Outlook of the Dynamic Behavior of Closed-Loop Control

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Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Outlook of the Dynamic Behavior of Closed-Loop Control through Open-Loop Analysis for Intensified Separation Processes Juliań Cabrera-Ruiz,*,† Ceś ar Ramírez-Maŕ quez,† Shinji Hasebe,‡ Salvador Hernań dez,† and J. Rafael Alcań tara Avila‡ †

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Universidad de Guanajuato, Campus Guanajuato, Departamento de Ingeniería Química, Noria Alta s/n, 36050 Guanajuato, Gto., México ‡ Department of Chemical Engineering, Kyoto University. Katsura Campus, Nishikyo-ku, Kyoto 615-8510 Japan S Supporting Information *

ABSTRACT: This work shows a comparative assessment of open-loop and closed-loop processes to select, among a set of alternatives, the best distillation scheme regarding its controllability. A new point of view to assess the controllability using the information from the singular value decomposition (SVD) technique is proposed. In the research, the techniques for comparing the openloop analysis and closed-loop analysis were standardized. The open-loop controllability index was calculated from the minimum singular value and the condition number of the certain areas in the frequency domain and a simplified index for measuring the singularity in SVD results. The derived controllability index then was compared with the normalized value of the Integral Absolute Error (IAE) of a proportional−integral (PI) controller obtained under closedloop operation. Several different frequency areas were investigated to calculate the SVD and were used to compare the open-loop and closed-loop behaviors. The selected best distillation scheme was the same despite the chosen approach. Through the simulation study, it is verified that the newly proposed controllability index based on the condition number derived from open-loop responses can be useful to predict the best performance under closed-loop operation. This prediction is suitable to use in an optimization algorithm, because of the structure of the proposed controllability index.

1. INTRODUCTION Distillation is a robust and flexible separation method that can be applied to various separation problems in the chemical industry. As the energy consumption by distillation processes is large, various energy-saving techniques have been proposed. They are thermal coupling, vapor recompression, heat integration among columns, internal heat integration, feed splitting, and combinations with other unit operations, such as the extractive distillation and reactive distillation, among others. These processes have complicated process structures, compared with the conventional distillation process. The distillation process has several degrees of freedom, which naturally increase as the structures increase in complexity. It indicates that we need much computational burden to obtain the optimal design of intensified processes, compared with the conventional processes. For intensified distillation processes, global optimization using a large number of design variables has been done by many works. For example, most of them have focused on the minimization of the utility cost or the total annual cost (TAC).1 The multiobjective global optimization combining TAC with greenhouse gas emission and/or eco-indicators have also been proposed. Most of the previous works have formulated the design problems based on steady-state optimization problems. By optimizing each process structure, and comparing the performance indexes of candidate process structures, it is possible to select the © XXXX American Chemical Society

process with the best intensified structure. However, it is not clear how the selected process behaves when the disturbances are added to the processes. If the selected process is hard to control, that process structure should be avoided even when it shows high energy efficiency. It indicates that the controllability should be considered at the design stage, especially for the cases where complicated process structures are treated. The ability to physically operate a process under various types of disturbances is an important factor as well as the low utility consumption and low TAC. There are several approaches to analyze the controllability of a process. They are classified into two groups: methods based on open-loop operation, and those based on closed-loop operation. The open-loop analysis uses dynamic responses of the output variables resulting from changes in each input variable that is used as a manipulated variable. On the other hand, in the closedloop analysis, dynamic responses of the output variables resulting from the changes in disturbances are used. The manipulated variables change, to keep the output constant. Although open-loop and closed-loop operations can complement each Received: Revised: Accepted: Published: A

August 28, 2018 November 12, 2018 November 13, 2018 November 13, 2018 DOI: 10.1021/acs.iecr.8b04164 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research other to assess the controllability of the process,2 in some cases, it is possible to attain open-loop stability, but closed-loop instability, and vice versa. In any case, the challenge is to obtain accurate dynamic information to determine the stability of the system by both open- and closed-loop simulations.3 Open-loop operations work with the natural dynamic behavior of a process and measure the responses, which acts freely to some disturbances. On the other hand, the closed-loop operation (or feedback control) is adequate for physical purposes, and it is expected for any process to react accurately to the disturbances and correct operating condition for a determined setpoint value. However, when simultaneous optimization of the design and feedback control is being tackled, the optimization of feedback controllers has been the main issue because the manipulation of the design variables affects all the optimization criteria (e.g., cost, energy consumption, controllability); in other words, the optimization problem appears as a double optimization loop with the influence of all design variables.1 To formulate and solve the optimization problem, two different types of algorithms can be selected: deterministic or stochastic algorithms.4 Any optimization problem requires the nontrivial selection of the specific objective functions and design variables. 1.1. The Design and Controllability of a Process. The mentioned global optimization algorithms using a closed-loop approach are not trivial, because the design variables are linked directly to models with increased complexity, especially for intensified processes. Yuan et al.1 reviewed many works that tackle the design-control optimization problems. Most of them optimized the design first via some optimization strategy (i.e., deterministic optimization or stochastic optimization)5 and used conventional indexes such as TAC, eco-indicators, etc. Then, a subsequent dynamics analysis (i.e., closed-loop or openloop) was performed.6−9 Indexes obtained from the open-loop calculations offer some information to estimate the dynamic behavior of a process, but the results are often not conclusive. First, the open-loop approach is useful in pairing input−output variables, which are necessary for feedback control. Typically, the best pairing is the one that corresponds to the physically closest input−output variables (i.e., the manipulation of the reflux ratio to control the purity of a component in the distillate).10 The Singular Value Decomposition (SVD) technique has been widely used to evaluate the theoretical control properties of separation processes11 or to set the pairings for the closed-loop analysis.12 Furthermore, Moore10 clarified how meaningful the results are from the SVD, and showed that the SVD does not require secondary inner optimization. However, the SVD technique has a remaining issue of selecting the frequencies that are suitable for evaluating the singular values. The frequency range plays a very important role when evaluating the difference between the open-loop and closed-loop behaviors.13 The optimization criteria at steady state (e.g., minimization of cost, energy consumption, or environmental impact) and the optimization criteria at nonsteady state (e.g., controllability index) are widely related to each other; nevertheless, because of the increased complexity of dynamic models, it is almost impossible to attain global optimization for the design variables at steady state and the control variables at nonsteady state. The necessary times for developing the optimization algorithm and solving the optimization problem are also critical. Thus, the controller design in the global optimization has been mainly limited to the parameter tuning of the controller. In some works,

such as those of Bansal et al.,14 and Flores-Tlacuahuac et al.,15 they solved the global simultaneous optimization problems by simplifying the models. Another way to solve the problem is to find index controllability criteria that allow one to keep the problem in its simplest form. The use of process simulators such as Aspen Plus allows us to analyze the process at steady and nonsteady state. Also, process simulators are advantageous because the model of a process has already been implemented. The combination of process simulators and global stochastic optimization techniques in complex distillation columns has been addressed previously,16 but the controllability was not included as optimization criterion, because of the complexity of implementing the closed-loop assessment, and the lack in defining a suitable controllability index for the closed-loop option. Vazquez-Castillo et al. solved the design and control problem of separation processes by using the condition number as one objective function17 and was followed by Contreras-Zarazua et al.18 Nevertheless, in those works, the condition number was calculated only by the gain at steady state (condition number at zero frequency) in all of the cases. The analysis of single frequency can be insufficient for predicting the dynamic behavior.13 Furthermore, there was an inherent disadvantage in the SVD technique used in assessing open-loop operations for a wider frequency range, because the results were rather qualitative than quantitative. To properly evaluate the dynamic performance of the selected process designs, it is desirable to compare the condition number and minimum singular value in the complete frequency range. However, it is difficult to optimize all design variables through closed-loop fashions because the dynamic process simulation is needed to evaluate the performance of a given design variable set, and it requires long computation time. Also, a possible disadvantage in dynamic optimization-based approaches can be related to the initialization values in the model at steady state, because there is steady-state multiplicity in an intensified process.1 In most cases, the economic criterion conflicts with the control criterion, thus the worsening of the economic criterion improves the dynamic closed-loop performance.19 This work aims to propose a new point of view that shows the relation between open-loop and closed-loop operations tests in an intensified distillation process. The proposed analysis uses the results from the minimum singular value and the condition number obtained from the SVD technique, to evaluate the openloop operation and then, their results are compared with IAE optimized values (tuned parameters) for the PI controller at closed-loop operation. The objective of the comparison between open-loop analysis and closed-loop analysis is to describe a suitable relationship (index) that allows us to predict the best closed-loop dynamic behavior through the open-loop analysis. Then, the optimal processes are designed by using the open-loop and closed-loop criteria proposed in this research, individually. The results show that the open-loop and closed-loop analysis derive the same or similar process structure as the best process scheme. Therefore, it will be possible to use the proposed controllability index embedded in a multiobjective global optimization problem. This relationship could remain slightly fuzzy in some cases, because, in fact, the selection of the controller pairing and parameter tunings is another extensive task. Nevertheless, the proposed index is useful to compare the dynamic controllability of different process structures quickly, and it can be used for screening B

DOI: 10.1021/acs.iecr.8b04164 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

Figure 4. Logarithmic values of the condition number of the CS M1F1 case.

Figure 1. Condition number lines for various distillation structures.

Figure 5. Derivate values (γ′) of logarithm condition number values and absolute slopes differences (|Δγ′|) of the CS M1F1 case.

Figure 2. Minimum singular values and their corresponding areas.

Figure 6. Graphics value for the main indexes; log(γ) dω for Aγ and γsmi for the smoothness of condition number (γsm) of the CS M1F1 case.

The singular values are important because they denote the maximum and minimum gains of the system output as the input is varied;21 σ* denotes the system invertibility, and the condition number (γ = σ*/σ*) measures the sensitivity of the system. Minimum singular values close to zero must be avoided because they imply that the system is not invertible, thus it cannot be controlled. Also, large condition number values must be avoided because they imply that the system is ill-conditioned, thus it is unrealistic or impossible to control the system, regardless of the control configuration.10 In other words, the changes (positive or negative) can modify the system and became not possible or very difficult to recover the original state.

Figure 3. Condition numbers and their corresponding areas.

the plausible process structures from a large number of feasible ones at the initial phase of process design. 1.2. Open-Loop Analysis. Contrary to the closed-loop operation, the open-loop fashion requires a different mathematical treatment to know its controllability as a natural response resulting from a disturbance. One of the extensively used techniques is the SVD technique.20 The SVD technique is a vectorial decomposition of the transfer function matrix of the process (G). In eq 1, Σ is a diagonal matrix that contains the singular values of the process. These values are arranged in decreasing order of the singular values: from the maximum singular value (σ*) toward the minimum singular value (σ*).

G = UΣVT

(1)

1.3. Closed-Loop Analysis. A feedback mechanism (i.e., controller) is necessary to control and to manipulate the process variables in chemical plants. The most widely used controller is the proportional−integral−derivative (PID) controller.22 However, other control configurations and mechanisms such as model predictive control (MPC) or soft sensors23 have been C


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2. METHODOLOGY It is necessary to assess open-loop and closed-loop analysis methods quantitatively to derive a possible relationship between open-loop and closed-loop operations by using rigorous simulations. The IAE minimization can be used to assess the closedloop performance where the proportional gain (KC) and reset time (τ) are tuned for a PI controller. The SVD results, σ* and γ, can be used to assess the open-loop analysis; however, their interpretation has been done qualitatively for previous works. In this work, a quantitative controllability index analogous to the IAE is proposed to evaluate the open-loop performance based on the SVD results. Figure 1 shows an example of the condition numbers of different process structures (these structures are explained in detail in section4). The control theory says that a low condition number is better a high one. Thus, the process structure depicted by the yellow line (ISP-II) shows the best controllability for lower frequency. However, this line becomes the worst for higher frequencies, then is it really the best? When the singular value is used instead of the condition number, the similar situation occurs. As explained in this example, it is almost impossible to make a decision between the lines, as well as where the best frequency range is if it is decided to take only a particular frequency range for the operation. If a system takes the lowest or lower condition numbers for a wide frequency range, such a system is expected to show better control performance. In this work, Aspen Plus was used to build the schemes in steady state and export to a dynamic state. Aspen Plus Dynamics is used for the dynamic simulations at open- and closed-loop operations. For the open-loop analysis, the process transfer function matrix typically obtained by fitting the responses of the output variables resulting from a 0.5% positive step change in the input variables.12,27,28 Similarly, for closed-loop analysis, the parameters for a PI controller were tuned in Aspen Dynamics to minimize the IAE for purity set-point changes of 0.5%. 2.1. Open-Loop Analysis. We can use Figures 2 and 3 to understand how the minimum singular value area and the condition number area of a system are obtained, respectively. As shown in these figures, these values are dependent on the frequency. If we compare the results at different frequency points, it is possible to appreciate that the scheme for a better controllability changes; for this reason, it is necessary to evaluate the condition number in an extensive way (range of frequencies). To quantitatively discuss the effect of minimum singular value and the condition number, the frequency range is divided into several sections, as shown in Figures 2 and 3 and the average value of these indexes is calculated. Equations 5 and 6 describe those values, where b is the upper limit of the range evaluated.

Figure 7. Conventional sequence (CS) and their control loops.

adopted recently; in any case, PID controllers are widely used in industrial practice, because of their simplicity and reliability. The tuning of the parameters of the controllers is another important issue, which is crucial because it directly impacts on the operability of a chemical plant. Although several tuning techniques for PID controllers have been proposed, it is still common that each industry uses its own tuning techniques (i.e., in-house tools). Moreover, it should be kept in mind that the feedback control structure (pairing) can also be set up under several options, and also as single-end or dual-end,24 as a consequence, the controller design can arise another extensive task. 1.4. PID Controller Tuning. Since the closed-loop analysis is the standard for industrial applications, it has received special attention, particularly for PID models, as has already been mentioned. The tuning problem has been solved in a deterministic way for several cases by using the dynamic optimization problem to achieve the accurate parameters for the controller,25 and also it can be done by reaction curve methods (e.g., Ziegler Nichols).26 The optimization of the tuning of the PID parameters (i.e., KC and τ) can be done by the integral of the absolute error criteria (IAE) defined in eq 2, where the error is calculated using eq 3, where yd is the set point, and y is the controlled variable. IAE ≡

∫0

∞

| ε( t ) | d t

b

(2)

ε(t ) = yd − y(t )

A σb+3 =

(3)

∫−4 log(σ*) dω̃ (4)

b−4

Table 1. Controller Parameters for the Extractive Cases and Their IAE Normalization Values Ethanol KC SSVR-GL CLR-EG CVR-GL CVR-EG

135 15 10 15

τ (min) 6.50 29.50 19.50 23.00

Norm with Maximum Value of Same Output Stream

Water KC

IAE −3

1.288 × 10 5.491 × 10−3 3.863 × 10−3 4.668 × 10−3

70 250 250 100

τ

IAE −4

2.10 4.575 × 10 1.00 1.993 × 10−4 1.00 2.008 × 10−4 13.00 4.297 × 10−3 MIN ± σ D

Norm with Maximum Value of the Set

IAEi

IAEi

IAE1

IAEk

0.235 1.000 0.704 0.850

0.106 0.046 0.047 1.000 0.341 ± 0.638

0.341 1.046 0.750 1.850

0.235 1.000 0.704 0.850

IAEk

IAE2

0.083 0.3180 0.036 1.0363 0.037 0.7401 0.783 1.6327 0.318 ± 0.553


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condition number, the result with the biggest value of Aσ and the lowest value of Aγ is expected to show the best dynamic behavior. The singular values and condition numbers sometimes show sharp spikes in some frequency ranges. Such a quick change of these values is not suitable from the control point of view. Although Aσi and Aγi provide us the average information on the singular value and the condition number, it is difficult to evaluate such quick changes of the values, because the curve is smoothed by the integration. To miss such a peak of the curve is not desirable from the viewpoint of stability of the dynamic singularities of the problem itself (see Figure 3, between frequencies of 1 to 100 rad h−1). In addition, the high-frequency oscillations of these values should also be avoided.13 To consider such oscillations, another factor using the difference of the values is introduced. A new factor (γsm) is defined from the slope of log(γ) and the difference of the obtained slope:

Table 2. Summary of Cases of Study group Sequence Type: Set A, Quaternary Mixture thermally coupled (three mixtures, two feed compositions)

Á lcantara-Á vila et al.29

Petlyuk-type (two feed compositions) Rong et al.30 Sequence Type: Set B, Extractive Distillation ́ Ramirez-Má rquez et al.31

name code CS TCDS-I TCDS-II TCDS-III TCDS-IV TCDS-V CSP-I CSP-II ISP-I ISP-II SSVR-GL CLR-EG CVR-GL CVR-EG

γi′ =

log(γ(ωi)) − log(γ(ωi − 1)) log(ωi) − log(ωi − 1)

(7)

i

A γb+3 =

∫−4 log(γ ) dω̃ b−4

ω̃ = log ω

(5)

|Δγi′| = |γi′+ 1 − γi′|

(8)

γsm = Δlog(ω)|Δγi′|

(9)

i

(6)

The large value of |Δγi′| means that the slope given by eq 8 changes rapidly. Seeking a smooth line, the differences between the slopes must be small; then, it has evaluated the difference between them (eq 9). Finally, to reduce those values in a proper order (see |Δγ′| in Figure 5) for calculated Aγ + γsm, it is obtained the product with the frequency difference range (eq 10), where γsm is the index that measures the condition number smoothness. In Figure 6, it can be noted that both criteria have the similar order, then none can be neglected by the other. Also, it is possible to see how γsm has the principal contribution when the condition number through the frequency domain changes

Note that both axes are measured by logarithms values; then, all of the values are calculated logarithmically (eqs 5, 6 and 7) then these values have lower magnitude orders as shown Figure 4. A value of 0.0001 rad h−1 is selected as the lower bound of the frequency. Values of 0.01, 0.1, 1, 10, and 100 rad h−1 are selected as the upper bounds of integration. The proper definition of the open-loop controllability index is analyzed from these average values of the minimum singular value areas (Figure 2) and the condition number areas (Figure 3). Since the best result is the one with the largest minimum singular value and the lowest

Figure 8. (a) TCDS-I and (b) TCDS-II. E


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Figure 9. (a) TCDS-III and (b) TCDS-IV.

rapidly or when the slope is quite steep. It has been mentioned in section 1.2 that this is an undesired behavior. A similar procedure has been applied for the minimum singular value. Because the minimum singular value area is a maximization (the bigger the best), then the singularity index must be subtracted (i.e., Aσ − σsm). It is important to remember that the results from the SVD analysis are dependent on the process transfer function matrix, which is obtained from disturbing each input variable of the design at dynamic operation. An advantage for these cases is that the gain is normalized by the step value; then there are no differences between this point and other calculated using a different step change. Also, the numerical issues of the magnitude of the step change are handled by the SVD.20 2.2. Closed-Loop Analysis. The same input−output pairing was chosen for open-loop and closed-loop analyses, and it was done between the nearest variables (LV pairing)10 in order to be able to compare both dynamic analyses. Also, the simplest pairing controller with the same disturbance step was kept. For example, Figure 7 shows the pairing of input-output variables for a sequence of conventional distillation columns where the input variables are the product streams purity, and the output variables are the reflux ratio and the reboiler duty (for further details about tuning procedure, see the Supporting Information). In addition, the derivative parameter from the PID controller had been avoided, because this became inherently noisy in some systems.26 In open-loop analysis, each input variable is disturbed independently, and Σ in eq 1 gives information about the entire process simultaneously. In contrast, under closed-loop conventional operation, each controller is tuned independently. Then, it became necessary to normalize the IAE for each scheme to compare the IAE results and the SVD areas. The normalization can be done in two ways: IAE1max = max{IAEi}

(10)

IAE 2max = max{IAEk}

(11)

i

k

Figure 10. TCDS-V.

where k belongs to the set of analyzed sequences and i does to the set of output variables in a sequence. Assuming that the output variables are equally important, the IAE normalized values (i.e., IAE1, i = IAEi/IAEmax and 1 max IAE 2, i = IAEi/IAE2 ) are added to define a normalized overall IAE values (IAE(1) = ∑iIAE1 and IAE(2) = ∑iIAE 2 ). Since the IAE is normalized, its standard deviation (σ) can usefully show the closed-loop controllability of each output variable in which small values mean that the controllability of all output streams are alike and large values mean that some output streams are more difficult to control than others. Once the normalized IAE and the standard deviation are known, the designs can be compared directly and quantitatively. Table 1 shows how this normalization has been done for the F


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Figure 11. (a) CSP-I and (b) CSP-II.

Figure 12. (a) ISP-I and (b) ISP-II.

3. CASES OF STUDY

extractive sequences of section 3. The closed-loop parameters for each output stream have been calculated by minimizing the IAE value for the PI controller. The values of the tuning parameters (i.e., KC and τ) for the extractive sequences were recalculated to represent the recycle-fresh feed streams in the simulation.

Two cases of study are considered in this work, and they cover a set of intensified distillation schemes. Table 2 shows the two cases, design, and thermodynamic information can be found as it is referred to the previous works. The first case G


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lability criteria were calculated by optimizing the interconnecting liquid and vapor flows. The nomenclature of the schemes for the conventional sequence (CS) and thermally coupled distillation sequences (TCDS) remains as in their original work, as well as the design specifications of the founded optimal solutions. Three quaternary mixtures were taken: n-butane, n-pentane, n-hexane, and n-heptane (denoted as M1); benzene, toluene, ethylbenzene, and o-xylene (denoted as M2); and i-butane, n-butane, i-pentane, and n-pentane (denoted as M3). In addition, two feed compositions were taken: F1 (0.40/0.10/0.10/ 0.40) and F2 (0.10/0.40/0.40/0.10). Figures 8−10 show the schemes in Set A. Rong et al.30 studied the second subgroup, which consisted of quaternary mixtures. It contains two conventional Petlyuk-type schemes (CSP) (Figure 11) and two intensified Petlyuk-type schemes (ISP) (Figure 12). Also, in this case, the design specifications for those schemes were taken as the ones in the original study. Two feed compositions were taken for the mixture (n-butane/n-hexane/n-octane/n-decane): 0.1/0.4/ 0.4/0.1; and 0.4/0.01/0.1/0.4, respectively. ́ The second case (Set B) was taken from Ramirez-Marquez et al.31 The nomenclature of the original work was kept, and it is dependent on the type of the recycle thermal condition in the streams: SSVR, CVR recycle vapor streams, and CLR recycled liquid streams (Figures 13 and 14, respectively). The suffixes -GL (glycerol) and -EG (ethylene glycol) refer to the solvent used.

Figure 13. SSVR.

(Set A) includes several distillation schemes to separate a quaternary mixture of aliphatic hydrocarbons, while the second case (Set B) includes several extractive distillation schemes to obtain fuel-grade ethanol. In addition, Set A is divided into two subgroups: (1) partially coupled schemes and (2) Petlyuktype schemes. Alcántara-Á vila et al.29 studied the first subgroup, which consisted of quaternary mixtures where economic and control-

4. RESULTS AND DISCUSSION This section presents the results of the overall normalized IAE and the values of the indexes Aσ − σsm and Aγ − γsm for some of the solutions. Because of the amount of the figures, full results are provided in Section S2 in the Supporting Information, in particular, figures for the minimum singular values, because this index shows lower prediction performance than condition

Figure 14. (a) CLR and (b) CVR. H


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Figure 15. Condition numbers for extractive sequences.

Figure 16. Condition numbers for partially thermal coupled schemes M1F1.

Table 3 shows the overall normalized IAE values of the Petlyuk schemes for the two feed compositions. The values in boldface font indicate the best scheme for each feed composition (lower IAE value). Both feeds, IAE (1) and IAE (2), match without a problem. However, it is also possible to observe how different it is to control each output. Even for the same output, we have “big differences”; this is visible in the standard deviation value. When comparing these values with the index (see Table 4), we observe that the lower frequencies of the SVD values chosen by the ISP-II scheme have the worst fit for the IAE criteria. Then, eventually, at higher frequencies, we consider the lack of smoothness of the index value, and finally, the CSP-I is selected by condition number values. We observe slightly different behavior in Feed 2 (see Table 5), where the best scheme switches between the CSP-I and CSP-II cases. It became important to notice that value

number. Some simulations showed numerical convergence problems to achieve the same purity necessary to be able to compare schemes. Therefore, their results are not shown. Figure 15 shows the condition number of the extractive distillation schemes. As previously mentioned, it was difficult to qualitatively determine the best solution, because there are singularities that are dependent on the frequency region where the area is integrated, the quantitative method will simplify the selection of the best process. It is even more difficult to find a solution in Figure 16 (Mixture 1, Feed 1) and Figure 17 (Mixture 2, Feed 1) for the case of the thermally coupled schemes. The easiest choice for the Petlyuk-type schemes is Feed 1 (Figure 18), where the ISP-II has the bigger minimum singular value and the lowest condition number, but this is not true in all frequency ranges since ISP-II worsened at frequencies of >1 rad h−1. Then it became necessary to be able to assess the best value in the simplest way. I


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Figure 17. Condition numbers for partially thermal coupled schemes M2F1.

Figure 18. Condition numbers for Petlyuk-type feed 1.

Table 3. Global IAE for Petlyuk-Type Schemes Feed 1

Feed 2

scheme

IAE (1)

IAE (2)

IAE (1)

IAE (2)

CSP-I CSP-II ISP-I ISP-II Min ± σ

1.4994 4.0000 2.0061 2.0117 1.499 ± 1.107

0.1802 1.5415 0.2641 0.2726 0.18 ± 0.653

1.4406 3.0803 1.7422 2.3756 1.441 ± 0.727

0.0365 0.2683 0.4339 1.0502 0.037 ± 0.434

differences between both schemes, from the IAE point of view and SVD values, are very close to each other. Sometimes there are very slight differences between the options. We show the general behavior with histograms icons to distinguish whether these differences have enough weight to choose the best and avoid the worst scheme. In Tables 6−9, we show the histograms of the values of each scheme, for IAE (1) and IAE (2) and the condition number indexes for the ranges up to 10 and 100 rad h−1. We show these two values because, when using only low frequencies, the percent matching is too low;

then, it becomes necessary to reach higher frequencies for decision making. For all case of studies, ∼85% accuracy has been reached, matching the best schemes versus the worst schemes. Also, it is more accurate to evaluate until 100 rad h−1 for that purpose. We can observe a few cases where this matching is completely contradictory; for example, in Table 8, the schemes TCDS II and TCDS III in M1F1 and M2F1 has conflicted values between the open- and closed-loop analyses. It is important to remember that we have been comparing SVD values with PI with LV pairing J


CSP-I CSP-II ISP-I ISP-II

−8.071 × 100 −9.894 × 100 −1.028 × 101 -4.927 × 100

Aσ − σsm

K

Aγ + γsm

2.547 × 101 2.618 × 101 3.793 × 101 2.733 × 101

Aσ − σsm

-2.347 × 101 −3.393 × 101 −3.554 × 101 −3.157 × 101

scheme

CSP-I CSP-II ISP-I ISP-II

Frequency = 0.01 rad h−1 -2.199 × 101 −3.147 × 101 −3.525 × 101 −3.438 × 101

Aσ − σsm 2.401 × 101 2.366 × 101 3.762 × 101 3.195 × 101

Aγ + γsm

Frequency = 0.1 rad h−1

1.072 × 101 1.289 × 101 1.389 × 101 8.671 × 100

Aγ + γsm


Table 5. SVD Index for Petlyuk-Type Schemes for Feed 2

Aγ + γsm

1.064 × 101 1.282 × 101 1.525 × 101 8.621 × 100

Aσ − σsm

−8.014 × 100 −9.820 × 100 −1.165 × 101 -4.901 × 100

scheme


Table 4. SVD Index for Petlyuk-Type Schemes for Feed 1 Aγ + γsm 1.079 × 101 1.254 × 101 1.310 × 101 8.848 × 100

-2.335 × 101 −2.893 × 101 −7.239 × 101 −8.331 × 101

Aσ − σsm

2.666 × 101 2.132 × 101 7.598 × 101 8.355 × 101

Aγ + γsm

Frequency = 1 rad h−1

−8.155 × 100 −9.588 × 100 −9.629 × 100 -5.187 × 100

Aσ − σsm

Frequency = 1 rad h−1 Aγ + γsm 1.082 × 101 1.206 × 101 1.285 × 101 1.110 × 101

−4.380 × 101 -3.324 × 101 −1.476 × 102 −2.125 × 102

Aσ − σsm

4.696 × 101 3.480 × 101 1.506 × 102 2.153 × 102

Aγ + γsm


−8.295 × 100 −9.561 × 100 −9.771 × 100 -7.850 × 100

Aσ − σsm

Frequency = 10 rad h−1 Aγ + γsm

1.009 × 101 1.667 × 101 1.291 × 101 1.240 × 101

−8.634 × 101 -3.382 × 101 −1.764 × 102 −3.292 × 102

Aσ − σsm

8.905 × 101 3.396 × 101 1.784 × 102 3.354 × 102

Aγ + γsm


−8.366 × 100 −1.035 × 101 −1.075 × 101 −1.003 × 101

Aσ − σsm


Industrial & Engineering Chemistry Research Article


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process, as it can be appreciated in Figures 15−18, this can change even in the same group of schemes (lines started with “sinusoidal” behavior). In order to standardize the evaluation procedure, it became important to have an additional index to provide information about this zone for the SVD values line. Then, for most of the cases, it is adequate to maintain the frequency of 100 rad h−1 to ensure comparability. In addition, in terms of providing information about the scheme type that shows better dynamic behavior, for all cases shown in this work, the scheme selected the best fit with the form of coupling (recycles) near to the reboiler. This is valid when the recycle is working with a significant amount of flow. If this flow became low, some difficulties to the column to stabilize the disturbances would appear. Generally, bigger interconnected flows provide better dynamic behavior, because it allows the entire column to address the disturbances. Generally, it is possible to consider an overall matching selection, the condition number of indexes being the best for this task. For the above-mentioned cases, it is possible to say that the index, if used, will select at least one of the groups to be the best and will avoid the worst ones. This is an important issue because, in the case of implementing it into an optimization algorithm, we must consider the selection of the best; a population algorithm can be the more convenient for this purpose. If a global optimization is done using this criterion, then we expect that the solution reached will have very good controllability behavior, and the only task that remains is to decide the type of closed-loop tuning and pairing to be used in it.

Table 6. Histogram Appreciation for the IAE Values and Condition Number Indexes of Petlyuk-Type Cases

Table 7. Histogram Appreciation for the IAE Values and Condition Number Indexes of Extractive Cases

only, which is, in fact, a very good method, but not necessarily the best for all cases. In particular, we can refer to the work of Alcantara-Avila et al.,29 who tried a new arrangement for schemes and mixtures, specifically using the RGA technique, and showed that the TCDS-III had lower IAE values than the original LV pairing. Therefore, switching results can be dependent on the selection of the best pairing type. Despite the minimum singular value index and condition number index show very similar scheme selection in most cases; this is due to the presence of the smoothness factor. Without the smoothness factor, the minimum singular value area usually gets lost in the selection of the best scheme. Nevertheless, the condition number index has better match selection, even without the smoothness factor; consequently, the condition number index generally has the best performance. The importance of the addition of those factors (γsm and σsm) rests on the fact that it is not possible to know with certainty what is the highest operability frequency. This frequency is particular for each

5. CONCLUSIONS The dynamic performance of open-loop and closed-loop operations for intensified distillation schemes was analyzed and compared to assess controllability. A new controllability index was proposed for open-loop processes by using the results from the SVD technique. The controllability index at closedloop operation was based on the minimization of integral absolute error by tuning a PI controller. Comparisons between both indices were made in open-loop processes for different frequency intervals and in closed-loop processes for two IAE normalization methods. The results show that both indices are adequate to find the scheme with the best controllability.

Table 8. Histogram Appreciation for the IAE Values and Condition Number Indexes of Thermally Coupled Cases Feed 1

Table 9. Histogram Appreciation for the IAE Values and Condition Number Indexes of Thermally Coupled Cases Feed 2

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Industrial & Engineering Chemistry Research The condition number index (Aγ + γsm) proved to be better in selecting the best scheme in open-loop analysis. Aγ + γsm showed sufficient evidence for selecting a scheme with good control properties in closed-loop analysis. Also, the value of Aγ + γsm until 100 rad h−1 frequencies shows better agreement between the solutions obtained by each index independently. It seems that the presence of stream recycles gives better agreement in selecting the best scheme between open-loop and closed-loop analyses. In particular, for the cases studied, a high flow near to the reboilers improves their controllability behavior. In any case, the selection of a scheme with a smaller Aγ + γsm allows one to obtain a scheme with better controllability in closed-loop analysis. These results justify the use of Aγ + γsm as a suitable objective function to assess the controllability in global optimization problems. For these studies, the traditional pairing of input− output variables was conducted, but it might be worth making a subsequent loop pairing analysis (e.g., RGA) to establish the control loops that are adequate for the selected scheme.

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MPC = model predictive control PI = proportional integral controller PID = proportional integral differential controller SSVR = two columns sequence with vapor recycle SVD = singular value decomposition TCDS = thermally coupled distillation sequence

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GREEK SYMBOLS γ = condition number γ′ = condition number slope (derivative) γsm = condition number, smoothness index Σ = singular values σ = standard deviation σ* = maximum singular value σ* = minimum singular value σsm = minimum singular value, smoothness index τ = reset time ω = frequency [rad h−1]

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ASSOCIATED CONTENT

S Supporting Information *

SUBSCRIPTS AND SUPERSCRIPTS i = output variable in an intensified sequence k = intensified sequence of a set

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.8b04164. Global normalized IAE and SVD index areas (Section 1); tuning flowchart and minimum singular values (Section 2) (PDF)

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REFERENCES

(1) Yuan, Z.; Chen, B.; Sin, G.; Gani, R. State-of-the-Art and Progress in the Optimization-Based Simultaneous Design and Control for Chemical Processes. AIChE J. 2012, 58 (6), 1640−1659. (2) Luyben, W. L. Process Modeling, Simulating and Control for Chemicals Engineers, 2nd Edition; McGraw−Hill: New York, 1990. (3) Huusom, J. K. Challenges and Opportunities in Integration of Design and Control. Comput. Chem. Eng. 2015, 81, 138−146. (4) González, T. F. Handbook of Approximation Algorithms and Metaheuristics; Sahni, S., Ed.; Chapman & Hall/CRC: Boca Raton, FL, 2007. (5) Gutiérrez-Antonio, C.; Gómez-Castro, F. I.; Hernández, S.; Briones-Ramírez, A. Intensification of a Hydrotreating Process to Produce Biojet Fuel Using Thermally Coupled Distillation. Chem. Eng. Process. 2015, 88, 29−36. (6) Acosta-Solórzano, A. D. A.; Guerrero-Farfán, O.; RamírezMárquez, C.; Gómez-Castro, F. I.; Segovia-Hernández, J. G.; Hernández, S.; Gutiérrez-Antonio, C.; Briones-Ramírez, A. Controllability Analysis of Distillation Sequences for the Separation of Bio-Jet Fuel and Green Diesel Fractions. Chem. Eng. Technol. 2016, 39 (12), 2273−2283. (7) Ross, R.; Bansal, V.; Perkins, J. D.; Pistikopoulos, E. N.; Koot, G. L. M.; van Schijndel, J. M. G. Optimal Design and Control of an Industrial Distillation System. Comput. Chem. Eng. 1999, 23, S875−S878. (8) Asteasuain, M.; Bandoni, A.; Sarmoria, C.; Brandolin, A. Simultaneous Process and Control System Design for Grade Transition in Styrene Polymerization. Chem. Eng. Sci. 2006, 61 (10), 3362−3378. (9) Lei, Z.; Yi, C.; Yang, B. Design, Optimization, and Control of Reactive Distillation Column for the Synthesis of Tert-Amyl Ethyl Ether. Chem. Eng. Res. Des. 2013, 91 (5), 819−830. (10) Moore, C. Application of Singular Value Decomposition to the Design, Analysis, and Control of Industrial Processes. In Proceedings of the 1986 American Control Conference, Seattle, WA, USA, 1986; pp 643−650 (). (11) Hernández, S.; Jiménez, A. Controllability Analysis of Thermally Coupled Distillation Systems. Ind. Eng. Chem. Res. 1999, 38 (10), 3957−3963. (12) Luyben, W. L. Distillation Design and Control Using Aspen Simulation, 2nd Edition; John Wiley & Sons: New York, 2006. (13) Morari, M.; Zafiriou, E. Robust Process Control, 1st Edition; Prentice Hall: Englewood Cliffs, NJ, 1989.

AUTHOR INFORMATION

Corresponding Author

*Tel.: +52 (473) 732 0006, ext. 8142. E-mail: j.cabreraruiz@ ugto.mx. ORCID

Julián Cabrera-Ruiz: 0000-0002-5144-5574 J. Rafael Alcántara Avila: 0000-0001-5974-2760 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the National Council of Science and Technology of Mexico (CONACyT). SYMBOLS Aσ = singular value area Aγ = condition number area G = transfer functions matrix KC = proportional gain IAE = integral absolute error IAE = normalized IAE U = left singular vector V = right singular vector

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ABBREVIATIONS CS = conventional sequence CSP = conventional Petlyuk-type sequence CLR = conventional sequence with liquid recycle CVR = conventional sequence with vapor recycle EG = ethylene glycol F = feed composition GL = glycerol ISP = intensified Petlyuk schemes M = mixture M


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Industrial & Engineering Chemistry Research (14) Bansal, V.; Perkins, J. D.; Pistikopoulos, E. N.; Ross, R.; van Schijndel, J. M. G. Simultaneous Design and Control Optimisation under Uncertainty. Comput. Chem. Eng. 2000, 24 (2−7), 261−266. (15) Flores-Tlacuahuac, A.; Biegler, L. T. A Robust and Efficient Mixed-Integer Nonlinear Dynamic Optimization Approach for Simultaneous Design and Control. Comput.-Aided Chem. Eng. 2005, 20, 67−72. (16) Cabrera-Ruiz, J.; Miranda-Galindo, E. Y.; Segovia-Hernández, J. G.; Hernández, S.; Bonilla-Petriciolet, A. Evaluation of Stochastic Global Optimization Methods in the Design of Complex Distillation Configurations. In Stochastic OptimizationSeeing the Optimal for the Uncertain; InTechOpen, Ltd.: London, 2011; Chapter 18 ( DOI: 10.5772/14449). (17) Vázquez-Castillo, J. A.; Segovia-Hernández, J. G.; Ponce-Ortega, J. M. Multiobjective Optimization Approach for Integrating Design and Control in Multicomponent Distillation Sequences. Ind. Eng. Chem. Res. 2015, 54 (49), 12320−12330. (18) Contreras-Zarazúa, G.; Vázquez-Castillo, J. A.; RamírezMárquez, C.; Segovia-Hernández, J. G.; Alcántara-Á vila, J. R. MultiObjective Optimization Involving Cost and Control Properties in Reactive Distillation Processes to Produce Diphenyl Carbonate. Comput. Chem. Eng. 2017, 105, 185−196. (19) Segovia-Hernández, J. G.; Hernández-Vargas, E. A.; MárquezMuñoz, J. A. Control Properties of Thermally Coupled Distillation Sequences for Different Operating Conditions. Comput. Chem. Eng. 2007, 31, 867−874. (20) Klema, V. C.; Laub, A. J. The Singular Value Decomposition Its Computation and Some Applications. IEEE Trans. Autom. Control 1980, 25 (2), 164−176. (21) Skogestad, S.; Morari, M. Undestanding the Dynamic Behavior of Distillation Columns. Ind. Eng. Chem. Res. 1988, 27 (10), 1848− 1862. (22) Bequette, B. W. Nonlinear Control of Chemical Processes: A Review. Ind. Eng. Chem. Res. 1991, 30, 1391−1413. (23) Kano, M.; Ogawa, M. The State of Art in Advanced Process Control in Japan. In IFAC Symposium ADCHEM, Istanbul, Turkey, 2009; pp 10−25. (24) Luyben, W. L. Distillation Control. In Distillation: Operation and Applications; Górak, A.; Schoenmakers, H., Ed.; Academic Press: Oxford, U.K., 2014. (25) Hang, C. C.; Astrom, K. J.; Wang, Q. G. Relay Feedback AutoTuning of Process Controllers- a Tutorial Review. J. Process Control 2002, 12, 143−162. (26) Svrcek, W. Y.; Mahoney, D. P.; Young, B. R. A Real-Time Approach to Process Control, 3rd Edition; John Wiley & Sons: Chichester, U.K., 2014. (27) Luyben, W. L. Derivation of Transfer Functions for Highly Nonlinear Distillation Columns. Ind. Eng. Chem. Res. 1987, 26 (12), 2490−2495. (28) Hovd, M.; Braatz, R. D.; Skogestad, S. SVD Controllers for H2−, H∞− and μ-Optimal Control. Automatica 1997, 33 (3), 433−439. (29) Alcántara-Á vila, J. R.; Cabrera-Ruiz, J.; Segovia-Hernández, J. G.; Hernández, S.; Rong, B. G. Controllability Analysis of Thermodynamically Equivalent Thermally Coupled Arrangements for Quaternary Distillations. Chem. Eng. Res. Des. 2008, 86, 23. (30) Rong, B.-G.; Errico, M.; Segovia-Hernández, J. G. New Intensified Distillation Systems for Quaternary Petlyuk. Comput.Aided Chem. Eng. 2014, 33, 97−102. (31) Ramírez-Márquez, C.; Segovia-Hernández, J. G.; Hernández, S.; Errico, M.; Rong, B.-G. Dynamic Behavior of Alternative Separation Processes for Ethanol Dehydration by Extractive Distillation. Ind. Eng. Chem. Res. 2013, 52, 17554−17561.

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Outlook of the Dynamic Behavior of Closed-Loop Control

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