Index Reduction and Discontinuity Handling using Substitute Equations

Several techniques exist for index reduction and consistent initialization of higher index DAEs. Many such techniques change the original set of equations by differentiation, substitution, and/or introduction of new variables. This paper introduces substitute equations as a new language element. By means of a substitute equation, the value of a continuous variable or its time derivative is specified by an expression. This expression is evaluated each time that the variable or its time derivative, respectively, is referenced in the model. The advantage of substitute equations is that they enable index reduction and consistent initialization of higher index DAEs without changing the original equations; no existing variables are removed and no new variables are introduced. Substitute equations can also be used to enable the use of general purpose numerical solvers for equations where one or more of the unknowns are discontinuous, and they can be used to prevent functions to be called outside of their domain.