An introduction to a dynamic plant-wide optimization strategy for an integrated plant

Computers and Chemical Engineering 29 (2004) 199–208

An introduction to a dynamic plant-wide optimization strategy for an integrated plant Thidarat Tosukhowonga , Jong Min Leea , Jay H. Leea,∗ , Joseph Lub a

School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, 311 Ferst Dr., Atlanta, GA 30332-0100, USA b Honeywell Hi-Spec Solutions, 16404 N. Black Canyon Highway, Phoenix, AZ 85053, USA Received 8 August 2003; received in revised form 22 March 2004; accepted 20 July 2004 Available online 11 September 2004

Abstract In this paper, we develop a dynamic real-time optimization (RTO) strategy for an integrated plant. The steady-state assumption in conventional RTOs severely limits frequency of optimization and precludes the use of dynamic degrees of freedom available in the plant (e.g. storage capacities), resulting in suboptimal economic performances. Other approaches that attempt to synchronize the frequency of the plant-level optimization to that of the local-unit-level model predictive controllers can be sensitive to local disturbances and model uncertainty in highfrequency parts of plant dynamics. We reason that a logical middle ground is to perform a dynamic optimization but at a rate significantly lower than the model predictive controllers in order to keep the modeling and computational requirements at a reasonable level for the optimization. We discuss the obtaining of a reduced-order model for a chosen optimization frequency and the interfacing of the real-time optimizer with unit controllers. Two examples are given to compare the various approaches. © 2004 Elsevier Ltd. All rights reserved. Keywords: Real-time optimization; Plant-wide control; Dynamic optimization; Economic optimization

1. Introduction With ever increasing need for improved process economics, efficiency, and quality in the globalized market environment, real-time optimization (RTO) has attracted the attention of process industries and has been adopted widely (Cutler & Perry, 1983). A typical RTO system is modelbased and implemented on top of unit-based multivariable controllers. The objective is to maintain the plant operation near an economic optimum in the face of disturbances and other external/internal changes. This RTO layer functions between the production planning, scheduling layer and the local control layer. Conventional RTO strategy is based on a steadystate model of the plant and calculates setpoints under constraints of controlled and manipulated variables of the multivariable controllers for various plant units, which then steer their respective units to the calculated steady-state conditions. ∗

Corresponding author. Tel.: +1 404 385 2148; fax: +1 404 894 2866. E-mail address: [email protected] (J.H. Lee).

0098-1354/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2004.07.028

While the steady-state model based RTO formulation is the current standard, most integrated plants have very long transient dynamics, lasting as long as several days, due to the presence of recycle loops, transportation delays, and large intermediate storage capacities. In such cases, the standard formulation can be extremely limited. First, once a change occurs, it may take a very long time for the plant to reach the new steady state, thus limiting execution frequency of the RTO. In fact, most plants would seldom be in steadystate conditions because additional changes would occur in the meantime. Second, optimal operating conditions calculated from a steady-state model may be suboptimal or even infeasible at the local units due to the transient dynamics, unit interactions, model errors, and disturbances. Third, dynamic degrees of freedom, such as those present in storage capacities of various units, may be left unexplored leading to suboptimal dynamic solutions. To overcome drawbacks exist in the steady-state RTO, many researchers have suggested the use of a dynamic RTO performing at a rate same as local unit controllers. One such

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approach is to combine the RTO and model predictive control (MPC) into a single layer by adding an economic objective function to the control objective function of MPC in a weighted average manner (Gouvˆea & Odloak, 1998; Vermeer, Pederson, & Canney, 1997). Nevertheless, the computational effort increases with the size of the problem, and the use of this idea for large-scale plant-wide dynamic RTO seems impractical. To reduce the computational requirement of the full-fledged dynamic RTO, Lu (2001) suggested a single-point dynamic RTO scheme, in which the plant-wide optimization is performed at some predicted future point (called ‘optimization point’), which may not necessarily be a steady state point. This approach has many similarities to the popular ‘coincidence-point’ approach for model predictive control (MPC), in which desired controlled variable values are specified at a single time point in the prediction horizon. The coincidence-point approach is known to have some disadvantages in terms of performance and stability. Since the optimization is performed at a high rate, we can expect that this method too will be sensitive to model errors in high frequency parts of plant dynamics, which are inevitable in practice. In fact, for many plants with large material and energy recycle loops, of which models tend to be very “stiff” and therefore ill-conditioned, it will be very difficult to obtain sufficiently accurate plant-wide models up to the execution frequency of MPCs. Motivated by this, we propose a logical middle ground wherein RTO based on a reduced-order slow-scale dynamic model of a plant is performed at a rate significantly lower than local-unit-level MPC controllers in order to dynamically track changes in optimal operating conditions. Such an approach would be more manageable from the modeling viewpoint because a slower rate model, which describes only dominant slow modes, should be better conditioned and easier to identify. It is also reasonable from a practical viewpoint since most changes relevant to plant economics as well as plant-wide interactions (e.g., those due to recycle loops) are low-frequency dominant in nature. The suggested method performs RTO at a frequency significantly lower than those of the local MPCs, but it does not have to wait for a plant to reach a steady state. This paper is organized as follows. In Section 2, the construction of a slow-scale plant-wide model and the interfacing of the RTO with the multivariable control layer are discussed. Section 3 illustrates a comparison study of different RTO strategies applied to two different examples: (A) an interactive plant involving two process units and (B) a reactor– storage–separator system connected via a large material recycle loop. Finally, this paper is concluded in Section 4.

on various factors, such as the bandwidths of RTO-relevant changes and interaction dynamics, expected accuracy of the plant-wide dynamic model in various frequency ranges, and computational feasibility. For example, one may obtain the bandwidth of the process dynamics by examining the eigenvalues of the system around a steady state. Those eigenvalues lying far away from zero correspond to fast modes of the process, and should be of interest in the unit-based control level, but not in the RTO. In contrast, those eigenvalues that are close to zero correspond to slow modes, which should be considered by the plant-wide optimizer. Therefore, one may choose a frequency that marks a time-scale separation as indicated by a large magnitude difference between two adjacent eigenvalues. Once the frequency of RTO is decided, one must develop a plant-wide model accurate within the chosen frequency range. This may be done using fundamental constitutive equations or using system identification. In the former case, one usually gets a very large set of differential algebraic equations (DAEs), which tend to be very stiff. Removing the illconditioning of such a model using the singular perturbation approach has been discussed in the literature (Kumar & Daoutidis, 1999), but the procedure can be extremely complicated for a large scale nonlinear dynamic system for which state variables are not explicitly separable in terms of time scales. A more practical scenario is to use linear/linearized models used in local MPCs and connect them through some ‘bridge’ dynamics to formulate a plant-wide model. The linear plant-wide model so obtained can be reduced through the procedure of residualization or a frequency-weighted model reduction technique (FWMR) (Enns, 1984), which minimizes the model truncation error within the chosen frequency range, i.e. ||Wo (G − Gr )Wi ||∞

(1)

where G is the original model, Gr is the reduced model, Wo and Wi are output and input weighting matrices, respectively. These weighting matrices consist of low-pass filters to remove the dynamics that are faster than the optimization frequency, ωopt = 2π/Topt , where Topt is the RTO optimization interval. Details of the procedure for FWMR can be found in Zhou, Doyle, and Glover (1996). Note that in the case of a highly nonlinear system that cannot be well approximated by a linearized model, nonlinear model reduction techniques such as the empirical eigenfuction method (Holmes, Lumley, & Berkooz, 1996; Bendersky & Christofides, 2000) coupled with residualization may be employed to reduce the system order.

2. Dynamic optimization using slow-scale model 2.1. Rationale and slow-scale model construction

2.2. Slow-scale dynamic optimization architecture

Choosing an appropriate dynamic optimization frequency is a critical step in this method. It should be decided based

Once a slow-rate reduced-order model is obtained, the dynamic optimization problem in Eq. (2) is solved at every

T. Tosukhowong et al. / Computers and Chemical Engineering 29 (2004) 199–208

Topt interval.

j

minf (Ug ) Ug

(2)

Ymin ≤ Y(k + 1|k) ≤ Ymax Umin ≤ Ug ≤ Umax

where Ug = [ug (k), ug (k + 1), . . . , ug (k + M − 1)]T denotes the global input vector with the control horizon M, which is constrained within lower and upper bounds Umin and Umax , respectively. Y(k + 1|k) = [y(k + 1|k), y(k + 2|k), . . . , y(k + P|k)]T is the predicted output vector of horizon P obtained by solving the model equation, and constrained within lower and upper bounds Ymin and Ymax , respectively. In practice, a RTO solution may not be feasible for a local controller due to model errors, local disturbances, or the fact that the local controller does not know future interactions from the other units. In light of this, Lu (2001) suggested passing the global setpoint to the following least-square coordination collar to find for each MPC a locally feasible setpoint closest to the global solution in a least-square sense:
j min [uls (k) − ugj ]2 j

uls (k) j

j

(3)

j

ymin ≤ yls ≤ ymax

j

j

201

j

umin ≤ uls (k) ≤ umax

RTO calculated for the jth local unit, and uls (k) is a vector of MVs recomputed by the coordination collar. As this coordination layer checks for feasibility of the setpoint at the end of MPC prediction horizon, the gain matrices in Eq. (4) are for the end of MPC prediction horizon. Note that it is also possible to formulate Eq. (3) to find the feasible CV values closest to the RTO solution. The choice of the minimized variables should be based on the given plant-wide objective. Furthermore, execution frequency of the coordination collar should be chosen according to dynamic interactions among the unit operations. That is, when the process interactions are rather fast, the coordination collar may need to be executed at a higher rate than the RTO. Nevertheless, in many cases the process interactions tend to be slow and the coordination collar can be executed at a same rate as the RTO. Architecture of the proposed scheme is illustrated in Fig. 1, where the feasible setpoints closest to the RTO solutions are sent from the coordination collar to each local MPC. At each MPC sample time, the MV profile is calculated and the first move is implemented. Then, the plant output is measured for MPC prediction update. With some appropriate filtering, this information is also used to update the state vector for the plant-wide optimizer at each MPC sample time. This process is repeated until the next execution time of the RTO is reached when the plant-wide state vector is then sampled and used in the optimization.

where j

j

j

yls = Gxj xj (k) + Guj uls (k) + Gd dj (k)

(4)

The superscript j denotes the index of local unit, and the subscript ls signifies that the vectors are computed by the j least-square coordination layer. Here, ug is the setpoint the

3. Illustrative examples In this section we provide two examples to compare performances of the conventional steady-state RTO, the singlepoint dynamic RTO suggested by Lu, and the proposed slow-

Fig. 1. Architecture of the slow-scale plant-wide dynamic RTO.

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Fig. 2. Transfer function model for example 1.

scale dynamic RTO scheme. The first example considers the process studied by Lu (2001), which has two units connected via bridge dynamics. Then, the second example involves a reactor, a storage tank, and a flash tank connected via a large recycle stream. This system is very stiff, and therefore adds further complexity to the control and optimization calculations. 3.1. Example 1 The transfer function model of this problem is given in Fig. 2. Each unit has two controlled variables (CVs), two manipulated variables (MVs), and one disturbance variable (DV). The controlled variables are denoted as CVij, where i represents the first or second unit operation and j indicates whether it is the first or the second variable in that unit. Similar indexing schemes are used for the MVs and the DVs. At the initial steady state, all variables are 0. The simulation begins at time 0 with the objective of maximizing the sum of CV12 and CV22 and increasing the setpoint of CV11 to 1 while maintaining the setpoint of CV21 at 0. 3.1.1. Unit-based control layer At the local-unit control level, an MPC controller was built for each unit. Its parameters are given in Table 1, where p is the prediction horizon, m is the control horizon, Q and R are weighting matrices for CVs and MVs, respectively. The local MPCs compute control actions every 1 min. 3.1.2. Plant-wide optimization layer Given the eigenvalues around the steady state of −0.0370, −0.04, −0.0455, −0.05, −0.05, −0.0667, −0.0667, −0.0714, −0.0833, −0.0833, −0.1, −0.1, −0.2, −0.2, −0.2, −0.25, −0.25, and −0.3333, the largest frequency difference

happens in the range between 0.1 and 0.2 rad/min, which corresponds to the time period around 30–60 min. Since this system model is rather simple and will not require a high computational load, we can choose shortest time period of 30 min as optimization interval. Any higher frequency dynamics were filtered by the weighting matrices Wo and Wi in Eq. (1), of which each diagonal element was a second order Bessel filter of the following form: ω2 (s/ωB )2 + 2ζω(s/ωB ) + ω2

(5)

where ω = 1.27, ζ = 0.87, and ωB = 0.21. The original model had 13 states and the FWMR scheme reduced the original system to an approximate model with 5 states. We used the prediction and the control horizons of 4 and 1, respectively. The economic optimization problem was formulated as a linear program (LP) of the following: max

MVij (k)

4


[CV12(k + ) + CV22(k + )]

(6)

=1

subject to −10 ≤ CVij(k + |k) ≤ 10  = 1, . . . , 4 −10 ≤ MVij(k|k) ≤ 10

(7)

Table 1 Parameters used in local MPCs for example 1 Unit

p

m

umax

uij

yij

1

20

10

0.3

[−10, 10]

[−10, 10]

2

100

20

0.3

[−10, 10]

[−10, 10]

Q  100 0  100 0

0 1 0 1



R 





30 0 0 30 30 0 0 30

 

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The calculated MV setpoints from the optimizer were sent to the coordination layer for feasibility check. Then, locally feasible output setpoint trajectories were generated from Eq. (4) and sent to the MPC. In addition, to compare economic performances of the different RTO schemes, the performance measure defined by Eq. (8) was used. E=

200


CV12(t) + CV22(t)

(8)

t=1

3.1.3. Simulation result The steady-state RTO executed only once at the initial steady state yielded a performance of 3281 as measured by Eq. (8). As the execution rate was increased to once every 30 min without waiting for the system to reach steady states, a worse performance of 3199 resulted. This is possible because there exists a mismatch between the steady-state model prediction and actual dynamic behavior. Therefore, it did not help running the steady-state RTO faster. For the single-point dynamic RTO scheme, the simulation plots are shown in Figs. 3 and 4, where solid lines and dotted lines represent MPC predictions and setpoint trajectories sent to the MPCs, respectively. Once the optimizer had been turned on, it optimized the future output based on a prediction at one future point and decided to increase CV12 and CV22 to their maximum values immediately. This demanded aggressive MV movements in unit 1 (as indicated by quick setpoint changes of MV11 and MV12 to −3.5 and +6.5, respectively), which later became large disturbances to unit 2 and decreased the value of CV22 during the time period of 60–150 min. On the other hand, the simulation results of the slow-scale dynamic RTO depicted in Figs. 5 and 6 show more gradual changes in the setpoints of CV12 and CV22 to minimize the adverse interactions between the units, and hence gave the best performance. Economic performances of the different RTO schemes are summarized in Table 2.

that the volatility of component 1 is much higher than that of component 2, and component 3 is nonvolatile. Hence, most of reactant 1 goes up the overhead, where it is completely condensed into liquid and sent back to the reactor. Since it is important to keep the selectivity of component 2 high, a single-pass conversion has to be kept low. High yields can still be achieved by maintaining a high ratio of the recycle flow to the fresh feed flow (D/F0 ). This is a challenging control problem as the system can exhibit a severe snowball effect (Luyben, Tyr´eus, & Luyben, 1999) if not properly controlled. That is, only a small change in the feed stream can cause a large variation in the process, especially when the recycleto-feed ratio is very high. For simplicity we assume a constant liquid density in every vessel and an isothermal operation for the entire process. Under this circumstance, the material balance consists of 12 equations as follows: ˙R = H

1 (F0 + D − FR ) ρAR

x˙ 1R =

F0 (x10 − x1R ) + D(x1D − x1R ) − k1 x1R ρAR HR

x˙ 2R =

−F0 x2R + D(x2D − x2R ) + k1 x1R − k2 x2R ρAR HR

x˙ 3R =

−(F0 + D)x3R + k2 x2R ρAR HR

˙M = H

1 (FR − FM ) ρAM

x˙ 1M =

FR (x1R − x1M ) ρAM HM

x˙ 2M =

FR (x2R − x2M ) ρAM HM

x˙ 3M =

FR (x3R − x3M ) ρAM HM

˙B = H

1 (FM − B − D) ρAB

x˙ 1B =

1 [FM (x1M − x1B ) − D(x1D − x1B )] ρAB HB

x˙ 2B =

1 [FM (x2M − x2B ) − D(x2D − x2B )] ρAB HB

x˙ 3B =

1 [FM (x3M − x3B ) + Dx3B ] ρAB HB

3.2. Example 2 Next, we consider a process with a CSTR, a storage tank, and a flash tank with a material recycle stream shown in Fig. 7. A fresh feed stream F0 consisting of pure component 1 is fed k1

k2

to the reactor, where two irreversible reactions 1→2→3 take place to produce a desired product 2 and an undesired product 3 at rates k1 = k2 . The reactor effluent stream FR consisting of components 1, 2, and 3 enters the storage tank, of which the downstream flow, FM , leads to the flash tank. We assumed Table 2 Economic performances of different RTO strategies applied to example 1 Strategy

E

One execution of the Steady-state RTO steady-state RTO executed every 30 min interval Single-point dynamic RTO Slow-scale dynamic RTO

3281 3199 3309 3570

203

where HR , HM , and HB denote the liquid levels in the reactor, the storage tank, and the flash tank, respectively. Here, xij de-

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Fig. 3. Output (top) and input (bottom) variables of unit 1 in example 1 using the single-point dynamic RTO (solid lines: measured outputs, dotted lines: setpoint trajectories).

Fig. 4. Output (top) and input (bottom) variables of unit 2 in example 1 using the single-point dynamic RTO (solid lines: measured outputs, dotted lines: setpoint trajectories).

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205

Fig. 5. Output (top) and input (bottom) variables of unit 1 in example 1 using the slow-scale dynamic RTO (solid lines: measured outputs, dotted lines: setpoint trajectories).

Fig. 6. Output (top) and input (bottom) variables of unit 2 in example 1 using the slow-scale dynamic RTO (solid lines: measured outputs, dotted lines: setpoint trajectories).

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Fig. 7. Schematic of the reaction-storage-separation network in example 2.

notes the molar liquid fraction of component i (i = 1, 2, 3) in the stream j (j = 0, R, M, B, D for feed, reactor, storage tank, flash tank, and recycle, respectively). The nominal values of the process and operating parameters are given in Table 3. The test scenario is to increase the production throughput by 20%, while keeping the product compositions and operating conditions within the constraints given in Table 4. In this problem, there are five MVs, including F0 , FR , FM , B, and D. However, as the liquid levels behave as integrators, some of these streams must be used to stabilize the levels. According to Richardson’s rule (Luyben et al., 1999), the largest stream should be selected to control the liquid level in the vessel. However, if we select FR , FM , and D to control the levels of the reactor, the storage tank, and the flash tank, respectively, the three levels are not independently controllable as the MVs are all internal flow variables. Instead, we used FR , FM , and B to stabilize the levels through P-only controllers. Although these flows are no longer available as manipulated variables for the plant-wide optimizer, the degree of freedom remains the same as the level setpoint of each vessel can be used as a MV. The following subsections provide more details on the MPC and plant-wide optimization formulations, which were obtained from the linearized model of the nonlinear plant in order to keep our strategy tractable and computationally manageable. Table 3 Nominal values for the process and operating parameters for example 2 Parameters

Value

Liquid density Volatility Rate constant Vessel area Vessel holdup Flowrate (h−1 )

ρ=1 α1 = 90 k1 = 0.0167 AR = 5 HR = 20 F0 = 1.667 B = 1.667 x10 = 1.00 x1R = 0.8861 x1M = 0.8861 x1B = 0.1139 x1D = 0.9295 KC,R = −10

Mole fraction

P-controller gains

3.2.1. Unit-based control layer The flowsheet was divided into two process units: unit 1 consists of the reactor and the intermediate tank, whereas unit 2 includes the flash tank. An MPC controller was built for each unit in order to steer the CVs to the setpoints, which are specified by the plant-wide optimizer, while respecting the constraints. A list of output and manipulated variables of each unit as well as their constraints are given in Table 4. The sample times of both MPCs are 6 min. As the transient dynamics last as long as 12 h, we used the model predictive control formulation for integrating dynamics (Lee, Morari, & Garcia, 1994; Lundstr¨om, Lee, Morari, & Skogestad, 1995), which allows the step-response models to be truncated well before the responses settle with little sacrifice in accuracy. The parameters for both MPCs are given in Table 5, where ttrnc is the truncation time of the step-response model, p is the prediction horizon, m is the control horizon, &y and &u are output and input weighting matrices, respectively. Note that in the MPC optimization problem, the output constraints were implemented as soft constraints to avoid computational  infeasibility. Nevertheless, a large penalty term (equal to 106 '2i , where 'i is a slack variable representing the magnitude of violation of the ith output) was added to the objective function to avoid output constraint violations. Table 4 Output and manipulated variables of unit 1 and 2 Unit 1 Output Variables Operating range

αB = 1 k2 = 0.0167 AM = 10 HM = 20 FR = 31.33 D = 29.67 x20 = 0 x2R = 0.1082 x2M = 0.1082 x2B = 0.7779 x2D = 0.0705 KC,M = −10

AB = 5 HB = 20 FM = 31.33 x30 = 0 x3R = 0.0058 x3M = 0.0058 x3B = 0.1082 KC,B = −5

Unit 2 Output Variables Operating range

1 2 3 4 5 6 7 8

x1R x2R x3R x1M x2M x3M FR FM

MV 1 2

Variables Operating range MV HR [10, 30] 1 HM [10, 30] 2

[0,1] [0,0.15] [0,0.02] [0,1] [0,0.15] [0,0.02] [8,47] [8,47]

1 2 3 4

x1B x2B x3B B

[0,0.15] [0.75,1] [0,0.15] [0.67,3]

Variables Operating range HB [10, 30] D [8, 45]

T. Tosukhowong et al. / Computers and Chemical Engineering 29 (2004) 199–208 Table 5 Parameters for local MPCs ttrnc

p

m

umax

MPC 1 8h

40

10

[0.3; 0.3]

MPC 2 8h

&y

 40

10

[0.04; 0.5]

0 0  0 0

the closest feasible MV setpoints. Then, the output setpoints j (ˆyls ) were computed from Eq. (4) and sent to the MPCs.

&u

  3 for x1M , y &ij = 7 for x2M ,  0 otherwise 0 1 0 0

0 0 0 0

 0 0  0 3





10 01

30 01





3.2.2. Plant-wide optimization layer Given the eigenvalues of −0.0097, −0.0167, −0.0167, −0.1567, −0.1567, −0.3133, −0.3279, −0.3458, −0.3458, −1, −1, and −2 around the steady state, we saw a time scale separation between the frequencies of 0.0167 and 0.1567 rad/min corresponding to the time period of 40–377 min. As a result, we selected the optimization period of 60 min and obtained a corresponding slow-scale model by the FWMR. The optimization is a quadratic program (QP) shown below. min(Bsp − B(k + 1|k))T &Ty &y (Bsp − B(k + 1|k)) Ug

+UTg &Tu &u Ug + 106 'T '

(9)

subject to ˆ + 1|k) ≤ Ymax Ymin ≤ Y(k Umin ≤ Ug ≤ Umax

207

(10)

where Bsp is the production target, B is the throughput prediction, and ' is a vector of slack variables. &y and &u are 100 and Inu ×nu , respectively. The optimization and the MV horizons were chosen as 8 and 2, respectively. The calculated MV setpoints, Ug , were sent to the coordination layer to compute

Fig. 8. Selected controlled variables (top) and manipulated variables (bottom) from the steady-state RTO scheme when there was no disturbance.

3.2.3. Simulation result The plant-wide objective we used in the simulation was to increase the production rate by 20% at time zero, i.e. from 1.667 to 2. Results from the slow-scale dynamic RTO were compared with those from the steady-state RTO executed every 10 h, and the single-point dynamic RTO executed every 6 min. The linearized plant model was used to formulate the prediction models for the MPCs and the RTO, while the nonlinear model was used to generate the plant output measurement. The simulation scenarios include cases when: (A) there is no disturbance, (B) there is a feed disturbance of +0.2 at time 700 min, and (C) there are parameter changes (+10% in k1 and −5% in αA ) at the time point of 700 min. Performances of the different RTO schemes were measured by integral squared error between the production target and the actual production rate. When there was no plant disturbance, the slow-scale dynamic RTO showed a superior performance over the other schemes. This is due to the fact that this scheme changed the MV setpoints more cautiously in order to prevent large process interactions among the units as shown in Fig. 10. Note that the dark area in the production rate plots are the results of actual product flow movement, not the MPC setpoint. This is because B was a small flow and used to control the holdup in the flash tank. Hence, the percentage change in this flow compared to the other streams can be large during the transient period. In contrast, the steadystate RTO and the single-point dynamic RTO optimized the plant based on a prediction at a single point and made aggressive MV setpoint changes as shown in Figs. 8 and 9. As a result, the production rates under the both RTO strategies were affected significantly. Besides, in the steady-state RTO case, the product concentration x2B violated the lower

Fig. 9. Selected controlled variables (top) and manipulated variables (bottom) from the single-point dynamic RTO scheme when there was no disturbance.

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Fig. 10. Selected controlled variables (top) and manipulated variables (bottom) from the slow-scale dynamic RTO scheme when there was no disturbance. Table 6 Economic loss defined by the integral square of error of the production rate 30-h period Simulation scenarios

Steady-state RTO

Single-point dynamic RTO

Slow-scale dynamic RTO

(a) No disturbance (b) Feed disturbance of 0.2 at t = 700 min (c) Parameter changes at t = 700 min (+10% in k1 and −5% in αA )

5.02 12.17

9.95 10.54

3.98 4.94

9.46

9.95

4.14

bound constraint early during transience, which resulted from the fact that the steady-state model did not take into account the transient behavior of the output. Since the other two dynamic RTO schemes had shorter sample intervals, though x2B moved toward the lower bound, the RTOs could reoptimize the plant to avoid the occurrence of constraint violation. In the cases for which disturbances were introduced to the plant at the time 700 min, the steady-state RTO showed the worst performance, since it had to wait for the plant to reach the steady state before the RTO could recalculate the setpoint. On the contrary, the single-point and the slow-scale dynamic RTOs had shorter sample intervals and could reject disturbance effects much faster. These results are summarized in Table 6.

4. Conclusion Dynamic optimization based on a slow-scale model can provide a computationally efficient plant-wide RTO solution than the steady-state RTO and single-point dynamic RTO schemes proposed earlier. The main questions for this method are how to obtain a slow-scale plant model accurate up to

a desired optimization frequency and how to exchange information between the plant-wide optimizer running at a slow rate and the local-unit-level multivariable controllers running at a faster rate. The first question is answered for a linear/linearized system, where a frequency-weighted model reduction provides a very satisfactory result. However, more study has to be done on how a nonlinear dynamic model may be reduced and used. The answer to the second question involves a two-way communication between the optimization layer and the control layer. For a downward route, the global solution from the RTO layer is sent to a coordination scheme, which tries to ensure feasibility of the RTO solution for the local controller. If not, a new feasible setpoint closest to the RTO solution is calculated and sent to the local controller. For an upward route, the plant-wide state vector is constructed and updated with filtered feedback errors at each MPC sampling time. The plant-wide optimizer then uses it for the prediction whenever a new optimization occurs. The suggested method is a promising alternative to the current steady-state or other single-point RTO schemes, which can be limited in terms of execution frequency and lacks robustness to model errors that can be expected in practice.

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