What country/region are you in? This forum has members from all over the world. I am in the USA, but from the voltage level I suspect you are not.
We use mostly 125VDC or 48 VDC ungrounded supplies for RTUs and protective relays, although some locations have a separate positive ground 48VDC battery for communications equipment. By solidly grounded, you mean that a short from the positive lead to ground will cause a fuse to blow? In our "ungrounded" applications, it is really capacitively grounded. Without a fault the capacitive current keeps the voltage measured positive/negative to ground at +65V/-65V, but a fault from positive to ground result in voltage measured positive/negative to ground at +0V/-125V.
Changing grounding types requires a careful analysis including:
1)That the devices are capable of floating operation and do not have the chassis connected to either positive/negative.
2)In large DC systems, a significant capacitive charging current can flow when a fault occurs. One of the IEEE guides on protection discusses how to design trip circuits such that the charging currents do not cause protective relay trips.
3)Ground fault monitoring must be in place and faults must be addressed promptly. Multiple unexpected grounds can allow unexpected currents to flow through the DC system.
4)Revised maintenance, testing and trouble shooting procedures.
On some of our more sensitive communication gear, we mount a simple low pass LC filter for each rack of equipment.