Synthesis of Plantwide Control Systems Using Optimization - Industrial

2984

Ind. Eng. Chem. Res. 1999, 38, 2984-2994

Synthesis of Plantwide Control Systems Using Optimization Thomas J. McAvoy Institute for Systems Research, Department of Chemical Engineering, University of Maryland, College Park, Maryland 20742

The design of plantwide control systems has received increasing attention in the past few years. This paper deals with how one can systematically synthesize plantwide control architectures from steady-state process models using optimization. The optimization problem solved is a mixedinteger linear programming (MILP) problem which aims at minimizing valve movement to compensate for a disturbance. Results are presented for its application to the Tennessee Eastman test-bed process. Introduction

Table 1. Plantwide Control Design Steps Taken from Luyben et al.7

The design of plantwide control systems has received increasing attention in the past few years. Tighter plant designs, due to such considerations as energy integration, are leading to more challenging control problems. Several authors have proposed approaches to the problem of configuring plantwide control systems. Narraway and Perkins1,2 discussed an approach based on economics which makes use of linear dynamic models and optimization. Ng3 has discussed a very comprehensive, systematic approach to the design of plantwide control strategies. Luyben and co-workers4-7 have presented heuristic approaches to selecting plantwide control structures. In their most recent publication7 a nine-step procedure is given and applied to three plantwide control problems. The nine steps are listed in Table 1. The first two steps are common to all plantwide control synthesis approaches. The approach discussed here is an alternative to steps 3-6, and it is based on optimization. The control system which results from this approach is called the base control system. For good plantwide control it is necessary that the base control system be sound; otherwise, operability problems can result. Once a base system is in place, then steps 7-9 can be carried out to complete the plantwide control system design. The design of the base control system is probably the most difficult aspect of designing a plantwide control system. In the approach proposed here several candidate base architectures are generated in a systematic manner using a steady-state process model. The approach can be extended and applied to a dynamic model if it is available.8 Recently, a number of test-bed problems for plantwide control have been published. The approach discussed in this paper is applied to the Tennessee Eastman testbed process which involves a plant with 41 measurements and 12 manipulated variables.9 A number of papers have appeared on how to control the Tennessee Eastman plant. Ng3 applied her systematic approach to the Tennessee Eastman process, and it is one of the three processes treated in ref 7. McAvoy and Ye10 used a steady-state approach coupled with dynamic simulation to develop a plantwide control strategy. In later papers11-13 McAvoy and co-workers proposed an improved control system as well as controllers that can be added on top of a decentralized PID system to

step

description

1 2 3 4 5

establish control objectives determine control degrees of freedom establish energy management control system determine production rate control system control product quality and handle safety, operational, and environmental constraints fix a flow in every recycle loop and develop an inventory control system control component balances control individual unit operations optimize economics or improve dynamic controllability

6 7 8 9

improve its performance. Recently, Ricker14 proposed a decentralized scheme and gave very detailed results to demonstrate its effectiveness over the entire range of the problem definition. Additional decentralized plantwide control systems and design approaches were proposed by Banerjee and Arkun,15 Price and Georgakis,16 Desai and Rivera,17 and Walsh et al.18 Several authors have used model predictive control (MPC) approaches on the problem. These include Ricker and Lee,19 Palavajjhala et al.,20 and Sriniwas et al.21 A detailed description of the Tennessee Eastman process, including typical disturbances and baseline operating conditions, is given in ref 9. The process involves the production of two products, G and H, from four reactants, A, C, D, and E. In addition, there are two side reactions that occur and an inert, B, essentially all of which enters with one of the feed streams. The optimization-based synthesis approach presented in this paper makes use of a steady-state gain matrix that includes integrating variables such as liquid levels. Arkun and Downs22 have discussed how such a gain matrix can be calculated from a dynamic process model. A recent paper23 discusses how one can obtain a very good approximation to the same gain matrix (median error 6.1% for the Tennessee Eastman process) using only a steady-state simulation. Both the exact and approximate gain matrices result in the same plantwide control architectures. Because one may not have a dynamic model for many processes, having a synthesis methodology that requires only steady-state information expands the number of processes to which one can apply the approach discussed in this paper. The gain matrix developed from a dynamic model is used in this paper.

10.1021/ie990023q CCC: $18.00 © 1999 American Chemical Society Published on Web 06/25/1999

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 2985

Figure 1. Tennessee Eastman process.

The structure of this paper is as follows. First, an overview of the synthesis methodology is given, and then a discussion of how to calculate the gain matrix is briefly reviewed. Next, the optimization aspects of the methodology are presented in detail. A plantwide design for the Tennessee Eastman plant is completed by applying steps 7 and 8 in Table 1. After completion of the design, simulation results for the most difficult to control step disturbances are given. Next, the issue of how to handle what has been called the “snowball effect” by Tyreus and Luyben4 is treated, and finally conclusions are drawn. Synthesis Methodology (a) Overview. The plantwide control synthesis methodology utilizes a number of basic components. The goal of the methodology is to develop several candidate plantwide control architectures. The various components include (1) development of a gain matrix that includes integrating variables; (2) use of optimization to select candidate control architectures for variables that must be controlled for safety reasons; (3) screening of the architectures for the safety variables using controllability tools; (4) use of optimization to select candidate control architectures for controlling product flow and quality; and (5) controlling chemical components and unit operations using standard approaches (steps 7 and 8 in Table 1). Components 2-4 are the primary focus of this paper. The discussion below gives results for applying the approach to the Tennessee Eastman process, which is shown in Figure 1. The proposed methodology does not assume specific loop pairings, so it can be used to select manipulated variables for traditional single-input single-out (SISO)

proportional-integral-derivative (PID) controllers as well as multivariable schemes. (b) Development of a Gain Model. To explain the gain matrix used for the synthesis methodology, it will first be assumed that a dynamic process model is available. Assume that the process under investigation has a nonlinear state space model given by

x3 ) f(x,u)

(1)

y ) g(x,u)

(2)

where x is the state vector, u is the vector of manipulated variables, and y is the vector of process measurements. The nonlinear functions f and g in eqs 1 and 2 can be linearized around a steady-state operating point using numerical differentiation to give

x3 ) Ax + Bu

(3)

y ) Cx + Du

(4)

where A, B, C, and D are constant matrixes. It is straightforward to use Laplace transforms in eqs 3 and 4 to determine the transfer function between y and u as

y(s) ) [C(sI - A)-1B + D]u(s)

(5)

From eq 5 the steady-state gain matrix between y and u can be obtained by setting s ) 0 to give

G(0) ) -CA-1B + D

(6)

However, if there are integrators in the process model, then the rate of change of some process states does not

2986 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999

Figure 2. Schematic representation of the gain matrix resulting from the Arkun and Downs approach.22

depend on the states themselves. For example, for some liquid levels the rate of change of the level is independent of the level itself. When integrators are present, the A matrix is singular and A-1 does not exist. Arkun and Downs22 proposed an approach to overcome this problem. They give expressions for the individual process transfer functions when integrating elements are present as

Gi,j(s) )

Gli,j(s) s

+ G0i,j(s)

(7)

The measurements, y, that are nonintegrating have 0 values for their Gl terms. Their steady-state gains can be calculated from their G0 terms as G0(0). For measurements, yl, that have integrating terms eq 7 can be rewritten to give

y3 l(s) ) (Gl + sG0)u(s)

(8)

If s is set to 0 in eq 8, the gains for y3 l can be calculated as Gl(0). The final gain matrix to be used for steadystate screening of plantwide control systems consists of the appropriate rows from G0(0) and Gl(0), as illustrated in Figure 2 for a particular case. The process gain matrix is given by

Y ) Ku

(9)

The measured variables for this matrix are y and y3 l, and the manipulated variables are u. As illustrated in Figure 2, the process measurement vector, Y, is made up of five subvectors, and three of them, y1, y3, and y5, are integrating. (c) Optimization-Based Design of Plantwide Control SystemssA Steady-State Approach. To use a gain matrix representation, the plant under consideration is assumed to have only integrating poles and no right halfplane poles. The right half-plane poles of the model of the Tennessee Eastman plant can be eliminated by closing a reactor cooling water temperature controller. Once this loop is closed, the plant has only integrating poles. The gain matrices used below in the synthesis procedure have the reactor cooling water temperature controller closed, and one of the manipulated variables is the set point for this controller. The candidate measurements to be controlled are split into three groups. The first group contains those variables that must be controlled for safety and/or environmental reasons. Included in this group are all liquid-level variables, because one cannot afford to have a spill in a plant or to have a supply run dry. If gas pressures are integrating in nature and a gas buildup can lead to an

operational problem, then these measurements would also be included in the group 1 variables. Finally, if energy integration is important to safe plant operation, then variables associated with energy integration would be included in group 1. The first group of variables can be labeled as must-control variables. The second group of variables to be treated involves the product flow and quality. Last, the remaining variables are placed in group 3, and they are controlled using steps 7 and 8 in Table 1. For control of groups 1 and 2, performance is measured by how well the control system performs in rejecting disturbances. To resolve conflicts where different architectures outperform one another for different disturbances, in principle one can use multi-objective optimization.24 Multi-objective optimization is not considered in this paper. If the group 1 measurements are designated as yM, then the fact that these measurements must be controlled can be expressed as

yM ) KMν + dj,M ) 0

(10)

where the gain matrix, KM, consists of the rows of K associated with yM, j is the disturbance being considered, and dj,M is the effect of the disturbance on yM. Equation 10 assumes perfect steady-state control, and it applies to designs in which the controllers have an integral mode. To anticipate that valve movement will be used in the optimization, ν is used in place of u in eq 10. For the Tennessee Eastman process two choices for the yM’s are considered: (1) the three liquid levels in the reactor, separator, and stripper reboiler and the reactor pressure, because the process shuts down if this pressure exceeds 3000 kPa; (2) the same variables as given in choice 1 plus the reactor temperature because it is constrained to be less than 175 °C. The first optimization problem to be solved to design a plantwide control architecture involves determining which manipulated ν’s should be used to control the yM’s for the disturbance. One measure of how good a particular architecture is can be taken to be how much of the manipulated valves it uses have to move from their original steady-state positions when dj,M occurs. A system that requires large changes in valve positions is deemed inferior to one requiring smaller changes. This metric for assessing control system performance is similar to the relative disturbance gain (RDG) introduced by Stanley et al.25 It should be noted that a system that requires large steady-state changes in the valves might have superior dynamic performance to one that requires small changes. This possibility is the reason dynamic performance may need to be assessed, either via simulation or through use of an approach based on dynamic models.8 In formulating an objective function for optimization, both positive and negative changes in valve movement are penalized. To account for both changes, ν is rewritten in terms of ν+ and ν- as

ν ) ν + - ν-

(11)

where the elements of ν+ and ν- are all positive. Then, an optimization problem to determine a plantwide architecture for the safety variables can be formulated


as shown in eq 12 for the choice 1 safety variables:

min

ν+,ν-,z+,z-

c+ν+ + c-ν-

s.t. KM(ν+ - ν-) + dj,M ) 0 for all values except the reactor cooling water valve + + ν+ k e zk νk,max (k ) 1-κ) - νk e zk νk,max (k ) 1-κ)

for the reactor cooling water valve + ν+ RCW e νRCW,max νRCW e νRCW,max +

-

KRCWT(ν - ν ) + dj,RCWT e

∆TRCW,max(zRCW

+

zRCW)

κ

∑ (z+k + z-k ) + (z+RCW + z-RCW) ) N k)1

(12)

+ c+ i , ci , νi , νi g 0 z+ k , zk ∈ {0, 1}

In eq 12 c+ and c- are positive weighting row vectors, + and νk,max are the maximum changes perand νk,max mitted for the valve movements away from their steadystate values, κ is equal to the number of valves used in the optimization minus 1 (κ does not include the reactor cooling water valve), N is equal to the number of must control variables, ∆TRCW,max is the maximum change for the reactor cooling water temperature, and KRCWT is the row of the gain matrix associated with the reactor cooling water temperature. In the objective function the weighted sum of the absolute values of the movements of all of the manipulated variables used for controlling the safety variables and the reactor cooling water temperature is minimized. In the results below all of the elements of c+ and c- are taken as 1.0. If one had information on the cost of moving each valve, then it could be used to determine different values for c+ and c-. The first constraint involves the steady-state process gain model (eq 10). The next four constraints deal with saturation of the manipulated variables. If a valve is selected (z+ k , zk * 0), then it is constrained not to saturate either open or closed. Because the reactor cooling water loop is always closed, the constraints involving saturation of the reactor cooling water valve do not contain its integer variables (z+ RCW, zRCW). If the Tennessee Eastman process did not have right halfplane poles, then no special treatment would be required for the reactor cooling water valve and the formulation of the synthesis problem would be simpler than that given in eq 12. The constraint involving ∆TRCW,max is required because the reactor cooling water cascade loop must be closed to eliminate the right half-plane poles, and the reactor cooling water valve is always used. If the reactor cooling water valve is not selected to control yM, its associated integer variables z+ RCW and zRCW will both be zero, and the constraint involving ∆TRCW,max results in the reactor cooling water temperature being controlled to its deviation set point, namely, 0. Because there are four choice 1 safety variables, N is set equal

to 4. The equality constraint involving the integer variables forces four manipulated variables to be selected to control the safety variables. In eq 12 the integer variables appear in a linear manner, and this fact allows the use of branch and bound solution techniques, as discussed by Quesada and Grossmann.26 Alternatively, for small problems one could simply enumerate all solutions and sort the answers as discussed below for the Tennessee Eastman process. The solution of eq 12 gives the four manipulated variables that have to move the least in order to keep the yM measurements at their set points. If enumeration is not used, then to generate additional candidate architectures for controlling yM, one can proceed as follows. One can add the following constraint on the integer variables that result from the optimum solution:

∑p (z+p + z-p ) + (z+RCW + z-RCW) e N - 1

(13)

where p is the optimum set of valves in addition to the reactor cooling water valve. After each solution one additional constraint is added to eq 12 to eliminate the most recent solution. Now consider which manipulated variables to include in the optimization. The Tennessee Eastman problem places constraints on how fast some of the process streams can be manipulated. These streams include the C feed, the D feed, the A feed, and the product. If one examines the four choice 1 safety variables, then two of them, namely, the reactor level and pressure, should be tightly controlled. The remaining two, namely, the separator and stripper levels, can be loosely controlled in an averaging manner27 to help filter the effect of step upsets on the product flow. For these latter two loops, the C feed, the D feed, the A feed, and the product are potential manipulated variable candidates, because averaging level control involves slow manipulation. However, the A feed, the C feed, and the D feed primarily affect the separator and stripper levels indirectly through their effect on conversion in the reactor. Furthermore, one of the important disturbances, IDV(6), involves the complete loss of the A feed.9 For controlling the yM variables one should not use a manipulated variable that is prone to failure. Thus, the C feed, the A feed, and the D feed are eliminated from the optimization, giving a value of κ ) 8. If there were no speed constraints placed on the feed streams, then the C feed and the D feed could be included in the optimization problem as potential candidates for controlling reactor level or pressure. When eq 12 is solved for the Tennessee Eastman process, the results shown in Table 2 are obtained for the choice 1 safety variables. Table 2 gives results for the most significant step disturbances given in the problem statement,9 namely, IDV(1), an upset in the A/C ratio in the C feed, IDV(2), an upset in the inert concentration of the C feed, and IDV(6). The remaining step disturbances are very easy to control.10 The results for the five best solutions are given in Table 2. To calculate these results, four of the nine possible manipulated variables are selected, and eq 10 is solved for the valve movements. The number of cases that are calculated is 9!/(5!4!) ) 126. Results from these 126 cases are sorted based on their performance indices. Then the constraints are checked to make sure they are not violated.

2988 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 Table 2. Optimization Results for Choice 1 Safety Variables rank

Σ|valve movement|

mv-1

mv-2

1 2 3 4 5

46.64 47.50 50.85 52.19 54.59

E feed E feed Sep E feed E feed

Sep Sep Prod Prod Purge

1 2 3 4 5

7.88 8.40 8.48 8.49 9.00

E feed E feed E feed E feed Sep

1 2 3 4 5

25.74 27.69 36.48 38.02 42.29

E feed E feed E feed Sep Purge

mv-3

mv-4

comments

IDV(1) Prod Re-CWT Re-CWT Re-CWT Sep

Re-CWT Cd-CWT Cd-CWT Cd-CWT Re-CWT

viable but one small RGA element not viable and poor RGA viable with some interaction not viable and poor RGA not viable and poor RGA

Sep Prod Sep Purge Prod

IDV(2) Prod Re-CWT Re-CWT Re-CWT Re-CWT

Re-CWT Cd-CWT Cd-CWT Cd-CWT Cd-CWT

viable but one small RGA element not viable and poor RGA not viable and poor RGA viable with some interaction viable with some interaction

Sep Prod Sep Prod Sep

IDV(6) Prod Re-CWT Re-CWT Re-CWT Re-CWT

Re-CWT Cd-CWT Cd-CWT Cd-CWT Cd-CWT

viable but one small RGA element not viable and poor RGA not viable and poor RGA viable with some interaction not viable and poor RGA

In most chemical plants operators would like to have a relatively noninteracting control system for each safety variable. To judge interaction, the relative gain array (RGA)28 and Niederlinski index (NI)29 can be used. In the approach below, SISO pairings are evaluated using the RGA. However, it is not necessary to implement the safety loops as SISO loops. If the safety control system is implemented as a multivariable system, then the RGA simply measures the amount of interaction in the system. Consider the best result for all disturbances, IDV(1), IDV(2), and IDV(6), in Table 2. The RGA for this result with suggested SISO pairings circled is

The RGA for the separator exit-separator level loop is 0.344 in eq 14. While this RGA value is low, it can be noted that the separator level response should be tuned for averaging behavior in order to help filter flow disturbances. For averaging level control, a small controller gain is used. As a result, the fact that controlling other variables in the system results in gain changes for the separator level-separator exit flow pairing has less significance than it does for a tightly controlled variable. The two tight variables in eq 14 have RGA’s of 0.853 and 1.582. Thus, this system is deemed as being viable. Now consider the second best result in Table 2 for IDV(1). The RGA for this result is

If the reactor cooling water is paired with the reactor pressure, then the only logical variable, based on engineering judgment, to control with the E feed is the reactor level. If these two pairings are made, then the separator level must be paired on a negative RGA element, which is not acceptable. Thus, solution 2 is deemed as being not viable, and this fact is listed in Table 1. A similar examination of each of the five best

solutions shows some of them to have problems when the RGA is used to evaluate them. An examination of the results in Table 2 indicates that, in addition to the E feed, separator, product, and reactor cooling water solution, the other scheme that appears viable involves the separator, product, and reactor and condenser cooling water. This scheme ranks third for IDV(1), fifth for IDV(2), and fourth for IDV(6). The RGA for this scheme is shown below with suggested pairings circled:

The separator level pairing has an RGA of 0.648, which is better than that for eq 14, but the RGA’s for the two tight variables are a little worse, namely, 1.859 and 1.823. The RGA’s given by eqs 14 and 16 are the best for all three major upsets. McAvoy and Miller23 have discussed how a very good approximation to the gain matrix given by eq 9 can be calculated using only steady-state information. The approximate gain matrix for the Tennessee Eastman plant yields the same plantwide control architectures as the rigorous gain matrix developed from a linearized dynamic model. However, one only gets the same architectures after RGA screening, because the solutions which are rejected by the RGA differ between the gain matrix developed from a dynamic model and that from a steady-state model. The approximate gain matrix does rank the E feed, separator exit, product, and reactor cooling water valves as the best choice for IDV(1), IDV(2), and IDV(6). The second best choice involves swapping the condenser cooling water for the E feed. However, there are a few more intervening solutions, which are rejected by the RGA, between the best and second best solutions when the approximate model is used. Next the methodology is applied to the choice 2 safety variables. When the resulting architectures are examined, almost all have problems. For example, for IDV(1) the optimum pairing involves using the E feed, separator exit, product, and reactor and condenser cooling water valves. This scheme has a valve movement of 98.4, which is roughly twice as large as the values given in Table 1 for the same upset. Because an extra


valve has been added, one can expect the sum of the absolute values of valve movements to increase. However, the large increase that occurs indicates that there may be an interaction problem with the choice 2 safety loops. The RGA for the optimum pairing is

The feasible loop pairings based on this RGA are not very good, and they do indeed indicate a great deal of interaction. In particular, the E feed-reactor level is paired on 3.79, the separator level-separator exit flow is paired on 0.37, and the reactor pressure-condenser cooling water is paired on 7.63. The other two pairings are good, 1.0 and 0.948. The poor reactor pressure and level pairings are troublesome because tight control is required for these variables. Because the choice 2 safety variables result in significantly more valve movement than the choice 1 safety variables, they are deemed inferior and not pursued further. Next the manipulated variables to control the quality and product flow variables, yQ, are selected. Equation 10 can be expanded to include these new variables as

[ ]

yM yQ ) KM,Qν + dj,M,Q ) 0

(18)

where KM,Q is the gain matrix for the safety and product variables, and the effect of disturbance j on these variables is dj,M,Q. In the Tennessee Eastman process there are two product variables to be controlled, namely, the production rate and the G/H ratio in the product. The gain for the G/H ratio can be calculated from the gains for G and H individually. The optimization problem to be solved for controlling the yQ variables is given by eq 19:

mim c+ν+ + c-ν-

ν+,ν-,z+,z-

s.t. KM,Q(ν+ - ν-) + dj,M,Q ) 0 + + ν+ k e zk νk,max (k ) 1-8) - νk e zk νk,max (k ) 1-8) + ν+ q e νq,max (q ) 1-4) νq e νq,max (q ) 1-4) 8

∑ (z+k + z-k ) ) N1 k)1

(19)

ν+ i , νi g 0 c+ i , ci g 0 z+ k , zk ∈ {0, 1}

In eq 19 q designates the four valves selected in the first step to control the yM variables, N1 is the number of

Table 3. Optimization Results for Product Flow and Composition Σ|valve scheme movement| mv-1 mv-2

mv-3

mv-4

mv-5

mv-6

1 2

63.89 80.17

IDV(1) E feed Sep Prod Re-CWT D feed C feed Sep Prod Re-CWT Cd-CWT D feed C feed

1 2

9.95 12.46


1 2

28.00 35.71


yQ variables, two for the Tennessee Eastman process, and k designates the remaining eight valves from which two are to be selected. Note that the reactor cooling water is included in the two viable candidates for the yM variables determined from the solution of eq 12. As a result the optimization problem (19) does not need to have a constraint involving ∆TRCW,max. Because integers are not used for the νq’s, these valves are automatically included in the objective function. The remaining eight valves are only included if they are selected. In solving eq 19, all valves were weighted equally. The results of solving eq 19 are given in Table 3. Again enumeration is used, and 8!/(2!6!) ) 26 cases are calculated. The only choice for controlling the product flow and composition involves the use of the C and D feeds. All other choices resulted in valve saturation for the three step upsets considered. Next the completion of the plantwide design using steps 7 and 8 in Table 1 is treated. Completing the Plantwide Design To complete the plantwide design, steps 7 and 8 in Table 1 need to be carried out. Step 7 involves the control of component balances. For scheme 1 in Table 3 the six manipulated variables that are not assigned are the A feed, purge, recycle, steam, condenser cooling water, and agitator. The A feed and purge directly affect component balances. For scheme 2 the condenser cooling water is assigned, but the E feed, which also affects component balances, is not assigned. For both schemes 1 and 2 the A feed can be used to control the composition of A or the A/C ratio at some point in the process. Similarly, for both schemes the purge can be used to control the composition of B in the purge, because the only place that B leaves is via the purge. For both schemes, the D/E ratio can be used to control the G/H ratio as discussed by McAvoy and Ye.10 For scheme 2, it appears that the E feed can be used to control the composition of E or the D/E ratio at some point in the process. However, in a recent paper30 Tyreus has shown that the E feed cannot be manipulated for such purposes. At this point step 7 is complete. Step 8 involves the control of individual unit operations, which is not plantwide in nature. For the stripper, the steam can be used to control the stripper temperature. If desired the temperature set point can be cascaded to the E impurity in the product stream. For the reactor cooling water an override is added to avoid overheating the reactor contents. When the reactor temperature reaches 125 °C, pressure control is sacrificed and the reactor cooling water is used to control reactor temperature until its set point falls below 125 °C. A simple means of constraining the reactor temperature is to cascade it to the reactor cooling water


Figure 3. Control system for the Tennessee Eastman process.

temperature loop and then use the reactor temperature set point to control reactor pressure in a double-cascade arrangement. Then, a low select can be used for the reactor temperature set point. For both schemes 1 and 2 the recycle and agitator are not used. For scheme 1 the condenser cooling water temperature is controlled, but its set point is not used to control another variable. One final multi-SISO plantwide control design for scheme 1 is shown in Figure 3. Except for the overrides, this design is the same as that presented by McAvoy et. al.11 but without a systematic approach to its development. It should be noted that there is no need to implement the control system for the yM and yQ variables and the variables arising from steps 7 and 8 as a multi-SISO system. In fact, for cases with complicated overrides, cases with strong interaction, and/or cases where constraint handling is important, a model predictive scheme is preferred. In ref 11 the focus was on improved reactor pressure control. One issue not addressed was how to deal with the loss of the A feed, IDV(6). When IDV(6) occurs and the A feed is lost, the process becomes rich in component C. In handling IDV(6), Ng3 recommended lowering all three levels in the process. Table 4 gives the results of steady-state calculations for the case where the reactor temperature is held at 125 °C and the production rate varies from 70% to 100% of its steady-state value. As can be seen, if the production rate is cut by 15-20%, the reactor pressure rises about 100 kPa when the A feed is lost. If in addition to reducing the production rate the levels in the reactor, separator, and stripper are all reduced to 10% above their lower limits, the reactor pressure rise is only about 20 kPa. In Figure 3 only the reduction in production rate is shown because a pres-

Table 4. Steady-State Results (Variables Controlled: G/H Ratio in Product ) 1.226, Product Flow, Reactor Temperature ) 125 °C, Reactor Level ) 75% and 60%, Separator and Stripper Levels ) 50% and 40%) reactor pressure product as reactor pressure product as (kPa): levels % of (kPa): levels 10% % of at steady state steady state above min steady state 2965.1 2886.7 2833.5 2805.1 2805.8 2845.7 2943.1

100 95 90 85 80 75 70

2887.5 2809.9 2756.0 2725.8 2723.8 2761.1 2859.0

100 95 90 85 80 75 70

sure of 2805 kPa is deemed to allow enough of a pressure safety margin compared with the shutdown limit of 3000 kPa. When IDV(6) occurs, the production set point is ramped down to 85% of its steady-state value in 1 h. A ramp is used to avoid a high-frequency upset to the upstream process supplying the C feed. Also, when IDV(6) occurs, the purge valve is opened to 100%. Figures 4-7 give results for the scheme shown in Figure 3 for IDV(1), IDV(2), IDV(6), and IDV(8), a stochastic disturbance in the composition of the C feed. To illustrate the underlying dynamic performance of the plantwide scheme, all measurement noise is removed from the simulation. All controllers are PI controllers, and the tuning constants for the outer loops are taken from ref 11. All of the tuning constants are given in Table 5. As can be seen from Figures 4-7, very good results are achieved. Figure 4 shows that the effects of the IDV(1) disturbance on the product take about 1000 min to die out, while Figure 5 shows that the effects of IDV(2) take about 1500 min to die out. Figure 6 gives results for IDV(6). The initial transients as the result


Figure 4. Response to IDV(1): change in the A/C ratio in the C feed.

Figure 5. Response to IDV(2): change in the B composition in the C feed.

of losing the A feed are small. At approximately 900 min, the reactor temperature set point reaches 125 °C and it is clamped there. From that point on the reactor pressure slowly rises in a monotonic manner to 2805. The stochastic upset, IDV(8), causes a continuous cycling of the process, but the product flow and composition are within the bounds set in the problem statement.9 Next, a brief discussion of how to handle the

Figure 6. Response to IDV(6): complete loss of the A feed.

Figure 7. Response to IDV(8): random fluctuations of compositions of A, B, and C in the C feed.

snowball effect discussed by Tyreus and Luyben4 is given. Handling the Snowball Effect Luyben and Luyben5 presented a control study of a three reactor/three distillation tower process that is shown in Figure 8. Most control schemes for this plant were found to be inoperable, and these schemes exhibited the snowball effect.4 The snowball effect arises


Figure 8. Luyben and Luyben process.5 Table 5. PI Controller Constants controlled

manipulated

Kc

A flow C flow D flow E flow purge separator exit product flow steam condenser cool water temp reactor cool water temp reactor temperature

Inner Cascade Controllers A valve C valve D valve E valve purge valve sep. exit valve product valve steam valve condenser cool water valve reactor cool water valve reactor cool water set pt.

150 (%/kscmh) 10 (%/kscmh) 0.026 (%/kg h-1) 0.017 (%/kg h-1) 200 (%/kscmh) 2.0 (%/m3 h-1) 3.0 (%/m3 h-1) 2.5 (%/kg h-1) -8.5 (%/°C) -10 (%/°C) 1.0

reactor level separator level stripper level reactor pressure stripper temp G/H prod. ratio product flow A/C reactor feed B in purge

Outer Cascade Controllers E feed set pt. sep. exit set pt. product set pt. reactor temp set pt. steam flow set pt. D/E ratio set pt. C feed set pt. A feed set pt. purge set pt.

500 (kg h-1/%) -2.5 (m3 h-1/%) -0.5 (m3 h-1/%) -0.1 (°C/kPa) 10.0 (kg h-1/°C) 0.05 0.08 (kscmh/m3 h-1) 1.0 (kscmh) -0.03 (kscmh/%)

when small changes in a variable result in extremely large changes in other process variables. In approaching the design of a plantwide control system for the process shown in Figure 8, if one only included liquid levels in the yM variables, then a snowball problem is almost certain. Part of item 6 in Table 1, namely, fixing a flow in every recycle, aims at developing plantwide architectures that avoid the snowball effect. The snowball effect is essentially a steady-state phenomenon, and thus whether it is likely to occur can be analyzed using a steady-state gain matrix that includes integrating variables. McAvoy and Miller23 developed such a gain matrix for the process shown in Figure 8. The resulting gain matrix is large, and it involves 524 variables. These include 515 mole fractions and 9 level derivatives. The

TR(min) 0.075 0.10 0.10 0.10 0.06 0.12 0.12 1.5 2.6 1.0 50 200 300 300 10 10 40 45 100 100

mole fractions include five for each tower tray and five for each reactor. The sixth mole fraction is not included because it is constrained by the fact that the sum of the mole fractions must equal 1. Only a small part of the gain matrix is presented here in eq 20:

In eq 20 z/1A and z/1B are the deviations in the mole


fractions of A and B in reactor 1, and l˙ /R1 is the rate of change of the level in reactor 1. The manipulated variables shown are the fresh feed, F/OA, the reactor exit flow, F/1, and the distillate streams from towers 1 and 3, D/1 and D/3. Equation 20 clearly indicates the potential for a snowball effect in the process. For example, a change in FOA equal to 1.0 (base case FOA ) 100) results in a change in the mole fraction of A equal to 305. The extremely large gains for both mole fractions z1A and z1B (variables that range between 0 and 1) indicate that if one is not careful, then valve saturation is almost certain. To avoid a snowball problem, one can proceed as follows. First, the process gain matrix is examined to assess if there are any large gain elements, such as those shown in eq 20. Then the measurements having the large gains are included in the must control variables, yM, in the synthesis methodology. Indeed, Luyben and Luyben5 found two control systems that avoid the snowball effect, and both involved controlling z1A and z1B. Without such a strategy, the process shown in Figure 8 cannot be controlled. The approach discussed so far makes use of only steady-state information. While dynamic models are not available for many processes today, this situation is changing. In the future one can anticipate that dynamic models will be much more widely available. The approach discussed here has been extended to cases where dynamic models are available.8 This dynamic extension is similar to well-established model predictive control approaches.

I ) identity matrix K ) gain matrix KRCWT ) row of K associated with reactor cooling water temperature lR1 ) reactor 1 level in the Luyben and Luyben process N ) number of safety variables to be controlled N1 ) number of product variables to be controlled p ) set of values, except reactor cooling water valve, resulting from solution to eq 12 q ) set of values resulting from solution of eq 19 Prod ) product flow Re-CWT ) reactor cooling water temperature Re-L ) reactor level Re-Pr ) reactor pressure Re-T ) reactor temperature s ) Laplace variable Sep ) separator exit flow Se-L ) separator level St-L ) stripper level u ) vector of manipulated variables x ) vector of states y ) vector of measurements Y ) vector of measurements which includes integrating variables z ) vector of integer variables used to select valves zi ) composition in the Luyben and Luyben process

Summary

Subscripts

This paper has presented an approach to synthesizing plantwide control architectures that makes use of steady-state models and optimization. The optimization problem solved is a MILP problem that aims at minimizing the absolute value of valve movements when a disturbance occurs. The first problem solved involves controlling variables that must be held constant for safety and other reasons. The results of this first problem are screened using controllability tools such as the relative gain and Niederlinski index. Then an optimization problem is solved to determine which manipulated variables move the least in controlling product flow and composition, subject to the same disturbances. The plantwide design is completed by controlling component balances and unit operations using standard approaches. Results for the application of the synthesis approach to the Tennessee Eastman process are presented. The synthesis approach can be extended to include dynamics.

A ) component A in the Luyben and Luyben process B ) component B in the Luyben and Luyben process j ) disturbance number k ) index l ) integrating M ) must control max ) maximum OA ) feed of A in the Luyben and Luyben process Q ) quality RCW ) reactor cooling water RCWT ) reactor cooling water temperature 0 ) nonintegrating

Greek Letters ∆T ) delta temperature κ ) number of valves available for safety control Λ ) relative gain matrix ν ) vector of valve movements

Superscripts * ) deviation variable ˘ ) derivative + ) open - ) close

Literature Cited Nomenclature A ) matrix in the linearized model B ) matrix in the linearized model c ) vector of weighting coefficients C ) matrix in the linearized model Cd-CWT ) condenser cooling water temperature d ) disturbance vector D ) matrix in the linearized model Di ) distillate flow in the Luyben and Luyben process E feed ) E feed f ) function Fi ) flow in the Luyben and Luyben process g ) function G ) transfer function

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Received for review January 7, 1999 Revised manuscript received May 6, 1999 Accepted May 10, 1999 IE990023Q

Synthesis of Plantwide Control Systems Using Optimization - Industrial

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