Real-Time Evolution for On-line Optimization of Continuous Processes

Ind. Eng. Chem. Res. 2002, 41, 1815-1825

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Real-Time Evolution for On-line Optimization of Continuous Processes Sebastia´ n Eloy Sequeira, Moise` s Graells, and Luis Puigjaner* Chemical Engineering Department, Universitat Polite` cnica de Catalunya, ETSEIB - Diagonal 647 Barcelona 08028, Spain

This paper introduces a new approach to on-line model-based optimization. The proposed approach differs from classical real-time optimization in that waiting for steady state is not necessary and novel optimization concepts are used in the solution procedure. Instead, current plant set points are periodically improved following periodic updates to the model. Thus, realtime evolution (RTE) is based on the continuous improvement of plant operation rather than on the optimization of a hypothetical future steady-state operation. Despite using a simpler scheme, the proposed strategy offers a faster response to disturbances, better adaptation to changing conditions, and smoother plant operation, regardless the complexity of the control layer. In this paper, RTE design principles are conceptually presented and illustrated with relevant examples. Case studies are considered to validate the proposed methodology and to discuss the results obtained. First, a classical CSTR benchmark is contemplated. Then, a real scenario is considered (a pilot plant at the UPC) where the RTE approach is currently implemented. I. Introduction Considerable effort has been made during the past decade in the field of model-based on-line optimization, namely, real-time optimization (RTO) of continuous processes.1-5 As a result, several implementations6-10 have appeared that have demonstrated quite attractive economical results.11,12 However, from a critical analysis of the RTO scheme behavior, some weaknesses arise. This paper briefly reviews the classical approach to RTO and its benefits and drawbacks. The attempts made to improve its performance are also discussed. Next, a novel approach for real-time optimization, real-time evolution (RTE), is described and discussed in detail. The proposed methodology is then validated in two demonstrative scenarios. Finally, the results are discussed, and the directions of our future work are indicated. II. The Real-Time Optimization Concept The main objective of a RTO system is to operate a plant, at every instant of time, as near to its optimum operating conditions as possible. To achieve this objective, a RTO system contains basically the components described in Figure 1. The RTO loop can be considered as an extension of a feedback control system. It consists of subsystems for measurement validation, steady-state detection, process model updating, model-based optimization, and command conditioning. Once the plant operation has reached steady state, plant data are collected and validated to avoid gross errors in the process measurements, and the measurements themselves might be reconciled using material and energy balances to ensure consistency of the data set used for model updating. Once validated, the measurements are used to estimate the model parameters to ensure that the model correctly represents the plant at the current operating point. Then, the optimum controller set points are calculated using * Author to whom correspondence should be addressed. Phone: +34-93.401.66.78. Fax: +34-93.401.71.50. E-mail: [email protected].

Figure 1. Schematic representation of the closed-loop RTO system functionality (CS ) control system). Adapted from Georgiou et al.7

the updated model and are transferred to the control system after a check by the command-conditioning subsystem. Qualitative RTO behavior is illustrated in Figure 2. From time t-ini to time t-end, some disturbance occurs in the process (the term “disturbance” in this paper refers to an uncontrolled variable that affects the process economy, such as an improvement in feed composition). As a consequence, when no adjustment is made to the set points, a change in the instantaneous objective function (IOF, i.e., profit) value to be optimized simultaneously takes place (dashed lines). However, the RTO system reaction to a disturbance only occurs at time t-ss, when the new steady state is detected and the optimization procedure is started. When the optimization is completed at time t-opt, the new set points are implemented, resulting in a better IOF after some stabilization time.

10.1021/ie010464l CCC: $22.00 © 2002 American Chemical Society Published on Web 03/07/2002

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Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002

Figure 2. Qualitative results of applying a RTO system (solid lines) compared with no RTO system (dashed lines).

Let us analyze in more detail and from the decisionmaking point of view the weaknesses of RTO systems.13 First, steady state must be achieved for optimization to be performed; this means that nothing can be done until this condition is satisfied. When the frequency of disturbances is higher than the process time constants, the RTO scenario cannot be contemplated. Furthermore, when several decision variables are involved, the optimization procedure might require too much time. Even worse, if the solver is not sufficiently robust, it might not converge. Also, to implement the set points resulting from the optimization, the plant must still be in the original steady state. Finally, the magnitudes of setpoint changes must be bounded for safety reasons. Thus, it is not clear which set points should be implemented when set-point changes resulting from optimization exceed the bounds. Some efforts have been made to avoid these RTO drawbacks. Besl et al.14 implemented a system that does not wait for steady state but is periodically optimized, performing data reconciliation only when steady state is achieved. Unfortunately, this approach cannot always be implemented because the quality of the optimization strongly depends on quality of the data and model.

Figure 3. RTO hierarchy.

Considering that the data and model are fitted only under steady state, this approach could lead to the optimization of an unsuitable model with unsuitable data, which could result in aggressive and profitless changing of set-point values. In addition, Cheng and Zafiriou15 presented an interesting approach in which model updating is performed implicitly during optimization and the optimization algorithm is directly applied over the plant in successive steps. This approach has some disadvantages as well. First, the updated model is not available for off-line studies. Second, several set-point changes are required (over the plant) for gradient evaluation, which can lead to undesirable plant behavior. Moreover, after each set-point change, steady state has to be reached to allow for an acceptable gradient approximation. Finally, in that work, the methodology was tested over a steady-state (ss) model, which is a very restrictive approach. The most interesting contribution has come from multivariable predictive control (MPC) research, which incorporates economic objectives into the controller to achieve an adequate transition between the current and desired operating points. One way of incorporating economy into the controller is to add a simplified economic objective to the standard quadratic regulatory objective, as proposed by several authors.9,16,17 However, the fact that all control/optimization requirements are translated into a single scalar performance index exposes a tradeoff between control and optimization, making the choice of relative weights quite difficult. Another alternative, widely used in industry,18-21 is to send the RTO results to a local linear programming (LP) or quadratic programming (QP) steady-state controller coupled to the MPC [termed LP(QP)-MPC]. Such a cascade control scheme continuously computes and updates the set points used by the lower-level MPC algorithm, producing an evolutionary transition with excellent control performance (Figure 3). However, in addition to the complexity of such systems, significant problems occur when both the LP(QP)-MPC and RTO layers have economic and performance objectives, which might not match.22 Additionally, it should also be considered that, for certain processes, the implementation of MPC might not be justified or might even not be the appropriate choice.20

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1817 Table 1. Functional sequences for RTO and RTE RTO (dog A)

RTE (dog B)

data acquisition data validation wait for steady sate data reconciliation

data acquisition data validation if steady state data reconciliation

model updating optimization of set points for current conditions using ss model check for steady state before implementing new sp’s

model updating

Figure 4. Dog A behavior (RTO).

if no steady state set-point improvement for current conditions using ss model

no check

Figure 5. Dog B behavior (RTE).

III. The RTE Concept Real-time evolution is next introduced as an alternative to current RTO approaches. The key idea is to obtain a continuous adjustment of set-point values, according to the current disturbance measurements (including also economical disturbances, such as changes in product price), the current operating conditions, and a steady-state model. An analogy is used to illustrate the proposed methodology. Consider a dog and his master. The position of the dog represents the current set points of the plant, the position of his master represents the plant optimal operating conditions, and the distance between the dog and his master represents how far the set points (sp’s) are from their steady-state optima. Solid lines represent the master trajectory, and dashed lines represent the dog trajectory. The dog objective is always to be close to his master, wherever his master is. Thus, a rational dog A (Figure 4), following the RTO scheme, will remain motionless until he acknowledges the position of his master. Hence, the sequence of the dog’s activity will be: wait for master to stop; determine his position; and then, if allowed, go to this position. This activity scheme results in periodic discrete movements and has all of the drawbacks mentioned in the previous section. Another dog (dog B) has the behavior shown in Figure 5. This dog does not wait for his master to be still. Instead, this dog corrects his position every few seconds according to the current position of his master. This behavior will lead to the dashed-line trajectory. By this analogy, we mean that, in the proposed strategy, the system’s set points are improved periodically regardless of the occurrence of steady state. In summary, the main differences between the behaviors of the two doge are: (1) Dog B does not seek optimization; he only improves his trajectory in such a way that, in the long run (we will show that it is not so long), his position is the “steady-state” optimum. (2) Dog B does not wait for his owner to be still to attempt improvement but rather is improving continuously. We call this kind of approach real-time evolution (RTE) instead of real-time optimization (RTO). Table 1

Figure 6. Qualitative results of applying a RTE system (solid lines) compared with no RT system (dashed lines).

summarizes the main aspects of the two methodologies. The steady-state information is used by RTE only for data reconciliation and model updating, while the core of the system is the recursive improvement, which does not require the process to be at steady state. The qualitative results of such a philosophy are shown in Figure 6. Immediately after the disturbance begins (t-ini), RTE will move the set point slightly in the direction that would offer a better objective function value at the corresponding steady state. In this way, RTE is continuously improving, and after the disturbance ceases (t-end), RTE reaches the optimal set points for the new conditions (t-rte). Regarding the exchange of information with the control system, there is a subtle difference between RTO and RTE. A RTO system uses process measurements to produce set points. By contrast, during the transient period, RTE uses disturbance measurements and current set points to update set-point values (Figure 7). To gradually reduce the available degrees of freedom, a

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Figure 7. RTE structure and information flows (SP ) set points, CV ) controlled variables, MV ) manipulated variables, Dp ) process disturbance with economic impact, De ) economical disturbance).

Figure 8. Direct search as improvement algorithm.

Figure 9. Linear approximation as improvement algorithm.

hierarchical structure of RTE layers can easily be used by selecting appropriate parameters (frequency, changes allowed, etc.). Advances in instrumentation, decreasing sensor prices, and also soft sensor techniques are making disturbance measurements,23 and hence the application of this approach, readily available. To understand how RTE works, the next section describes its components in more detail.

tive function values at the five points, and then the best one is selected. In this way, the corresponding set points are determined and sent to the control system. Another possibility is to approximate the objective function (linearly, for example) with the values obtained from the model at properly chosen points. Once the approximate objective function is obtained, the best point can be selected (Figure 9). The selection of the improvement algorithm becomes more important as the number of decision variables increases, because the number of points evaluated at each RTE step will determine the algorithm computing time. In the same way as for the RTO case, the optimization method strongly affects the algorithm calculation time. At this point, it should be noted that, given the improvement algorithm’s simplicity, it is very easy to deal with constraints (absolute and relative bounds in set-point values, undesirable conditions, etc.) because unfeasible points can be readily discarded. This is extremely significant because, for RTO, any additional constraint included in the optimization problem entails increased computational costs and/or a loss of robustness, which might not be affordable in a real-time environment.

IV. The RTE Components Four main aspects of RTE must be considered: the improvement algorithm, the exploration neighborhood, the time between two successive RTE calculations (∆t), and the model used. The improvement algorithm is any procedure used to obtain better plant operating conditions located inside the current operating point neighborhood. For example, Figure 8 shows the current decision variable values [DVx, DVy], which define the current operating point at steady state. Also shown in this figure are four possible new steady-state conditions located in the neighborhood of the current one [DVx ( δDVx, DVy ( δDVy]. A steadystate model is used to predict the corresponding objec-

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1819

Figure 11. Williams-Otto reactor. Figure 10. Framework for RT system validation and parameter tuning.

The next question is how to define the neighborhood. Although this can be done in many ways, in this work, the neighborhood was defined by bounding the changes allowed in the decision-variable values at each RTE step (δDV). Bounds for such changes are given by the capabilities of the control system and the desired optimization accuracy. Thus, minimum changes are bounded by the sensor accuracy and controller precision, whereas maximum changes are given by the accuracy allowed. In this work, changes of about 1-3% of the variation range of the process variables proved to give good performance, and consequently, they can be taken as preliminary heuristic values. It should be pointed out that bounds in the implementation of set-point changes can be implicitly taken into account in the definition of the neighborhood. An additional degree of freedom in a RTE system is ∆t. A very high value of ∆t will not likely improve plant operation. However, ∆t should be large enough to allow for calculation improvements when only one model is used. As a preliminary approximation, a ∆t value of about 1-7% of the process residence time was used in this work, producing satisfactory results. However, this parameter is strongly dependent on the disturbance rate and frequency. Finally, the last main component of the RTE system is the steady-state model. Unlike to direct search procedures,24,25 RTE is a model-based improvement procedure. It might be thought that, when the plant is far from the optimum operating point, there is no need for an exact model. Under such circumstances, the model needs only to reflect tendencies, allowing the opportunity for simplified models to be used. However, the quality of the system trajectory always depends on the model accuracy, and considering that the improvement algorithm is not time-consuming, the use of a rigorous model will always lead to better results. All of the above components must be tested and analyzed prior to implementation in the plant. This tuning can be achieved using dynamic simulation, because the RTE system needs to be adapted to the process complexity, process dynamics, disturbance rate, and disturbance frequency. However, it should be noted that there is no reason that these parameters constant must be kept constant. It is also possible to change the parameter values strategically and on-line. The proposed approach can be seen as a variant of the EVOP strategy.24 The differences are mainly that (1) RTE relies on a model rather than experimenting over the plant, which means that time, money, and safety are not spent in unprofitable trial moves, and (2) steady state is not awaited, as adequate tuning of the

RTE parameters allows the system to exhibit pseudosteady-state behavior and hence produces better economical performance even under continuous disturbances. V. Validation Studies The integration of the RT system (RTO or RTE) is tested using a dynamic model describing the plant. Figure 10 illustrates the scheme for validating a RT system and also for tuning the corresponding parameters. A dynamic first-principles model is used to emulate “on-line” data, which are subsequently validated, filtered, and reconciled. The RT system includes the following components: (1) the steady-state detector, used for model updating; (2) the steady-state process model and its corresponding economic model; (3) the solver, which, in RTO, is the optimization algorithm and, in RTE, is the improvement algorithm; and (4) the implementation block, which sends the generated set points, if they are acceptable, to the plant (only in RTO). Within this framework, two scenarios are presented. First, the benchmark of the Williams-Otto reactor (for details, see the Appendix) is used for RT implementation. Then, a second scenario is presented that consists of a pilot plant (PROCEL) at UPC where the authors are implementing the system. VI. Scenario I: Williams-Otto Reactor The Williams-Otto reactor is shown in Figure 11. A jacketed CSTR operating at temperature Tr is fed with reactants A and B, so that a six-component outlet stream (R) is obtained when the following reactions occur

A+BfC

k1 ) 1.660 × 106 exp

( ( (

C + B f P + E k2 ) 7.211 × 108 exp P+CfG

) ) )

-6667 Tr + 273

k3 ) 2.675 × 1012 exp

-8333 Tr + 273

-1111 Tr + 273

The objective is to maximize the IOF (Figure 12)

Profit ) 5554FrXp +125.9FrXe - 370.3Fa - 555.4Fb where Xi represents the mass fractions of the corresponding components. The main disturbance is the flow of stream A, Fa. The decision variables (and also set points) are the reactor temperature Tr and the flow of stream B, Fb.

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Figure 12. Objective function shape for the Williams-Otto reactor scenario (Fa ) 1.83 kg/s).

For these operating conditions, the optimum operating point corresponds to Fb ) 4.89 kg/s and Tr ) 89.7 °C, and the function is clearly convex. Once both models are developed (dynamic and steady-state), the system’s performance can be tested. In this case, the implementation was performed using Matlab in its Simulink environment. For the sake of simplicity, the first example assumes perfect control, and two experiments were carried out (simulating step and sinusoidal wave disturbances in Fa). Step in Fa. Starting from the optimum operating conditions, a step is simulated in Fa (from Fa ) 1.83 kg/s to Fa ) 1.7 Kg/s at t ) 300 s). The RTO system will react after steady-state detection, whereas the RTE is tuned to actuate every 4 s. The RTE parameters were steps of 0.2 °C for Tr and 0.02 kg/s for Fb, and direct search was used as the improvement algorithm. 1. RTO Response. Temporal values of Fa, Tr, Fb, and the instantaneous objective function (IOF) obtained using the RTO system are shown in the Figure 13. It can be seen that, at a time near t ) 1500 s (about 3.5 times the reactor residence time), the RTO system implements the optimal set points (Tr ) 89.7 °C and Fb ) 4.89 kg/s). The delay is mainly due to the steady-state detection and also to the time consumed by the subsequent optimization procedure. The system was considered steady when the variation of the variables was less than 7.2%/h. A plot of the IOF versus time clearly reflects the improvement attained by the RTO system (area under the curve). It is interesting to examine the effect of sudden changes in the set point on the IOF. First, the decrease in Fb and Tr is translated into an instantaneous decrement in IOF. This is because Fr is immediately reduced and, therefore, so is the produced quantity. However, after the CSTR stabilization time, the composition at the reactor output greatly compensates this temporary IOF decrement by an increase in the production quality. 2. RTE Response. Temporal values of Fa, Tr, Fb, and the instantaneous objective function (IOF) obtained using the RTE system are shown in the Figure 14. It can be seen that the RTE system does not wait for steady state but rather immediately reacts and changes set points. The two decision variables both proceed step-

Figure 13. (a) Step disturbance in Fa at t ) 300 s. (b, c) RTO response over the set points (Fb and Tr). (d) Instantaneous objective function (IOF) with (solid line) and without (dashed line) RTO.

by-step to their optimum values. In the case of Tr, RTE first tries to increment its value but later changes the direction toward the final optimum value. This means that RTE will allow Tr to increase until an acceptable product quality can be guaranteed according to the new Fa and Fb values, and only thereafter will Tr be allowed to reach its optimum value. Figure 15 shows comparative results for RTO, RTE, and no optimization with the IOF as the performance criterion. For both RTO and RTE, the final IOF is approximately $898/s $/s, whereas taking no action yields $891/s $/s. It can be seen that, after the process comes to a true steady state, the IOF value for RTO eventually approximates that for RTE as time proceeds. However, RTO and RTE should be compared not on the basis of the final IOF values but rather regarding the cumulative profit produced. Thus, a more meaning-


Figure 14. (a) Step disturbance in Fa at t ) 300 s. (b, c) RTE response over the set points (Fb and Tr). (d) IOF with RTE (solid line), with RTO (dashed line), and with no action (dot-dashed line).

Figure 15. MOF profile comparison between RTO (dot-dashed line), RTE (solid line), and no RT system (dashed line). (to ) 0 s.)

ful comparative plot (Figure 15) is obtained by using the mean objective function (MOF), which considers the history of the process and is defined as

MOF(t) )

∫tt IOF(t) dt o

(t - to)

where to and t represent the initial and current times, respectively. A deeper analysis of this graph reveals that, for a few seconds, the MOF values for the RTO and RTE systems are lower than those obtained when no action is applied. This happens because the RTO and RTE systems are based on steady-state models, which only guarantee long-term improvement when the steady state is achieved. Certainly, for the step disturbance case, both RTO and RTE reach steady state, and thus, the same IOF and MOF final values are expected when time tends to infinity. However, disturbances can arrive at any time, rather than infinity. Given that RTE produces faster improvement of the MOF than RTO, the latter should always lead to better overall performance if further disturbances occur (e.g., at t ) 2000 s). This fact

Figure 16. (a) Wave disturbance in Fa at t ) 300 s. (b, c) RTE response over the set points (Fb and Tr). (d) MOF with RTE (solid line) and no action (dashed line).

suggests that the RTE performance be tested when the disturbance consists of a continuous change, which is the subject of the following discussion. Wave in Fa. Starting from the optimum operating conditions, a wave was simulated in the feed flow [from Fa ) 1.83 kg/s to Fa ) 1.83 - 0.13 sin (t/6000) kg/s]. Figure 16a represents the shape of the disturbance. In this case, the RTO system could do nothing because there was no steady state. In contrast, RTE tuned the process every 4 s, producing the results shown in Figure 16b,c. The RTE parameters were the same as in the previous example. As one can see, RTE is able to make on-line adjustments of set points to improve process performance, thus adapting the plant performance to the changing external conditions. For the wave disturbance in Fa, a wavelike response was obtained for the variables Tr and Fb, as well as for the MOF. Regarding performance, it should be noted that, in both examples (step and wave disturbances), the shape of the MOF obtained using RTE closely follows the shape obtained for the process without optimization but shifted up and to the left. VII. Scenario II: PROCEL The following scenario consists of a real pilot plant (PROCEL at UPC). In this case, perfect control is not assumed, as it was in the previous example. The plant flowsheet is shown in Figure 17. Stream 1 is fed to a vessel (tank) provided with level and temperature controllers. The tank output is preheated (E1), heated (E-2), and finally cooled (E-3), with heat exchanger E-1 being used for heat recovery. Vessel V is used only to hold the heating utility (water). The process can be regarded as a pasteurization operation. The fluid to be pasteurized is first stored and later suddenly heated and cooled. The objective function to be maximized is the profit, defined as the difference between sales revenues and production costs, including a penalty for poor quality. Temperature indicates the raw material quality, whereas

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Figure 17. PROCEL flowsheet (continuous configuration).

the product quality depends on the process history, namely, the residence time in the vessel and the temperatures in the tank and streams 7 and 9 (see Figure 17). The decision variables are the operating conditions, that is, the set points of the controllers for the tank temperature (TC-tank, PI controller), tank level (LCtank, PI controller), stream 7 temperature (TC-7, PI controller), and output stream temperature (TC-9, PI controller). Finally, the disturbances are the quantity (flow) of raw material (stream 1), the quality (temperature) of raw material (stream 1), and the prices and costs (of the product, raw material, heating, cooling, etc.) Raw material flow continues throughout the residence time in the tank and, hence, influences the product quality. The raw material temperature affects the raw material cost and also dictates the recommended temperature in the tank (for highest quality). The influence of prices and costs on the objective function is obvious. The steady-state and dynamic models were developed using HYSYS.Plant. (For detailed simulation parameters, see http:\\g29.upc.es\procel.) To send on-line data to the RT system, the DCS interface of HYSYS was used. The architecture for the on-line optimization is the described in Sequeira et al.26 Again, two disturbance experiments were carried out, corresponding to a step and a wave disturbance. The starting point for both examples is given in Table 2 and corresponds to optimal operation for a given set of external conditions. Experiment 1 (Step Disturbance). In this example, the disturbance occurred at time t ) 10 min and consisted of a step in the inlet stream temperature from 20 to 13 °C. The RTE parameters used were calculated every minute, and step changes of 0.5 °C were considered for the temperatures. A direct search method was used in the improvement algorithm. The comparative results displayed in Figure 18 show the effect on the MOF when no action is taken and when the RTO and RTE systems are used. The MOF value will naturally decrease, but taking appropriate corrective actions according to both the RTO and RTE systems results in better performance than doing nothing. The calculation time spent during optimization was not taken into consideration, but even so, the RTE performance is slightly better than the RTO perfor-

Figure 18. MOF for RTE, RTO, and no action after a step disturbance. Table 2. Initial Conditions for Scenario II external conditions stream 1 flow (F-1) stream 1 temperature (T-1)

60 kg/h 20 °C

set-point values TC-tank LC-tank TC-7 TC-9

45 °C 50% 86 °C 48 °C

mance. The difference between the two curves is (mainly) due to the time required to achieve steady state. As mentioned before, a key issue in selecting this scenario is relaxation of the hypothesis of perfect control. Therefore, in addition to the objective function value, it is also relevant to consider the behavior of the operating conditions in both cases. Figure 19 shows process temperature profiles (set points and process variables). It can be seen that the differences between the set points (sp’s) and process variables (pv’s) are not significant when RTO is used (Figure 19b). The same plot for the RTE approach (Figure 19c) shows even smaller differences. This result is obvious if we consider that the changes in RTE are smoother than those in RTO, allowing the control system to keep the process variables values almost equal to the set-point values. Therefore, it can be concluded that poor controller tuning should have a more significant impact when RTO, rather than RTE, is used.


Figure 19. (a) Disturbance. (b, c) Set points (dashed line) and their corresponding process variables (solid line) for RTO and RTE.

and flow. The amplitudes were 8 kg/h for the flow and 6 °C for the temperature, with a 30-min interval for the flow and a 40-min interval for the temperature. The RTE parameters were the same as in the previous experiment. Again in this case, because there is no steady state, RTO can do nothing, whereas RTE produces the MOF shown in Figure 20. The RTE results provide a significant economic improvement, and the shapes of the two curves (flow and temperature) are clearly similar, which indicates the close follow-up to the process behavior. It can be seen that the MOF amplitude in both cases decreases as a result of the damping effect of integration (see MOF definition), which should result in limiting values for the MOF (in the cases of RTE and no action) when time tends to infinity. Furthermore, because corrective actions are performed throughout time, smooth plant behavior is observed, and negligible differences between the process variables and set points are obtained (Figure 21). This example enhances the differences between RTE and RTO and illustrates a more realistic situation (true steady state is difficult to reach in practice). A key point of the proposed method is that using a steady-state model improves the process performance, even though the steady state might not have been reached. This is mainly because of the known fact that many systems evolve as successive steady states (pseudo-steady states). This situation poses the additional issue of data reconciliation and subsequent model updating required by both RTO and RTE (Table 1). When steady state is not achieved, current dynamic data reconciliation techniques can be used to obtain instant reliable data. Furthermore, models can be periodically updated using average values and heuristic procedures. VIII. Conclusions

Figure 20. MOF for RTE after sinusoidal disturbances.

Figure 21. (a) Sine-wave disturbance in stream 1 temperature. (b) Set points and their corresponding process variables for RTE.

Experiment 2 (Wave Disturbance). The disturbance in this case occurred at time t ) 10 min and consisted of a sine wave in the inlet stream temperature

This paper presents a revised view of the philosophy behind on-line optimization objectives. As a result, a new methodology has been established, and the concept of real-time evolution (RTE) has been developed as an alternative to classical RTO systems. Tests carried out using the RTE system in a variety of scenarios show the adequacy of the RTE structure in keeping the plant at the optimum operating conditions when disturbance information is available. Once correctly tuned, the RTE system produces successive improvements in the plant performance, making the plant evolve continuously toward the set objectives. Although RTE requires more time than RTO to reach optimum operating conditions, it improves plant performance immediately after disturbances occur, thus resulting in an overall faster system that is able to deal with continuous changes. The algorithms involved in RTE (improvement) are simpler and faster than those used by RTO (optimization); this allows for the intensive use of available rigorous process models with little computational effort. Finally, results show that the RTE strategy is less affected than RTO when poor controller performance is encountered. Future work includes on-line validation of the proposed approach in a real plant scenario (PROCEL), contemplation of the case in which global optimization is required, and consideration of significant points such as noise and uncertainty in collected data. Acknowledgment The authors acknowledge the contribution of Prof. Antonio Espun˜a in offering valuable suggestions and

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stimulating discussion during the course of this work. One of the authors (S.E.S.) also acknowledges a grant from the Spanish “Ministerio de Ciencia y Tecnologı´a”. Finally, financial support from the Spanish Government and the European Community is gratefully acknowledged (Projects Realisstico QUI99-1091 and CHEM GIRD-CT-2001-0466). Appendix: Models Used for the Williams-Otto Reactor (Adapted from Williams and Otto27)

Kinetics k1 ) 1.660 × 106 exp k2 ) 7.211 × 108 exp

( ( (

) ) )

-6667 Tr + 273 -8333 Tr + 273

k3 ) 2.675 × 1012 exp

-1111 Tr + 273

Let R1 ) VFk1(Tr)XaXb R2 ) VFk2(Tr)XcXb R3 ) VFk3(Tr)XpXc Mass Balances During Steady State Global Fa + Fb ) Fr For A Fa - FrXa - R1 ) 0 For B Fb - FrXb - R1 - R2 ) 0 For C -FrXc - 2R1 - R2 - R3 ) 0 For E -FrXe - 2R2 ) 0 For P -FrXp + R2 - 0.5R3 ) 0 For G -FrXg - 1.5R3 ) 0 During Transitions Global Fa + Fb - Fr )

d(FV) )0 dt

For A Fa - FrXa - R1 ) FV

dXa dt

For B Fb - FrXb - R1 - R2 ) FV

dXb dt

For C -FrXc - 2R1 - R2 - R3 ) FV For E -FrXe - 2R2 ) FV

dXc dt

dXe dt

For P -FrXp + R2 - 0.5R3 ) FV For G -FrXg - 1.5R3 ) FV

dXp dt

dXg dt

where ki is the kinetic coefficient for reaction i (s-1),

Tr is the reactor temperature (°C), Fi is the mass flow of component i, Xi is the mass fraction of component i, F is the fluid density (assumed constant at 1 kg/L), and V is the reactor volume (assumed constant at 2105 L). Literature Cited (1) Marlin, T. E.; Hrymak, A. N. Real-Time Optimization of Continuous Processes. AIChE Symp. Ser. 1997, 316, 156. (2) Forbes, J. F.; Marlin, T. E.; MacGregor, J. F. Model Adequacy Requirements for Optimizing Plant Operations. Comput. Chem. Eng. 1994, 18 (6), 497. (3) Forbes, J. F.; Marlin T. E. Design Cost: A Systematic Approach to Technology Selection for Model-Based RealTime Optimization Systems. Comput. Chem. Eng. 1994, 20 (6/7), 717. (4) Perkins, J. D. Plant-Wide Optimization: Opportunities and Challenges. AIChE Symp. Ser. 1998, 94 (320), 15. (5) Zhang, Y.; Forbes, J. F. Extended Design Cost: Performance Criterion for Real-Time Optimization System. Comput. Chem. Eng. 2000, 24, 1829. (6) Lauks, U. E.; Vasbinder, R. J.; Valkenburg, P. J.; van Leeuween, C. On-line Optimization of an Ethylene Plant. Comput. Chem. Eng. 1992, 16S, S213. (7) Georgiou, A.; Sapre, A. V.; Taylor, P.; Galloway, R. E.; Casey, L. K. Ethylene Optimization System Reaps Operations and Maintenance Benefits. Oil Gas J. 1998, 96 (10), 46. (8) Krist, J. H. A.; Lape`re, M. R.; Wassink, S. G.; Koolen, J. L. A. System for Real Time Optimization. U.S. Patent 5,486,995, 1996. (9) Tvrzska´, M.; Odloak, D. One-Layer Real Time Optimization of LPG Production in the FCC Unit: Procedure, Advantages and Disadvantages. Comput. Chem. Eng. 1998, 22, S191. (10) van Wijk, R. A.; Pope, M. R. Advanced Process Control and On-line Optimization in Shell Refineries. Comput. Chem. Eng. 1992, 16S, S69. (11) White, D. C. On-line Optimization: What, Where and Estimating ROI. Hydrocarbon Process. 1997, Jun, 43. (12) Yoon, S.; Dasgupta, S.; Mijares, G. Real Time Optimization Boosts Capacity of Korean Olefins Plant. Oil Gas J. 1996, 94 (25), 36. (13) White, D. C. On-line Optimization: What Have We Learned? Hydrocarbon Process. 1998, Jun, 55. (14) Besl, H.; Kossman, W.; Crowe, T. J.; Caracotsios, M. Nontraditional Optimization for Isom Unit Improves Profits. Oil Gas J. 1998, 96 (19), 61. (15) Cheng, J.; Zafiriou, E. Robust Model-Based Iterative Feedback Optimization of Steady-State Plant Operations Ind. Eng. Chem. Res. 2000, 39, 4215. (16) Becerra, V. M.; Roberts, P. D.; Griffiths, G. W. Novel Developments in Process Optimization Using Predictive Control. J. Process Control 1998, 8 (2), 117. (17) Nath, R.; Alzein, Z. On-line dynamic optimization of olefin plants. Comput. Chem. Eng. 2000, 24, 533. (18) Sorensen, R, C.; Cutler, C. R. LP integrates economics into dynamic matrix control. Hydrocarbon Process. 1998, Sep, 57. (19) Rao, C. V.; Rawlings J. B. Steady States and Constraints in Model Predictive Control. AIChE J. 1999, 45 (6), 1266. (20) Qin, S. J.; Badgwell, T. A. An overview of industrial model predictive control technology. AIChE Symp. Ser. 1997, 316 (93), 232. (21) Ying, C.-M.; Joseph, B. Performance and Stability Analysis of LP-MPC and QP-MPC Cascade Control Systems. AIChE J. 1999, 45 (7), 1521. (22) Mizoguchi, A.; Marlin, T.; Hrymak, A. Operations Optimization and Control Design for a Petroleum Distillation Process. Can. J. Chem. Eng. 1995, 73, 896. (23) Harold, M. P.; Ogunnaike, B. A. Process engineering in the evolving chemical industry. AIChE J. 2000, 46 (11), 2123.

Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1825 (24) Box, G.; Draper, N. Evolutionary Operation: A Statistical Method for Process Improvement; Wiley: New York, 1969. (25) Marlin, T. E. Process Control; McGraw-Hill: New York, 1995. (26) Sequeira, S. E.; Graells, M.; Puigjaner, L. Synergic Use of the Available CAPE Tools: An Application to Real Time Optimization. Presented at the AIChE Annual Meeting, Los Angeles, CA, Nov 2000; Paper 234c.

(27) Williams, T. J.; Otto, R. E. A Generalized Chemical Processing Model for the Investigation of Computer Control. AIEE Trans. 1960, 79 (Nov), 458.

Received for review May 24, 2001 Revised manuscript received November 27, 2001 Accepted January 15, 2002 IE010464L

Real-Time Evolution for On-line Optimization of Continuous Processes

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