I've clicked around on a few of the related question. I am not using qgis or anything "fancier" than ArcMap 10.2.2 for desktop and Microstation V8i. Is there a way to successfully convert shapefiles to a dgn file? I've been able to convert and open a shapefile in a dgn, but the coordinate system is always off. Let me know if more information would be helpful!
You can do this in ArcGIS using the Export to CAD tool. The output type parameter has a dgn v8 option:
The CAD platform and file version of the output files. This value overrides any output_type values contained in the keyname column or alias column CADFile_type. Types include DGN_V8, DWG_R14, DWG_R2000
Before to load your shapefile convert your dataframe with the coordinate systems that you want to project.
Here is a workaround. Go to Microstation, Applications tab and click on Map. This will activate the Bentley Map application. Then go to File->Import->GIS Data Types. In the Interoperability pop-up window, right click on Import and select the first option. A new node should open up under the Import tree. Right click on the new node and select "Add File (the first option)". Then navigate to your shapefile. Set the level and appearance like you want it to be and then right click on the file in the tree and select "Import". This should bring the contents of the shapefile into the Microstation file. It may be slightly off the exact location. IT can be fixed automatically if the correct surface correction factor is know. Or, you can simply manually move it into the correct location. The scale should not be an issue. I have used this option successfully even in cases where the Microstation file is in state plane projection coordinates and the shapefile in GCS American NAD 1983.