Exceptional Opportunities in Process Control – Peak and Integrated Errors – Part 1

If you increase the controller gain by the same factor that you increase reset time (e.g. double the gain and the reset time), how does it affect key performance indicators such as quality, yield, on-stream time, and environmental costs? If you make the valve and measurement faster, how does it affect these same KPI? If you want to improve a KPI, what is the priority of solutions?

The equations for the peak (Ex) and integrated error (Ei) in terms of controller settings, shown on slide 1 of Effects-of-Loop-Tuning-and-Dynamics-on-KPI.pdf, provide an answer to many of these questions if you embrace your inner geekness as advocated in the Control Talk Jan 2010 issue “The Future is Now”

Both equations were derived in Appendix A and B of Tuning and Control Loop Performance (scheduled to be back in print by Momentum Press, 2010). The derivation of the equation for the integrated error was included in Appendix C of New Directions in Bioprocess Measurement and Control (ISA, 2007) along with a unification of controller tuning rules. This unification, which showed how all the major tuning rules give basically the same result for a controller gain to minimize peak error, was personally satisfying but possibly not for people who are adamant about the relative merits of personal favorite tuning rules.

Since the integrated error is inversely proportional to the controller gain and proportional to the reset time, doubling the controller gain and reset time cancel each other out. However, doubling the controller gain halves the peak error since reset time doesn’t appear in the equation of the peak error. Reset time has an effect on peak error but it is negligible unless the reset time is decreased to the point where it approaches the loop deadtime. This can happen for deadtime dominant systems, but the peak error here is basically the open loop (error with the controller in manual) as evident from the equations on slide 2 of Effects-of-Loop-Tuning-and-Dynamics-on-KPI.pdf.

Nearly all the process control literature focuses on integrated absolute error (IAE) as the measure of loop performance. The IAE is a good measure of product that is off-spec that can lead to reduce yield and the raw material or recycle processing to product cost ratio (euros per kg and dollars per lb). If the off-spec cannot be recycled or the feed rate cannot not be increased to compensate, there is also a loss in production rate. If the off-spec is not recoverable, there is an additional waste treatment cost.

What we usually don’t take into account is the filtering effect of back mixed volumes as indicated by the equation on slide 3 of Effects-of-Loop-Tuning-and-Dynamics-on-KPI.pdf . For chemical and pharmaceutical plants and refineries, there are large volmes that provide significant attenuation of oscillations. However, in other process industries, various pathways of variability do not have significant filtering and culminate in the final product. These processes are also more vulnerable to interactions because there is no smoothing of effect of one loop’s control valve movement on another loop’s process variable. This changes the whole view on how you tune controllers. For systems with little back-mixing, controllers are tuned to limit the transfer of variability from the controlled variable (controller PV) to the manipulated variable (controller output) to prevent interactions and to provide a smooth response. The controllers are also tuned for coordination by enforcing a closed loop time constant (Lambda). For pulp and paper plants, nearly all of the variability expressed by the IAE ends up in the sheet since most of the processing is done in pipes and inline or unagitated equipment. Lambda tuning has been exceptionally successful in optimizing the transfer of variability and the coordination of loops. The same requirements could occur for plastics and textiles, since the IAE in the polymer lines and extruders shows up in the yarns and webs. However, these plants may have extensive blend tanks that average out the plus and minus fluctuations in product quality.

I ran into a process control improvement (PCI) study, where after an hour of discussion and investigation it became obvious a reduction in the considerable variability observed in each textile line had no value because the product coming out of the huge blend tank was always in spec and the variable speed pumps were maxed out. My decision to move on to better opportunities was not well received, so we stayed for 2 days to confirm there were no PCI opportunities (reducing the size or inventory in the existing tank or replacing the pumps were considered accounting or process design improvements).

When loops are oscillating across the split range point (common case due to valve stick-slip and installed valve characteristics), there can be a cross neutralization of acids and bases or a cross compensation of hot and cold heat transfer fluids that increases reagent and energy costs. Here the IAE is important but an integration of individual reagent and heat transfer fluids is a better indication.

If there are appreciable back mixed volumes whose residence time is much larger than the control loop period, the integrated error (Ei) where the plus and minus errors cancel out for a disturbance can be a better indication of the effect on product quality. Taking into account that the integrated error is also the IAE for an over-damped or critically damped response, we realize the simplification of the relationship of off-spec to an integrated error offers considerable understanding as to the effect of tuning settings.

This topic will roam on for 4 parts. In part 2, I discuss the effect of the peak error on onstream time and environmental costs. In part 3, I cover how measurement and valve dynamics impacts both types of errors and hence KPI. In part 4, I conclude with some rules of thumb on the priority of PCI solutions for various scenarios.