Risk assessment of distribution network operation with scale distributed photovoltaics

In the context of large-scale photovoltaic integration into the distribution network, it is of great significance to comprehensively consider the uncertainty of photovoltaic output and load and scientifically and effectively evaluate the operational risks of the distribution network, in order to ensure the reliable and economic operation of the distribution network. At the same time, with the increase of distributed photovoltaic installed capacity, the small interference stability caused by it has an increasing impact on the operation of the distribution network. In this paper, we first establish a probabilistic model of photovoltaic output and load. Then, we propose a risk calculation method based on the non-sequential Monte Carlo method and probabilistic power flow calculation. Then, a risk analysis of large-scale photovoltaic access to the distribution network was carried out. A new small interference stability risk index was constructed to evaluate the risk of system oscillation caused by photovoltaic access. On this basis, the comprehensive risk value is calculated, and the risk levels are divided based on the voltage over-limit risk and power flow over-limit risk. Finally, the feasibility and effectiveness of the risk assessment model are verified through case analysis.


Introduction
At the general debate of the 75th United Nations General Assembly, General Secretary Xi Jinping proposed the "30ꞏ60 Double Carbon" goal: "China's carbon dioxide emissions will strive to peak before 2030 and achieve carbon neutrality before 2060" [1].The proposal of the "double carbon" goal points out the direction for energy conservation and emission reduction work in my country's energy industry.In the process of achieving the "double carbon" goal, building a new power system occupies an important position.The access to large-scale distributed photovoltaics with uncertainty and volatility has brought many uncertainties and potential risks to the safe and efficient operation of the distribution network, such as changes in power quality, changes in power flow distribution, etc. [2].Therefore, building a scientific and effective risk calculation model for distribution networks containing large-scale distributed photovoltaics and reasonably calculating their risk index values is of great significance to ensure the reliable and economic operation of distribution networks.
In the context of large-scale distributed photovoltaic integration into distribution networks, how to evaluate the operational risks of distribution networks has become a hot research topic.Ruan considered the uncertainty of photovoltaic output and load demand and proposed a distribution network operation risk assessment method based on improved quasi-Monte Carlo sampling [3].In response to the issue of the impact of flexible integration of distributed photovoltaics into the distribution network on its reliability, researchers considered the impact of different photovoltaic output scenarios, grid connection methods, and grid capacity on the distribution network.They conducted a reliability evaluation of the distribution network containing distributed photovoltaics grid connection [4].Zhang used the analytic hierarchy process and entropy weight analysis method to assign values to each risk indicator.He comprehensively considered the two assignment methods and finally quantified the weight of the indicators.He applied the fuzzy comprehensive evaluation method to build a complete comprehensive risk assessment model for photovoltaic power generation projects [5].Zhang et al. used the semiinvariant series expansion method to conduct stochastic power flow research on the grid-connected system, evaluate the risk factors during the system operation process, and also study the voltage out-oflimit risks and support caused by the grid-connected operation of distributed power sources on the system.The risk situation of road power flow exceeding the limit and the behavioral risk caused by branch road power flow [6][7].In [8], researchers completed the risk assessment of photovoltaic integrated power systems by calculating the probability and severity of exceeding limits for events such as overvoltage, undervoltage, overload, and thermal overload.
This paper starts with the analysis of the stochastic model of system components.It first establishes a probabilistic model of photovoltaic output and load.Then, it proposes a risk calculation model based on the non-sequential Monte Carlo method and probabilistic power flow calculation.Then, risk indicators for large-scale distributed photovoltaic grid connections were constructed.Risk values were calculated from three aspects: node voltage over-limit risk, branch power flow risk over-limit risk, and small interference stability risk.The risk index value is used to more comprehensively and reasonably evaluate the risk of distribution network operation using the risk level classification method.Finally, MATLAB was used to conduct case analysis to verify the feasibility and effectiveness of the risk assessment model.

Photovoltaic output probability model
The output of photovoltaic power plants is most related to the local light intensity.Without considering factors such as temperature and weather, the output power of photovoltaic power plants is directly proportional to the intensity of sunlight: (1) where is the actual output power of photovoltaic power plants; is the actual light intensity; is the total area of the photovoltaic system array; is the photoelectric conversion efficiency of the photovoltaic module; is the maximum light intensity; is the total photovoltaic power generation under the maximum light intensity, which is determined by the performance parameters of the photovoltaic system components.
According to relevant research, assuming that the light intensity approximately obeys the Beta distribution within a certain period (such as 11:00-12:00, 07:00-18:00, etc.), the distribution function is as follows: (2) where and represent the actual light intensity and the maximum light intensity, respectively; represents the Gamma function; and are the two shape parameters of the Beta distribution.
From the above two equations, the probability density function of the output power of the photovoltaic power generation system can be obtained: (3)

Load probability models
In actual situations, the power load demand in the distribution network is also constantly changing, which is reflected in two aspects: load value size and load fluctuation degree.This paper selects normal distribution to simulate the distribution of load demand.
Therefore, the load demand model is as follows: The probability distribution of load demand is expressed as: where L P is the actual load value; L u and L  are the average and standard deviation of the load, respectively.

Non-sequential Monte Carlo method
With the large-scale integration of new energy and the increasing scale and complexity of power systems, the Monte Carlo Method (MCM) is increasingly used in risk assessment.
The main process of the non-sequential Monte Carlo method is to assume that the states of each component in the system are independent and that there are only two component states: operating state and fault state.It can be extracted by extracting the values distributed in the interval [0,1].Random numbers are used to simulate the operating status of each component, and there is a formula for the component status: where represents the component state, is a random number uniformly distributed in [0,1], and represents the component failure probability.
Therefore, for a system containing N components, a set representing the overall state of the system can be obtained ( , , , , ) . The maximum number of samplings that meet the requirements of the large-scale photovoltaic power plant is used as the termination criterion.

Risk calculation method based on probabilistic power flow calculation
In order to conduct risk assessment on distribution networks affected by a large number of uncertain factors, the stochastic power flow algorithm based on Monte Carlo simulation sampling can be used.
The specific process of using a stochastic power flow algorithm to implement risk assessment is as follows:  Historical lighting intensity data and distribution network load data are classified and calculated for a certain area, and the expected values and standard deviations of the two are calculated. The input parameters for stochastic power flow calculation are determined, including the topology of the distribution network, resistance and reactance of each node, load and probability distribution of photovoltaic output, and other data. Based on the photovoltaic and load probability models, the non-sequential Monte Carlo method is used to randomly sample the photovoltaic output power and load status to form the system status. The forward-backward substitution method is used for deterministic power flow calculation for each sampled system state. After reaching the maximum sampling number, the statistical characteristics of the output results are calculated.The calculated values are substituted into the operational risk indicator expression to obtain the values of each operational risk indicator. The comprehensive risk indicator is calculated, and the risk level is judged based on the risk indicators.

Voltage exceeding limit risk indicator
The expression for the risk indicator of voltage exceeding the limit is: where , ( ) where ( ) i V t represents voltage per unit value of node i at time t, max V and min V are the upper and lower limits of voltage fluctuations allowed by the distribution grid, taken as 1.05 and 0.95 in this paper, respectively.

Tidal current exceeding limit risk indicator
The expression for the risk indicator of tidal exceeding limit is: where , ( ) R t is the risk indicator value of the tidal exceeding limit of branch l at time t, , ( ) is the upper limit of the carrying current allowed by the branch, taken as 0.9 in this paper.) and the impedance of photovoltaic grid-connected systems .Whether oscillation occurs depends on the relationship between the two impedances.Generally, the power grid will exhibit inductance, and the capacitive impedance of photovoltaic grid-connected inverters is the prerequisite for resonance to occur.A decrease in power grid strength or an increase in the number of parallel photovoltaic inverters will force the two impedance curves to intersect, and the intersection point will fall within the sub/hyper-synchronous range, which is the decisive condition for resonance to occur.Under these two conditions, if the control parameters, such as the inverter phase-locked loop or current loop are not set properly, it will result in a small phase margin at the intersection frequency of the amplitude-frequency characteristic curve, and there will be a risk of sub-synchronous oscillation.
According to the derivation in [9], the inverter impedance model is shown in Figure 1.When there is an intersection point between the amplitude-frequency characteristic curve of the grid impedance and the inverter impedance, if the phase margin at the intersection point is insufficient, there is a risk of oscillation.The oscillation risk of low-frequency oscillation is mainly related to the parameters of the converter and is also affected by the load and distributed power output.Without considering the fluctuations of distributed power sources and load, the risk of low-frequency oscillation and sub/hyper-synchronous oscillation instability is related to the system's short-circuit ratio.Therefore, oscillation instability can be considered as the most severe case, with the system short-circuit ratio as the reference value, and based on the impact of load and output fluctuations on low-frequency and sub/hyper-synchronous oscillation, the oscillation risk indicators within this frequency band can be obtained.The short circuit ratio SCR of a single feed system is defined as: where ac S is the short circuit capacity of the AC system, and B S is the rated capacity of power electronic equipment itself.When the system short-circuit ratio is greater than the SCR value, it indicates that the stronger the AC power grid connected to the power electronic equipment is, the greater the system stability margin will be.On the contrary, the smaller the SCR value is, the weaker the AC power grid connected to the power electronic equipment will be, and the more likely the system is to experience low-frequency oscillation and sub/hyper-synchronous oscillation.
According to research, when the local load level is high, the equivalent impedance of the power grid is smaller, the amplitude-frequency characteristic curve is not easy to intersect, and the relative phase margin is also greater.Therefore, the load level is negatively correlated with the risk of low-frequency, sub/hyper-synchronous oscillation.Meanwhile, the report also indicates that the fluctuation of irradiance has little impact on the stability of small disturbances.The stronger the overall irradiance is, the worse the stability of small disturbances is.Therefore, the severity of low-frequency oscillation and sub/hypersynchronous oscillation of node i is defined as: where k is all the nodes connected to photovoltaic inverters.

High-frequency oscillation indicator.
The risk assessment method for high-frequency oscillations caused by distributed photovoltaic grid connection is similar to low-frequency and sub/over frequency oscillations.However, it can be seen from Figure 1 that the dominant factor affecting the high-frequency impedance model of distributed power grid connection is the filter parameters.When considering the phase-locked loop, the impedance expression of the inverter is shown in Equation (13) [9]: where, due to the fact that the frequency of high-frequency oscillation is generally higher than 1000 Hz, when the value of s=jw is too large, the influence of the controller can be weakened, and the control of the phase-locked loop can be ignored.
The equivalent open-loop transfer function of the inverter system is shown in Equation ( 14): where 1 2   , L L and 1 C all are the parameters of photovoltaic grid connected filters, k is all nodes connected to the photovoltaic system, .
L is inductance between node k and node i.
The general system needs to control the damping ratio between 0.6 and 0.8.When the damping ratio is less than 0.6, the oscillation risk is considered.When the damping ratio is 0, the system oscillation becomes unstable.Therefore, the high-frequency oscillation risk is expressed as Equation ( 16): In summary, the normalization of oscillation risk is expressed as follows: Therefore, when there are both high-frequency harmonics, sub/hyper-synchronous, and lowfrequency harmonics, the distribution grid may experience both low-frequency oscillation and highfrequency oscillation.At this time, the small disturbance stability risk indicator is the maximum oscillation risk, as shown in Equation (18): where O R is the small disturbance stability risk indicator,

Comprehensive risk indicator
This article uses the Analytic Hierarchy Process combined with the Entropy Weight Method to determine the weights of each evaluation indicator.After program calculation, the weights of distribution grid risk indicators such as voltage exceeding limit risk , ( ) R t , tidal current exceeding limit risk , ( ) R t , and small disturbance stability risk O R are: 0.55, 0.33, and 0.12.
The comprehensive risk value requires a weighted summation of three indicators, and the calculation formula is as follows: where ( ) com R t is the comprehensive risk indicator value.The higher its value is, the higher the risk will be.i  represents the weights of three risk indicators, which can also be given by users based on the focus of their own needs.

Analysis plan design
According to the assessment model and assessment indexes, this paper integrates distributed photovoltaics in the IEEE-33 node system and conducts a study on the operational risk of distribution networks with large-scale distributed photovoltaics integration based on the Matlab R2022a simulation platform.The system consists of 33 nodes and 32 branches; its topology diagram is shown in Figure 2. The reference voltage of the system is 12.66 kV, and the total network load is 5084.26+i2547.32kV ꞏ A.
Firstly, the relevant parameters of the risk assessment model are set.In the probability model of photovoltaic output, the maximum illumination intensity

Analysis of calculation results
Considering the impact of distributed photovoltaics, risk assessment is conducted when a photovoltaic with a capacity of 1000 kW is connected to nodes 12 and 16.The voltage exceeding the limit of each node under photovoltaic output is shown in Figure 3. From Figure 3, it can be seen that the maximum per unit voltage value of each node in the distribution network is 1.0000 p.u., and the minimum is 0.7101 p.u. Due to the large load of the power grid during the evening peak period and the lack of photovoltaic power generation, there is no risk of voltage exceeding the upper limit in the distribution network, which will result in a greater risk of voltage exceeding the lower limit.Nodes 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 26, 27, 28, 29, 30, 31, 32, 33 have voltage values per unit less than 0.95 p.u.Among them, node 18 has the highest degree of voltage exceeding the lower limit, with a value of 0.7101 p.u. and a severity of 0.2399.
The risk of tidal current exceeding the limit for each branch of the distribution network is evaluated, and the risk of tidal current exceeding the limit for each branch is obtained as shown in Figure 4.It can be seen that except for branches 25, 29, 30, 31, and 32, all other branches have situations where the current exceeds the limit.Among them, branches 1, 2, 12, 16, 17, and 18 have a higher severity of tidal current exceeding the limit, while branch 17 has a maximum severity of 0.1634.The tidal current exceeding the limit mainly occurs during the noon period, as there is too much photovoltaic power generation and the load level is average, resulting in a serious problem of back-feeding of the tidal currents.Therefore, the branches close to the photovoltaic access location and connected to the higherlevel power grid are subject to a greater risk of exceeding the limit due to the reverse transmission of photovoltaic output power.
The risk of the small disturbance stabilization risk of the distribution network is evaluated and the risk value is obtained, as shown in Figure 5. From the figure, it can be seen that node 12 and node 16 at the photovoltaic access location have the highest risk of small disturbance stabilization, with values of 0.0260 and 0.0283.Next are nodes 1 and 2 connected to the higher-level power grid, as well as nodes 31, 32, and 33 near the end of the system.Taking into account the voltage exceeding limit risk indicator, tidal current exceeding limit indicator, and small disturbance stability risk indicator of the distribution network, the weighted method is used to calculate the comprehensive risk indicator.The comprehensive risk indicator can quickly reflect the overall risk of a certain location, and the obtained comprehensive risk indicator values are shown in Figure 6.On the basis of calculating the comprehensive risk indicator values, based on the risk level classification method, the comprehensive risk values of each node are divided into three risk levels: high risk, medium risk, and low risk, using unacceptable risk level lines and negligible risk level lines.As shown in Figure 7, the comprehensive risk values of nodes 15, 16, and 17 have all reached a high-risk level.Therefore, when photovoltaic power is connected to the distribution network, it is important to focus on the safety and stability of these three nodes.

Conclusion
This paper establishes a probability model for photovoltaic output and load.Based on this, it proposes a distribution grid operation risk assessment model that considers the uncertainty of photovoltaic and load.The non-sequential Monte Carlo method and probabilistic tidal current model were used to more accurately calculate the voltage exceeding limit indicator value, branch tidal current exceeding limit indicator value, small disturbance stability risk indicator value, and comprehensive risk value.MATLAB was used for example analysis.The impact of three risk indicators on the comprehensive risk value of the distribution grid was compared, and the magnitude of the risk value of each node was compared through risk level division.The research work provides a reference for the risk assessment of large-scale photovoltaic distribution grid operation.

4. 3 .
Small disturbance stability risk indicator 4.3.1 Sub/hyper-synchronous and low-frequency oscillation indicator.The determining factors of whether the system will experience sub-synchronous oscillations include power grid strength (i.e., grid impedance

Z
 are branch impedances connected to the access nodes, respectively. ,k i Z is the impedance between node k and node i, S and 0 S respectively represents irradiance and standard irradiance, their unit is 2 and node average load level, respectively.

C
are the risk values of low- frequency and high-frequency oscillations after normalization, respectively.
photovoltaic array

Figure 3 .
Figure 3. Severity of voltage exceeding limit for each node.

Figure 4 .
Figure 4. Severity of the tidal current exceeding limit for each branch.

Figure 5 .
Figure 5. Risk of small disturbance stability in distribution networks.

Figure 6 .
Figure 6.Values of various risk indicators.
is the current per unit value passing through branch l at time t, and max I l I t After calculation, the damping ratio of this node i is shown in Equation (15): 7