A Shortcut Method for Controlled Variable Selection and Its

Rama V. Mahajanam, Alex Zheng,* and J. M. Douglas. Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003-3110...
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Ind. Eng. Chem. Res. 2001, 40, 3208-3216

A Shortcut Method for Controlled Variable Selection and Its Application to the Butane Alkylation Process Rama V. Mahajanam, Alex Zheng,* and J. M. Douglas Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003-3110

The importance of selecting controlled variables appropriately for an entire process has long been recognized. However, because of its combinatorial nature, finding an optimal set of controlled variables requires solving a large number of nonconvex optimization problems, a computationally intensive task for any realistic chemical process. In this paper, we propose a shortcut method to eliminate poor choices and to generate and rank attractive alternatives without solving optimization problems. The method is based on scaling of all of the candidate controlled variables so that they have similar effects on the steady-state profit. The procedure is illustrated on a simplified butane alkylation process. 1. Introduction The importance of selecting the controlled variables appropriately (or more generally control structure design) has long been recognized7 and can be illustrated by the following simple example. Other illustrating examples can be found in ref 13. Consider a continuous stirred tank reactor (CSTR) with a cooling jacket. If the properties of the inlet stream(s) are fixed, then there is 1 degree of freedom at steady state. Fixing the cooling duty would determine the reactor effluent compositions and everything else (assuming that process dynamics is stable and that there is a unique steady state) but so does fixing the reactor temperature (or a reactor composition). Clearly, each strategy differs because of uncertainty (e.g., modeling error and unmeasured disturbances), and the question is, which one may be preferred? The answer depends on many factors. For example, if the reactor is open-loop unstable, then controlling the cooling duty, and for some complex kinetics even controlling the reactor temperature, may not be acceptable. For many chemical processes, the cost due to selectivity loss dominates. One may want to control the reactor temperature as low as possible, regardless of uncertainty, to improve selectivity. The simplest strategy that may achieve this goal is to set (or control) the cooling duty at its maximum value. The controlled variable selection problem has been formulated by various researchers over the last 2 decades. Findeison et al.6 first introduced the concept of “feedback optimizing control” to select a set of controlled variables so that a process operates nearly optimally in the presence of unmeasured disturbances. Arkun3 used this concept for the selection of controlled variables and applied it to a gasoline polymerization plant. Morari et al.9 and Arkun and Stephanopoulos4 presented a unified formulation for the problem of synthesizing control structures for chemical processes and discussed the controlled variable selection problem. Skogestad and co-workers11-13 extended the work to include the implementation error associated with the controlled variables and used the term “self-optimizing * To whom correspondence should be addressed. E-mail: [email protected]. Fax: (413) 545-1647. Voice: (413) 5452916.

control” to describe the concept. The concept of “partial control” suggested by Arbel et al.2 can be seen as a variation of feedback optimizing control, where dominating controlled variables are identified based not on optimization but on process insights. Because the number of possible sets of controlled variables grows combinatorially with the number of candidate variables, which is typically much bigger than the number of controlled variables, it implies that finding an optimal set of controlled variables involves solving a large number of nonconvex optimization problems, a computationally expensive task for any realistic chemical process. The purpose of this paper is to develop a shortcut method so that poor choices can be eliminated quickly and that attractive alternatives are generated for further considerations. For simplicity, we have considered only disturbances and measurement errors in this paper. Parametric uncertainty can be handled by treating it as an unmeasured disturbance. On the other hand, structural model uncertainty is difficult to handle because we know of no appropriate way of describing the structural model uncertainty for nonlinear systems. Currently, we are exploring various ways of describing the structural model uncertainty for nonlinear systems. 2. Problem Formulation and Issues Given a flowsheet with n steady-state degrees of freedom and m available variables to choose from (m g n), in the base case (i.e., no disturbances, measurement errors, and modeling error), it does not matter which n of the m variables are chosen, as long as they form an independent set. Optimal operation in the base case, however, does not imply optimal operation in the presence of unmeasured disturbances, modeling error, and measurement errors. We would like to select a set of controlled variables so that the process operation remains close to optimal conditions in the presence of unmeasured disturbances, measurement errors, and modeling error. Mathematically, we want to solve the following optimization problem:

min max [Φopt(d,e,∆) - Φ(d,e,∆,CVs)] CVs d,e,∆ subject to constraints (1)

10.1021/ie9907066 CCC: $20.00 © 2001 American Chemical Society Published on Web 06/09/2001

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where CVs denote the controlled variables and their setpoints, d the unmeasured disturbances, e the measurement errors associated with the controlled variables, and ∆ the modeling error. Φ ∆ Φ(d,e,∆,CVs) ) denotes the steady-state profit, while Φopt ∆ Φopt(d,e,∆) ) denotes the optimal steady-state profit for given values of d, e, and ∆ (i.e., Φopt ) maxCVs Φ). (Notice that the value of maxCVs Φ does not depend on the choice of controlled variables as long as they form a basis and maxCVs Φ actually optimizes over the setpoints of any basic set of controlled variables.) Essentially, we are trying to find a set of controlled variables so that the steady-state profit is as close to the optimum as possible in the worst case. Clearly, other objective functions can also be used (e.g., the average value of the loss function (Φopt - Φ)). Because the setpoints for any set of independent controlled variables can be adjusted optimally for a measured disturbance, a measured disturbance does not affect the choice of controlled variables and is not considered. Notice that the optimization problem (1) is constrained. The constraints ensure that the resulting system is stabilizable (i.e., there exists a control system to stabilize the system), that hard constraints corresponding to physical and safety limits are satisfied, etc. The solution would not be meaningful without imposing these constraints. While conceptually simple, solving the constrained optimization problem (1) may be difficult for any realistic process because it is combinatorial and nonconvex. The number of possible sets of controlled variables grows exponentially with the total number of candidate variables. Because the number of candidate controlled variables is generally much larger than the number of controlled variables that can be chosen (i.e., m . n), finding an optimal set of controlled variables involves solving a large number of minimax (nonconvex) problems, a difficult task for any realistic chemical process. For example, for the butane alkylation process to be discussed later, more than 19 000 minimax problems, each of which is nonconvex and large-scale and takes more than 24 h of CPU times on a Pentium II computer, would have to be solved to find the optimal solution. Recognizing this difficulty, Skogestad et al.11 proposed using second-order Taylor series expansion around the nominal optimal operating conditions to simplify the optimization problem. However, this Taylor series expansion has a limited range of validity. In this paper, we propose a shortcut method for eliminating poor sets of controlled variables and for generating and ranking attractive alternatives without solving the optimization problem (1). In section 4, the method is applied to a simplified butane alkylation process. 3. A Shortcut Method The basic idea behind the method is to significantly simplify the optimization problem (1) by making several reasonable assumptions and by scaling all of the candidate controlled variables so that they would have similar impacts on the profit function (i.e., Φ). The individual steps of the method are briefly described below. For brevity, we assume that the model is perfect (i.e., ∆ ) 0). (This assumption is clearly unrealistic. Extending the results in this paper to handle parameteric uncertainty is straightforward and can be done by simply treating the uncertainty as unmeasured disturbances. The difficulty arises when structural

model uncertainty is considered, for we know of no appropriate way of describing this type of uncertainty for nonlinear systems.) Step One: Determination of the Steady-State Degrees of Freedom. The number of variables that need to be controlled at steady-state equals the steadystate degrees of freedom. Step Two: Simplifying Assumptions and Variable Scaling. Two simplifying assumptions are made to significantly reduce the computational complexity of the optimization problem (1). Candidate variables are then scaled so that their impacts on the loss function can be directly compared without solving any optimization problem. Step Three: Optimal Set of Controlled Variables and Their Setpoints. Attractive alternative sets of controlled variables are determined, without solving the optimization problem (1). For each attractive set of controlled variables, their optimal setpoints may be determined by solving the optimization problem (1). Thus, instead of solving the optimization problem (1) for each set of controlled variables, the optimization problem (1) is solved only for the attractive ones. 3.1. Step One: Determination of the SteadyState Degrees of Freedom. To specify the number of variables we need to control at steady state for a given flowsheet, we need first to determine the steady-state degrees of freedom (DOF). The steady-state DOF can be determined via the following relations.

DOF ) number of variables number of independent equations This requires an exhaustive listing of all of the variables and governing equations of all of the units in the process. It is very easy to overlook a variable or a defining equation when computing the degrees of freedom by this method. Also, considering the large number of variables and equations involved for a realistic process, this is a very time-consuming approach. To overcome the above difficulties, several researchers have proposed simple procedures to determine the steady-state DOF. For example, Ponton10 developed a simple procedure to determine the number of potential adjustments which include the inventory manipulators. Luyben8 proposes to count the number of legitimate control valves in the process. Here we propose some heuristics in determining the steady-state DOF. We assume perfect inventory control (e.g., level and pressure control) because we are interested in the steadystate DOF; for instance, for a simple distillation column, with the assumption that the reboiler and reflux drum levels and the column pressure are perfectly controlled, for a fixed feed, we have two steady-state DOF. A process flowsheet can be decomposed into smaller subsystems based on the way individual units are connected to each other. The heuristics described below hold only for fixed feed streams. If a feed stream is not fixed, then we have up to nc + 2 additional DOF (i.e., feed composition, feed flow, feed temperature, and feed pressure). Here nc denotes the number of components in the stream. Interconnections between two units can be classified into three broad types (both units in Figure 1 may contain other streams which are not shown in Figure 1 for brevity; for example, a stream splitter can also be considered as a unit): Series Connections (Figure 1a). The output of one unit is the input to the next. The DOF for the overall

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Figure 1. Three types of interconnections of units.

system equals the sum of DOF for the two units, i.e., DOF ) DOF1 + DOF2. Parallel Connections (Figure 1b). Because the input is split into two, the DOF for the overall system is increased by 1, i.e., DOF ) DOF1 + DOF2 + 1. Recycle Connections (Figure 1c). The DOF for the overall system equals the sum of DOFs for the two units, i.e., DOF ) DOF1 + DOF2. Notice that this relation holds both for material recycle (i.e., mixing is involved) and for energy recycle (i.e., mixing may not be involved). By using the above three rules, we can successively simplify any flowsheet by combining two subsystems into one subsystem. We can thus determine the steadystate DOF for the whole flowsheet. However, because we have assumed that the properties for all of the feed streams are fixed and not all of the properties for the fresh feeds to the flowsheet may be fixed, we need to account for the DOF generated by the fresh feeds. 3.2. Step Two: Simplifying Assumptions and Variable Scaling. Because the inherent combinatorial and nonconvex nature of the optimization problem (1) cannot be changed, we make two simplifying assumptions (A-1 and A-2) to significantly reduce its size. The variable scaling allows us to rank each of the candidate controlled variables according to its economic impact on the loss function. Simplifying Assumption A-1: Disturbance Classification. For any realistic process, a large number of disturbances exist and often many of them have little impact economically. Depending on how a disturbance affects the optimal steady-state profit (i.e., Φopt), we can classify it as either dominant or insignificant. Morari et al.9 also classified disturbances similarly. Ignoring the insignificant disturbances would greatly simplify the max operation of the optimization problem. If we further assume that the worst case occurs at a corner, we only have to evaluate the max operation over the corner points formed by the dominant disturbances, i.e.,

max Φopt - Φ ddom∈{corner points},e It should be emphasized that the optimal solution is not necessarily a corner point. Simplifying Assumption A-2: Variable Association. Given n steady-state DOF and m candidate variables to choose from, the total number of possibili-

ties is Cm n ∆ m!/[n!(m-n)!] and grows exponentially with ) m. To solve the optimization problem (1), we need to m solve Cn nonconvex optimization problems. This may be impractical because, for a realistic process, m can be much larger than n. To resolve this difficulty, we decompose a system into subsystems and associate candidate controlled variables with a subsystem. While there is no unique way of decomposing a system, one can treat a unit operation as a subsystem. If desirable, one can further decompose a unit operation into several subsystems. This can be best illustrated by an example: Consider a simple distillation column, with 2 steady-state DOF and with the following candidate controlled variables: yd, RR, L, D, xb, VB, V, and B. Without associating any candidate variable with a subsystem, we have C82 ) 28 sets of controlled variables. However, if we divide the column into two subsystems (top and bottom) and associate yd, RR, L, and D with the top and xb, VB, V, and B with the bottom, then we would have C41 × C41 ) 16 sets of controlled variables. While the reduction in the number of possible sets of controlled variables is not very significant for this particular example, it can be significant for a realistic flowsheet. For example, for the butane alkylation process to be discussed later, this assumption reduces this number from 19 448 to 192. Candidate Controlled Variable Scaling. When a candidate controlled variable is associated with a subsystem, the number of minimax problems to be solved is greatly reduced. However, still a large number of minimax problems need to be solved. We now further simplify the optimization problem by scaling all of the candidate controlled variables so that they have similar impacts on the steady-state profit (Φ). Each candidate controlled variable is scaled via the following formula:

xˆ ) s(x - xj)

(2)

where x denotes the original variable, xˆ is the scaled variable, xj is the optimal setpoint for the variable in the base case, and s is the scaling for the variable. s is determined as follows: First the range of the variable is determined from the optimal values associated with Φopt for all possible combinations of the dominant disturbances (i.e., all corner points) and no measurement error. Then the maximum measurement error associated with each variable is added to the maximum value to get the upper bound for the variable (xmax) and is subtracted from the minimum value to give the lower bound for the variable (xmin). We then use this range to scale the variables by computing maxCVs Φ(0,0,0,CVs) when the variable is not chosen optimally but equals the two extremes (i.e., xmax and xmin) in the base case, i.e.,

J1 ) Φopt(0,0,0) -

max Φ(0,0,0,CVs) CVs,x)xmax

J2 ) Φopt(0,0,0) -

max Φ(0,0,0,CVs) CVs,x)xmax

Finally, s is calculated from the following formula:

s ) max

(|

||

|)

J1 J2 , xmax - xj xmax - xj

(3)

Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3211 Table 1. Kinetic Parameters parameter

rxn 1

rxn 2

preexponential activation energy (Btu/lb‚mol) heat of reaction (Btu/lb‚mol)

9.6 × 2.8 × 104 -3.9 × 104

2.4 × 1017 3.5 × 104 -2.9 × 104

1013

Table 2. Unmeasured Disturbances d1 d2 d3 Figure 2. Schematics of a simplified butane alkylation process.

3.3. Step Three: Optimal Set of Controlled Variables and Their Setpoints. With the variable scaling determined by (3), any scaled variable with a magnitude of 1 would have a similar impact on the steady-state profit. Ideally, we would like to choose a set of controlled variables so that their scaled variables have minimum impact on the loss function Φopt - Φ (i.e., Φ is as close to Φopt as possible). Thus, the variables whose scaled values are small may be preferred. Therefore, without solving the optimization problem (1), we can readily eliminate all of the candidate variables whose scaled values have large magnitudes from further consideration and generate (and rank) attractive alternatives based on the scaling. For each attractive alternative, we can solve the optimization problem (1) for optimal setpoints. If this optimization problem is too complex to be solved practically, we may choose the optimal value in the base case, xj, as the setpoint for a variable. Clearly, this value is not necessarily the optimal setpoint and may violate operating constraints in the presence of unmeasured disturbances and measurement errors. It is worthwhile to point out again that the optimization problem (1) is solved subject to hard operating constraints (e.g., flooding of columns), stability constraint, etc.

rxn #1:

i-butane + 1-butene f i-octane

rxn #2:

i-octane + 1-butene f n-dodecane

The kinetic parameters are given in Table 1. i-octane is the product, while n-dodecane is the undesirable

nominal

range

0.2 77 77

0.01-0.4 47-107 47-107

byproduct. The 1-butene feed has a considerable amount of propane as an impurity, which is removed in the first column. The unreacted reactants (mainly i-butane) are separated from the products in the second column and recycled to the reactor. The product is separated from the byproduct in the last column. The product purity must be at least 0.999 (i.e., 99.9%). The details for the flowsheet (e.g., equipment sizes) are given in Appendix A. The flowsheet is modeled in HYSYS v1.5 with the fluid package Peng-Robinson. While many disturbances exist, we consider only four of them (one measured and three unmeasured) for illustration purposes. They are the production rate (measured disturbance), the inlet temperatures for both fresh feeds, and the propane impurity in the 1-butene feed. Because the setpoints of the selected controlled variables can be adjusted optimally when a measured disturbance enters the process, the measured disturbance would have no impact on the objective function regardless of the choice of controlled variables. Thus, we would not consider the production rate change in the selection of controlled variables. The nominal values and range of each of the unmeasured disturbances are listed in Table 2. 4.2. Application of the Shortcut Method. The steady-state profit, Φ, is computed as follows:

profit ($/h) ) revenue from products raw material cost - operating cost ) $45Pi-C8 + $13.88Pn-C12 +

4. Illustrating Example: A Butane Alkylation Process The shortcut method developed in the previous section is applied to the simplified butane alkylation process shown in Figure 2. Notice we have assumed that the only impurity in the feeds is propane. Clearly, this is not the case in practice, and the feeds likely contain other impurities such as n-butane. However, those details are not essential because the main purpose of this example is to illustrate the basic ideas behind the method and its effectiveness. 4.1. Process Description. A series-parallel reaction sequence occurs in the liquid phase in a CSTR with a refrigeration system to provide the cooling. The main reactions, both second order and elementary, are as follows (clearly, other reactions (e.g., polymerization of 1-butene) are possible which we have ignored for brevity; again, we feel that those details are not essential for demonstrating the basic ideas behind the shortcut method and its effectiveness):

disturbance xC3 in 1-butene feed Tin of 1-butene feed (°F) Tin of i-butane feed (°F)

$3.82PC3fuel - $18.59Ri-C4 - $11.65R1-but (energy cost)(3)(condenser duty + reboiler duty) - (electricity cost) (pump power) - (refrigeration cost) (refrigeration duty) Energy cost is taken as $4.0/million Btu, electricity cost is at $8.28/million hp, and refrigeration cost is based on $6.94/million Btu from ref 1. The prices of raw materials and products are obtained from ref 5. 4.2.1. Step One: Steady-State DOF. We now apply the heuristics discussed in section 3.1 to determine the steady-state DOF for the flowsheet. First, the DOF for each unit are determined assuming fixed properties of feed streams: unit

DOF

unit

DOF

mixer reactor FEHE

0 1 0

C3 column C4 column P column

2 2 2

The DOF for the subsystem consisting of the reactor and the feed-effluent heat exchanger equal 1 (recycle connection). Combining this subsystem with the C3 column

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i.e.,

Table 3. Impact of Each Disturbance on Profit d1 d2 d3

disturbance

min %

max %

xC3 in 1-butene feed Tin of 1-butene feed Tin of i-C4 feed

32.44 -0.06 -0.02

-53.36 0.06 0.01

Table 4. Candidate Controlled Variables no.

candidate CV

measurement error (e)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

T (reactor) (°F) Q (reactor) (Btu/h) RR (C3 column) yd,i-C4 (C3 column) L (C3 column) xb,C3 (C3 column) VB (C3 column) RR (C4 column) yd,224 (C4 column) L (C4 column) D (C4 column) VB (C4 column) xb,i-C4 (C4 column) VB (product column) xb,224 (product column) Tot i-C4 MR

1 65 190 0.67 0.000 127 61 0.000 56 0.016 0.002 0.000 002 25 241 0.258 0.000 005 5 0.045 0.000 055 253 0.31

yields a new subsystem with 3 DOF (series connection). Thus, the DOF for the whole flowsheet equals 7, if the properties for the two feed streams are fixed. However, the flows of the two fresh feeds can be manipulated (i.e., not fixed). Thus, the whole flowsheet has 9 DOF. Notice if other properties of the fresh feed streams can be manipulated (e.g., temperature and compositions), we need to account for those DOF. For brevity, we assume that 2 DOF are used to control the production rate and the product purity at 0.999. Therefore, we need to choose seven additional controlled variables. We have chosen 17 candidate controlled variables (Table 4). (Note that this is not unique and that one may include many other variables (e.g., reactor effluent composition, column tray temperatures, etc.).) Tot i-C4 denotes the total i-C4 flow to the reactor, while MR is the molar ratio of i-butane to 1-butene for the reactor feed. There are C17 7 ) 19 448 possible sets of controlled variables. Thus, to solve the optimization problem (1) rigorously, we have to solve 19 448 minimax problems. However, without the simplifying assumptions discussed below, solving even one minimax problem for this example is computationally expensive (it takes more than 24 CPU h on a Pentium II computer). 4.2.2. Step Two: Simplifying Assumptions and Variable Scaling. We now apply the simplifying assumptions to reduce the complexity of the optimization problem (1) and variable scaling to generate attractive alternatives. Simplifying Assumption A-1: Disturbance Classification. The effect of each disturbance on the optimal steady-state profit is computed and given in terms of percent variation from the base case in Table 3. As one can see, d2 and d3 have minimal impacts on the optimal steady-state profit and are therefore ignored at this stage. It should be emphasized that they should be considered when we consider process dynamics. If we assume that the worst case occurs at a corner, then the max operation simplifies to

max [Φopt(d1) - Φ] d1){0.01,0.4} This is considerably simpler than the original problem,

max [Φopt(d1,d2,d3) - Φ] 0.01ed1e0.4, 47ed2,d3e107 Simplifying Assumption A-2: Variable Association. For the 17 candidate controlled variables (Table 4), there are 19 448 possible sets of controlled variables. The maximum measurement error associated with each variable is determined based on the following rules of thumbs: 1. Flow: (1%. 2. Composition: (1%. 3. Temperature: (1 °F. (Because the model used is nonlinear, it is clear that the magnitude of the potential measurement error would affect the ranking determined later. However, the quantitative effect would depend on the process and variability of the variable and is thus difficult to characterize in general. In the extreme case where the process is linear, then the ranking would not be affected. For this example, we have found the ranking to be insensitive to the magnitude of the measurement error.) For example, 1% measurement error in the C3 column reflux flow is 61 lb‚mol/h. (For brevity, we have set the measurement error for a variable based on the nominal value.) Thus, the reflux flow would vary from 6000 to 6122 lb‚mol/h even if we attempt to maintain it at 6061 lb‚mol/h. To associate candidate controlled variables, we divide the process flowsheet into the following subsystems and associate the available candidate controlled variables with them as follows. 1. Reactor: T (Q). (For this process, the reactor is open-loop unstable. Thus, controlling the duty (Q) would violate the constraint that the system is stabilizable. The temperature (T) is chosen as the controlled variable for this subsystem.) 2. C3 column top: RR, yd,i-C4, L. 3. C3 column bottom: xb,C3, VB. 4. C4 column top: RR, yd,224, L, D. 5. C4 column bottom: VB, xb,i-C4. 6. Product column bottom: VB, xb,224. 7. i-C4 inventory: Tot i-C4, MR. Each subsystem has 1 steady-state DOF. Thus, we need to select one variable from the list of candidate controlled variables available for each subsystem. There are a total of C11 × C31 × C21 × C41 × C21 × C21 × C21 ) 192 possible sets of controlled variables. Hence, the total sets of controlled variables to be considered for the minimization operation has been reduced from 19 448 to 192. Variable Scaling. We will illustrate how a variable is scaled by considering the reflux flow (L) for the C3 column. The optimal value for L in the base case is 6061 lb‚mol/h and varies from 275 to 12 370 when subject to d1, the only dominant unmeasured disturbance. With the measurement error of 61, the range becomes 21412 431. Thus, L h ) 6061 lb‚mol/h, Lmin ) 214 lb‚mol/h, and Lmax ) 12 431 lb‚mol/h. In the base case, L is chosen as the controlled variable for the subsystem C3 column top. For each of the other subsystems, one of the candidate controlled variables associated with it is chosen as the controlled variable. The optimal steady-state profit in the base case is $8300/h (i.e., Φopt(0,0,0) ) $8300/h). With L fixed at 214 lb‚mol/h, which is not optimal, the steady-state profit optimizing over the rest of controlled variables becomes -$14 400/h (i.e., J1 ) 8300 - (-14 400) ) 22 700). With

Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3213 Table 5. Optimum Operating Conditions

Table 7. Top 16 Alternatives (Shortcut Method) xopt(d)

subsystem

xnom

+C3

-C3

19.2 6061 0.0127 0.811 0.0565 0.1040 2508 2.07 × 10-4 2.411 × 104 12.64 0.000 55 0.0055 2.15 25 272 15.1 8287

14.8 12 370 0.007 65 1.06 0.0622 0.1036 2493 2.065 × 10-4 2.406 × 104 12.62 0.000 552 0.005 55 2.14 25 228 14.8 3865

24.3 275 0.0147 0.587 0.0561 0.1041 2518 2.07 × 10-4 2.418 × 104 12.69 0.000 545 0.005 48 2.16 25 346 15.2 10 975

variable

C3 column top

RR L yd,i-C4 C3 column VB bottom xb,C3 C4 column top RR L yd,224 D C4 column VB bottom xb,i-C4 product column xb,224 bottom VB i-C4 inventory Tot i-C4 MR profit ($/h)

Table 6. Scaled Candidate Controlled Variables C3 column top C3 column bottom C4 column top

C4 column bottom P column bottom i-C4 inventory

d1 ) 0.4

d1 ) 0.01

max |xˆ |

RR* L* yd,i-C4 VB* xb,C3 RR L yd,224 D xb,i-C4 VB xb,224 VB Tot i-C4 MR*

-191 22 700 -2.93 717 5.73 -1.17 -2.63 -0.23 -2.06 0.76 -4.19 1.33 -3.47 -6.06 -21.3

221 -20 800 1.16 -645 -0.43 0.29 1.75 0.00 2.89 -1.9 10.5 -0.53 1.33 10.2 7.9

221 22 700 2.93 717 5.73 1.17 2.63 0.23 2.89 1.9 10.5 1.33 3.47 10.2 21.3

L fixed at 12 431 lb‚mol/h, the optimal profit becomes $8100/h (i.e., J2 ) 8300 - 8100 ) 200). Thus, the scaling for L is

s ) max

(|

||

J1 J2 , Lmax - L h Lmin - L h

|)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

T T T T T T T T T T T T T T T T

C3 top

C3 bottom

C4 top

C4 P bottom bottom ∑|xˆ |

yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4

xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3

yd,224 RR yd,224 L D RR L D yd,224 RR yd,224 L D RR L D

xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB VB VB VB VB VB

xb,224 xb,224 VB xb,224 xb,224 VB VB VB xb,224 xb,224 VB xb,224 xb,224 VB VB VB

12.1 13.1 14.3 14.5 14.8 15.2 16.7 16.9 20.7 21.7 22.9 23.1 23.4 23.7 25.3 25.5

Table 8. Top 16 Alternatives (Rigorous Method)

CV

subsystem

ranking reactor

) 3.6

When subject to the unmeasured disturbance d1, the optimal values of the C3 reflux flow varies from 275 to 12 370 lb‚mol/h. Thus, the scaled variables vary from -20 800 to 22 700, respectively. The rest of the candidate controlled variables are scaled following the same procedure. The optimum operating conditions in the base case and for the dominant disturbance are listed in Table 5. They are scaled, and the resulting scaled values of all of the candidate controlled variables are listed in Table 6. A look at this table reveals that the variables marked with an asterisk can be safely eliminated from further consideration. With the elimination of the above variables, the number of possible combinations of controlled variables is further reduced to 16. 4.3. Step Three: Optimal Sets of Controlled Variables and Their Setpoints. With remaining variables, all alternatives are generated and ranked according to the sum of their scaled values (Table 7). Clearly, if we choose a variable with the smallest scaled value for each subsystem, the resulting set of controlled variables would have the smallest sum and can be explored further first. It should be emphasized that this ranking ignores the interaction among controlled variables and that the set may not be optimal. To get some feeling on how reasonable the results from the shortcut methods are, we have also solved the

SH ranking ranking reactor 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 4 2 5 8 3 6 7 9 12 11 15 13 10 16 14

T T T T T T T T T T T T T T T T

C3 top

C3 bottom

C4 top

yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4

xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3

yd,224 L RR D D yd,224 RR L yd,224 L yd,224 L D RR D RR

C4 P bottom bottom xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB VB VB VB VB VB

xb,224 xb,224 xb,224 xb,224 VB VB VB VB xb,224 xb,224 VB VB xb,224 xb,224 VB VB

EL (%) 0.57 0.57 0.6 0.6 0.6 0.6 0.6 0.6 0.62 0.62 0.62 0.65 0.65 0.65 0.68 0.68

Table 9. Optimal Setpoints subsystem reactor C3 bottom C3 top C4 top

i-C4 inventory C4 bottom P bottom

variable

final setpoint

T (°F) xb,C3 yd,i-C4 RR L D yd,224 Tot i-C4 flow xb,i-C4 xb,224 VB

20 0.056 07 0.009 0.1036 2493 2.406 × 104 0.000 206 5 25 228 0.000 55 0.0055 2.15

optimization problem (1) for all 192 possible sets of controlled variables by assuming A-1 (i.e., the worst case occurs at a corner formed by economically significant disturbances). The top 16 alternatives (a partial list is given in Appendix B), ranked by percent change in the profit from the optimal (i.e., maxd1 100(Φopt - Φ)/Φopt]) in the worst case, are listed in Table 8; notice that the next best alternative has a quite larger economic loss (2.7% versus 0.68%). A comparison of Tables 7 and 8 shows that all of the 16 alternatives are identical. This comparison shows that the shortcut method can be very effective in solving the controlled variable selection problem (1). Again, we emphasize that solving the optimization (1) is only feasible with assumption A-1 and takes a couple of CPU hours on a Pentium II computer with the HYSYS model. For each attractive set of controlled variables, we need to determine their optimal setpoints. While it is straightforward to solve the optimization problem (1) for a given set of controlled variables, we will discuss how to use process insights to reduce the size of the optimization

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Table 10. Worst [maxd1 (Φopt - Φ)/Φopt] and Average Profit Losses [avgd1 (Φopt - Φ)/Φopt] for Various Sets of Controlled Variables (nf ) Not Feasible) S no. reactor 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

C3 top

C3 bottom

C4 top

yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 yd,i-C4 L RR L RR yd,i-C4 L RR L RR yd,i-C4 L RR L RR yd,i-C4 L RR L RR L L L L L L L L L yd,i-C4 L yd,i-C4

xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 VB VB VB xb,C3 xb,C3 VB VB VB xb,C3 xb,C3 VB VB VB xb,C3 xb,C3 VB VB VB xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 VB VB VB

D yd,224 RR D yd,224 RR D yd,224 RR D yd,224 RR L L L L L L L L L L L L L L L L L L L L L L L L D yd,224 RR D yd,224 RR D yd,224 RR D D yd,224

C4 P bottom bottom worst ave S no. reactor xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB xb,i-C4 VB xb,i-C4 VB xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB VB VB xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB VB VB xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB VB VB VB xb,i-C4 xb,i-C4 xb,i-C4

xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 VB VB VB VB VB VB xb,224 xb,224 VB VB xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 VB VB VB VB VB VB VB VB VB VB xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 VB VB VB xb,224 xb,224 xb,224

0.6 0.57 0.6 0.65 0.62 0.65 0.6 0.6 0.6 0.68 0.62 0.68 0.57 0.62 0.6 0.65 3.34 6.54 nf nf nf nf nf nf nf nf 3.60 6.57 nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf 3.34 7.94 2.62

0.48 0.41 0.43 0.54 0.46 0.48 0.5 0.43 0.44 0.56 0.47 0.56 0.5 0.52 0.51 0.54 2.35 5.03 nf nf nf nf nf nf nf nf 2.52 5.07 nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf 2.35 5.14 2.31

problem. Let us focus our discussion on alternative #1 in Table 7. First, we look into the process constraints that may be violated in the presence of unmeasured disturbances. The two process constraints encountered are the maximum refrigeration duty and the column flooding limits. 1. The maximum refrigeration duty, dictated by the capacity of the compressor of the refrigeration system at the reactor, is 72 million Btu/h. Like many chemical processes, the operating cost for the butane alkylation process is dominated by the selectivity loss. To minimize the operating cost, we may want to minimize the cost due to selectivity loss. The selectivity loss can be minimized by operating the reactor at low temperatures, which can be achieved by setting the cooling duty at its maximum value. Unfortunately, this is not possible because the reactor is open-loop unstable. The reactor temperature needs to be controlled. The setpoint for the reactor temperature is chosen such that the cooling duty constraint is not violated in the worst case. The value is found to be 20 °F.

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76. 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

C3 top

C3 bottom

C4 top

L yd,i-C4 L yd,i-C4 L yd,i-C4 L yd,i-C4 L RR RR RR RR RR RR RR RR RR RR RR RR yd,i-C4 L yd,i-C4 L yd,i-C4 L yd,i-C4 L yd,i-C4 L yd,i-C4 L RR RR RR RR RR RR RR RR RR RR RR RR L L L

VB VB VB VB VB VB VB VB VB xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 xb,C3 VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB xb,C3 xb,C3 xb,C3

yd,224 RR RR D D yd,224 yd,224 RR RR D yd,224 RR D yd,224 RR D yd,224 RR D yd,224 RR D D yd,224 yd,224 RR RR D D yd,224 yd,224 RR RR D yd,224 RR D yd,224 RR D yd,224 RR D yd,224 RR D yd,i-C4 RR

C4 P bottom bottom worst ave xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB VB VB VB xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB VB VB VB xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB xb,i-C4 xb,i-C4 xb,i-C4 VB VB VB xb,i-C4 xb,i-C4 xb,i-C4

xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB VB xb,224 xb,224 xb,224 xb,224 xb,224 xb,224 VB VB VB VB VB VB VB VB VB

6.54 2.80 6.52 nf nf 4.67 nf 5.06 nf nf nf nf nf nf nf nf nf nf nf nf nf 3.55 8.10 2.96 6.62 2.88 6.70 nf nf 4.75 nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf

4.31 2.41 4.31 nf nf 3.21 nf 3.25 nf nf nf nf nf nf nf nf nf nf nf nf nf 2.53 5.24 2.51 4.36 2.47 4.42 nf nf 3.30 nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf nf

2. While the C3 and product columns operate well below their flooding limits, it was found to be profitable to operate the C4 column near its flooding limit. The reason for this is as follows: The selectivity to the desired product, isooctane, was found to be greater at lower conversions. Increasing selectivity implies increasing profitability. A lower conversion, however, implies higher recycle rates. Thus, we operate at high recycle rates (i.e., high distillate flow rates of the C4 column), limited only by the flooding limit of this column. Thus, the optimal operating conditions should occur at this constraint. It is found that, for certain values of the setpoints for xb,C3, yd,224, and Tot i-C4 flow, the C4 column flooding constraint is violated. Because the optimal point should occur at the constraint for this system, the minimum value of the optimum operating conditions listed in Table 5 is selected as the setpoint for each of these variables to ensure that flooding is not violated in the worst case. These are found to be xb,C3 ) 0.0561, yd,224 ) 2.065 × 10-4, and Tot i-C4 flow ) 25 228 lb‚mol/h, respectively.

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3. The product purity constraint places some constraint on the maximum amount of i-C4 in the product column feed. Because there is no incentive for overpurifying the product and i-butane is cheaper than the product, we want the product to contain as much i-butane as possible. Thus, the setpoint for xb,i-C4 is chosen to be 0.000 55, the maximum allowable i-butane in the product. (We have neglected the purity of C12 in the product. This is reasonable because the separation of C8 and C12 is relatively cheap.) With the setpoints for these five variables chosen based on process insights, the optimization problem is now much simpler and needs to be solved over the setpoints for the two remaining variables in this set, yd,i-C4 and xb,224. The optimal setpoints are found to be yd,i-C4 ) 0.009 and xb,224 ) 0.0055. 5. Conclusions A shortcut method has been developed for selecting controlled variables for a plantwide control problem. The method uses several reasonable assumptions and scales all of the candidate controlled variables so that they would have similar impacts on the steady-state profit. The method eliminates poor choices from further consideration and generates attractive alternatives without solving the optimization problem (1), which is impractical for any realistic process. We illustrated the method on a simplified butane alkylation process, compared the results from the shortcut method with actual solutions of (1) with assumption A-1, and found that the shortcut method gave almost identical results. It should be emphasized that, in the determination of economically attractive sets of controlled variables, process dynamics has been ignored; only system stabilizability has been enforced as a constraint. For each attractive set of controlled variables, we need to compute the cost associated with dynamic controllability. Thus, the most attractive set of controlled variable based on the steadystate consideration may not be optimal if process dynamics is considered. A method on how this may be done is described in ref 14. Acknowledgment Financial support from the National Science Foundation (Grant CTS-9713599) is gratefully acknowledged. Appendix A: Design and Nominal Parameters for the Butane Alkylation Process The equipment sizes and operating conditions are given below. 1. Fresh feeds (base values). 1.1. i-C4 feed. (i) Flow rate: 1163 lb‚mol/h. (ii) Composition: xi-C4 ) 1.0. (iii) Temperature: 77 °F. (iv) Pressure: 126 psia. 1.2. 1-Butene feed. (i) Flow rate: 1556 lb‚mol/h. (ii) Composition: x1-But ) 0.8, xC3 ) 0.2. (iii) Temperature: 77 °F. (iv) Pressure: 126 psia. 2. Reactor volume: 17 000 ft3. 3. Feed effluent heat exchanger. 3.1. Overall heat-transfer coefficient (U): 50 Btu/(h‚ ft2‚°F)

3.2. Heat-transfer area: 70 000 ft2. 4. C3 column. 4.1. Partial reboiler. 4.2. Total condenser. 4.3. Total number of trays (including the condenser and reboiler): 60. 4.4. Feed tray number (from the top): 14. 4.5. Condenser pressure: 237 psia. 4.6. Total pressure drop: 11.5 psia. 4.7. Tray efficiency: 100%. 5. C4 column. 5.1. Partial reboiler. 5.2. Total condenser. 5.3. Total number of trays (including the condenser and reboiler): 15. 5.4. Feed tray number (from the top): 8. 5.5. Condenser pressure: 126 psia. 5.6. Total pressure drop: 10 psia. 5.7. Tray efficiency: 100%. 6. Product column. 6.1. Partial reboiler. 6.2. Total condenser. 6.3. Total number of trays (including the condenser and reboiler): 10. 6.4. Feed tray number (from the top): 6. 6.5. Condenser pressure: 20 psia. 6.6. Total pressure drop: 5 psia. 6.7. Tray efficiency: 100%. Appendix B An exhaustive list of the top 96 of the 192 sets of controlled variables, the average and worst case economic penalty over all combinations of the dominant disturbance associated with each set, is listed below. The Tot i-C4 flow is controlled in all of these 96 cases. Nomenclature C3 ) propane C8 ) isooctane C12 ) n-dodecane d ) disturbance D ) distillate flow e ) measurement error i-C4 ) i-butane 1-C4 ) 1-butene L ) reflux flow MR ) molar ratio of i-C4 to 1-C4 for the reactor feed Q ) reactor duty RR ) reflux ratio T ) reactor temperature VB ) vapor boilup xb ) bottoms composition xb,C3 ) propane composition in the C3 column bottoms xb,i-C4 ) i-C4 composition in the C4 column bottoms xb,224 ) C8 composition in the product column bottoms yd ) distillate composition yd,i-C4 ) i-C4 composition in the C3 column distillate yd,224 ) C8 composition in the product column distillate ∆ ) modeling error Φ(d,e,∆,CVs) ) steady-state profit for a given set of controlled variables and their setpoints Φopt(d,e,∆) ) optimal steady-state profit

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(2) Arbel, A.; Rinard, I. H.; Shinnar, R. Dynamics and control of fluidized catalytic crackers. 3. designing the control system: Choice of manipulated and measured variables for partial control. Ind. Eng. Chem. Res. 1996 35, 2215-2233. (3) Arkun, Y. Design of Steady-State Optimizing Control Structures for Chemical Processes. Ph.D. Thesis, The University of Minnesota, Minneapolis, MN, 1979. (4) Arkun, Y.; Stephanopoulos, G. Studies in the synthesis of control structures for chemical processes: Part IV. design of steady-state optimizing control structures for chemical process units. AIChE J. 1980, 26 (6), 975-991. (5) Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill: New York, 1988. (6) Findeisen, W.; Bailey, F. N.; Brdys, M.; Malinowski, K.; Tatjewski, P.; Wozniak, W. Control and Coordination in Hierarchical Systems; Wiley & Sons: New York, 1980. (7) Foss, A. S. Critique of chemical process control theory. AIChE J. 1973, 19, 209-214. (8) Luyben, W. B. Design and control degrees of freedom. Ind. Eng. Chem. Res. 1996, 35, 2204-2214. (9) Morari, M.; Arkun, Y.; Stephanopoulos, G. Studies in the synthesis of control structures for chemical processes. part i.

formulation of the problem. process decomposition and the classification of the control tasks. analysis of the optimizing. AIChE J. 1980, 26, 220-259. (10) Ponton, J. W. Degrees of freedom analysis in process control. Chem. Eng. Sci. 1994, 49 (13), 2089-2095. (11) Skogestad, S.; Halvorsen, I. J.; Morud, J. C. Self-optimizing control: The basic idea and Taylor series analysis. AIChE Annual Meeting, Miami, FL, 1998. (12) Skogestad, S.; Larsson, T. A review of plantwide control. Preprint, 1998. (13) Skogestad, S.; Postlethwaite, I. Multivariable feedback control; John Wiley & Sons: New York, 1996. (14) Zheng, A.; Mahajanam, R. V.; Douglas, J. M. A hierarchical procedure for plantwide control. AIChE J. 1999, 45 (6), 12551265.

Received for review September 23, 1999 Revised manuscript received September 20, 2000 Accepted April 27, 2001 IE9907066