STUDY OF A NONLINEAR AUTOMATIC POWER CONTROL SYSTEM FOR AN ELECTRIC-ARC STEELMAKING FURNACE

B. A. Beinarovich, E. S. Astapenko, N. G. Volkov

(Presented by the scientific and technical seminar of the Research Institute of AEM)

In designing an automatic control system, the most difficult task is to obtain the specified dynamic properties, which are usually assessed from the form of the transient process in response to a standard input. For a power-control system of an electric-arc steelmaking furnace, the characteristic disturbance is a step-shaped form of disturbance. In linear automatic control systems, the problem of calculating the transient process and evaluating dynamic properties can be solved analytically, i.e., in a general form. At the same time, on the basis of known methods (for example, frequency methods), it is possible to determine the influence of individual parameters on the dynamics of the system, and also to select the parameters of correcting devices that ensure the specified dynamic properties, i.e., to carry out the synthesis of the automatic control system.

In nonlinear automatic control systems, calculation of transient processes cannot be carried out in general form. As a result, the possibilities for analytically evaluating the influence of individual parameters on the dynamics of an automatic control system are very limited, and the presence of several nonlinearities in an automatic control system practically excludes such a possibility. Therefore, the analysis of the influence of changes in the parameters of an automatic control system on its dynamic properties and the solution of engineering problems of synthesis of nonlinear automatic control systems are currently carried out with the aid of computing machines. Analog computers (ACs), which make it possible to solve nonlinear differential equations, are especially convenient for solving such problems. Work on an AC does not require extensive special training in computing technology and programming, especially when the structural modeling method is used. The electronic model makes it possible to give a full physical interpretation of the settings and processes. In view of the high speed of ACs, solutions are obtained quickly and are indicated on recording instruments, allowing visual monitoring. This makes it possible, in a short time, to examine and investigate the influence of many automatic-control-system parameters on the transient process.

The present work sets forth the methodology and results of the study and synthesis of the parameters of the automatic power-control system of an electric-arc steelmaking furnace of the DSV-3.5 type using an AC. The block diagram of the automatic control system is presented in Fig. 1. Figure 2 presents the diagram of its electronic model.

The measuring and converting elements, including the current and voltage transformers and the autotransformer for setting the power introduced into the furnace, are represented by linear links 2, a, 2, b (Fig. 1); the rectifiers and the capacitive smoothing filter are represented by an aperiodic—

Fig. 1.

Fig. 2.

...dynamic element 3 (Fig. 1). In the electronic model these elements are implemented by amplifier 2 (Fig. 2). The intermediate amplifier (element 5, Fig. 1; amplifiers 3, 4, Fig. 2) with correction circuits (integrating-differentiating element 6, Fig. 1; amplifiers 9—13, Fig. 2) and the thyristor converter (linear element 7, Fig. 1) correspond to the parameters of the complete reversible thyristor electric drive of the PTZR series. To move the electrodes, an electric motor with a smooth armature of type PGT-4 is used (element 8, Fig. 1; amplifiers 5—7, Fig. 2), together with a TG1 tachogenerator (element 9, Fig. 1) and a gear reducer with rack-and-pinion transmission (element 11, Fig. 1; amplifier 8, Fig. 2).

The APC system for the power of an arc steelmaking furnace contains substantially nonlinear elements:

1. A nonlinear negative feedback with respect to the speed of the electrode-travel motor (element 10, Fig. 1). The form of the feedback characteristic, set up on block BN8 (Fig. 2), is shown in Fig. 3.

Fig. 3.

1. Nonlinear electrical characteristics of the arc furnace (blocks BN1 and BN2 in Fig. 2), representing the dependence of the arc current \(I_{\partial}\) and arc voltage \(U_{\partial}\) on the arc length \(L\) (links 1a, 1b in Fig. 1); in this case the transfer coefficient with respect to arc voltage \(\beta\) may vary tenfold, from \(\beta=12\) V/mm at the beginning of melting to \(\beta=1.2\) V/mm at its end.

2. The nonlinear characteristic of link 12 (Fig. 1), representing backlash in the reducer-and-rack transmission (block BN7 in Fig. 2).

3. An adjustable dead zone with respect to the magnitude of the controlled parameter (link 4 in Fig. 1); block BN3 (Fig. 2).

4. The characteristic of the intermediate amplifier with saturation (block BN4 in Fig. 2).

5. Current limiting in the armature circuit of the motor (block BN5 in Fig. 2).

6. The static load of the motor, of the type of asymmetric dry friction (block BN6 in Fig. 2).

From potentiometer \(R_1\) (Fig. 2), a voltage corresponding to the nominal arc length \(L_{\text{nom}}\) is applied to the input of summing amplifier 1, while potentiometer \(R_2\) sets the magnitude of the disturbance due to the change in arc length \(\Delta L\). The modeling was carried out on an EMU-10 analog computer.

The investigations were carried out in order to determine the influence of nonlinearities of the type of backlash in the actuating mechanism, amplifier saturation, limiting of the armature current of the actuating motor, and the dead zone with respect to the controlled parameter, as well as to estimate the influence of the change, during the melting process, of the transfer coefficient with respect to arc voltage \(\beta\) on the dynamic properties of the arc-power regulator. Another task was to determine the optimal adjustment of the variable parameters of the regulator.

As the optimality criterion, the minimum time for compensating a stepwise decrease in arc length was adopted, with the overshoot limited to 20% (with a larger overshoot, arc extinction occurs).

The research procedure consisted in the following: with successive variation (within practically feasible limits) of the value of one of the adjustable parameters (the others being constant), the transient process was recorded and its character, regulation time, amount of overshoot, number of oscillations, and time to the first matching were determined. From the results of such investigations, the best combinations of the values of the adjustable parameters of the nonlinear characteristics were determined.

Figures 4a, b, c, d, e show the transient processes of changes in the motor speed \(\omega(t)\) and the arc length \(L(t)\) when stepwise decreases in the arc length are compensated in the automatic regulation system with various adjustments. Table 1 gives several combinations of the adjustable regulator parameters and the varying parameters of the controlled object, designated by serial numbers from 1 to 10, and Table 2 gives the indices of the dynamic properties of the automatic regulation system for these combinations.

From the results of the investigations the following conclusions may be drawn:

1. To increase the speed of response of the automatic regulation system and improve the electrical indices of the arc furnace, the setpoint of the current limit should have a value that is the maximum permissible for the type of motor used (Fig. 4a).

2. For the same purpose, the saturation level of the intermediate amplifier should have as high a value as possible (Fig. 4b).

3. The strongest influence on the dynamics of the automatic arc-power regulation system is exerted by the change, during the melting process, in the transfer coefficient of the object with respect to voltage (Fig. 4d). If at \(\beta=8\) the transient process has satisfactory indices (item 6 of Table 2), then at \(\beta=12\) the regulation time and the magnitude of overshoot increase considerably.

Fig. 4.

Visible labels in the figure:

• \(t\), sec; \(\omega\); \(L\)
• \(0.1\)
• a, b, c, d

For subfigure a:

1. \(I_a^{\max}=10 I_{a\,\mathrm{nom}}\)
2. \(I_a^{\max}=5 I_{a\,\mathrm{nom}}\)
3. \(I_a^{\max}=2.5 I_{a\,\mathrm{nom}}\)

For subfigure b:

1. \(U_{\max}=12.5\text{ V}\)
2. \(U_{\max}=10\text{ V}\)

For subfigure c:

1. backlash 0.5 revolution
2. backlash 0.5 revolution, dead zone \(+5\% I_{d\,\mathrm{nom}}\)
3. no backlash or dead zone

For subfigure d:

1. Object 1
2. Object 2

Object 1 \(>\) Object 2

For subfigure e:

1. \(\beta=12\)
2. \(\beta=8\)
3. \(\beta=4\)
4. \(\beta=1.2\)

Table 1

…oscillation (item 1, Table 2). When the coefficient \(\beta\) is decreased, the transient process is very prolonged; the settling time is large (items 7, 9, Table 2).

1. Backlash in the actuator mechanism of the regulator has a significant effect on the dynamics of the APC system in the case when the transient process is oscillatory in character (items 1, 2, Table 2), even more

Table 2

increasing the number of overshoots, their magnitude, and the total time for disturbance rejection. At large values of backlash (2 motor revolutions and more), undamped oscillations arise in the system. If the system has a transient process with a single overshoot of insignificant magnitude (less than 20%) or an aperiodic one, then the influence of backlash is manifested to a lesser degree (items 4, 5, Table 2). The rejection time increases due to the take-up of backlash at the beginning of acceleration and during reversal.

1. When the transfer coefficient of regulation with respect to arc voltage \(\beta\) is changed, the overall gain coefficient of the APC changes. To maintain it at a level that ensures an optimal transient process, it is necessary to change the gain coefficient of the regulator, for example, by changing the coefficient of negative feedback with respect to motor speed (Fig. 4, d). This also follows from comparison of transient processes recorded at different values of \(\beta\) with a constant feedback coefficient (items 1, 6, 7, 9, Table 2), with transient processes recorded at the same values of \(\beta\), but with a tuned feedback coefficient (items 4, 6, 8, 10, Table 2), which must be increased as \(\beta\) increases.

2. To reduce the influence of backlash in the direction of deterioration of the dynamic properties and stability margin of the APC when the plant gain coefficient \(\beta\) changes, the variable parameters of the system must be adjusted in such a way that the transient process differs little from the optimal one (items 1, 2 and 4, 5, Table 2).

3. Widening the dead zone with respect to the controlled parameter of the regulator within \(\pm 5\%\) of the nominal value reduces oscillation and the magnitude of overshoot (item 3, Table 2), but cannot fully compensate for the influence of backlash. The influence of backlash on the dynamics of the APC can be compensated by simultaneously widening the dead zone and changing the feedback coefficient with respect to motor speed.

4. As the coefficient \(\beta\) decreases during melting, the width of the regulator dead zone should be reduced, bringing it down to a minimum value by the end of melting (\(\pm 0.5\%\)), which leads to an increase in the accuracy of arc-power regulation while preserving the required dynamic characteristics of the APC.

Thus, in order to ensure an optimal transient process in the presence of backlash in the actuator mechanism of the regulator and changes

change during the melting process in the electrical characteristics of an electric-arc steelmaking furnace, the automatic arc-power control system must include a self-tuning loop which, as a function of the transfer coefficient of the controlled object with respect to voltage, adjusts the coefficient of negative feedback with respect to the speed of the electrode-travel motor and the width of the dead zone for the controlled parameter.

REFERENCES

1. A. M. Lebedev, V. A. Naidis, R. T. Orlova, A. V. Paltsev, V. F. Yuferov. Thyristor Tracking Electric Drive. Moscow, “Energiya,” 1972.

2. E. A. Arkhangel’skii, A. A. Znamenskii, Yu. A. Lukomskii, E. P. Chernyshev. Modeling on Analog Computing Machines. Leningrad, “Energiya,” 1972.