A Three-Stage Stochastic Framework for Smart Electric Vehicle Charging

As one of the most significant part of carbon neutralisation, the rapid growth of electric vehicle (EV) market in past few years has greatly expedited the transport electrification, which, however, has brought in new challenges to power system including isolated distribution network for commercial and industrial set up. Stochastic and complex EV behaviours would violate network permissible operation region and increase costs for system operators. To address these problems, a chance-constrained smart EV charging strategy in a DC microgrid (DCMG) supporting large office complex is proposed to minimize system cost from distribution network and fleet battery degradation cost from EVs providing ancillary service to the DCMG. When dealing with uncertainties from EVs, a Markov Chain Monte Carlo (MCMC) model is built to couple different parameters in load profiles and characterize the time series of likelihood of charging and discharging. A state-of-charge (SOC) space random walk method is then proposed to solve the resultant massive recursive probabilistic charging requirements. Based on that, a three-stage optimization framework is established to illustrate the work flow in system level. Numerical results verifying the effectiveness of the proposed method are also presented.

G j,k conductance between bus j and k. Jac Jacobian matrix. P t j power injection at time t, in bus j. P d,t j load power at time t, in bus j. P g,max j maximum generating power installed at bus j. P g,t j generating power at time t, in bus j. P t EV ,j EV charging/discharging power at time t at bus j. P max EV ,j maximum EV charging/discharging power of bus j. R v j virtual resistance of DG unit j. VOLUME   weighting factor for battery capacity losses. W s weighting factor for network system losses.

I. INTRODUCTION
The pace of EV commercialization is faster than expected. Number of EVs sold in 2021 doubled from the previous year to 6.6 million and the primary Leader China accounted for half of the growth [1]. At the end of June 2022, over 27% of all passenger vehicles sold in China were EVs. As a result, data from the traffic management bureau showed that as of end-June, Chinese EV fleet stood at 10.01 million, which is three years ahead of the target timeline of 2025 [2]. Meanwhile, leading battery manufactures BYD and CATL have launched their newest Cell-to-Pack (CTP) battery, which would extend EV travel range up to 1000 km with 3000 cycle [3], [4]. Additionally, the integration of EVs can also provide economic benefits to different players in the energy market [5]. All the information above shows that the era of EV has come, but whether the world is ready is doubtful. On one side, although the evolution of EVs and their related industrial chains is exceeding road map, challenges of large-scale EV adoption are becoming pressing. A detailed analysis evaluating the impact of EVs on distribution network including a deterministic sensitivity analysis on distribution network components and a probability-based stochastic analysis is presented in [6]. The study further reports user behaviour, which shows that EV charging could increase the risk of electrical component overheating, significant voltage drop, uncontrollable three phase unbalance and significant increase in system loss during power transfer [6]. Simulation in [7] demonstrates that EV penetration and location in power networks are main reasons for voltage drop. The voltage level may vary significantly at certain buses. Furthermore, even at relatively modest levels, the safe oper-ating region in both voltage and thermal loading levels are not secured. Moreover, EV quantity and penetration can also be limited by the capacity of the distribution transformers [8]. Thus, proper EV integration strategies including smart charging together with potential discharging will be necessary, otherwise distribution networks would hit the operating constraints [9].
The challenges from the perspective of EV drivers are more severe and urgent. Until now, hundreds of EV fire incidents were reported and traction battery fire during charging account for a substantial part of it, which makes charging a safety concern, and is becoming one of the biggest shackles of EV expansion [10]. Fault diagnosis for Lithium-Ion battery is multivariate and complex process, which is regarded as a black-box system in industries [11]. Although the global energy crisis in 2022 has encouraged many drivers to switch their gasoline fleet to electric, it is not promised that EV charging infrastructure will develop with drivers' needs at the forefront including charging hardware, various charging methods, customer service and roaming partnership [12]. Heatwaves in 2022 summer have pushed power loads of many countries to record highs. EV charging operator NIO, Tesla and Xpeng in Chengdu had to cut off most of its charging piles and battery swapping stations to adopt off-peak charging plans [13], [14]. Similar in California and Texas, Tesla also asked its drivers to voluntarily reduce their charging between 4pm and 9pm [15], [16]. It is easy to conclude that with the sustained growth of EV ownership, charging crisis would be deteriorating rapidly.
EV charging has put pressure on both EV drivers and Distribution Network Operators (DNOs). Therefore, smart charging strategies are introduced to address the problems from aspects of network support and charging demand management. A linear programming model considering EV charging cost and network demand variation is proposed in [17] which also included Voltage Unbalances Reduction (VUR) and Vehicle to Grid (V2G) to achieve better power quality. In [18], a set of smart charging scheduling methods was proposed to shave sudden peak demands and regulate under-voltage problem by deploying binary Particle Swarm Optimization (PSO). Rather than solving the problem locally, a two-stage EV charging coordination mechanism that frees the distribution system operator (DSO) from extra burdens of EV charging coordination is proposed in [19]. DC distribution network is introduced for optimal PHEV charging strategy considering traction battery charging and discharging in [20]. Additionally, voltage regulation and reverse power flow are also constructed as an online optimization model in this study. Furthermore, a DCOPF based multiobjective optimal charging model is proposed in [21] which considered system losses from distribution networks and battery degradation from vehicles as objectives. It achieved a better computational efficiency by modeling battery State-of-Charging (SOC) space as weighted single source shortest path problem and solving it using modified Dijkstra's algorithm.
Although varied objectives and methods for optimal charging had been studied, it is to be noted that EV user behaviour is another significant feature affecting implementations of the model. Most of the studies above built household travel patterns as their EV user profile. However, this is neither accurate nor realistic since the stochastic attribute of traffic [22]. Therefore, a data-driven EV behaviour analysis is needed to describe the status of vehicles from different aspects as detailed and as possible. Reference [23] created parking availability patterns by using a survey of resident mobility focusing attention on periods related to work that correspond to the majority of the trips, which was carried out by Spanish Ministry of Industry. However, such patterns are not detailed enough and these may vary with time, location and other factors. Reference [24] described EV charging behaviour from demographic-based statistical data in Gothenburg, Sweden, which indicated that different EV charging behaviors might affect the load demand and charging strategy. The study from [25] further confirms that parking location selection would make difference on network load curve and, in turn, affect EV charging strategies. In addition to the deterministic methods above, a stochastic programming method was used to achieve optimal charging strategy while taking stochastic user behavior into consideration in [26]. One of the most popular model describing stochastic charging pattern is Markov Decision Process (MDP), which can be further solved by deep reinforcement learning algorithms [27], [28]. Besides, robust optimization can also be used for uncertain user profiles from EV charging [29]. With more complex structure, a hierarchical stochastic predictive control scheme was proposed as an Energy Management System (EMS) in an island microgrid in China [30]. Wang and Xiao designed a hybrid centralized-decentralized charging strategy for massive EV integration [31]. Additionally, online optimization and Model Predictive Model (MPC) are also well studied to improve performance for specific date and network configuration [32], [33].
This study proposes a chance-constrained smart EV charging strategy in DCMGs minimizing system cost from distribution network and battery degradation cost while discharging. The contributions of this article can be summarized as follows.
1) A Markov Chain Monte Carlo (MCMC) model is proposed for EV load profile integrating Time-of-Arrival (TOA), Time-of-Departure (TOD) and Energyto-be-Charged (ETC) as a multi-variable probability distribution. 2) An EV battery SOC space is then designed which consists of optimal power flow (OPF) in DCMG, battery aging features and EV load profile from MCMC model, which is finally solved recursively by employing the random walk method.
3) A three-stage smart EV charging framework is established specifying the workflow from OPF and the load profile to chance-constrained probability level calculations. The rest of this article is organized as follows. Section II presents a mathematical formulation of optimal DC power flows and smart battery charging, and is then solved by interior point method (IPM). In Section III, a MCMC model is employed to analyze the relevance between TOA, TOD and ETC in EV load profiles. Additionally, a random walk in SOC space is designed to solve the proposed smart charging problem. Numerical results are presented and discussed in details in Section IV to examine the effectiveness of the proposed model. Finally, conclusions are given in Section V.

II. FORMULATION OF EV OPTIMAL CHARGING
In this Study, we consider a district DC distribution network working as DC microgrid containing substation, distributed energy resources (DERs) (e.g., Photovoltaic), network loads and community-level EV charging station. Microgrid is demonstrated in this work for its feasibility in meeting EV charging demand and mitigating the intermittency and volatility of EV integration into modern power networks [21]. DC interface in microgrids offers advantages like fewer conversions, skin-effect free and no requirement for reactive power control, which are more suitable for EV integration and DERs as compared with AC network [21]. The distributed system operator (DSO) will play the role of providing electricity to local loads and charging service to EV owners, which indicate distribution network operator (DNO) and EV aggregator separately. In this case, single EV is deployed with fixed location for better observation in other parameters in EV load profiles. We assume that local loads are non-controllable and DERs are embedded with energy storage system (ESS) which can be regarded as dispatchable generations (DGs). Besides, we assume that EV charging and discharging decision and corresponding power can be fully controlled while it is connected to the system, however its battery is required to be charged to a satisfied level before it leaves. Besides, the DCMG is assumed as a unipolar system in the network configuration and sampling time is set as 15 minutes.
As for EV fleets, this study has been performed with passenger EVs where lithium batteries are mostly popular in global market. Thus, the EV was modelled as a lithium battery in this study. Generally, battery pack can be modelled as battery equivalent circuits [34]. However, battery packs in EVs are controlled by battery management system (BMS) and connected to the grid via on-board chargers (OBC). Therefore, EVs are regarded as controllable DGs when discharging and controllable loads when charging.

A. DETERMINISTIC SMART CHARGING MODEL 1) OBJECTIVE FUNCTIONS
A deterministic optimization model was initially built to investigate an EV optimal charging problem in DC microgrid, which is to minimize the DSO cost from local loads and EV charging over 24 hours while satisfying physical constraints of distribution network and EV battery management system (BMS). DSO cost consists of two parts. One is system loss from the perspective of distribution network and the other is battery degradation cost from EV integration.
Finally, the objective function for the deterministic optimal charging model can be expressed as: where W indicates the weighting factor of the multiobjective optimization model and their specific functions and values can be found in previous work [21]. E s and E b represent system energy loss and battery capacity loss respectively. Notice that if the EV is not discharging, the objective function can be simplified as system cost only.
In DC network modelling, buses can be either modelled as load (P) buses for non-dispatchable DGs and uncontrollable loads without droop control deployed, or DG buses for dispatchable DGs and controllable loads with droop control deployed. Based DC droop control and DC power flow principle, the system energy loss had been derived in [21] and is expressed as: where E s indicates system energy loss, G j,k indicates the conductance between buses j and k, V j and V k indicate voltages of the buses j and k separately, n indicates the total number of branches in DCMG and t indicates step size which is 15 min in this study. Meanwhile, the concept of battery capacity losses, also named battery degradation cost, is developed based on EV charging/discharging principle with utilizing a linear aging coefficient η a to capture aging level for different battery technologies. In this study, η a has a value of 3 × 10 −4 for lithium-ion battery and the battery capacity losses E b can be expressed recursively as [21]: where h is the total number of buses, Cap nom ref ,j is the reference nominal battery capacity which is available from manufacturer and Cap t− t ref ,j is battery capacity of reference at previous step t − t. P t EV ,j is EV charging/discharging power at time t in bus j and it is assumed to be constant in time step t. P t EV ,j is set positive for charging and negative for discharging.

2) CONSTRAINTS
Constraints for smart charging model include the following: Jac =   Jac 11 Jac 12 Jac 13 Jac 14 Jac 21 Jac 22 Jac 23 Jac 24 Jac 31 Jac 32 Jac 33 Jac 34 where bus voltage V j , DG generated power P g j , virtual resistance R v j , and battery state-of-charge SOC t j are the state variables of the proposed optimal charging strategy model. The aim of the optimization is to minimize the cost of DSO while satisfying physical constraints and operational principles. Equality constraints are depicted in (4)- (8) and inequality constraints are formed in (9)- (14).
Specifically, (4) is power flow mismatch function for the j th bus; (5) describes the droop control method in DCMG; (6) gives a recursive expression for EV battery SOC. Notice that the total number of equality constraint equations is the sum of the number of equations in (4)-(6) and corresponding state variables set is configured in (7). (8) represents the Jacobian in terms of above state variable sets and the derivation of sub-matrices in J can be found in [21]. Finally, these equality constraints will be linearized using Newton-Raphson algorithm [35], [36].
Constraints (9)-(12) specify the bounds for bus voltage, DG output power, EV charging/discharging power and virtual resistance in DC droop. It is worth mentioning that in (11), the maximum EV discharging power is set as large as its charging power, which is true for most bi-direction DC-DC inverters. (13) indicates the battery operation zone where battery cannot be over-discharged lower than DOD and over-charged higher than 100%.
Expression (14) describes EV charging principle: battery is firstly connected to the network with initial state-of-charge SOC t a and reference capacity Cap t ref at step t a , then the battery is charged or discharged at rate P t EV ,j until the battery is charged to its desired level SOC des j at the end of process SOC t z . In this study, P t EV ,j is set positive during charging process and negative during discharging process. Note that from the beginning to the end, battery SOC may exceed SOC des j for more than one time, but as long as it does not offend the SOC upper bound in (13), these constraints can be satisfied.
Additionally, various ancillary services are available when V2G is enabled during smart EV charging. For instance, peak shaving is available in this case to whittle sudden load peaks from local demand. However, there won't be any compulsory peak shaving operation which would obstruct the observation on stochastic charging performance.
In this case, the method used to optimise the objective function (1), subject to constraints (4)- (14) is the interior point method (IPM) [37], which is a popularly adopted method in power system analysis of which detailed derivation can be found in [21].

B. CHANCE-CONSTRAINED SMART CHARGING MODEL
To study the impact of EV behaviour uncertainties on arrival and departure time and energy request, stochastic EV load profiles are introduced in this work, which convert the deterministic optimization model into a stochastic optimization problem [38]. Thus, EV smart charging mission (14), as the most significant constraint in this model, is rewritten as as a chance-constrained expression as below: where ∈ [0, 1] is the probability level which is a crucial parameter in this study and can be regarded as a trade-off between EV owners (EVO) and DSO. A higher probability level would untie the charging operation and allow EV discharge minimizing the costs consisting of system loss and battery degradation in this study. On the contrary, A lower probability level would bind the operation and discourage battery discharging which may lead to insufficient SOC at the end of the process.

III. STOCHASTIC PROGRAMMING WITH SMART CHARGING A. LOAD PROFILE WITH MARKOV CHAIN MONTE CARLO (MCMC) 1) MOTIVATION FOR MCMC
Apart from system configuration, load profiles from EVs are another significant part of the study -it is the key to realistic applications given its parameter complexity and the need for model compatibility. From previous studies, load profiles for smart charging strategies are usually obtained from the driving patterns [39] or trip chain theory [40], together with basic probability distributions like Poisson distribution [41] and normal distribution [42], [43]. This will lead to two problems: Firstly, travel patterns are made for local communities, districts or even larger areas which are not in the same scale of one distribution network. it is not guaranteed that the network is representative enough for the whole area. Secondly, parameters like time-of-arrival (TOA), time-of-departure (TOD) and energy request are strongly correlated, which cannot be simply expressed as individual probability density distributions. Thus, a high-dimensional parameter framework with network exclusive traffic flow deploying MCMC is proposed in this work.

2) INTRODUCTION TO MCMC
In order to generate a set of identically independently distributed samples x i with respect to the density π of this distribution, MCMC methods is proposed to sample from such a distribution which enables the estimation of the integral: of a function f : χ → R with respect to corresponding probability measure by 1 m m i f (x i ). In principle, the target density is a non-negative measurable function π : χ → R + , where χ ⊂ R d is the sample space with Lebesgue measure µ, corresponding to the probability measure (A) = A πdµ. Unusually, it is known that π is a multiplicative constant, that is it will be available to evaluateπ where π =π /Z for Z ∈ R + . For example, this is the case in Bayesian statistics, where the normalisation constant Z is model dependant, which is itself a complicated integral and not always in closed form [44]. Although the value of Z is given or available, sampling from π is still a complicated task, particularly in high dimensions where high probability regions are usually concentrated on small subsets of the sample space [45]. Samples can be generated in only a few densities.
MCMC algorithm is initially proposed in physics-based research [46] to order to solve these problems. The authors proposed a method which generates samples from an arbitrary random walk, and adds an accept/reject step to guarantee the samples originating from the target distribution. Later on, MCMC application showed up in the statistics [47]. The basic idea of MCMC methods [48] is to generate samples from the target π which are approximately i.i.d. by defining a Markov chain whose stationary density is π. Recall that a Markov chain is a sequence of random variables (X 0 , X 1 , . . .) where the distribution of X r depends only on X r−1 . A Markov chain starts from an initial density h 0 (x) for X 0 and a transition density T (x ← x) from which can be sampled. The density of X r is then defined by (17) π is called the stationary density of the Markov chain if whenever X r ∼ π then X r+1 ∼ π, that is π(x ) = T (x ← x)π(x)dx. In this situation, when Markov chain is ergodic, it will converge to its stationary distribution independently of initial distribution. The detailed balance condition is commonly used to guarantee π is the invariant density of the chain, which is shown below: It is easy to observe from the above equation that the probabilities of moving from state x to x and from x to x are equal, VOLUME 11, 2023 FIGURE 1. Initial EV changing and discharging SOC space.
however, detailed balance is a sufficient but not necessary condition [49].

3) GIBBS SAMPLER
The Gibbs Sampler constructs a Markov chain converging to the desired target π with a vector of parameters θ = (θ 1 , θ 2 , . . . , θ k ), and we want to estimate the joint posterior distribution p(θ|X ). Suppose we can find and draw random samples from all the conditional distributions: With Gibbs sampling, the Markov chain is constructed by sampling from the conditional distribution for each parameter θ i in turn, treating all other parameters as observed. When we have finished iterating over all parameters, we are said to have completed one cycle of the Gibbs sampler. Gibbs sampling is a type of random walk through parameter space, and hence can be thought of as a Metropolis-Hastings algorithm with a special proposal distribution. At each iteration in the cycle, we are drawing a proposal for a new value of a particular parameter, where the proposal distribution is the conditional posterior probability of that parameter. This means that the proposal move is always accepted. Hence, if we can draw samples from the conditional distributions, Gibbs sampling can be much more efficient than regular Metropolis-Hastings. The advantages of Gibbs sampling are obvious. It has no need for proposal distribution tuning and proposals will always be accepted, which are suitable for limited number of proposals and cause great computational efficiency. In contrast, it also has some drawbacks in implementation: firstly, this algorithm is enabled only when conditional probability distribution derivations and random samples from conditional probability distributions are available. Besides, sampling process would become slow when parameters are correlated, especially for high-dimensional implementations, because random walk steps are moved alongside each axis one by one rather than diagonal steps in Metropolis-Hastings algorithm. However, these features fit the EV load profiles well. Because for each set of EV load profile(generated from individual one-way travel pattern) samples are i.i.d. [50] which naturally works on Gibbs sampler and load profile probability distribution are not time sensitive since the overall traffic conditions will not change rapidly in certain districts. Thus, a high-dimension probability distribution for EV load profile including TOA, TOD and ETC is demonstrated in this study.

B. RANDOM WALK IN SOC SPACE
With the implementation of MCMC, each set of parameters in EV load profile at certain time instant becomes available and the upcoming step is to establish a SOC space corresponding to the EV load profile for smart scheduling problem. The concept of SOC is widely deployed in many EV optimal charging studies [28], which can be simply regarded as a Knapsack problem where given a set of items with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. Similarly, the basic idea of EV charging is to achieve target SOC in limited time while keep cost as little as possible.
A diagram for initial EV charging and discharging SOC space has been shown in Figure 1 where battery SOC limit (13) is not considered. A binary set S is named as battery state to indicate the charging/discharging states from initial step t 0 . At certain time step t, there are only two available states for the vehicle: discharging activated and discharging inactivated, based on the optimal power flow established in (4)- (8). When discharging is inactivated, system will operate conservatively to charge the battery as much as possible while minimizing the system loss and when discharging activated, batteries would discharge as DGs due to much lower system loss compared to EV charging, which release the flexibility for EV ancillary service. Thus, there will be a total of 2 n states at the end of the process and probability calculation of each state is shown in Figure 2. Figure 2(a) shows a sample slice of TOD for load profiles with TOA set as 8PM. The axis covers 32 hours from midday until 8PM the next day, and it shows one peak of departure at 6AM and another higher peak at 3PM. It is assumed that one vehicle connects to the network at 8PM, stays no longer than 24hours and will be fully charged at t f if battery starts charging instantly when connected without any stopping or discharging. At this instant, time step t is set 0 and battery state S is set null.
At initial step t = 0, state S = [] (null), P a indicates the probability of TOD between arrival time t a and earliest full charge time t f and sequentially P b indicates the probability of TOD between t f and must leave time on the next day t a . In this case, P a represents the probability that the vehicle cannot get fully charged before it leaves since the earliest full charging time would be t f , sequentially, P b represents the probability of fully charged. Therefore, fully charged probability P S can be derived from P a and P b as below: where . 0 in state set S indicates the battery discharging is inactivated from last step and vice versa. Since battery is charged from last step, the expected fully charged time will stay unchanged as t f but P a is reduced by P z which indicates TOD from last step. Then P S is calculated as follow:  where Figure 2(c) indicates a discharging scenario after Figure 2(a) with t = 1, S = [1]. Because the battery was discharged from last step, the earliest full time will be delayed to t f and P S calculation becomes: where In this way, all changing probabilities corresponding to the state S can be calculated recursively.

C. THREE-STAGE OPTIMISATION FRAMEWORK
After MCMC load profile and optimal charging model are established, a three-stage framework is proposed to accomplish this stochastic optimization model. A detailed figure on framework configuration is shown in Figure 3. MCMC plots are obtained from Stage one together with multi-objective optimal power flow from the Stage two as the SOC space and then solved by random walk method in Stage three. Finally, proper charging states would be selected to satisfy probability level of charging requirement.

IV. CASE STUDY A. TEST SYSTEM AND PARAMETERS 1) TEST SYSTEM
The test system shown in Figure 4 was used to verify the effectiveness of the proposed technique. In this network, bus 1, 2 and 3 were considered as dispatchable buses controlled by droop controller through virtual resistance. Buses 4, 5 and 6 were considered as time-series power (load) buses with IEEE European Low Voltage Test Feeder v2 load profiles having total duration of 24 hours and time interval of 10 minutes. These load profiles are identified as Loads 1, 2, 3 in this paper and are shown in later sections.

2) PARAMETERS
Initial battery SOC was set to 30% i.e. at the start of EV integration and desired SOC after the entire EV charging was limited to 100%. Maximum EV charging/discharging power was set to 4kW. The virtual resistance on droop buses varied between 0.2-ohms and 5.0-ohms. Power capping for peak shaving service is 0.3p.u. The DGs are considered as combined PV and BESS, like the arrangement in the microgrid supplying power to the company complex where the first author had worked in China prior to his commencement of PhD study in the UK. For the test network, two DGs were deployed with rating of 6.5kW and 7.0kW.

B. EV LOAD PROFILES
Original EV load charging profiles are coming from Caltech, Pasadena, CA. Charging data is collected from 54 charging points in one campus garage which is a public charging station for faculty, staff and students. Besides, this garage is the only charging station inside the campus and this charging station is fed by one single substation, which is assumed to be a part of the campus distribution network. But in this study, to avoid further geographical features that could increase the algorithm complexity, all charging data for this station has been aggregated as one charging point to guarantee sufficient samples based on good homogeneity in the same station. Further, data from entire calendar year of 2019 was selected to eliminate the impact of Covid from 2020. Finally, 9082 valid charging profiles were selected in this work. As MCMC algorithm with Gibbs sampling is proposed in previous section for its expansion capability in multidimension applications, TOA, TOD and ETC are conducted with data from Caltech. 3D pdf plots illustrating TOA-vs-TOD and TOA-vs-ETC in 7 days of the week are shown in Figure 5 and Figure 6 separately. In Figure 5, TOA is displayed from left hand side scaling from 00:00 to 24:00 of the day and TOD is displayed from right hand side sharing the same scaling with TOA. It seems that plots from Monday to Friday are highly similar. Only one significant peak can be observed from (a)-(e), where arrives at 9AM and departure at 7PM. However, another peak in (f) and (g) can be observed with TOA 6PM and TOD 10PM. Therefore, profiles can be easily aggregated as weekdays and weekends in terms of TOA and TOD distribution.
Similarly, TOA-vs-ETC plots are plotted in 7 days of the week in Figure 6 where sub-figures (a)-(g) indicate Monday to Sunday respectively. TOA is observed from left hand side as well and ETC is observed from right hand side. Plots in (a)-(e) show similar distribution shape where EV arrives in the morning with less energy request and plots in (f) and (g) show wider distribution on arrival time and higher energy request compared with (a)-(e). Thus, load profiles can be grouped as weekdays and weekends for better efficiency in EV behaviour analysis.
Aggregated load profiles are shown in Figure 7 and Figure 8. Each sub-plot shows similar shape compared with 662 VOLUME 11, 2023 their source plot from Figure 5 and Figure 6 but differentiated from each other, which are representative enough for SOC space construction in next stages.

1) NUMBER OF VALID STATES
In this study, time resolution is set as 15 min, so there will be a total of 144 steps for a 24 hour simulation. 7.6 × 10 6 valid states were observed from total number of 2 144 states. The number of valid states increases with total number of states but the curve is bumpy due to high system complexity. Every one additional step would double the total states, however, the number of new valid states from last step is affected by profiles from local load in networks and EV behaviours.

2) STATISTICS OF VALID STATES
Based on 7.6 × 10 6 valid states collected, a pie chart is drawn in Figure 9 to illustrate the relative ratio for these states, where sub-figure (a) and (b) indicate weekday and weekend  scenarios separately. Figure 9(a) shows that only a small proportion of states (0.13%) is leading 1% of the total states with P s higher than 99.91% and similar situation happens in 30% leading group with P s between 97.31% and 99.91%. On the other hand, 1% less group with P s between 91.02% and 91.11% occupies 63.95% of the total number of states and 30% less group with P s in zone 91.11% and 93.72% occupies 17.89% of those. Finally, the middle group 30% to 70% only takes 17.76% of the total share.
On the other chart, similar situation happens on the 1% more group which occupied 0.34% of the pie with P s higher than 99.94% but percentage for 30% group is much higher than that on weekdays which is 5.59% with P s between 98.11% and 99.94%. Besides, 30%-70% group with P s between 95.59% and 98.11% occupies 26.62% of the pie and 30% less group only takes a small proportion of 4.14% with P s between 93.76% and 95.59%. Finally, 63.31% of total valid states are aggregated as 1% group with P s between 93.70% and 93.76%. Compared between charts in both sides, combing load profile probability distribution in Figure 7 and Figure 8, it seems that a relative flatter peak or multi-peaks in load profiles may lead to smaller gaps in both aspect of state quantity and full charge rate aggregation level.

3) CHANCE-CONSTRAINED COMPARISON
Probability level in (15) can be divided into 5 groups according to data differentiation in Figure 9. Results in weekday scenario and weekend have been shown in Table 1 and 2,   respectively. From these statistics, it is shown that when high success rate is guaranteed, only few possible states are enabled for this task and with lower success rate P s required, possible operation flexibility will be released.

V. CONCLUSION AND FUTURE WORK
This article proposes a smart EV charging strategy with stochastic EV load profiles. In the context of uncertainties of EV users' behaviours, a MCMC model is firstly built to illustrate the relevance between different sectors in EV load profiles, and then utilized with a multi-objective DCOPF model considering system loss and battery degradation to battery SOC space. A random walk method is then employed to solve the proposed chance-constrained model. Simulation results suggest that EV charging task can be satisfied with high probability level (>91.02%). Further, a small portion of probability level can be traded for a large mount of valid operation states, which would provide considerable flexibility for V2G and related ancillary services. Optimal interaction design with massive EV scale in single network and benefit maximization based on tripartite transaction could be new interests in the future.