Global Optimization of
Nonconvex MINLP Problems
by Domain and Image
Partitioning: Applications to
Heat Exchanger Networks
Débora C. Faria and Miguel J. Bagajewicz*
University of Oklahoma
100 E. Boyd St, Room T335
Norman, OK 73019
USA
Keywords: Global Optimization, Heat Exchanger Networks.
(*) Corresponding Author: e-mail: [email protected], TE:+1-405-325-5458
1
Abstract
We extend a recently introduced methodology for global optimization of bilinear MINLP models
(Faria and Bagajewicz, 2011b). The method for bilinear MINLP problems is based on variable
partitioning and bound contraction eventually followed under certain conditions by a combination
of branch & bound and bound contraction at each node. We extend the method to full nonconvex
MINLP problems by partitioning the image of nonconvex functions in addition to partitioning
their domain. To illustrate the method we focus on the Heat Exchanger Network Stagewise
model.
2
Introduction
There are several global optimization methods, some even commercially available
like BARON (Sahinidis, 1996), COCOS, GlobSol, ICOS, LGO, LINGO, OQNLP,
Premium Solver, and others that are well-known like the αBB (Androulakis et al., 1995).
Many are described
in several books (Sherali and Adams, 1999; Floudas, 2000a;
Tawarmalani and Sahinidis, 2002; Horst and Tuy, 2003; Hansen and Walster, 2004) and
recent paper reviews (Floudas 2000b; Pardalos et al., 2000; Floudas et al., 2005; Floudas
and Gounaris, 2009).
Different approaches for global optimization exist: Lagrangean-based approaches
(Ben-Tal et al., 1994; Adhya et al., 1999; Kuno and Utsunomiya, 2000; Karuppiah and
Grossmann, 2008), disjunctive programming-based methods (Ruiz and Grossmann,
2010), Cutting plane methods (Westerlund et al., 1998), Intervals analysis arithmetic
(Moore, 1966; Hansen, 1979; Ratschek and Rokne; 1988; Moore et al., 1992;
Vaidyanathan and El-Halwagi, 1994; Byrne and I. D. L. Bogle, 1999), Branch-andreduce (Ryoo and Sahinidis, 1995), GMIN-αBB and SMIN-αBB (Adjiman et al., 1997),
spatial branch and bound (Smith and Pantelides, 1997), branch and cut (Kesavan and
Barton, 2000a) and branch and contract (Zamora and Grossmann 1998; Zamora and
Grossmann,
1999),
decomposition
through generalized
benders decomposition
(Geoffrion, 1972) or outer approximation (Duran and Grossmann, 1986; Fletcher and
Leyffer, 1994), both restricted to certain class of problems (Bagajewicz and
Manousiouthakis, 1991; Sahinidis and Grossmann, 1991) as well as others (Kesavan and
Barton, 2000b). The list is not exhaustive and we omit stochastic based methods as they
do not guarantee optimality.
In previous work (Faria and Bagajewicz, 2011a,b,c) we address problems that
contain bilinear terms and univariate concave terms. In particular (Faria and Bagajewicz,
2011b) uses some previously proposed partitioning methods to obtain lower bounds as
well as some other techniques: Sherali and Alamedinne, 1992; Meyer and Floudas,
3
2006); Misener and Floudas, 2009; Gounaris and Floudas, 2009; Karuppiah and
Grossmann (2006a); Karuppiah and Grossmann (2006b); Bergamini et al., (2005, 2008)
Wicaksono and Karimi (2008), Hasan and Karimi (2010), Pham et al. (2009).
While bilinear terms abound in process engineering because they represent
products of flowrates and concentrations in component balances, univariate concave
terms show up in cost functions. Although problems with only these type of terms
abound (water management, pooling problems, etc), there are several other problems
where additional functions are used. In particular, in this paper, we explore the use of
heat exchanger area calculations.
Following the ideas of these papers, we presented a variable partitioning
methodology to obtain lower bounds and a new bound contraction procedure. Our lower
bound model uses some modified versions of well-known over and underestimators
(some of which used in the literature review above), to obtain MILP models. Our
procedure differs from most of the previous approaches based on LB/UB schemes in that
it does not use a branch and bound methodology as the core of the method. Instead, we
first partition certain variables into several intervals and then use a bound contraction
procedure directly using an interval elimination strategy. Conceptually, the technique can
work if a sufficiently high number of intervals is used, but if that becomes
computationally too expensive, we allow a spatial branch and bound to be used.
The paper is organized as follows: We present the nonlinear model first, followed
by a description of the domain/image partition lower bound model. We then make a brief
review of the bound contraction procedure. Finally, examples are presented.
4
Domain/Image Partitioning
Consider the following problem
(1)
Min x0
∀x , y
s.t.
Ax + Bf ( x) + Cg ( x) ≤ d
(2)
x L ≤ x ≤ xU
(3)
where f ( x) is non-convex and g ( x) is convex. To isolate the nonconvexities, we
introduce a new variable y and write:
(4)
Min x0
∀x , y
s.t.
Ax + By + Cg ( x) ≤ d
(5)
y = f ( x)
(6)
x L ≤ x ≤ xU
(7)
Our proposed relaxation method is based on the local properties of the function
f ( x) and a partitioning of the independent variables (x). We start with the partitioning
by defining discrete values of xi as follows:
(
)
xˆi ,di ,k = xiL + di ,ki −1
i
(x
U
i
− xiL
Di
)
∀di ,ki = 1..,( Di + 1)
(8)
where d i , ki is an index pointing to the specific discrete value generated and Di the total
number of intervals between discrete values generated. The first discrete value is xiL and
the last is xiU . We next proceed to partition the domain for xi formally by writing:
5
∑r
i , ki
∀ki =1,...,Di
=1
(9)
ki
∑ xˆ
r
i , di ,ki i , ki
ki
≤ xi ≤ ∑ xˆi ,di ,k +1ri ,ki
i
(10)
i
Then, the relaxation of (6) consists of the following linear relations:
yi ≥ fi ( xˆ1,d
, xˆ2,d
yi ≤ fi ( xˆ1,d
, xˆ2,d
1,k1 + λi ,1,k1
1,k1 +1− λi ,1,k1
2,k2 + λi ,2,k2
,..., xˆm,d
2,k2 +1− λi ,2,k2
m ,km + λi ,1,km
,..., xˆm,d
m , km
⎛
⎞
) − M ⎜ m − ∑ ri ,ki ⎟
i
⎝
⎠
+1−λi ,1,km
∀i, ki
⎛
⎞
) + M ⎜ m − ∑ ri ,ki ⎟
i
⎝
⎠
∀i, ki
(11)
(12)
where M is a large enough value. The value of λi , j ,k j is set in such a way that λi , j ,k j =0
∂fi ( xˆ1,d1,k , xˆ2,d2,k ,..., xˆm,dm ,k )
when
1
2
∂x j ,d j ,k
≥ 0 and λi , j ,k =1 otherwise. We assume that the sign of
m
j
j
the derivative does not change in the box in corresponding to the set of binaries ri ,ki .
Thus, when all binary variables ri ,ki are equal to one, the value of yi will be bracketed
f i ( xˆ1, d
between
f i ( xˆ1, d
1,k1 +1− λi ,1,k1
1,k1 + λi ,1,k1
, xˆ2,d
2,k2
+1− λi ,2,k2
,..., xˆm ,d
m ,km
, xˆ2,d
+1− λi ,1,km
2,k2
+ λi ,2 ,k2
,..., xˆm , d
m ,km
+ λi ,1,km
)
and
).
With the above substitutions, the relaxed problem is a convex MINLP, which is
convenient. In addition, if g(x) is linear, then it is MILP. In both cases the problem is a
lower bound.
The methodology is not only directly applicable in situations where the modeling
of certain input-output relations for processes in flowsheets is given by a system of
nonlinear equations (distillation columns), but can also be used when f i ( x) represents
the output obtained integrating ordinary/partial differential equations (reactors). Indeed,
6
one can proceed to integrate the ODE for the discrete values of x given. We do not
attempt it here.
Bound Contraction
Once a lower bound is obtained, the interval elimination strategy presented by
Faria and Bagajewicz (2011b) can be implemented. This novel global optimization
strategy is now summarized as follows:
-
Construct a lower bound model partitioning variables in bilinear and quadratic
terms, thus relaxing the bilinear terms as well as adding piece-wise linear
underestimators of concave terms of the objective function.
-
The lower bound model is run identifying certain intervals as containing the
solution for specific variables that are to be bound contracted. These variables
need not be the same variables as the ones using to construct the lower bound.
-
Based on this information the value of the upper bound found by running the
original MINLP using the information obtained by solving the lower bound
model to obtain a good starting point.
-
If the objective function gap between the upper bound solution and the lower
bound solution is lower than the tolerance, the solution was found. Otherwise
go to the next step.
-
If the new problem is infeasible, or if feasible but the objective function is
higher than the current upper bound, then all the intervals that have not been
forbidden for this variable are eliminated. The surviving feasible region
between the new bounds is partitioned again.
-
Repeat the last 2 steps for all the other variables, one at a time.
-
Go back to the first step (a new iteration using contracted bounds starts).
-
Different options for bound contracting have been proposed (Faria and
Bagajewicz, 2011b): One-pass interval elimination, cyclic elimination, single
and extended interval forbidding, etc., all of which are detailed in the article
referenced.
7
-
The process is repeated with new bounds until convergence or until the
bounds cannot be contracted anymore.
-
If the bound contraction is exhausted, there are two possibilities to guarantee
global optimality:
o Increase the partitioning of the variables to a level in which the sizes
of the intervals are small enough to generate a lower bound within a
given acceptable tolerance to the upper bound; or,
o Recursively split the problem in two or more sub-problems using a
strategy such as the ones based on branch and bound procedure.
Note that to guarantee the optimality, not all of the lower bound models need to
be solved to zero gap. The only problems that need to have zero gap are the ones in
which the lower bound of the problem (or sub-problems) are obtained, which is done in
the first step. The LB models used to eliminate intervals can be solved to feasibility
between its lower bound and the current upper bound, which is set as the UB of the whole
procedure. Faria and Bagajewicz (2011b) applied the above scheme to bilinear problems,
where the relaxation rendered an MILP problem.
Heat Exchanger network Model
A comprehensive overview of advances in this field of research is given in
Furman and Sahinidis (2002). Among the superstructure-based models for HEN design,
the most popular one is a stage-wise superstructure approach (Yee and Grossmann,
1990).
Several approaches were proposed to obtain globally optimal solutions to the
Synheat model. Zamora and Grossmann (1998) proposed a global optimization algorithm
to rigorously optimize the Synheat model under the simplifying assumptions of linear
area cost functions and no stream splitting. The approach relies on the use of convex
underestimators for the heat transfer area. Later, the approach was extended to account
for the nonlinear area cost functions (Zamora and Grossmann, 1997). Adjiman et al.
8
(2000) solved the HEN synthesis problem under the assumption of linear area cost
functions, using SMIN-αBB algorithm. Björk and Westerlund (2002) covered stage-wise
models and solved HEN synthesis problems to global optimality under isothermal and
non-isothermal mixing assumption. The most recent attempt to solve the Synheat model
to global optimum is the one proposed by Bergamini et al. (2007). Their strategy is based
on an outer approximation methodology, employing piece-wise underestimators of nonconvex terms and physical insights.
The stage-wise superstructure of the model enables both parallel, as well as in
series decoupling of heat exchangers. A two stage superstructure for two hot and two
cold streams is presented in Figure 1.
Figure 1: Stage-wise superstructure.
Heat can be transferred between each pair of hot and cold streams in each stage,
and if the stream splits in a stage it is remixed isothermally before entering the next stage.
The isothermal mixing assumption eliminates the requirement for non-linear heat
9
balances around heat exchangers as well as non-linear heat mixing equations. In the
original formulation by Yee and Grossmann (1990) the feasible space is defined by
strictly linear constraints. Nonetheless, the model is non-convex due to the presence of
non-linear, non-convex terms in objective function related to area costs.
The non-convex MINLP model presented in this work differs slightly from the
one reported by Yee and Grossmann (1990). The differences are:
• The areas of heat exchangers are used explicitly in the objective function
(The original Synheat model uses the ratio of the heat transferred to the
log mean temperature difference).
• The area costs are assumed to be linearly dependant on the areas, thus
making the objective function linear.
• Because the areas of heat exchangers are explicitly defined in the
objective function, new constraints to calculate them are incorporated.
Although the second assumption can be conceptually challenged, Barbaro and
Bagajewicz (2005) argued that linear approximations of cost equations can be justified by
the fact that they already carry an inherent uncertainty.
We now present the model equations. The reader is referred to the paper by Yee
and Grossmann (1990) for explanations of each equation. We introduce explanations
about what is new.
•
Objective function
∑ ∑ ∑C
min
i∈HP j∈CP k∈ST
+
∑C
j∈CP
+∑
z
HE j HU j
∑ ∑C
i∈HP j∈CP k∈ST
z
HE i , j i , j ,k
+ ∑ CHE i zCU i
i∈HP
+ ∑ CCU qCU i +
i∈HP
A i, j
∑C
j∈CP
HU
Ai , j ,k + ∑ C A i ACU i +
i∈HP
(13)
qHU j
∑C
j∈CP
Aj
AHU j
As described above, the area cost has been linearized, although concave terms can be
used.
10
•
Overall heat balance for each stream
(T
H
H
− TOUT
i ) CFi =
(T
− TINC j ) CF jC =
H
IN i
C
OUT j
•
i , j ,k
+ qCU i
i ∈ HP
(14)
∑ ∑q
i , j ,k
+ qHU
j ∈ CP
(15)
k ∈ ST , i ∈ HP
(16)
k ∈ ST , j ∈ CP
(17)
k ∈ST i∈HP
j
Stage heat balance
(T
H
i ,k
(T
•
∑ ∑q
k∈ST j∈CP
C
j ,k
− TiH,k +1 ) CFiH =
− T jC,k +1 ) CF jC =
∑q
j∈CP
i , j ,k
∑q
i∈HP
i , j ,k
Superstructure inlet temperatures
TINH i = Ti H,1
i ∈ HP
(18)
TINC j = T jC,NOK +1
j ∈ CP
(19)
•
Feasibility of temperatures (monotonic decrease in temperatures)
Ti H,k ≥ Ti H,k +1
k ∈ ST ,i ∈ HP
(20)
H
Ti H,NOK +1 ≥ TOUT
i
i ∈ HP
(21)
T jC,k ≥ T jC,k +1
k ∈ ST , j ∈ CP
(22)
C
T jC,1 ≥ TOUT
j
j ∈ CP
(23)
•
Hot and cold utility load
H
qCU i = CFi H (Ti H,NOK +1 − TOUT
i)
C
C
qHU j = CFjC (TOUT
j − T j ,1 )
•
i ∈ HP
(24)
j ∈ CP
(25)
Approach temperatures
ΔT i , j ,k ≤ TiH,k − T jC,k + Γ (1 − zi, j ,k )
i ∈ HP, j ∈ CP, k ∈ ST
11
(26)
ΔT i , j ,k +1 ≤ TiH,k +1 − T jC,k +1 + Γ (1 − zi , j ,k )
i ∈ HP, j ∈ CP, k ∈ ST
(27)
ΔTCU i ≤ Ti H,NOK +1 − TOUT, CU + Γ (1 − zCU i )
i ∈ HP
(28)
ΔTHU j ≤ TOUT, HU − T jC,1 + Γ (1 − zHU j )
j ∈ CP
(29)
•
Minimum approach temperature (lower bounds)
ΔTi , j ,k ≥ EMAT
i ∈ HP , j ∈ CP , k ∈ ST
(30)
ΔTCU i ≥ EMAT
i ∈ HP
(31)
ΔTHU j ≥ EMAT
j ∈ CP
(32)
•
Logical constraints:
qi , j ,k − Ω zi , j ,k ≤ 0
i ∈ HP , j ∈ CP , k ∈ ST
(33)
qCU i − Ω zCU i ≤ 0
i ∈ HP
(34)
qHU j − Ω zHU j ≤ 0
j ∈ CP
(35)
•
Area calculations using Chen’s (1987) approximation for the logarithmic mean
temperature difference:
qi , j ,k − Ai , j ,kU i , j 3 ΔT i , j ,k ΔT i , j ,k +1
( ΔT
i , j ,k
+ ΔT i , j ,k +1 )
2
≤0
i ∈ HP , j ∈ CP , k ∈ ST (36)
H
⎫
⎡ ΔTCU i + (TOUT
⎤
i − TIN, CU ) ⎦
⎣
⎪
− TIN, CU )
≤0
qCU i − ACU iU i ,CU ΔTCU i (T
⎬
2
⎪
i ∈ HP, j ∈ CP, k ∈ ST
⎭
(37)
C
⎫
⎡ ΔTHU j + (TIN, HU − TOUT
⎤
j )⎦
⎣
⎪
qHU j − AHU jU j ,HU ΔTHU j (TIN, HU − T )
≤0
⎬
2
⎪
i ∈ HP, j ∈ CP, k ∈ ST
⎭
(38)
3
H
OUTi
3
C
OUTj
These are the new equations.
12
•
Additional variable bounds
H
H
H
TOUT
i ≤ Ti ,k ≤ TIN i
i ∈ HP
C
C
C
TOUT
j ≥ T j ,k ≥ TIN j
(39)
j ∈ CP
{
(40)
}
H
C
C
C
0 ≤ qi , j ,k ≤ min CFiH (TINH i − TOUT
i ) , CF j (TOUT j − TIN j )
i ∈ HP, j ∈ CP , k ∈ ST
(41)
H
0 ≤ qCU i ≤ CFi H (TINH i − TOUT
i)
i ∈ HP
(42)
C
C
0 ≤ qHU j ≤ CFjC (TOUT
j − TIN j )
j ∈ CP
(43)
Lower Bound Model
First we concentrate on the linearization of Chen’s approximation of logarithmic mean
D
temperature in Eq. (24). We discretize the temperature differences ΔlnTi , j ,k as follows:
ΔTi D, j ,k = ∑ ΔTi D, j ,k ,l yi , j ,k ,l
i ∈ HP, j ∈ CP, k ∈ ST, l ∈ L, m ∈ M ⊂ L
(44)
l
Then equation (24) can be rewritten as follows:
ΔlnTi D, j ,k = ∑∑ yi , j , k ,l ⋅yi , j , k +1, m
m
l
3
⎫
⎛
⎞
D
D
T
T
Δ
+
Δ
⎪
⎜ ∑ i , j ,k ,l ∑ i , j ,k +1, m ⎟
⎪
l
m
⎝
⎠
D
D
ΔTi , j ,k ,l +1ΔTi , j , k +1,m
⎬ (45)
2
⎪
i ∈ HP, j ∈ CP, k ∈ ST, l ∈ L, m ∈ M ⊂ L ⎪⎭
In turn equations (14) through (17) are rewritten as follows:
Ti H,k − T jC,k ≤ ∑ ΔTi D, j ,k ,l +1 yiL, j , k ,l + Γ (1 − zi , j , k )
i ∈ HP, j ∈ CP, k ∈ ST , l ∈ L, m ∈ M ⊂ L
(46)
l
Ti H, k − T jC, k ≥ ∑ ΔTi D, j ,k ,l yiL, j , k ,l − Γ (1 − zi , j , k )
i ∈ HP, j ∈ CP, k ∈ ST , l ∈ L, m ∈ M ⊂ L
(47)
l
Ti H, k +1 − T jC, k +1 ≤ ∑ ΔTi D, j ,k +1,m +1 yiM, j ,k , m + Γ (1 − zi , j , k ) i ∈ HP, j ∈ CP, k ∈ ST , l ∈ L, m ∈ M ⊂ L (48)
m
Ti H, k +1 − T jC,k +1 ≥ ∑ ΔTi D, j , k +1,m yiM, j ,k ,m − Γ (1 − zi , j , k ) i ∈ HP, j ∈ CP, k ∈ ST , l ∈ L, m ∈ M ⊂ L (49)
m
13
∑y
L
i , j , k ,l
= zi , j , k
(50)
i ∈ HP, j ∈ CP, k ∈ ST , l ∈ L, m ∈ M ⊂ L
l
∑y
M
i , j ,k ,m
i ∈ HP, j ∈ CP, k ∈ ST, l ∈ L, m ∈ M ⊂ L
= zi , j , k
(51)
m
Constraints (50) and (51) ensure that if match (i, j, k) exists only one partitioning
interval is selected.
We now rewrite equation (36) as follows:
qi , j , k
Ui, j
y
− Ai , j ,k ∑∑
l
m
L
i , j , k ,l
M
3
i , j , k +1, m
y
ΔT
D
i , j , k ,l +1
ΔT
D
i , j , k +1, m +1
( ΔT
D
i , j , k ,l +1
+ ΔTi D, j ,k +1, m +1 )
2
i ∈ HP, j ∈ CP, k ∈ ST, l ∈ L, m ∈ M ⊂ L
⎫
≤ 0⎪
⎬
⎪
⎭
(52)
Substituting the product of binaries ( yiL, j ,k ,l , yiM, j ,k +1,m ) and areas ( Ai , j ,k ) in equation
(52) with a new positive continuous variable ( hi , j ,k ,l ,m ), we obtain:
qi , j ,k
Ui, j
− ∑∑ hi , j ,k ,l ,m 3 ΔTi D, j ,k ,l +1ΔTiD, j ,k +1,m +1
l
( ΔT
D
i , j , k ,l +1
+ ΔTi D, j ,k +1,m+1 )
2
m
i ∈ HP, j ∈ CP, k ∈ ST, l ∈ L, m ∈ M ⊂ L
⎫
≤ 0 ⎪⎪
⎬
⎪
⎪⎭
(53)
To complement the above substitution valid, the following constraints are needed:
hi , j ,k ,l ,m − ΨyiL, j ,k ,l ≤ 0
i ∈ HP, j ∈ CP, k ∈ ST, l ∈ L, m ∈ M ⊂ L
(54)
hi , j ,k ,l ,m − ΨyiM, j ,k +1,m ≤ 0
i ∈ HP, j ∈ CP, k ∈ ST, l ∈ L, m ∈ M ⊂ L
(55)
hi , j , k ,l , m − Ai , j , k + ( 2 − yiL, j , k ,l − yiM, j , k +1, m ) ≥ 0 i ∈ HP, j ∈ CP, k ∈ ST, l ∈ L, m ∈ M ⊂ L (56)
hi , j , k ,l , m − Ai , j ,k ≤ 0
i ∈ HP, j ∈ CP, k ∈ ST, l ∈ L, m ∈ M ⊂ L
An analogous procedure is applied to linearize Eqs. (37) and (38).
14
(57)
Examples
Three examples of different sizes are presented in this section. The examples were
implemented in GAMS (Brooke et al., 1998) and solved using CPLEX as a MIP solver
and DICOPT (Viswanathan and Grossmann, 1990) a MINLP solver on a PC machine
(2.2 GHz, 2 GB RAM).
Example 1: The first example is a small two-stream stream example used to illustrate the
proposed approach in details. For this reason, a relatively large number (eight) of
partitioning intervals was chosen to ensure that only one elimination loop is needed to
satisfy the convergence criterion.
The data for the example is presented in Table 1. The fixed cost of units is
10 000 $, and the area cost coefficient is 350 $/m2. A single-stage superstructure was
used to solve this example. The lower-bounding MILP has 35 binary variables, of which
33 correspond to binary variables used for the partitioning, and 130 continuous variables.
Table 1: Example 1 data.
h
(kW / m 2K )
CF
kW/K
590 370
1.3
5
C1
395 670
0.5
15
CU
290 300
1.0
60
HU
680 680
5.0
120
Stream
TS
K
H1
TT
K
C
( $ / kWa )
EMAT = 5 K
According to the solution algorithm described in the previous section, the lowerbounding MILP is solved first to provide a lower bound, select the first set of partitioning
intervals, and initial point for the upper-bounding MINLP. The lower bound obtained is
467.330 k$.
15
In Figure 2, initial intervals for the partitioned variables (temperature differences)
are depicted. Denoted by a star are the intervals selected by a lower-bounding MILP.
Since this example consists of only two process streams and two utilities, only
one heat exchanger, one cooler, and one heater are possible in a one-stage superstructure.
Thus, only four temperature differences are variables which are subjected to partitioning.
These are: the temperature difference on a hot side (k = 1) of a heat exchanger ( ΔTH1,C1,1 ),
the temperature difference on a cold side (k = 2) of a heat exchanger ( ΔTH1,C1,2 ), the
temperature difference at the hot side of a cooler ( ΔTCU, H1 ), and the temperature
difference at the hot cold side of a cooler ( ΔTHU, C1 ).
Figure 2: Selection of intervals in lower-bounding MILP.
Next, the upper-bounding non-convex MINLP is solved, giving an upper bound
of 483.808 k$. The gap between the lower and upper bound equals 3.406 %. Because the
convergence criterion is not satisfied, elimination of intervals is performed in order to
tighten the feasible region of a lower-bounding problem.
In Figures 3–6 the one-pass interval elimination procedure is presented. The
procedure eliminates intervals – one partitioned variable at the time, starting with
ΔTHD1,C1,1 . First, the interval that was selected in the lower-bounding MILP (see Figure 2)
is forbidden to be selected. Solving the lower-bounding MILP under this constraint
produces a lower bound of 522.514 k$. Because it is higher than the current upper bound
16
(467.330 k$), all the intervals, except the one that was forbidden, can be eliminated. In
other words, the feasible space for variable ΔTH1,C1,1 is reduced from initial
ΔTH1,C1,1 ∈ [ EMAT , Γ ] to ΔTH1,C1,1 ∈ [115.6,152.5] . The surviving interval is re-partitioned
to the same number of intervals (eight).
Figure 3: Interval elimination for ΔTHD1,C1,1 .
The same procedure is repeated for the second partitioned variable ΔTH1,C1,2 ,
respectively. The available intervals are depicted in Figure 4. Note, that the available
discrete values for the temperature difference ( ΔTH1,C1,1 ) are the ones obtained after repartitioning in the previous elimination step (Figure 3).
Figure 4: Interval elimination for ΔTHD1,C1,2 .
17
The lower bound obtained equals 499.453 k$. Again, this lower bound is higher
than the current upper bound and all the allowed intervals ( ΔTHD1,C1,2 ≥ 41.9 ) are
eliminated. The remaining interval is re-partitioned.
The Figure 5 shows the available intervals for the third partitioned variable
( ΔTCU,H1 ). Unlike in the first two interval elimination steps, a feasible solution is obtained
with the objective value lower than the current upper bound.
D
Figure 5: Interval elimination for ΔTCU
,H1 .
Since the lower bound obtained in this elimination step (482.711 k$) is not higher
than the current upper bound (483.808 k$), none of the intervals can be eliminated.
D
Finally, the last partitioned variable ( ΔTHU
,C1 ) is subjected to the elimination
procedure. Figure 6 shows the available intervals.
18
D
Figure 6: Interval elimination for ΔTHU
,C1 .
Note, that none of the intervals for the first two partitioned variables were
selected, which means that this solution utilizes only a heater and a cooler. Thus,
forbidding the interval selected in the lower-bounding MILP produces a solution that has
a different HEN structure. The objective value for this problem is 627.592 k$ and as in
the first and the second elimination step the non-forbidden intervals are eliminated and
the forbidden one is re-partitioned. This concludes the first iteration.
In the second iteration, first, the lower-bounding MILP is solved, yielding a lower
bound of 479.607 k$. Next, the upper-bounding non-convex MINLP is solved. The upper
bound obtained equals 483.808 k$. The gap between lower and upper bound reduces
from 3.406 % (first iteration) to 0.868 %. Since the convergence criterion is met the
algorithm stops without triggering the second interval elimination loop.
The total CPU time needed to obtain the solution was 1.38 s, of which 60 %
correspond to MINLP resource usage. Solving the same example using NLP to obtain the
upper bound, reduces the total time needed to 0.65 s.
19
Example 2: The second example consists of two hot and two cold streams. The data is
given in Table 2. The fixed cost of units is 5 500 $, and the area cost coefficient is 150
$/m2. The globally optimal solution, depicted in Figure 7, has an annualized cost of
154.995 k$.
Table 2: Example 2 data.
h
(kW / m2K )
CF
kW/K
650 370
1.0
10
H2
590 370
1.0
20
C1
410 650
1.0
15
C2
350 500
1.0
13
CU
300 320
1.0
15
HU
680 680
5.0
80
Stream
TS
K
H1
TT
K
C
( $ / kWa )
EMAT = 10 K
Figure 7: Example 2 Solution
The example was solved using a two-stage superstructure, and different numbers of
partitioning intervals (2, 4, 6, 8, and 14).
20
Figure 8: Progression of lower bound (Example 2).
NOMENCLATURE
Sets:
L, M, N, O
discretization grid
HP
hot process streams
CP
cold process streams
ST
stages
I
hot process streams
J
cold process streams
K
stages
Parameters:
ΔTi D, j ,k ,l
discretized temperature difference in stage k; match (i , j)
D
ΔTCU
i ,n
discretized temperature difference on a hot side of a cooler; match (i, CU)
21
D
ΔTHU
j ,o
discretized temperature difference on a cold side of a heater; match (j, CU)
Ψ
upper bound on areas
β
area exponent
TIN
inlet temperature
TOUT
outlet temperature
CCU
cold utilities cost
CHU
cold utilities cost
CA
area cost parameter
CHE
fixed heat exchanger cost
NOK
number of stages
Ω
upper bound on exchanged heat
Γ
upper bound on temperature difference
EMAT
exchanger minimum approach temperature
Variables:
yiL, j ,k ,l
binary variable associated with partitioning interval l; match (i, j, k)
yiN,n
binary variable associated with partitioning interval n; match (i, CU)
y Oj ,o
binary variable associated with partitioning interval o; match (j, HU)
hi , j ,k ,l ,m
additional positive continuous variable; match (i, j, k)
hiCU
,n
additional positive continuous variable; match (i, CU)
hHU
j ,o
additional positive continuous variable; match (j, HU)
CF
heat capacity flow rate
ΔT i , j ,k
temperature difference in stage k for match between stream i and j
ΔTCU i
temperature difference for match between stream i and cold utility
ΔTHU j
temperature difference for match between stream j and hot utility
qi , j ,k
exchanged heat for ( i, j ) match in stage k
qCU i
cold utility demand for stream i
22
qHU j
hot utility demand for stream j
Ti H,k
temperature of hot stream i on the hot side of stage k
T jC,k
temperature of cold stream j on the hot side of stage k
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