  We deduce a procedure to apply balanced truncation to parameter-dependent
differential-algebraic systems. For that we solve multiple projected Lyapunov
equations for different parameter values to compute the Gramians that are
required for the truncation procedure. As this process would lead to high
computational costs if we perform it for a large number of parameters, we
combine this approach with the reduced basis method that determines a reduced
representation of the Lyapunov equation solutions for the parameters of
interest. Residual-based error estimators are then used to evaluate the quality
of the approximations. After introducing the procedure for a general class of
differential-algebraic systems we turn our focus to systems with a specific
structure, for which the method can be applied particularly efficiently. We
illustrate the efficiency of our approach on several models from fluid dynamics
and mechanics.
