# Two-Layer Linear MPC Approach Aimed at Walking Beam Billets Reheating Furnace Optimization

Authors: Silvia Maria Zanoli; Crescenzo Pepe
Journal: Journal of Control Science and Engineering
 (2017)

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## Abstract

In this paper, the problem of the control and optimization of a walking beam billets reheating furnace located in an Italian steel plant is analyzed. An ad hoc Advanced Process Control framework has been developed, based on a two-layer linear Model Predictive Control architecture. This control block optimizes the steady and transient states of the considered process. Two main problems have been addressed. First, in order to manage all process conditions, a tailored module defines the process variables set to be included in the control problem. In particular, a unified approach for the selection on the control inputs to be used for control objectives related to the process outputs is guaranteed. The impact of the proposed method on the controller formulation is also detailed. Second, an innovative mathematical approach for stoichiometric ratios constraints handling has been proposed, together with their introduction in the controller optimization problems. The designed control system has been installed on a real plant, replacing operators’ mental model in the conduction of local PID controllers. After two years from the first startup, a strong energy efficiency improvement has been observed.


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## Body

## 1. Introduction

In recent decades, process industries observed important changes and increasing need for technological improvements in the production units has been supported by an increasing level of automation. In this context, the migration toward more profitable control and optimization solutions has been registered, aimed at the research of optimal trade-offs between throughput and yields increasing, costs reduction, and product quality improvement [1].In particular, a profitable balance between the meeting of precise production and quality specifications and the achievement of energy efficiency, complying with rigorous environmental standards, has to be ensured. For this purpose, minimum payback time solutions are preferred by process engineers and managers. These smart solutions aim at a maximum exploitation of the already existent devices: software strategies are preferred to hardware modifications, introducing Advanced Process Control (APC) systems [2, 3].Among APC solutions, Model Predictive Control (MPC) architectures have shown their soundness and strong reliability in control and optimization of multivariable constrained industrial processes [4, 5]. MPC implementation in process industries is mainly due to its formulation flexibility: different cost functions, subject to different types of constraints, can be suitably designed [6, 7].In this paper, a developed APC framework for the control and the optimization of a walking beam billets reheating furnace located in an Italian steel plant is described. In the control literature, different control solutions for this control problem have been proposed. In [8], a computer control system for optimization of reheating furnaces has been developed which contains functions for fuel optimization, based on carpet diagrams and delay strategy multipliers. A hybrid optimization setpoint strategy for reheating furnaces temperatures is proposed in [9], based on steady state zone temperature optimization and dynamic optimization performed by PID controllers. In [10], a dynamic model of the reheating furnace is derived using material and energy balances and a multivariable predictive controller design procedure is proposed. In [11], the potential of nonlinear MPC techniques to improve the temperature control of the metal slabs in a hot mill reheat furnace is explored; in particular, a focus on energy consumption decreasing is presented. A Lyapunov-based MIMO state feedback controller is developed for slab temperatures in a continuous, fuel-fired reheating furnace in [12]. The controller modifies reference trajectories of furnace temperatures and is part of a cascade control scheme. In [13], a nonlinear MPC is designed for a continuous reheating furnace for steel slabs. Based on a first-principles mathematical model, the controller defines local furnace temperatures so that the slabs reach their desired final temperatures. In [14], an MPC based control method is developed for the accurate control of cooling temperature in a hot-rolled strip laminar cooling process exploiting a tailored extended Kalman filter to observe the spatial distribution of strip temperature in water cooling section.The proposed approach is based on a two-layer linear MPC architecture and addressed tailored control specifications. In particular, the linear approach allowed limiting the computational burden while the introduction of two main control modes ensured an effective control system [15].In this paper, two control requirements have been specifically addressed. First, a unified approach for the selection of the control inputs to exploit for control objectives fulfilment has been formulated for all process conditions. For this purpose, a suitable module has been introduced which tightly cooperates with a two-layer linearMPC block. Furthermore, a decoupling strategy developed by the authors in a previous work [16] has been suitably extended so to include a proper handling of all process conditions. Second, an innovative mathematical approach aimed at stoichiometric ratios constraints tightening has been proposed.Accurate simulations have proven the validity of the proposed approaches. The successive implementation of the designed controller on the industrial process successfully replaced the operators’ manual driving of local PID controllers.The paper is organized as follows: the analyzed walking beam billets reheating furnace is accurately described in Section2, together with the control specifications and the developed process modelling. Section 3 details the developed APC framework, explaining the functions of the various blocks and how the proposed solutions impact the formulations of the two MPC layers. Tuning procedures and simulation examples have been depicted in Section 4, while Section 5 describes observed field results. Finally, conclusions have been reported in Section 6.


## 2. The Studied Billets Reheating Furnace

This section discusses the analyzed industrial process, that is, a walking beam billets reheating furnace. This process represents the most significant phase of an Italian steel plant. Process features, together with the control requirements and the process modelling, are detailed.


### 2.1. Process Characterization

The production chain of the studied Italian steel industry is schematically depicted in Figure1.Figure 1
Workflow of the considered Italian steel industry.Raw materials, for example, waste steel products, are processed in the first phase, so as to obtain small steel bars at an intermediate stage of manufacture. These bars are calledbillets.In the investigated process, billets are characterized by a rectangular (0.20 [m] × 0.16 [m]) or a quadratic (0.16 [m] × 0.16 [m] or 0.15 [m] × 0.15 [m]) section and their length can be 4.5 [m] or 9 [m].The billets are then introduced in a reheating furnace. The considered reheating furnace measures 24 [m] and it can contain up to 80 billets simultaneously. When they enter the furnace, the billets can be characterized by very different temperatures. These input temperatures vary from 30[°C] to 700[°C]. The billets that are inside the furnace are reheated based on specific temperature profiles. In this way, at the furnace exit, billets temperature can vary, according to the specifications of the subsequent rolling phase. Typical billet exit temperatures vary in the range of 1000[°C]–1100[°C]. After their path along the furnace, the billets are moved toward the rolling mill stands: here they are subjected to a plastic deformation, suitably performed by stands cylinders. These cylinders deform the reheated billets according to the finished products specifications. Examples of finished products are angle bars, iron rods, or tube rounds.The present paper is focused on the reheating phase, that is, the crucial phase in a steel industry. The importance of this phase is due to the high energy amount required: in order to achieve energy efficiency, an optimal trade-off between conflicting requirements, that is, environmental impact decreasing, energy saving, and product and product quality increasing has to be ensured.The reheating phase in a furnace is schematically represented in Figure2: the billets are introduced in the considered reheating furnace from the left side and moved toward the outlet of the furnace by a revolving beam, according to the defined furnace production rate. This movement type names the considered reheating furnace aswalking beam [17, 18]. Within the furnace, air/fuel burners (located on the furnace walls) trigger combustion reactions: in this way, thermal energy is transferred to the billets during their permanence in the furnace by radiation, convection, and conduction phenomena. In the considered plant, the fuel used is the natural gas. Inside reheating furnaces, it is typical to distinguish three different areas, that is,Preheating,Heating, andSoaking area, characterized by different billets phases of the reheating process. In turn, areas are divided into zones. Starting from the furnace entrance (Figures 2 and 3, left side), the furnace areas in the case at issue are as follows:(i)
Preheating Area. This furnace area is characterized by the absence of burners. This area, exploiting hot gasses from downstream areas, performs a billets preheating process. It is constituted by a single zone, calledtunnel (Figure 3, “T”), with temperatures in the range 500[°C]–850[°C]. It is characterized by a length of 7 [m].(ii)
Heating Area. In this furnace area, the crucial phase of billets reheating process takes place. Here, the registered temperatures vary between 750[°C] and 1150[°C]. This area is divided into three zones, called (from the left side to the right side)zone 5,zone 4, andzone 3. Each zone is characterized by a length of 4 [m] and by the presence of 12 burners.(iii)
Soaking Area. In this last furnace area, the billets complete their reheating process. Here, temperatures vary approximately in the range 1100[°C]–1250[°C] and the total length is 5 [m]. The furnace area, equipped with 24 burners, is transversally divided into two zones, denoted aszone 2 andzone 1 (Figure 3).Figure 2
Representation of the considered walking beam billets reheating furnace.Figure 3
Schematic furnace zones disposition.Billets, during their traverse through the furnace, are subjected to monotonically increasing temperatures. At the beginning of thePreheating area is located a heat exchanger, denoted assmoke-exchanger.This allows the heat recover from the combustion smokes of the various furnace areas which is used for the preheating of the combustion air provided to each burner.


### 2.2. Process Instrumentation and Control Requirements

In the considered reheating furnace, measurements of the temperature of each furnace zone and of the smoke-exchanger are acquired by thermocouples suitably positioned within the furnace. TheHeatingandSoaking furnaceareas are equipped with flowmeters for fuel (natural gas) and air flow rates measurement. Air and furnace pressure are measured by manometers. The billets temperature at their entrance in the furnace are registered by an optical pyrometer. An additional pyrometer measures the temperature of the processed billets. Billets transition at the furnace inlet and outlet is detected by photocells.Before the introduction of the proposed APC system, the reheating furnace was regulated by plant operators’ manual conduction of local PID controllers. Operators, based on the furnace conditions and on the production requirements, set the zones temperature targets manually.A great amount of thermal energy is required in a steel billets reheating furnace. With the previous control system, the natural gas consumption, per year, was about 9 million [Sm3], related to a total steel production of approximately 35000 [ton]. The minimization of the fuel consumption, together with the furnace production rate maximization, clearly represents the crucial factor for energy efficiency achieving and improvement. In addition to these requirements, there is also the need to meet stringent quality standards of the finished products.After an initial study and analysis phase, repeated and targeted meetings with plant managers, engineers, and operators were conducted and the following main control specifications have been outlined for the considered APC design [19, 20]:(I)
Compliance with the required billets temperature constraints at their exit from the furnace, based on the specifications for the subsequent plastic deformation in the rolling mill phase.(II)
Meeting of stoichiometry constraints related to air/fuel ratios.(III)
Meeting of total air flow rate and smoke-exchanger temperature limits, so as to preserve furnace safety.(IV)
Furnace management in all predictable conditions.(V)
Minimization of the fuel (natural gas) consumption, optimizing all furnace conditions (e.g., with a varying furnace production rate).The simultaneous meeting of all the above requirements is not easily attainable by a manual conduction of the furnace. In the conduction of the reheating furnace, operators typically neglected the aspects more strictly tied to energy saving and environmental impact decreasing, being concentrated on assuring a suitable billets heating profile. So, with the previous furnace management, billets were often reheated more than needed by the subsequent rolling mill phase.


### 2.3. Modelling the Considered Process

In order to design an APC system for furnace energy efficiency improvement, an accurate process modelling has been performed. As widely used in APC applications, three main groups of process variables have been identified: Controlled Variables (CVs), Manipulated Variables (MVs), and Disturbance Variables (DVs). CVs are the process outputs (assumed all measured): their control is performed exploiting MVs and taking into account DVs information, in a feedforward way. In fact, MVs and DVs represent the measured process inputs: the APC system can only act on MVs, while DVs cannot be modified by the APC system. CVs have been included in ay[my×1] vector; MVs and DVs have been grouped in a u[lu×1] vector and in a d[ld×1] vector, respectively. Tables 1 and 2 depict MVs and DVs, while key CVs have been reported in Table 3. In the CVs group, the following process variables have been included: the temperature of each furnace zone (including tunnel), the difference between zone 1 and zone 2 temperatures, the smoke-exchanger temperature, the total air flow rate, and the fuel and air valves opening (per cent). In the DVs group, furnace production rate, air pressure, and furnace pressure have been included. Finally, in the MVs group, air and fuel flow rates of each furnace zone have been considered. In the plant configuration, fuel and air flow rates are regulated through PID controllers: the designed APC system will provide the PID low level controllers with suitable set-points. Their computation will be described in Section 3.Table 1
The Manipulated Variables.
Variable name Acronym [units] Zonei fuel flow rate (i=1,…,5) F u e l i [Nm3/h] Zonei air flow rate (i=1,…,5) A i r i [Nm3/h]Table 2
The Disturbance Variables.
Variable name Acronym [units] Furnace production rate Prod [t/h] Furnace pressure FurnPress [mm/H2O] Air pressure AirPress [mbar]Table 3
The main Controlled Variables.
Variable name Acronym [units] Tunnel temperature Tun [°C] Zonei (i=1,…,5) temperature Tempi [°C] Zone 1–zone 2 temp. difference Diff 12 [°C] Total air flow rate AirTot [Nm3/h] Smoke-exchanger temperature SE [°C]An identification procedure based on a black-box approach has been conducted, in order to obtain accurate dynamical models that could describe the selected process inputs (MVs/DVs) and outputs (CVs) behaviour [21–23]. First-order strictly proper asymptotically stable linear time invariant models without delays have been obtained. In these models, deviations of the process variables from consistent operating points have been included. Tables 4 and 5 symbolically represent the CVs-MVs and CVs-DVs gain matrices, Gyumy×lu and Gydmy×ld: the gain sign of the transfer functions that relate the MVs-DVs with the main CVs is here reported. Exploiting the identified models, billets temperature can be controlled with a suitable constrained control of the furnace zones temperature and of the temperature difference between the last two furnace zones (specification (I) in Section 2.2). Furthermore, smoke-exchanger temperature and total air flow rate can be controlled through fuel and air flow rates (specification (III) in Section 2.2). With regard to fuel and air flow rates usage, plant engineers have agreed on three additional requirements:(i)
The temperature of all the furnace zones, except tunnel, must be controlled exploiting only the related fuel flow rate.(ii)
The temperatures of the tunnel and of the smoke-exchanger must be controlled exploiting only zone 5 and zone 4 fuel flow rates.(iii)
The total air flow rate must be regulated using only zone 5 and zone 4 air flow rates.Table 4
The CVs-MVs mapping matrix.
Acronym Fuel5 Fuel4 Fuel3 Fuel2 Fuel1 Air5 Air4 Air3 Air2 Air1 Tun + + + + + Temp5 + + + + + − Temp4  + + + +  − Temp3   + + +   − Temp2    + +    − Temp1    + +     − Diff 12    − + AirTot      + + + + + SE + + + + +Table 5
The CVs-DVs mapping matrix.
Acronym Prod FurnPress AirPress Tun − + − Temp5 − + − Temp4 − + − Temp3 − + − Temp2 − + − Temp1 − + − Diff 12 AirTot SE − + −In order to fulfil these additional objectives, an extended decoupling strategy has been developed by the authors (Sections3.2 and 3.3).Each of the furnace zones equipped with burners must comply with stoichiometric constraints, in order to guarantee proper thermodynamic reactions. In the considered case study, five stoichiometric ratios have to be controlled. These variables have been grouped inz[mz×1] vector ofratio Controlled Variables (rCVs). Typical lower and upper constraints for the considered stoichiometric ratios have been summarized in Table 6. The stoichiometric ratios lower constraints represent safety conditions: their excessive violation can lead to a furnace stop. The upper constraints avoid air excess in the burners’ combustion, contributing to the achievement of furnace energy efficiency. Stoichiometric ratios constraints have been suitably handled by an innovative formulation in the MPC block (Section 3.5).Table 6
The ratio Controlled Variables constraints.
Variable name Acronym [units] Lower constraint Upper constraint Zone 5 stoichiometric ratio R 5   [ ] 10.8 12.3 Zone 4 stoichiometric ratio R 4   [ ] 10.5 12 Zone 3 stoichiometric ratio R 3   [ ] 10.5 12 Zone 2 stoichiometric ratio R 2   [ ] 9.8 11.3 Zone 1 stoichiometric ratio R 1   [ ] 9.8 11.3The proposed extended decoupling strategy, together with additional procedures, allows the optimization of all furnace conditions, searching for fuel minimization directions.


## 2.1. Process Characterization

The production chain of the studied Italian steel industry is schematically depicted in Figure1.Figure 1
Workflow of the considered Italian steel industry.Raw materials, for example, waste steel products, are processed in the first phase, so as to obtain small steel bars at an intermediate stage of manufacture. These bars are calledbillets.In the investigated process, billets are characterized by a rectangular (0.20 [m] × 0.16 [m]) or a quadratic (0.16 [m] × 0.16 [m] or 0.15 [m] × 0.15 [m]) section and their length can be 4.5 [m] or 9 [m].The billets are then introduced in a reheating furnace. The considered reheating furnace measures 24 [m] and it can contain up to 80 billets simultaneously. When they enter the furnace, the billets can be characterized by very different temperatures. These input temperatures vary from 30[°C] to 700[°C]. The billets that are inside the furnace are reheated based on specific temperature profiles. In this way, at the furnace exit, billets temperature can vary, according to the specifications of the subsequent rolling phase. Typical billet exit temperatures vary in the range of 1000[°C]–1100[°C]. After their path along the furnace, the billets are moved toward the rolling mill stands: here they are subjected to a plastic deformation, suitably performed by stands cylinders. These cylinders deform the reheated billets according to the finished products specifications. Examples of finished products are angle bars, iron rods, or tube rounds.The present paper is focused on the reheating phase, that is, the crucial phase in a steel industry. The importance of this phase is due to the high energy amount required: in order to achieve energy efficiency, an optimal trade-off between conflicting requirements, that is, environmental impact decreasing, energy saving, and product and product quality increasing has to be ensured.The reheating phase in a furnace is schematically represented in Figure2: the billets are introduced in the considered reheating furnace from the left side and moved toward the outlet of the furnace by a revolving beam, according to the defined furnace production rate. This movement type names the considered reheating furnace aswalking beam [17, 18]. Within the furnace, air/fuel burners (located on the furnace walls) trigger combustion reactions: in this way, thermal energy is transferred to the billets during their permanence in the furnace by radiation, convection, and conduction phenomena. In the considered plant, the fuel used is the natural gas. Inside reheating furnaces, it is typical to distinguish three different areas, that is,Preheating,Heating, andSoaking area, characterized by different billets phases of the reheating process. In turn, areas are divided into zones. Starting from the furnace entrance (Figures 2 and 3, left side), the furnace areas in the case at issue are as follows:(i)
Preheating Area. This furnace area is characterized by the absence of burners. This area, exploiting hot gasses from downstream areas, performs a billets preheating process. It is constituted by a single zone, calledtunnel (Figure 3, “T”), with temperatures in the range 500[°C]–850[°C]. It is characterized by a length of 7 [m].(ii)
Heating Area. In this furnace area, the crucial phase of billets reheating process takes place. Here, the registered temperatures vary between 750[°C] and 1150[°C]. This area is divided into three zones, called (from the left side to the right side)zone 5,zone 4, andzone 3. Each zone is characterized by a length of 4 [m] and by the presence of 12 burners.(iii)
Soaking Area. In this last furnace area, the billets complete their reheating process. Here, temperatures vary approximately in the range 1100[°C]–1250[°C] and the total length is 5 [m]. The furnace area, equipped with 24 burners, is transversally divided into two zones, denoted aszone 2 andzone 1 (Figure 3).Figure 2
Representation of the considered walking beam billets reheating furnace.Figure 3
Schematic furnace zones disposition.Billets, during their traverse through the furnace, are subjected to monotonically increasing temperatures. At the beginning of thePreheating area is located a heat exchanger, denoted assmoke-exchanger.This allows the heat recover from the combustion smokes of the various furnace areas which is used for the preheating of the combustion air provided to each burner.


## 2.2. Process Instrumentation and Control Requirements

In the considered reheating furnace, measurements of the temperature of each furnace zone and of the smoke-exchanger are acquired by thermocouples suitably positioned within the furnace. TheHeatingandSoaking furnaceareas are equipped with flowmeters for fuel (natural gas) and air flow rates measurement. Air and furnace pressure are measured by manometers. The billets temperature at their entrance in the furnace are registered by an optical pyrometer. An additional pyrometer measures the temperature of the processed billets. Billets transition at the furnace inlet and outlet is detected by photocells.Before the introduction of the proposed APC system, the reheating furnace was regulated by plant operators’ manual conduction of local PID controllers. Operators, based on the furnace conditions and on the production requirements, set the zones temperature targets manually.A great amount of thermal energy is required in a steel billets reheating furnace. With the previous control system, the natural gas consumption, per year, was about 9 million [Sm3], related to a total steel production of approximately 35000 [ton]. The minimization of the fuel consumption, together with the furnace production rate maximization, clearly represents the crucial factor for energy efficiency achieving and improvement. In addition to these requirements, there is also the need to meet stringent quality standards of the finished products.After an initial study and analysis phase, repeated and targeted meetings with plant managers, engineers, and operators were conducted and the following main control specifications have been outlined for the considered APC design [19, 20]:(I)
Compliance with the required billets temperature constraints at their exit from the furnace, based on the specifications for the subsequent plastic deformation in the rolling mill phase.(II)
Meeting of stoichiometry constraints related to air/fuel ratios.(III)
Meeting of total air flow rate and smoke-exchanger temperature limits, so as to preserve furnace safety.(IV)
Furnace management in all predictable conditions.(V)
Minimization of the fuel (natural gas) consumption, optimizing all furnace conditions (e.g., with a varying furnace production rate).The simultaneous meeting of all the above requirements is not easily attainable by a manual conduction of the furnace. In the conduction of the reheating furnace, operators typically neglected the aspects more strictly tied to energy saving and environmental impact decreasing, being concentrated on assuring a suitable billets heating profile. So, with the previous furnace management, billets were often reheated more than needed by the subsequent rolling mill phase.


## 2.3. Modelling the Considered Process

In order to design an APC system for furnace energy efficiency improvement, an accurate process modelling has been performed. As widely used in APC applications, three main groups of process variables have been identified: Controlled Variables (CVs), Manipulated Variables (MVs), and Disturbance Variables (DVs). CVs are the process outputs (assumed all measured): their control is performed exploiting MVs and taking into account DVs information, in a feedforward way. In fact, MVs and DVs represent the measured process inputs: the APC system can only act on MVs, while DVs cannot be modified by the APC system. CVs have been included in ay[my×1] vector; MVs and DVs have been grouped in a u[lu×1] vector and in a d[ld×1] vector, respectively. Tables 1 and 2 depict MVs and DVs, while key CVs have been reported in Table 3. In the CVs group, the following process variables have been included: the temperature of each furnace zone (including tunnel), the difference between zone 1 and zone 2 temperatures, the smoke-exchanger temperature, the total air flow rate, and the fuel and air valves opening (per cent). In the DVs group, furnace production rate, air pressure, and furnace pressure have been included. Finally, in the MVs group, air and fuel flow rates of each furnace zone have been considered. In the plant configuration, fuel and air flow rates are regulated through PID controllers: the designed APC system will provide the PID low level controllers with suitable set-points. Their computation will be described in Section 3.Table 1
The Manipulated Variables.
Variable name Acronym [units] Zonei fuel flow rate (i=1,…,5) F u e l i [Nm3/h] Zonei air flow rate (i=1,…,5) A i r i [Nm3/h]Table 2
The Disturbance Variables.
Variable name Acronym [units] Furnace production rate Prod [t/h] Furnace pressure FurnPress [mm/H2O] Air pressure AirPress [mbar]Table 3
The main Controlled Variables.
Variable name Acronym [units] Tunnel temperature Tun [°C] Zonei (i=1,…,5) temperature Tempi [°C] Zone 1–zone 2 temp. difference Diff 12 [°C] Total air flow rate AirTot [Nm3/h] Smoke-exchanger temperature SE [°C]An identification procedure based on a black-box approach has been conducted, in order to obtain accurate dynamical models that could describe the selected process inputs (MVs/DVs) and outputs (CVs) behaviour [21–23]. First-order strictly proper asymptotically stable linear time invariant models without delays have been obtained. In these models, deviations of the process variables from consistent operating points have been included. Tables 4 and 5 symbolically represent the CVs-MVs and CVs-DVs gain matrices, Gyumy×lu and Gydmy×ld: the gain sign of the transfer functions that relate the MVs-DVs with the main CVs is here reported. Exploiting the identified models, billets temperature can be controlled with a suitable constrained control of the furnace zones temperature and of the temperature difference between the last two furnace zones (specification (I) in Section 2.2). Furthermore, smoke-exchanger temperature and total air flow rate can be controlled through fuel and air flow rates (specification (III) in Section 2.2). With regard to fuel and air flow rates usage, plant engineers have agreed on three additional requirements:(i)
The temperature of all the furnace zones, except tunnel, must be controlled exploiting only the related fuel flow rate.(ii)
The temperatures of the tunnel and of the smoke-exchanger must be controlled exploiting only zone 5 and zone 4 fuel flow rates.(iii)
The total air flow rate must be regulated using only zone 5 and zone 4 air flow rates.Table 4
The CVs-MVs mapping matrix.
Acronym Fuel5 Fuel4 Fuel3 Fuel2 Fuel1 Air5 Air4 Air3 Air2 Air1 Tun + + + + + Temp5 + + + + + − Temp4  + + + +  − Temp3   + + +   − Temp2    + +    − Temp1    + +     − Diff 12    − + AirTot      + + + + + SE + + + + +Table 5
The CVs-DVs mapping matrix.
Acronym Prod FurnPress AirPress Tun − + − Temp5 − + − Temp4 − + − Temp3 − + − Temp2 − + − Temp1 − + − Diff 12 AirTot SE − + −In order to fulfil these additional objectives, an extended decoupling strategy has been developed by the authors (Sections3.2 and 3.3).Each of the furnace zones equipped with burners must comply with stoichiometric constraints, in order to guarantee proper thermodynamic reactions. In the considered case study, five stoichiometric ratios have to be controlled. These variables have been grouped inz[mz×1] vector ofratio Controlled Variables (rCVs). Typical lower and upper constraints for the considered stoichiometric ratios have been summarized in Table 6. The stoichiometric ratios lower constraints represent safety conditions: their excessive violation can lead to a furnace stop. The upper constraints avoid air excess in the burners’ combustion, contributing to the achievement of furnace energy efficiency. Stoichiometric ratios constraints have been suitably handled by an innovative formulation in the MPC block (Section 3.5).Table 6
The ratio Controlled Variables constraints.
Variable name Acronym [units] Lower constraint Upper constraint Zone 5 stoichiometric ratio R 5   [ ] 10.8 12.3 Zone 4 stoichiometric ratio R 4   [ ] 10.5 12 Zone 3 stoichiometric ratio R 3   [ ] 10.5 12 Zone 2 stoichiometric ratio R 2   [ ] 9.8 11.3 Zone 1 stoichiometric ratio R 1   [ ] 9.8 11.3The proposed extended decoupling strategy, together with additional procedures, allows the optimization of all furnace conditions, searching for fuel minimization directions.


## 3. APC Framework

This section describes the APC framework designed for the control and the optimization of the considered walking beam reheating furnace. The various blocks are differentiated, together with their functions. In particular, the proposed control solutions are detailed.


### 3.1. Detailing the Scheme

The APC architecture has been depicted in Figure4. Three main blocks can be noted:Plant & SCADA (P&S) block,Data Conditioning & Decoupling Selector (DC&DS), andMPC block.Figure 4
Architecture of the designed APC framework.At each control instantk,P&S block, based on a Supervisory Control and Data Acquisition (SCADA) system, supplies updated MVs, DVs, and CVs measurements (uk-1,dk-1,y(k)). Additionally, a status value (u-d-y-z status) for each process variable is forwarded.DC&DS block detects abnormal situations, exploiting field data conditioning information, and defines the final status value for each variable (u-d-y status, z status), suitably processingP&S status information. Depending on the variable final status, the variables set to be included in the controller formulation at the current control instant is accordingly modified. Details about status handling have been explained by the authors in [24]. In this work, an extension of a decoupling strategy proposed by the authors in a previous work ([16]) is formulated, assuring a straight relationship between the decoupling strategy and the MVs and CVs status values definition (Decoupling Matrix). In this way, a unified approach for the selection of MVs to exploit for the fulfilment of CVs specifications has been guaranteed in all process conditions (Section 3.2).The acquired information is forwarded to the linear Model Predictive Control (MPC) block characterized by a two-layer architecture. This block is constituted by the Targets Optimizing and Constraints Softening (TOCS) module at the upper level and the Dynamic Optimizer (DO) module at the lower lever, both supported by the Predictions Calculator module. These modules suitably cooperate, computing the MVs value u(k) to be applied to the plant at the current control instant.In both optimization problems, MVs constraints are assumed ashard constraints: they can never be violated and their feasibility has been suitably imposed. As widely used in MPC applications, CVs constraints are assumed assoft constraints: theirsoftening is admitted in critical situations through the introduction of suitable nonnegative slack variables vectors, denoted by ε. An ad hoc approach has been dedicated to rCVs constraints, as it will be shown in Section 3.5.


### 3.2. Data Conditioning & Decoupling Selector Module Tasks

An APC system must properly handle process variables in all process conditions. The control system design has to take into account all detectable situations. Certain MVs, DVs, CVs, or rCVs must be excluded from the controller formulation, based on abnormal situations (e.g., bad conditions or local control loop faults) detection or on specific plant needs.In the designed APC scheme,DC&DS block includes logics aimed at a proper modification of the initial status values supplied byP&S block (Figure 4,u-d-y-z status) as a consequence, for example, of spikes or freezing percentage violation [24]. These logics determine the final status values of all process variables (Figure 4, z status andu-d-y status).In this work, two status values have been considered:active andinactive. Anactive MV is used byMPC block for control purposes while aninactive MV cannot be moved byMPC block: for future predictions of the system behaviour, its value is frozen to its last plant measurement (Figure 4, u(k-1)). When a CV (or a rCV) isactive,MPC block acts for the satisfaction of the related control specifications, while, when a CV (or a rCV) isinactive, its control requirements must be ignored byMPC block.As stated in Section2, MVs/DVs-CVs relationships have been modelled with a linear time invariant asymptotically stable model without delays.In order to include status value information on the process model, aByumy×lu matrix has been introduced: a nonzero entry on the generic (i,j) element represents a nonzero relationship between the jth MV and the ith CV. Considering the gain matrix Gyu and defining with op≠(·,·) the element-wise inequality logical operation between two matrices, the binary matrix Byu is obtained:(1)Byu=op≠Gyu,0my×lu,where 0[my×lu] is a matrix with elements all equal to 0.For the genericith CV, if this CV isactive at the current control instant, allactive MVs tied to it (i.e., theactive MVs characterized by a 1 on the related position on the ith row of Byu matrix) can act for its specifications fulfilment. In order to admit additional degrees of freedom for the control specifications, in a previous work, the authors proposed a suitable decoupling strategy, not dependent on the tuning parameters. This strategy enables, for each CV, the selection of the MVs to exploit for its control purposes, that is, equivalently, the inhibition of given MVs.Two equivalent mathematical approaches have been formulated [16]. In this work, the formulation based on the definition of a Decoupling Matrix is considered and suitably extended.An initial Decoupling MatrixDE[my×lu] is supplied byP&S block (Figure 4). The generic (i,j) element of DE is set equal to 0 if both the following conditions hold:(i)
The(i,j) element of Byu matrix is equal to 1.(ii)
Thejth MV must not be used to control the ith CV.Otherwise, the(i,j) element is equal to 1.The generic(i,j) nonzero element of the initial DE matrix is zeroed byDC&DS block if at least one of the two following conditions holds:(i)
Theith CV isinactive.(ii)
Thejth MV isinactive.The finalDE matrix is then forwarded to TOCS and DO modules.In the considered case study, rCVs are represented by the ratio between the related air and fuel flow rates.DC&DS block processes the rCVs status value supplied byP&S block, taking into account the final status values of air and fuel flow rates (Figure 4,u-d-y status). In this way, the final status value associated with each rCV is obtained (Figure 4, z status). Generally, the ith stoichiometric ratio (i=1,…,5) is defined asinactive if the ith fuel flow rate isinactive. This means that if the ith fuel flow rate isinactive, the constraints of the ith stoichiometric ratio must be ignored byMPC block.


### 3.3. Predictions Calculator and TOCS Modules Tasks

As stated before,MPC block is composed of three modules. The two-layer linear MPC scheme exploits process variables predictions on aprediction horizon  Hp; these predictions are parametrized on a defined number of MVs moves (assumed here to be applied at the first Hu prediction steps), calledcontrol horizonand denoted with Hu(0<Hu≤Hp) [25]. At each control instant, a sequence of MVs values is computed, but only the first value (Figure 4, u(k)) is forwarded to the plant, repeating the entire control algorithm at the next control instant (receding horizon strategy) [26, 27].Predictions Calculator module computes CVsfree predictions over Hp (Figure 4, y free predictions), keeping MVs and DVs constant at their last field value (Figure 4, u(k-1) and d(k-1), resp.) and taking into account the last CVs value (Figure 4, y(k)) for plant-model mismatch inclusion. Future information about DVs is assumed unknown, so future DVs trends are assumed constant at d(k-1) value. In CVsfree predictions computation, CVs status values information is also included.For the genericvth CV, predictions over Hp are expressed by(2)y^vk+i∣k=y^freevk+i∣k+Δy^vk+i∣k(3)Δy^vk+i∣k=Fi,v·ΔUk(4)ΔUk=Δu^k∣kΔu^k+1∣k⋮Δu^k+Hu-1∣k,where y^freevk+i∣k is thefree prediction at the ith prediction instant, while Δy^vk+i∣k is the relatedforced component. ΔUklu·Hu×1 is a vector that contains the Hu MVs moves Δu^k+j∣k(j=0,…,Hu-1), assumed in the first Hu steps. Δu^k+j∣k terms are among the decision variables of DO optimization problem (Section 3.4). The effects of MVs moves on the ith prediction are modelled by Fi,v[1×lu·Hu] vector.In this work, the prediction horizonHp is assumed to be tuned so as to guarantee steady state reaching to the process model. Based on this assumption, the following relationship holds: (5)Gyu=FHp,11,j-1·lu+1:j·lu⋮FHp,my1,j-1·lu+1:j·luj=1,…,Hu,where FHp,v1,j-1·lu+1:j·lu(v=1,…,my) is a subvector of FHp,v of lu columns, from the (j-1·lu+1)th to the (j·lu)th.The two-layer linear MPC scheme performs two optimization problems: the first is solved by the upper layer, represented by TOCS module, while the second is related to the lower layer, that is, DO module. The formulation of the two modules is based on the same linear process model: TOCS module considers it in a steady state way, while both steady and transient states are taken into account by DO module.InTOCS module formulation, MVs steady state value, denoted by u^TOCS(k), is expressed by (6)u^TOCSk=uk-1+Δu^TOCSk,where Δu^TOCSk represents the MVs steady state move: this vector is among decision variables of TOCS optimization problem. CVs steady state value, indicated by y^TOCS(k), is expressed by(7)y^TOCSk=y^1k+Hp∣k⋮y^myk+Hp∣k=y^freek+Hp∣k+Δy^TOCSk(8)Δy^TOCSk=Gyu·Δu^TOCSk,where the CVs steady state forced component is represented by Δy^TOCSk.When introducing the proposed extended decoupling strategy in the CVs steady state expression, (8) is modified as follows:(9)Δy^TOCSk=Gyu∘DE·Δu^TOCSk,where ∘ indicates the element-wise product between matrices. In this way, information about MVs and CVs status values and additional control requirements are included. In this specific case, the additional control requirements are specifications (i)–(iii) of Section 2.3.At each control instant,TOCS module minimizes a linear cost function, subject to linear constraints, based on (6), (7), and (9):(10)VTOCSk=cuT·Δu^TOCSk+ρy_TOCST·εy_TOCSsubject to(11)ilbdu_TOCS≤Δu^TOCSk≤ubdu_TOCSiilbu_TOCS≤u^TOCSk≤ubu_TOCSiiilby_TOCS-ECRlby_TOCS·εy_TOCS≤y^TOCSk≤uby_TOCS+ECRuby_TOCS·εy_TOCSivTOCSzconstraintsvεy_TOCS≥0,where ubdu_TOCS and lbdu_TOCS represent the upper and lower constraints related to the MVs steady state move, while ubu_TOCS, lbu_TOCS and uby_TOCS, lby_TOCS are the upper and lower constraints for MVs and CVs steady state value, respectively. A set of ad hoc rCVs steady state linear constraints is represented by TOCSzconstraints. Their inclusion in TOCS module is detailed in Section 3.5. All these constraints are amongu-y-z constraints of Figure 4. cu[lu×1] vector weights MVs steady state move.InTOCS optimization problem for each CV, two slack variables have been assumed (upper and lower constraints are not dependent). εy_TOCS[2·my×1] is included in TOCS CVs constraints (11)(iii) suitably weighted by Equal Concern for the Relaxation (ECR) positive coefficients, while, in TOCS linear cost function (10), a positive column vector ρy_TOCS weights its effects.TOCS tuning parameters of Figure4 include cu, ρy_TOCS, ECRlby_TOCS, and ECRuby_TOCS terms.Information about MVs and CVs status values and about the considered additional control specifications has been included in CVs steady state prediction through (9). In order to include this information in the other crucial terms of TOCS Linear Programming (LP) problem, the following column vectors are defined:(12)pu=op≠∑i=1myDEi,:T,0lu×1,py=op≠∑j=1luDE:,j,0my×1,where DE(i,:) and DE(:,j) represent the ith row and the jth column of DE matrix, respectively. 0[n×1] represents a vector of zeros. pu is a [lu×1] vector, while py is a [my×1] vector. The initial cu vector is modified as follows:(13)cu=cu∘pu.Furthermore, consideringpu, when its jth generic element (j=1,…,lu) is equal to zero, all the upper and lower constraints related to the jth MV in (11)(i)-(11)(ii) are cut off by TOCS module. Similarly, taking into account py, when its ith generic element (i=1,…,my) is equal to zero, all the upper and lower constraints related to the ith CV in (11)(iii) are cut off by TOCS module.T O C S module, solving its LP problem, computes the decision variables vector, constituted by Δu^TOCSk and εy_TOCS. Through expressions (6), (7), and (9), u^TOCS(k) and y^TOCS(k) are obtained: these vectors represent MVs and CVs steady state targets which are consistent with TOCS computed constraints. They are forwarded to DO module (Figure 4,u-y target). In order to guarantee a coherency between CVs steady state targets and constraints, the initial CVs constraints provided byP&S block to TOCS module (Figure 4,u-y-z constraints) are suitably processed taking into account εy_TOCS values. The resulting CVs constraints over Hp are then forwarded to DO module (Figure 4, y constraints) [28].


### 3.4. DO Module Tasks

DO module represents the lower layer of the proposed MPC scheme. It solves the second optimization problem ofMPC block, computing the MVs value that will be applied to the plant at each control instant.In order to include the proposed extended decoupling strategy in the CVs predictions overHp, (3) is modified as follows: (14)Δy^vk+i∣k=Fi,v∘11×Hu⊗DEv,:·ΔUk,where 1[1×Hu] represents a vector of ones and ⊗ indicates the Kronecker product. In this way, the forced component of CVs dynamic expression (2) contains also information about MVs and CVs status values and additional control requirements.At each control instant, DO module minimizes a quadratic cost function, subject to linear constraints, based on (2), (4), and (14):(15)VDOk=∑i=1Hpy^k+i∣k-ytk+i∣kQi2+∑i=0Hp-1u^k+i∣k-utk+i∣kSi2+∑i=0Hu-1Δu^k+i∣kRi2+εDOkρDO2subject to(16)ilbdu_DOi≤Δu^k+i∣k≤ubdu_DOi,i=0,…,Hu-1iilbu_DOi≤u^k+i∣k≤ubu_DOi,i=0,…,Hu-1iiilby_DOi-ECRlby_DOi·εDOk≤y^k+i∣k≤uby_DOi+ECRuby_DOi·εDOk,i=1,…,HpivDOzconstraintsvεDOk≥0,where ubdu_DO(i) and lbdu_DO(i) represent the upper and lower constraints related to MVs future dynamic moves Δu^k+i∣k(i=0,…,Hu-1), while ubu_DO(i), lbu_DO(i), and uby_DO(i), lby_DO(i) are the upper and lower constraints for MVs and CVs future dynamic values u^k+i∣k(i=0,…,Hu-1) and y^k+i∣k(i=1,…,Hp), respectively. A set of ad hoc rCVs dynamic linear constraints is represented by DOzconstraints. Their inclusion in DO module is detailed in Section 3.5. All these constraints are amongu-z constraints and y constraints of Figure 4.In this work, inDO optimization problem, a single slack variable for each CV or rCV is assumed. εDO(k)[(my+mz)×1] is included in DO CVs constraints (16)(iii) through suitable Equal Concern for the Relaxation (ECR) positive coefficients, while, in DO linear cost function (15), a positive definite diagonal matrix ρDO weights its effects.Q i and S(i) in (15) are positive semidefinite diagonal matrices: they weight CVs and MVs future tracking errors, respectively. Future tracking errors are evaluated with respect to the future reference trajectories represented by ytk+i∣k and utk+i∣k, respectively. ytk+i∣k and utk+i∣k are suitably computed from CVs and MVs steady state targets, supplied by TOCS module. Finally, R(i) are positive definite diagonal matrices that weight MVs moves over Hu. Qi, S(i), R(i), ρDO, ECRlby_DOi, and ECRuby_DOi are among DO tuning parameters of Figure 4.Further details on the imposed cooperation and consistency betweenTOCS and DO modules formulations have been explained by the authors in [28].Similar toTOCS, in order to include MVs and CVs status values and the additional control specifications in the DO Quadratic Programming (QP) problem, the following diagonal matrices are defined:(17)Tu=diagpu,Ty=diagpy,where diag(x) indicates a [n×n] diagonal matrix with the element of x along the main diagonal. Tu is a [lu×lu] matrix and Ty is a [my×my] matrix. The initial Qi and S(i) matrices are modified as follows:(18)Qi=Qi∘TySi=Si∘Tu.A constraint cut-off procedure similar to that described forTOCS in the previous section is then applied.Through the solution of the QP problem,DO module computes Δu^k+j∣k(j=0,…,Hu-1) and εDOk terms. Only the first MVs move, represented by Δu^k∣k, is considered, obtaining u(k).


### 3.5. Stoichiometric Ratios Constraints Handling

In the previous subsections,TOCS and DO stoichiometric ratios constraints have been symbolically represented with TOCSzconstraints (11)(iv) and DOzconstraints (16)(iv).Inu^TOCS(k) and u^k+j∣k(j=0,…,Hu-1)[lu×1] vectors, the furnace zone i fuel and air flow rates (i=1,…,mz) represent the ith and (i+mz)th component (mz=5).The steady state formulation of the furnace zonei stoichiometric ratio (i=1,…,5) is the following:(19)z^TOCSik=u^TOCSi+5ku^TOCSik,where u^TOCSi+5(k) and u^TOCSi(k) represent the steady state values of furnace zone i air and fuel flow rates (nonnegative values). Denoting the steady state lower and upper constraints of furnace zone i stoichiometric ratio as lbz_TOCSi and ubz_TOCSi, TOCSzconstraints can be expressed by(20)lbz_TOCSi≤z^TOCSik≤ubz_TOCSi.Expression (20) can be recast as a set of linear inequality constraints on TOCS decision variables Δu^TOCSk, exploiting (6). In the proposed application, stoichiometric ratios steady state constraints are assumed ashard constraints: they can never be violated and their feasibility has been suitably imposed. Based on rCVs final status value, when the generic ith rCV isinactive, all the upper and lower constraints related to the ith rCV in (20) are cut off by TOCS module.The dynamic formulation of the generic furnace zonei stoichiometric ratio (i=1,…,5) at the jth prediction instant is the following: (21)z^ik+j∣k=u^i+5k+j∣ku^ik+j∣k,where u^i+5k+j∣k and u^ik+j∣k represent the dynamic values of furnace zone i air and fuel flow rates (nonnegative values). Denoting the dynamic lower and upper constraints of furnace zone i stoichiometric ratio at the jth prediction instant as lbz_DOij and ubz_DOij, DOzconstraints can be expressed by(22)lbz_DOij-ECRlbz_DOij·εDOk≤z^ik+j∣k≤ubz_DOij+ECRubz_DOij·εDOk.Expression (22) can be recast as a set of linear inequality constraints on DO decision variables Δu^k+j∣k(j=0,…,Hu-1) and εDOk, exploiting Hu definition. In the proposed application, stoichiometric ratios dynamic constraints are assumed assoft constraints: theirsoftening is admitted in critical situations thanks to the introduction of a suitable nonnegative slack variable for each stoichiometric ratio. These slack variables have been included in the εDOk vector. ECRlbz_DOij and ECRubz_DOij positive vectors contain the Equal Concern for the Relaxation (ECR) coefficients. These coefficients, in cooperation with the related elements of ρDO positive definite diagonal matrix, allow a suitable constraints relaxation ranking.Based on rCVs final status value, when the genericith rCV isinactive, all the Hu upper and lower constraints related to the ith rCV in (22) are cut off by DO module.


## 3.1. Detailing the Scheme

The APC architecture has been depicted in Figure4. Three main blocks can be noted:Plant & SCADA (P&S) block,Data Conditioning & Decoupling Selector (DC&DS), andMPC block.Figure 4
Architecture of the designed APC framework.At each control instantk,P&S block, based on a Supervisory Control and Data Acquisition (SCADA) system, supplies updated MVs, DVs, and CVs measurements (uk-1,dk-1,y(k)). Additionally, a status value (u-d-y-z status) for each process variable is forwarded.DC&DS block detects abnormal situations, exploiting field data conditioning information, and defines the final status value for each variable (u-d-y status, z status), suitably processingP&S status information. Depending on the variable final status, the variables set to be included in the controller formulation at the current control instant is accordingly modified. Details about status handling have been explained by the authors in [24]. In this work, an extension of a decoupling strategy proposed by the authors in a previous work ([16]) is formulated, assuring a straight relationship between the decoupling strategy and the MVs and CVs status values definition (Decoupling Matrix). In this way, a unified approach for the selection of MVs to exploit for the fulfilment of CVs specifications has been guaranteed in all process conditions (Section 3.2).The acquired information is forwarded to the linear Model Predictive Control (MPC) block characterized by a two-layer architecture. This block is constituted by the Targets Optimizing and Constraints Softening (TOCS) module at the upper level and the Dynamic Optimizer (DO) module at the lower lever, both supported by the Predictions Calculator module. These modules suitably cooperate, computing the MVs value u(k) to be applied to the plant at the current control instant.In both optimization problems, MVs constraints are assumed ashard constraints: they can never be violated and their feasibility has been suitably imposed. As widely used in MPC applications, CVs constraints are assumed assoft constraints: theirsoftening is admitted in critical situations through the introduction of suitable nonnegative slack variables vectors, denoted by ε. An ad hoc approach has been dedicated to rCVs constraints, as it will be shown in Section 3.5.


## 3.2. Data Conditioning & Decoupling Selector Module Tasks

An APC system must properly handle process variables in all process conditions. The control system design has to take into account all detectable situations. Certain MVs, DVs, CVs, or rCVs must be excluded from the controller formulation, based on abnormal situations (e.g., bad conditions or local control loop faults) detection or on specific plant needs.In the designed APC scheme,DC&DS block includes logics aimed at a proper modification of the initial status values supplied byP&S block (Figure 4,u-d-y-z status) as a consequence, for example, of spikes or freezing percentage violation [24]. These logics determine the final status values of all process variables (Figure 4, z status andu-d-y status).In this work, two status values have been considered:active andinactive. Anactive MV is used byMPC block for control purposes while aninactive MV cannot be moved byMPC block: for future predictions of the system behaviour, its value is frozen to its last plant measurement (Figure 4, u(k-1)). When a CV (or a rCV) isactive,MPC block acts for the satisfaction of the related control specifications, while, when a CV (or a rCV) isinactive, its control requirements must be ignored byMPC block.As stated in Section2, MVs/DVs-CVs relationships have been modelled with a linear time invariant asymptotically stable model without delays.In order to include status value information on the process model, aByumy×lu matrix has been introduced: a nonzero entry on the generic (i,j) element represents a nonzero relationship between the jth MV and the ith CV. Considering the gain matrix Gyu and defining with op≠(·,·) the element-wise inequality logical operation between two matrices, the binary matrix Byu is obtained:(1)Byu=op≠Gyu,0my×lu,where 0[my×lu] is a matrix with elements all equal to 0.For the genericith CV, if this CV isactive at the current control instant, allactive MVs tied to it (i.e., theactive MVs characterized by a 1 on the related position on the ith row of Byu matrix) can act for its specifications fulfilment. In order to admit additional degrees of freedom for the control specifications, in a previous work, the authors proposed a suitable decoupling strategy, not dependent on the tuning parameters. This strategy enables, for each CV, the selection of the MVs to exploit for its control purposes, that is, equivalently, the inhibition of given MVs.Two equivalent mathematical approaches have been formulated [16]. In this work, the formulation based on the definition of a Decoupling Matrix is considered and suitably extended.An initial Decoupling MatrixDE[my×lu] is supplied byP&S block (Figure 4). The generic (i,j) element of DE is set equal to 0 if both the following conditions hold:(i)
The(i,j) element of Byu matrix is equal to 1.(ii)
Thejth MV must not be used to control the ith CV.Otherwise, the(i,j) element is equal to 1.The generic(i,j) nonzero element of the initial DE matrix is zeroed byDC&DS block if at least one of the two following conditions holds:(i)
Theith CV isinactive.(ii)
Thejth MV isinactive.The finalDE matrix is then forwarded to TOCS and DO modules.In the considered case study, rCVs are represented by the ratio between the related air and fuel flow rates.DC&DS block processes the rCVs status value supplied byP&S block, taking into account the final status values of air and fuel flow rates (Figure 4,u-d-y status). In this way, the final status value associated with each rCV is obtained (Figure 4, z status). Generally, the ith stoichiometric ratio (i=1,…,5) is defined asinactive if the ith fuel flow rate isinactive. This means that if the ith fuel flow rate isinactive, the constraints of the ith stoichiometric ratio must be ignored byMPC block.


## 3.3. Predictions Calculator and TOCS Modules Tasks

As stated before,MPC block is composed of three modules. The two-layer linear MPC scheme exploits process variables predictions on aprediction horizon  Hp; these predictions are parametrized on a defined number of MVs moves (assumed here to be applied at the first Hu prediction steps), calledcontrol horizonand denoted with Hu(0<Hu≤Hp) [25]. At each control instant, a sequence of MVs values is computed, but only the first value (Figure 4, u(k)) is forwarded to the plant, repeating the entire control algorithm at the next control instant (receding horizon strategy) [26, 27].Predictions Calculator module computes CVsfree predictions over Hp (Figure 4, y free predictions), keeping MVs and DVs constant at their last field value (Figure 4, u(k-1) and d(k-1), resp.) and taking into account the last CVs value (Figure 4, y(k)) for plant-model mismatch inclusion. Future information about DVs is assumed unknown, so future DVs trends are assumed constant at d(k-1) value. In CVsfree predictions computation, CVs status values information is also included.For the genericvth CV, predictions over Hp are expressed by(2)y^vk+i∣k=y^freevk+i∣k+Δy^vk+i∣k(3)Δy^vk+i∣k=Fi,v·ΔUk(4)ΔUk=Δu^k∣kΔu^k+1∣k⋮Δu^k+Hu-1∣k,where y^freevk+i∣k is thefree prediction at the ith prediction instant, while Δy^vk+i∣k is the relatedforced component. ΔUklu·Hu×1 is a vector that contains the Hu MVs moves Δu^k+j∣k(j=0,…,Hu-1), assumed in the first Hu steps. Δu^k+j∣k terms are among the decision variables of DO optimization problem (Section 3.4). The effects of MVs moves on the ith prediction are modelled by Fi,v[1×lu·Hu] vector.In this work, the prediction horizonHp is assumed to be tuned so as to guarantee steady state reaching to the process model. Based on this assumption, the following relationship holds: (5)Gyu=FHp,11,j-1·lu+1:j·lu⋮FHp,my1,j-1·lu+1:j·luj=1,…,Hu,where FHp,v1,j-1·lu+1:j·lu(v=1,…,my) is a subvector of FHp,v of lu columns, from the (j-1·lu+1)th to the (j·lu)th.The two-layer linear MPC scheme performs two optimization problems: the first is solved by the upper layer, represented by TOCS module, while the second is related to the lower layer, that is, DO module. The formulation of the two modules is based on the same linear process model: TOCS module considers it in a steady state way, while both steady and transient states are taken into account by DO module.InTOCS module formulation, MVs steady state value, denoted by u^TOCS(k), is expressed by (6)u^TOCSk=uk-1+Δu^TOCSk,where Δu^TOCSk represents the MVs steady state move: this vector is among decision variables of TOCS optimization problem. CVs steady state value, indicated by y^TOCS(k), is expressed by(7)y^TOCSk=y^1k+Hp∣k⋮y^myk+Hp∣k=y^freek+Hp∣k+Δy^TOCSk(8)Δy^TOCSk=Gyu·Δu^TOCSk,where the CVs steady state forced component is represented by Δy^TOCSk.When introducing the proposed extended decoupling strategy in the CVs steady state expression, (8) is modified as follows:(9)Δy^TOCSk=Gyu∘DE·Δu^TOCSk,where ∘ indicates the element-wise product between matrices. In this way, information about MVs and CVs status values and additional control requirements are included. In this specific case, the additional control requirements are specifications (i)–(iii) of Section 2.3.At each control instant,TOCS module minimizes a linear cost function, subject to linear constraints, based on (6), (7), and (9):(10)VTOCSk=cuT·Δu^TOCSk+ρy_TOCST·εy_TOCSsubject to(11)ilbdu_TOCS≤Δu^TOCSk≤ubdu_TOCSiilbu_TOCS≤u^TOCSk≤ubu_TOCSiiilby_TOCS-ECRlby_TOCS·εy_TOCS≤y^TOCSk≤uby_TOCS+ECRuby_TOCS·εy_TOCSivTOCSzconstraintsvεy_TOCS≥0,where ubdu_TOCS and lbdu_TOCS represent the upper and lower constraints related to the MVs steady state move, while ubu_TOCS, lbu_TOCS and uby_TOCS, lby_TOCS are the upper and lower constraints for MVs and CVs steady state value, respectively. A set of ad hoc rCVs steady state linear constraints is represented by TOCSzconstraints. Their inclusion in TOCS module is detailed in Section 3.5. All these constraints are amongu-y-z constraints of Figure 4. cu[lu×1] vector weights MVs steady state move.InTOCS optimization problem for each CV, two slack variables have been assumed (upper and lower constraints are not dependent). εy_TOCS[2·my×1] is included in TOCS CVs constraints (11)(iii) suitably weighted by Equal Concern for the Relaxation (ECR) positive coefficients, while, in TOCS linear cost function (10), a positive column vector ρy_TOCS weights its effects.TOCS tuning parameters of Figure4 include cu, ρy_TOCS, ECRlby_TOCS, and ECRuby_TOCS terms.Information about MVs and CVs status values and about the considered additional control specifications has been included in CVs steady state prediction through (9). In order to include this information in the other crucial terms of TOCS Linear Programming (LP) problem, the following column vectors are defined:(12)pu=op≠∑i=1myDEi,:T,0lu×1,py=op≠∑j=1luDE:,j,0my×1,where DE(i,:) and DE(:,j) represent the ith row and the jth column of DE matrix, respectively. 0[n×1] represents a vector of zeros. pu is a [lu×1] vector, while py is a [my×1] vector. The initial cu vector is modified as follows:(13)cu=cu∘pu.Furthermore, consideringpu, when its jth generic element (j=1,…,lu) is equal to zero, all the upper and lower constraints related to the jth MV in (11)(i)-(11)(ii) are cut off by TOCS module. Similarly, taking into account py, when its ith generic element (i=1,…,my) is equal to zero, all the upper and lower constraints related to the ith CV in (11)(iii) are cut off by TOCS module.T O C S module, solving its LP problem, computes the decision variables vector, constituted by Δu^TOCSk and εy_TOCS. Through expressions (6), (7), and (9), u^TOCS(k) and y^TOCS(k) are obtained: these vectors represent MVs and CVs steady state targets which are consistent with TOCS computed constraints. They are forwarded to DO module (Figure 4,u-y target). In order to guarantee a coherency between CVs steady state targets and constraints, the initial CVs constraints provided byP&S block to TOCS module (Figure 4,u-y-z constraints) are suitably processed taking into account εy_TOCS values. The resulting CVs constraints over Hp are then forwarded to DO module (Figure 4, y constraints) [28].


## 3.4. DO Module Tasks

DO module represents the lower layer of the proposed MPC scheme. It solves the second optimization problem ofMPC block, computing the MVs value that will be applied to the plant at each control instant.In order to include the proposed extended decoupling strategy in the CVs predictions overHp, (3) is modified as follows: (14)Δy^vk+i∣k=Fi,v∘11×Hu⊗DEv,:·ΔUk,where 1[1×Hu] represents a vector of ones and ⊗ indicates the Kronecker product. In this way, the forced component of CVs dynamic expression (2) contains also information about MVs and CVs status values and additional control requirements.At each control instant, DO module minimizes a quadratic cost function, subject to linear constraints, based on (2), (4), and (14):(15)VDOk=∑i=1Hpy^k+i∣k-ytk+i∣kQi2+∑i=0Hp-1u^k+i∣k-utk+i∣kSi2+∑i=0Hu-1Δu^k+i∣kRi2+εDOkρDO2subject to(16)ilbdu_DOi≤Δu^k+i∣k≤ubdu_DOi,i=0,…,Hu-1iilbu_DOi≤u^k+i∣k≤ubu_DOi,i=0,…,Hu-1iiilby_DOi-ECRlby_DOi·εDOk≤y^k+i∣k≤uby_DOi+ECRuby_DOi·εDOk,i=1,…,HpivDOzconstraintsvεDOk≥0,where ubdu_DO(i) and lbdu_DO(i) represent the upper and lower constraints related to MVs future dynamic moves Δu^k+i∣k(i=0,…,Hu-1), while ubu_DO(i), lbu_DO(i), and uby_DO(i), lby_DO(i) are the upper and lower constraints for MVs and CVs future dynamic values u^k+i∣k(i=0,…,Hu-1) and y^k+i∣k(i=1,…,Hp), respectively. A set of ad hoc rCVs dynamic linear constraints is represented by DOzconstraints. Their inclusion in DO module is detailed in Section 3.5. All these constraints are amongu-z constraints and y constraints of Figure 4.In this work, inDO optimization problem, a single slack variable for each CV or rCV is assumed. εDO(k)[(my+mz)×1] is included in DO CVs constraints (16)(iii) through suitable Equal Concern for the Relaxation (ECR) positive coefficients, while, in DO linear cost function (15), a positive definite diagonal matrix ρDO weights its effects.Q i and S(i) in (15) are positive semidefinite diagonal matrices: they weight CVs and MVs future tracking errors, respectively. Future tracking errors are evaluated with respect to the future reference trajectories represented by ytk+i∣k and utk+i∣k, respectively. ytk+i∣k and utk+i∣k are suitably computed from CVs and MVs steady state targets, supplied by TOCS module. Finally, R(i) are positive definite diagonal matrices that weight MVs moves over Hu. Qi, S(i), R(i), ρDO, ECRlby_DOi, and ECRuby_DOi are among DO tuning parameters of Figure 4.Further details on the imposed cooperation and consistency betweenTOCS and DO modules formulations have been explained by the authors in [28].Similar toTOCS, in order to include MVs and CVs status values and the additional control specifications in the DO Quadratic Programming (QP) problem, the following diagonal matrices are defined:(17)Tu=diagpu,Ty=diagpy,where diag(x) indicates a [n×n] diagonal matrix with the element of x along the main diagonal. Tu is a [lu×lu] matrix and Ty is a [my×my] matrix. The initial Qi and S(i) matrices are modified as follows:(18)Qi=Qi∘TySi=Si∘Tu.A constraint cut-off procedure similar to that described forTOCS in the previous section is then applied.Through the solution of the QP problem,DO module computes Δu^k+j∣k(j=0,…,Hu-1) and εDOk terms. Only the first MVs move, represented by Δu^k∣k, is considered, obtaining u(k).


## 3.5. Stoichiometric Ratios Constraints Handling

In the previous subsections,TOCS and DO stoichiometric ratios constraints have been symbolically represented with TOCSzconstraints (11)(iv) and DOzconstraints (16)(iv).Inu^TOCS(k) and u^k+j∣k(j=0,…,Hu-1)[lu×1] vectors, the furnace zone i fuel and air flow rates (i=1,…,mz) represent the ith and (i+mz)th component (mz=5).The steady state formulation of the furnace zonei stoichiometric ratio (i=1,…,5) is the following:(19)z^TOCSik=u^TOCSi+5ku^TOCSik,where u^TOCSi+5(k) and u^TOCSi(k) represent the steady state values of furnace zone i air and fuel flow rates (nonnegative values). Denoting the steady state lower and upper constraints of furnace zone i stoichiometric ratio as lbz_TOCSi and ubz_TOCSi, TOCSzconstraints can be expressed by(20)lbz_TOCSi≤z^TOCSik≤ubz_TOCSi.Expression (20) can be recast as a set of linear inequality constraints on TOCS decision variables Δu^TOCSk, exploiting (6). In the proposed application, stoichiometric ratios steady state constraints are assumed ashard constraints: they can never be violated and their feasibility has been suitably imposed. Based on rCVs final status value, when the generic ith rCV isinactive, all the upper and lower constraints related to the ith rCV in (20) are cut off by TOCS module.The dynamic formulation of the generic furnace zonei stoichiometric ratio (i=1,…,5) at the jth prediction instant is the following: (21)z^ik+j∣k=u^i+5k+j∣ku^ik+j∣k,where u^i+5k+j∣k and u^ik+j∣k represent the dynamic values of furnace zone i air and fuel flow rates (nonnegative values). Denoting the dynamic lower and upper constraints of furnace zone i stoichiometric ratio at the jth prediction instant as lbz_DOij and ubz_DOij, DOzconstraints can be expressed by(22)lbz_DOij-ECRlbz_DOij·εDOk≤z^ik+j∣k≤ubz_DOij+ECRubz_DOij·εDOk.Expression (22) can be recast as a set of linear inequality constraints on DO decision variables Δu^k+j∣k(j=0,…,Hu-1) and εDOk, exploiting Hu definition. In the proposed application, stoichiometric ratios dynamic constraints are assumed assoft constraints: theirsoftening is admitted in critical situations thanks to the introduction of a suitable nonnegative slack variable for each stoichiometric ratio. These slack variables have been included in the εDOk vector. ECRlbz_DOij and ECRubz_DOij positive vectors contain the Equal Concern for the Relaxation (ECR) coefficients. These coefficients, in cooperation with the related elements of ρDO positive definite diagonal matrix, allow a suitable constraints relaxation ranking.Based on rCVs final status value, when the genericith rCV isinactive, all the Hu upper and lower constraints related to the ith rCV in (22) are cut off by DO module.


## 4. Simulation Results

This section illustrates some details on tuning procedures and reports simulation examples that show the efficiency of the proposed control solutions.


### 4.1. Controller Tuning Summary

Through a constrained control of zones temperature, the required billets temperature at the furnace exit can be ensured; thanks to the cooperative action ofTOCS and DO modules, optimal economic configurations of fuel flow rates inside the assigned process constraints can be obtained. The need of a two-layer architecture was motivated by the necessity of a steady state optimization, able to supply economic MVs targets. In addition, the developed cooperation feature between TOCS and DO modules avoids the process conduction toward dangerous operating points and/or the exhibition of inefficient transient state behaviours. A further optimization of the APC system that can deal with the various furnace conditions has been achieved exploiting a billets reheating first-principles model [28]. Two main control modes have been formulated: the first, which has been detailed in this paper, is based on the CVs/rCVs-MVs/DVs models, while the second exploits also the billets modelization. Online parameters estimation techniques have been developed and exploited in a linear parameter varying control scheme.In order to obtain the desired performances of the designed APC system, an accurate tuning phase ofTOCS and DO modules has been conducted. Setting the APC cycle time at 1 minute, a prediction horizon of 60 minutes and a control horizon of eight moves have been selected. In this way, the considered linear time invariant model reaches a steady state condition and the controller is equipped with a suitable number of decision variables for the optimization problems [29].InDO cost function (15), tracking objectives related to CVs have not been taken into account: Qi matrices are all zero matrices. In TOCS and DO cost functions (10) and (15), the highest priority has been assigned to the minimization of the involved slack variables, so as to comply with CVs and rCVs constraints. For this purpose, ρy_TOCS and ρDO terms have been suitably set. In order to define a CVs/rCVs priority ranking in constraints relaxation, a joint tuning of these weights (ρy_TOCS, ρDO) and ECR coefficients has been performed. For example, smoke-exchanger temperature, total air flow rate, and stoichiometric ratios constraints are of major importance with respect to furnace zones temperature constraints. Furthermore, a joint tuning of S(i) and R(i) initial matrices has been performed, in order to guarantee targets reaching for all the MVs while controlling moves magnitude.InTOCS cost function, cu elements related to fuel flow rates have been set as positive, in order to prefer the fuels minimization direction when possible. The remaining cu elements have been set to zero.Finally, the initial Decoupling MatrixDE provided byP&S block toDC&DS block has been set so as to satisfy the additional control specifications (i)–(iii) depicted in Section 2.3.In the following Sections4.2 and 4.3, two simulation examples are proposed showing the validity of the proposed control strategy. The proposed simulations refer to the control of zone 5 and zone 4 of the furnace. Tables 7 and 8 depict the considered process variables. Relationships between the considered MVs and CVs have been depicted in Table 9: relationships have been simply denoted with the gain sign of the related transfer function. Table 10 shows the initial Decoupling Matrix DE provided byP&S block toDC&DS block related to the MVs and CVs at issue.Table 7
Considered set of MVs.
Variable name Acronym [units] Zone 5 fuel Fuel5 [Nm3/h] Zone 4 fuel Fuel4 [Nm3/h] Zone 5 air Air5 [Nm3/h] Zone 4 air Air4 [Nm3/h]Table 8
Considered set of CVs/rCVs.
Variable name Acronym [units] Zone 5 temperature Temp5 [°C] Zone 4 temperature Temp4 [°C] Zone 5 stoichiometric ratio R 5   [ ] Zone 4 stoichiometric ratio R 4   [ ]Table 9
Reduced mapping matrix.
Acronym Fuel5 Fuel4 Air5 Air4 Temp5 + + − Temp4  +  −Table 10
Reduced Decoupling MatrixDE.
Acronym Fuel5 Fuel4 Air5 Air4 Temp5 1 0 0 1 Temp4 1 1 1 0


### 4.2. Zone 5 Control Simulation Example

The first simulation example considers to have only the furnace zone 5 under control of the proposed APC system. The other MVs, CVs, and rCVs are assumed asinactive, so they are not considered for control purposes;DC&DS block and the other modules perform all the operations related toinactive process variables as explained in the previous sections. In particular, all (initial) DE rows and columns related toinactive CVs and MVs are zeroed byDC&DS block. Theinactive MVs and all the DVs are assumed constant, so not influencing the proposed simulation.T O C S and DO modules have to ensure an optimal usage of zone 5 fuel and air flow rates, taking into account zone 5 temperature and zone 5 stoichiometric ratio constraints. In particular, thanks to the decoupling strategy, for zone 5 temperature constraints tightening only zone 5 fuel flow rate is used, thus meeting the related specification depicted in Section 2.3.The initial process operating point assumes that the two considered output variables (Figures5 and 6, blue line) satisfy their constraints: in the first 30 simulation instants, the cooperative action of TOCS and DO modules ensures zone 5 fuel flow rate minimization (Figure 7, blue line), saturating zone 5 stoichiometric ratio upper bound (Figure 6, red line) and meeting zone 5 temperature constraints (Figure 5, red lines). In this first part of the simulation, no TOCS and DO modules constraints softening is required (red, cyan, and green lines in Figures 5 and 6 overlap). A profitable operating point for zone 5 fuel and air flow rates is guaranteed, saturating zone 5 air flow rate lower bound (Figures 7 and 8, blue lines). Subsequently, at time instant 31, an operator request is simulated: zone 5 temperature upper and lower constraints are increased by 50[°C] and 40[°C], respectively. Thanks to the proposed decoupling strategy, only zone 5 fuel flow rate acts for the satisfaction of the updated temperature constraints, reaching the new TOCS target (Figure 7, black line). After a brief transient, requiring a temporary softening of its new lower constraint (Figure 5, green lines), zone 5 temperature is restored on its new lower bound. This corrective action of zone 5 fuel flow rate could lead to the violation of zone 5 stoichiometric ratio lower bound: in order to avoid this critical condition, also zone 5 air flow rate TOCS target is changed (Figure 8, black line). In this way, zone 5 stoichiometric ratio approaches its lower bound, after a negligible DO constraints softening (Figure 6, green lines).Figure 5
Zone 5 temperature trends.Figure 6
Zone 5 stoic. ratio trends.Figure 7
Zone 5 fuel flow rate trends.Figure 8
Zone 5 air flow rate trends.


### 4.3. Zones 4 and 5 Control Simulation Example

The second simulation example considers only the furnace zones 4 and 5 under control of the proposed APC system. The other MVs, CVs, and rCVs are assumed asinactive, so they are not considered for control purposes;DC&DS block and the other modules perform all the operations related toinactive process variables as explained in the previous sections. In particular, all (initial) DE rows and columns related toinactive CVs and MVs are zeroed byDC&DS block. Theinactive MVs and all the DVs are assumed constant, so not influencing the proposed simulation.T O C S and DO modules have to ensure an optimal usage of zones 4 and 5 fuel and air flow rates, taking into account zones 4 and 5 temperature and zones 4 and 5 stoichiometric ratios constraints. In particular, thanks to the decoupling strategy, for zone 5 temperature constraints tightening only zone 5 fuel flow rate is used, while for zone 4 temperature constraints meeting only zone 4 fuel flow rate is exploited. In this way, the related specification depicted in Section 2.3 is satisfied.The initial process operating point assumes that the four considered output variables (Figures9, 10, and 13, blue line) satisfy their constraints (zone 4 temperature has not been depicted for brevity): in the first 20 simulation instants, the cooperative action of TOCS and DO modules ensures zone 5 and zone 4 fuel flow rate minimization (Figures 12 and 14, blue line), saturating zone 5 stoichiometric ratio upper bound (Figure 10, red line) and meeting zone 4 stoichiometric ratio constraints (Figure 13, red lines). Furthermore, in this first part of simulation, zones 4-5 temperature meets the imposed constraints (Figure 9, red lines) and no TOCS and DO modules constraints softening is required (red, cyan, and green lines in Figures 9, 10, and 13 overlap). A profitable operating point for zones 4-5 fuel and air flow rates is guaranteed, saturating zone 5 air flow rate lower bound (Figures 11, 12, and 14, blue lines; zone 4 air flow rate has not been depicted for brevity: it remains constant at its initial value during the simulation). Subsequently, at time instant 21, an operator request is simulated: zone 5 temperature lower constraint is increased by 25[°C]. Thanks to the proposed decoupling strategy, only zone 5 fuel flow rate acts for the satisfaction of the updated temperature constraints, reaching the new TOCS target (Figure 12, black line). After a brief transient, requiring a temporary softening of its new lower constraint (Figure 9, green lines), zone 5 temperature is restored on its new lower bound. This corrective action of zone 5 fuel flow rate could lead to the violation of zone 5 stoichiometric ratio lower bound: in order to avoid this critical condition, also zone 5 air flow rate TOCS target is changed (Figure 11, black line). In this way, zone 5 stoichiometric ratio approaches its lower bound, after a negligible DO constraints softening (Figure 10, green lines). Note that the proposed constraint change does not modify zone 4 variables configuration.Figure 9
Zone 5 temperature trends.Figure 10
Zone 5 stoic. ratio trends.Figure 11
Zone 5 air flow rate trends.Figure 12
Zone 5 fuel flow rate trends.Figure 13
Zone 4 stoic. ratio trends.Figure 14
Zone 4 fuel flow rate trends.


## 4.1. Controller Tuning Summary

Through a constrained control of zones temperature, the required billets temperature at the furnace exit can be ensured; thanks to the cooperative action ofTOCS and DO modules, optimal economic configurations of fuel flow rates inside the assigned process constraints can be obtained. The need of a two-layer architecture was motivated by the necessity of a steady state optimization, able to supply economic MVs targets. In addition, the developed cooperation feature between TOCS and DO modules avoids the process conduction toward dangerous operating points and/or the exhibition of inefficient transient state behaviours. A further optimization of the APC system that can deal with the various furnace conditions has been achieved exploiting a billets reheating first-principles model [28]. Two main control modes have been formulated: the first, which has been detailed in this paper, is based on the CVs/rCVs-MVs/DVs models, while the second exploits also the billets modelization. Online parameters estimation techniques have been developed and exploited in a linear parameter varying control scheme.In order to obtain the desired performances of the designed APC system, an accurate tuning phase ofTOCS and DO modules has been conducted. Setting the APC cycle time at 1 minute, a prediction horizon of 60 minutes and a control horizon of eight moves have been selected. In this way, the considered linear time invariant model reaches a steady state condition and the controller is equipped with a suitable number of decision variables for the optimization problems [29].InDO cost function (15), tracking objectives related to CVs have not been taken into account: Qi matrices are all zero matrices. In TOCS and DO cost functions (10) and (15), the highest priority has been assigned to the minimization of the involved slack variables, so as to comply with CVs and rCVs constraints. For this purpose, ρy_TOCS and ρDO terms have been suitably set. In order to define a CVs/rCVs priority ranking in constraints relaxation, a joint tuning of these weights (ρy_TOCS, ρDO) and ECR coefficients has been performed. For example, smoke-exchanger temperature, total air flow rate, and stoichiometric ratios constraints are of major importance with respect to furnace zones temperature constraints. Furthermore, a joint tuning of S(i) and R(i) initial matrices has been performed, in order to guarantee targets reaching for all the MVs while controlling moves magnitude.InTOCS cost function, cu elements related to fuel flow rates have been set as positive, in order to prefer the fuels minimization direction when possible. The remaining cu elements have been set to zero.Finally, the initial Decoupling MatrixDE provided byP&S block toDC&DS block has been set so as to satisfy the additional control specifications (i)–(iii) depicted in Section 2.3.In the following Sections4.2 and 4.3, two simulation examples are proposed showing the validity of the proposed control strategy. The proposed simulations refer to the control of zone 5 and zone 4 of the furnace. Tables 7 and 8 depict the considered process variables. Relationships between the considered MVs and CVs have been depicted in Table 9: relationships have been simply denoted with the gain sign of the related transfer function. Table 10 shows the initial Decoupling Matrix DE provided byP&S block toDC&DS block related to the MVs and CVs at issue.Table 7
Considered set of MVs.
Variable name Acronym [units] Zone 5 fuel Fuel5 [Nm3/h] Zone 4 fuel Fuel4 [Nm3/h] Zone 5 air Air5 [Nm3/h] Zone 4 air Air4 [Nm3/h]Table 8
Considered set of CVs/rCVs.
Variable name Acronym [units] Zone 5 temperature Temp5 [°C] Zone 4 temperature Temp4 [°C] Zone 5 stoichiometric ratio R 5   [ ] Zone 4 stoichiometric ratio R 4   [ ]Table 9
Reduced mapping matrix.
Acronym Fuel5 Fuel4 Air5 Air4 Temp5 + + − Temp4  +  −Table 10
Reduced Decoupling MatrixDE.
Acronym Fuel5 Fuel4 Air5 Air4 Temp5 1 0 0 1 Temp4 1 1 1 0


## 4.2. Zone 5 Control Simulation Example

The first simulation example considers to have only the furnace zone 5 under control of the proposed APC system. The other MVs, CVs, and rCVs are assumed asinactive, so they are not considered for control purposes;DC&DS block and the other modules perform all the operations related toinactive process variables as explained in the previous sections. In particular, all (initial) DE rows and columns related toinactive CVs and MVs are zeroed byDC&DS block. Theinactive MVs and all the DVs are assumed constant, so not influencing the proposed simulation.T O C S and DO modules have to ensure an optimal usage of zone 5 fuel and air flow rates, taking into account zone 5 temperature and zone 5 stoichiometric ratio constraints. In particular, thanks to the decoupling strategy, for zone 5 temperature constraints tightening only zone 5 fuel flow rate is used, thus meeting the related specification depicted in Section 2.3.The initial process operating point assumes that the two considered output variables (Figures5 and 6, blue line) satisfy their constraints: in the first 30 simulation instants, the cooperative action of TOCS and DO modules ensures zone 5 fuel flow rate minimization (Figure 7, blue line), saturating zone 5 stoichiometric ratio upper bound (Figure 6, red line) and meeting zone 5 temperature constraints (Figure 5, red lines). In this first part of the simulation, no TOCS and DO modules constraints softening is required (red, cyan, and green lines in Figures 5 and 6 overlap). A profitable operating point for zone 5 fuel and air flow rates is guaranteed, saturating zone 5 air flow rate lower bound (Figures 7 and 8, blue lines). Subsequently, at time instant 31, an operator request is simulated: zone 5 temperature upper and lower constraints are increased by 50[°C] and 40[°C], respectively. Thanks to the proposed decoupling strategy, only zone 5 fuel flow rate acts for the satisfaction of the updated temperature constraints, reaching the new TOCS target (Figure 7, black line). After a brief transient, requiring a temporary softening of its new lower constraint (Figure 5, green lines), zone 5 temperature is restored on its new lower bound. This corrective action of zone 5 fuel flow rate could lead to the violation of zone 5 stoichiometric ratio lower bound: in order to avoid this critical condition, also zone 5 air flow rate TOCS target is changed (Figure 8, black line). In this way, zone 5 stoichiometric ratio approaches its lower bound, after a negligible DO constraints softening (Figure 6, green lines).Figure 5
Zone 5 temperature trends.Figure 6
Zone 5 stoic. ratio trends.Figure 7
Zone 5 fuel flow rate trends.Figure 8
Zone 5 air flow rate trends.


## 4.3. Zones 4 and 5 Control Simulation Example

The second simulation example considers only the furnace zones 4 and 5 under control of the proposed APC system. The other MVs, CVs, and rCVs are assumed asinactive, so they are not considered for control purposes;DC&DS block and the other modules perform all the operations related toinactive process variables as explained in the previous sections. In particular, all (initial) DE rows and columns related toinactive CVs and MVs are zeroed byDC&DS block. Theinactive MVs and all the DVs are assumed constant, so not influencing the proposed simulation.T O C S and DO modules have to ensure an optimal usage of zones 4 and 5 fuel and air flow rates, taking into account zones 4 and 5 temperature and zones 4 and 5 stoichiometric ratios constraints. In particular, thanks to the decoupling strategy, for zone 5 temperature constraints tightening only zone 5 fuel flow rate is used, while for zone 4 temperature constraints meeting only zone 4 fuel flow rate is exploited. In this way, the related specification depicted in Section 2.3 is satisfied.The initial process operating point assumes that the four considered output variables (Figures9, 10, and 13, blue line) satisfy their constraints (zone 4 temperature has not been depicted for brevity): in the first 20 simulation instants, the cooperative action of TOCS and DO modules ensures zone 5 and zone 4 fuel flow rate minimization (Figures 12 and 14, blue line), saturating zone 5 stoichiometric ratio upper bound (Figure 10, red line) and meeting zone 4 stoichiometric ratio constraints (Figure 13, red lines). Furthermore, in this first part of simulation, zones 4-5 temperature meets the imposed constraints (Figure 9, red lines) and no TOCS and DO modules constraints softening is required (red, cyan, and green lines in Figures 9, 10, and 13 overlap). A profitable operating point for zones 4-5 fuel and air flow rates is guaranteed, saturating zone 5 air flow rate lower bound (Figures 11, 12, and 14, blue lines; zone 4 air flow rate has not been depicted for brevity: it remains constant at its initial value during the simulation). Subsequently, at time instant 21, an operator request is simulated: zone 5 temperature lower constraint is increased by 25[°C]. Thanks to the proposed decoupling strategy, only zone 5 fuel flow rate acts for the satisfaction of the updated temperature constraints, reaching the new TOCS target (Figure 12, black line). After a brief transient, requiring a temporary softening of its new lower constraint (Figure 9, green lines), zone 5 temperature is restored on its new lower bound. This corrective action of zone 5 fuel flow rate could lead to the violation of zone 5 stoichiometric ratio lower bound: in order to avoid this critical condition, also zone 5 air flow rate TOCS target is changed (Figure 11, black line). In this way, zone 5 stoichiometric ratio approaches its lower bound, after a negligible DO constraints softening (Figure 10, green lines). Note that the proposed constraint change does not modify zone 4 variables configuration.Figure 9
Zone 5 temperature trends.Figure 10
Zone 5 stoic. ratio trends.Figure 11
Zone 5 air flow rate trends.Figure 12
Zone 5 fuel flow rate trends.Figure 13
Zone 4 stoic. ratio trends.Figure 14
Zone 4 fuel flow rate trends.


## 5. Field Results

The project for the realization of the proposed APC system began in August 2013 and ended in May 2014, through a profitable cooperation between Università Politecnica delle Marche and i.Process Srl. Together with an additional billets first principle model, the whole work has been introduced in proprietary software and it has been awarded with an Italian patent [15].The introduction of the APC system on the real plant dates back to early June 2014, replacing operators’ manual conduction of local PID controllers.Figure15 shows an example of the plant performances with the designed control system comparing them with the previous control system ones. The billets final temperature measured by the optical pyrometer is depicted for a total period of 15 hours. Similar furnace boundary conditions (e.g., billets input temperature and furnace production rate) are considered. The first 4 hours refer to the previous control system conduction (APC OFF): with this control solution, the billets temperature constraints at the exit of the furnace (straight lines, 1055[°C]–1080[°C]) are met, but energy saving and environmental impact decreasing aspects are neglected. In fact, final billets temperature often results in being very close to the imposed upper bound. In the following 11 hours, the controller is switched to the designed one (APC ON): as it can be noted, the billets final temperature approaches its lower bound thus leading to energy efficiency achievement.Figure 15
Billets final temperature trends and related constraints.In Figure16, the fuel specific consumption ([Sm3/ton]) before and after the introduction of the designed control system is depicted. The fuel specific consumption is computed considering natural gas usage and furnace production rate. A period of four and a half years is taken into account. The period from January 2011 to July 2013 (at the left side of Figure 16) refers to the previous control system, while the one from June 2014 to May 2015 (at the right side of Figure 16) is related to the proposed APC system. As shown in Figure 16, about 5% reduction is observed after the activation of the designed control system. Up to June 2016, the reduction has become greater than 6%. Furthermore, after two years from the first startup, a service factor greater than 95% has been registered.Figure 16
Fuel specific consumption trends and related expected values without and with the designed APC system.


## 6. Conclusions

In this work, the control and optimization of a walking beam billets reheating furnace located in an Italian steel plant have been addressed.An Advanced Process Control architecture, based on a two-layer linear Model Predictive Control strategy, has been developed. Suitable modifications on the mathematical formulations of the two layers have been proposed, in order to comply with the required control specifications, and two control solutions have been detailed. An extended decoupling strategy has been formulated assuring, in all process conditions and independently on tuning parameters, a unified approach to the problem of the selection of the input variables to exploit for the control of each single output variable. For this purpose, an ad hoc module has been introduced. In addition, a tailored stoichiometric ratios constraints formulation has been introduced into the two Model Predictive Control layers.Simulation results demonstrated the validity and the reliability of the proposed control solutions.Field results have shown the advantages of the introduction of the designed Advanced Process Control system on the real plant, substituting the previous operators’ manual conduction of local PID controllers: energy saving and environmental impact decreasing aspects have been optimized, thus leading to a major energy efficiency.The considered strategy has been introduced in a proprietary tool, obtaining an Italian patent.



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*Source: 5401616-2017-01-31.xml*