What happens if you connect both power supplies ?

@Cyberflake  You definitely won't damage anything, because there are certain system models where keeping the power adapter connected as well is explicitly recommended because the proper performance of those systems requires more power than the dock can supply.  I believe what will happen in that case is that the system will draw power from whichever source offers more wattage, but I may be wrong about that.  Out of curiosity though, why do you want to do this?  The WD19 can supply either 90W or 130W to attached Dell systems, depending on whether the WD19 itself has a 130W or 180W charger connected, and a quick check on the Vostro 5590's product page suggests that it comes with a 65W power supply, which means it's designed for a 65W power source.  That's easily within what the WD19 can supply, so there's no benefit to keeping a separate charger attached.