Double-Circuit Adaptive System of Fuzzy Phase-Autonomous and Energy-Efﬁcient Control of Arc Furnace Electric Modes

: The article provides an overview of the existing structures of fuzzy model systems for arc length control and optimal control of arc steel-melting furnaces (ASFs) electrical mode. The analysis of known adaptive optimal control strategies of alternating current steel-melting arc furnaces modes according to complex criteria of energy efﬁciency, energy saving and reliability was performed. Expedient system engineering solutions are proposed for the comprehensive improvement of energy efﬁciency and electromagnetic compatibility of arc furnaces and the power grid, which are implemented by improving automatic power and arc length control systems. Control principles, indicators of arc length control dynamics, energy efﬁciency and electromagnetic compatibility of various systems are analyzed. The expediency of using fuzzy control models for comprehensive improvement of dynamics, energy efﬁciency and electromagnetic compatibility indicators is sub-stantiated. The expediency of implementing simultaneous phase-independent control of arc lengths and fast-acting arc currents control is substantiated. A variant of the double-circuit structure of fuzzy arc lengths control system and implementation of the multi-criteria optimal control strategy of electrical modes (EM) is proposed. The design of fuzzy inference systems for the synthesis of the electrode movement control signal was carried out in order to improve the dynamic accuracy of the ASF electrical mode coordinates stabilization. A computer Simulink model of the developed double-circuit system was created, and studies of the dynamics and energy efﬁciency indicators when handling deterministic and random perturbations were performed. The obtained results of model studies showed an improvement of the control dynamics quality indicators under the inﬂuence of deterministic and random perturbations and an increase in the energy efﬁciency of ASF EM at various technological stages of melting. The expediency of the practical use of the proposed solutions for operating arc furnaces is substantiated.


Introduction
An analysis of the current steel market shows a steady increase in the share of steels and alloys smelted in arc furnaces. Amid this increase, the share of oxygen, convertor and, especially, open-hearth steels decreases accordingly. The share of total European Union steel production coming from electric arc furnaces exceeds 50%. Our times witness a continuous technical development of main and additional equipment and technologies in steelmaking, which is due to the higher energy efficiency and sustainability of melting steels and alloys in arc furnaces.
This trend introduces the main task of further improving the automatic control systems of electric arc furnaces to reduce energy consumption, increasing energy efficiency, electromagnetic compatibility and sustainability [1]. The most important way to achieve this is to improve the automatic control system (ACS) of electric mode (EM), arc length control systems, as well as melting technologies and arc furnace design [2].
In view of this, the world's leading manufacturers of electrical equipment for arc furnaces-Daniely Automation (Italy), Amec Spic (France), Siemens VAI and Ferrotron (Germany) [2,3] and others-focus on creating new efficient high-speed systems for regulating (stabilizing) arc lengths and hierarchical multi-circuit structures of control systems for electrical modes of arc furnace (AF). At the same time, they implement different system structures and laws (models) of high-speed arc length control and adaptive optimal control of AF electric modes [2,3].
Considerable attention in modern developments of automatic control systems of alternating current arc furnaces electric mode is paid to the implementation of adaptive optimal control strategies, in particular, based on the criteria of energy efficiency and energy saving [4][5][6][7][8][9][10].
In particular, papers [4,5] proposed an algorithm and concept of online optimal control synthesis of melting modes aimed at reducing the consumption of electric energy, minimizing the controlled consumption of other energy resources and maximizing the transfer of energy into the arc furnace melting space, i.e., in general, at minimizing operational costs. The presented results of model studies showed a decrease in energy consumption and a reduction in melting time.
In the article [6], an optimal prescriptive melting schedule for a high-power arc furnace was developed in order to minimize energy losses. Studies of this control on 19 experimental meltings showed a decrease in energy consumption and melting time by 4.5% and 4.6%, respectively. The main idea of the developed prescriptive schedule for the control of the technological process of melting was the proposed method of parameterization of the technological melting process optimal control vector by means of the appropriate voltage settings and arc impedance selection, which decisively affect the course of the technological melting process.
The model of operational synthesis of the technological regime optimal control in the arc furnace, proposed in [7], made it possible to reduce the consumption of electrical energy by 1-2%. This is achieved by taking into account additional technological coordinates of the melting process in the control synthesis algorithm. The practical use of the proposed synthesis methodology made it possible to implement predictable management of energy efficiency indicators, temperature and steel weight.
Optimization of the energy balance in the melting space of the arc furnace, in particular, the implementation of the optimal ratio of carbon and oxygen, taking into account the weight of the loaded scrap in the melting process, is proposed in [8]. The impact on the energy efficiency indicators of the weight and type of charge, non-productive pauses in the melting interval and the temperature of steel release was studied. The influence of the carbon-oxygen ratio during the charge melting on the electric arc furnace energy efficiency is also shown. The results of computer simulation of the proposed technological factors optimization model showed that the use of those solutions reduces operational costs and melting time.
A model of energy-efficient melting process control through the implementation of rational electrical modes, which are designed on the basis of the arc furnace electrical and technological characteristics control, is proposed in [9,10]. In these articles, the influence of the melting technological parameters, in particular, oxygen blowing, on the arc furnace energy efficiency indicators was studied, and the expediency of such control was shown.
But the optimization solutions and control models proposed in the analyzed articles [4][5][6][7][8][9][10] do not consider or do not fully take into account the complex impact of electrical coordinate settings and control principles that are implemented in automatic arc power (length) control systems on melting energy efficiency indicators and energy consumption. They are important factors in improving the mentioned indicators.
Therefore, the problem of improving system design and models of arc length control and electrical mode control at the present stage of circuit development and system engineer-Energies 2023, 16, 5350 3 of 19 ing in steel electrometallurgy is important and relevant. Solving the problem successfully and comprehensively will make it possible to save limited energy and material resources, enhance the electromagnetic compatibility of arc furnace and power grid modes, especially under the current conditions of intensifying their electrical modes, and reduce the negative impact of electrometallurgical production of high-alloy steels and precision alloys on the environment.

Overview of Technical Solutions
Modern arc steelmaking furnaces are equipped with electromechanical and electrohydraulic systems for moving electrode arc power regulators. Their main task is to ignite arcs and stabilize their lengths and coordinates of the electrical mode at the level of specified, in particular optimal, values [1,[11][12][13][14].
The qualitative realization of the above control tasks is complicated by continuous random parametric and coordinate disturbances of nonstationary nature. In addition, the electrode position control takes place under conditions of uncertainty and incompleteness about the state of the control object, since some of the coordinates of the melting process, in particular, resistances, arc voltages and temperature in the melting space, are not subject to precise operational control. These factors impinge on the stabilization quality of the electrical mode coordinates. Therefore, the dispersion of the EM coordinates in the melting process is significant, which affects energy efficiency and electromagnetic compatibility, particularly due to an increase in specific energy consumption, reduction in the productivity of the arc furnace, deterioration of the quality of the voltage on the furnace connection buses, etc. [15].
These features of disturbances and information support require new solutions and models for control strategies and regulation of the coordinates of AF EM, whose effectiveness is confirmed in the above-mentioned articles. The analysis of these and other recent publications shows that an appropriate approach to improve the dynamic accuracy of stabilizing the coordinates of AF EM is to employ fuzzy control models along with Neural Adaptive Control Technology [7-10].
The application of a fuzzy PI controller with parametric optimization by a genetic algorithm to control arc lengths has been proposed in [16]. The simulation results revealed an improvement in the dynamic accuracy of stabilizing arc lengths for the set values. However, the issue of prompt parameter adaptation of the proportional-integral model for generating the control signal U la c for electrode movement according to changes in the state of arcs is not addressed. It is also necessary to take into account the real conditions for the non-stationarity process of coordinate and parametric disturbances. This factor does not allow for utilizing the advantages and efficiency of fuzzy control to the full.
A comparative analysis of the dynamics of electrode position control in a three-phase electric arc furnace using a fuzzy integral controller and a typical proportional-integral control model has been performed [17,18]. The study was based on a model under the influence of deterministic (step) perturbations in arc lengths. The paper shows the advantages of using an intelligent fuzzy controller with rigid (unchanged) settings of its parameters. There is no operational adaptation of these parameters to the stochastic characteristics of disturbances during the melting process. Therefore, the performance indicators of electrode movement when regulating the main random disturbances will deviate from the optimal ones at most technological stages of melting.
The use of fuzzy and neural network models for synthesizing the vector of adaptive optimal control of the electrical mode of arc furnaces in multi-circuit structures has been proposed and studied in [19,20]. The obtained results demonstrate a comprehensive improvement in energy efficiency and electromagnetic compatibility based on the proposed generalized control quality functionalities. It is shown that an effective approach to improving these indicators under the influence of non-stationary random disturbances is to implement high-quality stabilization of the EM coordinates at the level of synthesized optimal settings and their adaptation to changing external influences and characteristics of disturbances. Various circuit solutions for high-performance thyristor voltage control of a furnace transformer make it possible to achieve a significant (by an order of magnitude or more) increase in the speed of controlling currents, arc powers and reactive power of an arc furnace. Using the appropriate multi-criteria optimal control functions, it is possible for such two-circuit ACS structures to undergo a comprehensive improvement in energy efficiency and electromagnetic compatibility, which is much higher than in existing singlecircuit electrical mode control structures that operate on the basis of electrode position control systems alone.
As regards the implementation of this approach, a system for fuzzy control of the position of the arc furnace electrodes based on a self-tuning proportional-integral-differential (PID) controller has been proposed [21]. The control is realized by the EM loop error signal, which is estimated on the basis of the deviation of the arc phase currents from the set value. The effectiveness of the solutions proposed in this paper was based on a simplified single-phase second-order model of the control object. Considering the interconnections between phase load modes in three-phase AFs, the indicators of the proposed arc current controller obtained using such a simplified model cannot be considered complete and general. The effectiveness of using a self-adaptive fuzzy PID controller for the electrical AF mode has been studied in [22]. The studies conducted in the Matlab/Simulink revealed better dynamic characteristics, performance and reliability compared to the traditional PID controller. However, the proposed adaptation is performed at the level of integral values of the EM coordinates, with no adaptation at the level of instantaneous changes in arcing states.
Models for the operational adaptation of the control signal for the process of moving the electrodes to changes in the arcing state have been proposed in [23][24][25]. However, the proposed solutions for the fuzzy adaptive control of the electrode position do not fully involve the nonlinearity of the external characteristic of arcing over the entire range of changes in arc voltages, which does not significantly improve the dynamic accuracy of stabilizing the EM coordinates, and in some cases can cause instability of the control system and impair the reliability of ignition of three-phase arcs. Their use reduces the negative impact of backlash, nonlinearities and gaps in the mechanical part of the electrode movement drive on the dynamics of electrode movement.
An appropriate approach to improving the dynamic accuracy of stabilizing arc lengths under conditions of random fluctuations is to use the principles of positional control of electrode movement based on fuzzy and neurofuzzy control models [26], in particular, the use of fuzzy PI and PID controllers for the position of electrodes in an arc steelmaking furnace. These papers introduce original adaptive fuzzy models of position controllers, which improve the performance of the electrical mode control system and reduce the dispersion of the electrical mode coordinates.
The paper under [27] introduces a solution for the simultaneous use of fuzzy models to improve the dynamics of electrode movement when adjusting arc lengths and extreme adaptive optimal control of the electrical mode. For this purpose, the fuzzy controller is included in the direct channel of the electromechanical system for controlling the position of the electrodes. This solution reduces the negative impact of nonlinearities of the automatic control system (ACS) of the electrode position, increases the speed and simultaneously reduces overregulation in the process of eliminating extreme disturbances (operational short circuits and arc breaks), i.e., in the modes of arc ignition, as well as reduces the dispersion of the EM coordinates when regulating random disturbances during melting. However, the law of formation of the control signal for electrode movement does not quickly adapt to changes in the state of three-phase arcs. Improving the dynamic accuracy by stabilizing the EM coordinates makes it possible to improve the efficiency of the extreme fuzzy control synthesized by the higher-level system.
The presented analysis of the applications of fuzzy models for controlling the position of electrodes (arc length control) shows an improvement in the electrode movement dynamics during perturbation. This is achieved due to the low sensitivity of fuzzy control models to fluctuations in the parameters of the control object, which are present in the power circuit of the furnace and elements of the kinematic scheme of the electrode movement mechanism.
Improving the electrode position control system in terms of optimizing and adapting the laws of electrode movement to the states of three-phase arcing is a relevant and important approach to enhancing the dynamic accuracy of stabilizing the EM coordinates for the setpoints. It should be noted that further significant improvement of the dynamics and speed of regulation (elimination) of disturbances by means of an electromechanical (electrohydraulic) system of electrode movement is limited by the actual mechanical strength and significant inertia of the elements of the mechanical part of the electrode drive.
The goals of the research are to develop solutions for improving the dynamics of regulation of random and deterministic disturbances and energy efficiency indicators of the arc furnace.
The novelty of the article is the fuzzy model of signal generation for controlling the movement of electrodes and system engineering solutions for high-speed current control based on the rapid formation and implementation of the corresponding artificial external characteristic (AEC) of the arc furnace, which will comprehensively enhance the integral indicators of the stabilization quality of the coordinates of the electric mode, energy efficiency and electromagnetic compatibility of the arc furnace and the power grid modes.

Developed Solutions
In order to meet the above requirements for automatic control systems of the EM AF, a two-circuit high-performance structure of the system for adaptive optimal control of EM AF was developed, which includes a subsystem for high-performance current control and a subsystem for arc length control with a fuzzy model for generating an EM loop error signal and a fuzzy model for generating a corrective signal for electrode movement ( Figure 1). The presented analysis of the applications of fuzzy models for controlling the position of electrodes (arc length control) shows an improvement in the electrode movement dynamics during perturbation. This is achieved due to the low sensitivity of fuzzy control models to fluctuations in the parameters of the control object, which are present in the power circuit of the furnace and elements of the kinematic scheme of the electrode movement mechanism.
Improving the electrode position control system in terms of optimizing and adapting the laws of electrode movement to the states of three-phase arcing is a relevant and important approach to enhancing the dynamic accuracy of stabilizing the EM coordinates for the setpoints. It should be noted that further significant improvement of the dynamics and speed of regulation (elimination) of disturbances by means of an electromechanical (electrohydraulic) system of electrode movement is limited by the actual mechanical strength and significant inertia of the elements of the mechanical part of the electrode drive.
The goals of the research are to develop solutions for improving the dynamics of regulation of random and deterministic disturbances and energy efficiency indicators of the arc furnace.
The novelty of the article is the fuzzy model of signal generation for controlling the movement of electrodes and system engineering solutions for high-speed current control based on the rapid formation and implementation of the corresponding artificial external characteristic (AEC) of the arc furnace, which will comprehensively enhance the integral indicators of the stabilization quality of the coordinates of the electric mode, energy efficiency and electromagnetic compatibility of the arc furnace and the power grid modes.

Developed Solutions
In order to meet the above requirements for automatic control systems of the EM AF, a two-circuit high-performance structure of the system for adaptive optimal control of EM AF was developed, which includes a subsystem for high-performance current control and a subsystem for arc length control with a fuzzy model for generating an EM loop error signal and a fuzzy model for generating a corrective signal for electrode movement ( Figure 1).  The system in each phase includes a current sensor CS and an arc voltage sensor VS, from whose outputs the averaged values of the arc current I a and U a are supplied to the inputs of the block for generating the loop error signal VS of the electrical mode, implemented on the basis of the Takagi-Sugeno fuzzy inference system FLC1. From its output, the loop error signal U r is supplied to the inputs of the CSFB control signal formation unit for electrode movement and the FLC2 fuzzy corrector, whose output signals U la cr and U la cc , respectively, are summed in the adder, forming a control signal for electrode movement U la c = U la cr + U la cc . The control signal U lc c is transformed by the electric drive ED and the MME mechanism into the electrode movement in each phase in accordance with the current deviation of the arc length from the set value. The voltage stage switching device VSS of the furnace transformer FT sets the corresponding secondary voltage U 2ph accordance with the signal N. The above elements constitute a local electromechanical system (EMSS), which performs the function of igniting three-phase arcs and stabilizing their lengths at a given level. This level is determined by the signals U * sa and I * sa , which are the arc voltage and current settings, respectively. The setpoints U * sa and I * sa , along with the number N of the secondary voltage stage of the furnace transformer FT, are variable control influences for the implementation of the selected strategy of adaptive optimal control of the electrical mode at each technological stage of melting by means of EMSS.
The use of artificial external characteristics of the arc furnace ( Figure 2) in the proposed structure of a double-circuit arcs power automatic control system ( Figure 1) makes it possible to implement various strategies of adaptive multi-criteria optimal control of the melting process electrical modes. But unlike the known solutions for this [11][12][13][14][15][16][17], which are based on the use of mainly technological influences or constructive solutions, variable artificial external characteristics are electrical control influences.
Energies 2023, 16, x FOR PEER REVIEW 6 of 20 The system in each phase includes a current sensor CS and an arc voltage sensor VS, from whose outputs the averaged values of the arc current and are supplied to the inputs of the block for generating the loop error signal VS of the electrical mode, implemented on the basis of the Takagi-Sugeno fuzzy inference system FLC1. From its output, the loop error signal is supplied to the inputs of the CSFB control signal formation unit for electrode movement and the FLC2 fuzzy corrector, whose output signals and , respectively, are summed in the adder, forming a control signal for electrode movement = + . The control signal is transformed by the electric drive ED and the MME mechanism into the electrode movement in each phase in accordance with the current deviation of the arc length from the set value. The voltage stage switching device VSS of the furnace transformer FT sets the corresponding secondary voltage in accordance with the signal . The above elements constitute a local electromechanical system (EMSS), which performs the function of igniting three-phase arcs and stabilizing their lengths at a given level. This level is determined by the signals * and * , which are the arc voltage and current settings, respectively. The setpoints * and * , along with the number of the secondary voltage stage of the furnace transformer FT, are variable control influences for the implementation of the selected strategy of adaptive optimal control of the electrical mode at each technological stage of melting by means of EMSS.
The use of artificial external characteristics of the arc furnace ( Figure 2) in the proposed structure of a double-circuit arcs power automatic control system ( Figure 1) makes it possible to implement various strategies of adaptive multi-criteria optimal control of the melting process electrical modes. But unlike the known solutions for this [11][12][13][14][15][16][17], which are based on the use of mainly technological influences or constructive solutions, variable artificial external characteristics are electrical control influences. They are characterized by higher speed for obtaining high dynamic accuracy of arc currents stabilization, as well as much wider functional properties for operational implementation of multi-criteria optimal control. These artificial characteristics are synthesized a priori or changed operationally according to the procedure of adaptation to the fluctuations of the perturbations stochastic characteristics parameters by correspondingly changing the coefficients of the arc furnace artificial external characteristics' analytical expres- They are characterized by higher speed for obtaining high dynamic accuracy of arc currents stabilization, as well as much wider functional properties for operational implementation of multi-criteria optimal control. These artificial characteristics are synthesized a priori or changed operationally according to the procedure of adaptation to the fluctuations of the perturbations stochastic characteristics parameters by correspondingly changing the coefficients of the arc furnace artificial external characteristics' analytical expressions I as = Ψ(U s , D Ia , N). The specified adaptation procedure is performed in the arc current setter ACS. Alternatively, information characterizing current disturbances can be provided by the value of the average arc currents dispersion D Ia .
The arc furnace artificial external characteristics 2-6, shown in Figure 2, are appropriate for control based on a compromise between such partial optimality criteria as the minimum arc currents dispersion D Ia → min , the minimum electrical power losses in the high-power network P el → min , the minimum arc furnace reactive power Q → min or maximum power factor cosϕ → max . In particular, the artificial external characteristic 5 ( Figure 2) is the most appropriate for the implementation of optimal control according to the partial criterion D Ia → min.
An adaptive fuzzy model is proposed to generate the loop error signal U r of the electrical mode which is based on the use of two partial control laws: voltage law U r1 (U a ) [28] and impedance law U r2 (U a , I a ) [29][30][31], each of which determines the estimate of the deviation of the EM from the set one, where k is the arc state coefficient that is quickly calculated by the fuzzy model; is an impedance model for estimating the deviation of the EM from the set one; Z a is the arc impedance value calculated from the current voltage U a and current I a of the arc; Z sa is an arc impedance setpoint; k u and k Z are the constant coefficients that define the stationary mode under partial control laws.
The peculiarity of the proposed fuzzy model (1) is the absence of the effect of the interconnection of phase loads on the process of arc length control in the neighborhood of point A of the stationary electrical mode (Figure 2), i.e., the phase-by-phase autonomy of arc length control, and the reliability of three-phase arc ignition in the event of extreme disturbances, i.e., in the modes of operational short circuits and arc breaks. This is achieved by using a fuzzy adaptation of the model of forming the EM mismatch signal to the current state of three-phase arc burning.
The adaptation is based on the use of a fuzzy model of the change in these partial laws to form estimates of the deviation of the ED from the specified one over the full range of changes in arc lengths (voltages). In the zone of average arc lengths (the neighborhood of point A, (2), the voltage law k·U r1 (U a ) is employed, which does not use the arc current signal I a (as in other laws of forming a loop error signal). This signal (CS output) in three-wire power supply circuits for three-phase arcs is noisy due to the action of disturbances (loads) of the other two phases, which is why the estimate of the loop error signal generated with its use will also be noisy, i.e., will not be appropriate to the actual deviation of the electrical mode from the specified one. The voltage law for calculating the loop error signal estimate does not have such an inaccuracy. It generates an accurate estimate of the electrical mode loop error signal in a phase that is invariant to disturbances in other phases. In the zones of short and long arcs, the impedance law U r2 (U a , I a ) is used to form an estimate of the EM loop error signal in (1). In these modes, the loop error signal is limited to the maximum value, and, therefore, the effect of interphase effects is leveled, i.e., interphase mutual influences are eliminated automatically.
The change in the laws of loop error signal formation in the functional (1) is realized on the basis of the Takagi-Sugeno fuzzy inference system. The functional block diagram of the implementation of the fuzzy adaptive model (1) for generating the loop error signal U r (U a , I a ) (1) of the electric mode of the arc furnace is shown in Figure 3.
The fuzzy adaptation of the process of regulating the EM disturbances to changes in the state of arcs makes it possible to realize phase-by-phase autonomous control of the process of moving electrodes, which, in turn, reduces the dispersion of EM coordinates and, accordingly, improves energy efficiency and electromagnetic compatibility.
Arc states are identified by the current value of the arc voltage U a , which is independent of arc currents and invariant to load fluctuations in the other two phases. This electrical mode coordinate is used as the input linguistic variable F1S1. The output signal of F1S1 is the coefficient k of the arc state, whose value is minimal under the influence of extreme disturbances (long and short arcs), k ∼ = 0, while in the zone of average arc lengths (the neighborhood of point A of the steady-state mode (Figure 2) takes the maximum value k ∼ = 1. Figure 4 shows the membership functions type gauss2mf of the input linguistic variable U a of the Gauss type. The fuzzy adaptation of the disturbance control process to changes in the state of arcs is performed on the basis of the Takagi-Sugeno fuzzy model. To implement it, the following fuzzy rule base has been compiled: If U a ∈ low, then k = 0; If U a ∈ medium, then k = 1; If U a ∈ high, then k = 0. (2) Energies 2023, 16, x FOR PEER REVIEW 8 of 20 The fuzzy adaptation of the process of regulating the EM disturbances to changes in the state of arcs makes it possible to realize phase-by-phase autonomous control of the process of moving electrodes, which, in turn, reduces the dispersion of EM coordinates and, accordingly, improves energy efficiency and electromagnetic compatibility.
Arc states are identified by the current value of the arc voltage , which is independent of arc currents and invariant to load fluctuations in the other two phases. This electrical mode coordinate is used as the input linguistic variable F1S1. The output signal of F1S1 is the coefficient k of the arc state, whose value is minimal under the influence of extreme disturbances (long and short arcs), k ≅ 0, while in the zone of average arc lengths (the neighborhood of point A of the steady-state mode (Figure 2) takes the maximum value k ≅ 1. Figure 4 shows the membership functions type gauss2mf of the input linguistic variable of the Gauss type. The fuzzy adaptation of the disturbance control process to changes in the state of arcs is performed on the basis of the Takagi-Sugeno fuzzy model. To implement it, the following fuzzy rule base has been compiled:
The adopted algorithmic and parametric degrees of freedom of FIS1 correspond to the dependence of its output signal, the arc state coefficient ( ), shown in Figure 5.  The fuzzy adaptation of the process of regulating the EM disturbances to changes in the state of arcs makes it possible to realize phase-by-phase autonomous control of the process of moving electrodes, which, in turn, reduces the dispersion of EM coordinates and, accordingly, improves energy efficiency and electromagnetic compatibility.
Arc states are identified by the current value of the arc voltage , which is independent of arc currents and invariant to load fluctuations in the other two phases. This electrical mode coordinate is used as the input linguistic variable F1S1. The output signal of F1S1 is the coefficient k of the arc state, whose value is minimal under the influence of extreme disturbances (long and short arcs), k ≅ 0, while in the zone of average arc lengths (the neighborhood of point A of the steady-state mode ( Figure 2) takes the maximum value k ≅ 1. Figure 4 shows the membership functions type gauss2mf of the input linguistic variable of the Gauss type. The fuzzy adaptation of the disturbance control process to changes in the state of arcs is performed on the basis of the Takagi-Sugeno fuzzy model. To implement it, the following fuzzy rule base has been compiled:
The adopted algorithmic and parametric degrees of freedom of FIS1 correspond to the dependence of its output signal, the arc state coefficient ( ), shown in Figure 5.  The adopted algorithmic and parametric degrees of freedom of FIS1 correspond to the dependence of its output signal, the arc state coefficient k(U a ), shown in Figure 5. To implement the fuzzy adaptive model (1) for generating the loop error signal, the component of the voltage law U r1 (U a ) is continuously generated in block 1 (Figure 3), as is the component of the impedance law U r2 (U a , I a ). in block 2. These signals in blocks 3 and 4 are multiplied by the normalizing factor of the arc state k and (1 − k), respectively, and according to (1), with the output loop error signal U r (U a , I a ) of the comparison unit CU being formed in the added 5 ( Figure 1).
In the modes of the neighborhood of a given EM (point A ( U a.set , I a.set ), (Figure 2), the control of the electrode movement is realized according to the voltage law U r1 (U a ) (k ∼ = 1). At the same time, phase-by-phase autonomous control of arc lengths is realized, i.e., the negative effect of the interfacial interconnection of modes is eliminated, and the dispersion of the EM coordinates is minimized. In the modes of extreme disturbances (short-circuit and arc extinction) and those close to them, the control of electrode movement is performed according to the impedance law, which ensures reliable arc ignition.
The nonlinearities, backlash and insensitivity zones in EMSS elements lead to a deviation from the actual laws of electrode movement when controlling disturbances from the desired optimal ones. This worsens the dynamic performance. To eliminate this, it is proposed to correct the output signal U l a c of the control signal generation unit CSFB. With this solution, the control signal U l a c for electrode movement at the input of the electric drive is calculated as follows: U l a c = U l a cr + U l a cc , where U l a cr is the main component of the control signal generated in the function of the loop error signal U r according to (1); U l a cc is the corrective signal generated at the output of the fuzzy corrector FIS2, which is represented by a fuzzy PI controller based on the Takagi-Sugeno model with a clear output.
The input signals of FLC2 are the signals U r (Figure 1) obtained at the output of FIS1 and its derivative U r . The output signal of the fuzzy PI corrector FIS2, according to the Takagi-Sugeno model, is formed by the following fuzzy rule model: where B k is a clear value of the FLC2 output for the corresponding ranges of the input signals U r and U r of FLC2. The algorithmic and parametric settings of the FIS2 corrector were obtained by comparing the real and desired (optimal) responses of the automatic control system of the electric arc furnace of DSP-200 type to test deterministic disturbances, including disturbances that cause operational short circuits and arc breaks, which made it possible to obtain the membership functions of the input linguistic variables U r and U r , as well as the output signals of the PI corrector (FIS2). Figure 6 shows the obtained membership functions type trimf and trapmf of the input linguistic variables U r and U r ', with the surface graph of the output signal U la cc (U r , U r ) of the fuzzy corrector FIS2 shown in Figure 7.    For the ACR arc current regulator, the expediency of using the proportional-integral link model was substantiated, and its parameters were obtained from the condition of tuning the dynamics of the ACRSS arc current control subsystems to the modular optimum.
Implementation of the PI-law of the high-speed electric ACRSS in the arc current controller ACR and the use of artificial external characteristics = Ψ( , , ) of type 2-6 ( Figure 2) together with the use of the fuzzy law (1) of the electrical mode mismatch signal ( , ) generation constitutes the essence of the proposed complex system approach. It reduces the arc current dispersion and improves the dynamic accuracy of other electrical mode coordinates stabilization (arc voltage, arc power, reactive power of the arc furnace, power factor, etc.) at the level of optimal values.

Research Results
The study of the effectiveness of the developed two-circuit structure of the adaptive fuzzy control system EM (Figure 1) was performed on the created Simulink model [19]. The effectiveness of the proposed solutions for phase-by-phase autonomous fuzzy adaptive control of arc lengths (model (1), FIS1, FIS2), high-speed control of arc currents and optimal control of the electrical mode based on the formation of artificial external characteristics of the arc furnace in the ACRSS structure corresponding to the selected quality criterion has been studied. At the same time, the quality indicators of the dynamics of controlling deterministic and random disturbances-fluctuations in arc lengths-have been studied. To observe correct comparison, the realizations of deterministic and random disturbances in the study of the dynamic indicators of the specified single-and two-circuit ACS were taken as the same. The obtained results of model studies into the dynamics indicators of the proposed system and structural solutions were compared with the corresponding indicators of a typical single-circuit structure of the electromechanical For the ACR arc current regulator, the expediency of using the proportional-integral link model was substantiated, and its parameters were obtained from the condition of tuning the dynamics of the ACRSS arc current control subsystems to the modular optimum.
Implementation of the PI-law of the high-speed electric ACRSS in the arc current controller ACR and the use of artificial external characteristics I as = Ψ(U s , D Ia , N) of type 2-6 ( Figure 2) together with the use of the fuzzy law (1) of the electrical mode mismatch signal U r (U a , I a ) generation constitutes the essence of the proposed complex system approach. It reduces the arc current dispersion and improves the dynamic accuracy of other electrical mode coordinates stabilization (arc voltage, arc power, reactive power of the arc furnace, power factor, etc.) at the level of optimal values.

Research Results
The study of the effectiveness of the developed two-circuit structure of the adaptive fuzzy control system EM (Figure 1) was performed on the created Simulink model [19]. The effectiveness of the proposed solutions for phase-by-phase autonomous fuzzy adaptive control of arc lengths (model (1), FIS1, FIS2), high-speed control of arc currents and optimal control of the electrical mode based on the formation of artificial external characteristics of the arc furnace in the ACRSS structure corresponding to the selected quality criterion has been studied. At the same time, the quality indicators of the dynamics of controlling deterministic and random disturbances-fluctuations in arc lengths-have been studied. To observe correct comparison, the realizations of deterministic and random disturbances in the study of the dynamic indicators of the specified single-and two-circuit ACS were taken as the same. The obtained results of model studies into the dynamics indicators of the proposed system and structural solutions were compared with the corresponding indicators of a typical single-circuit structure of the electromechanical control system for the electric mode of an arc furnace of DSP-200 type, which operates on the basis of a serial arc power controller ARDM-T-12 of this arc furnace. Figure 8 shows the processes of change in the EM coordinates of the ED obtained on the computer model when regulating a sequence of deterministic disturbances by a single-circuit ACS structure (EMSS) using the proposed fuzzy adaptive model of the control signal U l a c (t) generation for electrode displacement, while Figure 9 indicates the processes of change in the same EM coordinates using a typical single-circuit structure of the ACS with a differential law of the loop error signal generation, which is implemented in the power controller ARDM-T-12: where a, b are constant coefficients. Figures 8 and 9 show the processes of perturbation of the following phase-by-phase asymmetric deterministic disturbances by the specified ACS structures: a three-phase symmetrical operational short circuit, t ∈ [0.2, 1.2]; a short circuit in phase C, t ∈ [1.2, 2.2]; a short circuit in phase A and close to the arc break-in phase C, t ∈ [2.2, 3.5], a short circuit in phase B and close to the arc break-in phase C, t ∈ [3.5, 4.75] and a single-phase short circuit in phase B, t ∈ [4.75, 6]. the basis of a serial arc power controller ARDM-T-12 of this arc furnace. Figure 8 shows the processes of change in the EM coordinates of the ED obtained on the computer model when regulating a sequence of deterministic disturbances by a singlecircuit ACS structure (EMSS) using the proposed fuzzy adaptive model of the control signal ( ) generation for electrode displacement, while Figure 9 indicates the processes of change in the same EM coordinates using a typical single-circuit structure of the ACS with a differential law of the loop error signal generation, which is implemented in the power controller ARDM-T-12: where a,b are constant coefficients.  Figures 8 and 9 show the processes of perturbation of the following phase-by-phase asymmetric deterministic disturbances by the specified ACS structures: a three-phase Figure 8. Time dependencies of voltages U aj (t) (a) and currents I aj (t) (b) arcs, control signals U la cj (t) for electrode movement (c) undergoing deterministic disturbances U f j (t) (d) and with fuzzy models used for generating control signals U la cj (t) to move electrodes.
the basis of a serial arc power controller ARDM-T-12 of this arc furnace. Figure 8 shows the processes of change in the EM coordinates of the ED obtained on the computer model when regulating a sequence of deterministic disturbances by a singlecircuit ACS structure (EMSS) using the proposed fuzzy adaptive model of the control signal ( ) generation for electrode displacement, while Figure 9 indicates the processes of change in the same EM coordinates using a typical single-circuit structure of the ACS with a differential law of the loop error signal generation, which is implemented in the power controller ARDM-T-12: where a,b are constant coefficients. (c) (d) Figure 9. Time dependencies of voltages ( ( )) (a) and currents ( ( )) (b) arcs, control signals ( ( )) for electrode movement (c) undergoing deterministic disturbances ( ( )) (d) and with a differential model used for generating a loop error signal ( ( ); ARDM-T-12 controller). Figures 8 and 9 show the processes of perturbation of the following phase-by-phase asymmetric deterministic disturbances by the specified ACS structures: a three-phase Figure 9. Time dependencies of voltages (U aj (t)) (a) and currents (I aj (t)) (b) arcs, control signals (U la cj (t)) for electrode movement (c) undergoing deterministic disturbances (U f j (t)) (d) and with a differential model used for generating a loop error signal (U r (t); ARDM-T-12 controller).
A comparative analysis of the results of computing research of deterministic disturbance control is shown in Figures 8 and 9, where there is the proposed fuzzy law (1) for generating the loop error signal U r (t ) and the fuzzy model for generating the corrective signal U l a c , yield autonomous, i.e., phase-independent, perturbance: in phases with no disturbances, the control signal U c (t) ∼ = 0 and the position of the electrodes do not change. This reduces the disturbance control time by 15-25%. These conclusions about the obtained phase autonomy and higher performance are best illustrated by the time dependence of U c (t). This property of phase autonomy is achieved mainly due to the fuzzy localization of the use of the voltage law U r1 (U) in the vicinity of point A (Figure 2). At the same time, the implementation of reliable arc ignition in short-circuit and arc breakdown modes, which is inherent in the impedance law U r2 (U a , I a ), used in the modes of extreme perturbation.
However, deterministic, in particular extreme, disturbances in the melting process are not the main ones. They occur infrequently and only at the first technological stages of melting. The main ones are random coordinate disturbances-fluctuations in arc lengths, the statistical characteristics of which also change during the melting process. Therefore, to obtain a full assessment of the effectiveness of the proposed system and circuit solutions implemented in the structure shown in Figure 1, we studied the integral stochastic indicators of the quality of control of random disturbances f j (t)-fluctuations in the lengths of three-phase arcs (j = A, B and C)-namely, their mathematical expectations and variances. Figures 10-13 show the time dependences of voltages (U aj (t)), currents (I aj (t)), powers (P aj (t)) of arcs and power of electrical losses (P el (t)) in the short network of an arc furnace of the DSP-200 type when controlling random disturbances ( f j (t)) obtained in the Simulink model. The statistical characteristics of the disturbances correspond to the technological stages of melting wells in a solid charge in an arc furnace of DSP-200 type.
not change. This reduces the disturbance control time by 15-25%. These conclusions about the obtained phase autonomy and higher performance are best illustrated by the time dependence of ( ). This property of phase autonomy is achieved mainly due to the fuzzy localization of the use of the voltage law ( ) in the vicinity of point A ( Figure   2). At the same time, the implementation of reliable arc ignition in short-circuit and arc breakdown modes, which is inherent in the impedance law ( , ), used in the modes of extreme perturbation.
However, deterministic, in particular extreme, disturbances in the melting process are not the main ones. They occur infrequently and only at the first technological stages of melting. The main ones are random coordinate disturbances-fluctuations in arc lengths, the statistical characteristics of which also change during the melting process. Therefore, to obtain a full assessment of the effectiveness of the proposed system and circuit solutions implemented in the structure shown in Figure 1, we studied the integral stochastic indicators of the quality of control of random disturbances ( ) -fluctuations in the lengths of three-phase arcs (j = A, B and C)-namely, their mathematical expectations and variances. (a) (b) Figure 10. Time dependencies of voltages (U aj (t)) (a), currents (I aj (t)) (b), powers (P aj (t)) (c) arcs and electric loss power (P elj (t)) (d) undergoing random disturbances ( f j (t)) (e) with a differential model (5) of the loop error signal (U rj (t); ARDM-T-12 controller) used. Figure 10 shows the time dependencies of the indicated EM coordinates obtained when the differential law (5) operates in the single-circuit EMSS structure (power controller of the ARDM-T-12 type), while Figure 11 indicates the time dependencies of the same EM coordinates as in the first experiment, with the proposed fuzzy model of generating a control signal for electrode movement in the single-circuit structure of the ACSS EM (only EMSS operates) being used. Figure 12 shows the time dependencies obtained in the two-circuit ACS EM using the same fuzzy model in the structure of the ACS EMSS as in the second experiment ( Figure 11) and the implementation of the artificial external characteristic 3 (Figure 2) in the ACRSS structure. Figure 13 represents the time dependences of the same two-circuit ACS EM and with the same structural and system solutions for EMSS and ACRSS as in the third experiment ( Figure 12), with the implementation of the artificial external characteristic 7 (Figure 2) in ACRSS obtained by increasing the secondary FT voltage by 10% (U 2ph = 1.1U 2p f n ). This increase in U 2ph in the fourth experiment corresponds to the current global trend of increasing the specific power of power electrical AF equipment by 10-15% in order to intensify the melt of the charge in an arc furnace by increasing the secondary voltage of furnace transformers with the latest ladle furnace melting technologies [15,32].  . Time dependencies of voltages (U aj (t)) (a), currents (I aj (t)) (b), powers (P aj (t)) (c) arcs and electric loss power (P elj (t)) (d) undergoing random disturbances ( f j (t)) (e) with fuzzy models.
The results of modeling studies of the dynamics shown in Figures 10-13 are obtained when the corresponding structures of the ACS EM control the same random disturbance f j (t) (Figure 10e), whose statistical characteristics correspond to the final technological stage of the charge melting in the DSP-200 furnace. The integral stochastic quality indicators (given in Table 1) of the dynamics of controlling random disturbances f j (t) (Figure 10e This shows the average values of the mathematical expectations M and dispersions D of voltages U aj (t), currents I aj (t), powers P aj (t) of arcs and electrical losses P elj (t) in the short network of the DSP-200 arc furnace, calculated at the time intervals of process stationarity f (t) during the operation of the four studied ACS structures: two singlecircuit structures (EMSS) with a differential and a proposed fuzzy model of generating control signals U la cj (t) for electrode movement and two two-circuit ACS structures when implementing different dependencies of artificial external characteristics 5 and 7 (Figure 2) at the nominal (U 2phn ) and increased (U 2ph = 1.1U 2phn ) secondary voltage of the furnace transformer (FT).  Time dependencies of voltages (U aj (t)) (a), currents (I aj (t)) (b), powers (P aj (t)) (c) arcs and electric loss power (P elj (t)) (d) undergoing random disturbances ( f j (t)) (e) in the two-circuit ACS structure (AEC 5, Figure 5 in ACRSS) with fuzzy models. Analysis of the obtained results of computing research indicates the improvement of dynamic accuracy of stabilization (reduction of dispersion) of EM coordinates when implementing fuzzy models of electrode movement control in comparison with the ARDM-T-12 arc current controller. Dispersions of EM coordinates at different stages are reduced by 8-10%. Due to this, there is a decrease in the power of electric losses by 8-12% and an increase in arc power by 3-5%. As shown in [33][34][35][36], the obtained decrease in the dispersion of arc currents leads to a 5-7% decrease in the specific power consumption. In addition to the mentioned improvement of energy efficiency performance, the reduction of arc current dispersion improves electromagnetic compatibility indicators of arc furnace and power grid modes. A number of power quality indicators at the power buses of AFs are enhanced; in particular, fluctuations and sags of power voltage are reduced, as well as reactive power consumption, i.e., power factor increases, and the flicker dose reduces accordingly.  Figure 10 shows the time dependencies of the indicated EM coordinates obtained when the differential law (5) operates in the single-circuit EMSS structure (power controller of the ARDM-T-12 type), while Figure 11 indicates the time dependencies of the same EM coordinates as in the first experiment, with the proposed fuzzy model of generating a control signal for electrode movement in the single-circuit structure of the ACSS EM (only EMSS operates) being used. Figure 12 shows the time dependencies obtained in the two-circuit ACS EM using the same fuzzy model in the structure of the ACS EMSS as in the second experiment ( Figure  11) and the implementation of the artificial external characteristic 3 (Figure 2) in the ACRSS structure. Figure 13 represents the time dependences of the same two-circuit ACS EM and with the same structural and system solutions for EMSS and ACRSS as in the third experiment ( Figure 12), with the implementation of the artificial external characteristic 7 (Figure 2) in ACRSS obtained by increasing the secondary FT voltage by 10% ( = 1.1 ). This increase in in the fourth experiment corresponds to the current global trend of increasing the specific power of power electrical AF equipment by 10-15% in order to intensify the melt of the charge in an arc furnace by increasing the secondary voltage of furnace transformers with the latest ladle furnace melting technologies [15,32].
The results of modeling studies of the dynamics shown in Figures 10-13 are obtained when the corresponding structures of the ACS EM control the same random disturbance ( ) (Figure 10e), whose statistical characteristics correspond to the final technological Figure 13. Time dependencies of voltages (U aj (t)) (a), currents (I aj (t)) (b), powers (P aj (t)) (c), arcs and electric loss power (P elj (t)(d)) undergoing random disturbances ( f j (t)) (e) in the two-circuit ACS structure (AEC 7, Figure 5 in ACRSS) with fuzzy models.
Applying the high-speed subsystem of arc currents regulation ACRSS to the two-circuit ACS at U 2phn with AEC 5 (Figure 2) significantly reduces the dispersion of arc currents by a factor of 2-4 at various melting stages and reduces the power of electric losses by 5-10%, with the arc power decreasing by 4-6%. The latter is explained by the use of the AEC 5 (Figure 2), at which the highest dynamic accuracy of EM coordinate stabilization is achieved, while the average arc power is reduced due to the current limitation in accordance with the dependence of AEC 5 ( Figure 2).
To eliminate this disadvantage, it is proposed to use a higher arc voltage set point with an unchanged arc current set point I sa , i.e., to melt on long arcs. This is especially advisable to do in the first high-energy technological stages of melting. To do this, it is necessary to boost the PT power by increasing the secondary voltages of the furnace transformer stages. This is how the leading world manufacturers, such as Daniely Daniely Automation (Italy), Amec Spic (France), Siemens VAI and Ferrotron (Germany) and others, are designing new arc furnace power equipment or improving the existing ones.
To illustrate the effectiveness of this solution, the quality indicators of the dynamic regulation of the EM coordinates were investigated at the secondary voltage of FT increased by 10% in the structure of two-circuit ACS with the use of AEC 7 ( Figure 2) and fuzzy models in the CU and CSGU blocks (Figure 1) to control the movement of electrodes and electrical regulation of arc currents in the process of EM perturbation (Table 1).
Comparing the obtained quality indicators of two-circuit ACS (EMSS+ACRSS) with fuzzy models and U 2ph = 1.1U phn with three other investigated ACS structures has shown the improved performance of dynamics, energy efficiency and electromagnetic compatibility.
Thus, the two-circuit ACS with U 2ph = 1.1U phn has the lowest value of the dispersion D Ia =17 kA 2 , which is almost 3 times less than in the single-circuit ACS with the differential law U r3 (U a , I a ) (5) at almost invariable average arc currents. The highest arc power P a = 8.91 MW was obtained, up 11.2% in comparison to the single-circuit ACS and up 14.8% compared with the two-circuit ACS at U 2ph = U 2phn , yet at higher dispersion. The power of electric losses in the short circuit network is insignificantly different in the investigated ACS-the lowest in the two-circuit ACS at U 2ph = 1.1U phn , with its dispersion being by far the lowest among other ACSs, down 3.8 times than in the one-circuit ACS (differential law, ARDM-T-12 controller), which positively affects the deviation and fluctuation of power network violations and reduces specific power costs.
High speed, which is an order of magnitude higher than in the EMSS electromechanical subsystem of arc length control, and the presence of an integral component in the arc current control law (PI-structure of the ACR regulator) make it possible to obtain a close to astatic arc current control process. Accordingly, it minimizes the arc current dispersion and other coordinates of the arc furnace electric mode.
By using the complex structural and system solutions developed in the article, the intensity of mutual influence between the electromechanical EMSS and electrical ACRSS coordinate control subsystems of the arc furnace electrical mode is minimized. Thanks to this, the dynamic accuracy of the arc furnace electric mode coordinates stabilization at the synthesized optimal value levels is significantly increased. As a result, the indicators of energy efficiency, energy saving and electromagnetic compatibility of the arc furnace and the power grid are improved.
The given analysis of the research results shows that the use of vague models of arc length regulation increases the dynamic accuracy of EM coordinate stabilization, with various artificial external characteristics of arc furnace in the structures of two-circuit ACS a stimulus for implementing multi-criteria adaptive optimal EM control strategies of arc furnaces. These results point to some particular criteria of EM control quality being contradictory. That is why it is reasonable to implement Pareto-optimal control strategies, which will be examined in a separate study.
The following conclusions can be made based on the obtained results of model studies: 1.
The proposed fuzzy models for controlling the arc furnace electrode's movement make it possible to implement phase-by-phase autonomous control of arc lengths, which improves the dynamic accuracy of stabilization of electric mode coordinates at the level of set values. Dispersion of arc currents is reduced by 3-4 times, while an increase in arc power by 11-15% is achieved; 2.
A double-circuit high-speed structure of the automatic control system of arc furnace electric mode has been developed, which is characterized by wide functionality for implementing the multi-criteria optimal control strategy based on the formation of the appropriate artificial arc furnace external characteristics Ψ(U a , D I a , N); 3.
The implementation of multi-criteria optimal control strategies will make it possible to comprehensively improve indicators of energy efficiency, electromagnetic compatibility, ecology and quality of molten steel; 4.
A Simulink model of a double-circuit automatic electric mode control system for an arc furnace of the DSP-200 type was created with the implementation of fuzzy models of arc length control and a modification of the arc furnace's artificial external characteristic. Computer studies of the effectiveness of their use under the influence of deterministic and random perturbations were carried out on this numeric model;

5.
The obtained results of computer studies confirmed the feasibility of implementing phase-independent control of the arc lengths. This made it possible to reduce the time for handling deterministic, in particular, extreme, perturbations and to reduce the dispersion of electric mode coordinates when handling random perturbations; 6.
Changing the artificial external characteristics of the arc furnace in the structure of the double-circuit automatic electric mode control system is an effective controlling influence for the implementation of multi-criteria adaptive optimal control. This makes it possible to comprehensively improve the indicators of energy efficiency and electromagnetic compatibility of the modes of the arc furnace and the power grid, as well as to obtain high-quality stabilization of the electric mode coordinates at the given, in particular, optimal values; 7.
The analysis of the obtained integral indicators of the arc power control quality in the DSP-200 arc furnace showed the effectiveness and feasibility of practical use of the proposed strategy for improving the arc furnace electric mode automatic control system. It is advisable to implement the structural and system solutions developed for this purpose on alternating current steel-melting arc furnaces of different capacities; 8.
The system engineering solutions proposed in the article do not require large capital investments and can be implemented during the modernization of alternating current steel-smelting arc furnaces of any capacity.

Conclusions
As shown by the obtained results of computer studies, the use of a double-circuit structure for controlling the arc furnace modes, which combines the fast properties of the electric mode coordinates control and the fuzzy models of the arcs lengths control, makes it possible to: Funding: This research received no external funding.

Data Availability Statement:
Authors' own data obtained from calculations performed by the authors.

Conflicts of Interest:
The authors declare no conflict of interest.