A static voltage stability index based on P-V curve for the medium-voltage distribution network with distributed PV

With the rapid increase of distributed photovoltaic (DPV) generations, the static voltage stability (SVS) of distribution network (DN) is faced with more and more challenges. Considering the characteristics of DPV and the network structure, an appropriate static voltage stability index (SVSI) in the medium-voltage DN with DPV is studied in this paper. In terms of the limitation of the traditional SVSI that only reflects the load change, based on P-V curve, a novel voltage change rate - network load change rate index (M-index) is proposed. The M-index considers the impacts of double changes in DPV power output and load level. The model of classic 33-node network with a DPV source is established to investigate the proposed M-index. By changing the DPV penetration percentage and the DPV access point, and comparing with the existing Load-index, the validity and superiority of M-index is verified. Comparing with the traditional SVSI, the M-index is more propitious to evaluate the SVS for the medium-voltage DN with DPV.


Introduction
In the mode of photovoltaic (PV) power generation, distributed photovoltaic (DPV) has been vigorously developed [1].DPV has many advantages and technological superiorities, such as nearby power supply, clean and efficient, and convenient and flexible, etc.However, the influence of largecapacity DPV on the voltage stability of distribution network (DN) gradually appear, mainly owing to the power flow change in both directions and the randomness of DPV power output [2][3][4].
In the early research stage of DN's voltage stability, it is mainly manifested as a static instability process with a long time span, therefore, the static voltage stability (SVS) accounts for a large proportion.The static voltage stability index (SVSI) can reflect the voltage stability level in the current operating state of the network to a certain extent, and realize the voltage stability online-monitoring [5,6].
Many scholars have carried out in-depth researches on the SVSI.The authors in [7] analyzed the SVSI from the perspective of Jacobian matrix's determinant.The authors in [8] designed an online SVSI based on PMU (Phasor Measurement Unit); a Load-index considering the load change in the DN was proposed in [9], and a SVS assessment method with multi-index was proposed in [10].Aiming at the power grid with centralized PV, the authors in [11] proposed two quantitative indexes to characterize the SVS at the PV access point.The authors in [12] studies the influence of PV on the SVS margin of receiving-end grid from the perspective of the generalized short-circuit ratio and its critical value change.In [6], a NVSI (New Energy Voltage Stability Index) considering the power injection of new energy station was proposed so as to predict the SVS of PV access point.
At present, there are few researches on the SVSI suitable for the DN with DPV.Traditional SVSI only pays close attention to the influence of load change.With the integration of a large number of DPVs, consequentially, traditional SVSI will not be able to adapt to the randomness of DPV power output.In view of this, starting from the characteristics of DPV and the DN's structure, this paper proposes a novel voltage change rate -network load change rate index (M-index) based on the P-V curve of load node, so as to provide a reference for the SVS issue in the medium-voltage DN with DPV.

Traditional Load-index
In a general way, the traditional SVSI in the medium-voltage DN is calculated by the power flow solutions.The Load-index proposed in [9] is a practical SVSI for the load node in the traditional network, and it is defined as bellow: where V 0 and θ 0 represent the node's voltage and phase angle when its load power is 0, V L and θ L represent the node's voltage and phase angle when the load power is not 0. The Load-index shown as Formula (1) reflects the SVS of load node in the DN.If the Load-index is close to 0, the node voltage is stable; if the Load-index is close to 1, the node voltage is close to instability.

A novel M-index
The calculation of Load-index is fast and convenient for an online application, but it only focuses on the impact of load change on the SVS.Considering the impact of DPV power output on the SVS, it is necessary to establish an appropriate evaluation index for the DN with DPV.

P-V curve
Figure 1 shows the connection diagram of the DPV integrated into the DN, and the DN is a simplified expression.

Figure 1. Connection diagram of DPV integrated into the distribution network.
The apparent power at the receiving-end of the DN can be obtained from Figure 1: From Formula (2), we can get: Then V r is obtained: When Δ = 0, the DN operates at a critical point and the network just maintains the SVS, and the critical voltage V cr is: DPV can be regarded as a reversed load, and the DPV access point is usually a load node in the DN.The SVS can be analyzed by P-V curve method [5], and all load nodes have their own P-V curves.Figure 2 shows the P-V curve of the DPV access point, the horizontal coordinate is the load active power of the whole network (P) and the longitudinal coordinate is the voltage of the DPV access point (V r ).As can be seen from Figure 2, at the critical point, dP/dV r = 0, and P obtains a maximum value P cr .

Voltage change rate -network load change rate index
Based on the above analysis, the voltage change rate relative to the critical point of P-V curve is adopted to determine the weakest node in the SVS: where V 0i represents the voltage of node i (i=1, 2, …) when operating in the current state; V cri represents the voltage of node i at the critical point.The node corresponding to the maximum voltage change rate (VC max ) in the whole network is the weakest node, and the voltage collapse always starts from the weakest node and gradually spreads to the whole network.
When P increases to the maximum value P cr , the network reaches the critical state of maintaining its SVS.The network load change rate relative to the critical point is: where P 0 represents the load level in the current state.
The greater the load change rate LC, the greater the amount of load power increase, indicating that the network has better SVS.
See Figure 3, Curve 1 is the P-V curve of one load node in the DN without DPV, Curve 2 is the P-V curve of this node in the DN with DPV.
At the same load level, the critical voltage V cr.noPV of Curve 1 is usually less than the corresponding voltage V' in Curve 2, which indicates that after the integration of DPV, DN can withstand greater load growth and has larger SVS margin.Now define the voltage change rate in the DN with DPV:  From the definitions of VC' and LC, it can be seen that when M-index is close to 0, the network is stable, and when M-index tends to infinity (e.g. the current operating point is close to the critical point of Curve 2), the network will encounter a voltage collapse.Compared with the existing SVSI, the Mindex not only reflects the degree of voltage instability caused by overload, but also can more intuitively express the impacts of the load level and the DPV power output.

Example verification
Now the classic 33-node distribution network with DPV is modelled to verify the validity and superiority of M-index, by using the power system simulation tool PSAT (v2.1.10),as shown in Figure 4.The reference power and voltage are 10 MVA and 12.66 kV respectively, the total load active and reactive power are 3715 kW and 2300 kvar respectively.A DPV source is connected to one load node through a step-up transformer, and its rated ratio is 0.27 kV / 12.66 kV.
In order to verify the validity of the proposed M-index, the traditional Load-index is used as the comparison index.It is calculated from two aspects, DPV penetration percentage and DPV access point, to compare M-index with Load-index.

Different penetration percentage of DPV
See Figure 4, the DPV is connected to Node 33, the power factor of DPV remains unchanged (cosφ pv = 0.95); its penetration percentage is changed, and M-index and Load-index are calculated respectively.DPV penetration percentage is defined as below: The partial P-V curves of Node 18 (the weakest node) at different penetration percentage are shown in Figure 5.The curves of Load-index and M-index of Node 18 at different penetration percentage are shown in Figure 6.From Figure 5, it can be seen that with the increase in DPV penetration percentage, the critical point of the P-V curve gradually shifts to the right, illustrating that the integration of DPV is propitious to enlarge the SVS margin.
See Figure 6, with the increase of DPV penetration percentage, although Load-index and M-index have the same change trend, the Load-index shows an approximate linear variation in characterizing the SVS, but the M-index drops from steep to slow with the increase of DPV penetration percentage.The change trend of M-index indicates that the impact of different DPV penetration percentage is different, and oversize penetration percentage has minor impact on the SVS.
Through the above analyses, the proposed M-index can not only accurately indicate the SVS of DN at different DPV penetration percentage, but also comprehensively identifies the change trend of the SVS with the increase of DPV penetration percentage.M-index can overcome the one-sidedness of the traditional SVSI, and the calculation results are more suitable for the actual status in the DN with DPV.

Different access point of DPV
Now we respectively connect the DPV source to Node 7 ~ Node 18 in the 33-node distribution network with rated power of 500 kVA (cosφpv= 0.95).The P-V curves of Node 33 are shown in Figure 7, and the curves of Load-index and M-index of Node 33 are shown in Figure 8.  See Figure 7, the DPV source changes the access point in turn (from Node 7 to Node 18).In the process of moving the access point from the front node to the end node, the critical point of the P-V curve of Node 33 gradually shifts to the right, indicating that the DPV access point has a certain impact on the SVS margin.
However, from the curve of Load-index in Figure 8, we can see that the value of this index is keeping about 0.0186, indicating that Load-index cannot reliably evaluate the SVS.According to the calculation results of Load-index, the DPV access point has almost no impact on the SVS.From the curve of M-index in Figure 8, we can see that the M-index value gradually decreases with the changes of DPV access point, indicating that the SVS is gradually improved, which is consistent with the analysis result of Figure 7.
Combined with the above analyses, the traditional Load-index cannot accurately evaluate the SVS at different DPV access point, and the proposed M-index has better accuracy and sensitivity to estimate the SVS of DN with DPV.

Evaluation of M-index and Load-index
The evaluation effect of M-index and Load-index on the SVS of DN with DPV can be listed in Table 1, obviously, the M-index is better than the Load-index.

Conclusions
This paper proposes a novel SVSI, M-index, which is combined with voltage change rate and network load change rate based on the P-V curve of load node, to make a comprehensive evaluation for the SVS in the medium-voltage DN with DPV.By changing the DPV penetration percentage and the DPV access point in the classic 33-node distribution network, and comparing with the existing Load-index, the validity and superiority of the M-index is verified.The SVS evaluation results of M-index are more accurate and reliable, and it can be applied to the actual modern DN, instead of the traditional SVSI.

Figure 2 .
Figure 2. P-V curve of DPV access point.
.1088/1742-6596/2591/1/012042 4 where V 0 represents the node voltage in the DN with DPV when operating in the current state.

Figure 3 .
Figure 3. P-V curves of one load node.In summary, for the DN with DPV, considering the advantages of voltage change rate and network load change rate in the SVS analysis, a novel voltage change rate -network load change rate index can be defined to evaluate the SVS, and it is called M-index.The M-index is calculated by the following: '   VC M index LC(10)

Figure 4 .
Figure 4.The 33-node distribution network with a DPV source.

Figure 5 .
Figure 5. P-V curves of Node 18 at different DPV penetration percentage.

Figure 6 .
Figure 6.Curves of Load-index and M-index of Node 18 at different DPV penetration percentage.

Figure 7 .
Figure 7. P-V curves of Node 33 when DPV connects to different load node.

Figure 8 .
Figure 8. Curves of Load-index and M-index of Node 33 when DPV connects to different load node.

Table 1 .
Evaluation effect of M-index and Load-index on SVS. P/p.u.V/p.u.