Distributionally robust optimization method based on Wasserstein metric and kernel density estimation

Abstract

The present invention proposes a distributionally robust optimization method based on Wasserstein metric and kernel density estimation. First, the probability distribution of wind power prediction error is derived by kernel density estimation method. Then, the probability distribution fuzzy set of wind power prediction error is constructed by integrating Wasserstein metric and kernel density estimation. Finally, according to the probability distribution fuzzy set of wind power prediction error and comprehensive consideration of power system operation constraints, a distributionally robust unit combination model is established with the goal of minimizing the total operating cost of the power system under the worst wind power output. The present invention solves the unit combination optimization problem considering unstable wind power generation in the field of power dispatching, can effectively reduce the conservatism of wind power uncertainty characterization, further improve the economy of power system operation, and provide dispatchers of power systems with a dispatching strategy that can effectively handle wind power uncertainty and takes both robustness and economy into account.

Description

Technical Field

[0001] The present invention relates to the technical field of power system dispatching operation and optimization, and in particular to a distribution robust optimization method based on Wasserstein metric and kernel density estimation. Background Art

[0002] With the rapid increase in the proportion of wind power in the power grid, its uncertainty seriously threatens the safe operation of the power system. Therefore, it is very necessary to explore and study uncertain optimization methods for the problem of unit combination in power systems. In the past few decades, scholars have extensively studied stochastic optimization methods. This method assumes that the wind power or wind power forecast error obeys a certain distribution, and uses the scenario method or chance constraint method to solve the model. The defect of stochastic optimization is that it is often too radical, and the probability of the obtained results being infeasible is high. Robust optimization is another effective method to deal with uncertainty. This method predetermines a set of all possible situations of wind power output, and then solves the decision in the worst case. The defect of robust optimization is that it is too conservative, resulting in poor economic efficiency of the decision.

[0003] In view of the above-mentioned defects of stochastic optimization and robust optimization, scholars have proposed a distributionally robust optimization method in recent years. This method constructs a fuzzy set containing all possible probability distributions of wind power output, and then solves the decision under the worst probability distribution. Distributionally robust optimization takes into account the respective advantages of stochastic optimization and robust optimization, and avoids overly radical and conservative decisions. However, the general distributionally robust optimization method based on Wasserstein often uses empirical distribution for its central distribution, and the construction of the distribution is not accurate enough, resulting in the fuzzy set of probability distribution being still relatively conservative. In order to further reduce the conservatism and improve the economy of decision-making, the present invention proposes a distributionally robust optimization method based on Wasserstein metric and kernel density estimation. Summary of the invention

[0004] In view of the technical problem that the existing technology handles wind power uncertainty conservatively, resulting in poor economic efficiency of decision-making, the present invention proposes a distributionally robust optimization method based on Wasserstein metric and kernel density estimation, which is used to solve the unit combination optimization problem considering unstable wind power generation in the field of power dispatching, further reduce conservatism and improve the economic efficiency of decision-making.

[0005] In order to solve the above technical problems, the present invention adopts the following technical solution: a distribution robust optimization method based on Wasserstein metric and kernel density estimation, comprising the following steps:

[0006] Step S1: First, the probability density of the wind power prediction error is derived using the kernel density estimation method, and the probability distribution of the wind power prediction error is derived based on the probability density.

[0007] Step S2: constructing a confidence space of wind power prediction error with a confidence level according to the probability distribution of wind power prediction error.

[0008] Step S3: In the confidence space of wind power prediction error, with the probability distribution of wind power prediction error as the center, the Wasserstein distance is used to measure the distance between the probability distribution and the error distribution in the confidence space, and a fuzzy set of the probability distribution of wind power prediction error is constructed.

[0009] Step S4: Based on the probability distribution fuzzy set of wind power prediction error and taking into account the power system operation constraints, a distributed robust unit combination model is established with the goal of minimizing the total power system operation cost under the worst wind power output condition.

[0010] In step S1, the function expression of the probability density of the wind power prediction error is:

[0011]

[0012] Where: x represents the wind power prediction error; n represents the number of samples of wind power prediction error; x1, x2, …, x l ,…x n is the n samples of wind power prediction errors derived from the power grid database; R d represents the real number space of prediction errors; K h (·) represents the Gaussian kernel function with bandwidth parameter h.

[0013] Calculate the optimal value of the bandwidth parameter h of the Gaussian kernel function:

[0014]

[0015]

[0016]

[0017] Where: BW(h) represents the cross-validation function for calculating the bandwidth parameter h; The bandwidth parameter is The four-dimensional Gaussian kernel function of ; is the optimal value of the bandwidth parameter h, that is, the value of h that minimizes BW(h).

[0018] The probability distribution of wind power prediction error is derived according to the probability density function f(x), and the function expression is:

[0019]

[0020] The method for constructing the confidence space of the wind power prediction error in step S2 is: first constructing the α quantile function of the wind power prediction error according to the derived probability distribution function F(x) of the wind power prediction error:

[0021] F -1 (α)=min{x∈R|F(x)≥α} (6)

[0022] Then construct the confidence space of wind power prediction error with confidence level (1-2α):

[0023]

[0024] Where: F -1 (α) represents the α quantile function of F(x); R is the real number space; S is the confidence space; x min and x max are the lower and upper limits of the wind power prediction error x, respectively.

[0025] In step S3, the Wasserstein distance between the probability distribution of the wind power prediction error and all error distributions in its confidence space is:

[0026]

[0027] Where: F represents the probability distribution of wind power prediction error; represents the true distribution of wind power forecast error; x and They belong to the probability distribution F and the true distribution respectively Π refers to x and The joint distribution of

[0028] The probability distribution fuzzy set is:

[0029]

[0030] Where: represents the set of all error distributions on the confidence space S; r represents the radius of the probability distribution fuzzy set;

[0031] The radius r of the probability distribution fuzzy set is calculated as follows: First, the probability β of any distribution falling into the probability distribution fuzzy set is:

[0032]

[0033] The relationship between the probability β and the sample size n and the radius r of the probability distribution fuzzy set is:

[0034] β=1-exp(-nr 2 / c 2 ) (11)

[0035] Where: c is an auxiliary constant;

[0036] Therefore, the radius r of the probability distribution fuzzy set can be obtained as:

[0037] r(n)=c(ln((1-β) -1 ) / n) 0.5 (12)

[0038] The auxiliary constant c is:

[0039]

[0040] Where: λ is a real number greater than zero; is the sample mean of wind power prediction error.

[0041] The objective function of the distributed robust unit combination model in step S4 is:

[0042]

[0043] Where: SU i , SD i are the startup and shutdown costs of thermal power unit i; u it 、v it are 0-1 variables of the start-up state and shutdown state of thermal power unit i; E F is a probability distribution fuzzy set The expected cost under the corresponding distribution in ;

[0044] The expected cost E F for:

[0045]

[0046] Where: CU i , CD i are the upper reserve cost and lower reserve cost of thermal power unit i respectively; r it are the upper and lower reserves of thermal power unit i; a i , b i 、c i are the coefficients of the cost function of thermal power generation units; P it is the base power of thermal power unit i in period t; α it is the participation factor of thermal power unit i in dealing with wind power fluctuation power in period t; t It is the sum of wind power forecast errors in period t.

[0047] The power system operation constraints include:

[0048] ① Minimum start and stop time constraints of thermal power unit i:

[0049]

[0050] Where: TU i , TD i are the minimum operating time and minimum downtime of thermal power unit i; d it represents the 0-1 state quantity of thermal power unit i in time period t; k represents the kth time period; GG represents the number of thermal power units; T represents the total number of time periods for the thermal power units to operate;

[0051] ② Constraints on the start and stop status of thermal power unit i:

[0052]

[0053] ③Power system operation power balance constraints:

[0054]

[0055] Where: is the predicted power of wind power plant j in period t; is the prediction error of wind power plant j in period t; is the load of node b in period t; W and B are the number of wind power plants and nodes respectively;

[0056] ④ Base power constraint of thermal power unit i:

[0057]

[0058] Where: P i , are the minimum and maximum output of thermal power unit i respectively;

[0059] ⑤Climbing constraint of thermal power unit i:

[0060]

[0061] Where: is the ramp-up cost of thermal power unit i from shutdown to startup; RU i is the ramp cost during the continuous operation of thermal power unit i; is the landslide cost of thermal power unit i from startup to shutdown; RD i is the landslide cost during the continuous operation of thermal power unit i;

[0062] ⑥ Participation factor constraints of thermal power unit i:

[0063]

[0064] ⑦ Thermal power unit i bears the constraints of wind power fluctuation:

[0065]

[0066] Among them, the total wind power prediction error x t for:

[0067]

[0068] ⑧ Line transmission capacity constraints:

[0069]

[0070] Where: C mn is the transmission power limit from node m to node n; is the power transfer distribution factor from node m to node n caused by the net power of node b; G b are all thermal power units on node b; W b For all wind farms on node b.

[0071] The objective function is reconstructed into a single minimization problem. The reconstruction method is: first, the expected cost E of the max part of the objective function is F Converted to the sum of wind power prediction errors x t The function f(x t ),get:

[0072]

[0073] Where: is the function f(x t ), and:

[0074]

[0075] Thus, the expected cost E in the objective function is obtained F for:

[0076]

[0077] Where: θ is a real number greater than zero; f′(·) is the function f(x t )About x t The derivative of Therefore

[0078] Then replace formula (27) with formula (14) That is, the reconstructed objective function is obtained.

[0079] The present invention firstly uses the kernel density estimation method to derive the probability distribution of wind power prediction error, then integrates the Wasserstein metric and kernel density estimation to construct the probability distribution fuzzy set of wind power prediction error, and finally, according to the probability distribution fuzzy set of wind power prediction error and taking into account the operation constraints of the power system, a distributed robust unit combination model is established with the goal of minimizing the total operating cost of the power system under the worst wind power output condition, so as to solve the unit combination optimization problem considering unstable wind power generation in the field of power dispatching, effectively reduce the conservatism of wind power uncertainty characterization, further improve the economy of power system operation, and provide the dispatchers of the power system with a dispatching strategy that can effectively deal with wind power uncertainty and takes into account both robustness and economy. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0081] Figure 1 is a flow chart of the present invention;

[0082] Figure 2 The figure is a comparison diagram of the wind power prediction error boundary between the present invention and the general distributed robust optimization method, wherein (a) is the test result of the 57-bus power system, and (b) is the test result of the 118-bus power system;

[0083] Figure 3 It is a comparison chart of the total operating cost between the present invention and the general distributed robust optimization method, random optimization method and robust optimization method, wherein (c) is the test result of a 57-bus power system; (d) is the test result of a 118-bus power system. DETAILED DESCRIPTION

[0084] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0085] like Figure 1As shown, the present invention provides a distributionally robust optimization method based on Wasserstein metric and kernel density estimation. The probability distribution of wind power prediction error is derived by adopting the kernel density estimation method, and then the Wasserstein metric and kernel density estimation are integrated to construct the probability distribution fuzzy set of wind power prediction error. Finally, according to the probability distribution fuzzy set of wind power prediction error and taking into account the operation constraints of the power system, a distributionally robust unit combination model is established with the goal of minimizing the total operating cost of the power system under the worst wind power output. This not only solves the unit combination optimization problem considering unstable wind power generation in the field of power dispatching, but also effectively reduces the conservatism of wind power uncertainty characterization, further improves the economy of power system operation, and provides dispatchers of the power system with a dispatching strategy that can effectively deal with wind power uncertainty and takes into account both robustness and economy.

[0086] The present invention specifically comprises the following steps:

[0087] Step S1: First, the uncertainty of wind power prediction error is processed, and the probability density of wind power prediction error is derived by using the kernel density estimation method, and the function expression of the probability density of wind power prediction error is obtained as follows:

[0088]

[0089] Where: x represents the wind power prediction error; n represents the number of samples of wind power prediction error; x1, x2, …, x l ,…x n is the n samples of wind power prediction errors derived from the power grid database; R d represents the real number space of prediction errors; K h (·) represents the Gaussian kernel function with bandwidth parameter h.

[0090] The bandwidth parameter h has a significant impact on the probability density estimation effect. If its value is too large, the estimated probability density curve will be too smooth, and if its value is too small, the estimated probability density curve will be too volatile. In both cases, the estimated probability density curve will differ too much from the true probability density. In order to make the derivation of the probability distribution of wind power forecast error more accurate, the optimal value of the bandwidth parameter h of the Gaussian kernel function in the probability density function f(x) is calculated. The calculation method is:

[0091]

[0092]

[0093]

[0094] Where: BW(h) represents the cross-validation function for calculating the bandwidth parameter h; The bandwidth parameter is The four-dimensional Gaussian kernel function of ; is the optimal value of the bandwidth parameter h, that is, the value of h that minimizes BW(h). Substituting the calculated optimal value of the bandwidth parameter h into the probability density function f(x) can achieve the effect of accurately estimating the probability density function.

[0095] Then, the probability distribution of wind power prediction error is derived according to the probability density function f(x), and the function expression of the probability distribution is obtained as follows:

[0096]

[0097] Step S2: construct a confidence space of wind power prediction error with a certain confidence level according to the probability distribution F(x) of wind power prediction error. The specific construction method is as follows: first, according to the derived probability distribution function F(x) of wind power prediction error, construct the α quantile function of wind power prediction error:

[0098] F -1 (α)=min{x∈R|F(x)≥α} (6)

[0099] Then construct the confidence space of wind power prediction error with confidence level (1-2α):

[0100]

[0101] Where: F -1 (α) represents the α quantile function of F(x); R is the real number space; S is the confidence space; x min and x max are the lower and upper limits of the wind power prediction error x, respectively.

[0102] Step S3: In the obtained confidence space, with the probability distribution of wind power prediction error as the center, the Wasserstein distance is used to measure the distance between the probability distribution and all possible distributions of errors in the confidence space, and the result is:

[0103]

[0104] Where: F represents the probability distribution of wind power prediction error; represents the true distribution of wind power forecast error; x and They belong to the probability distribution F and the true distribution respectively Π refers to x and The joint distribution of .

[0105] All error distributions within a certain distance r constitute the probability distribution fuzzy set of wind power prediction error, and the probability distribution fuzzy set is obtained as follows:

[0106]

[0107] Where: represents the set of all error distributions on the confidence space S; r represents the radius of the probability distribution fuzzy set, which is related to the number of samples n;

[0108] The probability distribution fuzzy set The radius r is calculated as follows: First, the probability β of any distribution falling into the fuzzy set of probability distribution is:

[0109]

[0110] The relationship between the probability β and the sample size n and the radius r of the probability distribution fuzzy set is:

[0111] β=1-exp(-nr 2 / c 2 ) (11)

[0112] Where: c is an auxiliary constant.

[0113] Therefore, we can get the probability distribution fuzzy set The radius r is:

[0114] r(n)=c(ln((1-β) -1 ) / n) 0.5 (12)

[0115] The auxiliary constant c is:

[0116]

[0117] Where: λ is a real number greater than zero; is the sample mean of wind power prediction error.

[0118] Step S4: Based on the probability distribution fuzzy set of wind power prediction error The power system operation constraints are comprehensively considered, including the power balance constraints of the power system operation, the line capacity constraints to ensure power transmission, the power constraints of thermal power units, the ramp restrictions and the start-stop time restrictions, and other constraints. A distributed Bulu rod unit combination model is established with the goal of minimizing the total system operation cost under the worst wind power output. Its objective function is:

[0119]

[0120] Where: SU i , SD i are the startup and shutdown costs of thermal power unit i; u it 、v itare 0-1 variables of the start-up state and shutdown state of thermal power unit i; E F is a probability distribution fuzzy set The expected cost under the corresponding distribution in .

[0121] The expected cost E F for:

[0122]

[0123] Where: CU i , CD i are the upper reserve cost and lower reserve cost of thermal power unit i respectively; r it are the upper and lower reserves of thermal power unit i; a i , b i 、c i are the coefficients of the cost function of thermal power generation units; P it is the base power of thermal power unit i in period t; α it is the participation factor of thermal power unit i in dealing with wind power fluctuation power in period t; t It is the sum of wind power forecast errors in period t.

[0124] The power system operation constraints include:

[0125] ① Minimum start and stop time constraints of thermal power unit i:

[0126]

[0127] Where: TU i , TD i are the minimum operating time and minimum downtime of thermal power unit i; d it represents the 0-1 state quantity of thermal power unit i in time period t; k represents the kth time period; G represents the number of thermal power units; T represents the total number of time periods for the thermal power units to operate;

[0128] ② Constraints on the start and stop status of thermal power unit i:

[0129]

[0130] ③Power system operation power balance constraints:

[0131]

[0132] Where: is the predicted power of wind power plant j in period t; is the prediction error of wind power plant j in period t; is the load of node b in period t; W and B are the number of wind power plants and nodes respectively;

[0133] ④ Base power constraint of thermal power unit i:

[0134]

[0135] Where: P i , are the minimum and maximum output of thermal power unit i respectively;

[0136] ⑤Climbing constraint of thermal power unit i:

[0137]

[0138] Where: is the ramp-up cost of thermal power unit i from shutdown to startup; RU i is the ramp cost during the continuous operation of thermal power unit i; is the landslide cost of thermal power unit i from startup to shutdown; RD i is the landslide cost during the continuous operation of thermal power unit i;

[0139] ⑥ Participation factor constraints of thermal power unit i:

[0140]

[0141] ⑦ Thermal power unit i bears the constraints of wind power fluctuation:

[0142]

[0143] Among them, the total wind power prediction error x t for:

[0144]

[0145] ⑧ Line transmission capacity constraints:

[0146]

[0147] Where: C mn is the transmission power limit from node m to node n; is the power transfer distribution factor from node m to node n caused by the net power of node b; G b are all thermal power units on node b; W b For all wind farms on node b.

[0148] Since the objective function of the constructed distributed and robust unit combination model is in the form of min-max, which is not easy to solve, it is necessary to reconstruct the objective function into a single minimization problem for solution. The specific reconstruction method is: first, the expected cost E of the max part of the objective function is F Converted to the sum of wind power prediction errors x t The function f(x t ),get:

[0149]

[0150] Where: is the function f(x t ), and:

[0151]

[0152] Thus, the expected cost E in the objective function is obtained F for:

[0153]

[0154] Where: θ is a real number greater than zero; f′(·) is the function f(x t )About x t The derivative of Therefore

[0155] Then replace formula (27) with formula (14) That is, the reconstructed objective function is obtained.

[0156] The present invention uses a 57 bus power system and a 118 bus power system for testing, and the generator and power network data are derived from MATPOWER 6.0 software. The rated power of the wind power plant is 400MW, of which the 57 bus power system is connected to two wind power plants, and the 118 bus power system is connected to four wind power plants. The wind power data comes from TENNET (website https: / / www.tennet.eu / ?L=0#&panel1-1). The system standby price, unit startup and shutdown costs are taken as 50%, 80% and 40% of the linear coefficient in the cost curve, respectively. The test was carried out on a computer with a 2.9GHz, 8GB RAM Intel Core i7 processor.

[0157] The test results are as follows Figure 2As shown in (a) and (b), it can be seen that the prediction error boundary of the general distributionally robust optimization method gradually decreases with the increase of the sample size, while the prediction error boundary of the distributionally robust optimization method in the present invention is relatively small when the sample size is not large. As the sample size increases, the change in the boundary has only small fluctuations, which greatly reduces the conservatism of the model.

[0158] Tables 1 and 2 give the test results of the calculation time of the 57-bus power system and the 118-bus power system respectively:

[0159] Table 1

[0160]

[0161] Table 2

[0162]

[0163] It can be seen from Table 1 and Table 2 that as the sample size increases, the calculation time of robust optimization and random optimization is basically fixed, and the calculation time of the general distributional robust optimization method and the distributional robust optimization method in the present invention gradually increases with the increase of samples. The superiority of the distributional robust optimization method in the present invention is that the calculation time is almost the same as that of the general distributional robust optimization method, but the result is better. Figure 2 (a), (b) and Figure 3 The comparison results of (c) and (d) show that although the computation time of the distributed robust optimization method in the present invention is longer than that of general robust optimization and random optimization, it avoids the conservatism of robust optimization and the radicalism of random optimization.

[0164] Table 3 shows the reliability test results of the 57-bus power system and the 118-bus power system, that is, the probability that the obtained unit combination results can meet the constraint conditions under random wind power errors. Monte Carlo simulation is used to obtain the following results:

[0165] Table 3

[0166]

[0167] It can be seen from Table 3 that with the increase of sample size, the reliability of the general distribution robust optimization method gradually decreases, while the reliability of the distribution robust optimization method of the present invention fluctuates to a certain extent with the increase of samples, but still maintains a relatively high reliability level (above 98%).

[0168] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A distributed robust optimization method based on Wasserstein metric and kernel density estimation, characterized in that: The following steps are involved: Step S1: firstly, the probability density of the wind power prediction error is derived by using the kernel density estimation method, and the probability distribution of the wind power prediction error is derived according to the probability density; In step S1, the function expression of the probability density of the wind power prediction error is: Where: x represents the wind power prediction error; n represents the number of samples of wind power prediction error; x1, x2, …, x l ,…x n is the n samples of wind power prediction errors derived from the power grid database; R d represents the real number space of prediction errors; K h (·) represents the Gaussian kernel function with bandwidth parameter h; Calculate the optimal value of the bandwidth parameter h of the Gaussian kernel function: Where: BW(h) represents the cross-validation function for calculating the bandwidth parameter h; The bandwidth parameter is The four-dimensional Gaussian kernel function of is the optimal value of the bandwidth parameter h, that is, the value of h that minimizes the cross-validation function BW(h); Step S2: constructing a confidence space of wind power prediction error with a confidence level according to the probability distribution of wind power prediction error; Step S3: In the confidence space of wind power prediction error, with the probability distribution of wind power prediction error as the center, the Wasserstein distance is used to measure the distance between the probability distribution and the error distribution in the confidence space, and a fuzzy set of the probability distribution of wind power prediction error is constructed; In step S3, the Wasserstein distance between the probability distribution of wind power prediction error and all error distributions in its confidence space is: Where: F represents the probability distribution of wind power prediction error; represents the true distribution of wind power forecast error; x and They belong to the probability distribution F and the true distribution respectively Π refers to x and The joint distribution of The probability distribution fuzzy set is: Where: represents the set of all error distributions on the confidence space S; r represents the radius of the probability distribution fuzzy set; The radius r of the probability distribution fuzzy set is calculated as follows: First, the probability β of any distribution falling into the probability distribution fuzzy set is: The relationship between the probability β and the sample size n and the radius r of the probability distribution fuzzy set is: β=1-exp(-nr 2 / c 2 ) (8) Where: c is an auxiliary constant; Therefore, the radius r of the probability distribution fuzzy set is: The auxiliary constant c is: Where: λ is a real number greater than zero; is the sample mean of wind power prediction error; Step S4: Based on the probability distribution fuzzy set of wind power prediction error and taking into account the power system operation constraints, a distributed robust unit combination model is established with the goal of minimizing the total power system operation cost under the worst wind power output condition.

2. The distributed robust optimization method based on Wasserstein metric and kernel density estimation according to claim 1, characterized in that: The probability distribution of wind power prediction error is derived according to the probability density function f(x), and the function expression is:

3. The distributed robust optimization method based on Wasserstein metric and kernel density estimation according to claim 2, characterized in that: The method for constructing the confidence space of the wind power prediction error in step S2 is: firstly constructing the α quantile function of the wind power prediction error according to the derived probability distribution function F(x) of the wind power prediction error: F -1 (α)=min{x∈R|F(x)≥α} (12) Then construct the confidence space of wind power prediction error with confidence level (1-2α): Where: F -1 (α) represents the α quantile function of the probability distribution function F(x); R is the real number space; S is the confidence space; x min and x max are the lower and upper limits of the wind power prediction error x, respectively.

4. The distributed robust optimization method based on Wasserstein metric and kernel density estimation according to any one of claims 1 to 3, characterized in that: The objective function of the distributed robust unit combination model in step S4 is: Where: SU i , SD i are the startup and shutdown costs of thermal power unit i; u it 、v it are 0-1 variables of the start-up state and shutdown state of thermal power unit i; E F is a probability distribution fuzzy set The expected cost under the corresponding distribution in ; The expected cost E F for: Where: CU i , CD i are the upper reserve cost and lower reserve cost of thermal power unit i respectively; r it are the upper and lower reserves of thermal power unit i; a i 、b i 、c i are the coefficients of the cost function of thermal power generation units; P it is the base power of thermal power unit i in period t; α it is the participation factor of thermal power unit i in dealing with wind power fluctuation power in period t; t It is the sum of wind power forecast errors in period t.

5. The distribution robust optimization method based on Wasserstein metric and kernel density estimation according to claim 4, characterized in that: The power system operation constraints include: ① Minimum start and stop time constraints of thermal power unit i: Where: TU i , TD i are the minimum operating time and minimum downtime of thermal power unit i; d it represents the 0-1 state quantity of thermal power unit i in time period t; k represents the kth time period; G represents the number of thermal power units; T represents the total number of time periods for the thermal power units to operate; ② Constraints on the start and stop status of thermal power unit i: ③Power system operation power balance constraints: Where: is the predicted power of wind power plant j in period t; is the prediction error of wind power plant j in period t; is the load of node b in period t; W and B are the number of wind power plants and nodes respectively; ④ Base power constraint of thermal power unit i: Where: P i , are the minimum and maximum output of thermal power unit i respectively; ⑤Climbing constraint of thermal power unit i: Where: is the ramp-up cost of thermal power unit i from shutdown to startup; RU i is the ramp cost during the continuous operation of thermal power unit i; is the landslide cost of thermal power unit i from startup to shutdown; RD i is the landslide cost during the continuous operation of thermal power unit i; ⑥ Participation factor constraints of thermal power unit i: ⑦ Thermal power unit i bears the constraints of wind power fluctuation: Among them, the total wind power prediction error x t for: ⑧ Line transmission capacity constraints: Where: C mn is the transmission power limit from node m to node n; is the power transfer distribution factor from node m to node n caused by the net power of node b; G b are all thermal power units on node b; W b For all wind farms on node b.

6. The distribution robust optimization method based on Wasserstein metric and kernel density estimation according to claim 5, characterized in that: The objective function is reconstructed into a single minimization problem. The reconstruction method is: first, the expected cost E of the max part of the objective function is F Converted to the sum of wind power prediction errors x t The function f(x t ),get: Where: is the function f(x t ), and: Thus, the expected cost E in the objective function is obtained F for: Where: is a real number greater than zero; f′(·) is the function f(x t )About x t The derivative of Therefore Then replace formula (27) with formula (14) Get the reconstructed objective function.

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