Principles of Optimization of Parameters and Control Modes of Electric Motors at Variable Current Frequency

Yu. G. Meshcheryakov

Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering

In a number of installations—transport, road-construction, and other autonomous power installations, drives with frequency speed control, etc.—AC electric motors operate under conditions in which the current frequency varies within certain limits. In this case it is important to ensure the effective use of the motors over the entire range of frequency variation.

The technical-economic indicators of standard motors used at variable frequency are obtained by calculation for an unchanged frequency and, in the general case, cannot be optimal for a wide frequency range. As a result, the problem arises of finding the optimal variant of the technical-economic indicators of a motor intended for operation at variable frequency. This problem encompasses a broad range of tasks, many of which at present are only beginning to be considered and require comprehensive study. Among them are the tasks of optimizing the parameters and control modes of motors. In solving these tasks, questions of a methodological nature are of great importance; their study provides a scientific basis for developing new effective optimization methods.

This article sets forth the fundamental features that must be taken into account in the mathematical formulation of the indicated optimization problems. They are as follows.

Since a motor at variable frequency operates within a certain frequency range, optimization problems must be posed in such a way that the values of the motor’s technical-economic indicators are taken into account at each operating frequency of the specified range. The current frequency (or rotational speed) within the specified range may vary episodically or continuously, compulsorily or arbitrarily. The number of operating frequencies may be any: from two to a very large number. The operating time of the motor at individual frequencies may be distributed over the entire frequency range uniformly or according to any other law. Both deterministic and probabilistic cases of time distribution are possible. The voltage at the motor terminals may be set independently of the frequency or be connected with it by a definite dependence. When the frequency changes, the energy indicators, winding overheating, overload capacity, mechanical stresses, degree of saturation of the magnetic circuit, etc., do not remain constant. The magnitude and ratio of the motor’s electromagnetic loads change with frequency depending on the law

frequency control and the character of variation of the torque on the motor shaft. These factors must be taken into account in optimization.

Optimization of the parameters and operating modes of a motor may be considered as a controlled process. In accordance with the theory of optimal processes, optimization problems are formulated on the basis of the following two propositions. First, the effectiveness of optimization is always expressed in the form of some indicator (criterion), by means of which the most preferable variant can be selected. Second, the attainable values of the effectiveness indicator are limited by circumstances that prevent an unlimited increase in effectiveness. Such circumstances are conditions and constraints. For example, it is always required to ensure specified or permissible values of voltage, winding temperature, overload capacity, starting currents, etc.

Thus, the problem consists in finding an optimal control (single-stage or continuous) under which the effectiveness criterion reaches a maximum or minimum under specified conditions and constraints.

Various economic, energy, technical, ergonomic, and other indicators, or their aggregate, may be adopted as the effectiveness criterion. The specific nature of optimizing a controlled motor operating in conjunction with a frequency converter requires a multicriteria formulation of the problem. As a result of optimization, such motor characteristics should be obtained that do not worsen the technical, energy, weight-and-size, and cost indicators of the frequency converter, but contribute to improving the techno-economic indicators of the entire electric-drive installation as a whole. In this case, it is difficult to characterize the effectiveness of the motor characteristics by means of a single indicator. A number of additional indicators having different physical natures are also required. But a comprehensive evaluation simultaneously by several indicators contains an element of subjectivity and, as a rule, is very difficult. The difficulties of solving a multicriteria problem can be partially overcome by reducing it to a problem with a single effectiveness indicator. To this end, one indicator having primary importance is selected as the effectiveness criterion, and all the other indicators are introduced into the system of specified conditions and constraints. In the present case, when optimizing the parameters and control modes of motors, it is expedient to adopt energy or economic indicators (depending on the specific circumstances) as the effectiveness criterion, and to obtain the necessary technical, technological, and other motor characteristics by satisfying the specified conditions and constraints.

For a motor intended to operate at \(N\) frequencies, the effectiveness criterion should be represented as the sum of the cost indicators (the cost of electric-energy losses or total expenses) obtained for each operating current frequency:

\[ \sigma = \sum_{n=1}^{N} S_n, \]

where \(S_n\) is the effectiveness indicator at the \(n\)-th frequency.

The indicator \(S_n\) is a function of the control parameters \(u_1, u_2, \ldots, u_k\), the relative current frequency \(\alpha_n\), and the relative duration of motor operation \(\tau_n\) at frequency \(\alpha_n\):

\[ S_n = S_n(u_1,\ u_2,\ldots,\ u_k;\ \tau_n;\ \alpha_n), \]

where

\[ \alpha_n=\frac{f_n}{f_{\mathrm{n}}}, \quad \tau_n=\frac{t_n}{T}; \]

\(f_n\) is the absolute value of the \(n\)-th operating current frequency;
\(f_{\mathrm{n}}\) is the rated frequency of the motor;
\(t_n\) is the operating time of the motor at the \(n\)-th frequency;
\(T\) is the total operating time of the motor in the specified frequency range:

\[ T=\sum_{n=1}^{N} t_n . \]

The quantities \(\tau_n\) and \(\alpha_n\) are independent variables. In the general case, for adjustable motors (not necessarily at variable frequency), the independent variable may be the rotational speed.

As control parameters one takes parameters by selecting whose values it is possible to improve the efficiency index while keeping overheating, voltage, currents, overload capacity, and other phase coordinates within prescribed limits. (Phase coordinates are the minimum number of parameters by means of which the state of the system under study is characterized.) It is meaningful, using the presence of functional relations, to choose as control parameters those parameters that allow the simplest mathematical solution of the problem to be obtained. For example, it is economically expedient to optimize motor parameters without changing their geometrical dimensions, the quality of the electrical engineering materials, the type of windings, and other structural elements that it is desirable to keep the same as in the corresponding motors of series production. In this case, the optimization is applied to electromagnetic loads, useful power, torque, voltage, currents, and other parameters that may be classed as operational or functional. For asynchronous motors in such a case it is expedient to take, as control parameters, the absolute slip, the torque (and, if necessary, the number of turns in a phase of the stator winding).

When searching for optimal values of the control parameters, it is necessary to take into account the specified law of frequency regulation, the law of variation of the torque, the change in the degree of saturation of the magnetic circuit, the change in overheating of the windings, and other factors that may be specified in the form of constraint equations or in the form of an admissible region of parameter variation.

If one abstracts from the physical content of the optimized parameters and efficiency indices and passes to a mathematical description of the problem in general form, independently of the type of motor and the method of regulating its rotational speed, then the problem of optimizing the parameters and regulation modes of motors may be formulated as the problem of finding a set of optimal values of the control parameters \(u_{1\,\mathrm{opt}}, u_{2\,\mathrm{opt}}, \ldots, u_{K\,\mathrm{opt}}\), maximizing or minimizing the efficiency criterion \(\sigma\), which assumes definite values for any values of the control parameters \(u\) belonging to a specified region \(u\in U\), of the phase coordinates \(y\) belonging to a specified region \(y\in Y\), and of the independent variables \(\tau,\alpha\), under specified constraint equations between \(u,y,\tau,\alpha\):

\[ \sigma=\sum_{n=1}^{N} S_n(u_1,u_2,\ldots,u_K;\tau_n;\alpha_n)\to \mathrm{extr}; \]

\[ u_k\in U \quad \text{for } k=1,2,\ldots,K; \]

\[ y_{ln}=y_{ln}(u_1,u_2,\ldots,u_K;\tau_n;d_n)\in Y \tag{1} \]

for \(l=1,2,\ldots,L,\ n=1,2,\ldots,N\);

\[ \varphi_j(u_1,u_2,\ldots,u_K;y_l;\tau_n;a_n)=0 \]

for \(j=1,2,\ldots,J\),

\(K\) — number of control parameters;
\(L\) — number of phase coordinates;
\(N\) — number of operating frequencies;
\(J\) — number of constraint equations.

The stated problem can be solved by methods of nonlinear programming.

If the number of operating frequencies in the given range is large, one can pass to an integral criterion:

\[ I=\int_{\alpha_1}^{\alpha_2} S(u_1,u_2,\ldots,u_K;\tau(a);\alpha)\,da . \tag{2} \]

The principles considered here for the mathematical formulation of problems of optimizing the parameters of electric motors at variable frequency reflect the general case, when the number of control parameters may be arbitrary. A special case of parameter optimization is the optimization of motor operating modes, when the optimum value of a single control parameter is sought for a given magnitude of the torque. Hence it follows that optimization of the speed-control modes of motors is possible not only by optimizing the law of variation of the parameters \(u(\alpha)\to u_{\mathrm{opt}}(\alpha)\), but also by optimizing their nominal values \(u\to u_{\mathrm{opt}}\). Optimization of the nominal values of the parameters makes it possible to significantly increase the efficiency of using motors at variable frequency.

It is not difficult to show that the problem of optimizing the law of variation of the parameters follows from the general parameter-optimization problem stated here. For steady-state motor operating modes, this problem corresponds to a particular problem of the calculus of variations with a degenerate functional, whose solution in the absence of constraints leads to an objective function of the form

\[ \varepsilon=\varepsilon(u;\alpha). \]

The extremal value of this function is found by methods of differential calculus.

When optimizing the speed-control modes of a motor, it should be borne in mind that, from the standpoint of technical implementation of the optimum mode, the control parameters may have a different physical meaning, since they are chosen from considerations of obtaining the simplest and most reliable control system.

As an example, one may consider optimization of the operating mode of an induction motor in a given frequency range \([\alpha_1,\alpha_2]\) by the integral criterion (2), under the condition that the relative duration of motor operation at individual frequencies is distributed uniformly over the entire frequency range and that the torque \(M\) and the absolute slip \(\beta\) remain constant as the frequency changes. As an efficiency index, the specific annual reduced cost is used, equal to the ratio of the absolute value of the costs to the useful power of the motor. In structure, these costs do not differ from the costs incurred by a motor when it is supplied from an industrial network, and they are calculated according to the methodology generally accepted for electrical machines. According to this methodology, the capital and operating expenses for a motor include the cost of the motor and the losses of electric energy during its operation, as well as expenses associated—

associated with reactive-power compensation. The costs of reactive-power compensation depend to a certain extent on the circuit features of the power supply source (frequency converter). They have a comparatively small effect on the total amount of costs; therefore, in optimization it is advisable to take them into account formally, independently of the method of compensation, according to a generalized procedure. In this case the annual reduced costs may be taken as equal to

\[ Z_{\mathrm{gp}}=k_{\mathrm{p}}\sum P+k_{\mathrm{k}}Q_{\mathrm{k}}, \]

where \(\sum P, Q_{\mathrm{k}}\) are the power losses and the compensated reactive power;

\[ k_{\mathrm{p}} \]
— specific annual costs for covering losses of electrical energy, rubles/kW·year;

\[ k_{\mathrm{k}} \]
— specific annual costs for compensation of reactive power (taken with allowance for the normative coefficient of economic efficiency and annual depreciation charges), rubles/kvar·year.

The efficiency criterion (2), in the present case, for a uniform distribution of the motor operating time over individual frequencies, is expressed by the formula

\[ I=\frac{1}{\alpha_2-\alpha_1}\int_{\alpha_1}^{\alpha_2}\varepsilon\,d\alpha, \tag{3} \]

where

\[ \varepsilon=\frac{Z_{\mathrm{gp}}}{P_2} \]
is the specific annual reduced cost;

\[ P_2 \]
is the useful power of the motor;

\[ \alpha_2,\alpha_1 \]
are the maximum and minimum values of the frequencies in the specified range.

The quantity \(\varepsilon\) is expressed through the parameters \(\beta, M, \alpha\), and also the parameters of the motor equivalent circuit in accordance with [1, 2]. The required quantity is the optimal value of the nominal slip \(\beta_{\mathrm{opt}}\), satisfying the condition for the existence of a minimum of the efficiency criterion (3). For calculating \(\beta_{\mathrm{opt}}\) the following formula has been obtained:

\[ \beta_{\mathrm{opt}}= \sqrt{ \frac{ k_{\mathrm{p}}\left[ h_1\frac{\ln \frac{\alpha_2}{\alpha_1}}{\alpha_2-\alpha_1} +h_3+h_5(\alpha_2+\alpha_1) \right] +k_{\mathrm{k}}h_7\frac{\ln \frac{\alpha_2}{\alpha_1}}{\alpha_2-\alpha_1} }{ k_{\mathrm{p}}\left[ h_2\frac{\ln \frac{\alpha_2}{\alpha_1}}{\alpha_2-\alpha_1} +h_4+h_6(\alpha_2+\alpha_1) \right] +k_{\mathrm{k}}h_8\frac{\ln \frac{\alpha_2}{\alpha_1}}{\alpha_2-\alpha_1} } }, \]

where

\[ h_1=m_1 r_1' c_1 r_2'^2;\qquad h_2=m_1 r_1' c_1 x_{02}^2+m_1 r_2' c_2; \]

\[ h_3=k_{\mathrm{g}} f_{\mathrm{n}} c_3 r_2'^2;\qquad h_4=k_{\mathrm{g}} f_{\mathrm{n}} c_3 x_2'^2; \]

\[ h_5=\frac{1}{2}k_{\mathrm{v}} f_{\mathrm{n}}^2 c_3 r_2'^2;\qquad h_6=\frac{1}{2}k_{\mathrm{v}} f_{\mathrm{n}}^2 c_3 x_2'^2; \]

\[ h_7=m_1 c_1 r_2'^2\left[\alpha_2(x_1+x_0)-r_1'\tan\varphi_0\right]-r_2'^2 h_9; \]

\[ h_8=m_1 c_1 x_{02}^2(\alpha_2 x_1-r_1'\tan\varphi_0) +m_1 c_2(\alpha_2 x_2'-r_2'\tan\varphi_0)+ \]

\[ +x_2'^2(m_1 c_1\alpha_2 x_0-h_9); \]

\[ x_{02}=x_0+x_2';\qquad r_1'=r_1+r_0; \]

\[ h_{\mathrm{e}}=c_{3} f_{\mathrm{H}}\alpha_{2}\operatorname{tg}\varphi_{0}\left(k_{\mathrm{r}}+k_{\mathrm{v}} f_{\mathrm{H}}\alpha_{2}\right); \]

\[ c_{1}=\frac{\omega_{1\mathrm{H}}}{m_{1}r_{2}'x_{0}^{2}};\qquad c_{2}=\frac{\omega_{1\mathrm{H}}}{m_{1}r_{2}'}; \]

\[ c_{3}=\frac{\omega_{1\mathrm{H}}}{\left(c_{\mathrm{E}}f_{\mathrm{H}}\right)^{2}m_{1}r_{2}'};\qquad c_{\mathrm{E}}=4.44\,k_{w1}W_{1}, \]

\(k_{\mathrm{r}}, k_{\mathrm{v}}\) are the coefficients of iron losses due to hysteresis and eddy currents;

\(m_{1}\) is the number of phases;

\(r_{\mathrm{d}}\) is the coefficient of additional losses;

\(\operatorname{tg}\varphi_{0}\) is the tangent of the phase-shift angle that must be ensured in the electrical installation for economic considerations [3, 4];

\(\omega_{1\mathrm{H}}\) is the rated synchronous rotational speed of the magnetic field.

By \(r\) and \(x\) are denoted the active and inductive resistances of the motor equivalent circuit in accordance with [1].

For motor A02-42-4, the transition to optimal regulation with \(\beta=\beta_{\mathrm{opt}}\) in the range \(\alpha_{1}\div\alpha_{2}=0.1\div1\) is accompanied by a reduction in the integral value of annual reduced costs by approximately 10% in comparison with operation of the motor with a non-optimized rated slip.

REFERENCES

1. A. A. Bulgakov. Frequency Control of Asynchronous Electric Motors. Moscow, “Nauka,” 1955.

2. G. G. Meshcheryakov. Increasing the power factor of an asynchronous motor with regulated frequency. Izvestiya vuzov, “Energetika,” 1973, No. 1.

3. I. N. Charakhchyan, A. P. Voskresenskii. A method for the techno-economic assessment and selection of optimal general-purpose asynchronous electric motors. Proceedings of VNIIEM, vol. 7, TsINTI, Moscow, 1959.

4. P. G. Grudinskii, L. V. Litvak. Determination of the optimal power factor of industrial electrical installations. “Elektrichestvo,” 1960, No. 10.