1. Introduction
Short–term voltage instability is caused by fast-acting load components that tend to restore power consumption in the order of several seconds after a voltage drop induced by a contingency [
1]. Induction motor (IM) is one of typical such components [
2,
3]. To drive a constant mechanical power, the active power of an induction motor restores after a voltage drop. If the disturbance is such that the electrical power cannot overcome the mechanical load, the motor stalls, and thus gives rise to a further voltage drop and even a voltage collapse.
For the problem of short-term voltage instability, researchers have studied its essence from different levels and have drawn some important conclusions. In [
4], the authors explain the large-disturbance voltage instability using the
Q-
V curves, which present the network characteristics of the system to the load bus and generic dynamic load characteristics during the transient period. The basic theory in [
4] is such that reactive power balancing between the network and load is the essence that drives the states. In [
5], a conceptual understanding of the issues involved in the voltage stability for fast response loads has been provided through the use of “transient” system
P-
V curves. The author considers voltage instability can be attributed to the fact that the most stringent load is automatically adjusted to maintain a constant active power characteristic. Thus, active power balancing driving states are employed. Furthermore, using the “transient” system
P-
V curves, [
6] investigates the actions of control mechanisms that are intended to maintain constant voltage at the load supply point, such as load tap changer (LTC), distribution voltage regulators, thereby rendering any load constant power. Motivated by [
7], in which the authors propose an analytical method using the transient
P-
V curves of the targeted load bus, an improved analytical method for short-term voltage stability applying to disperse power generation is proposed [
8,
9,
10]. In these literatures, active power balance affected by voltage determines the motion state of the rotor speed, and thus determines the stability boundary. There are also some references that provide active control methods to control such short-term voltage instability [
11,
12,
13,
14,
15]. In [
11], an improved real-time, short-term voltage stability monitoring method is introduced, and a phase rectification method to eliminate the negative influence of oscillation is proposed. With the large integration of photovoltaic (PV) power generation systems into power systems, reactive power control using the inverters of PV systems is used to improve the short-term voltage stability in power systems [
12,
13]. In [
14,
15], researchers respectively apply an emergency-demand response based under speed load shedding scheme and a demand-response-based distributed preventive control to improve short-term voltage stability.
Based on different research perspectives, the conclusions of short-term voltage instability are quite different. In terms of the nature, the above researches are nothing more than studying voltage instability separately from the perspective of active power balancing or reactive power balancing. Actually, the balance of active power and the balance of reactive power are both necessary conditions for stable system operation. Moreover, the active power and the reactive power have mutual coupling effects in the actual system. Thus, whether the voltage is stable or not depends on the system’s active and reactive power balancing. However, to the best knowledge of the authors, very few studies have investigated the short-term voltage instability from the viewpoint of integrated active and reactive power balancing. Consequently, it is of vital importance to study the short-term voltage instability, encompassing active and reactive power balancing to universally explain the physical property behind this phenomenon.
Important insights into the mechanism of short-term voltage instability can be obtained through the use of a simplified model, incorporating only the elements that are dominant in controlling these mechanisms. Therefore, this paper analyzes the short-term voltage stability in a single-IM infinite-bus system. Active power balancing and reactive power balancing are respectively discussed based on this system. Besides, in order to facilitate application, a determination method based on active and reactive power joint balancing is presented to determine the short-term voltage stability. The rest of this paper is structured as follows.
Section 2 introduces the reactive power balancing, active power balancing, and a method used for determining the movement direction of operating point. In
Section 3, the constant slip curve in the
Q-
V plane is introduced for transient duration analysis.
Section 4 and
Section 5 elaborate the physical mechanisms of IM stalling and abnormally low voltage, and discuss the critical clearing voltage in the
Q-
V plane; relevant simulation studies are carried out to validate the correctness of the analyses.
Section 6 summarizes the conclusions.
2. Active and Reactive Power Joint Balancing
This section firstly introduces the reactive power balancing based on a single-IM infinite-bus system. From the viewpoint of active power balancing between mechanical power and air-gap power, this section also shows the acceleration and deceleration region of rotor in the Q-V plane. Finally, the integrated active power and reactive power balancing-based determination method is presented for deciding the movement direction of the operating point.
2.1. Reactive Power Balancing
Consider an infinite bus supplying power to an induction motor load through a long transmission line, shown in
Figure 1. The IM steady-state equivalent circuit is shown in
Figure 1a. Given an infinite bus and transmission line reactance, the power
Pe + jQe supplied by the network in
Figure 1 is
Since
xm is usually considerably larger than
xs, the IM steady-state equivalent circuit shown in
Figure 1a can be simplified to the system shown in
Figure 1b. Thus the power
Pe* +
jQe* demanded by the IM can be described by
Eliminating
θut from Equations (1) and (2) gives
Similarly, eliminating
sslip from Equations (3) and (4) obtains
In Equation (6), the minus sign corresponds to the low slip (high-speed) operating point, while the plus sign corresponds to the high slip (low-speed) operating point.
It can be seen from Equations (5) and (6) that the characteristics of the reactive power with respect to the terminal voltage magnitude Ut on the supply side are completely different from the demand side when transmitting a given active power. Note that Qe represents the reactive power features of the network supply, while Qe* represents the reactive power features of the IM demand. Although Equations (5) and (6) present different reactive power features, the terminal voltage magnitude Ut will be fixed when there is a reactive power balance between the IM demand and network supply.
Figure 2 plots
Qe and
Qe* with respect to terminal voltage magnitude. The system parameters are shown in the
Appendix A. As can be seen from the figure, the intersection points of the two kinds of reactive power curves are the operation points, and the right points are the stable operating points. That is, in the neighborhood of the stable operating point, the terminal voltage magnitude decreases when
Qe* > Qe and increases when
Qe* < Qe. In other words, when the IM-required reactive power is larger than the supplied reactive power, the unbalanced reactive power drives a decrease in terminal voltage magnitude. Conversely, if the IM-required reactive power is smaller than the supplied reactive power, the unbalanced reactive power will increase the terminal voltage magnitude. Therefore, we can determine the dynamic behavior of the terminal voltage magnitude subjected to the unbalanced reactive power.
2.2. Active Power Balancing—the Acceleration and Deceleration Region of Rotor in the Q-V Plane
The rotor motion in IM reflects the balance between mechanical power and air-gap power
where
Pe* represents the air-gap power that is the numeric equivalent of the IM-demanded active power [
16], and
Pm* is the mechanical power.
From Equation (7), rotor motion is the result of unbalanced active power. Driven by the unbalanced active power, rotor accelerates when Pm < Pe* and decelerates when Pm > Pe*.
Here, we firstly assume the mechanical power
Pm = 0.4 pu.
Figure 3 shows this active power balance. Note that constant stator rotating frequency is considered in this paper. Thus, the slip varies according to the rotor speed. In
sslip1, as an example, there exists a unique terminal voltage
Ut2 at which the remainder of the power is zero. If the grid-connected IM is subjected to a disturbance in the grid side, rotor speed remains constant due to the rotor inertia, while terminal voltage will be abruptly changed. From
Figure 3, it is obvious that under the constant point
sslip1, the remainder of the power is positive when the operating voltage is larger than
Ut2, while the remainder of power is negative when the operating voltage is smaller than
Ut2. Thus, driven by the unbalanced power, rotor will accelerate or decelerate.
For the range where
sslip varies from 0 to 1, we can calculate the terminal voltage which balances the mechanical power and air-gap power by Equations (3) and (7). The corresponding IM-required reactive power can also be calculated by Equation (6). With the data of IM-required reactive power and terminal voltage, the regions of rotor acceleration and deceleration where
Pm is the constant power can be painted in the
Q-
V plane, shown in
Figure 4.
The acceleration and deceleration regions of rotor in
Figure 4 can be understood in the following way. Given an operating voltage greater than
Ut2, for example
Ut3 in
Figure 3, it is clear that there are two intersections between air-gap power and mechanical power. When the slip lies between these two intersections, the remainder of power is positive and the rotor motion speeds up. Corresponding to
Figure 4, there are also two intersections when the operating voltage is
Ut3, and the reactive power between the two intersections corresponds to the positive remainder of power. Therefore, the acceleration and deceleration regions of rotor in the
Q-
V plane can be judged.
From
Figure 4, the red line splits the
Q-
V plane into the acceleration region of rotor and the deceleration region of rotor. If the operating point is located in the deceleration (acceleration) region of rotor, the remainder of active power is negative (positive) and drives the rotor to decelerate (accelerate). The operating point is stable only when it is located on the red line.
Similarly, the acceleration and deceleration regions of rotor in the
Q-
V plane where
Pm is modeled as a quadratic mechanical power characteristic can also be obtained, which is shown in
Figure 5. Equation (8) is the detailed expression of
Pm, where
P0 = 0.4.
With the acceleration and deceleration regions of rotor in the Q-V plane, it is clearly to judge the stability of the operating point and its direction of movement.
Comparing
Figure 4 with
Figure 5, different forms of mechanical power lead to the different acceleration and deceleration regions of rotor in the
Q-
V plane. The different regions of rotor acceleration and deceleration reflect the difference in motor operating characteristics, caused by different mechanical power. Besides, it should be noted that the acceleration and deceleration regions of rotor in the
Q-
V plane reflect the characteristics of the motor itself and are independent of the external grid environment.
With the acceleration and deceleration regions of rotor in Q-V plane, the operating voltage, which is determined by reactive power balance, also needs to be tested in the meantime to check whether its active power is balanced or not. Overall, the balance of active power and reactive power should be co-considered in large-disturbance voltage stability analysis.
2.3. Determination Method Based on Active and Reactive Power Joint Balancing
Based on the
Section 2.1 and 2.2, a determination method, which takes the balancing of active and reactive power into consideration separately, is presented in the following
Driven by unbalanced reactive power, terminal voltage Ut decreases when Qe* > Qe and increases when Qe* < Qe.
Driven by unbalanced active power, rotor accelerates when Pe* > Pm and decelerates when Pe* < Pm.
With the determination method, the movement direction of the operating point in the Q-V plane can be visually determined, and then it is possible to conduct an in-depth investigation into the instability mechanisms.
Two types of grid-connected IM load operating conditions after a disturbance, namely, abnormally low voltage and stalling, will be discussed in the following content. In a practical power system, there is typically low-voltage load-shedding protection installed on IM loads to disconnect the motor if the voltage falls below a given threshold. However, in order to analyze the mechanism behind this instability phenomenon, such low-voltage load-shedding protection will not be considered in this work. Similarly, protection that can avoid IM stalling is also not included in the following analysis.
3. Constant Slip Curve in the Q-V Plane in Transient
When the grid-connected IM is subjected to a disturbance in the grid side, rotor speed remains constant instantaneously due to the rotor inertia. At this moment, the operating point will move along the constant slip (constant rotor speed) curve. Therefore, before the mechanism analysis of stalling and abnormally low voltage, this section will introduce the constant slip (constant rotor speed) curve in the Q-V plane for transient duration analysis.
Given an active power, the reactive power characteristic of IM with respect to terminal voltage can be plotted by Equation (6).
Figure 6a shows this characteristic. The
sslip_m is obtained by the restriction in Equation (6) that
Pe* and
Ut should satisfy the following relationship
The solution to Equation (3), under the Constraint (9) = 0, yields
Given a slip, the reactive power characteristics of IM with respect to terminal voltage can also be plotted according to Equation (4). This characteristic is shown in
Figure 6b.
By integrating the above two figures together, there is
Figure 7. It reflects the change of slip on the
Q-
V characteristic curve. The black curve of
sslip =
sslip_m in
Figure 7 is the boundary line to these
Q-
V curves. Considering an active power of
Pe* = 0.4 pu, with the decrease of terminal voltage, the slip changes from 0 to
sslip_m along with the lower part of the
Q-
V curve. The direction of slip change is indicated by the arrow on the lower part of the curve in
Figure 7. Correspondingly, along the upper part of the
Q-
V curve, the terminal voltage increases and the slip changes from
sslip_m to 1. The direction of slip change is also indicated by the arrow on the upper part of the curve in
Figure 7.
4. IM Stalling after a Large Disturbance
The process in which the rotor speed of an IM decelerates to a complete stop is referred to as stalling. Since the quadratic mechanical power always intersects the electrical power at a stable operating point, we can assume a constant mechanical power of 0.4 pu in this case. Similar to the work described in [
16,
17], the mechanisms causing IM stalling for the system shown in
Figure 1 have been summarized as ST1 and ST2. However, here we will illustrate these mechanisms from a reactive power balancing standpoint, in terms of the balancing between air-gap power and mechanical power.
Similar to ST1 and ST2, IM stalling can be summarized as: (1) no operating points of intersection between the reactive power demand and supply, and no balance between the mechanical power and air-gap power; and (2) a lack of attraction for the operating points towards stable postfault reactive power equilibrium.
The large disturbance in this paper is set to the infinite voltage falling from 1.0 pu to 0.8 pu. The fault recovery is indicated as the infinite voltage recovering from 0.8 pu to 1.0 pu.
(1) No operating points of intersection between the reactive power demand and supply, and no balance between the mechanical power and air-gap power
Figure 8 provides the
Q-
V curves of IM-required and network supply in the normal and disturbance status. The
Q-
V curves in the normal status represent the
Q-
V curves before and after disturbance. In
Figure 8a, the infinite voltage is 1.0 pu, and there are two intersections (A
1 and B) between IM-demanded reactive power
Qe*_1 and network-supplied reactive power
Qe_1. Obviously, point A
1 is a stable operating point, while point B is an unstable operating point.
When the infinite voltage magnitude is greatly disturbed by −0.2 pu, the rotor speed cannot be abrupt due to the rotor inertia. At this moment, the characteristics of the IM are characterized by constant impedance, so the active power, reactive power, and terminal voltage are instantly affected by the disturbed voltage distribution. These are reflected as from
Figure 8b that under the distribution of disturbed infinite voltage, the
Q-
V characteristic of the IM is changed from
Qe*_1 to
Qe*_2, and the
Q-
V characteristic of the network is converted from
Qe_1 to
Qe_2. The stable intersection of these new curves is point A
2. In general, the total dynamic process of the moment is that the operating point changes from point A
1 to A
2 along the curve of constant rotor speed. As shown in
Figure 8b, point A
2 is located in the region of rotor deceleration, that is, at this voltage, the air-gap power is less than the mechanical power. Driven by the unbalanced active power, the operating point does not stay at point A
2 and moves in the direction of decreasing rotor speed.
Under the infinite voltage of 0.8 pu, the system still needs to provide mechanical power with 0.4 pu of active power. Therefore, as shown in
Figure 8c, the
Q-
V characteristic of the network changes from
Qe_2 to
Qe_3, and the
Q-
V characteristic required by the motor is recovered upward from
Qe*_2 to
Qe*_1. Since there is no intersection between the curves
Qe_3 and
Qe*_1, the terminal voltage magnitude
Ut will continuously drop, and then the motor eventually stalls.
(2) A lack of attraction for the operating point towards stable postfault reactive power equilibrium
Taking the terminal voltage
Ut_B as the boundary, as shown in
Figure 9a, if the fault is cleared at the point A
4, the rotor speed cannot be abruptly changed due to the rotor inertia. At the moment of fault removal, the
Q-V characteristic curve of the motor jumps from
Qe*_4 to
Qe*_1, and the
Q-
V characteristic curve of the network jumps from
Qe_4 to
Qe_1. Simultaneously, the operating point changes from point A
4 to B
4 along the constant rotor speed curve. Since the reactive power required by the motor at point B
4 is smaller than the reactive power supplied by the network, the terminal voltage rises. Moreover, the point B
4 is located in the region of rotor acceleration, that is, at this voltage, the air-gap power is greater than the mechanical power, and the rotor accelerates. As both the rotor speed and terminal voltage increase, the operating point will eventually return to the predisturbance steady-state operating point A
1.
If the fault is cleared at point A
5, as shown in
Figure 9b, similar to the analysis shown in above, the operating point will jump from point A
5 to B
5 along the curve of constant rotor speed. Since the reactive power required by the motor corresponding to point B
5 is greater than the reactive power supplied by the network, the terminal voltage decreases. Besides, the point B
5 is located in the region of rotor deceleration, that is, at this voltage, the air-gap power is less than the mechanical power, and the rotor decelerates. Both the rotor speed and terminal voltage continues to decrease, eventually causing the motor to stall.
Time-domain simulation results are shown in
Figure 10. A −0.2 pu step down disturbance is imposed at 2 s at the infinite voltage magnitude. Firstly, the system operates at the terminal voltage corresponding to point A
1. After the disturbance, the rotor speed and terminal voltage magnitude drop. Without clearing the fault, the fault remains on and the IM stalls. If the fault is cleared slightly larger than
Ut_B, the rotor speed and the terminal voltage magnitude restore to the initial value. If, on the other hand, the fault is cleared below
Ut_B, the rotor speed decelerates to stall. The simulation results have highly accordance with the analytical results above.