A New Method for Constructing the Static-Voltage-Stability Boundary with a Conversion Strategy

Featured Application

Voltage stability monitoring in a power grid.

Abstract

The static-voltage-stability region (SVSR) is an effective tool for monitoring the safe operation of grid voltage. Rapidly obtaining the static-voltage-stability-region boundary (SVSRB) of a power system is crucial for the application of SVSR technology. First, this paper reveals the mechanisms underlying the failure scenarios that may occur with conventional boundary tracking methods. Then, an improved method incorporating curve extrapolation and correction conversion strategies is proposed, which enhances the efficiency of constructing the SVSRB. Furthermore, the analytical expression of the SVSRB is derived from the intermediate information obtained during the predictor stage. Finally, simulation examples based on a simple power system, the IEEE 3-machine, 9-bus power system and IEEE 300-bus power system were developed to verify the accuracy and efficiency of the proposed method.

1. Introduction

With the rapid expansion of the power network and the ongoing development of renewable energy in the power grid, the power system exhibits more complex and various characteristics. These characteristics have changed the operating behavior of the power system [1,2,3], making the voltage operation more likely to approach the boundary or even lose stability [4,5], representing a significant risk to the safe operation of the power system [6]. Hence, characterizing the critical limit state of the system, especially on a background of uncertainty brought by widespread access to new energy resources, is of great significance to improve the safe operation level in terms of the static voltage stability of large power systems.

At present, the main methods for depicting the critical limit state of a power system include the maximum transmission capacity method [7,8], continuity power flow (CPF) method [9,10], fitting method [11,12], approximation method [13,14], etc. Once the critical-limit-state points are determined, the static-voltage-stability-region boundary (SVSRB) monitoring the system voltage can be constructed. In addition, the distance between the current operation points and the SVSRB is an effective index to evaluate the current system’s stability margin and can provide auxiliary decision-making information [15,16].

Many efforts have been made by researchers to construct the SVSRB [17,18]. Reference [19] detected the critical-voltage-limit points via the CPF method; thus, the load margin could be determined. To promote the robustness of the CPF method practically, some variants based on the CPF method and Newton’s technique were developed to estimate stability limits and load margins [20]. The nonlinear power injection variations are modeled as a piecewise linear approximation, and the effect of nonlinear load on the load margin, i.e., the SVSRB, was analyzed by the CPF method in reference [21]. Overall, the SVSRB constructed by the CPF method is achieved by continuously adjusting the direction of load change to acquire the boundary point, which is essentially a collection of several power flow calculations. In addition to the CPF method, combining the boundary morphology for effective fitting is also a good way to depict the SVSRB. In reference [22], a method based on the Taylor expansion technique is proposed to approximate the SVSRB in the presence of saddle node bifurcation (SNB), improving the efficiency of boundary construction. In reference [23], the SVSRB under reactive power limits is considered, and the global approximation of the SVSRB formulated in polynomial expression is given based on Galerkin’s method. Reference [24] formulated the construction of the SVSRB as a parametric problem, and a novel global approximation method is proposed to depict the shape of the SVSRB based on the polynomial analytical expression. These fitting approximation class methods are usually able to give an approximate expression of the SVSRB, which can provide a basis for the quantitative evaluation of stability, but those methods usually face the dilemma of complicated computation. Generally speaking, each of these methods has its advantages and disadvantages in constructing the SVSRB. The various methods are summarized in

Table 1. Summary and comparison of different SVSRB construction methods.

Methods	Principles	Advantages	Disadvantages	Future Directions
The maximum transmission capacity method	Calculate the maximum transmission capability of the power system	Provides numerical properties of stable boundary points Characterizes the stable boundary from a physical point of view	Requires a lot of numerical computation Defines the transmission capability of the system in advance	Develop a more practical numerical calculation method for maximum transmission power
The continuity power flow method	Carry out continuous power flow calculation until divergence	Provides numerical information near the divergent point of power flow Describes the stable boundary from the voltage perspective	Requires a lot of power flow calculations Difficulty in calculating boundary points.	Develop a more practical and robust continuous power flow algorithm
The fitting/approximation method	Fit the boundary of stability with approximate expression	Provides geometric estimates of boundary shapes Implements the analytic expression of the boundary.	The fitting effect is affected by the boundary shape The analytic expression of the boundary is too complex	Develop more practical and concise boundary fitting methods

below.

Recently, reference [25] presents the SVSRB construction method based on the continuous parameter tracking calculation, which uses any boundary point as the initial value and quickly migrates to the entire stable boundary. However, a large number of simulations have found that there is some divergent scenario happened occasionally. Therefore, this paper firstly analyzes the divergence of the method in the predictor and corrector stages mentioned in reference [25], respectively, and analyzes the mathematical mechanism of the calculation failure. Based on the causes of the divergent scenario, adaptive strategies, i.e., the alternate curve prediction strategy and orthogonal-arc-correction conversion strategy, are proposed to alleviate the computational burden and divergence difficulty. To sum up, the original contributions of this paper are listed below:

• Divergence mechanism: summarizing the reason for the tracking failure to construct the SVSRB in the predictor stage and corrector stage, respectively.
• Improving methods: proposing the curve extrapolation and correction conversion strategies to promote the efficiency of constructing the SVSRB.
• Analytical SVSRB expression: utilizing the intermediate data in the predictor stage to approximate the SVSRB piecewise.

The remainder of this paper is structured as follows: In Section 2, the basic definition of the SVSRB and the continuous parameter tracking method are reviewed. In Section 3, the solving process of the parameter tracking method and the typical failure scenario are presented. In Section 4, the improving boundary tracking method is proposed to promote the efficiency of constructing the SVSRB. In Section 5, the proposed method is verified in simple power system, the WECC 3-machine, 9-bus power system. Section 6 provides the conclusion.

2. Static Voltage Stability Boundary

2.1. The Definition of Static-Voltage-Stability Boundary

Without a loss of generality, the power flow equation can be written as follows:

$$\mathit{f}(\mathit{x},\mathit{y})=\mathbf{0},$$

(1)

where x is the voltage variable, i.e., amplitude and phase angle of the node voltage; y is the nodal injection power variable, i.e., the active and reactive power at different buses. The static-voltage-stability region (SVSR) Ω indicates the area where the power system can maintain stable operation, and its boundary $\partial \Omega$ is composed of several SNB points as follows:

$$\partial \Omega :=\left\{(\mathit{x},\mathit{y})|\mathit{f}(\mathit{x},\mathit{y})=\mathbf{0},{\displaystyle \frac{\partial \mathit{f}}{\partial \mathit{x}}}=\mathbf{0}\right\}.$$

(2)

Geometrically speaking, the SVSR $\Omega$ is a binary tuple (x, y) consisting of the power system state variable x and the control variable y. Given that high-dimensional boundaries are difficult demonstrated, dimensionality reduction is usually achieved through the boundary function h:

$$h:\mathit{y}\to \mathit{x},\mathit{y}\in \partial \Omega .$$

(3)

According to Equation (3), the SVSRB is equivalent to a stable boundary binary tuple (x, y) by characterizing it in the power injection space, i.e., the space of coordinate y, as shown in Figure 1.

As is shown in Figure 1, the SVSR under an active power injection space with the selected system key nodes i and j can be depicted by the shaded areas.

2.2. Stable Boundary Construction with Continuous Method

To achieve the construction of the SVSRB, referring to the continuous parameter tracking algorithm proposed in reference [25], the equation representing all SNB points in a two-dimensional power plane can be characterized as follows:

$$\Psi (S)=\left\{\begin{array}{l}f(x)+\eta {\lambda }_k{\overrightarrow{\mathit{b}}}_k(\alpha )=\mathbf{0}\\ J^{\mathrm{T}}(x)L=\mathbf{0}, L^{\mathrm{T}}L=\mathbf{1}\end{array}\right.,$$

(4)

where $S={[x,L,\eta ,\alpha ]}^{\mathrm{T}}\in {ℝ}^{2n+1}$ is the unknown variable, $\eta$ is the continuous parameter, and ${\lambda }_k$ is the load margin under the direction of ${\overrightarrow{\mathit{b}}}_k(\alpha )$ as follows:

$${\overrightarrow{\mathit{b}}}_k(\alpha )={\left[\mathbf{0},\cos ({\beta }_k+\alpha ),\mathbf{0},\sin ({\beta }_k+\alpha ),\mathbf{0}\right]}^{\mathrm{T}},$$

(5)

where $\alpha$ is the offset on angle ${\beta }_k$, and ${\beta }_k$ is the initial power-growth-direction angle, as shown in Figure 2.

As is shown in Figure 2, the initial power-growth-direction angle ${\beta }_k$ can be written as follows:

$${\beta }_k=\arctan (\Delta p_j/\Delta p_i),$$

(6)

where Δp_i and Δp_j are the power increments along with direction β_k, as shown in Figure 2. Notice that $\eta$ is the newly added continuous parameters used to characterize the quantitative impact on the load margin ${\lambda }_k$ when β_k is shifted to $\alpha$ in Equation (4). Figure 3 demonstrates the principle diagram of constructing the SVSRB using the continuous parameter tracking method, i.e., Equation (4).

The CPF simulation can be deployed in the b₀ direction to reach the first boundary point S₀. Thus, the initial values for the boundary construction in Equation (4) can be obtained. Notably, ${\eta }_0=1$ and $\alpha =0$ are satisfied at the boundary point S₀. By setting a continuous parameter $\alpha$ with an angle perturbation of size $\Delta \alpha$, the direction vector changes from b₀ to b₁. Correspondingly, the small change $\Delta {\eta }_0$ of continuous parameter ${\eta }_0$ is calculated, which is added to the previous calculation value $\eta$ to update ${\eta }_1$. The product of ${\eta }_1$ and the initial margin ${\lambda }_0$ can represent the load margin under the direction vector $b_1$. By continuously changing the parameter $\Delta \alpha$ and tracking the parameter $\Delta {\eta }_0$, all boundary points can be obtained; hence, the SVSRB is formed.

3. Analysis of Boundary Tracking Methods

3.1. The Solving Process of the Parameter Tracking Method

Equation (4) is usually solved using the predictor corrector technique, which is mainly divided into two stages: the tangent estimation variable $S^{pre}$ and normal hyperplane correction to the exact value $S^{cor}$. It is mainly described as follows:

(1)

Tangent estimation variable $S^{pre}$.

To obtain the tangent vector of $\Psi$ at the $S_0$ point in Equation (4), the incremental equation can be supplemented as follows:

$$\epsilon ·\Delta \alpha {|}_{s_0}=d,$$

(7)

where $\epsilon$ is a known numerical constant representing the relative magnitude of change in $\Delta \alpha$, and $d\in \{1,-1\}$ is the direction indication factor specifying the direction of change in $\Delta \alpha$.

Simultaneously using Equations (4) and (7), we have the following:

$$\left\{\begin{array}{l}\Psi (S)=\mathbf{0}\\ e·\Delta S=D\end{array}\right.,$$

(8)

where $e$ and $D$ are appropriate dimension vectors, that can be written as follows:

$$\left\{\begin{array}{l}\mathit{e}=[0,\cdots ,0,1]\\ \mathit{D}={[0,\cdots ,0,d]}^{\mathrm{T}}\end{array}\right..$$

(9)

Therefore, the tangent vector ΔS can be written via inversion operation from Equation (8) as follows:

$$\Delta S={\left[\begin{array}{l}{\displaystyle \frac{\partial \Psi }{\partial S}}\\ e\end{array}\right]}^{-1}·D.$$

(10)

Hence, the estimated value in the predictor stage can be obtained as follows:

$$S^{\mathrm{pre}}=S+\sigma ·\Delta S,$$

(11)

where $\sigma$ is the step-size control factor used to manipulate the relative distance between the estimated value and the initial value in the predictor stage.

(2)

Accurate value of hyperplane correction

To obtain the corrected value, the hyperplane equation passing through the ${\mathit{S}}^{\mathrm{pre}}$ point and perpendicular to vector $\Delta S$ can be written as follows:

$$\Delta S^{\mathrm{T}}·(S-{\mathit{S}}^{\mathrm{pre}})=0.$$

(12)

Simultaneously using Equations (4) and (12), we can obtain the following:

$$\Phi (S)=\left\{\begin{array}{l}\Psi (S)=\mathbf{0}\\ \Delta S^{\mathrm{T}}·(S-{\mathit{S}}^{\mathrm{cor}})=0\end{array}\right..$$

(13)

Notice that Equation (13) is a nonlinear system with multiple coupled variables. Hence, the Newton–Raphson method can be applied to solve it with $S^{\mathrm{pre}}$ as the initial value. Specifically, the iterative scheme can be written as follows:

$$S^{\mathrm{cor}}=S_0-{\left[{\displaystyle \frac{d\Phi }{dS}}\right]}^{-1}\Phi (S_0).$$

(14)

3.2. Typical Failure-Scenario Analysis

Many simulations have found that inefficient or non-convergent boundary-point tracking occurs during the calculation process, mainly in the following two scenarios:

(1)

Tangent estimation stage

In the process of tangent estimation, the calculation of the tangent involves the inverse operation of the gradient matrix. The dependent variable $\Delta S$ of the systems (10) exhibits high-dimensional characteristics, and the cost of inverse operation is relatively high. Furthermore, when the boundary presents a harsh form, it often consumes machine time and resources, as shown in Figure 4.

As is shown in Figure 4, due to the significant boundary shape changes at the boundary point S₃, the estimated value ${\overline{S}}_4^{pre-\tan }$ via the tangent method, i.e., Equation (11), is far from the true value of S₄, which can easily cause the phenomenon of difficult convergence in the correction stage. If the information regarding S₁, S₂ and S₃ is fully utilized in the prediction process and S₄ $S_4^{pre-fit}$ is fitted outside the boundary, the distance between the predicted point and the exact value is close and the divergent situation can be avoided.

(2)

Hyperplane correction stage

The correction stage of the hyperplane is the process of correcting the S^pre given in the tangent estimation stage to the boundary point $S^{\mathrm{cor}}$. The solution of this stage mainly depends on the initial value and the correction method. The initial value is mainly determined in the predictor stage. In certain scenarios, the phenomenon of tracking boundary points becomes ineffective due to the influence of correction methods. Suppose there is a convergence domain as follows:

$${\Theta }_k=\left\{\left.S\right||S-S_k|<\epsilon ,\Psi (S_k)=0\right\},$$

(15)

where $\epsilon$ is a small parameter constant characterized the shape of convergence domain. As is shown in Figure 5, the SNB saddle points along the P–V curve in the b₁ and b₂ directions are too far away, i.e., ${\Theta }_1\cap {\Theta }_2=\varnothing$, resulting in a divergence simulation from S₁ to S₂.

It is obvious that the corrected methods can affect the performance of computation. When the hyperplane in Equation (13) is applied, it is geometrically impossible to intersect with the boundary curve. Hence, Equation (13) cannot converge mathematically. Thus, more reliable correction methods need to be explored.

4. Improving Boundary Tracking Methods

4.1. Alternate Curve Prediction Strategy

The estimation stage mainly provides iterative initial value in the correction stage. The tangent estimated method, as depicted by Equation (11), can be ineffective during the prediction process mentioned in Section 3. As is mentioned in Equation (4), the essence of the continuous method can be interpreted as all variables being regarded as the function of a small parameter, as follows:

$$\varphi :\alpha \in \mathcal{E}\mapsto S=\left[\begin{array}{c}\begin{array}{c}\mathit{x}(\alpha )\\ \mathit{L}(\alpha )\\ \eta (\alpha )\end{array}\end{array}\right]\in {ℝ}^{2n+1},$$

(16)

where $\varphi$ is continuous mapping; $\mathcal{E}$ is the set of the continuous parameter $\alpha$. Consider that for the complicated inverse operation of the gradient matrix, an alternating estimation method can be proposed based on the changing shape of the boundary curve and the known boundary point data, as shown in Figure 6.

Based on the function relation revealed by Equation (16), the SVSRB can be approximated by a polynomial function locally. As is shown in Figure 6, S₀, S₁ S₂ and S₃ can be acquired via Equation (11) and T₀, T₁, T₂ and T₃ are the corresponding tangent vectors, respectively. θ₀₁, θ₁₂ and θ₂₃ are the angles between the corresponding tangent vectors. We can observe that S₄^pre can be obtained by the curve extrapolation method rather than the tangent method if θ₀₁, θ₁₂ and θ₂₃ are increasing monotonically, because the boundary morphology varies slowly, as shown in Figure 6.

Consider a series of points (α₀, x₀), (α₁, x₁), (α₂, x₂), …, (α_k, x_k) in the mapping of $\varphi$ based on Equation (16), where α_k is the value under the k-th time step, and x_k is the corresponding component based on Equation (16). The Lagrange interpolation polynomial can be formulated as follows:

$$P(\alpha )={\displaystyle \sum _{j=0}^k{x}_j}{L}_j(\alpha ),$$

(17)

where $x_j$ is the j-th of vector x_j, i.e., x_j = [x₁, x₂, …, x_n], k is the number of fitting points and L_j(α) is the Lagrange coefficient as follows:

$$\begin{array}{ll}{L}_j(\alpha )=& {\displaystyle \prod _{i=0,i\ne j}^k\left(\frac{\alpha -{\alpha }_i}{{\alpha }_j-{\alpha }_i}\right)}={\displaystyle \frac{\left(\alpha -{\alpha }_0\right)}{\left({\alpha }_j-{\alpha }_0\right)}}\times {\displaystyle \frac{\left(\alpha -{\alpha }_1\right)}{\left({\alpha }_j-{\alpha }_1\right)}}\times \cdots \\ & \times {\displaystyle \frac{\left(\alpha -{\alpha }_{j-1}\right)}{\left({\alpha }_j-{\alpha }_{j-1}\right)}}\times {\displaystyle \frac{\left(\alpha -{\alpha }_{j+1}\right)}{\left({\alpha }_j-{\alpha }_{j+1}\right)}}\times \cdots \times {\displaystyle \frac{\left(\alpha -{\alpha }_k\right)}{\left({\alpha }_j-{\alpha }_k\right)}}\end{array}$$

(18)

As is shown in Figure 6, (α₁, x₁), (α₂, x₂) and (α₃, x₃), i.e., S₁, S₂ and S₃, are known points, and the alternate strategy is satisfied: θ₀₁ < θ₁₂ < θ₂₃. Then, the next point, S₄^pre, can be written as follows:

$$\begin{array}{ll}{x}_4& ={\displaystyle \frac{\left({\alpha }_4-{\alpha }_2\right)\left({\alpha }_4-{\alpha }_1\right)}{\left({\alpha }_3-{\alpha }_2\right)\left({\alpha }_3-{\alpha }_1\right)}}{x}_3\\ & +{\displaystyle \frac{\left({\alpha }_4-{\alpha }_1\right)\left({\alpha }_4-{\alpha }_3\right)}{\left({\alpha }_2-{\alpha }_1\right)\left({\alpha }_2-{\alpha }_3\right)}}{x}_2\\ & +{\displaystyle \frac{\left({\alpha }_4-{\alpha }_2\right)\left({\alpha }_4-{\alpha }_3\right)}{\left({\alpha }_1-{\alpha }_2\right)\left({\alpha }_1-{\alpha }_3\right)}}{x}_1\end{array},$$

(19)

where α₄ = α₃ + Δα, Δα is an known step size. Therefore, the procedure of applying the tangent and extrapolation prediction method can be summarizes in

Table 2. Tangent and extrapolation prediction method.

Tangent and Extrapolation Prediction

Pre-requirements: current time step k, a series of the boundary point, i.e., SK = [xK, LK, η, α], and the corresponding tangent vector TK, where K = {k, k − 1, k − 2, k − 3}, step size Δα. STEP 1: Calculate the angle between the tangent vector TK, marked as θ(k−3)−(k−2), θ(k−2)−(k−1), θ(k−1)−(k). STEP 2: Judge whether θ(k−3)−(k−2) < θ(k−2)−(k−1) < θ(k−1)−(k). If satisfied, turn to STEP 3. If not satisfied, turn to STEP 4. STEP 3: Calculate the $S_{k+1}^{\mathrm{pre}}$ based on Equation (19). Calculate the angle θ(k)−(k+1) based on the newly predicted point $S_{k+1}^{\mathrm{pre}}$. STEP 4: Judge whether θ(k)−(k+1) > 0. If satisfied, turn to STEP 6. If not satisfied, turn to STEP 5. STEP 5: Calculate $S_{k+1}^{\mathrm{pre}}$ based on the tangent method. STEP 6: Output the predicted value $S_{k+1}^{\mathrm{pre}}$.

below.

4.2. Orthogonal-Arc-Correction Conversion Strategy

As is mentioned in Section 3, the hyperplane correction method may fail, which is caused by the fact that the normal hyperplane cannot cross the SVSRB. The series curve clusters in Figure 6 can be projected into the power injection space, as shown in Figure 7.

It can be observed that different correction methods can affect convergence. If Equation (13) is adopted, it is difficult to converge when the boundary curve cannot be intersected geometrically, as shown in Figure 7. Conversely, the arc-correction method can guarantee convergence because of the intersection with the SVSRB. Thus, the arc supplementary equation can be formulated as follows:

$$\begin{array}{ll}{\sigma }^2& ={\displaystyle \sum _{j=1}^n{(x_j^{cor}-x_j)}^2}+{\displaystyle \sum _{j=1}^n{(L_j^{cor}-L_j)}^2}\\ & +{({\alpha }^{cor}-{\alpha }_j)}^2+{({\eta }^{cor}-{\eta }_j)}^2\\ & \triangleq <{\mathit{S}}^{cor},{\mathit{S}}_j>\end{array},$$

(20)

where x_j is the j-th component of x_j, and L_j is the j-th component of L_j, and ${\mathit{S}}_j=[x_j,L_j,{\alpha }_j,{\eta }_j]$ in Equation (4). $\kappa$ is the arc radius; the symbol $<·,·>$ represents the inner product of two vectors, and the corrected variable can be written as ${\mathit{S}}_j^{cor}=[x_j^{cor},L_j^{cor},{\alpha }^{cor},{\eta }^{cor}]$.

Simultaneously using Equations (4) and (20), we can obtain the following:

$$\Phi ({\mathit{S}}^{cor})=\left\{\begin{array}{r}\Psi ({\mathit{S}}^{cor})=\mathbf{0}\\ 〈{\mathit{S}}^{cor},{\mathit{S}}_j〉-{\kappa }^2=0\end{array}\right.,$$

(21)

where the distance between S_j and S^pre can be set to $\kappa$, i.e., $\left|S_j{S}^{pre}\right|=\kappa$. And, Equation (21) can be illustrated with a graph as shown in Figure 8 below:

Notice that Equation (21) is a multivariable-coupling nonlinear system, which can be solved by the Newton–Raphson method with S^pre as the iterative initial value. Combining this with hyperplane correction in Equation (13), the arc-correction Equation (21) can be used as a backup scheme to ensure the correction process has a solution eventually. Thus, the orthogonal-arc-correction strategy proposed in this section is summarized in

Table 3. Orthogonal-arc-correction strategy.

Orthogonal-Arc-Correction Strategy

Pre-requirements: boundary point Sj, iterative initial value Spre, the maximum iteration number Ktol and iteration error Etol. STEP 1: Supplementary hyperplane Equation (13); construct Newton–Raphson iterative scheme based on Equation (14). STEP 2: Deploye numerical calculation, and obtain the number of iterations m and iteration error err. STEP 3: Judge whether m < Ktol. If satisfied, turn to STEP 4. If not satisfied, turn to STEP 4. STEP 4: Judge whether err < Etol. If satisfied, turn to STEP 7. If not satisfied, turn to STEP 5. STEP 5: Construct the arc-correction format based on Equation (21), and conduct the Newton–Raphson calculation. STEP 6: Output correction point Scor; this time-step correction ends.

below.:

4.3. Algorithm Flow

As is mentioned above, the boundary tracking algorithm can be divided into three: algorithm initialization, the alternate curve prediction strategy and the orthogonal-arc-correction conversion strategy. The detailed algorithm is formatted as shown in Figure 9.

5. Case Studies

5.1. Single-Machine, Single-Load System

The single-machine, single-load system and the system parameters are shown in Figure 10. The SVSRB is constructed as the active and reactive power of the load bus change.

The conventional boundary tracking method proposed in reference [25] is deployed to construct the SVSRB of the single-machine, single-bus power system. The coordinate of the ground state point N is set as [ΔP, ΔQ]^T = [0.8660, 0.5]^T. Thus, the initial power growth direction can be calculated as β = arctan(ΔP/ΔQ). The CPF method is applied to initiate the boundary point S₀ = [θ, V, l₀, l₁, 1, 0]^T. Starting from S₀ as the initial point, the proposed method in this paper and the conventional tracking method were used to search for each boundary point for comparison, as shown Figure 11.

As is shown in Figure 11, the voltage of the load bus can be monitored by the SVSR. Once the SVSRB is constructed, the voltage stability can be determined by the load consumption, i.e., the active and reactive power of the load. Moreover, it can be seen that the proposed method can effectively achieve accurate boundary fitting, and the boundary constructed by the traditional method is consistent in general, which further demonstrates the effectiveness of the proposed method. The detailed numerical result of some boundary points in Figure 11 are shown in

Table 4. Detailed numerical results of boundary points.

No.	S	θ	V	l0	l1	η	α
0	S0	−0.464	0.559	0.667	0.744	1	0
1	$S_1^{\mathrm{pre}}$	−0.381	0.541	0.383	0.928	0.852	0.300
	$S_1^{\mathrm{cor}}$	−0.382	0.538	0.373	0.927	0.855	0.302
2	$S_2^{\mathrm{pre}}$	−0.314	0.527	0.301	0.947	0.792	0.578
	$S_2^{\mathrm{cor}}$	−0.336	0.525	0.309	0.951	0.795	0.578
3	$S_3^{\mathrm{pre}}$	−0.090	0.501	0.090	0.996	0.903	1.404
	$S_3^{\mathrm{cor}}$	−0.089	0.502	0.089	0.995	0.906	1.403
4	$S_4^{\mathrm{pre}}$	0.001	0.499	0.001	1.00	1.035	1.641
	$S_4^{\mathrm{cor}}$	0.002	0.500	−0.002	0.999	1.038	1.639

.

Based on Equation (16), all variables can be regarded as the function of the continuous parameter. Figure 12 demonstrates the variation in x₁, x₂, L₁, L₂ and η with respect with continuous parameter α.

It can be seen from Figure 12 that the continuous tracking method itself can reflect the characteristics of continuous changes in other variables when the parameters continue to change, and it has good fitting characteristics. Hence, combined with the curve extrapolation technique, not only does it avoid the complex matrix inversion operation caused by traditional tracking methods in the predictor process, but the intermediate calculation results can also be effectively saved, providing an approximate expression for boundary stability in a linear estimation manner. The approximate linear expressions for some boundary curves are shown in

Table 5. Analytical expression of the hyperplane.

Hyperplane	Analytical Expressions	Q2
H1	1.600Q2 + P2 + 0.088 = 0	0.180 < Q2 < 0.205
H2	2.001Q2 + P2 + 0.260 = 0	0.205 < Q2 < 0.230
H3	5.200Q2 + P2 + 1.098 = 0	0.230 < Q2 < 0.240
H9	5.200Q2 + P2 − 0.366 = 0	0.040 < Q2 < 0.070
H10	1.010Q2 + P2 − 0.467 = 0	0.000 < Q2 < 0.040

.

As is illustrated in Table 5, the SVSRB can be fitted by linear expression. The system stability can be determined and quantified by the distance between the current operation point and the analytical linear expression. Furthermore, once the analytical boundary is given, it can provide quantitative analysis results for problems of uncertainty. Moreover, to investigate the adaptive mechanism of the algorithm in the face of harsh working condition, random disturbance is artificially applied to the system voltage to simulate the unpredictable operation conditions of a power system, and the SVSRB constructed by a different method under disturbed conditions is shown in Figure 13.

Both types of algorithms can construct the SVSRB under disturbed working conditions, as shown in Figure 13. In terms of conservative algorithms, taking the analytical boundary derived from reference [25] as the standard, the original method is more conservative compared with the proposed method, while the proposed algorithm has a good robust behavior in dealing with such working conditions. Generally speaking, the distance between the predicted value and the true value will greatly affect the number of iterations for solving the nonlinear equation in the correction stage. The number of iterations of the conventional tracking method proposed in reference [25] and the method presented in this paper in the correction stage of searching each boundary point are shown in Figure 14.

It can be seen from Figure 14 that, in most cases, the number of iterations of the proposed method is smaller than that of the conventional tracking method, indicating that the proposed algorithm has high robustness compared with the conventional method. Especially at S₀, S₁, S₃ and S₆, due to the curve extrapolation method, the distance between the predicted point and the exacted point is relatively close, which makes the number of iterations in the correction process less.

5.2. WECC 3-Machine, 9-Bus Testing System

The WECC 3-machine, 9-bus testing system was deployed to verify the accuracy and effectiveness of the proposed method for constructing the SVSRB, as shown in Figure 15. The detailed parameters of the WECC 3-machine, 9-bus testing system can be found in reference [26].

Bus 9 and bus 7 were selected as the key nodes to construct the corresponding SVSRB with the active power variations. The SVSRBs constructed by the conventional tracking method and the proposed method are shown in Figure 16. It can be observed that both methods can effectively track the boundary and highly overlap at the boundary points.

The accuracy of different methods in constructing the SVSRB were compared and analyzed. In theory, the SVSRB is composed of a series of SNB points. Hence, the determinant of the Jacobian matrix at each SNB boundary point system should be zero, and the minimum eigenvalue of the system is zero. Due to the inevitable numerical errors brought about by iterative computations, the Jacobian matrix at each boundary point cannot be strictly zero ideally. Thus, the accuracy of different methods can be measured using the minimum eigenvalue of the Jacobian matrix corresponding to each boundary point, as shown in Figure 17.

From Figure 17, it can be observed that in the WECC 3-machine, 9-bus power system, the conventional method and the proposed method maintain the same minimum eigenvalue of the Jacobian matrix at the initial tracking point S₁ (because they have the same initial boundary point). The minimum eigenvalue of the Jacobian matrix at each boundary point, except S₁, is smaller than that of the conventional tracking method, indicating that the accuracy of the proposed method in tracking boundary points is better than that of the conventional method.

To compare the timeliness of different methods in constructing the SVSRB, the timeliness was measured by the consumed computation time, as shown in Figure 18.

From Figure 18, the time difference for constructing the SVSRB in a single-machine, single-load system is relatively minor in general, but the proposed method still has advantages in timeliness. In complex systems, i.e., a 3-machine, 9-bus power system, as the size of the computational scale increases, the differences in time consumption become increasingly clear, and the proposed method has the advantage of saving computation time, indicating that the proposed method can construct boundaries quickly.

Compared with the traditional CPF method, the proposed method reduces the time complexity because it avoids the point-by-point calculation in the boundary construction process. To be specific, supposing that the boundary shape is known in advance as a standard circle, the number of two-dimensional boundary points can be set to n for discretization processing, as shown in Figure 19 below.

The time complexity of the proposed method is O(n) = n, while for the CPF method is O(n) = m·n$\approx$n², where m is the computation time performed in a CPF calculation along a particular direction, as shown in Figure 19, and $m>>n$ is satisfied in most cases. And, this time gap will be further widened in the construction of a three-dimensional SVSRB.

5.3. IEEE 300-Bus Test Power System

It can be seen from Section 5.1 and Section 5.2 that the proposed method has advantages both in accuracy and timeliness. In order to verify the feasibility of this method in the process of constructing the stability boundary of large-scale power systems, it was further verified on an IEEE 300-bus power system. The detailed parameters of the IEEE 300-bus power system can be found in reference [26].

As is shown in Figure 20, bus 25, bus 26 and bus 27, with dense loads in the network, are selected as key nodes, and the SVSRB is constructed as the active power varies. As can be seen from Figure 19, the proposed method can provide an estimate of the stable boundary in a large example scenario, which further demonstrates the robustness of the proposed method.

6. Conclusions

In this paper, a static-voltage-stability-domain boundary tracking method was proposed that considers curve extrapolation and a correction conversion strategy. This method avoids the iterative divergence phenomenon of conventional continuous methods in constructing stability domains, promoting the performance in constructing the SVSRB. The relevant conclusions are as follows:

• The typical failure scenario with the conventional continuous method is pointed out, and the reason for tracking failure to construct the SVSRB is analyzed.
• A boundary tracking method considering curve extrapolation and a correction strategy was proposed in the predictor and corrector stages, avoiding iterative divergence.
• The intermediate data in the predictor stage were utilized for providing the exact analytical expression of the boundary curve, achieving segmented analysis of the SVSRB.

This method can provide an effective voltage monitoring tool for the current power grid, helping to ensure the safe and stable operation of power grid voltage. The focus of follow-up work is to carry out research on voltage stability assessment and control decision-making in relation to large power grids based on the SVSRB technique, and propose a more practical voltage analysis method.