Rigorous Handling of Input Saturation in the Design of Dynamically

Rigorous Handling of Input Saturation in the Design of Dynamically. Operable Plants†. Rhoda Baker and Christopher L. E. Swartz*. Department of Chemi...
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Ind. Eng. Chem. Res. 2004, 43, 5880-5887

PROCESS DESIGN AND CONTROL Rigorous Handling of Input Saturation in the Design of Dynamically Operable Plants† Rhoda Baker and Christopher L. E. Swartz* Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7

The design of a plant can significantly affect its inherent ability to be satisfactorily controlled. Approaches for incorporating dynamic operability requirements within an optimal design framework have been proposed, where a dynamic model of the plant and its associated control system are included as constraints. This paper focuses on the inclusion of actuator saturation effects in formulations of this type. A mathematical description of actuator saturation that is suitable for incorporation within a simultaneous optimization framework is used. The optimization formulation is presented and its application demonstrated through two case studies. 1. Introduction Dynamic operability reflects the quality with which a plant can be controlled using feedback and is a function of both the design of the plant and its associated control system. A plant designed on the basis of steady-state considerations alone could exhibit poor dynamic characteristics, leading to a loss of economic performance and a reduced capacity to effectively handle safety and environmental constraints. This motivates the need for the development of quantitative techniques for dynamic operability assessment, as well as its incorporation into procedures for process plant design. Optimization-based approaches to dynamic operability assessment permit simultaneous consideration of performance-limiting factors of non-minimum-phase characteristics, input constraints, and model uncertainty and also provide considerable flexibility in the choice of performance criteria, decision variables, and constraints. Recent work has incorporated operability requirements as constraints within a single optimal plant design problem formulation.1-4 To date, optimization-based integrated design and control problems have focused almost entirely on linear controllers without consideration of actuator saturation effects. Young et al.5 describe the rigorous inclusion of input saturation effects in optimizing control. Actuator saturation introduces discontinuities in the system model, and to avoid potential problems using a sequential optimization approach, two alternative formulations were proposed for solving the problem within a simultaneous solution framework. Input saturation discontinuities were handled by the introduction of slack variables and their inclusion in either bilinear or mixedinteger constraints, resulting in a nonlinear or mixed† Part of this work was conducted by the authors at the Department of Chemical Engineering, University of Cape Town, Rondebosch 7700, South Africa. * To whom correspondence should be addressed. Tel.: (905) 525 9140. Fax: (905) 521 1350. E-mail: [email protected].

Figure 1. Relationship between the controller output and actuator output.

integer programming (MIP) problem, respectively. The formulations were applied to linear systems to find the economically optimal operating point for a controller with fixed structure and tuning when disturbance deviations are taken into account. It was shown that the assumption of a linear, closed-loop response in this case would lead to an overly conservative estimate of the feasible operating range and, consequently, a suboptimal operating point. In this paper, the actuator saturation formulation is extended to processes with nonlinear dynamics and applied to integrated plant and control system design. The paper is organized as follows. First, the phenomenon of input saturation is described, and a MIP formulation for rigorous input saturation handling is summarized. This is followed by the development of an optimal design framework for nonlinear dynamic systems that incorporates rigorous input saturation handling. Finally, two case studies demonstrating the application of the optimization formulation are presented. 2. Input Saturation All physical control systems have to deal with limitations on the control input. For example, a valve controlling the flow rate of cooling water to a reactor can only operate between being fully open or completely closed. When the controller attempts to push the actuator beyond its upper or lower limit, the actuator saturates, resulting in a discrepancy between the input to the process, u, and the controller output, uc, as shown in Figure 1. The manipulated input trajectory flattens against the constraint limit, leading to a nonlinear

10.1021/ie030528n CCC: $27.50 © 2004 American Chemical Society Published on Web 08/05/2004

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5881

closed-loop response even if the plant and controller dynamics are linear. The relationship between the controller output signal, uc, and the actuator output, u, with saturation accounted for may be described by

u ) sat(uc)

{

uL ) uc uU

if uc < uL if uL e uc e uU if uc > uU

(1)

where uL and uU are the lower and upper actuator constraint limits, respectively. Closed-loop optimization problems that attempt to take actuator limits into account by merely including constraints on the upper and lower bounds of the actuator are generally not making provision for input saturation effects. In the optimization strategy, the actuator signal would be calculated based on a linear relationship between the control signal and manipulated variable; thus, the input action will merely touch the limit but would not remain there, even if this would result in an improvement in the objective function. Saturation effects have to be explicitly accounted for in the process model. Two mathematical formulations for input saturation suitable for incorporation within a simultaneous optimization strategy have been described.5 These involve respectively (a) bilinear and (b) linear, but mixed-integer, constraints. To avoid potential difficulties due to nonconvexity introduced by the bilinear formulation, the mixed-integer approach is followed in this study. The mathematical formulation for the MIP formulation can be summarized as follows:

uc,k ) uk - SLk + SU k

(2)

uL - β(1 - zLk ) e uk e uL + β(1 - zLk ) U

u - β(1 -

zU k) L

U

e uk e u + β(1 -

zU k)

(3) (4)

U

u e uk e u

(5)

0 e SLk e βzLk

(6)

U 0 e SU k e βzk

(7)

where the subscript k refers to the time interval k, uc is the controller output, u is the actuator position, uLand uU are the lower and upper actuator bounds, SL and SU are the lower and upper slack variables, zL and zU are binary variables indicating actuator saturation at the lower and upper constraint limits, respectively, and β is a large positive constant. Constraints (2)-(7) may be shown to imply the logical conditions (1). To see this, consider the case where SLk > 0. Then, by eq 6, zLk ) 1 and, by eq 3, uk ) uL. We show now that the controller output, uc,k < uL so that this situation corresponds to the first of the cases in eq U 1. By eq 4, zU k ) 0, and eq 7 implies that Sk ) 0. Finally, we have from eq 2 that

uc,k < uk ) uL which completes what was set out to be shown. Representation of the other cases in eq 1 may be demonstrated

in a similar manner. This formulation corresponds to mixed-integer linear constraints where the decision variables now include the binary variables. It is important to recognize that the actuator saturation discontinuities may arise (a) regardless of whether the saturation logic is absorbed into the control law or considered as part of the process and (b) regardless of the inclusion of antireset windup schemes. The focus in this paper is on handling the equation discontinuity rather than on any specific control algorithm. While antireset windup schemes are not specifically considered here, they are readily accommodated within this framework. 3. Problem Formulation We consider the following general formulation for the optimal design of dynamically operable plants:

min Φ[x(t),w(t),u(t),d,z,pN,tf] subject to

d,z

x3 (t) ) f[x(t),w(t),u(t),d,z,p(t)] x(0) ) x0 h[x(t),w(t),u(t),d,z,p(t)] ) 0 g[x(t),w(t),u(t),d,z,p(t)] e 0

(P1)

where x(t) ∈ RX are differential state variables, w(t) ∈ RW are algebraic time-dependent variables, u(t) ∈ RU are input variables, d ∈ RD are continuous design and/ or operating variables, z ∈{0, 1}Z are binary variables, p(t) ∈ RP are uncertain parameters and disturbances whose nominal values are denoted as pN, and Φ is a scalar-valued objective function. f and h are the differential and algebraic equations respectively that describe the dynamic behavior of the plant and its associated control system and also relationships between design variables. The constraint set g includes path as well as time-invariant constraints. It is assumed that the system of dynamic algebraic equations has a maximum index of 1. We consider here a finite set of parameter values and disturbances. However, it may readily be extended to the more general setting of design under uncertainty in which the constraints are to be satisfied for infinitely many parameter realizations within specified bounds.1,6 In this case, the objective function is evaluated as the expected value of the cost function over the parameter distribution. The thrust of the present study is to incorporate a mechanism for actuator saturation. The set of continuous variables can be used to represent molar flow rates and compositions, steady-state values, and structural design parameters such as diameter or height, while binary variables are generally used to indicate the existence or nonexistence of units, thereby providing the means to create a plant superstructure that is able to represent alternative unit configurations. The above formulation includes dynamic equations, which we deal with by means of orthogonal collocation on finite elements. The result is a finite-dimensional, algebraic system that, within an optimization framework, can be solved using an ordinary nonlinear programming solver. Details of this approach may be found in Cuthrell and Biegler.7 Here, we assume piecewise constant control inputs over each of nCE control elements (sampling periods)

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and further subdivide these elements into nFE equally sized finite elements. The control element is distinguished from the finite element by representing it with the letter k. The finite elements are numbered from the start of the control element as shown in Figure 2. The states are approximated within each finite element i as nCOL

x˜ (t) ) x˜ k,i(τ) )

xk,ijφj(τ) ∑ j)0

where τ is the time ordinate scaled to the interval [0, 1] within each finite element, nCOL is the number of collocation points in each element, the φj(τ)’s are Lagrange polynomial basis functions, and the xk,ij’s are coefficients to be determined. Substitution into the differential equations gives a set of residuals that are made to vanish at the collocation points: nCOL

xk,ijφ˙ j(τn) - f(xk,in,wk,in,uk,d,z,pk,in) δ ) 0 ∑ j)0 k ) 1, ..., nCE i ) 1, ..., nFE n ) 1, ..., nCOL

N

yk,ij ) φ(xk,ij)

We include the controller parameters as decision variables in the optimization problem. In addition, it is possible to include the control loop pairing as part of the decision space to determine the best pairing through the inclusion of binary variables indicating whether a controlled variable is paired with a particular input.1,3,8 The input saturation constraints of the previous section, as well as constraints to ensure that the endpoint set-point error is within a specified tolerance, are included. The complete problem formulation thus takes the following form:

min

d,z,Kc,TI,Ek, uc,k,uk,yk,ij,xk,ij,wk,ij, L U SLk ,SU k ,Zk ,Zk xss,wss,yss,uss

Φ(xk,ij,wk,ij,uk,xss,wss,uss,d,z,pN,tf)

nCOL

(8)

xk,ijφ˙ j(τn) - f(xk,in,wk,in,uk,d,z,pk,in) δ ) 0 ∑ j)0 uc,k - [uc,k-1 + Kc(Ek - Ek-1 + TIEk∆t)] ) 0 uc,k - (uk - SLk + SU k) ) 0 h(xk,ij,wk,ij,uk,d,z,pk,ij) ) 0 g(xk,ij,wk,ij,uk,d,z,pk,ij) e 0 f(xss,wss,uss,d,z,pN) ) 0 h(xss,wss,uss,d,z,pN) ) 0 yk,ij - φ(xk,ij) ) 0 Ek - (yss - yk,10) ) 0

(9)

yss - φ(xss) ) 0

h(xss,wss,uss,d,z,p ) ) 0

(10)

x1,10 - xss ) 0

x1,10 ) xss

(11)

E0 ) 0 uc,0 - uss ) 0

where the subscript ss refers to the steady-state value. Continuity of the states across the finite-element boundaries within the control elements and across the control-element boundaries is also imposed. A multiloop formulation of the proportional-integral (PI) control law takes the form

uc,k ) uc,k-1 + Kc(Ek - Ek-1 + TIEk∆t)

uL - β(e - ZLk ) e uk e uL + β(e - ZLk ) U U uU - β(e - ZU k ) e uk e u + β(e - Zk )

u L e uk e uU

(12)

where Kc is a diagonal matrix of controller gains, TI is a diagonal matrix of reciprocal integral time constants, and ∆t is the sampling period. Here, the error term, , is defined as the difference between the steady-state value of the measured variable (which corresponds to the set point) and its current value

Ek ) yss - yk,10

(14)

subject to

Here, δ is the finite-element length (assumed to be uniform). In the sequel, we consider h to represent algebraic equations that form part of the plant model description. Other equality constraints, such as those representing the controller, are described separately. The initial values of the states, x0, coincide in our treatment with steady-state operating conditions we wish to determine and should be included as decision variables in the optimization problem. A corresponding set of initial value equations is included in the equality constraints in addition to steady-state balance equations. Thus, the following constraints should be added to the formulation:

f(xss,wss,uss,d,z,pN) ) 0

states

(13)

and the measured variables, y, are some function of the

0 e SLk e βZLk U 0 e SU k e βZk nCOL

xk,i0 )

xk,i-1,jφj(1) ∑ j)0

k ) 1, ..., nCE, i ) 2, ..., nFE

nCOL

xk,10 )

xk-1,nFE,jφj(1) ∑ j)0

k ) 2, ..., nCE (P2)

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5883

where

Table 1. Run-Specific Information for the CSTR Design

e ) [1, 1, ..., 1]T SLk ,

SU k ∈

R

U

U ZLk , ZU k ∈ {0, 1}

The resulting formulation is a mixed-integer nonlinear programming (MINLP) problem that rigorously accounts for actuator saturation. 4. Case Study 1: A Stirred Tank Reactor The first case study is concerned with the design of a single continuous stirred tank reactor (CSTR) using a dynamic model described in Schweiger and Floudas.8 The reaction taking place in this example is a firstorder, exothermic irreversible reaction (A f B). The rate of reaction in the CSTR is dependent on the reactor volume, and hence the height and diameter of the tank, as well as on the temperature and reactant concentration inside the reactor. The height and diameter of the tank are examples of structural design variables that can be included in the optimization problem as decision variables. It is assumed that the reactor contents are perfectly mixed; thus, the temperature of the mixture and the concentration of the reactants are uniform throughout the tank. A perfectly mixed, cylindrical cooling jacket surrounds the walls of the reactor to maintain the temperature inside the tank. It is also assumed that the volume and density of the reactor and jacket contents remain constant. The CSTR is illustrated in Figure 3. The variable to be controlled is the temperature of the CSTR, and the manipulated variable used to achieve this is the flow rate of cooling water through the cooling

parameter

value

description

t0 (h) tf (h) ∆t (h) td (h) ∆TR0 (K) tol nCE nFE nCOL nTOL β

0 10 0.1 0.5 35 1 × 10-6 100 1 2 10 1000

start of the time horizon end of the time horizon controller time step time of the disturbance step feed temperature step increase temperature error tolerance at end points number of control elements number of finite elements per CE number of collocation points per FE number of end tolerance points arbitrary large number

Table 2. Results for the CSTR Design with the L/D Ratio Restriction variable

no saturation

saturation

total cost ($) capital cost ($) utility cost ($) DR (m) HR (m) VR (m3) TR,ss (K) FJ,ss (m3/h) Kc τI

477 000 418 000 59 000 5.99 4.80 135.20 355.8 12.74 -21.20 2.191

418 000 359 000 59 000 6.76 3.38 121.14 357.0 12.74 -19.40 2.048

jacket. The disturbance to the process is a 35 °C step increase in the temperature of the feed to the reactor. The set point of the reactor temperature is taken as its steady-state value, which is included in the problem as one of the decision variables. The steady-state temperature, in turn, is determined by the steady-state flow rate of cooling water, which is also included as a decision variable. Other design variables included are the controller gain and controller time constant. The steady-state flow rate of cooling water is used to determine the utility cost of the CSTR over 4 years of operation. Together with the capital cost, which is a function of the diameter and height of the tank, this is used to calculate the total cost, which is the economic criterion to be minimized. In addition, constraints of the form

-tol e yss - y(k) e tol

Figure 2. Schematic illustrating collocation points, finite elements, and control elements.

are applied over the last nTOL control elements to force the trajectory to approach the set point at the end of the time horizon. Constraints on the upper and lower values of the reactor and jacket temperatures as well as the coolant flow rate are added to demonstrate the advantages of including input saturation:

333.3 K e TR e 361.1 K 333.3 K e TJ e 361.1 K 12.74 m3/h e FJ e 14.16 m3/h

Figure 3. Schematic of a jacketed stirred tank reactor.

A summary of the problem data is listed in Table 1. 4.1. Results and Discussion. The resulting optimization problem is first solved without input saturation and is then compared to the case where input saturation has been rigorously handled. In addition, the L/D ratio of the tank is constrained to lie between a lower limit of 0.5 and an upper limit of 2. The results are shown in Table 2, and a plot of the input and output trajectories in response to the disturbance can be found in Figures 4 and 5. The total cost of

5884 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004

Figure 4. Comparison of cooling water profiles with and without saturation with the L/D ratio restriction in effect.

Figure 6. Comparison of cooling water profiles with and without saturation with the L/D constraint removed. Table 3. Results for the CSTR Design without the L/D Ratio Restriction

Figure 5. Comparison of reactor temperature profiles with and without saturation with the L/D ratio restriction in effect.

the design without saturation is 14% higher than the case where saturation is rigorously handled. This shows that an assumption of strictly linear actuator behavior is overly conservative and that a cheaper design alternative, which also takes the dynamic performance into account, can be found if saturation is allowed. In this example, it is apparent that the limiting constraint in this case is the lower bound of the L/D ratio, which restricts the height of the tank from being smaller than 0.5 times its diameter. An interesting tradeoff occurs when the disturbance step is lowered to 25 °C and the L/D ratio constraint is removed. The results of the new problem are shown in Table 3. In this case a plant design that fails to account for saturation has a 25% higher total cost operating over the same period than one that does allow saturation. Because the restriction on the L/D ratio has been removed, this is no longer a limiting constraint and the objective is able to improve considerably. While the optimal design with saturation allowed for has a lower total cost than the linear actuator case, the operating cost is slightly higher. This can be understood as follows. The increase in the reactant feed temperature causes the reactor temperature to increase. The controller therefore increases the flow of cooling water through the jacket in order to reduce the temperature

variable

no saturation

saturation

total cost ($) capital cost ($) utility cost ($) DR (m) HR (m) VR (m3) TR,ss (K) FJ,ss (m3/h) Kc τI

367 000 308 000 59 000 11.19 1.43 140.28 359.6 12.74 -51.91 1.538

293 000 233 000 60 000 7.41 1.74 75.04 360.1 13.05 -10.55 1.757

in the reactor and returns it to its set point. However, the control valve for the cooling water has an operating range of 12.74-14.16 m3/h, which limits the available cooling water. The controller with the linear actuator is able to remain at its maximum for only an instant; thus, to reject the disturbance, the steady-state cooling water flow rate needs to be lower than would be required if the coolant flow were allowed to saturate at its upper limit. This means that an increased heat transfer area is required in order to achieve the same amount of cooling; therefore, the diameter and/or height of the tank is forced to increase. These effects are illustrated in the input and output response curves shown in Figures 6 and 7. The problems were coded within GAMS9 and solved on an Intel P4 3.0 GHz processor with 1 GB of RAM. Different initial values were chosen for the nonlinear optimization problem in which saturation was not allowed for, and the resulting solution was the same for all. The problem with saturation accounted for has 200 integer variables and was solved using SBB10 as the MINLP solver. A feasible solution was found after 300 s, with no improvement in the objective function value after the solver was allowed to continue the tree search for another 4 h. While the solution is not guaranteed to be the global minimum, it is a significant improvement over the linear actuator case, illustrating that inclusion of input saturation provides a less conservative design for cases where the dynamics of the design are important. An estimate of the approximation error may be obtained by computing the residual at a location other than a collocation point, such as the finite-element end point.11 The maximum absolute error over all of the

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5885 Table 4. Key Model Parameters for Binary Distillation Example

Figure 7. Comparison of reactor temperature profiles with and without saturation with the L/D constraint removed.

Figure 8. Schematic of a binary distillation column.

differential states and finite elements was found for this problem to be 8.12 × 10-6. An error control mechanism based on this construct could, in principle, be included as additional constraints within the optimization problem, with the interior element boundaries treated as additional optimization variables.11 This was, however, considered to be beyond the scope of this study. 5. Case Study 2: A Binary Distillation Column The second case study is the design of an ideal binary distillation column as illustrated in Figure 8. The purpose of the column is to separate a mixture at its bubble point into distillate and bottom products of a specified purity. The case study demonstrates how a distillation column that achieves specified product quality can be designed while taking into account dynamic operability considerations and the application of the formulation (P2) to a multi-input, multi-output system. It also demonstrates that allowing for the phenomenon of input saturation results in a less conservative economic objective. The diameter of the column is an example of a structural characteristic that will be designed by minimizing the annualized cost of the column over a specific payback period. The nominal steady-state reflux and vapor boilup rates are also included as decision variables. The liquid and vapor compositions and flow rates

parameter

value

parameter

value

relative volatility height above weir (m) trays

2.5 0.025 15

feed tray feed rate (kmol/min) feed composition

5 1 0.45

associated with each tray in the column are variable and time-dependent. The process is controlled using a multiloop PI control scheme, with the distillate composition controlled by the reflux rate and the bottoms composition by the vapor boilup. The disturbance considered is a step change in the feed composition to the column. The mathematical model for the binary distillation column used is that given by Schweiger and Floudas.8 The column has a single feed, a total condenser, and a partial reboiler, and a constant relative volatility is assumed. The liquid flows from the trays follow firstorder dynamics, with the time constant a function of the column diameter; equimolal overflow is obtained at steady state. The molar holdups are given by the Francis weir formula, with a constant height over the weir assumed. A minimum constraint on the column diameter as a function of the vapor flow rate is included in order to avoid flooding. A first-order lag between the calculated control move and the actual control inputs and a 5 min deadtime in the composition measurement of the bottoms and distillate are included. A fifth-order system is used to approximate this deadtime. Key model parameters are given in Table 4. The economic objective function used here is taken from Luyben and Floudas.12 It is an annualized cost comprising a capital cost estimate based on the number of trays and column diameter and utility costs for the condenser and reboiler. The disturbance considered is a step decrease of 0.09 in the feed composition, z, and constraints are imposed on the reflux rate and vapor boilup

1.36 e Vc e 1.70 0.50 e Vc e 1.1 the distillate composition

xD g 0.90 and the steady-state product compositions

xBss e 0.02 xDss g 0.95 Data pertaining to the solution of the problem can be found in Table 5. 5.1. Results and Discussion. Results for the design of a distillation column where input saturation has been rigorously handled and the case where saturation has not been accommodated are reported in Table 6. Graphs of the dynamic responses for the two cases are shown in Figures 9-12. In Table 6, it can be seen that the total cost for the case with saturation handling is lower than that where saturation has been disallowed. The improvement is gained by reducing the steady-state utility cost, which is achieved by reducing the steady-state vapor boilup

5886 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004

Figure 9. Bottoms composition response with and without input saturation handling.

Figure 11. Vapor boilup response with and without input saturation handling.

Figure 10. Distillate composition response with and without input saturation handling.

Figure 12. Reflux rate response with and without input saturation handling.

Table 5. Run-Specific Data for Binary Distillation Design

Table 6. Results for Binary Distillation Design

parameter

value

description

t0 (min) tf (min) ∆t (min) td (min) ∆z tol nCE nFE

0 150 2 6 -0.09 1 × 10-4 75 1

nCOL nTOL β

2 5 10

start of the time horizon end of the time horizon controller time step time of the disturbance step feed composition step composition error at end points number of control elements number of finite elements per control element number of collocation points per FE number of end tolerance points arbitrary large constant

rate, and the capital cost by decreasing the column diameter. To achieve the same purity specifications for the same disturbance, the vapor boilup rate saturates at the lower actuator bound near the start and end of the time horizon, while the reflux rate saturates at the upper actuator bound near the start of the time horizon. The problem was solved in GAMS using the SBB and CONOPT210 solvers on an Intel P4 3.0 GHz processor with 1 GB of RAM. An initial integer feasible solution was obtained by applying the input saturation handling constraints at points that were at an actuator limit and then re-solving the modified problem. This process was iterated until the objective function ceased to improve.

variable

no saturation

saturation

total cost ($) capital cost ($) utility cost ($) Dc (m) Rss (kmol/min) Vss (kmol/min) KV τV KR τR

39698 28096 11602 0.908 1.040 1.496 -6.194 4.341 0.010 0.127

37750 26226 11524 0.846 1.033 1.486 -5.823 4.199 0.010 0.101

The integer feasible solution was then used to set the branching priority for the full 300 binary variable problem. However, the solution found previously could not be improved upon within a reasonable amount of time. 6. Conclusion A formulation for optimal integrated plant and control system design has been extended to accommodate actuator saturation. The saturation formulation is, moreover, well suited for use in a simultaneous solution strategy. The case studies presented not only illustrate the application of the extended optimal design formulation but also compare the optimal solutions against

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5887

those obtained when input saturation is not allowed for, showing the superiority of the former. The comparison also highlights the fact that merely placing upper and lower bounds on the input trajectories does not make provision for saturation behavior: this has to be explicitly accounted for in the process model. A potential drawback of the mixed-integer description of actuator saturation is the well-documented worst-case rate of increase in solution time with the problem size. One strategy for alleviating this is strategic introduction of binary variables as illustrated in the second case study. An investigation into alternative solution strategies is currently in progress. Literature Cited (1) Mohideen, M. J.; Perkins, J. D.; Pistikopoulos, E. N. Optimal design of dynamic systems under uncertainty. AIChE J. 1996, 42 (8), 2251-2272. (2) Bahri, P. A.; Bandoni, A.; Romagnoli, J. A. Integrated flexibility and controllability analysis in design of chemical processes. AIChE J. 1997, 43 (4), 997-1015. (3) Bansal, V.; Perkins, J. D.; Pistikopoulos, E. N. A case study in simultaneous design and control using rigorous, mixed-integer dynamic optimization models. Ind. Eng. Chem. Res. 2002, 41 (4), 760-778. (4) Pistikopoulos, E. N.; Sakizlis, V. Simultaneous design and control optimization under uncertainty in reaction/separation systems. In Chemical Process ControlsVI; Rawlings, J. B., Ogunnaike, B. A., Eaton, J. W., Eds.; AIChE Symposim Series 326; CACHE and AIChE: New York, 2002; Vol. 98; pp 223-238.

(5) Young, J. C. C.; Baker, R.; Swartz, C. L. E. Input saturation effects in optimizing controlsinclusion within a simultaneous optimization framework. Comput. Chem. Eng. 2004, 28 (8), 13471360. (6) Halemane, K. P.; Grossmann, I. E. Optimal process design under uncertainty. AIChE J. 1983, 29 (3), 425-433. (7) Cuthrell, J. E.; Biegler, L. T. On the optimization of differential-algebraic process systems. AIChE J. 1987, 33 (8), 1257-1270. (8) Schweiger, C. A.; Floudas, C. A. Interaction of design and control: Optimization with dynamic models. In Optimal Control: Theory, Algorithms, and Applications; Hager, W. W., Pardalos, P. M., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1998; pp 388-435. (9) Brooke, A.; Kendrick, D.; Meerhaus, A.; Raman, R. GAMS. A User’s Guide; GAMS Development Corp.: Washington, DC, 1998. (10) GAMSsThe Solver Manuals; GAMS Development Corp.: Washington, DC, 2001. (11) Logsdon, J. S.; Biegler, L. T. Accurate solution of differential-algebraic optimization problems. Ind. Eng. Chem. Res. 1989, 28 (11), 1628-1639. (12) Luyben, M. L.; Floudas, C. A. Analyzing the interaction of design and controls1. A multiobjective framework and application to binary distillation synthesis. Comput. Chem. Eng. 1994, 18 (10), 933-969.

Received for review June 30, 2003 Revised manuscript received April 9, 2004 Accepted May 17, 2004 IE030528N