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Evaluation of Steady-State and Dynamic Soft Sensors for Industrial Crude Distillation Unit under Parametric Constraints
Proceedings, 10th IFAC International Symposium on Proceedings, 10th IFAC International Symposium on Proceedings, 10th IFAC International Symposium on Available online Proceedings, 10th of IFAC International Symposium on at www.sciencedirect.com Advanced Chemical Processes Advanced Control Control of Chemical Processes Proceedings, 10th of IFAC International Symposium on Advanced Control Chemical Processes Advanced Control of Chemical Processes Shenyang, Liaoning, China, July 25-27, 2018 Proceedings, 10th IFAC International Symposium on Shenyang, Liaoning, China, July 25-27, 2018 Advanced Control of China, Chemical Processes Shenyang, Liaoning, July 25-27, Shenyang,Control Liaoning, July 25-27, 2018 2018 Advanced of China, Chemical Processes Shenyang, Liaoning, China, July 25-27, 2018 Shenyang, Liaoning, China, July 25-27, 2018
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IFAC PapersOnLine 51-18 (2018) 566–571
Evaluation of Steady-State and Dynamic Soft Sensors for Industrial Crude Evaluation of Steady-State and Dynamic Soft Sensors for Industrial Crude Evaluation of Steady-State and Dynamic Soft Sensors for Industrial Crude Evaluation of Steady-State and Dynamic Soft Sensors for Industrial Crude Distillation Unit under Parametric Constraints Evaluation of Distillation Steady-State and Dynamic Soft Sensors for Industrial Crude Unit under Parametric Constraints Distillation Unit under Parametric Constraints Distillation Unit under Parametric Constraints Distillation Unit* under Parametric Constraints *,** *** *,** *** Andrei *,**,, Evgeny *** Anton Goncharov Goncharov*,** Evgeny Zhuravlev Zhuravlev*** Andrei Torgashov Torgashov***,, Anton
Andrei *,**, Evgeny Zhuravlev*** Anton Goncharov Goncharov Andrei Torgashov Torgashov*,, Anton *,**, Evgeny Zhuravlev*** Andrei Torgashov**, Anton Goncharov *,**, Evgeny Zhuravlev*** , Anton Goncharov , Evgeny Zhuravlev Andreiof Torgashov *Institute Automation and Control Process FEB RAS, 555 Radio and *Institute of Automation and Control Process FEB RAS, Radio St., St., and *Institute of Automation and Control Process FEB RAS, St., and *Institute of Automation and Control Process FEB RAS, 5 Radio Radio St.,Island and Far-Eastern Federal University, Bld. 12(E), Campus FEFU, Russky Far-Eastern Federal University, Bld. 12(E), Campus FEFU, Russky Island *Institute of Automation and Control Process FEB RAS, 5 Radio St., and Far-Eastern Federal University, Bld. 12(E), Campus FEFU, Russky Island *Institute of Automation and Control Process FEB RAS, 5 Radio St., and Far-Eastern Federal University, Bld. 12(E), Campus FEFU, Russky Island Vladivostok, Russia (Tel: +7-423-231-02-02; e-mail: [email protected]) Vladivostok, Russia (Tel: +7-423-231-02-02; e-mail: [email protected]) Far-Eastern Federal University, Bld. 12(E), Campus FEFU, Russky Island Vladivostok, (Tel: +7-423-231-02-02; Far-EasternRussia Federal University, Bld. 12(E), e-mail: [email protected]) FEFU, Russky Island Vladivostok, Russia (Tel: +7-423-231-02-02; e-mail: [email protected]) Vladivostok,**JSC Russia“Honeywell”, (Tel: +7-423-231-02-02; [email protected]) 777 Kievskaya St., Moscow, Russia. Kievskaya e-mail: St., Moscow, Russia. Vladivostok,**JSC Russia“Honeywell”, (Tel: +7-423-231-02-02; [email protected]) **JSC “Honeywell”, St., Russia. **JSC “Honeywell”, 7 Kievskaya Kievskaya e-mail: St., Moscow, Moscow, Russia. **JSC “Honeywell”, 7 Kievskaya St., Moscow, Russia. ***JSC “Gazprom neftekhim Salavat”, 30 Molodogvardeitsev st., Salavat, Russia. ***JSC “Gazprom neftekhim Salavat”, 30 Molodogvardeitsev st., **JSC “Honeywell”, 7 Kievskaya St., Moscow, ***JSC “Gazprom neftekhim Salavat”, 30 MolodogvardeitsevRussia. st., Salavat, Salavat, Russia. Russia. ***JSC “Gazprom neftekhim Salavat”, 30 Molodogvardeitsev st., Salavat, Russia. ***JSC “Gazprom neftekhim Salavat”, 30 Molodogvardeitsev st., Salavat, Russia. Abstract: The parametric identification problem for industrial crude distillation unit (CDU) is Abstract: The The parametric parametric identification identification problem problem for for industrial industrial crude crude distillation distillation unit unit (CDU) (CDU) is considered. considered. Abstract: is Abstract: The parametric identification problem for industrial crude distillation unit (CDU) is considered. considered. We take the a priori knowledge of the process into account by using a system of constraints for We take take The the parametric a priori priori knowledge knowledge of the the process into account account bydistillation using aa system system of constraints constraints for Abstract: identification problem for industrial crude unit (CDU) is considered. We the a of process into by using of for Abstract: The parametric identification problem for industrial crude distillation unit (CDU) is considered. We take the a priori knowledge of the process into account by using a system of constraints for parameters of soft sensors models. The identification problem is transformed into a constrained parameters ofa soft soft sensors models. The process identification problem isusing transformed into a constrained constrained We take theof priorisensors knowledge of the into account by is a system into of constraints for parameters models. The identification problem transformed a We take the a priori knowledge of the process into account by using a system of constraints for parameters of soft sensors models. The identification problem is transformed into a constrained optimization which we solved using the active set method. The static and dynamic sensors optimization problem, which we solved using the active set method. The static and dynamic soft sensors parameters ofproblem, soft sensors models. The identification problem is transformed into a soft constrained optimization problem, which we solved using the active set method. The static and dynamic soft sensors parameters of soft sensors models. The identification problem is transformed into a constrained optimization problem, which we solved using the active set method. The static and dynamic soft sensors are evaluated for industrial CDU located at JSC “Gazprom Salavat” It was found that are evaluatedproblem, for industrial CDU located at JSC “Gazprom neftekhim Salavat” refinery. It wassoft found that optimization which we solved using the active setneftekhim method. The staticrefinery. and dynamic sensors are for CDU located at JSC “Gazprom Salavat” refinery. It found that optimization problem, which wewhen solved using active setneftekhim method. The static and dynamic soft sensors are evaluated evaluated for industrial industrial CDU located atused JSCthe “Gazprom neftekhim Salavat” refinery. It was was found that the model performed better we the proposed constrained optimization approach for the model performed better when we used the proposed constrained optimization approach for are evaluated for industrial CDU located at JSC “Gazprom neftekhim Salavat” refinery. It was found that the model performed better when we used the proposed constrained optimization approach for are evaluated for industrial CDU located at JSC “Gazprom neftekhim Salavat” refinery. It was found that the model performed better when we used the proposed constrained optimization approach for identification instead of robust regression methods. identification instead of robust regression methods. the model performed better when we used the proposed constrained optimization approach for identification instead of robust regression methods. the model performed better when we used the proposed constrained optimization approach for identification instead of robust regression methods. identification instead of robust regression methods. Keywords: Soft sensing, identification, constrained parameters, optimization, distillation columns. Keywords: Soft sensing, identification, constrained parameters, optimization, distillation columns. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. identification instead of robust regression methods. Keywords: Keywords: Soft Soft sensing, sensing, identification, identification, constrained constrained parameters, parameters, optimization, optimization, distillation distillation columns. columns. Keywords: Soft sensing, identification, constrained parameters, optimization, distillation columns. Keywords: Soft sensing, identification, constrained parameters, optimization, distillation columns. and K-2 (Fig. The is located JSC and K-2 (Fig. 1). 1). The plant plant is located at at JSC “Gazprom “Gazprom and K-2 1). plant is at “Gazprom 1. INTRODUCTION 1. INTRODUCTION and K-2 (Fig. (Fig. 1). The The plant is located located at JSC JSC “Gazprom 1. neftekhim Salavat” refinery (Salavat, Russia). The feed flow neftekhim Salavat” refinery (Salavat, Russia). The feed flow and K-2 (Fig. 1). The plant is located at JSC “Gazprom 1. INTRODUCTION INTRODUCTION neftekhim Salavat” refinery (Salavat, Russia). The feed flow flow and K-2 (Fig. 1). The plant is located at JSC “Gazprom neftekhim Salavat” refinery (Salavat, Russia). The feed 1. INTRODUCTION (crude) comes comes from refinery the oil oil desalting desalting unit and enters enters underflow the In the present, researchers are paying more attention to the 1. INTRODUCTION In the present, researchers are paying more attention to the neftekhim Salavat” (Salavat, Russia). The feed (crude) from the unit and under the In the present, researchers are paying more attention to the th neftekhim Salavat” refinery (Salavat, Russia). The feed flow comes from theoverhead oil desalting unit and enters under the In the present,problems researchers are paying more attentionofto soft the (crude) th th of K-1. The products of K-1 are gas and 16 th tray identification of the model parameters tray of K-1. The overhead products of K-1 are gas and identification problems of the model parameters of soft (crude) comes from the oil desalting unit and enters under the In the present, researchers are paying more attention to the 16 th tray of The overhead products of are gas and identification problems of the model parameters of soft (crude) from theis oil desalting unit from and enters under the In the present, researchers are paying more attention the naphtha. of K-1. K-1. The overhead products of K-1 K-1 are gas and 16 identification problems ofal., the2010; model parameters ofto soft th traycomes The naphtha also withdrawn the top of K-2. sensors for CDU (Bolf et Dam and Saraf, 2006; naphtha. The naphtha is also withdrawn from the top of K-2. tray of K-1. The overhead products of K-1 are gas and sensors for CDU (Bolf et al., 2010; Dam and Saraf, 2006; 16 identification problems of the model parameters of soft th naphtha. The naphtha is also withdrawn from the top of K-2. sensors for CDU (Bolf et al., 2010; Dam and Saraf, 2006; tray of K-1. The overhead products of K-1 are gas and 16 identification problems of the model parameters of soft naphtha. The naphtha is also withdrawn from the top of K-2. sensors for CDU (Bolf et 2007; al., 2010; Dam and Saraf, 2011). 2006; The strippers the are Macias-Hernandez et al., Napoli and Xibilia, The bottom bottom products ofis the the strippers of offrom the column column K-2 are Theproducts naphthaof also333 withdrawn the top K-2 of K-2. Macias-Hernandez et al., Napoli and Xibilia, sensors for CDU (Bolf et 2007; al., 2010; Dam and Saraf, 2011). 2006; naphtha. The bottom products of the strippers of the column are Macias-Hernandez et al., Napoli and Xibilia, naphtha. The naphtha also3 withdrawn the(DOC). top K-2 of K-2. sensors forwidespread CDU (Bolf al., 2010; Dam Saraf, 2011). 2006; The bottom products ofis(KC) the strippers offrom thecut column K-2 are Macias-Hernandez et approach al.,et 2007; 2007; Napoli andand Xibilia, 2011). gasoline, kerosene cut and diesel oil The The most for soft sensor evaluation is gasoline, kerosene cut (KC) and dieselof oil oil cut (DOC). The The bottom productscut of (KC) the 3 and strippers thecut column K-2The are The most most widespread widespread approach for soft sensor sensor evaluation is gasoline, Macias-Hernandez et approach al., 2007; for Napoli and Xibilia, 2011). kerosene diesel (DOC). The soft evaluation is The bottom products of the 3 strippers of the column K-2 are Macias-Hernandez et al., 2007; Napoli and Xibilia, 2011). gasoline, kerosene cut (KC) and diesel oil cut (DOC). The The most widespread approach for soft sensor evaluation is product of the K-2 is desired 111 (DC1), based on partial (PLS) et main product of the column K-2 is desired cut (DC1), gasoline, kerosene cutcolumn (KC) and diesel oil cutcut (DOC). The basedmost on the the partial least-squares least-squares (PLS) method (Shang et al., al., The widespread approach for softmethod sensor (Shang evaluation is main main product of the column K-2 is desired cut (DC1), based on the partial least-squares (PLS) method (Shang et al., gasoline, kerosene cut (KC) and diesel oil cut (DOC). The The most widespread approach for soft sensor evaluation is main product of the column K-2 is desired cut 1 (DC1), based on the partial least-squares (PLS) method (Shang et al., which is the mixture kerosene and diesel oil cut 2015). The of the first principles which product is the mixture ofcolumn kerosene cutis(KC) (KC) and diesel oil cut of the of K-2cut desired cut 1 (DC1), 2015). The integration of the first principles (material and based on theintegration partial least-squares (PLS) method (material (Shang et and al., main is the kerosene (KC) and diesel oil 2015). The of the principles (material main product ofcutthe2 of column K-2cut desired 1 naphtha (DC1), based on theintegration partialand least-squares (PLS) methodof (Shang et and al., which which isDesired the mixture mixture of kerosene cutis (KC) and cut diesel oil cut cut 2015). The integration of the first first principles (material and (DOC). (DC2), which is the mixture of energy balances phase equilibrium) inferential (DOC). Desired cut 2 (DC2), which is the mixture of naphtha which is the mixture of kerosene cut (KC) and diesel oil cut energy balances and phase equilibrium) of inferential 2015). The integration of the first principles (material and (DOC). Desired cut 22 of (DC2), which is the mixture of naphtha energy balances and phase equilibrium) of inferential which is the mixture kerosene cut (KC) and diesel oil cut 2015). The integration of the first principles (material and (DOC). Desired cut (DC2), which is the mixture of naphtha energy balances and phase equilibrium) of inferential and gasoline. The atmospheric residue (AR) is withdrawn modeling can be found in Chatterjee and Saraf (2003), and gasoline. The atmospheric residue (AR) is withdrawn (DOC). Desired cut 2 (DC2), which is the mixture of naphtha modeling can be found in Chatterjee and Saraf (2003), energy balances and phase equilibrium) of inferential and gasoline. The atmospheric residue (AR) is withdrawn modeling can be found in Chatterjee and Saraf (2003), (DOC). Desired cut 2 (DC2), which is the mixture of naphtha energy balances and phase equilibrium) of inferential and gasoline. The atmospheric residue (AR) is withdrawn modeling can be found in Chatterjee and Saraf (2003), from the bottom part of column K-2. The main process Mahalec and and Fujii Yamamoto fromgasoline. the bottom of column K-2. The main process The part atmospheric residue (AR) is withdrawn Mahalec and Sanchez (2012), and Fujii and Yamamoto modeling can Sanchez be found(2012), in Chatterjee andand Saraf (2003), and the bottom part of K-2. The process Mahalec and (2012), and Yamamoto and The atmospheric residue (AR) isTable withdrawn modeling Johansen can Sanchez be found in considers Chatterjee andand Saraf (2003), from fromgasoline. the of bottom part of column column K-2. Theinmain main process Mahalec and Sanchez (2012), and Fujii Fujii and Yamamoto variables the industrial CDU are shown 111 and (2014). (1996) the more general variables of the industrial CDU are shown in Table and from the bottom part of column K-2. The main process (2014). Johansen (1996) considers the more general Mahalec and Sanchez (2012), and Fujii and Yamamoto variables of the industrial CDU are shown in Table (2014). Johansen (1996) considers the more general from the bottom part of column K-2. The main process Mahalec and Sanchez (2012), and Fujii and Yamamoto variables of the industrial CDU are shown in Table 1 and and (2014). Johansen (1996)available considers the information more general may be considered informative inputs of the soft sensor framework of integrating a priori into may be considered informative inputs of the soft sensor variables of the industrial CDU are shown in Table 1 and framework of integrating available a priori information into (2014). Johansen (1996) considers the more general may be considered informative inputs of the soft sensor framework of integrating available aa priori information into variables of the industrial CDU are shown in Table 1 and (2014). Johansen (1996) considers the more general may be considered informative inputs of the soft sensor framework of integrating available priori information into the problem. model.be considered informative inputs of the soft sensor may the identification identification problem. available a priori information into model. framework of integrating model. the identification problem. framework of integrating model.be considered informative inputs of the soft sensor the identification problem. available a priori information into may model. the identification problem. However, the methods developed from the previous research research model. the identification problem. However, the methods developed from the previous However, the developed from the research However, the methods methods developed fromobstacles the previous previous research do not completely deal with practical such as small do not completely deal with practical obstacles such as small However, the methods developed from the previous research do deal with practical such as However, the methods developed fromobstacles the previous research do not not completely completely deal with practical obstacles such as small small training datasets that cover all points, low training datasets that cover all operating points, low do not completely practical obstacles such as small training datasets deal that with cover all operating operating points, low do not completely deal with practical obstacles such as small training datasets that cover all operating points, low variability ranges for key (informative) input variables, and variabilitydatasets ranges for for keycover (informative) training thatkey all operating points, low input variables, and variability ranges (informative) training datasets that cover all operating points, low variability ranges for key (informative) input variables, and an unknown feed composition (i.e. the feed distillation curves an unknown unknownranges feed composition composition (i.e. the the feed feed distillation curves variability for key (informative) input variables, and an feed (i.e. distillation curves variability ranges for key (informative) input variables, and an unknown feed composition (i.e. the feed distillation curves as as TBP). TBP). an unknown feed composition (i.e. the feed distillation curves as TBP). an unknown feed composition (i.e. the feed distillation curves as TBP). as TBP). In order as In order to overcome the abovementioned difficulties, In TBP). order to to overcome overcome the the abovementioned abovementioned difficulties, difficulties, In order to overcome the abovementioned difficulties, Torgashov et al. (2016) has proposed the system Torgashov et al. al.overcome (2016) has hasthe proposed the use use of of the thedifficulties, system of of In order to abovementioned Torgashov et (2016) proposed the use of the system In order to overcome the abovementioned difficulties, Torgashov et al. (2016) has proposed the use of the system of of parametric constraints. The system of parametric constraints parametric et constraints. The system of ofthe parametric constraints Torgashov al. (2016) The has proposed use of theconstraints system of parametric constraints. system parametric Torgashov et al. (2016) has proposed the use of the system of parametric constraints. The system of parametric constraints is from first principle is derived derived constraints. from the the preliminary preliminary calibrated firstconstraints principle parametric The system calibrated of parametric is derived from the preliminary calibrated first principle parametric constraints. The system of parametric constraints is derived from the preliminary calibrated first principle (rigorous) (Torgashov and Zmeu, In (rigorous) distillation model (Torgashov and Zmeu, 2015). In is deriveddistillation from the model preliminary calibrated first 2015). principle (rigorous) distillation model (Torgashov and Zmeu, 2015). In is derived from we the consider preliminary calibrated first principle (rigorous) distillation model (Torgashov and of Zmeu, 2015). In the current work, the extension this technique the current work, we consider the extension of this technique (rigorous) distillation model (Torgashov and Zmeu, 2015). In the current work, we consider the extension of this technique (rigorous) distillation model (Torgashov and Zmeu, 2015). In the current work, we consider the extension of this technique in the case of CDU. We introduce and solve the statements of in the case of CDU. We introduce and solve the statements of the current work, we consider the extension of this technique in case CDU. introduce and statements of the current work, weWe consider the (steady-state) extension of this technique in the the case of of CDU. We introduce and solve solve the the statements of identification problems for static and dynamic identification problems static (steady-state) and dynamic in the case of CDU. We for introduce and solve the statements of identification problems for static (steady-state) and in the case ofusing CDU. We introduce and solve the statements of identification problems for staticoptimization (steady-state) and dynamic dynamic soft sensors the constraint technique. soft sensors using the constraint optimization technique. identification problems for static (steady-state) and dynamic soft the technique. identification problems for staticoptimization (steady-state) and dynamic soft sensors sensors using using the constraint constraint optimization technique. soft2.sensors using theCRUDE constraint optimization technique. INDUSTRIAL UNIT DESCRIPTION AND soft2. sensors using theCRUDE constraint optimization technique. INDUSTRIAL UNIT DESCRIPTION AND 2. CRUDE UNIT DESCRIPTION AND 2. INDUSTRIAL INDUSTRIAL CRUDE UNIT DESCRIPTION AND PROBLEM FORMULATION PROBLEM FORMULATION 2. INDUSTRIAL CRUDE UNIT DESCRIPTION AND PROBLEM FORMULATION 2. INDUSTRIAL CRUDE UNIT DESCRIPTION AND PROBLEM FORMULATION PROBLEM FORMULATION The unit considered PROBLEM FORMULATION The crude crude distillation distillation unit considered in in this this paper paper is is Fig. Fig. 1. 1. The The sequence sequence of of industrial industrial multicomponent multicomponent distillation distillation The crude distillation unit in paper is 1. sequence The crude by distillation unit considered considered in this this paperK-1 is Fig. Fig. 1. The The sequence of of industrial industrial multicomponent multicomponent distillation distillation represented two multicomponent distillation columns: columns (CDU). represented by two multicomponent distillation columns: K-1 The crude distillation unit considered in this paper is columns (CDU). Fig. 1. The sequence of industrial multicomponent distillation represented by two multicomponent distillation columns: K-1 columns (CDU). The crude by distillation unit considered in this paperK-1 is Fig. represented two multicomponent distillation columns: 1. The sequence of industrial multicomponent distillation columns (CDU). represented by two multicomponent distillation columns: K-1 columns (CDU). represented by two multicomponent distillation columns:Control) K-1 Hosting 2405-8963 © 2018, IFAC (International Federation of Automatic Elsevier Ltd. All rights reserved. columnsby(CDU).
Peer review© under of International Federation of Automatic Control. Copyright 2018 IFAC 560 Copyright © 2018 responsibility IFAC 560 Copyright © 560 10.1016/j.ifacol.2018.09.364 Copyright © 2018 2018 IFAC IFAC 560 Copyright © 2018 IFAC 560 Copyright © 2018 IFAC 560
2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018 Andrei Torgashov et al. / IFAC PapersOnLine 51-18 (2018) 566–571
We consider the plant with several measured inputs u1,…,ui,,…,uN and one output y(). We use the measured process variables (pressure, temperature, and flow) as inputs. A priori knowledge of the distillation process allows us to select the most informative variables (Table 1) from the thermodynamic essence.
We use the determination coefficient (a number that indicates the proportion of the variance in the dependent variable that is predictable from the independent variable) R2 1 i ( yi yi ) 2
2 3 4 5 6 7 8 9
Notation TIС6 PIC2 FIC13/ FIC5 TIC3 FIC9/ FIC5 FIC4/ FIC5 PIC1 TIC2 FIC10/ FIC5
10
FIC11/ FIC5
11
TI1
12
FIC12/ FIC5
Process variable, ui Temperature of the flow in the bottom stripper of column K-2, °С Top pressure of the column K-2, MPa The ratio of steam to feed flowrate for column K-2 Top temperature of the column K-2, °С The ratio of steam to feed flowrate for bottom stripper of column K-2 The ratio of bottom pumparound to feed flowrate of column K-2 Top pressure of column K-1, MPa Feed temperature of column K-2, °С The ratio of the product of the top stripper to the feed flowrate of column K-2 The ratio of the product of the middle stripper to the feed flowrate of column K-2 Feed temperature of the column K-1, °С The ratio of product of bottom stripper to feed flowrate of column K-2
RMSE
AIC M In
i
y a )2 ,
(3)
M i 1
( yi yi )2 / M
1/2
,
(4)
y y 2N 1 M InM , m 2 i
i
i
(5)
the Schwarz Bayesian Criterion: BSC M In
y y N 1InM M InM i
m 2 i
i
(6)
as identification criteria on a given time interval, where y i is the measured value of the output variable, yi is the value obtained based on the SS, y a is the mean value of the measured output variable, and M is the number of output measurements. The model is more consistent the closer to unity the value of the coefficient of determination R2 is or the closer to zero the value of the RMSE is or then less the value of the AIC and BSC are. The goal of the paper is to develop an approach for soft sensor model identification based on the industrial data while taking into account parametric constraints. These constraints can be derived from the rigorous modeling (Torgashov et al., 2016). The introduction of the system of constraints also allows us to overcome such difficulties as small training datasets and laboratory errors. The final boiling point (FBP) of DC1 and final boiling point (FBP) of DC2 are considered soft sensor outputs. 3. MAIN RESULTS
3.1 Steady-state model identification under constraints Let u [1, u1 ( ), u2 ( ),..., uN ( )]T be a combined vector of the measured input variables and b = [b0 , b1 ,..., bN ]T be a vector of coefficients of the same dimension, the components of which reflect the contributions of the corresponding input variables. Then the equation (1) takes the following form: y uT b . We form the vector Y of dimension q from the output value y dataset as
(1)
where bj is the j-th model coefficient, j=0,1,…,N, b0 is the constant term, N is the number of input variables, is the irregular time points of output measurement 1,2,3,…, i i 1 0 , i 2 ; 1 0 , 0 is the constant term; and is the random variable restricted in the given range.
Y ( y ( 1 ), y ( 2 ),..., y ( q ))T
The dynamic SS accounts for the influence the process dynamics have on the quality of the products. The predictive model is represented as a sum of convolutions of plant inputs and a finite impulse response (FIR) hi (discrete analogues of the first degree Volterra kernels):
and the matrix U, containing the measured inputs uj, corresponding to output value y from (1): U
y( ) h 1 h (k 1)u ( k ) 2 h2 (k 1)u2 ( k ) ... (2) n 1
i
the Akaike (1969) Information Criterion:
We consider the identification problem of the soft sensor (SS) evaluation, which is best for predicting the quality of the products of crude distillation process. We obtain the model for the soft sensor in the form of a linear regression model based on the following equation: y ( ) b0 b1u1 b2 u2 ... bN u N ,
(y
the root mean squared error (RMSE)
Table 1. Main process variables No 1
567
n 1
0 1 1 k 0 k 0
... kN0 hN (k 1)uN ( k ), n 1
1 u1 ( 1 ) u2 ( 1 ) ....uN ( 1 ) 1 u1 ( 2 ) u2 ( 2 ) ....uN ( 2 )
1 u1 ( q ) u2 ( q ) ...u N ( q )
We consider the multicollinearity case, which occurs when there is an almost linear relationship between inputs. In this case, the matrix C U T U is close to singular, so it is the smallest eigenvalue min 0 , the condition number is infinitely
where h0 is a constant term.
561
2018 IFAC ADCHEM 568 Andrei Torgashov et al. / IFAC PapersOnLine 51-18 (2018) 566–571 Shenyang, Liaoning, China, July 25-27, 2018
increased, and it causes the instability of the solution. If min 0 , then it corresponds to the strict multicollinearity. In order to obtain a stable solution, it is necessary to reduce the condition number of the matrix C by, for example, adding thereto a diagonal matrix B k I (k> 0). Then we find the solution in a class of ridge parameter estimates: b ( U T U k I ) 1 U T Y .
of Lagrange multipliers corresponding to the lower and upper active constraints. Below is the algorithm for searching the minimum point b* for each iteration k. 1. Find the starting point according to (7) in order to initialize the method of the active set.
(7)
2. Verify the performance of the stop conditions. (Reaching the performance errors of conditions (9), which are constraints on the number of iterations). 3. Select a logic branch. Does it make sense to remove any constraint from the list of the set of active constraints? The condition of performance 3 in (9) is checked. If the condition is not satisfied for some of the vector elements, the constraint is excluded from the list of active constraints. 4. Calculate the search direction pk. Equation (7) solves the problem min Y Ubk U FR p FR 2 . Calculate the non-zero
The quality, obtained using models (7), depends on the number of available output measurements. The length of the training sample is often insufficient to obtain reliable results. Also, the available data contains significant measurement errors in inputs and outputs, which are unmeasured influences. Taking into account constraints on the model coefficients bj allows us to avoid these problems. When taking into account constraints on the model parameters, we solve the problem of least squares with simple constraints on the variables: min Y Ub b
min
bb
2
max
N 1 tk , the dimensional vector p FR , and the direction
(8)
of search pk AT p FR , where tk is the number of active FR
.
constraints on k iterations. 5. Calculate the step length αk. We calculate the diagonal b p matrix Ψ from FR Ψ FR bˆ FR , bˆ FR consists of b p FR FR
The solution of the problem (8) is obtained by the active set numerical method (Gill et al., 1981). The given constraints are reduced to the form:
Ab bˆ ,
where
1 0 0 1 0 0 A 1 0 0 1 0 0
the elements bˆ , which aren’t active constraints. The elements bˆ , which are opposite boundary values in (8) for constraints in the active set, are excluded from bˆ FR . The k min Ψii is an available maximum positive step from
0 0 bmin ˆ 1 , b max . b 0 0 1
bk along pk. We remember the index j of minimum positive diagonal element Ψ. If k 1 , then k 1 ; otherwise, k k . 6. Add a constraint to the list of active constraints. If k k , then j constraint bˆ FR becomes active and necessary to add to the list. 7. Approximate a recalculation. After bk 1 bk k pk is calculated, return to step 2 of the algorithm.
Constraint aTi b bˆi is active in acceptable point b if aTi b bˆi , and inactive if aTi b bˆi , a T is the i-th row of A. The sufficient i
minimum conditions for simple constraints are as follows: * max 1. b m in b * b m a x bmin FR bFR bFR T * 2. UFR ( Y Ub ) 0 min T 3. λ U min ( Y Ub* ) λimin 0 i 1, , t min 4. λ max U Tmax ( Y Ub* ) , λ max 0 , i 1, , t max
3.2 Dynamic model identification under constraints (9)
Let u [1, u1 ( ),..., u1 ( n1 1),..., uN ( ),..., uN ( nN 1)]T be the
combined vector of the measured input variables of dynamic SS (DSS) with dimensionality q 1 kN 1 n k , where nk is a
i
5. UTFR U FR is the positive definite
number
of values of
the
k-th
input variable and
h (h0 , h1 (1), ..., h1 (n1 ), ..., hN (1), ..., hN (nN ))T is the vector FIR of the
where b* is the minimum point of the solution of problem (8); subscript FR indicates that in the vector and matrices, the elements and columns with index numbers corresponding to the index numbers of b elements that have not met the boundary values of (8) are used; the subscript min and max indicate that in the matrix, only the columns with index numbers corresponding to the index numbers of b elements that take the appropriate minimum or maximum boundary value are used. tmin and tmax refer to the numbers of active upper and lower limits, respectively; min and max are vectors
same dimension, the components of which reflect the contributions of the respective input variables of DSS. Then the equation (2) takes the following form: y uT h . We write the vector Y of dimension q from the output value y as Y ( y ( 1 ), y ( 2 ),..., y ( q ))T
and matrix U, containing the measured corresponding to output value y from (2) as
562
inputs
uj,
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1 u1 ( 1 ) u1 ( 1 n1 1) uN ( 1 ) 1 u ( ) u ( n 1) u ( ) 1 2 1 2 1 N 2 U 1 u1 ( q ) u1 ( q n1 1) uN ( q ) uN ( 1 ) uN ( 2 ) uN ( q )
.
s max j
uN ( 1 n N 1)) uN ( 2 n N 1)) uN ( q n N 1))
Wee write the m matrix equationn as Y U h . We introducce the error function: E Y Y Y Uh , wh here Y is the aactual measurrement of outtput, and miniimize thee following obbjective function: 2 Ψ E ( Y Uh h)2 .
T
s min j
s max j
s max ((1) 0 j s max (fl( a 1n j )) 0 j max max s j (fl(a1n j ) 1) b j a2 1 a2 1/ (n j fl(a1n j )) max max s j (fl(a1n j ) 2) b j a2 1 a2 2 / (n j fl(a1n j )) max s n ) b max ( j j j
,
(11)
s s1 (1), ..., s1 ( n1 ), ..., sN (1)), ..., sN ( nN )
.
min s min j (1) a3b j min s j (fl(a1 n j )) a3b min j min ) 1) b min 1 ( a3 1) 1 1/ (n j fl(a1 n j )) j s j (fl(a1 n j min 2) b min 1 ( a3 1) 1 2 / (n j fl(a1 n j )) s j (fl(a1 n j ) j ) b min s min ( n j j jj
The constraints on transient response comp ponents are w written as
wh here
s max (1) a3b max j j s max (fl(a1 n j )) a3b max j j max ) 1) b max 1 ( a3 1) 1 1/ (n j fl(a1 n j )) j s j (fl(a1 n j max ) 2) b max 1 ( a3 1) 1 2 / (n j fl(a1 n j )) s j (fl(a1 n j j s max ( n ) b max j j jj
For b j 0 :
(10)
smiin s smax ,
569
, smin s1min , ..., s min , N T
T
x s max s1max , ..., s max N . We obtain the constraaint (11) baseed on
thee value of bj ( j 0) of thee steady-state soft sensor m model That y ( ) b0 b1u1 ( ) ... b j u j ( ) ... bN u N ( ) . mo odel is deriveed prior basedd on the steaady-state induustrial datta of CDU. W We select the parametric co onstraints for each FIR R coefficient uusing the folloowing values of o a1, a2 and a 3. a1 is a fraction of the FIIR length n j from whichh the ax con nvergence of elements of s min and s ma to bmin , bmjmax is j j j
wherre fl means ro ounding to thee nearest integ ger number inn the direcction of -∞.
started.
0 ( b max 0 ) from a2 is i a fraction off b min m which the sm mooth j j
An example e for deetermining thee constraint sy ystem in term ms of
inccreasing (or ddecreasing) of o elements of o s min (or s max ) j j
We select s the paraameters a1=0.55, a2=0.8, and d a3=1.2.
max s min and s j off s j under b j 0 is show wn on the Figg. 2. j
beg gins.
0 ( b min 0 ) from whichh the a3 is a fractionn of bmax j j n mooth decreasiing (or increaasing) of elem ments of s min (or sm j m ) begins. s max j
Forr b j 0 , the ffollowing equuations are vallid for constraaints
n s min j
in s mi j (1) 0 s min (fl fl( a 1n j )) 0 j min b min a2 1 a2 1/ (n j fl(a1n j )) j s j (fl(a1n j ) 1) min b min a2 1 a2 2 / (n j fl(a1n j )) s j (fl(a1n j ) 2) j min n s ( ) b min j j j
Fig. 2. The assign nment of consstraints for eaach step respoonse namic soft sennsor model. function of the dyn
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4. Calculate the search direction pk. Use an equation like (7) 2 to solve the problem min Y Uhk UZk pZ . Calculate the
The coefficients of the finite step response (FSR) s j are related to the components of the FIR hj by the relations s j k i 1 h j (i ), k
j 1, 2 , , N ,
non-zero 1 kN1 nk tk
(12)
k 1, , n j .
matrix Ψ is calculated. The calculated k min Ψii is a minimum non-negative available step
diagonal
(13)
where
1 0 0 1 1 0 0 0 1 A 1 0 0 1 1 0 0 0 1
,
h1 (1) h1 ( n1 ) , h hN (1) h (n ) N N
from hk along pk, where i is the index number of the element, which is not an active constraint in (13) or an element of opposite boundary values (11) for constraints in the active set. Also, p k consists of elements of pk without the first element. We memorize the index j of the minimum positive diagonal element of Ψii. If k 1 , then k 1 ; otherwise, k k . 6. Add the constraint to the list of active constraints. If k k , then j constraint sˆ FR becomes active and Zk is recalculated. 7. Calculate the hk 1 hk k pk and return to step 2.
s min sˆ max . s
The sufficient minimum conditions are as follows for the current optimization problem (parametric identification of the dynamic model): 1. Ah sˆ , A ACT h sˆ ACT 2. Z T * U T ( Y Uh* ) 0 3. λ
A ACT A TACT
1
A ACT U T ( Y Uh * ) , λ i 0 ,
dimensional vector pZ and the
direction of search pk Zk pZ , where tk is the number of active constraints on the k iteration. 5. Calculate the step length k . From A h Ψp k sˆ , the
The constraints (11) can be written as
Ah sˆ ,
4. INDUSTRIAL CASE STUDY The CDU (Fig.1) is considered a case study for evaluating static (steady-state) and dynamic soft sensors based on the proposed identification algorithm under parametric constraints. The final boiling point temperature of the DC1 when we obtained a model on the training sample is considered. The number of output observations in the training sample is 70. The length of the test dataset is equal to 30 observations. Fig. 3 and Table 2 show the results of the performance of the static models on the test sample.
(14) i 1, , t
4. ZT U T U Z is a positive definite, where h * is the minimum point of the solution of problem (10) with constraints (13); subscript ACT indicates that in the vector matrix, we only use the rows with index numbers corresponding to the element index numbers of active constraint in (13); t is the number of active constraints; is the vector of Lagrange multipliers corresponding to the active constraints; Z is the matrix of the columns that are the basis of the feasible direction of the search for equality constraints (13). The matrix Z is formed by the variable-reduction technique (Gill et al., 1981).
Table 2. Results of the performance of the static models (test dataset) Without use constraints With use constraints
The search algorithm of minimum point h * for iteration k is as follows:
R2
RMSE
AIC
BSC
0,64
4,47
232,7
255,4
0,84
2,99
175,5
198,2
360 355
1. In order to start the method of the active set, it is necessary to determine the starting point (using a solution of the problem (10) without any constraints, with subsequent correction of coefficients hi that do not fall under the constraints (13)). 2. Verify the performance of the stop conditions (reaching the performance errors of conditions (14) and constraints on the number of iterations). 3. Select a logic branch. Does it make sense to remove any constraint from the set of active constraints list? The condition of performance of a condition 3 in (14) is checked. If the condition is not satisfied for some of the vector elements, constraint is excluded from the list of active constraints, and it is necessary to recalculate Zk.
350
FBP,°С
345 340 335 330 325 320 315
lab data w/t constrains with constrains 29.06.2017
06.07.2017 Time period
Fig. 3. Comparative study of static soft sensor model’s performance. 564
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The improvement of the prediction quality by the criterion RMSE of the identified static model obtained with the constraints on the parameters of SS is 100(4,47 – 2,99) / 4,47 33% compared to the case without constraints. In order to investigate the influence of constraints on the performance of the dynamic soft sensor model we obtained, we compared solutions of the optimization problem (10) without constraints and optimization problem (10) with constraints (13). We use the same values of the ridge coefficients. Fig. 4 and Table 3 show the results of the performance of the dynamic models on the test sample for the final boiling point temperature of DC2 when a model is obtained based on the training sample. The number of measurements in the training sample is 280. The size of the test dataset is 120 for the case with the dynamic model of the soft sensor.
that the reduction of the root mean square error on the test sample can be more than 33%. 7. ACKNOWLEDGEMENTS This work was partially supported by the Russian Foundation for Basic Research (Project No. 17-07-00235 A) and by the Ministry of Education and Science of the Russian Federation, through Government Contract No. 02.G25.31.0173. REFERENCES Akaike, H. (1969). Fitting Autoregressive Models for Prediction. Annals of the Institute of Statistical Mathematics, 21, pp. 243-247. Bolf, N., Galinec, G. and Baksa, T. (2010). Development of soft sensor for diesel fuel quality estimation. Chemical Engineering & Technology, 33, pp.405-413. Chatterjee, T., and Saraf, D. N. (2004). On-line estimation of product properties for crude distillation units. Journal of Process Control, 14, pp. 61–77. Dam M. and Saraf D.N. (2006) Design of neural networks using genetic algorithm for online property estimation of crude fractionator products. Computers & Chemical Engineering, 30 (4) pp. 722–729. Fujii K. and Yamamoto T. (2014). Development of a nonlinear soft sensor using a GMDH network for a refinery crude distillation tower, Electrical Engineering in Japan, Vol. 188, No. 3. pp. 31-38. Gill, P.E., W. Murray and Wright M.H. (1981). Practical Optimization. New York, USA: Academic Press. Johansen, T. (1996). Identification of non-linear systems using empirical data and prior knowledge-an optimization approach. Automatica, Vol. 32, No. 3, pp. 337–356. Macias-Hernandez J. J., P. Angelov and Zhou, X. (2007). Soft sensor for predicting crude oil distillation side streams using Takagi-Sugeno evolving fuzzy models. Proc. 2007 IEEE Intern. Conf. on Systems, Man, and Cybernetics, 7-10 Oct., 2007, Montreal, Canada, ISBN 1-4244-0991-8/07, pp.3305-3310 Mahalec, V. and Sanchez, Y. (2012). Inferential monitoring and optimization of crude separation units via hybrid models. Computers & Chemical Engineering, 45, pp. 1526. Napoli G. and Xibilia M.G. (2011). Soft sensor design for a topping process in the case of small data sets. Computers and Chemical Engineering, Vol. 35. No. 11, pp. 2447– 2456. Shang C., Yang F., Gao X. and Huang D. (2015) A comparative study on improved DPLS soft sensor models applied to a crude distillation unit. IFACPapersOnLine, Vol. 48. Issue 8, pp. 234–239. Torgashov A. and Zmeu K. (2015). Nonparametric soft sensor evaluation for industrial distillation plant. Computer Aided Chemical Engineering. Vol. 37, pp. 1487-1492. Torgashov A., Skogestad S. and Kozlov A. (2016). Comparative study of multicomponent distillation static estimators based on industrial and rigorous model datasets. IFAC-PapersOnLine. Vol. 49. Issue 7, pp. 1187-1192.
Table 3. Results of the performance of the dynamic models (test dataset)
R2
RMSE AIC BSC Without use 0,40 3,79 6418,4 6482 constraints With use 0,74 2,52 4453,2 4516,8 constraints The improvement of the prediction quality by the criterion RMSE of the dynamic model obtained using the constraints on the parameters of SS is 100(3,79 – 2,52)/ 3,79 33,5 % compared to the identification without constraints. in-line analyser w/t constrains with constrains
190
FBP,°С
185
180
175
170 04.05.17
Time period
571
05.05.17
Fig. 4. Comparative study of dynamic soft sensors. 6. CONCLUSIONS The introduction of a system of constraints into the identification algorithm improves the quality of derived models, especially on the test dataset. The reason is related to the integration of available a priori knowledge via the constraint identification problem statement and solution.The use of the active set method, taking into account constraints on the model coefficients, can improve the quality of the evaluated SS models, in particular in the case of small training datasets containing lab errors. The test of the proposed approach to solving the problem of obtaining a soft sensor model for industrial crude oil distillation unit showed 565
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Evaluation of Steady-State and Dynamic Soft Sensors for Industrial Crude Evaluation of Steady-State and Dynamic Soft Sensors for Industrial Crude Evaluation of Steady-State and Dynamic Soft Sensors for Industrial Crude Evaluation of Steady-State and Dynamic Soft Sensors for Industrial Crude Distillation Unit under Parametric Constraints Evaluation of Distillation Steady-State and Dynamic Soft Sensors for Industrial Crude Unit under Parametric Constraints Distillation Unit under Parametric Constraints Distillation Unit under Parametric Constraints Distillation Unit* under Parametric Constraints *,** *** *,** *** Andrei *,**,, Evgeny *** Anton Goncharov Goncharov*,** Evgeny Zhuravlev Zhuravlev*** Andrei Torgashov Torgashov***,, Anton
Andrei *,**, Evgeny Zhuravlev*** Anton Goncharov Goncharov Andrei Torgashov Torgashov*,, Anton *,**, Evgeny Zhuravlev*** Andrei Torgashov**, Anton Goncharov *,**, Evgeny Zhuravlev*** , Anton Goncharov , Evgeny Zhuravlev Andreiof Torgashov *Institute Automation and Control Process FEB RAS, 555 Radio and *Institute of Automation and Control Process FEB RAS, Radio St., St., and *Institute of Automation and Control Process FEB RAS, St., and *Institute of Automation and Control Process FEB RAS, 5 Radio Radio St.,Island and Far-Eastern Federal University, Bld. 12(E), Campus FEFU, Russky Far-Eastern Federal University, Bld. 12(E), Campus FEFU, Russky Island *Institute of Automation and Control Process FEB RAS, 5 Radio St., and Far-Eastern Federal University, Bld. 12(E), Campus FEFU, Russky Island *Institute of Automation and Control Process FEB RAS, 5 Radio St., and Far-Eastern Federal University, Bld. 12(E), Campus FEFU, Russky Island Vladivostok, Russia (Tel: +7-423-231-02-02; e-mail: [email protected]) Vladivostok, Russia (Tel: +7-423-231-02-02; e-mail: [email protected]) Far-Eastern Federal University, Bld. 12(E), Campus FEFU, Russky Island Vladivostok, (Tel: +7-423-231-02-02; Far-EasternRussia Federal University, Bld. 12(E), e-mail: [email protected]) FEFU, Russky Island Vladivostok, Russia (Tel: +7-423-231-02-02; e-mail: [email protected]) Vladivostok,**JSC Russia“Honeywell”, (Tel: +7-423-231-02-02; [email protected]) 777 Kievskaya St., Moscow, Russia. Kievskaya e-mail: St., Moscow, Russia. Vladivostok,**JSC Russia“Honeywell”, (Tel: +7-423-231-02-02; [email protected]) **JSC “Honeywell”, St., Russia. **JSC “Honeywell”, 7 Kievskaya Kievskaya e-mail: St., Moscow, Moscow, Russia. **JSC “Honeywell”, 7 Kievskaya St., Moscow, Russia. ***JSC “Gazprom neftekhim Salavat”, 30 Molodogvardeitsev st., Salavat, Russia. ***JSC “Gazprom neftekhim Salavat”, 30 Molodogvardeitsev st., **JSC “Honeywell”, 7 Kievskaya St., Moscow, ***JSC “Gazprom neftekhim Salavat”, 30 MolodogvardeitsevRussia. st., Salavat, Salavat, Russia. Russia. ***JSC “Gazprom neftekhim Salavat”, 30 Molodogvardeitsev st., Salavat, Russia. ***JSC “Gazprom neftekhim Salavat”, 30 Molodogvardeitsev st., Salavat, Russia. Abstract: The parametric identification problem for industrial crude distillation unit (CDU) is Abstract: The The parametric parametric identification identification problem problem for for industrial industrial crude crude distillation distillation unit unit (CDU) (CDU) is considered. considered. Abstract: is Abstract: The parametric identification problem for industrial crude distillation unit (CDU) is considered. considered. We take the a priori knowledge of the process into account by using a system of constraints for We take take The the parametric a priori priori knowledge knowledge of the the process into account account bydistillation using aa system system of constraints constraints for Abstract: identification problem for industrial crude unit (CDU) is considered. We the a of process into by using of for Abstract: The parametric identification problem for industrial crude distillation unit (CDU) is considered. We take the a priori knowledge of the process into account by using a system of constraints for parameters of soft sensors models. The identification problem is transformed into a constrained parameters ofa soft soft sensors models. The process identification problem isusing transformed into a constrained constrained We take theof priorisensors knowledge of the into account by is a system into of constraints for parameters models. The identification problem transformed a We take the a priori knowledge of the process into account by using a system of constraints for parameters of soft sensors models. The identification problem is transformed into a constrained optimization which we solved using the active set method. The static and dynamic sensors optimization problem, which we solved using the active set method. The static and dynamic soft sensors parameters ofproblem, soft sensors models. The identification problem is transformed into a soft constrained optimization problem, which we solved using the active set method. The static and dynamic soft sensors parameters of soft sensors models. The identification problem is transformed into a constrained optimization problem, which we solved using the active set method. The static and dynamic soft sensors are evaluated for industrial CDU located at JSC “Gazprom Salavat” It was found that are evaluatedproblem, for industrial CDU located at JSC “Gazprom neftekhim Salavat” refinery. It wassoft found that optimization which we solved using the active setneftekhim method. The staticrefinery. and dynamic sensors are for CDU located at JSC “Gazprom Salavat” refinery. It found that optimization problem, which wewhen solved using active setneftekhim method. The static and dynamic soft sensors are evaluated evaluated for industrial industrial CDU located atused JSCthe “Gazprom neftekhim Salavat” refinery. It was was found that the model performed better we the proposed constrained optimization approach for the model performed better when we used the proposed constrained optimization approach for are evaluated for industrial CDU located at JSC “Gazprom neftekhim Salavat” refinery. It was found that the model performed better when we used the proposed constrained optimization approach for are evaluated for industrial CDU located at JSC “Gazprom neftekhim Salavat” refinery. It was found that the model performed better when we used the proposed constrained optimization approach for identification instead of robust regression methods. identification instead of robust regression methods. the model performed better when we used the proposed constrained optimization approach for identification instead of robust regression methods. the model performed better when we used the proposed constrained optimization approach for identification instead of robust regression methods. identification instead of robust regression methods. Keywords: Soft sensing, identification, constrained parameters, optimization, distillation columns. Keywords: Soft sensing, identification, constrained parameters, optimization, distillation columns. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. identification instead of robust regression methods. Keywords: Keywords: Soft Soft sensing, sensing, identification, identification, constrained constrained parameters, parameters, optimization, optimization, distillation distillation columns. columns. Keywords: Soft sensing, identification, constrained parameters, optimization, distillation columns. Keywords: Soft sensing, identification, constrained parameters, optimization, distillation columns. and K-2 (Fig. The is located JSC and K-2 (Fig. 1). 1). The plant plant is located at at JSC “Gazprom “Gazprom and K-2 1). plant is at “Gazprom 1. INTRODUCTION 1. INTRODUCTION and K-2 (Fig. (Fig. 1). The The plant is located located at JSC JSC “Gazprom 1. neftekhim Salavat” refinery (Salavat, Russia). The feed flow neftekhim Salavat” refinery (Salavat, Russia). The feed flow and K-2 (Fig. 1). The plant is located at JSC “Gazprom 1. INTRODUCTION INTRODUCTION neftekhim Salavat” refinery (Salavat, Russia). The feed flow flow and K-2 (Fig. 1). The plant is located at JSC “Gazprom neftekhim Salavat” refinery (Salavat, Russia). The feed 1. INTRODUCTION (crude) comes comes from refinery the oil oil desalting desalting unit and enters enters underflow the In the present, researchers are paying more attention to the 1. INTRODUCTION In the present, researchers are paying more attention to the neftekhim Salavat” (Salavat, Russia). The feed (crude) from the unit and under the In the present, researchers are paying more attention to the th neftekhim Salavat” refinery (Salavat, Russia). The feed flow comes from theoverhead oil desalting unit and enters under the In the present,problems researchers are paying more attentionofto soft the (crude) th th of K-1. The products of K-1 are gas and 16 th tray identification of the model parameters tray of K-1. The overhead products of K-1 are gas and identification problems of the model parameters of soft (crude) comes from the oil desalting unit and enters under the In the present, researchers are paying more attention to the 16 th tray of The overhead products of are gas and identification problems of the model parameters of soft (crude) from theis oil desalting unit from and enters under the In the present, researchers are paying more attention the naphtha. of K-1. K-1. The overhead products of K-1 K-1 are gas and 16 identification problems ofal., the2010; model parameters ofto soft th traycomes The naphtha also withdrawn the top of K-2. sensors for CDU (Bolf et Dam and Saraf, 2006; naphtha. The naphtha is also withdrawn from the top of K-2. tray of K-1. The overhead products of K-1 are gas and sensors for CDU (Bolf et al., 2010; Dam and Saraf, 2006; 16 identification problems of the model parameters of soft th naphtha. The naphtha is also withdrawn from the top of K-2. sensors for CDU (Bolf et al., 2010; Dam and Saraf, 2006; tray of K-1. The overhead products of K-1 are gas and 16 identification problems of the model parameters of soft naphtha. The naphtha is also withdrawn from the top of K-2. sensors for CDU (Bolf et 2007; al., 2010; Dam and Saraf, 2011). 2006; The strippers the are Macias-Hernandez et al., Napoli and Xibilia, The bottom bottom products ofis the the strippers of offrom the column column K-2 are Theproducts naphthaof also333 withdrawn the top K-2 of K-2. Macias-Hernandez et al., Napoli and Xibilia, sensors for CDU (Bolf et 2007; al., 2010; Dam and Saraf, 2011). 2006; naphtha. The bottom products of the strippers of the column are Macias-Hernandez et al., Napoli and Xibilia, naphtha. The naphtha also3 withdrawn the(DOC). top K-2 of K-2. sensors forwidespread CDU (Bolf al., 2010; Dam Saraf, 2011). 2006; The bottom products ofis(KC) the strippers offrom thecut column K-2 are Macias-Hernandez et approach al.,et 2007; 2007; Napoli andand Xibilia, 2011). gasoline, kerosene cut and diesel oil The The most for soft sensor evaluation is gasoline, kerosene cut (KC) and dieselof oil oil cut (DOC). The The bottom productscut of (KC) the 3 and strippers thecut column K-2The are The most most widespread widespread approach for soft sensor sensor evaluation is gasoline, Macias-Hernandez et approach al., 2007; for Napoli and Xibilia, 2011). kerosene diesel (DOC). The soft evaluation is The bottom products of the 3 strippers of the column K-2 are Macias-Hernandez et al., 2007; Napoli and Xibilia, 2011). gasoline, kerosene cut (KC) and diesel oil cut (DOC). The The most widespread approach for soft sensor evaluation is product of the K-2 is desired 111 (DC1), based on partial (PLS) et main product of the column K-2 is desired cut (DC1), gasoline, kerosene cutcolumn (KC) and diesel oil cutcut (DOC). The basedmost on the the partial least-squares least-squares (PLS) method (Shang et al., al., The widespread approach for softmethod sensor (Shang evaluation is main main product of the column K-2 is desired cut (DC1), based on the partial least-squares (PLS) method (Shang et al., gasoline, kerosene cut (KC) and diesel oil cut (DOC). The The most widespread approach for soft sensor evaluation is main product of the column K-2 is desired cut 1 (DC1), based on the partial least-squares (PLS) method (Shang et al., which is the mixture kerosene and diesel oil cut 2015). The of the first principles which product is the mixture ofcolumn kerosene cutis(KC) (KC) and diesel oil cut of the of K-2cut desired cut 1 (DC1), 2015). The integration of the first principles (material and based on theintegration partial least-squares (PLS) method (material (Shang et and al., main is the kerosene (KC) and diesel oil 2015). The of the principles (material main product ofcutthe2 of column K-2cut desired 1 naphtha (DC1), based on theintegration partialand least-squares (PLS) methodof (Shang et and al., which which isDesired the mixture mixture of kerosene cutis (KC) and cut diesel oil cut cut 2015). The integration of the first first principles (material and (DOC). (DC2), which is the mixture of energy balances phase equilibrium) inferential (DOC). Desired cut 2 (DC2), which is the mixture of naphtha which is the mixture of kerosene cut (KC) and diesel oil cut energy balances and phase equilibrium) of inferential 2015). The integration of the first principles (material and (DOC). Desired cut 22 of (DC2), which is the mixture of naphtha energy balances and phase equilibrium) of inferential which is the mixture kerosene cut (KC) and diesel oil cut 2015). The integration of the first principles (material and (DOC). Desired cut (DC2), which is the mixture of naphtha energy balances and phase equilibrium) of inferential and gasoline. The atmospheric residue (AR) is withdrawn modeling can be found in Chatterjee and Saraf (2003), and gasoline. The atmospheric residue (AR) is withdrawn (DOC). Desired cut 2 (DC2), which is the mixture of naphtha modeling can be found in Chatterjee and Saraf (2003), energy balances and phase equilibrium) of inferential and gasoline. The atmospheric residue (AR) is withdrawn modeling can be found in Chatterjee and Saraf (2003), (DOC). Desired cut 2 (DC2), which is the mixture of naphtha energy balances and phase equilibrium) of inferential and gasoline. The atmospheric residue (AR) is withdrawn modeling can be found in Chatterjee and Saraf (2003), from the bottom part of column K-2. The main process Mahalec and and Fujii Yamamoto fromgasoline. the bottom of column K-2. The main process The part atmospheric residue (AR) is withdrawn Mahalec and Sanchez (2012), and Fujii and Yamamoto modeling can Sanchez be found(2012), in Chatterjee andand Saraf (2003), and the bottom part of K-2. The process Mahalec and (2012), and Yamamoto and The atmospheric residue (AR) isTable withdrawn modeling Johansen can Sanchez be found in considers Chatterjee andand Saraf (2003), from fromgasoline. the of bottom part of column column K-2. Theinmain main process Mahalec and Sanchez (2012), and Fujii Fujii and Yamamoto variables the industrial CDU are shown 111 and (2014). (1996) the more general variables of the industrial CDU are shown in Table and from the bottom part of column K-2. The main process (2014). Johansen (1996) considers the more general Mahalec and Sanchez (2012), and Fujii and Yamamoto variables of the industrial CDU are shown in Table (2014). Johansen (1996) considers the more general from the bottom part of column K-2. The main process Mahalec and Sanchez (2012), and Fujii and Yamamoto variables of the industrial CDU are shown in Table 1 and and (2014). Johansen (1996)available considers the information more general may be considered informative inputs of the soft sensor framework of integrating a priori into may be considered informative inputs of the soft sensor variables of the industrial CDU are shown in Table 1 and framework of integrating available a priori information into (2014). Johansen (1996) considers the more general may be considered informative inputs of the soft sensor framework of integrating available aa priori information into variables of the industrial CDU are shown in Table 1 and (2014). Johansen (1996) considers the more general may be considered informative inputs of the soft sensor framework of integrating available priori information into the problem. model.be considered informative inputs of the soft sensor may the identification identification problem. available a priori information into model. framework of integrating model. the identification problem. framework of integrating model.be considered informative inputs of the soft sensor the identification problem. available a priori information into may model. the identification problem. However, the methods developed from the previous research research model. the identification problem. However, the methods developed from the previous However, the developed from the research However, the methods methods developed fromobstacles the previous previous research do not completely deal with practical such as small do not completely deal with practical obstacles such as small However, the methods developed from the previous research do deal with practical such as However, the methods developed fromobstacles the previous research do not not completely completely deal with practical obstacles such as small small training datasets that cover all points, low training datasets that cover all operating points, low do not completely practical obstacles such as small training datasets deal that with cover all operating operating points, low do not completely deal with practical obstacles such as small training datasets that cover all operating points, low variability ranges for key (informative) input variables, and variabilitydatasets ranges for for keycover (informative) training thatkey all operating points, low input variables, and variability ranges (informative) training datasets that cover all operating points, low variability ranges for key (informative) input variables, and an unknown feed composition (i.e. the feed distillation curves an unknown unknownranges feed composition composition (i.e. the the feed feed distillation curves variability for key (informative) input variables, and an feed (i.e. distillation curves variability ranges for key (informative) input variables, and an unknown feed composition (i.e. the feed distillation curves as as TBP). TBP). an unknown feed composition (i.e. the feed distillation curves as TBP). an unknown feed composition (i.e. the feed distillation curves as TBP). as TBP). In order as In order to overcome the abovementioned difficulties, In TBP). order to to overcome overcome the the abovementioned abovementioned difficulties, difficulties, In order to overcome the abovementioned difficulties, Torgashov et al. (2016) has proposed the system Torgashov et al. al.overcome (2016) has hasthe proposed the use use of of the thedifficulties, system of of In order to abovementioned Torgashov et (2016) proposed the use of the system In order to overcome the abovementioned difficulties, Torgashov et al. (2016) has proposed the use of the system of of parametric constraints. The system of parametric constraints parametric et constraints. The system of ofthe parametric constraints Torgashov al. (2016) The has proposed use of theconstraints system of parametric constraints. system parametric Torgashov et al. (2016) has proposed the use of the system of parametric constraints. The system of parametric constraints is from first principle is derived derived constraints. from the the preliminary preliminary calibrated firstconstraints principle parametric The system calibrated of parametric is derived from the preliminary calibrated first principle parametric constraints. The system of parametric constraints is derived from the preliminary calibrated first principle (rigorous) (Torgashov and Zmeu, In (rigorous) distillation model (Torgashov and Zmeu, 2015). In is deriveddistillation from the model preliminary calibrated first 2015). principle (rigorous) distillation model (Torgashov and Zmeu, 2015). In is derived from we the consider preliminary calibrated first principle (rigorous) distillation model (Torgashov and of Zmeu, 2015). In the current work, the extension this technique the current work, we consider the extension of this technique (rigorous) distillation model (Torgashov and Zmeu, 2015). In the current work, we consider the extension of this technique (rigorous) distillation model (Torgashov and Zmeu, 2015). In the current work, we consider the extension of this technique in the case of CDU. We introduce and solve the statements of in the case of CDU. We introduce and solve the statements of the current work, we consider the extension of this technique in case CDU. introduce and statements of the current work, weWe consider the (steady-state) extension of this technique in the the case of of CDU. We introduce and solve solve the the statements of identification problems for static and dynamic identification problems static (steady-state) and dynamic in the case of CDU. We for introduce and solve the statements of identification problems for static (steady-state) and in the case ofusing CDU. We introduce and solve the statements of identification problems for staticoptimization (steady-state) and dynamic dynamic soft sensors the constraint technique. soft sensors using the constraint optimization technique. identification problems for static (steady-state) and dynamic soft the technique. identification problems for staticoptimization (steady-state) and dynamic soft sensors sensors using using the constraint constraint optimization technique. soft2.sensors using theCRUDE constraint optimization technique. INDUSTRIAL UNIT DESCRIPTION AND soft2. sensors using theCRUDE constraint optimization technique. INDUSTRIAL UNIT DESCRIPTION AND 2. CRUDE UNIT DESCRIPTION AND 2. INDUSTRIAL INDUSTRIAL CRUDE UNIT DESCRIPTION AND PROBLEM FORMULATION PROBLEM FORMULATION 2. INDUSTRIAL CRUDE UNIT DESCRIPTION AND PROBLEM FORMULATION 2. INDUSTRIAL CRUDE UNIT DESCRIPTION AND PROBLEM FORMULATION PROBLEM FORMULATION The unit considered PROBLEM FORMULATION The crude crude distillation distillation unit considered in in this this paper paper is is Fig. Fig. 1. 1. The The sequence sequence of of industrial industrial multicomponent multicomponent distillation distillation The crude distillation unit in paper is 1. sequence The crude by distillation unit considered considered in this this paperK-1 is Fig. Fig. 1. The The sequence of of industrial industrial multicomponent multicomponent distillation distillation represented two multicomponent distillation columns: columns (CDU). represented by two multicomponent distillation columns: K-1 The crude distillation unit considered in this paper is columns (CDU). Fig. 1. The sequence of industrial multicomponent distillation represented by two multicomponent distillation columns: K-1 columns (CDU). The crude by distillation unit considered in this paperK-1 is Fig. represented two multicomponent distillation columns: 1. The sequence of industrial multicomponent distillation columns (CDU). represented by two multicomponent distillation columns: K-1 columns (CDU). represented by two multicomponent distillation columns:Control) K-1 Hosting 2405-8963 © 2018, IFAC (International Federation of Automatic Elsevier Ltd. All rights reserved. columnsby(CDU).
Peer review© under of International Federation of Automatic Control. Copyright 2018 IFAC 560 Copyright © 2018 responsibility IFAC 560 Copyright © 560 10.1016/j.ifacol.2018.09.364 Copyright © 2018 2018 IFAC IFAC 560 Copyright © 2018 IFAC 560 Copyright © 2018 IFAC 560
2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018 Andrei Torgashov et al. / IFAC PapersOnLine 51-18 (2018) 566–571
We consider the plant with several measured inputs u1,…,ui,,…,uN and one output y(). We use the measured process variables (pressure, temperature, and flow) as inputs. A priori knowledge of the distillation process allows us to select the most informative variables (Table 1) from the thermodynamic essence.
We use the determination coefficient (a number that indicates the proportion of the variance in the dependent variable that is predictable from the independent variable) R2 1 i ( yi yi ) 2
2 3 4 5 6 7 8 9
Notation TIС6 PIC2 FIC13/ FIC5 TIC3 FIC9/ FIC5 FIC4/ FIC5 PIC1 TIC2 FIC10/ FIC5
10
FIC11/ FIC5
11
TI1
12
FIC12/ FIC5
Process variable, ui Temperature of the flow in the bottom stripper of column K-2, °С Top pressure of the column K-2, MPa The ratio of steam to feed flowrate for column K-2 Top temperature of the column K-2, °С The ratio of steam to feed flowrate for bottom stripper of column K-2 The ratio of bottom pumparound to feed flowrate of column K-2 Top pressure of column K-1, MPa Feed temperature of column K-2, °С The ratio of the product of the top stripper to the feed flowrate of column K-2 The ratio of the product of the middle stripper to the feed flowrate of column K-2 Feed temperature of the column K-1, °С The ratio of product of bottom stripper to feed flowrate of column K-2
RMSE
AIC M In
i
y a )2 ,
(3)
M i 1
( yi yi )2 / M
1/2
,
(4)
y y 2N 1 M InM , m 2 i
i
i
(5)
the Schwarz Bayesian Criterion: BSC M In
y y N 1InM M InM i
m 2 i
i
(6)
as identification criteria on a given time interval, where y i is the measured value of the output variable, yi is the value obtained based on the SS, y a is the mean value of the measured output variable, and M is the number of output measurements. The model is more consistent the closer to unity the value of the coefficient of determination R2 is or the closer to zero the value of the RMSE is or then less the value of the AIC and BSC are. The goal of the paper is to develop an approach for soft sensor model identification based on the industrial data while taking into account parametric constraints. These constraints can be derived from the rigorous modeling (Torgashov et al., 2016). The introduction of the system of constraints also allows us to overcome such difficulties as small training datasets and laboratory errors. The final boiling point (FBP) of DC1 and final boiling point (FBP) of DC2 are considered soft sensor outputs. 3. MAIN RESULTS
3.1 Steady-state model identification under constraints Let u [1, u1 ( ), u2 ( ),..., uN ( )]T be a combined vector of the measured input variables and b = [b0 , b1 ,..., bN ]T be a vector of coefficients of the same dimension, the components of which reflect the contributions of the corresponding input variables. Then the equation (1) takes the following form: y uT b . We form the vector Y of dimension q from the output value y dataset as
(1)
where bj is the j-th model coefficient, j=0,1,…,N, b0 is the constant term, N is the number of input variables, is the irregular time points of output measurement 1,2,3,…, i i 1 0 , i 2 ; 1 0 , 0 is the constant term; and is the random variable restricted in the given range.
Y ( y ( 1 ), y ( 2 ),..., y ( q ))T
The dynamic SS accounts for the influence the process dynamics have on the quality of the products. The predictive model is represented as a sum of convolutions of plant inputs and a finite impulse response (FIR) hi (discrete analogues of the first degree Volterra kernels):
and the matrix U, containing the measured inputs uj, corresponding to output value y from (1): U
y( ) h 1 h (k 1)u ( k ) 2 h2 (k 1)u2 ( k ) ... (2) n 1
i
the Akaike (1969) Information Criterion:
We consider the identification problem of the soft sensor (SS) evaluation, which is best for predicting the quality of the products of crude distillation process. We obtain the model for the soft sensor in the form of a linear regression model based on the following equation: y ( ) b0 b1u1 b2 u2 ... bN u N ,
(y
the root mean squared error (RMSE)
Table 1. Main process variables No 1
567
n 1
0 1 1 k 0 k 0
... kN0 hN (k 1)uN ( k ), n 1
1 u1 ( 1 ) u2 ( 1 ) ....uN ( 1 ) 1 u1 ( 2 ) u2 ( 2 ) ....uN ( 2 )
1 u1 ( q ) u2 ( q ) ...u N ( q )
We consider the multicollinearity case, which occurs when there is an almost linear relationship between inputs. In this case, the matrix C U T U is close to singular, so it is the smallest eigenvalue min 0 , the condition number is infinitely
where h0 is a constant term.
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increased, and it causes the instability of the solution. If min 0 , then it corresponds to the strict multicollinearity. In order to obtain a stable solution, it is necessary to reduce the condition number of the matrix C by, for example, adding thereto a diagonal matrix B k I (k> 0). Then we find the solution in a class of ridge parameter estimates: b ( U T U k I ) 1 U T Y .
of Lagrange multipliers corresponding to the lower and upper active constraints. Below is the algorithm for searching the minimum point b* for each iteration k. 1. Find the starting point according to (7) in order to initialize the method of the active set.
(7)
2. Verify the performance of the stop conditions. (Reaching the performance errors of conditions (9), which are constraints on the number of iterations). 3. Select a logic branch. Does it make sense to remove any constraint from the list of the set of active constraints? The condition of performance 3 in (9) is checked. If the condition is not satisfied for some of the vector elements, the constraint is excluded from the list of active constraints. 4. Calculate the search direction pk. Equation (7) solves the problem min Y Ubk U FR p FR 2 . Calculate the non-zero
The quality, obtained using models (7), depends on the number of available output measurements. The length of the training sample is often insufficient to obtain reliable results. Also, the available data contains significant measurement errors in inputs and outputs, which are unmeasured influences. Taking into account constraints on the model coefficients bj allows us to avoid these problems. When taking into account constraints on the model parameters, we solve the problem of least squares with simple constraints on the variables: min Y Ub b
min
bb
2
max
N 1 tk , the dimensional vector p FR , and the direction
(8)
of search pk AT p FR , where tk is the number of active FR
.
constraints on k iterations. 5. Calculate the step length αk. We calculate the diagonal b p matrix Ψ from FR Ψ FR bˆ FR , bˆ FR consists of b p FR FR
The solution of the problem (8) is obtained by the active set numerical method (Gill et al., 1981). The given constraints are reduced to the form:
Ab bˆ ,
where
1 0 0 1 0 0 A 1 0 0 1 0 0
the elements bˆ , which aren’t active constraints. The elements bˆ , which are opposite boundary values in (8) for constraints in the active set, are excluded from bˆ FR . The k min Ψii is an available maximum positive step from
0 0 bmin ˆ 1 , b max . b 0 0 1
bk along pk. We remember the index j of minimum positive diagonal element Ψ. If k 1 , then k 1 ; otherwise, k k . 6. Add a constraint to the list of active constraints. If k k , then j constraint bˆ FR becomes active and necessary to add to the list. 7. Approximate a recalculation. After bk 1 bk k pk is calculated, return to step 2 of the algorithm.
Constraint aTi b bˆi is active in acceptable point b if aTi b bˆi , and inactive if aTi b bˆi , a T is the i-th row of A. The sufficient i
minimum conditions for simple constraints are as follows: * max 1. b m in b * b m a x bmin FR bFR bFR T * 2. UFR ( Y Ub ) 0 min T 3. λ U min ( Y Ub* ) λimin 0 i 1, , t min 4. λ max U Tmax ( Y Ub* ) , λ max 0 , i 1, , t max
3.2 Dynamic model identification under constraints (9)
Let u [1, u1 ( ),..., u1 ( n1 1),..., uN ( ),..., uN ( nN 1)]T be the
combined vector of the measured input variables of dynamic SS (DSS) with dimensionality q 1 kN 1 n k , where nk is a
i
5. UTFR U FR is the positive definite
number
of values of
the
k-th
input variable and
h (h0 , h1 (1), ..., h1 (n1 ), ..., hN (1), ..., hN (nN ))T is the vector FIR of the
where b* is the minimum point of the solution of problem (8); subscript FR indicates that in the vector and matrices, the elements and columns with index numbers corresponding to the index numbers of b elements that have not met the boundary values of (8) are used; the subscript min and max indicate that in the matrix, only the columns with index numbers corresponding to the index numbers of b elements that take the appropriate minimum or maximum boundary value are used. tmin and tmax refer to the numbers of active upper and lower limits, respectively; min and max are vectors
same dimension, the components of which reflect the contributions of the respective input variables of DSS. Then the equation (2) takes the following form: y uT h . We write the vector Y of dimension q from the output value y as Y ( y ( 1 ), y ( 2 ),..., y ( q ))T
and matrix U, containing the measured corresponding to output value y from (2) as
562
inputs
uj,
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1 u1 ( 1 ) u1 ( 1 n1 1) uN ( 1 ) 1 u ( ) u ( n 1) u ( ) 1 2 1 2 1 N 2 U 1 u1 ( q ) u1 ( q n1 1) uN ( q ) uN ( 1 ) uN ( 2 ) uN ( q )
.
s max j
uN ( 1 n N 1)) uN ( 2 n N 1)) uN ( q n N 1))
Wee write the m matrix equationn as Y U h . We introducce the error function: E Y Y Y Uh , wh here Y is the aactual measurrement of outtput, and miniimize thee following obbjective function: 2 Ψ E ( Y Uh h)2 .
T
s min j
s max j
s max ((1) 0 j s max (fl( a 1n j )) 0 j max max s j (fl(a1n j ) 1) b j a2 1 a2 1/ (n j fl(a1n j )) max max s j (fl(a1n j ) 2) b j a2 1 a2 2 / (n j fl(a1n j )) max s n ) b max ( j j j
,
(11)
s s1 (1), ..., s1 ( n1 ), ..., sN (1)), ..., sN ( nN )
.
min s min j (1) a3b j min s j (fl(a1 n j )) a3b min j min ) 1) b min 1 ( a3 1) 1 1/ (n j fl(a1 n j )) j s j (fl(a1 n j min 2) b min 1 ( a3 1) 1 2 / (n j fl(a1 n j )) s j (fl(a1 n j ) j ) b min s min ( n j j jj
The constraints on transient response comp ponents are w written as
wh here
s max (1) a3b max j j s max (fl(a1 n j )) a3b max j j max ) 1) b max 1 ( a3 1) 1 1/ (n j fl(a1 n j )) j s j (fl(a1 n j max ) 2) b max 1 ( a3 1) 1 2 / (n j fl(a1 n j )) s j (fl(a1 n j j s max ( n ) b max j j jj
For b j 0 :
(10)
smiin s smax ,
569
, smin s1min , ..., s min , N T
T
x s max s1max , ..., s max N . We obtain the constraaint (11) baseed on
thee value of bj ( j 0) of thee steady-state soft sensor m model That y ( ) b0 b1u1 ( ) ... b j u j ( ) ... bN u N ( ) . mo odel is deriveed prior basedd on the steaady-state induustrial datta of CDU. W We select the parametric co onstraints for each FIR R coefficient uusing the folloowing values of o a1, a2 and a 3. a1 is a fraction of the FIIR length n j from whichh the ax con nvergence of elements of s min and s ma to bmin , bmjmax is j j j
wherre fl means ro ounding to thee nearest integ ger number inn the direcction of -∞.
started.
0 ( b max 0 ) from a2 is i a fraction off b min m which the sm mooth j j
An example e for deetermining thee constraint sy ystem in term ms of
inccreasing (or ddecreasing) of o elements of o s min (or s max ) j j
We select s the paraameters a1=0.55, a2=0.8, and d a3=1.2.
max s min and s j off s j under b j 0 is show wn on the Figg. 2. j
beg gins.
0 ( b min 0 ) from whichh the a3 is a fractionn of bmax j j n mooth decreasiing (or increaasing) of elem ments of s min (or sm j m ) begins. s max j
Forr b j 0 , the ffollowing equuations are vallid for constraaints
n s min j
in s mi j (1) 0 s min (fl fl( a 1n j )) 0 j min b min a2 1 a2 1/ (n j fl(a1n j )) j s j (fl(a1n j ) 1) min b min a2 1 a2 2 / (n j fl(a1n j )) s j (fl(a1n j ) 2) j min n s ( ) b min j j j
Fig. 2. The assign nment of consstraints for eaach step respoonse namic soft sennsor model. function of the dyn
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4. Calculate the search direction pk. Use an equation like (7) 2 to solve the problem min Y Uhk UZk pZ . Calculate the
The coefficients of the finite step response (FSR) s j are related to the components of the FIR hj by the relations s j k i 1 h j (i ), k
j 1, 2 , , N ,
non-zero 1 kN1 nk tk
(12)
k 1, , n j .
matrix Ψ is calculated. The calculated k min Ψii is a minimum non-negative available step
diagonal
(13)
where
1 0 0 1 1 0 0 0 1 A 1 0 0 1 1 0 0 0 1
,
h1 (1) h1 ( n1 ) , h hN (1) h (n ) N N
from hk along pk, where i is the index number of the element, which is not an active constraint in (13) or an element of opposite boundary values (11) for constraints in the active set. Also, p k consists of elements of pk without the first element. We memorize the index j of the minimum positive diagonal element of Ψii. If k 1 , then k 1 ; otherwise, k k . 6. Add the constraint to the list of active constraints. If k k , then j constraint sˆ FR becomes active and Zk is recalculated. 7. Calculate the hk 1 hk k pk and return to step 2.
s min sˆ max . s
The sufficient minimum conditions are as follows for the current optimization problem (parametric identification of the dynamic model): 1. Ah sˆ , A ACT h sˆ ACT 2. Z T * U T ( Y Uh* ) 0 3. λ
A ACT A TACT
1
A ACT U T ( Y Uh * ) , λ i 0 ,
dimensional vector pZ and the
direction of search pk Zk pZ , where tk is the number of active constraints on the k iteration. 5. Calculate the step length k . From A h Ψp k sˆ , the
The constraints (11) can be written as
Ah sˆ ,
4. INDUSTRIAL CASE STUDY The CDU (Fig.1) is considered a case study for evaluating static (steady-state) and dynamic soft sensors based on the proposed identification algorithm under parametric constraints. The final boiling point temperature of the DC1 when we obtained a model on the training sample is considered. The number of output observations in the training sample is 70. The length of the test dataset is equal to 30 observations. Fig. 3 and Table 2 show the results of the performance of the static models on the test sample.
(14) i 1, , t
4. ZT U T U Z is a positive definite, where h * is the minimum point of the solution of problem (10) with constraints (13); subscript ACT indicates that in the vector matrix, we only use the rows with index numbers corresponding to the element index numbers of active constraint in (13); t is the number of active constraints; is the vector of Lagrange multipliers corresponding to the active constraints; Z is the matrix of the columns that are the basis of the feasible direction of the search for equality constraints (13). The matrix Z is formed by the variable-reduction technique (Gill et al., 1981).
Table 2. Results of the performance of the static models (test dataset) Without use constraints With use constraints
The search algorithm of minimum point h * for iteration k is as follows:
R2
RMSE
AIC
BSC
0,64
4,47
232,7
255,4
0,84
2,99
175,5
198,2
360 355
1. In order to start the method of the active set, it is necessary to determine the starting point (using a solution of the problem (10) without any constraints, with subsequent correction of coefficients hi that do not fall under the constraints (13)). 2. Verify the performance of the stop conditions (reaching the performance errors of conditions (14) and constraints on the number of iterations). 3. Select a logic branch. Does it make sense to remove any constraint from the set of active constraints list? The condition of performance of a condition 3 in (14) is checked. If the condition is not satisfied for some of the vector elements, constraint is excluded from the list of active constraints, and it is necessary to recalculate Zk.
350
FBP,°С
345 340 335 330 325 320 315
lab data w/t constrains with constrains 29.06.2017
06.07.2017 Time period
Fig. 3. Comparative study of static soft sensor model’s performance. 564
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The improvement of the prediction quality by the criterion RMSE of the identified static model obtained with the constraints on the parameters of SS is 100(4,47 – 2,99) / 4,47 33% compared to the case without constraints. In order to investigate the influence of constraints on the performance of the dynamic soft sensor model we obtained, we compared solutions of the optimization problem (10) without constraints and optimization problem (10) with constraints (13). We use the same values of the ridge coefficients. Fig. 4 and Table 3 show the results of the performance of the dynamic models on the test sample for the final boiling point temperature of DC2 when a model is obtained based on the training sample. The number of measurements in the training sample is 280. The size of the test dataset is 120 for the case with the dynamic model of the soft sensor.
that the reduction of the root mean square error on the test sample can be more than 33%. 7. ACKNOWLEDGEMENTS This work was partially supported by the Russian Foundation for Basic Research (Project No. 17-07-00235 A) and by the Ministry of Education and Science of the Russian Federation, through Government Contract No. 02.G25.31.0173. REFERENCES Akaike, H. (1969). Fitting Autoregressive Models for Prediction. Annals of the Institute of Statistical Mathematics, 21, pp. 243-247. Bolf, N., Galinec, G. and Baksa, T. (2010). Development of soft sensor for diesel fuel quality estimation. Chemical Engineering & Technology, 33, pp.405-413. Chatterjee, T., and Saraf, D. N. (2004). On-line estimation of product properties for crude distillation units. Journal of Process Control, 14, pp. 61–77. Dam M. and Saraf D.N. (2006) Design of neural networks using genetic algorithm for online property estimation of crude fractionator products. Computers & Chemical Engineering, 30 (4) pp. 722–729. Fujii K. and Yamamoto T. (2014). Development of a nonlinear soft sensor using a GMDH network for a refinery crude distillation tower, Electrical Engineering in Japan, Vol. 188, No. 3. pp. 31-38. Gill, P.E., W. Murray and Wright M.H. (1981). Practical Optimization. New York, USA: Academic Press. Johansen, T. (1996). Identification of non-linear systems using empirical data and prior knowledge-an optimization approach. Automatica, Vol. 32, No. 3, pp. 337–356. Macias-Hernandez J. J., P. Angelov and Zhou, X. (2007). Soft sensor for predicting crude oil distillation side streams using Takagi-Sugeno evolving fuzzy models. Proc. 2007 IEEE Intern. Conf. on Systems, Man, and Cybernetics, 7-10 Oct., 2007, Montreal, Canada, ISBN 1-4244-0991-8/07, pp.3305-3310 Mahalec, V. and Sanchez, Y. (2012). Inferential monitoring and optimization of crude separation units via hybrid models. Computers & Chemical Engineering, 45, pp. 1526. Napoli G. and Xibilia M.G. (2011). Soft sensor design for a topping process in the case of small data sets. Computers and Chemical Engineering, Vol. 35. No. 11, pp. 2447– 2456. Shang C., Yang F., Gao X. and Huang D. (2015) A comparative study on improved DPLS soft sensor models applied to a crude distillation unit. IFACPapersOnLine, Vol. 48. Issue 8, pp. 234–239. Torgashov A. and Zmeu K. (2015). Nonparametric soft sensor evaluation for industrial distillation plant. Computer Aided Chemical Engineering. Vol. 37, pp. 1487-1492. Torgashov A., Skogestad S. and Kozlov A. (2016). Comparative study of multicomponent distillation static estimators based on industrial and rigorous model datasets. IFAC-PapersOnLine. Vol. 49. Issue 7, pp. 1187-1192.
Table 3. Results of the performance of the dynamic models (test dataset)
R2
RMSE AIC BSC Without use 0,40 3,79 6418,4 6482 constraints With use 0,74 2,52 4453,2 4516,8 constraints The improvement of the prediction quality by the criterion RMSE of the dynamic model obtained using the constraints on the parameters of SS is 100(3,79 – 2,52)/ 3,79 33,5 % compared to the identification without constraints. in-line analyser w/t constrains with constrains
190
FBP,°С
185
180
175
170 04.05.17
Time period
571
05.05.17
Fig. 4. Comparative study of dynamic soft sensors. 6. CONCLUSIONS The introduction of a system of constraints into the identification algorithm improves the quality of derived models, especially on the test dataset. The reason is related to the integration of available a priori knowledge via the constraint identification problem statement and solution.The use of the active set method, taking into account constraints on the model coefficients, can improve the quality of the evaluated SS models, in particular in the case of small training datasets containing lab errors. The test of the proposed approach to solving the problem of obtaining a soft sensor model for industrial crude oil distillation unit showed 565