Optimization-Based Framework for Designing Dynamic Flexible Heat

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Process Systems Engineering

Optimization-based Framework for Designing Dynamic Flexible Heat Exchanger Networks Siwen Gu, Linlin Liu, Lei Zhang, Yiyuan Bai, and Jian Du Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b04121 • Publication Date (Web): 18 Dec 2018 Downloaded from http://pubs.acs.org on December 20, 2018

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Industrial & Engineering Chemistry Research

Optimization-based Framework for Designing Dynamic Flexible Heat Exchanger Networks

Siwen Gua, Linlin Liua, Lei Zhanga, Yiyuan Baia, Jian Dua,b,*

a. Institute of Chemical Process Systems Engineering, School of Chemical Engineering, Dalian University of Technology, Dalian, 116024, Liaoning, China b. State Key Laboratory of Fine Chemicals, Dalian University of Technology, Dalian, 116024, Liaoning, China

* Corresponding Author E-mail: [email protected]

Abstract: With complex dynamic nature, Heat Exchanger Networks (HENs) should be operated successfully throughout the whole time horizon even facing the stochastic and the time-varying disturbances. In current studies, overdesigning HENs is a commonly adopted strategy to deal with the stochastic disturbances, and also the flexible design. However, it is not a good choice to find the trade-off between the dynamic flexibility and the total annual cost of HENs. In this paper, a new optimization-based framework for designing dynamic flexible HENs is presented. The key idea is to consider the ranges of variations in stream output temperatures to explore such trade-off. This allows a HEN to work under the stochastic and the time-varying disturbances without losing stream temperature targets while keeping the economically optimal energy integration. This work begins with the multiple disturbances and then 1

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dynamic flexibility analysis is employed to determine the Generalized Critical Operating Points (GCOPs) that are proposed to indicate the bottleneck of dynamic flexibility. As for each GCOP, the HEN retrofit is carried out for the capability of accommodating the stochastic and the time-varying disturbances. These are formulated as a superstructure-based Mixed Integer Non-Linear Programming (MINLP) model with the objective of minimizing the total annual cost. Three cases are given to demonstrate the application of the proposed framework. Dynamic simulation and quantitative measures show the overall economic performance and the capability of accommodating the multiple disturbances.

Keywords: Heat exchanger networks (HENs); Dynamic flexibility; Design; Disturbances

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1. INTRODUCTION Heat Exchanger Network (HEN) design is a key subject of relevance for various sectors in the chemical process industry, owing to its significant advantages of the intensive utilization of energy. It gives the best heat exchange matches of process streams and heat load distributions. However, the design is performed under the assumption of fixed parameters within nominal operating conditions. Once disturbances exist, especially, time-varying disturbances, feasibility and optimality of operation of HENs may fail to be ensured. Therefore, it is necessary to design dynamic flexible HENs to cope with such varying operating conditions. As a fundamental requirement of processes for maintaining feasibility, steady-state flexibility problems have been researched extensively in the literature. Swaney and Grossmann1 proposed a flexibility analysis method to explicitly consider stochastic disturbances in the evaluation of the given chemical processes, i.e. considering the changes with unknown (but bounded) magnitudes2. For which, flexibility index FI was defined as the maximum scaled deviation of that expected in positive and negative directions of the disturbances, such that FI ≥ 1 indicated the process had sufficient flexibility to meet the operating requirements. Otherwise, the bottleneck of flexibility was determined by identifying the critical realizations in the disturbances. Thus, it is widely used as the quantitative measure for the analysis, due to its adequate ability to evaluate a given process accommodating the stochastic disturbances. On basis of these pioneer contributions, extensive researches about the analysis have been launched. Recent reviews can be found in: 1) Applied to the HENs with both convex and non-convex feasible domains: Li et al.3; 2) Based on the robust optimization: Zhang et al.4; 3) For the processes having all the inequalities quadratic or liner: Jiang et al.5. Subsequently, based on the analyses, studies on flexible process design are to address the trade-off between total expenditure and flexibility. With this purpose, numerous work based on the 3



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optimization-based framework have appeared throughout the past years. Most of the frameworks reported in the literature follow the same key idea: implement the process retrofits in the presence of the critical realizations in the disturbances6. And recent studies dealing with such process retrofit investigate the solutions through mathematical programming techniques. Li et al.7 outlined a method for designing flexible HENs, which was sequentially implemented by two main steps: structure design and area optimization. Steimel et al.8-10 raised an environment for optimization-based process design that the degree of freedom was classified into design and operational variables, via two-stage stochastic programming. In addition to above, the studies considering multi-period are also worth highlighting. Recent reviews on this area can be found in Kang et al.11 and Isafiade12. Besides, probability distribution of disturbances is introduced to investigate their stochastic behaviors. Recent reviews on this area can be found in Banerjee et al.13-14. The above steady-state studies are based on the time-independent disturbances. However, some processes may be more sensitive to the time-varying disturbances, i.e. the ones who follow a certain class of time-dependent functions2, this may significantly deviate from the desirable performance, eventually failing to meet the required operation specifications despite the presence of well-designed controllers. Towards time-varying disturbances, Dimitriadis and Pistikopoulos15 proposed a dynamic flexibility analysis method. Similar with the steady-state problems stated above, the aim of the analysis was to determine whether a given process design could feasibly operate over the time-varying disturbances, or to quantitatively measure the capability of process accommodating the time-varying disturbances. The former was known as dynamic feasibility test problem, and the latter was known as dynamic flexibility index problem defined in their work15. That has inspired many of the later studies for featuring the capability of processes against time-varying disturbances. Adi and Chang16 proposed a temporal flexibility index to measure the cumulative effects of the disturbances on the dynamic feasibility. In their later work, Adi et 4


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al.17 raised a volumetric flexibility index and the corresponding computation strategy. And a new method based on rigorously deriving the Karush–Kuhn–Tucker conditions of a dynamic optimization model was developed by Wu and Chang18 for computing the dynamic and temporal flexibility indices. Recently, the similar work was also launched by Huang et al.19 by considering the disturbances would make the operating condition of batch reactor systems worse with time going. Then a delay tolerability index was proposed to find the maximum delay time that the process could accommodate. Until now, extensive methods and strategies have been launched to ensure processes dynamically operable and economic optimum under time-varying disturbances, known as the dynamic flexible design. Bahakim et al.20 raised a method via Power Series Expansions-based functions, which were employed to identify the variability due to the uncertain parameters using Monte Carlo sampling, such that to approximate process constraint functions and model outputs, and then the critical constraints were enforced to be satisfied all the times. Biegler21 reviewed a number of methods to solve the dynamic optimization problem and provided comprehensive summary of characteristics and advantages for future extension of the methods. Besides, influencing factors, such as initial conditions and controllers, were combined with the dynamic flexibility analysis to figure out operation stability, dynamic feasibility and flexibility of processes. Zhou et al.22 proposed a method based on optimizing the initial operating condition to improve dynamic flexibility of batch processes. Malcolm et al.23 applied dynamic flexibility analysis to integrate process design and control. In their later work, Moon et al.24 improved the embedded control optimization approach to reduce the combinatorial complexity of integrating design and control. This determined the optimal design specifications for optimal performance under uncertainty with reasonable control for dynamic feasible operations24. The similar work was also launched by Ricardez-Sandoval et al.25 by identification of an uncertain model and then estimating the set of disturbances that generated the 5



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worst-case variability. Towards the dynamic flexibility analysis, the nominal stability analysis and etc., Sánchez-Sánchez and Ricardez-Sandoval26 exploited an iterative decomposition framework for the integration between design and control of dynamic systems under uncertainty. Subsequently, they27 integrated these analyses in a single optimization formulation so as to reduce the costs evaluating the optimal design. Incorporating feasibility and stability analyses, Trainor et al.28 launched a new method to insure the processes dynamically operable under uncertainty. Koller and Ricardez-Sandoval29 proposed a dynamic optimization framework via an iterative algorithm that decomposes the integration of design, control and scheduling for multi-product processes into flexibility and feasibility analyses. The above stated studies have shown that the dynamic flexibility analysis is essential to design economically attractive chemical processes that can accommodate time-varying disturbances. For the same purpose, increasing attention has been paid to the integration of design and control under disturbances. Using Power Series Expansions, Mehta and Ricardez-Sandoval30 developed a new back-off method for simultaneous design and control, i.e. moving away from the variable values of the optimal steady-state design. Its extension was presented recently31. Towards integration of design, control, and scheduling, Koller et al.32 presented a back-off method via Monte Carlo sampling, such that to accommodate a specified variation range of disturbances and insure dynamic feasibility of operation. Accordingly, towards parametric uncertainty and process disturbances, the improved controller design method has received a lot of attention in recent years, such as Pantano et al.33 that aimed to achieve the tracking control of a given process and improve the control system response. Once the severe operational disturbances occur to an integrated process, it may impose control limitations, such as time delay and interactions. These limitations may make the control extremely difficult, even with complex controllers, significant efforts are still demanded. More important is that when 6


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lower order controllers are adopted, which are easier to understand and to implement in real operation, the actual flexibility of the integrated process may be limited by these controllers. As a result, it is difficult to ensure the feasibility and optimality of the operation. Thus, designing a process with appropriate dynamic flexibility could be a better choice rather than depending directly on controllers. Besides, enough overdesign is generally implemented to guarantee the process flexibility, especially for a HEN7. However, when this HEN operates in the nominal operating condition, an overneeded network configuration will be generated, such as the redundant heat exchanger area. Hence, the trade-off between the satisfactory dynamic flexibility and the economic optimality is particularly desirable in keeping the optimal energy integration in a practical operating environment. On the other hand, dynamic, non-linear and rigorous models solving dynamic flexible HEN design will easily become intractable numerically and computationally. All these factors keep such design problem fail to receive considerable attention. Therefore, the work on this topic is supposed to consider extending the present methods in a discrete and sequential strategy, rather than investing more efforts into the solving strategies. On basis of these pioneer contributions, extended research about dynamic flexible design applied to HENs are launched in this paper. We present an optimization-based framework, coupling the HEN retrofit stage with the dynamic flexibility analysis for addressing the problem of optimal design in terms of cost, in which dynamic flexibility consideration is simultaneously accomplished. The key idea is to consider the ranges of variations in the stream output temperatures. The remainder 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 the capability of accommodating stochastic and time-varying disturbances. The outline of the proposed framework and the mathematical formulation are presented in this section. Three cases studies are analyzed in Section 4, and several results of dynamic simulations are presented. Conclusions 7



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and final remarks are drawn in Section 5.

2. PROBLEM STATEMENT The aim of this paper is to design a HEN with minimum total annual cost that is able to ensure the dynamic flexibility under stochastic and time-varying disturbances. The problem to be addressed can be stated as follows. Given are: (1) stream data, (2) the specified ranges for stochastic disturbances, i.e. inlet stream temperatures and heat capacity flowrates, (3) time-varying disturbances, (4) a minimum temperature approach (△Tmin). Before the framework development, the following general assumptions are made: (1) pressure drop and further fluid dynamics considerations are neglected; (2) only bypasses are employed for control purpose, i.e. bypass fractions are regarded as the Manipulated Variables (MVs); (3) stream output temperatures are regarded as the Controlled Variables (CVs); (4) perfect control is considered during the design of dynamic flexible HENs, i.e. the controllers can be employed to compensate uncertainty parameters and no delays in the adjustments of the controlled variables are considered.

3. MATHEMATICAL FORMULATION 3.1 Framework for Designing Dynamic Flexible HENs

The previous optimization-based frameworks7-10 for designing flexible HENs generally contain three steps: initial HEN generation, flexibility analysis for identifying Critical Operating Points (COPs) under stochastic disturbances, HEN retrofits at all COPs. They feature more targeting flexibility and lower requirement on solution process. This study follows the similar route to achieve the capability of

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accommodating stochastic and time-varying disturbances, however the connotation of each step is different, as well as the framework. Firstly, as stochastic and time-varying disturbances are the two major types of disturbances usually occurring in real HEN operation, designing HEN merely having one type of disturbances involved would not be enough for both the desires of economically optimal energy integration and satisfactory dynamic flexibility. So in this study, both the kinds of disturbances are simultaneously considered to form a scenario of multiple disturbances. Then, the Generalized Critical Operating Points (GCOPs) are defined to indicate the bottleneck of dynamic flexibility for HENs that are described by sets of differential and algebraic equations and are subject to these disturbances. Another major difference of this framework from the previous ones is we propose the temperature control ranges to describe the ranges of the stream output temperature variations deviating from their set points. This gives spirit that the hyperrectangle desired by this study merely requires to close to rather than well locating at the boundary of the dynamic feasible region of the HEN. The profile in Figure 1 is given for the above statement in a clear manner, as shown, operating point P is assumed as the one locating at the boundary. It is also on a side of maximum scaled hyperrectangle (with black solid line) within the dynamic feasible region, but the HEN retrofitted in operating point P may be faced with the overneeded network configuration. This may be attributed to the fact that part of range of disturbance variations is typically handled within its controllable region. For other parts, the potential control difficulties may still arise. In contrast, the desired hyperrectangle (with red solid line) closes to the boundary. As results, the dynamic flexible HEN does not ever need to be that overdesigned, and the total expenditure of the HEN could be reduced as well. So temperature control ranges are investigated to make trade-off between the cost and dynamic flexibility. Besides, the optimum values of the controlled variables are relaxed from the initial HEN design under the nominal operating condition. But this will not modify these variables into the 9



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disturbances, which is typically evidenced by the optimized temperature control ranges and the given disturbances.

D2 Dynamic feasible region

D2 ,max D2N

Controllable region

D2 ,max

P

D1 ,max D1N

D1 ,max D1

Figure 1. Conceptual representation of hyperrectangle (the one with red solid line) within the dynamic feasible region.

Following the mentioned idea, outline of the proposed framework is shown in Figure 2 and the corresponding steps are listed as： (1) Design the network configuration in the nominal operating point as the initial HEN. It is achieved using the non-split two-stage superstructure involving all the possible bypasses, as shown in Figure 3, which is referred to the work of Yee and Grossmann34. Where the indexes i, j and k denote hot, cold streams and stages, respectively. The heat exchanger with ijk indicates the stream match between hot stream i and cold stream j in stage k. (2) Consider the multiple disturbances. (3) Implement dynamic flexibility analysis for the initial HEN under the above disturbances to determine GCOPs up to the number of streams, i.e. i+j. (4) HEN retrofitted in each GCOP to obtain a dynamic flexible HEN. The framework proposed here is intended to upload the initial HEN so as to identify the most promising network configuration, via 10


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retrofitting the existing bypasses, heat exchangers/utility units or inserting the new ones. These are achieved using the superstructure-based Mixed Integer Non-Linear Programming (MINLP) models. The detailed content of each step is stated in subsequent sections.

Design a HEN in nominal operating condition

Stochastic and timedependent disturbances

Dynamic flexibility analysis

1st generalized critical operating point

HEN retrofitted in 1st generalized critical operating point

……

i+j th generalized critical operating point

Rolling horizon optimization HEN retrofitted in i+j th generalized critical operating point ……

Dynamic flexible HEN

Figure 2. Proposed framework for designing a dynamic flexible heat exchanger network.

i=1 H1

H1-C1

i=2 H2

H1-C1

H1-C2

H2-C1

H2-C2

H1-C2

H1-CU

H2-C1

H2-C2

H2-CU

j=1

C1-HU

C1

j=2

C2-HU

C2

k=0

STAGE 1

k=1

STAGE 2

k=2

Figure 3. Non-split two-stage superstructure involving all the possible bypasses.

3.2 Process Disturbances

The previous method7 for designing flexible HENs was limited towards stochastic disturbances; while 11



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the time-varying disturbances following certain functions were involved in controllability and integration studies35-36. To provide a more general description of the disturbances in a real plant, this paper includes these two types of disturbances to form a scenario of multiple disturbances, as shown below. Assume that D is a set of the discretized disturbances going to happen to a HEN:

D   D1 , D2 ,..., Ds 

(1)

Where Ds is the sth discretized disturbance. Bahakim and Ricardez-Sandoval2 suggested that the stochasticity expressed in the discretization should be specified and then time was introduced. Hence, they proposed the stochastic disturbances to follow a user-defined probability distribution function:

Ds (t )  {Ds Ds ~PDF( s )}

(2)

Where  s is the parameter of the sth disturbance’s probability disturbance function PDF. In this way, Eq. (2) is capable to represent the stochastic behavior of the disturbances and used in solving flexible problems (e.g. Banerjee et al.14). However, due to the disturbance at any time t cannot be specified in advance, Bahakim and Ricardez-Sandoval2 also suggested stochastic disturbances’ time-dependence, as represented in Eq. (3): v

Ds (t )   Ds (v )

(3)

v 1

Where v indicates a sampling period and time t is obtained through the expression of t = v · △ t, in which △ t is the sampling interval. This relationship is valid in both the part period of and the whole period of operation. Eq. (3) is formulated to establish a scenario of multiple disturbances, however likely involves short intervals and periods, and consequently complicates the problem with resulting heavy computation load. To make simplification, Eq. (3) is modified into Eq. (4) in this content:

Dmds (t )   D1 (1), D2 (2),..., Ds (m) 

(4)

Eq. (4) indicates that the sth disturbance corresponds to moment m within time horizon. In terms of 12


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the stochastic disturbances’ time-dependence, the disturbances obtained from Eq. (4) are included in that from Eq. (3). In other words, the stochastic disturbances with specific moments given by the user are sequentially inserted in the time horizon. Referring to the relationship stated above, time t and moment m feature similar significance. In terms of the combination with time-varying disturbances, e.g. a step change in an inlet stream temperature, the desired multiple disturbances are achieved by placing the stochastic disturbances with time-dependence in time-varying disturbances. Thus, the set of the multiple disturbances, Dmds, is obtained, as denoted in Eq. (4).

Figure 4. Schematic representation for relationship between dynamic performance and stochasticity of disturbances.

In this way, the three-dimensional relationship lying in time, disturbances and the frequency given by probability disturbance function is obtained. A profile of such relationship is given in Figure 4 to show the application of Eq. (4). As indicated, time sequences tx, ty and tz over the time horizon have different moments (the ones with dash lines). Meanwhile, these correspond to different frequencies determined by the probability disturbance function, whose choice needs to be specified by the user, forming a scenario of multiple disturbances. As results, the information about the stochastic and dynamic characteristics of disturbances are available at the design stage. Moreover, the proposed framework provides more freedom 13



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in decision-making between conservative designs and economically attractive designs. Taking time as an example, the selection refers to a trade-off between attractive dynamic flexible HEN design and low cost computing. Which is to say that the time sequences with several moments are employed for the critical HENs so as to obtain conservative dynamic flexible HEN designs, whereas those with a few moments are introduced in order to reduce the computation cost.

3.3 Dynamic Flexibility Analysis

max DF 

s.t.

d( x(t ), x(t ), u(t ), D mds (t ), t )  0

h(x(t ), u(t ), Dmds (t ), t )  λ g(x(t ), u(t ), D mds (t ), t )  0 (P) (P)

x (t ) L  x (t )  x (t ) U u( t ) L  u( t )  u( t ) U λL  λ  λU t0  t  tf

x(t0 )  x 0 To overcome the issue that the steady-state flexibility analysis could not appropriately and efficiently indicate the bottleneck of the dynamic flexibility in HENs, taking our previous work3 as an example, this paper relaxes the commonly fixed stream output temperatures into reasonable ranges while the dynamic property of HENs and the occurrences of the “worst” disturbances are considered. In this way, differential and algebraic equations are established for the initial HEN. Then dynamic optimization model (P) towards the maximum deviations of stream output temperatures from their set points under the multiple disturbances obtained through the study stated above is developed and solved to implement the dynamic

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flexibility analysis over the initial HEN. In model (P), DF presents the summation in the deviations of stream output temperatures from their set points: (5)

N DF =  ThiOUT (tf )  ThiOUT, N   Tc OUT (tf )  Tc OUT, j j i

j

OUT

Where Thi

OUT

and Tc j OUT,N

and ThiOUT,N and Tc j

are the output temperatures of hot stream i and cold stream j, respectively,

are those of set points. x∈x, the state variable expressing output temperature

of a heat exchanger and x0 is the initial value. Considering the interaction among streams, therefore, the dynamic variability of state variables, caused by the multiple disturbances, are finally reflected by the deviations of stream output temperatures. Due to the multiple disturbances and such temperature deviations, the equality constraints h are reformulated into inequality constraints as indicated. And the limitation



is given on such constraint. d

and g are the differential and algebraic equations and the ordinary inequality constraints, respectively. Other indicators in the model are defined as, u∈u, the control variable forward to the optimum solution. Dmds, the set of the resulted disturbances. As denoted above, it means that all these disturbances are included in dynamic optimization model (P). Thus, the entire time-dependent and stochastic behaviors of the disturbances are captured, providing a more general description of the disturbances in the real world. Superscripts L and U are upper and lower bounds for the variables, respectively. To solve this type of the dynamic optimization problem, Dimitriadis and Pistikopoulos15 suggested to use the full discretization algorithm or the control parameterization algorithm. In this work the latter algorithm is employed and implemented with the Matlab optimal control package (DYNOPT). More details about this algorithm can be found in the work of Vassiliadis et al.37.

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D2 Feasible region

DN

Critical operating point that limit flexibility

D1 (a) Critical operating points

(b) Generalized critical operating points Figure 5. Critical operating points and generalized critical operating points.

According to above analyses it is conceivable that, large deviations from the set points of stream output temperatures indicate the incapability of this HEN withstanding stochastic and time-varying disturbances, meanwhile reaching the bottleneck of the dynamic flexibility. In this context, the GCOPs are regarded as the critical realization in the disturbances that produce the largest deviations in the whole period of operation. The schematic representations showing the COPs and the GCOPs are given in Figure 5. For the COPs, as described in Fig, 5(a), its identification is only related to the maximum hyperrectangle under stochastic disturbances. For the GCOPs, as described in Fig, 5(b), the colored contour denotes the dynamic feasible region, and the hyperrectangle within such region is represented by the cuboid with the black solid line. Obviously, the GCOPs exhibit not only the critical realizations in the disturbances but also the relevant moments. In summary, the GCOPs differ from the COPs that they are identified under dynamic consideration, as described below: 16


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δ  { (m) max ξ ~ max ξ} i

(6)

j

Where δ is the set of GCOPs consisting of the elements up to i+j. Obviously, these elements contain the critical realizations in the disturbances which correspond to the maximum temperature deviations from their set points. ξ is a vector of the deviations from the set points of the stream output temperatures, which is obtained by solving the model (P). It is noted that set δ is empty if all the deviations of the stream output temperatures are acceptable, which are checked by the lower bound

 min .

Further analyses for the effects of the GCOPs on a HEN are given on the basis of the worst-case performance tool27, as shown in Figure 6. When these exist GCOPs in a HEN, inequality relation is introduced into the formal equality constraints as discussed above. That is, when no disturbance occurs in some moments, constraints h follow equality relation. For these moments, the HEN has the maximum controlled operation space to handle the potential temperature deviations. In contrast, when a HEN is operated in a reduced controlled operation space, the physical limitation of MVs may arise, e.g. valve saturation. Hence, it is crucial to reduce the negative effects of the GCOPs on the feasibility and optimality of operation of HENs. These also reveal the importance of designing dynamic flexible HENs.

h(x(t),u(t),Ds(m),t)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Manipulated variable bound Maximum controlled operation space h(x(t),u(t),Ds(m),t)|δ

0

Time, t

Figure 6. The limitation of controlled operation space in a heat exchanger network caused by generalized critical operating points.

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3.4 HEN Retrofits through Rolling Horizon Optimization 1st HEN retrofitted in i+j th GCOP Current and future states

Current

Time

Future 2nd

Past

Current

Time

Future

…

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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i+j-1 th

Past

Moment m-1

Current

Moment m

Time

Future i+j th Moment m

Past

Current

Time Future

Figure 7. Heat exchanger network retrofits via rolling horizon optimization.

After the GCOPs identification of initial HEN, the HEN retrofits are implemented to eliminate the bottleneck of dynamic flexibility, such that to accommodate stochastic and time-varying disturbances. It is achieved through rolling horizon optimization38, with which the HEN retrofitted in the last GCOP will achieve the trade-off between economy target and capability of accommodating the multiple disturbances. The rolling horizon optimization to be employed aims to establish connection between the HENs retrofitted in adjacent moments, in which is presented by the stream output temperatures. This allows the retrofitted HEN to achieve the further optimization of the one retrofitted in the previous GCOP. As shown in Figure 7, each HEN contains both current and future states. The i+jth GCOP corresponds to moment m and the previous one corresponds to moment m-1. As for the HEN retrofitted in moment m-1, which is OUT,m-1

symbolled by HEN M-1, Thi

OUT,f(m-1)

and Thi

are both the stream output temperatures, but denoting

current and future states of this HEN, respectively. Superscript m-1 represents the current state of the HEN retrofitted in moment m-1 and superscript f(m-1) indicates the future states of this HEN. Thus, the current

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and future states of HENs are distinguished by different superscripts. Besides, subscripts i and j are involved for distinguishing different stream output temperatures in a HEN, for example, the output OUT,m-1

temperature of cold stream j of the HEN retrofitted in moment m-1 is described as Tc j

. As for the

HEN retrofitted in moment m, which is symbolled by HEN M, its current state is constrained by the future state of HEN M-1. That is, stream output temperature

ThiOUT,m

in HEN M is described as:

ThiOUT,m  ThiOUT,f(m-1) . And the deviations of stream output temperatures from the set points in future state are expected to be greater than that in its current state, gradually achieving the capability of HENs accommodating the multiple disturbances. These descriptions will be included in the superstructure-based MINLP models to implement the HEN retrofits in all the GCOPs. The non-split two-stage superstructure involving all the possible bypasses is employed in this work to conduct the rolling horizon optimization for the HEN retrofit, as shown in Figure 3. To retrofit the HEN in moment m, the related objective function TACm to be minimized consists of the capital cost in units and the operating cost in utility consumption. The constraints are composed by the process models and the HEN retrofit models. Objective function:

min TAC m = OP m +CAP m

(7)

Where TACm is the total annual cost for the HEN retrofitted in moment m. OPm and CAPm denote the operating and the capital costs, respectively. OPm is obtained as follow:

OP m =

C i

CU

QiCU,m   C HU Q HU,m j

(8)

j

CU,m

Where Qi

HU,m

and Q j

are the utility consumptions.

The capital costs CAPm is described by:

19



Page 20 of 105

CAP m = ( C zijkm   C HE ( ARijkm )  )  ( C HE ( ARijkmax,m )  ) i

j

k

i

j

 ( C z

 C

 ( C

max,CU,m  i

CU,m i

i

CU

CU

k

i

( AR

i

( AR

i

)  C z

CU,m  i

HU,m j

j

)  C

HU

( AR

j

k

(9)

) )   C HU ( AR HU,m j j

max,HU,m  j

) )

j

m

m

The first part represents the capital cost of the new heat exchangers. Where zijk and ARijk denote the existence and the corresponding area, respectively. The second part denotes the capital cost of the max,m

existing heat exchangers, which is only related to their areas. ARijk

represents the maximum value

between initial and retrofitted areas of a heat exchanger. Referring to the heaters and coolers, the last two parts in Eq. (9) are similar to the above. It is noted that, as shown in Eq. (7), the superstructure-based MINLP model is formulated for the HEN intended to retrofit in moment m, and developed on the basis of the information about the HEN retrofitted in the previous moment. For other moments/GCOPs, such models will be established again. Another feature to be noted is that the HEN obtained in the nominal operating condition, i.e. the initial HEN, is the previous HEN for the retrofit corresponding to the first GCOP. Constraints: (a) Process models As shown in Eq. (7), a superstructure-based MINLP model is developed for each retrofitted HEN. Thus, as part of the constraints, the process models used to characterize HENs should also be established for each HEN. Meanwhile, the characteristics about the HEN retrofitted in the previous moment are included to take the change of the network structure into account. All the process models are referred to Escobar et al.39 and established in the non-split two-stage superstructure of HENs including all the possible bypasses. For example, 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 20


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temperature are employed to ensure the temperature decreases (hot streams) or increases (cold streams) along the stages. The process models are given in Appendix A in Supporting Information and more details can also be found in the work of Escobar et al.39. (b) HEN retrofit models As the indispensable part of the constraints in the superstructure-based MINLP model, the HEN retrofit models are developed on the basis of the temperature control ranges and the relationship between the HENs retrofitted in the adjacent moments. The temperature control ranges are employed to explore the reasonable trade-off between the dynamic flexibility and the economic optimality of the HENs. The stream output temperatures is relaxed in the sense that they are able to vary around their set points rather than fixing as the nominal operating condition, which are enforced by the Eqs. (10)-(13):

ThiOUT,m  ThiOUT, N  ThiOUT, N   him

(10)

ThiOUT,m  ThiOUT, N  ThiOUT, N   him

(11)

OUT, N OUT, N m  Tc j  Tc j c j Tc OUT,m j

(12)

OUT, N N m  Tc j  Tc OUT, c j Tc OUT,m j j

Where  hi

m

and  c j

m

(13)

are defined as the control ranges of the stream output temperatures. These

are constrained by their upper and lower bounds,

 max,m

and

 min,m . In other words, each stream output

temperature is limited within a specific variation range to avoid the undesired target temperature deviation. Due to the temperature control ranges, sufficient heating or cooling of the streams in each retrofitted HEN is provided. As for the HEN retrofitted in the current moment, this is achieved using the constraints related to the heat duty of the one retrofitted in the previous moment, as shown in Eqs. (14)-(15).

Q

m i

i

  Qim-1

(14)

i

21



Q

Page 22 of 105

  Q m-1 j

m j

j

(15)

j

m

Where Qi

m-1

and Qi

are the heat duties of all the heat exchangers in hot stream i of the HENs m

m-1

retrofitted in moments m and m-1, respectively. Referring to cold stream j, Q j

and Q j

are similar to

the above. It is noted that the above descriptions may lead to the variation in stream matching in addition to the increase of the heat exchanger areas and the numbers of the bypasses. By the rolling horizon optimization, this paper proposes a generalized linear state-space model to establish the connection between the HENs retrofitted in the adjacent moments. It is developed based on the discrete-time linear state-space model which is the discretized form of the exact, continuous high-fidelity model40. Therefore, there is no mismatch between this and the original HEN40. Taking the output temperature of stream i as an example, the discrete-time linear state-space model can be described OUT,f(m)

as: Thi

 A  xhim  B  uhiIN,m . Where ThiOUT,f(m) is the stream output temperature of the HEN m

retrofitted in moment m and denotes the future state of this HEN. Current state variable xhi IN,m

the output temperatures of the heat exchangers and uhi

consists of

denotes the inputs for the HEN retrofitted in

moment m. A and B are the related elements in state matrix A and input matrix B, respectively. This discrete model correlates stream output temperatures of the future state with the inputs of the current state, as well as with the heat exchangers. In this way, the relationship between current and future states is OUT,f(m)

extracted from the model: Thi

 A*  ThiOUT,m  B*  uhim . Where uhim is the control action

(current state) for the output temperature of hot stream i in the HEN retrofitted in moment m, which will introduce the effects on the future state of this HEN. This paper adopts the bypasses as the MVs, so that *

the control actions are expressed by the variations in the bypass fractions. A

and B

*

represent the

effects of the stream output temperature and the related control action of the current state on the stream output temperature of the future state, respectively. Then, based on the description, the constraints of 22


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rolling horizon optimization are given for the generalized linear state-space model, so that this is reformulated as:

A**

ThiOUT,f(m)  A**  ThiOUT,f(m-1)  B**  uhif(m-1)  C **  uhim . Where

B**

and

denote the effects of the future state of the HEN retrofitted in the previous moment on the one in current moment. Referring to B

*

stated above, C

**

OUT,f(m-1)

features similar significance. Thi

f(m-1)

and uhi

represent the stream output temperature and the control action (future state) of the HEN retrofitted in moment m-1, respectively. However, this description is established to provide the constraint on the future state of the HEN retrofitted in moment m via the future state of the previous retrofitted HEN, rather than used to characterize HENs. In this context, the way for correlating stream output temperatures of the future state with the inputs of the current state, which is obtained from discrete-time linear state-space model, only stimulates the establishment of the relationship between the retrofitted HENs in adjacent moments. On the other hand, since the temperature control range is proposed to explore the trade-off, the variations, such as the stream output temperature deviation from its set point, are introduced in the above description. In this way, a generalized linear state-space model is obtained:

ξ f(m)  Λ m  ξ f(m-1)  Ψex,m  Δuf(m-1)  Ψ m  Δu m Where

ξ f(m)

(16)

denotes the vector of the stream output temperature deviations (future state) of the HEN

retrofitted in moment m, and

ξ f(m-1)

is that of the HEN retrofitted in moment m-1. Δu

f(m-1)

and Δu

m

are the vector of the variations of the related control actions for the HENs retrofitted in moments m-1 and m, respectively. Such variation is obtained by the difference between the control actions of current and the future states, meanwhile, it is assumed as a specific constant as this paper does not include the controller design. Referring to A** , Λ m is the vector of the configuration coefficients.

Ψm

and

Ψ ex,m

are the

vector of the control action coefficients, which are the structural measures of how direct the effects the control actions have on the HEN retrofitted in moment m. 23



Page 24 of 105

4. CASE STUDIES This section is given to demonstrate the application of the framework for designing dynamic flexible HENs. Both quantitative analyses and dynamic simulations are performanced to illustrate the proposed framework. The economic objective, the (control) degree of freedom and the disturbance intensity are analyzed quantitatively and compared to evaluate the resulted HEN directly. Then, the dynamic behaviors of the HENs are given to show their capability of accommodating the multiple disturbances. In this paper, dynamic simulations for tackling disturbances and set points are carried out to evaluate the relevant maximum deviations and settling times under various scenarios so as to indicate the control performance. And these dynamic behaviors are obtained by testing the relevant control performance. Further demonstration for the design results of this framework is given on the comparisons among the dynamic behaviors of the retrofitted HENs using the low order controllers (Proportional Integral Controller, PI)41 and using the complex controllers (Model Predictive Controller, MPC)2. In this way, three cases with respective content are studied to present this framework. In Case 1, the expected variations in inlet stream temperatures are assumed as the multiple disturbances that are concerned for the design of dynamic flexible HENs following the proposed procedure. Here, in addition to the analyses of the dynamic behaviors, the steady-state flexibility analysis of the resulted HEN is also introduced to quantitatively measure its capability of accommodating the stochastic disturbances. Then in Case 2, the stream output temperatures and the flowrates are both considered to develop a scenario of multiple disturbances. Further demonstration for the resulted trade-off between the total expenditure and the dynamic flexibility is presented by the quantitative analyses on the HENs. In Case 3, the multiple disturbances are similar to that in Case 2, moreover, the retrofitted HENs are compared with that of different variations in the control actions so that the importance of the proposed framework is revealed. 24


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The scenario of multiple disturbances, the dynamic flexibility analysis and the dynamic simulations are implemented in Matlab. While the models for HEN retrofits are formulated using GAMS and solved on an Intel Core 3.60 GHz machine with 16 GB memory. GAMS/BARON42 is used to solve these superstructure-based MINLP models.

4.1 Case 1 620K H1 1.5 kW·K-1

330kW

75kW

HE2

CU

20kW 583K HE1 H2 1.0 kW·K-1 563K 17.33m2 128.98m2 393K

350K 2

15.43m 240kW 323K HE3 388K C1 2.0 kW·K-1 53.12m2 313K C2 3.0 kW·K-1

TAC=38578 $·y-1

K1

Figure 8. Optimal heat exchanger network design for Case 1 under the nominal condition.

The stream data is taken from literature4 with two hot streams and two cold streams included. For all the heat exchange matches, △ Tmin sets as 10 (K) and U is 0.08 (kW·m-2·K-1). The costs of heating and cooling utilities are 147.4 ($·kW-1·y-1) and 52.1 ($·kW-1·y-1), respectively. An initial HEN obtained under the nominal operating condition is depicted in Figure 8, which is symbolled by HEN0. For positive (control) degree of freedom43, a bypass is added in the initial HEN. Note that this bypass with no fraction is randomly selected to some extent, such that further discussion is not given on the degree of freedom. And more details about the analysis of the degree of freedom will be raised in Case 2. The stochastic variations of ± 20 (K) for three inlet temperatures (streams H1, H2 and C1) are assumed and a known probability distribution function, i.e. normal (Gaussian), is selected. The time-varying disturbances are

25



Page 26 of 105

given as follows: the step change is of + 10 (K) in inlet temperature of stream C2.

CU

HE1

H1

H2

HE3

HE2

HU

C1

C2 K1

(a) In the first generalized critical operating point H1

HE5

HE1 HE2 K1

H2

HE3

HE4

CU

HU1

C1

HU2

C2 K2

(b) In the second generalized critical operating point Figure 9. Heat exchanger networks retrofitted in the generalized critical operating points.

According to the proposed framework, the multiple disturbances are given. These are obtained by IN

IN

employing the Matlab built-in random and time functions. And then the GCOPs for Th1 , Th2

and

Tc1IN are identified as (614.77, 567.45, 399.21), (603.86, 568.28, 405.68). For the dynamic flexibility analysis of the initial network HEN0, the total computing time is 2.8 (s). The HEN retrofits are carried out in all the GCOPs and the results are shown in Figure 9. A colored unit denotes the new one and the variations of the heat exchanger area in the existing units are expressed by the double circles. For retrofitting a HEN, taking the one retrofitted in the first GCOP as an example, the number of the equations is 313 and that of the binary variables is 17. The total computing time for this HEN retrofit is 857 (s).

26


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HEN1 and HEN2 symbol the HENs retrofitted in the first and second GCOPs, respectively. The general results for the retrofitted HENs are shown in Table 1.

Table 1. General results for retrofitted heat exchanger networks for Case 1

Heat exchanger area

HEN1

HEN2

HE1

70.46

9.61

HE2

27.34

73.39

HE3

59.95

27.34

HE4

---

38.85

HE5

---

57.69

CU

24.84

8.66

HU

2.18

(m2)

Heater or cooler area (m2)

13.76, 10.75

Bypass fraction

0.540

0, 0.071

TAC ($·y-1)

41411

75719

Table 2. Control loop design and model predictive control settings in Case 1

HEN

Controlled

Manipulated

Set point

variable

variable

(K)

OUT 2

Controlled nominal

Weight Q

Weight R

value (K)

C 222

323

323.00

1

0.1

HEN 0

Th

K

HEN 1

Th2OUT

C K 222

323

333.06

0.067

0.671

HEN 2

Th2OUT

C K 221

323

340.55

100

0.1

In contrast to HEN0, the HENs to be retrofitted aim to accommodate the stochastic and the time-varying disturbances. The flexibility index is employed to measure the steady-state flexibility on the above HENs. Using the analysis method3, the flexibility index of HEN0 is 0.9. Under the deviations of the stream output temperatures, that of HEN1 is 1.2 and the increase in the areas of the heat exchangers/heating or cooling utility also reveals the satisfactory flexibility, as the same as HEN2. However, this result does not imply large overdesign of the retrofitted HENs. The above comparison is 27



Page 28 of 105

only employed to illustrate that the retrofitted HENs can completely accommodate the expected stochastic disturbances.

Figure 10. Dynamic responses of stream output temperatures in initial and retrofitted heat exchanger networks.

The dynamic behaviors of the HENs are analyzed to further demonstrate the capability of the resulted HENs accommodating the stochastic and the time-varying disturbances. For the dynamic model see Appendix B in Supporting Information for a brief derivation. As for each HEN, the bypass fraction, whose location is the cold side of the heat exchanger closing to the end of stream H2, is selected to take control of the output temperature in this stream. Bypass K1 is selected for both HEN0 and HEN1, and bypass K2 is selected for HEN2, resulting in three control loops. And then the HENs including these control loops are shown in Figure S1 in Supporting Information. The MPC settings for the control loops in the above HENs are listed in Table 2. Where scalars Q and R are output and input weightings of MPC controller, respectively. And more details can also be found in the work of Bahakim and Ricardez-Sandoval2. Smooth response and small deviations to the set points appear on the retrofitted HENs, as shown in Figure 10. It is possible to conclude that HEN2 achieves the desired dynamic behaviors, although the initial value of the stream output temperature produces large gap to its set point. Additional improvements can be also given 28


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using tuning parameters of the controllers. But the main idea here is only to illustrate that the profitable HEN with the capability of accommodating the stochastic and the time-varying disturbances is revealed.

4.2 Case 2

The stream data is given by literature44 and the nominal HEN design is taken from our previous work7. It involves two hot streams and two cold streams, as shown in Figure S2. △Tmin, U and the costs of heating and cooling utilities are as the same as that in Case 1. The range of variations for the stochastic disturbances are listed in Table S1 in Supporting Information and a known probability distribution function, i.e. normal (Gaussian), is selected. The time-varying disturbances are given as follows: the step change is of + 10 (K) in the inlet temperature of stream C1.

7.10m2 22.90m2 46.50m2 323K HE2 HE3 CU1

Th1IN, N  583K H1 Fcph1N  1.4kW  K 1

2

27.00m 723K H2 HE1 2.0 kW·K-1 6.00m2 393K HU 553K TAC=39193 $·y-1

6.50m2 CU2

553K

313K C1 3.0 kW·K-1 Tc2IN, N  388K C2 Fcpc2N  2.0kW  K 1

Figure 11. Steady-state flexible heat exchanger network7.

In the literature7, the nominal HEN design was developed by the stage-wise superstructure considering stream split, and then a steady-state flexible HEN without stream split was obtained7, as shown in Figure 11. As this work uses the non-split two-stage superstructure, the retrofitted HENs in this case are based on the steady-state flexible HEN from the literature7. In this way, the comparisons among the

29



Page 30 of 105

steady-state flexible HEN and the retrofitted HENs are employed to illustrate the proposed framework.

HE3

H1

H2

CU

HE2

HE1

C1

C2

(a) In the first generalized critical operating point HE3

H1

H2

HE1

CU

HE4

HE2

C1

C2 K1

(b) In the second generalized critical operating point K1 HE3

H1

H2

HE1

CU

HE2

C1

C2 K2

(c) In the third generalized critical operating point Figure 12. Heat exchanger networks retrofitted in the generalized critical operating points.

According to the proposed framework, the GCOPs for Th1IN , Tc2IN , Fcph1 and Fcpc2 are found as (587.93, 383.10, 1.04, 2.13), (585.16, 390.41, 1.08, 1.70), (583.99, 387.85, 1.71, 2.24). Then the HEN retrofit step is executed and the results are shown in Figure 12. To be clear, HEN0, HEN1, HEN2 and HEN3 symbol the steady-state flexible HEN, the HENs retrofitted in the first, second and third GCOPs, respectively. The corresponding HEN details are summarized in Table S2. As this paper achieves the HEN 30


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retrofits by the rolling horizon optimization, HEN3 is expected to exhibit both the satisfactory dynamic flexibility and the overall economic performance. Hence, the previous retrofitted HEN, i.e. HEN2, is not involved in the following analyses, while HEN1 achieved further optimization on HEN0 needs to be included to demonstrate the proposed framework. Here, HEN0, HEN1 and HEN3 are selected for the following analyses. In terms of the total annual cost, the proposed framework gives HEN3 with 12.17% lower total annual cost than HEN0 and it produces an acceptable gap with that of the HEN designed under the nominal operating condition. Only increasing heat exchanger area and inserting new units were considered for improving HEN flexibility in the literature7, leading to a higher total annual cost. Whereas this paper also retrofits the existing bypasses or adds the new ones, and considers the relaxed stream output temperature to investigate the reasonable trade-off between the total expenditure and the capability of accommodating the multiple disturbances. Hence, the corresponding total annual cost is lower.

Table 3. Comparisons for Case 2

Number of heat exchangers Degree of freedom

HEN0

HEN1

HEN3

3 -4

3 -4

3 -2

In terms of the (control) degree of freedom, the comparison is listed in Table 3. The (control) degree of freedom in the HENs is described by the difference between the numbers of MVs and CVs43. This also reveals the difference with the formal degree of freedom which is related to the numbers of the independent equations and unknown variables in a model. Meanwhile, positive degree of freedom has been proved at offering insights about feasibility of the disturbance rejection accordingly45, which is to say that the larger value implies additional space for accommodating the multiple disturbances. From Table 3 it is

31



Page 32 of 105

founded that, the degree of freedom in the retrofitted HENs is improved under the same number of heat exchangers. In terms of the effects of disturbances on the HENs, the results are shown in Figure S3, which are implemented by determining the disturbance intensity in the entire HEN using our previous work46. We developed a method for identification and quantification of the disturbance propagations in a given HEN. In such method, the frequency was employed to express the disturbance intensity in a heat exchanger as the disturbances propagate through each heat exchanger. Accordingly, the disturbance intensity in each stream was obtained. To facilitate analyses, in this paper, the disturbance intensity of each stream is normalized, as denoted by the triangles. All the normalized disturbance intensity of each stream and the disturbance intensity of the entire HEN are employed in the following analyses. The disturbance intensity in the HENs is shown in Figure S3, taking HEN3 as an example to exhibit identification and quantification of the disturbance propagations. Considering disturbance intensity of the entire HEN, the disturbances are more intense, so that larger area may be assigned to each heat exchanger. In this case, the disturbance intensity from small to big is that of HEN0, HEN3 and HEN1. As mentioned above, the increases in both the heat exchanger areas and the number of the heaters or coolers lead to total annual cost of HEN0 far outweighs that of the retrofitted HENs. This indicates that there may exist further optimization in HEN0. In this context, the results combining with the dynamic behaviors of the HENs reflected in Figure 13(a) and Figure 13(b) are summarized. HEN0 with the minimum disturbance intensity exhibits poor dynamic behavior. HEN1 with the maximum disturbance intensity cannot achieve the satisfactory dynamic behavior. More cost-effective HEN3 has achieved the preferable dynamic behavior, even with intense disturbances. Considering disturbance intensity of each stream, the more intense disturbances exist, the harder it is to achieve their 32


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desired set points, meanwhile, the more aggressive control actions will exist. It is typically evidenced by the situation in which stream C2 featuring the maximum disturbance intensity in HEN1 does not show smooth response, as shown in Figure 13(c). In contrast, under the maximum disturbance intensity in HEN3, stream C2 shows smooth response, as shown in Figure 13(d). Therefore, it is possible to conclude that HEN3 achieves the optimal trade-off between the satisfactory dynamic flexibility and the economic optimality. More details regarding the dynamic behaviors are represented below. On the other hand, achieving the dynamic simulation of the HENs with the controllers is employed to analyze their dynamic behaviors. In each HEN, the bypass, which is located in the heat exchanger closing to the end of stream H1, is selected to take control of the output temperature in this stream. And then the HENs including the control loops are shown in Figure S4. PI and MPC controllers are both involved for different features: PI controllers are often used in real process operation because of its simplicity and wide range of applicability; MPC controllers can maintain the control objective, ensure optimal control action moves, and also handle constraints in the MVs and CVs, so that it is desired despite some of the computational challenge2. In this context, a few scenarios with the controllers tested to demonstrate the proposed framework are presented as below: with the same controllers (MPC or PI controllers); comparing the results with additional stochastic disturbances and different controllers. The specifications for the control loops in the retrofitted HENs are listed in Table S3. The dynamic behaviors of HEN0, HEN1 and HEN3 are analyzed by testing control performance, as presented in Figure 13(a) and Figure 13(b). As indicated, the dynamic behaviors of two retrofitted HENs still can be guaranteed with PI controllers even facing the time-varying disturbances. These results demonstrate the HENs after retrofit become capable to accommodate the stochastic and the time-varying disturbances. Beyond this, additional stochastic disturbances, i.e. ± 20 (K) in the inlet temperatures of 33



Page 34 of 105

streams H1 and C2, are added for the further analysis. The effects of taking control of the stream output temperature (stream H1) on the uncontrolled stream output temperature (stream H2) are investigated to comprehensively examine the dynamic behaviors. The results are shown in Figure 13(c) and Figure 13(d). As for HEN1, the smooth response of the output temperature in stream H2 cannot be maintained when the output temperature of stream H1 is handled using PI or MPC controllers. Conversely, the output temperature of stream H2 in HEN3 shows more smooth response. The above analyses indicate that not only the satisfactory dynamic behavior in HEN3 results from MPC, but also this HEN has the capability of withstanding the multiple disturbances.

(a) Initial and retrofitted heat exchanger networks

(b) Initial and retrofitted heat exchanger networks

34


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(c) Retrofitted heat exchanger networks in the first generalized critical operating point (under the additional stochastic disturbances)

(d) Retrofitted heat exchanger networks in the third generalized critical operating point (under the additional stochastic disturbances) Figure 13. Dynamic responses of stream output temperatures in initial and retrofitted heat exchanger networks.

4.3 Case 3 The stream data is taken from literature39 and it involves two hot streams and two cold streams. △ Tmin, U and the costs of heating and cooling utilities are as the same as that in Case 2. An initial HEN obtained under the nominal operating condition is shown in Figure 14. Expected variations in all the inlet temperatures of ± 10 (K) are assumed as the stochastic disturbances and a known probability distribution function, i.e. normal (Gaussian), is selected. The time-varying disturbances are given as follows: the step change is of + 10 (K) in output temperature of stream C1. According to the proposed framework, the retrofitted HENs are obtained, as shown in Figure 15. HEN0, HEN1 and HEN2 are employed to symbol 35



Page 36 of 105

the initial HEN, the retrofitted HENs in the first and second GCOPs, respectively. The general results for the retrofitted HENs are shown in Table S4.

220kW 144kW 323K CU HE3

583K H1 1.4 kW·K-1 723K H2 2.0 kW·K-1

35.58m2 10kW HE1

330kW

553K

HE2

10kW 393K HU 1m2

1m2

18.22m2

388K C2 2.0 kW·K-1

553K TAC=27769 $·y-1

313K C1 3.0 kW·K-1

25.00m2

Figure 14. Optimal heat exchanger network design for Case 3 under nominal condition.

CU

HE1

H1

H2

HE2

HU

C1

C2


HE1

H1

H2

CU

HE2

HU

C1

C2


Considering the output temperatures of streams H1 and C1, the comparisons are given among the dynamic behaviors of HEN0 and the retrofitted HENs. In each HEN, two additional bypasses are selected for control purposes. The bypasses located in cold sides of the same heat exchanger are selected to take the 36


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control of the output temperatures of the streams C1 in the HENs. That is, bypass K1 in cold side of heat exchanger HE1 is added in HEN0. As for HEN1 and HEN2, bypasses K2 and K1 in cold sides of heat exchanger HE2 are selected, respectively. However, the possible bypasses in this heat exchanger are not directly related to stream H1. In each HEN, therefore, the bypass, which is located in the heat exchanger closing to the end of stream H1, is selected to take the control of the output temperature in this stream. That is, bypass K2 in hot side of heat exchanger HE3 is added in HEN0. As for HEN1 and HEN2, bypasses K1 and K2 located in hot sides of heat exchangers HE1 and HE3 are selected, respectively. Thus, the HENs including the control loops are shown in Figure S5. Towards the output temperatures in streams H1 and C1, the dynamic behaviors of the HENs are detected in Figure 16.

(a) Output temperature of stream H1

(b) Output temperature of stream C1 Figure 16. Dynamic responses of stream output temperatures in initial and retrofitted heat exchanger networks. 37



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From Figure 16(a), the output temperature of stream H1 in HEN0 cannot achieve its desired set point, even there does not exist deviation in the initial value of this temperature from the set point. It is possible to conclude that this HEN fails to accommodate the multiple disturbances, as the same as HEN1. The further evidence is provided in large temperature control range in stream H1. Hence, the preferable dynamic behavior is expected to obtain under the situation in which the further optimization appears in the temperature control ranges. For this purpose, HEN2 achieves better dynamic behavior than the other two HENs. On the other hand, as shown in Figure 16(b), there may exist the control loop interaction as the output temperatures of streams H1 and C1 are both considered, leading to the negative effects on the control performance, as well as the dynamic behavior. Therefore, the further optimizations on the control actions also have to be considered for achieving the most satisfactory dynamic behavior. However, the controller design is not included in this work, the predicted variations of the control actions are only assumed as the fixed values. In this context, the analyses for optimizing the control actions are given as follows.

CU

HE1

H1

H2

HE2

HE3

HU

C1

C2 K3

Figure 17. A new retrofitted heat exchanger network.

As shown in Eq. (16),

Δu m

is the vector of the variations of the control actions for the HEN

retrofitted in moment m. It is expected to enable this HEN to achieve the preferable dynamic behavior. For

38


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which, smaller value should be found in the variation, so that the improvement takes place with: such variations are smaller than 0. A new retrofitted HEN, which is symbolled by HEN3, is obtained based on HEN1, as shown in Figure 17. The general results for HEN3 are shown in Table S4. The total annual cost of HEN3 is only 3.11% higher than that of HEN2 and produce an acceptable gap with that of the HEN0. The comparison of the retrofitted HENs (HEN1 and HEN3) is given. As the same as the above HENs, two additional bypasses are added in HEN3 for taking the control of the output temperatures in streams H1 and C1. Bypass K1 in the hot side of heat exchanger HE1 is added, as well as bypass K2 in the cold side of heat exchanger HE2. The retrofitted HEN including the control loops is shown in Figure S6. And the related dynamic behaviors of the HENs towards the dynamic responses of the output temperatures in streams H1 and C1 are denoted in Figure 18. As indicated, HEN3 shows smooth responses, especially instability expressed in output temperature of stream C1 in HEN1 is improved. The above analyses reveal the importance of this framework for the trade-off between the total annual cost and the capability of accommodating the multiple disturbances.


39



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(b) Output temperature of stream C1 Figure 18. Dynamic responses of stream output temperatures in retrofitted heat exchanger networks.

5. CONCLUSIONS The design of dynamic flexible HENs is addressed in this paper. A new optimization-based framework is to yield an economically attractive HEN with the capability of accommodating the stochastic and the time-varying disturbances. This work begins with the multiple disturbances, on basis of which, the GCOPs are proposed for bottleneck identification of dynamic flexibility. The HEN retrofits are subsequently developed to perform on all the GCOPs along time horizon, such that ensure the resultant HEN qualified capability of accommodating the multiple disturbances. Relaxing the commonly fixed stream output temperatures into reasonable ranges is the key basis of this study. By such temperature control range, the hyperrectangle desired by this study is merely required to close to rather than well locating at the boundary of dynamic feasible region of the HEN. This provides additional space for the optimal trade-off between the total annual cost and the dynamic flexibility. In the three cases, the proposed framework is evidenced by the comparisons in terms of the dynamic behavior and the quantitative information. By dynamic simulations, superior dynamic behaviors with following their set points very closely are revealed. And the total annual cost of a retrofitted HEN in the second case is 12.17% lower than that of the steady-state flexible HEN, meanwhile featuring an acceptable gap with that 40


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of the HEN designed under the nominal operating condition. Besides, the disturbances allow the dynamic feasibility to be closely related to the dynamic behavior, so that the ranges of variations in stream output temperatures will be employed to deeply investigate the coupling of the dynamic flexible design and the controller design. Hence, in the future work, the proposed framework will provide valuable insights to the integration of these two aspects, for the simultaneous achievement of the economic optimality, the satisfactory dynamic behavior and the low control complexity.

SUPPORTING INFORMATION Supporting information regarding the step-by-step implementation of the proposed framework into the case studies is included. This information is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author * Tel.: +86 411 84986301; Fax: +86-411-84986201; E-mail: [email protected] Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS The authors would like to thank Natural Science Foundation of China (No. 21576036, No. 21776035) 41



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for providing the research fund for this project.

ABBREVIATIONS HENs = heat exchanger networks MINLP = mixed-integer nonlinear programming MVs = manipulated variables CVs = controlled variables GCOPs = generalized critical operating points COPs = critical operating points PI = proportional integral controller MPC = model predictive controller

NOMENCLATURE m ARijk

Area for new heat exchanger of HEN retrofitted in moment m

ARiCU,m

Area for new cooler of HEN retrofitted in moment m

AR HU,m j

Area for new heater of HEN retrofitted in moment m

max,m ARijk

Maximum area for heat exchanger in the previous and the retrofitted HEN

ARimax,CU,m

Maximum area for cooler in the previous and the retrofitted HEN

AR max,HU,m j

Maximum area for heater in the previous and the retrofitted HEN

CAPm

Capital cost of the HEN retrofitted in moment m

DF

Summation in the deviations of stream output temperatures from their set points

Ds

Disturbances happening to HEN 42


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Fcpc mj

Heat capacity flowrate of cold stream j of HEN retrofitted in moment m

Fcphim

Heat capacity flowrate of hot stream I of HEN retrofitted in moment m

Fcpc HE,m j

Heat capacity flowrate of potential bypassed cold stream j of HEN retrofitted in

moment m

FcphiHE,m

Heat capacity flowrate of potential bypassed hot stream I of HEN retrofitted in

moment m H,m K ijk

Bypass fraction on the hot side of heat exchanger of HEN retrofitted in moment m

C,m K ijk

Bypass fraction on the cold side of heat exchanger of HEN retrofitted in moment m

OPm

Operating cost of the HEN retrofitted in moment m

QiCU,m

Heat load between hot stream i, and cold utility of HEN retrofitted in moment m

Q HU,m j

Heat load between hot stream j, and hot utility of HEN retrofitted in moment m

QijkHE,m

Heat load between hot stream i, and cold stream j, at stage k of HEN retrofitted in

moment m

Qim

Heat duty of all the heat exchangers in hot stream i of the HEN retrofitted in moment m

Q mj

Heat duty of all the heat exchangers in cold stream j of the HEN retrofitted in moment m

t

Time

TACm

Total annual cost of the HEN retrofitted in moment m

Tc IN,m j

Inlet temperature of cold stream j of HEN retrofitted in moment m

ThiIN,m

Inlet temperature of hot stream i of HEN retrofitted in moment m

Tc OUT,m j

Output temperature of cold stream j of HEN retrofitted in moment m 43



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ThiOUT,m

Output temperature of hot stream i of HEN retrofitted in moment m

ThiOUT,f(m)

Output temperature (future state) of hot stream i of HEN retrofitted in moment m

Tc OUT,N j

Set point for output temperature of cold stream j

ThiOUT,N

Set point for output temperature of hot stream i

TcijkIN,HE,m

Inlet temperature for heat exchanger between hot stream i, and cold stream j, at stage k

of HEN retrofitted in moment m

ThijkIN,HE,m



TcijkOUT,HE,m

Output temperature for heat exchanger between hot stream i, and cold stream j, at stage

k of HEN retrofitted in moment m

ThijkOUT,HE,m


k of HEN retrofitted in moment m mix,m Tcijk

Temperature of the mixing junction of cold side in a heat exchanger of HEN retrofitted

in moment m mix,m Thijk

Temperature of the mixing junction of hot side in a heat exchanger of HEN retrofitted

in moment m u

Control variable forward to the optimum solution of model (P)

v

Sampling period

x

State variable in model (P)

m zijk

Binary variable to existence of the match between hot stream i, and cold stream j, at stage k of HEN retrofitted in moment m

ziCU,m

Binary variable to existence of the match between hot stream i, and cold utility of HEN 44


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retrofitted in moment m

z HU,m j

Binary variable to existence of the match between cold stream j, and hot utility of HEN




Generalized critical operating point

 min

Lower bound for deviations of stream output temperatures from their set points

 c mj

Control range of output temperature of cold stream j of HEN retrofitted in moment m

 him

Control range of output temperature of hot stream i of HEN retrofitted in moment m

 max,m

Upper bound for control range of output temperature of cold stream j of HEN


 min,m

Lower bound for control range of output temperature of hot stream i of HEN retrofitted

in moment m

△Tmin

Minimum temperature approach

△t

Sampling interval

s

Parameter of the sth disturbance’s probability disturbance function



Limitation on constraints h

Indices i

Hot stream

j

Cold stream

k

Superstructure stage

m

Moment within time horizon

Sets/Vectors D

Set of discretized disturbances 45



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Dmds

Set of stochastic and time-varying disturbances

u

Vector of control variable forward to the optimum solution of model (P)

x

Vector of state variable in model (P)

δ

Set of generalized critical operating point

ξ

Vector of deviations of stream output temperatures from their set points

ξ f(m)

Vector of deviations of stream output temperatures (future state) from their set points

of the HEN retrofitted in moment m

m

Vector of configuration coefficient for the HEN retrofitted in moment m

 ex,m

Vector of control action coefficient for the HEN retrofitted in moment m

m


Δu m

Vector of variation of control action for the HEN retrofitted in moment m

Δu f(m-1)

Vector of variation of control action (future state) for the HEN retrofitted in moment

m-1

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(46) Gu, S.; Liu, L.; Zhang, L.; Bai, Y.; Wang, S.; Du, J. Heat exchanger networks synthesis integrated with flexibility and controllability. Chin. J. Chem. Eng. 2018, doi: 10.1016/j.cjche.2018.07.017.

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Clean version

Optimization-based Framework for Designing Dynamic Flexible Heat Exchanger Networks

Siwen Gua, Linlin Liua, Lei Zhanga, Yiyuan Baia, Jian Dua,b,*

a. Institute of Chemical Process Systems Engineering, School of Chemical Engineering, Dalian University of Technology, Dalian, 116024, Liaoning, China b. State Key Laboratory of Fine Chemicals, Dalian University of Technology, Dalian, 116024, Liaoning, China

* Corresponding Author E-mail: [email protected]

Abstract: With complex dynamic nature, Heat Exchanger Networks (HENs) should be operated successfully throughout the whole time horizon even facing the stochastic and the time-varying disturbances. In current studies, overdesigning HENs is a commonly adopted strategy to deal with the stochastic disturbances, and also the flexible design. However, it is not a good choice to find the trade-off between the dynamic flexibility and the total annual cost of HENs. In this paper, a new optimization-based framework for designing dynamic flexible HENs is presented. The key idea is to consider the ranges of variations in stream output temperatures to explore such trade-off. This allows a HEN to work under the stochastic and the time-varying disturbances without losing stream temperature targets while keeping the 53



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economically optimal energy integration. This work begins with the multiple disturbances and then dynamic flexibility analysis is employed to determine the Generalized Critical Operating Points (GCOPs) that are proposed to indicate the bottleneck of dynamic flexibility. As for each GCOP, the HEN retrofit is carried out for the capability of accommodating the stochastic and the time-varying disturbances. These are formulated as a superstructure-based Mixed Integer Non-Linear Programming (MINLP) model with the objective of minimizing the total annual cost. Three cases are given to demonstrate the application of the proposed framework. Dynamic simulation and quantitative measures show the overall economic performance and the capability of accommodating the multiple disturbances.

Keywords: Heat exchanger networks (HENs); Dynamic flexibility; Design; Disturbances

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1. INTRODUCTION Heat Exchanger Network (HEN) design is a key subject of relevance for various sectors in the chemical process industry, owing to its significant advantages of the intensive utilization of energy. It gives the best heat exchange matches of process streams and heat load distributions. However, the design is performed under the assumption of fixed parameters within nominal operating conditions. Once disturbances exist, especially, time-varying disturbances, feasibility and optimality of operation of HENs may fail to be ensured. Therefore, it is necessary to design dynamic flexible HENs to cope with such varying operating conditions. As a fundamental requirement of processes for maintaining feasibility, steady-state flexibility problems have been researched extensively in the literature. Swaney and Grossmann1 proposed a flexibility analysis method to explicitly consider stochastic disturbances in the evaluation of the given chemical processes, i.e. considering the changes with unknown (but bounded) magnitudes2. For which, flexibility index FI was defined as the maximum scaled deviation of that expected in positive and negative directions of the disturbances, such that FI ≥ 1 indicated the process had sufficient flexibility to meet the operating requirements. Otherwise, the bottleneck of flexibility was determined by identifying the critical realizations in the disturbances. Thus, it is widely used as the quantitative measure for the analysis, due to its adequate ability to evaluate a given process accommodating the stochastic disturbances. On basis of these pioneer contributions, extensive researches about the analysis have been launched. Recent reviews can be found in: 1) Applied to the HENs with both convex and non-convex feasible domains: Li et al.3; 2) Based on the robust optimization: Zhang et al.4; 3) For the processes having all the inequalities quadratic or liner: Jiang et al.5. Subsequently, based on the analyses, studies on flexible process design are to address the trade-off between total expenditure and flexibility. With this purpose, numerous work based on the 55



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optimization-based framework have appeared throughout the past years. Most of the frameworks reported in the literature follow the same key idea: implement the process retrofits in the presence of the critical realizations in the disturbances6. And recent studies dealing with such process retrofit investigate the solutions through mathematical programming techniques. Li et al.7 outlined a method for designing flexible HENs, which was sequentially implemented by two main steps: structure design and area optimization. Steimel et al.8-10 raised an environment for optimization-based process design that the degree of freedom was classified into design and operational variables, via two-stage stochastic programming. In addition to above, the studies considering multi-period are also worth highlighting. Recent reviews on this area can be found in Kang et al.11 and Isafiade12. Besides, probability distribution of disturbances is introduced to investigate their stochastic behaviors. Recent reviews on this area can be found in Banerjee et al.13-14. The above steady-state studies are based on the time-independent disturbances. However, some processes may be more sensitive to the time-varying disturbances, i.e. the ones who follow a certain class of time-dependent functions2, this may significantly deviate from the desirable performance, eventually failing to meet the required operation specifications despite the presence of well-designed controllers. Towards time-varying disturbances, Dimitriadis and Pistikopoulos15 proposed a dynamic flexibility analysis method. Similar with the steady-state problems stated above, the aim of the analysis was to determine whether a given process design could feasibly operate over the time-varying disturbances, or to quantitatively measure the capability of process accommodating the time-varying disturbances. The former was known as dynamic feasibility test problem, and the latter was known as dynamic flexibility index problem defined in their work15. That has inspired many of the later studies for featuring the capability of processes against time-varying disturbances. Adi and Chang16 proposed a temporal flexibility index to measure the cumulative effects of the disturbances on the dynamic feasibility. In their later work, Adi et 56


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al.17 raised a volumetric flexibility index and the corresponding computation strategy. And a new method based on rigorously deriving the Karush–Kuhn–Tucker conditions of a dynamic optimization model was developed by Wu and Chang18 for computing the dynamic and temporal flexibility indices. Recently, the similar work was also launched by Huang et al.19 by considering the disturbances would make the operating condition of batch reactor systems worse with time going. Then a delay tolerability index was proposed to find the maximum delay time that the process could accommodate. Until now, extensive methods and strategies have been launched to ensure processes dynamically operable and economic optimum under time-varying disturbances, known as the dynamic flexible design. Bahakim et al.20 raised a method via Power Series Expansions-based functions, which were employed to identify the variability due to the uncertain parameters using Monte Carlo sampling, such that to approximate process constraint functions and model outputs, and then the critical constraints were enforced to be satisfied all the times. Biegler21 reviewed a number of methods to solve the dynamic optimization problem and provided comprehensive summary of characteristics and advantages for future extension of the methods. Besides, influencing factors, such as initial conditions and controllers, were combined with the dynamic flexibility analysis to figure out operation stability, dynamic feasibility and flexibility of processes. Zhou et al.22 proposed a method based on optimizing the initial operating condition to improve dynamic flexibility of batch processes. Malcolm et al.23 applied dynamic flexibility analysis to integrate process design and control. In their later work, Moon et al.24 improved the embedded control optimization approach to reduce the combinatorial complexity of integrating design and control. This determined the optimal design specifications for optimal performance under uncertainty with reasonable control for dynamic feasible operations24. The similar work was also launched by Ricardez-Sandoval et al.25 by identification of an uncertain model and then estimating the set of disturbances that generated the 57



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worst-case variability. Towards the dynamic flexibility analysis, the nominal stability analysis and etc., Sánchez-Sánchez and Ricardez-Sandoval26 exploited an iterative decomposition framework for the integration between design and control of dynamic systems under uncertainty. Subsequently, they27 integrated these analyses in a single optimization formulation so as to reduce the costs evaluating the optimal design. Incorporating feasibility and stability analyses, Trainor et al.28 launched a new method to insure the processes dynamically operable under uncertainty. Koller and Ricardez-Sandoval29 proposed a dynamic optimization framework via an iterative algorithm that decomposes the integration of design, control and scheduling for multi-product processes into flexibility and feasibility analyses. The above stated studies have shown that the dynamic flexibility analysis is essential to design economically attractive chemical processes that can accommodate time-varying disturbances. For the same purpose, increasing attention has been paid to the integration of design and control under disturbances. Using Power Series Expansions, Mehta and Ricardez-Sandoval30 developed a new back-off method for simultaneous design and control, i.e. moving away from the variable values of the optimal steady-state design. Its extension was presented recently31. Towards integration of design, control, and scheduling, Koller et al.32 presented a back-off method via Monte Carlo sampling, such that to accommodate a specified variation range of disturbances and insure dynamic feasibility of operation. Accordingly, towards parametric uncertainty and process disturbances, the improved controller design method has received a lot of attention in recent years, such as Pantano et al.33 that aimed to achieve the tracking control of a given process and improve the control system response. Once the severe operational disturbances occur to an integrated process, it may impose control limitations, such as time delay and interactions. These limitations may make the control extremely difficult, even with complex controllers, significant efforts are still demanded. More important is that when 58


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lower order controllers are adopted, which are easier to understand and to implement in real operation, the actual flexibility of the integrated process may be limited by these controllers. As a result, it is difficult to ensure the feasibility and optimality of the operation. Thus, designing a process with appropriate dynamic flexibility could be a better choice rather than depending directly on controllers. Besides, enough overdesign is generally implemented to guarantee the process flexibility, especially for a HEN7. However, when this HEN operates in the nominal operating condition, an overneeded network configuration will be generated, such as the redundant heat exchanger area. Hence, the trade-off between the satisfactory dynamic flexibility and the economic optimality is particularly desirable in keeping the optimal energy integration in a practical operating environment. On the other hand, dynamic, non-linear and rigorous models solving dynamic flexible HEN design will easily become intractable numerically and computationally. All these factors keep such design problem fail to receive considerable attention. Therefore, the work on this topic is supposed to consider extending the present methods in a discrete and sequential strategy, rather than investing more efforts into the solving strategies. On basis of these pioneer contributions, extended research about dynamic flexible design applied to HENs are launched in this paper. We present an optimization-based framework, coupling the HEN retrofit stage with the dynamic flexibility analysis for addressing the problem of optimal design in terms of cost, in which dynamic flexibility consideration is simultaneously accomplished. The key idea is to consider the ranges of variations in the stream output temperatures. The remainder 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 the capability of accommodating stochastic and time-varying disturbances. The outline of the proposed framework and the mathematical formulation are presented in this section. Three cases studies are analyzed in Section 4, and several results of dynamic simulations are presented. Conclusions 59



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and final remarks are drawn in Section 5.

2. PROBLEM STATEMENT The aim of this paper is to design a HEN with minimum total annual cost that is able to ensure the dynamic flexibility under stochastic and time-varying disturbances. The problem to be addressed can be stated as follows. Given are: (1) stream data, (2) the specified ranges for stochastic disturbances, i.e. inlet stream temperatures and heat capacity flowrates, (3) time-varying disturbances, (4) a minimum temperature approach (△Tmin). Before the framework development, the following general assumptions are made: (1) pressure drop and further fluid dynamics considerations are neglected; (2) only bypasses are employed for control purpose, i.e. bypass fractions are regarded as the Manipulated Variables (MVs); (3) stream output temperatures are regarded as the Controlled Variables (CVs); (4) perfect control is considered during the design of dynamic flexible HENs, i.e. the controllers can be employed to compensate uncertainty parameters and no delays in the adjustments of the controlled variables are considered.

3. MATHEMATICAL FORMULATION 3.1 Framework for Designing Dynamic Flexible HENs

The previous optimization-based frameworks7-10 for designing flexible HENs generally contain three steps: initial HEN generation, flexibility analysis for identifying Critical Operating Points (COPs) under stochastic disturbances, HEN retrofits at all COPs. They feature more targeting flexibility and lower requirement on solution process. This study follows the similar route to achieve the capability of

60


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accommodating stochastic and time-varying disturbances, however the connotation of each step is different, as well as the framework. Firstly, as stochastic and time-varying disturbances are the two major types of disturbances usually occurring in real HEN operation, designing HEN merely having one type of disturbances involved would not be enough for both the desires of economically optimal energy integration and satisfactory dynamic flexibility. So in this study, both the kinds of disturbances are simultaneously considered to form a scenario of multiple disturbances. Then, the Generalized Critical Operating Points (GCOPs) are defined to indicate the bottleneck of dynamic flexibility for HENs that are described by sets of differential and algebraic equations and are subject to these disturbances. Another major difference of this framework from the previous ones is we propose the temperature control ranges to describe the ranges of the stream output temperature variations deviating from their set points. This gives spirit that the hyperrectangle desired by this study merely requires to close to rather than well locating at the boundary of the dynamic feasible region of the HEN. The profile in Figure 1 is given for the above statement in a clear manner, as shown, operating point P is assumed as the one locating at the boundary. It is also on a side of maximum scaled hyperrectangle (with black solid line) within the dynamic feasible region, but the HEN retrofitted in operating point P may be faced with the overneeded network configuration. This may be attributed to the fact that part of range of disturbance variations is typically handled within its controllable region. For other parts, the potential control difficulties may still arise. In contrast, the desired hyperrectangle (with red solid line) closes to the boundary. As results, the dynamic flexible HEN does not ever need to be that overdesigned, and the total expenditure of the HEN could be reduced as well. So temperature control ranges are investigated to make trade-off between the cost and dynamic flexibility. Besides, the optimum values of the controlled variables are relaxed from the initial HEN design under the nominal operating condition. But this will not modify these variables into the 61



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disturbances, which is typically evidenced by the optimized temperature control ranges and the given disturbances.

D2 Dynamic feasible region

D2 ,max D2N

Controllable region

D2 ,max

P

D1 ,max D1N

D1 ,max D1

Figure 1. Conceptual representation of hyperrectangle (the one with red solid line) within the dynamic feasible region.

Following the mentioned idea, outline of the proposed framework is shown in Figure 2 and the corresponding steps are listed as： (1) Design the network configuration in the nominal operating point as the initial HEN. It is achieved using the non-split two-stage superstructure involving all the possible bypasses, as shown in Figure 3, which is referred to the work of Yee and Grossmann34. Where the indexes i, j and k denote hot, cold streams and stages, respectively. The heat exchanger with ijk indicates the stream match between hot stream i and cold stream j in stage k. (2) Consider the multiple disturbances. (3) Implement dynamic flexibility analysis for the initial HEN under the above disturbances to determine GCOPs up to the number of streams, i.e. i+j. (4) HEN retrofitted in each GCOP to obtain a dynamic flexible HEN. The framework proposed here is intended to upload the initial HEN so as to identify the most promising network configuration, via 62


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retrofitting the existing bypasses, heat exchangers/utility units or inserting the new ones. These are achieved using the superstructure-based Mixed Integer Non-Linear Programming (MINLP) models. The detailed content of each step is stated in subsequent sections.

Design a HEN in nominal operating condition

Stochastic and timedependent disturbances

Dynamic flexibility analysis

1st generalized critical operating point

HEN retrofitted in 1st generalized critical operating point

……

i+j th generalized critical operating point

Rolling horizon optimization HEN retrofitted in i+j th generalized critical operating point ……

Dynamic flexible HEN

Figure 2. Proposed framework for designing a dynamic flexible heat exchanger network.

i=1 H1

H1-C1

i=2 H2

H1-C1

H1-C2

H2-C1

H2-C2

H1-C2

H1-CU

H2-C1

H2-C2

H2-CU

j=1

C1-HU

C1

j=2

C2-HU

C2

k=0

STAGE 1

k=1

STAGE 2

k=2

Figure 3. Non-split two-stage superstructure involving all the possible bypasses.

3.2 Process Disturbances

The previous method7 for designing flexible HENs was limited towards stochastic disturbances; while 63



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the time-varying disturbances following certain functions were involved in controllability and integration studies35-36. To provide a more general description of the disturbances in a real plant, this paper includes these two types of disturbances to form a scenario of multiple disturbances, as shown below. Assume that D is a set of the discretized disturbances going to happen to a HEN:

D   D1 , D2 ,..., Ds 

(1)

Where Ds is the sth discretized disturbance. Bahakim and Ricardez-Sandoval2 suggested that the stochasticity expressed in the discretization should be specified and then time was introduced. Hence, they proposed the stochastic disturbances to follow a user-defined probability distribution function:

Ds (t )  {Ds Ds ~PDF( s )}

(2)

Where  s is the parameter of the sth disturbance’s probability disturbance function PDF. In this way, Eq. (2) is capable to represent the stochastic behavior of the disturbances and used in solving flexible problems (e.g. Banerjee et al.14). However, due to the disturbance at any time t cannot be specified in advance, Bahakim and Ricardez-Sandoval2 also suggested stochastic disturbances’ time-dependence, as represented in Eq. (3): v

Ds (t )   Ds (v )

(3)

v 1

Where v indicates a sampling period and time t is obtained through the expression of t = v · △ t, in which △ t is the sampling interval. This relationship is valid in both the part period of and the whole period of operation. Eq. (3) is formulated to establish a scenario of multiple disturbances, however likely involves short intervals and periods, and consequently complicates the problem with resulting heavy computation load. To make simplification, Eq. (3) is modified into Eq. (4) in this content:

Dmds (t )   D1 (1), D2 (2),..., Ds (m) 

(4)

Eq. (4) indicates that the sth disturbance corresponds to moment m within time horizon. In terms of 64


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the stochastic disturbances’ time-dependence, the disturbances obtained from Eq. (4) are included in that from Eq. (3). In other words, the stochastic disturbances with specific moments given by the user are sequentially inserted in the time horizon. Referring to the relationship stated above, time t and moment m feature similar significance. In terms of the combination with time-varying disturbances, e.g. a step change in an inlet stream temperature, the desired multiple disturbances are achieved by placing the stochastic disturbances with time-dependence in time-varying disturbances. Thus, the set of the multiple disturbances, Dmds, is obtained, as denoted in Eq. (4).

Figure 4. Schematic representation for relationship between dynamic performance and stochasticity of disturbances.

In this way, the three-dimensional relationship lying in time, disturbances and the frequency given by probability disturbance function is obtained. A profile of such relationship is given in Figure 4 to show the application of Eq. (4). As indicated, time sequences tx, ty and tz over the time horizon have different moments (the ones with dash lines). Meanwhile, these correspond to different frequencies determined by the probability disturbance function, whose choice needs to be specified by the user, forming a scenario of multiple disturbances. As results, the information about the stochastic and dynamic characteristics of disturbances are available at the design stage. Moreover, the proposed framework provides more freedom 65



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in decision-making between conservative designs and economically attractive designs. Taking time as an example, the selection refers to a trade-off between attractive dynamic flexible HEN design and low cost computing. Which is to say that the time sequences with several moments are employed for the critical HENs so as to obtain conservative dynamic flexible HEN designs, whereas those with a few moments are introduced in order to reduce the computation cost.

3.3 Dynamic Flexibility Analysis

max DF 

s.t.

d( x(t ), x(t ), u(t ), D mds (t ), t )  0

h(x(t ), u(t ), Dmds (t ), t )  λ g(x(t ), u(t ), D mds (t ), t )  0 (P) (P)

x (t ) L  x (t )  x (t ) U u( t ) L  u( t )  u( t ) U λL  λ  λU t0  t  tf

x(t0 )  x 0 To overcome the issue that the steady-state flexibility analysis could not appropriately and efficiently indicate the bottleneck of the dynamic flexibility in HENs, taking our previous work3 as an example, this paper relaxes the commonly fixed stream output temperatures into reasonable ranges while the dynamic property of HENs and the occurrences of the “worst” disturbances are considered. In this way, differential and algebraic equations are established for the initial HEN. Then dynamic optimization model (P) towards the maximum deviations of stream output temperatures from their set points under the multiple disturbances obtained through the study stated above is developed and solved to implement the dynamic

66


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flexibility analysis over the initial HEN. In model (P), DF presents the summation in the deviations of stream output temperatures from their set points: (5)

N DF =  ThiOUT (tf )  ThiOUT, N   Tc OUT (tf )  Tc OUT, j j i

j

OUT

Where Thi

OUT

and Tc j OUT,N

and ThiOUT,N and Tc j

are the output temperatures of hot stream i and cold stream j, respectively,

are those of set points. x∈x, the state variable expressing output temperature

of a heat exchanger and x0 is the initial value. Considering the interaction among streams, therefore, the dynamic variability of state variables, caused by the multiple disturbances, are finally reflected by the deviations of stream output temperatures. Due to the multiple disturbances and such temperature deviations, the equality constraints h are reformulated into inequality constraints as indicated. And the limitation



is given on such constraint. d

and g are the differential and algebraic equations and the ordinary inequality constraints, respectively. Other indicators in the model are defined as, u∈u, the control variable forward to the optimum solution. Dmds, the set of the resulted disturbances. As denoted above, it means that all these disturbances are included in dynamic optimization model (P). Thus, the entire time-dependent and stochastic behaviors of the disturbances are captured, providing a more general description of the disturbances in the real world. Superscripts L and U are upper and lower bounds for the variables, respectively. To solve this type of the dynamic optimization problem, Dimitriadis and Pistikopoulos15 suggested to use the full discretization algorithm or the control parameterization algorithm. In this work the latter algorithm is employed and implemented with the Matlab optimal control package (DYNOPT). More details about this algorithm can be found in the work of Vassiliadis et al.37.

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D2 Feasible region

DN

Critical operating point that limit flexibility

D1 (a) Critical operating points

(b) Generalized critical operating points Figure 5. Critical operating points and generalized critical operating points.

According to above analyses it is conceivable that, large deviations from the set points of stream output temperatures indicate the incapability of this HEN withstanding stochastic and time-varying disturbances, meanwhile reaching the bottleneck of the dynamic flexibility. In this context, the GCOPs are regarded as the critical realization in the disturbances that produce the largest deviations in the whole period of operation. The schematic representations showing the COPs and the GCOPs are given in Figure 5. For the COPs, as described in Fig, 5(a), its identification is only related to the maximum hyperrectangle under stochastic disturbances. For the GCOPs, as described in Fig, 5(b), the colored contour denotes the dynamic feasible region, and the hyperrectangle within such region is represented by the cuboid with the black solid line. Obviously, the GCOPs exhibit not only the critical realizations in the disturbances but also the relevant moments. In summary, the GCOPs differ from the COPs that they are identified under dynamic consideration, as described below: 68


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δ  { (m) max ξ ~ max ξ} i

(6)

j

Where δ is the set of GCOPs consisting of the elements up to i+j. Obviously, these elements contain the critical realizations in the disturbances which correspond to the maximum temperature deviations from their set points. ξ is a vector of the deviations from the set points of the stream output temperatures, which is obtained by solving the model (P). It is noted that set δ is empty if all the deviations of the stream output temperatures are acceptable, which are checked by the lower bound

 min .

Further analyses for the effects of the GCOPs on a HEN are given on the basis of the worst-case performance tool27, as shown in Figure 6. When these exist GCOPs in a HEN, inequality relation is introduced into the formal equality constraints as discussed above. That is, when no disturbance occurs in some moments, constraints h follow equality relation. For these moments, the HEN has the maximum controlled operation space to handle the potential temperature deviations. In contrast, when a HEN is operated in a reduced controlled operation space, the physical limitation of MVs may arise, e.g. valve saturation. Hence, it is crucial to reduce the negative effects of the GCOPs on the feasibility and optimality of operation of HENs. These also reveal the importance of designing dynamic flexible HENs.

h(x(t),u(t),Ds(m),t)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Manipulated variable bound Maximum controlled operation space h(x(t),u(t),Ds(m),t)|δ

0

Time, t

Figure 6. The limitation of controlled operation space in a heat exchanger network caused by generalized critical operating points.

69



3.4 HEN Retrofits through Rolling Horizon Optimization 1st HEN retrofitted in i+j th GCOP Current and future states

Current

Time

Future 2nd

Past

Current

Time

Future

…

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 70 of 105

i+j-1 th

Past

Moment m-1

Current

Moment m

Time

Future i+j th Moment m

Past

Current

Time Future

Figure 7. Heat exchanger network retrofits via rolling horizon optimization.

After the GCOPs identification of initial HEN, the HEN retrofits are implemented to eliminate the bottleneck of dynamic flexibility, such that to accommodate stochastic and time-varying disturbances. It is achieved through rolling horizon optimization38, with which the HEN retrofitted in the last GCOP will achieve the trade-off between economy target and capability of accommodating the multiple disturbances. The rolling horizon optimization to be employed aims to establish connection between the HENs retrofitted in adjacent moments, in which is presented by the stream output temperatures. This allows the retrofitted HEN to achieve the further optimization of the one retrofitted in the previous GCOP. As shown in Figure 7, each HEN contains both current and future states. The i+jth GCOP corresponds to moment m and the previous one corresponds to moment m-1. As for the HEN retrofitted in moment m-1, which is OUT,m-1

symbolled by HEN M-1, Thi

OUT,f(m-1)

and Thi

are both the stream output temperatures, but denoting

current and future states of this HEN, respectively. Superscript m-1 represents the current state of the HEN retrofitted in moment m-1 and superscript f(m-1) indicates the future states of this HEN. Thus, the current

70


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and future states of HENs are distinguished by different superscripts. Besides, subscripts i and j are involved for distinguishing different stream output temperatures in a HEN, for example, the output OUT,m-1

temperature of cold stream j of the HEN retrofitted in moment m-1 is described as Tc j

. As for the

HEN retrofitted in moment m, which is symbolled by HEN M, its current state is constrained by the future state of HEN M-1. That is, stream output temperature

ThiOUT,m

in HEN M is described as:

ThiOUT,m  ThiOUT,f(m-1) . And the deviations of stream output temperatures from the set points in future state are expected to be greater than that in its current state, gradually achieving the capability of HENs accommodating the multiple disturbances. These descriptions will be included in the superstructure-based MINLP models to implement the HEN retrofits in all the GCOPs. The non-split two-stage superstructure involving all the possible bypasses is employed in this work to conduct the rolling horizon optimization for the HEN retrofit, as shown in Figure 3. To retrofit the HEN in moment m, the related objective function TACm to be minimized consists of the capital cost in units and the operating cost in utility consumption. The constraints are composed by the process models and the HEN retrofit models. Objective function:

min TAC m = OP m +CAP m

(7)

Where TACm is the total annual cost for the HEN retrofitted in moment m. OPm and CAPm denote the operating and the capital costs, respectively. OPm is obtained as follow:

OP m =

C i

CU

QiCU,m   C HU Q HU,m j

(8)

j

CU,m

Where Qi

HU,m

and Q j

are the utility consumptions.

The capital costs CAPm is described by:

71



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CAP m = ( C zijkm   C HE ( ARijkm )  )  ( C HE ( ARijkmax,m )  ) i

j

k

i

j

 ( C z

 C

 ( C

max,CU,m  i

CU,m i

i

CU

CU

k

i

( AR

i

( AR

i

)  C z

CU,m  i

HU,m j

j

)  C

HU

( AR

j

k

(9)

) )   C HU ( AR HU,m j j

max,HU,m  j

) )

j

m

m

The first part represents the capital cost of the new heat exchangers. Where zijk and ARijk denote the existence and the corresponding area, respectively. The second part denotes the capital cost of the max,m

existing heat exchangers, which is only related to their areas. ARijk

represents the maximum value

between initial and retrofitted areas of a heat exchanger. Referring to the heaters and coolers, the last two parts in Eq. (9) are similar to the above. It is noted that, as shown in Eq. (7), the superstructure-based MINLP model is formulated for the HEN intended to retrofit in moment m, and developed on the basis of the information about the HEN retrofitted in the previous moment. For other moments/GCOPs, such models will be established again. Another feature to be noted is that the HEN obtained in the nominal operating condition, i.e. the initial HEN, is the previous HEN for the retrofit corresponding to the first GCOP. Constraints: (a) Process models As shown in Eq. (7), a superstructure-based MINLP model is developed for each retrofitted HEN. Thus, as part of the constraints, the process models used to characterize HENs should also be established for each HEN. Meanwhile, the characteristics about the HEN retrofitted in the previous moment are included to take the change of the network structure into account. All the process models are referred to Escobar et al.39 and established in the non-split two-stage superstructure of HENs including all the possible bypasses. For example, 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 72


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temperature are employed to ensure the temperature decreases (hot streams) or increases (cold streams) along the stages. The process models are given in Appendix A in Supporting Information and more details can also be found in the work of Escobar et al.39. (b) HEN retrofit models As the indispensable part of the constraints in the superstructure-based MINLP model, the HEN retrofit models are developed on the basis of the temperature control ranges and the relationship between the HENs retrofitted in the adjacent moments. The temperature control ranges are employed to explore the reasonable trade-off between the dynamic flexibility and the economic optimality of the HENs. The stream output temperatures is relaxed in the sense that they are able to vary around their set points rather than fixing as the nominal operating condition, which are enforced by the Eqs. (10)-(13):

ThiOUT,m  ThiOUT, N  ThiOUT, N   him

(10)

ThiOUT,m  ThiOUT, N  ThiOUT, N   him

(11)

OUT, N OUT, N m  Tc j  Tc j c j Tc OUT,m j

(12)

OUT, N N m  Tc j  Tc OUT, c j Tc OUT,m j j

Where  hi

m

and  c j

m

(13)

are defined as the control ranges of the stream output temperatures. These

are constrained by their upper and lower bounds,

 max,m

and

 min,m . In other words, each stream output

temperature is limited within a specific variation range to avoid the undesired target temperature deviation. Due to the temperature control ranges, sufficient heating or cooling of the streams in each retrofitted HEN is provided. As for the HEN retrofitted in the current moment, this is achieved using the constraints related to the heat duty of the one retrofitted in the previous moment, as shown in Eqs. (14)-(15).

Q

m i

i

  Qim-1

(14)

i

73



Q

Page 74 of 105

  Q m-1 j

m j

j

(15)

j

m

Where Qi

m-1

and Qi

are the heat duties of all the heat exchangers in hot stream i of the HENs m

m-1

retrofitted in moments m and m-1, respectively. Referring to cold stream j, Q j

and Q j

are similar to

the above. It is noted that the above descriptions may lead to the variation in stream matching in addition to the increase of the heat exchanger areas and the numbers of the bypasses. By the rolling horizon optimization, this paper proposes a generalized linear state-space model to establish the connection between the HENs retrofitted in the adjacent moments. It is developed based on the discrete-time linear state-space model which is the discretized form of the exact, continuous high-fidelity model40. Therefore, there is no mismatch between this and the original HEN40. Taking the output temperature of stream i as an example, the discrete-time linear state-space model can be described OUT,f(m)

as: Thi

 A  xhim  B  uhiIN,m . Where ThiOUT,f(m) is the stream output temperature of the HEN m

retrofitted in moment m and denotes the future state of this HEN. Current state variable xhi IN,m

the output temperatures of the heat exchangers and uhi

consists of

denotes the inputs for the HEN retrofitted in

moment m. A and B are the related elements in state matrix A and input matrix B, respectively. This discrete model correlates stream output temperatures of the future state with the inputs of the current state, as well as with the heat exchangers. In this way, the relationship between current and future states is OUT,f(m)

extracted from the model: Thi

 A*  ThiOUT,m  B*  uhim . Where uhim is the control action

(current state) for the output temperature of hot stream i in the HEN retrofitted in moment m, which will introduce the effects on the future state of this HEN. This paper adopts the bypasses as the MVs, so that *

the control actions are expressed by the variations in the bypass fractions. A

and B

*

represent the

effects of the stream output temperature and the related control action of the current state on the stream output temperature of the future state, respectively. Then, based on the description, the constraints of 74


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rolling horizon optimization are given for the generalized linear state-space model, so that this is reformulated as:

A**

ThiOUT,f(m)  A**  ThiOUT,f(m-1)  B**  uhif(m-1)  C **  uhim . Where

B**

and

denote the effects of the future state of the HEN retrofitted in the previous moment on the one in current moment. Referring to B

*

stated above, C

**

OUT,f(m-1)

features similar significance. Thi

f(m-1)

and uhi

represent the stream output temperature and the control action (future state) of the HEN retrofitted in moment m-1, respectively. However, this description is established to provide the constraint on the future state of the HEN retrofitted in moment m via the future state of the previous retrofitted HEN, rather than used to characterize HENs. In this context, the way for correlating stream output temperatures of the future state with the inputs of the current state, which is obtained from discrete-time linear state-space model, only stimulates the establishment of the relationship between the retrofitted HENs in adjacent moments. On the other hand, since the temperature control range is proposed to explore the trade-off, the variations, such as the stream output temperature deviation from its set point, are introduced in the above description. In this way, a generalized linear state-space model is obtained:

ξ f(m)  Λ m  ξ f(m-1)  Ψex,m  Δuf(m-1)  Ψ m  Δu m Where

ξ f(m)

(16)

denotes the vector of the stream output temperature deviations (future state) of the HEN

retrofitted in moment m, and

ξ f(m-1)

is that of the HEN retrofitted in moment m-1. Δu

f(m-1)

and Δu

m

are the vector of the variations of the related control actions for the HENs retrofitted in moments m-1 and m, respectively. Such variation is obtained by the difference between the control actions of current and the future states, meanwhile, it is assumed as a specific constant as this paper does not include the controller design. Referring to A** , Λ m is the vector of the configuration coefficients.

Ψm

and

Ψ ex,m

are the

vector of the control action coefficients, which are the structural measures of how direct the effects the control actions have on the HEN retrofitted in moment m. 75



Page 76 of 105

4. CASE STUDIES This section is given to demonstrate the application of the framework for designing dynamic flexible HENs. Both quantitative analyses and dynamic simulations are performanced to illustrate the proposed framework. The economic objective, the (control) degree of freedom and the disturbance intensity are analyzed quantitatively and compared to evaluate the resulted HEN directly. Then, the dynamic behaviors of the HENs are given to show their capability of accommodating the multiple disturbances. In this paper, dynamic simulations for tackling disturbances and set points are carried out to evaluate the relevant maximum deviations and settling times under various scenarios so as to indicate the control performance. And these dynamic behaviors are obtained by testing the relevant control performance. Further demonstration for the design results of this framework is given on the comparisons among the dynamic behaviors of the retrofitted HENs using the low order controllers (Proportional Integral Controller, PI)41 and using the complex controllers (Model Predictive Controller, MPC)2. In this way, three cases with respective content are studied to present this framework. In Case 1, the expected variations in inlet stream temperatures are assumed as the multiple disturbances that are concerned for the design of dynamic flexible HENs following the proposed procedure. Here, in addition to the analyses of the dynamic behaviors, the steady-state flexibility analysis of the resulted HEN is also introduced to quantitatively measure its capability of accommodating the stochastic disturbances. Then in Case 2, the stream output temperatures and the flowrates are both considered to develop a scenario of multiple disturbances. Further demonstration for the resulted trade-off between the total expenditure and the dynamic flexibility is presented by the quantitative analyses on the HENs. In Case 3, the multiple disturbances are similar to that in Case 2, moreover, the retrofitted HENs are compared with that of different variations in the control actions so that the importance of the proposed framework is revealed. 76


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The scenario of multiple disturbances, the dynamic flexibility analysis and the dynamic simulations are implemented in Matlab. While the models for HEN retrofits are formulated using GAMS and solved on an Intel Core 3.60 GHz machine with 16 GB memory. GAMS/BARON42 is used to solve these superstructure-based MINLP models.

4.1 Case 1 620K H1 1.5 kW·K-1

330kW

75kW

HE2

CU

20kW 583K HE1 H2 1.0 kW·K-1 563K 17.33m2 128.98m2 393K

350K 2

15.43m 240kW 323K HE3 388K C1 2.0 kW·K-1 53.12m2 313K C2 3.0 kW·K-1

TAC=38578 $·y-1

K1

Figure 8. Optimal heat exchanger network design for Case 1 under the nominal condition.

The stream data is taken from literature4 with two hot streams and two cold streams included. For all the heat exchange matches, △ Tmin sets as 10 (K) and U is 0.08 (kW·m-2·K-1). The costs of heating and cooling utilities are 147.4 ($·kW-1·y-1) and 52.1 ($·kW-1·y-1), respectively. An initial HEN obtained under the nominal operating condition is depicted in Figure 8, which is symbolled by HEN0. For positive (control) degree of freedom43, a bypass is added in the initial HEN. Note that this bypass with no fraction is randomly selected to some extent, such that further discussion is not given on the degree of freedom. And more details about the analysis of the degree of freedom will be raised in Case 2. The stochastic variations of ± 20 (K) for three inlet temperatures (streams H1, H2 and C1) are assumed and a known probability distribution function, i.e. normal (Gaussian), is selected. The time-varying disturbances are

77



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given as follows: the step change is of + 10 (K) in inlet temperature of stream C2.

CU

HE1

H1

H2

HE3

HE2

HU

C1

C2 K1

(a) In the first generalized critical operating point H1

HE5

HE1 HE2 K1

H2

HE3

HE4

CU

HU1

C1

HU2

C2 K2


According to the proposed framework, the multiple disturbances are given. These are obtained by IN

IN

employing the Matlab built-in random and time functions. And then the GCOPs for Th1 , Th2

and

Tc1IN are identified as (614.77, 567.45, 399.21), (603.86, 568.28, 405.68). For the dynamic flexibility analysis of the initial network HEN0, the total computing time is 2.8 (s). The HEN retrofits are carried out in all the GCOPs and the results are shown in Figure 9. A colored unit denotes the new one and the variations of the heat exchanger area in the existing units are expressed by the double circles. For retrofitting a HEN, taking the one retrofitted in the first GCOP as an example, the number of the equations is 313 and that of the binary variables is 17. The total computing time for this HEN retrofit is 857 (s).

78


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HEN1 and HEN2 symbol the HENs retrofitted in the first and second GCOPs, respectively. The general results for the retrofitted HENs are shown in Table 1.

Table 1. General results for retrofitted heat exchanger networks for Case 1

Heat exchanger area

HEN1

HEN2

HE1

70.46

9.61

HE2

27.34

73.39

HE3

59.95

27.34

HE4

---

38.85

HE5

---

57.69

CU

24.84

8.66

HU

2.18

(m2)

Heater or cooler area (m2)

13.76, 10.75

Bypass fraction

0.540

0, 0.071

TAC ($·y-1)

41411

75719

Table 2. Control loop design and model predictive control settings in Case 1

HEN

Controlled

Manipulated

Set point

variable

variable

(K)

OUT 2

Controlled nominal

Weight Q

Weight R

value (K)

C 222

323

323.00

1

0.1

HEN 0

Th

K

HEN 1

Th2OUT

C K 222

323

333.06

0.067

0.671

HEN 2

Th2OUT

C K 221

323

340.55

100

0.1

In contrast to HEN0, the HENs to be retrofitted aim to accommodate the stochastic and the time-varying disturbances. The flexibility index is employed to measure the steady-state flexibility on the above HENs. Using the analysis method3, the flexibility index of HEN0 is 0.9. Under the deviations of the stream output temperatures, that of HEN1 is 1.2 and the increase in the areas of the heat exchangers/heating or cooling utility also reveals the satisfactory flexibility, as the same as HEN2. However, this result does not imply large overdesign of the retrofitted HENs. The above comparison is 79



Page 80 of 105

only employed to illustrate that the retrofitted HENs can completely accommodate the expected stochastic disturbances.

Figure 10. Dynamic responses of stream output temperatures in initial and retrofitted heat exchanger networks.

The dynamic behaviors of the HENs are analyzed to further demonstrate the capability of the resulted HENs accommodating the stochastic and the time-varying disturbances. For the dynamic model see Appendix B in Supporting Information for a brief derivation. As for each HEN, the bypass fraction, whose location is the cold side of the heat exchanger closing to the end of stream H2, is selected to take control of the output temperature in this stream. Bypass K1 is selected for both HEN0 and HEN1, and bypass K2 is selected for HEN2, resulting in three control loops. And then the HENs including these control loops are shown in Figure S1 in Supporting Information. The MPC settings for the control loops in the above HENs are listed in Table 2. Where scalars Q and R are output and input weightings of MPC controller, respectively. And more details can also be found in the work of Bahakim and Ricardez-Sandoval2. Smooth response and small deviations to the set points appear on the retrofitted HENs, as shown in Figure 10. It is possible to conclude that HEN2 achieves the desired dynamic behaviors, although the initial value of the stream output temperature produces large gap to its set point. Additional improvements can be also given 80


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using tuning parameters of the controllers. But the main idea here is only to illustrate that the profitable HEN with the capability of accommodating the stochastic and the time-varying disturbances is revealed.

4.2 Case 2

The stream data is given by literature44 and the nominal HEN design is taken from our previous work7. It involves two hot streams and two cold streams, as shown in Figure S2. △Tmin, U and the costs of heating and cooling utilities are as the same as that in Case 1. The range of variations for the stochastic disturbances are listed in Table S1 in Supporting Information and a known probability distribution function, i.e. normal (Gaussian), is selected. The time-varying disturbances are given as follows: the step change is of + 10 (K) in the inlet temperature of stream C1.

7.10m2 22.90m2 46.50m2 323K HE2 HE3 CU1

Th1IN, N  583K H1 Fcph1N  1.4kW  K 1

2

27.00m 723K H2 HE1 2.0 kW·K-1 6.00m2 393K HU 553K TAC=39193 $·y-1

6.50m2 CU2

553K

313K C1 3.0 kW·K-1 Tc2IN, N  388K C2 Fcpc2N  2.0kW  K 1

Figure 11. Steady-state flexible heat exchanger network7.

In the literature7, the nominal HEN design was developed by the stage-wise superstructure considering stream split, and then a steady-state flexible HEN without stream split was obtained7, as shown in Figure 11. As this work uses the non-split two-stage superstructure, the retrofitted HENs in this case are based on the steady-state flexible HEN from the literature7. In this way, the comparisons among the

81



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steady-state flexible HEN and the retrofitted HENs are employed to illustrate the proposed framework.

HE3

H1

H2

CU

HE2

HE1

C1

C2


H1

H2

HE1

CU

HE4

HE2

C1

C2 K1

(b) In the second generalized critical operating point K1 HE3

H1

H2

HE1

CU

HE2

C1

C2 K2

(c) In the third generalized critical operating point Figure 12. Heat exchanger networks retrofitted in the generalized critical operating points.

According to the proposed framework, the GCOPs for Th1IN , Tc2IN , Fcph1 and Fcpc2 are found as (587.93, 383.10, 1.04, 2.13), (585.16, 390.41, 1.08, 1.70), (583.99, 387.85, 1.71, 2.24). Then the HEN retrofit step is executed and the results are shown in Figure 12. To be clear, HEN0, HEN1, HEN2 and HEN3 symbol the steady-state flexible HEN, the HENs retrofitted in the first, second and third GCOPs, respectively. The corresponding HEN details are summarized in Table S2. As this paper achieves the HEN 82


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retrofits by the rolling horizon optimization, HEN3 is expected to exhibit both the satisfactory dynamic flexibility and the overall economic performance. Hence, the previous retrofitted HEN, i.e. HEN2, is not involved in the following analyses, while HEN1 achieved further optimization on HEN0 needs to be included to demonstrate the proposed framework. Here, HEN0, HEN1 and HEN3 are selected for the following analyses. In terms of the total annual cost, the proposed framework gives HEN3 with 12.17% lower total annual cost than HEN0 and it produces an acceptable gap with that of the HEN designed under the nominal operating condition. Only increasing heat exchanger area and inserting new units were considered for improving HEN flexibility in the literature7, leading to a higher total annual cost. Whereas this paper also retrofits the existing bypasses or adds the new ones, and considers the relaxed stream output temperature to investigate the reasonable trade-off between the total expenditure and the capability of accommodating the multiple disturbances. Hence, the corresponding total annual cost is lower.

Table 3. Comparisons for Case 2

Number of heat exchangers Degree of freedom

HEN0

HEN1

HEN3

3 -4

3 -4

3 -2

In terms of the (control) degree of freedom, the comparison is listed in Table 3. The (control) degree of freedom in the HENs is described by the difference between the numbers of MVs and CVs43. This also reveals the difference with the formal degree of freedom which is related to the numbers of the independent equations and unknown variables in a model. Meanwhile, positive degree of freedom has been proved at offering insights about feasibility of the disturbance rejection accordingly45, which is to say that the larger value implies additional space for accommodating the multiple disturbances. From Table 3 it is

83



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founded that, the degree of freedom in the retrofitted HENs is improved under the same number of heat exchangers. In terms of the effects of disturbances on the HENs, the results are shown in Figure S3, which are implemented by determining the disturbance intensity in the entire HEN using our previous work46. We developed a method for identification and quantification of the disturbance propagations in a given HEN. In such method, the frequency was employed to express the disturbance intensity in a heat exchanger as the disturbances propagate through each heat exchanger. Accordingly, the disturbance intensity in each stream was obtained. To facilitate analyses, in this paper, the disturbance intensity of each stream is normalized, as denoted by the triangles. All the normalized disturbance intensity of each stream and the disturbance intensity of the entire HEN are employed in the following analyses. The disturbance intensity in the HENs is shown in Figure S3, taking HEN3 as an example to exhibit identification and quantification of the disturbance propagations. Considering disturbance intensity of the entire HEN, the disturbances are more intense, so that larger area may be assigned to each heat exchanger. In this case, the disturbance intensity from small to big is that of HEN0, HEN3 and HEN1. As mentioned above, the increases in both the heat exchanger areas and the number of the heaters or coolers lead to total annual cost of HEN0 far outweighs that of the retrofitted HENs. This indicates that there may exist further optimization in HEN0. In this context, the results combining with the dynamic behaviors of the HENs reflected in Figure 13(a) and Figure 13(b) are summarized. HEN0 with the minimum disturbance intensity exhibits poor dynamic behavior. HEN1 with the maximum disturbance intensity cannot achieve the satisfactory dynamic behavior. More cost-effective HEN3 has achieved the preferable dynamic behavior, even with intense disturbances. Considering disturbance intensity of each stream, the more intense disturbances exist, the harder it is to achieve their 84


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desired set points, meanwhile, the more aggressive control actions will exist. It is typically evidenced by the situation in which stream C2 featuring the maximum disturbance intensity in HEN1 does not show smooth response, as shown in Figure 13(c). In contrast, under the maximum disturbance intensity in HEN3, stream C2 shows smooth response, as shown in Figure 13(d). Therefore, it is possible to conclude that HEN3 achieves the optimal trade-off between the satisfactory dynamic flexibility and the economic optimality. More details regarding the dynamic behaviors are represented below. On the other hand, achieving the dynamic simulation of the HENs with the controllers is employed to analyze their dynamic behaviors. In each HEN, the bypass, which is located in the heat exchanger closing to the end of stream H1, is selected to take control of the output temperature in this stream. And then the HENs including the control loops are shown in Figure S4. PI and MPC controllers are both involved for different features: PI controllers are often used in real process operation because of its simplicity and wide range of applicability; MPC controllers can maintain the control objective, ensure optimal control action moves, and also handle constraints in the MVs and CVs, so that it is desired despite some of the computational challenge2. In this context, a few scenarios with the controllers tested to demonstrate the proposed framework are presented as below: with the same controllers (MPC or PI controllers); comparing the results with additional stochastic disturbances and different controllers. The specifications for the control loops in the retrofitted HENs are listed in Table S3. The dynamic behaviors of HEN0, HEN1 and HEN3 are analyzed by testing control performance, as presented in Figure 13(a) and Figure 13(b). As indicated, the dynamic behaviors of two retrofitted HENs still can be guaranteed with PI controllers even facing the time-varying disturbances. These results demonstrate the HENs after retrofit become capable to accommodate the stochastic and the time-varying disturbances. Beyond this, additional stochastic disturbances, i.e. ± 20 (K) in the inlet temperatures of 85



Page 86 of 105

streams H1 and C2, are added for the further analysis. The effects of taking control of the stream output temperature (stream H1) on the uncontrolled stream output temperature (stream H2) are investigated to comprehensively examine the dynamic behaviors. The results are shown in Figure 13(c) and Figure 13(d). As for HEN1, the smooth response of the output temperature in stream H2 cannot be maintained when the output temperature of stream H1 is handled using PI or MPC controllers. Conversely, the output temperature of stream H2 in HEN3 shows more smooth response. The above analyses indicate that not only the satisfactory dynamic behavior in HEN3 results from MPC, but also this HEN has the capability of withstanding the multiple disturbances.

(a) Initial and retrofitted heat exchanger networks

(b) Initial and retrofitted heat exchanger networks

86


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(c) Retrofitted heat exchanger networks in the first generalized critical operating point (under the additional stochastic disturbances)

(d) Retrofitted heat exchanger networks in the third generalized critical operating point (under the additional stochastic disturbances) Figure 13. Dynamic responses of stream output temperatures in initial and retrofitted heat exchanger networks.

4.3 Case 3 The stream data is taken from literature39 and it involves two hot streams and two cold streams. △ Tmin, U and the costs of heating and cooling utilities are as the same as that in Case 2. An initial HEN obtained under the nominal operating condition is shown in Figure 14. Expected variations in all the inlet temperatures of ± 10 (K) are assumed as the stochastic disturbances and a known probability distribution function, i.e. normal (Gaussian), is selected. The time-varying disturbances are given as follows: the step change is of + 10 (K) in output temperature of stream C1. According to the proposed framework, the retrofitted HENs are obtained, as shown in Figure 15. HEN0, HEN1 and HEN2 are employed to symbol 87



Page 88 of 105

the initial HEN, the retrofitted HENs in the first and second GCOPs, respectively. The general results for the retrofitted HENs are shown in Table S4.

220kW 144kW 323K CU HE3

583K H1 1.4 kW·K-1 723K H2 2.0 kW·K-1

35.58m2 10kW HE1

330kW

553K

HE2

10kW 393K HU 1m2

1m2

18.22m2

388K C2 2.0 kW·K-1

553K TAC=27769 $·y-1

313K C1 3.0 kW·K-1

25.00m2

Figure 14. Optimal heat exchanger network design for Case 3 under nominal condition.

CU

HE1

H1

H2

HE2

HU

C1

C2


HE1

H1

H2

CU

HE2

HU

C1

C2


Considering the output temperatures of streams H1 and C1, the comparisons are given among the dynamic behaviors of HEN0 and the retrofitted HENs. In each HEN, two additional bypasses are selected for control purposes. The bypasses located in cold sides of the same heat exchanger are selected to take the 88


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control of the output temperatures of the streams C1 in the HENs. That is, bypass K1 in cold side of heat exchanger HE1 is added in HEN0. As for HEN1 and HEN2, bypasses K2 and K1 in cold sides of heat exchanger HE2 are selected, respectively. However, the possible bypasses in this heat exchanger are not directly related to stream H1. In each HEN, therefore, the bypass, which is located in the heat exchanger closing to the end of stream H1, is selected to take the control of the output temperature in this stream. That is, bypass K2 in hot side of heat exchanger HE3 is added in HEN0. As for HEN1 and HEN2, bypasses K1 and K2 located in hot sides of heat exchangers HE1 and HE3 are selected, respectively. Thus, the HENs including the control loops are shown in Figure S5. Towards the output temperatures in streams H1 and C1, the dynamic behaviors of the HENs are detected in Figure 16.


(b) Output temperature of stream C1 Figure 16. Dynamic responses of stream output temperatures in initial and retrofitted heat exchanger networks. 89



Page 90 of 105

From Figure 16(a), the output temperature of stream H1 in HEN0 cannot achieve its desired set point, even there does not exist deviation in the initial value of this temperature from the set point. It is possible to conclude that this HEN fails to accommodate the multiple disturbances, as the same as HEN1. The further evidence is provided in large temperature control range in stream H1. Hence, the preferable dynamic behavior is expected to obtain under the situation in which the further optimization appears in the temperature control ranges. For this purpose, HEN2 achieves better dynamic behavior than the other two HENs. On the other hand, as shown in Figure 16(b), there may exist the control loop interaction as the output temperatures of streams H1 and C1 are both considered, leading to the negative effects on the control performance, as well as the dynamic behavior. Therefore, the further optimizations on the control actions also have to be considered for achieving the most satisfactory dynamic behavior. However, the controller design is not included in this work, the predicted variations of the control actions are only assumed as the fixed values. In this context, the analyses for optimizing the control actions are given as follows.

CU

HE1

H1

H2

HE2

HE3

HU

C1

C2 K3

Figure 17. A new retrofitted heat exchanger network.

As shown in Eq. (16),

Δu m

is the vector of the variations of the control actions for the HEN

retrofitted in moment m. It is expected to enable this HEN to achieve the preferable dynamic behavior. For

90


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which, smaller value should be found in the variation, so that the improvement takes place with: such variations are smaller than 0. A new retrofitted HEN, which is symbolled by HEN3, is obtained based on HEN1, as shown in Figure 17. The general results for HEN3 are shown in Table S4. The total annual cost of HEN3 is only 3.11% higher than that of HEN2 and produce an acceptable gap with that of the HEN0. The comparison of the retrofitted HENs (HEN1 and HEN3) is given. As the same as the above HENs, two additional bypasses are added in HEN3 for taking the control of the output temperatures in streams H1 and C1. Bypass K1 in the hot side of heat exchanger HE1 is added, as well as bypass K2 in the cold side of heat exchanger HE2. The retrofitted HEN including the control loops is shown in Figure S6. And the related dynamic behaviors of the HENs towards the dynamic responses of the output temperatures in streams H1 and C1 are denoted in Figure 18. As indicated, HEN3 shows smooth responses, especially instability expressed in output temperature of stream C1 in HEN1 is improved. The above analyses reveal the importance of this framework for the trade-off between the total annual cost and the capability of accommodating the multiple disturbances.


91



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(b) Output temperature of stream C1 Figure 18. Dynamic responses of stream output temperatures in retrofitted heat exchanger networks.

5. CONCLUSIONS The design of dynamic flexible HENs is addressed in this paper. A new optimization-based framework is to yield an economically attractive HEN with the capability of accommodating the stochastic and the time-varying disturbances. This work begins with the multiple disturbances, on basis of which, the GCOPs are proposed for bottleneck identification of dynamic flexibility. The HEN retrofits are subsequently developed to perform on all the GCOPs along time horizon, such that ensure the resultant HEN qualified capability of accommodating the multiple disturbances. Relaxing the commonly fixed stream output temperatures into reasonable ranges is the key basis of this study. By such temperature control range, the hyperrectangle desired by this study is merely required to close to rather than well locating at the boundary of dynamic feasible region of the HEN. This provides additional space for the optimal trade-off between the total annual cost and the dynamic flexibility. In the three cases, the proposed framework is evidenced by the comparisons in terms of the dynamic behavior and the quantitative information. By dynamic simulations, superior dynamic behaviors with following their set points very closely are revealed. And the total annual cost of a retrofitted HEN in the second case is 12.17% lower than that of the steady-state flexible HEN, meanwhile featuring an acceptable gap with that 92


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of the HEN designed under the nominal operating condition. Besides, the disturbances allow the dynamic feasibility to be closely related to the dynamic behavior, so that the ranges of variations in stream output temperatures will be employed to deeply investigate the coupling of the dynamic flexible design and the controller design. Hence, in the future work, the proposed framework will provide valuable insights to the integration of these two aspects, for the simultaneous achievement of the economic optimality, the satisfactory dynamic behavior and the low control complexity.

SUPPORTING INFORMATION Supporting information regarding the step-by-step implementation of the proposed framework into the case studies is included. This information is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author * Tel.: +86 411 84986301; Fax: +86-411-84986201; E-mail: [email protected] Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS The authors would like to thank Natural Science Foundation of China (No. 21576036, No. 21776035) 93



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for providing the research fund for this project.

ABBREVIATIONS HENs = heat exchanger networks MINLP = mixed-integer nonlinear programming MVs = manipulated variables CVs = controlled variables GCOPs = generalized critical operating points COPs = critical operating points PI = proportional integral controller MPC = model predictive controller

NOMENCLATURE m ARijk

Area for new heat exchanger of HEN retrofitted in moment m

ARiCU,m

Area for new cooler of HEN retrofitted in moment m

AR HU,m j

Area for new heater of HEN retrofitted in moment m

max,m ARijk

Maximum area for heat exchanger in the previous and the retrofitted HEN

ARimax,CU,m

Maximum area for cooler in the previous and the retrofitted HEN

AR max,HU,m j

Maximum area for heater in the previous and the retrofitted HEN

CAPm

Capital cost of the HEN retrofitted in moment m

DF

Summation in the deviations of stream output temperatures from their set points

Ds

Disturbances happening to HEN 94


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Fcpc mj

Heat capacity flowrate of cold stream j of HEN retrofitted in moment m

Fcphim

Heat capacity flowrate of hot stream I of HEN retrofitted in moment m

Fcpc HE,m j

Heat capacity flowrate of potential bypassed cold stream j of HEN retrofitted in

moment m

FcphiHE,m

Heat capacity flowrate of potential bypassed hot stream I of HEN retrofitted in

moment m H,m K ijk

Bypass fraction on the hot side of heat exchanger of HEN retrofitted in moment m

C,m K ijk

Bypass fraction on the cold side of heat exchanger of HEN retrofitted in moment m

OPm

Operating cost of the HEN retrofitted in moment m

QiCU,m

Heat load between hot stream i, and cold utility of HEN retrofitted in moment m

Q HU,m j

Heat load between hot stream j, and hot utility of HEN retrofitted in moment m

QijkHE,m

Heat load between hot stream i, and cold stream j, at stage k of HEN retrofitted in

moment m

Qim

Heat duty of all the heat exchangers in hot stream i of the HEN retrofitted in moment m

Q mj

Heat duty of all the heat exchangers in cold stream j of the HEN retrofitted in moment m

t

Time

TACm

Total annual cost of the HEN retrofitted in moment m

Tc IN,m j

Inlet temperature of cold stream j of HEN retrofitted in moment m

ThiIN,m

Inlet temperature of hot stream i of HEN retrofitted in moment m

Tc OUT,m j

Output temperature of cold stream j of HEN retrofitted in moment m 95



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ThiOUT,m

Output temperature of hot stream i of HEN retrofitted in moment m

ThiOUT,f(m)

Output temperature (future state) of hot stream i of HEN retrofitted in moment m

Tc OUT,N j

Set point for output temperature of cold stream j

ThiOUT,N

Set point for output temperature of hot stream i

TcijkIN,HE,m



ThijkIN,HE,m



TcijkOUT,HE,m


k of HEN retrofitted in moment m

ThijkOUT,HE,m


k of HEN retrofitted in moment m mix,m Tcijk

Temperature of the mixing junction of cold side in a heat exchanger of HEN retrofitted

in moment m mix,m Thijk

Temperature of the mixing junction of hot side in a heat exchanger of HEN retrofitted

in moment m u

Control variable forward to the optimum solution of model (P)

v

Sampling period

x

State variable in model (P)

m zijk

Binary variable to existence of the match between hot stream i, and cold stream j, at stage k of HEN retrofitted in moment m

ziCU,m

Binary variable to existence of the match between hot stream i, and cold utility of HEN 96


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z HU,m j

Binary variable to existence of the match between cold stream j, and hot utility of HEN




Generalized critical operating point

 min

Lower bound for deviations of stream output temperatures from their set points

 c mj

Control range of output temperature of cold stream j of HEN retrofitted in moment m

 him

Control range of output temperature of hot stream i of HEN retrofitted in moment m

 max,m

Upper bound for control range of output temperature of cold stream j of HEN


 min,m

Lower bound for control range of output temperature of hot stream i of HEN retrofitted

in moment m

△Tmin

Minimum temperature approach

△t

Sampling interval

s

Parameter of the sth disturbance’s probability disturbance function



Limitation on constraints h

Indices i

Hot stream

j

Cold stream

k

Superstructure stage

m

Moment within time horizon

Sets/Vectors D

Set of discretized disturbances 97



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Dmds

Set of stochastic and time-varying disturbances

u

Vector of control variable forward to the optimum solution of model (P)

x

Vector of state variable in model (P)

δ

Set of generalized critical operating point

ξ

Vector of deviations of stream output temperatures from their set points

ξ f(m)

Vector of deviations of stream output temperatures (future state) from their set points

of the HEN retrofitted in moment m

m

Vector of configuration coefficient for the HEN retrofitted in moment m

 ex,m


m


Δu m

Vector of variation of control action for the HEN retrofitted in moment m

Δu f(m-1)

Vector of variation of control action (future state) for the HEN retrofitted in moment

m-1

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(46) Gu, S.; Liu, L.; Zhang, L.; Bai, Y.; Wang, S.; Du, J. Heat exchanger networks synthesis integrated with flexibility and controllability. Chin. J. Chem. Eng. 2018, doi: 10.1016/j.cjche.2018.07.017.

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TABLE OF CONTENTS

105


Optimization-Based Framework for Designing Dynamic Flexible Heat

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