paper design and optimizaton of steam distribution systems for steam power plants

Published: May 06, 2011

r 2011 American Chemical Society 8097 dx.doi.org/10.1021/ie102059n | Ind. Eng. Chem. Res. 2011, 50, 8097–8109

ARTICLE

pubs.acs.org/IECR

Design and Optimization of Steam Distribution Systemsfor Steam Power PlantsCheng-Liang Chen* and Chih-Yao Lin

Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, Republic of China

ABSTRACT: This paper presents a systematic methodology for the design of a steam distribution network (SDN) which satisfiesthe energy demands of industrial processes. A superstructure is proposed to include all potential configurations of steam systems,and a mixed-integer nonlinear programming (MINLP) model is formulated accordingly to minimize the total annualized cost. Theproposedmodel determines simultaneously (i) the structure and operational configuration of a steam system and (ii) the interactionbetween the steam system and the heat recovery system. A series of case studies are presented to demonstrate the feasibility andbenefit of the proposed approach.

1. INTRODUCTION

Steam power plants are the main energy supplier for runningchemical processing. Typically, a steam power plant consists ofvarious units including boilers, gas turbines, steam turbines,electric motors, steam headers, etc. In the plant, steam isconverted into two types of energy, specifically, electricity andmechanical power. Electricity demands are from the powerrequired to function process devices.Mechanical power demandsare from the requirement to drive process units. Steam demandsare from heat duties for the heat exchange network or heatsources for the reaction process.

The design of a steam power plant is a large and complexproblem, where the layout of all types of units and the operatingconditions must be optimized for efficient operation. The steamdistribution network (SDN) is an essential element in devisingthe energy management system of a steam power plant. A largevolume of related studies have already been published in theliterature. Basically, two distinct approaches were adopted inthese works: (a) the heuristics-based thermodynamic designmethod1,2 and (b) the model-based optimization method.3�5

The former networks were synthesized with thermodynamictargets for getting the maximum allowable overall thermalefficiency, while the latter were designed with mixed-integerlinear/nonlinear programs for attaining the minimum totalannualized cost (TAC).

The above-mentioned works were developed to address thedesign of an SDN assuming that all units operate at full load tosatisfy a single set of demands and conditions. However, in manyexisting chemical processes the common operational feature isvarying demands. This may be due to changing feed/productspecifications or changes of heat loss with seasonal variation inthe continuous operation plants, or changes in operations forbatch plants. For example, energy demands in peak season arehigher than in off peak season or steam power plants need moreheat demands in winter since the heat loss is higher.

Because of the limitations of these types of studies, capablemethodologies for the period-varying demands were developed.6�9

However, the research was only addressing operational problemsfor existing plants or design problems without simultaneously

optimizing unit sizes and loads as continuous functions. Morerecently, Aguilar et al.10,11 proposed a mixed-integer linearprogramming (MILP) model to address retrofit and operationalproblems for utility plants, considering structural and operationalparameters as variables to be optimized. The linear model wasrealized when some operating conditions of units (e.g., air flowrate or operating temperature of gas turbines) were prespecifiedor some of the entering streams (e.g., from boilers and a heatrecovery steam generator, HRSG) were already at the tempera-ture of the header (predetermined).

From a review of the current literature, there is a need todevelop a more comprehensive design method for SDNs. In thispaper, the main objective of the study is to develop a flexiblemodel for industrial problems. This model can address themultiperiod operating problem and can easily set up the linkbetween steam systems and heat recovery networks.

To illustrate the SDN design method developed in this work,the rest of this paper is organized as follows. The design problemis formally defined in section 2. The design concept developed byPapoulias and Grossmann3 is adopted and modified in thepresent study for a generalized SDN. The superstructure andcorresponding mixed-integer nonlinear programming (MINLP)model are described in sections 3 and 4, in which the perfor-mance model proposed by Aguilar et al.10 is utilized for the unitdesign while equipment is operating at different loads. Two caseson synthesis and design of the network are then presented insection 5 to demonstrate the feasibility and effectiveness of theproposed simultaneous optimization strategy. The discussionand the conclusion of the present studies are provided in sections6 and 7.

2. PROBLEM STATEMENT

The design problem addressed in this paper is stated asfollows: Given are a set of steam demands or a set of hot/cold

Received: October 10, 2010Accepted: May 6, 2011Revised: April 21, 2011

8098 dx.doi.org/10.1021/ie102059n |Ind. Eng. Chem. Res. 2011, 50, 8097–8109

Industrial & Engineering Chemistry Research ARTICLE

process streams to be cooled/heated in every period. Given alsoare the electricity demands and mechanical power needs ofchemical processing in every period. The objective is to developa systematic approach to manage the energy usage in anefficient way.

This work includes the following: (1) the multiperiod opera-tion problem with given steam demands and (2) the totalprocessing system design with given process stream data. Theformer is to synthesize a cost-optimal steam system that can fulfillenergy requirements. The latter is to design a steam distributionnetwork and heat recovery network (SDN�HEN) simulta-neously with a minimum TAC.

The given model parameters of this optimization probleminclude the following: (1) the design specifications of every boilerunit (i.e., its operating pressure, maximum operating tempera-ture, and the lower and upper bounds of steam flow rate), (2) thedesign specifications of every gas turbine unit (i.e., its operatingtemperature and its minimum and maximum heat loads), (3) thedesign specifications of every steam turbine unit (i.e., its lowerand upper bounds of steam flow rate), (4) the temperature levelsof cooling water, and (5) the design specifications of everyexchanger unit for the HEN design.

The resulting design includes the following: (1) the number ofboiler units and their throughputs in every period, (2) thenumber of gas turbine units and their throughputs in everyperiod, (3) the number of steam turbine units and theirthroughputs in every period, (4) the consumption rates offreshwater and the cooling water usage in every period, (5) theconsumption rates of fuel and rates of electricity import/export,(6) the steam header pressures/temperatures in every period,(7) the complete network configuration and the flow rate of anSDN, and (8) the complete network configuration and the flowrate of an HEN for the total processing problem.

3. SUPERSTRUCTURE

A superstructure of SDN is constructed to incorporate allpossible flow connections, as presented in Figure 1. More

detailed superstructures of all units concerned in the SDN areshown in Figure 2. Steam can be generated with either fired orheat recovery steam generator (HRSG) boilers which operate atconditions consistent with those of the steam headers. The firedboilers generate steam by providing heat from combustion of afuel or a fuel mixture. The HRSG can further utilize the exhaustsfrom gas turbines to heat water to generate steam.

Steam is collected and distributed to steam consumer units bythe steam header. Note that there are several steam headers in thegeneralized steam system. For the given pressure steam demands,

Figure 1. Steam distribution network superstructure.

Figure 2. Superstructures of SDN units: (a) boiler, (b) HRSG, (c) gasturbine, (d) single-stage steam turbine, (e) multistage steam turbine,(f) deaerator, and (g) steam header.



pressure levels are selected at saturated steam pressure corre-sponding to the minimum temperature requirements. Specifi-cally, values not less than their saturated temperatures aresuitable for choices of steam headers. The operating tempera-tures of headers are treated as decision variables to be optimized.For the simultaneous design of SDN and HEN, the level and theamount of steam demands are not given. Both pressures andtemperatures of headers are variables to be determined by theoptimization approach.

There are three types of power-generating devices in thesuperstructure for satisfying power demands, i.e., gas turbines,steam turbines, and electric motors. Turbines can generateelectricity or mechanical power. A gas turbine includes a com-pressor, a combustor, and a turbine, where the hot air exiting theturbine can further be used in HRSG to enhance the steamproduction. Both back-pressure steam turbines and condensingturbines are considered. For the back-pressure turbine, thepressure of exhaust steam is higher than atmospheric pressure.If the condensing turbine is chosen in the design, then acondenser has to be selected for its operation. Electric motorsare also used to meet the required shaft power demands.

A deaerator is installed to remove dissolved gases to providefeed water to the boilers to meet the process water demand.Demineralized water is added to compensate for plant losses.Pumps are included for the supply of boiler feed water andcooling water and for the return of condensate.

The stagewise superstructure12 is adopted for the design ofHEN, as shown in Figure 3. Nonisothermal mixing of streamsis considered for the flexible design. Each utility can be treatedas a process stream with unknown loads. By this superstruc-ture one can determine the loads of utilities for the givingprocess streams.

4. MODEL FORMULATION

Having introduced the superstructures of SDN and HEN, onecan then formulate the synthesis problem as a MINLP. Thematerial and energy balance equations associated with every unitare included as the constraints of the optimization problem. Thecorresponding equipment models adopted for SDN are takenfrom Aguilar et al.10

4.1. Steam Distribution Network. 4.1.1. Boilers. Equation 1states the mass balance of a boiler, where the feed boiler water

entering the boiler b in a certain period p (fbpbfw) equals the

steam to the steam header i (fbip) and the effluent of blowdownwater (fbip

bd ). Therein the blowdown is treated as a fixed jfraction of the boiler steam output and extracted at saturatedliquid conditions (see Figure 2a). An energy balance is neededto ensure that the enthalpy entering the boiler equals thatleaving (i.e., eq 2), where qbp denotes the heat absorbed by thewater stream.

f bfwbp ¼Xi ∈ I

fbip þXi ∈ I

f bdbip "b ∈ B , p ∈ P ð1Þ

f bfwbp Hdeaer þ qbp ¼Xi ∈ I

fbiphbip þXi ∈ I

f bdbipHsat, li "b ∈ B , p ∈ P

ð2Þ

f bdbip ¼ jfbip "b ∈ B , p ∈ P ð3ÞIn addition, depending on the capacity of units, the constraints

are imposed in the boiler formulation, i.e.

Ωbzbip e fbip e Ωbzbip "b ∈ B , i ∈ I , p ∈ P ð4Þ

where zbip is a binary variable and Ωhb and Ωhb are the minimum

and the maximum capacities, respectively. Notice that the steamflow rate is zero when the boiler unit is not selected. It should befurther considered that only one connection exists betweenboiler b and steam header i with the boiler b selected. Thus,the following constraints are imposed in this model.

zb ¼Xi ∈ I

zbi " b ∈ B ð5Þ

zbi g zbip "b ∈ B , i ∈ I , p ∈ P ð6Þ

zbi eXp ∈ P

zbip "b ∈ B , i ∈ I ð7Þ

There are two types of boilers used in this work, which aremultifuel boilers (b ∈ MB) and HRSGs (b ∈ HB). Theequipment models of boilers are adopted from Aguilar et al.10

4.1.2. Gas Turbines. For convenience, a set of gas turbinesg ∈ G is defined. Figure 2c shows the superstructure of a gasturbine. Since exhaust of gas turbine g is sent to the HRSG unit b

Figure 3. Heat exchanger network superstructure.



as heat input, the corresponding relations are derived below.

fgp ¼X

b ∈ HB

fgbp "g ∈ G , p ∈ P ð8Þ

Tgp ¼X

b ∈ HB

Tgbp "g ∈ G , p ∈ P ð9Þ

A gas turbine can generate electricity (g ∈ GE) or shaft power(g ∈ GS ). If the latter operation is chosen, then the producedshaft power wgp is selected to meet the shaft demand j.

wgp ¼Xj ∈ J

wgjp "g ∈ GS , p ∈ P ð10Þ

The following constraints are considered to reflect physical andpractical limitations of the equipment. Equations 11 and 12 definethe unit capacity thatmust complywith themaximumandminimumlimits. Equation 13 limits the operation range of a temperature.

Ωgzgbp e fgbp e Ωgzgbp "g ∈ G , b ∈ HB , p ∈ P ð11Þ

Γgzgjp e wgjp e Γgzgjp "g ∈ G , j ∈ J , p ∈ P ð12Þ

Φgzgbp e Tgbp e Φgzgbp "g ∈ G , b ∈ HB , p ∈ P ð13ÞSimilar to the relation between a boiler and a steam header,

eqs 14�18 describe the connection and operation between a gasturbine and a HRSG. If a gas turbine is chosen, a correspondingHRSG is automatically included, and vice versa.

zg ¼X

b ∈ HB

zgb " g ∈ G ð14Þ

zb ¼Xg ∈ G

zgb " b ∈ HB ð15Þ

zgb g zgbp "g ∈ G , b ∈ HB , p ∈ P ð16Þ

zgb eXp ∈ P

zgbp "g ∈ G , b ∈ HB ð17Þ

zgp ¼X

b ∈ HB

zgbp "g ∈ G , p ∈ P ð18Þ

Equations 19�22 are constraints to select gas turbine g for shaftdemand j. Notice a gas turbine is for one shaft demand.

zg ¼Xj ∈ J

zgj "g ∈ GS ð19Þ

zgj g zgjp "g ∈ GS , j ∈ J , p ∈ P ð20Þ

zgj eXp ∈ P

zgjp "g ∈ GS , j ∈ J ð21Þ

zgp ¼Xj ∈ J

zgjp "g ∈ GS , p ∈ P ð22Þ

4.1.3. Steam Turbines. Figure 2d,e shows a steam turbineoperating between higher pressure and lower pressure steam

headers. This device converts the energy of steam intopower. Equations 23 and 24 describe the constraints toensure that the operation does not exceed its design capa-cities. Moreover, it is possible to define a minimum partialload of units so that the actual output is not less than a givenspecified fraction.

Ω ii0tzii0tp e fii0tp e Ωii0tzii0tp "i, i0 ∈ I , i < i0,

t ∈ T , p ∈ P ð23Þ

Γ ii0tzii0tp e wii0tp e Γii0tzii0tp "i, i0 ∈ I , i < i0,

t ∈ T , p ∈ P ð24ÞThe steam turbine can generate electricity (t ∈ TE) or shaft

power (t∈TS ). If the latter operation is chosen, the shaft powerproduced by the turbine t is to meet the shaft demand j.

wii0tp ¼Xj ∈ J

wii0tjp "i, i0 ∈ I , i < i0, t ∈ TS , p ∈ P

ð25Þ

Γii0tzii0tjp e wii0tjp e Γii0tzii0tjp "i, i0 ∈ I , i < i0,

t ∈ TS , j ∈ J , p ∈ P ð26ÞAlso, operation for multiperiod problems should be consid-ered, which is like the boiler and gas turbine units discussedpreviously. Equations 27and 28 are the constraints of steamturbines.

zii0t g zii0tp "i, i0 ∈ I , i < i0, t ∈ T , p ∈ P ð27Þ

zii0t eXp ∈ P

zii0tp "i, i0 ∈ I , i < i0, t ∈ T ð28Þ

Similar to a gas turbine, constraints for the pair between steamturbines and shaft demands are presented in eqs 29�31.

zii0t ¼Xj ∈ J

zii0tj "i, i0 ∈ I , i < i0, t ∈ T ð29Þ

zii0tj g zii0tjp "i, i0 ∈ I , i < i0, t ∈ T , j ∈ J , p ∈ P

ð30Þ

zii0tj eXp ∈ P

zii0tjp "i, i0 ∈ I , i < i0, t ∈ T , j ∈ J ð31Þ

Single-stage steam turbines (t ∈ ST ) are adopted in this work.This steam turbine features only one inlet and a single outletthrough which the steam is discharged to a lower pressure level.Equation 32 is a logic constraint for single-stage steam turbines.

zt ¼X

i, i0 ∈ Ii < i0

zii0t " t ∈ ST ð32Þ

Multistage steam turbines (t∈MT ) can feature several inletsto take in or outlets to discharge steam at different pressure levels.The single-inlet steam turbines with several extractions areadopted in this work. Equations 33 and 34 are logic constraintsfor the multistage steam turbine, which means one steam inlet is



restricted to this unit.

zt g zii0t "i, i0 ∈ I , i < i0, t ∈ MT ð33ÞXi ∈ Ii < i0

zii0t e 1 "i0 ∈ I , t ∈ MT ð34Þ

The decomposition of a multistage steam turbine is shown inFigure 4, and the corresponding superstructure is shown inFigure 2e. A multistage steam turbine is decomposed into severalsingle-stage turbines which are connected in series so that theoriginal single-stage equipment performance model can be usedto determine its properties without newmodel. Equations 35 and36 describe themass balance of amultistage steam turbine, wherethe higher quality steam (f0ii0tp) flows through a multistageturbine t and then lower quality steam is delivered to a headeri0 (fii0tp) and to the next stage i00 (fii0 i0 0tp) as its steam input (f0ii0 0tp).

f0ii0tp ¼ fii0tp þ

Xi00 ∈ Ii < i0 < i00

fii0i00tp "i, i0 ∈ I , i < i0, t ∈ MT , p ∈ P

ð35Þ

f0ii00tp ¼

Xi0 ∈ I

i < i0 < i00

fii0i00tp "i, i00 ∈ I , i < i00, t ∈ MT , p ∈ P

ð36ÞThe following constraints are considered for this multistage

unit. Equations 37�40 are used to ensure that the stage i00 onlycan accept one steam stream from preceding stages and stage i0only can deliver steam to one next stage.

zii0tp g zii0 i00tp "i, i0, i00 ∈ I , i < i0 < i00, t ∈ MT , p ∈ P ð37ÞXi0 ∈ Ii < i0 < i00

zii0i00tp e 1 "i, i00 ∈ I , i < i00, t ∈ MT , p ∈ P

ð38ÞXi00 ∈ Ii < i0 < i00

zii0 i00tp e 1 "i, i0 ∈ I , i < i0, t ∈ MT , p ∈ P

ð39Þ

Ωzii0 i00tp e fii0i00tp e Ωzii0 i00tp "i, i0, i00 ∈ I , i < i0 < i00,

t ∈ MT , p ∈ P ð40Þ

4.1.4. Deaerator. Figure 2f shows a schematic representa-tion for a deaerator device. In this unit the inlet streamsmay come from low pressure steam, condensate return fromprocess, or treated water makeup. After water is treated andits dissolved gas is removed, the feed water is sent to the boileror to the let-down station. Equation 41 describes the massflow rate balance. Equation 42 is an energy balance toguarantee that enough steam is injected into the deaeratorso that the feed water leaving this unit is at saturated liquidconditions.

f wp þXi ∈ I

fip þ f cp ¼Xb ∈ B

f bfwbp þXi ∈ I

f ldip "p ∈ P ð41Þ

f wp Hwp þ

Xi ∈ I

fiphip þ f cp hcp

¼Xb ∈ B

f bfwbp þXi ∈ I

f ldip

!Hdeaer "p ∈ P ð42Þ

4.1.5. SteamHeaders. Figure 2g shows the stream balance for asteam header. The mass balance is given by eq 43. For the topsteam header, the highest pressure steam header, there are noinput streams from steam turbines or let-down stations, and thereis no output vented steam to the environment. The bottom steamheader is a condensing header, in which its flow rates of inputstreams from let-down stations or back-pressure steam tur-bines are zero and its output condensate water is sent to adeaerator. Equation 44 is an energy balance for a header toensure that the total amount of enthalpy entering the headerequals that leaving. It should be noted that a variable forenthalpy hip (hip = fn(Ti,Pi)) is employed because of theflexible consideration.

Xb ∈ B

fbip þXi0 ∈ I

i0 < i

Xt ∈ T

fi0 itp þXi0 ∈ I

i0 < i

fi0 ip þ f ldip þ f psip

¼Xi0 ∈ Ii0 > i

Xt ∈ T

fii0tp þXi0 ∈ Ii0 > i

fii0p þ fip þ f ventip þ f pdip

"i ∈ I , p ∈ P ð43ÞXb ∈ B

fbiphbip þXi0 ∈ I

i0 < i

Xt ∈ T

fi0 itphi0itp þXi0 ∈ I

i0 < i

fi0 iphi0p þ f ldip Hdeaer þ f psip H

psip

¼Xi0 ∈ I

i0 > i

Xt ∈ T

fii0tp þXi0 ∈ I

i0 > i

fii0p þ fip þ f ventip þ f pdip

0BBBBBBBB@

1CCCCCCCCAhip "i ∈ I , p ∈ P

ð44Þ

4.1.6. Power Balances. Equation 45 ensures that the actualpower delivered by all the drives attached to the common shaftmeets the corresponding demands in each operating period. Agas turbine g, a steam turbine t, and an electric motor m can be

Figure 4. Decomposition of the multistage steam turbine.



used to meet the required power demands.Xg ∈ GS

wgjp þX

i, i0 ∈ I

i0 < i

Xt ∈ TS

wii0tjp þX

m ∈ M

wmjp

¼ wdem, sjp "j ∈ J , p ∈ P ð45Þ

In this study, the electricity produced by the steam and/or gasturbine can be used to meet the needs of chemical process. Theoverall balance equation can be written accordingly by eq 46. Theleft-hand side of this expression accounts for the supply ofelectricity, while the terms of the right-hand side correspond tothe potential consumers.

Xg ∈ G E

wgp þX

i, i0 ∈ I

i < i0

Xt ∈ TE

wii0tp þ wimp, ep

¼ wdem, ep þ

Xm ∈ M

Xj ∈ J

wmjp

ηmþ wexp, e

p "p ∈ P ð46Þ

4.2. Heat Exchanger Network. 4.2.1. Overall Heat Balancefor Each Stream. An overall heat balance is included to ensureheat exchange for all process streams. The constraints specify thatthe overall heat of each hot process stream is removed with coldprocess streams or cold utilities. Similar constraints also apply forall cold streams, as stated in eqs 47 and 48:

ðT inhp � Tout

hp ÞFhp ¼Xk ∈ K

Xc ∈ C

qhckp þ qcuhp "h ∈ H , p ∈ P

ð47Þ

ðToutcp � T in

cpÞFcp ¼Xk ∈ K

Xh ∈ H

qhckp þ qhucp "c ∈ C , p ∈ P

ð48Þ

4.2.2. Heat Balance at Each Stream. Heat balances are alsoneeded in each stage for each stream, as shown in eqs 49 and 50.Note that the index k is used to represent the stage and thetemperature location in the superstructure. Stage location k = 1involves the highest temperatures. qhckp denotes the heat ex-change between hot process stream h and cold process streamc in stage k.

ðthkp � th, kþ1, pÞFhp ¼Xc ∈ C

qhckp "h ∈ H , k ∈ K , p ∈ P

ð49Þ

ðtckp � tc, kþ1, pÞFcp ¼Xh ∈ H

qhckp "c ∈ C , k ∈ K , p ∈ P

ð50Þ

4.2.3. Heat Balance for Each Unit. For each local exchangeunit, heat balances are needed, where fhckp and fhckp are split heat

capacity flow rates.

ðthkp � thc, kþ1, pÞfhckp ¼ qhckp "h ∈ H , c ∈ C , k ∈ K , p ∈ P

ð51Þ

ðtchkp � tc, kþ1, pÞfchkp ¼ qhckp "h ∈ H , c ∈ C , k ∈ K , p ∈ P

ð52ÞThe total flow balances for theses split heat capacity flow rates

in each stage k can be stated as follows.Xc ∈ C

fhckp ¼ Fhp "h ∈ H , k ∈ K , p ∈ P ð53Þ

Xh ∈ H

fchkp ¼ Fcp "c ∈ C , k ∈ K , p ∈ P ð54Þ

4.2.4. Assignment of Superstructure Inlet Temperatures. Thegiven inlet/outlet temperatures of hot and cold processes areassigned as the inlet/outlet temperatures to the superstructure. Forhot process streams, the inlet corresponds to the location k = 1,while for cold streams the inlet corresponds to location k = Kþ 1.

T inhp ¼ th, 1, p "h ∈ H , p ∈ P ð55Þ

T incp ¼ tc,Kþ1, p "c ∈ C , p ∈ P ð56Þ

4.2.5. Feasibility of Temperatures. The following constraints(eqs 57�60) are included to guarantee monotonic decrease of alltemperatures at successive stages.

thkp g th, kþ1, p "h ∈ H , k ∈ K , p ∈ P ð57Þ

tckp g tc, kþ1, p "c ∈ C , k ∈ K , p ∈ P ð58Þ

Touthp e th,Kþ1, p "h ∈ H , p ∈ P ð59Þ

Toutcp g tc, 1, p "c ∈ C , p ∈ P ð60Þ

4.2.6. Hot and Cold Utility Loads. Equations 61 and 62 areposed to calculate hot or cold utility loads needed for eachprocess stream.

ðth,Kþ1, p � Touthp ÞFhp ¼ qcuhp "h ∈ H , p ∈ P ð61Þ

ðToutcp � tc, 1, pÞFcp ¼ qhucp "c ∈ C , p ∈ P ð62Þ

4.2.7. Logic Constraints. Logic constraints and binary variables areneeded to determine the existence of streammatch (h, k) in stage k.zhckp, zhp

cu, and zcphu are binary variables for process streammatches, for

cold utility matches, and for hot utility matches, respectively.

qhckp �Ωzhckp e 0 "h ∈ H , c ∈ C , k ∈ K , p ∈ P

ð63Þ

qcuhp �Ωzcuhp e 0 "h ∈ H , p ∈ P ð64Þ



qhucp �Ωzhucp e 0 "c ∈ C , p ∈ P ð65Þ

4.2.8. Calculation of Approach Temperatures. For determin-ing the area requirement of the heat exchanger, approachtemperatures are used to calculate the log mean temperaturedifference (LMTD) approximated by using the Chen equation.13

The area requirement of each match will be incorporated in theobjective function. The constraints 66�69 are expressed asinequalities since the cost of the exchanger decreases with highervalues for the approach temperatures.

dthckp e thkp � tchkp þ Γð1� zhckpÞ "h ∈ H , c ∈ C ,

k ∈ K , p ∈ P ð66Þ

dthc, kþ1, p e thc, kþ1, p � tc, kþ1, p þ Γð1� zhckpÞ "h ∈ H ,

c ∈ C , k ∈ K , p ∈ P ð67Þ

dtcuhp e th,Kþ1, p � Tout, cu þ Γð1� zcuhpÞ "h ∈ H , p ∈ P

ð68Þ

dthucp e tout, hucp � tc, 1, p þ Γð1� zhucp Þ "c ∈ C , p ∈ P ð69Þ

4.3. Objective Function and MINLP Formulation. Theobjective function in the synthesis model is the TAC, whichincludes the sum of operating and the annualized capital costs.The former consists of the costs of fuels, cooling water, freshwater, and purchased electricity. The latter includes the fixedand variable costs of all units. There are two objectivesconsidered in this work, as shown in eqs 70 and 71. The firstobjective is the design of SDN for the given steam demands.The second objective is the simultaneous design of SDN andHEN for the given process streams, where their interactioncan be optimized. It should be mentioned that the hot utilityrequirement of HEN is satisfied with steam from the steamsystem.

minx1 ∈ Ω1

J1 ¼Xp ∈ P

ðCwp f

wp þ Ccw

p f cwp þ Cimp, ep wimp, e

p � Cexp, ep wexp, e

p

þXb ∈ B

Xu ∈ U

Cufbup þXg ∈ G

Xu ∈ U

CufgupÞthrsp

þXb ∈ B

ðzbCfixb þ Cvar

b Gγbb Þ þ

Xg ∈ G

ðzgCfixg þ Cvar

g Gγgg Þ

þXt ∈ T

ðztCfixt þ Cvar

t Gγtt Þ

þX

m ∈ M

ðzmCfixm þ Cvar

m Gγmm Þ

þXd ∈ D

ðzdCfixd þ Cvar

d Gγdd Þ ð70Þ

where x1 is a vector of variables, and Ω1 is a feasible searchingspace delimited by the constraints.

x1 �

f bfwbp ; fbip; f bdbip; fmaxb ; fbup; fgp; fgbp; fii0tp; f

0ii0tp

fii0 i00tp; f0ii00tp; f

wp ; fip; f

cp ; f

ldip ; fii0p; f

psip ; f

ventip ; f pdip ; f

maxd

hbip; hii0tp; hip; hcp;Tgp;Tgbp; qbp; qbup; qgp;wgp;wgjp;wii0tp

wii0tjp;wmaxg ;wmax

t ;wmjp;wmaxm ;wimp;e

p ;wexp;ep ; zb; zbp; zbi; zbip

zd; zg ; zgp; zgb; zgbp; zgj; zgjp; zm; zt; zii0t; zii0tp; zii0tj; zii0tjp; zii0 i00tp"b ∈ B ; d ∈ D ; g ∈ G ; i; i0; i00 ∈ I

j ∈ J ;m ∈ M ; p ∈ P ; t ∈ T ; u ∈ U

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

Ω1 ¼ fx1jeqs 1�46g

minx2 ∈ Ω2

J2 ¼Xp ∈ P

ðCwp f

wp þ Ccw

p f cwp þ Cimp, ep wimp, e

p � Cexp, ep wexp, e

p

þXb ∈ B

Xu ∈ U

Cufbup þXg ∈ G

Xu ∈ U

Cufgup þ qcuhpÞthrsp

þXb ∈ B

ðzbCfixb þ Cvar

b Gγbb Þ þ

Xg ∈ G

ðzgCfixg þ Cvar

g Gγgg Þ

þXt ∈ T

ðztCfixt þ Cvar

t Gγtt Þ þ

Xm ∈ M

ðzmCfixm þ Cvar

m Gγmm Þ

þXd ∈ D

ðzdCfixd þ Cvar

d Gγdd Þ

þXh ∈ H

Xc ∈ C

Xk ∈ K

ðzhckCfixhck þ Cvar

hckGγhckhck Þ

þXh ∈ H

ðzcuh Cfixh þ Cvar

h Gγhh Þ

þXc ∈ C

ðzhuc Cfixc þ Cvar

c Gγcc Þ ð71Þ

Table 1. Site Conditions

total working hours 8600 h/year

fuel oil no. 2 LHV 45 000 kJ/kg

natural gas LHV 50 244 kJ/kg

electric prices 0.07 $/kWh

fuel oil no. 2 price 0.19 $/kg

natural gas price 0.22 $/kg

raw water price 0.05 $/ton

Table 2. Demand Data (All in MW) for Case 1

period

1 2 3 4

HP steam demands (45 bar) 0 0 2 5

MP steam demands (17 bar) 20 16 22 10

LP steam demands (4.5 bar) 55 66 60 45

total steam demands 75 82 84 60

electricity demands 4.5 7.2 2.8 3.5

shaft power demand 1 1.2 2.0 1.3 1.8





where x2 is a vector of variables, and Ω2 is a feasible searchingspace delimited by the constraints.

x2 �

dthucp ; dthckp; dtcuhp; f

bfwbp ; fbip; f bdbip; f

maxb ; fbup; fchkp; fgp

fgbp; fhckp; fii0tp; f0ii0tp; fii0 i00tp; f

0ii00tp; f

wp ; fip; f

cp ; f

ldip ; fii0p

f psip ; fventip ; f pdip ; f

maxD ; hbip; hii0tp; hip; hcp; tchkp; tckp; t

out;hucp

thckp; thkp;Tgp;Tgbp; qbp; qbup; qhucp ; qgp; qhckp;wgp;wgjp;wii0tp

wii0tjp;wmaxg ;wmax

t ;wmjp;wmaxm ;wimp;e

p ;wexp;ep ; zb; zbp; zbi; zbip; zhucp

zg ; zgp; zgb; zgbp; zgj; zgjp; zhckp; zcuhp; zt; zii0t; zii0tp; zii0tj; zii0tjp; zii0 i00tp"b ∈ B ; c ∈ C ; d ∈ D ; g ∈ G ; h ∈ H ; i; i0; i00 ∈ Ij ∈ J ; k ∈ K ;m ∈ M ; p ∈ P ; t ∈ T ; u ∈ U

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

Ω2 ¼ fx2jeqs 1�69g

5. CASE STUDIES

In this section, two case studies are presented to demonstratethe application of the proposed MINLP model. In case 1, SDNdesign with the given steam demands is studied. The process dataare taken from the work of Bruno et al.,4 which was originallysolved for the single period operation only. The other perioddemands are added in the present example to facilitate a multi-period SDN design. In case 2, simultaneous design for SDN andHEN is studied, where the interaction between a steam systemand a heat recovery system can be optimized.

The site conditions for case studies are presented in Table 1.The optimization platform employed was the General AlgebraicModeling System (GAMS).14 The solver used was SBB15 for theMINLP model. An Intel Core 2 Duo CPU 2.53 GHz computerwith 1 GB of RAM was used.

5.1. Case 1. SDN Design Problem Associated with Multi-period Demands. Let us first consider the SDN design problemassociated with multiperiod demands in a chemical process. Theset of the energy demands is given in Table 2. As can be seen,there is a demand for electricity, three shaft power demands, anddemands for high, medium, and low pressure steam. The operat-ing pressures of steam headers and the corresponding saturatedtemperatures are shown in Table 3. In order to discuss the effectof the header temperatures, three scenarios are considered inthis case study. The investment cost functions are taken fromBruno et al.4 and are presented in Table 7 in the Appendix. Theannualized capital recovery factor adopted is 0.15.5.1.1. Scenario 1: Specified Steam Header Temperatures. In

this scenario, specified temperatures of steam headers are usedfor the design. More specifically, the temperatures are treated asgiven constants (not decision variables), and then an MINLPmodel is solved accordingly to synthesize the steam distributionnetwork. The specified header temperatures adopted in thisscenario are presented in Table 3. A two-period problem withequal operating time (50%) is studied. Note that electricityexport is not considered in this study.The optimal configuration obtained in the first scenario has a

TACof $13.97million year�1 and is shown in Figure 5. There areone boiler, three steam turbines, and one electric motor installedin the steam system. A high pressure (HP) boiler is chosen for thesteam production. A HP�medium pressure (MP) back-pressuresteam turbine is used for the electricity generation. Part ofelectricity import is required, which is 205 and 2319 kW forperiods 1 and 2, respectively. Shaft power demands 1 and 2 aresatisfied with steam turbines (two MP�low pressure (LP) back-pressure steam turbines). The remaining shaft power demand issatisfied with an electric motor.From the result, it is found that the specified header tempera-

ture strategy restricts the design of SDN. Some feasible structureor operating opportunities may be excluded due to the tempera-ture restriction.5.1.2. Scenario 2: Optimized Steam Header Temperatures. In

scenario 2, a more practical design strategy is proposed. Thesame problem is solved again without the specified temperatureconstraints. Each steam header temperature is treated as a

Table 3. Steam Header Conditions for Case 1

P (bar) saturated temp (�C) specified temp (�C)

45 257.4 369.0

17 204.3 265.0

4.5 147.9 148.0

Figure 5. Optimal SDN design for scenario 1 in case 1.



variable to be optimized. It is expected to find appropriatetemperatures for each steam header throughout all periods.Figure 6 shows the resulting network structure. One can see

that the optimal configuration is different from the result of

scenario 1. A HP�MP steam turbine and a MP�LP steamturbine are installed to meet the need of shaft demands 1 and 3,respectively, which replace the original MP�LP turbine and theelectric motor. An electric motor is installed for the shaft demand



Table 4. Comparative Economic Parameters for the MajorResults of Case 1

scenario 1 scenario 2 scenario 3

total annualized cost ($105) 139.72 134.75 120.63

overall fuel cost ($105) 121.75 123.78 119.90

overall electricity cost ($105) 7.60 0.00 �10.73

annualized capital cost ($105) 9.91 10.51 11.03

Table 5. Process Stream Data of Case 2

steam type and number CP (kW/�C) Tin (�C) Tout (�C)

H1 205 388 110

H2 152 210 60

C1 753 100 200

C2 377 140 255

C3 143 70 140



2. An HP�LP steam turbine is installed to meet the requirementof electricity. The electricity import decreases to 0 kW. Theoptimized temperatures of steam headers are 356.3, 284.9, and159.9 �C (period 1) and 399.8, 297.7, and 185.9 �C (period 2)for high, medium, and low pressure, respectively. The corre-sponding TAC is $13.47 million year�1. It is evident that theTAC can be reduced under the proposed flexible strategy.5.1.3. Scenario 3: Multiperiod Operating Design. In this

scenario, four periods with equal operating time (25%) areconsidered to show the capability for the design of multiperiodoperating. The set of demands is presented in Table 2. HP steamdemands are requested in period 3 and period 4. Note thatelectricity export is allowed and its price is assumed the same asthe import price.Figure 7 shows the network layout and the corresponding

operating state. The TAC of this design is $12.06 million year�1.

As can be seen, optimized header temperatures are determined.Clearly, the operating temperatures are not unique, which arechanged with varying demands. Mechanical demands are satis-fied with one steam turbine and two electric motors. The highpressure steam turbine exhausts mainly to the medium pressureheader for electricity generation. Steam systems tend to generatemore electricity during the first, second, and third time period,and therefore the excess electricity is exported. Electricity importis necessary only in the last period. Since the steam demands arelower than in other periods, the steam available to generate elec-tricity is lower, too. In this example the ability of the proposedmodel to choose the best option for electricity generation and theoptimal configuration of power generating devices, and theirinfluence on the operation, can be demonstrated.From the result of these scenarios, it is evident that if the

flexible model is considered for the SDN design, the lower TACcan be accomplished. More detailed information for thesescenarios is shown in Table 4.5.2. Case 2. Simultaneous SDN and HEN Design Problem.

Let us consider a chemical process in which the heat capacity flowrates (CP) of two hot streams and three cold streams areidentified (see Table 5). Single-period operation is considered.Electricity and mechanical power demands are required and are

Table 6. Demand Data (All in kW) of Case 2

power demands

electricity demand 4500

shaft power demand 1 2000

shaft power demand 2 1200

Figure 8. Optimal SDN design for case 2.

Figure 9. Optimal HEN design for case 2.



shown in Table 6. The annual cost is 1200[area (m2)]0.6 for allexchangers.12 The minimum temperature difference for thedesign of HEN is 10 K. In this case, the objective is to optimizethe interaction between SDN and HEN.The optimal SDN is presented in Figure 8. Three steam

headers are suggested for SDN, where their properties are HP(50.9 bar, 398.8 �C), MP (19.1 bar, 297.6 �C), and LP (4.7 bar,229.7 �C). It is mentioned that the steam levels are not specifiedpreviously, but are optimized by the proposed approach. Thissteam system provides the multiple utilities for the heat recoverynetwork. The steam level decisions are the trade-off results ofsimultaneous consideration for SDN and HEN. In SDN, one HPboiler and three back-pressure steam turbines are included. Twoturbines are used for the shaft power demands and one is forelectricity generation. A part of the electricity demand is satisfiedwith the steam system (2040 kW), and another part is from theimport (2707 kW). The optimal HEN is shown in Figure 9,where four heat exchangers, two heaters, and one cooler areincluded. Hot utilities are from an MP steam header (49.2 MW)and an LP steam header (3.1 MW). The corresponding TAC is$14.19 million year�1.

6. DISCUSSION

In scenario 1 of case 1, temperatures of steam headers arespecified before a network structure is available. On the other

hand, header temperatures are treated as temperature-indepen-dent and therefore operation and design possibilities may berestricted. In scenario 2, the header temperatures are consideredas decision variables and a one-step procedure is developed withthe ability to optimize the network structure and the operatingconditions simultaneously. It is appreciated that the better designand operation can be accomplished under this approach. Inscenario 3, four periods with electricity import and export arestudied. The result reveals that steam turbines for the electricitygeneration are preferred due to the higher operating flexibility.Thus, the steam system can maintain higher operating efficiencythroughout all periods.

In case 2, simultaneous design for SDN and HEN is studied.The proposedmodel can determine both themoderate operatingconditions and the corresponding network for the steam systemand the heat recovery system. The operating condition deter-mination can affect the operation efficiency for steam systemsand the heat recovery circumstance for the given chemicalprocess. In this work, the interaction between two systems canbe optimized.

7. CONCLUSION

Changes in specifications, composition of feed, and seasonalproduct demands may cause several process conditions withvariation in the energy requirements during an annual horizon. Inthe first part, an MINLP model, based on unit superstructures,has been developed to design a steam system with variable utilitydemands. Complex multiperiod scenarios were studied thattogether consider the design and operation of steam powersystems in an industrial plant. In the second part, a novelmethodology has been developed to address the design of asteam system and a heat recovery network. This work determinesthe optimal structure for both SDN and HEN, and also estimatesthe moderate operating conditions. The results from the casestudies demonstrate that better energy management and utiliza-tion can be realized with the proposed model.

’APPENDIX

The investment cost data according to Bruno et al.4 areitemized in Table 7.

’AUTHOR INFORMATION

Corresponding Author*Tel.: 886-2-33663039. Fax: 886-2-23623040. E-mail: [email protected].

’ACKNOWLEDGMENT

Financial support of the National Science Council of ROC(under Grants NSC98-3114-E-002-009 and NSC100-3113-E-002-004) is appreciated.

’NOMENCLATURE

Indicesb = index for boilersc = index for cold process or utility streamsg = index for gas turbinesh = index for hot process or utility streamsi = index for steam headers

Table 7. Investment Cost Data4

unit investment cost ($/year)

(1) field erected boiler (VHP) 22970F0.82fp1F, maximum steam flow rate (tons/h) fp1 = 0.6939 þ 0.1214P

� 3.7984�3P2

P, pressure (MPa)

(2) large package boiler 4954F0.77fp2F, maximum steam flow rate (tons/h) fp2 = 1.3794 � 0.5438P

þ 0.1879P2

P, pressure (MPa)

(3) heat recovery steam generator 941Ffg0.75

Ffg, maximum flue gas flow rate (tons/h)

(4) steam turbine 81594 þ 18.052Wst

Wst, maximum power (kW)

(5) gas turbine 321350 þ 67.618Wgt

Wgt, maximum power (kW)

(6) electric generator 8141 þ 0.6459Weg

Weg, maximum power (kW)

(7) electric motor 1601 þ 27.288Wel

Wel, maximum power (kW)

(8) deaerator 7271 þ 79.25FBFB, maximum BFW flow rate (tons/h)

(9) condensor 3977 þ 1.84FcFc, maximum cooling

water flow rate (tons/h)

(10) centrifugal pump (475.3 þ 34.95Pw

� 0.0301Pw2)fpw

Pw, power (kW)

fpw = 1 (<1.03 MPa)

fpw = 1.62 (1.03�3.45 MPa)

fpw = 2.12 (>3.45 MPa)



j = index for shaft power demandsk = index for stagesp = index for time periodst = index for steam turbinesu = index for fuels

SetsB = {b|b is a boiler, b = 1, ..., B} = MB ∪ HBMB = {b|b is a multifuel boiler, b = 1, ..., MB}HB = {b|b is a heat recovery steam generator, b = 1, ..., HB}C = {c|c is a cold process stream, c = 1, ..., C}G = {g|g is a gas turbine, g = 1, ..., G}GE = {g|g is a gas turbine for the generation of electricity, g = 1, ...,

GE}GS = {g|g is a gas turbine for the production of shaft power,

g = 1, ..., GS}H = {h|h is a hot process stream, h = 1, ..., H}I = {i|i is a steam header, i = 1, ..., I}J = {j|j is a shaft power demand, j = 1, ..., J}K = {k|k is a stage, k = 1, ..., K}P = {p|p is a time period, p = 1, ..., P}T = {t|t is a steam turbine, t = 1, ..., T} = ST ∪ MTST = {t|t is a single-stage steam turbine, t = 1, ..., ST}MT = {t|t is a multistage steam turbine, t = 1, ..., MT}TE = {t|t is a steam turbine for the generation of electricity,

t = 1, ..., TE}TS = {t|t is a steam turbine for the production of shaft power, t =

1, ..., TS}U = {u|u is a fuel, u = 1, ..., U}

ParametersC*fix = fixed coefficient function for units, where * = {b, c, d, g,

h, m, t}C*var = variable coefficient function for units, where * = {b, c, d, g,

h, m, t}Cpw = cost per unit mass of demineralized water makeup in time

period p, $ kg�1

Cpcw = cost per unit mass of cooling water in time period p, $ kg�1

Cpimp,e = specific cost of imported electricity in time period p,

$ kWh�1

Cpexp,e = specific cost of exported electricity in time period p,

$ kWh�1

Cu = cost per unit mass of fuel u, $ kg�1

Fcp = heat capacity flow rate for cold process stream c in period p,kW/�C

Fhp = heat capacity flow rate for hot process stream h in period p,kW/�C

G*γ* = coefficient function of units, where * = {b, c, d, g, h, m, t}

Hisat,l = enthalpy of saturated steam at steam header i level,

kJ kg�1

Hipps = enthalpy of steam supplied by processes and delivered at

header i in period p, kJ kg�1

Hpw = enthalpy of demineralized water makeup in period p,

kJ kg�1

Hdeaer = enthalpy of water leaving a deaerator, kJ kg�1

tphrs = number of operating hours in time period p, h period�1

Tcpin = inlet temperature of cold process stream c in period p, �C

Tcpout = outlet temperature of cold process stream c in period p, �C

Thpin = inlet temperature of hot process stream h in period p, �C

Thpout = outlet temperature of hot process stream h in period p, �C

wjpdem,s = shaft power demand j in time period p, kW

wpdem,e = total electricity demand in time period p, kW

Ωh = upper bound for heat exchange, kWΩhb,Ω

hb = upper and lower bounds of steam flow rate for boiler b,

kg s�1

Ωh g,Ωhg = upper and lower bounds of gas flow rate for gas turbine

g, kg s�1

Ωh ii0t,Ωhii0t = upper and lower bounds of steam flow rate for steam

turbine t, kg s�1

Ωh , Ωh= arbitrary very large value and very small value

Γh g, Γhg = upper and lower bounds of power generation for gas

turbine g, kWΓh ii0t,Γ

hii0t = upper and lower bounds of power generation for steam

turbine t, kWΦh g,Φ

hg =maximum andminimum operating temperatures for gas

turbine g, �Cj = fixed blowdown fraction for boilersηm = fixed efficiency for electric motors

Continuous Variablesdtcp

hu = temperature approach for the match of cold stream c andhot utility in period p, �C

dthckp = temperature approach for match (h, c) at temperaturelocation k in period p, �C

dthpcu = temperature approach for the match of hot stream h and

cold utility in period p, �Cfbmax = maximum steam flow rate for boiler b, kg s�1

fbip = steam output from boiler b to steam header i in time periodp, kg s�1

fbipbd = blowdown water for boiler b at pressure i in time period p,

kg s�1

fbpbfw = boiler feed water for boiler b in time period p, kg s�1

fbup = fuel u consumed in boiler b in time period p, kg s�1

fchkp = heat capacity flow rate for cold stream c at stage k in timeperiod p, kW/�C

fdmax = maximum water flow rate for deaerator, kg s�1

fgbp = exhaust gas from gas turbine g to HRSG b in time period p,kg s�1

fgp = gas turbine g exhaust mass flow rate, kg s�1

fgup = fuel u consumed in gas turbine g in time period p, kg s�1

fhckp = heat capacity flow rate for hot stream h at stage k in timeperiod p, kW/�C

fii0tp = steam flow rate from header i to header i0 through a steamturbine t in time period p, kg s�1

fii0tp0 = input steam flow rate from header i to header i0 through amultistage steam turbine t in time period p, kg s�1

fii0 i0 0tp0 = steam flow rate from stage i0 to stage i00 for a multistagesteam turbine t in time period p, kg s�1

fii0p = steam flow rate from header i to header i0 in time period p,kg s�1

fip = steam flow rate from header i to deaerator in time period p,kg s�1

fipld = desuperheating boiler feed water injected into header i in

time period p, kg s�1

fippd = steam process demand at header i in time period p, kg s�1

fipps = steam from process entering header i in time period p, kg s�1

fipvent = vented steam at header i in time period p, kg s�1

fpc = condensate return in period p, kg s�1

fpw = demineralized water makeup in time period p, kg s�1

fpcw = cooling water mass flow rate for condensers in time period

p, kg s�1

hbip = enthalpy of steam generated by boiler b entering header i inperiod p, kJ kg�1



hii0tp = enthalpy of a discharge by steam turbine t entering headeri0 in period p, kJ kg�1

hip = enthalpy of steam header i in period p, kJ kg�1

hpc = enthalpy of returning condensate from processes in period p,

kJ kg�1

qbp = heat added to the water in boiler b in time period p, kWqcphu = heat exchanged between hot utility and cold stream c in

time period p, kWqhckp = heat exchanged between hot stream h and cold stream c in

stage k in period p, kWqhpcu = heat exchanged between hot stream h and cold utility in

time period p, kWtchkp = temperature of cold stream c at stage k in time period p, �Ctckp = temperature of cold stream c at hot end of stage k in time

period p, �CTgbp = exhaust gas temperature for gas turbine g delivered to

HRSG b in time period p, �CTgp = exhaust gas temperature for gas turbine g, �Cthckp = temperature of hot stream h at stage k in time period p, �Cthkp = temperature of hot stream h at hot end of stage k in time

period p, �Cwgmax = design/maximum gas turbine g power output, kW

wgp = power produced by gas turbine g in time period p, kWwgjp = shaft power produced by gas turbine g to shaft demand j in

period p, kWwii0tp = power produced by steam turbine t in period p, kWwii0tjp = shaft power produced by steam turbine t to shaft demand

j in period p, kWwmmax = design/maximum power for electric motor m, kW

wmjp = shaft power produced by electricmotorm to shaft demandj in period p, kW

wtmax = design/maximum steam turbine t power output, kW

wpimp,e = electricity imported in time period p

wpexp,e = electricity exported in time period p

Binary Variableszb = denotes the presence of boiler bzbp = denotes the operating status of boiler bzbi = denotes the existence of the connection between boiler b

and header izbip = denotes the existence of steam flow from boiler b to header

i in time period pzcphu = denotes that hot utility exchanges heat with cold stream c in

period pzg = denotes the presence of gas turbine gzgb= denotes the existence of the connection between gas turbine

g and boiler bzgbp = denotes the existence of gas flow from gas turbine g to

boiler b in time period pzgj = denotes the existence of the connection between gas turbine

g and shaft demand jzgjp = denotes the existence of shaft demand j supplied by gas

turbine g in time period pzgp = denotes the operating status of gas turbine g in time period pzhckp = denotes the existence of match (h, c) in stage k in period pzhpcu = denotes that cold utility exchanges heat with hot stream h in

period pzt = denotes the presence of steam turbine tzii0t = denotes the existence of the connection of steam turbine t

between i and i0 headerszii0 i0 0tp = denotes the existence of steam flow from stage i0 to stage

i00 of a multistage steam turbine t in time period p

zii0tj = denotes the existence of the connection between steamturbine t and shaft demand j

zii0tjp = denotes the existence of shaft demand j supplied by steamturbine t in time period p

zii0tp = denotes the operating status of steam turbine t in timeperiod p

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