Path Constraint

Starting point Path Constraints T [K] Final point... 

One advantage in the sequential approach is that only the parameters that are used to discretize the control variable profile are considered as the decision variables. The optimization formulated by this approach is a small scale NLP that makes it attractive to apply for solving the optimal control with large dimensional systems that are modeled by a large number of differential equations. In addition, this approach can take the advantage of available IVP solvers. However, the limitation of the sequential method is a difficulty to handle a constraint on state variables (path constraint). This is because the state variables are not directly included in NLP. 

Note that for both strategies the reboiler holdup must not exceed the maximum capacity (constraint) at any time (path) within the entire operation period to avoid column flooding and this imposes a path constraint in the optimisation problem as discussed below. 

Mujtaba (1999) explained the BED operational constraint (path constraint) using two time intervals in a separation Task (producing a cut). To ensure a safe operation Mujtaba (1999) proposed the following treatment to the path constraint. The reboiler holdup profile (HTi, Figure 10.3) during any distillation Task is the solution of the total mass balance equation and can be expressed as ... 

First, two examples using both problems OP1 and OP2 are presented to explain the effects of different solvent feeding modes and path constraint on the operation. In these examples only Task 1 of Figure 10.6 is carried out where component 1 is recovered at a given purity. Then, example 3 using Multiperiod Optimisation Problem (OP) is presented, where all three Tasks of Figure 10.6 are carried out. 

Note that the path constraint does not appear in the optimisation problem for batch mode solvent feeding. 

We present below some easily implementable methods for improving the robustness and efficiency of feasible path dynamic optimization codes which have proved useful in our work. Here, we cover methods for preventing simulation error from disrupting optimization, representation of path constraints, and handling poor local approximations during the optimization. 

A path constraint that is inactive by even a small amount is invisible to the optimizer. 

A simple modification of the path constraint representation (40), is given by... 

Many optimizers use the feasibility test Z It l < optacc (cf. Eq. 36). The model generates a path constraint, x(rj), for which a value of 1 represents an insignificant constraint violation. Using... 

This modified path constraint representation was found to give a reduction of up to a factor of 2 in the number of iterations required for successful optimization. 

Local approximations (linear or quadratic) are often particularly poor in dynamic optimization problems. For instance, this situation is found to occur when taking the full step predicted from the local approximation, 6, causes a path constraint to become active or the system to become unstable. 

The dynamic models of chemical processes are represented by differential-algebraic equations (DAEs). Equation (2) and (3) define such a system. Equations (4), (5) and (6) are the path constraints on the state variables, control variables and algebraic variables respectively, while equation (7) represents the initial condition of the state variables. Obj is a scalar objective function at final time, tj. ... 

The first one is to decompose the dynamical system into the control and the state spaces. In the next step, only the control variables are discretized and remain as degrees of freedom for the NLP solver [5]. The method is called the sequential approach. The DAE system has to be solved at each NLP iteration. The disadvantages of the approach are problems of handling path constraints on the state variables, since these variables are not included directly in the NLP solver [5] the time needed to reach a solution can be very high in case the model of the dynamic system is too complex difficulties may arise while handling unstable systems [4]. 

In the recent years, a new approaeh has been developed for eliminating this disadvantage [5]. This approach is called the quasi-sequential approach and takes the advantages of both the sequential and the simultaneous approaches sinee both the control and the state variables are diseretized, the path constraints for the state variables can be handled the DAE system is integrated only once, so the computation becomes more effieient. 

V.S. Vassiliadis, R. W. H. Sargent, C. C. Pantelides, 1994, Solution of a class of multistage dynamic optimizations problems. 1-Problems without path constraints, Ind. Eng. Chem. Res, 33,2111-2122. 

Adiabatic-mixing pathways, where seawater (2°C) mixes into hydrothermal fluid (350°C), have been successfully used to model formation of sulfide minerals associated with venting hydrothermal solutions at mid-ocean ridges (9). Sulfate reduction can be quantitatively and isotopically important in such reactions (. Combinations of three types of isotopic path constraints discussed above have been examined, using the mixing reaction pathways calculated by Janecky and Seyfried ( ) for chemical equilibrium and initial sulfur isotopic compositions of 1 per mil for the hydrothermal solution and 21 per mil for seawater (Figure 1). 

Jones EG, Dias MB, Stentz A (2011) Time-extended multi-robot coordination for domains with intra-path constraints. Auton Robot 30(4) 41-56... 

A much better alternative is to explicitly discretize the DAE model as well as any additional path constraints at a finite number of points, and to use multiple shooting togeihei with an infeasible-path optimization method [4, 5, 8]. To this end, choose a multiple shooting mesh... 

Path constraints g represent conditions that must be fulfilled throughout the entire integration horizon. These inequality constraints augment the algebraic equations ft. 

Indirect or variational approaches are based on Pontryagin s maximum principle [8], in which the first-order optimality conditions are derived by applying calculus of variations. For problems without inequality constraints, the optimality conditions can be written as a set of DAEs and solved as a two-point boundary value problem. If there are inequality path constraints, additional optimality conditions are required, and the determination of entry and exit points for active constraints along the integration horizon renders a combinatorial problem, which is generally hard to solve. There are several developments and implementations of indirect methods, including [9] and [10]. 

In the optimization, the objective function is to maximize the product generated versus process time. The desired product was defined by endpoint equalities and inequalities, such as amount of unreacted components. In addition, the safety conditions required certain path-constraints for the state variables such as temperature. Unfortunately, we experienced optimization problems with the above formulation. The problems stem from getting stuck in infeasible regions due to complexity of the process and the nonlinearity of the objective function. At the moment, we are working to overcome these problems so that we can test the runaway behavior and cooler limitations with respect to optimization. 

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