Introduction

The term ``Real-Time Optimization'' in this work refers to a process control approach which uses rigorous, nonlinear process models, process measurements and current market information to calculate the setpoints for multivariable constraint controllers (MCC) on-line, in real-time. Its objective is to maintain the process operation as nearly as possible to the optimal steady-state operating conditions, so as to maximize process economic performance. Such RTO system can integrate the long-term planning/scheduling instructions into process units operation, and therefore it has been an appealing technique for process industries in recent years. It is reported that approximate 250 commercial on-line optimization system have been installed in the operating companies, and most of them basically consists of the following components (White, 1988, 1998): 1) Measurement Validation; 2) Data Reconciliation and Model Updating; 3) Model-based Optimization; and 4)Command Conditioning and Implementation.

In an RTO system, the kernel problem is to search feasible optimal operating points depending on the updated process models that describe the process steady-state behavior, as accurately as possible. However these optimal operating points calculated by RTO system, or RTO predictions, are severely affected by various uncertainties, which will be discussed with every detail in the next section. Due to these uncertainties acting on the RTO system, the RTO predictions will have a level of variability, which can be classified into two types (Miletic and Marlin, 1998) :

1)

Random variability; RTO prediction variance due to some common causes, such as measurement noises, high frequency, stationary process disturbances and so on, that may be occur during the normal operation of the plant.

2)

Gross variability; RTO prediction variance due to some special causes, such as the low-frequency, nonstationary plant disturbances that may lead to the changed process steady-state behavior.

Consider the different RTO prediction variance, some of them accurately follow the track of the optimal operating points, and should be downloaded to the low-level MCC and implemented for maximization of operating profit; on the contrary, some RTO prediction variance are only caused by propagation of process measurement noises, etc. around RTO system, and such unnecessary variance should be avoided since they may result in RTO profit loss and process instability (Forbes and Marlin, 1996).

In order to ensure that RTO predictions are both feasible and reasonable, results analysis, or more generally ``command conditioning'', is used to ``filter'' the optimizer outputs to achieve some certain goals. These goals are to reduce RTO operational variability, to decrease the closed RTO loop instability, and to increase process operating profit. Unfortunately, such an important issue for RTO implementation has received little attention from both academic research and industrial applications. The unique published method is developed by Miletic and Marlin (1998). This method applies multivariable statistical hypothesis test to decide when the RTO predictions are statistical significant from the current plant operation, and not the result of measurement noise transmission around the RTO loop. The simulation study indicates that, by using their method, only the statistical significant results are implemented in the plant, whereas most of unnecessary setpoint changes due to measurement noises are rejected and as a result, the increased profit can be derived.

However, Miletic and Marlin's method didn't address the inequality constraint activations that could be restricting the predicted change, for example, setpoints move limits (namely ``trust-region constraints'' in this work). In general, trust-region constraints are used in a vast amount of RTO and/or Advanced Process Control (APC) systems. They are some inequality constraints used to restrict the RTO/APC software from ``jumping'' the plant to the perceived optimum in time. Trust-region constraints exist for many reasons, such as system safety and stability, however they can also cause some problems. One of them is the restriction of actual plant progression toward the optimum. If the trust-region is small enough so as to be encompassed within the confidence region of RTO predictions, with the Miletic and Marlin's method, the operating points of the plant will never move. To determine wether the trust-region constraints imposed on the optimizer are actually restricting economic improvement, the level of activity of the these inequality constraints will have to be analyzed. An extension to the Miletic and Marlin's hypothesis testing which includes the Lagrangian multipliers for the active trust-region constraints is explored in this paper.

The paper begins by detailed discussion of RTO loop under uncertainties. A linear approximation is used to estimate the impact of these uncertainties on RTO predictions. Then an practical results analysis procedure is developed for RTO systems under uncertainties with trust-region constraints, and the Williams-Otto reactor case study is presented to illustrate the proposed method, where the performance of RTO system with results analysis procedure is evaluated by using a general RTO performance metric, Extended Design Cost (Zhang and Forbes, 1999).