Figure 2 (above). Loop interaction can be estimated from the slopes of the operating curves.
Figure 2 presents the choices available for a typical refinery column as determined by a four-component model. An example of a set of specifications appears in upper left corner. The feed is characterized as consisting of four components: light key (l) and lighter (ll), and heavy key (h) and heavier (hh). The split occurs between light and heavy key components; all the ll-key components exit overhead, and all the hh-key components leave the bottom. (There may be any number of actual components, but this model requires lumping them into these four groups.) Data representing desired operating conditions are entered into the blocks, and all the remaining values are then calculated by the model. The last item calculated is the relative volatility (α of the key components, which can be compared against its handbook value to test the validity of the model.
From this information, the operating curves are plotted, which cross at the specified product-composition coordinates. The curves relate the two controlled compositions in terms of key impurities: yh is the controlled heavy-key component in the distillate, and xl is the controlled light-key component in the bottoms. There are five independent MVs available to control the compositions, appearing left-to-right in the plot—all are treated as ratios by the model:
- Separation S, a function of the reflux/distillate (L/D) ratio
- Reflux L, actually reflux/feed (L/F) ratio
- Boilup V, actually boilup/feed (V/F) ratio
- Boilup/bottoms ratio R = V/B
- Distillate D, actually distillate/feed (D/F) ratio; bottoms B is dependent.
Each curve represents the x-y relationship when a selected MV is held constant while any other is varied. If two MVs produce the same curve, they are not independent—note that the L and V curves are almost identical.
Relative Gain Analysis
The magnitude of interaction between the composition (temperature) loops can be numerically estimated as the relative gain of any pair of MVs. RG is defined as the ratio of the open-loop gain with the other loop open to its value with the other loop closed. It can be expressed mathematically as the partial derivative of any controlled variable, ci, with respect to any manipulated variable, mj, with the other m held constant to that partial derivative evaluated with the other c held constant:
The RG λij is a dimensionless number with very definite properties. A value of 1.0 indicates no or only one-way interaction and is the best outcome. Positive values above 1.0 describe increasing levels of interaction caused by positive feedback through the other loop. Negative values indicate dominant positive feedback and must be avoided. Values approaching zero also indicate increasing interaction through negative feedback. Values in the range from 0.5-10 have the best prospects for control.
The relative gain values for a distillation column are readily calculated from the slopes of any two curves (1, 2) as they cross the operating point, where subscripts 1 and 2 represent the MVs selected to control y and x respectively:
The table in the lower left of Figure 2 gives RG values calculated from the operating curves for the example column. The three candidate MVs (my) for top composition control are S, L, and D, and the three (mx) for bottom composition are B, V, and R. Since D and B are mutually dependent, the relative gain for that combination is listed as ± ∞. As expected, the RG for the L-V combination is highest at 25.
A multivariable MPC is capable of decoupling if used in place of the TCs in Figure 1, instead of outside of them as shown. However, the higher the relative gain, the more sensitive the system is to error in the decoupler. For a structure having a relative gain of 25, a decoupler error (model mismatch) of as little as 2% can drive the decoupled relative gain from 1.0 to infinity (Shinskey, F. G., Process Control Systems, 4th ed., McGraw-Hill, New York, ’96, p. 269). In other words, a system having high RGs is not only difficult, but dangerous to decouple.
Fortunately, there is a way around the problem: restructuring the lower tier of control loops. There is no mandate that the product flows be under level control or that the temperature controllers manipulate reflux and boilup. Referring to Figure 2, we see that the RG can be dropped from 25 to 5.2 by manipulating reflux ratio instead of reflux flow to control top composition (temperature). A structure that accomplishes this is shown in Figure 3.
Figure 3 (above). Top temperature is now controlled by reflux ratio.
Reflux ratio is actually L/D, but holding any function of L/D constant will produce the S operating curve. In Figure 3, the top TC manipulates D/(L + D), which is 1/(1 + L/D). The level controller in the reflux drum manipulates the sum of reflux and distillate flows (L + D), and responds to a change in boilup V by moving them both at the same time while retaining their ratio intact.
Feed rate is used to set steam flow in ratio to it through a feed-forward loop. Following a change in feed rate, the dynamic compensator [t] lags and delays the flow signal, simulating the dynamics in the lower half of the column. Then the steam flow responds in ratio to the feed rate at the time the new feed rate reaches the column base. The resulting change in vapor flow travels up the column, is condensed, and raises the level in the reflux drum. The level controller then changes both L and D, maintaining the current ratio.