Analysis Method for Photovoltaic Absorption Capacity of Distribution Network Based on Risk Quantification and Network Reconstruction

The random characteristics of photovoltaic (PV) power will cause changes in node voltage and line power, which is an important factor affecting the photovoltaic absorption capacity of the distribution network (DN). So as to accurately evaluate the impact of photovoltaic randomness on the absorption capacity of DN, an analysis method of photovoltaic absorption capacity of DN is proposed based on risk quantification. Firstly, a photovoltaic absorption capacity evaluation model is proposed based on risk quantification, which considers the randomness of load and PV power and can improve the accuracy of photovoltaic absorption capacity assessment. Secondly, the photovoltaic absorption capacity DN is calculated with the network topology and photovoltaic access capacity of the DN as the optimization variables. The probabilistic power flow is performed by linear power flow, which effectively improves the calculation speed. Finally, the proposed PV absorption capacity analysis method is simulated and analyzed. The simulation results show that the proposed method can effectively take into account the impact of load and photovoltaic randomness and improve the photovoltaic absorption capacity of DN.


Introduction
Distributed power generation has the advantages of easy installation, no long-distance transmission, and a high utilization rate, which is an important way of new energy utilization.The distribution grid is the main carrier of wind, light and other distributed new energy, with large quantities of distributed photovoltaic grid operation, to achieve the safe and reliable operation of the DN to promote the economic and efficient use of renewable energy is of great significance.
The distributed power absorption capacity of the DN refers to the maximum capacity of the distributed power allowed to be accessed under the condition of meeting the safe and stable operation of the DN.It is important to evaluate the photovoltaic absorption capacity of the DN to improve the utilization efficiency of renewable energy and ensure the safe and reliable operation of the DN.[1] uses the Monte Carlo method to generate the DN operation scenarios after PV access and analyzes the photovoltaic absorption capacity of individual nodes and the PV absorption capacity of the whole DN.[2] uses voltage sensitivity to quickly assess the PV consumption capacity of the DN under random photovoltaic variations and uses reactive voltage control of photovoltaic grid-connected inverters to improve the overvoltage problem caused by grid-connected PV.In [3], a Monte Carlo modeling approach was used to analyze the effect of rooftop photovoltaic connection below the distribution grid on overvoltage, voltage imbalance, and equipment loading.
The DN contains a large number of contact switches, and the topology of the network can be adjusted by adjusting the opening and closing states of the switches to achieve the goals of reducing network loss, improving line overload, and improving power quality.In [4], an autonomic-constrained operation behavior model considering flexibility resources such as energy storage, flexible load, and network reconfiguration is proposed, and new energy consumption capacity assessment indexes and analysis methods are proposed.[5] proposed a dynamic reconfiguration method for DN with the objective of improving photovoltaic acceptance capacity and optimizing it using genetic algorithms.
The stochastic characteristics of the load and photovoltaic output can cause variations in node voltage and line power, which have an impact on the PV absorption capacity calculation results of the DN.In [6], considering the fluctuation interval of photovoltaic output power, the interval overvoltage probability index and the stochastic scenario simulation method of photovoltaic acceptance capacity based on interval overvoltage risk were proposed to improve the accuracy and adaptability of photovoltaic acceptance capacity analysis.[7] uses the Monte Carlo simulation method for probabilistic tide calculation after photovoltaic access to the DN, and analyzes the maximum access capacity of the DN to photovoltaic by calculating the probability of voltage crossing the limit and equipment power crossing the limit, but the assessment of photovoltaic acceptance capacity based on Monte Carlo simulation method requires a number of tide calculations, which increases the burden of the algorithm.
So as to accurately assess the impact of photovoltaic stochasticity on the DN acceptance capacity, this paper proposes a risk quantification-based PV absorption capacity analysis method for the DN, while considering the role of network reconfiguration in enhancing the PV absorption capacity of the DN.Its main contributions are as follows: 1) Considering the randomness of load and photovoltaic output, a photovoltaic acceptance capacity assessment model based on risk quantification is proposed for the DN to improve the accuracy of PV absorption capacity assessment.
2) Fully considering the effect of network topology reconstruction on the improvement of photovoltaic acceptance capacity, we optimize and calculate the maximum PV absorption capacity of the DN by taking the network topology and photovoltaic access capacity of the DN as variables, and tap the potential of the DN to accept photovoltaically.
3) In the proposed photovoltaic acceptance capacity calculation method for DN, probabilistic tide calculation using linear currents improves the calculation speed while effectively accounting for the effects of the randomness of load and photovoltaic output on photovoltaic acceptance capacity.

Photovoltaic Acceptance Capacity Assessment
After the photovoltaic is connected to the DN, it will have an influence on the voltage deviation, line and transformer transmission capacity, power quality, power supply reliability, and short-circuit current of the DN, all of which are important conditions for assessing the photovoltaic absorption capacity of the DN.In this study, the photovoltaic absorption capacity of the DN is analyzed by considering the voltage deviation constraint and the line and transformer transmission capacity constraint during the steady-state operation of the DN.
The active power PV P of the PV power system is expressed as: where P N is the number of PV panels in the PV power system; i S and i  are the area and conversion efficiency of the ith photovoltaic panel; L is the light intensity.
The DN shown in Figure 1 is used as an example to analyze the effect of photovoltaic access on the node voltage as along with the line and transformer power.When photovoltaic is installed at node 2 and node 4, ignoring the effect of line losses, the voltage magnitude at node 1 can be approximated as: where 1,0 U is the voltage of node 1 without photovoltaic; 1 U is the voltage of node 1 after photovoltaic connection; 1 R and 1 X are the resistance and reactance of the line; PV1 P and PV1 Q are the active and reactive power of photovoltaic 1; PV2 P and PV2 Q are the active and reactive power of photovoltaic 2. After connection to photovoltaic, the apparent transmitted power of the line 1 where L1,0 P and L1,0 Q are the active and reactive power of the line when it is not connected to photovoltaic.
Fig. 1 Diagram of an illustrative DN By Formula (2), the PV access will raise the node voltage, when the PV access capacity is too large will cause the node voltage over the limit; by Formula (3), when the PV output power exceeds the load power transmitted by the line before access to the photovoltaic, or cause the tide to reverse transmission, as the photovoltaic capacity continues to increase, it is possible to cause the line transmission power over the limit.
Considering the voltage deviation constraint during steady-state operation of the DN and the line and transformer transmission capacity constraints, the photovoltaic acceptance capacity model of the DN can be expressed as: where f is the sum of photovoltaic access capacity; P N is the set of nodes for photovoltaic access; ,PVC i P is the capacity of photovoltaic connected to node i; X is the maximum transmission power allowed at line l; T S is the transmission power of transformer; T,max S is the maximum transmission power allowed at the transformer.

photovoltaic Acceptance Capacity Assessment Based on Risk Quantification
The actual values of load and photovoltaic output usually have some errors with the predicted values, which cause variations in node voltage and equipment power and affect the capacity of the DN to accept photovoltaically.The random prediction errors of load and photovoltaic output can be described by a normal distribution.Taking photovoltaic as an example, the probability density function  (5) where PV P is the actual value of PV active power output; PV  is the mathematical expectation, taking the predicted value of the active output of PV power; PV  is the squared difference of the normal distribution of photovoltaic active power output.
So as to accurately assess the influence of load and photovoltaic randomness on the acceptance capacity of the DN, this study thinks about the degree and probability of voltage crossing limits, line and transformer transmission power crossing limits, and constructs a photovoltaic acceptance capacity assessment index based on risk quantification.
(1) Quantitative evaluation index of voltage overrun risk According to the random distribution characteristics of the load and PV output, the Monte Carlo method is used to generate a certain number of random scenes.In the sampled scenes, the more frequently the voltage exceeds the upper limit, the more seriously the voltage exceeds the upper limit, which means the more unreasonable the configured photovoltaic capacity is.Considering the probability of voltage exceeding the limit and the degree of exceeding the limit, the voltage exceeding the upper limit evaluation index is defined as u  : all u ,de all 1 where all N is the total number of samples; ,de c U is the maximum value of the voltage crossing degree of nodes in scenario c.
For node i, its nodal voltage crossing degree , ,de ic U can be expressed as: , (2) Quantitative evaluation index of line power overrun risk Taking into account the probability of line transmission power crossing the limit and the degree of crossing the limit, the line power crossing evaluation index 1 where , ,de cl S is the maximum value of the degree of line transmission power crossing limit in scenario c.The line l transmits power over the limit degree , ,de cl S in scene c can be expressed as: (3) Quantitative evaluation index of transformer power overrun risk Considering the transformer transmission power overrun probability and overrun degree, the transformer power overrun evaluation index tr  is defined as:

Analysis of Network Reconfiguration to Enhance Photovoltaic Acceptance Capacity
The network reconfiguration can significantly improve the problem of voltage crossing limit and partial line overload caused by the high percentage of PV access, and improve the photovoltaic acceptance capacity of the DN.Taking the DN shown in Figure 1 as an example, when line L4 in the network is disconnected, the photovoltaic power of both node 2 and node 4 will be transmitted through line L2, thus increasing the risk of line L2 power crossing the limit.When line L4 is closed and line L5 is disconnected, the photovoltaic power of node 2 is transmitted through line L2, and the photovoltaic power of node 4 is transmitted through line L4, thus reducing the transmission power of line L2 and avoiding overloading line L2.
It can be seen that the PV absorption capacity of the DN can be increased by network reconfiguration, so it is necessary to consider the role of network reconfiguration optimization when calculating the maximum PV absorption capacity of the DN.Therefore, this study proposes an analysis method for the photovoltaic absorption capacity of DN considering the optimization of network structure based on the stochasticity of photovoltaic and load.

Optimization Model
In the above proposed DN photovoltaic acceptance capacity analysis method, a typical day is selected according to the intra-day variation characteristics of photovoltaic output and load, the opening and closing state of DN line switches and photovoltaic access capacity are taken as optimization variables, and the maximum PV access capacity is taken as the optimization objective function, so as to improve PV access capacity through network reconfiguration optimization.The proposed objective function for the optimization of the maximum photovoltaic access capacity is: The optimization model needs to satisfy the following constraints: (1) DN topology constraints The DN operates radially, with no rings or isolated nodes in the system.( The set value of voltage crossing indicator, line and transformer power crossing indicator can be determined by considering the probability of crossing the limit and the degree of crossing the limit.For example, when the samples allowed to cross the voltage limit account for 4.56% of all samples, and the voltage crossing degree is 0.1 in the samples allowed to cross the voltage limit, the voltage crossing indicator setting value ,u,set t  is set to 0.00456.

Algorithm research 3.3.1. Chromosome coding
So as to improve the solution speed, this article combines the characteristics of photovoltaic capacity optimization and topology optimization and combines genetic algorithm and neighborhood search algorithm.The encoding of photovoltaic capacity and line opening and closing status are independent of each other, and decimal encoding is used for photovoltaic capacity.For ease of calculation, the photovoltaic capacity is optimized with a minimum unit of 10 kW.The decimal encoding method based on the random spanning tree is adopted for the open and closed states of the line.Taking the DN shown in Figure 1 as an example, randomly arranging the branches involved in the reconstruction and randomly generating photovoltaic capacity can obtain a chromosome, such as a chromosome L2-L3-L5-L4-100-50.
The closed circuits corresponding to this chromosome are L2, L3, and L5, and the capacities of Photovoltaic 1 and Photovoltaic 2 are 1000 kW and 500 kW.

Fitness calculation
For each chromosome, the proposed photovoltaic acceptance ability evaluation index is calculated based on its corresponding photovoltaic capacity and network topology structure.When the chromosome meets the constraint, the negative PV capacity is the fitness of the chromosome; When the chromosome does not meet the constraint, we set the fitness of the chromosome to 0.

Evolutionary operation
We implement population evolution through selection, crossover, and compilation operations.Among them, for genes representing photovoltaic capacity, a combination of mutation operation and neighborhood search is used to improve evolution speed.When a gene is randomly selected, a new gene is randomly generated within a certain neighborhood of its corresponding photovoltaic capacity.If the photovoltaic capacity corresponding to a certain gene is 1000 kW, increasing or decreasing the photovoltaic capacity by 100 kW is used as the neighborhood for its mutation, then the mutated gene is randomly generated within the range of [90 110].

Probabilistic power flow calculation based on linear power flow
Considering that accurate power flow calculation for each sample sampled by Monte Carlo will increase the computational workload when calculating the photovoltaic acceptance capacity evaluation index, this paper adopts a probabilistic power flow calculation method based on linear power flow for power flow calculation.Firstly, based on the network topology and photovoltaic capacity corresponding to each chromosome, power flow calculations are performed on deterministic loads and photovoltaics to obtain node voltages 0 U and Jacobian matrices 0 J .Secondly, the Monte Carlo method is used for sampling, with the voltage and Jacobian matrix under deterministic load and photovoltaic output as the initial values for power flow calculation.The correction amount of node voltage is calculated based on the difference between the node power value and the deterministic value in the sample.Therefore, the voltage amplitude U corresponding to the samples sampled by Monte Carlo can be expressed as:

Analysis method for photovoltaic absorption capacity of DN
Taking the IEEE33 (Figure 2) node DN as an example for analysis, the voltage rating is 12.66 kV.The daily generation curve of photovoltaics is shown in Figure 3.The range limits of voltage are 0.93 p.u. and 1.07 p.u., respectively.The photovoltaic power factor is 1.The number of reconstructions per day is set to 1.

Analysis of photovoltaic acceptance capability without considering the randomness
In scenario 1, the photovoltaic access nodes are set to nodes 5, 13, and 25, the capacity of the transformer and line L1 is 10 MW, and the maximum transmission capacity of other lines is set to 5 MW.
The DN photovoltaic acceptance capacity model represented by Formula ( 4) is used to analyze the role of topology optimization in improving photovoltaic acceptance capacity.Table 1 shows the maximum photovoltaic access capacity of the system under different topology structures.Among them, Scheme 1 does not adopt reconstruction, Scheme 2 adopts reconstruction with the constraint condition of minimizing network loss, and Scheme 3 adopts reconstruction with the goal of maximizing photovoltaic acceptance capacity.It can be seen that when network reconstruction is carried out with the goal of maximizing photovoltaic acceptance capacity, the maximum photovoltaic acceptance capacity is obtained.At the same time, the topology structure with the constraint condition of minimizing network losses can optimize power flow distribution and also improve the photovoltaic absorption capacity of the DN.We set the capacity of the transformer and line L1 to 15 MW and set the limit transmission capacity of other lines to 10 MW.Table 2 shows the maximum photovoltaic access capacity of the system under different topologies.It is observed that when the line and transformer capacities increase, the photovoltaic absorption capacity of the DN is increased, but the photovoltaic capacity increase is smaller than the increase in transformer and line capacity, indicating that the node voltage becomes a limiting condition for the photovoltaic absorption capacity of the DN at this time.

Photovoltaic Absorption Capacity Analysis Based on Risk Quantification
The stochastic nature of load and photovoltaic is further considered to analyze the role of topology optimization in enhancing the photovoltaic acceptance capacity.The squared difference of the normal distribution of the active output of load and photovoltaic generation is set to 10% of the predicted value, and the size of samples in the Monte Carlo method is 1000.According to the 3σ principle of normal distribution, the number of samples allowed to voltage or equipment power overrun is less than 4.56% of the total number of samples, and the voltage or equipment power overrun degree in the voltage or equipment power overrun samples is less than 0.1, and then the voltage overrun index, line, and transformer power evaluation index are set to 0.00456.Table 3 shows the largest PV admission capacity of the IEEE33 node arithmetic system under different topologies when the randomness of load and PV output is considered.Compared with Table 1, it is observed that the PV admission capacity of the DN decreases due to the effect of the randomness of load and PV generation on the node voltage and equipment power, which is considered.The DN tide distribution after considering the load and photovoltaic stochasticity is analyzed as an example from 13:00-14:00 when the photovoltaic output is high.The squared difference of the normal distribution of the active output of load and photovoltaic is set to 10% of the predicted value, and 1000 samples are obtained by using Monte Carlo method sampling.Figure 4 shows the maximum crossing of the line power and node voltage after the photovoltaic is connected according to the maximum access capacity under the topology adopted without considering randomness.Figure 5 shows the maximum crossing limits of node voltage and transformer power after the photovoltaic is connected according to the maximum access capacity under the topology adopted considering randomness.Table 4 shows the corresponding photovoltaic acceptance capacity evaluation index.It is observed that when the photovoltaic capacity is calculated without considering the randomness of load and photovoltaic output, the node voltage and line power have a higher probability of crossing the limit; when the randomness is considered to determine the photovoltaic capacity, the node voltage, and line power cross the limit within the set constraint value.Therefore, when using network reconfiguration to improve the photovoltaic absorption capacity of the DN, considering the randomness of load and photovoltaic is beneficial to ensure that the system voltage and equipment power is within a safe range.

Analysis of the accuracy of probabilistic trend calculations
The accuracy and rapidity of the proposed probabilistic tide calculation method are analyzed by taking the load power and PV output of the system at 13:00 hours as an example.The size of samples generated in the Monte Carlo method is 1000, the time required to perform the tide calculation for each of the 1000 samples independently is 10.05 s, and the time required for the probabilistic tide calculation using the linear tide model based on the linear tide model is 0.9 s. Figure 6 shows the maximum error of 0.068% for the node voltages for all scenarios with the exact tide calculation and the calculation using the linear tide model.It can be seen that the adopted probabilistic tidal current calculation method based on the linear tidal model improves the calculation speed while ensuring the calculation accuracy.

Conclusion
This study proposes an analysis method for the photovoltaic acceptance capacity of DN based on risk quantification and network reconfiguration.First, to accurately assess the influence of PV stochasticity on the acceptance capacity of DN, a photovoltaic absorption capacity assessment model based on risk quantification is proposed for DN; Second, a photovoltaic admittance capacity assessment calculation method considering network topology reconfiguration is proposed to optimize the computation of photovoltaic admittance capacity of DN with the network topology and photovoltaic access capacity of DN as variables and to improve the calculation speed by using linear currents for probabilistic current calculation.It can be seen from the simulation results that the proposed DN photovoltaic acceptance capacity analysis method can effectively measure and assess the operational risk of DN caused by the randomness of load and photovoltaic, and effectively account for the improvement effect of network reconfiguration on the photovoltaic acceptance capacity of DN.
and maximum values of voltage allowed at node i; l S is the transmission power of line l; ,max l S where ,T,de c S is the degree of transformer transmission power crossing limit in scenario c. ,T,de c S can be expressed as follows: Fig. 2 IEEE 33-bus distribution network

Fig. 4 Fig. 5
Fig. 4 Voltage and line power distribution under deterministic analysis results

Fig. 6
Fig. 6 Accuracy analysis of linear power flow 2) Voltage constraint For any moment t, the quantitative evaluation indicator ,u  are the differences between the active and reactive power values of nodes in the sampled samples and the deterministic values; U  and   are the corrections for voltage amplitude and phase angle.

Table 1 .
Maximum photovoltaic access capacity in scenario 1.

Table 2 .
Maximum photovoltaic access capacity in scenario 2.

Table 3 .
Maximum photovoltaic access capacity in scenario 2.

Table 4 .
Maximum photovoltaic access capacity in scenario 2.