Using Process Data for Finding Self-optimizing Controlled Variables

Transcription

1 Using Process Data for Finding Sef-optimizing Controed Variabes Johannes Jäschke and Sigurd Skogestad Norwegian University of Science and Technoogy, NTNU, Trondheim, Norway Abstract: In the process industry it is often not known how we a process is operated, and without a good mode it is difficut to te if operation can be further improved We present a data-based method for finding a combination of measurements which can be used for obtaining anestimateofhowwetheprocessisoperated,andwhichcanbeusedinfeedbackasacontroed variabe To find the variabe combination, we use past measurement data and fit a quadratic cost function to the data Using the parameters of this cost function, we then cacuate a inear combination of measurements, which when hed constant, gives near-optima operation Unike previousy pubished methods for finding sef-optimizing controed variabes, this method reies ony on past pant measurements and a few pant experiments to obtain the process gain It does not require a mode which is optimized off-ine to find the controed variabe Keywords: Process Optimization, Contro, Partia east squares, Empirica modeing, Sef-optimizing contro 1 INTRODUCTION Rising competition in a goba market, environmenta chaenges and governmenta reguations make it necessary to operate chemica pants cose to optimaity At the same time one is often faced with not knowing exacty how we or poory the pant is operated, and which options that may exist to systematicay improve operation If a suitabe first principe pant mode is avaiabe, it may be used onine for monitoring the performance or for rea-time optimization (RTO) Here a mathematica optimization probem is soved to find the optima operating parameters for the process [Marin and Hrymak, 1997] In this case the pant measurements are primariy used to update the mode parameters in the onine optimization probem Aternativey, the mode may be used offine to deveop a suitabe contro strategy, which resuts in an acceptabe oss, a sef-optimizing contro structure According to Skogestad [2000], Sef-optimizing contro is when we can achieve an acceptabe oss with constant setpoint vaues for the controed variabes (without the need to re-optimize when disturbances occur Another cosey reated concept is NCO tracking [Srinivasan et a, 2003a,b], where the necessary conditions for optimaity (NCO) are seected as the controed variabes Usuay the controed variabes in these approaches are found by optimizing a suitabe process mode However, often a good first principe mode is not avaiabe because it is prohibitivey expensive to deveop and maintain a mode which accuratey refects the process An attractive aternative is to use empirica data-based modes, such as regression modes and partia east squares This work was supported by the Norwegian Research Counci modes In virtuay a chemica pants, data is coected amost continuousy, and using this data to mode and subsequenty optimize the process seems very attractive In theory, it can ead to significant operationa savings, whie requiring a reativey imited effort to deveop and maintain the modes Not surprisingy, there is a arge body of genera iterature on empirica and data based modeing, see eg Box and Draper [1987], Esbensen [2004], and it is widey used in industry In the context of process contro, data-based approaches have often been used for soft-sensing appications (eg Sharmin et a [2006], Lin et a [2007], Facco et a [2009]), where avaiabe measurements are used to estimate an unmeasured variabe In view of onine process optimization, there have been many suggestions over the years, incuding evoutionary Operation (EVOP) [Box, 1957], dating back to the 1950s, and in more recent years McGregor and coworkers [Yacoub and MacGregor, 2004] In the context of using offine process optimization to find simpe operationa poicies, there has been much ess activity Data-based methods have been deveoped and used by Jäschke and Skogestad [2011b] and Skogestad et a [2011], where it is assumed that optima data is avaiabe, and it is used to find optima controed variabes Ye et a [2013] have aso used regression methods to find controed variabes However, they rey on a process mode to generate the data and they assume that a disturbances can be measured The contribution of this paper is to show how non-optima open-oop pant measurement data can be used to 1) obtain a quadratic mode of the cost function, and 2) further be used to find sef-optimizing controed variabes, which when controed at constant setpoints, keep the process cose to the optimum

2 d Controer Process c s = constant u c y c = Hy Fig 1 The idea of sef-optimizing contro: By controing c = Hy at a constant setpoint, the process shoud be kept cose to optima in presence of varying disturbances The paper is structured as foows In the next section, we present some reated background from sef-optimizing contro Section 3 describes how these resuts can be used to obtain sef-optimizing controed variabes from operating data In Section 4 we appy our approach to a case study of a CSTR and present some simuation resuts Finay, in Section 5 we discuss the resuts and draw concusions 2 SELF-OPTIMIZING CONTROL In this section, we briefy give some reevant background on sef-optimizing contro The genera idea is to find variabes which, when controed at a constant setpoint, resut in near-optima operation with acceptabe oss [Skogestad, 2000] A possibe impementation scheme of a sefoptimizing contro structure is given in Fig 1 The idea is that near-optima operation is achieved by controing the controed variabes c = Hy at constant setpoints We assume that the probem of optimay operating a process at steady state (or a sequence of steady states as d varies) can be formuated as a mathematica optimization probem, min u,x J(u,x,d) st (1) g(u,x,d) 0 h(u,x,d) 0 where the variabes u, x, and d denote the degrees of freedom, the state variabes, and the disturbances, respectivey The scaar function J denotes the cost function, g : R nu R nx R n d R ng the mode equations, and h : R nu R nx R n d R n h denotes the operationa and safety constraints In addition, we assume that we have a pant measurement mode y = f y (u,x,d), (2) wherey isthen y -dimensionavectorofmeasurements,and f y is the function mapping the variabes u,x and d to onto the measurement space The foowing poicy for impementing optima operation is suggested [Skogestad, 2000]: (1) Contro the active constraints at their optima vaues (2) Contro sef-optimizing variabes for the remaining unconstrained degrees of freedom With active constraints controed, the active constraints and the states x can be formay eiminated from the optimization probem (1) This enabes us to re-write the probem for the second part as an unconstrained owerdimensiona optimization probem, minj(u,d) (3) u Around the nomina operating point [u T, d T ], we approximate the cost function using a second-order Tayor expansion, where u = u u and = d d : J J +[J u J d ] [ u + 1 J 2 [ u ] uu Jud (4) u Jdu Jdd where Ju = J u, Jd = J d, and Juu = 2 J u,j 2 ud = Jdu T = J2 u d and Jdd = 2 J d are the first and second 2 derivatives, evauated at the nomina point Under optima nomina operation, we have that Ju = 0 Under the same assumptions used to obtain Eq (4), the gradient can be approximated around the optima nomina point (Ju = 0) as ] u (5) J u = Ju +[Juu Jud ] }{{} =0 For optima operation, the first-order optimaity conditions require that the gradient is zero[noceda and Wright, 2006], ie J u = [J uu J ud ] u = 0 (6) If we coud measure or evauate the gradient, it woud be the idea sef-optimizing controed variabe Unfortunatey, this is not the case in practice Instead, we propose to approximate the gradient in terms of measurements y ony, and use this as a sef-optimizing variabe [Jäschke and Skogestad, 2011a] To express the gradient (6) in terms of the pant measurements, we inearize the measurement mode (2) around the nomina operating point, and upon eiminating the state variabes, we obtain y = G y u+g y d = G y u (7) If there is a sufficient number of measurements avaiabe 1, ie n y n u +n d, we can use the measurement mode (7) to eiminate the unknowns u and d from the gradient, and thus obtain a controed variabe which is equivaent to the gradient Inverting (7), and inserting into (5) yieds J u = [J uu J ud ][ Gy ] y, (8) where ( ) denotes the pseudo-inverse of ( ) Defining 1 The degrees of freedom u are generay aso incuded in the measurement vector y

3 H = [Juu Jud ] [ Gy ] we have that the desired sef-optimizing controed variabe is c = H y (10) Controing c = c c to zero gives optima operation This method is equivaent to the previousy pubished nu-space method [Astad and Skogestad, 2007] Note that the above derivation of H does not take measurement noise into account, and assumes that there are sufficienty many independent measurements such that G y can be inverted If the measurement noise is arge, or if we have too few measurements, it wi not be possibe to make the gradient cose to zero, and there wi be an additiona oss These cases can be treated using methods described in detai by Astad et a [2009] However, for the purpose of this paper we assume that the measurement noise is negigibe, and that there is a sufficient number of independent measurements such that G y can be inverted 3 OBTAINING H FROM OPERATIONAL DATA In the previous section, we have shown how a ocay optima controed variabe combination c = H y, with H givenin(9),canbeobtainedfromainearprocessmode (7) and a quadratic approximation of the cost function (4) However, this assumes that we know J uu,j ud and G y, which may be difficut to obtain in practice In this section, we show how the H-matrix in (9) can be obtained from historica process operation data, in terms of y The idea is to express the cost function approximation in terms of the measurements y, and to use avaiabe measurement data to estimate the cost function parameters 31 Fundamenta reationships To obtain the desired controed variabe combination, we first express the quadratic cost function (4) in terms of the measurements Soving (7) for [ u ] T and inserting into the cost function (4) gives [ Gy ] J = J +[J u J d ] y } {{ } Jy + 1 [ Gy ] T J 2 yt uu Jud [ Gy ] Jdu Jdd y }{{} Jyy = J +J y y yt J yy y (9) (11) Inspecting the term Jyy coser, we see that H from (9) is contained in it: [ Jyy = ] T [Juu Jud ] G Gy [Jdu Jdd] G [ = ] (12) T H Gy [Jdu Jdd] G Since n y n u + n d, we have that Gy T G T = I, so the upper n u rows of Gy T Jyy are exacty the H-matrix given in equation (9), G yt Jyy H = [Jdu Jdd] (13) G For contro purposes we are primariy interested in H, so we do not need a eements in G y, but ony the first part, G y Thus, we obtain H by premutipying Jyy with [ G y ] T, 0 ny n d which yieds [ G y ] T 0 ny n d J H yy = (14) 0 nd n y In summary, given Jyy and the gain matrix G y, we can easiy cacuate the optima measurement combination H, and use it to contro the process Remark 1 To find controed variabes without a rigorous mode, it is an advantage that ony G y = y u is required (instead of the fu matrix G y = [G y G y d ]), because Gy can be easiy found using a few pant experiments On the other hand, obtaining G y d = y d from pant experiments is difficut, because it is not possibe to manipuate the disturbance d 32 Obtaining G y One approach to obtain the measurement gain matrix G y = y u = [g(1),,g (nu) ], is to perform step changes in the inputs u i and record the changes in the outputs y The rows i = 1n u of the gain matrix can then be cacuated by g (i) = ypert y, (15) u pert i u i where the subscript i of the input u denotes the i-th input, and the superscript pert denotes the perturbed vaue For better accuracy, one may perform severa pant experiments of this kind and use the average vaue of the gain 33 Obtaining J yy Gathering the data Before we proceed to gather data to find J yy, we make some assumptions: (1) The data is coected whie the process is operating in open oop (2) The number of independent measurements is greater or equa to the number of independent inputs and disturbances 2, n y n u +n d (3) The active constraints are controed and are not changing (4) A important disturbance changes are present in the data 3 (5) Thedataiscoectedwhenthepantisatsteadystate (6) The process data is samped in a region cose to the optimum, where the cost can be approximated by a quadratic cost function 2 This can be tricky, because one might not know what unmeasured disturbances may affect the pant 3 If the data is taken from a sufficienty ong period, it is reasonabe to assume that a reevant disturbances are present in the data

4 We coect a the raw measurement data in the matrix Y raw, Y raw = [ y (1) y (i) y (N)], (16) where the superscript (i) denotes the sampe number Preparing the data Before using using the data further, it shoud be centered by subtracting the mean, and scaed such that the variance of the measurements is equa Note that the data now is in form of deviation variabes In order to obtain a quadratic mode, we need to take and Eq (20) is aso the product of measurements into account This is done by augmenting the data by a second order terms, such that each coumn of the data matrix Y contains data corresponding to (where the for marking deviation variabes has been omitted): y aug = y 1 y ny y1 2 y 1 y 2 y 1 y ny y 2 y 2 y 2 y 3 y n 1 y n yn 2 T (17) In addition, we assume that the cost function can be measured at each sampe time, and we coect a cost data into a separate 4 vector: J m = [ J (1) J (i) J ] (N) T (18) Partia east squares regression If the measurements y were independent variabes, we coud simpy use regression to fit the measurements y to the quadratic cost function in order to find J,J y and J yy Unfortunatey, the measurements wi generay not be independent, and simpy fitting a cost function to y-data wi resut in an i-posed optimization probem, and give very poor resuts To obtain a sufficienty good estimate of the cost function parameters, we use a partia east squares (PLS) method [Geadi and Kowaski, 1986, Esbensen, 2004, Martens and Naes, 1992] This method enabes us to hande coinearity and inear dependence in the data we, and can be used to find a mode which describes the cost function we The basic idea of PLS is to find a inear transformation which expains the variation in the prediction variabes (in our system: Y) as we as the variation in the response variabes (in our case J m ) The PLS agorithm projects the Y and J m data onto a ower dimensiona space, which sti captures a the essentia correations: Y T = TP T +E 1 J m = UQ T +E 2 (19) The matrices T,P,U and Q are chosen such that the covariance between the data Y T and J m is maximized, and E 1,E 2 are the residuas Based on this decomposition, a regression factor β is determined, which predicts J as a function of y aug After appying the PLS agorithm, which is impemented eg in Matab, the prediction of the cost can be cacuated as J = [1 y T aug]β, (20) where the vector β is obtained from the PLS agorithm We do not further present detais of PLS here, instead we refer to the iterature [Esbensen, 2004, Martens and Naes, 1992] where the procedure is described in detai 4 Note that the measurements of the cost function must not be incuded in the data matrix Y, because this woud cause the mode to use J m to predict the cost, and we woud not obtain a quadratic mode Since y aug contains a the products of the measurements with each other, we can simpy re-arrange Eq (20) into the form of Eq (11) For exampe, in the case of 2 measurements, the augmented measurement vector is y aug = y 1 y 2 y 2 1 y 1 y 2 y 2 2 J = 1 y 1 y 2 y1 2 y 1 y 2 y2 2, (21) β 0 β 1 β 2 β 3 β 4 β 5 (22) This can be re-written as the quadratic form given in Eq (11), y1 J = β 0 +[β 1 β 2 ] + 1 y 2 2 [y 2β3 β 1 y 2 ] 4 y1, β 4 2β 5 y 2 (23) and comparing the coefficients in (23) with (11), we see that J = β 0, Jy = [β 1 β 2 ], Jyy 2β3 β = 4 β 4 2β 5 (24) This method yieds a parameters of the quadratic cost function (11), and in particuar J yy, which is required for cacuating H 4 EXOTHERMIC CSTR CASE STUDY 41 Process description To demonstrate our approach, we consider a simpe CSTR studied by Economou and Morari [1986], Astad [2005], Kariwaa [2007], Jäschke and Skogestad [2011a,b] A schematic diagram of the process is given in Fig 2, where aso the main variabes are introduced The feed stream containing mainy component A enters the reactor, where an equiibrium reaction A B takes pace The reactor effuent contains a mixture of A and B with the same concentrations as in the tank The manipuated variabe is the feed temperature T i, which can be adjusted to minimize the operating costs, which are cacuated as the cost for heating the feed minus the income generated by seing the product B, J = ( p B c B (p Ti T i ) 2) (25) From the mass and energy baances we obtain the foowing dynamic system: dc A = 1 dt τ (C A,in C A ) r (26) dc B = 1 dt τ (C B,in C B )+r (27) dt dt = 1 τ (T i T) 5r, (28) where the variabes C A and C B denote the concentrations of component A and B, respectivey, in the reactor The

5 F C A,in C B,in T i Tabe 2 Loss comparison without noise Controed variabe Average oss Worst case oss H minoss H data Tabe 3 Loss comparison with noise Fig 2 Simpe CSTR Tabe 1 CSTR parameters Symbo Parameter description Vaue p B Price for product c B 2009 $/mo p Ti Price for heating feed 1657e 3 $/K τ Time constant 1 min CA,in Nomina feed concentration A 1 mo/ CB,in Nomina feed concentration B 0 mo/ Ti Nomina input vaue K T C A C B temperature in the reactor is denoted by T, and the feed temperature is denoted by T i The feed concentrations of the two components are denoted by the variabes C A,in and C B,in Finay, the reaction rate r is cacuated as 1 r = 5000e T CA 10 6 e T CB (29) The vaues of the price parameters, the time constant τ and the nomina process vaues are given in Tabe 1 We assume that the process has four measurements, which are C A C y = B T, (30) T i and two unmeasured disturbances, namey the feed concentrations: CA,in d = (31) C B,in The exact vaues of the disturbances are not known under operation, but we assume that it is known that C A,in = 05 mo 15 mo, and C B,in = 0 mo 05 mo This process has an unconstrained optimum, because increasing the feed temperature wi ead to increased production of C B, which contributes to owering the cost At the same time a higher feed temperature contributes to increasing the cost function due to the price for heating Therefore, the optimum occur at some optima trade-off point The minimay required number of measurements for this process is n u + n d = = 3, so there are enough measurements to appy our approach 42 Simuations and Resuts To generate the measurement data we run the CSTR in open oop with random inputs T i in the range of K, and disturbances in the ranges given above We added some measurement noise to the data to make the case 1 Controed variabe Average oss Worst case oss H minoss H data study more reaistic The measurement noise on the concentrations is uniformy distributed with within ±001 mo, and the noise for the temperature measurements varies uniformy between ±05K A tota of 1000 sampes was taken The first 500 sampes were used to caibrate the mode, and the rest were used to vaidate the mode Based on inspection of the residuas for the two data sets, it was found that a PLS mode with 10 components reproduced both data sets reasonaby we The gain matrix G y was obtained by step testing as G y = 10057, (32) 10 and the data for our cost function obtained with the PLS regression is J = Jy = [ ] Jyy = (33) Appying our data based procedure from Section 3, we obtain the H-matrix as H data = [ ] (34) We compare the performance of the process when controing c = H y with the truy optima operation for 1000 random disturbances for the cases with and without measurement noise For comparison, we have aso simuated the CSTR with the controed variabe combination obtained from the mode based minimum oss method Astad et a [2009] H min,oss = [ ] (35) The average oss and the maximum oss for the simuations without measurement noise are given in Tabe 2 We observe that the oss when using the data method is about twice as arge as the oss when using the exact oca method However, the actua vaue of the cost function is around 08, so the reative oss is sti quite sma Athough the method is not designed to hande noise, we have aso tried the method on the case study with measurement noise incuded The resuts for the same data points (but this time with measurement noise), are given in Tabe 3 Here the trend is simiar to the noise-free case, but the absoute vaues are a itte bit arger

6 5 DISCUSSION AND CONCLUSION We have presented a method for finding a sef-optimizing controed variabe combination, which is based ony on data, and does not require a mode or disturbance measurements This can be very usefu in industria practice, where a good (first-principe) mode is rarey avaiabe As expected, the mode based minimum oss method gave better resuts than our data-based method However, when there is no mode avaiabe this aternative does not exist, and our approach can be used to obtain at east some approximation of the optima H The oss vaues for simuation case with noise and the case without noise have been shown to be very simiar, so we concude that our method can hande measurement noise to some degree The combination matrix H can be used for feedback contro, as we have done in our case study, but it may aso be used for monitoring purposes In this case, the process operators simpy monitor magnitude of the eements in c = H y, and use it as an indication of optimaity A topic which has not been discussed in this paper is the question of how many components to use in the PLS modewehavedividedtheavaiabedataintwo,andused onedatasetasavaidationsetthenumberofcomponents have been chosen such that the norm of the difference between the predictions and the actua vaue of the cost function for the vaidation data set has been minimized Directions for future work incude testing the approach on a arger case study, and systematicay studying how measurement noise can be handed REFERENCES Vidar Astad Studies on seection of controed variabes PhD thesis, Norwegian University of Science and Technoogy, Department of Chemica Engineering, 2005 Vidar Astad and Sigurd Skogestad Nu space method for seecting optima measurement combinations as controed variabes Industria & Engineering Chemistry Research, 46: , 2007 Vidar Astad, Sigurd Skogestad, and Eduardo Shigueo Hori Optima measurement combinations as controed variabes Journa of Process Contro, 19(1): , 2009 George E P Box 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