ConvergenceFieldDefaultTo specify1e-8X ToleranceAlternative tolerance on the manipulated variableIterationstops when the changeinthescaled manipulatedvariableislessthan X Tolerance.X Final on ErrorLast valueWhichvalue of manipulatedvariable to useas thefinal value when theconvergence blockencounters an errorOptions are Last value, Initial value, Minimum value of function, Lowerbound, and Upper bound.NoBracketIf the Secant algorithm should switch to a Bracketing algorithm.Bracketing attempts to find a variable range where the design specificationfunction changes sign and performs interval halving when Secant is notmaking progress.When Bracketisspecifiedas No,then Bracketing isnotused.Sincebracketing may add extra iterations, in some cases,particularly withanestedsecant loop,you might want to specify Bracketas No.When Bracketing is specifiedasYes,Bracketing istried if thefunctionisnot changing. The BracketYes option is useful for functions that are flat over a portion of the variable range.When Bracketis specified as Check Bounds, Bracketing is tried if thefunction is not changing orif the Secant algorithmhas moved to a variablebound. The Bracket= Check Bounds option is usefulfor functions that areflatovera portion of the variablerange.It can also beuseful for non-monotonic functions.This option ensures that if the Secant algorithmbecomesstuckatavariablebound,theothervariableboundwillalsobetried.Find Minimum FunctionNot checkedFind the minimum function value if bracketing fails to detect a sign change.Value if Bracketing Fails toDetect a Sign Change.BROYDENMethodTheBroyden method is amodification of Broyden's quasi-Newton method.TheBroydenmethod is similartotheNewtonmethod,but it usesapproximatelinearization.ThisapproximationmakesBroydenfaster,butoccasionallynotasreliable,as theNewtonmethod.17-12AspenPlusUserGuideVersion 10.1-0
17-12 Aspen Plus User Guide Version 10.1-0 Convergence Field Default To specify X Tolerance 1e-8 Alternative tolerance on the manipulated variable Iteration stops when the change in the scaled manipulated variable is less than X Tolerance. X Final on Error Last value Which value of manipulated variable to use as the final value when the convergence block encounters an error Options are Last value, Initial value, Minimum value of function, Lower bound, and Upper bound. Bracket No If the Secant algorithm should switch to a Bracketing algorithm. Bracketing attempts to find a variable range where the design specification function changes sign and performs interval halving when Secant is not making progress. When Bracket is specified as No, then Bracketing is not used. Since bracketing may add extra iterations, in some cases, particularly with a nested secant loop, you might want to specify Bracket as No. When Bracketing is specified as Yes, Bracketing is tried if the function is not changing. The Bracket = Yes option is useful for functions that are flat over a portion of the variable range. When Bracket is specified as Check Bounds, Bracketing is tried if the function is not changing or if the Secant algorithm has moved to a variable bound. The Bracket = Check Bounds option is useful for functions that are flat over a portion of the variable range. It can also be useful for nonmonotonic functions. This option ensures that if the Secant algorithm becomes stuck at a variable bound, the other variable bound will also be tried. Find Minimum Function Value if Bracketing Fails to Detect a Sign Change. Not checked Find the minimum function value if bracketing fails to detect a sign change. BROYDEN Method The Broyden method is a modification of Broyden's quasi-Newton method. The Broyden method is similar to the Newton method, but it uses approximate linearization. This approximation makes Broyden faster, but occasionally not as reliable, as the Newton method
Chapter17UseBroyden to converge tear streams,two or moredesign specifications, ortearstreams and design specifications simultaneously. Broyden is useful for multipletear streams and/ordesign specifications, tear variables thatarehighlyinterdependent, or recycle loops and design specifications so interrelated thatnesting is impractical.Whenconvergingboth tear streams anddesignspecifications,youcan specifythattear streamsbeconverged orpartiallyconverged first.The simultaneous convergence of both tear streams and designspecifications then follows.You can control the Broyden method by specifying:FieldDefaultTospecify30Maximum Flowsheet EvaluationsMaximum number of flowsheet evaluationsX Tolerance0.001Alternative tolerance on the manipulated variablesTheiteration stops when thechange in the scaledmanipulatedvariable is less than X Tolerance2WaitNumberof direct substitution iterationsbeforethefirstaccelerationiterationTear Tolerance (on AdvancedTear tolerance. Used if initializing tears by converging tears (tospecified tolerance)before design specifications are includedParametersdialogbox)Tear Tolerance Ratio (on AdvancedTear tolerance ratio. Used if initializing tears by converging tears (toParameters dialog box)a tolerance relative to the tear tolerance) before designspecificationsareincludedMaximum Iterations (on AdvancedMaximum numberof flowsheet iterations to solvetearsbeforeParameters dialog box)design specifications are included-5Lower Bound (on AdvancedMinimum value for the Wegstein acceleration parameter (q)Parameters dialog box)0Upper Bound (on AdvancedMaximum value for the Wegstein acceleration parameter (g)Parameters dialog box)AspenPlusUserGuide17-13Version10.1-0
Aspen Plus User Guide 17-13 Version 10.1-0 Chapter 17 Use Broyden to converge tear streams, two or more design specifications, or tear streams and design specifications simultaneously. Broyden is useful for multiple tear streams and/or design specifications, tear variables that are highly interdependent, or recycle loops and design specifications so interrelated that nesting is impractical. When converging both tear streams and design specifications, you can specify that tear streams be converged or partially converged first. The simultaneous convergence of both tear streams and design specifications then follows. You can control the Broyden method by specifying: Field Default To specify Maximum Flowsheet Evaluations 30 Maximum number of flowsheet evaluations X Tolerance 0.001 Alternative tolerance on the manipulated variables The iteration stops when the change in the scaled manipulated variable is less than X Tolerance Wait 2 Number of direct substitution iterations before the first acceleration iteration Tear Tolerance (on Advanced Parameters dialog box) Tear tolerance. Used if initializing tears by converging tears (to specified tolerance) before design specifications are included Tear Tolerance Ratio (on Advanced Parameters dialog box) Tear tolerance ratio. Used if initializing tears by converging tears (to a tolerance relative to the tear tolerance) before design specifications are included Maximum Iterations (on Advanced Parameters dialog box) Maximum number of flowsheet iterations to solve tears before design specifications are included Lower Bound (on Advanced Parameters dialog box) -5 Minimum value for the Wegstein acceleration parameter (q) Upper Bound (on Advanced Parameters dialog box) 0 Maximum value for the Wegstein acceleration parameter (q)
ConvergenceNEWTONMethodNEWTON is an implementation of the modified Newton method for simultaneousnonlinearequations.Derivativesarecalculated onlywhen therateofconvergence is not satisfactory.The implementation allows bounds on thevariables,andincludesalinesearchforimprovedstability.NEWTONisusefulwhentherecycleloopsand/ordesignspecificationsarehighlyinterrelated,butconvergenceisnotachieved usingtheBroydenmethod.Numericalderivativesarecalculatedfrequently.UseNEWTONfortear streams onlywhenthenumberofcomponents is small orwhen convergencecannotbeachievedbytheothermethods. When converging both tear streams and design specifications, you canspecifythattearstreamsbeconvergedorpartiallyconvergedfirst.Thesimultaneous convergence of both tear streams and design specifications thenfollows.When you use the Newton or Broyden methods to converge design specifications,andoneormoremanipulatedvariableshavereachedtheirlowerorupperlimitsasolutionisfoundthatminimizesthesumofsquaresofdesignspecificationandtear streamerrors,divided bytheirtolerances.Iterations stop whentherootmean square of the changes in the scaled manipulated variables is less thanX tolerance. Aspen Plus scales each manipulated variable, dividing it by theabsolute value of the lower or upper limit, whichever is larger.You can control the Newton method by specifying:FieldDefaultTo specify30Maximum Newton IterationsMaximum number of Newton iterations9999Maximum FlowsheetMaximum numberof flowsheet evaluationsEvaluations2WaitNumberofdirectsubstitutioniterationsbeforethefirstaccelerationiterationxTolerance0.0001Alternative tolerance on the manipulated variablesThe iteration stops when the change in the scaled manipulated variable islessthanXTolerance0.2Reduction FactorReduction factor which determines the number of Newton iterations usedbeforecalculatinganewJacobian(derivative)matrixWiththis option,theJacobian isreusedas longas itcontinuestodecreasetheerroreachiterationbytheReductionFactorlterations to ReuseNumber of iterations to reuse the Jacobian (derivative) matrixJacobianWiththisoption,theJacobian isreuseda setnumberof timesThe default is to base the reuse of the Jacobian on the Reduction FactorTear Tolerance (onTeartolerance.Usedif initializing tearsbyconverging tears (tospecifiedAdvanced Parameterstolerance) before design specifications are includeddialog box)Continued17-14Aspen Plus User GuideVersion 10.1-0
17-14 Aspen Plus User Guide Version 10.1-0 Convergence NEWTON Method NEWTON is an implementation of the modified Newton method for simultaneous nonlinear equations. Derivatives are calculated only when the rate of convergence is not satisfactory. The implementation allows bounds on the variables, and includes a line search for improved stability. NEWTON is useful when the recycle loops and/or design specifications are highly interrelated, but convergence is not achieved using the Broyden method. Numerical derivatives are calculated frequently. Use NEWTON for tear streams only when the number of components is small or when convergence cannot be achieved by the other methods. When converging both tear streams and design specifications, you can specify that tear streams be converged or partially converged first. The simultaneous convergence of both tear streams and design specifications then follows. When you use the Newton or Broyden methods to converge design specifications, and one or more manipulated variables have reached their lower or upper limits, a solution is found that minimizes the sum of squares of design specification and tear stream errors, divided by their tolerances. Iterations stop when the root mean square of the changes in the scaled manipulated variables is less than X tolerance. Aspen Plus scales each manipulated variable, dividing it by the absolute value of the lower or upper limit, whichever is larger. You can control the Newton method by specifying: Field Default To specify Maximum Newton Iterations 30 Maximum number of Newton iterations Maximum Flowsheet Evaluations 9999 Maximum number of flowsheet evaluations Wait 2 Number of direct substitution iterations before the first acceleration iteration X Tolerance 0.0001 Alternative tolerance on the manipulated variables The iteration stops when the change in the scaled manipulated variable is less than X Tolerance Reduction Factor 0.2 Reduction factor which determines the number of Newton iterations used before calculating a new Jacobian (derivative) matrix With this option, the Jacobian is reused as long as it continues to decrease the error each iteration by the Reduction Factor Iterations to Reuse Jacobian Number of iterations to reuse the Jacobian (derivative) matrix With this option, the Jacobian is reused a set number of times The default is to base the reuse of the Jacobian on the Reduction Factor Tear Tolerance (on Advanced Parameters dialog box) Tear tolerance. Used if initializing tears by converging tears (to specified tolerance) before design specifications are included Continued
Chapter17FieldDefaultTo specifyTear Tolerance Ratio (onTear toleranceratio.Used if initializing tears by converging tears (to atolerance relative to the tear tolerance) before design specifications areAdvanced Parametersdialog box)includedMaximum Iterations (onMaximum number of flowsheet iterations to solve tears before designAdvanced Parametersspecifications are includeddialog box)-5Lower Bound (on AdvancedMinimum value for the Wegstein acceleration parameter (g)Parameters dialog box)0Upper Bound (on AdvancedMaximum value for the Wegstein acceleration parameter (g)Parameters dialog box)COMPLEX MethodYou can usethe Complexmethod to converge optimization problems with boundson themanipulatedvariables and inequalityconstraints.COMPLEX is a directsearch method; it does not require numerical derivatives. It may be useful forsimpleproblems without recycleloops or equality constraints (designspecifications)SQP MethodYou can use the state-of-the-art sequential quadratic programming (SQP)method forflowsheet optimization for simultaneous convergence of optimizationproblems with constraints (equality or inequality)and/or tear streams.Thealgorithm generallyfollows an infeasiblepath (constraints and tear streams areconverged simultaneously with the optimization problem).But you can adjust ittofollowafeasiblepath(convergingthetearstreamsateachiterationoftheoptimization).SQP isused for system-generated optimization convergenceblocks.SQPisrecommendedforuser-generatedconvergenceblocks.SQP-Biegler is an SQPimplementation developedbyProfessor L.Biegler ofCarnegie-Mellon University and his students.AspenPlus UserGuide17-15Version 10.1-0
Aspen Plus User Guide 17-15 Version 10.1-0 Chapter 17 Field Default To specify Tear Tolerance Ratio (on Advanced Parameters dialog box) Tear tolerance ratio. Used if initializing tears by converging tears (to a tolerance relative to the tear tolerance) before design specifications are included Maximum Iterations (on Advanced Parameters dialog box) Maximum number of flowsheet iterations to solve tears before design specifications are included Lower Bound (on Advanced Parameters dialog box) -5 Minimum value for the Wegstein acceleration parameter (q) Upper Bound (on Advanced Parameters dialog box) 0 Maximum value for the Wegstein acceleration parameter (q) COMPLEX Method You can use the Complex method to converge optimization problems with bounds on the manipulated variables and inequality constraints. COMPLEX is a direct search method; it does not require numerical derivatives. It may be useful for simple problems without recycle loops or equality constraints (design specifications). SQP Method You can use the state-of-the-art sequential quadratic programming (SQP) method for flowsheet optimization for simultaneous convergence of optimization problems with constraints (equality or inequality) and/or tear streams. The algorithm generally follows an infeasible path (constraints and tear streams are converged simultaneously with the optimization problem). But you can adjust it to follow a feasible path (converging the tear streams at each iteration of the optimization). SQP is used for system-generated optimization convergence blocks. SQP is recommended for user-generated convergence blocks. SQP-Biegler is an SQP implementation developed by Professor L. Biegler of Carnegie-Mellon University and his students
ConvergenceYoucan control theSQPmethod by specifying:FieldDefaultTo specify30Maximum numberofSQPoptimization iterationsMaximumOptimization Iterations999Maximum Flowsheet EvaluationsMaximum number of flowsheet evaluationsEachperturbationstepfornumericalderivativesiscountedasone evaluation.2Additional lIterations when ConstraintsNumber of aditional iterations when constraints are notare not Satisfiedsatisfied after the convergencetest is satisfied.3Iterations toConvergeTearsforEachNumber of iterations to take toward converging tears at eachOptimization Iterationiteration of the optimization3Iterations to Enforce Maximum StepNumberofiterationstoenforcemaximumstepsizeontheSizemanipulated variablesTolerance0.001OptimizationconvergencetoleranceWait1Numberofdirectsubstitutioniterationsbeforethefirstacceleration iteration-5Lower BoundMinimum valuefortheWegstein acceleration parameter (g)0Upper BoundMaximum valuefortheWegsteinacceleration parameter (g)SQPWegsteinAccelerationParametersWhen the SQP method is used to converge tears and optimization problemssimultaneously,the algorithm is a hybrid of an infeasible path method (wherethe tears are not converged at each iteration but are converged at the optimum)and a feasible path method (where the tears are converged at each iteration ofthe optimization). You may control the degree to which the tears are convergedby specifying thenumber of iterations to take toward converging thetears(iterations To Converge Tears Each Optimization Iteration) and upper and lowerlimitsfortheWegstein accelerationparameterfortheWegsteiniterations(Upper Bound, Lower Bound).Specifying Convergence OrderYou can specify the calculation order of convergence blocks you define if you usemore than one user-defined convergence block. Specify the convergence order ontheConvOrder Specification or Sequence Specifications sheet.17-16Aspen Plus User GuideVersion 10.1-0
17-16 Aspen Plus User Guide Version 10.1-0 Convergence You can control the SQP method by specifying: Field Default To specify Maximum Optimization Iterations 30 Maximum number of SQP optimization iterations Maximum Flowsheet Evaluations 9999 Maximum number of flowsheet evaluations Each perturbation step for numerical derivatives is counted as one evaluation. Additional Iterations when Constraints are not Satisfied 2 Number of additional iterations when constraints are not satisfied after the convergence test is satisfied. Iterations to Converge Tears for Each Optimization Iteration 3 Number of iterations to take toward converging tears at each iteration of the optimization Iterations to Enforce Maximum Step Size 3 Number of iterations to enforce maximum step size on the manipulated variables Tolerance 0.001 Optimization convergence tolerance Wait 1 Number of direct substitution iterations before the first acceleration iteration Lower Bound -5 Minimum value for the Wegstein acceleration parameter (q) Upper Bound 0 Maximum value for the Wegstein acceleration parameter (q) SQP Wegstein Acceleration Parameters When the SQP method is used to converge tears and optimization problems simultaneously, the algorithm is a hybrid of an infeasible path method (where the tears are not converged at each iteration but are converged at the optimum) and a feasible path method (where the tears are converged at each iteration of the optimization). You may control the degree to which the tears are converged by specifying the number of iterations to take toward converging the tears (Iterations To Converge Tears Each Optimization Iteration) and upper and lower limits for the Wegstein acceleration parameter for the Wegstein iterations (Upper Bound, Lower Bound). Specifying Convergence Order You can specify the calculation order of convergence blocks you define if you use more than one user-defined convergence block. Specify the convergence order on the ConvOrder Specification or Sequence Specifications sheet