A framework for stochastic estimation of electric vehicle charging behavior for risk assessment of distribution networks

Abstract

Power systems are being transformed to enhance the sustainability. This paper contributes to the knowledge regarding the operational process of future power networks by developing a realistic and stochastic charging model of electric vehicles (EVs). Large-scale integration of EVs into residential distribution networks (RDNs) is an evolving issue of paramount significance for utility operators. Unbalanced voltages prevent effective and reliable operation of RDNs. Diversified EV loads require a stochastic approach to predict EVs charging demand, consequently, a probabilistic model is developed to account several realistic aspects comprising charging time, battery capacity, driving mileage, state-of-charge, traveling frequency, charging power, and time-of-use mechanism under peak and off-peak charging strategies. An attempt is made to examine risks associated with RDNs by applying a stochastic model of EVs charging pattern. The output of EV stochastic model obtained from Monte-Carlo simulations is utilized to evaluate the power quality parameters of RDNs. The equipment capability of RDNs must be evaluated to determine the potential overloads. Performance specifications of RDNs including voltage unbalance factor, voltage behavior, domestic transformer limits and feeder losses are assessed in context to EV charging scenarios with various charging power levels at different penetration levels. Moreover, the impact assessment of EVs on RDNs is found to majorly rely on the type and location of a power network.

Keywords

electric vehicles (EVs) / residential distribution networks (RDNs) / voltage unbalance factor (VUF) / state-of charge (SOC) / time-of-use (TOU)

Introduction

Present state

In recent years, rising concerns for climate change and imminent energy crisis demand for new requirements and prospects in power and transportation sectors. The environmental and energy issues should be reduced by new opportunities and technologies that can encounter higher standards set by clean sustainable energy systems. The transport sector is one of the highest contributors toward carbon emissions [¹]. Electrifying the transport sector with power networks is an evolving approach with an ultimate objective of a green transport system having an efficient fuel economy, a higher efficiency, and lesser CO₂ emissions. Deployment of electric vehicles (EVs) and advancement in EV technologies are attractive measures to achieve higher targets in green energy environment with new environmental policies [¹–³].

Introduction of distributed generation units (DGUs) into RDNs such as EVs can create potential issues for RDNs that may not be predicted when RDNs have been initially designed. According to the Global Sales Report for Plug-in EVs, sales in China in 2018 was much higher than that in the previous year and had gone beyond expectations [⁴]. Globally, the population of EVs on the road had reached three million units till 2018 [⁴] by introducing new public policies formulated by different governments. This led to the massive integration of EVs into RDNs. Currently, the EV charging technology permits Level 1 (120 Vac, 1.9 kW and 20 A) and Level 2 (240 Vac, 7.2 kW and 30A) charging at domestic locations [²,³]. Home-based EV charging is more convenient for customers. However, massive EV penetrations compel higher demands to conventional infrastructure of RDNs. Existing RDNs might not be prepared to handle such excessive electric demand due to massive integrations of EVs into RDNs. Operation and planning parameters of RDNs are significantly changed due to widespread adoption of EVs [²]. Moreover, consideration of a typical working day may result in a situation where the charging load of EVs overlaps with daily maximum demand, which leads to a heavy stress on power grid. Consequently, it is vital to explore and understand risks associated with EVs based on performance, voltage behavior, feeder losses, domestic transformer limits and unbalancing issues of RDNs. This will determine necessary upgrade requirements needed to restore reliability and stability profile of RDNs. Such studies are necessary for future planning and operations of RDNs in view of substantial issues raised from the massive integration of EVs [¹–³].

Literature review of EV studies

In earlier studies, reasonable assumptions were made to model the charging load of EV, while few significant aspects were not covered. For instance, fixed EV proportion was presumed in Ref. [⁵] whereas fixed EV power consumption was adopted in Refs. [⁶–⁸] with the assumption of certain mileage. The time zones priority charging approach was implemented in Ref. [⁹] to minimize total energy cost while maintaining limits of distribution parameters. However, stochastic EV parameters were not considered. The work presented in Ref. [¹⁰] estimated the daily EV mileage with its power consumption based on the predefined distribution curve. Reliability issues of a power system due to large-scale integration of EVs were well addressed through various significant approaches in Refs. [¹¹–¹³]. Charging scenarios with significant EV characteristics were considered with stochastic method for analyzing reliability parameters of a power network [¹¹]. The probabilistic approach was employed by utilizing the IEEE test system. However, stochastic aspects for EV charging pattern were limited to evaluate the reliability and unbalancing issues of a grid [¹¹–¹³]. In Ref. [¹⁴], the significance of EVs was emphasized by highlighting the support and valuable services provided by EVs in demand response programs. However, EV impacts on grid were not discussed. In Ref. [¹⁵], an optimal strategy was proposed to develop different management approaches for parking lots of EVs. The EV load profile was generated by using the Monte-Carlo method. Similarly, the charging load and discharging output of EVs were predicted by Monte-Carlo simulations in Ref. [¹⁶]. However, stochastic parameters were limited and gird impacts were not examined. The Roulette wheel mechanism along with the Monte-Carlo approach were used in Ref. [¹⁷] for modeling various uncertain parameters associated with supply and demand. Distributed energy resources with different penetration of renewables were optimally scheduled in view of environmental and economic aspects. However, detailed stochastic aspects were not covered for modeling EV charging demand along with unbalancing issue in low voltage (LV) networks.

Bidirectional charging and discharging scenarios were realized in Ref. [¹⁸] by adopting the power control methodology. However, the moment of charging was not defined for the development of the EV charging model [¹⁸,¹⁹]. In Ref. [²⁰], the charging load of EVs was estimated with limited adequacy to explore the impact of EVs on power networks as a substantial distributive charging load. Various EV charging approaches were recommended in Refs. [²¹–²³] to cope with harmful EV impacts on distribution networks (DNs). The study described in Ref. [²⁴] proposed a control algorithm to analyze EV impacts on low voltage RDNs. By implementing the centralized control approach, the charging points of EVs were managed with limited data to reduce technical issues of DNs. Similarly, technologies having lower carbon emissions were assessed in context to parameters of RDNs in Ref. [²⁵]. The capacity of various feeders was evaluated by integrating low carbon technologies into RDNs. The research reported in Ref. [²⁶] employed a new framework to evaluate EV impacts with massive penetrations into RDNs. The proposed framework was analyzed based on charging scenarios as well as the maximum level of EV penetration without any added upgradation in a DN.

In Ref. [²⁷], EV charging methods were assessed with and without renewable resources. It was concluded that the capacity of the weakest feeder to integrate EVs was improved by splitting the feeder. Two new feeders could accommodate more number of EVs. The work presented in Ref. [²⁸] addressed the issues of harmonic distortion in power networks resulted from the integration of EVs. The introduction of wind generators into the feeder network was recommended to lessen harmonics produced by EVs in a power grid. The work in Ref. [²⁹] developed an EV charging model by considering the maximum capability of existing DN. An economical plan was proposed to install vehicle charging stations in existing systems. The study conducted in Ref. [³⁰] suggested demand management programs and renewable based EV charging as a viable solution to reduce EV impacts on power networks. The work reported in Ref. [³¹] proposed a smart EV charging algorithm to improve the performance parameters of DN. The multi-objective algorithm based on the control charging process could reduce harmful impacts of EVs on DNs. Charging cost minimization was the main objective of optimization process. The impact assessment of uncontrolled home-based charging was performed in Ref. [³²] by developing highly resolved modeling of EV and residential power demands. It was concluded that peak load factor was substantially increased by applying Level 2 power charging.

Motivation and objectives

In view of DNs, previous technical studies in EV research area have investigated limitations regarding infrastructure of RDNs raised from massive integration of EVs in terms of stability, efficiency issues, and life of distribution components. RDNs can manage substantial EV levels by implementing restricted charging at off-peak hours. Several studies based on EVs impacts with the coordinated charging strategy have been performed with an emphasis on accommodating the maximum number of EVs within the limits of a power network. However, the model of EV is based on the deterministic approach. Besides, the stochastic model has not covered all realistic aspects. In terms of EVs impacts on RDNs, it is difficult to quantify the occurrence of per day charging events of EVs with their associated loads by implementing a deterministic method because multiple realistic aspects cannot be covered to develop a comprehensive model for EV mobility pattern. The charging load of EVs should be modeled as a random variable. The activity schedule of an individual with the driving pattern on daily basis is utilized to quantify the charging event of an EV and its related load consumption. Therefore, the stochastic modeling of EV covering all realistic factors for the actual estimation of an expected load on power grid is needed for upgradation of utility infrastructure, which can provide support to RDNs for massive EV penetrations in the near future.

Considering the foregoing discussion, various significant shortcomings are observed which can extremely affect the findings or results for effective impact analysis of EVs on RDNs: unrealistic assumptions are followed to develop the EV model without taking proper probability distributions; empty battery is assumed at home arrival time. However different technical aspects should be considered to determine EV initial SOC; battery SOC has uniform/random distribution; the same battery capacity is considered for all EVs; battery degradation is not considered (fully discharged battery at arrival); EVs are charged only at off-peak hours; circuits with medium voltage (>1kV) are considered to reduce EV impacts; some of the earlier analyses are performed based on vehicle and customer viewpoint, while gird prospective is not analyzed; only the price of EVs is considered in many previous studies when examining EV impacts on DNs; EVs uncertainties are not followed for technical impact assessment; balanced networks are implemented in various studies to evaluate EV charging scenarios. However, in reality, EV charging is majorly performed on unbalanced LV networks; in addition, the unbalance issue is often neglected, and network limitations are expressed as a whole, instead phase-by phase limitations should be considered. Consequently, there exists a knowledge gap in earlier studies based on the abovementioned shortcomings related to impact analysis of EVs into RDNs.

In this context, prospect of EV technology demands realistic scenarios to be considered for profound evaluation of EV impacts on existing infrastructure of RDNs to make necessary upgradation in the near future. Meticulous estimation of residential demand with stochastic charging demand of EVs is essential for utility operators for necessary risk evaluation of RDNs. It is necessary to develop a comprehensive EV model based on probabilistic scenarios with active demand of various domestic houses for effectual impact examination. Subsequently, the approach implemented this paper is undertaken with the target to overcome these shortcomings through the proposed framework by developing an EV model based on realistic aspects to evaluate significant performance parameters of RDNs.

Contributions

This paper aims to conduct risk assessment of RDNs by analyzing noteworthy power quality parameters including voltage behavior with respect to customers with voltage issues, domestic transformer limits, feeder losses, and voltage unbalance factor (VUF) due to random phase-wise house distribution with EVs at various feeders of the implemented benchmark system. Considering the abovementioned shortcomings in recent studies, the major contribution in terms of novelty is to develop a more comprehensive integrated probabilistic model incorporating various realistic factors with current EV market share for estimating EV charging demand, and detailed investigation of performance parameters of RDNs under several practical scenarios, which can provide support to utilities for planning stability and reliability attributes of future distribution networks. Besides, uncertainties related to EV charging demand with location as well as residential demand with location of domestic houses are well investigated. In addition, a realistic EV charging model is developed using the National Household Travel Survey (NHTS) data to account required probability distributions. Moreover, the mobility pattern is investigated with real time scenario. Furthermore, Monte-Carlo simulations are performed for probabilistic calculations.

The proposed framework comprises the salient contributions as follows:

To the best of the authors’ knowledge, there exists a knowledge gap in previous studies and a comprehensive EV model should be developed to estimate the charging demand of EVs with realistic scenarios to explore the possible improvements in various fundamental aspects of a power grid.

A stochastic model of EV charging pattern is developed to account for several realistic aspects comprising charging time, battery capacity, driving mileage, state of charge, traveling frequency, charging power, and time-of-use charging price in peak and off-peak charging strategies. Implemented model will cover all uncertainties associated with EV demand to provide a realistic charging model.

Profound examination of EV charging strategies (controlled and uncontrolled) are presented for analysis of performance parameters. The issues related to EV charging demand with location as well as residential power demand with location of domestic houses are well investigated.

The output of EV stochastic model obtained from Monte-Carlo simulations with power demand of various domestic housesis utilized to evaluate power quality parameters of RDNs including voltage unbalance factor, feeder losses, domestic transformers limits, and voltage behavior, which are assessed in context to EV charging scenarios with various charging power levels at different penetration levels.

Framework for evaluation

The proposed framework in this paper is illustrated in Fig. 1, which shows the complete schematic in evaluating the performance parameters of RDNs. Three main stages are included in the complete process of assessment. Stage-1 presents the steps involved in constructing the stochastic charging behavior of EVs. First, a detailed and realistic traveling data set is needed at this stage to account for various parameters including the times for departure and arrival, traveling frequency, mileage driven, driving duration, and type of EVs. These parameters are utilized to simulate the actual behavior of drivers. Afterwards, Monte-Carlo simulations are carried out to develop the stochastic charging profiles of EVs based on charging power levels, penetration level, charging strategies, and type of vehicles. Stage-2 presents the process of determining the average daily peak demand profiles of various domestic houses in summer and winter.

The outcomes obtained from Stage-1 and Stage-2 are utilized in Stage-3 for the assessment of performance parameters of RDNs. Stage-1 and Stage-2 are described in detail in Sections 3 and 4, while the assessment of performance parameters is presented in case study simulations.

Stochastic EV model

Realistic factors are essential and need to be considered for stochastic EV charging model, which include charging time, battery capacity, driving mileage, state-of-charge, traveling frequency, charging power, EV penetrations and time-of-use mechanism in peak and off-peak charging strategies. This paper considers these realistic aspects to determine the stochastic charge demand of EVs which are charged at residential premises only for impact evaluation.

The charging power levels used for EV charging in accordance with international standards are well described in Refs. [²,³]. The total charging time is directly influenced and determined by the selected EV charging level. In this paper, the residential charging demand of EVs is modeled by considering the SAE J1772 standard which defines charging power levels that have certain ranges according to the voltage and current levels as described in Table 1. However, in this paper, charging Level 1 with 120 Vac, 20 A and 1.9 kW (voltage, current and power ratings) and charging power Level 2 with 240 Vac, 30 A and 7.2 kW (voltage, current and power ratings) are used for determining the charging demands of EVs in residential areas [²,³].

EV transport market is growing rapidly due to widespread adoption, therefore, many car manufacturing companies are entering in the competition by introducing their EV models with different characteristics. To be more realistic, this paper considers five different EV models to perform EV impact analysis on RDNs as described in Table 2 [³,³⁰]. For analysis purpose, EV percentage is taken into account according to EV sales in the transportation market as mentioned in Table 2.

Two EVs per household are assumed for 1-kanal house and one EV per household is assumed for other three types of houses in modeling charging demand. At present, lithium-ion batteries are widely adopted as a power source in EVs due to their mature technology and key features including long life-time and high power as well as high energy density. Therefore, all EVs described in Table 2 have the lithium-ion battery technology.

Traveling behavior is necessary to analyze mobility pattern and to estimate charging demand of EVs. Normally, the EV parking time at home reflects the start time of EV charging. Therefore, arrival and departure times are crucial for analyzing charging impacts on RDNs. In this paper, a realistic EV charging model is developed using the National Household Travel Survey (NHTS) data [³³] to account for required probability density functions (PDFs) and to develop the stochastic charging pattern. The information related to starting time of the first trip, ending time of the last trip, driving mileage, traveling duration, and traveling frequency is obtained from NHTS data through statistical analysis. The significant parameters that are valuable for developing a real EV charging pattern are demonstrated in Fig. 2. Stochastic distributions are carefully chosen using the best-fit analysis in MATLAB, the distribution that best fits the data are selected for the analysis, and the start time of EV charging (Arrival time) is modeled as a normal distribution function, formulated as

(1)

where {\mu }_{\mathrm{A}\mathrm{r}\mathrm{r}} = 15.5 and {\sigma }_{\mathrm{A}\mathrm{r}\mathrm{r}} = 3.1. The values of standard deviation and mean are presented by \sigmaand µ respectively. The arrival time distribution defines the ending time of the last trip, which is described by PDF in Eq. (1).

The fitting results for the departure time are modeled as a normal distribution, formulated as

(2)

where {\mu }_{\mathrm{D}\mathrm{e}\mathrm{p}} = 7.6 and {\sigma }_{\mathrm{D}\mathrm{e}\mathrm{p}} = 2.2. The departure time distribution defines the starting time of the first trip, which is described by PDF in Eq. (2).

The per day energy consumption of an EV is linked with the driving mileage per trip, which can be changed on daily basis. The fitting results of EV driving mileage per trip are modeled as logarithmic distribution, formulated as

(3)f_{\mathrm{D}\mathrm{m}}(t\mathrm{)=}{\displaystyle \frac{1}{t{\sigma }_{\mathrm{D}\mathrm{m}}\sqrt{\mathrm{2\pi }}}}\exp \left[-{\displaystyle \frac{{(\ln t-{\mu }_{\mathrm{D}\mathrm{m}})}^2}{2{\sigma }_{\mathrm{D}\mathrm{m}}^2}}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}

where {\mu }_{\mathrm{D}\mathrm{m}} = 2.99 and {\sigma }_{\mathrm{D}\mathrm{m}} = 0.77. Similarly, the fitting results of EV daily travel frequency and driving duration per trip are modeled as logarithmic distributions and presented in Figs. 2(c) and 2(d)).

A higher EV efficiency is required in the future. Therefore, in this paper, EVs are expected to consume 0.15 kWh/km. Thus, the required energy in a day can be acquired by using driving mileage [²⁷], stated as

(4)E=\mathrm{0.15}{D}_{\mathrm{m}},

where the EV driven distance per day is presented by D_m. The required charging duration for each EV is estimated by

(5)T_{\mathrm{C}\mathrm{h}}={\displaystyle \frac{D_{\mathrm{m}}{E}_{\mathrm{100}}}{\mathrm{100}{P}_{\mathrm{C}\mathrm{h}}{\eta }_{\mathrm{C}\mathrm{h}}}},

where D_m is the daily traveling distance, E₁₀₀ is the energy consumption required for traveling 100 km, P_Ch is the charging power consumed by an EV at a particular power level, and η_Ch is the average charging efficiency. The PDF of required charging duration can be formulated as

(6)f_{T_{\mathrm{C}\mathrm{h}}}(t)={\displaystyle \frac{\mathrm{1}}{t{\sigma }_{\mathrm{D}\mathrm{m}}\sqrt{\mathrm{2}\pi }}}\exp \left[-{\displaystyle \frac{{(\ln t+\ln P_{\mathrm{C}\mathrm{h}}-\ln E_{\mathrm{100}}-{\mu }_{\mathrm{D}\mathrm{m}})}^{\mathrm{2}}}{\mathrm{2}{\sigma }_{\mathrm{D}\mathrm{m}}^{\mathrm{2}}}}\right]\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}

The next important parameter to estimate realistic EV charging pattern is the battery initial state-of-charge (SOC) which is obtained by PDF of daily driving mileage as expressed in Eq. (3). The initial battery SOC depends on the driving mileage per day and it is considered that EV SOC descents linearly with the traveling mileage, expressed as

(7)

where D_m is the daily driving mileage obtained from Eq. (21), and maximum EV range is presented by D_R. The PDF of initial SOC of an EV battery after one-day mileage can be obtained by using Eqs. (3) and (7) based on the transformation theorem of the random variable, formulated as

(8)

Considering the life-cycle of battery, it is suggested that battery SOC is kept in a range from 25% to 95%. For illustration, an initial battery SOC is plotted in Fig. 3 by using Eq. (8) and the parameters in Ref. [³⁴].

The 24 h of a day can be divided into different time periods based on a simple time-of-use (TOU) mechanism. This paper considers the valley (23:00–08:00) and peak times (16:00–23:00) to define controlled and uncontrolled EV charging strategies. The complete model flowchart to develop the stochastic EV charging pattern based on realistic aspects is presented in Fig. 4 where the parameters based on probability density functions are categorized as stochastic ones while other parameters including charging level, EV penetrations etc. are classified as deterministic ones. It is important to mention that the stochastic EV charging model developed in this paper is based on currently available popular EVs in today’s market. For consideration of a reasonable charging profile of an EV, three different levels of penetration (10%, 25%, and 50%) are realized to explore risks of RDNs. After determining required realistic parameters, Monte-Carlo simulations are performed for probabilistic estimation of the EV charging pattern. The stochastic EV charging pattern is well depicted in Fig. 5 for 10% EV penetration with controlled and uncontrolled charging strategies as well as Level 1 and Level 2 charging power levels.

Residential demand

In this paper, the data of historical load profiles, which are used for determining the average residential demand of various domestic houses are obtained from a grid station in Pakistan [³⁵]. Pakistan is a developing country, where the power demand in main cities is experiencing a sharp increase. Therefore, it is essential to fulfill the customer needs as well as appropriate planning for future load integration. In recent years, main cities like Lahore and Islamabad are experiencing a considerable socio-economic development. With this continuous extensive growth, large-scale deployment of EVs will make them a substantial residential load. Therefore, future load management is necessary to avoid significant impacts on parameters of RDNs. The area of a residential community in Pakistan consists of a combination of four different categories of houses called 5-Marla House (125 square-meters), 7-Marla House (175 square-meters), 10-Marla House (250 square-meters), and 1-Kanal House (500 square-meters) houses. “Kanal” and “Marla” are conventional units of land area used mainly in Pakistan and India. Residential houses are categorized according to their power consumption and sizes in a square meter.

The ten-year real-time data (2007–2017) considering an hourly load profiles are obtained from the National Transmission and Dispatch Company (NTDC) of Pakistan [³⁵]. The substation only supplies to a residential load consisting of various domestic houses. From the data, it is discovered that time series variations in load have a considerable dependency on seasonal and weather parameters. Considering the 24-h load pattern, it is also inferred from the data pattern that the summer morning peak appears from 07:00 to 10:00 whereas the evening peak occurs from 15:00 to 20:00. Similarly, in winter, the morning peak period occurs from 07:00 to 09:00 whereas the evening peak time is from 17:00 to 21:00. The load profile for weekends follows a different pattern in comparison with that of the weekdays. Monthly pattern examined from the data set for summer and winter load profiles show that consumer peak demand occurs in June for summer and in January for winter respectively. After complete analysis of the given data, a day with average peak demand in the month of June and January is selected for summer and winter as well. Figure 6 presents the residential peak demand profiles of various domestic houses in summer and winter, which are calculated according to the average peak demand profile in weekdays of winter and summer. Similar demand profiles can also be determined for different days of the year. However, as the summer peak of residential houses is higher in comparison with the winter peak, only summer demand is given to residential loads of DNs for the analysis.

Implemented system with voltage behavior and unbalance factor

Modeling of benchmark system

Structural modeling of unbalance residential distribution network (URDN) involves several distribution feeders, different types of domestic transformers, and various categories of houses with a variety of home appliances available in a residential community. In this paper, first, modeling of a distribution network based on IEEE benchmark is completed on a power system analysis tool [³⁶]. Afterwards, performance specifications of RDNs including transformer limits, losses, voltage unbalance factor and voltage behavior are assessed in context to EV charging scenarios with various charging power levels at different penetration levels. Issues related to EV charging demand with a location as well as residential power demand with a location of domestic houses are well investigated. Voltage behavior and VUF are analyzed due to random phase-wise house distribution as well as EV distribution at various feeders of URDN. Random phase-wise distribution is identified as a common problem in residential networks, which cause voltage behavior and VUF to surpass its permitted limit according to international standards. Quasi-dynamic (time-series) simulation is performed at user-defined time intervals, which is necessary for real-time analysis of a power network.

The benchmark network used for investigating the risks of DNs is the expanded version of IEEE 13-node test feeder [³⁷]. A simplified version of 13-node test feeder implemented in DigSilent Power Factory [³⁶] is shown in Fig. 7(a). The benchmark system contains 13-nodes/feeders, cables, and lines with different arrangement of phases, transformers, voltage regulator, shunt capacitor banks, and 9-unbalanced loads as well. Distributed and spot loads are included in the benchmark system. Different characteristics are observed by load including constant active power, constant impedance, and constant current load. Load connections are either delta or wye depending upon the requirement. The phasing for each feeder is different as mentioned in Fig. 7(a).

The specific information of a low voltage URDN is obtained from NTDC. The various feeders of the IEEE 13-node test feeder are scaled down to 0.4 kV by connecting different categories of domestic transformers. Some modifications in the benchmark system are implemented to comply with initial design stage requirements provided by NTDC i.e., 65% loading of various feeders of URDN. The output of EV stochastic charging behavior and peak residential demand obtained from Monte-Carlo simulations and historical load profiles respectively are utilized to evaluate power quality parameters of RDNs. The average demand of different residential households in the peak day of summer with estimated EV charging pattern is assigned as load duration curves (kW) to several types of domestic transformers available in a residential community.

For all distribution customers, only active power is considered for the modeling purpose with a power factor of 0.9. Each feeder is allotted with 113 houses of various kinds and loaded with a house demand of 529 kW as mentioned in Fig. 7(b). The maximum total demand of feeders is 5819 kW with a total of 1243 houses including 5-Marla, 7-Marla, 10-Marla, and 1-Kanal houses. Due to the unbalanced nature of the designed grid model, the house distribution on each phase (A, B, C) of a feeder is different. However, the total number of houses on each feeder is the same. The allocation of different categories of houses is performed as: 389 houses on Phase-A with 358, and 496 houses on Phases-B and-C respectively. This phase-wise distribution is identified as a utility problem due to the previous random ineffective planning of urban houses. This planning issue is a common problem for utilities in developing cities of Pakistan. The random distribution of EVs is performed as: Phase-A has 33% charging load of EV, while, Phases-B and-C bear 54% and 13% charging load respectively. The random phase-wise house distribution with random EV distribution cause voltage problems to customers and VUF to surpass its permitted limit according to IEC standard. Therefore, an analysis of voltage behavior and VUF is necessary for future load management and effective operation of RDNs. According to NTDC standards, detailed feeder-wise house distribution with several types of domestic transformers are given in Electronic Supplementary Material.

Load flow methodology

In this paper, the load flow methodology is based on the four-conductor current injection method (FCIM) to perform unbalanced load flows as discussed in Ref. [³⁸]. In unbalanced DNs, a reliable power flow solution is essential to find an appropriate solution for the three-phase power flow problem due to large-scale integration of storage devices, EVs, and renewable energy resources (RESs). The Newton-Raphson (NR) based algorithm provides a comprehensive solution to the three-phase power flow problem for both balanced and un-balanced DNs with radial, meshed topologies, DGUs, EVs, and RESs. The rectangular coordinates representation is used for formulating the power flow problem. The modeling of radial and meshed DNs can be achieved by the proposed formulation in Ref. [³⁸] with an explicit illustration of a neutral conductor, which is a fundamental part of unbalanced DNs. The method needs a certain level of information of a specific network such as network arrangement, nominal voltages, load types, and line impedances/admittances. During each iteration of the Newton-Raphson based the three-phase unbalanced load flow, the current mismatches are calculated by the known current’s injections through generators, loads, etc. at each bus of the power network. The elements involved in the nodal admittance matrix of a power network are used to obtain the Jacobian matrix. The inputs from the generators and different loads are iteratively updated by the Jacobian while voltage mismatches for each bus of a network are then computed by combining the Jacobian and the calculated current mismatches. A mismatch equation in the simplified form is given by Eq. (9), where the Jacobian matrix is presented as J, the voltage mismatch matrix is shown as ΔV; and ΔI is the current mismatch matrix. Equation (9) is linear. However, the Jacobians are not constant, and the iterative process is used to solve nonlinear equations.

(9)\Delta I=J\times \Delta V.

The current injection mismatches in real and imaginary terms at bus k phase d and bus k phase neutral are presented in Eqs. (10)-(13) [³⁸]. The contributions of current injection mismatches from lines and transformers (series elements of a power network) and generators and loads (shunt elements of a power network) are key components of these equations.

(10)\mathrm{\Delta }{I}_{{\mathrm{Re}}_k}^d={\displaystyle \frac{P_k^d(V_{{\mathrm{Re}}_k}^d-V_{{\mathrm{Re}}_k}^n)+Q_k^d(V_{{\mathrm{Im}}_k}^d-V_{{\mathrm{Im}}_k}^n)}{{(V_{{\mathrm{Re}}_k}^d-V_{{\mathrm{Re}}_k}^n)}^{\mathrm{2}}+{(V_{\mathrm{Im}}^d-V_{{\mathrm{Im}}_k}^n)}^{\mathrm{2}}}}-{\displaystyle \underset{t\in {\sigma }_p}{\mathrm{\Sigma }}(G_{kk}^{dt}{V}_{{\mathrm{Re}}_k}^t-B_{kk}^{dt}{V}_{{\mathrm{Im}}_k}^t)}-{\displaystyle \underset{\begin{array}{l}i\in {\Omega }_k\\ i\ne k\end{array}}{\mathrm{\Sigma }}{\displaystyle \underset{t\in {\sigma }_p}{\mathrm{\Sigma }}(G_{kk}^{dt}{V}_{{\mathrm{Re}}_i}^t-B_{ki}^{dt}{V}_{{\mathrm{Im}}_i}^t)}},\hspace{0.17em}\hspace{0.17em}

(11)\mathrm{\Delta }{I}_{{\mathrm{Im}}_k}^d={\displaystyle \frac{P_k^d(V_{{\mathrm{Im}}_k}^d-V_{{\mathrm{Im}}_k}^n)-Q_k^d(V_{{\mathrm{Re}}_k}^d-V_{{\mathrm{Re}}_k}^n)}{{(V_{{\mathrm{Re}}_k}^d-V_{{\mathrm{Re}}_k}^n)}^{\mathrm{2}}+{(V_{\mathrm{Im}}^d-V_{{\mathrm{Im}}_k}^n)}^{\mathrm{2}}}}\hspace{0.17em}-{\displaystyle \underset{t\in {\sigma }_p}{\mathrm{\Sigma }}(B_{kk}^{dt}{V}_{{\mathrm{Re}}_k}^t+G_{kk}^{dt}{V}_{{\mathrm{Im}}_k}^t)}\hspace{0.17em}-{\displaystyle \underset{\begin{array}{l}i\in {\Omega }_k\\ i\ne k\end{array}}{\mathrm{\Sigma }}{\displaystyle \underset{t\in {\sigma }_p}{\mathrm{\Sigma }}(B_{ki}^{dt}{V}_{{\mathrm{Re}}_i}^t+G_{ki}^{dt}{V}_{{\mathrm{Im}}_i}^t)}},\hspace{0.17em}

(12)\mathrm{\Delta }{I}_{{\mathrm{Re}}_k}^n=-\hspace{0.17em}(\hspace{0.17em}{I}_{{\mathrm{Re}}_{k,\hspace{0.17em}l\hspace{0.17em}g}}^a\hspace{0.17em}+\hspace{0.17em}{I}_{{\mathrm{Re}}_{k,\hspace{0.17em}l\hspace{0.17em}g}}^b\hspace{0.17em}+\hspace{0.17em}{I}_{{\mathrm{Re}}_{k,\hspace{0.17em}l\hspace{0.17em}g}}^c)-{\displaystyle \underset{t\in {\sigma }_p}{\mathrm{\Sigma }}(G_{kk}^{nt}{V}_{{\mathrm{Re}}_k}^t-B_{kk}^{nt}{V}_{{\mathrm{Im}}_k}^t)}-{\displaystyle \underset{\begin{array}{l}i\in {\Omega }_k\\ i\ne k\end{array}}{\mathrm{\Sigma }}{\displaystyle \underset{t\in {\sigma }_p}{\mathrm{\Sigma }}(G_{ki}^{nt}{V}_{{\mathrm{Re}}_i}^t-B_{ki}^{nt}{V}_{{\mathrm{Im}}_i}^t)}},\hspace{0.17em}

(13)\mathrm{\Delta }{I}_{{\mathrm{Im}}_k}^n=-\hspace{0.17em}(\hspace{0.17em}{I}_{{\mathrm{Im}}_{k,\hspace{0.17em}l\hspace{0.17em}g}}^a\hspace{0.17em}+\hspace{0.17em}{I}_{{\mathrm{Im}}_{k,\hspace{0.17em}l\hspace{0.17em}g}}^b\hspace{0.17em}+\hspace{0.17em}{I}_{{\mathrm{Im}}_{k,\hspace{0.17em}l\hspace{0.17em}g}}^c)-{\displaystyle \underset{t\in {\sigma }_p}{\mathrm{\Sigma }}(B_{kk}^{nt}{V}_{{\mathrm{Re}}_k}^t+G_{kk}^{nt}{V}_{{\mathrm{Im}}_k}^t)}-{\displaystyle \underset{\begin{array}{l}i\in {\Omega }_k\\ i\ne k\end{array}}{\mathrm{\Sigma }}{\displaystyle \underset{t\in {\sigma }_p}{\mathrm{\Sigma }}(B_{ki}^{nt}{V}_{{\mathrm{Re}}_i}^t+G_{ki}^{nt}{V}_{{\mathrm{Im}}_i}^t)}},\hspace{0.17em}\hspace{0.17em}

where d∈σ_d, t∈σ_p; σ_d is a set containing different phases of a DN {a, b, c}; σ_p is a set containing DN phases with neutral{a,b,c,n};{\Omega }_k is set containing buses which are directly connected to bus k; I_k^d\hspace{0.17em}=\hspace{0.17em}{I}_{R{e}_k}^d\hspace{0.17em}+\hspace{0.17em}j{I}_{I{m}_k}^dis the current injection of phase d at bus k; the active and reactive powers are presented by P_k^d\hspace{0.17em},\hspace{0.17em}{Q}_k^d at bus k phase d;V_k^d\hspace{0.17em}=\hspace{0.17em}{V}_{R{e}_k}^d\hspace{0.17em}+\hspace{0.17em}j{V}_{I{m}_k}^d is the voltage for phase d ground at bus k; and the conductance and susceptance are presented by G_{kk}^{dt},\hspace{0.17em}{G}_{kk}^{dt},\hspace{0.17em}{B}_{kk}^{dt},\hspace{0.17em}{B}_{ki}^{dt} and obtained from the nodal admittance matrix.

Sensitivities in distribution network

The sensitivities in terms of voltage deviations and loading profile of various parameters of a DN can be computed by performing a time series, three-phase unbalanced power flow due to the household loading and charging demands of EVs. DigSilent Power Factory [³⁶] is used for the evaluation of the network by giving time-based loading information of residential peak demand and stochastic charging demand EVs as mentioned in Figs. 5 and 6 respectively. The voltage sensitivities for the whole DN are calculated after integration of the EV load at various nodes with different charging levels and penetration levels. The voltage deviations at each bus are recorded after the incremental load addition of EVs with each time step. The change in voltage with this time variations is then used to compute the voltage sensitivity of the DN. The variations in voltage can be observed at every other bus of the DN by adding EV load to any bus of the network. The thermal loading of the domestic transformer is observed after the incremental load addition of EVs with each time variations. A sensitivity matrix of a DN is computed after the inversion of the Jacobin, which is related to the variations in real and imaginary terms of the current at each bus to the variations in real and imaginary terms of the voltage at all buses comprising itself. Basically, the matrix computed from the inverted Jacobian is a sensitivity matrix of a DN and offers the capability for the prediction of voltage deviations with respect to specific current changes in the network. The detailed model of the inverted Jacobian matrix is presented in Eqs. (14) and (15), and one block of the Jacobian matrix is further elaborated, respectively.

(14)J^{-\mathrm{1}}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{llll}\begin{array}{l}{\alpha }_{\mathrm{11}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\beta }_{\mathrm{11}}^{st}\\ {\mu }_{\mathrm{11}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\gamma }_{\mathrm{11}}^{st}\end{array}& \begin{array}{l}{\alpha }_{\mathrm{12}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\beta }_{\mathrm{12}}^{st}\\ {\mu }_{\mathrm{12}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\gamma }_{\mathrm{12}}^{st}\end{array}& \begin{array}{l}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\\ .\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}& \begin{array}{l}{\alpha }_{\mathrm{1}nb}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\beta }_{\mathrm{1}nb}^{st}\\ {\mu }_{\mathrm{1}nb}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\gamma }_{\mathrm{1}nb}^{st}\end{array}\\ \begin{array}{l}{\alpha }_{\mathrm{21}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\beta }_{\mathrm{21}}^{st}\\ {\mu }_{\mathrm{21}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\gamma }_{\mathrm{21}}^{st}\end{array}& \begin{array}{l}{\alpha }_{\mathrm{22}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\beta }_{\mathrm{22}}^{st}\\ {\mu }_{\mathrm{22}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\gamma }_{\mathrm{22}}^{st}\end{array}& \begin{array}{l}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\\ .\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}& \begin{array}{l}{\alpha }_{\mathrm{2}nb}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\beta }_{\mathrm{2}nb}^{st}\\ {\mu }_{\mathrm{2}nb}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\gamma }_{\mathrm{2}nb}^{st}\end{array}\\ \begin{array}{l}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\\ .\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\\ .\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}& \begin{array}{l}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\\ .\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\\ .\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}& \begin{array}{l}.\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}& \begin{array}{l}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\\ .\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\\ .\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}\\ \begin{array}{l}{\alpha }_{nb\mathrm{1}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\beta }_{nb\mathrm{1}}^{st}\\ {\mu }_{nb\mathrm{1}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\gamma }_{nb\mathrm{1}}^{st}\end{array}& \begin{array}{l}{\alpha }_{nb\mathrm{2}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\beta }_{nb\mathrm{2}}^{st}\\ {\mu }_{nb\mathrm{2}}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\gamma }_{nb\mathrm{2}}^{st}\end{array}& \begin{array}{l}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\\ .\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}& \begin{array}{l}{\alpha }_{nbnb}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\beta }_{nbnb}^{st}\\ {\mu }_{nbnb}^{st}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\gamma }_{nbnb}^{st}\end{array}\end{array}\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em},

(15){(J^{-\mathrm{1}})}_{\mathrm{11}}=\left[\begin{array}{llllllll}{\alpha }_{\mathrm{11}}^{aa}& {\alpha }_{\mathrm{11}}^{ab}& {\alpha }_{\mathrm{11}}^{ac}& {\alpha }_{\mathrm{11}}^{an}& {\beta }_{\mathrm{11}}^{aa}& {\beta }_{\mathrm{11}}^{ab}& {\beta }_{\mathrm{11}}^{ac}& {\beta }_{\mathrm{11}}^{an}\\ {\alpha }_{\mathrm{11}}^{ba}& {\alpha }_{\mathrm{11}}^{bb}& {\alpha }_{\mathrm{11}}^{bc}& {\alpha }_{\mathrm{11}}^{bn}& {\beta }_{\mathrm{11}}^{ba}& {\beta }_{\mathrm{11}}^{bb}& {\beta }_{\mathrm{11}}^{bc}& {\beta }_{\mathrm{11}}^{bc}\\ {\alpha }_{\mathrm{11}}^{ca}& {\alpha }_{\mathrm{11}}^{cb}& {\alpha }_{\mathrm{11}}^{cc}& {\alpha }_{\mathrm{11}}^{cn}& {\beta }_{\mathrm{11}}^{ca}& {\beta }_{\mathrm{11}}^{cb}& {\beta }_{\mathrm{11}}^{cc}& {\beta }_{\mathrm{11}}^{cn}\\ {\alpha }_{\mathrm{11}}^{na}& {\alpha }_{\mathrm{11}}^{nb}& {\alpha }_{\mathrm{11}}^{nc}& {\alpha }_{\mathrm{11}}^{nn}& {\beta }_{\mathrm{11}}^{na}& {\beta }_{\mathrm{11}}^{nb}& {\beta }_{\mathrm{11}}^{nb}& {\beta }_{\mathrm{11}}^{nn}\\ {\mu }_{\mathrm{11}}^{aa}& {\mu }_{\mathrm{11}}^{ab}& {\mu }_{\mathrm{11}}^{ac}& {\mu }_{\mathrm{11}}^{an}& {\gamma }_{\mathrm{11}}^{aa}& {\gamma }_{\mathrm{11}}^{ab}& {\gamma }_{\mathrm{11}}^{ac}& {\gamma }_{\mathrm{11}}^{an}\\ {\mu }_{\mathrm{11}}^{ba}& {\mu }_{\mathrm{11}}^{bb}& {\mu }_{\mathrm{11}}^{bc}& {\mu }_{\mathrm{11}}^{bn}& {\gamma }_{\mathrm{11}}^{ba}& {\gamma }_{\mathrm{11}}^{bb}& {\gamma }_{\mathrm{11}}^{bc}& {\gamma }_{\mathrm{11}}^{bn}\\ {\mu }_{\mathrm{11}}^{ca}& {\mu }_{\mathrm{11}}^{cb}& {\mu }_{\mathrm{11}}^{cc}& {\mu }_{\mathrm{11}}^{cn}& {\gamma }_{\mathrm{11}}^{ca}& {\gamma }_{\mathrm{11}}^{cb}& {\gamma }_{\mathrm{11}}^{cc}& {\gamma }_{\mathrm{11}}^{cn}\\ {\mu }_{\mathrm{11}}^{na}& {\mu }_{\mathrm{11}}^{nb}& {\mu }_{\mathrm{11}}^{nc}& {\mu }_{\mathrm{11}}^{nn}& {\gamma }_{\mathrm{11}}^{na}& {\gamma }_{\mathrm{11}}^{nb}& {\gamma }_{\mathrm{11}}^{nc}& {\gamma }_{\mathrm{11}}^{nn}\end{array}\right],

where s,t∈σ_p; σ_p is a set containing phases {a,b,c,n}; {\alpha }_{ki}^{st}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{\delta V_{{\mathrm{Re}}_k}^s}{\delta I_{{\mathrm{Im}}_i}^t}} is the sensitivity in real voltage term at bus k phase s to sensitivity in imaginary current term at bus i phase t; {\beta }_{ki}^{st}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{\delta V_{{\mathrm{Re}}_k}^s}{\delta I_{{\mathrm{Re}}_i}^t}} is the sensitivity in real voltage term at bus k phase s to sensitivity in real current term at bus i phase t; {\mu }_{ki}^{st}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{\delta V_{{\mathrm{Im}}_k}^s}{\delta I_{{\mathrm{Im}}_i}^t}} is the sensitivity in imaginary voltage term at bus k phase s to sensitivity in imaginary current term at bus i phase t; and {\gamma }_{ki}^{st}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{\delta V_{I{m}_k}^s}{\delta I_{{\mathrm{Re}}_i}^t}} is the sensitivity in imaginary voltage term at bus k phase s to sensitivity in real current term at bus i phase t.

Limits/Operating conditions of the network

The details referring to network conditions, which are taken into consideration are given below:

1) Power flow equations

(16)\{\begin{array}{l}{P}_{jk,t}-V_{j,t}{\displaystyle \underset{k=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{V}_{k,t}(G_{jk}\cos {\theta }_{jk,t}+B_{jk}\sin {\theta }_{jk,t})}=\mathrm{0},\\ Q_{jk,t}-V_{j,t}{\displaystyle \underset{k=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{V}_{k,t}(G_{jk}\sin {\theta }_{jk,t}-B_{jk}\cos {\theta }_{jk,t})}=\mathrm{0},\end{array}

where P_jk,t and Q_jk,t represent the intervals of active and reactive power, respectively with reference to branch jk at time t; V_j,t and V_k,t represent the intervals of voltage magnitude, respectively with reference to buses j and k at time t. For nodal admittance matrix, G_jk and B_jk represent the real and imaginary parts, respectively; and θ_jk,t represents the phase angle deviation of branch jk.

2) Voltage limits

The limits for the magnitude of bus voltage are developed by using the sensitivity matrix shown in Eqs. (14) and 15 respectively. The magnitude of the minimum and maximum voltage is presented byV_{\min }^d and V_{\max }^d. For the base case, V_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}^d shows the real and imaginary terms of voltage magnitude obtained after running the load flows.

(17)V_{\min }^d\le \left|V_{init}^d-{\displaystyle \underset{i=\mathrm{1}}{\overset{nb}{\mathrm{\Sigma }}}{\displaystyle \underset{t\epsilon {\sigma }_d}{\mathrm{\Sigma }}(({\alpha }_{ki}^{dt}+j{\mu }_{ki}^{dt})I_{{\mathrm{Im}}_i}^t+({\beta }_{ki}^{dt}+j{\gamma }_{ki}^{dt})I_{{\mathrm{Re}}_i}^t)}}\right|\le V_{\max }^d.

3) Apparent power

(18)P_{jk,t}\le P_{jk}^{\max },

where the maximum value of apparent power of branch jk is presented by P_{jk}^{\max }.

4) EV charging limits

(19)P_{E{V}_{\min }}^d\le P_{E{V}_k}^d\le P_{E{V}_{\max }}^d,

where P_{E{V}_k}^d=B_{{\max }_k}-B_{SO{C}_k} which represents the active power of EV at bus k phase d; P_{E{V}_{\max }}^d and P_{E{V}_{\min }}^d represent the upper and lower bounds of the EV active power, respectively; B_max represents the maximum energy capacity of batteries; and B_SOC represents the battery state of charge.

5) EV capacity

(20)\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{e}\mathrm{v}}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le \mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{e}\mathrm{v},t}\le \mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{e}\mathrm{v}}^{\max },

where \mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{e}\mathrm{v}}^{\min } and \mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{e}\mathrm{v}}^{\max } represent the minimum and maximum value of EV SOC, respectively.

6) Current limits

(21)\left|I_{\mathrm{M}\mathrm{C}}^d\right|\le I_{\mathrm{M}{\mathrm{C}}_{\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}}}.

The total current of the main cable, which flows through each of its phase is presented by I_{\mathrm{M}{\mathrm{C}}_{\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}}}and the current which flows through a certain phase of the main cable is presented by I_{\mathrm{M}\mathrm{C}}^d obtained by impedance matrix of the cable and the magnitude of voltages calculated after implementing Eq. (17).

7) Network capacity

The total customer load including the residential and EV load must be less than the maximum capacity of the network (S_{\mathrm{S}{\mathrm{C}}_{\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}}}).

(22)\left|P_k^d+P_{\mathrm{E}\mathrm{V}}^d\right|\le S_{\mathrm{S}{\mathrm{C}}_{\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}}},

where P_k^d represents the residential demand.

Voltage behavior and unbalance factor

In three-phase systems, the concept of voltage unbalance can be understood in terms of unequal phase angles and unequal magnitudes of voltages, which can be observed in urban as well as rural distribution networks. Heavy current unbalances can be expected due to the existence of negative sequence voltages. It may lead to tripping of overload protection circuits and may further cause weakening of cable insulation, which subsequently reduces the lifespan of distribution components. Other vulnerable impacts observed from voltage unbalance contain enhanced losses due to the extra heating and high probability of fault occurrence because of inappropriate feeding of loads in an unbalanced system. Consequently, it is vital to identify the existence of voltage unbalance for timely safe operation of DNs. Therefore, the assessment of VUF should be carried out.

As voltage behavior and VUF are substantial power quality issues, several formal definitions can be seen in various international standards including IEC 61000-2-12, IEC 61000-4-30, NEMA MG1-2009, ANSI C84.1-2011, and EN-50160. According to the IEC standard, VUF can be expressed as a percentage ratio of negative sequence component by considering positive component as a reference. Detailed expressions regarding VUF are stated in Eqs. (23)-(27).

(23)\mathrm{V}\mathrm{U}\mathrm{F}(\%)={\displaystyle \frac{V_{\mathrm{2}}}{V_{\mathrm{1}}}}\times \mathrm{100},

where V₂ and V₁ are negative and positive sequence voltages in a power network, which can be obtained by

(24)\left[\begin{array}{l}{V}_{\mathrm{0}}\\ V_{\mathrm{1}}\\ V_{\mathrm{2}}\end{array}\right]={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}\left[\begin{array}{lll}\mathrm{1}& \mathrm{1}& \mathrm{1}\\ \mathrm{1}& a& a^{\mathrm{2}}\\ \mathrm{1}& a^{\mathrm{2}}& a\end{array}\right]\left[\begin{array}{l}{V}_a\\ V_b\\ V_c\end{array}\right],

Where V_a, V_b, and V_care line-line voltages with V₀ as zero sequence voltage component; and a = 1<120^○, a² = 1<240^○.

Balanced components are stated as

(25)V_{\mathrm{1}}={\displaystyle \frac{V_{ab}+a{V}_{bc}+a^{\mathrm{2}}{V}_{ca}}{\mathrm{3}}}\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}{V}_{\mathrm{0}}={\displaystyle \frac{V_{ab}+a^{\mathrm{2}}{V}_{bc}+a{V}_{ca}}{\mathrm{3}}}.

The formulation of VUF according to the IEC standard is expressed as

(26)VUF(\%)=\sqrt{{\displaystyle \frac{\mathrm{1}-\sqrt{\mathrm{3}-\mathrm{6}\beta }}{\mathrm{1}+\sqrt{\mathrm{3}-\mathrm{6}\beta }}}}\times \mathrm{100},

where

(27)\beta ={\displaystyle \frac{V_{ab}^{\mathrm{4}}+V_{bc}^{\mathrm{4}}+V_{ca}^{\mathrm{4}}}{{(V_{ab}^{\mathrm{2}}+V_{bc}^{\mathrm{2}}+V_{ca}^{\mathrm{2}})}^{\mathrm{2}}}}.

According to the IEC 61000-4-30 standard, in DNs, the percentage limit of VUF should not surpass the limit of 2% while long-term effects with no duration longer than 10-min should be accounted as well. It is expressed as

(28)\mathrm{V}\mathrm{U}\mathrm{F}\le 2\%\mathrm{f}\mathrm{o}\mathrm{r}>95\%\mathrm{o}\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{10-}\min \mathrm{d}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{.}

According to EN-50160, the magnitude of voltage should be within a tolerance range of±10%, expressed as

(29)\mathrm{0.9}\le \left|V_{\mathrm{m}}\right|\le \mathrm{1.1}\hspace{0.17em}(p\mathrm{.u})\mathrm{f}\mathrm{o}\mathrm{r}>95\%\mathrm{o}\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{10-}\min \mathrm{d}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{.}

As the load distribution of houses and EVs are random on each phase of URDN, it may lead to a wider load difference between each phase. Consequently, the voltage drop is changed between different phases of URDN, and unbalances are observed at different feeders of URDN.

Case study simulation for impact assessment

Performance parameters of distribution network

The impact assessment of performance parameters of RDNs is realized by using the average peak daily load duration curves for all types of domestic houses in different seasons as presented in Fig. 6. As the summer peak of several residential houses is higher in comparison with the winter peak, only the summer demand is given to residential loads of URDN. As far as the charging load of EVs is concerned, the stochastic charging behavior estimated by Monte-Carlo simulations is used as mentioned in Fig. 8 for assessment of voltage behavior, transformer limits, feeder losses as well as VUF with different charging strategies and charging levels of EVs. To analyze power quality parameters, EV charging strategies (controlled and uncontrolled) are examined in context to charging power levels at different EV penetrations. The random ineffective planning of urban houses and random EV distribution are significant planning issues. Therefore, the analysis of performance parameters is necessary for future load management and effective operation of RDNs. Combined technical assessment of average peak residential demand as well as stochastic charging demand of EVs on RDNs is presented. The results of performance parameters are obtained by running unbalanced load flows and performing the quasi-dynamic simulation for real-time assessment. First, the thermal loading of transformer is presented in Fig. 8. For brevity, the results for charging power level 2 with 300 kVA and 250 kVA transformers are presented only. The obtained results of thermal limits of transformer with consideration of EV charging strategies (UC and CC), charging power level 2, and different penetration levels (PL:10%, 25%, 40%) of EVs are presented in Fig. 8. It can be seen that the thermal capacity of transformers exceed the threshold limit after EV penetrations are larger than 40% in the case of charging power Level 2 and uncontrolled charging strategy. However, it is under the maximum prescribed limit of loading for 40% controlled EV charging penetrations.

Secondly, the losses of the main feeder 650-632 is presented in Fig. 9 for different EV charging strategies (UC and CC), charging power levels (L1 and L2), and different penetration levels (PL: 10%, 25%, 40%) of EVs. The feeder losses are much higher in the case of power Level 2 and uncontrolled charging strategy.

Moreover, the issues related to EV charging demand with a certain location as well as residential power demand with a location of domestic houses are also illustrated. The distance among various nodes of URDN is presented in Fig. 7(a). The location of nodes is categorized as the near bus, mid bus, and far bus scenarios, respectively. The obtained results of voltage behavior with consideration of EV and household location (location at feeder), charging strategies (UC and CC), charging levels (L1 and L2) and different penetration levels (PL:10%, 25%, 40%) of EVs are presented in Fig. 10. It can be observed that voltage profiles due to the random phase-wise house and EV distribution at various feeders of the implemented benchmark system are under the limit prescribed by the EN-50160 standard for a scenario in which the loads of houses and EVs are connected at the near bus scenario (Fig. 10(a)). However, Phases-B and-C are more affected by charging power Level 2 with higher EV penetrations due to the higher household as well as EVs charging demand. No household customer is affected in the near bus scenario due to household demand and EV charging demand of up to 40% EVPL.

However, with increasing distance as presented in mid-bus scenario Fig. 10(b), the voltage behavior of Phase-A is under the limit suggested by EN-50150. The voltage behavior of Phase-B is approaching the regulatory voltage limit for 40% EVPL, L2 uncontrolled charging strategy. Besides, the voltage behavior of Phase-C goes below the voltage regulatory limit for 40% EVPL with charging level 1,2 and uncontrolled charging strategy. Therefore, 36 customers can be affected by voltage problems in Phase-C.

Similarly, with further increase in distance as presented in the far bus scenario Fig. 10(c), the voltage behavior of Phase-A is approaching the regulatory limit for 40% EVPL and L2 uncontrolled charging strategy. The voltage behavior is violating the standard voltage regulatory limit for Phase-B for 25% EVPL with charging level 1, 2 and uncontrolled as well as controlled charging. In addition, the voltage behavior goes below the voltage regulatory limit even for 10% EVPL for Phase-C with level 1,2 and uncontrolled as well as controlled charging strategies. Therefore, in the case of far bus scenario, 86 customers can be affected by voltage problems in phases-B and-C.

Obtained results of VUF with the consideration of EV and household location, charging strategies, charging levels, and different penetration levels of EVs are presented in Fig. 11. VUF is set to 2% according to the IEC 61000-4-30 standard. It can be observed that VUF due to random phase-wise house distribution at various feeders of implemented DN is under the limit prescribed by the IEC standard for a scenario in which the loads of houses are connected at the near bus scenario as presented in Fig. 11(a). However, due to random EV distribution, VUF is approaching the standard limit for 10% EVPL only in the case for the near bus scenario and violating the standard limit for 25% and 40% EVPL, whereas, with the increase in distance, VUF is approaching the standard limit due to random phase-wise house distribution. However, due to EV distribution, VUF is violating the standard limit for all levels of EV penetrations as proven in mid-bus scenario (Fig. 11(b)).

Similarly, with a further increase in distance, VUF in the far bus scenario is violating the standard limit and goes beyond the value of 2% for a random phase-wise house as well as random EV distribution as presented in Fig. 11(c). Therefore, feeder 680 is considered as the most vulnerable for further random house allocation as well as random EV integration. The obtained results reveal that random phase-wise house distribution with random EV integration cause VUF to exceed its tolerable limit in the far bus scenario, which can contribute to posing vulnerable issues for a power network. In RDNs, unbalancing issues are not observed by the level 2 charging demand because the charging load is evenly shared by two-split phases.

Computational aspects and convergence/validation

The three-stage proposed framework for risk assessment of distribution network is performed on a DELL machine with the PowerEdge R730 server and Windows operating system of 64-bit. The specifications of CPU include 4 Intel Xeon processors of 2.4 GHz along with a random access memory (RAM) of 32-GB. The total computational time needed to execute all the three stages (one case study) from generating EV charging profiles by Monte-Carlo simulations to evaluating performance parameters of RDNs is about 11 min. Stage-1 is performed on MTLAB 2018a to produce stochastic EV charging profiles for different charging strategies, whereas, Stage-3 is executed on DigSilent Power Factory for performing unbalanced load flows for 24-h time horizon and analyzing the parameters of RDNs.

Random sampling is used in the Monte-Carlo simulation method for estimation of a mathematical function. Statistical convergence is observed in the Monte-Carlo method where the fitting deviation is converged to a specific threshold. The convergence criteria adopted for the proposed simulation method is described as

(30){\gamma }_i={\displaystyle \frac{\sqrt{V_i(\bar{z})}}{\overline{Z_i}}}={\displaystyle \frac{{\sigma }_i(\bar{z})}{\overline{Z_i}}},

where variance coefficient is presented by {\gamma }_i at time interval i; the variance, standard deviation, and expectation are presented by V_i, σ_i, and \overline{Z_i} respectively. Monte-Carlo simulations are performed up to 100 times until the convergence criteria meet and coefficient of variance {\gamma }_i is restrained to less than 5% [¹⁶].

The coefficient of variance is tested for the convergence according to the set criteria for different charging strategies of EVs at the 25% level of penetration to validate the performance of the algorithm. Figure 12 demonstrates the convergence plots for model validation, where it is clearly observed that the criteria for the convergence is satisfied.

Conclusions

This paper proposed a detailed stochastic model of EV charging behavior based on realistic aspects to analyze the risks associated with RDNs. It provided a preemptive measure for utilities to tackle any undesirable situations related to the performance parameters of RDNs including voltage behavior, unbalancing issues, system losses, and thermal transformer limits, which could occur drastically with electric load variations in the near future. A combine technical assessment of average peak residential demand as well as EV stochastic charging demand on RDNs was presented in detail in terms of issues related to EV charging demand with a location as well as residential power demand with a location. Voltage behavior in terms of customers with voltage issues and VUF were analyzed as significant power quality issues due to random phase-wise household and EV distribution for stable and reliable operation of a power network. Random phase-wise distribution was identified as a common planning problem for utilities in residential networks, which caused performance parameters to surpass its permitted limit according to international standards.

First, a probabilistic EV charging model was developed to account for several realistic aspects comprising charging time, battery capacity, driving mileage, state-of-charge, traveling frequency, charging power, and time-of-use mechanism in peak and off-peak charging strategies. The mobility pattern was investigated with the real-time scenario. A realistic EV charging model was developed based on the results of NHTS data to account for required probability distributions. Monte-Carlo simulations were performed for required probabilistic calculations. Afterwards, the average peak daily residential demand for several domestic houses was obtained for different seasons by using the historical load data profiles from a utility. Moreover, an operational framework of a residential distribution grid was presented with the purpose of risk assessment of distribution networks.

Finally, the outputs of the EV stochastic model and average peak daily residential demand obtained from Monte-Carlo simulations and historical data respectively were utilized to evaluate power quality parameters of RDNs. Performance specifications of RDNs including thermal limits of transformers, feeder losses, voltage behavior, and VUF were assessed in context to EV charging scenarios with various charging power levels at different penetration levels. The results of voltage behavior and VUF were examined with the consideration of EV and household location (location at grid feeder), charging strategies (UC and CC), charging levels (L1 and L2), and different penetration levels (PL: 10%, 25%, 40%) of EVs. The obtained results reveal that thermal limits of the transformer surpass its maximum loading limit in the case of level 2 charging power level and uncontrolled charging. The results clearly indicate that random phase-wise house distribution with random EV integration causes voltage behavior as well as VUF to exceed its tolerable limit according to international standards in different scenarios as presented in case study simulations. Weak voltage profiles and higher VUF affect many residential customers and can contribute to posing vulnerable issues for a power network. It is intended that this paper can fill the gap between the traditional and future power networks by providing a framework to evaluate the risks associated with RDNs to manage harmful impacts with variations in electric load and large-scale EV penetration in the near future.

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