Iterative scenario based reduction technique for stochastic optimization using conditional value-at-risk

Abstract

In the last decades, several tools for managing risks in competitive markets, such as the conditional value-at-risk, have been developed. These techniques are applied in stochastic programming models primarily based on scenarios and/or finite sampling, which in case of large-scale models increase considerably their size according to the number of scenarios, sometimes resulting in intractable problems. This shortcoming is solved in the literature using (i) scenario reduction methods, and/or (ii) speeding up optimization techniques. However, when reducing the number of scenarios, part of the stochastic information is lost. In this paper, an iterative scheme is proposed to get the solution of a stochastic problem representing the stochastic processes via a set of scenarios and/or finite sampling, and modeling risk via conditional value-at-risk. This iterative approach relies on the fact that the solution of a stochastic programming problem optimizing the conditional value-at risk only depends on the scenarios on the upper tail of the loss distribution. Thus, the solution of the stochastic problem is obtained by solving, within an iterative scheme, problems with a reduced number of scenarios (subproblems). This strategy results in an important reduction in the computational burden for large-scale problems, while keeping all the stochastic information embedded in the original set of scenarios. In addition, each subproblem can be solved using speeding-up optimization techniques. The proposed method is very easy to implement and, as numerical results show, the reduction in computing time can be dramatic, and more pronounced as the number of initial scenarios or samples increases.

Appendix: Mathematical formulation of the case study

The detailed formulation of the self-scheduling problem of a power producer solved in Sect. 5 is shown below:

(7.1)

subject to

(7.2)

(7.3)

(7.4)

(7.5)

(7.6)

(7.7)

(7.8)

(7.9)

(7.10)

(7.11)

(7.12)

Variable $x_{{ti}}^{\mathrm{G}}$ is the power generated in period t and scenario i, $x_{{ti}}^{\mathrm{P}}$ is the power sold in the pool during period t and scenario i, and x ^C is the power sold through bilateral contracts. Binary variables v _t , y _t and z _t represent if unit is committed, is started-up and is shut-down in period t, respectively. Pool price in period t and scenario i is λ _ti , and λ ^C is the price of the bilateral contracts. Variable costs are represented through a quadratic function whose linear and quadratic coefficients are C ^L and C ^Q, respectively. Fixed cost is C ^F, shut-down cost is C ^SD and start-up cost is $c^{\mathrm{SU}}_{t}(\cdot)$, which is a function of the time the unit has been shutdown in period t, s _t . Minimum power output of the unit is $\underline{P}$ and capacity of the unit is $\overline{P}$. Available maximum power output in period t and scenario i is $x_{{ti}}^{\max}$. Ramp-up limit, ramp-down limit, start-up ramp limit and shut-down ramp limit are R ^U, R ^D, R ^SU and R ^SD, respectively. Minimum up and down time of the unit are T ^U and T ^D, respectively. The number of periods unit has been on (+) or off (−) at the end of period t is s _t . Finally, N _T is the number of considered time periods, and N is the number of considered scenarios.

The objective function (7.1) represents the tradeoff between risk and expected profit. The profit comprises revenues from selling energy as well as production costs, fixed costs, start-up costs and shut-down costs. Constraints (7.2) express the power generated by the unit. Constraints (7.3) state the minimum power that must be produced by the unit. Constraints (7.4) force the unit to work below its available maximum power output. Constraints (7.5) and (7.6) state that the available maximum power output at every period depends on ramp rate limits. Constraints (7.7) limit the power generated at every period depending on ramp rate limits. Constraints (7.8) and (7.9) enforce feasibility in terms of minimum up and down time constraints, respectively. Constraints (7.10) and (7.11) preserve the logic of running, start-up, and shut-down status changes. And finally, constraints (7.12) constitute binary variables declaration.

Note that additional equations are needed to compute the time the unit has been shutdown in period t (s _t ) and the number of periods the unit has been on or off at the end of period t (x _t ). For sake of clarity these equations are not included in this appendix, but more information of these expressions can be found in Arroyo and Conejo (2000).