Static Voltage Stability Zoning Analysis Based on a Sensitivity Index Reflecting the Influence Degree of Photovoltaic Power Output on Voltage Stability

Abstract

The large-scale integration of photovoltaic (PV) power can bring a greatly negative influence on the grid-connected system’s voltage stability. To study the static voltage stability (SVS) of PV grid-connected systems, the traditional SVS index, L-index, was re-examined. It was firstly derived and proved that the PV active output P_pv is proportional to the voltage phase angle of the PV station’s POI (Point of Interconnection), based on a simplified two-node system integrated with a PV station operating in PV (active power—voltage) mode or PQ (active power—reactive power) mode with unit power factor. Then a novel voltage stability sensitivity index LPAS-index was proposed that takes the derivative of the L-index with respect to the POI’s voltage phase angle, so as to reflect the influence degree of P_pv on the SVS of each load node. A SVS zoning analysis method for the PV grid-connected system was designed according to the classification results of load nodes based on the proposed LPAS-index, the power grid can be zoned into three kinds of areas that reflect different correlations between the SVS and P_pv: strong correlation, moderate correlation and weak correlation. Since the LPAS-index is less impacted by P_pv, the SVS zoning results are relatively unchanged. On the basis of a classic 14-node system with PV, the practicability of the zoning analysis method was verified. The simulation results show that the PV access point in general falls within the strongly or moderately associated area with P_pv. When most of the load nodes fall within the weakly associated area with P_pv, it is not necessary to consider the impact of P_pv and load power is still the main influencing factor on the SVS. In the multi-PV case, owing to the expansion of areas more affected by P_pv, an excessive P_pv can cause adverse influence on the SVS of the whole power grid; and an effective PV power-shedding measure is proposed to solve this problem. The proposed SVS zoning analysis method can be used for reference by power grid dispatchers.

1. Introduction

In order to alleviate problems such as energy shortage and environmental pollution and achieve the goal of ‘dual carbon’, China has made great efforts to develop photovoltaic (PV) power generation technology in recent years [1,2]. With the maturity of PV grid-connected technology and the continuous reduction of installation costs, PV power generation shows a development trend of large-scale grid-connected operation and becomes the main new-energy in the construction of modern power systems [3]. In China, according to the latest data released by the National Energy Administration, the installed capacity of grid-connected PV power generation reached 392.04 GW by the end of 2022, and centralized PV accounts for about 60 percent of the total installed capacity. The PV power output is greatly affected by natural and meteorological factors and has a strong random fluctuation and intermittence [4,5]. Meanwhile, the integration of large-scale PV intensifies the power electronic characteristics and reduces the power system inertia, and brings more effects and challenges to the static and dynamic stability of the power system [6,7,8], in particular the static-state (steady-state), small-disturbance, transient-state and long-term voltage stability [9,10,11].

Static voltage stability (SVS) refers to the voltage stability when the power system sustains various small disturbances (such as small changes in load power) without considering the dynamic characteristics of each electric element [12]. In recent years, the studies on the SVS of traditional AC power systems have been still a hot spot and various machine learning algorithms are applied to the SVS prediction issue [13,14,15].

In the context of the integration of large-scale PV power stations, what are the main factors affecting the SVS of the power grid? How to determine the fluctuation range of SVS critical point, SVS domain and SVS margin; how to analyze the impact of PV power fluctuation on the weakest SVS area; how to design a novel SVS index (or criterion) that can apply to the PV power fluctuation and how to use certain appropriate control strategies to improve the system’s SVS are all the main concerns.

The research on the SVS issue of the large-scale PV station grid-connected system has been reported in relevant references. In Reference [16], the impacts of multiple factors were studied on the SVS, including PV transmission line length, PV active output, and PV topological structure, etc. In Reference [17], the impacts of several factors such as solar irradiance, PV generation power factor, PV installed capacity and PV transmission line impedance, were further studied.

In Reference [18], the SVS margin of the IEEE 14-node system with PV under the two modes of unity power factor operation and constant voltage operation was calculated, and it is considered that the integration of large-scale PV is conducive to improving the SVS margin. However, it was pointed out that too much PV active output can decrease the SVS margin in Reference [9]. In Reference [19], an assessment for the probabilistic SVS margin was studied, by using the probabilistic model of PV/wind power and the Monte-Carlo simulation method.

Several novel indexes that are suited for the SVS analysis in the system with large-scale PV stations were proposed. In Reference [20], the influence of the PCC (Point of Common Coupling) of the PV station on the SVS was studied by defining a sensitivity index of load node voltage—PCC active power that can reflect the impact of PV penetration rate on the SVS. In Reference [21], an improved NVSI-index based on the traditional IVSI-index was used to measure the SVS of PCC, which directly considers the impact of PC/wind injection power. A short-circuit ratio index was proposed in Reference [22], which is more related to the SVS of a weak sending-end system with PV. In Reference [23], a synthetical application framework was proposed to measure the SVS of a PV grid-connected system, considering the critical eigenvalue by modal analysis, the reactive power margin by QV (reactive power—voltage) analysis and the line-loss by power flow analysis.

The SVS control for the system with a large-scale PV station was also explored. In Reference [24], a SVS fuzzy controller was designed, synthetically adopting the load node voltage and the load-margin index considering PV power fluctuation as the fuzzy input variables. In Reference [25], SSSC (Static Synchronous Series Compensator) was used to improve the SVS of weak nodes considering the integration of high penetration PV/wind.

In the above references, the influence law of PV power size and fluctuation on the SVS of load nodes, load areas and the whole power grid is not studied. In the SVS analysis of large-scale PV grid-connected systems, in addition to the frequently-used small disturbance condition of gradually increasing load power, the impact of PV power output should also be considered when the system load is at a certain level.

From the above concerns, a traditional SVS index, L-index [26], is re-examined in this paper. Firstly, the relationship between the L-index and PV active output P_pv is derived. Then a novel sensitivity index of L-index of each load node relative to P_pv is proposed. According to the numerical level of the proposed sensitivity index value, a zoning method for the SVS analysis is proposed and verified. Compared with previous studies, the proposed zoning analysis method can reveal the influence degree on the SVS of different areas in the power grid by P_pv, and some effective measures can be suggested to improve the SVS according to the zoning results.

2. A Novel Sensitivity Index of L-Index Relative to PV Active Output

2.1. The Traditional L-Index

The traditional L-index is a local SVS index and was first proposed by Kessel [26], which is used to monitor and evaluate the SVS of load nodes in the traditional power system.

For a large-scale PV grid-connected system, the node-voltage equation of the system is given below:

$$\left[\begin{array}{c}{\dot{\mathit{I}}}_{\mathbf{L}}\\ {\dot{\mathit{I}}}_{\mathbf{G}}\\ 0\end{array}\right]=\left[\begin{array}{ccc}{\mathit{Y}}_{\mathbf{LL}}& {\mathit{Y}}_{\mathbf{LG}}& {\mathit{Y}}_{\mathbf{LC}}\\ {\mathit{Y}}_{\mathbf{GL}}& {\mathit{Y}}_{\mathbf{GG}}& {\mathit{Y}}_{\mathbf{GC}}\\ {\mathit{Y}}_{\mathbf{CL}}& {\mathit{Y}}_{\mathbf{CG}}& {\mathit{Y}}_{\mathbf{CC}}\end{array}\right]\left[\begin{array}{c}{\dot{\mathit{V}}}_{\mathbf{L}}\\ {\dot{\mathit{V}}}_{\mathbf{G}}\\ {\dot{\mathit{V}}}_{\mathbf{C}}\end{array}\right]$$

(1)

In Equation (1):

Y_LL, Y_LG, Y_LC, Y_GL, Y_GG, Y_GC, Y_CL, Y_CG and Y_CC correspond to the sub-matrices of the grid-connected system’s node admittance matrix, respectively.

${\dot{\mathit{V}}}_{\mathbf{L}}$ and ${\dot{\mathit{I}}}_{\mathbf{L}}$ are the load nodes’ voltage and current vectors, respectively.

${\dot{\mathit{V}}}_{\mathbf{G}}$ and ${\dot{\mathit{I}}}_{\mathbf{G}}$ are the power nodes’ voltage and current vectors, respectively, including slack bus, PV (active power—voltage) bus of synchronous generator, and POI (Point of Interconnection) of PV or PV converter outlet bus.

${\dot{\mathit{V}}}_{\mathbf{C}}$ and ${\dot{\mathit{I}}}_{\mathbf{C}}$ are the contact nodes’ voltage and current vectors respectively, and the contact node is the node with neither power supply nor load demand.

By eliminating the contact nodes in Equation (1), we can get:

$$\left[\begin{array}{c}{\dot{\mathit{V}}}_{\mathbf{L}}\\ {\dot{\mathit{I}}}_{\mathbf{G}}\end{array}\right]=\mathit{H}\cdot \left[\begin{array}{c}{\dot{\mathit{I}}}_{\mathbf{L}}\\ {\dot{\mathit{V}}}_{\mathbf{G}}\end{array}\right]=\left[\begin{array}{cc}{\mathit{Z}}_{\mathbf{LL}}& {\mathit{F}}_{\mathbf{LG}}\\ {\mathit{K}}_{\mathbf{GL}}& {\mathit{Y}}_{\mathbf{GG}}\end{array}\right]\cdot \left[\begin{array}{c}{\dot{\mathit{I}}}_{\mathbf{L}}\\ {\dot{\mathit{V}}}_{\mathbf{G}}\end{array}\right]$$

(2)

where Z_LL, F_LG, K_GL and Y_GG are the block sub-matrices of the H-matrix, thereinto F_LG is the load participation factor sub-matrix.

Then the L-index of each load node can be given below:

$$L_j=\left|{\tilde{L}}_j\right|=\left|1-\frac{{\displaystyle \sum _{i\in a_{\mathrm{G}}}{\tilde{F}}_{ji}}\cdot {\dot{V}}_i}{{\dot{V}}_j}\right|$$

(3)

where i and j are the number of power nodes and load nodes, respectively; ${\dot{V}}_i$ and ${\dot{V}}_j$ are the voltage phasors of node i and node j respectively; α_G is the set of power nodes, ${\tilde{F}}_{ji}$ is the load participation factor (complex form); ${\tilde{L}}_j$ is the complex expression of L-index.

Virtually, the L-index is in the complex form ${\tilde{L}}_j$, and its modulus is taken as the practical index in order to measure the SVS of each load node. The range of L_j is between 0 and 1. When the value of L_j trends to 0, the SVS of load node j trends to more stability.

In general, the voltage fluctuations of load nodes and power nodes caused by the PV power fluctuation are very small, so the L-index value is less affected by the PV active output P_pv. The synchronous generator (set as PV bus) is the same.

2.2. The Relationship between PV Active Output and POI’s Voltage Phase Angle

In the SVS analysis, the PV power can be regarded as a negative load power, so that the Thevenin equivalent can be carried out from the POI of the PV station. Then a simplified PV station grid-connected two-node system is formed, as shown in Figure 1.

In Figure 1, P_pv and Q_pv are the active output and reactive output of PV, respectively; V_pv and δ are the voltage amplitude and phase angle of POI, respectively. Z is the impedance modulus of the equivalent power grid line and θ is the impedance angle. E is the potential of an equivalent electric source.

In this paper, all electrical quantities are in per-unit value (pu) and the phase angle’s unit is rad.

Then we can get:

$$\begin{array}{cc}-P_{\mathrm{pv}}-\mathrm{j}{Q}_{\mathrm{pv}}& =V_{\mathrm{pv}}\angle \delta {\left(\frac{E\angle 0-V_{\mathrm{pv}}\angle \delta }{Z\angle \theta }\right)}^{*}\\ & =\frac{E{V}_{\mathrm{pv}}}{Z}\cos \left(\delta +\theta \right)-\frac{V_{\mathrm{pv}}^2}{Z}\cos \theta +\mathrm{j}\left[\frac{E{V}_{\mathrm{pv}}}{Z}\sin \left(\delta +\theta \right)-\frac{V_{\mathrm{pv}}^2}{Z}\sin \theta \right]\end{array}$$

(4)

The followings can be obtained by arranging Equation (4):

$$\cos \left(\delta +\theta \right)=\frac{V_{\mathrm{pv}}}{E}\cos \theta -\frac{Z}{E{V}_{\mathrm{pv}}}{P}_{\mathrm{pv}}$$

(5)

$$\sin \left(\delta +\theta \right)=\frac{V_{\mathrm{pv}}}{E}\sin \theta -\frac{Z}{E{V}_{\mathrm{pv}}}{Q}_{\mathrm{pv}}$$

(6)

In normal operation of the grid-connected system, δ is a positive or negative value, and its absolute value is generally small (around 0 rad); while θ is generally large (close to π/2 rad), therefore (δ + θ) is greater than 0 rad and less than π rad.

The PV station’s operation mode can generally be divided into PV (active power—voltage) mode and PQ (active power—reactive power) mode [18,27]. PV mode is the constant voltage operation mode and the voltage amplitude of the POI or PV inverter outlet bus is set as a constant value. When P_pv changes, the POI’s voltage phase angle will change. PQ mode is the constant generation power factor operation mode. When P_pv changes, the voltage amplitude and voltage phase angle of POI will change.

2.2.1. PV Mode

In Figure 1, if the PV station operates in PV mode, V_pv is constant. According to Equation (5), with the increase of P_pv, cos(δ + θ) gradually decreases, (δ + θ) gradually increases. It can be known that δ gradually increases since θ is a constant, that is, δ is proportional to P_pv.

2.2.2. PQ Mode

If the PV station operates in PQ mode, the situation is more complicated. Now, we only consider that P_pv takes the unit power factor, that is, Q_pv = 0 pu. Considering the effect of equivalent line resistance, then θ is greater than 0 rad and less than π/2 rad.

From Equation (6), we can get:

$$\frac{d\left[\sin \left(\delta +\theta \right)\right]}{d{P}_{\mathrm{pv}}}=\frac{\sin \theta }{E}\frac{d{V}_{\mathrm{pv}}}{d{P}_{\mathrm{pv}}}$$

(7)

Since $\frac{d{V}_{\mathrm{pv}}}{d{P}_{\mathrm{pv}}}$ decreases monotonically from a positive value to a negative value (over 0) with the increase of P_pv (the derivation process is given in Appendix A), sin(δ + θ) first increases and then decreases monotonically. Whereas (δ + θ) is greater than 0 rad and less than π rad, δ changes from small to large, that is, δ is proportional to P_pv.

By summarizing the above results, it can be concluded that the active output P_pv of the PV station operating in PV mode or PQ mode with unit power factor is proportional to the voltage phase angle δ of POI.

2.3. A Novel Voltage Stability Sensitivity Index LPAS

Since the PV active output, P_pv is directly proportional to the POI’s voltage phase angle δ, we can take the derivative of L-index with respect to δ to reflect the influence degree of P_pv on the SVS of each load node.

Set ${\tilde{F}}_{ji}=F_{ji}\angle {\alpha }_{ji}$, $\stackrel{.}{V_i}=V_i\angle {\delta }_i$ and $\stackrel{.}{V_j}=V_j\angle {\delta }_j$, then Equation (3) can be expressed below:

$$L_j=\left|1-\frac{1}{V_j}{\displaystyle \sum _{i\in a_G}{F}_{ji}{V}_i[\cos ({\alpha }_{ji}+{\delta }_i-{\delta }_j)+j\sin ({\alpha }_{ji}+{\delta }_i-{\delta }_j)]}\right|$$

(8)

The sensitivity of the L-index of load node j relative to the voltage phase angle of power node i can be obtained by taking the partial derivative of the complex expression of L-index (${\tilde{L}}_j$) with respect to the voltage phase angle δ_i and taking the modulus value. According to the verification, the partial derivative of the modulus expression of L-index (L_j) with respect to δ_i is the same as it, so we can obtain:

$$\frac{\partial L_j}{\partial {\delta }_i}=\left|\frac{\partial {\tilde{L}}_j}{\partial {\delta }_i}\right|=\frac{F_{ji}{V}_i}{V_j}\left|\sin ({\alpha }_{ji}+{\delta }_i-{\delta }_j)-\mathrm{j}\cos ({\alpha }_{ji}+{\delta }_i-{\delta }_j)\right|=\frac{F_{ji}{V}_i}{V_j}$$

(9)

Equation (9) represents the coupling degree between the L-index of each load node and the voltage phase angle of each power node (including synchronous generators and PV stations). The influence of voltage phase angle change Δδ_i of each power node on the L-index of each load node can be expressed below:

$$\left[\begin{array}{c}\Delta L_1\\ \Delta L_2\\ ⋮\\ \Delta L_m\end{array}\right]=\left[\begin{array}{cccc}\frac{\partial L_1}{\partial {\delta }_1}& \frac{\partial L_1}{\partial {\delta }_2}& \cdots & \frac{\partial L_1}{\partial {\delta }_n}\\ \frac{\partial L_2}{\partial {\delta }_1}& \frac{\partial L_2}{\partial {\delta }_2}& \cdots & \frac{\partial L_2}{\partial {\delta }_n}\\ ⋮& ⋮& \cdots & ⋮\\ \frac{\partial L_m}{\partial {\delta }_1}& \frac{\partial L_m}{\partial {\delta }_2}& \cdots & \frac{\partial L_m}{\partial {\delta }_n}\end{array}\right]\left[\begin{array}{c}\Delta {\delta }_1\\ \Delta {\delta }_2\\ ⋮\\ \Delta {\delta }_n\end{array}\right]$$

(10)

where m is the maximum number of load nodes and n is the maximum number of power nodes.

For a PV station in the system, set the number of POI as k and its voltage amplitude as V_pvk. Then, a voltage stability sensitivity index (named LPAS-index), namely the derivative of L-index with respect to POI’s voltage phase angle can be defined below:

$$LPA{S}_j=\frac{F_{jk}{V}_{\mathrm{pv}k}}{V_j}$$

(11)

where j is the number of load nodes.

Equation (11) reflects the sensitivity of the L-index of each load node relative to POI’s voltage phase angle of each PV station, that is, the sensitivity of the L-index relative to P_pv. By calculating the LPAS-index, the relationship between the SVS of each load node and P_pv can be explored.

In a known power grid, if the power grid’s structure does not change, the load participation factor F_jk will remain unchanged. For a PV station operating in PV mode, since V_pvk is constant, the LPAS-index value is mainly affected by the load node voltage V_j. For a PV station operating in PQ mode with a unit power factor, the LPAS-index value is affected by the POI’s voltage V_pvk and the load node voltage V_j at the same time. It is noteworthy that the voltage fluctuations of load nodes and POI caused by the PV power fluctuation are very small normally, so the LPAS-index value is less affected by P_pv. The same is true for the synchronous generator (set as PV bus).

3. Static Voltage Stability Zoning Analysis Method

In the system with a large-scale PV station, if the PV station operates in PV mode or PQ mode with a unit power factor, for each load node, the sensitivity of the L-index relative to P_pv can be obtained by calculating the corresponding LPAS-index. According to the numerical level of the LPAS-index value, the whole system can be zoned into several areas which can reflect the correlations between the SVS and P_pv. On the other hand, the weakest SVS area and the weakest node can be determined by calculating the L-index value. Then according to the zoning results, the SVS analysis and control aiming at the weakest area and the whole power grid can be conducted.

Figure 2 is the flow chart of the SVS zoning analysis method based on the LPAS-index.

The main steps of the SVS zoning analysis method are as follows:

(1)

Get the sets of power nodes, load nodes and contact nodes by assessing the node types, and calculate the admittance matrix of the grid-connected system, find the sub-matrix F_LG of load participation factor and obtain the participation factor of each load node relative to the POI of each PV station.

(2)

According to Equations (3) and (11), the L-index and LPAS-index of each load node can be calculated. By ranking the L-index values of all load nodes, determine the weakest SVS area and the weakest node.

(3)

Load nodes are classified according to the numerical level of LPAS-index value, which can be generally classified into three types of nodes whose SVS is greatly, moderately and less affected by P_pv. According to the classification results for load nodes, the PV grid-connected system can be zoned into three kinds of SVS areas that are strongly, moderately and weakly associated with P_pv.

The numerical level of LPAS-index value can be classified according to the standard: greater than 50% (corresponding to the strong association area), from 20% to 50% (corresponding to the moderate association area), and less than 20% (corresponding to the weak association area). However, the classification standard is not unchangeable and should be determined according to the actual situation of the PV grid-connected system.

(4)

According to the classification results for load nodes, the PV grid-connected system can be zoned into three kinds of SVS areas that are strongly, moderately and weakly associated with P_pv. As the LPAS-index value of the load node is less affected by P_pv and other generators, the zoning results are relatively unchanged.

(5)

Analyze the SVS of the whole power grid and the weakest SVS area on the basis of the zoning results, and find out the rules that are affected by P_pv. Based on the analysis results, some useful control strategies can be proposed to improve the SVS of the weakest area and the whole system.

If most load nodes fall within the weakly associated area with P_pv, the impact of P_pv on the SVS can be neglected, and the change of load power is the main influencing factor on the SVS of the whole power grid. Similar to the traditional power grid, reactive power compensation or other effective measures can be used to improve the SVS.

If more load nodes fall within the strongly or moderately associated area with P_pv, the impact of P_pv on the SVS has to be valued. Load-shedding or other effective measures can be taken when necessary, and we will use a new PV power-shedding measure.

4. Simulation Verification for the Zoning Analysis Method

In order to verify the rationality and practicability of the above zoning analysis method, we take the IEEE 14-node system as the test example, as shown in Figure 3. The 14-node system is a sub-transmission system [18], and its voltage classes are 69 kV (including Node 1~Node 5), 13.8 kV (including Node 6, Node 7, and Node 9~Node 14), and 18 kV (Node 8).

See Figure 3, Node 1 is the slack bus (swing bus), Node 2 is a PV bus. SC1~SC3 are synchronous condensers. Set the base power as 100 MVA and the initial power of the total load as 2.59 + j0.814 pu. Set the voltage amplitude of Node 2 as 1.045 pu and the initial active output of generator Gen 2 (P_G2) as 1 pu.

See the dotted line in Figure 3, three plans about the centralized PV station integrated into the 14-node system will be adopted:

• Plan A: PV station A is integrated into Node 5 by a PV transmission line, and the corresponding POI is POI_A;
• Plan B: PV station B is integrated into Node 14 by a PV transmission line, and the corresponding POI is POI_B;
• Plan C: PV station B is integrated into Node 14, the generator Gen 2 is replaced with PV station C (operating in PV mode), and the corresponding POI is Node 2. This is a multi-PV case.

P_pvA, P_pvB and P_pvC represent the active output of PV stations A, B and C, respectively, and the maximum active output of them are 1 pu, 0.3 pu and 1 pu, respectively.

4.1. Verification for the Relationship between PV Active Output and Voltage Phase Angle of POI

Section 2.2 proved that the PV active output P_pv is proportional to the POI’s voltage phase angle δ, now we take Plan A as an example to verify it. When PV station A operates in PV mode, the PV inverter outlet bus voltage amplitude is set as 1.05 pu. When PV station A operates in PQ mode with a unit power factor, its reactive power is set as 0 pu. The P_pvA—δ_A (δ_A is the voltage phase angle of POI_A.) curve is shown in Figure 4, we can see that P_pvA is proportional to δ_A under the two operation modes.

4.2. Index Calculation

4.2.1. Plan A and Plan B

Firstly, Plan A and Plan B are investigated. Now take PV stations A and B operating in PV mode as an example. Set P_pvA = 1 pu, P_pvB = 0.3 pu (The capacity of PV station integrated into 13.8 kV node can’t be too large). Set the current load multiple of the whole system lm = 1 pu. Calculate the L-index values of all load nodes when the PV station adopts Plan A and Plan B, respectively, and does not integrate into the 14-node system, the results are listed in

Table 1. L-index values of load nodes under Plan A and Plan B (Load multiple lm = 1 pu).

Node Number	L-Index Value
	Plan A (PpvA = 1 pu)	Plan B (PpvB = 0.3 pu)	Without PV
14	0.0760	0.0297	0.0784
9	0.0646	0.0532	0.0681
10	0.0619	0.0526	0.0649
11	0.0351	0.0303	0.0366
13	0.0316	0.0209	0.0321
4	0.0293	0.0283	0.0307
12	0.0238	0.0185	0.0241
5	0.0197	0.0194	0.0209

.

It can be seen from Table 1 that when the PV station is not integrated into the system, Node 14 is the weakest node and Node 5 is the strongest node. Node 9, Node 10 and Node 14 compose the weakest SVS area.

When Plan A is carried out, the weakest SVS area is still composed of Node 9, Node 10 and Node 14, Node 14 and Node 5 are still the weakest node and the strongest node.

When Plan B is adopted, the weakest SVS area is composed of Node 9 and Node 10. The weakest node becomes Node 9, this is because the SVS of Node 14 is observably enhanced after integrating with PV station B. The strongest node is still Node 5.

It can be seen that the change of P_pv has a minor impact on the L-index from Figure 5 (Node 5 in Plan A and Node 14 in Plan B are taken as examples). According to the computing results, the variable amplitude of the L-index is less than 0.1%. Therefore, when P_pv changes, the determined weakest SVS area can remain consistent.

See Table 1, when a PV station is integrated into the system, compared with the system without a PV station, the L-index values of all load nodes are relatively reduced, indicating that the SVS of load nodes is improved to a certain extent. Moreover, when the PV station is integrated into different load nodes, the weakest SVS area and the weakest node can be changed.

As shown in

Table 2. LPAS-index values of load nodes under Plan A and Plan B (Load multiple lm = 1 pu).

Plan A (PpvA = 1 pu)		Plan B (PpvB = 0.3 pu)
Node Number	LPAS-Index	Node Number	LPAS-Index
5	23.17%	14	61.23%
4	14.18%	9	18.71%
9	5.95%	10	15.48%
10	4.92%	13	14.05%
14	3.78%	11	7.85%
11	2.49%	12	6.97%
13	0.85%	4	3.15%
12	0.42%	5	1.93%

, the LPAS-index values of all load nodes, respectively, relative to POI_A and POI_B are sorted from large to small under Plan A and Plan B.

Figure 6 shows the P_pv—LPAS-index curves (Node 5 in Plan A and Node 14 in Plan B are taken for examples). To the same load node, the change of P_pv has a minor impact on the LPAS-index value. Other active power supplies are similar such as a generator. In a general way, the variable amplitude of the LPAS-index is less than 1% when P_pv changes. Therefore, aiming at the changes of P_pv, the classification of load nodes and the SVS zoning results can remain unchanged.

In summary, the maximum active output of the PV station can be directly selected to calculate the LPAS-index and L-index of load nodes, so as to carry out the classification of load nodes and the determination of the weakest SVS area.

4.2.2. Plan C

In Plan C, PV station B has remained, and Gen 2 is replaced with PV station C. PV station C operates in PV mode, and the voltage of Node 2 remains as 1.045 pu. When P_pvB = 0.3 pu, P_pvC = 1 pu, the values of L-index and LPAS-index relative to Node 2 (namely POI of PV station C) and POI_B are listed in

Table 3. The values of L-index and LPAS-index of load nodes under Plan C (Load multiple lm = 1 pu).

Plan C (PpvB = 0.3 pu and PpvC = 1 pu)
Node Number	L-Index	LPAS-Index (to Node 2)	LPAS-Index (to POI_B)
5	0.0194	38.58%	1.93%
4	0.0283	38.18%	3.15%
9	0.0532	14.33%	18.71%
10	0.0526	11.86%	15.48%
11	0.0303	6.01%	7.85%
14	0.0297	3.99%	61.23%
13	0.0209	0.92%	14.05%
12	0.0185	0.45%	6.97%

. Node 9 is the weakest node and Node 12 is the strongest node. Node 9 and Node 10 compose the weakest SVS area.

Similarly, the active output of PV station C (P_pvC) has a minor impact on the values of the L-index and LPAS-index. For example, in Plan C, the LPAS-index values of Node 5 relative to Node 2 and Node POI_B are 0.3859 and 0.0193, respectively, when P_pvC = 0.3 pu and P_pvB = 0.3 pu, and are approximately equal to the corresponding LPAS-index values when P_pvC = 1 pu and P_pvB = 0.3 pu, as shown in Table 3.

4.3. SVS Zoning and Analysis

According to the data in Table 2 and Table 3, the load nodes can be classified and the 14-node system with PV can be zoned.

4.3.1. Plan A

See Table 2, in Plan A, only the LPAS-index value of Node 5 is greater than 20% and less than 50%, illustrating that the SVS of Node 5 is moderately affected by P_pvA. The LPAS-index values of other load nodes (including Node 4, Node 9, Node 10, Node 11, Node 12, Node 13 and Node 14) are all less than 20%, illustrating that the SVS of these nodes are all less affected by P_pvA.

Furthermore, the LPAS-index value of Node 5, namely the access point of PV station A, is not too big, indicating that the PV access point is not necessarily greatly affected by P_pv.

Figure 7 shows the zoning chart of Plan A. The system can be zoned as a moderately associated area (including Node 5) and a weakly associated area (including other load nodes) with P_pvA. Owing to the LPAS-index and L-index are less affected by P_pv and P_G2, the zoning results of Plan A are relatively unchanged.

See Figure 7, the weakest SVS area (composed of Node 9, Node 10 and Node 14) falls within the area that is weakly associated with P_pvA, so this area’s SVS is less affected by P_pvA. In fact, since the overwhelming majority of load nodes (except Node 5) are falls with the weakly associated area with P_pvA, the SVS of the whole power grid is less affected by P_pvA, and more attention should focus on the impact of load power change on the SVS.

As the impact of P_pv on the L-index is minimal, we select the system load-margin index I_LM to measure the SVS and verify the above conclusion. The load-margin index I_LM can be calculated by Equation (12):

$$I_{\mathrm{LM}}=1-\frac{1}{{\lambda }_{\mathrm{MAX}}}$$

(12)

where λ_MAX is the maximum load margin parameter that corresponds to the critical point of the SVS.

The value range of I_LM is 0~1. 0 means voltage collapse, and 1 means absolute voltage stability.

Now set P_G2 = 0.4 pu. The values of I_LM are given in

Table 4. Values of system load-margin index (Plan A, active output of Gen 2 P_G2 = 0.4 pu).

PpvA/pu	Load Multiple lm = 1 pu		Load Multiple lm = 1.5 pu
	λMAX/pu	ILM	λMAX/pu	ILM
0	2.5846	0.6131	1.7346	0.4235
0.1	2.5957	0.6147	1.7444	0.4267
0.2	2.6068	0.6164	1.7536	0.4297
0.3	2.6163	0.6178	1.7625	0.4326
0.4	2.6251	0.6191	1.7708	0.4353
0.5	2.6330	0.6203	1.7787	0.4378
0.6	2.6400	0.6212	1.7861	0.4401
0.7	2.6460	0.6221	1.7921	0.4420
0.8	2.6508	0.6228	1.7976	0.4437
0.9	2.6542	0.6232	1.8032	0.4454
1	2.6560	0.6235	1.8085	0.4471

. When the system load multiple lm is 1 pu, the system operates in a normal state (namely a low load level). When P_pvA changes from 0 to 1 pu, the I_LM values keep almost unchanged (Adjoining variation amplitude is less than 1%).

When the load multiple lm is 1.5 pu, the system operates in a heavy load (namely a high load level). See Table 4, with the change of P_pvA, the I_LM values still keep almost unchanged. However, compared to the normal operation state (lm = 1 pu), the I_LM values and the SVS margin significantly reduce, indicating that the SVS of the whole system (including the weakest area) is mainly affected by the load power. The load-shedding method can be used to improve the SVS [24].

4.3.2. Plan B

See Table 2, in Plan B, the LPAS-index value of Node 14 (the access point of PV station B) is greater than 50%, so the SVS of Node 14 is greatly affected by P_pvB. However, the LPAS-index values of other load nodes (including Node 4, Node 5, Node 9, Node 10, Node 11, Node 12 and Node 13) are all less than 20%, so the SVS of these nodes is less affected by P_pvB.

Figure 8 shows the zoning chart of Plan B. Node 14 can be set as a strongly associated area with P_pvB, and other load nodes can be set as a weakly associated area with P_pvB. The weakest SVS area (composed of Node 9 and Node 10) falls within the area that is weakly associated with P_pvB.

See

Table 5. Values of system load-margin index (Plan B, active output of Gen 2 P_G2 = 0.4 pu).

PpvB/pu	Load Multiple lm = 1 pu		Load Multiple lm = 1.5 pu
	λMAX/pu	ILM	λMAX/pu	ILM
0	2.5272	0.6043	1.6961	0.4104
0.1	2.5695	0.6108	1.7239	0.4199
0.2	2.6089	0.6167	1.7504	0.4287
0.3	2.6459	0.6221	1.7751	0.4367

, with the change of P_pvB, the variation amplitudes of I_LM value are still very small under normal state and heavy load, and the SVS of the whole power grid is mainly affected by the load power. It is similar to the case in Plan A, more attention should focus on the impact of load power change on the SVS.

4.3.3. Plan C

See Table 3, in Plan C, the LPAS-index values of Node 4 and Node 5 relative to P_pvC is greater than 20% and less than 50%, so the two nodes can compose a moderately associated area with P_pvC. The LPAS-index values of other load nodes relative to P_pvC are all less than 20%, so they can compose a weakly associated area with P_pvC. However, the LPAS-index value of Node 14 relative to P_pvB is greater than 50%, so we can neglect the impact of P_pvC and zone Node 14 as a strongly associated area with P_pvB. Ultimately, Node 9~Node 13 can be determined to compose a weakly associated area with P_pvB and P_pvC. The zoning chart of Plan C is shown in Figure 9.

The weakest SVS area (composed of Node 9 and Node 10) falls within the weakly associated area with P_pvB and P_pvC, so the impact of P_pv on the weakest area can be neglected.

See Figure 9, what is different from Plan A and Plan B is that the areas more affected by P_pv are significantly enlarged in Plan C. Especially, Node 4 and Node 5 (69 kV voltage class) fall within a moderately associated area with P_pvC, so the impact of P_pv on the SVS should be further investigated.

Now we set P_pvB = 0.3 pu, and P_pvC changes from 0 to 1 pu.

The I_LM values are listed in

Table 6. Values of system load-margin index (Plan C, active output of PV station B P_pvB = 0.3 pu).

PpvC/pu	Load Multiple lm = 1 pu		Load Multiple lm = 1.5 pu
	λMAX/pu	ILM	λMAX/pu	ILM
0	2.5846	0.6131	1.7378	0.4246
0.1	2.6138	0.6174	1.7505	0.4287
0.2	2.6343	0.6204	1.7608	0.4321
0.3	2.6451	0.6219	1.7692	0.4348
0.4	2.6459	0.6221	1.7751	0.4367
0.5	2.6321	0.6201	1.7787	0.4378
0.6	2.5960	0.6148	1.7791	0.4379
0.7	2.5171	0.6027	1.7761	0.4370
0.8	2.3401	0.5727	1.7685	0.4345
0.9	2.1311	0.5308	1.7557	0.4304
1	1.8507	0.4597	1.7343	0.4234

. It can be seen that when the system operates in a normal state (lm = 1 pu), with the gradual increase of P_pvC from 0 to 0.5 pu, the I_LM value gradually increases too, indicating that the SVS is gradual enhanced with the increase of P_pvC. However, with the gradual increase of P_pvC from 0.6 pu to 1 pu, the I_LM value gradually decreases and the SVS is gradually weakened. When P_pvC reaches the maximum (1 pu), the impact of P_pv on the I_LM value is very obvious (decreases to 0.4597), and the SVS becomes as weak as the status that the system operates in heavy load.

See Table 6, when the system operates under heavy load (lm = 1.5 pu), the I_LM value gradually increases with the increase of P_pvC from 0 to 0.6 pu, and gradually decreases with the increase of P_pvC from 0.7 pu to 1 pu, but the variation amplitude is very small.

Now we set P_pvC = 1 pu, and P_pvB changes from 0 to 0.3 pu. The I_LM values are listed in

Table 7. Values of system load-margin index (Plan C, active output of PV station C P_pvC = 1 pu).

PpvB/pu	Load Multiple lm = 1 pu		Load Multiple lm = 1.5 pu
	λMAX/pu	ILM	λMAX/pu	ILM
0	2.1028	0.5244	1.6760	0.4033
0.1	2.0303	0.5075	1.6978	0.4110
0.2	1.9583	0.4894	1.7173	0.4177
0.3	1.8507	0.4597	1.7343	0.4234

. It can be seen that excessive PV power can evidently weaken the SVS when the system operates in a normal state (lm = 1 pu). When the system operates under heavy load (lm = 1.5 pu), the increase of P_pvB only can slightly improve the SVS and has no negative impact on the SVS.

From the above analyses, it is concluded that in a multi-PV case, excessive PV active power can evidently weaken the SVS when the system operates in a normal state, and only slightly impact the SVS when the system operates under heavy load.

A PV power-shedding method can be used in time to maintain the SVS when the system operates at a low load level and an excessive PV power. For example, when lm = 1 pu, P_pvB = 0.3 pu and P_pvC = 0.95 pu, the value of I_LM is 0.4958 by calculation. Then the PV active power control system can execute a series of power-shedding operations and the shedding quantity is 0.1 pu each time, P_pvC will be reduced to 0.55 pu, and the value of I_LM will recover to 0.6180. The system’s SVS is improved effectively after power-shedding.

Moreover, reducing the rated installation capacity of PV station is also a valid solution to this problem.

5. Conclusions

Firstly, this paper derives and proves that the POI’s voltage phase angle of the PV station is proportional to the PV active output P_pv, based on a simplified two-node system with PV. From this, a novel sensitivity index LPAS-index of the traditional L-index relative to the voltage phase angle of POI is proposed, and the LPAS-index can be used to reflect the influence degree of P_pv on the SVS of load nodes. A PV grid-connected system can be zoned into different areas: strongly, moderately and weakly associated with P_pv for the SVS according to the numerical level of LPAS-index value. Based on the zoning results, the SVS analysis method is applied, by taking the classic 14-node system integrated with a centralized PV station as an example. The following conclusions can be obtained:

(1)

The LPAS-index value of one load node is less affected by P_pv or P_G (active output of generator), so the SVS zoning results are relatively unchanged. However, by changing the location of the PV access point or the numerical classification standard of the LPAS-index value, the SVS zoning results can change.

(2)

The access point of the PV station is not always greatly affected by P_pv, so it can fall within the strongly associated area with P_pv, and also fall within the moderately associated area with P_pv.

(3)

If most of the load nodes except for the PV access point fall within the weakly associated with P_pv, the impact of P_pv on the SVS can be neglected and more attention should be focused on the impact of load power.

(4)

In the multi-PV case, more load nodes may fall within the areas more affected by P_pv. If no excessive PV power flows into the power grid, the increase of P_pv can improve the SVS to a certain extent. However, excessive PV active power can evidently weaken the SVS when the system operates at a low load level, and PV power-shedding can make the system maintain the SVS. On the other hand, excessive PV active power has a minor impact on the SVS when the system operates under heavy load.

(5)

In addition, when the system load is constant, the change in P_pv has a minor impact on the L-index. It is necessary to design a novel SVS index for load nodes that can better adapt to the fluctuation of P_pv in the follow-up work.