A Grouping and Aggregation Modeling Method of Induction Motors for Transient Voltage Stability Analysis

Abstract

Induction motors are the most common type of motor in power systems, constituting approximately 70–90% of the dynamic loads, making them significant contributors to system dynamics. In transient voltage stability analysis, dynamic equivalent models are commonly used to simplify the representation of a group of induction motors. This paper presents a method for the grouping and aggregation of induction motors at a common bus. Firstly, grouping rules are provided for clustering induction motors into several subgroups based on the mechanical principles of rotor force and motion, and aggregation rules are provided for aggregating a motor subgroup into a single-unit model based on the relationship between voltage drop and power transmission in distribution networks. Secondly, guided by the grouping rules, high-speed remaining electromagnetic torque and low-speed remaining electromagnetic torque are defined as new clustering indicators, and an adaptive K-means clustering method using silhouette coefficient verification is introduced to obtain the optimal motor subgroups. Thirdly, guided by the aggregation rules, a dynamic equivalent method is further introduced to obtain the equivalent single-unit model from a motor subgroup. Lastly, a transient voltage stability simulation in a typical distribution network is presented to illustrate that the proposed clustering and equivalent methods are more reasonable, accurate, and effective than traditional methods, as the obtained model has better dynamic characteristics and can more accurately reproduce the process of voltage collapse.

1. Introduction

Power system stability simulation techniques are crucial for ensuring the secure operation of power systems today. Accurate dynamic representation of electric loads is essential for correct power system stability simulations [1,2]. For example, following the major power outage in North America in 1996, researchers found that the constant current model failed to accurately reproduce the voltage collapse waveform unless the induction motor model and static load model were combined [1,3,4]. Therefore, the accuracy of transient voltage stability calculations is directly affected by the precision of dynamic load models [5,6,7,8,9,10].

The purpose of dynamic load modeling research is to develop dynamic load models with performance characteristics of both sufficient accuracy and simple usability for power system simulations. In actual power systems, induction motors are the predominant dynamic loads, accounting for 70% to 90% of the total active power supplied by the power sources. Load modeling researchers refer to the collection of all induction motors (a total of N units) in a distribution system as an induction motor group. They use a component-based approach [9,11] to aggregate the induction motor group into a dynamic equivalent model with the correct structure and accurate parameters, aiming to reduce the model order and improve the accuracy of transient stability calculations.

The basic concept of the component-based approach is to treat the load as a set of various electrical devices. These electrical devices are first classified according to their typical mathematical models or characteristics, then the proportions of each type are statistically analyzed, and finally, a series of equivalent calculation formulas are used to derive an aggregated model that represents the actual devices. In the 1980s, the Electric Power Research Institute (EPRI) and General Electric Company (GE) conducted in-depth research on the aggregation of electrical devices such as induction motors [12,13,14,15,16,17,18]. The implementation steps are as follows: (1) survey and collect technical data of each device, including their types, characteristics, and parameters, and calculate the proportions of static and dynamic loads; (2) aggregate the characteristics of electrical devices of the same type to derive equivalent load characteristic parameters; and (3) convert these equivalent parameters into aggregated load models that are compatible with computer simulation software.

Early studies on the component-based approach employed a single-unit induction motor model to represent a group of induction motors [16,17,19,20]. These approaches had some drawbacks: the dynamic load model’s structure was overly simplified, resulting in “distortion” that could not accurately describe the actual induction motor group’s dynamic behavior. This could lead to overly conservative or risky voltage stability simulation results [21,22,23]. In subsequent studies [19,24,25,26,27,28,29,30,31,32], the subdivision and equivalent modeling of induction motor groups were further researched. Motors on the same bus may exhibit significant differences in dynamic characteristics such as mechanical motion, active power, and reactive power. To better preserve these differences’ impact on bus voltage recovery, some studies [1,32,33] proposed clustering induction motors into several sub-groups before they are aggregated into equivalent models. Current grouping methods for induction motors use various indicators: (1) damping ratio and attenuation rate [26,34,35,36]; (2) critical slip [25]; and (3) impedance parameters, inertia time constant, and slip at a steady state [28,37].

After dividing the induction motor group into several sub-groups, all motors in a sub-group need to be equivalently modeled as a single motor. Currently, weighted averaging methods are mainly used to aggregate induction motor sub-groups. These methods use different weighting factors: (1) rated power weighting factor [16,17,34]; (2) power weighting factor adjusted by motor locked-rotor reactance and open-circuit time constant [27]; and (3) power weighting factor adjusted by load factor and critical slip [29,38].

To achieve higher modeling accuracy, the above methods still need improvement [39]:

(1)

For the grouping methods of induction motors, during the entire voltage recovery process, individuals within a sub-group should exhibit similar patterns of resultant torque (electromagnetic torque and mechanical load torque) changes. However, the studies [26,34,35] considered local linearization near the actual operating point without accounting for the integral effect of resultant torque over time. The studies [25,37] focused only on electromagnetic torque parameters, neglecting mechanical torque parameters.

(2)

For the aggregation methods of each induction motor sub-group, the single-unit models should reflect the dynamic processes of active power, reactive power, and voltage exhibited by the sub-groups during voltage recovery. However, rhe studies [16,17] used weighting factors based on rated power rather than the active power under actual operating conditions. The studies [27,29,38] used weighting factors based on the power adjusted by load factors under actual operating conditions but still lacked consideration of the relationships among active power transmission, reactive power transmission, and voltage reduction.

To address the aforementioned shortcomings, this paper aims to improve the methods for grouping and aggregating induction motors:

(1)

It first defines and uses high-speed remaining torque and low-speed remaining torque as indicators to measure the similarity of rotor acceleration and deceleration processes. An adaptive K-means clustering method is introduced to identify sub-groups of induction motors, which facilitates a more reasonable reduction in the order of the dynamic model while ensuring the simplicity and usability of the load model.

(2)

It proposes using equal total power, equal total torque, and equal rotor stored kinetic energy as aggregation rules. An improved single-unit equivalent method for each induction motor sub-group is introduced, which enhances the accuracy of the reduced-order model and helps to prevent overly conservative or risky transient voltage stability simulation results.

(3)

Through transient voltage stability simulations in a typical distribution network, it is demonstrated that the induction motor model obtained using the proposed methods exhibits dynamic characteristics that are closer to those of the actual induction motor group compared to similar methods. Additionally, this confirms that the proposed methods more accurately reproduce the dynamic processes of transient voltage collapse.

2. Grouping and Aggregation Rules for Induction Motor Groups

2.1. Model and Parameters of an Induction Motor

2.1.1. Parameters of an Induction Motor Model

Dynamic loads primarily consist of a large number of induction motors. The T-shaped equivalent circuit of an induction motor is shown in Figure 1:

In Figure 1, $R_s$ represents the stator winding resistance and $R_r$ represents the rotor winding resistance. S denotes the rotor slip. $X_s={\omega }_0{L}_{0s}$ denotes the stator leakage reactance; $X_r={\omega }_0{L}_{0r}$ denotes the rotor leakage reactance; and $X_m={\omega }_0{L}_{ms}$ denotes the magnetizing reactance. Additionally, ${\omega }_0$ is the synchronous angular velocity, and $L_{0s}$, $L_{0r}$, and $L_{ms}$ are the stator leakage inductance, rotor leakage inductance, and magnetizing inductance, respectively. When using the dq-axis stator currents $i_{sd},i_{sq}$, rotor currents $i_{rd},i_{rq}$, and slip S as state variables, the T-shaped equivalent circuit in Figure 1 can be expressed as an electromagnetic transient model in the synchronous $dq$ coordinates:

$$\left\{\begin{array}{cc}\hfill & \left(L_{ss}{\displaystyle \frac{d{i}_{sd}}{dt}}+L_m{\displaystyle \frac{d{i}_{rd}}{dt}}\right)=-R_s{i}_{sd}+{\omega }_0\left(L_{ss}{i}_{sq}+L_m{i}_{rq}\right)+u_{sd}\hfill \\ \hfill & \left(L_{ss}{\displaystyle \frac{d{i}_{sq}}{dt}}+L_m{\displaystyle \frac{d{i}_{rq}}{dt}}\right)=-R_s{i}_{sq}-{\omega }_0\left(L_{ss}{i}_{sd}+L_m{i}_{rd}\right)+u_{sq}\hfill \\ \hfill & \left(L_m{\displaystyle \frac{d{i}_{sd}}{dt}}+L_{rr}{\displaystyle \frac{d{i}_{rd}}{dt}}\right)=-R_r{i}_{rd}+{\omega }_{0}S\left(L_m{i}_{sq}+L_{rr}{i}_{rq}\right)\hfill \\ \hfill & \left(L_m{\displaystyle \frac{d{i}_{sq}}{dt}}+L_{rr}{\displaystyle \frac{d{i}_{rq}}{dt}}\right)=-R_r{i}_{rq}-{\omega }_{0}S\left(L_m{i}_{sd}+L_{rr}{i}_{rd}\right)\hfill \end{array}\right.$$

(1)

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dS}{dt}}& =-{\displaystyle \frac{1}{J{\omega }_0}}\left(T_e-T_m\right)=-{\displaystyle \frac{{\omega }_0}{2H{P}_N}}\left(T_e-T_m\right)\hfill \\ \hfill T_e& =L_m\left(i_{sq}{i}_{rd}-i_{sd}{i}_{rq}\right)\hfill \\ \hfill T_m& =T_{m0}\left[\left(1-a\right){\left(1-S\right)}^2+a\right]\hfill \end{array}\right.$$

(2)

In Equation (1), $L_{ss}=L_{0s}+L_m$ denotes the stator self-inductance; $L_{rr}=L_{0r}+L_m$ denotes the rotor self-inductance; and $L_m={\displaystyle \frac{3}{2}}{L}_{ms}$ denotes the mutual inductance in the dq-axis. In Equation (2), $T_e$ is the electromagnetic torque, $T_m$ is the mechanical torque, J is the rotor’s moment of inertia, H is the rotor’s inertia time constant, $P_N$ is the rated power, and $T_{m0}$ and a are mechanical load coefficients.

2.1.2. Parameters of the Induction Motor Group

We assume that all induction motors connected to a power supply bus in an actual distribution network form a motor group. The parameters for each induction motor can be obtained by surveying the electrical devices or equipment in manufacturing enterprises [16] or by employing identification techniques [40,41,42]. These parameters serve as the basic data for this study. If the induction motors in the group are indexed by the serial number i, then the basic parameters for the i-th motor in the group are shown in

Table 1. Basic parameters obtained from statistical and parameter identification methods.

Category	Details
Rated Parameters	Rated Voltage $U_{Ni}$, Rated Power $P_{Ni}$
T-Equivalent Circuit Parameters	$R_{si},R_{ri},X_{si},X_{ri},X_{mi}$
Rotor Parameters	Rotor Inertia $J_i$ and Time Constant $H_i$
Mechanical Load Parameters	Mechanical Torque Coefficients $T_{m0i},a_i$

.

Based on the basic parameters provided in Table 1, the operational parameters necessary for the grouping and aggregation methods can be calculated as detailed in

Table 2. Operational parameters for the subsequent grouping and aggregating modeling method.

Category	Details
Starting Torque Parameters	Electromagnetic Torque $T_{ei\text{\_}st}$, Mechanical Torque $T_{mi\text{\_}st}$
Maximum Electromagnetic Torque Parameters	Critical Slip $S_{mi}$, Maximum Electromagnetic Torque $T_{ei\text{\_}max}$
Steady-State Torque Parameters	Slip Rate $S_{oi}$, Electromagnetic Torque $T_{eio}$, Mechanical Torque $T_{mio}$
Steady-State Electrical Parameters	Stator and Rotor Currents ${\dot{I}}_{si}$ and ${\dot{I}}_{ri}$, Absorbed Active and Reactive Power $P_i+j{Q}_i$, Electromagnetic Power $P_{ei}$

.

The parameters listed in Table 2 can be calculated using Equations (3)–(8). Assume that the actual voltage phasor of motor i under steady-state operating condition is:

$$\dot{U}={\displaystyle \frac{U_N}{\sqrt{3}}}+j0$$

(3)

The actual electromagnetic torque and mechanical torque are expressed as follows:

$$\left\{\begin{array}{cc}\hfill T_{ei}\left(S_i\right)& ={\displaystyle \frac{3U^2}{{\omega }_0}}·{\displaystyle \frac{\frac{R_{ri}}{S_i}}{{\left(R_{si}+\frac{R_{ri}}{S_i}\right)}^2+{\left(X_{si}+X_{ri}\right)}^2}}\hfill \\ \hfill T_{mi}\left(S_i\right)& =T_{m0i}\left[\left(1-a_i\right){\left(1-S_i\right)}^2+a_i\right]\hfill \end{array}\right.$$

(4)

Note that both $T_{ei}\left(S_i\right)$ and $T_{mi}\left(S_i\right)$ are functions defined on the interval $S_i\in \left(0,1\right]$. By using numerical methods, the parameters listed in Table 2 can be obtained as follows:

(1)

$T_{ei\text{\_}st}=T_{ei}\left(1\right)$ and $T_{mi\text{\_}st}=T_{mi}\left(1\right)$;

(2)

$T_{ei\text{\_}max}=max\left[T_{ei}\left(S_i\right)\right]$ and its corresponding critical slip ratio $S_{mi}$;

(3)

The intersection of $T_{ei}\left(S_i\right)$ and $T_{mi}\left(S_i\right)$ within the interval $S_i\in \left(S_{mi},1\right)$. This intersection, where $T_{eio}=T_{mio}$, represents the actual steady-state operating point $S_{oi}$ for motor i.

After determining the actual steady-state operating point $S_{oi}$, we can further calculate the stator current phasor, rotor current phasor, voltage phasor, active power, reactive power, and electromagnetic power of motor i using Equations (5) to (8):

$${\dot{I}}_{si}={\displaystyle \frac{\dot{U}}{R_{si}+j{X}_{si}+\frac{1}{\left(\frac{1}{\left(\frac{R_{ri}}{S_{oi}}+j{X}_{ri}\right)}+\frac{1}{j{X}_{mi}}\right)}}}$$

(5)

$${\dot{I}}_{ri}={\dot{I}}_{si}{\displaystyle \frac{j{X}_{mi}}{\frac{R_{ri}}{S_{oi}}+j{X}_{ri}+j{X}_{mi}}}$$

(6)

$$P_i+j{Q}_i=\dot{U}{\dot{I}}_{si}^{*}$$

(7)

$$P_{ei}=I_{ri}^2{\displaystyle \frac{R_{ri}}{S_{oi}}}$$

(8)

The above basic and operational parameters provide the essential information for deriving the grouping and aggregation methods, and their symbols (as denoted in Table 1 and Table 2) will be utilized in the following sections.

2.2. Grouping Rules and Aggregation Rules

2.2.1. Rotor Motion Characteristics and Grouping Rules

An induction motor group can be regarded as a set of parallel nonlinear impedances. These impedances respond to the voltage excitation from the power source by injecting current back into the circuit system through the feeder lines. Even under the same voltage, motors with different mechanical loads or different electromagnetic torque characteristics exhibit different dynamic impedance characteristics. For example, during a transient period when the supply bus voltage gradually recovers from a low level, the rotors of motors driving fans may easily return to their pre-transient speeds, resulting in a rapid increase in their impedance. In contrast, the rotors of motors driving pumps may struggle to return to their pre-transient speeds due to insufficient electromagnetic torque, leading to relatively low impedance being maintained for an extended period. If a larger proportion of motors can easily recover their speeds, the total impedance of the power load will increase more rapidly, aiding the recovery of the supply bus voltage. Conversely, if a larger proportion of motors struggle to recover their speeds, the total impedance of the power load will remain low for a longer period, making voltage recovery more difficult.

Therefore, voltage stability is primarily determined by the stability of the induction motor group, which, in turn, is governed by the rotor stability of each individual motor. In summary, if all motors connected to the same bus are indiscriminately represented by a single equivalent motor without distinguishing their individual characteristics, the dynamic features of different motors will be conflated. This can cause significant discrepancies between the simulation results by using the reduced-order equivalent model and the actual measured instability data of the actual motor group, leading to fundamentally incorrect assessments of transient voltage instability.

Therefore, to accurately assess the voltage stability of the supply bus, it is essential to group the induction motors based on the similarity of their rotor acceleration and deceleration processes before applying equivalent reduction to each group. From this, we can derive the following grouping rules (GR) for induction motor groups:

Grouping Rule 1 (GR1): Induction motors with similar rotor motion characteristics during the entire deceleration process following a sudden voltage drop should be grouped together.

Grouping Rule 2 (GR2): Induction motors with similar rotor motion characteristics during the entire acceleration process as the voltage gradually recovers should be grouped together.

2.2.2. Power Transmission Characteristics and Aggregation Rules

In power systems, the transmission of active power causes a lateral voltage drop along the line, while the transmission of reactive power causes a longitudinal voltage drop. With constant line impedance, the voltage deviation at the distribution bus depends on the active and reactive power drawn by the loads connected to the bus. Therefore, to study the transient voltage stability of the distribution system, it is essential to understand the active and reactive power characteristics of dynamic loads. To ensure that the transient voltage stability is neither too optimistic nor too conservative when using aggregated models, the instantaneous active and reactive characteristics of the actual motor group and the aggregated model must be as similar as possible [43].

More specifically, for the study of transient voltage stability, the aggregation of induction motor groups should follow these rules (aggregation rules, AR):

Aggregation Rule 1 (AR1): The active power, reactive power, stator and rotor losses, electromagnetic power, and mechanical power of the aggregated model should be equal to the total active power, total reactive power, total stator and rotor losses, total electromagnetic power, and total mechanical power of the actual induction motor group.

Aggregation Rule 2 (AR2): The mechanical load torque of the aggregated model should be equal to the total mechanical load torque of the actual induction motor group.

Aggregation Rule 3 (AR3): The stored kinetic energy of the aggregated model’s rotor should be equal to the actual total stored kinetic energy of all the rotors in the actual induction motor group.

3. Grouping Method for the Induction Motors

3.1. Grouping Indicators

To accurately assess the transient voltage stability of a supply bus using a simplified dynamic load model, the primary challenge is effectively and rationally grouping the induction motors. Grouping motors with known parameters is essentially a multidimensional data classification problem. Therefore, it is necessary to establish grouping indicators that meet GR1 and GR2.

As discussed in Section 2.2.1, GR1 and GR2 focus on the principles of rotor dynamics during voltage drops and recovery. Transient voltage stability is primarily determined by the rotor dynamics of individual motors. The timescale of the rotor’s mechanical transients under force $(T_e-T_m)$ is longer than that of the electromagnetic transients of the stator and rotor. Therefore, to emphasize the dominant role of rotor acceleration and deceleration in this slower mechanical dynamic, the fast electromagnetic dynamics included in the fifth-order model (2) can be neglected. Instead, a first-order mechanical model (9) should be used to derive more reasonable grouping indicators.

Equation (9) provides the first-order mechanical model for motor i, while Figure 2 shows the electromagnetic torque characteristic curve $T_e$ and the mechanical characteristic curve $T_m$ for Equation (9). From the form of rotor motion in Equation (9) and the curves ($T_e$ and $T_m$) in Figure 2, we can see that:

(1)

The internal factors determining the acceleration and deceleration rate of the motor rotor are its mechanical inertia and the resultant torque;

(2)

The external factors are the extent of the voltage drop and its duration.

To further illustrate the mechanisms of these internal and external factors, we use Equation (9) and Figure 2 to analyze the three periods—before the grid fault, during the fault, and after the fault is cleared:

(1)

Before the fault occurs in the power system, motor i operates normally with a constant speed. Therefore, in the interval $S_i\in \left[S_{oi},1\right]$, $T_{ei}\left(S_i,U^2\right)\ge T_{mi}\left(S_i\right)$ always holds true.

(2)

During a power system fault period, the voltage drops suddenly within a brief moment (less than 0.02 s). Therefore, in the interval $S_i\in \left[S_{oi},1\right]$, $T_{ei}\left(S_i,U^2\right)$ drops suddenly. As shown in Figure 2, the solid line $T_{ei}\left(S_i,U^2\right)$ drops to the dashed line $T_{ei}^{\prime }\left(S_i,U_{drop}^2\right)$. Since the mechanical torque $T_{mi}\left(S_i\right)$ depends on the resisting moment or inertia of mechanical load (such as pump or fan impeller) and cannot change suddenly, it can easily result in $T_{ei}^{\prime }\left(S_i,U_{drop}^2\right)<T_{mi}\left(S_i\right)$, causing the rotor to enter a braking and deceleration period. According to the equation, the greater the voltage drop and the smaller the moment of inertia $J_i$, the faster the rotor decelerates. Conversely, the smaller the voltage drop and the greater the moment of inertia $J_i$, the slower the rotor decelerates.

(3)

After the power system fault is cleared, the voltage gradually recovers, causing $T_{ei}\left(S_i,U^2\right)$ to rise gradually. In the interval $S_i\in \left[S_{oi},1\right]$, the relationship between $T_{ei}\left(S_i,U^2\right)$ and $T_{mi}\left(S_i\right)$ depends on the voltage change. According to the equation, the greater the voltage rise and the smaller the moment of inertia $J_i$, the faster the rotor accelerates. Conversely, the smaller the voltage rise and the greater the moment of inertia $J_i$, the slower the rotor accelerates.

$$\left\{\begin{array}{c}{\displaystyle \frac{d{S}_i}{dt}}=-{\displaystyle \frac{1}{J_i{\omega }_0}}\left[T_{ei}(S_i,U^2)-T_{mi}\left(S_i\right)\right]\hfill \\ T_{ei}(S_i,U^2)={\displaystyle \frac{3U^2}{{\omega }_0}}{\displaystyle \frac{R_{ri}/S_i}{{\left(R_{si}+\frac{R_{ri}}{S_i}\right)}^2+{\left(X_{si}+X_{ri}\right)}^2}}\hfill \\ T_{mi}\left(S_i\right)=T_{m0i}\left[(1-a_i){\left(1-S_i\right)}^2+a_i\right]\hfill \end{array}\right.$$

(9)

Based on the mechanical analysis results of the three periods above, the internal factors’ influence can be quantified using the inertia time constant $H_i$, high-speed remaining torque $K_{HSi}$, and low-speed remaining torque $K_{LSi}$ as grouping indicators:

$$K_{HSi}=1-{\displaystyle \frac{T_{mi\text{\_}HS}}{T_{ei\text{\_}max}}}$$

(10)

$$K_{LSi}=1-{\displaystyle \frac{T_{mi\text{\_}LS}}{T_{ei\text{\_}st}}}$$

(11)

In Equations (10) and (11),

$K_{HSi}$ represents the high-speed remaining torque, indicating the residual electromagnetic torque that motor i can output at high-speed operating points;

$K_{LSi}$ represents the low-speed remaining torque, indicating the residual electromagnetic torque that motor i can output at low-speed operating points;

$T_{mi\text{\_}HS}$ is the mechanical torque at the actual operating point $S_i=S_{oi}$, $T_{mi\text{\_}HS}=T_{mio}$, as shown in Table 2;

$T_{mi\text{\_}LS}$ is the mechanical torque at $S_i=1$, $T_{mi\text{\_}LS}=T_{mi\text{\_}st}$, as shown in Table 2;

$T_{ei\text{\_}max}$ and $T_{ei\text{\_}st}$ represent the maximum electromagnetic torque and starting torque, respectively, as shown in Table 2.

Under significant voltage variations, a larger $K_{HSi}$ indicates that the motor’s torque output performance is more robust near the actual high-speed operating point $S_i=S_{oi}$. Conversely, a smaller $K_{HSi}$ means that the motor’s torque output performance is less robust near $S_i=S_{oi}$. Similarly, a larger $K_{LSi}$ implies that the motor’s torque output performance is more robust at the low-speed point $S_i=1$. Conversely, a smaller $K_{LSi}$ means that the motor’s torque output performance is less robust at the low-speed point $S_i=1$.

3.2. Grouping Method Based on Adaptive K-Means Clustering

Based on the above analysis, using the inertia time constant $H_i$, high-speed remaining torque $K_{HSi}$, and low-speed remaining torque $K_{LSi}$ as grouping indicators can yield more rational sub-groups of the induction motor group. The following is an adaptive clustering algorithm with the steps outlined below:

(1) Initialize dataset. For an induction motor group with N individuals, where the basic parameters of individual i in the group are known (see Table 1) along with the operating parameters (see Table 2), the $K_{HSi}$ and $K_{LSi}$ can be calculated using Equations (10) and (11), respectively. This allows for the construction of a three-dimensional data point set $\mathit{Y}$:

$$\mathit{Y}=\left\{\begin{array}{c}{\mathit{y}}_{\mathit{i}}|{\mathit{y}}_{\mathit{i}}=\left(K_{HSi},K_{LSi},H_i\right),i=1,\cdots ,N\end{array}\right\}$$

(12)

Set a constant $Iterations\text{\_}max$ and initialize an empty set $S_{evaluation}=\{ \}$ to record the evaluation coefficients of clustering effectiveness.

(2) Initialize the number of clusters by setting $K=2$.

(3) Initialize the cluster centers by randomly selecting $\mathit{K}$ points from the dataset $\mathit{Y}$ as the centers ${\mathit{\mu }}_{\mathit{K}}=\left\{{\mu }_k|k=1,\cdots ,K\right\}$.

(4) Assign each data point to its nearest cluster center. Calculate the Euclidean distance from data point i to the cluster center ${\mathit{\mu }}_{\mathit{K}}$:

$${\mathit{d}}_{\mathit{iK}}=∥{\mathit{y}}_{\mathit{i}}-{\mathit{\mu }}_{\mathit{K}}∥$$

(13)

In Equation (13), ${\mathit{d}}_{\mathit{iK}}$ has the same dimensions as ${\mathit{\mu }}_{\mathit{K}}$. Calculate the index of the minimum value in the vector ${\mathit{d}}_{\mathit{iK}}$:

$${\mathit{I}}_i^K=arg\underset{k}{min}\left({\mathit{d}}_{\mathit{iK}}\right)$$

(14)

Apply Equations (13) and (14) to all points in the dataset $\mathit{Y}$ to obtain the data point assignments ${\mathit{I}}_{labels}^K=\left\{{\mathit{I}}_i^K|i=1,\cdots ,N\right\}$, which in turn provides the cluster assignment results ${\mathit{C}}_{\mathit{K}}=\left\{C_k|C_k=\mathit{Y}\left({\mathit{I}}_{labels}^K=k\right),k=1,\cdots ,K\right\}$.

(5) Update the cluster centers. Calculate the mean of all data points ${\mathit{y}}_{\mathit{i}}$ in each sub-cluster $C_k$, and use this mean as the new cluster center:

$${\mu }_k={\displaystyle \frac{1}{\left|C_k\right|}}\sum _{{\mathit{y}}_{\mathit{j}}\in C_k}{\mathit{y}}_{\mathit{j}}$$

(15)

where $\left|C_k\right|$ represents the total number of points in sub-cluster $C_k$.

(6) Check the convergence criteria. Verify if the cluster centers ${\mathit{\mu }}_{\mathit{K}}$ have stabilized, if the data point assignments ${\mathit{I}}_{labels}^K$ have ceased to change, and if the number of iterations has reached the maximum $Iterations\text{\_}max$. If convergence is achieved, proceed to step (7) to evaluate the clustering results ${\mathit{C}}_{\mathit{K}}$; otherwise, return to steps (4) and (5).

(7) Calculate the validity of clustering results. Compute the silhouette coefficient $s_K=s\left({\mathit{C}}_{\mathit{K}}\right)$ according to [44] and add $\left(k,s_K,{\mathit{C}}_{\mathit{K}}\right)$ to the evaluation set $S_{evaluation}=\left\{\left(K,s_K,{\mathit{C}}_{\mathit{K}}\right)\right\}$.

(8) Increment the number of clusters. Set $K=K+1$. If $K>N$, proceed to step (9); otherwise, return to steps (3) through (7).

(9) Determine the optimal clustering. Find the maximum silhouette coefficient in the evaluation set $S_{evaluation}$ and output the optimal clustering result ${\mathit{C}}_{\mathit{K}}=\left\{C_k\mid k=1,\cdots ,K\right\}$. The adaptive clustering process is complete.

4. Aggregation Method for the Induction Motor Sub-Group

4.1. Calculation of Equivalent Impedance Parameters

After obtaining the optimal grouping result ${\mathit{C}}_{\mathit{K}}=\left\{C_k\mid k=1,\cdots ,K\right\}$ for the induction motor group, each motor in the sub-group $C_k$ needs to be represented as a single-unit machine. Given the basic and operational parameters of each motor (as shown in Table 1 and Table 2), the equivalent results for each motor subgroup should satisfy the following relationship according to AR1:

$$P_{e\sum }=\sum _{i=1}^n{P}_{ei}$$

(16)

$$P_{cu1\sum }=3{\left|\sum _{i=1}^n{\dot{I}}_{si}\right|}^2{R}_{sE}=\sum _{i=1}^n\left[3{\left(I_{si}\right)}^2{R}_{si}\right]$$

(17)

$$P_{cu2\sum }=3{I_{rE}}^2{R}_{rE}=\sum _{i=1}^n\left(P_{ei}{S}_{oi}\right)$$

(18)

$$Q_{xs\sum }=3{\left|\sum _{i=1}^n{\dot{I}}_{si}\right|}^2{X}_{0sE}=\sum _{i=1}^n\left[3{\left(I_{si}\right)}^2{X}_{0si}\right]$$

(19)

In Equations (16)–(19), the subscript notation “E” denotes the T-equivalent circuit parameters of the aggregated motor, while the subscript notation “i” represents the T-equivalent circuit parameters of the individual motor. $P_{e\sum }$, $P_{cu1\sum }$, $P_{cu2\sum }$, and $Q_{xs\sum }$, respectively, represent the total electromagnetic power, total stator losses, total rotor losses, and total stator reactive power of the aggregated model.

For the equivalent T-circuit of the single-unit machine, let the excitation branch voltage be ${\dot{U}}_{mE}$. According to Kirchhoff’s Voltage Law (KVL), Kirchhoff’s Current Law (KCL), and Ohm’s Law, the following relationships hold:

$${\dot{U}}_{mE}=\dot{U}-\left(\sum _{i=1}^n{\dot{I}}_{si}\right)\left(R_{sE}+j{X}_{0sE}\right)$$

(20)

$${\dot{U}}_{mE}={\dot{I}}_{rE}\left(R_{rE}/S_{oE}+j{X}_{0rE}\right)$$

(21)

$${\dot{I}}_{mE}=\left(\sum _{i=1}^n{\dot{I}}_{si}\right)-{\dot{I}}_{rE}$$

(22)

$$G_{mE}+j{B}_{mE}={\displaystyle \frac{1}{j{X}_{mE}}}+{\displaystyle \frac{1}{R_{rE}/S_{oE}+j{X}_{0rE}}}=\left(\sum _{i=1}^n{\dot{I}}_{si}\right)/{\dot{U}}_{mE}$$

(23)

The electromagnetic power of the single-unit machine can also be expressed as:

$$P_{e\sum }=3U_{mE}^2{G}_{mE}=3I_{rE}^2{\displaystyle \frac{R_{rE}}{S_{oE}}}$$

(24)

The equality between the rotor leakage reactance and the stator leakage reactance is used as an additional condition:

$$X_{0rE}=X_{0sE}$$

(25)

Refs. [45,46] indicated that the proportional relationship between rotor and stator leakage reactances has a minimal effect on both the dynamic and steady-state characteristics of the motor. Therefore, it is both necessary and reasonable to assume that rotor and stator leakage reactances are equal.

By combining Equations (23), (24), and (25), we obtain:

$$P_{e\sum }={\displaystyle \frac{3U_{mE}^2\left(R_{rE}/S_{oE}\right)}{{\left(R_{rE}/S_{oE}\right)}^2+{X_{0rE}}^2}}$$

(26)

Equation (26) can be simplified to a quadratic equation in $R_{rE}/S_{oE}$:

$${\left(R_{rE}/S_{oE}\right)}^2-{\displaystyle \frac{U_{mE}^2}{P_{e\sum }/3}}\left(R_{rE}/S_{oE}\right)+{X_{0rE}}^2=0$$

(27)

Solving Equation (27) using the quadratic formula yields $R_{rE}/S_{oE}$:

$${\displaystyle \frac{R_{rE}}{S_{oE}}}={\displaystyle \frac{\left(3U_{mE}^2/P_{e\sum }\right)\pm \sqrt{{\left(3U_{mE}^2/P_{e\sum }\right)}^2-4{X_{0rE}}^2}}{2}}$$

(28)

Since the actual operating slip rate $S_o$ of each motor is usually less than 0.05, the value of $R_{rE}/S_{oE}$ and $X_m$ should be close to each other, and fall within the same range (to ensure that the power factor near $S_{oE}$ falls within the range of 0.8 to 0.85). Therefore, in Equation (28), the “±” on the right side should be taken as “+”, ensuring that $R_{rE}/S_{oE}$ yields the correct value.

Based on Equations (20)–(28), the excitation branch voltage ${\dot{U}}_{mE}$, excitation branch current ${\dot{I}}_{mE}$, and rotor branch current ${\dot{I}}_{rE}$ of the single-unit machine can be determined. This allows for the calculation of the equivalent machine’s stator resistance, rotor resistance, stator and rotor leakage reactance, excitation reactance, and operating slip:

$$R_{sE}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}\left[{\left(I_{si}\right)}^2{R}_{si}\right]}{{\left|{\displaystyle \sum _{i=1}^n}{\dot{I}}_{si}\right|}^2}}$$

(29)

$$R_{rE}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}\left(P_{ei}{S}_{oi}\right)}{3{I_{rE}}^2}}$$

(30)

$$X_{0sE}=X_{0rE}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}\left[{\left(I_{si}\right)}^2{X}_{0si}\right]}{{\left|{\displaystyle \sum _{i=1}^n}{\dot{I}}_{si}\right|}^2}}$$

(31)

$$X_{mE}=\mathrm{Im}\left[{\displaystyle \frac{{\dot{U}}_{mE}}{{\dot{I}}_{mE}}}\right]$$

(32)

$$S_{oE}={\displaystyle \frac{2{\displaystyle \sum _{i=1}^n}\left(P_{ei}{S}_{oi}\right)}{3{I_{rE}}^2\left[\left(3U_{mE}^2/P_{e\sum }\right)+\sqrt{{\left(3U_{mE}^2/P_{e\sum }\right)}^2-4{X_{0rE}}^2}\right]}}$$

(33)

where $\mathrm{Im}\left[·\right]$ denotes the imaginary part of a complex number. Using Equations (29) to (30), the T-equivalent circuit parameters $R_{sE},X_{0sE},R_{rE},X_{0rE},X_{mE}$ of the equivalent machine, as well as the slip rate $S_{oE}$ under steady-state operating conditions, can be determined.

4.2. Calculation of Equivalent Mechanical Parameters

According to AR1, under the actual steady-state operating conditions with a nearly constant mechanical load, the aggregated mechanical load power of the single-unit machine should equal the sum of the mechanical load powers of all individual motors in the group:

$$\begin{array}{cc}\hfill & T_{m0E}{\omega }_0\left[\left(1-a_E\right){\omega }_0{\left(1-S_{oE}\right)}^3+a_E{\omega }_0\left(1-S_{oE}\right)\right]\hfill \\ \hfill & =\sum _{i=1}^n{T}_{m0i}\left[\left(1-a_i\right){\left(1-S_{oi}\right)}^3+a_i\left(1-S_{oi}\right)\right]\hfill \end{array}$$

(34)

Since relationship (34) holds even when the speeds of the induction motors in the group are very close to the synchronous speed $\left(S_{oi}=0\right)$, the mechanical load coefficient $T_{m0E}$ can be calculated as follows:

$$T_{m0E}={\displaystyle \sum _{i=1}^n}{T}_{m0i}$$

(35)

According to AR2, the mechanical load coefficient $a_E$ can be calculated as follows:

$$a_E={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}{T}_{m0i}{a}_i\left(1-S_{oi}\right)}{T_{0E}\left(1-S_{oE}\right)}}$$

(36)

According to AR3, the stored kinetic energy of the single-unit machine’s rotor should be equal to the actual total stored kinetic energy of all the rotors in the sub-group:

$${\displaystyle \frac{1}{2}}{J}_E{\left({\omega }_r\right)}^2={\displaystyle \sum _{i=1}^n}\left[{\displaystyle \frac{1}{2}}{J}_i{\left({\omega }_{ri}\right)}^2\right]$$

(37)

Therefore, the rotor moment of inertia $J_E$ and time constant $H_E$ of the single-unit machine can be estimated as follows:

$$J_E={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}\left[J_i{\left({\omega }_{ri}\right)}^2\right]}{{\left({\omega }_r\right)}^2}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}\left[J_i{\left(1-S_{oi}\right)}^2\right]}{{\left(1-S_o\right)}^2}}$$

(38)

$$H_E={\displaystyle \frac{J_E{\left({\omega }_r\right)}^2}{2{\displaystyle \sum _{i=1}^n}{P}_i}}$$

(39)

Using Equations (16) through (39), the mechanical load torque coefficients $T_{m0E}$, $a_E$, moment of inertia $J_E$, and inertia time constant $H_E$ for the single-unit machine of the sub-group under the actual steady-state operating condition can be determined.

5. Simulation and Discussion

5.1. Simulation Setup

5.1.1. Distribution Network and Induction Motor Parameters

In this section, a typical 10 kV distribution system is used for transient voltage stability simulation. This distribution system has one incoming line and three feeders on the bus. The circuit structure and parameters are shown in Figure 3.

(1) Converting all parameters to the 10 kV side, the equivalent impedance of the transmission line on the power supply side is $x_l=j0.4644\mathsf{\Omega }$, and the short-circuit impedance parameter of the transformer (model SFZ10-40000/110kV) is $Z_T=0.0112+j1.0145\mathsf{\Omega }$;

(2) The line impedance of feeder 1 is $x_{l1}=0.085+j0.18\mathsf{\Omega }$, and the load at terminal bus Bus1 (converted to the 10 kV side) is $S_{L1}=914.57+j566.80kVA$;

(3) The line impedance of feeder 2 is $x_{l2}=0.17+j0.36\mathsf{\Omega }$, and the load at terminal bus Bus2 (converted to the 10 kV side) is $S_{L2}=4572.83+j2833.99kVA$;

(4) The line impedance of feeder 3 is $x_{l3}=0.17+j0.36\mathsf{\Omega }$. The load at terminal bus Bus3 (converted to the 10 kV side) includes a linear impedance $Z_{L3}=21.77+j13.49\mathsf{\Omega }$ and an induction motor group (parameters detailed in

Table 3. Impedance parameters and inertia time constants for 25 induction motors.

No.	${\mathit{P}}_{\mathit{N}}$ (kW)	${\mathit{U}}_{\mathit{N}}$ (kV)	Impedance (p.u.)					Mechanical Load
			${\mathit{R}}_{\mathit{s}}$	${\mathit{X}}_{\mathit{s}}$	${\mathit{R}}_{\mathit{r}}$	${\mathit{X}}_{\mathit{r}}$	${\mathit{X}}_{\mathit{m}}$	H (s)	${\mathit{K}}_{\mathit{L}}$ (%)	a
1	1000	10	0.0379	0.1139	0.0079	0.1139	1.4411	0.6593	40	0.53
2	900	10	0.0176	0.122	0.0108	0.122	1.9507	0.7294	70	0.53
3	1000	10	0.0299	0.1172	0.0107	0.1172	1.6506	0.7706	50	0.53
4	2000	10	0.0351	0.1221	0.0086	0.1221	5.7783	0.8939	70	0.53
5	800	10	0.0159	0.123	0.0109	0.123	2.0896	0.9001	60	0.95
6	200	10	0.0129	0.1245	0.0109	0.1245	2.2618	0.9334	40	0.55
7	450	10	0.0485	0.0886	0.0116	0.0886	2.4915	1.1508	50	0.53
8	2000	10	0.0127	0.1281	0.0057	0.1281	3.7853	1.381	80	0.82
9	450	10	0.0728	0.0761	0.0173	0.0761	2.3504	1.4572	60	0.53
10	750	10	0.0334	0.0957	0.0117	0.0957	2.958	1.5721	70	0.87
11	900	10	0.0836	0.0634	0.0203	0.0634	1.9818	1.6141	60	0.53
12	450	10	0.0671	0.0793	0.0144	0.0793	2.4316	1.644	60	0.53
13	2000	10	0.0213	0.1251	0.0057	0.1251	3.5895	2.1137	40	0.82
14	750	10	0.0236	0.1231	0.0033	0.1231	2.9099	2.2128	80	0.95
15	750	10	0.0252	0.1226	0.0033	0.1226	2.8846	2.2852	70	0.87
16	1000	10	0.0232	0.1227	0.0111	0.1227	2.6789	2.3327	50	0.69
17	450	10	0.0218	0.1243	0.0034	0.1243	3.2272	2.3427	90	0.95
18	315	10	0.0231	0.1239	0.0056	0.1239	3.2025	2.532	90	0.82
19	750	10	0.0242	0.1229	0.0033	0.1229	2.9003	2.6363	70	0.87
20	1000	10	0.0239	0.1225	0.0111	0.1225	2.6691	2.7026	40	0.69
21	1000	10	0.0209	0.1235	0.0111	0.1235	2.712	2.8616	40	0.69
22	800	10	0.0099	0.1243	0.0032	0.1243	2.0193	3.1924	80	0.95
23	900	10	0.0123	0.123	0.0048	0.123	1.884	3.1972	60	0.9
24	900	10	0.0082	0.1242	0.0048	0.1242	1.9184	3.4254	60	0.9
25	800	10	0.0049	0.1258	0.0033	0.1258	2.0657	3.7413	80	0.95

).

Using the component-based approach from [17,18,47] or the online parameter identification technique from [40], the parameters of the induction motors can be obtained, as shown in Table 3. Table 3 provides the impedance parameters and inertia time constants for 25 induction motors, as reported in [47]. In Table 3, $K_L$ represents the actual loading rate and a denotes the mechanical coefficient in Equation (2).

5.1.2. Simulation Arrangement

To demonstrate the advantages of the methods proposed in Section 3 and Section 4, a simulation model is built in MATLAB/Simulink 2021b using the network parameters in Figure 3 and the induction motor group parameters in Table 3. The transient voltage recovery process at Bus0 is observed under the same large disturbance for different induction motor models.

The specific simulation tests’ arrangements are shown in

Table 4. Arrangement of simulation tests.

No.	Disturbance Settings	Induction Motor Model on Bus3
		Model Description	Parameters
1	Disturbance 1: A temporary ground fault occurs on Bus0, lasting for 0.1 s.	Model 1: Single-unit model based on Ref. [32].	See Section 5.2.2
		Model 2: Two-machine model based on Ref. [47].	See Section 5.2.2
		Model 3: Models based on the proposed method in this paper.	See Section 5.2.2
		Motor Group: The actual group with 25 induction motors.	See Section 5.1.1
2	Disturbance 2: A temporary ground fault occurs on Bus0, lasting for 0.2 s.	Model 1: Single-unit model based on Ref. [32].	See Section 5.2.2
		Model 2: Two-machine model based on Ref. [47].	See Section 5.2.2
		Model 3: Models based on the proposed method in this paper.	See Section 5.2.2
		Motor Group: The actual group with 25 induction motors.	See Section 5.1.1

. As listed in Table 4, the purpose of the No.1 simulation test is to verify that the motor model established in this paper has dynamic characteristics closer to the actual induction motor group. Meanwhile, the purpose of the No.2 simulation test is to verify that the motor model established in this paper can accurately reproduce the transient voltage collapse process caused by the actual motor group. The modeling methods from [32,47] are replicated to obtain their aggregated models (Model 1 and Model 2), which serve as references for the subsequent simulations.

5.2. Simulation Results and Discussion

5.2.1. Grouping Results

For the 25 induction motors listed in Table 3, we applied the grouping method from Section 3 to calculate the remaining torque values $K_{HSi}$, $K_{LSi}$. Additionally, we normalized the inertia constant $H_i$ by taking its maximum value and rounding it up to the nearest ten as the base value, resulting in the dataset $\left\{\left({\overline{H}}_i,K_{HSi},K_{LSi}\right)\right\}$. We then performed adaptive K-means clustering, producing the silhouette coefficient curve shown in Figure 4 and the clustering scatter plot depicted in Figure 5.

From the silhouette coefficient curve in Figure 4, it is clear that the optimal number of clusters, corresponding to the maximum silhouette coefficient, is two. Thus, we applied the clustering method in Section 3.2 and obtained two clusters ($K=2$). Figure 5 shows that the motors marked with a “+” belong to Cluster C1, which represents the motors less likely to become unstable. The motors marked with an “o” belong to Cluster C2, which includes the more likely unstable motors: specifically, the motors numbered 2, 4, 8, 14, 15, 17, 18, 19, 22, and 25.

Compared to subgroup $C_1$, the motors in subgroup $C_2$ have a smaller low-speed remaining torque $K_{LSi}$. This means that, under the influence of a low voltage caused by the grounding fault, within the short period before the fault is cleared, the combined braking torque of all motors’ rotors in subgroup $C_2$ is larger, rotor velocity decelerates more quickly, and the slip rate $S_i$ increases more rapidly. With the $S_i$ increasing more rapidly, the ${\displaystyle \frac{R_{ri}}{S_i}}$ of a subgroup of motors may decrease more rapidly (as discussed in Section 2.2.1). Therefore, under the same duration of low voltage, the motors in subgroup $C_2$ are more prone to instability compared to those in subgroup $C_1$. If the grounding fault persists longer, the $S_i$ of the motors in subgroup $C_2$ will increase more quickly toward 1, while ${\displaystyle \frac{R_{ri}}{S_i}}$ will become much smaller (even close to $R_{ri}$) in a very short time, leading to exceeding the power transmission limit and ultimately causing transient voltage instability.

The silhouette coefficient can be used to quantitatively integrate the tightness and separation of each cluster [44]. The silhouette coefficient curve shown in Figure 4 represents the clustering of 25 motors with different numbers of clusters. Clustering with different numbers of clusters results in varying degrees of intra-cluster tightness and inter-cluster separation. The silhouette coefficient curve in Figure 4 shows a decreasing trend with an increasing number of clusters. Compared to a manually specified number of clusters, the silhouette coefficient introduces greater adaptability to the clustering method.

It should be noted that if the initial centers of K-means are selected randomly, different clustering results may occur, leading to silhouette coefficient curves that differ from those in Figure 4 and grouping results that differ from those in Figure 5. Additionally, the optimal silhouette coefficient may not necessarily be 2; it could be 3 or another number. However, by accepting the optimal number of clusters given by the silhouette coefficient curve, the constraints reflected by the remaining torque $K_{\mathrm{HSi}}$, $K_{\mathrm{LSi}}$, and the inertia constant $H_i$ can be utilized to adaptively place motors with similar acceleration and deceleration characteristics into the same cluster.

5.2.2. Equivalent Results

Based on the series of equations provided in Section 4 of this paper, the parameters of aggregation models for sub-groups (converted to the 10 kV side) are shown in

Table 5. Parameters of the models (Model-3) based on the proposed method.

No.	${\mathit{P}}_{\mathit{N}}$ (kW)	${\mathit{U}}_{\mathit{N}}$ (kV)	Impedance (Ω)					Mechanical Load
			${\mathit{R}}_{\mathit{s}}$	${\mathit{X}}_{\mathit{s}}$	${\mathit{R}}_{\mathit{r}}$	${\mathit{X}}_{\mathit{r}}$	${\mathit{X}}_{\mathit{m}}$	H (s)	${\mathit{K}}_{\mathit{L}}$ (%)	a
C1	12,800	10	0.0346	0.1175	0.0111	0.1175	2.2653	1.9345	50.55	0.72
C2	9515	10	0.0190	0.1327	0.0055	0.1327	3.0524	1.9084	76.18	0.79

:

Based on the method from [32], the parameters of the single-unit model are shown in

Table 6. Parameters of the single-unit model (Model-1) based on [32].

No.	${\mathit{P}}_{\mathit{N}}$ (kW)	${\mathit{U}}_{\mathit{N}}$ (kV)	Impedance (Ω)					Mechanical Load
			${\mathit{R}}_{\mathit{s}}$	${\mathit{X}}_{\mathit{s}}$	${\mathit{R}}_{\mathit{r}}$	${\mathit{X}}_{\mathit{r}}$	${\mathit{X}}_{\mathit{m}}$	H (s)	${\mathit{K}}_{\mathit{L}}$ (%)	a
1	22,315	10	0.0300	0.1207	0.0093	0.1262	2.5387	1.9235	61.48	0.74

.

Based on the method from [47], the parameters of the two-machine model are shown in

Table 7. Parameters of the two-machine model (Model-2) based on [47].

No.	${\mathit{P}}_{\mathit{N}}$ (kW)	${\mathit{U}}_{\mathit{N}}$ (kV)	Impedance (Ω)					Mechanical Load
			${\mathit{R}}_{\mathit{s}}$	${\mathit{X}}_{\mathit{s}}$	${\mathit{R}}_{\mathit{r}}$	${\mathit{X}}_{\mathit{r}}$	${\mathit{X}}_{\mathit{m}}$	H (s)	${\mathit{K}}_{\mathit{L}}$ (%)	a
1	20,515	10	0.0236	0.1205	0.0071	0.1205	2.5802	1.9534	61.61	0.77
2	1800	10	0.0784	0.0696	0.0182	0.0696	2.1665	1.5824	60.00	0.53

.

5.2.3. Transient Voltage Stability Results and Discussion

As shown in Figure 6, Disturbance 1 was initiated at 0.5 s. Once Disturbance 1 was cleared at 0.6 s, the distribution system underwent a transient process of gradual voltage recovery. Figure 6a–c illustrate the voltage curve at Bus0, the active power curve, and the reactive power curve of the transformer, respectively, throughout the entire transient voltage recovery period.

The black solid line in Figure 6a shows that during the grounding fault, the voltage dropped to zero at 0.5 s and remained there for 0.1 s. By 0.6 s, as the grounding fault was cleared, the system began to recover rapidly, with the voltages at Substation Bus0 and the power consumer side buses 1, 2, and 3 also recovering quickly.

In Figure 6a, the root mean square (RMS) deviation for model-1 is 0.0210, for Model-2 it is 0.0170, and for model-3 (this paper) it is 0.0105. The enlarged sub-figures in Figure 6a–c demonstrate that Model-3, developed using the proposed method, more accurately replicates the dynamic behavior of the actual motor group. In contrast, the single-unit model from [32] (Model-1) and the two-machine model from [47] (Model-2) exhibit greater deviations from the actual motor group’s dynamic behavior.

Therefore, compared to Model-1 from [32] and Model-2 from [47], Model-3 presented in this paper has higher precision, as its dynamic characteristics more closely resemble those of the actual motor group under the given simulation conditions.

As depicted in Figure 7, Disturbance 2 was initiated at 0.5 s. Once Disturbance 2 was cleared at 0.7 s, the distribution system underwent a transient period of gradual voltage collapse. Figure 7a–c illustrate the voltage curve at Bus0, the active power curve, and the reactive power curve of the transformer, respectively, throughout the entire transient voltage collapse period.

The black solid line in Figure 7a shows that during the grounding fault, the voltage at Bus0 dropped to zero at 0.5 s and remained at zero for 0.2 s. By 0.7 s, the system entered a transient recovery period. However, after 1 s, the voltage at Bus0 did not recover to its expected value, resulting in transient voltage instability. Accompanied by the voltage instability at Substation Bus0, the voltages at Bus1, Bus2, and Bus3 on the power consumer side also exhibited voltage instability. Additionally, a visual comparison of Figure 7a–c revealed the differences among Model-1, Model-2, and Model-3:

(1) The dynamic processes of Model-1 (single-unit model from [32]) and Model-2 (two-machine model from [47]) were represented by blue dashed lines and orange dotted lines, respectively, in Figure 7. In Figure 7b,c, the active and reactive power curves of both models diverged from those of the actual motor group, gradually returning to levels before the fault occurred. This resulted in their voltage curves in Figure 7a failing to accurately reproduce the voltage collapse at Bus0, constituting a significant misjudgment.

(2) The dynamic process of the model developed in this paper (Model-3) is represented by a magenta dotted line in Figure 7. In Figure 7b,c, its voltage, active power, and reactive power curves successfully replicated the voltage collapse at Bus0.

Therefore, under the given simulation conditions, the motor model established by the proposed method (Model-3) provides a more accurate assessment of transient voltage stability in the distribution system compared to Model-1 and Model-2.

From the above simulation results, we can conclude the following:

(1) The grouping rules (GR1 and GR2) and aggregation rules (AR1 to AR3) presented in Section 2.2 are correct and reasonable.

(2) The grouping indicators defined in Section 3.1, specifically the high-speed remaining torque and low-speed remaining torque, along with the adaptive clustering method proposed in Section 3.2, are effective. The introduction of the silhouette coefficient enhanced the adaptive capability of the K-means clustering algorithm, resulting in more reasonable induction motor grouping.

(3) The impedance parameter aggregation method derived in Section 4.1 and the mechanical parameter aggregation method derived in Section 4.2, for motor subgroups, are both accurate and effective.

6. Conclusions

This paper has analyzed the operating principles of induction motors and the mechanism of voltage drop caused by power transmission, providing specific grouping and aggregation rules. Guided by the grouping rules, the inertia constant H, high-speed remaining torque $K_{HS}$, and low-speed remaining torque $K_{LS}$ have been used as grouping indicators. An adaptive K-means clustering method incorporating the silhouette coefficient has been introduced to divide the induction motors into sub-groups rationally. Under the aggregation rules, impedance parameter and mechanical parameter equivalent methods have been proposed for the sub-groups of induction motors. Through simulations in typical distribution systems, the following conclusions have been drawn, compared with traditional single-unit and two-machine equivalent methods:

(1) The dynamic characteristics of the induction motor model from the proposed method are closer to those of the actual induction motor group, with a smaller root mean square (RMS) deviation of 0.0105 compared to 0.0210 and 0.0170 from other methods. This demonstrates that the proposed method can ensure a reasonable reduction in the order of the dynamic load model while maintaining simplicity and usability.

(2) The induction motor model from the proposed method can more accurately reproduce the bus voltage collapse process. This proves that the proposed method improves the accuracy of the dynamic load model, avoiding overly conservative or risky transient voltage stability simulation results. This is crucial for the security and stable operation of distribution systems.