If you have a site collection and want to upgrade from SharePoint 2010 to SharePoint 2013 with PowerShell, then this blog will give you step-by-step to do this a bit easier.
Go to Central Adminstration, create New Site and in Select Experience Version choose 2010. (Example: My site is Home).
The Home page is created in SharePoint 2010 as:
Open SharePoint 2013 Management Shell and Run as Administrator.
Enter command: Upgrade-SPSite http://server/sitecollection -VersionUpgrade (for example Upgrade-SPSite http://vngo15sql2/home -VersionUpgrade).
Wait until you see the following message:
Then, go to your site and launch it.