If you want to stay native powershell use the start-process cmdlet and you can specify the filepath (process to run) as powershell.exe and the -ArgumentList parameters as conditions for your new session. In my example below i'm setting ExecutionPolicy so you don't have to rely on the system level policy, NoProfile will make your script a bit more resiilent by not loading any customized profile on a system.
$Cred = (Get-Credential)
$ScriptLocation = "C:\Temp\TestScript.ps1"
Start-Process -FilePath "powershell.exe" -ArgumentList "-NoProfile -ExecutionPolicy Bypass -File $ScriptLocation" -Credential $Cred
You can see in the script getting a credential object (which you'll probably want to provide so the script will just run) then specifying the script location and executing.
As vonPryz mentioned, you can always troubleshoot by adding -NoExit to your Argument list so the window stays open after executing the script but keep in mind if that if the script location doesn't exist you'll still see the powershell host appear and close right away.
You can also add -WindowStyle Hidden to your argument list to hide any window from appearing at all.
-NoExitto Powershell's parameters. That will keep the process active and you should see the actual error.-ExecutionPolicy Bypassto your arguments.