A Decentralized Dynamic Pricing Model for Demand Management of Electric Vehicles

Transportation electrification is considered a green alternative to internal combustion vehicles. However, higher penetration of electric vehicles (EVs) can cause several technical challenges to the power systems, including local equipment overloading. This is especially challenging for distribution systems due to the direct connection of EVs with them. To mitigate equipment overloading, by managing EV loads locally, a dynamic pricing model is proposed in this study. First, a satisfaction function is devised for EVs considering the sensitivity of different EV owners to the battery state of charge and urgency of recharge. Then, a welfare function is developed for the EV fleet operator (EFO) considering the net load of EVs and the electricity price of the upstream grid. To this end, a welfare maximization problem is formulated considering the welfare of EVs and EFO. The problem is then decomposed into the EFO sub-problem and EV sub-problem to preserve privacy. Finally, a distributed mechanism is developed to solve the sub-problems iteratively without revealing private information. The performance of the proposed method is analyzed for a residential apartment complex in terms of load management and convergence for different day types (working days and holidays). Simulation results have shown the efficacy of the proposed method in mitigating the overloading of transformers during peak load hours.


INDICES t
Index for time interval, running from 1 to T . n Index for EV number, running from 1 to N . k Index for iteration number, running from 1 to K . VARIABLES x n,t Amount of power charged to EV n at time t. x soc n,t Amount of energy in EV n at time t. S n,t Satisfaction level of EV n at time t. SoC n,t SoC level of EV n at time t. ρ t Total EV charging price for interval t. ρ b t Base price for interval t. VOLUME

I. INTRODUCTION
Electric vehicles (EVs) are considered a viable option to reduce the emission of greenhouse gases from the transport sector. EVs are mobile energy storage systems, and they can support the grid during system contingencies via vehicle-togrid (V2G) service. In addition, EVs can provide power to homes during grid outages to survive the critical loads of the homes via V2G [1]. Several studies have investigated the potential of EVs in providing ancillary services to distribution systems. For example, peak load management [2], voltage and frequency support [3], and spinning reserves [4]. An overview of different ancillary services provided by EVs to distribution networks can be found in [5] and [6]. Therefore, the penetration of EVs has increased across the globe and it is expected to further increase in the coming future [7]. The increased penetration of EVs will necessitate the deployment of more charging infrastructure, especially fast charging stations to reduce the charging time of EVs. However, largescale deployment of charging stations will increase the load of power systems. The EV charging infrastructure is directly connected to the power distribution systems, and it can potentially overload the distribution system during peak charging intervals. It has been noted in [8] that the EV load is significant at the local transformer level as compared to the feeder or grid level. It has been concluded [9], after analyzing several distribution circuits, that distribution transformers and lines are at prime risk of getting overloaded. Similarly, the impact of EVs on power distribution systems is analyzed in [10] and several long-term solutions are proposed. It has been noted in [11] that fast charging stations can increase the on-site load by up to 250%. Therefore, several studies are conducted to manage the load of EVs, which are discussed in the following paragraphs.
Different charging and discharging methods for EVs, such as indirect controlled charging-discharging, intelligent charging-discharging, bidirectional charging-discharging, and hierarchical charging-discharging methods are analyzed in [12]. Demand management of EVs in the presence of renewables is discussed in [13] and the transportation system is also included in [13] during the infrastructure planning phase. A battery-swapping model is proposed in [14] to enhance the service capacity of charging stations while minimizing the cost. Similarly, competition among EV aggregators is considered in [15] via game theory and the non-ideal behavior of aggregators is also considered. A distributed trilayer framework is proposed in [16] to manage the load of EVs while preserving the privacy of EV owners. In addition, several researchers have proposed different pricing methods to manage the load of EVs, which are as follows.
Four well-known price-based demand response programs are analyzed in [17] and concluded that real-time pricing is the most suitable option for managing the load of EVs. Similarly, real-time pricing is proposed in [18] for managing the load of charging stations in residential complexes including renewables. Time-of-use pricing is proposed in [19] to minimize the peak-valley difference and reduce the charging cost of EVs. A charging pricing model is proposed in [20] considering the system dynamics to not exceed the regulated price of utilities. The EV charging pricing problem is formulated as Markov's decision process in [21] and as a deep reinforcement learning problem in [22]. The fairness in allocating financial responsibility for charging EVs during peak hours is considered in [23] and an energy pricing mechanism is proposed. Similarly, a block-chain assisted charging reservation mechanism is proposed in [24] while considering secure data transactions. However, most of these studies have considered the pricing mechanisms at the system level. This can result in a second peak during the early offpeak periods [10]. In addition, the EV charging patterns of different locations could be significantly different and thus local equipment overloading cannot be mitigated by using a general pricing mechanism.
Recently, several studies have focused on local pricing mechanisms to avoid local equipment overloading. A multiclass user traffic equilibrium is considered in [25] and elastic charging demand is formulated to manage the load of different charging stations. Individualized energy consumption pattern of EVs is considered in [26] and EV charg- ing price is formulated, which is non-discriminatory. The charging cost of EV users is minimized in [27] considering the degradation of EV batteries. Finally, a pricing mechanism is proposed in [28] to reduce the daily energy cost of charging EVs while reducing the stress on the distribution transformers.
However, none of these studies have considered the individual sensitivities of EV owners to different factors such as state-of-charge (SoC) and urgency of charge. These factors determine the satisfaction of EV owners during charging EVs at a given price level. In addition, the pricing mechanism also needs to consider the satisfaction of the EV fleet operator (EFO). Similarly, the capacity of the local transformer also needs to be included in the pricing mechanism to mitigate its overloading, which is also missing in most of the existing studies. It should be noted that the sensitivity of EV owners to different factors such as sensitivity to SoC and battery degradation is private. This information reflects individual preferences and charging behavior. This information can be used to manipulate the price if revealed. Therefore, it needs to remain within the EVs. Similarly, the capacity of the transformer is only known to the EFO. Therefore, a decentralized information exchange process is required to keep the private information of each entity to itself [29].
To address the issues mentioned in the existing literature, a dynamic pricing mechanism for managing the load of EVs is proposed in this study. The proposed method considers the sensitivity of different EV owners to the SoC level and the urgency of recharge. In addition, an iterative method is developed to solve the problem in a distributed manner without revealing the private information of EVs and EFOs, to keep the privacy of each entity. The major contributions of this study are as follows.
• A comprehensive satisfaction function (also known as utility function) is developed for the EVs considering different factors, such as sensitivities of EV owners to SoC level and urgency of recharge.
• A pricing mechanism for EFO is developed by considering the net EV load and the electricity price of the upstream grid.
• A welfare maximization problem is formulated and solved in a decentralized manner by decomposing it into the EFO sub-problem and EV sub-problem. The performance of the proposed method is analyzed for different day types and results have shown that the proposed method can effectively mitigate the transformer overload. Finally, convergence analysis has shown that the proposed method can converge to the desired prices after a limited number of iterations.
The remainder of the paper is organized as follows. The proposed pricing mechanism is discussed in section II, which includes the formation of welfare functions for EVs and EFO. In section III, the proposed decentralized solution method is discussed. The performance of the proposed method is analyzed for a selected weekday and a holiday in Section IV. The convergence analysis of the proposed method is discussed in Section V. Finally, the conclusions and future research directions are presented in section VI.

II. PRICING MECHANISM A. NETWORK CONFIGURATION
The proposed network for managing the load of EVs via dynamic pricing is comprised of four main entities, EFO, distribution system operator (DSO), EV owners, and building operator (BO). The BO and EFO could be a single entity for some buildings. However, for the sake of generalization, two different entities are considered in this study. An overview of the proposed framework is shown in Fig. 1. The DSO informs the EFO about the upstream electricity prices for different intervals of the day. Similarly, the BO informs the EFO and DSO about the electrical load of the building. VOLUME 11, 2023 The EFO is responsible for managing the load of the EV fleet and for determining the dynamic price signals for different intervals of the day. The proposed dynamic pricing mechanism for the EFO determines any additional price which needs to be imposed on the EVs for charging during equipment overload intervals. The EFO determines this signal by considering the remaining capacity of the local transformer, the net demand of EVs, and the load of the building. It should be noted that the EFO only needs to know the remaining capacity of the transformer for each interval (original capacity minus the building load). Each EV interacts with the EFO and exchanges the required demand information. The EFO then updates the price signal based on the net demand of EVs and this process continues till convergences. Details about this process are presented in the next section. The following are the key assumptions considered in this study.
• The proposed model is intended for residential apartments/complexes with shared parking.
• There is an agreement between the EFO and DSO on sharing selected equipment information to manage EV loads.
• The EFO and EV owners have an agreement on the upper and lower limits of per-unit electricity prices during overload intervals.

B. SATISFACTION OF EVS
The satisfaction of EVs depends on the current SoC level and the sensitivity of different EV owners to the SoC level. The satisfaction of gaining the same power for different EVs could be different since the sensitivity of each EV owner to the SoC level is different. For example, for two EVs with the same SoC level, the satisfaction of the EV with a higher sensitivity of SoC will be higher for the same amount of power, as compared to other EVs having a lower sensitivity. In addition, the satisfaction of EVs depends on the urgency of recharge. EVs having an upcoming critical task will have higher satisfaction in gaining the same amount of power as compared to EVs having no upcoming task. For example, assume two EVs, one with an upcoming critical task and the other without any upcoming task. The satisfaction level of the EV with an upcoming critical task will be higher for gaining x-amount of power as compared to the EV having no upcoming task (for gaining the same amount of power). Therefore, these two factors are considered in this study to model the satisfaction function of EVs, similar to [29]. The satisfaction function (S n,t ) of an EV for consuming (charging) x n,t amount of power for a duration of 1-hour can be formulated as where The parameters α n and β n in (1) represent the sensitivity of the EV to the SoC level. Similarly, the parameter γ n represents the sensitivity of the EV to the urgency of recharge. In (2), B max n is the capacity of the useable battery size of the EV in kWh and SoC n,t is the SoC level of the n th EV. It can be observed that the satisfaction function is modeled as a quadratic function. This is due to the ability of quadratic functions to mimic the satisfaction level of consumers [30]. Quadratic functions are widely used in literature for modeling the satisfaction level of consumers [31].

C. PRICING MECHANISM OF EV FLEET OPERATOR
The role of EFO is to devise a pricing mechanism for managing the load of EVs. The objective of the pricing mechanism is to increase the price during potential peak load intervals to mitigate local equipment overloading. However, during off-peak price intervals, it should not impose any additional price. Therefore, a two-factor pricing mechanism is proposed in this study, similar to [29]. The proposed price signal can be modeled as where, In (3), ρ b t is the base price and it refers to the cost of electricity during normal operation intervals (without system congestion). This price is decided by the utility grid for each interval of the day. The second part (ρ p t ) refers to the additional cost due to charging during system peak hours. The parameter δ is the penalty factor and it signifies the criticality of the equipment overloading. The value δ is set in accordance with the agreement between the EFO and EV owners, as discussed under assumptions. P cap t represents the reaming power capacity, which is determined by the EFO considering the capacity of local equipment such as transformers and building load. If the net EV load is lower than the remaining capacity, then the second part of (3) will be zero and it will have a positive value otherwise.

D. WELFARE FUNCTIONS
The welfare of an EV (W n,t ) is defined as the difference between the satisfaction gained for consuming x n,t amount of energy and the price paid for the same amount of energy (x n,t ). It can be mathematically represented as It can be observed from (4) that the price of EFO is used in computing the welfare of EVs and the satisfaction (S n,t ) of the n th EV can be computed using (1).
Similarly, the welfare of the EFO (W efo t ) can be formulated as the difference between the electricity price it charges from the EVs and the buying price of electricity from the grid. Mathematically, it can be formulated as where ρ b t refers to the per-unit price of electricity, defined by the upstream utility grid. It can be observed from (5) that the welfare of EFO is summed for the entire amount of energy it has sold to the EVs.

III. PROBLEM FORMULATION
The objective of the optimization problem is to maximize the welfare of all the entities (EVs and EFO). A centralized welfare maximization problem can be formulated by using the welfare of both EVs and EFOs.

A. WELFARE MAXIMIZATION
The centralized welfare maximization problem can be modeled in (6) and (7). The optimization problem maximizes the welfare of all EVs (W n,t ) and the EFO (W efo t ). However, the individual sensitivities of EVs to SoC and the urgency of recharge are the private information of EVs and cannot be directly disclosed to the EFO. Similarly, the remaining capacity of the transformer is only known to EFO, and it cannot be disclosed to the EVs.
Therefore, dual decomposition can be used to solve the problem in a decentralized manner [32]. The Lagrangian of the optimization problem (6), (7) can be written as It should be noted that the variable λ t represents the shadow price of the equality constraint (7). Equation (8) contains information specific to both EVs and the EFO. However, these terms can be separated, and the dual problem can be decomposed into two sub-problems. The EFO sub-problem can be written as Subject to: Equation (9) is the objective function of the EFO subproblem and (10) is the constraint. Equation (10) implies that the base price of EFO needs to be regulated between a lower (ρ min t ) and an upper bound (ρ max t ). It can be observed that this sub-problem contains only information about the EFO. Similarly, the EV sub-problem can be formulated as Subject to: x min n,t ≤ x n,t ≤ x max n,t , This sub-problem only contains information specific to EVs. Similarly, equation (11) is the objective function of EV, and (12) is the constraint. Equation (12) implies that the amount of power to be charged to any EV is regulated between the upper (x min n,t ) and the lower (x max n,t ) bounds. The lower bound generally is set to zero and the upper bound refers to the maximum amount of energy required by EV n.

Algorithm 1 EFO Problem-Solving Procedure
1: Get building load for the selected day 2: Get the remaining power capacity of the transformer 3: Initialize t (t=0) and set T=24 4: for all t < T do // Interval-wise operation

9:
Update net demand of EVs (16) 10: Solve EFO sub-problem (9), (10) The proposed pricing mechanism is comprised of two subproblems, which need to be solved by each entity (EVs and EFO). An overview of the EFO-subproblem-solving procedure is shown in Algorithm 1. First, the load profile of the building is obtained from the BO and the power capacity of the transformer from the DSO. The remaining capacity of the transformer (P The proposed problem can be solved in a decentralized manner since the optimal value of the parameter λ k t can be used to achieve the solution to the primal problem [29]. By using the gradient projection method, the value λ k t can be iteratively obtained as below The value λ k t is initialized first by the EFO and its value is updated using (15). Then, demand from all EVs is gathered by EFO using Algorithm 2. The net load (x s t ) is the sum of the individual EV loads as given below The EFO uses this information and solves the EFO subproblem. This process is repeated till the convergence of the parameter λ k t . VOLUME 11, 2023 Algorithm 2 EV Problem-Solving Approach 1: Initialize # of EVs (n=1) and total # of EVs (N ) 2: for all n < N do // For each EV

3:
Receive updated convergence parameter from EFO

4:
Receive updated net EV load from EFO

8:
Inform demand of the current round to EFO Similarly, the EV sub-problem is solved by all EVs individually. The step-by-step process is shown in Algorithm 2. Each EV receives the updated value of λ k t , the net load of all EVs, and the updated price from EFO. Then, each EV solves the EV sub-problem, and updated demand is sent to the EFO. This process is repeated by each EV till the convergence of the parameter λ k t .

IV. NUMERICAL SIMULATIONS A. EVALUATION OF EV SATISFACTION
In this section, the proposed satisfaction function for EVs is evaluated considering different parameters, which could impact the satisfaction of EV owners. It can be observed from equation (1) that the satisfaction of EVs is impacted by four parameters, which include three sensitivity factors (α, β, γ ) and the SoC. The first two parameters represent the sensitivity of any EV owner to the SoC level, and the last parameter represents the sensitivity of any EV owner to the urgency of recharge. The impact of all these four parameters is analyzed in this section. Figure 2 shows that the satisfaction of any EV has an inverse relationship with the SoC level. It implies that for the same amount of power, EVs having lower SoC will have a higher satisfaction level as compared to EVs having higher SoC. This is the desired characteristic for the power consumption of EVs. In addition, it can be observed from Fig. 2 that the designed satisfaction function is concave in nature, and it converges to a certain value and the level of satisfaction for users gradually gets saturated. This is also a desired characteristic for any satisfaction function.
The sensitivity parameters α and β collectively represent the sensitivity of any EV owner to the SoC level. It can be observed from Fig. 3 that the satisfaction of the EV changes exponentially with a change in the value of parameter α. However, the value of β results in a linearized change in the satisfaction function (Fig. 4). This can also be verified from equation (1) since parameter α is the coefficient of the square term and β is the coefficient of the linear term. It should be noted that higher values of the parameter α correspond to higher sensitivity of EV owners to the SoC level and vice    versa. Contrarily, lower values of the parameter β correspond to higher sensitivity and vice versa.
Finally, it can be observed from Fig. 5 that the change in the coefficient of the urgency of recharge (γ ) also results in a linearized change in the satisfaction function. This is also in accordance with the satisfaction function (1) since γ is the coefficient of the linear part of the equation. Higher values of the parameter γ correspond to higher sensitivity of EV owners to the urgency of recharge and vice versa.

B. INPUT DATA
The daily mileage of vehicles in Korea is estimated by considering the travel data of five years [33] and approximated using a lognormal function. Both commercial and private vehicles are considered during working days and holidays for the five years. Details about the data can be found in [33]. The probability density function (PDF) and cumulative density function (CDF) of the daily mileage are shown in Fig. 6. It can be observed that 90% of vehicles travel less than 70km a day. The daily mileage information is used with the energy efficiency of electric vehicles [34] to compute the amount of energy consumed daily. The amount of energy consumed by EVs (remaining SoC) is estimated using a lognormal function as shown in Fig. 7. A residential apartment complex is considered in this study, similar to [35] and the daily arrival and departure time of vehicles is estimated for weekdays and holidays. Daily arrival times are estimated using normal distribution functions. Figure 8 shows the daily arrival time of EVs in the residential complex during weekdays and Fig. 9 shows the arrival time during holidays. It can be observed that the arrival time during holidays has more spread as compared to weekdays. An EV fleet of 150 EVs is considered in this study. The number of EVs arriving home during different hours of the day (for weekdays and holidays) is shown in Fig. 10. It can be observed that the sum of all EVs (arriving during different hours of the day) is 150.
The per-unit load of the Korean power system is shown in Fig. 11 for a working day and a holiday. The capacity of the distribution transformer is taken as 200kVA and the remaining capacity of the transformer during different intervals of the day is shown in Fig. 12. It can be observed that the remaining capacity of the transformer is lower during afternoon and evening intervals due to higher building load during those intervals.

C. PERFORMANCE ANALYSIS
In this section, the performance of the proposed method is analyzed for two selected days of the year. Due to the difference in energy consumption and EV traveling behavior, a working day and a holiday are selected in this study. The price signals obtained by using the proposed method for the selected days are shown in Fig. 13. It should be noted that the proposed pricing mechanism provides the value of an additional price to reduce the EV load during system overload intervals. It can be observed from Fig. 13 that during off-peak load intervals, the pricing mechanism suggests no additional price (zero) for EV charging. Contrarily, an additional price is suggested during peak load intervals for both weekdays and holidays. Detailed analyses of the selected weekday and holidays are presented in the subsequent sections.

1) SELECTED WEEKDAY
The original load, optimized load, and the remaining capacity of the transformer for the selected weekday are shown in Fig. 14. Original load refers to the total load of the EVs, and   optimized load refers to the amount of EV load after applying the proposed pricing mechanism. Remaining capacity refers to the amount of capacity remaining in the transformer after serving the building load.
It can be observed from Fig. 14 that the original load surpasses the remaining capacity of the transformer during evening intervals. This is due to the consideration of a residential building in this study. It can be observed from Fig. 13 that the price corresponds to the severity of the transformer overloading. For example, during initial overload intervals (1-3 pm), the price is lower, and it hits the maximum value during interval 17 due to the highest EV load and maximum overloading during this interval.

2) SELECTED HOLIDAY
Similar to the previous case, the original load, optimized load, and the remaining capacity of the transformer during the selected holiday are shown in Fig. 15. It can be observed from Fig. 15 that the proposed method has successfully mitigated the overloading of the transformer in this case also. It can be observed from Fig. 13 that the price signals correspond to the    level of overloading of the transformer, similar to the previous case. It should be noted that the price signals are different for weekdays and holidays, i.e., dynamic price signals.
It can be concluded from the analysis in this section that the proposed method can mitigate the overloading of transformers via dynamic price signals. The proposed method has successfully mitigated the overloading issue for both weekdays and holidays.

V. DISCUSSION AND ANALYSIS
It has been mentioned in the problem formulation section that the proposed method is iterative. Therefore, in this section, the convergence analysis of the proposed method is presented. In addition, the impact of different sensitivity parameters and SoC levels is analyzed in detail by selecting one interval from the weekday and one from the holiday. The interval in each case is selected by considering the following factors.
• Intervals with transformer overload (intervals 13 to 21 for weekdays and 15 to 23 for holidays).
• Intervals having enough EVs to demonstrate the impact of different parameters (intervals 15 to 19 for both weekdays and holidays). For the sake of visualization, interval 17 (between 15 to 19) is considered for detailed analysis.

A. SELECTED WEEKDAY CONVERGENCE ANALYSIS
In this section, convergence analysis of the proposed method for weekdays is analyzed. Especially, interval 17 is selected and 12 EVs are present in the parking lot during this interval. The sensitivity parameters and SoC level of each EV are randomly generated within specified bounds and results are summarized in Table 1. The net EV load was 154kW and the transformer's remaining capacity was 80kW in this case.
It can be observed from Fig. 16 that the price has converged to 18.29KRW/kWh after a few iterations. Similarly, the capacity has also converged to 80kW, and it is the remaining capacity for that interval (Fig. 12). Similarly, Table 1 shows that demand for EVs having higher SoC and lower sensitivity reduces their charging demand when the price is higher. For example, EV 4 has reduced its demand to zero due to higher SoC and lower sensitivity values. Similarly, EVs 9-12 have also reduced to zero due to higher SoC and lower sensitivity to SoC and urgency of recharge. Contrarily, EVs 1,2, 5, and 6 have charged their full demand due to lower SoC and higher sensitivity to SoC and urgency of recharge.

B. SELECTED HOLIDAY CONVERGENCE ANALYSIS
A similar analysis is conducted for the selected holiday. Especially, interval 17 is selected for detailed analysis and there were 12 EVs in the parking lot during this interval (Fig. 10). The sensitivity parameters are randomly generated for each EV and are listed in Table 2.
It can be observed from Fig. 17 that the proposed pricing mechanism converges to a price of 8.48KRW/kWh and the capacity also converges to 88kW after a few iterations. The results in Table 2 show that only EVs 3 and 4 have reduced sensitivities. The reduced number of EVs with zero optimal demand is due to the lesser difference between the original load (144kW) and the available capacity (88kW) in this case.
It can be concluded from this analysis that the proposed method can converge to the desired price level within a limited number of iterations. Similarly, the proposed method   can mitigate transformer overload by managing EV load considering the sensitivities of EVs to different factors.

VI. CONCLUSION
A dynamic pricing model for managing the load of electric vehicles, in a decentralized manner, during peak load hours is proposed. The performance of the proposed mechanism is evaluated for different day types, such as working days and holidays. Simulation results have shown that the proposed method can effectively mitigate transformer overload by managing the load of electric vehicles locally. The proposed method determines the penalty price dynamically in accordance with the severity of the transformer overload. By using the proposed pricing mechanism, the remaining capacity limit of the transformer is always respected. In addition, the convergence analysis has shown that the proposed method can converge to the desired prices after a limited number of iterations. Finally, detailed analysis has shown that the proposed method can successfully manage the load of electric vehicles considering their sensitivities to different factors. These factors include state-of-charge level and urgency of recharge.
The proposed pricing model can be used as an interim strategy to mitigate transformer overload. In the long run, either utility need to upgrade their equipment or fleet operator can employ vehicle-to-vehicle services to manage electric vehicles' load locally. In addition, a mechanism for sharing information among distribution network operators and electric vehicle fleet operators is required, which is not discussed in this paper.