Collection of benchmark test problems for data reconciliation and gross error detection and identification

Accepted Manuscript

Collection of Benchmark Test Problems for Data Reconciliation and Gross Error Detection and Identification Edson Cordeiro do Valle, Ricardo de Araujo ´ Kalid, Argimiro Resende Secchi, Asher Kiperstok PII: DOI: Reference:

S0098-1354(18)30002-4 10.1016/j.compchemeng.2018.01.002 CACE 5991

To appear in:

Computers and Chemical Engineering

Received date: Revised date: Accepted date:

27 December 2016 13 December 2017 3 January 2018

Please cite this article as: Edson Cordeiro do Valle, Ricardo de Araujo ´ Kalid, Argimiro Resende Secchi, Asher Kiperstok, Collection of Benchmark Test Problems for Data Reconciliation and Gross Error Detection and Identification, Computers and Chemical Engineering (2018), doi: 10.1016/j.compchemeng.2018.01.002

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Highlights • A collection of benchmark test problems for data reconciliation (DR) and gross error detection and identification (GEDI) are provided.

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• A general overview of the challenges related to DR and GEDI are presented. • A guideline for selection of DR and GEDI methods is presented.

• The results of selected problems are presented to illustrate challenging examples.

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• All datasets and implementations are available at the Internet.

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• The collection will help researchers to develop new DR and GEDI techniques.

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Collection of Benchmark Test Problems for Data Reconciliation and Gross Error Detection and Identification Edson Cordeiro do Vallea,∗, Ricardo de Ara´ ujo Kalidc,∗, Argimiro Resende d,∗ Secchi , Asher Kiperstokb a

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Departamento de Engenharia Qu´ımica - Universidade Federal do Rio Grande do SulUFRGS - Rua Luiz Englert S/N Porto Alegre - RS - Brazil - CEP 90040-040 b Programa de Engenharia Industrial - Escola Polit´ecnica - Universidade Federal da Bahia - Rua Aristides Novis, 2, 6 sexto. andar, Salvador - BA - Brazil - CEP 40210-630 c Universidade Federal do Sul da Bahia, Centro de Forma¸c˜ ao em Ciˆencias, Tecnologias e Inova¸c˜ ao. Campus Jorge Amado, Rua Itabuna, s/n, Rod. Ilh´eus-Vit´ oria da Conquista, km 39, BR 415 - Itabuna - BA - Brazil - CEP 45613-204 d COPPE - Universidade Federal do Rio de Janeiro - Centro de Tecnologia, Bloco G, sala 115 Cidade Universit´ aria, Rio de Janeiro - RJ - Brazil - CEP 21941-972

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Abstract

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In an industrial scenario, one can find measured data that do not satisfy the mass and energy laws of conservation. This problem can be approached by applying data reconciliation (DR) and gross error detection and identification (GEDI) techniques, however, authors generally validate their methods using a reduced set of problems, restricting the application of the proposed methods to them. The objective of this work is to present a collection of benchmark problems for DR and GEDI to help the evaluation of these methods in different types of flowsheets. First, challenges issues related with DR and GED are presented with examples. Then, a general overview of the benchmark collection set is presented. In conclusion, it can be observed that this challenging research area needs a common problem set for validating DR and GEDI and this paper fills this gap, helping the validation of the methods. ∗

Corresponding author Email addresses: [email protected] (Edson Cordeiro do Valle), [email protected] (Ricardo de Ara´ ujo Kalid), [email protected] (Argimiro Resende Secchi)

Preprint submitted to Computers and Chemical Engineering

January 4, 2018

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Keywords: data reconciliation, gross error detection and identification, fault detection, benchmark problems

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1. Introduction

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In the chemical industry, measurement equipment plays a special role in process monitoring and control. Accurate process data are important for the application of advanced chemical process tool techniques, such as the estimation of unmeasured process variables, sensor fault identification, equipment leaking identification, process model identification, process optimization, steady-state identification, software sensors, advanced process control, and synthesis of mass and heat exchanger networks. However, some measurement equipment is subjected to climate oscillation and aggressive process conditions, which may degenerate their performance. In addition, it is not always possible to replace measurement equipment due to the type of the process to which it is attached: some chemical processes are large-scale and continuous and cannot be paused or shut down to replace a measurement sensor or instrument.

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Costs reduction also prevents the appropriate maintenance of measurement sensors. Consequently, process engineers must occasionally accept this performance degeneration or failure along the chemical process operation, which can be noticed when the mass and energy balances present discrepancies when it is calculated based on the measured sensor data. However, not all data oscillation is due to failures but may result from measurement equipment precision or leaking, which can also lead to mass and energy balance disagreement when it is calculated based on sensor measurements. To distinguish among sensor failure, leaking or data oscillation due to sensor variability, a procedure named data reconciliation (DR) and gross error detection and identification (GEDI) can be applied to process data and has been extensively studied since the 1960s in chemical and mineral processing. The main idea of DR is to adjust the measurements data to satisfy the mass, energy or momentum balance equations. Information about instrument uncertainty, as represented by the standard deviation (SD) or variance, is also considered in the DR problem solving. As a final result, the measurements are adjusted, thus satisfying the process balances imposed in the 3

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problem formulation.

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Steady-state DR is solved as an optimization problem, where the objective function is to minimize the difference between the measured variables and the adjusted ones, weighted by the reciprocal of the variance. The equality constraints of the optimization problem can be the either mass, energy or momentum balance equations, or a combination of them. In the DR problem formulation presented in Equation 1, it is assumed that all variables are measured or recorded from a steady-state process and that the measurement errors follow a Normal distribution with known variance. The objective function presented in Equation 1 is known as the weighted least squares function (WLS).

(1)

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 min (Y − X)T Σ−1 (Y − X)   n  X∈<   subject to   F (X) = 0    Xmin ≤ X ≤ Xmax

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where: Σ: Variance covariance matrix of measurements Y: Vector of measurements X: Vector of reconciled measurements Xmin : Vector of reconciled measurement lower bounds Xmax : Vector of reconciled measurement upper bounds F (X): Vector of process constraints related with mass, energy or momentum balance

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The WLS is deduced from the maximum likelihood method and assumes that the measurement errors have a normal distribution with known variance. If the measurement errors follow another distribution or the system presents measurement instrument failure or leaking, known as gross errors (GE), other objective functions can be applied, called robust functions (Arora ¨ and Biegler, 2001; Ozyurt and Pike, 2004) or robust estimators, presented in Appendix A. Robust functions may be used to either remove the effect of the GE in 4

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the DR or detect it if its parameters are appropriately tuned. The influence ¨ function (IF, see Appendix A) was used by Ozyurt and Pike (2004) and Alhaj-Dibo et al. (2008) to tune the robust estimators parameters for GEDI, whereas Arora and Biegler (2001) and Zhang et al. (2010) used the Akaike information criterion (Akaike, 1974). Analytically, the DR procedure with the WLS estimator without other method can not evaluate the presence of GEs; to perform this task, statistical hypothesis tests must be applied to the results of DR. These tests are formulated according to the hypotheses: H0 , no GE is present; and H1 , a GE is present. It is necessary to choose a significance level to accept or reject hypothesis H0 . The significance level is related to the probability of detecting a GE when it is not present (also known in statistics as Type I error) because the observed measurement can be caused by normal instrumentation variability. Some simple GEDI calculations, such as global test (GT), measurement test (MT) and nodal test (NT), are briefly presented in Appendix B. Several statistical tests with their respective techniques can be applied to perform GEDI and are presented in Table 1. Some methods to GEDI use an algorithm procedure combined with statistical tests in such task. Table 1: GEDI technique with the respective statistical tests.

Statistical Test Chi-squared

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GEDI Test Name Global (GT)

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Nodal (NT) or Equipment Measurement (MT) GLR

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Principal Components of Constraints Principal Components of Measurements

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Gross error type detected Detects a gross error in the mass balance as a whole, do not detect the location Detects a gross error in the equipment balance (leaking) Detects a gross error in the measurement Detects equipment leaking (GLRNT) or measurement bias (GLRMT) and also its magnitude Detects an equipment leaking Detects a gross error in the measurement

Although DR and GEDI have been studied since the 1960s, they still 5

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remains challenging and the reasons for proposing a collection of benchmark test problems are presented in the next section.

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2. Why is a collection of benchmark test problems for DR and GEDI needed ? A literature review

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2.1. DR Challenges 2.1.1. Effects of GE on Steady-State DR DR techniques applied to linear (total flow rate) steady-state flowsheets with all variables measured are described in the literature (Crowe et al., 1983; S´anchez and Romagnoli, 1996; Swartz, 1989; Mah et al., 1976; Veverka and Madron, 1997; Narasimhan and Jordache, 2000; Mah, 1990; Romagnoli and S´anchez, 1999) and have been solved by several analytical and numerical methods; thus, they do not pose any DR-related challenge. The challenges associated with DR problems begin to arise when gross errors (GE) are presented, since one of the requirements to apply DR techniques is that the variables’ errors follow a standard Normal distribution, i.e., only purely random error with known standard deviation is present. However, systems with measurement instrument failure or a leaking, GE, do not fulfill this requirement. Because the variables are linked by the mass balance, a GE spreads the effect of the wrong measurement to other variables, known as the ’smearing effect’. To handle this issue, other objective functions can be applied when the measurements do not exhibit a Normal distribution or in the presence of GE; these functions are called robust functions or estimators ¨ (Arora and Biegler, 2001; Ozyurt and Pike, 2004; Alhaj-Dibo et al., 2008; Llanos et al., 2015; Jin et al., 2012; Zhang et al., 2010). An appropriate robust function must have some characteristics to provide good results in DR, such as the following:

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• When the measurement errors are small, the effect of the measurement errors on the objective function is almost the same as WLS. • When the measurement errors are large, the effect of the measurement errors is reduced.

A demonstration of the ’semaring effect’ and its reduction through the appropriate tunning of the robust functions can be noticed in the following example, presented in Figure 1. The idea of this test is to promote a bias 6

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of 9.σ in the flow rate of stream 1 and evaluate its spread using a variety of estimators, such as WLS, Logistic and Quasi-Weighted. Table 2 presents the flow rates of streams 1 to 12 (rows), when a bias of 9.σ is added. In the column 2 of Table 2, the DR results (flow rates of streams 1 to 12) are presented when only random error are added to exact flow rates. Columns 3 to 7 present the differences observed when DR without bias and the selected estimator with bias are implemented. When a bias of 9.σ is added to stream 1, the resulting DR with WLS estimator shows spreading of the GE to other streams with the same equipment (streams 1, 2, 3 and 4), as can be observed in Table 2, column 3. Logistic and Quasi-Weighted estimators without any parameter tuning (CLo , and β, respectively) reduce this effect in the same streams, as presented in Table 2, column 4 and column 6, respectively. After tuning the Logistic and Quasi-Weighted estimators via trial and error, the errors are reduced in the first 4 streams but spread to other streams in the flowsheet, as presented in Table 2 column 5 and 7, respectively. A robust function may avoid the spread effect of the GE in the DR if its parameters are appropriately tuned. The appropriate selection and tuning of such robust functions represent interesting challenge in DR.

Figure 1: Flowsheet to demonstrate the ’semaring effect’.

2.1.2. Dynamic Systems and Change in Operating Points Another scenario that can be found in some industrial operations is the transition between operating points that, depending on the dynamic behavior 7

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Table 2: Comparison of streams flow rates between WLS without GE and other estimators for an error of 9.σ.

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flow rate W LS− QWGE β = 0.1

flow rate W LS− QWGE β= 2

7.6 10.2 7.3 7.6 2.5 -0.3 2.9 2.8 3.1 0.2 0.1 0.3

0.8 3.6 0.4 0.7 2.8 -0.3 3.3 3.2 3.5 0.2 0.1 0.3

34.4 35.2 34.3 34.3 0.8 -0.1 0.9 0.9 1.0 0.1 0.0 0.1

4.0 6.7 3.6 3.9 2.7 -0.3 3.1 3.0 3.3 0.2 0.1 0.3

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45.6 45.6 45.6 45.6 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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692.1 725.1 700.6 687.7 33.0 12.9 24.5 20.1 29.1 4.6 4.4 9.0

flow rate W LS− LogisticGE CLo = 0.1

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1 2 3 4 5 6 7 8 9 10 11 12

flow rate W LS− LogisticGE CLo = 0.602

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flow rate flow rate Stream WLS W LS− without W LSGE bias

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of the process, may occur gradually. The instrument may also present some permanent or instant failure, which will affect the process data recorded. These behaviors are presented in Figure 2, where it is possible to observe noisy data, smooth transitions between operating points because of dynamic system aspects (between 50 and 150 s) and a data bias (between 130 and 140 s). However, another requirement to apply DR using optimization methods presented by Equation 1, is that the process is at steady-state, which is not always true, leading to erroneous DR results. This issue can be also considered challenging in DR and GEDI. Approaches to reduce the dynamic effect in DR are to either detect the steady-state (Kelly and Hedengren, 2013; Jiang et al., 2003) and apply DR within this window or to use dynamic DR, known as DRPE (DR and Parameter Estimation), which can reconcile data and estimate state variables and parameters (Albuquerque and Biegler, 1996; Prata et al., 2009; Albuquerque and Biegler, 1995; Narasimhan and Jordache, 2000; Romagnoli and S´anchez, 1999).

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2.1.3. Data Filtering and DR Because typical industrial data contains random noise, it may be useful to apply some filtering techniques to reduce it when solving DR with process data. Narasimhan and Jordache (2000) presented some filtering techniques for data pretreatment. Soderstrom et al. (2001) reported DR and GEDI based on mixed integer optimization and the effect of the horizon window selection on the results. Arora and Biegler (2001) used a robust approach to DR, evaluating the effect of the horizon windows on one example. The selection of filters and their tuning (window selection and filter parameters) also constitute an interesting issue in DR: a well-tuned filter must filter the data appropriately without removing the GE (which may need to be identified) or the dynamic effect of the data. To present the challenges related to this topic a 2 tank with by-pass flowsheet, presented by Figure 3, is analysed. In the first example, a steady-state system was configured using a small time constant for the tanks. Stream 6, which is the target of this study, has only pure random noise, with σ = 0.5.kg.s−1 . Then, a GE of 7.σ was configured to be present at time 200s in stream 6, followed by a simulation, filtering and a DR. The plot of the raw data (random noise), filtered data with a window size or 40 points and a DR with filtered data is presented in Figure 4. The effect of the GE decreases in both the filtered and reconciled data and results in a smoothed GE. This effect may not be desirable because it is occasionally necessary to identify the source of GE. If the GE persists for a long time, 9

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Figure 2: Typical industrial data recorded by measurement equipment during a shift between operation points with measurement bias between 130 and 140 s.

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the filtering technique will provide a minor smoothing effect, and it could be possible to identify the source, but this smoothing effect is an interesting issue to study. The influence of the window size for the moving average filter is another subject to be studied, and this effect is presented in Figure 5. The effect of the increase in the time window causes a deformation in the curve, changing the behavior from a step response (window size equal to 5 and 10) to a ramp response (window size of 20 and 40 points). This deformation in the curve after the DR procedure can propagate to other variables because they are all linked by the balance equations and can lead to erroneous DR results or trigger a false GE in this variable. 2.1.4. Simultaneous Parameter Estimation and DR Another problem related with DR arise when the models to be reconciled are more complex and depend on measurements and unknown model 10

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Figure 3: Two-tank flowsheet example.

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parameters, which must be estimated simultaneously with DR. This class of problems is also known as error-in-variables (EVM) problems. The objective function in this case is different from that of the original DR problem and is presented in Equation 2.   min (X − Y)T Σ−1 (X − Y)   X,θ∈
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where: Σ: Variance covariance matrix of Y measurements θ: Vector of parameters to be estimated Y: Vector of measurements X: Vector of reconciled measurements Xmin : Vector of reconciled measurement lower bounds Xmax : Vector of reconciled measurement upper bounds F (X, θ): Vector of process constraints related with mass, energy or momentum balances

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Figure 4: Steady-state system with GE of magnitude of 7.σ at time 200 s in stream 6 with filtered (window size of 40) and unfiltered data.

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Figure 5: Steady-state system with GE of magnitude of 7.σ at time 200 s in stream 6 and the influence of the window size on data filtering before DR.

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In a typical DR problem, one dataset, snapshot, or windows average of the measured data is collected to be reconciled. Thus, the number of constraints equals the number of original models. In EVM, several datasets are available (one for each experiment), and the objective is to perform a reconciliation and a parameter estimation simultaneously. EVM problems increase the number of constraints because, for each data point, the number of constraints is multiplied by the number of datasets. For example, in a problem with 20 constraints and 20 data points, the number of constraints is 400. Consequently, EVM problems are quite difficult to solve. Furthermore, these constraints are, generally, implicit and cannot be isolated and placed in the objective function. Several authors have proposed techniques for solving this type of problem (Faber et al., 2003; Esposito and Floudas, 1998; Rod and Hancil, 1980; Gau and Stadtherr, 2000; Tjoa and Biegler, 1992; Biegler and Tjoa, 1993; Bazin et al., 2005; Macdonald and Howat, 1988). The challenges related to EVM problems can be exemplified by the results of the EVM method applied to the heat exchanger network proposed by Biegler and Tjoa (1993). This example is not trivial to solve because several data points are available. This may lead to some difficulties in solving using the traditional approach, which includes all constraints and data sets in the optimization problem. As presented in Table 3, the problem size and CPU time increase with the number data points, whereas the U A (product of the global heat transfer coefficient and the heat exchanger area) estimation does not change significantly from one run to another. The CPU time was recorded using an Intel Quad Core Q8400 4GB RAM machine (without parallelization). The scripts were executed in Scilab 5.4 and Ipopt 3.10 (W¨achter and Biegler, 2006) using an Ubuntu Linux 11.04. The EVM problem must not be confused with the DRPE, since EVM addresses a parameter-estimation problem where the measurement errors occur in both dependent and independent variables (Esposito and Floudas, 1998; Gau and Stadtherr, 2000) and the constraints are nonlinear equations while DRPE addresses dynamic models and can estimate both parameters and state variables with different methods (Albuquerque and Biegler, 1996; Prata et al., 2009; Albuquerque and Biegler, 1995; Narasimhan and Jordache, 2000; Romagnoli and S´anchez, 1999). 2.1.5. DR with Unmeasured Variables and Nonlinear Systems DR problems when unmeasured variables are present in a system are also challenging, especially when nonlinear constraints are present. Some analyti14

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Table 3: Comparison between runs with different data points using the problem proposed by Biegler and Tjoa (1993).

20 Data Points 4307.9 424 280 1300

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4.858 4.007 6.780 5.356

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12 Data Points 789.61 256 168 780

4.857 4.007 6.783 5.353

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CPU Time (sec) Number of Variables Number of Constraints Number of Nonzeros in Constraints Jacobian Number of Nonzeros in Lagrangian Hessian UA1 UA2 UA3 UA4

8 Data Points 372.83 172 112 520

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cal and numerical methods do not impose bounds on the reconciled variables (e.g., nonnegativity of flow rates or compositions), leading to unreal adjusted data. This type of problem complexity can be demonstrated by the flowsheet presented by Rao and Narasimhan (1996) where the authors applied DR to this flowsheet with a bilinear balance containing measured and unmeasured variables. The results compiled from Rao and Narasimhan (1996) were compared using an implementation of the interior point optimization method presented (Ipopt, W¨achter and Biegler, 2006) and are presented in Table 4. It can be seen that the original authors obtained some negative estimates for some variables when the authors applied DR in multicomponent processes and showed that some methods (Crowe et al., 1983; Simpson et al., 1988; Pai and Fisher, 1988) resulted in negative estimated compositions of some streams. In fact, these estimates have no meaning because they correspond to unobservable variables. Several authors have described DR for nonlinear system of equations (Bagajewicz and Cabrera, 2003; Rao and Narasimhan, 1996; Crowe, 1986; Kelly, 2004; Schraa and Crowe, 1998), which must be solved using numerical techniques and, depending on the nonlinearity involved, may result in more complex problems. This example illustrates the role of variable classification, especially in unmeasured streams and nonlinear systems, in DR problems that continue to be somewhat challenging in this 15

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field. Table 4: Comparison of the results from Rao and Narasimhan (1996) and those obtained by optimization with Ipopt.

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R: Reduntant; O: Observable; U: Unobservable

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Stream 1 flow Comp.(%) 1 7 Stream 2 flow Comp.(%) 1 7 Stream 3 flow Comp.(%) 1 3 7 Stream 4 flow Comp.(%) 7 Stream 8 flow Comp.(%) 5 Comp.(%) 7 Stream 9 flow Comp.(%) 2 Comp.(%) 7 Stream 10 flow Comp.(%) 3

Measurement Crowe Simpson Nonlinear Ipopt Variable or Estimate Classification1 3707 3694 3564 3564 3667 R 1.64 1.50 1.56 1.56 1.54 R 0.41 -0.30 -0.43 -0.42 0.36 O 1900 1887 1900 1829 1881 U 1.64 16.69 1.64 4.02 1.53 U 0.03 -0.61 15.42 2.03 0.35 U 1807 1807 1664 1736 1786 U 1.64 -14.36 1.48 -1.02 1.55 U 0.38 0.38 0.38 -2.21 0.39 U 0.03 0.03 -18.54 -2.99 0.37 U 2910 2929 2806 2807 2910 R 0.29 -0.49 -1.08 -1.06 0.20 O 105 105 111 112 104 R 0 8.18 0.00 -2.79 0.35 U 0 -19.08 -5.53 -2.74 0.33 U 2832 2824 2695 2696 2806 O 1.6.e−4 1.2.e−4 -0.002 -0.002 0 O 0.20 0.20 -0.90 -0.99 0.19 U 57.0 57.0 58.8 58.8 58.9 R 0 -2.97 0 -1.37 0.008 U

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2.1.6. Simplified Models for Nonlinear DR Another interesting topic to be explored in nonlinear DR is the selection of a rigorous or simplified model to represent each process. Several types of industrial equipment, such as distillation columns, flash separators, and pipelines, are represented by complex systems of nonlinear equations that are difficult to converge and may be a compute-intensive task. To reduce the solving complexity, and computation time these models are simplified either by linearization or polynomial adjustment (using simulation or plant data). 16

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• The effect of unmeasured variables in DR. • GE in the process.

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• DR involving nonlinear systems of equations.

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However, such simplifications may result in error propagation along the DR process, which represents another challenging issue requiring study that has not been extensively explored in the literature. For example, Bagajewicz and Cabrera (2003) proposed an iterative method for DR with simplified models and rigorous models obtained from a process simulator which resulted in a reduction on the execution time for the DR problem. In conclusion, the challenges related to DR techniques can be summarized as follows:

• Robust function selection and tuning of its parameters. • DR in process transitions and detecting the steady-state. • Data filtering before DR.

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• Simultaneous DR and parameter-estimation improvement.

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• Nonlinear DR with rigorous or simplified models.

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2.2. GEDI Challenges 2.2.1. Variables that Influence the GEDI The accurate identification of GEs is a challenging task because many factors affect the detection methods and algorithms and may lead to incorrect results. Previous studies performed by Rosenberg et al. (1987), Iordache et al. (1985), and S´anchez et al. (1999) indicated that the GEDI and MGEDI (Multiple GEDI) algorithms are influenced by many factors, including:

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• Magnitude of measurements and bounds. • Magnitude of GEs or the ratio between the GE and the respective measurement variance. • Position of the measurement containing GE. • Constraints and flowsheet structure. 17

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When considering the network structure, the accurate detection and identification of single or multiple GEs is a complex procedure because the redundancy needed to perform DR and GEDI (or MGEDI) may be degenerated in some flowsheets, which depreciates the results of the techniques. These redundant measures are named practically nonredundant measurements (Iordache et al., 1985) and, considering the total flow rates mass reconciliation, can be present in the following situations (Narasimhan and Jordache, 2000; S´anchez et al., 1999):

• Parallel streams. • Flowsheets with recycle.

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• Flow rates with small absolute standard deviations (or small orders of magnitude) in the same balance with streams with higher standard deviations (or higher orders of magnitude).

• Measured variables that appear in only one equation with one unmeasured variable.

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To demonstrate the challenge involved when parallel streams are present in a system, a process from Yang et al. (1995), presented in Figure 6, is analyzed. The stream flow rates and standard deviation units are given in kg.s−1 . The GEDI results for all measurement tests (MT and GLRMT) are presented in Table 5. The Overall Power (OP) measures the capacity of the test to correctly identify the position of the GE and it is applied using simulation studies where the position of the GE and its magnitude is previously known. In this example can be noticed that parallel streams results in the non-identifiability of streams 4 and 5 by MT and GLRMT. Parallel streams exhibit the same behavior as split streams (because split streams are also parallel) and are a serious limitation of the MT method, (mentioned by Iordache et al., 1985; Tamhane and Mah, 1985).This behavior also occurs in this type of mass balance with the GLR method because proportional columns result in an equal ”gross error signature vector”, which makes identification impossible (Narasimhan and Mah, 1987). This is the main reason why large flowsheets are not divided into small problems when applying these tests: the division would result in small flowsheets with proportional columns in the incidence matrix (Tamhane and Mah, 1985). Consequently, simultaneous or iterative GEDI strategies based on MT are limited for this type of flowsheet structure. 18

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Figure 6: Generic mass balance flowsheet with parallel streams.

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Table 5: Results of MT and GLR (testing measurement bias) for the measurement bias of problem presented by Yang et al. (1995)

Real Standard OP MT Value Deviation 98.7 0.994 0.774 41.1 0.641 0.596 78.9 0.888 0.784 30.2 0.550 0 109.1 1.045 0 19.8 0.445 0.374 57.6 0.759 0.71 37.8 0.615 0.626

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Stream Name 1 2 3 4 5 6 7 8

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OP GLR MT 0.774 0.596 0.784 0 0 0.374 0.711 0.627

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2.2.2. Multiple GEDI In real process operation and monitoring, the detection of a single GE itself does not provide enough information for process analysis. We want to identify multiple GEs and their types (measurement bias or leaking) and to estimate the error magnitude. Many strategies to detect and identify multiple GEs (MGE) are presented in the literature (Serth and Heenan, 1986; Iordache et al., 1985; Rollins and Davis, 1992; S´anchez et al., 1999; Rosenberg et al., 1987; Narasimhan and Mah, 1987; Harikumar and Narasimhan, 1993); some of them perform the identification of the MGE in one step, named simultaneous strategies (Narasimhan and Jordache, 2000). Other MGEDI techniques perform iterative GEDI testing by removing suspicious measurements or mass balances and repeating the test procedure until the appropriate set of GEs is identified. Gross error detection in nonlinear systems is also a challenging area that has not been studied as intensively as linear flowsheets. Mukherjee and Narasimhan (1996) applied leaking detection using the GLR method in networks of pipelines, whereas robust estimators in ¨ GEDI were used by Ozyurt and Pike (2004), Tjoa and Biegler (1991) and Arora and Biegler (2001).

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2.2.3. Tuning the Parameters of Optimization Methods The solving of DR problems, specially nonlinear, may require iterative constrained nonlinear optimization methods which generally requires some parameter tunning, such as absolute or relative convergence tolerance, constraint violation tolerance. The appropriate tunning of these parameters may be crucial for the problem solving or to reduce the computational effort needed for the DR problem-solving. The tuning of the parameters of the optimization method represents a special issue and were addressed by some authors (Chen et al., 2011; Audet and Orban, 2006; Adenso-Diiaz and Laguna, 2006; Hutter et al., 2009). Although the DR and GEDI research topic is not new, there are many challenges in this field (Crowe, 1996; Narasimhan and Jordache, 2000), specifically in flowsheets with measured and unmeasured variables, multiple GEs, nonlinear processes or a combination of them. The complexity of GEDI regarding the previously mentioned cases shows that the techniques proposed should be tested in a large number of test examples to make them more general, and to provide a suitable robustness, the parameters of the methods must be appropriately tunned.

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2.3. Validation of DR and GEDI techniques in a reduced problem set When a new or a modified DR, GEDI or MGEDI technique is proposed, it is compared with the existing techniques to validate the method itself. However, when authors propose new techniques for DR and GEDI (or MGEDI), only a few cases are used to validate the new methods, whereas the ideal case should involve testing in several cases (flowsheets) to verify and guarantee the generality of the proposed method. From a review of the DR and GEDI methods proposed in the literature, in many cases, only a few examples are studied, as presented in Table 6.

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2.4. Lack of an extensive benchmark collection for DR and GEDI technique validation Test problems are well accepted by the scientific community, and many research areas have consolidated benchmark sets. Such as optimization, (Hock and Schittkowski, 1981; Schittkowski, 1987; Bondarenko et al., 2004; Floudas and Pardalos, 1999), chemical process control (Downs and Vogel, 1993; Castro, 2004; Chen et al., 2003) and transport phenomena (Leong et al., 1998; Rajeshkanna, 2006). However, the field of DR and GEDI lacks a test collection for algorithm performance evaluation. In this context, the aim of this work is to collect and present challenging examples to help researchers test their DR and GEDI techniques.

3. Selection, Development and Implementation of Benchmark Problems

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In the present work, several examples were selected from a literature review or proposed. The main challenge related to each type of problem is presented. The structure of the problems set are presented in Figure 7. The problems were implemented in Scilab 5.4 and Ipopt 3.10 (W¨achter and Biegler, 2006) and are available in a GIT repository (Valle, 2011). Special care were taken in the implementation to improve modularity and to make the extension of the test sets as easy as possible. The data and flowsheets of these problems are described in the Supplement material. The test problems are divided in two groups: one group for DR and one group for GEDI. The problems were categorized into five types: • Steady-state linear DR and GEDI examples for robust functions or unmeasured variables testing (16 examples). 21

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Table 6: DR and GEDI techniques and the number of examples studied to validate them.

Brief Description

Romagnoli and Stephanopoulos (1981) Rosenberg et al. (1987) Narasimhan and Mah (1987) Reddy and Mavrovouniotis (1998) S´ anchez et al. (1999) Bagajewicz et al. (1999) Kongsjahju et al. (2000)

Algorithmic Method Proposed DMT and EMT Generalized Likelihood Ratio Neural Network Approach to GEDI MSEGE, MUBET and MGLR Evaluation of PCA for GEDI Modification of UBET for serially correlated systems MI approach to DR and GEDI Redescending Estimators for DR and GEDI Robust DR Improvement of Measurement and nodal test Likelihood and Bayesian Methods for DR and GEDI NT-MT for GEDI Robust DR using contaminated distributions DR and GEDI based on Regression Models Quasi-weighted least squares estimator Robust IMT Robust DR Robust DR

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Author

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¨ Ozyurt and Pike (2004) Wang et al. (2004)

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Soderstrom et al. (2001) Arora and Biegler (2001)

Devanathan et al. (2005)

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Mei et al. (2006) Alhaj-Dibo et al. (2008)

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Maronna and Arcas (2009) Zhang et al. (2010)

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Farias (2009) Llanos et al. (2015) Jin et al. (2012)

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Number of Examples Tested 3 7 2 1 2 1 1 4 3 7 1 1 2 1 2 4 2 2 2

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Figure 7: DR and GED problem set structure.

• Steady-state parameter estimation with DR (4 examples using the error-in-variables approach).

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• Data treatment and dynamic for DR and GEDI (one example with 5 test cases for DR and 2 examples with 10 test cases each for GED).

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• DR and GEDI with time-varying systems with steady-state and dynamic cases (2 example with 10 cases each).

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• Steady-state nonlinear examples with redundant, observable and unobservable variables for GEDI and DR (5 examples for DR and 8 examples for GEDI).

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3.1. Steady-state linear DR and GEDI In this class of problems, the 16 linear steady-state challenge test problems were selected via a literature review to cover a wide range of process flowsheets. In these problems, only total flow rates were selected, and all streams were measured and redundant. The measured dataset and standard deviations ( SD ), or variances, were either obtained from the original paper or, in some cases, when qthese data were not available, proposed in this work. The problems were selected to test the most common phenomena that negatively influence the performance of DR and GEDI algorithms, such as: 1. Magnitude of measurements and bounds. 23

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Effect of unmeasured data on DR and GEDI. Magnitude of GE or the ratio between the GE and the respective SD. Position of the measurement containing GE. Constraints and flowsheet structure. 5.1. Flow rates with small absolute standard deviations (or small order of magnitude) with other streams in the same balance. 5.2. Parallel streams. 5.3. Flowsheets with recycle.

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2. 3. 4. 5.

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In the presence of GEs or when the data do not follow a Normal Distribution, other objective functions must be selected, which are known as robust estimators. Several objective functions, mostly robust estimators, were implemented in Scilab (using Ipopt solver) to solve the problems: WLS, sum of absolute errors, Cauchy, Contaminated Normal, Fair Function,Hampel, Logistic, Lorenztian and Quasi-Weighted. Robust estimators have parameters that must be appropriately tuned to obtain a good ”cut-off” point between purely random errors and GE; this is one of the challenging issues associated with this category of problems. The parameters of the robust ¨ estimators were selected according to the literature (Ozyurt and Pike, 2004; Zhang et al., 2010), and they were not tuned individually. Details of the functions can be found in Appendix A. In this problem set, the user has the freedom to perform some customization: • Changing the bounds on measurements for DR and GEDI problems.

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• Changing the status of a measured stream to unmeasured and observing the impact on the data classification, DR and GEDI.

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• Changing the value of a measurement or adding a single or multiple GE.

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• Selecting the magnitude of GE or the ratio between the GE and the respective SD. • Selecting another objective function (robust estimator), proposing a new one (using an existing function as a template) or tuning its parameters. • Tuning the parameters of the existing optimization method. 24

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In the Scilab implementations, it is possible to test a very large combination of measured and unmeasured streams and also to add multiple measurement bias, multiple leakings, or a combination of them. Since, for large flowsheets, there is a large number of possible combinations of measured, unmeasured streams and GE, the authors did not fix its position, leaving this decision for future researchers.

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3.2. Steady-state Parameter Estimation with DR As previously mentioned, when DR must be performed together with parameter estimation, the number of equality constraints increases with the number of data points, and even for small problems, this type of optimization may be difficult to solve. The goal of this problem set is to present this type of challenge problems. The selected problems were chosen because they either exhibit substantial nonlinearity or several data points, making the problem relatively complex. Furthermore, the selected problems were the most commonly applied by the previously mentioned authors and represent a good basis for comparing methods. All the models are described in the Supplement material and listed below:

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• Kinetic constant estimation of a heterogeneous reaction.

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• Kinetic constant estimation of a CSTR reactor. • Estimation of thermodynamic constants from VLE data.

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• Estimation of heat exchanger parameters of a heat exchanger network. It is also possible to customize the 4 problems as follows:

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• Selecting, removing or adding new data points (adding GEs, for example). • Adding new inequality constraints to improve convergence rates.

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• Changing initial guesses and evaluating the resulting influence on the problem convergence and presence of local minima. • Tuning the parameters of the existing optimization method.

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1. 2. 3. 4. 5.

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3.3. Data Treatment and Dynamics The main challenge in the area of data filtering and DR is the proper selection and tuning of filters to reduce the noise before applying DR without removing the effects of GEs, which must be properly identified. Another problem associated with DR of dynamic systems is that the effect of the dynamics may spread along other variables that are not influenced by the dynamics. The objective of this benchmark category is to test the influence of data-filtering techniques on DR when measured and unmeasured variables and GE are present in steady-state and dynamic flowsheets. The proposed flowsheet consists of 2 parallel tanks with a by-pass valve and is the same as previously presented in Figure 3. For this flowsheet, 5 test cases were introduced: Steady-state flowsheet with all streams measured and no GEs. Steady-state flowsheet with some streams unmeasured and no GEs. Steady-state flowsheet with all streams measured and GEs. Dynamic flowsheet with all streams measured and no GEs. Dynamic flowsheet with all streams measured and GEs.

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These problems can be modified in the following ways:

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• Changing the operating point time and set-point amplitude. • Modifying the SDs of the measurements.

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• Adding single or multiple GEs and selecting the time and amplitude. • Removing measurements to evaluate the impact on the DR results.

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• In the dynamic case, changing the dynamic behavior (transfer function) of the system and adding or removing GEs (position, time and amplitude).

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• Tuning or implementing new filtering strategies for the data.

3.4. Time-varying GEDI Although several GEDI techniques proposed in the literature perform tests in datasets where only one operation point is considered, in industrial scenarios, the process frequently changes its operating conditions; consequently, the data obtained by sensors are time-varying. To test GEDI in this 26

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class of problems, two flowsheets were proposed: a two tanks with by-pass system and a simplified reactor-separator flowsheet, both of which are linear (total flow rate) with all streams measured. In each of the flowsheets both steady-state and dynamic behavior are present, with 10 cases for each problem. The objective of this benchmark category is to test specific failures in sensors and leaking individually and combined, such as: • Sensor bias with a fixed magnitude along the simulation.

• Drifting bias (with a time-varying magnitude) along the simulation.

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• Equipment leaking. • Sensor miscalibration.

• Multiple errors with a fixed bias and equipment leaking. • The influence of the dynamic behavior on GEDI for the aforementioned cases.

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• Data pretreatment before DR and the influence on GEDI.

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The first example for a time-varying system consists of 2 parallel tanks with a by-pass valve and 8 streams. The flowsheet is the same as previously presented in Figure 3. For this flowsheet, 5 test cases were introduced for both steady-state and dynamic systems:

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1. Sensor bias with a fixed magnitude along the simulation (8 cases, one for each stream). 2. Drifting bias (with a time-varying magnitude) along the simulation (8 cases, one for each stream). 3. Equipment leaking (4 cases). 4. Sensor miscalibration (8 cases, one for each stream). 5. Multiple GE with a fixed bias and equipment leaking (4 cases). The second problem consists of a simplified reaction-separation system (where only total flow rates are considered) with 3 pieces of equipment and 8 streams, as presented in Figure 8. For this flowsheet, 5 test cases were introduced for both steady-state and dynamic systems: 27

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Figure 8: Reactor-separator flowsheet example.

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1. Sensor bias with a fixed magnitude along the simulation (6 cases, one for each stream). 2. Drifting bias (with a time-varying magnitude) along the simulation (6 cases, one for each stream). 3. Equipment leaking (3 cases). 4. Sensor miscalibration (6 cases, one for each stream). 5. Multiple GEs with a fixed bias and equipment leaking (4 cases).

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Other details of the models are presented in the Supplement material. Although the dataset already provides challenging cases for GEDI techniques evaluation, these problems can be modified to test a wide range of scenarios in the following way:

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• Change the operating point time and set-point amplitude. • Modify the standard deviations of the measurements.

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• Change the start time, end time and magnitude of a single or multiple GEs.

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• Modify sensor behavior, such as drifting rates and drifting type (positive, negative or oscillatory).

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• Modify calibration parameters. • Remove measurements to evaluate the impact on the DR and GEDI results. • In the dynamic case, change the dynamic behavior (transfer function) of the system. • Tune or implement new filter strategies. 28

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3.5. Steady-state nonlinear problems In this class of problems, 5 steady-state nonlinear test problems were selected from literature or proposed, they are presented in Table 7.

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Table 7: Steady-state nonlinear problems references.

Source proposed Pai and Fisher (1988) Rao and Narasimhan (1996) Rao and Narasimhan (1996) Swartz (1989)

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Problem nonlin amonia nonlin pf88 nonlin rn96 1 nonlin rn96 2 nonlin sw89

The aim of this class of problems is to test DR and GEDI in more complex and nonlinear problems. Some features can be tested, such as: • How DR and GEDI methods behave with unobservable and observable unmeasured variables.

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• How to design and tune robust functions to the nonlinear cases described above.

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• How DR and GEDI are influenced by the parameters of the optimization method and how to detect appropriately global optima of the optimization problem. Customizations are also allowed for all nonlinear problems mentioned:

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• Change the measurement value or standard deviation or add a GE at a specific position.

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• Change the status of a measurement stream to unmeasured and determine the impact in the data classification, DR and GEDI. • Select another objective function (the same as included in the linear test set), propose a new one (using an existing function as a template) or tune its parameters. • Tune the parameters of the existing optimization method. 29

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3.6. Guidelines for problem solving Along the organization of this work, several papers and methods were revised. In this context, with the intention of helping the scientific community to select the appropriate DR or GEDI method, the authors organized a table with the suggested method for DR or GEDI problem solving. Table C.9 is presented in Appendix C and an excerpt is presented in Table 8. In these table, each row represents a paper, in the first column, the authors, the next columns relate each feature or challenge solved in the cited work. The number in the columns represent how many problems of each type is solved in the corresponding paper, and the number within parenthesis indicates how many of these problems belong to the proposed problem set. For example, in the first line of Table 8, Crowe et al. (1983) presented methods for linear DR and GEDI where the methods are applied to 3 test problems, where some problem challenges involved unmeasured streams (column 6), parallel streams (column 7) and recycle streams (column 8). Swartz (1989) presented a method for DR and GEDI for bilinear systems with unmeasured variables and the problem used by Swartz (1989) to validate the proposed method is included in the set of the present paper (number within parenthesis). Table C.9 can also be explored by columns and based on the specific challenge (e.g. DR with unmeasured streams) the reader can find relevant papers regarding this challenge where suggestions of how to overcome each challenge may be found. Also, exploring Table C.9 by columns allow the selection of publications by number of tested problems, by most recent publication or if belongs to the proposed problem set. The advantage of selecting a problem that is included in this set is that some basic methods for DR and GEDI are already implemented in Scilab. Table 8 is far away to become a practical guideline, however, it may be used as a starting guide in the studies of DR and GEDI methods. Table 8: Table excerpt of guidelines for problem solving

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Method used DR GEDI Linear Bilinear Unmeasured Parallel Recycle by literature Streams Streams Streams Crowe et al. (1983) 3 3 3 1 1 1 Swartz (1989) 1(1) 1(1) 1(1) 1(1) -

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4. Discussion

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The results and discussion of some selected problems can be inspected at the Supplement material. As presented in Table 6 and Table C.9, many authors test proposed DR methods with only a few problems. The main reason for this tendency may be historical because, in older studies from the 1970s and 1980s, the limited capacities of the available computational resources restricted the number of DR tests that could be performed. Furthermore, simulation tools and packages, which speed up the process simulation and optimization, such as Matlab, Scilab, Maple, and Mathcad, were not available in the past. The difficulty of validating DR methods for many case studies may also arise from the fact that test problems are dispersed throughout the literature, and time must be invested in selecting the benchmark test problems for the new proposed techniques. In the problems selected, special care was taken to cover the challenges related to DR reported in the literature. Although the test platform selection (computational software and optimization solver) and implementation of the problems in Scilab were not presented and discussed in great detail, the implementation modularity and the easy extension and customization of the tests, resulting in various possible test combinations, were highlighted. Regarding the GEDI tests, GT, MT or NT were chosen because they are the basis for several simultaneous or iterative strategies for GEDI. If GT, MT or NT do not present good GEDI performance (low OP), it may indicate that the techniques that use these tests as a base will also present difficulties in the correct GEDI. For this reason, this benchmark test set could help the scientific community drive their efforts in the selection or proposal of individual or collective statistical tests to perform simultaneously or iterative GEDI methods. In future works, the authors encourage the scientific community to engage in increasing the number of relevant and challenging problems available for the DR and GEDI field. The authors also encourage the industrial community to contribute with problems from their area since DR and GEDI methods can promote many benefits to industrial plants. 5. Conclusion The DR and GEDI techniques can be influenced by many factors, such as flowsheet connections, measurement variance, position of the GE, and un31

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measured variables. Consequently, GEDI is still a challenge even for fully measured, linear, steady-state diagrams (Crowe, 1996; Narasimhan and Jordache, 2000). Through the results of some of the presented problems, this fact was confirmed. In this work, special care was taken in the selection of the benchmark tests, it is presented the main characteristics of each problem, and the issues that the example intends to test are presented. The benchmark problems provide the reader with an idea of the particularities that appear when performing DR and GEDI techniques. The public availability (Valle, 2011) and the modular implementation of the problems and GEDI techniques in Scilab make the extension of this work (both for new problems and GEDI techniques) a simple task. Furthermore, several customizations are allowed to test other particularities of each problem, such as robust estimator selection, multiple GE data generation, and the removal of measured streams. The most important step in developing a new DR technique is validation, which, in most cases, is performed fairly non-rigorously because the methods are analyzed in a limited universe of potential test problems (as presented in Table C.9). Reviewing the literature revealed the need for a common data set for validating GEDI and DR techniques because of the complexity of these problems, this paper provides these data for the community, some of which were collected from the classical literature and others of which were proposed in this work, and can be considered the most complete benchmark set ever presented in the literature.

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6. Acknowledgement

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The authors thank Prof. Fernando Gonalves Amaral and Prof. Fernando Luiz Pellegrini Pessoa for suggestions in the paper structure, Prof. Zhengjiang Zhang, (for the flowsheets diagrams and streams exact values of linear problem ’P16’) and Prof. Christopher Swartz for providing the paper with the heat exchanger network problem (’nonlin sw89’). This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Appendix A. Robust Estimators ε :error definition: εi = (yi − xi ) /σi 32

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Influence Function ( ε . when |ε| is small c . when |ε| → ∞

WLS /2 .ε2i

Contaminated Normal h  2 −ε −ln (1 − pCN ) .exp 2 i + ε2i c2C



Lorenzian

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1 − 1+ ε2 /2c ( i 2CL )

Fair   |εi | 2 2cF cF − ln 1 +

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Logistic    − 2 ln 1 + exp cεLoi



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|εi | cF



Hampel’s redescending M-estimator

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aH .|εi | − 12 .a2H , aH < |εi | ≤ bH aH .bH −

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dρ(ε) ∝ dε

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Quasi-Weighted

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Appendix B. GEDI tests

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The global test (GT) uses the results of the DR, the variance matrix and the mass balance constraints matrix to perform statistical tests. First, the mass balance residuals matrix, R, is calculated based on measurements and the balance matrix, as presented in Equation B.1 (where A is the process constraints matrix). R = A.Y (B.1)

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In the next step, as proposed by Madron (1985) (apud Narasimhan and Jordache, 2000), the variance-covariance matrix is calculated by Equation B.2. V = cov(R) = A.Σ.AT

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(B.3)

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As γ collects in only one parameter the interactions of the whole mass balance, a multivariate statistical test χ2 with ν degrees of freedom is applied, where ν is the rank of matrix A. The GT can detect a GE in the flowsheet as a whole but cannot locate its position or type (leaking or measurement bias). The nodal test (NT) is applied to the balance residuals without considering its iteration (Mah et al., 1976). For each equipment or node, the zr,i is calculated by Equation B.4 and tested with a monovariate statistical hypothesis test z. | ri | zr,i = p | Vi,i | 34

(B.4)

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where: ri : Balance residual, or the ith line of the R matrix

Ad = Y − X

WAd = W1/2 (Y − X) ,

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Another test used to detect measurement bias is the measurement test (MT), which is based in the adjustment after the reconciliation (defined by Equation B.5). (B.5) (B.6)

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where W is the inverse of Σ matrix, or Σ−1 . With the calculation of Ad and WAd, it is possible to gain an intuitive idea of the worst measurements. To calculate the MT statistic, za,j is calculated based on values from the Ad calculation and W matrix , which is computed by Equation B.7. W = ΣAT V−1 AΣ

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Then, za,j is calculated by Equation B.8.

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The measurement test is performed using a monovariate statistical hypothesis test z based on the value of the computed za,j and can detect a GE in the measurement j. Appendix C. Guidelines for problem solving

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In Table C.9, a guideline for DR and GEDI method selection is presented, each row represents a paper, in the first column, the authors, the next columns relate each feature or challenged solved in the cited work. The numbers in the columns represent how many problems of each type is solved in the revised paper, and the number within parenthesis indicates how many of these problems belong to the proposed problem set (this work). For example, in the first line of Table C.9, Crowe et al. (1983) presented methods for

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linear DR and GEDI where the methods are applied to 3 test problems, where some problem challenges involves unmeasured streams, parallel streams and recycle streams. Swartz (1989) presented a method for DR and GEDI for bilinear systems with unmeasured variables, and one problem used by Swartz (1989) to validate the proposed method is included in the set of the present paper (number within parenthesis). Gau and Stadtherr (2000) presented a method for nonlinear DRPE and applied to 3 problems, being 2 of them included in the set of the present work.

Table C.9: Guidelines for problem solving.

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DR GEDI Linear Bilinear Nonlinear Unmeasured Parallel Recycle DRPE Dynamic Filtering GE hidden Streams Streams Streams DR Techniques by holdups 3 3 3 1 1 1 1(1) 1(1) 1(1) 1(1) 1(1) 1 1(1) 1(1) 1 1 2 2 2 1 1 2(2) 2(2) 1(1) 1(1) 1(1) 1(1) 1(1) 7(3) 7(3) 2(2) 2(1) 3(1) 5(2) 4(1) 4(1) 4(3) 4(3) 4(3) 4(3) 4(3) 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 2 2 2 2 2 2 1 1 2 2 2 1 1 1 1 1 1(1) 1(1) 1(1) 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 2 2 1(1) 1(1) 2(2) 1(1) 1(1) 1(1) 5(1) 5(1) 5(1) 3(2) 3(2) 7(2) 7(2) 4(2) 4(2) 3(2) 3(2) 1 1 1 1 1 1 1 1 1 1 3(2) 3(2) 1(1) 1(1) 1(1) 1(1) 1(1) 1 1 9(4) 6(3) 1(1) 4(3) 2 7(4) 7(4) 7(3) 2 1(1) 1 1 1 1 2(2) 1(1) 1(1) 1(1) 1(1) 1(1) 3(1) 3(1) 1(1) 2 3(1) 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1(1) 1(1) 1(1) 1(1) 1(1) 3 3 3 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1(1) 1(1) 1(1) 1(1) 1(1) 2(2) 2(2) 2(2) 2(2) 2(2) 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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Method used by literature Crowe et al. (1983) Swartz (1989) Kelly (1999) Kelly (1998) Llanos et al. (2015) ¨ Ozyurt and Pike (2004) Zhang et al. (2010) Alhaj-Dibo et al. (2008) Jin et al. (2012) Schraa and Crowe (1998) S´ anchez and Romagnoli (1996) Simpson et al. (1988) Pai and Fisher (1988) Crowe (1986) Kelly (2004) Bagajewicz and Cabrera (2003) Tjoa and Biegler (1991) Wongrat et al. (2005) Tjoa and Biegler (1992) Gau and Stadtherr (2000) Esposito and Floudas (1998) Kim et al. (1990) Gau and Stadtherr (2002) Ijaz et al. (2013) Kim et al. (1991) Arora and Biegler (2001) Narasimhan and Jordache (2000) Rollins et al. (2002) Llanos et al. (2017) Albuquerque and Biegler (1996) Prata et al. (2009) Leibman et al. (1992) McBrayer and Edgar (1995) Kim et al. (1991) Zhang and Chen (2014) Zhang and Chen (2015) S´ anchez et al. (1999) Romagnoli and Stephanopoulos (1981) Jiang and Bagajewicz (1999) Rollins and Davis (1992) Devanathan et al. (2005) Reddy and Mavrovouniotis (1998) Tong and Crowe (1996) Rollins et al. (1996) Mei et al. (2006) Bagajewicz and Jiang (1998) Bhat and Saraf (2004)

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List of Symbols and Nomenclature

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Average Error of Estimation Akaike Information Criterion Average Number of Type 1 Errors Dynamic Measurement Test Data Reconciliation Data Reconciliation and Parameter Estimation Extended Measurement Test Gross Error Detection Gross Error Detection and Identification Generalized Likelihood Ratio Iterative Measurement Test Multiple Gross Error Detection and Identification Modified Generalized Likelihood Ratio Modified Simultaneous Estimation of Gross Error Modified Unbiased Estimation Technique Mixed Integer Overall Power Principal Components Analysis Standard deviation Simultaneous Estimation of Gross Error Unbiased Estimation Technique Weighted Least Squares Incidence Matrix (Jacobian matrix of the constraints) Adjustment line of the Ad matrix Parameter of Logistic Robust Estimator Vector of process constraints related to the mass, energy or momentum balance Hypotheses for the statistical test for GEDI Mass balance residuals matrix Balance residual, or the ith line of the R matrix Product of the global heat transfer coefficient and the heat exchanger area Covariance matrix of residuals matrix, R Weighted Adjust Inverse of Σ matrix The j, j element of the W matrix Vector of reconciled measurements Vector of reconciled measurement lower bounds Vector of reconciled measurement upper bounds Vector of measurements MT statistics

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AEE AIC AVT1 DMT DR DRPE EMT GED GEDI GLR IMT MGEDI MGLR MSEGE MUBET MI OP PCA SD SEGE UBET WLS A Ad aj : The j th CLo F (X) H0 ,H1 R ri UA V W ad W wj,j X Xmin Xmax Y za,j

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ρ σ σi χ2 ν θ

NT statistics Parameter of Quasi-Weighted Robust Estimator Value of the Global Test statistics Residual of measured and reconciled data, weighted by the reciprocal of the standard deviation The generic objective function of DR Standard deviation Standard deviation of measurement i Chi-squared test Degrees of freedom for Chi-squared test Vector of parameters to be estimated

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zr,i β γ εi

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References

Adenso-Diiaz, B., Laguna, M., 2006. Fine-Tuning of Algorithms Using Fractional Experimental Designs and Local Search. Operations Research 54 (1), 99–114. Akaike, H., 1974. A New Look at the Statistical Model Identification. IEEE Transactions on Automatic Control AC-19 (6), 716.

ED

M

Albuquerque, J. a. S., Biegler, L. T., Oct. 1996. Data reconciliation and gross-error detection for dynamic systems. AIChE Journal 42 (10), 2841– 2856.

PT

Albuquerque, J. S., Biegler, L. T., 1995. Decomposition algorithms for online estimation with nonlinear models. Computers & Chemical Engineering 19 (10), 1031–1039.

CE

Alhaj-Dibo, M., Maquin, D., Ragot, J., Feb. 2008. Data reconciliation: A robust approach using a contaminated distribution. Control Engineering Practice 16 (2), 159–170.

AC

Arora, N., Biegler, L. T., 2001. Redescending estimators for data reconciliation and parameter estimation. Computers and Chemical Engineering 25, 1585–1599. Audet, C., Orban, D., 2006. Finding Optimal Algorithmic Parameters Using DerivativeFree Optimization. SIAM Journal on Optimization 17 (3), 642– 664.

39

ACCEPTED MANUSCRIPT

Bagajewicz, M., Jiang, Q., S´anchez, M., Jun. 1999. Performance evaluation of PCA tests for multiple gross error identification. Computers & Chemical Engineering 23 (Supplement 1), S589–S592.

CR IP T

Bagajewicz, M. J., Cabrera, E., Oct. 2003. Data Reconciliation in Gas Pipeline Systems. Industrial & Engineering Chemistry Research 42 (22), 5596–5606.

Bagajewicz, M. J., Jiang, Q., Nov. 1998. Gross error modeling and detection in plant linear dynamic reconciliation. Computers & Chemical Engineering 22 (12), 1789–1809.

AN US

Bazin, C., Hodouin, D., Zouadi, M., Nov. 2005. Data reconciliation and equilibrium constant estimation: Application to copper solvent extraction. Hydrometallurgy 80 (1-2), 43–53. Bhat, S. a., Saraf, D. N., Jul. 2004. Steady-State Identification, Gross Error Detection, and Data Reconciliation for Industrial Process Units. Industrial & Engineering Chemistry Research 43 (15), 4323–4336.

ED

M

Biegler, L. T., Tjoa, I.-B., Dec. 1993. A parallel implementation for parameter estimation with implicit models. Annals of Operations Research 42 (1), 1–23. Bondarenko, A., Bortz, D., Dolan, L., Merritt, M., Mor´e, J., Munson, T., 2004. COPS: Large-Scale Optimization Problems.

CE

PT

Castro, J., Apr. 2004. A pulp mill benchmark problem for control: application of plantwide control design. Journal of Process Control 14 (3), 329– 347.

AC

Chen, R., Dave, K., McAvoy, T. J., Luyben, M., 2003. A nonlinear dynamic model of a vinyl acetate process. Industrial & Engineering Chemistry Research 2003 (42), 4478–4487. Chen, W., Shao, Z., Wang, K., Chen, X., Biegler, L. T., 2011. Random sampling-based automatic parameter tuning for nonlinear programming solvers. Industrial and Engineering Chemistry Research 50 (7), 3907–3918. Crowe, C. M., 1986. Reconciliation of process flow rates by matrix projection, Part II: The nonlinear case. AIChE Journal 32, 616–623. 40

ACCEPTED MANUSCRIPT

Crowe, C. M., 1996. Data reconciliation - progress and challenges. Journal of Process Control 6 (2/3), 89–98.

CR IP T

Crowe, C. M., Campos, Y. G., Hrymak, A., 1983. Reconciliation of Process Flow Rates by Matrix Projection, Part I: Linear Case. AICHE Journal 29 (6), 881–888.

Devanathan, S., Vardeman, S. B., Rollins Sr., D. K., Rollins, D. K., Dec. 2005. Likelihood and Bayesian Methods for Accurate Identification of Measurement Biases in Pseudo Steady-State Processes. Chemical Engineering Research and Design 83 (12), 1391–1398.

AN US

Downs, J. J., Vogel, E., 1993. A plant-wide industrial problem. Computers and Chemical Engineering 17 (3), 245–255.

Esposito, W. R., Floudas, C. a., 1998. Global optimization in parameter estimation of nonlinear algebraic models via the error-in-variables approach. Industrial & Engineering Chemistry Research 37 (5), 1841–1858.

M

Faber, R., Li, P., Wozny, G., Nov. 2003. Sequential Parameter Estimation for Large-Scale Systems with Multiple Data Sets. 1. Computational Framework. Industrial & Engineering Chemistry Research 42 (23), 5850–5860.

ED

Farias, A. C., 2009. Avalia¸ca˜o de Estrat´egias para Reconcilia¸ca˜o de Dados e Detec¸ca˜o de Erros Grosseiros. Disserta¸ca˜o de mestrado, Universidade Federal do Rio Grande do Sul.

PT

Floudas, C. A., Pardalos, P. M., 1999. Handbook of test problems in local and global optimization. Nonconvex optimization and its applications. Kluwer Academic Publishers.

CE

Gau, C.-y., Stadtherr, M. A., 2000. Reliable nonlinear parameter estimation using interval analysis: An error-in-variable approach. Computers & Chemical Engineering 24, 631–637.

AC

Gau, C.-Y., Stadtherr, M. A., 2002. Deterministic Global Optimization for Error-in-Variables Parameter Estimation. AICHE Journal 48 (6), 1192– 1197. Harikumar, P., Narasimhan, S., 1993. A Method to Incorporate Bounds in Data Reconciliation and Gross Error Detection - II. Gross Error Detection Strategies. Computers & Chemical Engineering 17 (11), 1121–1128. 41

ACCEPTED MANUSCRIPT

Hock, W., Schittkowski, K., 1981. Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems (187).

CR IP T

Hutter, F., Hoos, H. H., Leyton-Brown, K., St¨ utzle, T., 2009. ParamILS: An automatic algorithm configuration framework. Journal of Artificial Intelligence Research 36, 267–306. Ijaz, H., Ati, U. M., Mahalec, V., 2013. Heat exchanger network simulation, data reconciliation & optimization. Applied Thermal Engineering 52 (2), 328–335.

AN US

Iordache, C., Mah, R. S. H., Tamhane, A. C., 1985. Performance Studies of the Measurement Test for Detecting Gross Errors in Process Data. AIChE Journal 31, 1187–1201.

Jiang, Q., Bagajewicz, M. J., May 1999. On a Strategy of Serial Identification with Collective Compensation for Multiple Gross Error Estimation in Linear Steady-State Reconciliation. Industrial & Engineering Chemistry Research 38 (5), 2119–2128.

ED

M

Jiang, T., Chen, B., He, X., Stuart, P., 2003. Application of steady-state detection method based on wavelet transform. Computers & Chemical Engineering 27 (4), 569–578.

PT

Jin, S., Li, X., Huang, Z., Liu, M., 2012. A New Target Function for Robust Data Reconciliation. Industrial & Engineering Chemistry Research 51 (30), 10220–10224.

CE

Kelly, J., Jun. 1999. Reconciliation of process data using other projection matrices. Computers & Chemical Engineering 23 (6), 785–789. Kelly, J. D., Oct. 1998. On finding the matrix projection in the data reconciliation solution. Computers & Chemical Engineering 22 (11), 1553–1557.

AC

Kelly, J. D., Nov. 2004. Techniques for solving industrial nonlinear data reconciliation problems. Computers & Chemical Engineering 28 (12), 2837– 2843. Kelly, J. D., Hedengren, J. D., 2013. A steady-state detection (SSD) algorithm to detect non-stationary drifts in processes. Journal of Process Control 23 (3), 326–331. 42

ACCEPTED MANUSCRIPT

Kim, I.-W., Liebman, M. J., Edgar, T. F., 1990. Robust error- in-variables estimation using nonlinear programming techniques. AIChE Journal 36 (7), 985–993.

CR IP T

Kim, I.-W., Liebman, M. J., Edgar, T. F., 1991. A Sequential Error-inVariables Method for Nonlinear Dynamic Systems. Computers & Chemical Engineering 15 (9), 663–670. Kongsjahju, R., Rollins, D. K., Bascu˜ nana, M. B., 2000. Accurate Identification of Biased Measurements Under Serial Correlation. Trans IChemE 78 (Part A), 1010–1018.

AN US

Leibman, M. J., Edgar, T. F., Lasdon, L. S., 1992. Efficient data reconciliation and estimation for dynamic processes using nonlinear programming techniques. Computers & Chemical Engineering 16 (1011), 963–986. Leong, W., Hollands, K., Brunger, a., Dec. 1998. On a physically-realizable benchmark problem in internal natural convection. International Journal of Heat and Mass Transfer 41 (23), 3817–3828.

ED

M

Llanos, C. E., Sanch´ez, M. C., Maronna, R. A., 2015. Robust Estimators for Data Reconciliation. Industrial & Engineering Chemistry Research 54 (18), 5096–5105.

PT

Llanos, C. E., S´anchez, M. C., Maronna, R. A., 2017. Classification of Systematic Measurement Errors within the Framework of Robust Data Reconciliation. Industrial & Engineering Chemistry Research 56 (34), 9617–9628.

CE

Macdonald, R. J., Howat, C. S., 1988. Data Reconciliation and Parameter EStimation in Plant Performance Analysis. AIChE Journal 34 (1), 1–8.

AC

Madron, F., 1985. A New Approach to the Identification of Gross Error in Chemical Engineering Measurements. Chemica Engineering Science 40, 1855–1860. Mah, R. S. H., 1990. Chemical process structures and information flows. Butterworths series in chemical engineering. Butterworths. Mah, R. S. H., Stanley, G. M., Downning, D. M., 1976. Reconciliation of Process Flow and Invetory Data. Ind. & Eng. Chem. Proc. Des. Dev, Vol. 15 (1), 175–183. 43

ACCEPTED MANUSCRIPT

Maronna, R., Arcas, J., 2009. Data reconciliation and gross error diagnosis based on regression. Computers & Chemical Engineering 33 (1), 65–71.

CR IP T

McBrayer, K. F., Edgar, T. F., 1995. Bias detection and estimation in dynamic data reconciliation. Journal of Process Control 5 (4), 285–289.

Mei, C., Su, H., Chu, J., Oct. 2006. An NT-MT Combined Method for Gross Error Detection and Data Reconciliation. Chinese Journal of Chemical Engineering 14 (5), 592–596.

AN US

Mukherjee, J., Narasimhan, S., Jan. 1996. Leak Detection in Networks of Pipelines by the Generalized Likelihood Ratio Method. Industrial & Engineering Chemistry Research 35 (6), 1886–1893. Narasimhan, S., Jordache, C., 2000. Data Reconciliation and Gross Error Detection: An Intelligent Use of Process Data, 1st Edition. Gulf Publishing, Houston.

M

Narasimhan, S., Mah, R. S. H., 1987. Generalized Likelihood Ratio Method for Gross Error Identification. AIChE Journal 33, 1514–1521.

ED

¨ Ozyurt, D. B., Pike, R. W., 2004. Theory and practice of simultaneous data reconciliation and gross error detection for chemical. processes. Computers and Chemical Engineering 28, 381–402.

PT

Pai, C. C. D., Fisher, G. D., 1988. Application of Broyden’s method to reconciliation of nonlinearly constrained data. AIChE Journal 34, 807–817.

CE

Prata, D. M., Schwaab, M., Lima, E. L., Pinto, J. C., Sep. 2009. Nonlinear dynamic data reconciliation and parameter estimation through particle swarm optimization: Application for an industrial polypropylene reactor. Chemical Engineering Science 64 (18), 3953–3967.

AC

Rajeshkanna, P., Oct. 2006. Conjugate heat transfer study of backwardfacing step flow A benchmark problem. International Journal of Heat and Mass Transfer 49 (21-22), 3929–3941. Rao, R. R., Narasimhan, S., Apr. 1996. Comparison of Techniques for Data Reconciliation of Multicomponent Processes. Industrial & Engineering Chemistry Research 35 (4), 1362–1368.

44

ACCEPTED MANUSCRIPT

Reddy, V. N., Mavrovouniotis, M. L., May 1998. An Input-Training Neural Network Approach for Gross Error Detection and Sensor Replacement. Chemical Engineering Research and Design 76 (4), 478–489.

CR IP T

Rod, V. V., Hancil, V., 1980. Iterative estimation of model parameters when measurements of all variables are subject to error. Computers & Chemical Engineering 4, 33–38. Rollins, D. K., Cheng, Y., Devanathan, S., 1996. Intelligent Selection of Hypothesis Tests to Enhance Gross Error Identification. Comp. & Chem. Engineering’ 20 (5), 517–530.

AN US

Rollins, D. K., Davis, J. F., 1992. Unbiased Estimation of Gross Error in Process Measurements. AIChE Journal 38, 563–572.

Rollins, D. K., Devanathan, S., Bascun, M. V. B., 2002. Measurement bias detection in linear dynamic systems. Chemical Engineering 26, 1201– 1211.

M

Romagnoli, J., S´anchez, M. C., 1999. Process Systems Engineering - Volume 2 - Data Processing and Reconciliation for Chemical Process Operations.

ED

Romagnoli, J., Stephanopoulos, G., 1981. Rectification of Process Measrement Data in the Presence of Gross Errors. Chemical Engineering Science 36 (11), 1849–1863.

PT

Rosenberg, J., Mah, R. S. H., Iordache, C., 1987. Evaluation of Schemes for Detecting and Identifying Gross Errors in Process Data. Ind. & Eng. Chem. Proc. Des. Dev, Vol. 26, 555–564.

CE

S´anchez, M., Romagnoli, J., 1996. Use of orthogonal transformations in data classification-reconciliation. Computers & Chemical Engineering 20 (5), 483–493.

AC

S´anchez, M., Romagnoli, J., Jiang, Q., Bagajewicz, M., jul 1999. Simultaneous estimation of biases and leaks in process plants. Computers & Chemical Engineering 23 (7), 841–857. S´anchez, M., Romagnoli, J. J., Jiang, Q., Bagajewicz, M., Sa, M., Jul. 1999. Simultaneous estimation of biases and leaks in process plants. Computers & Chemical Engineering 23 (7), 841 – 857.

45

ACCEPTED MANUSCRIPT

Schittkowski, K., 1987. More Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems (282).

CR IP T

Schraa, O. J., Crowe, C. M., 1998. The numerical solution of bilinear data reconciliation problems using unconstrained optimization methods. Computers & Chemical Engineering 22 (9), 1215–1228. Serth, R. W., Heenan, W. A., 1986. Gross error detection and data reconciliation in steam-metering systems. AIChE Journal 32, 733–747.

AN US

Simpson, D. E., Everett, M. G., Voller, V. R., 1988. Reducing the Number of Unknowns in a Constrained Minimization Problem - An Application to Material Balances. Applied Mathematical Modeling 12, 204–212.

Soderstrom, T. A., Himmelblau, D. M., Edgar, T. F., Aug. 2001. A mixed integer optimization approach for simultaneous data reconciliation and identification of measurement bias. Control Engineering Practice 9 (8), 869–876.

M

Swartz, C. L. E., 1989. Data Reconciliation for Generalized Flowsheet Applications. In: American Chemical Society National Meeting.

ED

Tamhane, A. C., Mah, R. S. H., 1985. Data Reconciliation and Gross Error Detection in Chemical Process Networks. Technometrics 24 (4), 409–422.

PT

Tjoa, I., Biegler, L. T., 1992. Reduced Successive Quadratic Programming Strategy for Errors-in-variables Estimation. Computers & Chemical Engineering 16 (6), 523–533.

CE

Tjoa, I. B., Biegler, L. T., 1991. Simultaneous strategies for data reconciliation and gross error detection of nonlinear systems. Computers Chemical Engineering 15, 679–690.

AC

Tong, H., Crowe, C. M., 1996. Detecting Persistent Gross Errors by Sequential Analysis of Principal Components. Computers and Chemical Engineering 20 (Suppl.), 733–738. Valle, E. C., 2011. Data Reconciliation and Gross Error Detection Benchmark Problems.

46

ACCEPTED MANUSCRIPT

Veverka, V., Madron, F., 1997. Material and Energy Balancing in the Process Industries: From Microscopic Balances to Large Plants, first edit Edition. Elsevier B.V., Amsterdan.

CR IP T

W¨achter, A., Biegler, L. T., 2006. On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming. Mathematical Programming 106 (1), 25–57.

Wang, F., Jia, X.-p., Zheng, S.-q., Yue, J.-c., Oct. 2004. An improved MTNT method for gross error detection and data reconciliation. Computers & Chemical Engineering 28 (11), 2189–2192.

AN US

Wongrat, W., Srinophakun, T., Srinophakun, P., Apr. 2005. Modified genetic algorithm for nonlinear data reconciliation. Computers & Chemical Engineering 29 (5), 1059–1067.

Yang, Y., Ten, R., Jao, L., Jun. 1995. A study of gross error detection and data reconciliation in process industries. Computers & Chemical Engineering 19 (Supplement 1), 217–222.

ED

M

Zhang, Z., Chen, J., 2014. Simultaneous data reconciliation and gross error detection for dynamic systems using particle filter and measurement test. Computers & Chemical Engineering 69, 66–74.

PT

Zhang, Z., Chen, J., 2015. Correntropy based data reconciliation and gross error detection and identification for nonlinear dynamic processes. Computers & Chemical Engineering 75, 120–134.

AC

CE

Zhang, Z., Shao, Z., Chen, X., Wang, K., Qian, J., Feb. 2010. Quasi-weighted least squares estimator for data reconciliation. Computers & Chemical Engineering 34 (2), 154–162.

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