The Importance of Nominal Operating Point Selection in Self

Article pubs.acs.org/IECR

The Importance of Nominal Operating Point Selection in SelfOptimizing Control Eduardo S. Schultz,* Jorge O. Trierweiler, and Marcelo Farenzena Group of Intensification, Modeling, Simulation, Control and Optimization of Processes (GIMSCOP), Chemical Engineering Department, Universidade Federal do Rio Grande do Sul (UFRGS), Rua Luiz Englert s/n, Porto Alegre, Rio Grande do Sul 90035-191, Brazil ABSTRACT: A self-optimizing control (SOC) structure is when we can achieve an acceptable loss with constant setpoints for controlled variables. In the original paper (Skogestad, J. Process Control, 2000, 10 (5), 487−507), the operating point is selected first and then the controlled variables, using the optimal values for nominal disturbance as setpoints. In this work, the importance of nominal operating point selection is evaluated, showing that setpoints obtained by optimization along the disturbance region can provide a better self-optimizing structure. A simultaneous procedure is proposed where controlled variables are selected together with the nominal operating point. Despite the more expensive optimization procedure, this new policy leads to a better controlled variables set. This is corroborated by a reactor− distillation case study and a cumene unit case study where the operating point selected decreases significantly the worst-case and mean losses for the process.

1. INTRODUCTION The selection of the best set of controlled variables (CVs) in chemical units is an important decision when designing the control system. Most of the time this decision is made based on operating parameters for keeping the process safe and stable.1 However, as proposed by Morari et al.,2 the control structure may provide several economic benefits, becoming a powerful optimization tool beginning from the regulatory layer. This idea was improved by Skogestad3,4 with the concept of selfoptimizing control (SOC) to select the best set of CVs: “Selfoptimizing control is when we can achieve an acceptable loss L with constant setpoint values cs for the controlled variables.” Although a constant setpoints policy results in a not optimal operation when there is a disturbance in the plant, it provides a simple and cheap way for unit optimization when compared to real-time optimization (RTO) technologies. Engell5 has made some considerations about the concept of using feedback control with optimization purpose, showing the benefits and disadvantages of each technique, and he proposed some refinements. Several papers demonstrate the usefulness of SOC methods when applied in real units as studied by Larsson et al.6 in the Tennessee Eastman process, by Araujo et al.7,8 in an HDA unit, or by Sayalero et al.9 in a diesel hydrodesulfurization plant. In recent years, several methodologies were developed to find the best set of CVs and for loss calculation, using the SOC approach. Skogestad3 proposed qualitative criteria for CVs selection and a minimum singular value rule for loss evaluation, calculating the loss for all possible set of CVs and choosing the better one with acceptance loss. Halsorven et al.10 proposed an exact local method for worst-case loss calculation that can be applied for selecting variables and for linear combination of © XXXX American Chemical Society

measurement definition. The synthesis of CVs based on linear combination of measurements was also proposed by Alstad and Skogestad11 with the proposition of the null space method, later improved.12 Kariwala developed a new method with the same objective with the eigenvalue decomposition approach to minimize the worst-case loss13 and later for the average loss.14 François et al. proposed the use of measurements for enforcing necessary conditions of optimality,15 and a new method was proposed by Ye et al. based on this idea where it used the approximating necessary conditions of optimality as controlled variables allowing the use of nonlinear combination of measurements.16 The definition of a plant control structure based on an SOC concept involves two separate steps: The first one is to define the nominal operating point where the optimal values for nominal disturbance are usually used, and the second is to choose the variables that must be controlled at that point with the objective of loss minimization. Although Skogestad has already mentioned that CVs’ setpoints could be subject to optimization during CVs selection;3 all papers select the setpoints as the nominal optimal values. Govatsmark and Skogestad17 proposed to change the operating point and use robust setpoints to avoid infeasibility in the active constraint but without an optimization purpose and not to enable more variable sets. Using a fixed optimal nominal operating point during the CVs selection procedure, some variables may be excluded because the system cannot keep their Received: June 4, 2015 Revised: May 23, 2016 Accepted: June 3, 2016

A

DOI: 10.1021/acs.iecr.5b02044 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research ⎧ min J(u , x , d) ⎪ u ⎪ ⎨ f (u , x , d ) = 0 ⎪ ⎪ g (u , x , d ) ≤ 0 ⎩

values constant along the entire disturbance region. Therefore, setpoints influence the available sets of CVs, and for each set of values, there may be a different group of self-optimizing variables optimal for the problem. The main scope of this paper is to analyze the influence of operating point on the best control structure obtained by SOC. This idea is based on the premise that CVs selection and setpoint definition are not two decoupled optimization procedures. Therefore, for each operating point there may be a different optimal set of CVs that minimizes the loss along the disturbance region. The original requisite of ensuring null loss in the nominal point for the process without noise may not be optimal for SOC, restricting the number of possible CVs, and when this assumption is relaxed, the number of possible CVs increases, allowing reduction of (worst-case or mean) loss. This makes the problem more complex and difficult numerically and requires more computational effort to solve it. A clear understanding about the process disturbance behavior and its distribution is also necessary, considering that this may make the optimization problem even more complex. However, better solutions can be provided, and the losses are reduced. Additionally, this approach can be extended to the actual methodologies for calculating the best variables combination. Furthermore, two simplified procedures are proposed to improve the gain of SOC by changing the operating point during SOC procedure. These formulations are proposed for optimizing the process along disturbance region instead of for just the nominal point. In the first one, the values of manipulated variables (MVs) are optimized before CV selection, and in the second, the values of the setpoints of selected controlled variables are optimized. This paper is segmented as follows: Section 2 briefly describes the classical methodology of SOC. Next, the impact of the operating point in the CVs selection is studied and then two simplified alternatives are presented to reduce process loss. In section 4, an example is given of a reactor and a distillation column with recycle that will be used to show the gains of the new approach and the simplified procedures. Another case study, based on a cumene production process, is presented in section 5, for evaluating the proposed procedure in a larger unit. In these case studies, the results obtained by using the different operating points will be compared. The paper ends with some concluding remarks.

(1)

where f represents the process model, g represents the inequality constraints, and x represents the process states. The solution of this problem provides the optimal inputs (uopt(x, d)) and cost value (Jopt(x, d)) for each disturbance (d). When the process is not operating at optimal point, a loss is generated in the process, and its value can be calculated by L(u , x , d) = J(u , x , d) − Jopt (uopt(x , d), x , d)

(2)

For each value of disturbance there will be an inherent loss that depends only on the operating point u. When there is a change in the disturbance value, there will be another optimal point of operation, and if the values of manipulated variables are not changed, then there will be a loss in the process. The main idea of SOC is to select controlled variables that minimize process loss without changing their setpoints. The first step during SOC procedure is to define which operating point will be used because this will be the setpoint of the CVs. It is usually made an optimization for nominal disturbance, and the result is applied as constant setpoints. However, there is no guarantee that this point will be optimal for the entire disturbance region. The disturbance region is composed of all possible values of disturbance allowed in the process, so the number of combined values increases with the number of disturbance variables of the process. During the procedure of CVs selection, the loss along the region of allowed disturbance and implementation error is computed, where worstcase loss, average loss, or other formulation for loss measurement can be used along this region such as Monte Carlo evaluation.3 The worst-case loss (Lworst) is defined by the maximum value of the loss for all disturbances and implementation errors for a specific operating point, defined by u and x, and can be represented by14 Lworst(u , x , D , N ) = max max(J(u , d , n) − Jopt (uopt(x , d), x , d)) d∈D n∈N

(3)

where D represents the disturbance region, n represents the implementation error, and N represents the allowed values for n. The average loss can be calculated by the following expression:14

2. SELF-OPTIMIZING CONTROL The SOC concept consists of finding the best set of controlled variables that minimizes the loss with constant setpoints. Thus, it is important to define the concept of loss since it is the main variable used in this procedure. In the context of SOC, loss means the difference between the profit obtained by solving an optimization problem for every disturbance realization and the profit seen by holding the controlled variables with fixed setpoints. The best operating point of a process can be defined as a minimization problem where the objective function can be the process cost (J) subject to equality and inequality constraints according to a process model and their physical limits. Solving this problem involves the search for values of manipulated variables (u) that provide the lowest value of J for a defined disturbance d. This problem can be represented mathematically as

Lavg (u , x , D , N ) =

1 |D||N |

∫D ∫N (J(u , d , n) − Jopt (uopt(x , d), x , d)) dn dd (4)

It is important to mention that it is not usual to find analytical expressions for evaluating the loss (J − Jopt). Thus, it is normally necessary to discretize the region of D and N and evaluate the loss for each point. After this, the mean loss may be calculated by the average of the points, and the worst-case loss is the largest value. There are expressions for loss evaluation based on approximating cost function by second-order Taylor series and process model linearization,10,14 but the results are just valid locally. The SOC variable selection problem can be solved using a mixed-integer nonlinear optimization (MINLP) algorithm. The available measurements are Y = {Y1, Y2, ..., Yk}, where k is the number of variables available to be controlled; the available B


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Industrial & Engineering Chemistry Research manipulated variables are U = {U1, U2, ..., Um}, where m is the number of degrees of freedom available for optimization. The entire set of variables (v) is v = {Y ∪ U} because manipulated variables can also be selected as constants. An auxiliary Boolean variable vb (vb = {0, 1}k+m) is used to set the variables whose values are kept constant during the optimization procedure. Thus, the SOC variable selection problem can be written as follows, disregarding the implementation error: ⎧ min Loss(vb , u , x , D) ⎪ vb ⎪ s.t. ⎪ ⎪ f (u , x , d ) = 0 ⎪ ⎪ g (u , x , d ) ≤ 0 ⎪ ⎪ vb ∈ {0, 1}k + m ⎪ ⎨ vi = vi ,set , if vb(i) = 1; i = 1, 2, ..., k + m ⎪ ⎪ k+m ⎪ ∑ vbi = m ⎪ i=1 ⎪ ⎪ d ∈ + nd ⎪ ny ny ⎪ vi ∈  , vi ,set ∈  ⎪ nu nx ⎩u ∈  , x ∈ 

Figure 2. Behavior of process profit along disturbance region for different setpoints.

3. BEST OPERATION POINT FOR SELF-OPTIMIZING CONTROL (BOPSOC) The design of the best control structure involves not only the selection of the controlled variables but also the best setpoints that minimize loss. This problem can be solved separately, as discussed previously, when the first nominal optimal point is computed for nominal disturbance, and then the CVs are selected considering zero loss at this point. However, it is not ensured that a null loss in the nominal point will result in the minimum loss in the entire disturbance region. This difference is illustrated in Figure 1. This example shows the behavior of operation cost (J) for three scenarios: (1) where the system is reoptimized whenever there is a change in the disturbance; (2) where the setpoints (SPs) are optimal for nominal disturbance (d*); and (3) where SPs and CVs are chosen in the same optimization problem analyzing the loss for the entire region (D). It is possible to see in Figure 1 that a flat curve along disturbance region may provide smaller losses than the optimal nominal when the entire region is considered. The nominal setpoints may be optimal for a narrow disturbance region, but as a general rule, it is necessary to choose their values simultaneously with CVs, which can allow reducing the loss provided by constant setpoints along a disturbance region when compared to nominal operating point. Figure 2 illustrates the importance of the previous setpoint selection. If SP1 is chosen, then the profit will be maximum at the nominal point (d*); however, the curve behavior is more abrupt than SP2 and SP3. Despite the loss in the specific nominal point, its profit in the entire disturbance region is larger. In this paper, a modification in the original methodology is proposed: The setpoints will be computed together with the controlled variables selection. This method is called best operating point self-optimizing control (BOPSOC). The new optimization problem is stated as follows, and a mixed-integer nonlinear programing algorithm (MINLP) is required to solve the problem.

(5)

where vi represents the values of controlled variables candidates, vi,set represents their setpoints, nd represents the number of disturbances, nx represents the number of states, ny represents the number of available measurements, and nu represents the number of available degrees of freedom. The Loss (L) may be computed as the worst-case loss calculated by eq 3 or the mean loss calculated by eq 4 and depends on the chosen disturbance region and the selected operating point (u, x). The choice of each loss to be used depends on what is more important for the process since each one may provide a different optimal set of CVs. When using local formulations for loss calculation, the set of CVs that minimizes the average loss also minimizes the worst case loss, which is not true when the nonlinear model is used. In the original procedure proposed by Skogestad,3,4 initially eq 1 is solved to define the variable setpoints and then eq 5 to define which variables will be controlled where not only CVs but also MVs are candidates. Moreover, the disturbance value (d) has a specific value.

Figure 1. Loss imposed by different setpoints during SOC procedure. C


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Industrial & Engineering Chemistry Research ⎧ min Loss(vb , u , x , D) ⎪ vb , vset ⎪ ⎪ s.t. ⎪ f (u , x , d ) = 0 ⎪ ⎪ g (u , x , d ) ≤ 0 ⎪ ⎪ vb ∈ {0, 1}k + m ⎪ ⎨ vi = vi ,set , if vb(i) = 1; i = 1, 2, ..., k + m ⎪ ⎪ i=1 ⎪ ∑ vbi = m ⎪ k+m ⎪ ⎪ d ∈ + nd ⎪ ny ny ⎪ vi ∈  , vi ,set ∈  ⎪ nu nx ⎩u ∈  , x ∈ 

change can also be inserted as a constraint of the optimization problem such as robust setpoints discussed by Govatsmark and Skogestad.17 3.1. Relaxed Solutions. The MINLP problem associated with the procedure here proposed may be very expensive computationally for large or even medium size problems. Two simplified policies will be subsequently proposed to relax the problem, which provide better results than the nominal optimal operating point. In both, the variable selection problem is run separately from the operating point selection; however, the approach to solve the last problem is distinct. The first simplified policy is to change the operating point before defining the set of CVs. Instead of using the best operating point for nominal disturbance calculated by eq 1, it is proposed to find an operating condition that minimizes the loss along a disturbance region. This results in a modification of the initial problem defined by eq 1 using the loss as a new objective function. With this change, a better behavior is expected since the operation is not optimized for a single one point. This new operating point can be calculated by an optimization formulation where the manipulated variable free for optimization U = {U1, U2, ..., Um} should have optimal values u = {u1, u2, ..., um} that minimize the loss while respecting process constraints, as defined by the following NLP problem:

(6)

It is clear that this new problem is more complex than the original because the variables set as constants are determined together with their values. It involves solving several optimization problems that usually need stochastic algorithm to be solved inside the major optimization of mean loss or worst-case loss, what is classified as MINLP. The solution of this optimization problem may seem intractable for medium or large systems, but simplified procedures may become feasible its application as discussed further in this work. Moreover, this problem will not be solved online, only in the control loop design. Thus, the “cost” for solving this more complex problem will be “paid” by a more profitable structure. Moreover, the worst result of this new optimization procedure will be the nominal operating point itself, with the set of CVs provided by any SOC procedure, which may be provided as an initial guess for the optimization algorithm. An important result of this change is the increase of the number of available sets of CVs that can be kept with constant setpoints along a disturbance region. When a specified condition is defined for the process and all variables must keep those values, it is impossible for some of them to be controlled at that setpoint for all disturbance values, so these sets of variables are rejected during SOC procedure. However, when the values are not previously defined and it analyzes the best operation point for each set of CVs, new possible structures appear using different values of setpoints. Another implication of this improvement is related to the impact of active constraints. The classical concept considers that all of them must be controlled and just the remaining unconstrained degrees of freedom should be used for optimization. The control of active constraints is only advisable when the constraints are active for the entire disturbance region, and in the cases where there are changes in the actives constraints, it is necessary to use different techniques such as proposed by Cao18 and Hu et al.19 However, using the proposed procedure, where the operating point is not defined in the beginning, it is not necessary to restrict the degrees of freedom, and all possible sets of CVs must be available for optimization during a BOPSOC procedure. Using this strategy, there are more sets to be evaluated and more possibilities of control structures for the loss minimization. The change in active constraints does not affect the system since the optimization procedure covers all disturbances when local loss evaluation is not used, and during the optimization procedure, all constraints must be satisfied for all disturbance values. Alternatively, policies to avoid an active set

⎧ min Loss(u , x , D) ⎪ u ⎪ s.t. ⎪ ⎪ f (u , x , d ) = 0 ⎨ ⎪ g (u , x , d ) ≤ 0 ⎪ ⎪ d ∈ + nd ⎪ nu nx ⎩u ∈  , x ∈ 

(7)

where the objective function Loss may be calculated using eq 3 for mean loss or eq 4 for worst-case loss, with discrete points along a disturbance region. The main difference between eqs 7 and 1 is that in the first the loss is minimized throughout the entire disturbance region which is different from the second where the loss is null in the nominal point when none implementation error is considered. After the solution of eq 7, the optimal values of u are used to calculate the setpoints, using the stationary model of the process and the nominal values of the disturbances (d*). It is important to highlight that the worst result of the optimization procedure is the nominal operating point itself. Although it is usually used the nominal disturbance for evaluating the setpoints in SOC, it would be possible to optimize the value of d using a new optimization procedure. The new values for setpoints are used for finding the best set of CVs following the original SOC procedure. This proposition can lead the process to a flatter loss behavior when compared with a null loss point, which may provide a better control structure that decreases the loss with less computational effort when compared to BOPSOC. It is important to note that it may not be possible to find values for the degree of freedom feasible for an entire disturbance region when considering process constraints. However, it will be always possible to calculate a feasible point for the process with constraints only in the manipulated variable. In the second simplified policy, first the set of CVs is selected using the original SOC concept. Then, their setpoints are optimized to reach a minimum loss over the entire disturbance D


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show the minor variations of their optimal values along disturbance region. The set of variables with minor variations and their average setpoints may be used as an initial guess for the optimization algorithm applied to solve BOPSOC. This is an important step for complex systems where it is unfeasible to optimize the setpoints for every possible set of CVs for finding the global optimal. Step 5.1: BOPSOC. Considering that the disturbance region D is defined, it is necessary to use a MINLP algorithm to find the best set of CVs and their setpoints, based on eq 6. For simple cases, it may be tested all possible combinations of CVs, evaluating the optimal setpoints for each set and finding the global optimal. An alternative is to restrict the analysis to the variables with minor optimal values variation along disturbance region. Step 5.2: Relaxed Solution. Next, we calculate the optimal degrees of freedom values along disturbance region, using eq 7. On the basis of these optimal values and with the nominal disturbance values, d*, we evaluate the setpoints for all CVs candidates and find the set of CVs that provided the smaller loss using an optimization algorithm or testing all possible combination. Using eq 8 optimizes the setpoints of the selected set of CVs. Step 6: Validation. Using the nonlinear model, it is necessary to check whether the selected CVs and their setpoints may be kept with constant values for the entire disturbance region. If any linear model or expression was used to evaluate the loss, then this analysis is even more important because most of the linearization methods may provide results that are only valid near the linearization point, which in SOC means that are valid only near the nominal disturbance. Additional analysis may be done using several statistical distribution and discretization patterns.

region. The main difference between this approach and the one discussed before is that the previous calculates the best values of manipulated variables and with these values an SOC procedure is run for variables selection whereas in the last the setpoints are changed after CVs selection, without affecting the SOC procedure. The change in the setpoints can be represented as an optimization problem where the decision variables are the setpoints of the selected CVs and the objective function is the loss calculated by eq 3 or 4 along a discretized disturbance region. The formulation for setpoints optimization is presented in eq 8, which can be considered an NLP problem. In this case, vb is constant during optimization procedure since the CVs are already selected. ⎧ min Loss(vb , u , x , D) ⎪ vset ⎪ s.t. ⎪ ⎪ f (u , x , d ) = 0 ⎪ ⎪ g (u , x , d ) ≤ 0 ⎪ ⎨ vi = vi ,set , if vb(i) = 1; i = 1, 2, ..., k + m ⎪ ⎪ d ∈ + nd ⎪ ⎪ vb ∈ {0, 1}k + m ⎪ ny ny ⎪ vi ∈  , vi ,set ∈  ⎪ nu nx ⎩u ∈  , x ∈ 

(8)

3.2. Procedure Summary. In this section, the proposed method is summarized for better understanding. Step 1: Previous Work. Steps 1 (degrees of freedom analysis), 2 (cost functions and constraints), 3 (identify the most important disturbances), 4 (optimization), and 5 (identify candidate controlled variables) of Skogestad3 procedure must be accomplished. After these steps are defined the cost function of the process, the process constraints, degrees of freedom, and the main disturbances. It is also assumed that the nominal operating point, the nominal disturbance, the candidate controlled variables, and (if there are any) changes in the active constraints are known. Step 2: Disturbance Behavior Analysis. At this point, it is necessary to define the possible values of the disturbances (D) and their behavior over time. On the basis of the behavior of this variable, it is possible to define the better discretization method to be used in the loss evaluation. For example, if a process where one of the disturbances is the inlet temperature and its range is between 90 and 130 °C and for 90% of the operating time its value is between 105 and 115 °C, then it is better to discretize the disturbance region with more values between the values of 105 and 115 °C. However, if the operating time that the system operates at 90 °C is near the time the system operates at 130 and 110 °C, then the best option should be a uniform discretization. Step 3: Loss Evaluation. This step defines how the loss will be calculated during the optimization procedure, considering the process model complexity. The loss may be evaluated using the nonlinear model using eqs 3 or 4 and the discrete points defined in step 2. An alternative is to use expressions obtained by the linear model10,14 to calculate the loss using just one evaluation, usually used for variable prescreening. Step 4: Setpoints Analysis. On the basis of the optimization analysis done in step 1, it is possible to check which variables

4. CASE STUDY−CSTR WITH SEPARATION COLUMN AND RECYCLE A system was considered that was composed of a CSTR reactor and a separation column with recycle as shown in Figure 3. The process has four components (A−D) and only component A is fed. Inside the reactor, there is a reaction following the kinetics of Van de Vusse described by the following reaction (more details about this reactor were discussed by Trierweiler).1

Figure 3. Flowsheet for the reactor−distillation−recycle system. E


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disturbance distribution; however, this will not be addressed in this case study. 4.1. Results. The main objective of SOC is to define which variables must be controlled with fixed setpoints for minimizing the loss. The first step is to calculate the optimal operating point for nominal disturbance. The optimization problem described by eq 1 was used for finding this point where eq 10 was the objective function. The model equations presented in Appendix A were used as equality constraints, and eqs 11−13 were used as inequality constraints. The calculation was performed using MATLAB R2009 using the function fmincon, and the nominal optimal values are presented in Table 2. The nominal optimal cost is −2.182 $/h.

The most valuable product is B. The lighter component is considered to be A, and it is completely removed and recycled to the reactor by column top stream. Therefore, component A does not leave the column from the bottom stream. Heavy components C and D are also completely separated in the column and removed in the bottom of column. Component B is the only one that can be distributed in both streams as a function of distillation in operation. A simplified version of the column model is used that is described by an empirical operating curve proposed by Muller.20 This system is similar to the one studied by Govatsmark,17 but there are differences in the reaction, in the available variables for control optimization, and in the disturbances studied. The process variables are described in Table 1, and the model equations are shown in Appendix A.

Table 2. Optimal Nominal Operating Point

Table 1. Model Variables and Units variable

description

unit

Cain Fin Car Cbr Fr Ca Cb Cc Cd F Ya Vr T Qr

feed concentration of A feed flow rate recycle concentration of A recycle concentration of B recycle flow rate concentration of A from reactor concentration of B from reactor concentration of C from reactor concentration of D from reactor flow rate from reactor Car/(Car + Cbr) reactor volume reactor temperature reboiler duty

kmol/m3 kmol/h kmol/m3 kmol/m3 kmol/h kmol/m3 kmol/m3 kmol/m3 kmol/m3 kmol/h − m3 °C kJ/h

P = 3.53(F ·Cb − Fr ·Cbr ) + 1.1F ·Cc − 0.5F ·Cd

and the cost function may be calculated by J=−P

(9)

(10)

The disturbance of the process is the temperature of the reactor and it is considered that concentration of A in the feed is constant. The set of manipulated variables (U) available for optimization are U = {Ya, Vr, F}. The available measurements (v) for SOC are the following, including the set of manipulated variables: v = [Fin , Car , Cbr , Fr , Ca , Cb , Cc , Cd , F , Ya , Vr ]

The process has the following constraints:

3 ≤ F ≤ 30

(11)

0.1 ≤ Ya ≤ 1

(12)

0.5 ≤ Vr ≤ 5

(13)

optimal value

unit

Fin Car Cbr Fr Ca Cb Cc Cd F Ya Vr

9.413 1.843 0.0570 20.59 1.265 0.668 0.702 0.135 30.0 0.971 2.82

kmol/h kmol/m3 kmol/m3 kmol/h kmol/m3 kmol/m3 kmol/m3 kmol/m3 kmol/h − m3

Table 2 shows that F is an active constraint at nominal point, and for this problem it will be considered controlled. Thus, two degrees of freedom remain available for optimization. It is important to clarify that the control of active constraints is only advisable when the constraint is active for the entire disturbance region. In this case, there are 10 candidate variables that can be grouped in sets of two, totaling 45 possibilities for CVs, considering that only single variables will be controlled, i.e., measurements combinations will not be studied. The worst-case loss was measured using eq 3, and the average loss was calculated with eq 4, using the nonlinear process model. The use of the nonlinear model was possible because the process is simple and easily modeled; however, for a complex system, it may be necessary to use local expression for loss measurement to prescreen the alternatives that should be corroborated against the nonlinear model. For this case, only 2 sets of CVs can be maintained with constant setpoints for all disturbance values, and the set that minimizes the worst-case and average loss is the open loop operation. The calculated losses are Lavg = 3.33 $/h and Lworst = 10.9 $/h, and it is clear that these values are not acceptable when compared to nominal process profit. Using the BOPSOC concept, the best setpoints were calculated for each available set of CVs without considering F controlled since the nominal point will not be used. The optimization of setpoints and CVs were accomplished with MATLAB R2009 using the function patternsearch. The number of available combinations rises from 45, when F is controlled, to 165 according to the combination of 11 variables in sets of 3. For each of the possible sets, the setpoints were optimized for minimizing the average and the worst-case loss using the same discretization of disturbance region described previously. The relaxed solutions were also applied, using the same software and optimization function of BOPSOC, where in the first case an

The profit of this process is given by

− 1.05Fin − 10Qr

variable

The nominal disturbance value is T = 110 °C, and Cain is kept constant with the value of 5.1 kmol/m3. The SOC methods will be evaluated for the case where the temperature varies from 90 to 130 °C, and the loss evaluation will be done using discrete points along disturbance region with a step of 10 °C uniformly distributed. Although a uniform step is used for disturbance discretization, it could be used a discretization based on the F


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different best control structures, and by using the values optimized along the disturbance region, it was possible to reduce the maximum loss from 10.9 to 1.27 $/h (reduction of 88.3%) and the average loss from 3.33 to 0.271 $/h (reduction of 91.9%). The results for the relaxed solutions were not as good as the ones provided by BOPSOC, but they were better than using the nominal optimal operating point. Therefore, they can be used as a simplified way to reduce the loss without requiring much computational effort. Furthermore, BOPSOC provided better results than the use of linear combination of variables in this case study, which indicates that it is possible to reduce the process loss without increasing control system complexity. In the literature, the disturbance impact is not evaluated for SOC problems. However, it is clear that depending on its distribution, the control structure can be distinct. Thus, we evaluated the average and worst-case losses for each method using 100 disturbance realizations, according to different statistic distributions, using the structures described in Table 3. The distributions analyzed were: uniform, normal with average of 110 and standard deviation of 7, Poisson with lambda 4, and leptokurtic and platykurtic normal distributions. The results for each set of CVs and each distribution are presented in Table 4 for average loss and in Table 5 for worstcase loss. They show that the disturbance has stronger impact on average loss than on worst-case loss and that structures generated based on BOPSOC provide a better result even when compared to linear combination of variables. On the basis of the results, where the loss differed up to 90% when compared to uniform distribution, it is possible to corroborate that the statistic model used in disturbance optimization is an important definition of the procedure, and a different distribution may provide a new best set of CVs and setpoints. Another important decision is the best loss function for the process since the worst-case loss and the average loss may provide different sets of CVs, as shown in the example. Both approaches are important and should be used in different types of process. This decision should be made considering what the critical loss for the process is because this will be the optimization criterion for BOPSOC procedure. In the example, there is a set of CVs that minimizes the average loss and another set that is optimal for worst loss, what justifies the importance of this decision. Considering the worst-case problem, the cost and loss behaviors along the disturbance region are shown in Figures 4 and 5, respectively. These figures show that the selected set of controlled variables using BOPSOC provides better results when compared with all other methods. Moreover, it is shown that the relaxed procedures here proposed give slightly poorer results. This analysis may be made for an average loss case where similar

optimal operating point was calculated along the disturbance region and then CVs are chosen, denominated here by BOP → SOC, using eq 7. In the second case, the setpoints of the selected variables were reoptimized, denominated SOC → BOP, using eq 8. The best control structures obtained by each study are presented in Table 3, where “avg” was used for average loss and “worst” for the worst-case loss. Table 3. Comparison of Losses Provided by Different Methods method

CVs

setpoints

Lworst ($/h)

Lavg ($/h)

SOCworst and SOCavg

{F, Vr, Ya}

10.9

3.33

BOP → SOCavg

{Fr, Vr, Ya}

5.12

1.80

BOP → SOCworst

{Fr, Vr, Ya}

3.34

2.02

SOC → BOPavg

{Fr, Vr, Ya}

4.68

1.76

SOC → BOPworst

{Fr, Vr, Ya}

3.30

2.03

BOPSOCavg

{Cbr, Fr, Ya} {Car, Cd, F} {F, (14)}

{30.0; 2.82; 0.971} {3.25; 0.500; 0.954} {7.12; 0.500; 0.954} {4.14; 0.500; 0.957} {7.20; 0.500; 0.954} {0.0536; 20.43; 0.972} {1.78; 0.107; 25.3} {30.0; 0; 0}

2.63

0.271

1.27

0.936

2.02

0.875

BOPSOCworst CVlin

Additionally, we generated the best set of linear combination of measurements based on the method proposed by Ye,16 approximating the necessary conditions of optimality as controlled variables and keeping their values null. In this case, F was considered controlled at its optimal nominal value, and it used 1331 points uniformly distributed along disturbance and degree of freedom regions for data regression of controlled variables C1 = ∂J/∂Vr and C2 = ∂J/∂Va. All available measurements were used in the linear combinations, except the manipulated variables and the disturbance, resulting in the CVs of eq 14 with R2 of 0.8435 and 0.9001, respectively. For this structure, the losses are presented in Table 3. ⎧ C1 = 6.01Fin + 22.83Car − 55.63Cbr ⎪ + 13.44Fr − 215.0Ca − 89.36Cb ⎪ ⎪ ⎪ − 56.66Cc − 20.14Cd + 1.644 CVlin = ⎨ 2 62.12Fin + 90.88Car − 304.64Cbr = − C ⎪ ⎪ − 1.18Fr − 7.78Ca + 155.0Cb ⎪ ⎪ ⎩ + 354.6Cc + 925.2Cd + 0.4357 (14)

The results presented in Table 3 show that the operating point has a direct effect on the best process control structure obtained by SOC. In this case study, the different operating points provide

Table 4. Comparison of Average Loss for 100 Disturbance Realizations Using Different Statistic Distributions method

uniform

normal

Poisson

leptokurtic

platykurtic

SOCworst and SOCavg BOP → SOCavg BOP → SOCworst SOC → BOPavg SOC → BOPworst BOPSOCavg BOPSOCworst CVlin

4.14 2.29 2.49 2.24 2.51 0.263 1.14 0.690

1.33 1.97 2.20 1.89 2.21 0.122 1.15 0.326

4.31 2.79 2.49 2.62 2.50 0.132 1.18 0.618

2.06 2.26 2.28 2.14 2.30 0.149 1.16 0.386

1.25 1.96 2.19 1.89 2.21 0.080 1.15 0.282

G


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Industrial & Engineering Chemistry Research Table 5. Comparison of Worst-Case Loss for 100 Random Points Using Different Statistic Distributions method

uniform

normal

Poisson

leptokurtic

platykurtic

SOCworst and SOCavg BOP → SOCavg BOP → SOCworst SOC → BOPavg SOC → BOPworst BOPSOCavg BOPSOCworst CVlin

10.76 5.08 3.30 4.64 3.27 2.27 1.27 1.97

8.25 4.46 2.88 4.03 2.93 1.37 1.25 1.90

10.94 5.12 3.34 4.68 3.31 2.63 1.27 2.02

10.94 5.12 3.34 4.68 3.31 2.63 1.17 2.02

3.89 3.39 2.55 3.02 2.60 0.122 1.21 0.583

nominal operating point, just 2 sets of CVs were feasible for the entire disturbance region. When the setpoints are changed, this number rises to 32 sets. Thus, a list of “optimal candidates” can be built as shown in Table 6 for worst-case loss, and the decision Table 6. Controlled Variables Sets Rank for Analysis on the Basis of Worst-Case Loss number

CVs

setpoints

Lworst ($/h)

1 2 3 4

{Car, Cd, F} {Fr, Ca, Cd} {Cd, F, Ya} {Car, Fr, Vr}

{1.78; 0.107; 25.3} {19.7; 1.39; 0.104} {0.104; 28.9; 0.908} {1.74; 11.8; 0.699}

1.27 1.30 2.42 2.43

Figure 4. Cost along disturbance for the different procedures.

of the control structure can be taken by the engineer, considering not only Loss but also operating criteria. Figure 6 presents the loss curve for each set of CVs of Table 6. This comparison may be done for average loss with similar results.

Figure 5. Comparison of the loss along disturbance using the different operating points.

results are seen, expect for the BOPSOC curve that is located below CVlin curve most of time. On the basis of Figures 4 and 5, the improvement brought by the optimization of operating point is clear. The loss curves for the procedures here proposed are flatter when compared with that of the SOC, despite SOC providing a null loss in the nominal operating point. Even when compared to linear combination of measurements, BOPSOC provided a lower maximum loss. The proposed procedure not only provided smaller losses for the example studied but was also robust related to the change in active constraints. There are two changes in active constraints along the disturbance region for temperatures smaller than 105 °C in variables F and Vr. It is possible to see that this behavior increased the loss in the region, but it does not have significant effect for the set optimized for worst-case loss. It may indicate that a change in the operating point can fix some open issues in SOC methodologies generated when there are changes in active constraint of the process. Another consequence for changing the operating point is an increase in the number of sets of CVs that can keep constant setpoints for the entire disturbance region. While using the

Figure 6. Loss along disturbance for different sets of BOPSOC variables compared to SOC.

As shown in Figure 6, even when the second, third, and fourth sets of CVs are used, the results are better than when the nominal optimal values are used. Therefore, when some sets of variables are not available to be controlled due to operational issues, a change in the operating point may provide a larger rank of sets that may be used. An issue that may change the results of selected controlled variables and their setpoints is the discretization procedure of disturbance region. In the previous results, the disturbance region (temperature) was discretized using five discrete points. In this study, we evaluated the same procedure using a thinner and bigger mesh, using three and nine disturbance discrete points. The best controlled variables and their setpoints obtained by BOPSOC for each discretization are shown in Table 7 corroborating the impact of disturbance discretization in the BOPSOC results. H


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Industrial & Engineering Chemistry Research Table 7. Best Sets of Controlled Variables for Different Discretization

Table 8. Process Profit for Different Time Steps for Each Optimization Method

discretization

procedure

CVs

setpoints

time step

WO ($/h)

RTO ($/h)

SOC ($/h)

3 points 3 points 5 points 5 points 9 points 9 points

BOPSOCavg BOPSOCworst BOPSOCavg BOPSOCworst BOPSOCavg BOPSOCworst

{Cd, F, Ya} {Car, Cd, F} {Cbr, Fr, Ya} {Car, Cd, F} {Cbr, Cb, Vr} {Car, Cd, F}

{0.113; 25.09; 0.914} {1.78; 0.107; 25.3} {0.0536; 20.43; 0.972} {1.78; 0.107; 25.3} {0.107; 0.667; 1.227} {1.78; 0.107; 25.3}

4 10 30 50 70 100

−1.332 −1.464 −1.538 −1.554 −1.560 −1.565

−1.332 −1.464 0.651 1.229 1.490 1.731

1.196 1.497 1.530 1.535 1.537 1.537

4.2. Dynamic Analysis. The application of self-optimizing control may seem limited by the profit of a static RTO when it is analyzed just by the static loss. However, real processes are dynamic, and there is a nonstationary operation when the disturbance changes its value. During dynamic operation, the static RTO system does not change the operating point, and the operation is not optimal. However, SOC works during both steady-state and dynamic operation, keeping the controlled variable at their specified setpoints. Thus, SOC may provide better results along the time than a static RTO system, considering dynamic operation. To compare the behavior of both techniques during dynamic operation, the reactor temperature was changed according to a square wave with the following values: [110 120 130 120 110 100 90 100 110] °C. Each value is kept constant during the same time period, until the temperature has completed all values of the square wave. For example, for a time step of 3 h, the disturbance starts with a value of 110 °C and keeps this value for 3 h. After this time, the value is changed to 120 °C and is kept for the next 3 h. Thus, the procedure is repeated until the last value. More details of the procedure are presented in Schultz and Farenzena.21 The presented case study has an average time constant of 2.8 h, and Figure 7 shows the profit of the process along the time for

bring any improvement because the steady-state is never reached. This result is particularly interesting because a cheaper solution (SOC) can bring better or equivalent results when compared with a more expensive tool (RTO).

5. CASE STUDY−CUMENE PROCESS One of the main concerns of an optimization method is the viability of the method be applied in complex and large industrial units. Thus, we analyzed the application of SOC and BOPSOC in the cumene process, using the process described by Luyben.23 The plantwide control of this process was previously studied by Gera et al.,24,25 but the process analyzed was slightly different than the one studied by Luyben, with a distillation column instead of a flash vessel for separation of gas stream. The process flowsheet used in this work is the same as that presented by Luyben,23 where the feed is composed of a propylene and propane stream which is mixed with a benzene stream. The main product is a stream composed of cumene 99.9% pure, and there are other two product stream, GAS and B2, with a reduced commercial value. The process consists of PFR where two reactions take place: a reaction of propylene and benzene, generating cumene, and a second between cumene and propylene, generating isopropylbenzene (PBI). Then, the reactor is followed by a separation section composed of a flash tank and two distillation columns. The top product of column C1 is composed mainly of unreacted benzene, and it is mixed to feed streams as a reflux. The equipment size, kinetics data, and economic values were taken from Luyben.23 The process was modeled using Aspen Plus, according to the flowsheet presented in Figure 8. It is required an extensive number of simulations to find the optimal points and for evaluating the loss along disturbance region. For this reason, we chose to create a black box model for the unit using neural networks. The neural networks were created in Matlab R2009 and used the data generated in a set of Aspen Plus simulations for the training. In this case study, we considered that the process has five degrees of freedom for optimization, of which two are related to the first column, two are related to the second column, and the fifth is the reactor temperature (Treactor). For the columns, we used the boiler heat (QrebC1 and QrebC2) and reflux ratio (RRC1 and RRC2) as freedom degrees. We considered that benzene fresh flow rate is fixed 99.81 kmol/h and that there are two disturbances in process: the flow rate of fresh C3 (FreshC3), with values between 105 and 115 kmol/h, and its composition (Zf reshpropyl), whose molar concentration of propylene varies from 0.92 to 0.98%. The profit of the process was calculated by eq 15, where the economic variables and their values are described in Table 9 and process variables are described in Table 10:

Figure 7. Profit along the time for time step of 50 h.

time steps of 50 h. It represents the behavior of three different situations: WO, for the process without any optimization technique where the degrees of freedom are kept at optimal nominal operating point; RTO, for the process with a static optimization every time the process achieves a steady-state operation; and SOC, where we controlled the best set of CVs obtained by BOPSOC for worst-case loss optimization. The average process profit for different time steps are shown in Table 8. Details of this implementation are discussed by Schultz.22 Table 8 shows that SOC provides better profit for faster disturbances dynamics, whereas static RTO has better results for slower step times. In the first step time (4 h), the RTO does not I


Article


Figure 8. Flowsheet of cumene process modeled in Aspen Plus.

Table 9. Economic Variables Used in Profit Function

Table 10. List of Variables of Profit Function

variable

description

values

unit

variable

description

unit

$cumpure $C3 $benzpure $benz $cum $propyl $propa $PBI $Eger

commercial value of cumene 99.9% commercial value of C3 stream commercial value of pure benzene commercial value of impure benzene commercial value of impure cumene commercial value of impure propylene commercial value of impure propane commercial value of impure PBI commercial value of generated energy in the reactor commercial value of high pressure steam used in the process commercial value of electrical energy used in the process

132.5 34.30 68.60 27.90 26.60 10.40 10.90 36.50 27.91

$/kmol $/kmol $/kmol $/kmol $/kmol $/kmol $/kmol $/kmol $/Gcal

41.13

$/Gcal

70.29

$/Gcal

Fprod FreshC3 Freshbenz Fgas FB2 Zgasbenz Zgascum Zgaspropy Zgaspropa ZgasPBI ZB2cum ZB2PBI Qreactor Qboiler QrebC1 QrebC2 QHX1 QHX2

flow rate of main product generated flow rate of fresh C3 in the feed stream flow rate of fresh benzene in the feed stream flow rate of gas that leaves the flash vessel flow rate of the bottom of C2 molar fraction of benzene in gas stream molar fraction of cumene in gas stream molar fraction of propylene in gas stream molar fraction of propane in gas stream molar fraction of PBI in gas stream molar fraction of cumene in B2 stream molar fraction of PBI in B2 stream generated heat in reactor boiler duty reboiler duty of C1 reboiler duty of C2 heat exchanged in HX1 heat exchanged in HX2

kmol/h kmol/h kmol/h kmol/h kmol/h kmol/kmol kmol/kmol kmol/kmol kmol/kmol kmol/kmol kmol/kmol kmol/kmol Gcal/h Gcal/h Gcal/h Gcal/h Gcal/h Gcal/h

$HP $EE

P = FProd· $Cumpure − FreshC3· $C3 − Freshbenz · $ benzpure + FGas(Zgas benz · $ benz + Zgascum · $cum + Zgaspropy · $propy + Zgaspropa · $propa + ZgasPBI ·$PBI ) + FB2(ZB2cum ·$cum + ZB2 PBI · $PBI ) + Q reactor· $Eger − (Q boiler + Q rebC1 + Q rebC2)

1.01 Gcal/h ≤ Q rebC2 ≤ 1.67 Gcal/h

(19)

$HP − (Q HX1 + Q HX2)$EE

0.2 ≤ RR C2 ≤ 1.2

(20)

Zprodcum ≥ 0.999

(21)

(15)

The first step in SOC procedure consists in finding the optimal operating point of the process for the nominal disturbance. In this study, we considered as nominal disturbance the flow rate of fresh C3 of 110 kmol/h with a molar fraction of propylene of 0.95%. The constraints of optimization are described by eqs 16−21 where Zprodcum is the molar fraction of cumene in the final product stream. The optimization procedure was solved using Matlab R2009, using the ga function, based on genetic algorithm. Table 11 shows the nominal optimal point of the process. 360°C ≤ Treactor ≤ 390°C

(16)

1.31 Gcal/h ≤ Q rebC1 ≤ 1.97 Gcal/h

(17)

0.37 ≤ RR C1 ≤ 1

(18)

Initially, we studied 46 process variables as candidate CVs plus the degrees of freedom, but after a prescreening, we excluded the concentrations with null values or values close to zero because it Table 11. Optimal Nominal Point of the Process

J

variable

nominal optimal value

Treactor QrebC1 RRC1 QrebC2 RRC2 Zprodcum Profit

371.3 °C 1.40 Gcal/h 0.37 1.11 Gcal/h 0.71 0.999 3407.5 $/h DOI: 10.1021/acs.iecr.5b02044 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article


constraint for all disturbances, but RRC1 is active for just 65% of the region. In this case, it is clear that Zprodcum must be controlled, but as discussed before, RRC1 may not be the best option. Therefore, in this study just Zprodcum will be considered controlled, leaving four degrees of freedom available for SOC. The procedure for selecting variables was accomplished in two steps, where in the first we used local loss expressions10,14 to evaluate the loss for all possible combination of CVs and in the second we evaluated the loss for the best sets using a nonlinear model in 25 disturbances values uniformly distributed. The possible set of CVs was divided in two groups, where in the first group were the variables related to the process until column C1, with three degrees of freedom available, and the second group were the variables of column C2, with just one degree of freedom available, considering Zprodcum controlled at 0.999. This group division was made to avoid the control of variables of C2 manipulating variables that are not related to this column and to avoid the impossible control of variable in the reflux loop by manipulation of C2 degrees of freedom. We considered an implementation error of 1% for all variables during the entire procedure. The best set of CVs using SOC method was composed of Treactor, RRC1, F6, RRC2, and Zprodcum with an average loss of 2.23% and a maximum loss of 6.90%. This set of CVs showed the best results for mean loss and for worst-case loss. It is important to highlight that even reflux ratio of C1 was not considered controlled because of nominal active constraint; its selection provided a smaller loss than other variables. Considering that BOPSOC method requires many more calculations than when a preselected operating point is used, it was necessary to improve the prescreening of the candidates. This was done on the basis of the analysis of the behavior of optimal value of each candidate for all disturbances values. We calculated the standard deviation of the optimal values divided by its mean optimal values, and we chose the candidates that presented a minor variation of its optimal point. The new set of candidates is T15, F15, T14, T16, F6, Zref benz, F22, T22, Z22cum, Fprod, Tprod, TB2, and T18C2. Additionally, the degrees of freedom were included as candidates. Considering the groups previously mentioned, there are 12 candidates for three degrees of freedom of part 1, and 6 candidates for the degree of freedom of part 2. We analyzed all possible combinations of candidates using the average optimal operating point of each variable as setpoint using de nonlinear model. After a first evaluation, the setpoints were optimized for worst-case loss and mean loss using eq 8, and the result is presented in Table 13 with a comparison to the nominal point: As shown in Table 13, it was possible to reduce the worst-case loss by 53.8% and the average loss by 28.7% via changing the operating point and finding a better set of CVs. The values of the loss along disturbance region for SOC and BOPSOC are shown in Figures 9 and 10, respectively. Note that the structure obtained by nominal operating point keeps the loss around 3%, reducing this value just near the middle of the graph, whereas BOPSOC keeps the loss less than 2% for almost the entire region.

is difficult to measure and control this type of variable and because they may not respond to manipulated variables when their values achieve the null value. The list of variables and their optimal nominal values are shown in Table 12. Table 12. List of Variables of Cumene Process name T15 F15 T14 Z14benz Z14cum T16 QHX1 Qboiler QHX2 T6 F6 Z6propa Z6benz Fgas F20 Z20benz Z20cum Fref ZRef benz ZRef propy Tref QcondC1 T7C1 F22 T22 Z22cum Fprod FB2 Tprod TB2 QcondC2 DFC2 T18C2 Zprodcum

nominal point

variable

unit

temperature of stream 15 flow rate of stream 15 temperature of stream 14 molar fraction of benzene in stream 14 molar fraction of cumene in stream 14 temperature of stream 16 heat of HX1 heat of vaporizer (boiler) heat of HX2 temperature of stream 6 flow rate of stream 6 molar fraction of propane in stream 6 molar fraction of benzene in stream 6 flow rate of GAS stream flow rate of stream 20 molar fraction of benzene in stream 20 molar fraction of cumene in stream 20 flow rate of reflux stream molar fraction of benzene in reflux stream molar fraction of propylene in reflux stream temperature of reflux stream condenser duty of C1 temperature of plate 7 of C1 flow rate of stream 22 temperature of stream 22 molar fraction of cumene in stream 22 flow rate of product stream flow rate of B2 stream temperature of product stream temperature of B2 stream condenser duty of C2 ratio distillate−feed of C2 temperature of plate 18 of C2 molar fraction of cumene in product stream

°C kmol/h °C kmol/kmol

421.4 304.8 423.6 0.438

kmol/kmol

0.490

°C Gcal/h Gcal/h Gcal/h °C kmol/h kmol/kmol

275.1 −0.539 3.131 −2.640 33.3 297.5 0.352

kmol/kmol

0.590

kmol/h kmol/h kmol/kmol

10.43 189.9 0.375

kmol/kmol

0.568

kmol/h kmol/kmol

89.9 0.930

kmol/kmol

0.0617

°C Gcal/h °C kmol/h °C kmol/kmol

58.12 −1.03 116.3 100.3 177.0 0.964

kmol/h kmol/h °C °C Gcal/h kmol/kmol °C kmol/kmol

85.83 3.560 152.0 207.3 −1.307 0.960 213.2 0.999

At the optimal nominal operating point, this unit has two active constraints: Zprodcum and RRC1. Thus, the disturbance region was discretized uniformly in 25 points, 5 in each direction, to analyze the behavior of the optimal operating point along the disturbance region. It was seen that Zprodcum is an active Table 13. Comparison of Losses Provided by BOPSOC and SOC method

CVs

setpoints

Lworst (%)

Lavg (%)

SOC BOPSOC

Treactor, RRC1, F6, RRC2, and Zprodcum QrebC1, T15, Z22cum, RRC2, and Zprodcum

{371.3; 0.37; 297.5; 0.71; 0.999} {1.625; 422.0; 0.945; 1.00; 0.999}

6.90 3.19

2.23 1.59

K


Article


82.3% and the mean loss by 91.9%. The number of possible sets of CVs that can keep constant setpoints along disturbance regions was increased from 2 to 32. A second study case based on a cumene process was also considered. It was necessary to reduce the number of candidate variables by analyzing the ones that keep their optimal values almost unchanged along disturbance region, reducing the number of possibilities to be evaluated. By using a different operating point, it was possible to reduce the worst-case loss by 53.8% and the mean loss by 28.7%, corroborating the gain of the ideas here proposed and feasibility of application in larger study cases.

■

Figure 9. Loss along disturbance region for SOC.

APPENDIX A

⎛ 9758.3 ⎞⎟ k1 = k 2 = 1.2870 × 1012 exp⎜ − ⎝ T + 273.15 ⎠

(22)

⎛ 8560.0 ⎞⎟ k 3 = 4.5215 × 109 exp⎜ − ⎝ T + 273.15 ⎠

(23)

Fin· Cain + Fr· Car − F ·Ca − Vr(k1·Ca + 2· k 3· Ca2) = 0 (24)

Figure 10. Loss along disturbance region for BOPSOC.

Fr· Cbr + F · Cb − Vr(k 2·Cb − k1·Ca) = 0

(25)

F · Cc − Vr·k 2·Cb = 0

(26)

F · Cd − Vr· k 3· Ca 2 = 0

(27)

F = Fr − Fin

(28)

Fr(Ca + Cb + Cc + Cd) = F(Car + Cbr )

(29)

Car(1 − Ya) = Ya · Cbr

(30)

Car·Fr = Ca ·F

(31)

Za =

6. CONCLUSIONS The main contribution of this paper is an analysis of the impact that the operating point has during the CVs selection procedure using SOC. A different optimization procedure was proposed to determine the best operating point for SOC where setpoints and CVs are computed together in the same optimization problem (BOPSOC). This new approach may provide different sets of controlled variables, reducing significantly the average and worstcase losses and increasing the number of CVs that can be kept with constant setpoints for all the disturbance values. It also had a good performance when there were changes in active constraints of the process, indicating that a change in the operating point may handle this problem; however, a more rigorous study is required. The main disadvantage of the proposed method is the computational effort required to solve the resultant MINLP problem. Simplified formulations were also proposed for changing the operating point aiming at loss reduction along the disturbance region. The simplified techniques allow an intermediate gain compared to BOPSOC, but can reduce the process loss with almost the same computational effort requirements for the traditional SOC method. A reactor−distillation case study was used to evaluate the gains of changing the operating point, reducing the worst-case loss by

Ca Ca + Cb + Cc + Cd

(32)

⎛ 10−5 ⎞ Qr = F ⎜0.141 + 0.448Za(Ya − 1.09) + 1.679 ⎟ 1 − Ya ⎠ ⎝

■

(33)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

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The Importance of Nominal Operating Point Selection in Self

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