Abstract
Active distribution grids cause bi-directional power flow between transmission system (TS) and distribution system (DS), which not only affects the optimal cost but also the secure operation of the power system. This paper proposes a hybrid coordination method to solve the risk-aware distributed optimal power flow (RA-DOPF) problem in coordinated TS and DS. For operation risk evaluation, the weather-based contingencies are considered in both TS and DS. A hybrid coordination method is developed that entails analytical target cascading (ATC) and Benders decomposition (BD). Moreover, the risk-aware optimal power flow (RAOPF) in TS and risk-based security-constrained optimal power flow in DS have been performed using the BD method considering basic optimal power flow as a master problem, whereas and contingencies are considered as sub-problems. Different case studies are performed using the IEEE 30-bus system with generation reserves as a TS and a 13-bus system as a DS. The results demonstrate the efficacy of the proposed method.
THE participation of distribution networks in power system balancing and cost optimization has been increased due to the exponential escalation in distributed energy resources (DERs) such as diesel generators (DG), wind farm, and solar farm, etc. Thus, a distribution system (DS) partakes along with the transmission system (TS) in the process of matching supply and demand to achieve the minimum cost. Therefore, previous researchers have investigated the distributed unconstrained economic dispatch (ED) [
Previous researches have studied mostly TS risk and DS risk separately, but have not considered the mutual effect of contingencies. TSs mainly face contingencies related to line outages [
Previously, researchers have solved risk assessment and cost assessment separately for coordinated TS and DS [
1) In previous studies, the impact of emergency occurrence in the TS on transmission and distribution coordinated system has not been explored. In future power grids, ADN will not only participate in minimizing the overall system cost, but also play a key role in reducing system risk. As shown in

Fig. 1 Physical structure of coordinated TS and DS with contingency locations. (a) Line outage contingency case in TS. (b) Line outage contingency in DS causing islanding.
2) Furthermore, previous researchers have not studied the impact of the DS contingencies such as the impact of isolated islands on the coordinated cost of TS and DS. TS risks not only originate from TS contingencies but also DS contingencies. In
3) Besides, previous researchers have not investigated the optimization of transmission and distribution coordination systems under simultaneous contingent situations in both systems. System risks can arise from simultaneous contingencies in both TS and DS. Even in this case, the DS can help the TS reduce risks and optimize the synergy costs. Likewise, DERs available in DSs can reduce the island risk.
Further, the above-discussed cases are considered as basic problems in this research. To solve the above-mentioned issues, it is requisite to investigate the minimum cost re-evaluation based on contingent situations of TS and DS. Hence, it paves the way to study on the risk-aware distributed OPF (RA-DOPF) problem, which focuses on and contingencies in the TS and contingency in the DS. Besides, and contingencies fall under the category of prevention and correction risk analysis, respectively.
In smart grids, switch operations are automated in DS and can handle the contingency problem efficiently. However, for contingency in DS, DERs play a key role and can mitigate the risk. Due to the complexity of the proposed algorithm, only contingency after contingency is considered for simplification.
In this paper, an RA-DOPF algorithm with transmission and distribution coordination has been proposed. Additionally, the RA-DOPF solution is utilized to protect the system from the expected emergency state while maintaining the minimum cost. The contributions of this paper are summarized as follows.
1) The RA-DOPF algorithm has been applied to the coordinated TS and DS while considering: ① and contingencies in TS only and no contingency in DS; ② contingency in DS only, which causes islanding and no contingency in TS; ③ and contingencies in TS and contingency in DS simultaneously.
2) A two-level hierarchical model is utilized for RA-DOPF involving TS and DS for evaluating both risk and cost. Furthermore, analytical target cascading (ATC) has been utilized as TS and DS risk-averse coordination algorithm, and the Benders decomposition (BD) as an algorithm for solving the RAOPF and risk-based security-constrained OPF (RBSCOPF) problem.
The rest of the paper is organized as follows. Section II presents a hierarchical coordination framework. Section III explains the problem formulation and solution. Section IV describes different case studies. Section V concludes the paper and gives future directions.
The proposed framework consists of two sections: optimization and coordination. Initially, the problems of TS and DS are optimized, and then coordinated to achieve the optimal state of coordinated operation.

Fig. 2 TSO-DSO framework for RAOPF- and RBSCOPF-based coordination.
At this point, the weather-based contingencies are considered as the main reason for contingent situations in TS and DS [
As noted earlier, system risks include risks from TS and DS. Therefore, both TS and DS require separate risk assessments for line outages. Additionally, the contingencies in the two cases are different. In this paper, and contingencies are considered in TS and contingency in DS.
In most cases, the main causes of transmission line interruption are line overload, weather changes, etc. These unexpected events can cause the system to go from a normal state to an emergency state [
The three-state weather model is utilized for evaluation of the impact of weather conditions such as normal, adverse, and extreme conditions. This model is also developed in [
(1) |
(2) |
(3) |
The distribution of forced outage probability of overhead line within a short time can be calculated as:
(4) |
Moreover, the probabilities of each common-mode outages are evaluated using (5), while the severity of each possible contingency on system security is quantified by the load interruption cost, which is the product of VOLL and as shown in (6). In addition, the operation risk is calculated using the severity and probability of all contingencies [
(5) |
(6) |
(7) |
Equations (
The outage event in DS divides DS into various islands. This causes the loads within the islands to disconnect from the main bus, and curtailment may occur. However, if DGs are available in those islands, the loads can survive during a contingency. Consequently, curtailment risk needs to be re-evaluated considering loads that survive with the help of DGs. Thus, DGs are necessary for the survival of loads.
The problem at hand is complex considering the risk awareness. Further, OPF in TS is referred to direct current OPF (DCOPF) because the resistance is negligible in TS. Therefore, the voltage magnitude at every transmission and distribution node is assumed to be 1.0 p.u., and only active power is considered. Conversely, OPF in DS is referred to as alternating current OPF (ACOPF) because the ratio of line resistance to line reactance is large. Hence, the voltage at distribution terminals can vary between 0.95 p.u. and 1.05 p.u.. Additionally, the risk assessment based on load curtailment is considered, so the DS shares active power information and risk information, while reactive power sharing information is not included. To model the reactive power mismatch at the border between TS and DS, the voltage at TS boundary terminal is considered to be 1.0 p.u., while the boundary terminal voltage of DS can vary between 0.95 p.u. and 1.05 p.u., though the reactive power mismatch is reduced by bringing the boundary terminal voltage of DS to approximately 1 p.u.. Further, the reactive power at the boundary of DS is considered to be 50% of the assigned boundary active power (load or generator), which is sent as a shared variable by TSO to the relevant DSO. In this way, the DSOs will only need to take up the assigned active power and reactive power (half of assigned active power shared by TSO). Here, the assigned active power means the shared variable from TSO. Since the shared variable does not involve the reactive power variable, DSOs will take up 50% of active power (load or generator) shared by TSO, while the remaining reactive power is taken up by TSO. It should be noted that the scope of this paper will not involve the evaluation of voltage stability.
The proposed problem has a hierarchical structure where the TS operator (TSO) acts as an upper-level problem, while the DS operator (DSO) acts as a lower-level problem as shown in
TSO will serve as the upper level in the hierarchy to solve RA-DOPF. TSO will evaluate its RAOPF at TS boundaries considering weather-based line outages as shown in
DS acts as a lower-level section in the hierarchical structure of the problem. DSOs will evaluate the RBSCOPF and receive the targeted power generation from DERs. contingency after contingency is considered in DS. But it will be dealt with as contingency (RBSCOPF) in the formulation because the is already handled by the automated operation of switches. DS can help TS reduce both the total system cost and total system risk.
The RAOPF and RBSCOPF are formulated for TS and DS, respectively, and solved by using BD. However, the coordination problem between TS and DS are handled using ATC. The following sub-sections explain the problem formulation and solution.
RAOPF and RBSCOPF are different from basic OPF in terms of pre- and post-contingency constraints. The base-case constraints are taken as pre-contingency constraints, while and contingency constraints are taken as post-contingency constraints. Initially, this paper has solved RAOPF for TS, which is a convex non-linear problem (NLP) as shown below:
(8) |
s.t.
(9) |
(10) |
(11) |
The objective and basic constraints of OPF for TS are defined as:
(12) |
s.t.
(13) |
(14) |
(15) |
Here, for TS. Since DS has utilized ACOPF, (12)-(15) can be rewritten as:
(16) |
s.t.
(17) |
(18) |
(19) |
(20) |
(21) |
(22) |
In RBSCOPF, the boundary power for DS. To reduce reactive power mismatch at the boundary, the voltage angles on the transmission side are considered to be 1 p.u., while on the distribution side, they are considered within the range of 0.95-1.05 p.u..
The operation constraints given below are obligatory for the pre-contingency state of the system:
(23) |
(24) |
(25) |
(26) |
(27) |
(28) |
Equations (
Besides, the transmission loss is not negligible in DS. Therefore, the power loss in each line should be added. Equations (
(29) |
(30) |
(31) |
In the case of contingency, the preventive dispatching of generation units controls the branch flow violations to meet the criteria.
(32) |
(33) |
Here, (32) and (33) are used as the constraints for satisfying branch flow limits in contingent situations. The RAOPF and RBSCOPF both involve preventive constraints.
In RBSCOPF, (32) and (33) are rewritten as (34) and (35), which involve transmission losses in the evaluation of OPF.
(34) |
(35) |
In the case of contingencies caused by common-mode outages, the corrective measures such as generation re-dispatching and load curtailment eliminate post-fault branch overflows.
(36) |
(37) |
(38) |
(39) |
(40) |
The corrective measures must maintain the balance in the system (36) by generation re-dispatching and load curtailment, and bring the branch flows back within their nominal limits (37)-(38). Moreover, (39) allows the re-dispatching of generation units within the imposed limits by the scheduled reserves in the base case. Constraint (40) imposes the maximum limit of lost load j at bus b due to common-mode failure. It should be noted that RAOPF involves corrective constraints, while RBSCOPF does not evaluate corrective constraints. Accordingly, RAOPF involves the objective (12) with constraints (11), (13)-(15), (23)-(28), (32), (33), and (36)-(40). Conversely, RBSCOPF involves the objective (16) with constraints (11), (17)-(22), (26)-(31), (34), and (35).
For restricting the risk exposure of TS to its tolerable limit, it is enforced to follow the probabilistic constraint to avoid unnecessary load shedding, and balance the anticipated unsupplied demand that occurs due to common-mode outage failures:
(41) |
The constraint (41) can manage and mitigate system risk. Here, the system risk is divided into TS and DS risks. Here, VOLL is in $/MWh while EN

Fig. 3 Risk matrix under normal, adverse and extreme weather-based contingency conditions.
To avoid critical risk in the power system, system operators should select and follow for the optimized system operation. One should select the risk threshold keeping in view of the above-explained severity and probability levels in
By choosing risk-averse behavior and risk threshold properly, the proposed algorithm can achieve an optimized risk-based cost assessment.
This paper utilizes BD for solving the problem. For TS, the RAOPF problem is divided into master OPF problem and and sub-problems, while for DS, the RBSCOPF problem is divided into master OPF problem and sub-problem.
The master OPF problem consists of basic OPF objective function (12) and constraints (13)-(15). The constraints from (23)-(28) are added as base-case constraints.
Furthermore, the proposed algorithm adds more constraints in each iteration, which act as feasibility and optimality cuts from and sub-problems, as shown in (32)-(35) and (36)-(40), respectively. The BD-based RAOPF solution is shown in

Fig. 4 BD framework with master problem and sub-problems for evaluation of RAOPF and RBSCOPF.
For solving contingency violations, the preventive security of sub-problem is assessed to satisfy the violation of power flow limits:
(42) |
(43) |
(44) |
Here, (42) shows the objective of preventive security check, (43) and (44) shows the constraints for preventive measures, (45) shows the feasibility cuts that the proposed algorithm generates for preventive measures and adds to the master problem.
(45) |
For the sub-problem evaluation in DS, variable is added to (43) and (44) to obtain (46) and (47) with the objective (45) remaining the same.
(46) |
(47) |
To solve the common-mode contingencies that lead to power flow violations and unsupplied demand, the proposed algorithm evaluates corrective security sub-problem to calculate the required generation re-dispatching and load curtailment for system power balance.
(48) |
(49) |
(50) |
(51) |
(52) |
(53) |
(54) |
The master problem deals with these cuts as constraints, and optimizes iteratively to achieve the minimum cost for the required generation re-dispatching and load curtailment.
The hierarchical structure of RA-DOPF involves the objective functions of TSO and DSO. The coordination method is shown in

Fig. 5 Coordination of TSO and DSOs.
TSO is responsible for solving its RAOPF problem with common-mode contingencies due to the weather-based transmission outages. TSO also generate the targets, which are sent to DSOs. The following formulation (55) is utilized for the evaluation of coordinated TSO RAOPF, subject to (13)-(15), (23)-(28), (45), (54), and (56)-(59).
(55) |
(56) |
(57) |
(58) |
(59) |
The symbol is the Hadamard product for element-wise multiplication. If the contingency risk is located within TS, is considered as the risk threshold . On the other hand, is considered as overall system risk. When achieves less value than with less cost, the value of is updated to . In this way, the cost can be reduced further beyond the risk threshold. Although, an operator can decide whether to allow the algorithm to search for further least cost with less risk, or stop the algorithm at the initial risk threshold .
In modern power systems, ADNs play a vital role in the system cost optimization. DSO needs to optimize its cost objective function given in (60), subject to (11), (17)-(22), (26)-(31), and (61)-(63).
(60) |
(61) |
(62) |
(63) |
The TSO utilizes the DCOPF, thus, , and (61) can take care of the reactive power mismatch at the boundary of TS and DSs. If there is no contingency risk in the DS, TS risk will be considered as the overall system risk.
Due to the contingencies in DS, we need to investigate the islanding risk for the secure operation of the power system. For this reason, (64) has been utilized for evaluating the islanding-risk-based security-constrained OPF.
(64) |
DSO needs to evaluate separately the islanded section, which is disconnected from the main transformer bus. It is worth mentioning that will be added with to calculate the system risk for the simultaneous occurrence of islanding problem in DS and contingent problem in TS. This will increase the overall system risk, and a better optimized operation state can be achieved by reducing the risk.

Fig. 6 Flowchart of coordinated risk-based OPF of TSO and DSO.
Step 1: at the upper level, TSO evaluates the RAOPF problem and targets . Then, TSO communicates them to all DSOs. The risk calculated in the RAOPF problem is set to be a risk response for the system.
Step 2: at lower levels, each DSO receives targets from TSO and solves its RBSCOPF problem along with curtailment load calculation with and without islanding, respectively. DSOs send back the responses to TSO. Additionally, each DSO calculates the curtailed load for the cases with and without the support of DERs in DS, respectively. In the proposed algorithm, DSOs share this information with TSO for the evaluation of risks, i.e., expected system risk and reduced system risk. When the coordinated OPF of coupled TS and DSs is evaluated, the total evaluated risk using (7) is called the expected system risk. RAOPF and RBSCOPF are risk corrective methods. When these OPF methods are applied to coupled TS and DSs, the total evaluated risk is called reduced system risk.
Step 3: after receiving data from all DSOs, TSO solves its RAOPF problem to generate new targets for the next iteration while evaluating the minimum risk that is achievable.
Step 4: in the IL, TSO checks whether the following stopping rules are satisfied:
(65) |
(66) |
(67) |
Step 5: in the OL, the following sufficient conditions are checked for convergence:
(68) |
(69) |
If the above conditions are satisfied, the iteration is ended. Otherwise, the multipliers are updated by using the following equations:
(70) |
(71) |
(72) |
(73) |
Step 6: the algorithm repeats Steps 2-4 until the stopping criteria are met.
In this paper, we propose an RA-DOPF model for solving the risk problem with the minimum cost for coupled TS and DSs. Further, ATC is utilized as a coordination algorithm whose convergence and optimality has been proven in [
1) There will be communication between the upper- and lower-level sub-systems.
2) The communication between the same-level sub-systems is not allowed.
3) At the
To show the superiority of ATC, it is necessary to discuss the capability of the ATC algorithm to solve non-convex problems such as DCOPF with losses and ACOPF. The augmented Lagrange method is used as a penalty method, which involves a linear term and a quadratic term. As discussed in [
In this paper, a modified test system is used for the TS and DS. The IEEE 30-bus system with reserves acts as TS while the IEEE 13-bus radial DS with five DERs acts as DS. It should be noted that the total demand Pd of the system is scaled proportionally on all buses of TS. Likewise, for a particular DS, the load at relevant boundary bus of TS system connected to the DS is scaled proportionally on all buses of the DS. This means that a total load of the DS will be equal to the total load at the relevant bus of the TS system. Further, the impedance of the line, which connects the boundary bus of every DS with TS, is set to be p.u.. Also, the simulation has been performed on Core i7 2.7 GHz with 8 GB RAM, with MATLAB R2018a. Matpower has been used for OPF analysis and for obtaining the IEEE system datasets [

Fig. 7 System model of IEEE 30-bus system as TS and ten IEEE 13-bus radial DSs (ADNs) on shown buses of TS.
This paper involves contingency after contingency in the DS considering the automated switch operation. Thus, contingency in the DS utilizes the formulation of RBSCOPF, as contingency is already tackled by automated switch operation. To reduce the complexity of the problem, only DGs are considered as DERs. Henceforth, the uncertainty due to wind farms or solar power is not involved in this paper.
Here, the evaluated results are represented as system cost and system risk. The system risk is also the same as reduced system risk when a comparison between expected system risk and reduced system risk is provided. Further, expected system risk means the evaluation of risk without involving risk awareness measures given in the proposed algorithm.
Usually, DSOs are needed to share necessary data for defining the observability area of TSO, but any such request by TSO is subject to Article 4.2 of the system operation (SO) regulation [
In this case, RAOPF has been performed on TS with reserves while DCOPF is performed on DS. The coordinated risk-aware cost has been evaluated. The line outage in TS can cause the generation resources or load to be cut off from the main system. In this way, TS reserves and DERs in DS are used in a coordinated manner to reduce the total risk and cost. Also, the proposed algorithm allows TS and DS to adjust their generation source outputs to mitigate the projected risk of system and attain a state with low risk and optimal cost.
However,

Fig. 8 Comparison of reduced system cost and system risk in case 1.
Furthermore,

Fig. 9 Comparison of reduced system risk and expected system risk in case 1.
Some TSOs need a larger observability area including different interconnected systems due to structure and operation conditions of the power system [
This case study considers the DS islanding impact on TS risk evaluation. Besides, DCOPF is performed in TS while RBSCOPF is applied in DS to evaluate the risk due to islanding. When a line outage occurs in DS, it can create islands within DS. Thus, the islands connected to the main transformer bus will remain intact, and no load loss is expected. While other islands disconnected from the transformer bus will face the problem of curtailment. In this case, DERs in these islanded sections can be handy and help reduce the load curtailment. Besides, large-scale DERs are considered. Subsequently, the load curtailment is the minimum if the proposed algorithm utilizes DERs efficiently. In other words, DERs in DS can help reduce the total system risk by taking up the expected curtailed load. The proposed method will evaluate the risk due to islanding in DS and adjust risk within the threshold limits. Likewise, the proposed method also optimizes the cost with reasonable risk.
Further,

Fig. 10 Comparison of reduced system cost and system risk in case 2.

Fig. 11 Comparison of reduced system risk and expected system risk in case 2.
For a system with no such constraints on TS and DS described in earlier case studies, both TS and DS can share data for risk evaluation. Consequently, in case 3, the contingencies in both TS and DS are evaluated. This case study deals with a specific case of simultaneous line outage in TS and DS, which causes common-mode outages in TS and islanding in DS. Such risk is less probable, but it is necessary to study and evaluate such a problem because of its criticality. In this case, RAOPF is performed in TS while RBSCOPF in DS. The islanding in DS can divide it into different sections: one is connected to the main transformer bus while other sections act as an island. TS risk can be reduced using DER generation, which is available in the DS section connected to the main transformer bus. While DERs available in the islanded section can take up the load in that section, and, henceforth, reduce the load curtailment. It should be noticed that the overall system risk is the sum of TS risk and DS risk. Therefore, the overall system risk is greater than that evaluated in cases discussed above.

Fig. 12 Comparison of reduced system cost and system risk in case 3.

Fig. 13 Comparison of reduced system risk and expected system risk in case 3.
1) Case 1: and contingencies in the section of the IEEE 30-bus system while no contingency in the section of IEEE 13-bus radial DS.
2) Case 2: no contingency in the section of the IEEE 30-bus system while contingency in the section of the IEEE 13-bus radial DS.
3) Case 3: and contingencies in the section of the IEEE 30-bus system and contingency in the section of the IEEE 13-bus radial DS.
The truncated diagonal quadratic approximated (TDQA) algorithm [
Since ACOPF is utilized in DSOs for the evaluation of risk-based cost, the bus voltage optimization should be presented for the proposed algorithm. Case 3 involves higher risk than cases 1 and 2. Therefore, more voltage violations are expected in case 3. Hence, in
Furthermore, the boundary voltages in TS and DSs have a quite negligible mismatch as shown in
Generally, the cost is increased for the power system whenever the risk-averse measures are taken. The same phenomenon has been depicted in

Fig. 14 Trade-off relationship between risk and system cost. (a) Case 1 with risk threshold of $1800. (b) Case 2 with risk threshold of $250. (c) Case 3 with risk threshold of $2000.
Since the probability of emergencies under extreme weather conditions is much more than normal and adverse weather, the operation cost of the system increases so as to control such a high-risk threshold. Accordingly, in the proposed method, the preventive generation and reserve allocation are adjusted dynamically to limit the contingency risk.

Fig. 15 Dynamic adjustment of generation dispatching of TS generators under different weather conditions.
This paper has presented an RA-DOPF model for coordinated TS and DS. The proposed method has emphasized that the RAOPF in coupled TS and DS is inevitable. In previous research, the coordinated cost optimization in these systems mostly leads to a high-risk problem that this paper has addressed efficiently. On one hand, TS and DS both contribute to risk origination. On the other hand, the proposed method efficiently utilizes TS sources and DERs in DS to mitigate the risk coordinately.
Different case studies have described different categories of possible contingencies within both TS and DS. In the case of TS, the preventive and corrective measures are taken into account for mitigating and contingencies. Also, contingencies are considered in DS and its preventive measures are evaluated. Different case studies give a brief view of contingencies in different system boundaries and their effect on the total risk. Besides, the proposed method is evaluated for all cases and its efficacy is demonstrated. The RA-DOPF helps both systems to reduce their risk within their limits along with the optimized cost and less load curtailment. In the future, this research can be enhanced using the probabilistic coordination methods for the sources with uncertainty such as wind farms and photovoltaic. Moreover, the coordination of voltage stability between TS and DS could be considered in future studies.
REFERENCES
I. Kouveliotis-Lysikatos and N. Hatziargyriou, “Fully distributed economic dispatch of distributed generators in active distribution networks considering losses,” IET Generation, Transmission & Distribution, vol. 11, no. 3, pp. 627-636, Feb. 2017. [百度学术]
C. Lin, W. Wu, Z. Li et al., “Decentralized economic dispatch for transmission and distribution networks via modified generalized benders decomposition,” in Proceedings of 2017 IEEE PES General Meeting, Chicago, USA, Jul. 2017, pp. 1-5. [百度学术]
Y. Wen, C. Y. Chung, and X. Liu, “Hierarchical interactive risk hedging of multi-TSO power systems,” IEEE Transactions on Power Systems, vol. 33, no. 3, pp. 2962-2974, Aug. 2018. [百度学术]
Z. Li, Q. Guo, H. Sun et al., “Coordinated transmission and distribution AC optimal power flow,” IEEE Transactions on Smart Grid, vol. 9, no. 2, pp. 1228-1240, Jun. 2016. [百度学术]
A. Kargarian, M. Mehrtash, and B. Falahati, “Decentralized implementation of unit commitment with analytical target cascading: a parallel approach,” IEEE Transactions on Power Systems, vol. 33, no. 4, pp. 3981-3993, Dec. 2018. [百度学术]
M. Bragin, Y. Dvorkin, and A. Darvishi, “Toward coordinated transmission and distribution operations,” in Proceedings of 2018 IEEE PES General Meeting, Portland, USA, Aug. 2018, pp. 1-5. [百度学术]
A. Kargarian and Y. Fu, “System of systems based security-constrained unit commitment incorporating active distribution grids,” IEEE Transactions on Power Systems, vol. 29, no. 5, pp. 2489-2498, Mar. 2014. [百度学术]
A. Nawaz, H. Wang, Q. Wu et al., “TSO and DSO with large-scale distributed energy resources: a security constrained unit commitment coordinated solution,” International Transactions on Electrical Energy Systems, vol. 30, no. 3, pp. 1-26, Mar. 2020. [百度学术]
ENTSO-E. (2015, Nov.). General guidelines for reinforcing the cooperation between TSOs and DSOs. [Online]. Available: https://docstore.entsoe.eu/Documents/Publications/Position%20papers%20and%20reports/entsoe_pp_TSO-DSO_web.pdf#search=tso%2Ddso [百度学术]
M.-S. Kim, R. Haider, G.-J. Cho et al., “Comprehensive review of islanding detection methods for distributed generation systems,” Energies, vol. 12, pp. 1-21, Mar. 2019. [百度学术]
P. Mahat, C. Zhe, and B. Bak-Jensen, “Review of islanding detection methods for distributed generation,” in Proceedings of 2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies, Nanjing, China, Apr. 2008, pp. 2743-2748. [百度学术]
A. Nawaz and H. Wang, “Stochastically coordinated transmission and distribution system operation with large-scale wind farms,” CSEE Journal of Power and Energy Systems, doi: 10.17775/CSEEJPES.2020.02150. [百度学术]
Z. Li, Q. Guo, H. Sun et al., “A new LMP-sensitivity-based heterogeneous decomposition for transmission and distribution coordinated economic dispatch,” IEEE Transactions on Smart Grid, vol. 9, no. 2, pp. 931-941, Jul. 2017. [百度学术]
ENTSO-E. (2015, Mar.). Towards smarter grids: developing TSO and DSO roles and interactions for the benefit of consumers. [Online]. Available: https://docstore.entsoe.eu/Documents/Publications/Position%20papers%20and%20reports/150303_ENTSO-E_Position_Paper_TSO-DSO_interaction.pdf [百度学术]
A. Olson, A. Mahone, E. Hart et al., “Halfway there: can California achieve a 50% renewable grid?” IEEE Power and Energy Magazine, vol. 13, pp. 41-52, Jun. 2015. [百度学术]
I. Olivine. (2014, Jan.). Distributed energy resources integration summarizing the challenges and barriers. [Online]. Available: http://www.caiso.com/Documents/OlivineReport_DistributedEnergyResourceChallenges_Barriers.pdf [百度学术]
R. Billinton, C. Wu, and G. Singh, “Extreme adverse weather modelling in transmission and distribution system reliability evaluation,” in Proceedings of 14th Power System Calculation Conference, Seville, Spain, Jun. 2002, pp. 1-7. [百度学术]
W. Li and R. Billinton, “Common cause outage models in power system reliability evaluation,” IEEE Transactions on Power Systems, vol. 18, no. 2, pp. 966-968, May 2003. [百度学术]
M. Papic, S. Agarwal, R. Allan et al., “Research on common-mode and dependent (CMD) outage events in power systems: a review,” IEEE Transactions on Power Systems, vol. 32, no. 2, pp. 1528-1536, Jul. 2016. [百度学术]
E. Ciapessoni, D. Cirio, G. Kjølle et al., “Probabilistic risk-based security assessment of power systems considering incumbent threats and uncertainties,” IEEE Transactions on Smart Grid, vol. 7, no. 6, pp. 2890-2903, Mar. 2016. [百度学术]
H. M. Kim, W. Chen, and M. M. Wiecek, “Lagrangian coordination for enhancing the convergence of analytical target cascading,” AIAA Journal, vol. 44, pp. 2197-2207, Sept. 2006. [百度学术]
N. Ming, J. D. McCalley, V. Vittal et al., “Online risk-based security assessment,” IEEE Transactions on Power Systems, vol. 18, no. 1, pp. 258-265, Feb. 2003. [百度学术]
N. Michelena, H. Park, and P. Y. Papalambros, “Convergence properties of analytical target cascading,” AIAA Journal, vol. 41, pp. 897-905, May 2003. [百度学术]
S. Tosserams, L. Etman, P. Papalambros et al., “An augmented Lagrangian relaxation for analytical target cascading using the alternating direction method of multipliers,” Structural and Multidisciplinary Optimization, vol. 31, pp. 176-189, Feb. 2006. [百度学术]
R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MATPOWER: steady-state operations, planning, and analysis tools for power systems research and education,” IEEE Transactions on Power Systems, vol. 26, no. 2, pp. 12-19, Jun. 2011. [百度学术]
2 August 2017 Establishing a Guideline on Electricity Transmission System Operation, Commission Regulation (EU) 2017/1485, Aug. 2017. [百度学术]
A. Mohammadi, M. Mehrtash, and A. Kargarian, “Diagonal quadratic approximation for decentralized collaborative TSO + DSO optimal power flow,” IEEE Transactions on Smart Grid, vol. 10, no. 3, pp. 2358-2370, Jan. 2018. [百度学术]
A. Kargarian, J. Mohammadi, J. Guo et al., “Toward distributed/decentralized DC optimal power flow implementation in future electric power systems,” IEEE Transactions on Smart Grid, vol. 9, no. 4, pp. 2574-2594, Oct. 2016. [百度学术]