MULTIVARIABLE MODEL PREDICTIVE CONTROL OF AN INDUSTRIAL FLUID CATALYTIC CRACKING (FCC) UNIT O. A. Z. SOTOMAYOR1, ✫, D. ODLOAK1, E. ALCORTA-GARCIA2, J. F. TISZA-CONTRERAS3 1

Department of Chemical Engineering, Polytechnic School of the University of São Paulo Av. Prof. Luciano Gualberto trav.3, n.380, 05508-900 São Paulo-SP, BRAZIL 2 Faculty of Mechanical and Electrical Engineering, Autonomous University of Nuevo León Pedro de Alba S/N, 66450 San Nicolás de los Garza-NL, MEXICO 3 Department of Electronic Engineering, Technological University of Peru Av. Petit Thouars 116, Lima 1, PERU E-mails: [email protected], [email protected], [email protected], [email protected] Abstract  The fluid catalytic cracking (FCC) unit is the functional heart, and the most profitably critical component, of a petroleum refinery. The purpose of this paper is to investigate the performance of an novel infinite-horizon model preditive control (IHMPC) algorithm for the multivariable control of an industrial FCC unit. Linear state-space process models are obtained by using subspace identification methods. These models are then converted to extract coefficients of the step response function, which are required by the IHMPC controller. From the control simulation, carried out on a well-defined FCC benchmark, it was shown the effectivenes of the proposed control strategy under realistic process perturbations. Keywords  FCC control, Model predictive control, Process control, Subspace identification, MIMO systems.

1

Introduction

Fluid catalytic cracking (FCC) is one of most important processes in the petroleum refining industry. The FCC unit is operated to convert large quantities of heavy petroleum fractions (ranging from 15000 – 95000 BPSD, approximately 40% of the overall refinery capacity) into light and more valuable hydrocarbon products, such as cracked naphtha (gasoline) and liquefied petroleum gas (LPG). The FCC unit is characterized by being physically and chemically complex and its operation is very difficult. It involves a highly nonlinear and multivariable process, with stiff dynamics, many cross-coupling and high amount of recycle, subject to a number of operational constraints, and working at high temperatures and pressures, with a slate of possible fires and explosions arising from malfunctions. This complex characteristic together with its economic significance makes the FCC unit a potential candidate and a challenging problem to the application of control systems and optimization algorithms (Moro and Odloak, 1995; Puebla et al., 2003). There are many studies in the literature addressing the problem of controlling FCC unit, covering from conventional PI(D) control, including cascade control, to advanced control systems, as nonlinear control, robust control, optimal control and model predictive control (MPC). If the stabilization of the system is the only purpose of the control, a regulatory control system, based on conventional ✫

Corresponding author.

PI(D) controllers, can be sufficient to achieve this goal. But, the increasing complexity of the processes and, mainly, the market globalization have motived the high automatization in refining and petrochemical plants, with the uses of more sophisticated and efficient control system, like MPC (Seborg, 1999). MPC control has several advantages over other control techniques because it incorporates an explicit process model into the control calculation and because of its close relationship with on-line optimization, which allows it to deal with multivariable process, coupling, inverse response, time-delays, process constraints, modelling uncertainties and measurement errors. MPC controllers are commonly found in the supervisory level of a hierarchical (plant-wide) control scheme (Skogestad, 2004). The FCC control problem is by definition a multivariable one rather than a multiloop problem and, hence, MPC control seems the best choice for this purpose. As a consequence, there are plenty of papers on MPC algorithms and their application to FCC units. For instance see Moro and Odloak (1995), Han et al. (2001), Jia et al. (2003) and Cristea et al. (2002, 2003). A recent survey conducted by Qin and Badgwell (2003) reveals about 2500 MPC applications in the petrochemical and refining industries. The objective of this paper is to study the performance of the infinite-horizon MPC algorithm (IHMPC) for the control of an industrial FCC unit. Because MPCs are based on linear process models identified by single step tests, the control performance may be seriously deteriorated when applied to a highly nonlinear process. In this work,

subspace identification methods are used aiming to obtain suitable linear state-space process models. These models are then converted to extract coefficients of the step response function, which are required by the IHMPC. The control performance is evaluated through simulations in which the IHMPC controller, under realistic perturbations, is applied to the FCC RECOPE benchmark. 2 The FCC Unit 2.1 Process Overview Before presenting the dynamic model of the process, the FCC process is briefly described. As a reference, it is considered the FCC unit Kellog Orthoflow model F at the Petrobras’s Henrique Lajes (REVAP) refinery in São José dos Campos. A simplified schematic diagram of this process is shown in Figure 1.

2.2 Dynamic Process Model The dynamic equations for the FCC RECOPE benchmark were adopted from the extensive mathematical model provided by Moro and Odloak (1995), which was validated with industrial data from the REVAP refinery. The benchmark, constituted by a 9x9 MIMO system, with 26 ODE’s and 74 linear and nonlinear algebric equations, was written in language C and implemented in Simulink/Matlab platform by the RECOPE Process Control Group, and it has become a standard for validation of FCC control structures in the Petrobras’s refineries. Model equations and process parameters are here omitted but they can be found in the aforementioned paper. For the stable operation of the FCC unit, the benchmark includes necessarily a conventional regulatory control system, based on three PI controllers (not shown in Fig. 1), aiming to keep the differential pressure between reactor and regenerator, the catalyst inventory in the reactor, and the suction pressure of the wet gas compressor at 0.65 kgf/cm2, 90 ton. and 1.0 kgf/cm2, respectively. The complete benchmark can be downloaded from the FCC Home Page in . 3 The MPC Control System

Figura 1. FCC Unit Kellog Orthoflow model F (See Table 1 for definition of the variables).

The process consists mainly of three major sections: the reactor, the regenerator an the riser. Preheater total oil feed (gasoil + deasphalted heavy oil), with a small amount of low pressure steam, is injected at the bottom of the riser along with the hot catalyst from the regenerator. This totally vaporizes the feed. The hot catalyst provides the heat of vaporization and the thermal energy necessary to carry out the endothermic catalytic cracking reactions. In the riser, by-product coke is deposited on the catalyst surface deactivating it. The spent catalyst is separated from the vapor in the reactor cyclone and falls into the stripping zone where entrained hydrocarbons are removed by stripping steam. The stripped spent catalyst is reactivated in the regenerator by an exothermic reaction, i.e. the coke is burned off the catalyst surface by the heater air blown into the bed. Reactor hydrocarbon gas stream is sent to the main fractionator for heat recovery and separation into various products, such as gasoline, LPG, fuel gas, heavy and light fuel oil and cycle oil.

MPC, or model-based predictive control (MBPC), has become an attractive control strategy especially for linear but also nonlinear systems subject to input, state and output constraints. It incorporates ideas from system theory, system identification and optimization. Through the use of an explicit process model, the MPC algorithm attempts to compute, at each control interval, an open-loop sequence of manipulated variable adjustments in order to optimize future plant behavior. Only the first input in the optimal sequence is applied to the plant, characterizing a moving (receding) horizon control. In this paper it is used the nominal version of the novel robust infinite-horizon MPC (IHMPC) controller proposed by Rodrigues and Odloak (2003). 3.1 The IHMPC Algorithm The IHMPC is based on a modified discrete-time state-space model of the process in which the output is represented as a continuous function of time, as:

[ x] k +1 = A[ x] k + B∆u k [ y (kT + t )] k = C(t)[ x] k

(1)

where u ∈ R nu is the input vector, ∆u k = u k − u k −1 is the input increment vector, y ∈ R ny is the output

[

]

T

x = ( x s ) T ( x d ) T ∈ C nx , is the state vector, nx = ny (1 + nu ⋅ na) , na is the model order,

vector,

where x s is related to the predicted steady state of the system output and x d is related to the dynamic modes of the system, k is the discrete sampling instant, T is the sampling time and t is the continuous time and matrices A , B and C(t) , with appropriate dimensions, are defined as: I 0  nx x nx , A=  ∈C 0 F  

C(t) = [I Ψ(t )]∈ C

 D0  nx x nu B= d  ∈C D FN 

and the d 0 ’s and d d ’s are coefficients of the step response obtained by partial fractions expansion of the system transfer functions. Taking advantage of this particular model form (1), the cost function is defined in terms of the integral of the output error along an infinite prediction horizon, as (Rodrigues and Odloak, 2003): m

and

n =1 ∞

T

{e(kT + t ) + δ k } Q{e(kT + t ) + δ k }dt + ( n −1)T

J k ,∞ = ∑ ∫

nT

T ∫mT {e(kT + t ) + δ k } Q{e(kT + t ) + δ k }dt +

ny x nx

∆u T R∆u + δ k T Sδ k

where r1 ,1,1⋅T

r1 ,1, na⋅T

r1 , nu ,1⋅T

r1 , nu , na⋅T

...e ...e ...e ... F = diag(e rny ,1,1⋅T rny ,1, na⋅T rny , nu ,1⋅T rny , nu , na⋅T ...e ...e ...e ...e ) ∈ C nd x nd ,

nd = ny ⋅ nu ⋅ na,

D d = diag (d1d,1,1 ...d 1d,1, na ...d 1d,nu ,1 ...d1d,nu ,na ... d d d d nd x nd ...d ny ,1,1 ...d ny ,1, na ...d ny , nu ,1 ...d ny , nu , na ) ∈ C

0 0 L 0 0 L M M L 0 0 L 1 0 L 1 0 L M M L 1 0 L M 0 0 L 0 0 L M M L 0 0 L

0 0 M  0 0  0 M  0   1  1 M  1

J i ∈ R nu ⋅na x nu 0 Φ1 (t )  0 Φ 2 (t ) Ψ (t ) =   M M  0  0

[

[

∆u = ∆u kT

∆u kT+1 L ∆u kT+ m −1

]

T

∈ R m⋅nu , m is

the control horizon, δ k ∈ R ny is a slack variable

 d10,1 L d10,nu    0 D = M O M  ∈ R ny x nu , 0 d 0  L d ny , nu   ny ,1

1 1  M  1 0   J1  0 J  2  nd x nu  ∈R , J i =  M N=  M  0    J ny   0  0 M  0

(2) where e(kT + t ) = y (kT + t ) − r (t ) is the output prediction error, r is the set-point, and

0  L 0  L ∈ C ny x nd , with O M   L Φ ny (t )

]

Φ i (t ) = e ri ,1,1⋅t ...e ri ,1,na⋅t ...e ri ,nu ,1⋅t ...e ri ,nu ,na⋅t ∈ C nd , ∀i = 1,2,.., ny . with the identity and zero matrices of suitable sizes. In these equations, the r ’s are the poles of the system

vector and Q ∈ R ny x ny , R ∈ R nu x nu and S ∈ R ny x ny are positive definite weighting matrices. The slack variables are introduced to overcome the problem of an unbounded cost due to offset in the controlled outputs or unfeasibility of the infinite-horizon control problem. For stable systems represented by the state-space model defined in (1), the IHMPC law is found by minimizing the cost function (2) subject to linear constraints on ∆u . Re-arranging (2) in terms of ∆u , the IHMPC may be formulated as:

[∆u

min ∆u ,δ

T

]

 ∆u 

s.t. − ∆u max ≤ ∆u k + j ≤ ∆u max , j −1

u min ≤ u k −1 + ∑ ∆u k + i ≤ u max , i =0

∆u k + j = 0,

[e ] s

k

∆u 

δ k T H   + 2c Tf   + c δ k  δ k 

+ δ k + D 0m ∆u = 0

j = 1, L , m   j = 1, L , m    j≥m  

where + H 11,∞ + R H12  H H =  11,m , T H m Q T + S 12 

[

c Tf = c Tfm + c Tf∞

with m

{

c Tfδ

]

H 11, m = ∑ D 0n QD 0n T + n =1

T

T

2D 0n Q(G1 (n) − G1 (n − 1))Wn Z +

}

Z T Wn T (G 2 (n) − G 2 (n − 1)) Wn Z H 11,∞ = Z T Wm T (G 2 (∞ ) − G2 (m)) Wm Z

(3)

(4)

[

n =1 m

{

T

[

]

]

c fm = ∑ [e s ] Tk Q D 0n T + (G1 (n) − G1 (n − 1)) Wn Z + n =1

[

[ x d ] Tk (G1 (n) − G1 (n − 1)) T QD 0n + (G 2 (n) − G 2 (n − 1)) Wn Z]}

c Tf∞ = [ x d ]Tk (G2 (∞) − G 2 (m)) Wm Z

Table 1. Manipulated and Controlled Variables for the MIMO Control System

Manipulated

m

H 12 = ∑ D 0n QT + Z T Wn T (G1 (n) − G1 (n − 1)) T Q

m

c Tfδ = m[e s ] Tk QT + [ x d ] Tk ∑ (G1 ( n) − G1 (n − 1))Q

{

Controlled

m

n =1

c = ∑ [e s ] Tk Q[e s ] k T + n =1

2[e s ] Tk Q[G1 (n) − G1 (n − 1)][ x d ] k +

}

[ x d ] Tk [G 2 (n) − G 2 (n − 1)][ x d ] k + [ x d ] Tk [G 2 (∞) − G 2 (m)][ x d ] k

(*)

Variable Total combustion air flowrate Regenerated catalyst valve position

Tag

Nominal Value

u1

221 ton/h

u2

82 %

Total oil feed flowrate

u3

9700 m3/d (*)

Temperature of the total oil feed Temp. regenerator 1st. stage dense phase Temp. regenerator 2nd stage dense phase Estimated cracking reaction severity

u4

235 °C

y1

670.14 °C

y2

700.88 °C

y3

77.5 %

Riser (reactor) temp.

y4

542.2 °C

≈ 61000 BPSD.

and D 0n

n 444  64447  8 0 0  = D D L D 0 0 L 0 ∈ R ny x m⋅nu    

[

]

Wn = I F −1 L F − ( n −1)

D N 0  0 Dd N Z=  M M  0  0 d

nT

G1 (n) =

L 0   L 0  ∈ C m⋅nd x m⋅nu O M   L D d N 

∫ Ψ(t )dt , G2 (n) = 0

0 L 0 ∈ C nd x m⋅nd

nT

∫ Ψ(t )

T

QΨ(t )dt ,

0

[e s ] k = [ x s ] k − r . The above minimization problem (3) subject to constraints (4) is solved by using nonlinear programming (NLP) solvers or linear matrix inequalities (LMI) solvers in the LMI Control Toolbox/Matlab. Note that only the first change of the computed ∆u is used. More details about the IHMPC can be found in Rodrigues and Odloak (2003). 3.2 Multivariable Control Structure Selection The control performance of a FCC process highly depends on the control structure selection (Hovd and Skogestad, 1993). This problem has been addressed by several authors, mainly focusing on possible combinations of manipulated and controlled variables for exact and partial control. In this paper, the control structure is based on the original version of the MPC controller proposed for the FCC unit of the REVAP refinery. From this scheme we consider only a 4x4 MIMO control system, where the manipulated and controlled variables, and its respective nominal steady-state values, are shown in Table 1.

3.3 Controller Model Identification In MPC control of processes with complex dynamics and predominant constraints, such as FCC unit, it is important that the process model be able to provide realistic predictions of relevant process variables over a relatively long horizon (Ljungquist et al., 1993). Typically, the model is developed from a series of open-loop experiments. In the industrial practice, it is common to use the step test to identify simple SISO linear models or to directly determine the step response coefficient for the MPC controller. However, according to Kalra and Georgakis (1996), the step test contains inadequate power to excite frequencies of interest to closed-loop control. Here, sixteen 2nd-order SISO models are obtained by individually exciting the manipulated variables with m-level pseudo-random signals, varying around ±2% of their nominal values, and using the MOESP subspace identification method (Verhaegen and Dewilde, 1992) to obtain approximate models and then to improve them by the PEM identification method (Ljung, 1999). Coefficients of the step response function are obtained from these models, which are required by the IHMPC. Figure 2 shows the step responses of the process and identified models. It can be seen the strong correlation between outputs y1 and y 2 and between y 3 and y 4 . 4 Control Simulations and Results

In most applications in refinery processing, the goal is largely to keep the process at a steady-state (regulator problem), rather than moving rapidly from one operating point to another (servo problem). The control objective considered in this FCC unit is to provide stable operation without large upsets, that could otherwise lead to a violation of operational, safety and environmental constraints, minimizing interactions effects and maintaining the

catalyst valve position ( u 2 ) that is governed directly by the MPC. However, in the present study, it is considered that the PI controllers and its corresponding inner-loop dynamics can be ignored in the simulations, i.e. the variables are assumed to reach the specified set-point instantaneously. To verify the MPC performance, the following potential FCC negative events were investigated: 4.1 Disturbance Rejection

Figure 2. Comparison of the Step Response for the FCC unit: Process (solid), Model (dashed). The Vertical Axis gives the Controlled Variables and the Horizontal Axis gives the Manipulated Variables (as Listed in Table 1).

controlled variables close to its nominal values where, usually, optimal operation conditions lie. Fine tuning of MPC controllers has an iterative character and the control performance enhancement can be performed, in a great extent, by recursive simulation. Several tuning guidelines have been proposed in the literature and they have concentrated on such aspects as stability and robustness. The IHMPC applied here is developed such that we can guarantee nominal stability. Heuristic rules and trial and error procedure were used for tuning it, aiming to achieve good FCC control performance. The selected controller tuning parameters are shown in Table 2.

At t = 50 min, a change in the feed composition produced by 5% step increase in the coking formation factor. This disturbance occurs regularly affecting severely the overall FCC operation and leads to violation of almost all operating constraints. More coke in the feed results in a rapid increase of coke deposition in the riser and concentration of coke on spent catalyst. This additional coke is transported to the regenerator resulting in higher combustion rates and thus an increase in regenerator temperature. To reject this disturbance and in order to stabilize the system, the IHMPC is able to decrease the total air flowrate, the amount of hot regenerated catalyst and the total oil feed flowrate. As can be seen in Figure 3, the output variables are well-controlled providing appropriate control performance.

Table 2. Tuning parameters for the IHMPC controller

Description Sampling time Control horizon Output prediction error Weighting matrix Input increment Weighting matrix Slack variables Weighting matrix Number of inputs Number of outputs Order of the model Lower limits of the Inputs (*) Upper limits of the Inputs (*) Upper limits of the input increments

Value

T = 3 min. m=2 Q = diag (1, 1, 1, 1) R = diag (0.01, 0.01, 0.01, 0.01)

S = diag (500, 500, 500, 500)

nu = 4 ny = 4 na = 2

Figure 3. Response of the FCC Process with MPC Control for a 5% Step Increase in the Coke Formation Factor (solid) and Disturbed Process without Control (dashed).

u min = [-11, - 0.37, - 50, - 15]T

4.2 Fault Tolerance

u max = [19, 0.13, 50, 20] T

According to Maciejowski (2002), MPC has certain fault tolerant properties even in the absence of any knowledge of the failure, i.e. by itself it constitutes a passive fault tolerant MPC control. At t = 50 min, a fault in the air pre-heater system produces 20% step decrease in the combustion air temperature. This fault reduces the regenerator temperature and increases the coke concentration in the regenerated catalyst. A lower regenerated catalyst temperature is transported to the riser, decreasing the endothermic reactions and leading to a decrease in the riser (reactor) temperature.

∆u max = [2.5, 0.05, 5, 4]T

(*) Related to its nominal steady-state values, respectively.

In the real case, the MPC of the FCC unit was designed in a two-level hierarchical control scheme, acting at the top level by cascading conventional PI controllers in the lowest level, such that all the manipulated variables are set-points to the conventional PI controllers except for the regenerated

To accommodate this fault by compensating its effects, the IHMPC controller increases the total air flowrate and the temperature of the oil feed in order to augment the regenerator temperature. The total oil feed flowrate is reduced until it reaches its lower limit value. This input limitation does not seem to have a negative impact on the controlled variables, as can be observed in Figure 4 and, therefore, the IHMPC succeeds in maintaning a good control performance.

Figure 4. Response of the FCC Process with MPC Control for a 20% Step Decrease in the Combustion Air Temperature (solid) and Faulty Process without Control (dashed).

5 Conclusions

The control performance of the IHMPC controller was investigated by simulations for an industrial FCC unit under realistic perturbations. The perturbations are considered unexpected and unmeasured, and their effects are not feedforwarded to the controller. The use of a conservative tuning of the IHMPC resulted in a FCC well-controlled for both investigated cases, disturbance rejection and fault tolerance, showing good potential for real-life applications. IHMPC control performance under failure occurrence in FCC control devices not have been investigated. These faults can be treated through redundancies, along with reliable fault detection and isolation (FDI) system. The integration FDI-IHMPC provides the framework for an active fault tolerant MPC control. This is our proposal for future works. Acknowledgements

The authors would like to thank the finantial support from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), under grant N°02/08119-2. References

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