Linear models help refiners develop RFG recipes

Bruce T. Naman
Bonner & Moore Associates Inc.
Houston

Linear methods to predict TOX (toxic emissions), NO _x (nitrogen oxides), and VOCs (volatile organic compounds) of a blend can help refiners calculate the likelihood that their blend recipes will meet requirements of Phases I and II complex models for reformulated gasoline (RFG).

Multiple regression techniques to predict TOX, NO_x, and VOCs are based on the following variables:

• Oxygenates content, wt % oxygen
• Reid vapor pressure (Rvp), psi
• Sulfur content, ppm
• Aromatics content, vol %
• Olefins content, vol %
• E200, % fuel evaporated at 200° F.
• E300, % fuel evaporated at 300° F.
• Benzene content, vol %.

The method can be incorporated into a multiperiod, multiproduct, linear programming (LP) model to obtain optimal recipes. An accurate model will push the critical specifications to their limits and free the refiner from setting more restrictive limits in advance of running the LP model.

Refiners can achieve better fit of regression equations by restricting the range of values for each input to operational limits based on historical blends.

Error in this method is well within the error as a result of laboratory measurement of the inputs. In fact, the method is reasonably accurate even if the values for the inputs vary over broad ranges.

Several approaches

Although a nonlinear approach can be used to find optimal recipes, it is very rigorous. It may produce optimal recipes according to a mathematical model, but it may not give practical recipes if multiproduct and multiperiod constraints are included in the model. Also, not every blender has a nonlinear tool available.

Another approach is to identify the limits on the measurable qualities that are input to the complex model that will ensure compliance-for example, limits on Rvp, sulfur, or benzene. Using the the Environmental Protection Agency (EPA) spreadsheet containing the complex model equations, blenders can determine these limits through trial and error. The disadvantage of this approach, however, is that blending flexibility may be reduced, and more economical blends may be missed.

A more flexible and easier-to-use method can be employed by linear approximation with a spreadsheet. This procedure uses multiple linear regression techniques to predict TOX, NO_x, and VOCs based on measurable inputs to the complex model.

The author uses three cases to demonstrate the abilities of this method:

1. Case 1: Linear fit over an RFG blend with methyl tertiary butyl ether (MTBE)
2. Case 2: Linear fit over a broad range of input values
3. Case 3: Linear fit with practical limits applied to variables.

Blender's challenge

Since Jan. 1, 1998, refiners who sell gasoline in nonattainment areas of the U.S. have been obliged to meet RFG emission quality standards based upon the EPA complex model, Phase I. Today, refiners are preparing for more-stringent Phase II requirements, scheduled to take effect on Jan. 1, 2000.

To meet these requirements, every refiner who blends RFG and conventional gasoline will find that a varying number of quality specifications-including those mandated by the EPA-are critical. A critical specification is one which is at its maximum or minimum limit-for example, at minimum octane or maximum Rvp limits.

Depending on the availability of blend components, TOX, NO_x, and VOCs may also be critical specifications.

Typically, two or three quality specifications will be limiting while the other specifications will not be at their limits. This situation indicates a condition of quality giveaway.

Optimizing techniques help the refiner prepare blend recipes. They tell the refiner which qualities should and can be blended to their limits-that is, to the qualities that are the critical specifications. Indeed, the critical specifications for one grade of gasoline may not be the same critical specifications for another grade of gasoline. And as component volumes and qualities available to the blender change, in addition to the change in demand for the products, the critical specifications may also change.

Ideally, the blender attempts to develop recipes for each grade of gasoline using the lowest-cost blending component to meet all specifications and to minimize giveaway for the critical specifications. To help develop these recipes, the blender may use a spreadsheet tool, such as Microsoft Excel and its solver routines, or an LP tool that calculates optimal blend recipes. A tool which also considers demand of multiple products over time is another choice.

Laboratory-measurement error

Table 1 [30,868 bytes] shows the average qualities of an RFG blend using MTBE as the oxygenate. The laboratory-measurement error of each quality is also listed. Laboratory-measurement error will vary by laboratory, test method, condition of equipment, and laboratory operator.

By using these values and assuming that the measurement errors are distributed normally (a reasonable assumption), a set of laboratory readings (100, in this case) can be generated, simulating 100 measurements of a blend sample. Based on the 100 simulated measurements, the average and standard deviation for each quality, as well as the high and low values can be calculated (Table 2 [43,537 bytes]).

Inputting the 100 sample quality readings for the blend into the complex model generates values for TOX, NO_x, and VOC. Table 3 [57,501 bytes] shows the average and standard deviations of these values.

In this case, the complex model equations for Phase I, summer, and Area Class B were used. Even though NO_x is not a critical quality for summer gasoline, it is used here for measuring the accuracy of this linear predictive method.

The 100 sets of inputs and corresponding outputs using the complex model can then be used to generate multiple linear regression equations for each of the three emission qualities. In this example, Microsoft Excel's statistical package generated the coefficients (a, b, ..., z, aa) for the measured variables. The following equations illustrate the form of the linear equations developed to approximate the complex models:

1. TOX = a + b*Oxygen + c*Sulfur + d*Rvp + e*E200 + f*E300 + g*Aromatics + h*Olefins + i*Benzene

2. NO_x = j + k*Oxygen + l*Sulfur + m* Rvp + n*E200 + o*E300 + p*Aromatics + q*Olefins + r*Benzene

3. VOC = s + t*Oxygen + u*Sulfur + v* Rvp + w*E200 + x*E300 + y*Aromatics + z*Olefins + aa*Benzene

The resulting average and standard deviation for each quality, based on the linear regression equations, are shown in Table 3. Table 3 also calculates the difference between the EPA complex model values and that for the regression equations.

Predictions from the regression equations are unbiased, the average errors are all zero, and the standard deviations are nearly identical to those resulting from the complex model. The standard deviations of the differences, a measure of the accuracy of the regression equations, are extremely small when compared to the deviations as a result of laboratory-measurement error. Table 4 [27,123 bytes] uses the data in Table 3 to show this.

Broad input range

What if the complex model input qualities vary over a broad range of values? How accurate will the resulting multilinear regression equations be?

In Table 5 [29,202 bytes], the acceptable range of values (minimum and maximum) for each input quality is shown. Using a uniform distribution, qualities for 100 simulated blends were randomly generated. Table 6 [46,632 bytes] shows the averages, standard deviations, and minimum and maximum values for the 100 simulated blends.

The range of qualities in a real blending situation will not likely see this sort of variability for a grade of gasoline. Each of these simulated blends was then input to the complex model to generate values for TOX, NO_x, and VOC (Table 7 [45,237 bytes]).

In this case, the values correspond to the complex model settings of Phase II, winter, Area Class C.

Table 8 [27,231 bytes] summarizes the key results of the accuracy of linear regression equations. Even under extreme conditions in which the measurable blend qualiites vary over broad ranges, the predictive errors are still within 20% of the error as a result of variability of the complex model.

More importantly, the predictive errors are only about 1% of the values being predicted.

Practical limits

Each refiner producing RFG will, of course, have practical limits on the qualities affecting TOX, NO _x, and VOCs as a result of his/her crude slate, processing capabilities, and blending flexibility.

These limits can be used for each quality to generate simulated blends that will represent the range of blends to be expected in actual operations. By using the multiple linear regression technique on these blends, the refiner can develop very accurate predictor equations to support blend decisions.

To illustrate, the data in Table 9 [45,931 bytes] are associated with a refiner in Area Class C blending RFG with ethanol (EtOH) during the winter season. Note that the averages and range of values for input qualities to the complex model do not vary as widely as in Case 2.

Table 10 [56,601 bytes] shows the resulting complex-model output values. For winter seasonal blends, the VOC values are not relevant, but are included here for analyzing the accuracy of this predictive method.

The results of the regression analysis and predictive error are shown in Table 10. This table shows that the method is most accurate for TOX and NO_x calculations. VOC is not as accurate because, for winter blends, many of the values generated by the complex model are out of range and therefore extrapolated. n

The Author