Multi-Level Cooperative Scheduling Based on Robust Optimization Considering Flexibilities and Uncertainties of ADN and MG

: This paper develops the coordination structure and method for utilizing ﬂexibilities in a Micro-Grid (MG), an Active Distribution Network (ADN) and a Transmission Grid (TG), which can play an essential role in addressing the uncertainties caused by renewable energy power generation (REPG). For cooperative dispatching, both ﬂexibilities and uncertainties on the interface of MG–ADN and ADN–TG are portrayed in uniﬁed forms utilizing robust optimization (RO), based on the modiﬁed equipment-level model of ﬂexible resources. The Constraint-and-Column Generation method is adopted to solve the RO control problems. Simulations on the modiﬁed IEEE case-6 and case-33 systems are carried out. The results suggest that the proposed algorithm can exploit ﬂexible resources in both an MG and an ADN, improving the economy and promoting REPG consumption within each level (MG, ADN and TG) while reducing uncertainties and providing ﬂexibilities for superior operators.


Introduction
The continuous growth of renewable energy power generation (REPG) represented by wind power (WP) and photovoltaics (PV) is an important support for peaking carbon dioxide emissions before 2030 and achieving carbon neutrality [1]. REPGs are injecting into power systems at various levels, including a Micro-Grid (MG), an Active Distribution Network (ADN) and a Transmission Grid (TG), causing problems of multi-level scheduling for REPG consumption and economic operation [2,3].
Dealing with the uncertainties of REPG utilizing various kinds of flexible resources plays a significant role in improving REPG consumption and optimal operation, despite the different emphases of scheduling in multiple levels [4,5]. The scenario-based stochastic method used to be the most popular way to cope with the uncertainties in optimization. For optimal dispatching in an MG, the operating economy is generally the primary goal, which is frequently integrated with indicators related to reliability and the environment [6]. A stochastic model for the coordinated scheduling of renewable and thermal units, including fuel cell units and hydrogen storage, was proposed in [7]. A two-layer control scheme operating at two different timescales was illustrated in [8] for the energy management of an MG based on stochastic model predictive control. The main feature of optimal scheduling in an ADN compared to it in an MG is the necessity to handle the power flow, which can be nonlinear [9,10]. In studies [11,12], the high penetration of plug-in electric vehicles, demand response and energy storage were considered as flexible resources, and network loss was included in the optimization goals. For optimization in a TG, unit commitment is the key issue. Stochastic methods were adopted in [13][14][15] to comprise fluctuations of WP and load.

1.
Based on RO, considering equipment-level models, the flexibilities and uncertainties of MGs are uniformly expressed through the signs and magnitudes of the variables interacting with an ADN, allowing transactions for both power and flexibilities between an ADN and MGs.

2.
Whether an ADN has flexibilities or uncertainties is determined utilizing RO methods. The power in the root node of an ADN will result in controllable intervals if the ADN has flexibilities. Otherwise, it will turn to uncertain budgets. This method provides convenience for a TG to perceive uncertainties and employ flexibilities of the ADN. 3.
The Constraint-and-Column Generation (C&CG) method is chosen to solve the RO problems for reliability and efficiency. 4.
To minimize the cost of flexibilities, a bi-level optimization of an MG and an ADN is developed based on a price adjustment mechanism. To begin the process, RO is carried out in the MG, considering the uncertainties internal REPG and regular load (RL). The purpose is to exploit the flexibilities and redu the uncertainties for the and, while the operating costs of the MG are cut down. Af optimization, the intervals of flexibilities or uncertainties will be sent to the ADN. addition, the flexibilities or uncertainties of multiple MGs cooperate with resour directly controlled by the ADN using RO, to optimize the controllability of power in t root node of the ADN. Modulations of price for flexibility/uncertainty are perform according to the optimization results before being fed back to MGs. Once the iteration h been completed, a flexible/uncertain range of power in the root node of the ADN will reported to the TG. Finally, the TG utilizes flexibilities provided by ADNs and therm power plants to deal with uncertainties, aiming at reducing the operating cost a promoting the consumption of REPG. To begin the process, RO is carried out in the MG, considering the uncertainties of internal REPG and regular load (RL). The purpose is to exploit the flexibilities and reduce the uncertainties for the and, while the operating costs of the MG are cut down. After optimization, the intervals of flexibilities or uncertainties will be sent to the ADN. In addition, the flexibilities or uncertainties of multiple MGs cooperate with resources directly controlled by the ADN using RO, to optimize the controllability of power in the root node of the ADN. Modulations of price for flexibility/uncertainty are performed according to the optimization results before being fed back to MGs. Once the iteration has been completed, a flexible/uncertain range of power in the root node of the ADN will be reported to the TG. Finally, the TG utilizes flexibilities provided by ADNs and thermal power plants to deal with uncertainties, aiming at reducing the operating cost and promoting the consumption of REPG.

Optimization Models
The section below will propose RO models for the MG, ADN and TG. The number of time intervals is represented by T, and the length of each interval is donated by ∆t.

MG
The optimization goal of the MG is as follows:  Turning now to the constraints. The following formula reveals the power balance constraint of the MG considering flexibilities and uncertainties: plants to deal with uncertainties, aiming at reducing the operating cost and promoting the consumption of REPG.

Optimization Models
The section below will propose RO models for the MG, ADN and TG. The number of time intervals is represented by T, and the length of each interval is donated by t  .

MG
The optimization goal of the MG is as follows:   ,PV  MG,GT  MG,BD  ,  ,  ,  ,  ,  ,  ,  ,   MG,base  MG,down  MG,L  MG,FL  MG,BC  MG,PV  MG,GT  MG,BD  ,  ,  ,  ,  ,  ,  ,  ,   MG,up  MI,up  MO,up  , , ,     refer to the upward/downward FRPs of PV, BS, GT and FL inside the MG i, respectively. The relevant constraints will be discussed in detail below.
• PV PV can provide upward/downward FRPs by increasing/reducing the output power. However, it is restricted by the limitation of PV output. . The constraints are as follows: , which can be depicted by the following formula: where s MG,BCD i,t are 0-1 variables. When s MG,BDC i,t = 1, the BS is switched from a charging state to a discharging state. Otherwise, its state remains.
A downward FRP of BS can be modeled in the same way: The following is the modified constraints of GT considering the flexibility. u MG,GT The constraints on GT off/on time are as follows: where T MG,GT i,down,min /T MG,GT i,run,min are the minimum off/on times of GT. • FL Three kinds of FLs are examined in this paper, including transferable load, interruptible load and reducible load [30]. The constraints are discussed below.
Firstly, the transferable load cannot be interrupted once it is started. However, the integral working time can be advanced or delayed, as follows: represents the running power. Since the transferable load is disabled to adjust power continuously, it is not employed to provide an FRP.
Secondly, the interruptible load of which power can be regulated only has requirements for cumulative absorbed energy during the acceptable time range represents the power value to be determined. The interruptible load provides upward/downward FRPs through a decreasing/increasing running power. The total energy absorbed during the schedulable period would be restricted to more/less than the minimum/maximum value E MG,FL,b Thirdly, the reducible loads running power is adjustable, but the operating time cannot be changed. It is supposed to have n c discrete operating points, P MG,FL,c is the power value to be determined. The constraints are as follows: To sum up, the running power and FRP of FL can be described in the following formula:

Parameters and Variables from MG
In the RO of the ADN, the absorbed power values of the MG, which purchases an FRP from the ADN, are regarded as uncertain parameters. In contrast, the power values of the MG, which sells an FRP to the ADN, are treated as control variables. As shown below, four conditions can be categorized according to p The internal flexible resources of the MG i are sufficient to cover its uncertainties, and the spare flexibilities are sold to the ADN. The power values p MG j,t of the MG i are seen as control variables that satisfy the following constraints: where j is the node number of the ADN to which the MG i is connected.
Under this condition, the MG i fails to handle uncertainties itself and needs to achieve an FRP from the ADN. p MG j,t is considered as an uncertain parameter, which belongs to the uncertain set P MG In this case, the MG i has both flexibilities and uncertainties towards the ADN. It sells an upward FRP to the and, while it buys a downward FRP from the ADN. P MG j,t meets the following constraints: It is similar to case c. However, the MG i sells a downward FRP to the and, while it buys an upward FRP from the ADN: Energies 2021, 14, 7376 9 of 23

Power Flow and Other Constraints
Power flow constraints can be relaxed based on second-order cone programming (SOCP), of which validity has been demonstrated in [9]. The power flow constraints in the standard form of SOCP are as follows, and the subscript t is omitted for brevity: where j, k and l are the node numbers of the ADN. P jk and Q jk are the active and reactive power at the head of branch jk. R jk and X jk refer to the resistance and reactance of branch jk, respectively. P k /Q k are the injection of active/reactive power in node k. u(k) is defined as the set of head nodes, of which branches end at k. Similarly, v(k) is the set of end nodes, of which branches start at k. U j is used to describe the square of voltage amplitude in node j, while L jk represents the square of current amplitude in branch jk.
The safety constraints of node voltage and branch current are as follows: Suppose the ADN is equipped with continuous reactive power compensation devices, the corresponding constraints are as follows: where Q com i,min /Q com i,max are the minimum/maximum reactive power of compensation device in node j.
For the nodes to which MGs are connected, the injected power is as follows: where ϕ j is the power factor angle. RL, FL, GT, BS and PV directly regulated by the ADN have the same constraints as the corresponding elements in the MG; therefore, they will not be a repeat.

Optimization Objectives and Criteria for Flexibilities
Since the ADN generally has a radial structure, it usually affects the TG through the power of its root node, p ADN root,t . The number of ADNs is ignored for simplicity. Following the idea of RO, this paper intends to propose the criterion for flexibility/uncertainty of the ADN based on the following two optimizations: where p P, ADN and P P, ADN are uncertain parameters and their collections in the ADN, respectively. The weight coefficient ω m will decrease with iteration, to ensure convergence. The goal of (21) is to find the robust minimum value of p ADN root,t considering the network loss, while (22) is to find the robust maximum value.
Since RO premeditates the degradation of the result owing to uncertainties, the solutions of (21) and (22) may appear in the following two situations, as shown in Figure 3.
ADN based on the following two optimizations: where P , A D N p and P, ADN P are uncertain parameters and their collections in the ADN, respectively. The weight coefficient m ω will decrease with iteration, to ensure convergence. The goal of (21) is to find the robust minimum value of ADN root ,t p considering the network loss, while (22) is to find the robust maximum value.
Since RO premeditates the degradation of the result owing to uncertainties, the solutions of (21) and (22) may appear in the following two situations, as shown in Figure 3.  Uncertainties will cause the degradation of the optimization results, in other words, make the maximum value smaller but the minimum value larger. However, RO takes this degradation into account and adopts flexible resources to counteract it; therefore, the robust solution is stable and controllable. If the robust maximum value p ADN,max root,t is less than the robust minimum value p ADN,min root,t , as shown in Figure 3a, the flexible resources inside the ADN are not enough to cover its uncertainties. As a result, p ADN root,t , the power of the ADN is supposed to be an uncertain parameter that belongs to p ADN,max root,t , p ADN,min root,t for the TG. The flexibility rises with the P ADN, F t . If P ADN, F t ≥ 0, the ADN has a flexible adjustment range for the TG. On the contrary, the ADN has uncertainty toward the TG when P ADN, F t < 0.

TG
As discussed above, flexibilities/uncertainties due to RL, REPG, BS, etc., are integrated into unified expressions in the root node of the ADN (interface with the TG). Regarding optimization in the TG, a unit commitment is carried out considering the flexibilities/uncertainties of ADNs and concentrated REPGs (represented by WP).

Optimization Goal
The objective of unit commitment is to minimize both the power generation cost C G and the reserve cost C R as follows: where P wind is the uncertain set for output power of WP and P ADN is the uncertain set for the ADN. G is the number of the thermal generators and P G g,t is the output power of generator g during time interval t. R U g,t /R D g,t are the upward/downward reserves provided by generator g, respectively. k 0 g , k 1 g and k 2 g refer to the generation cost coefficients and q U g and q D g describe the reserve cost coefficients. Since the cost of WP is not counted in, the economic optimization will autonomously result in the increase in WP output power within the feasible range.

•
Power balance considering the reserve where N W is the number of wind farms and P W w,t is the output power of wind farm w during time interval t. N refers to the total number of ADNs. R ADN,U where P G,max g /P G,min g are the maximum/minimum power of generator g, of which r u g /r d g are the upward/downward reserves.
where T max

A Solution Method for RO Models
This paper adopts the C&CG algorithm to solve the multi-level scheduling for the MG, ADN and TG, which are RO problems in nested form and have an NP-hard nature. The core idea of C&CG is to gradually approach the solution of RO through the iteration of the master problem (MP) and the sub problem (SP) [31]. For the sake of simplicity, let y be the first stage decision variables, concerning BS, FL and thermal generators, which need to be determined in real time. Let x be the second stage variables that can be determined after the uncertain parameters are further clarified. x includes variables related to PV, WP and GT. Now we turn to the algorithm as follows:

Iteration between ADN and MGs
As previously stated, the prices for FRPs are distributed to MGs from the ADN. After ROs, according to the prices, have been carried out in MGs, the results will be sent back to the ADN. Hence, the following iteration steps are designed:

1.
Set the minimum/maximum prices C FRP min /C FRP max for the FRP of the MG and send to each MG at the initial prices. The optimization is carried out in the ADN based on the results feedback from the MGs. The iteration will be terminated if one of the following two situations occurs: or prices for FRPs in all the time intervals reach the maximum limit. Otherwise, ω m and the prices for FRPs will be amended as follows and return to step two: where C FRP t,m includes prices for both upward/downward FRPs. , of which signs can be used to judge whether the MG is flexible or uncertain. Then, each ADN carries out RO to determine if p ADN root,t is adjustable and the flexible/uncertain range. Finally, the unit commitment will be conducted in the TG, and the results will be fed back to the ADN.

Case Studies
In this section, the proposed RO models are validated in the multi-level power depicted in Figure 5. There is a modified case-6 TG and two modified case-33 ADN which parameters (line impedance, basic load, etc.) can be obtained from MATPOW MG1~MG4 are located in ADN1, and MG5~MG7 are located in ADN2. The elemen each MG are presented in the figure below. T = 4h, Δt = 15min and the optimizations at 8:00. The convergence tolerance 0 ε and 1 ε are both set to 10 −2 . The initial weigh efficient 1 ω is 0.5. The maximum load L in bus-5 of the TG is set to 220 MW. The det parameters and uncertain sets of the multi-level grid are provided in the Append Table 1 displays the time-of-use electricity prices for the MGs. The minimum/maxi prices for upward FRPs are 0.14/0.37 (USD/kW), and the minimum/ maximum price downward FRPs are 0.01/0.06 (USD/kW).

Case Studies
In this section, the proposed RO models are validated in the multi-level power grid, depicted in Figure 5. There is a modified case-6 TG and two modified case-33 ADNs, of which parameters (line impedance, basic load, etc.) can be obtained from MATPOWER. MG1~MG4 are located in ADN1, and MG5~MG7 are located in ADN2. The elements in each MG are presented in the figure below. T = 4 h, ∆t = 15 min and the optimizations start at 8:00. The convergence tolerance ε 0 and ε 1 are both set to 10 −2 . The initial weight coefficient ω 1 is 0.5. The maximum load L in bus-5 of the TG is set to 220 MW. The detailed parameters and uncertain sets of the multi-level grid are provided in the Appendix A. Table 1 displays the time-of-use electricity prices for the MGs. The minimum/maximum prices for upward FRPs are 0.14/0.37 (USD/kW), and the minimum/ maximum prices for downward FRPs are 0.01/0.06 (USD/kW).  All the experiments were carried out on a personal computer with an Intel Core i7-9700 CPU and 32GB of RAM using MATLAB 2019a with MOSEK.

Flexibilities/Uncertainties of MGs and ADNs
The purpose of this section is to illustrate the results of proposed scheduling strategies on excavating the flexibilities and portraying the uncertainties of the ADN/MG. The results of RO optimizations of MGs are displayed in Figure 6, including the base powers and the power boundaries considering flexibilities/uncertainties. MG2, MG3 and MG5 converge because the optimization results are stable, while MG1, MG4 and MG6 stop iterating because the FRP price limits are reached. MG1, MG2 and MG4 sell FRPs to the ADN: when the MG reduces its power, it provides an upward FRP to the ADN. On the other hand, when the MG increases its power, a downward FRP is achieved by the ADN. The adjustable ranges reach 69.03, 18.96 and 16.82% of their base powers, respectively. These results may be due to the abundance of their internal resources. MG5, which only has PV and BS, mainly has a downward FRP to be exploited. MG3 has to purchase an FRP from the ADN since it contains fewer flexible resources, which cannot cover the uncertainties with rather big basic loads. However, the RO reduces its demand for outside an FRP to lower than 17.25% of its basic power. The situation of MG6 is a bit special. Flexibilities and uncertainties simultaneously exist, but the controllable and uncontrollable ranges are both small, which are less than 2.25% of its basic power.  All the experiments were carried out on a personal computer with an Intel Core i7-9700 CPU and 32GB of RAM using MATLAB 2019a with MOSEK.

Flexibilities/Uncertainties of MGs and ADNs
The purpose of this section is to illustrate the results of proposed scheduling strategies on excavating the flexibilities and portraying the uncertainties of the ADN/MG. The results of RO optimizations of MGs are displayed in Figure 6, including the base powers and the power boundaries considering flexibilities/uncertainties. MG2, MG3 and MG5 converge because the optimization results are stable, while MG1, MG4 and MG6 stop iterating because the FRP price limits are reached. MG1, MG2 and MG4 sell FRPs to the ADN: when the MG reduces its power, it provides an upward FRP to the ADN. On the other hand, when the MG increases its power, a downward FRP is achieved by the ADN. The adjustable ranges reach 69.03, 18.96 and 16.82% of their base powers, respectively. These results may be due to the abundance of their internal resources. MG5, which only has PV and BS, mainly has a downward FRP to be exploited. MG3 has to purchase an FRP from the ADN since it contains fewer flexible resources, which cannot cover the uncertainties with rather big basic loads. However, the RO reduces its demand for outside an FRP to lower than 17.25% of its basic power. The situation of MG6 is a bit special. Flexibilities and uncertainties simultaneously exist, but the controllable and uncontrollable ranges are both small, which are less than 2.25% of its basic power. ADN1 and ADN2 converged after 11 and 13 iterations, respectively. The flexible/uncertain power ranges of ADN1 and ADN2 are shown in Figure 7. The robust minimum values are smaller than the robust maximum values of power in the ADN1 root node. This means that ADN1 can provide the TG with a flexible power range of about 10MW after overcoming its uncertainties. However, the flexible resources in the ADN2 are not enough to offset the influence of internal uncertainties on the optimization results, resulting in an uncertain range (about 3% of its own basic power) to the TG. MG1, MG4~MG6, ADN1 and ADN2 contain REPG, of which the consumption rates are above 95% after optimization.

Sensitivity Analysis to Price
To assess the effect of electricity purchase prices on the optimal dispatch, experiments were carried out under proportionally changed prices. Figure 8a displays the impact of electricity prices on the operating costs of MGs, of which the initial values are marked on the right. Additionally, the coordinate on the horizontal axis represents the multiple of the current price relative to the initial price. Since MG1~4 and MG6 are all purchasers of electricity, the operational costs rise along with electricity prices. The growth rate of operational expense increases with the increase in the electricity price as well. A possible explanation for this might be that the flexible resources have reached their output limitation and cannot alleviate the impact of higher electricity prices further. MG5 sells electricity to ADN2. Its operating income rises significantly with the electricity price. Figure 8b shows ADN1 and ADN2 converged after 11 and 13 iterations, respectively. The flexible/uncertain power ranges of ADN1 and ADN2 are shown in Figure 7. The robust minimum values are smaller than the robust maximum values of power in the ADN1 root node. This means that ADN1 can provide the TG with a flexible power range of about 10 MW after overcoming its uncertainties. However, the flexible resources in the ADN2 are not enough to offset the influence of internal uncertainties on the optimization results, resulting in an uncertain range (about 3% of its own basic power) to the TG. MG1, MG4~MG6, ADN1 and ADN2 contain REPG, of which the consumption rates are above 95% after optimization. ADN1 and ADN2 converged after 11 and 13 iterations, respectively. The flexi certain power ranges of ADN1 and ADN2 are shown in Figure 7. The robust mi values are smaller than the robust maximum values of power in the ADN1 root nod means that ADN1 can provide the TG with a flexible power range of about 10M overcoming its uncertainties. However, the flexible resources in the ADN2 are not e to offset the influence of internal uncertainties on the optimization results, resultin uncertain range (about 3% of its own basic power) to the TG. MG1, MG4~MG6, and ADN2 contain REPG, of which the consumption rates are above 95% after op tion.

Sensitivity Analysis to Price
To assess the effect of electricity purchase prices on the optimal dispatch, exper were carried out under proportionally changed prices. Figure 8a displays the im electricity prices on the operating costs of MGs, of which the initial values are mar the right. Additionally, the coordinate on the horizontal axis represents the mul the current price relative to the initial price. Since MG1~4 and MG6 are all purcha electricity, the operational costs rise along with electricity prices. The growth rate o ational expense increases with the increase in the electricity price as well. A poss planation for this might be that the flexible resources have reached their output lim and cannot alleviate the impact of higher electricity prices further. MG5 sells electr

Sensitivity Analysis to Price
To assess the effect of electricity purchase prices on the optimal dispatch, experiments were carried out under proportionally changed prices. Figure 8a displays the impact of electricity prices on the operating costs of MGs, of which the initial values are marked on the right. Additionally, the coordinate on the horizontal axis represents the multiple of the current price relative to the initial price. Since MG1~4 and MG6 are all purchasers of electricity, the operational costs rise along with electricity prices. The growth rate of operational expense increases with the increase in the electricity price as well. A possible explanation for this might be that the flexible resources have reached their output limitation and cannot alleviate the impact of higher electricity prices further. MG5 sells electricity to ADN2. Its operating income rises significantly with the electricity price. Figure 8b shows the influence of electricity prices on the moderate demand of MGs for an outside FRP (∑ t∈T (p MO,up i,t + p MO,down i,t )/T) during the scheduling period, and the initial values are also marked on the right. The average demands of MG1~2 and MG4~6 under the initial electricity price are less than 0, while the demand of MG5 is greater than 0. The demands for an external FRP increase and the signs of MG2 and MG6 are changed due to the growth of electricity price. These results are likely to be related to the flexible resources that are employed to reduce the growth of operational costs under high electricity prices, resulting in the decline of flexibilities.
) during the scheduling period, and the initial values are also marked on the right. The average demands of MG1~2 and MG4~6 under the initial electricity price are less than 0, while the demand of MG5 is greater than 0. The demands for an external FRP increase and the signs of MG2 and MG6 are changed due to the growth of electricity price. These results are likely to be related to the flexible resources that are employed to reduce the growth of operational costs under high electricity prices, resulting in the decline of flexibilities.
(a) influence on operating costs (b) influence on outside FRP The P ADN,F root,t and network losses of ADN1 and ADN2 in different iterations are presented in Figure 9. Eleven and thirteen iterations are performed on ADN1 and ADN2, respectively. The prices for an FRP in ADN2 reach the upper limitations. P ADN,F root,t of ADN1 are lower than 0 under the initial prices for an FRP and eventually, converge to values above 0 with the growth of prices. The P ADN,F root,t of ADN2 are still below 0 under the maximum price for an FRP; however, the increases in charges for an FRP reduce its uncertainties. As shown in Figure 10, owing to the rise of the weight coefficient, the convergence of the network losses is accelerated, and the minimum values are achieved after five to six iterations. Table 2 compares the optimizations of ADNs in different limit ranges for FRP prices. Since the iteration of ADN1 is terminated due to the trigger of the condition that the changes of powers are extremely insignificant, the changes of the price limit range do not affect the scheduling results of ADN1. However, the lower minimum price brings a wider iteration range, which leads to the increase in the calculation time. The rise of the maximum price results in a restricted improvement of ADN2 scheduling effects but also brings a higher computational burden that may limit online applications.
are lower than 0 under the initial prices for an FRP and eventually, converge to values above 0 with the growth of prices. The ADN,F root,t P of ADN2 are still below 0 under the maximum price for an FRP; however, the increases in charges for an FRP reduce its uncertainties. As shown in Figure 10, owing to the rise of the weight coefficient, the convergence of the network losses is accelerated, and the minimum values are achieved after five to six iterations.
(a) iterations of ADN1 (b) iterations of ADN2   Table 2 compares the optimizations of ADNs in different limit ranges for FRP Since the iteration of ADN1 is terminated due to the trigger of the condition th changes of powers are extremely insignificant, the changes of the price limit range affect the scheduling results of ADN1. However, the lower minimum price brings a iteration range, which leads to the increase in the calculation time. The rise of the mum price results in a restricted improvement of ADN2 scheduling effects but also a higher computational burden that may limit online applications.

Effectiveness for TG
The purpose of the experiments below are to verify the application effects of the proposed dispatching strategy for the TG: Case one: the strategy proposed in this paper is adopted in both MGs and ADNs. The RO in the TG is conducted after ADN1 uploads the controllable ranges, while ADN2 reports the uncertain ranges to the TG.
Case two: the strategy proposed in this paper is not employed in MGs and ADNs. The RO in the TG is carried out assuming that the powers in the root nodes of ADN1 and ADN2 fluctuate within 10% of their base values. Table 3 displays the comparison between optimization results of case one and case two in the TG: For the TG, the application of the proposed strategy can reduce the need for generator reserve. The consumption rate of WP is improved while the operating cost of the TG declines.

Conclusions
This study set out to exploit the flexibilities of a multi-level power grid, which can play a crucial role in dealing with the uncertainties caused by the widely developed REPGs. A multi-level scheduling framework for the MG, ADN and TG has been established. Unified models for flexibilities/uncertainties of the MG and ADN have been developed based on RO. The iterations between the ADN and MGs have been carried out to maximize the utilization of flexibilities within different participants considering the economic costs. These studies provide new insights into awareness and improvement strategies for the flexibilities of multi-level scheduling. Based on the proposed methods, the ADN's flexibilities were expanded while the uncertainties were curtailed, supplying new resources for the unit commitment with REPGs in the TG.
Simulations on cases including a modified 6-bus TG, two modified 33-bus ADNs and several MGs have been carried out to verify the effectiveness of the strategy proposed in this paper. This study had identified that when the ADN had sufficient flexible resources, the robust minimum powers at the root node were less than the robust maximum values, which brought controllable ranges for the TG. Otherwise, the ADN should be treated as uncertain loads toward the TG. The results demonstrated that the awareness of flexibilities and reduction in uncertainties in the ADNs are favorable to the unit commitment for the TG. The operating cost had decreased by 6.28%, and the consumption rate of WP had risen by 6.21%. The comparative examples have shown that high electricity prices may limit flexibility. Moreover, reasonable price ranges for FRPs are suggested to be employed to balance calculation time with optimization results. Further research might explore the combination of scenario and robust methods to reduce the conservatism, and a more practical price mechanism needs to be investigated.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to business secret.    Figure A1. Interval forecast of PV output, WT output and load (per unit).