Solving Nonsmooth and Discontinuous Optimal Power Flow problems via interior-point ℓp-penalty approach

Abstract

The Nonsmooth and Discontinuous Optimal Power Flow (ND-OPF) is a large-scale, nonsmooth, discontinuous, nonconvex and multi-modal problem. In this problem, discontinuity is related to the representation of Prohibited Operating Zones (POZ), whereas nonsmoothness is related to the representation of Valve-Point Loading Effect (VPLE) in the fuel costs. Due to such features, all approaches that have been proposed for solving this problem are based on heuristics or on mixed integer nonlinear programming reformulations. In this paper, we propose the Piecewise Polynomial Interpolation (PPI) function approach for handling discontinuities related to the POZ constraints, which consists in replacing such constraints by an equivalent set of smooth and continuous equality and inequality constraints. The PPI function is of class C1 and assumes null values at all allowed operating zones and non-null values at all forbidden zones. The model resulting from the PPI approach is the Equivalent Intermediate Model (EIM). For handling nonsmoothness on the EIM, we recast it as an equivalent model, where the VPLE term in the objective function becomes linear, and nonlinear box inequality constraints are introduced. The optimization model that results from such recasts is the Equivalent Smooth and Continuous OPF(ESC-OPF) model proposed, which can be solved by strictly gradient-based methods. Finally, we propose a primal–dual interior point ℓp-penalty approach (with 0<p≤1) for solving the ESC-OPF model, where inequality constraints are penalized by using the modified log-barrier function of Jittortrum–Osborne–Meggido and the equality constraints associated with the PPI function approach are penalized via ℓp lower-order exact penalty functions. Numerical results involving systems with up to 2007 buses, with 282 generating units and 846 POZ have shown that the proposed approach has been able to solve large-scale ND-OPF problems, with acceptable computation times.