Abstract
Owing to potential regulation capacities from flexible resources in energy coupling, storage, and consumption links, central energy stations (CESs) can provide additional support to power distribution network (PDN) in case of power disruption. However, existing research has not explicitly revealed the emergency response of PDN with leveraging multiple CESs. This paper proposes a decentralized self-healing strategy of PDN to minimize the entire load loss, in which multi-area CESs’ potentials including thermal storage and building thermal inertia, as well as the flexible topology of PDN, are reasonably exploited for service recovery. For sake of privacy preservation, the co-optimization of PDN and CESs is realized in a decentralized manner using adaptive alternating direction method of multipliers (ADMM). Furtherly, bilateral risk management with conditional value-at-risk (CVaR) for PDN and risk constraints for CESs is integrated to deal with uncertainties from outage duration. Case studies are conducted on a modified IEEE 33-bus PDN with multiple CESs. Numerical results illustrate that the proposed strategy can fully utilize the potentials of multi-area CESs for coordinated load restoration. The effectiveness of the performance and behaviors’ adaptation against random risks is also validated.
THE energy dilemma and environmental pollution issues have expedited the revolution of energy utilization [
Recently, frequent occurrences of emergencies such as natural disasters and hostile attacks bring out tremendous operation loss to energy system, and these events have the characteristics of low probability and large destruction [
Significant efforts have been conducted on the self-healing scheduling of PDN under extreme events. The utilization of various controllable resources such as distributed generations (DGs), network reconfiguration, and demand-side approaches can contribute to the service recovery effect [
As energy systems are undergoing a transition from separated power supply pattern to multi-energy and multi-link collaboration, the coordination potentials of heterogeneous resources for electricity service recovery will be exploitable [
The above researches mainly focus on the centralized scheduling manner that ignores the obstacles of information exchange. Different utilities may have autocephalous energy management systems (EMSs) and they are usually operated independently with privacy-preserving [
Moreover, extra risks will be evoked regarding various uncertainties, which may derive from renewable energy sources (RESs), multiple demands, and some others. To decrease the negative influence, handling methods such as robust optimization [
To the best of our knowledge, the uncertainties from outage duration have not attracted much attention in the service recovery of PDN. Facing indeed existing duration disturbance in PDN, it still lacks efficient self-healing strategy due to the time-series relevance of system status for uncertain scenarios. Especially, with deep coupling between PDN and CESs, the auxiliary service potentials of CESs for emergency response need to be well exploited; meanwhile, the integration of CESs raises the difficulty of reasonable risk-based restoration due to the requirements of multi-energy coordination and spatio-temporal resource utilization. Furthermore, the operations of PDN and CESs are often independent along with private information preserving; the optimal self-healing strategy considering endogenous uncertainty from outage duration is more challenging.
To deal with the above issues, this paper proposes a decentralized risk-based self-healing strategy for PDN considering the support of multiple CESs. The main contributions are summarized as follows.
1) A self-healing recovery strategy for PDN is proposed considering topology reconfiguration and multiple regulation potentials of multi-area CESs. For dispersive CESs, emergency response from GTs and thermal storage, as well as building thermal inertia are well coordinated for load restoration. Furthermore, the model is tackled as a mixed-integer second-order cone programming (MISOCP) problem.
2) Bilateral risk management with CVaR assessment for PDN and margin constraints for CESs is employed to cope with operation risks caused by uncertain outage duration. CVaR criteria are introduced to measure the load shedding risk of PDN while guaranteeing the supply of CESs within permissible margin for risk controllability. Better risk management effects are realized.
3) The self-healing strategy is conducted in a decentralized manner, in which consensus-based ADMM algorithm is adopted for reducing information exchange and preserving privacy between PDN and CESs. Meanwhile, the iteration process is expedited by adaptive ADMM algorithm, showing better convergence performance.
The remainder of this paper is organized as follows. Section II builds the mathematical model of system operation, as well as the conic relaxation method. Section III describes the bilateral risk-based self-healing scheduling strategy. The solution methodology of decentralized risk-based self-healing scheduling is presented in Section IV. Case studies are conducted in Section V to verify the performance of the proposed strategy. Finally, conclusions are drawn in Section VI.
The DistFlow branch model with considering flexible topology is adopted to describe the PDN. Equations (
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
(9) |
(10) |
The radical topology should be maintained in the formed islands, which is described as:
(11) |
(12) |
(13) |
(14) |
where is the system reference voltage.
This research mainly focuses on the self-healing scheduling for space cooling and electricity during the cooling period. And the cooling devices can be divided into electricity-driven and gas-driven categories.
Popular electricity-driven devices include ground source heat pump (HP), conventional water-cooled chiller (WC), and cold water tank (CWT).
The mathematical model of HP is depicted in (17) and (18), where COP denotes coefficient of performance.
(17) |
(18) |
The operation constraints of WC are given in (19) and (20).
(19) |
(20) |
CWTs can store the cooling energy from HPs and WCs, and the energy storage constraints, cooling-storage constraints, and capacity constraints are formulated as follows:
(21) |
(22) |
(23) |
Gas-driven devices can be GT and absorption chiller (AC). GTs burn natural gas with electricity and heating generation, and electricity and heating have a certain ratio relationship, as shown in (24) and (25). And (26) presents the constraints of output power.
(24) |
(25) |
(26) |
In case of disruption at the root node of PDN, complete energy loss will occur. Facing this, both active and reactive power supports should be carried out for effective fault restoration. As the coupling point of PDN and natural gas system, converter-based GTs in CESs can serve as controllable distributed generators to provide active and reactive support when electricity emergency takes place [
(27) |
(28) |
The output power constraints, absorbed power, as well the maximum output constraints of AC are shown as follows:
(29) |
(30) |
(31) |
The thermal inertia characteristic of buildings will provide more flexibilities for system operation, and buildings can be regarded as virtual storages. The mathematical thermal inertia model of buildings in the cooling season can be stated as [
(32) |
Lots of nonconvex terms exist in the operation model. To expedite the solution, the nonconvex model is converted into an MISOCP formulation.
Auxiliary variables and are introduced to replace and . Thus, (1), (2), (6), (7), (15), and (16) can be linearized:
(35) |
(36) |
(37) |
(38) |
(39) |
(40) |
For (3), it can be further relaxed as a standard second-order cone constraint, which can be expressed as:
(41) |
In the section, the bilateral risk-based self-healing model is introduced to realize service recovery and risk measures.
The objective function F is to minimize the load losses of PDN and CESs, which is:
(43) |
(44) |
(45) |
can be calculated as:
(46) |
where equals .
The general expression of CVaR model can be described as follows [
(47) |
(48) |
(49) |
where is greater than in scenario .
Based on the above CVaR theory and system optimization model, the risk-management model of self-healing strategy can be reformulated as (50).
(50) |
Except for the original constraints for system operation, the following constraints are supplemented to risk management model.
(51) |
(52) |
In general, we describe the strategy with as risk-averse preference, the strategy with as risk-seeking preference, and the strategy with as risk-neutral preference. To accommodate diverse risk preferences, different values of risk weight factors can be considered. As the weight factor increases from 0 to 1, the scheduling preference turns from risk seeking to risk aversion.
For the essential requirement of CESs, the indoor temperature and the ramping rates of buildings should be maintained within the comfort range, which are expressed as:
(53) |
(54) |
Due to the unpredictability of outage duration time, the fault recovery schedule will keep consistent for each possible duration. Therefore, the schedules including charging-discharging power of thermal storage, building temperature in dispersive CESs, and reconfiguration topology for PDN, as well as the load restoration state, will be issued to the local control system for execution.
Since PDN and CES usually belong to different entities, only restricted operation information can be exchanged with each other, resulting in the absurdity of centralized scheduling.
In this section, a decentralized method is introduced to achieve private information preserving and independent operation of each subsystem through adaptive ADMM algorithm.
The energy system is divided into PDN and CES subsystems. The fault restoration of each subsystem is carried out independently, and each operator has complete information of itself, and the shared information with others is only the active and reactive injections for PDN.
In this case, consensus variables and are introduced to describe the boundary parameters between them, as depicted in (55).
(55) |
The augmented Lagrangian function is constructed for the load restoration of PDN subproblem, as expressed in (56) and (57):
(56) |
s.t.
(57) |
The augmented Lagrangian function for CES is expressed as:
(58) |
s.t.
(59) |
The combination of for multiple CESs will form vectors .
With the consensus-based ADMM algorithm, the unified self-healing model can be decomposed into several subproblems, which can be solved separately by consensus interaction.
The consensus-based ADMM for decentralized recovery is illustrated in
Algorithm 1 : consensus-based ADMM for decentralized recovery |
---|
1. Input parameters for each subsystem, including system topology, load, environment information, and equipment parameters |
2. Initialize algorithm parameters, including , ,, , , , , convergence thresholds , , and the maximum iteration |
3. for |
4. Perform decentralized self-healing optimization for each subsystem |
PDN optimization |
Objective function: (56) |
Constraints:(4), (5), (8)-(14), (35)-(41) |
CES optimization |
Objective function: (58) |
Constraints:(17)-(27), (29)-(34), (42) |
5. Exchange coupling variables and update consensus variables: |
, |
6. Calculate the primal residual and dual residual : |
,
|
|
|
7. Check the stopping criteria if & Output the self-healing scheduling results and break else Update the Lagrangian multipliers for each subsystem |
|
|
|
|
8. |
9. end |
The convergence efficiency of ADMM is significantly affected by the value of step size. Conventional ADMM is conducted with fixed value, leading to the deterioration of algorithm performance in the last stage of iteration. One effective method to facilitate convergence speed is to adjust parameters for each iteration. As for the issue, a self-adaptive step size method for ADMM (adaptive ADMM) is utilized to improve the algorithm performance, in which the penalty parameter is dynamically modified with less dependence on the initial value, shown as follows [
(60) |
where and .
Based on the energy structure and forecasting data of the whole system, the EMSs for CESs and the distribution network operator (DSO) for PDN generate self-governed schedules separately after evaluating the outage duration probability in case of power disruption. The optimal risk-based self-healing strategy for PDN and CESs can be obtained in a decentralized way by limited information exchange and iterative optimization. Then, the corresponding schedules will be issued to each device for execution. The proposed self-healing framework provides a novel decentralized risk-based load recovery for PDN by uncertainty evaluation and CVaR-based management, and the flexibilities of CESs are fully utilized with the self-healing capacity significantly facilitated.
In this section, the rationality and effectiveness of the decentralized self-healing strategy with risk management are verified on the distribution network, which is composed of a modified IEEE 33-bus PDN integrated with multiple CESs. Case studies are carried out on Intel CPU i9-10900K and 32 GB RAM-based PC with MATLAB 2020b platform. The self-healing strategy is solved in YALMIP toolbox and optimized by linking CPLEX 12.1 solver [
The structure of the modified IEEE 33-bus PDN with multiple CESs is shown in

Fig. 1 Structure of modified IEEE 33-bus PDN with multiple CESs.

Fig. 2 Configuration and energy flows of CESs.
The scheduling interval is 0.5 hour and a typical day in cooling season is selected for case analysis. The electricity load profile of PDN and the outdoor temperature for the typical cooling day are presented in

Fig. 3 Electricity load profile of PDN and outdoor temperature for typical cooling day.
The duration of electricity disruption can be 2, 2.5, 3, 3.5, and 4 hours, and the probability of the corresponding scenarios are 0.15, 0.2, 0.3, 0.2, and 0.15, respectively. Thus, the comprehensive probabilities of each period during 02:00-04:00 are 0.333, 0.283, 0.217, 0.117, and 0.050, respectively. Unit penalty costs of curtailed electricity loads in PDN and cooling loads in CESs are 100 CNY/kWh and 5 CNY/kWh, respectively. For risk management parameters, the confidence level is set to be 0.8. As for the weight factor , it can be changed from 0 to 1; and lower value denotes risk-seeking schedule, while higher value represents risk-averse schedule. Especially, 0.7 is assigned to for concrete analysis.
In the ADMM optimization procedure, the initial penalty parameter is set to be 1.0. Step size adjustment parameter is set to be 2, where the coefficient is 6. The maximum iteration is supposed to be 200 and convergence thresholds of both primary and dual residuals are set to be 0.5. For CPLEX solver, it is implemented with default settings and the optimality gap is .
It is assumed that line 1-2 has a permanent three-phase fault at 09:30, and loads of bus 2 to bus 33 are completely out of service. After fault isolation, the risk-based decentralized self-healing operation is conducted for fault restoration.
The reconfiguration strategy of PDN during 09:30-13:00 is presented in

Fig. 4 Reconfiguration strategy of PDN during 09:30-13:00.

Fig. 5 Active and reactive power control strategy of GTs in multi-zone CESs.
The variation of indoor temperature of buildings in multiple CESs is depicted in

Fig. 6 Indoor temperature of buildings in multiple CESs.

Fig. 7 Stored energy variation of thermal storages in CESs.
Benefiting from flexibilities of multi-area CESs, including the active/reactive support, thermal storage, and building demand response, as well as the flexible topology in PDN, more regulation capacities are exploited and the out-of-service demands can be recovered as much as possible with considering risk preferences. Self-healing oriented fault restoration results of PDN for different outage durations are listed in
Outage duration (hour) | Total load (kWh) | Unrecovered load (kWh) |
---|---|---|
2.0 | 6334.1 | 3026.0 |
2.5 | 7355.7 | 3499.2 |
3.0 | 8470.2 | 3972.0 |
3.5 | 9677.6 | 4444.9 |
4.0 | 11052.1 | 5009.8 |
Different resource utilization can affect the fault restoration effects significantly. The comparison of different potential combination scenarios for CESs is depicted in
Scenario | Thermal storage | Building thermal inertia | Expected loss of PDN (kWh) |
---|---|---|---|
1 | × | √ | 4155.0 |
2 | √ | × | 4764.9 |
3 | √ | √ | 3985.8 |
Note: √ means with considerarion and × means without considerarion.
Compared with scenario 1, the thermal storage devices, i.e., CWTs, are considered in scenario 3. With timely energy charging-discharging behaviors of CWTs, additional flexibilities are provided for spatio-temporal emergency response in case of power disruption. Thus, more out-of-service loads are recovered with the risk-based self-healing strategy.
In scenario 2, the thermal inertia of building in CESs is ignored in contrast with scenario 3. By considering building thermal inertia, the indoor temperature can be regulated within reasonable range along with more adjustable margin for energy coordination and multi-area complementation. And the load recovery effect can be effectively improved.
By comparing the results of different potential combinations, it can be noted that the self-healing strategy can fully exploit the regulation potentials from CESs for facilitating self-healing capacity, achieving better fault restoration effect.
To indicate the effectiveness of the proposed decentralized strategy, the convergence processes of primary and dual residuals for adaptive and standard ADMM are illustrated in

Fig. 8 Convergence processes of primary and dual residuals for adaptive and standard ADMM. (a) Residual convergence for adaptive ADMM. (b) Residual convergence for standard ADMM.
To further demonstrate the accuracy of the decentralized strategy, the restoration results of conventional centralized and adaptive ADMM are shown in
ADMM | Expected loss of PDN (kWh) | Expected loss of CESs (kWh) | Optimal goal (CNY) | Solution time (s) |
---|---|---|---|---|
Centralized | 3029.7 | 2416.8 | 129248.2 | 60.4 |
Adaptive | 3030.9 | 2416.8 | 129262.6 | 1771.0 |
The variation profiles of load loss value and CVaR with different weight factors are presented in

Fig. 9 Load loss value and CVaR with different weight factors.
As can be observed, with the increase of weight factor, the CVaR decreases while the expected load loss value increase simultaneously; and system operation varies from risk-seeking to risk-averse preferences. In other words, the lower operation risk can be obtained along with poorer fault-restoration effect, and vice versa. In actual operation, the operator needs to select the appropriate weight factor to pursue the utmost service restoration on the premise of satisfying their specific risk preference.
Without loss of generality, the operation results with different weight factors of 0.2, 0.6, and 1.0 are listed in
Value of weight factor | Load loss value (CNY) | CVaR (CNY) |
---|---|---|
0.2 | 305723.7 | 58736.0 |
0.6 | 314580.8 | 49620.4 |
1.0 | 512733.0 | 47237.3 |

Fig. 10 Schedules of CES1 with different weight factors. (a) Operation state of CES1 when . (b) Operation state of CES1 when . (c) Operation state of CES1 when .
It is obvious that the risk-seeking strategy (
To further demonstrate the effectiveness of risk management strategy, three schemes are constructed for performance comparison.
1) Scheme 1: the proposed strategy, i.e., CVaR-based self-healing scheduling, is adopted for service restoration.
2) Scheme 2: the stochastic optimization is conducted for service restoration, i.e., multiple scenarios with risk weight factor .
3) Scheme 3: the deterministic operation for the worst-case scenario is conducted for service restoration, i.e., the outage duration time is 4 hours.
The variations of stored energy in CWTs for different schemes are shown in

Fig. 11 Variation of stored energy in CWTs for different schemes.

Fig. 12 Unrecovered load of PDN for each scheduling period.
Incorporating
The comparisons of operation results for different schemes are listed in
Scheme | CVaR (CNY) | Load loss value (CNY) |
---|---|---|
1 | 49617.1 | 314582.5 |
2 | 74994.0 | 305358.4 |
3 | 62955.0 | 316728.4 |
This paper presents a decentralized risk-based self-healing strategy for PDN. The regulation potentials of multiple CESs, including active and reactive power support of GTs, as well as emergency response of thermal storage and building thermal inertia, are fully utilized for load restoration in case of power disruption. In terms of inherent outage duration uncertainty, bilateral risk management with CVaR for PDN and essential constraints for CESs is implemented for operation analysis considering risk preference. Furthermore, an adaptive ADMM is introduced to achieve decentralized optimization.
Case studies are conducted using the modified IEEE 33-bus PDN with multi-point CESs. It is indicated that the strategy can give full play to the flexible support capacities of multiple resources in CESs to restore out-of-service loads as much as possible. By applying bilateral risk measures with CVaR, the PDN load shedding of each period can be reasonably coordinated for effective operational risk control, guaranteeing the indispensable supply of CESs. Besides, the consensus-based ADMM solution is carried out to conduct decentralized optimal scheme of PND and CESs. The results are in accordance with that of the centralized strategy, and limited information interaction and privacy protection can be achieved. With the application of adaptive ADMM, convergence performances can be effectively improved.
In conclusion, the proposed risk-based decentralized self-healing strategy can realize better emergency service recovery, with tough risk management ability under unpredictable outage duration. And the decentralized strategy is more applicable for privacy-safety scheduling under the independent operation of subsystems.
Nomenclature
Symbol | —— | Definition |
---|---|---|
A. | —— | Symbols |
—— | Probability with outage duration | |
, H | —— | Cooling and heating power |
—— | Specific heat capacity of air | |
—— | Injected cooling for building | |
, | —— | Unit penalty costs for electricity and cooling loads |
Fn | —— | Surface area of building |
fs | —— | Objective value |
, | —— | Current magnitude and its square of branch |
Kn | —— | Equivalent heat dissipation coefficient |
Ns | —— | Number of scenarios |
, | —— | Active and reactive power |
, | —— | Active and reactive injections into power distribution network (PDN) of coupling points |
, | —— | Active and reactive outputs of gas turbines (GTs) in central energy stations (CESs) |
ps | —— | Probability of scenario s |
pt | —— | Comprehensive probability of period t |
, | —— | Resistance and reactance of branch |
—— | Inverter capacity | |
—— | Indoor temperature of buildings | |
Tt,out | —— | Outdoor temperature |
Tref | —— | Reference indoor temperature |
tin | —— | The minimum value of inevitable outage duration |
tout | —— | Outage duration time |
, v | —— | Voltage magnitude and its voltage square of bus |
V0 | —— | System reference voltage |
Vn | —— | Volume of building |
—— | Cooling energy stored | |
Zp, Zq | —— | Consensus variables |
B. | —— | Greek Symbols |
—— | Binary variable (1: branch is connected; 0: otherwise) | |
—— | Binary variable (1: node is the parent of bus ; 0: otherwise) | |
—— | Power factor of gas turbines | |
—— | Scheduling interval | |
—— | Ramping limit | |
, | —— | Heat loss rate and load recovery coefficient |
—— | Value at risk | |
—— | Efficiency of gas-driven device | |
, | —— | Vectors of Lagrangian multipliers for PDN |
, | —— | Vectors of Lagrangian multipliers for CESs n |
, | —— | Step size adjustment parameters |
, | —— | Non-negative values |
—— | Penalty parameter | |
—— | Density of air | |
—— | Heat-electricity ratio | |
—— | Confidence level | |
—— | Weight factor of risk | |
—— | Set of specified elements | |
C. | —— | Superscripts |
HP, WC, CWT, GT, AC | —— | Heat pump, water-cooled chiller, cold water tank, gas turbine, and absorption chiller |
—— | Index of iteration number | |
, | —— | Load and tie line |
, | —— | The maximum and minimum values |
, | —— | Cooling storage and releasing |
D. | —— | Subscripts |
, b | —— | Branch and bus |
—— | Indices of buses | |
, jh | —— | Indices of branches |
, s | —— | Indices of CESs and scenarios |
—— | Index of time periods | |
, v | —— | Tie switch and voltage support bus |
Appendix
Item | Capacity | COP or efficiency | Loss rate |
---|---|---|---|
HP |
1000 kW for CES1 1000 kW for CES2 | 5.38 | |
WC |
1000 kW for CES1 1000 kW for CES2 | 5.13 | |
CWT |
10000 kWh for CES1 15000 kWh for CES2 | 0.001 | |
GT |
900 kW for CES1 800 kW for CES2 |
0.35 for electricity 0.40 for heating | |
AC |
1200 kW for CES1 1000 kW for CES2 | 1.20 |
location | Surface area ( | Volume ( | Dissipation coefficient () |
---|---|---|---|
CES1 | 200000 | 280000 | 1.2 |
CES2 | 250000 | 320000 | 1.2 |
References
M. A. Bagherian and K. Mehranzamir, “A comprehensive review on renewable energy integration for combined heat and power production,” Energy Conversion and Management, vol. 224, p. 113454, Nov. 2020. [Baidu Scholar]
L. Chen, Q. Xu, Y. Yang et al., “Community integrated energy system trading: a comprehensive review,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 6, pp. 1445-1458, Nov. 2022. [Baidu Scholar]
C. Lv, H. Yu, P. Li et al., “Coordinated operation and planning of integrated electricity and gas community energy system with enhanced operational resilience,” IEEE Access, vol. 8, pp. 59257-59277, Mar. 2020. [Baidu Scholar]
A. Ghanbari, H. Karimi, and S. Jadid, “Optimal planning and operation of multi-carrier networked microgrids considering multi-energy hubs in distribution networks,” Energy, vol. 204, p. 117936, Aug. 2020. [Baidu Scholar]
X. Zhang, C. Liang, M. Shahidehpour et al., “Reliability-based optimal planning of electricity and natural gas interconnections for multiple energy hubs,” IEEE Transactions on Smart Grid, vol. 8, no. 4, pp. 1658-1667, Jul. 2017. [Baidu Scholar]
D. Feng, F. Wu, Y. Zhou et al., “Multi-agent-based rolling optimization method for restoration scheduling of distribution systems with distributed generation,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 4, pp. 737-749, Jul. 2020. [Baidu Scholar]
A. A. Bajwa, H. Mokhlis, and S. Mekhilef, “Enhancing power system resilience leveraging microgrids: a review,” Journal of Renewable and Sustainable Energy, vol. 11, no. 3, p. 035503, Apr. 2019. [Baidu Scholar]
S. Poudel and A. Dubey, “Critical load restoration using distributed energy resources for resilient power distribution system,” IEEE Transactions on Power Systems, vol. 34, no. 1, pp. 52-63, Jul. 2018. [Baidu Scholar]
F. Shen, Q. Wu, and Y. Xue, “Review of service restoration for distribution networks,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 1, pp.1-14, Jan. 2019. [Baidu Scholar]
E. Hossain, S. Roy, N. Mohammad et al., “Metrics and enhancement strategies for grid resilience and reliability during natural disasters,” Applied Energy, vol. 290, p. 116709, May 2021. [Baidu Scholar]
Y. Li, J. Xiao, C. Chen et al., “Service restoration model with mixed-integer second-order cone programming for distribution network with distributed generations,” IEEE Transactions on Smart Grid, vol. 10, no. 4, pp. 4138-4150, Jul. 2018. [Baidu Scholar]
J. Jian, P. Li, 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]
H. Ji, C. Wang, P. Li et al., “SOP-based islanding partition method of active distribution networks considering the characteristics of DG, energy storage system and load,” Energy, vol. 155, pp. 312-325, Jul. 2018. [Baidu Scholar]
H. Zhao, Z. Lu, L. He et al., “Two-stage multi-fault emergency rush repair and restoration robust strategy in distribution networks,” Electric Power Systems Research, vol. 184, p. 106335, Mar. 2020. [Baidu Scholar]
J. Liu, C. Qin, and Y. Yu, “A comprehensive resilience-oriented FLISR method for distribution systems,” IEEE Transactions on Smart Grid, vol. 12, no. 3, pp. 2136-2152, May 2021. [Baidu Scholar]
X. Liu, H. Wang, Q. Sun et al., “Research on fault scenario prediction and resilience enhancement strategy of active distribution network under ice disaster,” International Journal of Electrical Power & Energy Systems, vol. 135, p. 107478, Feb. 2022. [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]
F. Hafiz, B. Chen, C. Chen et al., “Utilising demand response for distribution service restoration to achieve grid resiliency against natural disasters,” IET Generation, Transmission and Distribution, vol. 13, no. 14, pp. 2942-2950, Jul. 2019. [Baidu Scholar]
X. Wang, X. Li, X.-J. Li et al., “Soft open points based load restoration for the urban integrated energy system under extreme weather events,” IET Energy Systems Integration, vol. 4, pp. 335-350, Feb. 2022. [Baidu Scholar]
F. Vazinram, M. Hedayati, R. Effatnejad et al., “Self-healing model for gas-electricity distribution network with consideration of various types of generation units and demand response capability,” Energy Conversion and Management, vol. 206, p. 112487, Feb. 2020. [Baidu Scholar]
C. Li, P. Li, H. Yu et al., “Optimal planning of community integrated energy station considering frequency regulation service,” Journal of Modern Power Systems and Clean Energy, vol. 9, no. 2, pp. 264-273, Mar. 2021. [Baidu Scholar]
M. Yan, Y. He, M. Shahidehpour et al., “Coordinated regional-district operation of integrated energy systems for resilience enhancement in natural disasters,” IEEE Transactions on Smart Grid, vol. 10, no. 5, pp. 4881-4892, Sept. 2019. [Baidu Scholar]
C. Wang, W. Wei, J. Wang et al., “Robust defense strategy for gas-electric systems against malicious attacks,” IEEE Transactions on Power Systems, vol. 32, no. 4, pp. 2953-2965, Jul. 2016. [Baidu Scholar]
H. Cong, Y. He, X. Wang et al., “Robust optimization for improving resilience of integrated energy systems with electricity and natural gas infrastructures,” Journal of Modern Power Systems and Clean Energy, vol. 6, no. 5, pp. 1066-1078, Sept. 2018. [Baidu Scholar]
X. Li, X. Du, T. Jiang T et al., “Coordinating multi-energy to improve urban integrated energy system resilience against extreme weather events,” Applied Energy, vol. 309, p. 118455, Mar. 2022. [Baidu Scholar]
C. Shao, M. Shahidehpour, X. Wang et al., “Integrated planning of electricity and natural gas transportation systems for enhancing the power grid resilience,” IEEE Transactions on Power Systems, vol. 32, no. 6, pp. 4418-4429, Nov. 2017. [Baidu Scholar]
C. Lv, R. Liang, W. Jin et al., “Multi-stage resilience scheduling of electricity-gas integrated energy system with multi-level decentralized reserve,” Applied Energy, vol. 317, p. 119165, Jul. 2022. [Baidu Scholar]
X. Jiang, J. Chen, M. Chen et al., “Multi-stage dynamic post-disaster recovery strategy for distribution networks considering integrated energy and transportation networks,” CSEE Journal of Power and Energy Systems, vol. 7, no. 2, pp. 408-420, Oct. 2020. [Baidu Scholar]
R. Hemmati, H. Mehrjerdi, S. M. Nosratabadi, “Resilience-oriented adaptable microgrid formation in integrated electricity-gas system with deployment of multiple energy hubs,” Sustainable Cities and Society, vol. 71, p. 102946, Aug. 2021. [Baidu Scholar]
T. Qian, X. Chen, Y. Xin et al., “Resilient decentralized optimization of chance constrained electricity-gas systems over lossy communication networks,” Energy, vol. 239, p. 122158, Jan. 2022. [Baidu Scholar]
P. Wang, Q. Wu, S. Huang et al., “ADMM-based distributed active and reactive power control for regional AC power grid with wind farms,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 3, pp. 588-596, Jun. 2021. [Baidu Scholar]
S. Xu, Z. Yan, D. Feng et al., “Decentralized charging control strategy of the electric vehicle aggregator based on augmented Lagrangian method,” International Journal of Electrical Power & Energy Systems, vol. 104, pp. 673-679, Jan. 2019. [Baidu Scholar]
X. Zhou, Q. Ai, and M. Yousif, “Two kinds of decentralized robust economic dispatch framework combined distribution network and multi-microgrids,” Applied Energy, vol. 253, p. 113588, Nov. 2019. [Baidu Scholar]
C. He, L. Wu, T. Liu et al., “Robust co-optimization scheduling of electricity and natural gas systems via ADMM,” IEEE Transactions on Sustainable Energy, vol. 8, no. 2, pp. 658-670, Apr. 2016. [Baidu Scholar]
J. Wei, Y. Zhang, J. Wang et al., “Decentralized demand management based on alternating direction method of multipliers algorithm for industrial park with CHP units and thermal storage,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 1, pp. 120-130, Jan. 2022. [Baidu Scholar]
N. Jia, C. Wang, W. Wei et al., “Decentralized robust energy management of multi-area integrated electricity-gas systems,” Journal of Modern Power Systems and Clean Energy, vol. 9, no. 6, pp. 1478-1489, Nov. 2021. [Baidu Scholar]
W. Lin, X. Jin, H. Jia et al., “Decentralized optimal scheduling for integrated community energy system via consensus-based alternating direction method of multipliers,” Applied Energy, vol. 302, p. 117448, Nov. 2021. [Baidu Scholar]
G. Li, K. Yan, R. Zhang et al., “Resilience-oriented distributed load restoration method for integrated power distribution and natural gas systems,” IEEE Transactions on Sustainable Energy, vol. 13, no. 1, pp. 341-352, Sept. 2021. [Baidu Scholar]
X. Xu, W. Hu, W. Liu et al., “Robust energy management for an on-grid hybrid hydrogen refueling and battery swapping station based on renewable energy,” Journal of Cleaner Production, vol. 331, p. 129954, Jan. 2022. [Baidu Scholar]
Z. Li, S. Su, X. Jin et al., “Stochastic and distributed optimal energy management of active distribution network with integrated office buildings,” CSEE Journal of Power and Energy Systems. doi: 10.17775/CSEEJPES [Baidu Scholar]
S. Su, Z. Li, X. Jin et al., “Energy management for active distribution network incorporating office buildings based on chance-constrained programming,” International Journal of Electrical Power & Energy Systems, vol. 134, p. 107360, Jan. 2022. [Baidu Scholar]
X. Yu and D. Zheng, “Cross-regional integrated energy system scheduling optimization model considering conditional value at risk,” International Journal of Energy Research, vol. 44, no. 7, pp. 5564-5581, Jun. 2020. [Baidu Scholar]
A. Xuan, X. Shen, Q. Guo et al., “A conditional value-at-risk based planning model for integrated energy system with energy storage and renewables,” Applied Energy, vol. 294, p. 116971, Jul. 2021. [Baidu Scholar]
J. Wang and Y. Song, “Distributionally robust OPF in distribution network considering CVaR-averse voltage security,” International Journal of Electrical Power & Energy Systems, vol. 145, p. 108624, Feb. 2023. [Baidu Scholar]
Z. Liu, S. Liu, Q. Li et al., “Optimal day-ahead scheduling of islanded microgrid considering risk-based reserve decision,” Journal of Modern Power Systems and Clean Energy, vol. 9, no. 5, pp. 1149-1160, Sept. 2021. [Baidu Scholar]
A. S. G. Langeroudi, M. Sedaghat, S. Pirpoor et al., “Risk-based optimal operation of power, heat and hydrogen-based microgrid considering a plug-in electric vehicle,” International Journal of Hydrogen Energy, vol. 46, no. 58, pp. 30031-30047, Aug. 2021. [Baidu Scholar]
J. Zhao, M. Zhang, H. Yu et al., “An islanding partition method of active distribution networks based on chance-constrained programming,” Applied Energy, vol. 242, pp. 78-91, May 2019. [Baidu Scholar]
Y. Li, C. Wang, G. Li et al., “Improving operational flexibility of integrated energy system with uncertain renewable generations considering thermal inertia of buildings,” Energy Conversion and Management, vol. 207, p. 112526, Mar. 2020. [Baidu Scholar]
D. Xiao, H. Chen, C. Wei et al., “Statistical measure for risk-seeking stochastic wind power offering strategies in electricity markets,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 5, pp. 1437-1442, Sept. 2022. [Baidu Scholar]
Z. Li, P. Li, Z. Yuan et al, “Optimized utilization of distributed renewable energies for island microgrid clusters considering solar-wind correlation,” Electric Power Systems Research, vol. 206, p. 107822, May 2022. [Baidu Scholar]
IBM. (2022, Dec.). IBM ILOG CPLEX Optimization Studio. [Online]. Available: https://www.ibm.com/products/Ilog-cplex-optimization-studio [Baidu Scholar]
P. Li, H. Ji, C. Wang et al., “Coordinated control method of voltage and reactive power for active distribution networks based on soft open point,” IEEE Transactions on Sustainable Energy, vol. 8, no. 4, pp. 1430-1442, Oct. 2017. [Baidu Scholar]