Brazilian Journal of Chemical Engineering
Print version ISSN 01046632Online version ISSN 16784383
Braz. J. Chem. Eng. vol.17 n.47 São Paulo Dec. 2000
http://dx.doi.org/10.1590/S010466322000000400061
ROBUST PREDICTIVE CONTROL OF A GASOLINE DEBUTANIZER COLUMN
E.Almeida Neto^{1}, M.A.Rodrigues^{2} and D.Odloak^{2 }^{1}Petrobras, Advanced Control Group, ABASTREF, Rio de Janeiro.
^{2}Chemical Engineering Department, University of São Paulo, C.P. 6148, 05448970,
São Paulo  Brazil, Email: odloak@usp.br
(Received: ; Accepted)
Abstract  This paper studies the application of Model Predictive Control to moderately nonlinear processes. The system used in this work is an industrial gasoline debutanizer column. The paper presents two new formulations of MPC: MMPC (MultiModel Predictive Controller) and RSMPC (Robust Stable MPC). The approach is based on the concepts of Linear Matrix Inequalities (LMI), which have been recently introduced in the MPC field. Model uncertainty is considered by assuming that the true process model belongs to a convex set (polytope) of possible plants. The controller has guaranteed stability when a Lyapunov type inequality constraint is included in the MPC problem. In the debutanizer column, several nonlinearities are present in the advanced control level when the manipulated inputs are the reflux flow and the reboiler heat duty. In most cases the controlled outputs are the contents of C_{5}^{+} (pentane and heavier hydrocarbons) in the LPG (Liquefied Petroleum Gas) and the gasoline vapor pressure (P_{VR}). In this case the QDMC algorithm which is usually applied to the debutanizer column has a poor performance and stability problems reflected in an oscillatory behavior of the process. The new approach considers several process models representing different operating conditions where linear models are identified. The results presented here show that the multimodel controller is capable of controlling the process in the entire operating window while the conventional MPC has a limited operating range.
Keywords: debutanizer column, model predictive control, robust control, linear matrix inequalities, gasoline stabilization
INTRODUCTION
The debutanizer column is an important part of several process units of the oil refinery. It is usually utilized to remove the light components from the gasoline stream producing LPG (Liquefied Petroleum Gas) and the stabilized gasoline that can be included in the gasoline pool. In a typical Brazilian refinery, gasoline is produced in several process units, namely: Crude Atmospheric Distillation Unit, Fluid Catalytic Cracking Unit and Delayed Coking Unit. The gasoline production scheme is represented in Fig.1. The main controlled variables of the debutanizer column are the C_{5}^{+} vol.% LPG stream and the Reid Vapor Pressure (P_{VR}) of the stabilized gasoline, which is produced at the bottom of the column. In Brazilian refineries, instead of the C_{5}^{+} vol.% in the LGP it is usually controlled the Weather (W_{LPG}) that is a property closely related to the C_{5}^{+} vol.% as shown in Fig. 2. The Weather corresponds to the temperature at which 95% of the LPG volume is evaporated at atmospheric pressure.
Applications of model predictive control have been fully reported in the literature (Prett and Gillete, 1980; Zanin and Odloak, 1992; Moro and Odloak, 1995: Magalhães and Odloak, 1995; Almeida et al., 1996) and this practice has been extended to the debutanizer column without much concern. However, a substantial part of model predictive controllers based on conventional approaches show poor performance when applied to debutanizers of atmospheric crude units. Apparently this is related to the small fraction of LPG normally available in the feed of the column. The LPG volume is normally under 10% of the total feed and, as we will show later, this brings severe nonlinearities to the process model. The result is that when MPC is applied to this system, the response is sluggish or oscillatory depending on the adopted tuning parameters.
Several authors have already studied robustness of MPC. Zafiriou (1990) and Zafiriou and Marchal (1991) have studied the robust stability of QDMC with model uncertainty and output constraints. Tvrzská de Gouvêa and Odloak (1997) have presented a general framework to analyze stability and performance of MPC without constraints. Kothare et al. (1996) proposed an infinity horizon MPC written as an LMI optimization problem where robust stability was provided by an upper bound function for the output square error. Some of the ideas proposed in that paper are utilized here.
This paper is organized as follows: in the next section the debutanizer process is described and some of the nonlinear aspects of the system are studied with the help of a rigorous process simulator. Then, the new predictive controller based on the LMI optimization problem is presented. The performance of the proposed controller is compared to QDMC for different operating conditions of the system. In the sequel a robust version of the new controller is presented and the paper is concluded.
Process Description
The debutanizer column considered in this study is located at the crude distillation unit of Cubatão Refinery where LPG can be produced with low W_{LPG} values (2^{o}C to 2^{o}C) and incorporated into the refinery LPG pool. Alternatively the LPG can be produced with high W_{LPG} values (up to 15^{o}C) and sent to the FCC unit to be reprocessed. Depending on the adopted operating strategy, completely different process conditions are obtained and consequently the system follows different models. Then, a robustness problem is created and the control system must deal with it. The schematic representation of the debutanizer is shown in Fig.3. The feed stream comes from the top of the crude preflash column at a temperature of 40^{o}C and pressure of 8 kgf/cm^{2}. It is preheated in the heat exchanger E01 where the hot fluid is the stabilized gasoline that leaves the bottom of the debutanizer at approximately 163 ^{o}C. At the exit of E01 the temperature of the feedstream is approximately 136^{o}C and it is 6% vaporized. The feed is introduced in the debutanizer column (T01) at stage 17. The column has 30 stages with a Murphree efficiency of approximately 75%. Measured points of the temperature profile are the top (about 54^{o}C), bottom (about 163^{o}C) and stage 5 (about 67^{o}C). The top condenser has a heat duty of approximately 1.0 MMcal/h that corresponds to a distillate flow of about 108 m^{3}/d and a reflux flow of about 400 m^{3}/d. A small flow of fuel gas is also produced.
In the control scheme adopted in this study the manipulated variables are the reboiler heat duty (Q_{REB}) and the reflux flow rate (V_{RT}) which operating window is represented in Fig.4. This operating window results from several process constraints that are defined in Table 1.
Uncertainties in the Process Gains
Here it is studied the static behavior of the debutanizer column using HYSYS^{TM} to evaluate the effects of the manipulated variables on the controlled variables at different points of the operating window of the process. Fig. 5 shows the relations between C_{5}^{+} vol.% in the LPG and gasoline vapor pressure (P_{VR}) with the manipulated inputs, which are the reflux flow (V_{RT}) and reboiler duty (Q_{REB}). Observe that the C_{5}^{+} in LPG has a strong nonlinear relation with V_{RT} and Q_{REB} for C_{5}^{+} values below 5 %. Figures 5c and 5d show that P_{VR} has a more linear behavior in terms of the manipulated variables.
The effects of these nonlinearities are also reflected in the step responses of the process. To illustrate this point considers the step responses at several operating points in the operating window of the process. These points are depicted in Fig.6 and numbered from 1 to 10.
The process step responses obtained for each of these points are shown in Fig. 7. It can be seen that the dynamic response of the process is quite different depending on the operating point considered. Thus, becomes clear the reason why conventional MPC that uses a single prediction model cannot cope with the different operating conditions of the debutanizer column.
MultiModel Predictive Controller (MMPC)
In this section we describe a new formulation of the predictive controller where a set of possible process models are simultaneously considered in the control problem. The MPC optimization problem is converted into a conventional LMI problem and can be solved with the available tools of Linear Matrix Inequalities (LMI). For this aim consider the objective function of the conventional model predictive controller which can be written as follows:
(1) 
where is the available predicted error on the system output at time instant k
Du is the control move to be calculated
S_{mn} is a matrix of step response coefficients considering a control horizon m and a prediction horizon n
G and L are weighting matrices
Since the purpose of the controller is to minimize J_{k} the objective function can also be formulated as follows:
(2) 
where g >0 and the objective becomes to minimize g .
Combining Eq.(1) and Eq. (2) results:
(3) 
where
Since (3) is a quadratic inequality, with the help of Schur complement, the MPC optimization problem can be formulated as the following standard LMI problem (Rodrigues and Odloak, 2000a):
Problem P1)
subject to
g > 0
Problem P1) defines the conventional MPC without constraints. In this case matrix S_{mn} is built solely with the nominal or ideal model of the process and no considerations are made about uncertainties on this model. As shown before the debutanizer column is a process where large changes in the process model can occur if operating conditions are changed. To take into account model variations, suppose that the system can be represented in the state space form as follows:
(4) 
where matrices A and B are not exactly known but belong to a convex set defined by a limited number of possible plant models (polytopic set). In this study we assume that

(5) 
The strategy of MMPC is to minimize J_{k} for the worst case in terms of l_{1}, l_{2}, ... l . This means to solve Problem P1) for all convex combinations of l_{i} and select the worst case (minmax strategy). However, since the objective and the constraints involved in Problem P1) are linear, it can be shown that the search can be limited to the vertices of the polytopic set. Then, the multimodel predictive control problem can be formulated as Problem P2) below:
Problem P2)
, i =1, 2, ... ,L g > 0
where
Simulation and Results
In this study the process dynamic simulator HYSYSä is used to represent the real system and the predictive controller is programmed and executed using the facilities of MATLABä . The dynamic simulator contains all the geometric parameters of the system, the structure of the regulatory control level and the corresponding tuning parameters of the PID controllers.
Process Simulator and Controller Interface
The exchange of information between MATLABä and HYSYSä is performed through a DDE (Dynamic Data Exchange) synchronous interface. The schematic representation of this interface is presented in Fig.8. In the advanced control scheme, the process computer would represent MATLABä and the real plant and the regulatory instrumentation would represent HYSYSä .
Comparing the Performance of MMPC and QDMC
Here it is compared the performance of two classes of controllers. The first one, which is called conventional, is represented by QDMC that is one of the most successful control algorithms in industry. The second one that can be called robust is represented by MMPC. The name robust is used here rather freely in the sense that the controller tries to take into account the uncertainties present in the model used to predict the system output. Later on we will be more specific about the robust stability of the controller and the necessary modifications on Problem P2) to guarantee stability of the system. Initially the comparison focuses on the performance of the two controllers at several operating points of the system. For this purpose it is considered setpoint moves along the operating range of the debutanizer column. It is also included in this section a study on the sensitivity of each the controllers to their tuning parameters particularly the input weight.
In this study the tuning parameters are assumed the same for both controllers except the input weights that will be commented in the sequel. The sampling period is equal to 1 min, the control horizon is 2 and the output weights for W_{LPG} and P_{VR} are respectively 5 and 1. These weights are usually adopted in the plant since the control of W_{LPG} is more strict than the control of the gasoline that is sent to a blending system where there is more flexibility to correct any erroneous operating procedure. Model 1 was adopted as the nominal model and models 1 to 10 are included in MMPC. The sampling time instants considered in the optimization horizon for both controllers are: 1, 2, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90, 100, 150, 200. Table 2 shows the three operating points considered here and the adopted input weights for each case. Case 1 corresponds to the most common operating condition, while Case 2 corresponds to a situation when we have high reflux flow and high reboiler heat duty. Case 3 corresponds to a condition with low reflux flow and low reboiler heat duty. The controllers should be tuned for Case 1 and the disturbances considered here corresponds to small setpoint changes around each operating point.
Figs 9 and 10 show that for Case 1 both controllers can stabilize the system. However QDMC tends to be unstable for low input weights and needs large input weights to be stable. The effect of these large weights is that offset can appear in the less important variable which is the gasoline vapor pressure (Figs 9b and 10b). Since for MMPC the performance is satisfactory even with small input weight, offset in P_{VR} can be made negligible. For Case 2 (high reflux rate and heat duty), MMPC and QDMC have similar responses when the input weight are the same for both controllers (L =[10 10]). It is clear that the two controllers perform satisfactorily and this can be explained by the almost linear behavior of the process for the conditions corresponding to this case.
Case 3 corresponds to the operating conditions related to minimum energy consumption and consequently these are the conditions that eventually can be sought by the process optimizer operating in an upper level of the control structure. It is shown in Fig. 12 that in this case QDMC can not stabilize the system even with large input weights. On the contrary MMPC stabilizes the debutanizer and brings the controlled variables to their setpoints smoothly. Then it becomes clear the advantage of MMPC over the conventional MPC. The inclusion of several models that cover all possible model variations can substantially enlarge the stabilizing capability of the controller and consequently improve its performance.
ROBUST STABLE MODEL PREDICTIVE CONTROL (RSMPC)
As shown above, when compared to the conventional MPC, MMPC has an expanded capability of stabilizing uncertain systems. However, there is no guarantee that the inclusion of multiple models in the controller will stabilize the system. Thus, if guaranteed stability is the issue, the adopted approach has to be extended. This can be achieved by a suitable modification of Problem P2). As shown by Rodrigues and Odloak (2000a) robust stability of the closed loop system is equivalent to the existence of matrices K_{mpc} and (P = P^{T} > 0) that satisfy the following Lyapunov type inequality:
i = 1, ... , L  (6) 
where L represents the number of vertices of polytope. It is assumed that the MPC optimization problem results in a control law represented by:
Du(k) = K_{mpc} e(k)
Inequality (6) can be included in the control problem, generating Problem P3) below:
Problem P3)
, i =1, 2, ... ,L
i =1, 2, ... ,L
P = P^{T} > 0
g > 0
This is a nonlinear problem since both K_{mpc} and P are unknown. Different solutions of Problem P3) have been proposed in the literature (Rodrigues and Odloak, 2000a,b) but they can be time consuming for problems of moderate size.
For the debutanizer column studied here the system responses for RSMPC, with a prediction horizon shorter than that used in MMPC, coincide with the responses of MMPC since the inclusion of the Lyapunov condition is not necessary to stabilize the system. However in the general case the solution of Problem P3) becomes essential to a stable and secure operation of the system.
CONCLUSION
This paper shows that the application of model predictive control can be successful for systems ranging from moderate to heavily nonlinear if a set of models representing the operating window of the process is included in the control optimization problem. The system studied here was the debutanizer column of atmospheric distillation units. In industry this has shown to be not efficiently controlled by the existing predictive controllers. Results obtained with the new controller (MMPC or RSMPC) show that a much better performance can be obtained in comparison to the already installed control packages.
ACKNOWLEDGMENTS
Support for this work was provided by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under grants 96/080870 and 98/143843 and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under grant 300860/979.
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