Switching between ODE and DAE behavior is controlled by a structural parameter called enableDAEModeParameter.

A structural parameter (causality = structuralParameter) may only be set during Configuration Mode or Reconfiguration Mode. This constraint ensures that the importer can re-examine the model structure from the manifest and reinitialize its solver before any computation takes place.

The variable enableDAEModeParameter is declared in the modelDescription.xml as a Boolean under <ModelVariables>. The name and valueReference is assigned freely by the FMU exporter; the importer locates the parameter by its valueReference declared in the <EnableDAEParameter> element of fmi-ls-manifest.xml. The description attribute is optional.

An example of how the enableDAEModeParameter is declared:

Because enableDAEModeParameter is a structural parameter, the importer must change it inside a configuration or reconfiguration scope.

Using the semi-explicit formulation defined in Section 2.2, the differential equations are

with

and the algebraic equation is

with

The initial values in this notation are

The Jacobian of the residual equation with respect to the algebraic variable is

which is non-singular. Hence, the algebraic equation can be locally solved for \(\mathbf{a}_c\), and the Robertson problem is an index-1 DAE in the representation supported by this layered standard.

Assume the FMU has the following variables:

Here, \(y_1\) and \(y_2\) are the components of \(\mathbf{x}_c\), while \(y_3\) is the single component of \(\mathbf{a}_c\). The residual \(\mathbf{f}_{\mathit{res}}\) is represented by the variable \(r\).

The DAE-specific manifest elements are:

Because the residual equation is locally solvable for \(\mathbf{a}_c\), the algebraic variable can be eliminated to obtain an explicit ODE.

For fixed \(\mathbf{x}_c\), \(\mathbf{u}_c\), \(\mathbf{p}\), and \(t\), consider the mapping

Its inverse with respect to \(\mathbf{a}_c\), evaluated at residual zero, gives

For the Robertson problem,

and therefore

Substituting this inverse into the differential equations gives the explicit ODE

For the Robertson problem, this becomes

The quantity corresponding to the algebraic variable in the DAE formulation can still be computed in the ODE representation as

The model is then represented as an ODE Model Exchange FMU. Using the value references of the Robertson FMU, the relevant variables are:

In the ODE representation, \(y_3\) is computed from \(y_1\) and \(y_2\) and is not listed as an <AlgebraicVariable>.

The corresponding regular ODE ModelStructure available in the modelDescription.xml is:

The generic fully-implicit DAE, or implicit DAE, formulation:

where \(z(t)\) represents state and additional variables and \(t\) is time.

A similar and sometimes more useful representation:

with dynamic variables \(x(t)\), algebraic variables \(y(t)\) and time \(t\).

Note that \(x(t)\) is not necessarily the state variable, more on this in Section 4.1.7.2.

The semi-explicit index 1, also called Hessenberg index 1, DAE representation

with \(g\) solvable for \(y\), \(det \left( \frac{\partial}{\partial y}g\right) \neq 0\).

Here \(f\) is called the differential equations and \(g\) the algebraic equations.

A fully implicit DAE

can always be rewritten as a semi-explicit DAE by introducing a new algebraic variable \(x^{\prime}\)

The semi-explicit DAE representation

with \(g^1\) solvable for \(y\), \(det \left( \frac{\partial}{\partial y}g^1\right) \neq 0\) and \(\left\{ \frac{d}{dt} g_i^k \right\} \subseteq \{ g_i^{k-1}\}, \; k > 1\).

where mass matrix \(M\) is singular.

An equivalent explicit formulation:

See [cui2020mass].

where matrix \(E\) is singular.

There are multiple, sometimes closely related, definitions of indices for DAE systems. Usually these indices are a measure of how far from an ODE the DAE is.

A simple initial-value problem

with variables \(x,y,z \in \mathbb{R}\) and start condition

at time \(t_0\). For this specific example a start condition \(x_0\) is sufficient to calculate consistent start values. The independent variable (usually time) \(t\) is omitted for simplicity.

The Modelica model Modelica.Electrical.Analog.Examples.CauerLowPassAnalog from the Modelica Standard Library ^[3]

has structural singularities.

To produce a complete and consistent set of initial conditions for the DAE, exporting tools usually use a symbolic index reduction algorithm.

The model has nine state candidates

and four constraint equations

Modelica tools that want to export an ODE ModelExchange FMU need to reduce the perturbation index to index 1 to end up with an ODE representation of the DAE system.

One option to achieve this is the so called dummy derivative method. From the set of state candidates, five state variables are chosen (e.g. C1.v, C3.v, C5.v, L1.i, L2.i) and the constraint equations

are differentiated

which results in four additional dummy states ($DER.L1.v, $DER.L2.v, $DER.C2.v, $DER.C4.v). Here the Modelica keyword stateSelect influences how variables are picked.

For simulation models representing power grids on a national or larger scale, traditional ODE solvers could struggle with convergence and performance. Also, the time needed to perform symbolic transformation from DAE to ODE is growing with the number of equations.

See for example Modelica models ScalableTestSuite.Electrical.TransmissionLine.ScaledExperiments.TransmissionLineModelica_N_10 to ScalableTestSuite.Electrical.TransmissionLine.ScaledExperiments.TransmissionLineModelica_N_1280 from ScalableTestSuite ^[4].

The simulation was performed using OpenModelica v1.27-dev-92 with the following settings:

Since OpenModelica does not support FMI 3.0 or FMI-LS-DAE at the time of writing, the export and simulation times were measured using OpenModelica’s default C export and simulation.