The active distribution network (ADN) structure with the MG is shown in Figure 1. In addition to fixed topology parameters, the DN also includes local loads and DG. The distribution network is connected to the microgrid through a point of common coupling (PCC). The MG directly combines the DG with local user loads in a certain area. The DG will prioritize supplying power to the local load within the MG. When the DG in the MG is excessive or insufficient, the MG will interact with the DN for power. The MG can include diesel generators (DEG), photovoltaics (PV), wind power generation (WG), energy storage (ES), and local load. In addition, the local load can be flexibly adjusted to participate in demand response.

At the same time, DNs are interconnected through SOPs. Power interaction between the DNs is realized by controlling the flexible interconnection devices. The interconnection architecture of the distribution network is shown in Figure 2. DNs can be divided into several sub-DNs. Flexible interconnection between sub-DNs can be achieved through SOPs.

In this paper, the DistFlow model is applied to constrain the power flow of the DN: U min 2 ≤ V i , t ≤ U max 2 , i ∈ R . (1) 0 ≤ l i j , t ≤ l i j , ⁡ max . (2) P i j , t + ∑ i DEG = 1 N DEG κ j P t , i DEG = r i j l i j , t + p j , t + ∑ j : j → k P j k , t + ∑ i S O P = 1 N S O P υ j P t , i SOP . (3) Q i j , t + ∑ i DEG = 1 N DEG κ j Q t , i DEG = x i j l i j , t + q j , t + ∑ j : j → k Q j k , t . (4) V i , t + l i j , t r i j 2 + x i j 2 = 2 r i j P i j , t + x i j Q i j , t + V j , t . (5) 2 P i j , t 2 Q i j , t V i , t − l i j , t 2 ≤ V i , t + l i j , t . (6)

Here, R is the node set of the DN; i and j are the node ID of the DN; t is the time interval; P i j , t is the active power of branch i j in t ; Q i j , t is the reactive power of branch i j in t ; V i , t is the square of voltage amplitude; U max and U min are maximum and minimum values of voltage amplitude, respectively; l i j , t is the square of the current amplitude; l i j , ⁡ max is the maximum values of l i j , t ; r i j is the resistance value of the branch i j ; x i j is the reactance value of the branch i j ; N DEG is the number of DEGs; P t , i DEG is the active power of DEGs; Q t , i DEG is the reactive power of DEGs; κ j is a binary variable that indicates whether the DEG is installed at node j ; p j , t is the active load at node j ; q j , t is the reactive load at node j ; υ j is a binary variable that indicates whether the SOP is installed at node j ; P t , i SOP is the active power of the SOP.

At the same time, the DEG needs to meet the constraints of power upper and lower limits and the constraints of ramp rate: P i DEG , min ≤ P t , i DEG ≤ P i DEG , max . (7) Q i DEG , min ≤ Q t , i DEG ≤ Q i DEG , max . (8) P t , i DEG − P t − 1 , i DEG ≤ ε DEG . (9)

Here, P i DEG , max and P i DEG , min are the maximum and minimum values of DEG active power output, respectively; Q i DEG , max and Q i DEG , min are the maximum and minimum values of DEG reactive power output, respectively; ε DEG is the ramp rate of the DEG.

The MG can include DEG, PV, WG, ES, and local electricity loads. Among them, the local electricity load can be adjusted flexibly, which can be powered by time-series translation. Therefore, the operational constraints of the MG are as follows: P t DEG + P t P V + P t W G + P t g r i d = P t E S + P t l o a d , (10)

where P t DEG is the active power of DEG in the MG; P t P V is the active power of PV in the MG; P t W G is the active power of WG in the MG; P t g r i d is the electricity purchased from the DN; P t E S is the active power of ES in the MG; P t l o a d is the load of the MG. P l o a d = ∑ t = 1 T P t l o a d . (11) P t , min l o a d ≤ P t l o a d ≤ P t , max l o a d . (12)

Here, P l o a d is the total load of the MG within a day; P t , max l o a d and P t , min l o a d are the maximum and minimum values of P t l o a d , respectively.

The constraints of DEG in MGs are similar to those of DNs and will not be elaborated here. − P t , max E S ≤ P t E S ≤ P t , max E S . (13) S min E S ≤ S t E S ≤ S max E S . (14) S t E S = S t − 1 E S + P t E S . (15)

Here, P t , max E S is the maximum value of P t E S ; S t E S is ES capacity; S max E S and S min E S are the maximum and minimum values of S t E S , respectively.

The DNs are flexibly interconnected through SOPs with DC charging devices. The upper control center can achieve power flow direction control through voltage source converters (VSCs) on both sides of the SOP. In order to realize the regional division, the DC charging device and DN are the same optimized area. In addition, information exchange is carried out point-to-point with the adjacent DN side. The decoupled interaction power relationship is as follows: P t v s c i − P t , s a l e i = P ′ t , v s c i . (16) P t v s c i + 1 = P ′ t , v s c i + 1 , (17)

where P t v s c i is the power on the VSC AC side at the boundary of the DN; P t , s a l e i is the power of the DC charging load; P ′ t , v s c i and P ′ t , v s c i + 1 are the amounts of power interaction information between adjacent active distribution networks; P t v s c i + 1 is the power on the VSC AC side at the boundary of the DN.

At the same time, the upper limit constraint of interaction power is as follows: P t v s c i − P t , s a l e i ≤ P t , s o p max , (18)

where P t , s o p max is the maximum value of P t , v s c i ′ .

The MG and DN are connected through a PCC. They are coupled and interconnected. In order to achieve partition calculation, the DN area in the coupling area between the DN and MG is defined as region H, and the MG area is defined as region L. The PCC will be decoupled into two independent regions. The relationship between the variables decoupled from two independent regions is as follows: P t , D N = P t , D N , H ′ . (19) P t , M G = P t , M G , L ′ . (20)

Here, P t , M G is the power value of the MG area at the PCC; P t , D N is the power value of the DN area at the PCC; P t , D N , H ′ is the replication area variable of the DN; P t , M G , L ′ is the replication area variable of the DN.

Sub-problem 1 can be decomposed into two local sub-problems; namely, the optimization of DN _i and DN _i+1. The decision variables within the DN can be optimized by solving local sub-problems using the ADMM. The SOP serves as the boundary between two DN areas, and its injection power is optimized as an interaction variable during the calculation process.

For DN _i L D N i A D M M = C D N i − C s a l e i + λ i ∑ t = 1 T P t V S C i − P t , s a l e i − P ′ t , V S C i + 1 + ρ 2 P t V S C i − P t , s a l e i − P ′ t , V S C i + 1 2 2 . (26)

For DN _i+1: L D N i + 1 A D M M = C D N i + 1 + λ i ∑ t = 1 T P t V S C i + 1 − P ′ t , V S C i + ρ 2 P t V S C i + 1 − P ′ t , V S C i 2 2 . (27)

Here, L D N i A D M M and L D N i + 1 A D M M are the Lagrangian functions.

The original residual error, dual residual error, and iterative update mechanism for sub-problem 1 are as follows: k i r = P t V S C i , r − P t , s a l e i r − P ′ t , V S C i + 1 r 2 2 k i + 1 r = P t V S C i + 1 , r − P ′ t , V S C i r 2 2 . (28) d i r = P t V S C i + 1 , r − P t , s a l e i r − P t V S C i + 1 , r − 1 − P t , s a l e i r − 1 2 2 d i + 1 r = P t V S C i + 1 , r − P t V S C i + 1 , r − 1 2 2 . (29) λ i r + 1 = λ i r + ρ 2 P t V S C i , r − P t , s a l e i r − P ′ t , V S C i + 1 r λ i + 1 r + 1 = λ i + 1 r + ρ 2 P t V S C i + 1 , r − P ′ t , V S C i r . (30)

Here, r is the number of iterations; k is the original residual; d is the dual residual.

L M G j , i A D M M = C M G j , i + λ j , i ∑ t = 1 T P t , M G j , i − P ′ t , D N i + ρ 2 P t , M G j , i − P ′ t , D N i 2 2 . (31) k j , i r = P t , M G j , i r − P ′ t , D N i r 2 2 . (32) d j , i r = P t , M G j , i r − P t , M G j , i r − 1 2 2 . (33) λ j , i r + 1 = λ j , i r + ρ 2 P t , M G j , i r − P ′ t , D N i r . (34)

Here, L M G j , i A D M M is the Lagrangian function.

The selection of the penalty coefficient will affect the convergence speed of the algorithm. By balancing the original residual error and the dual residual error, the penalty coefficient can be adaptively adjusted according to the residual error, which can accelerate the convergence speed of the algorithm. ρ r + 1 = ρ r 1 + ⁡ lg d r k r k r < 0.1 d r ρ r 1 + ⁡ lg k r d r k r > 10 d r ρ r others . (35)

The decision variables are divided into two parts: 1) the interaction variables within the flexible interconnection devices of the DN; 2) the interaction variables between the PCC coupling part of the DN and MG. The optimization process is shown in Figure 4.

The detailed optimization process is as follows:

Step 1: input the initial data. Input the TOU for regional purchase and sale, DC charging price for EVs, user flexible load, operating parameters of the DN and MG, and rolling prediction data for the PV and WG.

Step 2: initialize the number of iterations r = 0 and penalty coefficient ρ 0 . Set the convergence values of the original residual and the dual residual ( φ o r i and φ d u a l ). The initial settings for dual variables, original residuals, and dual residuals are all set to 0. Set the initial value of the global variables ( P ′ t , V S C i , P ′ t , V S C i + 1 , and P ′ t , D N i ).

Step 3: update the number of iterations ( r = r + 1 ).

Step 4: solving sub-problems.

(1) Solve sub-problem 1. At the flexible interconnection device SOP, the interaction data received by DN _i from DN _i+1 are p ′ t , V S C i + 1 r + 1 = P t V S C i + 1 , r . The interaction data received by DN _i+1 from DN _i are p ′ t , V S C i + 1 r + 1 = P t V S C i + 1 , r − P t , s a l e i r .

Step 5: Update Lagrange multipliers and penalty coefficients.

Step 6: Residual update and iteration termination determination. When k ≤ φ o r i and d ≤ φ d u a l , the algorithm ends and outputs the result. Otherwise, return to step 3 to continue iterating until the convergence condition is met.

In order to verify the effectiveness of the optimization strategy, this paper selected two IEEE 33 node systems and two MGs as a study case. The topology structure is shown in Figure 5, and the load data of the DN are shown in Figure 6.

In this paper, a multiple time-scale rolling optimization strategy is proposed to achieve a full-day optimization calculation. TOU is applied for purchasing and selling electricity between the regions, and the charging price for EVs is set based on TOU. The TOU is shown in Figure 7, and the charging price is shown in Figure 8.

The PV output is shown in Figure 9.

The WG output is shown in Figure 10.

The MG load curve is shown in Figure 11, and the adjustable range of the flexible load is shown in Figure 12 and Figure 13. The WG parameters and ES parameters are shown in Table 1, Table 2, Table 3, Table 4.

The comparison of the scenario settings is shown in Table 5. For scenario 1, SOP and flexible load are available. For scenario 2, SOP is available, but flexible load is not available. For scenario 3, flexible load is available, but SOP is not available. For scenario 4, flexible load and SOP are not available. The SOP parameters are shown in Table 6.

The operating costs for DN1 and DN2 for different scenarios are shown in Figure14 and Figure 15. For DN1, when the SOP is coordinated with flexible loads, the operating cost of DN1 is the lowest. Compared with scenario 2, the operating cost of DN1 has significantly decreased. Compared with scenario 3, due to the lack of an SOP interconnection between DN1 and DN2, there is no power interaction. These two DNs will operate independently of each other and lack collaborative assistance, which, to some extent, increases the operating cost of the DN. Comparison of scenario 2 with scenario 4 shows that due to the lack of an SOP interconnection between DN1 and DN2, the two networks are unable to interact with each other based on their operating status. The operating cost of DN1 is increased. On comparing scenario 3 with scenario 4, when the internal load of the MG can be flexibly controlled, the load can be adjusted based on TOU, and the operating cost of the DN is reduced.

The operating cost of DN2 in scenario 1 is ¥291.894 yuan higher than that of DN2 in scenario 2. The operating cost of DN2 has increased, but the operating cost of DN1 has decreased by ¥504.8324. The overall operating cost of the DN has reduced.

The comparison of the operating costs between MG1 and MG2 is shown in Figure 16 and Figure 17, respectively. According to Figure 16, it can be seen that the overall load of MG2 is greater than the overall load of MG1. Therefore, the operating costs of MG2 are higher than those of MG1. In scenario 2, due to the non-flexible regulation of load, the MG will purchase electricity from the DN according to the original plan during the peak load period of the DN. When the amount of load access is adjusted flexibly, MGs can adjust the time shift access of the load to reduce the operating costs and meet the demand for load access. In scenario 4, due to the lack of power interaction with the upper level DNs, MGs will increase the output power of the DEG to meet the load demand of the microgrid. When scenario 3 is adopted, the MG load can achieve time-shift access to reduce the purchase of electricity from the DN during weak DN periods and the peak time of TOU. At the same time, the power generation of DEGs is adjusted to optimize the operating costs of the MG.

The flexible load regulation curves of MG1 and MG2 are shown in Figure 18 and Figure 19, respectively. MG1 increases load during low load periods and reduces the load during peak load periods. The load fluctuations have been suppressed. The operational pressure during peak load periods has been alleviated. MG2 has a larger load access volume than that of MG1. That will be more evident in flexible load regulation. MG2 will increase load access volume during periods of low electricity prices in the DN, and MG2 will reduce load access volume during periods of high electricity prices to meet the demand for flexible loads.