Non-linear models

Many processes occurring in a plant exhibit non-linear behavior. These cannot always be modeled using standard Base + Delta modeling techniques.

Look at the plot below showing the relationship between feed Conradson Carbon (CCR) and product yield. This clearly shows a positive linear relationship between the two.

The following plot shows instead the relationship between feed sulfur and product yield. An increase in sulfur also tends to lead to an increase in product yield, but at higher sulfur levels there also seems to be a wider range of yield values. The increase in yield with respect to sulfur must also be coupled to the change in another property.

Another way of plotting the data is to arrange it against Conradson Carbon again, but separating the data points into low, medium and high sulfur. Plotting the chart this way shows a general linear increase in product yield with feed CCR, but also a small increase in product yield with sulfur.

The plots together suggest that there is a coupled relationship between Conradson Carbon and sulfur content, and this relationship cannot be adequately expressed by a single linear model.

Regressing the data as a simple linear model, it is found that a reasonable fit can be found to correlate the change in Conradson Carbon to the change in distillate yield. The Linear vs. Non-Linear plot below shows the match between real measured data (on the x axis) and the modeled value derived from a linear model (on the y axis). The real data and measured data exhibit some curvature on this plot, indicating the model does not fit the data exactly. However the error is not significant and may be acceptable.

The Regression graph shows how the change in input property value is correlated to the change in output value:

• A strong correlation between input driver and output property appears as a significant gradient in the plot. In this case there is a significant gradient, so distillate yield is strongly correlated to Conradson Carbon.

• If the model fits well, the data points are randomly scattered close to the line. This appears to be the case in our example. There is however more error the further the input Conradson Carbon moves away from the base value.

  

As sulfur was also measured on the input, we can examine the response of the model with respect to sulfur. There is only a single Linear vs Non-Linear graph (as this displays the end result of the base delta model). The regression graph for sulfur also shows a gradient, indicating some correlation between feed sulfur and distillate yield. However for this regression graph the error is significant, and the model does not closely fit the measured data.

Therefore this simple linear model, relating distillate yield to changes in input Conradson Carbon and sulfur is not particularly effective as there is significant error in the model, especially as the input feed qualities move away from the base values.

Another way to plot the data could be by separating the input data into groups according to the input sulfur content. The measured data could be grouped as low, medium or high sulfur. Grouping the data this way, it can be seen that there is a positive correlation between Conradson Carbon and yield, and also between the sulfur content and yield. While a single model for the whole data set does not match the data effectively, we could create three models for the different sulfur levels to match the data better.

Multiple bases

The conditions within a process unit can often be changed by varying an operating parameter of that unit. The yields and qualities of the products produced by that unit are affected. Typically, the operating parameter can be varied continuously over a range.

Example: A Fluid Catalytic Cracker (FCC) cracks VGO feed into lighter (lower boiling) products. The degree of cracking can be controlled by varying the temperature of a riser through which material in the FCC passes. Increasing the riser temperatures causes the high boiling feed to be cracked further into lower boiling gasoline type material. A lower riser temperature does not crack the feed as much, so the resultant product has a higher boiling point and is more useful for diesel production.

A process model for such a unit should respond to changes across the whole range of the operating parameter, as well as changes in the input feed qualities to the unit. This is often achieved by breaking the process model into smaller pieces known as bases. Each base models the process unit at a different value of the operating parameter. If the actual value of the operational parameter is somewhere between the bases, a blend of the different base models is used to represent the unit performance.

Example: The FCC model might have two bases, for the FCC riser set to a temperature of 500°C and 530°C, respectively. The base model describing the unit performance at 500°C produces relatively more diesel; the base model describing the unit performance at 530°C produces relatively more gasoline.
When the riser temperature is set to 500°C, the base model for 500°C is used. If instead the FCC riser temperature was set to 515°C, 50% of the 500°C base and 50% of the 530°C base would be used. The predicted yields of products would be halfway between maximizing gasoline production and maximizing diesel production.

When the operational parameter is not fixed and the case is being optimized, AVEVA Unified Supply Chain will also optimize the value of the operational parameter within the allowed constraints of the problem.

Going back to the relationship between feed Conradson Carbon (CCR) and product yield, by breaking the original data set into the three different sulfur groups, and independently regressing each of these, it is possible to fit the data better.

Convexity in multi-base models

You should try to ensure that any multi-base models are not convex in their behavior. That is, the prediction values should be monotonically increasing or decreasing, with no inflection in any of the predictions.

For example, in the above figure an FCC naphtha yield of 14% corresponds to multiple values of the FCC riser temperature. Several combinations of bases can all be used which result in this solution value, but the solution is only valid where neighboring bases are used.

Note: When using multiple bases, it may be necessary to invoke mixed integer programming for the support of special ordered sets. Special ordered sets ensure that only neighboring bases are used in multi-base models. The use of integer programming may add substantially to solution time, but the problem is adjusted to avoid the use of special ordered sets if not necessary, so in the majority of cases run time will not be impacted.