An Integrated Framework for Robust and Flexible Process Systems

Ind. Eng. Chem. Res. 1999, 38, 133-143

133

PROCESS DESIGN AND CONTROL An Integrated Framework for Robust and Flexible Process Systems Michael C. Georgiadis and Efstratios N. Pistikopoulos* Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College, London SW7 2BY, U.K.

This paper presents developments toward a unified framework for incorporating both process flexibility and robustness (in terms of product quality) criteria in process optimization under uncertainty. A robust design methodology, known as the Taguchi approach, is discussed in the context of uncertainty, and some limitations are identified. Taguchi’s method is then extended to take into account process constraints and probabilistic uncertainty. A framework for establishing the interactions and synergistic benefits between the two operability objectives is proposed, based on an expected measure, where product quality losses are taken into account explicitly. Tradeoffs between stochastic flexibility and a robust criterion are explored in order to depict optimal operating policies in the presence of uncertainty; extensions toward design optimization are also briefly discussed. A number of examples is presented to illustrate the applicability of the proposed framework. Introduction The problem of accounting for uncertainty at the design stage of chemical plants is clearly a problem of great practical significance, due to the large number of technical and commercial parameters which are subjected to significant uncertainty. These uncertainties can arise from variations either in external parameters such as the quality of the feed streams, product demand, environmental conditions, and economic cost data, or from internal process parameters such as transfer coefficients, reaction constants, and physical properties. If the technology is new, there are additional uncertainties due to limited performance data.1 The potential effect of variability on process decisions regarding process design and operations constitutes a challenging problem. In this context, devising suitable numerical techniques and algorithms through the applications of which one could analyze and quantify the effect of uncertainty is a major research goal.2 In most recent studies, evaluation and design techniques of stochastic flexibility (e.g., the probability of feasible operation), where the uncertain parameters are described by (joint) probability distribution functions, have been developed.3,4 Extensions to consider the availability of each piece of equipment have also been presented. The problem of selecting an optimal design, in the presence of stochastic uncertain conditions, was mathematically posed as a stochastic nonlinear optimization problem.5 The solution of this problem was based on a two-stage stochastic programming design strategy where the objective was to determine the design that maximizes an expected profit or revenue while simultaneously measuring design feasibility. Extending this * To whom all correspondence should be addressed. Tel.: +44(0)171 5946620. Fax: +44(0) 171 5946606. E-mail: [email protected].

work, mixed-integer stochastic optimization based algorithms and computational studies for the solution of process synthesis problems under stochastic uncertainty have been presented by Acevedo and Pistikopoulos.6 In the context of optimal design and operation under uncertainty, process flexibility is one operability objective. Clearly, there are other aspects to the operability of a plant such as controllability, reliability, and robustness which are equally important. The need and importance for incorporating different operability measures in an integrated framework has been widely recognized.2 In this context, Mohideen et al.7 introduced a method for incorporating both flexibility and closedloop controllability in the optimal design of dynamic systems under uncertainty.7 Thomaidis and Pistikopoulos8 and recently Vassiliadis and Pistikopoulos9 addressed the problem of simultaneously accounting for flexibility, reliability, and maintenance policies during the design of processing systems. One key issue which must be addressed when operating and designing a process under uncertainty is whether a robust product quality can be ensured in the face of continuous uncertain parameters. This is an issue of major practical importance due to pressing market requirements for high-quality products. Robust quality methods were introduced to Japan in the late 1940s as an off-line quality control method, by G. Taguchi.10 This approach is a systematic and efficient way for designing cost optimal high-quality products and processes which are insensitive (“robust”) to uncertainty and variability, based on the principles of quality engineering. Taguchi’s approach has been applied in many engineering fields and the literature, especially in the mechanical engineering area, is extensive. Charteris11 discussed the benefits of this approach when applied in the food industry in conjunction with rigorous data analytical methods. Anand12 presented applications of the Taguchi method in a number of cases

10.1021/ie980440n CCC: $18.00 © 1999 American Chemical Society Published on Web 12/08/1998

134 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999

Figure 1. Quadratic quality loss function.

studied from the Indian industry. The concept of robust design methodology was also applied by Shang13 for the design and optimization of a flexible manufacturing system. In process systems engineering, the Taguchi robust design methodology is a relatively new concept. Straub and Grossman14 presented a product quality metric (based on one of the forms of the Taguchi’s quadratic loss function) which is evaluated over the uncertain region similar to stochastic flexibility. Since this metric may not yield satisfactory results because of its inability to handle hard constraints, they defined a reward function which is dependent on the choice of a parameter and exploited tradeoffs between stochastic flexibility and this product quality metric. Diwekar and Rubin15,16 proposed a parameter design methodology for chemical processes using a simulator. Their approach is loosely based on the minimization of the variance of the quality characteristic, where a Latin hypercube sampling technique is used for the collection of the samples. Diwekar and Kalagnanam17 presented a new sampling technique based on Hammersley points. This technique was applied in the robust design concept in the context of a batch distillation column design operating under internal and external uncertainties. The computational efficiency was proved to be significant, which is an important development for stochastic optimization problems. Recently, Ahmed and Sahinidis18 presented an optimization framework for the problem of long-range planning accounting for robustness of the resource costs through the use of a suitable variability criterion. Samsatli et al.19 proposed some performance measures to account for different robustness objectives in dynamic systems under parametric uncertainty. Bernardo and Saraiva20 developed a robust optimization framework leading to the simultaneous definition of optimal operation, parameters, and tolerance design. The main objectives of this work which extends and completes earlier works by the authors Georgiadis21 and Georgiadis and Pistikopoulos22 are (i) to extend Taguchi’s robust design approach, taking into account design and operating constraints and probabilistic uncertainty, (ii) to investigate the behavior of robustness with respect to process flexibility and an expected profit criterion, and (iii) to establish a systematic design framework optimizing an expected criterion and featuring an optimal degree of stochastic flexibility and robustness. A number of example problems are used to illustrate the applicability of the proposed approach. 2. Taguchi’s Robust Design Methodology The traditional method for ensuring product quality has been testing and inspection. These techniques,

including acceptance sampling, were used prior to the introduction of quality as a separate field of study and continue to be applied today. By the late 1950s the quality movement in Japan began to shift toward the product development area with specific emphasis given to product planning, design, trial manufacturing, and testing. Today’s perspective of product development quality techniques is “Quality must be built into each design and each process”.23 Taguchi’s method has become increasingly popular as a method for developing a engineering products principle because of its ability to increase the quality of an engineered product via simple changes in the method by which engineers perform their usual design tasks.10,24 According to the Taguchi approach product quality is defined as the total loss to society due to functional variation and harmful side effects. To quantify the quality losses, Taguchi10 proposed the use of the quadratic loss function. This function is applicable whenever the quality characteristic has a finite value, usually nonzero, and the quality loss is symmetric on either side of the target. It is ideally suited for evaluating the quality level of a product as it is shipped by a supplier to a customer. The quadratic loss function has the following form:

L(y) ) k(y - m)2

(1)

where k is a constant called the quality loss coefficient. The above equation is plotted in Figure 1. We observe that the loss L(y) increases slowly when the quality characteristic is near m, but as it goes further from m, the loss increases more rapidly. Qualitatively, the quality loss function is exactly the kind of behavior we would like the quality characteristic to have. It is important to determine the constant k so that eq 1 can approximate the actual loss within the region of interest. This is a rather difficult, though important, task. Because of the uncertain parameters, the quality characteristic y of a product varies from time to time. If y1, y2, ..., yn are n representative measurements of the quality characteristic y, then the average quality loss Q, resulting from this product, is given by25,26

Q ) k[(µ - m)2 + σ2]

(2)

where µ and σ2 are the mean and variance of y, respectively, as computed by

µ)

σ2 )

1

1

n

∑ yi n i)1

(3)

n

(yi - µ)2 ∑ n - 1 i)1

(4)

Thus, the average quality loss function has the following two components: (1) k(µ - m)2 resulting from the deviation of the average value of y from the target; (2) kσ2 resulting from the mean-squared deviation of y around its own mean. Between these two components of quality loss sometimes it is common practice to ignore the first one. Reducing the second component requires decreasing the variance which is more difficult.25 We should emphasize here that a decision based on the use of eq 2 as the objective is influenced not only by the sensitivity to noise (σ2) but also by the deviation from the target mean (µ - m). Often such a decision is heavily

Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 135

influenced, if not dominated, by the deviation from the target mean. As a result, there is a high risk of not minimizing the sensitivity of the quality characteristic to uncertainty. This, of course, is clearly undesirable. On the basis of the Taguchi approach, Dehnad25 proposed a new methodology known as robust design. In the robust design methodology the signal-to-noise (S/ NR) is used as the objective function to be optimized.26,27 This is a predictor of the quality losses which isolates the sensitivity of the product’s characteristic to noise factors. A number of benefit arises from using the S/NR for optimizing a product or product design: (1) Optimization does not depend on the target mean value as in the quadratic quality loss function. Thus, the design can be reused in other applications where the target is different. In fact, most of the problems in the process industry do not depend on the target of the quality characteristic. (2) there is no need for the quality characteristic and the quality losses to be symmetric on either side of the quality target. Common types of product quality problems and their respective S/N Ratio are given in Table 1 as adopted by Phadke.27 Note that the n observations of the quality characteristic under different noise conditions are denoted by y1, y2, ..., yn. So far, there has been little research to compare Taguchi’s technique to other methods, either analytically or experimentally, except for comparisons with experimental design techniques from which Taguchi’s method is derived. In addition, there has been little research to attempt to improve the method itself. Song28 presented a methodology for the design of processes/ products for the case of multiple product quality characteristics. Lee et al.29 posed Taguchi’s approach as an unconstrained optimization problem considering design variables noises. They illustrated their approach with small motivating examples. However, none of the above methods address process or product constraints which are commonly encountered in the process systems engineering problems (e.g., due to safety or environmental requirements). Furthermore, probabilistic requirements due to uncertainty are not fully taken into account. According to Taguchi’s approach, the control factors are chosen to take only one value. However, from a process engineering point of view the control variables can in principle be adjusted so that the process meets its operating constraints and, if possible, increase robustness. In this paper, we first extend Taguchi’s method to incorporate constraints and probabilistic uncertainty. To do so, this method is posed as a nonlinear stochastic optimization problem as described in the next section. 3. Stochastic Optimization Procedure for Robust Product Quality Motivating Example. Consider a motivating example taken from Pistikopoulos and Grossmann.30 The specifications of a design are represented by the following inequalities:

f1 ) z - θ1 + 0.5θ2 + d1 - 3d2 e 0

(5)

f2 ) -z - θ1/3 - θ2 + d2 + 1/3 e 0

(6)

f3 ) z + θ1 - θ2 - d1 - 1 e 0

(7)

cost(d) ) d1 + d2 e R

(8)

Figure 2. Feasible region and placement of quadrature points.

These inequalities involve a single control variable z, two design variables, d1 and d2, and two uncertain parameters, θ1 and θ2. The uncertain parameters are assumed to follow normal distributions functions; N(20,4) for θ1 and N(10,2) for θ2. The constant R represents an upper bound for a cost function. A product quality characteristic (QC) (e.g., product flow rate or composition) is assumed to be a function of the uncertain and control variables:

QC ) z + 2θ1 + θ2

(9)

The key question which must be addressed is whether the quality characteristic will remain close to each target specification in the face of the uncertain parameters. From the conceptual mathematical point of view this can be stated as is there a control variable z, for a fixed design, which can be readjusted for every possible realization of the uncertain parameters in order for the quality characteristic to be robust? According to Taguchi’s approach, the selection of a proper quality characteristic and S/NR is an important decision. The “nominal the better” type is the most appropriate for the chemical engineering problems. It is clear that, in order to quantify the S/NR, statistical data are required to compute the mean and the variance. One way to derive the statistical data is to use the Gaussian quadrature or some other integration scheme to generate quadrature/sample points. This is the key idea of the proposed procedure which is described as follows: (1) For a fixed design place the quadrature/sample points like those in Straub and Grossman14 or Acevedo and Pistikopoulos6. (2) Use the quadrature points as a source of statistical data, since at each point there is a value of the quality characteristic. (3) Compute the mean and the variance of the quality characteristic. (4) Solve the optimization problem at which the S/NR is maximized. More efficient sampling techniques such as the one presented by Diwekar and Kalagnanam17 can in principle be employed to define the required statistical data; other integration schemes, such as the ones discussed in Acevedo and Pistikopoulos,6 can also be easily incorporated into our proposed framework. 3.1. Mathematical Formulation. The systems of interest are modeled mathematically with a set of equality and inequality constraints:


h(d,z,x,θ) ) 0

(10)

g(d,z,x,θ) e 0

(11)

where d is the vector of design variables (physical parameters as volumes, sizes, etc.), z and x are the vectors of control and state variables (operating conditions), and θ represents the vector of uncertain parameters. Here, we assume that the vector of control variables z does not involve any uncertainty and all the uncertainty of the system is described by the vector θ. The mathematical formulation for a fixed design d and for two uncertain parameters is given as follows:

()

µ2 max S/NR ) 10 log10 2 σ zq1,q2

(P1)

q1 min q1 θq11 ) 0.5[θmax 1 (1 + v1 ) + θ1 (1 - v1 )]

q2 min q2 θq21q2 ) 0.5[θmax 2 (1 + v2 ) + θ2 (1 - v2 )]

hq31,q2(d,z q1,q2, x q1,q2, θq11, θq21,q2) ) 0 gq31,q2(d,z q1,q2, x q1,q2, θq11, θq21,q2) e 0 q1 ) 1, ..., Q1 q2 ) 1, ..., Q2

µ)

σ2 )

1

Q1

Q2

∑ ∑

Q1‚Q2 q1)1 q2)1

1 (Q1‚Q2

Q1

yq1,q2

Q2

(y q ,q ∑ ∑ - 1) q )1 q )1 1

1

2

- µ)2

2

max θmin e θ(‚) 2 1 e θ2 max θmin e θ(‚) 2 2 e θ2

The quality characteristic at each quadrature point is denoted by y q1,q2 and can be expressed in terms of state variables. The identification of the feasible region of operation and the quadrature points is based on the following approach. First, quadrature points θq11 are generated for a fixed choice of Q1 nodes q1 ) 1, ..., Q1. Next, quadrature points θq21,q2 are generated in θ2 space for a fixed choice of Q2. The procedure continues until the quadrature points at each parameter space are determined (see Figure 2). For more details about different integration schemes and computational issues, the reader can refer to Straub and Grossman14 and Acevedo and Pistikopoulos.6 vq11 and vq22 correspond to the location of the quadrature points in the [-1,1] interval. The value of stochastic flexibility (SF) can easily be calculated by incorporating the following known formula in the above problem:

SF )

L θU 1 - θ1

2

Q1

∑

q1)1

Wq11

q1 (θU - θL2 q1) 2

2

×

Q2

∑

q2)1

Wq22 j(θq11,θq21,q2) (12)

where Wq11, Wq22, are weights corresponding to each quadrature point.31 For the sake of clarity the above formulation was only restricted to two-dimensional problems, since the extension to higher dimensions is straightforward. The signal-to-noise ratio selected here corresponds to the “nominal the best” type which is the most appropriate for the process systems problems. The ideal value for the quality characteristic is both nonzero and finite and its observation range runs from zero to infinity. Other types of S/NR can be similarly employed. The solution of problem [P1] will determine an optimal operating policy which maximizes product robustness. Note that problem [P1], because of the presence of nonconvex terms, yields local optimal solutions. The recent developments of global optimization algorithms32,33 provide enough hope for addressing special classes of nonconvex problems for integrating flexibility and robustness. Furthermore, a reformulation is presented in Appendix A which can guarantee global optimality for problems described by a convex feasible region. Application to the Motivating Example. The proposed formulation is applied to the motivating example described above, and the problem is solved parametrically in terms of the cost bound R. The S/NR as a function of this constant is depicted in Figure 3. We observe that for a small design the quality characteristic is very robust with S/NR equal to 100 and the variance 10-7. And, as R increases, the S/NR decreases and the process is less robust corresponding to a design with a value of R equal to 20 while the variance of the quality characteristic is about 60. Also, stochastic flexibility as a function of R is shown in Figure 4. A comparison with Figure 3 reveals that tradeoffs exist between the two operability measures. The physical explanation behind this tradeoff may be attributed to the following fact: since the main objective of the Taguchi approach is to minimize the variance of the quality characteristic, this implies that the distribution of the uncertain parameters in the space of the state variables (quality characteristic) must not be scattered. This, however, leads to an operation with low SF. Having imposed Taguchi’s approach as a stochastic optimization problem and identified general tradeoffs between the two operability measures, a new integrated approach for addressing both robustness and flexibility is proposed in the next section. 4. Integrated Approach for Flexibility and Robustness In general, the operation of a system can be described by the vector of constraints (10) and (11). As illustrated in Figure 5, the key question from a feasibility point of view (fixed design d) is whether a control vector z exists for which the operation is feasible for every realization of the vector of uncertain parameters θ. Also, from a product quality point of view, the question is whether, for the same design d, a control policy z exists for which product quality (which is generally a function of z and


QL ) kσ2

(13)

where constant k is the quality loss coefficient (see also section 2). Since this constant is difficult to be determined at this stage, the quality losses should be calculated parametrically in terms of this. In fact, k can be viewed as a penalty to the profit because of the quality losses. Therefore, the net profit value, NPV, is introduced corresponding to the difference between the expected profit and the quality losses as follows: NPV ) EP - QL. Note that the mean of the quality characteristic is not

max NPV ) EP - QL zq1,q2 EP )

L θU 1 - θ1

2

Figure 3. S/NR vs design for the motivating example.

Q1

∑

q1)1

Wq11

1 1 (θUq - θLq 2 2 )

2

(P2)

×

Q2

Wq2 ∑ q )1

2

P(d,z q1,q2,x q1,q2,θq11,θq21,q2)J(θq11,θq21,q2)

2

subject to QL ) kσ2 θL1 ) arg{min θ1|h1(d,z,x,θ1,θ2) ) 0 g1(d,z,x,θ1,θ2) e 0} θUP 1 ) arg{max θ1|h1(d,z,x,θ1,θ2) ) 0 g1(d,z,x,θ1,θ2) e 0} q1 L q1 θq11 ) 0.5[θUP 1 (1 + v1 ) + θ1 (1 - v1 )]

q1 ) 1, ..., Q1 Figure 4. Stochastic flexibility vs R for the motivating example.

θL2 q1 ) arg{min θ2|h2(d,z,x,θq11,θ2) ) 0 g2(d,z,x,θq11,θ2) e 0} q1 ) arg{max θ2|h2(d,z,x,θq11,θ2) ) 0 θUP 2

g2(d,z,x,θq11,θ2) e 0} q1 ) 1, ..., Q1 q1 θq21,q2 ) 0.5[θUP (1 + vq22) + θL1 q1(1 - vq22)] 1

hq31,q2(d,z q1,q2, x q1,q2, θq11, θq21,q2) ) 0

Figure 5. Integrated framework for flexibility and robustness representation.

θ) is robust (according to some Signal-to-noise ratio criterion) for all realizations of the vector of uncertain parameters θ. Therefore, the main objective for the integrated approach is to find the best settings for the control variables so that the process may be as robust as possible (close to the target quality specifications) while ensuring the corresponding level of flexibility. An expected profit criterion is used as the objective function to additionally account for quality losses. These quality losses according to Phadke27 are mainly due to variance. Therefore, a function that can be used in order to quantify the quality losses is given as follows:

gq31,q2(d,z q1,q2, x q1,q2, θq11, θq21,q2) e 0 q1 ) 1, ..., Q1 q2 ) 1, ..., Q2 fixed (as proposed by Taguchi’s approach) but may be

SF )

L θU 1 - θ1

2

Q1

∑

q1)1

Wq11

1 1 (θUq - θLq 2 2 )

2

× Q2

Wq2 ∑ q )1 2

2

j(θq11, θq21,q2)


optimally selected by the optimization algorithm so as to rigorously account for the effect of the mean product quality value on the plant’s profitability. However, if desirable, a fixed value of the mean is also allowed in the formulation, as a special case. The mathematical formulation of the integrated approach is given by the following NLP problem [P2] for a fixed design d, and two uncertain parameters. Here, the expected profit is evaluated within the feasible region of operation (which is fixed and can be determined in this case for constant d).

()

S/NR ) 10 log10

µ)

σ2 )

1

Q1

Q2

∑ ∑

Q1‚Q2 q1)1 q2)1 1

Q1

µ2 σ2

y q1,q2 Figure 6. Expected profit vs S/NR (minimum requirement) for the illustrative example.

Q2

∑ ∑

(Q1‚Q2 - 1) q1)1 q2)1

(y q1,q2 - µ)2

max θmin e θL1 e θUP 1 1 e θ1 UPq1 1 θmin e θLq e θmax 2 2 e θ2 2 , q1 ) 1, ..., Q1 max θmin e θ(‚) 2 2 e θ2

S/NR g δ where P is a given profit function. All the other variables are denoted as those in the previous section. The solution of [P2], which is a local optimum, provides a strategy z which properly accounts for the interactions of flexibility and robustness (by varying the parameter δ). Motivating Example Revisited. Consider again the motivating example described in the previous section. Assume that the profit is a function of the quality characteristic QC and the control and design variables as follows:

P ) 4z + 4QC - 8d1 - 4d2

Figure 7. Quality losses vs S/NR for the illustrative example.

(14)

For the analysis we select the design which maximizes the expected profit (following the approach of Pistikopoulos2). This corresponds to a design vector, d1 ) 12.2, d2 ) 1.8, while the value of stochastic flexibility is 0.75. We first solve problem [P2] by imposing a minimum requirement δ for the S/NR:

S/NR g δ

(15)

The parametric solution of [P2] without any quality losses (EP ) NPV) is depicted in Figure 6. Note that as the product quality becomes better, the EP decreases. This is expected since, as the product quality increases, it reflects a tighter (more conservative) control policy around the mean. Introducing quality losses by assuming a quality loss coefficient equal to 0.3 leads to results depicted in Figure 7. Note that quality losses decrease as the S/NR increases. By combining Figures 6 and 7, constructing the NPV function leads to the results shown in Figure

Figure 8. Net profit vs S/NR for the illustrative example.

8. Note that there is an optimal operating point at which the NPV is maximized under the optimal degree of robustness (S/NR ) 12, with a variance of 30) and the corresponding stochastic flexibility value is 0.75. This is exactly the direct solution of problem [P2] (including


Figure 11. A simple chemical reaction with a recycle flow sheet.

ibility (different designs) and product quality variance is illustrated. Figure 9. Net profit vs R for the illustrative example.

6. Process Examples Two examples are presented in this section to illustrate the basic features of the proposed framework. Eight quadrature points have been used for each of the uncertain parameters (the effect of the number of quadrature points on the performance of the algorithm is discussed in ref 6. Example 1. The simple chemical process with recycle in Figure 11 is considered in this example taken from ref 34. It consists of a CSTR of volume V (m3), with perfect mixing, where the type of the reaction is assumed to be Denbigh’s reaction which comprises firstorder elementary reactions as shown below.

Figure 10. Pareto curve of SF vs variance for the illustrative example.

the minimum requirement for the S/NR) for k ) 0.3. A similar analysis can be performed for other values of k. 5. Design Optimization So far, the integrated approach in problem [P2] has been based on a fixed design. Extensions to design optimization can be made based on the approach described in ref 6. Here, we explore only the impact of the design variables by parametrically varying their values. Application to the Motivating Example. Problem [P2] is solved parametrically in terms of the cost upper bound R. The results are shown in Figure 9. Note that an optimal design exists which maximizes the net profit. This design corresponds to a value of R equal to 10 (d1 ) 9.5 and d2 ) 0.45) and the optimal degree of SF is 0.52. At the same time it features an optimal degree of robustness which in terms of product quality variance is 11.6. It is worth noticing that this design is different from the one that corresponds to the maximization of the expected profit without robust quality considerations (R ) 14). It seems, under the specified conditions, that some flexibility is sacrificed (0.52 from 0.75) for the benefit of improving product robustness (from 31 to 11.6). Another interesting tradeoff is depicted in Figure 10 where the pareto optimal curve of stochastic flex-

The outlet flow of the CSTR is F, with fractions of molar concentrations xA, xB, xR, xX, and xY. R is the desired product, and this is produced from fresh feed containing pure A, whose flow and concentration are FA0 (mol/h) and CA0 (mol/m3), respectively. The effluent flow of the reactor F (mol/h) is assumed to be separated into a product flow containing only R at the top and another flow containing A, B, X, and Y at the bottom. X and Y are undesirable byproducts. At the bottom, the fraction R of the components A and B and the fraction β of the components X and Y are recycled in order to be mixed with the inlet stream of the pure A. The uncertain parameters are taken to be the rate constants kB and kR following Gaussian distribution. The first has a of mean 0.1 h-1 with a standard deviation of 0.02 h-1 while kR has a mean of 0.4 h-1 and a standard deviation of 0.08 h-1. With a level of confidence of 99% the expected parameter deviations are (4σ. Note that these two rates have a greater impact in the quality characteristic than the other two according to the reaction scheme. The quality characteristic is taken to be the product rate which is given by the product (FxR). The design variable is the volume of the reactor V. Performing a degree of freedom analysis in this problem, it becomes apparent that we must specify three variables in order to be able to solve the problem. These are the recycle fractions R and β and the outlet flow of the reactor F. Finally, the state variables of this problem are the molar fractions xA, xB, xR, xX, and xY and the volume of the liquid in the reactor V op.


Figure 12. Expected profit vs minimum requirement in S/NR for example 1.

For steady-state and isothermal operation the process model can be described by the following system of nonlinear equations.

FA0 - xAF (1 - R) - V op(kB + kX)CA0xA ) 0

(16)

-F xB (1 - R) + V opCA0[kBxA - (kR + kY)xB] ) 0 (17) -F ‚ xX (1 - β) + V opCA0kXxA ) 0

(18)

-F ‚ xY (1 - β) + V opCA0kYxB ) 0

(19)

-F ‚ xR + V opCA0kRxB ) 0

(20)

xA + xB + xR + xX + xY ) 1

(21)

where V op is the reactor holdup which must satisfy the following feasibility constraint:

V op e V

(22)

Finally, the minimum amount of R that has to be produced is FR yielding

FR - FxR e 0

(23)

The values of the system parameters are given in Table 2. The profit function includes a revenue term (from product’s sells) and a cost factor (annualized capital and operating cost) and has the following form:

P ) 491FRxR - (12500 + 160V 2 + 0.9848FR(R(xA + xB) + β(xX + xY))) (24) The product quality losses are a function of variance according to the following expression:

QL ) 20 ‚ σ2

(25)

The S/NR is taken to be the nominal the best type, where the mean value of the quality characteristic is optimally selected by the problem. The case of maximizing the expected profit by imposing a minimum requirement for the S/NR is first investigated and the results are shown in Figure 12. Note that as the product quality increases, the EP decreases, indicating that tradeoffs between robustness

and the expected criterion exists. For the design value that corresponds to the maximization of the EP (V ) 10.5 m3) the net profit value (NPV) as a function of S/NR is depicted in Figure 13. We observe that an optimal operating point exists at which NPV is maximized under the optimal degree of robustness (S/NR ) 21.5) and the corresponding flexibility (SF ) 0.81). The NPV as a function of the design, V, is shown in Figure 14. It is revealed that an optimal design exists which maximizes NPV under the optimal degree of flexibility and robustness. This design (V ) 10.2 m3) corresponds to a SF of 0.77 and S/NR of 22.5 (product quality variance ) 31). Compared to the optimal design of the EP maximization problem (V ) 10.5 m3), we note that a small flexibility is sacrificed for the benefit of improving product robustness (S/NR from 21.5 to 22.5). The general tradeoffs between the two operability objectives in a pareto form are depicted in Figures 15 and 16. The GAMS/CONOPT NLP solver has been used for the solution and the computational time corresponding to the maximization of NPV is approximately 200 CPU/s on a Sun Ultra Workstation. Example 2. The reactor system considered in ref 35 is shown in Figure 17. It consists of a reactor and a cooler, where a first-order exothermic reaction A f B takes place. The design variables are the reactor volume V (m3) and the area of the heat exchanger A. Two uncertain variables are considered following the normal probability distribution: the feed flow F0 with a mean of 45.36 kmol/h and standard deviation of 7 kmol/h and the temperature of the feed stream T0 with a mean of 333 K and standard deviation of 7 K. For a confidence level of 99% the expected deviations are (4σ. The control variables are the reactor temperature T1, the temperature T2 of the recycle stream, and the outlet temperature and flow rate of the cooling water Tw2 and Fw (see Figure 17). The following bounds are imposed in the control variables:

311 e T1 e 389 K

(26)

311 e T2 e 389 K

(27)

301 e Tw2 e 355 K

(28)

The specification constraints, along with heat and mass balance and design equations are presented in Table 3; the values of all the system parameters are shown in Table 4. The quality loss coefficient is taken to be equal to 2 × 106. Here, the quality characteristic is taken to be the conversion of the raw material A and the S/NR is the nominal the best type. The problem is to determine the optimal operating point at which the NPV is maximized while accounting for quality losses (optimal product robustness). The solution of problem [P2] leads to an optimal operating policy and the results are shown in Figure 18. This optimal operating point corresponds to an S/NR value equal to 40 with a variance of the quality characteristic equal to 4 × 10-5. The corresponding degree of process flexibility is 0.81. 7. Concluding Remarks Theoretical developments toward a formal theory and unified framework for incorporating both process flexibility and robustness (in terms of product quality) in process optimization and design under uncertainty have


Figure 13. Net profit vs minimum requirement in S/N ratio for example 1. Figure 15. Pareto curve of SF vs S/NR for example 1.

Figure 14. Net profit vs design for example 1.

been presented in this work. Taguchi’s robust design methodology was first discussed and extensions have been proposed to account for process constraints and probabilistic uncertainty. The signal-to-noise ratio criterion was used as the objective function and the required statistical data for its computation were exploited through an integration scheme. General tradeoffs between process flexibility and robustness were identified and a novel optimization formulation was also

Figure 16. Pareto curve of SF vs variance for example 1.

proposed to determine the optimal operating policy which defines an optimal degree of robustness. The objective is a net expected profit criterion where product quality losses are explicitly taken into account. Note that the computational cost for the solution of these problems may be quite high. This strongly de-

Table 1. S/NR for Product Quality Problems problem type smaller-the-better type

nominal the best type

range for observations 0eye∞ 0eye∞

ideal value

S/N ratio

0

η ) -10 log10 nonzero, finite

1

∞

n i)1

µ2 σ2

∑y 1

i n

∑ (y - µ)

-∞ e y e ∞

finite, usually 0

( )

η ) 10 log10

1

n

1

1

∑y n

η ) -10 log10 σ σ2 )

2

n - 1 i)1

i)1

signed target

y21

n

n i)1

σ2 ) 0eye∞

n

()

η ) 10 log10 µ)

larger-the-better type

(∑ ) 1

2 i

2

n

∑ (y - µ) n-1 i)1

2


formed independently, a factor which must be further explored to enhance computational performance as in ref 6. Moreover, improved integration schemes based on cubatures and specialized quadrature integration formulas, as discussed in refs 33 and 36 can also provide the basis for addressing large-scale problems. Acknowledgment Financial support by EPSRC is gratefully acknowledged. Nomenclature Figure 17. Reactor-cooler system. Table 2. Values of the Process Parameters for Example 1 parameters

values

FA0 mol/h CA0 mol/m3 FR mol/h kX (h-1) kY (h-1)

100 100 70 0.02 0.01

Table 3. Model Equations and Specification Inequalities for Example 2 model equations F0(CA0 - CA1/CA0 ) Vk0 × exp[T1 - T0]CA1 (-∆H)F0(CA0 - CA1/CA0 ) F0Cp(T1 - T0) + QHE QHE ) F1Cp(T1 - T2) QHE ) FwCpw(Tw2 - Tw1) QHE ) AU/∆Tln ∆Tln ) f(T1,T2,Tw1,Tw2)

specification inequalities Vd g V (CA0 - CA1)/CA0 g 0.8 311 e T1 e 389 T 1 - T2 g 0 Tw2 - Tw1 g 0 T1 - Tw1 g 11.1 T2 - Tw1 g 11.1

Table 4. Values of the Process Parameters for Example 2 parameters

values

Tw1 (K) k0 (h-1) U (kJ‚m-2‚h-1‚K-1) E/R (K) CA0 (kmol/m3) -∆H (kJ/kmol) Cp (kJ/kmol‚K)

300 12 1635 555.6 32.04 23260 167.4

d ) vector of design variables x ) vector of state variables z ) vector of control variables SF ) stochastic flexibility SN/R ) signal-to-noise ratio k ) quality loss coefficient L ) quadratic loss function m ) target value for the quality characteristic y ) quality characteristic µ ) mean value of the quality characteristic σ ) standard deviation of the uncertain parameter σ2 ) variance of the quality characteristic J ) joint probability function EP ) expected profit NPV ) net profit value Wi ) weights corresponding to quadrature points v ) parameters for the location of the quadrature points θ ) vector for the uncertain parameters θL1 ) lower bound for the first uncertain parameter ) maximum value of the first uncertain parameter θmax 1 ) minimum value of the first uncertain parameter θmin 1 θUP ) upper bound for the first uncertain parameter 1 θL2 q1 ) lower bound for the second uncertain parameter corresponding to the q1 point of the first uncertain parameter θmax ) maximum value of the second uncertain parameter 2 ) minimum value of the second uncertain parameter θmin 2 q1 θUP ) upper bound for the second uncertain parameter 2 corresponding to the q1 point of the first uncertain parameter Greek Letters R ) design parameter or recycle fraction β ) recycle fraction for example 1 δ ) minimum requirement for the signal-to-noise ratio

Appendix: Reformulation for Global Optimality For problems described by a convex feasible region the following reformulation of problem [P1] can guarantee global optimal solutions.

max S/NR ) 10 log10(µ2) z q1,q2 σ2e q1 min q1 θq11 ) 0.5[θmax 1 (1 + v1 ) + θ1 (1 - v1 )]

Figure 18. Net profit vs S/NR for example 2.

pends on the number of quadrature points for the Gaussian quadrature approximation and increases exponentially with the number of uncertain parameters. Nevertheless, the algorithm has a a highly parallel structure, since most optimization tasks can be per-

q2 min q2 θq21q2 ) 0.5[θmax 2 (1 + v2 ) + θ2 (1 - v2 )]

hq31,q2(d, z q1,q2, x q1,q2, θq11, θq21,q2) ) 0 gq31,q2(d, z q1,q2, x q1,q2, θq11, θq21,q2) e 0 q1 ) 1, ..., Q1 q2 ) 1, ..., Q2

(P3)


µ)

Q1

Q2

∑ ∑

Q1‚Q2 q1)1 q2)1

1

2

σ )

1

(Q1‚Q2

Q1

yq1,q2

Q2

(y q ,q ∑ ∑ - 1) q )1 q )1 1

1

2

- µ)2

2

max θmin e θ(‚) 2 1 e θ2 max θmin e θ(‚) 2 2 e θ2

where is a suitable constant. It can easily be proved that µ2 is a concave function and σ2 a convex function. The above problem [P3] is parametric (in ) NLP. Literature Cited (1) Grossmann, I. E.; Straub, D. A. Recent developments in the evaluation and optimization of flexible chemical processes. In Computer-Oriented Process Engineering; Puigjaner, L., Espuna, A., Eds.; Elsevier: Amsterdam, 1991. (2) Pistikopoulos, E. N. Uncertainty in process design and operations. Comput. Chem. Eng. 1995, S19, S553. (3) Pistikopoulos, E. N.; Mazzuchi, T. A. A novel flexibility analysis approach for processes with stochastic parameters. Comput. Chem. Eng. 1990, 14, 991. (4) Straub, D. A.; Grossmann, I. E. Integrated stochastic metric of flexibility for systems with discrete and continuous parameters uncertainties. Comput. Chem. Eng. 1990, 14, 967. (5) Pistikopoulos, E. N.; Ierapetritou, M. G. A novel approach for optimal process design under uncertainty. Comput. Chem. Eng. 1995, 19, 1089. (6) Acevedo, J.; Pistikopoulos, E. N. Stochastic optimization based algorithms for procerss synthesis and design under uncertainty. Comput. Chem. Eng. 1998, 22, 647. (7) Mohideen, M. J.; Perkins, J. D.; Pistikopoulos, E. N. Optimal design of dynamic systems under uncertainty. AIChE J. 1996, 42, 2251. (8) Thomaidis, T. V.; Pistikopoulos, E. N. Integration of flexibility reliability and maintenance in process synthesis and design. Comput. Chem. Eng. 1994, 18, S259. (9) Vassiliadis, C. G.; Pistikopoulos, E. N. On the interaction of chemical-process design under uncertainty and maintenanceoptimization. In Annual Reliability and Maintenability Symposium; Anaheim, CA, Jan 19-22, 1998; IEEE: New York, 1998. (10) Taguchi, G. Introduction to quality engineering. Designing quality into products and processes; American Supplier Institute: Dearbon, MI, 1986. (11) Charteris, W. Taguchi’s systems of experimental design and data analysis: a quality engineering technology for the food industry. J. Soc. Dairy Technol. 1992, 45, 33. (12) Anand, K. N. Optimization of design parameters using Taguchi method of off-line quality controlsa few Indian case studies. J. Eng. Design. 1995, 6, 91. (13) Snang, J. S. Robust design and optimization of material handling in an fms. Int. J. Prod. Res. 1995, 33, 2437. (14) Straub, D.; Grossmann, I. E. Design optimization of stochastic flexibility. Comput. Chem. Eng. 1993, 13, 339. (15) Diwekar, U. M.; Rubin, E. S. Parameter design methodology for chemical processes using a simulator. Ind. Eng. Chem. Res. 1994, 33, 292.

(16) Diwekar, U. M.; Rubin, E. S. Stochastic modelling of chemical processes. Comput. Chem. Eng. 1991, 15, 105. (17) Diwekar, U. M.; Kalagnanam, J. R. Efficient sampling technique for optimization under uncertainty. AIChE J. 1997, 43, 440. (18) Ahmed, S.; Sahinidis, N. V. Robust process planning under uncertainty. Ind. Eng. Chem. Res. 1998, 37, 1883. (19) Samsatli, N. J.; Papageorgiou, L. G.; Shah, N. Robustness metrics for dynamic optimization models under parameter uncertainty. AIChE J. 1988, 44, 1993. (20) Bernardo, F.; Saraiva, P. A robust optimization framework for process parameter and tolerance design. AIChE J. 1998, 44, 2007. (21) Georgiadis, M. C. An integrated design framework for robust and flexible process systems. Master’s Thesis, University of London, Imperial College of Science, Technology and Medicine, 1995. (22) Georgiadis, M. C.; Pistikopoulos, E. N. Flexibility and robustness issues in process optimization under uncertainty. In 5th IFAC Symposium on Dynamics and Control of Process Systems; Georgakis, C., Ed.; IFAC: Corfu, Greece, 1998. (23) Osborne, D. M.; Armacost, R. L. Review of techniques for optimizing multiple quality characteristics in product development. Comput. Chem. Eng. 1996, 31, 107. (24) Barker, T. B. Quality by Experimental Design; Marcel Dekker: New York, 1994. (25) Dehnad, K. Quality control, robust design and the Taguchi methods; Bell Laboratories: New York, 1989. (26) Shoemaker, R. V.; Kacker, R. N. Performance measures independent of adjustment. an explanation and extension of Taguchi’s signal-to-noise ratios. Technometrics 1987, 29, 253. (27) Phadke, M. S. Quality Engineering Using Robust Design; Prentice Hall: Englewood Cliffs, NJ, 1989. (28) Song, A. A. Design of process parameters using robust design techniques and multiple criteria optimization. Trans. Systems, Man and Cubernetics 1995, 25, 1437. (29) Lee, K.-H.; Eom, In-Sup.; Park, G. J.; Lee, W. I. Robust design for unconstrained optimization problems using the Taguchi method. AIAA J. 1996, 34, 1059. (30) Pistikopoulos, E. N.; Grossmann, I. E. Stochastic optimization of flexibility in retrofit design of linear systems. Comput. Chem. Eng. 1998, 12, 1215. (31) Carnahan, B. Applied Numerical Methods; Wiley, New York, 1969. (32) Adjiman, C. S.; Androulakis, I. P.; Floudas, C. A. A global optimization method, Rbb, for process design. Comput. Chem. Eng. 1996, S20, S419. (33) Epperly, T. G. W.; Ierapetritou, M. G.; Pistikopoulos, E. N. On the global and efficient solution of stochastic batch plant design problems. Comput. Chem. Eng. 1997, 21, 1411. (34) Ostrovsky, G. M.; Volin, Y. M.; Barit, E. I.; Senyavin, M. M. Flexibility analysis and optimization of chemical plants with uncertain parameters Comput. Chem. Eng. 1994, 18, 755. (35) Halemane, K. P.; Grossmann, I. E. Optimal design under uncertainty. AIChE J. 1983, 29, 425. (36) Bernardo, F.; Saraiva, P.; Pistikopoulos, E. N. Integration and computational issues in stochastic design and planning. Submitted for publication in Ind. Eng. Chem. Res. 1999.

Received for review July 9, 1998 Revised manuscript received October 13, 1998 Accepted October 14, 1998 IE980440N

An Integrated Framework for Robust and Flexible Process Systems

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