Recently, we have made major advances in optimization in power systems. This includes time-varying optimization with non-linear constraints of the alternating-current optimal power flows, as well as mixed-integer optimization in unit commitment.

Similar to cars, which are increasingly becoming "computers on wheels," many processes in power systems are increasingly digitized and involve solvers for optimization problems. The solvers both find feasible solutions, which may be hard to see for dispatchers even on the intraday production planning of larger power producers, the less in structured offers on EU's Internal Electricity Balancing Market, and find ever improving solutions with guarantees of their quality. Similar to the gains within Computer Vision related to the advent of  "computers on wheels," power systems have driven much recent work in mathematical optimization.

The applications of mathematical optimization in power systems is varied: from continuous-valued problems of the alternating-current optimal power flows, through various problems in intraday trading, towards unit commitment problems of power producers, where discrete aspects dominate. Also, many forecasting problems are, in the end, solved using solvers for optimization problems.

There are also common characteristics. Notably, many applications of mathematical optimization in power systems are instances of model predictive control (MPC). For each time horizon, there are regularly arriving instances of a problem with the same structure, but different coefficients. In this sense, most problems are "time-varying". Up until recently, however, there were guarantees available for tracking the trajectory of optimal solutions in time-varying optimization only for convex, unconstrained cases.

Recently, we have made major advances in time-varying continuous optimization with constraints. This includes non-convex non-linear constraints of the alternating-current optimal power flows. There, traditional textbooks consider only local methods (Gauss-Seidl Newton-Raphson), which can produce arbitrarily bad solutions, when the initial point is poor. With the intermittent generation from renewable energy sources, a great initial point is harder to obtain, though. As an alterntive, one can consider successive convex overapproximations, which are known as convex relaxations. Such convex relaxations guarantee convergence to the global optimum, independent of the initial point. For the first time, we have shown the structure of the trajectory of the optimal solutions for a broad class of such convex relaxations (semidefinite programming, SDP) and algorithms for the same, with guarantees on the distance between the trajectory of solutions produced and the optimal trajectory, depending on the number of arithmetic operations performed between two updates.

Independently, we have worked on mixed-integer optimization. In mixed-integer optimization, such as unit commitment problems, textbooks mostly focus on straight-forward applications of off-the-shelf solvers, without considering time-varying aspects. In contrast, considering the time-varying aspects makes it possible to produce solutions with good guarantees faster. These aspects have been developed in the so-called Optimization Module, which governs unit commitment in CEZ a.s., the largest utility in Central and Eastern Europe. This is replacing the heuristics previously used by the dispatchers there.

We believe that the techniques of time-varying optimization will have many further applications in power systems. Indeed, the digitalization of power systems requires new solvers -- and the research in mathematical optimization requires new challenges.

Text credit: Jakub Marecek, Technicall Podzim 2021,

Translation: Jakub Marecek

Image credit: Jakub Marecek. The figure shows how one can track the trajectory of optimal solutions to an alternating-current optimal power flow (ACOPF) in a well-known benchmark instance.