Calculation of Functional Parameters of an Induction Motor under Optimization by an Integral Efficiency Indicator in a Specified Frequency Range
Abstract Generated abstract
The paper develops a calculation method for selecting functional parameters of a variable-frequency asynchronous induction motor so as to minimize an integral indicator of specific annual reduced costs over a prescribed frequency range. Using an equivalent-circuit expression that accounts for magnetic saturation, the authors reduce the optimization variables to electromagnetic torque and absolute slip, with stator turns determined subsequently from the terminal voltage, and impose a stator-winding overheating constraint through a Lagrange multiplier formulation. Closed-form relations are derived for computing the optimal torque and slip under constant-torque and constant-slip operation, with heating checked at the limiting frequencies. An example for an A02-42-4 motor operating from 5 to 50 Hz gives optimal values of 26 N m for torque and 0.021 for absolute slip, after which standard technical and economic motor indicators can be calculated.
Full Text
CALCULATION OF FUNCTIONAL PARAMETERS OF AN ASYNCHRONOUS ELECTRIC MOTOR UNDER OPTIMIZATION BY AN INTEGRAL EFFICIENCY INDICATOR IN A SPECIFIED FREQUENCY RANGE
Yu. G. Meshcheryakov, E. V. Kononenko
Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering
In the development of motors intended for operation at variable frequency, the problem arises of optimizing their parameters. Here a method is considered for calculating the optimal parameters of an asynchronous motor for the geometric dimensions, winding type, quality of materials, and cooling conditions corresponding to motors of the A02 series. The variant corresponding to the minimum of annual reduced costs [1] is considered optimal. As the sought quantities it is advisable to take the functional parameters—the electromagnetic torque \(M\) and the absolute slip \(\beta\), as well as the number of turns in a phase of the stator winding \(W_1\). In order to reduce the number of variables, the number of turns \(W_1\) may be excluded from their set, calculating it after finding the optimal values of \(M\) and \(\beta\), proceeding from the specified value of the voltage at the motor terminals. In this case it is required to determine the optimal values of the parameters \(M\) and \(\beta\), minimizing the integral value of the specific annual reduced costs [2], which is expressed through the parameters of the motor equivalent circuit in accordance with [3], with allowance for saturation of the magnetic circuit according to [4], as follows:
\[ I_1=\frac{1}{\alpha_2-\alpha_1}\int_{\alpha_1}^{\alpha_2}\varepsilon(M,\beta,\alpha)\,d\alpha , \tag{1} \]
where
\[ \varepsilon(M,\beta,\alpha)=\frac{C+E_{\mathrm{n}}K}{M\omega} = \frac{ b_1\left(\dfrac{k_{\mathrm{p}}' r_1}{\alpha}+b_3\right) }{ \dfrac{H\beta}{b_2}-M } + \]
\[ +\beta\left(\frac{b_4}{\alpha}+b_5\right) +\frac{b_6}{\beta}\left(k_{\mathrm{r}}+k_{\mathrm{v}}f_{\mathrm{n}}\alpha\right) +\left(\frac{k'}{\omega_{1\mathrm{n}}\alpha}+b_7\alpha\right)\frac{1}{M} -b_8; \]
\[ b_1=\frac{m_1Gk_i}{\omega_{1\mathrm{n}}}; \qquad b_2=\frac{m_1r_2'}{c_M^2\omega_{1\mathrm{n}}}; \qquad b_3=k_k'\left(\frac{x_{0\mathrm{n}}}{k_1}+x_1\right); \]
\[ b_4=k_{\mathrm{p}}'\left(\frac{r_1k_1}{r_2'}+1\right); \qquad b_5=\frac{k_k'}{r_2'}(x_1k_1+x_2'); \]
\[ b_6=\frac{k_{\mathrm{p}}'b_2f_{\mathrm{n}}}{\omega_{1\mathrm{n}}}; \qquad b_7=\frac{k_{\mathrm{p}}'k_M}{\omega_{1\mathrm{n}}}; \qquad b_8=k_k'\tan\varphi_0. \]
$C$—annual operating costs; $E_{\mathrm{n}}$—standard coefficient of economic efficiency; $K$—one-time capital investments; $M$—electromagnetic torque; $\omega$—rotor rotational speed; $m_1$—number of phases; $\omega_{1\mathrm{n}}$—nominal synchronous rotational speed of the magnetic field; $f_{\mathrm{n}}$—nominal frequency; $k_1=1.09$; $G$ and $H$—coefficients obtained in approximating the magnetization curve: $G=1.68 I_{\mathrm{on}}^2$; $H=2.51\Phi_{\mathrm{n}}^2$; $I_{\mathrm{on}}$ and $\Phi_{\mathrm{n}}$—nominal values of the magnetizing current and magnetic flux; $\operatorname{tg}\varphi_0=0.484$; $k_{\mathrm{M}}$—coefficient of mechanical losses; $k_{\mathrm{g}}$ and $k_{\mathrm{v}}$—coefficients of losses due to hysteresis and eddy currents; $\alpha=\dfrac{f}{f_{\mathrm{n}}}$—current frequency;
\[ c_{\mathrm{M}}=\frac{1}{\sqrt{2}}pm_1 k_{w1} W_1; \]
$p$—number of pole pairs;
\[ \overline{k}_{\mathrm{p}}'=s_{\mathrm{e}}t-k'\operatorname{tg}\varphi_0; \]
$s_{\mathrm{e}}$—specific expenditures to cover electricity losses; $t$—motor operating time during the year;
\[ k_{\mathrm{k}}'=(E_{\mathrm{k}}+E_{\mathrm{n}})k_{\mathrm{uk}}; \]
$E_{\mathrm{k}}$—annual depreciation rate for the compensating device; $k_{\mathrm{uk}}$—specific capital investments in the compensating device; by $r$ and $x$ are denoted the parameters of the motor equivalent circuit in accordance with [3]. The problem is solved under the condition $M=\mathrm{const}$, $\beta=\mathrm{const}$.
The minimum of function (1), in the absence of constraints and with a specified permissible overheating of the stator windings, is found by solving the problem for a conditional extremum:
\[ \frac{\partial H_1}{\partial \beta}=0;\qquad \frac{\partial H_1}{\partial M}=0;\qquad \Theta=\Theta_{\mathrm{d}}, \tag{2} \]
where
\[ H_1=I_1+\lambda_1(\Theta_{\mathrm{d}}-\Theta), \]
$\lambda_1$—Lagrange multiplier; $\Theta_{\mathrm{d}}$—permissible magnitude of overheating of the stator winding; $\Theta$—overheating of the stator winding, expressed as a function of $M$, $\beta$, $\alpha$ for motors of the A02 series by the simplified formula:
\[ \Theta=\frac{P_{\mathrm{g}}}{S_0\left(W_0+k_0\sqrt{\alpha}\right)}; \]
\[ P_{\mathrm{g}}= \frac{m_1 r_1 k_1 G}{\dfrac{H\beta}{b_2M}-1} + \left[ \left(\frac{r_1 k_1}{r_2'}+1\right)\omega_{1\mathrm{n}}\beta + k_{\mathrm{ca}} f_{\mathrm{n}} b_2\frac{\alpha}{\beta} \right]M + k_{\mathrm{M}}\alpha^2, \]
$P_{\mathrm{g}}$—losses causing heating of the motor; $S_0$—surface area of the active part [5]; $w_0$—heat-transfer coefficient in still air; $k_0$—coefficient of blowing intensity [6]. It is sufficient to calculate the overheating $\Theta$ for the values $\alpha=\alpha_1$ and $\alpha=\alpha_2$, where $\alpha_2$ and $\alpha_1$ are the maximum and minimum frequencies in the specified range.
The optimal values of $M$ and $\beta$ are determined as a result of solving the system of equations (2) and are calculated by means of the following formulas:
\[ \frac{ \left(a_1-\dfrac{a_2}{\beta^2}-\dfrac{a_3}{\varphi_1}\right) \left[ \dfrac{g_1}{\varphi_1} - \dfrac{g_4}{\left(\varphi_2-\sqrt{\varphi_2^2-\varphi_3}\right)^2} \right] \left(\varphi_2-\sqrt{\varphi_2^2-\varphi_3}\right) }{ \left[ \left(\dfrac{a_3}{\varphi_1}+a_1\right)\beta+\dfrac{a_2}{\beta} \right] \left( g_2-\dfrac{g_3}{\beta^2}-\dfrac{g_5}{\varphi_1} \right) } =1; \tag{3} \]
\[ M=\varphi_2-\sqrt{\varphi_2^2-\varphi_3}, \tag{4} \]
where
\[ a_1=\omega_{1\mathrm{n}}\left(\frac{r_1 k_1}{r'_2}+1\right);\qquad a_2=k_{c a} f_{\mathrm{n}} b_2 \alpha; \]
\[ a_3=m_1 r_1 k_1 G\,\frac{H}{b_2};\qquad a_4=\frac{a_1 H}{b_2}; \]
\[ a_5=k_M a^2-\frac{a_2 H}{b_2}-\frac{a_3 b_2}{H} -\Theta_\partial S_0\left(W_0+k_0\sqrt{a}\right); \]
\[ a_6=\frac{H}{b_2}\left[\Theta_\partial S_0\left(W_0+k_0\sqrt{a}-k_M a^2\right)\right]; \]
\[ \varphi_1=\left(\frac{H\beta}{b_2}-\varphi_2+\sqrt{\varphi_2^2-\varphi_3}\right)^2;\qquad \varphi_2=\frac{(a_4\beta^2-a_5)\beta}{2(a_1\beta^2+a_2)}; \]
\[ \varphi_3=\frac{a_6\beta^2}{a_1\beta^2+a_2};\qquad g_1=b_1\left[\frac{\ln\frac{\alpha_2}{\alpha_1}}{(\alpha_2-\alpha_1)}\,k'_{\mathrm{p}} r_1+b_3\right]; \]
\[ g_2=b_5+\frac{\ln\frac{\alpha_2}{\alpha_1}}{(\alpha_2-\alpha_1)}\,b_4;\qquad g_5=\frac{g_1 H}{b_2}; \]
\[ g_3=b_6\left[\frac{(\alpha_2^2-\alpha_1^2)}{2(\alpha_2-\alpha_1)}\,k_{\mathrm{B}} f_{\mathrm{n}}+k_{\mathrm{r}}\right]; \]
\[ g_4=\frac{k'\ln\frac{\alpha_2}{\alpha_1}}{\omega_{1\mathrm{n}}(\alpha_2-\alpha_1)} +\frac{b_7(\alpha_2^2-\alpha_1^2)}{2(\alpha_2-\alpha_1)}. \]
The optimal value of the slip \(\beta\) is found from equation (3). The optimal value of the torque \(M\) is calculated by formula (4). For the A02-42-4 motor in the frequency range 5–50 Hz, with \(M=\mathrm{const}\) and \(\beta=\mathrm{const}\), the following optimal values are obtained: \(M=26\) N·m; \(\beta=0.021\). If it is not known which of the frequencies \(\alpha_1\) or \(\alpha_2\) is critical with respect to heating, then the calculation by (3), (4) should be carried out for \(\alpha_1\) and \(\alpha_2\), and the optimal variant with the smaller torque value should be selected, which is necessary to ensure permissible overheating. From the values found for the torque and absolute slip, all the technical-economic indicators of the motor can be determined in the usual way [1, 3, 4, 5]: power, currents, overload capacity, overheating, efficiency, and other quantities.
REFERENCES
-
I. N. Charakhch’yan, A. P. Voskresenskii, A Method for the Technical-Economic Evaluation and Selection of Optimal General-Purpose Asynchronous Electric Motors. Proceedings of VNIIEM, vol. 7, Moscow, TsINTI, 1959.
-
Yu. G. Meshcheryakov. Principles of optimization of parameters and control modes of electric drives with variable-frequency current. Present collection.
-
A. A. Bulgakov. Frequency Control of Asynchronous Electric Motors. Moscow, “Nauka,” 1955.
-
E. V. Kononenko, Yu. G. Meshcheryakov. Laws of economical frequency control for an asynchronous motor with a saturated magnetic circuit. Electrical Engineering Industry, series “Electrical Machines,” issue 9(31), Moscow, 1973.
-
D. I. Sannikov, V. A. Zhadan. Calculation of stator-winding overheating in the design of an optimal asynchronous motor. Izvestiya TPI, vol. 212, Tomsk, TSU Press, 1971.
-
A. E. Alekseev. Design of Electrical Machines. Leningrad, GEI, 1958.