Inherent Dynamic Problems with On-Demand Control Structures

Ind. Eng. Chem. Res. 1999, 38, 2315-2329

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PROCESS DESIGN AND CONTROL Inherent Dynamic Problems with On-Demand Control Structures William L. Luyben† Department of Chemical Engineering, Lehigh University, Iacocca Hall, Bethlehem, Pennsylvania 18015

There are two basic control structures for chemical plants: fixed feeds and fixed products. In the former, liquid levels are controlled by manipulating flows leaving individual units down through a series of reactors, tanks, distillation columns, etc. In the latter, which is called “ondemand” control, the product flow rate leaving the last unit is fixed (by a downstream user), and liquid levels are controlled by manipulating streams fed to each individual unit sequentially from its upstream unit. Buckley called these structures “material balance in the direction of flow” and “material balance in the reverse direction of flow”. This paper points out that the on-demand structure has several inherent dynamic disadvantages compared to the more conventional approach of setting the feed streams to a process. The most important disadvantage is that level control loops typically have more dynamic lags with this structure, which makes level controller tuning more difficult and degrades dynamic performance. A second problem is that this structure can produce interaction between level and composition loops that is not present in the conventional structure. Another disadvantage is that the on-demand structure tends to produce larger changes in process time constants as throughput changes, making the plant more nonlinear. In addition, the propagation of disturbances in the on-demand control structure is more complex than that in the conventional control structure because flow disturbances move upstream while composition disturbances move both upstream and downstream. Several of these features are shown mathematically using a simple stripper process. Then they are illustrated on two processes of increasing complexity. The first is a binary system with the reaction A f B and a plant topology of one reactor, one stripping column, and one recycle stream. The second is a ternary system with the reaction, A + B f C and a flowsheet containing one reactor, two distillation columns, and two recycle streams. Dynamic simulations demonstrate that the ondemand structure introduces larger disturbances into the system, which results in more variability in product quality. 1. Introduction The control structure used on a chemical plant has a profound impact on the performance of the plant. Structure is much more important than the choice of the control algorithm (P, PI, MPC, etc.) or the choice of tuning method (Ziegler-Nichols, Cohen-Coon, etc.). Control structure is second only to plant design in importance for effective dynamic control. Buckley1 discussed two basic structures for laying out the “material balance” control system (the choice of manipulated variables to hold liquid levels and gas pressures in process units). By far the most common structure uses material balance in the direction of flow. By this we mean that the liquid level or pressure in an individual unit is controlled by manipulating a stream leaving that unit. This stream is then fed to a downstream unit as a feed stream. There is a cascade of units in series, with the flow rates changing sequentially as changes in throughput or compositions occur. Disturbances propagate downstream from unit to unit. Of course, the presence of material or energy recycle streams can make disturbances propagate upstream † Telephone: 610-758-4256. Fax: 610-758-5057. E-mail: [email protected].

also and cycle through the process. We will use the term “conventional structure” for this scheme. The alternative “on-demand” control structure uses the opposite approach. The product flow rate from the last unit is fixed. The liquid level or pressure in an individual unit is controlled by manipulating a stream feeding that unit. Thus, flow-rate changes propagate upstream sequentially from the last unit to the first. The popularity of the on-demand structure has increased in recent years because of the interest in “as needed” production to reduce intermediate inventories and in “agile” manufacturing to be responsive to customer demands. Price et al.2 studied several other alternatives in which production is set at an intermediate unit in the process and material balance control propagates both upstream and downstream from this unit. Luyben et al.3 give both conventional and on-demand control structures for the Eastman process and for the vinyl acetate process. In this paper we compare these two alternative control structures quantitatively by considering both simple and complex processes. Mathematical analysis of a simple stripping column is used to reveal the underlining differences between the conventional and

10.1021/ie990044k CCC: $18.00 © 1999 American Chemical Society Published on Web 05/12/1999

2316 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 Table 1. Ultimate Controller Gains for On-Demand Level Control

Figure 1. Conventional and on-demand level control structures.

the on-demand control structures. We use the terminology that CS1 is the conventional control structure (level control in the direction of flow) and CS2 is the ondemand control structure.

ultimate controller gain Ku

10 15 20 25

3.90 2.50 1.82 1.42

structure controls base holdup by manipulating the bottoms flow rate; the feed flow rate is the load or disturbance variable and is set by the upstream unit. The on-demand control structure controls base holdup by manipulating the feed flow rate to the stripper. Now the bottoms flow rate is the load or disturbance variable and is set by the downstream unit or consumer. It is clear from this simple example that the dynamics of the two level control structures are very different. In the conventional structure, the open-loop transfer function relating the controlled variable (MB) and the manipulated variable (B) is a simple integrator.

GM(s) )

2. Mathematical Analysis 2.1. Hydraulic Lag Effects. Figure 1 shows a simple stripping column with feed flow rate F (mol/s) and bottoms flow rate B (mol/s). The column has N trays, and each tray has a liquid hydraulic time constant of τH (s). Assuming equimolal overflow and counting trays from the bottom of the column, the transfer function relating the flow rate of liquid leaving the nth tray Ln to the flow rate of liquid entering the tray Ln+1 is

Ln 1 ) Ln+1 τHs + 1

no. of trays N

(1)

The liquid holdup in the base of the stripper is MB (mol) and must be controlled. The conventional control

Figure 2. Conventional and on-demand level control performance.

MB 1 ) B s

(2)

In the on-demand structure, the open-loop transfer function relating the controlled variable (MB) and the manipulated variable (F) is an integrator in series with N first-order lags.

GM(s) )

MB 1 ) F s(τHs + 1)N

(3)

It is obvious that the closed-loop performances of these two systems are different. We know from basic control theory that the presence of additional lags inside a feedback control loop degrades performance, so the

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2317

on-demand structure is expected to be inherently inferior to the conventional structure in this type of process. The stripping column example is chosen because its topology is typical of many chemical processes. Fresh feeds enter a reactor, and the reactor effluent is fed into a staged separation unit from which products are removed. The separation unit is frequently a distillation column, and it is often fed a liquid stream. If the product is the bottoms from the column, we have a system similar to that shown in Figure 1. The situation is different if the product from the column is the distillate stream instead of the bottoms. It is also different if the feed to the column is vapor instead of liquid. These cases are discussed later in this paper. Hydraulic time constants in the distillation column are normally about 3-6 s, so a 20-tray section can introduce a 1-2 min delay in the response of base level to a change in the liquid fed to the top. The more trays, the larger the delay and the poorer the dynamic performance of the on-demand structure is expected to be. To analyze the closed-loop system mathematically, let us assume that proportional-only level controllers are used.

GC ) Kc

(4)

With the conventional control structure, the closedloop characteristic equation is

1 + GMGC ) 1 + Kc/s

(5)

Controller tuning is trivial in this first-order system. The controller gain is selected (typically Kc ) 2) to keep the level between some maximum and minimum limits. The steady-state offset in the level can be reduced by increasing controller gain but at the expense of faster changes in exit flow rates. See Figure 2. There is, in theory, no ultimate gain, and the step response is always monotonic with no overshoot or oscillation. With the on-demand control structure, the closed-loop characteristic equation is

1 + GMGC ) 1 +

Kc

(6)

s(τHs + 1)N

Now the system is of order N + 1, and there is an ultimate gain beyond which the closed-loop system is unstable. As the gain approaches this value, the response becomes underdamped and overshoots in level and flow can occur. See Figure 2. Equation 6 can be used to solve for the ultimate gain by equating the arg(GMGC) to -180° and solving for the ultimate frequency ωu.

arg(GMGC) ) -π/2 - N arctan(ωuτH) ) -π ωu )

1 π tan τH 2N

( )

(7)

Then the ultimate gain Ku is found from eq 8, which gives the magnitude of GMGC.

|GMGC| )

1 ωu

(x

1

1 + (ωuτH)

)

N

2

)

1 Ku

(8)

Table 1 gives values of the ultimate gain for different numbers of trays. Figure 2 compares the step responses of the conventional and the on-demand control structures for a 10-tray stripper with a hydraulic time constant of 2.77 s. Three values of controller gain are used in the on-demand case: Ku/3, Ku/2, and Ku/1.5. These results illustrate the underdamped responses possible in the on-demand structure. Figure 3 shows how the dynamics of the on-demand structure are affected by changes in the number of trays, the hydraulic time constant, and the controller gain. In the time domain, the step responses of base holdup MB are shown for unit step changes in the bottoms flow rate. Also shown are the log modulus plots for the closed-loop regulator transfer function between the controlled variable and the load (MB/B). As the number of trays and the hydraulic time constant increase, level loop closedloop time constants increase (frequencies decrease) and the amount of overshoot increases (damping coefficient decreases and the closed-loop resonant peak increases). Notice in Figure 2 that the holdup MB increases at higher throughputs in the conventional structure, but the opposite is true for the on-demand structure. This means that the residence times in vessels in the conventional structure tend to not change much with throughput: larger throughput gives larger holdup. Therefore, composition dynamics will not change much. However, in the on-demand structure, larger throughputs give smaller holdups, which results in larger changes in composition dynamics. This results in a process that is more nonlinear and therefore more difficult to control. 2.2. Effect of PI Control. In the analysis above, proportional-only level control was assumed. The simple P algorithm is widely used because it achieves flow smoothing. However, it produces a steady-state offset in levels, which may be undesirable in a few systems. To illustrate that the results of the comparison between conventional and on-demand structures are not dependent on the controller algorithm employed, the use of P and PI controllers for the on-demand structure is briefly discussed below. Figure 4 compares P and PI control for a stripping column with 10 trays and a hydraulic time constant of 2.77 s. The ultimate gain of the level controller is Ku ) 3.90, and the ultimate period is Pu ) 1.82 min. The P and PI controllers are tuned by using the ZieglerNichols tuning rules: for P, Kc ) Ku/2, and for PI, Kc ) Ku/2.2 and τI ) Pu/1.2. In the P-only control case, a steady-state offset in holdup occurs. In the PI control case, the holdup is driven back to the setpoint value (zero perturbation from steady state). However, the flow rate of the manipulated variable (F) varies much more in the PI control case, as expected. Note that the maximum deviation in holdup is about the same in the two cases. This illustrates that using a PI algorithm does not permit the use of smaller holdups. We return to this issue later in this paper when a generic rule about on-demand structures is proposed. 2.3. Interaction of Level and Composition Loops. Figure 5 gives block diagrams for the open-loop and closed-loop systems when both base holdup and bottoms composition are controlled. We assume that a reboiler generates vapor boilup V, which is used to control product composition xB, using composition controller CC. At the same time, a second controller LC is used to control base holdup MB. The open-loop system is shown

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Figure 3. (A) On-demand responses, Kc ) Ku/3. (B) On-demand responses, Kc ) Ku/2.

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Figure 4. Closed-loop P and PI step response; on-demand control.

at the top of Figure 5. There are three inputs to the system: V, B, and F. Vapor boilup is one of the manipulated variables in this 2 × 2 system. The other manipulated variable is either B or F, depending on what control structure is used. The load disturbance is the remaining input variable, either F or B in the two structures. It is important to note that both F and V affect xB, but B does not. Changing the bottoms flow rate reduces the base level, but it does not affect what is going on up in the column, so compositions are not directly affected. The absence of a transfer function element between xB and B has an important impact on the issue of pairing variables. In the conventional structure CS1, the level is controlled by manipulating B. There is no interaction between the level controller and the composition controller because there is no path for changes in the level controller output to affect composition. Changes in the composition loop act as disturbances for the level loop because V does affect MB, but there is no feedback in the other direction. In the on-demand CS2 control structure, the two loops are fully interacting. Tuning of controllers cannot be done independently, as can be done in the conventional case. We give a quantitative demonstration of this problem in the next section. We now consider more realistic process examples and present quantitative comparisons between conventional control schemes and on-demand control schemes. The first process has two chemical components, one reactor, one column, and one recycle stream. The second process has three components, one reactor, two columns, and two recycle streams. 3. Binary Process 3.1. Process Description. Figure 6A gives a sketch of the flowsheet with the steady-state values of param-

eters and variables. This is a slight variation of the recycle process first studied by Papadourakis.4 Numerical parameter values are taken from Luyben.5 The system contains two chemical components: reactant A and product B. The isothermal, irreversible reaction A f B occurs in the liquid phase in a CSTR with holdup VR )2500 lb mol. The fresh feed stream, with flow rate F0 ) 239.5 (lb mol)/h and composition z0 ) 0.9 mole fraction A, is fed into the reactor. The composition in the reactor is z (mole fraction A), and the reaction rate is first-order ((lb mol)/h of A consumed).

R ) kzVR

(9)

The specific reaction rate is k ) 0.340 86 h-1, and the steady-state concentration in the reactor is z ) 0.25 mole fraction A. The flow rate of the reactor effluent is F ) 646.7 (lb mol)/h, and it is fed to the top tray of a 17-tray stripping column. The bottoms product leaving the column has a flow rate B ) 239.5 (lb mol)/h and a composition xB ) 0.0105 mole fraction A. Vapor boilup is produced in a partial reboiler, and overhead vapor is condensed and recycled back to the reactor at a flow rate D ) 407.2 (lb mol)/h and with a composition xD ) 0.3809 mole fraction A. 3.2. Control Structures. Parts A and B of Figure 6 give the conventional and on-demand control structures for this simple binary process. In both schemes, the overhead accumulator level is controlled by manipulating the recycle flow rate, using a proportional controller with Kc ) 2. Also in both schemes, bottoms purity is controlled by manipulating the vapor boilup (V). A 3-min deadtime is used in this composition loop. The relayfeedback test was used to get the ultimate gain and ultimate period, and the Tyreus-Luyben tuning constants are used in a PI controller (Kc ) 1.33 and τI ) 0.73 h). This test was done with the level controller on manual; i.e., both F and B are constant. These composition controller settings are used in both the CS1 and

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Figure 5. Block diagrams for CS1 and CS2.

CS2 control structures. We will return to this tuning question later in this section. Note that neither of the control structures shown in Figure 6A,B have flow

controllers somewhere in the recycle loop, so problems with “snowballing” could be experienced.3 Our purpose in this paper is not to explore the snowball problem but to compare on-demand control structures with conventional control structures, and the control schemes shown in Figure 6A,B permit this fair comparison. The span of the composition transmitter is 10 mol %, and the span of the vapor-boilup flow transmitter is twice the steady-state value. Liquid holdup in the column base is 53.9 lb mol, and that in the overhead accumulator is 33.9 lb mol. The tray holdup is 1.65 lb mol, and the tray liquid hydraulic time constant is 2.77 s. A proportional controller is used to control the reactor level with Kc ) 5 in both schemes but using different manipulated variables. In the conventional control structure, fresh feed is flow controlled, the reactor level sets the reactor effluent flow, and the base level sets the bottoms flow. A proportional controller is used to control the base level with Kc ) 2. In the on-demand control structure, the bottoms product is flow controlled, the base level sets the reactor effluent flow, and the reactor level sets the fresh feed flow. A proportional controller is used to control the base level. The value of gain was determined by using a relay-feedback test to get the ultimate gain (Ku ) 20) and setting the controller gain to one-third of the ultimate gain. Dynamic simulation results are shown in Figures 7-9. In Figure 7 the responses of the conventional control structure are shown for two disturbances: a +20% step change in the fresh feed flow rate (the curves labeled F0) and a step change in the fresh feed composition from 0.9 mole fraction A to 1 mole fraction A (the curves labeled z0). The increase in fresh feed works its way down through the system, with F, MB, and B all gradually increasing. The slow change in the feed to the stripper does not disturb the product composition loop much, so fairly tight control of xB is achieved. In Figure 8 the responses of the on-demand control structure are given for the same disturbance in the fresh feed composition but for a +20% step change in the bottoms product flow rate (the curves labeled B). The base level drops rapidly, which causes a rapid increase in feed to the column. This produces a large upset in the xB composition loop. Figure 9 gives a direct comparison of the two control structures for the two disturbances. The dynamic superiority of CS1 is clearly shown, particularly for throughput changes. It should be noted that the performance of the CS2 on-demand structure could be improved by reducing the gain in the base level loop, which would bring in feed to the stripper more gradually. However, this would require a larger base holdup. This appears to be a generic rule for the on-demand control structure: Larger surge volumes are needed in on-demand control structures. Note that the brief study of PI control in the previous section illustrated that this rule is not dependent on the use of P-only level control. 3.3. Loop Interaction with On-Demand Structure. The effect of holdup and the problem of level/ composition loop interaction in the CS2 control structure can be easily demonstrated in this process. The basecase holdup is 53.9 lb mol. In the results presented above, the tuning of the composition controller is the same in both CS1 and CS2. However, as Figure 5 clearly shows, the level loop impacts the composition loop in

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2321

Figure 6. (A) Binary process with conventional control. (B) Binary process with on-demand control.

the CS2 structure. Therefore, two cases are studied: the first with a base holdup of 53.9 lb mol and the second with a base holdup of 10 lb mol. These correspond to having holdup times of 5 and 0.9 min, respectively. The level loop is first tuned by setting the controller gain equal to one-third of the ultimate gain, which is determined from a relay-feedback test in each case. This ultimate gain changes as holdup changes, as shown in Table 2. As holdup decreases, the level controller gain decreases. Then, using the appropriate level controller gain, the level controller is put on automatic, and a relay-feedback test is run on the composition controller. Composition controller tuning is different from that previously obtained with the level controller on manual. Table 2 shows that for the 53.9 lb mol case the composition controller gain is larger and the reset time is smaller. The results for the 10 lb mol case are also given in Table 2. The smaller holdup results in smaller gains and reset time in the composition controller. Figure 10 compares the performance of CS1 and CS2 for the large and small holdup cases. The disturbance is a +20%

change in throughput, a step change either in F0 or in B at zero time. These results show that the dynamic performance of the CS2 on-demand control structure degrades as holdups decrease. The effect of holdup in the conventional CS1 structure is much less. 3.4. Changes in Residence Times. A comparison of Figures 7 and 8 shows that the base holdup increases as production rate increases in CS1, while base holdup decreases in CS2. This illustrates that residence times can change quite significantly in the on-demand control structure. For the specific numerical example, at the initial steady state the product flow rate is 239.5 (lb mol)/h and the base holdup is 53.9 lb mol, giving a residence time of 0.22 h. At the 20% higher production rate, base holdup in the CS1 structure increases by about 10% to 59 lb mol, giving a slightly smaller residence time of 0.20 h. However, holdup in the CS2 structure decreases by about 10% to 49 lb mol, giving a much smaller residence time of 0.17 h. In the next section a more realistically complex process is explored.

2322 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999

Figure 7. Binary process with CS1; +20% B and z0 ) 0.9 f 1.

Figure 8. Binary process with CS2; +20% F0 and z0 ) 0.9 f 1.

4. Ternary Process 4.1. Process Description. The process explored is the same as that studied in previous papers.6,7 Figure 11 gives the flowsheet with steady-state conditions. The two fresh feed streams of reactants A and B (F0A and F0B) are fed into the reactor. The isothermal irreversible reaction A + B f C takes place in a single stirred-tank reactor with holdup VR and concentrations of zA, zB, and zC. Reactor effluent, with flow rate F, is fed into the first distillation column. The relative volatilities are in the following order: RA > RC > RB (4/2/1). Two columns are needed to separate the two

reactants from the intermediate-boiling product. The indirect separation sequence (“heavy-out-first”) is used to be consistent with previous work. The heaviest component B is recycled from the bottom of the first column in stream B1. The lightest component A is recycled from the top of the second column in D2. Base and reflux drum holdups (MBn and MDn) in both columns are 100 lb mol. Both columns have 40 trays, are fed on tray 20, and have 3 lb mol holdup per tray and a hydraulic time constant of 6 s. The product impurity specifications are xB2(A)