Recent advances in mathematical programming techniques for the optimization of process systems under uncertainty

Optimization under uncertainty has been an active area of research for many years. However, its application in Process Systems Engineering has faced a number of important barriers that have prevented its effective application. Barriers include availability of information on the uncertainty of the data (ad-hoc or historical), determination of the nature of the uncertainties (exogenous vs. endogenous), selection of an appropriate strategy for hedging against uncertainty (robust/chance constrained optimization vs. stochastic programming), large computational expense (often orders of magnitude larger than deterministic models), and difficulty of interpretation of the results by non-expert users. In this paper, we describe recent advances that have addressed some of these barriers for mostly linear models.     

Solving online dynamic time-linkage problems under unreliable prediction

Dynamic time-linkage optimization problems (DTPs) are a special class of dynamic optimization problems (DOPs) with the feature of time-linkage. Time-linkage means that the decisions taken now could influence the problem states in future. Although DTPs are common in practice, attention from the field of evolutionary optimization is little. To date, the prediction method is the major approach to solve DTPs in the field of evolutionary optimization. However, in existing studies, the method of how to deal with the situation where the prediction is unreliable has not been studied yet for the complete Black-Box Optimization (BBO) case. In this paper, the prediction approach EA + predictor, proposed by Bosman, is improved to handle such situation. A stochastic-ranking selection scheme based on the prediction accuracy is designed to improve EA + predictor under unreliable prediction, where the prediction accuracy is based on the rank of the individuals but not the fitness. Experimental results show that, compared with the original prediction approach, the performance of the improved algorithm is competitive.     

H∞ control of continuous-time Markov jump linear systems with detector-based mode information†

This paper features new results on H_∞ analysis and control of linear systems with Markov jump disturbances, in a scenario of partial observations of the jump process. We consider the situations in which the jump process can only be measured through a suitable detector. A distinctive feature of the approach here is that it is general enough to encompass particular scenarios such as that of perfect information, no information (mode independent) and cluster observations of the Markov jump process. The main results, comprising a new bounded real lemma and a condition for state feedback synthesis, are expressed via linear matrix inequality-based optimisation problems. The method devised for the design of H_∞ controllers is applied to the control of an unmanned aerial vehicle model.     

Synthesis of multi-variate Volterra systems by a topological assemblage scheme

The Volterra series expansion is widely employed to represent the input–output relationship of nonlinear dynamical systems. Such a representation is based on the Volterra frequency-response functions (VFRF), which can be calculated from the governing equations of the system by the harmonic probing method. This operation is straightforward for simple systems, but may become very complicated for multi-variate or high-order systems. An alternative technique for the evaluation of the VFRFs of multi-variate systems is presented here generalizing a previously reported technique that focused on uni-variate cases. Additionally, it is extended to derive, within the same framework, a set of associated linear equations (ALE). Examples of a 2-dof system with polynomial nonlinearities, a wind-excited cable and a 2-dof system subjected to ocean waves are utilized to demonstrate applications of the proposed technique.     

Stability of Karhunen–Loève expansion for the simulation of Gaussian stochastic fields using Galerkin scheme

The paper investigates the problem of numerical stability of the Karhunen–Loève expansion for the simulation of Gaussian stochastic fields using Galerkin scheme. The instability is expressed as loss of positive definiteness of covariance matrix and is the result of modifications of standard exponential covariance functions that are commonly applied to increase the sparsity of the covariance matrix. The loss of positive definiteness of covariance matrix limits the use of efficient eigenvalue solvers that are needed for the solution of the resulting generalized eigenvalue problem. Two modifications of the shape of covariance function to avoid instability problems and at the same time to raise the numerical efficiency of Karhunen–Loève expansion by increasing the sparsity of the covariance matrix are proposed. The effects of the proposed modifications are demonstrated on numerical examples.     

Global sensitivity analysis of a feedback-controlled stochastic process model

The purpose of the study was to investigate whether global sensitivity analysis can be utilised in concurrent process and control design to gain insight into the process and its control. This paper addresses the issue of sensitivity of a control law performance to its parameters in a dynamic, hybrid deterministic–stochastic process model. The control law under investigation is a collection of single-input, single-output type tower level controllers in a papermaking process. Global sensitivity analysis is shown to attribute higher importance to certain key parameters, thus providing valuable insight for the designer.     

Stochastic frequency-security constrained energy and reserve management of an inverter interfaced islanded microgrid considering demand response programs

This paper addresses a novel security constrained energy management system of a microgrid which considers the steady-state frequency. Microgrid frequency as a key control variable, continuously exposes to be excursed of its nominal value due to unpredictable intermittencies arise from renewable sources and/or load consumptions. Moreover, great utilization of inertia-less inverter-interfaced distributed energy resources intensifies potential frequency excursions. As a result, energy and reserve resources of a microgrid should be managed such that the microgrid frequency lies within secure margins. To that end, a new objective function on the basis of the frequency dependent behavior of droop-controlled distributed generations is formulated using a mixed integer linear programming. It is aimed to optimize the microgrid frequency according to the economic and environmental policies. Besides, to seek the active participation of the consumers into proposed frequency management approach, a linearized ancillary service demand response program is also proposed. In addition, to properly model the impacts of microgrid various uncertainties in the frequency management approach, a two-stage stochastic optimization algorithm is employed. Simulations are performed in a typical microgrid which operates in the islanded mode during a 24 h scheduling time horizon. The numerical results show the impressiveness of the proposed frequency aware energy management system while concurrently managing the microgrid security and economical aspects. Furthermore, it is demonstrated that utilization of demand response programs economizes the microgrid frequency management approach.     

Cash farming in building and construction: a stochastic analysis

Cash flow management is a significant issue in the management of a building or construction firm. This paper steps back from the well researched area of poor cash management and its relationship with failure, to focus on the funds which are generated through operations, and the positive benefits which can follow in a well managed organization. A stochastic model is developed which illustrates how an average of 16% of turnover can be available for reinvestment. This is sufficient to allow investment in non-liquid assets, provided that this is managed carefully and precautions are taken against a severe reduction in turnover. This level of funds is sufficient to encourage firms to enter the industry with the motivation of generating funds, rather than a desire to build. This has implications for large clients and for government when dealing with the industry.     

Urine separation as part of a real-time control strategy

Considering wastewater from household sources, urine accounts for approximately 80% of the ammonia loads but for only 1% of the hydraulic load. Where classical urine separation deals with the separation and subsequent removal of urine from the wastewater stream, the method promoted here (urine separation for waste design) aims to (1) separate urine from the wastewater stream, (2) store it at the household and (3) release it in a controlled manner into the drainage system. The ultimate goal is on the one hand to average the daily dynamics in the ammonia load to the wastewater treatment plant (WWTP) and on the other hand to reduce ammonia emissions towards receiving waters. All process steps from urine production, separation, harvesting and controlled release of urine have been modelled on a micro-level where all toilets in the system are balanced separately. As a case study the integrated urban drainage system of Vils/Reutte in Tyrol was used where a total of 11 different control strategies – ranging from simple to highly complex real-time control (RTC) applications – have been developed and tested. The evaluation (taking into account both aims as expressed above) showed that the simplest RTC applications are the most effective. The optimal performance achieved in the case study resulted in an ammonia reduction of 62% (load based) compared to the status quo. As such a method would require considerable systems changes, an incremental implementation of associated improvements were also tested in this work. An important outcome was that a large proportion of the final improvement is gained already with the first 25% of installed separation toilets.