A decomposition approach to the two-stage stochastic unit commitment problem

The unit commitment problem has been a very important problem in the power system operations, because it is aimed at reducing the power production cost by optimally scheduling the commitments of generation units. Meanwhile, it is a challenging problem because it involves a large amount of integer variables. With the increasing penetration of renewable energy sources in power systems, power system operations and control have been more affected by uncertainties than before. This paper discusses a stochastic unit commitment model which takes into account various uncertainties affecting thermal energy demand and two types of power generators, i.e., quick-start and non-quick-start generators. This problem is a stochastic mixed integer program with discrete decision variables in both first and second stages. In order to solve this difficult problem, a method based on Benders decomposition is applied. Numerical experiments show that the proposed algorithm can solve the stochastic unit commitment problem efficiently, especially those with large numbers of scenarios.     

A stochastic programming model for the optimal electricity market bid problem with bilateral contracts for thermal and combined cycle units

This paper develops a stochastic programming model that integrates the most recent regulation rules of the Spanish peninsular system for bilateral contracts in the day-ahead optimal bid problem. Our model allows a price-taker generation company to decide the unit commitment of the thermal and combined cycle programming units, the economic dispatch of the bilateral contract between all the programming units and the optimal sale bid by observing the Spanish peninsular regulation. The model was solved using real data of a typical generation company and a set of scenarios for the Spanish market price. The results are reported and analyzed.     

Electricity market clearing with improved scheduling of stochastic production

In this paper, we consider an electricity market that consists of a day-ahead and a balancing settlement, and includes a number of stochastic producers. We first introduce two reference procedures for scheduling and pricing energy in the day-ahead market: on the one hand, a conventional network-constrained auction purely based on the least-cost merit order, where stochastic generation enters with its expected production and a low marginal cost; on the other, a counterfactual auction that also accounts for the projected balancing costs using stochastic programming. Although the stochastic clearing procedure attains higher market efficiency in expectation than the conventional day-ahead auction, it suffers from fundamental drawbacks with a view to its practical implementation. In particular, it requires flexible producers (those that make up for the lack or surplus of stochastic generation) to accept losses in some scenarios. Using a bilevel programming framework, we then show that the conventional auction, if combined with a suitable day-ahead dispatch of stochastic producers (generally different from their expected production), can substantially increase market efficiency and emulate the advantageous features of the stochastic optimization ideal, while avoiding its major pitfalls.     

Scenario reduction for stochastic programs with Conditional Value-at-Risk

In this paper we discuss scenario reduction methods for risk-averse stochastic optimization problems. Scenario reduction techniques have received some attention in the literature and are used by practitioners, as such methods allow for an approximation of the random variables in the problem with a moderate number of scenarios, which in turn make the optimization problem easier to solve. The majority of works for scenario reduction are designed for classical risk-neutral stochastic optimization problems; however, it is intuitive that in the risk-averse case one is more concerned with scenarios that correspond to high cost. By building upon the notion of effective scenarios recently introduced in the literature, we formalize that intuitive idea and propose a scenario reduction technique for stochastic optimization problems where the objective function is a Conditional Value-at-Risk. Numerical results presented with problems from the literature illustrate the performance of the method and indicate the cases where we expect it to perform well.     

Problem-based optimal scenario generation and reduction in stochastic programming

Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem.

     

Stochastic Lagrangian Relaxation Applied to Power Scheduling in a Hydro-Thermal System under Uncertainty

A dynamic (multi-stage) stochastic programming model for the weekly cost-optimal generation of electric power in a hydro-thermal generation system under uncertain demand (or load) is developed. The model involves a large number of mixed-integer (stochastic) decision variables and constraints linking time periods and operating power units. A stochastic Lagrangian relaxation scheme is designed by assigning (stochastic) multipliers to all constraints coupling power units. It is assumed that the stochastic load process is given (or approximated) by a finite number of realizations (scenarios) in scenario tree form. Solving the dual by a bundle subgradient method leads to a successive decomposition into stochastic single (thermal or hydro) unit subproblems. The stochastic thermal and hydro subproblems are solved by a stochastic dynamic programming technique and by a specific descent algorithm, respectively. A Lagrangian heuristics that provides approximate solutions for the first stage (primal) decisions starting from the optimal (stochastic) multipliers is developed. Numerical results are presented for realistic data from a German power utility and for numbers of scenarios ranging from 5 to 100 and a time horizon of 168 hours. The sizes of the corresponding optimization problems go up to 200000 binary and 350000 continuous variables, and more than 500000 constraints.     

Stability Properties of a Bond Portfolio Management Problem

The bond portfolio management problem is formulated as a multiperiod two-stage or multistage stochastic program based on interest rate scenarios. These scenarios depend on the available market data, on the applied estimation and sampling techniques, etc., and are used to evaluate coefficients of the resulting large scale mathematical program. The aim of the contribution is to analyze stability and sensitivity of this program on small changes of the coefficients – the (scenario dependent) values of future interest rates and prices. We shall prove that under sensible assumptions, the scenario subproblems are stable linear programs and that also the optimal first-stage decisions and the optimal value of the considered stochastic program possess acceptable continuity properties.     

Second-order scenario approximation and refinement in optimization under uncertainty

When solving scenario-based stochastic programming problems, it is imperative that the employed solution methodology be based on some form of problem decomposition: mathematical, stochastic, or scenario decomposition. In particular, the scenario decomposition resulting from scenario approximations has perhaps the least tendency to be computationally tedious due to increases in the number of scenarios. Scenario approximations discussed in this paper utilize the second-moment information of the given scenarios to iteratively construct a (relatively) small number of representative scenarios that are used to derive bounding approximations on the stochastic program. While the sizes of these approximations grow only linearly in the number of random parameters, their refinement is performed by exploiting the behavior of the value function in the most effective manner. The implementation SMART discussed here demonstrates the aptness of the scheme for solving two-stage stochastic programs described with a large number of scenarios.This paper was presented at the IFIP Workshop onStochastic Programming: Algorithms and Models, Lillehammer, Norway, January 1994.     

The sample average approximation method for empty container repositioning with uncertainties

One of the challenges faced by liner operators today is to effectively operate empty containers in order to meet demand and to reduce inefficiency in an uncertain environment. To incorporate uncertainties in the operations model, we formulate a two-stage stochastic programming model with random demand, supply, ship weight capacity, and ship space capacity. The objective of this model is to minimize the expected operational cost for Empty Container Repositioning (ECR). To solve the stochastic programs with a prohibitively large number of scenarios, the Sample Average Approximation (SAA) method is applied to approximate the expected cost function. To solve the SAA problem, we consider applying the scenario aggregation by combining the approximate solution of the individual scenario problem. Two heuristic algorithms based on the progressive hedging strategy are applied to solve the SAA problem. Numerical experiments are provided to show the good performance of the scenario-based method for the ECR problem with uncertainties.     

A new multiobjective allocator of capacitor banks and distributed generations using a new investigated differential evolution

This article has investigated a new multiobjective allocation of optimal sizing and sitting of distributed generation (DG) units and capacitor banks in simultaneous mode to improve reliability and reduce energy losses. The proposed method consists of four objectives, that is, cost of energy not supplied, system average interruption duration index, costs of energy loss and investment. A novel structure differential evolution has been suggested to solve this nonlinear complex problem and its results are compared with related values of genetic algorithm and simple differential evolutionary algorithm. In addition to the novel objective function, the other contribution of this article is proposing a new model for load and energy cost. Three types of DGs, that is, wind turbine, solar cell, and diesel generator have been used in placement process. To verify the comprehensiveness of the proposed function, three scenarios have been introduced: scenario i: first, placement of DGs, then capacitor banks, scenario ii: first, placement of capacitor banks, and then DGs, and scenario iii: simultaneous placement of DGs and capacitor banks. Simulations have been carried out on one part of practical distribution network in Metropolitan Tabriz in North West of Iran. The results of simulations have been discussed and analyzed using the five novel indices. The obtained simulation results using proposed function shows that the simultaneous placement of DGs and capacitor banks results in more reduction of the energy losses and increase improvements of reliability indices as well as voltage profile. © 2013 Wiley Periodicals, Inc. Complexity 19: 40–54, 2014     

Cost/risk balanced management of scarce resources using stochastic programming

We consider the situation when a scarce renewable resource should be periodically distributed between different users by a Resource Management Authority (RMA). The replenishment of this resource as well as users demand is subject to considerable uncertainty. We develop cost optimization and risk management models that can assist the RMA in its decision about striking the balance between the level of target delivery to the users and the level of risk that this delivery will not be met. These models are based on utilization and further development of the general methodology of stochastic programming for scenario optimization, taking into account appropriate risk management approaches. By a scenario optimization model we obtain a target barycentric value with respect to selected decision variables. A successive reoptimization of deterministic model for the worst case scenarios allows the reduction of the risk of negative consequences derived from unmet resources demand. Our reference case study is the distribution of scarce water resources. We show results of some numerical experiments in real physical systems.     

On air traffic flow management with rerouting. Part II: Stochastic case

We present a framework for modeling multistage mixed 0-1 problems for the air traffic flow management problem with rerouting (ATFMRP) under uncertainty in the airport arrival and departure capacity, the air sector capacity and the flight demand. The model allows for flight cancelation, if necessary. It considers several types of objective functions to minimize, namely, total ground and air holding cost, penalization of the alternative routes to the scheduled one for each flight, delay cost for the flights to arrive to the airports and the air sector nodes, and penalization for advancing the arrival of the flights to the airport over the scheduled period. A scenario tree based scheme is used to represent the Deterministic Equivalent Model (DEM) of the stochastic mixed 0-1 program with full recourse. The nonanticipativity constraints that equate the so named common 0-1 and continuous variables from the same group of scenarios in each period are implicitly satisfied in the compact representation of DEM. Some computational experience is reported for medium-scale instances. The model is so tight that none of the instances of the testbed but two of them requires to execute the branch-and-cut phase of the MIP optimization engine of choice.     

Scenario tree modeling for multistage stochastic programs

An important issue for solving multistage stochastic programs consists in the approximate representation of the (multivariate) stochastic input process in the form of a scenario tree. In this paper, we develop (stability) theory-based heuristics for generating scenario trees out of an initial set of scenarios. They are based on forward or backward algorithms for tree generation consisting of recursive scenario reduction and bundling steps. Conditions are established implying closeness of optimal values of the original process and its tree approximation, respectively, by relying on a recent stability result in Heitsch, Römisch and Strugarek (SIAM J Optim 17:511–525, 2006) for multistage stochastic programs. Numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management.     

A dual strategy for the implementation of the aggregation principle in decision making under uncertainty

A solution method for stochastic programs is proposed based on the aggregation principle, which allows one to find the solution of a stochastic program by aggregating the solutions of individual deterministic scenario problems. The method concentrates on finding good estimates of the dual variables associated with the non-anticipativity constraints.     

Scenario generation and stochastic programming models for asset liability management

In this paper, we develop and test scenario generation methods for asset liability management models. We propose a multi-stage stochastic programming model for a Dutch pension fund. Both randomly sampled event trees and event trees fitting the mean and the covariance of the return distribution are used for generating the coefficients of the stochastic program. In order to investigate the performance of the model and the scenario generation procedures we conduct rolling horizon simulations. The average cost and the risk of the stochastic programming policy are compared to the results of a simple fixed mix model. We compare the average switching behavior of the optimal investment policies. Our results show that the performance of the multi-stage stochastic program could be improved drastically by choosing an appropriate scenario generation method.     

Scenario optimization

Uncertainty in the parameters of a mathematical program may present a modeller with considerable difficulties. Most approaches in the stochastic programming literature place an apparent heavy data and computational burden on the user and as such are often intractable. Moreover, the models themselves are difficult to understand. This probably explains why one seldom sees a fundamentally stochastic model being solved using stochastic programming techniques. Instead, it is common practice to solve a deterministic model with different assumed scenarios for the random coefficients. In this paper we present a simple approach to solving a stochastic model, based on a particular method for combining such scenario solutions into a single, feasible policy. The approach is computationally simple and easy to understand. Because of its generality, it can handle multiple competing objectives, complex stochastic constraints and may be applied in contexts other than optimization. To illustrate our model, we consider two distinct, important applications: the optimal management of a hydro-thermal generating system and an application taken from portfolio optimization.     

Financial scenario generation for stochastic multi-stage decision processes as facility location problems

The quality of multi-stage stochastic optimization models as they appear in asset liability management, energy planning, transportation, supply chain management, and other applications depends heavily on the quality of the underlying scenario model, describing the uncertain processes influencing the profit/cost function, such as asset prices and liabilities, the energy demand process, demand for transportation, and the like. A common approach to generate scenarios is based on estimating an unknown distribution and matching its moments with moments of a discrete scenario model. This paper demonstrates that the problem of finding valuable scenario approximations can be viewed as the problem of optimally approximating a given distribution with some distance function. We show that for Lipschitz continuous cost/profit functions it is best to employ the Wasserstein distance. The resulting optimization problem can be viewed as a multi-dimensional facility location problem, for which at least good heuristic algorithms exist. For multi-stage problems, a scenario tree is constructed as a nested facility location problem. Numerical convergence results for financial mean-risk portfolio selection conclude the paper.     

Knowledge‐based scenario tree generation methods and application in multiperiod portfolio selection problem

A scenario tree is an efficient way to represent a stochastic data process in decision problems under uncertainty. This paper addresses how to efficiently generate appropriate scenario trees. A knowledge‐based scenario tree generation method is proposed; the new method is further improved by accounting for subjective judgements or expectations about the random future. Compared with existing approaches, complicated mathematical models and time‐consuming estimation, simulation and optimization problem solution are avoided in our knowledge‐based algorithms, and large‐scale scenario trees can be quickly generated. To show the advantages of the new algorithms, a multiperiod portfolio selection problem is considered, and a dynamic risk measure is adopted to control the intermediate risk, which is superior to the single‐period risk measure used in the existing literature. A series of numerical experiments are carried out by using real trading data from the Shanghai stock market. The results show that the scenarios generated by our algorithms can properly represent the underlying distribution; our algorithms have high performance, say, a scenario tree with up to 10,000 scenarios can be generated in less than a half minute. The applications in the multiperiod portfolio management problem demonstrate that our scenario tree generation methods are stable, and the optimal trading strategies obtained with the generated scenario tree are reasonable, efficient and robust. Copyright © 2013 John Wiley & Sons, Ltd.     

Mittelfristige Produktionsplanung bei ungewissen Szenarien

In medium term production planning at a highly aggregated level the uncertainty about future demand plays a central role. A widely used method to take the uncertainty into account is to investigate the same model with different scenarios. This approach produces only suboptimal results. In the first part of this paper some principles of optimality are formulated where forecasting is incorporated and future scenarios are treated as a stochastic process. The resulting models are of the type of a Markovian decision process. They have the property of actualization of forecasts (adaption), of looking ahead production smoothing (anticipation) and of efficient risk balancing. The different models are formulated in view of some typical situations occuring in practice. As a byproduct it is shown that the separation of long term forecasting and short term production planning may be disadvantageous. The theory developed so far will then be applied to a concrete situation in the automotive industriy. In particular the problem investigated is how to control the production rate throughout an imminent period of recession of unknown severity and duration. The computational results demonstrate that the model with a stochastic scenario yields smoother production lines than the model with a fixed scenario. This is due to an additional cost minimizing inertia caused by the stochastic law of motion.     

Adaptive discretization of convex multistage stochastic programs

We propose a new scenario tree reduction algorithm for multistage stochastic programs, which integrates the reduction of a scenario tree into the solution process of the stochastic program. This allows to construct a scenario tree that is highly adapted on the optimization problem. The algorithm starts with a rough approximation of the original tree and locally refines this approximation as long as necessary. Promising numerical results for scenario tree reductions in the settings of portfolio management and power management with uncertain load are presented.