New bounding and decomposition approaches for MILP investment problems: Multi-area transmission and generation planning under policy constraints

We propose a novel two-phase bounding and decomposition approach to compute optimal and near-optimal solutions to large-scale mixed-integer investment planning problems that have to consider a large number of operating subproblems, each of which is a convex optimization. Our motivating application is the planning of power transmission and generation in which policy constraints are designed to incentivize high amounts of intermittent generation in electric power systems. The bounding phase exploits Jensen's inequality to define a lower bound, which we extend to stochastic programs that use expected-value constraints to enforce policy objectives. The decomposition phase, in which the bounds are tightened, improves upon the standard Benders' algorithm by accelerating the convergence of the bounds. The lower bound is tightened by using a Jensen's inequality-based approach to introduce an auxiliary lower bound into the Benders master problem. Upper bounds for both phases are computed using a sub-sampling approach executed on a parallel computer system. Numerical results show that only the bounding phase is necessary if loose optimality gaps are acceptable. However, the decomposition phase is required to attain optimality gaps. Use of both phases performs better, in terms of convergence speed, than attempting to solve the problem using just the bounding phase or regular Benders decomposition separately.