A Data-Driven Fault Diagnosis Method for Marine Steam Turbine Condensate System Based on Deep Transfer Learning

Abstract

Accurate fault diagnosis in marine steam turbine condensate systems is challenged by insufficient real fault samples and dynamic operational conditions. To address this limitation, DTL-DFD, a novel framework integrating digital twins (DTs) and deep transfer learning (DTL), is proposed, wherein a high-fidelity physics-constrained digital twin model is constructed through the systematic injection of six diagnostic classes (1 normal + 5 faults), including insufficient circulation water flow.Through an innovative all-layer parameter initialization with a partial fine-tuning (ALPT-PF) strategy, all weights and biases from a pre-trained one-dimensional convolutional neural network (1D-CNN) were fully transferred to the target model, which was subsequently fine-tuned via a hierarchical learning rate mechanism to adapt to real-world distribution discrepancies. Experimental results demonstrate 94.34% accuracy on cross-distribution test sets with a 4.72% improvement over state-of-the-art methods, confirming significant enhancements in generalization capability and diagnostic stability under small-sample conditions with significant real data reduction, thereby providing an effective solution for the intelligent operation and maintenance of marine steam turbine systems.

1. Introduction

The condensate system, as a fundamental component of the marine steam turbine system, maintains the thermodynamic cycle performance through efficient heat exchange, and its reliability directly impacts propulsion efficiency and navigation safety [1,2]. Failures in the condensate system can set off a series of reactions, such as aberrant turbine backpressure, deviation in lubricant temperature from threshold values, and even more serious consequences such as engine shutdown. Furthermore, undetected pipeline leaks may cause coolant contamination, accelerate subsystem corrosion [3]. Therefore, a critical technical challenge in the field of marine engineering is the development of intelligent fault diagnosis methods with fault monitoring and early warning capabilities [4,5].

The academic community has developed three main diagnostic paradigms in response to this difficulty, but the unique nature of marine steam turbine systems still limit their application. Fault diagnosis has made use of knowledge-based reasoning methods [6,7], which determine faults by constructing a domain-specific rule database. Model-driven approaches [8,9] have begun to be applied in fault diagnosis and state prediction for condensate systems as a result of developments in numerical simulation technologies. Zhang et al. [10] proposed a fouling prediction method for the condensate system based on differential modeling. To optimize the nuclear-renewable energy hybrid system, Watanabe et al. [11] developed a Modelica-based digital twin (DT) framework, demonstrating how integrating energy storage and hydrogen technologies may enhance efficiency. In order to monitor tool wear in intricate dynamic machining processes, Liu et al. [12] proposed a DT-based anomaly detection framework that dynamically models the vibration signal characteristics of machining processes in real time and extracts nonlinear output frequency response functions as model frequency features.

In recent years, data-driven methods [13,14] have garnered significant attention because of their robust nonlinear mapping capabilities. In order to accurately diagnose low-vacuum defects in an operational condensate system, Chen et al. [15] built a diagnostic model based on BP neural networks. An automatic machine learning-based fault diagnosis method for chiller units was proposed by Tian, which significantly improved diagnostic performance under low fault severity and complex fault scenarios [16]. Nie et al. [17] presented relief-recursive feature elimination based on a cross validation–support vector machine (SVM) model for heating, ventilation, and air conditioning fault diagnosis, demonstrating high accuracy and substantially reduced diagnostic times when validated on datasets. However, a persistent problem that has not been addressed is that models are prone to overfitting and insufficient generalization across operating conditions due to the limited number of fault samples in real ship operations, which are usually just 0.1–1% of normal data [18].

While transfer learning mitigates data scarcity by reusing knowledge from source domains through knowledge transfer across domains [19,20], the integration of multiple data points for diagnostics [21,22] has arisen as a novel research direction in response to the constraints of individual methodologies. By developing five transfer strategies and validating them on two types of chiller unit data, Chen et al. [23] proposed a fault diagnosis method for cross-building energy systems based on transfer learning and Shapley additive explanations at the feature level. Using scenario-based transfer network architectures and weights, Yang et al. [24] adopted a transfer learning strategy for a deep learning prognostics and health management algorithm, validating its efficacy in bearing fault diagnosis through scenario-based transfer network structures and weights. Solís M and Colvo-Valverde L.A [25] compared deep learning models with and without transfer learning alongside traditional methods based on time series datasets from the M4 and M3 competitions, revealing that models that integrated transfer learning performed better in terms of prediction. Jamil et al. [26] proposed a deep boosted transfer learning method based on the Case Western Reserve University bearing and wind turbine datasets.

Three main issues plague current research on diagnosing marine condensate systems: first, diagnostic models are predicated on the assumption that training and testing data follow independent and identically distributed conditions, which are not representative of real-world scenarios; second, there is a severe lack of available fault data, and some high-risk faults cannot be reproduced through physical experiments; third, the mapping relationship between fault characteristics and causes is often unclear, and single sensor signals are insufficient for precise root cause tracing.

To address these challenges, this paper proposes to integrate DTs and deep transfer learning (DTL) for fault diagnosis and takes the marine turbine condensate system as the research subject. There are two main innovations.

First, a high-fidelity physical model replicating the marine steam turbine condensate system is developed. By injecting five fault-type parameters (e.g., insufficient coolant flow), it generates over 12,000 labeled fault simulations, overcoming data scarcity.

Second, the ALPT-PF strategy enables the full transfer of weights and biases from the pre-trained 1D-CNN to the target model. This comprehensive parameter migration, combined with hierarchical learning rate fine-tuning, effectively bridges simulation-to-reality distribution gaps while minimizing overfitting risks in small-sample scenarios.

This paper’s next sections are organized as follows: In Section 2, the defect diagnosis problem is defined and the fundamental ideas of transfer learning are further explained. The structure, models, and procedures of the proposed DTL-based DT-assisted fault diagnosis (DTL-DFD) are thoroughly explained in Section 3. Section 4 uses case study experimental design, findings, and analyses to verify the efficacy of the proposed approach. Section 5 concludes with a summary of the work, findings, and future research opportunities.

2. Problem Description

In this study, simulation data from the marine steam turbine condensate system and operational data are designated the source domain $D_s$ and target domain $D_t$, respectively. The source domain contains sufficient labeled fault samples, while the target domain includes only limited labeled samples and abundant unlabeled samples. Crucially, feature distributions differ significantly across domains due to simulation–actual mismatches: ${\chi }_s={\chi }_t$, $p\left(X_s\right)\ne p\left(X_t\right)$, $T_s=T_t$, where X denotes multi-sensor fault features. However, both domains share identical fault modes, enabling knowledge transfer for the same diagnostic task. The small-sample challenge is formally defined under three conditions:

(1)

The simulation model replicates the physical structure, fault modes, and operating conditions of the actual system while matching its signal sampling frequency.

(2)

For the simulation model, a statistically sufficient volume of labeled samples for all fault categories is available.

(3)

For the actual system, extreme data scarcity exists—limited labeled samples (typically <0.5% of operational data) are obtainable.

The objective is to transfer diagnostic knowledge from simulation to operational environments. We resolve this by exploiting shared feature representations between domains, enabling DTL to train a domain-invariant classifier $f_T\left(·\right)$ using joint knowledge from $D_s$ and $D_t$.

3. Methods

3.1. The High-Fidelity Physical Model of the Condensate System

3.1.1. Condenser

Figure 1 illustrates the structural schematic of the condenser and the heat and the mass transfer model encompassing the tube side, heat exchange wall surface, and shell side of the condenser. The surface condenser consists of cooling tubes, a water chamber, a hot well, and steam/cooling water inlets/outlets. During operation, a circulating pump drives cooling water through the tubes. Turbine exhaust steam enters the shell side via the steam inlet, flows over the tube walls, and releases latent vaporization heat as it cools and condenses into water (collected in the hot well). Heat transfers through the tubes to the cooling water, which is then discharged. This cycle maintains high vacuum in the condenser, enabling continuous steam condensation. Under steady-state or dynamic operating conditions, each control volume unit on the tube side exchanges heat with the shell side through the metal wall surface. For the heat and mass transfer processes involving the tube side, heat exchange wall surface, and shell side of the condenser, dynamic differential equations can be formulated for each segment based on mass conservation, momentum conservation, and energy conservation laws.

(1)

Tube side

All parallel heat exchange tubes on the condenser tube side can be equivalent to a single heat exchange tube. The mass conservation equation for the fluid in each control volume element on the tube side is expressed as

$${\displaystyle \frac{d{m}_i}{dt}}={\dot{m}}_{in,i}-{\dot{m}}_{out,i}$$

(1)

where $m_i$ is the fluid mass of control volume element i, kg; ${\dot{m}}_{in,i}$ is the mass flow rate of fluid entering control volume element i, kg/s; ${\dot{m}}_{out,i}$ is the mass flow rate of fluid exiting control volume element i, kg/s; and t is time, s.

The momentum conservation equation for the fluid in each control volume element on the condenser tube side is expressed as

$${\displaystyle \frac{1}{A}}{\displaystyle \frac{d{\dot{m}}_i}{dt}}{l}_i=p_{in,i}-p_{out,i}-{\displaystyle \frac{{\lambda }_i}{D}}{\displaystyle \frac{v_i^2}{2}}{\rho }_i-{\rho }_{i}g{h}_i$$

(2)

where ${\dot{m}}_i$ is the fluid mass flow rate of control volume element i, kg/s; $p_{in,i}$ and $p_{out,i}$ are the pressures at the boundaries of control volume element i, Pa; ${\lambda }_i$ is the frictional resistance loss coefficient of control volume element i; D is the pipe diameter, m; $v_i$ is the fluid velocity in control volume element i, m/s; ${\rho }_i$ is the density of control volume element i, kg/m³; g is the gravitational acceleration, m/s²; and $h_i$ is the height difference between the inlet and outlet of control volume element i, m.

The energy conservation equation for the fluid in each control volume element on the condenser tube side is expressed as

$${\displaystyle \frac{d\left(m_i·u_i\right)}{dt}}={\dot{m}}_{in,i}{h}_{in,i}-{\dot{m}}_{out,i}{h}_{out,i}+Q_{w,in,i}$$

(3)

where $u_i$ is the specific internal energy of control volume element i, J/kg; $h_{in,i}$ is the specific enthalpy of the fluid entering control volume element i, J/kg; $h_{out,i}$ is the specific enthalpy of the fluid exiting control volume element i, J/kg; and $Q_{w,in,i}$ is the heat exchange between control volume element i and the tube wall, W.

The heat exchange $Q_{w,in,i}$ between control volume element i and the tube wall is calculated by

$$Q_{w,in,i}={\alpha }_i{S}_i(T_i-T_{w,in,i})$$

(4)

where ${\alpha }_i$ is the convective heat transfer coefficient between the fluid in control volume element i and the wall; $S_i$ is the heat exchange area between control volume element i and the tube wall, m²; $T_i$ is the fluid temperature of control volume element i, K; and $T_{w,in,i}$ is the wall temperature of control volume element i, K.

(2)

Shell side

According to the law of mass balance, the mass of steam on the condenser shell side is calculated as follows:

$${\displaystyle \frac{d{m}_{s,i}}{dt}}={\dot{m}}_{s,in,i}-{\dot{m}}_{s,out,i}-{\dot{m}}_{l,i}$$

(5)

where $m_{s,i}$ is the steam mass of the control volume element i, kg; ${\dot{m}}_{s,in,i}$ is the steam mass flow rate entering the control volume element i, kg/s; ${\dot{m}}_{s,out,i}$ is the steam mass flow rate exiting the control volume element i, kg/s; and ${\dot{m}}_{l,i}$ is the condensation amount in the control volume element i, kg/s.

The momentum conservation equation for the fluid in each control volume element on the condenser shell side is expressed as

$$p_{s,in,i}-p_{s,out,i}={\displaystyle \frac{{\lambda }_{s,i}}{D_s}}{\displaystyle \frac{v_{s,i}^2}{2}}{\rho }_{s,i}+{\rho }_{s,i}g{h}_i$$

(6)

where ${\dot{m}}_{s,i}$ is the fluid mass flow rate of control volume element i, kg/s; $p_{s,in,i}$ is the inlet pressure of control volume element i, Pa; $p_{s,out,i}$ is the outlet pressure of control volume element i, Pa; ${\lambda }_{s,i}$ is the frictional resistance loss coefficient of shell-side control volume element i; $D_s$ is the shell-side diameter, m; $v_{s,i}$ is the steam velocity in control volume element i, m/s; ${\rho }_{s,i}$ is the fluid density in control volume element i, kg/m³; and g is the gravitational acceleration, m/s².

The energy conservation equation for the fluid in each control volume element on the condenser shell side is expressed as

$${\displaystyle \frac{d\left(m_{s,i}·u_{s,i}\right)}{dt}}={\dot{m}}_{s,in,i}{h}_{s,in,i}-{\dot{m}}_{s,out,i}{h}_{s,out,i}-{\dot{m}}_{l,out,i}{h}_{l,out,i}+Q_{w,out,i}+Q_{l,i}$$

(7)

where $u_{s,i}$ is the specific internal energy of the control volume element i, J/kg; $h_{s,in,i}$ is the specific enthalpy of steam entering control volume element i, J/kg; $h_{s,out,i}$ is the specific enthalpy of steam exiting control volume element i, J/kg; $h_{l,out,i}$ is the specific enthalpy of condensed water, J/kg; $Q_{w,out,i}$ is the heat exchange between control volume element i and the outer wall of the tube, W; $Q_{l,i}$ is the heat released by steam condensation in control volume element i, W.

The heat exchange $Q_{w,out,i}$ between shell-side control volume element i and the tube wall is calculated by

$$Q_{w,out,i}={\alpha }_{s,i}{S}_{s,i}(T_{s,i}-T_{w,out,i})$$

(8)

where ${\alpha }_{s,i}$ is the convective heat transfer coefficient between the fluid in control volume element i and the wall; $S_{s,i}$ is the heat exchange area between control volume element i and the outer wall of the tube, m²; $T_{s,i}$ is the fluid temperature of control volume element i, K; and $T_{w,out,i}$ is the outer wall temperature of control volume element i, K.

(3)

Exchange Wall

The heat exchange wall in the condenser is usually cylindrical, and the two fluids are separated by the wall transfer heat through this cylindrical wall. The dynamic energy balance equation on the wall is expressed as

$$M_{w,i}·c_{p,w}·{\displaystyle \frac{d{T}_{w,i}}{dt}}=Q_{w,out,i}-Q_{w,in,i}$$

(9)

where $M_{w,i}$ is the mass of the wall in control volume element i, kg; $c_{p,w}$ is the specific heat capacity of the wall at constant pressure; and $T_{w,i}$ is the wall temperature of control volume element i, K.

The heat flow through the wall is calculated using Fourier’s law in cylindrical coordinates. The Fourier equation for the inner wall is expressed as

$$Q_{w,in,i}={\lambda }_w{\displaystyle \frac{2\pi ·l_i}{ln\left(\right(e+D)/D)}}(T_{w,i}-T_{w,in,i})$$

(10)

where e is the wall thickness, m; ${\lambda }_w$ is the thermal conductivity of the wall; and $T_{w,i}$ is the wall temperature of the control volume element i, K.

The Fourier equation for the outer wall is expressed as

$$Q_{w,out,i}={\lambda }_w{\displaystyle \frac{2\pi ·l_i}{ln\left(\right(2·e+D)/(e+D\left)\right)}}(T_{w,i}-T_{w,out,i})$$

(11)

where $T_{w,i}$ is the wall temperature of control volume element i, K.

3.1.2. Jet Air Ejector

The steam jet air ejector continuously extracts air and non-condensable gases leaking into the condenser through the jet power generated by working steam, maintaining a high-vacuum environment in the condenser while compressing and boosting the pressure of low-pressure gases. Based on the fluid dynamics characteristics of the nozzle diffuser, combined with the law of energy conservation and Dalton’s law of partial pressures, the energy equation and partial pressure equation of the steam–gas mixture are established.

The nozzle outlet flow equation is expressed as

$${\dot{q}}_{m0}=0.648·f_{min}·\sqrt{p_0/v_0}$$

(12)

where ${\dot{q}}_{m0}$ is the mass flow rate of working steam, kg/s; $f_{min}$ is the nozzle throat area, m²; $p_0$ is the inlet pressure of working steam, Pa; and $v_0$ is the inlet specific volume of working steam, m³/kg.

The air extraction port energy equation is expressed as

$${\dot{q}}_{ms}·h_s={\dot{q}}_{mv}·h_v+{\dot{q}}_{mair}·h_{air}$$

(13)

where ${\dot{q}}_{ms}$ is the mass flow rate of the mixed working fluid at the air extraction port, kg/s; ${\dot{q}}_{mv}$ is the steam mass flow rate at the air extraction port, kg/s; ${\dot{q}}_{mair}$ is the air mass flow rate at the air extraction port, kg/s; $h_s$ is the specific enthalpy of the mixed working fluid, J/kg; $h_v$ is the specific enthalpy of steam at the air extraction port, J/kg; and $h_{air}$ is the specific enthalpy of air at the air extraction port, J/kg.

The air extraction port partial pressure equation is expressed as

$$p_{1v}=p_1·{\displaystyle \frac{\frac{{\dot{q}}_{mv}}{M_{H_{2}O}}}{\frac{{\dot{q}}_{mv}}{M_{H_{2}O}}+\frac{{\dot{q}}_{mair}}{M_{air}}}}$$

(14)

$$p_{1air}=p_1-p_{1v}$$

(15)

where $p_1$ is the total pressure at the air extraction port, Pa; $p_{1v}$ is the steam partial pressure at the air extraction port, Pa; $p_{1air}$ is the air partial pressure at the steam port, Pa; $M_{H_{2}O}$ is the molar mass of water vapor, kg/mol; and $M_{air}$ is the molar mass of air, kg/mol.

The diffuser inlet energy equation is expressed as

$$({\dot{q}}_{m0}+{\dot{q}}_{ms})·h_2=({\dot{q}}_{m0}+{\dot{q}}_{mv})·h_{2v}+{\dot{q}}_{mair}·h_{2air}$$

(16)

$$({\dot{q}}_{m0}+{\dot{q}}_{mv})·h_{2v}={\dot{q}}_{m0}·h_1+{\dot{q}}_{mv}·h_v$$

(17)

where $h_2$ is the specific enthalpy of the mixed working fluid at the diffuser inlet, J/kg; $h_{2v}$ is the specific enthalpy of steam at the diffuser inlet, J/kg; $h_{2air}$ is the specific enthalpy of air at the diffuser inlet, J/kg; and $h_1$ is the specific enthalpy of steam at the nozzle outlet, J/kg.

The diffuser inlet partial pressure equation is expressed as

$$p_{2v}=p_1·{\displaystyle \frac{\frac{({\dot{q}}_{m0}+{\dot{q}}_{mv})}{M_{H_{2}O}}}{\frac{({\dot{q}}_{m0}+{\dot{q}}_{mv})}{M_{H_{2}O}}+\frac{{\dot{q}}_{mair}}{M_{air}}}}$$

(18)

$$p_{2air}=p_2-p_{2v}$$

(19)

where $p_2$ is the total pressure at the ejector inlet, Pa; $p_{2v}$ is the steam partial pressure at the ejector inlet, Pa; and $p_{2air}$ is the air partial pressure at the ejector inlet, Pa.

The injector outlet partial pressure equation is expressed as

$$p_{4v}=p_4·{\displaystyle \frac{p_{1v}}{p_1}}$$

(20)

$$p_{4air}=p_4-p_{4v}$$

(21)

where $p_4$ is the ejector outlet pressure, Pa; $p_{4v}$ is the steam partial pressure at the ejector outlet, Pa; and $p_{4air}$ is the air partial pressure at the ejector outlet, Pa.

3.1.3. Condensate Pump

The condensate pump converts mechanical energy into fluid kinetic energy and potential energy, thereby increasing fluid pressure. The operating characteristics of the pump are described by polynomial-fitted head characteristic equations and efficiency characteristic equations, and a complete pump model is constructed by combining power equations, temperature equations, and torque equations.

The head characteristic equation is expressed as

$${\displaystyle \frac{H}{H_0}}=\sum _{k=1}^5{A}_k{m}^k$$

(22)

The efficiency characteristic equation is expressed as

$${\displaystyle \frac{\eta }{{\eta }_0}}=\sum _{k=1}^5{B}_k{m}^k$$

(23)

where $A_1$~$A_5$ are the coefficients of the head characteristic polynomial; $B_1$~$B_5$ are the coefficients of the efficiency characteristic polynomial; H is the actual head, m; $H_0$ is the design point head, m; $\eta$ is the actual efficiency; ${\eta }_0$ is the design point efficiency; and m is the ratio of the actual converted flow rate to the design point flow rate.

$$m={\displaystyle \frac{{\dot{q}}_{m,z}}{{\dot{q}}_{m,0}}}={\displaystyle \frac{{\dot{q}}_m}{{\dot{q}}_{m,0}}}·{\displaystyle \frac{N_0}{N}}$$

(24)

where ${\dot{q}}_m$ is the actual flow rate, kg/s; ${\dot{q}}_{m,0}$ is the design point flow rate, kg/s; ${\dot{q}}_{m,z}$ is the flow rate after speed conversion, kg/s; N is the actual rotational speed, rpm; and $N_0$ is the design point rotational speed, rpm.

The power equation is expressed as

$$P_p={\displaystyle \frac{\Delta p_p{\dot{q}}_m}{{\eta }_p}}$$

(25)

$$\Delta p_p=p_L-p_E=\rho gH$$

(26)

where ${\eta }_p$ is the actual efficiency of the pump; $\rho$ is the fluid density, kg/m³; $p_L$ is the outlet pressure, Pa; $p_E$ is the inlet pressure, Pa; $\Delta p_p$ is the pressure difference between inlet and outlet, Pa; and g is the gravitational acceleration, taken as 9.81 m/s².

The outlet temperature rise equation is expressed as

$$\Delta T=T_L-T_E={\displaystyle \frac{(1-{\eta }_0)g{H}_0}{{\eta }_0{c}_p}}$$

(27)

where $T_L$ is the outlet temperature, K; $T_E$ is the inlet temperature, K; and $c_p$ is the specific heat capacity at constant pressure, J/(kg·K).

The pump torque equation is expressed as

$${\tau }_p={\displaystyle \frac{30P_p}{\pi N}}$$

(28)

3.1.4. The System of the Condensate System

A high-fidelity mathematical model for the condensate system in MST systems was developed using a Modelica-based multidisciplinary modeling approach to obtain sufficient data. This proposed modeling methodology was implemented via MWORKS—a Modelica-compliant platform developed by Suzhou Tongyuan—to construct the physics-based condensate system model, as illustrated in Figure 2. For details of the construction process, please refer to the authors’ published work [27].

3.2. DTL-DFD Method Framework

DT technology provides comprehensive immersive solutions for fault diagnosis in complex systems by enabling dynamic virtual–physical interactions [28,29]. While it facilitates precise state mapping of physical entities through real-time data iteration [30,31], early-stage operational data scarcity and domain distribution shifts caused by dynamic conditions pose significant challenges. Traditional deep learning methods struggle under these constraints. A cross-domain knowledge transfer mechanism between virtual simulation data (source domain) and physical monitoring data (target domain) enables the rapid generalization optimization of diagnostic models under small-sample conditions, as shown in Figure 3.

To address this, we propose DTL-DFD, the cross-domain knowledge transfer framework shown in Figure 4, which enables rapid diagnostic generalization under small-sample conditions through virtual-to-physical domain adaptation. First, a simulation model of the condensate system is constructed using Suzhou Tongyuan’s Modelica platform MWORKS, calibrated with measured operational data to ensure model accuracy. Based on this validated model, various faults are injected to perform batch simulations, generating sufficient health-state and fault-state data. Subsequently, both the simulated data and limited measured field data undergo preprocessing for 1D-CNN-based classification training. Finally, a DTL diagnostic model integrating 1D-CNN with a parameter migration strategy is established. This model leverages 1D-CNN for feature extraction and transfers knowledge acquired from simulation scenarios to real-world applications through migration and fine-tuning strategies, achieving intelligent fault diagnosis in condensing water systems under limited samples. DTL-DFD aims to address the weak diagnostic capability of traditional data-driven methods caused by insufficient training data during the early operational stage.

3.3. The Fault Diagnosis Model of the Condensate System

To address the need for signal temporal feature extraction in fault diagnosis tasks for marine steam turbine condensate systems, 1D-CNN is adopted as the foundational architecture. In this application, the raw temporal vibration signals are acquired from key components such as the condensate pump, ejector, and condenser tubes under various fault scenarios (F1—insufficient coolant flow; F2—tube fouling; F3—tube blockage; F4—vacuum leakage; F5—ejector fault). Local feature patterns in these vibration signals—such as amplitude spikes from tube blockage (F3) or frequency shifts due to vacuum leakage (F4)—are efficiently captured by the local perception and weight-sharing capabilities of 1D-CNN. The network structure comprises two sets of convolution–pooling modules, each consisting of a convolutional layer, a batch normalization layer, and a max-pooling layer. In the convolutional layer, feature extraction is achieved through sliding window operations, and Equation (29) illustrates the computation process:

$${\mathit{y}}^{l(i,j)}={\mathit{K}}_i^{l\left(j\right)}\ast {\mathit{X}}^{l\left(rj\right)}=\sum _{j^{\prime }=0}^{W-1}{\mathit{K}}_i^{l\left(j^{\prime }\right)}{\mathit{X}}^{l(j+j^{\prime })}$$

(29)

where ∗ denotes the convolution operation, $K_i^{l\left(j^{\prime }\right)}$ represents the $j^{\prime }$-th weight parameter of the i-th convolutional kernel in layer l, ${\mathit{X}}^{l\left(r^j\right)}$ denotes the j-th convolutional region in layer l, and W is the kernel width.

Following each convolutional layer, a max-pooling layer performs feature dimensionality reduction, that is, retaining the most prominent vibration peaks caused by tube blockage (F3)—and a batch normalization layer is applied to speed up convergence:

$${\mathit{y}}^{l(i,j)}=max\left({\mathit{X}}^{l\left(rj\right)}\right)$$

(30)

The flattened features are then passed through two fully connected layers with dropout regularization, enabling more complex feature fusion, critical for distinguishing between faults with similar spectral patterns, such as F2 (tube fouling) and F3 (tube blockage). Finally, a softmax classification layer outputs the probability distribution over fault classes:

$$\mathit{y}=\mathrm{softmax}\left({\mathit{W}}^T\mathit{x}\right)={\displaystyle \frac{e^{{\mathit{W}}^T\mathit{x}}}{{\mathbf{1}}_c^T{e}^{{\mathit{W}}^T\mathit{x}}}}$$

(31)

The output presents probabilities for different fault categories, where $\mathit{W}$ denotes the weight matrix, and ${\mathbf{1}}_C$ represents a vector of ones. Using multi-level nonlinear transformations, this architecture accomplishes end-to-end fault diagnosis and makes it possible to map raw signals to fault classes.

By applying multi-level nonlinear transformations to the fault-specific vibration sequences of the condensate system, this architecture maps raw signals to fault categories (F1–F5) in an end-to-end manner. The ALPT-PF strategy is employed: all pre-trained CNN parameters, encapsulating diagnostic knowledge from simulation data of the marine condensate system (including ±15 % load variations and multi-condition fault injections) are migrated to initialize the target diagnostic model. Full-parameter transfer across convolutional and dense layers ensures that learned representations—such as flow fluctuation patterns in F1 or resonance peaks in F4—are retained, enabling adaptation to real-world conditions with minimal target-domain data.

3.4. Implementation Process

Within the proposed DTL-DFD, the 1D-CNN diagnostic model undergoes pre-training using extensive simulated datasets. Subsequently, the optimized network parameters are transferred to the target diagnostic network and fine-tuned with sparse real-world samples. Crucially, this approach directly processes raw time-domain signals as inputs to the 1D-CNN, eliminating manual feature engineering or spectral transformations. This integrated architecture synergistically combines fault identification and feature extraction within a unified computational framework. The implementation workflow of the DTL-DFD methodology is detailed in Figure 5.

Step 1: Data acquisition

To generate a robust set of operational signals representative of diverse equipment states, including various fault modes, batch simulations are performed on the marine steam turbine condensate model. Measured operational signals from the target system are also gathered in parallel. The effective configuration of these simulations requires detailed knowledge of subsystem/component models, fault categories, operational scenarios, and the sampling frequency of the operational signals associated with the system under consideration for diagnosis. Crucially, simulations for each fault mode must incorporate a spectrum of severities within the safe operational envelope to ensure the developed diagnostic model possesses strong generalization capability.

Step 2: Data preprocessing

Owing to dynamic coupling effects during operation, the condensate system signals in turbine units exhibit high-dimensional, nonlinear, and non-stationary characteristics. The inherent dimensional heterogeneity among variables may lead to erroneous comparisons during signal analysis. To mitigate feature-scale discrepancies and enhance model convergence stability, all operational data samples are standardized using zero- mean normalization:

$$\widehat{x}={\displaystyle \frac{x-\mu }{\sigma }}$$

(32)

where, $\widehat{x}$ is the normalized sample value, x is the original sample value, $\mu$ is the sample mean, and $\sigma$ is the sample standard deviation.

Step 3: Pre-trained 1D-CNN diagnostic model

The preprocessed simulation samples underwent stratified random partitioning into training (70%), validation (20%), and test (10%) sets to maintain class distribution integrity. The 1D-CNN diagnostic model was trained with parameter optimization using the training set, while the validation set guided hyperparameter tuning and early stopping. The optimal model checkpoint based on validation performance was retained. The final evaluation of diagnostic accuracy was conducted on the held-out test set using performance metrics.

Step 4: Transferring pre-trained 1D-CNN network parameters

To address the scarcity of real-world operational data, a transfer learning approach was implemented. All learnable parameters (weights and biases) from the simulation-trained 1D-CNN were transferred to initialize the target diagnostic model. This parameter transplantation encapsulates the feature representations learned from simulated fault patterns, enabling the target model to leverage diagnostic knowledge acquired during simulation training. Consequently, the model overcomes data limitation constraints while preserving the feature extraction capability developed on simulated domains.

Step 5: Fine-tuning the pre-trained 1D-CNN diagnostic model

A class-balanced, small-sample dataset was created, comprising both a training set and a test set for domain-specific fine-tuning. The pre-trained source-domain model was fine-tuned using the training set, while the number of unfrozen layers and the fine-tuning learning rate were systematically optimized to maximize adaptation to the target domain. Finally, the optimized 1D-CNN was evaluated as the diagnostic model on the target domain test set to assess its performance.

4. Experiment and Analysis

4.1. Case Description

This case study focuses on the condensate system critical for turbine exhaust steam condensation and vacuum maintenance. Based on operational data from the marine turbine condensate system, five primary fault modes are identified: (1) Reduced circulating water flow, which diminishes cooling capacity and elevates exhaust steam temperature, progressively lowering vacuum levels; (2) Condenser tube blockage, indicated by abnormal terminal temperature difference and excessive condensate sub-cooling, compromising heat transfer efficiency; (3) Accumulation of non-condensable gases from air ejector failure, inducing vacuum system oscillations and reducing the heat transfer coefficient; (4) Copper tube fouling, increasing thermal resistance and impairing internal/external heat transfer; (5) Vacuum leaks permitting air ingress, which disrupts pressure stability and heat transfer processes. Collectively, these faults destabilize the condenser’s thermal equilibrium, triggering a systemic decline in unit thermal efficiency and operational stability, necessitating preemptive load reduction during severe events to preserve safety margins.

4.2. Experimental Design

4.2.1. Fault Modes and Simulation Test Design

Fault injection testing was performed within a DT framework to generate comprehensive fault datasets, supported by the simulation analysis of the condensate system physical model. Core failure modes, including condensate pump cavitation and heat transfer tube leakage, were identified along with their injection methodologies through integrated analysis of historical cases, operational processes, and expert assessments. A minimal feature parameter matrix was constructed by allocating calibrated variations to key component characteristics using expert-informed data, as shown in

Table 1. Fault modes and corresponding parameters.

Serial No.	Fault Mode	Operating Parameters & Ranges	Fault Injection Method	No. of Conditions
F1	Insufficient circulating water	Main steam pressure: 0.186–0.466 MPa Main steam enthalpy: 2632.16--2670.86 kJ/kg Cooling water flow: 2166 kg/s Cooling water temperature: 5–35 °C	Cooling water flow: 1500, 1000	180
F2	Condenser copper tube blockage		Blockage coefficient: 0.03, 0.06	180
F3	Condenser copper tube fouling		Fouling thermal resistance: 0.005, 0.0008	180
F4	Air extractor fault		Air extraction coefficient: 0.5, 0	180
F5	Vacuum system leakage		Leaked air volume: 0.2, 0.5	180

, enabling the simulation of multi-severity fault conditions through parametric adjustments. Diverse fault scenarios were simulated under operational modes to capture critical fault signatures and dynamic evolution patterns, supporting the development of high-precision diagnostic algorithms.

A sensitivity analysis was conducted using the condensate system’s high-fidelity physical model to identify fault-sensitive parameters and their valid operating bounds, which vary during fault conditions. Comprehensive fault simulations revealed significant fluctuations in output characteristic parameters across diverse failure modes. The correlation between fault signals and failure mechanisms was established through structured fault injection, validating the parametric sensitivity thresholds as shown in

Table 2. The influence of fault modes on characteristic parameters.

	${\mathit{P}}_{\mathbf{cw},\mathbf{in}}$	${\mathit{P}}_{\mathbf{cw},\mathbf{out}}$	${\mathit{Q}}_{\mathbf{cw}}$	${\mathit{T}}_{\mathbf{cw}}$	$\mathbf{\Delta }{\mathit{T}}_{\mathbf{cw}}$	$\mathbf{\Delta }{\mathit{P}}_{\mathbf{cw}}$	$\mathbf{\Delta }{\mathit{T}}_{\mathbf{su},\mathbf{cw}}$	$\mathbf{\Delta }{\mathit{P}}_{\mathbf{su},\mathbf{ej}}$	$\mathbf{\Delta }{\mathit{T}}_{\mathbf{ej},\mathbf{cw}}$	${\mathit{V}}_{\mathbf{cond}}$	${\mathit{R}}_{\mathbf{cond},\mathbf{w}}$	$\mathbf{\Delta }{\mathit{T}}_{\mathbf{sub},\mathbf{cond}}$
F1	—	↓	↓	↑	—	—	↑	—	—	↓	—	—
F2	↓	—	—	↓	↓	—	↑	—	—	↓	↑	—
F3	—	—	—	—	—	↓	—	—	↓	↓	↑	—
F4	—	—	—	—	↓	—	—	—	—	↓	—	↑
F5	—	—	—	—	—	—	—	↓	—	↓	—	↑

.

In the table, the first column represents the fault modes of the condensate system, and the first row represents the characteristic variables. The symbols of characteristic variables and their corresponding meanings are listed as follows: $P_{\mathrm{cw},\mathrm{in}}$ is the circulating water inlet pressure, $P_{\mathrm{cw},\mathrm{out}}$ is the circulating water outlet pressure, $Q_{\mathrm{cw}}$ is the circulating water flow rate, $T_{\mathrm{cw}}$ is the circulating water temperature, $\Delta T_{\mathrm{cw}}$ is the circulating water inlet–outlet temperature difference, $\Delta P_{\mathrm{cw}}$ is the circulating water inlet–outlet pressure difference, $\Delta T_{\mathrm{su},\mathrm{cw}}$ is the temperature difference between extracted air at the suction port and cooling water inlet, $\Delta P_{\mathrm{su},\mathrm{ej}}$ is the pressure difference between the condenser suction port and ejector inlet, $\Delta T_{\mathrm{ej},\mathrm{cw}}$ is the temperature difference between extracted steam from the ejector and circulating cooling water inlet, $V_{\mathrm{cond}}$ is the condenser vacuum degree, $R_{\mathrm{cond},\mathrm{w}}$ is the condenser water resistance, and $\Delta T_{\mathrm{sub},\mathrm{cond}}$ is the condensate subcooling degree. ↑ indicates an increase, ↓ indicates a decrease, and − indicates no change.

4.2.2. Dataset Construction and Partition Strategy

A deep transfer learning-based fault diagnosis model was developed using a one-dimensional convolutional neural network architecture for the marine steam turbine condensate system. This three-layer convolutional architecture accepts raw signals with dimensionality 10 by 1 at the input layer. Sequential convolutional layers employ 256, 128, and 64 filters, respectively, utilizing a kernel size of 3 and a stride of 2. Feature dimensionality reduction was achieved through max-pooling operations with a kernel size of 3 and a stride of 2. The classification output layer uses a softmax function corresponding to five fault categories. Two fully connected layers containing 128 and 64 neurons incorporate dropout regularization at a rate of 0.3 and L2 weight decay with a coefficient of 1 times 10 to the power of minus 4 to mitigate overfitting. Model training employed Adam optimization with a batch size of 64, 500 training epochs, and an initial learning rate of 0.001, as shown in Figure 6.

Transfer learning methodology was implemented through pre-training on simulated data until convergence was achieved in the source domain. During domain transfer, parameters in the first two convolutional layers remained frozen, while the final two layers underwent hierarchical learning rate adaptation. Specifically, the learning rate for the latter layers was maintained at five times that applied to the initial frozen layers. This parameter configuration yielded 94.34% classification accuracy on the test set. All experiments were executed in MWORKS 2025 developed by Suzhou Tongyuan, Suzhou, China.

4.2.3. Algorithm Comparison and Performance Evaluation Metrics

To verify the proposed DTL-DFD approach’s applicability without physical data, the experiment used Support Vector Regression (SVR), Time Series Fuzzy Neural Network (TS-FNN), and 1D-CNN as comparative methods. The experimental dataset was constructed exclusively using simulated data. A multi-operating-condition simulation model of the marine power condensate system generated all data, encompassing five common fault classes. The complete dataset comprises 1000 standardized one-dimensional time series samples (200 per fault class), with each sample containing 10 data points acquired at 10 kHz. Data partitioning followed a 7:2:1 ratio for the training, validation, and test sets, serving the respective functions of initial model training, hyperparameter tuning, and final performance evaluation.

To validate diagnostic robustness, an extensive hierarchical evaluation framework was developed in this study to assess the DTL-DFD approach. Let x denote the samples in the test set $D_{test}$, $y_i$ denote the actual labels of the test samples, and ${\widehat{y}}_i$ denote the predicted values output by the diagnostic model. Consequently, the accuracy $Acc$ can be written as follows:

$$ACC={\displaystyle \frac{\left|\left\{x\mid x\in D_{\mathrm{test}}\cap {\widehat{y}}_i=y_i\right\}\right|}{\left|\left\{x\mid x\in D_{\mathrm{test}}\right\}\right|}}$$

(33)

To quantitatively measure and compare the degree of performance improvement for different source domains in addressing practical diagnostic issues, two metrics, $I_{ACC}$ and $I_{Std}$, are used, defined as follows:

$$I_{\mathrm{Acc}}={\mathrm{Acc}}_{(D_s\to R_{\mathrm{ft}})}-{\mathrm{Acc}}_{(R\to R)}$$

(34)

$$I_{\mathrm{Std}}=\mathrm{std}\left({\mathrm{Acc}}_{(R\to R)}\right)-\mathrm{std}\left({\mathrm{Acc}}_{(D_s\to R_{\mathrm{ft}})}\right)$$

(35)

where $\mathrm{std}\left(·\right)$ represents the standard deviation of $\left(·\right)$, and $D_s$ denotes distinct source domain data, $D_s\in \{S,R_1\}$.

The metric $I_{\mathrm{ACC}}$ represents the difference in the accuracy of two diagnostic tasks, while the $I_{\mathrm{STD}}$ metric indicates the difference in the stability of the identification performance of two diagnostic tasks across multiple repeated experiments. A stronger improvement effect is indicated by a higher value for both metrics.

4.3. Results and Analysis

4.3.1. Accuracy of Different Algorithms

Based on the performance comparison curves of the training process shown in Figure 7, this study systematically analyzes the dynamic performance across three models. The experimental results demonstrate that DTL-DFD exhibits significant performance advantages. Its accuracy steadily increases throughout 100 training epochs, surpassing 90%—representing relative improvements of 5.01%, 8.02%, and 13.8% over standard 1D-CNN, FNN, and Support Vector Machine baselines, respectively. Regarding convergence characteristics, DTL-DFD achieves optimal loss minimization, stabilizing below 0.3 with a smooth optimization trajectory that outperforms both conventional 1D-CNN and Support Vector Machines.

Notably, DTL-DFD demonstrates superior generalization capabilities, maintaining higher validation accuracy than training accuracy throughout the learning process. By contrast, the standard 1D-CNN exhibits validation loss rebound after 300 epochs, indicating overfitting susceptibility. Traditional Support Vector Machines display substantial performance oscillations attributable to fixed kernel function limitations. These findings empirically validate DTL-DFD’s cross-domain feature transfer effectiveness, establishing a robust methodology for small-sample intelligent diagnostic applications.

4.3.2. Confusion Matrices of Different Algorithms Showing the Specific Diagnostic Conditions of Each Category

Based on the comparative analysis of confusion matrices presented in Figure 8, this study systematically evaluates the fault diagnosis performance across four models: 1D-CNN, SVM, FNN, and the DTL-DFD framework. Experimental results demonstrate DTL-DFD’s superior classification capability, achieving 94.34% overall accuracy—significantly outperforming the standard 1D-CNN (89.62%), FNN (86.79%), and SVM (81.13%) benchmarks.

Critically, DTL-DFD achieves 100% recognition accuracy for all normal operating states (Label 6) while maintaining exceptionally low misclassification rates for critical fault conditions: 3.03% (Fault 1), 5.83% (Fault 2), and 3.33% (Fault 3). Crucially, no faulty states were misclassified as normal, eliminating hazardous diagnostic misinterpretations. In stark contrast, the SVM model exhibits severe performance degradation for complex states, demonstrating a 50% misclassification rate for Fault 3 while erroneously categorizing 12.7% of fault conditions as normal states.

Visual analysis of confusion matrix heatmaps further confirms DTL-DFD’s robustness, revealing intense concentration along the primary diagonal and minimal off-diagonal color block density. This pattern visually corroborates the model’s sharp classification boundaries and decision certainty. These findings empirically validate that transfer learning mechanisms enhance cross-domain feature reuse, establishing DTL-DFD as a reliable solution for industrial equipment diagnosis under small-sample constraints.

4.3.3. Scatter Plots of Different Algorithms Showing Specific Diagnostic Conditions of Each Category

Figure 9 visualizes two-dimensional PCA projections of features learned by distinct models. While Figure 9a–c (1D-CNN, SVM, and FNN, respectively) exhibit clustering overlap between fault conditions—consistent with their confusion matrix performance—the proposed DTL-DFD model (Figure 9d) demonstrates superior discriminative capability. Its features form six distinct, well-separated clusters corresponding precisely to each diagnostic category, with minimal inter-class overlap. This geometric isolation confirms the model’s capacity to extract condition-specific representations across operational domains, validating its cross-domain feature learning efficacy.

4.3.4. Comparison of Diagnostic Results with Different Parameters

Figure 10 shows the heatmap of diagnostic accuracy as a function of learning rate and number of migrated layers, which identifies the optimal parameter range across multiple faults and operating scenarios, highlighting that deeper transfers and medium learning rates improve robustness. This study examines the influence of transfer depth and learning rate on cross-domain fault diagnosis performance, covering the five fault modes listed in Table 1 (F1–F5) and dynamic operating conditions such as ±15% load fluctuations, coolant flow reduction, and fouling thermal resistance changes. The results show that deep transfer (7–9 layers) with medium learning rates (0.0022–0.01) achieves the best adaptability, with a peak accuracy of 94.34% at 9-layer transfer and $\eta$ equals 0.01, representing a 16.63% improvement over shallow configurations and significantly enhancing the detection of high-risk faults such as F3 (tube blockage). In contrast, shallow transfer (1–3 layers) requires much lower learning rates (0.0002–0.0005) but reaches only 77.71% accuracy due to limited feature transfer capacity.

5. Conclusions

The DTL-DFD method, integrating digital twin (DT) and deep transfer learning (DTL), effectively addresses critical challenges in fault diagnosis for marine steam turbine condensate systems. Key outcomes are the following:

(1)

Overcoming small-sample constraints: Leveraging a high-fidelity DT model and full-parameter transfer from a pre-trained 1D-CNN, the method achieves reliable diagnosis even with extremely limited real fault data (<0.5% of operational data), avoiding overfitting issues of traditional approaches.

(2)

Enhancing cross-domain stability: Through hierarchical parameter transfer and fine-tuning, the method attains 94.34% accuracy on cross-distribution test sets—4.72% higher than state-of-the-art methods—with a low standard deviation ($I_{\mathrm{Std}}$ = 0.5%), demonstrating strong adaptability to dynamic operating conditions.

(3)

Establishing critical principles for industrial applications: These findings establish three critical principles for industrial applications: (1) transfer depth should be at least 7 layers, (2) $\eta$ = 0.01 maximizes feature reuse, and (3) shallow transfer suffers inherent performance limitations. The resulting parameter optimization framework provides a practical reference for multi-fault, cross-domain diagnostics.

Future research could further investigate optimizing the generalization capability of DTL-DFD in more complex industrial scenarios. Additionally, integrating real-time data-driven mechanisms could enhance early warning accuracy, thereby promoting deeper application of intelligent fault diagnosis technology in marine engineering and broader industrial domains.