Abstract
The volatility of increasing distributed generators (DGs) poses a severe challenge to the supply restoration of active distribution networks (ADNs). The integration of power electronic devices represented by soft open points (SOPs) and mobile energy storages (MESs) provides a promising opportunity for rapid supply restoration with high DG penetration. Oriented for the post-event rapid restoration of ADNs, a bi-level supply restoration method is proposed considering the multi-resource coordination of switches, SOPs, and MESs. At the upper level (long-timescale), a multi-stage supply restoration model is developed for multiple resources under uncertainties of DGs and loads. At the lower level (short-timescale), a rolling correction restoration strategy is proposed to adapt to the DG and load fluctuations on short timescales. Finally, the effectiveness of the proposed method is verified based on a modified practical distribution network and IEEE 123-node distribution network. Results show that the proposed method can fully utilize the coordination potential of multiple resources to improve load restoration ratio for ADNs with DG uncertainties.
DISTRIBUTION networks play a crucial role in the power supply for users, necessitating high reliability. Therefore, it is crucial to realize the rapid post-event supply restoration of distribution networks [
The integration of power electronic devices represented by soft open points (SOPs) and mobile energy storages (MESs) provides a promising opportunity for rapid supply restoration with high penetration of DGs. ADNs provide voltage and power support to outage regions by intelligently switching control modes of flexible resources, providing an opportunity for rapid post-event restoration with high penetration of DGs [
SOPs, which replace traditional tie switches, play an important role in ADNs [
Supply restoration methods for ADNs based on SOPs have been extensively studied. A bi-level interval robust optimization model was proposed in [
The integration of flexible resources into DC link has also been investigated to improve the regulatory capability of SOPs. Reference [
Traditional mobile emergency generators (e.g., mobile diesel generators) can be connected to the grid via AC modular to provide energy to the outage regions. Reference [
There are extensive research works on the operational optimization strategies for MESs in ADNs. A two-step optimal allocation model is proposed to achieve the optimal allocation of MESs and reduce the annual cost of ADNs in [
Supply restoration with multi-resource coordination has been investigated to fully exploit the potential of flexible resources. Reference [
The multi-stage concept has been introduced to achieve the coordination of multiple types of resources on multiple timescales. Reference [
To address these problems, a bi-level supply restoration method for ADNs that considers multi-resource coordination is proposed, as shown in

Fig. 1 Schematic of proposed bi-level supply restoration method for ADNs.
The main contributions are summarized as follows.
1) A bi-level supply restoration method of multi-terminal SOP with MES integration is proposed. The SOP mode is selected to ensure the coordination with MESs in ADNs.
2) At the upper level, a multi-stage supply restoration model with uncertainty of DGs and loads is developed for multiple resources. The strategies of switch operation and selection of SOP control mode considering the uncertainties of DGs and loads are determined to maximize load restoration on a long timescale.
3) At the lower level, a rolling correction restoration strategy is proposed for supply restoration of ADN. The power outputs of SOPs and MESs are corrected to adapt to DG and load fluctuations and varied traffic conditions, thereby improving the efficiency of short-timescale supply restoration.
The remainder of this paper is organized as follows. In Section II, the bi-level supply restoration method of multi-terminal SOP with MES integration is developed. Section III proposes a multi-stage supply restoration model at the upper level. A rolling correction restoration strategy at the lower level is proposed in Section IV. Case studies on a modified practical distribution network are presented in Section V. Finally, conclusion is given in Section VI.
In this section, the bi-level supply restoration method of multi-terminal SOP with MES integration is proposed to achieve the coordination between SOP and MES. Considering the impact of the peak intervals of the transport network, the MES operation model is modified to ensure the effectiveness of the proposed method.
Based on a fully-controlled voltage source converter (VSC), each SOP terminal can be operated with multiple control modes. The SOP control method without MES integration has been analyzed in [
Fault scenario | Control mode | |
---|---|---|
VSC 1 | VSC 2 | |
Fault on VSC 1 side | ||
Fault on VSC 2 side | ||
Faults on both sides |
As presented in

Fig. 2 Schematic of control modes of SOP terminals under faults.
As can be seen in
Based on the above analysis, the restoration method of multi-terminal SOP with MES integration is modeled as:
(1a) |
(1b) |
where and are the sets of MESs and nodes with SOP terminals, respectively; is the set of MES integration points; and indicate whether the SOP terminal provides voltage support in time interval ; is the binary variable indicating the integration status of MES at the DC link of SOP in time interval ; and are the voltage square of node in time interval and the voltage reference square value, respectively; and is a large constant for modeling.
(2a) |
(2b) |
(2c) |
(2d) |
(2e) |
(2f) |
where and are the active and reactive power outputs of SOP at node in time interval , respectively; is the active power output of MES in time interval ; and are the active power losses of SOP at node i and MES in time interval ; , , and are the loss coefficients of SOP terminals, respectively; and are the minimum and maximum active power outputs of SOP, respectively; and are the minimum and maximum reactive power outputs of SOP, respectively; and is the maximum capacity of SOP.
The power transfer constraint among the terminals of SOP is denoted by (2a) with MES integration. Equations (
(3a) |
(3b) |
The MES operation model includes the charging/discharging and mobility constraints [
(4a) |
(4b) |
(4c) |
(4d) |
(4e) |
where is the rated power of MES; is the state of charge (SOC) of MES in time interval ; is the binary variable indicating the travelling status of MES in time interval ; is the MES power consumption per unit distance; is the speed of the MES; is the unit time interval; and and are the minimum and maximum SOCs of MES m, respectively.
Equations (
(5a) |
(5b) |
(5c) |
(5d) |
where is the initial position of MES ; is the maximum number of MESs connected; is the travelling time from node to node in time interval ; and is the number of time intervals in the operating cycle.
Additionally, the peak and off-peak intervals of the transport network affect the MES scheduling strategies. The travelling time of the MES is modified by:
(6) |
where and are the travelling time of MESs in peak and off-peak time intervals of transport network, respectively, which can be obtained by Dijkstra algorithm; and are the start-of-the-peak and off-peak time intervals, respectively; and and are the sets of peak and off-peak time intervals, respectively.
In this section, the multi-stage supply restoration model of ADN is formulated at the upper level. Considering the uncertainties of DGs and loads, the strategies of switch operation and the selection of SOP control mode are obtained.
Before stage division, the ADN is divided into several regions based on the positions of faults and switches to reduce the number of binary variables. The regional division method is described in [
(7a) |
(7b) |
(7c) |
(7d) |
where is the set of nodes in region ; is the set of candidate fictitious source nodes; is the set of switches; and are the sets of all nodes and substations, respectively; and are the upstream and downstream regions of region , respectively; is the binary variable indicating energized status of region at stage ; and are the fictitious flow variables of line and at stage , respectively, for the energized status constraints; indicates whether the node located in region provides voltage support at stage ; and is the binary variable indicating the status of switch at stage .
Equations (
As indicated in (7a)-(7d), the energized status of outage regions is related to switch status and SOP control mode. Considering the slow action of switches, the supply restoration process is divided into several stages based on switch operation intervals. The time-stage mapping constraints are given by:
(8a) |
(8b) |
where is the set of restoration stages; is the binary variable indicating whether time interval is related to stage ; is the starting time of stage ; and is the operation time of switches .
A multi-stage supply restoration model is established with coordinated operation of flexible devices.
The objective function of the multi-stage supply restoration model is given by:
(9a) |
(9b) |
(9c) |
(9d) |
(9e) |
where and are the sets of time intervals and lines, respectively; is the cost of the unrestored load; is the cost of power loss; is the flexible resource operation cost; is the operation cost of switching action; is the restoration ratio of each node in time interval ; , , and are the unit prices of load loss, power purchasing, and switch operation, respectively; is the resistance of line ; and is the current square of line in time interval .
The total cost during the supply restoration of ADN is represented by (9a).
The modified fictitious flow constraints are adopted to ensure the radial operation of ADN, as shown in (10a)-(10h).
(10a) |
(10b) |
(10c) |
(10d) |
(10e) |
(10f) |
(10g) |
(10h) |
where and are the total numbers of outage regions and substations, respectively; , , and are the fictitious flow variables of line , , and at stage , respectively; and is the binary variable indicating whether node is a fictitious source node at stage . The fictitious source nodes are used to ensure the radial operation of the ADN and have no actual physical meaning. If , node is the fictitious source node at stage . Otherwise, node is a regular node.
Equations (
Considering the limitation of (10a), one node from each region can be selected as the candidate fictitious source node. is used to represent the set composed of the candidate fictitious source nodes. As a result, the number of binary variables is reduced to the number of regions, decreasing the total number of binary variables. To ensure the reliable operation of the system, the nodes capable of providing voltage support are selected as candidate fictitious source nodes. If none of nodes in the region can provide voltage support, any node can be selected as the candidate fictitious source node. The detailed explanation of (10a) is shown in Supplementary Material A.
The DistFlow model is used to describe power flow constraints of ADN [
(11a) |
(11b) |
(11c) |
(11d) |
(11e) |
(11f) |
(11g) |
(11h) |
(11i) |
(11j) |
(11k) |
(11l) |
(11m) |
where is the set of faulty lines; and are the active and reactive power flows of line in time interval , respectively; and are the active and reactive power outputs of DG at node in time interval , respectively; and are the restored active and reactive power at node in time interval , respectively; is the reactance of line ; is the maximum current of line ; and and are the minimum and maximum voltages of node , respectively.
Equations (
The load and DG constraints are given in (12a)-(12c).
(12a) |
(12b) |
(12c) |
where is the predicted active power generated by DG in time interval ; and is the power factor of DG.
The restoration ratios of nodes are constrained in (12a). Equations (
The SOP operation constraints with MES integration are shown in (1a)-(3b).
(13) |
where is the binary variable indicating whether the SOP terminal provides voltage support at stage .
The operation constraints of MES are shown in (4a)-(5d). Thus, the proposed multi-stage supply restoration model can be expressed as (14), subject to (2a)-(8b), (10a)-(13).
(14) |
The strategies of switch operation and selection of SOP control mode can be derived by solving the above model on the long timescale. The operation variables used to derive the lower-level restoration model are , , and .
Uncertainties of DGs and loads will affect the effectiveness of strategies. Thus, the uncertainty set is established based on the typical scenarios to describe the uncertainties of DGs and loads, as shown in (15).
(15) |
where is the element of the uncertainty set ; is the number of scenarios; is the probability of scenario ; is the predicted probability of scenario ; is the probability volatility coefficient in scenario ; and and are the upper and lower limits of the volatility coefficient in scenario , respectively.
Based on the uncertainty set, a min-max bi-layer model is formulated in (16).
(16) |
where and are the constraint sets of variables and , respectively; and is the objective function of scenario .
The model can be further linearized by duality theory [
(17a) |
s.t.
(17b) |
(17c) |
(17d) |
(17e) |
(17f) |
(17g) |
where , , , , , , and G are the coefficient matrices of constraints; , , , , , and are constant column vectors; is the set of fictitious flow variables and power flow variables in scenario ; is the binary variable vector for switch status and selection of SOP control mode; is the set of variables related to SOPs, MESs, and DGs in scenario ; and is the load demand in scenario .
Equations (
Based on the uncertainty handling method, the uncertainties of DGs and loads in ADNs are modeled to enhance the effectiveness of the restoration method.
Note that the solution of multi-stage supply restoration model may be time-consuming. To ensure timely strategy determination in practice, the strategies of switch operation and selection of SOP control mode at stage 1 can be determined based only on the current system state. Subsequently, the multi-timescale strategies of the subsequent stages can be derived based on the results of stage 1.
In this section, the rolling correction restoration strategy is proposed at the lower level. The restoration strategies of SOPs and MESs are corrected by dynamically updating the information of the transport network and ADN in each time interval.
The framework of the rolling correction restoration strategy is illustrated in

Fig. 3 Framework of rolling correction restoration strategy.
The restoration model for each rolling correction is shown in (18) without uncertainties, subject to (2a)-(6), (11a)-(12c). The operation variables required to be output are , , , , , and .
(18) |
Based on the switch status and SOP control mode derived from the upper level, the power transfer of flexible resources and the travelling route of MES can be timely obtained at the lower level.
However, the determined travelling route of MES cannot be easily changed during the correction horizon. To ensure the effectiveness of MES scheduling, the uncertainty within prediction horizon also needs to be resolved by (17), and becomes the binary variable vector for the MES position.
The update of the transport network contains two aspects: the update of the current traffic conditions of each road and the update of the current MES positions. When the MES is travelling on the transport network, the current MES position cannot be obtained only by (5a)-(5d). Consequently, a virtual traffic node is introduced to represent the real-time MES position in mobility status, as shown in
(19) |

Fig. 4 Transport network with virtual traffic node.
where is the current rolling moment; is the current MES position; and indicates that the MES is in a virtual node at current time interval.
The flowchart of the rolling correction restoration strategy considering the update of transport network is shown in

Fig. 5 Flowchart of rolling correction restoration strategy.
Based on the updated information, the transport network matrix is calculated for MES travelling route selection. The scheduling strategies of SOPs and MESs can be rolling corrected by solving the restoration model in current time interval.
The proposed bi-level supply restoration model is a mixed-integer nonlinear programming (MINLP) model essentially, which is difficult to solve directly. Based on the second-order cone transformation and convex relaxation, the MINLP model is transformed into a mixed-integer second-order cone programming (MISOCP) model, which can be effectively solved by commercial solvers. Besides, the is adopted to evaluate the accuracy of the result, as shown in (20). The model is considered to be solved exactly when the value is close to 0 [
(20) |
In this section, the effectiveness of the proposed bi-level supply restoration method is verified on a modified practical distribution network. The proposed method is implemented in the YALMIP optimization toolbox with MATLAB R2020b and solved by Gurobi 10.0.1. The numerical experiments are performed on a computer with an Intel Core i7-11700 CPU @ 2.50 GHz and 16 GB of RAM.
The structure of the modified practical distribution network along with the transport network is shown in

Fig. 6 Structure of modified practical distribution network with transport network.
The photovoltaic (PV) units and wind turbines (WTs) are integrated into the modified practical distribution network separately. The power factor of DG is set to be 1.0, and the specific information is shown in
Type | Node | Capacity |
---|---|---|
WT | 4, 5, 41, 47, 48 | 800 kVA |
PV | 2, 3, 12, 13, 61, 62, 63 | 600 kWp |
To consider the impact of the uncertainties of the DGs and loads, 1000 scenarios are generated based on normal state, and three typical scenarios are obtained via clustering. The predicted DG and load prediction output curves are given in Fig. SC1 of Supplementary Material C.
In addition, three SOPs are installed in the distribution network. The capacity of each terminal is set to be 3.5 MVA with a loss coefficient of 0.01, and MES can be integrated with SOP through the DC link. The maximum MES charging or discharging power is 1 MW with a capacity of 1 MWh. The upper and lower limits of SOCs are designed to be 95% and 20%, respectively, and the initial SOC is set to be 60%. It is assumed that MES consumes 0.75 kWh of electricity per unit time interval while travelling.
Part of the traffic network in
Assuming that a fault occurs in the main transformer on the left side, the fault restoration time is from 10:00 to , and nodes 1-41 are de-energized. After the fault occurs, switches S10-S14 are opened to isolate the fault.
The proposed bi-level supply restoration method is conducted to minimize the outage loss during the restoration process.
The restoration of multiple resources is analyzed based on the proposed bi-level supply restoration method. The diagram of restoration configuration in outage regions is presented in

Fig. 7 Restoration configuration in outage regions.
Note that nodes 22 and 33 are connected through SOP to enable the flexible interconnection between Feeders 3 and 4. Considering the flexibility of the SOP, the loop structure formed by the SOP is permitted in distribution networks. Therefore, the topology illustrated in

Fig. 8 Supply restoration for SOPs.

Fig. 9 Supply restoration of MESs.
The relaxation deviation value is shown in Fig. SC3 of Supplementary Material C, and the error is within tolerance. The model has been exactly solved.
The following three schemes are used to verify the effectiveness of the proposed supply restoration method.
1) Scheme I: single-stage supply restoration method is conducted without multi-stage restoration of ADN.
2) Scheme II: multi-stage supply restoration model adopted in [
3) Scheme III: the proposed bi-level supply restoration method for ADN is conducted.
The operation costs of the three schemes are detailed in
Scheme | Total cost ($) | Unrestored load cost ($) | Operation power loss cost ($) | MES loss cost ($) | SOP loss cost ($) | Switch action cost ($) |
---|---|---|---|---|---|---|
I | 431994.52 | 431964.60 | 11.38 | 0.30 | 17.24 | 1 |
II | 160368.90 | 160318.26 | 23.19 | 1.67 | 21.78 | 4 |
III | 24232.80 | 24178.72 | 29.60 | 1.62 | 18.86 | 4 |
The restoration ratios and operation strategies of SOP at different stages are shown in
Scheme | Stage | Restoration ratio (%) | Operation strategy |
---|---|---|---|
I | 1 (10:00-14:00) | 48.76 |
1) Close T5. 2) Terminals at both ends of the SOP-1 adopt control mode. 3) SOP-2 and SOP-3 adopt - control mode. |
II | 1 (10:00-10:30) | 63.53 |
1) Close T2 and T5. 2) Terminals at both ends of SOP-1 adopt control mode. 3) SOP-2 adopts - control mode. 4) SOP-3 adopts - control mode. |
2 (10:31-14:00) | 82.92 |
1) Close T3 and T4. 2) Terminal at node 22 adopts control mode and terminal at node 33 adopts control mode 3) SOP-2 and SOP-3 adopt - control mode. | |
III | 1 (10:00-10:30) | 70.58 |
1) Close T2 and T5. 2) Terminals at both ends of SOP-1 adopt control mode. 3) SOP-2 adopts - control mode. 4) SOP-3 adopts - control mode. |
2:(10:31-14:00) | 100.00 |
1) Close T3 and T4. 2) Terminal at node 22 adopts control mode and terminal at node 33 adopts control mode. 3) SOP-2 and SOP-3 adopt - control mode. |
The specific node restoration ratios of the above three schemes in fault regions are shown in Figs.

Fig. 10 Restoration ratio of nodes in Scheme I.

Fig. 11 Restoration ratio of nodes in Scheme II.

Fig. 12 Restoration ratio of nodes in Scheme III.
The restoration strategies for SOPs and MESs in Scheme I and Scheme II are given in Figs. SC4-SC7 of Supplementary Material C, respectively. The above results show that the proposed bi-level supply restoration method reduces the impact of uncertainty and improves supply restoration efficiency.
Scheme | Level | Computation time (s) |
---|---|---|
I | 13.09 | |
II | 37.56 | |
III | Upper level | 93.30 |
Lower level | 38.65 |
The computation time of Scheme I is the shortest. However, the strategies derived from Scheme I ignore the sequential coordination of multiple resources in ADN, where a serious loss of load is exhibited. The sequential coordination of multiple resources at different timescales is realized in Scheme II, and the capability of restoration is greatly improved by the multi-stage supply restoration model. Nevertheless, there are more binary variables and related constraints, and the solution time is about half a minute.
Scheme III adopts the proposed bi-level supply restoration method. At the upper level, the multi-stage action strategies for switches on the long timescale are derived. Due to the consideration of the uncertainties, the solution of the multi-stage supply restoration model is time-consuming. Thus, the initial strategies of switch operation and selection of SOP control mode at stage 1 are determined only based on the current system state, and the computation time of stage 1 is 1.05 s. The duration of stage 1 is 30 min. The solution time for the switch operation and selection of SOP control mode at stage 2 is 92.25 s, which is within 30 min, ensuring the timely determination of strategies at stage 2.
At the lower level, the rolling correction restoration strategy is proposed to adapt to DG and load fluctuations and varied traffic conditions, and the rolling optimization interval is set to be 15 min. Besides, the mean computation time for each rolling optimization is approximately half a minute, which fulfills the requirements in practice.
The proposed bi-level supply restoration method is further implemented on the modified IEEE 123-node distribution network in

Fig. 13 Structure of modified IEEE 123-node distribution network.
According to the optimization results, the duration of stage 1 is from 14:00 to 14:30, and the duration of stage 2 is from 14:31 to 18:00. The strategies of switch operation and selection of SOP control mode are also shown in
Scheme | Total cost ($) | Unrestored load cost ($) | Operation power loss cost ($) | MES loss cost ($) | SOP loss cost ($) | Switch action cost ($) |
---|---|---|---|---|---|---|
I | 9276.21 | 9238.75 | 27.00 | 3.34 | 6.12 | 1 |
II | 942.67 | 886.80 | 43.34 | 1.18 | 10.35 | 1 |
III | 422.99 | 369.40 | 41.33 | 1.18 | 10.08 | 1 |
As for the computation performance in IEEE 123-node distribution network, the computation time of the restoration strategies at stage 1 is less than 3 s at the upper level. As the scale of the test case expands, the number of related variables increases, resulting in longer computation time. The computation time of stage 2 is approximately 331 s, which also ensures the timely implementation of the strategy.
At the lower level, the mean computation time for each rolling optimization is approximately 1.5 min. The lower-level model includes binary variables only related to the MES, and the computation time is less affected by the scales of ADNs. Hence, the computation time grows approximately linearly.
In summary, the solution time of the proposed method can meet the practical application and improve the performance of supply restoration.
This paper proposes a bi-level supply restoration method for ADNs considering multi-resource coordination. As validated in test cases, the SOP restoration with MES integration achieves the coordination of SOP and MES in spatial and temporal dimensions. At the upper level, the multi-stage supply restoration model can derive the reliable switch and SOP control method on the long timescale. At the lower level, the rolling correction restoration strategy responds to the fluctuations of loads and DGs on the short timescale. The proposed method fully explores the potential of multiple resources for ADN restoration and further improves the restoration efficiency of ADN.
Several directions can be outlined for future research. The efficiency of supply restoration in distribution networks is affected by the positions and capacity of MESs and SOPs, so it is important to realize the coordinated planning of MESs and SOPs. In addition, considering the uncertainties from the fault locations and the SOC of MES, the uncertainty model will be further studied to enhance the effectiveness of the restoration method. Moreover, supply restoration based on data-driven methods should be further investigated.
References
A. Bosisio, A. Berizzi, D. Lupis et al., “A tabu-search-based algorithm for distribution network restoration to improve reliability and resiliency,” Journal of Modern Power Systems and Clean Energy, vol. 11, no. 1, pp. 302-311, Jan. 2023. [Baidu Scholar]
Q. Zhao, W. Liao, S. Wang et al., “Robust voltage control considering uncertainties of renewable energies and loads via improved generative adversarial network,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 6, pp. 1104-1114, Nov. 2020. [Baidu Scholar]
J. Zhao, Z. Zhang, H. Yu et al., “Cloud-edge collaboration-based local voltage control for DGs with privacy preservation,” IEEE Transactions on Industrial Informatics, vol. 19, no. 1, pp. 98-108, Jan. 2023. [Baidu Scholar]
X. Yang, C. Xu, J. Wen et al., “Cooperative repair scheduling and service restoration for distribution systems with soft open points,” IEEE Transactions on Smart Grid, vol. 14, no. 3, pp. 1827-1842, May 2023. [Baidu Scholar]
A. Azizi, B. Vahidi, and A. F. Nematollahi, “Reconfiguration of active distribution networks equipped with soft open points considering protection constraints,” Journal of Modern Power Systems and Clean Energy, vol. 11, no. 1, pp. 212-222, Jan. 2023. [Baidu Scholar]
A. Tao, N. Zhou, Y. Chi et al., “Multi-stage coordinated robust optimization for soft open point allocation in active distribution networks with PV,” Journal of Modern Power Systems and Clean Energy, vol. 11, no. 5, pp. 1553-1563, Sept. 2023. [Baidu Scholar]
C. Wang, J. Sun, M. Huang et al., “Two-stage optimization for active distribution systems based on operating ranges of soft open points and energy storage system,” Journal of Modern Power Systems and Clean Energy, vol. 11, no. 1, pp. 66-79, Jan. 2023. [Baidu Scholar]
Z. Yin, S. Wang, and Q. Zhao, “Sequential reconfiguration of unbalanced distribution network with soft open points based on deep reinforcement learning,” Journal of Modern Power Systems and Clean Energy, vol. 11, no. 1, pp. 107-119, Jan. 2023. [Baidu Scholar]
W. Liu, M. Fu, M. Yang et al., “A bi-level Interval robust optimization model for service restoration in flexible distribution networks,” IEEE Transactions on Power Systems, vol. 36, no. 3, pp. 1843-1855, May 2021. [Baidu Scholar]
P. Li, J. Ji, H. Ji et al., “Self-healing oriented supply restoration method based on the coordination of multiple SOPs in active distribution networks,” Energy, vol. 195, p. 116968, Mar. 2020. [Baidu Scholar]
T. Zhang, X. Yu, Y. Mu et al., “Multiple sources restoration for soft open points in distribution networks with a two-stage accelerated algorithm,” IEEE Transactions on Sustainable Energy, vol. 14, no. 2, pp. 892-905, Apr. 2023. [Baidu Scholar]
J. Zhao, J. Qu, H. Ji et al., “Robust operation of flexible distribution network with large-scale EV charging loads,” IEEE Transactions on Transportation Electrification, vol. 10, no. 1, pp. 2207-2219, Mar. 2024. [Baidu Scholar]
S. Zhang, Y. Fang, H. Zhang et al., “Maximum hosting capacity of photovoltaic generation in SOP-based power distribution network integrated with electric vehicles,” IEEE Transactions on Industrial Informatics, vol. 18, no. 11, pp. 8213-8224, Nov. 2022. [Baidu Scholar]
C. Wang, J. Sun, M. Huang et al., “Two-stage optimization for active distribution systems based on operating ranges of soft open points and energy storage system,” Journal of Modern Power Systems and Clean Energy, vol. 11, no. 1, pp. 66-79, Jan. 2023. [Baidu Scholar]
W. Sun, W. Liu, J. Zhang et al., “Bi-level optimal operation model of mobile energy storage system in coupled transportation-power networks,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 6, pp. 1725-1737, Nov. 2022. [Baidu Scholar]
Z. Wang, Q. Shi, K. Fan et al., “Analytical modeling of disaster-induced load loss for preventive allocation of mobile power sources in urban power networks,” Journal of Modern Power Systems and Clean Energy, vol. 12, no. 4, pp. 1063-1073, Jul. 2024. [Baidu Scholar]
K. Lai and L. Zhang, “Sizing and siting of energy storage systems in a military-based vehicle-to-grid microgrid,” IEEE Transactions on Industry Applications, vol. 57, no. 3, pp. 1909-1919, May-Jun. 2021. [Baidu Scholar]
M. A. Masrur, A. G. Skowronska, J. Hancock et al., “Military-based vehicle-to-grid and vehicle-to-vehicle microgrid – system architecture and implementation,” IEEE Transactions on Transportation Electrification, vol. 4, no. 1, pp. 157-171, Mar. 2018. [Baidu Scholar]
X. Jiang, J. Chen, W. Zhang et al., “Two-step optimal allocation of stationary and mobile energy storage systems in resilient distribution networks,” Journal of Modern Power Systems and Clean Energy, vol. 9, no. 4, pp. 788-799, Jul. 2021. [Baidu Scholar]
Z. Lu, X. Xu, Z. Yan et al., “Multistage robust optimization of routing and scheduling of mobile energy storage in coupled transportation and power distribution networks,” IEEE Transactions on Transportation Electrification, vol. 8, no. 2, pp. 2583-2594, Jun. 2022. [Baidu Scholar]
S. Yao, P. Wang, X. Liu et al., “Rolling optimization of mobile energy storage fleets for resilient service restoration,” IEEE Transactions on Smart Grid, vol. 11, no. 2, pp. 1030-1043, Mar. 2020. [Baidu Scholar]
S. Lei, C. Chen, Y. Li et al., “Resilient disaster recovery logistics of distribution systems: co-optimize service restoration with repair crew and mobile power source dispatch,” IEEE Transactions on Smart Grid, vol. 10, no. 6, pp. 6187-6202, Nov. 2019. [Baidu Scholar]
Y. Wang, Y. Xu, J. He et al., “Coordinating multiple sources for service restoration to enhance resilience of distribution systems,” IEEE Transactions on Smart Grid, vol. 10, no. 5, pp. 5781-5793, Sept. 2019. [Baidu Scholar]
J. Liu, Z. Jiao, C. Zhang et al., “Joint scheme for electric locomotive evacuation and distribution system restoration with faults of external power supply systems,” IEEE Transactions on Transportation Electrification, vol. 10, no. 2, pp. 4252-4267, Jun. 2024. [Baidu Scholar]
L. Zhang, C. Wang, J. Liang et al., “A coordinated restoration method of hybrid AC/DC distribution network for resilience enhancement,” IEEE Transactions on Smart Grid, vol. 14, no. 1, pp. 112-125, Jan. 2023. [Baidu Scholar]
B. Chen, Z. Ye, C. Chen et al., “Toward a synthetic model for distribution system restoration and crew dispatch,” IEEE Transactions on Power Systems, vol. 34, no. 3, pp. 2228-2239, May 2019. [Baidu Scholar]
B. Chen, Z. Ye, C. Chen et al., “Toward a MILP modeling framework for distribution system restoration,” IEEE Transactions on Power Systems, vol. 34, no. 3, pp. 1749-1760, May 2019. [Baidu Scholar]
H. Sekhavatmanesh and R. Cherkaoui, “A multi-step reconfiguration model for active distribution network restoration integrating DG start-up sequences,” IEEE Transactions on Sustainable Energy, vol. 11, no. 4, pp. 2879-2888, Oct. 2020. [Baidu Scholar]
J. Jian, P. Ji, H. Yu et al., “Multi-stage supply restoration of active distribution networks with SOP integration,” Sustainable Energy, Grids and Networks, vol. 29, p. 100562, Mar. 2022. [Baidu Scholar]
F. Fan, R. Zhang, Y. Xu et al., “Robustly coordinated operation of an emission-free microgrid with hybrid hydrogen-battery energy storage,” CSEE Journal of Power and Energy Systems, vol. 8, no. 2, pp. 369-379, Mar. 2022. [Baidu Scholar]
X. Wang, Q. Guo, C. Tu et al., “A two-layer control strategy for soft open points considering the economical operation area of transformers in active distribution networks,” IEEE Transactions on Sustainable Energy, vol. 13, no. 4, pp. 2184-2195, Oct. 2022. [Baidu Scholar]
S. Lei, C. Chen, H. Zhou et al., “Routing and scheduling of mobile power sources for distribution system resilience enhancement,” IEEE Transactions on Smart Grid, vol. 10, no. 5, pp. 5650-5662, Sept. 2019. [Baidu Scholar]
Y. Yang, W. Yang, H. Yang et al., “Electrolyte design principles for low-temperature lithium-ion batteries,” eScience, vol. 3, no. 6, p. 100170, Dec. 2023. [Baidu Scholar]
J. Jian, J. Zhao, H. Ji et al., “Supply restoration of data centers in flexible distribution networks with spatial-temporal regulation,” IEEE Transactions on Smart Grid, vol. 15, no. 1, pp. 340-354, Jan. 2024. [Baidu Scholar]
R. Jabr, “Segregated linear decision rules for inverter watt-var control,” IEEE Transactions on Power Systems, vol. 36, no. 3, pp. 2702-2708, May 2021. [Baidu Scholar]
R. Wang, H. Ji, P. Li et al., “Multi-resource dynamic coordinated planning of flexible distribution network,” Nature Communications, vol. 15, no. 1, p. 4576, May 2024. [Baidu Scholar]