Distributed Auction-Based Incentive Mechanism for Energy Trading Between Electric Vehicles and Mobile Charging Stations

With the increasing number of electric vehicles, deploying fixed charging stations (FCSs) has been a widely adopted solution for providing charging services to EVs. However, the charging requirement of EVs being near overload FCSs and/or in areas of inadequate charging infrastructures such as highways and rural areas will surpass the capabilities of FCSs. To address this challenge, the realized peer-to-peer (P2P) energy trading between electric vehicles (EVs) and mobile charging stations (MCSs) can be utilized to relieve the overload on FCSs, leverage under-utilized energy resources among cities and achieve trading benefits. By deploying this energy trading model, EVs purchase available energy from MCSs instead of FCSs to fulfill their charging demands. However, such an energy trade requires an incentive mechanism to ensure fair trading and prevent personal gain, which motivates us to study the incentive mechanism design for the energy trading model. In this paper, first, we consider an energy trading system involving multiple MCSs and EVs. Then, we formulate the incentive mechanism between MCSs and EVs as an auction game, in which the MCSs are auctioneers and EVs are bidders. In the formulated problem, each EV secretly submits its bid to MCSs, and each MCS distributively decides the winners without the knowledge of other MCSs. To achieve this, we design a distributed action-based energy trading mechanism that ensures fairness of providing charging service, validity in trading price’s resource determination, and nothing incentivizing the EVs to cheat on payment decisions. The proposed energy trading scheme achieves certain critical properties, including truthfulness, individual rationality, budget balance, and computational efficiency for both buyers and sellers. Finally, we perform simulation experiments to verify the proposed scheme’s effectiveness compared with the baseline in the literature.


I. INTRODUCTION
The proliferation of renewable energy technologies has played an essential role in developed countries due to an increasing trend in green energy and economics. Moreover, The associate editor coordinating the review of this manuscript and approving it for publication was Chandan Kumar . electric vehicles (EVs), with the number expected to grow to 125 million by 2030 [1], have been seen as cutting-edge technology to reduce air-polluting emissions [2]. Therefore, with the increasing awareness of climate change, the number of EVs will increase significantly in the foreseeable future. Although the utilization of EVs provides significant advantages to eco-friendly life, it also comes with constraints in terms of energy. EVs are limited by the capacity of batteries, which can be best depicted with the range anxiety problem, worry that EVs will not have sufficient energy to reach their desired destinations.
One approach to mitigate the limitations is to charge periodically at the charging spots. Innovative charging strategies are one solution to effectively utilize charging spots, which has become an open challenge for researchers interested in energy trading between EVs and charging spots. A few methods are considered as viable strategies to improve charging infrastructures producing energy for EV users, such as fixed charging stations (FCSs), mobile charging stations (MCSs), and battery swapping stations (BSSs).
FCS is proposed to provide a charging service to EVs via multiple charging options [3]- [5]. For example, Alternating Current (AC) level 2 and Direct Current Quick Charging (DCQC) are the two most common charging options [5]. EVs using level 2 for charging spend 4 to 6 hours to completely fulfill a depleted battery, while it takes 20 to 40 minutes using DCQC. There are three main disadvantages of FCS: 1) As the number of EV users increases, the waiting time for charging at FCS increases; 2) Simultaneous charging of numerous EVs results in a negative impact on the grid, e.g., voltage fluctuations, increase in the likelihood of blackouts, and decrease in system efficiency [6]; 3) Both driving routes and the convenience of charging of EVs users, which are essential concerns of EVs user, can be affected.
On the other hand, MCSs have been identified as a beneficial and convenient approach to supplying better charging service to EV owners compared with FCSs [7], [8]. Similar to FCSs, MCSs can provide faster-charging service to a few EVs simultaneously; however, they can be moved from one place to another place requiring high demand of charging, e.g., Volkswagen [7]. Therefore, MCSs have emerged as a key solution for charging strategies. The reasons are as follows.
• MCSs are beneficial not only to EVs but also to FCSs and the power grid. MCSs provide faster-charging services for EVs by reducing charging and waiting times. Equipped with fast charging poles [9], [10], MCSs can provide charging services with an average charging time of around 17 minutes [7]. MCSs can be transferred to areas with high charging demand to reduce charging time, thus relieving FCS's stress [9] and power grids' workloads, allowing the power grid to maintain voltage steadily [11]. Due to the mobility, MCSs can be distributed to different areas, then EVs will be navigated to MCSs instead of to the high demand FCSs.
• MCSs play a vital role in harvesting and utilizing energy efficiently. MCSs can economically harvest renewable energy from renewable resources, e.g., wind and solar. They also can arrange to re-charge at a specific time, e.g., the low peak demand [12]- [14]. Inspired by the above observations, many research efforts have been devoted to providing EVs with charging services by MCSs [5], [8]- [10], [13]- [20]. Studies conducted in [9], [13], [15]- [18] focus on finding the optimal number of MCSs to cover EVs' charging requests and minimizing the expected energy costs to mitigate challenges and cost burdens due to building charging infrastructures. The studies in [9], [15] optimize the number of MCSs and charging piles at each MCS to reduce the service delay. In [16], the authors propose a nonlinear flow-refueling location model to optimize MCS placements. Similarly, the study in [17] develops an algorithm to determine the optimal placements of MCSs concerning the charging demand and maintenance cost. In addition, the authors a novel heterogeneous network model to improve the communications between EVs and MCSs by using macro cells and small cells. In [18], mobile charging station architecture is studied to charge EVs at designated areas such as urban and resorts. In [13], an optimal dynamic programming (DP) solution is developed to minimize EVs' waiting times.
Nevertheless, several critical issues such as MCSs' price strategy, maximizing profits while achieving a high reputation in service quality, and EV satisfaction have not been properly studied in most of the current research works. MCSs will not sacrifice their resources to provide charging services to EVs. Furthermore, MCSs are equipped with limited energy storage and several charging ports. In addition, as MCSs move around the cities to provide charging services, getting charged at MCSs can help EVs reduce the distance to charging providers, thus reducing the amount of energy spent. So naturally, the requiring less energy to approach MCSs, the increasing EVs wanting to get charged at MCSs rapidly increases, and thereby EVs have to compete with others to get charging service at MCSs. Therefore, an optimal decision on energy supply for EVs needs to be researched to utilize the limited power resource of MCSs efficiently and fairly, considering the mobility of charging providers. This problem is called an energy trading strategy between EVs and MCSs. The study in [5] provides a Lyapunov-based optimization algorithm to determine the optimal power management strategy, which can maximize the long-term profits of MCSs. However, this work only considers the power supply and dynamic of EV users' arrival. For charging prices, the authors assume that it is fixed and given value, which is inappropriate for the natural energy trading problems where the prices are changeable. In [14], the authors propose offline heuristic algorithms for assigning MCSs to charge EVs. In this study, there is no incentive mechanism that can encourage MCSs to provide their surplus energy to EVs. Without practical incentive mechanisms, MCSs may be unwilling to offer energy. Therefore, designing an effective incentive is a critical challenge for energy trading between EVs and MCSs. In literature, auctions mechanisms [8], [10], and blockchain techniques [19], [20] are promising approaches to solve energy trading problems. However, in these proposals, the authors model the energy trading problems as one-to-one trading markets where one seller can be matched with at most one buyer and vice versa. It is unsuitable for a situation when sellers have multiple available resources that can simultaneously provide various buyers. That scenario is known as many-to-one trading markets where one seller can be matched with multiple buyers. The energy trading problem between EVs and MCSs belongs to the many-to-one trading markets since each MCSs can simultaneously provide charging services to multiple EVs.
Motivated by the challenges mentioned above, in this paper, we design a distributed auction-based incentive mechanism to enable energy trading between EVs and MCSs. Through the proposed model, the efficiency of the energy trading between MCS and EVs is improved by selecting appropriate MCSs for EVs satisfying the limited energy resource of MCS. Our contributions are summarized below.
• We propose an energy trading system model where EVs ask MCSs for charging services without the knowledge of other EVs. Then, we formulate an energy trading problem enabling a many-to-one resource trading market, where each EV gets the required energy from only one MCS while MCSs is able to serve multiple EVs depending on their resource capacities.
• We propose a distributed action-based incentive scheme for energy trading between EVs and MCSs where EVs act as buyers and MCSs act as sellers. The action process can be performed round-by-round to recruit EVs to join the energy trading. Through our proposed mechanism, we can solve the energy trading problem in a distributed manner.
• We disclose how the distributed auction-based energy trading between EVs and MCS works and prove the properties of the proposed incentive mechanism.
• We conduct the numerical simulations and obtain results to the effectiveness of our analysis and the proposed method. The performance of our proposal outperforms the baseline, TIM, in literature [21]. The remainder of this paper is organized as follows. In Section II, we briefly review some related works. Section III introduces the system models, analyzes demander and supplier models for energy trading between EV users and MCSs, and formulates the energy trading problem as an auction game. In Section IV, we propose an algorithm to determine winners and payment of the auction problem and prove the properties of the proposed mechanism. The experimental results are given in Section V. Finally, we make some brief concluding remarks in Section VI.

II. RELATED WORK
When a numerous number of EVs require charging is a major challenge for both scientists and companies. The goal is to reduce the impact on the power grid by solving the charging problem [3], [22]- [28]. One such approach is MCSs, known as a new type of EV charging equipment that offers charge service at any location and time requested [8], [29]. One main advantage of MCSs is that they can harvest idle energy from the grid efficiently by being charged from substations during off-peak hours or renewable resources [8], [10]. The reason is that the energy of MCSs can be supplied from traditional power and renewable power. Moreover, a Lyapunov-based online distributed algorithm is used to maximize the MCSs' profit by finding out the optimal control power variable of MCSs and optimal EVs' demand for EVs under the fixed energy price and the fluctuation of renewable power [5]. However, the fixed price-based charging services are no longer suitable for the present context of the energy trading system.
Recently, most resource trading systems have been operated as ecosystems (i.e., multi-access edge computing [30], caching [31], HetNet [32]), wherein users have to compete with each other to get resources. Therefore, a new market model that enables energy trading among EVs and MCSs is necessary to design an improve system efficiency to reduce system operation costs. A novel peer-to-peer (P2P) energy trading system among EVs is an energy trading arrangement helping to reduce peak demand, lowering reserve requirements, and curtailing network loss in the distribution power system [33]- [35]. There are two layers in the P2P network: the virtual layer and the physical layer. The virtual layer provides a secured connection for participants to decide on their energy trading parameters, while the physical layer facilitates the electricity transfer from sellers to buyers once the financial settlements in the virtual platform are completed [36]. The existing approaches for the P2P energy trading schemes can be classified into four general techniques: 1) game theory [37], [38], 2) auction theory [39], [40], 3) constrained optimization [41], [42], and 4) blockchain [43], [44]. However, there are open challenges in the P2P energy trading system that requires more investigation. One challenge is that the system should be scalable to handle a large number of EVs in modeling transactions in both the virtual and the physical layers of the network [36].
As a promising approach, auction mechanisms in energy trading markets have been studied in the literature [19], [20], [45], [46]. In [19], a blockchain-based double auction is proposed for peer-to-peer energy trading. In [45], an auction mechanism that allows participants to keep their own information locally is proposed. The main purpose of the energy trading in both [19] and [45] is to maximize the benefit of buyer and seller. However, the factor of local electricity balancing is not taken into account. The work in [46] proposes not only an auction mechanism for energy trading but also an unsupervised learning-based clustering algorithm for balancing the distribution of energy in each local energy market. Similar to [19], [20] also adopts the block-chain technique for solving energy trading problems. However, the work in [20] takes balanced characteristics into account to balance the electricity demand and supply of each local energy market. Despite the ability to deal with a set of sellers and buyers, the auction mechanisms mentioned above are confined to one-to-one resource trading markets; and thus cannot satisfy the resource requirements of many-to-one resource trading markets. Meanwhile, our work considers the many-to-one resource trading markets, where one seller can provide the energy to several buyers in a practical energy trading system as energy trading between EVs and MCSs.  EVs are moving on roads and on low battery, and thus they need charging services. Moreover, our system model is focused on an environment in which EVs are unable to get charges at FCSs due to various reasons, such as FCSs are facing overwhelming charging requests; EVs are on highways or rural areas, but they are insufficient battery to reach FCSs. From this standpoint, MCSs are proposed to be an alternative solution for providing charging services to these EVs. They are distributed in regions requiring high charging demand and/or no nearby FCSs to help EVs overcome the range anxiety. MCSs are equipped with limited energy storage and some onboard charging ports. Therefore, EVs have to compete with each other to get charging at MCSs. Under the proposed model, we examine the supplying charging requests to EVs by MCSs as an energy trading process between demanders -EVs and suppliers -MCSs. Let d i ∈ D, and s j ∈ M represent a demander i and a supplier j, respectively. From now on, we use d i and i; s j and j interchangeably. In addition, we assume that there is no centralized control entity in this studied charging network. Therefore, EVs and MCSs have to perform energy trading in a decentralized manner. Table 1 summarizes the main notations used throughout the paper. Without loss of generality, we assume that time is slotted, and we study the system for one time period.

B. MODEL OF DEMANDERS
For each EV that belongs to the set of N demanders D, we assume EV's required energy amount differs among EVs. Let denote the energy amount required by demander d i as e i . Each EV owner has its own financial ability, and thus it asks MCSs to supply them with charging services by sending their bidding price. The bidding price is the highest price willing to pay for one unit of energy supplied by MCSs. The bidding price profile of an EV d i is given by where b ij represents the highest price a demander d i is willing to pay for MCS s j . In such a competitive energy market, these bidding prices are utilized to finalize winners. Meanwhile, when a buyer wins an energy auction, it has to contribute a real final payment to obtain an agreed purchased energy amount. Let p d i represent the final buying price per unit acquired energy of buyer d i . In addition, each EV evaluates profit from getting charged at a MCS. This profit is often related to several factors such as the distance between EVs and MCSs, charging quality, and the charging rate. We Since it is assumed all MCSs have the same charging rate, we generally model v ij as a function of the expense of EV d i for approaching a MCS s j : v ij = ( ij , ψ i ), where ij is the distance between EV d i and MCS s j ; ψ i is the energy consumed by EV per unit driving range (for example, one kilometer). Both the buying price and the valuation price are the price of one unit of energy. For a convenient trip, a demander EV tends to make the energy trading with a MCS that can let it reach its destination as soon as possible. However, the calculation of the bid valuation of EVs is out of scope in our paper.

C. MODEL OF SUPPLIERS
For each MCS belongs to the set of S suppliers S, they are able to sell energy to demanders EVs with different limited resources, including the number of chargers (known as quota) and the sellable energy amount. We denote the number of available chargers owned by MCS s j as c j , and the maximum available energy capacity of the supplier MCS s j as E j .
Moreover, due to unstable charging demand, a MCS will be relocated to other places requiring a high request for charging service. This dynamic mobility of MCSs will significantly impact the quality of the charging service as charging completion time. Therefore, before starting energy trading with demanders, each supplier needs to estimate its remaining time being in the current location to avoid interrupting the charging service. We use , presents the least staying time of supplier s j before it moves to another location. We assume c j , E j , and T j are constant in the one slot of interest, and this local information is disclosed to EVs at the beginning of each trading period. Given the above information, a supplier MCS will either choose to participate or reject the energy trading request of EVs. In the case of MCS admitting to supplying energy to EVs, it independently sets a price per unit of energy that it can get from trading energy service paid by EVs named as a seller's price.
It is worth recalling that there are three price variants involved in the energy trading process: The bidding price of buyer EV d i submits to a seller s j to express the highest payment it can pay for one traded energy unit. • p d i : The final buying price paid by buyer i as it becomes the eventual winner of the energy trading process.
• p t j : The seller's price is informed by seller s j as it grants energy to EVs.

D. PROBLEM FORMULATION
In this section, we model our energy trading problem using the auction framework. In this trading market, EVs acting as bidders (also known as buyers) initialize the trading process by submitting their bid to MCSs in secret, e.g., no buyer has any knowledge of others, while MCSs acting as auctioneers (also known as sellers) separately judge who should they trade with based on the submitted bids. The objective is to provide matched buyer-seller pairs so that both buyers and sellers satisfy their desirable subjects.
First, we define the energy trading assignment matrix as X = x i,j N ×M , where the binary variable x ij indicates whether EVs d i is assigned to MCSs s j or not.
The energy trading assignment matrix must satisfy the following constraints: i) Each EV can finally trade energy with only one MCS expressed through the equation below, ii) The total number of EVs served by a MCS can only up to its maximum available chargers shown as follows, iii) Each MCS can only sell its available energy resource up to its maximum available energy resource represented as follows, Whenever MCS s j satisfies the equation (4), we say the MCS s j in supply surplus. Otherwise, we say MCS s j in demanded surplus if the energy demand for MCS s j exceeds the supply of MCS s j Next, we define the utilities of a buyer EV d i ∈ D, denoted as U d i , as the difference between the profit and the final buying price is evaluated as follows: Similarly, we define the utilities of a seller MCS s j ∈ S, denoted as U s j , as the total payment acquired providing charging service to EVs in energy trading, calculated as following:

IV. THE DISTRIBUTED AUCTION-BASED INCENTIVE MECHANISM FOR ENERGY TRADING
In this section, we propose a distributed auction scheme to deal with energy trading between EVs and MCSs. The proposed mechanism can solve the energy trading problem in a distributed manner and achieve optimal solutions.

A. FRAMEWORK OF THE ENERGY TRADING MECHANISM
In this paper, we contemplate the energy trading between EVs and MCSs as a two-sided market. MCSs act as auctioneers who sell their surplus energy to EVs, while EVs act as bidders who buy energy from MCSs. Our energy trading mechanism aims to maximize the utility of both MCSs and EVs based on auction theory. The auction theory is an efficient method for solving a matching problem by allowing market competition. We now present the framework of our proposed distributed auction-based incentive mechanism for energy trading between EVs and MCS, wherein the energy trading process is continuously exploited round-by-round. The detailed energy trading process of each round is shown in Fig. 2, which consists of five steps as followings: 1) Initially, each buyer requests a charging service by broadcasting their bids, including the requested demand and bidding, to the MCSs in the proximity. 2) Each seller locally processes a winning bid determination task. Specifically, the seller gathers the bids and then forms its winning buyer candidates. In addition, the seller also determines the corresponding seller's prices for its candidates. 3) Sellers inform their auction decision to their local winning buyers and the corresponding required payments. 4) Each buyer locally handles a final seller determination.
Specifically, each buyer gathers the sellers' decisions.
Since the sellers determine their buyers' candidates in a distributed manner, one buyer can be chosen by more than one seller; however, at a specific time, one buyer can only get the charging service at one seller. Therefore, each buyer evaluates seller candidates on obtained utilities to elect the best seller as its actual final seller to participate in the trading. 5) Finally, each winner buyer -the corresponding selected MCS pair is settled, and the energy trading transaction of each pair is carried out. At a specific action round detailed above, the seller MCSs provide their available to energy matched buyer EVs. Naturally, after the trade, there exist sellers who still have sufficient energy; thereby, they can invite unmatched buyer EVs to re-bid. By doing that, a new round is initiated. Consequently, the aforementioned one-round action is operated round-byround and terminated as if no new matched buyer-seller pair is established or no MCS has sufficient energy to process their new claims.

B. PROCESS OF ENERGY TRADING MECHANISM IN EACH ROUND
In this subsection, we outline the detailed energy trading process, which is divided into three main tasks.

1) BIDDING SUBMISSION
EVs execute a bidding submission phase by submitting their bidding strategy. The strategy of EV i is presented as a tuple B i = (b ij , e i , t i ). Recall that b ij is the bidding price at which EV i is willing to pay for one energy unit offered by j; e i is the required energy amount of EV i; t i is the minimum required charging time of EV i.

2) WINNING BID DETERMINATION
This stage is made locally at each seller MCS with the goal is to determine the set of winning bids and the seller's price for each seller. We define an auctioned indicator, x temp ij ∈ {0, 1} to capture whether a seller s j selects a buyer d i as its winning buyer candidate or not. Initially, we set x temp ij = 0, and p t j = 0, ∀d i ∈ D, and ∀s j ∈ S. If the buyer d i is accepted as the winning buyer candidate of seller s j , then x temp ij = 1. Moreover, when the buyer d i temporarily wins its bid at the seller s j , then the buyer d i pays the auctioned buying price, p d i for per energy unit getting from EV s j . The winning bid determination is conducted in the following steps: Step 1) Determine the set of feasible buyers: Each seller s j collects the bidding submission of all buyers. Only buyers who fulfill energy trading constraints will participate in the auction process. Each seller discovers its own satisfied buyers to form its own set of feasible buyers. Let W temp j be the set of feasible buyers of seller s j and be computed as follows, Step 2) Sort the feasible buyers in non-increasing order of the buyer bid value: Each MCS s j arranges all buyers d i ∈ W temp j in non-increasing order of their bids to prioritize the needs of high-bid buyers. Considering a seller s j with K = W temp j bidders, the re-sort of W temp j is noted as W order j as follows: Step 3) Determine the set of winning buyer candidates and the seller's price: Since each MCS s j has limited energy and charger resources, it cannot guarantee that it would serve all buyers in the re-sort of feasible buyer set, W order j .
Regarding to the charger resources constrains, each MCS s j only selects up to |c j | bidders having the highest bids (i.e., W cons j ) depicted as follows: Regarding to the providable energy resource constrains, each MCS s j computes the total energy demand of all EV d i ∈ W cons j , and then determines the set of candidate of winner denoted by W can j as follows: otherwise.

26:
27: i) Case 1: the seller s j is in supply surplus, The auctioneer, seller, s j temporarily sets: • The set of winning buyer candidates: W can j = W cons j .
• The auctioned indicator: ii) Case 2: the seller j is in demanded surplus, In this case, the auctioneer j determines: • The winning bid determination is detailed in Algorithm 1, line 4-12.

3) FINAL SELLER DETERMINATION
According to the winning bid determination stage, each seller j selects the set of its winning buyer candidates in a distributed way. As a result, one buyer can be selected by many sellers violating the constraint in (2)   ii) Case 2: There is one seller candidate for buyer d i , This mean the buyer d i is selected by only one buyer s j , and thus we set Furthermore, we set the final buying price iii) Case 3: There is more than one seller candidates for buyer d i , This means the buyer d i is selected by many sellers, MCSs. Therefore, the buyer d i seeks the one among all the seller candidates contributing the largest utility to it and elects that seller as its final seller. Let U d ij be the utility gained by EV i if it complies with the energy trade of MCS j, which is computed as bellows: Then, the EV i figures the best seller for it as follows: We then set Moreover, we set the final buying price p d i = p t j * . The detail of the final seller determination is presented in Algorithm 1, line 13-27.

C. A WALK-THROUGH EXAMPLE
In this subsection, we give an example to illustrate clearly how the distributed auction-based energy trading between EVs and MCSs works. In this example, there are 4 MCSs, S = {s 1 , s 2 , s 3 , s 4 }, and 5 EVs, Corresponding the biddings submitted by each EV are given in Table 2. The available energies at MCSs and the request energies of EVs are shown in Table 3 and Table 4, respectively. To ease presentation, we assume a constraint T j ≥ t i is satisfied for all d i ∈ D and s j ∈ S. In other words, the immovable time of each MCS is longer than or equal to the minimum required charging time of all EVs. We also assume that the quota of each MCS is c j = 3.

1) WINNING BID DETERMINATION
Initially, x i,j = 0, p t j = 0.p d i = 0, ∀i, j. Each MCS decides the candidate winners locally. In particular, we have: Because it has c j = 3 chargers and available energy at s 1 is 38kWh, W can

D. PROOF OF PROPERTIES
In this subsection, we tend to present the main properties of the proposed incentive mechanism and prove the following properties, including budget balance, individual rationality, computational efficiency, and truthfulness thereafter.
1) Budget balance: The proposed auction scheme meets the budget balance when there is no seller who loses money during the energy trading. In other words, the total payment the MCSs charges the EVs equals the total price the EVs pays the MCSs, i.e., M i=1 x ij e i p t j = M i=1 x ij e i p d i . Theorem 1: The distributed auction-based energy trading between EVs and MCSs is budget balance.
The proof is provided in Appendix A.
2) Individual rationality: The proposed auction scheme is individual rationality for both buyers and sellers if no one gets a non-negative payoff if it joints the auction. In other words, for each buyer EV d i , the final buying price p d i charged by seller MCS s j is lower than its bids at MCS s j , b ij , i.e.,

Theorem 2: The distributed auction-based energy trading between EVs and MCSs is individual rationality for the buyers and sellers.
The proof is provided in Appendix B.
3) Computational efficiency: The proposed auction scheme is computational efficiency if it can be executed within polynomial time.
Theorem 3: The distributed auction-based energy trading between EVs and MCSs is computationally efficient.
The proof is provided in Appendix C. 4) Truthfulness: The distributed auction-based energy trading between EVs and MCSs is truthful when each buyer EV i cannot increase its utility by submitting an untruthful bid b i .
Theorem 4: The distributed auction-based energy trading between EVs and MCSs is truthful.
The proof is provided in Appendix D.
With Theorems 1 to 4, we have Theorem 5 as follows. Theorem 5: The distributed auction-based energy trading between EVs and MCSs is budget balance, individually rational, computationally efficient, and truthful.

V. SIMULATION
In this section, we conduct a numerical analysis to validate our distributed auction-based incentive mechanism. For our experiment, we randomly set the required charging energy in a range from 10kW to 45kW [47]. We assume the bid value of each EV follows the uniform distribution within [0.10$/kWh, 0.23$/kWh] [48]. We also generate the available energy of MCSs by randomly choosing from [252 kWh, 360 kWh]. We assume that the least staying time of all MCSs is less than the duration of one round. In this simulation, we use Python 3.8 to run our proposed scheme. In the following subsections, first, we investigate the critical properties of our proposal by varying various parameters to justify its results. Finally, we evaluate the performance of our proposed incentive mechanism.

A. PROPERTIES OF THE PROPOSED MODEL
In this subsection, we verify our proposed energy trading algorithm's properties via simulations as seen in the proofs in Section IV-D.

1) INDIVIDUAL RATIONALITY
In this simulation, we consider a scenario that includes 50 EVs and 10 MCSs. We choose 20 winning EVs to verify EVs' individual rationality among the winning EVs. In Fig. 3, we compare the bid submitted by each EV and the final payment of EVs to the corresponding MCSs. Clearly, from Fig. 3, we can see that the final payment paid by each EV is no greater than the bid submitted by this EV. It means the proposed scheme is individually rational.
Furthermore, we validate the individual rationality by varying different numbers of EVs, increasing from 50 to 100. From Fig 4, we can observe that the sum of bid values of winning EVs is constantly larger than the total payment these winning EVs need to pay for MCSs. In other words, all buyers, EVs, get non-negative utility which is evident that the proposed incentive mechanism guarantees the individual rationality. Moreover, the result shows that as the number of EVs increases, the sum of EVs' bidding values and the VOLUME 10, 2022  total payments of EVs to MCS increase. The main reason for this increase is that as the number of EVs grows, that is synonymous with the rising number of bidders, thus, resulting in the sum of bidding value increments. Similarly, as the more bidders EVs participate in the energy trading process, the more energy MCSs can sell, which leads to the growth of the total EVs' payment.

2) TRUTHFULNESS
To verify the truthfulness of our proposal, we examine two scenarios in the simulations illustrated in Figs. 5 and Figs. 6. Here, the x-axis shows the submitted bid ratio, which is observed as the proportion of the submitted bid b ij to the truthful valuation v ij . In the first scenario shown in Fig. 5, we explore the utility of EV d i on a specific MCS s j with respect to the varying value of the submitted bid ratio. Fig. 5a shows a case when the 1st EV wins with its truthful bid at the 2nd MCS, and it is unable to improve its utility at other untruthful bids. Similarly, Fig. 5b shows a case when the 1st EV sends its truthful to the 3rd MCS but loses the auction. In this case, the 1st EV is unable to enhance its utility no matter what bid values it submits. Therefore, from the above results illustrated in Fig. 5a and Fig. 5b, we can see that for both cases, winning or losing bids, the EV cannot improve its utility if it submits untruthful bids. This result again asserts that no buyer can improve profits by submitting dishonest bids. Furthermore, this rigging may lead to a decrease in buyers' utility. In the second scenario demonstrated in Fig. 6, we inspect the EV's utility, the 2nd EV, obtained at all the MCSs in different ratios of submitted bid. From Fig. 6, we can see that the EV only reaches a maximum utility if and only its placed bid is truthfulness, e.g., the ratio = 1. Besides, when it submits an untruthful bid smaller than the truthful valuation, the ratio < 1, it still has a chance to improve its utility by raising its submitted bid; However, when it submits untruthful bids exceeding the truthful valuation, the ratio > 1, the larger bid it places, the smaller utility it gets. Results illustrated in in Figs. 5 and Figs. 6 are consistent with the analysis of Algorithm 1, confirming the truthfulness property of our proposed mechanism.
Moreover, we investigate the truthfulness aspect of the auction under the assumption that specified EVs will behave in a dishonest manner. Specifically, in Fig. 7, we present the changing of total utilities of the 1st EV achieved at all MCSs when the 1st EV and the 2nd EV vary their submitted bids to demonstrate the impact of the 2nd EV's dishonest actions on the 1st EV. The x-axis shows the submitted bid ratio of the 1st EV. For the 2nd EV, we choose three submitted bid ratios, i.e., 0.5, 1, and 1.5, shown as in the legend. As depicted in Fig. 7, the 1st EV consistently achieves the highest total  utility when it honestly offers truthful bids disregarding the untruthful behavior of the 2nd EV. This shows that if one buyer wants to maximize its utility, it simply submits honest bid values. The dishonest behavior of other buyers cannot affect its trading process. In summary, the proposed scheme ensures truthfulness.

3) BUDGET BALANCE
We validate the budget balance property by varying the parameter of charge demand of EVs, which is randomly chosen in the range [10kW, A kW], where A goes from 35 to 75. As depicted in Fig. 8, MCSs receive a payment equal to the amount that EVs give to MCSs under varying values of charging demand. In other words, our proposed auction scheme achieves the budget balance property under various sizes of total EVs' charge demand. Besides, Fig. 8 indicates that the payment of EVs and the receiving of MCS increase when the maximal charging demand of EVs increases. The main reason for this growth is that as charge demand rises, the overall quantity of energy that MCSs supply to EVs becomes increases, thus, resulting in more money MCSs can get.
Furthermore, we evaluate the budget balance with respect to different submitted bid ratios. Fig. 9 illustrates the overall  payment of EVs and the receiving of MCSs of the 1st EV under different submitted bid ratios. We can see that the total payment of EVs and receiving of MCSs only get the maximum values when the submitted bid ratio is equal to 1, indicating that this EV provides a true bid value, as shown in Fig. 9. This result is compatible with the one presented in Section V-A2. Furthermore, the results in Fig. 9 reflect that the total payment of EVs and receiving of MCSs are always the same regardless of submitted bid ratios. It can be inferred that the proposed mechanism assures the budget balance property even when EVs engage in dishonest behavior.

B. SYSTEM PERFORMANCE
In this subsection, we provide experimental results to examine the system performance in terms of the number of EVs that are satisfied with their energy demand. Specifically, the more satisfied EVs the system obtains, the better performance the system expresses.

1) IN ONE-ROUND SETTING
In this setting, the proposed approach is compared with two baseline algorithms: TIM (Truthful Incentive Mechanism) is VOLUME 10, 2022 proposed in [21] and optimal energy trading scheme (OPT), which are introduced as follows: • TIM: This scheme aims to match one seller to at the most one buyer such that the result satisfies the individual rationality, the budget balance, and the truthfulness of the bidders.
• Optimal energy trading scheme: This scheme aims to produce an optimal assignment between EVs and the MCSs, which is subjected to maximizing the number of satisfied EVs without considering the cheating of EVs on bid activities. We obtain the optimal result by solving the following Integer Linear Programming (ILP) problem formulated as belows: In this study, we utilize Gurobi Python 1 to solve the above ILP. In Fig. 10, we examine the system efficiency under varying the number of MCS within [10,22] with the step of 2, each MCS is equipped with three chargers, and the number of EVs is 50. As seen in Fig. 10, the OTP scheme gets the highest average number of satisfied EVs because it is the optimal solution. However, the optimal match may contain sell-buyer pairs in which sellers win the energy trading due to relying on the cheat on bid. Therefore, focusing on the system performance alone is not sufficient. Regarding the TIM scheme, the gap between the TIM and our proposed mechanism is quite large, as shown in Fig. 10. Although the TIM can guarantee the truthfulness of the buyers, it provides lower system efficiency than our incentive mechanism. The reason is because our proposal allows one MCS can serve simultaneously multiple EVs as long as the MCS has sufficient energy and chargers, while the TIM grants each MCS to serve at most one EV. Thus, our proposed mechanism is better than the TIM scheme pertaining to the system efficiency and 1 https://www.gurobi.com/ guarantees truthfulness for EVs, which is unreachable by the OPT scheme. Besides, as shown in Fig. 10, the system efficiencies for all the schemes grow as the number of sellers goes up. This is mainly because the more MCSs the system has, the more EVs the system can serve. In summary, our proposed energy trading scheme is able to not only achieve better system efficiencies but also control the possible untruthful activities of EVs.
In Fig. 11, we present the system efficiency when the quota of MCSs varies from 2 to 10. In the OPT scheme, when the quota of MCSs is up to 6, the number of successful trades is stable because all demanding EVs are satisfied. In the TIM's case, each MCS provides the service to up to 1 EV; therefore, it cannot boost the system efficiency even by raising the quota of MCSs. On the other hand, our proposal makes efficient use of the limited resources of MCSs. As shown in Fig. 11, under our proposed scheme, the number of satisfied EVs increases as the quota of MCSs increases since more EVs can be served simultaneously.

2) IN MULTI-ROUND SETTING
In this setting, we consider a dynamic energy trading environment where demanding EVs may join the system at any point in time. For more details, we assume that i) The EVs' arrival time follows the Poisson distributions; ii) The proposed incentive mechanism is executed round by round where each round refers to a unit time slot; iii) Every requiring charging finishes before a new round starts. Fig. 12 shows the overall system efficiency with EV's arrival rate λ = 15, λ = 20, and λ = 25 as varying the number of available MCSs from 10 to 22. We can observe that as the number of available MCSs increases, the system efficiency increases for all schemes. The main reason for this growth is that as the available MCSs raise, more resources are accessible, thereby improving the system performance. Moreover, it can also be seen that once the arrival rate of EVs grows, MCSs have more customers to serve, resulting in a higher number of satisfied EVs. Fig. 13 provides the analysis of the system efficiencies with the quota of MCSs is 3 and 5 by varying the parameters 56342 VOLUME 10, 2022 of the EV's arrival rate and running the simulation up to 10 rounds. In Fig. 13, we see that an increased number of MCSs' quotas (i.e., from quota = 3 to quota = 5) will lead to higher chances of providing charging service to EVs, and thereby the system efficiencies obtained when the quota of MCSs is 5 outperforms the system efficiencies obtained when the quota of MCSs is 3 in both cases: the number of rounds (NoR) <= 6 and NoR<= 10. Moreover, we can infer that when the arrival rate of EVs changes in the range [6,12], the number of satisfied EVs grows in all four considering cases. This is because MCSs have sufficient energy to trade with EVs. Additionally, we can observe that as the arrival rate is from 12 to 20, the number of satisfied EVs increases with NoR <= 6, while the achieved result does not rise with the NoR <= 10. The reason behind this trend is that as the NoR is within six rounds, MCSs still have adequate energy to supply EVs' demand resulting in more EVs being served. Meanwhile, as the NoR <= 10 and the quota = 5, MCSs are out of energy, which results in unsuccessful energy trading with EVs. For the case of the NoR <= 10 and the quota = 3, the number of satisfied EVs is quite stable, despite an increase in the EV's arrival rate. This is because the fewer EVs served in each round (i.e., quota = 3 < quota = 5 in the case of quota = 5, NoR <= 10), the more remaining energy for other rounds.
In Fig. 14, we investigate the expected number of rounds that MCSs can sell their available energy up under various values of the EVs' arrival rate. When the arrival rate accumulates (i.e., more EVs demand energy), increasing the energy supply of one round, thereby, the number of rounds lowers. In addition, we observe the number of rounds increases with respect to a decline of the MCSs' quota. This is because as the quota decreases, inducing more MCSs refuse EV's charging requests due to a lack of charger resources, even if the energy resource of MCSs is still in excess. Therefore, MCSs can continue execution in the following rounds to provide their residual energy resource.
Furthermore, we demonstrate the charging scheduling of MCS under different time-slots illustrated in Fig. 15. We choose the arrival rate of EVs at each time slot as λ = 15 for this simulation. As depicted in Fig. 15, the remaining energy and the number of satisfied EVs tend to go down over time-slots. The reason for that is that the MCSs providable energy will be reduced after each time slot, which leads to a lowering in the number of EVs served. In Fig. 15, the number of satisfied EVs in the case of quota = 5 is higher than the one in the case of quota = 3 at the first five time-slots; however, this tendency is reversed at the last five time-slots. This is because, in the first five time-slots, MCSs with quota = 5 have more available chargers to serve EV than MCSs with quota = 3. However, the more energy MCSs serve, the less energy they have left. Therefore, the remaining energy of MCSs and the number of satisfied EVs when quota = 5 are smaller than when quota = 3 for the last five time-slots.

VI. CONCLUSION
In this paper, we investigated an energy trading problem between EVs and MCSs from a distributed auction perspective, in which MCSs act as sellers while EVs act as buyers. First, we conduct a distributed auction-based incentive mechanism enabling competition between EVs for available energy at MCSs by submitting their own private bids to MCSs. Simultaneously, MCSs distributively determine the seller's price based on the submitted bids and allocate energy to EVs. Next, we prove that our distributed auction-based energy trading scheme is individual rationality, truthfulness, and computational efficiency through theoretical analysis. Besides, we experiment with a series of simulations to validate our theoretical results. The simulation results illustrate that critical properties of our proposed energy trading mechanism are achievable, which is unable under the OTP scheme. The results also indicate that our proposal significantly enhances the system's efficiency by up more than two times compared to the existing scheme, TIM. In future works, we would extend our study by investigating complicated scenarios that would take charging and energy trading problems into account. In addition, another future work would be to develop a contract theory-based incentive mechanism to encourage crowdsourced EVs to share their surplus energy.

APPENDIX A PROOF OF THEOREM 1
Proof: When a EV d i wins its bid at MCS s j , then EV d i pays p d i for each unit energy, and p d i = p t j as described in Algorithm 1. Therefore, we have Obviously, Theorem 1 is proven.

APPENDIX B PROOF OF THEOREM 2
Proof: If the worst comes to the worst, an EV d i ∈ D loses the distributed auction, and the final payment p d i is zero. Whereas, the bid value is always larger than zero. Thus, we have U d i ≥ 0 . If EV d i wins the distributed auction at MCS s j , then according to Algorithm 1, d i has the opportunity to become winning buyer candidates at MCS s j , only if it is one of the first h buyers with the highest bidding price among all the buyers in W cons j , where h is decided as (11). Thus, we get that Therefore, the utility of EV d i is given From the above analysis, the non-negative utility can be achieved by each buyer d i . Thus, Theorem 2 is proved.

APPENDIX C PROOF OF THEOREM 3
Proof: The complexity of the proposed incentive mechanism depends on two processes as follows: • For the winning bid determination process: Recall that, this process is conducted at each seller MCS s j ∈ S by running two sub-processes. First, each MCS s j determines and arranges its feasible buyers with a consumed complexity is O (N log 2 N ). Next, the MCS s j computes d i ∈W cons j e i and elects its expected partners, which requires complexity O(N ). Thus, the complexity of this sub-process at each MCS is O(N log 2 N ).
• For the final seller determination process: Recall that, this process is operated at each buyer EV d i ∈ D to compute s j ∈S x ij and then decide the suitable value for each x ij . These task consume O(M 2 ). Therefore, the complexity of this sub-process at each EV is O(M 2 ). By way of consequences, we confirm that the proposed energy trading mechanism is polynomial, which shows the computational efficiency of our proposal. Therefore, the complexity of the proposed auction mechanism is polynomial and thus computationally efficient. instead of p d i to indicate the final buying price of d i paid for the MCS s j' . The same applies to p d ij , andp d ij' . Then, we evaluate two possible sub-cases as following: • The EV d i ∈ W can j' but d i / ∈ W f j when it is honest about its bid: This situation only happens when the obtained utility at the MCS s j' is smaller than the obtained utility at the MCS s j . It means we have In addition, whatever the bid's value submitted by d i , the EV d i pays the same price for its final seller. Hence, we always havep d ij' = p d ij ' . Combining with the equation (20), we can achievē • The EV d i / ∈ W can j' when it is honest about its bid: There is only one reason why the EV d i belongs to this case, that is v ij' ≤ b j' h+1 j' , where h = |W can j' |. Nonetheless, the EV d i successfully cheats on the auction to dishonestly get d i ∈ W f j' . As the results, we realize the following equations b ij' ≥ b j' h+1 j' , andp d ij' = b j' h j' orp d ij' = b j' h+1 j' . Based on the above analyses, we get Case 2) b ij = v ij and d i ∈ ∪ s j ∈S W can j : ∀d i ∈ D, this situation expresses that d i failed its truthful bid at all MCSs, leading to U d i = 0. Assume the EV d i submits an untruthful bid, two possible observation points can happen examined as follows: Observation 2.
Consequently, we can get: Next, when the EV d i is the bid-winner at the MCS s j in a dishonestly way, its offered bid b ij must satisfy the condition b ij ≥ b j h+1 ,j . Furthermore, recall that, as d i ∈W temp j e i ≥ E j , the final buying price at this case must be eitherp d j = b j h ,j or p d j = b j h+1 ,j based on Algorithm 1. Thus, we achieve: From (25) and (26), we obtain: In summary, whenever an EV d i ∈ D cheats on its bid, the proposed incentive mechanism can counteract its action by decreasing its utility. Therefore, buyers EVs must be honest about their bids with sellers MCSs to guarantee maximum utility. To this point, we complete the proof.