REAL TIME OPTIMIZATION OF a FCC REACTOR USING LSM DYNAMIC IDENTIFIED MODELS IN LLT PREDICTIVE CONTROL ALGORITHM
|
|
- Jasper Jacobs
- 5 years ago
- Views:
Transcription
1 REAL TIME OTIMIZATION OF a FCC REACTOR USING LSM DYNAMIC IDENTIFIED MODELS IN LLT REDICTIVE CONTROL ALGORITHM Durask, R. G.; Fernandes,. R. B.; Trerweler, J. O. Secch; A. R. federal unversty of Ro Grande do Sul, Abstract: Ths work shows a new way to perform Real-Tme Optmzaton usng two mportant control technology. The frst one s the LLT nonlnear model predctve controller and the second one s the LSM dynamc model dentfcaton method. Ths two concepts assocated consttute a complete technology for Real-Tme Optmzaton usng nonlnear models. The applcaton of ths technology was tested on the FCC (fuel catalytc crackng) reactor whch shows the advantages and dsadvantages of ts applcaton. Keywords: Real Tme Optmzaton,.nonlnear dentfcaton, nonlnear predctve control, Advanced Control. 1. INTRODUCTION The control process has been each tme more developed n dfferent levels due to the desre of havng the best results of a ndustral process. Increasng the level of control of a ndustral process, the concept of Real-Tme Optmzaton becomes very mportant, snce, n ths level, the results are more vsble and t can be easly transformed n proft. The process called Fuel Catalytc Crackng (FCC) represents, n proftable terms, the most mportant ndustral process n a petroleum refnery. Through ths process, all the most valuable products of the petroleum refnery, lke gasolne and Lquefed petroleum gas (LG), are produced, and, for ths reason, t s usual ths process receves a specal attenton n the ndustral envronment, whch make the applcaton of Real-Tme Optmzaton becomes strongly justfed. The man goals of ths optmzaton s to reach the maxmum converson of the feed flow n products, the maxmum producton of valuable products as gasolne and LG, maxmum proft and maxmum producton. In ths work, a frst prncple model was used to smulate the ndustral process behavour, n order to perform an analyss of the applcaton possblty of the two technques (LSM and LLT) n real process. for ths study a mono-layer optmzaton structure was developed on the LLT algorthm whch produce a dynamc optmzaton of the process. THE FUEL CATALITC CRACKING ROCESS (FCC)
2 The FCC process s a mportant perod n petroleum refne, It uses a fracton of the dstllaton called gasol and transforms ths feed flow n gasolne (NC), lquefed petroleum gas (LG), lght crackng ol (LCO), clarfed ol (OCLA), fuel gas (GC) and coke (CK). The gasol s loaded n the converser together wth the catalyst, prevously warm. Ths mxture gets up through the converser generatng the products whch are separated on the top of the converser. In ths separaton, gaseous products are separated from the catalyst and the coke, and sent to dstllaton column. The catalyst covered by coke turns back to the system, beng loaded n the regenerator, where the coke s fred and generates energy to the reacton. Ths energy keeps accumulated on the catalyst whch s ntroduced back to the converser together wth gasol. The model used to represent ths process was developed by (Secch et. al 1) and consst n a frst prncples model wth 65 states, outputs, and 8 nputs. Ths model descrbes de converser as a plug-flow reactor (FR) n whch, the axal varable was dscretzed n twenty stages. 3 THE LOCAL LINEARIZATION ON THE TRAJECTORY (LLT) ALGOTITHM The LLT algorthm (Durask, 1) s a knd of sequental predctve control method whch conssts n a sequence of teratve steps. The frst step n the control acton desgn s to predct the up-to-dated trajectory of the system. Ths trajectory s determned usng the current value of the nputs of the process appled to the nonlnear model. In ths frst trajectory, lnearzed models are obtaned usng a dynamc lnearzaton. The set of models obtaned on each pont of the trajectory s grouped performng one devaton model from the orgnal trajectory, whch s used n the optmzaton step. Ths optmzaton wll generate a set of control actons to the system, whch, not necessarly, wll be the best set of control actons, once that t was obtaned usng a lnearzed devaton model. Therefore, ths set of control actons obtaned are appled to the nonlnear model and a new trajectory s generated. On ths trajectory, a new set of lnearzed models s obtaned and a new optmzaton performed. Ths sequence of steps keep gong on untl the nonlnear predcted trajectory converges to the setpont and no more varatons are observed n the control actons desgned. 3.1 Optmzaton Layer The orgnal LLT algorthm do not allow optmzaton and, therefore, a addtonal mplementaton was necessary to do such work. In predctve control, a closed shape of objectve functon can be obtaned, snce the goals of the control s very well defned. Ths concluson makes that the algorthm use hessan and gradent matrces prevously defned based on the model equatons. When ncludng a optmzaton layer such matrces can not be prevously defned because the expresson of objectve functon can take any shape. Ths fact makes necessary evaluate the hessan and gradent of the objectve functon n each teraton snce the system s solved by a quadratc programmng n each step. The gradent vector n ths mplementaton s been estmated by numercal approach usng conventonal perturbaton of varables. The hessan uses an estmaton suggested by used n BFGS optmzaton method. A lnesearch step was ncluded n ths algorthm to mprove the qualty of the soluton. Ths mplementaton had to be done because the nature of the objectve functon have change consderably. n the orgnal formulaton of the LLT algorthm the objectve functon was completely quadratc whch turns possble the use of successve Q solutons wthout any correcton. usng ths new formulaton of objectve functon, a lnesearch turns necessary snce the new objectve functon can take many nonlnear shapes, and the successve quadratc approach ndcates only the drecton of the search. The orgnal objectve functon of the LLT algorthm was transformed n the followng expresson:
3 ( γ ( y r )) + = M ( λ ( u u 1 ) + = = ( ϕ s ) α ( Lucro ) = J = mn s, δu % conv M β FCG = 1 ηgl µ FCG = 1 η NC ν FCG = 1 η LCO π FCG = 1 M B ( ψ ( δu + u 1) z ) ) + (1) where s the predcton horzon; M s the control horzon; γ s the weght of the error n the controlled varables; y s the value of the controlled varables; r s the value of setponts to the controlled varables; λ s the weght for varaton of the control actons; δu s the varaton of the control actons related to the reference trajectory; u s the absolute value of the control actons; ψ s the weght to the error n the proposed target for manpulated varables; z s the target establshed to the manpulated varables;φ s the weght for volaton of softconstrans set to the controlled varables; s s the volaton of the softconstrans of the controlled varables; α s the weght for the proft n the objectve functon; β s the weght of the converson degree;%conv s the converson degree of the gasol n products; FCG s the gasol feed flow; µ s the mportance of the LG producton; η GL s the mass fracton of LG; ν s the weght of the producton of gasolne; η NC s the mass fracton of gasolne; π s the weght of the mportance of the producton of LCO; η LCO s the mass fracton of LCO n the output; lucro s the expresson of the proft of the process gven by equaton : - fuel gas flow (GC) (product); - lquefed petroleum gas flow (GL) (product); - gasolne flow (NC) (product); - Lght cvcle ol flow(lco) (product); - coke produced (CK) (product); - clarfed ol flow (OCLA) (product); ROD s the prce of the product (postve) or nput (negatve). t tme nterval between two consecutve nstants of the predcton. 4 THE LINEARIZATION ON THE STATIONARY MANIFOLD (LSM) The LSM Algorthm developed by (Fernandes 5) s a dentfcaton method that can descrbe nonlnear models as a approach of multple lnear models by developng a nterpolaton rule to ths models. Ths nterpolaton generates a matrx A of the state space representaton dependent of the nputs of the system. The fnal shape of ths model can be represented n the equaton 3: dx dt ( x SS( )) = A( u) u (3) where A(u) s the nterpolaton of the matrces A of the multple models; SS(u) s the steady states as a functon of nputs, whch are also obtaned by nterpolaton; x s the current state. 5 THE ALICATION OF RTO IN THE FCC REACTOR A sensblty analyss was performed. to decde whch ponts was mportant to perform the dentfcaton of the system. Lucro F prod = 1 + F prod t prod () F ROD s the feed or output é rate of some of the followng product or nputs: - Catalyst (CAT) (nput); - Gasol (CG) (nput); - otênca do soprador (SO) (nput); Fg 1: Sensblty analyss for the temperature on the rser
4 Although the other results of sensblty analyss are not shown here, the fgure 1 shows the man behavor of nterest of the process. In ths fgure s shown that the temperature s almost lnear wth gasol feed flow, however t presents a nonlnear behavor wth the ar flow. As the man nonlnearty s ths one produced by the ar flow the lnear models dentfed were dstrbuted n fve ponts of operaton varyng only the ar flow rate. Two of then n the regon of postve gan, two n the regon of negatve gan and one n the regon near by gan zero. The nterpolaton performed n ths case for the A matrx has used a lnear rule whch was the most approprated form, snce the negatve sgnal of the poles have to be kept through all space. The steady state functon follow a quadratc nterpolaton, because t gave to descrbe the nverson of the sgn of the gan of the system. Although the results of the sensblty analyss were not presented here, t s known that, n FCC reactor, the converson of the products are straght related to the temperature of the rser and the behavor of ths varables are equal to the temperature behavor. 6 RESULTS AND DISCUSSION Optmzatons of the FCC reactor were performed usng dfferent weghts for each term of the objectve functon. Ths usage of dfferent weghts looks for show the behavor of the optmzer when goals are changed. The frst smulaton were performed usng a weght 1 for the proft term and for the other ones (except move suppresson whch were kept wth weght equal to 1). Fgure shows the control actons taken by the optmzer when the goal was maxmum proft. n ths fgure s shown that the ar flow rate ncreases untl a value related to the maxmum temperature of the rser. The value reached for the temperature can not be ncreased by the ar flow rate at ths pont and for ths reason the total feed flow has to be decreased to allow that converson n the rser be effectve, as shown n fgure 3, and the most valuable products were produced, reducng the output of gasol from the rser. Fg : Control Actons taken by the optmzer Fg 3: Converson of gas ol n products The same sequence of control actons can be seen when the objectve of the optmzaton s the maxmum converson. Ths goal mples n reducng the feed flow rate to allow the best converson possble snce the lmt of the equpment s reached. Fg 4. Responses of the system optmzaton usng maxmzaton of LG as the goal of the optmzer
5 (Fernandes 5). R. B. Fernandes, Contnuous Nonlnear Identfcaton Usng arameterzed Local Models ; hd. Thess, Unversty of Dortmund; (5) Fg 5. Responses of the system optmzaton usng maxmzaton of LCO as the goal of the optmzer A comparson between other two goals of optmzaton s shown n fgures 4 and 5. Fgure 4 shows the responses of the system optmzaton usng maxmzaton of LG as the goal of the optmzer and fgure 5 shows the responses of the system optmzaton usng maxmzaton of LCO as the goal of the optmzer. Comparng ths two fgures, t s possble to see that the mass fracton of each component changes accordng wth the man objectve of optmzaton. 7 CONCLUSIONS The optmzer used n ths paper shows a good soluton to perform real-tme optmzaton of contnuous dynamc process although, s stll needs some adjusts. As presented n prevous secton, t s possble to see that the optmzer takes the process to a new condton as close as possble of the man goal of optmzaton. t s stll possble to perform so mprovements n ths tool snce some problems were found n ts applcaton on the model of FCC reactor. In some specfc cases oscllatory behavor were captured when the plant reaches the optmum. whch mples n a more detaled analyss of the problem to arse ts possble reasons. 8 REFERENCES (Durask 1) R.. G. Durask.; Controle redtvo Não Lnear Utlzando Lnearzações ao Longo da Trajetóra ; M. Sc. Thess, Unversdade Federal do Ro Grande do Sul, Brazl. (1) (Secch et. al 1) A. R. Secch, M. G. Santos,G. A. Neumann, J. O. Trerweler; A Dynamc Model for FCC UO Stacked Converter Unt; Computer and Chemcal Engneerng; V 5; (1); pp
Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More information4DVAR, according to the name, is a four-dimensional variational method.
4D-Varatonal Data Assmlaton (4D-Var) 4DVAR, accordng to the name, s a four-dmensonal varatonal method. 4D-Var s actually a drect generalzaton of 3D-Var to handle observatons that are dstrbuted n tme. The
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationDesign and Optimization of Fuzzy Controller for Inverse Pendulum System Using Genetic Algorithm
Desgn and Optmzaton of Fuzzy Controller for Inverse Pendulum System Usng Genetc Algorthm H. Mehraban A. Ashoor Unversty of Tehran Unversty of Tehran h.mehraban@ece.ut.ac.r a.ashoor@ece.ut.ac.r Abstract:
More informationSimultaneous BOP Selection and Controller Design for the FCC Process
Smultaneous BOP Selecton and Controller Desgn for the FCC Process Benjamn Omell & Donald J. Chmelewsk Department of Chemcal & Bologcal Engneerng Outlne Motvatng Example Introducton to BOP Selecton and
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationEVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES
EVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES Manuel J. C. Mnhoto Polytechnc Insttute of Bragança, Bragança, Portugal E-mal: mnhoto@pb.pt Paulo A. A. Perera and Jorge
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationmodeling of equilibrium and dynamic multi-component adsorption in a two-layered fixed bed for purification of hydrogen from methane reforming products
modelng of equlbrum and dynamc mult-component adsorpton n a two-layered fxed bed for purfcaton of hydrogen from methane reformng products Mohammad A. Ebrahm, Mahmood R. G. Arsalan, Shohreh Fatem * Laboratory
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationChapter Newton s Method
Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve
More informationDETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationParametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010
Parametrc fractonal mputaton for mssng data analyss Jae Kwang Km Survey Workng Group Semnar March 29, 2010 1 Outlne Introducton Proposed method Fractonal mputaton Approxmaton Varance estmaton Multple mputaton
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationChapter - 2. Distribution System Power Flow Analysis
Chapter - 2 Dstrbuton System Power Flow Analyss CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More information829. An adaptive method for inertia force identification in cantilever under moving mass
89. An adaptve method for nerta force dentfcaton n cantlever under movng mass Qang Chen 1, Mnzhuo Wang, Hao Yan 3, Haonan Ye 4, Guola Yang 5 1,, 3, 4 Department of Control and System Engneerng, Nanng Unversty,
More informationCinChE Problem-Solving Strategy Chapter 4 Development of a Mathematical Model. formulation. procedure
nhe roblem-solvng Strategy hapter 4 Transformaton rocess onceptual Model formulaton procedure Mathematcal Model The mathematcal model s an abstracton that represents the engneerng phenomena occurrng n
More informationLOW BIAS INTEGRATED PATH ESTIMATORS. James M. Calvin
Proceedngs of the 007 Wnter Smulaton Conference S G Henderson, B Bller, M-H Hseh, J Shortle, J D Tew, and R R Barton, eds LOW BIAS INTEGRATED PATH ESTIMATORS James M Calvn Department of Computer Scence
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationCONTRAST ENHANCEMENT FOR MIMIMUM MEAN BRIGHTNESS ERROR FROM HISTOGRAM PARTITIONING INTRODUCTION
CONTRAST ENHANCEMENT FOR MIMIMUM MEAN BRIGHTNESS ERROR FROM HISTOGRAM PARTITIONING N. Phanthuna 1,2, F. Cheevasuvt 2 and S. Chtwong 2 1 Department of Electrcal Engneerng, Faculty of Engneerng Rajamangala
More informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationStatistics for Economics & Business
Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationMarkov Chain Monte Carlo Lecture 6
where (x 1,..., x N ) X N, N s called the populaton sze, f(x) f (x) for at least one {1, 2,..., N}, and those dfferent from f(x) are called the tral dstrbutons n terms of mportance samplng. Dfferent ways
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationA novel mathematical model of formulation design of emulsion explosive
J. Iran. Chem. Res. 1 (008) 33-40 Journal of the Iranan Chemcal Research IAU-ARAK www.au-jcr.com A novel mathematcal model of formulaton desgn of emulson explosve Mng Lu *, Qfa Lu Chemcal Engneerng College,
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More informationSecond Order Analysis
Second Order Analyss In the prevous classes we looked at a method that determnes the load correspondng to a state of bfurcaton equlbrum of a perfect frame by egenvalye analyss The system was assumed to
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationPrediction of steady state input multiplicities for the reactive flash separation using reactioninvariant composition variables
Insttuto Tecnologco de Aguascalentes From the SelectedWorks of Adran Bonlla-Petrcolet 2 Predcton of steady state nput multplctes for the reactve flash separaton usng reactonnvarant composton varables Jose
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationEEL 6266 Power System Operation and Control. Chapter 3 Economic Dispatch Using Dynamic Programming
EEL 6266 Power System Operaton and Control Chapter 3 Economc Dspatch Usng Dynamc Programmng Pecewse Lnear Cost Functons Common practce many utltes prefer to represent ther generator cost functons as sngle-
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationTracking with Kalman Filter
Trackng wth Kalman Flter Scott T. Acton Vrgna Image and Vdeo Analyss (VIVA), Charles L. Brown Department of Electrcal and Computer Engneerng Department of Bomedcal Engneerng Unversty of Vrgna, Charlottesvlle,
More informationAn identification algorithm of model kinetic parameters of the interfacial layer growth in fiber composites
IOP Conference Seres: Materals Scence and Engneerng PAPER OPE ACCESS An dentfcaton algorthm of model knetc parameters of the nterfacal layer growth n fber compostes o cte ths artcle: V Zubov et al 216
More informationCIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M
CIS56: achne Learnng Lecture 3 (Sept 6, 003) Preparaton help: Xaoyng Huang Lnear Regresson Lnear regresson can be represented by a functonal form: f(; θ) = θ 0 0 +θ + + θ = θ = 0 ote: 0 s a dummy attrbute
More informationA LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS. Dr. Derald E. Wentzien, Wesley College, (302) ,
A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS Dr. Derald E. Wentzen, Wesley College, (302) 736-2574, wentzde@wesley.edu ABSTRACT A lnear programmng model s developed and used to compare
More informationOn a direct solver for linear least squares problems
ISSN 2066-6594 Ann. Acad. Rom. Sc. Ser. Math. Appl. Vol. 8, No. 2/2016 On a drect solver for lnear least squares problems Constantn Popa Abstract The Null Space (NS) algorthm s a drect solver for lnear
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More informationj) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1
Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons
More informationFUZZY APPROACHES TO THE PRODUCTION PROBLEMS: THE CASE OF REFINERY INDUSTRY
FUZZY APPROACHES TO THE PRODUCTION PROBLEMS: THE CASE OF REFINERY INDUSTRY Mustafa GÜNES Unv. Of Dokuz Eylül Fac. Of Econ. & Adm. Scences Department of Econometrcs Buca Izmr TURKEY mgunes@sfne.bf.deu.edu.tr
More informationWinter 2008 CS567 Stochastic Linear/Integer Programming Guest Lecturer: Xu, Huan
Wnter 2008 CS567 Stochastc Lnear/Integer Programmng Guest Lecturer: Xu, Huan Class 2: More Modelng Examples 1 Capacty Expanson Capacty expanson models optmal choces of the tmng and levels of nvestments
More informationNormally, in one phase reservoir simulation we would deal with one of the following fluid systems:
TPG4160 Reservor Smulaton 2017 page 1 of 9 ONE-DIMENSIONAL, ONE-PHASE RESERVOIR SIMULATION Flud systems The term sngle phase apples to any system wth only one phase present n the reservor In some cases
More informationSOLVING CAPACITATED VEHICLE ROUTING PROBLEMS WITH TIME WINDOWS BY GOAL PROGRAMMING APPROACH
Proceedngs of IICMA 2013 Research Topc, pp. xx-xx. SOLVIG CAPACITATED VEHICLE ROUTIG PROBLEMS WITH TIME WIDOWS BY GOAL PROGRAMMIG APPROACH ATMII DHORURI 1, EMIUGROHO RATA SARI 2, AD DWI LESTARI 3 1Department
More informationChapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
More informationDesign Equations. ν ij r i V R. ν ij r i. Q n components. = Q f c jf Qc j + Continuous Stirred Tank Reactor (steady-state and constant phase)
Desgn Equatons Batch Reactor d(v R c j ) dt = ν j r V R n dt dt = UA(T a T) r H R V R ncomponents V R c j C pj j Plug Flow Reactor d(qc j ) dv = ν j r 2 dt dv = R U(T a T) n r H R Q n components j c j
More informationAn Algorithm for Tuning NMPC Controllers with Application to Chemical Processes
Ths s an open access artcle publshed under an ACS AuthorChoce Lcense, whch permts copyng and redstrbuton of the artcle or any adaptatons for non-commercal purposes. Artcle pubs.acs.org/iecr An Algorthm
More informationPolynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
More informationNote 10. Modeling and Simulation of Dynamic Systems
Lecture Notes of ME 475: Introducton to Mechatroncs Note 0 Modelng and Smulaton of Dynamc Systems Department of Mechancal Engneerng, Unversty Of Saskatchewan, 57 Campus Drve, Saskatoon, SK S7N 5A9, Canada
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
More information10.34 Numerical Methods Applied to Chemical Engineering Fall Homework #3: Systems of Nonlinear Equations and Optimization
10.34 Numercal Methods Appled to Chemcal Engneerng Fall 2015 Homework #3: Systems of Nonlnear Equatons and Optmzaton Problem 1 (30 ponts). A (homogeneous) azeotrope s a composton of a multcomponent mxture
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationCorrelation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationCOEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN
Int. J. Chem. Sc.: (4), 04, 645654 ISSN 097768X www.sadgurupublcatons.com COEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN R. GOVINDARASU a, R. PARTHIBAN a and P. K. BHABA b* a Department
More informationOn the Multicriteria Integer Network Flow Problem
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of
More informationModeling of Dynamic Systems
Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationConstitutive Modelling of Superplastic AA-5083
TECHNISCHE MECHANIK, 3, -5, (01, 1-6 submtted: September 19, 011 Consttutve Modellng of Superplastc AA-5083 G. Gulano In ths study a fast procedure for determnng the constants of superplastc 5083 Al alloy
More informationMATH 5630: Discrete Time-Space Model Hung Phan, UMass Lowell March 1, 2018
MATH 5630: Dscrete Tme-Space Model Hung Phan, UMass Lowell March, 08 Newton s Law of Coolng Consder the coolng of a well strred coffee so that the temperature does not depend on space Newton s law of collng
More informationCHAPTER 4. Vector Spaces
man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear
More informationFUZZY GOAL PROGRAMMING VS ORDINARY FUZZY PROGRAMMING APPROACH FOR MULTI OBJECTIVE PROGRAMMING PROBLEM
Internatonal Conference on Ceramcs, Bkaner, Inda Internatonal Journal of Modern Physcs: Conference Seres Vol. 22 (2013) 757 761 World Scentfc Publshng Company DOI: 10.1142/S2010194513010982 FUZZY GOAL
More informationCHAPTER III Neural Networks as Associative Memory
CHAPTER III Neural Networs as Assocatve Memory Introducton One of the prmary functons of the bran s assocatve memory. We assocate the faces wth names, letters wth sounds, or we can recognze the people
More informationINTRODUCTION TO CHEMICAL PROCESS SIMULATORS
INTRODUCTION TO CHEMICAL PROCESS SIMULATORS DWSIM Chemcal Process Smulator A. Carrero, N. Qurante, J. Javaloyes October 2016 Introducton to Chemcal Process Smulators Contents Monday, October 3 rd 2016
More informationPh.D. Qualifying Examination in Kinetics and Reactor Design
Knetcs and Reactor Desgn Ph.D.Qualfyng Examnaton January 2006 Instructons Ph.D. Qualfyng Examnaton n Knetcs and Reactor Desgn January 2006 Unversty of Texas at Austn Department of Chemcal Engneerng 1.
More informationOn the Interval Zoro Symmetric Single-step Procedure for Simultaneous Finding of Polynomial Zeros
Appled Mathematcal Scences, Vol. 5, 2011, no. 75, 3693-3706 On the Interval Zoro Symmetrc Sngle-step Procedure for Smultaneous Fndng of Polynomal Zeros S. F. M. Rusl, M. Mons, M. A. Hassan and W. J. Leong
More informationTurbulence classification of load data by the frequency and severity of wind gusts. Oscar Moñux, DEWI GmbH Kevin Bleibler, DEWI GmbH
Turbulence classfcaton of load data by the frequency and severty of wnd gusts Introducton Oscar Moñux, DEWI GmbH Kevn Blebler, DEWI GmbH Durng the wnd turbne developng process, one of the most mportant
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationApplication of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems
Mathematca Aeterna, Vol. 1, 011, no. 06, 405 415 Applcaton of B-Splne to Numercal Soluton of a System of Sngularly Perturbed Problems Yogesh Gupta Department of Mathematcs Unted College of Engneerng &
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationParameter Estimation for Dynamic System using Unscented Kalman filter
Parameter Estmaton for Dynamc System usng Unscented Kalman flter Jhoon Seung 1,a, Amr Atya F. 2,b, Alexander G.Parlos 3,c, and Klto Chong 1,4,d* 1 Dvson of Electroncs Engneerng, Chonbuk Natonal Unversty,
More informationTime-Varying Systems and Computations Lecture 6
Tme-Varyng Systems and Computatons Lecture 6 Klaus Depold 14. Januar 2014 The Kalman Flter The Kalman estmaton flter attempts to estmate the actual state of an unknown dscrete dynamcal system, gven nosy
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationRELIABILITY ASSESSMENT
CHAPTER Rsk Analyss n Engneerng and Economcs RELIABILITY ASSESSMENT A. J. Clark School of Engneerng Department of Cvl and Envronmental Engneerng 4a CHAPMAN HALL/CRC Rsk Analyss for Engneerng Department
More informationClock-Gating and Its Application to Low Power Design of Sequential Circuits
Clock-Gatng and Its Applcaton to Low Power Desgn of Sequental Crcuts ng WU Department of Electrcal Engneerng-Systems, Unversty of Southern Calforna Los Angeles, CA 989, USA, Phone: (23)74-448 Massoud PEDRAM
More informationDUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More informationA PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.
Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationAn Interactive Optimisation Tool for Allocation Problems
An Interactve Optmsaton ool for Allocaton Problems Fredr Bonäs, Joam Westerlund and apo Westerlund Process Desgn Laboratory, Faculty of echnology, Åbo Aadem Unversty, uru 20500, Fnland hs paper presents
More informationThe Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method
Journal of Electromagnetc Analyss and Applcatons, 04, 6, 0-08 Publshed Onlne September 04 n ScRes. http://www.scrp.org/journal/jemaa http://dx.do.org/0.46/jemaa.04.6000 The Exact Formulaton of the Inverse
More informationInfluence Of Operating Conditions To The Effectiveness Of Extractive Distillation Columns
Influence Of Operatng Condtons To The Effectveness Of Extractve Dstllaton Columns N.A. Vyazmna Moscov State Unversty Of Envrnmental Engneerng, Department Of Chemcal Engneerng Ul. Staraya Basmannaya 21/4,
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationLossy Compression. Compromise accuracy of reconstruction for increased compression.
Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost
More information