General polynomial chaos in the current-voltage formulation of the optimal power flow problem

Mathematical optimization techniques play a key role in enabling the power system transition to sustainable energy and are used for a variety of applications such as scenario analysis, optimal planning and operational decision making. Power flow optimization, a.k.a., optimal power flow, is a building block for many applications in network operations and planning. This paper discusses the treatment of general polynomial chaos expansion for the current-voltage formulation of the optimal power flow problem. The power flow equations of the current-voltage formulation are linear, making their Galerkin projection significantly more tractable compared to formulations in the power-voltage space, while still being exact. Furthermore, auxiliary variables and quadratic constraints enable chance constraints as second-order-cone constraints. An additional advantage of this approach is that the Galerkin projection of the quadratic constraints is significantly less complex compared to those of non-linear constraints with a polynomial degree higher than two, as would be needed for expressing the original variables' variance without the auxiliary variables. On average, the current-voltage formulation using auxiliary variables shows more than an order of magnitude speed-up with respect to the power-voltage formulation without auxiliary variables.