During the entire restoration process of TSs and DSs, various uncertainties such as wind and solar power output forecasts, load predictions, and buses and lines restoration failure probabilities exist. If these uncertainties are not addressed properly, it may lead to restoration strategies that do not meet the security requirements of the system’s restoration. Therefore, this paper employs CVaR theory to assess the conditional risk value of multiple uncertainties during the entire restoration process.

CVaR is defined as the average loss incurred when the risk loss of an investment portfolio exceeds the value at risk (VaR) at a given confidence level within a certain investment horizon (Rockafellar and Stanislav, 2002), whose specific expression is represented by V CVaR = E f R Z , λ f R Z , λ > V VaR (1) where Z is the optimization variable matrix, which corresponds to the restoration strategy in this work; V CVaR and V VaR are the CVaR and VaR, respectively; λ is continuous random variables that represent multiple uncertainties; E ⋅ is the mathematical expectation; f CVaR B is the net restoration benefit of the TSs and DSs considering CVaR of multiple uncertainties, whose specific expression is presented by f CVaR B = f B − ∑ t = 1 T step p t RES Δ P t RES + p t L Δ P t L + p t line ∑ l = 1 N line ρ l , t RF , line + p t bus ∑ n = 1 N bus ρ l , t RF , bus (2) where f B is the restoration benefit of TSs and DSs without considering CVaR, and its specific expression will be illustrated in Section 2.2; T step is the number of restoration time step; p t RES and p t L are the unit restoration risk costs associated with wind and solar power output forecasts errors and load prediction errors, respectively; Δ P t RES and Δ P t L are the errors in wind and solar power output forecasts and load predictions during the restoration process, respectively; p t line and p t bus are the unit restoration risk costs resulting from line and node restoration failures, respectively; ρ l , t RF , line and ρ l , t RF , bus are the restoration failure probabilities of lines and buses, respectively; N line and N bus are the total numbers of lines and buses, respectively.

On this basis, the CVaR and VaR for the multiple uncertainties during the restoration process can be simultaneously obtained by solving Eq. 3. min   V CVaR = V VaR + ∑ n = 1 N SCE P r , n f CVaR , n B − V VaR , 0 + 1 − β (3) where β CVaR is the confidence level of CVaR; f CVaR , n B − V VaR , 0 + is the maximum between f CVaR , n B − V VaR and 0; N ^SCE is the number of operation scenarios; P _r,n is the occurred probability of the nth scenario.

The objective function of the proposed model is to maximize the entire-process generation and load power restoration benefits of the TSs and DSs, as shown in (4). f B = max   ∑ z = 1 N TDS ∑ n = 1 N bus z ∫ 0 T step P NBS , n , t z d t + b L , n z ∑ t = 1 T step P L , n , t z Δ t (4) where N TDS is the number of TSs and DSs; N bus z is the number of buses in the zth power system; b L , n z is the unit restoration benefit of the nth bus in the zth power system; P NBS , n , t z and P L , n , t z are the restored generation and load power of the nth bus in the zth power system at time step t; Δ t is the duration of each time step. Note that the linearization method for generation power restoration refers to reference (Zhao et al., 2018), and the linearized expression is represented by (5). P NBS , n z , max T NBS , n z , rp − P NBS , n z , st T NBS , n z , rp + T NBS , n z , sp 2 + P NBS , n z , max − P NBS , n z , st T step − T NBS , n z , sp − T NBS , n z , rp − T NBS , n z , st P NBS , n z , max − P NBS , n z , st (5) where P NBS , n z , max and P NBS , n z , st are the rated power and auxiliary consumption power of non-black-start units at the bus n in the zth power system, respectively; T step is the number of time steps; P NBS , n z , st , T NBS , n z , sp and T NBS , n z , rp are the start time, start duration and ramp duration, respectively.

The constraints of the proposed model consist of the entire-process restoration constraints for the TSs and DSs. The unique and general constraints for the TSs and DSs are described as follows.

1) Unique constraints of the TS

The TS typically includes large thermal power units that are not black-start capable, so when a major power outage occurs, these units need to be black-started initially using black-start units. Therefore, the unique constraints for the TSs encompass the black-start-related constraints. More specifically, the relationship between the restoration power and time step for non-black-start units is expressed as 6); the relationship between the start-up time step of non-black-start units and bus restoration status is presented as 7); the constraints on the start-up time step of non-black-start units concerning cold start and hot start time step are expressed as 8). P NBS , n , t TS = 0 , t < t 1 t − t 1 R NBS , n TS − P NBS , n TS , st , t 1 ≤ t < t 2 P NBS , n TS , max − P NBS , n TS , st , t ≥ t 2 (6) ∑ t = 1 T step 1 − B bus , n , t TS ≤ T NBS , n TS , st (7) T NBS , n TS , st , min ≤ T NBS , n TS , st ≤ T NBS , n TS , st , max (8) where P NBS , n , t TS , P NBS , n TS , st , and P NBS , n TS , max are the restored generation power, auxiliary consumption power, and rated power of non-black-start units at the bus n in time step t, respectively; R NBS , n TS is the ramping rate of non-black-start units; t ₁ and t ₂ are the time steps when non-black-start units restart and reach their rated power, respectively; B bus , n , t TS is a binary variable indicating the restoration status of the bus where the non-black-start unit is located; T NBS , n TS , st , max and T NBS , n TS , st , min are the upper and lower bounds on the black-start time of non-black-start units.

2) Unique constraints of the DS

DSs typically do not have large non-black-start units, but they directly manage the loads connected to them. Therefore, the unique constraints for DSs encompass load restoration-related constraints. More specifically, restored non-flexible loads cannot be disconnected again, as represented by Eq. 9). Flexible loads can be subject to demand response and partial disconnection after being restored, as represented by Eq. 10). Considering practical factors like power flow limits and system frequency security, the load restored in each time step should not exceed a certain value, as expressed in Eq. 11). The load at bus n can only be restored after its bus has been restored, as expressed in Eq. 12). Note that some buses in the TS also have high-voltage level loads connected to them, so they also need to follow these load restoration constraints as well. 0 ≤ P n , t − 1 DS , L ≤ P n , t DS , L ≤ P n DS , L , max , ∀ n ∈ Ω bus DS , NFL (9) 0 ≤ P n , t DS , L ≤ P n DS , L , max , ∀ n ∈ Ω bus DS , FL (10) P n , t DS , L − P n , t − 1 DS , L ≤ Δ P n , t DS , L , max (11) B bus , n , t DS , L ≤ B bus , n , t − 1 DS (12) where P n , t DS , L and P n DS , L , max are the already restored load at bus n and the maximum load, respectively; Ω bus DS , NFL and Ω bus DS , FL are sets of non-flexible and flexible load buses; Δ P n , t DS , L , max is the upper limit of the restorable load for each time step, which is related to power flow limits and system frequency security; B bus , n , t DS , L is the load restoration state variable for the bus n.

3) General constraints for the TSs and DSs

The general constraints for the TSs and DSs include network restoration constraints and power balance constraints. More specifically, the restored buses and lines will not experience another power outage, as represented by Eqs 13, 14), respectively; the necessary but not sufficient condition for line restoration is that both of its start and terminal buses have been restored, as represented by Eq. 15); the necessary condition for bus restoration is that at least one connected line has been restored, as represented by Eq. 16); considering practical factors such as manual operations and line charging time, if the adjacent lines of a specific line have not been restored in a given time step, that line cannot be restored in that time step, as represented by (17); the operation constraints of energy storage systems refer to reference (Chen C. et al., 2023). B bus , n , t T D S ≥ B bus , n , t − 1 T D S (13) B line , l , t T D S ≥ B line , l , t − 1 T D S (14) B bus , n , t T D S ≥ B line , l , t T D S , ∀ l ∈ Ω line , bus n T D S (15) B bus , n , t T D S ≤ ∑ l ∈ Ω line , bus n T D S B line , l , t T D S (16) B line , p , t T D S ≤ ∑ q ∈ Ω line , line p T D S B line , q , t − 1 T D S (17) where B bus , n , t T D S and B line , l , t T D S are Boolean variables reflecting the restoration status of bus n and line l, respectively, with a value of 1 indicating that the bus or line has been restored; Ω line , bus n T D S is the set of lines connected to bus n, while Ω line , line p T D S is the set of lines connected to line l.

Both TSs and DSs need to consider power balance constraints, as represented by Eqs 18, 19), respectively. It can be seen from Eqs 18, 19 that the power balance constraints in the DS are less complex than those in the TS, as they do not involve the power of black-start and non-black-start units. Besides, power interaction between TSs and DSs can occur through the transmission-distribution coupling buses. ∑ n ∈ Ω bus TS P NBS , n , t TS + P BS , n , t TS + P RES , n , t TS + P ES , n , t TS = ∑ n ∈ Ω bus TS P L , n , t TS + P DS , n , t TS (18) P DS , n , t TS + ∑ x ∈ Ω bus DS , n P RES , x , t DS , n + P ES , x , t DS , n = ∑ x ∈ Ω bus DS , n P L , x , t DS , n (19) where Ω bus TS and Ω bus DS , n are the sets of buses in the TS and its connected DS under bus n, respectively; P BS , n , t TS , P RES , n , t TS , and P ES , n , t TS are the power generation of black-start units, renewable energy units, and energy storage devices at bus n in the TS, respectively; P L , n , t TS is the load power at bus n in the TS; P DS , n , t TS is the interaction power provided by the TS to the connected DS at its bus n, with positive values indicating supply from the TS to the DS; P RES , x , t DS , n , P ES , x , t DS , n , and P L , x , t DS , n are the power generation of renewable energy units and energy storage devices, as well as the load power at bus x in the TS connected under bus n in the TS.

Both TSs and DSs are subject to power flow constraints. More specifically, the TS typically has a meshed structure, so AC power flow constraints as shown in Eqs 20, 23) need to be considered. On the other hand, the DS is typically radial in structure, allowing the use of DistFlow DC power flow constraints as in Eqs 24–29). Since both TSs and DSs power flow constraints involve non-convex and non-linear terms, they are linearized separately using the approximate linearization methods proposed in references (Zhao et al., 2021; Chen et al., 2022b). P i , t TS = ∑ i , j ∈ Ω line TS P i , j , t TS (20) Q i , t TS = ∑ i , j ∈ Ω line TS Q i , j , t TS (21) P i , j , t TS = U i , t TS 2 G i , j , t TS − U i , t TS U j , t TS G i , j , t TS ⁡ cos ⁡ θ i , j , t TS + B i , j , t TS ⁡ sin ⁡ θ i , j , t TS (22) Q i , j , t TS = − U i , t TS 2 B i , j , t TS − U i , t TS U j , t TS G i , j , t TS ⁡ sin ⁡ θ i , j , t TS − B i , j , t TS ⁡ cos ⁡ θ i , j , t TS (23) ∑ p ∈ Ω line , n + DS P p , t DS − ∑ q ∈ Ω line , n − DS P q , t DS − I q , t DS 2 R q , j DS = P n , t , out DS (24) U j , t DS = U i , t DS − I i , j , t DS R i , j DS (25) P i , j , t DS = U i , t DS I i , j , t DS (26) P i , j , t DS ≤ P i , j DS , max (27) I i , j , t DS ≤ I i , j DS , max (28) 0 ≤ U i , t DS ≤ U i DS , max (29) where P i , t TS and Q i , t TS are the total injected active and reactive power at bus i, respectively; P i , j , t TS and Q i , j , t TS are the active and reactive power on line ij, respectively; Ω line TS is the set of lines in the TS; U i , t TS is the voltage at bus i. G i , j , t TS , B i , j , t TS , and θ i , j , t TS are the conductance, susceptance, and phase angle of line ij in the TS, respectively; Ω line , n + DS and Ω line , n − DS are the set of lines in the DS that start from and end at bus n, respectively; P q , t DS , I q , t DS , and R q , j DS are the power, current, and resistance of bus i in the DS, respectively; P n , t , out DS is the power flowing out of bus n in the DS; U i , t DS is the voltage at bus i in the DS; P i , j DS , max , I i , j DS , max , and U i DS , max are the upper limits for line power, current, and bus voltage in the DS, respectively.

Considering the objective function and constraints of the unified restoration optimization model for the TSs and DSs shown in Eq. 4), the augmented Lagrangian function is introduced to formulate the distributed restoration optimization models for the TSs and DSs, as represented by (30) and (31), respectively. f TS = max   f CVaR TS , B + ∑ x = 1 N bus , DS TS ∑ t = 1 T step λ DS , x , t TS P DS , x , t TS − P TS , t DS , x + ∑ x = 1 N bus , DS TS ρ DS , x TS 2 ∑ t = 1 T step P DS , x , t TS − P TS , t DS , x 2 (30) s.t. (6)–(8), (13)–(18), (20)–(23) f DS , n = max   f CVaR DS , n , B + ∑ t = 1 T step λ TS , t DS , n P DS , n , t TS − P TS , t DS , n + ρ TS DS , n 2 ∑ t = 1 T step P DS , n , t TS − P TS , t DS , n 2 (31) s.t. (9)–(12), (13)–(18), (24)–(29).

where f TS and f DS , n are the augmented objective functions for the TS and the nth DS, respectively; λ DS , x , t TS and ρ DS , x TS are the Lagrange multipliers and penalty factors for the TS, while λ TS , t DS , x and ρ TS DS , x are the Lagrange multipliers and penalty factors for the nth DS, respectively; f CVaR TS , B and f CVaR DS , n , B are the restoration benefits for the TSs and DSs considering CVaR, and their specific expressions are represented by Eqs 32, 33), respectively. f CVaR TS , B = ∑ n = 1 N bus TS ∫ 0 T step P NBS , n , t TS d t + b L , n TS ∑ t = 1 T step P L , n , t TS Δ t (32) f CVaR DS , n , B = ∑ m = 1 N bus DS , n b L , m DS , n ∑ t = 1 T step P L , m , t DS , n Δ t (33)

The adaptive ADMM-based distributed solving process for the previously constructed distributed optimization models for the TSs and DSs is utilized, as outlined below.

Step 1

Set the maximum iteration number k max , primal residual convergence threshold δ P , dual residual convergence threshold δ D , and initial iteration counter k = 1.

Step 2

The TS collects the desired transmission-distribution interaction power strategy matrix P TS , t DS , n , k from each DS. Then, it solves the distributed restoration model for the TS as represented by (30) to obtain the desired transmission-distribution interaction power strategy matrix P DS , n , t TS , k + 1 for the TS. Set n = 1.

Step 3

The nth DS receives P DS , n , t TS , k + 1 from the TS. Then, it solves the distributed restoration model for the nth DS as represented by (31) to obtain the desired transmission-distribution interaction power strategy matrix P TS , t DS , n , k + 1 . n = n+1.

Step 4

Check if n > N bus , DS TS ? If yes, proceed to Step 5 , otherwise, return to Step 3 .

Step 5

According to the previous description, the utilized adaptive ADMM can adjust the step size dynamically according to the primal residue and dual residue during the iterative process, thereby achieving a faster convergence rate than that of standard ADMM. More specifically, according to the dual-update acceleration iteration strategy, if the primal residual is greater than a certain threshold, update the penalty factors according to (34)–(35) and update the Lagrange multipliers according to (36)–(37); otherwise, when the primal residual is equal to or less than a certain threshold at iteration k * , keep the penalty factors unchanged and update the Lagrange multipliers according to (38)–(39). ρ DS , n TS , k + 1 = ρ DS , n TS , k / 1 + lg d DS , n TS , k , D p DS , n TS , k , P , d DS , n TS , k , D δ D ≥ 10 p DS , n TS , k , P δ P ρ DS , n TS , k 1 + lg p DS , n TS , k , P d DS , n TS , k , D , p DS , n TS , k , P δ P ≥ 10 d DS , n TS , k , D δ D ρ DS , n TS , k , otherwise (34) ρ TS DS , n , k + 1 = ρ TS DS , n , k / 1 + lg d TS DS , n , k , D p TS DS , n , k , P , d TS DS , n , k , D δ D ≥ 10 p TS DS , n , k , P δ P ρ TS DS , n , k 1 + lg p TS DS , n , k , P d TS DS , n , k , D , p TS DS , n , k , P δ P ≥ 10 d TS DS , n , k , D δ D ρ TS DS , n , k , otherwise (35) λ DS , n , t TS , k + 1 = λ DS , n , t TS , k + ρ DS , n TS , k + 1 P DS , n , t TS , k + 1 − P TS , t DS , n , k + 1 (36) λ TS , t DS , n , k + 1 = λ TS , t DS , n , k + ρ TS , t DS , n , k + 1 P DS , n , t TS , k + 1 − P TS , t DS , n , k + 1 (37) λ DS , n , t TS , k + 1 = λ DS , n , t TS , k + ρ DS , n TS , k + 1 P DS , n , t TS , k + 1 − P TS , t DS , n , k + 1 + K D P DS , n , t TS , k + 1 − P TS , t DS , n , k + 1 − P DS , n , t TS , k − P TS , t DS , n , k + K I ∑ m = k * k P DS , n , t TS , m + 1 − P TS , t DS , n , m + 1 (38) λ TS , t DS , n , k + 1 = λ TS , t DS , n , k + ρ TS , t DS , n , k + 1 P DS , n , t TS , k + 1 − P TS , t DS , n , k + 1 + K D P DS , n , t TS , k + 1 − P TS , t DS , n , k + 1 − P DS , n , t TS , k − P TS , t DS , n , k + K I ∑ m = k * k P DS , n , t TS , m + 1 − P TS , t DS , n , m + 1 (39) where K D and K I are the integral and double residual control parameters, respectively; p DS , n TS , k , P = p TS DS , n , k , P and d DS , n TS , k , D = d TS DS , n , k , D are the primal and dual residuals for the kth iteration, whose specific expressions are represented by Eqs 40, 41). p DS , n TS , k , P = p TS DS , n , k , P = ∑ t = 1 T step P DS , n , t TS , k + 1 − P TS , t DS , n , k + 1 2 (40) d DS , n TS , k , D = d TS DS , n , k , D = max   ∑ t = 1 T step P DS , n , t TS , k + 1 − P DS , n , t TS , k 2 , ∑ t = 1 T step P TS , t DS , n , k + 1 − P TS , t DS , n , k 2 (41)

Step 6

Determine the convergence of the primal and dual residuals according to (42). If (42) is satisfied or k ≥ k max , terminate the iterations. Otherwise, k = k+1 and return to Step 2. p DS , n TS , k , P ≤ δ P ∩ d DS , n TS , k , D ≤ δ D = 1 (42)

According to the above process, it can realize distributed solving for the TSs and DSs and obtain their respective optimal restoration strategies. The distributed optimization models are established using the YALMIP platform in MATLAB R2021a software and the distributed solving can be realized through the GUROBI 10.0 solver.