A Composite Model Predictive Control Strategy for Furnaces

Hao Zang,Hongguang Li*,Jingwen Huang,JiaWang

Automation Department,Beijing University of Chemical Technology,Beijing 100029,China

A Composite Model Predictive Control Strategy for Furnaces

Hao Zang,Hongguang Li*,Jingwen Huang,JiaWang

Automation Department,Beijing University of Chemical Technology,Beijing 100029,China

A R T I C L E I N F O

Article history:

Received 15 June 2013

Received in revised form 20 November 2013 Accepted 29 December 2013

Available on line 19 June 2014

Furnace

Tracking nonlinear model predictive control

Economic nonlinear model predictive control

Distributed model predictive control

Tube furnaces are essential and primary energy in tensive facilities in petrochemical plants.Operational optimization of furnaces could not only help to improve product quality but also benefit to reduce energy consumption and exhaust emission.Inspired by this idea,this paper presents a composite model predictive control(CMPC) strategy,which,taking advantage of distributed model predictive control architectures,combines tracking nonlinear model predictive control and economic nonlinear model predictive control metrics to keep process running smoothly and optimize operational conditions.The controllers connected with two kinds of communication net works are easy to organize and maintain,and stable to process interferences.A fast solution algorithm combining interior point solvers and New ton's method is accommodated to the CMPC realization,with reasonable CPU computing time and suitable on line applications.Simulation for industrial case demonstrates that the proposed approach can ensure stable operations of furnaces,improve heat efficiency,and reduce the emission effectively.

©2014 Chemical Industry and Engineering Society of China,and Chemical Industry Press.All rights reserved.

1.Introduction

Nowadays,oil energy be com es increasingly scarce and the amount of greenhouse gas emissions is getting huge.Technological improve ments of control strategies for petroleum refining plantsare recognized as potentially effective solutions.Practically,heating various hydrocarbon compounds by burning fuels,tube furnaces consume a significant amount of energy and generate huge exhaust emission.It is reported that two ways are available to improve operational conditions of tube furnaces.One is to maintain or rep lace old production facilities such as using high efficient heat exchange systems,insulation walls and burners,which is no doubt rather expensive and time consuming[1]. Another is to apply advanced control strategies,which can effectively increase thermal efficiencies of furnaces by the optimization of operation condition.

Kalogirou[2]applied artificial intelligence methods in combustion processes.Mercedes[3]introduced fuzzy cascade control to furnace outlet temperature and achieved good results.Th rough experiments and case study,Lee and Jou[4,5]presented numerical relationship of flue gas residual oxygen concentration,air preheat temperature and furnace thermal efficiency,pointing out that appropriate excess air oxygen concentration and air preheat temperature can reduce fuel consumption and pollution emissions.Lu et al.[6]proposed an intelligent self-searching optimization algorithm for thermal efficiency,giving satisfactory simulations.However,existing optimization methods concerning the thermal efficiency of furnace are usually insensitive to process disturbances,easily leading to malfunctions,or even causing accidents,which discourage their applications.

Being capable of dealing with process dynamics and constraints for multi-input and multi-output systems,nonlinear model predictive control(NMPC)has been widely circulated in academia and industry[7-9]. Tracking nonlinear model predictive control(TNMPC)is commonly used to formulate target tracking problems,in which the cost functions are assumed to be positive definite with respect to a certain set-point or trajectory to be tracked.However,this basic assumption does not hold for all cases,particularly for optimizing process economic objectives [10].In response,economic nonlinear model predictive control (ENMPC)approaches have been developed,where generic cost functions are used instead.TNMPC demonstrates good dynamic performance and robustness in strongly nonlinear systems such as furnace, but in the absence of optimization in formation in objective functions, its applications are limited to the traditional two-layer control structure (real-time optimization+model predictive control).In this context, ENMPC is rather competent but suffers complex optimization models, longer control cycles,and slow response to perturbations.Novel control structures for both product qualities and economic objectives are demanded.

Taking advantage of distributed model predictive control(DMPC), communications between different NMPC strategies can be established. Several DMPC methods[11-13]have been circulated in literature, though most relevant articles only highlight DMPC schemes conceptually.Motivated by these observations,this paper introduces a CMPC strategy involving TNMPC and ENMPC based on rigorous nonlinear mathematical models,which is easy to implement,enjoys good stability, and optimizes quickly strongly nonlinear constrained complex systems.The CMPC strategy is compared with conventional control performance through an industrial example.

2.Furnace Models

Furnaces are recognized as one of the most crucial facilities in petrochemical plants,which heat hydrocarbon mixtures rapidly to a desired temperature by the combustion of fuel gas or exhaust gas. Fig.1 shows a schematic of a vertical furnace with a radiation chamber and a convection chamber.

To formulate generic first-principle dynamic models of furnaces,the following assumptions are made.

(1)Flue gas and process variables distribute uniformly in the chambers.

(2)The furnace is a multi-fuel(fuel gas and exhaust gas)burning stove.

(3)Heat loss is negligible.

(4)Flue gas temperature in the convection chamber equals that in the radiation chamber and distributes uniformly.

(5)The mole change of vapor during combustion is negligible.

Models for furnace temperature are based on the feed energy balance of fuel gas,exhaust gas and flue gas at the outlet,respectively,U is the average heat transfer coefficient,A is the heat transfer area of furnace,QLmfrep resents the low heating value of mixed fuel,γrepresents the combustion rate of mixed fuel,andαis the excess air coefficient.Heat capacities ρfVfCfand ρfgVfgCfgare ad justed by correcting factors to fit the real time trend.The parameters are obtained by using equipment dimensions and fitting to measurement data.

where Kgand Kegdenote low heating value coefficients of fuel gas and exhaust gas.

Dynamic characteristics of the fuel gas circuit is

where τgis the time constant of fuel gas circuit,and Fs,gis the set-points of Fg.

Models for furnace flue gas and air system are as follows

where Cpand COare the capacity factor of the chamber at negative pressure and residua lO2of flue gas,respectively,P is the chamber negative pressure,Ofgis the residua lO2concentration of flue gas,Fais the volumetric flow rate of air,and Amfis the theoretical air-fuel ratio of mixed fuel,which can be calculated as follows.

where ρf,Vfand Cfare the density,volume and specific heat of the feed in tubes,respectively,ρfg,Vfgand Cfgare those of the flue gas in the chamber,Ti,f,To,f,Tfg,and To,fgare the temperatures of feed at the inlet and outlet,flue gas in the chamber,and flue gas at the outlet,respectively,Fg,Feg, and Fo,fgare temperature-pressure compensated volumetric flow rates

Fig.1.A simplified schematic of furnaces.

where Agand Aegare the theoretical air-fuel ratio to fuel gas and exhaust gas,respectively,Frgand Fregare the volume fractions of fuel gas and exhaust gas,respectively,and Yjis the volume fraction of material j in the vapor fuel.

In addition,we have the model for air preheater

where Vapis the volume of flue gas in the air preheater,Uapand Aapstand for the average heat transfer coefficient and area of air preheater, respectively,and Ti,ais the temperature of the inlet air.

Thermal efficiency of the furnace is evaluated by

Substituting Eqs.(1)and(2)in to Eq.(12),we have

where η is the thermal efficiency of furnace and Q is the furnace load.

3.CMPCApp roaches

3.1.Control strategies

Recently,the combination of optimization techniques and NMPC be comes a hot research topic.For instance,a two-layer architecture presents several advantages and has extensive applications,which involves a control layer and an optimization layer associated with different time scales in a plant[14-18].The NMPC locates at the control layer accounting for multi-variable coordination and constraint treatments, while a real-time optimization(RTO)system using steady-state models locates at the optimization layer to provide optimum set-points for NMPC loops.However,applications of this method are usually limited by unreachable set-points,in feasible soft-constraints and other in tractable issues.In response to these difficulties,some researchers have introduced a one-layer approach term ed economic model perceive control to rep lace the quadratic tracking cost function.Several merits of this approach over conventional methods have been demonstrated, such as responding quickly to disturbances,implementing constraints effectively with measured variables,avoiding inconsistencies of models and using full manipulating degrees for optimizations,even during process transients[19-21].

In this context,the CMPC strategy can be regarded as a novel NMPC method concerning both the stability and economic performance.It is also recognized as a distributed architecture composed of TNMPC and ENMPC formulations,as shown in Fig.2,where TNMPC ones are connected by a bidirectional communication network(network 1) and ENMPC ones are connected by an unidirectional communication network(network 2).The in form ation in network 1 is delivered to network 2 through an alternative unidirectional communication network (network 3).TNMPC formulations calculate a group of optimization input trajectories through an iteration algorithm,while each ENMPC one evaluates the input trajectory separately in sequence.The number of TNMPC controller,n,and that of ENMPC ones,m,are assigned according process specifications.

An implementing procedure of the CMPC strategy is as follows.

Step 1 At time tk,all TNMPC and ENMPC controllers receive measurement χ(tk)from process sensors.

Step 2 For i=1 to m+1

2.1 If i＞1,go to Step 2.5.

2.2 At iteration j(j≥1),each TNMPC controller evaluates its ow n future input trajectory based onχ(tk)and the latest received input trajectories of all other TNMPC ones(u1,…,un)from network 1.

2.3 The controllers calculate their values of objective functions based on their future input trajectories received from network 1 and pass back to network 1.

2.4 If a terminating condition is satisfied,each controller delivers its future input trajectories corresponding to the smallest value of the cost function to the system,otherwise go to Step 2.2 for a next iteration(j←j+1).

2.5 ENMPC i receives the entire future input trajectories of ui(i=1, 2,…,n)from network 3 and ui(i=n+1,…,n+i−2)fromnetwork 2.Based on the received future input trajectories it evaluates the future input trajectory.

Fig.2.The CMPC architecture.

2.6 ENMPC i sends the first step input value of uito the system and the entire future input trajectories of ui(i=n+1,…,n+i−1) to network 2.

Step 3 When a new measurement is received,set k=k+1,go to Step 1.

3.2.CMPC details

In regard to the furnace,CMPCin tends to stabilize the outlet temperature,chamber negative pressure and improve the thermal efficiency with reducing energy consumption and exhaust emission.To facilitate the discussion,we present the CMPC strategy for the furnace in Fig.3. The CMPC architecture is made of two TNMPC and one ENMPC controllers,where TNMPC controllers connected by network 1 controlling the feed outlet temperature and the chamber negative pressure,while the ENMPC strategy is used to optimize the furnace thermal efficiency. Network 1 is a bidirectional communication network for the storage of future input trajectories and cost functions calculated by TNMPC controllers,and network 2 is a unidirectional communication network used to deliver the input trajectories to ENMPC.

Fig.3.The CMPCscheme of furnaces.

With conventional NMPC,the control law is obtained by rolling optimization in the context of exact process models,in which nonlinear programming(NLP)or intelligent optimization algorithm s are employed to solve the following nonlinear equations pertaining to a specific dynamic optimization problem. where t is the scalar time dimension,N is the horizon length,χ(t)is a vector of state variables,y(t)is a vector of output variables,and u(t)is a vector of manipulated variables.Eqs.(14b)-(14c)give rigorous description of the plant,and Eqs.(14d)-(14 f)are bound constraints for the states,output and manipulated variables,respectively.

Similarly,the objective functions and control structures involved in CMPC strategies can be represented as follows.

The objective function of TNMPC 1 is

where To,f(t)is controlled variables and Ts,f(t)is the set-point of feed outlet temperature.Detailed descriptions associated with TNMPC1 are illustrated in Table 1.

The objective function of TNMPC2 is where P(t)is the chamber negative pressure and Ps(t)is the set-point of the chamber negative pressure.Detailed descriptions associated with TNMPC 2 are illustrated in Table 2.

ENMPC is recognized as a dynamic real time optimization(D-RTO) techno logy,which uses a one-layer structure instead of a traditional two-layer optimized control structure with a RTO layer and a MPC layer.It can not only overcome some of the problems stemmed from the two-layer structure such as unreachable set-points,but also respond quickly and operate optimally.For the furnace,the ENMPC is utilized to optimize thermal efficiency.

It is reported that thermal efficiency is mainly affected by outlet temperature and residua lO2concentration of flue gas,and it is difficult to express their relationship in an analytical model.This paper presents and solves a new economic objective function to optimize the thermal efficiency indirectly.

The objective function of ENMPC 1 is

where Oref,fgis the reference values of flue gas residua lO2concentration given by the operation experience,and Yeand Uedenote weights specified as100 and 10,respectively.The detailed descriptions of ENMPC 1 are illustrated in Table 3.

3.3.CPMC solutions

Apparently,solving the optimization problem characterized by the first-principle dynamic models of CMPC is rather complicated,especially for those of ENMPC,which inevitably suffers very long CPU time.In order to meet on line requirements of CMPC,a fast solving strategy [21-23]for dynamic optimization problems is strongly advisable.In w hat follows,we incorporate the interior point solver(IPOPT)algorithm with New ton's method to deal with the dynamic optimization of CMPC.

Table 1In formation about TNMPC 1

Table 2In formation about TNMPC 2

In this sense,Eq.(16)can be expressed asa general parametric NLP problem of the form

with variable w,objective function F,constrain t function c,and parameter p.Notice the implicit dependence of variables on a particular value of parameter.In the context of NMPC,this parameter is the initial condition χ(k).IPOPT handles the bound constraints implicitly through logarithmic barrier term s added to the objective function[23],

where μ＞0 is a barrier parameter and wi(p)denotes the ith component of vector w(p).The solution of Eqs.(19a)and(19b)converges to the solution of the original NLP(Eqs.(18a),(18b)and(18c))as the barrier parameter approaches zero.

To solve each barrier problem,IPOPT applies New ton's method to the Karush-Kuhn-Tucker(KKT)conditions,which result in following large-scale linear system[23]solved at each iteration j

where λ and ν are the Lagrange multipliers for the equality constraints and bounds,respectively,H:=H(w(p),p)is the Hessian of the Lagrange function L=F(w(p),p)+c(w(p),p)Tλ(p)−v(p)Tw(p),A:=A(w(p),p) is the constrain t Jacobian,Z:=diag(w(p)),and V:=diag(ν(p)).Because the KKT matrix on the left hand side of Eq.(20)is identical to the New ton iteration matrix used in IPOPT,which is already available in factorized form.Hence,once the new state w(p)is known,the change p0→p is noted and the desired approximate solution can be obtained with a single on-line solver.This on-line step usually requires less than 1%of the dynamic optimization calculation.

Table 3Information about ENMPC

4.Case Study

Dynamic models of an industrial furnace with 6.430×107kJ·h-1rated load,4-inlet and 4-outlet feeds are built in the HYSYS environment.Two cases with different disturbances are used to demonstrate the performance of CMPC strategy.For comparison,we apply the traditional control strategy,centralized ENMPC and CMPC strategies.In the traditional control structure,the control layer utilizes PID control strategy to ad just the outlet temperature and residua lO2concentration of flue gas.In the CMPC strategy,thermal efficiency is the goal of optimization,with the flow rate of fuel gas,F.D.fan and I.D.fan as the manipulated variables.For this example we specify 10 sampling time in the tracking predictive horizon and 20 sampling time in the economic predictive horizon.The differential algebraic equation model is transformed to a dscrete time model using collocation.

4.1.Step changes

The furnace starts from a nominal steady-state.At t=1 min,feed flow rate is reduced from 400 t·h-1to 320 t·h-1instantaneously and goes back to the original value at t=11m in.Fig.4 presents the profiles of the outlet temperature,thermal efficiency,residua lO2concentration and furnace negative pressure with different control strategies.After the step change of feed flow rate,the outlet temperature deviates from the set-point value,the thermal efficiency is lower,and the relevant para meters gradually go back to the normal state aft era period of time.For the control effect of key variables,CMPC controller yields very good sensitivity approximations,while the other two control strategies have poor dynamic performance.For the optimization of thermal efficiency, CMPC controller and centralized ENMPC controller are much betterthan PID controller.Thus the CMPC controller presents good control performance and optimization performance despite relatively large disturbances.

Fig.4.Controlled variable profiles.Dot-dashed line:centralized ENMPC;dashed line: traditional control;solid line:CMPC.

4.2.Ram p changes

The furnace runs smoothly at t=0;at t=1～2m in,feed flow rate is reduced from 400 t·h-1to 320 t·h-1through a ram p change;at t=11～12m in it recovers to the original value.Assuming no additional immeasurable disturbances involved,we apply PID and CMPC strategies.

Fig.5 presents profiles of the outlet temperature and the thermal efficiency with different control strategies.When the feed flow rate changes the control target suffers substantial shocks.After the flow rate recovers,the optimized parameters change little for all control strategies.The objectives demonstrate small amplitudes with the CMPC strategy,so the furnace runs relatively smoothly.

In industrial processes,the residua lO2concentration cannot achieve closed-loop control due to various reasons.Since manual operation does not always ad just air flow rate for frequently varying load,operators often give an air volume much larger than that needed,wasting large amounts of energy.

5.Conclusions

A majority of study on furnace operation optimization deals with energy shortages and environmental pollutions,but the strong nonlinearity and heavy uncertainty associated with furnace control greatly discourage the application of PID and conventional nonlinear model predictive control strategies.Thus we develop a CMPC control strategy combining TNMPC and ENMPC approaches.Simulations demonstrate good control performance in reducing energy consumption and pollution emissions.

Fig.5.Controlled variable profiles.Dot-dashed line:centralized ENMPC;dashed line: traditional control;solid line:CMPC.

Nevertheless,it is still a challenge to deal with more complicated processes involving process uncertainty or model inaccuracy as well as soft constraints,which provides a guideline for our future work.Industrial applications of the proposed approaches are expected.

Nomenclatu re

A average heat transfer area of furnace,m2

Aapaverage heat transfer area of air preheater,m2

Aegtheoretical air-fuel ratio of exhaust gas

Aftheoretical air-fuel ratio

Agtheoretical air-fuel ratio of fuel gas

Amftheoretical air-fuel ratio of mixed fuel

Cfspecific heat of feed,k J·kg−1·k−1

Cfgspecific heat of flue gas,k J·kg−1·k−1

COcapacity factor of flue gas residua lO2

Cpcapacity factor of the chamber negative p ressu re

Favolumetric flow rate of air,m3·s−1

Fegvolumetric flow rate of exhaust gas,m3·s−1

Fgvolumetric flow rate of fuel gas,m3·s−1

Fo,fgvolumetric flow rate of flue gas at the outlet,m3·s−1

Fs,gset-point of Fg,m3·s−1

Fregvolumetric fraction of exhaust gas

Frgvolume fraction of fuel gas,m3·s−1

Ofgflue gas residua lO2concentration

P chamber negative pressure,kPa

Q furnace load,k J·s−1

QLmflow heating value of mixed fuel,k J·s−1

Tfgtemperature of flue gas in the chamber,K

Ti,atemperature of air at the inlet

Ti,ftemperature of feed at the inlet,K

To,ftemperature of feed at the outlet,K

To,fgtemperature of flue gas at the outlet,K

U average heat transfer coefficient of furnace

Uapaverage heat transfer coefficient of air preheater

Vfvolume of feed,m3

Vfgvolume of flue gas,m3

Yjvolume fraction of material j in vapor fuel

α excess air coefficient

γ combustion rate of mixed fuel

τgtime constant of fuel gas circuit,s

η thermal efficiency of furnace

ρfdensity of feed,kg·m−3

ρfgdensity of flue gas,kg·m−3

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E-mailaddress:lihg@m ail.buct.edu.cn(H.Li).