The main objective in RTO design is to maximize a given measure of process
performance, which is usually expressed in terms of economics. One simulation
result of profit trajectories for System A, B, and C is
shown in Figure (
). Clearly, in the presence of trust-region
constraints, the System B results in the poor RTO performance,
whereas System A and System C obtain the almost same
operating profit.
Figure: Profit Trajectory for RTO Systems (1) without Results Analysis, (2)
with Miletic and Marlin's Method, and (3) with Practical Results Analysis
Procedure
In order to analyze the RTO performance systematically, a general RTO performance metric, namely Extended Design Cost (Zhang and Forbes, 1999), is applied to System A, B, and C in this paper. Extended Design Cost () is defined as the total loss of performance relative to perfect optimization within a pre-specified performance evaluation period, that is:
where: are the true plant optimum operation, are the RTO predictions for the manipulated variables, P is the plant profit function in the reduced space, and indicate start and end of RTO performance evaluation period, respectively. It can be considered to consist of three parts: 1) the performance loss due to persistent offset in steady-state from the true plant optimum operation; 2) the performance loss due to the initial transient behavior of RTO system; and 3) the performance loss due to unnecessary variance of predicted optimum values for the manipulated variables. These three parts are given by Zhang and Forbes (1999),
where is the reduced Hessian of the plant profit surface.
They are defined as Bias Cost (), Transition Cost () and
Variance Cost (), respectively. Bias Cost is a static performance
index, which represents the ability to correctly identify the optimal
operation policy. Transition Cost is a dynamic performance index, which
represents the ability to quickly converge to some plant operating conditions.
Finally, Variance Cost is a performance index that describes the ability of an
RTO system to ``filter'' out the unwanted effects of common cause variance
transmission around the closed RTO loop. By the definition given in Equation
(), an RTO design is expected to obtain the minimum
the Extended Design Cost.
In this case study, the Extended Design Cost for RTO systems with different results analysis procedures is given by Table 1 with the pre-specified performance evaluation period 30 RTO . Clearly, System B exhibits the largest Bias Cost due to no setpoint changes, and this lead to the poor RTO design (i.e., the largest Extended Design Cost among three RTO systems). Comparing the System A and System C, we can see that System A eventually converge to the true plant optimum operating point and as a result, takes the smallest Bias Cost, however, the price for that is fairly large Variance Cost; System C reduces the Variance Cost significantly by applying practical results analysis procedure, but this result in the larger Transition Cost than System A.
Table 1. Extended Design Cost for Different RTO Systems (Based on Simulations)
>From the view point of Extended Design Cost, Table 1 also shows that System C exhibits the smallest Extended Design Cost, and as a result the conclusion can be drawn that the increased operating profit can be derived by using practical results analysis procedure in RTO systems. Note that in fact any RTO performance is due to the trade-off between Bias Cost, Transition Cost and Variance Cost. The impact of practical results analysis procedure on RTO systems is only to reduce the Variance Cost.