# Analysis of Security Operational Characteristics of Distribution Network considering Electric Vehicle Load Access Forecast

Authors: Lei Sun; Yong Ma; Jinsong Zhu; Xiaohan Li; Peng Wu; Xiangyu Bao
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In recent years, electric vehicles have been widely promoted by virtue of environmental protection and pollution-free advantages. In the future, they are expected to become a mainstream means of transportation. However, as new services and business models become more and more closely related to the distribution network, the traditional AC distribution district has limited ability to respond to the large-scale access of electric vehicles. Therefore, it is becoming increasingly important to improve the ability to respond to electric vehicle access, reduce the negative impact of large-scale electric vehicle access on the grid, and predict and calculate the uncertainty caused by electric vehicle load access. Therefore, this paper proposes the concept of flexible distribution district, constructs the security operational boundary model for flexible distribution district, and uses “security margin percentage” as the measurement mechanism to provide dispatchers with an intuitive system operating status. In addition, this paper also proposes an effective uncertainty quantification method for the problem that it is difficult to measure the changes of bus voltage and line current at network nodes when large-scale electric vehicles are connected to the network. Numerical results demonstrate both the extensibility and the practicality of the proposed security operational boundary. In addition, the proposed uncertainty quantification method can reflect the impact of load changes on the characteristics of the power grid, which is helpful to study the improvement significance of adjusting electric vehicle load changes to distribution network optimization.


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## Body

## 1. Introduction

With the excessive consumption of fossil energy, the global energy crisis has intensified and environmental damage has become increasingly serious. Under the dual pressure on tight natural resources and increased environmental pollution, electric vehicles (EVs), which have advantages in energy conservation and emission reduction in urban transportation systems, have received widespread attention. With the continuous maturity of related technologies, the large-scale application of EVs will surely become an important feature of the future urban transportation system and an effective way to solve urban environmental problems and alleviate the energy crisis [1, 2]. China has also formulated a development plan suitable for national conditions to promote the industrialization of EVs and planned to basically build an infrastructure system that meets the charging needs of more than 5 million EVs by 2020.The access to the charging load of large-scale EVs has brought new challenges to the grid. In the traditional slow charging mode, the charging load of EVs overlaps with the peak period of residential electricity consumption, which is likely to cause problems such as overload of grid components, voltage fluctuations, increased line losses, and increased three-phase imbalance in the distribution network. Under the circumstances of fast charging mode, the randomness of EV access, the strong impact of the aggregation effect of fast charging piles on the grid, and the current limitation of the distribution transformer capacity will all affect the safe operation of the distribution network in the fast charging mode [3, 4]. Through the AC/DC converter, the EV is connected to the active distribution network, and the minimum line current harmonic can be obtained. Ai et al. [5] proposed a flexible AC and DC power supply system topology combining a multi-time scale system control strategy to realize the energy interactive control method of EV charging and discharging units on different time scales, which can ensure the safety, stability, and reliability of the system operation under different operating conditions and realize the low-carbon operation of the system. Shen et al. [6] proposed an orderly charging and discharging control method for EVs based on BP control and neural network regulation, which can improves the output stability. B. Sun et al. [7] proposed adopts a modular multilevel commutation control method, combined with the analysis of AC side voltage and IGBT switching frequency parameters, which can effectively reduce the system oscillation and realize the adaptive control of orderly charging and discharging of EVs. Takagi M et al.[8]used a bidirectional converter to control the charging and discharging state of electric vehicles and control the bidirectional flow of energy in the DC microgrid and the large grid based on the bidirectional AC/DC converter, which can effectively control the stability of the DC bus voltage and improve the stability of the system operation.In addition, EV-related services and business models are becoming increasingly closely connected with the distribution network, and it is becoming increasingly closely significant to incorporate load uncertainty into forecasting calculations.In terms of reducing the impact on EV charging with the grid, Gao et al. [9] proposed a pricing strategy for EV charging stations based on a noncooperative game in order to reduce the adverse effects of large-scale disorderly charging of EVs on the safe and stable operation of the distribution network. M.Wang et al. [10] proposed the idea of an optimization model that minimizes daily load fluctuations and uses a binary particle swarm optimization algorithm to solve the model, outputting an orderly charging strategy for EVs so as to achieve the goal of reducing the peak-valley difference in daily load. Jin et al. [11] used the deep reinforcement learning method to realize the EV charging scheduling considering a variety of random factors, which can significantly reduce the user’s mileage worry and charging cost in the fast charging mode. In the study of G. Gruosso et al. [12], from the perspective of suppressing the load fluctuation of the distribution network and reducing the three-phase phase imbalance, centralized control of the EV charging in the regional cluster is carried out, and the charging time is optimized and adjusted.Wang et al. [13] proposed a multi-time scale, multiobjective control strategy, which can not only meet the charging requirements of EVs but also adjust the grid voltage. Chen et al. [14] proposed a multi-time scale optimal scheduling scheme for EVs to provide optimal voltage operation control for the distribution network.In terms of predicting the load uncertainty caused by EVs connected to the distribution network, people have realized that uncertainty calculation tools can play a key role in the design and real-time control of distribution circuits. Such tools can provide a comprehensive view of the entire network. They can predict bus voltage and line current changes at network nodes that are difficult to measure [15]. The mainstream network uncertainty analysis method uses probabilistic models for loads and uses repeated deterministic load flow (LF) analysis in the Monte Carlo (MC) iteration process. This method is usually called probabilistic load flow (PLF) and can be very time-consuming [16].The actual load modeling is uncertain and is usually based on statistical analysis of available data, which is collected and analyzed for multiple network areas and utility types (such as residential, commercial and industrial, and EVs). In order to solve the interaction between many independent uncertain loads, a large number of MC runs are required to obtain a satisfactory statistical description. In this case, the statistical information about the average value and variance of the electrical variable is not enough to describe it correctly, and a detailed probability density function (PDF) model is needed for further inference. Using the MC method to accurately determine the PDF may require tens of thousands of repeated LF analysis, so it is very time-consuming [17].Based on the concept of “operation region” [15, 18], this paper proposes a security operational boundary model of flexible distribution district and, considering the influence of large-scale EV load, proposes an uncertainty quantification method based on generalized polynomial chaos (GPC) extension and random testing (ST) algorithm. Dispatchers can implement flexible scheduling of EV charging loads according to the security boundary, adjust the number of EVs connected to each station, and prevent high-risk operation in the distribution district from the source.The innovations of this paper are as follows:(1)
A security operational boundary model and a security margin percentage as a measurement mechanism are proposed to facilitate the dispatcher to quickly judge the security status of the system(2)
An uncertainty quantification method based on GPC + ST is proposed to predict the statistical distribution and variable interval of relevant observable variables in distribution network due to load uncertaintyThe methods proposed in this paper help to control and guide the load access of EVs online and have a certain improvement in optimizing the operation of the distribution network.


## 2. Safe Operation of Flexible Distribution District

### 2.1. Structure of the Flexible Distribution District

The traditional EV access method will increase the peak-to-valley difference of the distribution network load and reduce the effective utilization rate of the transformer. In addition, the different stations of the AC distribution network operate in an open loop and the maximum connectable load of each station will be limited by the capacity of the station’s transformer. In this mode, when EVs are connected on a large scale, the expansion of the transformer can only be used to meet the load growth demand. In this case, a single transformer will be at risk of long-term overload operation, which is not conducive to the safety of the distribution district. On the other hand, if an N-1 contingency occurs in a transformer in a certain distribution district because the load transfer cannot be realized in the distribution district, it will face the risk of load shedding, which will reduce the reliability of power supply.In order to solve the above-mentioned problems, this paper proposes the concept of flexible distribution districts by interconnecting several AC districts through voltage source converter (VSC) to realize the cooperative operation and control of multiple station areas, the topology of which is shown in Figure1.Figure 1
Structure of the flexible distribution districts. Security operational boundary of flexible distribution district.The DC branch of the flexible distribution district can adapt to the fluctuation of the charging load faster and more widely so that the district is within the security operational boundary, and this mode will not increase the short-circuit current in the districts. To sum up, the flexible distribution district can realize the optimal transfer and balanced allocation of loads, avoid the heavy load or overload of a single transformer, delay the expansion of the transformer, reduce construction costs, and improve the reliability of power supply in the district.The security operational boundary of a flexible distribution district refers to the maximum/minimum injected power of a specific node in the district under a certain safety condition. In this paper, the AC and DC charging piles in the district are selected as the research objects. On the premise of considering the optimal load transfer and balanced allocation, the maximum charging load that can be connected to the AC and DC charging piles in each period is calculated to obtain the security operational boundary of the district. In addition, when calculating the maximum and minimum access load of a charging pile, keep the access loads of the other charging piles unchanged. The specific model for obtaining the security operational boundary of the flexible distribution district is as follows:(1)maxPi,tacPi,tdc,(2)Ctr=Cfe+Ccu,(3)Cfe=∑t=1T∑i=1nLife,(4)Ccu=∑t=1T∑i=1nLicuβi,t2,(5)βi,t=Pi,ttrCica,(6)Pi,ttr=Pi,tcity+Pi,tac+Pi,tvsc,(7)βi,t≤βmax,(8)−Pivsc,max≤Pi,tvsc≤Pivsc,max,(9)∑i=1nPi,tvsc=Ptdc+Pi,tess.Equation (1) is the objective function of the security operational boundary model of the flexible distribution district, where Pi,tac and Pi,tdc represent the AC load and DC load of the EV at time t, respectively. Ctr, Cfe, and Ccu represent the operating loss, iron loss, and copper loss of transformer, respectively; T represents the total number of periods and n is the number of distribution districts; Life and Licu, respectively, represent rated iron loss and rated copper loss of transformer; βi,t and Pi,ttr, respectively, represent the load factor and power of the transformer at time t; Cica is the capacity of the transformer. Equation (6) represents the power balance on the AC side, where Pi,ttr is the power on the AC side of the transformer at time t, and Pi,tvsc is the power flowing through the ith VSC at time t ( Pi,tvsc >0 indicates that the VSC is in the rectifying state and the power flows from the transformer to the DC bus; Pi,tvsc <0 indicates that the VSC is in the inverter state and the power flows from the DC bus to the transformer), and Pi,tcity means the residential load power at time t. βmax represents the upper limit of the transformer load rate, which is predesignated by the dispatcher. Pivsc,max is the maximum power of the ith distribution district converter. (1) represents the power balance equation on the DC side, where Pi,tess is the energy storage power at time t.Equation (1)–(9) are linear models. The existing commercial solution software can efficiently and accurately obtain the global optimal solution, which provides the possibility of real-time construction of the security operational boundary.


### 2.2. Security Operational Boundary of Flexible Distribution District

Based on the security operational boundary obtained in the previous section, the “security margin percentage” can be defined to guide the operation of dispatchers, so that dispatchers can intuitively judge the safety of the current operating status and ensure the safe operation of the flexible distribution district. The security margin percentage refers to the ratio of the power difference between the current operational point and the operational boundary point to the power at the operational boundary point. The security margin percentages of AC charging stations and DC charging stations are defined as follows:(10)Si,tac=Pi,tac∗−Pi,tacPi,tac∗×100%,(11)Si,tdc=Pi,tdc∗−Pi,tdcPi,tdc∗×100%.Among them,Si,tac and Si,tdc refer to the security margin of the AC/DC charging station and Pi,tac∗ and Pi,tdc∗, respectively, represent the operational boundary point power of the AC/DC distribution district.The above security distance can indicate the security margin of the corresponding charging stations and the ability to respond to the uncertain access of EVs under the current load state.


## 2.1. Structure of the Flexible Distribution District

The traditional EV access method will increase the peak-to-valley difference of the distribution network load and reduce the effective utilization rate of the transformer. In addition, the different stations of the AC distribution network operate in an open loop and the maximum connectable load of each station will be limited by the capacity of the station’s transformer. In this mode, when EVs are connected on a large scale, the expansion of the transformer can only be used to meet the load growth demand. In this case, a single transformer will be at risk of long-term overload operation, which is not conducive to the safety of the distribution district. On the other hand, if an N-1 contingency occurs in a transformer in a certain distribution district because the load transfer cannot be realized in the distribution district, it will face the risk of load shedding, which will reduce the reliability of power supply.In order to solve the above-mentioned problems, this paper proposes the concept of flexible distribution districts by interconnecting several AC districts through voltage source converter (VSC) to realize the cooperative operation and control of multiple station areas, the topology of which is shown in Figure1.Figure 1
Structure of the flexible distribution districts. Security operational boundary of flexible distribution district.The DC branch of the flexible distribution district can adapt to the fluctuation of the charging load faster and more widely so that the district is within the security operational boundary, and this mode will not increase the short-circuit current in the districts. To sum up, the flexible distribution district can realize the optimal transfer and balanced allocation of loads, avoid the heavy load or overload of a single transformer, delay the expansion of the transformer, reduce construction costs, and improve the reliability of power supply in the district.The security operational boundary of a flexible distribution district refers to the maximum/minimum injected power of a specific node in the district under a certain safety condition. In this paper, the AC and DC charging piles in the district are selected as the research objects. On the premise of considering the optimal load transfer and balanced allocation, the maximum charging load that can be connected to the AC and DC charging piles in each period is calculated to obtain the security operational boundary of the district. In addition, when calculating the maximum and minimum access load of a charging pile, keep the access loads of the other charging piles unchanged. The specific model for obtaining the security operational boundary of the flexible distribution district is as follows:(1)maxPi,tacPi,tdc,(2)Ctr=Cfe+Ccu,(3)Cfe=∑t=1T∑i=1nLife,(4)Ccu=∑t=1T∑i=1nLicuβi,t2,(5)βi,t=Pi,ttrCica,(6)Pi,ttr=Pi,tcity+Pi,tac+Pi,tvsc,(7)βi,t≤βmax,(8)−Pivsc,max≤Pi,tvsc≤Pivsc,max,(9)∑i=1nPi,tvsc=Ptdc+Pi,tess.Equation (1) is the objective function of the security operational boundary model of the flexible distribution district, where Pi,tac and Pi,tdc represent the AC load and DC load of the EV at time t, respectively. Ctr, Cfe, and Ccu represent the operating loss, iron loss, and copper loss of transformer, respectively; T represents the total number of periods and n is the number of distribution districts; Life and Licu, respectively, represent rated iron loss and rated copper loss of transformer; βi,t and Pi,ttr, respectively, represent the load factor and power of the transformer at time t; Cica is the capacity of the transformer. Equation (6) represents the power balance on the AC side, where Pi,ttr is the power on the AC side of the transformer at time t, and Pi,tvsc is the power flowing through the ith VSC at time t ( Pi,tvsc >0 indicates that the VSC is in the rectifying state and the power flows from the transformer to the DC bus; Pi,tvsc <0 indicates that the VSC is in the inverter state and the power flows from the DC bus to the transformer), and Pi,tcity means the residential load power at time t. βmax represents the upper limit of the transformer load rate, which is predesignated by the dispatcher. Pivsc,max is the maximum power of the ith distribution district converter. (1) represents the power balance equation on the DC side, where Pi,tess is the energy storage power at time t.Equation (1)–(9) are linear models. The existing commercial solution software can efficiently and accurately obtain the global optimal solution, which provides the possibility of real-time construction of the security operational boundary.


## 2.2. Security Operational Boundary of Flexible Distribution District

Based on the security operational boundary obtained in the previous section, the “security margin percentage” can be defined to guide the operation of dispatchers, so that dispatchers can intuitively judge the safety of the current operating status and ensure the safe operation of the flexible distribution district. The security margin percentage refers to the ratio of the power difference between the current operational point and the operational boundary point to the power at the operational boundary point. The security margin percentages of AC charging stations and DC charging stations are defined as follows:(10)Si,tac=Pi,tac∗−Pi,tacPi,tac∗×100%,(11)Si,tdc=Pi,tdc∗−Pi,tdcPi,tdc∗×100%.Among them,Si,tac and Si,tdc refer to the security margin of the AC/DC charging station and Pi,tac∗ and Pi,tdc∗, respectively, represent the operational boundary point power of the AC/DC distribution district.The above security distance can indicate the security margin of the corresponding charging stations and the ability to respond to the uncertain access of EVs under the current load state.


## 3. Uncertainty Quantification of Generalized Polynomial Chaos

### 3.1. The Model of Load Changing

The type of load considered in this paper is the active power absorbed by the EV charging station. Figure2 shows some typical time evolution of the EV power curve PA(t) measured in literature [16]. The blue line represents the average value, and the shaded area represents the range of variation.Figure 2
Three-phase load changes in EV charging stations.The distribution of EVs is randomly divided into three groups and connected to the three-phase line. In order to reproduce the influence of EV load and its statistical uncertainty, its active power is(12)Pnt=PnAt1+σnAξj,where ξj represents a zero-mean Gaussian distribution statistical variable with one-variance, that is, 〈ξj〉 = 0 and〈ξj2〉 = 1, and σAn is the degree of variability in the distribution of EVs. The distribution of EVs has a higher continuity over time and therefore consumes more power.


### 3.2. Generalized Chaos Polynomial

The uncertainty of the load power distribution can be described bym random parameters ξr to describe the active power variability, and the random parameters are represented by vectors ξ⟶=ξ1,ξ2,…,ξm. The GPC method uses generalized polynomial chaos to expand variables, which can be all node voltages in the network or a subset thereof. These voltages are usually represented as complex phasors, including magnitude and phase. In some cases, the variables that need to be expanded are limited to the number to be monitored: certain node voltages, line currents, or assumed peak or minimum values within a certain period of time, and one of these variables is represented by Vt,ξ⟶. Under a moderate assumption that Vt,ξ⟶ has finite variance, it can be approximated as an β-order truncated sequence, consisting of Na multivariate basis functions Hi(ξ⟶) weighted by the unknown polynomial chaotic coefficient ci(t).(13)Vt,ξ⟶≈∑i=1NacitHiξ⟶.The expression for each multivariate basis functionHi(ξ⟶) is as follows:(14)Hiξ⟶=∏r=1mϕi,rξr,where φir(ξr) is a univariate orthogonal polynomial of ir whose form depends on the density function of the rth parameter ξr.For a given number of parametersm and series expansion truncation order β, the order ir of the univariate polynomial forms Hi(ξ⟶), where r = 1,...,m, and their relational expression is as follows:(15)∑r=1mir≤β.The number of GPC basis functionsNa is(16)Na=β+m!β!m!.


### 3.3. Calculating GPC Coefficient

In formula (4), ST is usually used to calculate the GPC expansion coefficient [21]. By selecting test points ξB appropriately, unknown coefficients ci(t) can be calculated and Vct=Vt,ξ⟶c can be calculated in deterministic power flow analysis in random space. At each test point, the series expansion (13) is enforced to accurately fit Vc(t). Mathematically, this leads to the following linear system:(17)Mc⟶t=V⟶t,where c⟶t=c1t,…,cNatTis the column vector of unknown coefficients and V⟶t=V1t,…,VNstT is the column vector of node voltage values.Na-order square matrix M=ac,i=Hiξ⟶c, which collects the GPC basis functions evaluated at the test point(18)M=H1ξ⟶1⋯HNaξ⟶1⋮⋱⋮H1ξ⟶Ns⋯HNaξ⟶Ns.It is worth noting that the matrixM only depends on the selected basis function and test points, so it can be precalculated, inverted, and taken t = tm, as follows:(19)c⟶tm=M−1V⟶tm.


### 3.4. Complex Postprocessing Calculations

Once the coefficientcj(t) is calculated, the average value and standard deviation of the observable variables Vt,ξ⟶ can be easily determined. More information can be further derived through complex postprocessing calculations. Two examples of complex calculations related to PLF applications are described as follows.(1)
Peak voltage duration Td: first, what counts is determining the probability that the node voltage exceeds the safety limit VL within the duration Td. Consider the generic time window of duration Td=nwΔtnw≪Nt, which corresponds to the time subgrid tm+1,…,tm+nw, and the relevant samples of the observable node voltage:(20)Vtm+1,ξ⟶,…,Vtm+nw,ξ⟶,(21)αmξ⟶=minVtm+1,ξ⟶,…,Vtm+nw,ξ⟶,(22)VTdpeakξ⟶=maxαmξ⟶︸m∈0,…,Nt−nW,(23)∫VL∞fTdVdV.The minimum value of this voltage is expressed as follows:The node voltage under consideration exceeds (or equals) the value ofαm during the entire duration Td. The maximum value of αm calculated over all possible time windows of duration Td is as follows:The probability density functionfTd(V) can be calculated by the GPC + ST method and VTdpeakξ⟶ , and then integrated(2)
Voltage unbalance factor (VUF): the probability fluctuation of the node voltage will cause some imbalances between the line phases and affect the quality of service. VUF is a measure of voltage quality degradation, which is defined as the ratio of the negative voltage sequence componentVn to the positive voltage sequence component Vp:(24)VUF=VnVp⋅100,(25)Vn=VAB+a2⋅VBC+a⋅VCA3,(26)Vp=VAB+a⋅VBC+a2⋅VCA3,where VAB, VBC, and VCA are the phasors of unbalanced line voltage, a = exp(j120°), and a2 = exp(j240°). Through the GPC + ST method and postprocessing expression (13), the detailed probability density function of the maximum VUF within a given observable time window can be determined. In addition, by combining (24) with (21) and (22), the probability density function of the maximum VUF VUFWpeak can be inferred; if it lasts for a long enough time Td, the quality of service will be reduced.


## 3.1. The Model of Load Changing

The type of load considered in this paper is the active power absorbed by the EV charging station. Figure2 shows some typical time evolution of the EV power curve PA(t) measured in literature [16]. The blue line represents the average value, and the shaded area represents the range of variation.Figure 2
Three-phase load changes in EV charging stations.The distribution of EVs is randomly divided into three groups and connected to the three-phase line. In order to reproduce the influence of EV load and its statistical uncertainty, its active power is(12)Pnt=PnAt1+σnAξj,where ξj represents a zero-mean Gaussian distribution statistical variable with one-variance, that is, 〈ξj〉 = 0 and〈ξj2〉 = 1, and σAn is the degree of variability in the distribution of EVs. The distribution of EVs has a higher continuity over time and therefore consumes more power.


## 3.2. Generalized Chaos Polynomial

The uncertainty of the load power distribution can be described bym random parameters ξr to describe the active power variability, and the random parameters are represented by vectors ξ⟶=ξ1,ξ2,…,ξm. The GPC method uses generalized polynomial chaos to expand variables, which can be all node voltages in the network or a subset thereof. These voltages are usually represented as complex phasors, including magnitude and phase. In some cases, the variables that need to be expanded are limited to the number to be monitored: certain node voltages, line currents, or assumed peak or minimum values within a certain period of time, and one of these variables is represented by Vt,ξ⟶. Under a moderate assumption that Vt,ξ⟶ has finite variance, it can be approximated as an β-order truncated sequence, consisting of Na multivariate basis functions Hi(ξ⟶) weighted by the unknown polynomial chaotic coefficient ci(t).(13)Vt,ξ⟶≈∑i=1NacitHiξ⟶.The expression for each multivariate basis functionHi(ξ⟶) is as follows:(14)Hiξ⟶=∏r=1mϕi,rξr,where φir(ξr) is a univariate orthogonal polynomial of ir whose form depends on the density function of the rth parameter ξr.For a given number of parametersm and series expansion truncation order β, the order ir of the univariate polynomial forms Hi(ξ⟶), where r = 1,...,m, and their relational expression is as follows:(15)∑r=1mir≤β.The number of GPC basis functionsNa is(16)Na=β+m!β!m!.


## 3.3. Calculating GPC Coefficient

In formula (4), ST is usually used to calculate the GPC expansion coefficient [21]. By selecting test points ξB appropriately, unknown coefficients ci(t) can be calculated and Vct=Vt,ξ⟶c can be calculated in deterministic power flow analysis in random space. At each test point, the series expansion (13) is enforced to accurately fit Vc(t). Mathematically, this leads to the following linear system:(17)Mc⟶t=V⟶t,where c⟶t=c1t,…,cNatTis the column vector of unknown coefficients and V⟶t=V1t,…,VNstT is the column vector of node voltage values.Na-order square matrix M=ac,i=Hiξ⟶c, which collects the GPC basis functions evaluated at the test point(18)M=H1ξ⟶1⋯HNaξ⟶1⋮⋱⋮H1ξ⟶Ns⋯HNaξ⟶Ns.It is worth noting that the matrixM only depends on the selected basis function and test points, so it can be precalculated, inverted, and taken t = tm, as follows:(19)c⟶tm=M−1V⟶tm.


## 3.4. Complex Postprocessing Calculations

Once the coefficientcj(t) is calculated, the average value and standard deviation of the observable variables Vt,ξ⟶ can be easily determined. More information can be further derived through complex postprocessing calculations. Two examples of complex calculations related to PLF applications are described as follows.(1)
Peak voltage duration Td: first, what counts is determining the probability that the node voltage exceeds the safety limit VL within the duration Td. Consider the generic time window of duration Td=nwΔtnw≪Nt, which corresponds to the time subgrid tm+1,…,tm+nw, and the relevant samples of the observable node voltage:(20)Vtm+1,ξ⟶,…,Vtm+nw,ξ⟶,(21)αmξ⟶=minVtm+1,ξ⟶,…,Vtm+nw,ξ⟶,(22)VTdpeakξ⟶=maxαmξ⟶︸m∈0,…,Nt−nW,(23)∫VL∞fTdVdV.The minimum value of this voltage is expressed as follows:The node voltage under consideration exceeds (or equals) the value ofαm during the entire duration Td. The maximum value of αm calculated over all possible time windows of duration Td is as follows:The probability density functionfTd(V) can be calculated by the GPC + ST method and VTdpeakξ⟶ , and then integrated(2)
Voltage unbalance factor (VUF): the probability fluctuation of the node voltage will cause some imbalances between the line phases and affect the quality of service. VUF is a measure of voltage quality degradation, which is defined as the ratio of the negative voltage sequence componentVn to the positive voltage sequence component Vp:(24)VUF=VnVp⋅100,(25)Vn=VAB+a2⋅VBC+a⋅VCA3,(26)Vp=VAB+a⋅VBC+a2⋅VCA3,where VAB, VBC, and VCA are the phasors of unbalanced line voltage, a = exp(j120°), and a2 = exp(j240°). Through the GPC + ST method and postprocessing expression (13), the detailed probability density function of the maximum VUF within a given observable time window can be determined. In addition, by combining (24) with (21) and (22), the probability density function of the maximum VUF VUFWpeak can be inferred; if it lasts for a long enough time Td, the quality of service will be reduced.


## 4. Uncertainty Quantification of Generalized Polynomial Chaos

The structure of flexible distribution district adopted in this section is shown in Figure3 to verify the effect of the proposed flexible distribution district security boundary. Four AC distribution districts (T1-T4) are connected to the DC side through four VSCs. DC and AC charging stations are, respectively, connected to the DC bus and AC T1 distribution district. The DC charging station includes three 120 kW DC charging piles. The AC charging station includes three 60 kW AC charging piles. The VSC has a power upper limit of 120kVA and a working efficiency of 98.5%. The relevant parameters of the transformers in each distribution district are shown in Table 1. The capacity of the energy storage device is set as 80kWh, and the upper limit of charge and discharge power is 120 kW. Figure 3 shows the charging load curve of EV. The security boundary model is solved by CPLEX in GAMS.Figure 3
Topological structure of demonstration project in flexible distribution district.Table 1
Transformer parameter settings.
t1t2t3t4Capacity (kVA)630630800800Rated iron loss (kW)1.21.21.41.4Rated copper loss (kW)6.26.27.57.5Load rate (%)100100100100Figure4 shows the charging load of EVs at different times of the day. In this scenario, the EV users are not motivated by any incentives, and they start charging when the EV is connected to the grid when it is parked. Since the time when users are connected to the power grid is mainly concentrated around 11 : 30 and 20 : 30, the charging load at these two times increases sharply, and the load curve has a peak.Figure 4
EV charging load curve. Security boundary analysis of AC charging station.


### 4.1. Security Boundary Analysis of AC Charging Station

The intraday security boundary of an AC charging station based on the given DC charging load of EV is shown in Figure5. Combining the EV charging load curve with the obtained results combined with EV charging load curve, it can be seen that the security boundary curve of AC charging station is negatively correlated with the trend of EV charging load curve. Specifically, due to the low DC charging load in the early morning (24 : 00 to 6 : 00 the next day), the AC charging station can connect to the maximum 662 kW charging load. As the DC charging load increases, the charging load that can be accessed by AC charging stations decreases, especially at 20 : 00 during the evening peak hours (19 : 00 to 21 : 00), only 164 kW can be accessed.Figure 5
Security boundary of AC charging station.The black curve in Figure5 represents the actual AC charging load connected, and the red dotted line and solid line represent the security operational boundary when the security index is 30 kW and 50 kW, respectively. It can be seen from the figure that when the security index is 30 kW, due to the low security index, the actual access volume at 19 : 00, 20 : 00, 21 : 00, and 22 : 00 is greater than the maximum accessible volume. According to formula (3), the security margin percentage at the above time is negative, indicating that the access of a large number of charging loads under this security index is likely to cause the operating point to be outside the security boundary, resulting in overloaded operation in the distribution district. When the security index is 50 kW, it can be seen from the figure that at 20 : 00 and 21 : 00, the actual access quantity is very close to the maximum accessible quantity, and the security margin percentage is only 2.4% and 6.07%. During this period, the flexible distribution district has limited ability to cope with the random access of EV. If the charging management of EV in AC charging station is improper, the operation point is likely to be located outside the security boundary, resulting in overload operation of the distribution district.


### 4.2. Security Boundary Analysis of AC Charging Station

The intraday security boundary of DC charging station based on the given AC charging load of EV is shown in Figure6. Unlike AC charging stations, the security boundary of DC charging stations is always maintained at 680 kW from 23 : 00 to 18 : 00 of the next day and has nothing to do with intraday fluctuation of AC charging load. This is because the maximum accessible amount of DC charging station is limited by the capacity of energy storage and four VSCs, which will not be affected by the above load fluctuation in this period. Therefore, the security boundary of DC charging station remains unchanged in the corresponding period. From 19 : 00 to 22 : 00, the AC charging load reaches its peak, and it is necessary to avoid overloading of a single transformer through load transfer. Therefore, part of the capacity of the four VSCs will be used to support the function of load transfer, resulting in the maximum accessible load of the DC charging station being directly affected by the fluctuation of the above load, forming the curve as shown in Figure 6.Figure 6
Security boundary of DC charging station.The black curve in Figure6 represents the actual connected DC charging load, and the red dotted line and solid line represent the security boundary when the safety index is 30 kW and 50 kW, respectively. It can be seen from Figure 6 that when the security index is 30 kW, due to the low security index, the actual access volume at 19 : 00, 20 : 00, 21 : 00, and 22 : 00 is greater than the maximum accessible volume. According to formula (2), the security margin percentage at the above time is negative, indicating that the access of a large number of charging loads under this security index is likely to cause the operating point to be outside the security boundary, resulting in overloaded operation in the distribution district. When the security index is 50 kW, it can be seen from the figure that at 20 : 00 and 21 : 00, the actual access quantity is very close to the maximum accessible quantity, and the security margin percentage is only 4.67% and 10.29%. During this period, the flexible distribution district has limited ability to cope with the random access of EV. If the charging management of EV in DC charging station is improper, the operation point is likely to be located outside the security boundary, resulting in overload operation of the distribution district.The operating loss of transformers and converters is positively correlated with the power that flows through them. Therefore, when the safety index is low, the operating power of transformers and converters will be limited and the accessible load in the flexible distribution district will be significantly reduced.


### 4.3. Load Uncertainty Prediction of EVs after Access

In this paper, Figure7 shows the topology of the IEEE low-voltage European test feeder used in this study as a benchmark example to analyse and predict the load distribution after the EVs connecting to the distribution network. Loads are applied to the nodes using circle markers (red = phase A, black = phase B, and green = phase C).Figure 7
Topology of the base distribution network.The charging of the vehicle under consideration is carried out according to the schedulingestablished by the vehicle manager. Single-phase and slow charging can only be considered, and it is assumed that the power change is small and the uncertainty of the load is small.In the implementation process, the GPC + ST code written with Matlab is connected to the power flow deterministic solver OpenDSS.According to Formula (3), it is assumed that the active power Pev(t) of all charging stations with a given phase line is distributed through the same Gaussian statistical parameter. For all charging stations, the variable degree is fixed as equal and set as σev1 = σev2 = σev3 = 0.08, which means that the statistical change is less than 8%. Figure 8 shows the boxplot of the B-phase voltage waveform for each GPC sample, with the red-cross representing the outliers and the blue box representing the range of variation with the relevant median.Figure 8
Changes of B-phase voltage waveform drawn for GPC sample.The peak distribution comparison in the two conditions is shown in Figure9. At this stage, the current value induces a higher voltage. All lines in the test network are described by self-inductance and mutual inductance, while phase B is the uncharged phase (in this example, the load is inserted into phase A and the EV into phase C). In this case, there is no associated voltage drop in this phase, but there is an induced voltage due to the coupling between the phases, and it is strictly related to the current change caused by the EV charging.Figure 9
Comparison of daily voltage peak distribution of B-phase at observable nodes under the two conditions.Considering the relatively regular trend of power absorption of EVs, it can be seen that in this case, as shown in Figure10, the instantaneous value of voltage imbalance is low and far lower than that calculated on the window, indicating that the distribution network has higher voltage quality in this case.Figure 10
Distribution of voltage imbalanceVUFTdpeak values at observable node in the case of EV 396 charging for three different time windows.Finally, the number of simulations for the proposed variability analysis for increasing statistical parameters and a fixed expansion orderβ = 3 is reported in Table 2. For m = 3, a deterministic power flow analysis takes about 12 seconds, so a variability analysis using the GPC method takes about 4 minutes. The same analysis with the same precision with the exact Monte Carlo method takes more than 16 hours.Table 2
Simulation times.
Number of variables236Number of simulations102084Total simulation time (s)1132411455Single simulation time (s)111217


## 4.1. Security Boundary Analysis of AC Charging Station

The intraday security boundary of an AC charging station based on the given DC charging load of EV is shown in Figure5. Combining the EV charging load curve with the obtained results combined with EV charging load curve, it can be seen that the security boundary curve of AC charging station is negatively correlated with the trend of EV charging load curve. Specifically, due to the low DC charging load in the early morning (24 : 00 to 6 : 00 the next day), the AC charging station can connect to the maximum 662 kW charging load. As the DC charging load increases, the charging load that can be accessed by AC charging stations decreases, especially at 20 : 00 during the evening peak hours (19 : 00 to 21 : 00), only 164 kW can be accessed.Figure 5
Security boundary of AC charging station.The black curve in Figure5 represents the actual AC charging load connected, and the red dotted line and solid line represent the security operational boundary when the security index is 30 kW and 50 kW, respectively. It can be seen from the figure that when the security index is 30 kW, due to the low security index, the actual access volume at 19 : 00, 20 : 00, 21 : 00, and 22 : 00 is greater than the maximum accessible volume. According to formula (3), the security margin percentage at the above time is negative, indicating that the access of a large number of charging loads under this security index is likely to cause the operating point to be outside the security boundary, resulting in overloaded operation in the distribution district. When the security index is 50 kW, it can be seen from the figure that at 20 : 00 and 21 : 00, the actual access quantity is very close to the maximum accessible quantity, and the security margin percentage is only 2.4% and 6.07%. During this period, the flexible distribution district has limited ability to cope with the random access of EV. If the charging management of EV in AC charging station is improper, the operation point is likely to be located outside the security boundary, resulting in overload operation of the distribution district.


## 4.2. Security Boundary Analysis of AC Charging Station

The intraday security boundary of DC charging station based on the given AC charging load of EV is shown in Figure6. Unlike AC charging stations, the security boundary of DC charging stations is always maintained at 680 kW from 23 : 00 to 18 : 00 of the next day and has nothing to do with intraday fluctuation of AC charging load. This is because the maximum accessible amount of DC charging station is limited by the capacity of energy storage and four VSCs, which will not be affected by the above load fluctuation in this period. Therefore, the security boundary of DC charging station remains unchanged in the corresponding period. From 19 : 00 to 22 : 00, the AC charging load reaches its peak, and it is necessary to avoid overloading of a single transformer through load transfer. Therefore, part of the capacity of the four VSCs will be used to support the function of load transfer, resulting in the maximum accessible load of the DC charging station being directly affected by the fluctuation of the above load, forming the curve as shown in Figure 6.Figure 6
Security boundary of DC charging station.The black curve in Figure6 represents the actual connected DC charging load, and the red dotted line and solid line represent the security boundary when the safety index is 30 kW and 50 kW, respectively. It can be seen from Figure 6 that when the security index is 30 kW, due to the low security index, the actual access volume at 19 : 00, 20 : 00, 21 : 00, and 22 : 00 is greater than the maximum accessible volume. According to formula (2), the security margin percentage at the above time is negative, indicating that the access of a large number of charging loads under this security index is likely to cause the operating point to be outside the security boundary, resulting in overloaded operation in the distribution district. When the security index is 50 kW, it can be seen from the figure that at 20 : 00 and 21 : 00, the actual access quantity is very close to the maximum accessible quantity, and the security margin percentage is only 4.67% and 10.29%. During this period, the flexible distribution district has limited ability to cope with the random access of EV. If the charging management of EV in DC charging station is improper, the operation point is likely to be located outside the security boundary, resulting in overload operation of the distribution district.The operating loss of transformers and converters is positively correlated with the power that flows through them. Therefore, when the safety index is low, the operating power of transformers and converters will be limited and the accessible load in the flexible distribution district will be significantly reduced.


## 4.3. Load Uncertainty Prediction of EVs after Access

In this paper, Figure7 shows the topology of the IEEE low-voltage European test feeder used in this study as a benchmark example to analyse and predict the load distribution after the EVs connecting to the distribution network. Loads are applied to the nodes using circle markers (red = phase A, black = phase B, and green = phase C).Figure 7
Topology of the base distribution network.The charging of the vehicle under consideration is carried out according to the schedulingestablished by the vehicle manager. Single-phase and slow charging can only be considered, and it is assumed that the power change is small and the uncertainty of the load is small.In the implementation process, the GPC + ST code written with Matlab is connected to the power flow deterministic solver OpenDSS.According to Formula (3), it is assumed that the active power Pev(t) of all charging stations with a given phase line is distributed through the same Gaussian statistical parameter. For all charging stations, the variable degree is fixed as equal and set as σev1 = σev2 = σev3 = 0.08, which means that the statistical change is less than 8%. Figure 8 shows the boxplot of the B-phase voltage waveform for each GPC sample, with the red-cross representing the outliers and the blue box representing the range of variation with the relevant median.Figure 8
Changes of B-phase voltage waveform drawn for GPC sample.The peak distribution comparison in the two conditions is shown in Figure9. At this stage, the current value induces a higher voltage. All lines in the test network are described by self-inductance and mutual inductance, while phase B is the uncharged phase (in this example, the load is inserted into phase A and the EV into phase C). In this case, there is no associated voltage drop in this phase, but there is an induced voltage due to the coupling between the phases, and it is strictly related to the current change caused by the EV charging.Figure 9
Comparison of daily voltage peak distribution of B-phase at observable nodes under the two conditions.Considering the relatively regular trend of power absorption of EVs, it can be seen that in this case, as shown in Figure10, the instantaneous value of voltage imbalance is low and far lower than that calculated on the window, indicating that the distribution network has higher voltage quality in this case.Figure 10
Distribution of voltage imbalanceVUFTdpeak values at observable node in the case of EV 396 charging for three different time windows.Finally, the number of simulations for the proposed variability analysis for increasing statistical parameters and a fixed expansion orderβ = 3 is reported in Table 2. For m = 3, a deterministic power flow analysis takes about 12 seconds, so a variability analysis using the GPC method takes about 4 minutes. The same analysis with the same precision with the exact Monte Carlo method takes more than 16 hours.Table 2
Simulation times.
Number of variables236Number of simulations102084Total simulation time (s)1132411455Single simulation time (s)111217


## 5. Conclusion

Aiming at the limited coping ability of traditional distribution district, this paper proposes the concept of flexible distribution district, establishes the security operational boundary model of flexible distribution district, and uses “security margin percentage” as the measurement mechanism to provide dispatchers with an intuitive system operating state. In addition, after the large-scale access of electric vehicles to the distribution network, it is difficult to measure the changes of bus voltage and line current at the network nodes, this paper proposes an effective uncertainty quantification method based on the GPC + ST method, which can predict the probability distribution and variable interval of the relevant observable variables in the distribution network due to load uncertainty. The simulation results show that the flexible distribution district can realize the coordinated operation and control of multiple stations, the security operational boundary can take into account the security of the system, and the scalability is strong. In addition, the proposed uncertainty quantification method is helpful to understand the influence of load changes on the characteristics of the power grid, which can achieve faster computing speed and has better implementation effect. Dispatchers can adjust the number of EVs connected to each district to optimize the operation of the distribution network according to the changes in the characteristic quantities and the security operational boundary and prevent the high-risk operation of the district from the source.



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*Source: 9751132-2022-07-22.xml*