Summary of fault detection results (case 1).
\r\n\tIC offer services both at the macro-scale (country) and at the micro-scale (cities). The stability and the protection of these networks on both scales are a significant task, entrusted to the Operators who operate CI in a concession mandate. Due to the high level of interdependency of CI, which exchange services to each other for their functioning, their management and protection cannot be carried out by a "linearized" strategy (each infrastructure managed and protected independently on the others) due to the presence of tight links which connect each other. CI protection through provision of "smart" properties such as resilience, has become a complex task which must be tackled not only by deploying advanced technological means but also by triggering new management strategies, enabling holistic and global management policies.
\r\n\r\n\tThe book aims at providing an overview of the understanding of complex phenomena taking place on interdependent networks and of the advanced technical solutions related to management, risk analysis and resilience enhancement of networks, either from a theoretical and operational (i.e. with solution related to real or realistic cases) points of view. A large emphasis is provided to the capability opened by the use of field and remote sensing tools for monitoring and assessing risks on CI. The use of comprehensive data set, the access to big data is going to open the way to the realization of new tools for supporting the decision making process needed for both daily and emergency management.
",isbn:"978-1-83962-621-0",printIsbn:"978-1-83962-620-3",pdfIsbn:"978-1-83962-628-9",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"7cfcd62bae8c99be207e18bb73e2a7b1",bookSignature:"Dr. Vittorio Rosato and Dr. Antonio Di Pietro",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10256.jpg",keywords:"Complex systems, Interdependence, Resilience, Cascading effects, Event prediction, Emergency management, Decision support, AI, Field sensors, Remote sensing, IoT, GIS",numberOfDownloads:525,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"June 15th 2020",dateEndSecondStepPublish:"July 6th 2020",dateEndThirdStepPublish:"September 4th 2020",dateEndFourthStepPublish:"November 23rd 2020",dateEndFifthStepPublish:"January 22nd 2021",remainingDaysToSecondStep:"8 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Supervisor and Project Evaluator for EU, for the Italian Ministry of University and Research, and that of Economic Development; Consultant for several Italian Regions and the Italian Ministry of Defense; Coordinator of several National Projects; Co-founder of two SMEs active in software engineering and biotechnology; Author of more than 140 scientific papers in peer reviewed journals and conference proceedings.",coeditorOneBiosketch:"A full researcher at ENEA (Italian Energy, New Technology and Environment Agency) since 2007 and a joined member to the Laboratory for the Analysis and Protection of Critical Infrastructures (APIC) from 2015. Dr. Di Pietro took part in several European and Italian national research projects and acted as an advisor for certain Evaluation Studies commissioned by the EU.",coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"27002",title:"Dr.",name:"Vittorio",middleName:null,surname:"Rosato",slug:"vittorio-rosato",fullName:"Vittorio Rosato",profilePictureURL:"https://mts.intechopen.com/storage/users/27002/images/system/27002.jpg",biography:"Vittorio Rosato received the Laurea degree (M.Sc.) in Physics from the University of Pisa (Italy) and a Ph.D. in Condensed Matter Physics from the University of Nancy (France). He has extensively been working in Computational Physics, particularly in Condensed Matter and Material Science in his positions as Research Assistant at the University College of Wales in Aberystwyth (UK) and at the Centre d'Etudes Nucleaires in Saclay (France). Staff Scientist at ENEA (Italian National Agency for New Technologies, Energy and Sustainable Economic Development) since 1990, he is currently Head of the Laboratory of Analysis and Protection of Critical Infrastructures and Manager of the Italian Node of the European Infrastructure Simulation and Analysis Centre (EISAC.it).\nHis current research activities span from risk analysis to the design of Decision Support Systems for the management of complex technological networks. He acts as Supervisor and Project Evaluator for EU, for the Italian Ministry of University and Research, and that of Economic Development; he is also consultant for several Italian Regions and the Italian Ministry of Defense. He is and has been Coordinator of several National Projects. He is co-founder of two SMEs active in software engineering and biotechnology. He is author of more than 140 scientific papers in peer reviewed journals and conference proceedings.",institutionString:"ENEA (Italian National Agency for New Technologies, Energy and Sustainable Economic Development)",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"0",institution:null}],coeditorOne:{id:"284589",title:"Dr.",name:"Antonio",middleName:null,surname:"Di Pietro",slug:"antonio-di-pietro",fullName:"Antonio Di Pietro",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bReF6QAK/Profile_Picture_1581328351906",biography:"Antonio Di Pietro received the Laurea degree (M.Sc.) in Informatics Engineering from Sapienza University of Rome (Italy) and a Ph.D. in Methodologies for Emergency Management in Critical Infrastructures from Roma Tre University (Rome).\nHe has been working as a full researcher at ENEA (Italian Energy, New Technology and Environment Agency) since 2007 and in 2015 he joined the Laboratory for the Analysis and Protection of Critical Infrastructures (APIC) in the same institution.\nHis research interests include modelling and simulation of critical infrastructures and the development of Decision Support Systems integrating seismic and meteorological natural threat modeling. He is also an Unmanned Aerial Vehicles (UAV) ENAC-certifed pilot to perform critical operations involving aerial photogrammetry tasks for biological and Infrastructure monitoring applications. He took part in several European and Italian national research projects and acted as an advisor in some Evaluation Studies commissioned by the EU. He has also been advisor of several M.Sc. students and also a teacher in several professional courses on Software Engineering and Databases.",institutionString:"ENEA (Italian National Agency for New Technologies, Energy and Sustainable Economic Development)",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"0",institution:null},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"11",title:"Engineering",slug:"engineering"}],chapters:[{id:"74122",title:"Risk Analysis in Early Phase of Complex Infrastructure Projects",slug:"risk-analysis-in-early-phase-of-complex-infrastructure-projects",totalDownloads:87,totalCrossrefCites:0,authors:[null]},{id:"74493",title:"Flood Risk Analysis for Critical Infrastructure Protection: Issues and Opportunities in Less Developed Societies",slug:"flood-risk-analysis-for-critical-infrastructure-protection-issues-and-opportunities-in-less-develope",totalDownloads:50,totalCrossrefCites:0,authors:[null]},{id:"74123",title:"Resilience in Critical Infrastructures: The Role of Modelling and Simulation",slug:"resilience-in-critical-infrastructures-the-role-of-modelling-and-simulation",totalDownloads:78,totalCrossrefCites:0,authors:[null]},{id:"73984",title:"Validation Strategy as a Part of the European Gas Network Protection",slug:"validation-strategy-as-a-part-of-the-european-gas-network-protection",totalDownloads:35,totalCrossrefCites:0,authors:[null]},{id:"74174",title:"Defects Assessment in Subsea Pipelines by Risk Criteria",slug:"defects-assessment-in-subsea-pipelines-by-risk-criteria",totalDownloads:44,totalCrossrefCites:0,authors:[null]},{id:"74240",title:"Analyzing the Cyber Risk in Critical Infrastructures",slug:"analyzing-the-cyber-risk-in-critical-infrastructures",totalDownloads:85,totalCrossrefCites:0,authors:[null]},{id:"74141",title:"Italian Crisis Management in 2020",slug:"italian-crisis-management-in-2020",totalDownloads:49,totalCrossrefCites:0,authors:[null]},{id:"74668",title:"A Strategy to Improve Infrastructure Survivability via Prioritizing Critical Nodes Protection",slug:"a-strategy-to-improve-infrastructure-survivability-via-prioritizing-critical-nodes-protection",totalDownloads:34,totalCrossrefCites:0,authors:[null]},{id:"74143",title:"Resilience of Critical Infrastructures: A Risk Assessment Methodology for Energy Corridors",slug:"resilience-of-critical-infrastructures-a-risk-assessment-methodology-for-energy-corridors",totalDownloads:66,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"205697",firstName:"Kristina",lastName:"Kardum Cvitan",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/205697/images/5186_n.jpg",email:"kristina.k@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3621",title:"Silver Nanoparticles",subtitle:null,isOpenForSubmission:!1,hash:null,slug:"silver-nanoparticles",bookSignature:"David Pozo Perez",coverURL:"https://cdn.intechopen.com/books/images_new/3621.jpg",editedByType:"Edited by",editors:[{id:"6667",title:"Dr.",name:"David",surname:"Pozo",slug:"david-pozo",fullName:"David Pozo"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"68233",title:"Fault Detection of Single and Interval Valued Data Using Statistical Process Monitoring Techniques",doi:"10.5772/intechopen.88217",slug:"fault-detection-of-single-and-interval-valued-data-using-statistical-process-monitoring-techniques",body:'\nCurrent technological advancements allow data to be collected from a number of different sources. The availability of abundant data collected from different sensors is beneficial, as they can be utilized in order to observe trends between and within different measured process variables. This allows process models to be developed in order to help identify if different processes or applications are behaving as expected [1]. Additionally, with industrial growth present in many developing countries, efficient process monitoring is essential for newer and more complex processes. Monitoring of these processes is required in order to ensure process safety, maintain product quality, increase economic benefits, and also to ensure that the process adheres to strict environmental regulation standards [2].
\nStatistical process monitoring methods can be classified into three broad categories: quantitative model based methods, qualitative model based methods, and process history based methods [3, 4, 5]. Quantitative model based methods require detailed knowledge of a process in order to construct a model that can be used for monitoring, for example, Kalman filters [3], while qualitative model based methods require the presence of process engineering experts in order to develop monitoring procedures or tasks, for example, fault trees [4]. In the absence of these two requirements, and due to the complexity of many processes that require monitoring, data-based techniques are often commonly used by the industry for various applications from drug design, to drinking water treatment [5, 6, 7].
\nPrincipal component analysis (PCA) is a powerful, linear data analysis technique widely used in research and industrial applications [8], for fault detection and isolation, data modeling and reconstruction, feature extraction, and noise filtration. PCA is useful for the extraction of dominant underlying information from a dataset, without any previous knowledge of the model. An example of the practical application of PCA has been discussed in [8], where data gathered from parallel sensors are used to quantify the quality of a given food sample. PCA is used to reduce the dimensionality of a dataset, whilst filtering out variability caused by noise [9]. The PCA model has been utilized in order to monitor a wide variety of processes, and has seen many extensions [10, 11, 12, 13]. Two main fault detection statistics are typically utilized with a PCA model: Hotelling’s T2 statistic, and the Q statistic [10]. Variations captured by the principal component space are monitored using the T2 statistic, while variations in the residual space are monitored using the Q statistic [14].
\nOn the other hand, statistical hypothesis testing methods function by using statistical techniques in order to determine if observations collected from a given process follow the null hypothesis, that is, operating under normal operating conditions, or alternate hypothesis, that is, operating under abhorrent or faulty operating conditions [15]. These faults can be of different types, such as shifts in the mean, variance, or both. The generalized likelihood ratio (GLR) technique has received a lot of attention in process monitoring literature [10, 11, 13, 16]. The GLR method aims to maximize the detection rate for a fixed false alarm rate [15]. Therefore, an objective of this work is to provide a comparative review of the different GLR charts by utilizing examples such as the benchmark Tennessee Eastman Process (TEP) [17].
\nData utilized in the construction of a PCA model may be of two types depending on the application being monitored: single-valued, and interval-valued. Single-valued data can be directly obtained from sensors measuring particular variables in a process, while interval-valued data is aggregated or artificially generated from batch single-valued measurements, thereby resulting in a range of possible measurement values for a given process variable at one time instant. The use of interval data in fault detection was originally introduced in order to reduce large datasets to a more manageable size [18], without compromising the integrity of the dataset. In addition, the use of interval data is beneficial because of its inherent ability to deal with missing values in samples, which may happen due to malfunctioning sensors or varying sampling frequencies between variables [19].
\nHowever, in cases where reducing the dataset may not be a viable option, due to a relatively limited sample size or sampling frequency, the use of interval data can be applied using a moving window aggregation method. This is also true of applications where batch process monitoring is not a viable option, thereby necessitating the need for real-time online monitoring of samples. The benchmark TEP example will be used once more in order to analyze the benefit of using moving window interval aggregation on the fault detection performance of PCA and GLR.
\nThe rest of this chapter will be organized as follows. In Section 2, a more detailed introduction to PCA is provided along with a quick overview of the fault detection statistics used to examine the fault detection performance of the methods discussed in this paper. Section 3 will introduce hypothesis testing methods and the different GLR charts. In Section 4, the moving window interval aggregation method is explained, as well as its integration with PCA and GLR for the purposes of fault detection. Section 5 then presents illustrative examples using simulated synthetic data and TEP using a PCA-based GLR technique, used to demonstrate the effect that using GLR and interval data has on the fault detection performance. Conclusions are then presented in Section 6.
\nPrincipal component analysis (PCA) is a linear dimensionality reduction tool used to reduce the number of variables in a dataset, whilst retaining most of the data’s variability. PCA finds a new set of variables, called principal components, using a linear combination of the dataset’s original cross-correlated variables [9]. The algorithm for PCA is summarized below.
\nGiven a \n
Find the correlation matrix \n
Find the column eigenvectors matrix \n
Retain \n
Find the predictive transformation matrix, \n
Find the residual transformation matrix, \n
\n\n
The training dataset \n
Knowing the optimal number of eigenvectors or principal components to retain, fault detection is then carried out by evaluating the PCA model’s residuals using any detection statistic. This section will focus on briefly introducing the two most well-known statistics in literature: The Q and T2 statistics.
\nThe Q-statistics of a \n
\n\n
The fault detection performance is tabulated by comparing \n
FAR is the average percentage of samples that were wrongfully declared as faults. The detection rate is the average percentage of samples that were rightfully declared as faults. It is desirable to maximize DR, for a fixed FAR, in order to have a better fault detector.
\nAlternatively, the Hotelling T2 statistic, which measures variations in the principal component space can be used, is computed as follows [22]:
\nwhere, \n
Hypothesis testing methods such as the generalized likelihood ratio (GLR), have received a lot of attention in recent literature [10, 13, 23]. Hypothesis testing methods utilize fundamental statistical theory in order to determine if given data conforms to a targeted distribution, that is, a null hypothesis, or deviates from this distribution, and follows an alternative distribution, that is, an alternate hypothesis [15]. In process monitoring terms, the parameters of the null and alternate hypotheses are defined using data from normal and abhorrent operating conditions, respectively [1].
\nThe generalized likelihood ratio (GLR) technique defines the alternate hypotheses by parameters that can assume an infinite number of values, and is therefore called a composite hypothesis. An efficient point estimation method that utilizes the concept of maximum likelihood estimates (MLEs) is employed in order to estimate the required parameters.
\nThe univariate GLR chart uses the concept of maximum likelihood estimates in order to maximize the detection rate for a fixed false alarm rate. The GLR process is accomplished through the following steps [15]:
The null and alternate hypotheses are defined, and their respective likelihood functions are derived.
Any unknown parameters in the alternate hypothesis are computed from the testing data using their MLEs, for example, the mean and/or variance.
The log likelihood ratio of the alternate to null hypotheses is then computed, and its maximum value is calculated, which maximizes the detection rate.
Univariate GLR charts can be designed based on the type of the fault that needs to be detected. Most processes experience shifts in the mean, and/or shifts in the variance, and three of these GLR charts will be explained next.
\nFor the case when residuals are collected from processes under normal operating conditions, the likelihood function derived from a random normal distribution can be defined as follows [24]:
\nwhere \n
If a shift in the mean has occurred at time \n
Since the magnitude of the new mean is unknown, its MLE can be computed using testing data as follow [24]:
\nThe GLR statistic designed to specifically monitor a shift in the mean can now be computed by taking the log-likelihood ratio of (Eqs. (3) and (4)) [24]:
\nThe authors in [24] state that it is not necessary to store the entire length of previous historical data in order to compute the MLEs, but a window length of about 400 is sufficient to provide reliable results. Therefore, a window length of 400 was utilized throughout this work for all GLR charts.
\nIf only a shift in the variance has occurred from at time \n
From a quality control standpoint we are only concerned with increases in variance, as larger variations imply that product is being manufactured with quality further away from the targeted amount, and since the magnitude of the new variance is unknown, its MLE can be computed using testing data as follows [25]:
\nThe GLR statistic designed to specifically monitor a shift in the variance can now be computed by taking the log-likelihood ratio of (Eqs. (3) and (7)) [25]:
\nSince it is possible for most processes to experience both shifts in the mean and variance, a GLR statistic that is capable of detecting either type of shift can be designed. The likelihood function of the alternate hypothesis for this case is defined as follows [26]:
\nThe MLE of the mean can be computed from the testing data using (Eq. (5)). However, the variance now has to be computed utilizing the MLE for the mean as well [26]:
\nAs previously stated, from a quality control standpoint only an increase in the variance is of concern, and the MLE for the variance can be computed as follows [26]:
\nIf there are no shifts in the mean for testing data, the variance is computed as follows [26]:
\nIn this case, the GLR statistic designed to simultaneously monitor both shifts in the mean and variance, and can be computed by taking the log-likelihood ratio of (Eqs. (3) and (10)) resulting in the following equation [26]:
\nIt is important to note that for this particular GLR method, two parameters, that is, the mean and the variance have to be estimated using their MLE, since the type of shift is unknown.
\nSince using a univariate GLR chart may not always be practical, Wang and Reynolds [27] introduce the multivariate GLR chart, designed to specifically monitor shifts in the process mean for multivariate applications. In this case, the GLR statistic is defined as follows:
\nWhere \n
The PCA method introduced in Section 2 is commonly utilized by many industries. Therefore, it is necessary to integrate the simplicity of the PCA method with the advantages brought forward by the GLR charts, so that it can be easily applied to monitor processes online. \nFigure 1\n illustrates the fault detection algorithm utilized in this work.
\nPCA-based GLR fault detection algorithm.
PCA is utilized in order to model available data. The different GLR charts can then be applied on the residuals produced by the PCA model in order to determine if the process is operating under normal or faulty conditions. The fault detection threshold limits are obtained from an empirical distribution of the GLR statistic computed under normal operating conditions. The residual space is typically better able at detecting faults of smaller magnitude [10].
\nData utilized in the construction of a PCA model may be of two types depending on the application being monitored: single-valued, and interval-valued. Single-valued data can be directly obtained from sensors measuring particular variables in a process, while interval-valued data is aggregated or artificially generated from batch single-valued measurements, thereby resulting in a range of possible measurement values for a given process variable at one time instant [18].
\nAn interval is defined using a lower and upper bound, such as [a, b], where \n
Initially, the use of interval data is motivated by the need to quickly and efficiently monitor large datasets [28], in addition to its ability to deal with missing values without the need to remove entire samples. Generating intervals by aggregation is a form of batch processing, which may not always be ideal. The ability to monitor faults in real-time is typically much more desirable from a quality and safety standpoint. It also becomes impractical to use batch aggregation when discussing processes with a low sample size or low sampling frequency.
\nAs a result, interval data aggregation must be adapted for real-time monitoring purposes. One way to do that would be to use a moving window aggregation technique, such that any observed sample is aggregated with previously gathered samples, if any, in the defined window size. This allows for the generation and processing of interval data in real-time, without the need to wait for multiple samples to be observed before processing.
\nAs expected, however, this method suffers from some drawbacks relative to its batch aggregation counterpart. The moving window approach may cause smearing along the detection statistic, leading to higher false alarms and lower detection rates. This is especially true for large window sizes, as is the case for most methods which apply that approach. The problem can be mitigated by limiting the window size to reasonable limits, whilst also adjusting the threshold in order to meet the desired false alarm rates of the process.
\nInterval principal component analysis (IPCA) methods are an extension to the classical PCA method, and they have been explored in literature for fault detection and isolation examples [29, 30]. In this work, three IPCA methods will be briefly introduced, before discussing our proposed method of integrating the moving window interval approach to the PCA-based GLR technique.
\nCenters IPCA (CIPCA) was introduced by Cazes et al. [31], where the idea was to only apply PCA to the matrix of interval centers. This method focuses on the variation between the intervals of a dataset, rather than the variations within them [18, 32]. Midpoint-Radii IPCA (MRIPCA) was developed by Lauro et al. [33, 34, 35, 36], where PCA models are separately generated for the centers and radii matrices of the interval training dataset. Finally, the Symbolic Covariance IPCA (SCIPCA) method was introduced by Le-Rademacher et al. [18, 32] as a way to better represent the range and variability found in interval data.
\nIn this paper, the integration of the moving window aggregation to PCA-based GLR will be as follows. After generating an interval sample for each single-valued sample, the single-valued matrices of interval centers and radii are extracted. The matrices are then concatenated along the variables dimension, so as to maintain the number of samples, but double the number of variables. This is similar to the MRIPCA method, except it avoids the need to apply PCA twice, eliminating any additional processing complexity.
\nThis section evaluates the performance of the three PCA-based GLR charts described in Section 3, and the moving window aggregation method discussed in Section 4. The PCA-based GLR charts are evaluated under different fault scenarios, and this is done through two illustrative examples: a simulated synthetic data set, and the benchmark Tennessee Eastman Process (TEP). Three fault detection metrics are used to evaluate the performance of each univariate chart: missed DR (which is equal to 100-DR), FAR, and average out-of-control run length (ARL1). Finally, the moving window interval aggregation method, in tandem with the PCA-based multivariate GLR chart, are analyzed using the benchmark TEP process, and the results are tabulated and compared to the single-valued multivariate GLR chart.
\nThe purpose of this example is to utilize a simple linear model to compare and evaluate the performance of the difference PCA-based univariate GLR charts. The linear data set can be generated using the following model [37]:
\nwhere, \n
The linear model is used to generate 6000 observations, split into training and testing data sets of 3000 observations each. The training data are used to train the PCA model, while the testing data are used to evaluate the performance of all techniques using three cases of faults: a shift in the mean, a shift in the variance, and a simultaneous shift in both.
\nFive charts are evaluated and compared: the PCA-based T2 and Q charts, and the three different PCA-based univariate GLR charts. The faulty region is highlighted in light blue for all figures, and the fault detection threshold limits for all charts are represented by the red dotted line. For each case a Monte-Carlo simulation of 1000 realizations is carried out in order to obtain meaningful results, so that conclusions can be drawn.
\nFor this case, a shift in the mean of \n
As can be seen through \nFigure 2\n, the T2 and Q charts are unable to detect the entirety of the fault. In contrast, two GLR charts (\nFigure 3a\n and \nc\n), are able to detect most of the fault, while the GLR chart designed to monitor a shift in the variance (\nFigure 3b\n) could not detect that a shift in the mean was present.
\nPCA-based T2 and Q charts (case 1).
PCA-based GLR charts (case 1).
Examining the summary of the fault detection results (\nTable 1\n), it can be observed that the GLR chart designed to monitor shifts in the mean (\nFigure 3a\n) provided the lowest missed DR and ARL1 values, compared to all other charts.
\n\n | PCA-based T2\n | \nPCA-based Q | \nPCA-based GLR (to monitor mean) | \nPCA-based GLR (to monitor variance) | \nPCA-based GLR (to monitor mean and/or variance) | \n
---|---|---|---|---|---|
Missed DR (%) | \n95.3 | \n94.5 | \n00.4 | \n85.1 | \n31.5 | \n
FAR (%) | \n05.2 | \n05.5 | \n05.3 | \n05.8 | \n04.6 | \n
ARL1\n | \n20.1 | \n16.6 | \n04.8 | \n81.8 | \n05.0 | \n
Summary of fault detection results (case 1).
The relatively high missed DR of the GLR chart designed to simultaneously monitor shifts in both the mean and variance (\nFigure 3c\n) can be attributed to the fact that two parameters need to be estimated from available data while maximizing the GLR statistic, thereby making it difficult to predict a shift in a single parameter as efficiently.
\nFor this case, an increase in the variance (double that of the training data) was introduced between observations 1501:3000 in \n
As can be seen through \nFigure 4\n, the T2 and Q charts are unable to detect the entirety of the fault. In contrast, two GLR charts (\nFigure 5b\n and \nc\n) were able to detect most of the fault, while the GLR chart designed to monitor a shift in the mean (\nFigure 5a\n) could not detect it as well. Examining the summary of the results (\nTable 2\n), it can be observed that the GLR chart designed to monitor a shift in the variance (\nFigure 5b\n) provided the lowest missed DR and ARL1 values, compared to other charts.
\nPCA-based T2 and Q charts (case 2).
PCA-based GLR charts (case 2).
\n | PCA-based T2\n | \nPCA-based Q | \nPCA-based GLR (to monitor mean) | \nPCA-based GLR (to monitor variance) | \nPCA-based GLR (to monitor mean and/or variance) | \n
---|---|---|---|---|---|
Missed DR (%) | \n90.2 | \n88.6 | \n47.5 | \n00.7 | \n33.0 | \n
FAR (%) | \n05.3 | \n05.4 | \n05.0 | \n04.8 | \n04.8 | \n
ARL1\n | \n10.1 | \n8.3 | \n07.9 | \n04.5 | \n05.6 | \n
Summary of fault detection results (case 2).
For this case, a simultaneous shift in the mean of \n
As can be seen through \nFigure 6\n, the T2 and Q charts are unable to detect the entirety of the fault once more. Although it might seem that all three GLR charts (\nFigure 7\n) are able to detect most of the fault, upon closer inspection of the results summarized in \nTable 3\n, it can be observed that the GLR charts designed to independently detect a shift in the mean (\nFigure 7a\n), and variance (\nFigure 7b\n), are able to provide significantly lower missed DR and ARL1 values compared to the chart designed to monitors shifts in both (\nFigure 7c\n).
\nPCA-based T2 and Q charts (case 3).
PCA-based GLR charts (case 3).
\n | PCA-based T2\n | \nPCA-based Q | \nPCA-based GLR (to monitor mean) | \nPCA-based GLR (to monitor variance) | \nPCA-based GLR (to monitor mean and/or variance) | \n
---|---|---|---|---|---|
Missed DR (%) | \n86.7 | \n84.5 | \n00.4 | \n00.4 | \n24.2 | \n
FAR (%) | \n05.2 | \n05.2 | \n04.9 | \n05.3 | \n05.5 | \n
ARL1\n | \n07.5 | \n06.0 | \n03.2 | \n03.9 | \n04.9 | \n
Summary of fault detection results (case 3).
The main conclusion from this example is that if a process is expected to experience shifts in both the mean and/or variance, it is more beneficial to run the PCA-based GLR charts designed to independently monitor shifts in the mean and variance as two parallel charts, rather than utilizing the GLR chart designed to simultaneously monitor both. Based on this conclusion, only the former two GLR charts will be utilized for the next example.
\nIn order to assess the feasibility of using two separate GLR charts to monitor shifts in the process mean and variance, their performance has to be evaluated using real data. Many authors utilize the Tennessee Eastman Process (TEP) in order to evaluate the performance of their techniques [17, 38, 39]. The Tennessee Eastman Process is a realistic simulation of an actual chemical process that consists of a reactor, condenser, stripper, compressor, and separator, and is widely accepted as a benchmark for fault detection [17].
\nThe Tennessee Eastman Process contains a bank of pre-defined faults that can be utilized by authors in order to assess the performance of their developed fault detection algorithms. More information on the Tennessee Eastman Process, the process description, and the available bank of faults is available in literature [10, 17, 21, 38, 39].
\nTwo fault scenarios will be examined in this work: IDV 3 and IDV 11 [39]. IDV 3 is a shift in the mean of the temperature of Feed D, while IDV 11 is random variation in the reactor cooling water inlet temperature [39]. These two fault scenarios were selected because the conventional techniques are unable to provide the best possible detection. For both scenarios, the fault is introduced after 800 observations of normal operation. The performance of four charts are evaluated: PCA-based T2 and Q charts, and the PCA-based univariate GLR charts designed to independently monitor shifts in the mean and variance. The faulty region is highlighted in light blue in all figures.
\nFor the case where there is a shift in the mean of the temperature of Feed D, the PCA-based T2 and Q charts, and the PCA-based univariate GLR charts are illustrated in \nFigures 8\n and \n9\n respectively, and the fault detection results are summarized in \nTable 4\n.
\nPCA-based T2 and Q charts (IDV 3).
PCA-based GLR charts (IDV 3).
\n | PCA-based T2\n | \nPCA-based Q | \nPCA-based GLR (to monitor mean) | \nPCA-based GLR (to monitor variance) | \n
---|---|---|---|---|
Missed DR (%) | \n97.6 | \n92.8 | \n07.9 | \n70.9 | \n
FAR (%) | \n04.8 | \n04.5 | \n05.0 | \n05.4 | \n
ARL1\n | \n02.0 | \n86.0 | \n84.0 | \n84.00 | \n
Summary of fault detection results (IDV 3).
From \nFigure 8\n it can be observed that the T2 and Q charts are unable to detect the entirety of the fault, while the GLR chart designed to monitor shifts in the mean (\nFigure 9a\n) is able to detect the most of the fault, and provides the lowest missed DR (\nTable 4\n). Although, the T2 chart returns a low ARL1 value, it does not detect the fault efficiently, and the low ARL1 value can be attributed to random noise.
\nFor the case where there is random variation in the reactor cooling water inlet temperature, the T2 and Q charts, and the GLR charts are illustrated in \nFigures 10\n and \n11\n respectively, and the fault detection results are summarized in \nTable 5\n.
\nPCA-based T2 and Q charts (IDV 11).
PCA-based GLR charts (IDV 11).
\n | PCA-based T2\n | \nPCA-based Q | \nPCA-based GLR (to monitor mean) | \nPCA-based GLR (to monitor variance) | \n
---|---|---|---|---|
Missed DR (%) | \n09.9 | \n22.3 | \n02.3 | \n01.9 | \n
FAR (%) | \n05.1 | \n05.0 | \n05.0 | \n05.4 | \n
ARL1\n | \n20.0 | \n24.0 | \n28.0 | \n24.0 | \n
Summary of fault detection results (IDV 11).
Although it might seem like the T2 and Q charts (\nFigure 10\n) are able to detect most of the fault, they still have higher missed DR than both GLR charts (\nFigure 11\n). The GLR chart designed to monitor shifts in the variance provides the lowest missed DR from the charts that were compared.
\nFrom this example we can conclude that the PCA-based GLR charts are able to provide improved fault detection results over the conventional PCA-based T2 and Q charts. The improved results can be attributed to the use of MLEs to estimate the values of the unknown parameters used to maximize the GLR statistic, allowing for the best possible DR to be achieved for a fixed FAR. This example also demonstrates that the GLR charts can be easily designed and utilized to monitor chemical processes, such as the TEP.
\nFor the final case study, the moving window interval aggregation method is tested for the same fault scenarios tested previously for the TEP: IDV 3 and IDV 11. A smaller sample window size of 10 samples is used for the multivariate GLR chart in order to highlight the difference between using single and interval-valued data more clearly.
\nThe interval aggregation window size was set at 10 samples. The IDV 3 and IDV 11 scenarios for both data types are shown in \nFigures 12\n and \n13\n, and the metrics for each method are tabulated in \nTable 6\n.
\nPCA-based multivariate GLR charts (IDV 3).
PCA-based multivariate GLR charts (IDV 11).
\n | IDV 3 Single-valued multivariate GLR | \nIDV 3 Interval-valued multivariate GLR | \nIDV 11 Single-valued multivariate GLR | \nIDV 11 Interval-valued multivariate GLR | \n
---|---|---|---|---|
Missed DR (%) | \n15.1 | \n00.0 | \n02.0 | \n00.0 | \n
FAR (%) | \n05.0 | \n05.0 | \n05.0 | \n05.0 | \n
Summary of fault detection results (single vs. interval data) for α = 5%.
There are two major observations to be made from the results. First, the use of the multivariate GLR chart allowed for a more stable FAR for all cases due to the presence of a single statistic to monitor for all variables, as opposed to the one for each variable when using the univariate GLR charts. Second, the missed DR when using interval data was significantly lower than that for single-valued data, reaching perfect performance levels of zero missed DR for both scenarios.
\nThe latter observation is attributed to interval data, especially the method of generation, where the centers and radii are used as independent variables in the same dataset. This method of aggregation helps the PCA model account for shifts in the mean and variance respectively, similar to the univariate GLR chart outline in Section 3.1.3. However, it does so without the need to tune any extra parameters, due to the fact that a fault in the centers is likely to be caused by a shift in the mean, while a fault in the radii is likely to be caused by a shift in the variance.
\nIn this chapter, the performance of GLR charts were compared to conventional fault detection statistics, specifically the Q and T2 statistics, and the integration of interval-valued data into real-time process monitoring was explored. The performance of different PCA-based univariate GLR charts were examined using single-valued data through two illustrative examples: simulated synthetic data, and the Tennessee Eastman Process. The performance of the moving window interval aggregation method was evaluated alongside that of single-valued data for the multivariate GLR chart as well.
\nThe results demonstrate that in order to monitor processes that may experience both shifts in the mean and/or variance, the best performance is achieved by implementing the two respective univariate GLR charts separately in parallel, rather than the single chart designed to simultaneously detect shifts in both, as the simultaneous estimation of two parameters is unable to provide the best possible fault detection performance. Moreover, the moving window interval aggregation method, when combined with the multivariate GLR chart, was able to provide a perfectly stable statistic, with an unwavering false alarm rate, in addition to the best possible performance in detecting shifts in the mean and variance for two scenarios of the Tennessee Eastman Process.
\nThis work was made possible by NPRP grant NPRP7-1172-2-439 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. The statements herein are solely the responsibility of the authors.
\nStability constant of the formation of metal complexes is used to measure interaction strength of reagents. From this process, metal ion and ligand interaction formed the two types of metal complexes; one is supramolecular complexes known as host-guest complexes [1] and the other is anion-containing complexes. In the solution it provides and calculates the required information about the concentration of metal complexes.
Solubility, light, absorption conductance, partitioning behavior, conductance, and chemical reactivity are the complex characteristics which are different from their components. It is determined by various numerical and graphical methods which calculate the equilibrium constants. This is based on or related to a quantity, and this is called the complex formation function.
During the displacement process at the time of metal complex formation, some ions disappear and form a bonding between metal ions and ligands. It may be considered due to displacement of a proton from a ligand species or ions or molecules causing a drop in the pH values of the solution [2]. Irving and Rossotti developed a technique for the calculation of stability constant, and it is called potentiometric technique.
To determine the stability constant, Bjerrum has used a very simple method, and that is metal salt solubility method. For the studies of a larger different variety of polycarboxylic acid-, oxime-, phenol-containing metal complexes, Martel and Calvin used the potentiometric technique for calculating the stability constant. Those ligands [3, 4] which are uncharged are also examined, and their stability constant calculations are determined by the limitations inherent in the ligand solubility method. The limitations of the metal salt solubility method and the result of solubility methods are compared with this. M-L, MLM, and (M3) L are some types of examples of metal-ligand bonding. One thing is common, and that is these entire types metal complexes all have one ligand.
The solubility method can only usefully be applied to studies of such complexes, and it is best applied for ML; in such types of system, only ML is formed. Jacqueline Gonzalez and his co-worker propose to explore the coordination chemistry of calcium complexes. Jacqueline and et al. followed this technique for evaluate the as partial model of the manganese-calcium cluster and spectrophotometric studies of metal complexes, i.e., they were carried calcium(II)-1,4-butanediamine in acetonitrile and calcium(II)-1,2-ethylendiamine, calcium(II)-1,3-propanediamine by them.
Spectrophotometric programming of HypSpec and received data allows the determination of the formation of solubility constants. The logarithmic values, log β110 = 5.25 for calcium(II)-1,3-propanediamine, log β110 = 4.072 for calcium(II)-1,4-butanediamine, and log β110 = 4.69 for calcium(II)-1,2-ethylendiamine, are obtained for the formation constants [5]. The structure of Cimetidine and histamine H2-receptor is a chelating agent. Syed Ahmad Tirmizi has examined Ni(II) cimetidine complex spectrophotometrically and found an absorption peak maximum of 622 nm with respect to different temperatures.
Syed Ahmad Tirmizi have been used to taken 1:2 ratio of metal and cimetidine compound for the formation of metal complex and this satisfied by molar ratio data. The data, 1.40–2.4 × 108, was calculated using the continuous variation method and stability constant at room temperature, and by using the mole ratio method, this value at 40°C was 1.24–2.4 × 108. In the formation of lead(II) metal complexes with 1-(aminomethyl) cyclohexene, Thanavelan et al. found the formation of their binary and ternary complexes. Glycine,
Using the stability constant method, these ternary complexes were found out, and using the parameters such as Δ log K and log X, these ternary complex data were compared with binary complex. The potentiometric technique at room temperature (25°C) was used in the investigation of some binary complex formations by Abdelatty Mohamed Radalla. These binary complexes are formed with 3D transition metal ions like Cu2+, Ni2+, Co2+, and Zn2+ and gallic acid’s importance as a ligand and 0.10 mol dm−3 of NaNO3. Such types of aliphatic dicarboxylic acids are very important biologically. Many acid-base characters and the nature of using metal complexes have been investigated and discussed time to time by researchers [7].
The above acids (gallic and aliphatic dicarboxylic acid) were taken to determine the acidity constants. For the purpose of determining the stability constant, binary and ternary complexes were carried in the aqueous medium using the experimental conditions as stated above. The potentiometric pH-metric titration curves are inferred for the binary complexes and ternary complexes at different ratios, and formation of ternary metal complex formation was in a stepwise manner that provided an easy way to calculate stability constants for the formation of metal complexes.
The values of Δ log K, percentage of relative stabilization (% R. S.), and log X were evaluated and discussed. Now it provides the outline about the various complex species for the formation of different solvents, and using the concentration distribution, these complexes were evaluated and discussed. The conductivity measurements have ascertained for the mode of ternary chelating complexes.
A study by Kathrina and Pekar suggests that pH plays an important role in the formation of metal complexes. When epigallocatechin gallate and gallic acid combine with copper(II) to form metal complexes, the pH changes its speculation. We have been able to determine its pH in frozen and fluid state with the help of multifrequency EPR spectroscopy [8]. With the help of this spectroscopy, it is able to detect that each polyphenol exhibits the formation of three different mononuclear species. If the pH ranges 4–8 for di- or polymeric complex of Cu(II), then it conjectures such metal complexes. It is only at alkaline pH values.
The line width in fluid solutions by molecular motion exhibits an incomplete average of the parameters of anisotropy spin Hamilton. If the complexes are different, then their rotational correlation times for this also vary. The analysis of the LyCEP anisotropy of the fluid solution spectra is performed using the parameters determined by the simulation of the rigid boundary spectra. Its result suggests that pH increases its value by affecting its molecular mass. It is a polyphenol ligand complex with copper, showing the coordination of an increasing number of its molecules or increasing participation of polyphenol dimers used as ligands in the copper coordination region.
The study by Vishenkova and his co-worker [8] provides the investigation of electrochemical properties of triphenylmethane dyes using a voltammetric method with constant-current potential sweep. Malachite green (MG) and basic fuchsin (BF) have been chosen as representatives of the triphenylmethane dyes [9]. The electrochemical behavior of MG and BF on the surface of a mercury film electrode depending on pH, the nature of background electrolyte, and scan rate of potential sweep has been investigated.
Using a voltammetric method with a constant-current potential sweep examines the electrical properties of triphenylmethane dye. In order to find out the solution of MG and BF, certain registration conditions have been prescribed for it, which have proved to be quite useful. The reduction peak for the currents of MG and BF has demonstrated that it increases linearly with respect to their concentration as 9.0 × 10−5–7.0 × 10−3 mol/dm3 for MG and 6.0 × 10−5–8.0 × 10−3 mol/dm3 for BF and correlation coefficients of these values are 0.9987 for MG and 0.9961 for BF [10].
5.0 × 10−5 and 2.0 × 10−5 mol/dm3 are the values used as the detection limit of MG and BF, respectively. Stability constants are a very useful technique whose size is huge. Due to its usefulness, it has acquired an umbrella right in the fields of chemistry, biology, and medicine. No science subject is untouched by this. Stability constants of metal complexes are widely used in the various areas like pharmaceuticals as well as biological processes, separation techniques, analytical processes, etc. In the presented chapter, we have tried to explain this in detail by focusing our attention on the applications and solutions of stability of metal complexes in solution.
Stability or formation or binding constant is the type of equilibrium constant used for the formation of metal complexes in the solution. Acutely, stability constant is applicable to measure the strength of interactions between the ligands and metal ions that are involved in complex formation in the solution [11]. A generally these 1-4 equations are expressed as the following ways:
Thus
K1, K2, K3, … Kn are the equilibrium constants and these are also called stepwise stability constants. The formation of the metal-ligand-n complex may also be expressed as equilibrium constants by the following steps:
The parameters K and β are related together, and these are expressed in the following example:
Now the numerator and denominator are multiplied together with the use of [metal-ligand] [metal-ligand2], and after the rearranging we get the following equation:
Now we expressed it as the following:
From the above relation, it is clear that the overall stability constant βn is equal to the product of the successive (i.e., stepwise) stability constants, K1, K2, K3,…Kn. This in other words means that the value of stability constants for a given complex is actually made up of a number of stepwise stability constants. The term stability is used without qualification to mean that the complex exists under a suitable condition and that it is possible to store the complex for an appreciable amount of time. The term stability is commonly used because coordination compounds are stable in one reagent but dissociate or dissolve in the presence of another regent. It is also possible that the term stability can be referred as an action of heat or light or compound. The stability of complex [13] is expressed qualitatively in terms of thermodynamic stability and kinetic stability.
In a chemical reaction, chemical equilibrium is a state in which the concentration of reactants and products does not change over time. Often this condition occurs when the speed of forward reaction becomes the same as the speed of reverse reaction. It is worth noting that the velocities of the forward and backward reaction are not zero at this stage but are equal.
If hydrogen and iodine are kept together in molecular proportions in a closed process vessel at high temperature (500°C), the following action begins:
In this activity, hydrogen iodide is formed by combining hydrogen and iodine, and the amount of hydrogen iodide increases with time. In contrast to this action, if the pure hydrogen iodide gas is heated to 500°C in the reaction, the compound is dissolved by reverse action, which causes hydrogen iodide to dissolve into hydrogen and iodine, and the ratio of these products increases over time. This is expressed in the following reaction:
For the formation of metal chelates, the thermodynamic technique provides a very significant information. Thermodynamics is a very useful technique in distinguishing between enthalpic effects and entropic effects. The bond strengths are totally effected by enthalpic effect, and this does not make any difference in the whole solution in order/disorder. Based on thermodynamics the chelate effect below can be best explained. The change of standard Gibbs free energy for equilibrium constant is response:
Where:
R = gas constant
T = absolute temperature
At 25°C,
ΔG = (− 5.708 kJ mol−1) · log β.
The enthalpy term creates free energy, i.e.,
For metal complexes, thermodynamic stability and kinetic stability are two interpretations of the stability constant in the solution. If reaction moves from reactants to products, it refers to a change in its energy as shown in the above equation. But for the reactivity, kinetic stability is responsible for this system, and this refers to ligand species [14].
Stable and unstable are thermodynamic terms, while labile and inert are kinetic terms. As a rule of thumb, those complexes which react completely within about 1 minute at 25°C are considered labile, and those complexes which take longer time than this to react are considered inert. [Ni(CN)4]2− is thermodynamically stable but kinetically inert because it rapidly exchanges ligands.
The metal complexes [Co(NH3)6]3+ and such types of other complexes are kinetically inert, but these are thermodynamically unstable. We may expect the complex to decompose in the presence of acid immediately because the complex is thermodynamically unstable. The rate is of the order of 1025 for the decomposition in acidic solution. Hence, it is thermodynamically unstable. However, nothing happens to the complex when it is kept in acidic solution for several days. While considering the stability of a complex, always the condition must be specified. Under what condition, the complex which is stable or unstable must be specified such as acidic and also basic condition, temperature, reactant, etc.
A complex may be stable with respect to a particular condition but with respect to another. In brief, a stable complex need not be inert and similarly, and an unstable complex need not be labile. It is the measure of extent of formation or transformation of complex under a given set of conditions at equilibrium [15].
Thermodynamic stability has an important role in determining the bond strength between metal ligands. Some complexes are stable, but as soon as they are introduced into aqueous solution, it is seen that these complexes have an effect on stability and fall apart. For an example, we take the [Co (SCN)4]2+ complex. The ion bond of this complex is very weak and breaks down quickly to form other compounds. But when [Fe(CN)6]3− is dissolved in water, it does not test Fe3+ by any sensitive reagent, which shows that this complex is more stable in aqueous solution. So it is indicated that thermodynamic stability deals with metal-ligand bond energy, stability constant, and other thermodynamic parameters.
This example also suggests that thermodynamic stability refers to the stability and instability of complexes. The measurement of the extent to which one type of species is converted to another species can be determined by thermodynamic stability until equilibrium is achieved. For example, tetracyanonickelate is a thermodynamically stable and kinetic labile complex. But the example of hexa-amine cobalt(III) cation is just the opposite:
Thermodynamics is used to express the difference between stability and inertia. For the stable complex, large positive free energies have been obtained from ΔG0 reaction. The ΔH0, standard enthalpy change for this reaction, is related to the equilibrium constant, βn, by the well thermodynamic equation:
For similar complexes of various ions of the same charge of a particular transition series and particular ligand, ΔS0 values would not differ substantially, and hence a change in ΔH0 value would be related to change in βn values. So the order of values of ΔH0 is also the order of the βn value.
Kinetic stability is referred to the rate of reaction between the metal ions and ligand proceeds at equilibrium or used for the formation of metal complexes. To take a decision for kinetic stability of any complexes, time is a factor which plays an important role for this. It deals between the rate of reaction and what is the mechanism of this metal complex reaction.
As we discuss above in thermodynamic stability, kinetic stability is referred for the complexes at which complex is inert or labile. The term “inert” was used by Tube for the thermally stable complex and for reactive complexes the term ‘labile’ used [16]. The naturally occurring chlorophyll is the example of polydentate ligand. This complex is extremely inert due to exchange of Mg2+ ion in the aqueous media.
The nature of central atom of metal complexes, dimension, its degree of oxidation, electronic structure of these complexes, and so many other properties of complexes are affected by the stability constant. Some of the following factors described are as follows.
In the coordination chemistry, metal complexes are formed by the interaction between metal ions and ligands. For these type of compounds, metal ions are the coordination center, and the ligand or complexing agents are oriented surrounding it. These metal ions mostly are the transition elements. For the determination of stability constant, some important characteristics of these metal complexes may be as given below.
Ligands are oriented around the central metal ions in the metal complexes. The sizes of these metal ions determine the number of ligand species that will be attached or ordinated (dative covalent) in the bond formation. If the sizes of these metal ions are increased, the stability of coordination compound defiantly decreased. Zn(II) metal ions are the central atoms in their complexes, and due to their lower size (0.74A°) as compared to Cd(II) size (0.97A°), metal ions are formed more stable.
Hence, Al3+ ion has the greatest nuclear charge, but its size is the smallest, and the ion N3− has the smallest nuclear charge, and its size is the largest [17]. Inert atoms like neon do not participate in the formation of the covalent or ionic compound, and these atoms are not included in isoelectronic series; hence, it is not easy to measure the radius of this type of atoms.
The properties of stability depend on the size of the metal ion used in the complexes and the total charge thereon. If the size of these metal ions is small and the total charge is high, then their complexes will be more stable. That is, their ratio will depend on the charge/radius. This can be demonstrated through the following reaction:
An ionic charge is the electric charge of an ion which is formed by the gain (negative charge) or loss (positive charge) of one or more electrons from an atom or group of atoms. If we talk about the stability of the coordination compounds, we find that the total charge of their central metal ions affects their stability, so when we change their charge, their stability in a range of constant can be determined by propagating of error [18]. If the charge of the central metal ion is high and the size is small, the stability of the compound is high:
In general, the most stable coordination bonds can cause smaller and highly charged rations to form more stable coordination compounds.
When an electron pair attracts a central ion toward itself, a strong stability complex is formed, and this is due to electron donation from ligand → metal ion. This donation process is increasing the bond stability of metal complexes exerted the polarizing effect on certain metal ions. Li+, Na+, Mg2+, Ca2+, Al3+, etc. are such type of metal cation which is not able to attract so strongly from a highly electronegative containing stable complexes, and these atoms are O, N, F, Au, Hg, Ag, Pd, Pt, and Pb. Such type of ligands that contains P, S, As, Br and I atom are formed stable complex because these accepts electron from M → π-bonding. Hg2+, Pb2+, Cd2+, and Bi3+ metal ions are also electronegative ions which form insoluble salts of metal sulfide which are insoluble in aqueous medium.
Volatile ligands may be lost at higher temperature. This is exemplified by the loss of water by hydrates and ammonia:
The transformation of certain coordination compounds from one to another is shown as follows:
A ligand is an ion or small molecule that binds to a metal atom (in chemistry) or to a biomolecule (in biochemistry) to form a complex, such as the iron-cyanide coordination complex Prussian blue or the iron-containing blood-protein hemoglobin. The ligands are arranged in spectrochemical series which are based on the order of their field strength. It is not possible to form the entire series by studying complexes with a single metal ion; the series has been developed by overlapping different sequences obtained from spectroscopic studies [19]. The order of common ligands according to their increasing ligand field strength is
The above spectrochemical series help us to for determination of strength of ligands. The left last ligand is as weaker ligand. These weaker ligand cannot forcible binding the 3d electron and resultant outer octahedral complexes formed. It is as-
Increasing the oxidation number the value of Δ increased.
Δ increases from top to bottom.
However, when we consider the metal ion, the following two useful trends are observed:
Δ increases with increasing oxidation number.
Δ increases down a group. For the determination of stability constant, the nature of the ligand plays an important role.
The following factors described the nature of ligands.
The size and charge are two factors that affect the production of metal complexes. The less charges and small sizes of ligands are more favorable for less stable bond formation with metal and ligand. But if this condition just opposite the product of metal and ligand will be a more stable compound. So, less nuclear charge and more size= less stable complex whereas if more nuclear charge and small in size= less stable complex. We take fluoride as an example because due to their smaller size than other halide and their highest electro negativity than the other halides formed more stable complexes. So, fluoride ion complexes are more stable than the other halides:
As compared to S2− ion, O22− ions formed more stable complexes.
It is suggested by Calvin and Wilson that the metal complexes will be more stable if the basic character or strength of ligands is higher. It means that the donating power of ligands to central metal ions is high [20].
It means that the donating power of ligands to central metal ions is high. In the case of complex formation of aliphatic diamines and aromatic diamines, the stable complex is formed by aliphatic diamines, while an unstable coordination complex is formed with aromatic diamines. So, from the above discussion, we find that the stability will be grater if the e-donation power is greater.
Thus it is clear that greater basic power of electron-donating species will form always a stable complex. NH3, CN−, and F− behaved as ligands and formed stable complexes; on the other hand, these are more basic in nature.
We know that if the concentration of coordination group is higher, these coordination compounds will exist in the water as solution. It is noted that greater coordinating tendency show the water molecules than the coordinating group which is originally present. SCN− (thiocynate) ions are present in higher concentration; with the Co2+ metal ion, it formed a blue-colored complex which is stable in state, but on dilution of water medium, a pink color is generated in place of blue, or blue color complex is destroyed by [Co(H2O)6]2+, and now if we added further SCN−, the pink color will not appear:
Now it is clear that H2O and SCN− are in competition for the formation of Co(II) metal-containing complex compound. In the case of tetra-amine cupric sulfate metal complex, ammonia acts as a donor atom or ligand. If the concentration of NH3 is lower in the reaction, copper hydroxide is formed but at higher concentration formed tetra-amine cupric sulfate as in the following reaction:
For a metal ion, chelating ligand is enhanced and affinity it and this is known as chelate effect and compared it with non-chelating and monodentate ligand or the multidentate ligand is acts as chelating agent. Ethylenediamine is a simple chelating agent (Figure 1).
Structure of ethylenediamine.
Due to the bidentate nature of ethylenediamine, it forms two bonds with metal ion or central atom. Water forms a complex with Ni(II) metal ion, but due to its monodentate nature, it is not a chelating ligand (Figures 2 and 3).
Structure of chelating configuration of ethylenediamine ligand.
Structure of chelate with three ethylenediamine ligands.
The dentate cheater of ligand provides bonding strength to the metal ion or central atom, and as the number of dentate increased, the tightness also increased. This phenomenon is known as chelating effect, whereas the formation of metal complexes with these chelating ligands is called chelation:
or
Some factors are of much importance for chelation as follows.
The sizes of the chelating ring are increased as well as the stability of metal complex decreased. According to Schwarzenbach, connecting bridges form the chelating rings. The elongated ring predominates when long bridges connect to the ligand to form a long ring. It is usually observed that an increased a chelate ring size leads to a decrease in complex stability.
He interpreted this statement. The entropy of complex will be change if the size of chelating ring is increased, i.e., second donor atom is allowed by the chelating ring. As the size of chelating ring increased, the stability should be increased with entropy effect. Four-membered ring compounds are unstable, whereas five-membered are more stable. So the chelating ring increased its size and the stability of the formed metal complexes.
The number of chelating rings also decides the stability of complexes. Non-chelating metal compounds are less stable than chelating compounds. These numbers increase the thermodynamic volume, and this is also known as an entropy term. In recent years ligands capable of occupying as many as six coordination positions on a single metal ion have been described. The studies on the formation constants of coordination compounds with these ligands have been reported. The numbers of ligand or chelating agents are affecting the stability of metal complexes so as these numbers go up and down, the stability will also vary with it.
For the Ni(II) complexes with ethylenediamine as chelating agent, its log K1 value is 7.9 and if chelating agents are trine and penten, then the log K1 values are 7.9 and 19.3, respectively. If the metal ion change Zn is used in place of Ni (II), then the values of log K1 for ethylenediamine, trine, and penten are 6.0, 12.1, and 16.2, respectively. The log βMY values of metal ions are given in Table 1.
Metal ion | log βMY (25°C, I = 0.1 M) |
---|---|
Ca2+ | 11.2 |
Cu2+ | 19.8 |
Fe3+ | 24.9 |
Metal ion vs. log βMY values.
Ni(NH3)62+ is an octahedral metal complex, and at 25 °C its log β6 value is 8.3, but Ni(ethylenediamine)32+ complex is also octahedral in geometry, with 18.4 as the value of log β6. The calculated stability value of Ni(ethylenediamine)32+ 1010 times is more stable because three rings are formed as chelating rings by ethylenediamine as compared to no such ring is formed. Ethylenediaminetetraacetate (EDTA) is a hexadentate ligand that usually formed stable metal complexes due to its chelating power.
A special effect in molecules is when the atoms occupy space. This is called steric effect. Energy is needed to bring these atoms closer to each other. These electrons run away from near atoms. There can be many ways of generating it. We know the repulsion between valence electrons as the steric effect which increases the energy of the current system [21]. Favorable or unfavorable any response is created.
For example, if the static effect is greater than that of a product in a metal complex formation process, then the static increase would favor this reaction. But if the case is opposite, the skepticism will be toward retardation.
This effect will mainly depend on the conformational states, and the minimum steric interaction theory can also be considered. The effect of secondary steric is seen on receptor binding produced by an alternative such as:
Reduced access to a critical group.
Stick barrier.
Electronic resonance substitution bond by repulsion.
Population of a conformer changes due to active shielding effect.
The macrocyclic effect is exactly like the image of the chelate effect. It means the principle of both is the same. But the macrocyclic effect suggests cyclic deformation of the ligand. Macrocyclic ligands are more tainted than chelating agents. Rather, their compounds are more stable due to their cyclically constrained constriction. It requires some entropy in the body to react with the metal ion. For example, for a tetradentate cyclic ligand, we can use heme-B which forms a metal complex using Fe+2 ions in biological systems (Figure 4).
Structure of hemoglobin is the biological complex compound which contains Fe(II) metal ion.
The n-dentate chelating agents play an important role for the formation of more stable metal complexes as compared to n-unidentate ligands. But the n-dentate macrocyclic ligand gives more stable environment in the metal complexes as compared to open-chain ligands. This change is very favorable for entropy (ΔS) and enthalpy (ΔH) change.
There are so many parameters to determination of formation constants or stability constant in solution for all types of chelating agents. These numerous parameters or techniques are refractive index, conductance, temperature, distribution coefficients, refractive index, nuclear magnetic resonance volume changes, and optical activity.
Solubility products are helpful and used for the insoluble salt that metal ions formed and complexes which are also formed by metal ions and are more soluble. The formation constant is observed in presence of donor atoms by measuring increased solubility.
To determine the solubility constant, it involves the distribution of the ligands or any complex species; metal ions are present in two immiscible solvents like water and carbon tetrachloride, benzene, etc.
In this method metal ions or ligands are present in solution and on exchanger. A solid polymers containing with positive and negative ions are ion exchange resins. These are insoluble in nature. This technique is helpful to determine the metal ions in resin phase, liquid phase, or even in radioactive metal. This method is also helpful to determine the polarizing effect of metal ions on the stability of ligands like Cu(II) and Zn(II) with amino acid complex formation.
At the equilibrium free metal and ions are present in the solution, and using the different electrometric techniques as described determines its stability constant.
This method is based upon the titration method or follows its principle. A stranded acid-base solution used as titrate and which is titrated, it may be strong base or strong acid follows as potentiometrically. The concentration of solution using 103− M does not decomposed during the reaction process, and this method is useful for protonated and nonprotonated ligands.
This is the graphic method used to determine the stability constant in producing metal complex formation by plotting a polarograph between the absences of substances and the presence of substances. During the complex formation, the presence of metal ions produced a shift in the half-wave potential in the solution.
If a complex is relatively slow to form and also decomposes at measurable rate, it is possible, in favorable situations, to determine the equilibrium constant.
This involves the study of the equilibrium constant of slow complex formation reactions. The use of tracer technique is extremely useful for determining the concentrations of dissociation products of the coordination compound.
This method is based on the study of the effect of an equilibrium concentration of some ions on the function at a definite organ of a living organism. The equilibrium concentration of the ion studied may be determined by the action of this organ in systems with complex formation.
The solution of 25 ml is adopted by preparing at the 1.0 × 10−5 M ligand or 1.0 × 10−5 M concentration and 1.0 × 10−5 M for the metal ion:
The solutions containing the metal ions were considered both at a pH sufficiently high to give almost complete complexation and at a pH value selected in order to obtain an equilibrium system of ligand and complexes.
In order to avoid modification of the spectral behavior of the ligand due to pH variations, it has been verified that the range of pH considered in all cases does not affect absorbance values. Use the collected pH values adopted for the determinations as well as selected wavelengths. The ionic strengths calculated from the composition of solutions allowed activity coefficient corrections. Absorbance values were determined at wavelengths in the range 430–700 nm, every 2 nm.
For a successive metal complex formation, use this method. If ligand is protonate and the produced complex has maximum number of donate atoms of ligands, a selective light is absorbed by this complex, while for determination of stability constant, it is just known about the composition of formed species.
Bjerrum (1941) used the method stepwise addition of the ligands to coordination sphere for the formation of complex. So, complex metal–ligand-n forms as the following steps [22]. The equilibrium constants, K1, K2, K3, … Kn are called stepwise stability constants. The formation of the complex metal-ligandn may also be expressed by the following steps and equilibrium constants.
Where:
M = central metal cation
L = monodentate ligand
N = maximum coordination number for the metal ion M for the ligand
If a complex ion is slow to reach equilibrium, it is often possible to apply the method of isotopic dilution to determine the equilibrium concentration of one or more of the species. Most often radioactive isotopes are used.
This method was extensively used by Werner and others to study metal complexes. In the case of a series of complexes of Co(III) and Pt(IV), Werner assigned the correct formulae on the basis of their molar conductance values measured in freshly prepared dilute solutions. In some cases, the conductance of the solution increased with time due to a chemical change, e.g.,
It is concluded that the information presented is very important to determine the stability constant of the ligand metal complexes. Some methods like spectrophotometric method, Bjerrum’s method, distribution method, ion exchange method, electrometric techniques, and potentiometric method have a huge contribution in quantitative analysis by easily finding the stability constants of metal complexes in aqueous solutions.
All the authors thank the Library of University of Delhi for reference books, journals, etc. which helped us a lot in reviewing the chapter.
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