High-Precision Main Shaft Displacement Measurement for Wind Turbines Using an Optimized Position-Sensitive Detector

Abstract

The main shaft of a wind turbine is a critical component that ensures the normal operation of the turbine, and its axial displacement directly impacts its efficiency and safety. The inaccurate measurement of axial displacement may lead to severe issues such as shaft fractures, causing turbine shutdowns. Correcting measurement errors related to axial displacement is essential to prevent potential accidents. This study proposes an improved error correction method for measuring the axial displacement of wind turbine main shafts. Using a position-sensitive detector (PSD) and laser triangulation, the axial and radial displacements of the main shaft are measured to address environmental interference and cost constraints. Additionally, a Sparrow Search Algorithm- Backpropagation (SSA-BP) model is constructed based on operational data from the wind turbine’s main shaft to correct the system’s nonlinear errors. The Sparrow Search Algorithm (SSA) is employed to optimize the weights and thresholds of the Backpropagation (BP) neural network, enhancing prediction accuracy and model stability. Initially, a main shaft displacement measurement system based on a precision displacement stage was developed, and system stability tests and displacement measurement experiments were conducted. The experimental results demonstrate that the system stability error is ±0.025 mm, which is lower than the typical error of 0.05 mm in contact measurement. After model correction, the maximum nonlinear errors of the axial and radial displacement measurements are 0.83% and 1.29%, respectively, both of which are lower than the typical measurement error of 2% in contact measurements. This indicates that the proposed model can reliably and effectively correct the measurement errors. However, further research is still necessary to address potential limitations, such as its applicability in extreme environments and the complexity of implementation.

Full text

Citation: Zhang, W.; Wang, L.; Li, G.;
Zheng, H.; Pang, C. High-Precision
Main Shaft Displacement
Measurement for Wind Turbines
Using an Optimized Position-
Sensitive Detector. Electronics 2024,13,
5055. https://doi.org/10.3390/
electronics13245055
Academic Editors: Yeongsu Bak and
Sesun You
Received: 28 November 2024
Revised: 16 December 2024
Accepted: 19 December 2024
Published: 23 December 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Article
High-Precision Main Shaft Displacement Measurement for Wind
Turbines Using an Optimized Position-Sensitive Detector
Weitong Zhang 1, Lingyun Wang 1,* , Guangxi Li 2, Huicheng Zheng 1and Chengwei Pang 1
1School of Optoelectronic Engineering, Changchun University of Science and Technology,
Changchun 130022, China; 2022100364@mails.cust.edu.cn (W.Z.); 2022100361@mails.cust.edu.cn (H.Z.);
2022100396@mails.cust.edu.cn (C.P.)
2College of Electronic Information Engineering, Changchun University, Changchun 130022, China;
ligq@ccu.edu.cn
*Correspondence: 1999800022@cust.edu.cn
Abstract: The main shaft of a wind turbine is a critical component that ensures the normal operation
of the turbine, and its axial displacement directly impacts its efficiency and safety. The inaccurate
measurement of axial displacement may lead to severe issues such as shaft fractures, causing turbine
shutdowns. Correcting measurement errors related to axial displacement is essential to prevent
potential accidents. This study proposes an improved error correction method for measuring the
axial displacement of wind turbine main shafts. Using a position-sensitive detector (PSD) and laser
triangulation, the axial and radial displacements of the main shaft are measured to address environ-
mental interference and cost constraints. Additionally, a Sparrow Search Algorithm- Backpropagation
(SSA-BP) model is constructed based on operational data from the wind turbine’s main shaft to correct
the system’s nonlinear errors. The Sparrow Search Algorithm (SSA) is employed to optimize the
weights and thresholds of the Backpropagation (BP) neural network, enhancing prediction accuracy
and model stability. Initially, a main shaft displacement measurement system based on a precision
displacement stage was developed, and system stability tests and displacement measurement ex-
periments were conducted. The experimental results demonstrate that the system stability error is
±
0.025 mm, which is lower than the typical error of 0.05 mm in contact measurement. After model
correction, the maximum nonlinear errors of the axial and radial displacement measurements are
0.83% and 1.29%, respectively, both of which are lower than the typical measurement error of 2% in
contact measurements. This indicates that the proposed model can reliably and effectively correct
the measurement errors. However, further research is still necessary to address potential limitations,
such as its applicability in extreme environments and the complexity of implementation.
Keywords: PSD; main shaft displacement; laser triangulation; SSA-BP neural network
1. Introduction
The main shaft, as a key component connecting the blades and the generator, is an
essential transmission element in wind turbine systems. In China, most wind turbines are
located in regions prone to low temperatures, strong winds, sandstorms, or high humidity
and coastal areas with severe corrosion, where the main shaft is particularly susceptible to
fractures, potentially leading to accidents [1–4].
In January 2011, a wind turbine in the Dali Dafengba Wind Farm operated by Hua-
neng witnessed a main shaft fracture accident on turbine No. 60. The reason for this was
that excessive main shaft displacement caused by improper machining led to fatigue fail-
ure [
4
]. Reports indicate that certain models of wind turbines have encountered premature
main shaft fractures after just four years of operation [
5
]. These failures have resulted in
significant economic losses for wind farms. According to information from wind power
stations, dozens of wind turbines have suffered main shaft fractures during the early stages
Electronics 2024,13, 5055. https://doi.org/10.3390/electronics13245055 https://www.mdpi.com/journal/electronics

Electronics 2024,13, 5055 2 of 20
of use. Such premature failures not only directly damage the equipment but also lead
to unexpected shutdowns, further disrupting power generation schedules. This, in turn,
incurs additional costs for emergency repairs and results in revenue losses [
6
,
7
]. Studies
suggest that main shaft failures in wind turbines may contribute to millions of dollars in
losses annually, with some individual wind farms reporting losses in the range of tens to
hundreds of thousands of dollars per incident [
8
]. Operation and maintenance (O&M) costs
for wind turbines can account for 10–20% of the total cost of a wind energy project, and
this percentage can rise to 35% by the end of a turbine’s service life [9].
The main shaft displacement is a critical indicator used to assess the health of the
main shaft. It directly affects the stability of turbine operation and the reliability of the
transmission system. Variations in the shaft displacement reflect the relative motion
between the main shaft and other key components, such as bearings and the gearbox.
To ensure the safe and stable operation of the wind turbine, precise monitoring of the
main shaft displacement is necessary [
10
,
11
]. Excessive displacement can lead to poor
alignment between the main shaft and other critical components, such as bearings and
the gearbox, increasing friction and wear. This accelerates bearing damage and may
even cause bearing failure, impacting the entire turbine’s normal operation. Furthermore,
excessive displacement can cause vibrations and noise, raising the risk of mechanical
failure, reducing the efficiency of the turbine’s transmission system, and decreasing power
generation efficiency. In extreme cases, it may even lead to catastrophic failures, such as the
main shaft.
In recent years, domestic and international researchers have continuously explored
monitoring the state of wind turbine main shafts. Traditional monitoring methods largely
rely on contact sensors, with vibration monitoring being the predominant technique [12].
Zimroz R et al. [
13
] installed accelerometer vibration sensors on a wind turbine to
monitor the vibration of the main shaft and detect potential faults. Cheng J et al. [
14
]
developed an online monitoring system using ultrasonic excitation and reception devices
to achieve the real-time monitoring of main shaft quality. However, these systems are often
affected by noise, which compromises their reliability and accuracy. Jiang Z et al. [
15
] used
infrared sensors for the fault detection of turbine main shafts, but the high cost of infrared
equipment limits its widespread application. Selvaraj Y et al. [
16
] proposed a wind turbine
main shaft monitoring method combining IoT and machine learning technologies, which
improves fault prediction capabilities through real-time data collection and intelligent
analysis. However, its high computational requirements for large-scale data processing
limit its applicability in resource-constrained environments. To address the issues of noise
interference and high equipment costs in the aforementioned methods, this paper designs
a non-contact, high-precision detection method for main shaft axial displacement. This
method not only maintains high monitoring accuracy in complex environments but also
significantly reduces the overall system operational cost.
During the operation of wind turbines, environmental noise caused by high-speed
rotation, airflow disturbances, and mechanical vibrations significantly interfere with mea-
surements made by traditional contact sensors. To address this issue, this study adopts an
optical non-contact sensor and utilizes a PSD as the core component. PSD position sensors
are a type of photoelectric distance measurement device that determines the position by
detecting the photocurrent corresponding to the energy center of the light spot on the
sensor’s photosensitive area. Compared to other non-contact technologies, PSDs offer
significant advantages: first, PSD measurements are solely related to the geometric center
of the light spot and are not strictly dependent on its shape, enabling stable measurements
even when the light spot shape changes. Second, PSDs have fast response speeds, allowing
real-time detection and feedback of positional information. Furthermore, PSDs have a high
resolution, enabling precise position measurements and ensuring measurement accuracy
in complex environments [
17
,
18
]. Similarly, compared to capacitive sensors, which are
sensitive to environmental humidity and material properties, PSD technology is more

Electronics 2024,13, 5055 3 of 20
stable and less affected by environmental factors. This makes PSDs particularly suitable for
dynamically changing environments, enabling continuous and accurate measurements [
19
].
Laser Triangulation is a non-contact displacement measurement method based on op-
tical principles, commonly used for high-precision distance or displacement measurements.
The basic principle involves projecting a laser beam onto the surface of an object, with the
reflected light being received by a detector. The displacement of the object is measured
by analyzing the angular changes in the reflected light [
20
,
21
]. When combined with a
PSD, the performance of laser triangulation is significantly enhanced. PSDs feature high
sensitivity and fast response, enabling more precise detection of changes in the position of
the reflected light, thereby improving the accuracy of displacement measurements. More-
over, the real-time feedback capability of PSDs further optimizes the measurement process,
allowing the system to adapt better to complex measurement environments, enhance its
resistance to interference, and achieve more efficient signal processing [22].
However, despite the numerous advantages of PSD sensors, certain nonlinear errors
persist in practical applications, particularly when the light spot approaches the edge of
the sensor, where the output signal may deviate from linearity. To overcome these errors
and improve measurement accuracy, this paper proposes the use of a Sparrow Search
Algorithm (SSA) to optimize a BP neural network. Since BP neural networks are often
prone to being trapped in local optima, which limits significant improvements in prediction
accuracy, the SSA is used to optimize the weights and thresholds of the BP network. By
introducing an advanced error correction algorithm, the aim is to effectively reduce the
impact of environmental interference and equipment nonlinearity on measurement results.
This will enable the precise monitoring of the main shaft displacement, providing reliable
technical support for the health management and safe operation of wind turbine systems.
2. The Measurement Principle of Wind Turbine Main Shaft Displacement
The internal structure of a wind turbine, as shown in Figure 1, mainly includes the
blades, hub, pitch system, main shaft, gearbox, generator, nacelle and other components [
23
].
The main shaft, as a critical load-bearing part, is responsible for transferring the kinetic
energy generated by the blades to the gearbox.
Electronics 2025, 14, x FOR PEER REVIEW 4 of 22
Figure 1. Internal structure of a wind turbine.
Figure 2. Analysis of the forces acting on the main shaft of the wind turbine.
Axial displacement of the main shaft is directly transferred to the input shaft of the
gearbox, causing both the planetary gears and the carrier to move axially. This can lead to
wear on the gear surfaces of the planetary gears and potentially damage other compo-
nents within the gearbox, ultimately causing gearbox failure and turbine shutdown.
Therefore, axial displacement has a detrimental effect on the bearings supporting the plan-
etary gear carrier in the gearbox.
Radial displacement of the main shaft can result in bearing damage, gearbox failures,
and uneven wear of other mechanical components, reducing their service life. The me-
chanical imbalance caused by displacement increases generator vibration and noise. Pro-
longed radial displacement may lead to mechanical failures or even significant damage to
the generator, impacting the reliability of the entire wind turbine [25].
To prevent axial and radial displacement from interfering with the operation of wind
turbine systems, this paper uses the laser triangulation method to measure the axial and
radial displacement of the wind turbine main shaft.
2.1. Working Principle
Laser triangulation includes both direct laser triangulation and oblique laser triangu-
lation. Compared to direct laser triangulation, oblique laser triangulation offers a larger
measurement range and higher stability. In this method, the laser beam is directed at a
specific angle to the object, allowing the reflected light to be more evenly distributed,
Figure 1. Internal structure of a wind turbine.
As shown in Figure 2, during operation, the horizontal force
Fxn
on the main shaft
mainly results from the overturning force and centrifugal force from the blades, as well as
the braking force applied to the main shaft during the activation of the mechanical brake
system. The radial force
Fzn
on the main shaft primarily comes from the gravitational forces
acting on the main shaft, gearbox, and other components. Therefore, during the operation
of the wind turbine main shaft, in addition to axial displacement, radial displacement also
occurs [
24
]. These two types of displacement can impact the stability of the main shaft and
the overall performance of the wind turbine.

Electronics 2024,13, 5055 4 of 20
Electronics 2025, 14, x FOR PEER REVIEW 4 of 22
Figure 1. Internal structure of a wind turbine.
Figure 2. Analysis of the forces acting on the main shaft of the wind turbine.
Axial displacement of the main shaft is directly transferred to the input shaft of the
gearbox, causing both the planetary gears and the carrier to move axially. This can lead to
wear on the gear surfaces of the planetary gears and potentially damage other compo-
nents within the gearbox, ultimately causing gearbox failure and turbine shutdown.
Therefore, axial displacement has a detrimental effect on the bearings supporting the plan-
etary gear carrier in the gearbox.
Radial displacement of the main shaft can result in bearing damage, gearbox failures,
and uneven wear of other mechanical components, reducing their service life. The me-
chanical imbalance caused by displacement increases generator vibration and noise. Pro-
longed radial displacement may lead to mechanical failures or even significant damage to
the generator, impacting the reliability of the entire wind turbine [25].
To prevent axial and radial displacement from interfering with the operation of wind
turbine systems, this paper uses the laser triangulation method to measure the axial and
radial displacement of the wind turbine main shaft.
2.1. Working Principle
Laser triangulation includes both direct laser triangulation and oblique laser triangu-
lation. Compared to direct laser triangulation, oblique laser triangulation offers a larger
measurement range and higher stability. In this method, the laser beam is directed at a
specific angle to the object, allowing the reflected light to be more evenly distributed,
Figure 2. Analysis of the forces acting on the main shaft of the wind turbine.
Axial displacement of the main shaft is directly transferred to the input shaft of the
gearbox, causing both the planetary gears and the carrier to move axially. This can lead to
wear on the gear surfaces of the planetary gears and potentially damage other components
within the gearbox, ultimately causing gearbox failure and turbine shutdown. Therefore,
axial displacement has a detrimental effect on the bearings supporting the planetary gear
carrier in the gearbox.
Radial displacement of the main shaft can result in bearing damage, gearbox failures,
and uneven wear of other mechanical components, reducing their service life. The mechan-
ical imbalance caused by displacement increases generator vibration and noise. Prolonged
radial displacement may lead to mechanical failures or even significant damage to the
generator, impacting the reliability of the entire wind turbine [25].
To prevent axial and radial displacement from interfering with the operation of wind
turbine systems, this paper uses the laser triangulation method to measure the axial and
radial displacement of the wind turbine main shaft.
2.1. Working Principle
Laser triangulation includes both direct laser triangulation and oblique laser triangu-
lation. Compared to direct laser triangulation, oblique laser triangulation offers a larger
measurement range and higher stability. In this method, the laser beam is directed at a
specific angle to the object, allowing the reflected light to be more evenly distributed, which
reduces the impact of variations in reflection direction on measurement accuracy. Addition-
ally, the oblique method effectively avoids issues such as the overlap of the incident and
sensor-received light, making it more suitable for complex measurement environments.
It is particularly effective for measuring objects with smooth or irregular surfaces. This
system adopts the oblique laser triangulation method for measurements.
The basic principle for measuring the axial and radial displacement of the main shaft
involves fixing a reflector to the surface of the main shaft. A laser beam is projected through
a converging lens directly onto the surface of the reflector. The reflected laser beam is then
projected onto the photosensitive surface of a PSD. By reading the geometric centroid of
the light spot and using a signal processing unit to convert this into a voltage signal, the
data are transmitted to an A/D acquisition chip for further processing. The measurement
principle is illustrated in Figure 3.

Electronics 2024,13, 5055 5 of 20
Electronics 2025, 14, x FOR PEER REVIEW 5 of 22
which reduces the impact of variations in reflection direction on measurement accuracy.
Additionally, the oblique method effectively avoids issues such as the overlap of the inci-
dent and sensor-received light, making it more suitable for complex measurement envi-
ronments. It is particularly effective for measuring objects with smooth or irregular sur-
faces. This system adopts the oblique laser triangulation method for measurements.
The basic principle for measuring the axial and radial displacement of the main shaft
involves fixing a reflector to the surface of the main shaft. A laser beam is projected
through a converging lens directly onto the surface of the reflector. The reflected laser
beam is then projected onto the photosensitive surface of a PSD. By reading the geometric
centroid of the light spot and using a signal processing unit to convert this into a voltage
signal, the data are transmied to an A/D acquisition chip for further processing. The
measurement principle is illustrated in Figure 3.
Figure 3. Schematic of main shaft displacement measurement principle.
The solid line represents the position of the wind turbine main shaft before displace-
ment. The laser strikes point A on the main shaft surface, is reflected by the mirror, and
lands at point B on the PSD photosensitive surface.
The dashed line 1 represents the position after axial displacement, where the laser
hits point A1 on the main shaft surface, is reflected by the mirror, and lands at point B1 on
the PSD surface. The displacement between points C and C1, denoted as 1
x
, represents
the axial displacement of the main shaft, while the displacement between points B and B1,
denoted as
P
SD
x
, represents the displacement of the light spot on the PSD surface.
The dashed line 2 represents the position after radial displacement, where the laser
strikes point D on the main shaft surface, is reflected by the mirror, and lands at point B2
on the PSD surface. The displacement between points C and C2, denoted as 1
y
, represents
the radial displacement of the main shaft, while the displacement between points B and
B2, denoted as
P
SD
y
, represents the displacement of the light spot on the PSD surface.
The angles 1
θ
, 2
θ
, 3
θ
and 4
θ
are defined as follows: 1
θ
is the angle between the
main shaft and the ground, 2
θ
is the angle between the reflector and the radial direction of
the main shaft, 3
θ
is the angle between the laser’s incident direction and the reflector, and
4
θ
is the angle between the laser’s incident direction and the surface normal of the main shaft.
From geometric relationships, it can be determined that 3
θ
= 2
θ
− 1
θ
and 4
θ
=
3
2
πθ
−. Based on the trigonometric relationship in Equation (1), the linear relationship
between the axial displacement 1
x
and the horizontal displacement of the light spot on
the PSD photosensitive surface
P
SD
x
can be derived, as shown in Equation (2).
Figure 3. Schematic of main shaft displacement measurement principle.
The solid line represents the position of the wind turbine main shaft before displace-
ment. The laser strikes point Aon the main shaft surface, is reflected by the mirror, and
lands at point Bon the PSD photosensitive surface.
The dashed line 1 represents the position after axial displacement, where the laser
hits point A
1
on the main shaft surface, is reflected by the mirror, and lands at point B
1
on
the PSD surface. The displacement between points Cand C
1
, denoted as
x1
, represents
the axial displacement of the main shaft, while the displacement between points Band B
1
,
denoted as xPSD, represents the displacement of the light spot on the PSD surface.
The dashed line 2 represents the position after radial displacement, where the laser
strikes point D on the main shaft surface, is reflected by the mirror, and lands at point B
2
on the PSD surface. The displacement between points Cand C
2
, denoted as
y1
, represents
the radial displacement of the main shaft, while the displacement between points Band B
2
,
denoted as yPSD , represents the displacement of the light spot on the PSD surface.
The angles
θ1
,
θ2
,
θ3
and
θ4
are defined as follows:
θ1
is the angle between the main
shaft and the ground,
θ2
is the angle between the reflector and the radial direction of the
main shaft,
θ3
is the angle between the laser’s incident direction and the reflector, and
θ4
is
the angle between the laser’s incident direction and the surface normal of the main shaft.
From geometric relationships, it can be determined that
θ3
=
θ2−θ1
and
θ4
=
π−
2
θ3
.
Based on the trigonometric relationship in Equation (1), the linear relationship between
the axial displacement
x1
and the horizontal displacement of the light spot on the PSD
photosensitive surface xPSD can be derived, as shown in Equation (2).





x2=x1
tan θ2
x3=x2sin θ2
sin(θ2−θ1)
xPSD =x3tan(2θ2−2θ1)
(1)
which can be simplified to
x1=−xPSD
sin(θ2−θ1)
cos θ2tan(2θ2−2θ1)(2)
Similarly, based on the trigonometric relationship in Equation (3), the linear relation-
ship between the radial displacement
y1
, and the horizontal displacement of the light spot
on the PSD photosensitive surface, yPSD can be derived, as shown in Equation (4).





y2=y1
y3=y2sin θ2
sin(θ2−θ1)
yPSD =y3tan(2θ2−2θ1)
(3)

Electronics 2024,13, 5055 6 of 20
which can be simplified to:
y1=−yPSD
sin(θ2−θ1)
sin θ2tan(2θ2−2θ1)(4)
In these equations,
x1
represents the axial displacement of the main shaft,
y1
represents
the radial displacement of the main shaft,
θ1
is the angle between the main shaft and the
ground,
θ2
is the angle between the reflector and the radial direction of the main shaft,
xPSD
is the horizontal displacement of the light spot on the PSD for axial displacement, and
yPSD
is the horizontal displacement of the light spot on the PSD for radial displacement.
After simplification, it can be concluded that the axial and radial displacement of the
wind turbine main shaft is linearly related to the displacement of the light spot on the PSD
photosensitive surface, and this relationship only depends on the angle between the main
shaft and the ground
θ1
and the angle between the reflector and the radial direction of the
main shaft θ2.
2.2. Overall System Structure
The wind turbine main shaft displacement measurement system consists of a 2D
PSD, PSD signal processing circuit, main controller (STM32F429), and high-speed A/D
conversion chip (AD7606). The overall structure is shown in Figure 4The 2D PSD receives
the light spot reflected from the mirror and converts it into a current signal. This signal is
processed by the PSD signal processing circuit and converted into a voltage signal suitable
for processing by the AD7606. The main controller, STM32F429, is responsible for displace-
ment calculations and determining the main shaft’s displacement after processing the data.
The system uses USB communication combined with RL-USB and USB-Host protocols,
allowing the STM32’s Micro-USB interface to connect to a Micro-SD card for storing main
shaft displacement data. The system utilizes RL-TCPnet to upload the collected displace-
ment data to a remote computer. The remote computer, using the FlashFxp 5.1.0 client,
retrieves and manages the displacement data from the Micro-SD card over long distances.
Electronics 2025, 14, x FOR PEER REVIEW 7 of 22
Figure 4. Structure of the wind turbine main shaft displacement measurement system.
3. Factors Affecting PSD Measurement Error
During the operation of a wind turbine, various environmental factors can lead to
measurement errors, especially when measuring main shaft displacement. First, errors in
the signal processing unit may arise from factors such as circuit noise and stray light in-
terference, which are particularly significant during high-speed operation or under com-
plex operating conditions. Vibration or electromagnetic interference from the turbine’s
operation can affect the sensor signals, leading to measurement inaccuracies. Secondly,
the nonlinear characteristics of the PSD sensor can also contribute to measurement errors.
Due to material inhomogeneity or edge effects, the sensor’s response may not be entirely
linear, which is especially noticeable during turbine operation, particularly with low dis-
placements or high-frequency variations, potentially leading to cumulative errors. Finally,
errors related to testing conditions and sensor assembly are common sources of error dur-
ing turbine operation. Factors such as the sensor’s installation position, orientation, and
stability can be influenced by turbine vibrations, temperature changes, and the installation
environment, all of which can cause the sensor’s output to become unstable, thereby af-
fecting the accuracy of displacement measurements [26]. Therefore, these environmental
factors collectively influence the measurement of main shaft displacement during turbine
operation, potentially leading to certain measurement errors.
The output electrodes of a two-dimensional PSD have a specific size and shape, and
their dimensions are not perfectly geometrically symmetrical. As a result, this can lead to
nonlinear output errors in the PSD measurements. To address this, the PSD photosensitive
surface is typically divided into two zones, A and B, as shown in Figure 5. The positional
accuracy in zone A is higher than in zone B. Typically, zone B exhibits larger nonlinear
errors, which can reach up to 3% in some cases, while the typical nonlinear error in zone
A usually ranges from 0.1% to 2%. To obtain more accurate measurements for the main
shaft displacement momentum, and to make the measurement data in Area B more accu-
rate, it is necessary to correct the PSD measurement values.
The nonlinear errors of the PSD are much larger than those caused by circuit errors.
These nonlinear errors are primarily due to the non-uniformity of the resistivity across the
PSD photosensitive surface. The larger the area of the photosensitive surface with non-
uniform resistivity distribution, the higher the nonlinear error in the measurement data.
This nonlinear error worsens as the non-uniformity of the PSD surface resistivity in-
creases, and its error characteristics cannot be corrected through hardware compensation.
Furthermore, complex environmental factors during turbine operation, such as strong vi-
brations, temperature fluctuations, and electromagnetic interference, can also affect the
performance of the sensor, further exacerbating the non-compensable nature of the non-
linear errors. For example, during high-speed operation of the turbine main shaft, small
Figure 4. Structure of the wind turbine main shaft displacement measurement system.
3. Factors Affecting PSD Measurement Error
During the operation of a wind turbine, various environmental factors can lead to
measurement errors, especially when measuring main shaft displacement. First, errors
in the signal processing unit may arise from factors such as circuit noise and stray light
interference, which are particularly significant during high-speed operation or under
complex operating conditions. Vibration or electromagnetic interference from the turbine’s
operation can affect the sensor signals, leading to measurement inaccuracies. Secondly, the
nonlinear characteristics of the PSD sensor can also contribute to measurement errors. Due
to material inhomogeneity or edge effects, the sensor’s response may not be entirely linear,
which is especially noticeable during turbine operation, particularly with low displacements
or high-frequency variations, potentially leading to cumulative errors. Finally, errors related

Electronics 2024,13, 5055 7 of 20
to testing conditions and sensor assembly are common sources of error during turbine
operation. Factors such as the sensor’s installation position, orientation, and stability can be
influenced by turbine vibrations, temperature changes, and the installation environment, all
of which can cause the sensor’s output to become unstable, thereby affecting the accuracy
of displacement measurements [
26
]. Therefore, these environmental factors collectively
influence the measurement of main shaft displacement during turbine operation, potentially
leading to certain measurement errors.
The output electrodes of a two-dimensional PSD have a specific size and shape, and
their dimensions are not perfectly geometrically symmetrical. As a result, this can lead to
nonlinear output errors in the PSD measurements. To address this, the PSD photosensitive
surface is typically divided into two zones, A and B, as shown in Figure 5. The positional
accuracy in zone A is higher than in zone B. Typically, zone B exhibits larger nonlinear
errors, which can reach up to 3% in some cases, while the typical nonlinear error in zone A
usually ranges from 0.1% to 2%. To obtain more accurate measurements for the main shaft
displacement momentum, and to make the measurement data in Area B more accurate, it
is necessary to correct the PSD measurement values.
Electronics 2025, 14, x FOR PEER REVIEW 8 of 22
vibrations or temperature fluctuations may occur, leading to instability in the PSD sensor
output and increasing the sources of error.
Figure 5. Photodetector surface distribution diagram of PSD.
To improve the output performance and measurement accuracy of the PSD over a
wide range, this paper utilizes an SSA-BP neural network to correct the measurement er-
rors of the PSD. This approach helps mitigate the measurement errors caused by the wind
turbine’s operating environment and sensor nonlinearity to some extent.
4. SSA-BP Neural Network Error Correction Method
With the development of computer network technology and machine learning meth-
ods, a large number of scientific approaches with strong nonlinear processing capabilities
and real-time learning abilities have emerged. Compared to traditional methods such as
table lookup and least squares, BP neural networks offer superior adaptability and nonlin-
ear modeling capabilities. The table lookup method relies on pre-established tables, making
it difficult to handle complex input–output relationships, and updates to the table require
manual intervention. Although the least squares method can fit linear models, it performs
poorly when dealing with nonlinear data. In contrast, BP neural networks can adaptively
learn to address complex nonlinear problems, without relying on predefined tables or linear
assumptions [27–29]. This enables more accurate predictions and error correction, making
BP neural networks especially suitable for large-scale data or dynamic systems.
4.1. BP Neural Network
The BP neural network is a classic multilayer feedforward neural network model,
consisting of an input layer, hidden layer(s), and output layer. Neurons in each layer are
only connected to neurons in the adjacent layers, with no connections between neurons in
the same layer [30,31]. As shown in Figure 6, the BP neural network structure includes
two main stages: forward signal propagation and backward error propagation [32]. The
design of a BP neural network involves several key steps: preprocessing sample data, de-
termining the number of hidden layers, selecting the number of neurons, and seing neu-
ron thresholds and weights.
To accelerate the network’s convergence, the training sample data for PSD are nor-
malized. This process maps the theoretical calculated position data from PSD and the ac-
tual main shaft displacement data to a range between 0 and 1. The network’s output data
must then be denormalized to restore them to their original scale. The normalization for-
mula is shown in Equation (5).
Figure 5. Photodetector surface distribution diagram of PSD.
The nonlinear errors of the PSD are much larger than those caused by circuit errors.
These nonlinear errors are primarily due to the non-uniformity of the resistivity across
the PSD photosensitive surface. The larger the area of the photosensitive surface with
non-uniform resistivity distribution, the higher the nonlinear error in the measurement
data. This nonlinear error worsens as the non-uniformity of the PSD surface resistivity
increases, and its error characteristics cannot be corrected through hardware compensation.
Furthermore, complex environmental factors during turbine operation, such as strong
vibrations, temperature fluctuations, and electromagnetic interference, can also affect
the performance of the sensor, further exacerbating the non-compensable nature of the
nonlinear errors. For example, during high-speed operation of the turbine main shaft, small
vibrations or temperature fluctuations may occur, leading to instability in the PSD sensor
output and increasing the sources of error.
To improve the output performance and measurement accuracy of the PSD over a
wide range, this paper utilizes an SSA-BP neural network to correct the measurement errors
of the PSD. This approach helps mitigate the measurement errors caused by the wind
turbine’s operating environment and sensor nonlinearity to some extent.
4. SSA-BP Neural Network Error Correction Method
With the development of computer network technology and machine learning meth-
ods, a large number of scientific approaches with strong nonlinear processing capabilities
and real-time learning abilities have emerged. Compared to traditional methods such as ta-
ble lookup and least squares, BP neural networks offer superior adaptability and nonlinear
modeling capabilities. The table lookup method relies on pre-established tables, making it

Electronics 2024,13, 5055 8 of 20
difficult to handle complex input–output relationships, and updates to the table require
manual intervention. Although the least squares method can fit linear models, it performs
poorly when dealing with nonlinear data. In contrast, BP neural networks can adaptively
learn to address complex nonlinear problems, without relying on predefined tables or linear
assumptions [
27
–
29
]. This enables more accurate predictions and error correction, making
BP neural networks especially suitable for large-scale data or dynamic systems.
4.1. BP Neural Network
The BP neural network is a classic multilayer feedforward neural network model,
consisting of an input layer, hidden layer(s), and output layer. Neurons in each layer are
only connected to neurons in the adjacent layers, with no connections between neurons
in the same layer [
30
,
31
]. As shown in Figure 6, the BP neural network structure includes
two main stages: forward signal propagation and backward error propagation [
32
]. The
design of a BP neural network involves several key steps: preprocessing sample data,
determining the number of hidden layers, selecting the number of neurons, and setting
neuron thresholds and weights.
Electronics 2025, 14, x FOR PEER REVIEW 9 of 22
min
max min
k
k
XX
XXX
−
=− (5)
In the formula, k
X represents the normalized input data for training, while max
X
and min
X are the maximum and minimum values of the input training data, respectively.
Figure 6. BP neural network structure.
The number of hidden layers is critical to the accuracy of the model’s training. Too
few hidden layers may lead to insufficient system accuracy, while too many can cause
instability in the error gradient, making the network more prone to geing stuck in local
minima. In this system, a single hidden layer structure is adopted.
The number of neurons in the network includes those in the input layer, hidden
layer(s), and output layer. The number of neurons in the hidden layer significantly im-
pacts the predictive accuracy of the neural network [33,34]. An empirical formula for esti-
mating the number of hidden layer neurons is provided in Equation (6).
LMNA=++
(6)
In the formula, L represents the number of hidden layer neurons, M is the number of
input layer neurons, N is the number of output layer neurons, and A is a constant ranging
between 0 and 10.
The learning rate is a coefficient that determines the adjustment of weights in a BP
neural network, and it directly influences the network’s convergence ability and speed. In
a typical BP neural network, the thresholds and weights of the neurons are initialized
randomly. These random values can affect the recognition accuracy and operating speed,
leading to poor optimization performance after training, slower convergence, and poten-
tially limiting the generalization ability of the model [35,36].
To mitigate the influence of random thresholds and weights during the optimization
process, this paper proposes combining the Sparrow Search Algorithm (SSA) [37] with the
BP neural network to optimize the network’s thresholds and weights. The Sparrow Search
Algorithm (SSA), introduced in 2020, is a novel bio-inspired optimization algorithm based
on the foraging and anti-predation behaviors of sparrows. It effectively addresses issues
such as poor optimization performance and slow convergence speed.
4.2. Sparrow Search Algorithm (SSA)
The SSA is a novel bio-inspired optimization algorithm introduced in 2020, based on the
foraging and anti-predation behavior of sparrows. It effectively addresses issues such as in-
sufficient optimization capability and slow convergence speed. Compared to other
population-based intelligent algorithms, such as Particle Swarm Optimization (PSO). SSA is a
novel population-based optimization algorithm that outperforms traditional algorithms [38].
The foraging behavior of sparrows can be categorized into two main types: predation
and anti-predation. Predatory behavior consists of two roles—producers and scroungers.
Figure 6. BP neural network structure.
To accelerate the network’s convergence, the training sample data for PSD are normal-
ized. This process maps the theoretical calculated position data from PSD and the actual
main shaft displacement data to a range between 0 and 1. The network’s output data must
then be denormalized to restore them to their original scale. The normalization formula is
shown in Equation (5).
Xk=Xk−Xmin
Xmax −Xmin
(5)
In the formula,
Xk
represents the normalized input data for training, while
Xmax
and
Xmin are the maximum and minimum values of the input training data, respectively.
The number of hidden layers is critical to the accuracy of the model’s training. Too
few hidden layers may lead to insufficient system accuracy, while too many can cause
instability in the error gradient, making the network more prone to getting stuck in local
minima. In this system, a single hidden layer structure is adopted.
The number of neurons in the network includes those in the input layer, hidden
layer(s), and output layer. The number of neurons in the hidden layer significantly impacts
the predictive accuracy of the neural network [
33
,
34
]. An empirical formula for estimating
the number of hidden layer neurons is provided in Equation (6).
L=√M+N+A(6)
In the formula, Lrepresents the number of hidden layer neurons, Mis the number of
input layer neurons, Nis the number of output layer neurons, and Ais a constant ranging
between 0 and 10.
The learning rate is a coefficient that determines the adjustment of weights in a BP
neural network, and it directly influences the network’s convergence ability and speed.
In a typical BP neural network, the thresholds and weights of the neurons are initial-
ized randomly. These random values can affect the recognition accuracy and operating

Electronics 2024,13, 5055 9 of 20
speed, leading to poor optimization performance after training, slower convergence, and
potentially limiting the generalization ability of the model [35,36].
To mitigate the influence of random thresholds and weights during the optimization
process, this paper proposes combining the Sparrow Search Algorithm (SSA) [
37
] with the
BP neural network to optimize the network’s thresholds and weights. The Sparrow Search
Algorithm (SSA), introduced in 2020, is a novel bio-inspired optimization algorithm based
on the foraging and anti-predation behaviors of sparrows. It effectively addresses issues
such as poor optimization performance and slow convergence speed.
4.2. Sparrow Search Algorithm (SSA)
The SSA is a novel bio-inspired optimization algorithm introduced in 2020, based
on the foraging and anti-predation behavior of sparrows. It effectively addresses issues
such as insufficient optimization capability and slow convergence speed. Compared to
other population-based intelligent algorithms, such as Particle Swarm Optimization (PSO).
SSA is a novel population-based optimization algorithm that outperforms traditional
algorithms [38].
The foraging behavior of sparrows can be categorized into two main types: predation
and anti-predation. Predatory behavior consists of two roles—producers and scroungers.
Producers possess more resources and guide the population’s search by setting the direction,
while scroungers follow the producers, aiding in the collective foraging effort. As they
follow, scroungers also increase their own predation rate and, in some cases, monitor the
producers to compete for food or forage nearby. The roles of producers and scroungers are
dynamic, with scroungers who accumulate more resources transitioning into producers.
However, the overall ratio of producers to scroungers remains constant across iterations.
Anti-predation behavior is driven by early-warning agents that alert the population to
the presence of a predator or any potential danger. When the warning signal exceeds the
safe threshold, the producer leads the group to migrate to safer areas. Sparrows at the
edge of the group move faster to the safe zones, while those in the middle of the group
move randomly to stay with the population. The corresponding mathematical model is
as follows:
Producer location updates:
Xt+1
i,j=(Xt
i,j·exp(−i
α·itemmax )i f R2<ST
Xt
i,j+Q·L i f R2≥ST (7)
Let
Xt+1
i,j
represent the position of the ith sparrow in the jth dimension at the tth
iteration.
T
is the maximum number of iterations, while
Q
is a random number drawn
from a standard normal distribution.
α
is a uniformly distributed random number in the
range (0, 1], and
L
is a matrix of size 1
×
d, with all elements equal to 1.
R2
is the warning
value, and when this value is reached, it indicates that the sparrow population is facing
danger and must take protective actions (
R2∈
[0, 1]). On the other hand,
ST
represents
the safety threshold, which defines the condition under which the sparrow population can
continue its movement without threat (ST ∈[0.5, 1]).
Xt+1
i,j=






Q·exp(Xworst−Xt
i,j
i2)i f i >n/2
Xt+1
p+Xt
i,j−Xt+1
p·A∗·Lotherwise
(8)
Let
Xt+1
p
denote the position of the best producer at the (t + 1)th iteration, and
Xt
worst
represent the global worst position at the tth iteration.
Q
is a random number sampled
from a standard normal distribution. A is a 1
×
d matrix where each element is randomly
assigned either 1 or −1, and A+=AT(AAT)−1.

Electronics 2024,13, 5055 10 of 20
Early-warning agent location updates:
Xt+1
i,j=




Xt
best +β·Xt
i,j−Xt
besti f fi>fg
Xt
i,j+K·(Xt
i,j−Xt
worst
(fi−fw)+ε)i f f i=fg
(9)
Let
Xbest
denote the current global optimal position, which is a random value following
a normal distribution. Krepresents the movement direction and step control parameters for
the sparrow, with each element of
K
being (
−
1, 1)
fi
is the adaptation value of the current
individual sparrow,
fg
is the adaptation value of the current global optimal position, and
fw
is the adaptation value of the current global worst position. A small constant is included
to prevent division by zero.
4.3. SSA-BP Neural Network
First, the data undergo preprocessing, including denoising, normalization, and stan-
dardization, to ensure that the input data to the neural network falls within an appropriate
range. The purpose of data preprocessing is to enhance the stability and accuracy of model
training. In the optimization of the BP neural network, the “position” of each sparrow
individual is considered as the weights and thresholds of the neural network.
Initially, the sparrow population is randomly initialized, with each sparrow’s posi-
tion representing a candidate solution for the neural network (i.e., a set of weights and
thresholds). In each iteration, the BP neural network is trained using the current sparrow’s
weights and thresholds, and the error between the network’s output and the target value
is calculated, serving as the sparrow’s fitness. Based on the update rules of the Sparrow
Search Algorithm (SSA), the position of each sparrow is updated, which in turn updates
the weights and thresholds of the BP neural network. The updated weights and thresholds
are then used to train the BP neural network, with the parameters adjusted through the
backpropagation algorithm. Unlike traditional BP training methods, the training process
here is driven by the initial solutions and parameter updates provided by the Sparrow
Search Algorithm.
This process is repeated, executing the SSA search process until a stopping criterion is
met (such as the maximum number of iterations or achieving a certain fitness threshold).
After several iterations, the SSA outputs the optimal weights and thresholds, and the
corresponding BP neural network becomes the optimal model. The flowchart for SSA-BP
neural network-based main shaft displacement prediction is shown in Figure 7.
Electronics 2025, 14, x FOR PEER REVIEW 11 of 22
4.3. SSA-BP Neural Network
First, the data undergo preprocessing, including denoising, normalization, and
standardization, to ensure that the input data to the neural network falls within an appro-
priate range. The purpose of data preprocessing is to enhance the stability and accuracy
of model training. In the optimization of the BP neural network, the “position” of each
sparrow individual is considered as the weights and thresholds of the neural network.
Initially, the sparrow population is randomly initialized, with each sparrow’s position
representing a candidate solution for the neural network (i.e., a set of weights and thresholds).
In each iteration, the BP neural network is trained using the current sparrow’s weights and
thresholds, and the error between the network’s output and the target value is calculated,
serving as the sparrow’s fitness. Based on the update rules of the Sparrow Search Algorithm
(SSA), the position of each sparrow is updated, which in turn updates the weights and thresh-
olds of the BP neural network. The updated weights and thresholds are then used to train the
BP neural network, with the parameters adjusted through the backpropagation algorithm.
Unlike traditional BP training methods, the training process here is driven by the initial solu-
tions and parameter updates provided by the Sparrow Search Algorithm.
This process is repeated, executing the SSA search process until a stopping criterion
is met (such as the maximum number of iterations or achieving a certain fitness threshold).
After several iterations, the SSA outputs the optimal weights and thresholds, and the cor-
responding BP neural network becomes the optimal model. The flowchart for SSA-BP
neural network-based main shaft displacement prediction is shown in Figure 7.
Figure 7. SSA-BP neural network flowchart.
4.4. Algorithm Validation and Analysis
In intelligent optimization algorithms, the fitness function plays a crucial role. Neural
networks use the mean squared error (
M
SE ), mean absolute error (
M
AE ), and coefficient
of determination ( 2
R) as model evaluation metrics. Smaller
M
SE and
M
AE values in-
dicate a smaller gap between the predicted and actual values, reflecting beer model per-
formance. After each iteration, the
M
SE and
M
AE values of the neural network are cal-
culated, and the parameters of the neural network are adjusted accordingly to continu-
ously optimize the model’s performance. The coefficient of determination ( 2
R) reflects the
model’s goodness of fit, with values closer to 1 indicating a higher level of fit.
2
1
1n
ii
n
M
SE y y
n
∧
=

=−


 (10)
Figure 7. SSA-BP neural network flowchart.

Electronics 2024,13, 5055 11 of 20
4.4. Algorithm Validation and Analysis
In intelligent optimization algorithms, the fitness function plays a crucial role. Neural
networks use the mean squared error (
MSE
), mean absolute error (
MAE
), and coefficient
of determination (
R2
) as model evaluation metrics. Smaller
MSE
and
MAE
values indicate
a smaller gap between the predicted and actual values, reflecting better model performance.
After each iteration, the
MSE
and
MAE
values of the neural network are calculated, and
the parameters of the neural network are adjusted accordingly to continuously optimize the
model’s performance. The coefficient of determination (
R2
) reflects the model’s goodness
of fit, with values closer to 1 indicating a higher level of fit.
MSE =1
n
n
∑
n=1yi−∧
yi2
(10)
MAE =1
n
n
∑
i=1yi−∧
yi(11)
R2=
n
∑
n=1
(yi−y)2
n
∑
n=1
(∧
yi−y)2(12)
In this formula, nrepresents the number of test samples;
yi
denotes the predicted
values from the model,
∧
yi
represents the actual values, and
y
is the mean of the actual values.
To verify the reliability and practicality of the SSA-BP algorithm, a dataset of 90 training
samples and 10 testing samples was used for the SSA-BP neural network. The Sparrow
Search Algorithm was employed to optimize the initial thresholds and weights of the model.
The optimization algorithm uses the mean squared error of both the training and testing
sets as the fitness value for SSA, with the fitness function defined as shown in Equation (8).
f itness =argmin(MSETrain +MSETest )(13)
In this formula,
MSETrain
and
MSETest
represent the predicted mean squared error for
the training and testing sets, respectively.
To verify the reliability and high performance of the SSA-BP algorithm, the BP neural
network, PSO-BP neural network, and SSA-BP neural network were used to train sample
sets for both axial and radial main shaft displacement. Due to the different PSD photosensi-
tive surface measurement ranges for axial and radial displacement, separate training was
conducted for the respective sample sets. A total of 90 training samples and 10 test samples
were used as the neural network dataset. MATLAB 2021was utilized for programming
the experiments.
First, training was performed for the axial displacement. As shown in Figure 8a, the
conventional BP neural network required approximately 80 iterations to reach a minimum
mean squared error (
MSE
) of 0.0347 mm
2
. The PSO-BP neural network required about
62 iterations to achieve an
MSE
of 0.0156 mm
2
, while the SSA-BP neural network achieved
a minimum MSE of 0.0123 mm2in approximately 51 iterations.
Next, training was conducted for the radial displacement. As shown in Figure 8b, the
conventional BP neural network required approximately 78 iterations to reach a minimum
MSE
of 0.0212 mm
2
. The PSO-BP neural network required about 70 iterations to achieve
an
MSE
of 0.0195 mm
2
, whereas the SSA-BP neural network achieved a minimum
MSE
of
0.0086 mm2in around 53 iterations.
As shown in Table 1, the accuracy of the SSA-BP model significantly surpasses that of
the BP and PSO-BP models. The
MSE
and
MAE
of the SSA-BP model are much closer to 0,
while the R2value is closer to 1 compared to the other two models.

Electronics 2024,13, 5055 12 of 20
Electronics 2025, 14, x FOR PEER REVIEW 12 of 22
1
1n
ii
i
M
AE y y
n
∧
=
=−
 (11)
2
21
2
1
()
()
n
i
n
n
i
n
yy
R
yy
=
∧
=
−
=
−

 (12)
In this formula, n represents the number of test samples; i
y
denotes the predicted val-
ues from the model, i
y
∧
represents the actual values, and y is the mean of the actual values.
To verify the reliability and practicality of the SSA-BP algorithm, a dataset of 90 training
samples and 10 testing samples was used for the SSA-BP neural network. The Sparrow Search
Algorithm was employed to optimize the initial thresholds and weights of the model. The
optimization algorithm uses the mean squared error of both the training and testing sets as
the fitness value for SSA, with the fitness function defined as shown in Equation (8).
argmin( )
Train Test
fitness MSE MSE=+
(13)
In this formula, Train
MSE and Test
M
SE represent the predicted mean squared error
for the training and testing sets, respectively.
To verify the reliability and high performance of the SSA-BP algorithm, the BP neural
network, PSO-BP neural network, and SSA-BP neural network were used to train sample
sets for both axial and radial main shaft displacement. Due to the different PSD photosen-
sitive surface measurement ranges for axial and radial displacement, separate training
was conducted for the respective sample sets. A total of 90 training samples and 10 test
samples were used as the neural network dataset. MATLAB 2021was utilized for pro-
gramming the experiments.
First, training was performed for the axial displacement. As shown in Figure 8a, the
conventional BP neural network required approximately 80 iterations to reach a minimum
mean squared error (
M
SE ) of 0.0347 mm2. The PSO-BP neural network required about 62
iterations to achieve an
M
SE of 0.0156 mm2, while the SSA-BP neural network achieved
a minimum
M
SE of 0.0123 mm2 in approximately 51 iterations.
Next, training was conducted for the radial displacement. As shown in Figure 8b, the
conventional BP neural network required approximately 78 iterations to reach a minimum
M
SE of 0.0212 mm2. The PSO-BP neural network required about 70 iterations to achieve
an
M
SE of 0.0195 mm2, whereas the SSA-BP neural network achieved a minimum
M
SE
of 0.0086 mm2 in around 53 iterations.
Figure 8. Neural network iteration: (a) training results of neural network for axial displacement; (b)
training results of neural network for radial displacement.
Figure 8. Neural network iteration: (a) training results of neural network for axial displacement;
(b) training results of neural network for radial displacement.
Table 1. Comparison of the evaluation index of each model.
Model MSE MAE R2
BP 0.0347 0.1863 0.96485
PSO-BP 0.0156 0.1249 0.98345
SSA-BP 0.0123 0.1109 0.99975
BP 0.0212 0.1456 0.96357
PSO-BP 0.0195 0.1396 0.98454
SSA-BP 0.0086 0.0927 0.99987
From the training results for both models, it is evident that the SSA-BP network
converges faster, offers higher measurement accuracy, and demonstrates superior correction
capability compared to the BP neural network.
After multiple experiments, the optimized weights and thresholds from SSA training
were integrated into the model to predict the training samples. For the SSA-BP model of
axial displacement, the number of hidden layer nodes was set to 7, with a learning rate
of 0.001. The SSA parameters included an initial population size of 30, 51 iterations, a
producer ratio of 0.3, and a scrounger ratio of 0.7. Similarly, for the SSA-BP model of radial
displacement, the hidden layer nodes were set to 7, the learning rate was 0.001, and the
initial population size was 30; 53 iterations were performed, with a producer ratio of 0.3
and a scrounger ratio of 0.7.
5. Experiment Design and Analysis
5.1. Experimental Model Design and Construction
Due to the inability to directly access the top of wind turbine towers, for actual
measurements, a laboratory model was used to simulate the main shaft of a wind turbine.
The overall structure of the PSD-based main shaft displacement monitoring platform
designed in this paper is shown in Figure 9. The structure consists of a simulated wind
turbine main shaft model, a laser, a mirror, a PSD measuring box, and a motion controller
(WNSC6000). The PSD, STM32, and AD7606 are all integrated within the measurement
box, and the collected displacement data of the main shaft are displayed on a remote
computer. The motion controller WNSC6000 controls the electric displacement stages to
move in specified directions and displacement values. The experimental model simulating
the wind turbine main shaft consists of two angular displacement stages and two electric
displacement stages.
A precision displacement stage was employed to simulate the displacement of the
wind turbine main shaft. The precision displacement stage provides high-accuracy control,
with axial and radial adjustment precision reaching the millimeter level or better, offering
a reliable method for replicating the actual displacement of the main shaft. During the
simulation process, the displacement parameters of the precision stage were configured
based on an analysis of the displacement range of wind turbine main shafts under various
operating conditions. The motion patterns of the displacement stage can be flexibly adjusted
to mimic the displacement characteristics of the main shaft during different phases, such as

Electronics 2024,13, 5055 13 of 20
startup, steady operation, variable-speed operation, and shutdown. For instance, during
the simulation of the startup phase, the displacement stage can be programmed to gradually
accelerate its movement, replicating the transition of a real main shaft from rest to its rated
speed, thereby effectively reproducing its motion characteristics.
Electronics 2025, 14, x FOR PEER REVIEW 14 of 22
Figure 9. Experimental platform structure.
However, it is important to note that laboratory simulations cannot fully account for
the uncertainties encountered by real wind turbines operating in outdoor environments.
Extreme weather conditions, such as heavy rain, lightning, and sandstorms, may signifi-
cantly impact the structural integrity and lubrication state of the main shaft, thereby al-
tering its displacement characteristics. Additionally, due to experimental limitations, cer-
tain adverse environmental factors—such as wind and humidity—were not considered in
this study, although these factors are critical in real-world wind turbine operations. For
example, strong winds can exert additional lateral forces on the main shaft, causing vari-
ations in displacement, while high humidity may affect the electrical performance of sen-
sors, reducing measurement accuracy.
While these environmental factors have relatively minor effects on displacement
measurements during actual operation—since wind turbines are typically designed with
environmental adaptability in mind and are equipped with measures to mitigate extreme
conditions—components such as the main shaft and sensors are specially engineered and
protected to withstand interference from harsh environments. Nevertheless, these factors
should not be overlooked. Future research will aim to replicate more realistic operating
conditions in the laboratory. For example, controllable fans can be used to generate artifi-
cial wind fields that simulate lateral wind loads on the main shaft, while humidity control
systems can be employed to investigate the impact of high-humidity environments on
sensor performance. Careful consideration of these factors in future studies and design
efforts will ensure the reliability of the system and the accuracy of displacement measure-
ments under varying operational conditions.
Statistical analysis of a large amount of measured data shows that during normal
operation, over 98% of the axial displacement data fall within the ±10 mm range, while
over 97% of the radial displacement data lie within the ±5 mm range. When the axial dis-
placement exceeds ±10 mm or the radial displacement exceeds ±5 mm, the alignment be-
tween the main shaft and other key components may fail severely, leading to structural
instability. Therefore, the electric displacement stages are set to simulate an axial displace-
ment of 20 mm and a radial displacement of 10 mm. Electric Displacement Stage 1 simu-
lates the axial displacement of the main shaft. It is model WN26TA30H, manufactured by
Micro Nano Optics Instruments, Beijing, Chinawith a stroke of 30 mm and a resolution of
5 µm. Electric Displacement Stage 2 simulates the radial displacement and is model
WNKS330-10C, also from Beijing Micro-Nano Optics Instruments, with a stroke of 20 mm
and a resolution of 5 µm.
According to Equations (2) and (4), the axial and radial displacement momentum of
the wind turbine main shaft depend only on the angle θ
1
between the shaft and the
Figure 9. Experimental platform structure.
However, it is important to note that laboratory simulations cannot fully account
for the uncertainties encountered by real wind turbines operating in outdoor environ-
ments. Extreme weather conditions, such as heavy rain, lightning, and sandstorms, may
significantly impact the structural integrity and lubrication state of the main shaft, thereby
altering its displacement characteristics. Additionally, due to experimental limitations,
certain adverse environmental factors—such as wind and humidity—were not considered
in this study, although these factors are critical in real-world wind turbine operations.
For example, strong winds can exert additional lateral forces on the main shaft, causing
variations in displacement, while high humidity may affect the electrical performance of
sensors, reducing measurement accuracy.
While these environmental factors have relatively minor effects on displacement
measurements during actual operation—since wind turbines are typically designed with
environmental adaptability in mind and are equipped with measures to mitigate extreme
conditions—components such as the main shaft and sensors are specially engineered
and protected to withstand interference from harsh environments. Nevertheless, these
factors should not be overlooked. Future research will aim to replicate more realistic
operating conditions in the laboratory. For example, controllable fans can be used to
generate artificial wind fields that simulate lateral wind loads on the main shaft, while
humidity control systems can be employed to investigate the impact of high-humidity
environments on sensor performance. Careful consideration of these factors in future
studies and design efforts will ensure the reliability of the system and the accuracy of
displacement measurements under varying operational conditions.
Statistical analysis of a large amount of measured data shows that during normal
operation, over 98% of the axial displacement data fall within the
±
10 mm range, while
over 97% of the radial displacement data lie within the
±
5 mm range. When the axial
displacement exceeds ±10 mm or the radial displacement exceeds ±5 mm, the alignment
between the main shaft and other key components may fail severely, leading to struc-
tural instability. Therefore, the electric displacement stages are set to simulate an axial
displacement of 20 mm and a radial displacement of 10 mm. Electric Displacement Stage 1
simulates the axial displacement of the main shaft. It is model WN26TA30H, manufactured
by Micro Nano Optics Instruments, Beijing, Chinawith a stroke of 30 mm and a resolution
of 5
µ
m. Electric Displacement Stage 2 simulates the radial displacement and is model

Electronics 2024,13, 5055 14 of 20
WNKS330-10C, also from Beijing Micro-Nano Optics Instruments, with a stroke of 20 mm
and a resolution of 5 µm.
According to Equations (2) and (4), the axial and radial displacement momentum
of the wind turbine main shaft depend only on the angle
θ1
between the shaft and the
ground, and the radial angle
θ2
between the mirror and the main shaft. The two angular
displacement stages are responsible for simulating the angles
θ1
(between the shaft and
the ground) and
θ2
(between the mirror and the shaft). Both angular displacement stages,
model WN06GM30, have a range of ±20◦and a resolution of 1◦.
During the experiment, there may be several potential sources of error, including
deviations in the simulation model, inaccuracies in the sensors, and interference from the
motion controller. These factors could affect the accuracy and reliability of the experimental
results. To enhance the credibility of the research findings, it is essential to identify and
briefly discuss these sources of error. For example, deviations in the simulation model may
arise from simplifying assumptions or imprecise model parameters, which can be mitigated
by more accurate modeling and validation. Inaccuracies in the sensors may result from
limitations in sensor precision or changes in environmental conditions, and can be reduced
by using higher-precision PSD sensors or by performing calibration. Interference from
the motion controller may stem from delays in system response or instability in control
signals, which can be minimized by optimizing the control system design and improving
hardware stability.
5.2. System Stability Testing
To ensure the system’s resistance to interference and long-term stability, conducting
stability testing is essential. System stability testing is crucial for the wind turbine main
shaft displacement monitoring system, as even small positional errors in the main shaft can
significantly affect the overall performance of the system.
The statistical analysis of extensive operational data indicates that a 12 h period is
sufficient to capture the various stability characteristics that may arise under sustained
operational stress during a complete operational cycle. During this time, critical factors
such as the thermal stability of electronic components, the long-term consistency of the data
acquisition system, and the reliability of algorithms under prolonged operation can be ef-
fectively assessed. The choice of a 30 min measurement interval was made after a thorough
evaluation of the system’s dynamic response characteristics and data reliability. On the one
hand, the displacement of the wind turbine main shaft during normal operation typically
changes at a relatively slow rate (excluding sudden fault scenarios). Excessively frequent
measurements could introduce unnecessary noise, hindering an accurate assessment of the
system’s stability. On the other hand, a 30 min interval strikes a balance by allowing for the
timely detection of gradual changes over an extended period, such as displacement varia-
tions caused by structural thermal expansion due to changing environmental temperatures.
At the same time, it ensures efficient data processing and analysis, avoiding an excessive
data burden that could strain computational resources.
To evaluate the system’s stability, the experimental model was kept stationary, and
measurements were taken from the moment the system was started. Data were recorded
every 30 min for a continuous 12 h period. As shown in Figure 10, during the initial
phase of operation, measurement errors were relatively large due to the instability of the
laser intensity and uneven light spots. However, once the laser intensity stabilized, the
measurement data remained relatively consistent. The fluctuation range of the light spot
was approximately ±0.024 mm, which meets the system’s accuracy requirements.
As shown in Table 2, the precision displacement table was adjusted to different
positions, and stability tests were conducted again to measure the position drift at these
various points. The position drift remained within
±
0.024 mm across all tests, with no
significant increase observed at 1 mm and 17 mm near the edge of the photosensitive surface
of the PSD. Compared to the axial displacement range (
±
10 mm) and radial displacement
range (
±
5 mm) of the wind turbine main shaft during normal operation, this fluctuation

Electronics 2024,13, 5055 15 of 20
range is significantly smaller than the actual displacement range of the main shaft, far
exceeding the
±
0.05 mm to
±
0.08 mm of contact-based methods. This indicates that
the system’s measurement accuracy under laboratory conditions is far superior to the
displacement requirements of the main shaft during normal operation, demonstrating
its capability to accurately capture the displacement characteristics of the main shaft in
standard operating conditions.
Electronics 2025, 14, x FOR PEER REVIEW 16 of 22
exceeding the ±0.05 mm to ±0.08 mm of contact-based methods. This indicates that the
system’s measurement accuracy under laboratory conditions is far superior to the dis-
placement requirements of the main shaft during normal operation, demonstrating its ca-
pability to accurately capture the displacement characteristics of the main shaft in stand-
ard operating conditions.
Figure 10. System stability test.
Table 2. Position displacement for different positions.
Spot Position/mm ΔX/mm
1 ±0.024
5 ±0.020
9 ±0.019
13 ±0.021
17 ±0.025
5.3. Displacement Measurement Experiment
Absolute error, mean squared error (
M
SE ), and maximum nonlinearity error are cru-
cial parameters in monitoring the displacement of wind turbine main shafts. These metrics
not only provide a quantitative evaluation of measurement accuracy but also directly im-
pact the reliability of operational state assessments and maintenance decisions. Under-
standing and minimizing these errors ensure precise displacement measurements, which
are essential for maintaining the stability and efficiency of wind turbine systems.
Absolute error is the absolute value of the difference between the measured value
and the true value, directly reflecting the degree to which the measurement deviates from
reality. In wind turbine main shaft displacement measurements, smaller absolute errors
indicate that the measured values are closer to the true displacement of the shaft, which
is critical for accurately assessing the operating status of the main shaft. For instance, if
the absolute error is too large, it could lead to a misjudgment of whether the shaft is in its
normal operating position. In actual operations, if the measured absolute error of the axial
displacement of the main shaft reaches a certain level, maintenance personnel may mis-
takenly believe that the shaft has undergone severe misalignment, prompting unneces-
sary shutdowns and inspections. This would not only increase maintenance costs but also
reduce the operational efficiency of the wind turbine.
The mean squared error (
M
SE ) is the average of the squared differences between
measured and true values, reflecting both the magnitude and distribution of errors. In
wind turbine main shaft displacement measurements, a lower
M
SE indicates higher ac-
curacy and stability, making the results more reliable. Monitoring changes in
M
SE over
time can help evaluate the long-term performance of the measurement system. A gradual
increase in
M
SE may signal sensor degradation or environmental interference, requiring
Figure 10. System stability test.
Table 2. Position displacement for different positions.
Spot Position/mm ∆X/mm
1±0.024
5±0.020
9±0.019
13 ±0.021
17 ±0.025
5.3. Displacement Measurement Experiment
Absolute error, mean squared error (
MSE
), and maximum nonlinearity error are
crucial parameters in monitoring the displacement of wind turbine main shafts. These
metrics not only provide a quantitative evaluation of measurement accuracy but also
directly impact the reliability of operational state assessments and maintenance decisions.
Understanding and minimizing these errors ensure precise displacement measurements,
which are essential for maintaining the stability and efficiency of wind turbine systems.
Absolute error is the absolute value of the difference between the measured value
and the true value, directly reflecting the degree to which the measurement deviates from
reality. In wind turbine main shaft displacement measurements, smaller absolute errors
indicate that the measured values are closer to the true displacement of the shaft, which
is critical for accurately assessing the operating status of the main shaft. For instance,
if the absolute error is too large, it could lead to a misjudgment of whether the shaft is
in its normal operating position. In actual operations, if the measured absolute error of
the axial displacement of the main shaft reaches a certain level, maintenance personnel
may mistakenly believe that the shaft has undergone severe misalignment, prompting
unnecessary shutdowns and inspections. This would not only increase maintenance costs
but also reduce the operational efficiency of the wind turbine.
The mean squared error (
MSE
) is the average of the squared differences between
measured and true values, reflecting both the magnitude and distribution of errors. In wind
turbine main shaft displacement measurements, a lower
MSE
indicates higher accuracy
and stability, making the results more reliable. Monitoring changes in
MSE
over time can
help evaluate the long-term performance of the measurement system. A gradual increase in
MSE
may signal sensor degradation or environmental interference, requiring maintenance
or calibration.
MSE
is also useful for comparing different measurement methods; the

Electronics 2024,13, 5055 16 of 20
approach with the lowest
MSE
ensures optimal performance. In wind farm operations,
MSE
serves as a key indicator for assessing system performance. If
MSE
exceeds a set
threshold, corrective actions such as replacing sensors, optimizing the environment, or
improving algorithms should be taken to ensure accurate measurements and reliable
maintenance decisions.
Maximum nonlinearity error is the largest deviation of measured values from the
ideal linear response, indicating the system’s degree of nonlinearity. In wind turbine main
shaft displacement measurements, this error can result from sensor defects, environmental
factors, or system constraints. Significant nonlinearity error introduces systematic bias,
especially in large displacement measurements, affecting the accuracy of shaft position
or motion assessments and reducing reliability. For instance, during critical startup or
shutdown phases, high nonlinearity error may distort the shaft’s true motion trajectory.
Under extreme conditions, excessive nonlinearity error can amplify deviations, leading
to incorrect maintenance decisions. To address this, calibration techniques, nonlinear
compensation algorithms, or high-performance sensors should be used. Minimizing this
error ensures consistent measurement accuracy across the displacement range, which is
vital for real-time monitoring and long-term wind turbine performance evaluation.
To verify the feasibility of the SSA-BP algorithm for error correction, the main shaft
model was kept stationary while motorized displacement stages 1 and 2 were controlled
to move in 1 mm increments. As indicated by the statistical analysis of measured data
discussed earlier, most axial displacement data during normal operation falls within the
±
10 mm range, while radial displacement data are typically within the
±
5 mm range. To
simulate extreme conditions that exceed these normal ranges, an axial displacement of
20 mm and radial displacement of 10 mm were simulated, and the main shaft displacement
was measured using three approaches: without a neural network model, with the BP neural
network model, and with the SSA-BP neural network model trained earlier.
Figure 11 presents the measurement results of the main shaft displacement without
model correction. Figure 11a shows the displacement position comparison data, where
the maximum absolute error for axial displacement is 0.72 mm, with a mean squared error
(
MSE
) of 0.0787 mm
2
. For radial displacement, the maximum absolute error is 0.251 mm,
and the
MSE
is 0.0249 mm
2
. Figure 11b illustrates the position error distribution across the
measurement points. Calculations show that the maximum nonlinearity error rate for the
axial displacement measurement system, with a total range of 20 mm, is 3.6%, while the
radial displacement measurement system, with a total range of 10 mm, exhibits a maximum
nonlinearity error rate of 2.5%. These results indicate that the system error is relatively
large without model correction, making it insufficient for high-precision measurement
requirements, and highlighting the necessity of error correction.
Electronics 2025, 14, x FOR PEER REVIEW 17 of 22
maintenance or calibration.
M
SE is also useful for comparing different measurement
methods; the approach with the lowest
M
SE ensures optimal performance. In wind farm
operations,
M
SE serves as a key indicator for assessing system performance. If
M
SE ex-
ceeds a set threshold, corrective actions such as replacing sensors, optimizing the environ-
ment, or improving algorithms should be taken to ensure accurate measurements and re-
liable maintenance decisions.
Maximum nonlinearity error is the largest deviation of measured values from the
ideal linear response, indicating the system’s degree of nonlinearity. In wind turbine main
shaft displacement measurements, this error can result from sensor defects, environmen-
tal factors, or system constraints. Significant nonlinearity error introduces systematic bias,
especially in large displacement measurements, affecting the accuracy of shaft position or
motion assessments and reducing reliability. For instance, during critical startup or shut-
down phases, high nonlinearity error may distort the shaft’s true motion trajectory. Under
extreme conditions, excessive nonlinearity error can amplify deviations, leading to incor-
rect maintenance decisions. To address this, calibration techniques, nonlinear compensa-
tion algorithms, or high-performance sensors should be used. Minimizing this error en-
sures consistent measurement accuracy across the displacement range, which is vital for
real-time monitoring and long-term wind turbine performance evaluation.
To verify the feasibility of the SSA-BP algorithm for error correction, the main shaft
model was kept stationary while motorized displacement stages 1 and 2 were controlled
to move in 1 mm increments. As indicated by the statistical analysis of measured data
discussed earlier, most axial displacement data during normal operation falls within the
±10 mm range, while radial displacement data are typically within the ±5 mm range. To
simulate extreme conditions that exceed these normal ranges, an axial displacement of 20
mm and radial displacement of 10 mm were simulated, and the main shaft displacement
was measured using three approaches: without a neural network model, with the BP neu-
ral network model, and with the SSA-BP neural network model trained earlier.
Figure 11 presents the measurement results of the main shaft displacement without
model correction. Figure 11a shows the displacement position comparison data, where
the maximum absolute error for axial displacement is 0.72 mm, with a mean squared error
(
M
SE ) of 0.0787 mm2. For radial displacement, the maximum absolute error is 0.251 mm,
and the
M
SE is 0.0249 mm2. Figure 11b illustrates the position error distribution across
the measurement points. Calculations show that the maximum nonlinearity error rate for
the axial displacement measurement system, with a total range of 20 mm, is 3.6%, while
the radial displacement measurement system, with a total range of 10 mm, exhibits a max-
imum nonlinearity error rate of 2.5%. These results indicate that the system error is rela-
tively large without model correction, making it insufficient for high-precision measure-
ment requirements, and highlighting the necessity of error correction.
Figure 11. Position measurement results: (a) position data comparison; (b) position error distribution.
Figure 11. Position measurement results: (a) position data comparison; (b) position error distribution.
The trained BP and SSA-BP neural network models were implemented on an STM32
microcontroller for measurement. Figure 12 presents the measurement results after correc-
tion using BP neural networks and SSA-BP neural networks. As shown in Figure 12a, for
axial displacement, the BP neural network correction yields a maximum absolute error of
0.385 mm, a maximum nonlinearity error of 1.925%, and a mean squared error (
MSE
) of

Electronics 2024,13, 5055 17 of 20
0.0357 mm
2
. In contrast, the SSA-BP neural network reduces the maximum absolute error
to 0.165 mm, the maximum nonlinearity error to 0.825%, and the
MSE
to 0.0197 mm
2
. Com-
pared to the uncorrected results, the SSA-BP correction decreases the maximum absolute
error by 0.555 mm and reduces the nonlinearity error by 2.77%. For radial displacement,
as shown in Figure 12b, the BP neural network correction results in a maximum absolute
error of 0.187 mm, a maximum nonlinearity error of 1.87%, and an
MSE
of 0.0148 mm
2
.
After correction using the SSA-BP neural network, the maximum absolute error decreases
to 0.129 mm, the maximum nonlinearity error to 1.29%, and the
MSE
to 0.0071 mm
2
. Com-
pared to the uncorrected results, the SSA-BP correction reduces the maximum absolute
error by 0.122 mm and decreases the nonlinearity error by 1.21%.
Electronics 2025, 14, x FOR PEER REVIEW 18 of 22
The trained BP and SSA-BP neural network models were implemented on an STM32
microcontroller for measurement. Figure 12 presents the measurement results after cor-
rection using BP neural networks and SSA-BP neural networks. As shown in Figure 12a,
for axial displacement, the BP neural network correction yields a maximum absolute error
of 0.385 mm, a maximum nonlinearity error of 1.925%, and a mean squared error (
M
SE )
of 0.0357 mm2. In contrast, the SSA-BP neural network reduces the maximum absolute
error to 0.165 mm, the maximum nonlinearity error to 0.825%, and the
M
SE to 0.0197
mm2. Compared to the uncorrected results, the SSA-BP correction decreases the maximum
absolute error by 0.555 mm and reduces the nonlinearity error by 2.77%. For radial dis-
placement, as shown in Figure 12b, the BP neural network correction results in a maxi-
mum absolute error of 0.187 mm, a maximum nonlinearity error of 1.87%, and an
M
SE
of 0.0148 mm2. After correction using the SSA-BP neural network, the maximum absolute
error decreases to 0.129 mm, the maximum nonlinearity error to 1.29%, and the
M
SE to
0.0071 mm2. Compared to the uncorrected results, the SSA-BP correction reduces the max-
imum absolute error by 0.122 mm and decreases the nonlinearity error by 1.21%.
The results clearly demonstrate that the SSA-BP neural network outperforms both
the BP neural network and the uncorrected state. The maximum nonlinearity errors for
both axial and radial displacements after SSA-BP correction are below 2%, significantly
improving the system’s linearity and ensuring the measurement results are closer to the
ideal linear model. Additionally, the substantial reduction in
M
SE after SSA-BP correc-
tion indicates improved anti-interference capabilities and enhanced stability of the sys-
tem. Furthermore, the SSA-BP algorithm exhibits stronger error correction and prediction
performance in complex measurement environments, effectively enhancing the system’s
accuracy and reliability.
In conclusion, the SSA-BP neural network demonstrates superior performance in cor-
recting the measurement of main shaft displacement. It successfully reduces measurement
errors, improves system linearity, stability, and precision, and validates the algorithm’s
effectiveness and practicality. This provides robust technical support for the high-preci-
sion monitoring of main shaft displacement.
Figure 12. Comparison of position errors before and after correction: (a) axial displacement error;
(b) radial displacement error.
5.4. Model Generalization Validation
To verify the generality of the SSA-BP algorithm, another simulated experiment was
conducted using the trained BP and SSA-BP models. According to the experimental re-
sults, the maximum nonlinearity error for axial displacement measurement after correc-
tion was 0.79%, while for radial displacement measurement, it was 1.16%. As shown in
Figure 13, a comparison of the correction values between the BP neural network and the
SSA-BP network demonstrates that the SSA-BP algorithm provides effective optimization
for the nonlinear correction of main shaft displacement, significantly improving the over-
all measurement accuracy of main shaft displacement.
Figure 12. Comparison of position errors before and after correction: (a) axial displacement error;
(b) radial displacement error.
The results clearly demonstrate that the SSA-BP neural network outperforms both
the BP neural network and the uncorrected state. The maximum nonlinearity errors for
both axial and radial displacements after SSA-BP correction are below 2%, significantly
improving the system’s linearity and ensuring the measurement results are closer to the
ideal linear model. Additionally, the substantial reduction in
MSE
after SSA-BP correction
indicates improved anti-interference capabilities and enhanced stability of the system.
Furthermore, the SSA-BP algorithm exhibits stronger error correction and prediction perfor-
mance in complex measurement environments, effectively enhancing the system’s accuracy
and reliability.
In conclusion, the SSA-BP neural network demonstrates superior performance in cor-
recting the measurement of main shaft displacement. It successfully reduces measurement
errors, improves system linearity, stability, and precision, and validates the algorithm’s
effectiveness and practicality. This provides robust technical support for the high-precision
monitoring of main shaft displacement.
5.4. Model Generalization Validation
To verify the generality of the SSA-BP algorithm, another simulated experiment was
conducted using the trained BP and SSA-BP models. According to the experimental results,
the maximum nonlinearity error for axial displacement measurement after correction was
0.79%, while for radial displacement measurement, it was 1.16%. As shown in
Figure 13
,
a comparison of the correction values between the BP neural network and the SSA-BP
network demonstrates that the SSA-BP algorithm provides effective optimization for the
nonlinear correction of main shaft displacement, significantly improving the overall mea-
surement accuracy of main shaft displacement.
After successfully validating the versatility of the SSA-BP algorithm in main shaft
displacement measurement and its outstanding nonlinear correction capabilities, we are
optimistic about its potential applications in a broader range of displacement measurement
scenarios for shaft systems. Taking the displacement measurement of the shaft system
in large hydropower generators as an example, the shaft system must endure significant
hydraulic impacts, mechanical vibrations, and temperature fluctuations during operation.

Electronics 2024,13, 5055 18 of 20
The displacement characteristics of the shaft system are highly complex and play a critical
role in ensuring the safe and stable operation of the generator. Traditional measurement
methods often struggle to accurately capture displacement variations under such harsh
conditions and are highly susceptible to environmental interference, resulting in significant
measurement errors.
Electronics 2025, 14, x FOR PEER REVIEW 19 of 22
Figure 13. Comparison of errors before and after system calibration. (a) axial displacement error;
(b) radial displacement error.
After successfully validating the versatility of the SSA-BP algorithm in main shaft
displacement measurement and its outstanding nonlinear correction capabilities, we are
optimistic about its potential applications in a broader range of displacement measure-
ment scenarios for shaft systems. Taking the displacement measurement of the shaft sys-
tem in large hydropower generators as an example, the shaft system must endure signifi-
cant hydraulic impacts, mechanical vibrations, and temperature fluctuations during op-
eration. The displacement characteristics of the shaft system are highly complex and play
a critical role in ensuring the safe and stable operation of the generator. Traditional meas-
urement methods often struggle to accurately capture displacement variations under such
harsh conditions and are highly susceptible to environmental interference, resulting in
significant measurement errors.
The SSA-BP algorithm demonstrated in this study, with its strong adaptability and
high-precision correction capabilities, offers a novel technical approach to addressing
these challenges. We envision applying a similar measurement and correction system to
the shaft system monitoring of hydropower generators. By deploying an array of PSD
sensors to comprehensively capture displacement information at critical points in the
shaft system, and feeding the data into an SSA-BP model specifically optimized through
targeted training, the algorithm’s exceptional ability to handle nonlinear errors can be lev-
eraged to accurately reconstruct the true displacement state of the shaft system under
complex operating conditions. This approach would enable the timely identification of
potential issues such as shaft misalignment and bearing wear, providing robust data sup-
port for predictive maintenance and the safe, stable operation of the generator.
6. Conclusions
This paper presents a correction method for measuring the displacement error of
wind turbine main shafts based on PSD technology. By combining PSD with laser trian-
gulation, the axial and radial displacements of the wind turbine main shaft are measured,
and the errors are corrected using an SSA-BP neural network. This approach achieves high
stability and precision in displacement measurement.
Experimental results show that the system’s stability reaches ±0.024 mm, far exceeding
the ±0.05 mm to ±0.08 mm of contact-based methods. After correction, the maximum abso-
lute error for axial displacement is reduced to 0.165 mm, and for radial displacement, it is
reduced to 0.129 mm. The maximum linear errors are 0.83% and 1.29%, both below the typ-
ical 2% nonlinearity error. Within the normal operating range of the wind turbine main
shaft, compared to contact-based measurement methods, this approach offers significant
advantages in terms of accuracy, stability, and adaptability to complex environments.
The contribution of this research lies in addressing the challenges of traditional non-
contact measurement techniques, particularly their limitations in accuracy and stability
under complex environmental conditions. It highlights the advantages of combining the
Figure 13. Comparison of errors before and after system calibration. (a) axial displacement error;
(b) radial displacement error.
The SSA-BP algorithm demonstrated in this study, with its strong adaptability and
high-precision correction capabilities, offers a novel technical approach to addressing
these challenges. We envision applying a similar measurement and correction system to
the shaft system monitoring of hydropower generators. By deploying an array of PSD
sensors to comprehensively capture displacement information at critical points in the shaft
system, and feeding the data into an SSA-BP model specifically optimized through targeted
training, the algorithm’s exceptional ability to handle nonlinear errors can be leveraged
to accurately reconstruct the true displacement state of the shaft system under complex
operating conditions. This approach would enable the timely identification of potential
issues such as shaft misalignment and bearing wear, providing robust data support for
predictive maintenance and the safe, stable operation of the generator.
6. Conclusions
This paper presents a correction method for measuring the displacement error of wind
turbine main shafts based on PSD technology. By combining PSD with laser triangulation,
the axial and radial displacements of the wind turbine main shaft are measured, and the
errors are corrected using an SSA-BP neural network. This approach achieves high stability
and precision in displacement measurement.
Experimental results show that the system’s stability reaches
±
0.024 mm, far exceeding
the
±
0.05 mm to
±
0.08 mm of contact-based methods. After correction, the maximum
absolute error for axial displacement is reduced to 0.165 mm, and for radial displacement,
it is reduced to 0.129 mm. The maximum linear errors are 0.83% and 1.29%, both below the
typical 2% nonlinearity error. Within the normal operating range of the wind turbine main
shaft, compared to contact-based measurement methods, this approach offers significant
advantages in terms of accuracy, stability, and adaptability to complex environments.
The contribution of this research lies in addressing the challenges of traditional non-
contact measurement techniques, particularly their limitations in accuracy and stability
under complex environmental conditions. It highlights the advantages of combining the
SSA-BP algorithm with PSD and laser triangulation, demonstrating strong generalization
capabilities. The proposed method offers a cost-effective and precise solution for main
shaft displacement measurement, providing theoretical foundations and technical support
for research in the field of wind turbine main shaft displacement monitoring.
Although the experimental results validate the effectiveness of the proposed method,
certain limitations remain to be addressed. For instance, in real-world operations, the main
shaft of a wind turbine may experience simultaneous axial and radial displacements. This
can be addressed by developing an improved model capable of measuring and analyzing

Electronics 2024,13, 5055 19 of 20
both types of displacement simultaneously. Additionally, we plan to conduct a series of
specifically designed experiments to investigate the effects of extreme environmental factors,
including wind and humidity. Through these future research efforts, we aim to further
enhance the reliability and applicability of the main shaft displacement measurement
methods for wind turbines, thereby improving their performance in practical applications.
Author Contributions: Conceptualization, W.Z., L.W. and G.L.; methodology, W.Z. and L.W.; soft-
ware, W.Z.; validation, W.Z., L.W. and H.Z.; formal analysis, W.Z.; investigation, W.Z. and L.W.;
resources, W.Z.; data curation, W.Z.; writing—original draft preparation, W.Z.; writing—review and
editing, W.Z., L.W., G.L. and C.P.; visualization, H.Z.; supervision, L.W. and G.L.; project administra-
tion, W.Z. and L.W.; funding acquisition, L.W. All authors have read and agreed to the published
version of the manuscript.
Funding: This research was funded by the Science and Technology Development Plan of Jilin Province
of China under Grant 20220201089GX.
Data Availability Statement: The data are contained within the article.
Conflicts of Interest: The authors declare no conflicts of interest.
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