I haven't tried finding the controller gains in Dymola but this can be implemented in Simulink using the Control Toolbox. You can even connect your Dymola model with Simulink using the Functional Mock-up Interface standard and the FMI Toolbox.
If you want ot use your original nonlinear model, you may tune the control parameters by using optimization (e.g. minimize step-time, overshoot, steady-state error, control power etc.) To apply traditional linear methods for control design, you can linearize the Dymola model (with the linearize command). Then, you can import the linearized model into MATLAB, generate a state space model and use any method for control design of linear models.
Here are some hints: http://www.ep.liu.se/ecp/043/091/ecp09430112.pdf
One option would be to identify the PID paramters using the Optimization library that comes as a licensed add-on option to Dymola. There are a couple of examples showing how controller parameters can be identified that you could look at for inspiration. Furthermore, for DOE/parameter studies, you could use the Design library (which also contains basic optimization functionality).
Furthermore, for system analysis, you could use the tools in the Linear Analysis menu (and even more in the complete Modelica_LinearSystems2 library).