Modelica & Dymola Introduction - Lunds universitetctr.maths.lu.se/na/courses/FMNN05/course_media/material/Modelica... · Lund University | Numerical Analysis | FMNN05 | 2016-11-08

Lund University | Numerical Analysis | FMNN05 | 2016-11-08

Modelica & Dymola IntroductionFMNN05 – EMIL FREDRIKSSON & CLAUS FÜHRER


Modelica


Introduction

Modelica is a free, object-oriented, equation based language

for modelling of complex systems.

Initiated in 1996 by Hilding Elmqvist and now used in

industry all over the world.

Companies: Audi, BMW, Daimler,

Ford, Toyota, VW, ABB, Siemens


Introduction

Modelica is a multi-domain language, not geared towards

any specific domain.

Easily couple models containing for example, mechanical

and electrical components.

Declarative, the order of the

equations does not matter.


Introduction

• Object-oriented, models are classes and thus can easily

be extended using ordinary object-oriented features.

• Modelica allow acausal modelling, i.e. the

input-output causality is not fixed.

• Promotes model reuse!

• The language elements are mapped to

differential, algebraic and discrete

equations. NO support for partial

differential equations.

model FirstReal x;

equationder(x) = -1;

end First;


Introduction

Modelica is designed for fully implicit ordinary differential

equations,

𝟎 = 𝑭(𝒕, 𝒙, ሶ𝒙)


Language VS Tool

Commercial Open-source

Dymola JModelica.org

SimulationX OpenModelica

MapleSim

Modelica → Compiler → Simulation Environment

Language Tool


Modelica Standard Library (MSL)

Free library consisting of over 1200 models and about 900

functions in many engineering domains.

• Mechanics

• Electrical

• Fluid

• Thermal

• Etc...


Modelica Basics


Modelica, Hello World

How would the equation,

look in the Modelica language?

ሶ𝑥 = −𝑥, 𝑥 𝑡0 = 1


Modelica, Hello World



ሶ𝑥 = −𝑥, 𝑥 𝑡0 = 1

model Basic

Real x(start=1);

equation

der(x) = -x;

end Basic;


Modelica Model

model Example

…

…

equation

…

end Example;

Model body

• Declaration of constants /

parameters / states / inputs

Equation section

• Declaration of model

equations

Name of the modelType of the object (i.e a model)

Start of equations


Modelica Basics

• A key feature is that it is equation based, consider:

• x=y+1

• NOT an assignment in Modelica, a relation! The compiler

takes care of manipulating the equations.

• 0 = y+1-x is identical!


Modelica, Hello World (again)



ሶ𝑥 = −𝑥, 𝑥 𝑡0 = 1

model Basic

Real x(start=1);

equation

0 = - der(x) - x ;

end Basic;

Note that this is identical

to the previous example!


Modelica Basics (Syntax)

The important variable types are:

Real

Integer

Boolean

Additionally there is:

String

Enumeration

Real p;Integer y;Real x;



Constants

– Known by the compiler

and do not change values.

Parameters

– Known before the

simulation starts and

remain constant

throughout the simulation

Variability

• Constants

• Parameters

• Time-varyings

• Inputs

• Outputs

parameter Real p1 = 1;constant Integer y = 10;Real x;



To specify derivatives the keyword der() is to be used.

der(x) → 𝑑𝑥

𝑑𝑡

Only first order derivatives! der(der(x))


Example, Pendulum

𝑚𝑑2𝑥

𝑑𝑡2=𝐹

𝐿𝑥

𝑚𝑑2𝑦

𝑑𝑡2=𝐹

𝐿𝑦 −𝑚𝑔

𝑥2 + 𝑦2 = 𝐿2

L is the length (parameter) of the Pendulum and g is the

gravity (constant), x and y are the coordinates and F is the

force.

Pendulum equations in Modelica code?



Start attributes for time-varying variables may be set

using the modifier start.

Real x(start=1.0); x has (guess) start value 1.0!

NOTE, during initialization this value may change!

Real x(start=1.0, fixed=true); x must have start value 1.0!

Forces the initialization algorithm to use 1.0 as start value.



model ChangedInitValuesReal x(start = 1);Real y(start = -1);

equationder(x) = -y;y = 1+x;

end ChangedInitValues;

• Consider the

following model,

• What happens

here?

•

If the guessed

values are

used:

• What actually happens is that ONE of the

guessed values are used to calculate the other.

𝑦 = 1 + 𝑥 → −1 = 1 + 1 → −1 ≠ 2


Pendulum

model Pendulumconstant Real g=9.81 ”Gravity";

parameter Real L=1 ”Length";parameter Real m=1.0 ”Mass";

Real x(start=0.1);Real y(start=-0.9);Real vx;Real vy; Real F;

equationder(x) = vx;der(y) = vy;x^2 + y^2 = L^2;m*der(vx) = (F/L)*x;

m*der(vy) = (F/L)*y - m*g;

end Pendulum;


Example

The Van der Pol oscillator

• Parameter declaration

• State declaration

• Equations declaration

model VDP

// State start values

parameter Real x1_0 = 0;

parameter Real x2_0 = 1;

// The states

Real x1(start = x1_0);

Real x2(start = x2_0);

// The control signal

input Real u;

equation

der(x1) = (1 - x2^2) * x1 - x2 + u;

der(x2) = x1;

end VDP;NOTE: the equations may just as well

be specified as a residual!

ሷ𝑥 = 1 − 𝑥2 ሶ𝑥 − 𝑥 + 𝑢


Modelica Features

In Modelica you can additionally specify:

• Functions

• Algorithms

• External functions (calling C or FORTRAN)

• Hybrid constructs (if and when clauses)


Modelica (connect statement)

In order to support model reuse, Modelica has a key

component, connect, for describing the interaction

between models.

connect(resistor2.n, resistor1.p)

In the electric case:

• The current should sum up to zero

• The voltage should be the same at

both connections

Different meaning in other domains.


Electrical Circuit

Consider this Simple Circuit model:

model SimpleCircuit

parameter Real C = 0.001;

parameter Real R = 100;

parameter Real V0 = 5;

// The states

Real i;

Real uC(start=0);

equation

V0 – uC = R*i;

der(uC)*C = i;

end SimpleCircuit;


Electrical Circuit, Decomposed

• In order to model large systems, decomposing the system

into smaller components which are then connected is

essential.

• Trying this for our simple circuit:

model Resistor

parameter R;

Real i1, v1;

Real i2, v2;

Real i,u;

equationu = v2 - v1;

i1 + i2 = 0;

i = i2;

u = R*i;

end Resistor;

model Capacitator

parameter C;

Real i1, v1;

Real i2, v2;

Real i,u;


i1 + i2 = 0;

i = i2;

der(u)*C = i;

end Capacitator;


Electrical Circuit, Decomposed

model ConstVoltage

parameter V0;

Real i1, v1;

Real i2, v2;

Real i,u;


i1 + i2 = 0;

i = i2;

u = V0;

end ConstVoltage;

model Ground

Real i,u;

equationv = 0;

end Ground;

Voltage Source: Ground Model:


Electrical Circuit, Reassembled

• The missing equations

are added in the top-level

model.

• This has to be done

manually and is thus error

prone.

• In order to simplify this,

Modelica introduces the

connector class.

model SimpleCircuit

Ground G;

Resistor R(R=100);

Capacitor C(C=0.001);

ConstVoltage S(V0=5);

equation

S.v2 = R.v1S.i2 + R.i1 = 0;R.v2 = C.v1R.i2 + C.i1 = 0;C.v2 = S.v1C.v2 = G.vC.i2 + S.i1 + G.i = 0;

end SimpleCircuit;


Connect Statement

In the electric case, we want a connector with the

properties:

• The current should sum up to zero

• The voltage should be the same at both connections

connector Pin

Real v;

flow Real i;

end Pin;

connect(pin1, pin2)pin1.v = pin2.v

pin1.i + pin2.i = 0


Resistor Revisited

model Resistor

parameter R;

Real i1, v1;

Real i2, v2;

Real i,u;


i1 + i2 = 0;

i = i2;

u = R*i;

end Resistor;

model Resistor

parameter R;

Pin n, p;

Real i,u;

equationu = p.v – n.v;

p.i + n.i = 0;

i = p.i;

u = R*i;

end Resistor;

Using Pins


Electrical Circuit, Reassembled (Again)

model SimpleCircuit

Ground G;

Resistor R(R=100);

Capacitor C(C=0.001);

ConstVoltage S(V0=5);

equation

connect(G.p, S.n);connect(S.p, R.n);connect(R.p, C.n);connect(C.p, G.p);

end SimpleCircuit;


Modelica Graphical Representation

Modelica is a textual language. However, there are language

features for specifying graphical representation of models.

Note, the equations in each model can always be viewed and

that the structure of the physical system is kept!


Modelica References

In charge of developing the Modelica language is the

non-profit organization Modelica Association

https://www.modelica.org/

Modelica by Example, free book about Modelica,

http://book.xogeny.com/

Cheat Sheet

http://modref.xogeny.com/

https://www.modelica.org/

http://book.xogeny.com/

http://modref.xogeny.com/


Dymola


Dymola

State of the art modelling and simulation tool based on the

Modelica language.

“Dymola, Dynamic Modeling Laboratory, is a complete tool

for modeling and simulation of integrated and complex

systems for use within automotive, aerospace, robotics,

process and other applications.”

Developed since 1996 and now used in

industries such as, SCANIA, Toyota, BMW,

Ford and Tetra Pak.


Dymola (modelling layer)

• Modelica

standard library

• Toggle between

viewing textual or

objects layer

• Modelling area

• Toggle between

the modelling and

simulation tab


Dymola (modelling layer)

DEMO (MSL)

DEMO (Modelica Grammar)


Dymola (simulation layer)

• Translate the

model

• Simulate the model

• Simulation setup

• Browse the model

variables,

parameters and

constants


Dymola (simulation information)

Retrieving information

after a simulation:

• Goto, Simulation

• Choose, Show Log

A log showing the

simulation statistics will

appear.


Dymola

DEMO (Simple dx)


More Information from Dymola

See under the

Translation tab

and the Debug

tab in Simulation

Setup.


Integrators in Dymola

Integrators are found under

General in Simulation

Setup.

• Switch integrator

• Dassl

• Euler

• Radau

• more...

• Change tolerance


Pendulum, revisited

𝑚𝑑2𝑥

𝑑𝑡2=𝐹

𝐿𝑥

𝑚𝑑2𝑦

𝑑𝑡2=𝐹

𝐿𝑦 −𝑚𝑔

𝑥2 + 𝑦2 = 𝐿2

L is the length (parameter) of the Pendulum and g is the

gravity (constant), x and y are the coordinates and F is the

force.

Write the Pendulum equations in Modelica code:


Phase plots

In the Variable

Browser:

• Right click on a

variabel

• Choose to use as

independent

variable


Installation

• Download and install Dymola

• Download and install Visual Studio Express

– https://www.microsoft.com/en-

us/download/details.aspx?id=44914

https://www.microsoft.com/en-us/download/details.aspx?id=44914


Installation

• Goto simulation tab (bottom right corner)

– Goto Simulation menu (top, third from the left)

– Goto Setup

– Goto Compiler

– Choose Visual Studio and press OK.

• Next, goto Help menu (top, last from the left)

– Goto License

– Goto Setup

– Choose a local license file and point it to the provided license file.

– Press OK and restart!


Modeling Issues


Starting Point

Before starting on the projects:

• Check the many examples in the Modelica standard

library!

• Read the information about the different parts and objects!


Initial Values

Real x(start = 0.9, fixed = true);

The initial guess If the initial guess is fixed or not


Initial Values

model Pendulumconstant Real g=9.81 ”Gravity";

parameter Real L=1 ”Length";parameter Real m=1.0 ”Mass";

Real x(start=0.1);Real y(start=-0.9);Real vx;Real vy; Real F;

equationder(x) = vx;der(y) = vy;x^2 + y^2 = L^2;m*der(vx) = (F/L)*x;

m*der(vy) = (F/L)*y - m*g;

end Pendulum;

What happens if you specify

inconsistent initial values?

What happens if you do NOT

fully specify the initial values?

What happens if you force the

initial values?


Initial Values


Initial Values

Cases (Pendulum):

• Not fully specified (x=0.1)

– Dymola, tries to find a consistent set. May fail!

• Fixed inconsistent values (x=0.9, y=0.9, fixed=true)

– Error, overspecified system

• Guessed inconsistent values (x=0.9, y=0.9, fixed=false)

– Dymola, changes one of the values!

• Guessed inconsistent values (x=1.1, y=0.9, fixed=false)

– Error, cannot recover

Documents

Modelica & Dymola Introduction - Lunds universitetctr.maths.lu.se/na/courses/FMNN05/course_media/material/Modelica... · Lund University | Numerical Analysis | FMNN05 | 2016-11-08