03a - Initialization

8/13/2019 03a - Initialization

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Initialization of Thermofluid modelsInitialization of Thermofluid models

Francesco CasellaFrancesco Casella([email protected])([email protected])

Dipartimento di Elettronica e InformazioneDipartimento di Elettronica e Informazione

Politecnico di MilanoPolitecnico di Milano


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Introduction

Initialization is critical for models with dynamics

solution of ODE/DE depends on initial conditions

if consistent initial conditions cannot !e found" the model is useless

#hermofluid models often stron$ly nonlinear

%teady&state initialization in'ol'es sol'in$ lar$e" stron$ly nonlinear

systems of e(uations

sensiti'ity to initial $uess 'alues

sensiti'ity to selection of iteration 'aria!les

tracin$ root of pro!lem difficult

$uess 'alues)

errors in the model *e+$+ wron$ parameters,)

#rou!leshootin$ is not user friendly

understandin$ of the sym!olic/numeric methods understandin$ the e(uations of the ori$inal model *not encrypted-,

e(uations are rearran$ed / sym!olically sol'ed

no a'alia!le Modelica tool currently $i'es user friendly .I support

unpredicta!le time&to&$o


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Specifying initial conditions - defaults

Each 'aria!le in Modelica has a startattri!ute

0y default" Modelica tools use start 'alues to assi$n initial 'alues to state

'aria!les

It is possi!le to lo$ these additional initial e(uations*%imulation 1 %etup 1 #ranslation 1 2o$ selected default initial conditions,

modelSimpleSystem

parameterTime tau;

parameterReal x_start = 2; Real x(start = x_start); Real u;equation

u = if time < 1 then 1 else 2;

tau * der(x) = - x^2 + u;endSimpleSystem


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Specifying initial conditions fixed attribute

%ettin$ the fi3ed attri!ute to true one indicates e3plicitly that the initial

'alue of the 'aria!le is the initial 'alue

modelSimpleSystem2

parameterTime tau;

parameterReal x_start = 2; Real x(start = x_start fixe! = true); Real y;

Real u;equation


tau * der(x) = - x^2 + u; y = "*x^2;endSimpleSystem2


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Specifying initial conditions initial equations

In Modelica 4+3" initialiation was mainly thou$ht as a tool issue

Dymola had a run&time interface to specify initial conditions a define free

and fi3ed 'aria!les *usin$ data from fi3ed and start attri!utes,

%ince Modelica 5+6 *5665,7 e3plict initial e(uations with ar!itrary structure

*the old fi3ed 8 true mechanism is still a'aila!le,

Initial e(uations are com!ined with dynamic e(uations

9onlinear sol'ers use start attri!ute for $uess 'alues

modelSimpleSystem

parameterTime tau;

parameterReal x_start = 2; Real x(start = x_start); Real y; Real u;equation


tau * der(x) = - x^2 + u; y = "*x^2;initial equation

x = x_start;endSimpleSystem

modelSimpleSystem

parameterTime tau;

parameterReal x_start = 2; Real x(start = x_start); Real y; Real u;equation



der(x) = #;

endSimpleSystem


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Specifying initial conditions Unknown parameters

Parameters can !e freed and determined !y further initial e(uations

*!eware of initial e(uation count+++,

Fi3ed attri!utes can !e chan$ed at instantiation

*%how Component feature in the Dymola .I,

modelSimpleSystem

parameterTime tau;

parameterReal x_start = 2;

parameterReal y_start = 1#;

parameterReal u_start(fixe! = false start = 2); Real x(start = x_start); Real y;

Real u;equation

u = if time < 1 then u_start else u_start + 1;


der(x) = #; y = y_start;endSimpleSystem

System S(u_start(start = 2 fixe! = false) y(start = " fixe! = true));


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eaningful initial conditions for thermofluid systems

In some cases" the initial state of a thermofluid system

con'eniently e3pressed !y $i'in$ the 'alues of initial states

E3ample7 direct steam $eneration solar plant :ust !efore dawn temperature distri!ution in the collector pipes 8 am!ient temperature

$i'en temperatures in the stora$e 'olumes"

dependin$ on len$th of shutdown period

zero *or ne$li$i!le, flows *!eware of sin$ular conditions,

In other conte3ts" startin$ from an ;off&state< is pro!lematic/not re(uired

E3ample7 power plant model for load chan$e or primary fre(uency

studies

ran$e of 'alidity =6&466>

pipin$ and instrumentation for startup not included in the model

*tur!ine !ypass systems" steam 'ents" etc+,

In these cases" the typical transient is the response to a distur!ance

startin$ from an e(uili!rium state *steady&state,

Fi3in$ appro3imated initial states can lead to unphysical transients

out&of&ran$e mass ? heat flow 'alues

flow re'ersal

trip conditions +++


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!uick re"iew of odelica model transformations

%teps of model transformation

flattenin$

inde3 reduction

elimination of tri'ial e(uations / alias 'aria!les

02# transformation

dynamic pro!lem e(uations *initial 'alues and parameters,

initialization pro!lem *deri'ati'es" al$e!raic 'aria!les,

#earin$ applied to implicit systems of e(uations to reduce the num!erof iteration 'aria!les

E(uations sol'ed as e3plicit assi$nments @ no pro!lem

Implicit linear e(uations @ sol'ed without pro!lem if non&sin$ular

Implicit nonlinear e(uations @ iterati'e methods @ reasona!le $uesses

re(uired for the iteration 'aria!les

.uess 'alues pro'ided in Modelica !y start attri!utes

defined !y types

defined !y modifiers in li!rary models" possi!ly throu$h parameters

defined !y direct modifiers on the simulation model


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!uick re"iew of odelica model transformations #cont$d%

.uess 'alues pro'ided in Modelica !y start attri!utes

Defined !y types

Defined !y modifiers in li!rary models" possi!ly throu$h parameters

Defined !y direct modifiers on the simulation model

type Temperature = Real(unit = $%$ start = "##);

S&'ressure p_start;

S&'ressure p(start = p_start);

modellantithStartttriutes

extendslant(

turine(p_in(start = 1'2,e)) !rum.(p(start = 1'"2e)));endlantithStartttriutes;

S&'ressure p_start;

S&'ressure p(start = p_start);


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Tearing & 'lias "ariables

small fraction of model 'aria!les re(uire a $uess 'alue to ensure sol'er

con'er$ence

AB6> alias 'aria!les *3 8 y,

A=6> assi$ned in assi$nment section of torn systems

%election of alias and iteration 'aria!les in torn systems not uni(ue-

lias 'aria!le selection in Dymola7

hi$hest priority to direct modifiers"

intermediate priority to modifiers in li!raries

lowest priority to type&defined start 'alues

#earin$ 'aria!le selection

optimal selection7 9P&complete pro!lem

proprietary heuristics

chan$es w/ small modifications of model

chan$es w/ tool 'ersion

9ot really o!:ect oriented-


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Understanding initialization failures in (ymola

Interacti'e demo

dsmodel+mof7 how to $enerate" how to read

dslo$+t3t7 how to $enerate" how to read


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Strategies for steady-state initialization

Di'ide and con(uer

!rea plant in smaller modules

initialize each model w/ suita!le !oundaries

mae sure each model is correctly parameterized

Chec dsmodel+mof/dslo$+t3t and impro'e start 'alues

Fi3 initial states" run transient until e(uili!rium

re(uires asymptotically sta!le e(uili!rium

*if not" use simple feed!ac controllers to sta!ilize,

transient mi$ht $o wron$ !ecause of a!normal mass and ener$y

flows due to unphisical 'alues of initial states

e3ample7 e'aporator with hi$h heat transfer coefficient

small error in delta fluid&wall @ lar$e error in heat flow @

condensation/e'aporation @ !ad flow rates *possi!ly flow re'ersal,

Important remar7 if the parametrization of the model is incorrect"

a steady&state solution mi$ht not e3ist-


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'n example of model without steady-state solution

9atural circulation steam $enerator

9ominal operatin$ data7

drum pressure 45 !ar

feedwater pressure 4= !ar

steam 'al'e outlet pressure 6 !ar

power from flue $as7 466 MG

steam flow7 H6 $/s

%team flow rate rou$hly proportional tosteam 'al'e C' and drum pressure

ssume now C' is :ust 46> smaller than

the correct 'alue

in order to e'acuate H6 $/s

drum pressure should !e 46 !ar

feedwater cannot flow from source at lower pressure-

%olution does not e3ist-


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Troubleshooting

%et initial 'alues of pressures and

enthalpies close enou$h to nominal

operatin$ point #he system has an asymptotically sta!le

e(uili!rium thans to le'el controller

%tart the simulation

Drum pressure increases steadily

%team flow JJ feedwater flow

#oo small steam 'al'e

Increase steam 'al'e C'

#o find the correct C'7

set C'*fi3ed 8 false,

set steamKflow*start 8 H6" fi3ed 8 true,


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)omotopy-based initialization* oti"ations

%ettin$ start 'alues

tedious

selection of iteration 'aria!les can chan$e

how to determine $ood enou$h 'alues)

not o!:ect oriented

not ro!ust

re(uires hi$h&end sills to trou!leshoot models

more ro!ust method re(uired

$uarantee con'er$ence

use only well&nown desi$n data

reasona!le defaults for iteration 'aria!les

no need of directly settin$ start 'alues o!:ect&oriented

Lomotopy&!ased initialization


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)omotopy-based initialization* basic ideas

%implified pro!lem sol'ed first

same 'aria!les" sli$htly modified e(uations

close enou$h to the actual one

steady&state *otherwise structure can chan$e too much-,

simpler to sol'e *;less nonlinar< @ less sensisti'e to $uess 'alues,

$uess 'alues possi!ly pro'ided !y default

%ol'e con'e3 com!ination of simplified and actual pro!lem

Ca'eat7 the pro!lem chan$es continuously" the solution mi$ht not turnin$ points *can !e handled !y

specialized sol'ers,

asymptotes *failure,

!ifurcations *re(uires interaction

with e3pert user,

%pecification in Modelica =+57

homotopy operator

h/m/t/py(a0tual simplifie!)

durin$ solution of initialization returns

actual N *4 )simplified


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Simplified models for thermofluid systems

2inear momentum !alance e(uations with constant coefficients !ased on

nominal operatin$ point

pressure loss components

static head term

steam tur!ine

pump7 linearized characteristic around nominal operatin$ point

ressure l/ss 0/mp/nent

pin - p/ut = homotopy(sm//th(1 %f*suareRe3(44n/m*4nf))rh/ !pn/m4n/m*4) 56l/4 0hara0teristi0s5;

7al8e f/r in0/mpressile flui!4 = homotopy(6l/49har(theta)*8*srt(rh/)*srtR(!p) thetathetan/m*4n/m!pn/m*!p);

ump

function!f_! = der(fl/49hara0teristi0 _fl/4);hea! = homotopy( (nn#)^2*fl/49har(*n#(n+n_eps)) !f_!(#)*(-#) + (2n#*fl/49har(#) - #n#*!f_!(#))*(n-n#) + hea!#);

Turine4 = homotopy(%t*partialr0*srt(p_in*rh/_in))*srtRe3(1-(1R)^2) 4n/mpn/m*p_in);


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Simplified models for thermofluid systems #cont$d%

Ener$y !alance e(uation7 nonlinear wh terms

use nominal flow rate for ener$y !alance in 4D models for heat

e3chan$ers $ets rid of most wh terms in a typical model

pstream specific enthalpy dependin$ on flow direction

assume desi$n direction of flow

Flow&dependent heat transfer coefficients

assume nominal h+t+c+ 'alue

#emperature&specific enthalpy relationship 9ot too nonlinear in most cases *sa'e phase chan$es,

Do not chan$e *difficult to mae it consistent across models,

%et rou$h start 'alues

li(uid / two&phase / 'apour for water

within 466&566 for ideal $ases

hi = homotopy(if n/t all/46l/4Re8ersal then inStream(inlet'h_/utfl/4) else a0tualStream(inlet'h_/utfl/4) inStream(inlet'h_/utfl/4));

4all'3amma: = h/m/t/py(3amma_n/m*n/8ent(as(infl'm_fl/44n/m)^>4) 3amma_n/m);


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Simplified models for thermofluid systems #cont$d%

Controllers actin$ on flows and controllin$ ener$y&related (uantities

may introduce stron$ nonlinear couplin$ !etween hydraulic and

thermal e(uations simplified model7 eep control 'aria!le fi3ed at nominal 'alue

*opened feed!ac loop,

Controllers with saturations and anti&windup

%aturations are stron$ nonlinearities

If nown to !e in linear re$ime7 remo'e saturations

Else7 fi3 at 6> or 466>

Control 'al'es

ssume flow is nominal flow openin$

*re$ardless of density and pressure drop,

Initialization not at nominal operatin$ point

%implified model has all setpoints at nominal 'alues

Lomotopy !rin$s the setpoints to the desired 'alue

Different from rela3ation transient7 (uasi&static transformation

*less pro!lems if initial $uess not accurate no weird transients,


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+erforming the homotopy transformation

%uita!le continuation sol'ers can !e used

can trac turnin$ points

efficient interpolation&e3trapolation

Ideally

no turnin$ points

no other sin$ularities

%in$ularities mi$ht arise if wron$ parametrization

split system into su!systems

di'ide and con(uer


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)omotopy-based initialization* tool support

%uccessful e3periments at Politecnico and EDF

e3perimental sol'er lined to Dymola

!y M+%ielemann" !ased on 9O/2OC sol'er

definitely not user&friendly

%mall&size use cases

multi!ody systems with multiple steady&state confi$urations analo$ electronic circuits

hydrauliccircuits

air conditionin$ systems

2ar$e&sized use cases

complete com!ined&cycle power plant models

up to B6 iteration 'aria!les *-,

no need of manual settin$ of start 'alues

Preliminary implementation in Dymola + *not $ood enou$h yet,

Lopefully a'aila!le soon as standard feature in all Modelica tools


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DI%C%%IO9


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,eferences

M+ %ielemann" F+ Casella" M+ Otter" C+ Clauss" Q+ E!orn" %+E+ Mattsson"

L+ Olsson7 Ro!ust Initialization of Differential&l$e!raic E(uations sin$

Lomotopy+ Proceedin$s of HthInternational Modelica Conference"Dresden" .ermany" 56&55 March 5644+

F+ Casella" M+ %ielemann" 2+ %a'oldelli7 %teady&state Initialization of

O!:ect&Oriented #hermo&Fluid Models !y Lomotopy Methods"

Proceedin$s of HthInternational Modelica Conference" Dresden"

.ermany" 56&55 March 5644+

03a - Initialization

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