Journal of Control Science and Engineering
Volume 2017 (2017), Article ID 5401616, 15 pages
https://doi.org/10.1155/2017/5401616

Research Article

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

Silvia Maria Zanoli and Crescenzo Pepe

Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, Ancona, Italy

Correspondence should be addressed to Silvia Maria Zanoli; s.zanoli@univpm.it

Received 23 September 2016; Revised 18 November 2016; Accepted 21 December 2016; Published 31 January 2017

Academic Editor: Shuyou Yu

Copyright © 2017 Silvia Maria Zanoli and Crescenzo Pepe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 linear MPC 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 Section 2, 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 Figure 1.

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 called billets. 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 Figure 2: 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 as walking 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, and Soaking 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, called tunnel (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, and zone 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 as zone 2 and zone 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 the Preheating area is located a heat exchanger, denoted as smoke-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. The Heating and Soaking furnace areas 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 a vector; MVs and DVs have been grouped in a vector and in a 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]

Zone fuel flow rate
[Nm³/h]

Zone air flow rate
[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]

Zone () 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, and : 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 (Sections 3.2 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 in vector of ratio 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 block (Section 3.5).

Table 6: The ratio Controlled Variables constraints.

Variable name
Acronym [units]
Lower constraint
Upper constraint

Zone 5 stoichiometric ratio

10.8
12.3

Zone 4 stoichiometric ratio

10.5
12

Zone 3 stoichiometric ratio

10.5
12

Zone 2 stoichiometric ratio

9.8
11.3

Zone 1 stoichiometric ratio

9.8
11.3

The 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 Figure 4. Three main blocks can be noted: Plant & SCADA (P&S) block, Data Conditioning & Decoupling Selector (DC&DS), and MPC block.

Figure 4: Architecture of the designed APC framework.

At each control instant , P&S block, based on a Supervisory Control and Data Acquisition (SCADA) system, supplies updated MVs, DVs, and CVs measurements (). 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, status), suitably processing P&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 to be applied to the plant at the current control instant.

In both optimization problems, MVs constraints are assumed as hard constraints: they can never be violated and their feasibility has been suitably imposed. As widely used in MPC applications, CVs constraints are assumed as soft constraints: their softening is admitted in critical situations through the introduction of suitable nonnegative slack variables vectors, denoted by . 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 by P&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, status and u-d-y status).

In this work, two status values have been considered: active and inactive. An active MV is used by MPC block for control purposes while an inactive MV cannot be moved by MPC block: for future predictions of the system behaviour, its value is frozen to its last plant measurement (Figure 4, ). When a CV (or a rCV) is active, MPC block acts for the satisfaction of the related control specifications, while, when a CV (or a rCV) is inactive, its control requirements must be ignored by MPC block.

As stated in Section 2, 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, a matrix has been introduced: a nonzero entry on the generic element represents a nonzero relationship between the th MV and the th CV. Considering the gain matrix and defining with the element-wise inequality logical operation between two matrices, the binary matrix is obtained:

where is a matrix with elements all equal to 0.

For the generic th CV, if this CV is active at the current control instant, all active MVs tied to it (i.e., the active MVs characterized by a 1 on the related position on the th row of 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 Matrix is supplied by P&S block (Figure 4). The generic element of is set equal to 0 if both the following conditions hold:

(i)The element of matrix is equal to 1.(ii)The th MV must not be used to control the th CV.

Otherwise, the element is equal to 1.

The generic nonzero element of the initial matrix is zeroed by DC&DS block if at least one of the two following conditions holds:

(i)The th CV is inactive.(ii)The th MV is inactive.

The final 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 by P&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, status). Generally, the th stoichiometric ratio () is defined as inactive if the th fuel flow rate is inactive. This means that if the th fuel flow rate is inactive, the constraints of the th stoichiometric ratio must be ignored by MPC 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 a prediction horizon  ; these predictions are parametrized on a defined number of MVs moves (assumed here to be applied at the first prediction steps), called control horizon and denoted with [25]. At each control instant, a sequence of MVs values is computed, but only the first value (Figure 4, ) is forwarded to the plant, repeating the entire control algorithm at the next control instant (receding horizon strategy) [26, 27].

Predictions Calculator module computes CVs free predictions over (Figure 4, free predictions), keeping MVs and DVs constant at their last field value (Figure 4, and , resp.) and taking into account the last CVs value (Figure 4, ) for plant-model mismatch inclusion. Future information about DVs is assumed unknown, so future DVs trends are assumed constant at value. In CVs free predictions computation, CVs status values information is also included.

For the generic th CV, predictions over are expressed by

where is the free prediction at the th prediction instant, while is the related forced component. is a vector that contains the MVs moves , assumed in the first steps. terms are among the decision variables of optimization problem (Section 3.4). The effects of MVs moves on the th prediction are modelled by vector.

In this work, the prediction horizon is assumed to be tuned so as to guarantee steady state reaching to the process model. Based on this assumption, the following relationship holds:

where is a subvector of of columns, from the th to the 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.

In module formulation, MVs steady state value, denoted by , is expressed by

where represents the MVs steady state move: this vector is among decision variables of optimization problem. CVs steady state value, indicated by , is expressed by

where the CVs steady state forced component is represented by .

When introducing the proposed extended decoupling strategy in the CVs steady state expression, (8) is modified as follows:

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, module minimizes a linear cost function, subject to linear constraints, based on (6), (7), and (9):

subject to

where and represent the upper and lower constraints related to the MVs steady state move, while , and , 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 . Their inclusion in module is detailed in Section 3.5. All these constraints are among u-y-z constraints of Figure 4. vector weights MVs steady state move.

In optimization problem for each CV, two slack variables have been assumed (upper and lower constraints are not dependent). is included in CVs constraints (11)(iii) suitably weighted by Equal Concern for the Relaxation (ECR) positive coefficients, while, in linear cost function (10), a positive column vector weights its effects.

TOCS tuning parameters of Figure 4 include , , , and 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 Linear Programming (LP) problem, the following column vectors are defined:

where and represent the th row and the th column of matrix, respectively. represents a vector of zeros. is a vector, while is a vector. The initial vector is modified as follows:

Furthermore, considering , when its th generic element () is equal to zero, all the upper and lower constraints related to the th MV in (11)(i)-(11)(ii) are cut off by module. Similarly, taking into account , when its th generic element () is equal to zero, all the upper and lower constraints related to the th CV in (11)(iii) are cut off by module.

module, solving its LP problem, computes the decision variables vector, constituted by and . Through expressions (6), (7), and (9), and are obtained: these vectors represent MVs and CVs steady state targets which are consistent with computed constraints. They are forwarded to module (Figure 4, u-y target). In order to guarantee a coherency between CVs steady state targets and constraints, the initial CVs constraints provided by P&S block to module (Figure 4, u-y-z constraints) are suitably processed taking into account values. The resulting CVs constraints over are then forwarded to module (Figure 4, constraints) [28].

3.4. DO Module Tasks

DO module represents the lower layer of the proposed MPC scheme. It solves the second optimization problem of MPC 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 over , (3) is modified as follows:

where 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):

subject to

where and represent the upper and lower constraints related to MVs future dynamic moves , while , , and , are the upper and lower constraints for MVs and CVs future dynamic values and , respectively. A set of ad hoc rCVs dynamic linear constraints is represented by . Their inclusion in module is detailed in Section 3.5. All these constraints are among u-z constraints and constraints of Figure 4.

In this work, in optimization problem, a single slack variable for each CV or rCV is assumed. 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 weights its effects.

and 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 and , respectively. and are suitably computed from CVs and MVs steady state targets, supplied by module. Finally, are positive definite diagonal matrices that weight MVs moves over . , , , , , and are among DO tuning parameters of Figure 4.

Further details on the imposed cooperation and consistency between and DO modules formulations have been explained by the authors in [28].

Similar to , 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:

where indicates a diagonal matrix with the element of along the main diagonal. is a matrix and is a matrix. The initial and matrices are modified as follows:

A constraint cut-off procedure similar to that described for in the previous section is then applied.

Through the solution of the QP problem, module computes and terms. Only the first MVs move, represented by , is considered, obtaining .

3.5. Stoichiometric Ratios Constraints Handling

In the previous subsections, and stoichiometric ratios constraints have been symbolically represented with (11)(iv) and (16)(iv).

In and vectors, the furnace zone fuel and air flow rates () represent the th and th component ().

The steady state formulation of the furnace zone stoichiometric ratio () is the following:

where and represent the steady state values of furnace zone air and fuel flow rates (nonnegative values). Denoting the steady state lower and upper constraints of furnace zone stoichiometric ratio as and , can be expressed by

Expression (20) can be recast as a set of linear inequality constraints on decision variables , exploiting (6). In the proposed application, stoichiometric ratios steady state constraints are assumed as hard constraints: they can never be violated and their feasibility has been suitably imposed. Based on rCVs final status value, when the generic th rCV is inactive, all the upper and lower constraints related to the th rCV in (20) are cut off by module.

The dynamic formulation of the generic furnace zone stoichiometric ratio () at the th prediction instant is the following:

where and represent the dynamic values of furnace zone air and fuel flow rates (nonnegative values). Denoting the dynamic lower and upper constraints of furnace zone stoichiometric ratio at the th prediction instant as and , can be expressed by

Expression (22) can be recast as a set of linear inequality constraints on decision variables and , exploiting definition. In the proposed application, stoichiometric ratios dynamic constraints are assumed as soft constraints: their softening 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 vector. and positive vectors contain the Equal Concern for the Relaxation (ECR) coefficients. These coefficients, in cooperation with the related elements of positive definite diagonal matrix, allow a suitable constraints relaxation ranking.

Based on rCVs final status value, when the generic th rCV is inactive, all the upper and lower constraints related to the th 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 of 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 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 of 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].

In cost function (15), tracking objectives related to CVs have not been taken into account: matrices are all zero matrices. In 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, and terms have been suitably set. In order to define a CVs/rCVs priority ranking in constraints relaxation, a joint tuning of these weights (, ) 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 and initial matrices has been performed, in order to guarantee targets reaching for all the MVs while controlling moves magnitude.

In cost function, elements related to fuel flow rates have been set as positive, in order to prefer the fuels minimization direction when possible. The remaining elements have been set to zero.

Finally, the initial Decoupling Matrix provided by P&S block to DC&DS block has been set so as to satisfy the additional control specifications (i)–(iii) depicted in Section 2.3.

In the following Sections 4.2 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 provided by P&S block to DC&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

Zone 4 stoichiometric ratio

Table 9: Reduced mapping matrix.

Acronym
Fuel₅
Fuel₄
Air₅
Air₄

Temp₅
+
+
−

Temp₄

+

−

Table 10: Reduced Decoupling Matrix .

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 as inactive, so they are not considered for control purposes; DC&DS block and the other modules perform all the operations related to inactive process variables as explained in the previous sections. In particular, all (initial) rows and columns related to inactive CVs and MVs are zeroed by DC&DS block. The inactive MVs and all the DVs are assumed constant, so not influencing the proposed simulation.

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 (Figures 5 and 6, blue line) satisfy their constraints: in the first 30 simulation instants, the cooperative action of 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 and 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 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 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 as inactive, so they are not considered for control purposes; DC&DS block and the other modules perform all the operations related to inactive process variables as explained in the previous sections. In particular, all (initial) rows and columns related to inactive CVs and MVs are zeroed by DC&DS block. The inactive MVs and all the DVs are assumed constant, so not influencing the proposed simulation.

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 (Figures 9, 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 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 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 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 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.

Figure 15 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 Figure 16, 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.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

http://www.honeywellprocess.com

M. Bauer and I. K. Craig, “Economic assessment of advanced process control—a survey and framework,” Journal of Process Control, vol. 18, no. 1, pp. 2–18, 2008.

P. L. Latour, J. H. Sharpe, and M. C. Delaney, “Estimating benefits from advanced control,” ISA Transactions, vol. 25, no. 4, pp. 13–21, 1986.

C. R. Cutler and B. L. Ramaker, “Dynamic matrix control—a computer control algorithm,” in Proceedings of the Dynamic matrix controla computer control algorithm, 1980.

S. J. Qin and T. A. Badgwell, “A survey of industrial model predictive control technology,” Control Engineering Practice, vol. 11, no. 7, pp. 733–764, 2003.

Y. G. Xi, D. W. Li, and S. Lin, “Model predictive control—current status and challenges,” Acta Automatica Sinica, vol. 39, no. 3, pp. 222–236, 2013.

M. L. Darby and M. Nikolaou, “MPC: current practice and challenges,” Control Engineering Practice, vol. 20, no. 4, pp. 328–342, 2012.

B. Lede, “A control system for fuel optimization of reheating furnaces,” Scandinavian Journal of Metallurgy, vol. 15, no. 1, pp. 16–24, 1986.

Z. Wang, T. Chai, S. Guan, and C. Shao, “Hybrid optimization setpoint strategy for slab reheating furnace temperature,” in Proceedings of the American Control Conference (ACC '99), pp. 4082–4086, San Diego, Calif, USA, June 1999.

H. S. Ko, J.-S. Kim, T.-W. Yoon, M. Lim, D. R. Yang, and I. S. Jun, “Modeling and predictive control of a reheating furnace,” in Proceedings of the American Control Conference, pp. 2725–2729, June 2000.

L. Balbis, J. Balderud, and M. J. Grimble, “Nonlinear predictive control of steel slab reheating furnace,” in Proceedings of the American Control Conference (ACC '08), pp. 1679–1684, Seattle, Wash, USA, June 2008.

A. Steinboeck, D. Wild, and A. Kugi, “Feedback tracking control of continuous reheating furnaces,” in Proceedings of the 18th IFAC World Congress, pp. 11744–11749, Milano, Italy, September 2011.

A. Steinboeck, D. Wild, and A. Kugi, “Nonlinear model predictive control of a continuous slab reheating furnace,” Control Engineering Practice, vol. 21, no. 4, pp. 495–508, 2013.

Y. Zheng, N. Li, and S. Li, “Hot-rolled strip laminar cooling process plant-wide temperature monitoring and control,” Control Engineering Practice, vol. 21, no. 1, pp. 23–30, 2013.

L. Barboni, G. Astolfi, C. Pepe, and F. Cocchioni, “Metodo Per Il Controllo Di Forni Di Riscaldo,” Italian Patent n. 0001424136 awarded by Ufficio Italiano Brevetti e Marchi (UIBM), 2014 http://www.uibm.gov.it/uibm/dati/Avanzata.aspx?load=info_list_uno&id=2253683&table=Invention&

S. M. Zanoli, C. Pepe, and L. Barboni, “Application of advanced process control techniques to a pusher type reheating furnace,” Journal of Physics: Conference Series, vol. 659, no. 1, 2015.

W. Trinks, M. H. Mawhinney, R. A. Shannon, R. J. Reed, and J. R. Garvey, Industrial Furnaces, John Wiley & Sons, New York, NY, USA, 2004.

A. von Starck, A. Muhlbauer, and C. Kramer, Handbook of Thermoprocessing Technologies, Vulkan, Essen, Germany, 2005.

A. Martensson, “Energy efficiency improvement by measurement and control: a case study of reheating furnaces in the steel industry,” in Proceedings of the 14th National Industrial Energy technology Conference, pp. 236–243, 1992.

F. G. Shinskey, Energy Conservation Through Control, Academic Press, New York, NY, USA, 1978.

L. Ljung, System Identification: Theory for the User, Prentice-Hall, Englewood Cliffs, NJ, USA, 1987.

Multivariable system identification for process control, Elsevier, 2001.

T. Liu, Q.-G. Wang, and H.-P. Huang, “A tutorial review on process identification from step or relay feedback test,” Journal of Process Control, vol. 23, no. 10, pp. 1597–1623, 2013.

C. Pepe and S. M. Zanoli, “A two-layer model predictive control system with adaptation to variables status values,” in Proceedings of the 17th IEEE International Carpathian Control Conference (ICCC '16), pp. 573–578, High Tatras, Slovakia, May 2016.

J. Maciejowski, Predictive Control with Constraints, Prentice-Hall, Harlow, UK, 2002.

E. F. Camacho and C. A. Bordons, Model Predictive Control, Springer, London, UK, 2007.

J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design, Nob Hill, Madison, Wisc, USA, 2013.

S. M. Zanoli and C. Pepe, “The importance of cooperation and consistency in two-layer Model Predictive Control,” in Proceedings of the 17th IEEE International Carpathian Control Conference (ICCC '16), pp. 825–830, High Tatras, Slovakia, May 2016.

J. L. Garriga and M. Soroush, “Model predictive control tuning methods: a review,” Industrial & Engineering Chemistry Research, vol. 49, no. 8, pp. 3505–3515, 2010.