Voltage Stability Estimation Considering Variability in Reactive Power Reserves Using Regression Trees ^†

Abstract

The rapid integration of renewable energy sources, such as photovoltaic power systems, has reduced the necessary for synchronous generators, which traditionally contributed to grid stability during disturbances. This shift has led to a decrease in reactive power reserves (RPRs), raising concerns about voltage stability. Real-time monitoring of voltage stability is crucial for transmission system operators to implement timely corrective actions. However, conventional methods, such as continuation power flow calculations, are computationally intensive and unsuitable for large-scale power systems. Machine learning techniques using data from phasor measurement units have been proposed to estimate voltage stability. However, these methods do not consider changes in generator operating conditions and fluctuating RPRs. As renewable energy generation increases, the operating conditions of generators vary, which leads to significant changes in system RPRs and voltage stability. In this paper, a voltage stability margin is proposed using regression trees with RPRs varying based on generator operation conditions. Simulations based on the IEEE 9-bus system demonstrate that the proposed approach provides an accurate and efficient voltage stability estimation.

1. Introduction

The large-scale integration of renewable energy sources, including photovoltaic power generation [1], has significantly reduced the number of operational conventional rotating generators in use. As a result, the stability of the power system is expected to decline. Currently, research on Grid-Forming Inverters (GFM) is progressing [2]. However, they are still in the research stage and have not yet been widely adopted. Moreover, it remains unclear to what extent voltage stability can be improved when GFM is widely deployed. Thus, a more precise monitoring of stability conditions in the system is required.

To mitigate the risk of large-scale blackouts resulting from the diminished grid stability, power system operators worldwide employ wide-area measurement systems (WAMSs) with phasor measurement units (PMUs), which provide time-synchronized power system data using the Global Positioning System (GPS) [3]. Optimal PMU placement strategies have been investigated to minimize installation costs while maximizing the effectiveness of system monitoring [4,5,6,7,8]. Concurrently, power system stability estimation using PMU data has garnered considerable attention [9,10,11].

Reactive power reserves (RPRs) could be a key factor in maintaining voltage stability. Its appropriate management is a critical challenge for transmission system operators (TSOs). The decline in RPRs is primarily caused by three factors.

• Reduction in the number of synchronous generators: Fewer operational generators reduce the total amount of reactive power that can be supplied during disturbances.
• Decreased reactive power supply capability per generator: To achieve efficient and economical operation, generators are often operated near their capacity limits, which restricts their ability to supply reactive power.
• Expansion of PV power generation: The large-scale integration of photovoltaic systems introduces significant reverse power flow and voltage fluctuations in the system [12]. Control methods, such as constant power factor operation for photovoltaic systems, have been proposed to prevent voltage deviations [13]. This method consumes reactive power by the photovoltaic system to mitigate voltage rises caused by reverse power flow, and thereby prevents voltage violations. However, such measures further reduce the RPRs, exacerbating voltage stability issues.

In power systems where RPRs have declined and the voltage stability has deteriorated, disturbances such as transmission line faults or generator outages may cause voltage collapse and large-scale blackouts in low-voltage stability systems [14,15,16].

As real-time voltage stability monitoring has become more essential, various static methods for voltage stability assessment have been proposed. The voltage stability margin (VSM) is one of the most useful indicators, typically derived from active power–voltage (P–V) curves, which can be obtained by numerous methods. Tiranuchit and Thomasra [17] introduced a method that iteratively uses conventional power flow analysis techniques, such as the Newton–Raphson and Gauss–Seidel methods. However, near voltage collapse points, the singularity of the Jacobian matrix deteriorates the convergence, and reduces both computational speed for and accuracy in voltage collapse point identification. Gao et al. [18] proposed a method based on eigenvalues of the Jacobian matrix to analyze the degree of voltage instability.

Ajarapu and Christy [19] proposed a continuation power flow (CPF) method to improve the convergence by avoiding singularities near voltage collapse points. Chiang et al. [20] further developed CPFLOW, a tool for the analysis of steady-state power system behaviors under various parameter changes. These studies have served as a basis for further advancements in CPF methods.

Lee et al. [21] introduced a novel voltage stability analysis method using P–V–Q curves, which can be used to calculate the maximum active and reactive power consumed by a load while maintaining the minimum voltage. However, this approach to identifying unstable operating points causes an increase in computational burdens.

Mori et al. [22] proposed a novel CPF method incorporating a nonlinear predictor to reduce the computational load. However, even with this approach, the computation time for a large-scale power system is approximately 18 s, making the real-time application challenging.

Machine learning has recently attracted attention as a promising technique for the estimation of voltage stability indices due to its ability to provide high-speed inference once the model is trained. Although model training is time-intensive, a well-trained model enables real-time estimation of voltage stability based on input information. This realizes rapid and efficient grid monitoring and operation. Machine learning techniques used for voltage stability estimation include random forest regression (RFR), support vector machine (SVM), artificial neural network (ANN), and decision tree (DT) methods.

In studies [23,24], methods using random forests are examined. Dharmapal et al.’s proposed method to improve the robustness of voltage stability index (VSI) estimation has been further developed [24]. In addition, SVM-based algorithms for VSM evaluation have been proposed [25]. Furthermore, ANN-based techniques for voltage stability estimation have been extensively studied [26,27,28,29,30,31,32,33]. Inspired by biological neural networks, ANNs provide improved estimation accuracy. For example, Chakrabarti and Jeyasurya [27] proposed multiple ANN models tailored to specific load levels, while Zhou et al. [28] accounted for PMU measurement errors to optimize PMU placement. Hybrid intelligent systems combining DTs and ANNs have also been investigated [29]. DT-based data selection in the systems improves the estimation accuracy. Radial basis function (RBF) neural networks have been explored [30], and efficient model training techniques, such as particle swarm optimization (PSO) for ANN hyperparameter tuning [31], and loss functions following the P–V curve [32], have been proposed. And Su and Hong [33] proposed a new estimation method based on Extreme Learning Machine (ELM).

Advanced machine learning techniques such as ANNs and SVMs, as black-box models, obscure the interpretability of the estimation process [34]. In contrast, simpler methods such as DTs have a high interpretability of the model as an important factor for estimation. Another advantage of DTs are their low computational load and ability to determine critical system information and thresholds for stability maintenance. Accordingly, the use of DTs for VSM estimation has been investigated [34,35,36,37]. For example, a DT branching analysis has been employed to identify critical PMU placements for an accurate voltage stability estimation [34]. Similarly, Jia et al. [35] proposed a rule-based approach for the maintenance of voltage stability derived from tree structures. A classification approach has been proposed [36], with classification trees to assess and categorize voltage stability based on its degree. And Cai et al. [37] proposed a new method using the LightGBM (Light Gradient Boosting Machine). These methods have high estimation accuracy and interpretability.

Recent studies on RPR-related voltage stability estimation [24,28,30,32,33] have assumed constant reactive power supply limitations regardless of generator operating conditions. However, the reactive power supply limit varies with the generator operating point and thus depends on the renewable energy output. If the reactive power limit change is not considered, the estimated voltage stability may significantly deviate from the actual stability. Thus, it is essential to consider generator operating conditions to accurately estimate the voltage stability.

The authors have studied a voltage stability estimation method using regression trees, which offer high interpretability and low computational burden, under conditions where RPRs varied due to changes in generator operating conditions [38]. However, this paper does not examine the impact of RPR variations on voltage stability. Furthermore, the discussion on the results of the proposed method is insufficient. Therefore, the contributions of this study are presented as follows:

• demonstration of the relationship between system-wide RPRs and voltage stability in the system;
• clarification of the impact of RPRs variation on the accuracy of voltage stability estimation under reactive power supply limitations;
• development of a method to address changes in reactive power limitations to improve the estimation accuracy.

The remainder of this article is structured as follows. In Section 2, the authors examine the relationship between the VSM and RPRs. In Section 3, the authors evaluate the limitations of conventional methods for estimating voltage stability estimation. In Section 4, the authors propose estimation methods and validate their effectiveness through simulations. Section 5 concludes the paper with a summary of key findings and future research directions.

2. Relationship Between the VSM and RPRs

2.1. VSM

This study is focused on the estimation of the VSM, a critical indicator of voltage stability, in real time through the application of machine learning techniques. Figure 1 presents the P–V curve, which shows the relationship between load active power and voltage. The VSM is defined as the distance to the voltage collapse point ($P_{L\_max}$) from the operating point ($P_L$). The definition of VSM is expressed in Equation (1), where the nominal value corresponds to the rated capacity of the load bus.

$$\mathrm{V}\mathrm{S}\mathrm{M}={\displaystyle \frac{P_{L\_max}-P_L}{Nominalpower}}$$

(1)

2.2. RPRs

The primary role of RPRs is to maintain voltage levels by supplying reactive power from sources in response to various disturbances, and thereby ensure a stable operation of the power system. Several studies have focused on the relationship between RPRs and voltage stability [39,40]. Sources of reactive power include reactors, capacitors, generators, and synchronous condensers. System operators must effectively utilize these resources to maintain adequate RPRs. Generally, RPRs are closely related to the VSM, and a lack of RPRs can result in a reduction in VSM [39].

Among the sources of reactive power, synchronous generators contribute to the majority of RPRs. Synchronous generators typically increase their active power output in line with load increase, and the terminal voltage of the generator decreases. The excitation current is increased to supply more reactive power to maintain the terminal voltage. However, the reactive power supply of synchronous generators is limited by its capability curve, shown in Figure 2. This limitation can lead to a lack of RPRs, and, as a result, the power system is more prone to voltage instability.

As shown in Figure 2, the available reactive power supply is determined by the operating point. The RPR for each generator is defined by Equation (2) [39], where ${RPR}_i$ represents the reactive power reserve of generator $i$, $Q_{{max}_i}$ denotes the maximum reactive power supply capability of the generator, and $Q_{g_i}$ is the current reactive power output. RPR values differ by operation point (e.g., ${RPR}_A$ at Point A and ${RPR}_B$ at Point B), as shown in Figure 2. Moreover, as ${RPR}_A$ is smaller than ${RPR}_B$, the RPR value in the operation point with a high output decreases under heavy load conditions due to current capacity limitations.

$${RPR}_i=Q_{{max}_i}-Q_{g_i.}$$

(2)

Therefore, with the increasing use of renewable energy sources, the decreased number of generators operating in parallel leads to a reduction in RPRs. Additionally, as the RPR values may fluctuate in real time depending on PV generation, TSOs need to manage operations while carefully considering RPRs.

2.3. Impact of RPR Changes on the P–V Curve

In this section, the authors explain the impact of changes in RPRs on the P–V curve. Figure 3 shows the system diagram. The system consists of one generator bus, one load bus, and an infinite bus representing a large-scale grid. The P–V curves are obtained through CPF. CPF is a method based on incrementally increasing the load from the base system state. It sequentially solves equations to plot the P–V curve. This approach enables the computation of accurate P–V curves in a shorter time, as it reduces the computational load and improves the convergence near the voltage collapse point. In this study, MATLAB PSAT (ver.2.1.11) is used to perform the continuation power flow calculations [41].

In the calculations, the load bus is designated as PQ bus, and the infinite bus is set as a swing bus. Synchronous generators are typically configured as PV bus. However, when the reactive power output reaches its upper limit, it switches to PQ bus. To evaluate the impact of RPR changes on the P–V curve, we examine the following cases; by varying the value of $Q_{max}$ in each case, the RPRs are observed:

• Case 1: Without Q limit;
• Case 2: With a Q limit ($Q_{max}=1.0\left[pu\right]);$
• Case 3: With a Q limit ($Q_{max}=0.5\left[pu\right]).$

The upper limit of reactive power output for the generator significantly narrows the P–V curve. This shows that, when the synchronous generator reaches the upper limit of reactive power supply, the terminal voltage drops, causing a rapid decline in voltage. Additionally, in the case of $Q_{max}=0.5$ [pu], the P–V curve is further narrowed compared to that at $Q_{max}=1.0$ [pu] or to those in cases without limitations on reactive power supply. For the case of $Q_{max}=0.5$ [pu], the load level at the voltage collapse point corresponds to a voltage of approximately 0.95 [pu] under the other two conditions.

This observation highlights that the system remains stable when the generator possesses a sufficient reactive power supply capacity, but becomes unstable in its absence. Moreover, as illustrated in Figure 4b in the absence of reactive power supply limitations on the generator, the reactive power output exhibits a significant escalation near the voltage collapse point. This behavior suggests a substantial deviation from the generator’s feasible operational range. As a result, the P–V curve is unrealistic. Consequently, the necessity of incorporation of RPRs into voltage stability estimations is emphasized to achieve a reliable power system operation. Furthermore, the VSM in practical power systems is likely to be influenced not only by RPRs, but also by the overall system conditions. Thus, it is imperative to obtain comprehensive insights into both RPRs and VSM to ensure an accurate assessment and enhanced system stability.

3. Assessment of Voltage Stability Estimation by Conventional Methods

In this section, the conventional method is used to clarify the impact of changes in generator operating conditions under reactive power supply limitations on voltage stability estimation.

3.1. Regression Tree Overview

A regression tree is employed to quantitatively estimate the VSM in this study. Regression trees are a type of DTs, which are a type of supervised machine learning method which combine simple identification rules to obtain complex identification boundaries. DTs are extremely fast in providing output results, because the conditions relating to an “if statement” are judged only a few times in the process from data input to estimation.

3.2. Flowchart

Figure 5 shows the flowchart for the creation of the estimation model. The flowchart has two major stages, the data generation stage and estimation model creation stage.

In the first stage, the load bus for the estimation model is selected and the system status, such as regards load level and the rated capacity of the generator, is changed. Data are generated by varying the load at the selected load bus while holding other conditions. The generated data are divided into training and test data.

In the second stage, each model for all load buses learns using the training data. The MATLAB function ‘fitrtree’ is used for training and adjusts the tree depth from cross-validation of the results to prevent overfitting. Test data are run through the model to evaluate the estimation performance.

3.3. Test Power System

Figure 6 shows a diagram of the developed test system based on the IEEE 9 bus system. The test system consists of three generators, three loads, and six transmission lines. The load characteristics are assumed to be constant power load. Bus 1 is set as a swing bus, while bus 2 and bus 3 are set as PV buses in CPF. The reactive power is limited based on a power factor of 0.9 for the rated capacity. The PV bus connected to the generator changes to the PQ bus when the reactive power supply reaches the limit.

3.4. Data Generation Method

To develop estimation models and obtain training data for their training, as well as test data for evaluation, it is necessary to prepare various system conditions. Then, P–V curves are drawn based on these system conditions. The P–V curves are obtained using CPF by performing simulations on the test system, as shown in Figure 6. The system conditions are varied as described below. The training data consists of 20,000 cases from CPF, while 2000 cases are prepared for testing.

3.4.1. Initial Load Conditions

Demand fluctuates continuously according to factors such as time. Additionally, demand also varies depending on power generation from renewable energy sources. To address these factors, the load values were set randomly. The active power value of each load bus is randomly varied in the range of ±50% from the rated value, except for the bus load, for which the estimation model was created. Notably, the power factor of each load bus is kept constant. The load bus related to the estimation model is initially set to 50% of the value shown in Figure 6.

3.4.2. Initial Conditions of the Generator

Depending on power generation from renewable energy sources, the operational status of the generators changes. Therefore, the generator capacity varies as shown in

Table 1. Range of the generator’s rated capacity.

	X [MVA]	Y [MVA]	Z [MVA]
Maximum value	495.0	384.0	256.0
Minimum value	247.5	192.0	128.0

. The active power output of each generator is dispatched based on the ratio of its capacity to the total load.

3.4.3. Reactive Power Limit

The reactive power limit for each generator is set to a power factor of 0.9, based on the randomly assigned capacities shown in Table 1.

3.5. Estimation Model Development Method (Conventional Method)

This section explains the general model creation method for Margin estimation. Although the machine learning techniques and assumed conditions used in references [22,23,24,25,26,27,28,29,30,31,32,33,34] differ, they largely share common features. First, training data are generated through CPF calculations. Then, multiple sets of power system information and the corresponding Margin values for various operating points are prepared to construct the estimation model. In this study, it is assumed that PMUs are installed on all buses, and the estimation model is created using the power system information listed below.

• Voltage magnitude at each bus;
• Phase at each bus (referenced to the swing bus);
• Active and reactive power output of each generator;
• Flow of active and reactive power on each transmission line;
• Active and reactive power at each load bus.

Additionally, Figure 7 illustrates the input and output information of the estimation model. This power system information corresponds to Figure 7. In this study, a regression tree with low computational burden and high interpretability is used.

3.6. Estimation Results with the Conventional Method

The investigation is conducted by varying the conditions relative to the reactive power supply limit. The analysis compares cases where the reactive power limits are either constant or varied.

The evaluation indicators for the evaluation of VSM estimation accuracy include root mean square error (RMSE) and maximum error (MAX), which are formulated as shown in Equations (3) and (4). N is the number of samples, while $y_i$ and $\widehat{y_i}$ are the estimated value and true value, respectively.

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{(y_i-\widehat{y_i})}^2}$$

(3)

$$\mathrm{M}\mathrm{A}\mathrm{X}=\max \left|\left(y_i-\widehat{y_i}\right)\right|$$

(4)

Figure 8 compares the estimation results under the different conditions. The estimation accuracy in the case with changes in the reactive power limits is significantly lower than that when the reactive power limit is constant. The estimation error for Bus 8 is particularly significant. This is believed to be due to it being the most affected by the variations in the generator’s reactive power supply limitations. This is because it is located between Bus 2 and Bus 3, which are generator buses with reactive power supply constraints. Such estimation errors are considered to potentially affect the stable supply of electricity.

Therefore, in this paper, this method is referred to as the conventional method, and a new method is developed by creating an estimation model to improve estimation accuracy, considering the changes in the operating conditions of generators with reactive power supply limitations.

4. Proposed Methods and Evaluation Results

In this section, an estimation modeling method is proposed to mitigate the significant estimation errors caused by the failure to account for the varied reactive power limitations, as identified in the previous section. Additionally, the results of application of the proposed method to Bus 8, where the largest estimation error was exhibited, are presented.

4.1. Proposed Method

Figure 9 illustrates the overall framework of this method. The proposed method consists of two main components. First, estimation models are trained using data that includes the RPR information of generators. Second, the method employs an adaptive estimation model that switches based on the operational state of the generators.

4.1.1. Addition of the RPR Information of the Generator

The information used for the creation of the estimation model is enhanced by adding RPR information to the data previously used in conventional method. As the RPR information, obtained by Equation (2), is added to each generator bus except the swing bus, the authors incorporate the RPR information for Bus 2 and Bus 3. This is expected to improve the estimation accuracy due to the strong relationship between RPRs and VSM.

4.1.2. Multiple Estimation Models

This method constructs capacity-segmented estimation models according to the total capacity of generators at Bus 2 and Bus 3, where the reactive power output is constrained. In this study, the capacities of the generators at Bus 2 and Bus 3 are randomly varied to account for changes in reactive power limits. The estimation models are developed by dividing the generator capacity into N segments, ranging from the minimum value (320 MVA) to the maximum value (640 MVA), in accordance with the total capacity. Each estimation model is trained based on the data for the corresponding capacity range segment. The estimation process in the testing phase obtains the VSM with a model in the segment to which the total capacity belongs.

4.2. Optimal Number of Estimation Models

In this section, the authors examine the estimation performance with respect to the number of segments in the proposed method. Figure 10 illustrates the relationship between RMSE and MAX corresponding to each number of segments in the conventional method (N = 1), which does not use RPR information. Estimation accuracy is improved by increasing the number of segments. Thus, the construction of estimation models based on the operational states of generators is an effective approach.

However, the improvement in estimation accuracy becomes negligible after 10 segments. Therefore, in this study, the number of segments is set to 10, which balances the trade-off between accuracy enhancement and modeling complexity.

A total of 10 models are assigned sequentially, starting with Model 1 for PV buses with the smallest total capacity (

Table 2. Model assignments for the total capacity range.

Model Number	Range [MVA]	Model Number	Range [MVA]
Model 1	320–352	Model 6	480–512
Model 2	352–384	Model 7	512–544
Model 3	384–416	Model 8	544–576
Model 4	416–448	Model 9	576–608
Model 5	448–480	Model 10	608–640

). Based on the total capacity of Bus2 and Bus3 shown in Table 2, create separate estimation models and switch between them during the testing phase.

4.3. Evaluation Results

Figure 11 compares the results between the conventional method and the proposed method. The proposed method significantly improves the estimation accuracy compared to the conventional method. Specifically, the RMSE improves by 81.65%, and the MAX is reduced by 80.24%.

The origin of the improvement in estimation accuracy of the proposed method is discussed below. To illustrate the discussion, the proposed method is categorized as Method 1 and Method 2. The key points of each method are shown in

Table 3. Key Points of Each Method.

	Addition of RPR Information of Generator	Multiple Estimation Models
Method 1	Applied	Not applied
Method 2	Not applied	Applied
Proposed method	Applied	Applied

. Method 1 is an approach in which the training data are included with RPR information and the estimation model is not switched based on the operating status of the generators (method with N = 1). Method 2 is an approach in which 10 estimation models (N = 10) are trained without RPR information for the generators.

Figure 12 compares the proposed method to Method 1 and Method 2. All methods significantly improve the estimation accuracy compared to the conventional method. The estimation results of Method 2 are better than those of Method 1. Therefore, it is more important to use multiple estimation models rather than to incorporate the RPR information into the training data. This indicates that, by incorporating RPR information for generators into the training data, the estimation models for each generator capacity are improved, which leads to better performance.

Figure 13 shows the estimation results of each model for the proposed method and Method 2. Regarding the results of Method 2, Model 1 shows the lowest performance in terms of both RMSE and MAX. Therefore, it is difficult to obtain an accurate estimate when the system has a smaller total generator capacity. This indicates that the simple switching of models (Method 2) is insufficient under conditions with stringent reactive power supply limits.

According to the comparison of the results of the proposed method to those of Method 2, the addition of RPR information to the training data improves the estimation accuracy. Notably, the estimation performance of Model 1, with the worst accuracy in terms of both RMSE and maximum error, is significantly improved. Therefore, under system conditions with stringent reactive power supply constraints, the simple switching of estimation models is insufficient, and the incorporation of RPR information from generators is necessary.

Thus, RPR information from generators is a critical factor for voltage stability estimation, particularly under system conditions with a limited reactive power supply. The use of different estimation models according to the operational status of the generators leads to an enhanced estimation accuracy.

Furthermore,

Table 4. Computational times required for a single result.

	CPU Time
Proposed method	0.13 μs
CPF	0.361 s

presents a comparison of the computation time required to estimate the VSM at a single operating point. The simulation in this study is performed using a computer which is TSUKUMO eX.computer (JAPAN) with a 13th Gen Intel(R) Core(TM) i9-13900K 3.00 GHz processor. The results indicate that the proposed machine learning-based method significantly reduces computational burden. Additionally, the time required for the development of the estimation model is 1.58 s. Notably, the use of regression trees contributes to the method’s computational efficiency, highlighting a key advantage of the proposed approach.

The authors consider that this method can be applied to large-scale power systems, such as actual power grids. Incorporating the generator’s RPR as input information can be easily implemented in actual power grids. This enables the estimation model to clearly understand the relationship between system conditions and voltage stability. Furthermore, switching estimation models based on the operational status of generators can also be introduced in actual power grids. While the method of switching estimation models may require further consideration, it is expected that accuracy can be improved by switching models according to the number of operating generators, as proposed in this method.

5. Conclusions

In this study, the authors elucidated the relationship between RPR and voltage stability. The small RPRs significantly narrow the P–V curve due to the reactive power limitations of generators. The results of the estimations under varying reactive power limit conditions revealed that estimations using conventional methods are prone to significant errors.

The authors developed a method to estimate voltage stability with improved estimation accuracy by considering the changes in generator operating conditions. By including RPR information for generators in the training data, the estimation accuracy was significantly improved compared to the conventional method. Furthermore, the estimation using multiple estimation models based on the total generator capacity was found to be highly useful. The proposed method, combining these factors, improved RMSE by 81.65% and reduced the maximum error by 80.24%.

Voltage stability estimation methods in power systems under disturbances such as transmission line faults and tripping of generators could be subjects of future studies.