The main task of SCUC is to determine the start-stop state of the unit in each period of the scheduling cycle, and its goal is to minimize the operating cost under the premise of satisfying the coupling constraints of the system and the physical constraints of the single unit (Ding et al., 2021). The system coupling constraints include power balance constraints, system rotation reserve constraints, transmission line capacity constraints, etc. The single unit operation physical constraints include unit start-stop logic constraints, ramp constraints, and minimum start-stop time constraints (Ju et al., 2023). The model is presented as follows: min ∑ t = 1 T ∑ i ∈ G e n C i u u i , t + C i d d i , t + ∑ s ∈ S ∑ i ∈ G e n π s f i p i , t , s , x i , t (1) ∑ i ∈ G e n p i , t , s + ∑ i ∈ R e P i , t , s re = ∑ b ∈ B u s D b , t (2) ∑ i ∈ G e n x i , t P i max ≥ ∑ i ∈ G e n p i , t , s + P t res , ∀ t ∈ 1 , T Z (3) ∑ b ∈ B u s ∑ i ∈ G e n H l , t PTDF p i , t , s + ∑ b ∈ B u s ∑ i ∈ R e H l , t PTDF P i , t , s re − ∑ b ∈ B u s H l , b PTDF D b , t ≤ F l max , l ∈ L i n e s , ∀ t ∈ 1 , T Z (4) x i , t − x i , t − 1 = u i , t − d i , t , i ∈ G e n , t ∈ 2 , . . . , T (5) ∑ τ = t − T i on + 1 t u i , τ ≤ x i , t , t ∈ T i on , . . . , T (6) ∑ τ = t − T i off + 1 t d i , τ ≤ 1 − x i , t − T g off   , t ∈ T i off , . . . , T (7) x i , t P i min ≤ p i , t , s ≤ x i , t P i max , i ∈ G e n , t ∈ 1 , . . . , T , s ∈ S (8) p i , t , s − p i , t − 1 , s ≥ − x i , t R i down − d i , t R i shut , i ∈ G e n , t ∈ 2 , . . . , T , s ∈ S (9) p i , t , s − p i , t − 1 , s ≤ x i , t R i up + u i , t R i start , i ∈ G e n , t ∈ 2 , . . . , T , s ∈ S (10)

Equation 1 constitutes the objective function, targeting the minimization of aggregate start-up and shut-down costs for generating units alongside the expectation of operational expenses across diverse scenarios, aiming for their absolute minimum. The objective function f is typically a quadratic function of the generator output and does not include cross-terms with the start-up and shut-down variables of the units. During the computation, this function is generally piecewise linearized. Equations 2–4 delineate the system constraints, encompassing the requirement for power balance, the stipulation for system reserves, and the static transmission line limitations, respectively. Equation 5 articulates the logical constraints that dictate the interrelation between the start-up and shut-down states of the units. Equations 6, 7 serve to uphold the condition that the uninterrupted operation period of each unit surpasses the stipulated minimum durations. Equation 8 imposes that the output levels of the units remain confined within predefined upper and lower bounds, concurrently adhering to the constraints imposed by their operational statuses. Lastly, Equations 9, 10 are dedicated to ensuring that the ramp-up and ramp-down rates of the units do not transgress the established maximum limits, and that the generation output during these transitional periods remains within the prescribed safety margins.

As the foundational elements for the DLR model and the superquantile regression model, rather than the focal points of this paper, the heat balance equation and polynomial regression are described in the Supplementary Appendix.

Although QR is already versatile and widely used for quantifying risk, its capabilities can be expanded by using super quantile regression (SQR) to estimate the cumulative tail behavior of a random variable. Unlike QR, which focuses on quantifying the risk based on the probability of undesirable events, SQR provides insight into the magnitude of these events. This feature is particularly valuable in applications where understanding the consequences of undesirable events is necessary.

The α-super quantile function of a continuous random variable y is defined as its tail expectation. In relation to the quantile function (46), the upper-tail α-super quantile function can be expressed in Equation 11: Q α s y = E y | y ≥ Q α y = 1 1 − α ∫ α 1 Q β y d β (11)

Similar to other regressions, the linear SQR model is an estimator of the conditional α-super quantile of y in Equation 12: y α = Q α s y | x = x T β α (12) where y ^α is the estimated upper-tail α-super quantile; α is the super quantile level of the random variable; β _α is the model parameter vector. The lower tail α-super quantile function and its corresponding regression model can be defined similarly. As SQR is evolved from QR, it benefits from the ability to determine its model parameter vector β _α by solving a linear programming problem.

Based on the concept of HBE and several regression models, the dynamic capacity increase model is constructed. In the modeling process, a quantile regression model is used to quantify the temporal characteristics of conductor temperature. This model mainly forecasts the α-quantile T _t ^α of conductor temperature at time period t based on the input vector of environmental parameters for the previous k time periods. The input vector consists of weather-related parameters, including wind speed W _s , wind direction angle W _a , ambient temperature T _a , solar radiation Q _s and conductor temperature I _t . Specifically, the input vector has the following form: x t = 1 , I 2 , I , W s , W a , T a , Q s T W s ∘ W a , W s ∘ T a , W s ∘ Q s , W a ∘ T a , W a ∘ Q s , T a ∘ Q s T W s ∘ W s , W a ∘ W a , T a ∘ T a , Q s ∘ Q s T (13)

In Equation 13, I (Xu et al., 2024), I , W _s, W _a , T _a and Q _s are all row vectors representing the current and previous k time periods’ conductor current and quadratic terms, wind speed, wind direction angle, ambient temperature, and solar radiation, respectively, as shown below: I 2 = I t − 1 2 , … , I t − k 2 I = I t − 1 , … , I t − k W s = W s , t − 1 , … , W s , t − k W a = W a , t − 1 , … , W a , t − k T a = T a , t − 1 , … , T a , t − k Q s = Q s , t − 1 , … , Q s , t − k (14)

It can be seen that Equation 13 includes several cross-product terms of row vectors. These cross-product terms can better explore the dependencies between input parameters. By more effectively modeling the correlations between these parameters, these cross-terms enhance the influence of hyper-parameters on estimation.

Based on this, the construction of the big data-based model follows similar steps. The quantile regression model for a given risk level is obtained by solving an optimization model. The input vector x t of the model is constructed based on past predicted data of weather parameters and conductor current, as shown in Equation 13, with the actual conductor temperature values y _i = T _i being regressed as the output value. Afterwards, to estimate the super-quantile of conductor temperature at time t, the input vector is substituted into the traditional regression model to calculate the α-super-quantile of predicted conductor temperature. The resulting super-quantile model has the closed form shown in Equation 15: T t α = β α 0 + ∑ j = 1 k β α t − j , 2 I t − j 2 + ∑ j = 1 k β α t − j , 1 I t − j + ∑ j = 1 k F t − j , ∀ t ∈ k , T Z (15)

T _t ^α is the estimated α -quantile temperature of the transmission line at time t, and the remaining variables correspond to the variables in Equations 13, 14. According to the mathematical model, the conductor temperature is not only related to the current passing through the transmission line in the current time period, but also to the current in the previous time periods, which reflects the temporal variation characteristics of the conductor’s capacity.

When using the equation to represent a specific transmission line which requires dynamic capacity increase, the model takes the following form: T l , t , s = β α 0 + ∑ j = 1 k β t − j , 2 I l , t − j , s 2 + ∑ j = 1 k β t − j , 1 I l , t − j , s + ∑ j = 1 k F l , t − j , s , ∀ t ∈ k , T Z , ∀ l ∈ L i n e d , s ∈ S (16) where T _l,t,s is the α-hyperquantile of the estimated temperature of the transmission line at the time period t, and the remaining variables correspond to the variable contents of the Equations 13, 14. The model describes the functional relationship between current and temperature of conductor and provides model support for the SCUC model.

On the basis of the transmission line temperature prediction model shown in Equation 16, it is modified to embed dynamic transmission line constraints into SCUC model. The resulting dynamic transmission line constraint is as follows. β α 0 + ∑ j = 1 k β t − j , 2 p l , t − j , s 2 + ∑ j = 1 k β t − j , 1 p l , t − j , s + ∑ j = 1 k F l , t − j , s ≤ T l max , ∀ t ∈ k , T Z , ∀ l ∈ L i n e d , ∀ s ∈ S (17) where T _l ^max is the maximum temperature of the selected transmission line l, and the meaning of other variables is the same as that of Equation 16 From the DC power flow theory and the resulting power transfer distribution factor (PTDF), the relationship between the wire current and the transmission line power can be calculated as follows: p l , t , s = ∑ b ∈ B u s ∑ i ∈ G e n H l , t PTDF p i , t , s + ∑ b ∈ B u s ∑ i ∈ R e H l , t PTDF P i , t , s re − ∑ b ∈ B u s H l , b PTDF D b , t (18)

Above all, the dynamic security-constrained unit commitment model (D-SCUC) is described in the following form, which combines the dynamic line capacities with the SCUC model: min ∑ t = 1 T ∑ i ∈ G e n C i u u i , t + C i d d i , t + ∑ s ∈ S ∑ i ∈ G e n π s f i p i , t , s , x i , t (19) s . t .   2 − 4 , 5 − 10 (20) 17 − 18 (21)

As evidenced by Equations 19–21, the optimization model encompasses both integer and discrete variables, with quadratic components featured in both the objective function and the dynamic transmission line constraints. This configuration presents significant computational difficulties when applied to large-scale systems. In response to these challenges, we propose a solution algorithm predicated on an enhanced Lagrangian relaxation methodology.

Based on the aforementioned D-SCUC model, it is evident that the system constraints—power balance, reserve requirements and line flow limits—couple all unit variables together. Conversely, unit-specific constraints are separable by unit. Thus, the application of the Lagrangian Relaxation method becomes feasible for relaxing these system constraints. This relaxation transforms the original problem into a collection of decoupled single-unit subproblems, each pertaining to an individual generator without inter-unit dependencies.

Dynamic transmission line constraints involve quadratic terms of line flows, making it difficult to decouple them using conventional methods. Therefore, this paper proposes an ADMM-based approach to address the issue of coupling caused by transmission line constraints, enabling the iterative solution of the D-SCUC model proposed in this article.

In the research conducted herein, the IEEE 118-bus system was selected as the test case, encompassing a network topology featuring 118 nodes, inclusive of 54 generation units, interconnected via 177 transmission lines, and serving 91 loads, thereby furnishing the dataset for the SCUC model. Aiming to investigate the impact of the dynamic capacity augmentation model on system operation expenses and the alleviation of transmission line congestion, a prototypical single-scenario representation of renewable energy from a provincial context was employed. This scenario entailed the modification of select conventional thermal power units from the baseline case to renewable energy installations, with their respective power generation levels being predetermined.

Additionally, an analytical computation is performed on the SCUCmodel that incorporates dynamic transmission line capacities. Parameters pertinent to transmission lines within this dynamic capacity augmentation framework are derived through the resolution of a planning model, leveraging historical data specific to the region. In the course of practical computations, dynamic capacity constraints are enforced on transmission lines encompassed within the set denoted as Lined, whereas static transmission line constraints persist for alternative conductors. A selection of various upper bound values for transmission line temperatures is made, with the ensuing outcomes delineated in Table 2.

The quartet of columns—T _max, f _val , ∑P _line and n—respectively denote the maximum designated transmission line temperature, the computed operational expenditure, the cumulative power across transmission lines, and the aggregate count of intervals wherein transmission line loading surpasses the static constraint threshold. As per the tabulated insights, within the temperature setting spectrum from 75°C to 100°C, the operational cost exhibits a sequential diminution from 2,142,780 to 2,132,563, paralleled by a progressive escalation in the total transmission line load summation from 173,515 to 176,603, and a rise in the tally of congestion instances from 3 to 44. Moreover, under the temperature settings delineated above, a heatmap encompassing both congested conduits and those benefiting from dynamic capacity enhancements is illustrated in Figure 1. The interpretation of grid squares mirrors that of Figure 2, wherein a deepening hue signifies an elevation in power magnitude. A comparative analysis with Figure 2 reveals that an upsurge in the prescribed temperature correlates with a marked proliferation of time segments characterized by elevated transmission line loads, concomitant with an augmentation in transmission line loads themselves.

In this section, we address the proposed Lagrangian Relaxation algorithm to validate its efficacy in resolving the D-SCUC model that incorporates dynamic transmission line capacity enhancements. The IEEE-300 bus system serves as the platform for simulation calculations, featuring a network topology comprising 69 generating units, 60 transformers, 304 transmission lines, and 185 load nodes. Following the approach outlined previously, a portion of the conventional thermal power units are substituted with renewable energy counterparts. A selection of various representative renewable energy scenarios from a specific locale is employed for the computation of the D-SCUC model under multiple scenarios. The parameters for the transmission line models are obtained through training with historical data.

Initially, for test cases encompassing a scenario count spanning from 1 to 8, computations are executed employing both the Gurobi Mixed Integer Programming (MIP) solver and the Lagrangian Relaxation framework. Concerning the Lagrangian Relaxation algorithm, a dual gap threshold of 0.01 is utilized, signifying that once the dual gap dips beneath this threshold, a solution of sufficient quality is deemed to have been attained. The magnitudes of the calculated operating costs and corresponding relative errors are delineated in Table 3. The quartet of columns within the table signifies the scenario quantity, the outcomes derived from the MIP solver, the results emanating from the Lagrangian Relaxation framework, and the relative error, respectively. As evidenced, under differing counts of renewable energy scenarios, the discrepancies between solutions procured via the Lagrangian Relaxation algorithm and those yielded by the solver remain confined within a margin of 0.5%, indicative of the high-caliber approximations delivered by the proposed Lagrangian Relaxation algorithm.

Building upon the validation of algorithmic accuracy, this study contrasts the temporal performance of two algorithms and explores their respective solution velocities in relation to the quantity of scenarios. Pertaining to the aforementioned eight resolution contexts, graphical representation delineates the correlation between scenario count and computational duration for both algorithms, illustrated in Figure 3.

From the graph, it can be observed that when the number of scenarios is small, the Lagrangian Relaxation algorithm takes more time than the Mixed Integer Programming (MIP) method. However, as the number of scenarios increases, the time complexity of the MIP method rises significantly with the increase in scenarios, whereas, in contrast, the solution time for Lagrangian Relaxation grows at a relatively slower pace. This phenomenon aligns with the characteristics of both solving algorithms.

In Gurobi’s implementation of the MIP method, branch-and-cut techniques are typically employed, which involve partitioning and contracting the feasible region to find a more precise range for the problem’s feasible domain, thus facilitating the solving process. Since this algorithm involves the generation of branches, its solution space grows exponentially with the increase in variables, resulting in slower solution times when dealing with a large number of scenarios.

On the other hand, the Lagrangian Relaxation algorithm decouples the D-SCUC subproblems across multiple scenarios. While an increase in the number of scenarios does add complexity to the subproblems, they remain greatly simplified compared to their original form, and thus the solution time does not drastically increase. Theoretically, it has been proven that the iteration time for the Lagrangian Relaxation algorithm is approximately linearly related to the number of variables. Consequently, the Lagrangian Relaxation algorithm is better suited for solving large-scale, multi-scenario SCUC problems.

Lastly, with regard to the iterative procedure of the Lagrangian algorithm, its iteration curve is illustrated in Figure 4.

As depicted in the chart, following a specific number of iterations, both the values of the original (primal) problem and its dual counterpart converge to a particular interval. Nonetheless, there is n’t a strict inequality with the primal being greater than the dual. This is attributable to the nature of the subgradient algorithm, which during its iterative process, cannot ensure a strict adherence to a descent direction. This characteristic is directly influenced by the choice of the step size at each iteration.