Distributed security state estimation based on homomorphic encryption for privacy-preserving consensus in cloud environment

As data sharing is essential in applications such as distributed state estimation of cyber-physical power systems (CPPSs), the issue of data privacy leakage among individual regional system operators is increasingly concerned. To solve the issue, this paper proposes a new distributed security state estimation (DSSE) method based on homomorphic encryption for privacy-preserving consensus. First, considering that regional measurement data managed by individual regional system operators could be attacked by false data injection attacks (FDIAs), a new active attack detection method based on the statistical characteristics of the watermarking signal before and after FDIAs is proposed to improve detection proactivity and accuracy, and it is found that the detection accuracy is positively correlated with the watermarking intensity when the signal to interference plus noise ratio (SINR) is greater than 10db. Second, considering that each regional system operator needs to exchang intermediate data under the premise of protecting data privacy to guarantee the consistency process of distributed state estimation, a homomorphic encryption (HE)-based privacy-preserving consensus method is proposed, where a hash function-based dual verification mechanism is presented to prevent ciphertext data from being tampered by FDIAs. Third, according to the detection results and data compensation mechanism, a local secure state estimation model is proposed, and it is proved that the upper and lower bounds of the reconstructed estimation error covariance are not only related to system parameters and external noise but also negatively related to the compensation error. Furthermore, according to a Lyapunov function including privacy-preserving consensus, sufficient condition for consistency is proven, which depends on Laplacian matrix of the system and the iteration step size. Finally, experimental results demonstrate the feasibility and effectiveness of the proposed dynamic watermarking-based active attack detector and distributed secure state estimation method for CPPSs. Moreover, the computational overhead of incorporating advanced IND-CPA countermeasures (i.e., ciphertext re-randomization and branchless arithmetic) is quantified, which confirms the feasibility of practical deployment.