This document presents a methodology for optimizing the design of steam distribution networks (SDNs) for steam power plants. The methodology formulates the problem as a mixed-integer nonlinear programming (MINLP) model to minimize total annualized cost. The model determines the optimal structure, configuration, and operation of the SDN as well as its interaction with the heat recovery system. Case studies are used to demonstrate the feasibility and benefits of the proposed simultaneous optimization approach.

1. 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 Systems
for Steam Power Plants
Cheng-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 satisfies
the 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. The
proposed model determines simultaneously (i) the structure and operational configuration of a steam system and (ii) the interaction
between the steam system and the heat recovery system. A series of case studies are presented to demonstrate the feasibility and
benefit of the proposed approach.
1. INTRODUCTION
Steam power plants are the main energy supplier for running
chemical processing. Typically, a steam power plant consists of
various units including boilers, gas turbines, steam turbines,
electric motors, steam headers, etc. In the plant, steam is
converted into two types of energy, specifically, electricity and
mechanical power. Electricity demands are from the power
required to function process devices. Mechanical power demands
are from the requirement to drive process units. Steam demands
are from heat duties for the heat exchange network or heat
sources for the reaction process.
The design of a steam power plant is a large and complex
problem, where the layout of all types of units and the operating
conditions must be optimized for efficient operation. The steam
distribution network (SDN) is an essential element in devising
the energy management system of a steam power plant. A large
volume of related studies have already been published in the
literature. Basically, two distinct approaches were adopted in
these works: (a) the heuristics-based thermodynamic design
method1,2
and (b) the model-based optimization method.3À5
The former networks were synthesized with thermodynamic
targets for getting the maximum allowable overall thermal
efficiency, while the latter were designed with mixed-integer
linear/nonlinear programs for attaining the minimum total
annualized cost (TAC).
The above-mentioned works were developed to address the
design of an SDN assuming that all units operate at full load to
satisfy a single set of demands and conditions. However, in many
existing chemical processes the common operational feature is
varying demands. This may be due to changing feed/product
specifications or changes of heat loss with seasonal variation in
the continuous operation plants, or changes in operations for
batch plants. For example, energy demands in peak season are
higher than in off peak season or steam power plants need more
heat demands in winter since the heat loss is higher.
Because of the limitations of these types of studies, capable
methodologies for the period-varying demands were developed.6À9
However, the research was only addressing operational problems
for existing plants or design problems without simultaneously
optimizing unit sizes and loads as continuous functions. More
recently, Aguilar et al.10,11
proposed a mixed-integer linear
programming (MILP) model to address retrofit and operational
problems for utility plants, considering structural and operational
parameters as variables to be optimized. The linear model was
realized when some operating conditions of units (e.g., air flow
rate or operating temperature of gas turbines) were prespecified
or some of the entering streams (e.g., from boilers and a heat
recovery steam generator, HRSG) were already at the tempera-
ture of the header (predetermined).
From a review of the current literature, there is a need to
develop a more comprehensive design method for SDNs. In this
paper, the main objective of the study is to develop a flexible
model for industrial problems. This model can address the
multiperiod operating problem and can easily set up the link
between 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 problem
is formally defined in section 2. The design concept developed by
Papoulias and Grossmann3
is adopted and modified in the
present study for a generalized SDN. The superstructure and
corresponding 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 unit
design while equipment is operating at different loads. Two cases
on synthesis and design of the network are then presented in
section 5 to demonstrate the feasibility and effectiveness of the
proposed simultaneous optimization strategy. The discussion
and the conclusion of the present studies are provided in sections
6 and 7.
2. PROBLEM STATEMENT
The design problem addressed in this paper is stated as
follows: Given are a set of steam demands or a set of hot/cold
Received: October 10, 2010
Accepted: May 6, 2011
Revised: April 21, 2011

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Industrial & Engineering Chemistry Research ARTICLE
process streams to be cooled/heated in every period. Given also
are the electricity demands and mechanical power needs of
chemical processing in every period. The objective is to develop
a systematic approach to manage the energy usage in an
efficient way.
This work includes the following: (1) the multiperiod opera-
tion problem with given steam demands and (2) the total
processing system design with given process stream data. The
former is to synthesize a cost-optimal steam system that can fulfill
energy requirements. The latter is to design a steam distribution
network and heat recovery network (SDNÀHEN) simulta-
neously with a minimum TAC.
The given model parameters of this optimization problem
include the following: (1) the design specifications of every boiler
unit (i.e., its operating pressure, maximum operating tempera-
ture, and the lower and upper bounds of steam flow rate), (2) the
design specifications of every gas turbine unit (i.e., its operating
temperature and its minimum and maximum heat loads), (3) the
design specifications of every steam turbine unit (i.e., its lower
and upper bounds of steam flow rate), (4) the temperature levels
of cooling water, and (5) the design specifications of every
exchanger unit for the HEN design.
The resulting design includes the following: (1) the number of
boiler units and their throughputs in every period, (2) the
number of gas turbine units and their throughputs in every
period, (3) the number of steam turbine units and their
throughputs in every period, (4) the consumption rates of
freshwater and the cooling water usage in every period, (5) the
consumption 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 an
SDN, and (8) the complete network configuration and the flow
rate of an HEN for the total processing problem.
3. SUPERSTRUCTURE
A superstructure of SDN is constructed to incorporate all
possible flow connections, as presented in Figure 1. More
detailed superstructures of all units concerned in the SDN are
shown in Figure 2. Steam can be generated with either fired or
heat recovery steam generator (HRSG) boilers which operate at
conditions consistent with those of the steam headers. The fired
boilers generate steam by providing heat from combustion of a
fuel or a fuel mixture. The HRSG can further utilize the exhausts
from gas turbines to heat water to generate steam.
Steam is collected and distributed to steam consumer units by
the steam header. Note that there are several steam headers in the
generalized 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) gas
turbine, (d) single-stage steam turbine, (e) multistage steam turbine,
(f) deaerator, and (g) steam header.

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Industrial & Engineering Chemistry Research ARTICLE
pressure levels are selected at saturated steam pressure corre-
sponding to the minimum temperature requirements. Specifi-
cally, values not less than their saturated temperatures are
suitable 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 the
amount of steam demands are not given. Both pressures and
temperatures of headers are variables to be determined by the
optimization approach.
There are three types of power-generating devices in the
superstructure for satisfying power demands, i.e., gas turbines,
steam turbines, and electric motors. Turbines can generate
electricity or mechanical power. A gas turbine includes a com-
pressor, a combustor, and a turbine, where the hot air exiting the
turbine can further be used in HRSG to enhance the steam
production. Both back-pressure steam turbines and condensing
turbines are considered. For the back-pressure turbine, the
pressure of exhaust steam is higher than atmospheric pressure.
If the condensing turbine is chosen in the design, then a
condenser has to be selected for its operation. Electric motors
are also used to meet the required shaft power demands.
A deaerator is installed to remove dissolved gases to provide
feed 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 and
cooling water and for the return of condensate.
The stagewise superstructure12
is adopted for the design of
HEN, as shown in Figure 3. Nonisothermal mixing of streams
is considered for the flexible design. Each utility can be treated
as a process stream with unknown loads. By this superstruc-
ture one can determine the loads of utilities for the giving
process streams.
4. MODEL FORMULATION
Having introduced the superstructures of SDN and HEN, one
can then formulate the synthesis problem as a MINLP. The
material and energy balance equations associated with every unit
are included as the constraints of the optimization problem. The
corresponding equipment models adopted for SDN are taken
from Aguilar et al.10
4.1. Steam Distribution Network. 4.1.1. Boilers. Equation 1
states the mass balance of a boiler, where the feed boiler water
entering the boiler b in a certain period p (fbp
bfw
) equals the
steam to the steam header i (fbip) and the effluent of blowdown
water (fbip
bd
). Therein the blowdown is treated as a fixed j
fraction of the boiler steam output and extracted at saturated
liquid conditions (see Figure 2a). An energy balance is needed
to ensure that the enthalpy entering the boiler equals that
leaving (i.e., eq 2), where qbp denotes the heat absorbed by the
water stream.
fbfw
bp ¼
X
i ∈ I
fbip þ
X
i ∈ I
fbd
bip "b ∈ B, p ∈ P ð1Þ
fbfw
bp Hdeaer
þ qbp ¼
X
i ∈ I
fbiphbip þ
X
i ∈ I
fbd
bipH
sat,l
i "b ∈ B, p ∈ P
ð2Þ
fbd
bip ¼ jfbip "b ∈ B, p ∈ P ð3Þ
In addition, depending on the capacity of units, the constraints
are imposed in the boiler formulation, i.e.
Ωb
zbip e fbip e Ωbzbip "b ∈ B, i ∈ I , p ∈ P ð4Þ
where zbip is a binary variable and Ω
h
b and Ωhb are the minimum
and the maximum capacities, respectively. Notice that the steam
flow rate is zero when the boiler unit is not selected. It should be
further considered that only one connection exists between
boiler b and steam header i with the boiler b selected. Thus,
the following constraints are imposed in this model.
zb ¼
X
i ∈ I
zbi " b ∈ B ð5Þ
zbi g zbip "b ∈ B, i ∈ I , p ∈ P ð6Þ
zbi e
X
p ∈ P
zbip "b ∈ B, i ∈ I ð7Þ
There are two types of boilers used in this work, which are
multifuel boilers (b ∈ MB) and HRSGs (b ∈ HB). The
equipment models of boilers are adopted from Aguilar et al.10
4.1.2. Gas Turbines. For convenience, a set of gas turbines
g ∈ G is defined. Figure 2c shows the superstructure of a gas
turbine. Since exhaust of gas turbine g is sent to the HRSG unit b
Figure 3. Heat exchanger network superstructure.

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Industrial & Engineering Chemistry Research ARTICLE
as heat input, the corresponding relations are derived below.
fgp ¼
X
b ∈ HB
fgbp "g ∈ G , p ∈ P ð8Þ
Tgp ¼
X
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 produced
shaft power wgp is selected to meet the shaft demand j.
wgp ¼
X
j ∈ J
wgjp "g ∈ GS , p ∈ P ð10Þ
The following constraints are considered to reflect physical and
practical limitations of the equipment. Equations 11 and 12 define
the unit capacity that must complywiththe maximumand minimum
limits. Equation 13 limits the operation range of a temperature.
Ωg
zgbp e fgbp e Ωgzgbp "g ∈ G , b ∈ HB, p ∈ P ð11Þ
Γg
zgjp e wgjp e Γgzgjp "g ∈ G , j ∈ J , p ∈ P ð12Þ
Φg
zgbp e Tgbp e Φgzgbp "g ∈ G , b ∈ HB, p ∈ P ð13Þ
Similar to the relation between a boiler and a steam header,
eqs 14À18 describe the connection and operation between a gas
turbine and a HRSG. If a gas turbine is chosen, a corresponding
HRSG is automatically included, and vice versa.
zg ¼
X
b ∈ HB
zgb " g ∈ G ð14Þ
zb ¼
X
g ∈ G
zgb " b ∈ HB ð15Þ
zgb g zgbp "g ∈ G , b ∈ HB, p ∈ P ð16Þ
zgb e
X
p ∈ P
zgbp "g ∈ G , b ∈ HB ð17Þ
zgp ¼
X
b ∈ HB
zgbp "g ∈ G , p ∈ P ð18Þ
Equations 19À22 are constraints to select gas turbine g for shaft
demand j. Notice a gas turbine is for one shaft demand.
zg ¼
X
j ∈ J
zgj "g ∈ GS ð19Þ
zgj g zgjp "g ∈ GS , j ∈ J , p ∈ P ð20Þ
zgj e
X
p ∈ P
zgjp "g ∈ GS , j ∈ J ð21Þ
zgp ¼
X
j ∈ J
zgjp "g ∈ GS , p ∈ P ð22Þ
4.1.3. Steam Turbines. Figure 2d,e shows a steam turbine
operating between higher pressure and lower pressure steam
headers. This device converts the energy of steam into
power. Equations 23 and 24 describe the constraints to
ensure that the operation does not exceed its design capa-
cities. Moreover, it is possible to define a minimum partial
load of units so that the actual output is not less than a given
specified fraction.
Ωii0t
zii0tp e fii0tp e Ωii0tzii0tp "i, i0
∈ I , i < i0
,
t ∈ T , p ∈ P ð23Þ
Γii0t
zii0tp e wii0tp e Γii0tzii0tp "i, i0
∈ I , i < i0
,
t ∈ T , p ∈ P ð24Þ
The steam turbine can generate electricity (t ∈ TE) or shaft
power (t ∈ TS ). If the latter operation is chosen, the shaft power
produced by the turbine t is to meet the shaft demand j.
wii0tp ¼
X
j ∈ J
wii0tjp "i, i0
∈ I , i < i0
, t ∈ TS , p ∈ P
ð25Þ
Γii0t
zii0tjp e wii0tjp e Γii0tzii0tjp "i, i0
∈ I , i < i0
,
t ∈ TS , j ∈ J , p ∈ P ð26Þ
Also, operation for multiperiod problems should be consid-
ered, which is like the boiler and gas turbine units discussed
previously. Equations 27and 28 are the constraints of steam
turbines.
zii0t g zii0tp "i, i0
∈ I , i < i0
, t ∈ T , p ∈ P ð27Þ
zii0t e
X
p ∈ P
zii0tp "i, i0
∈ I , i < i0
, t ∈ T ð28Þ
Similar to a gas turbine, constraints for the pair between steam
turbines and shaft demands are presented in eqs 29À31.
zii0t ¼
X
j ∈ 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 e
X
p ∈ 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 outlet
through which the steam is discharged to a lower pressure level.
Equation 32 is a logic constraint for single-stage steam turbines.
zt ¼
X
i,i0
∈ I
i < i0
zii0t " t ∈ ST ð32Þ
Multistage steam turbines (t ∈ MT ) can feature several inlets
to take in or outlets to discharge steam at different pressure levels.
The single-inlet steam turbines with several extractions are
adopted in this work. Equations 33 and 34 are logic constraints
for the multistage steam turbine, which means one steam inlet is

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restricted to this unit.
zt g zii0t "i, i0
∈ I , i < i0
, t ∈ MT ð33Þ
X
i ∈ I
i < i0
zii0t e 1 "i0
∈ I , t ∈ MT ð34Þ
The decomposition of a multistage steam turbine is shown in
Figure 4, and the corresponding superstructure is shown in
Figure 2e. A multistage steam turbine is decomposed into several
single-stage turbines which are connected in series so that the
original single-stage equipment performance model can be used
to determine its properties without new model. Equations 35 and
36 describe the mass balance of a multistage steam turbine, where
the higher quality steam (f0
ii0
tp) flows through a multistage
turbine t and then lower quality steam is delivered to a header
i0
(fii0
tp) and to the next stage i00
(fii0
i00
tp) as its steam input (f0
ii00
tp).
f
0
ii0tp ¼ fii0tp þ
X
i00
∈ I
i < i0
< i00
fii0i00tp "i, i0
∈ I , i < i0
, t ∈ MT , p ∈ P
ð35Þ
f
0
ii00tp ¼
X
i0
∈ I
i < i0
< i00
fii0i00tp "i, i00
∈ I , i < i00
, t ∈ MT , p ∈ P
ð36Þ
The following constraints are considered for this multistage
unit. Equations 37À40 are used to ensure that the stage i00
only
can accept one steam stream from preceding stages and stage i0
only can deliver steam to one next stage.
zii0tp g zii0i00tp "i, i0
, i00
∈ I , i < i0
< i00
, t ∈ MT , p ∈ P ð37Þ
X
i0
∈ I
i < i0
< i00
zii0i00tp e 1 "i, i00
∈ I , i < i00
, t ∈ MT , p ∈ P
ð38Þ
X
i00
∈ I
i < i0
< i00
zii0i00tp e 1 "i, i0
∈ I , i < i0
, t ∈ MT , p ∈ P
ð39Þ
Ωzii0i00tp e fii0i00tp e Ωzii0i00tp "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 streams
may come from low pressure steam, condensate return from
process, or treated water makeup. After water is treated and
its dissolved gas is removed, the feed water is sent to the boiler
or to the let-down station. Equation 41 describes the mass
flow rate balance. Equation 42 is an energy balance to
guarantee that enough steam is injected into the deaerator
so that the feed water leaving this unit is at saturated liquid
conditions.
fw
p þ
X
i ∈ I
fip þ fc
p ¼
X
b ∈ B
fbfw
bp þ
X
i ∈ I
fld
ip "p ∈ P ð41Þ
fw
p Hw
p þ
X
i ∈ I
fiphip þ fc
p hc
p
¼
X
b ∈ B
fbfw
bp þ
X
i ∈ I
fld
ip
!
Hdeaer
"p ∈ P ð42Þ
4.1.5. Steam Headers. Figure 2g shows the stream balance for a
steam header. The mass balance is given by eq 43. For the top
steam header, the highest pressure steam header, there are no
input streams from steam turbines or let-down stations, and there
is no output vented steam to the environment. The bottom steam
header is a condensing header, in which its flow rates of input
streams from let-down stations or back-pressure steam tur-
bines are zero and its output condensate water is sent to a
deaerator. Equation 44 is an energy balance for a header to
ensure that the total amount of enthalpy entering the header
equals that leaving. It should be noted that a variable for
enthalpy hip (hip = fn(Ti,Pi)) is employed because of the
flexible consideration.
X
b ∈ B
fbip þ
X
i0
∈ I
i0
< i
X
t ∈ T
fi0itp þ
X
i0
∈ I
i0
< i
fi0ip þ fld
ip þ f
ps
ip
¼
X
i0
∈ I
i0
> i
X
t ∈ T
fii0tp þ
X
i0
∈ I
i0
> i
fii0p þ fip þ fvent
ip þ f
pd
ip
"i ∈ I , p ∈ P ð43Þ
X
b ∈ B
fbiphbip þ
X
i0
∈ I
i0
< i
X
t ∈ T
fi0itphi0itp þ
X
i0
∈ I
i0
< i
fi0iphi0p þ f ld
ip Hdeaer
þ f
ps
ip H
ps
ip
¼
X
i0
∈ I
i0
> i
X
t ∈ T
fii0tp þ
X
i0
∈ I
i0
> i
fii0p þ fip þ fvent
ip þ f
pd
ip
0
BB
B
B
B
B
B
B
@
1
CC
C
C
C
C
C
C
A
hip "i ∈ I , p ∈ P
ð44Þ
4.1.6. Power Balances. Equation 45 ensures that the actual
power delivered by all the drives attached to the common shaft
meets the corresponding demands in each operating period. A
gas turbine g, a steam turbine t, and an electric motor m can be
Figure 4. Decomposition of the multistage steam turbine.

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used to meet the required power demands.
X
g ∈ GS
wgjp þ
X
i,i0
∈ I
i0
< i
X
t ∈ TS
wii0tjp þ
X
m ∈ M
wmjp
¼ w
dem,s
jp "j ∈ J , p ∈ P ð45Þ
In this study, the electricity produced by the steam and/or gas
turbine can be used to meet the needs of chemical process. The
overall balance equation can be written accordingly by eq 46. The
left-hand side of this expression accounts for the supply of
electricity, while the terms of the right-hand side correspond to
the potential consumers.
X
g ∈ G E
wgp þ
X
i,i0
∈ I
i < i0
X
t ∈ TE
wii0tp þ wimp,e
p
¼ wdem,e
p þ
X
m ∈ M
X
j ∈ J
wmjp
ηm
þ wexp,e
p "p ∈ P ð46Þ
4.2. Heat Exchanger Network. 4.2.1. Overall Heat Balance
for Each Stream. An overall heat balance is included to ensure
heat exchange for all process streams. The constraints specify that
the overall heat of each hot process stream is removed with cold
process streams or cold utilities. Similar constraints also apply for
all cold streams, as stated in eqs 47 and 48:
ðTin
hp À Tout
hp ÞFhp ¼
X
k ∈ K
X
c ∈ C
qhckp þ qcu
hp "h ∈ H , p ∈ P
ð47Þ
ðTout
cp À Tin
cpÞFcp ¼
X
k ∈ K
X
h ∈ H
qhckp þ qhu
cp "c ∈ C , p ∈ P
ð48Þ
4.2.2. Heat Balance at Each Stream. Heat balances are also
needed 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 the
temperature location in the superstructure. Stage location k = 1
involves the highest temperatures. qhckp denotes the heat ex-
change between hot process stream h and cold process stream
c in stage k.
ðthkp À th,kþ1,pÞFhp ¼
X
c ∈ C
qhckp "h ∈ H , k ∈ K , p ∈ P
ð49Þ
ðtckp À tc,kþ1,pÞFcp ¼
X
h ∈ H
qhckp "c ∈ C , k ∈ K , p ∈ P
ð50Þ
4.2.3. Heat Balance for Each Unit. For each local exchange
unit, heat balances are needed, where fhckp and fhckp are split heat
capacity flow rates.
ðthkp À thc,kþ1,pÞfhckp ¼ qhckp "h ∈ H , c ∈ C , k ∈ K , p ∈ P
ð51Þ
ðtchkp À tc,kþ1,pÞfchkp ¼ qhckp "h ∈ H , c ∈ C , k ∈ K , p ∈ P
ð52Þ
The total flow balances for theses split heat capacity flow rates
in each stage k can be stated as follows.
X
c ∈ C
fhckp ¼ Fhp "h ∈ H , k ∈ K , p ∈ P ð53Þ
X
h ∈ H
fchkp ¼ Fcp "c ∈ C , k ∈ K , p ∈ P ð54Þ
4.2.4. Assignment of Superstructure Inlet Temperatures. The
given inlet/outlet temperatures of hot and cold processes are
assignedasthe inlet/outlettemperatures tothe superstructure.For
hot process streams, the inlet corresponds to the location k = 1,
while for cold streams the inlet corresponds to location k = K þ 1.
Tin
hp ¼ th,1,p "h ∈ H , p ∈ P ð55Þ
Tin
cp ¼ tc,Kþ1,p "c ∈ C , p ∈ P ð56Þ
4.2.5. Feasibility of Temperatures. The following constraints
(eqs 57À60) are included to guarantee monotonic decrease of all
temperatures 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
Tout
hp e th,Kþ1,p "h ∈ H , p ∈ P ð59Þ
Tout
cp g tc,1,p "c ∈ C , p ∈ P ð60Þ
4.2.6. Hot and Cold Utility Loads. Equations 61 and 62 are
posed to calculate hot or cold utility loads needed for each
process stream.
ðth,Kþ1,p À Tout
hp ÞFhp ¼ qcu
hp "h ∈ H , p ∈ P ð61Þ
ðTout
cp À tc,1,pÞFcp ¼ qhu
cp "c ∈ C , p ∈ P ð62Þ
4.2.7. Logic Constraints. Logic constraints and binary variables are
needed to determine the existence of stream match (h, k) in stage k.
zhckp, zhp
cu
, and zcp
hu
are binary variables for process stream matches, for
cold utility matches, and for hot utility matches, respectively.
qhckp À Ωzhckp e 0 "h ∈ H , c ∈ C , k ∈ K , p ∈ P
ð63Þ
qcu
hp À Ωzcu
hp e 0 "h ∈ H , p ∈ P ð64Þ

7. 8103 dx.doi.org/10.1021/ie102059n |Ind. Eng. Chem. Res. 2011, 50, 8097–8109
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qhu
cp À Ωzhu
cp e 0 "c ∈ C , p ∈ P ð65Þ
4.2.8. Calculation of Approach Temperatures. For determin-
ing the area requirement of the heat exchanger, approach
temperatures are used to calculate the log mean temperature
difference (LMTD) approximated by using the Chen equation.13
The area requirement of each match will be incorporated in the
objective function. The constraints 66À69 are expressed as
inequalities since the cost of the exchanger decreases with higher
values for the approach temperatures.
dthckp e thkp À tchkp þ Γð1 À zhckpÞ "h ∈ H , c ∈ C ,
k ∈ K , p ∈ P ð66Þ
dthc,kþ1,p e thc,kþ1,p À tc,kþ1,p þ Γð1 À zhckpÞ "h ∈ H ,
c ∈ C , k ∈ K , p ∈ P ð67Þ
dtcu
hp e th,Kþ1,p À Tout,cu
þ Γð1 À zcu
hpÞ "h ∈ H , p ∈ P
ð68Þ
dthu
cp e tout,hu
cp À tc,1,p þ Γð1 À zhu
cp Þ "c ∈ C , p ∈ P ð69Þ
4.3. Objective Function and MINLP Formulation. The
objective function in the synthesis model is the TAC, which
includes the sum of operating and the annualized capital costs.
The former consists of the costs of fuels, cooling water, fresh
water, and purchased electricity. The latter includes the fixed
and variable costs of all units. There are two objectives
considered in this work, as shown in eqs 70 and 71. The first
objective is the design of SDN for the given steam demands.
The second objective is the simultaneous design of SDN and
HEN for the given process streams, where their interaction
can be optimized. It should be mentioned that the hot utility
requirement of HEN is satisfied with steam from the steam
system.
min
x1 ∈ Ω1
J1 ¼
X
p ∈ P
ðCw
p fw
p þ Ccw
p fcw
p þ Cimp,e
p wimp,e
p À Cexp,e
p wexp,e
p
þ
X
b ∈ B
X
u ∈ U
Cufbup þ
X
g ∈ G
X
u ∈ U
CufgupÞthrs
p
þ
X
b ∈ B
ðzbCfix
b þ Cvar
b G
γb
b Þ þ
X
g ∈ G
ðzgCfix
g þ Cvar
g G
γg
g Þ
þ
X
t ∈ T
ðztCfix
t þ Cvar
t G
γt
t Þ
þ
X
m ∈ M
ðzmCfix
m þ Cvar
m Gγm
m Þ
þ
X
d ∈ D
ðzdCfix
d þ Cvar
d G
γd
d Þ ð70Þ
where x1 is a vector of variables, and Ω1 is a feasible searching
space delimited by the constraints.
x1
f
bfw
bp ; fbip; fbd
bip; fmax
b ; fbup; fgp; fgbp; fii0tp; f
0
ii0tp
fii0i00tp; f
0
ii00tp; fw
p ; fip; fc
p ; fld
ip ; fii0p; f
ps
ip ; fvent
ip ; f
pd
ip ; fmax
d
hbip; hii0tp; hip; hc
p; Tgp; Tgbp; qbp; qbup; qgp; wgp; wgjp; wii0tp
wii0tjp; wmax
g ; wmax
t ; wmjp; wmax
m ; w
imp;e
p ; w
exp;e
p ; zb; zbp; zbi; zbip
zd; zg; zgp; zgb; zgbp; zgj; zgjp; zm; zt; zii0t; zii0tp; zii0tj; zii0tjp; zii0i00tp
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 1À46g
min
x2 ∈ Ω2
J2 ¼
X
p ∈ P
ðCw
p fw
p þ Ccw
p fcw
p þ Cimp,e
p wimp,e
p À Cexp,e
p wexp,e
p
þ
X
b ∈ B
X
u ∈ U
Cufbup þ
X
g ∈ G
X
u ∈ U
Cufgup þ qcu
hpÞthrs
p
þ
X
b ∈ B
ðzbCfix
b þ Cvar
b G
γb
b Þ þ
X
g ∈ G
ðzgCfix
g þ Cvar
g G
γg
g Þ
þ
X
t ∈ T
ðztCfix
t þ Cvar
t G
γt
t Þ þ
X
m ∈ M
ðzmCfix
m þ Cvar
m Gγm
m Þ
þ
X
d ∈ D
ðzdCfix
d þ Cvar
d G
γd
d Þ
þ
X
h ∈ H
X
c ∈ C
X
k ∈ K
ðzhckCfix
hck þ Cvar
hckG
γhck
hck Þ
þ
X
h ∈ H
ðzcu
h Cfix
h þ Cvar
h G
γh
h Þ
þ
X
c ∈ C
ðzhu
c Cfix
c þ Cvar
c Gγc
c 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
shaft power demand 2 1.5 1.0 1.1 0.9
shaft power demand 3 0.7 0.6 0.5 0.8

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where x2 is a vector of variables, and Ω2 is a feasible searching
space delimited by the constraints.
x2
dthu
cp ; dthckp; dtcu
hp; f
bfw
bp ; fbip; fbd
bip; fmax
b ; fbup; fchkp; fgp
fgbp; fhckp; fii0tp; f
0
ii0tp; fii0i00tp; f
0
ii00tp; fw
p ; fip; fc
p ; fld
ip ; fii0p
f
ps
ip ; fvent
ip ; f
pd
ip ; fmax
D ; hbip; hii0tp; hip; hc
p; tchkp; tckp; tout;hu
cp
thckp; thkp; Tgp; Tgbp; qbp; qbup; qhu
cp ; qgp; qhckp; wgp; wgjp; wii0tp
wii0tjp; wmax
g ; wmax
t ; wmjp; wmax
m ; w
imp;e
p ; w
exp;e
p ; zb; zbp; zbi; zbip; zhu
cp
zg; zgp; zgb; zgbp; zgj; zgjp; zhckp; zcu
hp; zt; zii0t; zii0tp; zii0tj; zii0tjp; zii0i00tp
b ∈ B; c ∈ C ; d ∈ D; g ∈ G ; h ∈ H ; i; i0
; i00
∈ I
j ∈ J ; k ∈ K ; m ∈ M ; p ∈ P ; t ∈ T ; u ∈ U
8
:
9
=
;
Ω2 ¼ fx2jeqs 1À69g
5. CASE STUDIES
In this section, two case studies are presented to demonstrate
the application of the proposed MINLP model. In case 1, SDN
design with the given steam demands is studied. The process data
are taken from the work of Bruno et al.,4
which was originally
solved for the single period operation only. The other period
demands are added in the present example to facilitate a multi-
period SDN design. In case 2, simultaneous design for SDN and
HEN is studied, where the interaction between a steam system
and 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 Algebraic
Modeling System (GAMS).14
The solver used was SBB15
for the
MINLP model. An Intel Core 2 Duo CPU 2.53 GHz computer
with 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 problem
associated with multiperiod demands in a chemical process. The
set of the energy demands is given in Table 2. As can be seen,
there is a demand for electricity, three shaft power demands, and
demands for high, medium, and low pressure steam. The operat-
ing pressures of steam headers and the corresponding saturated
temperatures are shown in Table 3. In order to discuss the effect
of the header temperatures, three scenarios are considered in
this case study. The investment cost functions are taken from
Bruno et al.4
and are presented in Table 7 in the Appendix. The
annualized 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 used
for the design. More specifically, the temperatures are treated as
given constants (not decision variables), and then an MINLP
model is solved accordingly to synthesize the steam distribution
network. The specified header temperatures adopted in this
scenario are presented in Table 3. A two-period problem with
equal operating time (50%) is studied. Note that electricity
export is not considered in this study.
The optimal configuration obtained in the first scenario has a
TAC of $13.97 million yearÀ1
and is shown in Figure 5. There are
one boiler, three steam turbines, and one electric motor installed
in the steam system. A high pressure (HP) boiler is chosen for the
steam production. A HPÀmedium pressure (MP) back-pressure
steam turbine is used for the electricity generation. Part of
electricity import is required, which is 205 and 2319 kW for
periods 1 and 2, respectively. Shaft power demands 1 and 2 are
satisfied with steam turbines (two MPÀlow pressure (LP) back-
pressure steam turbines). The remaining shaft power demand is
satisfied with an electric motor.
From the result, it is found that the specified header tempera-
ture strategy restricts the design of SDN. Some feasible structure
or 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. The
same problem is solved again without the specified temperature
constraints. 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.

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variable to be optimized. It is expected to find appropriate
temperatures 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 HPÀMP steam turbine and a MPÀLP steam
turbine are installed to meet the need of shaft demands 1 and 3,
respectively, which replace the original MPÀLP turbine and the
electric motor. An electric motor is installed for the shaft demand
Figure 7. Optimal SDN design for scenario 3 in case 1.
Figure 6. Optimal SDN design for scenario 2 in case 1.
Table 4. Comparative Economic Parameters for the Major
Results 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 À10.73
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

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2. An HPÀLP steam turbine is installed to meet the requirement
of electricity. The electricity import decreases to 0 kW. The
optimized temperatures of steam headers are 356.3, 284.9, and
159.9 °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 yearÀ1
. It is evident that the
TAC 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%) are
considered to show the capability for the design of multiperiod
operating. The set of demands is presented in Table 2. HP steam
demands are requested in period 3 and period 4. Note that
electricity export is allowed and its price is assumed the same as
the import price.
Figure 7 shows the network layout and the corresponding
operating state. The TAC of this design is $12.06 million yearÀ1
.
As can be seen, optimized header temperatures are determined.
Clearly, the operating temperatures are not unique, which are
changed with varying demands. Mechanical demands are satis-
fied with one steam turbine and two electric motors. The high
pressure steam turbine exhausts mainly to the medium pressure
header for electricity generation. Steam systems tend to generate
more electricity during the first, second, and third time period,
and therefore the excess electricity is exported. Electricity import
is necessary only in the last period. Since the steam demands are
lower than in other periods, the steam available to generate elec-
tricity is lower, too. In this example the ability of the proposed
model to choose the best option for electricity generation and the
optimal configuration of power generating devices, and their
influence 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 TAC
can be accomplished. More detailed information for these
scenarios 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 flow
rates (CP) of two hot streams and three cold streams are
identified (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.

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Industrial Engineering Chemistry Research ARTICLE
shown in Table 6. The annual cost is 1200[area (m2
)]0.6
for all
exchangers.12
The minimum temperature difference for the
design of HEN is 10 K. In this case, the objective is to optimize
the 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 specified
previously, but are optimized by the proposed approach. This
steam system provides the multiple utilities for the heat recovery
network. The steam level decisions are the trade-off results of
simultaneous consideration for SDN and HEN. In SDN, one HP
boiler and three back-pressure steam turbines are included. Two
turbines are used for the shaft power demands and one is for
electricity generation. A part of the electricity demand is satisfied
with the steam system (2040 kW), and another part is from the
import (2707 kW). The optimal HEN is shown in Figure 9,
where four heat exchangers, two heaters, and one cooler are
included. 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 yearÀ1
.
6. DISCUSSION
In scenario 1 of case 1, temperatures of steam headers are
specified 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 be
restricted. In scenario 2, the header temperatures are considered
as decision variables and a one-step procedure is developed with
the ability to optimize the network structure and the operating
conditions simultaneously. It is appreciated that the better design
and operation can be accomplished under this approach. In
scenario 3, four periods with electricity import and export are
studied. The result reveals that steam turbines for the electricity
generation are preferred due to the higher operating flexibility.
Thus, the steam system can maintain higher operating efficiency
throughout all periods.
In case 2, simultaneous design for SDN and HEN is studied.
The proposed model can determine both the moderate operating
conditions and the corresponding network for the steam system
and the heat recovery system. The operating condition deter-
mination can affect the operation efficiency for steam systems
and the heat recovery circumstance for the given chemical
process. In this work, the interaction between two systems can
be optimized.
7. CONCLUSION
Changes in specifications, composition of feed, and seasonal
product demands may cause several process conditions with
variation in the energy requirements during an annual horizon. In
the first part, an MINLP model, based on unit superstructures,
has been developed to design a steam system with variable utility
demands. Complex multiperiod scenarios were studied that
together consider the design and operation of steam power
systems in an industrial plant. In the second part, a novel
methodology has been developed to address the design of a
steam system and a heat recovery network. This work determines
the optimal structure for both SDN and HEN, and also estimates
the moderate operating conditions. The results from the case
studies 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
are
itemized in Table 7.
’ AUTHOR INFORMATION
Corresponding Author
*Tel.: 886-2-33663039. Fax: 886-2-23623040. E-mail: CCL@
ntu.edu.tw.
’ 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
Indices
b = index for boilers
c = index for cold process or utility streams
g = index for gas turbines
h = index for hot process or utility streams
i = index for steam headers
Table 7. Investment Cost Data4
unit investment cost ($/year)
(1) field erected boiler (VHP) 22970F0.82
fp1
F, maximum steam flow rate (tons/h) fp1 = 0.6939 þ 0.1214P
À 3.7984À3
P2
P, pressure (MPa)
(2) large package boiler 4954F0.77
fp2
F, maximum steam flow rate (tons/h) fp2 = 1.3794 À 0.5438P
þ 0.1879P2
P, pressure (MPa)
(3) heat recovery steam generator 941Ffg
0.75
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 79.25FB
FB, maximum BFW flow rate (tons/h)
(9) condensor 3977 þ 1.84Fc
Fc, maximum cooling
water flow rate (tons/h)
(10) centrifugal pump (475.3 þ 34.95Pw
À 0.0301Pw2
)fpw
Pw, power (kW)
fpw = 1 (1.03 MPa)
fpw = 1.62 (1.03À3.45 MPa)
fpw = 2.12 (3.45 MPa)

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j = index for shaft power demands
k = index for stages
p = index for time periods
t = index for steam turbines
u = index for fuels
Sets
B = {b|b is a boiler, b = 1, ..., B} = MB ∪ HB
MB = {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 ∪ MT
ST = {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}
Parameters
C*
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}
Cp
w
= cost per unit mass of demineralized water makeup in time
period p, $ kgÀ1
Cp
cw
= cost per unit mass of cooling water in time period p, $ kgÀ1
Cp
imp,e
= specific cost of imported electricity in time period p,
$ kWhÀ1
Cp
exp,e
= specific cost of exported electricity in time period p,
$ kWhÀ1
Cu = cost per unit mass of fuel u, $ kgÀ1
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}
Hi
sat,l
= enthalpy of saturated steam at steam header i level,
kJ kgÀ1
Hip
ps
= enthalpy of steam supplied by processes and delivered at
header i in period p, kJ kgÀ1
Hp
w
= enthalpy of demineralized water makeup in period p,
kJ kgÀ1
Hdeaer
= enthalpy of water leaving a deaerator, kJ kgÀ1
tp
hrs
= number of operating hours in time period p, h periodÀ1
Tcp
in
= inlet temperature of cold process stream c in period p, °C
Tcp
out
= outlet temperature of cold process stream c in period p, °C
Thp
in
= inlet temperature of hot process stream h in period p, °C
Thp
out
= outlet temperature of hot process stream h in period p, °C
wjp
dem,s
= shaft power demand j in time period p, kW
wp
dem,e
= total electricity demand in time period p, kW
Ωh = upper bound for heat exchange, kW
Ωhb, Ω
h
b = upper and lower bounds of steam flow rate for boiler b,
kg sÀ1
Ωhg, Ω
h
g = upper and lower bounds of gas flow rate for gas turbine
g, kg sÀ1
Ωhii0
t, Ω
h
ii0
t = upper and lower bounds of steam flow rate for steam
turbine t, kg sÀ1
Ωh, Ω
h
= arbitrary very large value and very small value
Γhg, Γ
h
g = upper and lower bounds of power generation for gas
turbine g, kW
Γhii0
t, Γ
h
ii0
t = upper and lower bounds of power generation for steam
turbine t, kW
Φhg, Φ
h
g = maximum and minimum operating temperatures for gas
turbine g, °C
j = fixed blowdown fraction for boilers
ηm
= fixed efficiency for electric motors
Continuous Variables
dtcp
hu
= temperature approach for the match of cold stream c and
hot utility in period p, °C
dthckp = temperature approach for match (h, c) at temperature
location k in period p, °C
dthp
cu
= temperature approach for the match of hot stream h and
cold utility in period p, °C
fb
max
= maximum steam flow rate for boiler b, kg sÀ1
fbip = steam output from boiler b to steam header i in time period
p, kg sÀ1
fbip
bd
= blowdown water for boiler b at pressure i in time period p,
kg sÀ1
fbp
bfw
= boiler feed water for boiler b in time period p, kg sÀ1
fbup = fuel u consumed in boiler b in time period p, kg sÀ1
fchkp = heat capacity flow rate for cold stream c at stage k in time
period p, kW/°C
fd
max
= maximum water flow rate for deaerator, kg sÀ1
fgbp = exhaust gas from gas turbine g to HRSG b in time period p,
kg sÀ1
fgp = gas turbine g exhaust mass flow rate, kg sÀ1
fgup = fuel u consumed in gas turbine g in time period p, kg sÀ1
fhckp = heat capacity flow rate for hot stream h at stage k in time
period p, kW/°C
fii0
tp = steam flow rate from header i to header i0
through a steam
turbine t in time period p, kg sÀ1
fii0
tp
0
= input steam flow rate from header i to header i0
through a
multistage steam turbine t in time period p, kg sÀ1
fii0
i00
tp
0
= steam flow rate from stage i0
to stage i00
for a multistage
steam turbine t in time period p, kg sÀ1
fii0
p = steam flow rate from header i to header i0
in time period p,
kg sÀ1
fip = steam flow rate from header i to deaerator in time period p,
kg sÀ1
fip
ld
= desuperheating boiler feed water injected into header i in
time period p, kg sÀ1
fip
pd
= steam process demand at header i in time period p, kg sÀ1
fip
ps
= steam from process entering header i in time period p, kg sÀ1
fip
vent
= vented steam at header i in time period p, kg sÀ1
fp
c
= condensate return in period p, kg sÀ1
fp
w
= demineralized water makeup in time period p, kg sÀ1
fp
cw
= cooling water mass flow rate for condensers in time period
p, kg sÀ1
hbip = enthalpy of steam generated by boiler b entering header i in
period p, kJ kgÀ1

13. 8109 dx.doi.org/10.1021/ie102059n |Ind. Eng. Chem. Res. 2011, 50, 8097–8109
Industrial Engineering Chemistry Research ARTICLE
hii0
tp = enthalpy of a discharge by steam turbine t entering header
i0
in period p, kJ kgÀ1
hip = enthalpy of steam header i in period p, kJ kgÀ1
hp
c
= enthalpy of returning condensate from processes in period p,
kJ kgÀ1
qbp = heat added to the water in boiler b in time period p, kW
qcp
hu
= heat exchanged between hot utility and cold stream c in
time period p, kW
qhckp = heat exchanged between hot stream h and cold stream c in
stage k in period p, kW
qhp
cu
= heat exchanged between hot stream h and cold utility in
time period p, kW
tchkp = temperature of cold stream c at stage k in time period p, °C
tckp = temperature of cold stream c at hot end of stage k in time
period p, °C
Tgbp = exhaust gas temperature for gas turbine g delivered to
HRSG b in time period p, °C
Tgp = exhaust gas temperature for gas turbine g, °C
thckp = temperature of hot stream h at stage k in time period p, °C
thkp = temperature of hot stream h at hot end of stage k in time
period p, °C
wg
max
= design/maximum gas turbine g power output, kW
wgp = power produced by gas turbine g in time period p, kW
wgjp = shaft power produced by gas turbine g to shaft demand j in
period p, kW
wii0
tp = power produced by steam turbine t in period p, kW
wii0
tjp = shaft power produced by steam turbine t to shaft demand
j in period p, kW
wm
max
= design/maximum power for electric motor m, kW
wmjp = shaft power produced by electric motor m to shaft demand
j in period p, kW
wt
max
= design/maximum steam turbine t power output, kW
wp
imp,e
= electricity imported in time period p
wp
exp,e
= electricity exported in time period p
Binary Variables
zb = denotes the presence of boiler b
zbp = denotes the operating status of boiler b
zbi = denotes the existence of the connection between boiler b
and header i
zbip = denotes the existence of steam flow from boiler b to header
i in time period p
zcp
hu
= denotes that hot utility exchanges heat with cold stream c in
period p
zg = denotes the presence of gas turbine g
zgb = denotes the existence of the connection between gas turbine
g and boiler b
zgbp = denotes the existence of gas flow from gas turbine g to
boiler b in time period p
zgj = denotes the existence of the connection between gas turbine
g and shaft demand j
zgjp = denotes the existence of shaft demand j supplied by gas
turbine g in time period p
zgp = denotes the operating status of gas turbine g in time period p
zhckp = denotes the existence of match (h, c) in stage k in period p
zhp
cu
= denotes that cold utility exchanges heat with hot stream h in
period p
zt = denotes the presence of steam turbine t
zii0
t = denotes the existence of the connection of steam turbine t
between i and i0
headers
zii0
i00
tp = denotes the existence of steam flow from stage i0
to stage
i00
of a multistage steam turbine t in time period p
zii0
tj = denotes the existence of the connection between steam
turbine t and shaft demand j
zii0
tjp = denotes the existence of shaft demand j supplied by steam
turbine t in time period p
zii0
tp = denotes the operating status of steam turbine t in time
period p
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