by the wave loading. Additionally, low expansion quality is observed for the brace elements during both idling and
operational conditions. The expansion quality in these cases is low because the modes included in the basic setup do not adequately
represent the wave-induced vibrations or the local brace modes.
To alleviate the noted shortcomings, the extended expansion setup is suggested in this paper. The setup includes sub-sea sensors
and a wave radar sensor, which allow for extraction of, respectively, local sub-sea brace vibration modes and static wave modes.
Inclusion of these modes in the expansion improves the expansion quality significantly.
For both setups, a systematic reduction in expansion quality is observed for the brace elements when the wind speed increases. It is
contended that this decrease arises because the control system renders the structural behaviour non-linear and time-variant, which is
not accounted for in the employed expansion method. This will be addressed explicitly in a separate publication by the authors.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.

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Acknowledgements

The work presented herein is financially supported by Ramboll Foundation, Ramboll Energy, and Innovation Fund Denmark. The
financial support is highly appreciated. The first author would like to thank colleagues from Ramboll for their continuous support.

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