Multi-period Market Operation of Transmission-Distribution Systems Based on Heterogeneous Decomposition and Coordination

: The integration of shiftable / curtailment distribution generators (DGs) along with quick-response storage has not only increased the transaction’s ﬂexibility but also puzzled the bidding willingness of transmission-connected market players (TMPs). In this paper, the method of heterogeneous decomposition and coordination (HGDC) is applied to decompose the integrated transmission-distribution market framework into a bi-level problem with a transmission wholesale market master problem and several distribution retail market subproblems in a decentralized organization structure. The price-based bidding willingness of demand-side resources’ (DSRs’) aggregator is simulated considering the relation between distribution system operator’s (DSO’s) operation modes and its equivalent market transactive price. Besides the traditional mixed-integer linear programming (MILP) model, the active reconﬁguration model of DSOs based on mixed-integer second-order conic programming (MI-SOCP) is proposed to rearrange its operation switch status and elaborate its operation cost according to the market transaction. Multi-period optimal operation modes could be obtained through an HGDC-based iteration process by coordinating the transmission system operator (TSO) with DSOs and considering the market energy equilibrium and reserve requirements for security considerations. Karush-Kuhn-Tucker (KKT) conditions are used to testify the optimality and convergence of the bi-level model in theory. The T5-3D33 case is employed to illustrate the e ﬃ ciency of the proposed model and method.


Background and Motivation
Distribution system operators (DSOs) have traditionally been responsible for the reliable and economic operation of power distribution systems which has been independent from the transmission system operator (TSO) [1]. From the traditional monopolized power industry to the gradually decentralized developed context of the competitive electricity market with advanced ICT facilities, the integrated ISO/RTO market could be decomposed into the wholesale market in the transmission system and retail markets in the distribution systems [2,3]. Accordingly, a more volatile and unpredictable operation pattern is imposed to DSO, acting as an intermediary [4] and coping with transaction First, a G-MSS method based on heterogeneous decomposition and coordination (HGDC) is for its first time put into application in the field of the decentralized transmission-distribution market operation (TDMO). A bi-level model of TDMO based on the HGDC algorithm is proposed to implement energy transaction and reserve sharing through all available flexibility resources accessible in the integrated system, which is different from the classic LR method. The HGDC algorithm does not have a parameter tuning issue and does not require a central coordinator which is compatible with the current EMS software. It has been proven optimality and the convergence properties are not limited to only convex problems. It is computationally efficient with a lower communication burden and reduced iterations. Different from Reference [13] is that the method used in the market operation needs a coordinator to set bids/offers and its decision making of the market players both in TSO and DSOs. As the network and generator parameters of the TSO and DSOs are currently always non-observable to each other for scale calculation consideration, the HGDC algorithm is ideal for TDMO which is testified for its local optimization and fast convergence characteristics in theory, regardless of what the network model transmission or distribution system would use. • Second, the strategic behavior of DSOs, including the DSRs aggregators' bidding willingness and DSOs' physical network reconfiguration, is implemented and simulated in detail. Under the assumption that only the energy market is implemented, the strategic bidding willingness of DSRs' aggregators is constructed as constraints and simulated in the TSO wholesale market based on the optimization method, which differs from game theory in many of the research articles about microgrid's bidding. By contrast with the mixed-integer linear programming (MILP) model, an extra MI-SOCP model [30] of DSO is applied to elaborate its power loss and its open-loop network reconfiguration according to the specific transaction. • Lastly, the benefit of HGDC-based TDMO is demonstrated in cases from different aspects to testify its effects, which broadens the application of this method. To tackle with the influence of market equilibrium uncertainties, flexibility in distribution systems, including DGs' shiftable/curtailment offers and DSO's open-loop network reconfiguration, could be fully implemented and simulated to thereby replace high-cost centralized units with less costly generation and defend (N-1) contingencies from congestion in the transmission system. It indicates that the TDMO mode based on the HGDC algorithm is more economic and reasonable than the independent transmission operation mode which, for system operators, means an improved cost-effectiveness of the integrated market, the elimination of huge price spikes arising from unnecessary congestion through relatively reasonable LMPs, a more feasible redistribution of power flow after the (N-1) contingence and a more market-wise decision-making strategy to follow for market players.

Paper Organization
The rest of the paper is organized as follows. The proposed framework and its formulation for the multi-period integrated transactive market of the transmission-distribution system are illustrated in Section 2. The MILP formulation of the network model applied in TSO and DSOs along with the optimal market-wise strategic bidding willingness of DSRs' aggregators are presented in Section 2 including the extra MISOCP-based reconfiguration model of DSOs. The price-based iteration convergence and its HGDC-based solution are also illustrated in Section 2. Case simulation results and analysis are presented in Section 3 and a brief conclusion is provided in Section 4.

Joint Transactive Formulation of TSO and DSOs
The proposed model for the joint operation of energy and reserves considering the interaction between TSO and DSOs is shown in Figure 1. In this framework, the transmission-connected market players (TMPs) in the transmission grid level include DSRs' aggregators operated by DSOs, transmission-connected large units (TLUs, including traditional units and large-scale renewable energy resources, like wind power) and large loads (TLLs, like direct power purchaser). DSOs interact flexibly with both TSO and DSRs within local distribution areas (LDAs). LDAs includes fixed/flexible DR resources like DGs and load aggregators (LAs).  By contrast with the centralized ISO, in this paper, the maximum function of DSO is considered. Each flexible load area is represented by one DSO as an equivalent prosumer (producer/consumer). DSO aggregates and coordinates all DSRs within each LDA. DSO submits a single bid to TSO through the equivalent individual T-D interface (substation) which is named the pricing-node [3]. The wholesale transactive market is operated by the TSO at the transmission level. TSO receives bids/offers from all TMPs and set the transaction prices of energy (LMPs) and reserves (MCPs) at pricing nodes. DSOs operates retail transactive markets which feed back the bids of the total demand dispatch to TSO. The iterative and flexible interactions between the TSO and DSOs are modeled as a bi-level formulation which focuses on both minimizing the system operators' integrated purchasing cost and satisfying the integrated reserve requirement based on HGDC.
In the TSO level, the DSRs aggregator within each LDA operated by DSO are modeled as equivalent load-shifting and load-shedding DG entities with a flexible bidding willingness to engage in the energy market. To account its maximum of payoff, DSOs are under a market option whether to decrease PT-D to transact energy with LDAs or increase PT-D to transact energy with TSO. In the DSO level, TSO is modeled as a power injection source at the pricing node from the DSO's point of view.
To simulate the optimal transaction of TSO and DSOs, the formulation of network models is proposed, and it is assumed that every market player (TMP or DMP) is bidding close to their marginal clearing price with a price-taking strategy regulated in each iteration. Bidding is assumed to both reflect the market participants' true costs and represent its willingness of market behavior within its flexibility.

TSO Wholesale Market Simulation Model
The master problem is for TSO to be responsible for deciding energy T-LMPs and reserves T-MCPs for each TMP. The TSO wholesale market allows bidding for multiple products, namely, energy and regulation up/down reserve including DSOs' DR-based bids. The wholesale market model could be composed of the classical TSO SCMC model and DSOs' market-wise behavior constraints which is a MILP and non-convex problem.  By contrast with the centralized ISO, in this paper, the maximum function of DSO is considered. Each flexible load area is represented by one DSO as an equivalent prosumer (producer/consumer). DSO aggregates and coordinates all DSRs within each LDA. DSO submits a single bid to TSO through the equivalent individual T-D interface (substation) which is named the pricing-node [3]. The wholesale transactive market is operated by the TSO at the transmission level. TSO receives bids/offers from all TMPs and set the transaction prices of energy (LMPs) and reserves (MCPs) at pricing nodes. DSOs operates retail transactive markets which feed back the bids of the total demand dispatch to TSO. The iterative and flexible interactions between the TSO and DSOs are modeled as a bi-level formulation which focuses on both minimizing the system operators' integrated purchasing cost and satisfying the integrated reserve requirement based on HGDC.
In the TSO level, the DSRs aggregator within each LDA operated by DSO are modeled as equivalent load-shifting and load-shedding DG entities with a flexible bidding willingness to engage in the energy market. To account its maximum of payoff, DSOs are under a market option whether to decrease P T-D to transact energy with LDAs or increase P T-D to transact energy with TSO. In the DSO level, TSO is modeled as a power injection source at the pricing node from the DSO's point of view.
To simulate the optimal transaction of TSO and DSOs, the formulation of network models is proposed, and it is assumed that every market player (TMP or DMP) is bidding close to their marginal clearing price with a price-taking strategy regulated in each iteration. Bidding is assumed to both reflect the market participants' true costs and represent its willingness of market behavior within its flexibility.

TSO Wholesale Market Simulation Model
The master problem is for TSO to be responsible for deciding energy T-LMPs and reserves T-MCPs for each TMP. The TSO wholesale market allows bidding for multiple products, namely, energy and regulation up/down reserve including DSOs' DR-based bids. The wholesale market model could be composed of the classical TSO SCMC model and DSOs' market-wise behavior constraints which is a MILP and non-convex problem.
subject to In the TSO wholesale market, the objective in Equation (1) is to minimize the purchasing costs of both energy and reserve services in the transmission system. The power dispatch at boundary buses is sent from DSOs at each iteration. It is subject to transmission operational constraints, such as the system power balance constraint (Equation (2)), the generator minimum ON/OFF time and its ON/OFF status constraints (Equations (3)-(6)), the generator startup and shutdown cost constraints (Equations (7) and (8)), the generator ramp-rate limit constraint (Equation (9)), the line capacity constraints (Equation (10)), the generator reserve power constraints (Equation (11)), the system reserve power balance constraints (Equation (12)) and the feasibility-check constraint of anti-peak regulation with wind power (Equation  The DSRs aggregator participates in the energy transaction with TSO as a DR-based prosumer while meeting the regulation reserve requirement. In the energy-only market, it is assumed that all MPs are bidding their prices according to cost characteristics or bidding their quantities according to operation modes. Thus, the DSRs aggregator's DR-based bidding willingness could be simulated. In Equation (14), by contrast with the last iteration (q -1)-th results, the decreased value of the T-D k boundary power in q-th iteration is ∆P D k ,q,t (the positive value means decreased). In Equation (15), LMP D k q−1,t is the known LMP at the pricing node of the DSO's side in the last (q -1)-th iteration, LMP T q−1,t is the known LMP at pricing node of the TSO's side in the last (q -1)-th iteration. DSRs aggregator's bidding willingness behavior coefficient σ E T−D k ,q,t in the q-th iteration has its limits, as shown in Equation (16). The bidding willingness coefficient is decided by the coordination of the coupling relationship between the variation value of the T-D k boundary power and the energy LMP gap on the pricing node of the TSO and k-th DSO side. In Equation (15), if energy willing bidding price of the DSRs aggregator in the next iteration exceeds T-LMP when the DSRs aggregator is treated as a price-taker in the last iteration (the bidding coefficient equals to 1), then the TSO would decrease its T-D k boundary power which means ∆P D k ,q,t > 0 because of the DSRs aggregator's increased bidding willingness, and vice versa.

DSO Retail Market Model
The slave problem is to minimize the purchasing cost of DSOs based on T-LMPs on the pricing nodes and D-LMPs of DSOs in the last iteration, as shown in Equation (17). Iteratively, each DSO provides a flexible demand response which is a dynamic priced-based coordination with the TSO for energy transaction and reserve sharing considering the game of benefit between the wholesale market and retail markets. Different modeling methods could be used in DSOs only if the clearing price in TSO is properly evaluated.

MILP-Based DSO Model
Besides the normal start-up/shut-down constraints of DGs which is the same formulation (Equations (3)- (9) and (11)) with X = {D}, the model is also subject to the nodal power balance equation constraint at the boundary nodes (Equation (18)), the nodal power balance equation constraint at inner nodes (Equation (19)), the T-D k power limitation constraints (Equation (20)), the line power equation with its limits constraint (Equations (21) and (22)), the DGs' dispatch variation at iteration q (Equation (23)), the maximum DGs' load curtailment constraint in iteration q within the permitted time periods with the help of storage installed as DR (Equation (24)), the maximum demand time interval ramping constraint of DSO (Equation (25)), the zonal regulation-up/regulation-down reserve constraints of the k-th DSO (Equations (26) and (27)), the DGs' generation and their regulation reserve limitation constraints (Equations (28) and (29)

MISOCP-Based DSO Open-Loop Reconfiguration Model
To simulate the characteristics of open-loop operation topology and its active reconfiguration with remotely controllable switches as DR, the MI-SOCP model of DSO is proposed to elaborately rearrange its operation modes according to the transaction levels and contingency circumstances, if any. The MI-SOCP model in Reference [30] is simplified and applied in the distribution systems with enhancing advantages of reducing purchasing cost, eliminating transmission congestion and elaborating its power loss. The reactive power is simplified according to a fixed power factor.
The optimal DSO topology in the market environment is the transaction-oriented equivalent spanning tree. The necessary condition for network radiality is shown in Equation (30), which could be derived directly from below; Equations (31)-(35).
To reformulate the power flow constraints in terms of its continuous variables as a convex SOCP, auxiliary variables are introduced in Equations (36)-(38) with their upper bound and lower bound constraints in Equations (39)-(41), which are rewritten from the voltage limits constraints. When the power flow in line (i,j) inverses, T ij could be in its opposite direction. T ij = −T ji .
The original line power constraints (Equation (21)) is replaced by its SOCP-based reformulation (Equation (42)). The original line ampacity limits (Equation (22)) is replaced by the squared line current ampacity expression (Equation (43)). The rotated conic quadratic constraint is given by the constraint constituted of auxiliary variables (Equation (44)). i j ∈ set of lines = 1, · · · , N D l .
By contrast with the MILP model, the MI-SOCP model gives a more vivid and active illustration of the DSO network with controllable line switches as DR.

HGDC-Based T-D Iteration Convergence and its Optimality Testification
There is an intricate coupling relationship among the iteration-oriented T-D boundary variables. The slave problem of the DSO in the q-th iteration could be summarized as shown in Equation (45). The master problem of TSO in the (q + 1)-th iteration could be illustrated in Equation (46). The iteration-oriented LMP relationship between TSO and DSOs could be given in Equation (47).
Through the iteration calculation method, the results of market operation could be coordinated to its optimization when the T-LMPs(q) and T-LMPs(q + 1) are almost the same at convergence. More specifically, by solving the KKT conditions about P E T−D k ,t and P Z D k −T,t in the centralized and decentralized model, we can conclude that if LMP at convergence, the HGDC solution must satisfy the optimization condition according to Reference [13]. This optimal solution is testified in theory to meet the KKT condition of the centralized TDMO problem. If any security-based re-dispatch circumstances caused by contingencies occur, its optimal solution could be changed.

Decentralized Solution Based on HGDC Algorithm
For practical applications, estimating a reasonable value of boundary power is usually simpler than estimating T-LMPs. Thus, the iterative algorithm beginning from the TSO master problem may be more suitable for most cases. The initial guess of the boundary power could be represented as the total loads of the DSO if DGs are at the small scale or the difference between the DSO's total loads and generation forecast of the DGs if DGs are in the large scale. The decentralized solution is as follows.

1.
Initialize P Solve the TSO master problem and send the obtained LMPs and MCPs to all TMPs, including DSOs.

3.
With the updated LMP T−D k q,t and MCP T−D k Z,q,t with price-takers' bidding strategies, solve the slave problem of k-th DSO to obtain P T−D k q+1,t , which is sent to TSO.

4.
Make a convergence judgment.
. . . , T h }, then exit the iteration. Otherwise, q = q + 1, if q is no bigger than the maximum iteration number, go to step 1 to continue the iteration. If q exceeds the maximum iteration number, exit the iteration.

5.
End the iteration with the obtained optimal transaction results and DSRs aggregator's strategic bidding willingness coefficient.

Basic Idea for Transaction Cases
The basic data case is T5-3D33, which means the modified PJM 5-bus system (T5) [32] is used as the transmission system data, which are displayed in Figure 2 and three of same IEEE 33-bus systems (D33) in Reference [33] are used as the distribution system data. The controllable DGs with storage units installed as the DR in DSO are listed in the bus set: {3 7 8 12 18 20 25 31}. The maximum capacity of the storage is the same as that of DGs. All the loads of DGs in DSOs have shiftable/curtailment characteristics. The wind power and load profiles are simulated in TSO as shown in Figure 3, while the DSOs are coordinating with transmission-connected GenCos to cope with the power variation from both the load and wind in the wholesale market. The flexible transaction in DSOs' retail markets should satisfy the integrated regulation reserve requirement.  The wind power and load profiles are simulated in TSO as shown in Figure 3, while the DSOs are coordinating with transmission-connected GenCos to cope with the power variation from both the load and wind in the wholesale market. The flexible transaction in DSOs' retail markets should satisfy the integrated regulation reserve requirement. There are five cases which could be used to validate the proposed method and model as efficient to obtain the day-ahead 24 h optimal TDMO mode.
• Case 1: The TSO is independently operated from DSOs. The wholesale market is based on costbased energy bidding while considering the integrated reserve requirement. This is nominated as the independent transmission operation mode (IT-OM).

Comparison of the HGDC-Based TD-OM with IT-OM
The convergence situation of relative T-LMP errors at pricing nodes in Case 2 could be seen in Figure 4. By comparison of Case 1 with Case 2, it is testified that the economic operation welfare of TSO could be improved in an HGDC-based mode by $1359.95 (4857.67-3497.72) as shown in Table 1. There are five cases which could be used to validate the proposed method and model as efficient to obtain the day-ahead 24 h optimal TDMO mode.

•
Case 1: The TSO is independently operated from DSOs. The wholesale market is based on cost-based energy bidding while considering the integrated reserve requirement. This is nominated as the independent transmission operation mode (IT-OM).

•
Case 2: Based on Case 1, the TSO and DSOs are iteratively coordinated with a node power forecast of (P D -P dg ) as the first-iteration estimation of the T-D boundary power where P dg is the forecasted value of the DGs' total output in the distribution system. The estimation of the T-D power mismatch in the following iteration is zero. This is nominated as the transmission-distribution operation mode (TD-OM).

Comparison of the HGDC-Based TD-OM with IT-OM
The convergence situation of relative T-LMP errors at pricing nodes in Case 2 could be seen in Figure 4. By comparison of Case 1 with Case 2, it is testified that the economic operation welfare of TSO could be improved in an HGDC-based mode by $1359.95 (4857.67-3497.72) as shown in Table 1. By contrast with IT-OM, the HGDC-based operation mode could utilize the DSOs as a demand response to reduce the generation bidding-based cost OE and the expensive TSO generation units are preferentially less used or shut down along with a more economic set of unit commitment, inducing a lower OU. The more expensive G3 and G4 units are shut down or kept at a low generation level as is manifested in Table 2 Meanwhile, the inexpensive DGs with storages increase their output as shown in Figures 5 and 6. If we run Case 1, the IT-OM, in an iteratively way like Case 2, then the LMP price spike phenomenon could be eliminated, as seen in Figure 7. Therefore, the demand prices could be reasonably reflected by the HGDC-based algorithm.
preferentially less used or shut down along with a more economic set of unit commitment, inducing a lower OU. The more expensive G3 and G4 units are shut down or kept at a low generation level as is manifested in Table 2 Meanwhile, the inexpensive DGs with storages increase their output as shown in Figures 5 and 6. If we run Case 1, the IT-OM, in an iteratively way like Case 2, then the LMP price spike phenomenon could be eliminated, as seen in Figure 7. Therefore, the demand prices could be reasonably reflected by the HGDC-based algorithm.

Impact of Cost-Based Bidding Reserve on Energy Market
In an analysis of Case 1 and 2, the reserve cost is set as around 20$/MWh. While in Case 3, the units bid their basic reserve cost at 25$/MWh, which is generally above the nodal energy prices in Case 2. As illustrated in Table 3 and Figure 8, the purchasing cost of TSO increases in Case 3 by

Impact of Cost-Based Bidding Reserve on Energy Market
In an analysis of Case 1 and 2, the reserve cost is set as around 20$/MWh. While in Case 3, the units bid their basic reserve cost at 25$/MWh, which is generally above the nodal energy prices in Case 2. As illustrated in Table 3 and Figure 8, the purchasing cost of TSO increases in Case 3 by

Impact of Cost-Based Bidding Reserve on Energy Market
In an analysis of Case 1 and 2, the reserve cost is set as around 20$/MWh. While in Case 3, the units bid their basic reserve cost at 25$/MWh, which is generally above the nodal energy prices in Case 2. As illustrated in Table 3 and Figure 8, the purchasing cost of TSO increases in Case 3 by

Impact of Cost-Based Bidding Reserve on Energy Market
In an analysis of Case 1 and 2, the reserve cost is set as around 20$/MWh. While in Case 3, the units bid their basic reserve cost at 25$/MWh, which is generally above the nodal energy prices in Case 2. As illustrated in Table 3 and Figure 8, the purchasing cost of TSO increases in Case 3 by increasing the output of the relatively lower pricing units of GenCos (like G1) in TSO. Meanwhile, the regulation-down reserve of the units is decreased, and all units bid their downward reserve at relatively high pricing intervals. In total, with a higher reserve cost-based bidding price, the purchasing cost of the energy generation increases and that of the units' reserve decreases, leading to a higher total cost of the system. increasing the output of the relatively lower pricing units of GenCos (like G1) in TSO. Meanwhile, the regulation-down reserve of the units is decreased, and all units bid their downward reserve at relatively high pricing intervals. In total, with a higher reserve cost-based bidding price, the purchasing cost of the energy generation increases and that of the units' reserve decreases, leading to a higher total cost of the system.

Impact of Branch Outage on IT-OM and TD-OM
This part compares the differences between the (N-1)-OM based on the IT-OM/TD-OM power flows after the disturbance and the base TD-OM power flow before disturbance. Suppose that the disturbance in Case 4a,b is an outage in line 3-4 whose original power flow is −0.75 (p.u.). The new steady system state after the disturbance occurs is a redistribution of the original power flow. The IT-OM only uses the transmission controls, while the TDZO-OM could use all available transmission and distribution controls.
As shown in Figure 9, after the line goes through an outage, the incremental MW-flow resulting in Case 4 could be obtained by reflecting the different redistributions of the original line power flow. Different from Case 4a, the incremental MW-flow on line 1-5 in Case 4b is much smaller, which may alleviate the transmission congestion of line 1-5 and 1-2. Additionally, Case 4b also has more an equally distributed power transfer among the network. It is concluded that power flow is more reasonably redistributed, avoiding huge power transfers in critical power lines.

Impact of Branch Outage on IT-OM and TD-OM
This part compares the differences between the (N-1)-OM based on the IT-OM/TD-OM power flows after the disturbance and the base TD-OM power flow before disturbance. Suppose that the disturbance in Case 4a,b is an outage in line 3-4 whose original power flow is −0.75 (p.u.). The new steady system state after the disturbance occurs is a redistribution of the original power flow. The IT-OM only uses the transmission controls, while the TDZO-OM could use all available transmission and distribution controls.
As shown in Figure 9, after the line goes through an outage, the incremental MW-flow resulting in Case 4 could be obtained by reflecting the different redistributions of the original line power flow. Different from Case 4a, the incremental MW-flow on line 1-5 in Case 4b is much smaller, which may alleviate the transmission congestion of line 1-5 and 1-2. Additionally, Case 4b also has more an equally distributed power transfer among the network. It is concluded that power flow is more reasonably redistributed, avoiding huge power transfers in critical power lines. increasing the output of the relatively lower pricing units of GenCos (like G1) in TSO. Meanwhile, the regulation-down reserve of the units is decreased, and all units bid their downward reserve at relatively high pricing intervals. In total, with a higher reserve cost-based bidding price, the purchasing cost of the energy generation increases and that of the units' reserve decreases, leading to a higher total cost of the system.

Impact of Branch Outage on IT-OM and TD-OM
This part compares the differences between the (N-1)-OM based on the IT-OM/TD-OM power flows after the disturbance and the base TD-OM power flow before disturbance. Suppose that the disturbance in Case 4a,b is an outage in line 3-4 whose original power flow is −0.75 (p.u.). The new steady system state after the disturbance occurs is a redistribution of the original power flow. The IT-OM only uses the transmission controls, while the TDZO-OM could use all available transmission and distribution controls.
As shown in Figure 9, after the line goes through an outage, the incremental MW-flow resulting in Case 4 could be obtained by reflecting the different redistributions of the original line power flow. Different from Case 4a, the incremental MW-flow on line 1-5 in Case 4b is much smaller, which may alleviate the transmission congestion of line 1-5 and 1-2. Additionally, Case 4b also has more an equally distributed power transfer among the network. It is concluded that power flow is more reasonably redistributed, avoiding huge power transfers in critical power lines.   As illustrated in Table 4, by contrast with IT-OM-based (N-1)-OM, the TD-OM-based (N-1)-OM has more economic controls on unit commitment and generation schedule based on the HGDC algorithm.

MI-SOCP Model of DSO
The MI-SOCP model of DSO could be illustrated in three parts: power loss, the detailed simulation of the designated DSRs aggregator's bidding willingness in the wholesale market and network reconfiguration. To be simplified in Table 5, two cases are illustrated to show the impact of retailers' price bidding on the market performance of DSO and its strategic behavior of DSRs aggregator. The strategic bidding willingness coefficient for the DSRs aggregator in Case 5 is given in Figure 10, which varies as wind power varies. DSO's bidding-based payoff has a positive correlation with the retailers' price bidding. The lower the supply-demand ratio is, the higher the retail price bidding is, so the more willingness DSO would gain through a price-based drive to pursue its maximum payoff in the wholesale energy market. When the retail bidding price (25$/MWh) distinctly exceeds T-LMPs (average value is like 23$/MWh), DSO will buy more energy from TSO. Thus, the higher retailers' price bidding could expand the T-D market liquidity between the wholesale and retail market. As illustrated in Table 4, by contrast with IT-OM-based (N-1)-OM, the TD-OM-based (N-1)-OM has more economic controls on unit commitment and generation schedule based on the HGDC algorithm.

MI-SOCP Model of DSO
The MI-SOCP model of DSO could be illustrated in three parts: power loss, the detailed simulation of the designated DSRs aggregator's bidding willingness in the wholesale market and network reconfiguration. To be simplified in Table 5, two cases are illustrated to show the impact of retailers' price bidding on the market performance of DSO and its strategic behavior of DSRs aggregator. Table 5. Convergent ratio of local marginal prices (LMP).

Case Number Supply-Demand Ratio Retail Bids ($/MWh) Convergence Ratio
Case 5a  The strategic bidding willingness coefficient for the DSRs aggregator in Case 5 is given in Figure  10, which varies as wind power varies. DSO's bidding-based payoff has a positive correlation with the retailers' price bidding. The lower the supply-demand ratio is, the higher the retail price bidding is, so the more willingness DSO would gain through a price-based drive to pursue its maximum payoff in the wholesale energy market. When the retail bidding price (25$/MWh) distinctly exceeds T-LMPs (average value is like 23$/MWh), DSO will buy more energy from TSO. Thus, the higher retailers' price bidding could expand the T-D market liquidity between the wholesale and retail market.  Furthermore, as seen in Figure 11, the lower the supply-demand ratio is and the higher the price bidding of retailers (e.g., DGs), the bigger the power loss the DSO would have. Thus, the power loss could be decreased by improving the system's supply-demand ratio and imposing restrictions on the maximum retail prices. As the distribution network is always close-looped while in construction and open-looped while in operation, Figure 12 shows the branch switch reconfiguration status in the market operation environment under different retail bidding prices in which the 24 different colors mean the day-ahead 24 h. As the retail price decreases, the day-ahead switching frequency changes from 126 to 106 with a 15.9% ratio drop. Thus, by imposing restrictions on the maximum retail price, the remote switching frequency and the network loss could be decreased and the market liquidity between the wholesale and retail market could be enhanced. Meanwhile, the switch reconfiguration in the distribution system, together with the SCUC/SCED in the demand side, could be an efficient means of DR to reduce the purchasing cost of system operators. Furthermore, as seen in Figure 11, the lower the supply-demand ratio is and the higher the price bidding of retailers (e.g., DGs), the bigger the power loss the DSO would have. Thus, the power loss could be decreased by improving the system's supply-demand ratio and imposing restrictions on the maximum retail prices. As the distribution network is always close-looped while in construction and open-looped while in operation, Figure 12 shows the branch switch reconfiguration status in the market operation environment under different retail bidding prices in which the 24 different colors mean the day-ahead 24 h. As the retail price decreases, the day-ahead switching frequency changes from 126 to 106 with a 15.9% ratio drop. Thus, by imposing restrictions on the maximum retail price, the remote switching frequency and the network loss could be decreased and the market liquidity between the wholesale and retail market could be enhanced. Meanwhile, the switch reconfiguration in the distribution system, together with the SCUC/SCED in the demand side, could be an efficient means of DR to reduce the purchasing cost of system operators.   4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Conclusions
In this paper, a G-MSS method based on HGDC is applied as a decentral solution to multi-period integrated TDMO for its first time. With the idea of HGDC, the demand side retail markets in distribution systems are decomposed from the wholesale market in the transmission system, which means the TSO coordinates with several DSOs. The local optimization and convergence properties for this application are proven, which indicate that HGDC is an efficient algorithm with limited communication burdens to solve the decentralized TDMO problem. Besides the optimal transactive dispatch, the DSRs aggregator's strategic bidding willingness coefficient and its market-wise decision-making for reserve sharing responsibility are simulated in the wholesale market. What is Furthermore, as seen in Figure 11, the lower the supply-demand ratio is and the higher the price bidding of retailers (e.g., DGs), the bigger the power loss the DSO would have. Thus, the power loss could be decreased by improving the system's supply-demand ratio and imposing restrictions on the maximum retail prices. As the distribution network is always close-looped while in construction and open-looped while in operation, Figure 12 shows the branch switch reconfiguration status in the market operation environment under different retail bidding prices in which the 24 different colors mean the day-ahead 24 h. As the retail price decreases, the day-ahead switching frequency changes from 126 to 106 with a 15.9% ratio drop. Thus, by imposing restrictions on the maximum retail price, the remote switching frequency and the network loss could be decreased and the market liquidity between the wholesale and retail market could be enhanced. Meanwhile, the switch reconfiguration in the distribution system, together with the SCUC/SCED in the demand side, could be an efficient means of DR to reduce the purchasing cost of system operators.   4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Conclusions
In this paper, a G-MSS method based on HGDC is applied as a decentral solution to multi-period integrated TDMO for its first time. With the idea of HGDC, the demand side retail markets in distribution systems are decomposed from the wholesale market in the transmission system, which means the TSO coordinates with several DSOs. The local optimization and convergence properties for this application are proven, which indicate that HGDC is an efficient algorithm with limited communication burdens to solve the decentralized TDMO problem. Besides the optimal transactive dispatch, the DSRs aggregator's strategic bidding willingness coefficient and its market-wise decision-making for reserve sharing responsibility are simulated in the wholesale market. What is

Conclusions
In this paper, a G-MSS method based on HGDC is applied as a decentral solution to multi-period integrated TDMO for its first time. With the idea of HGDC, the demand side retail markets in distribution systems are decomposed from the wholesale market in the transmission system, which means the TSO coordinates with several DSOs. The local optimization and convergence properties for this application are proven, which indicate that HGDC is an efficient algorithm with limited communication burdens to solve the decentralized TDMO problem. Besides the optimal transactive dispatch, the DSRs aggregator's strategic bidding willingness coefficient and its market-wise decision-making for reserve sharing responsibility are simulated in the wholesale market. What is more, by contrast with the MILP model, the MI-SOCP model of DSO is applied to elaborate its power loss and simulate its active open-loop reconfiguration as a means of DR. The enhancing advantages of the method and model to reduce the purchasing cost and eliminate transmission congestion are also testified in the cases.
To tackle the uncertainties rising from the forecast of bidding prices and power transaction, in future work, a stochastic/robust method is needed to cope with these dynamic characteristics to keep its fast convergence in both theory and field work. Funding: This paper is supported by the scientific and technology program of "Key technology research and application of spot market with abundant renewable energy at inter-regional and inter-provincial areas" from State Grid Corporation of China and the program "Smart grid applications to the design and operation of electricity grids" from Illinois Institute of Technology in U.S.A.

Conflicts of Interest:
The authors declare no conflict of interest.

Sets and indices: t
Index for multi-period hours from t = 1 to t = T h . (i, j) Index for line with from bus i and to bus j. q Index for iteration number of HGDC.
X Index X∈{T, D k , T-D k , D k -T}, where T is for transmission system operator (TSO), D k for the k-th distribution system operator (DSO), T-D k for boundary variables with direction from TSO to D k and D k -T for boundary variables with direction from D k to TSO.
Y Index Y T ∈{g, w, D, td} for market players of TSO, e.g., GenCos(g), wind generators(w), DSOs(D). and transmission load(td). Index Y D ∈{dg, dd} for market players of DSOs, e.g., distribution generators(dg), distribution demand(dd). Purchasing cost of services for system operators. P Decision variables for service Z transaction dispatch (MW) of market player Y in X.

F Z X,Y
Bidding-based cost of market player Y in X for service Z which is the multiply of bidding price and its service deployment quantity.
σ Z X,Y Bidding willingness coefficient of market player Y for service Z which equals to 1 for price takers.