By Greg McMillan, Sridhar Dasani and Dr. Prakash Jagadeesan
The tuning of level controllers can be challenging because of the extreme variation in the process dynamics and tuning settings. Control systems studies have shown that the most frequent root cause of unacceptable variability in the process is a poorly tuned level controller. The most common tuning mistake is a reset time (integral time) and gain setting that are more than an order of magnitude too small.
In this article we first provide a fundamental understanding of how the speed and type of level responses varies with volume geometry, fluid density, level measurement span and flow measurement span for the general case of a vessel and the more specific case of a conical tank. Next we clarify how tuning settings change with level dynamics and loop objectives. Finally, we investigate the use of an adaptive controller for the conical tank in a university lab and discuss the opportunities for all types of level applications.
General Dynamics for Vessel Level
There have been a lot of good articles on level control dynamics and tuning requirements. However, there often are details missing on the effect of equipment design, process conditions, transmitter calibration and valve sizing that are important in the analysis and understanding. Here we offer a more complete view with derivations in Appendix A, available on the ControlGlobal website (www.controlglobal.com/1002_LevelAppA.html).
Frequently, the flows are pumped out of a vessel. If we consider the changes in the static head at the pump suction to have a negligible effect on pump flow, the discharge flows are independent of level. A higher level does not force out more flow, and a lower level does not force out less flow. There is no process self-regulation, and the process has an integrating response. There is no steady state. Any unbalance in flows in and out causes the level to ramp. When the totals of the flows in and out are equal, the ramp stops. For a setpoint change, the manipulated flow must drive past the balance point for the level to reach the new setpoint. If we are manipulating the feed flow to the volume, the feed flow must be driven lower than the exit flow for a decrease in setpoint. The ramp rate can vary by six orders of magnitude from extremely slow rates (0.000001%/sec) to exceptionally fast rates (1%/sec). The ramp rate of level in percent per second for a 1% change in flow is the integrating process gain (%/sec/% = 1/sec). The integrating process gain (Ki) for this general case of level control, as derived in Appendix A, is:
Since the PID algorithm in nearly all industrial control systems works on input and output signals in percent, the tuning settings depend upon maximums. The flow maximum (Fmax) and level maximum (Lmax) in Equation 1 must be in consistent engineering units (e.g. meters for level and kg/sec for flow). The maximums are the measurement spans for level and flow ranges that start at zero. Most of the published information on process gains does not take into account the effect of measurement scales and valve capacities. The equation for the integrating process gain assumes that there is a linear relationship between the controller output and feed flow that can be achieved by a cascade of level to flow control or a linear installed flow characteristic. If the controller output goes directly to position a nonlinear valve, the equation should be multiplied by the slope at the operating point on the installed characteristic plotted as percent maximum capacity (Fmax) versus percent stroke.
Normally, the denominator of the integrating process gain that is the product of the density (ρ), cross-sectional area (A) and level span (mass holdup in the control range) is so large compared to the flow rate that the rate of change of level is extremely slow. For horizontal tanks or drums and spheres, the cross-sectional area varies with level. In these vessels, the integrating process gain is lowest at the midpoint (e.g. 50% level) and highest at the operating constraints (e.g. low- and high-level alarm and trip points).
Most people in process automation realize that a controller gain increased beyond the point at which oscillations start can cause less decay (less damping) of the oscillation amplitude. If the controller gain is further increased, the oscillations will grow in amplitude (the loop becomes unstable). Consequently, an oscillatory response is addressed by decreasing the controller gain. What most don’t realize is that the opposite correction is more likely needed for integrating processes. Most level loops are tuned with a gain below a lower gain limit. We are familiar with the upper gain limit that causes relatively fast oscillations growing in amplitude. We are not so cognizant of the oscillations with a slow period and slow decay caused by too low of a controller gain. The period and decay gets slower as the controller gain is decreased. In other words, if the user sees these oscillations and thinks they are due to too high a controller gain, he or she may decrease the controller gain, making the oscillations worse (more persistent). In the section on controller tuning, we will see that the product of the controller gain and reset time must be greater than a limit determined by the process gain to prevent these slow oscillations.