Modeling and Control of the Oxygen Concentration in a Post Combustion Chamber of a Gas-Fired Furnace*

Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 2017 The International Federation of Automatic Control Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 13766–13771 Modeling and Control of the Oxygen Modeling and Control of the Oxygen Modeling and Control of the Oxygen Concentration in a Post Combustion Modeling and Control of the Oxygen Concentration in a Post Combustion Concentration in a Post Combustion ⋆⋆ Chamber of a Gas-Fired Furnace Concentration in a Post Combustion Chamber of a Gas-Fired Furnace ⋆⋆ Chamber Furnace Chamber of of a a Gas-Fired Gas-Fired Furnace ∗ ∗∗

Stephan ∗ Christoph Froehlich, ∗∗ Stephan Strommer, Strommer, ∗ Christoph Froehlich, ∗∗ ∗ ∗∗ ∗∗ Stephan Strommer, Christoph Froehlich, Martin Niederer, ∗ Andreas∗ Steinboeck, ∗∗ Andreas ∗∗ Kugi ∗∗ MartinStephan Niederer, Strommer, Christoph Froehlich, ∗ Andreas Steinboeck, ∗∗ Andreas Kugi ∗∗ Martin Niederer, ∗ Andreas Steinboeck, ∗∗ Andreas Kugi ∗∗ Martin Niederer, Andreas Steinboeck, Andreas Kugi ∗ ∗ Complex Dynamical Systems, Austrian Institute of Technology, ∗ Complex Dynamical Systems, Austrian Institute of Technology, ComplexDonau-City-Strasse Dynamical Systems,1, Institute of Technology, 1220 Austria. ∗ Donau-City-Strasse 1,Austrian 1220 Vienna, Vienna, Austria. Complex Dynamical Systems, Institute of Technology, Donau-City-Strasse 1,Austrian 1220 Vienna, Austria. { stephan.strommer, martin.niederer }@ait.ac.at { stephan.strommer, martin.niederer }@ait.ac.at Donau-City-Strasse 1, 1220 Vienna, Austria. ∗∗ { stephan.strommer, martin.niederer }@ait.ac.at and Vienna of ∗∗ Automation and Control Control Institute, Institute, Vienna University University of Technology, Technology, {Gusshausstrasse stephan.strommer, martin.niederer }@ait.ac.at ∗∗ Automation Automation and Control Institute, Vienna University of Technology, 27–29, 1040 Vienna, Austria. ∗∗ Gusshausstrasse 27–29, 1040 Vienna, Austria. Automation and Control Institute, Vienna University of Technology, 27–29, 1040 Vienna, Austria. froehlich, kugi }@acin.tuwien.ac.at {{Gusshausstrasse froehlich, steinboeck, steinboeck, kugi }@acin.tuwien.ac.at 27–29, 1040 Vienna, Austria. {Gusshausstrasse froehlich, steinboeck, kugi }@acin.tuwien.ac.at { froehlich, steinboeck, kugi }@acin.tuwien.ac.at Abstract: Gas-fired industrial Abstract: Gas-fired industrial furnaces furnaces are are used used for for heat-treatment heat-treatment of of semi-finished semi-finished steel steel Abstract:The Gas-fired industrial furnaces are gas-fired used forburners, heat-treatment ofsupplied semi-finished products. required energy is provided by which are by and products. The required energy is provided by gas-fired burners, which are supplied by fuel fuelsteel and Abstract: Gas-fired industrial furnaces are used for heat-treatment of semi-finished steel products. The required energy is provided by gas-fired burners, which are supplied by fuel and air. The combustion is often realized fuel rich to avoid scale formation at the product surface. air. The combustion is often realized fuel rich to avoid scale formation at the product surface. products. The required energy is provided by which gas-fired burners, which are supplied by chamber fuel and air. The combustion is often realized fuel rich to avoid scale formation at the product surface. Thus, the flue gas contains unburnt products, are oxidized in a post combustion Thus, thecombustion flue gas contains unburnt products, are scale oxidized in a post combustion air. The is often realized fuel volume richwhich to avoid formation at combustion the productchamber surface. Thus, the flue gas contains unburnt products, which are oxidized in a post combustion by adding fresh air. The control of the flow of air to the post chamber by adding fresh The control ofproducts, the volume flow air to the post chamber Thus, the flue gasair. contains unburnt which areof in acontain post combustion combustion chamber bya air. The the control of the volume ofoxidized air to not the post combustion chamber is crucial task flue leaving the furnace must unburnt is aadding crucial fresh task because because the flue gas gas leaving the flow furnace must not contain unburnt products. products. by adding fresh air. The control of the volume flow of air to the post combustion chamber is a crucial task because the flue gas leaving the furnace must not contain unburnt products. For this control task, a two-degrees-of-freedom control strategy based on differential flatness For this control task, a two-degrees-of-freedom control strategy based on differential flatness is a crucial task because the flue gas leaving the furnace must not contain unburnt products. For this control task, a two-degrees-of-freedom control strategy based on differential flatness in combination with a MIMO-PI controller is proposed. The basis for the control design is in combination with a aMIMO-PI controller is proposed. The basis for the control design is aa For this control task, two-degrees-of-freedom control strategy based on differential flatness in combination with a MIMO-PI controller is proposed. The basis for the control design is a first-principles mathematical model of the air supply circuit and the combustion of flammable first-principles mathematical model of the airis supply circuit and thefor combustion of design flammable in combination with ais MIMO-PI controller proposed. The basis the control is a first-principles mathematical model of the air supply circuit and the combustion of flammable products. The model validated by means of measurement data from a real plant. products. The model is validated by of means of supply measurement data fromcombustion a real plant. first-principles mathematical model the air circuit and the of flammable products. The model is validated by means of measurement data from a real plant. products. The(International model is validated byofmeans of measurement data a real © 2017, IFAC Federation Automatic Control) Hosting by from Elsevier Ltd. plant. All rights reserved. Keywords: chemical engineering, engineering, delay delay systems, systems, model model validation, validation, nonlinear control, Keywords: chemical nonlinear process process control, Keywords: chemical engineering, delay systems, model validation, nonlinear process control, process control applications, process modeling and identification, tracking process control applications, process modeling identification, Keywords: chemical engineering, delay systems,and model validation, tracking nonlinear process control, process control applications, process modeling and identification, tracking process control applications, process modeling and identification, tracking 1. INTRODUCTION INTRODUCTION as 1. as the the air air supply supply line. line. The The model model should should capture capture the the 1. INTRODUCTION as the air supply line. The model should capture the essential dynamic behavior of the considered process essential dynamic behavior of model the considered process and and 1. INTRODUCTION as the air supply line. The should capture the essential dynamic behavior of the considered process and it is computationally inexpensive. 1.1 Objective of this work it is computationally inexpensive. 1.1 Objective of this work essential dynamic behavior of the considered process and it is computationally inexpensive. 1.1 Objective of this work it is computationally inexpensive. 1.1 Objective of this work In gas-fired industrial furnaces (cf. Fig. 1), the combustion 1.2 Gas-fired Gas-fired furnace furnace with with air air supply supply line line In gas-fired industrial furnaces (cf. Fig. 1), the combustion 1.2 1.2 Gas-fired furnace with air supply line In gas-fired industrial furnaces (cf. Fig. 1), the combustion of natural gas is often realized fuel rich to ensure that of natural gas is often realized fuel rich to ensure that 1.2 Gas-fired furnace with air supply line In gas-fired industrial furnaces (cf.fuel Fig.rich 1), the combustion of natural is not often realized to would ensure that Figure the flue gas gasgas does not contain oxygen, which would cause the flue does contain oxygen, which cause Figure 11 shows shows the the considered considered gas-fired gas-fired furnace, furnace, where where of natural gas is often realized fuel rich to ensure that the flue gas scale does formation not contain which wouldDue cause undesirable scale formation at oxygen, the product product surface. Due to Figure 1strip shows the considered gas-fired furnace, where aa steel is heated by the hot flue gas. The gas undesirable at the surface. to steel strip is heated by the hot flue gas. The flue flue gas the flue gas does not contain oxygen, which would cause Figure 1 shows the considered gas-fired furnace, where undesirable scale formation at the product surface. Due to the fuel-rich combustion, the flue gas may contain unburnt a steel strip is heated by the hot flue gas. The flue gas the fuel-rich combustion, the flue gas may contain unburnt undesirable scale formation atflue the product surface. Due to a steelPost Funnel chamber stripcombustion is heated by the hot flue gas. The flue gas the fuel-rich combustion, the gas may contain unburnt products like carbon monoxide and hydrogen, which must Funnel Post combustion chamber products likecombustion, carbon monoxide and which must Air O Funnel the fuel-rich the flue gashydrogen, may unburnt Post combustion Air intake intake chamber -sensor O22 -sensor products like carbon monoxide and hydrogen, which must not be emitted emitted into the the environment. Thus,contain they are burnt Funnel Post combustion chamber not be into environment. Thus, they are burnt Air intake Air supply line -sensor O products like carbon monoxide and hydrogen, which must 2 Air supply line notaabepost emitted into the environment. Thus, are burnt in post combustion chamber (PCC) (PCC) bythey adding fresh Air intake -sensor O 2 in combustion chamber by adding fresh Air supply line notabe emitted into Thus, areaburnt ff in post combustion chamber (PCC) bythey adding air. The amount of the air environment. is controlled controlled to guarantee guarantee fuelO2 Air supply line x air. The amount of air is to a fresh fuel2 in a post combustion chamber (PCC) by adding fresh xO f air. The amount of air is controlled to guarantee a fuellean gas atmosphere in the PCC, i.e., the flue gas leaving O2 xO2 lean gas atmosphere in the PCC, i.e.,to theguarantee flue gas leaving f air. PCC The amount ofexcess air is oxygen. controlled a fuellean gas atmosphere in the PCC, i.e., flue gas leaving x the contains In the strip annealing h Fuel-lean gas atmosphere Gas flow the PCC contains excess oxygen. In the stripgas annealing h Fuel-rich Fuel-lean gas Strip atmosphere Gas flow gas atmosphere lean gas atmosphere in the PCC, i.e., the flue leaving motion the PCCconsidered contains excess In the stripflow annealing h Fuel-rich Fuel-lean gas Strip atmosphere Gas flow gas atmosphere furnace considered in this this oxygen. paper, the the volume flow of motion furnace in paper, volume of air air thecontrolled PCCconsidered contains excess oxygen. Incontrollers. the stripflow annealing h Fuel-rich Fuel-lean gas Strip atmosphere Gas flow gas atmosphere motion furnace in this paper, the volume of air is by decentralized PI However, is controlled by decentralized PI controllers. However, Inert gas Fuel-rich gas atmosphere Strip motion furnace considered in this paper, the volume flow of air Inert gas is controlled by decentralized PI able controllers. However, the PI controllers controllers are not not always always able to ensure ensure fuelInert gas the PI are to aa fuelis controlled by decentralized PI controllers. However, i the PI controllers are not always able to ensure a fuellean gas atmosphere in the PCC, especially in transient Inert gas i lean gascontrollers atmosphere innot thealways PCC, especially in transient the PI are able to ensure a fueli lean gas atmosphere in the PCC, especially transient operating situations (changing (changing heating power).inTherefore, Therefore, Burner i operating situations heating power). lean gasoperators atmosphere in the PCC, especially inTherefore, transient Burner Steel operating situations heating power). furnace use aa high value of oxygen Heating zone: cc b aa Steel strip strip Burner furnace operators use(changing high set-point set-point value of the the oxygen Heating zone: d d b operating situations (changing heating power). Therefore, Steel strip Burner furnace operators use a high set-point value of the oxygen concentration to guarantee a fuel-lean gas atmosphere in Heating zone: d c b a concentration to guarantee aset-point fuel-leanvalue gas atmosphere in Fig. 1. Heating Steel strip furnace operators usethis a high of the zone: d with c b a concentration to guarantee a fuel-lean atmosphere in Fig. 1. Gas-fired the PCC. However, practice may gas reduce the oxygen energy Gas-fired furnace furnace with post post combustion combustion chamber. chamber. the PCC. However, this practice may reduce the energy concentration to guarantee a fuel-lean gas atmosphere in 1. Gas-fired furnace with post combustion chamber. the PCC. of this(Strommer practice may reduce the energy Fig. efficiency ofHowever, the furnace furnace (Strommer et al., al., 2013). efficiency the et 2013). Fig. 1. Gas-fired furnace with post combustion chamber. the PCC. ofHowever, this(Strommer practice may reduce the energy efficiency theadvanced furnace et al., 2013). composition varies varies in in different different sections sections of of the the furnace. furnace. Therefore, an advanced model-based control strategy is composition Therefore, an model-based control strategy is composition varies in different sections of the furnace. Inert gas consisting of hydrogen and nitrogen enters the efficiency of the furnace (Strommer et al., 2013). Therefore, an advanced model-based control strategy is Inert gas consisting of hydrogen and nitrogen enters the proposed in this paper, which is based on a validated composition varies in different sections of the furnace. proposed inanthis paper, model-based which is based on astrategy validated Inert gas consisting of hydrogen and nitrogen enters the furnace at point . In the heating zones a–d, natural gas i Therefore, advanced control is proposed in this paper, which is based on a validated furnace at point . In the heating zones a–d, natural gas i mathematical model of the combustion process as well Inert gas consisting of hydrogen and nitrogen enters the mathematical model of the combustion process as well furnace at point . In the heating zones a–d, natural gas i is burnt fuel rich. If the burners of a heating zone (HZ) proposed in this paper, which is based on a validated mathematical model of the combustion process as well furnace is burntatfuel rich. If the burners of a heating zone (HZ) point . Inthe the heating zones a–d, natural gas ithey ⋆ is burnt fuel rich. If burners of a heating zone (HZ) are switched off, are flushed with nitrogen to cool mathematical model of the combustion as well The authors kindly express their gratitude to process the industrial re⋆ areburnt switched off, they areburners flushedofwith nitrogen to (HZ) cool The authors kindly express their gratitude to the industrial reis fuel rich. If the a heating zone ⋆ are switched off, they are flushed with nitrogen to cool the burner nozzles. The flue gas leaving the HZs contains search partner voestalpine Stahl GmbH. The partner authors voestalpine kindly express gratitude to the industrial rethe burner nozzles. Theare flue gas leaving HZs contains search Stahltheir GmbH. ⋆ are switched off, they with the nitrogen to cool The partner authors voestalpine kindly express gratitude to the industrial rethe burner nozzles. The flueflushed gas leaving the HZs contains search Stahltheir GmbH. the burner nozzles. The flue gas leaving the HZs contains search partner voestalpine Stahl GmbH.

Copyright © 2017 IFAC 14330 Copyright © 2017, 2017 IFAC 14330 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 responsibility IFAC 14330 Peer review©under of International Federation of Automatic Control. Copyright © 2017 IFAC 14330 10.1016/j.ifacol.2017.08.2569

Proceedings of the 20th IFAC World Congress Stephan Strommer et al. / IFAC PapersOnLine 50-1 (2017) 13766–13771 Toulouse, France, July 9-14, 2017

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from the environment, and two control valves b and p. These two valves are used to control the volume flow V˙ pa of air and the pressure pt at the branch pipe tee t . Here, a desired constant operating pressure is established to avoid a reverse flow from the PCC to the branch pipe tee. By means of the volume flow V˙ pa , a desired oxygen concentration in the PCC is adjusted. The pressure at the points e and b is the ambient pressure p∞ . The pressure at the point p corresponds to the known pressure pF in the PCC, which is feedback controlled.

Taylor et al., 2010). They work fine for steady-state operating situations but they may be inefficient for transient situations. A two-degrees-of-freedom (2-DOF) controller of a fuel supply line was developed by Strommer et al. (2014b). The feedforward part utilizes the property of differential flatness. In the feedback part, a simple PI controller is used. G¨olles et al. (2014) designed a controller based on input-output linearization for a biomass furnace, which also comprises an air supply line. The combustion processes inside a boiler was investigated by Havlena and Findejs (2005). They suggested a linear model predictive controller to optimize the efficiency of the boiler by adjusting the air-fuel ratio. However, the performance of this linear controller is limited due to the nonlinear behavior of the underlying process. To avoid the emission of unburnt products into the environment, accurate control of the oxygen concentration in the PCC is indispensable. Moreover, the efficiency of the furnace may be improved by an accurate controller because the set point value of the oxygen concentration can be reduced (smaller safety margin). Thus, less unnecessary air has to be heated up in the PCC and the enthalpy saved in this way can be used, e.g., for preheating the steel strip. Though simple PI controllers are feasible for this task, they do not exploit the mathematical structure of the underlying process. By applying a model-based controller, e.g., 2-DOF control (Strommer et al., 2014b), the furnace may be operated in a safe way closer to its limits. For realizing such a concept, especially the feedforward part, a precise mathematical model is needed. Such a model can be also used for other purposes, e.g., observer design, analysis of the energy consumption, operator training, and improvement of the efficiency of the process.

1.3 Existing models

1.4 Contents

carbon monoxide and hydrogen, see point h in Fig. 1. These harmful products are burnt in the PCC by adding air via an air intake. The amount of air is controlled to realize a fuel-lean gas atmosphere in the PCC. That is, the oxygen concentration xO2 , which is measured by an O2 -sensor, should be xO2 > 0, see point f in Fig. 1. The additional air is provided by an air supply line shown in Fig. 2. It features a compressor, which sucks fresh air p∞

b Post combustion chamber

Valve b

βb , β˙ b V˙ ba

p∞ V˙ ca

βp , β˙ p Valve p pt

t V˙ pa p Branch pipe tee Compressor Flue gas from heating zones

e

pF

Fig. 2. Air supply line of the post combustion chamber.

For modeling and control of such a system, several approaches can be found in the literature. In the following, a short overview is presented. A common way of modeling combustion processes is the use of computational fluid dynamics (CFD). In general, CFD-models are characterized by high accuracy and a high system dimension (Trivellato and Labiscsak, 2015). The computational effort of these models is also high. Thus, they are time consuming and not appropriate for real-time applications. Mathematical models based on first principles often provide a good balance between accuracy and computational effort. Strommer et al. (2014a, 2016) analyzed the combustion processes inside a strip annealing furnace. The combustion air supplied to the PCC is provided by an air supply line. G¨ olles et al. (2014) derived a model of the air supply of a biomass furnace. They assumed the air to be incompressible. In (Strommer et al., 2014b), a model of the fuel supply of gas-fired burners was developed. This model also captures the compressibility of the gas. A more detailed model of the air and fuel supply of an industrial gas-fired furnace was presented by Froehlich et al. (2016). The model consists of submodels capturing the orifice flow, the valve dynamics, a pressure reducing valve as well as the pressure drop and the gas temperature in a long pipeline. Usually, the media supply of gas-fired burners is controlled by decentralized PI controllers (Bhowmick and Bera, 2012;

In Sec. 2, the mathematical model of the process is presented. It consists of two submodels capturing the combustion process and the air supply line. The control strategy is introduced in Sec. 3, and the performance of the concept is demonstrated by an example problem in Sec. 4. Conclusions are drawn in Sec. 5. 2. MATHEMATICAL MODEL The mathematical model includes the combustion of flammable products, i.e., natural gas and hydrogen, and the air supply line. The most important process variables are the flue gas composition in the PCC, the volume flows of air, and the pressure at the branch pipe tee of the air supply line, see Figs. 1 and 2. 2.1 Combustion of flammable products Inert gas (HNx) consisting of hydrogen H2 and nitrogen N2 enters the furnace at the point i , see Fig. 1. Hydrogen has to be burnt completely before the flue gas leaves the furnace. Moreover, the natural gas supplied to the HZs a–d has to be oxidized inside the furnace. Natural gas mainly consists of methane CH4 . The flue gas flows into the PCC, where additional air is supplied. Finally, the flue gas leaving the PCC towards the funnel contains only carbon dioxide CO2 , water H2 O, oxygen O2 , and nitrogen N2 . The

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abbreviations of these components are summarized in the set S = {CO2 , H2 O, O2 , N2 }. The combustion is considered to be a steady-state process. The global chemical reaction, which occurs in the furnace, is defined as   � f � a,O 2  n˙ j CH4 + n˙ HNx,H2 H2 +  n˙ j 2 + n˙ a,O O2 P CC j∈H

j∈H



n˙ HNx,N2 +

n˙

CO2

��

j∈H

�



2  2 n˙ a,N + n˙ sj + n˙ a,N N2 −→ j P CC

(1)

CO2 + n˙ H2 O H2 O + n˙ O2 O2 + n˙ N2 N2

with the set H = {HZa, HZb, HZc, HZd}. In (1), the molar flows are defined in the form ρCH4 (2a) n˙ fj = ¯ CH V˙ jf ∀j ∈ H M 4 ρν = ¯ ν xa,ν lst λj V˙ jf ∀j ∈ H, ν ∈ {O2 , N2 } (2b) n˙ a,ν j M ρν a,ν ˙ a (2c) Vp ∀ν ∈ {O2 , N2 } n˙ a,ν P CC = ¯ ν x M ρν (2d) n˙ HNx,ν = ¯ ν xHNx,ν V˙ HNx ∀ν ∈ {H2 , N2 } M ρ N2 n˙ sj = ¯ N V˙ js ∀j ∈ H (2e) M 2 ¯ ν of the with the density ρν and the molar mass M component ν ∈ {CH4 , H2 , N2 , O2 }. Moreover, lst is the so-called stoichiometric air-fuel ratio, λj denotes the airfuel equivalence ratio, and V˙ jf is the volume flow of fuel to the HZ j ∈ H (Turns, 2006). The product lst λj V˙ jf is the net volume flow of air, which enters the furnace via the burners. The volume flow of air supplied to the PCC is V˙ pa . The mole fractions of oxygen and nitrogen in the air are xa,O2 = 0.21 and xa,N2 = 1 − xa,O2 = 0.79. The volume flow of the inert gas is V˙ HNx , the mole fraction of hydrogen in the inert gas is xHNx,H2 , and the mole fraction of nitrogen is thus xHNx,N2 = 1 − xHNx,H2 . V˙ js denotes the volume flow of nitrogen due to flushing of switched-off burners. The densities and the volume flows are defined at technical standard reference conditions (Beater, 2007). The quantities V˙ jf , λj , V˙ js , V˙ HNx , and xHNx,H2 constitute time-variant input parameters, which are known from measurements. The remaining parameters in the righthand side of (2) are constant. The molar flows n˙ ν , ν ∈ S, can be determined based on the mole balances � f n˙ j = n˙ CO2 (3a) C: j∈H

H:

�

2n˙ fj + n˙ HNx,H2 = n˙ H2 O

(3b)

j∈H

O: N:

1 2 2 n˙ a,O + n˙ a,O ˙ CO2 + n˙ H2 O + n˙ O2 j P CC = n 2 j∈H � � � a,N 2 n˙ j 2 + n˙ sj + n˙ HNx,N2 + n˙ a,N ˙ N2 . P CC = n �

(3c) (3d)

j∈H

The PCC is equipped with a sensor for the oxygen concentration xO2 in the flue gas, i.e., the mole fraction. The mole fraction of a component ν ∈ S is defined in the form

2.2 Air supply line

xν = �

n˙ ν σ∈S

n˙ σ

.

(4)

Figure 2 shows the air supply line of the considered PCC. It consists of a compressor and two butterfly valves. The compressor and the valves are used to control the volume flow V˙ pa of air and the pressure pt at the branch pipe tee. The pipe flow is considered to be a one-dimensional plug flow. It is assumed that the air is a compressible medium and an ideal gas. Moreover, an isothermal fluid flow is assumed due to the small temperature variations. Therefore, a constant temperature Tt , which has been identified by means of measurements, is used in the thermodynamic state equations. The air supplied to the PCC is compressed by a radial fan. This compressor is driven by an electric motor without speed control. Generally, the so-called pressure head ∆p = pt −p∞ depends on the volume flow V˙ ca , i.e., ∆p = ∆p(V˙ ca ). In this work, an exponential relation between the pressure head ∆p and the volume flow V˙ ca is used  �2 χ � ˙a V c  , ∆p = pt − p∞ = ∆p0 1 − (5) V˙ 0

with the pressure head ∆p0 for zero flow, the maximum volume flow V˙ 0 , and an exponent χ > 0. These three parameters were identified using the optimization prob�2 � � lem min∆p0 ,V˙ 0 ,χ i ∆pi − ∆ˆ pi (V˙ca,i ) with the meai sured pressure head ∆p and volume flow V˙ ca,i at the sampling instances ti . The estimated pressure head ∆ˆ pi was calculated from (5). The control valves b and p are pneumatically actuated. Each valve is equipped with an underlying position controller. The valve is assumed to be a sharp-edged orifice. The volume flow V˙ ja of air with j ∈ V = {b, p} is modeled by the orifice formula for a compressible fluid flow (Beater, 2007), i.e., � T0 a Ψ(Πj ), j ∈ V. (6) V˙ j = Cj pt Tt Here, pt and Tt are the pressure and the constant temperature at the branch pipe tee, point t in Fig. 2. Cj is the discharge coefficient and T0 denotes the reference temperature. The pressure Πj is defined as Πj = pj/pt . The � ratio �2 � Πj −b is a normalized approximaterm Ψ(Πj ) = 1 − 1−b tion of the discharge function for b < Πj ≤ 1 with the soκ called critical pressure ratio b := Πcrit = (2/(κ + 1)) κ−1 . Here, κ is the adiabatic index of air. In the considered air supply line, Πj > b is always valid. Therefore, the socalled choked-flow (0 < Πj < b) is not required. Generally, the discharge coefficient Cj is difficult to determine because it depends on several process parameters, e.g., the pressure ratio Πj and the opening angle βj of the valve j ∈ V (Beater, 2007). By rearranging (6), the discharge coefficient at a sampling instance ti follows from V˙ a,i i � j � = Cj,meas � � Πij −b 2 pit TT0i 1 − 1−b

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Proceedings of the 20th IFAC World Congress Stephan Strommer et al. / IFAC PapersOnLine 50-1 (2017) 13766–13771 Toulouse, France, July 9-14, 2017

with Πij = pj/pit and j ∈ V. In (Froehlich et al., 2016), an approximation of the discharge coefficient Cj was developed in the form of a multidimensional polynomial Cj = aj + bj Πj + cj Πj βj + dj βj . The parameters aj , bj , cj , and dj were identified based on measured data. It is assumed that the discharge coefficients Cj also incorporate the pressure drops caused by friction in the pipes. To account for the dynamic behavior of the closed-loop butterfly valve, a mass-spring-damper system is considered with the mass m, the spring coefficient kc , and the viscous damping coefficient kd . It is assumed that the forces due to the fluid flow are negligibly small. The valves are actuated by a spring-return pneumatic cylinder. A deadtime behavior between the reference signal βjsp and the real opening angle βj can be observed. Thus, the dynamics of the opening angle βj of the control valve j ∈ V is defined by the state-space model        d βj βj 0 1 0 = + 2 βjsp (t − τj ). (7) −ωj2 −2Dj ωj β˙ j ωj dt β˙ j       bj

 ωj = kc,j/mj , Dj is the Here, ωj is the natural frequency √ damping ratio Dj = kd,j/2 mj kc,j , and τj is the dead time. These three parameters were identified using an optimization problem that minimizes the difference between the angle βj predicted by (7) and the corresponding measured signal. 2.3 Assembling the system The assembled mathematical model can be written in the form    sp    βb (t − τb ) bb 0 Ab 0 (8a) x+ x˙ = 0 Ap 0 bp βpsp (t − τp )       A

βjsp (◦ ) ∆p (mbar)

j∈H

2.4 Validation of the model The mathematical model (8) is validated by means of measurement data from a real plant. Figure 3a) shows the volume flows of fuel supplied to the four heating zones. Figure 3b) shows the reference signals βbsp and βpsp of the valves b and p. These signals are given from

0 60

b) Reference signals of control valves Valve b Valve p

30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t (h)

the existing control scheme. The air-fuel ratios λj have a constant value of λj = 0.92, j ∈ H. The volume flows of nitrogen of the hzs a and b are zero. If the hzs c and d f f = V˙ HZd = 0, they are flushed are switched off, i.e., V˙ HZc with a constant amount of nitrogen. At point i , inert gas with a constant volume flow of V˙ HNx = 925 m3 /h enters the furnace. The corresponding mole fraction xHNx,H2 of hydrogen is 3.5 % and at the time 8 h, xHNx,H2 changes to 3.2 %. Figure 4 shows a comparison between the simulated and the measured oxygen concentration xO2 in the PCC as well as the pressure head ∆p = pt − p∞ at the branch pipe tee. In general, the proposed model achieves a good

T  and the iniwith the state vector x = βb , β˙ b , βp , β˙ p T  tial condition x(0) = x0 . The output y = xO2 , pt contains the oxygen concentration xO2 in the PCC and the pressure pt . The input parameters are summarized in  T ˙ f )T , λT , (V ˙ s )T , V˙ HNx , xHNx,H2 the vector θ = (V with     ˙ f = V˙ f ˙ s = V˙ s V , λ = [λj ]j∈H , and V . j j j∈H

100

Fig. 3. Certain input parameters.

(8b)

The right-hand side of (8a) follows from (7). The output (8b) results from (3)–(6). Moreover, the mass balance at the branch pipe tee yields V˙ ca = V˙ ba + V˙ pa . Considering the composition of the flue gas, one  equation of (4) can be omitted because of the identify ν∈S xν = 1. For an implementation on a real-time system, the model (8) has to be discretized with respect to time. The explicit Euler method is used for this purpose.

HZa HZb HZc HZd a) Volume flow of fuel to heating zones a–d

200

0

B

y = h (x, θ)

300

90

xO2 (%)

Aj

V˙ f ( m3 /h)

i

13769

2 1.5 1 0.5 0 45

a) Oxygen concentration in the PCC

Measurement Simulation b) Pressure head at the branch pipe tee

40 35 30

Measurement Simulation 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t (h)

Fig. 4. Comparison between measured and simulated process variables. accuracy. Only at the time 6 h, a model-plant mismatch in terms of xO2 and ∆p can be observed. This mismatch can be explained by the identified discharge coefficient Cj , j ∈ V. During the measurements for the identification of Cj , the opening angle βjsp mainly varied in the range [30◦ , 70◦ ]. Outside this range, the approximation may thus be imprecise. At the time 6 h, the valve b is completely open, i.e., βbsp = 90◦ . 3. CONTROL STRATEGY The main control objective is that the oxygen concentration xO2 in the PCC follows a desired trajectory. For this, a

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Proceedings of the 20th IFAC World Congress 13770 Stephan Strommer et al. / IFAC PapersOnLine 50-1 (2017) 13766–13771 Toulouse, France, July 9-14, 2017

certain amount of air is required, which is provided by the air supply line, see Fig. 2. Moreover, the control strategy should ensure that the pressure pt is constant to avoid a reverse flow from the PCC to the branch pipe tee. These two control valves govern the volume flow V˙ pa of air and the pressure pt at the branch pipe tee. A model-based 2DOF control strategy is developed in order to fulfill the control objectives. The control variables u = uff + uf b are the set-point values βbsp and βpsp of the control valves, T  i.e., u = βbsp , βpsp . Here, uff is the feedforward part and uf b the feedback part. In Fig. 5, the block diagram of the proposed control strategy is shown. Here, e is the error yr Trajectory planner

Feedforward uff control

¨d yd , y˙ d , y

Feedforward part

u uf b

Furnace & air branch

y

Feedback control

e yd

Feedback part

Fig. 5. Block diagram of the 2-DOF control strategy. between the desired output yd and the measured output  T y = xO2 , pt and yr are the reference values. 3.1 Feedforward control

The feedforward controller is derived exploiting the differential flatness property of the considered system (Fliess et al., 1995). The model (8) is characterized by a dead-time behavior. Therefore, the design of the feedforward control law uff is the same as proposed in (Mounier and Rudolph, 1998) for delay systems. For the considered system (8), the oxygen concentration xO2 and the pressure pt constitute flat outputs. The parametrization is given by   x = Φx yd , y˙ d , θ, θ˙ (9a)  ¨ d , δ p yd , δ p y˙ d , δ p y ¨d, u = Φu δ b yd , δ b y˙ d , δ b y  (9b) ˙ δ b θ, ¨ δ p θ, δ p θ, ˙ δpθ ¨ δ b θ, δ b θ,

with the inverse delay operators δ b and δ p , i.e., δ b yd (t) = yd (t + τb ) and δ p yd (t) = yd (t + τp ). The derivation of (9) is essentially the same as outlined in (Strommer et al., 2014b) and thus omitted here. By means of the measured input parameters θ and a Golay filter, the derivatives ¨ are obtained. The trajectories yd , y˙ d , and y ¨d θ˙ and θ are generated by a linear filter from the reference inputs yr . To obtain (9b), the reference inputs have to be noncausal. The reference input of the oxygen concentration ˙ f of fuel. These also depends on the volume flows V volume flows follow from a superordinate nonlinear model predictive controller (Niederer et al., 2016) and thus are known in advance. The reference input of the pressure is constant and defined by the operator. The feedforward control law uff directly follows from (9b). 3.2 Feedback control In general, the feedforward controller does not exactly ensure the desired behavior because of model-plant mis-

matches and disturbances. Thus, a feedback controller is required to compensate for the remaining deviations. In the feedback part, a MIMO-PI controller x˙ c = yd − y (10a) (10b) uf b = KI xc + KP (yd − y) is used. The matrices KI and KP have to be adequately designed to achieve good tracking of the desired trajectories yd . The following design of KI and KP is given in (Lunze, 1989). For simplicity, the dead-time behavior is neglected in the design of KI and KP and thus, the error dynamics is given by e˙ x = Aex + Buf b (11a) (11b) e = Cex with ex = xd − x and e = yd − y. Here, xd is the desired state trajectory. Equation (11b) follows from y = h (x, θ) = h (xd − ex , θ) ≈ h(xd , θ) − (∂h/∂x)|{xd ,θ} ex . To obtain a time-invariant system with C = (∂h/∂x)|{xd ,θ} , h is linearized with respect to the initial values xd (0) and θ(0) rather than the trajectories xd (t) and θ(t). The error due to this approximation is negligible for the design of KI and KP . The transfer function G(s) of (11) in the Laplace domain is given by G(s) = C(sI − A)−1 B with the Laplace variable s and the identity matrix I. Calculating the limit s → 0, the static transfer matrix Ks = lims→0 G(s) = −CA−1 B is obtained. The matrices KI and KP are calculated by KI = aK−1 s KP = bK−1 s . Based on this choice, the MIMO-PI controller (10) consists of a static decoupling matrix K−1 and two decentralized s PI controllers with user-defined tuning parameters a and b. Moreover, an anti-windup strategy is implemented (Hippe, 2006). 4. EXAMPLE PROBLEM The performance of the proposed control strategy is demonstrated in a simulation study using the validated mathematical model (8) as a replacement of the plant. In the simulation scenario, the same input parameters θ are used as in Sec. 2.4. After the time of 10 h, the volume flows V˙ jf of fuel, j ∈ H, used in the feedforward controller are assumed to be 10 % higher than in the simulated plant. This manually introduced deviation should verify the robustness of the proposed control concept. Figure 6 shows a comparison of the desired and the simulated oxygen concentration and pressure head. The simulation values are given for both pure feedforward control and the 2DOF control concept. In case of pure feedforward control, an error occurs due to the simulated model-plant mismatch after 10 h. This error is compensated by the feedback controller (10) in the 2-DOF control strategy. The control inputs uff and uf b are shown in Fig. 7. Because of the sudden rise of the volume flows V˙ jf of fuel, j ∈ H, at the sp sp decreases and βp,ff increases to realize a time 10 h, βb,ff a ˙ higher volume flow Vp of air to the PCC. In the feedback sp sp part, βb,f b increases and βp,f b decreases to compensate the erroneous action of the feedforward controller, see Fig. 7. Operators of the considered plant typically choose a high

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a) Oxygen concentration in the PCC

2 1.75

∆p (mbar)

xO2 (%)

2.25

1.5 41

b) Pressure head at the branch pipe tee

40 39

Desired

Pure feedforward

2-DOF

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t (h)

Fig. 6. Closed-loop behavior of the oxygen concentration in the PCC and the pressure head at the branch pipe tee.

sp ◦ βj,f b ( )

sp βj,ff (◦ )

70

a) Control variables of feedforward controller

50 30 10 10 5 0 −5

Valve b

Valve p

b) Control variables of feedback controller Valve b Valve p

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t (h)

Fig. 7. Control inputs of feedforward and feedback controller. set point value for the oxygen concentration xO2 to have a safety margin that always ensures a fuel-lean gas atmosphere in the PCC. Due the encouraging simulation results achieved by the proposed 2-DOF controller, it seems reasonable to implement the proposed control strategy in the 2 real plant and to reduce the set point values xO d in favor of a reduced energy consumption of the system. For analyzing the energy savings, an additional simulation study with a 2 = 0.5 % was performed. In this set point value of xO d scenario, the energy stored in the flue gas leaving the PCC is approximately 5 % higher than in the original scenario. 5. CONCLUSIONS In this work, a first-principles model of the air supply line and the combustion of flammable products in a gas-fired furnace was presented. The good accuracy of the model is demonstrated by a comparison with measurement data from a real plant. The model constitutes the basis for the design of a control strategy comprising a flatness-based feedforward and a MIMO-PI-feedback controller including an anti-windup strategy. Simulation results of the validated model demonstrate the good control performance achieved by the proposed concept. REFERENCES Beater, P. (2007). Pneumatic Drives: System Design, Modelling and Control. Springer, Berlin, Heidelberg, 1st

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edition. Bhowmick, M. and Bera, S. (2012). An approach to optimum combustion control using parallel type and crosslimiting type technique. Journal of Process Control, 22, 330–337. Fliess, M., L´evine, J., Martin, P., and Rouchon, P. (1995). Flatness and defect of non-linear systems: Introductory theory and examples. International Journal of Control, 61(6), 1327–1361. Froehlich, C., Strommer, S., Steinboeck, A., Niederer, M., and Kugi, A. (2016). Modeling of the media-supply of gas burners of an industrial furnace. IEEE Transactions on Industry Applications, 52(3), 2664–2672. G¨olles, M., Reiter, S., Brunner, T., Dourdoumas, N., and Obernberger, I. (2014). Model based control of a smallscale biomass boiler. Control Engineering Practice, 22, 94–102. Havlena, V. and Findejs, J. (2005). Application of model predictive control to advanced combustion control. Control Engineering Practice, 13, 671–680. Hippe, P. (2006). Windup in Control - Its Effects and Their Prevention. Springer, London, 7th edition. Lunze, J. (1989). Robust Multivariable Feedback Control. Prentice-Hall, New Jersey, 1st edition. Mounier, H. and Rudolph, J. (1998). Flatness based control of nonlinear delay systems: A chemical reactor example. International Journal of Control, 71, 871–890. Niederer, M., Strommer, S., Steinboeck, A., and Kugi, A. (2016). Nonlinear model predictive control of the strip temperature in an annealing furnace. Journal of Process Control, 48, 1–13. Strommer, S., Niederer, M., Steinboeck, A., and Kugi, A. (2013). Analysis of energy comsumption in a direct-fired continuous strip annealing furnace. In Proceedings of the 9th International and 6th European Rolling Conference, 1–14. Venice, Italy. Strommer, S., Niederer, M., Steinboeck, A., and Kugi, A. (2014a). A mathematical model of a direct-fired continuous strip annealing furnace. International Journal of Heat and Mass Transfer, 69, 375–389. Strommer, S., Niederer, M., Steinboeck, A., and Kugi, A. (2016). Combustion processes inside a direct-fired continuous strip annealing furnace. In Proceedings of the 17th IFAC Symposium on Control, Optimization and Automation in Mining, Mineral and Metal Processing, 222–227. Vienna, Austria. Strommer, S., Steinboeck, A., Begle, C., Niederer, M., and Kugi, A. (2014b). Modeling and control of gas supply for burners in gas-fired industrial furnaces. In Proceedings of the IEEE Conference on Control Applications (CCA), 210–215. Antibes, France. Taylor, J., Sinopoli, B., and Messner, W. (2010). Nonlinear modeling of butterfly valves and flow rate control using the circle criterion bode plot. In Proceedings of the American Control Conference, 1967–1972. Baltimore, USA. Trivellato, F. and Labiscsak, L. (2015). The postcombustion chamber of steelmaking plants: Role of ambient air in reactant exhaust fumes. Applied Mathematical Modelling, 39, 19–35. Turns, S. (2006). An Introduction to Combustion. McGraw-Hill, Singapore, 2nd edition.

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