Optimal operation of heat exchanger networks

Computers and Chemical Engineering 23 (1999) 509–522

Optimal operation of heat exchanger networks

B. Glemmestad a,1, S. Skogestad b,*, T. Gundersen b

a Telemark Institute of Technology, Porsgrunn, Norwayb Norwegian Uni6ersity of Science and Technology, 7034 Trondheim, Norway

Received 11 February 1998; received in revised form 3 August 1998

Abstract

The paper discusses optimal operation of a general heat exchanger network with given structure, heat exchanger areas andstream data including predefined disturbances. A formulation of the steady state optimization problem is developed, which iseasily adapted to any heat exchanger network. Using this model periodically for optimization, the operating conditions thatminimize utility cost are found. Setpoints are constant from one optimization to the next, and for implementing the optimalsolution special attention is paid to the selection of controlled variables such that the operation is insensitive to uncertainties(unknown disturbances and model errors). This is the idea of self-optimizing control. In addition to heat exchanger networks, theproposed method may also be applied to other processes where the optimum lies at the intersection of constraints. © 1999Elsevier Science Ltd. All rights reserved.

1. Introduction

Methods for heat exchanger network (HEN) synthe-sis have been developed during the last decades andthese methods aim to design a HEN that yields areasonable trade-off between capital and operating costin the nominal case. Since the mid 1980s several au-thors have also investigated flexibility of HENs, e.g.Kotjabasakis and Linnhoff (1986) who introduced sen-sitivity tables to find which heat exchanger areas shouldbe increased in order to make a nominal design suffi-ciently flexible. In Papalexandri and Pistikopoulos(1994), HEN synthesis and flexibility are consideredsimultaneously using mathematical programming.

The total design effort (on a systems level) requiredfor a HEN typically involves the following three stages:a. Nominal design. Synthesize one or more networks

with good properties for nominal stream data.b. Flexibility and controllability. Investigate the net-

works with regard to flexibility and controllability,and possibly introduce some modifications (e.g. in-creased area) such that at least one HEN showssatisfactory results.

c. Operation. Design a control system to operate theHEN properly. This involves control structure selec-tion and possibly some method for on-lineoptimization

For each step, some networks may be rejected or thedesigner may go back to the preceding step to findother alternatives. The steps are usually carried out in asequential manner, however, the design may also be ofa more simultaneous character, depending on the meth-ods used.

Compared to synthesis of nominal and flexibleHENs, much less effort has been dedicated to findmethods for the operation of HENs (step c). Mathisen,Skogestad and Wolff (1992) investigated bypass selec-tion for control of HENs, without considering theutility consumption. In Mathisen, Morari and Skoges-tad (1994) method for operation of HENs that mini-mizes utility consumption is proposed. The method isbased on structural properties of the network, however,the variable control configuration may result in poordynamic performance. A method based on repeatedsteady state optimization is suggested by Boyaci, Uz-turk, Konukman and Akman (1996), but their focus isnot on the control structure for closed loop implemen-tation. Recently, Aguilera and Marchetti (1998) pro-posed a method for on-line optimization and control ofHENs. They also discussed degrees of freedom withrespect to optimization of HENs during operation.

* Corresponding author. Fax: +47-7359-4080.E-mail address: [email protected] (S. Skogestad)1 Present address: Norsk Hydro Research Centre, N-3901 Pors-

grunn, Norway.

0098-1354/99/$ - see front matter © 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 9 8 - 1 3 5 4 ( 9 8 ) 0 0 2 8 9 - 0

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522510

In this paper, we consider optimal operation ofHENs. This is normally done in two steps:1. Obtaining the optimal solution at regular time inter-

vals (normally using steady state data).2. Implementing the optimal solution by specifying the

optimal values (setpoints) for some variables withthe aim of achieving ‘self-optimizing control’.

Step 1 is usually solved by numerically optimizing thedegrees of freedom in the model, and in Section 5 wepresent a simple formulation of the optimization prob-lem where the bypass fractions do not explicitly appear.This makes the problem less nonlinear. An alternativeway of solving the optimization problem, which worksin some cases, is to use structural information only.This procedure is discussed in Mathisen et al. (1994)and Glemmestad (1997), and is not discussed any fur-ther here.

The importance of selecting the right variables in step2 is often overlooked. It will be shown that the choiceof optimization variables affects the steady state perfor-mance of the (controlled) HEN when unknown distur-bances are present, and a procedure for optimalselection of these variables is presented. The procedureextends ideas for selection of controlled outputs pre-sented in Skogestad and Postlethwaite (1996).

In the following, it is assumed that the stream data(heat capacity flowrates and supply/target tempera-tures), network structure and heat exchanger areas aregiven and that the HEN is sufficiently flexible. Tomanipulate the network it is assumed that utility dutiescan be adjusted and that a variable bypass is placedacross each process-to-process heat exchanger. In caseof stream splits, we may also assume that split fractionscan be varied. Variable split fractions introduce nonlin-earities in the steady state optimization problem.

The remaining part of the paper is organized asfollows: First, the complete method is outlined. InSection 3 the procedure for selection of optimizationvariables will be described in detail, and robust opti-mum is explained in Section 4. The steady state opti-mization model is presented in Section 5, and thecomplete method is applied to an example in Section 6.Finally some conclusions are drawn in Section 7.

2. Outline of method

With the term optimal operation, we mean that thefollowing two goals are fulfilled:� Primary goal: Satisfy targets (usually outlet

temperatures).� Secondary goal: Minimize operating cost.

A prerequisite for performing a meaningful on-lineoptimization is that there is at least one extra degree offreedom during operation, and most HENs have thisfeature. As an example, consider the network in Fig. 1

where there are four manipulations (two bypasses andheater and cooler duties) to control the three outlettemperatures to their targets (primary goal). Hence wehave one manipulation ‘in excess’ which implies onedegree of freedom. This extra degree of freedom can beused to minimize utility cost, i.e. to achieve the sec-ondary goal.

Note that the number of degrees of freedom duringoperation is different from the synthesis stage. Withinthe ‘synthesis terminology’, the HEN in Fig. 1 hasminimum number of units and no degrees of freedom(constraints on DTmin, etc. have no relevance duringoperation). In some cases the degrees of freedom thataffects the objective during operation may be less thanthe number of excess manipulations, however, this isnot discussed any further in this paper (see Glemmes-tad, 1997).

Figure 2a shows a schematic block diagram of themethod that will be described. The optimizer contains ascalar objective function (criterion) J which indicateshow well the HEN is operated and a steady state modelof the HEN. As objective function we will typically usetotal utility cost which is a linear function, see Eq. (4)in Section 5. The model is optimized regularly andreference values for the optimization variables arepassed to the controller K2. The reference values (set-points) are constant in the period between eachoptimization.

All inputs (manipulations) u and outputs (measure-ments) y are separated into u= [u1 u2]T and y= [y1 y2]T,respectively. y1 are those outputs which have giventarget (reference) values and u1 are those manipulationsdedicated to keep y1 at their target values. Satisfyingthe targets for y1, is simply the primary goal of optimaloperation. Now, we close the control loops for theprimary outputs with controller K1 (‘base control’) andassume integral action in the loops (no steady statecontrol error). This leads to Fig. 2b where focus is onthe remaining part of the system, i.e. the secondaryvariables (u2, y2 and setpoints r2) and the optimizer. Itis simply assumed that the base control is implementedand that it works. The problem can be formulated as

minu 2

J(x, d)

f(x, d)=0

Fig. 1. Simple HEN with one degree of freedom during operation.

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522 511

Fig. 2. General optimizing control structure.

g(y2)50

where the latter equations represent the constraints. Asexplained above we assume that the constraints on y1

are satisfied using the base control system and thus willenter the model equations f(x, d)=0 (as equalityconstraints).

We want to focus on the secondary goal of optimaloperation; utility cost minimization (variables associ-ated with this goal have index 2). u2 is the ‘excess’manipulation(s) which represent the degree(s) of free-dom that we will use to minimize utility cost. Of course,one could compute optimal values for u2 and applythese directly (open-loop implementation) as indicatedby the dashed line in Fig. 2a and b. Alternatively, theoptimizer could pass reference values for some ‘extra’measurements y2 (closed-loop implementation). If thedisturbance d was perfectly known (and constant), itwould not matter (at steady state) which variables werechosen. However, from the explanation below it will beclear that the selection of which variables that arepassed from the optimizer down to the control levelaffects how close to optimum the HEN can beoperated.

The variables (setpoints) r2 that are passed from theoptimizer to the control level will be denoted optimiza-tion variables.

Let the disturbance d be partitioned into the follow-ing two contributions:

d=d0+du (1)

where d0 is the information that the optimizer hasabout the disturbances when it performs an optimiza-tion, and du (unknown disturbances) are all deviationsfrom d0 and the real disturbance until a new optimiza-tion is carried out. That is, du consists of, for example,unknown disturbances and model errors in addition tochanges of the disturbances in the period between twooptimizations (optimization interval). Measurement/es-timation errors will not be handled explicitly in this

paper, but these errors may be included in du andtreated as any other uncertainty.

Since the optimizer has no specific information aboutdu, the optimization is based on d=d0. In practice,however, du may vary within some known (or selected)bounds. The effect of du"0 should be taken care of inthe optimizer in order to avoid that the HEN becomesinfeasible (primary goal cannot be satisfied) for somedisturbances. Figure 3 shows a typical situation for ageneral plant with one degree of freedom (one extramanipulation) and an objective function J that shouldbe minimized. The plant has one disturbance input andtwo candidate measurements A and B (y2= [y2,A y2,B]T)that can be controlled to a desired value using the extramanipulation u2 (since subscripts 1 and 2 are used todistinguish between the primary and secondary sets ofinputs and outputs, we uses letters A, B, etc. to denoteindividual elements of u and y). Also, remember thatbase control to keep primary outputs at fixed setpointsis already implemented. See Fig. 1 where two candi-dates for secondary controlled temperatures (y2,A andy2,B) are indicated. The primary controlled measure-ments are the outlet temperatures of the three streams.Notice, however, that the HEN in Fig. 1 actually is aconstrained process and the curves in Fig. 4 are morerealistic (see below).

Figure 3a shows J as a function of y2,A with thedisturbance as a parameter. The solid line is for du=0,and the two dashed lines represent the extremes for du.Figure 3b shows similar curves as a function of y2,B.Since we have to base our optimal values on du=0, wecan choose to keep either y2,A:0.5 or y2,B:0.4 usingfeedback control. From the figure, however, we see thatwhen keeping y2,B constant, J is less sensiti6e to bothvariations in y2 (control error) and to unknown distur-bances, than when keeping y2,A constant. Therefore, weprefer to keep y2,B constant between the optimizations.This simple example illustrates how the choice of opti-mization variables affects the objective function for anunconstrained process. Figure 4 shows similar curves as

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522512

in Fig. 3 for a process where the optimum is con-strained, which is typical for most HENs (minimumutility consumption corresponds to maximum utiliza-tion of process-to-process exchangers which againmeans that some bypasses are closed).

In Fig. 4a the process is infeasible when y2,A isspecified too small (marked with ‘+ ’) and in Fig. 4bwhen y2,B is too large. In a HEN this typically happenswhen a bypass saturates such that a target temperatureno longer can be met. As an example see Fig. 1. If y2,A

is specified too low the outlet temperature of stream C1will not be reached even if bypass around the processheat exchanger to the right is fully closed. Thus theHEN operation becomes infeasible. This represents aserious problem when implementing the optimal solu-tion. For example, if we compute the nominally optimalvalue of y2,A (with du=0), then we see from Fig. 4athat this will lead to infeasibility if the disturbanceincreases (e.g. to du,max). Two simple approaches can beused to avoid the infeasibility:1. ‘Back-off’ from the nominal optimum by imple-

menting the value where b is the back-off. Thisapproach has been suggested by Arkun andStephanopoulos (1980) and Narraway, Perkins andBarton (1991). The value of b should be such thatthe solution is feasible for all expected disturbances(as indicated by the vertical lines in Fig. 4.

2. Introduce safety margin on the constraints duringthe nominal optimization, i.e. replace g(u, y)50 byg(u, y)5o where again the safety margin o shouldbe such that the solution is feasible for all expecteddisturbances.

These two approaches only require that the nominalproblem (with du=0) is solved on-line as b or o areassumed to be precomputed. A third more computa-tionally demanding approach is also possible:

3. Find the robust optimal solution (the trulyoptimal value of y2 taking into account all possibledisturbances). This approach is discussed in Section 4.

The following steps summarize the main parts of thecomplete procedure for on-line optimization of HENs(the notation is as in Fig. 2):

1. Determine which manipulations (u1) that should beused to control the primary outputs y1 and design acontrol configuration and controllers for the pri-mary goal (base control).

2. For each excess manipulation u2 choose a measure-ment y2 (among all candidates) such that the opera-tion is insensitive to disturbances (see more detailsin next section). The additional constraints (safetymargins) on u1 are also found. Design decentralizedcontrollers for control of y2.

3. Implement the steady state model including theconstraints found in step 2 in the optimizer.

These three steps are carried out prior to operation.With the steady state optimizer from step 3 imple-mented, the optimizer computes setpoints for the opti-mization variables and apply these to the controller K2

(see Fig. 2) at regular intervals during operation. Stepone (base control) is considered rather trivial in mostcases and is not treated in detail in this paper, see e.g.step 1 in the example in Section 6. The important stepof selecting optimization variables (secondary measure-ments, step 2) is treated in the section below.

3. Selection of controlled outputs

This section describes a procedure for selection of thecontrolled variables (step 2 in the complete methodgiven above). The selection of outputs for optimizingcontrol is discussed by Skogestad and Postlethwaite(1996), chapter 10) where a method based on choosingoutputs that maximize s(G22) (smallest singular value)for a properly scaled system is proposed. In this papera more direct method is applied (which is also men-tioned in Skogestad & Postlethwaite, 1996). In thissection, we assume that the HEN is capable of reachingall targets (primary goal of optimal operation) for alldisturbances that are encountered, and we also requirethat targets are satisfied for prespecified bounds on theunknown disturbances du. Before the procedure is pre-sented, the following notation is introduced:

Fig. 3. Unconstrained process.

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522 513

Fig. 4. Constrained process (typical for HENs).

y2,cand is a vector containing all candidates to y2

is the optimal value of y2,cand for a givenyopt

du

is a fixed value of y2,cand such that theyopts

objective function is minimized while thenetwork is feasible for all du

J s is J(yopts ) for a given value of du

Du1 is the constraint imposed on u1 such thatan optimization problem based on du=0gives feasibility for all du within prespe-cified bounds.

The steps in the procedure are listed below and some ofthe points will be further explained. For simplicity, wewill assume there is only one degree of freedom (oneoptimization variable).a. Select (i) minimum and maximum values for du; (ii)

the objective function J ; (iii) the entries of y2,cand;(iv) the values for du that should be included in thecomputations; and (v) define the type of Jmean thatwill be used for choosing optimization variable. Thislast step consists in selecting whether Jmean shouldbe based on arithmetic mean or some weightedaverage of J for the different unknown disturbancesdu.

b. Compute yopt and Jopt for ‘all’ cases of du (i.e. thevalues from step (iv) in the previous point), seeTable 1. This table may also include row(s) for u2,opt

(open-loop implementation). Note that du,j is case jof du while yopt,i denotes element i in yopt.

c. Keep y2,cand,j=yopt,is for each output candidate, and

evaluate Jis(du,j) and the resulting Jmean. In general,

the setpoint y sopt,i should be optimized in order to

minimize Jmean, but for constrained processes it willbe some extreme value from Table 1 (to ensurefeasibility for all du, see remark 2 at the end ofSection 6 for an explanation for the example). No-tice that for the Monte Carlo method described inthe next section, y s

opt,i will not be any extreme valuefrom Table 1.

d. Choose the variable that gives the smallest Jmean

from the last column in Table 2 as optimizationvariable, i.e. this measurement should be controlledto a setpoint which is updated periodically by theoptimizer.

We have now found the best optimization variables. Tosimplify the on-line optimization we may want to useonly the nominal disturbance set, du=0. To ensure thatwe find the correct value of y s

opt (which ensures feasibil-ity for all disturbances), we may impose some con-straint (‘safety margin’) for the optimizer, e.g. ul]Du1.This will be explained in more detail for a simpleexample in Section 6 (see also remark 1 at the end ofthis section). The ‘safety margin’ on u1 should of coursenot be implemented in the regulatory control level.

Until now we have only considered one degree offreedom. If there were two degrees of freedom, twoelements of y2,cand would have to be fixed at a time.Tables 1 and 2 would need as many rows as there arepossibilities to pick two variables out of the totalnumber of candidate measurements, and each rowwould contain two parameter values. For example, ifthere are six candidate measurements and two degreesof freedom, the number of combinations is 6!/2!4!=15.

For cases with more than one degree of freedom, itmay be difficult to pick the optimal parameter valuesbased on reasoning. In such cases, the optimal parame-ter values may be found as follows: for each distur-bance du,j in Table 1, fix the parameters to the optimalvalue yopt(du,j) and let a computer evaluate the steady

Table 1yopt and Jopt for all cases of du

du,jdu,1 du,2

yopt,A yopt, A(du,j)yopt, A(du,1) yopt, A(du,2)yopt, B(du,1)yopt,B yopt, B(du,j)yopt, B(du,2)

yopt, i(du,2)yopt,i yopt, i(du,j)yopt, i(du,1)Jopt(du,2) Jopt(du,j)Jopt(du,1)Jopt

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522514

Table 2J s for all cases of du

du,2 du,jdu,1 Jmean

JAs (du,j)JA

s (du,2)JAS (du,1) Jmean,AJa

s

Jmean,BJBs (du,j)JB

s JBS(du,1) JB

s (du,2)Jmean,iJ s

i (du,j)J si (du,2)J s

i (du,1)J si

y2,A=0.55. It is clear that the nominal optimal valuecannot be applied since this will result in infeasiblity forsome unknown disturbances.

Mathematically, if we have an objective function J(x,d) where the value of the variable (argument) x can beselected/manipulated and d represent the disturbances,we can write

Nominal optimum: Jnopt=opt(J(x, d=d0))Jropt=opt(J(x, d�D))Robust optimum:

In operation, we are not interested in the objectivevalue directly, however, the argument (e.g. manipula-tions or setpoints) that minimize the objective is morerelevant:

Nominally optimal argument:

xnopt=arg optx

( f(x, d=d0))

Robust optimal argument:

xropt=arg optx

( f(x, d�D))

The set D represents the possible values of the un-known disturbances. D may also represent a probabilitydistribution of the unknown disturbances. The distur-bances may for instance be assumed to be normallydistributed with a given mean (nominal) value and agiven standard deviation. If the probability distributionis not bounded, we cannot require the process to befeasible for all possible disturbances. Instead, we couldrequire feasibility with a specified probability, such asrequiring feasibility in 99% of the operating time. Weshall, however, use a different approach where thedisturbances are specified by probability distributions:When a HEN is announced infeasible during operation,this normally does not imply that the HEN cannot beoperated. Typically, the reason for infeasibility is that itis impossible to reach the targets for the primary mea-surements (outlet temperatures). Instead of requiringthat the targets are satisfied, we choose to penalize thedeviation (control error in primary control loop). Inthis way, the constrained problem is transformed intoan unconstrained problem, however, the new objectivefunction will often be asymmetric. This implies that therobust optimum is different from the nominaloptimum.

In the search for the robust optimum, several ap-proaches may be used depending on the assumptionsmade and the external conditions. Here, we shall focuson two methods: (1) a Monte Carlo method where theunknown disturbances are specified as probability dis-tributions; and (2) requiring that targets are satisfied forsome corner points of the unknown disturbances, i.e.similar to the assumptions made in the previous section.These two methods are described below.

state equations describing the HEN for all unknowndisturbances du,j. The parameter values that result infeasibility for all du,j should be used to generate Table 2in step c. This procedure is repeated for all (sets of)candidate measurements. It is emphasized that indus-trial HENs usually do not have more than one (or insome cases, two) degrees of freedom available for opti-mization after regulatory control is implemented. Thus,problems involving many degrees of freedom will notappear frequently in real applications.

Remark 1. It is clear that the value of Dul maydepend on d0, i.e. the value of the safety margin imple-mented in the optimizer to ensure feasibility may de-pend on the operating point. This may be due tononlinearities in the model or due to the fact that theactive constraints may not the same for all operatingpoints. We assume that this change in Du1 is small andthat the value can be used for all d0. In practice, oneshould carry out the procedure for selection of opti-mization variables at different operating points to ver-ify that Du1 does not change too much. The worst casevalue should be chosen if it is not acceptable to violatethe primary goal, while a mean value can be used if theresulting violation of the targets is tolerable.

4. Robust optimum

This section introduces the term robust optimum. Itwill be clear that, when we have made a selection ofvariable(s) for y2, we seek the value of this variable thatcorresponds to the robust optimum. The robust opti-mal value is the optimal value when unknown distur-bances and model errors are encountered. The robustoptimum will often be different from the nominal opti-mum, where no disturbances (only nominal disturbancevalues) or model errors are considered.

For the unconstrained and smooth objective functionin Fig. 3, the value of y2 resulting in robust optimum isapproximately equal to the value resulting in nominaloptimum. For the constrained objective function in Fig.4, the situation is different. We require that the processis feasible for the unknown disturbances. When y2,A isselected, the nominal optimal value is y2,A=0.30,whereas the value resulting in robust optimum is y2,A=0.50. When y2,B is selected, the nominal optimal value isy2,B=0.65, whereas robust optimum is achieved for

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522 515

4.1. Monte Carlo method

In this case, the unknown disturbances are given asprobability distributions. If a disturbance is, for exam-ple normally distributed, we cannot require that targetsare satisfied for all disturbances since this would requireinfinite back-off. Instead of requiring targets to besatisfied, control error is penalized. Since the con-straints on the outputs (targets) are removed, there willalways exist a feasible solution. The constraints on themanipulations are still present, but control errors re-sulting from satisfying these targets will be penalized inthe objective function J. To find an (estimate) for therobust optimal value, Monte Carlo simulations areapplied. That is, random disturbances are generatedfrom the given probability distribution. For each set ofrandom disturbances the objective J is computed as afunction of the selected secondary variable. This isrepeated for a large set of randomly generated distur-bances, and the robust optimum is the optimum of themean value of J for the set of disturbances. All in-stances of the randomly generated values, includingthose that result in control error is used for the compu-tation of robust optimum.

An example is shown in Fig. 5 below. There are twodifferent disturbances and they are both normally dis-tributed. The thin solid line shows the objective J as afunction of the selected secondary measurement y2 forthe nominal disturbances. Starting from a high value ofy2, the objective decreases when y2 decreases until thenominal optimum where y2=150. Decreasing y2 fur-ther, the primary outputs can no longer be kept attarget. This is penalized in the objective function result-ing in a steep increase of the objective.

The four dashed lines show the objective for fourdifferent sets of the disturbances. The location wherethe primary targets are violated and objective increasessteeply, depends on the value of the disturbances. Themost interesting curve in Fig. 5, however, is the thick

solid curve. This curve is the average of 1000 randomlygenerated sets of disturbances. While each fixed distur-bance results in a sharp break of the correspondingcurve, the average curve is smooth. To reach the robustoptimum, a setpoint of y2:153 should be applied. Therobust optimum is slightly above 150, which is higherthan the nominal optimum of J=145. However, apply-ing the nominally optimal values of y2=150, the ex-pected value of the objective is roughly J:170. Sincewe do not know the exact value of the disturbances,only the probability distribution, the optimal setpointfor y2 is the value corresponding to the robust optimum(y2:153).

The curves in Fig. 5 are actually based on the resultsfor the HEN in the example in Section 6, when thedisturbances are normally distributed. The supply tem-perature of stream H1 has a mean value of 190°C anda standard deviation of 3°C, while CPC2 has a meanvalue of 0.50 kW/°C and a standard deviation of 0.01kW/°C. The variable selected for y2 is the setpoint fortemperature T1 (in Fig. 7) and the objective function is:

J=utility consumption (kW)+25�TC2o −TC2

t �+25�y2

−r2�That is, a deviation of 1°C in the bypass controlled

temperature or in T1, is assumed to ‘cost’ the same as25 kW. This type of objective function is suitable if thecontrol error (deviation from target) corresponds ap-proximately linearly to increase in operating cost. Insuch cases the weights should be selected to reflect thereal change in operating cost. In other cases the rela-tionship between control error and operating cost willbe strongly nonlinear and difficult to estimate. As anexample, a control error in an outlet temperature fromthe HEN corresponding the inlet temperature of areactor may strongly affect the conversion or evenextinguish the reaction. In such cases where the detailedcorrelation between control error and operating cost isnot known, the control error may be given a quadraticpenalty and the weights will have to be based on thebest engineering judgment available.

4.2. Method requiring target satisfaction

This method is similar to the assumptions made inSection 3. That is, a prespecified (discrete) set of un-known disturbances is defined (typically corner-points),and target satisfaction is demanded for this discrete setof disturbances. If this results in infeasibility for somedisturbances (during step b of the procedure) someassumptions such as the magnitude of unknown distur-bances need to be relaxed. Alternatively, the plant willhave to be modified.

This second method has some disadvantages com-pared to the Monte Carlo method:Fig. 5. Robust optimum.

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522516

1. The worst case may not be at the corner points forthe disturbances, thus finding the worst case distur-bances may be a difficult task itself.

2. Even a corner point check may be time consumingand even prohibitive when there are many indepen-dent disturbances.

3. If there are more than just a few disturbances, theprobability that all have the worst case values at thesame time is very small.

Taking these three points into account, it is assumedthat Monte Carlo simulations will give reasonable re-sults for most practical applications. Using the MonteCarlo approach, it is also straightforward to includedependencies between different disturbances when suchinformation is available. Despite this preference for theMonte Carlo approach in real applications, the exam-ple in Section 6 requires that the HEN is feasible at thecorner points for the two disturbances. Further, requir-ing targets to be satisfied also forms the basis for thesteady state optimization model presented in the nextsection.

A disadvantage with the Monte Carlo method is thata large number of disturbance sets may have to begenerated in order to get good results, and it is difficultto say in advance how many disturbance sets that arenecessary. However, as it is explained in the previoussection, the robust optimum is only determined once(or for a few cases) and this is done off-line. Thedifference in the nominal and robust optimal values areused to find a constraint on the primary manipulationsu1. With this constraint, the (approximate) robust opti-mum is found periodically from the measured andinaccurate values of the disturbances.

5. A steady state formulation of the optimizationproblem

This section presents a simple steady state formula-tion of the optimization problem that can be adapted toany HEN. The formulation has the advantage of notexplicitly including the bypass fractions, thus a majorsource to nonlinearities in the model is avoided. Inaddition, the formulation is based on heat balancesaround each heat exchanger in a way that is simple tounderstand and implement. It is developed primarilyfor implementation in the optimizer, however, it mayalso be used in the procedure for selection of optimiza-tion variables (to generate Tables 1 and 2).

Before we present the general formulation, considerthe two alternatives to model a single heat exchangerwith bypass, see Fig. 6. The first alternative is to useEq. (2a) and Eq. (2b) while the second alternative is touse Eq. (3).

At steady state it is of no consequence whether thebypass is placed across the hot side or cold side, and

Fig. 6. Single heat exchanger with bypass.

the choice in Fig. 6 is arbitrary. The temperature driv-ing force DTm(·) may be the logarithmic mean or someapproximation to it. Note particularly the differencebetween Eq. (2a) and Eq. (3) regarding the argumentsof DTm(·).

Q=UA DTm(Thot,in, Tcold,in, T*hot,out, Tcold,out) (2a)

Thot,out=uThot,in+ (1−u)T*hot,out (2b)

Q5UA DTm(Thot,in, Tcold,in,Thot,out, Tcold,out) (3)

Eqs. (2a) and (2b) includes the hot exit temperaturebefore it is mixed with the bypass stream and thisresults in bilinearities in Eq. (2b). The inequality in Eq.(3) expresses a constraint on Q when the boundary isplaced outside the bypass splitter and mixer. The bypassfraction u does not even occur in Eq. (3) but theequality part of Eq. (3) corresponds to u=0. In theoptimization model, we choose the second alternativefor each heat exchanger since this eliminates the bilin-earities in the bypass mixer. If u is needed, it can befound after the optimization of the network by solvingone nonlinear equation for each bypass fraction. Thisequation can be found from solving Eq. (2b) for T*hot,out

and inserting this expression into Eq. (2a), which issolved for u through iteration (solving one unknown inone nonlinear equation n times is much simpler thansolving n unknowns in n nonlinear equations simulta-neously). As it will be shown, the value of u is often notrequired explicitly as it normally is the manipulatedinput in a feedback control loop.

The steady state formulation for a general HEN usesthe following sets of heat exchangers:

PHX: set of all process-to-process heat exchangers.HBT: subset of PHX with hot side outlet directlyentering a bypass controlled target.CBT: subset of PHX with cold side outlet directlyentering a bypass controlled target.HUT: subset of PHX with hot side outlet entering autility controlled target (through a cooler).CUT: subset of PHX with cold side outlet entering autility controlled target (through a heater).HS: subset of PHX with hot side inlet directly enter-ing from a (hot) supply.

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522 517

CS: Subset of PHX with cold side inlet directlyentering from a (cold) supply.The general HEN formulation shown below (Eqs.

(4)–(12b) is an NLP problem The variable c in Eq.(4) denotes the cost (pr. energy unit) for the utilities.

min� %

i�HUT

c icoolersQi

coolers+ %j�CUT

c jheatersQj

heaters� (4)

subject to equalities (5) to (9)

Qi=CPicold(Ti

cold,out−Ticold,in) i�PHX (5a)

Qi=CPihot(Ti

hot,in−Tihot,out) i�PHX (5b)

Qicooler=CPi

hot(Tihot,out−Ti

t) i�HUT (6a)

Qiheaters=CPi

cold(Tit−Ti

cold,out) i�CUT (6b)

Tihot,out=Ti

t i�HBT (7a)

Ticold,out=Ti

t i�CBT (7b)

Tihot,in=Ti

S i�HS (8a)

Ticold,in=Ti

S i�CS (8b)

Interconnection equations (problem specific) (9)

Inequalities, Eqs. (10), (11), (12a) and (12b)

Qi5ai Ui Ai DTmi i�PHX (10)

Qi50 i�PHX (11)

Qicoolers]0 i�HUT (12a)

Qiheaters]0 i�CUT (12b)

Note that the index denotes heat exchangers and notstreams (which is common in many other models),and that DTm denotes the temperature driving forceoutside the bypass stream as in Eq. (3). As an exam-ple, the network in Fig. 7 will lead to the followingsets: PHX={A, B}, HUT={B}, CUT={A},HBT=¥, CBT={B}, HS={A} and CS={A, B},and the only interconnection Eq. (9) is TA

hot,out=TB

hot,in.During each optimization, Tt, Ts, CP and UA for

each heat exchanger are treated as constants. Themodel is valid without modifications for networkswith fixed stream split fractions since CP denotes heatflow capacity in each heat exchanger. For networks

with variable stream splits, CP in the split streamscan be regarded as variables, and equations that pre-serve the mass balance in the splitter(s) and energybalance in the mixer(s) must be included. During op-eration, variable stream splits can be used as manipu-lated inputs.

When the model above is used during operation, itis important to ensure a feasible solution for all pos-sible cases that may be encountered. This may beimplemented by a hierarchical strategy for constraintsatisfaction. For example, if infeasibility is encoun-tered, then the least critical target in Eqs. (7a) and(7b) is removed and a new optimization is performed.If this problem also result in infeasibility, the secondleast critical target is removed and so on until feasi-bility is achieved. Alternatively, all constraints in Eqs.(7a) and (7b) may be removed and control error maybe penalized in the objective function (Eq. (4)). Thisis similar to the Monte Carlo approach in the previ-ous section.

The constant a in Eq. (10) is a factor that maylimit the duty of a heat exchanger somewhat belowits theoretical maximum. This is simply the way thatthe constraint on u1, is implemented. Instead of im-plementing u1]Du1, directly (which is impossiblesince the model does not include u1), the correspond-ing value for a has to be computed. This is doneseparately for each heat exchanger that controls aprimary output. The example below explains how thiscan be done. Notice that the relationship between Du1

and a may depend on the operating point. For heatexchangers associated with u2, we have a=1.

The model does not include any upper constraintson the duty of the utility exchangers, and this impliesthe assumption that these are designed to handle therequired duty. If this is not the case, additional con-straints have to be added to the model, e.g. an upperlimit on the duty.

The only possible source of nonlinearities in themodel (for networks without variable splits) is theterm DTm in Eq. (10). In other words, if arithmeticmean (as opposed to logarithmic mean) is used as thetemperature driving force, the model can be solved asan LP problem. The following procedure for solvingthe model has proven to be reliable: First, use arith-metic mean in Eq. (10) for all exchangers and solvethe corresponding LP problem. Second, replace arith-metic mean with logarithmic mean (or e.g. Patersonor Chen approximations, see Paterson, 1984 andChen, 1987) and solve the NLP problem using the LPsolution as the initial value. In the case of one ormore variable split fractions, the steady state model(Eqs. (4)–(12b)) have to be modified to incorporatethis. Then the problem becomes nonlinear even ifDTm is approximated by arithmetic mean.Fig. 7. Heat exchanger network used in example.

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522518

Table 3Values for yopt and Jopt for all cases of du in the examplea

Case 3 Case 4Case 1 Case 2 Case 5

du,1=�0

0

ndu,3=

� −3

+0.01

ndu,5=

� +3

+0.01

ndu,2=

� −3

−0.01

ndu,4=

� +3

−0.01

n151.9151.0 151.9149.0T1,opt 150.0

104 0 107.4T2,opt 106.7 105.4 107.494.9 98.0T3,opt 95.0 95.1 95.8

0.0000.292 0.0000.105uA,opt 0.0000.000 0.000 0.038 0.011uB,opt 0.000

149.0 146.9Jopt 145.0 147.0 144.7

a Case 1 is the nominal disturbance.

6. Example

The HEN used in the example is shown in Fig. 7.The primary outputs are the outlet temperatures ofeach stream which should be controlled to their targetvalues of 30, 160 and 130°C for streams H1, C1 andC2, respectively. That is, we have

y1= [TH1o TC1

o TC2o ]T

where superscript ‘o’ denotes outlet temperature. Thereis a total of four manipulations (two bypasses and twovariable utility duties) which gives

u= [uA uB qc qh]T

There are two disturbances; 910°C in the supply tem-perature of stream H1 and 90.05 kW/°C in the CP ofstream C2. These values represent the maximum varia-tions d that may be present. The smaller variations/er-rors (du) that may occur within the optimizationinterval is defined in step 2a of the procedure. UA forheat exchangers A and B are 0.523 and 1.322 kW/°C,respectively. For simplicity, it is assumed that the utilityexchangers are able to deliver sufficient duty for allpossible cases. With this assumption and the givenUA-values, all target temperatures can be reached forall combinations of disturbances mentionedabove.Applying the procedure step by step yields:

Step 1. Assign primary manipulations.We use the main rule for selection of manipulations inHENs which is to choose the manipulation closest tothe measurement (e.g. Mathisen, 1994, chapter 4). Thisimplies that the primary manipulations u1 become qc, qh

and uB and these control the outlet temperatures ofstreams H1, C1 and C2, respectively.

Step 2. Selection of optimization variable.There is one excess manipulation, u2=uA, and the steps(a) to (d) below illustrate the selection of optimizationvariable.

(2a) We assume:

i. du= [93°C, 90.01 kW/°C]T (maximum variations/errors of the disturbances within the optimizationinterval).

ii. The objective function is J=qc+qh (utilityconsumption).

iii. Possible candidates to y2 are y2,cand= [T1 T2 T3 uA]T

(see Fig. 7). Note that the open-loop implementa-tion (uA) is an alternative.

iv. The computations are done for the four ‘cornerpoints’ of du, in addition to du=0. Jmean, is thearithmetic mean of these five cases. (We require thattarget temperatures have to be reached for the fivecases).

(2b) yopt and Jopt for different du are shown in Table3. The table is generated for d0= [0 0]T, i.e. for nominalvalues of the disturbances (190°C and 0.5 kW/°C). Alsoa row for uB,opt is included for extra information.

(2c) Table 4 shows J for optimal fixed values ofy2,cand. Note that in this example, the values for y2,cand

can be found without optimization, but simply fromTable 3 and physical insight (see remark 2 at the end ofthis section). If there is a possibility that the optimum isnot constrained one would have to resort to conven-tional optimization.

(2d) From the last column of Table 4 it is clear thatkeeping T1, constant is preferred.

Step 3. Implementation of optimizer.The model (including the sets and connection equa-tions) was described in the previous section The con-straint (‘safety margin’) that should be included in theoptimizer is uB]0.025. We will explain how this valueis obtained, but first we explain the details in theimplementation of this constraint. To implement theconstraint, we first find qB=55 kW for du=0 (55 kW isthe deficit heat of stream C2). Then we find aB=0.946from qB=aBUABDTm,B, where the last term is thelogarithmic mean for heat exchanger B for du=0 andT1=151.9°C. Implementing aB=0.946 (and aA=1.0)in Eq. (10) will ensure the required safety margin on uB

when unknown disturbances du are present.

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522 519

Table 4J s for the possible choices of measurement and for all cases of du in the example

Case 5 JmeanCase 3 Case 4Case 1 Case 2

148.9144.8J(T s1=151.9) 148.9 153.0 150.8 147.0

153.0155.0159.2J(T s2=104 0) 149.0153.0 151.2

155.1 146.9 149.1J(T s3=98 0) 151.0151.0 152.9

149.0 153.2J(u sA=0.292) 151.0151.1 151.1151.2

Note that all values for uB (and the constraint DuB)differ from the values given in Glemmestad, Skogestadand Gundersen (1997) for the same example. The val-ues in that paper actually refer to the bypass fractionson the hot side of heat exchanger B (which is inconsis-tent with Fig. 7), while the correct values for thebypass fractions on the cold side are given here.

The actual value for the safety margin (DuB=0.025)is obtained as follows: The values of uA and uB for thefive cases in Table 4 corresponding to T1=151.9°C(first row in Table 4) are given in Table 5. For cases 4and 5, uA saturates at zero which implies that uA just iscapable of maintaining T1=151.9°C for these distur-bances. The computations in the optimizer is based ondu=0 (case one, see Table 3) where uB takes the valueof 0.025, see Table 5. Thus, in order to handle cases 4and 5 without violating T1=151.9°C, a safety marginof DuB=0.025 has to be used by the optimizer. Notethat if we accepted that T1 deviated from its setpoint(due to saturation in uA) it would be possible to furtherreduce utility consumption somewhat. Then the set-point for T1, could be reduced slightly below 151.9°Cuntil uB saturated for some disturbance. In this exam-ple we require that setpoints for secondary measure-ments have to be satisfied.

The reason for implementing the ‘safety margin’ onuB as an inequality constraint is that other values of d0

(i.e. other operating points) may give uB,opt\0.025.Requiring uB=0.025 in such cases will result ininfeasibility.

The value for Jmean of 148.9 kW in Table 4 shouldbe compared to the mean value of Jopt from Table 3which is 146.5 kW. That is, it costs 1.6% of the utilityconsumption to guarantee feasibility for the worst caseunknown disturbance.

Figure 8 shows the results for the example when T1

is selected as secondary measurement (optimizationvariable). Figure 8a shows that To

C2 can be controlledto its setpoint for all unknown disturbances around thenominal operating point d0= [0 0]T. Note from Fig. 8band c, that the time up to zero corresponds to case 1 ofdu. from 0 to 20 min corresponds to case 2, from 20 to40 min is case 3 and so on. Bypass fractions are shownin Fig. 8d, and uA drops to close to zero after 40minutes (case 4 and 5). The perhaps most importantcurves are shown in Fig. 8e. The steady state values for

the utility consumption corresponds to the values inTable 4 (upper row). The optimal values (when du isperfectly known) from the lower row of Table 3 is alsoplotted for comparison. The utility consumptionshould be compared with the ‘traditional’ scheme with-out optimization also shown in Fig. 8e. The latter isimplemented by fixing uA at a value such that thenetwork is just feasible for all possible disturbances, i.e.d= [910, 90.05]T, using uB only (this requirementgives uA=0.680). From the results given in Fig. 8 andTable 4, it is clear that the main reduction in utilityconsumption compared to the traditional case is due tothe periodic optimization (about 13% nominally),whereas the selection of optimization variable consti-tutes 2.75% (between best and worst case).

Figure 8 has shown results around the nominal oper-ating point corresponding to the cases 1–5 in Tables 3and 4. Only one optimization is done prior to thesimulations, and the optimal setpoint for T1 found hereis maintained during the simulation (see Fig. 8f). Fig-ure 9 shows similar results, but for a larger part of theoperating region (with respect to disturbances) andwith optimizations updating the setpoint for T1 at 0, 20and 40 min.

At tB0, we have T sH1=200°C and CPC2=0.45 kW/

°C (i.e. d= [10 −0.05]T). For this operating state uA issaturated at zero. For the steady state optimizationcarried out before the simulations started the con-straint on uB was not active. From Fig. 9d we see thatuB (:0.13) is larger than the safety margin of 0.025for tB0, thus uB will not saturate if unknown distur-bances (within the prespecified bound of du5 [93 90.01]T) should occur. At time equal to zero, nominaloperating conditions are encountered and an optimiza-tion is performed immediately after. Figure 9f showshow T1 is controlled to its new setpoint (which nowhas the same value as in Fig. 8). At t=20, the (known)disturbance d= [−7 0.04]T is applied and a new opti-mization is done. At t=40, an unknown disturbance of

Table 5Values of uA and uB when T1=151.9°C

Case 1 Case 2 Case 3 Case 4 Case 5

0.207uA 0.354 0.354 0 00.0380.025uB 0.0120.0380.012

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522520

Fig. 8. Results for example. (a) Controlled output TC2o with setpoint 130°C. (b) Disturbance T s

H1. (c) Disturbance CPC2. (d) Bypass uA (thick line)and bypass uB (thin line). (e) Utility consumption. (f) Temperature T1 (secondary measurement), thick line shows actual value while thin line showssetpoint (constant at 152°C).

du= [−3 0.01]T is applied which brings the operatingstate to the opposite corner of where we started. Sincethe new disturbance is unknown, the optimizer nowreturns the same setpoint for T1 as it computed att=20. The steady state value for uB at this corner pointis close to zero.

Note that the utility consumption at the last part ofthe simulation is similar for the ‘traditional’ approach(uA is fixed at 0.680) and the proposed method. This isbecause this corner point is limiting uA for the tradi-tional approach, thus this approach is optimal for thiscorner of the operating region. After 40 min (the ex-treme corner point) the traditional approach actuallyhas lower utility consumption than the proposedmethod. This can be explained from the requirement wehave made in this example that also the setpoint for T1

should be satisfied when unknown disturbances arepresent. As mentioned above, relaxing this requirementcould reduce the utility consumption somewhat furtherfor this example.

Remark 2. From Fig. 7, it is clear that decreasing T1,T3 (by decreasing uA) or uA will reduce utility consump-tion (J), i.e. optimal values for these variables in Table

3 are minimum values (smaller values will violate theprimary goal). Therefore, the case with the largest valuehas to be chosen as this is the smallest value feasible forall du. For T2, a similar (but opposite) argument leadsto choosing the smallest value in Table 3.

7. Conclusions

A method for optimal operation of heat exchangernetworks based on periodic steady state optimization isproposed. A fixed control structure for the outlet tem-peratures (primary measurements) is selected prior tooperation. Thus, all outlet temperatures are usuallycontrolled by the heat exchanger located immediatelyupstream, and a fast response is obtained. The periodicsteady state optimization concerns the setpoints formeasurements internally in the HEN, which are con-trolled by the remaining manipulations. An importantissue is to make the implementation insensitive to un-certainty (self-optimizing control): which outputs (mea-surements) should be kept constant between eachoptimization. The outputs are selected such that thesensitivity to unknown disturbances and model errors is

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522 521

Fig. 9. Results for example. The different curves show the same as Fig. 8 but for a larger part of the operating region, and with steady stateoptimization done at 0, 20 and 40 min.

as small as possible. Optimal operating conditions forheat exchanger networks are normally located at theintersection of constraints. Implementing the setpointscomputed by the optimizer without considering uncer-tainty (nominal optimum) will cause infeasible opera-tion for certain disturbances. This infeasibility isavoided by introducing constraints in the steady stateformulation in the optimizer. It is proposed to computethe constraints such that the optimizer finds the robustoptimal solution, i.e. the optimal setpoints taking intoaccount all possible uncertainties. This provides anoptimal back-off from the nominal optimum.

A steady state formulation for heat exchanger net-works avoiding the nonlinearities due to bypass frac-tions is proposed. This general model concerns heatexchanger networks in particular, however, the method-ology comprising periodic steady state optimizationcombined with self-optimizing control is well suitedalso for other processes than heat exchanger networks.

8. Notation

c cost data

heat capacity flowrateCPdisturbancedcontrol erroreelement in transfer function/matrixg

G process transfer functionobjective functionJtransfer function for controllerK

Q heat load (duty) of heat exchangerreference (setpoint)rtemperatureTtimetmanipulated input (bypass fraction)u

y output (measurement)

SuperscriptsbypassBPutilityUactual output or outlet (temperature)o

s supply (temperature)t target or reference (temperature)

Subscripts0 nominal

primary1secondary2

B. Glemmestad et al. / Computers and Chemical Engineering 23 (1999) 509–522522

Ci cold stream IHj hot stream jm mean valuenopt nominal optimum

robust optimumroptu unknown

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