Real-time optimization under parametric uncertainty: a probability constrained approach
Introduction
Real-time optimization (RTO), or on-line operations optimization, has a wide appeal in the process industries because of its promise for improving process profitability. Unfortunately, many of these systems have poor long-term service factors due, at least partially, to inadequate optimization robustness [1]. RTO is a model-based process control approach that uses current process information (i.e. process model and economic data) to predict the optimal operating policy for a process unit during the next RTO interval. Although most RTO systems attempt to improve model accuracy through model updating, there is a large number of uncertainty sources in the RTO problem and these uncertainties can seriously degrade the performance of any RTO system. A key issue in RTO is ensuring that any predictions will result in feasible operation given the recognized uncertainty in the RTO problem. In this work the term robustness of a model-based RTO system speaks to the ability of the system to predict an “optimal” operating policy that will be feasible at some level of probability. Then, a robust RTO system will usually provide a more conservative solution than a conventional RTO system to ensure that process operating constraints are unlikely to be violated. Such a robust RTO system “trades-off” some potential profitability for an assurance of the feasibility of the predictions.
Uncertainty is inherent in most process control and optimization problems. It is well recognized that uncertainty in the problem data (e.g. noisy, incomplete, unmeasurable or erroneous data) is an essential consideration in both process design/synthesis and process operation optimization. Such uncertainty was classified by Kraslawski [2] into two types: (1) ambiguity, if its value cannot be definitely established; and (2) imprecision, if its value is not sufficiently determined with respect to a given scale. In that work, the author discussed the effect of various uncertainties in process modeling, control and synthesis. Uncertainty in process design/planning was also investigated by Ierapetritou et al. [3].
A number of approaches have been proposed to deal with the uncertainties in engineering optimization applications, including: (1) solution sensitivity/stability analysis; (2) robust optimization, and (3) stochastic optimization. Sensitivity analysis was originally employed to measure the impact of changes in the problem data (i.e. parametric uncertainties) on the optimal solution. In order to achieve solutions that are less sensitive to variations in the problem data, Becker et al. [4] developed a robust optimization approach by augmenting the objective function of the optimization problem with a penalty term intended to minimize parametric sensitivity. Robust optimization [5], [6] is used to optimize the expected value of a chosen performance index for a given level of risk and includes formulations based on expectation, weighted mean-variance, worst-case and so forth. Alternatively, stochastic optimization [7], [8] attempts to directly solve a problem given uncertainty in the data. It handles these uncertainties using recourse functions, chance constraints and so forth, to transform the stochastic optimization problem to an equivalent deterministic optimization problem, which can be solved using a number of readily available optimization techniques. Since stochastic optimization deals directly with uncertainty in an explicit, “natural” manner, it is a good candidate technology for use in RTO.
Although there has been considerable effort in the areas of off-line optimization under uncertainty, making robust decisions under uncertainty in real-time has received little attention. Consider an RTO system that includes components for detecting steady-state operation, gathering and validating plant data, updating the process model, estimating optimum plant operation, analyzing optimization results, and transferring optimal operating policies to advanced controllers, there are often many sources of uncertainty around this closed RTO loop, which influence the model-based RTO predictions. This paper begins with discussing four sources of uncertainties (i.e. process uncertainty, market uncertainty, measurement uncertainty and model uncertainty) in the RTO system and their influences on RTO predictions. This is followed by a brief overview of the existing approaches to deal with these uncertainties. Then, a robust RTO formulation is presented that is able to account for various uncertainties in the closed RTO loop, to make robust decisions under such uncertainty in real-time, as well as to increase the operating profit of the plant. The proposed approach is limited to problems where parameters enter the constraints nonlinearly and the economic uncertainty enters the objective function linearly. This formulation addresses a substantial fraction of the RTO problems encountered in industry. Although the main focus of this paper is the development of a practical formulation for the Robust RTO problem and a solution method for this formulation, other contributions include a clear discussion of how the process operating constraints are naturally cast as probabilistic constraints and how these constraints can either be treated independently or jointly within the optimization problem. Further, the benefits and effects of treating the probabilistic constraints as independent or correlated are compared. The proposed approach is illustrated by a gasoline blending example that is taken from Forbes and Marlin [10] and modified slightly for the purposes of this paper.
Section snippets
RTO under uncertainty
A typical model-based RTO loop is shown in Fig. 1. First, the plant data (z) are gathered, validated to avoid gross errors in the process measurements, and may be reconciled using material and energy balances to ensure the data set used for model updating is self-consistent. These validated measurements (z′) are used to estimate some set of adjustable model parameters (β) to ensure the model represents the plant, as accurately as possible, at the current operating point. Then, the optimum
Robust RTO
Most RTO problems can be cast as:where: is a vector of decision variables; is a vector of dependent variables; is a vector representing the uncertain economic information (market uncertainty) in the objective function; is a vector representing the parametric uncertainty (process uncertainty, parametric model uncertainty, measurement uncertainty, etc.) in the process constraints; J0 is a nonlinear objective function and is often
Solution method
As previously developed, a robust RTO problem can be cast as either IPC or JPC problem when the probability constraints are employed. It is worth noting that the inequality constraints represented by , which are contained within the probabilistic constraints in problems (11) and (12), may be either linear or nonlinear. Currently, there is very little stochastic optimization literature for situations where the uncertain parameters enter nonlinearly into the problem statement. The method
Gasoline blending case study
The example considered here is a gasoline blending process that is taken from [9] and modified slightly for the purpose of this paper. The process, shown in Fig. 3, involves simultaneously blending two grades (i.e. regular and premium) of automotive gasoline using five feedstocks: LSR naphtha, reformate, n-butane, catalytic gas, and alkylate. The optimization objective is to determine the feedstock flow rates at each RTO interval that maximize the profit, while satisfying the product
Summary and conclusions
Various sources of uncertainty, including market uncertainty, process uncertainty, measurement uncertainty and model uncertainty, exist in any RTO systems, and influence RTO performance. Then the challenge for any well-designed RTO system is to make robust decisions under such uncertainty in real-time. The work presented here developed a stochastic optimization approach to robust RTO that deals effectively with parametric uncertainty. The key contribution is the incorporation of uncertainty
Acknowledgements
The authors are grateful for the financial support of Imperial Oil Ltd., Sunoco Inc., and Natural Sciences and Engineering Research Council of Canada.
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