The two-stage robust UC model with DC flow constrained is established in this section. The first stage [constraints (4)–(5)] aims to optimize the UC decisions for conventional units, and the second stage [constraints (6)–(11)] optimizes the economic dispatch decisions of all units. From a physical perspective, the second stage explores all scenarios of renewable energy resources and obtains the decision results for the worst case. The first stage then adjusts the UC decisions based on the outcomes from the second stage.

The objective function (1) is the total cost, which includes the first-stage UC cost of conventional units and the second-stage dispatch cost under the worst-case scenario.

The first-stage cost (2) includes the startup cost and shutdown cost for conventional units. The relationship between binary variables is modeled in (4), which ensures the correctness of conventional units’ operation statuses during startup and shutdown periods (Zhang et al., 2017). Constraint (5) describes the minimum up and down time limits for each conventional unit (Yuan et al., 2022).

The second-stage cost (3) includes the dispatch costs of all units. The uncertain parameters in the robust solution are typically set at the upper or lower limits of their uncertainty range (Wu et al., 2008), and the uncertainty budget is commonly employed to mitigate conservative result. The uncertainty set and uncertainty budget are modeled in (6). Constraint (7) represents the maximum and minimum active power output limit for each conventional unit (Cao et al., 2021). Constraint (9) enforces the power output range for renewable units. Constraint (8) imposes ramping-up and ramping-down rate constraints for conventional units. System active power balance is ensured by (10). The nodal phase angle range is defined by (11).

1) Linearized branch flow model with Q and v

The original nonlinear power flow equations are expressed as follows: f i j , t  P = g i j v i , t 2 − v i , t v j , t cos ⁡ θ i j , t − b i j v i , t v j , t sin ⁡ θ i j , t : ∀ l ∈ Ω L , i , j = B + l , B − l , t ∈ T (12) f i j , t  Q = − b i j v i , t 2 − v i , t v j , t cos ⁡ θ i j , t − g i j v i , t v j , t sin ⁡ θ i j , t : ∀ l ∈ Ω L , i , j = B + l , B − l , t ∈ T (13)

The phase angle θ i , t and the square of the magnitude v i , t 2 of the nodal voltage are chosen as independent variables to fully linearize the AC power flow model, where the auxiliary variable v i j , t s is defined as follows: v i j , t s : = v i , t − v j , t 2 (14)

A Taylor second-order expansion for the sine and cosine terms of θ i j , t , and an approximation and mathematical transformation for the nonlinear terms of v i , t are used (Yang et al., 2016): sin ⁡ θ i j , t ≈ θ i j , t cos ⁡ θ i j , t ≈ 1 − θ i j , t s 2 v i , t = v j , t ≈ 1 v i , t s − v i , t v j , t = v i j , t s 2 + v i , t s − v j , t s 2 (15)

After above analysis, the linearized AC power flow model can be obtained as follows: f i j , t P , A = g i j v i , t s − v j , t s 2 − b i j θ i j , t + f l , t P , L f i j , t Q , A = − b i j v i , t s − v j , t s 2 − g i j θ i j , t + f l , t Q , L : ∀ l ∈ Ω L , i , j = B + l , B − l , t ∈ T (16)

The branch losses f l , t P , L and f l , t Q , L in (16) can be further linearized through an approximation process and a first-order Taylor expansion. The detailed derivation is provided in Yang et al. (2017b), and the final expression is as follows: f l , t P , L = g i j θ i j , t 0 θ i j , t + g i j v i , t 0 − v j , t 0 v i , t 0 + v j , t 0 v i , t s − v j , t s − g i j 2 θ i j , t s , 0 + v i j , t s , 0 f l , t Q , L = − b i j θ i j , t 0 θ i j , t − b i j v i , t 0 − v j , t 0 v i , t 0 + v j , t 0 v i , t s − v j , t s + b i j 2 θ i j , t s , 0 + v i j , t s , 0 : ∀ l ∈ Ω L , i , j = B + l , B − l , t ∈ T (17)

2) Other network constraints with Q and v under AC power flow linearization

With the introduction of the linearized AC power flow model, the operational constraints of the second stage must also consider the reactive power output range of conventional units (18), the balance of system reactive power (19), nodal voltage magnitude range (20), and the limits of branch flow (21). − Q g T x g , t T ≤ q g , t T ≤ Q g T x g , t T ; ∀ g ∈ Ω T G , t ∈ T (18) ∑ g ∈ Ω T G i q g , t T − ∑ i , j ∈ Ω L i f i j , t Q , A = ∑ d ∈ Ω D i q d , t D ; ∀ i ∈ Ω B , t ∈ T (19) v i 2 ≤ v i , t s ≤ v ¯ i 2 ; ∀ i ∈ Ω B , t ∈ T (20) f i j , t P , A 2 + f i j , t Q , A 2 ⩽ F ¯ l 2 ; ∀ l ∈ Ω L , i , j = B + l , B − l , t ∈ T (21)

Further, the ACRUC model is formulated, with the objective function defined by Eqs 1–3. The constraints of the first stage include (4)–(11), while the constraints of the second stage include (16)–(17) and (18)–(21).

Figure 1 illustrates the proposed model architecture in this paper. Firstly, a robust UC model is employed to fully consider the output power uncertainty of renewable energy units. Then, a linearized AC power flow model is introduced to reflect the changes in nodal voltage and reactive power flow during the system operation. Based on the above, the ACRUC model is established.

However, direct solving of the ACRUC model is difficult due to the existence of stochastic variables. Therefore, this paper adopts the C&CG method (Zeng and Zhao, 2013) to transform the first stage into the master problem and the second stage into the feasibility subproblem under uncertain scenarios for iterative solving. The subproblem incorporates constraints and decision variables for worst-case scenarios into the master problem based on the results obtained from solving the master problem. In addition, the outer approximation method (Bertsimas et al., 2012) is used to handle the economic dispatch subproblem during the solving process.

Note that the branch flow limits (21) is a quadratic constraint. Directly adding them to the UC model will make it a mixed-integer quadratic programming problem, that is, difficult to solve and yields poor scalability. However, it can be observed that these constraints essentially describe a security operating region of AC branches on the P-Q complex plane. Therefore, they can be approximated by circular linearization instead of quadratic constraints to obtain a fully linearized AC network constraint. Furthermore, the branch limits (21) can be approximated by a group of linear constraints as follows: − K l , m f f i j , t P , A − f i j , t Q , A ⩾ B l , m f K l , M f + 1 − m f f i j , t P , A − f i j , t Q , A ⩾ B l , M f + 1 − m f K l , M f + 1 − m f f i j , t P , A + f i j , t Q , A ⩾ B l , M f + 1 − m f − K l , m f f i j , t P , A + f i j , t Q , A ⩾ B l , m f (22)

As shown in Figure 2, each quadrant is segmented into M f parts, and there exist errors when applying the circular linearization. The linearization error is quantified by calculating the ratio of the difference between the sector’s area and the triangle’s area to the sector’s area. According to error estimation in (23), increasing the number of linearization segments can effectively reduce the linearization error, making the linearized branch flow constraint a more exact approximation of the original one. ϵ f ≈ 1 − cos π 4 M f (23)

Note that the scale of the network constraint Eq 21 is quite large, and it is necessary to add corresponding constraints to each branch at each time period in the scheduling horizon. After linearizing the AC power flow, the number of constraints will also double according to the number of linearization segments. For example, if M linearization segments are required in one quadrant, a single branch at a single time period will have at least 4 × M branch flow limit constraints. In a UC model with about N branches and a time scale of 24 h, there will be 96 × M × N such constraints. This brings pressure to the solution efficiency of our problem and may cause the model to fall into the dilemma of exhausting computing resources or unacceptable solution time after several iterations. Furthermore, in the iterative solution process using the C&CG method, the scale of constraints added by sub-problems to the main problem will also increase sharply with the number of iterations.

Inspired by RCI technique under the DC power flow model (Zhai et al., 2010; Yang et al., 2021), this paper proposes a customized RCI method for the linearized AC power flow model. Its main idea is to estimate conservatively branch flow limits in actual operation under given load level, renewable energy prediction results, and uncertainty error by solving multiple relatively simple optimization subproblems. Hence, branch flow limits that will not be overloaded are eliminated.

1) Identification method in “cold start” mode

The first step in the “cold start” mode is to identify network constraints to determine the power limits for each branch during each time period. To obtain the power loss for each branch, a Taylor expansion is used based on the base case system operating condition that in the “cold start” mode, in which nodal voltage magnitudes are 1 and nodal voltage phase angles are 0. In this case, the network power loss is zero and the first end branch flow is equal to the second end branch flow, so the two-end branch flow limits do not need to be considered separately. Based on this, the RCI model in “cold start” mode is established, which includes an objective function defined by Eq. 24 and constraints such as system power balance constraints (10) and (19), linearized AC branches flow model (25), branch flow limits (26), conventional units output range (7) and (18), renewable energy units output range (9), nodal voltage phase angle limit (11), and nodal voltage magnitude constraints (20). max   f l ′ , t P , A 2 + f l ′ , t Q , A 2 (24) f l , t P , A = g i j v i , t s − v j , t s 2 − b i j θ i j , t f l , t Q , A = − b i j v i , t s − v j , t s 2 − g i j θ i j , t (25) f l , t P , A 2 + f l , t Q , A 2 ⩽ F ¯ l 2 : ∀ l ∈ Ω L \ l ′ (26)

Since the constraints set of RCI model is a subset of the constraints set of the ACRUC model, its feasible region is greater than or equal to that of the ACRUC model. This ensures that the maximum value of the calculated branch flow is greater than or equal to the actual maximum value of the branch flow during operation. Therefore, if the RCI method finds that the constraint is not violated, then this constraint must be redundant in actual operation.

Assuming that f l ′ , t P , A * , f l ′ , t Q , A * is the optimal solution of the identification method problem under the “cold start” mode, if there is no feasible solution, no optimal solution, or the branch flow limit is exceeded, i.e., if f l ′ , t P , A * 2 + f l ′ , t Q , A * 2 > F ¯ l ′ 2 , it is necessary to add a branch flow limit constraint for this branch at this time period. Otherwise, the constraint can be considered redundant and ignored without affecting the model solution.

2) Identification method in “warm start” mode

The identification model for the “warm start” mode is slightly different from that of the “cold start” mode. In the “warm start” mode, the base case system operating condition for branch power loss in the Taylor expansion is obtained from the nodal voltage magnitudes and voltage phase angles calculated by ACRUC with RCI method in the “cold start” mode. In this case, the network power loss on the branch cannot be ignored, resulting in different power flows at the first end and second end of the branch. Hence, the identification method must be carried out separately for the first end and second end of the branch. The objective function is expressed by Eq. 27 when identifying the first end branch flow, and by Eq. 28 when identifying the second end branch flow. The constraints for two-end of the branch flow include system power balance constraints (10) and (19), linearized AC branch flow model (16)–(17), conventional units output range (7) and (18), renewable energy units output range (9), nodal voltage phase angle range (11), and nodal voltage magnitude range (20). The branch flow limit constraint is expressed by Eqs. 29, 31 for the first end branch flow identification and Eqs. 29, 30 for the second end branch flow identification. max   f i ′ j ′ , t P , A 2 + f i ′ j ′ , t Q , A 2 (27) max   f j ′ i ′ , t P , A 2 + f j ′ i ′ , t Q , A 2 (28) f i j , t P , A 2 + f i j , t Q , A 2 ⩽ F ¯ l 2 f j i , t P , A 2 + f j i , t Q , A 2 ⩽ F ¯ l 2 ; ∀ l ∈ Ω L ⧵ l ′ , i , j = B + l , B − l (29) f i ′ j ′ , t P , A 2 + f i ′ j ′ , t Q , A 2 ⩽ F ¯ l 2 ;   i ′ , j ′ = B + l ′ , B − l ′ (30) f j ′ i ′ , t P , A 2 + f j ′ i ′ , t Q , A 2 ⩽ F ¯ l 2 ;   i ′ , j ′ = B + l ′ , B − l ′ (31)

Assuming that f i ′ j ′ , t P , A * , f i ′ j ′ , t Q , A * is the optimal solution for the RCI model of the first end branch flow, and f j ′ i ′ , t P , A * , f j ′ i ′ , t Q , A * is the optimal solution for the RCI model of the second end branch flow. If either of the RCI models for the two-end branch flows has no feasible solution, no optimal upper bound, or exceeds the branch flow limit, it is necessary to add a branch flow limit constraint for this branch at time period t . Otherwise, the constraint is considered as redundant and can be ignored without affecting the solution result of the model.

3) The process of proposed RCI method

The computational process of the RCI method under the linearized AC power flow model is shown in Figure 3. The linearized AC power flow network constraints on each branch at each time period are split into RCI sub-problems, and the calculations are completed using solvers separately. Then, the necessary branch flow limit constraints are added to the constraint set of the ACRUC model.

The RCI method based on linearized AC power flow can effectively reduce the scale of the constraints. In addition, since the models based on the RCI method for each branch and each time period are decoupled from each other, the solution can be independent and parallel. Therefore, the method has good potential for parallel computing and can fully utilize the computing resources of hardware devices.

The modified NERL-118 system includes 118 buses, 186 branches, and 54 thermal units. In addition, there are five different capacity wind farms located at bus 24, 27, 31, 82, and 100, and 14 different capacity photovoltaic plants located at bus 15, 18, 19, 32, 54, 55, 69, 76, 92, 100, 104, 105, 110, and 112. The case has large number of buses and branches, with renewable energy sources accounting for 38.52% of the total installed capacity. The complexity of the UC model is relatively high, which makes it difficult to solve. Therefore, to reduce the complexity of the robust UC model, the test system has merged the same type of renewable energy sources on same bus, treating them as equivalent renewable energy plants.

This paper presents case studies to analyze the rationality and effectiveness of linearized AC power flow network constraints by comparing the operation statuses of thermal units, the load conditions of overload branches, and the system operational cost. Additionally, the effectiveness of the proposed RCI method is verified by comparing the problem scale and solving efficiency with and without RCI method. The case study designed in this section is presented in Table 1.

The number of segments used to linearize the network constraints is set to 6. The renewable energy forecasting error is assumed to be 30%, and the uncertainty budget for renewable energy units Γ g are 6, 12, and 24. The comparison results between Case 2–1 and Case 3–2, as well as between Case 2–2 and Case 4–2 (as shown in Table 2; Table 3), demonstrate that the proposed RCI method correctly selects redundant network constraints. In both “cold start” mode and “warm start” mode, constraints related to branches that may exceed the power transmission security limit during operation are retained, effectively limiting the transmission power of those branches within the security range.

The comparison results of the system operating cost are shown in Table 4. Comparing Case 3–1 and Case 3–2, it can be observed that in the “cold start” mode, when Γ g is set to 6, 12, and 24, the relative errors of operating costs with and without RCI method are 0.14%, 0.13%, and 0.03%, respectively. Comparing Case 4–1 and Case 4–2, it can be observed that in the “warm start” mode, when Γ g is set to 6, the relative error of operating costs with and without RCI method is 0.09%. Therefore, the RCI has little effect on the optimal value of the objective function, ensuring its correctness.

Figure 4 illustrate UC solutions of conventional units for Case 1 under different renewable energy uncertainty budgets ( Γ g is set to 6, 12, and 24). The comparison demonstrates that, as the level of renewable energy uncertainty increases, the unit commitment of conventional units become more frequent. This is mainly because that the generation scheduling is needed to ensure sufficient system flexibility to cope with high uncertainty.

Figure 5 illustrates the number of online thermal units for Case 1, Case 3–2 and Case 4–2 under the same uncertainty budget of renewable energy ( Γ g is set to 6). The comparison reveals that, unlike Case 3–2 and Case 4–2, Case 1 uses a DC power flow model and fails to activate enough thermal units to mitigate the impact of reactive power flow and voltage changes. However, Case 4–2 considers both linearized AC power flow and network losses, requiring the activation of a larger number of thermal units and demanding a higher level of capacity adequacy for the dispatchable resources.

Figure 6 illustrates the distribution of voltage magnitude for Case 2–1, Case 2–2, Case 3–1, Case 3–2, Case 4–1, and Case 4–2. These cases all adopt the AC power flow model, which reflects the variation range and distribution of the system’s voltages. In contrast, the voltage is assumed to be constant in the DC power flow model. Therefore, the AC power flow model more accurately reflects the operational conditions of the system. Furthermore, by comparing Case 3–1 and Case 3–2, as well as Case 4–1 and Case 4–2 ( Γ g is set to 6), we observe that the voltage distributions obtained with and without considering RCI are very similar. This finding further validates the correctness of the RCI method.

The dispatch results of all units for Case 3–1, Case 3–2, Case 4–1 and Case 4–2 ( Γ g is set to 6) are shown in Figure 7. By comparing Case 3–1 and Case 4–1, as well as Case 3–2 and Case 4–2, we observe that the warm-start AC model takes network losses into account. Therefore, the warm-start AC model provides a more accurate portrayal of the system’s operational conditions compared to the cold-start AC model.

Table 5 and Table 6 demonstrate the improvement of the RCI method in both “cold start” and “warm start” modes. The term “original problem constraints” refers to the sum of the constraints in the robust model first stage and second stage. “Total solving time” refers to the total time spent on RCI and model solving. “N/A” indicates that the model was considered unsolvable due to excessive computational time in the solving environment described in the article. It can be seen that, under the premise of ensuring correctness, the identification method can effectively improve the efficiency of model solving.

As shown in Table 5, the “cold start” mode reduced the original problem constraints by 60.58% before and after using the RCI method. When the renewable energy uncertainty budgets Γ g are set to 6, 12, and 24, using the RCI method in the “cold start” mode can reduce the solving time by 70.54%, 38.1%, and 33.87%, respectively. This demonstrates that when the power system has a large network scale, the RCI method can significantly reduce the number of constraints and the original problem’s size, and improve the solving efficiency.

The effect of solution efficiency improvement is more significant in the “warm start” mode when the renewable energy uncertainty budgets Γ g is set to 6, as shown in Table 6, where the size of the original problem constraints is reduced by 69.45%, and there is also a corresponding improvement in the solving time.