Optimization of the lining wear coefficient on the basis of control of three-phase current electric arcs characteristics

The problems of increasing the efficiency of industrial enterprises are now becoming increasingly relevant [1, 2], while the task of optimizing the activity of industrial units and machines is of particular importance. There are a large number of industrial units, the basis of which is the use of energy created by an electric arc. The lining layer present in such units during operation is subjected to thermal and electromagnetic effects created by the arc. A quantitative measure of this effect is the lining wear coefficient (LWC), one of the most important technological characteristics of modern multi-electrode arc furnaces. The most significant effect on the LWC is exerted by the arc power, the distance from the arc axis to the lining, and the applied voltage. In most studies, the influence of the described factors on the LWC is not taken into account. At the same time, heat fluxes affecting all elements of the furnace structure, including the lining, depend on the shape of the arc.

The aim of the work is to study this effect and find the parameters that provide optimization of the LWC. To achieve this goal, an algorithm for calculating the shape of arcs in an AC steelmaking furnace has been developed, which will be used to optimize the objective function.

Analysis of the shape of alternating current electric arcs parallel burning in the direction of the horizontal surface during their electromagnetic interaction and the influence of this shape on the lining wear coefficient

The electromagnetic interaction of the arcs is determined by the equilibrium equation of the arc column element [3–6]

Краткие сообщения

—-            —-           —-

^dF 1 - 2 + dF^ + dF i - ц = 0 ,

—                                                                                                                                                                                                                     — where dF1-2 is the force of interaction of an arc element with another arc (or other arcs), dF1_1 is the force of interaction of an arc element with itself, dF1_ц is centrifugal force seeking to straighten an arc column.

Electromagnetic interaction of three arcs through which harmonic currents with phase shift flow

• 2п     .     .   .f 2 п 2л) .    .     . f 2 п 2 л) ,

2п/ 3: i1 = i 01Sin у т , 12 = i02 sin I у T + у I и i3 = 103sin I у T-у I, where i01, i02, i03 are amPli- tude values of currents [4]. Consider the case when the current value of the currents is the same id1 = id 2 = id 3 = id .

In the electromagnetic interaction of three alternating currents, two repulsive forces act on the element dl 1 : dF 1 - ₂ = x • 1 1 1 ₂ d1 1 , dF 1 - ₃ = x • i ₁ i ₃ d1 1 . Direction of the resulting force dF ₂ 3 changes over time and

—                  — coincides at any time with the vector dF1-2 + dF1-3. The modulus of the resulting force is equal to dF1-2-3

= Vs dF _{1 - 2} = Vs F ₀

• 2n sin— т • dL , T ¹

where F _o = ^X i'¹ 0 2

.

Since the instantaneous strength values dF _{1 - 2 - 3} are continuously changing during each half-cycle, the electrodynamic forces acting on the arc element change in magnitude and direction. We find the average integral force over half a period that determines the average position of the arc

T dF    =-2д.Z^ 2sin(2nT)dт-d1x =-—id2 • d1x «-0,551-id2 • d1x.                        (3)

T 2 0 T          ¹ п ^{d 1}              д ¹

This force is directed from the center of decay of the electrodes.

Consider the electromagnetic interaction of an arc element dl ₁ and direct current from the side of the arc. Vector dF_{1 - 1} is in the plane yOz , is perpendicular to dl 1 and oppositely directed to the vector dF _{1 - 2} , and its module [4]

2 1 1 d VV- ■y «- y ( z■ )) ^

I dF1 1= f —----------5--------------- dZ • d11 = X1112 • d11.(4)

'       1     4п 0

f 2 п     ,)..,..,„

In the electromagnetic interaction of a harmonic current i 1 = i ₀₁ sin I у т + ф I with itself average integral value of force

T            2T                 22

dF - 1 = Z 1 J i 2 d т = ^X2 - 0 ¹ j l sin-^ т + ф| d т^ d1 1 =^X- ⁰¹ • d1 1 = X 1 i2 • d1 1 .

0              0 ^

It is seen that the electromagnetic interaction of direct current with itself is the same as an alternating current with the same effective value.

In the general case, equation (1) of the equilibrium of an element of an arc column during its electromagnetic interaction with one or two arcs of constant or alternating currents can be written as kx dF.1-2 + dFH + dFX-ц = 0.                                                                       (5)

Based on the algorithms proposed in [3, 4, 6–12], one can solve equation (5) and obtain an approximate description of the shape of the arc.

The most important indicator of the thermal load of the furnace lining is the lining wear coefficient

3 PU

(LWC) [13-15]. LWC is defined as lwc = —d-d -, where P _d is the arc power, U _d is the arc voltage, a is a ²

a distance from the arc axis to the lining [13–15].

To more accurately determine the heat perception of the side wall, we modify the LWC calculation formula taking into account the shape of the axis of the arc column found when solving equation (5). To do this, we will find the LWC at the minimum distance from the wall from the arc. From physical considerations, we can conclude that the minimum distance from the arc to the lining is achieved at the point of contact of the arc with the surface of the melt. Based on this, it is possible to formulate a control problem for finding the optimal coefficient of wear of the lining, changing within acceptable limits the parameters that determine the operation of the furnace. As such parameters, the applied voltage and the distance from the surface of the melt to the end of the electrode can act, which can be adjusted during each stage of the process. To solve the problem of optimizing the LWC value, we use an approximately determined dependence of the shape of the arc on the control parameters. Since this dependence cannot be described analytically, the use of gradient optimization methods seems inappropriate. Therefore, for optimization, evolutionary methods are used.

For optimization, a genetic algorithm was chosen [16] with a population size of 17 individuals, a simple single-point crossover, a mutation with a probability of 0.2 was implemented. The software implementation is done using freely distributed Python packages. It was found that at currents of about 50 kA and arc powers of about 20 MW, the lining wear coefficient varies from 1630 to 1750 MW • V/ m².

Conclusions

Based on the shape of the arcs, algorithmic and software have been developed to optimize the wear coefficient of the lining from three arcs of three-phase alternating current burning in the direction of a horizontal surface, make decisions on the effective heat transfer of the radiation of the arcs to the charge, depending on the shape of the arcs, determined by the power and voltage of the arc, the distance from the arc to the axis of the electrode.