Model Equations
The equation section of the model description is preceded by the keyword equations:
followed by a number of assignments.
The keyword must be on a separate line.
The assignments can be to intermediate variables, but must finally be to the residuals.
All residuals must be assigned.
A simple example:
The available arithmetic operators are:
| operators | |
|---|---|
| + | addition |
| - | subtraction |
| * | multiplication |
| / | division |
| ^ | exponentiation, or to the power of |
Normal operator precedence rules apply with some caveats:
Subtraction and unary minus come before addition.
Division comes before multiplication.
Exponentiation is left-associative, so 2^1^2 = 4.
Also note that the unary minus is treated as a subtraction from zero, so -2^2 = -4.
Use parentheses ( )
where required or for disambiguation and readability.
The available special functions are:
| functions | |
|---|---|
| sin( ) | sine function |
| cos( ) | cosine function |
| tan( ) | tangent function |
| asin() | arc sine function |
| acos( ) | arc cosine function |
| atan( ) | arc tangent function |
| sinh( ) | hyperbolic sine function |
| cosh( ) | hyperbolic cosine function |
| tanh( ) | hyperbolic tangent function |
| exp( ) | exponential function |
| erf( ) | error function |
| ln( ) | natural logarithm function, base e |
| log( ) | natural logarithm function, base e (alt) |
| log10( ) | common logarithm function, base 10 |
| sqrt( ) | square root function |
| abs( ) | absolute value function |
| sign( ) | sign function, returns 1 or -1 |
Conditional evaluation is also supported:
if (condition)
assignments
fi
or:
if (condition)
assignments
else
assignments
fi
All keywords must be on a separate line.
When assigning variables, care must be taken that they receive a value in every conditional branch of the model. The used-before-assigned errors are only caught during runtime, which means that there are no actionable diagnostics available. The simulator and optimizer will simply skip the regions, and the data points respectively, where these errors occur. The same holds true for the residuals.
The available conditional operators are:
| conditional operators | |
|---|---|
| == | equal to |
| != | unequal to |
| > | larger than |
| < | smaller than |
| >= | larger than or equal to |
| <= | smaller than or equal to |
| ! | not |
| not( ) | not (alt) |
| & | and |
| | | or |
The operator precedence rules are: comparison before not before and, or.
Error exit:
When the evaluation of the model can't continue under certain conditions,
this can be signaled by calling error or assert.
These functions take a conditional expression as their argument.
The call error(condition) will exit when the conditional expression evaluates to true,
while assert(condition) will exit when the conditional expression evaluates to false.
For backward compatibility, the old syntax is still supported:
if (condition)
error(1)
fi
or for assert:
if (condition)
assignments
else
error(1)
fi
The available special numbers are:
Special numbers (and physical constants) start with an underscore and can be prefixed with a number as a multiplier.
| special numbers | |
|---|---|
| _pi | π |
| _pi_2 | π/2 |
| _pi_4 | π/4 |
| _1_pi | 1/π |
| _2_pi | 2/π |
| _sqrtpi | √π |
| _sqrt2pi | √2π |
| _1_sqrtpi | 1/√π |
| _2_sqrtpi | 2/√π |
| _e | e |
| _ln2 | ln(2) |
| _ln10 | ln(10) |
| _log10e | log10(e) |
| _sqrt2 | √2 |
| _sqrt1_2 | √1/2 |
| physical constants | |
|---|---|
| _k | Boltzman constant |
| _q | elementary charge |
| _c | light speed in vacuum |
| _G | gravitational constant |
| _h | Planck constant |
| _F | Faraday constant |
| _R | gas constant |
| _NA | Avogadro number |
| _0C | 0°C in Kelvin = 273.15 K |
| _eps0 | electric constant |
| _mu0 | magnetic constant |
A constant of 2π can be specified as: 2_pi.