11 March 2009ICVT, Universitat Stuttgart

MAX−PLANCK−INSTITUT

TECHNISCHER SYSTEMEMAGDEBURG

DYNAMIK KOMPLEXER

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DIANA — A short introduction

Michael Krasnyk

Max Planck Institute for Dynamics of Complex Technical Systems, PSPD groupOtto-von-Guericke-University, IFAT

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Motivation

High demand for first principle modeling of chemical processes

Complexity of processes and models

Structure of implemented models

Goals of computer-based modeling

computer-aided process engineering

systematical modeling approaches

reusable and transparent models

Physically motivated concepts for structuring of balance based models

Ponton (1991), Marquardt (1996), Gilles (1997), Mangold (2002)

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Modeling tool ProMoT

Equation-based modelingdifferential-algebraic systems (differential index 1)Petri networks

object-oriented concepts for modelsmultiple inheritanceaggregation

model implementationtext-based in modeling language MDLgraphical modeling with GUI

Equation analysis and optimizationelimination of explicit algebraic relationsstructure analysis of the complete systemfor solvability

Symbolic differentiation in ProMoT

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Simulation tool Diana

Promot

Diana

Python

C++ model

(define-module :class "Test" ... )

SWIG

CapeDAESOCapeDAESOSolverPetriNetworksContinuation

import dianaimport solvermod=Model("Test") ...

Data

Model descriptionInitial valuesSimulation results

UserXML

Make

Create

Run

GUI

PyGTKNumPySciPy

Current activity

Future plans

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Diana architecture

C++ Model

- encapsulates model data: - Parameters - State Variables- Provides interface to - Equations/Help-Vars - Event Functions - Petri network - Assertions

Model Loader

- loads a model from a file

- calls model-specific initialization routines

Result Files

Solver Factory

- loads numerical algorithms from file

Diana Numerical Algorithms

Integrators

Solvers

Event Handlers

Optimizers

Continuators

Sensitivity Analysers

Parameter Analysers

Numerical Libraries

LAPACKBLAS

SUNDIALS

Umfpack Harvell

ARPACK

Python Scripting- Control of simulator activity- Model Data Access- Callback Functions

SciPy/ Numeric- Scientific Programming in Python- Matlab-like Functionality

Python Python-embedded shared libraries

Swig-Wrapping

Online Plotter

Gnuplot / wxPython- Online / Offline Plots

- Selection of Variables- Generation of standard Plots

Output Service

- collection of data during calculation- Access by scripts/plotting- Output to Simulation-Log

Model FilesPromot

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Diana dependencies

Required dependenciesgcc (version > 3.3.1)python (version > 2.4)swig (version > 1.3.31)xerces (version > 2.7.0)cppunit (version > 1.11.0)

External numerical libraries:Required linear algebra libraries

BLAS http://www.netlib.org/blas/LAPACK http://www.netlib.org/lapack/UMFPACK http://www.cise.ufl.edu/research/sparse/umfpack/

Optional linear algebra libraries

ARPACK http://www.cse.scitech.ac.uk/nag/hsl/Harwell Subroutine Library http://www.cse.scitech.ac.uk/nag/hsl/

Differential algebraic solvers

IDA/Sundials http://acts.nersc.gov/sundialsDASPK http://www.cs.ucsb.edu/~cse/software.html

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Diana installation

Latest sources can be found here:http://promottrac.mpi-magdeburg.mpg.de/dist/

Doxygen-generated documentation for the Diana:http://promottrac.mpi-magdeburg.mpg.de/doc/Diana/

Build and install:

./ config/bootstrap

./ configure --prefix=<installation prefix > \--with -ufsparse=<ufsparse installation dir > \--with -arpack=<arpack lib path > \--with -blas=<blas lib path > \--with -lapack=<lapack lib path > \--with -sundials=<sundials installation dir > \--with -cppunit -prefix=<cppunit installation dir > \--with -xerces=<xerces installation dir > \--with -mdl2diana=<mdl2diana path >

makemake checkmake install

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Simulation models

CAPE-OPEN interface extension IDianaDAESO defines a dynamical model asa DAE system in implicit form

f (t, x , x , ν) = 0, f : R × Rn × R

n × Rp → R

n ∈ C∞

extension interface gives an access to derivatives of the model

∂k+l+m f (t, x , x , ν)

∂kx ∂ l x ∂mν, k, l , m > 0,

that are obtained with help of CAS Maxima

ProMoT command mdl2diana produces C++ model code

mdl2diana <module > [-f <mdl -file >] [-d <diana -name >] [-c][-g <generation -dir >] [-sd <triples of integers >][-sd -sing] [-sd -sens]

Diana command dianac compiles a model to shared library

dianac <model_name > [--clean] [--rebuild] [-O <OPT >]

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Simulation model interface

Interface methodsGet/Set(All)[Variables|Derivatives|Parameters] access methods to thestate vector x , derivatives vector x and parameters vector ν.

Get(All)Residuals methods return the residual vector f

Get[Lower|Upper]Bounds methods return user defined maximal and minimalvalues for the state variables

Get/SetIndependentVar are access methods to the independent variable t

Get(All)(Diff|Par)JacobianValues methods return values of the Jacobianmatrices ∂f /∂x , ∂f /∂x or ∂f /∂ν

GetHighOrderJacobian method returns values of the higher order Jacobian ma-

trix ∂(k)f /∂{x , x , ν}(k)

Save/LoadState save and restore a state of the ESO instance

ESO additional parametersSymbolicJacobian shows whether a symbolically or numerically computed deriva-tives will be returned by Get(*)JacobianValues

FDPartition, FDOrder, FDEpsilon numerical differentiation parameters

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Model: Continuous Stirred Tank Reactor

cin

qin

c

T , c

The mass balances of the modelread [Zeyer et al., 1999]

cH2O2= qin/V (cH2O2,in − cH2O2

) − (r1 + r2 + r3)

cCH3CHO = qin/V (cCH3CHO,in − cCH3CHO) + (r1 − r2)

cCH3COOH = qin/V (cCH3COOH,in − cCH3COOH) + r2

ccat = qin/V (ccat,in − ccat) − (r4 − r5)

Reaction rates ri , i = 1, . . . , 5 are2

6

6

6

6

4

k1e−E1/(RT )ccat cH2O2

k2e−E2/(RT )ccat cH2O2

cCH3CHO

k3e−E3/(RT )ccat cH2O2

k4e−E4/(RT )ccat

√cCH3CHO

k5e−E5/(RT )(cF ,ges − ccat)

3

7

7

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5

The energy balance isV ρcpT = ρcp qin(Tin − T ) + (UA)cool(Tcool − T ) + V

P3i=1 ri (−∆hR)i

VcoolρcpTcool = ρcp qcool(Tcool,in − Tcool) + (UA)cool(T − Tcool)

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MDL simulation model

(define-module :class"Hafke_Reactor" :super-classes("module" )

:documentation "Ruehrkesselreaktor zur Oxidation von Ethanol zu Essigsaeure"

:variables(

("NC" :documentation "Anzahl der (C)Komponenten"

:system-theoretic "structure-parameter" :value "4")

("i" :system-theoretic "index")

("Tkzu" :documentation "Zulauftemperatur Kuehlmittel [K]"

:system-theoretic "real-parameter" :value "314.0")

("qknormal" :documentation "Zulaufstrom Kuehlmittel [m^3/s]"

:system-theoretic "real-parameter" :value "4.44 d-05" )

...

("c_" :documentation "Konzentration der Komponenten P E S F [mol/m^3]"

:is-a "variable" :system-theoretic "state"

:indices ((: index "i" :lower "1" :upper "NC"))

:value "0.1" :minimum "0.0" :maximum "3000.0" :absolute-error "1.0 D-09")

("Temp" :documentation "Temperatur Reaktionsgemisch [K]" ... )

("Tk" :documentation "Temperatur Kuehlmittel [K]" ... )

("R_reak" :documentation "Reaktionsgeschwindigkeit der Teilreaktion"

:indices ((: index "i" :lower "1" :upper "NR"))

:is-a "variable" :system-theoretic "help"

:value "k_reak[i] * :cond(i==1, cP_cF ,

i==2, cP_cF * c_[2],

i==3, cP_cF ,

i==4, c_[4] * :sqrt(c_[3]),

c_F_ges - c_[4])")

... )

:equations(

("KMB_Eq" :is-a "equation"

:indices ((: index "i" :lower "1" :upper "NC"))

:relation ":diff <t> c_[i] == qVR * (c_zu[i] - c_[i]) + RCumsatz[i]")

("Temp_Eq" :is-a "equation"

:relation ":diff <t> Temp == qVR*(Tzu - Temp) + kappa/VR/rho_cp *(Tk - Temp)

+ :sum <i,1,NSUM >( R_reak[i]*DHR[i])/ rho_cp")

("Tk_Eq" :is-a "equation"

:relation ":diff <t> Tk == qVk*(Tkzu - Tk) + kappa/Vk/rho_cp *(Temp - Tk)" ) ) )

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Simulation models in Python

Diana interpreter is launched with

diana [<python arguments >] [<script name >] [<arguments >]

Diana module import and initialization

import dianamain=diana.GetDianaMain ()

model loaded as a shared library

mm=main.GetModelManager ()model=mm.CreateModel (CAPE_CONTINUOUS , "HafkeReactor.so")

acquiring equation set object (ESO) with references to states and parameters

eso=model.GetActiveESO ()evar=eso.GetStateVariables ()epar=eso.GetRealParameters ()

access to ESO data

eso.GetAllVariables ()epar[’tkzu’]. GetValue ()evar[’tk’]. SetValue (300.0)eso.GetAllJacobianValues ()

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Dynamic simulation

Dynamic solvers find solutions of the Cauchy problem ϕ(t, x0, ν), such that

f (t, ϕ, ϕ, ν) ≡ 0, s.t. ϕ(t0, x0, ν) = x0

Dynamic simulation is presented by the following solvers

IDASolver and DASPKSolver for implicit DAE systemswith differential index 1 (libraries ida.so and daspk.so)

OdessaSolver for ODE systems with ∂f /∂x = I (library odessa.so)

The linear systems in the integrators are solved by the direct dense LAPACK orsparse UMFPACK linear algebra solvers

For integrating a DAE initial-value problem, an important requirement is that thepair of vectors x0 and x0 are both initialized to satisfy the DAE residual

f (t0, x0, x0, ν) = 0

For semi-explicit differential index-one systems, IDA provides a routine thatcomputes consistent initial conditions from a user’s initial guess

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Dynamic solver interface

Interface methodsGetParameters method returns collections of solver parameters

Solve method starts the solution of an ESO that is associated with the solver

GetSolution method returns the solution vector x

Solver parametersStart parameter specifies the start of a new simulation (True) or continuationof the current one (False)

CalcIC parameter controls whether consistent initial conditions are computed atthe initial time (True) or not (False)

T is the current value of the independent variable t

T0 is the starting value of the independent variable

Tend is the final value of the independent variable

Intermediate parameter with the True value tells the solver to take one internalstep and to return the solution at the point reached by that step, otherwiseintegration proceeds to the parameter value Tend without interruption

VerboseLevel controls a verbosity level of the solver

LASolver specifies whether the dense or sparse linear algebra solver will be used

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Dynamic simulation in Python

loading of IDA solver form SUNDIALS package

sf=main.GetSolverFactory ()solver=sf.CreateSolver(CAPE_DAE , model , ’ida.so’)solver.Initialize ()spar=solver.GetParameters ()

initializing of a reporting interface and linking to the solver

ri=main.CreateReportingInterface (’basic ’)solver.SetReportingInterface (ri)ri.Add(spar[’T’])ri.Add(epar[’qknormal ’])

setting of solver parameters

spar[’VerboseLevel ’]. SetValue (2)spar[’T0’]. SetValue (0)spar[’Tend’]. SetValue (1000.0)

performing of a simulation and storing of output data to ’out.m’

epar[’qknormal ’]. SetValue (1e-4)solver.Solve ()ri.WriteDataMatlab (’DynamicSimulation /Example.m)

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Dynamic simulation

plotting of the dynamic simulation in Matlab

run(’DynamicSimulation/Example.m’)figure; hold on; box on; grid on;plot(data1 (:,1), data1 (:,2))figure; hold on; box on; grid on;plot(data1 (:,1), data1 (:,7))

results of the tutorial case tutorial/hafke/simulation

0 1 2 3 4 5 6×1040

0.5

1

1.5×10−4

t, [s]

qcool,

[m3/s]

0 1 2 3 4 5 6×104300

320

340

360

380

400

420

t, [s]

T.[K

]

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Parameter continuation and nonlinear analysis

Parameter continuation is presented by the following

continuation of steady-state points and analysis of local stability (library sstate.so)

continuation of limit point curves and analysis of singularities of such points(library sanalyser.so)

continuation of Hopf points (library hopf.so)

continuation of periodic solutions and analysis of local stability based on Flo-quet multipliers (library periodic.so)

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Continuation solver interface

Interface methodsSolve method solves a nonlinear task f (x , ν) = 0 for constant parameter νContinuate method performs a continuation with respect to parameter νAdd/RemoveFreeParameter method adds or removes λ to the nonlinear system

Continuation solver parametersParametrization parameter specifies the parametrization type (PseudoArclengthor Local)Predictor is the predictor type (Tangent or Chord)

StepSize is the current step size σ(k)

InitialStepSize is the initial step size σ(1)

InitialDirection is the initial direction of a continuationMinStepSize is the minimal step size σmin

MaxStepSize The is the maximal step size σmax

MaxStepsAmount is the maximal number of steps kmax

Tol relative tolerance in the argument spaceSteady-state continuation solver parameters

StabilityCheck, Stability parameters specify local stability check and stabilityof the current pointConditionCheck check for limit point SteadyStateZCE or Hopf point SteadyStateZREconditions

Singularity analysis solver parametersConditionCheck check for zeros of test functions (SingularityG[x|xx|p|xp])Gx,Gxx,Gp,Gxp values of test function derivativesConditionEquations set of adjoint test function equations

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Parameter continuation of steady-states

create model with higher order derivatives (mdl2diana option -sd-sing)

loading of continuation solver

sf=main.GetSolverFactory ()conti=sf.CreateSolver(diana.CAPE_CONTI , model , "sstate.so")conti.Initialize ()cpar=conti.GetParameters ()

initializing of a reporting interface and linking to the solver

setting of solver parameters

cpar[’VerboseLevel ’]. SetValue (0)cpar[’MaxStepsAmount ’]. SetValue (5000)cpar[’MaxStepSize ’]. SetValue (5.0)cpar[’StabilityCheck ’]. SetValue(True)cpar[’ConditionCheck ’]. SetValue(diana.SingularityNone )

adding of a continuation parameter

conti.AddFreeParameter("tkzu", 250.0, 400.0)

performing of a continuation

conti.Continuate ()

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Parameter continuation of steady-states

Results of the tutorial case tutorial/hafke/continuation

285 290 295 300 305 310290

300

310

320

330

340

350

360

Tcool,in [K]

T[K

]

290 292 294 296 298 30030

35

40

45

50

55

60

65

Tcool,in [K]

qcool[l/h]

0.5 1 1.5 2 2.5 3 3.5300

305

310

315

320

325

330

335

qcool [l/h]

T[K

]

4.6 4.8 5 5.2 5.4 5.6 5.8×10−7310

312

314

316

318

qcool [l/h]

T[K

]

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Optimization Problems Supported in Diana

Optimization of explicit functions

Parameter EstimationMaximum likelihood approach for Gaussian white noise

Optimal Experimental DesignSigma point approach

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Global Nonlinear Continuous Optimization Packages

Global Random Search:

Genetic

Quite flexible implementation of Genetic Algorithms with self-tuning of ran-domization parameters

Statistical Global Optimization:

DIRECT v2.0

Dividing rectangular global optimization method by D. R. Jones and J. Gablon-sky

BBOWDA

Black box global optimization method with data analysis by K. Kofler

GMFL

routine bayes1

Bayesian global optimization methodroutine unt

The global method of extrapolation type by A. Zilinskasroutine lbayes

The local Bayesian method by J. Mockus

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Local Nonlinear Continuous Optimization Packages

Non-gradient Methods

routine NMSimplex

Nelder-Mead downhill simplex method implementation by D. E. Shaw

Gradient Based Methods

routine L-BFGS-B

Limited-memory quasi-Newton code for large-scale bound-constrained or un-constrained optimization by C. Zhu and J. Nocedal.

IPOpt

Package for large-scale nonlinear optimization of continuous systems, imple-ments a primal-dual interior point method, and uses line searches based onFilter methods and Hessian approximation using BFGS update.

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Capabilities of optimizers in Diana

FeatureSO-library

gen

etic

direc

t

bbow

da

bay

es1

unt

lbay

es

nm

sim

ple

x

lbfg

sb

ipopt

Global search X X X X XStochastic objectives X X X X X X

RequirementsNeed of gradient X X

Support of constraints handlingbound constraints X X X X X X X Xdirect constraints X Xinequality constraints X X X Xequality constraints X X

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Running Parameter Estimation

Create optimizer

sfactory = main.GetSolverFactory ()solverPE = sf.CreateSolver(diana.CAPE_NLP , None , ’direct ’)

Load measured data from specified file (details [Schenkendorf et al., 2009])

md = DianaMeasuredData (model , ["c_x", "c_s"])md.load(’./ observations.dat’)

Specify estimated parameters

sps = [diana.DianaNLPRealParameterSpec ("k_s", "k_s", 1., 1., 3.),diana.DianaNLPRealParameterSpec ("mu_max", "mu_max", 4., 4., 6.)]

Create parameter fitting task

taskPE = diana.ParameterFittingTask (main , ’ida.so’, model , md , sps)

Assign task to optimizer and run optimization

solverPE.SetReportingInterface (report );solverPE.SetNLPTask(taskPE );solverPE.Initialize ();solverPE.Solve ()

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Running Optimal Experimental Design

Describe design variables

colDesignVars = [diana.DianaNLPRealParameterSpec ("q_0", "q_0", 0.09, 0.07, 0.08)

]

Create OED task

taskOED = nlptaskfactory .CreateOEDTask (taskPE , solverPE , ...colDesignVars , ’oedsigmapoint ’)

oedpar=taskOED.GetParameters ()# use E* optimality criterionoedpar["OptimalityCriterion "]. SetValue(diana.DIANA_OED_EStar )taskOED.Initialize ()

Assign task to optimizer and run optimization

solverOED.SetReportingInterface (report );solverOED.SetNLPTask(taskOED );solverOED.Initialize ();solverOED.Solve ()

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Contact

Promot/Diana homepagehttp://www.mpi-magdeburg.mpg.de/projects/promot/

ProMoT and network theory questionsMichael Mangold mangold@mpi-magdeburg.mpg.de

ProMoT and GUI questionsSebastian Mirschel mirschel@mpi-magdeburg.mpg.de

Optimization in DianaSergiy Gogolenko gogolenk@mpi-magdeburg.mpg.de

Diana questionsMichael Krasnyk miha@mpi-magdeburg.mpg.de

Thank you for your attention!

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References

Schenkendorf, R., Kremling, A., and Mangold, M. (2009).Optimal experimental design with the sigma point method.IET Systems Biology, 3(1):10–23.

Zeyer, K. P., Mangold, M., Obertopp, T., and Gilles, E. D. (1999).The iron(III)-catalyzed oxidation of ethanol by hydrogen peroxide: a thermoki-netic oscillator.Journal of Physical Chemistry, 103A(28):5515–5522.

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