A Coordinate-Descent Algorithm for Tracking Solutions in Time-Varying Optimal Power Flows

Abstract

Consider a polynomial optimisation problem, whose instances vary continuously over time. We propose to use a coordinate-descent algorithm for solving such time-varying optimisation problems. In particular, we focus on relaxations of transmission-constrained problems in power systems.On the example of the alternating-current optimal power flows (ACOPF), we bound the difference between the current approximate optimal cost generated by our algorithm and the optimal cost for a relaxation using the most recent data from above by a function of the properties of the instance and the rate of change to the instance over time. We also bound the number of floating-point operations that need to be performed between two updates in order to guarantee the error is bounded from above by a given constant.

Timescale Separation in Autonomous Optimization

“…The historic roots of the approach pursued in this paper can be traced back to the study of communication networks where congestion control algorithms have been analyzed from an optimization perspective [12]- [14]. Similar ideas have recently attracted a lot of interest in power systems, where feedbackbased optimization schemes have been proposed for voltage control [15], [16], frequency control [17]- [19], or general power flow optimization [20]- [23]. For a survey see [24].…”

“…The application of Theorem V.1 hinges on Assumption V.3 and therefore on the existence explicit Lyapunov function for the dynamics (23). This question, however, remains open and has only recently become a topic of intense study [39]- [41].…”

Pursuit of Low-Rank Models of Time-Varying Matrices Robust to Sparse and Measurement Noise

“…Second, we do not make restrictive assumptions on RIP properties, incoherence, identical covariance matrices, independence of all outlier supports, or initialisation. Broadly speaking, such analyses of time-varying non-convex optimisation (Liu et al 2018;Tang et al 2018;Fattahi et al 2019;Massicot and Marecek 2019), seems to be an important direction for further research.…”

Projections onto the Set of Feasible Inputs and the Set of Feasible Solutions

“…The method can plausibly be adapted to other families nonlinear optimisation problems, and perhaps analysed at that level of generality. One may also wonder how to extend this to the on-line setting [35]. There is hence a considerable scope for further work.…”