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Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng

An interactive multi-objective approach to heat exchanger network synthesis Timo Laukkanen a,∗ , Tor-Martin Tveit a , Vesa Ojalehto b , Kaisa Miettinen b , Carl-Johan Fogelholm a a b

Aalto University, School of Science and Technology, Department of Energy Technology, P.O. BOX 14400, FI-00076 Aalto, Finland Department of Mathematical Information Technology, P.O. Box 35 (Agora), FI-40014 University of Jyväskylä, Finland

a r t i c l e

i n f o

Article history: Received 24 June 2009 Received in revised form 22 December 2009 Accepted 5 January 2010 Available online 15 January 2010

a b s t r a c t In this work we present a multi-objective approach to heat exchanger network synthesis. The approach solves a modiﬁed version of the Synheat model using an interactive multi-objective optimisation method, NIMBUS, which is implemented in GAMS. The results obtained demonstrate the potential of interactive multi-objective optimisation. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Pareto optimality Synheat model Multi-objective optimisation MINLP NIMBUS GAMS

1. Introduction

solving an optimisation problem of the form:

The heat exchanger network synthesis problem is a very important problem for the efﬁcient use of energy, and a lot of research work has been put into developing models for these network synthesis problems. One important methodology for heat exchanger network synthesis is the pinch technology (see for instance Hohmann, 1971; Linnhoff, Mason, & Wardle, 1979). In addition to the pinch methodology, mathematical programming has also been widely applied in these problems. The common mathematical formulations of the heat exchanger network synthesis problem are hard to solve, and have been proven to belong to the class of NP-hard problems by Furman and Sahinidis (2001). Mathematical programming models can roughly be divided into sequential methods and simultaneous methods. The sequential methods decompose the problems into sub-problems. The most widely used sequential method consists of three optimisation models that are solved in a series, where the ﬁrst model minimises the utility cost Cerda, Westerberg, Mason, & Linnhoff, 1983 and Papoulias and Grossmann (1983), the second model minimises the number of units (Cerda & Westerberg, 1983) and the third model minimises the investment cost related to the size of the heat exchangers (Floudas, Ciric & Grossmann, 1986). This leads to

min s.t

∗ Corresponding author. Tel.: +358 0 451 3637; fax: +358 9 451 3418. E-mail address: [email protected]ﬁ (T. Laukkanen). 0098-1354/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2010.01.002

(cost of heat exchanger surface area) min(number of units) s.t. min(utility consumption).

(1)

In order to better take trade-offs into account, several simultaneous methods have been developed, of which the Synheat model by Yee and Grossmann (1990) is one. Another example is the MINLP model by Ciric and Floudas (1991), where the utility load is an explicit variable, heat is allowed to ﬂow through the pinch and the optimal structure is selected from a hyperstructure containing all alternative matches and network conﬁgurations. A good review of the literature on heat exchanger network synthesis can be found in the work by Furman and Sahinidis (2002). The objective for most of the research done on the heat exchanger network synthesis problems can be formulated as follows: The objective is to design a heat exchanger network that minimises the total annualised cost, given sets of hot streams, cold streams, hot utilities and cold utilities. Each hot and cold stream has a speciﬁc heat capacity ﬂowrate, a start- and target temperature. In this formulation, the multi-objective nature of the heat exchanger network synthesis problem has been transformed into a single-objective problem using annualised cost. However, the true nature of the problem can be better captured by formulating several individual objectives in the modelling phase instead of only one. The resulting problem with conﬂicting objectives can then be solved by using multi-objective optimisation

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(see, e.g. Miettinen, 1999) methods. Multi-objective optimisation enables consideration of interdependencies between the conﬂicting objectives and one can in that way learn about the problem considered. On the other hand, it is often also easier to formulate individual objectives because annualised cost does not need to be estimated. In multi-objective optimisation problems, we can identify socalled Pareto optimal solutions where none of the objectives can be improved without impairing at least one of the others. An optimal solution is called weakly Pareto optimal, if improving all of the objectives of the solution is not possible simultaneously. If optimised objectives are conﬂicting, there usually exist several (weakly) Pareto optimal solutions. Which of them is to be selected as the most preferred one depends on the preferences of a decision maker (DM). Both of the previously mentioned simultaneous mathematical programming approaches for HENS use the objective stated earlier, and subsequently treat the heat exchanger synthesis problem in principle as a single objective problem. Even though most authors acknowledge that the problem actually has multiple objectives, there has been very little work on fully treating the problem as a multi-objective optimisation problem. Published work in the open literature on multi-objective optimisation of heat exchanger network synthesis, are the works by Björk and Westerlund (2002b); Chen and Hung (2005) and Agarwal and Gupta (2008). In their work, Björk and Westerlund (2002b) discuss the limitations of using linear weights to obtain the Pareto set for heat exchanger network synthesis problems. Chen and Hung (2005) developed a multiobjective mixed-integer linear programming model for ﬂexible heat exchanger network synthesis, that simultaneously considers minimum utility consumption, maximum source–stream temperature ﬂexibility and minimum number of matches. They applied a fuzzy multi-objective decision-making method to deal with the objectives of their model. Agarwal and Gupta (2008) used a genetic algorithm to generate a set of solutions approximating Pareto optimal solutions of heat exchanger network synthesis problems. All of the multiobjective HENS works described above use a posteriori methods, where the DM is presented with a set of Pareto optimal solutions after its generation. For a heat exchanger network synthesis problem this generation is computational expensive and the DM has to select one solution from a very large set of solutions, which can be cognitively very demanding. These two problems associated with a posteriori methods, can be avoided using interactive methods, which is why we have chosen to focus on an interactive method in this work. To be more speciﬁc, in this paper we will use NIMBUS (Miettinen, & Mäkelä, 1999; Miettinen, & Mäkelä, 2006), a multi-objective optimisation method, to solve a multi-objective formulation of the heat exchanger network synthesis problem based on the Synheat model by Yee and Grossmann (1990). The rest of this paper is organized as follows. Next in Section 1.1 the Synheat model is presented. In Section 2 the multiobjective formulation of the Synheat model and the interactive multiobjective NIMBUS method are presented. In Section 3 the proposed tool, called A–GAMS–NIMBUS-tool, is presented together with the needed models in different steps of the integrated method. In Section 4 two simple HENS examples obtained from the literature are solved with the presented approach. Finally in Section 5 the main conclusions are given.

wise representation and the assumption of isothermal mixing at the end of each stage. The advantage of this superstructure is that the dimensionality of the MINLP model is reduced and that the constraints deﬁning the feasible region are linear. The tradeoffs between operational costs (utility cost in this case) and the investment costs are taken into account by annuity factors. The nonlinearities in the model are all in the objective function and are due to the area cost of the heat exchangers. The objective function of the Synheat model is given in Problem (2) and the constraints are given in Eqs. (3) through (9): min total cost = cold utility cost + hot utility cost + ﬁxed cost per units + area cost

(2)

s.t.energy balance for each stream

(3)

energy balance for each stage

(4)

calculation of hot and cold utility need

(5)

assignment of inlet temperatures

(6)

feasibility of temperatures

(7)

logical constraints for matches

(8)

calculation of approach temperatures

(9)

The mathematical formulation of the objective function is given in full in Appendix A. As can be seen from Problem (A.1), the optimisation problem can be written in the form of min

k

wi fi (x) +

i=1

s.t. x ∈ S,

m

wj fj (x)ˇj

(10)

j=1

where S is the feasible region deﬁned by Eqs. (3) through (9). The cold utility cost, hot utility cost and ﬁxed cost per units are included in

k

m

wi fi (x) and area cost is j=1 wj fj (x)ˇj . The functions fi and fj are the cold utility consumption, the hot utility consumption, the number of units and the heat exchanger surface area. By considering the functions fi and fj as objective functions, instead of the cost related to them, the formulation in Problem (10) can be seen as a multiobjective optimisation problem transformed into a single-objective function by associating weighting coefﬁcients to the objective functions. The formulation described is known as a weighting method (see, e.g. Miettinen, 1999; Steuer, 1986). Unfortunately, this method has several shortcomings. Firstly, it is not necessarily easy to get desirable solutions by setting weights, because their role and actual meaning is not understandable to the DM. They are often claimed to represent relative preferences of objectives but this is not clear in practice. For example, if (some of) the objectives correlate, very undesirable solutions may be obtained (Steuer, 1986). On the other hand, it is not possible to ﬁnd so-called unsupported Pareto optimal solutions with the weighting method. For example, in nonconvex problems there typically are Pareto optimal solutions that cannot be found no matter how the weights are set. Thus, it is advisable to use other methods that can guarantee that any Pareto optimal solution can be found and that utilize preference information that is understandable for the DM. i=1

2. Methods 1.1. Synheat model The Synheat model was developed to simultaneously take into account the trade-offs between the utility cost, ﬁxed cost of units and the cost related to the size of the heat exchangers. The main idea is to formulate a simpliﬁed superstructure by using a stage-

Let us next consider the Synheat model from a multi-objective optimisation perspective. In a multi-objective optimisation formulation, the original objective function (2) is replaced by four objective functions, namely the cold utility consumption, the hot utility consumption, the number of units and the heat exchanger surface

T. Laukkanen et al. / Computers and Chemical Engineering 34 (2010) 943–952

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area. These objective functions are identical to the functions found in Eqs. (A.2)–(A.5), with the exceptions that all the cost parameters (CCU, CHU, CF, C and ˇ) are set to the value of 1. Thus, we have four objective functions to be minimised simultaneously as shown in Problem (11). We shall solve this problem with the interactive NIMBUS method. It is worth noticing that the method used is independent of the chosen objective functions and the number of objective functions: min cold utility consumption min hot utility consumption min number of heat exchanger units min total heat exchanger surface area s.t. x ∈ S.

(11)

Here S is the feasible region that is deﬁned by (3) through (9). NIMBUS (Miettinen, 1999; Miettinen, & Mäkelä, 1999, 2006) is an interactive multi-objective optimisation method. Being interactive means that it operates iteratively and needs preference information from the decision maker (DM) at each iteration. In this way, it is possible to ﬁnd the solution that best satisﬁes the DM without too much cognitive or computational load. NIMBUS is applied here because it has been successfully used, for example, in optimal shape design of ultrasonic transducers (Heikkola, Miettinen, & Nieminen, 2006), optimal control in continuous casting of steel (Miettinen, Mäkelä, & Männikkö, 1998), problems in chemical engineering (Hakanen, Miettinen, Mäkelä, & Manninen, 2005), separation of glucose and fructose (Hakanen, Kawajiri, Miettinen, & Biegler, 2007) and intensity modulated radiotherapy treatment planning (Ruotsalainen, Boman, Miettinen, & Tervo, 2009). NIMBUS is based on classiﬁcation of objectives. At each iteration, the DM sees the current Pareto optimal objective vector consisting of objective function values calculated at the current Pareto optimal solution, and is asked to classify these functions into ﬁve different classes in order to get a more satisfactory solution. After the classiﬁcation, the original objective functions and the preference information speciﬁed by classiﬁcation are converted into scalarised problems that can be solved by appropriate single objective solvers. Based on these scalarised problems, the so-called synchronous NIMBUS method (Miettinen, & Mäkelä, 2006) generates 1–4 new Pareto optimal solutions that follow the classiﬁcation in slightly different ways. The DM can decide how many new solutions she/he wishes to consider. In this way, the DM can see how feasible her/his desired changes were. Because the method uses objective function values as preference information, they are understandable to the DM and it is easy to compare the desired and attained solutions. For further information, see Miettinen and Mäkelä (2006). The NIMBUS method has two implementations: • WWW-NIMBUS (http://nimbus.it.jyu.ﬁ) operating via the Internet (free for academic use) (Miettinen, & Mäkelä, 2000, 2006) • IND-NIMBUS (http://ind-nimbus.it.jyu.ﬁ) (Ojalehto, Miettinen, & Mäkelä, 2006) for industrial applications. The latter can be connected with different simulation and modelling tools. We have integrated the IND-NIMBUS user interface with the modeling environment GAMS and in this paper we utilize this new tool. With this tool we increase the versatility of GAMS. To be more speciﬁc, it is now possible to use GAMS for solving multiobjective optimisation problems. On the other hand, the variety of single objective solvers of GAMS is now available for NIMBUS. In the next section we describe the application created by integrating NIMBUS and GAMS.

Fig. 1. Outline of integrated A–NIMBUS–GAMS-tool.

3. NIMBUS and GAMS integration IND-NIMBUS is a multi-platform desktop application for implementing the NIMBUS method. It can be connected to an external simulation or modeling software in order to be used for solving any multi-objective optimisation problems. In this paper, we introduce an integrated tool, called A–GAMS–NIMBUS-tool, where the DM can use the graphical user interface developed for the IND-NIMBUS software to solve multi-objective optimisation problems expressed with the GAMS modeling language, utilizing single objective solvers provided in the GAMS software.

3.1. NIMBUS method implementation with GAMS The procedure of the A–GAMS–NIMBUS-tool can be seen in Fig. 1. When the application is started, the NIMBUS software requests GAMS to calculate the initialisation step. During the initialisation, some additional information and the ﬁrst initial solution are calculated and the objective vectors are shown to the DM. She/he is then asked to make the ﬁrst classiﬁcation using the NIMBUS user interface. In order to use single objective optimisation methods included in GAMS, parts of the NIMBUS method need to be implemented with the GAMS modeling language. These are the initialisation step and the classiﬁcation step. The steps are presented in the following sections.

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Fig. 2. IND-NIMBUS classiﬁcation window.

3.1.1. Step 1: initialisation As a ﬁrst step of NIMBUS, estimated upper and lower bounds of objective values in the set of Pareto optimal solutions are calculated. For a minimisation problems, they are called nadir and ideal objective vectors, respectively. The SYNHEAT model is solved for all four objectives separately so that one objective at a time is minimised and the values of the other objectives are calculated, but not optimised. The ideal objective value zi∗ for a speciﬁc objective is the value that is obtained when that speciﬁc objective is minimised. The components of the nadir objective vector zinad are then estimated using a payoff table (see, e.g. Miettinen (1999)). We deﬁne zi∗∗ = zi∗ − ε for each i, where ε > 0 is a small scalar. Next, the ﬁrst (weakly) Pareto optimal solution must be calculated for the ﬁrst classiﬁcation step. This can be done by solving the following scalarised Problem (12): min ˛ subject to ˛ ≥ x ∈ S.

fi (x) − zi∗

zinad − zi∗∗

,

for all i functions

(12)

Solving the initialisation phase provides a ﬁrst (weakly) Pareto optimal solution. 3.1.2. Step 2: classiﬁcation In the ﬁrst classiﬁcation step, the initial solution is presented to the DM. Then the DM has to decide into which classes the different objective functions are assigned. As mentioned earlier there are ﬁve classes into which objectives can be classiﬁed. For a minimisation problem, the different classes are: functions fi : • whose value should be minimised as much as possible, • I ≤ whose value should be minimised till a desired aspiration level zˆi , • I ≥ whose value can increase till a speciﬁed upper bound i , • I = whose value is acceptable and • I ♦ whose value can be changed freely.

improve in value. Thus, classes I < or I ≤ and classes I ≥ or I ♦ cannot be empty. Additionally, if an objective function is assigned in the class I ≤ , the DM has to give an aspiration level zˆi (which is lower than the current objective value) and if an objective function is in class I ≥ , the DM has to give the upper bound value i (which is higher than the current objective value). As mentioned, the so-called synchronous NIMBUS algorithm uses altogether four different scalarised problems to generate several new Pareto optimal solutions for a given classiﬁcation information. In the presented A–GAMS–NIMBUS-tool, only one scalarisation problem, Problem (13), is used.

min ˛ subject to fi (x) − zi∗ , ˛≥ zinad − zi∗∗ ˛≥

fi (x) − zˆi zinad − zi∗∗ j∗

fi (x) ≤ fi , fi (x) ≤ i ,

,

for all i ∈ {I < } for all i ∈ {I ≤ }

for all i ∈ {I < , I ≤ , I = } for

(13)

all i ∈ {I ≥ }

f1 (x) = amount of units f2 (x) = total heat exchanger area f3 (x) = total hot utility f4 (x) = total cold utility equations (3)–(9)

I<

It is naturally possible to use only some of the ﬁve classes available. In any case, because the solution to be classiﬁed is Pareto optimal, when making a classiﬁcation the DM must allow some of the objectives to get worse in order to allow some others to

Here, the index j* refers to current objective function values. For j∗ example, for the ﬁrst classiﬁcation step, fi represents the value of fi obtained by solving the Problem (12) during the initialisation step. After solving the scalarised Problem (13), the values of the different objective functions are provided to the DM. If the DM is satisﬁed with the result, the optimisation procedure stops. If not, the DM provides new classes (and aspiration levels and upper bounds if needed) and Problem (13) is solved again.

T. Laukkanen et al. / Computers and Chemical Engineering 34 (2010) 943–952 Table 3 Heat exchanger and utility cost parameters for Example 1.

Table 1 Solvers for the optimisation problems used in the examples. Problem type

Solver(s):

MINLP NLP MILP

DICOPT a CONOPT3 b CPLEX c

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a Engineering Design Research Center (EDRC) at Carnegie Mellon University. b ARKI Consulting and Development A/S. c ILOG CPLEX Division.

Tin (◦ C)

Tout (◦ C)

Fcp (kW/K)

h (kW/m2 K)

H1 H2 C1 C2 Hot utility Cold utility

180 240 40 120 325 25

75 60 230 300 325 40

30 40 35 20 -

0.15 0.10 0.20 0.10 2.00 0.50

USD/(a m2 )

CFij = 15000 CFi,CU = 15000 CFi,HU = 15000

Cij = 30 Ci,CU = 30 Ci,HU = 60

USD/(a kW) ˇij = 0.8 ˇi,CU = 0.8 ˇi,HU = 0.8

CCU = 10 CHU = 110

which the value can then be edited. If the current objective function value is deemed suitable, the arrow pointing downwards may be clicked. If the objective value is allowed to change freely, one can leave it unclassiﬁed, that is, the DM does not click any part of the function bar. After the DM has expressed one’s preferences, the IND-NIMBUS user interface assigns the correct NIMBUS classiﬁcation for each objective function. This information, and the current solution, is then communicated to the GAMS model. Then the IND-NIMBUS software starts the GAMS software, which is used to solve optimisation Problem (13), using the previously given information. After the GAMS solution process is ﬁnished, the new (weakly) Pareto optimal solution is then passed back to IND-NIMBUS. Using the IND-NIMBUS user interface the DM can either use this solution as the basis for a new classiﬁcation, or continue with a new classiﬁcation from some other solution. In IND-NIMBUS, the DM has access to all Pareto optimal solutions generated during the interactive solution process. Therefore, any solution can be selected as a starting solution for a new classiﬁcation at any time. In this way, the DM can generate a representation of Pareto optimal solutions according to her/his preferences for further study. In order to collate the solutions generated, the DM can create different ﬁlters, for example to show only those solutions where the ﬁrst objective function has values above zero. There is also a possibility to view only those solutions obtained from the last classiﬁcation. The DM can also store the best solution candidates to the “best candidates” panel during the solution process. Furthermore, undesirable solutions can be deleted from the solution list. There exists also tools for visualizing generated solutions, with several different kinds of graphics (e.g. with 2D and 3D bar graphs, value paths, spider-web charts or petal diagrams; Miettinen, 2003).

Table 2 Stream data for Example 1 taken from Table 1 in Björk and Westerlund (2002a). Stream

USD/a

3.2. IND-NIMBUS optimisation interface As described earlier, the initialisation and classiﬁcation steps of the NIMBUS algorithm are implemented with the GAMS modeling language. On the other hand, NIMBUS is an interactive method so a user interface is needed for its use. In the A–GAMS–NIMBUStool, previously implemented IND-NIMBUS software is used as an interface for the interactive optimisation. An example of the classiﬁcation of objectives with the INDNIMBUS user interface is given in Fig. 2. Here, each bar on the left of the window represents an objective function. Objective function values and their ranges in the Pareto optimal set are shown under each bar. Solutions previously calculated with the software are shown in the right side of the ﬁgure, numbered from one (the initial solution) to six (in this case). Instead of numbering, solutions can also be renamed, if the DM so desires. The DM can also view numerical objective function values instead of the graphical presentation given here. When making a new classiﬁcation, the DM simply clicks different parts of the objective function bar depending on how she/he wishes to change the current objective function value. If the value is desired to be decreased as much as possible, the arrow pointing to the left is clicked. If the DM desires to use some upper or lower bound, the actual bar can be clicked and the corresponding value is then shown in the edit box next to the objective function bar, in

4. Examples The presented A–GAMS–NIMBUS-tool has been used in solving two examples. The problems have been implemented in and solved using GAMS (Brook, Kendrick, Meeraus, & Raman, 2008). The Synheat superstructure and model are used in both

Table 4 Results of iterations of Example 1. Iteration k

Issue

Units [−]

Area (process exchangers) [m2 ]

HU [kW]

CU [kW]

1 (Init.)

Ideal Nadir Sol.

4 8 6

2385(874) 382781(367365) 258474(257107)

1756 7100 5354

1856 7200 5454

2

Classes Bounds Opt. values

I≥ UP=7 7

I<

I<

I♦

14492(14429)

1933

2033

3

Classes Bounds Opt. values

I= FX=7 7

I<

I≥ UP=2500 2500

I♦

≤

7475(6660.6) <

4

Classes Bounds Opt. values

I LO=5 5

I

5

Classes Bounds Opt. values

I= FX=5 5

I≥ UP=8000 8000(7076)

7475(6454)

2600

≥

I UP=3000 3000

I♦

I<

I♦

2443

2543

3100

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T. Laukkanen et al. / Computers and Chemical Engineering 34 (2010) 943–952

Fig. 3. Final ﬂowsheet of Example 1.

Table 5 Stream data for Example 2 taken from Linnho and Ahmad (1990). Stream

Tin (◦ C)

Tout (◦ C)

Fcp (kW/K)

h (kW/m2 K)

H1 H2 H3 H4 C1 C2 C3 C4 C5 Hot utility Cold utility

327 220 220 160 100 35 85 60 140 330 15

40 160 60 45 300 164 138 170 300 250 30

100 160 60 400 100 70 350 60 200 -

0.50 0.40 0.14 0.30 0.35 0.70 0.50 0.14 0.60 0.50 0.50

examples. The solvers used together with GAMS are shown in Table 1. All the mathematical programming problems were solved on a laptop with a mobile Intel Core 2 Duo T8300 NV 2.4 GHz processor. 4.1. Example 1 Example 1 is taken from Björk and Westerlund (2002a). The stream data can be found in Table 2. The heat exchanger cost parameters for the example are shown in Table 3, even though they are not used in this case (are used with the basic single-objective Syn-

heat model for comparison). The number of stages is 2 and tmap , the minimum allowed approach temperature, is 0.1. Table 4 shows values of the objective functions in different NIMBUS iterations. First, the nadir and ideal objective values are calculated for all four objectives (units, area, hot utility and cold utility). With this information and using Problem (12) the initial solution is calculated and the values of the four objectives are shown to the DM. Next (iteration 2), the DM is asked to classify the objectives. In this case the DM allows the units objective to increase from the current value 6 to at most 7. The DM wants that objectives area and HU are both in class I < , and hence they should be minimised as much as possible. The fourth objective CU is in class I ♦ and can vary freely in this iteration and also in all subsequent iterations. This is mainly because the DM understands from the calculations so far and from his previous knowledge of HENS that CU is very strongly coupled with the objective HU, and in this case, the DM decided that the objective HU is more important than CU. In general, hot and cold utility consumption are highly correlated and these two objectives could be united into a single objective and the presented approach could solve the problem with three objectives as well. But because there might be situations where the DM wants to keep both the hot and cold utility as objectives, all four objectives are kept in this example. With this classiﬁcation information Problem (13) is solved. As can be seen, the values of Units, HU and CU are all quite close to their ideal values, but area is quite far. For this reason, the DM decides to

Table 6 Results of iterations of Example 2. Iteration k

Issue

Units [−]

Area (process exchangers) [m2 ]

HU [kW]

CU [kW]

1 (Init.)

Ideal Nadir Sol.

9 32 12

9885(1136) 289450(206990) 46350(37877)

15300 58030 19772

23020 65750 21492

2

Classes Bounds Opt. values

I≤ LO=10 10

I<

I≥ UP=27000 27000

I♦

≥

17086(10508) ≤

≥

34720.00

3

Classes Bounds Opt. values

I UP=11 11

I LO=15000 16247(9642)

I UP=29000 29000

I♦

4

Classes Bounds Opt. values

I= FX=11 11

I≥ UP=17000 17000(10144)

I≤ LO=25000 26251

I♦

36720

33971

T. Laukkanen et al. / Computers and Chemical Engineering 34 (2010) 943–952

improve the value of objective area in the next iteration. In iteration 3 the objective units should not increase from its current level, but the objective HU can increase to 2500. The objective area should be minimised as much as possible. Once again Problem (13) is solved and the results are presented to the DM. This procedure continues

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until after iteration 5 the DM decides to stop the optimisation procedure being pleased with the results. At this time the DM felt that she/he knew enough of the problem and the results were reasonable close to their ideal values indicating that big improvements are not possible any more.

Fig. 4. Final ﬂowsheet of Example 2.

950

T. Laukkanen et al. / Computers and Chemical Engineering 34 (2010) 943–952

Fig. 3 shows the ﬁnal ﬂowsheet of Example 1. The ﬂowsheet shown was drawn with HeVi (Nykopp, Laukkanen, Tveit, & Fogelholm, 2008), a software which can be downloaded from http://eny.tkk.ﬁ/hevi/ for free. As can be seen from the ﬁgure, no stream splits are needed. When Example 1 is solved with the original single-objective Synheat model (objective function (2) and constraints (3) through (9), with the same parameters (number of stages allowed in the superstructure equal to 2, tmap equal to 0.1 and with the same solvers) and with cost parameters given in Table 3, the minimised total annual cost is egual to 429.4 kEuro. We can compare this result with the result of the ﬁnal iteration shown in Table 4 obtained by using the A–GAMS–NIMBUS-tool. If we use these values of the objective functions to calculate the total annual cost with the same cost parameters (Table 3), after the optimisation, the total annual cost for the A–GAMS–NIMBUS-tool solution is 424.0 kEuro, which is slightly better than with the single-objective Synheat model. Although the reason for this might be the use of local solvers, it can also be for the reason that the single-objective Synheatmodel cannot ﬁnd unsupported Pareto optimal solutions (Steuer, 1986) and anyway this certainly gives a strong indication of the potential of the presented A–GAMS–NIMBUS-tool. Important is to note, that in the A–GAMS–NIMBUS-tool the total annual cost was not optimised, but calculated after the optimisation with the values of the optimised objective functions. Also, if the cost parameters would slightly vary, the ﬁnal network obtained with the A–GAMS–NIMBUS-tool would not necessarily change, unless the DM wants to change it. 4.2. Example 2 Example 2 is taken from Linnhoff and Ahmad (1990). The stream data for the example is given in Table 5. Altogether 4 hot streams and 5 cold streams are present. The cost parameters are equal for all the heat exchangers, namely in this case CF = 2000 USD/a , C = 70 USD/(a m2 ) and ˇ = 1.0. The cold and hot utility costs are 6 and 60 USD/(a kW), respectively. In our formulation the cost parameters are not needed, because the objectives to be minimised are units, area, hot utility and cold utility. They are given here because these cost parameters are used in the basic single-objective Synheat model, whose results are compared to the ones obtained with the A–GAMS–NIMBUS-tool used in this paper. Altogether 5 stages are allowed in the superstructure and tmap , the minimum allowed approach temperature, is 1. Table 6 shows values of the objective functions in different NIMBUS iterations and Fig. 4 shows the ﬁnal ﬂowsheet. In this case the DM was happy with the results already after 4 iterations, because the DM felt that no major improvements are possible any more. The ﬁnal ﬂowsheet needs only 3 stages and no stream splits are needed. Altogether 11 heat exchangers, of which 5 are utility exchangers, are needed. When Example 2 is solved with the original single-objective Synheat model (objective function (2) and constraints (3) through (9), with the same parameters (number of stages allowed in the superstructure equal to 5, tmap equal to 1 and with the same solvers) and with the given cost parameters, the minimised total annual cost is equal to 4.4 MEuro. When comparing this result with the results of the ﬁnal iteration shown in Table 4 obtained by using the A–GAMS–NIMBUS-tool, and using these values of the objective functions to calculate the total annual cost with the same cost parameters, after the optimisation, the total annual cost is 2.99 MEuro, which is substantially better than with the singleobjective Synheat model, indicating once again the potential of the presented A–GAMS–NIMBUS-tool. The reason for this might be that the single-objective Synheat model cannot ﬁnd unsupported Pareto

optimal solutions. In this speciﬁc problem another reason is that for some reason (probably due to initial variable values) the result of the single-objective Synheat model is very bad.

5. Conclusions This paper presents a new approach for heat exchanger network synthesis (HENS) based on a formulation as a multi-objective optimisation problem. The method used is based on an interactive multi-objective optimisation method, NIMBUS, developed at the University of Jyväskylä. The word interactive means that the method operates iteratively and needs preference information from the decision maker (DM) at each iteration. In this way, the DM is in control of the decision making process and can direct the search into areas of most interest. In HENS, this means that the method’s ability to control the synthesis process resembles pinch technology, while still being able to optimise different objectives simultaneously at each iteration. Interactivity also means that it is possible to ﬁnd the solution that best satisﬁes the DM without too much cognitive or computational load, which is very important in MINLP-type problems as HENS. Multiobjectivity means that cost information of the different objectives is not necessarily needed, although it can be used. Typically the different objectives are weighted with their annual costs and these annual costs are summed to one objective that is minimised. With the presented approach this is not necessary which makes the modelling phase easier and the results less sensitive to uncertainties in the cost parameters. The NIMBUS method is based on the classiﬁcation of objectives. At each iteration, the DM sees the current Pareto optimal objective function values, and is asked to tell how the objective values should change in order to get a more satisfactory solution. There are ﬁve different classes available, of which some allow the objective function values to improve, some to impair, some to be unchanged and some to vary freely. Because Pareto optimal solutions are generated at each iteration step, it is necessary to let some objective function values to impair in order to improve others. In the presented paper, an implementation of NIMBUS, called IND-NIMBUS, is integrated with the GAMS modelling system. This enables using GAMS for solving multi-objective optimisation problems without using weighting factors. On the other hand, the variety of single objective solvers of GAMS is now available for INDNIMBUS, the industrial software implementation of the NIMBUS method. For example, NIMBUS can be used to solve MINLP-models with deterministic solvers. This A–GAMS–NIMBUS tool can easily be applied to ﬁelds outside HENS. The Synheat model is used as a superstructure generating model for HENS in this paper, although other simultaneous optimisation models could also have been applied as well. Two examples were solved with the presented tool and the Synheat model. In these examples four objectives: amount of heat exchanger units, total amount of heat exchanger area, total amount of hot utility and total amount of cold utility were the chosen objective functions which all need to be minimised. The presented HENS method is ﬂexible also regarding the number of objective functions and no changes to the method are needed to cope with more or less objective functions. This is an important issue especially if additional complexity (e.g. optimisation of network ﬂexibility) is added to the optimisation model. The results of the two examples show that the method is capable to solve HENS problems efﬁciently and to satisfactory results. If comparing these results with results obtained by using the basic single-objective Synheat-model, better total annual cost values were obtained using the presented method, even though the total annual cost was not optimised, but calculated after the optimisation with the optimised objective function values. one reason for the differences might be that the single-objective Synheat-model

T. Laukkanen et al. / Computers and Chemical Engineering 34 (2010) 943–952

cannot ﬁnd unsupported Pareto optimal solutions. This gives a strong indication of the potential of the presented method. The examples were solved with solvers that can only provide locally optimum results for nonconvex problems. This is an important issue especially when solving the ideal objective values, because these values effect subsequent iterations. GAMS has also deterministic algorithms that can quarantee a global optimum. In the examples they were not used, although they could have been used, because generally the HENS problems are much bigger with many more process streams causing the solution times needed in the global optimum algorithms to increase too much. This deserves further study.

Area cost =

+

i ∈ HP

+

(CCU · qcui )

(A.2)

(CHU · qhuj )

(A.3)

i ∈ HP

Hot utility cost =

j ∈ CP

Fixed cost per unit =

(CFij · zijk ) +

i ∈ HP j ∈ CP k ∈ ST

+

(CFi,HU · zhui )

(A.4)

ˇij

qijk Uij · [(1/2) · (dtijk dtijk+1 ) · (dtijk + dtijk+1 )]

(1/3)

ˇi,CU

qcui Ui,CU · [(1/2) · dtcui (TOUTi − TINCU )(dtcui + (TOUTi − TINCU ))]

(1/3)

Cj,HU ·

j ∈ CP

(CFi,CU · zcui )

i ∈ HP

j ∈ CP

Ci,CU ·

Cij ·

i ∈ HP j ∈ CP k ∈ ST

Cold utility cost =

951

ˇj,HU

qhuj Uj,HU · [(1/2) · dthuj (TOUTj − TINHU )(dthuj + (TOUTj − TINHU ))]

(1/3)

The indices, sets, parameters and variables in the equations are given in the following sections.

In the future, the presented A–GAMS–NIMBUS tool will be developed further. The NIMBUS method has already been developed to improve the calculation procedure. These improvements in the NIMBUS method as well as all of its features should be integrated to the A–GAMS–NIMBUS-tool now that the applicability of the tool has been tested. With the developed ﬂexible multiobjective optimisation tool more complex HENS problems could and should be solved. Issues like network ﬂexibility, pressure drop, etc., could be taken into account.

A.1. Indices The indices in table in this appendix are used in Eq. (A.1). Index i j k

Hot streams Cold streams Superstructure stage

A.2. Sets Acknowledgments The sets in table in this appendix are used in Eq. (A.1). The GAMS–NIMBUS tool development has been a part of the MASIT01 projects supported by TEKES – the Finnish Funding Agency for Techonoly and Innovation. The research was partly supported by the Academy of Finland, grant number 128495 (on the part of Vesa Ojalehto).

Set = {j | jis a cold stream} = {i | iis a hot stream} = {k | kis a superstructure stage}

CP HP ST

Appendix A. Objective function for the Synheat model The objective function of the Synheat model is given in Eq. (A.1). The notation used is similar to the one used by Biegler, Grossmann, and Westerberg (1997):

⎛ min ⎝

(CCU · qcui ) +

i ∈ HP

+

i ∈ HP j ∈ CP k ∈ ST

+

i ∈ HP

+

j ∈ CP

Ci,CU ·

Cj,HU ·

Cij ·

j ∈ CP

CHU · qhuj +

CFij · zijk +

i ∈ HP j ∈ CP k ∈ ST

Uij · [(1/2) · (dtijk dtijk+1 ) · (dtijk + dtijk+1 )]

CFi,CU · zcui +

i ∈ HP

CFi,HU · zhui

j ∈ CP

ˇij

qijk

(A.5)

(1/3)

ˇi,CU

qcui Ui,CU · [(1/2) · dtcui (TOUTi − TINCU )(dtcui + (TOUTi − TINCU ))]

(1/3)

qhuj Uj,HU · [(1/2) · dthuj (TOUTj − TINHU )(dthuj + (TOUTj − TINHU ))]

(1/3)

ˇj,HU ⎞ ⎠

(A.1)

952

T. Laukkanen et al. / Computers and Chemical Engineering 34 (2010) 943–952

A.3. Parameters The parameters in table in this appendix are used in Eq. (A.1). Parameter CCU CHU CF C TIN TOUT U ˇ

Cost of cold utility Cost of hot utility Fixed cost for a heat exchanger Area cost coefﬁcient Stream inlet temperature Stream outlet temperature Overall heat transfer coefﬁcient Exponent for area cost

A.4. Variables The variables table in this appendix are used in Eq. (A.1). Variable dtijk dtcui dthuj qijk qcui qhuj zijk zcui zhui

Temperature approach for the match (i, j) in temperature location k Temperature approach for the match hot stream i and cold utility Temperature approach for the match cold stream j and hot utility Heat exchanged between hot stream i and cold stream j in stage k Heat exchanged between hot stream i and cold utility Heat exchanged between cold stream j and hot utility Binary variable to denote existence of match (i,j) in stage j Binary variable to denote cold utility exchange with hot stream i Binary variable to denote hot utility exchange with cold stream j

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