Modeling and Control

I thought of naming this series “Truth or Consequences” or the “The End of the Innocence.”

Level loops look so simple, they must be easy to tune. Also, what is so important about level? Don’t you just need to keep the vessel from running dry or overflowing?

Did you know that fast surge or feed tank level control is the most common source of variability in a plant? Did you know that slow distillate receiver or recycle tank level control frequently undermines the performance of important unit operations? Did you know that all of these loops are probably oscillating?

You might not know you can have too low of a controller gain for a level loop and that if you decrease the controller gain, the oscillations get worse. The common quick fix for getting rid of slow rolling level oscillations is to increase the controller gain for distillation columns and recycle tanks and increase the reset time for surge and feed tanks by an order of magnitude (10x) or more. If you want to know why or want to do a better job of tuning, read on but be aware you might experience “The End of the Innocence” or at least the inspirational influence of this album by Don Henley that I am listening to.

There is a window of allowable controller gains for loops that do not line out (reach a steady state) when the controller is in manual. These processes are called non-self-regulating. The most common example is level. If you put a level loop in manual, it will eventually if not immediately ramp away. Level is an integrating process, where the process output (level) is the integral of the process inputs (inlet and outlet flows). The change in % level ramp rate per % change in controller output is the integrating process gain. For 99% of the applications, the integrating gain is low (e.g. < 0.01% PV per sec per % delta OUT). For big tanks (e.g. storage tanks), and some horizontal tanks (e.g. distillate receivers), the ramp rate is very slow and consequently the integrating process gain is incredibly small (e.g. < 0.0001% PV per sec per % delta OUT). Since the controller gain is inversely proportional to this integrating process gain, it is the rare bird that has a level loop has a controller gain that is too high. You would be hard pressed to make a level loop go unstable due to too high of controller gain. You would be way beyond your comfort level (e.g. controller gains > 10), the controller output would be spiking due to measurement noise, and if you ever made a set point change, the controller output would most likely step to its output limit.

This brings to mind an example of how far below the upper controller gain stability limit we operate. Around 1980, analog controllers on a distillation column were replaced with DCS controllers on a test basis to see if there was any benefit to this new technology. The analog level controller setting of 100% proportional band was used in the DCS level controller. Fortunately, the configuration person didn’t know how to convert from proportional band to controller gain. The DCS level controller gain was set to 100 on the distillate receiver and the column performed better than ever due to a tight enforcement of the column’s material balance. The DCS was deemed truly wonderful technology and the rest is history.

Many distillation columns show slow rolling oscillations that are not readily evident unless you look over several shifts. These oscillations are always damped but never die out because the chance of a disturbance once per day even if it is just the day to night temperature change is high.

The product of the controller gain and reset time must be greater than 4 divided by the integrating process gain. Thus, you can increase the controller gain or the reset time to get rid of the slow rolling oscillations. For surge and feed tanks, the level just needs to be kept within the alarm limits and sudden changes in the manipulated flow show up as disruptive changes in feed to important unit operations (e.g. columns, crystallizers, evaporators, extruders, and reactors). For these applications, the reset time should be increased in most cases. For recycle tanks where the change in make-up reactant feed manipulated by a level controller matches the change in recycle reactant flow, tighter level controller generally helps maintain the correct material balance of reactant in the recycle tank. Here the level controller gain should be increased to get rid of the slow oscillations. This is also the case for column distillate receivers. For fed-batch reactors where one of the byproducts is an off-gas, tight level control by manipulation of reactant feed helps match the reactant feed to the reaction rate. This brings to mind that many reactor pressure loops have an integrating response and that tight pressure control by manipulation of a gaseous reactant feed helps keep the reactant concentration and hence the reaction rate constant. Many furnaces and incinerators have an integrating response but here the integrating process gain can be quite high since the allowable pressure range is in inches of water column. In these applications it is possible to encounter a controller gain that is too high in terms of instability or excessive amplification of noise.

There are many types of integrating loops. Nearly every fed-batch reactor has a non-self regulating response for concentration and temperature control. There are even the extreme examples of runaway reactors whose divergence may start out as a ramp but then accelerates. The tuning of these reactor temperature controllers and the point of no return is the subject for next week.

“People don’t run out of dreams, they just run out of time” in “River of Dreams” by Glen Frey. I have this dream that process design and configuration engineers and users and suppliers understand and communicate about controller tuning and loop performance, but in this hectic work place with expertise attrition, we are all running out of time.

The heat of reaction can raise the temperature and hence the reaction rate of exothermic reactions so fast the temperature accelerates upscale. There is a point of no return reached where the cooling rate is not enough to keep the temperature from rising. If the relief valves and flare stack are sized properly, the day is saved. Polymerization reactions are the most notorious runaways (particularly batch reactors where there is no self-regulation from a discharge stream).

The temperature controllers on runaways need rate action to react to acceleration and compensate for thermal lags in the heat transfer surface and the temperature sensor. Reset action is dangerous because it has no sense of direction. For this reason, some batch polymerization reactors have proportional plus rate controllers (no integral action).

These controllers cannot be put in manual for the type of step testing normally done for loops. The relay auto tuner oscillation may work if the step size is large enough to force the temperature back when it starts to accelerate. The tuning method that generally works best is where the controller is kept in auto, the reset time is increased by a factor of 100, the controller gain is increased and small set point changes are made until the control is fast enough or there is the start of a small oscillation. The controller gain is then halved and the rate time is set equal to the total delay time between a set point change and the start of the change in temperature. If any reset time is used, it is set at least 10 times larger than the rate time. This procedure, which involves approaching but not getting too close to the ultimate gain, is just a general guide and exceptions are to be expected. Tests and new tuning settings must be closely monitored.

It is critical to realize that there is a window of allowable gains where too low besides too high of a controller gain causes instability and loss of control. If the total loop dead time or the thermal lags exceed the positive feedback time constant of the runaway, the window of allowable controller bands is closed and there are no tuning settings that will stabilize the reactor. Slide 14 in the attached lecture I used at Washington University, provides an equation for the ultimate period that details the problem. The lecture introduces frequency response and how this leads to the equations for the ultimate gain and period for self-regulating, integrating, and runaway processes.
WU ChE462 Lecture on Ultimate Gains and Periods

If the pressure set point is a fraction of an inch of water column, you have a high integrating process gain. The response is often a high speed ramp in the control region. For a waste incinerator and a phosphorous furnace I worked on decades ago, the pressure could ramp off scale in 0.2 and 5 seconds, respectively. Trying to control the incinerator and furnace pressure was reported to be like trying to control an explosion when there was a shutdown or a slag slide, respectively, and a corresponding burst of vapors and gases. Needless to say these pressure loops could never go to manual and open loop tuning methods were down right dangerous. In the old days I used a modified ultimate oscillation method and a high speed recorder. As with runaway reactors, the reset time (e.g. sec/repeat) was increased by 100x to make reset action negligible and the controller gain was increased until there was the start of an oscillation. The reset time was set equal to the period of the damped oscillation and the controller gain was halved. A set point change was then made and if the response was more oscillatory than dictated by valve limit cycles from stick-slip or deadband, the controller gain was decreased. If damped oscillations persisted and got worse or slower, then the reset time was increased until the oscillations period and decay rate were faster. This test was repeated and the gain decreased or the reset time increased until the response was sufficiently smooth.

Before we go further, one should realize that the original ultimate oscillation method asked the user to increase the controller gain until there were equal amplitude oscillations. This was too exciting and gave controller tuning settings that were too oscillatory especially if there was an increase in the loop dead time or process gain or a decrease in the process time constant. The damped oscillations mentioned here are rapidly decaying where each succeeding peak is less than ¼ the previous peak.

The damped oscillation period is larger than the ultimate oscillation period and the damped oscillation controller gain is smaller than the ultimate oscillation gain and the factors of 1.0 for period and 0.5 for controller gain are not per the textbook definitions of the Ziegler-Nichols ultimate oscillation method. Using the text version of the closed loop (ultimate oscillation) or open loop (reaction curve) Ziegler-Nichols tuning method and thinking that tuning settings with more than one significant digit are practical, is a great way to reject the pioneering work of Ziegler and Nichols and to glorify new tuning methods. What I found early in my career is a simple change of using damped oscillations instead of ultimate oscillations and using easy to remember rounded off factors, gave me the proportional mode action needed for these loops that lack self-regulation and can be headed for a trip point. I also quickly realized that the nonlinear and non-stationary nature of chemical processes and valve stick-slip and backlash meant that the long term tuning setting accuracy of better than 50% was wishful thinking.

Today, integrated online adaptive tuning tools that look at set point changes, such as DeltaV Insight, should be able to automatically identify tuning settings of most fast integrating processes. However, some pressures can be so fast (e.g. the cited incinerator) digital delays must be eliminated and tuning tools that directly connect to the I/O, such as those used by EnTech, are needed. It is important that the module execution time, the tuning tool, and the trend chart update time not cause aliasing or an extra observed dead time. The controller, final element, and pressure sensor must also be extremely fast. Finally, it is particularly critical to test and observe new tuning settings for these and other types of loops that require aggressive feedback control.

If you want to get more details on the importance of making the loop fast enough, check out the chapter “Pressure Control: Without Deadtime I Might be Out of a Job” in the free E-book A Funny Thing Happened on the Way to the Control Room on pages 31-41:
http://www.easydeltav.com/controlinsights/FunnyThing/default.asp

If you are interested in how sample time delay affects PID tuning and performance, you might want to check out Advanced Application Note 005. AdvancedApplicationNote005 This note provides quantitative guidelines as to what point additional sample time delay causes a noticeable deterioration in performance and excessive oscillations for a given controller tuning. Alternatively, the note predicts new tuning settings for smooth control. Not emphasized in the note is that while the control may be smooth after retuning a PID for an additional delay time, the integrated absolute error (IAE) per Equation 3-1 in the note will be accordingly larger. Also, if the controller is tuned for maximum load rejection, the peak and integrated errors are proportional to the total loop dead time and dead time squared, respectively per references 2 through 4 in the note. The total loop dead time is the summation of all time delays in the loop including the control module execution time.

This leads to some rules of thumb that the additional delay from sampling should be less than 1/10 of the total loop dead time and process time constant. For more normal tuning, the sample delay should be less than 1/5 of the loop dead time and process time constant. For liquid pressure control, most of the loop dead time is the module execution time. The measurement sample time should be less than 0.04 seconds for a module execution time of 0.2 seconds. For flow control, the controller tuning can often be slowed down (Lambda increased to 4 or more) to accommodate the additional delay and sample times of several seconds are OK. For most level loops, the integrating process gain is so small (Lambda is so large) that measurement sample delays are inconsequential. For furnace and incinerator pressure control with set points in inches of water column, the response is so fast - the integrating process gains are so large (Lambda is so small), that the sample time requirement is similar to liquid pressure control. For gas pressure control of large header or vessel volumes with set points in psi, the allowable sample time can be substantially larger (e.g. seconds) because the process time constant or integrating process gain is usually slow. For temperature and composition control of vessels, quite a large sample time (e.g. minutes) is tolerated because the process time constant or integrating process gain are so slow. However, for runaway reactors, sample times approaching the runaway time constant are disastrous. For bioreactor and distillation column composition control, sample times in hours may be acceptable.