Mixed-integer quadratically constrained quadratic program- ming (MIQCQP) is a general class related to several important problems such as polynomial, semidefinite, and conic programming. Moreover, MIQCQP is a natural way to model many important problems in chemical engineering applications. Our motivation is the refinery operational planning problem (ROPP) under uncertainty, which has a large-scale deterministic equivalent MIQCQP. We tackle this problem proposing a primal-dual decomposition algorithm named the $p$-Lagrangian method, which combines a bundle-method inspired Lagrangian decomposition with MIP-based relaxations. These relaxations are obtained using the normalised multiparametric desegregation technique (NMDT) and can be made arbitrarily precise by means of a precision parameter $p$. We present enhancements for the NMDT-based relaxation and how to effectively employ them in the decomposition algorithm. The proposed method was tested on a real-world ROPP and compared with the commercial solver BARON in terms of performance. The numerical results obtained illus- trate the efficiency of the method for several instances.