The requirement of ADNs’ optimal dispatching is to reduce its power supply cost while meeting its load. The ADN needs to decide its electricity procurement schedule and the operation state of its controllable unit. The ADN’s comprehensive cost F A D N can be calculated as Eq. 1, and the subentry cost of Eq. 1 can be calculated as Eqs. 2–5 min F A D N = C D G + C g r i d + C l o s s + C e x (1) C D G = ∑ t = 1 T ∑ i = 1 | N D G | [ a i D G ( P i , t D G ) 2 + b i D G P i , t D G + c i D G ] Δ t   i ∈ N D G (2) C g r i d = ∑ t = 1 T c t g r i d P t g r i d Δ t (3) C l o s s = ∑ t = 1 T ∑ j = 1 | N b | c l o s s P j , t l o s s Δ t (4) C e x = ∑ t = 1 T ∑ i = 1 | N L A | ( c D R P i , t D R − c t D N P i , t b ) Δ t   i ∈ N L A (5) C D G is the cost function of controllable DG in the ADN and expressed as the quadratic function of their active output P i , t D G . N D G is the set of nodes with controllable DG. The function | S | represents the number of elements in the set S. C g r i d is the electricity purchase cost. c t g r i d is the time-of-use (TOU) price of the main grid. P t g r i d is the injected power from the main grid. C l o s s is the network loss cost and c l o s s is the unit network loss cost. P t l o s s is the network losses. N b is the set of branch lines. C e x is the interaction cost with LAs, and contains two parts: DR compensation for LAs participating in the DR program and the profit from selling electricity to LAs. The term c D R P i , t D R represents the DR compensation for LAs participating in the DR program. P i , t D R is the shedding power of the LA at node i. The calculation of P i , t D R and the mechanism of DR are illustrated in Constraints of the Active Distribution Network Optimal Dispatching. c D R is the DR incentive for unit shedding power and is set by the ADN. The term c t D N P i , t b represents the profit from selling electricity to LAs. P i , t b is the purchasing power of node i and c t D N is the electricity price inside the ADN. N L A is the set of LA nodes.

(1) Constraints of controllable DG

The constraints of power output Eq. 6, ramp rate Eq. 7, and running time Eq. 8 are considered: { P i , min D G ⋅ u i , t D G ≤ P i , t D G ≤ P i , max D G ⋅ u i , t D G     Q i , min D G ⋅ u i , t D G ≤ Q i , t D G ≤ Q i , max D G ⋅ u i , t D G   i ∈ N D G (6) − r i , max ≤ P i , t D G − P i , t − 1 D G ≤ r i , max   i ∈ N D G (7) { ∑ k = 1 T i , o n D G − 1 u i , t + k D G ≥ ( u i , t + 1 D G − u i , t D G ) ⋅ T i , o n D G ∑ k = 1 T i , o f f D G − 1 ( 1 − u i , t + k D G ) ≥ ( u i , t D G − u i , t + 1 D G ) ⋅ T i , o f f D G     i ∈ N D G (8) u i , t D G is the binary variable representing the running state of controllable DG at node i equal to 1 when controllable DG is on, while 0 means the DG is off. P i , max D G and P i , min D G ( Q i , min D G , Q i , max D G ) are its maximum and minimum active (reactive) output. r i , max is the maximum ramp rate. T i , o n D G and T i , o f f D G are the minimum continuous working time and the minimum off time.

(2) Constraints of renewable DG

The cubic set is adopted to define the uncertain output of PV and WT (Ding et al., 2017). { P i , t P V = P ^ i , t P V + μ t P V ξ t , max P V   i ∈ N P V P i , t W T = P ^ i , t W T + μ t W T ξ t , max W T   i ∈ N W T | μ t P V | ≤ 1 ,   | μ t W T | ≤ 1    (9) P i , t P V and P i , t W T are the actual outputs of PV and WT at node i. P ^ i , t P V and P ^ i , t W T are the predicted outputs. ξ t , max P V and ξ t , max W T are the maximum prediction errors of PV and WT. μ t P V and μ t W T are the uncertain variables used to adjust the range of uncertain prediction error. N P V and N W T are the sets of nodes installed with PV and WT.

(3) Power flow constraints

A distribution network is normally configured to be a radial/tree-like topology, which means that each network node has only one parent node. Figure 2 shows a line diagram of a radial power network. The power flows corresponding to Figure 2 can be described by DistFlow branch equations. However, the traditional DistFlow model is nonlinear, which makes the problem difficult to solve. To make relevant problems computationally tractable and meanwhile guarantee an acceptable calculation result, the linearized DistFlow model is adopted here (Song et al., 2019). { P n , t b r = P n + 1 , t b r + P n , t n d Q n , t b r = Q n + 1 , t b r + Q n , t n d V n + 1 , t = V n , t − ( r n + 1 P n + 1 , t b r + x n + 1 Q n + 1 , t b r ) V 0 (10) P j , t b r and Q j , t b r are the active and reactive power at the sending end of branch j, while r j and x j are the resistance and reactance of the same branch line. P i , t n d and Q i , t n d are the total active and reactive load at node i, while V i , t is the voltage magnitude. V 0 is the rated voltage magnitude of the distribution system.

According to the definition of robust optimal scheduling, the voltage security must be ensured as the prediction errors of renewable DG change, which is expressed as follows: { max μ V n , t ( P , Q , μ ) ≤ V max min μ V n , t ( P , Q , μ ) ≥ V min (11) Except for the voltage safety constraint, the branch current constraint is also considered in some studies. However, the current carrying capacity of the branch line is usually two to three times larger than its rated current. Besides, the voltage drop will increase as the branch current increase. Therefore, the branch current constraint will also be satisfied if the voltage security constraint is satisfied.

For node i, its total active and reactive load can be calculated as follows: { P i , t n d = P i , t b − P i , t r − P i , t P V − P i , t W T − P i . t D G Q i , t n d = Q i . t b − Q i . t D G (12) Since the network losses are much smaller than line flow terms P n , t b r and Q n , t b r , the node voltage is insensitive to the network loss terms which can be neglected in the voltage constraints Eq. 10 of the linear DistFlow model (Zhong et al., 2019). However, to accurately calculate the cost of the ADN, this article considers the network losses in the objective function of the ADN. By using the approximation V n , t ≃ V 0 , the network loss power of the ADN can be expressed as Eq. 13, which is a commonly used expression with the employment of the adopted linear DistFlow constraint (Fu and Chiang, 2018). P j . t l o s s ≈ r j ( P j , t b r ) 2 + x j ( Q j , t b r ) 2 V 0 2   j ∈ N b (13)

(4) DR constraint

When the TOU price of the main grid is higher than the selling price inside the ADN, the higher cost will be caused by the higher power purchasing price from the main grid. Therefore, the ADN hopes to reduce its cost by compensating and encouraging LAs to reduce their power consumption in the above period or to transfer their power consumption time to other periods. In this article, the ADN releases DR incentive in the corresponding period to encourage LAs to participate in the DR program. After LAs reduce their power consumption in the DR period set by the ADN, they will get DR compensation according to their reduction. The DR mechanism is shown in Eq. 14 P i , t D R = u t D R ⋅   ( P i , t 0 − P i , t b )   i ∈ N L A (14) u t D R is the binary variable which represents the DR period set by the ADN, while 1 means the DR program is implemented in the period t. P i , t 0 is the load of LA node i before DR. Eq. 14 describes the DR mechanism: LAs will only gain DR compensation by reducing their power consumption in the DR period set by the ADN, but will not gain any DR compensation if they reduce their power consumption in other periods. This DR mechanism encourages LAs to reduce their power in the DR period or transfer their load to other periods, which will reduce the cost of the ADN by the means of peak shifting and valley filling.

There is a product form of the binary variable and continuous variable in Eq. 14. We apply the Big-M method to transfer Eq. 14 into a linear constraint Eq. 15 where M is a large enough constant. { 0 ≤ P i , t D R ≤ M ⋅ u t D R P i , t 0 − P i , t b − M ⋅ ( 1 − u t D R ) ≤ P i , t D R ≤ P i , t 0 − P i , t b + M ⋅ ( 1 − u t D R ) (15) Thus, in the optimal scheduling model for ADNs, the decision variables are x A D N = [ P i , t D G , Q i , t D G , u i , t D G , P t g r i d , u t D R , P i , t D R , P i , t b ] . The objective is to minimize Eq. 1 while satisfying the constraints Eqs. 6–13, 15.

Based on the electricity selling price and the DR incentive of the ADN, the LA adjusts its power consumption plan of the flexible load to minimize its comprehensive cost Eq. 16, which includes electricity purchasing cost, DLC cost and the profit from participating in the DR program. The subentry cost can be calculated as Eqs. 17–19 min F L A , i = C b , i + C D L C , i − C D R , i (16) C b , i = ∑ t c t D N P i , t b Δ t (17) C D L C , i = ∑ t [ a i l , i ( P i , t i l ) 2 + b i l , i P i , t i l ] Δ t + ∑ t [ a s h , i | P i , t s h − P i , t s h 0 | 2 + b s h , i | P i , t s h − P i , t s h 0 | ] Δ t (18) C D R , i = ∑ t c D R P i , t D R Δ t (19) P i , t i l is the shedding power of the interruptible load. P i , t s h 0 and P i , t s h the power of the shiftable load before and after DR. a i l , i and b i l , i ( a s h , i , b s h , i ) are the quadratic and linear cost coefficient of the interruptible load (shiftable load). It is assumed that all LAs have reached their optimal scheduling plan without DR incentive, which means that LAs cannot further reduce their cost by changing its power consumption plan without DR incentive (Guo et al., 2020). As a result, the linear cost coefficients of the interruptible load and shiftable load in Eq. 18 must obey the following constraint: { b i l , i ≥ c t D N b s h , i ≥ max { | c t 1 D N − c t 2 D N | }    t 1 , t 2 ∈ [ 1 , T ] (20)

(1) Constraints of interruptible load

In order to meet the basic demand of users, the shedding power of the interruptible load cannot be larger than the maximum shedding power. 0 ≤ P i , t i l ≤ α i l ⋅ P i , t i l , 0 (21) P i , t i l , 0 is the consumption power of the interruptible load before DR. α i l is the maximum shedding ratio.

(2) Constraints of shiftable load

Thus, in the optimal scheduling model for LAs, the decision variables are x L A , i = [ P i , t i l , P i , t s h , Δ P i , t s h , P i , t D R , P i , t b ] . The objective is to minimize Eq. 16 while satisfying the constraints Eqs. 21–27.

Figure 6 shows the output of controllable DG. DG1 is off during 1–7 h and t = 24 h, as the TOU price in the main grid during these periods are lower than the unit cost of DG1. Due to the ramping rate constraints, DG1 gradually increases its output since t = 6 h until it reaches its maximum value before the peak period. Compared to DG1, DG2 starts increasing its output at t = 2 h due to its higher capacity. DG1 and DG2 both maintain high output during 10–21 h and gradually decrease their output since t = 21 h, with the end of the peak period. Seen from the DGs’ marginal price in Figure 7, the marginal cost of controllable DGs is higher than the TOU price in some period. This is because the controllable DGs have to generate more power to protect the ADN from voltage violation caused by the fluctuating output of renewable DG. With the increasing output of DG, the voltage drop on the distribution line will be decreased due to the less power transmitted through the distribution line.

According to the result, u t D R is equal to 1 during 7–23 h, as the trading prices between the ADN and the LA during these periods are lower than the TOU price in the main grid. Denote Δ P i , t L A as the variation of the LA’s purchasing power after the DR program. The calculation of Δ P i , t L A is given as follows: Δ P i , t L A = P i , t b − P i , t 0     ( i ∈ N L A ) (41) Figure 8 shows Δ P i , t L A of each LA. Under the DR incentive, the LA reduces its power consumption in the flat and peak periods and transfers part of the load from the DR period to the valley period, which demonstrates LA’s role in peak shifting and valley filling under DR incentive. The LA can obtain additional income by participating in DR, which will also help the ADN reduce its higher power supplying cost in peak and flat periods and make extra electricity selling profit in the valley period.

Figure 9 illustrates the convergence performance of the ATC algorithm for the ADN-LA power exchange. The applied distributed optimization scheduling method is stably convergent after 112 times iteration, which means it does not require many computing resources. By applying the ATC algorithm, the optimization times of the LA and the ADN and the communication burden between them can be reduced. Besides, the LA and the ADN only need to send their expected purchasing power to each other in the iteration process. The privacy data inside the LA, such as the cost function of the LA, is unknown to the ADN. As a result, the privacy and security of users’ data can be guaranteed.

To further demonstrate the effectiveness of the proposed method, three scheduling models are used to calculate the operating costs of the ADN and each LA. The three scheduling models are as follows:

(1) Centralized DLC model: assuming that the ADN can directly control the flexible load of users and that the ADN and LAs are regarded as the same stakeholder, the total cost of the ADN and LAs is taken as the objective function, and the centralized optimization method is applied to solve the problem.

The results of different scheduling models are compared in Table 3. The total cost of the centralized DLC method is the lowest among Table 3, but the cost of each LA is higher than that of other methods, which means the total cost is decreased at the expense of each LA. Besides, the ADN needs to collect the relevant parameters of the flexible load under the centralized DLC model, which makes it difficult to guarantee the privacy of users. The benefits of each LA can be ensured under the “source changing with load” model. However, the cost of the ADN and total cost of this model are the highest due to the lack of enough interaction between the ADNs and LAs. Under the distributed optimization model based on ATC where the optimal scheduling problems of the ADNs and LAs are decoupled and solved independently, the economic benefits of LAs and the ADNs can be reconciled, and the privacy and security of users’ electric power data can be guaranteed. Although it takes more time to gain the results through distributed optimization due to its iterative process, the distributed method based on ATC is still fast enough to be applied in the day-ahead optimal scheduling and is more applicable than the centralized optimization for its advantage in guarantying the security of the private data inside each subject.

To further study the influence of robust level on the optimization results, uncertainty analysis is presented in this article. We denote the violation rate to quantitatively measure the influence of uncertain parameters on voltage security. The violation rate is calculated as Eq. 42 through the Monte Carlo simulation where all uncertain parameters are assumed to follow uniform distribution. α v i o = | N v i o | | N t o t a l | × 100 % (42) N t o t a l is the set of 1,000 renewable DGs’ output scenarios generated by the Monte Carlo simulation. The randomly generated output of renewable DG and the optimal scheduling results are substituted into the topology of the ADN for power flow calculation to obtain the power flow distribution of the ADN. N v i o is the set of scenarios where the voltage safety constraint is not satisfied.

When Γ takes different values, different solutions with different levels of robustness can be obtained (compared in Table 4). For the ADN, different levels of robustness make it possible to make trade-offs between the economy and voltage level.

It can be seen from Table 4 that the cost of the ADN increases with the increasing Γ. The rise in cost was mainly due to the increasing cost of controllable DG. Controllable DG needs to increase its output and reduce the transmission power through distribution network lines, which indicates the role of controllable DG in voltage supporting under uncertain environment. When Γ is less than 1.2, the violation rate goes on declining rapidly, while the cost of the ADN rises with the increasing Γ. This result demonstrates that the ADN has to cost more to improve its power quality under an uncertain environment. However, declining speed of the violation rate is close to 0 when Γ is larger than 1.2, which means the ADN has to cost much more to reach a higher robust level when the current robust level is relatively high.

The robust optimization model proposed in this article is equivalent to the deterministic optimization model when Γ is equal to 0. The cost of the ADN is the least in Table 4 when no uncertainties are considered. However, it does not mean that the solution obtained by deterministic optimization is better than that by robust optimization. The voltage violation rate of deterministic optimization exceeds 90%, which results in the poor power quality caused by insufficient voltage support. The robust optimization model is equivalent to the completely robust optimization model when Γ is equal to 2 in which all uncertainty is considered. While short time voltage violation is allowed in the practical operation of distribution network, the result of completely robust optimization model is too conservative for considering every possible situation, resulting in its highest cost. However, the completely robust optimization model is applicable in the system with high reliability and quality requirements. Therefore, the ADN can select the appropriate robust level according to the practical operation requirements for power supplying.

Table 5 shows the effect of DR under different DR incentive. With the increase of DR incentive, the total DR power is also increasing, which means that LAs are more willing to participate in the DR program with higher incentive. LAs can help the ADN reduce power supply cost by peak shifting and valley filling. When the c D R is less than 0.4, the operating cost of the ADN will decrease with the increasing DR incentive. Although the ADN has to pay more compensation to LAs with the higher c D R , more benefit will be brought to the ADN by the higher participation of LAs in the DR program. However, when c D R is larger than 0.4, the benefit brought by the higher DR participation is less than the compensation. As a result, the cost of the ADN starts to increase as c D R increases from 0.4. Therefore, the ADN can further reduce its operation cost by making the appropriate DR incentive according to the response of LAs.