## Figure 162

Signal flow diagram for simple level control system

dQp d0c

proportional controller so that the tank level is midscale on the level transmitter. This leaves the top 25 percent of the transmitter span for overrides. Then:

### Proportional-Reset Control

Although the proportional-only control system is simple, inexpensive, and almost foolproof (at least when implemented with fixed-gain relays), it has limitations:

1. If tank size is specified, the flow smoothing is limited by th because one cannot safely use Kj, < 1.

2. Because mechanics often do not calibrate valves precisely, use of Ka, = 1 is risky; the valve may not be closed when the level is at zero. Consequendy we usually specify K^ = 2.

3. If tank size is not specified but is to be calculated from equation (16.9a) or (16.9c), a large tank will be required since for a given th, Vt is proportional to Kj,.

Although in theory there exists a number of controllers that permit one to use Ka, < 1, the PI controller has been most popular. Since the unenhanced PI controller with K& < 1 does not ensure that the tank will not run dry or overflow, past practice has been to have high and low alarms or high and low interlocks. In recent years, however, the PI controller enhanced or augmented with auto overrides has provided an almost foolproof way of keeping liquid within the vessel. It provides, under most circumstances, much more flow smoothing for a given size vessel than will a proportional-only controller.10

Before exploring the theory, let us make some additional design assumptions:

1. Nozzle-to-nozzle spacing is so chosen that process operation will be satisfactory with the level at any location between the nozzles. "Nozzles" here refers to those used for connecting the level-measuring device to the vessel.

2. Normal set point for the PI level controller is midscale of level transmitter span, that is, MiT/2. This is required for proper functioning of the auto overrides. Note that the level transmitter span is usually less than the nozzle-to-nozzle spacing. This allows for some variation in liquid specific gravity.

3. Level control is cascaded to flow control. With floating pressure columns and with the trend toward small control-valve pressure drops for energy conservation, this is virtually mandatory to counteract the effect of control valve up- and downstream pressure variations. The flow measurement must be linear; if an orifice flow meter is used, it must be followed by a square-root extractor.

For level control cascaded to flow control:

'max

A schematic for PI level control on a simple tank is given in Figure 16.3. As a result of a number of studies (unpublished), we have concluded that auto overrides with gain 2 and a controller tuned for a damping ratio of one are optimum for most situations. We have also found for most cases that the PI level control system so designed functions in a linear manner for step changes in input flow of up to 10 percent of span of the manipulated flow if K^ 3= 0.25. 0 This means that the controller output at its maximum value verges on being taken over by an auto override.

Other overrides are shown in both the level controller output signal path to the flow controller set point and in the flow controller output signal path to the control valve. The latter arrangement (overrides in the signal path to the control valve) has been far more common, but the former permits more accurate, quantitative design. The former also implies that whenever the flow control station is not switched to "remote auto," the overrides are out of service. This may or may not be desirable. But regardless of override location (other than auto overrides), we provide a switching design that causes the level controller reset to be bypassed (i.e., level controller has very fast reset) whenever the flow control station is not switched to "remote auto." This virtually eliminates "bumping" when the flow controller switches from either "manual" or "local auto" to "remote auto." A primary control station is not necessary, but a level indicator is desirable.

In the analysis that fallows we will ignore the role of overrides and will assume that the PI level controller is always "in command." This assumption permits us to use Laplace transforms and frequency response. If it is desired to predict system behavior when forced by disturbances large enough to cause an override to take over, we must resort to digital simulation.

The transfer function for a PI controller is:

where tr is the reset time in minutes. Substituting this into equation (16.5) and replacing dQ^/dQ, by 1 /K^-, we get:

where tr is the reset time in minutes. Substituting this into equation (16.5) and replacing dQ^/dQ, by 1 /K^-, we get:

TjiTsi2 + rfii+ 1

Similarly, from equation (16.6):

ThTR

Now the denominator of equations (16.12) and (16.13) has some interesting characteristics not widely appreciated. It is a quadratic whose damping ratio is:

Let us also define:

From equations (16.14) and (16.15) we can see that if tr is fixed, then as one decreases K^ two things happen:

1. The damping ratio, approaches zero and the control loop becomes very resonant, approaching instability.

### 2. tq becomes very large.

The loop therefore becomes slower and less stable at the same time. This resonance is sometimes called a "reset cycle" since it would not exist if the controller did not have automatic reset.

If, on the other hand, one increases K^ while holding tr constant, tq becomes small, transmitter and valve dynamics become significant, and the loop eventually becomes unstable. This is commonly called a "gain cycle" since it is caused by excess gain. Since the loop approaches instability for both very large and very small values of K^ we say that it is conditionally stable.

In designing a level control system with a proportional-reset controller several practical considerations must be kept in mind:

1. The damping ratio preferably should be at least unity. A low damping ratio, as shown by equations (16.12) and (16.13), causes severe peaking in the frequency response in the vicinity of the closed-loop natural frequency, 1 /tq. Flow and level regulation in that frequency range will be very poor.

2. Adjacent or related process controls must be designed with closed-loop natural frequencies much different from that of the level control; usually they are designed to be much faster.

3. For a given damping ratio, reset time must be increased as KA is decreased. It frequently happens that for a desired tq and specified K^, one cannot readily obtain the necessary tr with a particular commercial controller. The major instrument manufacturers can usually furnish modification kits or modified controllers with a larger tr.

For £ = 1 it can be seen from equation (16.14a) that:

One of the major disadvantages of PI controllers for liquid level is that they always cause the manipulated flow change temporarily to be greater than the disturbance flow change. For example, if the system we have been considering is subjected to a step change in inflow, we obtain the following:

The fact that the outflow swings more than the inflow can create serious problems if the process is running close to capacity. For such applications we should choose £ = 2.0 or use a proportional-only controller. For most other applications, £ = 1 should suffice and places less of a burden on available controller settings.

At this point it may be appropriate to note that a viable alternative to PI level control is PL level control.6'7 It has transfer functions very similar to those of the PI level control; for a damping ratio of unity, the transfer functions are identical if one reduces the PL level controller gain, by a factor of 2. For Kef, < 1, it requires auto overrides just as the PI controller does. It has the feature, useful in some circumstances, of not needing antireset windup. It is not a standard commercial item but usually can be assembled with various standard devices.

### Augmented PI Controllers

A plain PI controller, even if tuned for £ = 1, cannot guarantee that level will be held within the vessel. To protect upper and lower permissible level limits, we have found two approaches useful:

1. Auto overrides for pneumatics. When the level becomes too high or too low, a proportional-only controller (usually a fixed-gain relay) takes over through a high- or low-selector.2 As shown by Figure 16.3, the high-level gain 2 auto override is so biased that its output is 3.0 psig when the input (level transmitter signal) is 9.0 psig. Then its output is 15.0 psig when its input is 15 psig. Correspondingly the low-level gain 2 auto override is so biased that when the input is 9.0 psig, its output is 15 psig; if its input goes down to 3.0 psig (zero level), its output is also 3.0 psig.

2. Nonlinear PI controllers for electronics.4 A preferred version has long reset time and a small Kj, in the vicinity of the set point. As the level deviates significandy from set point, K& increases and tr decreases. Although in theory this is not quite as foolproof as auto overrides, our studies show that it rarely permits excessive deviations of level, and then only by a small amount. This design does not provide flow smoothing quite as good as that of the PI plus override scheme. This is so because manipulated flow changes more rapidly so that level moves away from the top or bottom of the tank more quickly.

### Effect of Installed Valve Flow Characteristic

If the PI level control system (or proportional-only) is not a cascade level-flow system, then it is desirable to have a control valve with a linear installed flow characteristic, that is, dQ,,/d\$c = constant. Then control-loop dynamics would be independent of flow rate and dQc OSJmax ft3/min mc=U--pü- <orsPm/Psl)

where (£)„)max is the flow through the valve in its wide-open position.

If the valve has an equal-percentage installed characteristic, then from Table 15.1, reference 1:

Since kEP = 3.9 for a 50:1 equal-percentage value (kEP = In a where a = 50):

dQo Q0

de0 12 3

Referring to equation (16.14), we see that if a controller is set up correctly with £ = 1 atjg = Qps, then:

a. £ = 0.7 when Q0 = (&s/2), and b. C = 1.4 when Q0 = 2

Thus we see that, with an equal-percentage installed flow characteristic, the relative stability is decreased at low flow and increased at high flow. For this case it is best to find controller settings for £ = 1 at the minimum expected flow.