Validating Sample Average Approximation Solutions with Negatively Dependent Batches

Sample-average approximations (SAA) are a practical means of finding approximate solutions of stochastic programming problems involving an extremely large (or infinite) number of scenarios. SAA can also be used to find estimates of a lower bound on the optimal objective value of the true problem which, when coupled with an upper bound, provides confidence intervals for the true optimal objective value and valuable information about the quality of the approximate solutions. Specifically, the lower bound can be estimated by solving multiple SAA problems (each obtained using a particular sampling method) and averaging the obtained objective values. State-of-the-art methods for lower-bound estimation generate batches of scenarios for the SAA problems independently. In this paper, we describe sampling methods that produce negatively dependent batches, thus reducing the variance of the sample-averaged lower bound estimator and increasing its usefulness in defining a confidence interval for the optimal objective value. We provide conditions under which the new sampling methods can reduce the variance of the lower bound estimator, and present computational results to verify that our scheme can reduce the variance significantly, by comparison with the traditional Latin hypercube approach

Assessing the quality of convex approximations for two-stage totally unimodular integer recourse models

We consider two types of convex approximations of two-stage totally unimodular integer recourse models. Although worst-case error bounds are available for these approximations, their actual performance has not yet been investigated, mainly because this requires solving the original recourse model. In this paper we assess the quality of the approximating solutions using Monte Carlo sampling, or more specifically, using the so-called multiple replications procedure. Based on numerical experiments for an integer newsvendor problem, a fleet allocation and routing problem, and a stochastic activity network investment problem, we conclude that the error bounds are reasonably sharp if the variability of the random parameters in the model is either small or large; otherwise, the actual error of using the convex approximations is much smaller than the error bounds suggest. Moreover, we conclude that the solutions obtained using the convex approximations are good only if the variability of the random parameters is medium to large. In case this variability is small, however, typically sampling methods perform best, even with modest sample sizes. In this sense, the convex approximations and sampling methods can be considered as complementary solution methods. Moreover, as required for our applications, we extend our approach to derive new error bounds dealing with deterministic second-stage side constraints and relatively complete recourse, and perfect dependencies in the right-hand side vector

Integration of Design and Control under Uncertainty: A New Back-off Approach using PSE Approximations

Chemical process design is still an active area of research since it largely determines the optimal and safe operation of a new process under various conditions. The design process involves a series of steps that aims to identify the most economically attractive design typically using steady-state optimization. However, optimal steady-state designs may fail to comply with the process constraints when the system under analysis is subject to process disturbances (e.g. the composition of a reactant in a feed stream) or parameter uncertainty (e.g. the activation energy in a chemical reaction). Moreover, the practice of overdesigning a process to ensure feasibility under process disturbances and parameter uncertainty has been proven to be costly. Therefore, a new methodology for simultaneous design and control for dynamic systems under uncertainty has been proposed. The proposed methodology uses Power Series Expansions (PSE) to obtain analytical expressions for the process constrains and cost function. The key idea is to use the back off approach from the optimal steady state design to address the simultaneous process and design problem in an efficient systematic manner using PSE approximations. The challenge in this method is to determine the magnitude of the back-off needed to accommodate the transient and feasible operation of the process in presence of disturbances and parameter uncertainty. In this approach, PSE functions are used to obtain analytical expressions of the actual process constraints and are explicitly defined in terms of system’s uncertain parameter and the largest variability in a constraint function due to time-varying changes in the disturbances. Also, the PSE approximation for each constraint is developed around a nominal point in the optimization variables and for each realization considered for the uncertain parameters. The PSE-based constraint represents the actual process constraint and can be evaluated faster since it is explicitly defined in the terms of the optimization variables. The work focuses on calculating various optimal design and control parameters by solving various sets of optimization problems using mathematical expressions obtained from power series expansions. These approximations are used to determine the direction in the search of optimal design parameters and operating conditions required for an economically attractive, dynamically feasible process. The proposed methodology was tested on an isothermal storage tank and a step by step procedure to develop the methodology has been presented. The methodology was also tested on a non-isothermal CSTR and the results were compared with the formal integration process. Effect of tuning parameter, which is a key parameter in the methodology, have been studied and the results show that the quality of the results improves when smaller values of tuning parameter are used but at the expense of higher computational costs. The effect of the order of the PSE approximation used in the calculations has also been studied and it shows that the quality in the results is improved when higher orders in the PSE approximations are used at the expense of higher computational costs. The methodology was also tested on a large-scale Waste Water treatment plant. A comparison was made for different values of tuning parameters and the most feasible value was chosen for the case study. Effects of different disturbances profiles such as step and ramp changes were also studied. The studies concluded that a lower cost value is obtained when ramps are used as disturbance profile when compared with step changes. The methodology was also tested when parameter uncertainty was introduced and the results show a higher cost is required when uncertainty is present in the system when compared with no uncertainty. The results show that this method has the potential to address the integration of design and control of dynamic systems under uncertainty at low computational costs

Operational model for empty container repositioning

Ph.DDOCTOR OF PHILOSOPH

Efficient Ranking-Based Methodologies in the Optimal Design of Large-Scale Chemical Processes under Uncertainty

Chemical process design is still an active area of research since it largely determines the optimal and safe operation of a new process under various conditions. The design process involves a series of steps that aims to identify the most economically attractive design typically using steady-state optimization. However, optimal steady-state designs may fail to comply with the process constraints when the system under analysis is subject to uncertainties in the inputs (e.g. the composition of a reactant in a feedstream) or in the system’s parameters (e.g. the activation energy in a chemical reaction). This has motivated the development of systematic methods that explicitly account for uncertainty in optimal process design. In this work, a new efficient approach for the optimal design under uncertainty is presented. The key idea is to approximate the process constraint functions and outputs using Power Series Expansions (PSE)-based functions. A ranking-based approach is adopted where the user can assign priorities or probabilities of satisfaction for the different process constraints and process outputs considered in the analysis. The methodology was tested on a reactor-heat exchanger system, the Tennessee Eastman plant, which is an industrial benchmark process, and a post-combustion CO2 capture plant, which is a large-scale chemical plant that has recently gained attention and significance due to its potential to mitigate CO2 emissions from fossil-fired power plants. The results show that the present method is computationally attractive since the optimal process design is accomplished in shorter computational times when compared to the stochastic programming approach, which is the standard method used to address this type of problems. Furthermore, it has been shown that process dynamics play an important role while searching for the optimal process design of a system under uncertainty. Therefore, a stochastic-based simultaneous design and control methodology for the optimal design of chemical processes under uncertainty that incorporates an advanced model-based scheme such as Model Predictive Control (MPC) is also presented in this work. The key idea is to determine the time-dependent variability of the system that will be accounted for in the process design using a stochastic-based worst-case variability index. A case study of an actual wastewater treatment industrial plant has been used to test the proposed methodology. The MPC-based simultaneous design and control approach provided more economical designs when compared to a decentralized multi-loop PI control strategy, thus showing that this method is a practical approach to address the integration of design and control while using advanced model-based control strategies

Elektromobile Flotten im lokalen Energiesystem mit Photovoltaikeinspeisung unter Berücksichtigung von Unsicherheiten

In diesem Buch wird ein Modell entwickelt, welches zur Identifizierung der Lastverschiebepotenziale von elektromobilen Flotten unter Berücksichtigung der Integration von Photovoltaik-Erzeugung und Unsicherheiten nutzbar ist. Es werden unterschiedliche Ansätze unter der Anwendung von Simulation, deterministischer und stochastischer Optimierung entwickelt, um den Ladevorgang von drei verschiedenen Elektrofahrzeugflotten an einer gemeinsam genutzten Ladeinfrastruktur unter Unsicherheit zu planen