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