A strategy for MINLP synthesis of flexible and operable processes

Computers and Chemical Engineering 28 (2004) 1105–1119

A strategy for MINLP synthesis of flexible and operable processes Zorka Novak Pintariˇc, Zdravko Kravanja∗ University of Maribor, Faculty of Chemistry and Chemical Engineering, Smetanova 17, SI-2000 Maribor, Slovenia

Abstract This paper presents a sequential two-stage strategy for the stochastic synthesis of chemical processes in which flexibility and static operability (the ability to adjust manipulated variables) are taken into account. In the first stage, the optimal flexible structure and optimal oversizing of the process units are determined in order to assure feasibility of design for a fixed degree of flexibility. In the second stage, the structural alternatives and additional manipulative variables are included in the mathematical model in order to introduce additional degrees of freedom for efficient control. The expected value of the objective function is approximated in both stages by a novel method, which relies on optimization at the central basic point (CBP). The latter is determined by a simple set-up procedure based on calculations of the objective function’s conditional expectations for uncertain parameters. The feasibility is assured by simultaneous consideration of critical vertices. The important feature of the proposed stochastic model is that its size depends mainly on the number of design variables and not on the number of uncertain parameters. The strategy is illustrated by two examples for heat exchanger network synthesis. © 2003 Elsevier Ltd. All rights reserved. Keywords: Synthesis; Process; MINLP; Operability; Flexibility; Controllability; Steady-state Model; Central basic point

1. Introduction During the last few decades, mathematical programming has proved to be an excellent tool for the design, synthesis and retrofit of chemical processes and numerous methods have been successfully implemented into the existing Computer Aided Process Engineering (CAPE) tools. In order to carry forward the solutions obtained by means of these methods into the plants that are actually implemented in practice, operability issues (like flexibility, controllability, reliability and safety) should be considered during the design, synthesis and retrofit of chemical processes. There have been many attempts recently to introduce operability aspects into those systematic methods from which flexibility has been most widely addressed. Interactions between process design and process control have been shown to have a great impact on the economic optimality of a process design. Procedures for the optimization and synthesis of chemical processes under uncertainty have, therefore, to assure the flexibility and operability of ∗ Corresponding author. Tel.: +386-2-22-94-481; fax: +386-2-252-7774. E-mail addresses: [email protected] (Z.N. Pintariˇc), [email protected] (Z. Kravanja).

0098-1354/$ – see front matter © 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2003.09.010

optimal solutions. Two main tasks which such procedures have to accomplish are firstly, to obtain optimal oversizing of process units to ensure feasibility of the solutions over the whole space of uncertainty and secondly, to ensure sufficient degrees of freedom for the operability of derived process structures. Both tasks are closely linked with the problem of accurately estimating the objective function’s expected value. Accurate estimation of the expected objective function is computationally expensive as integration over a multi-dimensional space of uncertainty is required. Different integration schemes have been presented in literature, e.g. the Gaussian quadrature formula and Monte Carlo simulation (Acevedo & Pistikopoulos, 1998). Recently, a specialized cubature technique (Bernardo, Pistikopoulos & Saraiva, 1999) was developed, suitable for integrating normally distributed parameters. Novak Pintariˇc and Kravanja (1999) proposed a method for the approximation of expected objective function based on a linear combination of the objective function’s values at the extreme points. Schmidt and Grossmann (2000) presented multiple integration techniques for uncertain parameters described by continuous distribution. There is also a hybrid parametric stochastic technique (Hené, Dua & Pistikopoulos, 2002) that avoids NLP optimization at the integration points.

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Nomenclature a A Aov Areq b B c ct cz C CBP CF CV d EC fB g gd h m M N NC ND NG NP t
T
ln T U v x xbp xc y yt yt∗ yz z za

lower bound of uncertain parameter heat exchanger area (m2 ) oversized heat exchanger area (m2 ) required heat exchanger area (m2 ) upper bound of uncertain parameter beta function coefficients of the zeros of Legendre polynomial fixed cost of topological alternatives in the first-stage superstructure fixed cost of additional structural alternatives in the second-stage superstructure economic objective function central basic point heat capacity flow rate (kW/K) critical vertices vector of design variables (sizes of process units) expected value of the economic objective function beta density function vector of inequality constraints vector of design constraints vector of equality constraints mode value large positive constant number of discrete points for numerical integration number of critical vertices number of design variables number of zeros of Legendre polynomial (Gaussian quadrature points) number of uncertain parameters zeros of the Legendre polynomial in the interval (−1,1) temperature difference (K) logarithmic mean temperature difference (K) overall heat transfer coefficient (kW/m2 K) weights of Gaussian quadrature points vector of dependent (state) variables fraction of bypassed stream Gaussian quadrature points in the interval (0,1) vector of binary variables vector of binary variables for topological alternatives in the first stage optimal topology of the first stage vector of binary variables for additional structural alternatives in the second stage vector of independent (control) variables vector of additional independent (control) variables

Indices i index of uncertain parameters k index of discrete points Greek letters α positive parameter of beta density function β positive parameter of beta density function Φ heat flow rate (kW) µ mean value θ vector of uncertain parameters θB central basic point θc critical vertex nominal value of uncertain parameter θ nom

Some authors have also addressed the problem of operability, e.g. Kotjabasakis and Linnhoff (1986) introduced sensitivity tables for the design of flexible processes. Mathisen (1994) comprehensively investigated the design and control of heat exchanger networks. Several authors have developed systematic methods for the synthesis of process schemes, which are flexible to operate under uncertain conditions and operable for disturbances (e.g. Papalexandri & Pistikopoulos, 1994; Glemmestad, Skogestad & Gundersen, 1997; Mizsey, Hau, Benko, Kalmar & Fonyo, 1998; Tantimuratha, Asteris, Antonopoulos & Kokossis, 2001). In this paper, we present a sequential two-stage approach which can handle flexibility and some aspects of operability by the synthesis of chemical processes with mixed-integer nonlinear programming (MINLP). The term ‘operability’ in this work denotes steady-state resilience (Morari, 1983), i.e. the ability of a plant to handle static disturbances (e.g. changes in feedstocks, operating conditions, economic parameters, etc.). Dynamic (transient) issues, like control of dynamic disturbances, tuning of the controllers and quality of control, are not taken into account. In the first stage of the proposed approach the expected value of the objective function is optimized in order to obtain optimal process topology and optimal oversizing of process units to assure the desired flexibility. In the second stage structural deficiencies of the selected structure are corrected with the optimal selection of additional structural alternatives and manipulated variables, in order to assure efficient control. This procedure does not enable automated identification of additional manipulated variables and structural alternatives for a particular problem. This remains a creative task for an engineer based on his or her knowledge, experiences and intuition. It is the purpose of this work to study some static issues of operability and their interactions with the design of flexible processes. This paper is organized as follows. Section 2 briefly introduces a novel approximate method for stochastic optimization under uncertainty which was applied efficiently during this work for estimation of the expected value. Section 3 presents a two-stage strategy for simultaneous consideration

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of flexibility and steady-state operability by the optimization and/or synthesis of chemical processes. This strategy is illustrated by the small example in Section 4, while a larger example of flexible and operable HEN synthesis is given in Section 5. Finally, conclusions on this work are presented.

2. Approximate stochastic optimization and/or synthesis for flexibility The stochastic optimization and synthesis of chemical processes denotes the use of specific methods and strategies in those cases where the values of some parameters are not known exactly but are given by probabilistic models instead. These models describe uncertainty in the terms of different probabilistic functions. The solutions of such problems are obtained through optimization of the expected value rather than the nominal value of the objective function. The expected value is usually obtained by discretization of the uncertain space and numerical integration over this space: min EC =

xk ,zk ,d

N


C(xk , zk , d, θk )vk

k=1

s.t. hk (xk , zk , d, θk ) = 0 gk (xk , zk , d, θk ) ≤ 0 d ≥ gd,k (xk , zk , θk ) xk ∈ X, zk ∈ Z, θk ∈ TH d∈D

        

(P1) k = 1, 2, . . . , N

EC represents the expected value of the economic objective function, C. Vectors of independent (manipulated) and dependent (state) variables (e.g. flow rate, temperature, pressure) are denoted as z and x, respectively, and d represents a vector of design variables which determine the sizes of process units (e.g. area, diameter, height, power, etc.). Vector of uncertain parameters, θ, comprises different parameters that can change the values during the operation (e.g. operating parameters and economic data) and the parameters whose exact values are unknown (e.g. internal process parameters, like heat transfer coefficients, catalyst efficiency). Vectors of equality and inequality constraints are denoted by h and g, respectively, while gd represents correlations for the estimation of design variables. The expected value is determined over N discrete values of uncertain parameters which correspond to the zeros of the Legendre polynomial if applying the Gaussian integration method or, if applying Monte Carlo simulation, the points are selected randomly according to the density functions. The values obtained are then used for the estimation of the expected value by using weights, v, obtained from the density functions. Note, that manipulated variables, z, and state variables, x, as well as the constraints in the model (P1) are defined for all N discrete points, while design variables, d, are not. The reason is that manipulated variables can be ad-

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justed during the operation in order to control the state variables when uncertain parameters change their values, while design variables have to obtain single values for the whole space of uncertainty. Inequality in design expressions assures a suitable selection of design variables for every realization of uncertain parameters. In this way, the simultaneous one-stage procedure can be applied for solving the model (P1) instead of the more common two-stage procedures. Both integration methods mentioned above require evaluation of the objective function at the comprehensive set of points to achieve sufficient confidence limit. To overcome this problem the approximate methods should be developed for easier and more efficient estimation of the expected value. In addition, distribution functions should be used that can be obtained easily, e.g. from historic data or practical experiences on similar processes. 2.1. Approximate method for stochastic optimization with a central basic point The approximate method for reduced dimensional stochastic (RDS) optimization that we developed earlier (Novak Pintariˇc & Kravanja, 1999) approximates the expected value by a linear combination of the objective values in a reduced set of extreme points (basic vertices), determined in the preceding set-up procedure. The drawback of the RDS method is that the selection of basic vertices is not unique and many suboptimal solutions can be generated. The aim of this work is to improve and to simplify the procedure so that a more accurate approximation of the expected value can be obtained more easily. To accomplish this task a novel method for stochastic optimization and synthesis has been developed called the central basic point (CBP) method. We will present the CBP method briefly in the following subsections, while more details are given elsewhere (Novak Pintariˇc & Kravanja, 2003). The main idea of the CBP method is to determine just one central basic point in which the value of the objective function would be close to the expected value of the objective function. The identification of this point is performed through the following steps. 2.1.1. Critical vertices In this work, we have defined the critical vertices as those extreme realizations of uncertain parameters that require the largest overdesign of process units. A set of critical vertices can be identified through the sequential optimization of an original problem at the extreme points. This assumption is strictly valid only for the convex problems, nevertheless, it may also be valid in some cases with nonconvex constraints (Grossmann, Halemane & Swaney, 1983). If the number of vertices is small and the size of the model is limited, the procedure can also be performed simultaneously. After optimization the largest values of design variables obtained are identified and corresponding vertices compose the set of critical vertices, CV = {θkc |k = 1, 2, . . . , NC }. In this

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way, the maximal number of critical vertices, NC , is equal to the number of design variables, ND , which often represents significant reduction in a problem’s dimensionality. 2.1.2. Conditional expected values A multi-dimensional evaluation of the expected value can be extremely exhaustive, on the other hand, one-dimensional integration is much easier. Conditional expectations of the objective function for uncertain parameters, EC(θ i ), i = 1, 2, . . . , NP , are estimated over a significantly reduced one-dimensional region and provide information about the influence of each uncertain parameter on the objective function. When applying the Gaussian method for integration over the interval of one uncertain parameter, optimization of the problem is performed simultaneously at NG Gaussian quadrature points of this particular uncertain parameter, while others are fixed at the nominal values (θjnom ). To assure feasibility the optimization is also performed simultaneously at NC critical vertices. min

xi,k ,zi,k ,di

EC(θi ) =

NG
k=1

nom )v C(xi,k , zi,k , di , θi,k , θj|j =i i,k

s.t. nom ) = 0  hk (xi,k , zi,k , di , θi,k , θj|j  =i   g (x , z , d , θ , θ nom ) ≤ 0  k

i,k

i,k

i

i,k

j|j=i nom )  k = 1, . . . , NG di ≥ gd,k (xi,k , zi,k , θi,k , θj|j =i    (SU1)i xk ∈ X, zk ∈ Z, θk ∈ TH j = 1, 2, . . . , NP  hk (xi,k , zi,k , di , θkc ) = 0    gk (xi,k , zi,k , di , θkc ) ≤ 0  k = 1, . . . , NC di ≥ gd,k (xi,k , zi,k , θkc )     xk ∈ X, zk ∈ Z, θkc ∈ CV d∈D

In each model (SU1)i , i = 1, 2, . . . , NP , the first group of constraints is written at Gaussian quadrature points (k = 1, 2, . . . , NG ) of the particular parameter, θ i,k , while other nom ). uncertain parameters are held at the nominal values (θj|j =i

The values obtained are then used for estimation of the conditional expected objective function, EC(θ i ), by using coefficients, v, obtained from the density functions. In order to consider feasibility constraints simultaneously, the second group of constraints is written at the critical vertices, θkc , k = 1, 2, . . . , NC , determined in Section 2.1.1. In order to obtain the conditional expected values for all uncertain parameters, the problems (SU1)i , i = 1, 2, . . . , NP , have to be solved sequentially NP -times, i.e. independently for each uncertain parameter. Hence it follows, that the sizes of the models (SU1)i are independent of the number of uncertain parameters. Since five Gaussian quadrature points are sufficient, in most cases, for efficient one-dimensional integration, the original mathematical model is multiplied by (NC + 5)-times in each model (SU1)i . The problem’s size may eventually become too large if the model comprises a large number of design variables and thus, a large number of the critical vertices. 2.1.3. Central basic point Based on the optimal values of the objective function obtained by solving each model (SU1)i simultaneously at the NG points (usually 5), the approximate functions C (θ i ) are developed for each uncertain parameter by a simple curve fitting (Fig. 1). From these functions it is then possible, using a simple back calculation, to predict the value of uncertain parameter, θiB , which would then produce a value of the objective function equal to the corresponding conditional expected value C (θiB ) = EC(θi )

i = 1, 2, . . . , Np

Fitting procedure and back calculation of θiB with Eq. (1) are performed sequentially for all uncertain parameters. The calculated value of θiB represents the component of a vector θ B which serves as a central basic point (CBP). In this way the CBP is determined, at which the expected value will be approximated over the whole region of uncertain parameters as C(θ B ).

C( θi ) Values obtained by optimizaton of (SU1) i

Conditional expected value obtained by optimization of (SU1)i

Approx. function C’ ( θi )

EC( θi )

θi,1

θi,2

(1)

θiB θi,3

θi,4

θi,5

Fig. 1. Graphical presentation of CBP determination.

θi

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2.1.4. Lower bounds on design variables Since in this work we deal with one single basic point instead of many quadrature points, the economic trade-off over the entire uncertain space is not considered and the optimization at CBP would produce optimistic results. This can be overcome if the trade-off is rebalanced by enforcing design variables to the values obtained when calculating the conditional expectations (as described in Section 2.1.2), where a valid one-dimensional trade-off is obtained for each uncertain parameter. The most obvious way to enforce design variables to these values is to limit their space by lower bounds, dLO , that are equal to the maximal values obtained by calculation of the conditional expectations. It is assumed that ‘conditional overdesign’ obtained in this way is a fair approximation of the exact overdesign that would be obtained by simultaneous optimization at quadrature points over the whole region of uncertainty. Approximate stochastic optimization is finally performed at the CBP and critical vertices with lower bounds on design variables, as will be shown in Section 3. In this way the expected value is determined at one central point rather than at many extreme points, as in more rigorous methods. 2.2. The beta distribution and Gaussian numerical integration In general, every distribution function could be used by applying the CBP method, however, in this paper beta distribution is applied. It is assumed that a feasible solution can be obtained for every realization of uncertain parameters in the prespecified region of uncertainty. Uncertain parameters are often described by means of normal and uniform distributions which are both symmetric. Skewed distributions are more realistic from the practical point of view. Beta distribution is one of the suitable nonsymmetrical density functions which is defined over a finite interval and may have its modal value (mode) anywhere in the interval. The approximate density function can be obtained rather easily by estimation of only three values: lower bound, a, upper bound, b, and the value with the greatest probability of occurring—mode, m. Mean value, µ, can then be approximated by the following formula (Nahmias, 1997) µ=

a + 4m + b 6

(2)

Analytic expressions for the mean and mode values of beta distribution in the interval (0,1) are given by Eqs. (3) and (4) α µ= (3) α+β m=

α−1 α+β−2

(4)

where α and β are positive parameters which determine the density function of beta distribution. The system of two equations is obtained and solved for unknowns α and β

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by substituting Eq. (2) into Eq. (3). Based on these two values a density function of the beta distribution can be evaluated:  α−1  x (1 − x)β−1 0≤x≤1 (5) fB (x) = B(α, β)  0 otherwise where B is the beta function defined with the following integral: 1 uα−1 (1 − u)β−1 du α > 0, β > 0 B(α, β) = (6) 0

Values of density functions can be easily obtained for various distributions by means of commercial mathematical programs, e.g. MathCAD. Gaussian integration is performed at the points which are the roots of the Legendre polynomial. Zeros of the kth degree polynomial, tk , are located in the interval (−1,1) and can be found in mathematical handbooks together with corresponding coefficients, ck . In order to apply Eq. (5), the points tk are converted into the Gaussian quadrature points within the interval (0,1): tk + 1 xck = (7) , k = 1, 2, . . . , NG 2 Finally, weights, v, for Gaussian quadrature points are calculated by Eq. (8). vk = 21 ck fB (xck )

k = 1, 2, . . . , NG

(8)

These weights are then used for the approximation of the expected value in the objective functions of problems (P1) and (SU1)i . 3. Two-stage strategy for flexibility and operability Two important facts should be considered when performing the optimization and synthesis of flexible and controllable process schemes: firstly, the process units should be properly (optimally) oversized to assure the desired flexibility and secondly, the process structure should have sufficient degrees of freedom for efficient control. There are two ways to handle the aspects of flexibility and operability within the superstructural approach. The first way is by simultaneous one-stage procedure and the other is by sequential two-stage procedure. One-stage procedure handles flexibility and operability issues simultaneously, therefore, the original superstructure has to comprise sufficient manipulated variables and structural alternatives for desired flexibility and efficient control. The resulting mathematical model usually comprises a huge number of alternatives and its solution can become a large combinatorial problem. In principle, one-stage procedure can be applied only for small-scale problems. The proposed strategy is, therefore, divided in two stages. In the first stage, conventional superstructural approach is applied for optimal selection of process units

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and determination of design variables to assure flexibility for the whole region of uncertainty. In addition, the optimal values of operating variables are determined for each particular realization of uncertain parameters. The solutions of this stage are flexible but may have structural deficiencies and, thus, insufficient degrees of freedom for efficient control. In the second stage, the optimal structure of the first stage is extended with additional structural alternatives and manipulated variables in order to ensure sufficient degrees of freedom. Since the topological alternatives are already chosen in the first stage, the superstructure of the second stage is considerably smaller. The optimal selection between additional structural alternatives is performed in the second stage together with optimization of corresponding manipulated variables. Mathematical models of both stages are given in the following subsections. 3.1. First stage—flexibility In the first stage, MINLP model (P2) is developed where a vector of discrete variables, yt , represents discrete decisions between topology alternatives in the original superstructure. Model (P2) is solved simultaneously in a union of the critical vertices and the central basic point with lower bounds on design variables determined as described in Section 2.1. The expected objective function is approximated with objective function, C, calculated at the single central basic point. Fixed costs associated with process topology, ct , can also be included. min

yt ,xk ,zk ,d

ctT yt + C(xk , zk , d, θk )k∈CBP

s.t.  hk (yt , xk , zk , d, θk ) = 0  

gk (yt , xk , zk , d, θk ) ≤ 0 k ∈ CV ∪ CBP   d ≥ gd,k (xk , zk , θk )

(P2)

d ≥ d LO xk ∈ X, zk ∈ Z, θk ∈ TH; mt

d ∈ D, yt ∈ {0, 1}

k ∈ CV ∪ CBP

, CBP ∈ {θ B }

The constraints h, g and gd in model (P2) are multiplied over the union of critical vertices and CBP. The maximal number of critical vertices, NC , is equal to the number of design variables, ND , and the union CV ∪ CBP is comprised of the most NC + 1 points. This indicates that the number of uncertain parameters has no influence on the size of the model (P2). Optimization of problem (P2) yields a solution with optimal process structure and optimally oversized design variables, which enables the desired flexibility for the process schemes. 3.2. Second stage—operability In order to determine optimal oversizing of process units which is feasible for the whole range of uncertain param-

=150 kW 100 C

50 C

Tfix = 60 C

30 C

CF (kW/K) 3

=150 kW 100 C

50 C

68.75 C

5

Tfix = 60 C

30 C

2

A ov =8.5 m

(a)

30 C

x bp=22.5 %

(b)

Fig. 2. Oversized process heat exchanger unit without (a) and with (b) bypass.

eters, design correlations in the model (P2) are written as inequalities, d ≥ gd,k (xk , zk , θk ). The optimal structures obtained during the first stage, although potentially flexible, may not have enough manipulated variables because the sizes of the process units (design variables) cannot usually be manipulated during the operation. This may cause some output variables to be unable to meet the specified values (e.g. purity of products, outlet temperatures, etc.) at certain realizations of uncertain parameters. In order to make the design operable, additional degrees of freedom in the form of additional structural alternatives and manipulated variables, za , have to be introduced into the optimal structure of the first stage. In this way the design (x , z , za , θ ) for constraints can become active, d = gd,k k k k k all realizations of uncertain parameters from the prespecified intervals. 3.2.1. Motivating problem Before we present the general model for the second stage, consider the heat exchanger unit in Fig. 2a. Outlet temperatures of hot and cold streams should be controlled to their target values of 50 and 60 ◦ C, respectively, while inlet temperatures and heat capacity flow rates, CF, may vary. Assume that the oversized exchanger area is determined, Aov = 8.5 m2 and the overall heat transfer coefficient is given, U = 0.7 kW/(m2 K). For particular realization of inlet temperatures and heat capacity flow rates, as given in Fig. 2a, the calculated logarithmic mean temperature difference is,
ln T = 28.9 K and the required exchanger area is obtained Φ 150 Areq = (9) = = 7.4 m2 U
ln T 0.7 × 28.9 The calculated exchanger area is obviously smaller than the actual (oversized) value: Φ Aov = 8.5 > (10) = Areq U
ln T Since actual oversized area, Aov , cannot be manipulated during the operation and Φ and U are given, the only way to make the constraint (10) active, is to modify
ln T: Φ Aov = 8.5 = (11) U
ln T which gives Φ 150
ln T = = = 25.21 K (12) Aov U 8.5 × 0.7

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Supposing that the outlet temperature of the hot stream remains unchanged, the outlet temperature of the cold stream should increase above 60 ◦ C, accurately to 68.75 ◦ C. Because of the larger increase in temperature, the heat capacity flow rate of the cold stream through the exchanger decreases from 5 kW/K to F 150 CF = = = 3.87 kW/K (13)
T 68.75 − 30 This result indicates that part of the cold stream should bypass the exchanger, exactly CFbp = (5 − 3.87)kW/K = 1.13 kW/K or 22.5%. This can be achieved by providing an additional structural alternative (a controlled bypass) on the cold stream as shown in Fig. 2b and manipulate the fraction of a bypassed stream. The latter enables compensation for the difference (8.5 − 7.4)m2 = 1.1 m2 in design variable and to attain the desired outlet temperatures. 3.2.2. General mathematical model In order to derive a general mathematical model, optimal flexible topology from the first stage, yt∗ , is extended with additional structural alternatives and manipulated variables, za . In the reformulated model (P3) a new vector of binary variables, yz , is assigned to additional structural alternatives. Modified design constraints are written as equalities, gd , in which one or more additional manipulated variables, za , are assigned to each design variable. Additional manipulated variables, za , are used, as necessary, to make design constraints active for all realizations of uncertain parameters. For those realizations of uncertain parameters at which the required value of design variable meets exactly the actual oversized value determined in the first stage, an additional manipulated variable is not required (za = 0), while for realizations where oversized design variable exceeds the required value, the manipulated variable obtains a positive value (za > 0) which compensates for the difference between the oversized and required values. In this way additional degrees of freedom are introduced. min a ctT yt∗ + C(xk , zk , zak , d, θk )k∈CBP + czT yz

of an additional structural alternative and manipulated variable. If an alternative is required at the particular point, its binary variable, yz , is set to 1 and the logical relationship allows a nonzero value for the manipulated variable. Otherwise, if an additional manipulated variable is not needed, its binary variable will be set to zero. Solving the MINLP model (P3) leads to the optimal selection of additional manipulated variables and structural alternatives and, thus, to the optimal flexible and operable structures. The model (P3) as written above can be applied to various problems in the process industry, however, the identification of additional manipulated variables and corresponding structural alternatives still remains a creative task for engineers, and depends on the particular problem.

4. Illustrating example The methodology is outlined on the small example of HEN optimization (Fig. 3). The uncertain parameters are the inlet temperatures of cold stream C1 and hot stream H2 and the heat capacity flow rate of cold stream C2. The lower, modal and upper values [ai , mi , bi ] for three uncertain parameters (i = 1, 2, 3) are given as follows: TIN,C1 [378, 391.3, 398] K, TIN,H2 [573, 586.3, 593] K, CFC2 [2.9, 3.03, 3.1] kW/K. The mathematical model comprises heat balances, the restrictions on temperatures and the expressions for heat transfer area. The objective is to minimize the sum of annual investment and operating cost. A more detailed model is given elsewhere (Novak Pintariˇc & Kravanja, 1999). A fixed structure as shown in Fig. 3, was used for this example to shorten the first synthesis step. 4.1. First stage—flexibility In the first stage, the proposed set-up procedure is performed for evaluation of the central basic point. The expected value of the objective function is then approximated by performing optimization of the model (P2) at this point.

yz ,xk ,zk ,zk ,d

s.t.

 hk (yt∗ , xk , zk , d, θk ) = 0    g (y∗ , x , z , d, θ ) ≤ 0  k

d=

k (x , z , za , θ ) gd,k k k k k

t

k

zak ≤ Myz

k

   

H1 H2

k ∈ CV ∪ CBP

d ≥ d LO xk ∈ X, zk ∈ Z, zak ∈ Z, θk d ∈ D, yz ∈ {0, 1}mz , CBP

∈ TH;

CW

620 K 586.3 K

1 2

4

k ∈ CV ∪ CBP

∈ {θ B }

If the introduction of an additional manipulated variable requires the physical structural alternative, fixed cost, cz , is included in the objective function. M is a large positive constant in a logic relationship for the potential (non)existence

CF / (kW/K) 350 K

1.5

T 323 K

3

600 K

(P3)

1111

391.3 K

5 393 K Steam

313 K

1.0

C1

2.0

C2

3.03

Cost of exchangers and coolers ($/yr):1846⋅A

0.65

Cost of heaters ($/yr): 2350⋅A Cost of CW: 20 $/(kW⋅yr) Cost of Steam: 230 $/(kW⋅yr) U1=U2= U3 =U4= 0.7 kW/(m 2 ⋅K) U5= 1 kW/( m2⋅K)

0.65

Fig. 3. Heat exchanger network of illustrating example.

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Table 1 Zeros and weights of Legendre polynomial of the fifth degree

represented with the following equations:

k

t

c

xc

fB (xc)

v

1 2 3 4 5

−0.90618 −0.53847 0 0.53847 0.90618

0.23693 0.47863 0.56889 0.47863 0.23693

0.04691 0.23076 0.5 0.76923 0.953089

0.02517 0.49157 1.5 1.63859 0.51135

0.002981 0.117638 0.426667 0.392137 0.060576

EC(TIN,C1 ) =

5
k=1

EC(TIN,H2 ) =

5


5
k=1

4.1.1. Set-up procedure

4.1.1.2. Conditional expected values. Conditional expected values are determined for each particular uncertain parameter by means of Gaussian integration at five quadrature points while the remaining two parameters are fixed at the nominal (mode) values. The parameters of the beta functions are assumed to be α = 3 and β = 2 for all three uncertain parameters. Zeros of the Legendre polynomial of the fifth degree, t, and corresponding coefficients, c, are given in Table 1. Quadrature points in the interval (0,1), xc, and weights, v, are calculated as described in Section 2.2. while probability densities, fB (xc), are obtained by MathCAD. Five quadrature points for each uncertain parameter are calculated by Eq. (14) and are shown in Table 2. θi,k = ai + xck (bi − ai ) k = 1, 2, . . . , 5 i = 1, 2, 3

(14)

The problem (SU1) is formulated and solved simultneously at five quadrature points for each of the three uncertain parameters while the other two parameters are held at the nominal (modal) values. Critical vertices are considered simultaneously to assure feasibility of design. The objective functions of models (SU1)i , i = 1, 2, 3, are

(15)

k C(TIN,H2|T )vk IN,C1 =391.3,CFC2 =3.03

(16)

C(CFkC2|TIN,C1 =391.3,TIN,H2 =586.3 )vk

(17)

k=1

EC(CFC2 ) =

4.1.1.1. Critical vertices. The original problem comprises eight combinations of the extreme values of uncertain parameters. Mathematical model was solved simultaneously at these vertices. The largest values of the design variables were observed at the two vertices which compose the set of critical vertices: CV = {(378 K, 573 K, 3.1 kW/K), (398 K, 593 K, 2.9 kW/K)}.

k C(TIN,C1|T )vk IN,H2 =586.3,CFC2 =3.03

The values of the objective function obtained at the quadrature points are shown in Table 2 (columns 3, 5 and 7). The last row in Table 2 presents the conditional expected values obtained by Eqs. (15)–(17). 4.1.1.3. Central basic point. Based on the values of the objective function at five quadrature points (Table 2), regression curves were constructed for each uncertain parameter and the abscissae of the conditional expected values were calculated. This procedure is graphically presented in Fig. 4. The values obtained compose the vector of the central basic point, θ B = (TIN,C1 = 388.87 K, TIN,H2 = 584.63 K, CFC2 = 3.045 kW/K). Note, that two solutions for CFC2 exist (2.970 and 3.045) kW/K, however, the higher value is chosen since the density of distribution function is concentrated in its neighborhood. 4.1.2. Stochastic optimization at CBP Finally, the model (P2) is solved as an NLP problem at the central basic point subject to the feasibility constraints defined at the two critical vertices. Lower bounds on design variables are set to the maximal values obtained at the calculations of the conditional expectations in Section 4.1.1., ALO = (15.736, 2.511, 6.300, 2.290, 2.478) m2 . The GAMS/CONOPT optimization solver was used for NLP optimization. Minimal cost obtained is 45 825 US$ per year. Two other techniques were also applied in order to estimate the accuracy of the CBP method. Firstly, rigorous three-dimensional Gaussian integration was performed simultaneously at 53 = 125 quadrature points in order to

Table 2 Quadrature points and corresponding values of the objective function k

1 2 3 4 5 EC

i =1

i =2

i =3

TIN,C1 (K)

C (TIN,C1 ) (US$ per year)

TIN,H2 (K)

C (TIN,H2 ) (US$ per year)

CFC2 (kW/K)

C (CFC2 ) (US$ per year)

378.938 382.615 388.000 393.385 397.062

45952.4 45577.4 45028.3 44716.0 44863.1

573.938 577.615 583.000 588.384 592.062

47613.7 46768.0 45529.5 44467.2 44293.2

2.909 2.946 3.000 3.054 3.091

47690.8 46374.5 44561.8 45552.3 46229.0

44963.2 TIN,H2 = 586.3 K, CFC2 = 3.03 kW/K;

45190.0 TIN,C1 = 391.3 K, CFC2 = 3.03 kW/K;

45273.8 TIN,C1 = 391.3 K, TIN,H2 = 586.3 K.

Z.N. Pintariˇc, Z. Kravanja / Computers and Chemical Engineering 28 (2004) 1105–1119 C ($/yr)

1113

C ($/yr) 48000

46000 47000 45500 46000 EC=45190 $/yr

45000 EC=44963 $/yr

45000

B TIN1,C1 = 388.87 K

44500 380

B

390

385

TIN1,H2= 584.63 K

44000 395

575

585

580

590

(K)

(K)

(a)

(b) C ($/yr) 48000 47000

46000 EC= 45273 $/yr

45000 B CF C2 = 3.045

44000 2.90

2.95

3.00

3.05

3.10 (kW/K)

(c) Fig. 4. Calculation of the central basic point.

obtain the most accurate stochastic result. Secondly, the model was solved simultaneously at the nominal (mode) point and the critical vertices. The results of all three techniques are presented in Table 3. It was assumed that the rigorous Gaussian method gives a correct solution and the comparison of the results is based on deviation of the approximative results from the Gaussian one. Relative deviation between the Gaussian method and CBP method is only +0.27% and is much smaller than the relative deviation between nominal (mode) point and Gaussian method (−2.08%).

4.2. Second stage—operability The main goal of the second stage is to ensure the operability of the first stage’s optimal solution. Since the model of the first stage does not contain enough manipulated variables, additional degrees of freedom have to be introduced in the HEN structure. This is achieved by new structural elements from which manipulated variables are introduced to the problem. In the HEN example three bypasses are added to the cold streams of the process exchangers (1, 2 and 3) as structural alternatives, while the fractions of the bypassed streams are corresponding manipulated variables (Fig. 5). The bypass can be placed either across the hot or

Table 3 Stochastic optimization for flexibility (first stage)

EC (US$ per year) A1 (m2 ) A2 (m2 ) A3 (m2 ) A4 (m2 ) A5 (m2 ) CPUa (s) a

Gaussian integration

Mode + critical vertices

CBP + critical vertices

45700 14.965 2.609 6.543 2.147 2.456

44750 15.793 1.953 6.072 2.313 2.459

45825 16.417 2.511 7.051 2.290 2.478

13

0.25

Pentium III/866 MHz (Windows XP Professional).

0.93

H1

CW

620 K

1

4 CW

H2

586.3 K

2

3

6

CF/ (kW/K) 350 K T 323 K

1.5 1.0

Steam

600 K 393 K

5

391.3 K

7

313 K

C1

2.0

C2

3.03

Steam

Fig. 5. Superstructure with additional structural alternatives for operability.

1114

Z.N. Pintariˇc, Z. Kravanja / Computers and Chemical Engineering 28 (2004) 1105–1119

cold streams because a steady-state model with static disturbances is applied. Besides bypasses, an additional cooler is placed on the hot stream H2 and an additional heater on the cold stream C2. It is assumed that utility exchangers are controlled by adjusting the utility flows. In this way a trade-off between bypass placement and utilities consumption is introduced into the optimization and a new superstructure is generated. The mathematical model (P3) of the extended superstructure is given in the Appendix A and comprises five binary variables; three for the selection of bypasses and two for additional heater and cooler. Bypasses are charged by fixed cost (1300 US$ per year) in the objective function and the additional heater and cooler by corresponding cost correlations. Three new vectors of continuous variables are introduced in the model (DT1 , DT2 , DT3 ) in order to allow an increase in cold stream temperatures in the process exchangers above the marked stream temperatures (T4 + DT1 , T8 + DT2 and T12 + DT3 ). Such an increase in temperature compensates for the difference between the required and oversized process exchanger area, as discussed in Section 3.2.1. Note, that the fractions of bypassed streams (xBP,1 , xBP,2 and xBP,3 ) do not need to be expressed explicitly as optimization variables, but can be calculated after optimization. The upper limit of the bypassed streams is fixed at 30% of the total heat flow rate. The conditional expectations are calculated for three uncertain parameters, EC = (47804, 48008, 48019) US$ per year and the basic point is determined to be (389.85, 584.79 K and 3.046 kW/K). MINLP optimization of the extended superstructure at the basic point with flexibility constraints at two critical vertices was performed with a GAMS/DICOPT algorithm (CONOPT, Cplex). The expected cost of optimal solution amounts to 48 402 US$ per year which is higher than the one of the first stage (45 825 US$ per year) because the optimal structure (Fig. 6) additionally comprises two bypasses and one heater on cold stream C2 and some additional heating utility. This problem is also solved simultaneously at 125 Gaussian quadrature points yielding the solution 48 433 US$ per year (Table 4), which indicates that good approximation of the expected value is obtained by the proposed CBP procedure.

CF / (kW/K)

CW

H1 H2

620 K 586.3 K

600 K 393 K

Steam

5 7

1 2

3

4

350 K T 323 K 391.3 K 313 K

1.5 1.0

C1

2.0

C2

3.03

Steam

Fig. 6. Optimal flexible and operable heat exchanger network.

Table 4 Stochastic optimization for operability (second stage) CBP + critical vertices

Gaussian integration EC (US$ per year) A1 (m2 ) A2 (m2 ) A3 (m2 ) A4 (m2 ) A5 (m2 ) A6 (m2 ) A7 (m2 ) xBP,1 xBP,2 xBP,3

48 433 14.818 2.640 6.461 2.140 2.282 0 0.057 0.261 0 0.152

CPUa (s) a

48 402 16.296 2.460 6.970 2.219 2.328 0 0.057 0.3 0 0.294

185

6.3

Pentium III/866 MHz (Windows XP Professional).

5. HEN synthesis An example of HEN synthesis with three hot and four cold streams was considered in this section (Yee & Grossmann, 1990). General superstructure for the synthesis problem with m hot streams, n cold streams and k stages is given in Fig. 7. This picture illustrates clearly the combinatorial complexity of the mathematical model in which binary variables are assigned to all matches between process streams in all stages, as well as to the matches between process streams and utilities. Data for the example are shown in Table 5. Firstly, the synthesis of the network was performed with certain input parameter values and with minimum approach temperature,
min T = 2 K. The total cost of optimal nonflexible structure amounts to 158 × 1000 US$ per year. 5.1. Definition of uncertain parameters In this example, six uncertain parameters are defined: inlet temperatures of the streams H1 and C3, heat capacity flow rate of the stream H2, prices of cooling water and steam, and heat transfer coefficients. The first five parameters are described by beta distribution functions with the lower, mode and upper values as given in Table 6. Parameters Table 5 Data for HEN synthesis Stream

TIN (K)

TOUT (K)

CF (kW/K)

h (kW/ (m2 K))

Price (US$ per (kW year))

H1 H2 H3 C1 C2 C3 C4 CW St

626 620 528 497 389 326 313 293 650

586 519 353 613 576 386 566 308 650

9.802 2.931 6.161 7.179 0.641 7.627 1.690 – –

1.25 0.05 3.20 0.65 0.25 0.33 3.20 3.50 3.50

– – – – – – – 20 80

Cost of exchangers (US$ per year): 8600 + 670A0.83 (A in m2 ).

Z.N. Pintariˇc, Z. Kravanja / Computers and Chemical Engineering 28 (2004) 1105–1119

Stage 1

Stage 2

1115

Stage k

1,1,1

1,1,2

1,1,k

1,2,1

1,2,2

1,2,k

.. . 1,n,1

.. .

1,n,2

1,n,k

2,1,1

2,1,2

2,1,k

2,2,1

2,2,2

2,2,k

.. . 2,n,1

.. .

2,n,2

2,n,k

.. .. .

.. .. .

.. .. .

Steam

. .. . . . .. . .

m,1,1

m,1,2

m,1,k

m,2,1

m,2,2

m,2,k

.. .

.. .

Hm

m,n,1

C2

.. .

. .. . .

H2

. .. . .

Steam

C1

.. .

Cn

CW

CW

CW

.. .

. .. . .

H1

. .. . .

Steam

m,n,2

m ,n,k

Fig. 7. General superstructure of HEN synthesis problem (first stage).

CF (kW/K) H1 H2 H3

626 K

586 K

1 3

620 K

9.802

1.25

2.931

0.05

6.161

3.20

C1

7.179

0.65

C2

0.641

0.25

C3

7.627

0.33

C4

1.690

3.20

519 K

2

CW

6

528 K

4

7

5

h (kW/(m 2 ⋅K))

353 K

Steam

613 K

497 K

8

576 K

389 K

386 K

326 K

566 K

313 K

EC = 176 k$/yr Fig. 8. Optimal flexible structure of the first stage.

α and β are determined as described in Section 2.2. Heat transfer coefficients, h, are assumed to decrease linearly from the upper (1.15h) to the lower values (0.85h). Since heat transfer coefficients affect design variables (exchanger area) monotonously, only the ‘worst’ (lower) values are

CW

H1 H2 Table 6 Data for uncertain parameters distributions

H3

626 K

m

b

α

β

613 K

TIN,H1 (K) TIN,C3 (K) Price of CW (US$/(kW year)) Price of steam (US$/(kW year)) CFH2 (kW/K)

613 320 18.5

626 326 20.0

635 339 21.1

3.363 2.263 3.308

2.636 3.737 2.693

576 K

2.6

2.931

84 3.2

2.333

3.666

3.206

2.793

6 4

5

386 K

7

586 K 519 K 353 K

497 K

8

566 K 80

10

2

528 K

a

78

3

620 K

Uncertain parameter

9

1

389 K

11

326 K

12

313 K

13 Steam

Fig. 9. Superstructure of the second stage.

C1 C2 C3 C4

1116

Z.N. Pintariˇc, Z. Kravanja / Computers and Chemical Engineering 28 (2004) 1105–1119

Fig. 10. Optimal flexible and operable structure of the second stage.

considered in the synthesis (85% of the values given in Table 5). It should be noted that economical uncertain parameters (prices) have an influence on the synthesis of optimal structure and determination of optimal oversizing, while uncertain process parameters are also important for control. If the aspects of flexibility and operability are considered simultaneously within the synthesis of HEN, the superstructure in Fig. 7 has to be extended with the bypasses on all process heat matches, and additional binary variables have to be assigned for the selection between bypasses. Since such a model is combinatorially very complex and difficult to solve, the proposed sequential two-stage procedure is applied. 5.2. First stage In the first stage, synthesis is performed by applying a modified outer approximation/equality relaxation algorithm automated within a computer code MIPSYN—a Mixed Integer Process SYNthesizer (the successor of PROSYN/MINLP by Kravanja & Grossmann, 1994). The central basic point is determined for the selected HEN structure at each main iteration followed by simultaneous optimization at this point and 32 (25 ) vertices, to obtain the approximate expected cost of the structure. Table 7 Summary of HEN synthesis problem First stage EC × 1000 (US$ per year) Number Number Number Number Solver CPUa a

of of of of

heat transfer units constraints variables binary varibales

176

5.3. Second stage In the second stage, the optimal structure of the first stage is extended with bypasses on the process exchangers, additional heaters on streams C2, C3, C4 and additional coolers on streams H1, H2 in order to obtain sufficient degrees of freedom for control. The additional structural alternatives of the second-stage superstructure (Fig. 9) are shown with bold lines. The upper limit for fractions of bypassed streams is fixed at 22%. The second stage model comprises 11 binary variables for the selection of bypasses and additional utility exchangers. The mathematical model of the extended structure is smaller for one order of magnitude than the first stage model (Table 7). The final structure is obtained by MINLP optimization applying the code GAMS/DICOPT (CONOPT, Cplex) and comprises six exchanger units from which four have bypasses (Fig. 10). Heaters are placed on all cold streams and coolers on all hot streams. Annual cost of flexible and operable structure amounts to 229 × 1000 US$ per year. Total CPU time for the first and second stage models indicates that the solutions of nontrivial problems can be obtained within a reasonable period of time by the proposed approach.

Second stage 229

8 10500 7500 55 MIPSYN

13 1835 785 11 GAMS/DICOPT

Approximately 15 min

Approximately 200 s

Intel Pentium/1600 MHz (Windows 2000).

Optimal topology is obtained in the second main iteration with six exchanger units, cooler on the stream H3 and heater on the stream C1 (Fig. 8). Total expected cost of the flexible structure amounts to 176 × 1000 US$ per year. The size of the first stage model and CPU time for its solution is substantial (Table 7).

6. Conclusion In this paper we have presented a two-stage strategy for the synthesis of flexible and operable chemical processes under uncertainty. In the first stage, we performed stochastic MINLP optimization for a given process superstructure in order to select optimal topology and optimal oversizing of process units.

Z.N. Pintariˇc, Z. Kravanja / Computers and Chemical Engineering 28 (2004) 1105–1119

The optimal oversizing of process units is just a necessary condition to assure flexibility of the resulting solutions. The structures would have to contain enough manipulated variables in order to obtain feasible solutions for all realizations of uncertain parameters. If there are insufficient manipulated variables for the given oversized design variables, some process specifications cannot be reached at certain realizations of uncertain parameters from the specified intervals. Therefore, in the second stage, alternatives for additional manipulated variables have to be identified and added to the optimal structure of the first stage in order to assure sufficient degrees of freedom for efficient control. MINLP optimization of the second superstructure is performed to select an optimal combination of additional structural alternatives and manipulated variables. It should be noted that the proposed strategy can be performed, in principle, in one single stage on condition that the number of additional structural alternatives is limited. This paper also presents a novel approximate method for evaluation of the expected objective function. The latter is calculated at the central basic point which is determined in the preceding set-up procedure with feasibility constraints considered simultaneously at the critical vertices. This method was used efficiently for stochastic optimization and synthesis in the first and second stages of the procedure. The problems solved so far show fair agreement between the approximations and the results of more rigorous methods. More examples will be presented elsewhere to illustrate the efficiency of the method (Novak Pintariˇc and Kravanja, 1993), while systematic statistical analyses are under way.

1117 CF/ (kW/K)

620 K H1

5

2 T8

620 K

393 K

7

620 K

T6

T8+DT2

T11

3

T4

T4+DT1

xBP,2

T12

T12+DT3

xBP,1

313 K C2

3.03

Fig. 11. Notation used in the second stage of illustrating example.

This strategy enables the evolution of flexible and operable process schemes and, thus, bridges the gap between the synthesis and final operation of flexible and on-line optimized processes.

Acknowledgements The authors are grateful for the financial support of Slovenian Ministry of Education, Science and Sport (Project L2-3498-0794-01, Contract No. PP-0794-99-502).

Appendix A MINLP mathematical model of the second (operability) stage of the illustrating example is given. Notation used in the model is presented in Fig. 11. Uncertain parameters (T3 , T5 , CFC2 ) are marked by the dashed lines.

k∈CBP

Heat balances: Φ1,k = 1.5(620 − T2,k ) = 2(T4,k − T3,k ) Φ2,k = 1.0(T5,k − T6,k ) = 2(T8,k − T4,k ) Φ3,k = 1.0(T6,k − T11,k ) = CFC2,k (T12,k − 313) T10,k = Φ4,k /6 + 298

350 K

xBP,3

0.65 min C = 1846(A0.65 + A0.65 + A0.65 + A0.65 + A0.65 + A0.65 7 ) + 1300(y1 + y2 + y3 ) 1 2 3 4 6 ) + 2350(A5

+ 20(Φ4,k + Φ6,k ) + 230(Φ5,k + Φ7,k ) k∈CBP

298 K

1.5 4 T10 298 K T7 323 K 6 1.0 T13 T3=391.3 K 2.0 C1

1

T5=586.3 K H2 600 K

T2

Φ4,k = 1.5(T2,k − 350) Φ5,k = 2(600 − T8,k ) Φ6,k = 1.0(T11,k − T7,k ) Φ7,k = CFC2,k (393 − T12,k )

T13,k = Φ6,k /6 + 298 Restrictions on temperatures: 620 − T4,k ≥ 1 T2,k − T3,k ≥ 1 T2,k − T10,k ≥ 1 T11,k − T13,k ≥ 1 T5,k − T8,k ≥ 1 T6,k − T4,k ≥ 1 620 − T8,k ≥ 1 T7,k − 298 ≥ 1 k ∈ CV ∪ CBP T6,k − T12,k ≥ 1 T11,k − 313 ≥ 1 620 − T12,k ≥ 1 T7,k ≤ 323 Area calculations: A1 =

Φ1,k (U1
ln T1,k )


ln T1,k =

[620 − (T4,k + DT1,k )] − [T2,k − T3,k ] ln(620 − (T4,k + DT1,k ))/(T2,k − T3,k )

A2 =

Φ2,k (U2
ln T2,k )


ln T2,k =

[T5,k − (T8,k + DT2,k )] − [T6,k − T4,k ] ln(T5,k − (T8,k + DT2,k ))/(T6,k − T4,k )

1118

Z.N. Pintariˇc, Z. Kravanja / Computers and Chemical Engineering 28 (2004) 1105–1119

A3 =

Φ3,k (U3
ln T3,k )


ln T3,k =

[T6,k − (T12,k + DT3,k )] − [T11,k − 313] ln(T6,k − (T12,k + DT3,k ))/(T11,k − 313)

A4 ≥

Φ4,k (U4
ln T4,k )


ln T4,k =

[T2,k − T10,k ] − [350 − 298] ln(T2,k − T10,k )/(350 − 298)

A5 ≥

Φ5,k (U5
ln T5,k )


ln T5,k =

[620 − 600] − [620 − T8,k ] ln(620 − 600)/(620 − T8,k )

A6 ≥

Φ6,k (U6
ln T6,k )


ln T6,k =

[T11,k − T13,k ] − [T7,k − 298] ln(T11,k − T13,k )/(T7,k − 298)

A7 ≥

Φ7,k (U7
ln T7,k )


ln T7,k =

[620 − 393] − [620 − T12,k ] ln(620 − 393)/(620 − T12,k )

Restrictions on bypasses :

 1 1.5(620 − T2,k ) xBP,1,k = 2− ≤ 0.3 2 T4,k + DT1,k − T3,k

xBP,2,k =

1 2

xBP,3,k =

1 CFC2,k

1.0(T5,k −T6,k ) T8,k +DT2,k −T4,k

2−

CFC2,k −

Logical relations : DT1,k ≤ 100y1 DT2,k ≤ 100y2

≤ 0.3

1.0(T6,k −T11,k ) T12,k +DT3,k −313

≤ 0.3

DT3,k ≤ 100y3 Φ6,k ≤ 270y4 Φ7,k ≤ 250y5 A6 ≤ 100y4 A7 ≤ 100y5

 T2,k , T4,k , T6,k , T7,k , T8,k , T10,k , T11,k , T12,k , T13,k , DT1,k , DT2,k , DT3,k      Φl,k ,
ln Tl,k , Al   l = 1, 2, . . . , 7 ∈ R; k ∈ CV ∪ CBP    y1 , y2 , y3 , y4 , y5 ∈ {0, 1}     U1 = U1 = U3 = U4 = 0.7, U5 = 1 2 LO 2 LO 2 LO 2 LO 2 LO LO 2 ALO 1 = 15.63 m , A2 = 2.46 m , A3 = 6.56 m , A4 = 2.21 m , A5 = 2.32 m , A6 = 0, A7 = 0.056 m

CBP = {(T3 , T5 , CFC2 )} = {(389.85, 584.79, 3.046)} CV = {(T3 , T5 , CFC2 )} = {(378, 573, 3.1), (398, 593, 2.9)}

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