1 Privacy and optimality of distributed schemes for secondary frequency regulation in power networks Kanwal Khan, Andreas Kasis, Marios M. Polycarpou and Stelios Timotheou Abstract— The increasing participation of local generation and controllable demand units within the power network motivates the use of distributed schemes for their control. Simultaneously, it raises two issues; achieving an optimal power allocation among these units, and securing the privacy of the generation/demand profiles. This study considers the problem of designing distributed optimality schemes that preserve the privacy of the generation and controllable demand units within the secondary frequency control timeframe. We propose a consensus scheme that includes the generation/demand profiles within its dynamics, providing guarantees that those cannot be inferred from the communicated signals, even when eavesdrop- pers possess knowledge of the underlying system dynamics. For the proposed scheme, we provide analytic stability and optimality guarantees and show that the secondary frequency control objectives are satisfied. The presented scheme is dis- tributed, locally verifiable and applicable to arbitrary network topologies. Our analytic results are verified with simulations on a 9-bus system, where we demonstrate that the proposed scheme enables an optimal power allocation and preserves the privacy of the generation/demand and the stability of the power network. I. I NTRODUCTION Motivation and literature survey: The increasing pene- tration of renewable sources of generation is expected to cause more frequent generation-demand imbalances within the power network, which may harm power quality and even cause blackouts [1]. Controllable demand is considered to be a means to address this issue, since loads may provide a fast response to counterbalance intermittent generation [2]. However, the increasing number of such active units makes traditionally implemented centralized control schemes expensive and inefficient, motivating the adoption of dis- tributed schemes. Such schemes offer many advantages, such as scalability, reduced expenses associated with the necessary communication infrastructure and enhanced reliability due to the absence of a single point of failure. The introduction of controllable loads and local renewable generation raises an issue of economic optimality in the power allocation. In addition, the introduction of smart me- ters for the monitoring of generation and demand units poses a privacy threat for the citizens, since readings may be used to expose customers daily life and habits, by inferring the users energy consumption patterns and types of appliances [3]. For example, this issue led the Dutch Parliament to prohibit the deployment of smart meters until the privacy concerns are resolved [4]. These concerns motivate the de- sign of distributed schemes that will simultaneously achieve an optimal power allocation and preserve the privacy of local prosumption profiles. Kanwal Khan and Andreas Kasis are first authors who contributed equally. Kanwal Khan, Andreas Kasis, Marios M. Polycarpou and Stelios Tim- otheou are with the KIOS Research and Innovation Center of Excellence and the Department of Electrical and Computer Engineering, University of Cyprus, Cyprus; e-mails: khan.kanwal@ucy.ac.cy, kasis.andreas@ucy.ac.cy, mpolycar@ucy.ac.cy, timotheou.stelios@ucy.ac.cy. This work was funded by the European Union’s Horizon 2020 research and innovation program under grant agreements No. 891101 (SmarTher Grid) and No. 739551 (KIOS CoE), and from the Republic of Cyprus through the Directorate General for European Programs, Coordination, and Development. In recent years, various studies considered the use of decentralized/distributed control schemes for generation and controllable demand with applications to both primary [5], [6], [7] and secondary [8], [9] frequency regulation, where the objectives are to ensure generation-demand balance and that the frequency attains its nominal value at steady state respectively. In addition, the problem of obtaining an optimal power allocation within the secondary frequency control timeframe has received broad attention in the literature [10], [11]. These studies considered suitably constructed optimization problems and designed the system equilibria to coincide with the solutions to these problems. In many studies, the control dynamics were inspired from the dual of the considered optimization problems [12], [13]. Such schemes, usually referred to in the literature as Primal-Dual schemes, yield an optimal power allocation and at the same time allow operational constraints to be satisfied. Alternative distributed schemes, which ensure that frequency attains its nominal value at steady state by using the generation outputs, have also been proposed [14]. However, the use of real-time knowledge of the prosumption in the proposed schemes may result in privacy issues. The topic of preserving the privacy of generation and demand units has recently attracted wide attention in the liter- ature. Different types of privacy concerns, resulting from the integration of information and communication technologies in the smart grid, are mentioned in [15]. In addition, [16] ana- lyzes various smart grid privacy issues and discusses recently proposed solutions for enhanced privacy, while [17] proposes a privacy-preserving power request scheme. Moreover, [18] uses the differential privacy framework to provide privacy guarantees and [19] studies the effect of differential privacy on smart metering data. A privacy-preserving aggregation scheme is proposed in [20] which considers various security threats. The use of energy storage units to preserve the privacy of user consumption has been considered in [21] and [22]. Furthermore, [23] and [24] aim to simultaneously preserve the privacy of individual agents and enable an optimal power allocation using homomorphic encryption and differential privacy respectively. Both approaches result in suboptimal allocations, which suggests a trade-off between optimality and privacy. Several existing techniques that aim at preventing disclosure of private data are also discussed in [25]. Although the problems of preserving the privacy of power prosumption and obtaining an optimal power allocation in power networks have been independently studied, the problem of simultaneously achieving these goals has not been adequately investigated. In addition, to the authors best knowledge, no study has considered the impact of such schemes on the stability and dynamic performance of the power grid. This study aims to jointly consider these objectives within the secondary frequency control timeframe. Contribution: This paper studies the problem of pro- viding optimal frequency regulation within the secondary frequency control timeframe while preserving the privacy of generation and controllable demand profiles. We first propose an optimization problem that ensures that secondary