A stochastic model for the daily coordination of pumped storage hydro plants and wind power plants

We propose a stochastic model for the daily operation scheduling of a generation system including pumped storage hydro plants and wind power plants, where the uncertainty is represented by the hourly wind power production. In order to assess the value of the stochastic modeling, we discuss two case studies: in the former the scenario tree is built so as to include both low and high wind power production scenarios, in the latter the scenario tree is built on historical wind speed data covering a time span of one and a half year. The Value of the Stochastic Solution, computed by a modified new procedure, shows that in scenarios with low wind power production the stochastic solution allows the producer to obtain a profit which is greater than the one associated to the deterministic solution. In-sample stability of the optimal function values for increasing number of scenarios is reported.     

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.     

R&D pipeline management: Task interdependencies and risk management

Maintaining a rich research and development (R&D) pipeline is the key to remaining competitive in many industrial sectors. Due to its nature, R&D activities are subject to multiple sources of uncertainty, the modeling of which is compounded by the ability of the decision maker to alter the underlying process. In this paper, we present a multi-stage stochastic programming framework for R&D pipeline management, which demonstrates how essential considerations can be modeled in an efficient manner including: (i) the selection and scheduling of R&D tasks with general precedence constraints under pass/fail uncertainty, and (ii) resource planning decisions (expansion/contraction and outsourcing) for multiple resource types. Furthermore, we study interdependencies between tasks in terms of probability of success, resource usage and market impact. Finally, we explore risk management approaches, including novel formulations for value at risk and conditional value at risk.     

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.     

Stochastic model for energy commercialisation of small hydro plants in the Brazilian energy market

Computational Management Science - This paper presents a stochastic model for energy commercialisation strategies of small hydro plants (SHPs) in the Brazilian electricity market. The model aims to...     

Workforce planning at USPS mail processing and distribution centers using stochastic optimization

Service organizations that operate outside the normal 8-hour day and face wide fluctuations in demand constantly struggle to optimize the size and composition of their workforce. Recent research has shown that improved personnel scheduling methods that take demand uncertainty into account can lead to significant reductions in labor costs. This paper addresses a staff planning and scheduling problem that arises at United States Postal Service (USPS) mail processing & distribution centers (P&DCs) and develops a two-stage stochastic integer program with recourse for the analysis. In the first stage, before the demand is known, the number of full-time and part-time employees is determined for the permanent workforce. In the second stage, the demand is revealed and workers are assigned to specific shifts during the week. When necessary, overtime and casual labor are used to satisfy demand. This paper consists of two parts: (1) the analysis of the demand distribution in light of historical data, and (2) the development and analysis of the stochastic integer programming model. Using weekly demand for a three-year period, we first investigate the possibility that there exists an end-of-month effect, i.e., the week at the end of month has larger volume than the other weeks. We show that the data fail to indicate that this is the case. In the computational phase of the work, three scenarios are considered: high, medium, and low demand. The stochastic optimization problem that results is a large-scale integer program that embodies the full set of contractual agreements and labor rules governing the design of the workforce at a P&DC. The usefulness of the model is evaluated by solving a series of instances constructed from data provided by the Dallas facility. The results indicate that significant savings are likely when the recourse problem is used to help structure the workforce. This work was supported in part by the National Science Foundation under grants DMI-0218701 and DMI-0217927.     

Two-phase optimization of coal strategies at a power station

Development of optimal coal mining and stockpiling strategies for a new 960 MW thermal power station depends strongly on the operation of New Zealand's mixed hydro/thermal power system. To overcome the large dimensionality of the problem the optimization proceeds in two phases for each future scenario studied. Phase 1 finds optimal hydro operating strategies with thermal generation in merit order of marginal fuel costs. It uses stochastic dynamic programming. The hydro generating strategies are used as input to the phase 2 stochastic simulation of the coal mining and stockpiling operation at the new thermal station. Finally, a strategy is identified which is a priori best over all future scenarios of industrial development and oil prices studied.     

A genetic algorithm for multi-mode resource constrained project scheduling problem

This article considers a general class of nonpreemptive multi-mode resource-constrained project scheduling problems in which activity durations depend on committed renewable resources (multi-mode time resource tradeoff). We propose a genetic algorithm for these problems and compare it with a stochastic scheduling method proposed by Drexl and Gruenewald. Computational results show that the proposed genetic algorithm is superior to the stochastic scheduling method.     

Single-Machine Scheduling with Exponential Processing Times and General Stochastic Cost Functions

We study a single-machine stochastic scheduling problem with n jobs, in which each job has a random processing time and a general stochastic cost function which may include a random due date and weight. The processing times are exponentially distributed, whereas the stochastic cost functions and the due dates may follow any distributions. The objective is to minimize the expected sum of the cost functions. We prove that a sequence in an order based on the product of the rate of processing time with the expected cost function is optimal, and under certain conditions, a sequence with the weighted shortest expected processing time first (WSEPT) structure is optimal. We show that this generalizes previous known results to more general situations. Examples of applications to practical problems are also discussed.This work was partially supported by the Research Grants Council of Hong Kong under Earmarked Grants No. CUHK4418/99E and No. PolyU 5081/00E.     

When greediness fails: examples from stochastic scheduling

The purpose of this paper is to present examples for the sometimes surprisingly different behavior of deterministic and stochastic scheduling problems. In particular, it demonstrates some seemingly counterintuitive properties of optimal scheduling policies for stochastic machine scheduling problems.     

Chance constrained programming models for refinery short-term crude oil scheduling problem

The main objective of this work is to put forward chance constrained mixed-integer nonlinear stochastic and fuzzy programming models for refinery short-term crude oil scheduling problem under demands uncertainty of distillation units. The scheduling problem studied has characteristics of discrete events and continuous events coexistence, multistage, multiproduct, nonlinear, uncertainty and large scale. At first, the two models are transformed into their equivalent stochastic and fuzzy mixed-integer linear programming (MILP) models by using the method of Quesada and Grossmann [I. Quesada, I E. Grossmann, Global optimization of bilinear process networks with multicomponent flows, Comput. Chem. Eng. 19 (12) (1995) 1219–1242], respectively. After that, the stochastic equivalent model is converted into its deterministic MILP model through probabilistic theory. The fuzzy equivalent model is transformed into its crisp MILP model relies on the fuzzy theory presented by Liu and Iwamura [B.D. Liu, K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets Syst. 94 (2) (1998) 227–237] for the first time in this area. Finally, the two crisp MILP models are solved in LINGO 8.0 based on scheduling time discretization. A case study which has 267 continuous variables, 68 binary variables and 320 constraints is effectively solved with the solution approaches proposed.     

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.     

A survey of data envelopment analysis in energy and environmental studies

Data envelopment analysis has gained great popularity in energy and environmental (E&E) modeling in recent years. In this paper, we present a literature survey on the application of data envelopment analysis (DEA) to E&E studies. We begin with an introduction to the most widely used DEA techniques, which is followed by a classification of 100 publications in this field. The main features observed are summarized. Issues related to the selection of DEA models in E&E studies are discussed.     

Safe scheduling: Setting due dates in single-machine problems

We consider single-machine stochastic scheduling models with due dates as decisions. In addition to showing how to satisfy given service-level requirements, we examine variations of a model in which the tightness of due-dates conflicts with the desire to minimize tardiness. We show that a general form of the trade-off includes the stochastic E/T model and gives rise to a challenging scheduling problem. We present heuristic solution methods based on static and dynamic sorting procedures. Our computational evidence identifies a static heuristic that routinely produces good solutions and a dynamic rule that is nearly always optimal. The dynamic sorting procedure is also asymptotically optimal, meaning that it can be recommended for problems of any size.     

Bounding Distributions for Stochastic Logic Networks

A stochastic logic network is defined as a connected set of logic and time delay elements. Each of the latter elements has an associated probability distribution describing the nature of that element's delay. When used, for example, in project planning and scheduling, combinations of logic and time delay elements in such networks may represent conditions for the starting of project activities which are themselves represented by time delay elements. It is at present not known how to calculate the probability distributions for the events in such a network. This paper shows how to obtain upper and lower bounds for these probability distributions. The method is not a simulation technique; rather, it is a straightforward computational scheme derived from elementary probability theory. An example is given where the method is applied to a stochastic project scheduling network in which alternative ways exist for carrying out one of the jobs in the network.     

A mixed R&D projects and securities portfolio selection model

The business environment is full of uncertainty. Allocating the wealth among various asset classes may lower the risk of overall portfolio and increase the potential for more benefit over the long term. In this paper, we propose a mixed single-stage R&D projects and multi-stage securities portfolio selection model. Specifically, we present a bi-objective mixed-integer stochastic programming model. Moreover, we use semi-absolute deviation risk functions to measure the risk of mixed asset portfolio. Based on the idea of moments approximation method via linear programming, we propose a scenario generation approach for the mixed single-stage R&D projects and multi-stage securities portfolio selection problem. The bi-objective mixed-integer stochastic programming problem can be solved by transforming it into a single objective mixed-integer stochastic programming problem. A numerical example is given to illustrate the behavior of the proposed mixed single stage R&D projects and multi-stage securities portfolio selection model.     

Medium-term hydro-thermal coordination via hydro and thermal subsystem decomposition

Summary This paper addresses the medium-term hydro-thermal coordination problem in an electric energy system. That is, the problem of finding the energy production of every power plant (hydro or thermal) in every subperiod of a given planning period, so that the customer load is supplied at minimum cost. The planning horizon is typically one to two months and the first week of this planning period is modeled in detail. The solution method proposed decomposes the problem in two subproblems corresponding to the hydro and thermal subsystems. These two subproblems are coordinated using a coordinating function for every subperiod. The coordinating function of a given subperiod expresses total production cost in that subperiod as a function of the total hydro production in that subperiod. The decomposition proposed makes it possible to use specialized algorithms to solve the hydro and thermal subproblems. This results in a very efficient computational procedure. From an experimental point of view the coordinating mechanism is robust. A case study is provided. It considers 61 thermal plants, a hydro system including 8 cascaded hydro plants and a 48 subperiods planning period.     

First-order inertial algorithms involving dry friction damping

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

     

Time complexity of single machine scheduling with stochastic precedence constraints

This paper deals with single machine scheduling problems with stochastic precedence relations (so calledGERT networks). Until now most investigations on such problems, dealt with algorithms running in polynomial time. On the other hand, for scheduling problems with deterministic precedence relations exist a lot of results about time complexity. Therefore, the object of this paper is to consider time complexity of scheduling problems with stochastic precedence constraints and to describe the boundary between theNP-hard problems and those which can be solved in polynomial time.     

Retail store scheduling for profit

In spite of its tremendous economic significance, the problem of sales staff schedule optimization for retail stores has received relatively scant attention. Current approaches typically attempt to minimize payroll costs by closely fitting a staffing curve derived from exogenous sales forecasts, oblivious to the ability of additional staff to (sometimes) positively impact sales. In contrast, this paper frames the retail scheduling problem in terms of operating profit maximization, explicitly recognizing the dual role of sales employees as sources of revenues as well as generators of operating costs. We introduce a flexible stochastic model of retail store sales, estimated from store-specific historical data, that can account for the impact of all known sales drivers, including the number of scheduled staff, and provide an accurate sales forecast at a high intra-day resolution. We also present solution techniques based on mixed-integer (MIP) and constraint programming (CP) to efficiently solve the complex mixed integer non-linear scheduling (MINLP) problem with a profit-maximization objective. The proposed approach allows solving full weekly schedules to optimality, or near-optimality with a very small gap. On a case-study with a medium-sized retail chain, this integrated forecasting–scheduling methodology yields significant projected net profit increases on the order of 2–3% compared to baseline schedules.