Study of distribution network loss allocation with high photovoltaic penetration ratio

With the rapid development of renewable energy, the penetration ratio of photovoltaic (PV) in the distribution network is getting higher and higher. The fair and reasonable allocation of high loss caused by high PV penetration in the distribution network is of great significance to the healthy development of distribution systems. In order to study the loss allocation in distribution networks with high PV penetration, based on the cooperative game theory, combined with the power flow algorithm, the power flow and loss of the 72-node active distribution network in agricultural and pastoral areas were analyzed. Three scenarios were included: no PV, low PV penetration, and high PV penetration. The allocated loss of the distribution network with high PV penetration and multiple distributed generators was computed. The results show that the loss allocation method, which takes into account the power as well as the network topology, is applicable to distribution networks with multiple distributed generators. In distribution networks with high PV penetration, distributed generators are the main cause of high loss. If the network contains the extreme reverse power flow concerned in this paper, the loss allocated to DGs accounts for more than 95% of the total network power loss.


Introduction
In recent years, with the popularity of the concepts of new power systems, "carbon peak carbon neutral" and the promotion of construction tasks of rooftop PV, the distributed PV generation in the distribution network has been sustained and rapidly developed.In 2022, the new installed PV capacity in China was 87.41 million kilowatts(kW), and the cumulative installed capacity reached 392.61 million kW, a yearon-year increase of 28.6%.The additional 51.114 million kW of distributed PV are connected to the grid [1].Distributed PV continues to develop rapidly.
In the case of high PV penetration in the distribution network, if there is no configuration of energy storage, reverse power flow may occur in the distribution network [2] (Reverse Power Flow, RPF).It can result in a temporal mismatch between the photovoltaic power generation and the electricity consumed by loads.When there is sunshine in the daytime, the PV outputs more power.RPF can occur and may result in excessive loss and voltage overshooting.
Current research on RPF focuses on improving the protection strategy of relay protection devices and methods to suppress RPF [3][4][5].There are also some studies on loss allocation in distribution networks with distributed generators (DGs).For the distribution network loss allocation with distributed generators, an improved average network loss method was proposed [6], but it lacks fairness in practical application.To address this issue, a new method of network loss sharing for radial distribution networks with DG was proposed [7], which allocates losses without considering any additional assumptions and approximations.However, the impact of reactive power transmission on network loss is not taken into account.Several studies have proposed loss allocation methods based on cooperative game theory [8][9][10][11][12], such as those based on Shapley value (SV), Nucleolus value (NV), Aumann Shapley (AS), and τvalue loss allocation.For example, the method for network loss apportionment based on Shapley value and circuit theory was proposed [8].However, the workload is large and the computational burden is high when there are more loads and DGs.High loss caused by the high PV penetration is not considered in the current loss allocation methods and the quantitative research on the responsibility of distributed generators for high loss of distribution networks is still blank.The study on reasonable loss allocation in the active distribution network with a high PV penetration ratio can clarify the responsibility of high loss borne by each power generator under the background of "double carbon".The basis can be provided for the subsequent development of dynamic feed-in tariffs for distributed PV.
In this paper, the situation of an active distribution network with high PV penetration is considered, based on the theory of cooperative game.The power flow and loss are analysed for a 72-node active distribution network in agricultural and pastoral areas containing high PV penetration.Finally, the loss allocation is analysed in active distribution networks at typical moments.

Loss allocation model
The problem of loss allocation in the distribution network is modelled as a cooperative game problem.In this problem, the loads and distributed generators at different nodes in the distribution network are considered as the players involved in the game.The coalition in which all the players participate is the largest coalition which is denoted as N.The largest coalition's total gain is the total loss summed all line losses in the distribution network.The gain is denoted as v.In this paper, the value of the eigenfunction v is calculated directly by using the method of forward and backward for power flow calculation.
Taking a radial distribution network containing ML loads and MG distributed generators as an example, the τ values of the loads and distributed generators are calculated separately, and the detailed calculation steps are as follows.
It is assumed that the maximum union of the loads connected in this distribution network is represented by { 1 , 2 , 3 , ,} LL NM < Κ .The upper limit value of the load i is calculated as Formula (1).
where Ploss is the network loss when all the loads are connected in the network and Ploss-i is the network loss after removing only the load i from the network.
The minimum clearance value of load i is calculated as Formula (2).( , ) () where ploss-i is the network loss when only load i is connected in the network.We make { 1 , 2 , 3 , ,} GG NM < Κ a set of distributed generators connected to the distribution network.

The upper limit value ( ,)
iG XNv and the minimum clearance value G i κ of each distributed generator can be calculated according to the procedure for calculating ( ,) iL XNv and L i κ of the load.Finally, the L  and G  of loads and distributed generators are calculated according to Formula (3), and the τ values of loads and distributed generators are calculated according to Formula (4), respectively.

∋ ( ∋ (
Since the sum of line losses in the network with only loads and distributed generators is not equal to the line losses generated by connecting both loads and distributed generators in the actual distribution network, the calculated τ value is not directly used as the allocated loss of loads and distributed generators.To effectively and fairly allocate loss, the loss allocated to the loads is equal to the corresponding τ values calculated when only the loads are connected, while the losses allocated to the distributed generators should be equal to the losses of both loads and generations connected to the network minus the losses allocated to the loads.

Simulation analysis
The algorithm procedures, including the part of the forward and backward calculation of the power flow and the part of the calculation of the loss allocation based on cooperative game theory, are implemented in a 64-bit environment with MATLAB R2022.Based on the actual 10 kV network in agricultural and pastoral areas of Qinghai Province shown in Figure 1, the distribution network with high PV penetration is studied and analyzed.

Analysis of distribution network losses with a high PV penetration ratio
The RPF in the network is used to describe the PV penetration.The higher the PV penetration, the higher the RPF in the network.When the PV power is more than 10 times the power required by the total load, there is an extreme RPF in the radial distributed network.The distribution network concerned in this paper consists of 72 nodes with 71 branches and a PV distributed generator connected at Node 72.The PV power curve is the actual PV power recorded in one day.One sample is taken every 1 hour.This curve is used as the standard PV power curve.The moment at 0:00 AM is specified as the first moment.There are 24 moments in one day.The actual maximum generation of this PV power in the simulation is more than 10 times the actual required power of the radial network.To compare the network loss and loss allocation when the PV penetration ratio in the network is different, three simulation cases are set up.The standard PV power generation is reduced at the same rate at different moments by multiplying by a factor r. The simulation results are shown in Figure 2. Case 1: r=0.05, the network does not have reverse power flow during the day.Case 2: r=0.1, the network has a low reverse power flow at noon.Case 3: r=1, there is an extreme reverse power flow in the network at noon.In Case 1, the generation power at the slack node in the network is always positive throughout the day and there is no RPF.The network loss is very small during the day.In Case 2, there is a very small RPF only from 12:00-18:00, and the network loss is slightly larger at this time than at other times of the day.In Case 3, the generation power at the slack node of the network is negative from 10:00-18:00, there is a large RPF in the network, and the loss of the network is much higher in this period than in other periods of the day.The losses allocated to the DG are consistent with the network loss profile, and the high losses due to high PV penetration are borne by the DG.

Loss analysis of distribution network with high PV penetration at a typical moment
A typical moment of the day is selected for analysis.At that moment, the actual operating load of the distribution network is 241.71 kW of total active power and 48.70 kvar of total reactive power, and the total loss in the network without DG is 0.16 kW.In order to simulate loss allocation in the scenario with high PV penetration, four distributed generators DG1-DG4 are connected to the distribution network.The losses allocated to DGs are shown in Figure 3.They have a certain positive correlation with the output active and reactive powers of DGs.The loss allocation model not only takes the active and reactive powers of DGs into account but also the location of DGs in the distribution network.The method is still applicable in the case of multiple DGs with a high penetration ratio.The loss allocated to loads is shown in Figure 4. Network loss is not allocated to nodes with no load such as Nodes 12, 20, and 24.Although the required power at Node 53 is smaller than that at Node 2, the loss allocated to Node 53 is larger than that at Node 2. It is because Node 53 is further away from the slack node.It is reasonable for loss allocation.

Conclusions
Based on the cooperative game loss allocation method, the impact of extreme reverse power flow has been studied on power loss and loss allocation in active distribution networks.The conclusions are as follows: (1) In distribution networks with high PV penetration, if there is extreme reverse power flow in the network concerned in this paper, the loss allocated to DGs accounts for more than 95% of the total network loss.
(2) In radial distribution networks, high PV penetration leads to extreme RPF in the network.The network loss rises and the economics and stability of the distribution network are degraded.
.1088/1742-6596/2741/1/012069 3 where j denotes L or G. N is the grand coalition of all loads and all distributed generators in the distribution network.

Figure 2 .
Figure 2. The simulation results.(a) Generation power at the slack node; (b) Total power loss in the distribution network; (c) Loss allocated to the DG.

Figure 3 .
Figure 3. Proportion of DG generation power in total power generated and proportion of allocated loss in total loss allocated to DGs.

Figure 4 .
Figure 4. Loss allocated to the load of each node at noon.