  Transmission-constrained problems in power systems can be cast as polynomial
optimization problems whose coefficients vary over time. We consider the
complications therein and suggest several approaches. On the example of the
alternating-current optimal power flows (ACOPFs), we illustrate one of the
approaches in detail. For the time-varying ACOPF, we provide an upper bound for
the difference between the optimal cost for a relaxation using the most recent
data and the current approximate optimal cost generated by our algorithm. This
bound is a function of the properties of the instance and the rate of change of
the coefficients over time. Moreover, we also bound the number of
floating-point operations to perform between two subsequent updates to ensure a
bounded error.
