Optimizing control steady-state optimization problem

Bahri, P Bandoni, A., and Romagnoli, J. A. (1996). Effect of disturbances in optimizing control Steady-state open-loop backoff problem. AIChE J. 42,983-995. 

Many HVAC system engineering problems focus on the operation and the control of the system. In many cases, the optimization of the system s control and operation is the objective of the simulation. Therefore, the appropriate modeling of the controllers and the selected control strategies are of crucial importance in the simulation. Once the system is correctly set up, the use of simulation tools is very helpful when dealing with such problems. Dynamic system operation is often approximated by series of quasi-steady-state operating conditions, provided that the time step of the simulation is large compared to the dynamic response time of the HVAC equipment. However, for dynamic systems and plant simulation and, most important, for the realistic simulation... 

Other synonyms for steady state are time-invariant, static, or stationary. These terms refer to a process in which the values of the dependent variables remain constant with respect to time. Unsteady state processes are also called nonsteady state, transient, or dynamic and represent the situation when the process-dependent variables change with time. A typical example of an unsteady state process is the operation of a batch distillation column, which would exhibit a time-varying product composition. A transient model reduces to a steady state model when d/dt = 0. Most optimization problems treated in this book are based on steady state models. Optimization problems involving dynamic models usually pertain to optimal control or real-time optimization problems (see Chapter 16)... 

Although, as explained in Chapter 9, many optimization problems can be naturally formulated as mixed-integer programming problems, in this chapter we will consider only steady-state nonlinear programming problems in which the variables are continuous. In some cases it may be feasible to use binary variables (on-off) to include or exclude specific stream flows, alternative flowsheet topography, or different parameters. In the economic evaluation of processes, in design, or in control, usually only a few (5-50) variables are decision, or independent, variables amid a multitude of dependent variables (hundreds or thousands). The number of dependent variables in principle (but not necessarily in practice) is equivalent to the number of independent equality constraints plus the active inequality constraints in a process. The number of independent (decision) variables comprises the remaining set of variables whose values are unknown. Introduction into the model of a specification of the value of a variable, such as T = 400°C, is equivalent to the solution of an independent equation and reduces the total number of variables whose values are unknown by one. 

In this paper, the advective control model for groundwater plume capture design is described, algorithmic requirements to accommodate unconfined aquifer simulation are presented, and two- and three- dimensional example problems are used to demonstrate the optimization model capabilities and design implications. The model is applicable for designing long-term plume containment systems and as such assumes steady-state flow and time-invariant pumping. 

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, Stephanopoulos and Morari (1978) have added a new twist to control system synthesis. They developed the theory and then demonstrated on two example problems how to move from one control point to another for a chemical process. They note that the desirable control point is likely at the intersection of a number of inequality constraints, the particular set being those that give optimal steady-state performance for the plant. Due to process upsets or slow changes with time, the point may move at which one wishes to operate. Also, the inequality constraints themselves may shift relative to each other. Arkun, Stephanopoulos and Morari provide the theory to decide when to move, and then develop alternative paths along which to move to the new... 

In heat recovery applications there can be a large number of feasible plant configurations. After the configuration is optimized based on steady state considerations (which may not be an easy problem), the evaluation of the effectiveness of various control schemes can be performed. The dynamic plant operation must be evaluated in terms of economics, regulation, reliability, and safety over a broad range of operating regimes. 

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). 

This last class of methods provides a way of avoiding the repeated optimization of a process model by transforming it into a feedback control problem that directly manipulates the input variables. This is motivated by the fact that practitioners like to use feedback control of selected variables as a way to cormteract plant-model mismatch and plant disturbances, due to its simphcity and reliability compared to on-line optimization. The challenge is to find functions of the measured variables which, when held constant by adjusting the input variables, enforce optimal plant performance [19,21]. Said differently, the goal of the control structure is to achieve a similar steady-state performance as would be realized by an (fictitious) on-line optimizing controller. 

The monitoring uses formulas that take into account feed flow rates, targets calculated by the optimization layer of multivariable control, controlled variables upper and lower limits and other parameters. The economic benefits are based on the degrees of freedom and the active constraints at the steady state predicted by the linear model embedded in the controller. In order to improve the current monitoring, parameters dealing with process variability will be incorporated in the formulas. By doing this, it will be also possible to quantify external disturbances that affect the performance of the advanced control systems and identify regulatory control problems. 

With weighting-faetor methods, the basie idea is to form a multiobjective optimization problem in whieh some faetor related to dynamie eontrollability is added to the traditional steady-state eeonomie faetors. These two faetors are suitably weighted, and the sum of the two is minimized (or maximized). The dynamic controllability factor can be some measure of the goodness of eontrol (integral of the squared error), the eost of the eontrol effort, or the value of some controllability measure (sueh as the plant eondition number, to be diseussed in Chapter 9). One real problem with these approaehes is the diffieulty of determining suitable weighting faetors. It is not elear how to do this in a general, easily applied way. 

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