Optimal deterministic and robust selection of electricity contracts

Abstract

We address the Electricity Contract Selection Problem (ECSP), of finding best parameters of an electricity contract for a client based on his/her past records of electricity consumption over a fixed time period. The objective is to optimize the electricity bill composed by some fixed cost, the cost of the subscription of the electricity contract and penalties due to overpowering when consumption exceeds subscribed power. The ECSP can be formulated as a convex separable optimization problem subject to total order constraints. Due to this special structure, ECSP is a special case of two well known classes of convex separable optimization problems, namely the minimum network flow under convex separable cost and minimizing convex separable functions under chain constraints. Both classes are well treated in the litterature and can be solved in polynomial time (Ahuja and Orlin in Oper Res 49(5):784–789, 2001; Ahuja et al. in Manag Sci 49(7):950–964, 2003; Best et al. in SIAM J Optim 10:658–672, 2000; Karzanov and McCormick in SIAM J Comput 26(4):1245–1275, 1997; Minoux in Eur J Oper Res 18(3):377–387, 1984; Minoux, in Gallo and Sandi (eds) Netflow at pisa, mathematical programming studies, Springer, Berlin, 1986). In particular, the algorithm in Ahuja and Orlin (2001) achieves the best theoretical time complexity assuming that computing the objective function value at one specific point can be done in constant time. However, when we work on a big amount of historic data as in ECSP, the time required for evaluating the objective function cannot be assumed to be O(1) anymore. In this paper, we propose a new algorithm for ECSP which is specially designed to reduce the computational effort over large scale historical data. We present numerical results showing that our algorithm outperforms the algorithm in Ahuja and Orlin (2001) when applied to consumption data of various types of clients. A robust version of ECSP based on a Seasonal and Trend decomposition approach for modelling consumption uncertainty is also investigated. The resulting worst-case cost minimization problem is shown to be efficiently solvable using the same algorithm as for deterministic ECSP.

Appendix

We state in this appendix the reformulation of ${\mathcal {P}}$ as a separable convex cost network flow problem.

The graph to be considered contains K nodes numbered $1,2,\ldots , K$ and a set of $2K-1$ arcs, decomposed into K ’primary arcs’ and $K-1$ ’secondary arcs’.

• the set of primary arcs contains the $K-1$ arcs of the form $(i,i+1)$ for $i=1,\ldots ,K-1$, together with a ’return arc’ (K, 1). The flow value $x_i$ on each of these arcs (including $x_K$ for the return arc) has to meet the bound constraints $0\le x_i \le C$ (and $0 \le x_K \le C$) and the corresponding cost function is $F_i$ ($F_K$ for the return arc);

• the set of secondary arcs is composed of $K-1$ arcs of the form (1, j) for $j=2,\ldots ,K$. The flow value $s_j$ on each of these arcs has to meet the bound constraints $0\le s_j \le C$, and the associated cost function is identically 0.

Problem ${\mathcal {P}}$ then reduces to determining a minimum cost circulation on the above graph. We observe that the flow values $s_j$ on the secondary arcs play the role of slack variables for the constraints $x_{j-1} \le x_{j}$ for $j=2,\ldots ,K$.