A converter arrangement has a line-commutated converter with an AC voltage terminal to be connected to an AC voltage grid via at least one phase line. The converter arrangement has at least one switching module branch that is arranged in series in the at least one phase line and that includes a series connection of switching modules at whose terminals bipolar voltages that sum to give a branch voltage are in each case able to be generated. A bypass branch is arranged in a parallel connection to the switching module branch. At least one switching device is arranged in the bypass branch. The switching device includes activatable semiconductor switches that are connected in antiparallel. There is also described a method for starting up the converter arrangement.