Optimal Design with Optimal Uncertainty in Parameters and Control Variables A.W. Longley and P.L. Douglas* Department of Chemical Engineering, University of Waterloo Waterloo,Ontario, CANADAN2L 3GI This paper presents a method for determining the optimal process design while simultaneously choosing the optimal levels of parameter and decision variable uncertainty. The optimal level of uncertainty is valuable in accessing steady state control systems and justifying {or refuting) further laboratory or bench-scale investigations of process parameters. Development of the method begins with the classical approach to optimal design, then progresses to include parameter uncertainty. Next, the development incorporates decision variable uncertainty into the optimization. Finally, parameter and decision variable uncertainties are optimized. The development is accompanied by a case study of a continuous stirred tank reactor (CSTR). Introduction Several different and often competing factors must be considered in the design of a chemical plant. The necessary decisions in plant design are often complex and beyond the capabilities of human visualization. The design problem is frequently Author for correspondence (Email: pdouglas@cape.uwaterloo.ca). 39 A. W.Longley and P.L. Douglas resolved by marhematical modelling and optimization. Most design optimization is performed in a deterministic fashion with empirical overdesign factors added to the optimal design. In reality, the model parameters and control variables are random values with varying degrees of uncertainty. These uncertainties may have a profound effect on the final design and should not be excluded from the design optimization. Furthermore, the level of uncertainty is often chosen by the designer based on experimentation, experience, and/or rules of thumb. Deciding whether further investigations are necessary to reduce design uncertainty is frequently the designer's responsibility. To aid in this decision. the uncertainty levels can be included as decision variables in a design optimization. The optimal levels of uncertainty are then compared to the present uncertainty levels, providing insight into possible parameter investigationsand process control systems. These design methods offa an alternative to the blind and possibly costly use of overdesign factors. The pioneering work in this field was undertaken by Grossman and Halemane (1982)and Halemane and Grossman (1983). Applications of these techniques to synthesize flexible process flowsheets have been repaed by Saboo and Morari (1984)and Westerburg and Chen (1986). Wagler and Douglas (1988)and Douglas et al(l991) simplified these techniques and applied them to large scale systems, in particular to a distillation sequence with and without thermal integration. No work has been r-ed in the literature in the use of these techniques to study the effect a€ uncertainty on chemical reactor design. The purpose of this paper is to illustrate the CoIlCepts of design optimization, decision variable uncertainty, and optimal uncmainty as applied to a chemical reactor. The chemical reactor is the centre of many chemical processes and the CSTR is often the reactor of choice (Hill, 1977). In addition to the standard decision variables in reactor design (e.g. volume, flowrate, and temperature),variables can be 40 Optimal Design with Optimal Uncertainty in Parameters and Control Vahbles introduced to represent the level of uncertainty or tolerance of the inlet flowratesand the reactor temperature. The opposing forces in choosing the levels of the temperature and flowrateuncertainties are the cost of reactor overdesign and the cost of the control system. The uncertainty in model parameters (e.g. reaction kinetic parameters) is accounted for by selecting critical parameter values that ensure feasibility of the design. The level of unertainty in these parameters can also be included in the design optimization. The opposing forces here are the cost of overdesign and the cost of bench scale investigation of model parameters. Reducing uncertainty will decrease overdesign but increase the cost of parameter investigation. CSTR Model and Objective Function The process under consideration employs a CSTR to convert the liquids A and B to the more valuable product C. A process flow diagram is presented in Figure 1. The reaction mechanism is simple: ?he reaction occurs at high temperature and will require a steam-heatedjacket. For simplicity, the design will not include a dynamic heat analysis; we assume that a residence time of 15 minutes will allow for proper heat dissipation. The isothermal steady-state behaviour of the system is modelled by three mass balance equations: F~ - rV- BFo = 0 rV- m0 = 0 41 A. W.Longley and P.L. Douglas . Fo Fb -a- Volume (V) Temperature (T) Figure 1. Processflow diagram. Assuming there is no volume change upon reaction,the outlet flomte must ‘equal the sum of the inlet flowrates: where the coefficients are the mole-volume conversion factors. Equations (I), (2), and (3) are combined to produce the reactor model: FA - ye-BmTABV- A(O.~FA+ 0 . 1 F ~ )= 0 y e-p’RTABV- C(O.1FA + 0 . 1 F ~ )= 0 Assume p = 42 that the values of the kinetic parameters (y = 3x10’ l/mol/tnin, 3x104 calhol) were determined from bench-scale investigations and have a Oprimal Design wirh Optimal Uncertaintyin Parameiers and Control Variclbles -+25%tolerance. Furthermore, assume that the decision variables (FA,FB, T) can be regdami by a process control system to within a 25% tolerance. Assumed market research for the fictitiousproduct C dictates that a production rate of 6OOO kmol of C per day is desired, however, occasional deviations below this production rate are acceptable. To satisfy the prospective customers, the product must be 80%pure. The design objective is to determine the reactor volume and operating conditions that meet product requirements at a minimum cost The decision variables in this problem are reactor volume, V, inlet flowrates, (FA,FB),and reactor temperature, T. The state variables in this problem are functions of the decision variables and are therefore not included as decision variables. The terms of the cost function, which includes both capital and operating costs, are: Costl = CReactor] + [A inlet pump] + [B inlet pump] + [Outlet pump] + [Heater] + [Feeds] Costl = [200Voy] + [FA + 2F:Dm] + [2F, + 4F:61] + [1.5Fo + 3F,Om] + [lOT + 3Toa] + [2FA + 3FB] Additional costs used in the objective function are the cost of reactor control systems and the cost of further parameter investigations: Cost2 = [ A flow control] + + [B flow control] + [ y investigation] [P investigation] + [Temperature control1 cost2 = [F][F][y] [ ; . .+ [ + + + 27 x l@ 60000 30s 43 A. W. Longley and P.L. Douglas These cost terns are discussed later. Classical Optimization Approach The classical approach to optimal steady-state design is to represent the problem as a nonlinear programming problem (Grossmann and Sargent, 1978): min Cost = f(va.0) (7) "lx s.t h(v,x,B) = 0 gwxB) Io The problem is solved at the nominal values of the uncertain parameters, e=@,and the over-design factors are added to the solution. The use of over- design factors can be detrimental to a design, possibly resulting in a process never having an opportunity to prove its economic value (Peters and Timmerhaus, 1991). Furthermore, no rational method exists for including over-design factors therefore the resulting design is usually not optimal and may be infeasible. Including uncertainty in Equation 7 would provide the designer with a clearer picture of the final design, but increases the problem size and complexity. Applying the classical optimization approach to the CSTR design problem detailed above yields the following optimization problem: min Cost = f(v$A$B,T) VJA.FB,T SL h(VFA&,T) = O g, = 15Fo- v s o g, = 6 x lo6 - 1440FoCI0 g3 = 0.80A + 0.80B - 0.2OC 0 where h is the vector of model equations (4). Problem (8) was solved in MATLAB using the optimization toolbox routine CONSTR which uses a sequentialquadratic programming method (Grace, 1990). Multiplying the optimal equipment design values obtained from problem (8) by an over-design factor of 44 Optimal Design with Optimal Uncertainry in Parameters and Control Variables 1.2 yields the fmal design recorded in Table 1. Readers should note that the reactor volume and feed flows increase by a factor of 1.2. However, the temperature increases by a factor determined from a solution of model equations at the new flowram and reactor volume. This design method generates a process cost at least 10% higher than the other design methods. In addition, this approach requires a sensitivity analysis to provide any insight into the flexibility requirements of the design variables. Parameter Uncertainty Grossmann and Sargent (1978) included parameter uncertainty in problem (7) and made an important distinction between design variables. The design variable v is divided into fixed design variables d (e.g. volume) and control variables z (e.g. temperature). The control variables are manipulated to have optimal performance in light of the actual parameter values realized during process operation. Assuming the uncertain parameters are continuous bounded variables and the expectation of the cost function is adequately represented by a finite weighted sum, problem (7) becomes: 45 A. W.Longley and P.L. Douglas Table 1. Design Optimization Decision Variable Results No Uncmainty' optimal Plus Panmeter Unccpinty PanmcterBiDccision Uncertainty Uncertainty a58 Dedsion' Values Overdesign 0) CQSl (13) (22) cost (25) 81,394 94,312 81.409 83,750 86,140 85,361 12.856 15,427 12,860 13,391 13,373 13,212 4204 5.165 4306 4.473 4,468 4,471 4,301 4,47 1 4,464 4,470 4,312 4,476 4,469 4,455 4,267 4.454 4.448 4,337 4,271 4.456 4,451 4,338 4260 4.45 1 4.447 4,340 1,050 1,056 1.06, 1,106 1,338 1,342 1,352 1,393 777 786 791 836 5,558 5545 10,001 5,511 5,496 1,197 310 314 1,894 6.23 10'3 6.25(10") 6.25(10*3 3.93(10'3 6.25(lb) 6.25(1@) 6.25(1@) 6.23(1@) 10.11,12 13,14,15 10.11.12 13.14.15 10,11,12 13,14,15 10,11,13,14 15.16,lS 4,266 1,051 43 i No C a t 5.1 19 1,216 cost of unartainty not included in objective function ' cost of mcertainty, as defined in quation (6). added to objective function 46 Optimal Design with Optimal Uncertaintyin Parameters and Control Variables N min Cost = b’r’ qf(d.2 ,’x ’,ek) k.1 When the inequality constraints in Equation 9 are monotonic in the uncertain parameters (e), the maximization subproblem can be solved apriori (Grossmann and Sargent, 1978). The solution of this maximization provides an initial set of N critical points in the parameter space that the design must satisfy. Theoretically, these initial N points may not be sufficient to guarantee feasibility for the entire parameter space (Grossmann et al, 1983). However, these N points are sufficient to enhance insight into the final design. Given these N critical points, problem (9) becomes: N min &‘a‘ S.t. Cost = akf(d,z,x‘ ‘,e’> k-1 h(d,z ’.x ’,@) = 0 k = l;-,N g(d,z k , ’,@) ~ I0 This method can easily be extended to account for uncertainty in parameters external to the process (Grossmann et al., 1983), but cannot account for 47 A. W.Longley and P.L Douglas Uncertainties in the decision variables themselves. For example, consider a process with one feed stream containing multiple components. Assume that the feed composition is fuced and the design p b l e m must determine the optimal feed flowrate. The feed composition is an external parameter, therefore problem (10) can account for composition uncertainty, but the feed flowrate is a decision variable, therefore problem (10) can not account for flowrate uncertainty. Applying this method to the CSTR design problem the uncertainty in the kinetic parameters (y,p) is now included in problem (8). To include parameter uncertainty assume the parameters are bounded at the minimum and maximum values. If no information is available to identify the upper and lower bounds assume that the given parameter tolerance equals three standard deviations (3a=tol). Therefore: eL = pe eu = Pe - 30~ + 30, To determine the critica points in the parameter space, examine the constraints in problem (8). The first constraint, g,, is independent of the parameters and therefore is not affected by the maximization subproblem in problem (9). The other constraints, (g2,g,), are both monotonic in the parameters and are always maximized at the extreme value (?,pu), Assume that this point the only one that must be satisfied to ensure feasible design of the reactor. To generate a more representative cost function, two more points or vertices in the parameter space are also considered. The three points and the associated cost function weightings are: 48 Optimal Design with Optimal Uncertainty in Parameters and Control Variables 83 = (+73"]* aj = 0.10 The decision variables (V,FA&T) are divided into design variable, d = (V), and control variables, z = (FA,FB,T). With these modifications, problem (8) becomes: - 1440F:C' I 0 g t = 0.80A ' + 0.80B ' - 0.2OC ' I 0 8," = 6 x 106 Each of the constraints must now be satisfied at the three parameter points which increases the total number of constraints from 6 to 18. The solution to problem (13) is recorded in Table 1. Including parameter uncertainty in the design has revealed that the reactor temperature will require a wide operating range, from T, = 777K to T, = 1338K. Using over-design factors, the process would have been designed for a temperature of 1261K resulting in infeasible operation at vertices 2 and 3. Parameter and Decision Variable Uncertainty Including uncertainty in the decision variables (d, 2, x') requires a probabilistic formulation of problem (10). Assuming that the equality constraints, h, in 49 A. W.Longley and P.L. Douglas Equation 10 represent the process model, a probabilistic formulation transforms the inequality constraints into probabilistic constraints, resulting in a chance- consfraind model (Wagner, 1975): N min Cost = b’s’ s.t C ~&,z”,x~,fY) ”-1 h&,z k,x ’,0L) = 0 k = l,-*-,N ~r(g,.(d,rk,xk,8t) I01 Ipi i = I,***,i gj(d,zk k,P) I0 , ~ j = I + l,-;t 4 ‘2‘,x 9: (p,o’> If the probability distribution functions (PDF) of the random variables (8, 2,x’) and probabilistic constraints are known, then the probabilistic constraints in problem (14) can be transformed into deterministic constraints. The new constraints are derived from the means (psi> and standard deviation (osJof the original constraints (gi): pli + C,o,, I0 The parameter 4 is determined from the PDF of the constraint i and the associated minimum acceptable probability, pi. Often, the PDF of the random variables and/or the probabilistic constraints are unknown or impractical to determine. For example, even if the PDFs of the decision variables in problem (13) were know,determining the PDF of the quality constraints,g3, would result in a large and complex integration problem. Solving this integration problem 50 Optimal Design with Optimal Uncertainty in Parameters and Control Variables would be a labourous task and probably not worth the effon The parameter 5 can be determined without the PDF using Chebyshev's inequality,providing that the PDF of the probabilistic consnaints have finite mean and standard deviation (Harr, 1987). Chebyshev's inequality is: where ty - 4 is the absolute value of the deviation fiom the mean. Applying (16) to the probabilistic constraints in problem (14) yields: Thus, problem (14) becomes: N min baa' s.t. Cost = okf(d,zk ',w , ~ k-1 h(d,z ',x ',@) = 0 k = l;-,N p&(dJ',x ',@? + o,(d,z (18) ,'@) k , ~ I0 j = I + l;-,I Id '3 ',x '1: (p,* 51 A. W.Longley and P.L Douglas The statistical moments of the constraints &,,a& can be determined using first-order second-moment (FOSM)analysis, (Curi and Ponnambalam, 1992). This method approximates the fmt two moments of a function of random variables from the first two moments of those random variables. This approximation may contain a large error for highly non-linear constraints and/or high standard deviationsof the random variables. The FOSM analysis equations are: where y is the subset of random variables (da,z’, xa). If the random variables are independent, equation set (19) is reduced to: Returning to the CSTR problem, the conEol variables {FA,F,, T) are 52 Optimal Design with Optimal Uncertainty in Parameters and Control Variables uncertain in the sense that they may vary due to external disturbances and the lack of an adequate control system. To include this uncertainty requires the transformation of problem (13) into a chanceconstrainedproblem. Constraint gf is a convenient simplificationof the heat analysis and enhancing this constraint ,(: with a probabilistic fornulation is senseless. The constraints g g3y are formulated probabilistically using judgement to transform the fuzzy market requirements into numbers: Pr {Production 2 6.0 x lo6}2 0.90 h {Conversion 2 0.80)1 0.99 Applying Equation (17) to the above constraints, problem (14) becomes: s.t h(VPi,F,k,T ',@) = 0 g,L = 15Ft - V I0 pg2. + 3 . 1 6 2 ~ ~ I .0 .,p + loa,. I 0 where the statistical moments of the constraints (pg,c& are estimated using Equation 20. The solution to problem (22), shown in Table 1, reveals a slightly modified design. The required reactor volume is larger than the design with only parameter uncertainty, problem (13). Also,including control variable uncertainty 53 A. W.Lungley and P.L. Douglas has increased the optimal values of the control variables and the projected cost compared to problem (13). However, the reacm volume and expected cost is at least 11% less than the method using overdesign factors. Problem (22)was solved again with the additional cost function terms of equation (6)added to the objective function. These terms are representative of the cost of parameter and control variable uncertainty. The resulting design changes less than 1% (see Table 1). The new projected cost increases by $2239 but is still 9% ($8172)less than the cost obtained using over-design factors. The tolerances used and associated costs are recorded in Table 2, for example the annual cost Quantity Tolerancc' (preset)' Cost (preset tol.)' TolClaQCC' (as decision)' Cost (decision tol.)' of the reactor heater control system with a temperatue tolerance of 25.0% is !§4424/year,while the annualized cost of investigating the kinetic parameter j3 to a tolerance of i25.0 is $14l/year. Uncertainty as a Decision Variable When solving problem (18) in conjunction with Equations (20), the designer 54 Optimal Design with Optimal Uncertaintyin Parameiers and Control Variables supplies the parameter and decision variable Uncertainty. Further insight into the design problem is achieved by determining the optimal values of uncertainty. With uncertainty as a decision variable, problem (18) becomes: N min 4da 'r ' qf(d.a',z k,x k,BL) Cost = k-1 S.t. h(d,z ',x '.ek) = 0 k p, (d,d,z 'sc ',OY = l,-*,N o,(d,$,z ,'x k,Bk) I 0 i = 1;-J + g,(d,z k,x '$37 I0 j = I + l;-,t d ',z ',x '1 : ( p d ) where d is the vector of parameter and decision variable variances. Often the critical parameter values, ek, chosen in the maximization subproblem of problem (9). are at the extreme values of the parameters, (eL,eu). Thus, the critical parameter values can be represented by the parameter means and variances: fY= p&+TO& where (24) is known, O, is a decision variable, and z is chosen to achieve the desired probability of the parameter bounds. The choice of parameter uncertainty in the CSTR problem (22) is based on the available parameter &ta obtained from laboratory and bench scale investigations. The selection of control variable uncertainty in problem (22) arises h m the selection of process control system. Clearly, the designer has some control over parameter and control variable uncertainty in CSTR design. 55 A. W.Longley and P.L.Douglas Two logical questions are: (a) Is the available parameter data sufficient; and (b) How tight should the process be controlled at steady state operation? Some insight into these two questions is achieved by including parameter and control variable uncertainties (8) as decision variables. Thus, problem (22)becomes: 3 min V,a'P.PJ k-1 S.t. h(V,F:,F,k,T k,@) g,L = 15F: + =0 - V I0 pg2t + 3.1620,. Po. q f ( V , a 2 , F ~ , F ~',8L> ,T Cost = I0 looc3.5 0 The moments of the inequality constraints (g2, g3)are functions of the parameter and control variable uncertainty. This functionality stems ftom the FOSM analysis of Equation 20. Including uncertainty as a decision variable, problem (25). yields a slightly more economical design as shown in Table 1. The most notable change is the increased reactor temperature. More interesting are the optimal tolerances in Table 2. The tolerance of the control variables (FA,F,, T) represent the performance of the control system and the uncertainty in external disturbances. The tolerance of the kinetic parameters (y$) represents the uncertainty arising from the kinetic investigations (i.e. laboratory and bench-scale experimentation). The optimal control variable tolerances are useful for steady state control system decisions. For example, the inlet flowrate of B requires tight control. The design warrants an expenditure of $3400 to achieve this flow control. However, the 56 Optimal Design with Optimal Uncertainty in Parameters and Control Variables reactor temperature requires little control. Operator control of the reactor temperature may be sufficient to maintain the ~ 1 1 . 8 %optimal design tolerance. The optimal variances of the kinetic parameters are valuable in justifying or refuting further bench or pilot-plant investigations. Assuming the present parameter data has tolerances of +25%, the optimal tolerance of -+25% p is at least therefore no further investigation of this parameter is justified. The optimal tolerance of y is +19.82%, which is 5% less than the present value of -+25%. The design justifies an annualized expenditure of $59 to improve the parameter tolerance by 5%. Conclusions Accounting for decision variable uncertainty gives the designer greater insight into the final design than traditional design optimization methods. Combined with uncertainty in the parameters, this method can reduce the reliance on empirical over-design factors which usually yield non-optimal solutions and may result in an infeasible design. Including uncertainty as a decision variable yields information about the optimal level of uncertainty in the final design. This information, when compared to actual uncertainty, provides insight into the necessary steady state control systems and insight into the benefit of further kinetic studies. Acknowledgements The authors would like to thank MRCO (ManufacturingResearch Corporation of Ontario) and NSERC (National Sciences and Engineering Research Council) for financial support of this project. 57 A. W.Lnngley and P.L. Douglas Nomenclature A B C FA FB F. I N R T v d d' g gl g2 g3 h k Pi r t V X X' Y 2 2' P Y 0 eN s"4 d 2 0, 58 Concenuations of species A [molfl] Concenavltions of species B [mol/ll Concentrations of species C [mol/l] Inlet flowrate of species A [mol/min] Inlet flowrate of B [moVminl Outlet flowrate [ lhnh] Number of probabilisticconstraints Number of points in the parameter space that the design must satisfy Gas canstant [cal/mol/K] Reactor temperature [K] Reactor volume [ 11 Fixed design variables Subset of design variables which contain uncertainty Vector of inequality consuaints Residence time constraint Production constraint Quality constraint Vector of Equality constraints including the process model Reaction rate constant [I/moVmin] Minimum acceptableprobability that constraint i is satisfied Reaction rate [mol/l/min] Number of consuaints Design variables State variables Subset of state variables which contain uncertainty Subset of random variables (d', z', x'] Control variables Subset of control variables which contain uncertainty Arrhenius exponential factor [calhol] Arrhenius preexponentialfactor [l/mol/min] Uncertain parameters Nominal value of the uncertain parameters Mean of x Chebyshev parameter Standard deviation of x Vector of parameter and decision variable variances Parameterchosen to achieve the desired probability of the UIlCenainty bounds Weights representative of the likelihoodof the associated parameter point Optimal Design with Optimal Uncertainty in Parameters and Control Variables References Curi, W.F., and Ponnambalam, K. 1993. Intrductory Probabilistic and Stochastic Analysis and Design of Circuits. IEEE Trans. Educ., 36( l), 5 1-56. Douglas, PL., Mallick, S.K., Wagler R.M. 1991. Synthesis of Flexible Thermally Integrated Distillation Sequences,Trans. IChemE, 69(4), 483-491. Grace, A. 1990. Optimization toolbox for use with MATLAB User's Guide. The Math Works, South Matick, Massachusetts,USA. Grossmann IE., and Halemane K.P. 1982. Decomposition Strategy for Designing Flexible Chemical Plants, AICheJ, 28(4), 686694. Grossmaun, I.E.,Halemane, K.P., and Swaney, W. 1983. Optimization Strategies for Flexible Chemical Processes,Comput. Chem. Eng., 7 (4), 439-462. Grossman, IE., and Sargent, R.WH. 1978. Optimal Design of Chemical Plants with Uncertain Parameters. AIChE J., 24 (6). 1021-1028. Halemane, K.P., and Grossnann, IE. 1983. Optimal Process Design under Uncertainty, AICE J., 29(3), 425-433. Han, ME. 1987. "Reliability-Based Design in Civil Engineebg". McGraw-Hill, - Toronto. Hill, C.G. 1977. "An Innoduction to Chemical Engineering Kinetics and Reactor Design",John Wiley, Toronto. Peters, M.S., and Timmerhaus, K.D. 1991 "Rant Design and Economics for Chemical Engineers",4th ed.,McGraw-Hill, New York. Saboo, AX., and Morari, M., 1984, Design of Resilient Processing Plants - W, Some new results on Heat Exchanger Network Synthesis, Chem. Eng. Sci., 39(3), 579-592. Wagler, R.M., and Douglas, PL., 1988, A Method for the Design of Flexible Distillation Sequences,Can.J. Chem. Eng., 66,579-590. Wagner, H.M. 1975. "Principles of Operations Research", 2nd ed.,Prentice-Hall. Westerberg, A.W., and Chen B., 1986, Structural Flexibility for Heat Integrated Distillation Columns - II Synthesis, Chem. Eng. Sci., 41(2), 365-377. Received: 15 April 1995; Accepted afier revision:27 February 1996. 59

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