Smart meters play an important role in modern power grids by providing fine-grained power consumption data and enabling services such as dynamic pricing and demand-side management. The smart metering devices are firmware-driven, where it is important that the devices be able to securely update their firmware on a regular basis to fix bugs, and improve as well as add services. In this paper, we propose a new privacy-aware secure firmware-updating framework called PRSUF (Privacy-aware Reconfigurable Secure-Firmware Updating Framework) to securely update the firmware in smart metering devices. The proposed the framework allows a hardware intrinsic secret to being updated and stored in a secure and efficient way. One of its key differentiating features is that, unlike existing mechanisms, the proposed scheme does not require storing any keys in the meter's non-volatile memory (NVM), thereby making it is secure against a number of physical and side-channel attacks. As compared to state-of-the-art solutions, the proposed security framework has notable features such as reconfigurability, protection against cloning and downgrading, detection of theft of services and tampering with the firmware and the hardware, etc.