Energy supply planning and supply chain optimization under uncertainty

In energy supply planning and supply chain design, the coupling between long-term planning decisions like capital investment and short-term operation decisions like dispatching present a challenge, waiting to be tackled by systems and control engineers. The coupling is further complicated by uncertainties, which may arise from several sources including the market, politics, and technology. This paper addresses the coupling in the context of energy supply planning and supply chain design. We first discuss a simple two-stage stochastic program formulation that addresses optimization of an energy supply chain in the presence of uncertainties. The two-stage formulation can handle problems in which all design decisions are made up front and operating parameters act as ‘recourse’ decisions that can be varied from one time period to next based on realized values of uncertain parameters. The design of a biodiesel production network in the Southeastern region of the United States is used as an illustrative example. The discussion then moves on to a more complex multi-stage, multi-scale stochastic decision problem in which periodic investment/policy decisions are made on a time scale orders of magnitude slower than that of operating decisions. The problem of energy capacity planning is introduced as an example. In the particular problem we examine, annual acquisition of energy generation capacities of various types are coupled with hourly energy production and dispatch decisions. The increasing role of renewable sources like wind and solar necessitates the use of a fine-grained time scale for accurate assessment of their values. Use of storage intended to overcome the limitations of intermittent sources puts further demand on the modeling and optimization. Numerical challenges that arise from the multi-scale nature and uncertainties are reviewed and some possible modeling and numerical solution approaches are discussed.     

Results analysis for trust constrained real-time optimization

In real-time optimization (RTO), results analysis is used to ensure that RTO predictions can be implemented and are not the result of the unnecessary variance transmission around the RTO loop. Miletic and Marlin [2] proposed a statistical framework for analyzing RTO results; however, their method cannot effectively deal with inequality constraints. Many industrial RTO implementations include bounds on the changes that the RTO system can make to the process operation (i.e. trust-region constraints). Such trust-region constraints can seriously degrade the performance of existing results analysis methods. In this paper, a results analysis procedure is proposed that incorporates statistical testing on both the primal and dual variables of the optimization problem to effectively analyze steady-state RTO results in the presence of trust-region constraints. The proposed method is illustrated using two small case studies, one of which is the same Williams and Otto [11] reactor example used in [2].     

Semiglobal stabilization of linear systems using constrained controls: a parametric optimization approach

A bounded feedback control for asymptotic stabilization of linear systems is derived. The designed control law increases the feedback gain as the controlled trajectory converges towards the origin. A sequence of invariant sets of decreasing size, associated with a (quadratic) Lyapunov function, are defined and related to each of them, the corresponding possible highest gain is chosen, while maintaining the input bounded. Gains as functions of the position are designed by explicitly solving a c-parameterized programming problem. The proposed method allows global asymptotic stabilization of open-loop stable systems, with inputs subject to magnitude bounds and globally bounded rates. In the general case of linear systems that are asymptotic null controllable with bounded input, the semiglobal stabilization is also addressed taking into account the problem of semiglobal rate-limited actuators. The method is illustrated with the global stabilization of an inertial navigator, and the stabilization of a nonlinear model of a crane with hanging load. Copyright © 1999 John Wiley & Sons, Ltd.     

Combinatorial optimization under uncertainty

In this paper, the general problem of Euclidean combinatorial optimization under uncertainty is formulated for the first time and the concepts of a stochastic multiset, a multiset of fuzzy numbers, a stochastic Euclidean combinatorial set, and general Euclidean combinatorial set of fuzzy stochastic numbers that combines the properties of both types of uncertainty are introduced. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 35–44, September–October 2008.     

基于二阶段随机规划的不确定条件下过程优化研究

在基于二阶段随机规划的不确定条件下过程优化研究中,Ierapetritou and Pistikopoulos(1994)提出了可行域求解策略,Liu and Sahinidis(1996)在此基础上用蒙特卡洛积分策略代替了高斯积分策略,但对于可行域的限定条件尚有欠缺。本文分析和比较了前人的工作,将蒙特卡罗积分策略与基于对偶理论的可行域限定条件相结合,提出了新的求解策略,不仅避免了可行域求解策略中求解一系列子问题而引起的计算负荷随不确定参数数目呈指数增加的不足,而且使蒙特卡洛积分策略算法中的可行域限定条件更加合理,应用文献中的算例进行了仿真实验,证明了该算法的有效性。     

Sequential integer programming methods for stress constrained topology optimization

This paper deals with topology optimization of load carrying structures defined on a discretized design domain where binary design variables are used to indicate material or void in the various finite elements. The main contribution is the development of two iterative methods which are guaranteed to find a local optimum with respect to a 1-neighbourhood. Each new iteration point is obtained as the optimal solution to an integer linear programming problem which is an approximation of the original problem at the previous iteration point. The proposed methods are quite general and can be applied to a variety of topology optimization problems defined by 0-1 design variables. Most of the presented numerical examples are devoted to problems involving stresses which can be handled in a natural way since the design variables are kept binary in the subproblems.     

Hierarchical stochastic modeling and optimization for petroleum field development under geological uncertainty

Real-world simulation optimization (SO) problems entail complex system modeling and expensive stochastic simulation. Existing SO algorithms may not be applicable for such SO problems because they often evaluate a large number of solutions with many simulation calls. We propose an integrated solution method for practical SO problems based on a hierarchical stochastic modeling and optimization (HSMO) approach. This method models and optimizes the studied system at increasing levels of accuracy by hierarchical sampling with a selected set of principal parameters. We demonstrate the efficiency of HSMO using the example problem of Brugge oil field development under geological uncertainty.     

The use of metamodeling techniques for optimization under uncertainty

Metamodeling techniques have been widely used in engineering design to improve efficiency in the simulation and optimization of design systems that involve computationally expensive simulation programs. Many existing applications are restricted to deterministic optimization. Very few studies have been conducted on studying the accuracy of using metamodels for optimization under uncertainty. In this paper, using a two-bar structure system design as an example, various metamodeling techniques are tested for different formulations of optimization under uncertainty. Observations are made on the applicability and accuracy of these techniques, the impact of sample size, and the optimization performance when different formulations are used to incorporate uncertainty. Some important issues for applying metamodels to optimization under uncertainty are discussed.     

A heuristic approach for resource constrained project scheduling with uncertain activity durations

In this paper, we address the resource constrained project scheduling problem with uncertain activity durations. Project activities are assumed to have known deterministic renewable resource requirements and uncertain durations, described by independent random variables with a known probability distribution function. To tackle the problem solution we propose a heuristic method which relies on a stage wise decomposition of the problem and on the use of joint probabilistic constraints.     

Ordinal optimization approach to rare event probability problems

In this paper we introduce a new approach to rare event simulation. Because of the extensive simulation required for precise estimation of performance criterion dependent on rare event occurrences, obstacles such as computing budget/time constraints and pseudo-random number generator limitations can become prohibitive, particularly if comparative study of different system designs is involved. Existing methods for rare events simulation have focused on simulation budget reduction while attempting to generate accurate performance estimates. In this paper we propose a new approach for rare events system analysis in which we relax the simulation goal to the isolation of a set of good enough designs with high probability. Given this relaxation, referred to as ordinal optimization and advanced by Ho et al. (1992), this paper's approach calls instead for the consideration of an appropriate surrogate design problem This surrogate problem is characterized by its approximate ordinal equivalence to the original problem and its performance criterion's dependence not on rare event occurrences, but on more frequent events. Evaluation of such a surrogate problem under the relaxed goals of ordinal optimization has experimentally resulted in orders of magnitude reduction in simulation burden.     

A comparative analysis of genetic algorithms and directed grid search for parametric optimization

In this paper, a comparative analysis of the performance of the Genetic Algorithm (GA) and Directed Grid Search (DGS) methods for optimal parametric design is presented. A genetic algorithm is a guided random search mechanism based on the principle of natural selection and population genetics. The Directed Grid Search method uses a selective directed search of grid points in the direction of descent to find the minimum of a real function, when the initial estimate of the location of the minimum and the bounds of the design variables are specified. An experimental comparison and a discussion on the performance of these two methods in solving a set of eight test functions is presented.     

Convex parametric piecewise quadratic optimization: Theory and algorithms

In this paper we study the problem of parametric minimization of convex piecewise quadratic functions. Our study provides a unifying framework for convex parametric quadratic and linear programs. Furthermore, it extends parametric optimization algorithms to problems with piecewise quadratic cost functions, paving the way for new applications of parametric optimization in explicit dynamic programming and optimal control with quadratic stage cost.     

Accelerating Benders stochastic decomposition for the optimization under uncertainty of the petroleum product supply chain

This paper addresses the solution of a two-stage stochastic programming model for an investment planning problem applied to the petroleum products supply chain. In this context, we present the development of acceleration techniques for the stochastic Benders decomposition that aim to strengthen the cuts generated, as well as to improve the quality of the solutions obtained during the execution of the algorithm. Computational experiments are presented for assessing the efficiency of the proposed framework. We compare the performance of the proposed algorithm with two other acceleration techniques. Results suggest that the proposed approach is able to efficiently solve the problem under consideration, achieving better performance in terms of computational times when compared to other two techniques.     

Location based treatment activities for end of life products network design under uncertainty by a robust multi-objective memetic-based heuristic approach

Rapid growth in world population and recourse limitations necessitate remanufacturing of products and their parts/modules. Managing these processes requires special activities such as inspection, disassembly, and sorting activities known as treatment activities. This paper proposes a capacitated multi-echelon, multi-product reverse logistic network design with fuzzy returned products in which both locations of the treatment activities and facilities are decision variables. As the obtained nonlinear mixed integer programming model is a combinatorial problem, a memetic-based heuristic approach is presented to solve the resulted model. To validate the proposed memetic-based heuristic method, the obtained results are compared with the results of the linear approximation of the model, which is obtained by a commercial optimization package. Moreover, due to inherent uncertainty in return products, demands of these products are considered as uncertain parameters and therefore a fuzzy approach is employed to tackle this matter. In order to deal with the uncertainty, a stochastic simulation approach is employed to defuzzify the demands, where extra costs due to opening new centers or extra transportation costs may be imposed to the system. These costs are considered as penalty in the objective function. To minimize the resulting penalties during simulation's iterations, the average of penalties is added to the objective function of the deterministic model considered as the primary objective function and variance of penalties are considered as the secondary objective function to make a robust solution. The resulted bi-objective model is solved through goal programming method to minimizing the objectives, simultaneously.     

Multiobjective process line optimization under uncertainty applied to papermaking

Multiobjective optimization methodology for the development of the papermaking process is considered. The aim is to find efficient and reliable solution procedures for the process line model consisting of sequential unit-process models; some of them based on physics, whereas others on experimental data. By the consequence of modeling procedures, nonphysical states or inherited from modeling data in statistical case, the unit-process models may suffer from undesired unreliability. To control the uncertainty resulting from the unit-process models, a new multiobjective optimization approach is introduced where both the papermaking targets as well as the uncertainty related unit-process models are simultaneously taken into account. We illustrate the solution process by numerical examples related to the quality of the produced paper.     

A GRASP based solution approach to solve cardinality constrained portfolio optimization problems

In the current work, a solution methodology which combines a meta-heuristic algorithm with an exact solution approach is presented to solve cardinality constrained portfolio optimization (CCPO) problem. The proposed method is comprised of two levels, namely, stock selection and proportion determination. In stock selection level, a greedy randomized adaptive search procedure (GRASP) is developed. Once the stocks are selected the problem reduces to a quadratic programming problem. As GRASP ensures cardinality constraints by selecting predetermined number of stocks and quadratic programming model ensures the remaining problem constraints, no further constraint handling procedures are required. On the other hand, as the problem is decomposed into two sub-problems, total computational burden on the algorithm is considerably reduced. Furthermore, the performance of the proposed algorithm is evaluated by using benchmark data sets available in the OR Library. Computational results reveal that the proposed algorithm is competitive with the state of the art algorithms in the related literature.     

不确定条件下控制分配问题的鲁棒优化方法

针对具有冗余执行机构的过驱动系统, 在考虑控制效率不确定性的条件下, 提出了一种基于鲁棒优化理论的控制分配算法. 研究了原始不确定鲁棒优化模型的建立和基于椭球不确定集的鲁棒对等式的转化问题, 并推广到可由锥二次不等式表示的不确定集的情况. 讨论了鲁棒优化控制分配算法的求解方法及其计算复杂度. 最后, 针对多操纵面飞机的最优控制分配问题与传统算法进行了仿真比较, 结果表明鲁棒优化算法能有效降低控制效率不确定性的影响, 使分配结果更为合理, 从而具有更好的鲁棒性, 同时能有效提高操纵面故障情况下闭环系统的控制重构能力, 很好地改善了飞控系统的性能.     

Seasonal scale modeling of river inflows in stratified reservoirs: Structural vs. parametric uncertainty in inflow mixing

The extent to which vertical transport in reservoirs is accurately represented by one-dimensional (1D) lake models is linked to the quality of the algorithms representing the fate of inflows. Recent findings suggest that gravity currents in stratified environments not only entrain ambient water but they also detrain water as they flow down-slope. Here, we analyze whether the differences between the entrainment and the entrainment–detrainment approaches of river inflow mixing implemented in 1D models could affect the seasonal-scale simulations of vertical transport in a reservoir. We demonstrate that the structural uncertainty associated with using different inflow modeling paradigms is statistically significant in terms of temperature and salinity predictions, as well as phosphorous loads available to the lake surface; and is of similar order of magnitude as the parametric uncertainty errors arising from the unknown value of model parameters and forcing.     

Limit load and shakedown analysis of plastic structures under stochastic uncertainty

Problems from plastic analysis are based on the convex, linear or linearised yield/strength condition and the linear equilibrium equation for the stress (state) vector. In practice one has to take into account stochastic variations of several model parameters. Hence, in order to get robust maximum load factors, the structural analysis problem with random parameters must be replaced by an appropriate deterministic substitute problem. A direct approach is proposed based on the primary costs for missing carrying capacity and the recourse costs (e.g. costs for repair, compensation for weakness within the structure, damage, failure, etc.). Based on the mechanical survival conditions of plasticity theory, a quadratic error/loss criterion is developed. The minimum recourse costs can be determined then by solving an optimisation problem having a quadratic objective function and linear constraints. For each vector a(·) of model parameters and each design vector x, one obtains then an explicit representation of the “best” internal load distribution F^∗. Moreover, also the expected recourse costs can be determined explicitly. Consequently, an explicit stochastic nonlinear program results for finding a robust maximal load factor μ^∗. The analytical properties and possible solution procedures are discussed.