Siwen Gu ,Lei Zhang ,Yu Zhuang ,Weida Li ,Jian Du,*,Cheng Shao

1 Institute of Chemical Process Systems Engineering,School of Chemical Engineering,Dalian University of Technology,Dalian 116024,China

2 Jiangsu Collaborative Innovation Center for Cultural Creativity,School of Photoelectric Engineering,Changzhou Institute of Technology,Changzhou 213032,China

3 Institute of Advanced Control Technology,School of Control Science and Engineering,Dalian University of Technology,Dalian 116024,China

Keywords:Heat exchanger networks (HENs)Controllability Two-tier control structure Manipulated variable pairing Control pairing

ABSTRACT Because of its paramount importance in the successful industrial control strategy of a given heat exchanger network (HEN),the control structure designs for providing appropriate manipulated variable (MV)and controlled variable pairings have received considerable attention.However,quite frequently HENs with such control structures face the problem of hard constraints,typically holding the HENs at less controlled operating space.So both the MV pairings and the above control pairings should be considered to design a control structure.This paper investigates the systematic incorporation of the two pairings,and presents a methodology for designing such two-tier control structure.This is developed based on the sequential strategy,coupling an indirect-tier with direct-tier control structure design,wherein the intention is realized in the former stage and the latter is implemented for further optimization.The MV identification and pairing are achieved through variations in heat load of heat exchangers to design the indirect-tier control structure.Then the direct-tier control structure is followed the relative gain array pairing rules.With the proposed methodology,on the one hand,it generates an explicit connection between the MV pairings and the HEN configuration,and the quantitative interaction measure is improved to avoid the multiple solutions to break the relationship among all the control pairings into individuals;on the other hand,a two-tier control structure reveals control potentials and control system design requirements,this may avoid complex and economically unfavourable control and HEN structures.The application of proposed framework is illustrated with two cases involving the dynamic simulation analysis,the quantitative assessment and the random test.

1.Introduction

A heat exchanger network (HEN) usually plays an important role in process systems and the synthesis is widely recognized as the most effective approach for intensive utilization of energy.The control system design of HENs has been also an active area of research since twenty years[1-3].This problem is more difficult to be completed than the other chemical processes in the sense that the thermal output condition of several process streams must be controlled without reducing heat integration.Strong interactions in the process streams caused by the combinatorial nature in HENs leads to the potential difficulties in attaining high control quality and low control complexity.In this context,the term controllability is often referred to the ease with a process can be controlled [4].Therefore,before investigating the process control of HENs,better controllability should be achieved using the suitable control structures [4].

A control structure is generally formed by the whole set of the control loops which denote the control pairings among the manipulated variables(MVs)and the controlled variables(CVs).The control strategy for HENs in face of disturbances with the minimum performance loss is heavily dependent on such control structure used [5-9].However,the number of possible control structures grows very rapidly with respect to the number of available MVs and CVs [8].WhennyMVs are selected amongnupotential MVs and assigned to pair withnyCVs,Kookos [10] described the number of the alternatives:

Due to a considerable amount of the alternatives,the control structure design problem for providing control loops is particularly challenging.Controllability metrics,usually assessed by linear transfer function models,aim to find what control performance can be expected,or whether a specific closed loop performance can be achieved[4].Thus,much attention has been paid to controllability metrics based methods.Kookos and Perkins [10] proposed a control structure design method based on the relative gain array number (RGA number) [11,12].The value of RGA number reflects the sensitivity of process system towards control requirement,which depends on the relationship between MVs and CVs.The method was expressed by a mixed-integer nonlinear programming(MINLP) model with the objective of minimizing the control loop interaction and the disturbance sensitivity.Escobaret al.[4] further extended the above work by defining an effective flexibility index,implementing the synthesis of flexible and controllable HENs.In the recent literature,a number of methodologies have been proposed for the generation of promising control structures.These methodologies range pure mathematical programming based methods [13-15] to heuristic-based methods [16,17].A characteristic shared by most of these methodologies is still the controllability metrics for searching control structures,especially popular RGA number which can effectively decouple interactions between control loops [18].

The above works for control structure design were introduced to help build and affirm the final control strategy.However,the difficulties in finding proper control structures arouse as a demanding research subject,because hard constraints on MVs emerge as a natural and frequent part of the control system design problem [19-21].Especially when its operating condition changes,or during shut down,this will lead to a reduction in the controlled operation space which denotes flexibility in control from HEN margin aspect[14,19,20].Therefore,considering primary and auxiliary MVs,Giovanini and Marchetti [19] proposed a low-level flexible-structure control strategy using heuristic analysis to move away saturated variations from MVs so as to ensure feasibility of MV operation,achieving the capability of handling hard constraints.Their work with a focus on control strategy has caught more attention.

After that,such issue in control structure design has already been found,which is related to the whole set of the pairings between primary and auxiliary MVs.As the same as the previous works for control pairings,most of the recent literature dealing with this control structure design problem investigates the solutions through the mathematical programming techniques.Lersbamrungsuket al.[22] proposed a control structure design method to determine the best set of the MV pairings.Identifying saturated MVs in a given operating window was implemented using parametric programming,and pairing MVs was sequentially achieved by solving an integer-linear programming (ILP) model with two objective functions.The first objective was used to minimize unnecessary relationships between primary and auxiliary MVs,and the number of heat exchangers in possible control pairings taken as the second objective was used to select the most controllable control structure.With special emphasis on the interconnection among MVs and CVs,their work aimed at identifying all possible regions with different set of active constraints and then attempted to find an operation policy for switching between regions.Recent developments in active constraint switching related to control structures also present some heuristic methods[23-25].

From the abovementioned introduction,on one hand,the previous methods for providing control pairings were developed on the basis of RGA number,but introduced the nonunique solution,so that the additional controllability analyses were considered to find a desired control structure in these multiple solutions.On the other hand,for a given HEN,the previous methods for providing MV pairings involved the simplified structural measures of how direct an effect a MV has on a CV and the unnecessary relationships between MVs,which were based on the HEN configuration but independent of the heat load variations related to MV adjustments.This may lead to process networks with difficulties in terms of integrated synthesis and control.And the computational complexity was clearly enhanced by the need to consider disturbances that influence the process operation.The internal relationship between the MV identification and pairing was also not adequately processed,so that such step-wise nature imposes restrictions on optimization ability.Moreover,the combination of MV and control pairings is little touched in the previous works,even though this is quite available to ensure controllability and more practical importance.The potential mainly lies in the unclear connection between the two pairing,and the extremely large number of possible combinations among MVs and CVs even in a small-scale HEN.

This paper takes the challenge and investigates the MV and control pairings and their combination.A methodology for designing this desired control structure which is defined as a two-tier control structure is proposed.It is developed based on the sequential strategy,coupling an indirect-tier with direct-tier control structure design.The rest of the paper is organized as follows.Section 2 provides the problem statement of this study.Section 3 discusses the main features and conditions for a two-tier control structure.The outline of the proposed methodology and the mathematical formulation are also presented in this section.Two case studies are analysed in Section 4,and several results of the dynamic simulations are presented.Conclusions and final remarks are drawn in Section 5.

2.Problem Statement

Strong interactions among streams profoundly complicate a system’s inherent properties and further alter the plantwide process dynamics.So the formal control structure design is generally performed through considering control loop interaction.However,the obtained control structure of precise positioning would fail to maintain the production controllable once encountering hard constraints on MVs,resulting in the stream output temperatures deviating seriously from the targets.Therefore,this paper proposes a new methodology for designing two-tier control structures.

Through the theoretical analyses from the proposed mathematical model and the case studies for designing two-tier control structures,this paper is to show respectively that:(1) in terms of MV pairings,using an indirect-tier control structure,it is possible to avoid hard constraint violation for disturbances;(2) in terms of control pairings,using a direct-tier control structure,it is possible to decouple interactions between control loops.This study focuses on improving the HEN with its inherent property,namely the controllability,while the problems of how to control,which controllers to use and the dynamic responses of controllers are not discussed.The two-tier control structure design problem to be addressed in this paper can be briefly stated as follows.

Given are:(1) a given HEN,such as heat exchange matches between process streams,the initial consumption of external utilities,and operation and equipment results of each exchanger(such as stream flowrates,inlet and outlet temperatures,heat duties and transfer areas);(2) the expected disturbances.

The desired design result is a given HEN with its MV and control pairings.So in addition to the general contents of HEN,the bypass fractions by heat exchangers are also essential with their locations and paring with CVs.

To reduce the design difficulties,the general assumptions listed below are applied:

(1) Pressure drop and further fluid dynamics considerations,e.g.,variation of heat transfer coefficients due to the change in mobility,are neglected.

(2) A bypass can be placed over any heat exchanger.

(3) Only bypasses are used for adjustment during operation,that is,bypass fractions are regarded as MVs.

(4) Stream output temperatures are regarded as CVs.

3.Methodology

3.1.Methodology framework

Framework of the proposed methodology is presented in Fig.1.

Fig.1.Outline of proposed methodology for designing a two-tier control structure of heat exchanger networks.

For designing an indirect-tier control structure to be provided desired MV pairings,three steps are generally included:(1)Ensuring the set of active constraints,i.e.,identifying all the possible saturated MVs in a given operating window,(2) Finding the MV pairings and (3) Deciding the primary MVs [22-25].However,the previous methods were based on the HEN configuration but independent of the heat load variations related to MV adjustments.This may lead to process networks with difficulties in terms of integrated synthesis and control.The internal relationship between the MV identification and pairing was also not adequately processed,so that these were isolated to some extent.In this context,the variations in heat load of heat exchangers and in the relevant bypass fractions are employed in this paper to improve the method from Lersbamrungsuket al.[22].An explicit connection between the MV pairings and the HEN configuration is generated and these steps can be concurrently considered by comparing each bypass fraction,which is implemented by examining the variations of bypass fractions under the influences of the disturbances on each heat exchanger.First,a MV pairing is confirmed when these two ones can be used together to deal with the variation of a stream output temperature,while ensuring the minimum fraction variation.Second,the bypass identification aims to determine whether a bypass can take control of a stream output temperature independently,finding the bypasses with unsaturated fractions,and this step is dependent on the other two steps.Third,the expected disturbances are merely to offer possibility of comparing the bypass fraction with each other,and also develop the scenario with stream outlet temperatures deviating from the targets.

After that,the direct-tier control structure design focuses on the control loop interaction.This paper extends the work of Escobaret al.[4] in which the control loop interaction is explicitly considered by RGA number.However,the connotation of each step is different,as well as the method.First,we propose the improved RGA number by Frobenius norm in order to avoid multiple solutions.Second,breaking all the control loops into individuals are exploited,while assessing the feasibility of achieving the stability.

Another important issue is to construct the relationship between the indirect-and direct-tier control structures.The resulted bypass fraction pairings and the bypass fractions with unsaturated variations are expected to take complete control of the relevant stream output temperatures.Therefore,these two control structures are incorporated by optimizing the relationship among the unsaturated bypass fractions,the pairings and the stream output temperatures.The potential control pairings between the stream output temperature and the saturated bypass fraction are still considered when the degree of freedom(potential manipulations for control) is limited.It is noted that applying the bi-level optimization to design a two-tier control structure will be restricted,due to the computational complexity associated with the mix of the steady-state and the dynamic models.Especially,as shown in Eq.(1),the number of the potential control pairings makes it a challenging problem.In this way,with the sequential strategy,such relationship-based investigation can generate useful insights regarding the incorporation of the MV and control pairings in the control structure design.

3.2.Indirect-tier control structure design

In a given HEN,streams experience different disturbance intensity and require bypasses to give the relevant control actions,so that the bypasses can also be divided into different roles by their influences on stream output temperatures.In all the possible bypasses,the inactive ones are strongly affected by the hard constraints.Once the fraction of an inactive bypass is employed for control purpose,its variation becomes saturation,limiting the controlled operation space of HENs.To alleviate such negative effects,therefore,the fraction of each inactive bypass should pair with another,forming an active pairing.In which has intensive variation is the primary bypass fraction,and it is certain inactive.Similarly,the fraction of an active bypass is expected to take complete control of a stream output temperature,and it may also be involved in the active pairing.Thus,both the bypasses in a pairing may be inactive,or formed by the active and inactive ones.

Taking a HEN as an example,as shown in Fig.A1 (Supplementary Material),bypass K1 is assumed as an inactive one and an active pairing may be formed with the fraction of bypass K2.The variations in heat load of the heat exchangers caused by disturbances are employed for the relevant fraction variations to generate an explicit connection among the HEN configuration,the bypass identification and pairing.In this way,an indirect-tier control structure,which is formed by the whole set of the fractions of the active bypasses and active pairings,is the one with minor variations of bypass fractions.

Differing from the HEN synthesis,this study is developed on a given HEN.But considering all the possible bypasses,a superstructure-based MINLP model is developed and the heat exchangers are fixed.As shown in Fig.2,the superstructure is similar to the one proposed by Yee and Grossmann[26].The objective functionJto be minimized consists of the variations in the bypass fractions and the corresponding utility consumptions.The constraints include the process models for a given HEN,the models for the heat load variations of heat exchangers and the logical constraints for bypasses.

Objective function:

Constraints:

(1) Models for heat load variations in heat exchangers

When disturbances occur to the HEN,heat load variation of a heat exchanger is strongly dependent on the variation of the temperature difference between its inlet and outlet temperatures.And such heat variation propagates to the other heat exchangers along the downstream path[27].Similarly,the heat load variation of a stream is also influenced by the temperature difference between its inlet and outlet temperatures.And such variation implies that the deviation from the set point has already existed in this stream output temperature due to the disturbances,and it should be removed by providing control actions.In this context,heat load variation of a stream is equal to the summation of the heat load variations in the heat exchangers related to the bypass fraction adjustment for this stream,which is restricted as follows:

Heat exchangers related to the bypass fraction adjustments are also identified by using the downstream path strategy.As for a stream output temperature,different fraction adjustments are confronted with different related heat exchangers.Taking the HEN in Fig.3 as an example,when the fraction of bypass K1 is employed to take control of outlet temperature in stream C2,heat load variations of the heat exchangers related to this fraction adjustment will propagate through stream interaction and finally to the end temperature of stream C2.For a given HEN,therefore,all possible heat exchangers related to the fraction adjustments are found in advance.In this way,the number of nonlinear constraints is reduced even without reformulation of the equations,such as big-M formulation.The heat load variations of these heat exchangers are described below:

Fig.2.Non-split two-stage superstructure of heat exchanger network involving all the possible bypasses.

Fig.3.Heat exchangers related to the bypass fraction adjustments.

(2) Models for primary bypass fractions

In an active pairing,the bypass fraction with greater variation is determined as the primary one,which is described as follows:

(3) Logical constraints of bypasses

To avoid negative degree of freedom,a stream output temperature can only be controlled by a maximum of two bypass fractions and a similar constraint is also given by the number of streams.Then,the number of bypasses is restricted:

The following constraint represents that a bypass cannot be shared by several active pairings to avoid an increase of the potential interaction in streams:

(4) Process models for a given HEN

All the process models are referred to Escobaret al.[4]and then established by the non-split two-stage superstructure of HENs involving all the possible bypasses.For instance,the heat balance model for each stream is employed to ensure sufficient heating or cooling so that the stream output temperatures reach its desired set points at the end of the superstructure;the energy balance model is added to define the duty of the utilities,and the constraints of feasible temperature are employed to ensure the temperature decreases (hot streams) or increases (cold streams)along the stages.The process models are given in Appendix B and more details can also be found in the work of Escobaret al.[4].

3.3.Direct-tier control structure design

For a given HEN with the fractions of the active bypasses and the primary bypasses determined,the direct-tier control structure is designed in this section.As discussed above,several papers contributed in the control structure design and developed different quantitative measurements for controllability.Due to highly integrated process network,this work mainly employs two metrics:(1) RGA number as a quantitative measurement of control loop interaction and (2) μ-interaction measure (μ-IM) as a measurement of the feasibility of achieving the stability [28].In this way,a desired direct-tier control structure is the one with low control loop interaction and with feasibility of achieving the stability.

This paper proposes an improved RGA number to avoid multiple solutions.Instead of measuring the difference between all the RGA elements and 1 using 1-norm,Frobenius norm is adopted to determine the direct-tier control structure with the minimum control loop interaction.Although the improved RGA number will introduce nonlinearity to the models in contrast to the formal one,the resulted indirect-tier control structure can be employed to alleviate these numerical difficulties.On the other hand,due to highly integrated process network,certain control loop interactions can still be found in the control structure determined by the formal or the improved RGA number.μ-IM is introduced in the objective function for assessing the feasibility of stabilizing the system through independent design of individual control loops,following the μ-IM rule that is close to one as possible [28].And then a weighting factor,which is developed by the quantification of the dynamic behavior in the HEN without controllers,is to assign different contributions for the two parts in the objective function with linear relationship.

For designing a direct-tier control structure,the HEN needs to be described by a transfer function matrix with:Y（S）=G）（s）·U（s）.Where Y(s) is the Laplace transform of the CVs.U(s)is the Laplace transform of the MVs.The matrix G relates the set of potential MVsu∈U={1,2,···,u} with the CVs over sety∈Y={1,2,···,y},denoted in Eq.(14).Wheregyupoints out the stationary gain element of matrix G(0).With this linearized model,it is possible to estimate the impact of the disturbances on the CVs,and also select control structures with the best controllability properties based only in static knowledge [4].

Control structurecsis expressed by control pairing matrix Z.As shown in Eq.(15),elementszyuof matrix Z are the binary variables associated with the following logical statements:when stream output temperatureypairs with bypassu,zyu=1,otherwise,zyu=0.One feature to be noted is that setYcorresponds to the outlet temperatures of the hot and cold streams,which can also be formulated as:Y={1,2,i,1,2,···,j}.Furthermore,setUis not equal to sets HL and CL.It is only dependent on the existence of the active and primary bypass fractions from the resulted indirect-tier control structure,as shown in Eq.(16).This is also employed to restrict the elements of matrix Z:only the fractions of the active bypasses or the primary bypasses can be selected to pair with the stream output temperatures.

Meanwhile,both the bypasses in an active pairing should be restricted by:when bypass fractionuis independent of stream output temperaturey,i.e.gyu=0,the relevant pairing does not exist,zyu=0,as shown in Eq.(17).Elements λyuof the RGA matrix are expressed in Eq.(18).Whereis the element of the transpose of the inverse of matrix G(0).

In the objective function,the first part represents the control loop interaction in a given HEN.Rcsis the Frobenius norm of matrix Dcsobtained by the difference between RGA matrix and Z,as well as refers to the improved RGA number,which is defined in Eq.(19).Where elementdyuof matrix Dcsis described in Eq.(20).The distance between the elements of RGA matrix and Z is considered so that the improved RGA number is employed to effectively measure the control loop interaction.The improvement increases the possibility of avoiding multiple solutions to directly determine the direct-tier control structure.

In the objective function,the second part represents the metric for feasibility of achieving the stability so as to reduce the negative influences of control loop interactions on the controllability.All the corresponding models are referred to Kariwala and Cao[28],which are expressed in Eqs.(21)-(23).Ncsis the difference of μ-IM value with 1.Where μ denotes the structured singular value computed with a diagonal structure Δ [12].G is permuted that the chosen control loops lie along the diagonal elements and then the matrix is regarded asis the matrix consisting of the diagonal elements of

Besides,weighting factorPcsis expressed as the quantification of the matrix ψcs,yu,denoted in Eq.(24).Where ψcs,yuis established as a matrix of the same size with the above matrices.The element ψyuof matrix ψcs,yuis defined in Eq.(25).Whereis settling time.is the stream output temperature andis the desired set point.For the dynamic model see Appendix C for a brief derivation[4].

Constraints (26)-(27) denote the fact that one bypass fraction,which may be the primary or active one,must be assigned to each stream output temperature.This work adopts the pseudo inverse(Pinv) of the nonsquare matrices,in which can be given withWhereis the pseudo inverse of transfer function matrix.

4.Case Studies

Two cases are employed to demonstrate the application of the methodology for designing two-tier control structures.Dynamic simulation and quantitative measures are employed to illustrate the proposed methodology.In Case 1,comparing to the directtier control structure design method and its controllability analysis in the literature[4],the ability of the proposed method to avoid the multiple solutions is shown in the HEN involving the resulted control structure.The further demonstration for such ability is given by a random test,which includes a HEN with twenty streams and twenty bypasses.In Case 2,the closed loop performance from dynamic simulations is illustrated.These include the dynamic responses of the stream output temperatures and the corresponding bypass fractions.It is noted that the main focus here is just to show that the design can be implemented in practice.Hence,additional improvements in the control performance made by different tuning parameters of controllers are not considered in this paper.

The models for designing an indirect-tier control structure are formulated in GAMS.The solver BARON is employed to solve this MINLP model [4].And the models for designing a direct-tier control structure are implemented in Matlab.The matrix operations and dynamic simulations are also performed in Matlab.Both of the case studies are solved on an Intel Core 3.60 GHz machine with 4 GB memory.

4.1.Case 1

The HEN configuration is taken from the literature [4],which involves two hot streams and two cold streams,as shown in Fig.4.Escobaret al.[4] proposed a method for the HEN synthesis considering flexibility and controllability.The minimum control loop interaction was involved to determine the control structure,which was quantified by the formal RGA number.Here,in terms of RGA number,this case study is used to introduce our proposed method with the comparison from Escobaret al.[4].

Fig.4.Heat exchanger network from the literature.

Escobaret al.[4] suggested that the cooler provided a direct effect on output temperature of stream H1.To compare with Escobaret al.[4],therefore,the cooler is assumed as the MV for its end temperature and such control loop is fixed in the following analyses.The other CVs are the outlet temperatures of streams H2,C1 and C2.Then the fractions of the bypasses in the hot and cold sides of heat exchangers HE1,HE3 and HE4 are regarded as the MVs respectively.In this way,there are twenty-four possible directtier control structures.According to Escobaret al.[4],four promising control structures are chosen for the comparison,which is shown in Fig.5.The alternatives are presented by different dashed line,which can be described as:

According to the proposed methodology,the values of the improved RGA number andNcsfor these four control structures are obtained.The comparison with the results of Escobaret al.[4] is listed in Table 1.The first column is taken from Escobaret al.[4].However,due to multiple solutions,the method in the literature cannot directly give a control structure with the minimum control loop interaction.Then,Escobaret al.[4] introduced other controllability metrics to analyze these four control structures,as shown in the middle columns in Table 1,including disturbance sensitivity,condition number and minimum singular value.The minimum singular value must be the largest possible,so we can avoid problems with saturation,whereas the condition number and the disturbance sensitivity must be the lowest possible [4].And Escobaret al.[4] gave a final decision based on these additional controllability analyses.Conversely,this paper directly determines a direct-tier control structure,as shown in the last two columns in Table 1.And the resulted control structure is the same as that of Escobaret al.[4]:it is possible to conclude that CS3 is the most promising control structure,which is also shown in Fig.6.

Fig.5.Four promising direct-tier control structures for Case 1.

The above analyses show the ability to avoid the multiple solutions.This can be further evidenced by the distribution of the values of the formal and the improved RGA number,that is the objective function values corresponding to the potential control structures,such as the above control structures:CS1,CS2,CS3 and CS4.When the values are well distributed,a specific sparsity will be ensured.Then,the most promising control structure will be determined directly,rather than by using the additional controllability analyses.Only four promising control structures are chosen in the above analyses,therefore,a further demonstration by using a random test is given below.

As a continuation of the above analyses,a new HEN is given to form a random test.Considering this HEN,as well as the given control structures,the results of the test by using the proposed methodology are compared with that of Escobaret al.[4].In this way,the test is presented as follows:

Random Test:A HEN involving twenty streams,twenty bypasses and ten given control structures.

The effectiveness of this methodology is examined through a randomly generated matrix regarded as the transfer function matrix of the HEN,whose elements are also randomly generated from 1 to 2.It is assumed that this HEN involves ten control structures randomly selected.In the presence of the given control structures,the comparison is shown in Fig.7 and Table 2.RCS denotes the control structure randomly selected.It can be seen that the values of the objective function obtained by the proposed methodology are well distributed,even to the situation in which the values possibly feature similarity caused by a narrow range of random elements.Therefore,the superior capability of the proposed methodology is further illustrated.

4.2.Case 2

The problem data is taken from the literature [29],as listed in Table 3,and ΔTminis 10 K.The cost of heating and cooling utility is 147.4 USD·kW-1·a-1and 52.1 USD·kW-1·a-1,respectively.The nominal HEN design is obtained based on the aforementioned superstructure.All the possible bypasses are introduced in this HEN for positive degree of freedom,as shown in Fig.8.The symbols of bypass locations are employed to simply denote the corresponding fractions,e.g.bypass K1 corresponds to its fractionThe variations of the disturbances whose expected values in inlet stream temperatures are assumed to be -10 K.These disturbances may cause the variations of heat load in the heat exchangers.These propagate through stream interactions to the downstream,resulting in the product quality and process safety.The bypass fractions are assumed to take complete control of stream output temperatures,reducing or removing the temperature deviation.Here,the HEN should receive sufficient controlled operating space and ensure that the control loop is relatively independent.According to the proposed methodology,a two-tier control structure is developed.Initially,a MINLP model is developed to determine the primary bypass fractions,the active bypass fractions and pairings,consequently designing an indirect-tier control structure.After that,by the active pairings and the primary bypass fractions,a direct-tier control structure is obtained with the objective of theminimum control loop interaction while ensuring the feasibility of achieving the stability.

Table 1 Comparison with the literature results

Fig.6.Final direct-tier control structure for Case 1.

For designing an indirect-tier control structure,317 equations and 28 binary variables are involved,so that the computing time is 145 s.The active bypass fractions and pairings are listed in Table 4.Symbols*between two bypass fractions denote that a pairing exists.Where the bypass fractions without combining others are active:K4 and K6.There exist inactive bypass fractions among K1,K2,K3 and K5,and then they have to pair with others.Bypass fractions K3 and K5 form an active pairing and the primary one is K3.Also,bypass fractions K1 and K2 form an active pairing and both of them can be selected as the primary one.The HEN involving the active pairings is depicted in Fig.9.

Table 2 Comparisons under the given control structures

Table 3 Problem data for Case 2

Table 4 Active bypass fractions and active pairings

Fig.7.Comparison of the values of objective functions under the given control structures.

Fig.8.Nominal heat exchanger network design.

After that,a direct-tier control structure is designed,which is shown in Table 5.Symbols*in brackets indicate the active pairings,such as both bypass fractions K3 and K5 will be employed to take control of outlet temperature in stream H1.The HEN involving two-tier control structure CS1 is shown in Fig.10.The computing time of only designing a direct-tier control structure is 135.37 s.Whereas,based on the indirect-tier control structure,the computing time of designing the relevant direct-tier control structure is 2.24 s.This indicates that the proposed methodology yields a two-tier control structure,while enabling focusing on the more complex control structure design problem more efficiently.

In terms of the HEN configuration,the disturbance intensity in each bypass is employed to analyze the desirability of the resulted active bypass fractions and pairings.The strategy for analyzing such disturbance intensity can be found in Guet al.[30,31].For a given HEN,the disturbances propagate through each heat exchanger to each bypass,so that the numbers of the heat exchangers in the disturbance propagations indicate the disturbance intensity in each bypass.Taking bypass K3 as an example,the disturbances propagated to this bypass are shown in Fig.11.The dashed lines indicate the potential disturbances propagating from each heat exchanger to bypass K3.To facilitate the analysis,the disturbance intensity in each bypass is normalized and denoted by the triangles,which are listed in Table 6.The bypass fraction with small disturbance intensity is not the primary one in an active pairing or may be the active bypass fraction with slight variation.Conversely,the bypass fraction with intense disturbances is the primary one and also may be the active one.Beside the bypass fractions with the minimum disturbance intensity(K1 and K2),therefore,K5 with minor disturbance intensity is not the primary one.Accordingly,bypass fraction K3 with the maximum disturbance intensity is the primary one and an active pairing can be formed by them.Bypass fractions K1 and K2 have the minimum disturbance intensity,but no downstream path(an unbroken connection).Hence,K1 and K2 form an active pairing and then both of the other two bypass fractions are the active ones.The results according to the above analysis is the same as the ones shown in Fig.10,revealing the effectiveness of proposed methodology.

On the other hand,the comparison in the closed loop performance of HENs with different control structures can be used as a tool to further demonstrate the proposed methodology.However,the main focus here is just to show that the design can be implemented in practice.Hence,additional improvements in the control performance made by different tuning parameters of controllers are not considered in this paper.For the dynamic model see Appendix C for a brief derivation[4].For each loop,a Proportional Integral (PI) controller is designed.In this paper,for each output a closed loop performance is specified in terms of rise time and maximum overshoot allowed.

This comparison occurs in the HENs with the control structures CS1 and CS2 obtained from the proposed method and the literature[10] respectively.Control structure CS2 is designed as:In order to illustrate the closed loop performance some dynamic simulations are made.The expected disturbance scenario for the following dynamic simulation is given as follows:2 s later set point of outlet temperature of stream H1 changes from 323 K to 313 K;4 s later the set point changes to 333 K,and after another 4 s the set point changes to 303 K.Control loop CL1 in control structure CS1 consists of bypass fractionsand outlet temperature of stream H1,as well as control loop CL3 includes bypass fractionand outlet temperature of stream C1.Accordingly,control loop CL2 in control structure CS2 consists of bypass fractionsand outlet temperature of stream H1,as well as control loop CL4 includes bypass fractionand outlet temperature of stream C1.The control loop design and its specifications are listed in Table 7.

Dynamic simulations for tracking disturbances and set points are finally carried out to evaluate the relevant maximum deviations and settling times under the disturbance scenario to indicate the closed loop performance of the HEN with the two-tier control structure.The temperature responses are shown in Fig.12.The corresponding control actions are shown in Fig.13 and the variations of the MVs under disturbance scenario are shown in Table 8.

Table 5 Direct-tier control structure for Case 2

Fig.9.Heat exchanger network containing active pairings.

Fig.10.Heat exchanger network involving a two-tier control structure.

For control structure CS2,the worst performance is observed during the time period between the first and third load changes,most notably on outlet temperature of stream H1.The reasons can be found by observing the control actions of the corresponding bypass fractions.Large negative effects of the physical limitation may exist in these bypasses,e.g.valve saturation.As shown in the last first and third columns of Table 8,the control loops CL2 and CL4 become saturated manipulation at lower or upper bound.

Table 6 Disturbance intensity in all the bypass fractions

Table 7 Specifications of control loops for Case 2

Table 8 Variations of the MVs under the disturbance scenario

Fig.11.Possible disturbances propagating to bypass K3.

For control structure CS1,such effect is effectively taken out of operation by the bypass fractions in an active pairing.Under the nominal operating condition,bypass fractions K3 and K5 are completely closed,so that the fraction changes in control loops CL1 and CL3 are not very smooth at the beginning of the responses.However,both the temperature responses achieve stationary conditions quickly.Furthermore,although bypass fraction K3 is placed on the hot side of the heat exchanger closing to the end of the stream,which is expected to give a direct effect,bypass fraction K4 can still provide fast response and positive effects on handling this end temperature.This indicates that bypass fraction K4,as an active bypass fraction,is expected to give perfect control actions in face of disturbances and set point changes.

Figs.12 and 13 show the input saturation problem is solved by searching and pairing MVs,and the control loop interaction is considered.The ability of the resulted two-tier control structure totrack all target temperatures at the desired values even under the saturation of some manipulations is found.The above analyses are employed to highlight the importance of considering two-tier control structures in HENs.

Fig.12.Temperature responses to the set point changes.

Fig.13.Control actions of the bypasses in different control structures.

5.Conclusions

When only control pairings are involved,optimal control structure of HENs can be formulated as a linear programming implying the operation lies always at some MV constraints.This motivates the need of a control structure with the ability to break the control loops into individuals and to handle MV constraints under the change of operating condition simultaneously.In this work the two-tier control structure design has been addressed,integrating the MV and control pairings as a whole.Indirect-and direct-tier control structure design stages constitute the whole methodology,wherein the combination intention is realized by the active pairings and the primary bypass fractions.In the first stage,the HEN configuration and the heat load variations related to MV adjustments are considered to implement the identification and pairing of the inactive bypass fractions simultaneously.Then,the control loop interaction is discussed and the quantitative manner is improved to avoid multiple solutions to directly determine the most promising direct-tier control structure.This results in a desired two-tier control structure which can be found by solving a two-stage mixed-integer nonlinear and dynamic programming.

In Case 1,for revealing the ability to avoid the multiple solutions in the resulted direct-tier control structure,the given promising control loop design is adopted to compare with the literature.And then,a HEN involving twenty streams,twenty bypasses and ten given control structures are employed to deal with a random test,wherein the values of the objective function obtained by the proposed methodology are well distributed.In Case 2,a two-tier control structure is developed and its ability is evidenced by the quantitative assessment and the dynamic simulation.The active pairings according to the quantitative analysis is the same as the resulted control structure.PI controller is designed to implement dynamic simulation.The temperature and control action responses show that the near optimizing nature of the HEN is obtained by involving the resulted two-tier control structure that enables the controllers to provide the adequate control actions to relief the disturbance that is creating the problem.In this way,the proposed methodology has the ability to deliver reasonable good controllability.In the future work,dynamic flexibility and controllability aspects will be both considered in the HEN synthesis to achieve the economically optimal energy integration in a practical operating environment.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to gratefully acknowledge financial support from Jiangsu Collaborative Innovation Center for Cultural Creativity (XYN1911),the National Natural Science Foundation of China (22 008023;21776035) and Natural Science Foundation of Jiangsu Education Department (20KJB510041).

Nomenclature

Subscripts

cl index over the set of CL

cs index over the set of control structures

hl index over the set of HL

iindex over the set of hot streams

jindex over the set of cold streams

rindex over the set of superstructure stages

uindex over the set ofU

yindex over the set ofY

Appendix A.Schematic of a HEN with bypasses

See Fig.A1

Appendix B.Process models for a given HEN

(a) Heat balance for each heat exchanger:

(b) Overall heat balance for each stream:

(c) Heat balance at each stage in superstructure:

(d) Non-isothermal streams mixing:

Fig.A1.Schematic of active pairing,active and inactive bypass fractions in a HEN.

(e) Flowrate balance for each heat exchanger:

(f) For logical constraints:

(g) For approach temperatures:

(h) Assignment of inlet temperature in superstructure:

(i) Feasibility constraints for temperatures:

Appendix C.Dynamic modeling for a given HEN

The dynamic models are based on the situation that there existnheat exchangers in a HEN.These models can be found in Escobaret al.[4],which are described by the following equations: