Ind. Eng. Chem. Res. 1988,27, 611-615 would always have a “sensible” solution. That is, the process constraints can be satisfied without a severe economic penalty. The goal of our control system is then to drive the process toward ita optimum operating conditions for all disturbances. In part 3 of this series, we consider the optimum steady-state control of the plant.
Conclusions and Significance The flow-sheet decomposition procedure of Douglas (1985) was used to specify some of the alternative operating and equipment overdesign policies that satisfy the process constraints at each level for the complete range of the anticipated disturbances. In addition, nearly optimum overdesign factors were selected for each piece of equipment to minimize the effect of disturbances on the total process costa. Often, overdesign policies corresponding to either “worsbcase” or “base-case”operating conditions are appropriate. More detailed operability analyses are only recommended when (1)neither the “worst-case” nor the “base-case” alternative is obviously favorable and (2) both policies significantly reduce the profitability of the process. With this approach, the appropriate equipment overdesign factors can be quantitatively related to the magnitude of the anticipated disturbances entering the process. Acknowledgment The authors are grateful to the National Science
61 1
Foundation for providing financial support under Grant CPE-8105500.
Literature Cited Douglas, J. M. AZChE J . 1985, 31, 353. Fisher, W. R. Ph.D. Thesis, University of Massachusetts at Amherst, 1985. Fisher, W. R.; Douglas, J. M. Comp. Chem. Eng. 1985, 9, 449. Fisher, W. R.; Doherty, M. F.; Douglas, J. M. Znd. Eng. Chem. Process Des. Deu. 1985, 24, 593. Grossmann, I. E.; Morari, M. “Operability, Resiliency, and Flexibility-Process Design Objectives for a Changing World”. 2nd Int. Cong. of Found. of Computer-Aided Process Design, Snowmass, CO, June 19-24, 1983. Grossmann, I. E.; Halemane, K. P.; Swaney, R. E. Comput. Chem. Eng. 1982, 7, 151. Halemane, K. P.; Grossmann, I. E. AZChE J . 1983,29, 425. Kittrell, J. R.; Watson, C. C. Chem. Eng. Prog. 1966, 62(4), 79. Morari, M.; Lenhoff, A. M.; Marselle, D. F.; Rudd, D. F. “Deseign of Resilient Energy Integrated Processing Sysems”. 72nd Annual AIChE Meeting, San Francisco, 1979, paper 28c. Morari, M.; Marselle, D. F.; Rudd, D. F. “Synthesis of Resilient Energy Management Systems”. 73rd Annual AIChE Meeting, Chicago, 1980, paper 3f. Swaney, R. E.; Grossmann, I. E. AZChE J . 1985a, 31, 621. Swaney, R. E.; Grossmann, I. E. AZChE J. 198513, 31, 631.
Received for reuiew July 15, 1986 Revised manuscript received July 23, 1987 Accepted August 17, 1987
The Interface between Design and Control. 3. Selecting a Set of Controlled Variables Wayne R. Fisher, Michael F. Doherty, and James M. Douglas* Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003
By solving the optimum steady-state control problem in terms of the significant disturbances and manipulative variables, we often find that the optimum values of some of the operating and/or manipulative variables lie at constraints. If we select these constrained variables as controlled variables, the resulting feedback system will have near optimal performance without the need for measuring all the disturbances or for calculating the entire optimum steady-state control policy on-line. In many cases it is possible to propose heuristics to identify the constrained controlled variables. In parts 1 and 2 of this series, we have presented systematic procedures for identifying the significant disturbances, for ensuring that there are an adequate number of manipulative variables to be able to satisfy the active process constraints and to optimize the significant operating variables, and to ensure that there is an adequate amount of overdesign to obtain optimum steady-state control while satisfying the process constraints. Moreover, we considered modifications of the flow sheet (we intend to apply the procedures at the conceptual stage of a process design, so that flow-sheet modifications are still possible) in order to accomplish these goals. Our goal now is to select a set of controlled variables. We require that the control system to be developed satisfies the active process constraints and, in addition, gives close to the optimum steady-state performance after dynamic transients have decayed. Thus, it seems reasonable to select controlled variables that correspond to the solution of the optimum steady-state control problem. If we can accomplish this goal, it will not be essential to solve the optimum steady-state control problem for these variables on-line. 0888-5885/88/2627-0611$01.50/0
The primary control objective is the profitable operation of the process. Other control objectives include product quality specifications, production goals, safety and environmental regulations, and other process constraints. Many of these control objectives for the process are well defined even at the conceptual design stage. In recent years, synthesis strategies have been introduced for determining the interconnections between manipulated and controlled variables. These methods generally assume that the process flow sheet is fixed and the controlled and manipulated variables are specified. The Relative Gain Array approach developed by Bristol (1966) has been applied extensively to distillation columns and other simple processes. Govind and Powers (1982) present a nonnumerical algorithm which generates alternative control structures for complete plants based on a causeand-effect representation of the process. In a similar spirit, Morari and Stephanopoulos (1980a) developed structural controllability as a basis for generating alternative feasible control structures. The selection of the best set of controlled variables from the hundreds of process state variables is not a trivial task. 8 1988 American Chemical Society
612 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988
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PURGE COMPOSITION
Figure 1. Comparison of the design and control optimization problem for purge composition in the HDA process.
Unfortunately, much of the literature concerned with control structure synthesis does not address this important topic. In this paper, the selection of the controlled variables is considered to be a primary goal. Our approach differs from the very detailed studies of Morari and coworkers (Morari et al., 1980; Morari and Stephanopoulos, 1980a,b)in that we use the results of optimum steady-state control analyses to generate heuristics for plant control in much the same way that heuristics have been developed for setting the values of certain design variables to aid in flow-sheet design and synthesis.
Optimum Steady-State Control On-line calculation of the optimum steady-state control is rapidly becoming state-of-the-art in several companies (ICI, Shell, etc.). The optimum values of the manipulative variables are calculated as a function of the disturbances, and these results are used to adjust the set points in a previously specified set of control loops. However, this strategy implies that the “best” sets of controlled variables and control loops were selected by some other procedure. Our approach is to use the results from an optimum steady-state control study to select the best set of controlled variables. The optimum steady-state control problem is quite different from the optimum steady-state design problem. In the design problem, the dominant trade-offs for each design variable normally involve both capital and operating costs. However, after the equipment sizes have been fixed, we can only minimize the operating costs. If we examine the trade-offs for any operating variable, the minimum operating cost corresponds to a shift in the optimum value in one direction or the other (see Figure 1for a case where the minimum operating cost is less than the minimum total annual cost, but the curves may be reversed). Because of the fixed equipment sizes, however, we can only shift the value of the operating variable until we encounter an equipment constraint. Some examples of this type are discussed below. Purge Composition. If a process contains a gas recycle and a purge stream, for the design problem there is a trade-off between large raw material losses at high purge compositions of reactant balanced against high gas recycle costs (both capital and operating); usually the gas recycle
compressor or the reactor heating/cooling system dominates the recycle costs. However, for control we still trade off raw material costs against either or both the power cost of the compressor and/or the fuel for the furnace. Thus, for the control problem we would like to move to lower purge compositions of reactants, but eventually we encounter a constraint on either or both the compressor and the furnace. Thus, we expect that the optimum purge composition will fall on a constraint. This result implies that we should normally use the maximum (overdesign) capacity of a gas recycle compressor, even if we turn down the production rate of a plant. Conversion. For processes where there are complex reactions and the product distribution can be correlated against conversion, the dominant economic trade-offs for the design problem normally involve large selectivity losses of reactants to byproducts and large reactor costs at high conversions balanced against large recycle costs (both capital and operating) at low conversions (which often are dominated by the recycle column and/or the reactor heating/cooling system). For control, the trade-offs change to balancing the selectivity losses against the operating costs for the recycle stream. Since raw materials are normally more costly than utilities, the optimum operating policy will favor lower conversions (i.e., higher recycle flows), and we continue to decrease the conversion until we encounter an equipment constraint. Molar Ratio. For processes where the product distribution of a set of complex reactions is improved by increasing the molar ratio of a reactant to the limiting reactant at the reactor inlet, the optimum design problem involves a trade-off between large selectivity losses of reactant to byproduct at low molar ratios vs large recycle costs (both capital and operating) to recover and recycle the excess material (the recycle column and/or the reactor heating/cooling costs normally are the dominant factors in the recycle costs). For control, we trade off selectivity losses vs the recycle operating costs only. Therefore, the optimum control action will correspond to larger molar ratios, which normally means larger recycle flows. Thus, we increase the molar ratio (ie., recycle flow) until we encounter a constraint. Reactor Temperature. For processes where the product distribution is improved by decreasing the reactor temperature, in the design problem there is a trade-off between large selectivity losses of reactant to byproduct at high temperatures vs large reactor and reactor cooling costs (both capital and operating) at low temperatures. For control, we trade off the selectivity losses vs reactor cooling costs only, so that for optimum control we reduce the reactor temperature until we encounter the constraint for the reactor cooling system. Flash Temperature. For design, the optimum temperature of a phase splitter (flash drum) involves a trade-off between either large losses in a purge stream or high loads on a vapor recovery system balanced against the capital and operating costs of either a partial condenser preceding the flash or the cooling system for the flash drum. For steady-state control, we no longer consider the capital cost of the cooling system, and the optimum shifts to running the flash drum as cold as possible until we encounter a cooling system constraint. Product Recovery Fraction. The product composition (say the distillate) is assumed to be fixed, but the optimum fractional recovery for the design problem involves a trade-off between a large cost of incremental trays in the stripping section of the column at high recoveries vs either the high cost of recycling the product back
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 613 through the reactor or large product losses in a waste or fuel stream. For control, the distillate and bottoms compositions become coupled by the reflux ratio. If the distillate (product) composition is held constant, the fractional recovery trades off large heating and cooling costs for large reflux ratio (i.e., vapor rate) vs either the operating cost for recycling the product or product loss to a waste or fuel stream. Fisher et al. (1985a) noted that the optimum operating conditions are approximately equal to those at the design stage, even though quite different tradeoffs are involved. Thus, dual composition control of distillation columns is close to the optimum control. Minimum Operating Costs. The discussion above indicates that we can often expect the minimum operating costs to correspond to constrained operation. We would like to use this insight to simplify the problem of selecting a set of controlled variables. Selecting Controlled Variables In order to implement an on-line optimum steady-state control system, it is necessary to measure all the important disturbances and to solve the multivariable optimization problem for the manipulative variables. However, if these solutions normally correspond to constraints on the recycle flows or some other quantity, we can use these contrained optimization variables as the controlled variables. Thus, we select the controlled variables to correspond to the active process constraints and the constrained operating (or manipulative) variables, if possible. Since we desire to maintain these variables constant, we will obtain regulatory controllers, which simplifies the dynamic control analysis. If any of the operating variables (the number of operating variables is equal to the design degrees of freedom minus the number of equipment constraints) are not specified as controlled variables by this procedure and if the operating costs are sensitive to an optimization of these variables, then we need to use an on-line optimizer. However, our procedure makes it possible to decrease the size and complexity of the on-line optimization analysis. Heuristics for P l a n t Control Design heuristics (i.e., R / R , = 1.2 for distillation, L/mG = 1.4 for gas absorbers, fractional recoveries in columns greater than 99%, approach temperatures of 10 K in heat exchangers, etc.) correspond to the solutions of optimization problems, where the optimum solutions are very insensitive to any of the design or cost parameters. Since our optimum steady-state control solutions normally lie on constraints, for all values of the disturbances, we can propose heuristics for plant control and/or the control of an individual unit. These heuristics are simply restatements of the constraints identified by examining the economic trade-offs that we discussed earlier. Some heuristics of this type are presented in Table I (Fisher et al., 1985b). Example: Hydrodealkylation of Toluene To Produce Benzene (HDA Process) The controllability and operability of the HDA process have been discussed in the first two papers in this series. Now we consider the optimum steady-state control of the process. The significant disturbances are the production rate, the feed composition of hydrogen, and the cooling water inlet temperature to the partial condenser; see Figure 6, part 1. The process constraints that we must satisfy are the product composition, the production rate, the hydrogen-to-aromatics ratio at the reactor inlet, and the quench
Table I. Some Heuristics for P l a n t Control 1. Production rate: compensate changes in the production rate by manipulating the fresh feed rate of the limiting reactant. 2. Processes with a gas recycle and purge: keep the gas recycle flow constant at its maximum value. 3. Reactor heating/cooling: depending on the process a. If the product distribution is improved by operating at low temperatures, operate the reactor cooling system to achieve the lowest possible temperature. b. For complex reactions where the product distribution is insensitive to temperature, adjust the reactor temperature to maintain the recycle flow of the limiting reactant constant at its maximum possible value. c. For single reactions and liquid feeds, adjust the reactor temperature to achieve the maximum recycle flow of the limiting reactant (if steam for the recycle column is cheaper than incremental fuel) or maximize the temperature to obtain the largest possible conversion (if incremental fuel is cheaper than incremental steam). 4. Feed rate of the nonlimiting reactant: depending on the process a. Adjust the feed rate to satisfy a molar ratio constraint at the reactor inlet. b. If the product distribution improves as the molar ratio at the reactor inlet increases, adjust the feed rate to keep the recycle flow of the nonlimiting reactant constant at its largest possible value. c. If the product distribution is independent of the molar ratio and the nonlimiting reactant is a liquid, use a ratio controller to the limiting reactant. 5. Flash drum temperature: maximize the cooling water return temperature. Table 11. Preliminary Steady-State Control S t r u c t u r e variables variables operating manipulated operating manipulated XD,RC xB,SCa xB.PC Tflssh
FCW
xD,SC xD,PCa
RPC
RRC
vsc VPC
Rsc
Treated as noneconomic controlled variables.
temperature at the reactor exit (we eliminated the constraint on the reactor exit temperature by overdesigning the reactor, and in this analysis we neglect the constraint on the cooling water return temperature from the partial condenser in order to assess the importance of this constraint). The manipulative variables that we have available are listed in Table I1 (see also part 1). For the optimum steady-state control problem, we simply find the values of the manipulative variables that minimize the total operating costs for various values of the disturbances. Thus, we do not define a set of controlled variables. Instead we attempt to use the results of the analysis to define the controlled variables. The results from an optimum steady-state control analysis are presented in Table I11 for some of the manipulative variables. The results indicate that the recycle gas flow, RG,is always constant, which corresponds to the constraint on the gas recycle compressor capacity. Hence, we choose the gas recycle flow as one of our controlled variables. (If we choose a state variable as the controlled variable, it would be necessary to compute its optimum value on-line and then to change the controller set point; see Table IV.) The results, see Table IV, also indicate that when the production rate is greater than 90% of its design capacity, the toluene recycle column is driven to operate at its maximum vapor boilup (i.e., a constraint is encountered, which is the maximum heat duty available in the reboiler).
614 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 Table 111. Optimum Steady-State Control Results disturbances
optimum manipulated variables
PBQ
YFH
TCW
FfuelD
FG
FCWn
RPC
RGb
100.0 120.0 80.0 120.0 80.0 100.0 100.0 100.0 100.0
0.95 0.90 1.00 0.95 0.95 0.90 1.00 0.95 0.95
294.4 305.6 283.3 294.4 294.4 294.4 294.4 305.6 283.3
111.0 143.0 80.0 144.0 79.0 112.0 111.0 110.0 113.0
190.0 260.0 134.0 240.0 142.0 206.0 177.0 190.0 190.0
112.0 211.0 72.0 130.0 91.0 112.0 112.0 182.0 91.0
1.40 1.35 1.46 1.34 1.43 1.41 1.41 1.40 1.41
2600.0 2600.0 2600.0 2600.0 2600.0 2600.0 2600.0 2600.0 2600.0
In percent of design value. *Operation a t maximum compressor load.
Table IV. ODtimum Steady-State Control Results disturbances pea YFH TCW 100.0 120.0 80.0 120.0 80.0 100.0 100.0 100.0 100.0
0.95 0.90 1.00 0.95 0.95 0.90 1.00 0.95 0.95
optimum state variables YPH Tflash
X
0.654 0.688 0.615 0.688 0.631 0.648 0.648 0.648 0.648
294.4 305.6 283.3 294.4 294.4 294.4 294.4 305.6 283.3
disturbances pBa
100.0 120.0 80.0 120.0 80.0 100.0 100.0 100.0 100.0
Y FH 0.95 0.90 1.00 0.95 0.95 0.90 1.00 0.95 0.95
295.2 306.6 283.9 295.9 294.8 295.2 295.2 305.9 284.4
0.295 0.326 0.259 0.328 0.252 0.297 0.300 0.298 0.298
XB,PC
0.0054 0.0052 0.0059 0.0060 0.0060 0.0054 0.0054 0.0063 0.0053
optimum state variables
T,,
FGIFkT
RT
294.4 305.6 283.3 294.4 294.4 294.4 294.4 305.6 283.3
1.53 1.74 1.35 1.61 1.43 1.66 1.43 1.53 1.53
66.0 67.0 62.0 67.0 58.0 67.0 67.0 67.0 67.0
Tcwh 322.0 322.0 320.0 322.0 322.0 322.0 322.0 322.0 318.0
In percent of design value.
This directly translates into operating the recycle column at maximum distillate flow (i.e., maximum recycle toluene flow). If we choose the toluene recycle flow as a controlled variable and always maintain this value constant at its maximum value, we only pay a $1000/year penalty (for a rectangular distribution in the disturbances) as compared to the optimum operating cost. Hence, we choose the toluene recycle flow as a controlled variable. We also find that the reflux ratio in the product column, see Table 111, is very nearly constant, and if we maintain this as a constant, we only increase the heating and cooling costs of the column by 2%. However, we still prefer to use the product composition as a controlled variable in order to ensure that we can always meet our product specifications. The other manipulative variables shown in Table I11 all vary quite substantially with the disturbances. Hence, if we select any of these variables as controlled variables, it would be necessary to measure the most significant disturbances and to solve the optimum steady-state control problem on-line in order to adjust their set points. The optimum solutions for some of the state variables are shown in Table IV. We see that the bottoms composition from the product column (xe,pc)is essentially constant (there is only a small penalty if we maintain it exactly constant) and that the cooling water exit temperature from the partial condenser (Tcw,out) is normally at its constrained value. The end compositions for the other columns also are essentially constant (although these results are not included in Table IV). Hence, we also choose these as controlled variables. The optimum values of the conversion, purge composition, and the temperature of the flash
Table V. Proposed Steady-State Control Structure variables manipulated controlled maintain hydrogen/aromatics = 5 at reactor hydrogen feed inlet gas recycle operate compressor at maximum load furnace fuel keep toluene recycle flow rate constant or the toluene flow leaving the reactor constant cooling water keep return temperature at constraint quench flow rate keep reactor quench temperature constant reflux ratio (PC) keep product composition constant vapor rate (PC) keep bottoms composition constant vapor rate (RC) keep bottoms composition constant reflux ratio (SC) keep distillate composition constant vapor rate (SC) operate at maximum required load
drum all change with the disturbances. Hence, we avoid using these variables as controlled variables. Our new list of controlled and manipulative variables is given in Table V. We have not fixed the structure of the control system as yet, but we have established the variables that we want to include in the control system. That is, we expect that a regulatory control system which uses these variables will perform almost at the optimum steady-state conditions.
Completing the Synthesis of the Control System Now that we have developed systematic procedures for selecting sets of manipulative and controlled variables, we still need to synthesize a control structure and evaluate its dynamic performance. As a starting point we would assume that single-loop controllers will be installed be-
Ind. Eng. Chem. Res., Vol. 27, No. 4,1988 615 tween the manipulated variables and the controlled variables given in Table V. This option can be evaluated by determining the interaction between the loops using either the relative gain array (Bristol, 1966) or singular value decomposition (Klema and Laub, 1980, Roat et al., 1986). More advanced, decentralized multivariable control structures between the variables given in Table V can be synthesized by using the block relative gain array developed by Manousiouthakis et al. (1985). However, ratios of the manipulated variables could also be used in a control system structure if the simpler systems did not prove to be satisfactory. With these techniques we can identify poor control structures that we would want to discard and also identify promising candidates that should be considered when using a dynamic analysis. It is essential to evaluate the dynamic response and to proceed through the hierarchy of the control system synthesis procedure that was discussed in part 1of this series. Evaluating the dynamic response and stability of complete chemical plants with a control structure in place is a difficult task.
Conclusions The economic trade-offs that describe the optimum steady-state control problem (i.e., select the values of the manipulative variables that minimize the total operating costs for a given set of disturbances) often correspond to constrained optimizations (Le., the solution corresponds to one or more pieces of equipment operating at maximum capacity). By selecting controlled variables to correspond to these constraints, as well as the active process constraints, we obtain a set of controlled variables that we desire to maintain constant and that will cause the process to have almost the same performance as the optimum steady-state control action (for these variables), without the need for solving the entire optimum steady-state control problem on-line or measuring all of the disturbances. In addition, by understanding the nature of the economic trade-offs, it is often possible to develop heuristics for identifying the constrained controlled variables. Acknowledgment The authors are grateful to the National Science Foundation for providing financial support under Grant CPE-8105500.
Nomenclature F,, = cooling water flow rate to partial condenser FFT = toluene feed flow rate FG = hydrogen feed flow rate FQ= reactor quench flow rate H2/ T = hydrogen-to-tolueneratio at reactor inlet HDA = hydrodealkylation of toluene-to-benzeneprocess PB= production rate of benzene PC = product column R = reflux ratio RC = recycle column RG = gas recycle flow rate R, = minimum reflux ratio RT = toluene recycle flow rate SC = stabilizer column TAC = total annualized cost for the process Tcond= column condenser operating temperature T,, = cooling water inlet temperature Tfleeh = flash drum operating pressure TQ = reactor outlet quench temperature Treb= column reboiler operating temperature V = column vapor flow rate x = reactor conversion xB = bottoms composition xD = distillate composition YFH = hydrogen feed composition yPH = hydrogen purge composition Literature Cited Bristol, E. H. ZEEE Trans. Auto Control 1966,AC-11, 133. Fisher, W. R.; Doherty, M. F.; Douglas, J. M. Znd. Eng. Chem. Process Des. Deu. 1985a,24, 593. Fisher, W. R.; Doherty, M. F.; Douglas, J. M. Proceedings of the 1985 American Control Conference, Boston, MA, June, 1985b; p 293. Govind, R.; Powers, G. J. AZChE J. 1982,28,60. Klema, V. C.; Laub, A. J. ZEEE Trans. Auto. Control 1980, AC-25, 164. Manousiouthakis, B.; Savage, B.; Arkun,Y. Proceedings of the 1985 American Control Conference, Boston, MA, June, 1985; p 299. Morari, M.; Arkun, Y.; Stephanopoulos, G. AZChE J. 1980,26,220. Morari, M.; Stephanopoulos, G. AZChE J. 1980a,26, 232. Morari, M.; Stephanopoulos, G. AZChE J. 1980b,26, 247. Nishida, N.; Stephanopoulos, G.; Westerberg, A. W. AZChE J. 1981, 27, 321. Roat, S. D.; Downs, J. J.; Vogel, E. F.; Doss, J. E. Proceedings of Chemical Process Control 3, Asilomar, CA, Jan 1986.
Received for review July 15, 1986 Revised manuscript received July 23, 1987 Accepted November 19,1987
