Optimal steady state

Latour, P., Use of steady-state optimization for computer control in the process industries. In On-line Optimization Techniques in Industrial Control (Kompass, E. J. and Williams, T. J., eds.). Technical Publishing Company, 1979. 

Timm, Gilbert, Ko, and Simmons O) presented a dynamic model for an isothermal, continuous, well-mixed polystyrene reactor. This model was in turn based upon the kinetic model developed by Timm and co-workers (2-4) based on steady state data. The process was simulated using the model and a simple steady state optimization and decoupling algorithm was tested. The results showed that steady state decoupling was adequate for molecular weight control, but not for the control of production rate. In the latter case the transient fluctuations were excessive. 

The targets for the MPC calculations are generated by solving a steady-state optimization problem (LP or QP) based on a linear process model, which also finds the best path to achieve the new targets (Backx et al., 2000). These calculations may be performed as often as the MPC calculations. The targets and constraints for the LP or QP optimization can be generated from a nonlinear process model using a nonlinear optimization technique. If the optimum occurs at a vertex of constraints and the objective function is convex, successive updates of a linearized model will find the same optimum as the nonlinear model. These calculations tend to be performed less frequently (e.g., every 1-24 h) due to the complexity of the calculations and the process models. 

Process design modifications usually have a bigger impact on operability (dynamic resilience). Dynamic resilience depends on controller structure, choice of measurements, and manipulated variables. Multivariable frequency-response techniques have been used to determine resilience properties. A primary result is that closed-loop control quality is limited by system invertability (nonmin-imum phase elements). Additionally, it has been shown that steady-state optimal designs are not necessarily optimal in dynamic operation. 

This problem is obviously a large one in that it includes all the problems of optimal control with uncertain parameters as well as embedding in synthesis. Two example problems are given, with one illustrating that the minimax structure may well be different from the steady-state optimal structure. 

Arkun, Y., Stephanopoulos, G. and Morari, M., "Design of Steady-State Optimizing Control Structures for Chemical Processes," AlChE 71st Annual Meeting, Miami Beach, FL., November 12-lb, 1978. 

The Steady State Optimization Problem. For a set of slowly varying external disturbances, we assume that the process is at pseudo-steady state. Then, the following static optimization problem can be formulated ... 

Since the plant is decomposed to its subsystems, the steady-state optimization problem is characterized by a multiechelon structure where the subsystem optimizing controllers communicate with a coordinator. For further details the reader is referred to the work by Arkun (20). 

Steady State Optimizing Control of a Fluid Catalytic Cracker. The process model used in this example can be found in (21) while the design parameters are given in (20). The important constraints are T e reactor temperature 930°F, T = regenerator... 

For continuous processes operating at steady state, optimization typically consists in determining the operating point that minimize or maximize some performance of the process (such as minimization of operating cost or maximization of production rate), while satisfying a number of constraints (such as bounds on process variables or product specifications). In mathematical terms, this optimization problem can be stated as follows ... 

Most industrial RTO applications consider steady-state optimization. Thus, measurements should represent steady-state plant behavior when used to update the RTO model. The steady-state detection module determines if a steady state has been reached based... 

Bamberger, W. Hermann, R. Adaptive on-line steady-state optimization of slow dynamic processes. Automatica 1978, 14, 223-230. 

By calculating TACs for a range of values of Vr and Nj, the minimum steady-state optimal plant turns out to have a reactor holdup of 3000 Ib-mol and a stripper with 19 trays. With no consideration of dynamic controllability, this is the best plant. 

Use the remaining control valves for either steady-state optimization (minimize energy, maximize yield, etc.) or to improve dynamic controllability. A common example is controlling purities of recycle streams. Even though these streams... 

Nominal design values of the spatial structure of the column for the synthesis of MTBE (i.e. column diameter and reboiler and condenser heat exchange areas) are firstly estimated. They are obtained by solving a steady-state optimization problem, which minimizes the total annualized cost of the RD unit in the absence of disturbances. The following constraints are included to the problem formulation (i) the column diameter is bounded by flooding conditions (ii) the heat exchange areas of the condenser and reboiler are estimated by bounded values of outlet temperatures of the hot and cold utilities, and (m) the molar fraction of MTBE at the top and bottom stream is constrained to values lower than 0.1% and 99% respectively. Thus, the statics optimization problem results in,... 

This theory will now be applied to the multiobjective design cases for the MTBE column. The reference design used for the sake of comparison is the structure previously introduced in chapter 7. This structure corresponds to a reactive distillation column for the synthesis of MTBE. The spatial design variables were obtained by steady-state optimization of the unit s economic performance. For background information, the reader is referred to section 6.2. The relevant parameters and schematic representation of the optimized design are given in table 8.4 and figure 8.7, respectively. 

Chapter 2 treats the topic of steady-state optimization. Necessary conditions for extrema of functions are derived using variational principles. These steady-state optimization techniques are used for the determination of optimal setpoints for regulators used in supervisory computer control. 

---