Data Driven Comprehensive Performance Evaluation of Aeroengines: A Network Dynamic Approach

Yuting Wang; Feng Liu; Feng Xi; Bofei Wei; Dongli Duan; Zhiqiang Cai; Shubin Si

doi:10.1016/j.eng.2024.11.024

Engineering ›› 2025, Vol. 46 ›› Issue (3) :307 -321. DOI: 10.1016/j.eng.2024.11.024

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Data Driven Comprehensive Performance Evaluation of Aeroengines: A Network Dynamic Approach

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Abstract

Aeroengines, often regarded as the heart of aircraft, are crucial for flight safety and performance. Comprehensive performance evaluation of aeroengines supports Prognostics and Health Management (PHM) and aeroengine digital engineering. Due to their highly integrated nature, aeroengines present challenges in performance evaluation because their test-run data are high-dimensional, large-scale, and exhibit strong nonlinear correlations among test indicators. To solve this problem, this study proposes a unified framework of the comprehensive performance evaluation of aeroengines to assess performance objectively and globally. Specifically, the network model and the dynamics model of aeroengine performance are constructed driven by test-run data, which can explain the patterns of system state changes and the internal relationship, and depict the system accurately. Based on that, three perturbations in the model are used to simulate three fault modes of aeroengines. Moreover, the comprehensive performance evaluation indexes of aeroengines are proposed to evaluate the performance dynamically from two dimensions, the coupling performance and the activity performance. Thirteen test-run qualified and four test-run failed aeroengines are used to validate and establish the qualified ranges. The results demonstrate that the comprehensive evaluation indexes can distinguish test-run qualified and test-run failed aeroengines. By changing the dynamic parameters, the comprehensive performance under any thrust and inlet guide vanes (IGV) angle can be estimated, broadening the test-run scenarios beyond a few typical states. This novel approach offers significant advancements for the comprehensive performance evaluation and management of aeroengines, paving the way for future PHM and aeroengine digital engineering developments.

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Keywords

Comprehensive performance evaluation / Aeroengine performance / Network / Resilience

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Yuting Wang, Feng Liu, Feng Xi, Bofei Wei, Dongli Duan, Zhiqiang Cai, Shubin Si. Data Driven Comprehensive Performance Evaluation of Aeroengines: A Network Dynamic Approach. Engineering, 2025, 46(3): 307-321 DOI:10.1016/j.eng.2024.11.024

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1. Introduction

1.1. Motivation

As the central component of aircraft, the comprehensive performance of aeroengines is crucially linked to both flight safety and performance. According to statistics, from 1963 to 1975, US Air Force fighter jets were involved in a total of 3824 flight accidents, with 1664 attributed to aeroengine issues, accounting for 43.5% of the total. Between 1989 and 1993, global air transportation experienced 279 major aviation accidents, with aeroengine failure responsible for over 20% [1]. According to the International Civil Aviation Organization Aircraft Accident and Investigation Report, from 2008 to 2017, 67 aviation accidents were caused by aeroengine failure, accounting for 30% of all aviation accidents, resulting in significant economic losses and casualties [2]. In addition, aeroengines are highly complex engineering systems, and their maintenance and operation costs are very expensive, accounting for approximately 72% of the total life cycle cost [3]. Thus, the aeroengine stands as one of the most critical components of aircraft, necessitating comprehensive performance evaluation.

The comprehensive performance evaluation of aeroengines refers to the global and holistic assessment of aeroengines which are typical complex systems. This process involves applying specific methods to assign an evaluation value (or index) to each aeroengine based on given conditions, encompassing all of its attributes, and making decisions accordingly. It serves as a foundational component for Prognostics and Health Management (PHM). Precise evaluation can accurately reflect equipment performance status, minimize downtime due to performance degradation, and facilitate proactive maintenance to prevent accidents or losses [4]. More importantly, comprehensive performance evaluation is pivotal in aeroengine digital engineering. Aeroengine digital engineering uses digital technology in a virtual reality environment to design, assemble, and test aeroengine systems. It supports the entire lifecycle of aeroengines, from planning and design to maintenance, using digital tools and methods. Innovations in big data analysis, cognitive technology, and advanced computing further enhance these practices, facilitating real-time interaction and decision-making among stakeholders [5]. The comprehensive performance evaluation of aeroengines is a key part of testing, use, and maintenance. Accurate and quantitative comprehensive performance evaluation of aeroengines facilitates intelligent decision-making, effectively meeting aeroengine requirements, and supporting preliminary programs and project planning. Therefore, conducting a comprehensive performance evaluation of aeroengines is crucial.

Presently, aeroengine performance evaluation is extensively utilized in practical engineering. Airlines typically rely on a single parameter, such as the exhaust gas temperature (EGT) indicator or the remaining EGT margin indicator, to assess the performance of aeroengines [6]. Aeroengine manufacturers have a complete test-run process to evaluate the condition of aeroengines. In addition to basic inspections, they conduct performance tests on aeroengines under several typical operating conditions during the test-run. Several typical states are simulated to record test indicators by setting different thrusts, high-pressure compressor (HPC) rotor speed, the angle of the inlet guide vane (IGV) of the HPC (IGV angle), or throttle lever position. Focus on the thrust, HPC rotor speed, temperature, and fuel consumption rate, and draw the performance curves between them to verify compliance with specifications. Furthermore, monitoring specific temperatures and overall vibration under the aforementioned typical states is essential.

In general, there are notable shortcomings in the performance evaluation of aeroengines in practical engineering applications. The single-parameter method used by airlines is easy to operate and can quickly provide a certain basis for maintenance decisions. However, as a complex system, the aeroengine has numerous parameters affecting its performance, so relying solely on a single parameter cannot fully reflect its actual performance. For the test-run process used by aeroengine manufacturers, it is obvious that this process can help manufacturers carry out the basic inspection of aeroengines, but the isolated consideration of certain key test indicators and their relationships cannot represent their comprehensive performance. Besides, the operational status of aeroengines in actual use is constantly changing, and evaluating only a few typical states is incomplete. Therefore, further research is necessary to enhance the comprehensive performance evaluation of aeroengines in practical engineering, addressing these limitations.

1.2. Related work

To date, much research has been dedicated to the comprehensive performance evaluation of aeroengines, categorized broadly into three domains: traditional methods and their enhancements, artificial intelligence methods, and model-based approaches.

Traditional methods and their enhancements for comprehensive performance evaluation of aeroengines, guided by the decomposition-integration paradigm, aim to provide a thorough assessment of aeroengines, typical complex systems in a big data environment. Those methods, including analytic hierarchy process (AHP) [7], evidential reasoning [8], [9], principal component analysis (PCA) [10], [11], data envelopment analysis [12], [13], [14], the technique for order preference by similarity to an ideal solution (TOPSIS) [15], [16], [17], among others, have been extensively studied and shown promising results. Applebaum [18] utilized a fuzzy diagnostic strategy in developing total health usage and monitoring systems for evaluating performance and ensuring the safety of aeroengines. Hou et al. [19] introduced a global optimization method using chaotic variables to calculate weight coefficients for diverse parameters in the comprehensive performance evaluation of aeroengines. This approach enhances sensitivity compared to expert survey methods, accurately reflecting overall performance. Demirci et al. [20] established an automated engine health monitoring system for commercial aircraft by implementing fuzzy logic that relied on expert knowledge and online data. This system can evaluate the aeroengine performance during operation and generate output values that represent whether the aeroengine has malfunctioned. Wang et al. [21] presented an aeroengine performance evaluation model, which integrated the fuzzy AHP, fuzzy preference programming, and TOPSIS. Specifically, the fuzzy AHP and fuzzy probability analysis are used to determine the relative weights of multiple evaluation indicators, synthesize the rating of aeroengines, and then TOPSIS is applied to make the final decision on the overall performance of each alternative. Huang et al. [22] employed multi-linear regression to construct a health index for assessing turbofan engine performance. Cai et al. [23] proposed a similarity-based state assessment method using the kernel two-sample test to query analogous run-to-failure profiles, followed by preliminary remaining useful life predictions and Weibull analysis for probability distribution. However, the gaps of this type of method that are difficult to bridge include: Firstly, human-set evaluation indexes and index systems are subjective and may not fully capture the overall system characteristics; Secondly, the evaluation results are mainly obtained through linear operation of indicators, which can only describe the quantitative changes of the system, and limit their ability to reflect the qualitative changes and emergence patterns of the system which caused by the nonlinear relationships and fusion of indicators. This makes it difficult to accurately evaluate the aeroengine performance states, potentially leading to erroneous estimations.

Artificial intelligence technology, which is rapidly advancing, has also been applied to the comprehensive performance evaluation of aeroengines. The genetic algorithm serves as an effective optimization tool, deriving a set of component parameters via a nonlinear gas turbine model to obtain the prediction that best matches the measured values [24], [25]. Bettocchi et al. [26] conducted extensive research on neural network methods, proposing algorithms and structures tailored for assessing the aerodynamic performance of aeroengines, yielding accurate evaluation. Additionally, they explored fuzzy neural network theory in depth, devising effective methods for aeroengine evaluation [27]. Zhou et al. [28] used the support vector machine to simulate the nonlinear characteristics of the aeroengine, describing the deviation degree from normal operating states at the whole machine level. Ma et al. [29] developed a framework for aeroengine state-of-health assessment that uses density-distance clustering for pseudo labels and a fuzzy Bayesian risk model for weight assignment, which introduces two indicators to enhance the comprehensiveness of performance evaluations. Xiang et al. [30] used multicellular long short-term memory (LSTM) artificial neural networks and deep neural networks to fuse multiple performance parameters, extracting health indicators for aeroengines. De Pater and Mitici [31] developed an LSTM autoencoder to generate health indicators for aircraft systems and subsequently predicted the remaining useful life. Pan et al. [32] utilized a Bayesian neural network trained on flight data and failure rates to evaluate aeroengine operational states and utilized Shapley additive explanations values calculated with the light gradient boosting machine to quantify the effect of each feature on the operating state. While these recent artificial intelligence methods compensate for the shortcomings of traditional and enhanced methods to a large extent, it is difficult to explain the mechanism of aeroengine state change and internal relationship scientifically, which limits its broader application.

In order to achieve objective and interpretable comprehensive performance evaluation of aeroengines, many scholars put forward model-based methods. The most prominent one is gas path analysis (GPA). GPA, a linear model method, was introduced for the first time in 1967 by Urban [33]. GPA utilizes a method based on a fault coefficient matrix to analyze the correlation between aeroengine component degradation and gas path parameter deviations. This method has found extensive applications in studies conducted by Nieden and Fiedle [34] and Simani et al. [35]. It is evident that aeroengines are nonlinear and complex thermodynamic systems, and linear models cannot provide accurate aeroengine performance and biases. Escher and Singh [36] sought to enhance the accuracy of this method by employing the Newton–Raphson iterative approach to resolve nonlinear relationships between aeroengine performance parameters and measurement indicators. This method is limited to evaluating the performance at a small scale, and the inaccurate coefficient matrix as well as the correlation between measurement indicators seriously affect the accuracy. Li and Ying [37] proposed a diagnostic method for the GPA under transient conditions, employing an equivalent cooling flow processing technique for thermodynamic modeling and a local optimization strategy for steady-state performance evaluation. However, classic linear models, such as GPA, although uncomplicated, tend to be insufficiently precise. Alternatively, the nonlinear model-based methods involve complex iterative processes that are resource-intensive and do not always guarantee convergence.

Aeroengines are typical complex systems, and in fact, a large number of complex systems can be described and analyzed as a whole through complex networks. The complex network model originated from the idea of graph theory. In the mid-19th century, Erdös and Rényi [38] proposed the Erdös–Rényi (ER) random graph model. In 1967, American social psychologist Stanley Milgram discovered the phenomenon of six degrees of separation in social networks through the email experiment [39]. At the end of the last century, the discovery of the small world feature and the scale-free feature of the Internet sparked a wave of research in network science. Due to the fact that the complex network can help us better understand and analyze the characteristics, rules, and mechanisms of complex systems such as society, biology, information, and transportation, it has become an important supporting theory and method of complex system management. Resilience is the ability of a system to adjust its own behavior to maintain its basic functionality in the event of errors, failures, and environmental changes, which is an inherent attribute of complex systems [40]. Over the past 50 years, resilience has been increasingly utilized in science and engineering, making it a multidisciplinary concept [41], [42]. Owing to the usual obscure intrinsic dynamics and the inherent limitations of analytical tools, much research has concentrated on analyzing low-dimensional systems or time series data without explicit modeling [43], [44], [45]. Extracting these functions from complex systems requires precise system wiring diagrams and nonlinear dynamic descriptions of their interactions. The emergence of network science provides a powerful tool for characterizing the structure of large-scale complex networks [46], [47]. In the past two decades, important topological structures of real-world networks have been discovered and characterized [48]. In addition, the accumulation of massive data and the rapid development of computational methods make it possible to directly identify and predict the exact form of dynamic models from empirical data [49], [50], [51]. These two prerequisites make it possible to develop analytical tools for the resilience of large-scale complex networks. In 2016, Gao et al. [40] proposed an analytical framework for identifying the natural control and state parameters of large-scale complex systems, helping to deduce the effective one-dimensional (1D) dynamic and accurately predict the critical point at which the system loses its resilience. The proposed analytical framework enables us to systematically separate the effects of system dynamic and system topology. Based on the theoretical tools of low-dimensional and large-scale networks mentioned above, as well as advanced data analysis techniques, it is possible to study the resilience of real networks in various fields from the natural world to the artificial world [52], [53], [54]. However, there is still a lack of application of complex network and resilience theory in the comprehensive performance evaluation of complex equipment, such as aeroengines.

To sum up, the comprehensive performance evaluation of aeroengines still faces challenges in practical applications that need to be solved:

(1)The first aspect is how to use the collected data to evaluate the comprehensive performance of aeroengines from a systemic perspective. Aeroengines are highly integrated complex systems, including many components and subsystems, such as the compressor, combustion chamber, and turbine; these components interact with each other. Multiple performance parameters of aeroengines can reflect their performance to some extent, and these performance parameters have complex nonlinear correlations with each other. With the continuous development of sensors and big data technology, more and more data are collected which tend to be high-dimensional, multi-source, multi-modal, and coupled. The above reasons make the comprehensive performance evaluation of aeroengines exceptionally complex and urgently require a new method to conduct research from the perspective of the whole system.

(2)The second aspect pertains to the necessity for the comprehensive performance evaluation of aeroengines to be objective and interpretable. This implies that methods involving the artificial and arbitrary setting of weights for indicators or solely considering linear relationships within the system are no longer suitable. In addition, the comprehensive performance evaluation method and obtained results should describe and explain system state changes and the nonlinear and coupled relationship between internal indicators.

(3)The third aspect is that a method for dynamically evaluating aeroengine performance needs to be proposed. To achieve PHM and advance aeroengine digital engineering, real-time evaluation based on operational data is necessary [5]. Previous comprehensive evaluations often yielded single-value or qualitative conclusions. However, the aeroengine state is dynamic, and the comprehensive performance is not unchanged but a dynamic curve.

Therefore, this study undertakes the comprehensive performance evaluation of aeroengines through the lens of complex network and resilience theory, offering promising solutions to these challenges.

1.3. Contribution

To address the above issues, a unified framework of the comprehensive performance evaluation of aeroengines is proposed as illustrated in Fig. 1. Firstly, viewed through the lens of the complex network and driven by the test-run data, the positive and negative correlations and strengths between test indicators are described and the aeroengine performance network is constructed. Secondly, a differential equation of aeroengine performance dynamics is established to depict the dynamic evolution of the system. Three types of perturbation of the model are employed to simulate three fault modes of the aeroengine and elucidate the practical implications. Thirdly, utilizing the dimensionality reduction method of network dynamics and the Frobenius norm, the comprehensive performance evaluation indexes of aeroengines are proposed. The scientific rigor and effectiveness of this approach are validated using data from 13 test-run qualified and four test-run failed aeroengines. The results indicate that the fitting errors of the proposed dynamics are predominantly less than 1%, demonstrating that the comprehensive performance of aeroengines at each stage can be well characterized. Moreover, the proposed comprehensive evaluation indexes effectively distinguish between qualified and faulty aeroengines at each stage. Additionally, by adjusting dynamic parameters, the comprehensive performance of aeroengines under varying thrusts and IGV angles can be estimated. This methodology can offer a novel approach for comprehensive performance evaluation and management decision-making of aeroengines, opening up new possibilities for future PHM and aeroengine digital engineering.

The primary contributions of this study are summarized as follows:

(1)A comprehensive performance evaluation framework of aeroengines is proposed for this typical complex system. This framework enables objective and global evaluation of the highly integrated, internally coupled, and nonlinear aeroengine system.

(2)The construction of the aeroengine performance network model and the aeroengine performance dynamics model driven by test-run data. This integration can accurately depict system states at various stages and explain the relationships between internal test indicators and patterns of system state changes. That means it is possible to explore the quantitative change, qualitative change and emergence patterns of the aeroengine which is a typical complex system from the perspective of network structures and dynamics.

(3)The comprehensive performance evaluation indexes of aeroengines are proposed, assessing performance in two dimensions: coupling and activity. These indexes can dynamically describe the aeroengine state, differentiate between qualified and faulty units, and provide production guidance.

2. Materials and methods

2.1. Aeroengine delivery process and test-run data description

To ensure the performance, reliability, safety, and lifespan requirements, a strict quality control mode is adopted during the aeroengine delivery process by manufacturers. Fig. 2 provides an overview of the aeroengine delivery process. Initially, an aeroengine is assembled from components and must satisfy various assembly requirements. Subsequently, the aeroengine undergoes a test-run lasting 2–15 hours under artificially set combinations of thrust and IGV angle. Sensors monitor test indicators such as pressure, temperature, speed, and flow rate to verify compliance. If all key test indicators meet requirements, the aeroengine will be evaluated as test-run qualified; otherwise, it undergoes reassembly and retesting until compliance is achieved.

In this paper, a specific type of aeroengine is selected to be checked under six states during the test-run. The six states are as follows (a and b represent two specific numbers respectively):

•State 1: The thrust is artificially set to a pounds, and the IGV angle is maintained at the initial value;

•State 2: The thrust is artificially set to a-1000 pounds, and the IGV angle is maintained at the initial value;

•State 3: The thrust is artificially set to a-2000 pounds, and the IGV angle is maintained at the initial value;

•State 4: No thrust setting and the IGV angle is artificially set to b degrees;

•State 5: No thrust setting and the IGV angle is artificially set to b-7.5 degrees;

•State 6: No thrust setting and the IGV angle is artificially set to b-15 degrees.

Through the above test-run, the test-run data is collected which is the time series data of 28 test indicators including fuel flow, low-pressure compressor (LPC) rotor speed, HPC rotor speed, turbine outlet temperature, oil temperature, oil pressure, and so forth. The test indicators used in this study, along with their meanings and selection methods, are detailed in Section S1 in Appendix A. To be specific, Fig. 2 shows the test-run data of the specific type of aeroengine, with the horizontal coordinate being time and the vertical coordinate being the value of each test indicator, clearly delineating six distinct stages and highlighting variations in performance across different operational conditions. This study collected test-run data from 17 groups of aeroengines following the described procedure, with 13 test-run qualified aeroengines and 4 test-run failed aeroengines. These test-run failed aeroengines exhibited a common issue: the vibration of the entire machine exceeding permissible limits during the test-run, significantly impacting aeroengine performance.

2.2. Aeroengine performance network model

The complex network models serve to understand real-world complex systems. Various complex systems, including infectious disease spread, power grids, and aviation networks, can be analyzed using complex network models. These models all abstract entities in the complex systems into nodes and relationships between entities into connections for modeling and analysis. The aeroengine test-run process evaluates the assembly capability and quality of aeroengines. From the complex network perspective, test indicators from 2–15 hours of test-run data are abstracted as nodes, and their relationships are abstracted as connections to build the aeroengine performance network model, in order to explore and mine the evolution laws of aeroengine performance contained in test-run data.

Assuming that the coupling relationships between test indicators remain stable throughout a complete test-run, since the internal structure of an aeroengine will not undergo significant changes within several hours. The relationship between two test indicators

(x_{i}, x_{j})

is built based on the correlation coefficient of their change rates throughout the entire test-run, calculated as follows

(1)

ρ (Δ x_{i}, Δ x_{j}) = c o v (Δ x_{i}, Δ x_{j}) / (σ_{Δ x_{i}} σ_{Δ x_{j}})

where

ρ (Δ x_{i}, Δ x_{j})

is the correlation coefficient between the change rates of test indicators

Δ x_{i}

and

Δ x_{j}

c o v (Δ x_{i}, Δ x_{j})

is the covariance between

Δ x_{i}

and

Δ x_{j}

σ_{Δ x_{i}}

and

σ_{Δ x_{i}}

are the standard deviations of

Δ x_{i}

and

Δ x_{j}

, separately. This paper uses a threshold of 0.2, meaning that the interaction between two test indicators is not considered when the correlation coefficient between them is less than 0.2. The detailed discussion on threshold determination is presented in Section S2 in Appendix A. Therefore, the elements of the entire weighted adjacency matrix

W

(2)

W_{ij} = \{\begin{cases} {\hat{W}}_{ij} = ρ (Δ x_{i}, Δ x_{j}), i f ρ (Δ x_{i}, Δ x_{j}) \geq 0.2 \\ 0, i f |ρ (Δ x_{i}, Δ x_{j})| < 0.2 \\ {\bar{W}}_{ij} = - ρ (Δ x_{i}, Δ x_{j}), i f ρ (Δ x_{i}, Δ x_{j}) \leq - 0.2 \end{cases}

that is, the edge exists when

W_{ij} \neq 0

, otherwise, it does not exist. So, the entire weighted adjacency matrix can be written as

(3)

W = \hat{W} + \bar{W}

\hat{W}

and

\bar{W}

are the positive and negative weighted adjacency matrices of the network, respectively.

The entire weighted adjacency matrix

W

is symmetric, meaning that

W_{ij} = W_{ji}

, which indicates that two test indicators

(x_{i}, x_{j})

mutually influence each other equally. The plus or minus of

W_{ij}

reveals whether the influence between two test indicators

(x_{i}, x_{j})

is positive or negative. A positive correlation indicates that an increase in one test indicator stimulates an increase in the other, while a negative correlation suggests that an increase in one stimulates a drop in the other. In actual aeroengine operation, the interactions between test indicators are extensive. For example, as the LPC rotor speed increases, the HPC rotor speed also increases, while the vibration value decreases. Additionally, the absolute value of

W_{ij}

reflects the extent of interaction between two test indicators

(x_{i}, x_{j})

. For example, the

W_{ij}

between the LPC rotor speed and the HPC rotor speed is approximately 0.8, which is structurally attributed to their connection through the same shaft. The entire weighted adjacency matrix

W

serves as the core of the aeroengine performance network model since it encapsulates all test indicators and the interactions among them. Fig. 3 illustrates the network performance models and corresponding topological properties of an aeroengine under different service times. Specifically, three network performance models of three complete test-run processes are constructed as shown in Figs. 3(b)–(d).

2.3. Aeroengine performance dynamics model

Through the construction of the network model, two types of relationships are identified between test indicators, positive or negative. Given the prevalence of these relationships in the constructed network model, differential equations are employed to simulate dynamic changes among test indicators. The genomic regulatory network aims to identify underlying relationships within a dataset, mirroring the operation of gene regulatory mechanisms. Regulatory dynamics are represented using the Michaelis–Menten equation to model gene regulatory interactions [55], [56], and can be caught by

(4)

\frac{d x_{i}}{d t} = - B x_{i} + R_{1} \sum_{j = 1}^{N} {\hat{A}}_{ij} \frac{x_{j}^{h}}{1 + x_{j}^{h}} + R_{2} \sum_{j = 1}^{N} {\bar{A}}_{ij} \frac{1}{1 + x_{j}^{h}}

where the parameter h is the Hill coefficient which we set to h = 1,

B

is the death rate,

N

is the number of nodes that are not isolated,

R_{1}

and

R_{2}

represent the excitation and inhibition strength, separately.

{\hat{A}}_{ij}

and

{\bar{A}}_{ij}

are the active and suppressive adjacency matrices of the gene regulatory network, respectively. So, the entire adjacency matrix can be written as

A = \hat{A} + \bar{A}

The test-run state of the aeroengine is determined by the artificially set combinations of thrust and IGV angle. For each state, we introduce the thrust parameters and the IGV angle parameters into the dynamic equation by using the idea of gene regulatory dynamics. The entire weighted adjacency matrix

W

demonstrates the interactions among test indicators; however, these interactions are regulated by the set thrust and the set IGV angle. In reality, the influence of test indicators on the others varies, so we assume that parameters

B

R_{1}

, and

R_{2}

do not assume the same value during the fitting procedure. Thereby,

R_{1 i}

is introduced to represent the influence of engine thrust, and the corresponding

R_{2 i}

represents the influence of IGV angle. Here,

R_{1 i}

is composed of the excitation term

{\hat{R}}_{1 i}

and the inhibition term

{\bar{R}}_{1 i}

, and

R_{2 i}

is composed of the excitation term

{\hat{R}}_{2 i}

and the inhibition term

{\bar{R}}_{2 i}

. Meanwhile, to reduce fitting errors, the weighted adjacency matrix

W

is used instead of the adjacency matrix

A

to provide more accurate interactions. Therefore, the dynamics could be changed into

(5)

\begin{array}{l} \frac{d x_{i}}{d t} = - B_{i} x_{i} + {R_{1}}_{i} (\sum_{j = 1}^{N} {\hat{W}}_{ij} \frac{x_{j}}{1 + x_{j}} + \sum_{j = 1}^{N} {\bar{W}}_{ij} \frac{1}{1 + x_{j}}) + {R_{2}}_{i} (\sum_{j = 1}^{N} {\hat{W}}_{ij} \frac{x_{j}}{1 + x_{j}} + \sum_{j = 1}^{N} {\bar{W}}_{ij} \frac{1}{1 + x_{j}}) \\ = - B_{i} x_{i} + {\hat{R}}_{1 i} \sum_{j = 1}^{N} {\hat{W}}_{ij} \frac{x_{j}}{1 + x_{j}} + {\bar{R}}_{1 i} \sum_{j = 1}^{N} {\bar{W}}_{ij} \frac{1}{1 + x_{j}} + {\hat{R}}_{2 i} \sum_{j = 1}^{N} {\hat{W}}_{ij} \frac{x_{j}}{1 + x_{j}} + {\bar{R}}_{2 i} \sum_{j = 1}^{N} {\bar{W}}_{ij} \frac{1}{1 + x_{j}} \\ = - B_{i} x_{i} + R_{E i} \sum_{j = 1}^{N} {\hat{W}}_{i j} \frac{x_{j}}{1 + x_{j}} + R_{I i} \sum_{j = 1}^{N} {\bar{W}}_{i j} \frac{1}{1 + x_{j}} \end{array}

where

B_{i}

is the proportion of node

i

returning to normal range.

{\hat{R}}_{1 i}

{\bar{R}}_{1 i}

{\hat{R}}_{2 i}

, and

{\bar{R}}_{2 i}

are the excitation and inhibition strength of node

i

under the set thrust and the set IGV angle,

(6)

\begin{array}{l} R_{1 i} = \{\begin{cases} {\hat{R}}_{1 i} = ρ (Δ x_{t h r u s t}, Δ x_{i}), i f ρ (Δ x_{t h r u s t}, Δ x_{i}) \geq 0 \\ {\bar{R}}_{1 i} = - ρ (Δ x_{t h r u s t}, Δ x_{i}), i f ρ (Δ x_{t h r u s t}, Δ x_{i}) \leq 0 \end{cases} \\ R_{2 i} = \{\begin{cases} {\hat{R}}_{2 i} = ρ (Δ x_{I G V}, Δ x_{i}), i f ρ (Δ x_{I G V}, Δ x_{i}) \geq 0 \\ {\bar{R}}_{2 i} = - ρ (Δ x_{I G V}, Δ x_{i}), i f ρ (Δ x_{I G V}, Δ x_{i}) \leq 0 \end{cases} \end{array}

R_{E i}

and

R_{I i}

are used to represent the excitation term and the inhibition term, which means

R_{E i} = {\hat{R}}_{1 i} + {\hat{R}}_{2 i}

and

R_{I i} = {\bar{R}}_{1 i} + {\bar{R}}_{2 i}

respectively.

Δ x_{t h r u s t}

represents the change rate of the thrust of the aeroengine, while

Δ x_{I G V}

denotes the change rate of the IGV angle.

However,

B_{i}

of each node is an unknown parameter and needs to be estimated. When the system reaches a steady state, the Eq. (5) must satisfy

(7)

\frac{d x_{i}}{d t} = 0

which can be written as

(8)

B_{i} x_{i} = R_{E i} \sum_{j = 1}^{N} {\hat{W}}_{ij} \frac{x_{j}}{1 + x_{j}} + R_{I i} \sum_{j = 1}^{N} {\bar{W}}_{ij} \frac{1}{1 + x_{j}}

From this, N equations about

B_{i}

can be obtained

(9)

B_{i} = R_{E i} \sum_{j = 1}^{N} {\hat{W}}_{ij} \frac{x_{j}}{x_{i} (1 + x_{j})} + R_{I i} \sum_{j = 1}^{N} {\bar{W}}_{ij} \frac{1}{x_{i} (1 + x_{j})}

Hence, the unknown parameter

B_{i}

of each node can be obtained by solving Eq. (9) simultaneously.

In general, the aeroengine dynamics model employs 28 differential equations to simulate the dynamic variations of test indicators respectively, as indicated in Eq. (5), implying that each test indicator

x_{i}

has its unique dynamic model

\frac{d x_{i}}{d t}

. This unique dynamic model, essentially a differential equation, consists of two parts: the self-dynamic

- B_{i} x_{i}

and the interaction dynamic

R_{E i} \sum_{j = 1}^{N} {\hat{W}}_{ij} \frac{x_{j}}{1 + x_{j}} + R_{I i} \sum_{j = 1}^{N} {\bar{W}}_{ij} \frac{1}{1 + x_{j}}

, and these two parts work together to determine the state of the test indicator

x_{i}

. The self-dynamic captures the inherent changes of the test indicator, while the interaction dynamic describes the changes induced by other test indicators in the system. The strengths of these changes are controlled by the dynamic parameters that represent the system characteristics, specifically

B_{i}

R_{E i}

, and

R_{I i}

. In this paper, the set thrust and the set IGV angle are regarded as significant determinants of system characteristics in the aeroengine dynamics model, as aeroengine manufacturers evaluate whether aeroengines meet the requirements under consistent conditions with varying combinations of thrust and IGV angle during the test-run process. Consequently, we utilize test-run data to calculate the corresponding

R_{1 i}

and

R_{2 i}

of test indicator

x_{i}

under the current thrust and IGV angle through Eq. (6), thereby obtaining

R_{E i}

and

R_{I i}

. Next, we estimate the corresponding

B_{i}

using the steady state of the complex system. In this way, the construction of the aeroengine performance dynamics model is completed. These test indicators and corresponding dynamic parameters are vital in the comprehensive performance evaluation of aeroengines: An increase in the overall levels of the test indicators

x_{i}

and the corresponding parameters

B_{i}

suggests that the system possesses a stronger self-regulation ability, that is, the activity performance defined below; an increase in the overall levels of interaction strengths among the test indicators, reflected in

R_{E i}

R_{I i}

, and

W

, indicates that the system has a stronger internal coupling ability, that is, the coupling performance defined below.

2.4. Comprehensive performance evaluation indexes of aeroengines

Eq. (5) can be recoupled into a 1D equation to obtain the average activity of the system by calculating the expectations of both sides of Eq. (8). Accordingly, the relationship between the average behavior and the structure of the system can be further obtained as follows

(10)

〈 B 〉 〈 x 〉 = 〈 R_{E} 〉 ∙ (1 - p) ∙ 〈 w 〉 \frac{〈 x 〉}{1 + 〈 x 〉} + 〈 R_{I} 〉 ∙ p ∙ 〈 w 〉 \frac{1}{1 + 〈 x 〉}

where

〈 B 〉

〈 R_{E} 〉

and

〈 R_{I} 〉

is the average of parameters

B_{i}

R_{E i}

, and

R_{I i}

, and

〈 x 〉 = \frac{' 1}{N} \sum_{i = 1}^{N} x_{i}

. Besides,

〈 w 〉 = \frac{1}{N} \sum_{i = 1}^{N} \sum_{j = 1}^{N} W_{ij}

is the average weighted degree of the network, and we have the average weighted degree of

\hat{W}

\sum_{j = 1}^{N} {\hat{W}}_{ij} = (1 - p) 〈 w 〉

and the average weighted degree of

\bar{W}

\sum_{j = 1}^{N} {\bar{W}}_{ij} = p 〈 w 〉

. The proportion of edges

p

that represent negative correlations between nodes can be denoted as

(11)

p = \frac{\sum_{i = 1}^{N} \sum_{j = 1}^{N} {\bar{W}}_{ij}}{\sum_{i = 1}^{N} \sum_{j = 1}^{N} W_{ij}} = \frac{\sum_{i = 1}^{N} \sum_{j = 1}^{N} {\bar{W}}_{ij}}{\sum_{i = 1}^{N} \sum_{j = 1}^{N} ({\hat{W}}_{ij} + {\bar{W}}_{ij})}

Since

〈 B 〉 〈 x 〉

is not equal to zero in the model we constructed, we have rewritten Eq. (10) as matrix multiplication,

(12)

(\begin{matrix} 〈 R_{E} 〉 (1 - p) 〈 w 〉 & 〈 R_{I} 〉 p 〈 w 〉 \end{matrix}) ∙ (\begin{matrix} \frac{1}{〈 B 〉 (1 + 〈 x 〉)} \\ \frac{1}{〈 B 〉 〈 x 〉 (1 + 〈 x 〉)} \end{matrix}) = 1

Here we use

P_{C}

and

P_{A}

to represent the above two matrices

P_{C} = (\begin{matrix} 〈 R_{E} 〉 (1 - p) 〈 w 〉 & 〈 R_{I} 〉 p 〈 w 〉 \end{matrix})

(13)

P_{A} = (\begin{matrix} \frac{1}{〈 B 〉 (1 + 〈 x 〉)} \\ \frac{1}{〈 B 〉 〈 x 〉 (1 + 〈 x 〉)} \end{matrix})

Thus, Eq. (12) can be written as

(14)

P_{C} ∙ P_{A} = 1

Since the matrix

P_{C}

contains parameters including

〈 R_{E} 〉

〈 R_{I} 〉

p

, and

〈 w 〉

which represent the coupling between test indicators within the system, we believe that its size reflects the coupling performance of the system. Similarly, since the matrix

P_{A}

contains parameters including

〈 B 〉

and

〈 x 〉

which represent the activity of test indicators themselves, we believe that its size reflects the activity performance of the system. Here, the Frobenius norms

{‖ P_{C} ‖}_{F}

and

{‖ P_{A} ‖}_{F}

are used to measure the sizes of

P_{C}

and

P_{A}

respectively. In order to better fit the actual situation, we use

{‖ P_{C} ‖}_{F}

to measure the coupling performance and the reciprocal of

{‖ P_{A} ‖}_{F}

to measure the activity performance. Therefore, the coupling performance index and the activity performance index of aeroengines can be expressed as follows

(15)

\begin{array}{l} {‖ P_{C} ‖}_{F} = {\{{[〈 R_{E} 〉 (1 - p) 〈 w 〉]}^{2} + {(〈 R_{I} 〉 p 〈 w 〉)}^{2}\}}^{1 / 2} \\ {‖ P_{A} ‖}_{F}^{- 1} = {\{{[〈 B 〉 (1 + 〈 x 〉)]}^{- 2} + {[〈 B 〉 〈 x 〉 (1 + 〈 x 〉)]}^{- 2}\}}^{- 1 / 2} \end{array}

It is evident that the greater the coupling effect within the aeroengine performance network, the higher the corresponding coupling performance index. Similarly, increased activity within the network correlates with a larger activity performance index.

3. Results

3.1. Network construction and topology properties

Based on the test-run data and the correlation of test indicators, we construct the aeroengine performance networks consisting of 28 nodes. Three networks are presented in Figs. 3(b)–(d), including:

•The delivery network, representing the network model of the aeroengine that passed the test-run after assembly and was delivered;

•The first overhaul network, representing the network model of the aeroengine that ran out of 650 hours of service life, completed the first overhaul, passed the test-run and was delivered;

•The second overhaul network, representing the network model of the aeroengine that ran out of another 400 hours, completed the second overhaul, passed the test-run and was delivered.

Without considering isolated nodes, the topological properties of the above three networks are analyzed and radar charts are presented as depicted in Figs. 3(e)–(g). Analysis of changes in the delivery network and the first overhaul network reveals that as service time increases the average path length of the network decreases indicating a sparser network structure. Both the average weighted degree and average degree of the network decrease, which suggests that the correlation between test indicators decreases, leading to the disappearance of edges between test indicators. This gradual weakening suggests diminished control ability between components. As service time increases, the number of aging components rises, limiting the operational capacity of remaining normal components, thereby reducing aeroengine performance and impacting overall aircraft performance. Conversely, differences in average path length, diameter, natural connectivity, average weighted degree, and average degree between the first overhaul network and the second overhaul network are minimal, indicating that maintenance effectively maintains test indicators and their relationships at a stable level. Notably, there is a significant decrease in the proportion of negative edges, denoted as p, suggesting a reduction in negative excitations within the aeroengine performance network as service time increases, a trend not mitigated or improved by maintenance.

The function and performance of a complex network hinge on its invulnerability, the ability of the network to maintain connectivity even when nodes are removed or compromised. The underlying concept is that even if the network is disrupted, the more redundant paths the network has, the more likely it is that the connection between the nodes will still be maintained [57]. For aeroengines, it is essential to assess the degree of performance degradation resulting from failures. To quantify this impact, we use natural connectivity as a measure to simulate the random removal of a certain number of nodes, in this case simulating the loss of correlation between nodes and other nodes, to obtain the natural connectivity of different networks. As shown in Fig. 3(h), we randomly remove 1–20 nodes step by step for each network. The natural connectivity of the three aeroengine performance networks decreases steadily with the removal of nodes. In addition, regardless of how many nodes are removed, the natural connectivity of the delivery network is higher than the other two networks, and the natural connectivity of the first overhaul network and the second overhaul network are basically the same due to the maintenance efforts.

3.2. Aeroengine performance dynamics

The aeroengine test-run can be divided into six stages. The state transition is achieved by switching between the thrust and the IGV angle, and reaches the steady state after a period of time. For each stage, due to the correlation between test indicators in the test-run data, an increase in one test indicator may lead to an increase or decrease in related test indicators. Therefore, the positive and negative coupled dynamics model can effectively characterize this system. In Section 2.3, we present the modified aeroengine performance dynamics model Eq. (5) and detail the parameter estimation method. In this way, we fit the six stages of an aeroengine using this dynamics model which simulates the regulation interactions between test indicators. Naturally, if a test indicator has no interaction with any test indicator, that is, no correlation, it should be removed. For each stage, the deviation between real and simulated values of each test indicator is depicted in Fig. 4, with the mean relative error (MRE) provided in the upper right corner indicating high accuracy, mostly under 1% across six stages. This dynamics model is highly suitable for analyzing the aeroengine test-run process, so it can be assumed to be the aeroengine performance dynamics.

Consider three types of perturbations, including node removal, weighted edge removal, and weight reduction, to simulate the isolation of test indicators due to component failures, the loss of correlation between test indicators due to the failure of the connectors, and the overall weakening of the network correlation and due to the aging of aeroengine structure (Fig. 5). For node removal, 1–28 nodes are randomly removed from the network and the simulations are conducted 100 times. Results indicate that the removal of any node has a negative impact on the system, as the average level of test indicators

〈 x 〉

decreases as the removal ratio increases. Upon removing 60% of nodes, the system enters the least desired low state

x^{L}

(

〈 x 〉

= 0). Weighted edge removal experiments involve randomly removing 0 to 100% of edges across 100 simulations. Removing 0 to 60% of weighted edges causes a rapid decline in

〈 x 〉

. Subsequently,

〈 x 〉

is slowly decay from the intermediate state to the low state

x^{L}

. This indicates that the failure of any component and connector of the aeroengine causing node loss and weighted edge loss can impact aeroengine performance significantly, particularly in the early phase. Weight reduction simulations involve reducing edge weights by 0 to 100%. Results show a rapid deterioration in aeroengine performance when the overall weight is reduced by 10%. Thus, regular maintenance to mitigate aging effects on aeroengine structures is crucial.

3.3. Comprehensive performance evaluation of aeroengines

Parameters in Eq. (5) dynamics, including

B_{i}

({\hat{R}}_{1 i} + {\hat{R}}_{2 i})

({\bar{R}}_{1 i} + {\bar{R}}_{2 i})

{\hat{W}}_{i j}

, and

{\bar{W}}_{i j}

, as well as fitting errors of dynamics, can be obtained by fitting six stages of the aeroengine. Combining these parameters with the dynamic form, each state of the aeroengine can be accurately described. Gao et al. [40] innovatively proposed using resilience to evaluate the system state through dynamic derivation, which is essentially the average level of items and the degree of the network. This is of great significance for evaluating the state of a system and preventing its collapse. In this study, the comprehensive evaluation of aeroengine performance essentially compares the states of different systems of the same type, accounting for both positive and negative interactions within each system. However, merely considering average level and degree may not effectively differentiate between different aeroengines. Using the same dynamic formulation, parameters of dynamics can reflect the system’s state to a certain extent. Therefore, integrating these parameters into evaluation metrics through dynamic derivation can better achieve this goal. Therefore, we recast Eq. (5) into a 1D equation and express it in matrix multiplication form. Two indexes in Eq. (15), utilizing the Frobenius norm of the matrix, are proposed to quantify the comprehensive performance of aeroengines. The comprehensive performance across six stages, encompassing coupling and activity indexes, is calculated for a test-run qualified aeroengine and a test-run failed aeroengine due to the whole machine vibration exceeding the limit value, and is depicted in Fig. 6.

Fig. 6 shows that the comprehensive performance of aeroengines is dynamic and varies with their operational states, constituting a dynamic process. This feature facilitates real-time evaluation based on operational data. In addition, the coupling performance index and the activity performance index of the test-run failed aeroengine are significantly higher than the normal one at all stages. Rotor imbalance is a common cause of the whole machine vibration in newly assembled aeroengines. This imbalance arises from uneven mass distribution within the rotor or misalignment of its rotation axis with the geometric centerline during operation, which causes generating the unbalanced centrifugal force during high-speed rotation of the rotor. Rotor imbalance will generate additional vibration during the operation of the rotor, which varies with the rotation of the rotor and is transmitted to the entire aeroengine structure, causing the whole machine vibration. In this complex system, the extent of coupling between test indicators will markedly increase, which explains why the coupling performance of the aeroengine experienced the whole machine vibration is significantly higher than the test-run qualified one. In addition, the whole machine vibration can induce relative motion among internal aeroengine components, potentially increasing friction and collisions. These effects can elevate various test indicators, including localized temperature, vibration sensor readings, fuel consumption, and others, significantly increasing the activity performance of aeroengines experiencing whole machine vibration.

By fitting the dynamics of 13 test-run qualified aeroengines at each stage, the parameters and the errors are obtained. By plotting the relationship between the coupling performance and the activity performance of test-run qualified aeroengines under varied parameters and accounting for errors through Eqs. (10) and (15), a distinction range is provided to distinguish between qualified and faulty aeroengines, as depicted in Fig. 7. Meanwhile, the coupling performance and the activity performance of 13 qualified aeroengines and 4 faulty aeroengines are calculated by Eq. (15) and plotted in the graphs for verification. As previously mentioned, the coupling performance and the activity performance of aeroengines experiencing the whole machine vibration are significantly higher than that of test-run qualified aeroengines, which can be intuitively observed in the figures. Regarding the whole machine vibration, aeroengine manufacturers stipulate that the acceptance limit for vibration at the specific measurement point during the test-run is 76.2 microns, exceeding 100 microns necessitates manual intervention and emergency braking. The figures indicate four aeroengines (F1 to F4) exhibiting varying levels of whole machine vibration. F1 and F4 experience the most severe whole machine vibration, with monitoring values at some stages reaching up to 98 microns. That is the reason that F1 and F4 are always in the upper right corner of the graphs. In contrast, F2 and F3 experience relatively mild whole machine vibration and do not experience vibration at all stages. Even when there is vibration, the measured values are only around 85 microns.

As previously noted, the current test-run process used by aeroengine manufacturers only evaluates typical states with specific combinations of thrust and IGV angle to determine aeroengine qualification. However, in practical operation, the thrust of the aeroengine varies constantly, and each aeroengine has a unique IGV angle setting. In the constructed aeroengine performance dynamics model,

R_{1 i}

and

R_{2 i}

are employed to characterize the excitation and inhibition effects of thrust and IGV angle on various test indicators, as illustrated in Eq. (5). The coupled effects of these excitations and inhibitions ultimately manifest in the comprehensive performance of aeroengines. By applying dimensionality reduction techniques, the 28 differential equations for 28 test indicators are recoupled to derive Eq. (10), which reflects the average activity of the aeroengine, a typical complex system. Here,

〈 R_{E} 〉

and

〈 R_{I} 〉

represent the internal average intensities of excitation and inhibition, dictated by the corresponding thrust and IGV angle. Overall,

〈 R_{E} 〉

and

〈 R_{I} 〉

can quantify the average strengths of positive and negative interactions within the aeroengine system for different combinations of thrust and IGV angle, and the corresponding comprehensive performance can be obtained through Eq. (15).

By adjusting

〈 R_{E} 〉

and

〈 R_{I} 〉

in Eq. (15), we give the comprehensive performance indexes of aeroengines under different combinations of thrust and IGV angle to guide the practical applications (Fig. 8). In the actual operation of aeroengines, the correlation coefficients between the current thrust and the IGV angle with the 28 test indicators can be calculated, and the corresponding

〈 R_{E} 〉

and

〈 R_{I} 〉

can be obtained by taking the averages for positive and negative interactions respectively. Then, the comprehensive performance of the aeroengine can be calculated through Eq. (15), as illustrated in Fig. 8. In this way, the comprehensive performance of aeroengines under various combinations of thrust and IGV angle can be obtained, as shown in Fig. 8.

4. Discussion

As a vital part of the aircraft, the performance of the aeroengine is directly linked to flight safety. The comprehensive performance evaluation of aeroengines aids in accurately reflecting the equipment conditions and serves as the basis for the PHM and aeroengine digital engineering of aeroengines. In current practical applications, the single-parameter method and the test-run process are commonly employed, yet clear limitations are present. Firstly, relying on a single or a limited set of indicators fails to capture the true states of aeroengines. Additionally, concentrating exclusively on performance in typical conditions is insufficient, since parameters such as thrust vary continuously in real scenarios. Consequently, there is an urgent need to refine current evaluation methods to more comprehensively account for various factors influencing aeroengine performance, thereby improving assessment accuracy and effectiveness.

In this study, a unified framework of the comprehensive performance evaluation of aeroengines is proposed to address persistent challenges in practical engineering applications drawing from complex networks and network dynamics. Based on aeroengine test-run data, a performance network model with 28 test indicators is constructed to describe the state of the aeroengine from the perspective of the network structure. Meanwhile, thrust and IGV angle parameters are introduced in the performance dynamics model of the aeroengine, achieving MREs of less than 1% mostly and accurately characterizing the aeroengine as a typical complex system. Combining the network model and the dynamics model, three typical fault modes of aeroengines are simulated with three perturbations, and the corresponding practical meanings are explained. Moreover, a new measure, the comprehensive performance evaluation indexes of aeroengines, is defined which can evaluate the real-time performance in terms of the coupling performance and the activity performance. Those indexes provide qualified ranges for fault diagnosis, distinguishing between test-run qualified and failed aeroengines. The results indicate that the coupling performance and the activity performance of the aeroengine with whole machine vibration are significantly higher. By adjusting dynamic parameters, the relationship between various combinations of thrust and IGV angle and the comprehensive performance evaluation indexes of aeroengines can be quantitatively analyzed.

The network dynamic approach proposed in this paper has three outstanding advantages, which demonstrate its effectiveness and potential contribution to the comprehensive performance evaluation of aeroengines.

Firstly, one of the primary advantages of this method is that the proposed framework for the comprehensive performance evaluation of aeroengines allows for an objective and comprehensive assessment. This assessment considers a wide range of test indicators recorded by sensors during the operation of aeroengines and the high-dimensional, heterogeneous, and nonlinear characteristics of real-world data. Traditional methods and their enhancements for comprehensive performance evaluation of aeroengines often struggle to effectively address these complexities. For instance, while Wang et al. [21] present an evaluation model that integrates fuzzy AHP and TOPSIS, their approach relies on expert-derived coefficients and subjective judgments, which can introduce bias and limit the overall objectivity of the evaluation process. Additionally, Huang et al. [22] employ substantial sensor data to derive a health indicator for engine degradation, focusing on a comprehensive assessment of aeroengine conditions. However, their method primarily addresses linear relationships through multiple linear regression, which fails to capture the intricacies of real-world data. In contrast, the objectivity of the framework proposed in this paper is based on calculations derived from actual test-run data, which incorporates the properties of complex systems alongside the mechanisms of aeroengines. Furthermore, the framework proposed in this paper effectively integrates the complexity of test-run data with complex networks and network dynamics, offering a more comprehensive evaluation of aeroengine performance.

Additionally, the aeroengine performance network model and the aeroengine performance dynamics model can accurately describe the system states and offer valuable interpretability. Given the complexity of real-world aeroengine operation data, artificial intelligence methods have been widely applied in the performance evaluation of aeroengines with promising results. However, these methods often function as black boxes, which poses challenges for scientifically explaining the mechanisms that influence changes in aeroengine states and inherent relationships. Recently, some researchers have sought to enhance the interpretability. Xiao et al. [58] proposed a deep learning model based on LSTM that incorporates the physical topology of aeroengines to predict performance through EGT, achieving good accuracy. However, EGT is a single parameter that cannot comprehensively reflect the overall state of the aeroengine. Model-based methods, offering good interpretability, require the development of precise aeroengine models and accurate component characteristics. Obtaining these characteristics can be challenging. Furthermore, as noted by Loboda et al. [59], there remains a notable gap between various model-based methods and the limited algorithms utilized in actual monitoring systems. For instance, using GPA, a typical linear model, to evaluate aeroengine performance is often inadequate. Additionally, the enhanced nonlinear GPA method faces limitations due to the number of parameters involved and the computational resources demand. The approach proposed in this paper directly addresses these challenges by ensuring the capability to handle complex data while maintaining interpretability. By merging the network model with the dynamics model, a quantitative depiction of the aeroengine condition and the interactions between test indicators is achieved using graph theory and differential equations, aiding in the exploration of the quantitative and qualitative changes in the comprehensive performance evaluation of aeroengines.

Moreover, this approach enables a dynamic description of comprehensive performance of aeroengines from two dimensions: activity performance and coupling performance. The IGV angle of the aeroengine varies, and thrust changes continuously during operation, with performance fluctuating under different operating conditions. In previous studies, the dynamic performance evaluation of aeroengines was also considered necessary, and corresponding methods have been conducted on it. Ullah et al. [60] introduced a new method based on LSTM that utilizes various monitoring indicators for dynamic health evaluation of aeroengines. However, in addition to the poor interpretability of deep learning, a significant disadvantage is the reliance on a single parameter, EGT, to characterize performance. Lu et al. [61] proposed a state-propagation extreme learning machine using a Kalman filter for health assessment dynamically. However, this method only selected three indicators from the gas path for health assessment in applications which cannot reflect its comprehensive performance. On the contrary, this paper quantifies the impact of thrust and IGV angle on the comprehensive performance evaluation of aeroengines through dynamic parameters and combines changes in test indicators and their interactions to capture the dynamic characteristics of comprehensive performance under different conditions. This extends the comprehensive performance evaluation of aeroengines beyond several typical conditions.

The network dynamic approach as employed in this study, integrating the network model and the dynamics model, has demonstrated significant applicability and validity in assessing complex system states. Gao et al. [40] made a notable theoretical contribution by pioneering a framework specifically designed for the evaluation of complex systems considering positive interactions and applied it to multiple complex systems such as ecosystems, making significant theoretical contributions. This innovative approach provides a foundational understanding of how such complex systems can be analyzed theoretically. Additionally, Hou et al. [52] proposed a comprehensive assessment model that utilizes real-world blood test data from 25 indicators to diagnose hepatitis B virus-related diseases, and simulated state transitions among chronic hepatitis B, liver cirrhosis, hepatocellular carcinoma, acute-on-chronic liver failure, and death. In practical applications, this method achieved a diagnostic accuracy of 84.74%, significantly surpassing the 55.64% accuracy attained by physicians relying solely on clinical experience, which demonstrates the approach’s capability to comprehensively evaluate patients’ liver conditions. However, it is important to note that the blood test data is static rather than time series, limiting its capacity for dynamic monitoring. To sum up, the network dynamic approach proposed in this paper, which can depict the coupling effect of positive and negative excitation within the system and dynamically evaluate the comprehensive performance of aeroengines based on time series data, can be applied to the evaluation and diagnosis in real time of various complex systems, including complex equipment represented by aeroengines and ecosystems with both positive and negative interactions, and even applied to the diagnosis of diseases and real-time human health assessment.

This paper proposes a novel method to evaluate the comprehensive performance of aeroengines, which contributes to the research field as well as the practical engineering of aeroengines. By offering a systematic approach that combines the network model and the dynamics model, this method enhances the understanding and evaluation of comprehensive performance in intricate equipment. Its applicability also offers a robust foundation for enhancing understanding and managing the comprehensive performance evaluation of aeroengines, laying the groundwork for future advancements in PHM and aeroengine digital engineering.

5. Conclusions

This study proposes a unified framework for the comprehensive performance evaluation of aeroengines to address persistent challenges in practical engineering applications. We develop a performance network model based on 28 test indicators from aeroengine test-run data, characterizing the system with MRE below 1%. By integrating network and dynamics models, we simulate typical fault modes, elucidating their practical significance in affecting overall engine performance. We introduce new performance evaluation indexes that assess real-time coupling performance and activity performance, aiding in fault diagnosis. Our results indicate that these indexes can effectively distinguish between qualified and failed aeroengines, thereby facilitating targeted maintenance and operational decisions. Additionally, the relationship between thrust, IGV angle, and performance indexes is characterized, broadening the evaluation scope. This approach enhances the understanding of aeroengine comprehensive performance and supports future advancements in PHM and aeroengine digital engineering.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (72231008, 72171193, and 72071153), the Science and Technology Innovation Group Program of Shaanxi Province (2024RS-CXTD-28), and the Open Fund of Intelligent Control Laboratory (ICL-2023-0304).

Compliance with ethics guidelines

Yuting Wang, Feng Liu, Feng Xi, Bofei Wei, Dongli Duan, Zhiqiang Cai, and Shubin Si declare that they have no conflict of interest or financial conflicts to disclose.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.eng.2024.11.024.

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