Moving horizon scheme

In the first part of this chapter (Section 9.2), we present an uncertainty conscious scheduling approach that combines reactive scheduling and stochastic scheduling by using a moving horizon scheme with an uncertainty conscious model. In this approach, it is assumed that decisions are made sequentially and that the effect of the revealed uncertainties can be partially compensated by later decisions. The sequence of decisions and observations is modeled by a sequence of two-stage stochastic programs. 

Fig. 9.2 Moving horizon scheme only the current decision jf,-(t,) is applied to the plant.

The performance of the scheduler can be significantly improved by the use of a stochastic model. The stochastic model used here considers not only the probability distribution of the uncertain parameters but also the structure of decisions and observations that result from the moving horizon scheme. 

The sequence of decisions obtained from the scheduler (Figure 9.4) has a tree structure. This structure results from the scenario tree of the uncertain demand parameters (Figure 9.3). Due to the moving horizon scheme, the decisions and the observations alternate at each period and the decisions are functions of the observations. Each point in time where a decision is made is called a stage. The result is a multi-stage tree where each stage corresponds to a period. 

However, the description of the tree structure of a multi-stage model leads to complicated constraints. To simplify the original multi-stage model, it is approximated by a model with two stages. It consists of only one sequence of decisions-observation-decisions. The two-stage structure leads to considerably simpler optimization problems. It is also adequate from a practical point of view in the moving horizon scheme, only the first decision x is applied to the plant while all the remaining variables are used to compute the estimated performance only. 

Fig. 9.6 Moving horizon scheme with two-stage stochastic models the first decisions >r,(t ) are applied to the plant (compare to Figure 9.2).

The moving horizon scheme using the two-stage model is shown in Figure 9.6. In contrast to the deterministic scheduler which uses the expected value of the demands dls and ds i (see Section 9.2.2), the stochastic scheduler updates the demand in form of the distribution given in Table 9.1 d, df, d]+1, and dj+1. 

An uncertainty conscious scheduling approach for real-time scheduling was presented in this chapter. The approach is based on a moving horizon scheme where in each time period a two-stage stochastic program is solved. For the investigated example it was found that the stochastic scheduler improved the objective on average by 10% compared to a deterministic scheduler. 

The AFM algorithm can be easily incorporated into an MFC scheme, where in each time step k a rigorous nonlinear optimization problem is formulated. The objective is to calculate the optimal values of the manipulated variables v over a control horizon M, so that the error between the RBF model predictions and the desired set-point over a prediction horizon N is minimized. As soon as the optimization problem is solved, the first control move (k) is implemented, and then the RBF model is updated using the AFM algorithm. The procedure is shown in figure 2. Assuming one controlled variable, the optimization problem can be described by the following set of equations ... 

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