Distributionally robust multi-stage stochastic programming for mid- and long-term cross-regional power markets

: So far the study of stochastic programs with recourse has been limited to the case (called by G. Dantzig programming under uncertainty) when only the right-hand sides or resources of the problem are random. In this paper the authors extend the theory to the general case when essentially all the parameters involved are random. This generalization immediately raises the problem of attributing a precise meaning to the stochastic constraints. They examine a probability formulation (satisfying the constraints almost surely) and a possibility formulation (satisfying the constraints for all values of the random parameters in the support of their joint distribution) and show them equivalent under a rather weak but curious W-condition. Finally, they prove that without restriction the equivalent deterministic form of a stochastic program with recourse is a convex program for which we obtain some additional properties when some of the parameters of the original problem are constant. The applications of the theoretical results of this paper to certain classes of stochastic programs which have arisen from practical problems will be presented in a separate paper: 'Stochastic Programs with Recourse: Special Forms.' (Author)

A robust approach for planning electric power systems associated with environmental policy analysis

Robust stochastic fuzzy possibilistic programming for environmental decision making under uncertainty

In the setting of public transportation system, improving the service quality as well as robustness against uncertainty through minimizing the total waiting times of passengers is a real issue. This study proposed robust multi-objective stochastic programming models for train timetabling problem in urban rail transit lines. The objective is to minimize the expected value of the passenger waiting times, its variance and the penalty cost function including the capacity violation due to overcrowding. In the proposed formulations, the dynamic and uncertain travel demand is represented by the scenario-based time-varying arrival rates and alighting ratio at stops. Two versions of the robust stochastic programming models are developed and a comparative analysis is conducted to testify the tractability of the models. The effectiveness of the proposed stochastic programming model is demonstrated through the application to line 5 of Tehran underground railway. The outcomes validate the benefits of implementing robust timetables for rail industry. The computational experiments shows significant reductions in expected passenger waiting time of 21.27 %, and cost variance drop of 59.98 % for the passengers, through the proposed robust mathematical modeling approach.

A comparison of four approaches from stochastic programming for large-scale unit-commitment

Long-term reservoir management often uses bounds on the reservoir level, between which the operator can work. However, these bounds are not always kept up-to-date with the latest knowledge about the reservoir drainage area, and thus become obsolete. The main difficulty with bounds computation is to correctly take into account the high uncertainty about the inflows to the reservoir. In this article, we propose a methodology to derive minimum bounds while providing formal guarantees about the quality of the obtained solutions. The uncertainty is embedded using either stochastic or robust programming in a model-predictive-control framework. We compare the two paradigms to the existing solution for a case study and find that the obtained solutions vary substantially. By combining the stochastic and the robust approaches, we also assign a confidence level to the solutions obtained by stochastic programming. The proposed methodology is found to be both efficient and easy to implement. It relies on sound mathematical principles, ensuring that a global optimum is reached in all cases.

The traditional power grid is confronted with great challenges brought by the integration of renewable power sources (such as solar and wind) for their uncertain, volatile, and intermittent characteristics. This paper investigates the unit commitment problem with stochastic solar power integration and makes the following major contributions. First, the scheduling problem is formulated as a two-stage stochastic programming. In the first stage the unit commitment, economic dispatch, and solar power scheduling decisions are made based on the day-ahead solar power prediction and in the second stage the solar power is rescheduled with real-time solar power generation as each decision instant approaching. Second, in the rescheduling, cost for buying reserve and penalty for curtailing solar power are considered for higher penetration and better utilization of solar power. Third, the problem is reformulated as a stochastic mix-integer linear programming to facilitate computation and the influences of spinning reserve price, penalty for curtailing solar power, and solar power uncertainty are analyzed and discussed. The performance of the proposed method is compared with a deterministic programming, a robust programming, and a stochastic programming through a modified six-bus system and the results demonstrate that the proposed method can better accommodate the fluctuation of solar power.

This paper presents a new and high performance solution method for multistage stochastic convex programming. Stochastic programming is a quantitative tool developed in the field of optimization to cope with the problem of decision-making under uncertainty. Among others, stochastic programming has found many applications in finance, such as asset-liability and bond-portfolio management. However, many stochastic programming applications still remain computationally intractable because of their overwhelming dimensionality. In this paper we propose a new decomposition algorithm for multistage stochastic programming with a convex objective and stochastic recourse matrices, based on the path-following interior point method combined with the homogeneous self-dual embedding technique. Our preliminary numerical experiments show that this approach is very promising in many ways for solving generic multistage stochastic programming, including its superiority in terms of numerical efficiency, as well as the flexibility in testing and analyzing the model.

An interval two-stage robust stochastic programming approach for steam power systems design and operation optimization under complex uncertainties

Medium-Term Production Scheduling of a Large-Scale Steelmaking Continuous Casting Process under Demand Uncertainty

Optimization under uncertainty: state-of-the-art and opportunities

: Random outcomes can often produce significant effects on planning decisions that consider several time periods. Multistage stochastic programs can model these decisions but implementations are generally restricted to a limited number of scenarios in each period. We present an alternative approximation scheme that can obtain lower and upper bounds on the optimal objective value in these stochastic programs. The method is based on building response functions to future outcomes that depend separably on the variation of random parameters around the limited set of scenarios that is initially provided. For stochastic linear programs, the resulting optimization problem involves an objective with a limited number of nonlinear terms subject to linear constraints. The method can be incorporated into various alternative procedures for solving multistage stochastic linear programs with finite numbers of scenarios. Section 2 discusses the basic model and alternative approaches. Section 3 then discusses the basic properties of piecewise linear response functions. The fourth section presents a basic model for a single scenario and randomness restricted to constraint levels. The fifth section extends this to multiple scenarios with varying scenario ranges and to possibilities for randomness among the constraint vectors.

Multi-stage stochastic programs are typically extremely large, and can be prohibitively expensive to solve on the computer. In this paper we develop an algorithm for multistage programs that integrates the primal-dual row-action framework with proximal minimization. The algorithm exploits the structure of stochastic programs with network recourse, using a suitable problem formulation based on split variables, to decompose the solution into a large number of simple operations. It is therefore possible to use massively parallel computers to solve large instances of these problems. The algorithm is implemented on a Connection Machine CM-2 with up to 32K processors. We solve stochastic programs from an application from the insurance industry, as well as random problems, with up to 9 stages, and with up to 16392 scenarios, where the deterministic equivalent programs have a half million constraints and 1.3 million variables.

A model of distributionally robust two-stage stochastic convex programming with linear recourse

Today, organisations have focused on improving their supply chain performance to achieve sustainable profit and proceed in volatile markets. The nature of today's volatile markets imposes parametric uncertainty to optimisation problems particularly in strategic decision making problems such as supply chain network design (SCND) problem. Two-stage stochastic programming (TSSP) and robust stochastic programming (RSP) approaches are widely used to deal with the uncertainty of optimisation problems. In this paper, the performance of these two approaches in a SCND problem is evaluated through conducting a case study in Iran and performing realisation process. The main objectives of this study are optimising three stage SCND problems under uncertainty and evaluating the performance of TSSP and RSP methods in optimising SCND problem under uncertainty. The results show that the RSP method leads to more robust solution than TSSP method. Also, the RSP method has more degree of flexibility to deal with the uncertainty according to DM preferences.