optimal design of experiments a case study approach

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Optimal Design of Experiments: A Case Study Approach

ISBN: 978-1-119-97401-7

Bradley Jones , Peter Goos

"It's been said: 'Design for the experiment, don't experiment for the design.' This book ably demonstrates this notion by showing how tailor-made, optimal designs can be effectively employed to meet a client's actual needs. It should be required reading for anyone interested in using the design of experiments in industrial settings." — Christopher J. Nachtsheim , Frank A Donaldson Chair in Operations Management, Carlson School of Management, University of Minnesota 

This book demonstrates the utility of the computer-aided optimal design approach using real industrial examples. These examples address questions such as the following:

• How can I do screening inexpensively if I have dozens of factors to investigate?
• What can I do if I have day-to-day variability and I can only perform 3 runs a day?
• How can I do RSM cost effectively if I have categorical factors?
• How can I design and analyze experiments when there is a factor that can only be changed a few times over the study?
• How can I include both ingredients in a mixture and processing factors in the same study?
• How can I design an experiment if there are many factor combinations that are impossible to run?
• How can I make sure that a time trend due to warming up of equipment does not affect the conclusions from a study?
• How can I take into account batch information in when designing experiments involving multiple batches?
• How can I add runs to a botched experiment to resolve ambiguities?

While answering these questions the book also shows how to evaluate and compare designs. This allows researchers to make sensible trade-offs between the cost of experimentation and the amount of information they obtain.

Bradley Jones , Senior Manager, Statistical Research and Development in the JMP division of SAS, where he leads the development of design of experiments (DOE) capabilities in JMP software. Dr. Jones is widely published on DOE in research journals and the trade press. His current interest areas are design of experiments, PLS, computer aided statistical pedagogy, and graphical user interface design.

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Optimal Design of Experiments: A Case Study Approach 1st Edition, Kindle Edition

"It's been said: 'Design for the experiment, don't experiment for the design.' This book ably demonstrates this notion by showing how tailor-made, optimal designs can be effectively employed to meet a client's actual needs. It should be required reading for anyone interested in using the design of experiments in industrial settings." — Christopher J. Nachtsheim , Frank A Donaldson Chair in Operations Management, Carlson School of Management, University of Minnesota 

This book demonstrates the utility of the computer-aided optimal design approach using real industrial examples. These examples address questions such as the following:

• How can I do screening inexpensively if I have dozens of factors to investigate?
• What can I do if I have day-to-day variability and I can only perform 3 runs a day?
• How can I do RSM cost effectively if I have categorical factors?
• How can I design and analyze experiments when there is a factor that can only be changed a few times over the study?
• How can I include both ingredients in a mixture and processing factors in the same study?
• How can I design an experiment if there are many factor combinations that are impossible to run?
• How can I make sure that a time trend due to warming up of equipment does not affect the conclusions from a study?
• How can I take into account batch information in when designing experiments involving multiple batches?
• How can I add runs to a botched experiment to resolve ambiguities?

While answering these questions the book also shows how to evaluate and compare designs. This allows researchers to make sensible trade-offs between the cost of experimentation and the amount of information they obtain.

• ISBN-13 978-0470744611
• Edition 1st
• Sticky notes On Kindle Scribe
• Publisher Wiley
• Publication date June 28, 2011
• Language English
• File size 9223 KB
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"There is simply no other integral narrative history of the Jews in the 'Bohemian lands.' Prague and Beyond will fill lacunae on reading lists in European and Jewish history, history of the Holocaust, and Central European political science."—Moshe Rosman, Bar-Ilan University

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From the Inside Flap

"This is an engaging and informative book on the modern practice of experimental design. The authors' writing style is entertaining, the consulting dialogs are extremely enjoyable, and the technical material is presented brilliantly but not overwhelmingly. The book is a joy to read. Everyone who practices or teaches DOE should read this book." — Douglas C. Montgomery , Regents Professor, Department of Industrial Engineering, Arizona State University

"This book is the compelling story of two consultants in dialog as they show their clients how to leave the roads of textbook experimental design and fly the direct route of optimal design as enabled by computer-based methods." — John Sall , Executive Vice President and Cofounder, SAS Institute

"This book puts cutting-edge optimal design of experiments techniques into the hands of the practitioner. Ten real-world design scenarios, which Goos and Jones present as consulting session conversations with clients, easily engage and absorb the reader. A behind-the-scenes look at various technical treasures accompanies each scenario." — Marie Gaudard , Professor Emeritus, University of New Hampshire

"Each chapter begins with a realistic experimental situation being informally discussed on site by local engineers and statistical consultants. Next an optimal experimental design is constructed and the data with full detailed analysis provided. Statisticians and para-statisticians alike should enjoy this book. Clearly a new day is dawning in the art and practice of experimental design." — J. Stuart Hunter , Professor Emeritus, Princeton University

From the Back Cover

"This is an engaging and informative book on the modern practice of experimental design. The authors' writing style is entertaining, the consulting dialogs are extremely enjoyable, and the technical material is presented brilliantly but not overwhelmingly. The book is a joy to read. Everyone who practices or teaches DOE should read this book." ― Douglas C. Montgomery , Regents Professor, Department of Industrial Engineering, Arizona State University

"This book is the compelling story of two consultants in dialog as they show their clients how to leave the roads of textbook experimental design and fly the direct route of optimal design as enabled by computer-based methods." ― John Sall , Executive Vice President and Cofounder, SAS Institute

"This book puts cutting-edge optimal design of experiments techniques into the hands of the practitioner. Ten real-world design scenarios, which Goos and Jones present as consulting session conversations with clients, easily engage and absorb the reader. A behind-the-scenes look at various technical treasures accompanies each scenario." ― Marie Gaudard , Professor Emeritus, University of New Hampshire

"Each chapter begins with a realistic experimental situation being informally discussed on site by local engineers and statistical consultants. Next an optimal experimental design is constructed and the data with full detailed analysis provided. Statisticians and para-statisticians alike should enjoy this book. Clearly a new day is dawning in the art and practice of experimental design." ― J. Stuart Hunter , Professor Emeritus, Princeton University

About the Author

Bradley Jones , Senior Manager, Statistical Research and Development in the JMP division of SAS, where he leads the development of design of experiments (DOE) capabilities in JMP software. Dr. Jones is widely published on DOE in research journals and the trade press. His current interest areas are design of experiments, PLS, computer aided statistical pedagogy, and graphical user interface design.

Product details

• ASIN ‏ : ‎ B005DIAPC2
• Publisher ‏ : ‎ Wiley; 1st edition (June 28, 2011)
• Publication date ‏ : ‎ June 28, 2011
• Language ‏ : ‎ English
• File size ‏ : ‎ 9223 KB
• Text-to-Speech ‏ : ‎ Enabled
• Screen Reader ‏ : ‎ Supported
• Enhanced typesetting ‏ : ‎ Enabled
• X-Ray ‏ : ‎ Not Enabled
• Word Wise ‏ : ‎ Not Enabled
• Sticky notes ‏ : ‎ On Kindle Scribe
• Print length ‏ : ‎ 459 pages
• #264 in Science Experiments & Projects
• #893 in Scientific Experiments & Projects
• #1,555 in Industrial Engineering (Kindle Store)

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Optimal Design of Experiments: A Case Study Approach, by Peter Goos and Bradley Jones. John Wiley & Sons, Ltd, Hoboken, NJ, 2011, 288 pp., $95.00, ISBN: 978-0-470-74461-1.

I MUST make a disclaimer in this review. I have not been too keen in the past on the philosophy of optimal experimental design. I've had the idea that classical design of experiments have proven to work in a variety of situations over many years, so why mess with this indispensable tool? What I have previously read about optimal design has the feel of a salesman saying something that is too good to be true. "Look, you can use this handy software to create a design with a minimum amount of runs and answer all your questions guaranteed!"

Imagine my surprise when reading this timely book, to find myself being more convinced of the utility of optimal design. This book uses a different presentation approach than standard statistics textbooks. Each chapter consists of a fictitious case study (based on...

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Optimal Design of Experiments: A Case Study Approach by Peter Goos and Bradley Jones

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Optimal Design of Experiments: A Case Study Approach

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Design of experiments is a powerful tool for understanding systems and processes with many applications for efficient and effective information gathering.

Authors Peter Goos and Bradley Jones use conversational case studies to show how to make a design match the process or system under study using available resources – and still optimize the information obtained from the experiment.

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In Chapter 7, titled “A Response Surface Design in Blocks,” you will learn:

• When to “block an experiment,” or perform an experiment in a group of runs.
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• When to use optimal blocking as an alternative.

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T1 - Optimal Design of Experiments: A Case-Study Approach

AU - Goos, PP

AU - Jones, B

SN - 9780470744611

BT - Optimal Design of Experiments: A Case-Study Approach

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• Published: 02 September 2024

Fault diagnosis method for oil-immersed transformers integrated digital twin model

• Haiyan Yao 1 ,
• Xin Zhang 2 ,
• Qiang Guo 1 ,
• Yufeng Miao 1 &
• Shan Guan 3  

Scientific Reports volume  14 , Article number:  20355 ( 2024 ) Cite this article

Metrics details

• Electrical and electronic engineering
• Mechanical engineering

To address the problems of low accuracy in fault diagnosis of oil-immersed transformers, poor state perception ability and real-time collaboration during diagnosis feedback, a fault diagnosis method for transformers based on the integration of digital twins is proposed. Firstly, fault sample balance is achieved through Iterative Nearest Neighbor Oversampling (INNOS), Secondly, nine-dimensional ratio features are extracted, and the correlation between dissolved gases in oil and fault types is established. Then, sparse principal component analysis (SPCA) is used for feature fusion and dimensionality reduction. Finally, the Aquila Optimizer (AO) is introduced to optimize the parameters of the Kernel Extreme Learning Machine (KELM), establishing the optimal AO-KELM diagnosis model. The final fault diagnosis accuracy reaches 98.1013%. Combining transformer digital twin models, real-time interaction mapping between physical entities and virtual space is achieved, enabling online diagnosis of transformer faults. Experimental results show that the method proposed in this paper has high diagnostic accuracy and strong stability, providing reference for the intelligent operation and maintenance of transformers.

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Introduction.

The transformer, as the hub of power systems, its health status directly impacts the stability and reliability of the electrical system's operation. Therefore, the precise management of a transformer's health status is paramount to ensuring the steadfast and secure operation of the power grid 1 .

Presently, the technology of Dissolved Gas Analysis (DGA) is extensively employed in the monitoring and identification of faults within oil-insulated transformers 2 , 3 , primarily encompassing: the IEC triad ratio method 4 , the Rogers quadruple ratio method 5 , and the DUVAL triangle technique 6 . Despite their simplicity of operation, these approaches lack the depth of representation for fault characteristics and are limited by their capabilities, resulting in a blurred and indistinct encoding boundary, thereby leading to a low accuracy rate in fault recognition 7 . With the rapid advancement of artificial intelligence, eminent scholars have integrated machine learning with DGA technology, achieving notable results in the field of transformer fault detection. The literature 8 optimizes the support vector machine parameters through the refinement of the scalar search algorithm, thereby augmenting both the convergence velocity and the diagnostic precision of the methodology. The literature 9 proffers an SE-ELM diagnostic method, whose efficacy was validated through the verification across various datasets. The literature 10 enhances the particle swarm optimization algorithm through the dynamic adjustment of inertial weights and acceleration factors, iteratively optimizing the parameters of XGBoost, thereby augmenting the model's classification acumen. Additionally, methods such as Convolutional Neural Networks 11 , 12 , Long Short-Term Memory Networks 13 , 14 , 15 , LightGBM 16 , and the Capsule Network 17 are extensively employed.

With the advancement of big data and the Internet of Things (IoT) technologies, the Digital Twin (DT) 18 technology has paved a new path for enhancing the efficiency of equipment health management. The core concept is to construct a holographic virtual twin model in the digital realm, utilizing advanced technologies such as intelligent sensing and data transmission, which accurately, comprehensively, and in real-time reflect the evolution of physical devices, achieving intelligent control over entities 19 , 20 , 21 . This technology has been extensively utilized in various sectors including aerospace, manufacturing, and healthcare.

In the field of transformer fault diagnosis, scholars both domestically and internationally have carried out extensive research. Referencing 22 , the study proposed a method for constructing a dual-driving twin model integrating data and models, focusing on 10 kv oil-immersed transformers. This approach enables the synchronization between the actual operating conditions of the transformer and the digital twin center. Referencing 23 , a digital twin fault diagnosis model was constructed based on the mechanism model and data model of transformers. Five characteristic gases extracted from DGA data were selected as input feature vectors for a CNN. Experimental results showed that the 1D-CNN model established in this study responded rapidly, had a short training time, and achieved high accuracy, thus validating the effectiveness of the model. Referencing 24 , a fault diagnosis model based on digital twin was constructed for transformers, taking into account their structural characteristics and operational traits. By optimizing the smoothing factor δ in a probabilistic neural network through differential evolution algorithm, the diagnostic accuracy reached an impressive 96.7%, enabling precise monitoring of the transformer's actual operating state. Reference 25 conducts a statistical analysis of the operating data and state information quantity of power transformers, proposes a framework for a state evaluation system and fault detection system based on GCA-CNN, and verifies with 2000 real data cases that the model has higher accuracy and evaluation and detection effects. The literature 26 establishes a high-fidelity simulation model of transformers to accurately simulate winding currents and the temperatures of different components, which can be used for the identification of early faults. However, the aforementioned research is only focused on a single dissolved gas in oil or vibration signal as the basis for fault diagnosis, but there are many factors affecting transformer faults. In the future, it may be possible to combine multi-source data for comprehensive judgment.

In light of the above context, this paper proposes a fault diagnosis method for oil-immersed transformers that integrates a digital twin model. The main contributions of the paper are divided into several parts. Part 1 mainly elaborates on the research background of the paper and the future research direction. Part 2 establishes a transformer digital twin framework, based on geometric, physical, behavioral, and rule models, to achieve interaction mapping between the virtual entity and the physical entity. Part 3 introduces the methods used in the paper, providing theoretical support for the establishment of an accurate and efficient fault diagnosis model. Part 4 addresses the issue of imbalanced small sample data that can easily lead to misjudgment of minority class samples, deeply explores the correlation between dissolved gases in oil and fault types, and eliminates the 'dimensionality catastrophe' problem, using instance data to obtain diagnostic results. Part 5 discusses and analyzes different sampling methods, different features, and different diagnostic models. Part 6 summarizes the entire paper.

Transformer fault diagnosis model fusing digital twin

Transformer digital twin framework.

This article takes a 400kV oil-immersed transformer as the research object and establishes a transformer digital twin integrated digital twin technology. The constructed digital twin framework mainly includes: physical space, twin body, twin data layer and application service layer 27 , as shown in Fig.  1 .

Transformer digital twin framework.

In the process of building a digital twin, the geometric model is the foundation for creating the digital twin model. Three-dimensional software such as UG and SolidWorks are used to comprehensively describe the solid model in terms of geometric dimensions, material properties, and assembly relationships. Based on prior knowledge, physical properties, and operating mechanisms, the geometric model is analyzed and tested for magnetic field, structure, and other modeling aspects, fully reflecting the intrinsic nature and operating mechanism of the transformer. Heterogeneous data from multiple sources, such as dissolved gas in oil and acoustic vibration signals, are collected using state-aware devices. Artificial intelligence algorithms integrated in the behavior model are used for processing and analysis. The derived data generated from simulation calculations are fed back to the mechanism model in real-time. At the same time, simulation data, state-aware data, as well as transformer's full life cycle process data, maintenance records, and computed derived data collectively form the twin database. Through data communication protocols and interfaces, real-time updates and interactive control between the physical entity and the digital twin are achieved, enabling visual description, real-time monitoring, analysis, diagnosis, and intelligent decision-making for the physical transformer. This provides new ideas for improving the safety and reliable operation of power transmission and transformation equipment.

The five-dimensional model of digital twin

The present work is founded on the five-dimensional model proposed by Tao Fei from Beijing Aerospace University 28 , culminating in the creation of a digital twin for transformers, as exemplified by Eq. ( 1 ).

where: PE denotes the physical entity of the transformer, VE represents the virtual entity, SS signifies data, algorithms and models of the digital twin, DD stands for the twinning data of the transformer, and CN symbolizes the interaction and communication among the various components.

The acronym PE stands for transformer physical entity, an ensemble of components including the core, windings, tap-changer, and cooling equipment, it caters to the perception of contact or non-contact by state-sensing devices, embodying the interactive and responsive essence of an objective presence.

The SS represents the process of integrating data and models generated by the digital twin transformer system, thereby facilitating comprehensive monitoring of entities, diagnostic analysis of equipment failures, and predictive maintenance.

VE represents the twin model of the virtual realm, establishing the fundamental groundwork for mapping the virtual to the real. The specific composition is delineated by the formula ( 2 ) shown:

where: Gv represents the geometric model, which uses 3D modeling software to create a comprehensive description of the geometric features of physical entities; Pv represents the physical model, which describes the physical properties and operating mechanisms of electrical equipment; Bv represents the behavior model, which combines artificial intelligence algorithms to create Bv; Rv represents the rule model, which mainly includes expert experience and rule inference based on processed historical data for optimization and deduction.

DD represents twin data, which dynamically stores relevant data of PE/VE/SS, and is an important prerequisite for ensuring intelligent operation and maintenance of transformers. The specific representation is shown in formula ( 3 ):

where: Dp refers to the dynamic factor data collected through the state-aware device; Dv refers to the running parameters in the virtual model; Ds mainly refers to the functional and business service data; Dk includes expert experience, industry rules in the transformer field, and usage guidelines, etc. Df refers to the integrated transformation, interactive fusion, and other derived data of the above-mentioned data.

CN represents the data connection part, which is crucial for ensuring the interaction and updating of the elements in the digital twin model. Through data interfaces, communication protocols, etc., efficient transmission and utilization of data in the digital twin system can be achieved, enabling seamless communication and connectivity among different parts of the model. The interactive relationships of the five dimensions in the digital twin model are shown in Fig.  2 .

Transformer digital twin five-dimensional model connection relationship.

Transformer fault diagnosis model based on optimized extreme learning machine

Iterative nearest neighbor oversampling algorithm.

The iterative neighborhood oversampling 29 algorithm is a sampling method designed to tackle class imbalance issues, with its principal tenet being the selection of a multitude of class-specific samples as neighbors, and then traversing all k data points within this category, scouring for the most recent unlabeled instance within each label data subset of said category until the dataset balances out or approaches close to it. Here follow the specific steps:

Assume the samples in the dataset for each tag to be \({\text{r}} = \left\{ {r_{1} ,r_{2} , \cdots ,r_{j} , \cdots ,r_{a} } \right\}\) , with \(r_{j} \left( {j = 1,2, \cdots a} \right)\) denoting the number of samples contained within category j . Define the sample set's imbalance factor, utilizing the standard deviation \({\text{var}} \left( r \right)\) to symbolize the dispersal of various types of samples within the dataset, as illustrated in Eq. ( 4 ):

where: \(\mathop r\limits^{ - } = \frac{1}{a}\sum\limits_{j = 1}^{a} {r_{j} }\) .

Based on the philosophy of greedy search, endeavor to identify a multitude of particular sub-samples, with the process detailed in formula ( 5 ):

where: \(x_{j}\) represents the labeled data in category j . If \(x_{\max k}\) is the classification boundary, remove it and select the next nearest neighbor. Then, label it as category j , remove it from the unlabeled data set \(X_{U}\) , add it to the labeled data set \(X_{L}\) , and set \(r_{j} = r_{j} + 1\) . Recalculate the imbalance degree until the preset value is reached, and stop iterating.

Extreme learning machine algorithm

The Kernel Extreme Learning Machine (KELM) 30 is based on a single hidden layer feedforward neural network. It introduces a kernel function on top of the ELM algorithm, which maps low-dimensional data to a high-dimensional feature space, resulting in a model with stronger generalization and robustness. The specific steps are as follows:

Assume we are provided with N samples represented as \(\left\{ {\left( {{\text{x}}_{{\text{i}}} ,t_{i} } \right)} \right\}_{i = 1}^{N}\) , where \(x_{i} = \left[ {x_{i1} ,x_{i2} , \cdots ,x_{in} } \right]^{T} \in R^{n}\) and \(t_{i} = \left[ {t_{i1} ,t_{i2} , \cdots ,t_{im} } \right]^{T} \in R^{n}\) denote the input vector and output function of the model respectively. In the context of a neural network with k hidden layers and an activation function \(g\left( x \right)\) , the number of hidden nodes is L , and the ELM model can be articulated by the formula shown in Eq. ( 6 ):

where: \(\beta_{j} = \left[ {\beta_{j1} ,\beta_{j2} , \cdots ,\beta_{jL} } \right]^{T} \left( {j = 1,2, \cdots ,L} \right)\) denotes the output weight value connecting the j th implicit layer node with the output layer node. Among these, \(H = \left\{ {h_{ij} } \right\}\left( {i = 1,2, \cdots ,N;j = 1,2, \cdots ,L} \right)\) represents the output matrix of the hidden layer, and H denotes the jth column of the input \(x_{1} ,x_{2} , \cdots ,x_{n}\) corresponding to the jth hidden layer node. Within H, the jth row corresponds to the output vector of \(x_{i}\) .

Using the least squares method to obtain the output weight values, as shown in formula ( 7 ):

In the formula, \(H{\prime}\) represents the generalized inverse matrix of the hidden layer output matrix H .

Introducing the kernel function mitigates the issue of randomly generated input weights and bias values, exemplified by the KELM weight output formula ( 8 ):

The KELM output function as expressed in formula ( 9 ):

When \(h\left( x \right)\) remains unknown, the kernel function matrix is represented by formula ( 10 ):

In the equation, \(K\left( {x_{i} ,x_{j} } \right)\) denotes the nuclear function, represented as:

The KELM model's output function expression is delineated in formula ( 12 ):

Sparse principal component analysis

The sparse principal component analysis 31 is a method that builds upon the principal component analysis algorithm by incorporating the LASSO penalty term, thereby enabling the matrix to be sparsely populated. By solving the regression coefficient matrix, it further transforms PCA into an optimization problem aimed at finding the optimal set of coefficients for regression. Compared to traditional PCA, SPCA excels in effectively managing the sparsity within high-dimensional data, yielding results that are more interpretative.

The SPCA algorithm is resolve into two segments: the first entails calculating the principal components via PCA; the second entails enhancing the LASSO penalty term to render the obtained solution sparse. Here follow the specific steps:

Given a \({\text{n}} \times m\) -variant dataset X, the feature decomposition upon normalization treatment, as expounded upon in formula ( 13 ):

In the equation, \(\Lambda \in R^{m \times m}\) represents a diagonal matrix of eigenvalues, arranged in descending order. \(\Lambda \in R^{m \times m}\) is a unitary matrix with column vectors as load vectors.

Select the first k columns of the load matrix \(P \in R^{m \times k}\) , compute the score matrix T , as shown in Eq. ( 14 ):

Projecting T onto X yields a new matrix \(\mathop X\limits^{ \wedge }\) that encompasses information from the corresponding principal component; the difference with X is denoted as E , as illustrated in formula ( 15 ), ( 16 ):

The solution of the SPCA first reverts to the PCA model. The formula ( 15 – 16 ) yields the expression ( 17 ):

Ensure the main component is as near to the original data as possible, that is,it mandates E'sminimalism. Therefore, the principal component seeks resolution through formula ( 18 ):

In the equation, \(\mathop P\limits^{ \wedge }\) is the solution to the minimum value of the principal matrix P .

The vectors sought by PCA are all non-zero; thus, the sparse solution is achieved by incorporating the LASSO penalty term, thereby mitigating the overfitting issue in PCA. The solution formula for sparse principal components, as displayed in formula ( 19 ), is illustrated:

In this equation, matrix A denotes the expected demand matrix to be sought, while matrix B represents the demand matrix expected under the regression problem. A and B represent the \(m \times k\) matrix, \(\mathop A\limits^{ \wedge }\) and \(\mathop B\limits^{ \wedge }\) the matrices to be solved for minimizing values of A and B; they are subject to the constraints \(b_{j} \propto P_{j}\) , \(\lambda\) and \(\lambda_{1,j}\) being the penalty coefficients, and must adhere to \(\lambda > 0\) . The adjusted variance, as expressed in formula ( 20 ), is indicative of:

In the equation, the diagonal matrix interpreting variance is delineated, with \(\mathop P\limits^{ \wedge }\) representing the load matrix following the coefficients. Model contribution lies articulated in formula ( 21 ):

Transformer fault diagnosis model process

This article, established on the premise of transformer fault imbalance within small sample sets, aims at achieving real-time and precise diagnosis through the establishment of a diagnostic model and a determined diagnostic process. The specific diagnostic process is illustrated in Fig.  3 . The article employs the AO-KELM model as the diagnostic model, erecting a diagnostic process that integrates offline model training with online fault identification.

Transformer fault diagnosis model based on optimized kernel extreme learning machine.

⑴ Train the model offline

The article delves into the offline model training segment from three perspectives: data preprocessing, feature extraction, and model recognition.

Step 1: the preprocessing segment encompasses data INNOS's oversampling and normalization treatment. Collect the gathered DGA samples through INNOS for augmenting the minority class samples, followed by normalization treatment.

Step 2: the feature extraction section encompasses the establishment of ratio signature generation and the integration of SPCA for fusion dimensionality reduction. First, construct a multidimensional discriminant signature, delving deeply into the correlation between the ratio of dissolved gas content in oil and the type of fault. Subsequently, employ SPCA for feature fusion to acquire the optimal principal component, thereby removing redundant information, and divide the training set, validation set, and test set proportionally.

Step 3: the model identification segment encompasses the training and validation of the model. Utilizing the AO algorithm to optimize the regularization parameters C and the kernel functions within the KELM model, one verifies the model's accuracy through validation set on each iteration. Should the discrepancy between consecutive training sessions fall beneath 5%, the model training continues; otherwise, the model retraining commences anew until the prerequisite conditions are met. The ultimate establishment of the AO-KELM optimal diagnostic model.

⑵ Online fault diagnosis

Normalize the samples collected in real-time to handle and construct multi-dimensional features, employing an unencoded ratio method to input into an optimal diagnosis model directly following optimal principal component projection, thereby achieving swift recognition of transformer fault. Although the computational time for offline model training is accordingly elevated, it is merely necessary to undergo training once, with the aim of achieving online recognition and diagnosis of transformer faults as data from real-time monitoring continues to be inputted.

Case study analysis

Data source and normalization processing.

Transformer insults are exacerbated by thermal electrochemical action, causing the decomposition of internal insulating materials and the dissolution of various hydrocarbon gases within the insulation oil. Distinct characteristics of gas dissolved in oil under varying fault types exist; research has demonstrated that diagnostic and classification of faults can be achieved through the use of DGA techniques 32 . Consequently, these five gas contents are utilized as a basis for transformer fault diagnosis in this article.

The article selected a comprehensive sample of 337 monitoring data from a particular power supply company, dividing the operating status of transformers into categories such as normal, moderate heat overload, high temperature overload, high energy discharge, low energy discharge, and local discharge, each represented by labels 1 through 6. Each type of fault is augmented with specific characteristic gases including H 2 , CH 4 , C 2 H 4 , C 2 H 6 , and C 2 H 2 ; the exact number of samples for each category is detailed in Table 1 . The data reveals that the majority of samples fall into the category of normal, comprising 35.63% of the total. Low-energy discharge and local discharge types account for 5.55% and 9.78% respectively, with a maximum disparity reaching 5.1:1. Such imbalanced data is prone to misidentifying samples of the minority class as normal, thereby impacting recognition accuracy. Therefore, this paper employs the INNOS algorithm to augment the minority class samples, achieving a balance in sample categories.

To manifest the disparities between data prior to and after sampling, a principal component analysis is conducted upon the sample data from before and after said sampling process. Subsequently, the first two principal components are selected for visualizing the data of various types both before and after said sampling, as illustrated in Fig.  4 . In Fig.  4 , it becomes apparent that the data distribution trends for various types of faults, prior to and after the adoption of the INNOS sampling method, are identical, thereby underscoring the viability of the INNOS sampling approach.

Scatter plot of INNOS samples.

Transformer malfunction signature composition

Considering the substantial disparities among the various volatile gases, a preliminary normalization is required for each gas's abundance, as illustrated in Eq. ( 22 ):

In the equation: \(x_{i}\) and \(x_{{\text{i}}}^{*}\) represent features pre-normalized; \({\text{x}}_{{{\text{i}}\max }}\) and \({\text{x}}_{{{\text{i}}\min }}\) indicate the original minimal and maximum values.

The method of unencoded ratio analysis 33 is but one among numerous techniques widely employed, utilizing the percentage ratio of key gases to either the total gas or the hydrocarbon concentration can profoundly illustrate the interconnectedness between characteristic gases and types of failures. For instance, the ratio of a singular gas to the total hydrocarbon concentration provides a more conclusive indicator of the interplay between diverse fault types; the concentrations of C 2 H 4 and CH 4 can effectively demarcate local discharge from discharge with overheating diagnosis; the percentage composition of C 2 H 2 can determine whether a transformer has experienced thermal failure, among other determinations. The construction of this paper is predicated on the integration of pertinent literature, establishing a nine-dimensional candidate ratio signature for transformer fault diagnosis 31 , as delineated in Table 2 , wherein THC = CH 4  + C 2 H 4  + C 2 H 6  + C 2 H 2 , and ALL = H 2  + CH 4  + C 2 H 4  + C 2 H 6  + C 2 H 2 .

Dimensionality reduction through feature parameter fusion

To avoid the redundancy of fault-related feature information within the samples and to enhance the efficiency and precision of the diagnostic model, the SPCA method was employed for the integration of the derived rational features. The cumulative explicable variance contribution rate of each principal component is depicted in Fig.  5 . It is evident from Fig.  5 that the cumulative variance contribution rate for the first six principal components reaches 90.4419%, indicating that the first five principal components can achieve more than 90% of the ability expressed by all the principal components. Hence, selecting these five principal components as inputs for the transformer fault diagnosis model is warranted.

Cumulative variance contribution rate.

Transformer malfunction diagnosis outcomes

The fused features derived from the SPCA extraction are delineated in a ratio of 6:2:2 to be divided into training, testing, and validation datasets. The regularization parameters C within KELM determine the learning capacity of the model and its diagnostic precision; in this paper, we employ the AO optimization algorithm to optimize C, with an introduction of the AO algorithm as delineated in literature 34 , 35 , culminating in the establishment of a diagnostic model based on SPCA-AO-KELM. Figure  6 delineates the confusion matrix diagram of the transformer fault diagnosis. It is evident from Fig.  6 that within the test set of 158 samples, 155 were correctly diagnosed, representing a total correct rate of 98.1013%. The accuracy rates for normal, high-temperature overheating, and low-energy discharge diagnoses are 100%, one case of misjudgment was found in medium–low temperature overheating, high-energy discharge, and partial discharge.

Transformer fault diagnosis results.

However, the precision of diagnostic accuracy alone cannot comprehensively nor efficaciously evaluate the impact of rare class faults on classification performance 36 , 37 . In this paper, we introduce classification model performance evaluation metrics derived from confusion matrices, employing accuracy (R), precision (P), and F1-score as the core components of our evaluation system. The veracity of diagnostic models for identifying various faults is assessed by the accuracy rate, the sensitivity of the model in recognizing a variety of faults is evaluated by the coverage rate, while the F1 score derived from the amalgamation of precision and recall reflects the model's classification performance amidst sample imbalance, with specific formulas denoted in the literature displayed here. The model's precision, recall, and F1-score derived from the computed graph in Fig.  6 respectively stand at 0.9816, 0.9825, and 0.9820, further underscoring the model's high fault detection accuracy and its stable nature.

Results and discussions

Comparison and analysis of different sampling methods.

To verify the effectiveness of the new samples synthesized based on INNOS in improving the accuracy of transformer fault diagnosis, this paper uses unbalanced data set, random oversampling, SMOTE, and ADASYN oversampling algorithms for sample augmentation, and the diagnostic results are shown in Fig.  7 . The red dots in the figure represent the samples that are correctly classified in the test set, while the circles represent the samples of the true class, and the scattered dots represent the samples that are misclassified as other classes. The more scattered sample points, the higher the misclassification rate. In Fig.  7 d, the diagnostic accuracy of the original unbalanced data set without balancing processing is only 88.4058%, indicating that due to the imbalance of data in each fault category, the training of the diagnostic model is insufficient, and it is easy to misclassify minority class samples as majority class samples during classification recognition. After balancing the data set using different sampling methods, the misclassification rate of the samples decreases. The sampling method used in this paper improves the diagnostic accuracy by 7.7967%, 2.5316%, and 1.8987% compared to ADASYN, SMOTE, and random oversampling, respectively, indicating that the INNOS sampling method can effectively solve the problem of low diagnostic accuracy caused by data imbalance.

Diagnostic results under different sampling methods.

Qualitative and quantitative analysis with integrated features

To demonstrate the effectiveness of the SPCA feature fusion method, this study conducted analysis from two perspectives: qualitative observation and quantitative analysis. Firstly, PCA, KPCA, and SPCA were used to extract features from the constructed ratio signs. The cumulative variance contribution rate threshold was set at 90%, and the obtained principal component information is detailed in Table 3 . LASSO penalty term was introduced based on PCA to constrain some loading vectors to zero, resulting in a loss of variance contribution rate. From the data in the table, it can be seen that the contribution rate of SPCA principal components is slightly lower than that of PCA and KPCA, effectively removing redundant information in the ratio features and providing a valid data foundation for subsequent classification and recognition.

Furthermore, for the above feature extraction methods, quantitative calculations were performed. The fused features extracted by the 9-dimensional joint feature, PCA, KPCA, and SPCA were input into the diagnostic model for comparative analysis, as shown in Fig.  8 . From Fig.  8 a–d, it can be observed that the diagnostic accuracy is significantly improved after feature extraction. Figure  8 a has a higher accuracy compared to Fig.  8 b and c, which validates the superiority of the SPCA feature extraction method.

Diagnostic outcomes under various characteristics.

Analysis of contrastive diagnostic models

To explore the diagnostic performance of the models, three diagnostic models, ELM, KELM, and AO-ELM, were constructed for horizontal comparison. The diagnostic results are shown in Table 4 . From the perspective of a single model, the introduction of a kernel function improved the diagnostic accuracy and evaluation indicators of ELM. From the perspective of optimization algorithms, the diagnostic capability of fault recognition was effectively improved after parameter optimization using the AO algorithm.

On the other hand, the extracted integration features are respectively inputted into the POA-SVM model proposed in Literature 38 , the SSA-ELM model suggested in Literature 39 , and the PSO-BiLSTM model introduced in Literature 40 for longitudinal comparison. To circumvent the chances of chance, each model is subjected to ten-fold cross-validation, as manifested in Table 5 . It is evident from Table 5 that, under conditions where the input features remain identical, the AO-KELM outperforms both the POA-SVM and POA-SVM by elevating the average accuracy by 3.23% and 2.64%, respectively, while the PSO-BiLSTM lags behind with a mere 1.8% increase in accuracy. This clearly signifies the robust stability of the AO-KELM model and its formidable classification capabilities.

The paper introduces an oil-immersed transformer fault diagnosis method that integrates digital twin models, providing validation through case studies, leading to the conclusions below:

Build a twin mechanism model based on geometric, physical, rule, and behavior models, use real-time data to drive the fusion of data and mechanism models, complete real-time mapping between physical entities and virtual entities, and use visualization technology to express the twin in multiple dimensions, achieve intelligent diagnosis, health monitoring, and optimization decision-making for the transformer entity.

Proposed a transformer fault diagnosis model based on optimized kernel extreme learning machine, which solves the problem of misjudgment of minority class samples caused by unbalanced small samples, effectively extracts fusion features, establishes the optimal AO-KELM classifier, and achieves an accuracy of 98.1013%. By comparing with different diagnostic models, the classification performance and stability of the proposed method are verified.

Data availability

The datasets generated and/or analysed during the currentstudy are not publicly availabledue [REASON WHY DATA ARENOT PUBLlC] but are availablefrom the corresponding authoron reasonable request. E-mail:[email protected].

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Project supported by Jilin Provincial Development and Reform Commission innovation capacity construction fund (2020C022-6).

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Haiyan Y designed the experiments and contributedmaterials/analysis tools; Xin Zhang analyzed the data and its visualization; Qiang Guo and Yufeng Miao M guided the data analysis; Shan Guan wrote the paper; All authors have reviewed the manuscript.

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Yao, H., Zhang, X., Guo, Q. et al. Fault diagnosis method for oil-immersed transformers integrated digital twin model. Sci Rep 14 , 20355 (2024). https://doi.org/10.1038/s41598-024-71107-w

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A novel approach for plunging using plungers of predefined geometry

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• Mohammad A. Omari   ORCID: orcid.org/0000-0001-9501-7531 1 ,
• Mahmoud M. Rababah 2 &
• Ma’mon M. AlFakeh 1  

Plunge milling is a very efficient roughing process. Optimizing the tool path of the plungers will enhance the plunge process. A novel approach is developed and introduced in this paper to optimize the tool path of the plungers using plungers of predefined geometry. For this purpose, maximizing the covered plunging area and minimizing the number of the plunging cycles are required. Unlike the current methods that demand specific geometry for the plungers, the proposed approach starts from predefined geometry for these plungers according to the availability in the market; eliminating the constrains imposed by the current methods. The developed approach combines two basic approaches, namely: The medial axis transformation approach and the offset techniques approach in order to generate data points within the contour of the required shape. Data points are then studied for possible selection to ensure an optimal plunging process. The developed approach is validated by simulation and compared with previous work, its superiority over the current methods is confirmed through many case studies. For instance, a flower shape with a multi-regions contour and the human hand shape with some narrow and complicated regions. The studied case studies revealed the possibility of reducing the number of plungers used and increasing the plunging area. The effectiveness of the approach for narrow shapes and its orientation-independence is obvious in comparison with the current methods; qualitatively through figures and quantitatively by reduce the number of plungers to 37 instead of 72, and the plunging ration is improved to 92.3, surpassing the previous method’s ratio of 72.26 as shown in Table 2. The last comparison includes a complex pocket with island, our findings indicate an improvement over the previous method in two key aspects: reducing the number of tools and expanding the covered area.

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Omari, M.A., Rababah, M.M. & AlFakeh, M.M. A novel approach for plunging using plungers of predefined geometry. Int J Interact Des Manuf (2024). https://doi.org/10.1007/s12008-024-02079-4

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Optimization of laser cladding parameters for high-entropy alloy-reinforced 316l stainless-steel via grey relational analysis.

1. Introduction

2. materials and methods, 3.1. analysis of microhardness, 3.2. analysis of dilution rate, 3.3. analysis of average contact angle, 3.4. analysis of mean difference of contact angles, 3.5. multi-response grey relational analysis, 3.6. processing parameters optimization and experimental validation, 3.7. economic, energy, and sustainability analyses, 4. conclusions.

• The analysis of microhardness as the sole response variable indicates a notable decrease with higher powder feed rates. Moreover, variations in laser power, scanning speed, and substrate tilt angle initially lead to a reduction in microhardness, followed by subsequent improvements and subsequent declines.
• The analysis of the dilution rate indicates a significant decrease with higher powder feed rates and an increase with greater laser power. As scanning speed increases, the dilution rate initially rises and then declines. Conversely, as the substrate tilt angle increases, the dilution rate first decreases and then rises. Regarding the average contact angle, there is a progressive increase with higher powder feed rates, while an increase in substrate tilt angle leads to a decrease. The addition of HEA initially raises the average contact angle and subsequently reduces it, whereas higher laser power consistently enhances it. The mean difference in contact angles shows an initial sharp increase followed by a slight decrease with increasing laser power. Increases in the HEA addition rate, scanning speed, and powder feed rate result in an initial rise followed by a decline in the mean difference of contact angles.
• Grey relational analysis effectively discerns the processing parameters that optimize coating quality. The optimal combination comprises a 15% HEA addition rate, 750 W laser power, 7 mm/s scanning speed, 2 r/min powder feed rate, and a 30° substrate tilt angle. Coatings produced under these conditions exhibit markedly enhanced microhardness and average contact angle compared to those achieved using the optimal parameters identified in the orthogonal test.
• The validation experiment, conducted using optimal parameter settings identified through grey relational analysis, demonstrates a minimal error of 0.95% relative to the predicted value, underscoring the efficacy of GRA in optimizing laser cladding parameters. The preparation of laser cladding coatings utilizing the optimal parameters derived through GRA demonstrates enhanced economic efficiency and superior environmental sustainability compared to conventional methods. The methodologies presented in this study can also be applied to optimize laser processing parameters in selective laser melting of titanium alloys and laser powder bed fusion of aluminum alloys. These methodologies offer a robust framework for advancing laser cladding technology in industrial applications, thereby facilitating the fabrication of wear-resistant coatings and the enhancement or repair of component surfaces. Specifically, laser cladding coatings optimized through GRA can be used for corrosion resistance treatment of ships in the maritime industry, surface enhancement of specialized parts in the aerospace industry, and service protection of cooling tower components in the power industry.

Author Contributions

Institutional review board statement, informed consent statement, data availability statement, conflicts of interest.

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Gao, S.; Fu, Q.; Li, M.; Huang, L.; Liu, N.; Cui, C.; Yang, B.; Zhang, G. Optimization of Laser Cladding Parameters for High-Entropy Alloy-Reinforced 316L Stainless-Steel via Grey Relational Analysis. Coatings 2024 , 14 , 1103. https://doi.org/10.3390/coatings14091103

Gao S, Fu Q, Li M, Huang L, Liu N, Cui C, Yang B, Zhang G. Optimization of Laser Cladding Parameters for High-Entropy Alloy-Reinforced 316L Stainless-Steel via Grey Relational Analysis. Coatings . 2024; 14(9):1103. https://doi.org/10.3390/coatings14091103

Gao, Senao, Qiang Fu, Mengzhao Li, Long Huang, Nian Liu, Chang Cui, Bing Yang, and Guodong Zhang. 2024. "Optimization of Laser Cladding Parameters for High-Entropy Alloy-Reinforced 316L Stainless-Steel via Grey Relational Analysis" Coatings 14, no. 9: 1103. https://doi.org/10.3390/coatings14091103

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