Design Margin and Control Performance Analysis ... - ACS Publications

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Design Margin and Control Performance Analysis of a Fluid Catalytic Cracking Unit Regenerator under Model Predictive Control Feng Xu, Xionglin Luo,* and Rui Wang Department of Automation, China University of Petroleum, Beijing 102249, China S Supporting Information *

ABSTRACT: The design margin is defined as the value added on the nominal value of a design variable, which must be determined not only for process uncertainties but also for dynamic control, but its size cannot be too large in consideration of device and operation costs. In the fluid catalytic cracking unit (FCCU), the catalyst inventory and the air flow rate of regenerator are important design variables, so it is necessary to analyze the relationship between their margins and control performance. In this paper, the design margins of catalyst inventory and air flow rate under model predictive control are solved via dynamic optimization. A linear model predictive control is used based on the linear state space model obtained by linearization of the steady-state nominal operating point. The relationship between the control performance and the design margin is discovered by changing the prediction horizon. It can be found that improving the control performance requires more air flow rate margin but less catalyst inventory margin for the process. An inflection point on the relationship curve exists between control performance and air flow rate margin. The prediction horizon of the model predictive control should be determined based on the inflection point to improve the process control performance significantly with a lower air flow rate margin cost.

1. INTRODUCTION The margin of process design is defined as the value added on the nominal value of the design variable from the process uncertainties. It is always hoped that an ideal designed process will achieve optimal economic performance and simultaneously meet all process constraints. Therefore, the steady-state operating point of the ideal designed process generally depends on process constraints. However, process uncertainties may perturb the plant from the desired steady-state optimum operating point and result in violations of active constraints. Sufficient margins must be added onto the design variables in the presence of these uncertainties to prevent process variables from breaking process constraints. The operating point will then move into the feasible region. Proper distances between the current operating point and active constraints exist. The process variables will vary dynamically over time when disturbances occur. It often happens that the steady-state value is within the range of constraint but some of the dynamic values during the transient process are outside this range. Therefore, the design margins must be determined not only for the steadystate uncertainties but also for the dynamic response relevant to disturbances and the control system. At the same time, its size should not be too large considering the device cost. Considering that the process control system must have the ability to realize dynamic control and overcome external disturbances, the design margins and control system should be determined simultaneously through dynamic optimization. Hence, the optimal process design considering process dynamic performance, also called as simultaneous design and control,1−4 has been developed to achieve good control performance and low-cost process design. People have attempted to design economically optimal processes that could operate in an efficient dynamic mode within an envelope around the nominal point. Dynamic optimization is thus © XXXX American Chemical Society

employed to determine the most economic process design and control system satisfying all dynamic operability constraints. Most of the methodologies developed for the simultaneous design and control were conventional feedback proportional− integral (PI) controllers;1,2,5,6 some other control algorithms such as optimal control,7,8 internal model control,9 and robust control10−12 were also applied. Optimization algorithms have developed from conventional mixed integer nonlinear programming13 to a heuristic particle swarm optimization (PSO) algorithm.14 Now model predictive control (MPC) is mature enough after almost three decades of implementation.15−17 MPC ensures optimal control through a rolling optimization and robustness through feedback correction, which can provide better control performance than conventional proportional− integral−derivative (PID) feedback controllers. Therefore, it is desired to incorporate MPC in the simultaneous design and control methodology despite some of the computational challenges. Previous works that implemented MPC control algorithms in the simultaneous design and control are available.18−20 The authors integrated margin analysis into the simultaneous process and control design.21,22 It is proposed that the design margin should be divided into the steady-state and dynamic margin, in which the dynamic margin is necessary for control and is related to the control system. A higher requested control performance requires more dynamic margins. A process with more margins is easily controllable but involves more cost, whereas the process with fewer margins is difficultly controllable but entails less cost. We also analyzed how the design Received: January 16, 2014 Revised: August 20, 2014 Accepted: August 21, 2014

A

dx.doi.org/10.1021/ie5026002 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

Article

2.1. Steady-State Design Model of Regenerator. In this work, we suppose that the design of the reactor has been finished and the operation condition of the reactor is fixed, so we only discuss the design of the regenerator. The process data of the reactor are shown in Table 1.

margins influence the economic benefit of operation optimization and control performance to find the bottlenecks of process design.23 A fluid catalytic cracking unit (FCCU) is the main unit of oil deep processing in a modern refinery, and the most important and complex part is the reactor/regenerator system. Many researchers have established the dynamic model of different FCCU units for operation analysis and control design.24−27 The reactor possesses great operating flexibility, so the design margin is mainly concentrated on the regenerator that mainly comprises catalyst inventory and the air flow rate. The authors analyzed the relationship between the manipulation margin (air flow rate) and control performance of the FCCU under MPC.28 On this basis, further study on the relationship between the control performance and the corresponding required margins including catalyst inventory and air flow rate will be performed under MPC. The margins of air flow rate and catalyst inventory are derived by solving a dynamic optimization problem. We obtain the relationship between control performance and design margins by changing the parameters of MPC, from which the proper margins can be determined.

Table 1. Process Data of Reactor name

value

units

press. of reactor top temp of reactor top fresh feed flow rate recycle oil flow rate recycle slurry flow rate catal circulation flow rate yield of C1−C4 vs fresh feed oil yield of naphtha vs fresh feed oil yield of diesel vs fresh feed oil yield of coke vs fresh feed oil

245 250 773.15 75 000 12 750 7250 410 000 14.1 42.3 38.6 5.0

Pa K kg·h−1 kg·h−1 kg·h−1 kg·h−1 wt % wt % wt % wt %

From the operation condition of the reactor in Table 1, we can obtain the amount of carbon that has to be burned off. Because sufficient CO promoters are added to achieve the complete combustion of coke, we assume no CO existed in the regenerator, and the reactions in the regenerator are

2. MATHEMATICAL MODEL OF FCCU WITH A SINGLE-STAGE REGENERATOR An FCCU with a single-stage regenerator under complete combustion is adopted as a study case in this work. The schematic diagram of the FCCU is illustrated in Figure 1, from

C + O2 → CO2 H+

1 1 O2 → H 2O 4 2

Based on the steady-state design model, the carbon burnoff rate and O2 in the flue gas can set the air flow rate and catalyst inventory. The single-stage regenerator usually consists of a dense phase section and a dilute phase section. Usually there is 80% catalyst inventory in the dense phase section and 20% catalyst inventory in the dilute phase section. We suppose that the burning of carbon mainly occurs in the dense phase section and there is only gas−solid separation in the dilute phase section. The dense bed can be simulated by two virtual gas−solid continuous stirred-tank reactors (CSTRs) in series for the single-stage regenerator. We suppose that hydrogen has been burned up in the first CSTR due to the higher reaction rate of hydrogen than that of carbon and there is only carbon burning reaction in the second CSTR. Based on the mass and heat balance, the steadystate design model can be given as follows:

Figure 1. Schematic diagram of FCCU.

⎛ E ⎞ ⎛ C ⎞ GCst⎜ st − Crg1⎟ − 0.4Wrgk C0 exp⎜⎜ − C ⎟⎟Crg1prg yrg1 ⎝1 + η ⎠ ⎝ RTrg1 ⎠

which it is shown that the FCCU is mainly composed of a reactor and a regenerator. The feed oil of the FCCU is mainly vacuum gas oil, and the annual processing capacity of FCCU is 600 000 tons. The regenerator is a single-stage regenerator with sufficient CO promoters added to achieve the complete combustion of coke. In this work, we only consider the design margin of the regenerator that mainly consists of catalyst inventory and the air flow rate. However, given the catalyst circulation between the reactor and regenerator in FCCU that builds the organic link between two devices, we need to consider the reactor and regenerator as a whole to discuss the air flow rate and catalyst inventory margins of the regenerator.

(1)

=0

Vrg(0.21 − yrg1) − − =0 B

⎛ C ⎞ 1 GCst⎜ st − Crg1⎟ 12 ⎝1 + η ⎠

⎛ ηCst ⎞ 1 GCst⎜ ⎟ 4 ⎝1 + η ⎠ (2) dx.doi.org/10.1021/ie5026002 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX


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1−10 are solved to calculate the air flow rate Vrg and catalyst inventory Wrg; then the design of the regenerator can be finished. The process data of the designed regenerator are shown in Table 3. The air flow rate and catalyst inventory in Table 3 will be the basis of margin analysis in this work.

GCstcps(Tst − Trg1) + Vrgcpg(Tg,inlet − Trg1) ⎛ C ⎞ ηCst + GCst⎜ st − Crg1⎟( −ΔHC) + GCst ( −ΔHH) 1+η ⎝1 + η ⎠ (3)

=0 GCst(Crg1 − Crg2) − 0.4Wrgk C0

⎛ E ⎞ exp⎜⎜ − C ⎟⎟Crg2prg yrg2 = 0 ⎝ RTrg2 ⎠

Table 3. Process Data of Designed Regenerator

(4)

Vrg(yrg1 − yrg2 ) − GCst

Crg1 − Crg2 12

=0

(5)

GCstcps(Trg1 − Trg2) + Vrgcpg(Trg1 − Trg2) + GCst(Crg1 − Crg2)( −ΔHC)

where prg = prgf +

0.2Wrgg

units Pa K mol % wt % kmol·h−1 kg m m

Ωdi

+

2.2. Dynamic Simulation Models of Riser/Regenerator. 2.2.1. Reactor. The model of the reactor adopted in this work is derived from the model proposed by Luo in the appendix of ref 27. The reactor can be mainly divided into two parts: a riser reactor where the cracking reactions occur and a stripper where the products and catalyst are separated. A distributed parameter model with the assumption of gas− solid plug flow and no slip is built for the riser reactor. The dynamic response time of gas−solid material can be ignored because of the short residence time of gas−solid material. Thus, the quasi-steady-state ordinary differential equations are used for material balance of the feed oil mass fraction, product oil mass fractions (diesel, naphtha, and gas), and coke mass fraction. However, the reaction temperature must be modeled as a dynamic partial differential equation since the heat capacity of the thermal insulating layer of riser influences the heat balance. A five-lump model is adopted for the cracking kinetic model as follows:

0.4Wrgg Ωde

(7)

According to the design specifications of regenerator, we set the superficial gas velocity of the dense phase section as ude = 1.1 m s−1 and the superficial gas velocity of the dilute phase section as udi = 0.6 m s−1. When the air flow rate Vrg and catalyst inventory Wrg are set, we can calculate the crosssectional area of the dense phase section and the dilute phase section through the following equations.

Ωde =

value 274 680 973.15 3.0 0.1 2030 23 600 4.7 6.0

(6)

=0

Ωdi =

name press. of regenerator top temp of regenerator O2 mole fraction in flue coke mass fraction of regenerated catal air flow rate catal inventory diam of dense phase section diam of dilute phase section

⎞ RTrg2 1 ⎛ − 0.1Wrgg ⎟ ⎜1000Vflue prgf ⎝ udi ⎠ 1 prgf +

0.2Wrgg Ωdi

(8)

⎛ ⎞ RTrg2 − 0.4Wrgg ⎟ ⎜1000Vflue ude ⎝ ⎠

(9)

rA

where

gas oil (A) → υADdiesel (D) + υAN naphtha (N)

Vflue = Vrg +

ηCst 1 GCst 4 1+η

+ υAG gas (G) + υACcoke (C)

(10)

rD

diesel (D) → υDN naphtha (N) + υDG gas (G)

In eqs 1−6, the spent catalyst circulation rate GCst, the coke mass fraction of spent catalyst Cst, and the temperature of spent catalyst Tst can be obtained from the operation condition of the reactor. When the air flow rate Vrg and catalyst inventory Wrg of the regenerator are set, the coke mass fraction of regenerated catalyst Crg2, the temperature of regenerated catalyst Trg2, and the O2 mole fraction of flue gas yrg2 can be calculated through solving eqs 1−10. The design constraints of the regenerator are shown in Table 2. If we set the O2 mole fraction in flue yrg2 = 3 mol % and the coke mass fraction of regenerated catalyst Crg2 = 0.1 wt %, eqs

+ υDCcoke (C) rN

naphtha (N) → υ NG gas (G)

All the cracking reactions combined with a catalyst deactivation reaction are of first order. According to the mass balance of differential elements, the following differential equations are given.

Table 2. Design Constraints of Regenerator name temp of regenerator O2 mole fraction in flue coke mass fraction of regenerated catal

⎛ GCrg ⎞ = S T⎜ ⎟rAp φ dX ⎝ FO ⎠ ra

(11)

⎛ GCrg ⎞ = S T⎜ ⎟(υADrA − rD)pra φ dX ⎝ FO ⎠

(12)

⎛ GCrg ⎞ = S T⎜ ⎟(υANrA + υDNrD − rN)pra φ dX ⎝ FO ⎠

(13)

dyA

dyD

lower bound

upper bound

units

953.15 2.0 0

993.15 5.0 0.15

K mol % wt %

dyN

) C


Industrial & Engineering Chemistry Research ⎛ GCrg ⎞ = S T⎜ ⎟(υAGrA + υDGrD + υ NGrN)pra φ dX ⎝ FO ⎠

Article ρL Ω LcpL

dyG

cpO +

where ⎛ E ⎞ rA = −kA0 exp⎜ − A ⎟yA ⎝ RTra ⎠

(16)

⎛ E ⎞ rN = −kN0 exp⎜ − N ⎟yN ⎝ RTra ⎠

(17)

rD = −kD0

cpO +

1 k OFO2 (18) 2 where Δpdis is the constant static pressure drop of the dilute phase of the disengager, kO is the resistance coefficient of feed oil flow. yN |X = 0 = 0,

(20)

The catalyst deactivation function is given by

YC = GCrg (22)

boundary conditions:

YD = (23)

YN =

∂Tra 1 ⎛ 1 ⎞ ∂Tra ⎜ ⎟ + ST ⎝ 1 + Γ ⎠ ∂X ∂t ⎛ Λ ⎞⎛ GCrg ⎞ ⎟⎜ = −⎜ ⎟[r ΔH + rDΔHDR + rNΔHNR ⎝ 1 + Γ ⎠⎝ FO ⎠ A AR

pw

(27)

β FO,freshCCR,fresh + γ GCrg

(28) (29)

(30)

Cst − Crg2 FO,fresh

·100%

yD |X = 1 FO − FOyA |X = 1 − GCrg (Cst − Crg2) FO,fresh

yD |X = 1 + yN |X = 1 + yG |X = 1

(31)

·100%

yN |X = 1 FO − FOyA |X = 1 − GCrg (Cst − Crg2) FO,fresh

yD |X = 1 + yN |X = 1 + yG |X = 1

·100% (33)

YG =

yG |X = 1 FO − FOyA |X = 1 − GCrg (Cst − Crg2) FO,fresh

yD |X = 1 + yN |X = 1 + yG |X = 1

·100% (34)

(24)

2.2.2. Regenerator. For the single-stage regenerator, we suppose that the burning of carbon mainly occurs in the dense phase section and there is only gas−solid separation in the dilute phase section. The dense bed can be simulated by two virtual gas−solid continuous stirred-tank reactors (CSTRs) in series. The residence time of flue gas is much shorter than that of solid catalyst, so the dynamic response time of the gas mole fraction can be neglected. Thus, quasi-steady-state algebraic equations are used for the O2 mole fraction of flue gas, but dynamic differential equations are still used for the coke mass fraction of catalyst and temperature. The equations of the FCCU regenerator are given as follows:

boundary conditions: Tra|X = 0 ∑ FO, icpO, iTO, i + GCrg cpsTrg2 + Fwcp wTw − FO, iΔHV, i ∑ FO, icpO, i + GCrg cps + Fwcp w

i = fresh, hco, slurry

Fw FO

(32)

According to the heat balance of differential elements, the following partial differential equations about reaction temperature are given.

− (υACrA + υDCrD)ΔHAA ]pra φ

ps

FO

The product yield (coke, diesel, naphtha, and gas) of the reactor can be calculated by

(21)

φ|X = 0 = 1 − 50Crg2

( )c + ( )c

dWst = GCrg − GCst dt

boundary conditions:

⎛ Eφ ⎞ dφ = −STkφ0 exp⎜ − ⎟y p φ dX ⎝ RTra ⎠ A ra

(26)

where β is the ratio of the carbon residue of the feed converted to coke; γ is the amount of unstripped hydrocarbon which depends on the stripping condition. The dynamics of the inventory is considered since it is an important variable for control of the FCCU. The dynamics of the catalyst inventory is determined by

yG |X = 0 = 0 (19)

Cra|X = 0 = Crg2

pw

Tst = Tra|X = 1 − ΔTst

The catalytic coke from catalytic cracking reactions is calculated by dCra = ST(υACrA + υDCrD)pra φ dX

GCrg

Cst = Cra|X = 1 +

boundary conditions: yD |X = 0 = 0,

Fw FO

ps

FO

The stripper is modeled as a well-mixed tank with the assumption that no further reactions occurred. The coke of spent catalyst is from the oil cracking reaction, feed oil, and unstripped hydrocarbon. The outlet temperature of the stripping section is equal to the outlet temperature of the riser minus a constant temperature difference. The equations of the stripping section are given as follows:

pra = praf + Δpdis +

yA |X = 0 = 1,

GCrg

( )c + ( )c 1

Λ=

(15)

⎛ E ⎞ exp⎜ − D ⎟yD ⎝ RTra ⎠

=

ρO, i Ω ra

Γ=

(14)

(25)

where Γ and Λ are the heat capacity correction coefficients for the effects of the thermal insulating layer and reaction temperature, respectively. D



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⎛ E ⎞ ⎛ C ⎞ G = ⎜ st − Crg1⎟ Cst − k C0 exp⎜⎜ − C ⎟⎟Crg1 ⎝1 + η ⎠ 0.4Wrg ⎝ RTrg1 ⎠

dCrg1 dt

prg yrg1

where KCst is a static pressure constant of spent catalyst circulation and ΔpCst is the pressure drop of the spent catalyst slide valve.

(35)

ΔpCst = 9722.7

⎞ ⎛ C 1 GCst⎜ st − Crg1⎟ 0 = Vrg(0.21 − yrg1) − 12 ⎠ ⎝1 + η − dTrg1 dt

⎛ ηCst ⎞ 1 GCst⎜ ⎟ 4 ⎝1 + η ⎠

=

GCst

0.4[ρs (1 − ε)cps + ρg εcpg ]Wrg

[GCstcps(Tst − Trg1)

dt

dTrg2 dt

K Crg + prgf +

=

ρs (1 − ε) 0.4[ρs (1 − ε)cps + ρg εcpg ]Wrg

(41)

dt

(42)

(43)

1000RT

= GCst − GCrg

(44)

2.2.3. Pressure Balance. The pressure balance is necessary for the FCCU to ensure the catalyst circulation between the reactor and the regenerator. The catalyst circulation rates can be calculated through the relation that the driving force equals the resistance. For the spent catalyst circulation, driving force = resistance, that is

= prgf +

GCrg 2 ρCrg (vCrgzCrg)2

(49)

2.2.4. Necessary PID Controllers of FCCU. Because of the steady-state multiplicity of the exothermic reaction system,29−31 the operating point of the FCCU is often not open-loop stable. To ensure the stability of the FCCU, the catalyst inventory in the stripper and the pressure of the regenerator are controlled necessarily.27 Due to the quick dynamic response time of flue gas, we consider the pressure of the regenerator ideally controlled. Therefore, the necessary closed-loop PID controller added in the dynamic model of the FCCU is the controller of the catalyst inventory in stripper Wst, whose manipulated variable is the opening of the spent catalyst slide valve zCst. The dynamic equations of the PID controller are described as follows:

prg

Wg + st Ωst 0.2Wrgg Ωdi

+

(48)

(50)

ρg Ωde

K Cst + praf

Ωde

ηC

Vrg + 4 GCst 1 +stη

The catalyst inventory of the regenerator is given by

dWrg

0.4Wrgg

1/2 0.2Wrgg 0.4Wrgg ⎛ 2⎞ ⎜ K Crg + (prgf − praf ) + Ωdi + Ωde − k OFO ⎟ ⎜ ⎟ 9722.7 ⎜ ⎟ + k Crg ρCrg (vCrgz Crg)2 ⎝ ⎠

where

ρg =

Ωdi

+

GCrg =

(40)

1

⎟ ⎟⎟ ⎠

Hence, the regenerated catalyst circulation rate is calculated as follows:

[GCstcps(Trg1 − Trg2)

+ Vrgcpg(Trg1 − Trg2) +GCst(Crg1 − Crg2)( −ΔHC)]

ude =

0.2Wrgg

ΔpCrg = 9722.7

(39)

ε = f (ude)

Ωde

where KCrg is a static pressure constant of regenerated catalyst circulation and ΔpCrg is the pressure drop of the regenerated catalyst slide valve.

Crg1 − Crg2 12

0.8Wrgg ⎞1/2

= praf + ΔpCrg + k CrgGCrg 2 + k OFO2

(38)

0 = Vrg(yrg1 − yrg2 ) − GCst

0.2Wrgg Wstg ⎛ ⎜ K Cst − (prgf − praf ) + Ωst − Ωdi − =⎜ 9722.7 ⎜ + k Cst ρCst (vCstz Cst)2 ⎝

For the regenerated catalyst circulation, driving force = resistance, that is

⎛ E ⎞ G = (Crg1 − Crg2) Cst − k C0 exp⎜⎜ − C ⎟⎟Crg2prg 0.4Wrg ⎝ RTrg2 ⎠ yrg2

(46)

(47)

⎛ C ⎞ + Vrgcpg(Tg,inlet − Trg1) + GCst⎜ st − Crg1⎟ ⎝1 + η ⎠ ηCst ( −ΔHC) + GCst ( −ΔHH)] 1+η (37)

dCrg2

ρCst (vCstzCst)2

Hence, the spent catalyst circulation rate is calculated as follows: (36)

ρs (1 − ε)

GCst 2

0.8Wrgg Ωde

⎡ 1 u(t ) = KP⎢e(t ) + TI ⎣

∫t

t

e(t ) + TD 0

de(t ) ⎤ ⎥ + u0 dt ⎦

(51)

In this work, unlike MPC, the PID controller is the necessary part of the dynamic simulation model of the FCCU, which cannot be removed owing to the stability. Otherwise, the running rates of PID and MPC controllers are also different; the sampling period of the PID controller is within milliseconds but the minimum sampling period of MPC is 0.5 min. Therefore, the PID controller can be treated as continuous

+ ΔpCst + k CstGCst 2 (45) E



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control but the MPC controller should be treated as discrete control. 2.2.5. Constraints of Dynamic Simulation Model. The process constraints comprise the production requirements and operating constraints, such as the outlet temperature of the riser, product yield, temperature of the regenerator, coke mass fraction of regenerated catalyst, oxygen mole fraction of flue gas, and so on. The process constraints when the FCCU is running are shown in Table 4.

We use eqs 55 and 56 in eq 54 at the nominal steady-state operating point δxd(0) = 0, and then transform the control variables into their control increments as follows: δycor ̂ (k + P) = δy(k) + S(P)Δu(k) k

+

temp of reactor top yield of coke to fresh feed oil yield of naphtha to fresh feed oil yield of diesel to fresh feed oil yield of C1−C4 to fresh feed oil temp of regenerator O2 mole fraction in flue coke mass fraction of regenerated catal opening of spent catal slide valve opening of regenerated catal slide valve press. drop of spent catal slide valve press. drop of regenerated catal slide valve superficial gas velocity of dense phase section in regenerator superficial gas velocity of dilute phase section in regenerator

lower bound

upper bound

units

763.15 4.5 40 20 10 953.15 2.0 0 10 10 19 600 19 600 1.0

783.15 6.0 60 40 20 993.15 5.0 0.15 90 90 39 200 39 200 1.2

K wt % wt % wt % wt % K mol % wt % % % Pa Pa m s−1

0.6

0.7

m s−1

∑ij=1

where Δu(k) = δu(k − 1) = u(k − 1) and S(i) = CG H, Δu(k) are the control increments, and S(i) is the step response at the ith sampling period based on the discrete state space model. We define δysp = ysp − y0 and the objective function of optimal control by min ∥δysp − δycor(k + P)∥Q + ∥Δu(k)∥R. The optimum control increments of predictive control can then be obtained

(52)

δy(k) = C δxd(k) + D δu(k)

(53)

k

{ysp − y(k) −

(58)

3. DYNAMIC OPTIMIZATION FOR DESIGN MARGINS OF REGENERATOR UNDER MPC On the basis of the steady-state design model of the regenerator, the dynamic continuous simulation model of the FCCU, and the dynamic discrete controller model of MPC, we can give the mathematical description of dynamic optimization to calculate the air flow rate and catalyst inventory margins. For the steady-state design model of the regenerator, we define x0,rg = [Trg1(0) Trg2(0) Crg1(0) Crg2(0) yrg1(0) yrg2 (0)]T

as state variables d = [Wrg(0) Vrg,max ]T

as design variables u 0,rg = [Vrg(0)]T

as manipulated variables p = [k C0]T

as uncertain parameter variables, and

P

δycor ̂ (k + P) = CG δxd̂ (k) + (∑ CG

i−1

H + D)

ysp,rg = [Trg2(0) yrg2 (0)]T

i=1

δu(k) + δy(k) − δyP̂ (k)

as set points of controlled variables. The steady-state design model equations and constraints in section 2.1 can be tranformed into

(54)

where k

δyP̂ (k) = CGk δxd(0) +

∑ CGi− 1H δu(k − i) + D i=1

δu(k)

δxd̂ (k) = G δxd(0) +

f0 (x0,rg , d , u 0,rg , p) = 0

(59)

ysp,rg = h0(x0,rg , d , u 0,rg , p)

(60)

g0(x0,rg , d , u 0,rg , p) ≤ 0

(61)

Vrg(0) ≤ Vrg,max

(62)

(55) k

k

∑ [S(P + i) − S(i)]Δu(k − i)} i=1

where δxd(k) = xd(k) − xd,0, δu(k) = u(k) − u0, and δy(k) = y(k) − y0. A single-valued predictive control algorithm is used to predict the future estimated values of controlled output variables at the prediction horizon based on the discrete state space model in eqs 52 and 53. When the current values of controlled output variables are introduced for feedback correction, the corrected future values of controlled variables are given as follows: P

∑G i=1

j−1

Δu(k) = [ST(P)QS(P) + R ]−1 ST(P)Q

2.3. Dynamic Discrete Controller Model of MPC. The discrete mathematical model of MPC is obtained by linearization of the nominal operating point of steady-state design, that is δxd(k + 1) = G δxd(k) + H δu(k)

(57)

i=1

Table 4. Process Constraints of FCCU name

∑ [S(P + i) − S(i)]Δu(k − i)

i−1

H δu(k − i) (56) F



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Figure 2. Dynamic trend of disturbances.

For the dynamic continuous simulation model of FCCU, we define

G = e ATs ,

xd(t ) = [Tra(t ) Trg1(t ) Trg2(t ) Crg1(t ) Crg2(t ) Wst(t ) Wrg(t )]T

xa(t ) = [yA (t ) Tst(t )

Cst(t )

yD (t )

yN (t )

yrg1(t )

yG (t )

Cra(t )

H=

⎛ ∂f ⎞ A=⎜ d⎟ ⎝ ∂xd ⎠

φ (t )

⎛ ∂f ⎞ B= ⎜ d⎟ ⎝ ∂u ⎠

xa,0 = xa(0), x0 =

x0 , u0

⎛ ∂fa ⎞ ⎜ ⎟ ⎝ ∂xa ⎠

(68) −1

x0 , u0

⎛ ∂fa ⎞ ⎜ ⎟ ⎝ ∂xd ⎠

x0 , u0

(69)

u 0 = u(0),

x0 , u0

⎛ ∂f ⎞ − ⎜ d⎟ ⎝ ∂xa ⎠

x0 , u0

⎛ ∂fa ⎞ ⎜ ⎟ ⎝ ∂xa ⎠

−1

x0 , u0

⎛ ∂fa ⎞ ⎜ ⎟ ⎝ ∂u ⎠

x0 , u0

(70)

⎛ ∂h ⎞ C=⎜ ⎟ ⎝ ∂xd ⎠

y(t ) = [Tra|X = 1(t ) Trg2(t ) yrg2 (t )]T

y0 = y(0),

e At B dt

yrg2 (t )]

v(t ) = [ FO,fresh(t ) CCR,fresh(t ) praf (t ) Tg,inlet(t )]T

T [ xd,0

⎛ ∂f ⎞ − ⎜ d⎟ ⎝ ∂xa ⎠

Ts

T

u(t ) = [ zCrg(t ) Vrg(t )]T

xd,0 = xd(0),

x0 , u0

∫0

v0 = v(0),

x0 , u0

⎛ ∂h ⎞ −⎜ ⎟ ⎝ ∂xa ⎠

x0 , u0

⎛ ∂fa ⎞ ⎜ ⎟ ⎝ ∂xa ⎠

−1

x0 , u0

⎛ ∂fa ⎞ ⎜ ⎟ ⎝ ∂xd ⎠

x0 , u0

(71)

T T ] xa,0

⎛ ∂h ⎞ D=⎜ ⎟ ⎝ ∂u ⎠

The dynamic continuous simulation model of FCCU in section 2.2 can be transformed into

x0 , u0

⎛ ∂h ⎞ −⎜ ⎟ ⎝ ∂xa ⎠

x0 , u0

⎛ ∂fa ⎞ ⎜ ⎟ ⎝ ∂xa ⎠

−1

x0 , u0

⎛ ∂fa ⎞ ⎜ ⎟ ⎝ ∂u ⎠

x0 , u 0

(72)

dxd(t ) = fd (xd(t ), xa(t ), u(t ), v(t ), p) dt

(63)

fa (xd(t ), xa(t ), u(t ), v(t ), p) = 0

(64)

y(t ) = h(xd(t ), xa(t ), u(t ), v(t ), p)

(65)

(73)

g (xd(t ), xa(t ), u(t ), v(t ), p) ≤ 0

(66)

The objective function of dynamic optimization for minimum margin is given by

Vrg(t ) ≤ Vrg,max

(67)

Based on the optimum control of MPC in section 2.3, the discrete dynamic model of the multivariable predictive controller can be summarized as fc (ysp , y(k), u(k), u(k − 1), ..., u(0), A , B , C , D) = 0

J = c1MW + c 2MV

(74)

where c1 and c2 are the total cost coefficients involving the equipment investment cost and operating cost for catalyst inventory and the air flow rate, respectively. MW and MV are the relative percentage margins of catalyst inventory and the air flow rate.

When the steady-state design of the regenerator is achieved, the nominal steady-state operation point of the FCCU will be determined. The mathematical model of MPC can be obtained through linearization on the steady-state nominal operating point. G


Industrial & Engineering Chemistry Research MW =

MV =

Wrg(0) − Wrg ̅ Wrg ̅ Vrg,max − Vrg̅ Vrg̅

Article N

·100%

IAE = (75)

N

∑ Q |ysp − y(k)| + ∑ R|Δu(k)| k=1

k=1

(78)

Low IAE implies better control performance. The prediction horizon is the most important control parameter in MPC. A smaller prediction horizon implies more powerful control of manipulated variables if the closed-loop system is stable, resulting in better control performance. We reduced the prediction horizon gradually to improve the control performance index IAE. Under different prediction horizons, we solved the dynamic optimization problem to obtain the air flow rate margin and the inventory margin of the regenerator, and at the same time the control performance index IAE was calculated. We then drew the relationship curves of the air flow rate margin, the regenerator inventory margin, and the control performance index IAE to the prediction horizon (Figure 3).

·100% (76)

where W̅ rg and V̅ rg are the base values of catalyst inventory and the air flow rate in Table 3. Process uncertainties include parameter uncertainties, variations of the operating point, and process disturbances. The parameter uncertainties are considered as steady-state change, and their influences on the dynamic process can be ignored. The variations of the operating point can be regarded as step changes. The variation mode of process disturbances is more complicated: the dynamic model is given by v(t ) = v0 + v1 sin(ω1t + ϕ1) + v2(t − t 2) + v3(1 − e−t / T3)

(77)

where v0 is the nominal value, v1 is the amplitude of the sinusoidal variation, v2 is the amplitude of the step change, and v3 is the amplitude of the first-order inertial change. In this paper, the reaction rate constant of burning carbon kC0 is regarded as an uncertain parameter, which decreases by 10% of its nominal value. For the variations of the operating point, it is considered that the temperature set point of the riser reactor Tra,sp step increases by 3 K and the flow rate of fresh feed oil FO,fresh step increases by 2.5 t·h−1. For the process disturbances, the reactor top pressure praf is treated as a sinusoidal variation with an amplitude of 3000 Pa and a period of 60 min, the inlet air temperature of regenerator Tg,inlet is treated as a sinusoidal variation with an amplitude of 5 K and a period of 600 min, the residual carbon mass fraction CCR,fresh is treated as the output of a first-order inertia unit with an amplitude of 0.04% and a time constant of 20 min due to the volume of the fresh feed oil tank, and a sinusoidal flow fluctuation with an amplitude of 0.5 t·h−1 and a period of 120 min is added on the nominal value of the flow rate of fresh feed oil. The dynamic trend of disturbances is shown in Figure 2. We use the software gPROMS to solve the dynamic optimization for minimum margins. gPROMS is an equationoriented dynamic process simulation and optimization software and can solve the dynamic optimization problem. The nonlinear programming solver used is sequential quadratic programming (SQP). In this work, the differential equations and algebraic equations of the steady-state design model of the regenerator and the dynamic simulation model of the FCCU are written in gPROMS language. The MPC controller is programmed using the foreign object interface of gPROMS. The optimization variables are MW and MV. After the optimization variables and objective function are assigned, gPROMS can solve the dynamic optimization problem automatically.

Figure 3. (a) Air flow rate margin, (b) regenerator inventory margin, and (c) control performance with different values of prediction horizon.

4. DESIGN MARGIN AND CONTROL PERFORMANCE ANALYSIS We set the whole simulation horizon of dynamic optimization at 720 min to ensure that the simulation horizon contains the entire dynamic process. The integral absolute error (IAE) is a measure index of control performance and represents the weighed integral of absolute errors between set points and measurements of controlled variables, that is

Reducing the prediction horizon of model predictive control decreases the IAE and yields better control performance. We required more air flow rate margin but less catalyst inventory margin to realize the corresponding control performance. From Figure 3, with the reduction of the prediction horizon, the air flow rate margin is increasing but the regenerator inventory margin is decreasing, while the control performance H



Article

can see that the air flow rate margin inversely correlates with the control performance index IAE while the catalyst inventory margin directly correlates with it. Improving the control performance requires more air flow rate margin but less catalyst inventory margin. Furthermore, a marked inflection point on the relationship curve exists between the control performance and the required air flow rate margin. If the prediction horizon is longer than 4 min, we can significantly improve the control performance by increasing only a small amount of margin. Otherwise, a massive amount of margin is required if we want to improve the control performance. Therefore, we should select the MPC parameter based on the inflection point when considering both control performance and the margin required for control.

is improved. When the prediction horizon is reducing, the regulation capability of MPC is enhanced; with the better control performance, the MPC needs more scope of manipulated variables, so the air flow rate margin is increasing. With the control performance improved, the fluctuations caused by disturbances are restrained, so less regenerator inventory margin is needed to prevent the process variables from breaking the process constraints. Moreover, an obvious inflection point exists in Figure 3a, where the prediction horizon is 4 min. If the prediction horizon is less than 4 min, a significant amount of margin is required, so the extremely small prediction horizon is inappropriate for MPC. Figure 4 shows the relationships of the air flow rate and regenerator inventory margins with respect to the control performance index IAE.

5. CONCLUSIONS The design margin of an FCCU regenerator under MPC is solved via dynamic optimization, and the influence of control performance on catalyst inventory margin and air flow rate margin is studied from the perspective of operation and control. It can be found that there is a difference between the device margin and the manipulated margin when influenced by the control performance. Improving the control performance requires a larger area in which the manipulated variables can be adjusted. A better control performance requires more air flow rate margins. However, this improvement implies less fluctuation of process variables and yields less catalyst inventory margins needed to hold the process fluctuation. At the same time, a marked inflection point exists on the relationship curve between the control performance and the air flow rate margin needed. When the prediction horizon is gradually reduced to improve the control performance, before the inflection point increasing the air flow rate margin by only a small amount can significantly improve the control performance, but for values beyond the inflection point a very higher required air flow rate margin is necessary if we want to further improve the control performance. Therefore, we cannot select a significantly small prediction horizon from the trade-off between the design margin and the control performance.

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Figure 4. (a) Air flow rate and (b) regenerator inventory margins with varying control performances.

ASSOCIATED CONTENT

S Supporting Information *

Information that supports the design margin and control performance analysis of FCCU is presented, in which the optimization results showing the relationship between the prediction horizon and the control performance index IAE, the air flow rate margin, and inventory margin of regenerator are listed. This material is available free of charge via the Internet at http://pubs.acs.org.

From Figure 4, we can find the difference between the device margin and the manipulated margin. The catalyst inventory margin belongs to the device margin, but the air flow rate margin belongs to the manipulated margin. The device margin provides the size of the device variable, which is fixed after process design and cannot be changed during the process while the device is running. More process disturbances that cause fluctuations of process variables imply the need for more device margins to prevent the process variables from breaking the process constraints. The manipulated margin determines the range of the manipulated variable used by the controller that corresponds to the regulation capability of this variable. More manipulated margins imply a more powerful regulation capability of this variable. Improving the control performance indicates less fluctuation of process variables but more powerful regulation capability of the manipulated variable, so this improvement requires less device margin but more manipulated margin. In Figure 4, we

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-10-89733277. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (21006127) and the National Basic Research Program of China (2012CB720500). I



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Article

NOMENCLATURE

MW = relative percentage margins of catalyst inventory MV = relative percentage margins of air flow rate P = prediction horizon of predictive control p = uncertainties pra = pressure of reactor, Pa praf = pressure of reactor top, Pa prg = pressure of regenerator, Pa prgf = pressure of regenerator top, Pa ΔpCrg = pressure drop of regenerated catalyst slide valve, Pa ΔpCst = pressure drop of spent catalyst slide valve, Pa Δpdis = constant static pressure drop of dilute phase of disengager, Pa Q = error weighting coefficient matrix of controlled variables R = weighting coefficient matrix of control variables rA = reaction rate of crude oil, Pa−1·s−1 rD = reaction rate of diesel, Pa−1·s−1 rN = reaction rate of naphtha, Pa−1·s−1 S = coefficient matrix of step response ST = riser reaction time, s Tg = temperature of air, K TO = temperature of oil, K Tra = temperature of riser reactor, K Trg1 = temperature in first CSTR of regenerator, K Trg2 = temperature in second CSTR of regenerator, K Tst = temperature in stripping section, K Tw = temperature of water vapor, K ΔTst = temperature drop of stripping section, K u = manipulated variables ude = superficial gas velocity of dense phase section, m s−1 udi = superficial gas velocity of dilute phase section,m s−1 Vflue = air flow rate of flue, kmol·s−1 Vrg = air flow rate of regenerator, kmol·s−1 v = disturbances vCrg = slide valve core area of regenerated catalyst, m2 vCst = slide valve core area of spent catalyst, m2 Wrg = catalyst inventory of regenerator, kg Wst = catalyst inventory of stripping section, kg X = dimensionless length of riser x0 = steady-state variables xa = algebraic variables xd = differential variables YC = product yield of coke vs fresh feed, % YD = product yield of diesel vs fresh feed, % YG = product yield of C1−C4 vs fresh feed, % YN = product yield of naphtha vs fresh feed, % y = controlled variables yA = mass percentage fraction of feed oil, % yD = mass percentage fraction of diesel, % yG = mass percentage fraction of C1−C4, % yN = mass percentage fraction of naphtha, % yrg1 = mole percentage fraction of O2 in the first CSTR of regenerator, % yrg2 = mole percentage fraction of O2 in the second CSTR of regenerator, % zCrg = opening of regenerated catalyst slide valve, % zCst = opening of spent catalyst slide valve, %

Symbols

A = system matrix of continuous linear state space model B = input matrix of continuous linear state space model C = output matrix of linear state space model CCR = carbon residue in oil, % Cra = coke mass percentage fraction of catalyst in riser, % Crg1 = coke mass percentage fraction of catalyst in first CSTR of regenerator, % Crg2 = coke mass percentage fraction of catalyst in second CSTR of regenerator, % Cst = coke mass percentage fraction of catalyst in stripping section, % cpg = specific heat capacity of gas flue in regenerator, kJ· kmol−1·K−1 cpL = specific heat capacity of thermal insulating layer, kJ· kg−1·K−1 cpO = specific heat capacity of oil, kJ·kg−1·K−1 cps = specific heat capacity of solid catalyst, kJ·kg−1·K−1 cpw = specific heat capacity of water vapor, kJ·kg−1·K−1 D = direct transfer matrix of linear state space model d = design variables EA = reaction activation energy of crude oil cracking, kJ· kmol−1 EC = reaction activation energy of carbon burning, kJ·kmol−1 ED = reaction activation energy of diesel cracking, kJ·kmol−1 EN = reaction activation energy of naphtha cracking, kJ· kmol−1 Eφ = reaction activation energy of catalyst deactivation, kJ· kmol−1 FO = flow rate of feed oil, kg·s−1 FW = flow rate of water vapor, kg·s−1 f 0 = steady-state design equations fa = algebraic equations of process model fc = discrete equations of MPC controller fd = differential equations of process model G = system matrix of discrete linear state space model GCst = spent catalyst circulation rate, kg·s−1 GCrg = regenerated catalyst circulation rate, kg·s−1 g = equations of process constraints H = input matrix of discrete linear state space model h = output equations of controlled variables ΔHAA = coke adsorption heat, kJ·kg−1 ΔHAR = cracking reaction heat of crude oil, kJ·kg−1 ΔHC = burning heat of carbon in regenerator, kJ·kg−1 ΔHDR = cracking reaction heat of diesel, kJ·kg−1 ΔHH = burning heat of hydrogen in regenerator, kJ·kg−1 ΔHNR = cracking reaction heat of naphtha, kJ·kg−1 ΔHV = vaporization heat of oil, kJ·kg−1 J = objective function of optimal design KCrg = static pressure constant of regenerated catalyst circulation, Pa KCst = static pressure constant of spent catalyst circulation, Pa kA0 = reaction rate constant of crude oil, Pa−1·s−1 kC0 = reaction rate constant of carbon burning, Pa−1·s−1 kCrg = resistance coefficient of regenerated catalyst flow kCst = resistance coefficient of spent catalyst flow kD0 = reaction rate constant of diesel, Pa−1·s−1 kN0 = reaction rate constant of naphtha, Pa−1·s−1 kO = resistance coefficient of feed oil flow kφ0 = reaction rate constant of catalyst deactivation, Pa−1

Greek Symbols

β = ratio of carbon residue in oil to carbon in catalyst Γ = correction coefficient of heat capacity for riser tube γ = mass percentage fraction of stripping hydrocarbon, % δ = variation to nominal steady-state operating point ε = average porosity in dense bed of regenerator, % η = hydrogen−carbon ratio

J



Article

Λ = correction coefficient of heat capacity for oil υAC = stoichiometric coefficient of oil−coke reaction υAD = stoichiometric coefficient of oil−diesel reaction υAG = stoichiometric coefficient of oil−gas reaction υAN = stoichiometric coefficient of oil−naphtha reaction υDC = stoichiometric coefficient of diesel−coke reaction υDG = stoichiometric coefficient of diesel−gas reaction υDN = stoichiometric coefficient of diesel−naphtha reaction υNG = stoichiometric coefficient of naphtha−gas reaction ρCrg = density of regenerated catalyst in circulation pipe, kg· m−3 ρCst = density of spent catalyst in circulation pipe, kg·m−3 ρg = average gas mole density in dense bed of regenerator, kmol·m−3 ρL = density of thermal insulating layer, kg·m−3 ρs = catalyst density, kg·m−3 φ = activity of catalyst, % Ωde = cross-sectional area of dense phase section, m2 Ωdi = cross-sectional area of dilute phase section, m2 ΩL = cross-sectional area of thermal insulating layer, m2 Ωra = cross-sectional area of riser, m2 Ωst = cross-sectional area of stripping section, m2

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Subscripts and Superscripts

A = crude oil C = coke Cst = spent catalyst circulation Crg = regenerated catalyst circulation cor = corrected predictive value D = diesel de = dense phase section di = dilute phase section fresh = fresh oil G = C1−C4 g = gas H = hydrogen hco = heavy recycle oil inlet = inlet of regenerator L = thermal insulating layer max = maximum N = naphtha O = oil P = predictive value ra = reactor raf = reactor top rg = regenerator rgf = regenerator top s = solid slurry = recycle slurry sp = set point st = stripping section 0 = nominal steady-state operating point

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