Static Stability of Synchronous Reluctance Machines at Variable Supply Network Frequency

E. V. Kononenko, G. I. Lukyanov

Synchronous reluctance motors (SRM) are used in automated electric drives where a strictly synchronous rotational speed is required. The use of such motors is promising in drives with speed regulation by varying the frequency of the supply network. Therefore, the study of the static stability of synchronous reluctance machines (SRM) at variable supply-network frequency is a highly relevant problem.

Violation of the static stability of an SRM is associated with a change in the rotor rotational speed and constitutes an electromechanical transient process. The behavior of an SRM in this mode at variable supply-network frequency is described by the following system of nonlinear differential equations

\[ \left. \begin{aligned} U_d &= p\psi_d + r i_d - \psi_q(\alpha - p\Theta);\\ U_q &= p\psi_q + r i_q + \psi_d(\alpha - p\Theta);\\ \psi_d &= x_d(p)i_d;\\ \psi_q &= x_q(p)i_q;\\ M &= M_{\mathrm{st}} - Hp^2\Theta, \end{aligned} \right\} \tag{1} \]

where

\[ U_d=-\gamma U_{\mathrm{n}}\sin\Theta;\qquad U_q=\gamma U_{\mathrm{n}}\cos\Theta; \]

\[ x_d(p)=\frac{x_d+p x_d^{\prime\prime}T_d^{\prime\prime}}{1+pT_d^{\prime\prime}};\qquad x_q=\frac{x+p x_q^{\prime\prime}T_q^{\prime\prime}}{1+pT_q^{\prime\prime}}; \]

\[ \gamma=\frac{U}{U_{\mathrm{n}}};\qquad \alpha=\frac{f}{f_{\mathrm{n}}}; \]

\[ M=\psi_d i_q-\psi_q i_d. \]

In deriving equations (1), a system of relative units was used. It was assumed that the rate of change of the supply-network frequency is significantly less than the rate at which mechanical transients proceed (i.e., at any instant of time the condition \(\alpha=\mathrm{const}\) is valid).

All inductive reactances entering equations (1) are determined for the base (rated) frequency of the supply network. The frequency variation is taken into account only by the coefficient \(\alpha\).

One of the main issues in frequency control of an SRM is the choice of the law for regulating the voltage as a function of the supply frequency.

network [1]. In order to ensure the constant overload capacity of a motor operating over a wide range of rotational-speed variation at \(M_{\mathrm{st}}=\mathrm{const}\), it is necessary to find the dependence of \(\gamma\) on the parameters of the SRM and on the coefficient \(\alpha\). This dependence can be obtained from the following considerations.

The maximum electromagnetic torque developed by an SRM in synchronous operation at supply voltage \(U=\gamma U_{\mathrm{n}}\) and frequency \(f=\alpha f_{\mathrm{n}}\) is determined by the expression

\[ M_m = \frac{\gamma^2 U_{\mathrm{n}}^2 (x_d-x_q)} {2(\alpha^2 x_d x_q+r^2)^2} \left[ \pm \sqrt{(\alpha^2 x_d^2+r^2)(\alpha^2 x_q^2+r^2)} - r\alpha (x_d-x_q) \right], \tag{2} \]

where the sign \((+)\) corresponds to motor operation, and the sign \((-)\) to generator operation.

The maximum torque in the nominal mode \(M_{mn}\) is found from expression (2) by substituting \(\gamma=\alpha=1.0\). Then, from the equality \(M_m=M_{mn}\), we determine

\[ \gamma = \frac{1}{U_{\mathrm{n}}} \sqrt{ \frac{ \sqrt{(\alpha^2 x_d^2+r^2)(\alpha^2 x_q^2+r^2)} + r\alpha (x_d-x_q) }{ \sqrt{(x_d^2+r^2)(x_q^2+r^2)} + r(x_d-x_q) } }. \tag{3} \]

Expression (3) shows how it is necessary to regulate the voltage supplied to the SRM when the frequency changes, so that the maximum electromagnetic torque of the motor remains constant and equal to its value in the nominal mode.

In practice, the most widespread method is to vary the voltage in proportion to the frequency. Such an expression is obtained from (3) by substituting \(r=0\). In this case

\[ \gamma=\alpha. \tag{4} \]

Loss of static stability of SRMs operating from a network with variable voltage and frequency, under the adopted assumptions, may be of two kinds.

For certain ratios of parameters and load, an aperiodic loss of stability, or “slipping,” is possible; this characterizes the limit of the machine’s static overload capacity.

In some operating cases, self-excited periodic oscillations of the rotor arise. Such a loss of static stability is known as self-swinging.

In order to determine the capabilities of an SRM under frequency control, and also to design motors that operate stably for a specified frequency variation, it is necessary to investigate the influence of frequency, as well as of various motor parameters, on the occurrence of self-swinging. To solve this problem, it is sufficient to consider stability under small disturbances. Then the analytical investigation of static stability can be carried out on the basis of the linearized equations.

When linearizing the equations, we assume that, as a consequence of small harmonic oscillations of the rotor, all variables (except \(\alpha\) and \(\gamma\)) acquire increments. Therefore, in the original equations (1) it is necessary to make the substitution

\[ \begin{aligned} U_d &= U_{d0}+\Delta U_d; \qquad & U_q &= U_{q0}+\Delta U_q;\\ i_d &= i_{d0}+\Delta i_d; \qquad & i_q &= i_{q0}+\Delta i_q;\\ \psi_d &= \psi_{d0}+\Delta \psi_d; \qquad & \psi_q &= \psi_{q0}+\Delta \psi_q;\\ \Theta &= \Theta_0+\Delta \Theta; \qquad & p\Theta &= p\Delta \Theta, \end{aligned} \tag{5} \]

where quantities with the subscript “0” characterize the initial steady-state condition; quantities with the symbol “\(\Delta\)” are due to small oscillations of the rotor.

Substituting (5) into (1), excluding the terms corresponding to the established operating condition, and neglecting products of increments, the system of equations of the SRM, linearized at the point of the initial operating condition, can be represented in the form

\[ \left. \begin{aligned} &(\gamma U_{\mathrm{n}}\cos\Theta_0+x_q i_{q0}p)\Delta\Theta +[p x_d(p)+r]\Delta i_d-\alpha x_q(p)\Delta i_q=0;\\ &(\gamma U_{\mathrm{n}}\sin\Theta_0-x_d i_{d0}p)\Delta\Theta +[p x_q(p)+r]\Delta i_q+\alpha x_d(p)\Delta i_d=0;\\ &Hp^2\Delta\Theta+[x_d(p)-x_q]i_{q0}\Delta i_d +[x_d-x_q(p)]i_{d0}\Delta i_q=0. \end{aligned} \right\} \tag{6} \]

The stability analysis of the SRM can be carried out by solving equations (6). Forming and equating to zero the determinant of the system of equations (6), after the corresponding transformations we obtain the characteristic equation

\[ a_0p^6+a_1p^5+a_2p^4+a_3p^3+a_4p^2+a_5p+a_6=0, \tag{7} \]

where

\[ a_0=HT_dT_qx_d''x_q''; \]

\[ a_1=H\left[T_qx_q''(T_dr+x_d)+T_dx_d''(T_qr+x_q)\right]; \]

\[ a_2=H\left[(r^2+\alpha^2x_d''x_q'')T_dT_q +r(x_q+x_d'')T_d+r(x_d+x_q'')T_q+x_dx_q\right]+ \]

\[ +Ax_qx_q''T_q i_{q0}^2+Bx_dx_d''T_d i_{d0}^2; \]

\[ a_3=H\left[r^2(T_d+T_q)+r(x_d+x_q)+\alpha^2(x_d''x_qT_d+x_dx_q''T_q)\right]+ \]

\[ +\alpha(Bx_d''x_qT_d-Ax_dx_q''T_q)i_{d0}i_{q0} +Ax_q''T_q\gamma U_{\mathrm{n}}\cos\Theta_0\,i_{q0}- \]

\[ -Bx_d''T_d\gamma U_{\mathrm{n}}\sin\Theta_0\,i_{d0} +\left[A(T_qr+x_q)-(x_q-x_q'')x_q''T_q\right]x_q i_{q0}^2+ \]

\[ +\left[B(T_dr+x_d)+(x_d-x_q)x_d''T_d\right]x_d i_{d0}^2; \]

\[ a_4=H(r^2+\alpha^2x_dx_q)+\alpha Bx_d''T_d\gamma U_{\mathrm{n}}\cos\Theta_0\,i_{d0} +\alpha Ax_q''T_q\gamma U_{\mathrm{n}}\sin\Theta_0\,i_{q0}+ \]

\[ +\alpha\left[(B-A)x_dx_q+(x_d-x_q)(x_d''x_qT_d+x_dx_q''T_q)\right]i_{d0}i_{q0}+ \]

\[ +\left[A(T_qr+x_q)-(x_d-x_q)T_qx_q''\right]\gamma U_{\mathrm{n}}\cos\Theta_0\,i_{q0} +\left[Ar-(x_d-x_q)\times\right. \]

\[ \left.\times(T_qr+x_q)\right]x_q i_{q0}^2 -\left[B(T_dr+x_d)+(x_d-x_q)T_dx_d''\right]\gamma U_{\mathrm{n}}\sin\Theta_0\,i_{d0}+ \]

\[ +\left[Br+(x_d-x_q)(T_dr+x_d)\right]x_d i_{d0}^2; \]

\[ a_5=\alpha\left[Bx_d-(x_d-x_q)x_d''T_d\right]\gamma U_{\mathrm{n}}\cos\Theta_0\,i_{d0} +2\alpha(x_d-x_q)x_dx_q i_{d0}i_{q0}+ \]

\[ +\alpha\left[Ax_q-(x_d-x_q)x_q''T_q\right]\gamma U_{\mathrm{n}}\sin\Theta_0\,i_{q0} +r(x_d-x_q)(x_d i_{d0}^2-x_q i_{q0}^2)+ \]

\[ +\left[Ar-(x_d-x_q)(T_qr+x_q)\right]\gamma U_{\mathrm{n}}\cos\Theta_0\,i_{q0} -\left[Br+(x_d-x_q)\times\right. \]

\[ \left.\times(T_dr+x_d)\right]\gamma U_{\mathrm{n}}\sin\Theta_0\,i_{d0}; \]

\[ a_6=-(x_d-x_q)\gamma U_{\mathrm{n}} \left[\alpha x_d\cos\Theta_0-r\sin\Theta_0\right]i_{d0} -\left(\alpha x_q\sin\Theta_0+r\cos\Theta_0\right)i_{q0}; \]

\[ A=(x_q-x_d'')T_d;\qquad B=(x_d-x_q'')T_q. \]

\[ i_{d0}=\frac{\alpha x_q\cos\Theta_0-r\sin\Theta_0}{\alpha^2x_dx_q+r^2}\,\gamma U_{\mathrm{n}}; \]

\[ \tag{8} \]

\[ i_{q0}=\frac{\alpha x_d\sin\Theta_0+r\cos\Theta_0}{\alpha^2x_dx_q+r^2}\,\gamma U_{\mathrm{n}}. \]

In the preceding equations, the value of the load is determined by the angle \(\Theta_0\). From the physical point of view, it is more correct to determine the load by the ang—

...by the angle between the rotor pole axis and the axis of the magnetomotive force of the stator winding \(\beta_0\) [2]. In the case where stability is calculated for a specified value of the angle \(\beta_0\), in equations (7) and (8) the angle \(\Theta_0\) is determined from the formula

\[ \tan \Theta_0 = \frac{a x_q \sin \beta_0 - r \cos \beta_0} {a x_d \cos \beta_0 + r \sin \beta_0}. \tag{9} \]

The regions of stable and unstable operation of the linearized system of equations can be determined from an analysis of the roots of the characteristic equation. In practice, when determining the boundaries of the region of static stability, it is expedient to use special criteria that make it possible to determine the stability of the system without resorting to computation of the roots of the characteristic equation. In studying an SRM, it is more convenient to use the Routh criterion because of the uniformity of the calculations. In addition, its application makes it possible to calculate the stability boundaries on a digital computer rather simply.

The Routh criterion for equation (7) may be written in the form of Table 1.

Table 1

The system corresponding to the characteristic equation (7) is stable in those cases when the coefficients of the first column of the Routh table are positive. If at least one of these coefficients becomes negative, stable operation of the SRM is lost. Consequently, the boundary of stable operation of the SRM will correspond to those values of the parameters entering equation (7) for which one of the coefficients of the first column of Table 1 becomes zero.

The boundaries of stable operation of an SRM at variable frequency of the supply network are expediently found in the form of the dependence \(M=f(\alpha)\).

The electromagnetic torque in the established synchronous mode is determined by the expression

\[ M=(x_d-x_q)i_{d0}i_{q0}, \tag{10} \]

and the coefficient \(\alpha\) is an independent variable. The coefficient \(\gamma\) is calculated from formula (3).

The calculation of the boundaries of static stability of an SRM is performed in the plane of two parameters, \(\beta_0\) and \(\alpha\), with all other machine parameters being taken as known and constant. For specified values of \(\beta_0\) and \(\alpha\), the coefficients of Table 1 are calculated. Varying \(\alpha\) with \(\beta_0=\mathrm{const}\), the calculations are repeated until one of the coefficients of the first column of the table changes sign. In this case two cases are possible.

The first to pass through zero and become negative is \(a_6\). This indicates that the limiting slip point has been found [2]. If the coefficient \(c_{16}\) is the first to pass through zero, then this characterizes crossing the self-oscillation boundary. It is possible that the coefficient \(c_{15}\) or \(c_{14}\) will be the first to change sign. This means that the step of variation of \(\alpha\) has been chosen too large and must be reduced.

In practical calculations, the slip boundary in the coordinates \(M\) and \(\alpha\) is more simply determined with the aid of equation (2), since the maximum value of the torque characterizes the limit of the machine’s static overload capacity. The self-oscillation boundary is determined by the Routh criterion using the method described above. The calculation begins at the value of the angle \(\beta_0 = 0\), which corresponds to ideal no-load operation of the SRM, while \(\alpha\) is decreased from 1.0 until two points of transition across the self-oscillation boundary are determined (Fig. 1). For the points of transition across the self-oscillation boundary, the electromagnetic torque is determined from equation

Fig. 1. Region of stable and unstable operation of an SRM:
I — region of stable operation; II — region of self-oscillation; III — slip region

(10). By assigning different values of \(\beta_0\), the calculations are repeated. The step of variation of \(\alpha\) and \(\beta\) in the calculations must be sufficiently small so that the region of stable operation is found with sufficient accuracy. All stability calculations whose results are presented in this article were performed on the “Minsk-1” digital computer.

In the case when the SRM parameters are constant, they were taken equal to the following values: \(x_d = 2.5\); \(x_q = 0.5\); \(r = 0.06\); \(T_d = 48\) el. sec.; \(T_q = 12\) el. sec.; \(x_d'' = 0.2\); \(x_q'' = 0.2\); \(H = 100\) el. sec.; \(U_n = 1.0\). The regions of stable and unstable operation of an SRM having such parameters are presented in Fig. 1. It follows from this figure that, when the frequency of the supply network is decreased (decrease of \(\alpha\)), the maxi-

the maximum torque in the motor mode remains constant. In the frequency range corresponding to \(\alpha = 0.17 \div 0.40\), self-oscillation arises. With increasing load, the possibility of self-oscillation

Fig. 2. Boundaries of the region of stable operation of an SRM:
a) for different values of the mechanical constant \(H\);
b) for different values of the active resistance in the stator-winding circuit \(r\)

In (a):
\(1 — H = 50\) el. sec.;
\(2 — H = 100\) el. sec.;
\(3 — H = 150\) el. sec.;
\(4 — H = 150\) el. sec.

In (b):
\(1 — r = 0.03\);
\(2 — r = 0.06\);
\(3 — r = 0.1\);
\(4 — r = 0.2\).

decreases. At values of the shaft torque equal to \(M = (0.75 \div 1.0)M_m\), the SRM operates stably over the entire range of frequency variation from \(f_n\) to zero.

The magnitude of the moment of inertia of the rotating masses has a substantial influence on the boundaries of the self-oscillation region. In Fig. 2,a the boundaries of stable operation are presented for variation of the mechanical constant \(H\). From the curves shown it is evident that, as the moment of inertia of the rotating masses decreases, the self-oscillation region expands and shifts toward higher frequencies.

In an SRM, as in synchronous machines, self-oscillation is caused by the presence of active resistance in the stator-winding circuit. The boundaries of stable operation, calculated for different values of \(r\), are given in Fig. 2,b. It is clear from the figure that an increase in \(r\) substantially expands the frequency range in which self-oscillation occurs. For some values of \(r\), stable operation of the SRM becomes

Fig. 3. Boundaries of stable operation of an SRM for different ratios
\[ \frac{x_d}{x_q} \]
and \(x_d=\mathrm{const}\):

\[ 1 — \frac{x_d}{x_q}=6.0;\qquad 2 — \frac{x_d}{x_q}=5.0;\qquad 3 — \frac{x_d}{x_q}=4.0; \]

\[ 4 — \frac{x_d}{x_q}=3.5 \]

impossible already at the nominal frequency. Consequently, to expand the region of stable operation, an SRM must be designed with the minimum possible value of the active resistance of the stator winding.

The values of the synchronous inductive reactances \(x_d\), \(x_q\), and their ratio have a great influence on the region of stable operation (Figs. 3, 4). As is known, the ratio

\[ \frac{x_d}{x_q} \]

is determined by the rotor design. At present there is a tendency, by improving the rotor design, to create SRMs with an increased ratio-

Fig. 4. Boundaries of stable operation of an SRM for various values of \(x_d\) and a constant ratio

\[ \frac{x_d}{x_q} = 5.0. \]

... by the ratio $\dfrac{x_d}{x_q}$ and, consequently, with improved energy indicators. From Fig. 3 it is seen that, as the ratio $\dfrac{x_d}{x_q}$ increases, the self-oscillation region increases. At $\dfrac{x_d}{x_q}=3.0$ and the adopted parameter values, self-oscillation does not arise. The boundaries of the stability region

Fig. 5. Boundaries of the region of stable operation at different laws of voltage regulation as a function of the supply-network frequency. Solid curves are calculated taking expression (3) into account; dashed curves, taking equality (4) into account.

of operation at a constant ratio $\dfrac{x_d}{x_q}=5.0$ and different values of $x_d$ are shown in Fig. 4. It follows from these curves that, as $x$ decreases, the self-oscillation region expands. The values of the synchronous inductive reactances are inversely proportional to the size of the air gap. Consequently, as the air gap increases, the self-oscillation region expands. From the data considered it may be concluded that changes in the rotor design aimed at increasing the maximum power of the SRM lead to an expansion of the self-oscillation region.

The boundaries of the region of stable operation also depend on the law of voltage regulation as a function of the supply-network frequency, as this...

follows from Fig. 5. The calculations showed that not only the law of voltage variation, but also the value of \(U_{\mathrm{n}}\), have a substantial influence on the region of stable operation of the SRM. To confirm this, Fig. 6 gives the boundaries of the stability region for different values of \(U_{\mathrm{n}}\),

Fig. 6. Boundaries of stable operation, calculated for different \(U_{\mathrm{n}}\) and for voltage varying in direct proportion to frequency.

calculated under the condition that the voltage varies in direct proportion to frequency. Analysis of the influence of the voltage magnitude on the region of stable operation indicates that, by an appropriate change in voltage, stable operation of the synchronous reluctance motor can be ensured over the entire range of frequency regulation.

LITERATURE

1. E. M. Berlin, B. A. Egorov, V. D. Kulik, I. S. Kosyrev: Systems of Frequency Control of Synchronous Reluctance Motors. “Energiya,” 1968.
2. E. V. Kononenko. Static stability of synchronous reluctance machines; “Elektrichestvo,” 1967, No. 10.