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Ind. Eng. Chem. Res. 2008, 47, 4170–4183
Process Flexibility for Multivariable Systems Sau M. Lai and Chi-Wai Hui* Department of Chemical Engineering, The Hong Kong UniVersity of Science and Technology, Clear Water Bay, Hong Kong
Process flexibility is important to ensure that an operation can be both profitable and manipulative to changes in internal or external process parameters. A new flexibility metric (FIV) is defined based on the hypervolume ratio of the feasible region and the region containing all combinations of expected uncertain parameters. The FIV evaluation requires a hypervolume determination which may not be straightforward for multidimensional systems or systems with nonconvex feasible regions. Thus, an auxiliary vector approach is developed to estimate the feasible space’s size. Compared to the existing methodologies, this new method is able to provide a reliable and facile flexibility evaluation. In addition, this approach can also offer indications of the possible future changes to enhance the overall flexibility. This information is useful for making reasonable and rational engineering decisions to retrofit the designs. Introduction With swift changes in internal and external conditions and requirements, process parameters are usually uncertain. Because of the possible deviations, the traditional process design approach based on process parameters’ nominal values is not enough to provide realistic solutions. The incapability of the design to handle uncertainties may either lead to unsatisfied customer demand1 or deprivation of the process’s original economic efficiency.2 Therefore, it is essential to find ways to solve this problem. To tackle process uncertainties, chemical process design should not only focus on economic considerations but also its operability. One of the key concerns in process operability is flexibility which refers to the ability of the process to handle changes when process parameters vary. Flexibility focuses on steady state operation changes while dynamic changes are usually taken care of by dynamic resiliency.3–6 It is necessary to obtain a flexible option when designing a process so as to handle process parameters’ deviations with the production economics being optimized. To evaluate how flexible a process is, two approaches are developed. One is to analyze whether feasible operation is achievable for specific discrete conditions. A possible method to evaluate this is by conducting a feasibility test as suggested by Grossmann and co-workers.3,7 Another approach is to measure the degree of flexibility in a process. Various metrics have been defined to evaluate this. In early 1980s, Grossmann and co-workers developed flexibility index (FIG).3,5,6 It quantifies the smallest percentage of the expected deviation for all uncertain parameters that the process can feasibly handle. This index can be visualized as half the length of the biggest inscribable hypercube inside the feasible region in the uncertain space with the process’s nominal point located at its center. On the other hand, Saboo et al. introduced the resilience index (RI).8 It is the largest disturbance load, independent of the disturbance direction, a network can withstand without becoming infeasible. The RI can be visualized as half the diagonal length of the largest hyper-rhombus, whose diagonals parallel to the principal axes in the uncertain space, that can be inserted within the feasible region. Since the length of the inserted hypercube or hyper-rhombus will be larger with a bigger feasible space, the * To whom correspondence should be addressed. E-mail: kehui@ ust.hk.
larger the FIG or RI can effectively represent a process with higher flexibility. FIG and RI determinations are generally restricted to systems with the nominal point located within the feasible region. Though an alternative approach has been introduced to evaluate FIG for systems with infeasible nominal point,9 the FIG obtained fails to provide meaningful flexibility information. Moreover, these indices only account for process flexibility based on the critical parameters. When the nominal point is located on the constraint boundary, the evaluated indices will equal to 0 no matter how large the actual feasible space is. Therefore, these metrics are not sufficient to provide comprehensive insight into the process and may easily lead to process misinterpretation. The simplest methodology to formulate FIG is by vertex searching which gives the critical point of the design. However, when the feasible region is not one-dimensional quasi-convex,5 this FIG determination method is not applicable. Like other optimization problems, the presence of nonconvex constraints is usually difficult to handle. In order to evaluate FIG for systems with such constraints, a convexification step is needed to convert the nonconvex constraints so as to obtain an underestimating function which satisfies the Karush-Kuhn-Tucker optimization condition. This step creates a new optimization problem that can be solved to obtain FIG.10 These complicated procedures show the difficulties in process evaluation for systems with nonconvex feasible space. Due to the problems associated with RI and FIG, Pistikopoulos and Mazzuchi11 defined another process flexibility measurement called stochastic flexibility (SF). It is a measurement of the cumulative probability within the feasible region. It predicts the probability for feasible operation without process constraint violation as the uncertain parameters vary. Unlike FIG, it is not necessary to have a known nominal point in SF determination. In addition, the whole feasible region is being considered, and therefore, SF is able to provide a comprehensive flexibility measurement. More complicated process operability characteristics can also be evaluated based on SF. For example, Straub and Grossmann12,13 accounted for the overall expected process flexibility using the expected stochastic flexibility (E(SF)) which is a combined process flexibility and equipment reliability measurement and Thomaidis and Pistikopoulos14,15 introduced the combined flexibility-reliability index (FR) and the com-
10.1021/ie070183z CCC: $40.75 2008 American Chemical Society Published on Web 05/28/2008
Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008 4171
(i.e., FIV ) Vf/Vo). For systems with higher dimensionality, the FIV is determined by the hypervolume ratio. A larger FIV reflects that more combination of the uncertain parameters can be feasibly handled while a process with FIV ) 0 represents that it is impossible to run the process without constraint violation for all the expected operating conditions. To determine FIV, the hypervolumes of Sf and So have to be measured. Since θiU and θiL are generally independent, So is usually a hyper-rectangular prism and its volume for system with N uncertain parameters can be determined by N (θ - θ ) Vo ) ∏i)1 iU iL
Figure 1. Schematic diagram of the uncertain space for a system with three uncertain parameters and the flexibility index (FIV) determination method.
bined flexibility-availability index (FA) to simultaneously optimize multiple process operabilities. Although SF provides a comprehensive process flexibility evaluation, its computation requires probability distribution information for the uncertain parameters. Such information may not be available at the process design stage. Even though the probability distribution is obtainable, computation of SF is usually tedious. Different SF computation approaches have been developed,1,11–13,16–18 but these methods are still complex and requirelengthycomputationeffortcomparedwithFIG determination. As discussed above, FIG is easy to compute but it is unable to provide comprehensive process information while SF can evaluate process flexibility with sound physical meaning, but its calculation is not facile. It is necessary to develop a new flexibility metric that can be determined practically with substantial implication of the process flexibility. This paper aims at introducing a new flexibility metric which gives the percentage of the region spanned by the expected deviation in the process’s uncertain parameter space that can be feasibly handled. The definition of the new metric is introduced first which is followed by the examples demonstrating the determination of process flexibility for problems with multiple uncertain parameters. New Flexibility Metric Definition. The uncertain space with three uncertain parameters (θ1, θ2, and θ3) is shown in Figure 1. The expected upper and lower limits of the parameters (θiU and θiL) are shown by the dotted lines in the figure. The region bounded by these lines is the whole space (So) which contains all expected combinations of the uncertain parameters. The constraint boundary is represented by the surface of the ellipsoid outlined. This boundary surface separates the uncertain space into the constrained space (Sc) which is the region within the ellipsoid and the infeasible space which is the region outside. The feasible space (Sf) is the intersection of So and Sc (i.e., So ∩ Sc) representing the combinations of uncertain parameters that can be feasibly handled by the process within their expected deviations. This region is outlined by the dashed line in the figure. The new flexibility index (FIV) is defined as the volume ratio of Sf to So
However, Sf is usually irregular in shape whose volume (Vf) determination is not straightforward. To estimate Vf, a constructed space (Se, the region outlined by the thick solid line), whose volume determination is less difficult, can be inscribed inside Sf. With a careful selection of the shape of Se, its volume (Ve) can be used as a close estimate of Vf. Therefore, the estimated value of FIV can be determined by Ve/Vo. As illustrated in the figure, Se can be constructed by first picking a reference point (PR), which is not necessarily the nominal point, within Sf. The uncertain space can then be scaled based on this reference point and the expected parameter limits. Auxiliary vectors Vfj with selected directions can be radiated from PR. The interception points (Pj) of Vfj and the feasible space boundary are obtained. The Se can then be constructed by joining these Pj points according to their positions in space. Since different Se can be generated by different auxiliary vector direction selection schemes, estimation accuracy of Vf and FIV will depend on the selection scheme employed. The general formulation for the auxiliary vectors’ positions in a threedimensional space is as follows: objective fuction: max Ve(xj, yj, zj) by varying xj, yj, zj (1a) constraints: fk(θij) e 0 and 0 e xj, yj, zj e 1 θxj ) Vxjxj∆xj + θRx such that
θyj ) Vyjyj∆yj + θRy , ∆ij ) θzj ) Vzjzj∆zj + θzR
{
(1b)
∆+θi (Vij g 0) (1c) |∆-θi| (Vij < 0)
where Ve(xj, yj, zj) ) generalized volume determination function for Se; fk (θij) ) kth constraint function evaluated at Pj (θij); θij ) coordinates of the jth interception point, (θxj, θyj, θzj); xj, yj, zj ) variables for auxiliary vector position index at Pj which gives the vector’s pointing direction; Vxj, Vyj, Vzj ) auxiliary vector direction index at Pj which restricts the region that the vector can point into; ∆ij ) expected deviation index of the ith uncertain parameter for Vij; ∆+θi ) expected positive deviation for the ith uncertain parameter; |∆-θi| ) expected negative deviation for the ith uncertain parameter; and θiR (θRx ,θRy ,θzR) ) coordinates of PR. By imposing more restrictions on xj, yj, and zj, the shape of the constructed space can be manipulated. For example, by restricting xj ) yj ) zj, the constructed space generated will be a hypercube centered at the PR. Confining Se with a simpler shape can effectively ease the Ve determination. The objective function employed above is to maximize the volume of the constructed space by varying the position indices without restricting the shape of Se. With such an arbitrary Se, generalized Ve(xj, yj, zj) may not be available and it may not be used at all time to determine the interception points. Other objective functions will be needed and this will be further discussed in
4172 Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008
Figure 2. Schematic diagram showing the procedures in the proposed flexibility evaluation method.
the examples below. The systematic approach to perform the flexibility evaluation is illustrated in Figure 2. Another metric based on the volume ratio of the feasible space and the expected deviation space has been introduced by Ierapetritou.19 Such a volume ratio is called feasible convex hull ratio (FCHR). Because of the difficulties in the size determination of the feasible space, its evaluation is conducted by joining points on the constraint boundary using the Quickhull algorithm so that a convex space within the feasible region is obtained. The hypervolume of the generated convex hull is then determined via Delaunay triangulation. After the first introduction of FCHR, the group focuses on enhancing the volume estimation accuracy by utilizing multiple vertices based on the simplicial approximation approach. On the other hand, a convex polytope enclosing the feasible region is also constructed to give the upper volume bound of the feasible space.20 This method was further extended for nonconvex feasible space.21 With the group’s continuous development in the hypervolume estimation methods, the size of the feasible space can be accurately approximated. However, the research misses out on the importance of the physical meaning of such a defined metric and its significance in providing information for design improvement. Before going into the details of our current methodology for high dimensional systems and its importance in process design, the two flexibility measurements based on size evaluation are first compared via the following examples to evaluate their advantages and disadvantages. Example 1. A system with three uncertain parameters (θx, θy, θz) studied by Goyal and Ierapetritou21 is selected as an
example to compare the two approaches. The uncertain space is illustrated in Figure 3a with the set of constraints shown below: f1 ) θx - 3 e 0 f2 ) θy - 3 e 0 f3 ) θz - 3 e 0
(2)
f4 ) -(θx2 + θy2 + θz2 - 1) e 0 f5 ) θx, θy, θz g 0 The nominal point located at θx ) θy ) θz ) 1.5 (shown by the dot in the figure) and the expected positive deviations for all three parameters are ∆+θi ) 2.5, and the expected negative deviations are |∆-θi| ) 1.5. The constraint surfaces are outlined by the solid lines, and the expected upper and lower limits are indicated by the dashed lines. To construct Se, the nominal point is picked as PR. Eight auxiliary vectors are radiated from PR into the eight octants. The computation algorithm of the auxiliary vectors is shown in Table 1. Two objective functions which maximize the summation of the scaled distances between Pjs and PR (i.e., max[Σ(xj2 + yj2 + zj2)]) and maximize the summation of the absolute values of the final position index of each Pj (i.e., max[Σ(|xj| + |yj| + |zj|)] for all 8 interception points are used. Both objective functions have the physical meaning in maximizing the distance between each Pj and PR so as to attain Se with the largest possible volume. In the table, the auxiliary vector direction index, Vij, indicates the confined portion of the octant that the auxiliary vector can run into. For example, at P1, Vx1, Vy1, and Vz1 are all equal to 1, this indicates that Vf1 is located within the octant in which the
Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008 4173 Table 1. Computation Algorithm for Example 1 θx, θy, and θz θiN ) 1.5 where i ) x, y, or z ∆+θi ) 2.5 and |∆-θi| ) 1.5
uncertain parameter nominal value expected deviation
auxilary vector direction index Vxj Vyj Vzj objective function variable
P1
P2
1 1 1
1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 -1 1 -1
P4
P5
P6
P7
P8
(1) max Σ(|xj| + |yj| + |zj|) or (2) max Σ(xj2 + yj2 + zj2) for j ) 1-8 xj, yj, and zj
auxiliary vector position index Ixj Iyj Izj constraint
P3
P1
P2
P3
P4
P5
P6
P7
P8
x1 y1 z1
x2 y2 z2
x3 y3 z3
x4 y4 z4
x5 y5 z5
x6 y6 z6
x7 y7 z7
x8 y8 z8
fk(θij) e 0 for k ) 1-4; θij g 0 and 1 g xj, yj, zj g 0 θxj ) VxjIxj∆xj+θxN where θyj ) VyjIyj∆yj+θyN ∆ij ) θzj ) VzjIzj∆zj+θzN
{
∆+θi (Vij g 0) |∆-θi| (Vij < 0)
Table 2. Result Summary of the Interception Points, Pj, in Example 1 P1
Figure 3. (a) Uncertain space for example 1. (b) Sketch of the constructed space, Se, for volume determination.
uncertain parameters deviated from PR in the positive directions. Alternatively, P8 with Vx8, Vy8, and Vz8 all equal to -1 indicated that Vf8 is within the octant with all uncertain parameters deviating from PR in their negative directions. The only set of variables in the algorithm is the value of auxiliary vector position indices, Iij. This reflects how far the auxiliary vector can be extended away from PR with its end point being feasible. The optimized results of the Iijs obtained from both objective functions are the same and summarized in Table 2. Using the Vij and Iij, the coordinates of the interception points Pj(θxj, θyj, θzj) can be determined and these points can be plotted (indicated by the solid squares) in the figure with vector direction shown by the dotted lines. The obtained Pjs are joined together based on their positions and Se is constructed as outlined by the thick dotted lines. The So and Se are reproduced in the three-dimensional uncertain space without the process constraints shown in Figure 3b to demonstrate how the volume is determined. Vo is the volume of the dashed outer cube with 4 units long edges. Thus, Vo equals to 64 cubic units. Ve is calculated by subtracting the volume of the two pyramids (labeled as I and II) from the volume of the inner cube. The edge of the inner cube is 3 units long, and its volume is equal to 27 cubic units. The two pyramids are of equivalent shapes. Their bases are right-angled isosceles triangles sharing the same edge as the inner cube (i.e.,
P2
P3
P4
P5
Ixj Iyj Izj
0.60 0.60 0.60
0.60 0.60 1.00
0.60 1.00 0.60
0.60 1.00 1.00
1.00 0.60 0.60
1.00 0.60 1.00
1.00 1.00 0.60
0.62 0.62 0.62
θxj θyj θzj
3 3 3
3 3 0
3 0 3
3 0 0
0 3 3
0 3 0
0 0 3
0.58 0.58 0.58
1 -1 -1
-1 -1 1
P8
1 1 1
1 -1 1
-1 1 -1
P7
Vxj Vyj Vzj
1 1 -1
-1 1 1
P6
-1 -1 -1
3 units) and the top vertex is 0.58 units away from the base. Therefore, the volume of each pyramid is 0.87 cubic units. The Ve is equal to 27 - (2 × 0.87) ) 25.26 cubic units, and the estimated FIV is equal to 25.26/64 ) 0.395. The volume of the feasible region obtained from simplicial convex hull approach equals 25.88 units.21 Compared with the upper limit of the feasible region’s volume, which equals 26.45 units,21 the current proposed method gives a volume 95.5% of this upper limit. As clearly shown in this example, the feasible region size determination using the proposed auxiliary vector method is very close to that obtained using the simplicial convex hull approach. As expected, with fewer interception points used, the proposed methodology results in a less accurate approximation (with Ve found to be 2.4% less than the value obtained from the simplicial convex hull method). However, the computation based on auxiliary vectors is less tedious compared with the simplicial convex hull approach which requires iteration steps to optimize the number of boundary points for the best volume estimate. The current method applies a fixed number of interception points for the approximation, and this easier computation may compensate for the minute discrepancies in the estimation accuracy between the two approaches. Furthermore, by inspecting the Iij values of the auxiliary vectors, uncertain parameters restricted by the process can be evaluated and possible changes to enhance flexibility can be identified. As discussed above, Iij indicates how far the auxiliary vector can be extended away from PR. An auxiliary vector with
4174 Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008
by the infeasible region from the expanded convex hull. However, as discussed, the simplicial approach required iterative steps to give the optimal number of boundary points for the best volume approximation, evaluation of the various volumes (both the expanded convex hull and the infeasible region) can be tedious especially when the number of the nonconvex constraints is large. Flexibility of a system with nonconvex feasible region is re-evaluated using the proposed approach to examine its capability in handling such process conditions in the following example. Example 2. This example with two uncertain parameters compares the newly proposed approach with the simplicial convex hull approach for a nonconvex system. The set of constraints is as shown below, and the uncertain space is plotted in Figure 4. f1 ) θ2 - 2θ1 - 15 e 0 Figure 4. Uncertain space for example 2: (i) dashed line representing results from objective 1 and (ii) grey dashed line representing results from objective 2.
all its Iij ) 1 indicates that this vector can reach the corner of the So boundary, while any Iij of a vector smaller than 1 represents that this vector is restricted by some process constraints and the expected limits cannot be reached. The smaller the Iij value, the more critical will be the restriction imposed by a specific constraint. The Iij and Vij values give the direction at which the uncertain parameters are constrained. By evaluating the set of Iij for each auxiliary vector, the directions restricted by the process can be sorted out. Corresponding measures can then be performed to relax the restriction and enhance the overall flexibility. It is not always essential to relax the most critical constraint in the first step, the actual process enhancement should be made depending on the final profit that can be generated from the improved process. As shown in Table 2, none of the vectors have all Iij ) 1.Thus, the uncertain parameters in this example are restricted in all eight octants evaluated. The parameters are most critically restricted when they are all positively deviated from PR (i.e., P1) with all Ii1 ) 0.6. As shown in the figure, the first three constraints (f1-f3), which are restricting the design in the suggested direction, are the most critical constraints. The second smallest set of Iij is from P8 with all uncertain parameters deviated in the negative direction which is confined by f4 as shown. Therefore, in order to enhance the process flexibility, modifications on the design should attempt to release restrictions imposed by constraints f1-f4. The final decision on the modifications should be made based on the ease to perform the changes and the corresponding increase in profit generated. Flexibility Determination for Nonconvex Process Constraint The degree of flexibility is usually difficult to determine when the feasible spaces of the processes are nonconvex. In order to obtain a reliable evaluation, Goyal et al.21 developed a modified feasible space size evaluation methodology. By identifying the nonconvex constraints, determination of the volumes of the infeasible spaces enclosed by these constraints and that of the expanded convex hull (which only considers the convex constraints) via simplicial approximation can be conducted and the feasible space’s volume can be calculated. The feasible space’s volume is determined by deducting the volume occupied
θ12 + 4θ1 - 5 - θ2 e 0 2 (θ1 - 4)2 θ22 e0 f3 ) 10 5 0.5 f4 ) θ2 - 15 e 0 f5 ) θ2(6 + θ1) - 80 e 0 f2 )
(3)
As shown in the figure, f3 and f5 are the nonconvex constraints. The nominal point is located at θ1 ) -2.5, θ2 ) 0 (shown by the hollow square), and the expected deviations of the uncertain parameters are ∆+θ1 ) 7.5, |∆-θ1| ) 7.5, ∆+θ2 ) 15, and |∆-θ2| ) 15 (indicated by the dotted lines). The constraints are represented by solid lines, and the space bounded by these lines is Sf (as the whole constrained space is within S0). The area of the feasible region is found to be 129.69 square units with the area upper bound equals to 156.66 square units.21 The estimated feasible region is 83% of the upper bound calculated. The size of the feasible region is re-estimated using the auxiliary vector approach suggested above and the computation algorithm is shown in Table 3. In this 2-D example, eight auxiliary vectors are used for Se construction. Since the nominal point is not within the feasible region, θ1NO ) -5, θ2NO ) 0 (represented by the hollow dot) is picked as the initial PR. For a system with high dimensionality, it is impossible to judge the feasibility of a point visually and PR can be evaluated using the feasibility test suggested by Grossmann et al.3,7 The position of PR(θ1NO,θ2NO) is a variable, and it is optimized together with the values of Iij to give the maximum summation of the absolute values of the auxiliary vector position indices (shown in objective 1 in Table 3). Using the results obtained, the Pjs are plotted (solid diamonds) in the figure. Only seven interception points are shown since P6 and P7 are so close in space that they are not distinguishable in the figure. The constructed Se is shown by the dashed line and its area, Ve, is found equal to 148.03 square units while the area bounded by the expected limits (Vo) is 450 square units. The FIV of this system is 0.329. As depicted, Se covers part of the infeasible region (marked as I) while it fails to account for parts of the feasible region (such as region marked as II). The overestimated and underestimated regions balance each other in this example, and this gives a Vf estimation 14% larger than the Ve from the simplicial approach and 94% of the estimated area upper bound. In a 2-D system, the generalized determination function for Ve is readily available (shown in Table 3). The maximization of Ve(xi, yi) is employed as a second objective function to evaluate the process’s flexibility. The results obtained are also revealed in Figure 4. Clearly shown in the figure, the second objective function will generate an Se (outlined by the gray
Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008 4175 Table 3. Computation Algorithm for Example 2 θ1 and θ2 θ1N ) -2.5 and θ2N ) 0 ∆+θ1 ) 7.5, |∆-θ1| ) 7.5, ∆+θ2 ) 15, and |∆-θ2| ) 15
uncertain parameter nominal value expected deviation
auxiliary vector direction index v1j v2j
P2
P3
P4
P5
P6
P7
P8
1 1
0 1
-1 1
-1 0
-1 -1
0 -1
1 -1
1 0
(1) max Σ(|xj| + |yj|) or (2) max Ve(xj, yj)* for j ) 1-8 xj, yj, θ1NO and θ2NO (the initial nominal value θ1NO ) -5 and θ2NO ) 0)
objective function variable
auxiliary vector position index I1j I2j constraint
P1
P1
P2
P3
P4
P5
P6
P7
P8
x1 y1
0 y2
x3 y3
x4 0
x5 y5
0 y6
x7 y7
x8 0
fk(θ1j, θ2j) e 0; θ1N + ∆+ θ1 g θ1j g θ1N - |∆-θi| and θ2N + ∆+θ2 g θ2j g θ2N - |∆-θ2| where
θ1j ) V1jI1j∆1j+θNO 1 θ2j )
V2jI2j∆2j+θNO 2
∆ij )
{
∆+θi + θiN - θiNO (Vij g 0) θiN - |∆-θi| + θiNO (Vij < 0)
for i ) 1 or 2 and k ) 1-5 Remark: *Ve(xj,yj) ) (x1y2 + x2y3 + x3y4 + x4y5 + x5y6 + x6y7 + x7y8 + x8y1) - (x2y1 + x3y2 + x4y3 + x5y4 + x6y5 + x7y6 + x8y7 + x1y8)
dashed line) with larger volume. The Ve based on this objective is 155.75 square units which is 5% larger than the Ve from the first objective function, and the corresponding FIV is 0.346. The obtained Ve is comparable to the upper volume bound determined from simplicial approach and 20% larger than the Ve estimated from the simplicial approach. The differences in the estimation results from these two objective functions illustrated that different objective functions or confinements in Se construction will result in different degrees of estimation accuracy and different, but comparable, flexibility measurement can be obtained. As illustrated, the developed approach using auxiliary vectors are able to provide an estimate comparable to that obtained from the simplicial convex hull approach for system with nonconvex feasible region. It should be noted that the simplicial approach accounting for both volumes of the feasible and infeasible region generated by the nonconvex constraints can usually provide a better Vf estimate. However, in general cases with nonconvex constraints (such as in this example), the auxiliary vector approach can still provide a close and reliable volume approximate of the feasible space with relatively easy computation steps. Flexibility Determination for High Dimensional System Size determination of Se is needed in determining FIV. Its calculation is not tedious for systems with two or three uncertain parameters since the uncertain space can be easily sketched and visually evaluated. However, when the dimensionality of the uncertain space increases, the uncertain space cannot be visualized and the hypervolume determination may not be simple. For all Se constructed, the Delaunay triangulation algorithm can always be used to evaluate Ve. Alternatively, in most chemical processes, not all uncertain parameters are restricted by the process. By careful evaluation of the values of Iij, the uncertain parameters which can be feasibly handled by the process for all situations can be identified. Appropriate cuts on the multidimension space can be performed to give subspaces with lower dimensionality and this allows easier hypervolume evaluation. By integrating the information of these cuts or subspaces, the overall hypervolume can be determined. The
following examples illustrate flexibility evaluation for systems with multiple uncertain parameters and the possible calculation methods for hypervolumes in high dimensional uncertain space. In addition, the use of auxiliary vectors to facilitate decision making in process modifications to enhance process flexibility is demonstrated. Example 3sFlexibility Evaluation. A heat exchanger network (HEN) with four uncertain parameters (T1, T3, T5, and T8) (Figure 5a) is selected from the work of Grossmann and Floudas7 as an example. The inequality constraints are shown below: f1 f2 f3 f4 f5
) ) ) ) )
-0.67Qc + T3 - 350 e 0 -T5 - 0.75T1 + 0.5Qc - T3 + 1388.5 e 0 -T5 - 1.5T1 + Qc - 2T3 + 2044 e 0 -T5 - 1.5T1 + Qc - 2T3 - 2T8 + 2830 e 0 T5 + 1.5T1 - Qc + 2T3 + 3T8 - 3153 e 0
(4)
where Ti ) uncertain temperature with i ) 1, 3, 5, or 8 (uncertain parameter) and Qc ) cooling load in the cooler (control parameter). The nominal values of the uncertain parameters (TiN) are shown in the figure, and the expected positive and negative deviations of all uncertain parameters are 10 K. The uncertain parameters are scaled based on their expected positive and negative deviations using the equations below: δi+ ) δi- )
Ti - TiN ∆+Ti Ti - TiN |∆-Ti|
for Ti g TiN
(5a)
for Ti < TiN
(5b)
where TiN ) the nominal value of the uncertain parameter Ti; δi+ ) the scaled parameter of Ti in the positive deviation direction; δi- ) the scaled parameter of Ti in the negative deviation direction; |∆-Ti| ) the expected negative deviation of the uncertain parameter Ti (i.e. 10 K); ∆+Ti ) the expected positive deviation of the uncertain parameter Ti (i.e., 10 K). The upper and lower limits of all δi are equal to 1 and -1 in the scaled uncertain space. Since there are four uncertain
4176 Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008
Figure 5. HEN for system with four uncertain parameters in example 3: (a) original design, (b) after 1st process modification step, (c) after 2nd process modification step.
Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008 4177 Table 4. Computation Algorithm for Example 3 uncertain parameter expected deviation scaled uncertain parameter nominal value for scaled space expected deviation for scaled space auxiliary vector direction index
Ti for i ) 1, 3, 5, and 8 ∆+Ti ) 10 and |∆-Ti| ) 10 δi for i ) 1, 3, 5, and 8 δiN ) 0 ∆+δi ) 1 and |∆-δi| ) 1 16 combinations of Vij with Vij ) 1 or -1
objective function
(1) max Σi,j|xij| or (2) max Σi,jxij2 for i ) 1, 3, 5, and 8 and j ) 1-16 xi,j and Qc
variable
I1j auxiliary vector position index constraint
I3j
Pj x1j x3j fk(Qc, Tij) e 0; -Qc e 0 and 0 e Iij e 1 where Tij ) TiN + VijIij∆ij,
∆ij )
{
I5j
I8j
x5j
x8j
∆+Ti (Vij g 0) |∆-Ti| (Vij < 0)
for i ) 1, 3, 5, and 8, j ) 1-16, and k ) 1-5 Table 5. Result Summary of the Interception Points, Pj, of the Original and Modified Designs in Example 3 P5
P6
1 1 1 -1
1 1 -1 -1
1 1 1 0.33
1 1 1 1
1 1 1 1
1 1 1 0.33
1 1 1 1
after 1st modification step I1j 1 1 1 I3j 1 1 1 I5j 1 1 1 I8j 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
after 2nd modification I1j 1 1 I3j 1 1 I5j 1 1 I8j 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
V1j V3j V5j V8j
P1
P2
P3
1 1 1 1
-1 1 1 1
1 -1 1 1
original design I1j 1 I3j 1 I5j 1 I8j 1
1 1 1 1
1 1 1 1
step 1 1 1 1
P4 1 1 -1 1
P7 1 -1 -1 1
P8
P10
P11
P12
-1 -1 1 1
-1 1 1 -1
1 -1 -1 -1
1 1 1 0.33
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 0.42
1 1 1 0.91
1 1 0.91 0.36
0.88 1 1 0
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 0.42
1 1 1 0.91
1 1 0.91 1
0.88 1 1 0
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 -1 1 -1
parameters, the dimensionality of the problem is four and the scaled uncertain space cannot be sketched. Although there is no way to visualize the problem, the above auxiliary vector approach can still be employed. The nominal point, which is the origin in the scaled space, is picked as PR, and 16 auxiliary vectors running along the specific regions in the hyperspace are used. Again, due to the unavailability of a generalized hypervolume equation, two objective functions maximizing the absolute values of the vector position indices of the interception points and the distances between Pj and PR in the scaled space (i.e., maxΣij|xij| and maxΣijx2ij, respectively) are employed. Computation algorithm of Pj is summarized in Table 4. It was found that both objective functions result into the same set of interception points, and Table 5 summarizes the Vij and Iij of Pj obtained for this original design. Evaluating the value of Iij for the auxiliary vectors shows that all values of I3j equal to 1. This represents that T3 is not being restricted in all the expected combinations of the uncertain parameters. By performing two cuts in the uncertain space (with T3 fixed at its upper and lower limits), two three-dimensional spaces in scaled uncertain parameters, δ1, δ5, and δ8, can be constructed and they are sketched in Figure 6a and b. Ve at δ3 ) 1 equals 6.27 cubic units, and Ve at δ3 ) -1 equals 5.87 cubic units. The overall Ve in four dimensional space equals 12.14 units,
P9 -1 1 -1 1
P13
P14
-1 1 -1 -1
-1 -1 1 -1
P15 -1 -1 -1 1
P16 -1 -1 -1 -1
the hypervolume of Vo is equal to 24 ) 16 units, and, thus, the estimated FIV ) 0.76. To evaluate the accuracy of the Vf estimation, a space feasibility evaluation approach which can provide a more accurate hypervolume evaluation is used. The whole So is divided into 10 000 hypercubic subspaces with equal sizes. Each center of these subspaces is checked with its feasibility, and the number of feasible subspace is counted. The degree of flexibility FIVS can then be calculated with the equation, FIVS )
Nfs Ns
(6)
where Nfs ) the amount of feasible subspace among the Ns hypercubic subspaces evaluated and Ns ) the amount of subspace evaluated () 10 000 in this example). This evaluation method based on the feasibility evaluation on discrete points in the whole space is able to give a closer estimate of Vf when the subspace’s size is small enough. The determined FIVS is found to be 0.78. The FIV estimated by the auxiliary vectors as proposed above is 97% of the FIVS obtained. As illustrated, the proposed approach using auxiliary vectors can already provide a very close estimate of the process flexibility requiring significantly less computation steps. According to the literature,7 the FIG of the system is 0.50 which is
4178 Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008
provide enough cooling load to the system. Therefore, a possible modification of the system is to insert a cooler into the position as shown in Figure 5b. When the inlet temperatures of T8 and T5 are unfavorable, the hot stream H2 can be further cooled down after exchanging heat at H2-C2 to supply additional cooling load to the HEN. By doing so, extra freedom was introduced into the process. The new set of process constraints is shown as follows: f1 ) -0.67Qc + T3 - 350 e 0 f2 ) -T5 - 0.75T1 + 0.5Qc - T3 + 1388.5 e 0 f3 ) -T5 - 1.5T1 + Qc - 2T3 + 2044 e 0 f4 ) -T5 - 1.5T1 + Qc - 2T3 - 2T8 + 2830 e 0 f5 ) T5 + 1.5T1 - Qc + 2T3 + 3T8 - 3153 - QCM e 0
Figure 6. Uncertain space of the system in reduced dimensionality with T3 fixed at (a) δ3 ) 1 (upper limit) and (b) δ3 ) -1 (lower limit) with (i) Se of the original design (solid line), (ii) Se after the 1st process modification step (dashed line showing the new boundary from the new interception points), and (iii) Se after the 2nd modification step (equal to the whole space). Table 6. Flexibility Metrics Evaluated for Example 3 process flexibility FIG FIV FIVS
original design
1 modification step
2 modification steps
0.50 0.76 0.78
0.56 0.92
1.00 1.00
only 66% of the FIV obtained. The process flexibility metrics evaluated by the various methods are summarized in Table 6. Example 3sProcess Modification. In order to improve process flexibility, Iij of the original design in Table 5 is reinspected. The relatively small I8j values at P4, P7, P9, and P15 show that T8 is the most restricted uncertain parameter in the HEN. At all these points, V5j are all negative and V8j are all positive, this indicates that as T5 deviates from its nominal value in the negative direction, T8 will be restricted in its positive direction. This implies that when T5 is at its lower limit, T8 cannot be higher than a certain value (e.g., 3.3 K higher than the nominal value at P4); otherwise, the process constraints cannot be all satisfied. This suggests that when T8 is too high and T5 is too low, the cold stream, C2 is sometimes unable to
(7)
where QCM ) cooling load provided by the additional cooler. With this modification, the process flexibility was re-evaluated and the corresponding Iij of Pj are summarized in Table 5. The plot of the uncertain space at fixed values of δ3 is shown in Figure 6a and b with the dashed line showing the new boundary of the constructed space. VeM at δ3 ) 1 equals 7.61 cubic units, and VeM at δ3 ) -1 equals 7.14 cubic units. The overall VeM in four dimensional space equals 14.75 units, and thus, the estimated FIV ) 0.92. This shows that the process flexibility is enhanced by 16%. Though the above modification can successfully enhance process flexibility, the FIV obtained is still less than 1 which may still be unacceptable. In order to achieve a completely flexible process, Table 5 is further studied to identify any additional modifications required. As shown, the Iij of the last four points in the table are still less than 1 after the above modification step; thus, the process constraints are confining the process along these four directions. Negative values of V1j and V5j of P13, and P16 indicate that when the inlet temperatures of the two hot streams (T1 and T5) are smaller than their nominal values, the value of T8 is restricted in its negative deviation direction. This means that when the cold stream (C2) is too cold and the entering temperatures of the hot streams are not high enough, there will be a lack of heat available in the system to attain the required process conditions. In order to ease this restriction, a heater can be added into the process as shown in Figure 5c to heat up the cold stream C1 so as to avoid deficient in heat when T8 is too low. With this modification, the new set of process constraints is shown below: f1 ) -0.67Qc + T3 - 350 e 0 f2 ) -T5 - 0.75T1 + 0.5Qc - T3 + 1388.5 - Qh e 0 (8) f3 ) -T5 - 1.5T1 + Qc - 2T3 + 2044 - Qh e 0 f4 ) -T5 - 1.5T1 + Qc - 2T3 - 2T8 + 2830 - Qh e 0 f5 ) T5 + 1.5T1 - Qc + 2T3 + 3T8 - 3153 - QCM + Qh e 0 where Qh ) heating load provided by the additional heater. Process flexibility of the modified process is evaluated. With this additional process freedom introduced, Iij for the sixteen auxiliary vectors are all equal to 1 (Table 5), and thus, the final FIV equals 1.00. This indicates that all combinations of the uncertain parameters within So can be feasibly handled. An overall flexibility enhancement of 24% is achieved. The flexibility metrics evaluated after this two modification steps are shown in Table 6. As demonstrated, two modification steps will be needed to establish a completely flexible process. The installation of a heater, which heats up C1 when T1, T5, and T8 are too cold, and
Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008 4179
a cooler, which cools down H2 when T8 is too hot, are needed. Unless other parts of the process can take up these extra heat or cold available or it is acceptable for the process to violate some process constraints during operation, additional facilities are required to improve the process flexibility. There can be other process modifications that are able to achieve the same degree of process flexibility improvement. Other possible alternatives in the HEN modifications are not being considered in depth in this paper. As expected, extra investment is required to enhance the process flexibility, whether these changes are applicable and practically favorable will depend on the cost required for the modifications and the overall process profitability when the suggested changes are employed. The best process modification will be always dependent on the degree of profit associated after any change is made. Therefore, it is essential to strike the balance between process flexibility and process economics. Example 4sFlexibility Evaluation. In the above example, process flexibility evaluation is facilitated by performing appropriate cuts to lower the dimensionality of the uncertain space and the evaluated subspaces from these cuts are integrated to study the overall feasible space. However, when the dimensionality of the uncertain space is getting high, a lot of cuts will be needed to split the whole uncertain space into various three-dimensional subspaces and the integration of these evaluated cuts can be abstract. This example aims at analyzing the process’s flexibility for high dimensional system without performing cuts to visualize the uncertain space. The flexibility evaluation is further extended for chemical processes with more uncertain parameters. The heat exchanger network with seven uncertain temperatures is studied. The structure of the HEN with nominal values of the uncertain temperatures is shown in Figure 7a, and the expected positive and negative deviations of these temperatures are 10 K. The determined FIG of this system is 0.75 according to the literature.7 Flexibility of this HEN is re-evaluated by determining the size of the feasible space. The uncertain space is first scaled using eqs 5a and 5b above. As there are 7 uncertain parameters, 27 () 128) auxiliary vectors are radiated from the origin (PR) of the scaled uncertain space into specific regions (determined by Vij) in the hyperspace. The summation of the absolute values of the position indices, Iij, of each Pj is employed as the objective function (i.e., max Σi|xij|). Inspecting the position indices (Iij) of the 128 vectors, 112 of them can intercept with the corners of the hypercubic So (i.e., with all Iij ) 1). This indicates that the whole 7-D hypercubic subspace with these auxiliary vectors running along the diagonals is not restricted by any process constraint and these 7-D hypercubic subspace is feasible. The total hypervolume of these 112 hypercubes is 112 units. However, the remaining 16 vectors are bounded by some process constraints and the VijIij of these Pjs are summarized in Table 7. Among these 16 vectors, only the values of V4jI4j are not equal to 1 when the V4jI4j, V5jI5j, and V6jI6j are all positive. This indicates that T4 is the only restricted parameter when T4, T5, and T6 are having values larger than their nominal values. Thus, the Se subspaces in these restricted portions are 7-D hyperrectangular prisms with 6 edges of 1 unit long and the remaining one equal to 0.4. The volume of each hyper-rectangular prism is 0.4 units. The whole Se should have an overall hypervolume of 0.4 × 16 + 112 ) 118.4 units, and the hypervolume of So is 27 ) 128 units. Thus, the FIV is 0.925. The subspace feasibility evaluation is used to estimate the hypervolume. The 7-D hypercubic So is divided into 78 125
identical hypercubic subspaces. The feasibility of each center of these subspaces is evaluated, and the overall number of feasible subspaces is counted. The FIVS determined based on this method equals to 0.99. The FIV evaluated by the auxiliary vector approach is 93.4% of the FIVS estimated by the subspace feasibility test while FIG is only 76% of the FIVS value. The process flexibility metrics evaluated by the various methods for this original design are summarized in Table 8. Example 4-Process Modification. As discussed above, T4 is the only restricted parameter when T4, T5, and T6 are larger than the nominal values. When the temperatures T5 and T6 are both at their expected upper limits, the value of T4 should not be higher than its nominal value by 4 K to guarantee process feasibility. This means that when the entering temperatures of T5 and T6 are too high, T4 cannot be too hot to provide sufficient cooling load to satisfy the process requirements. To enhance the process flexibility, an addition cooler can be inserted into the position as shown in Figure 7b that introduces additional cooling into the process when necessary. The process flexibility was re-evaluated after this modification and the determined flexibility metrics after the modification is shown in Table 8. Iij of the 128 vectors are all equal to 1 and FIV ) 1.00. This indicates that the whole So can be feasibly handled by the process after the modification. With the extra cooler, the overall process flexibility is improved by 7.5%. Example 5-Flexibility Evaluation. A flow problem with five uncertain parameters from Grossmann and Floudas7 is reexamined. In this example, a centrifugal pump is used to transport liquid at a flow rate m from its original pressure P1 through a pipe to meet the desired outlet pressure P2′. The flow rate m, desired pressure P2′, pump efficiency η, pressure drop constant in the pipe k, and liquid density F are treated as uncertain parameters. From the details of the design parameters and the uncertain parameters listed in the literature, the nonlinear solution of the flexibility index (FIG) is found to be 0.4656. The degree of flexibility is re-evaluated using our proposed method. The nominal point is picked as the reference point and 32 vectors pointing into different directions are used to obtain the interception points to generate the constructed space (Se). The interception points are obtained by the following objective function: max Πi|xij|. This objective function aims at obtaining an interception point that can generate a hyperrectangular subspace with the largest size within the feasible region in a particular direction and the total size of the constructed space is the summation of all these subspaces (i.e.Σj max Πi|xij|). From the computation results, the summation of all these subspaces is 23.81 and the flexibility index (FIV) is equal to 23.81/32 ) 0.744. The flexibility index is also evaluated by the subspace feasibility test using 32 768 cubic subspaces. The FIVS is found to be 0.913. This shows that the flexibility index determined by the proposed auxiliary vector approach is 81.5% of that from the subspace feasibility test and the FIG is only 51.0% of it. Although the estimation accuracy from the proposed method is only accounting for about 80% of the actual feasible space evaluated from the subspace feasibility test, the feasibility test method is using more than 30 000 points within the whole space while the proposed method is just using 32 points (i.e., less than 0.1% of the points used in the feasibility test approach). Example 5-Process Modification. By evaluating the size of the subspaces, there are 20 deviation directions that have the interception points located at the corner of the whole space.
4180 Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008
Figure 7. HEN for the system with seven uncertain parameters in Example 4: (a) original design and (b) with process modifications.
This indicates that these directions are not restricted by any process constraints. However, the process constraints are restricting the remaining 12 deviation directions and corresponding measures should be performed to enhance the process flexibility. The interception points of these 12 auxiliary vectors are summarized in Table 9. As shown in the table, all the restricted auxiliary vectors are pointing into positive deviation of the mass flow rate m. This shows that the process is restricted when the mass flow rate, m, is larger than the nominal value. The first eight vectors show that when the mass flow rate is larger than the
nominal value, the final pressure of the process P2′ cannot be too high. Even though the target pressure is below the nominal value, the flow system is also restricted when the pump efficiency is too low. Therefore, it is essential to increase the pump driver power and the pump head so as to handle these process conditions. It is suggested to replace the pump by a new one (having the driver power 50% and the pump head 25% more than the specification of the current pump). The new driver power and pump head of the modified system are 1.25 and 1.5 times the existing values. The process flexibility of this new system is 1.00, and this means that all
Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008 4181 Table 7. Result Summary of the 16 Interception Points, Pj, which are Restricted by the Process Constraints in Example 4
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16
V1jI1j
V2jI2j
V3jI3j
V4jI4j
V5jI5j
V6jI6j
V7jI7j
1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1
1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1
1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1
0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
Table 8. Flexibility Metrics Evaluated for Example 4 process flexibility
original design
with modification
FIG FIV FIVS
0.75 0.93 0.99
1.00 1.00
Table 9. Result Summary of the Interception Points of the Auxiliary Vectors Restricted by the Process Constraints in Example 5 VmjImj VP2′jIP2′j P1 P2 P3 P4 P5 P6 P7 P8 P13 P14 P15 P16
0.651 0.317 0.805 0.415 0.400 0.317 0.457 0.400 0.500 0.500 0.500 0.500
0.605 0.283 0.704 0.347 0.834 0.283 1 0.360 -1 -1 -1 -1
VηjIηj
VkjIkj
VFjIFj
size of subspace, Πj|xij|
1 1 1 1 -1 -1 -0.904 -1 -0.833 -0.833 -0.833 -0.833
1 1 -1 -1 1 1 -1 -1 1 1 -1 -1
1 -0.822 1 -1 1 -0.822 1 -1 1 -1 1 -1
0.394 0.074 0.567 0.144 0.334 0.074 0.414 0.144 0.417 0.417 0.417 0.417
the expected combinations of the uncertain parameters can be feasibly handled by the modified system. Discussions As shown in the above examples, process flexibility evaluated using FIV and FIG are not the same. The difference between these two metrics is caused by the dimensionality of the feasible space considered. The feasible space determined using FIG is the distance between the nominal point and the constraint boundary. Alternatively, FIV is a multidimensional hypervolume evaluation of the whole feasible space that compares the size of the feasible operation region with that of the expected operating region. Therefore, FIG is sometimes unable to account for the whole feasible region, especially when the nominal point is tilting away from the center of the feasible region. The proposed flexibility measurement, FIV, signifies the process flexibility in a multidimensional manner. It is assumed that the combinations of uncertain parameters outside So will never be attained (i.e., cumulative probability within So ) 1) and every single point within So have the same chance to take place (i.e., uniform probability distribution within So). The probability distribution at each point equals to 1/Vo and the FIV () Vf/Vo) reflects the cumulative probability of the process to be feasibly handled without any constraint violation. Thus, in this case, FIV gives the same measurement as SF. Though the probability distribution of the processes may not
always be uniform, when the probability information is unavailable which usually happens at the early process design stage, uniform probability distribution can always be a reliable initial guess for process evaluation. FIV can, therefore, be useful in assessing the chance of feasible operation. Three feasible space hypervolume evaluation approaches are demonstrated. They are: simplicial approximation, auxiliary vector approach and subspace feasibility test. Among these three methods, the subspace feasibility evaluation approach can theoretically provide the best estimate of the actual hypervolume when the size of each subspace evaluated is small enough. However, when the dimensionality of the uncertain space increases, the number of subspaces to be evaluated will be significantly increased and this is obviously not computationally favorable. The simplicial approximation can also give an accurate volume evaluation of the feasible space, especially with its ability to evaluate the size of the nonconvex space. However, this approach requires iteration steps to generate optimal number of boundary points that give the best approximation. Such an iterative computation may lower the attractiveness of the approach, especially when there exists other approximation alternatives with similar level of estimation accuracy but simpler computation method (such as the auxiliary vector approach). As compared in the above examples, the auxiliary vector approach is able to give flexibility evaluation with accuracy comparable to the other methods. Also, it is relatively simple to compute with the use of few fixed number of auxiliary vectors. This makes it attractive in practice. Furthermore, the computed auxiliary vectors do not only help in process flexibility evaluation, they also provide indicators on the possible directions of design changes so as to enhance the final process flexibility. The presence of nonconvex process constraints is not easy to handle in process evaluation. Although Ierapetritou and coworkers21 developed strategies to tackle such process problems, the method requires multistep hypervolume evaluation of both the feasible and infeasible spaces to provide accurate flexibility evaluation. Computation of such a problem can sometimes be tedious especially when the number of nonconvex constraints is huge. The auxiliary vector approximation approach can provide an alternative method. It has been demonstrated to provide a reliable process flexibility analysis with simple computation and comparable estimation accuracy for the general type of nonconvex constraint problem. Although it is not possible to visualize the uncertain space when the number of uncertain parameters is larger than four, careful evaluation of the auxiliary vectors’ position and direction indices may provide valuable information to identify the unrestricted uncertain parameters or unconstrained subspaces. The information can then allow reduction in uncertain space dimensionality or decrement in the number of subspaces for hypervolume evaluation. Thus, the complexity of the hypervolume evaluation can be significantly reduced and this allows the proposed method to be useful for systems with multiple uncertain parameters without vigorous computation requirements. For systems in which the space evaluation cannot be simplified using the above schemes, the Delaunay triangulation can always be employed to measure the hypervolume obtained from the evaluated Pjs. Another advantage of the auxiliary vector approach is its ability to identify process bottlenecks and provide possible process enhancement directions. The values of auxiliary vector indices, Iij, and the direction indices, Vij, can indicate directions which are restricted by the process constraints. This suggests the list of process constraints which should be released and
4182 Ind. Eng. Chem. Res., Vol. 47, No. 12, 2008
identifies the possible process improvement directions. Therefore, this evaluation approach does not only provide process flexibility information but also guides design engineers for possible solutions in process enhancement. Although the cost evaluation step which will lead to the final modification decision is not demonstrated in the current study, the abilities of the auxiliary vector approach to measure the process flexibility and to pinpoint possible changes enhancing the overall process flexibility are illustrated. Conclusions A new flexibility metric (FIV) is developed. This index is capable to handle systems with nonconvex process constraints or multiple uncertain parameters. This ability of the metric is important as processes with such characteristics are usually difficult to tackle using the traditional process flexibility evaluation methodologies. The FIV determination requires analysis of the hypervolume of feasible space that may not be easy to be calculated exactly. An estimation approach based on the use of auxiliary vectors to construct an approximation space is proposed. This method is found capable to provide reliable estimate comparable to the existing methodologies (such as simplicial approximation) with relatively mild computation requirements. In addition to the evaluation of the degree of process flexibility, the position indices of the auxiliary vectors can provide additional process information that allows engineers to identify limits of the existing design. Thus, possible future modification directions can be pinpointed to relax the constraints and enhance the process performance. Although cost evaluation is not conducted to assess the economic feasibility of the suggested modifications in this study, it is always essential to consider the best utilization of the investment so as to maximize the overall process profitability. Studies on the use of FIV in utility production systems are underway to demonstrate the importance and application of this index for process design with the process flexibility and economics optimization. Acknowledgment The authors gratefully acknowledge financial support from Hong Kong RGC grant (No. 614005) and DAG05/06.EG23. Notation Ci ) heat capacity of the ith cold stream, kW/K E(SF) ) expected stochastic flexibility fk(θij) ) kth constraint function evaluated at Pj (θij) FA ) combined flexibility-availability index FCHR ) feasible convex hull ratio FIG ) flexibility index defined by Grossmann et al.3,7 FIV ) flexibility index based on hypervolume ratio FIVS ) flexibility index based on subspace feasibility test FR ) flexibility-reliability index HEN ) heat exchanger network Hi ) heat capacity of the ith hot stream, kW/K Iij ) auxiliary vector position index for θi at Pj i ) dummy variable indicating uncertain parameter j ) dummy variable indicating interception point k ) dummy variable indicating constraint function k ) pressure drop constant in the pipe in Example 5, (kPa)(m)5.16/ (kg/s)1.84 m ) mass flow rate in Example 5, kg/s Nfs ) number of feasible subspace Ns ) number of subspace evaluated Pj ) jth interception point
Pj(θxj, θyj, θzj) ) jth interception point with coordinates (θxj, θyj, θzj) PR ) reference point PR(θ1NO, θ2NO) ) reference point with coordinates (θ1NO, θ2NO) P1 ) initial pressure of the flow problem in Example 5, kPa P2′ ) desired outlet pressure in Example 5, kPa Qc ) cooling load in cooler QCM ) cooling load provided by the additional cooler Qh ) heating load provided by the additional heater RI ) resilience index SF ) stochastic flexibility Sc ) constrained space Se ) constructed space Sf ) feasible space So ) whole space Ti ) ith uncertain temperature TiN ) nominal value of Ti Tij ) Ti at Pj ∆-Ti ) expected negative deviation of Ti ∆+Ti ) expected positive deviation of Ti Ve ) hypervolume of Se VeM ) hypervolume of Se of the modified design Ve(xj, yj, zj) ) generalized volume determination function for Se Vf ) hypervolume of Sf Vo ) hypervolume of So Vij ) auxiliary vector direction index of θi at Pj VijIij ) product of Vij and Iij f Vj ) jth auxiliary vector xj, yj, zj ) variable for auxiliary vector index of θx, θy, θz at Pj xij ) variable for auxiliary vector index of θi at Pj Greek Letters θi ) ith uncertain parameter θij ) ith uncertain parameter at Pj θiU ) expected upper limit of θi θiL ) expected lower limit of θi θiNO ) optimized nominal value of θi θiR ) θi at PR ∆+θi ) expected positive deviation of θi ∆-θi ) expected negative deviation of θi δi+ ) scaled Ti in positive deviation direction δi- ) scaled Ti in negative deviation direction δi ) scaled uncertain parameter of Ti ∆+δi ) expected positive deviation of δi ∆-δi ) expected negative deviation of δi ∆ij ) expected deviation index of the ith uncertain parameter for Vij F ) liquid density in Example 5, kg/m3 η ) pump efficiency in Example 5
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ReceiVed for reView January 31, 2007 ReVised manuscript receiVed February 4, 2008 Accepted February 25, 2008 IE070183Z
