A restriction and feasible direction algorithm for a capacity expansion problem (Q1063517)

This paper examines the problem of optimally expanding existing capacity in order to meet an expected load in the context of an electric utility. The problem studied is a static single-period model which may be stated as follows: \[ \quad Minimize\quad \sum^{N}_{i=1}c_ iy_ i+\sum^{N}_{i=1}f_ iE_ i, \] \[ subject\quad to\quad \sum^{N}_{i=1}(y_ i+z_ i)=f(0),\quad y_ i\geq 0,\quad 0\leq z_ i\leq b_ i\quad (i=1,...,N), \] where N denotes the number of equipment types being considered, and for equipment type i, \(c_ i({\$}/Mw)\) denotes the (annualized) capital cost per unit capacity (if purchased for expansion), \(f_ i({\$}/Mw\) yr.) denotes the operating cost per unit energy generated, \(b_ i(Mw)\) denotes the existing capacity, and \(E_ i(Mw\) yr.) denotes the energy generated by equipment type i in a merit order loading scheme. Namely, if one assumes that \(f_ 1<f_ 2<...<f_ N\), then in terms of the decision variables \(z_ i\) and \(y_ i\), which respectively denote the existing and the new capacities of equipment type i used, we have \[ E_ i=\int^{x_ i}_{x_{i-1}}F(\ell)d\ell,\quad where\quad x_ j=\sum^{j}_{k=1}(y_ k+z_ k). \] Here, F(\(\cdot)\) is the anticipated inverse load duration curve for the period under consideration. This single-period model arises as an important subproblem in algorithms for multi-period models, and also serves a role in providing preliminary planning estimates. For this problem, a two phase procedure is developed. A preoptimization (Phase I) analysis easily determines (a) the capacities of existing equipments which will be used in an optimal solution; (b) the optimal (nonnegative) capacities of a subset of the new equipments to be purchased, and (c) a good quality starting solution. Having thus restricted a part of the solution to its optimal value, the problem is transformed into one of minimizing a convex, differentiable function, subject to a single generalized upper bounding constraint along with nonnegativity restrictions. An efficient specialization of a feasible directions algorithm (Phase II) is presented to solve this problem. The algorithm is versatile in that it provides a preview of whether or not all existing equipment capacity will be used in the light of available equipments, and which new equipments may be used in the optimal expansion plan. The algorithm can also solve the problem which enforces the use of all existing equipment capacity. Furthermore, Phase I, which is the principal part of this algorithm, provides the user with insightful sensitivity, and marginal cost duality information. A numerical problem is analyzed to illustrate the effectiveness of the procedure.