Abstract
The goal of flow rates reconciliation was to adjust measured values and estimate unmeasured streams so as to balance both measured and unmeasured values, identify gross errors and detect leaks and losses. Thus, data reconciliation plays a key role in the monitoring of industrial plants for the early detection of critical events which might cause environmental and economic damages, and it is,therefore, an essential component of any clean technology process. Consequently, any method that improves the accuracy of the reconstructed data by considering more realistic assumptions on the statistical nature of the data can add considerably to the overall reliability of the process. The reconciliation procedure is statistical in nature and requires adequate information on the structure of the random errors of the flow rates measured. A frequent assumption is the homoscedasticity and the independence of the errors affecting different streams. This assumption leads to efficient algorithms based on advanced linear algebra decompositions, such as QR or Singular Value Decomposition, but it frequently leads to biased estimates, especially when the values of flow rates vary over two or more orders of magnitude. The goal of this article was to show the importance of considering general heteroscedasticity when reconciling flow rates. Errors are supposed to be normally distributed according to \(\varepsilon_{\text{i}} \cong N(0,\,\sigma_{0} L_{\text{i}}^{ 2\eta } )\), where \(L_{\text{i}}\) is the measurement of the ith flow rate and \(\theta\) = \(\left\{ {\theta |\sigma_{0} ,\eta } \right\}\) is a set of two parameters to be estimated along with the adjustments to the measured flow rates. Therefore, the overall variance–covariance is characterised by 3 parameters \(\sigma_{0} ,\eta\) and the correlation factor among measurement errors ρ. The algorithm here proposed is based on conditional optimality, and it carries out the whole optimisation in terms of the parameters \(\theta\) only, the unknown adjustments being expressed at each iteration as functions of \(\theta\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- a :
-
Adjustments of measured flow rates (kg h−1)
- A, A 1 ,A 2 :
-
Matrices that describe the topological structure of flow rates
- c :
-
Accurately measured flow rates (kg h−1)
- e u :
-
Vector with all components equal to 0, except e u(u) = 1
- E :
-
Expectation operator
- g :
-
Defined as \(- Y^{\text{T}} \left[ {A_{0} c + A_{1} x} \right]\)
- G :
-
Defined as \(VA_{1}^{\text{T}} YH^{ - 1} Y^{\text{T}} A_{1} V\)
- H :
-
Defined as \(\left[ {Y^{\text{T}} A_{1} VA_{1}^{\text{T}} Y} \right]\)
- M :
-
Defined as \(A_{1}^{\text{T}} Y\left[ {Y^{\text{T}} A_{1} VA_{1}^{\text{T}} Y} \right]^{ - 1} Y^{\text{T}} A_{1}\)
- N :
-
Normal distribution
- q :
-
Vector of measured and unmeasured flow rates (kg h−1)
- q m :
-
Measured flow rate in the numerical example
- q homo :
-
Reconstructed flow rate value using homoscedastic model
- q het :
-
Reconstructed flow rate value using heteroscedastic model
- Q 1 , Q 2 , R 1 , R 2 :
-
Matrices resulting from the QR decomposition of A2
- R :
-
Defined as \(Y^{\text{T}} A_{1} VA_{ 1}^{\text{T}} Y\)
- V :
-
Variance–covariance matrix of the flow rate measurement errors
- x’ :
-
Measured flow rates (kg h−1)
- x * :
-
Theoretical unknown flow rates (kg h−1)
- Y :
-
Defined as \(A_{2}^{\text{T}} Y = 0\)
- δ ij :
-
Kronecker delta function
- ε :
-
Flow rate measurement errors (kg h−1)
- η :
-
Exponent parameter of heteroscedastic variance
- θ :
-
Parameters of heteroscedastic variance–covariance matrix
- λ :
-
Defined as \(- \left[ {Y^{\text{T}} A_{1} VA_{1}^{\text{T}} Y} \right]^{ - 1} Y^{\text{T}} \left[ {A_{0} c + A_{1} x^{\prime}} \right]\)
- ρ :
-
Correlation factor of flow rate measurement errors
- σ 0 :
-
Homoscedastic variance or pre-exponential parameter of heteroscedastic variance
- Φ,Ψ :
-
Objective functions
- χ 2 :
-
Chi-squared distribution (Chi-squared values)
- ω :
-
Defined as \(\left( {e_{\text{s}}^{\text{T}} VA_{1}^{\text{T}} Y\lambda } \right)^{\text{T}}\)
- Ω :
-
Defined as \(VA_{1}^{\text{T}} YH^{ - 1} Y^{\text{T}} A_{1} V\)
References
Al-Mutairi EM, El-Halwagi MM (2010) Environmental-impact reduction through simultaneous design, scheduling, and operation. Clean Tech Environ Policy 12:537–545
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Du Croz J, Greenbaum A, Hammarling S, McKenney A, Sorensen D (1999) LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia
Bagatin R, Klemeš JJ, Reverberi AP, Huisingh D (2014) Conservation and improvements in water resource management: a global challenge. J Clean Prod 77:1–9
Bard Y (1974) Nonlinear parameter estimation. Academic Press, Orlando
Bhatt NP, Mitna A, Narasimhan S (2007) Multivariate calibration of non-replicated measurements for heteroscedastic errors. Chemom Intell Lab 85:70–81
Choi YJ, Han SK, Chung ST, Row KH (2004) Chromatographic separation of bupivacaine racemate by mathematical model with competitive Langmuir isotherm. Korean J Chem Eng 21:829–835
Crowe CM (1986) Reconciliation of process flow rates by matrix projection. Part II: the nonlinear case. AIChE J 32:616–623
Crowe CM, Garcia Campos YA, Hrymak A (1983) Reconciliation of process flow rates by matrix projection Part I: Linear case. AIChE J 29:881–888
De Rademaeker E, Suter G, Pasman HJ, Fabiano B (2014) A review of the past, present and future of the European loss prevention and safety promotion in the process industries. Process Saf Environ. doi:10.1016/j.psep.2014.03.007
Dovì VG, Reverberi AP, Maga L (1997) Reconciliation of process measurements when data are subject to detection limits. Chem Eng Sci 52:3047–3050
Fabiano B, Reverberi AP, Del Borghi A, Dovì VG (2012) Biodiesel production via transesterification: process safety insights from kinetic modeling. Theor Found Chem Eng 46:673–680
Fabiano B, Currò F, Reverberi AP, Palazzi E (2014) Coal dust emissions: from environmental control to risk minimization by underground transport. An applicative case-study. Process Saf Environ 92:150–159
Farsang B, Gomori Z, Horvath G, Nagy G, Nemeth S, Abonyi J (2013) Simultaneous validation of online analyzers and process simulators by process data reconciliation. Chem Eng Trans 32:1303
IPLOM S.p.A., Busalla, Italy (2014); www.iplom.it. Cited on 22 Jun 2014
Kong M, Chen B, He X (2002) Wavelet-based regulation of dynamic data reconciliation. Ind Eng Chem Res 41:3405–3412
Mah RSH, Tamhane AC (1982) Detection of gross errors in process data. AIChE J 28:828–830
Manenti F, Grottoli MG, Pierucci S (2011) Online data reconciliation with poor-redundancy systems. Ind Eng Chem Res 50:14105–14114
Murtagh BA, Sargent RWH (1970) Computational experience with quadratically convergent minimization methods. Comput J 13:185–194
Pascariu V, Avadanei O, Gasner P, Stoica I, Reverberi AP, Mitoseriu L (2013) Preparation and characterization of PbTiO3-epoxy resin compositionally graded thick films. Phase Transit 86:715–725
Reverberi AP, Cerrato C, Dovì VG (2011) Reconciliation of flow rate measurements in the presence of solid particles. Ind Eng Chem Res 50:5248–5252
Tagliabue M, Reverberi AP, Bagatin R (2014) Boron removal from water: needs, challenges and perspectives. J Clean Prod 77:56–64
Tao L-Y, Zhang M, Li Z-D (2013) Study on the product quality control in small batch trial process. Adv Mat Res 694–697:3507–3511
Acknowledgments
The financial support from the EC’s FP7 EFENIS Project (Contract No: ENER/FP7/296003—Efficient Energy Integrated Solutions for Manufacturing Industries) is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vocciante, M., Mantelli, V., Aloi, N. et al. Process data reconciliation in the presence of non-uniform measurement errors. Clean Techn Environ Policy 16, 1287–1294 (2014). https://doi.org/10.1007/s10098-014-0824-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10098-014-0824-6