Short-term voltage stability constrained ORPF for real-time voltage control

Abstract

In power systems with high renewable energy penetration, short-term voltage stability (STVS) has become an increasingly critical issue. In future real-time automatic voltage control (AVC), sufficient STVS margins need to be incorporated into the optimization process. However, the strong coupling between the STVS and steady-state features makes it challenging to formulate suitable constraints for real-time optimization. This paper proposes a short-term voltage stability constrained optimal reactive power flow (STVSC-ORPF) model for real-time AVC. First, a short-term voltage stability index is employed, and it is mapped into device-level reactive power-related features and system-level aggregated features to approximate transient voltage responses within the state space. Then, a partitioned constraint construction method is proposed to achieve dimension reduction. Finally, feature coefficients are estimated from offline datasets via the least squares method. The constraints are embedded into the ORPF model and solved efficiently with gradient-based optimization algorithms. Case studies on the CSEE-VS 66-bus system demonstrate that the proposed strategy effectively enhances STVS while ensuring real-time computational efficiency.

1 Introduction

With the increasing penetration of power electronic devices and renewable energy, a large number of synchronous generators have been replaced by renewable units. As a result, the reactive power reserves from synchronous machines have decreased markedly (Jiang et al., 2021). The reduction in reactive power reserves weakens the ability to resist disturbances. The voltage becomes more sensitive, and the risk of short-term voltage instability issues increases (Yang et al., 2022). One of the causes of the “4.28” blackout in Spain was the failure of scheduled voltage support resources to provide effective control, which led to insufficient reactive power support, thereby triggering cascading instability and widespread outages (European Network of Transmission System, 2025). Ensuring that voltage and reactive power control solves current operation problems without creating new security risks has become a core challenge in power system with high renewable energy penetration (Sun et al., 2019). At present, voltage and reactive power control relies mainly on automatic voltage control (AVC) technology to ensure secure and economic operation (Ge et al., 2016). AVC can be implemented through OPF-based frameworks, including security-constrained OPF and voltage stability constrained OPF. Existing studies improve system security or stability by constructing relationships between steady-state operating variables and steady-state security or stability margins. However, short-term voltage stability (STVS) is characterized by post-fault voltage recovery behavior, and an explicit relationship between steady-state controllable variables and STVS has not yet been established. Consequently, existing OPF-based AVC frameworks cannot directly incorporate STVS into the optimization process. This limitation becomes more significant under high renewable penetration, where insufficient voltage support may lead to severe post-fault voltage recovery problems. Therefore, advanced control strategies are needed to improve STVS.

The problem of voltage and reactive power control considering short-term voltage stability constraints (STVSCs) is formulated as a high-dimensional optimization problem involving continuous and integer variables under nonlinear differential-algebraic equations. The key challenge is the construction of stability constraints. Early studies mainly adopted model-based analytical approaches, including sensitivity methods and numerical discretization methods. The sensitivity method evaluates system responses by calculating the numerical sensitivities of stability index with respect to control variables (Tang and Sun, 2017; Xia et al., 2014; Li et al., 2011). The numerical discretization method reformulates transient swing equations into sets of inequality constraints through discretization (Gan et al., 2000; Jiang and Geng, 2010; Jiang and Huang, 2010). These approaches impose excessive computational burdens, rendering them unsuitable for real-time AVC applications.

To overcome the limitations of conventional model-based analytical methods, recent studies have introduced data-driven strategies for constructing stability constraints. Existing studies can generally be classified into evaluation-based approaches and modeling-based approaches. The evaluation-based approaches rely mainly on machine learning and intelligent algorithms to assess stability margins implicitly under different operating conditions. In (Zhu et al., 2021), neural networks were applied to map operating conditions to stability margins, resulting in “black-box” constraints. These methods avoid the complexity of analytical modeling while capturing dynamic response characteristics. However, their inherent drawback is that the constructed constraints remain in “black-box” form without explicit formulation. The “black-box” form makes constraints unsuitable for embedding into gradient-based optimization algorithms (Costa et al., 2018). Moreover, their reliance on intelligent algorithms restricts online computational efficiency (Zhou et al., 2018).

Another category of modeling approaches attempts to directly construct stability constraints in the parameter space, representing stability boundaries as explicit algebraic constraints. Representative studies mainly include polynomial fitting (Xia et al., 2019; Wu et al., 2021) and security-region construction (Ge et al., 2020). In terms of polynomial fitting, a polynomial approximation method was proposed in (Xia et al., 2019), where system dynamic equations were expanded in the parameter space and the stability boundaries for transient rotor angles were characterized using low-order explicit polynomials. In (Wu et al., 2021), sensitivity index was introduced to refine the polynomial expressions, thereby improving their ability to capture transient characteristics in the parameter space. However, the high dimension of the parameter space in polynomial-based approaches significantly increases the computational complexity. To avoid high dimension of the parameter space, a limited set of aggregated features are used in practical applications. Such aggregated features fail to adequately capture the differentiated responses and regulation capabilities of various voltage support resources during transient processes. In security-region construction, voltage levels are maintained within acceptable ranges by setting limits on bus voltage magnitude and reactive power outputs (Ge et al., 2020). While this method provides explicit descriptions of stability boundaries, it often relies on strict voltage magnitude and reactive power limits. As a result, the feasible region is significantly reduced, and the combined impacts of various sources on STVS are not fully represented.

Overall, the STVS boundary is a highly nonlinear surface formed by the collective dynamic responses of numerous sources. Accurately characterizing this boundary is therefore challenging. In evaluation-based approaches, constraints are constructed without an explicit analytical form, which makes them unsuitable for embedding into optimization models. In model-based approaches, nonlinear constraints are typically adopted, where the formulation is restricted to aggregated features and device-level responses are not captured. In contrast, linear constraints avoid these issues and allow the inclusion of more device-level features, making them more suitable for real-time AVC applications. However, a single linear constraint cannot capture the nonlinear STVS boundary, leading to a conservative feasible region. The challenge is to construct constraints that reflect nonlinear stability characteristics while remaining computationally efficient.

To address the aforementioned issues, a STVSCs modeling approach for real-time AVC is proposed in this paper. Specifically, a linear relationship is first established between the short-term voltage stability index (STVSI) and key features, including system loads, renewable generation, reactive power reserves of dynamic reactive power sources (such as synchronous generators, synchronous condensers, and SVGs), and reactive power outputs of static reactive power compensators (such as capacitors and reactors). Then, a partitioned constraint construction method is proposed to achieve dimension reduction and to improve the efficiency of linear constraints formulation. Finally, the coefficients of the selected features are estimated using the least squares method based on an offline dataset. Compared with existing approaches, the proposed method provides explicit constraints that can be embedded into gradient-based optimization frameworks, making the proposed approach more suitable for real-time AVC applications.

The remainder of this paper is organized as follows. Section 2 presents the research motivation and overall technical route of the proposed STVSC-ORPF. Section 3 presents the construction of the STVSCs. Section 4 presents the model of the STVSC-ORPF. Section 5 provides case studies to validate and analyze the proposed method. Finally, Section 6 concludes this paper and discusses future research directions.

2 Motivation and technical route of the proposed STVSC-ORPF

The optimal reactive power flow (ORPF) constitutes an essential component of the AVC. In conventional power systems dominated by synchronous generators, the ability to provide reactive power is relatively strong, and the STVS issue is not significant. As a result, the STVSCs typically lie outside the feasible region, as illustrated in Figure 1a. Under most operating conditions, the optimization problem can therefore be solved without incorporating STVSCs.

FIGURE 1

In power system dominated by renewable energy, the reactive power support capability of the system is relatively limited, and the STVS issue becomes more critical. In this case, the STVSCs become active constraints, leading to a reduction in the feasible region, as shown in Figure 1b. It becomes necessary to incorporate the STVSCs into the OPRF model.

When STVSCs are represented in an explicit analytical formulation, they can be incorporated directly into ORPF algorithms, such as primal-dual interior-point algorithms. However, current methods fail to provide STVSCs in such a formulation. The “black-box” constraints obtained from data-driven techniques rely on intelligent algorithms and therefore cannot satisfy the real-time computational requirements of AVC (Zhu et al., 2021). The large number of device-level features that must be considered in AVC makes nonlinear constraint formulations impractical (Xia et al., 2019). These limitations highlight the need for an explicit analytical formulation that remains compatible with practical AVC implementations while preserving device-level features.

To address these limitations, a linear formulation of STVSCs is proposed in this paper. Real-time computational requirements of AVC can be satisfied through an explicit analytical form. Device-level features can be preserved through linear constraint formulation. Based on the STVSCs, the STVSC-ORPF model is proposed, through which dynamic reactive reserves are scheduled appropriately and STVS is improved during optimization.

The flowchart of the proposed STVSC-ORPF framework is illustrated in Figure 2. The framework consists of two processes: offline constraints construction and online real-time AVC optimization.

FIGURE 2

In the offline stage, the initial power grid data are first processed. The steady-state features and the corresponding STVSI are combined as one sample. All generated samples constitute the offline training dataset. The STVSCs are constructed using a regression-based data-driven approach. The least squares method (LSM) is applied to estimate the coefficients of a linear formulation, linking the STVSI with features such as reactive power reserves, reactive power outputs, system load, and renewable generation.

In the online stage, the STVSCs are incorporated into the ORPF model and participate in the real-time AVC. Since AVC operates on a short computation cycle and constraints reconstruction requires additional time, the update of STVSCs follows specific rules. Under normal operating conditions, system data are collected weekly to update the STVSCs. When the network topology or operating condition is expected to change, such as during line maintenance, the STVSCs need to be reconstructed. Essentially, these coefficients are updated periodically, acting as time-varying system parameters that remain constant over stable operating intervals.

3 Construction of short-term voltage stability constraints

3.1 Short-term voltage stability index

To quantify the severity of the STVS, the STVSI based on the multiple-two-element notation criterion (Xue et al., 2018) is adopted in this paper. The dimension of the voltage curve is reduced by the STVSI through the integration of the voltage–time trajectory. The complex transient process is compressed into a single scalar that effectively characterizes the severity of voltage dips or rises. For bus i under contingency p, a two-element notation criterion [U_cr, T_cr] is defined, consisting of the voltage magnitude threshold U_cr and the maximum allowable duration T_cr that the voltage magnitude can remain below the threshold. The corresponding STVSI is given in Equation 1:Where Ʌ_i,p is the STVSI of bus i under contingency p; S_m is the integrated area of voltage dips or rises; S_cr,m is the threshold defined by the two-element notation criterion; t_p and τ_p represent the time when the voltage first violates and then returns within the threshold, respectively; U_i(t) is the voltage curve of bus i; U_N is the rated voltage. The system is considered unstable when Ʌ_i,p > 1, and a larger STVSI indicates more severe STVS in the system. Contingencies that cause the system to be unstable are referred to as instability contingencies. A quantifiable measure of STVS is provided by the STVSI, enabling the establishment of analytical relationships with steady-state operating features. The threshold parameters {[0.75p.u.,1s], [0.9p.u.,10s]} are specified according to the standard of the China Southern Power Grid (China Southern Power Grid, 2009).

3.2 Features selection based on AVC and stability mechanisms

System loads have a big influence on STVS. When a contingency occurs and the voltage magnitude decline, induction motors rapidly draw additional reactive power, further deteriorating STVS. Therefore, system loads are regarded as important features.

Renewable generation is included as important feature. Variations in renewable generation substantially change the steady-state operating conditions and the voltage support capability of the system. These changes may further affect the transient response when a contingency occurs.

To meet the requirements of AVC, the reactive power reserves of dynamic reactive power sources and the outputs of static reactive power compensators under steady-state operating conditions are selected as key features.

The reactive power reserve of the i-th dynamic reactive power source is defined in Equation 2:Where Q_i,max and Q_i are the maximum reactive power and the current reactive power output of the i-th dynamic reactive power source. The reactive power reserve of dynamic sources is regarded as an effective indicator of the STVS level in the system (Sun et al., 2017). When a contingency occurs and the voltage magnitude decline, rotating machines are driven into the overexcitation state by the excitation system, allowing a large amount of reactive power to be supplied within a short time to support the transient process. Static var generators (SVGs) can possess overcurrent capability and deliver reactive power outputs that exceed their rated capacities during transient process. If the reactive power reserve is low, sufficient reactive power may not be provided in time to maintain the STVS, which could lead to severe voltage fluctuations or instability. In contrast, the outputs of static reactive power compensators vary with the square of the voltage magnitude. Since these compensators cannot provide active support during transient process, making them another critical factor that influences STVS (Jiang et al., 2021).

It should be noted that the dynamic reactive power support capability of renewable energy plants strongly depends on inverter control modes and grid-forming capabilities. Accurately modeling such characteristics is beyond the scope of this work and remains a challenging problem. Therefore, to focus on the proposed STVSC-ORPF framework without affecting the validity of the main conclusions, the reactive power output limit of each renewable energy plant is assumed to be equal to its apparent power capacity. The upper and lower bounds of the reactive power output for the i-th renewable unit in the system are given in Equations 3, 4:

Where S_new,i and P_new,i are the apparent power and the active power of the i-th renewable unit, respectively. The reactive power support capability of renewable unit is provided by the inverter, and its available reactive power reserve is constrained by the current active power output. During scenarios of high renewable generation, the reactive power reserve significantly decreases, making it essential to include this characteristic as an independent feature in the STVS analysis.

The schematic of mapping system features to STVS in the power injection space is shown in Figure 3.

FIGURE 3

3.3 Grid partitioning for feature dimension reduction

For a large-scale power system, the number of both reactive devices and the selected features can be large, which makes data-driven approaches inefficient. Moreover, for each instability contingency, the impact on the system is limited in scope and primarily concentrated within a local area. The dominant features influencing STVS therefore tend to be confined to a particular region rather than being distributed across the entire power system. Grid partitioning is thus adopted to restrict the linear constraint construction to buses with similar transient behavior.

Basically, grid partitioning involves the following two steps: defining an “electrical distance” that describes the degree of “difference” between two buses, and then merging the buses into clusters according to “electrical distance”. Specifically, a vector-type STVS feature of each bus is constructed based on the STVSI under a predefined contingency. Cosine similarity is then employed to quantify the similarity of voltage dynamic characteristics between buses, which is treated as the transient electrical distance. Buses with similar transient responses are clustered into the same partition using the method in (Wang et al., 2016).

Grid partitioning confines the modeling of STVSCs within each partition, thereby significantly reducing feature dimension. It also enhances the rationality and efficiency of constraint formulation.

3.4 Linear formulation of STVSCs

The STVS boundary is determined by the dynamic interactions of numerous devices and is inherently nonlinear. Particularly in power systems with high renewable energy penetration, the shape of STVS boundary is strongly influenced by the nonlinear control characteristics of renewable energy converters, making accurate characterization highly challenging.

In constraint construction, neither linear nor nonlinear constraints are inherently superior; the choice between them represents a trade-off. In theory, nonlinear constraints provide higher approximation accuracy and are therefore widely used for stability assessment and boundary fitting. However, as the feature dimension increases, the number of coefficients to be identified grows rapidly due to the presence of cross terms, resulting in a significant increase in computational burden. To alleviate the curse of dimensionality, nonlinear methods often rely on a limited number of aggregated features. Such aggregated features are generally unable to capture the differences in dynamic responses and regulation capabilities among various types of reactive power support devices.

By contrast, linear constraints sacrifice part of the approximation accuracy, but the number of coefficients to be identified is substantially reduced under the same feature set. More features can be incorporated into the constraint construction, making linear formulations more suitable for ROPF applications. Nevertheless, linear constraints alone are generally unable to closely represent the highly nonlinear STVS boundary of power systems. As a result, the estimated secure operating region may be underestimated, leading to conservative optimization results.

The STVS boundary within each partition can be represented by a linear approximation. Multiple linear constraints are then used to piecewise fit the nonlinear boundary. In this way, computational efficiency is preserved while maintaining the ability to characterize dynamic behaviors of systems. A schematic illustration of the proposed partitioned constraint construction method is shown in Figure 4.

FIGURE 4

Distinct STVS boundaries are observed under different operating conditions due to variations in renewable generation and system load characteristics. These differences are reflected by the selected system-level features, including system loads and renewable generation. Meanwhile, the grid partitioning is adopted according to the voltage support provided by static and dynamic reactive power sources. It reduces the dimension of features and the number of constraints. As shown in Figure 4, multiple linear constraints are used to approximate the nonlinear STVS boundary across different partitions under different scenarios.

Within each operating scenario, all considered contingencies are first simulated. For partition n, the most severe contingency is defined as the contingency that produces the largest STVSI within the partition. The most severe STVSI is defined in Equation 5:

Where N_c, N_bus,n, N_area are the total numbers of contingency, buses within partition n (n=1,2,…,N_area), system partitions, respectively; Λ_i,p,n is the STVSI of bus i under contingency p within partition n. Λ_severe,n is selected as the target variable for linear regression.

The final linear STVSC is expressed in Equation 6:Where Λ_set,n is the threshold of STVSI within partition n, which can be determined by the system operator based on practical system conditions; ω_i,n, ω_j,n, ω_k,n, ω_l,n, ω_m,n are the coefficients corresponding to the reactive power reserves or outputs of synchronous generators, synchronous condensers, SVGs, renewable units, and capacitors within partition n, respectively; N_g,n, N_sc,n, N_svg,n, N_new,n, N_com,n are the number of synchronous generators, synchronous condensers, SVGs, renewable units, and capacitors within partition n, respectively; Q_res,g,i,n, Q_res,sc,i,n, Q_res,svg,i,n, Q_res,new,i,n are the reactive power reserves within partition n, respectively; Q_com,m,n is the reactive power output of the capacitor within partition n; α_n, β_n, γ_n are the coefficients of active power loads, reactive power loads and renewable generation within partition n, respectively; P_L,n, Q_L,n, P_NEW,n are the total active power loads, reactive power loads, and the renewable generation within partition n, respectively; b_n is other intrinsic factor within the partition that affect STVS.

The coefficients obtained in the linear constraints can be used to quantitatively evaluate the impact of a unit change in the corresponding features on STVS.

3.5 Coefficient fitting for STVSCs

In this paper, the features are manually selected based on the AVC and stability mechanisms to reveal the differentiated roles of various types of reactive power sources during transient processes. For this reason, regression methods with regularization, such as lasso or ridge regression, are not adopted. These methods perform feature selection by introducing penalty terms. Penalty terms may weaken or even eliminate physically meaningful features that are critical to STVS, which conflicts with the modeling objective of this paper. Therefore, the least squares method (LSM) is employed to fit the linear relationship between the STVSI and key features. The coefficients are obtained by minimizing the sum of squared errors between the predicted and actual values of each sample. The formulation for the regression coefficients is expressed in Equation 7:Where β is the vector of regression coefficients of the features; y is the output STVSI vector corresponding to each sample; X is the feature matrix of the samples.

4 STVSC-ORPF mathematical formulation

The objective function aims to minimize the total active power losses of the system, as formulated in Equation 8:Where Ω_s is the set of system buses, i, j∈ Ω_s; G_ij is the mutual conductance between buses i and j; V_i and V_j are the voltage magnitudes at buses i and j, respectively; θ_ij is the difference in voltage phase angle between buses i and j.

Power flow equality constraints can be formulated in Equations 9, 10:where P_g,i and Q_g,i are the active and reactive power of synchronous generator at bus i, respectively; P_new,i and Q_new,i are the active and reactive power of renewable unit at bus i, respectively; P_L,i and Q_L,i are the active power and reactive power loads at bus i, respectively; Q_svg,i, Q_sc,i, Q_com,i are the reactive power from SVG, synchronous condenser, and capacitor at bus i, respectively; B_ij is the mutual susceptance between buses i and j.

Bus voltage magnitude constraints can be formulated in Equation 11:Where V_i,max and V_i,min are the upper and lower bounds of voltage magnitude at bus i.

Reactive power constraints can be formulated in Equations 12–16:Where the superscripts min and max indicate the upper and lower bounds of the respective reactive power sources.

The partitioned STVSCs are given in Equation 6. The proposed STVSC-ORPF model involves discrete control variables, such as capacitor switching states, which cannot be directly handled by conventional interior-point methods. This challenge is typically addressed through relaxation or mixed-integer extensions. In this study, a quadratic penalty function is incorporated into the optimization model, following method described in (Liu et al., 2002), to achieve a continuous relaxation. The relaxed problem is subsequently solved using the primal-dual interior-point method.

5 Example analysis

5.1 Case system description

To show the effectiveness of the proposed method, the CSEE-VS 66-bus standard system (Zhao et al., 2023) is adopted. The system topology (500 kV buses only) is shown in Figure 5. This system is an AC/DC hybrid network incorporating photovoltaic and wind power plants, with a renewable penetration level exceeding 50%. The reactive power sources include synchronous generators (Gens) with reactive power limits of [−150, 300] MVar, synchronous condensers (SCs) with limits of [−150, 300] MVar, SVGs with limits of [−100, 100] MVar, and 17 buses equipped with capacitor banks (Caps).

FIGURE 5

To cover the diversity of typical operating conditions in high-renewable power systems, a dataset is constructed that incorporates uncertainties in renewable generation, load levels, and reactive power sources. Renewable generation is randomly sampled between 80% and 120% of the rated capacity based on historical fluctuations. The proportion of induction motors in the load model is assumed to vary between 30% and 60% to reflect the impact of different load compositions on STVS.

For reactive power support modeling, the voltage setpoints of synchronous generators and synchronous condensers are uniformly sampled within [0.95, 1.05] p. u. The reactive power outputs of SVGs are randomly selected between 0% and 80% of their rated capacity. Capacitors are discretely adjusted according to the reactive power demand of adjacent buses to reflect typical operating conditions. In total, 6,008 operating scenarios are generated, with 20 N-1 contingencies considered for each scenario. Each contingency is set as a three-phase short-circuit contingency occurring at t = 1 s and cleared after 0.1 s. To further illustrate the diversity of the generated dataset, the operating scenarios are classified into low-, medium-, and high-renewable penetration groups according to the renewable generation level. Specifically, scenarios with renewable penetration below 40% are defined as low-penetration cases, scenarios between 40% and 50% are defined as medium-penetration cases, and scenarios above 50% are defined as high-penetration cases. The corresponding numbers of operating scenarios are 1,462, 2,996, and 1,550, respectively. Considering the N-1 contingencies for each scenario, a total of 120,160 valid samples are obtained and divided into training, validation, and testing sets with ratios of 7:2:1, corresponding to 84,120, 24,040, and 12,000 samples, respectively. This distribution ensures that the dataset covers a broad range of renewable penetration levels and provides sufficient samples for evaluating the proposed STVSCs under different operating conditions.

All case studies are performed on a computer with an AMD Ryzen 7 5800H 3.2 GHz CPU, 16 GB RAM, and an NVIDIA GeForce RTX 3070 8 GB Laptop GPU.

5.2 Comparative analysis

To show the effects of both feature representation and constraint formulation, four STVSC construction methods are investigated, including linear constraints with device-level features, linear constraints with aggregated features, polynomial constraints with device-level features, and polynomial constraints with aggregated features. For the device-level features, the key features identified in Section 3.2 are adopted, including the reactive power reserves of synchronous generators, synchronous condensers, SVGs, and renewable units, the reactive power outputs of static reactive power compensators, the active and reactive loads and renewable generation within each partition. For the aggregated features, the sums of the reactive power reserves of dynamic reactive power sources and the outputs of static reactive power compensators within each partition are chosen as key features. The same training, validation, and testing datasets are adopted for all methods. Their fitting performances are compared using MAE, MSE, ACC, FA and MA. Statistical metrics can be formulated in Equations 17–21:

Table 1 shows that both the feature representation and the constraint formulation influence the fitting performance of the STVSCs. From the perspective of feature representation, the device-level features consistently outperform the aggregated features under both linear and polynomial formulations. This shows that preserving the individual reactive power support characteristics of different voltage support resources is beneficial for accurately capturing the STVS boundary. Under the linear formulation, the device-level features reduce the MAE from 8.46% to 7.28% and improve the ACC from 82.04% to 86.39%. Meanwhile, both FA and MA are significantly reduced, demonstrating a better capability to distinguish stable and unstable operating conditions. Similar trends can also be observed under the polynomial formulation.

From the perspective of constraint formulation, the polynomial constraints provide only marginal improvements over the linear constraints. When device-level features are adopted, the polynomial constraints reduce the MAE from 7.28% to 7.05% and the MSE from 9.48 × 10⁻³ to 9.02 × 10⁻³. However, the ACC decreases from 86.39% to 86.25%. This is because MAE and MSE evaluate the overall regression accuracy, whereas ACC depends on whether the predicted STVSI crosses the predefined stability threshold. Although the polynomial formulation improves the global fitting accuracy, it may shift a small number of samples located near the stability boundary to the opposite side of the threshold, resulting in a slight decrease in accuracy. Therefore, the improvement in regression performance does not necessarily translate into a higher ACC. A similar phenomenon is observed for the aggregated-feature cases. Although the polynomial constraints improve all statistical metrics, the improvements remain relatively limited. Specifically, the ACC increases only from 82.04% to 82.58%, while the MAE decreases by merely 0.03 percentage points.

Regarding computational efficiency, polynomial constraints require substantially longer construction and optimization times than their linear counterparts. For the device-level feature set, the constraint construction time increases from 0.029 s to 0.082 s, while the online ORPF optimization time increases from 22 s to 71 s. Similar trends can be observed for the aggregated-feature cases. This increase is mainly attributed to the higher computational complexity introduced by polynomial cross terms, which result in additional nonzero Hessian terms and larger KKT systems during the interior-point iterations. In contrast, the proposed linear STVSCs maintain a simple analytical form with zero second-order derivatives, leading to significantly lower online computational burden while preserving device-level information.

Overall, the linear constraints with device-level features provide the most balanced solution. Compared with aggregated features, device-level features achieve higher fitting accuracy and lower misclassification rates. Although polynomial formulations can improve the accuracy, the improvement is limited and comes with additional computational complexity. Therefore, for real-time AVC applications, preserving device-level reactive power support information is more beneficial than introducing higher-order polynomial formulations. Consequently, the proposed linear STVSCs with device-level features achieve a favorable balance among accuracy, interpretability, and computational efficiency.

For the STVSC-ORPF analysis, only two representative schemes are selected for dynamic response comparison, the polynomial STVSCs with aggregated features (PSTVSCs) and the proposed linear STVSCs with device-level features (LSTVSCs). The former represents a typical nonlinear constraint formulation adopted in existing studies, while the latter represents the proposed constraint construction framework. The constraints of the PSTVSCs and LSTVSCs are given in Tables 2, 3.

The fitted coefficients not only represent the sensitivities of STVSI to the corresponding features, but also provide insights into the mechanisms of STVS. Specifically, the coefficients associated with dynamic reactive power reserves of synchronous generators, synchronous condensers, and SVGs are predominantly negative, indicating that larger reactive power reserves generally reduce STVSI and improve voltage recovery capability under contingency.

The coefficients associated with load-related features are generally positive, suggesting that higher loading levels tend to increase STVSI. This is because increased load demand leads to higher reactive power consumption. Similarly, renewable generation-related features exhibit positive coefficients in several partitions, reflecting the fact that higher renewable penetration may weaken system voltage support capability and increase the difficulty of post-fault voltage recovery.

The proposed device-level STVSCs explicitly distinguish the contributions of different reactive power resources. Therefore, the fitted coefficients provide quantitative information regarding the relative influence of synchronous generators, synchronous condensers, SVGs, static reactive compensators, loads, and renewable generation on STVS. Compared with conventional black-box machine learning models, this improves the interpretability of the constructed constraints and facilitates engineering analysis of voltage support mechanisms. From an engineering perspective, the proposed device-level STVSCs enable the contributions of different reactive power resources to be explicitly distinguished. The fitted coefficients provide quantitative information regarding the relative effectiveness of synchronous generators, synchronous condensers, SVGs, and static reactive power compensators in supporting post-fault voltage recovery.

The analysis results are shown in Figures 6, 7.

FIGURE 6

FIGURE 7

Figures 6, 7 show the voltage responses following PSTVSC-ORPF and LSTVSC-ORPF, respectively. For both methods, the minimum voltage magnitudes, voltage recovery times, and final voltage levels are very similar. The minimum voltage magnitude of the 500 kV buses remains approximately 0.2 p.u., while the voltage recovers above 0.9 p.u. within about 1.5 s. Similar recovery characteristics are observed for the 220 kV buses. Consequently, the resulting STVSI values obtained by the two methods are close, indicating that both PSTVSCs and LSTVSCs can effectively improve STVS.

However, differences can be observed in the post-fault voltage curves. Under PSTVSC-ORPF, oscillations appear after voltage recovery, particularly in the 500 kV buses, where persistent fluctuations can be observed during the later stage of the simulation. By contrast, the voltage curves obtained by LSTVSC-ORPF are smoother. This phenomenon can be attributed to the fact that PSTVSCs are constructed using aggregated features, which cannot distinguish the individual regulation effects of different reactive power sources. As a result, the optimized reactive power allocation may not achieve sufficient coordination among various dynamic and static reactive power resources. In contrast, LSTVSCs employ device-level features and explicitly capture the contributions of different reactive power sources, enabling a more coordinated allocation of reactive power support. Therefore, although both methods achieve comparable voltage recovery performance and STVSI reduction, LSTVSC-ORPF provides better damping characteristics and smoother post-fault voltage responses.

To further justify the adopted feature set, an ablation study is conducted by removing one feature group at a time while keeping all other settings unchanged. The results are summarized in Table 4.

Among all ablation cases, removing static reactive compensation features leads to the most significant performance degradation. The ACC decreases from 86.39% to 71.37%, while the MAE and MSE increase from 7.28% to 9.48 × 10⁻³–11.28% and 18.64 × 10⁻³, respectively. Meanwhile, the FA sharply increases from 30.25% to 71.00%. This indicates that static reactive power compensation features are essential for characterizing the STVS boundary. Although static reactive power compensators cannot actively increase reactive power output during severe voltage depressions, their pre-fault compensation affects the reactive power distribution and the available voltage support margin of the system. Removing dynamic reactive reserve features also causes performance degradation. The ACC decreases from 86.39% to 82.47%, and the MA increases from 5.63% to 6.62%. This shows that the reactive power reserves of synchronous generators, synchronous condensers, SVGs, and renewable units are also important for STVS characterization, because they directly determine the reactive power output during severe voltage depressions. The removal of load-related features and renewable-generation-related features results in relatively smaller degradation. Specifically, the ACC decreases to 83.42% and 84.01%, respectively. These features mainly describe the operating condition of the system rather than directly providing voltage support. Nevertheless, they still influence power flow distribution, reactive power demand, and the available reactive power reserves. Therefore, excluding these features also reduces the accuracy of the constructed STVSCs.

Overall, the ablation results show that static reactive compensation and dynamic reactive reserve features are the two most critical feature groups. This confirms that the proposed STVSC construction should preserve both the steady-state reactive compensation level and the dynamic voltage support capability.

The sensitivity of STVSC-ORPF results to the number of partitions is shown, as illustrated in Figure 8.

FIGURE 8

The results show that the partitioning strategy is not merely a dimension-reduction technique. It also improves the capability of the constructed STVSCs to characterize regional stability characteristics. Quantitatively, under the initial operating state, the voltage drops to approximately 0.05 p.u. after the contingency and fails to recover above 0.9 p.u. within the 10 s simulation period. When the entire system is represented by a single STVSC, the post-fault voltage recovery is improved, but the voltage remains below 0.9 p.u. for most of the simulation period and only approaches this level near the end. In contrast, with three partitioned STVSCs, the voltage recovers above 0.9 p.u. within about 1 s and remains close to 1.0 p.u. This demonstrates that the partitioned constraints improve post-fault voltage recovery.

When the entire system is represented by a single STVSC, the constraint is constructed using features distributed across the whole system. The resulting constraint reflects only the overall stability tendency of the system and cannot capture local voltage support characteristics. By introducing grid partitioning, buses with similar transient characteristics are grouped together and independent STVSCs are constructed for each partition. Grid partitioning allows the STVS boundary to be represented more effectively.

To show the effectiveness of the proposed STVSC-ORPF, three typical operating scenarios are selected. The renewable generation and load characteristics for three scenarios are shown in Figure 9.

Then, overall optimization results of the ORPF and the STVSC-ORPF are analyzed, with the number of iterations and optimization outcomes summarized in Table 5.

Compared with the ORPF optimization results, the feasible solution space is reduced by the incorporation of additional STVSCs in the STVSC-ORPF. As a result, the optimal solution corresponds to higher power losses than does the ORPF, achieving a greater stability margin at the expense of reduced economic efficiency. During the iterations, both the ORPF and the STVSC-ORPF converged within 30 iterations, demonstrating the rapid convergence property of the interior-point method. Even with the inclusion of STVSCs, the optimal solution can still be obtained within an acceptable computation time, thereby satisfying the practical requirement of rapid computation for AVC. The reactive power adjustments of controllable sources after STVSC-ORPF optimization are summarized in Table 6.

FIGURE 9

The results in Table 6 show that the impacts of the proposed STVSC-ORPF vary among device types, partitions, and operating scenarios. In the fault-near partitions (Partitions 1 and 2), the reactive power outputs of several capacitors are reduced after optimization. Meanwhile, many dynamic reactive power sources, including synchronous generators, SCs, and SVGs, are dispatched toward operating points with larger available reactive power reserves. However, this trend is not universal for all devices or scenarios, since the final operating points are jointly determined by network conditions, reactive power requirements, and device constraints.

It should be noted that reactive power output and reactive power reserve represent different concepts. The optimization does not necessarily increase the reactive power output of every dynamic device. Instead, it aims to improve the overall transient voltage support capability of the system by coordinating the steady-state reactive power outputs and the available reactive power reserves of different resources. Consequently, a balance between static reactive compensation and dynamic reactive support can be achieved, leading to improved post-fault voltage recovery performance.

To show the adaptability of the proposed method, the optimization results of ORPF and STVSC-ORPF under different scenarios are compared, as shown in Figure 10.

FIGURE 10

For a given operating scenario, the observed voltage curves of different buses exhibit similar trends. Therefore, the STVSI of a representative bus is selected for quantitative analysis, and the results are shown in Figure 11.

FIGURE 11

When a contingency occurs, the voltage curves of the initial operating state exhibit instability, characterized by low voltage magnitude and prolonged voltage depression. This behavior is primarily attributed to insufficient reactive power support under high demand for system reactive power. Limited reactive power reserves, delayed or constrained dynamic reactive power responses, and increased reactive power absorption by loads jointly weaken the ability of the system to maintain STVS.

In different scenarios, the traditional ORPF is optimized based on steady-state voltage distributions without accounting for STVS. The optimization tends to rely more on static reactive power compensators, whose reactive power output is proportional to the square of the voltage. Under severe voltage depressions, the reactive power support capability of these devices decreases. By contrast, after implementing the STVSC-ORPF, the share of static reactive power compensators is reduced, while a larger proportion of reactive power is allocated from dynamic reactive power sources.

Under the conventional ORPF, the STVSI is reduced by only 13.3% and 16.8% in Scenarios A and B, respectively. Although a larger reduction of 52.6% is achieved in Scenario C, the optimized STVSI remains above the stability threshold. The proposed STVSC-ORPF reduces the STVSI by 95.5%, 89.8%, and 91.1% in Scenarios A, B, and C, respectively. The optimized STVSI values are all reduced below the stability threshold of 1.

These results show that the constructed STVSCs guide the allocation of reactive power resources toward improving transient voltage support capability rather than merely satisfying steady-state voltage requirements. The simulation results validate the effectiveness of the proposed STVSC-ORPF in enhancing STVS under different operating conditions.

On this basis, further analyses are conducted by considering different renewable penetration levels to examine the performance of the proposed STVSC-ORPF. Starting with the basic operating condition (Baseline), part of the renewable generation near bus B19 is sequentially replaced with an equivalent output from synchronous generators. As a result, three comparative operating conditions are obtained, as summarized in Table 7.

The voltage responses before and after optimization under different penetration levels are compared in Figure 12. The STVSI of representative bus under different renewable penetration levels are shown in Figure 13.

FIGURE 12

FIGURE 13

With increasing renewable energy penetration, a clear deterioration of system voltage stability is observed. On the one hand, renewable units are predominantly connected through power electronic converters and cannot provide the same level of transient voltage support as synchronous generators. On the other hand, the replacement of synchronous generators reduces the available dynamic reactive power reserves, thereby weakening the capability for rapid voltage recovery following disturbances.

The quantitative results are shown in Figure 13. Under the Case A, the initial STVSI reaches 3.265, indicating a severe short-term voltage instability risk. Traditional ORPF reduces the STVSI to 2.222, corresponding to a reduction of 31.9%. However, the optimized STVSI remains higher than the stability threshold. In contrast, the proposed STVSC-ORPF reduces the STVSI to 0.967, achieving a 70.4% reduction. When the operating point is close to the STVS boundary, conventional ORPF may even introduce new stability risks. As shown in Case C, the STVSI increases from 1.606 to 1.816 after ORPF optimization, representing a deterioration of 13.1%. Similarly, in Case D, the STVSI increases from 1.023 to 1.369, corresponding to a deterioration of 33.8%. These results show that traditional ORPF may sacrifice STVS. In contrast, the proposed STVSC-ORPF reduces the STVSI to 0.487 and 0.191 in Cases C and D, corresponding to reductions of 69.7% and 81.3%, respectively.

For Cases C and D, the STVSI after conventional ORPF increases compared with the initial operating point. This is because ORPF primarily improves reactive power dispatch without explicitly considering STVS. In low renewable penetration scenarios, the system already possesses sufficient voltage support, and the optimization may reduce the available dynamic reactive power reserve of critical devices. Consequently, although the steady-state operating condition is improved, the STVS margin may deteriorate slightly.

In practical power systems, the operating conditions may differ from those represented in the offline dataset. Therefore, additional unseen operating conditions are constructed to evaluate the generalization capability of the proposed STVSCs. Comparisons of STVSI before and after STVSC-ORPF under N-1 and N-2 contingency are given in Table 8. Voltage curves of unseen operating condition are shown in Figures 14, 15.

FIGURE 14

FIGURE 15

As shown in Figure 14a, the voltage response of the red curve is different from those of the other monitored buses. This is because the corresponding bus is electrically closest to the B03-B05 contingency. It suffers the most severe voltage depression during the fault and becomes the weakest bus in the post-fault recovery process. Under the initial operating state, the voltage of this bus drops to 0 p.u. during the fault period and only recovers to 0.61 p.u. at 10 s. In contrast, the other monitored buses recover to 0.80 p.u. by the end of the simulation.

As shown in Figure 14b, the voltage recovery performance is improved after applying the proposed STVSC-ORPF. The red curve recovers above 0.9 p.u. within 1 s and finally stabilizes close to 1.0 p.u. The other monitored buses also recover to approximately 1.0 p.u., and the voltage differences among buses are significantly reduced. These results show that the proposed STVSC-ORPF effectively improves the voltage recovery of the weakest bus in unseen operating condition under N-1 contingency.

As shown in Figure 15a, B03-B05 and B05-B08 contingencies are more severe than the N-1 contingency. Compared with the N-1 contingency, the voltage recovery of all monitored buses becomes slower. Under the initial operating state, the red curve drops to nearly 0 p.u. during the fault and only recovers to 0.60 p.u. at 10 s. The other monitored buses recover to approximately 0.77p.u., which is also slower than the N-1 case. This indicates that the N-2 contingency causes a wider and more severe voltage support deficiency.

As shown in Figure 15b, after applying the proposed STVSC-ORPF under the N-2 contingency, most monitored buses recover above 0.9 p.u. within 2 s and finally stabilize around 1.0 p.u. However, the red curve still exhibits a slow recovery process. Its voltage magnitude remains below 0.9 p.u. before 5 s. This result is expected because the proposed STVSCs are designed to enhance voltage support capability rather than to eliminate the impact of all severe disturbances. For the unseen N-2 contingency, the optimized operating point provides stronger reactive power support and improves voltage recovery performance. However, the disturbance severity exceeds the voltage support margin available under the current operating condition, leading to a residual slow voltage recovery process.

Finally, the adaptability of STVSC to different system topologies is further shown. Assuming the line connecting B06 to B05 and B08 is disconnected, the system topology is modified, and the simulation results are presented in Figure 16.

FIGURE 16

Quantitatively, under the original topology shown in Figure 16a, the minimum post-fault voltage is 0.2 p.u., and all monitored buses recover above 0.95 p.u. within 1 s. The voltage curves remain close to the pre-fault operating point.

When the system topology changes and the original STVSCs are directly applied, as shown in Figure 16b, the minimum voltage magnitude remains similar. However, the post-fault voltage recovery performance deteriorates. Several buses exhibit a gradual voltage decline after the recovery stage, and the final voltage levels decrease from 0.97–1.02 p.u. to 0.82-0.95 p.u. The resulting optimization fails to provide sufficient voltage support during the transient process. This is because the constraint coefficients are obtained from the steady-state to transient-state mapping relationship under the original topology. Once the system topology changes, the underlying mapping relationship is altered, leading to a mismatch between the fitted constraints and the actual system.

After reconstructing the STVSCs using the updated topology, the voltage recovery performance is restored, as shown in Figure 16c. All monitored buses recover above 0.9 p.u. within 2 s and converge to voltage levels close to 1.0 p. u. These results show that the proposed method relies on a fixed network topology and that the STVSCs should be updated when significant topology changes occur.

6 Conclusion

With the growing penetration of renewable energy, STVS has been regarded as one of the most critical challenges because of the reduced availability of reactive power support. In this paper, a STVSC-ORPF framework for AVC is proposed to enhance system resilience against disturbances in power systems with high renewable penetration. Within this framework, a linear functional is established between the STVSI and key features under steady-state operating conditions to represent the stability constraints. The STVSCs are embedded into the ORPF model, ensuring compatibility with primal-dual interior-point algorithms and satisfying the computational requirements of AVC. Nevertheless, several aspects deserve further investigation. Future work will focus on improving the adaptability of the proposed framework under changing operating conditions and validating its effectiveness on large-scale practical power systems.

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