Study of the Static Stability of Synchronous Reluctance Machines

E. V. Kononenko

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

In studying the operating modes of synchronous reluctance machines (SRM), special attention must be given to the investigation of static stability. The loss of static stability of SRMs operating from a network with constant voltage and frequency may be of two kinds.

For certain ratios of parameters and load, an aperiodic loss of stability, or pull-out, is possible; this characterizes the limit of the static overload capability of an SRM.

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

The operation of an SRM in the general case is described by a system of nonlinear Park–Gorev differential equations [1]. In investigating static stability, it is necessary to divide the parameter space of these machines into regions corresponding to stable and unstable operation. To solve this problem it is sufficient to consider stability under small disturbances. Then an analytical investigation of the static stability of an SRM can be carried out on the basis of linearized equations, as is customary in the analysis of ordinary synchronous machines [2, 3, 4].

Under generally accepted assumptions and with a constant resisting torque, the system of equations of an SRM having a complete winding on the rotor, linearized at the point of the initial operating mode, may be represented in the form:

\[ \begin{aligned} &(U\cos\theta_0 + x_q i_{q0}p)\Delta\theta + [p x_d(p)+r]\Delta i_d - x_q(p)\Delta i_q = 0;\\ &(U\sin\theta_0 - x_d i_{d0}p)\Delta\theta + x_d(p)\Delta i_d + [p x_q(p)+r]\Delta i_q = 0; \tag{1}\\ &Hp^2\Delta\theta + [x_d(p)-x_q]i_{q0}\Delta i_d + [x_d - x_q(p)]i_{d0}\Delta i_q = 0. \end{aligned} \]

Here the quantities with subscript zero characterize the initial steady-state operating condition; the quantities with the sign \(\Delta\) are due to small oscillations of the rotor, and

\[ x_d(p)=\frac{x_d + p x''_d T_d}{1+pT_d}; \qquad x_q(p)=\frac{x_q + p x''_q T_q}{1+pT_q}. \]

Equations (1) were obtained under the condition that the electromagnetic torque is positive in the motor mode of operation, the \(q\)-axis leads the \(d\)-axis by \(90^\circ\), and a per-unit system is used.

Forming and setting equal to zero the determinant of the system of equations (1), after the corresponding transformations, the characteristic equation can be represented in the following form:

\[ a_0 p^6 + a_1 p^5 + a_2 p^4 + a_3 p^3 + a_4 p^2 + a_5 p + a_6 = 0, \tag{2} \]

where

\[ \begin{aligned} a_0 &= H x_d'' x_q'' T_d T_q,\\ a_1 &= H\left[r\left(x_d''+x_q''\right)T_dT_q+x_d''x_qT_d+x_dx_q''T_q\right];\\ a_2 &= H\left[\left(r^2+x_d''x_q''\right)T_dT_q+r\left(x_d''+x_q\right)T_d \right.\\ &\quad \left. + r\left(x_d+x_q''\right)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\left(T_d+T_q\right)+r\left(x_d+x_q\right)+x_d''x_qT_q+x_dx_q''T_q\right] \\ &\quad +\left(Bx_d''x_qT_d-Ax_dx_q''T_q\right)i_{d0}i_{q0} +Ax_q''T_q U\cos\theta_0\,i_{q0} \\ &\quad -Bx_d''T_d U\sin\theta_0\,i_{d0} +\left[A\left(T_qr+x_q\right)-\left(x_d-x_q\right)x_q''T_q\right]x_q i_{q0}^2 \\ &\quad +\left[B\left(T_dr+x_d\right)+\left(x_d-x_q\right)x_d''T_d\right]x_d i_{d0}^2;\\ a_4 &= H\left(x_dx_q+r^2\right)+Bx_d''T_dU\cos\theta_0\,i_{d0} +Ax_q''T_qU\sin\theta_0\,i_{q0}+;\\ &\quad +\left[(B-A)x_dx_q+\left(x_d-x_q\right)\left(x_d''x_qT_d+x_dx_q''T_q\right)\right]i_{d0}i_{q0} \\ &\quad +\left[A\left(T_qr+x_q\right)-\left(x_d-x_q\right)x_q''T_q\right]U\cos\theta_0\,i_{q0} \\ &\quad +\left[Ar-\left(x_d-x_q\right)\left(T_qr+x_q\right)\right]x_q i_{q0}^2 \\ &\quad -\left[B\left(T_dr+x_d\right)+\left(x_d-x_q\right)x_d''T_d\right]U\sin\theta_0\,i_{d0} \\ &\quad +\left[Br+\left(x_d-x_q\right)\left(T_dr+x_d\right)\right]x_d i_{d0}^2;\\ a_5 &= \left[Bx_d+\left(x_d-x_q\right)x_d''T_d\right]U\cos\theta_0\,i_{d0} +2\left(x_d-x_q\right)x_dx_q i_{q0}i_{d0} \\ &\quad +\left[Ax_q-\left(x_d-x_q\right)x_q''T_q\right]U\sin\theta_0\,i_{q0} +r\left(x_d-x_q\right)\left(x_d i_{d0}^2-x_q i_{q0}^2\right) \\ &\quad +\left[Ar-\left(x_d-x_q\right)\left(T_qr+x_q\right)\right]U\cos\theta_0\,i_{q0} \\ &\quad -\left[Br+\left(x_d-x_q\right)\left(T_dr+x_d\right)\right]U\sin\theta_0\,i_{d0};\\ a_6 &= \left(x_d-x_q\right)U\left[\left(x_d\cos\theta_0-r\sin\theta_0\right)i_{d0} -\left(x_q\sin\theta_0+r\cos\theta_0\right)i_{q0}\right];\\ A &= \left(x_q-x_d''\right)T_d;\qquad B=\left(x_d-x_q''\right)T_q;\\ i_{d0} &= \frac{Ux_q\cos\theta_0-Ur\sin\theta_0}{x_dx_q+r^2};\qquad i_{q0}=\frac{Ux_d\sin\theta_0+Ur\cos\theta_0}{x_dx_q+r^2}. \end{aligned} \]

The regions of stable and unstable operation of the system of equations under consideration may be determined from an analysis of the roots of the characteristic equation.

When crossing the boundary of the stability region, two cases are possible [5].

a) One root may become equal to zero. This is possible if, in equation (2), the constant term becomes equal to zero \((a_6=0)\). In this case, upon leaving the stability region, one positive root appears and aperiodic instability sets in.

b) A pair of roots may become purely imaginary. In this case, when crossing the stability boundary, oscillatory instability arises.

Consequently, the boundary of the pull-out region can be determined from the condition

\[ a_6=0. \tag{3} \]

From equation (2) it is evident that \(a_6\) depends only on the parameters of the steady-state operating mode of the SRM. It is not difficult to establish that \(a_6\) is proportional to the synchronizing torque \(M_{s0}\).

Indeed, the torque developed by the SRM in the synchronous steady-state operating mode is equal to

\[ M_0=\psi_{d0}i_{q0}-\psi_{q0}i_{d0}=\left(x_d-x_q\right)i_{d0}i_{q0}. \tag{4} \]

The synchronizing torque is determined as

\[ M_{so}=\frac{dM_o}{d\theta_o} =\frac{(x_d-x_q)U}{x_d x_q+r^2} \left[(x_d\cos\theta_o-r\sin\theta_o)i_{do} -(x_q\sin\theta_o+r\cos\theta_o)i_{qo}\right]. \tag{5} \]

Comparing (5) with the expression for \(\alpha_6\), we see that

\[ \alpha_6=(x_d x_q+r^2)M_{so}. \tag{6} \]

It follows from equation (6) that \(\alpha_6\) will be equal to zero only in the case when \(M_{so}=0\). This indicates that the boundary of the pull-out region represents the limit of the static overload capability of an SRM.

In practice, when determining the boundary of the region of static stability, it is expedient to use the Hurwitz or Routh criteria, since this eliminates the need to find the roots of the characteristic equation.

In an SRM, as in synchronous machines, self-excitation is caused by the presence of active resistance in the stator-winding circuit \((r)\). In the case when \(r=0\), and the rotor of the SRM has a starting short-circuited winding, the phenomenon of self-excitation cannot arise.

Let us assume that on the rotor there is a short-circuited winding only along the transverse axis. Then the characteristic equation (2) for \(r=0\) can be represented in the form:

\[ (p^2+1)\left\{ H x_d x_q'' T_q p^3 +H x_d x_q p^2 +U^2 T_q\left[(x_d-x_q'')\cos^2\theta_o -(x_d-x_q)\frac{x_q''}{x_q}\sin^2\theta_o\right]p +(x_d-x_q)U^2\cos 2\theta_o \right\}=0. \tag{7} \]

It follows from equation (7) that the characteristic equation in this case has a pair of conjugate purely imaginary roots \((p_{1,2}=\pm j)\). This indicates that, in addition to pull-out and self-excitation, undamped oscillations with synchronous frequency also arise in an SRM when \(r=0\). As calculations have shown, and also the investigations of A. A. Gorev [2], these oscillations have no practical significance because of the negligible magnitude of their amplitude and their rapid damping in any real machine when \(r\ne 0\). However, the presence of these oscillations makes it necessary to carry out all calculations of static stability taking into account the actual values of the active resistances in the stator-winding circuit.

The real parts of the other roots of characteristic equation (7) will be negative if all the coefficients and the Hurwitz determinant \((\Delta_r)\) of the equation enclosed in braces are positive.

In the present case, after the corresponding transformations, the Hurwitz determinant can be represented in the form

\[ \Delta_r=H x_d^2 T_q(x_q-x_q'')U^2\cos^2\theta_o. \tag{8} \]

An analysis of the coefficients of equation (7) and expression (8) shows that, when the load angle \(\theta_o\) varies within the limits from \(0^\circ\) to \(\pm45^\circ\), all coefficients and \(\Delta_r\) are positive. At \(\theta=45^\circ\), the constant term of equation (7) becomes equal to zero, which characterizes the boundary of the pull-out region.

In the case when on the rotor there is a short-circuited winding only along the direct axis, characteristic equation (2) for \(r=0\) will have the form:

\[ (p^2+1)\left\{ H x_d'' x_q T_d p^3 +H x_d x_q p^2 +U^2T_d\left[(x_q-x_d'')\sin^2\theta_o +(x_d-x_q)\frac{x_q''}{x_d}\cos^2\theta_o\right]p +(x_d-x_q)U^2\cos 2\theta_o \right\}=0. \tag{9} \]

Analysis of equation (9) shows that all the coefficients of the expression in braces, when the angle \(\Theta_0\) varies within the limits from \(0^\circ\) to \(\pm 45^\circ\), are positive. At \(\Theta_0=\pm 45^\circ\) the constant term of (9) becomes equal to zero, which characterizes the boundary of the pull-out region. The Hurwitz determinant of the equation under consideration is equal to

\[ \Delta_r' = H x_q'' T_d U^2 (x_d - x_d'') \sin^2 \theta_0 . \tag{10} \]

Within the range of variation of the angle \(\Theta_0\) under consideration, \(\Delta_r' > 0\). At \(\Theta_0=0^\circ\) determinant (10) is equal to zero. This means that the rotor winding along the direct axis, when the synchronous reluctance machine operates at no load, has no effect on static stability.

It follows from the above that, if the active resistance in the stator-winding circuit is neglected, the phenomenon of self-swinging in a synchronous reluctance machine cannot be detected, since the latter is caused by a negative asynchronous torque due to active resistance.

The presence of a short-circuited winding on the rotor of a synchronous reluctance machine expands the region of stable operation. It is not difficult to prove that, in the absence of windings on the rotor, stable operation of a synchronous reluctance machine is impossible.

When determining the boundary of the region of static stability, it is more convenient to use the Routh criterion, because of the uniformity of the computations. In addition, application of this criterion makes it possible to carry out stability calculations on digital computers quite simply.

Fig. 1. Boundaries of static stability of the synchronous reluctance machine for different values of the load: \(\Theta_0=0^\circ\) (curve 1); \(\Theta_0=+20^\circ\) (curve 2); \(\Theta_0=-20^\circ\) (curve 3); \(\Theta_0=+5^\circ\) (curve 4). Solid lines correspond to the boundary between stable operation and self-swinging; dashed lines to the boundary between stable operation and pull-out.

Figure 1 presents curves determining the stability boundaries of the synchronous reluctance machine as a function of the magnitude of the time constant of the rotor winding in the direct-axis circuit, for different load values. The calculation was carried out on the Minsk-1 electronic digital computer for a synchronous reluctance machine having the following parameters: \(x_d=2.33\); \(x_q=0.45\); \(x_d''=0.2\); \(x_q''=0.2\); \(H=100\) electrical seconds; \(T_d=\frac14 T_q\). From the data given it follows that, with increasing \(T_d\), the region of stable operation of the synchronous reluctance machine initially increases sharply and then, having reached a maximum, decreases. The latter is explained by the fact that, as \(T_d\) increases beyond a certain value, the positive asynchronous torque due to the rotor winding decreases. At an infinitely large value of \(T_d\)

(which corresponds to a very small value of the active resistance of the rotor winding), the rotor winding will not exert a damping influence on self-oscillation, and stable operation of the SRM will be impossible.

The regions of stable and unstable operation of the SRM, plotted in the plane of the parameters \(\Theta_0\) and \(r\), are shown in Fig. 2. The dotted curve corresponds to ideal no-load operation and is the boundary between the motor and generator modes of operation of the SRM.

Fig. 2. Regions of stable and unstable operation, calculated for an SRM having the following parameters:
\(x_d = 2.33;\ x_q = 0.45;\ x''_d = 0.2;\ x''_q = 0.2;\ H = 200\) el. sec.; \(T'_d = 40\) el. sec.; \(T'_q = 10\) el. sec.
I — region of stable operation,
II — region of self-oscillation,
III — region of slipping.

REFERENCES

1. E. V. Kononenko, A. L. Kislitsyn, A. F. Fink. Investigation of self-oscillations during operation from a regulated generator, Izvestiya TPI, vol. 132, 1965.
2. A. A. Gorev. Transient Processes of a Synchronous Machine, GEI, 1950.
3. A. I. Vazhnov. Fundamentals of the Theory of Transient Processes of a Synchronous Machine, GEI, 1960.
4. V. A. Venikov, N. D. Anisimova, A. I. Dolginov, D. A. Fedorov. Self-Excitation and Self-Oscillation in Electrical Systems, Vysshaya Shkola Publishing House, 1964.
5. A. A. Feldbaum. Electrical Systems of Automatic Control, Oborongiz, 1957.