HEATING AND LIMITS OF FREQUENCY CONTROL OF AN INDUCTION MOTOR OPERATING WITH MINIMUM ANNUALIZED COSTS

E. V. Kononenko, Yu. G. Meshcheryakov

When regulating the frequency and voltage at the terminals of an induction motor, it is possible to ensure its operation in a mode having the best economic indicators. This is accomplished by an economically justified choice of the operating mode.

As a result of studies carried out at the Tomsk Polytechnic Institute, it has been established that the most economical mode of operation of an induction motor corresponds to the following law of variation of absolute slip, obtained from the condition of ensuring a minimum of annualized costs (see the appendix):

\[ \beta = \sqrt{ \frac{ k_{\mathrm{p}}\left(a_3+a_4 k_{c\alpha}\right) + k_{\mathrm{k}}\left[\alpha c_2-\left(a_3+a_4 k_{c\alpha}\right)\tg\varphi_0\right] }{ k_{\mathrm{p}}\left(1+a_1+a_2 k_{c\alpha}\right) + k_{\mathrm{k}}\left[\alpha c_1-\left(a_1+a_2 k_{c\alpha}\right)\tg\varphi_0\right] } }, \tag{1} \]

where

\[ \beta=\frac{\omega_1-\omega_2}{\omega_{1\mathrm{n}}}, \qquad \omega_1=\frac{2\pi f_1}{p} \]
— the synchronous rotational speed of the magnetic field;

\[ \omega_2 \]
— the rotational speed of the rotor;

\[ \alpha=\frac{\omega_1}{\omega_{1\mathrm{n}}} \]
— the relative frequency of the stator current;

\[ a_1=\frac{\left(r_0^2+x_{02}^2\right)k_r}{r_2' z_0^2}; \qquad a_2=\frac{x_2'^2}{c^2 E_{1\mathrm{n}}^2 m_1 r_2'}; \qquad a_3=\frac{r_2' k_r}{z_0^2}; \]

\[ a_4=\frac{r_2'}{c^2 f_{1\mathrm{n}}^2 m_1}; \qquad k_r=r_1+\frac{k_0}{m_1}; \qquad k_{c\alpha}=k_r+k_{\mathrm{v}} f_{1\mathrm{n}}\alpha; \]

\[ c_1=\frac{x_2'}{r_2'} \left[ \frac{x_0 x_2'}{z_0^2} + \frac{\left(r_0^2+x_{02}^2\right)x_1}{z_0^2 x_2'} +1 \right]; \qquad c_2=\frac{r_2'(x_0+x_1)}{z_0^2}; \]

\(x_1, x_2', x_0, r_0, r_1, r_2'\) — parameters of the T-shaped equivalent circuit of the induction motor;

\[ z_0^2=x_0^2+r_0^2; \qquad x_{02}=x_0+x_2'; \qquad m_1 \]
— the number of phases;

\(k\) — coefficient of additional losses;

\(k_{\mathrm{r}}\) and \(k_{\mathrm{v}}\) are the loss coefficients in steel for hysteresis and eddy currents;

\(c_E\) is a constant including the number of turns and the winding coefficient;

\(k_{\mathrm{p}}\) and \(k_{\mathrm{k}}\) are quantities equal to the annualized costs for 1 kW of power losses, rubles/year·kW, and for compensation of 1 kvar of reactive power, rubles/year·kvar.

\[ k_{\mathrm{p}}=s_{\mathrm{e}}t;\qquad k_{\mathrm{k}}=s_{\mathrm{k}}t-(E_{\mathrm{k}}+E_{\mathrm{n}})k_{\mathrm{uk}}+p_{\mathrm{uk}}s_{\mathrm{e}}t; \]

\(s_{\mathrm{e}}\) is the specific cost of covering energy losses, rubles/kWh;

\(t\) is the operating time of the motor per year, h/year;

\(s_{\mathrm{k}}\) is the specific cost of compensation of reactive energy, rubles/kvar·h;

\(E_{\mathrm{k}}\) is the annual depreciation rate for the compensating device, \(1/\)year;

\(E_{\mathrm{n}}\) is the normative coefficient of economic efficiency, \(1/\)year;

\(k_{\mathrm{uk}}\) is the specific capital investment in compensating devices, rubles/kvar;

\(p_{\mathrm{uk}}\) is the specific active losses for compensation, kW/kvar.

To make possible the use of mode (1), it is necessary to have an idea of the thermal state of the motor, according to which the limits of control are established and its power is selected. This problem is solved for enclosed fan-cooled motors of the AO2 series.

According to [1], the temperature rise of the stator winding is

\[ \theta=\frac{P_{\mathrm{r}}}{S_0 W}, \tag{2} \]

where

\(P_{\mathrm{r}}\) are the losses causing heating of the motor;

\(S_0=\pi D_j l+\frac{1}{2}\pi D_j^2\) is the surface area of the active part;

\(D_j\) and \(l\) are the outside diameter and length of the stator;

\(W\) is the conditional heat-transfer coefficient, equal to the density of the heat flux through unit surface area \(S_0\), referred to one degree of the average overheating of the stator winding.

For calculating \(W\), a simplified formula is used, obtained on the basis of [1, 2] with experimental verification in the speed range

\[ \nu=\frac{n}{n_{\mathrm{n}}}=0\div 2: \]

\[ W=W_0+k_\theta\sqrt{\nu}. \tag{3} \]

Here \(W_0\) is the heat-transfer coefficient in still air [3]. The quantity \(k_\theta\) is calculated from the condition of equality of \(W\) and the motor heat-transfer coefficient according to [1], \(\alpha_0\), in the range \(\nu=0.7\div 2\) for several values of speed \(\nu_i\):

\[ k_\theta=\frac{1}{k}\sum_{i=1}^{k}\frac{\alpha_0-W_0}{\sqrt{\nu_i}}. \]

The necessary data for the calculation are taken from [1, 2].

Direct use of the methods [1, 2] for calculating overheating gives accurate results in the range \(\nu=0.7\div 2\), but a noticeable decrease-

the reduction in calculation accuracy for low rotational speeds and the complexity of the mathematical expression for \(\theta\) make it necessary to obtain a simplified formula that is valid over the entire range \(v=0\div 2\). Use of formula (3) makes it possible, with satisfactory approximation, to determine the overheating and to calculate the limits of speed control without the use of special computing facilities.

The losses \(P_{\mathrm{g}}\) are taken as equal to

\[ P_{\mathrm{g}}= P_{\mathrm{e1n}} i_1^2+ P_{\mathrm{e2n}} i_2'^2+ P_{\mathrm{gn}} a\varphi^2+ P_{\mathrm{vn}} a^2\varphi^2+ P_{\mathrm{dn}} i_1^2+ P_{\mathrm{mn}} v^2 , \tag{4} \]

where

\[ P_{\mathrm{e1n}},\quad P_{\mathrm{e2n}} \]

are the rated losses in the stator and rotor windings;

\[ P_{\mathrm{gn}}\ \text{and}\ P_{\mathrm{vn}} \]

are the rated losses in the steel due to hysteresis and eddy currents;

\[ P_{\mathrm{dn}}\ \text{and}\ P_{\mathrm{mn}} \]

are the rated additional and mechanical losses;

\[ i_1=\frac{I_1}{I_{1\mathrm{n}}};\qquad i_2'=\frac{I_2'}{I_{2\mathrm{n}}'};\qquad \varphi=\frac{\Phi}{\Phi_{\mathrm{n}}} \]

are the multiples of the currents in the stator and rotor windings and of the magnetic flux.

The currents and magnetic flux are written in terms of the shaft torque \(\mu \dfrac{M}{M_{\mathrm{n}}}\), the absolute slip \(\beta\), and the parameters of the equivalent circuit:

\[ i_1^2=\mu k_{1\mathrm{n}} \left(\frac{x_{0\mathrm{n}}}{x_0}\right)^2 \left(\frac{r_2'^2}{\beta}+x_{02}^2\beta\right); \tag{5} \]

\[ k_{1\mathrm{n}}= \frac{1}{ \dfrac{r_2'^2}{\beta_{\mathrm{n}}}+ \left(x_{0\mathrm{n}}+x_2'\right)^2\beta_{\mathrm{n}} }; \]

\[ i_2'^2=\frac{\mu\beta}{\beta_{\mathrm{n}}}; \tag{6} \]

\[ \varphi^2=\mu k_{\mathrm{sn}} \left(\frac{r_2'^2}{\beta}+x_2'^2\beta\right); \tag{7} \]

\[ k_{\mathrm{sn}}= \frac{1}{ \dfrac{r_2'^2}{\beta_{\mathrm{n}}} +x_2'^2\beta_{\mathrm{n}} }, \]

\[ \beta_{\mathrm{n}} \]

is the rated slip;

\[ x_{0\mathrm{n}} \]

is the rated mutual-inductance reactance.

Taking (3)–(7) into account, the overheating formula (2) is written in the form

\[ \theta= \frac{ \mu k_{1\mathrm{n}} \left(\dfrac{x_{0\mathrm{n}}}{x_0}\right)^2 \left(P_{\mathrm{e1n}}+P_{\mathrm{dn}}\right) \left(\dfrac{r_2'^2}{\beta}+x_{02}^2\beta\right) + P_{\mathrm{e2n}}\dfrac{\mu\beta}{\beta_{\mathrm{n}}} + }{S_0\left(W_0+k_\theta\sqrt{v}\right)} \]

\[ \to \frac{ + k_{\mathrm{sn}}\left(P_{\mathrm{gn}}+P_{\mathrm{vn}}a\right)a\mu \left(\dfrac{r_2'^2}{\beta}+x_2'^2\beta\right) + P_{\mathrm{mn}}v^2 }{} . \tag{8} \]

According to formula (8), the overheating can be calculated for any operating mode of the motor. For this it is sufficient to substitute for \(\beta\) the value corresponding to the adopted control law. Substitution of function (1) into (8) will give the overheating formula for the mode with minimum costs.

Calculation by formulas (1) and (8) for low rotational speeds is accompanied by an error that increases as the speed decreases. For the speed range permissible with respect to heating, the state of the parameters of AO2-series motors of the third and fourth frame sizes, operating in the mode with minimum costs, was analyzed. The principal cause affecting the accuracy of the calculation is saturation of the motor magnetic circuit along the path of the mutual-induction flux, which increases as the speed decreases.

Saturation is taken into account by the nonlinear dependence of the magnetizing current \(i_0=\dfrac{I_0}{I_{0\mathrm{n}}}\) on the magnetic flux \(\varphi=\dfrac{\Phi}{\Phi_{\mathrm{n}}}\), in accordance with the magnetization curve of the asynchronous machine

\[ i_0^2=\frac{g_k}{\dfrac{h_k}{\varphi^2}-1}. \tag{9} \]

The coefficients \(g_k\) and \(h_k\) are 1.68 and 2.51 for the motors indicated above.

Taking (9) into account, the following expression was obtained for the law of variation of the slip, satisfying the condition of motor operation with minimum annual reduced costs:

\[ \beta=e_1+k_{\mathrm{e}1}\mu+(e_2-k_{\mathrm{e}1}\mu)\sqrt{\frac{\nu}{2}}; \tag{10} \]

\[ e_1= \sqrt{ \frac{ g_k k_{\mathrm{cn}} r_2^2 I_{0\mathrm{n}}^2 }{ \dfrac{h_k}{\beta_{\mathrm{n}}} \left( I_{2\mathrm{n}}^2+ \dfrac{P_{\mathrm{e}2\mathrm{n}} I_{1\mathrm{n}}^2}{P_{\mathrm{e}1\mathrm{n}}+P_{\mathrm{dn}}} \right) } }; \]

\[ k_{\mathrm{e}1}=\frac{k_{\mathrm{cn}}r_2^2}{h_k}; \qquad k'=k_{\mathrm{r}}+2k_{\mathrm{B}}t_{\mathrm{1n}}; \]

\[ e_2= \sqrt{ \frac{ k_{\mathrm{p}}(a_3+2a_4k')+ k_k\left[2c_2-(a_3+2a_4k')\operatorname{tg}\varphi_{\mathrm{n}}\right] }{ k_{\mathrm{p}}(1+a_1+2a_2k')+ k_k\left[2c_1-(a_1+2a_2k')\operatorname{tg}\varphi_{\mathrm{n}}\right] } }\cdot e_1 . \]

The optimal value of the absolute slip of the saturated motor (10) depends on the shaft torque, the rotor rotational speed, the design parameters of the motor, and economic factors (the costs of compensation of reactive energy and of energy losses, the annual operating time, the optimal \(\operatorname{tg}\varphi_{\mathrm{n}}\), etc.).

The overheating of the stator winding in the mode with minimum costs is calculated by substituting formula (10) into (8). In formula (8), the change in \(x_0\) is taken into account with the aid of the formula following from (9),

\[ x_0=\frac{c_E f_{1\mathrm{n}}\Phi_{\mathrm{n}}}{I_{0\mathrm{n}}} \sqrt{\frac{h_k-\varphi^2}{g_k}} . \]

The graph of the dependence of stator-winding overheating on rotor rotational speed for the AO2-42-4 motor with nominal shaft torque, calculated by formulas (8) and (10), is given in Fig. 1.

Accounting for the change in \(x_0\) excessively complicates the formula for calculating the limits of speed control in the mode with minimum costs. For slip values corresponding to (10), it may be assumed

in (8) \(x_0=\mathrm{const}\). On this basis, from (8), taking (10) into account, an approximate formula was obtained for calculating the limiting speed values for a given \(\theta\) and \(\mu=\mathrm{const}\) under control with minimum costs:

\[ \nu_{\mathrm{p}}=(d_1\pm\sqrt{d_1^2-d_2})^2, \tag{11} \]

\[ d_1= \frac{ \dfrac{1}{2}\theta S_0(W_0c_{1\mathrm{e}}+kc_{0\mathrm{e}})-\mu c_{0\mathrm{e}}c_{1\mathrm{e}}a' }{ \mu c_{1\mathrm{e}}^2a' + \mu r_2'^2P_{\mathrm{sn}}k_{\mathrm{sn}}+P_{\mathrm{mn}}(c_{0\mathrm{e}}+c_{1\mathrm{e}})-\theta S_0k_\theta c_{1\mathrm{e}} }; \]

\[ d_2= \frac{ \mu r_2'^2(P_{\mathrm{e1n}}+P_{\mathrm{dn}})k_{1\mathrm{n}}+\mu c_{0\mathrm{e}}^2a'-\theta S_0W_0c_{0\mathrm{e}} }{ \mu c_{1\mathrm{e}}^2a' + \mu r_2'^2P_{\mathrm{sn}}k_{\mathrm{sn}}+P_{\mathrm{mn}}(c_{0\mathrm{e}}+c_{1\mathrm{e}})-\theta S_0k_\theta c_{1\mathrm{e}} }; \]

\[ c_{0\mathrm{e}}=e_1+k_{\mathrm{e1}}\mu;\qquad c_{1\mathrm{e}}=\frac{e_2-k_{\mathrm{e1}}\mu}{\sqrt{2}}; \]

\[ a'=x_{02}^2(P_{\mathrm{e1n}}+P_{\mathrm{dn}})k_{1\mathrm{n}}+\frac{P_{\mathrm{e2n}}}{\beta_{\mathrm{n}}}. \]

The lower control limit corresponds to the smaller value of \(\nu_{\mathrm{p}}\) in (11). The upper limiting speed value is obtained greater than the synchronous speed of a two-pole machine; therefore it need not be calculated.

Formula (11) gives speed values reduced by 15–25% when calculating the lower control limit. More accurately, \(\nu_{\mathrm{p}}\) can be determined by calculating the overheating for speeds close to those computed from (11).

To compare law (10) with the frequency-control laws most widely used in practice, a calculation of overheating and limiting speed values was made for the mode \(\beta=\mathrm{const}\), which coincides at \(\mu=\mathrm{const}\) with the mode \(\varphi=\mathrm{const}\). The overheating was calculated by formula (8) at \(\beta=\beta_{\mathrm{n}}\), \(\mu=1\), and the graph \(\theta(\nu)\) for motor AO2-42-4 is shown in Fig. 1.

Fig. 1. Overheating as a function of rotational speed in modes with nominal slip (curve 1) and with minimum costs (curve 2)

The quantity \(\nu_{\mathrm{p}}\) for \(\beta=\mathrm{const}\), \(\mu=\mathrm{const}\) is calculated by the formula

\[ \nu_{\mathrm{p}}=(b_1\pm\sqrt{b_1^2-b_2})^2; \]

\[ b_1= \frac{ \dfrac{1}{2}\theta S_0k_\theta }{ \mu r_2'^2P_{\mathrm{sn}}k_{\mathrm{sn}}\dfrac{1}{\beta}+P_{\mathrm{mn}} }; \]

\[ b_2= \frac{ \mu r_2'^2k_{1\mathrm{n}}(P_{\mathrm{e1n}}+P_{\mathrm{dn}})\dfrac{1}{\beta}+\mu\beta a'-\theta S_0W_0 }{ \mu r_2'^2P_{\mathrm{sn}}k_{\mathrm{sn}}\dfrac{1}{\beta}+P_{\mathrm{mn}} }. \]

According to Fig. 1, the regime with minimum costs is characterized by lower overheating than the regime \(\beta=\beta_{\mathrm{n}}\), and by broader limits of speed control. For the AO2-42-4 motor at \(\mu=1\), the limiting values of \(\nu_{\mathrm{p}}\) for the maximum permissible overheating \(\Theta=80^\circ\) are 0.24 in the regime \(\beta=\beta_{\mathrm{n}}\) and 0.11 in the regime with minimum costs. The difference between the indicated values of \(\nu_{\mathrm{p}}\) arises due to the smaller magnitude of losses in regime (10). The upper speed limit in both regimes is practically unrestricted.

This comparison confirms the technical and economic advantages of regime (10), especially in the case of deep downward speed control from the rated speed, when the use of regime (10) not only reduces the total costs but also makes it possible to reduce the installed power of the motor.

Conclusion

The studies carried out indicate the advisability of using regime (10) for an induction motor with regulated frequency. The use of the indicated regime makes it possible to improve the economic performance of the motor while simultaneously expanding the limits of speed control.

Appendix

The economy of the motor is determined by the value of the annualized costs [4, 5]

\[ Z=C+E_{\mathrm{n}}K, \tag{12} \]

where

\(Z\) — annualized costs, rubles/year;
\(C\) — annual operating expenses;
\(K\) — one-time capital investments;
\(E_{\mathrm{n}}\) — normative coefficient of economic efficiency.

For an induction motor, the formula for annualized costs (12) takes the form:

\[ Z=S_{\mathrm{e}}t\left(\sum P+p_{\mathrm{uk}}Q_{\mathrm{k}}\right)+(E_{\mathrm{k}}+E_{\mathrm{n}})K_{\mathrm{k}}, \tag{13} \]

where

\(K_{\mathrm{k}}=K_{\mathrm{uk}}Q_{\mathrm{k}}\) — capital costs for the compensating device;

\(Q_{\mathrm{k}}=Q_1-P_1\tg \varphi_{\mathrm{p}}\) — power of the compensating device;

\(Q_1\) and \(P_1\) — reactive and active powers consumed by the motor;

\(\sum P\) — power losses in the motor; the remaining designations correspond to those given for formula (1).

Investigation of the minimum of function (12), expressed through slip \(\beta\), torque \(\mu\), and frequency \(\alpha\), leads to the derivation of formula (1).

When the motor is controlled in accordance with (1), costs (13) are reduced due to maintaining, over the entire control range, the optimal ratio between the values of losses and the reactive power consumed by the motor. Owing to this, a noticeable ...

reduction of costs in comparison with the known laws of frequency control, such as \(\gamma=\alpha\sqrt{\mu}\); \(\beta=\beta_{\mathrm{n}}\); \(\Phi=\Phi_{\mathrm{n}}\); \(\sum P=\sum P_{\min}\); \(I_1=I_{\min}\), etc.

REFERENCES

1. D. I. Sannikov, V. A. Zhadan. Calculation of overheating of the stator winding in designing an optimal induction motor. Izvestiya TPI, vol. 212, 1971.

2. V. A. Zhadan, Yu. V. Kopylov, D. I. Sannikov. A method for thermal calculation of enclosed low-power induction motors. Izvestiya TPI, vol. 212, 1971.

3. A. E. Alekseev. Design of Electrical Machines. GEI, 1958.

4. Basic methodological principles for techno-economic calculations in power engineering. Elektrichestvo, 1959, No. 10.

5. A. Ya. Berger. Economic Issues in the Design of Electrical Machines. Moscow, “Vysshaya Shkola,” 1967.