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\begin{document}

\title{Digital twin-driven online intelligent assessment of wind turbine
drivetrain}



\author[1]{Yadong Zhou}%
\author[1]{Zhou Jianxing}%
\author[1]{Quanwei Cui}%
\author[1]{Jianmin Wen}%
\author[1]{Xiang Fei}%
\affil[1]{Xinjiang University}%


\vspace{-1em}



  
  \date{June 16, 2024}


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\selectlanguage{english}
\begin{abstract}
Condition monitoring and evaluation of wind turbine drivetrain hold
great importance. However, the implementation of real-time monitoring
often faces challenges in efficiency and accuracy, as the drivetrain
typically operates under harsh conditions. In order to resolve this,
this paper proposes a vibration-based damage monitoring digital twin
(VBDM-DT) that enables the online intelligent evaluation of wind turbine
drivetrain. The VBDM-DT integrates a random wind load model, a
high-fidelity dynamics model, and a fatigue damage model. The random
wind load model takes the wind speed from the supervisory control and
data acquisition (SCADA) as input to estimate the input torque of the
drivetrain in real time. Simultaneously, VBDM-DT uses the vibration
signals from the condition monitoring system (CMS) to intelligently
calibrate the dynamics model, allowing it to be continuously adjusted
and optimized in response to actual vibrations. And the fatigue damage
model takes the real-time dynamic load estimated by the high-fidelity
dynamics model as input to realize real-time fatigue damage monitoring
of key components of the drivetrain. The VBDM-DT model is applied to a 2
MW wind turbine drivetrain to verify the effectiveness of the proposed
method. In addition, a visualization platform is developed to vividly
and intuitively display the real-time operating information, dynamic
loads, and damage levels of the key components of wind turbine.%
\end{abstract}%



\sloppy


Digital twin-driven online intelligent assessment of wind turbine
drivetrain

Yadong Zhou\textsuperscript{1}, Jianxing Zhou\textsuperscript{1},
Quanwei Cui\textsuperscript{1}, Jianmin Wen\textsuperscript{1,2}, Xiang
Fei\textsuperscript{1}

\textsuperscript{1}School of Mechanical Engineering, Xinjiang
University, Urumqi 830047, China.

\textsuperscript{2}Xinjiang~Goldwind Science \& Technology Co., Ltd.,
Urumqi 830026, China.

\textbf{Correspondence}

Jianxing Zhou, School of Mechanical Engineering, Xinjiang University,
Urumqi 830047, China.

Email: xju\_zhjx@xju.edu.cn

\textbf{Abstract:}Condition monitoring and evaluation of wind turbine
drivetrain hold great importance. However, the implementation of
real-time monitoring often faces challenges in efficiency and accuracy,
as the drivetrain typically operates under harsh conditions. In order to
resolve this, this paper proposes a vibration-based damage monitoring
digital twin (VBDM-DT) that enables the online intelligent evaluation of
wind turbine drivetrain. The VBDM-DT integrates a random wind load
model, a high-fidelity dynamics model, and a fatigue damage model. The
random wind load model takes the wind speed from the supervisory control
and data acquisition (SCADA) as input to estimate the input torque of
the drivetrain in real time. Simultaneously, VBDM-DT uses the vibration
signals from the condition monitoring system (CMS) to intelligently
calibrate the dynamics model, allowing it to be continuously adjusted
and optimized in response to actual vibrations. And the fatigue damage
model takes the real-time dynamic load estimated by the high-fidelity
dynamics model as input to realize real-time fatigue damage monitoring
of key components of the drivetrain. The VBDM-DT model is applied to a 2
MW wind turbine drivetrain to verify the effectiveness of the proposed
method. In addition, a visualization platform is developed to vividly
and intuitively display the real-time operating information, dynamic
loads, and damage levels of the key components of wind turbine.

\textbf{Keywords:} wind turbine drivetrain, digital twin, intelligent
calibration, damage assessment

1. Introduction

Wind power, being a source of renewable energy, holds a significant
position in today's global energy scenario.\textsuperscript{1} It
contributes to the reduction of carbon emissions and lessens the
reliance on fossil fuels, bringing about positive effects on both the
Earth's environment and energy security. With the global demand for wind
energy continuously rising, the number of wind turbines has seen a rapid
increase.\textsuperscript{2,3} However, wind turbines often operate
under harsh conditions, heightening the risk of potential
failures.\textsuperscript{4} Based on data presented in the referenced
literature,\textsuperscript{5} the electrical system has the highest
failure rate, yet it is straightforward to maintain. In contrast, the
gearbox drivetrain takes the longest time to repair and incurs high
maintenance costs. Hence, exploring the monitoring and assessment of
wind turbine drivetrains proves to be essential and beneficial.
Efficient monitoring contributes to the turbines' safety and
reliability, minimizes downtime, and cuts maintenance expenses,
ultimately extending their operational life.

The SCADA system provides comprehensive real-time operational
information,\textsuperscript{6} while the CMS system records a wealth of
vibration signals. Currently, the monitoring and life prediction methods
for wind turbine drivetrains primarily involve extracting features
related to the main drivetrain's operational status from SCADA or CMS
data. These features are then used to construct monitoring and
assessment models. Zhang et al.\textsuperscript{7,8} employed various
methods such as artificial neural networks (ANN) and support vector
machines (SVM) to analyze SCADA data, establishing state monitoring
models for wind turbines. Additionally, Pan et al.\textsuperscript{9}
used the maximum mean difference algorithm to extract vibration data
features under multiple conditions, achieving gearbox state monitoring.
Guo et al.\textsuperscript{10} introduced a data-driven multiscale
sparse model to isolate bearing impact elements from vibration signals,
enhancing the fault diagnosis of wind turbine bearings. In these
data-driven modeling methods, there's often a need for extensive
historical data support. They rely more on empirical models and manually
set thresholds. Applying these models to new systems may lead to
undetected changes or frequent false alarms, limiting the observation
results.

Digital twin (DT) technology offers a novel approach for monitoring and
life prediction of wind turbine drivetrain systems. The essence of DT is
to create a digital model that can capture data from a physical entity
in real-time and possesses self-learning and evolutionary capabilities,
thereby reflecting the evolution of the physical entity's health status
more accurately.\textsuperscript{11}In recent years, with advancements
in sensor technologies, software and hardware capabilities, and computer
computational performance, DT is extensively utilized across a diverse
range of fields.\textsuperscript{12} Zhang et al.\textsuperscript{13}
presented a DT framework utilizing transfer learning for the fault
diagnosis of rolling bearings. This approach effectively ascertains the
health status of actual rolling bearings, even in the absence of
abundant fault data. Luo et al.\textsuperscript{14} utilized the
particle filter algorithm to merge sensor data with the digital model,
predicting the lifespan of machine tool cutters in CNC machines. Ye et
al.\textsuperscript{15}integrated diverse models and datasets into a
dynamic Bayesian network (DBN), grounded in the DT framework, to
forecast the durability of spacecraft structures. Chetan et
al.\textsuperscript{16} created a multi-fidelity digital twin
representation for wind turbine blades, aimed at confirming the
efficiency of dual blades in operation. Wind turbines typically need to
operate for extended periods without failure to ensure consistent power
generation. Thus, developing an online intelligent assessment for wind
turbine drivetrain systems based on DT has significant practical
importance.

However, the current application of digital twin technology for online
monitoring and performance evaluation of wind turbine drivetrain still
faces some critical challenges. The first challenge is the real-time
estimation of the input torque. Due to the constant change of wind
speed, which results in the drivetrain usually operating under changing
operating conditions, accurate estimation of the input torque is crucial
to realize the real-time monitoring of the drivetrain. The second
challenge lies in constructing a high-fidelity dynamics model. The
structure of the wind turbine drivetrain system is complex, typically
composed of planetary gears and parallel-stage gears, including
components such as shafts, gears, and bearings. For each component, some
characteristics, like damping coefficients and stiffness parameters,
vary based on operational conditions. Therefore, ensuring that the
dynamics model genuinely reflects the response of the wind turbine
drivetrain system presents a challenge. The third challenge is the
application of twin models in the remaining useful life (RUL) monitoring
of critical components. For RUL monitoring, real-time performance is
critical because timely failure warning and maintenance decisions can
reduce downtime and maintenance costs. These issues hinder the
widespread application of DT technology in wind turbines. In response to
the outlined challenges, this paper introduces the VBDM-DT model to
implement an online intelligent assessment for wind turbine drivetrain
systems. The core focuses and contributions of this paper are
encapsulated in the following segments:

\begin{itemize}
\tightlist
\item
  Integrating the DT model with the SCADA system, the input torque of
  the wind turbine drivetrain can be estimated in real time using the
  recorded wind speed. This alignment guarantees that the DT model's
  input parameters are in sync with the actual operational conditions of
  the wind turbine.
\item
  To refine the precision of the DT model, the measured vibration
  signals captured by the CMS are employed for intelligent calibration
  of the dynamics model, a process guided by the multi-objective
  optimization algorithm. This allows the DT model to be continuously
  adjusted and optimized based on actual vibrations, ensuring that the
  dynamic load of the components estimated by the dynamics model is
  highly accurate.
\item
  Based on the Palmgren-Miner hypothesis of linear damage and the S-N
  curve, a fatigue damage model is established. The real-time dynamic
  load estimated by the high-fidelity dynamics model is then used as
  input for the fatigue damage model, allowing for the estimation of the
  real-time cumulative damage of key components in the wind turbine
  drivetrain system.
\end{itemize}

In summary, the VBDM-DT model developed in this paper integrates SCADA,
CMS, and the multi-objective optimization algorithm. Unlike most models
that are currently data-driven, this model adopts a primary
mechanism-based approach with data correction as supplementary. This
approach ensures that the results of the mechanism analysis provide
theoretical support for the choice of variables in data analysis, and
the outcomes of the data analysis serve as crucial resources for
refining the parameters of the dynamics model. VBDM-DT can effectively
estimate the dynamic loads of key components in the wind turbine
drivetrain system in real time and monitor their RUL.

The remainder of this paper is organized as follows: Section 2
introduces the construction method of VBDM-DT, including the
establishment of the random wind load model, high-fidelity dynamics
model, and fatigue damage model. Section 3 evaluates the effectiveness
of the proposed VBDM-DT model and its application in RUL monitoring of
drivetrain components. Section 4 summarizes the research work discussed
in this paper.

2. Method of VBDM-DT

2.1. Framework of VBDM-DT Model

The VBDM-DT model framework proposed in this paper is shown in Figure 1,
including physical space, twin space. In this framework, the physical
space mainly contains the wind turbine, SCADA and CMS. the SCADA system
monitors various operational data of the wind turbine in real time, such
as generator output power, wind speed, main shaft speed, etc.; the CMS
primarily monitors the vibration signals from the bearings and gears
within the wind turbine's drive system.\textsuperscript{17} The twin
space can utilize TCP/IP communication to obtain the real-time relevant
operating parameters of the physical wind turbine and the vibration
signals of the components from SCADA and CMS.VBDM-DT mainly consists of
three parts: 1) the random wind load model estimates the input torque of
the wind turbine drivetrain in real time from measured wind speeds; 2)
the high-fidelity dynamics model, established through the measured
vibrations in the physical space, estimates the dynamic loads of key
components in the wind turbine drivetrain; and 3) a fatigue damage model
is established based on the linear damage Palmgren-Miner hypothesis and
S-N curve, to evaluate the extent of damage to key components of the
wind turbine drivetrain.

\textbf{Hosted file}

\verb`image1.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 1.} Basic framework of the developed VBDM-DT model

Using the above method, the real-time wind turbine operation data and
vibration signals collected in the physical space are input into the
twin space, enabling online intelligent assessment of the damage extent
of key components. A visualization platform is also developed to
visualize and render the real-time data of wind turbine, which vividly
and intuitively displays the real-time operation information, dynamic
load and damage degree of key components of wind turbine. The
visualization platform can not only monitor the operation status of wind
turbine in real time, but also provide valuable information about the
degradation of key components, so as to carry out reasonable and
predictive maintenance and replacement, in order to ensure the safe
operation of the wind power transmission system. The construction
process of the VBDM-DT model is described in detail below.

2.2. Random wind load model

The input torque of wind turbine drivetrains is significantly affected
by the random fluctuations in wind speed. In VBDM-DT, this is estimated
utilizing real-time random wind speed data from SCADA. According to the
aerodynamic theory, Equation (1) describes the input power of the wind
turbine.\textsuperscript{18,19}

where P\textsubscript{in} is the input power, \selectlanguage{greek}\emph{ρ} \selectlanguage{english}is the air
density, \emph{R} is the blade radius, and \emph{v} is the wind speed.
The wind turbine power coefficient \emph{C\textsubscript{p}} is a
function of the tip-speed ratio \selectlanguage{greek}\emph{λ} \selectlanguage{english}and the blade pitch
angle\selectlanguage{greek}\emph{β} ,\selectlanguage{english}\textsuperscript{20} which reflects the wind energy
utilization:

where \selectlanguage{greek}\emph{ω} \selectlanguage{english}is the angular speed.

Figure 2 shows a two-dimensional representation of the power coefficient
\emph{C\textsubscript{p}} . When the wind speed is below the rated
speed, the pitch angle \selectlanguage{greek}\emph{β} \selectlanguage{english}= 0. Controlling the rotor speed
\selectlanguage{greek}\emph{ω} \selectlanguage{english}to ensure the tip-speed ratio\selectlanguage{greek}\emph{λ} \selectlanguage{english}remains at
\selectlanguage{greek}\emph{λ\textsubscript{opt}}\selectlanguage{english} enables the wind turbine to operate
consistently at the peak power coefficient C\emph{\textsubscript{pmax}}
. Observe that the optimal power coefficient,
C\emph{\textsubscript{pmax}} , is valued at 0.48, and is attained when
\selectlanguage{greek}\emph{λ} \selectlanguage{english}= \selectlanguage{greek}\emph{λ\textsubscript{opt}}\selectlanguage{english} = 8.03.

\textbf{Hosted file}

\verb`image6.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 2.} The wind turbine \emph{C\textsubscript{p}} -
\selectlanguage{greek}\emph{λ}\selectlanguage{english}curves

Wind speed divides the operational range of a wind turbine into three
distinct regions,\textsuperscript{21} as depicted in Figure 3. In region
1, the wind speed falls below the turbine's cut-in speed
(\emph{v\textsubscript{min}} ), leaving the blades stationary. In region
2, the wind speed value is less than the rated value
(\emph{v\textsubscript{rate}} ), the wind turbine maintains stability at
the maximum power coefficient(C\emph{\textsubscript{pmax}} ). In region
3, where the wind speed surpasses the rated speed yet remains below the
cut-out speed, the wind turbine maintains operation at its rated power
(P\emph{\textsubscript{rate}} ).\textsuperscript{22}

\textbf{Hosted file}

\verb`image7.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 3.} Schematic diagram of the operating regions of a wind
turbine

In Equations (1), P\textsubscript{in} denotes the input power. And the
input torque(T\textsubscript{in}) developed by a wind turbine can be
written as:

Based on Equations (1) and (5), the relationship between wind speed and
input torque can be expressed as:

where T\textsubscript{rate} is the rated torque of the wind turbine.

2.3. High-fidelity dynamics model

It is difficult to directly measure the dynamic loads of components in a
commercial wind turbine, therefore, in this paper, a high-fidelity
dynamics model is constructed based on the digital twin framework by
integrating the dynamics model and CMS. The high-fidelity dynamics model
is able to generate the real-time dynamic loads of each component of the
wind turbine drivetrain system in a low-cost and computationally
efficient way in the twin space. The main innovation of this method is
to update the parameters in the dynamics model based on the vibration
signals in the CMS. The construction processes of the high-fidelity
dynamics model are introduced as follows.

2.3.1. Modeling the dynamics of the wind turbine drivetrain.

Figure 4 illustrates the schematic representation of a 2.0 MW wind
turbine gearbox. It consists of a first-stage planetary gear system and
two stages of parallel gear transmission systems connected in series.
The planetary stage features straight tooth gearing, and the parallel
stages feature helical gears. The symbols in Figure 4 are defined as
follows: s denotes the sun gear, r is the ring gear, c is the carrier,
and p\textsubscript{n} (n=1,2,3) represents the nth planets.
g\textsubscript{1} and g\textsubscript{2} refer to the intermediate
stage gears, and g\textsubscript{3} and g\textsubscript{4} to the
high-speed stage gears. T\textsubscript{in} represents the input torque,
as estimated by the random wind load model.

\textbf{Hosted file}

\verb`image10.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 4.} The schematic diagram of the wind turbine gearbox

This paper employs the lumped-parameter method to formulate the dynamics
model of the planetary transmission system.\textsuperscript{23,24} Each
gear within the planetary transmission system is characterized as a spur
gear, as illustrated in Figure 5. The matrix form of the dynamic
equation of the planetary transmission system can be written
as\textsuperscript{25}:

Where \emph{M\textsubscript{P}} , \emph{C\textsubscript{P}}
,\emph{K\textsubscript{P}} , \emph{F\textsubscript{P}} ,
and\emph{x\textsubscript{p}} represent the planetary stage's mass
matrix, damping matrix, stiffness matrix, excitation vector, and
displacement column vector, respectively.

\textbf{Hosted file}

\verb`image13.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 5.} Schematic of the planetary transmission dynamics mode

The transmission shaft is primarily used to support rotating parts and
transfer power during operation. Using the lumped-parameter method for
the dynamic modelling of a two-stage parallel gear transmission system
fails to account for shaft deformation, resulting in a lack of model
fidelity. The finite element method allows the dynamics modeling of gear
trains including flexible shafts, bearings and
gears.\textsuperscript{26} Therefore, in this paper, the finite element
method is used for dynamic modeling of a two-stage parallel gear train
as shown in Figure 6. The two-stage gear transmission system includes
three shafts, six bearings, and two helical gear pairs. The first shaft
is divided into 11 elements, the second shaft is divided into 8
elements, and the third shaft is divided into 8 elements, with each
element containing two nodes. Each node of every element is numbered,
dividing the finite element model of the two-stage gear transmission
system into 31 nodes. The intermediate stage driving gear's meshing
stiffness and mass matrix link to node 7, while the driven 's
counterparts attach to node 16. Together, they engage to constitute an
intermediate stage helical gear meshing element. The high-speed stage
driving wheel and driven wheel are connected to nodes 19 and 28,
respectively, meshing with each other to form a high-speed stage helical
gear meshing element. Each shaft contains two bearings, forming a
shaft-bearing coupling element.

In this paper, the motion differential equation of the shaft is
established using the Timoshenko beam element. Given that the two-stage
parallel gear transmission system is characterized as a helical
cylindrical gear transmission, it's essential to consider axial
excitation. We define the nodes corresponding to
the\emph{i\textsuperscript{th}} element as \emph{j} and \emph{j+1} ,
with each node accounting for translations in the \emph{x} , \emph{y} ,
and\emph{z} directions and torsion in the \emph{z} direction, resulting
in four degrees of freedom. The shaft element's motion differential
equation can be articulated as follows:

where \emph{M\textsubscript{e}} , \emph{C\textsubscript{e}}
and\emph{K\textsubscript{e}} denote mass matrix, damping matrix and
stiffness matrix of shaft element. \emph{x\textsubscript{e}} represents
the generalized coordinates of the shaft element.

The dynamics model of the helical gear meshing element in a two-stage
parallel gear transmission system is shown in Figure 7, and its dynamics
equation can be expressed as follows:

Where \emph{m\textsubscript{p}} and \emph{m\textsubscript{g}} are the
masses of pinion and gear respectively, \emph{I\textsubscript{p}}
and\emph{I\textsubscript{g}} are the rotational inertia of pinion and
gear,\emph{k\textsubscript{m}} is the time-varying meshing stiffness of
the meshing elements, and \emph{c\textsubscript{m}} is the meshing
damping.\emph{x\textsubscript{p}} , \emph{y\textsubscript{p}}
,\emph{y\textsubscript{p}} , \selectlanguage{greek}\emph{θ\textsubscript{p}}\selectlanguage{english}
,\emph{x\textsubscript{g}} , \emph{y\textsubscript{g}}
,\emph{y\textsubscript{g}} , and \selectlanguage{greek}\emph{θ\textsubscript{g}}\selectlanguage{english} are the
vibrational displacements and torsion angles of pinion and gear
respectively, \selectlanguage{greek}\emph{α} \selectlanguage{english}denotes the pressure angle, \selectlanguage{greek}\emph{β} \selectlanguage{english}is the
helical angle of the helical gear, \emph{r\textsubscript{p}}
and\emph{r\textsubscript{g}} are the radius of pinion and gear.

\textbf{Hosted file}

\verb`image18.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 6.} The dynamics model of parallel shaft gear drive in
the wind turbine system

\textbf{Hosted file}

\verb`image19.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 7.} Helical gear meshing element

Within the shaft-bearing coupling element, the time-varying stiffness of
the bearing in the dynamics model is ascertained through Equation
(13).\textsuperscript{27}

where \emph{k\textsubscript{a}} is bearing static stiffness,
and\emph{k\textsubscript{0}} is the bearing stiffness fluctuation
amplitude. \emph{f\textsubscript{b}} denotes the bearing ball passing
frequency. \selectlanguage{greek}\emph{β\textsubscript{b}}\selectlanguage{english} represents the bearing phase angle.

The 1st node in the two-stage parallel shaft gear transmission system is
connected to the planetary transmission system to couple the two
subsystems. Since the planetary stage involves spur gears and the
parallel stage involves helical gears, an axial force exists during the
operation of helical gears but is absent in spur gears. During system
coupling, it's assumed that the axial force generated by helical gears
is absorbed by the gearbox casing. After the two subsystems are coupled,
the overall stiffness of the system is as illustrated in Figure 8. The
system generalized coordinate vector can be expressed as:

The dynamics equation of the wind turbine gearbox transmission system is
given as:

where \emph{M} , \emph{C} and \emph{K} are the mass matrix, damping
matrix and stiffness matrix of wind turbine gearbox transmission
system.\emph{F} represents the vector of externally applied force, while
the damping matrix \emph{C} adopts the definition of Rayleigh damping,
articulated as follows:

where \selectlanguage{greek}\emph{η} , \selectlanguage{english}\selectlanguage{greek}\emph{ζ} \selectlanguage{english}are the mass matrix and stiffness matrix
coefficient.

\textbf{Hosted file}

\verb`image24.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 8.} Schematic diagram of the overall stiffness matrix of
the gear transmission

\emph{2.3.2. Intelligent calibration of the dynamics model}

Some parameters in the dynamics model of the wind turbine gearbox
transmission system are determined by empirical formulas or analysts'
experience. However, during actual operation, the values of certain
parameters can vary with changing conditions, such as the damping
coefficient. The operating conditions of wind turbines are highly
dependent on environmental factors, especially wind speed. Variations in
wind speed directly influence the input rotation speed and load of the
gearbox transmission system. To ensure the accuracy of the dynamics
model, it is integrated with CMS within the digital twin framework for
online intelligent calibration. As illustrated in Figure 9, the
intelligent calibration process involves comparing the measured
vibration data from the wind turbine's CMS with the simulated vibration
from the dynamics model and extracting quantitative similarity
indicators. If these indicators fall below a threshold, a
multi-objective optimization algorithm is introduced for dynamic
calibration, updating the optimal values of the dynamics model
parameters. This process guarantees a significant alignment between the
simulated and measured vibration accelerations.

In the vibration signal time domain, the root mean square (RMS) value of
the vibration signal reflects the vibration intensity and energy level.
The frequency spectrum shows the amplitude distribution of the vibration
signal at different frequencies, and the variation rule of the
fundamental frequency and its harmonic amplitude in the frequency
spectrum is very important for the health monitoring of the gear
transmission system. The correlation between the measured vibration and
the simulated vibration fundamental frequency and its harmonic amplitude
can be characterized by the Pearson correlation coefficient, and when
the Pearson correlation coefficient is greater than 0.8, it indicates
that there is a strong positive correlation between the simulated
response of the dynamics model and the measured response of the CMS,
which suggests that the dynamics model captures the characteristics of
the actual vibration signals very well. In the VBDM-DT model, in order
to realize the online intelligent calibration of the dynamical model,
three similarity quantitative indicators are proposed to monitor the
model performance, and the mathematical expressions for these similarity
quantitative indicators are articulated as follows:

Where the superscript \emph{MOD} denotes the simulated values from the
dynamics model, while \emph{CMS} represents the measured values from the
condition monitoring system. \emph{X\textsubscript{rms}}
and\emph{Y\textsubscript{rms}} are the RMS values of the simulated and
measured time domain waveforms, respectively. \emph{correlation} denotes
Pearson's correlation coefficient, \emph{amp} denotes the amplitude of
the fundamental frequency and its harmonics in the spectrogram, with the
subscript \emph{mean} indicating the average.\emph{n} is the number of
the fundamental frequency and its harmonics. When the quantitative
similarity indicator surpasses the threshold, the VBDM-DT model
activates an optimization algorithm to calibrate the dynamics model.

\textbf{Hosted file}

\verb`image26.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 9.} Intelligent calibration of dynamics model

In this paper, the values for the similarity quantitative
indicators\emph{f\textsubscript{1}} , \emph{f\textsubscript{2}}
and\emph{f\textsubscript{3}} are set at 0.5, 0.2 and 0.2
respectively.\emph{f\textsubscript{1}} primarily compares the RMS values
of the time domain waveforms between the simulated and measured
accelerations. Given that the measured vibration signal's time domain
waveform contains some noise, the preset threshold for
\emph{f\textsubscript{1}} is 0.5. The indicator
\emph{f\textsubscript{2}} represents the changing trend of the primary
frequency observed in the spectrogram. When it falls below the preset
threshold of 0.2, the Pearson correlation coefficient between the
amplitudes of the simulated and measured primary frequencies exceeds
0.8, indicating a strong positive correlation between the two. The
indicator \emph{f\textsubscript{3}} mainly characterizes the difference
in the amplitude of the main frequency in the spectrogram, and is
evaluated by using the mean absolute percentage error (MAPE), which is a
kind of index used to measure the degree of error between the simulated
signal and the measured signal, and when \emph{f\textsubscript{3}} is
lower than the preset threshold value of 0.2, it is considered that the
two errors are small.

Within the VBDM-DT framework, the employed optimization algorithm is the
multi-objective grasshopper optimization algorithm (MOGOA), renowned for
its superior search efficiency and swift convergence rate. GOA is a
meta-heuristic bionic optimization algorithm that simulates the group
behavior of grasshoppers in nature to find the optimal
solution.\textsuperscript{28} MOGOA improves on GOA to adapt to the
characteristics of multi-objective problems. Specifically, the MOGOA
algorithm introduces the concept of Pareto domination, and evaluates the
advantages and disadvantages of solutions by comparing their Pareto
dominance relationships. The algorithm finds better Pareto optimal
solutions by continuously updating and improving the solutions. The
mathematical model proposed by MOGOA is as follows:

Where \emph{X\textsubscript{i}} represents the position of the
i\textsuperscript{th} grasshopper after the algorithm's iterative
update, \emph{ub} and \emph{lb} are the upper and lower bounds of the
optimization parameters, \emph{N} denotes the number of grasshoppers in
the population,
and\emph{\textbar{}x\textsubscript{j}-x\textsubscript{i}\textbar{}} is
the distance between the current ith grasshopper and the
j\textsuperscript{th} grasshopper. \emph{s(r)} represents the social
behavior among the grasshoppers. Based on the recommendation from
reference,\textsuperscript{28} \emph{m} = 0.5 and \selectlanguage{greek}\emph{μ} \selectlanguage{english}=
1.5.\emph{T\textsubscript{temp}} is the current optimal solution. The
purpose of introducing the parameter c is to decrease the global search
range as the number of iterations increases but to enhance the local
precision search around the target. Its mathematical expression is as
follows:

Where \emph{l} is the current iteration number and \emph{L} is the
maximum iteration number, \emph{c\textsubscript{min}}
and\emph{c\textsubscript{max}} are the minimum and maximum values of the
parameter \emph{c} . In this paper, we set \emph{c\textsubscript{min}} =
0.00004 and \emph{c\textsubscript{max}} = 1.

Within the VBDM-DT model, in order to ensure that the optimal parameters
can still be obtained when the time-domain waveforms and frequency
spectra of the vibration signals have certain noise or errors, the
vibration signals of the two measurement points in the CMS are utilized
to calibrate the dynamics model, and when the similarity quantification
indicator of either measurement point is lower than the threshold value,
the MOGOA is invoked to update the parameters. When updating the
dynamics parameters, the time-frequency domain characteristics of both
measurement points are compared, and the mathematical expression of the
objective function is consistent with the similarity quantization
indicator in Eq. (18), which ensures that the simulated response of the
two measurement points matches well with the actual response.

In summary, the VBDM-DT framework facilitates the online intelligent
calibration of the dynamics model, enhances the fidelity of the dynamics
model, and provides real-time estimates of the dynamic meshing force in
the gears and dynamic loads in the bearings of the wind turbine
drivetrain system. This is beneficial for component performance
evaluation and RUL monitoring.

2.4. Fatigue damage model

The dynamic load of the key components of the wind turbine gearbox
transmission system can be estimated in real time by using the
high-fidelity dynamics model. The dynamic load calculated through this
high-fidelity model cannot be directly applied to fatigue life
prediction methods. Instead, this load must be processed into stress,
providing a stress spectrum for fatigue analysis of the gears and
bearings in the transmission chain. Subsequently, cumulative damage is
estimated based on the linear damage Palmgren-Miner hypothesis and the
S-N curve. In the following, this paper details these processes, using
gear contact fatigue damage as an example.

\emph{2.4.1. Gear contact stress}

Utilizing the estimated torque from the random wind load model and the
real-time rotational speed measured by SCADA as inputs for the
high-fidelity dynamics model, it becomes feasible to estimate the
real-time dynamic load on the gears of the wind turbine transmission
system. Based on the ISO 6336-2 standard, the contact stress of the gear
is further computed \textsuperscript{29}:

Where \emph{u} represents the gear transmission
ratio,\emph{d\textsubscript{1}} and \emph{b} are reference diameter and
face width of the pinion, \emph{F\textsubscript{t}} is the gear meshing
force calculated by the high-fidelity dynamics model. Other parameters
are discussed in reference.\textsuperscript{29}

\emph{2.4.2. Stress cycle counting method}

After calculating the contact stress of the gear, the contact stress
needs to be counted and transformed into the contact stress fatigue
stress spectrum. The dynamic meshing force of the gear is based on the
entire gear, while fatigue damage analysis is based on the gear tooth.
This means the stress cycle count of the gear tooth is related to the
gear rotation speed, and the contact stress on the tooth surface varies
from zero to amplitude. Therefore, the traditional rainflow counting
method cannot be used to statistically analyze the contact stress on the
tooth surface. Referring to the method recommended in ISO
6336-6,\textsuperscript{30} the irregular contact stress load on the
gear is statistically analyzed. As shown in Figure 10, assuming the
stress-time load process calculated between
\selectlanguage{greek}\emph{σ\textsubscript{i-1}}\selectlanguage{english}and \selectlanguage{greek}\emph{σ\textsubscript{i}}\selectlanguage{english} takes the
time\emph{t\textsubscript{1}+t\textsubscript{2}+t\textsubscript{3}} ,
then the stress duration at the stress level \selectlanguage{greek}\emph{σ\textsubscript{i}}\selectlanguage{english}
is\emph{t\textsubscript{1}+t\textsubscript{2}+t\textsubscript{3}} . The
number of stress cycles at each stress level depends on the duration of
the stress and the gear rotation speed, and it is calculated by the
following formula:

Where \emph{n\textsubscript{i}}represents the number of stress cycles
within the \emph{i\textsuperscript{th}}stress bin. In this paper, with
the sun gear meshing with three planetary gears at the planetary stage,
the number of stress cycles needs to be multiplied by 3. The duration of
the\emph{i\textsuperscript{th}} stress bin, denoted
as\emph{t\textsubscript{i}} , is measured in seconds and can be
referenced from the time series presented in Figure 10.
Additionally,\emph{w\textsubscript{i}} indicates the gear's rotational
speed in rpm.

\textbf{Hosted file}

\verb`image32.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 10.} Process of creating stress bins

\emph{2.4.3. Gear contact fatigue damage analysis}

The contact fatigue damage of each gear can be calculated according to
the Palmgren-Miner linear damage accumulation theory with the following
formula:

Where \emph{D\textsubscript{c}} represents the cumulative fatigue
damage. \emph{N\textsubscript{b}} indicates the total number of stress
bins within the stress spectrum. \emph{n\textsubscript{i}} signifies the
count of stress cycles within the \emph{i\textsuperscript{th}} stress
bin. \emph{N\textsubscript{ci}} designates the requisite number of
stress cycles for the gear to achieve contact fatigue failure at the
stress magnitude corresponding to the \emph{i\textsuperscript{th}}stress
bin. This is typically derived from the S-N curve, as delineated in
reference \textsuperscript{30}:

Where \selectlanguage{greek}\emph{σ} \selectlanguage{english}denotes the equivalent fatigue stress, expressed in
MPa.\emph{N} represents the number of stress cycles until fatigue
failure is reached; \emph{m} and \emph{C} are material-specific
constants, determined by the material itself.

To enhance the accuracy of the analysis, this study adopts the
dual-slope S-N curve method recommended in ISO 6336-6. For values
of\emph{N} below \emph{N\textsubscript{0}} , the curve's slope is
defined as \emph{-1/m} ; for values greater than
\emph{N\textsubscript{0}} , the slope becomes -1/(2\emph{m} -1).
Specifically, for gear contact fatigue evaluations,
\emph{N\textsubscript{0}} = 5\selectlanguage{ngerman}×10\textsuperscript{7}.

3. Validation and application of the VBDM-DT

To achieve real-time intelligent assessment of wind turbine transmission
systems, this section presents tests and simulations of the proposed
VBDM-DT model using a 2.0 MW wind turbine from a wind farm in Urumqi,
Xinjiang, China as a case study. The comprehensive specifications of the
wind turbine are provided in Table 1, and the structural parameters of
the gearbox gear train are shown in Table 2.

\textbf{Table 1.} Wind turbine parameters.\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
Parameter & Value & Parameter & Value & Parameter & Value\tabularnewline
Rated power & 2MW & Cut-in wind speed & 4m/s & Rotor radius &
41m\tabularnewline
Rated torque & 1.23\selectlanguage{ngerman}×10\textsuperscript{6}N\selectlanguage{english}*m & Cut-out wind speed &
20m/s & Transmission ratio & 103\tabularnewline
Rated speed & 14.8RPM & Rated wind speed & 12m/s & Air density &
1.255kg/m\textsuperscript{3}\tabularnewline
\bottomrule
\end{longtable}

\textbf{Table 2.} Structural parameters of the wind turbine gearbox
transmission.\selectlanguage{english}
\begin{longtable}[]{@{}llllllll@{}}
\toprule
Gear stage & Modulus (mm) & Pressure angle (\selectlanguage{ngerman}°) & Component & Number of
teethe & Tooth width (mm) & Mass (kg) & Moment of
inertia(kg.m\textsuperscript{2})\tabularnewline
Planetary gear stage & 17 & 20 & Ring & 99 & 370 & 2652 &
1252\tabularnewline
& & & Planet & 39 & 370 & 994 & 266\tabularnewline
& & & Sun & 21 & 370 & 286 & 15\tabularnewline
Inter-mediate stage & 14 & 20 & Gear g\textsubscript{1} & 92 & 300 &
2724 & 1025\tabularnewline
& & & Gear g\textsubscript{2} & 21 & 300 & 102 & 2\tabularnewline
High-speed stage & 14 & 20 & Gear g\textsubscript{3} & 107 & 170 & 2096
& 1067\tabularnewline
& & & Gear g\textsubscript{4} & 26 & 170 & 101 & 3\tabularnewline
\bottomrule
\end{longtable}

3.1. Validation of the random wind load model.

The load applied to the wind turbine drivetrain is aerodynamic torque.
The comparison between the actual aerodynamic torque and the estimated
aerodynamic torque is shown in Figure 11. The estimated torque is
obtained through the online estimation using the model proposed in
Section 2.2, while the actual torque is acquired via a rotary torque
sensor. The MAPE is used as an evaluation strategy to assess the
discrepancy between the estimated and actual values, and its formula
definition is as follows:

Where \emph{n} is the number of samples;
\emph{y\textsubscript{i}}denotes the torque estimated by the model.
\selectlanguage{polish}\emph{ŷ?}\selectlanguage{english} represents the actual torque, and \selectlanguage{romanian}\emph{ȳ} is the mean value
of the actual torque. The data in Figure 11 shows the operating torque
of the wind turbine on a particular day, and the error between the model
estimate and the actual value is 8.34\%.

\textbf{Hosted file}

\verb`image36.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 11.} Comparison of estimated torque and actual torque

3.2. Validation of high-fidelity dynamics model

To validate the effectiveness of the dynamics model intelligent
calibration module in the developed VBDM-DT model, vibration signals
from two measurement points in the CMS were used to calibrate the
dynamic model. The selected measurement points are the high-speed shaft
bearing seat and the intermediate shaft bearing seat, which correspond
to measurement point 1 and measurement point 2 in Figure 12,
respectively.

\textbf{Hosted file}

\verb`image37.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 12.} The sensors arrangement of wind turbine gearbox

When the wind speed is 6.75m/s, the input rotational speed of the wind
turbine drivetrain is 8.33RPM, and the input torque is 361747N.m. Under
this condition, the similarity quantitative indicator is higher than the
preset threshold, and the VBDM-DT model starts invoking the optimization
algorithm for dynamic calibration. During the dynamic model calibration
process, MOGOA simultaneously extracts the similarity quantification
indicators for measurement points 1 and 2 and then updates the dynamic
parameters. The similarity quantization
indicators\emph{f\textsubscript{1}} , \emph{f\textsubscript{2}} ,
and\emph{f\textsubscript{3}} are obtained according to Equation (17)
discussed in Section 2.3.2. The ultimate goal is to obtain a good match
between the simulated and measured accelerations so that the similarity
quantization indicators are below a preset threshold. The comparison
before and after calibration for measurement point 1 is shown in Figure
13, and for measurement point 2 in Figure 14. The post-calibration
simulated response shows high consistency with the CMS measured
response. At measurement point 1, the primary frequency components are
around 372 Hz, 744 Hz, and 1,116 Hz, corresponding to the fundamental
frequency, second harmonic, and third harmonic of the high-speed stage.
The peak values of the primary frequency at test point 2 are consistent
with the situation of measurement point 1. The comparison results before
and after calibration are shown in Table 3. Before calibration, the
value of \emph{f\textsubscript{2}} for measurement point 1 is 0.006, but
both \emph{f\textsubscript{1}} and \emph{f\textsubscript{3}} exceed the
preset threshold, indicating that the trend of the primary frequency of
the simulated signal at measurement point 1 matches the measured signal,
but the amplitude differs significantly from the measured vibration.
Before calibration, the value of \emph{f\textsubscript{1}} for
measurement point 2 is 0.01, but both \emph{f\textsubscript{2}}
and\emph{f\textsubscript{3}} are above the preset threshold, indicating
that the time-domain waveform of the simulated acceleration at
measurement point 2 is consistent with the measured results. However,
the variation pattern and amplitude of the main frequency in the
frequency domain differ significantly from the measured signal. After
calibration, the similarity quantification indices for both measurement
points 1 and 2 are below the preset threshold, demonstrating the
effectiveness of the intelligent calibration module for dynamics in the
VBDM-DT model.

\textbf{Table 3.} Comparison of similarity quantification metrics.\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
Testing position & Point 1 & Point 1 & Point 2 & Point 2\tabularnewline
& Before intelligent updating & After intelligent updating & Before
intelligent updating & After intelligent updating\tabularnewline
f\textsubscript{1} ([?]0.5) & 2.12 & 0.36 & 0.01 & 0.40\tabularnewline
f\textsubscript{2} ([?]0.2) & 0.006 & 0.006 & 0.81 & 0.01\tabularnewline
f\textsubscript{3} ([?]0.2) & 3.90 & 0.13 & 0.41 & 0.18\tabularnewline
\bottomrule
\end{longtable}

\textbf{Hosted file}

\verb`image38.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 13.} Comparison before and after intelligent calibration
of measurement point 1

\textbf{Hosted file}

\verb`image39.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 14.} Comparison before and after intelligent calibration
of measurement point 2

\textbf{3.3. Fatigue damage estimation}

Utilizing the random wind load model validated earlier, along with the
high-fidelity dynamics model, we can carry out real-time estimations of
dynamic loads on each gear pair. Further, using the program in the
fatigue damage model, we can estimate the contact stress of the gear
pairs and compile their load spectra. Figure 15 displays the stress
range and corresponding cycle counts for the first-stage sun gear on a
particular day.

\textbf{Hosted file}

\verb`image40.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 15.} Stress range and corresponding cycle counts for the
sun gear

The material of the sun gear in the gearbox of the 2 MW wind turbine
studied in this paper is 17CrNiMo6, and \emph{m} = 17.035 and \emph{C} =
3090.3 are obtained according to ISO 6336-2,\textsuperscript{29} and the
S-N curve of the contact fatigue of the sun gear is determined using the
method discussed in Section 2.4.3, as shown in Figure 16. Utilizing the
VBDM-DT model proposed in this paper, we conducted an online intelligent
evaluation of the wind turbine drivetrain. The cumulative damage of the
sun gear after one day of operation due to contact fatigue is
illustrated in Figure 17, with the accumulated damage approximated to be
1.023e-05. The growth rate of contact fatigue damage is lower when the
wind turbine is operated at less than the rated torque. On the contrary,
when the wind turbine is operated close to the rated torque, the contact
stress level of the gears increases significantly, which leads to a
rapid growth rate of contact fatigue damage.

\textbf{Hosted file}

\verb`image41.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 16.} S-N curve of the sun gear

\textbf{Hosted file}

\verb`image42.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 17.} Expected accumulated damage of the sun gear over one
day of operation

The design life of wind turbines is approximately 20 years. However, the
remaining life information of key components in wind turbines is very
limited during operation. Therefore, this study estimated the cumulative
contact fatigue damage of various gears in the wind turbine gearbox
based on historical operating state information provided by SCADA and
CMS, combined with the methods proposed in the VBDM-DT model. This helps
predict the life of the gears and determines when maintenance or gear
replacement is needed. The wind turbine studied in this paper has been
in operation for 11 years, and the current cumulative damage of the sun
gear is about 0.415. The fatigue damage threshold is usually set at 1,
and it is considered to have experienced fatigue failure when it exceeds
this threshold.

\textbf{3.4. Digital twin visualization platform}

To intuitively display the interaction between the physical space and
the digital space, this paper has developed a visualization platform for
wind turbine digital twins. The data-driven functions of this platform
are implemented based on the Unity platform. As the operating state of
the wind turbine in the physical space constantly changes, the digital
model continuously obtains real-time status data and transforms it,
achieving a virtual-real mapping of the wind turbine drivetrain state,
ensuring synchronization between the virtual and physical wind turbines.
Furthermore, this visualization platform, based on the VBDM-DT model,
can capture real-time and high-precision dynamic load changes of key
wind turbine components. It then further estimates the contact stress of
key components and conducts statistical analysis, realizing quantitative
assessment of cumulative contact fatigue damage to various key
components, achieving the goal of precisely mapping the operating state
of the wind turbine.

\textbf{Hosted file}

\verb`image43.emf`  available at \url{https://authorea.com/users/725323/articles/708623-digital-twin-driven-online-intelligent-assessment-of-wind-turbine-drivetrain}

\textbf{Figure 18.} Digital twin visualization platform for wind
turbines.

Figure 18 displays the visualization interface of this platform. Figure
18(a) provides a clear view of the main operating parameters of the wind
turbine digital twin system (wind speed, input rotational speed, input
torque, generator power, etc.). Figure 18(b) shows the interface for the
intelligent calibration of the dynamics model. Figure 18(c) displays the
real-time dynamic load time-domain waveform and spectrum graph for each
key component. Figure 18(d) shows the contact stress characterization
interface and the real-time cumulative damage quantitative assessment
interface for each key component.

4. Conclusions

The performance and reliability of the wind turbine system directly
affects the stability and efficiency of power generation, and it is of
great importance to monitor the operation of the system in real time as
well as to evaluate its degradation status in real time. In this paper,
a digital twin model named VBDM-DT is proposed, which can be used to
carry out online intelligent assessment of wind turbine drivetrain
system. By integrating SCADA and CMS systems, the model can estimate the
input torque in real time and intelligently calibrate the dynamics model
in order to build a high-fidelity dynamics model to reveal the dynamic
response of each key component in the wind turbine drivetrain. In
addition, the model has the ability to estimate the cumulative fatigue
damage of the drivetrain components in real time, which provides
valuable information for monitoring the health status of the wind
turbine system. The VBDM-DT model of a wind turbine with a rated power
of 2.0 MW is used as a case study to verify the validity of the model.
In addition, the developed VBDM-DT can also be used in other gear
trains, such as construction machinery and helicopters, to promote the
wide application of digital twin technology in the health management of
key components of gear trains. In future research, we will further
consider other degradation mechanisms, such as wear monitoring of gears,
in order to extend the functionality and applicability areas of the
model.

Acknowledgements

This work was supported by the Natural Science Foundation of Xinjiang
Province (No. 2021D01C050) and Xinjiang Province key research and
development (2022B01017-1).

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\end{enumerate}

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