Controllable regulation resources are defined as power units with strong regulation capabilities, whose output can be flexibly adjusted within allowable limits in response to dispatch instructions. They are characterized by strong ramping capabilities, rapid response, and high controllability, and are regarded as key flexible resources in power system dispatch and operational optimization.

Semi-controllable regulation resources, including wind and solar power, exhibit inherent uncertainty, which is captured using a probabilistic model to characterize their output variability. The probability of each scenario is determined by the proportion of original unreduced scenarios it represents.

The stochastic characteristics of wind and solar power are modeled using the Weibull distribution for wind speed and the Beta distribution for solar irradiance. Representative samples are generated through Latin Hypercube Sampling (LHS), and redundant scenarios are reduced with the Backwards Reduction (BR) algorithm to improve computational efficiency. The corresponding power outputs are then obtained using wind-speed-to-power conversion models and empirical solar power estimation formulas, as shown in Equations 8, 9. P w , t wt = 0   v < v in P rated   v n ⩽ v ⩽ v off v − v in v n − v in P rated   v in ⩽ v < v n 0   v > v off (8) P p , t pv = η pv A pv x pv (9) where v , v in , v n , and v off denote the regional wind speed, cut-in speed, rated speed, and cut-out speed of the wind turbine, respectively. P rated is the turbine’s rated power. For solar power, η pv is the conversion efficiency and A pv is the array area.

In addition, a probability-weighted method is used to evaluate the importance of each scenario, ensuring that the generated scenarios capture the uncertainty of semi-controllable resources, as shown in Equation 10: p s = N s N total (10) where P s is the probability of scenario s . N s is the number of original samples mapped to scenario s . N total is the total number of original scenarios.

To characterize typical daily load patterns, historical load data are fitted using a maximum likelihood estimation (MLE)-based method. The normal distribution is selected for each time interval due to its ability to capture the central tendency of load variations and its convenient analytical properties. For each interval, multi-day load data are fitted via MLE, and the estimated mean is taken as the representative load. Aggregating these interval-wise means yields the typical daily load profile.

Power loads are classified into fixed, variable, and transferable types. Fixed loads are inflexible and cannot be curtailed without affecting user comfort or operational safety. Variable loads offer short-term adjustability and can be modified in response to grid conditions with limited impact on user behavior. Transferable loads exhibit temporal flexibility, allowing their operation to shift across time periods without changing total energy consumption. Accordingly, the total load at any time is expressed as the sum of fixed, variable, and transferable components, as shown in Equation 11. L t = L t f + L t v + L t s L t f = α f L t typ 0 ≤ L t v ≤ α v L t typ ∑ t = 1 T L t s = ∑ t = 1 T L t typ − α f L t typ − α v L t typ L min s ≤ L t s ≤ L max s (11) where L t f , L t v and L t s denote the fixed, variable, and transferable loads, respectively. α f is the proportion of fixed load in the typical daily load, and α v is the maximum allowable proportion of variable load. L min s and L max s denote the lower and upper bounds of the transferable load, respectively.

To evaluate the supply capability of coordinated T&D networks under component failures, typical fault scenarios are constructed using the first-order failure modeling principle. Major components, including generation units, storage systems, renewable resources, and interface lines, are modeled as independent fault sets. For each set, both the no-fault case (0 order) and the single-fault case (1 order) are considered. The Cartesian product of these sets generates the complete fault scenario set, capturing system behavior under structural vulnerabilities.

Assuming independent failure events across components and lines, a joint Bernoulli-based probability model is used to assign probabilities to each fault scenario. The probability of a scenario is computed as the product of the failure probabilities of the failed components and the operational probabilities of the remaining components, as shown in Equation 12. p ω = ∏ i ∈ F ω dev λ i · ∏ j ∉ F ω dev 1 − λ j · ∏ l ∈ F ω line λ l · ∏ k ∉ F ω line 1 − λ k (12) where F ω dev and F ω line denote the sets of devices and interface lines that fail in fault scenario ω . The parameters λ i and λ l are the failure rates of device i and line l . This approach constructs a first-order joint fault scenario space that includes all critical system components and interface lines, resulting in a consistent scenario probability distribution. The distribution supports subsequent evaluations of power supply capability under extreme operating conditions.

To enhance adaptability under complex operating conditions, the T&D network scheduling model is reformulated in a compact form, as shown in Equations 13–20. max y f y = c T y (13)

The constraints are as follows: A x + b ≤ 0 (14) G x + N ξ ^ ≤ 0 (15) L x + M z ≤ 0 (16) Q x + F s ≤ 0 (17) J z + u ≤ 0 (18) I x + H y ≤ 0 (19) D y + q ≤ 0 (20)

In the above optimization model, the total operational revenue is expressed by the objective function f y . The variable set x contains continuous power-related variables in the T&D network, including P M , t , P D n , t , P g , t gen , P h , t hyd , P e , t ch , P e , t dis , P w , t wt , P p , t pv . ξ ^ represents continuous renewable-related variables under uncertainty, including v , x pv . s denotes the energy storage levels S e , t . y contains load-related variables L t f , L t v , L t s , Δ L . Finally, z contains binary decision variables u g , t gen , u e , t ch , u e , t dis . The variable x is constrained by (14), corresponding to constraints (1), (27), (5)-(30), and (32). Constraint (15) correlates x with ξ ^ , corresponding to constraint (8) and (9). Constraint (16) links x with z , corresponding to constraint (4) and (31). Constraint (17) models storage dynamics, corresponding to constraint (6). The variable z is restricted by (18), corresponding to constraints (28) and (33). Constraint (19) correlates x and y , corresponding to constraints (2)-(3). Constraint (20) imposes load limits, corresponding to constraint (11).

In the framework, fault scenarios are assumed to be nonrecoverable, meaning that a failed device remains unavailable throughout the study period. Faults influence the key operational constraints, including (14), (15), and (16). To consistently capture these effects, binary fault state variables δ are introduced into the original constraints to model component failures, as shown in Equations 21–23: A x + b δ ≤ 0 (21) G x + N ξ ^ δ ≤ 0 (22) L x + M z δ ≤ 0 (23) where δ ∈ 0 , 1 indicates the fault status of critical resources.

The CVaR is used to evaluate the power supply risk of the coordinated T&D networks. Two scenario sets are considered: the fluctuation scenario set Ω r , which represents renewable output variability, and the one-order joint fault scenario set Ω f , which covers key components and interface lines. The joint uncertainty scenario set is then formed as the Cartesian product Ω = Ω r × Ω f .

For each joint scenario ω ∈ Ω , the total load shedding Δ L and its probability π ω can be obtained. Based on this, the CVaR is used to measure the extreme load curtailment risk at confidence level α : Δ L = ∑ t Δ L T , t + ∑ t ∑ n Δ L D n , t (24) CVaR Δ L ω , α = min VaR + 1 1 − α ∑ ω ∈ Ω π ω · σ ω (25) Δ L ω − σ ω − VaR ≤ 0 σ ω ≥ 0 (26)

Equation 24 computes the total load curtailment under different fault scenarios. Equation 25 defines the CVaR-based load shedding risk. Equation 26 introduces the auxiliary variable σ ω to linearize the CVaR formulation. Adopting the CVaR-based modelling strategy enables a quantitative assessment of supply reliability under multiple uncertainties and supports informed dispatch decisions.

This study investigates a coordinated T&D network consisting of a single transmission network interconnected with three distribution networks. In this context, “lines” refer to the tie lines connecting the transmission and distribution networks. An energy flow-based modeling framework is adopted to represent the hierarchical energy exchange relationships. Wind and solar output scenarios are generated from typical daily wind speed and irradiance data, with wind characterized by the Weibull parameters k and c and solar characterized by the Beta parameters α and β , as shown in Figure 2. The remaining system parameters are provided in Supplementary null Appendix.

The dispatch cycle used in the study is 24 h with a step size of 1 h. Other system parameters are provided in the Appendix. The case study is implemented in MATLAB R2024b, modeled by Yalmip, and solved using Gurobi 12.0.0. All the numerical simulations are implemented on a PC with 32 GB RAM and a 2.1 GHz Intel Core i7 processor.

Figure 3 illustrates the aggregated response of heterogeneous adjustable resources under different disturbance levels. By jointly coordinating multiple types of adjustable resources, the proposed method enables effective flexibility aggregation across time scales. Under small disturbances, the aggregated flexibility is sufficient to fully eliminate load shedding, demonstrating strong local balancing capability. Under large disturbances, although load shedding cannot be completely avoided, the coordinated response significantly mitigates its magnitude by redistributing regulation among different resource types. These results highlight that, compared with conventional methods that rely on single or loosely coordinated resources, the proposed approach enhances system resilience by explicitly aggregating multisource flexibility and enabling differentiated responses to disturbance severity.

To evaluate the impact of coordinated T&D network optimization on system load supply capability under typical renewable generation and fault scenarios, the following case studies are designed and implemented:

Case 1

Independent optimization of the transmission network and each distribution network. This serves as a widely adopted benchmark;

Case 2

Coordinated optimization between the transmission network and Distribution Network 1, while Distribution Networks 2 and 3 are optimized independently;

Case 3

Coordinated optimization between the transmission network and Distribution Networks 1 and 2, while Distribution Network 3 is optimized independently;

Case 4

Coordinated optimization across the transmission network and all three distribution networks.

Figure 4 presents the total load shedding under different case settings. The results show that higher levels of T&D coordination consistently reduce total load shedding, demonstrating that coordinated optimization enhances overall supply reliability.

Figures 5, 6 illustrate the distributions of transferable and variable loads under different cases. As shown in Figure 5, Case 4 exhibits a more uniform temporal allocation of transferable loads, rather than concentrating them during periods of high renewable output, indicating enhanced scheduling flexibility enabled by coordinated optimization. Figure 6 further shows that higher coordination levels increase the utilization of variable loads, allowing a larger portion of total load demand to be satisfied.

This section evaluates the impact of critical equipment failures on the power supply capability of coordinated T&D networks. Based on typical wind and solar output scenarios, representative fault combinations are constructed, and the CVaR at a 95% confidence level is used to quantify supply risk. The resulting CVaR value of 645.5 MWh indicates that, in the most severe 5% of scenarios, the system experiences an average curtailment of 645.5 MWh.

Tables 1, 2 summarize the multi-scenario simulations and the corresponding impacts of different component fault types. The results indicate that interface line outages and failures of conventional units exert the most significant influence on system supply capability, with fault type C2 isolating Distribution Network D1 and causing 2061.81 MWh of load shedding. Similarly, thermal and hydropower unit failures (fault types C3 and C5) reduce the regulation capacity of the transmission network, resulting in 1953.94 MWh and 4247.67 MWh of load shedding, respectively. By contrast, local renewable generation faults in distribution networks (such as C9 and C11) mainly increase reliance on transmission support, but inter-network coordination enables stable operation without causing large-scale curtailment.

To evaluate the impact of typical faults on the energy balance of aggregated adjustable resources, three representative scenarios are considered: C2 (interface line failure), C3 (thermal unit failure), and C9 (wind power failure in D1). Figure 7 shows that transferable loads are mainly scheduled during periods of high renewable output. For C2, the interface line outage isolates D1 and its limited renewable generation cannot meet peak demand, leading to significant load shedding. In contrast, the transmission network is supported by more abundant resources, enabling it to effectively compensate for local deficits.

Figures 8, 9 examine the C3 and C9 scenarios. For C3, load shedding occurs in the first few hours after the thermal unit outage because the failure combined with low renewable output intensifies pressure on adjustable resources. For C9, the wind power failure in D1 reduces local renewable availability and shifts load scheduling to periods of low transmission utilization. This indicates that local renewable outages reshape regional resource allocation and reinforce the need for stronger transmission-distribution coordination.