Increasing the Level of Autonomy of Control of the Electric Arc Furnace by Weakening Interphase Interactions

Abstract

Steelmaking is one of the most energy-intensive industries, so improving control efficiency helps to reduce the energy used to produce a tonne of steel. Mutual influences between the phases of an electric arc furnace in available electrode movement control systems cause unproductive electrode movements as a reaction to the redistribution of currents among the phases of a three-phase power supply system due to changes in arc length in one of the phases. The nonlinearity of the characteristics of an electric arc furnace significantly complicates the ability to provide autonomous electrode movement control. The approach proposed in this paper, based on the formation of a matrix of mutual influences with variable coefficients, significantly improves the per-phase autonomy of the electrode movement control system. Nonlinear dependences of the mutual influence coefficients as a function of the current increment in the phase in which the disturbance occurred are obtained. Thus, it is possible to practically eliminate unproductive electrode movements in existing control systems by avoiding the traditional use of a dead zone, which reduces the control quality in the zone of small disturbances. The complex of experiments performed using the mathematical model demonstrate that the mutual influence improves the dynamic properties of the electrode movement system in certain operating modes.

1. Introduction

Manufacturing industries are leading consumers of energy as well as emitters of carbon-based greenhouse gases, but these industries are working to overcome this problem, finding new and better ways to reduce emissions [1,2]. The steel industry, one of the largest industrial sectors, has been under considerable pressure recently to reduce emissions and its reliance on coal and coke. Coal currently meets around 75% of the energy and feedstock demand of the sector. The decarburization of scrap-based steelmaking is advancing, with solutions such as substituting natural gas-firing with hydrogen, biomass, and biocarbon or substituting charge and injection carbon with biogenic alternatives [3,4]. The metallurgic industry seeks to increase the energy efficiency of its production processes [5]. A significant amount of energy is being lost in most of the electric arc furnace (EAF) because of inaccurate control [6].

Competition in the product market increases the requirements for the quality of process control, which results in the development of new approaches to the design of control influences. Taking into account the nonlinearity of objects, the theory of nonlinear control has been significantly developed. Methods such as feedback linearization [7,8,9] and back-stepping control [10,11,12] are well known. In both approaches, the control influence contains a component that compensates the nonlinearity existing in the system. In the case of nonlinear systems with mutual influences between subsystems, which are typical for multi-engine systems of modern electric vehicles, power grids containing wind and solar power plants, etc., the optimal setting of each subsystem may not ensure optimal performance of the whole system. Taking into account the mutual influences between different subsystems of a complex nonlinear system significantly complicates the synthesis of control influences [13,14,15,16]. Quite simply, the problem of compensating the mutual influences is solved by the method of energy-forming control [17,18]. Thus, when applying IDA passivity-based control [19,20], the control influences are obtained by forming the desired interconnections and damping. The choice of the matrix of interconnections and damping ensures the target correction of processes in the controlled object. A common limitation of the above approaches is the complexity of synthesizing the parameters of the control system. As shown in [20,21], the control system influences synthesized on the basis of the classical theory of optimal control are similar to those obtained by applying the method of energy-forming control or feedback linearization to a linear system.

Methods of fuzzy set theory play an important role in the development of nonlinear control theory [22,23]. The use of fuzzy modeling makes it possible to represent a model of a nonlinear object as a family of linear dynamic models [24]. The availability of such models significantly simplifies the procedure for the synthesis of control influences and allows the application of classical control theory methods. Thus, the use of fuzzy controllers synthesized on the basis of a family of dynamic linear systems enables the ability to provide high quality control in nonlinear systems.

In addition to nonlinearity and interconnections between subsystems, asymmetry is an additional factor in electrical systems that impacts the quality of control [25,26]. Asymmetric load distributions between phases can also be caused by the peculiarities of the object functioning. An example of a nonlinear object with mutual influences between different subsystems and asymmetric load distribution between phases is an electric arc furnace [27,28,29]. The scrap melting stage is characterized by sharpness and random changes in the arc gap of the electric arc furnace, which leads to changes in the distribution of currents in the three-phase system. Moreover, the interphase relationship is nonlinear. This period is characterized by higher values of current dispersion and arc power, which causes unproductive energy consumption and increases the cost of melted steel. Particularly high current oscillations are observed at the start of the melting process, which is characterized by numerous scrap collapses. At the end of the scrap melting stage, it is necessary to reduce the arc power to a suitable level. In this case, the arc burning is characterized by a smoother mode. The change in the current distribution causes activation of the electrode movement system. Effective automatic electrode control is one of the most important factors in efficient arc furnace operation since the energy consumption, melting time, electrode and refractory consumption, and the load factor presented to the supply system depends largely on the speed and precision with which the electrodes are controlled and positioned in the furnace [30]. Wrong movements of electrodes may occur in phases in which the arc length has not changed, and the variation in the current is due to the mutual influence among the phases of the three-phase system [31]. When a high-speed current loop based on an adjustable inductor is used in the system, there is mutual influence between the mentioned loop and the electrode movement control system [32,33]. Stabilization of the current using a high-speed circuit leads to a reduction in the speed of the disturbance response by the electrode movement circuit in the phase where the disturbance occurred, but false electrode movements are reduced in other phases. The complexity of the processes in the controlled object, the presence of mutual influences, and asymmetric operating modes are the main reasons that constrain the use of modern control methods in electric arc furnaces.

According to references [34,35,36], the current most commonly used primary methods of controlling the electrical mode of an electric arc furnace (EAF) in commercial applications are based on the differential control law (1):

$$U_{arc}\left(U_{arc}, I_{arc}\right)=\alpha \cdot U_{arc}-\beta \cdot I_{arc},$$

(1)

where $U_{arc}, I_{arc}$ are arc voltage and current, respectively, and α and β are constant coefficients.

In addition, impedance control laws are employed [37,38]:

$$U_c\left(U_{arc}, I_{arc}\right)=k\cdot \left(\frac{U_{arc}}{ I_{arc}}-\frac{U_{arc\_set}}{I_{arc\_set}}\right),$$

(2)

where $U_{arc\_set},I_{arc\_set}$ are the set values of arc voltage and current, respectively, and k is the gain of the proportional regulator.

In addition to the above mentioned two basic laws, a modified differential law is occasionally used [39]:

$$U_c\left(U_{arc}, I_{arc}\right)=\alpha \cdot U_{arc}-\beta \cdot \Delta I_{arc},$$

(3)

where $\Delta I_{ar{c}^{\prime }}=I_{arc\_set}-I_{arc}$. Moreover, control of the arc voltage variation from the given value is expressed as follows:

$$U_c\left(U_{arc}\right)=k\cdot \left(U_{arc\_set}-U_{arc}\right)$$

(4)

This equation attempts to implement a control system based on the arc length variation, assuming a linear relationship between these values.

The effectiveness of the mentioned control laws, as indicated in [39], varies depending on the melting stage and arc length. A significant number of publications are focused on the adaptation of the above-mentioned control laws to the electrode movement system to account for the nonlinear dependence of the arc current from the arc length [34,40,41,42] and tune the coefficients of proportional or proportional-integral controllers through the use of fuzzy set theory [43,44,45,46] and the application of fractional order control [47] or model predictive control [48].

During the synthesis of control influences and the investigation of the efficiency of the proposed control systems, the authors applied different models of an EAF [49,50,51]. However, for synthesis of control influences, the system of electrode movement of an EAF is limited to the application of a single-phase model [34,52]. Such an approach eliminates the possibility of taking into account the mutual influences between the phases of an electric arc furnace at the stage of synthesis of the control system. The existence of mutual influences between the phases of an EAF requires the use of MIMO approaches to the synthesis of effective control influences, which further complicates an already difficult task. In [53], researchers used an RBF network and nearest-neighbor clustering to realize the real-time decoupling of an inverse MIMO control system of electrode movement of a three-phase EAF into a pseudo-linear SISO system. This transformation allowed the authors to apply classical control theory approaches to the synthesis of control influences. The elimination of mutual influences between the phases of an EAF in the process of control signal synthesis in the electrode movement system, as shown in [54], is possible by using arc length control.

Since most existing systems to control the movement of electrodes in EAFs are based on differential or impedance control principles, the problem of eliminating mutual influences between phases in the process of the control influence design remains relevant. Solving this problem would avoid wrong movements of electrodes caused by redistribution of currents due to the disturbance and simplify the application of known algorithms for maintaining operation at the point of maximum power consumption. As a result, it would help to stabilize the arc gap power at a given level and thereby improve the performance and energy efficiency of the electric arc furnace.

The novelty of this paper lies in the proposed approach to eliminate interphase interactions in the process of designing the control influences in the system of electrode movement in an EAF.

2. Overview of the Proposed Approach

In the case of a disturbance in one of the phases of an EAF, currents are redistributed among the phases (Figure 1). Each of the phases receives an increase in the current caused by both the change in the situation in this phase and the mutual influence among the phases. In general, it can be written as follows:

$$\begin{array}{c}{I}_A=I_{A0}+\Delta I_{AA}+k_{AB}{I}_{BB}+k_{AC}{I}_{CC}\\ I_B=I_{B0}+\Delta I_{BB}+k_{BA}{I}_{AA }+k_{BC}{I}_{CC}\\ I_C=I_{C0}+\Delta I_{CC}+k_{CA}{I}_{AA }+k_{CB}{I}_{BB}\end{array}$$

(5)

where $\Delta I_{AA},\Delta I_{BB},\Delta I_{CC}$ are the current increments due to the arc length disturbance in phases A, B, and C, respectively; $I_A,I_B,I_C$ represent phase currents of an electric arc furnace; $I_{A0},I_{B0},I_{C0}$ represent values of phase currents that determine the set point of the electric mode of an electric arc furnace; and $k_{AB},k_{AC},k_{CB},k_{CA},k_{BA}$ represent coefficients that take into account mutual influences among the phases of the electric arc furnace.

Writing Equation (5) in matrix form, we obtain the following:

$$\left[\begin{array}{c}{I}_A\\ I_B\\ I_C\end{array}\right]=\left[\begin{array}{c}{I}_{A0}\\ I_{B0}\\ I_{C0}\end{array}\right]+\left[\begin{array}{ccc}1& k_{AB}& k_{AC}\\ k_{BA}& 1& k_{BC}\\ k_{CA}& k_{CB}& 1\end{array}\right]\cdot \left[\begin{array}{c}\Delta I_{AA}\\ \Delta I_{BB}\\ \Delta I_{CC}\end{array}\right]$$

(6)

From Equation (6), we determine the current increments caused by disturbances in the corresponding phase:

$$\left[\begin{array}{c}\Delta I_{AA}\\ \Delta I_{BB}\\ \Delta I_{CC}\end{array}\right]={\left[\begin{array}{ccc}1& k_{AB}& k_{AC}\\ k_{BA}& 1& k_{BC}\\ k_{CA}& k_{CB}& 1\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}{I}_A-I_{A0}\\ I_B-I_{B0}\\ I_C-I_{C0}\end{array}\right]$$

(7)

The recalculated phase current values in the case that there was no mutual influence among the phases of the EAF are obtained using the following equation:

$$\left[\begin{array}{c}{I}^{*}{}_A\\ I^{*}{}_B\\ I^{*}{}_C\end{array}\right]=\left[\begin{array}{c}{I}_{A0}\\ I_{B0}\\ I_{C0}\end{array}\right]+{\left[\begin{array}{ccc}1& k_{AB}& k_{AC}\\ k_{BA}& 1& k_{BC}\\ k_{CA}& k_{CB}& 1\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}{I}_A-I_{A0}\\ I_B-I_{B0}\\ I_C-I_{C0}\end{array}\right]$$

(8)

This equation provides phase-by-phase control autonomy and avoids undesired electrode movements when applying traditional control laws (1)–(3) in the electrode movement system.

3. Mathematical Model for Conducting Research

To identify the coefficients of mutual influences between the phases of an electric arc furnace, we use the model of the system described in [32,50]. The adequacy of this model is confirmed in [32,50] based on the coincidence of the obtained results with the results of experimental studies and the results of theoretical calculations obtained by other authors. To create a mathematical model of the elements of the power circuit of the system “power supply—electric arc furnace”, we applied the method of average voltages at the integration step AVIS [55], according to which an electric branch with inductance, electric power source e(t), active resistance, and a nonlinear element (arc) is described by the following equation:

$$\frac{1}{\Delta t}{\displaystyle \underset{t_0}{\overset{t_0+\Delta t}{\int }}\left(u+e-u_R-u_L-u_{arc}(i)\right)dt}=0,$$

(9)

where e, u_R, u_L, and u_arc represent instantaneous values of emf, voltage on resistance, inductance voltage, and arc voltage, respectively; u is the voltage applied to the branch; t₀ represents the time value at the beginning of the integration step; and Δt is the step of numerical integration.

By representing the instantaneous values of voltages on the resistance and on the arc as the sum of the initial values u_R0 and u_arc0 in the step with increments Δu_R and Δu_arc, we obtain the following equation:

$$U+E-u_{R0}-u_{arc0}-\frac{1}{\Delta t}{\displaystyle \underset{t_0}{\overset{t_0+\Delta t}{\int }}\left(\Delta u_R-u_L-\Delta u_{arc}(i)\right)dt}=0,$$

(10)

where $U=\frac{1}{\Delta t}{\displaystyle \underset{t_0}{\overset{t_0+\Delta t}{\int }}udt}$ and $E=\frac{1}{\Delta t}{\displaystyle \underset{t_0}{\overset{t_0+\Delta t}{\int }}edt}$ are the average values at the numerical integration step of the voltage and emf applied to the branch.

The voltage increments at the numerical integration step in Equation (10) using the Taylor series are equal:

$$\Delta u_R={\displaystyle \sum _{k=1}^m\frac{d^{\left(k\right)}{u}_{R0}}{d{t}^{\left(k\right)}}\cdot \frac{\Delta t^k}{k!}}=R{\displaystyle \sum _{k=1}^m\frac{d^{\left(k\right)}{i}_{arc0}}{d{t}^{\left(k\right)}}\cdot \frac{\Delta t^k}{k!}},\Delta u_{arc}={\displaystyle \sum _{k=1}^m\frac{d^{\left(k\right)}{u}_{arc0}}{d{t}^{\left(k\right)}}\cdot \frac{\Delta t^k}{k!}}$$

(11)

where $\frac{d^{\left(k\right)}{i}_{arc0}}{d{t}^{\left(k\right)}}$ and $\frac{d^{\left(k\right)}{u}_{arc0}}{d{t}^{\left(k\right)}}$ are the k-th derivatives of branch current and arc voltage at the start of the step, respectively; m is the number taken into account derivatives, which determines the order of the AVIS method; and Δt is the integration step.

Given that $\frac{d{u}_{arc}}{di}=R_d$ represents dynamic arc resistance, which depends on the arc current and arc length and is determined at each numerical integration step and $u_L(t)=L\frac{di(t)}{dt}$ is the inductance voltage of the branch, we obtain the algebraic equation for the circuit with an arc and the arc voltage at the numerical integration step for the AVIS method of the first order (m = 1) by substituting (11) into (10),

$$i_1=i_0+{\left(\frac{R+R_{d0}}{2}+\frac{L}{\mathrm{\Delta }t}\right)}^{-1}\left(U-R{i}_0-u_{arc0}\right),u_{arc1}=u_{arc0}+R_d\left(i_1-i_0\right)$$

(12)

The arc dynamic resistance is a function of the arc current and arc length and is determined based on arc volt–ampere characteristics, for which the following approximation Equations (13) and (14) were obtained in [50]:

If $I_{arc}\cdot \frac{d{I}_{arc}}{dt}>0$, then

$$R_d\left(I_{arc},l_{arc}\right)=\frac{0.4337}{\sqrt{\sigma \left(l_{arc}\right)}}\cdot a_{11}\left(l_{arc}\right)\cdot \left(1-{\left(\frac{I_{arc}}{\sigma \left( l_{arc}\right)\cdot a_{21}\left(l_{arc}\right)}\right)}^2\right)exp\left(-a_{31}\left(l_{arc}\right)\cdot {\left(\frac{I_{arc}}{\sigma \left( l_{arc}\right)}\right)}^{2\cdot a_{41}\left(l_{arc}\right)}\right)$$

(13)

If $I_{arc}\cdot \frac{d{I}_{arc}}{dt}<0$, then

$$R_d\left(I_{arc},l_{arc}\right)=a_0\left(l_{arc}\right)\cdot \left(1-\left(th{\left(\frac{I_{arc}}{\sigma \left(l_{arc}\right)\cdot 0.5}\right)}^2\right)\right)$$

(14)

where $a_{11}\left(l_{arc}\right)=0.9658\cdot l_{arc}-0.1282$; $a_{21}\left(l_{arc}\right)=2.0477\cdot {10}^{-5}\cdot l_{arc}{}^3-0.0005\cdot l_{arc}{}^2+0.0019\cdot l_{arc}+0.3992$; $a_{31}\left(l_{arc}\right)=-0.0026\cdot l_{arc}{}^3+0.0939\cdot l_{arc}{}^2-0.3429\cdot l_{arc}+14.0293$; $a_{41}\left(l_{arc}\right)=2$; $\sigma \left(l_{arc}\right)=\left(0.0093\cdot l_{arc}{}^2-0.4513\cdot l_{arc}+9.2986\right)\cdot 10000$; and $a_0\left(l_{arc}\right)=\left(0.0004\cdot l_{arc}{}^4-0.0124\cdot l_{arc}{}^3+0.1468\cdot l_{arc}{}^2+0.0081\cdot l_{arc}+0.1418\right)/1000$. Here, I_arc = i—arc current, which is equal to the current of the branch with the arc, namely, l_arc—arc length.

The second-order AVIS method (m = 2) provides better calculation accuracy with a high value for the numerical integration step; however, it requires additional calculations for the branch current derivative and the branch dynamic resistance derivative. Therefore, we use the first-order AVIS method.

The system mathematical model as a whole is formed from the models of its separate elements, represented in the form of multipoles and connected according to the calculation scheme shown in Figure 2.

The mathematical models of the elements (power grid, transformer, and electric circuits with an arc) are formed on the basis of Equation (12), in which we substitute u_arc0 = 0 and R_d = 0 for the power grid and transformer, respectively. These models, the model of the electrode movement system with a DC motor, and the calculation algorithm are described in detail in [50].

The information from the electrical part model is transferred to the model module of the electrode movement control system, and the information on the arc length as a result of the electrode movement during disturbance processing is returned to the electrical part model to determine the dynamic arc resistance.

Using the model, the current distribution of an electric arc furnace was investigated for different conditions in the arc gap of the EAF. As an example, in Figure 3, the surfaces of change in the current of phase C are shown. Similar surfaces were obtained for the currents of other phases of the electric arc furnace.

The obtained surfaces reflect the current value of the phase, taking into account the mutual influences of other phases, according to Equation (6). Each of the surface points of the phase C current change can be obtained as a result of changing the arc length in phase A, phase B, and phase C. Furthermore, the order of changing arc lengths in the phases of an electric arc furnace may be different.

It is important to note that the nonlinear nature of the EAF’s characteristics causes a different current distribution in response to the same disturbance, depending on the initial conditions of the EAF (Figure 4). An especially complex nature of mutual influences is observed in the case when the arc lengths of the phases are close to the short circuit or arc breakdown mode. The surfaces of coefficient changes shown in Figure 5 are obtained through current increases determined based on phase currents (Figure 3), which, in accordance with (5), also contain influences from disturbances in other phases. These mutual influences are differently manifested at different arc lengths, which, therefore, introduces an error in determining the coefficients of the interconnections. Such a different change in the values of the phase currents in response to the same disturbance will lead to different values of the mutual influence coefficients $k_{AC},k_{BC}$ depending on the initial conditions.

4. The Results of Research and Analysis of the Proposed Approach’s Effectiveness

To solve the problem of reducing mutual influences in existing systems that regulate the electrode movement in an electric arc furnace, we analyze the change in the coefficients of interphase influences from the arc length disturbance when all three phases are at a given point of the operating mode $I_{phase(A,B,C)}=I_{phase(A,B,C) 0}$. For this purpose, by changing the arc length value in one of the phases up and down from the set value, the current increments in each of the phases were determined. On the basis of the obtained correlations of phase current increments, the dependences of changes of the coefficients of mutual influences between the phases of an electric arc furnace as a function of the increment of the current of the phase in which the disturbance occurred were determined (Figure 6).

The obtained dependencies of the coefficients as a function of the incremental change in current in the phase where the disturbance occurred are nonlinear. These dependences are used to recalculate the currents in the case of a disturbance in phase A and its processing using a classical control system type (1). The results given in Figure 7 and Figure 8 confirm the effectiveness of the proposed approach for enhancing the autonomy of the control of the electrode movement in EAF.

Next, we investigate the effectiveness of using the coefficients of interphase influences determined on the basis of Figure 6 for the case of different initial conditions of the electric arc furnace. Under the action of the same disturbances, we analyze the operation of the electrode movement control system based on the differential control law and with the proposed phase autonomy.

The dependencies reported in Figure 9 demonstrate that the interconnection coefficients determined on the basis of Figure 6 quite successfully solve the task of phase-to-phase autonomy for synthesis of the control under different initial conditions. The electrode movement control system processes a disturbance that occurs in a phase and generally does not react to mutual influences from other phases (Figure 10).

At the same time, phase autonomy, as shown in Figure 10, may cause a slight deviation in the dynamic characteristics of the electrode movement system. A change in the phase current in a traditional control system due to the effects of a disturbance in another phase is shown in Figure 10. This will cause an increase in the mismatch signal under the differential control law and lead to rapid attainment of the steady-state value. However, the further action of this disturbance will cause the electrode to move incorrectly.

The mismatch signals in the case of differential and impedance control laws without and with the use of current recalculation for phase autonomy are shown in Figure 11 and Figure 12.

The obtained results make it possible to conclude that the proposed approach to the phase-by-phase autonomy of the electrode movement system provides a significant reduction in the mutual influence between the phases of an electric arc furnace, regardless of the situation in the arc gap of the EAF at the time of disturbance.

5. Discussion

Mutual influences between the phases of an electric arc furnace in the case that classical control algorithms cause incorrect electrode movements as a reaction to the redistribution of currents in the three-phase power supply system of the EAF due to a disturbance in the arc length in one of the phases. On the other hand, mutual influences in some cases improve the disturbance response of the electrode movement system. Thus, the compensation of mutual influences between the phases of the EAF in available electrode movement systems requires further research. At the same time, in our opinion, obtaining information about the current values without taking into account the increments from mutual influences due to disturbances in other phases can significantly simplify the development of a system for electrode movement control based on arc length identification. The effectiveness of the proposed approach for electric arc furnaces of different capacities requires further research as well as a comparative analysis of the effectiveness of the proposed approach with the approach based on the use of a high-speed current loop in the control system, which introduces a significant asymmetry in the power supply system and thus changes the distribution of currents under the perturbation.

6. Conclusions

The coefficients of mutual influences between the phases of an EAF are nonlinear functions of the current increment in the phase in which the disturbance occurred.

The application of the proposed approach on the basis of the generated matrix of mutual influences with variable coefficients makes it possible to significantly (by an order or more), and in some cases to reduce almost to zero (see Figure 11 and Figure 12), the influence of the situation in other phases of the electric arc furnace on the formation of the controlling influence of the electrode movement system and therefore increase the accuracy of control and maintain the specified power mode to the arc steelmaking furnace.

The phase-by-phase autonomy of the arc furnace electrode movement control system makes it possible to practically avoid unproductive electrode movements caused by disturbances in other phases, which, under certain conditions in the EAF, can lead to a negative impact on the dynamic performance of the disturbance (see Figure 9).

The proposed approach to reducing mutual influences between phases during control signal generation can be easily implemented in existing control systems for the movement of electrodes in an arc steelmaking furnace.