A robust optimal scheduling method for microgrids considering segmented multi-interval uncertainty sets

Abstract

This invention provides a robust optimization scheduling method for microgrids that considers segmented multi-interval uncertainty sets. Based on a Gaussian mixture model, the source-load power and error data for wind, solar, and load are piecewise fitted. This is combined with a box-type interval uncertainty set to establish a multi-interval uncertainty set for each power segment, used to describe the uncertainties of wind, solar, and load. With the goal of minimizing microgrid operating costs and carbon emission costs, a robust optimization scheduling model for microgrids that considers segmented multi-interval uncertainty sets and carbon emissions is constructed. A column and constraint generation algorithm is used to solve the robust optimization scheduling model for microgrids. This invention addresses uncertainty modeling and carbon emission issues in microgrids, reduces the conservatism of robust optimization scheduling strategies, and improves the economic efficiency of microgrid scheduling.

Description

Technical Field

[0001] The present invention relates to the technical field of microgrid planning, and in particular to a robust optimization scheduling method for a microgrid considering segmented multi-interval uncertainty sets. Background Art

[0002] Microgrids (MGs), composed of renewable energy (RE), energy storage systems, and loads, play a vital role in the grid's transition to a low-carbon, green, and intelligent future. MGs create significant economic benefits for the grid. However, the uncertainty of RE output power and load demand poses significant challenges to the reliable operation of MGs. Addressing this uncertainty in MG sources and loads and ensuring their economical and safe operation is a worthy research topic.

[0003] Currently, scholars at home and abroad have conducted in-depth research on the optimal scheduling of MGs, taking into account the impact of uncertainty factors, with the goals of operational economics, renewable energy absorption capacity, and carbon emission reduction. Current approaches to address uncertainty include scenario analysis, stochastic optimization (SO), robust optimization (RO), and distributed robust optimization (DRO). Both scenario analysis and SO rely on the probability distribution of uncertain variables, requiring a large amount of sample data, increasing the complexity of the problem and potentially overlooking some extreme scenarios. RO, however, does not rely on the probability distribution of uncertain variables. Instead, it represents the uncertainty of variables in the form of uncertainty sets, enabling the solution of the optimal system scheduling under extreme scenarios. Therefore, it has been widely used in MG optimal scheduling. Existing literature has constructed a two-stage robust MG energy management model, using polyhedron sets to describe source and load uncertainty. Other literature has developed a two-stage RO model with binary recourse variables based on polyhedron sets of periodic budgets. However, these literatures all use simple uncertainty sets to describe system uncertainty, resulting in the inclusion of many scenarios that are impossible in practice, making the scheduling results overly conservative and reducing system operational efficiency.

[0004] To overcome the specificity of SO and the conservatism of RO, DRO extracts probability information from historical data and constructs fuzzy sets containing the probability distribution of uncertain variable factors, thereby solving the minimum expected cost under the worst-case scenario. Depending on the method used to establish the fuzzy set, DRO can be divided into two types: moment information-based and distance metric-based. Moment information-based DRO methods derive probability distribution parameters of uncertain parameters from historical data. Distance-based DRO methods establish the confidence set of the fuzzy set by introducing a statistical distance between two probability distributions. However, these methods convert the DRO model into a semidefinite programming or second-order conic programming problem to improve numerical tractability, which can easily lead to suboptimal solutions.

[0005] While the aforementioned literature has achieved significant results in reducing RO conservatism, most studies have overlooked the direct relationship between uncertainty parameter datasets and prediction errors. Exploiting the prediction error information contained in source-load datasets and combining it with uncertainty set modeling to improve the quality of robust scheduling results remains a challenge. Existing literature combines least-squares fitting of straight lines with polyhedral uncertainty sets to construct uncertainty sets that encompass the spatiotemporal correlations of PV generation. Other literature utilizes multi-interval uncertainty sets to address the adaptive RO scheduling problem in mobile electricity generation (MG), to some extent avoiding the worst-case scenario always falling within the boundaries of the prediction interval. Other literature utilizes Dirichlet process mixture models and Bayesian nonparametric methods to fit uncertainty sets, eliminating the discrepancy between traditional polyhedral sets and historical scenarios. Existing literature selects certain extreme scenarios from historical data and combines them with traditional convex uncertainty sets to describe the spatiotemporal correlations of uncertainty sets and demonstrates their applicability. Summary of the Invention

[0006] The purpose of the present invention is to provide a robust optimization scheduling method for microgrids considering segmented multi-interval uncertainty sets, which can solve the uncertainty modeling and carbon emission problems of microgrids, reduce the conservatism of robust optimization scheduling strategies and improve the economy of microgrid scheduling.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is: a robust optimization scheduling method for microgrid considering segmented multi-interval uncertainty sets, comprising the following steps:

[0008] Step 1: Based on the Gaussian mixture model, the source-load power and error data of wind, solar, and load are segmented and fitted. Combined with the box-type interval uncertainty set, a multi-interval uncertainty set for power segmentation is established to describe the uncertainty of wind, solar, and load.

[0009] Step 2: With the goal of minimizing the microgrid operation cost and carbon emission cost, a robust optimization scheduling model for the microgrid is constructed that considers the segmented multi-interval uncertainty set and carbon emissions.

[0010] Step 3: Use the column and constraint generation algorithm to solve the microgrid robust optimization scheduling model, including the following steps:

[0011] Step 31, dividing the microgrid robust optimization scheduling model into a main problem and a sub-problem;

[0012] Step 32: Convert the max-min subproblem into a single-level max problem using the Lagrange duality method, and convert the dual problem of the subproblem into a mixed integer linear programming problem using the big-M method: linearize the product of binary variables and continuous variables by introducing auxiliary variables and related constraints.

[0013] Step 33: Alternately iterate and solve the main and subproblems. After the derivation and transformation in step 32, the proposed two-stage robust optimization scheduling model has been transformed into the form of a mixed integer linear programming problem. The main problem and subproblems are solved iteratively through the column and constraint generation algorithm until the difference between the upper bound LB and the lower bound UB is reduced to the convergence threshold ξ, and the final optimized daily stage scheduling decision is obtained.

[0014] Preferably, the actual values ​​of wind, solar and load power in step 1 are represented by the sum of their power prediction values ​​and prediction errors, and the power prediction error is:

[0015] Where: Unit ψ includes wind turbine WT, photovoltaic PV and load Load; e ψ,t is the power prediction error of unit ψ during period t; P ψ,t is the actual power of unit ψ during period t; is the power prediction value of unit ψ during period t;

[0016] The source-load power prediction value is divided into three intervals, and the power prediction error in each interval is fitted using a Gaussian mixture model.

[0017] Preferably, establishing the multi-interval uncertainty set of the power partition in step 1 includes the following steps:

[0018] Step 11: Obtain the probability distribution of the corresponding prediction error according to the source-load prediction power segmentation situation;

[0019] Step 12: Combining the multi-interval partitioning of the power segmentation uncertainty set to analyze the construction method of the power segmentation error multi-interval uncertainty set;

[0020] Step 13: Based on the size and probability of the prediction error, combined with the uncertain budget parameter Γ ψ The error single interval Divided into N AS intervals, satisfying the following formula:

[0021]

[0022] Where: N SP is the number of segments of the power prediction value; and is the predicted upper and lower deviation values ​​of the unit ψ in the power section s error interval j during time period t; Γ ψ,j,s is the uncertain time budget parameter of unit ψ in the error interval j of power segment s, and the degree of uncertainty can be changed by adjusting its size; and is the predicted upper and lower deviation values ​​of unit ψ in a single interval during period t; Γ ψis the single interval uncertain time budget parameter; ρ ψ,j is the probability of occurrence of unit ψ in interval j, and its value can be obtained according to the fitting result of the corresponding probability distribution, ρ ψ is the probability of occurrence of unit ψ in a single interval;

[0023] Step 14: Based on the constructed data, the segmented multi-interval source-load uncertainty set is driven. The wind, solar, and load uncertainties in the microgrid MG are shown as follows:

[0024]

[0025] Where: U ψ The piecewise multi-interval uncertainty set representing the unit ψ; It is a Boolean variable, indicating whether the predicted deviation variable is located in the upper or lower half of the j-th interval.

[0026] Preferably, the microgrid includes traditional distributed power sources, renewable distributed power sources, an energy storage system and loads.

[0027] Preferably, in step 2, a two-stage robust optimization scheduling model is established with the goal of minimizing the microgrid operating cost and environmental cost. The constraints that need to be met include power generation power constraints, energy storage system constraints, demand response load constraints, and interactive power constraints with the power grid. The objective function is as follows:

[0028]

[0029] Preferably, the microgrid robust optimization scheduling model constructed in step 2 considering segmented multi-interval uncertainty sets and carbon emissions is as follows:

[0030]

[0031] Corresponding to the power generation constraint, energy storage system constraint, demand response load constraint, and interactive power constraint with the grid, the inner layer 'max' is to find the worst scenario with concentrated uncertainty, and 'min' is to minimize the microgrid operating cost under the worst scenario; the outer layer 'min' is to optimize the day-ahead scheduling decision by the corresponding scheduling power under the worst scenario; c represents the coefficient column vector corresponding to the objective function (28); B and I represent the equality constraint coefficient matrix of the day-ahead deterministic scenario; e represents the constant column vector of the equality constraint of the day-ahead deterministic scenario; A, C, D, and F represent the inequality constraint coefficient moments of the day-ahead deterministic scenario; d, f, and h represent the constant column vector of the inequality constraint of the day-ahead deterministic scheduling scheme;

[0032] x and y represent the optimization variables of the day-ahead scenario, and their specific expressions are as follows:

[0033]

[0034] Preferably, the main question in step 31 is written in the following form:

[0035]

[0036] Where: U w is the set of all worst cases searched; for U w For each worst case us in the , generate the corresponding hourly phase variable y k and its constraints, the mean value umeean of the uncertainty range is selected as the initial worst uncertainty scenario u0*; the sub-problem is expressed as:

[0037]

[0038] Preferably, in step 32, the subproblem max-min is transformed into a single-layer max problem by the Lagrange duality method, and Ω(x,u) represents the feasible region of the optimization variable y when a set of (x,u) is given. The specific expression is as follows:

[0039]

[0040] Where: σ, ω, τ, θ and ρ are the dual variables corresponding to the inner constraints respectively;

[0041] When (x,u) is given, the inner layer of the subproblem is a linear problem. The dual problem of the subproblem can be obtained according to the corresponding relationship of the following formula:

[0042]

[0043] Preferably, in step 32, the dual problem of the subproblem is transformed into a mixed integer linear programming problem using the big M method:

[0044]

[0045] Where: Δu j+ , Δu j- is the power prediction deviation; is the introduced continuous auxiliary variable; π j+ , π j- is a binary variable; M is the upper bound of the dual variable, which is a sufficiently large positive integer.

[0046] Preferably, the alternating iterative solution between the main and sub-problems in step 33 includes the following steps:

[0047] Step 331: Given a set of uncertain variable values ​​u meean , as the initial worst scenario, the upper bound UB = +∞, the lower bound LB = -∞, the number of iterations k = 1, and the convergence gap is set to ξ = 0.01;

[0048] Step 332: u meean Substitute into the main problem and get the optimal solution of the problem (x k ,y k ,λ k ), update the objective function value of the main problem as the new lower bound LB = λ k ;

[0049] Step 333: x k Substitute into the subproblem and get the optimal solution f of the subproblem k (x k ) and the corresponding worst scenario u k+1 , update the sub-problem objective function value as the new upper bound UB=min{UB,f k (x k )};

[0050] Step 334: If UB-LB≤0.01, stop the iteration and return to the optimal solution (x k ,y k ), otherwise, increase the variable y k+l and the following constraints:

[0051]

[0052] Let k=k+1, and return to step 332 until the algorithm converges.

[0053] The beneficial effects of the present invention are:

[0054] This solution addresses the uncertainty of sources and loads in microgrids, considers the direct relationship between the uncertainty parameter data set and the prediction error, and establishes a robust optimization scheduling method for microgrids considering segmented multi-interval uncertainty sets. The following conclusions are drawn from the case analysis:

[0055] 1) Comprehensively considering demand response, carbon emission costs and source-load uncertainty can fully tap the dispatching potential of the source-load side, effectively smooth out wind and solar power and load fluctuations in the worst scenarios, and reduce system operating costs.

[0056] 2) By describing the uncertainty of wind, solar and load using segmented multi-interval uncertainty sets, the conservatism of the microgrid scheduling scheme can be effectively reduced, thereby reducing the total cost of system scheduling.

[0057] 3) When the deterministic budget parameter is set small, the system's scheduling cost is low, but its robustness is weak. When the uncertain parameter is set large, the system's scheduling results are more robust, but the scheduling cost is higher. Therefore, when formulating a scheduling plan, appropriate uncertain budget parameters can be selected according to system requirements to flexibly adjust the robustness and cost-effectiveness of the scheduling plan.

[0058] This paper focuses on constructing an uncertainty set model for wind, solar, and load uncertainties in traditional microgrids and solves a robust optimal scheduling model for microgrids based on this uncertainty set. Future research will explore and analyze the various complex uncertainties in integrated energy microgrids and multi-microgrid systems, further investigating their robust optimal scheduling.

[0059] The main contributions of this solution are summarized as follows:

[0060] 1) A new power segmentation multi-interval uncertainty set modeling method is proposed. To reduce the conservatism of the robust optimization scheduling strategy and improve the economic efficiency of MG scheduling, a Gaussian mixture model is used to fit the prediction errors within the source and load power intervals based on historical predicted power and error data of uncertain parameters. Then, based on the error distribution fitting results corresponding to different power intervals, a new power segmentation multi-interval uncertainty set modeling method is proposed.

[0061] 2) A two-stage robust optimization scheduling model for microgrids is established, which takes into account both source-load uncertainty and environmental costs. This model considers the economic efficiency of microgrid operation, environmental protection, and the conservatism of optimization results.

[0062] 3) The proposed model is linearized using the Big M method and Lagrangian duality theory, and an improved CCG algorithm is used to solve the MILP. Four case studies are designed for comparative analysis, demonstrating that the proposed model balances the economic and environmental benefits of MG operation while ensuring the robustness of the dispatch strategy and the absorption of renewable energy. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0064] Figure 1 Schematic diagram of the microgrid system structure of the present invention.

[0065] Figure 2 This is a scatter plot of the wind power prediction value and its prediction error of the present invention.

[0066] Figure 3 This is a probability density fitting curve diagram of the wind power prediction error of the present invention.

[0067] Figure 4 Schematic diagram of multi-interval uncertainty sets in existing literature.

[0068] Figure 5 This is a multi-interval partitioning diagram of the power segment uncertainty set of the present invention.

[0069] Figure 6 Schematic diagram of the segmented multi-interval uncertainty set of the present invention.

[0070] Figure 7 This is a graph showing the predicted source-load power curve of the microgrid of the present invention.

[0071] Figure 8 This is the micro-network electricity sales plan diagram of the present invention.

[0072] Figure 9 This is the day-ahead scheduling plan diagram of the DG and ESS of the present invention.

[0073] Figure 10 This is the actual power consumption plan diagram of the DR of the present invention. DETAILED DESCRIPTION

[0074] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0075] The present invention discloses a robust optimization scheduling method for a microgrid considering segmented multi-interval uncertainty sets. The embodiment includes the following steps:

[0076] Step 1: Based on the Gaussian mixture model, the source-load power and error data of wind, solar, and load are segmented and fitted. Combined with the box-type interval uncertainty set, a multi-interval uncertainty set for power segmentation is established to describe the uncertainty of wind, solar, and load.

[0077] Step 2: With the goal of minimizing the microgrid operation cost and carbon emission cost, a robust optimization scheduling model for the microgrid is constructed that considers the segmented multi-interval uncertainty set and carbon emissions.

[0078] Step 3: Use the column and constraint generation algorithm to solve the microgrid robust optimization scheduling model, including the following steps:

[0079] Step 31, dividing the microgrid robust optimization scheduling model into a main problem and a sub-problem;

[0080] Step 32: Convert the max-min subproblem into a single-level max problem using the Lagrange duality method, and convert the dual problem of the subproblem into a mixed integer linear programming problem using the big-M method: linearize the product of binary variables and continuous variables by introducing auxiliary variables and related constraints.

[0081] Step 33: Alternately iterate and solve the main and subproblems. After the derivation and transformation in step 32, the proposed two-stage robust optimization scheduling model has been transformed into the form of a mixed integer linear programming problem. The main problem and subproblems are solved iteratively through the column and constraint generation algorithm until the difference between the upper bound LB and the lower bound UB is reduced to the convergence threshold ξ, and the final optimized daily stage scheduling decision is obtained.

[0082] like Figure 1 The figure shows a typical grid-connected microgrid system. The system includes traditional distributed power sources (MTs), renewable distributed power sources (PVs, WTs), an energy storage system, and loads. MGs exchange energy with the upper-level distribution network via centralized power lines. While considering the economic and environmental performance of the microgrid, the control center optimizes the scheduling of each flexible unit in the system to improve economic efficiency and reliability.

[0083] Current MG robust scheduling ignores the important parameter characteristics of historical source and load data, thus severing the connection between day-ahead prediction and scheduling. Therefore, to improve the overly conservative shortcomings of RO, first, the forecast errors within different power intervals are fitted using the probabilistic characteristics of wind and solar output and the load itself. This confirms that the power segment fitting method can more accurately reflect the true distribution of source and load forecast errors. Then, based on the error distribution fitting results corresponding to different power intervals, the traditional single-interval uncertainty set is divided into multiple intervals. Finally, the uncertainty budget parameters are allocated according to the distribution of forecast errors in each power interval, making the defined multi-interval uncertainty set more consistent with the actual situation and effectively reducing the conservatism of the scheduling scheme.

[0084] The actual values ​​of wind, solar and load power in step 1 are represented by the sum of their power prediction values ​​and prediction errors. The power prediction error is:

[0085]

[0086] Where: Unit ψ includes wind turbine WT, photovoltaic PV and load Load; e ψ,t is the power prediction error of unit ψ during period t; P ψ,t is the actual power of unit ψ during period t; is the power prediction value of unit ψ in period t.

[0087] Some literature points out that the error probability distribution corresponding to the power forecast values ​​in different sections is different. Taking wind power as an example, the wind power output and error data of Elia power grid in 2022 are selected for analysis. The data are normalized by wind power installed capacity. The scatter plot of wind power forecast value and its forecast error is shown in the figure below. Figure 2 As shown, from Figure 2It can be seen that the prediction errors corresponding to low power prediction values ​​are more concentrated and have smaller values, while the prediction errors corresponding to high power prediction values ​​are more dispersed and have larger values. This indicates that the distribution characteristics of power prediction errors are correlated with the size of the power prediction values, and it is necessary to explore these characteristics to better describe source-load uncertainty.

[0088] The Gaussian mixture model has the advantage of high applicability in describing distributions of different characteristics and can effectively handle the probability distribution fitting problem of wind and solar power and load forecast errors with strong uncertainty, as shown in formula (2).

[0089]

[0090] Where: f i (ΔP) represents the probability density function of the power prediction error ΔP in the i-th power segment; ω k is the weight coefficient of the kth normal distribution; is the probability density function of the kth normal distribution; μ k and are the expectation and variance respectively; m is the number of normal distributions in the Gaussian mixture model.

[0091] In this embodiment, the source load power prediction value is divided into three intervals, specifically [0, 0.3], [0.3, 0.6] and [0.6, 1], and the Gaussian mixture model is used to fit the power prediction error in each interval. The probability distribution of the prediction error in each interval and the probability distribution of the overall error when no segmentation is performed are as follows: Figure 3 As shown. Figure 3 It can be seen that the probability distribution of prediction errors in different intervals is significantly different: when When it is less than 0.3, the fitting curve shows a peak characteristic; when When it is between 0.3 and 0.6, the fitting curve is symmetrical; When it is greater than 0.6, the fitting curve is right-skewed. Therefore, the method of fitting the prediction error by segment of the predicted power can more realistically reflect the distribution of the prediction error.

[0092] Robust optimization scheduling models often use a single interval uncertainty set to describe source-load uncertainty, but the optimization results are quite conservative. To this end, some literature has proposed multiple interval uncertainty sets, such as Figure 4 As shown. Figure 4 It can be seen that although the multi-interval source-load uncertainty set can reduce the conservatism of robust optimization to a certain extent, it ignores the relationship between the source-load predicted power and the prediction error, resulting in the uncertainty set covering many extreme scenarios that are impossible to occur in practice.

[0093] In order to effectively eliminate extreme scenarios that are unlikely to occur in the uncertainty interval, a more realistic multi-interval uncertainty set for power segments is constructed by considering the relationship between historical predicted power and prediction error. The establishment of a multi-interval uncertainty set for power segments in step 1 includes the following steps:

[0094] Step 11: Obtain the probability distribution of the corresponding prediction error according to the source-load prediction power segmentation situation;

[0095] Step 12: Take the prediction error probability distribution of the power prediction standard value in the interval [0.6,1] as an example, combined with Figure 5 The construction method of multi-interval uncertainty set of power segmentation error is further analyzed;

[0096] Step 13: Based on the prediction error size and probability in the figure, combined with the uncertain budget parameter Γ ψ The error single interval Divided into N AS intervals, satisfying formula (3)

[0097]

[0098] Where: N SP is the number of segments of the power prediction value; and is the predicted upper and lower deviation values ​​of the unit ψ in the power section s error interval j during time period t; Γ ψ,j,s is the uncertain time budget parameter of unit ψ in the error interval j of power segment s, and the degree of uncertainty can be changed by adjusting its size; and is the predicted upper and lower deviation values ​​of unit ψ in a single interval during period t; Γ ψ is the single interval uncertain time budget parameter; ρ ψ,j is the probability of occurrence of unit ψ in interval j, and its value can be obtained according to the fitting result of the corresponding probability distribution, ρ ψ is the probability of occurrence of unit ψ in a single interval;

[0099] Step 14: Based on the constructed data, the uncertainty set of source and load in the segmented multi-interval is driven. The uncertainty of wind, solar and load in MG is shown in formula (4). The uncertainty scenario is as follows: Figure 6 As shown:

[0100]

[0101] Where: P ψ,t,s is the actual power of unit ψ in power section s during period t; U ψ The piecewise multi-interval uncertainty set representing the unit ψ; is a Boolean variable, indicating that the predicted deviation variable is located in the upper and lower half of the jth interval. and are the predicted upper and lower deviation values ​​of unit ψ in time period t; T represents the total number of scheduling time periods.

[0102] Compared to the uncertainty sets proposed in existing literature, the uncertainty set proposed in this embodiment determines the prediction errors corresponding to power values ​​in different segments based on fitted probability distribution parameters, making it more realistic and, in turn, reducing the conservatism of the robust optimization results. This is because this embodiment not only considers the temporal correlation of the source and load, but also the distribution characteristics of the prediction errors corresponding to their day-ahead predictions.

[0103] Traditional distributed power generation: The cost of traditional distributed power generation in microgrid is shown in formula (5), which includes operation and maintenance cost and fuel consumption cost.

[0104]

[0105] Where: C MT (t) is the power generation cost of the MT machine in period t; b MT is the operation and maintenance cost coefficient; P MT (t) is the output power of MT during period t; c MT is the unit price of fuel; η MT is the operating efficiency of MT, which is 40%; Δt is the scheduling step, which is 1h; T is the scheduling period, which is 24h.

[0106] Since the power response speed of MT is faster than that of hourly scheduling, its ramp rate constraint is not considered, and only the power generation constraint is considered:

[0107] P MT,min ≤P MT (t)≤P MT,max (6)

[0108] Where: P MT,min 、P MT,max They are the upper and lower limits of the MT's output power, which are limited by its rated power and minimum load rate.

[0109] Renewable distributed generation: Operation and maintenance costs can be defined as:

[0110]

[0111] Where: C j (t) is the operation and maintenance cost of WT and PV in period t; b j is the operation and maintenance cost coefficient; P j (t) is the output power of wind and solar power during period t.

[0112] Energy storage system: ESS charge and discharge state constraints:

[0113] O ch (t)+O dis (t)≤1 (8)

[0114] Where: O ch (t) represents the charging status of ESS in period t (“1” represents charging, “0” represents not charging), dis (t) represents the discharge state of ESS in period t (“1” represents discharge, “0” represents no discharge).

[0115] ESS charging and discharging constraints:

[0116] 0≤P dis (t)≤O dis (t)P dis,max (9)

[0117] 0≤P ch (t)≤O ch (t)P ch,max (10)

[0118] P ESS (t) = P dis (t)-P ch (t) (11)

[0119] Where: P ESS (t) is the output power of ESS during period t; P dis (t) and P ch (t) are the charge and discharge power of ESS during period t.

[0120] State of Charge (SOC) constraints of ESS:

[0121]

[0122] SOC(t=0)=SOC(t=T) (13)

[0123] SOC min ≤SOC(t)≤SOC max (14)

[0124] Where: Equation (12) represents the energy constraint of ESS; η ch and η dis are the charge and discharge efficiency of ESS respectively; Pr is the rated capacity of ESS; SOC min and SOC max are the lower and upper limits of the ESS state of charge, respectively. Equation (13) ensures that the energy storage capacity is equal at the beginning and end of the dispatch, which is conducive to the cyclic dispatch of energy storage.

[0125] ESS in MGs often uses lead-acid batteries and lithium batteries. Battery life depends on the depth of charge / discharge. As the depth of charge / discharge increases, the battery life decreases. The aging cost of ESS can be expressed as:

[0126]

[0127] Where: C ESS (t) is the aging cost of ESS in period t; c ESS The cost of aging.

[0128] Demand response load: The implementation of demand response (DR) is to encourage users to transfer non-essential loads in order to obtain benefits.

[0129] P DR (t) = P up (t)-P down (t) (16)0≤P up (t)≤O up (t)P up,max (17)

[0130] 0≤P down (t)≤O down (t)P down,max (18)

[0131] O up (t)+O down (t)≤1 (19)

[0132]

[0133] Where: P DR (t) is the actual dispatch power of DR in period t, that is, the amount of power load transferred; P up (t) and P down (t) is the power regulation of DR up / down during period t. The power limits for upward and downward regulation are given by equations (17) and (18). Equation (19) ensures that the power load cannot be regulated downward and upward at the same time. up (t) represents the upward adjustment state of the power load in the t period, O down (t) represents the downward adjustment state of the power load in period t. Formula (20) ensures that the total power consumption of the load remains unchanged during the scheduling period.

[0134] The scheduling cost of implementing DR can be expressed as:

[0135]

[0136] Where: C DR (t) is the implementation cost of DR in period t; cDR is the scheduling cost coefficient of DR.

[0137] Interaction power with the grid: When there is a shortage or surplus of energy supply within the MG system, it is necessary to interact with the grid to adjust the power balance and realize the local consumption of renewable energy.

[0138] In order to avoid the situation where the microgrid purchases and sells electricity from the grid at the same time, the constraints are set as follows:

[0139] O buy (t)+O sell (t)≤1 (22)0≤P buy (t)≤O buy (t)P buy,max (twenty three)

[0140] 0≤P sell (t)≤O sell (t)P sell,max (twenty four)

[0141] Where: O buy (t) represents the state of MG purchasing electricity from the grid during period t, O sell (t) represents the state of MG selling electricity to the grid during period t; P buy (t) and P sell (t) are the power purchased from and sold to the grid during period t.

[0142] The power balance constraint is:

[0143] P buy (t)-P sell (t)+P ESS (t)+P PV (t)+P WT (t)+P MT (t) = P DR (t)+P RL (t)+P EV (t) (25)

[0144] Where: P RL (t) is the non-dispatchable load in the power grid during period t.

[0145] The cost of interacting with the main grid to buy and sell electricity is:

[0146]

[0147] Pollutant treatment costs: The cost of microgrid environmental pollution control, including the environmental compensation cost of the MT and the environmental compensation cost of fossil energy combustion in the distribution network. The pollutants considered in this example mainly include CO2, SO2, and NOX. Their costs are described as follows:

[0148]

[0149] Where: C emi (t) is the pollutant treatment cost of the microgrid in period t; q MT,p The emission coefficient of type p pollutants generated by traditional distributed power sources; b p is the treatment cost of the pth type of pollutant; q G,p Generate the p-th type pollutant emission coefficient for the distribution network.

[0150] In step 2, a two-stage robust optimization scheduling model is established with the goal of minimizing the microgrid's operating cost and environmental cost. The constraints that need to be met include equations (6), (8-14), (16-20), and (22-25). The objective function is shown in equation (28):

[0151]

[0152] Where: C MT (t) is the power generation cost of the MT machine in period t; C j (t) is the operation and maintenance cost of WT and PV in period t; C ESS (t) is the aging cost of ESS in period t; C DR (t) is the implementation cost of DR in period t; C G (t) is the cost of electricity exchange with the main grid; C emi (t) is the pollutant treatment cost of the microgrid in period t.

[0153] The scheduling results of deterministic optimization models depend largely on the accuracy of the forecast data, but the uncertainty of the MG significantly impacts the accuracy of the scheduling results. Therefore, uncertainty should be considered when formulating day-ahead scheduling plans. The two-stage "min-max-min" robust optimization model constructed in this embodiment is shown in Equation (29). Equation (29) corresponds to constraints (6), (8-14), (16-20), and (22-25).

[0154]

[0155] Where: the inner layer 'max' is to find the worst scenario with concentrated uncertainty, and 'min' is to minimize the microgrid operating cost under the worst scenario. The outer layer 'min' is to optimize the day-ahead scheduling decision by using the corresponding scheduling power under the worst scenario. c represents the coefficient column vector corresponding to the objective function (28); B and I represent the equality constraint coefficient matrix of the day-ahead deterministic scenario; e represents the constant column vector of the equality constraint of the day-ahead deterministic scenario; A, C, D, and F represent the inequality constraint coefficient moments of the day-ahead deterministic scenario; d, f, and h represent the constant column vector of the inequality constraint of the day-ahead deterministic scheduling scheme.

[0156] x and y represent the optimization variables of the day-ahead scenario, and their specific expressions are as follows:

[0157]

[0158] The two-stage robust optimization problem cannot be solved directly. Therefore, this embodiment uses the column and constraint generation C&CG algorithm to solve the MG robust optimization problem. First, the proposed two-stage RO model (29) needs to be divided into a main problem (31) and a subproblem (32). Then, the main and subproblems are solved alternately. The main problem is written as follows:

[0159]

[0160] Where: U w is the set of all worst cases found. w For each worst case us in the , generate the corresponding hourly phase variable y k and its constraints. Select the mean u of the uncertainty range meean As the initial worst scenario of uncertainty u0*.

[0161] The sub-problem can be expressed as:

[0162]

[0163] The inner layer of the subproblem contains only continuous variables. This paper transforms the subproblem max-min into a single-layer max problem through the Lagrangian duality method. Ω(x,u) represents the feasible region of the optimized variable y given a set (x,u). The specific expression is as follows:

[0164]

[0165] Where: σ, ω, τ, θ and ρ are the dual variables corresponding to the inner constraints respectively.

[0166] When (x,u) is given, the inner layer of the subproblem is a linear problem. According to the corresponding relationship of formula (33), the dual problem of the subproblem can be obtained:

[0167]

[0168] However, the bilinear term uTρ in the subproblem poses a challenge to solving the inner problem. Considering that the optimization result of u must be the vertex of each subinterval of the uncertainty set, the deviation variable is 0 or 1. Then, the big M method is used to transform Equation (34) into a mixed integer linear programming (MILP) problem: the product of binary variables and continuous variables will appear, and it is linearized by introducing auxiliary variables and related constraints.

[0169]

[0170] Where: Δu j+ , Δu j- is the power prediction deviation; φ j+ 、φ j-为 The continuous auxiliary variable introduced; π j+ , π j- is a binary variable; M is the upper bound of the dual variable, which is a sufficiently large positive integer.

[0171] After the above derivation and transformation, the proposed two-stage RO model has been transformed into a mixed integer linear programming problem. The main problem and subproblems can be solved iteratively using the C&CG algorithm until the difference between LB and UB is reduced to the convergence threshold ξ, and the final optimized daily stage scheduling decision is obtained. The solution process is as follows:

[0172] Step 331: Given a set of uncertain variable values ​​u meean , as the initial worst scenario, the upper bound UB = +∞, the lower bound LB = -∞, the number of iterations k = 1, and the convergence gap is set to ξ = 0.01;

[0173] Step 332: u meean Substitute into the main problem and get the optimal solution of the problem (x k ,y k ,λ k ), update the objective function value of the main problem as the new lower bound LB = λ k ;

[0174] Step 333: x k Substitute into the subproblem and get the optimal solution f of the subproblem k (x k ) and the corresponding worst scenario u k+1 , update the sub-problem objective function value as the new upper bound UB=min{UB,f k (x k )};

[0175] Step 334: If UB-LB≤0.01, stop the iteration and return to the optimal solution (x k ,yk ), otherwise, increase the variable y k+l and the following constraints:

[0176]

[0177] Let k=k+1, and return to step 332 until the algorithm converges.

[0178] For case studies and analysis, all simulations were run on a personal computer equipped with an Intel Core i7-6700 CPU and 16GB of RAM. The robust optimization scheduling model was built using MATLAB 2018a and YALMIP, and the CPLEX solver was used to solve the optimization problem.

[0179] Simulation parameter setting: In order to verify the effectiveness of the proposed microgrid robust optimization scheduling model and method, based on Figure 1 The microgrid framework described in [1] was simulated. The simulation parameters are shown in Table 1. The pollutant emission coefficients and treatment costs are shown in Table 2. The day-ahead transaction price for power exchange between the distribution grid and the microgrid is a tiered residential electricity price, as shown in Table 3. The price of electricity sold to the main grid is set at 40% of the purchase price.

[0180] Table 1 Parameter settings

[0181]

[0182] Table 2 Pollutant emission coefficients and treatment costs

[0183]

[0184] Table 3. Grid day-ahead time-of-use electricity prices

[0185]

[0186] According to the analysis, the specific distribution parameters of the source-load segment power prediction error are shown in Tables 3 to 5. According to the specific distribution parameters corresponding to the source-load prediction power error in different sections, the wind and solar power generation uncertainty set is divided into three intervals with deviations of μ±σ, μ±2σ and μ±3σ, and the load uncertainty set is divided into two intervals with deviations of μ±1.5σ and μ±3σ. According to the probability of occurrence of each interval, the total interval Γ ψ Divide, can be set as: (Γ pv,1 ,Γ pv,2 ,Γ pv,3 )=(4,1,1),(Γ wt,1 ,Γ wt,2 ,Γ wt,3 )=(8,3,1) and (Γ L,1 ,Γ L,2) = (9, 3). The number and size of intervals depend on the amount of historical data available from actual projects. With more information about the dominant uncertainty, the uncertainty set can be divided into finer intervals, which not only aligns with reality but also significantly reduces the conservatism of robust optimization. Figure 7 Predicted source-load power curve for the microgrid. The shaded area indicates the range of the uncertainty set considered.

[0187] Table 3 Probability distribution parameters of wind power prediction error

[0188]

[0189] Table 4. Probability distribution parameters of photovoltaic power prediction error

[0190]

[0191] Table 5 Load power forecast error probability distribution parameters

[0192]

[0193]

[0194] Results and Analysis: The microgrid is optimized and dispatched according to the two-stage robust optimization method of the microgrid proposed in this embodiment. The interactive power distribution between the system and the main grid under the worst scenario obtained by optimization is as follows: Figure 8 As shown, the charging and discharging power of energy storage is as follows Figure 9 The demand response results are shown in Figure 10 As shown in the figure, the ESS discharges power to the microgrid, the power is positive, and the charging power is negative; the microgrid purchases power from the upper grid, the power is positive, and the power sold is negative.

[0195] Depend on Figure 8-Figure 9As can be seen, wind and photovoltaic power generation is fully absorbed, maximizing the utilization of clean energy. However, when photovoltaic power generation is zero at night, renewable energy generation cannot meet energy balance requirements. To ensure stable operation of the microgrid system, the system adopts measures such as purchasing electricity from the grid, generating power through micro gas turbines, and generating power through the ESS. During this period, the MT generation cost exceeds the grid price, so the MG primarily purchases power from the main grid, and the MT operates at minimum power output (24 hours to 7 hours the next day). During the rest of the time, the MT operates at maximum power to increase the amount of electricity sold to the distribution network, effectively increasing sales revenue, enhancing energy interaction between the MG and the grid, and reducing the MG's operating costs. The participation of the DG effectively ensures that the microgrid sells electricity to the grid at maximum power during high electricity price periods, effectively reducing operating costs. Furthermore, the ESS charges during low electricity price periods and low load demand, and discharges during high electricity price periods and high load demand, achieving peak load shifting for the grid. The participation of the ESS effectively increases the absorption of renewable power generation and reduces the operating costs of the MG, fully demonstrating the flexibility of the ESS in optimizing scheduling. The power grid plays a crucial role in ensuring the stable operation of the microgrid. MGs purchase electricity from the grid and charge the ESS during low-price periods. MGs sell electricity to the grid at maximum power during high-price periods.

[0196] After implementing demand response, electricity load demand has changed. Figure 10 This is the power load demand before and after implementing demand response. As can be seen, while meeting total power demand and power consumption constraints, the MG system shifts power load demand during high-price periods to low-price periods (1-7 hours), which helps reduce power purchase costs from the main grid. During high-price periods, the MG increases the power output of the MT units to reduce power purchase costs, further optimizing system efficiency.

[0197] Comparison with Different Robust Optimization Models: Considering the uncertainty of the source and load, a two-stage robust optimization model for MG was established. To intuitively describe the accuracy of the proposed uncertainty model, this example constructed four models for comparative experiments. The corresponding models are as follows:

[0198] Model 1 (baseline): The uncertainty caused by source and load power is ignored in day-ahead scheduling, and day-ahead forecast data is used. This model is also called a deterministic scheduling model. Model 2: The uncertainty set is captured by a normal distribution, and the corresponding mean and variance are obtained from 1000 historical sampling data. This model can be called a two-stage chance-constrained optimization model. Model 3: The uncertainty is represented by a multi-interval uncertainty set. Model 4 (proposed): The uncertainty set is described by a data-driven piecewise multi-interval uncertainty set.

[0199] Table 6 Cost comparison of different models

[0200]

[0201] The optimization results of the four models are shown in Table 6. As can be seen from Table 6, Model 1 achieves the lowest scheduling cost because it does not consider the impact of uncertainties. However, due to the influence of uncertainties during actual system operation, the resulting scheduling results are less robust. Model 3 increases the total system scheduling cost by 224.2 yuan compared to Model 2. This is because the two-stage chance-constrained optimization model, compared to the multi-interval uncertainty set, further considers the prediction of uncertain parameters, which better copes with changes in uncertainty and reduces the model's scheduling cost. Compared to Model 3, Model 4 considers the historical data distribution characteristics of the uncertainty parameters, providing a more accurate description of the uncertainty parameters and making them more realistic. As a result, the total cost decreases by 126.4 yuan. This demonstrates that the proposed model maintains the robustness of the day-ahead scheduling results while reducing the model's conservatism. Although Model 4's strategy has a higher total cost than Models 2 and 1, it ensures that the dispatchable units within the MG operate within the safe range, ensuring the safe operation of the system. Models 1 and 2, however, may cause dispatchable units to operate outside the safe operating range when uncertainties emerge, leading to system crashes. Therefore, in order to ensure a balance between economy and risk, the proposed model 4 is more suitable for actual engineering.

[0202] Comparison of Microgrid Operation Costs with Different Uncertain Budget Parameters: To analyze the impact of uncertain budget parameters on microgrid operation costs, four different uncertain budget parameters were set for comparison: 0%, 25%, 50%, and 100%. The system dispatch costs under different uncertain budget parameters are shown in Table 7.

[0203] Table 7 Influence of uncertain adjustment parameters on optimization results

[0204]

[0205]

[0206] As can be seen from Table 7, when the uncertain budget parameters are all set to 0, the robust optimization model is transformed into a deterministic optimization model, and the total system scheduling cost is the lowest. As the uncertain budget parameters continue to increase, the total system scheduling cost also continues to increase, and the scheduling plan gradually tends to be conservative. ψ,j Larger values ​​mean more periods during which the load power reaches the maximum value of the forecast interval and the PV output reaches the boundary of the forecast interval. Consequently, the uncertainty faced by the microgrid's power balance increases, leading to a more conservative MG scheduling plan. Therefore, when formulating a scheduling plan, appropriate uncertainty budget parameters can be set as needed to flexibly adjust the economic and robustness of the scheduling plan.

[0207] It should be noted that the parts not described in detail in the above embodiments are all prior art.

[0208] The above description is merely a preferred embodiment of the present invention and does not constitute any form of limitation to the present invention. Although the present invention has been disclosed as a preferred embodiment, it is not intended to limit the present invention. Any technician familiar with the present profession can make slight changes or modifications to equivalent embodiments using the technical contents disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention should be covered by the protection scope of the present invention.

Claims

1. A robust optimization scheduling method for microgrids considering segmented multi-interval uncertainty sets, characterized by: The following steps are involved: Step 1: Based on the Gaussian mixture model, the source-load power and error data of wind, solar, and load are segmented and fitted. Combined with the box-type interval uncertainty set, a multi-interval uncertainty set for power segmentation is established to describe the uncertainty of wind, solar, and load. Step 2: With the goal of minimizing the microgrid operation cost and carbon emission cost, a robust optimization scheduling model for the microgrid is constructed that considers the segmented multi-interval uncertainty set and carbon emissions. Step 3: Use the column and constraint generation algorithm to solve the microgrid robust optimization scheduling model, including the following steps: Step 31, dividing the microgrid robust optimization scheduling model into a main problem and a sub-problem; Step 32: Convert the max-min subproblem into a single-level max problem using the Lagrange duality method, and convert the dual problem of the subproblem into a mixed integer linear programming problem using the big-M method: linearize the product of binary variables and continuous variables by introducing auxiliary variables and related constraints. Step 33: Alternately iterate and solve the main and sub-problems. After the derivation and transformation in step 32, the proposed two-stage robust optimization scheduling model has been transformed into a mixed integer linear programming problem. The main problem and sub-problems are solved iteratively through the column and constraint generation algorithm until the difference between the upper bound LB and the lower bound UB is reduced to the convergence threshold ξ, and the final optimized daily stage scheduling decision is obtained. The actual values ​​of wind, solar, and load power in step 1 are represented by the sum of their power prediction values ​​and the prediction error. The power prediction error is: Where: Unit ψ includes wind turbine WT, photovoltaic PV and load Load; e ψ,t is the power prediction error of unit ψ during period t; P ψ,t is the actual power of unit ψ during period t; is the power prediction value of unit ψ during period t; The source-load power prediction value is divided into three intervals, and the power prediction error in each interval is fitted using a Gaussian mixture model; the establishment of a multi-interval uncertainty set for power segmentation in step 1 includes the following steps: Step 11: Obtain the probability distribution of the corresponding prediction error according to the source-load prediction power segmentation situation; Step 12: Combining the multi-interval partitioning of the power segmentation uncertainty set to analyze the construction method of the power segmentation error multi-interval uncertainty set; Step 13: Based on the size and probability of the prediction error, combined with the uncertain budget parameter Γ ψ The error single interval Divided into N AS intervals, among which is the power prediction value of unit ψ in power section s during time period t, and satisfies the following formula: Where: N SP is the number of segments of the power prediction value; and is the predicted upper and lower deviation values ​​of the unit ψ in the power section s error interval j during time period t; Γ ψ,j,s is the uncertain time budget parameter of unit ψ in the error interval j of power segment s, and the degree of uncertainty can be changed by adjusting its size; and is the predicted upper and lower deviation values ​​of unit ψ in a single interval during period t; Γ ψ is the single interval uncertain time budget parameter; ρ ψ,j is the probability of occurrence of unit ψ in interval j, and its value can be obtained according to the fitting result of the corresponding probability distribution, ρ ψ is the probability of occurrence of unit ψ in a single interval; Γ ψ,j is the uncertain time budget parameter of unit ψ in error interval j; Step 14: Based on the constructed data, the segmented multi-interval source-load uncertainty set is driven. The wind, solar, and load uncertainties in the microgrid MG are shown as follows: Where: P ψ,t,s is the actual power of unit ψ in power section s during period t; U ψ The piecewise multi-interval uncertainty set representing the unit ψ; is a Boolean variable, indicating whether the predicted deviation variable is located in the upper or lower half of the jth interval; and are the predicted upper and lower deviation values ​​of unit ψ in time period t; T represents the total number of scheduling time periods.

2. A robust optimization scheduling method for microgrids considering segmented multi-interval uncertainty sets according to claim 1, characterized in that: The microgrid includes traditional distributed power sources, renewable distributed power sources, an energy storage system and loads.

3. The robust optimization scheduling method for microgrids considering segmented multi-interval uncertainty sets according to claim 2 is characterized by: In step 2, a two-stage robust optimization scheduling model is established with the goal of minimizing the microgrid's operating cost and environmental cost. The constraints that need to be met include power generation constraints, energy storage system constraints, demand response load constraints, and interactive power constraints with the grid. The objective function is as follows: Where: C MT (t) is the power generation cost of the MT machine in period t; C j (t) is the operation and maintenance cost of WT and PV in period t; C ESS (t) is the aging cost of ESS in period t; C DR (t) is the implementation cost of DR in period t; C G (t) is the cost of electricity exchange with the main grid; C emi (t) is the pollutant treatment cost of the microgrid in period t.

4. The robust optimization scheduling method for microgrids considering segmented multi-interval uncertainty sets according to claim 3 is characterized by: The robust optimization scheduling model of the microgrid considering the segmented multi-interval uncertainty set and carbon emissions in step 2 is as follows: Corresponding to power generation constraints, energy storage system constraints, demand response load constraints, and interactive power constraints with the power grid, the inner layer 'max' is to find the worst scenario with concentrated uncertainty, and 'min' is to minimize the microgrid operating cost under the worst scenario; the outer layer 'min' is to optimize the day-ahead scheduling decision by the corresponding scheduling power under the worst scenario; c represents the coefficient column vector corresponding to the objective function (28); B and I represent the equality constraint coefficient matrix of the day-ahead deterministic scenario; e represents the constant column vector of the equality constraint of the day-ahead deterministic scenario; A, C, D, and F represent the inequality constraint coefficient moments of the day-ahead deterministic scenario; d, f, and h represent the constant column vector of the inequality constraint of the day-ahead deterministic scheduling scheme; U ψ The piecewise multi-interval uncertainty set representing the unit ψ; x and y represent the optimization variables of the day-ahead scenario, and their specific expressions are as follows: Where: O buy (t) represents the state of MG purchasing electricity from the grid during period t, O sell (t) represents the state of MG selling electricity to the grid during period t; up (t) represents the upward adjustment state of the power load in the t period, O down (t) represents the downward adjustment state of the power load in the t period; ch (t) represents the charging status of ESS in time period t, "1" represents charging, "0" represents not charging, dis (t) represents the discharge state of ESS in time period t, "1" represents discharge, and "0" represents no discharge; P MT (t) is the output power of MT during period t; P DR (t) is the actual dispatch power of DR in period t, that is, the amount of power load transferred; P buy (t) and P sell (t) are the power purchased and sold from the grid during period t; P ch (t) and P dis (t) are the charge and discharge power of ESS during period t.

5. The robust optimization scheduling method for microgrids considering segmented multi-interval uncertainty sets according to claim 4 is characterized by: The main problem in step 31 is written as follows: Where: k is the number of iterations; U w is the set of all worst cases searched; for U w For each worst case us in the , generate the corresponding hourly phase variable y k and its constraints, select the mean u of the uncertainty range meean As the initial worst scenario of uncertainty u0*; The sub-problem is expressed as:

6. The robust optimization scheduling method for microgrids considering segmented multi-interval uncertainty sets according to claim 5 is characterized by: In step 32, the subproblem max-min is transformed into a single-layer max problem by using the strong Lagrangian duality method. Ω(x,u) represents the feasible region of the optimization variable y when a set (x,u) is given. The specific expression is as follows: Where: σ, ω, τ, θ and ρ are the dual variables corresponding to the inner constraints respectively; When (x,u) is given, the inner layer of the subproblem is a linear problem. The dual problem of the subproblem can be obtained according to the corresponding relationship of the following formula:

7. The robust optimization scheduling method for microgrids considering segmented multi-interval uncertainty sets according to claim 6 is characterized by: In step 32, the dual problem of the subproblem is transformed into a mixed integer linear programming problem using the big M method: Where: Δu j+ , Δu j- is the power prediction deviation; is the introduced continuous auxiliary variable; π j+ , π j- is a binary variable; M is the upper bound of the dual variable, which is a sufficiently large positive integer.

8. The robust optimization scheduling method for microgrids considering segmented multi-interval uncertainty sets according to claim 7 is characterized by: The alternating iterative solution between the main and sub-problems in step 33 includes the following steps: Step 331: Given a set of uncertain variable values ​​u meean , as the initial worst scenario, the upper bound UB = +∞, the lower bound LB = -∞, the number of iterations k = 1, and the convergence gap is set to ξ = 0.01; Step 332: u meean Substitute into the main problem and get the optimal solution of the problem (x k ,y k ,λ k ), update the objective function value of the main problem as the new lower bound LB = λ k ; Step 333: x k Substitute into the subproblem and get the optimal solution f of the subproblem k (x k ) and the corresponding worst scenario u k+1 , update the sub-problem objective function value as the new upper bound UB=min{UB,f k (x k )}; Step 334: If UB-LB≤0.01, stop the iteration and return to the optimal solution (x k ,y k ), otherwise, increase the variable y k+l and the following constraints: Let k=k+1, and return to step 332 until the algorithm converges.

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