Capitalize "planning department" if that's the name of the department; don't if it's a description of the department. It seems unlikely that the department is Charlotte's except in the sense that she works there. On the other hand, you would write "Eric Holder's Department of Justice," because that's the official name of the US DoJ, and traditionally we use the possessive with the guy (or now gal) who runs it.
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Anonymousin Charlotte's planning department.I'd say it's fine as shown above, but I don't think any city uses a possessive in the official name of its departments. That's why you can't do Charlotte's Planning Department. More likely they're done with an of-phrase: The Planning Department of [Austin / Albany / Bloomington / Denve