Motor gasoline is blended at the refinery to specifications that include octane rating, typically grades of 87 and 93 octane; at the pump these may be further blended to produce an intermediate grade such as 89 octane. An engineer I met was especially proud of how well octane number was controlled at his refinery—but he always filled his car at a competitor's station, although the price per gallon was the same. His rationale was that the competitor's control over octane number was more variable, but still had to meet the minimum standard of 87 for regular grade—hence its average octane rating was higher by that variability. The competitor's refinery was giving away octane, and he was willing to accept it. The analysis below can be used to describe the cost of giving away any valuable ingredient, or any operating cost associated with the variability of control.
The cost to the refinery in excess octane giveaway is a direct function of its variability, with two possibilities shown in the step-load response curves of Fig. 1. The upset could either drive the controlled variable in the "safe"direction shown in the black curve, or the "unsafe"direction, shown in red. Here, "safe"is meant to infer that the controlled variable is driven toward the favorable side of the specification limit, in this case above the 87-octane specification. Safe-side variations do not require any action either by the operator or any automatic response, because product specifications are not thereby violated. (Although there is a slight overshoot of setpoint on the return trajectory and therefore a momentary violation, any downstream capacity at all would absorb this brief transient, and it would not appear in the final product.) Safe-side variations simply result in an economic loss that is quantifiable.
If the setpoint in Fig. 1 is also the specification limit, and the measured octane rating rises due to the upset before returning to setpoint, the octane giveaway is related to the integrated error of the controlled variable from setpoint. Integrating this deviation or error e between controlled octane and its setpoint over time t gives the Integrated Error IE in terms of octane-minutes. Subsequent multiplication by the current product flow rate in gal/min yields octane-gal given away during that interval. Multiplying that result by the cost difference in the dollars/gallons/octane number converts the integrated error into dollars lost.
The cost function may not be linear. For example, the price difference at the pump between 87 and 89 octane, and between 89 and 93 octane may both be $0.10 per gallon, which probably reflects market forces rather than production cost. Nevertheless, small variations in the function around a given specification can be assumed to be linear, allowing IE to be a useful criterion in evaluating the cost of poor control.
When using PI or PID control over any variable, integrated error is directly related to the size of the disturbance and the controller settings. The ideal PID controller is simply represented in Eq. (1):
where m is the variable manipulated by the controller, and P, I and D are its proportional band in percent and integral and derivative time settings respectively. This formula may be evaluated both before and after an upset, when the deviation e and its time derivative are both zero. If the controller output has changed between those two steady states, there will be an integrated error:
Equation (3) applies equally to interacting and non-interacting PID controllers and PI controllers as well. There are some obvious conclusions to be drawn from it: To minimize the cost associated with IE, minimize the feedback control effort Δm required to respond to an upset (by using feedforward), and reduce the P and I settings (without sacrificing stability).
When the controlled variable wanders to the unsafe side of the specification limit, decisions have to be made. In failing to meet requirements upon which the product is to be sold, it must be set aside for blending with better-than-acceptable product, rejected to a less-valuable use or rerun. All of these operations are costly and cause disruptions affecting production rate. Operators avoid them if at all possible.