Strategic Planning, Design and Development of the Shale Gas Supply Chain Network

Transcription

1 Carnegie Mellon Universiy Research CMU Deparmen of Chemical Engineering Carnegie Insiue of Technology Sraegic Planning, Design and Developmen of he Shale Gas Supply Chain Nework Diego C. Cafaro Universidad Nacional de Lioral Ignacio E. Grossmann Carnegie Mellon Universiy, grossmann@cmu.edu Follow his and addiional works a: hp://reposiory.cmu.edu/cheme Par of he Chemical Engineering Commons Published In AIChE Journal, 60, 6, This Aricle is brough o you for free and open access by he Carnegie Insiue of Technology a Research CMU. I has been acceped for inclusion in Deparmen of Chemical Engineering by an auhorized adminisraor of Research CMU. For more informaion, please conac research-showcase@andrew.cmu.edu.

2 Sraegic Planning, Design and Developmen of he Shale Gas Supply Chain Nework Diego C. Cafaro 1 and Ignacio E. Grossmann 2* 1 INTEC (UNL CONICET), Güemes 3450, 3000 Sana Fe, ARGENTINA 2 Deparmen of Chemical Engineering, Carnegie Mellon Universiy, Pisburgh, PA 15213, U.S.A. ABSTRACT The long-erm planning of he shale gas supply chain is a relevan problem ha has no been addressed before in he lieraure. This paper presens a mixed-ineger nonlinear programming (MINLP) model o opimally deermine he number of wells o drill a every locaion, he size of gas processing plans, he secion and lengh of pipelines for gahering raw gas and delivering processed gas and by-producs, he power of gas compressors, and he amoun of freshwaer required from reservoirs for drilling and hydraulic fracuring so as o maximize he economics of he proec. Since he proposed model is a large-scale non-convex MINLP, we develop a decomposiion approach based on successively refining a piecewise linear approximaion of he obecive funcion. Resuls on realisic insances show he imporance of heavier hydrocarbons o he economics of he proec, as well as he opimal usage of he infrasrucure by properly planning he drilling sraegy. KEYWORDS Shale gas, supply chain, sraegic planning, MINLP, soluion algorihm * Corresponding auhor. Tel.: ; fax: address: grossmann@cmu.edu (I.E. Grossmann). 1

3 1. INTRODUCTION Naural gas is he cleanes-burning fossil fuel. Naural gas exraced from dense shale rock formaions has become he fases-growing fuel in he U.S. and has he poenial of becoming a significan new global energy source. Over he pas decade, he combinaion of horizonal drilling and hydraulic fracuring has allowed access o large volumes of shale gas ha were previously uneconomical o produce. The producion of naural gas from shale formaions has reinvigoraed he naural gas and chemical indusries in he U.S. The Energy Informaion Adminisraion proecs U.S. shale gas producion o grow from 23% o almos 50% of he oal gas producion in he nex 25 years. [1] Shale gas is found in plays conaining significan accumulaions of naural gas, sharing similar geologic and geographic properies. A decade of producion has come from he Barne Shale play in Texas. Experience gained from Barne Shale has improved he efficiency of shale gas developmen around he counry. Today, one of he mos producive plays is he Marcellus Shale in he easern U.S., mainly in Pennsylvania. Regarding boh economic and environmenal impacs, he long-erm planning and developmen of he shale gas supply chain nework around each play is a very relevan problem. However, o he bes of our knowledge, i has no been addressed before in he lieraure. The raw gas exraced from shale formaions is ranspored from wellbores o processing plans hrough pipelines. The processing of shale gas consiss of he separaion of he various hydrocarbons and fluids from he pure gas (mehane) o produce wha is known as pipeline qualiy dry naural gas. [2] This means ha before he naural gas can be ranspored by midsream disribuors, i mus be purified o mee he requiremen for pipeline, indusrial and commercial uses. The associaed hydrocarbons (ehane, propane, buane, penanes and naural gasoline) known as Naural Gas Liquids (NGLs), are valuable byproducs afer he naural gas has been purified and fracionaed. These NGLs are sold separaely (usually hrough dedicaed pipelines) and have a variey of differen uses, including enhancing oil recovery in wells, providing raw maerials for oil refineries or perochemical plans, and as sources of energy. [3] One of he mos criical issues in he design and planning of he shale gas supply 2

4 chain nework is he sizing and locaion of new shale gas processing and fracionaion plans (as well as fuure expansions) due o heir high cos. On he oher hand, he number of wells drilled in each locaion can dramaically influence coss and he ecological fooprin of naural gas operaions. [4] The abiliy o drill muliple wells from a single locaion (or pad ) is seen as a maor echnological breakhrough driving naural gas developmen as for insance has happened in he Marcellus Shale. The uilizaion of muli-well pads also has large environmenal and socio-economic implicaions given ha as many as 20 or more naural gas wells and associaed pipeline infrasrucure can be concenraed in a single locaion. Furhermore, he oal amoun of indusrial aciviy can be compressed as hese wells can be drilled in rapid-succession and he echnology now exiss o perform hydraulic fracuring simulaions on muliple wells simulaneously. Hence, anoher key decision ackled by his paper is he drilling sraegy, i.e. how many wells o se up or add on exising well pads a every ime period. Anoher criical aspec in he shale gas producion is waer managemen. Shale gas producion is a highly waer-inensive process, wih a ypical well requiring around 5 million gallons of waer normally over a 3-monh period o drill and fracure, depending on he basin and geological formaion. [5] The vas maoriy of his waer is used during he fracuring process, wih large volumes of waer pumped ino he well wih sand and chemicals o faciliae he exracion of he gas. Alhough increasing amouns of waer are being recycled and reused, freshwaer is sill required in high quaniies for he drilling operaions as flowback waer usually only represens abou 25-30% of he waer ineced ino he well. The need for freshwaer is an issue of growing imporance, especially in waerscarce regions and in areas wih high cumulaive demand for waer, leading o pressure on sources and compeiion for waer wihdrawal permis. Therefore, a long-erm planning model for he developmen of shale gas fields should also accoun for waer availabiliy. The goal of his paper is o develop a mixed-ineger nonlinear programming (MINLP) model for he susainable long-erm planning and developmen of shale gas supply chains, which should opimally deermine: (a) he number of wells o drill on new/exising pads; (b) he size and locaion of new gas 3

5 processing plans (as well as fuure expansions); (c) he secion, lengh and locaion of new pipelines for gahering raw gas, delivering dry gas, and moving NGLs; (d) he locaion and power of new gas compressors o be insalled, according o he flowrae a every line; and (e) he amoun of freshwaer coming from available reservoirs used for well drilling and fracuring, so as o maximize he economic resuls (NPV-based approach) over a planning horizon comprising 10 years. Lieraure Review Some of he firs papers in he opimal design and planning of supply chains were published in he lieraure 40 years ago. [6] A complee review on more recen developmens in supply chain opimizaion problems can be found in he work of Melo e al. [7] Regarding he sraegic planning of naural gas supply chains, Durán and Grossmann [8] propose a supersrucure represenaion, an MINLP model and a soluion sraegy for he opimal synhesis of gas pipelines, deciding on he gahering pipeline sysem configuraion, compressors power and pipeline pressures. Iyer e al. [9] propose a muliperiod MILP model for he opimal planning and scheduling of offshore oilfield infrasrucure invesmen and operaions. Since he resuling model becomes inracable due o he large-scale, nonlinear reservoir equaions are approximaed hrough piecewise linear funcions. Van den Heever and Grossmann [10] propose a muliperiod generalized nonlinear disuncive programming model for oilfield infrasrucure planning, whose opimal soluion is found hrough a bilevel decomposiion mehod. In his model, he number of wells is given beforehand hrough a fixed drilling plan. More recenly, Gupa and Grossmann [11] address some new feaures of he same problem, accouning for all hree componens (oil, waer, and gas) explicily in he formulaion. They also incorporae more accurae esimaions of he nonlinear reservoir behaviour, variable number of wells for each field (o capure drill rig limiaions) and faciliy expansions, including heir lead imes. On he oher hand, some work has also been repored on he opimizaion of he operaion of shale gas fields. Rahman e al. [12] presen an inegraed opimizaion model for hydraulic fracuring design, accouning for fracure geomery, maerial balances, operaional limiaions, characerisics of he gas 4

6 formaion, and producion profiles. By combining geneic algorihms and evoluionary echniques, improved hydraulic fracuring designs reduce he reamen (simulaion) coss up o 44% a he expense of a 12% reducion in he gas producion. Knudsen e al. [13] propose a Lagrangean relaxaion approach for scheduling shu-ins imes in igh formaion muli-well pads, so as o simulae he shale gas producion in differen wells o comply wih he gas raes required by he disribuion company. In ha work, a proxy model capures he physics during shu-ins operaions. Based on he proxy model resuls, he ime domain is discreized ino daily ime periods, and an MILP model is hen solved using Lagrangean relaxaion echniques. Finally, few recen publicaions deal wih he sraegic and operaional managemen of waer resources and oher environmenal concerns in he developmen of shale gas plays. Mauer e al. [14] argue ha sraegic planning by boh companies and regulaory agencies is criical o miigae he environmenal impacs of unconvenional exracion. Rahm and Riha [15] aemp o deermine waer resource impacs of shale gas exracion, from regional, collecive based perspecives, seeking o balance he need for developmen wih environmenal concerns and regulaory consrains. Yang and Grossmann [16] presen an MILP formulaion whose main obecive is o schedule he drilling and fracuring of well pads o minimize he ransporaion, reamen, and freshwaer acquisiion coss, as well as reamen infrasrucure, while maximizing he number of well sages o be compleed wihin he ime frame. The goal is o find an opimal shor-erm fracuring schedule, he waer recycling raio, and he need for addiional impoundmen and reamen capaciy. Like mos enerprise-wide opimizaion (EWO) problems, he sraegic planning of he shale gas supply chain has grea economic poenial. Considerable effor has been spen owards he soluion of EWO problems during he las 20 years, paricularly in he field of oil and gas producion. [17] Bu none of hem has been focused on he shale gas supply chain. The shale gas producion has is own peculiariies, and is a problem of very recen developmen. [18] In fac, one of he maor barriers is he size and complexiy of compuaional opimizaion models for achieving he goal of EWO. [19] The sraegic planning of shale gas infrasrucure consiss on he design of large supply chains, including 5

7 well-pads, processing plans, compressors, produc delivery nodes, and he complex pipeline nework ransporing shale gas and he resuling hydrocarbons. As concluded by Oliveira e al., [20] careful evaluaion of he invesmen opions in his kind of problems has paricular imporance, and he use of efficien decision-making ools ha capure he problem complexiy becomes crucial. Poenial Sies for Well Pads (i) Poenial Sies for Juncion Nodes () Ehane Demand Nodes (l) Liquid Pipelines p-l Transmission Pipelines p-k Fresh Waer Sources (f) Gas Demand Nodes (k) Flow Pipelines i- Gahering Pipelines -p Poenial Sies for Gas Processing Plans (p) Waer supplies Figure 1. A simplified supersrucure of he shale gas supply chain (for he sake of clariy, only few arcs of each ype are drawn). 2. PROBLEM DESCRIPTION We address he problem of deermining he opimal design for a shale gas supply chain nework, he well drilling and hydraulic fracuring sraegy over he planning horizon, ogeher wih he size and locaion of gas separaion plans, compressors and pipeline infrasrucure, in order o maximize he ne presen value of he proec. This problem can be formally saed as follows. 6

8 A comprehensive shale gas supply chain nework supersrucure like he one depiced in Figure 1 is given. I includes: (a) poenial or exising well pads where new wells can be drilled and hydraulically fracured over he planning horizon (nodes i I), (b) poenial or exising uncion nodes where shale gas flows coming from nearby well pads converge (nodes J), (c) poenial or exising flow pipelines connecing nodes i and, (d) candidae sies for he insallaion/expansion of a new/exising shale gas processing plans (nodes p P), (e) poenial/exising gahering pipelines connecing uncion nodes wih plan sies (f) demand nodes for dry naural gas (nodes k K) and ehane (nodes l L), (g) poenial/exising ransmission and liquid pipelines connecing plan sies p wih nodes k and l, respecively, and (h) freshwaer source nodes from where he waer required for drilling and fracuring new wells can be supplied. A sraegic long erm planning horizon is considered. In his paper, a planning horizon of 10 years is considered, and is divided ino 40 ime periods (quarers). The reasons for his ime discreizaion is as follows: (1) Gas prices normally exhibi a seasonal behavior wih a high peak in he winer. (2) The drilling and compleion of wells normally akes beween 50 and 90 days, plus he following 20 days during which he well does no produce a seady sream of gas, bu a flowback of waer ha is capured and sored for furher reamen. Overall, approximaely 90 days (hree monhs) are required since he well-pad is se up and wells sar o be drilled unil hey begin o produce a seady flow of shale gas. (3) Freshwaer availabiliy in some waer-scarce regions is srongly seasonal, and can be a criical issue if high cumulaive demand for waer leads o pressure on sources and compeiion for waer wihdrawal permis. Besides he nework supersrucure and he ime horizon, he produciviy profile of every well a any locaion is assumed o be deerminisic and known beforehand. Dry and semi-dry shale gas wells exhibi many of he same characerisics: an early peak in he gas rae from he sudden release of gas sored in pores and naural fracure neworks, followed by a long ransien decline in he producion rae. Such decline in he rae is caused boh by pressure loss and he inherenly low permeabiliy of shale rocks. In his problem, he well produciviy (measured in Mm 3 /day) is represened by a piecewise consan 7

9 funcion of he well age. The parameer pw i,τ sands for he producion rae of a shale gas well of age τ (given in quarers) drilled in locaion i (see Figure 2). Moreover, he shale gas composiion, and paricularly is weness (% of hydrocarbons ohers han mehane), are assumed o be known and independen of boh he well sie and is age. This assumpion can be relaxed as will be discussed laer in his paper. Well Prduciviy (Mm 3 /day) Time Periods since Drilling (quarers) Figure 2. Piecewise consan well produciviy profile. Regarding he pipeline infrasrucure, gas and liquid pipelines mus be considered separaely. On he one hand, gas pipelines (ransporing eiher raw or processed gas) are assumed o handle an ideal mixure of ideal gases. Raw gas pipelines connecing nodes i o (well pads o uncion nodes) and o p (uncion nodes o plans) operae a medium-low pressures, while ransmission pipelines p-k supplying gas demand nodes from processing plans operae a higher pressures. For simpliciy, gas sucion/discharge pressures a every node of he nework are assumed o be given consan values. These are as follows: (i) shale gas discharge pressure a he well pads is Pd i, (ii) uncion nodes receive he shale gas a a pressure of Ps < Pd i, (iii) compressor saions insalled a uncion nodes increase he pressure from Ps o Pd, o make he gas flow owards processing plans, (iv) he shale gas pressure a he inle of processing plans is Pi p < Pd, (v) processing plans deliver dry gas a a pressure of Po p, (vi) compressor saions insalled a he oule of processing plans increase he dry gas pressure from Po p = 8

10 Ps p o Pd p before sending flows o markes, and (vii) gas demand nodes receive dry gas a a pressure of Pr k < Pd p. By fixing such values, he maximum flow of a gas pipeline is direcly proporional o he pipeline diameer raised o he power of 2.667, and he proporionaliy facor depends on he gas properies, he inpu/oupu pressures and he pipeline lengh. [8][21] Moreover, compressors are assumed o be adiabaic and heir power is direcly proporional o he gas flow, since he compression raio is a given parameer. More deails are given in he Appendix. On he oher hand, liquid pipelines ranspor hydrocarbons like ehane, propane, buane, penanes and naural gasoline (NGLs) in liquid sae from separaion plans o eiher perochemical plans or LPG (liquefied peroleum gases) disribuion faciliies. In his problem, all NGLs excep ehane are assumed o be separaely sold o cusomers near he processing plans, while ehane is coninuously delivered o perochemical plans by dedicaed pipelines. The maximum flow in liquid pipelines is assumed o be direcly proporional o he pipeline secion since a maximum mean velociy is imposed. i1 Well Pads (i) i2 (A) (B) Juncion Node () 1 Pd i1 = 2.1 MPa Pd i2 = 2.1 MPa Ps 1 = 1.4 MPa Pd 1 = 2.1 MPa Pd i1 = 2.1 MPa Pd i2 = 2.1 MPa Pi p = 1.4 MPa Pi p = 1.4 MPa Liquid Demand Node (l) Plan (p) Gas Demand Node (k) Ps p = 4.0 MPa Pd p = 6.0 MPa Pr k = 4.0 MPa Ps p = 4.0 MPa Pd p = 6.0 MPa Pr k = 4.0 MPa Figure 3. Simplified nework supersrucure and alernaive nework designs. An illusraive example comparing wo nework designs is presened in Figure 3. The shale gas produced a wo differen well pads i1 and i2 is sen o a processing plan in wo alernaive ways: (A) hrough an inermediae uncion node 1, or (B) direcly, hrough separae lines. Typical values for he sucion and discharge pressures a each node are given in he figure. The example also reveals one of he rade-offs o be deermined by he model. Opion (A) requires a compressor saion a node 1, bu pipelines are smaller in diameer and shorer han in opion (B) which does no require a compressor. 9

11 Pipeline and compressor coss wih regards o heir size (usually deermined by economies of scale funcions) are he key o deermine which opion is he mos convenien one. Finally, freshwaer consumpion, mainly for hydraulic fracuring, is considered o be a fixed amoun required during he drilling period, which depends on he well-pad locaion and he possibiliy of reusing he flowback waer. The selecion of opimal sources for waer supply is a key model decision, bu no deails on he waer ransporaion logisics are considered a his planning level. Oher operaional issues like flowback waer capure, reamen and final disposal, as well as planning shu-ins and well simulaions are also ou of he scope of his work. Given all he iems described above, he goal is o opimally deermine: (a) he number of wells o drill on new/exising pads a every rimeser; (b) he size and locaion of new gas processing plans (as well as fuure expansions); (c) he secion, lengh and locaion of new pipelines for gahering raw gas, delivering dry gas, and ransporing NGLs; (d) he locaion and power of new gas compressors o be insalled, and (e) he amoun of freshwaer from available reservoirs for well drilling and fracuring so as o maximize he Ne Presen Value (NPV) of he proec. Assumpions The main assumpions have already been discussed and can be summarized as follows: (1) Shale gas is assumed o be an ideal mixure of ideal gases. (2) The composiion, and paricularly he shale gas weness, are known consans independen of he well locaion. The relaxaion of his assumpion is discussed afer he model presenaion. (3) The planning horizon is discreized in ime periods, commonly quarers. (4) Muliple wells can be drilled in a single pad over one ime period, alhough no necessarily a he same ime. I is assumed ha all of hem are hydraulic fracured and compleed wihin he same ime period hey are drilled. 10

12 (5) Wells sar o produce shale gas in he period following he drilling period. Once he wells are compleed, heir producion canno be delayed by shuing hem in. This assumpion can be relaxed as shown in he Appendix. (6) Afer he well is compleed, is produciviy rae is a piecewise consan funcion in erms of he well age. In oher words, a decreasing funcion as depiced in Figure 2 is assumed o be given. (7) Muli-well pads can be se u and muliple wells in he same pad can be drilled, fracured and compleed during he same period. However, an upper bound is given due o echnology limiaions. Moreover, he oal number of wells ha can be drilled in he same pad over he given ime horizon is also bounded. (8) The pressure a pipelines ransporing raw gas from well pads o uncion nodes decreases from Pd i o Ps as a funcion of heir lengh [21] (for furher deails see he Appendix). The same is also valid for pipelines ransporing raw gas from uncion nodes o processing plans (from Pd o Pi p ), and dry gas from plans o demand nodes (from Pd p o Pr k ). (9) All he gas pressures (Pd i a he oule of pad i; Ps a he inle of he uncion node ; Pd a he oule of ; Pi p a he inle of he plan p; Po p = Ps p a he oule of plan p; Pd p a he oule of he p-compressor saion; and Pr k a he gas demand node k) are given. Relaxing his assumpion would imply solving a much more complex opimizaion problem. [8] Alhough pressure opimizaion is ou of he scope for his model, i will be shown laer ha varying pressure levels wihin normal values does no lead o maor changes in he opimal soluion. (10) The liquid pipeline flow is bounded by a maximum mean velociy (commonly, 1.5 m/s). (11) Cenrifugal pumps have negligible coss compared o processing plans, pipelines and gas compressors. (12) Shale gas processing plans separae NGLs (namely ehane, propane, buane, penanes and naural gasoline) from he shale gas (mehane), also removing H 2 S, CO 2, N 2 and H 2 O; and finally delivering he mehane o consumer markes. All NGLs excep ehane are sold o nearby markes, while ehane is sen o chemical plans by dedicaed pipelines. 11

13 (13) Concave cos funcions of he form f(x) = c x r (wih 0 < r < 1 and c > 0) are assumed for: (a) he cos f a (x a ) of a shale gas processing plan wih a capaciy of x a MMm 3 /day, [22] (b) he cos f b (x b ) of a pipeline of diameer x b, (c) he cos f c (x c ) of a compressor saion of power x c, [23] and (d) he cos f d (x d ) of drilling and hydraulically fracuring x d wells during he same quarer year. (14) Pipeline diameers are reaed as coninuous variables, bu afer he soluion hey are rounded up o he closes commercial diameer. A rigorous model would explicily handle discree size diameers, bu his is ou of scope of his work. 3. MATHEMATICAL FORMULATION The opimizaion problem for he long-erm planning, design and developmen of he shale gas supply chain is formulaed in erms of a mixed-ineger nonlinear programming (MINLP) model described in he following secions. 3.1 Model Consrains The feasible region of he model is deermined by a se of linear consrains. They are grouped ino five blocks: Shale Gas Producion; Flow Balances; Plans, Pipelines and Compressors Sizing; Plan, Pipelines and Compressors Cosing; Waer Supplies; and Maximum Demands Shale Gas Producion Number of Wells Drilled in a Pad. The number of wells drilled, fracured and compleed in he muliwell pad i during he period is represened by he variable N i,. Is value is deermined by eq. (1) in erms of 0-1 variables y i,,n, one of which is equal o one o make N i, = n. The index n sands for an ineger number greaer or equal o zero and lesser or equal o n i, where n i is he maximum number of wells ha can be drilled during a single quarer in pad i. For he examples solved in he resuls secion, he value of n i varies from 2 o 4. Moreover, he oal number of wells ha can be drilled in a pad over he given planning horizon is bounded by eq. (3) o a maximum of N i. The curren rend in shale gas producion is o increase his number as much as possible o reduce he environmenal impac. [4] 12

14 n i N i, = n yi, n, i I, T (1) n= 0 n i n= 0 yi, n, = 1 i I, T (2) T N, N i I (3) i i Shale Gas Producion a Every Well-Pad. As saed in he model assumpions, he oal producion of shale gas (including mehane, ehane and oher NGLs) in a well-pad i a a cerain period depends on he age of every acive well a ha ime. If pw i,a is a model parameer sanding for he produciviy (in Mm 3 of shale gas per day) of a well drilled in pad i, a quarers before he curren ime period, hen he oal daily producion coming from all he wells in pad i can be deermined hrough eq. (4). Noe ha a ime, he age of a well drilled in ime period τ < is a = τ. Moreover, wells of age 0 (being drilled and fracured) do no produce gas unil he following period (see Figure 2). 1 τ = 1 Ni, τ pwi, τ = SPi, i I, > 1 (4) Mehane, Ehane and oher NGLs Produced a Well Pads. Since he shale gas composiion a every well is assumed o be he same (uniform gas weness ), he producion of such fuels wihin he shale gas sream coming from each well pad can be easily deermined from eqs. (5), (6) and (7). SP = gc SP i I, G i, i, > 1 (5) SP E i, = ec SP i I, > 1 i, (6) SP = lc SP i I, L i, i, > 1 (7) where gc is he volume mehane composiion, ec is he ehane composiion, and lc is he remaining hydrocarbons composiion. If hese parameers become dependen on he well locaion, he model 13

15 srucure mus be modified in order o preserve he lineariy of model consrains. This will be discussed in a laer secion Flow Balances Sream Flows from a Well Pad o Juncion Nodes. Shale gas producion a a cerain pad during a ime period is sen o one or more uncion nodes (depending on he nework design), which is conrolled by eq. (8). SPi, = FPi,, i I, > 1 (8) J The model variable FP i,, sands for he daily shale gas flowing from pad i o uncion node during period. By simple exension of eqs. (5), (6) and (7), individual hydrocarbon flows in he shale gas sream (FP G i,,, FP E i,,, FP L i,,) can be easily obained. Flow Balances a Juncion Nodes. Eq. (9) saes ha he sum of incoming shale gas flows a a cerain uncion node equals he sum of ougoing sreams sen o one or more processing plans, depending on he nework design. Under he given assumpions, flow spliing a well pads and uncion nodes are boh allowed. i I FPi,, = GP, J, > 1 (9) p P Similarly o variable FP i,,, individual fuel flows can also be derived from he shale gas sream flowing beween nodes and p (GP,, GP G,, GP E,, GP L,). Flow Balances a Separaion Plans. Assuming ha all he mehane from he shale gas flows processed a plan p is separaed and sen o one or more dry gas demand nodes k, eq. (10) is added o he formulaion. TP k, is he flow of dry gas (mehane) ranspored hrough pipeline p-k during period. J GP G, = TPp, k, p P, > 1 k K (10) 14

16 The same also applies for ehane flows, which are received wih he shale gas, separaed and pumped o one or more perochemical plans l in liquid sae hrough dedicaed pipelines p-l, a a rae of LP l, ons per day, during he enire period. E E sg GP, = LPp, l, p P, > 1 (11) J l L s g E is he specific graviy of ehane in sandard condiions, given in on/mmm 3. Finally, oher NGLs from uncion nodes are processed, and sold o nearby markes a a rae of NP ons per day as saed by eq. (12). L L sg GP, = NP p P, > 1 (12) J Plans, Pipelines and Compressors Sizing Separaion Plans. The oal processing capaciy of a plan p a ime (SepCap ) is given in MMm 3 of shale gas per day, and can be calculaed from is capaciy a he previous period ( 1) plus he capaciy expansion sared a he beginning of period ( τs), i.e. SepIns -τs. In oher words, i is assumed ha separaion plans insallaions/expansions ake τs ime periods, as saed in eq. (13). SepCap = SepCap 1 + SepIns τ s p P, > 1 (13) Upper Bound on he Shale Gas Flows Converging o a Separaion Plan. The sum of he shale gas flows coming from one or several uncion nodes o a single separaion plan during every period, should no exceed is processing capaciy as expressed by eq. (14). J GP, SepCap p P, > 1 (14) Insallaion of Gas Pipelines. As shown in he Appendix, given he gas inle and oule pressures, he fluid properies and he pipeline lengh, maximum gas flows are direcly proporional o he pipeline 15

17 diameer o he power of I is also assumed ha boh raw and dry gases are ideal mixures of ideal gases. In order o preserve lineariy in he consrains, he diameer of he pipeline insalled beween a pair of nodes during a cerain ime period (a model decision) is subsiued by a variable ha sands for such diameer raised o he power of In oher words, he model variables DFP i,,, DGP, and DTP k, sand for he diameers of he pipelines insalled a period beween nodes i-, - and p-k, respecively, raised o he power of In summary, he gas pipeline flows wih regards o pipeline diameers are calculaed from eqs. (15), (16) and (17). 0.5 FPFlowi = k l DFP i I, J, 1 (15),, i, i, i,, > 0.5 GPFlow, = k, p l, p DGP, J, p P, > 1 (16) 0.5 TPFlow = k l DTP p P, k K, 1 (17), k, k, k, k, > Due o assumpion 9, parameers k i,, k,p and k k ake fixed values ha can be calculaed as shown in he Appendix. Disances beween every pair of nodes (l i,, l,p and l k ) are also given daa. Maximum Gas Flow beween a Pair of Nodes. The maximum gas flow beween every pair of nodes depends on he size of he pipelines insalled in previous periods, plus he addiional flow capaciy added due o a recen pipeline consrucion, as saed in eqs. (18), (19) and (20). I is assumed ha pipelines are insalled from period ( τp) o ( 1) and are no able o ranspor gas unil he period, wih τp being he pipeline consrucion lead ime in quarers. FPCapi,, = FPCapi,, 1 + FPFlowi,, τ p i I, J, > 1 (18) GPCap g,, = GPCap g,, 1 + GPFlow, τ p J, p P, > 1 (19) TPCap k, = TPCap k, 1 + TPFlow k, τ p p P, k K, > 1 (20) 16

18 Finally, shale gas and dry gas flows a every ime period are bounded by he flow capaciy connecing every pair of nodes, as enforced by eqs. (21), (22) and (23). FPi FPCap i I, J, 1 (21),, i,, > GP p GPCap p J, p P, 1 (22),,,, > TPp k TPCap p k p P, k K, 1 (23),,,, > Insallaion of Liquid Pipelines. By assumpion 10, a maximum mean velociy is imposed o liquid flows o make sure ha head losses remain a specified values. In liquid pipeline nework design, a ypical value used is v max = 1.5 m/s. Under such an assumpion, liquid flows are direcly proporional o he pipeline secion, and by exension, direcly proporional o he pipeline diameer raised o he power of 2. As for gas pipelines, he diameer of a liquid pipeline insalled beween a gas processing plan p and a perochemical plan l during a cerain ime period (a model decision) is subsiued by an analogous variable, which sands for such diameer o he power of 2 (variable DLP l, ). As a resul, gas pipeline flows (given in ons per day) wih regards o pipeline diameers are calculaed by eq. (24). LPFlow p,, = k l DLPp, l, p P, l L, > 1 (24) where k l = ρ π v l max / 4, ρ is he liquid (ehane) densiy given in on/m 3, and v l max he maximum mean velociy, in m/s. Maximum Flow in Liquid Pipelines. Similarly o gas pipelines, he model decides when o insall a new pipeline and is corresponding size. Eq. (25) deermines he flow capaciy of each liquid pipeline a every ime period (in on/day), while consrain (26) imposes such value as an upper bound on he liquid flow from p o l during period. LPCap l, = LPCap l, 1 + LPFlow l, τ p p P, l L, > 1 (25) 17

19 LPp l LPCap p P, l L, 1 (26),, l, > Power of Compressors. If he sucion and discharge pressures are given (see assumpion 9), and assuming ha compressors are adiabaic, a simple expression can be derived in order o calculae he required compression power (in kw) as shown in he Appendix. Under such assumpions, he required power is direcly proporional o he oal flow of gas being compressed. In he case of raw gas, compressed a uncion nodes and sen o processing plans, he oal power insalled up o ime (JCP, ) mus be greaer or equal han he power demanded by he oal flows of raw gas compressed by a ime, as expressed by eq. (27). JCP kc GP J, 1 (27),, > p P Similarly, he power of compressors insalled a he oule of he processing plan p up o ime (PCP ) sending dry gas o demand nodes k (e.g., gas disribuion companies) is bounded from below by consrain (28). PCPp kc p TPp k p P, 1 (28),,, > k K Moreover, compressor saions can be expanded in he planning horizon by insalling new compressors a he same node. Eqs. (29) and (30) deermine he oal power of compressors insalled up o ime a nodes and respecively, and where τc is he compressor insallaion lead ime in quarers. JCP, = JCP, 1 + JCIns, τ c J, > 1 (29) PCPp, = PCPp, 1 + PCIns τ c p P, > 1 (30) Waer Supplies Waer Demand for Drilling and Fracuring Wells. As explained before, a large amoun of freshwaer is required in he shale gas indusry for he hydraulic fracuring of new wells. This model assumes ha 18

20 he oal amoun of waer required by a single well during he drilling, fracuring and compleion processes (wr i ) is known (ypically, 20 Mm 3 /well) bu may depend on he well locaion. Eq. (31) saes ha he oal number of wells drilled, fracured and compleed in pad i during period deermines he oal waer requiremen of ha pad a ha period, and such amoun should be supplied from one of more freshwaer sources f. The amoun of freshwaer supplied by source f for drilling and fracuring new wells in pad i during period is a key model decision represened by he coninuous variable WS f,i,. In addiion, if he well-pad i has he infrasrucure for flowback waer reamen and reuse, a reuse facor rf i (usually below 20%) can reduce he need for freshwaer, as shown in he LHS of eq. (31). N i, wri / (1 + rf i ) = WS f, i, i I, T (31) f F Waer Availabiliy. Every freshwaer resource (rivers, lakes, underground waer, ec.) usually has an upper limi on he amoun of waer ha i can provide o he shale gas indusry, ofen given by a seasonal profile. If he parameer fwa f, sands for he maximum volume of freshwaer ha source f can supply o he drilling and fracuring of new wells during he whole period, he oal amoun supplied from f o every pad i should be bounded by above as in consrain (32). i I WS f, i, fwa f, f F, T (32) Maximum Demands A criical model decision is where o sell boh he dry gas and he ehane flows produced by he shale gas processing plans. Every poenial marke (or demand node) is assumed o consume a maximum amoun of produc (dry gas for gas disribuors, ehane for perochemical plans) based on heir own ransporaion or processing capaciies. Moreover, such demand profile can be seasonal, especially in dry gas markes. Consrains (33) and (34) resric he oal flow of dry gas and ehane ha can be sen from processing plans o each demand node during every period of he planning horizon. 19

21 p P TPp, k, gasdemk, k K, > 1 (33) p P LPp, l, ehdeml, l L, > 1 (34) 3.2 Obecive Funcion The obecive funcion of he model is o maximize he Ne Presen Value (NPV) of he long-erm planning proec as expressed in eq. (35). NPV = T (1 + dr / 4) [ p P k K i I p P J gasp k, shgc i, nd ks SepIns nd TP FP SepExp k, i,, + p P l L ehp l, nd LP l, + p P lpgp nd NP n i i I n= 1 kd n WellExp y i, n, i I J kp l i, DFP GasPipeExp i,, J p P kp l, p DGP GasPipeExp, p P k K J kp l k kc JCIns DTP CompExp, GasPipeExp k, p P p P l L kc PCIns kp l l CompExp DLP LiqPipeExp l, ( fix f + varf l f, i ) WS f, i, ] f F i I (35) The obecive funcion comprises posiive and negaive erms for every period of he planning horizon, discouned back o is presen value by he annual discoun rae of he proec, dr. Posiive erms are dry gas sales income, ehane sales income, and NGL oher han ehane sales income. Negaive erms are shale gas acquisiion cos (including producion, ransporaion and oher operaing coss), he cos of drilling, hydraulic fracuring and compleion of shale gas wells, he cos of insalling/expanding shale gas processing capaciy a separaion plans, he cos of consrucing new pipelines eiher for 20

22 gahering raw gas or disribuing dry gas and ehane, he cos of insalling new compressor saions a uncion nodes and processing plans, and freshwaer acquisiion and ransporaion coss for drilling and fracuring purposes. I should be noiced ha insead of using linear coss wih fixed charges, nonlinear expressions are used o represen economies of scale funcions in some of he negaive erms of eq. (35), feauring exponens beween 0 and 1. Hence, he obecive funcion can be classified as non-convex, wih sricly concave separable erms. However, all consrains are linear as was shown in he previous secions. 3.3 Cos Esimaion Special aenion mus be paid o he equipmen cosing in he obecive funcion (35). Regarding shale gas separaion plans and gas compressors, ypical values for he exponens SepExp and CompExp vary from 0.60 o [24] However, a paricular case arises in his model for pipeline consrucion. By assumpion 13, he cos of pipelines also follows an economy of scale funcion wih regards o he pipeline diameer, wih a ypical exponen of However, i should be noiced ha pipeline diameers are no direcly considered in he model bu hrough he subsiued variables DFP i,,, DGP,, DTP k, (for gas pipelines) and DLP l, (for liquid pipelines). In fac, such variables accoun for he diameers raised o he power of in he case of gas pipelines, and power 2 in he case of liquid pipelines. Therefore, if 0.60 is considered as exponen for he economy of scale regarding pipeline consrucion, he values of he exponens GasPipeExp and LiqPipeExp in he obecive funcion (35) will be 0.60/2.667 = and 0.60/2 = 0.30, respecively (see Appendix). 3.4 Model Adapaion o Accoun for Shale Gas Composiion Variaions Relaxing he assumpion of uniform shale gas composiion so ha he gas weness is dependen of he well locaion significanly complicaes he naure of he consrains. In order o precisely race he composiion of shale gas flows, he criical poins in he proposed nework supersrucure are he uncion nodes. If i is required o spli flows o more han one separaion plan, he model involves 21

23 bilinear equaions so ha he composiion of all he ougoing flows akes a common value given ha uncions are mixing-spliing nodes as saed by eqs. (36), (37) and (38). G GComp, GP, = GP, p, J, p P, > 1 (36) E EComp, GP, = GP, p, J, p P, > 1 (37) L LComp, GP, = GP, p, J, p P, > 1 (38) Eqs. (36), (37) and (38) include he addiional variables GComp,, EComp,, LComp, (no dependen on he index p) which are he volume composiions of mehane, ehane and LPG (hydrocarbons oher han mehane and ehane) in he shale gas, forcing all he flows deparing from he uncion node (a mixing-spliing node) o have he same composiion. Moreover, individual componen flow balances are incorporaed o he formulaion hrough eqs. (39), (40) and (41). i I FP G i G,, = GP, J, > 1 p P (39) i I FP = GP J, E E i,,, > p P 1 (40) i I FP L i L,, = GP, J, > 1 p P (41) I can be easily seen ha equaions (36), (37) and (38) involve bilinear erms ha add significan difficuly o he MINLP model, especially because he feasible region can no longer be modeled wih linear consrains. However, he nex secion presens a paricular case in which no bilinear erms have o be added when incorporaing shale gas composiion variaions Shale Gas Flows Converging o a Single Processing Plan If he model is inended o selec only one of he given locaions o insall a separaion plan (as i is expeced due o he high cos of his kind of plans), linear expressions hold since no spliing occurs a 22

24 uncion nodes. The endency of he model o selec only one plan locaion is demonsraed in he resuls secion wih Example 1. Under his assumpion he model modificaions necessary o comply wih shale gas composiion variaions are as follows. Firs, we include a new binary variable w p, represening wheher he locaion p is seleced o insall he plan. As a resul, he single plan condiion leads o consrains (42) and (43). SepCap p sepmax w p P, 1 (42), p > p P w 1 (43) p where sepmax is an upper bound on he capaciy of a single gas processing plan. Noe ha alhough he plan locaion mus be unique, i may be insalled and expanded in differen ime periods. In his way, upper bounds on he individual produc flows emerging from every plan are imposed by consrains (44), (45) and (46) in place of eqs. (10), (11) and (12). max { gci} GP, TPp, k, p P, > 1 i I J k K (44) E sg max { eci} GP, LPp, l, p P, > 1 (45) i I J l L L sg max { lci} GP, NPp, p P, > 1 (46) i I J gc i, ec i and lc i are he volume composiions of mehane, ehane and LPG in he shale gas produced a pad i. Noe ha from eq. (14), if he plan is no seleced (zero capaciy) no shale gas flows can be sen o i, and he LHS of he las inequaliies is zero. Finally, individual componen balances are given by eqs. (47), (48) and (49). Eq. (47) for he mehane balance is illusraed hrough he simple example depiced in Figure 4, comprising wo well-pads, one uncion node, he processing plan and he gas demand node. 23

25 i1 Well Pads i2 SP i1 = 0.5 MMm 3 /day gc i1 = 75% CH 4 1 SP i2 = 0.5 MMm 3 /day gc i2 = 85% CH 4 GP 1,p1 = 1.0 MMm 3 /day Plan p1 Demand Node k1 TP p1,k1 = 0.5 x x 0.85 = 0.8 MMm 3 /day Figure 4. Mixing flows a a single processing plan. i I gc i SPi, = TPp, k, (47) p P k K, = (48) E s g eci SPi LPp, l, i I p P l L, = (49) L s g lci SPi NPp, i I p P In summary, he modified MINLP model accouning for shale gas composiion variaions according o he well sie, assuming ha a unique processing plan locaion is o be seleced, seeks o minimize equaion (35) subec o consrains (1)-(4), (8), (9), (13)-(34), (42)-(49). 4. SOLUTION STRATEGIES Solving he MINLP models described in he previous secions is a very challenging ask due o hree main reasons: (1) he size of he model is large, (2) he obecive funcion involves non-concave erms accouning for equipmen coss (processing plans, pipelines and compressors), and (3) such non-linear funcions have unbounded derivaives a zero values. The las wo feaures direcly follow from he economies of scale, usually used o model he equipmen cos variaion wih regards o he equipmen 24

26 size. In principle he MINLP model can be solved o global opimaliy wih a spaial branch and bound search mehod wih he use of convex envelopes for he concave erms in he obecive (he secan). However, given he large size of he MINLP, he problem is inracable wih such mehods like he ones implemened in BARON, LINDOGLOBAL and COUENNE. [25] Therefore, in his secion a ailored sraegy is described for solving he large-scale MINLP problem. 4.1 Plan and Equipmen Cos Esimaions Nonconvex power law expressions of he form f(x) = c x r wih exponens less han one as in Biegler e al. [23] are commonly handled wih wo approaches: (a) approximae he concave funcion by a piecewise linear funcion, [26] and (b) adding a small value ε o he variable x hus slighly displacing he curve so as o ensure non-zero argumen in he funcion. Approximaion (a) is compuaionally cosly, bu can be useful for generaing global upper bounds for he maximizaion problem by solving approximae MILP problems wih piecewise linear approximaions (underesimaions) of he concave equipmen cos funcions. [27][28] On he oher hand, approximaion (b) is mean o avoid unbounded derivaives bu can have drawbacks, especially if he exponens are raher small. [29] To overcome his problem, a simple expression of logarihmic form is used here. In he following secions, boh he piecewise linear approach and he logarihmic approximaion used are briefly presened. 4.2 Piecewise Linear Approximaion of Concave Cos Funcions Given he nonlinear concave cos funcions in eq. (1) (generically referred o as f(x)), accouning for he cos f(x) of a processing plan, pipeline, or compressor of size x X, i is simple o demonsrae ha piecewise linear approximaions like he one depiced in Figure 5 provide valid underesimaions of f(x). Tha is achieved by pariioning he domain of variable x ino inervals (X = [a 0 ; a 1 ] [a 1 ; a 2 ] [a m- 1; a m ] and inroducing binary variables z v o deermine o wha inerval he seleced value of x belongs. A every inerval v, funcion f(x) is approximaed by: φ(x) = f(x v-1 ) + (x x v-1 ) (f(x v ) f(x v-1 )) / (x v x v-1 ). According o Padberg, [30] such a piecewise linearizaion can be modeled hrough wo formulaions: δ and λ. In his case, we adop he δ-formulaion ha leads o: 25

27 26 m v z m v y a a y m v z a a y m v z a a y a a a f a f y a f x y a x v v v v v v v v v v m v v v v v v m v v 1... {0,1} ) ( ) ( ] [ / )] ( ) ( [ ) ( ) ( = = = = + = + = + = = φ (50) Noe ha if z v = 0 (wih v > 1), y v = 0 because of he fourh consrain in (50). From ha, z v+1 is also zero o saisfy he hird consrain in (50), given ha (a v a v-1 ) is always greaer han zero. In oher words, z v = 0 implies z v+1 = 0, and equivalenly, z v+1 = 1 implies z v = 1, for v > 1. To illusrae he meaning of he variables in (50) reconsider he example given in Figure 5. Assume ha x = 6.5. From he consrains described above, i follows ha z 2 = z 3 = z 4 = 1, y 1 = a 1 a 0 = 2 (because z 2 = 1), y 2 = a 2 a 1 = 2 (because z 3 = 1), y 3 = a 3 a 2 = 2 (because z 4 = 1), and y 4 = 0.5. Figure 5. Concave cos funcion and piecewise linear underesimaion.

28 By replacing he nonlinear erms in he obecive funcion (35) wih he piecewise linear approximaions given in (50), he MINLP model reduces ino an MILP model yielding valid upper bounds for he global opimum of he original problem. A key decision is how o divide he variable domain, i.e. how many inervals o consider. The finer he domain discreizaion, he closer is he upper bound o he acual obecive value of he MINLP, bu also he higher is he CPU ime required by he MILP since he number of ineger variables increases significanly. In his case, such a radeoff is managed hrough he successive refining sraegy presened in Secion 4.4 based on he ideas of You and Grossmann [31][32] dealing wih he nonlinear concave funcion x. 4.3 Logarihmic Approximaion of Concave Cos Funcions In order o avoid unbounded derivaives and esimaion errors when solving NLP subproblems in he MINLP model, he alernae approximaion funcion g(x) for f(x) by Cafaro and Grossmann [29] is used: r f ( x) = c x g( x) = k ln( bx + 1) (51) where x is he size of he equipmen, f(x) is he acual cos of he equipmen of size x, g(x) is he esimaed cos of he equipmen of size x, and k, b > 0 are parameers seleced o fi f(x) as closely as possible. Furher deails of his approximaion are given in Cafaro and Grossmann. [29] The proposed funcion has wo main advanages wih regards o he classic ε-approximaion f ( x) h( x) c ( x ε ) r = + : (1) he cos of x = 0 is exacly zero: g(0) = k ln(b 0 +1) = k ln(1) = 0, and (2) he derivaives of g(x) for all x 0 are bounded posiive values given by g'(x) = b k / (b x +1). In paricular a he origin (x = 0), g'(x) = b k. These properies are paricularly useful when dealing wih concave cos funcions wih small exponens, like he cos of pipelines (exponens and 0.300, for gas and liquid pipelines, respecively). Appropriae values for parameers k and b in funcion g(x) can be found relaively easily for liquid and gas pipelines, and he logarihmic approximaion leads o very good resuls (less han 0.50 % error) in he calculaion of pipeline coss in all of he case sudies ackled in Secion 5. 27

29 4.4 Soluion Algorihm: Branch-Refine-Opimize (BRO) Sraegy To find he global opimum of he nonconvex MINLP model presened in Secion 3, a wo-level branch-and-refine procedure is proposed (see Figure 6). In he upper level we successively solve MILP approximaions of he original MINLP problem following wo purposes: (1) provide valid (and increasingly igher) upper bounds of he global opimum, and (2) propose efficien supply chain nework configuraions. Once he MILP approximaion is solved, he corresponding supply chain nework design is fixed by removing all he nodes and arcs of he original supersrucure no acive in he MILP soluion so as o define he lower level opimizing procedure. 28

30 Iniializaion Se iniial piecewise linear pariion (P 1,1 ) for he plans, pipelines and compressors size domains. Se p = 1, GUB = +, GLB = - MILP: Global Piecewise Linear Approximaion Solve he MILP approximaion of he original MINLP problem using he incumben piecewise linear esimaion o deermine a global upper bound. Ineger Soluion (x p, z p ) z p < GLB? NO GUB = min{gub, z p } YES Sop Fixing Nework Configuraion Remove all he nodes and arcs no acive in he soluion x p from he problem supersrucure. Se q = 1, x q = x p, RUB = GUB, RLB = - Refine plans, pipelines and compressors size domain pariion (P p-1,1 P 1 ). Add an ineger cu o previous MILP o avoid configuraions already ried. MINLP: Reduced Problem Opimizaion Solve he MINLP model wih a non-global solver o deermine a lower bound for seleced nework, saring from x q using he logarihmic approximaion of he concave cos erms. Ineger Soluion (ẋ q, ż q ) RLB = max{rlb, ż q } (RUB-RLB)/RLB < ε 1? YES GLB = max{glb, RLB} NO Ineger Soluion (x q, z q ) RUB = min{rub, z q } MILP: Reduced Problem Piecewise Linear Approximaion Deermine an upper bound for he seleced nework. Refine plans, pipelines and compressors size domain pariion (P q-1 P q ) based on ẋ q q = q + 1 Inner Loop Ouer Loop p = p + 1 NO (GUB-GLB)/GLB < ε 2? Sop Figure 6. Branch-Refine-Opimize (BRO) algorihm. The aim of he lower level of he algorihm is o find he global opimal soluion of a reduced MINLP problem (or subproblem) focused only on equipmen sizing (plan, pipelines and compressors) and he 29

31 drilling sraegy (ineger variables n i, ) as he nework srucure is fixed. Since he reduced MINLP problem is nonconvex, is global opimal soluion is found by solving on he one hand he reduced MINLP wih a non-global solver (DICOPT, SBB) [25] o deermine a lower bound for he seleced nework, and hen successively pariioning he equipmen size domains and recursively solving piecewise linear approximaions of he obecive funcion o deermine igher upper bounds (inner loop). Finally, he global opimal soluion of he reduced MINLP is a feasible soluion of he original MINLP, and is obecive value provides a valid global lower bound of he problem. Noe ha supply chain nework designs proposed by he upper level a previous ieraions are excluded wih ineger cus in order o reduce he enumeraion effor. Such ineger cus are similar o hose proposed by Durán and Grossmann, [8] which eliminae paricular binary combinaions accouning for nework configuraions already analyzed. The cus are derived from he values of he binary variables z v used by he piecewise linear approximaion of he concave cos erms in he obecive funcion of he MILP (see Secion 4.2). As a resul, if he approximae soluion obained by he upper level in a new ieraion is worse han he bes soluion found (or global lower bound), he algorihm auomaically sops. Oherwise, he ouer loop refines he piecewise linear approximaion of he original problem and migh improve he nework srucure so ha he global opimal soluion can be obained afer a finie number of ieraions. In summary, he proposed soluion algorihm is as follows: Sep 1: Iniializaion. A one-piece linear underesimaion (secan) is used for all he concave cos erms of laer periods (for insance, > 10), while in earlier periods he saring piecewise linearizaion comprises wo o four inervals. The global upper bound is se o GUB = +, and he global lower bound GLB = -. Sep 2. Global Piecewise Linear Approximaion. Solving he incumben MILP approximaion of he original MINLP problem (as shown in Secion 4.2) provides a global upper bound GUB. Since all he consrains in he MINLP are linear, he opimal soluion of he MILP is also a feasible soluion of he MINLP problem. Thus, a global lower bound (GLB) can be direcly obained by subsiuing he 30