Operators and Expressions¶

The lexical units are combined to form even larger building blocks such as expressions according to the rules given by the expression part of the Modelica grammar in TODO: Appendix B.

This chapter describes the evaluation rules for expressions, the concept of expression variability, built-in mathematical operators and functions, and the built-in special Modelica operators with function syntax.

Expressions can contain variables and constants, which have types, predefined or user defined. The predefined built-in types of Modelica are Real, Integer, Boolean, String, and enumeration types which are presented in more detail in Section predefined-types. [The abbreviated predefined type information below is given as background information for the rest of the presentation.]

Expressions¶

Modelica equations, assignments and declaration equations contain expressions.

Expressions can contain basic operations, +, -, *, /, ^, etc. with normal precedence as defined in Table 1 and the grammar in Section modelica-concrete-syntax. The semantics of the operations is defined for both scalar and array arguments in TODO: Section 10.6.

It is also possible to define functions and call them in a normal fashion. The function call syntax for both positional and named arguments is described in TODO: Section 12.4.1 and for vectorized calls in TODO: Section 12.4.4. The built-in array functions are given in TODO: Section 10.1.1 and other built-in operators in Section Built-in Intrinsic Operators with Function Syntax.

Operator Precedence and Associativity¶

Operator precedence determines the order of evaluation of operators in an expression. An operator with higher precedence is evaluated before an operator with lower precedence in the same expression.

The following table presents all the expression operators in order of precedence from highest to lowest, as derived from the Modelica grammar in TODO: Appendix B. All operators are binary except the postfix operators and those shown as unary together with expr, the conditional operator, the array construction operator {} and concatenation operator [], and the array range constructor : which is either binary or ternary. Operators with the same precedence occur at the same line of the table:

Table 1 Operators.¶
Operator Group	Operator Syntax	Examples
postfix array index operator	`[]`	`arr[index]`
postfix access operator	`.`	`a.b`
postfix function call	`funcName(function-arguments)`	`sin(4.36)`
array construct/concat	`{expressions}` `[expressions]` `[expressions; expressions, ...]`	`[2,3; 7,8]` `{2,3}` `[5,6]`
exponentiation	`^`	`2^3`
multiplicative and array elementwise multiplicative	`* / .* ./`	`23 2/3` `[1,2;3,4].[2,3;5,6]`
additive and array elementwise additive	`+ - +expr -expr .+ .-`	`a+b, a-b, +a, -a` `[1,2;3,4].+[2,3;5,6]`
relational	`< <= > >= == <>`	`a<b, a<=b, a>b, ...`
unary negation	`not expr`	`not b1`
logical and	`and`	`b1 and b2`
logical or	`or`	`b1 or b2`
array range	`expr : expr` `expr : expr : expr`	`1:5` `start:step:stop`
conditional	`if expr then expr else expr`	`if b then 3 else x`
named argument	`ident = expr`	`x = 2.26`

The conditional operator may also include elseif-clauses. Equality = and assignment := are not expression operators since they are allowed only in equations and in assignment statements respectively. All binary expression operators are left associative, except exponentiation which is non-associative. The array range operator is non-associative.

[The unary minus and plus in Modelica is slightly different than in Mathematica and in MATLAB [1], since the following expressions are illegal (whereas in Mathematica and in MATLAB these are valid expressions):

2*-2 // = -4 in Mathematica/MATLAB; is illegal in Modelica

--2 // = 2 in Mathematica/MATLAB; is illegal in Modelica

++2 // = 2 in Mathematica/MATLAB; is illegal in Modelica

2--2 // = 4 in Mathematica/MATLAB; is illegal in Modelica

Non-associative exponentation and array range operator:

x^y^z Not legal, use parenthesis to make it clear.

a:b:c:d:e:f:g Not legal, and scalar arguments gives no legal interpretation.

]

Evaluation Order¶

A tool is free to solve equations, reorder expressions and to not evaluate expressions if their values do not influence the result (e.g. short-circuit evaluation of Boolean expressions). If-statements and if-expressions guarantee that their clauses are only evaluated if the appropriate condition is true, but relational operators generating state or time events will during continuous integration have the value from the most recent event.

If a numeric operation overflows the result is undefined. For literals it is recommended to automatically convert the number to another type with greater precision.

Example: Guarding Expressions Against Incorrect Evaluation¶

[Example. If one wants to guard an expression against incorrect evaluation, it should be guarded by an if:

  Boolean v[n];
  Boolean b;
  Integer I;
equation
  x=v[I] and (I>=1 and I<=n); // Invalid
  x=if (I>=1 and I<=n) then v[I] else false; // Correct

To guard square against square root of negative number use noEvent:

der(h)=if h>0 then -c*sqrt(h) else 0; // Incorrect
der(h)=if noEvent(h>0) then -c*sqrt(h) else 0; // Correct

]

Arithmetic Operators¶

Modelica supports five binary arithmetic operators that operate on any numerical type:

`^`	Exponentiation
`*`	Multiplication
`/`	Division
`+`	Addition
`-`	Subtraction

Some of these operators can also be applied to a combination of a scalar type and an array type, see Section TODO: 10.6.

The syntax of these operators is defined by the following rules from the Modelica grammar:

arithmetic_expression =
  [ add_op ] term { add_op term } ;

add_op =
  "+" | "-" ;

term =
  factor { mul_op factor } ;

mul_op =
  "*" | "/" ;

factor =
  primary [ "^" primary ] ;

Equality, Relational, and Logical Operators¶

Modelica supports the standard set of relational and logical operators, all of which produce the standard boolean values true or false.

`>`	greater than
`>=`	greater than or equal
`<`	less than
`<=`	less than or equal to
`==`	equality within expressions
`<>`	Inequality

A single equals sign = is never used in relational expressions, only in equations (TODO: Chapter 8, Section 10.6.1) and in function calls using named parameter passing (TODO: Section 12.4.1).

The following logical operators are defined:

`not`	negation, unary operator
`and`	logical and
`or`	logical or

The grammar rules define the syntax of the relational and logical operators.

logical_expression =
  logical_term { or logical_term } ;
logical_term =
  logical_factor { and logical_factor } ;
logical_factor =
  [ not ] relation ;
relation =
  arithmetic_expression [ rel_op arithmetic_expression ] ;
rel_op =
  "<" | ( "<=" ) | ">" | ( ">=" ) | ( "==" ) | ( "<>" ) ;

The following holds for relational operators:

Relational operators <, <=, >, >=, ==, <>``, are only defined for scalar operands of simple types. The result is Boolean and is true or false if the relation is fulfilled or not, respectively.
For operands of type String, str1 op str2 is for each relational operator, op, defined in terms of the C-function strcmp as strcmp(str1,str2) op 0.
For operands of type Boolean, false<true.
For operands of enumeration types, the order is given by the order of declaration of the enumeration literals.
In relations of the form v1 == v2 or v1 <> v2, v1 or v2 shall, unless used in a function, not be a subtype of Real. [The reason for this rule is that relations with Real arguments are transformed to state events (see Events, Section 8.5) and this transformation becomes unnecessarily complicated for the == and <> relational operators (e.g. two crossing functions instead of one crossing function needed, epsilon strategy needed even at event instants). Furthermore, testing on equality of Real variables is questionable on machines where the number length in registers is different to number length in main memory].
Relations of the form v1 rel_op v2, with v1 and v2 variables and rel_op a relational operator are called elementary relations. If either v1 or v2 or both variables are a subtype of Real, the relation is called a Real elementary relation.

Miscellaneous Operators and Variables¶

Modelica also contains a few built-in operators which are not standard arithmetic, relational, or logical operators. These are described below, including time, which is a built-in variable, not an operator.

String Concatenation¶

Concatenation of strings (see the Modelica grammar) is denoted by the + operator in Modelica [e.g. "a" + "b" becomes "ab"].

Array Constructor Operator¶

The array constructor operator { ... } is described in TODO: Section 10.4.

Array Concatenation Operator¶

The array concatenation operator [ ... ] is described in TODO: Section 10.4.2.

Array Range Operator¶

The array range constructor operator : is described in TODO: Section 10.4.3.

If-Expressions¶

An expression

if expression1 then expression2 else expression3

is one example of if-expression. First expression1, which must be boolean expression, is evaluated. If expression1 is true expression2 is evaluated and is the value of the if-expression, else expression3 is evaluated and is the value of the if-expression. The two expressions, expression2 and expression3, must be type compatible expressions (TODO: Section 6.6) giving the type of the if-expression. If-expressions with elseif are defined by replacing elseif by else if. [Note: elseif has been added for symmetry with if-clauses.] For short-circuit evaluation see Section Evaluation Order.

[Example:

Integer i;
Integer sign_of_i1=if i<0 then -1 elseif i==0 then 0 else 1;
Integer sign_of_i2=if i<0 then -1 else if i==0 then 0 else 1;

]

Member Access Operator¶

It is possible to access members of a class instance using dot notation, i.e., the . operator.

[Example: R1.R for accessing the resistance component R of resistor R1 . Another use of dot notation: local classes which are members of a class can of course also be accessed using dot notation on the name of the class, not on instances of the class.]

Built-in Variable time¶

All declared variables are functions of the independent variable time. The variable time is a built-in variable available in all models and blocks, which is treated as an input variable. It is implicitly defined as:

input Real time (final quantity = "Time", final unit = "s");

The value of the start attribute of time is set to the time instant at which the simulation is started.

[Example:

encapsulated model SineSource
  import Modelica.Math.sin;
  connector OutPort=output Real;
  OutPort y=sin(time); // Uses the built-in variable time.
end SineSource;

]

Built-in Intrinsic Operators with Function Syntax¶

Certain built-in operators of Modelica have the same syntax as a function call. However, they do not behave as a mathematical function, because the result depends not only on the input arguments but also on the status of the simulation.

There are also built-in functions that depend only on the input argument, but also may trigger events in addition to returning a value. Intrinsic means that they are defined at the Modelica language level, not in the Modelica library. The following built-in intrinsic operators/functions are available:

Mathematical functions and conversion functions, see Section Numeric Functions and Conversion Functions below.
Derivative and special purpose operators with function syntax, see Section Derivative and Special Purpose Operators with Function Syntax below.
Event-related operators with function syntax, see Section Event-Related Operators with Function Syntax below.
Array operators/functions, see Section TODO: 10.1.1.

With exception of built-in operator String(...), all operators in this section can only be called with positional arguments.

Numeric Functions and Conversion Functions¶

The following mathematical operators and functions, also including some conversion functions, are predefined in Modelica, and are vectorizable according to Section 12.4.6, except for the String function. The functions which do not trigger events are described in the table below, whereas the event-triggering mathematical functions are described in Section Event Triggering Mathematical Functions.

`abs(v)`	Is expanded into `noEvent(if v >= 0 then v else –v)`. Argument v needs to be an Integer or Real expression.
`sign(v)`	Is expanded into `noEvent(if v>0 then 1 else if v<0 then –1 else 0)`. Argument v needs to be an Integer or Real expression.
`sqrt(v)`	Returns the square root of `v` if `v>=0`, otherwise an error occurs. Argument `v` needs to be an Integer or Real expression.
`Integer(e)`	Returns the ordinal number of the expression e of enumeration type that evaluates to the enumeration value E.enumvalue, where Integer(E.e1)``=1, ``Integer(E.en)``=n, for an enumeration ``type E=enumeration(e1, ..., en). See also Section TODO 4.8.5.2.
`String(b, <options>)` `String(i, <options>)` `String(r, significantDigits=d, <options>)` `String(r, format=s)` `String(e, <options>)`	Convert a scalar non-String expression to a String representation. The first argument may be a `Boolean b`, an `Integer i`, a `Real r` or an `Enumeration e` (Section TODO: 4.8.5.2). The other arguments must use named arguments. The optional <options> are: Integer minimumLength=0: minimum length of the resulting string. If necessary, the blank character is used to fill up unused space. Boolean leftJustified = true: if true, the converted result is left justified in the string; if false it is right justified in the string. For Real expressions the output shall be according to the Modelica grammar. Integer significantDigits=6: defines the number of significant digits in the result string. [Examples: `"12.3456"`, `"0.0123456"`, `"12345600"`, `"1.23456E-10"`]. The format string corresponding to options is: for Reals: `(if leftJustified then "-" else "")+String(minimumLength)+"."+ String(signficantDigits)+"g"`, for Integers: `(if leftJustified then "-" else "")+String(minimumLength)+"d"`. Format string: According to ANSI-C the format string specifies one conversion specifier (excluding the leading %), may not contain length modifiers, and may not use “*” for width and/or precision. For all numeric values the format specifiers f, e, E, g, G are allowed. For integral values it is also allowed to use the d, i, o, x, X, u, and c-format specifiers (for non-integral values a tool may round, truncate or use a different format if the integer conversion characters are used). The x,X-formats (hexa-decimal) and c (character) for Integers does not lead to input that agrees with the Modelica-grammar.

Event Triggering Mathematical Functions¶

The built-in operators in this section trigger state events if used outside of a when-clause and outside of a clocked discrete-time partition (see Section 16.8.1). [ If this is not desired, the noEvent function can be applied to them. E.g. noEvent(integer(v)) ]

`div(x,y)`	Returns the algebraic quotient x/y with any fractional part discarded (also known as truncation toward zero). [Note: this is defined for / in C99; in C89 the result for negative numbers is implementation-defined, so the standard function div() must be used.]. Result and arguments shall have type Real or Integer. If either of the arguments is Real the result is Real otherwise Integer.
`mod(x,y)`	Returns the integer modulus of x/y, i.e. mod(x,y)=x-floor(x/y)y. Result and arguments shall have type Real or Integer. If either of the arguments is Real the result is Real otherwise Integer. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously. Examples* mod(3,1.4)=0.2, mod(-3,1.4)=1.2, mod(3,-1.4)=-1.2]
`rem(x,y)`	Returns the integer remainder of x/y, such that div(x,y)y + rem(x, y) = x. Result and arguments shall have type Real or Integer. If either of the arguments is Real the result is Real otherwise Integer. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously. Examples* rem(3,1.4)=0.2, rem(-3,1.4)=-0.2]
`ceil(x)`	Returns the smallest integer not less than x. Result and argument shall have type Real. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously.]
`floor(x)`	Returns the largest integer not greater than x. Result and argument shall have type Real. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously.].
`integer(x)`	Returns the largest integer not greater than x. The argument shall have type Real. The result has type Integer. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously.].

Todo

This is using an alternative to using a table: headings. There are other alternatives (like definition lists). Using headers makes it easy to get permanent links to the text.

div(x,y)¶

div(x,y) returns the algebraic quotient x/y with any fractional part discarded (also known as truncation toward zero). [Note: this is defined for / in C99; in C89 the result for negative numbers is implementation-defined, so the standard function div() must be used.]. Result and arguments shall have type Real or Integer. If either of the arguments is Real the result is Real otherwise Integer.

mod(x,y)¶

mod(x,y) returns the integer modulus of x/y, i.e. mod(x,y)=x-floor(x/y)*y. Result and arguments shall have type Real or Integer. If either of the arguments is Real the result is Real otherwise Integer. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously. Examples mod(3,1.4)=0.2, mod(-3,1.4)=1.2, mod(3,-1.4)=-1.2].

rem(x,y)¶

rem(x,y) returns the integer remainder of x/y, such that div(x,y)*y + rem(x, y) = x. Result and arguments shall have type Real or Integer. If either of the arguments is Real the result is Real otherwise Integer. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously. Examples rem(3,1.4)=0.2, rem(-3,1.4)=-0.2].

ceil(x)¶

ceil(x) returns the smallest integer not less than x. Result and argument shall have type Real. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously.].

floor(x)¶

floor(x) returns the largest integer not greater than x. Result and argument shall have type Real. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously.].

integer(x)¶

integer(x) returns the largest integer not greater than x. The argument shall have type Real. The result has type Integer. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously.].

Todo

This is using an alternative to using a table: definition lists.

div(x,y): returns the algebraic quotient x/y with any fractional part discarded (also known as truncation toward zero). [Note: this is defined for / in C99; in C89 the result for negative numbers is implementation-defined, so the standard function div() must be used.]. Result and arguments shall have type Real or Integer. If either of the arguments is Real the result is Real otherwise Integer.
mod(x,y): returns the integer modulus of x/y, i.e. mod(x,y)=x-floor(x/y)*y. Result and arguments shall have type Real or Integer. If either of the arguments is Real the result is Real otherwise Integer. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously. Examples mod(3,1.4)=0.2, mod(-3,1.4)=1.2, mod(3,-1.4)=-1.2].
rem(x,y): returns the integer remainder of x/y, such that div(x,y)*y + rem(x, y) = x. Result and arguments shall have type Real or Integer. If either of the arguments is Real the result is Real otherwise Integer. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously. Examples rem(3,1.4)=0.2, rem(-3,1.4)=-0.2].
ceil(x): returns the smallest integer not less than x. Result and argument shall have type Real. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously.].
floor(x): returns the largest integer not greater than x. Result and argument shall have type Real. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously.].
integer(x): returns the largest integer not greater than x. The argument shall have type Real. The result has type Integer. [Note, outside of a when-clause state events are triggered when the return value changes discontinuously.].

Built-in Mathematical Functions and External Built-in Functions¶

The following built-in mathematical functions are available in Modelica and can be called directly without any package prefix added to the function name. They are also available as external built-in functions in the Modelica.Math library.

`sin(x)`	sine
`cos(x)`	cosine
`tan(x)`	tangent (x shall not be: ..., -π/2, π/2, 3π/2, ...)
`asin(x)`	inverse sine (-1 ≤ x ≤ 1)
`acos(x)`	inverse cosine (-1 ≤ x ≤ 1)
`atan(x)`	inverse tangent
`atan2(y, x)`	the `atan2(y, x)` function calculates the principal value of the arc tangent of `y/x`, using the signs of the two arguments to determine the quadrant of the result
`sinh(x)`	hyperbolic sine
`cosh(x)`	hyperbolic cosine
`tanh(x)`	hyperbolic tangent
`exp(x)`	exponential, base e
`log(x)`	natural (base e) logarithm (`x > 0`)
`log10(x)`	base 10 logarithm (`x > 0`)

Derivative and Special Purpose Operators with Function Syntax¶

The following derivative operator and special purpose operators with function syntax are predefined:

Todo

This is using a definition list for the operators. There are other alternatives (tables are not that nice). Definition lists can be referred to using links, like der(expr), but the link will not be visible in the HTML. Could possibly be themed into the HTML in some way.

der(expr)

The time derivative of expr. If the expression expr is a scalar it needs to be a subtype of Real. The expression and all its subexpressions must be differentiable. If expr is an array, the operator is applied to all elements of the array. For non-scalar arguments the function is vectorized according to Section TODO: 10.6.12. [For Real parameters and constants the result is a zero scalar or array of the same size as the variable.]

delay(expr,delayTime, delayMax), delay(expr,delayTime)

Returns: expr(time–delayTime) for time>time.start + delayTime and expr(time.start) for time <= time.start + delayTime. The arguments, i.e., expr, delayTime and delayMax, need to be subtypes of Real. DelayMax needs to be additionally a parameter expression. The following relation shall hold: 0 <= delayTime <= delayMax, otherwise an error occurs. If delayMax is not supplied in the argument list, delayTime need to be a parameter expression. See also Section delay. For non-scalar arguments the function is vectorized according to Section TODO: 10.6.12.

cardinality(c)

[This is a deprecated operator. It should no longer be used, since it will be removed in one of the next Modelica releases.] Returns the number of (inside and outside) occurrences of connector instance c in a connect-equation as an Integer number. See also Section cardinality (deprecated).

homotopy(actual=actual, simplified=simplified)

The scalar expressions “actual” and “simplified” are subtypes of Real. A Modelica translator should map this operator into either of the two forms:

Returns “actual” [a trivial implementation].
In order to solve algebraic systems of equations, the operator might during the solution process return a combination of the two arguments, ending at actual, [e.g., actual*lambda + simplified*(1-lambda), where lambda is a homotopy parameter going from 0 to 1].

The solution must fulfill the equations for homotopy returning actual. See also Section homotopy. For non-scalar arguments the function is vectorized according to Section TODO: 12.4.6.

semiLinear(x, positiveSlope, negativeSlope): Returns: if x>=0 then positiveSlope*x else negativeSlope*x. The result is of type Real. See Section semiLinear [especially in the case when x = 0]. For non-scalar arguments the function is vectorized according to Section TODO: 10.6.12.
inStream(v): The operator inStream(v) is only allowed on stream variables v defined in stream connectors, and is the value of the stream variable v close to the connection point assuming that the flow is from the connection point into the component. This value is computed from the stream connection equations of the flow variables and of the stream variables. The operator is vectorizable. For more details see Section TODO: 15.2.
actualStream(v): The actualStream(v) operator returns the actual value of the stream variable v for any flow direction. The operator is vectorizable. For more details, see Section TODO: 15.3.
spatialDistribution(in0, in1, x, pv, iP, iV): The spatialDistribution(...) operator allows approximation of variable-speed transport of properties, see Section spatialDistribution.
getInstanceName(): Returns a string with the name of the model/block that is simulated, appended with the fully qualified name of the instance in which this function is called, see Section getInstanceName.

A few of these operators are described in more detail in the following.

delay¶

[The delay() operator allows a numerical sound implementation by interpolating in the (internal) integrator polynomials, as well as a more simple realization by interpolating linearly in a buffer containing past values of expression expr. Without further information, the complete time history of the delayed signals needs to be stored, because the delay time may change during simulation. To avoid excessive storage requirements and to enhance efficiency, the maximum allowed delay time has to be given via delayMax.

This gives an upper bound on the values of the delayed signals which have to be stored. For real-time simulation where fixed step size integrators are used, this information is sufficient to allocate the necessary storage for the internal buffer before the simulation starts. For variable step size integrators, the buffer size is dynamic during integration. In principle, a delay operator could break algebraic loops. For simplicity, this is not supported because the minimum delay time has to be give as additional argument to be fixed at compile time. Furthermore, the maximum step size of the integrator is limited by this minimum delay time in order to avoid extrapolation in the delay buffer.]

spatialDistribution¶

[Many applications involve the modelling of variable-speed transport of properties. One option to model this infinite-dimensional system is to approximate it by an ODE, but this requires a large number of state variables and might introduce either numerical diffusion or numerical oscillations. Another option is to use a built-in operator that keeps track of the spatial distribution of z(y, t), by suitable sampling, interpolation, and shifting of the stored distribution. In this case, the internal state of the operator is hidden from the ODE solver.]

The spatialDistribution() operator allows to approximate efficiently the solution of the infinite-dimensional problem:

${\partial z(y,t) \over \partial t} + v(t) {\partial z(y,t) \over \partial y} = 0.0$

$z(0.0, t) = in_0(t) \text{ if } v >= 0$

$z(1.0, t) = in_1(t) \text{ if } v < 0$

where z(y, t) is the transported quantity, y is the normalized spatial coordinate (0.0 ≤ y ≤ 1.0), t is the time, v(t)=der(x) is the normalized transport velocity and the boundary conditions are set at either y = 0.0 or y = 1.0, depending on the sign of the velocity. The calling syntax is:

(out0, out1) = spatialDistribution(in0, in1, x, positiveVelocity,
  initialPoints = {0.0, 1.0},
  initialValues = {0.0, 0.0});

where in0, in1, out0, out1, x, v are all subtypes of Real, positiveVelocity is a Boolean, initialPoints and initialValues are arrays of subtypes of Real of equal size, containing the y coordinates and the z values of a finite set of points describing the initial distribution of z(y, t0). The out0 and out1 are given by the solutions at z(0.0, t) and z(1.0, t); and in0 and in1 are the boundary conditions at z(0.0, t) and z(1.0, t) (at each point in time only one of in0 and in1 is used). Elements in the initialPoints array must be sorted in non-descending order. The operator can not be vectorized according to the vectorization rules described in section 12.4.6. The operator can be vectorized only with respect to the arguments in0 and in1 (which must have the same size), returning vectorized outputs out0 and out1 of the same size; the arguments initialPoints and initialValues are vectorized accordingly.

The solution, z(..), can be described in terms of characteristics:

= z(y, t), for all, as long as staying inside the domain.

This allows to directly compute the solution based on interpolating the boundary conditions.

The spatialDistribution operator can be described in terms of the pseudo-code given as a block:

Listing 2 Pseudo-code for spatialDistribution in terms of a block.¶

block spatialDistribution
  input Real in0;
  input Real in1;
  input Real x;
  input Boolean positiveVelocity;
  parameter Real initialPoints(each min=0, each max=1)[:] = {0.0, 1.0};
  parameter Real initialValues[:] = {0.0, 0.0};
  output Real out0;
  output Real out1;
  protected
  Real points[:];
  Real values[:];
  Real x0;
  Integer m;
algorithm
  if positiveVelocity then
    out1:=interpolate(points, values, 1-(x-x0));
    out0:=values[1]; // similar to in0 but avoiding algebraic loop
  else
    out0:=interpolate(points, values, (x-x0));
    out1:=values[end]; // similar to in1 but avoiding algebraic loop
  end if;
  when /* acceptedStep */ then
    if x>x0 then
      m:=size(points,1);
      while (if m>0 then points[m]+(x-x0)>=1 else false) then
        m:=m-1;
      end while;
      values:=cat(1, {in0}, values[1:m], {interpolate(points, values,1-(x-x0))} );
      points:=cat(1, {0}, points[1:m] .+ (x1-x0), {1} );
    elseif x<x0 then
      m:=1;
      while (if m<size(points,1) then points[m]+(x-x0)<=0 else false) then
        m:=m+1;
      end while;
      values:=cat(1, {interpolate(points, values, 0-(x-x0))},values[m:end],{in1});
      points:=cat(1, {0}, points[m:end] .+ (x1-x0), {1});
    end if;
    x0:=x;
  end when;
initial algorithm
  x0:=x;
  points:=initialPoints;
  values:=initialValues;
end spatialDistribution;

[The infinite-dimensional problem stated above can then be formulated in the following way:

der(x) = v;
(out0, out1) = **spatialDistribution**\ (in0, in1, x, v>=0, initialPoints, initialValues);

Events are generated at the exact instants when the velocity changes sign – if this is not needed, noEvent() can be used to suppress event generation.

If the velocity is known to be always positive, then out0 can be omitted, e.g.:

der(x) = v;
(,out1) = spatialDistribution(in0, 0, x, true, initialPoints, initialValues);

Technically relevant use cases for the use of the spatialDistribution() operator are modeling of electrical transmission lines, pipelines and pipeline networks for gas, water and district heating, sprinkler systems, impulse propagation in elongated bodies, conveyor belts, and hydraulic systems. Vectorization is needed for pipelines where more than one quantity is transported with velocity v in the example above.]

cardinality (deprecated)¶

[The cardinality operator is deprecated for the following reasons and will be removed in a future release:

Reflective operator may make early type checking more difficult.
Almost always abused in strange ways
Not used for Bond graphs even though it was originally introduced for that purpose.

]

[The cardinality() operator allows the definition of connection dependent equations in a model, for example:

connector Pin
  Real v;
  flow Real i;
end Pin;

model Resistor
  Pin p, n;
equation
  assert(cardinality(p) > 0 and cardinality(n) > 0, "Connectors p and n of Resistor must be connected");
  // Equations of resistor ...
end Resistor;

]

The cardinality is counted after removing conditional components. and may not be applied to expandable connectors, elements in expandable connectors, or to arrays of connectors (but can be applied to the scalar elements of array of connectors). The cardinality operator should only be used in the condition of assert and if-statements – that do not contain connect (and similar operators – see section 8.3.3).

homotopy¶

[During the initialization phase of a dynamic simulation problem, it often happens that large nonlinear systems of equations must be solved by means of an iterative solver. The convergence of such solvers critically depends on the choice of initial guesses for the unknown variables. The process can be made more robust by providing an alternative, simplified version of the model, such that convergence is possible even without accurate initial guess values, and then by continuously transforming the simplified model into the actual model. This transformation can be formulated using expressions of this kind:

lambda*actual + (1-lambda)*simplified

in the formulation of the system equations, and is usually called a homotopy transformation. If the simplified expression is chosen carefully, the solution of the problem changes continuously with lambda, so by taking small enough steps it is possible to eventually obtain the solution of the actual problem.

The operator can be called with ordered arguments or preferably with named arguments for improved readability.

It is recommended to perform (conceptually) one homotopy iteration over the whole model, and not several homotopy iterations over the respective non-linear algebraic equation systems. The reason is that the following structure can be present:

w = f₁(x) // has homotopy operator

0 = f₂(der(x), x, z, w)

Here, a non-linear equation system f₂ is present. The homotopy operator is, however used on a variable that is an “input” to the non-linear algebraic equation system, and modifies the characteristics of the non-linear algebraic equation system. The only useful way is to perform the homotopy iteration over f₁ and f₂ together.

The suggested approach is “conceptual”, because more efficient implementations are possible, e.g. by determining the smallest iteration loop, that contains the equations of the first BLT block in which a homotopy operator is present and all equations up to the last BLT block that describes a non-linear algebraic equation system.

A trivial implementation of the homotopy operator is obtained by defining the following function in the global scope:

function homotopy
  input Real actual;
  input Real simplified;
  output Real y;
algorithm
  y := actual;
annotation(Inline = true);
end homotopy;

Example 1:

In electrical systems it is often difficult to solve non-linear algebraic equations if switches are part of the algebraic loop. An idealized diode model might be implemented in the following way, by starting with a “flat” diode characteristic and then move with the homotopy operator to the desired “steep” characteristic:

model IdealDiode
  // ...
  parameter Real Goff = 1e-5;
  protected
  Real Goff_flat = max(0.01, Goff);
  Real Goff2;
equation
  off = s < 0;
  Goff2 = homotopy(actual=Goff, simplified=Goff_flat);
  u = s*(if off then 1 else Ron2) + Vknee;
  i = s*(if off then Goff2 else 1 ) + Goff2*Vknee;
  // ...
end IdealDiode;

Example 2:

In electrical systems it is often useful that all voltage sources start with zero voltage and all current sources with zero current, since steady state initialization with zero sources can be easily obtained. A typical voltage source would then be defined as:

model ConstantVoltageSource
  extends Modelica.Electrical.Analog.Interfaces.OnePort;
  parameter Modelica.SIunits.Voltage V;
equation
  v = homotopy(actual=V, simplified=0.0);
end ConstantVoltageSource;

Example 3:

In fluid system modelling, the pressure/flowrate relationships are highly nonlinear due to the quadratic terms and due to the dependency on fluid properties. A simplified linear model, tuned on the nominal operating point, can be used to make the overall model less nonlinear and thus easier to solve without accurate start values. Named arguments are used here in order to further improve the readability.

model PressureLoss
  import SI = Modelica.SIunits;
  // ...
  parameter SI.MassFlowRate m_flow_nominal "Nominal mass flow rate";
  parameter SI.Pressure dp_nominal "Nominal pressure drop";
  SI.Density rho "Upstream density";
  SI.DynamicViscosity lambda "Upstream viscosity";
equation
  // ...
  m_flow = homotopy(actual = turbulentFlow_dp(dp, rho, lambda),
  simplified = dp/dp_nominal*m_flow_nominal);
  // ...
end PressureLoss;

Example 4:

Note that the homotopy operator **shall not* be used to combine unrelated expressions, since this can generate singular systems from combining two well-defined systems.*

model DoNotUse
  Real x;
  parameter Real x0 = 0;
equation
  der(x) = 1-x;
initial equation
  0 = homotopy(der(x), x - x0);
end DoNotUse;

The initial equation is expanded into

0 = lambda*der(x) + (1-lambda)*(x-x0)

and you can solve the two equations to give

x = (lambda+(lambda-1)*x0)/(2*lambda - 1)

which has the correct value of x0 at lambda = 0 and of 1 at lambda = 1, but unfortunately has a singularity at lambda = 0.5.

]

semiLinear¶

(See definition of semiLinear). In some situations, equations with the semiLinear() function become underdetermined if the first argument (x) becomes zero, i.e., there are an infinite number of solutions. It is recommended that the following rules are used to transform the equations during the translation phase in order to select one meaningful solution in such cases:

Rule 1: The equations

y = semiLinear(x, sa, s[1]);
y = semiLinear(x, s[1], s[2]);
y = semiLinear(x, s[2], s[3]);
// ...
y = semiLinear(x, s[n], sb);
// ...

may be replaced by

s[1] = if x >= 0 then sa else sb
s[2] = s[1];
s[3] = s[2];
// ...
s[n] = s[n-1];
y = semiLinear(x, sa, sb);

Rule 2: The equations

x = 0;
y = 0;
y = semiLinear(x, sa, sb);

may be replaced by

x = 0
y = 0;
sa = sb;

[For symbolic transformations, the following property is useful (this follows from the definition):

semiLinear(m_flow, port_h, h);

is identical to :

-semiLinear(-m_flow, h, port_h);

The semiLinear function is designed to handle reversing flow in fluid systems, such as

H_flow=semiLinear(m_flow, port.h, h);

i.e., the enthalpy flow rate H_flow is computed from the mass flow rate m_flow and the upstream specific enthalpy depending on the flow direction.

]

getInstanceName¶

Returns a string with the name of the model/block that is simulated, appended with the fully qualified name of the instance in which this function is called.

[Example:

package MyLib
  model Vehicle
  Engine engine;
  ...
  end Vehicle;

  model Engine
  Controller controller;
  ...
  end Engine;

  model Controller
  equation
  Modelica.Utilities.Streams.print("Info from: " + getInstanceName());
  end Controller;

end MyLib;

If MyLib.Vehicle is simulated, the call of getInstanceName() returns:”Vehicle.engine.controller”

]

If this function is not called inside a model or block (e.g. the function is called in a function or in a constant of a package), the return value is not specified.

[The simulation result should not depend on the return value of this function. ]

Event-Related Operators with Function Syntax¶

The following event-related operators with function syntax are supported. The operators noEvent, pre, edge, and change, are vectorizable according to Section 12.4.6

initial()

Returns true during the initialization phase and false otherwise [thereby triggering a time event at the beginning of a simulation].

terminal()

Returns true at the end of a successful analysis [thereby ensuring an event at the end of successful simulation].

noEvent(expr)

Real elementary relations within expr are taken literally, i.e., no state or time event is triggered. See also Section noEvent and smooth and Section 8.5.

smooth(p, expr)

If p>=0 smooth(p,expr) returns expr and states that expr is p times continuously differentiable, i.e.: expr is continuous in all real variables appearing in the expression and all partial derivatives with respect to all appearing real variables exist and are continuous up to order p. The argument p should be a scalar integer parameter expression. The only allowed types for expr in smooth are: real expressions, arrays of allowed expressions, and records containing only components of allowed expressions. See also Section noEvent and smooth.

sample(start,interval)

Returns true and triggers time events at time instants start + i*interval(i=0,1, ...). During continuous integration the operator returns always false. The starting time start and the sample interval interval need to be parameter expressions and need to be a subtype of Real or Integer. |

pre(y)

Returns the “left limit” y(t^pre) of variable y(t) at a time instant t. At an event instant, y(t^pre) is the value of y after the last event iteration at time instant t (see comment below). The pre() operator can be applied if the following three conditions are fulfilled simultaneously:

variable y is either a subtype of a simple type or is a record component
y is a discrete-time expression
the operator is not applied in a function class.

[Note: This can be applied to continuous-time variables in when-clauses, see Section Discrete-Time Expressions for the definition of discrete-time expression.] The first value of pre(y) is determined in the initialization phase. See also Section pre.

edge(b)

Is expanded into (b and not pre(b)) for Boolean variable b. The same restrictions as for the pre() operator apply (e.g. not to be used in function classes).

change(v)

Is expanded into (v<>pre(v)). The same restrictions as for the pre() operator apply.

reinit(x, expr)

In the body of a when clause, reinitializes x with expr at an event instant. x is a Real variable (or an array of Real variables) that is implicitly defined to have StateSelect.always [so must be selected as a state, and it is an error, if this is not possible]. expr needs to be type-compatible with x. The reinit operator can only be applied once for the same variable - either as an individual variable or as part of an array of variables. It can only be applied in the body of a when clause in an equation section. See also Section TODO: 8.3.6.

A few of these operators are described in more detail in the following.

pre¶

A new event is triggered if at least for one variable v pre(v) <> v after the active model equations are evaluated at an event instant. In this case the model is at once reevaluated. This evaluation sequence is called “event iteration”. The integration is restarted, if for all v used in pre-operators the following condition holds: pre(v) == v.

[If v and pre(v) are only used in when-clauses, the translator might mask event iteration for variable v since v cannot change during event iteration. It is a “quality of implementation” to find the minimal loops for event iteration, i.e., not all parts of the model need to be reevaluated.

The language allows mixed algebraic systems of equations where the unknown variables are of type Real, Integer, Boolean, or an enumeration. These systems of equations can be solved by a global fix point iteration scheme, similarly to the event iteration, by fixing the Boolean, Integer, and/or enumeration unknowns during one iteration. Again, it is a quality of implementation to solve these systems more efficiently, e.g., by applying the fix point iteration scheme to a subset of the model equations.]

noEvent and smooth¶

The noEvent operator implies that real elementary expressions are taken literally instead of generating crossing functions, Section TODO: 8.5. The smooth operator should be used instead of noEvent, in order to avoid events for efficiency reasons. A tool is free to not generate events for expressions inside smooth. However, smooth does not guarantee that no events will be generated, and thus it can be necessary to use noEvent inside smooth. [Note that smooth does not guarantee a smooth output if any of the occurring variables change discontinuously.]

[Example:

Real x,y,z;
parameter Real p;
equation
x = if time<1 then 2 else time-2;
z = smooth(0, if time<0 then 0 else time);
y = smooth(1, noEvent(if x<0 then 0 else sqrt(x)*x));
// noEvent is necessary.

]

Variability of Expressions¶

The concept of variability of an expression indicates to what extent the expression can vary over time. See also Section TODO: 4.4.4 regarding the concept of variability. There are four levels of variability of expressions, starting from the least variable:

constant variability
parameter variability
discrete-time variability
continuous-time variability

For an assignment v:=expr or binding equation v=expr, v must be declared to be at least as variable as expr.

The right-hand side expression in a binding equation [that is, expr ] of a parameter component and of the base type attributes [such as start] needs to be a parameter or constant expression.
If v is a discrete-time component then expr needs to be a discrete-time expression.

Constant Expressions¶

Constant expressions are:

Real, Integer, Boolean, String, and enumeration literals.
Variables declared as constant.
Except for the special built-in operators initial, terminal, der, edge, change, sample, and pre, a function or operator with constant subexpressions as argument (and no parameters defined in the function) is a constant expression.

Components declared as constant shall have an associated declaration equation with a constant expression, if the constant is directly in the simulation model, or used in the simulation model. The value of a constant can be modified after it has been given a value, unless the constant is declared final or modified with a final modifier. A constant without an associated declaration equation can be given one by using a modifier.

Parameter Expressions¶

Parameter expressions are:

Constant expressions.
Variables declared as parameter.
Except for the special built-in operators initial, terminal, der, edge, change, sample, and pre, a function or operator with parameter subexpressions is a parameter expression.

Discrete-Time Expressions¶

Discrete-time expressions are:

Parameter expressions.
Discrete-time variables, i.e., Integer, Boolean, String variables and enumeration variables, as well as Real variables assigned in when-clauses
Function calls where all input arguments of the function are discrete-time expressions.
Expressions where all the subexpressions are discrete-time expressions.
Expressions in the body of a when-clause, initial equation, or initial algorithm.
Unless inside noEvent: Ordered relations (>, <, >=, <=) if at least one operand is a subtype of Real (i.e. Real elementary relations, see Section Equality, Relational, and Logical Operators) and the functions ceil, floor, div, mod, rem. These will generate events if at least one subexpression is not a discrete-time expression. [In other words, relations inside noEvent() , such as noEvent(x>1), are not discrete-time expressions].
The functions pre, edge, and change result in discrete-time expressions.
Expressions in functions behave as though they were discrete-time expressions.

For an equation expr1 = expr2 where neither expression is of base type Real, both expressions must be discrete-time expressions. For record equations the equation is split into basic types before applying this test. [This restriction guarantees that the noEvent() operator cannot be applied to Boolean, Integer, String, or enumeration equations outside of a when-clause, because then one of the two expressions is not discrete-time]

Inside an if-expression, if-clause, while-statement or for-clause, that is controlled by a non-discrete-time (that is continuous-time, but not discrete-time) switching expression and not in the body of a when-clause, it is not legal to have assignments to discrete variables, equations between discrete-time expressions, or real elementary relations/functions that should generate events. [This restriction is necessary in order to guarantee that there all equations for discrete variable are discrete-time expressions, and to ensure that crossing functions do not become active between events.]

[Example:

model Constants
  parameter Real p1 = 1;
  constant Real c1 = p1 + 2; // error, no constant expression
  parameter Real p2 = p1 + 2; // fine
end Constants;

model Test
  Constants c1(p1=3); // fine
  Constants c2(p2=7); // fine, declaration equation can be modified
  Boolean b;
  Real x;
equation
  b = noEvent(x > 1) // error, since b is a discrete-time expr. and
  // noEvent(x > 1) is not a discrete-time expr.
end Test;

]

Continuous-Time Expressions¶

All expressions are continuous-time expressions including constant, parameter and discrete expressions. The term “non-discrete-time expression” refers to expressions that are not constant, parameter or discrete expressions.

[1]	MATLAB is a registered trademark of MathWorks Inc. Mathematica is a registered trademark of Wolfram Research Inc.

Operators and Expressions¶

Expressions¶

Operator Precedence and Associativity¶

Evaluation Order¶

Example: Guarding Expressions Against Incorrect Evaluation¶

Arithmetic Operators¶

Equality, Relational, and Logical Operators¶

Miscellaneous Operators and Variables¶

String Concatenation¶

Array Constructor Operator¶

Array Concatenation Operator¶

Array Range Operator¶

If-Expressions¶

Member Access Operator¶

Built-in Variable time¶

Built-in Intrinsic Operators with Function Syntax¶

Numeric Functions and Conversion Functions¶

Event Triggering Mathematical Functions¶

div(x,y)¶

mod(x,y)¶

rem(x,y)¶

ceil(x)¶

floor(x)¶

integer(x)¶

Built-in Mathematical Functions and External Built-in Functions¶

Derivative and Special Purpose Operators with Function Syntax¶

delay¶

spatialDistribution¶

cardinality (deprecated)¶

homotopy¶

semiLinear¶

getInstanceName¶

Variability of Expressions¶

Constant Expressions¶

Parameter Expressions¶

Discrete-Time Expressions¶

Continuous-Time Expressions¶

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