The availability of high-fidelity energy networks brings significant value to academic and commercial research. However, such releases also raise fundamental concerns related to privacy and security as they can reveal sensitive commercial information and expose system vulnerabilities. This paper investigates how to release the data for power networks where the parameters of transmission lines and transformers are obfuscated. It does so by using the framework of Differential Privacy (DP), that provides strong privacy guarantees and has attracted significant attention in recent years. Unfortunately, simple DP mechanisms often result in AC-infeasible networks. To address these concerns, this paper presents a novel differentially private mechanism that guarantees AC-feasibility and largely preserves the fidelity of the obfuscated power network. Experimental results also show that the obfuscation significantly reduces the potential damage of an attack carried by exploiting the released dataset.