New MINLP Formulations for Flexibility Analysis for Measured and Unmeasured Uncertain Parameters

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1 Anton A. Kiss, Edwin Zondervan, Richard Lakerveld, Leyla Özkan (Eds.) Proceedings of the 29 th Eropean Syposi on Copter Aided Process Engineering Jne 16 th to 19 th, 219, Eindhoven, The Netherlands. 219 Elsevier B.V. All rights reserved. New MINLP Forlations for Flexibility Analysis for Measred and Uneasred Uncertain Paraeters Maria Paz Ochoa, Ignacio E. Grossann, Departent of Cheical Engineering, Carnegie Mellon University, Pittsbrgh, PA 15213, U.S.A. Abstract In this paper, we forlate the flexibility analysis with easred and neasred paraeters as a rigoros lti-level optiization proble. First, we propose to recrsively reforlate the inner optiization probles by the Karsh-Khn-Tcker conditions and with a ixed-integer representation of the copleentarity conditions to solve the reslting ltilevel optiization proble. Three types of probles are addressed and solved with the proposed strategy: 1) linear prograing proble, 2) nonlinear prograing proble with onotonic variation of neasred ncertain, and finally 3) nonlinear prograing proble. We illstrate the new forlations with a heat exchanger network proble with ncertain heat capacity flowrates. Keywords: extended flexibility analysis, optiization nder ncertainty, MINLP forlation. 1. Introdction Traditionally, the approach to handle ncertainty in the paraeters of a odel is to consider noinal conditions in plant operation, and se overdesign to copensate for the potential ipact of the ncertainty. In contrast, flexibility analysis addresses the garanteed feasibility of operation of a plant over a range of conditions, with the ltiate goal being on how to design a process for garanteed flexible operation (Grossann et al., 214). The flexibility test proble only deterines whether a design does or does not eet the flexibility target. To deterine how ch flexibility can be achieved in a given design, the flexibility index is defined as the largest vale of δ sch that the odel ineqalities hold over the ncertain paraeter range (Swaney and Grossann, 1985). However, these forlations are based on the assption that aniplated variables can copensate for any variation in the ncertain paraeter set and that dring operation stage ncertain paraeters can be easred with precision to take the corrective action. Ostrovsky et al. (23) and Rooney and Biegler (23) extended the analysis by taking into accont the level of paraetric ncertainty in the atheatical odels at the operation stage, by groping the ncertain paraeters, θ Θ, into two types, easred and neasred paraeters. The flexibility constraint was then extended to accont for odel paraeters, θ, that cannot be easred.

2 2 M.P. Ochoa et al. In this paper, we propose a new reforlation of the extended flexibility analysis where the innerost probles are recrsively replaced by their optiality conditions and the copleentarity conditions are expressed with a discrete representation. 2. Matheatical Model The basic odel for the flexibility analysis involves design variables, d, control variables, z, and ncertain paraeters, θ. One of the ain probles addressed in the flexibility analysis is the flexibility test proble. It consists in deterining whether by proper adstent of the control variables the process constraints g (d,z,θ), ϵ J, hold for any realization of ncertain paraeters for a given design (Haleane and Grossann, 1983). This stateent can be expressed with the logic expression (1), and is reforlated by the se of in and ax operators as shown in Eq. (2). z ( J [g (d,z, ) ]) (1) d axin axg (d,z, ) z J (2) The ain difference between the design and control variables is that the design variables are fixed dring the operation stage, while the control variables can be adsted in order to satisfy process constraints. In fact, to solve the flexibility constraint it is reqired to have accrate estiation of the ncertain paraeters. This can only be achieved if there is enogh process data for precise deterination of all ncertain paraeter vales. However, this assption is restrictive, and is often not et in practice. To address these liitations, two grops of ncertain paraeters are identified. The first grop of ncertain paraeters contains paraeters whose vales can be deterined to within any desired accracy at the operation stage, naely the easred ncertain paraeters, θ. Meaning that appropriate sensors are available to deterine accrate vales of all the ncertain paraeters by direct easreent or by solving paraeter estiation probles. Therefore, recorse action can be taken in order to copensate for their variation. Exaples of this type of paraeters inclde process conditions sch as feed flowrates, pressres, teperatres, concentrations, and inpt variables sch as prodct deands. The second grop incldes the neasred ncertain paraeters, θ, whose estiation cannot be perfored dring the operation stage, conseqently no control actions can be applied to the. This distinction has been taken into accont and the flexibility constraint was then extended to Eq. (3) and reforlation as the ltilevel optiization proble described by Eq. (4) by Ostrovsky et al. (23) and Rooney and Biegler (23). z (, J [g (d,z,, ) ]) (3) d ax in ax ax g (d,z,, ) z J To solve the extended flexibility analysis, Ostrosvky et al. (23) sggested an algorith for calclation of the flexibility fnction based on the branch and bond strategy, while partitioning the ncertain set into sbregions. On the other hand, Rooney and Biegler (23) proposed an extension to the approach presented by Raspanti et al. (2), which involved the se of the KS sooth fnction (Krelsseleler and Steinhaser, 1983) that (4)

3 New MINLP Forlations for Flexibility Analysis for Measred and Uneasred Uncertain Paraeters 3 aggregated all of the odel ineqality constraints, and the KKT derivation together with a sooth approxiation of the copleentarity conditions for the inner optiization probles. Therefore, the extended flexibility constraint reslted in a nonlinear progra. In this work, we reforlate the extended flexibility constraint by developing the optiality conditions for each nested probles. In addition, in order to ake the forlation tighter, the bonds of the nonnegative Lagrange ltipliers related to the ineqality constraints and the bonds on the slack variables are treated as constraints of the following level optiization proble. Finally, we express the copleentarity conditions with a ixed-integer representation and asse that the Haar condition holds, which states that the nber of active constraints is eqal to the diension of the control variables pls one. This condition holds tre provided the Jacobian is fll rank (Grossann and Flodas, 1987). In the following sbsections, we derivate the forlation of three different cases, linear proble, non-linear proble with onotonic variation of neasred paraeters with respect to odel constraints, and non-linear probles Special Case 1: Linear Prograing proble The order of the inner ax operators in Eq. (4), are interchangeable and can be eqivalently expressed as follows. d ax (d, ) s.t (d, ) in (d,z, ) z s.t: (d,z, ) ax ax g (d,z,, ) J We consider that the inner proble is described by the linear ineqality constraints: g(d,z,, ) = a d b z c d, J (6) Constraints g in Eq. (6) vary onotonically with respect to θ. Hence, the soltion of the innerost proble st lie in one of the extree points of its range of variation, depending on the sign of the derivative, dg /dθ,. (5) ax g ( ),, (7) The bilevel proble M1 described by Eq. (8) is obtained by replacing Eq. (7) in Eq. (5). To obtain a single level optiization proble, the innerost proble of Eq. (8) is replaced by its optiality conditions with a ixed integer representation of the copleentarity condition following the active constraint set strategy (Grossann and Flodas, 1987). This yields the MILP proble: d ax (d, ) s.t: (d, ) in( g (d,z,, ) = a d b z c, d, J) (8) z, Where is a scalar variable that represents the worst constraint violation.

4 4 M.P. Ochoa et al Special Case 2: Non-Linear Prograing proble with onotonic variation of neasred ncertain paraeters If the set of fnctions g (d,z,θ,θ) varies onotonically with respect to the neasred ncertain paraeters, then the relationship expressed by Eq. (7) holds tre. Therefore, the soltion of the innerost proble lies at an extree point of the range of variation of θ. Analogosly, the following bilevel prograing proble M2 is obtained and then reforlated in the sae way as the previos case, bt leading to the MINLP proble. d ax (d, ) (9) s.t: (d, ) in( g (d,z,, ), J), z, 2.3. General Case: Non-Linear Prograing proble The extended flexibility constraint can be eqivalently expressed as the following ltilevel optiization proble M3. d ax (d, ) s.t (d, ) in (d,z, ) z s.t: (d,z, ) ax (d,z,, ) s.t: (d,z,, ) in( g (d,z,, ), J) In order to solve Eq. (1), we propose to replace the inner probles by their optiality conditions in a recrsive fashion. De to space liitations, we cannot report the coplete reforlation. To illstrate the soltion strategy, we will replace the innerost proble of Eq. (1) by its KKT conditions (Eqns. (11) and (12)) and copleentarity conditions (Eq. (13)). Bonds of the Lagrange ltipliers, λ, and slack variables, s, are added as odel constraints of the next level optiization proble, described by Eqns. (14) and (15), in order to tighten the forlation. 1- (11) g(d,z,, )- s J (12), s J (13) J (14) J (15) s Following this procedre, we obtain a single level optiization proble. The copleentarity conditions are then expressed with a ixed integer representation, the Haar conditions is assed, leading to an MINLP proble. 3. Nerical Exaple A well-known exaple in the flexibility analysis literatre is the heat exchanger network, shown in Figre 1. Grossann and Flodas (1987) sed this exaple to introdce the active set strategy, which is able to find non-vertex soltions. After the eliination of the state variables, the redced odel consists of for constraints, and three variables: the (1)

5 New MINLP Forlations for Flexibility Analysis for Measred and Uneasred Uncertain Paraeters 5 cooling load (Q c) is the control variable and the heat capacity flowrate of streas 1 and 2 (F H1 and F H2) are the ncertain paraeters. We solve a odified version the proble for three cases. First, considering both ncertain paraeters as neasred, so no recorse actions can be taken. Second, considering θ1 as a easred ncertain paraeter and θ2 as an neasred ncertain paraeter, where control actions can only be adsted to copensate for variations in θ1, solved with the proposed forlation. Finally, we consider both as easred ncertain paraeters like in the traditional flexibility analysis. (a) Figre 1. (a) Heat exchanger network schee. (b) Feasibility diagra for fixed vale of FH2=2 Nerical reslts are sarized in Table 1Table 1. As we can see, we obtain positive vales of for all the cases, indicating an infeasible design. The worst constraint violation (=138.5) is obtained for the case of no recorse actions. This vale can be redced p to certain degree (=2) when control variables can copensate for the variations in θ1. Frtherore, this can be redced (=7.8) when recorse actions can copensate for variation in both ncertain paraeters. It is also iportant to note, that non-vertex critical points are obtained for the second and third cases. The different probles are ipleented in GAMS (GAMS Developent Corporation, 218) and solved with BARON (Kilinc and Sahinidis, 218) in an Intel i7 achine with 16 Gb of RAM. The tolerance of the solver is set to.1 and the big M vale is 6. Table 1. Worst constraint violation, critical paraeters vales for three exaples and odel size. Both neasred ncertain paraeters θ=θ1, θ2 Cobined type of ncertain Paraeters θ=θ1,θ=θ2 Both easred ncertain Paraeters θ=θ1, θ θ1= F H θ2= F H #bin var #cont. var #constraints CPU tie For a fixed vale of z. 4. Conclsion In order to obtain ore realistic reslts when dealing with operation nder ncertainty, a distinction of the ncertain paraeters can ade between the easred and neasred ncertain paraeters. Ths, the traditional flexibility constraint has been extended. In this (b)

6 6 M.P. Ochoa et al. work, we have proposed new reforlations of the reslting ltilevel optiization proble, which involves the replaceent the innerost proble by its optiality conditions in a recrsive fashion and the introdction of a ixed-integer representation of the copleentarity conditions, reslting in an MINLP proble. We have developed the forlation of three different cases, linear proble, non-linear proble with onotonic variation of neasred paraeters with respect to odel constraints, and non-linear probles. A particlar featre of the first two cases is that the worst constraint violation lies at a vertex of the neasred ncertain paraeter set; where the forlation can be siplified, leading to a siilar forlation as the one obtained by applying the active set constraint strategy, naely the traditional flexibility analysis (Grossann and Flodas, 1987). An exaple of a heat exchanger network has been presented to illstrate the proposed reforlation of the general case and copared to cases with different degree of recorse actions. Acknowledgeents The athors acknowledge the financial spport fro Eli Lilly and Copany and the Center for Advanced Process Decision-aking fro Carnegie Mellon University. References GAMS Developent Corporation, 218, General Algebraic Modeling Syste (GAMS) Release Washington, DC, USA. I.E. Grossann, B. Calfa and P. Garcia-Herreros, 214, Evoltion of concepts and odels for qantifying resiliency and flexibility of cheical processes, Copters and Cheical Engineering, vol. 7, pp I.E. Grossann and C.A. Flodas, 1987, Active Constraint strategy for flexibility analysis in cheical processes, Copters and Cheical Engineering, vol. 6, pp K. Haleane and I.E. Grossann, 1983, Optial Process Design nder Uncertainty, AlChE Jornal, vol. 29, no. 3, pp M. Kilinc and N.V. Sahinidis, 218, Exploiting integrality in the global optiization of ixedinteger nonlinear prograing probles in BARON, Optiization Methods and Software, vol 33, pp G. Kreisseleier and R. Steinhaser, 1983, Application of vector perforance optiization to a robst control loop design for a fighter aircraft, International Jornal of Control, vol. 37, pp G. Ostrovsky, I. Datskov, L. Achenie and Y. Volin, 23, Process Uncertainty: Case of Insfficient Process Data at the Operation Stage, AIChE Jornal, vol. 49, no. 5, pp C. Raspanti, J. Bandoni and L. Biegler, 2, New Strategies for Flexibility Analysis and Design Under Uncertainty, Copters and Cheical Engineering, vol. 24, pp W. Rooney and L. Biegler, 23, Optial Process Design with Model Paraeter Uncertainty and Process Variability, AIChE Jornal, vol. 49, no. 2, pp R. Swaney and I.E. Grossann, 1985, An index for operational flexibility in cheical process design. Part I. Forlation and Theory, AIChE Jornal, vol. 31, pp