New Algorithm for the Flexibility Index Problem of Quadratic Systems

Transcription

1 New Algorthm or the Flexblty Index Problem o Quadratc Systems Hao Jang a, Bngzhen Chen a*, Ignaco E. Grossmann b* a Insttute o Process Systems Engneerng, Dept. o Chemcal Engneerng, Tsnghua nversty, Beng , P.R. Chna b Center or Advanced Process Decson-makng, Dept. o Chemcal Engneerng, Carnege Mellon nversty, Pttsburgh, PA 15213,.S.A. Abstract: In ths work, a new lexblty ndex algorthm or systems under uncertanty and represented by quadratc nequaltes s presented. Inspred by the outer-approxmaton algorthm or convex mxed-nteger nonlnear programmng, a smlar teratve strategy s developed. The subproblem, whch s a nonlnear programmng, s constructed by xng the vertex drectons snce ths class o system s proved to have a vertex soluton the entres on the dagonal o the Hessan matrx are non-negatve. By overestmatng the nonlnear constrants, a lnear mn-max problem s ormulated. By dualzng the nner maxmzaton problem, and ntroducng new varables and constrants, the master problem s reormulated as a mxed-nteger lnear programmng. By teratvely solvng the subproblem and master problem, the algorthm can be guaranteed to converge to the lexblty ndex. Numercal examples ncludng a heat exchanger network, a process network, and a unt commtment problem are presented to llustrate the computatonal ecency o the algorthm. Topcal headng: Process Systems Engneerng Keywords: lexblty ndex, quadratc systems, overestmaton * Correspondence concernng ths artcle should be addressed to I. E. Grossmann at grossmann@cmu.edu and B. Chen at dcecbz@tsnghua.edu.cn 1

2 1. Introducton Snce Grossmann and Sargent addressed the optmum desgn wth uncertan parameters n 1978, 1 the area o lexblty analyss has evolved or about 40 years n the communty o process systems engneerng (PSE). For a hstorcal perspectve on the evoluton o ths area and the relatons between lexblty, reslency and robust optmzaton, please reer to the revew paper by Grossmann et al., 2 and the paper by Zhang et al. 3 as a trbute to proessor Roger Sargent, the poneer n the PSE communty. The basc dea o lexblty analyss s to explctly consder uncertantes n the evaluaton and desgn o chemcal processes. The uncertantes can be thermodynamc and knetc parameters, concentraton and temperature o nlet streams, demand and prce o products, etc. In general, lexblty analyss problems apply to the case where the desgn o a chemcal process s xed. It s an evaluaton to determne whether the desgn can tolerate the uncertantes n a speced range, or to quanty the range o uncertantes that the desgn can tolerate. The ormer s known as lexblty test problem, and the latter s known as lexblty ndex problem. 4 The mathematcal model o a chemcal process comprses a set o equatons to descrbe mass and energy balances and a set o nequaltes to descrbe operatng lmts and product speccatons: 4 h( d, z, x, ) 0 (1a) g( d, z, x, ) 0 (1b) where nd nz d are desgn varables, nx z are control varables, x are state 2

3 n varables wth the same dmenson as h( d, z, x, ), and are uncertan parameters. From Eq. (1a), x can be expressed as a uncton o d, z, and θ by x x( d, z, ). Equaton (1b) can then be reormulated as g( d, z, x( d, z, ), ) ( d, z, ) 0 (2) Thereore, a chemcal process can be descrbed by a set o nequaltes as shown n Eq. (2). For a xed desgn d (ths s always the case n ths paper unless noted) and every realzaton o uncertan parameters θ n a regon R, a set o control varables can be ound such that Eq. (2) s satsed, then ths process s easble n R, the easble regon proected n the θ-space. Dene the ollowng easblty uncton: 4,5 whch s equvalent to, ( d, ) mn max ( d, z, ) (3) z J ( d, ) mn u uz, s.t. ( d, z, ) u, J (4) Hence, the easble regon R can be expressed as, R{ ( d, ) 0} (5) The easble regon R s dcult to obtan or hgh-dmensonal systems, and the shape s qute derent or derent processes. Thereore, a hyperrectangle nscrbed n R s ntroduced to smply the evaluaton and comparson o tolerance or uncertan parameters. The hyperrectangle s centered at the nomnal pont N, and the lengths o ts sdes are proportonal to the expected postve and negatve devatons, denoted by and, respectvely. A non-negatve scalar δ s used to dene the hyperrectangle T(δ): N N T( ) (6) 3

4 The lexblty ndex F can be dened as the maxmum value o δ such that T( ) R, whch means Eq. (2) s satsed or every realzaton o n T(δ) wth approprate adustment o z. Ths abstract denton can be converted to an equvalent mathematcal programmng problem: 4,5 T ( ) Fd ( ) max s.t. max mn max ( d, z, ) 0 z J N N T( ) (7) Smlar to the aorementoned lexblty ndex problem, the lexblty test problem s to determne whether by adustng the control varables z, Eq. (2) holds or all N N T(1) { }. Ths paper does not consder the lexblty test problem explctly snce t s equvalent to determne whether the lexblty ndex s greater than or equal to one, although one may take advantage o ts smpler ormulaton to devse a more ecent algorthm. The geometrc nterpretaton o the lexblty ndex s to nd the largest hyperrectangle nscrbed n the easble regon R. Accordng to the Propertes 1 and 2 proved by Swaney and Grossmann, 4 ( d, z, ), J, are contnuous n z and θ, then the crtcal pont (the soluton * o problem (7)) les at the boundary o R. I the crtcal pont corresponds to a vertex o the hyperrectangle, then problem (7) can be solved by drect search or mplct enumeraton algorthms. 6 I there s a nonvertex crtcal pont, there are also several algorthms. Grossmann and Floudas 7 proposed an actve constrant strategy (actve-set) n whch the problem s reormulated as a mxed-nteger lnear programmng (MILP) or mxed-nteger nonlnear programmng (MINLP) by applyng the Karush-Kuhn-Tucker (KKT) condtons to 4

5 problem (4). Ostrovsky et al. 8,9 used a branch and bound method to obtan nonvertex solutons or the lexblty uncton. Bansal et al. 10 solved the lexblty analyss problem o lnear systems by parametrc programmng, and then generalzed t to nonlnear systems. 11 Floudas et al. 12 proposed a global optmzaton algorthm to solve the lexblty test and lexblty ndex problems o nonconvex systems, n whch the problem s convexed and solved to global optmalty by the αbb algorthm to obtan a lower bound, and the orgnal problem s solved by the actve-set method to obtan an upper bound. By branchng on the space o θ and z, the gap s closed wthn a gven tolerance. As stated by Ierapetrtou and coworkers, 13,14 the lexblty ndex has lmtatons to quanty the degree o uncertan parameters a process can tolerate. Sometmes, t s overly conservatve snce t underestmates the sze o the easble regon. However, compared to other lexblty measures, e.g., the easble convex hull rato 13 and the smplcal approxmaton, 14 the lexblty ndex provdes a good balance between the accuracy o approxmaton o the easble regon and ease o nterpretaton. Thereore, n ths paper we use the tradtonal lexblty ndex to quanty the ablty o a process to tolerate uncertan parameters. In ths work, we consder a specal class o systems, where all the nequaltes are quadratc or lnear n θ, and lnear n z. Quadratc terms o uncertan parameters are common n chemcal processes, such as the product o nlet temperature and heat capacty lowrate n heat exchanger networks (HEN), the product o lowrate and contamnant concentraton n water networks, and the product o demand and prce o product n plannng problems. Although general algorthms can solve the lexblty ndex problem o these systems, one cannot expect they 5

6 wll take advantage o the specal structure to accelerate the computaton. Thereore, a more ecent teratve procedure, whch s qute derent rom the exstng algorthms, s proposed n ths paper to solve the lexblty ndex problem or ths class o systems. The proposed algorthm n ths paper s nspred by the work o Zhang et al, 3 n whch a dualty-based method s proposed to solve the lexblty analyss problem o lnear systems that s much more ecent than the actve-set method. 7 In ths work, we extend ths method to nonlnear systems, whch are lnearzed, yeldng a lower bound problem whch can be ecently solved by the dualty-based method. An upper bound s easly obtaned snce the crtcal pont o ths specal class o system les at a vertex as proved n the ollowng secton. By teratvely solvng the upper bound and lower bound problems, the gap s closed wthout the need o branchng. Although the proposed method s or a lmted class o nonlnear systems, the ecent computatonal perormance on several examples ncludng HEN, process networks and unt commtment, shows that t s a promsng method. The remander o ths artcle s organzed as ollows. In the next secton, the problem s stated mathematcally and the property o vertex soluton s proved. The new lexblty ndex algorthm s developed n the subsequent secton. In secton 4, we apply ths new algorthm to our numercal examples, and compare t wth the actve-set method. Fnally, the conclusons are presented n secton Problem statement The nequaltes o ths class o system can be wrtten as: T T T Q a b z c 0, J (8) 6

7 where the Hessan matrx Q s an n n matrx, J {1,2,..., m} s the set o constrants, a and b are column vectors, and c are constants. I all the elements o Q are zero, then Eq. (8) becomes a lnear system. I, J, are convex,.e., Q are postve semdente, the lexblty ndex problem o ths system has a vertex soluton. Below we derve a weaker condton, under whch the vertex soluton can also be guaranteed as shown n the ollowng proposton. Proposton 1. I q 0, I, J, then the lexblty ndex problem o (8) wll have a vertex soluton. Proo. Based on Theorems 2 and 3 proved by Swaney and Grossmann, 4 all the constrants, J, are ontly quas-convex n z and one-dmensonal quas-convex (1-DQC) n θ, then the soluton o the lexblty ndex problem wll le at a vertex o the hyperrectangle T(δ). Obvously, s quas-convex n z. Thereore, we only need to prove that s 1-DQC n θ. T Dene 1,...,,...,, 1,...,,..., 1 2 T 1 n n e, where β s a scalar, and e s the th unt vector,.e., the th column o the dentty matrx. For xed d and z, s 1- DQC n θ and only d z d z d z max,, 1,,, 2,, 1 1 2, [0,1] (9) Snce the lnear and constant terms are convex, they are 1-DQC. Thereore, only the quadratc term needs to be consdered. The quadratc term o the rght-hand sde (QRHS) o Eq. (9) can be wrtten as ollows: 7

8 T e Q e QRHS 1 1 T e Q e T 1 1T T T Q Q e e Q e Q e T 1 1T T Q Q e e Q q (10) I Q Q, the quadratc term o the let-hand sde (QLHS) o Eq. (9) wll be 1T 1 2T 2. By substtutng 1 T 1 Q 2 1 e nto Q Q, we get 1T 1 2T 2 Q e e Q Q 0. Snce and q 0, the ollowng nequalty 1T T 1 2 holds: T T T T Q e e Q q Q e 1 e Q 1 q 0 (11) Thereore, the ollowng nequalty s obtaned: QLHS QRHS (12) 1T 1 Q whch means Eq. (9) holds. It s smlar when Q Q. 1T 1 2T 2 Thereore, vertex. s ontly quas-convex n z and 1-DQC n θ, and hence the soluton les at a Note that, q 0, I, are merely necessary condtons or Q to be postve semdente. Thereore, t s not as restrctve as the condton that s convex, and urthermore t may n act correspond to a nonconvex uncton. The ollowng parts wll restrct the dagonal elements o Q, J, to be non-negatve to guarantee a vertex soluton. 3. New algorthm or the lexblty ndex problem 3.1 Basc dea To better understand the basc dea o ths algorthm, the dualty-based lexblty ndex 8

9 algorthm or lnear systems s ntroduced rst. Consder the ollowng lnear nequalty constrants: A Bz C 0 (13) or whch the lexblty ndex problem has a vertex soluton. Thereore, the lexblty ndex problem can be stated n an alternatve way: F mn max x z, s.t. A Bz C 0 N x 1 x, I x {0,1} n (14) The nterpretaton o ths ormulaton s that, or every vertex drecton,.e., every possble combnaton o x, the dstance rom the nomnal pont to the boundary o the easble regon s maxmzed. Among these dstances, the shortest s chosen to make the hyperrectangle be totally nscrbed n the easble regon. 15 For a gven vertex drecton x, and by substtutng nto Eq. (13), the nner maxmzaton problem can then be replaced by ts dual, whch s a mnmzaton problem and can be merged wth the outer mnmzaton problem. The obtaned sngle-level problem s shown as ollows: T N F mn ( A C) x, T s. t. B 0 x 1 x a 1 (15) J I 0 x {0,1} n Problem (15) s an MINLP snce there are blnear terms x n the constrant. However, the blnearty o bnary and contnuous varables can be elmnated by replacng x by a new non-negatve contnuous varable together wth the ollowng constrants: 16 9

10 M M x M x (16) where M s a bg-m parameter. I x 0, then 0 ; x 1, then, whch s exactly equvalent to the result o x. Thereore, problem (15) s equvalent to the ollowng MILP: T N F mn ( A C) x,, T s. t. B 0 J I 1 M M x, I, J, I, J M x, I, J 0, 0 x {0,1} a n (17) However, ths reormulaton s not applcable to the lexblty ndex problem o quadratc systems as shown n the ollowng snce the nner maxmzaton problem s nonlnear n δ. F mn max x, z T T T s.t. Q a b z c 0, J N x 1 x, I x {0,1} n (18) We are nspred by the outer-approxmaton (OA) algorthm or convex MINLP, 17 n whch the nonlnear constrants are replaced by ther supportng hyperplanes to construct a master MILP to provde a lower bound, and subsequently, the nteger varables are xed to the optmal soluton o the master MILP to construct a nonlnear programmng (NLP) subproblem to provde an upper bound. By teratvely solvng the master problem and the subproblem, they wll converge to the optmal soluton. Smlarly, we can develop the subproblem and master 10

11 problem or ths nonlnear mn-max problem, where however the NLP subproblem may have multple local optma. I all the bnary varables x are xed, meanng xed vertex drecton, problem (18) s an NLP, and provdes an upper bound o the lexblty ndex, smlar to the subproblem n the OA algorthm. However, the master problem s qute derent rom that used n the OA algorthm n two aspects: (a) the constrants are nonconvex n the mn-max problem, OA wll cut o some o the easble regon; (b) or the case where the constrants are convex, OA wll yeld an upper bound o the lexblty ndex nstead o a lower bound snce the nner problem s a maxmzaton problem. On the contrary, we use an overestmaton o the uncton to replace OA,.e., underestmaton o the uncton, the master problem wll yeld a lower bound o the lexblty ndex. Here, we use a nonlnear convex uncton to compare OA and overestmaton. Consder the ollowng constrant: (19) N The nomnal pont s (1.5, 5), and the expected devatons are 1 1, 2 2, respectvely. Frstly, we apply OA at two ponts, 1 (0, 2) and 2 (3, 5). Equaton (19) s thereore replaced by two lnear nequaltes, whch are the tangents o at the two ponts, as seen n Fgure 1(a) (20) For comparson, the range n whch we overestmate the constrant s 01 3,

12 Thereore, the constrant s overestmated by the ollowng nequalty, n whch the nonlnear term s replaced by ts secant, as seen n Fgure 1(b) (21) The correspondng easble regons are shaded n Fgure 1(a) and Fgure 1(b), n whch the sold and dotted rectangles are the lexble regons or the orgnal and lnearzed systems, respectvely. The easble regon o the second case s derved rom Eq. (21) together wth the bounds o the uncertan parameters. It s clear that ater applyng OA to the constrant, the easble regon s enlarged. Thereore, we wll obtan an upper bound o the lexblty ndex we solve the lexblty ndex problem or the lnearzed system. On the contrary, the constrant s overestmated, the easble regon s totally nscrbed n the orgnal one. Thereore, as expected, we obtan a lower bound o the lexblty ndex. Detals about the subproblem, master problem, propertes o the master problem, and the algorthm are gven n the ollowng our subsectons. Fgure 1. Feasble regons o the lnearzed systems. (a) OA and (b) overestmaton. 3.2 Subproblem The subproblem, whch wll provde an upper bound, s the nner maxmzaton problem wth 12

13 xed x, as shown below. In derence to the subproblem n the OA algorthm, ths subproblem s always easble as long as the nomnal pont s easble, and ths s always the case. * ( x) max, z 2 T T (S) s. t. q q q a b z c 0, J k k k k 1 N x 1 x, I As requred by the assumpton o the alternatve ormulaton o the lexblty ndex, and gven that (S) s nonconvex, (S) has multple local solutons, the smallest one s chosen to guarantee the easblty o the entre path rom the nomnal pont to the boundary. For some systems, t can be dcult to solve because the constrants are nonconvex n δ, and we need to guarantee the smallest local maxmum to be obtaned. But or ths quadratc system, although nonconvex, we can add some extra constrants to make the soluton o (S) to be unque and satsy the requrement. By substtutng nto, becomes a uncton o x, δ and z, and hence can be denoted by x,, z. For xed x, s quadratc and/or lnear n δ and lnear n z. Hence, (S) can be rewrtten n compact orm as: * ( x) max, z q p b z c J 2 T s.t. 0, (22) where the coecents q, p, and c are unctons o x: 13

14 1 q x q x x q q x 1 x x 1 x k k kk k k k 1 N 2 1 p x q x x N N q q x 1 x x 1 x k 1 a x 1 x 2 k k k kk k k 2 c x q q q a c N N N N k k k k 1 (23) 0 Fgure 2. The range o δ or derent constrants For xed control varables z, the range o δ or every nequalty n (22) can be derved analytcally snce all the nequaltes are quadratc. Assume L and are roots o 0 and, the range o δ or the th nequalty s L q 0, and L L or q 0. These ranges can be dvded nto three typcal cases as shown n Fgure 2. The rst case corresponds to constrants wth postve q whch are convex, and the easble regon s a contnuous nterval. The second case corresponds to constrants wth negatve q whch are nonconvex and rght-hand hal plane symmetry axs, then the easble regon s two dsunctve hal open ntervals and wll lead to multple local optma. The last case s smlar to the second one but wth let-hand hal plane symmetry axs, and wll not lead to multple local 14

15 optma. The lnear constrants ( q 0 ) are neglected n the gure. It s clear rom Fgure 2 that there are two maxma o δ, among whch δ1 s a local maxmum and δ2 s the global maxmum. As requred by the assumpton o the alternatve ormulaton o the lexblty ndex problem, we need to guarantee that the soluton o (22) s the smallest maxmum δ1. The only reason that problem (22) has multple local maxma les n the constrants wth negatve q and rght-hand hal plane symmetry axs. Thereore, or those constrants we can add some extra constrants as shown below to make sure δ does not cross the ntermedate neasble regon. The vertcal red thck lne n Fgure 2 gves a graphcal llustraton. p p, J, q 0, 0 2q 2q (24) Note that there exsts some such that constrant (24) s actve ater solvng the problem, then the actve constrant should be removed and the problem needs to be re-solved. Ths s because constrant (24) s actve, the correspondng s always less than zero and the ntermedate neasble nterval vanshes. Thereore, wll not lead to multple optma. 3.3 Master problem As mentoned n Secton 3.1, the nonlnear constrants need to be overestmated to derve the master problem. Thereore, we need to derve a lnear or pecewse lnear uncton l (the superscrpt l means lnear), such that l n the hyperrectangle L TF ( ) { }, where F denotes the upper bound o the lexblty ndex F, and L, are gven by L N F N F (25) 15

16 In ths hyperrectangle, the convex quadratc term n (18) can be overestmated by ts secant: 2 L 2 2 L 2 L L L L (26) The overestmaton o the blnear term s based on the McCormck relaxaton as shown below: 18,19 L L L L mn(, ) (27a) k k k k k k k L L L L max(, ) (27b) k k k k k k k By keepng Eq. (27a) unchanged and multplyng -1 to Eq. (27b), we get, p L L L L w mn(, ) (28a) k k k k k k k k n L L L L w mn(, ) (28b) k k k k k k k k Thereore, p w k and n w k are the overestmatons o k and k, respectvely. By replacng the postve and negatve blnear terms n (18) together wth ther sgns by p w k and n w k, respectvely, the master problem or (18) wth pecewse lnear constrants s shown n (M1). The upper bound o δ s explctly added to make the problem tghter, although ths constrant s not essental. F L s.t. mn max x p n, z, w, w q a q q w l L p k k k p (, k ) I n (, k ) I n T L q q w b z q c 0, J k k k w mn,, (, k) I w p L L L L p k k k k k k k mn,, (, k) I n L L k k k N x 1 x, I F x {0,1} n L L n k k k k (M1) 16

17 The sets p I and n p I are dened by I {(, k), k I, k, J, q k q k 0} and n I {(, k), k I, k, J, q q 0}, respectvely. By substtutng k k p w k and n w k nto l, l becomes a uncton o θ and z. Snce θ s a uncton o x and δ, l can also be a uncton o x, δ and z. Note that ths s derent rom the tradtonal use o McCormck relaxaton, n whch every blnear term s replaced by w, and w s bounded by the ollowng nequaltes: w L L,, k I, k (29a) k k k k w L L,, k I, k (29b) k k k k w L L L L,, k I, k (29c) k k k k w,, k I, k (29d) k k k k The easble regon s enlarged ater usng the tradtonal McCormck relaxaton. Thereore, t wll provde an upper bound o the lexblty ndex nstead o a lower bound. The nner maxmzaton problem o (M1) ncludes mn operators o two lnear unctons. A conventonal technque s to ntroduce bnary varables, and then use dsunctons to represent the choce o the two lnear unctons. However, ths wll lead to a mxed-nteger blevel lner programmng wth bnary varables n both levels, whch s much harder to solve than blevel lnear programmng wth bnary varables only n the outer level. Actually, the result o the mn operators n the nner problem s determned by the outer level varables x. To smply the presentaton, we consder the case o and (see Appendx A or other cases). The results o the mn operators are gven n k k Propostons 2 and 3. 17

18 Proposton 2. For the blnear terms n Eq. (28a) wth postve coecents, x 0, then w ; x 1, then p L L k k k k w, regardless o x k, < k. p L L k k k k Proo. Let us consder x 0 rst. Snce x s xed to zero, but x k s not xed, and k can be expressed as unctons o δ: N N k k k Together wth Eq. (25), the rst term n the mn operator o (30) p w k n (M1) mnus the second one can be wrtten as L L L L k k k k k k L L L L k k k k k N N F k k F k N N N N F k F k F k F k 2 F k k F k F k k (31) Recall that we assume and k k, then Eq. (31) s equal to 0 xk 0,.e., ; and can be smpled to the ollowng equaton x 1,.e., N k k k k. N k k k F (32) F k k k It s clear that 0 always holds x 0. Thereore, p w k s equal to the rst term o the mn operator x 0,.e., w. p L L k k k k The proo or x 1 s smlar, and s omtted or smplcty. Proposton 3. For the blnear terms n Eq. (28b) wth negatve coecents, x 0, then w ; x 1, then n L L L L k k k k w, regardless o x k, < k. n k k k k Proo. Smlar to the proo o Proposton 2. 18

19 Based on Propostons 2 and 3, the mn operators n (M1) can be rewrtten as the ollowng dsunctons: x x, (, k) I p L L p L L wk k k k wk k k k p x x, (, k) I n L L L L n wk k k k wk k k k n (33) (34) The dsunctons n Eqs. (33) and (34) are reormulated as mxed-nteger constrants wth the bg-m method. 20 Then, (M1) can be reormulated as the ollowng more tractable mn-max programmng wthout the mn operators n the nner level. F L s.t. mn max x p n, z, w, w q a q q w L p k k k p (, k ) I n (, k ) I n T L q q w b z q c 0, J k k k Mx w a b d Mx, s s s s k 1k 1k k 1k s { p, n}, (, k) I s s s s M 1 x wk a2k b2kk d2k M 1 x, s { p, n}, (, k) I N x 1 x, I F x {0,1} n s s (M2) s s s where a, b, d, h {1,2}, denote the correspondng coecents and constants o the hk hk hk lnear unctons n Eqs. (33) and (34) or s { p, n}, whle s w k represents the aggregaton o p w k and n w k n (M1). Smlar to the dual-based method or lnear systems ntroduced n secton 3.1, we can obtan the ollowng MINLP by replacng the nner maxmzaton problem by ts dual. The varables λ, μ, υ, and η are the multplers assocated wth the nequaltes n (M2). 19

20 L L N L F mn q a q c,,,, x s.t. s h s (, k ) I a b d s s s 1k 2k 2k s s (, k ) I s s (, k ) I s s (, k ) I s h s (, k) I s s N s N s hk hk hk k hk Mx M Mx a b d s s N s N s hk hk hk k hk Mx M Mx F s s s 1k 2k 2k s s s s (, k ) I s (, k ) I s (, k ) I L q a x s h s (, k ) I a x b x s s s s hk hk hk k k k hk s a b s h s (, k ) I s h s (, k ) I s h s (, k ) I s s hk hk k n a x b x s s s a b 1 T B 0 q q 0, (, k) I n q q 0, (, k) I hk s s s s hk hk hk k k k hk hk hk k hk p p p k k hk hk h h k k hk hk h h,,, 0 x 0,1 n n (M3) T where B [ b1, b2,..., b n ]. It should be noted that although there are bnary varables n the nner problem, t can be rgorously converted to ts dual snce the bnary varables belong to the outer problem. An llustratve example s presented n Appendx B to explan ths. Wth the technque n secton 3.1, by ntroducng auxlary varables, (M3) s reormulated as the ollowng MILP, whch s the nal verson we actually solve. 20

21 L L N L F mn q a q c,,,,,,, x s.t. s h s (, k ) I a b d s s N s N s hk hk hk k hk M M M s s s 1k 2k 2k s s (, k ) I s s (, k ) I s s (, k ) I a b d s h s (, k ) I s s N s N s hk hk hk k hk M M M F s s s 1k 2k 2k s s (, k ) I s s (, k ) I s s (, k ) I L a q s h s (, k ) I b a s s s s hk hk hk k k khk a b s h s (, k ) I s h s (, k ) I s h s (, k ) I s s s hk hk k hk n 21 b a s s s a b 1 T B 0 s s s s hk hk hk k k khk hk hk k hk q q 0, (, k) I p p p k k hk hk h h n q q 0, (, k) I k k hk hk h h M M x I, J M x s s hk hk s hk s hk s hk s s hk hk s hk s hk s hk s s khk hk k s khk s khk s hk k s s khk hk k s khk s khk M M x M x M M x Mx M M x M x M M x s,,,,,, 0 x {0,1} hk Mx n k s { p, n}, h {1,2}, (, k) I n s (M4)

22 3.4 Propertes o the master problem There are our versons o master problems, rom (M1) to (M4). It s nterestng to nvestgate ther relatons. In most nstances, all the our problems are equvalent, but there are some exceptons as lsted n Table 1. The reason or ths lack o equvalence s two-old: one les n ntroducng bg-m parameters or an unbounded problem, and the other s dualzng an neasble problem. Table 1. Exceptons when reormulatng the master problems I (M1) s unbounded,.e., the nner maxmzaton problem o (M1) to (M2) (M1) s unbounded or every x, then the result o (M1) s plus nnty. But (M2) has a nte postve optmum snce M s nte. I or some (or all) vertex drectons, the nner maxmzaton (M2) to (M3) problem o (M2) s neasble, then (M2) has a postve optmum (or neasble). But (M3) s unbounded and the result s mnus nnty snce the dual o an neasble problem s unbounded. I (M3) s unbounded, then the result o (M3) s mnus nnty. (M3) to (M4) But (M4) has a nte negatve optmum, snce Mλ, Mμ, and Mυ are nte. Snce (M4) s the master problem we actually solve, we only need to consder the exceptons between (M1) and (M4). It can be concluded rom Table 1 that, (1) I (M1) s unbounded, then ts optmum s plus nnty. But (M4) has a nte postve optmum snce (M2) has a nte postve optmum. 22

23 (2) I or some (or all) vertex drectons, the nner maxmzaton problem o (M1) s neasble, then t has a postve optmum (or neasble). But (M4) has a negatve optmum. For all the other scenaros except these two exceptons, (M1) and (M4) have the same result. Let x * denote the optmal soluton o (M4), and * * ( x ) the optmal value o the subproblem L (S). The relatons o F, F, F, and * * ( x ) are gven n Propostons 4 and 5. Let X + denote the set o x n whch * ( x) F, X = the set o x n whch * ( x) F, and X - the set o x n whch * ( x) F. As shown n Appendx C, the easble regon o (M1) or a xed desgn d wll look lke ths: (1) It s nscrbed n the ntersecton o TF ( ) and the easble regon o the orgnal problem. N (2) I x X X, the vertex F x (1 x) o ( TF ) s on the boundary o the easble regon. I x X, the vertex N * ( x) x (1 x) s outsde o the easble regon. Proposton 4. I F F, then L F F F. Proo. I F F, then X, and hence X X contans all the vertex drectons. Accordng to the aorementoned summary, all the vertces o TF ( ) are on the boundary o the easble regon o (M1). Thereore, or every x, the nner maxmzaton problem o (M1) has the same optmum F, and hence the optmum o (M1) s F,.e., F. Snce the nner maxmzaton problem o (M1) s bounded and easble or every vertex, (M1) and (M4) have the same optmum. Thereore, L F F F 23

24 Proposton 5. I F F, then L * * F F ( x ) F. Proo. Snce l s an overestmaton o n TF ( ), and l only occurs at the vertces/boundares o TF ( ). Thereore, n a smaller T( F ), l s always satsed. We can then obtan the ollowng nequalty: max mn max max mn max 0 (35) l T ( F ) z J T ( F ) z J I the nomnal pont s easble n (M1), then ts lexblty ndex must be less than F, and (M1), (M4) have the same optmum. On the other hand, the nomnal pont s neasble n (M1), then (M4) has a negatve optmum. In ether case, L F F always holds. Next, we only need to prove * * ( x ) F snce * * ( x ) F always holds. Based on the conclusons n Appendx C, x X X, ( d, x, F ) 0 ; x X *, d x x,, ( ) 0. I the nner maxmzaton problem o (M1) s easble or every x X, the soluton x * must be n the set X - snce we need to choose the smallest δ such that ( dx,, ) 0. In ths case, (M1) and (M4) have the same soluton. Thereore, * * ( x ) F. On the other hand, there are some x X that are neasble n the nner maxmzaton problem o (M1), the optmum o (M4) wll be negatve, and one o these x wll be the soluton o (M4). In ether case, * * ( x ) F always holds. 3.5 Algorthm The proposed dualty-based algorthm s shown n Fgure 3, and the procedure s as ollows: Step 1: Choose an ntal vertex drecton x 0, such as settng all elements to zero (.e., vertex drecton or lower bounds). Choose a stoppng crteron ε. Step 2: se x 0 to solve the subproblem (S) wth the constrant n Eq. (24), and the result 24

25 s * ( x ). Set 0 F * ( x0). Step 3: Set L N F, and N F. se these parameters to construct and solve the master problem (M4), the optmal results are L F and x *. Step 4: I L F F, stop; otherwse, set x0 * x, and return to step 2. The convergence o the proposed algorthm s proved n the ollowng proposton. Proposton 6. The proposed algorthm wll converge to the optmal soluton n a nte number o teratons regardless o the choce o x 0. Proo. Let F denote the lexblty ndex, and F x denote the drecton that the lexblty ndex s obtaned,.e., * ( x F ) F. I x0 x F, then n the rst teraton F F. Accordng to Proposton 4, L F F F, thereore, the algorthm converges to F n one teraton. I x0 x F, and let the superscrpt n the parentheses denote the number o teraton. In the rst teraton, (1) * F ( x0) F L. Accordng to Proposton 5, (1) * *(1) (1) F F x F. In the second teraton, (2) * *(1) (1) F x F, meanng F s strctly decreasng. Snce the number o vertces s nte, F wll nally be equal to F; and the correspondng L F wll also be equal to F. Thereore, the algorthm wll converge to F n a nte number o teratons regardless o x 0. 25

26 Start x 0 = [0, 0,...0] Solve the subproblem (S), and set F = δ*(x 0 ) x 0 = x* L N F N F Solve the master problem (M4) N F -F L ε? Y F = F End Fgure 3. Flowchart o the proposed dualty-based lexblty ndex algorthm 4. Numercal examples In the ollowng, our examples are presented to compare the tradtonal actve-set method wth the proposed dualty-based method. The rst one s a small problem to show the detals o the proposed method, the second one s a small HEN, the last two are a process network problem, and a securty constraned unt commtment problem to compare the computatonal perormance o derent methods. All models are solved n GAMS , the MINLP n the actve-set method s solved by BARON , 21 and the NLP and MILP n the proposed dualty-based method are solved by CONOPT 3 and CPLEX 12.7, respectvely. The 26

27 computaton s carred out on an Intel Core TM laptop at 2.30 GHz wth 8GB RAM. 4.1 Small problem Ths problem s desgned to show some dagrams to llustrate the detals o the algorthm. Assume that there are two uncertan parameters 1 and 2, no control varables. The nomnal N N values and expected devatons are 1 7, 2 8, 1 4, 2 6, respectvely. The ollowng two constrants must be satsed (36) Problem (S) s solved wth our derent x, and the solutons o our vertex drectons are shown n Fgure 4. Note that, when solvng problem (S) wth x = (0, 0), an extra constrant lke the one n Eq. (24) should be added to guarantee the smallest optmum s obtaned. N N (37) The two curves are the boundary o the easble regon, and the our arrows represent the maxmum dstance rom the nomnal pont to the boundary. The lexblty ndex F 0.51 corresponds to the smallest vertex soluton, and ts correspondng lexble regon s marked by the sold rectangle. The dotted rectangle s the reerence rectangle,.e., the expected lexble regon. 27

28 Fgure 4. Vertex solutons and the easble regon The proposed algorthm s carred out step by step to show the detals. The bg-m parameters used n ths problem are M = 100, M M M 1, and these bg-m parameters are also used n the subsequent three examples. It takes two teratons to obtan the optmal soluton, and the results o the subproblem (S), master problem (M4) and some ntermedate results are gven n Table 2. Table 2. Results o the small problem wth x0 = (0, 0) Iter subproblem (S) * x0 F L F master problem (M4) * x 1 (0, 0) (1, 1) 2 (1, 1) (1, 1) 1 Note: 1 Ths optmal value can be any combnaton o x, snce every drecton yelds the same result n the last teraton. In the rst teraton, we start rom x0 = (0, 0) to solve the subproblem, and the upper bound o the lexblty ndex s The range n whch s overestmated s L L, 3.546,10.454, and 2 2, , We can then construct 28

29 the master problem n the orm o (M1), as shown below: F L mn max p 12 n 12 x p n, w12, w12 s. t. w w w w p L L L L n L L L L mn, mn, 7 4 x 4 1 x 8 6 x 6 1 x F x, x {0,1} (38) Its correspondng easblty uncton s stated n the ollowng: l L L L L 1 mn , , l L L L L 2 mn , ( ) max L (39) The easble regon s depcted n Fgure 5, n whch the black dot n the center s the nomnal pont, the dotted rectangle s the regon n whch the constrant s overestmated,.e., the L hyperrectangle TF ( ). The shaded area and the sold rectangle wth F are the easble regon and lexble regon or the master problem, respectvely. We also depct the easble regon or the orgnal problem n Fgure 5, correspondng to the regon between the two blue sold curves. In the rst teraton, X + ={(0, 1), (1, 0)}, X = ={(0, 0)}, X - ={(1, 1)}. The graph s consstent wth the aorementoned conclusons about the easble regon n secton 3.4. The three vertces o TF ( ) along the drectons n X X are on the boundary, and the vertex correspondng to the maxmum devaton along the drecton n X - n the orgnal problem s 29

30 outsde o the easble regon. There s only one element n X -, whch s exactly the crtcal drecton or the orgnal problem. Thereore, the soluton wll converge n the next teraton. Fgure 5. Feasble regon n the rst teraton wth x0 = (0, 0) Fgure 6. Feasble regon n the second teraton wth x0 = (0, 0) In the second teraton, we use the optmal soluton x * o the rst teraton as the vertex drecton to solve the subproblem, and the new upper bound o the lexblty ndex s 0.51, whch s exactly the lexblty ndex. Thereore, X - s empty, and all the vertces o TF ( ) are on the boundary o the easble regon, whch means TF ( ) s exactly the same as the 30

31 easble regon o the master problem as shown n Fgure 6. Hence, the optmum o the master problem, 0.51, s equal to F, and the algorthm converges n two teratons. I we start rom x0 = (0, 1), the results are gven n Table 3. In the rst teraton, the master problem (M4) has a negatve optmum whch means that the constrants are greatly overestmated, such that even the nomnal pont s neasble n the lnearzed system as shown n Fgure 7. Strctly speakng, ths s not a lexblty ndex problem snce the nomnal pont s neasble. I we solve problem (38) drectly, the only soluton s x * = (0, 1), and the algorthm cannot converge. But n practce, we solve (M4) nstead, whch dualzes the nner maxmzaton problem o (M1). As dscussed n secton 3.4, there are neasble drectons n (M1), the optmum o (M4) wll be negatve and the optmal value o x wll be one o these neasble drectons. Table 3. Results o the small problem wth x0 = (0, 1) Iter x0 subproblem (S) * F L F master problem (M4) * x 1 (0, 1) (1, 1) 2 (1, 1) (1, 1) The conclusons about the easble regon o the master problem are stll vald. In ths case, X + s empty, X = ={(0, 1)}, and X - ={(0, 0), (1, 0), (1, 1)}. We can only exclude the sngle drecton n X =. But ortunately, the optmal soluton x * s exactly the crtcal drecton or the orgnal problem. Thereore, the algorthm converges n the next teraton, ust lke startng rom x0 = (0, 0). 31

32 Fgure 7. Feasble regon n the rst teraton wth x0 = (0, 1) 4.2 Heat exchanger network Ths example o a small HEN s a modcaton o the example ntroduced by Grossmann and Floudas. 7 There are two hot streams, two cold streams, three heat exchangers and one cooler n the HEN, as shown n Fgure 8. The uncertan parameters are the nlet temperatures o the our streams and the heat capacty low rates o H1 and C2, namely, T1, T3, T5, T8, FH1, and FC2. All the nomnal values are shown n Fgure 8, the expected devatons or the nlet temperatures are ±5 K, and the expected devatons or the heat capacty lows o H1 and C2 are assumed to be ±0.3 kw/k and ±0.5 kw/k, respectvely. The only control varable les n the heat load o the cooler (Qc). 32

33 H1 H2 C1 H1-C1 H2-C1 C2 H2-C2 Fgure 8. HEN wth sx uncertan parameters and one control varable The nequalty constrants are gven as ollows: 350F F T Q 0 1 H1 H1 3 c T T 175F 0.5F T 0.5Q H1 H1 1 c 2T T 350F F T Q H1 H1 1 c 2T T T 350F 393F F T F T Q H1 C 2 H1 1 C2 8 c 2T T 350F 393F F T F T Q H1 C 2 H1 1 C 2 8 c T, T, T, T, F , F, Q 0 H1 C2 c (40) The tradtonal actve-set method and the proposed dualty-based method are appled to analyze the lexblty ndex o ths system. The results, whch yeld F = 0.174, are gven n Table 4. Although the MILP problem n the proposed dualty-based method has more varables than the MINLP problem n the actve-set method, the dualty-based method requres less tme to solve ths small example (0.094 s vs s). 33

34 Table 4. Flexblty ndex results o the HEN Method # o bn. varables # o cont. varables # o constrants # o teratons 2 F Soluton tme [s] Actve-set Dualty 0/6 1 2/ / Notes: 1 The rst number denotes the subproblem, the second denotes the master problem 2 The number o teratons n the dualty-based method 3 Summaton o all tme used n the solvers It s clear rom Table 4 that the actve-set method has much ewer constrants and contnuous varables than the MILP problem n the dualty-based method, but t takes longer to solve. The reasons are two-old. On one hand the actve-set method has to solve a complex MINLP problem wth nonconvex blnear terms, whle the dualty-based method only needs to solve some small NLPs wth nz 1 varables (where n z s the number o control varables) and easer MILPs; on the other hand, the MINLP n the actve-set method usually has more bnary varables than the MILP n the dualty-based method, because the ormer has n (number o constrants) bnary varables, whle the latter has n (number o uncertan parameters) bnary varables, and there are usually more constrants than uncertan parameters. The perormance o the dualty-based method depends on the ntal vertex drecton used to solve the subproblem. I the optmal vertex drecton s used as the ntal vertex drecton, only one teraton s needed. Here, the dualty-based method s tested wth all possble ntal vertex drectons as shown n Table 5. There are sx uncertan parameters. Thereore, the number o 34

35 possble ntal vertex drectons s 2 6 = 64. All the runs yeld the correct result, and on average, they take 2.4 teratons and 0.16 seconds to converge. Table 5. Statstcs o dualty-based method wth all possble ntal vertex drectons Intal Iteratons Tme [s] Intal Iteratons Tme [s]

36 Plannng o process network Consder a process network consstng o 6 processes and 10 chemcals as shown n Fgure The ollowng plannng model o ths process network s the one used by Zhang et al. 3 t 0 mn Q Pt Pt Wt D t Q, J, t T t1 Iˆ I (41a) t 0 max Q Pt Pt Wt Dt Q, J, t T t1 Iˆ I (41b) 36

37 P max t P, I, t T (41c) W W D, J, t T (41d) max t t t J P 0, I, t T (41e) t W 0, J, t T (41) t Acetylene (1) Process 2 Acetaldehyde (5) HCN (2) Process 1 Acrylontrle (6) Process 3 Propylene (3) Process 6 Isopropanol (7) Benzene (4) Process 4 Process 5 Phenol (8) Acetone (9) Cumene (10) Fgure 9. Process network superstructure In ths model, I, J, and T are sets o processes, chemcals, and tme perods, respectvely. 0 Q, mn Q, and max Q are the ntal, mnmum and maxmum nventores, respectvely. P t s the total amount o chemcals produced by process n tme perod t. denotes the converson racton o chemcal by process wth respect to the total amount. Thereore, the amount o chemcal consumed or produced by process n tme perod t s gven by Pt. I ˆ and I are sets o processes producng and consumng chemcal, respectvely. W t denotes the purchased amount o raw materal n tme perod t, and D t denotes the demand or product n tme perod t. J s the set o raw materals (chemcals 1 to 4). The data s avalable n the supplementary materal. Equatons (41a) and (41b) ensure that the nventory level les wthn the bounds at every 37

38 pont n tme. Equaton (41c) corresponds to the capacty constrant. Equaton (41d) s the purchase lmt or raw materals. As stated by Zhang et al., 3 t s assumed that the purchase lmt o chemcal depends on the demand o products consumng, denoted by the set J. The qualtatve explanaton s that as the demand ncreases, the avalablty o ts correspondng raw materals decreases. For example, the purchase lmt o acetylene s aected by the demand o acetaldehyde and acrylontrle. The coecent descrbes how much the avalablty s aected by the demand. Equatons (41e) and (41) are nonnegatvty constrants. In ths model, the coecent and the demand D t or products (chemcals 5 to 10) are consdered to be uncertan parameters, and P t and W t are the control varables. For derent number o tme perods, N T, the number o uncertan parameters, control varables, and constrants are 6NT 7, 10N T, and 40N T, respectvely. The lexblty ndex problem or ths process network wth N T uncertan parameters varyng rom 1 to 4 s solved by the actve-set method and the proposed dualty-based method, as shown n Table 6. The tme lmt or each case s one hour. In the smallest nstance ( NT 1), both methods can solve the lexblty ndex problem to optmalty qute ecently. The actve-set method s slghtly aster than the dualty-based method snce the latter needs to solve 2 NLPs and 2 MILPs, whch are not sgncantly easer than the small MINLP n the ormer. As the problem sze ncreases, the number o bnary varables and blnear terms n the actve-set method ncreases quckly. Thereore, the dualtybased method becomes superor to the actve-set method. In the larger nstances ( NT 3, NT 4 ), the actve-set method can obtan the correct result, but t cannot prove optmalty. The 38

39 lower bound does not mprove (remans zero) durng the computaton. However, the proposed dualty-based method can solve all the nstances wthn one mnute. Table 6. Flexblty ndex results o the process network problem N T Method # o bn. varables # o cont. varables # o constrants # o Iteratons F Gap [%] Soluton tme [s] Actve-set Dualty 0/13 24/646 53/ Actve-set Dualty 0/19 40/ / Actve-set Dualty 0/25 56/ / Actve-set Dualty 0/31 72/ /15, Another observaton s that, t takes only a ew teratons to converge to the optmum n the dualty-based method, regardless o the number o uncertan parameters and the choce o the ntal vertex drecton. There s no theoretcal guarantee o ths property, but the drecton correspondng to the smallest devaton n the orgnal problem tends to also have the smallest devaton n the master problem. 4.4 Securty constraned unt commtment problem In ths example, we consder the unt commtment model, whch arses n electrc power systems. 23 It deals wth the schedulng o I power generators over T tme perods n a power 39

40 generaton plant. The MINLP model, whch nvolves uncertantes n the power generated, conssts o the ollowng constrants: I T 2 a b pt c pt ytsd zts C (42a) 1 t1 I I t t 1 1 p R p t (42b) L p p p, t (42c) t t t I T EC pt E (42d) 1 t1 R R R t (42e) L t t t p p R t (42) t, t1, p p RD t (42g) t, t1, where constrant (42a) s the convex obectve uncton n the orgnal schedulng problem solved by Nknam et al. 24, but we ntroduce an upper lmt o cost n the lexblty ndex problem. Equaton (42b) s the spnnng reserve constrant. Equaton (42c) sets the power lmt or every generator at every tme perod. Equaton (42d) ensures that the carbon doxde emssons satsy the emsson lmt. Equaton (42e) sets the spnnng reserve lmt or every tme perod. Equatons (42) and (42g) are the ramp rate lmts. In ths model, the spnnng reserve R t are the control varables, the uncertan parameters correspond to the generated power p t, and all the others are constants. In partcular, y t and z t are bnary varables n the orgnal schedulng problem, but n the lexblty ndex problem they are xed to the optma o the schedulng problem. All the constrants except Eq. (42a) are lnear, and there are only quadratc terms wth postve coecents n the nonlnear 40

41 constrant. Thereore, the proposed dualty-based method s applcable. We consder a case consstng o 10 unts and 24 tme perods. The data s avalable rom the webste The correspondng lexblty ndex problem s a very large problem wth 240 uncertan parameters, 24 control varables, and 1034 constrants. The nomnal operatng pont s the optmal soluton o the schedulng problem, n whch all the generators are orced to operate durng all tme perods. Ths can smply the ormulaton o the lexblty ndex problem, snce we do not need to deal wth the specal case where some generators are shut down. The expected devaton s set to be 5 percent o the nomnal value, and the spnnng reserve can only be adusted wthn 10 percent o the nomnal value. To avod the lexblty ndex beng zero, the upper lmt o the total cost C s 700,000 $/h, whch s greater than the optmal cost. The emsson lmt E s 200,000 g hgher than the value (15,500,000 g) used n the orgnal schedulng problem. The power lmt p o every generator s 10 MW hgher than the orgnal value. The ramp rate lmts R and RD are 10 MW/h hgher than the orgnal values. The lexblty ndex problem s solved by the actve-set method and the proposed dualtybased method, and the results are shown n Table 7. Table 7. Flexblty ndex results o the unt commtment problem Method # o bn. varables # o cont. varables # o constrants # o Iteratons F Soluton tme [s] Actve-set Dualty 0/240 25/249, /744,

42 Both methods yeld the same result (F = 0.258), but the dualty-based method s sgncantly aster than the actve-set method (51.5 s vs s). The reason s the same as the one explaned n the second example or HEN. The master problem o the dualty-based method can be very large, snce we have to ntroduce new varables and new constrants or every combnaton o bnary varables and constrants to elmnate the nonlnearty. However, t only takes less than 30 seconds to solve the large MILP snce the relaxed MILP (RMIP) has the same optmum as the MILP, although the bnary varables are not ntegral n the RMIP. In contrast, the optmum o the relaxed MINLP o the actve-set method s zero. 5. Conclusons In ths work, we have presented a new lexblty ndex algorthm or systems n whch all the nequaltes are quadratc or lnear n θ, and lnear n z. Ths class o system s proved to have a vertex soluton all the dagonal elements o Q, J, are non-negatve. Based on ths property, the subproblem, whch provdes an upper bound o the lexblty ndex, s easly obtaned. Smlar to the dea o OA n convex MINLP, the master problem s constructed by overestmatng the nonlnear constrants, provdng a lower bound o the lexblty ndex. Ater elmnatng the mn operators n the nner maxmzaton problem, dualzng the nner maxmzaton problem, and ntroducng new varables and constrants, the master problem s reormulated as a tractable MILP. By teratvely solvng the subproblem and the master problem, the algorthm can be guaranteed to converge to the lexblty ndex. Four computatonal studes, whch nclude a small example HEN, a process network and unt commtment, are presented to llustrate ts applcablty n solvng lexblty ndex problem. 42

43 The results show that the proposed algorthm s more ecent than the actve-set method, especally or large-scale problems wth more constrants than uncertan parameters. Fnally, ths algorthm s not restrcted to quadratc systems. It can be drectly extended to nequaltes wth unvarate convex nonlnear terms, snce they can be overestmated by ther secants ust lke the quadratc terms. For general nonlnear systems wth vertex solutons, ths algorthm also provdes a promsng approach. I one can develop the overestmaton satsyng these condtons: (1) the overestmaton s lnear or pecewse lne, (2) the overestmaton s equal to the orgnal uncton at all the vertces o TF ( ), and greater than the orgnal uncton nsde o TF ( ), (3) the maxmum o the overestmaton n TF ( ) les at a vertex, the algorthm n ths work can also be appled to the general systems, but the master problem must be reormulated n a smlar way. Acknowledgment The authors grateully acknowledge the nancal support rom the Chna Scholarshp Councl and the Center or Advanced Process Decson-makng at Carnege Mellon nversty. Notaton a, b, c = coecents o quadratc cost uncton o unt C = upper lmt o the total cost E = emsson lmt or the system EC = emsson coecent or unt p t = generated power o unt at tme t L p = lower power lmt o unt 43