Figure 12.2. General Input/Output Map of Neural Network
12.2.3. Dynamic Optimisation Framework Using First Principle Model
In the past many authors considered Equation 12.1 as the true representative of the actual dynamic process and developed dynamic optimisation and solution algorithms for such processes. These are presented in Chapter 5.
12.2.4. Dynamic Optimisation Framework Using Hybrid Model
Figure 12.4 illustrates a general optimisation framework (developed by Mujtaba and Hussain, 1998) to obtain optimal operation policies for dynamic processes with process-model mismatches.
Dynamic sets of process-model mismatches data is generated for a wide range of the optimisation variables (z). These data are then used to train the neural network. The trained network predicts the process-model mismatches for any set of values of z at discrete-time intervals. During the solution of the dynamic optimisation problem, the model has to be integrated many times, each time using a different set of z. The estimated process-model mismatch profiles at discrete-time intervals are then added to the simple dynamic model during the optimisation process. To achieve this, the discrete process-model mismatches are converted to continuous function of time using linear interpolation technique so that they can easily be added to the model (to make the hybrid model) within the optimisation routine. One of the important features of the framework is that it allows the use of discrete process data in a continuous model to predict discrete and/or continuous mismatch profiles.
The development and training of the neural network estimators for mismatches requires both the state variables (predicted by the model) and the mismatches at discrete points for a wide range of each optimisation variables. The number of sets of state variable and mismatch data for each type of state variable depends on the non-linearity and complexity of the system concerned.
The state variable profiles of the model are assumed to be continuous and are obtained by integration of the DAEs over the entire length of the time. Also efficient integration methods (as available in the literature) are based on variable step size methods and not on fixed step size method where the step sizes are dynamically adjusted depending on the accuracy of the integration required. Therefore, the discrete values of the state variables are obtained using linear interpolation technique. For example, if the values of a state variable predicted by the model are xdk and Xd,k+i at tk and tk+l, then at any discrete i„ which lies within [tk, tk+1], the state variable value (xd i) is calculated using the following expression:
Usually discrete points are of equal length (A = ti+1 - r,), which usually represents the sampling time of the actual process. Now, if the state variable of the actual process at discrete time f„ is given by x,ti, the discrete mismatch at f, will therefore be exli , = x^ - x^.
220.127.116.11. Continuous Mismatch Profiles During Optimisation
The mismatch estimator of the neural networks estimates mismatches only at fixed discrete points. Therefore, to use the optimisation framework presented in Figure 12.4 requires estimation of mismatches at variable discrete points (these points should coincide with those by the DAE integrator). This is again achieved by interpolation techniques. For example, if the values of a mismatch predicted by the estimator are exdi and exdi+1 at discrete points f, and ti+1 (fixed A = tM - ti) then at any variable discrete point (by the integrator) th which lies within [i„ f,+;], the mismatch value (exdk) is calculated using the following expression:
Note, for highly non-linear profiles of state variables, switching from continuous to discrete or from discrete to continuous using linear interpolation technique may not be efficient and non-linear interpolation technique may need to be employed.
12.2.5. Hybrid Model Development for Pilot Batch Distillation Column
Mujtaba and Hussain (1998) implemented the general optimisation framework based on the hybrid scheme for a binary batch distillation process. It was shown that the optimal control policy using a detailed process model was very close to that obtained using the hybrid model.
In Greaves et al. (2001) and Greaves (2003), instead of using a rigorous model (as in the methodology described above), an actual pilot plant batch distillation column is used. The differences in predictions between the actual plant and the simple model (Type III and also in Mujtaba, 1997) are defined as the dynamic process-model mismatches. The mismatches are modelled using neural network techniques as described in earlier sections and are incorporated in the simple model to develop the hybrid model that represents the predictions of the actual column.
The pilot-plant description is given in section 3.3.4. in Chapter 3. Methanol-Water system was considered with an initial charge of 900 ml of Methanol and 2100 ml of Water giving a total of 85.04 gmol of the mixture with <0.25, 0.75> molefractions for Methanol and Water respectively.
Experimental Vapour Flowrate, mol/min
Distillate rate, mol/min Reflux rate = VeXp, mol/min Experimental Accumulated Distillate Hold-up, mol Accumulated Distillate Composition, mole fraction Instant Distillate Composition, mole fraction
Figure 12.5. Schematic of Batch Distillation Column
18.104.22.168. Relation Between Experimental Reflux Ratio (/?exp) and Model Reflux Ratio (R^m)
Many industrial users of batch distillation (Chen, 1998; Greaves, 2003) find it difficult to implement the optimum reflux ratio profiles, obtained using rigorous mathematical methods, in their pilot plants. This is due to the fact that most models for batch distillation available in the literature treat the reflux ratio as a continuous variable (either constant or variable) while most pilot plants use an on-off type (switch between total reflux and total distillate operation) reflux ratio controller. In Greaves et al. (2001) a relationship between the continuous reflux ratio used in a model and the discrete reflux ratio used in the pilot plant is developed. This allows easy comparison between the model and the plant on a common basis.
For the sake of convenience, Figure 3.11 is presented here again. The reflux in the column (Figure 12.5) is produced by a simple switching mechanism that is controlled by a solenoid in a cyclic pattern (on-off). The valve is open for a fixed period of time (to withdraw distillate) and is closed for a fixed period of time (to return the reflux to the column). In this column the valve is always open (top) for 2 seconds and then closed (tciose) for 2x(Rexp) second, where Rexp is the reflux setting.
It was shown in section 3.3.4 that the discrete experimental reflux ratio Rexp in the cyclic operation could be viewed as the average continuous external reflux ratio
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