Kriging model

King James Bible, Mk 12 41-44, 247 Kriging modeling, ethnological studies, Palmarejo, 221-228... 

Using kriging models for real-time process optimisation... 

In this work, new developments were achieved through the use of new examples, one of which the optimisation of a real crude distillation unit involving 19 decision variables. The performance of the metamodel-based optimisation is compared with results obtained with the optimisation based on a first-principles model, embedded in a sequential-modular process simulator. It is shown that metamodel-based optimisation with adaptation of the metamodels during the optimisation procedure provides results with good accuracy and significant reduction of computational effort. The performance comparison between neural networks and kriging models for chemical processes is another contribution of this work. 

One of the most used families of reduced models that have been used as metamodels [2] are the kriging models. 

The kriging model structure presented here is the most frequently used in the literature (for more details, refer to [3]). Let the set of functions y(jc,M) be a rigorous mathematical description of a process, where x are independent variables and u model parameters. 

The kriging models that approximate the rigorous one are built from a set of design... 

Using Kriging Models for Real-Time Process Optimisation... 

For kriging models, the structure is defined by the set of independent variables selected - including quadratic terms - and the selection of the correlation model. The parameter estimation is performed by a maximum likelihood procedure. For neural nets, the activation function to be used is defined a priori. The structure is completed by the selection of the number of neurons in the hidden layer. A backpropagation procedure has been used for training. 

Kriging models and neural nets were generated for each of the 39 dependent variables required for the computation of the objective function and the constraints. Table 2 presents the main characteristics of the metamodels (for more details, refer to [5]). 

A new strategy for Real-Time Optimisation combining metamodels and a Sequential Approximate Optimisation (SAO) procedure has been proposed. This methodology is based on automatic procedures, aiming its use in real-time applications. Kriging models and neural nets were used as metamodels. 

Selection of the implicit models to be substituted by a kriging metamodel. Given a flowsheet to optimize, identify the unit operations that introduce noise or are very CPU time consuming in simulation. These will be the units to be substituted by a a kriging model. And estimate the noise introduced by dependent variables (especially if the noise introduced is important). Note that the accuracy of the final optimum point cannot be higher than the noise, but should be as close as possible. 

Fit all the surfaces using kriging and validate the model. Once all the variables (surfaces) have been estimated by kriging, it is important validate the metamodel, i.e. using cross validation that allows us to asses the accuracy of the model without extra sampling [2], A kriging model can be considered correct if all the errors in cross validation are inside the interval [-3,+3] standard errors. 

Refinement without updating kriging parameters. It is possible to improve the solution obtained in point 4 by adding this new point to the kriging model without reoptimize the parameters. With the new point added we re-optimize the model. This re-optimization is expected to be very fast because the initial point should be near the optimal solution. The procedure is repeated until two successive results are inside a pre-specified tolerance. 

From a simulation point of view, the two thermal couples introduce two recycles.. In this case, recycles introduces two complications 1. Good initial values are needed to converge the flowsheet, and 2. The time to converge the flowsheet considerably increases. It would be possible to decompose the system in individual columns and develop a kriging for each column, but the dimensionality of the kriging model would be larger, because we would have to deal with the recycles explicitly. Therefore, it is better to keep the divided wall column as an entity that can be efficiently substituted by a kriging metamodel. 

Simpson, T. W., Korte, J. J., Mauery, T. M. and Mistree, F. (1998). Comparison of response surface and kriging models for multidisciplinary design optimization, in Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multid,isciplina,ry Analysis Optimization, AIAA-98-4755 (St. Louis, MO). 

A way to overcome this problem is to generate an approximation of complex analysis code that describes the process accurately, but at a much lower cost. Metamodels offer an approximation in that they provide a model of the model . Clarke et al. (2005) [58] suggested metamodelhng techniques, namely response surface methodology (RSM), radial basis function (RBF), kriging model and multivariate adaptive regression sphnes (MARS) as potentially useful approaches. Computer deterministic experiments have been addressed by Charles et al. (1996) [59], Simpson et al. (1998) [60], CappeUeri et al. (2002) [61] and Aguire et al. (2(X)7) [62],... 

Comparison of response surface and kriging models for multidisciplinary design optimization Design of a PZT bimorph actuator using a metamodel-based approach... 

Simpson TW, Mauery TM, Korte JJ, Mistree F. Comparison of response smface and kriging models for multidisciphnary design optimization. AIAA J 1998 1-11. 

Kandathil SM, Fletcher XL, Yuan Y, Knowles J, Popelier PLA (2013) Accuracy and tractability of a Kriging model of intramolecular polarizable multipolar electrostatics and its application to histidine. J Comput Chem 34 1850-1861... 

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