Design Methodology for an Induction Drive of a Mine Hoisting Machine
Abstract Generated abstract
This paper proposes a coordinated design methodology for asynchronous drives of mine hoisting machines, addressing the interdependent selection of the speed diagram, starting rheostat, magnetic station, and motor. The method derives rotor resistance stages from actual hoist operating conditions, relates acceleration and braking regimes to motor overload capacity and static load, and includes procedures for constructing mechanical characteristics, estimating currents, and checking motor heating. An example calculation for a Karaganda skip hoist shows an effective current within the permissible limit and indicates that operation below the designed acceleration reduces efficiency and output, causing substantial excess energy consumption.
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Design Methodology for an Asynchronous Drive of a Mine Hoist Machine
V. D. Petunov
(Presented at the scientific and technical conference of the Mining Faculty)
The established practice of designing asynchronous drives for mine hoisting machines, without sufficient mutual coordination, resolves questions connected with the choice of the speed diagram, the starting rheostat, and the motor. The proposed design methodology takes into account the actual operating conditions of hoisting machines and makes it possible to calculate the speed diagram as a function of the rotor resistances and the overload capacity of the motor.
A motor is first selected with a rotational speed corresponding to the maximum speed of motion of the hoisting vessels, and with a rated power
\[ P_{н} \approx \frac{\rho k Q v_{м}}{102 \eta_{р}}, \tag{1} \]
where
\(\rho\) — constant of the dynamic regime;
\(k\) — coefficient of mine resistances;
\(Q\) — weight of the useful load, kg;
\(v_{м}\) — maximum speed of motion of the hoisting vessels, m/sec;
\(\eta_{р}\) — efficiency of the reducer.
The constant of the dynamic regime depends on the speed and the height of hoisting. For the Karaganda and Kuznetsk coal basins, the most common speeds are from 4 to 7.5 m/sec. At such speeds we recommend adopting, depending on the hoisting depth, the following values of the constant of the dynamic regime:
| Hoisting height, m | up to 200 | 200–300 | 300–400 | 400–500 | over 500 |
|---|---|---|---|---|---|
| Constant of the dynamic regime | 1.45 | 1.40 | 1.35 | 1.30 | 1.25 |
The magnetic station is selected depending on the overload capacity of the motor, so that during starting there are identical moment surges at the time of switching the rheostat stages. Therefore we recommend adopting the following number of contactors:
| Motor overload capacity | 1.8–2.0 | 2.1–2.2 | 2.3–2.5 | over 2.5 |
|---|---|---|---|---|
| Minimum number of contactors | 6 | 7 | 8 | 9 |
Taking into account the conditions of smooth starting, magnetic stations with fewer than six contactors should not be used. More than 9 contactors should likewise not be recommended, since bulky magnetic stations are unreliable in operation.
The resistances of the preliminary stages are calculated from the actual operating conditions of the hoisting installation. The first stage must ensure kinematic tightening of the system, as well as a sufficiently smooth transition to the counter-switching braking mode. The resistance of this stage is
\[ R_1=\frac{R_{\mathrm{n}}}{0.3:0.5}, \tag{2} \]
where \(R_{\mathrm{n}}\) is the nominal rotor resistance, ohms.
In the case of skip hoisting, the second stage is intended for pulse drawing-up of the vessel at the end of the hoisting cycle without applying the mechanical brake. The resistance of the second stage is
\[ R_2=\frac{R_{\mathrm{n}}\left(v_0-0.5\,v_{\mathrm{in}}\right)} {v_0\left(\mu j_{\mathrm{in}}+\gamma_{\mathrm{st.k}}\right)}, \tag{3} \]
where \(v_0\) is the hoisting speed corresponding to the synchronous speed of the motor, m/sec;
\(v_{\mathrm{in}}\) is the adopted speed at which the skip enters the unloading curves, m/sec;
\(\mu\) is a coefficient characterizing the reduced mass;
\(j_{\mathrm{in}}\) is the acceleration during motion in the curves at the end of the hoisting cycle, m/sec\(^2\);
\(\gamma_{\mathrm{st.k}}\) is the static load during motion in the curves at the end of the hoisting cycle, in relative units.
The coefficient characterizing the reduced mass of the moving parts of the hoisting installation is
\[ \mu=\frac{mD_{\mathrm{b}}}{2M_{\mathrm{n}}i\eta_{\mathrm{r}}}, \tag{4} \]
where \(m\) is the mass of the moving parts of the hoisting installation, reduced to the radius of the winding element, kg·sec\(^2\)/m;
\(D_{\mathrm{b}}\) is the drum diameter, m;
\(M_{\mathrm{n}}\) is the nominal torque of the motor, kg·m;
\(i\) is the gear ratio of the reducer.
The third stage must ensure that the skip leaves the unloading curves at the prescribed speed without applying the mechanical brake. Its resistance is
\[ R_3=\frac{R_{\mathrm{n}}\left(v_0-0.5\,v_{\mathrm{out}}\right)} {v_0\left(\mu j_{\mathrm{out}}+\gamma_{\mathrm{st.n}}\right)}, \tag{5} \]
where \(v_{\mathrm{out}}\) is the adopted departure speed, m/sec;
\(j_{\mathrm{out}}\) is the acceleration during motion of the skip in the curves at the beginning of the hoisting cycle, m/sec\(^2\);
\(\gamma_{\mathrm{st.n}}\) is the static load during motion in the curves at the beginning of the hoisting cycle, in relative units.
We recommend taking the calculated values of the departure speed as 1.0 to 1.5 m/sec and the entry speed as 0.6–1.0 m/sec. The acceleration during motion in the curves is determined by the formulas
\[ j_{\mathrm{in}}=\frac{v_{\mathrm{in}}^2}{2h_{\mathrm{r}}}, \tag{6} \]
\[ j_{c.x}=-\frac{v_{c.x}^{2}}{2h_p}, \tag{7} \]
where \(h_p\) is the travel of the conveyance in the unloading curves, m.
On the basis of experimental work carried out on the skip hoist of Mine No. 8 of the Prokopyevskugol Trust (Kuzbass), we recommend that, when the skip moves in the unloading curves (because of the imbalance of the dead weights of the hoisting conveyances and the additional resistances), the static load be taken as 15–50% greater than when moving outside the curves. Thus, we have
\[ \gamma_{\text{st.k}}=(1.15 \div 1.20)\frac{(kQ-pH)D_b}{2M_n i\eta_p}, \tag{8} \]
\[ \gamma_{\text{st.k}}=(1.15 \div 1.20)\frac{(kQ+pH)D_b}{2M_n i\eta_p}, \tag{9} \]
where \(p\) is the weight of one running meter of rope, kg/m;
\(H\) is the full hoisting height, m.
The calculation of the preliminary stages for cage hoisting is also recommended to be made on the basis of the specific conditions of their operation.
All subsequent main stages are calculated according to the law of a geometric progression
\[ R_{z+1}=R_z\varphi, \]
\[ R_{z+2}=R_z\varphi^2, \]
\[ \cdots\cdots\cdots \]
\[ R_n=R_z\varphi^{\,n-z}, \tag{10} \]
where \(R_z\) is the resistance of the last preliminary stage, ohm;
\(\varphi\) is the resistance coefficient, equal to the denominator of the geometric progression;
\(R_n\) is the resistance of the last stage of the rheostat, ohm.
The resistance coefficient is equal to
\[ \varphi=\sqrt[n-z+1]{\frac{R_{\text{rot}}}{R_z}}, \tag{11} \]
where \(R_{\text{rot}}\) is the resistance of one phase of the motor rotor, ohm.
To determine the magnitude of the resistance of the individual sections of the starting rheostat, it is necessary to determine the differences in resistance of adjacent stages, for example,
\[ \Delta R_{z+2}=R_{z+1}-R_{z+2}. \tag{12} \]
The sections of the preliminary stages should be assembled from boxes with Fechral elements, the resistance of which is practically independent of heating temperature. The sections of the main stages may be assembled from boxes with cast-iron elements. With such a selection of resistance boxes, the effect of heating should not affect the operation of the hoisting machine.
For the correct choice of the principal acceleration we specify the utilization of the motor overload capacity
\[ \lambda_1=\frac{\gamma_1}{\gamma_M}, \tag{13} \]
where \(\gamma_1\) is the maximum starting torque of the motor in relative units;
\(\gamma_M\) is the overload capacity of the motor.
The relative slip (the ratio of the calculated slip to the critical slip) for the maximum starting torque will be
\[ \sigma_1=\frac{1\mp\sqrt{1-\lambda_1^2}}{\lambda_1}. \tag{14} \]
Here and below, the minus sign is for operation with slip less than the critical value, and the plus sign for operation with slip greater than the critical value.
The switching torque in relative units is
\[ \gamma_2=\frac{2\sigma_1\varphi\,\gamma_M}{1+\sigma_1^2\varphi^2}. \tag{15} \]
Having determined the limits of the starting torques, we find the value of the principal acceleration during acceleration outside the curves and compare it with the permissible acceleration
\[ j_1=\frac{0.5(\gamma_1+\gamma_2)-\gamma_{st.p}}{\psi}\leq j_{dop}, \tag{16} \]
where \(\gamma_{st.p}\) is the static resisting torque during starting, in relative units.
In this formula the average starting torque is taken as the arithmetic mean of the limits of the starting torque. The static resisting torque is considered constant for the entire starting time and is determined by formula (9) without the correction factor.
Hoisting installations of the mines of Karaganda and Kuzbass are distinguished by a large load capacity and a comparatively small hoisting height; therefore, for the principal deceleration one should use the free-running mode
\[ j_{fr}=\frac{\gamma_{st.stop}}{\psi}\leq j_{dop}, \tag{17} \]
where \(\gamma_{st.stop}\) is the static resisting torque during stopping (it is considered constant and is determined by formula (8) without the correction factor).
If the deceleration value obtained is very small, then one should be guided by braking deceleration; in this case
\[ j_m=\frac{\gamma_{st.stop}+\gamma_m}{\psi}\leq j_{dop}, \tag{18} \]
where \(\gamma_m\) is the braking torque in relative units. Equating the braking and permissible decelerations from formula (18), one can determine the required braking torque.
If, however, the magnitude of the deceleration during free run-out proves to be large, then we orient the calculation toward motor (rheostatic) deceleration; in this case
\[ j_{dv}=\frac{\gamma_{st\cdot ost}-\gamma_{dv}}{\psi}\leq j_{dop}, \tag{19} \]
where \(\gamma_{dv}\) is the average motor torque in relative units. Analogously to the preceding case, the motor torque is determined and, if necessary, a special rheostatic characteristic is calculated.
After selecting the basic acceleration and deceleration, the elements of kinematics and dynamics are calculated by generally known formulas.
If, for any reason, an acceleration has been selected that differs from that determined by formula (16), then the limits of the starting torques must be determined. This problem can be solved analytically from a system of two equations
\[ \gamma_1+\gamma_2=2\left(\gamma_{st\cdot n}+\psi j_1\right), \]
\[ \gamma_2\left(\gamma_M-\sqrt{\gamma_M^2-\gamma_1^2}\right) = \varphi\gamma_1\left(\gamma_M-\sqrt{\gamma_M^2-\gamma_2^2}\right). \tag{20} \]
However, solving this system of equations is difficult, and therefore the limits of the starting torques are more simply found by a graphical method (Fig. 1). For this purpose, any two basic rheostatic characteristics are plotted and the average starting torque is laid off,
\[ \gamma_{p\cdot sr}=\gamma_{st\cdot n}+\psi j_1. \tag{21} \]
Fig. 1. Graphical determination of the limits of the starting torque.
After this, a switching line is constructed, parallel to the torque axis, such that the line of the average starting torque divides the switching line into equal parts. The points of intersection of the switching line with the rheostatic characteristics give the required starting torques.
To construct the mechanical characteristic, the critical slip is determined:
\[ s_k=\frac{s_{k\cdot e}R}{R_{om}} = \frac{R\left(\gamma_M+\sqrt{\gamma_M^2-1}\right)}{R_n}, \tag{22} \]
where \(S_{k.c}\) is the critical slip for the natural characteristic; \(R\) is the resistance of the stage for which the critical slip is determined, ohms.
The mechanical characteristics are constructed according to the formula
\[ \tau=\frac{2\tau_m S S_k}{S^2+S_k^2}. \tag{23} \]
After the driving torques have been determined for all periods of the hoisting cycle, we determine the relative currents
\[ I^*=I_k^*\sqrt{\,1+\sqrt{1-\left(\frac{\tau}{\tau_m}\right)^2}\,}, \tag{24} \]
where
\[ I_k^*=\sqrt{\tau_m\left(\tau_m+\sqrt{\tau_m^2-1}\right)} \]
is the relative value of the current at critical slip.
It should be noted that for the period of main acceleration the current is determined separately for the maximum starting torque and for the switching torque. In the case where the relative torque is less than unity, the relative values of current and torque may be equated.
The preliminarily selected motor is checked for heating:
\[ I_{\text{eff}}^*=\sqrt{\frac{\sum I^{*2}t}{t_{\text{eff}}}}\le 1, \tag{25} \]
where \(t_{\text{eff}}\) is the effective time, sec.
If the effective current is greater than unity, a new motor should be selected, or a smaller acceleration should be used with the selected motor, and all calculations should be repeated.
We recommend determining the effective time from the conditions of normal cooling of the motor during its rotation, while taking into account the deterioration of heat transfer during the pause by an empirically selected coefficient:
\[ t_{\text{eff}}=\frac{H}{v_m}+q t_p, \tag{26} \]
where \(q\) is a coefficient that takes into account the deterioration of heat transfer of an immobile motor; it is recommended to take it equal to \(2/3\) for skip hoists and \(1/3\) for cage hoists.
When a liquid rheostat is used, all calculations are performed much more simply, since it is not necessary to determine the limits of the starting torques.
At each hoisting installation it is necessary to install active-energy meters and counters for the number of hoists. This will make it possible to evaluate correctly the work of the machinists and to determine the efficiency of the hoisting installation. In addition, efforts must be made to ensure that the machinist learns to carry out a hoisting cycle corresponding to the designed speed diagram. For this purpose, from time to time it is necessary to monitor the work of machinists by recording speed and current diagrams with self-recording instruments.
To illustrate the proposed design methodology, let us consider the skip hoist of mine No. 86 of the “Stalingol” trust (Karaganda). The hoisting machine is \(2\times5\times2.3\); the motor is of type AT 18 B 10—20,
720 kW, 295 rpm; a single-stage reduction gear with a gear ratio of 10.5. Total hoisting height 510 m; skip with a sector gate, capacity 6 t; weight per running meter of rope 6.2 kg/m; reduced mass
\[ 7{,}200 \frac{\mathrm{kg}\cdot \mathrm{s}^2}{\mathrm{m}}; \]
the magnetic station is eight-contactor.
The results of the calculations are summarized in Table 1. The graphical determination of the limits of the starting torques is shown in Fig. 1. The effective current is 0.92; the efficiency of the hoisting installation is 0.60.
Table 1
Results of calculation of kinematic and dynamic elements
| Periods of the hoisting cycle | Acceleration, m/s² | Speed, m/s | Time, s | Relative torque, upper limit | Relative torque, lower limit | Relative current, upper limit | Relative current, lower limit |
|---|---|---|---|---|---|---|---|
| Skip departure from curves | 0.3 | 0—1.2 | 4.0 | 1.18 | 1.18 | 1.20 | 1.20 |
| Acceleration outside curves | 0.8 | 1.2—7.35 | 7.6 | 1.85 | 1.31 | 2.10 | 1.35 |
| Uniform travel | — | 7.35 | 58.5 | 0.97 | 0.43 | 0.97 | 0.43 |
| Main deceleration | −0.6 | 7.35—1.0 | 10.2 | — | — | — | — |
| Crawling | 0.25 | 1.0—0 | 4.0 | 0.34 | 0.34 | 0.34 | 0.34 |
In reality the hoist operates with smaller accelerations, as is evident from the speed diagram (Fig. 2). The efficiency of the hoisting installation is 0.47, and the output is 25% lower than possible. Such a reduction in efficiency leads to an excessive expenditure of energy of approximately 0.5 million kWh per year.
Fig. 2. Speed diagram of the skip hoist of mine No. 86.