Event constraint unit commitment robust optimization solution method based on primary function adaptive chaotic polynomial expansion

Abstract

The invention discloses an event constraint unit commitment robust optimization solving method based on primary function adaptive chaos polynomial expansion, and the method comprises the following steps: 1) building an event constraint unit commitment robust optimization model which comprises a day-ahead unit commitment model and a state evaluation model considering a composite scene; 2) carrying out approximation on the state evaluation model by adopting base self-adaptive sparse chaos polynomial expansion, and constructing a state evaluation agent model; 3) replacing a state evaluation model in the unit commitment robust optimization model with a state evaluation agent model; 4, the unit commitment robust optimization model after replacement is solved, and a unit commitment plant.The state evaluation model is reconstructed through the relaxation technology to improve the smoothness of model response, on this basis, an agent model of the state evaluation model is established based on BASPCE, and the precision of the agent model is remarkably improved.

Description

Technical Field

[0001] This invention relates to the field of uncertainty analysis in power systems, specifically a robust optimization solution method for event-constrained unit combinations based on basis function adaptive chaotic polynomial expansion. Background Technology

[0002] In recent years, power system uncertainty analysis methods based on surrogate models have attracted widespread attention for their efficiency and accuracy in balance calculations. Among them, Basis-Adaptive Sparse Polynomial Chaos Expansion (BASPCE) is currently a research hotspot. However, most existing BASPCE-based optimization models do not consider operational adjustment measures such as load shedding, making it difficult to accurately simulate the system's response behavior under faults or extreme conditions. Therefore, their application in safety and reliability assessments remains somewhat limited.

[0003] In system state assessment and risk analysis, load shedding indicators often exhibit non-smooth characteristics under uncertain environments, especially in scenarios such as small-capacity unit failures. Their probability distribution clusters around zero values, leading to an unsmooth transition between zero and positive values. This discontinuity in distribution poses a challenge to the direct application of chaotic polynomial expansion models, as the accuracy of such methods depends on the smoothness of the output response. In reality, decision-dependent uncertainties in power system operation (such as unit start-up and shutdown, day-ahead generation output, and reserve allocation) are high-dimensional and complex, significantly affecting state assessment results. Existing surrogate model methods do not consider this factor.

[0004] For uncertainty optimization scenarios that simultaneously consider source load uncertainty and component failure scenarios, existing research mostly focuses on stochastic optimization frameworks. These frameworks lack sufficient consideration for conservatism in extreme scenarios, making it difficult to ensure adequate robustness of the decision. Furthermore, stochastic optimization often suffers from excessive computational burden due to scenario combinatorial explosion. Robust optimization is another uncertainty optimization method that seeks the most severe uncertainty scenario to guarantee the robustness of the final solution. Traditional robust optimization employs iterative methods, such as the Column & Constraint Generation (C&CG) algorithm. However, these iterative algorithms are time-consuming when dealing with large uncertainty scenarios.

[0005] As mentioned above, existing technologies have limitations in handling high-dimensional uncertainty, non-smooth model output, and decision-dependent uncertainty, and lack efficient solution methods for robust optimization models of unit combinations under a large number of complex scenarios. Summary of the Invention

[0006] The purpose of this invention is to provide a robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions, comprising the following steps:

[0007] Step 1) Establish an event-constrained robust optimization model for unit combination, including a day-ahead unit combination model and a state assessment model considering complex scenarios;

[0008] Step 2) The state evaluation model is approximated by a basis-adaptive sparse chaotic polynomial expansion to construct a state evaluation proxy model;

[0009] Step 3) Replace the state assessment model in the unit combination robust optimization model with a state assessment proxy model;

[0010] Step 4) Solve the robust optimization model of the replaced unit combination to obtain the unit combination scheme.

[0011] Furthermore, in step 1), the optimization objective of the unit combination robust optimization model is to minimize the cost under the worst-case scenario;

[0012] The objective function of the unit combination robust optimization model is shown below:

[0013] (1)

[0014] In the formula, z and y represent the sets of decision variables for the day-ahead unit combination model and the condition assessment model, respectively; f UC Ω represents the operating cost function of the system in its ground state. C and Ω S Let k and s represent the set of system component failure scenarios and the set of source load prediction error scenarios, respectively, with k and s being their indices; (k, s) represents a composite scenario composed of failure scenarios and source load prediction error scenarios, whose state space is represented as Ω. C ×Ω S ; This represents the load shedding cost function of the system in the composite scenario (k,s); g k and l k These represent the state matrices of the system units and lines under fault scenario k, respectively, where 1 indicates that the component is in the operating state and 0 indicates that the component is in the fault shutdown state. and These represent the load forecasting error and the new energy forecasting error, respectively, under forecasting error scenario s.

[0015] Furthermore, the objective function of the current unit combination model is as follows:

[0016] (2)

[0017] In the formula, f OC fSSC and f SRC These represent the generator's operating cost, start-up and shutdown cost, and spinning reserve cost, respectively; a Q,g a L,g and a C,g Represent the quadratic, linear, and constant terms of the power generation cost coefficient of generator g, respectively; auxiliary variable C SU,g,t and C SD,g,t These represent the costs incurred by generator g at time t due to starting and stopping, respectively; a R,g The generator represents the standby cost factor of g; Ω B Ω G and Ω L Let i be the set of buses, generators, and lines, and g be the index of each. Let t and k be the indices of the scheduling period and the system component failure scenario, respectively. z = {u g,t ,P G,g,t ,R g,t C SU,g,t C SD,g,t , θ i,t | }, u g,t This represents the start / stop state of generator g at time t, where 1 indicates the generator is running and 0 indicates it is stopped; P G,g,t and R g,t θ represents the output and reserve capacity of generator g at time t, respectively; i,t This represents the phase angle of bus i at time t.

[0018] Furthermore, the constraints of the current-day unit combination model are as follows:

[0019] (3)

[0020] (4)

[0021] (5)

[0022] (6)

[0023] (7)

[0024] (8)

[0025] (9)

[0026] (10)

[0027] (11)

[0028] (12)

[0029] (13)

[0030] In the formula, constraints (3) to (5) represent DC power flow constraints; P D,i,t P represents the predicted active power load of bus i at time t. ij,t and θ ij,t Let X represent the active power and phase angle difference of line ij at time t, respectively. ij This represents the reactance of line ij. This indicates the maximum active power allowed to pass through line ij, with the subscript 'ref' indicating the slack node; constraint (6) represents the upper and lower limits of generator output, where, and These represent the minimum and maximum output of the generator, respectively; constraint (7) represents the generator's ramp and landslide constraints, r DN,g and r UP,g Let represent the landslide rate and ramp rate of generator g, respectively; constraint (8) is the standby capacity constraint of the generator; constraints (10) and (11) are the minimum start-stop time constraints of the generator, DT g and UT g Let represent the minimum downtime and minimum running time of generator g, respectively, and τ be an auxiliary variable; constraints (12) and (13) represent the start-up cost and downtime cost of generator g at time t, respectively.

[0031] Furthermore, for , The objective function of the state assessment model considering the complex scenario is as follows:

[0032] (14)

[0033] In the formula, a LS,i Represents the load shedding cost coefficient for bus i; y={ | },in , and Let g represent the output of generator g, the load on bus i, and the phase angle, respectively, in the composite scenario (k, s). This represents the load sheared by bus i at time t in the composite scenario (k, s).

[0034] Furthermore, the constraints of the state assessment model under complex scenarios are as follows:

[0035] (15)

[0036] (16)

[0037] (17)

[0038] (18)

[0039] (19)

[0040] (20)

[0041] In the formula, constraints (15)~(17) represent DC power flow constraints, where, and In the composite scenario (k, s), the active power flow and phase angle difference of line ij; constraint (18) is the active power load constraint of bus i; constraint (19) is the upper and lower limits constraint of generator active power output; constraint (20) is the ramp constraint of generator.

[0042] Furthermore, in step 2), the step of approximating the state evaluation model using a basis-adaptive sparse chaotic polynomial expansion includes:

[0043] Step 2.1) Determine the random inputs to the condition assessment model, including day-ahead unit combination decisions and source-load forecasting errors; day-ahead unit combination decisions include unit start-up and shutdown status, unit output, and unit standby;

[0044] Step 2.2) Set the probability density distribution of the day-ahead unit combination decision, and use the Gaussian-copula model to transform the historical samples of source-load prediction errors that follow arbitrary distribution into experimental design samples for building the surrogate model, and determine the probability density distribution of the experimental design samples.

[0045] Step 2.3) Select orthogonal basis functions to approximate the random input based on the probability density distribution type of the random input; use the orthogonal basis functions to approximate the random input, thereby relaxing the state to evaluate the model;

[0046] Wherein, if the probability density distribution is uniform, the orthogonal basis function is the Legendre polynomial; if the probability density distribution is normal, the orthogonal basis function is the Hermite polynomial; if the probability density distribution is Gamma, the orthogonal basis function is the Laguerre polynomial; and if the probability density distribution is Beta, the orthogonal basis function is the Jacobi polynomial.

[0047] Step 2.4) Relax the upper bound of the node load to obtain:

[0048] (twenty one)

[0049] In the formula, It is much larger than The number of nodes is used to relax the node load;

[0050] Step 2.5) Construct a state assessment proxy model, namely:

[0051] (twenty two)

[0052] (twenty three)

[0053] In the formula, D k This represents the set of proxy models under the k-th fault scenario, including proxy models for each time period. ; k and s represent the fault scenario vector and the source load prediction error scenario vector, respectively.

[0054] Furthermore, in step 2.2), the unit's start-up / shutdown state u and the unit's output P are... G Construct composite variable u·P G And set the composite variable u·P G Obey [0, The uniform distribution of R; for the unit's standby R G Set the unit's standby R G Obey [0, The uniform distribution of ].

[0055] Furthermore, in step 2.2), the step of transforming the historical samples of source load prediction errors that follow an arbitrary distribution into experimental design samples for constructing the surrogate model includes:

[0056] Step 2.2.1) Calculate the Spearman rank correlation coefficient matrix C based on historical samples of the random variable. S,O ;

[0057] The Spearman correlation coefficient matrix in standard Gaussian space satisfies the following equation:

[0058] (twenty four)

[0059] Step 2.2.2) Based on the transformation relationship between Pearson correlation coefficient and Spearman correlation coefficient in Gaussian variables, construct the corresponding Pearson correlation coefficient matrix C in standard Gaussian space. P,Z ,Right now:

[0060] (25)

[0061] Step 2.2.3) Analyze the Pearson correlation coefficient matrix C P,Z Performing Cholesky decomposition, we obtain:

[0062] (26)

[0063] In the formula, L is a lower triangular matrix;

[0064] Step 2.2.4) Independent samples generated by Latin hypercube sampling =Construct relevant standard Gaussian samples ,Right now:

[0065] (27)

[0066] Step 2.2.5) Transform the relevant Gaussian samples using inverse probability transformation. Mapped to the original domain, generating original domain samples ,Right now:

[0067] (28)

[0068] In the formula, Φ represents the standard Gaussian cumulative distribution function. Let be the inverse cumulative distribution function of the m-th random variable in the original domain.

[0069] Furthermore, in step 4), the steps for solving the robust optimization model of the replaced unit combination include:

[0070] Step 4.1) Add a Constrained Cost Variable (CCV) constraint for each failure scenario to the day-ahead unit combination model:

[0071] (29)

[0072] (30)

[0073] Step 4.2) Reconstruct the robust optimization model of unit combination into a min-max problem, i.e.:

[0074] (31)

[0075] In the formula, d represents the set of auxiliary variables;

[0076] Step 4.3) Introduce an independent set of source-load scenario decision variables s for each fault scenario k. k The min-max two-level optimization problem is reconstructed into a single-level minimization problem, namely:

[0077] (32)

[0078] In the formula, s kLet S represent the set of decision variables for source load prediction error under fault scenario k, and let S represent the set of box-shaped uncertainties for source load prediction error.

[0079] Step 4.4) Solve the single-layer minimization problem using the gurobi solver to obtain the unit combination scheme under the most severe fault scenario and the continuous source load uncertainty scenario.

[0080] The technical effects of this invention are undeniable, and its beneficial effects are as follows:

[0081] 1) To address the issue of unsmooth response in the state assessment model, this patent uses relaxation techniques to reconstruct the state assessment model to improve the smoothness of the model response. Based on this, a surrogate model for the state assessment model is established using BASPCE, which significantly improves the accuracy of the surrogate model.

[0082] 2) A robust optimization method for unit combination based on BASPCE event constraints is proposed. By embedding the constructed BASPCE surrogate model into the robust optimization framework using relaxation techniques, the overall solution speed is significantly improved. Attached Figure Description

[0083] Figure 1 A schematic diagram of a proxy model for constructing a state evaluation model using relaxation techniques;

[0084] Figure 2 This is a flowchart. Detailed Implementation

[0085] The present invention will be further described below with reference to embodiments, but it should not be construed that the scope of the present invention is limited to the following embodiments. Various substitutions and modifications made based on ordinary technical knowledge and common practices in the art without departing from the above-described technical concept of the present invention should be included within the scope of protection of the present invention.

[0086] Example 1:

[0087] See Figures 1 to 2 A robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions includes the following steps:

[0088] Step 1) Establish an event-constrained robust optimization model for unit combination, including a day-ahead unit combination model and a state assessment model considering complex scenarios;

[0089] Step 2) The state evaluation model is approximated by a basis-adaptive sparse chaotic polynomial expansion to construct a state evaluation proxy model;

[0090] Step 3) Replace the state assessment model in the unit combination robust optimization model with a state assessment proxy model;

[0091] Step 4) Solve the robust optimization model of the replaced unit combination to obtain the unit combination scheme.

[0092] Example 2:

[0093] A robust optimization solution method for event-constrained unit combination based on basis function adaptive chaotic polynomial expansion, with the same technical content as in Embodiment 1, further wherein, in step 1), the optimization objective of the unit combination robust optimization model is to minimize the cost under the worst-case scenario.

[0094] The objective function of the unit combination robust optimization model is shown below:

[0095] (1)

[0096] In the formula, z and y represent the sets of decision variables for the day-ahead unit combination model and the condition assessment model, respectively; f UC Ω represents the operating cost function of the system in its ground state. C and Ω S Let k and s represent the set of system component failure scenarios and the set of source load prediction error scenarios, respectively, with k and s being their indices; (k, s) represents a composite scenario composed of failure scenarios and source load prediction error scenarios, whose state space is represented as Ω. C ×Ω S ; This represents the load shedding cost function of the system in the composite scenario (k,s); g k and l k These represent the state matrices of the system units and lines under fault scenario k, respectively, where 1 indicates that the component is in the operating state and 0 indicates that the component is in the fault shutdown state. and These represent the load forecasting error and the new energy forecasting error, respectively, under forecasting error scenario s.

[0097] Example 3:

[0098] A robust optimization solution method for event-constrained unit combination based on adaptive chaotic polynomial expansion of basis functions is provided. The technical content is the same as any one of Embodiments 1-2. Furthermore, the objective function of the current-day unit combination model is as follows:

[0099] (2)

[0100] In the formula, f OC f SSC and f SRC These represent the generator's operating cost, start-up and shutdown cost, and spinning reserve cost, respectively; a Q,g a L,g and a C,gRepresent the quadratic, linear, and constant terms of the power generation cost coefficient of generator g, respectively; auxiliary variable C SU,g,t and C SD,g,t These represent the costs incurred by generator g at time t due to starting and stopping, respectively; a R,g The generator represents the standby cost factor of g; Ω B Ω G and Ω L Let i be the set of buses, generators, and lines, and g be the index of each. Let t and k be the indices of the scheduling period and the system component failure scenario, respectively. z = {u g,t ,P G,g,t ,R g,t C SU,g,t C SD,g,t , θ i,t | }, u g,t This represents the start / stop state of generator g at time t, where 1 indicates the generator is running and 0 indicates it is stopped; P G,g,t and R g,t θ represents the output and reserve capacity of generator g at time t, respectively; i,t This represents the phase angle of bus i at time t.

[0101] Example 4:

[0102] A robust optimization solution method for event-constrained unit combination based on adaptive chaotic polynomial expansion of basis functions is provided. The technical content is the same as any one of embodiments 1-3. Furthermore, the constraints of the day-ahead unit combination model are as follows:

[0103] (3)

[0104] (4)

[0105] (5)

[0106] (6)

[0107] (7)

[0108] (8)

[0109] (9)

[0110] (10)

[0111] (11)

[0112] (12)

[0113] (13)

[0114] In the formula, constraints (3) to (5) represent DC power flow constraints; P D,i,t P represents the predicted active power load of bus i at time t. ij,t and θ ij,t Let X represent the active power and phase angle difference of line ij at time t, respectively. ij This represents the reactance of line ij. This indicates the maximum active power allowed to pass through line ij, with the subscript 'ref' indicating the slack node; constraint (6) represents the upper and lower limits of generator output, where, and These represent the minimum and maximum output of the generator, respectively; constraint (7) represents the generator's ramp and landslide constraints, r DN,g and r UP,g Let represent the landslide rate and ramp rate of generator g, respectively; constraint (8) is the standby capacity constraint of the generator; constraints (10) and (11) are the minimum start-stop time constraints of the generator, DT g and UT g Let represent the minimum downtime and minimum running time of generator g, respectively, and τ be an auxiliary variable; constraints (12) and (13) represent the start-up cost and downtime cost of generator g at time t, respectively.

[0115] Example 5:

[0116] A robust optimization solution method for event-constrained unit combinations based on basis function adaptive chaotic polynomial expansion, with technical content identical to any one of embodiments 1-4, further, for , The objective function of the state assessment model considering the complex scenario is as follows:

[0117] (14)

[0118] In the formula, a LS,i Represents the load shedding cost coefficient for bus i; y={ | },in , and Let g represent the output of generator g, the load on bus i, and the phase angle, respectively, in the composite scenario (k, s). This represents the load sheared by bus i at time t in the composite scenario (k, s).

[0119] Example 6:

[0120] A robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions is provided. The technical content is the same as any one of embodiments 1-5. Furthermore, the constraints of the state assessment model under the complex scenario are as follows:

[0121] (15)

[0122] (16)

[0123] (17)

[0124] (18)

[0125] (19)

[0126] (20)

[0127] In the formula, constraints (15)~(17) represent DC power flow constraints, where, and In the composite scenario (k, s), the active power flow and phase angle difference of line ij; constraint (18) is the active power load constraint of bus i; constraint (19) is the upper and lower limits constraint of generator active power output; constraint (20) is the ramp constraint of generator.

[0128] Example 7:

[0129] A robust optimization solution method for event-constrained unit combinations based on basis function adaptive chaotic polynomial expansion, with the same technical content as any one of embodiments 1-6, further comprising the following steps in step 2), which involves approximating the state evaluation model using basis adaptive sparse chaotic polynomial expansion:

[0130] Step 2.1) Determine the random inputs to the condition assessment model, including day-ahead unit combination decisions and source-load forecasting errors; day-ahead unit combination decisions include unit start-up and shutdown status, unit output, and unit standby;

[0131] Step 2.2) Set the probability density distribution of the day-ahead unit combination decision, and use the Gaussian-copula model to transform the historical samples of source-load prediction errors that follow an arbitrary distribution into experimental design samples for building the surrogate model, and determine the probability density distribution of the experimental design samples; Step 2.2) is used to generate the experimental design samples required for building the surrogate model.

[0132] Step 2.3) Select an orthogonal basis function to approximate the random input based on the probability density distribution type of the random input; use the orthogonal basis function to approximate the random input, thereby relaxing the state to evaluate the model; Step 2.3) is used to relax the state to evaluate the model, so as to improve the approximation accuracy of the surrogate model.

[0133] Wherein, if the probability density distribution is uniform, the orthogonal basis function is the Legendre polynomial; if the probability density distribution is normal, the orthogonal basis function is the Hermite polynomial; if the probability density distribution is Gamma, the orthogonal basis function is the Laguerre polynomial; and if the probability density distribution is Beta, the orthogonal basis function is the Jacobi polynomial.

[0134] Step 2.4) Relax the upper bound of the node load to obtain:

[0135] (twenty one)

[0136] In the formula, It is much larger than The number of nodes is used to relax the node load;

[0137] Step 2.5) Construct a state assessment proxy model, namely:

[0138] (twenty two)

[0139] (twenty three)

[0140] In the formula, D k This represents the set of proxy models under the k-th fault scenario, including proxy models for each time period. k and s represent the fault scenario vector and the source load prediction error scenario vector, respectively. k and s are vectors used to simplify the representation of the model input, where k represents g. k and l k The set of vectors formed, s represents and The set of vectors formed.

[0141] Example 8:

[0142] A robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions is provided. The technical content is the same as any one of embodiments 1-7. Further, in step 2.2), for the unit start-up / shutdown state u and the unit output P... G Construct composite variable u·P G And set the composite variable u·P G Obey [0, The uniform distribution of R; for the unit's standby RG Set the unit's standby R G Obey [0, The uniform distribution of ].

[0143] Example 9:

[0144] A robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions, with the same technical content as any one of embodiments 1-8, further comprising the following steps in step 2.2), which involves converting historical samples of source load prediction errors following arbitrary distributions into experimental design samples for constructing surrogate models:

[0145] Step 2.2.1) Calculate the Spearman rank correlation coefficient matrix C based on historical samples of the random variable. S,O ;

[0146] The Spearman correlation coefficient matrix in standard Gaussian space satisfies the following equation:

[0147] (twenty four)

[0148] Step 2.2.2) Based on the transformation relationship between Pearson correlation coefficient and Spearman correlation coefficient in Gaussian variables, construct the corresponding Pearson correlation coefficient matrix C in standard Gaussian space. P,Z ,Right now:

[0149] (25)

[0150] Step 2.2.3) Analyze the Pearson correlation coefficient matrix C P,Z Performing Cholesky decomposition, we obtain:

[0151] (26)

[0152] In the formula, L is a lower triangular matrix;

[0153] Step 2.2.4) Independent samples generated by Latin hypercube sampling =Construct relevant standard Gaussian samples ,Right now:

[0154] (27)

[0155] Step 2.2.5) Transform the relevant Gaussian samples using inverse probability transformation. Mapped to the original domain, generating original domain samples ,Right now:

[0156] (28)

[0157] In the formula, Φ represents the standard Gaussian cumulative distribution function. Let be the inverse cumulative distribution function of the m-th random variable in the original domain.

[0158] Example 10:

[0159] A robust optimization solution method for event-constrained unit combination based on basis function adaptive chaotic polynomial expansion, with the same technical content as any one of embodiments 1-9, further comprising, in step 4), the steps for solving the replaced robust optimization model of unit combination include:

[0160] Step 4.1) Add a Constrained Cost Variable (CCV) constraint for each failure scenario to the day-ahead unit combination model:

[0161] (29)

[0162] (30)

[0163] Step 4.2) Reconstruct the robust optimization model of unit combination into a min-max problem, i.e.:

[0164] (31)

[0165] In the formula, d represents the set of auxiliary variables;

[0166] Step 4.3) Introduce an independent set of source-load scenario decision variables s for each fault scenario k. k The min-max two-level optimization problem is reconstructed into a single-level minimization problem, namely:

[0167] (32)

[0168] In the formula, s k Let S represent the set of decision variables for source load prediction error under fault scenario k, and let S represent the set of box-shaped uncertainties for source load prediction error.

[0169] Step 4.4) Solve the single-layer minimization problem using the gurobi solver to obtain the unit combination scheme under the most severe fault scenario and the continuous source load uncertainty scenario.

[0170] Example 11:

[0171] A robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions is proposed, with the following steps:

[0172] Step S1: Establish an event-constrained unit commitment (R-CCUC) model.

[0173] Step S2: Establish a state assessment agent model based on BASPCE.

[0174] Step S3: Establish an R-CCUC solution model based on the BASPCE surrogate model.

[0175] The characteristics of step S1 are as follows:

[0176] The optimization objective of R-CCUC is to minimize the cost under the worst-case scenario. This cost includes operating costs and worst-case load shedding costs, where operating costs represent generator fuel costs, start-up and shutdown costs, and standby costs for the day-ahead unit combination. The worst-case scenario is determined by a composite scenario generated jointly by source-load prediction errors and system failure scenarios. R-CCUC includes a day-ahead unit combination model and a state-of-the-art performance evaluation model under the composite scenario.

[0177] (1)

[0178] In the formula, z and y represent the sets of decision variables for the day-ahead unit combination model and the condition assessment model, respectively; f UC Ω represents the operating cost function of the system in its ground state. C and Ω S Let k and s represent the set of system component failure scenarios and the set of source load prediction error scenarios, respectively, with k and s being their indices; (k, s) represents a composite scenario composed of failure scenarios and source load prediction error scenarios, whose state space is represented as Ω. C ×Ω S ; This represents the load shedding cost function of the system in the composite scenario (k,s); g k and l k These represent the state matrices of the system units and lines under fault scenario k, respectively, where 1 indicates that the component is in the operating state and 0 indicates that the component is in the fault shutdown state. and These represent the load forecasting error and the new energy forecasting error, respectively, under forecasting error scenario s.

[0179] 1) Establish a day-ahead unit combination model

[0180] The objective function is:

[0181] (2)

[0182] The constraints are:

[0183] (3)

[0184] (4)

[0185] (5)

[0186] (6)

[0187] (7)

[0188] (8)

[0189] (9)

[0190] (10)

[0191] (11)

[0192] (12)

[0193] (13)

[0194] In the formula, f OC f SSC and f SRC These represent the generator's operating cost, start-up and shutdown cost, and spinning reserve cost, respectively; a Q,g a L,g and a C,g Represent the quadratic, linear, and constant terms of the power generation cost coefficient of generator g, respectively; auxiliary variable C SU,g,t and C SD,g,t These represent the costs incurred by generator g at time t due to starting and stopping, respectively; a R,g The generator represents the standby cost factor of g; Ω B Ω G and Ω L Let i be the set of buses, generators, and lines, and g be the index of each. Let t and k be the indices of the scheduling period and the system component failure scenario, respectively. z = {u g,t ,P G,g,t ,R g,t C SU,g,t C SD,g,t , θ i,t | }, where u g,t This represents the start / stop state of generator g at time t, where 1 indicates the generator is running and 0 indicates it is stopped; P G,g,t and R g,tθ represents the output and reserve capacity of generator g at time t, respectively; i,t Let P represent the phase angle of bus i at time t; constraints (3) to (5) represent DC power flow constraints, where P D,i,t P represents the predicted active power load of bus i at time t. ij,t and θ ij,t Let X represent the active power and phase angle difference of line ij at time t, respectively. ij This represents the reactance of line ij. This indicates the maximum active power allowed to pass through line ij, with the subscript 'ref' indicating the slack node; constraint (6) represents the upper and lower limits of generator output, where, and Let r represent the minimum and maximum output of the generator, respectively; constraint (7) represents the ramp and landslide constraints of the generator, where r DN,g and r UP,g Let represent the slope rate and ramp rate of generator g, respectively; constraint (8) is the standby capacity constraint of the generator; constraints (10) and (11) are the minimum start-stop time constraints of the generator, where DT g and UT g Let represent the minimum downtime and minimum running time of generator g, respectively, and τ be an auxiliary variable; constraints (12) and (13) represent the start-up cost and downtime cost of generator g at time t, respectively.

[0195] 2) Considering the state assessment model under complex scenarios

[0196] for , The objective function of the state assessment model is:

[0197] (14)

[0198] The constraints are:

[0199] (15)

[0200] (16)

[0201] (17)

[0202] (18)

[0203] (19)

[0204] (20)

[0205] In the formula, a LS,i Represents the load shedding cost coefficient for bus i; y={ | },in , and Let g represent the output of generator g, the load on bus i, and the phase angle, respectively, in the composite scenario (k, s). This represents the load shedding amount of bus i at time t under the composite scenario (k, s). Constraints (15) to (17) represent DC power flow constraints, where, and In the composite scenario (k, s), the active power flow and phase angle difference of line ij; constraint (18) is the active power load constraint of bus i; constraint (19) is the upper and lower limits constraint of generator active power output; constraint (20) is the ramp constraint of generator.

[0206] 3) Traditional solution method for R-CCUS

[0207] R-CCUC is a three-stage min-max-min structure and cannot be solved directly. The uncertainty set Ω C ×Ω S It is a hybrid of discrete and continuous problems, with a massive or even infinite number of scenarios, making it impossible to solve by enumeration. Column & Constraint Generation (C&CG) is a mainstream approach to solving this problem. This algorithm transforms the three-stage problem into a series of solvable main problems and subproblems. The main problem seeks the lowest-cost robust plan under a finite number of worst-case scenarios, while the subproblems find new, more costly worst-case scenarios for the current plan and feed them back to the main problem. Through iteration between the subproblems of the main problem, C&CG can gradually approach the optimal solution of the original problem, ensuring robustness while avoiding the curse of dimensionality caused by directly handling a large number of scenarios.

[0208] The R-CCUC subproblem solved based on C&CG is state assessment under uncertain scenarios. Since the composite scenario consists of discrete fault scenarios and continuous source load uncertainty scenarios, in order to improve the solution efficiency of the model, the composite scenario is processed in layers, and the scenario set Ω of source load uncertainty is assumed. S Represented by box-type indeterminate sets, i.e.:

[0209] (twenty one)

[0210] The outer-layer max problem identifies the most severe failure scenario through enumeration. For the inner-layer max-min problem, duality theory is used to transform the two-layer max-min problem into an equivalent, solvable single-layer max problem. The output of this equivalent model represents the worst-case source load uncertainty scenario. and the corresponding minimum load shedding cost .

[0211] The main problem of R-CCUC based on C&CG is to introduce the worst-case combined scenario (k) into the objective function of the day-ahead unit combination. * , s * The auxiliary variable η for load shedding cost is updated through iterative solutions to the main problem and subproblems. Specifically, a composite scenario (k) is added to the main problem. * , s * The power balance constraints (24), line power flow constraints (25), node active load constraints (26), unit ramping constraints (27), unit reserve constraints (28), and C&CG optimality cut plane constraints of auxiliary variables in the state assessment model (29) are as follows:

[0212] The objective function is:

[0213] (twenty two)

[0214] The constraints are:

[0215] (twenty three)

[0216] (twenty four)

[0217] (25)

[0218] (26)

[0219] (27)

[0220] (28)

[0221] (29)

[0222] The characteristics of step S2 are as follows:

[0223] The traditional iterative solution process for R-CCUC is time-consuming. This study proposes an efficient solution method for R-CCUC based on a surrogate model. The method uses a basis-adaptive sparse polynomial Chaos Expansion (BASPCE) to approximate the state assessment model in R-CCUC, and combines the constructed surrogate model with the day-ahead unit combination model to form a unified solution for R-CCUC.

[0224] 1) Basic principles of BASPCE

[0225] According to the theory of Generalized Polynomial Chaos (GPC), a surrogate model constructed based on orthogonal basis functions can be used to approximate an original model (or original function) with M-dimensional independent random inputs:

[0226] (30)

[0227] In the formula, ξ=[ξ1,…,ξ M ] is a vector composed of M random variables; K represents the number of terms in the polynomial; F and D represent the original model and the surrogate model, respectively. The key to constructing this surrogate model is to determine the orthogonal basis functions Ψ and their expansion coefficients a. The orthogonal basis functions are determined based on the probability density functions of the random variables. Table 1 lists the correspondence between the distribution types of random variables and the orthogonal basis functions. Multivariable orthogonal polynomials are constructed through tensor products of univariate variables:

[0228] Table 1. Correspondence between variable distribution types and optimal univariate orthogonal polynomial basis functions

[0229]

[0230] (31)

[0231] In the formula, The tensor product operator is represented; Φ represents a single-variable orthogonal polynomial basis function; the subscript i m Let represent the i-th order of the m-th random variable. In GPC, the truncation standard parameter p is set by constraints, where... The expansion terms are truncated to ensure that the total number of non-zero terms in the surrogate model's expansion after truncation satisfies K = (M + p)! / (M!p!). Clearly, K increases exponentially with the dimension M of the random variable and the order p of the maximum polynomial, leading to the curse of dimensionality in high-dimensional or high-order cases. Research shows that the interactions of basis functions in low-order univariate polynomials are often more statistically significant than those in higher-order terms, giving rise to sparse PCE. Sparse PCE employs the hyperbolic truncation scheme shown below to prioritize low-order interactions, thereby significantly reducing the number of polynomials in the expansion terms.

[0232] (32)

[0233] In the formula, i represents the multidimensional index; q is the truncation norm used to adjust sparsity. The smaller q is, the greater the restriction on the interaction of higher-order terms, and the higher the sparsity of the polynomial.

[0234] The expansion coefficients in the surrogate model can be solved using least squares regression. To further suppress higher-order interaction terms, a penalty term is added to the least squares objective function.

[0235] (33)

[0236] In the formula, N represents the sample size; X = {x1, ..., x} N} represents the set of training samples, also known as the experimental design; Y={F(x1),…,F(x2)} N )} represents the model output of the experimental design, also known as the response; λ is a penalty factor that takes a positive value. The above optimization problem can be solved using the least angle regression algorithm, which reduces the number of expansion terms by iteratively constructing a series of sparse regression subproblems.

[0237] Before constructing the surrogate model, a value for p must be given. The approximation model is then built through three steps: determining orthogonal basis functions, selecting a truncation strategy, and calculating the expansion coefficients. However, it is usually impossible to know in advance the value of p corresponding to a surrogate model with acceptable accuracy. Therefore, based on sparse PCE and combined with adaptive algorithms, the BASPCE method selects the most suitable p value through leave-one-out cross-validation error as shown below:

[0238] (34)

[0239] In the formula, e LOO D represents the cross-validation error; -l This represents the surrogate model constructed after removing the l-th sample from the experimental design sample set. By presetting the maximum value of p and combining it with the above process, the most suitable expansion order can be adaptively selected.

[0240] 2) Source load prediction error probability modeling

[0241] Based on the fundamental principles of BASPCE, the probability density distribution of random variables is crucial for constructing the surrogate model, and the use of regression to solve for the multinomial coefficients requires that the variables in the experimental design sample be mutually independent. The random inputs to the state assessment model include day-ahead unit combination decisions and source-load forecasting errors. For unit combination decisions, since their probability distribution is practically difficult to obtain, the following measures are adopted: for unit start-up / shutdown states u and unit output P... G , construct u·P G Composite variables, and assume they follow the order [0, ...]. The uniform distribution of R; for the unit's standby R G Assume it follows [0, The uniform distribution of source load prediction errors is used. For source load prediction errors that follow any distribution and are correlated, the Gaussian-copula model is used to generate experimental design samples for constructing the surrogate model. The specific steps are as follows:

[0242] Step 1: Determine the correlation matrix in standard Gaussian space. First, calculate the Spearman rank correlation coefficient matrix C based on historical samples of the random variable. S,O After the probability integral transformation, the rank correlation coefficient remains unchanged. Therefore, the Spearman correlation coefficient matrix in standard Gaussian space satisfies:

[0243] (35)

[0244] Subsequently, based on the transformation relationship between Pearson correlation coefficient and Spearman correlation coefficient in Gaussian variables, the corresponding Pearson correlation coefficient matrix C in standard Gaussian space is obtained. P,Z :

[0245] (36)

[0246] Step 2: Generate relevant standard Gaussian samples. For the Pearson correlation coefficient matrix C... P,Z Perform Cholesky decomposition:

[0247] (37)

[0248] In the formula, L is a lower triangular matrix. Then, independent samples are used... (e.g., using Latin hypercube sampling) Construct relevant standard Gaussian samples :

[0249] (38)

[0250] Step 3: Transform the relevant samples to the original domain. This is done by using inverse probability transformation to transform the relevant Gaussian samples... Mapping to the original domain:

[0251] (39)

[0252] In the formula, Φ represents the standard Gaussian cumulative distribution function. Let be the inverse cumulative distribution function of the m-th random variable in the original domain. This function can be estimated based on historical samples using the kernel density estimation method. Thus, it is possible to generate relevant original domain samples from historical samples of source load prediction errors that follow arbitrary distributions. and its corresponding independent Gaussian experimental design samples This provides a foundation for building a proxy model for the state assessment model of BASPCE.

[0253] 3) Constructing a surrogate model for the state assessment model using relaxation techniques.

[0254] The approximate accuracy of BASPCE is highly correlated with the smoothness of the original model's response. However, the state assessment model exhibits significant non-smoothness under uncertain inputs. This is primarily due to two model characteristics: first, the model's output, i.e., the cost of load shedding, is non-negative; second, the model outputs zero in most scenarios, such as when unit outage capacity is small or source load prediction errors are small, because the system has sufficient reserves to achieve power balance, thus eliminating the need for load shedding. These two factors cause a significant clustering of the probability distribution of the state assessment model's response at zero, and the non-smooth transition between zero and positive values ​​results in an unsmooth model response, thereby limiting the direct application of BASPCE. The following discussion focuses on the construction of a surrogate model for the state assessment model based on relaxation techniques.

[0255] According to the state assessment model, the upper limit of node load is constrained by the maximum node load, resulting in a non-negative load shedding value for the system. The upper bound of node load is relaxed as shown in the following equation:

[0256] (40)

[0257] In the formula, It is much larger than The numbers are used to relax the node load. (40) together with (14)~(17), (19) and (20) constitute the reconfigured state assessment model. Figure 1 As shown, due to the clustering at the zero value, the frequency distribution histogram of the load shedding output from the original state assessment model exhibits a significant abrupt increase at the zero value, resulting in a non-smooth response to uncertain inputs. In the reconstructed state assessment model, the clustering at the zero value is transformed into a series of negative, smooth distributions, thus smoothing the model's response. Based on this, BASPCE is used to approximate the relaxed state assessment model for each fault scenario. Since the predicted load and renewable energy output differ across time periods, a corresponding surrogate model needs to be established for each time period. Summing the surrogate models for all time periods yields an approximation of the relaxed state assessment model.

[0258] (41)

[0259] (42)

[0260] In the formula, D k This represents the set of proxy models for k different time periods under fault scenarios, containing proxy models for each time period. ; k and s represent the fault scenario vector and the source load prediction error scenario vector, respectively. It is important to note that the constructed surrogate model cannot completely replace the original state assessment model, because the surrogate model may have negative outputs, corresponding to the relaxed portion of the original model that takes a value of 0. The relaxed portion of the surrogate model is restored based on the relaxation theory of the Constrained Cost Variable (CCV). An auxiliary variable d is defined to represent the output of the surrogate model; the lower bound of d is constrained by both 0 and the surrogate model output. Furthermore, d is incorporated into the objective function of R-CCUC, thereby restoring the relaxed portion. Figure 1 The reduction process was demonstrated. In short, this study constructed a BASPCE surrogate model for the state evaluation model through three steps: relaxation, approximation, and reduction, laying the foundation for subsequent implementation of R-CCUC solution based on the surrogate model.

[0261] (43)

[0262] (44)

[0263] According to the event-constrained unit combination robust optimization solution method based on basis function adaptive chaotic polynomial expansion, step S3 has the following characteristics:

[0264] R-CCUC is a min-max-min structure. After constructing a BASPCE-based proxy model for the innermost min problem, i.e. the state evaluation model, the original R-CCUC problem (constructed through (22)~(29)) is reconstructed into the following min-max problem:

[0265] (45)

[0266] In the formula, d represents the set of auxiliary variables. The inner max of the above model refers to the most severe composite scenario (k...). * ,s * The following steps describe how to reconstruct the max-min problem into a single-layer min problem based on a proxy model.

[0267] Step 1: Identify the worst-case failure scenario. Based on the surrogate model, add CCV constraints for each failure scenario to the day-ahead unit combination model:

[0268] (46)

[0269] (47)

[0270] By tightening the lower bound of the auxiliary variable d representing the load shedding under the most severe fault scenario in the above formula, the most severe fault scenario k in the max problem can be realized. * The search yields the following R-CCUC problem, which can be reconstructed as follows:

[0271] (48)

[0272] Step 2: Identify the worst-case source-load prediction error scenario. The uncertainty of system load and renewable energy output prediction errors is represented by a continuous box set S. For each fault scenario k, an independent set of source-load scenario decision variables s is introduced. k The min-max bilayer optimization problem shown in (48) is reconstructed into a single-layer minimization problem as shown below:

[0273] (49)

[0274] In the formula, s k This represents the set of decision variables for source load prediction error under fault scenario k. During the optimization process, the optimizer selects a variable s for each fault scenario k. k To maximize And in all the failure scenarios, the one that makes The scenario with the largest value (k) * , s k* This is the most severe complex scenario.

Claims

1. A robust optimization solution method for event-constrained unit combinatorial systems based on adaptive chaotic polynomial expansion of basis functions, characterized in that, Includes the following steps: Step 1) Establish an event-constrained robust optimization model for unit combination, including a day-ahead unit combination model and a state assessment model considering complex scenarios; Step 2) The state evaluation model is approximated by a basis-adaptive sparse chaotic polynomial expansion to construct a state evaluation proxy model; Step 3) Replace the state assessment model in the unit combination robust optimization model with a state assessment proxy model; Step 4) Solve the robust optimization model of the replaced unit combination to obtain the unit combination scheme.

2. The robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions as described in claim 1, characterized in that, In step 1), the optimization objective of the unit combination robust optimization model is to minimize the cost under the worst-case scenario; The objective function of the unit combination robust optimization model is shown below: ；（1） In the formula, z and y represent the sets of decision variables for the day-ahead unit combination model and the condition assessment model, respectively; f UC Ω represents the operating cost function of the system in its ground state. C and Ω S Let k and s represent the set of system component failure scenarios and the set of source load prediction error scenarios, respectively, with k and s being their indices; (k,s) represents a composite scenario composed of both failure scenarios and source load prediction error scenarios, whose state space is represented as Ω. C ×Ω S ; This represents the load shedding cost function of the system in the composite scenario (k,s); g k and l k These represent the state matrices of the system units and lines under fault scenario k, respectively, where 1 indicates that the component is in the operating state and 0 indicates that the component is in the fault shutdown state. and These represent the load forecasting error and the new energy forecasting error, respectively, under forecasting error scenario s.

3. The robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions as described in claim 1, characterized in that, The objective function of the current unit combination model is shown below: ；（2） In the formula, f OC f SSC and f SRC These represent the generator's operating cost, start-up and shutdown cost, and spinning reserve cost, respectively; a Q,g a L,g and a C,g Represent the quadratic, linear, and constant terms of the power generation cost coefficient of generator g, respectively; auxiliary variable C SU,g,t and C SD,g,t These represent the costs incurred by generator g at time t due to starting and stopping, respectively; a R,g The generator represents the standby cost factor of g; Ω B Ω G and Ω L Let i be the set of buses, generators, and lines, and g be the index of each. Let t and k be the indices of the scheduling period and the system component failure scenario, respectively. z = {u g,t ,P G,g,t ,R g,t C SU,g,t C SD,g,t ,θ i,t | }, u g,t This represents the start / stop state of generator g at time t, where 1 indicates the generator is running and 0 indicates it is stopped; P G,g,t and R g,t θ represents the output and reserve capacity of generator g at time t, respectively; i,t This represents the phase angle of bus i at time t.

4. The robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions as described in claim 1, characterized in that, The current constraints of the unit combination model are as follows: ；（3） ；（4） ；（5） ；（6） ；（7） ；（8） ；（9） ；（10） ；（11） ；（12） ；（13） In the formula, constraints (3) to (5) represent DC power flow constraints; P D,i,t P represents the predicted active power load of bus i at time t. ij,t and θ ij,t Let X represent the active power and phase angle difference of line ij at time t, respectively. ij This represents the reactance of line ij. This indicates the maximum active power allowed to pass through line ij, with the subscript 'ref' indicating the slack node; constraint (6) represents the upper and lower limits of generator output, where, and These represent the minimum and maximum output of the generator, respectively; constraint (7) represents the generator's ramp and landslide constraints, r DN,g and r UP,g Let represent the landslide rate and ramp rate of generator g, respectively; constraint (8) is the standby capacity constraint of the generator; constraints (10) and (11) are the minimum start-stop time constraints of the generator, DT g and UT g Let represent the minimum downtime and minimum running time of generator g, respectively, and τ be an auxiliary variable; constraints (12) and (13) represent the start-up cost and downtime cost of generator g at time t, respectively.

5. The robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions according to claim 1, characterized in that, for , The objective function of the state assessment model considering the complex scenario is as follows: ；（14） In the formula, a LS,i This represents the load shearing cost coefficient for bus i; y = { | },in , and Let g represent the output of generator g, the load on bus i, and the phase angle, respectively, in the composite scenario (k,s). This represents the load sheared by bus i at time t in the composite scenario (k,s).

6. The robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions according to claim 1, characterized in that, The constraints of the state assessment model under the complex scenario are as follows: ；（15） ；（16） ；（17） ；（18） ；（19） ；（20） In the formula, constraints (15)~(17) represent DC power flow constraints, where, and In the composite scenario (k,s), the active power flow and phase angle difference of line ij; constraint (18) is the active load constraint of bus i; constraint (19) is the upper and lower limits constraint of generator active power output; constraint (20) is the ramp constraint of generator.

7. The robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions according to claim 1, characterized in that, Step 2), the steps of approximating the state evaluation model using basis-adaptive sparse chaotic polynomial expansion, include: Step 2.1) Determine the random inputs to the condition assessment model, including day-ahead unit combination decisions and source-load forecasting errors; day-ahead unit combination decisions include unit start-up and shutdown status, unit output, and unit standby; Step 2.2) Set the probability density distribution of the day-ahead unit combination decision, and use the Gaussian-copula model to transform the historical samples of source-load prediction errors that follow arbitrary distribution into experimental design samples for building the surrogate model, and determine the probability density distribution of the experimental design samples. Step 2.3) Select orthogonal basis functions to approximate the random input based on the probability density distribution type of the random input; use the orthogonal basis functions to approximate the random input, thereby relaxing the state to evaluate the model; Wherein, if the probability density distribution is uniform, the orthogonal basis function is the Legendre polynomial; if the probability density distribution is normal, the orthogonal basis function is the Hermite polynomial; if the probability density distribution is Gamma, the orthogonal basis function is the Laguerre polynomial; and if the probability density distribution is Beta, the orthogonal basis function is the Jacobi polynomial. Step 2.4) Relax the upper bound of the node load to obtain: ；（21） In the formula, It is much larger than The number of nodes is used to relax the node load; Step 2.5) Based on the construction of the state evaluation agent model, namely: ；（22） ；（23） In the formula, D k This represents the set of proxy models under k fault scenarios, including proxy models for each time period. ; k and s represent the fault scenario vector and the source load prediction error scenario vector, respectively.

8. The robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions according to claim 7, characterized in that, In step 2.2), the unit's start-up / shutdown state u and the unit's output P are... G Construct composite variable u·P G And set the composite variable u·P G Obey [0, The uniform distribution of R; for the unit's standby R G Set the unit's standby R G Obey [0, The uniform distribution of ].

9. The robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions according to claim 7, characterized in that, Step 2.2), the step of transforming historical samples of source load prediction errors following an arbitrary distribution into experimental design samples for constructing the surrogate model, includes: Step 2.2.1) Calculate the Spearman rank correlation coefficient matrix C based on historical samples of the random variable. S,O ; The Spearman correlation coefficient matrix in standard Gaussian space satisfies the following equation: ；（24） Step 2.2.2) Based on the transformation relationship between Pearson correlation coefficient and Spearman correlation coefficient in Gaussian variables, construct the corresponding Pearson correlation coefficient matrix C in standard Gaussian space. P,Z ,Right now: ；（25） Step 2.2.3) Analyze the Pearson correlation coefficient matrix C P,Z Performing Cholesky decomposition, we obtain: ；（26） In the formula, L is a lower triangular matrix; Step 2.2.4) Independent samples generated by Latin hypercube sampling Construct relevant standard Gaussian samples ,Right now: ；（27） Step 2.2.5) Transform the relevant Gaussian samples using inverse probability transformation. Mapped to the original domain, generating original domain samples ,Right now: ；（28） In the formula, Φ represents the standard Gaussian cumulative distribution function. Let be the inverse cumulative distribution function of the m-th random variable in the original domain.

10. The robust optimization solution method for event-constrained unit combinations based on adaptive chaotic polynomial expansion of basis functions according to claim 1, characterized in that, Step 4) involves solving the robust optimization model for the replaced unit combination, including the following steps: Step 4.1) Add constraint cost variables for each failure scenario to the day-ahead unit combination model: ；（29） ；（30） Step 4.2) Reconstruct the robust optimization model of unit combination into a min-max problem, i.e.: ；（31） In the formula, d represents the set of auxiliary variables; Step 4.3) Introduce an independent set of source-load scenario decision variables s for each fault scenario k. k The min-max two-level optimization problem is reconstructed into a single-level minimization problem, namely: ；（32） In the formula, s k Let S represent the set of decision variables for source load prediction error under fault scenario k, and let S represent the set of box-shaped uncertainties for source load prediction error. Step 4.4) Solve the single-layer minimization problem using the gurobi solver to obtain the unit combination scheme under the most severe fault scenario and the continuous source load uncertainty scenario.

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