Figure 1 shows that the hierarchical transactive power exchange system on expressways includes the control center, the power grid, BSSs, and EVs. EVs are dispatched to BSSs according to the BSS energy storage situation, EV remaining energy, and moving energy consumption. In BSSs, the power exchange among the power grid and batteries is performed. When the BSS operates at an exorbitant cost, some EVs are encouraged to release energy in the BSS and supplement energy in other BSSs.

The control center collects information from EVs, BSSs, and the power grid (e.g., SOC of batteries and forecast electricity price). According to the collected information, the power exchange schemes are optimized in the control center, and the results are sent to EVs and BSSs. When BSSs operate at prohibitive costs, EVs will be considered the energy prosumers in the optimized process.

The control center collects information from EVs. It increases the risk of privacy leakage. Privacy protection could be performed by increasing the difficulty and cost of privacy leakage. The following measures can be adopted:

1) Hidden EV user’s information. It increased the difficulty and the time cost of privacy leakage (Efthymiou et al., 2010; Su et al., 2019).

To reduce BSSs’ operation and power costs, the process of EV supplementing energy is hierarchical. At the lower level, EVs are dispatched to BSSs for satisfying users’ driving requirements. For providing sufficient swappable batteries economically in each BSS, the power exchange among batteries and the power grid is performed at the middle level. At the upper level, some EVs release energy in the BSS which operates at exorbitant costs and supplement energy in other BSSs. It is worth noting that the power exchange scheme at the upper level will cause the updated schemes at lower and middle levels. The power exchange model is shown as follows (1): C E V = P E V u p ⋅ E P P ω + C B D − P E V d o w n ⋅ E S P ω , (1) C E V = C 1 , C 2 , ⋯ , C N L ω , C B D = C B D , 1 , C B D , 2 , ⋯ , C B D , N L ω , (2) E P P = a p , 1 , a p , 2 , ⋯ , a p , N L ω , E S P = a s , 1 , a s , 2 , ⋯ , a s , N L ω , P E V u p = Q E V ⋅ Δ S O C 1 u p , Δ S O C 2 u p , ⋯ , Δ S O C N L u p ω , P E V d o w n = Q E V ⋅ Δ S O C 1 d o w n , Δ S O C 2 d o w n , ⋯ , Δ S O C N L d o w n ω , (3) where C _EV represents the cost matrix of EVs. P E V u p and P E V d o w n represent the supplement energy and the releasing energy matrixes of EVs, respectively. C _BD represents the BDC matrix of EVs. EPP ^ω and ESP ^ω represent the transposition of energy purchasing and selling price matrixes, respectively. a _p,l and a _s,l represent the electricity purchasing and selling prices, respectively. Q _EV represents the battery capacity. Δ S O C 1 u p and Δ S O C 1 d o w n represent the increasing and decreasing SOC of an EV battery, respectively. N _L represents the number of EVs at lower power exchange.

In (1), the cost matrix of EVs includes the power purchase cost and BDC. The power purchase cost and the power selling income depend on the real-time electricity prices and power cost. The cost at the middle level is reflected by electricity prices and the number of dispatched EVs.

At the lower level, EVs are dispatched to various BSSs. The dispatching optimization model is established as follows: min   ∑ l = 1 N L C l = ∑ l = 1 N L a p , l ⋅ Δ S O C l u p ⋅ Q E V + C B D , l , (4) s . t . C B D , l = α 1 P l t β 1 + α 2 1 − S O C l t β 2 , (5) 0 < v l t ≤ v max , 0 ≤ Δ S O C l u p ≤ Δ S O C max , Δ S O C max = S O C max − S O C min , ⇒ 0 < P l t ≤ v max ⋅ ψ l , S O C min ≤ S O C l t ≤ S O C max , (6) a p , l = ∑ b = 1 N b a p , b ⋅ Δ S O C l , b u p ⋅ Q E V / ∑ b = 1 N b Δ S O C l , b u p ⋅ Q E V , (7) Q E V ⋅ S O C l t − S O C min ≥ D a d , l ⋅ ψ l , D a d , l = min D 1 , l , D 2 , l , ⋯ , D n , l , ⋯ , D N B , l ≥ 0 , (8) where C _l represents the cost of the lth EV. C _BD,l represents the BDC of the lth EV battery. P _l represents the power consumption of the EV. α ₁, α ₂, β ₁, and β ₂ represent the battery degradation cost parameters. SOC _l represents the SOC value of the EV battery. v _l and v _max represent the speed and maximum speed of the moving EV, respectively. SOC _max and SOC _min represent the maximum and minimum SOC of the battery, respectively. D _n,l and D _ad,l represent the distance and minimum distance between the BSS and the lth EV, respectively. N _B represents the number of batteries in the BSS. ψ _l represents the unit moving energy cost of the EV.

In (4), the optimization object is the minimum EV cost. The number of EVs dispatched to each BSS is considered the decision variable. EV’s moving energy consumption accelerates the cycle aging of batteries, and the BDC is formulated as shown in (5). In (6), the real-time output power is constrained within the rational range, and the real-time SOC of batteries is constrained for preserving batteries from over-discharge. In BSSs, the energy price for EVs fluctuates, as shown in (7). The remaining energy of EV batteries is constrained, as shown in (8).

At the middle level, the power exchange among batteries and the power grid is performed in each BSS. The optimization model is established as follows: min   C b = ∑ b = 1 N b C B D , b + ∑ b = 1 N b − 1 + 1 a c h , b ⋅ P b , (9) s . t . a c h , b = ∑ t 0 t e n d a p , g r i d t ⋅ P b − g r i d t + ∑ t 0 t e n d a p , g r i d t ⋅ P b − b ′ t + C B D , b ′ ∑ t 0 t e n d P b − g r i d t + ∑ t 0 t e n d P b − b ′ t , (10) C B D , b = ∑ t 0 t α 1 P b t β 1 + α 2 1 − S O C b t β 2 , P b = ∑ t 0 t P b t = ∑ t 0 t ∑ b ′ = 1 N B − 1 + 1 P b − b ′ t , t 0 ≤ t ≤ t e n d , (11) P b − b ′ t = min P b , r e m t , − P b ′ , r e m t , P b , r e m t = P b t − ∑ b ′ = 1 n b P b − b ′ t , P b ′ , r e m t = P b ′ t − ∑ b = 1 n b P b ′ − b t P G r i d t = P b , r e m t n b = N b − 1 + P b ′ , r e m t n b = N b − 1 , ⇒   P b t = P b − g r i d t + ∑ b ′ = 1 N b − 1 P b − b ′ t , P b − g r i d t ≥ 0 , P b ′ t = P g r i d − b t + ∑ b = 1 N b − 1 P b ′ − b t , P g r i d − b ′ t ≥ 0 , P G r i d t = ∑ b = 1 N b P b t ≥ 0,0 ≤ P b t ≤ P c h , max , − P d i s , max ≤ P b ′ t ≤ 0 , (12) S O C b t e n d ≥ S T b = S O C max − S O C d e v , S O C min ≤ S O C b t ≤ S O C max , (13) S T b − S O C b t Q E V ≤ ∑ t t e n d P b t ≤ S O C max − S O C b t Q E V , (14) S O C b ′ t ≥ S O C min S O C b ′ t ≤ S O C max ⇒ ∑ t t e n d P b ′ t ≥ S O C min − S O C b ′ t Q E V , (15) S O C b t + Δ t = S O C b t + P b t Q E V ≥ S O C b t , S O C b ′ t + Δ t = S O C b ′ t + P b ′ t Q E V ≤ S O C b ′ t , (16) N b ≥ N b , l t ⇒ ∑ b = 1 N B N b ≥ N L t , n b t | S O C ≥ S O C max − S O C d e v ≥ N b , l t , (17) where C _b represents the operation cost of the bth BSS. a _ch,b and a _p,grid represent the electricity prices of charging the battery and the power grid, respectively. P _b represents the charging power of the battery. P _Grid represents the output power of the power grid. P _b-grid represents the power that battery charges from the grid. P _b-b’ represents the B2B power. P _b,rem represents the battery’s unmatched optimization power. ST _b represents the desired SOC of the battery. SOC _dev represents the allowable SOC deviation of the battery. N _b,l represents the number of EVs dispatched to the lth BSS. N _B represents the number of BSSs. b and b' represent the indexes for the number of batteries and discharging batteries, respectively. P _ch,max and P _dis,max represent the maximum charging and discharging power of batteries, respectively. Δt represents the time interval. t ₀ and t _end represent the initial and end of a certain period, respectively.

The power exchange performed in each BSS aims to reduce the operation cost, and the BSS’s minimum operation cost is the optimization object, as shown in (9). The total charging power of the BSS is considered the decision variable. In (10), the electricity price of BSSs is defined, and it depends on the power cost and BDC. The BDC is defined in (11). Each battery could match with multiple objects. In (12), the exchanged power is limited by the smaller power of matchable objects. The charging power and discharging power are both limited in the rational range. The SOC of the battery is constrained, as shown in (13) and (15). During the charging period, the real-time SOC of the charging batteries is constrained for preventing over-charging of the battery, and the battery’s power range is concluded as (14). During the discharging period, the real-time SOC of the battery is constrained for preventing over-discharging of the battery, and the battery’s power range is concluded as (15). The real-time SOC of the battery is updated and limited, as shown in (16). At the middle level, each BSS satisfies users’ driving requirements. Therefore, the number of batteries with the designated SOC value is constrained, as shown in (17).

At the upper level, some EVs release energy in BSSs, which operate at exorbitant operation costs, and supplement energy in other BSSs. The optimization model is established as follows: max   ∑ u = 1 N u R u ′ = λ ∑ u = 1 N u u = λ R B S S > 0 , (18) s . t . R B S S = C n | μ − 1 + C n + 1 | μ + ⋯ + C N B | μ − C n | μ , (19) Δ S O C u u p + Δ S O C u d o w n − ψ u ⋅ D n , u / Q E V ≥ 0 , (20) S O C min ≤ S O C m , u ≤ S O C max , S O C min ≤ S O C m ′ , u ≤ S O C max , (21) S O C m ′ , u − S O C min ≥ D a d , u ⋅ ψ u Q E V , (22) where R _u represents the decreasing cost of BSSs due to the uth EV releasing energy. R _u ’ represents the income of the uth EV. R _BSS represents the decreasing cost of the BSS. C _n | _u represents the cost of the nth BSS that u EVs join in the upper power exchange. N _u represents the number of EVs at the upper power exchange. λ represents the operation cost ratio of BSSs.

In the optimization model, the maximum profit of users is the objective, and the number of EVs dispatched to BSSs is considered the decision variable. In (18), the users’ reward is dependent on the decreased cost of BSSs. The upper power exchange could avoid fast charging of some batteries and reduce BDC. When the electric price at the period is high, the upper power exchange could reduce the charging power and the power cost. The power cost and BDC in the BSS are decreased and slightly increased in other BSSs. This is because an acceptable number of EVs are dispatched to other BSSs. The fast-charging power in the BSS is decreased, and the charging power in other BSSs is slightly increased. Therefore, the total cost of BSSs is decreased. At the upper power exchange, the BSSs decreased the cost due to EVs releasing energy, and EVs’ extra cost due to supplementing energy in other BSSs is considered, as shown in (19). In (20), EV moving loss is considered, and the residual energy of EV batteries is constrained to ensure that EVs could drive to supplement energy. In (21) and (22), the SOC of batteries is limited in a reasonable range, and the residual energy of EVs is limited by the adjacent BSS’s distance.

At the lower power exchange, EVs are dispatched to BSSs for supplementing energy. In this process, the number of EVs dispatched to each BSS is considered the decision variable, and the particle swarm optimization algorithm is applied. The optimization result will have an impact on the optimization process at the middle level.

At the middle level, the power exchange among batteries and the power grid is performed in each BSS for reducing the operation cost. This process is considered a power assignment problem. The particle swarm optimization offers several advantages, including a fast rate of convergence, high precision, and simplicity in implementation. On the other hand, the Hungarian algorithm is a combinatorial optimization algorithm used to solve assignment problems efficiently within polynomial time. These algorithms are applied to solve the power assignment problem. The decision variable in this problem is the total charging power of the BSS, and the particle swarm optimization algorithm is employed to obtain the optimal solution. In the power assignment process, the Hungarian algorithm is applied to obtain the batteries’ optimal power exchange scheme at each particle (Zeng et al., 2020). The procedure of the designed algorithm is shown in Figure 3. C P S O t = c 11 t , c 12 t   …   c 1 N b t c 21 t , c 22 t   …   c 2 N b t ⋮ c N b 1 t , c N b 2 t   …   c N b N b t , (23) c i j = c i j − min c i 1 , c i 2 , ⋯ , c i N b , i ∈ 1,2 , ⋯ , N b , c i j = c i j − min c 1 j , c 2 j , ⋯ , c N b j , j ∈ 1,2 , ⋯ , N b , (24) c i j = c i j − min E u n , c i j ∈ E u n , c i j ′ = c i j ′ + min E u n , c i j ′ ∈ E c o , (25) where E _un and E _co represent the uncovered elements and the elements covered by two lines, respectively. N _cell represents the number of assigned cells. N _row and N _column represent the row and column number of the matrix, respectively.

In the optimization process, the operation cost matrix of the BSS is expressed as (23). In the matrix, the number of rows represents that of batteries. The number of columns represents that of the divided power. c _nn in the element represents the corresponding cost generated by the battery charging/discharging. In (24), the calculation rule ensures zero elements in each row and column. In (25), the calculation rule updates the cost matrix.

At the upper power exchange, EVs release energy in the BSS which operates at an exorbitant cost. In the optimization process, the number of EVs dispatched to the BSS is considered the decision variable, and the particle swarm optimization algorithm is applied. After EVs release energy, the number of EVs that supplement energy is changed, and the optimized schemes at the other two levels should be updated.

The simulation model parameters are presented in Table 2 (Song et al., 2017; Kim et al., 2018; Zhang et al., 2019). At the upper level, EVs release energy in the first BSS and supplement energy in other BSSs. For simplifying the computation, a _ch,b is assumed to be equal to a _p,l .

The initial number of EVs that supplement energy and a _p,grid is dynamic, as shown in Figure 4 (Song et al., 2017). To simplify the simulation process, the BDC resulting from fast charging is assumed as shown in (26): C B D , b = C B D , b | P b t = P c h , max ⋅ P b t − P c h , max P c h , max , i f   P b t > P c h , max . (26)

The purchasing power from the power grid using three different methods is shown in Figure 5. During some periods, such as the 01st–07th hours, P _Grid measured using three different methods is equal. Batteries in BSSs are charged from the power grid due to low electricity prices. The low operating cost of the BSS results in few EVs releasing energy. In the 09th hour, P _Grid using M2 is lesser than that using M1 in BSS3. This is because a _p,grid increases, and there is some B2B power in M2. Due to the increased number of dispatched EVs, almost all batteries are charged in BSS1. Therefore, there is less B2B power in BSS1, and P _Grid using M1 and M2 seems to be equal. In BSS2 and BSS3, the distinction between M1 and M2 is imperceptible due to the few dispatched EVs. P _Grid using M3 is significantly less than that using other methods in BSS1 and more than that using other methods in other BSSs. This is because some EVs release energy in BSS1, and the fast-charging power is decreased. However, this process increases the number of EVs that supplement energy in BSS2 and BSS3. EVs centralize supplement energy in BSS1 for reducing the BDC, resulting from EV moving.

The power purchasing cost measured using three different methods is shown in Figure 6. The cost calculated using M1 and M2 is almost identical due to low electricity prices and less B2B power. The excessive EVs supplement energy in BSS1, resulting in few dischargeable batteries. Few EVs supplement energy in BSS2 and BSS3, resulting in low charging power and low power cost. In M3, the fast-charging power of BSS1 is reduced due to EVs releasing energy. It results in the decreased power cost in BSS1 and increased power cost in BSS2 and BSS3.

The power exchange schemes result in the inconsistent BDC, as shown in Figure 7. The BDC in BSS1 is more than that in BSS2 and BSS3. It is because EVs centralize supplement energy in BSS1 and fast charging in some batteries takes place. The BDC variance between M1 and M2 is not remarkable. This is because almost all batteries in BSS1 have to charge for excess EVs supplementing energy. With fewer EVs driving to BSS2 and BSS3, BSSs could schedule an economic and elastic charging scheme. The BDC using M3 in BSS1 is less than that using M1 and M2 by reducing the fast-charging power. Meanwhile, it increases the number of EVs that supplement energy in BSS2 and BSS3. The BDC using M3 is a little more than that using M1 and M2 due to an acceptable number of EVs being dispatched to BSS2 and BSS3.

The detailed cost performance is presented in Table 3, and the power purchasing cost is referred to as PPC. According to the analysis shown in Figures 5–7, the power purchasing cost using M2 is less than that using M1 in BSS3 due to the slight B2B power. The BDC using M2 is only a little more than that using M1 in BSS3 due to few discharging batteries. The power purchasing cost using M3 in BSS1 is less than that using M1 and M2 and more in other BSSs. This is because the fast-charging power in BSS1 is decreased and the charging power in other BSSs is increased. The BDC using M3 in BSS1 is less than that using M1 and M2 and more in other BSSs. This is because the fast-charging power in BSS1 is reduced and the charging power in other BSSs is increased.

Figure 8 shows the chosen battery location in each BSS. The charging power and SOC curve of the battery in the locations are presented with the three methods. In BSS1, due to some EVs releasing energy at the upper power exchange, the purchasing power using M3 is reduced significantly during the 14th–17th hours. In BSS2 and BSS3, the purchasing power using M3 increased remarkedly during the 12th–13th hours due to the additional EVs with low SOC batteries generated at the upper level. Batteries in BSS2 and BSS3 need more charging power for meeting owners’ driving requirements. In BSS3, during the 10th hour, the battery using M2 discharged because of the high electric price and idle batteries. The battery charging from other batteries could reduce the BSS’s operation cost. During the 11th hour, the purchasing power with M3 decreased relatively to that with M2. This is because the upper power exchange results in the varying number of EVs driving to BSS3 and the re-optimization for the lower and middle power exchange schemes.

At the upper power exchange, the cost reduction of the randomly chosen EVs is shown in Table 4. EVs swap lower SOC batteries in BSS1 and supplement energy in BSS2 or BSS3. It can be seen from Table 4 that the initial SOC of the EV battery has little impact on the BSSs’ cost. The cost reduction of EV1 is approximately equal to that of EV4, and the cost reduction of EV3 is much more than that of EV4. The initial SOC of the EV1 battery is equal to that of the EV3 battery. This is because the cost reduction is dependent on the BSS’s operation conditions. When the BSS operates at an exorbitant cost, the responsive EVs at the upper power exchange will receive optimistic rewards. The BSS’s cost reduction is considered the user’s contribution, which directly affects their reward.