Stochastic Dual Dynamic Programming Algorithm Research Articles

Evaluating policies in risk-averse multi-stage stochastic programming

We consider a risk-averse multi-stage stochastic program using conditional value at risk as the risk measure. The underlying random process is assumed to be stage-wise independent, and a stochastic dual dynamic programming (SDDP) algorithm is applied. We discuss the poor performance of the standard upper bound estimator in the risk-averse setting and propose a new approach based on importance sampling, which yields improved upper bound estimators. Modest additional computational effort is required to use our new estimators. Our procedures allow for significant improvement in terms of controlling solution quality in SDDP-style algorithms in the risk-averse setting. We give computational results for multi-stage asset allocation using a log-normal distribution for the asset returns.

Generation expansion planning under uncertainty with emissions quotas

Generation expansion planning for hydro-thermal power systems aims to find optimal investment decisions among a set of possible power plant projects. For any given investment plan, expansion planning models must account for investment costs as well as expected operational costs. Additionally, when modeling hydro-thermal power systems, one must consider the inherent uncertainty in hydro inflows because variations in inflows can significantly impact operational decisions. To model the expansion planning problem for a hydro-thermal power producer, we propose a novel decomposition algorithm, based on Benders decomposition. The Benders master problem computes the investment decisions while the separation problems are emissions-constrained, least-cost, and stochastic hydro-thermal scheduling problems. The separation problems are solved using a standard stochastic dual dynamic programming (SDDP) algorithm. To demonstrate the effectiveness of the approach, we present a case study for Panama's power system. The computational results allow us to price carbon dioxide emissions reductions, aiding the evaluations of environmental policies.

The economic value of coordination in large‐scale multireservoir systems: The Parana River case

The coordination of reservoir operation is critical for water systems' efficiency. Improved coordination requires sharing information, demanding a clear understanding of the potential gains and its distribution among the users to motivate engagement in coordinated operations and bearing of transaction costs. In a multiuser, multireservoir system, the evaluation of the potential coordination gains is not trivial because it requires the simultaneous evaluation of numerous trade offs. This paper presents a methodology to identify the likely upper and lower bounds in multireservoir system benefits, providing a reference framework for analyzing the economic value of coordination. The methodology is applied to a large‐scale multireservoir system in Brazil. The methods rely on the comparison between two management scenarios. The first one mimics typical system operation based on individually designed rule curves, which are likely to perform on the lower bound. This is compared with fullscale system‐wide optimization through an Stochastic Dual Dynamic Programming algorithm to represent fully coordinated reservoir operation (upper bound). For our case study, results indicate that better coordination reduced spills and improved releases timing according to reservoirs characteristics and location, allowing overall gains between 3% and 8% in energy and 7.9% in revenues, with revenues mostly improved by coordination in dry years. Larger reservoirs presented the highest gains in absolute terms, while the smaller ones presented the highest relative increases. By indicating individual gains at each reservoir, valuable information is produced to support future negotiations and benefit sharing among different agents, being water agencies or power companies.

In This Issue

In This Issue

SDDP for some interstage dependent risk-averse problems and application to hydro-thermal planning

We consider interstage dependent stochastic linear programs where both the random right-hand side and the model of the underlying stochastic process have a special structure. Namely, for equality constraints (resp. inequality constraints) the right-hand side is an affine function (resp. a given function b t ) of the process value for the current time step t. As for m-th component of the process at time step t, it depends on previous values of the process through a function h tm . For this type of problem, to obtain an approximate policy under some assumptions for functions b t and h tm , we detail a stochastic dual dynamic programming algorithm. Our analysis includes some enhancements of this algorithm such as the definition of a state vector of minimal size, the computation of feasibility cuts without the assumption of relatively complete recourse, as well as efficient formulas for sharing optimality and feasibility cuts between nodes of the same stage. The algorithm is given for both a non-risk-averse and a risk-averse model. We finally provide preliminary results comparing the performances of the recourse functions corresponding to these two models for a real-life application.

SDDP for multistage stochastic linear programs based on spectral risk measures

We consider risk-averse formulations of multistage stochastic linear programs. For these formulations, based on convex combinations of spectral risk measures, risk-averse dynamic programming equations can be written. As a result, the Stochastic Dual Dynamic Programming (SDDP) algorithm can be used to obtain approximations of the corresponding risk-averse recourse functions. This allows us to define a risk-averse nonanticipative feasible policy for the stochastic linear program. Formulas for the cuts that approximate the recourse functions are given. In particular, we show that some cut coefficients have analytic formulas.

Stochastic Hydro-Thermal Scheduling Under ${\rm CO}_{2}$ Emissions Constraints

Despite the uncertainty surrounding the design of a mechanism which is ultimately accepted by nations worldwide, the necessity to implement regulations to curb emissions of greenhouse gases on a global scale is consensual. The electricity sector plays a fundamental role in this puzzle and countries may soon have to revise their operating policy directives in order to make them compatible with additional constraints imposed by such regulations. We present a modeling approach for greenhouse gas emissions quotas which can be incorporated into a stochastic dual dynamic programming algorithm, commonly used to solve the hydro-thermal scheduling problem. Our approach is flexible and capable of accommodating a detailed representation of emissions and related constraints. A case study based on the Guatemalan power system exemplifies the potential effects of considering these restrictions.

Sampling-Based Decomposition Methods for Multistage Stochastic Programs Based on Extended Polyhedral Risk Measures

We define a risk-averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it. The approach is based on dynamic programming equations written for a risk-averse formulation of the problem. This formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures. Dual representations of such risk functionals are given and used to derive conditions of coherence. In the one-period case, conditions for convexity and consistency with second order stochastic dominance are also provided. The risk-averse dynamic programming equations are specialized considering convex combinations of one-period extended polyhedral risk measures such as spectral risk measures. To implement the proposed policy, the approximation of the risk-averse recourse functions for stochastic linear programs is discussed. In this context, we detail a stochastic dual dynamic programming algorithm which converges to the optimal value of the risk-averse problem. (A corrected PDF is attached to the original pdf.)

Sampling strategies and stopping criteria for stochastic dual dynamic programming: a case study in long-term hydrothermal scheduling

The long-term hydrothermal scheduling is one of the most important problems to be solved in the power systems area. This problem aims to obtain an optimal policy, under water (energy) resources uncertainty, for hydro and thermal plants over a multi-annual planning horizon. It is natural to model the problem as a multi-stage stochastic program, a class of models for which algorithms have been developed. The original stochastic process is represented by a finite scenario tree and, because of the large number of stages, a sampling-based method such as the Stochastic Dual Dynamic Programming (SDDP) algorithm is required. The purpose of this paper is two-fold. Firstly, we study the application of two alternative sampling strategies to the standard Monte Carlo—namely, Latin hypercube sampling and randomized quasi-Monte Carlo—for the generation of scenario trees, as well as for the sampling of scenarios that is part of the SDDP algorithm. Secondly, we discuss the formulation of stopping criteria for the optimization algorithm in terms of statistical hypothesis tests, which allows us to propose an alternative criterion that is more robust than that originally proposed for the SDDP. We test these ideas on a problem associated with the whole Brazilian power system, with a three-year planning horizon.

Analysis of stochastic dual dynamic programming method

In this paper we discuss statistical properties and convergence of the Stochastic Dual Dynamic Programming (SDDP) method applied to multistage linear stochastic programming problems. We assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original (true) distribution is generated and consequently the SDDP algorithm is applied to the constructed Sample Average Approximation (SAA) problem. Then we proceed to analysis of the SDDP solutions of the SAA problem and their relations to solutions of the “true” problem. Finally we discuss an extension of the SDDP method to a risk averse formulation of multistage stochastic programs. We argue that the computational complexity of the corresponding SDDP algorithm is almost the same as in the risk neutral case.