Generalized Two-Stage Mixed Integer Programming Solving Method and Transmission Network Planning Method

Abstract

The present invention proposes a method for solving generalized two-stage mixed-integer programming. By constructing a master problem and solving it to obtain a lower bound, obtaining the optimal value of the second stage and adding the value of the first stage to obtain an upper bound. When the upper bound and the lower bound do not converge to a preset threshold, adding a proportional cut generated by the row to the master problem, adding a primal cut to the master problem using column generation, and maintaining the master problem, and continuously iterating until the upper bound and the lower bound converge to the preset threshold, then returning the optimal solution of the first stage and the upper bound UB, and the algorithm ends. The present invention can accurately and quickly solve the two-stage mixed-integer programming problem. Specifically, when applied to power transmission network planning, it can accurately and efficiently solve the power transmission system planning problem.

Description

Technical Field

[0001] The present invention relates to the technical field of operations research, and particularly to a method for solving a generalized two-stage mixed integer programming and a method for transmission network planning. Background Art

[0002] In actual production and life, there are many practical problems. For example, optimization problems in medical care, energy, logistics, etc. all involve uncertainty and integer decision variables. Then the modeling method for dealing with such optimization problems is the generalized two-stage mixed integer recourse model. The first stage corresponds to long-term planning or investment, while the second stage often corresponds to short-term decision-making or real-time operating costs.

[0003] So far, scientific researchers have conducted a large number of studies on two-stage mixed integer programming problems, but there are still some problems. For example: low solution efficiency, and only some special cases can be solved, etc. Such drawbacks make the existing algorithms unable to solve many practical problems, such as transmission network planning problems. Therefore, there is an urgent need for new algorithms to solve this problem. Summary of the Invention

[0004] The purpose of the present invention is to at least solve one of the deficiencies of the prior art, and provide a method for solving a generalized two-stage mixed integer programming and a method for transmission network planning.

[0005] To achieve the above purpose, the present invention adopts the following technical solutions.

[0006] Specifically, a method for solving a generalized two-stage mixed integer programming is proposed, including the following:

[0007] Adding the variables representing the second stage to the first stage to construct a master problem, and solving the master problem to obtain a lower bound LB;

[0008] Solving the second-stage problem to obtain the objective function value of the second stage, and adding it to the objective function value of the first stage to obtain an upper bound UB;

[0009] Judging whether the upper bound UB and the lower bound LB converge to a preset threshold epsilon. If so, return the objective function value of the second stage and the upper bound UB. If not, continuously iterate until the upper bound UB and the lower bound LB converge.

[0010] Furthermore, specifically, the continuous iteration includes the following:

[0011] Obtaining a proportional cut by the row generation algorithm and adding it to the master problem, obtaining a primal cut by the column generation algorithm and adding it to the master problem. Under the above conditions, solving the master problem to obtain a lower bound LB, solving the second stage, obtaining the objective function value of the second stage, and adding it to the objective function value of the first stage to obtain an upper bound UB, and continuously iterating in this way.

[0012] Furthermore, specifically, the proportional cut obtained through the row generation algorithm includes the following:

[0013] Obtained by solving the inner optimization problem. Take the value of the variable obtained by solving the master problem as the initial value, substitute the initial value into the solution of CGMP to obtain α ω , β ω , τ ω . Substitute the values of α ω , β ω , τ ω into CGSP for solution, and check whether the objective function value of CGSP meets the termination condition. If not, it is necessary to add x, θ, y obtained by solving CGSP ω to generate new constraints and add them to CGMP, and enter the next iteration of the inner optimization problem until the objective function value of CGSP meets the termination condition. This scenario solution ends, and return the values of (α ω , β ω , τ ω ). Solve (α ω , β ω , τ ω ) of the next scenario until all scenarios are solved, and generate constraints.

[0014] Furthermore, specifically, the original cut obtained through the column generation algorithm includes the following:

[0015] Find the scenario with the largest weighted objective function value in the second stage, or the scenario with the largest objective function value in the second stage, or the scenario with the largest weight in the second stage. If it is the first iteration, then multiply the objective function of the obtained scenario by the coefficient d and add it to the master problem, and add the constraint conditions of the scenario to the master problem constraint conditions. If it is not the first iteration, then multiply the objective function of the obtained scenario by the coefficient d to replace the scenario added to the master problem objective function in the previous iteration, and replace the constraint conditions of the scenario with the constraint conditions added to the master problem in the previous iteration.

[0016] The present invention also proposes a transmission network planning method, which applies the generalized two-stage mixed integer programming solution method, including:

[0017] Input relevant data in the power system and establish a transmission system planning model.

[0018] Objective function:

[0019]

[0020] Among them, c1 and c0 are the coefficients of the real-time operating cost of the generator set, a represents the planning stage, T is the number of overall planning stages. Here, let T = 2, p g,ais the power generation of the generator in stage a, c k is the investment cost of newly built lines, z ij，a is a binary variable in stage a used to determine whether to build a new line;

[0021] The constraint conditions include:

[0022] Power balance constraint

[0023]

[0024] p g,a represents the power generation of the generator at the node in stage a, p d,i,a represents the connected load at the node in stage a, p ij,a represents the line power in stage a, represents the node, represents the generator sets connected to the system

[0025] Node power angle constraint

[0026]

[0027] Among them, θ i，a represents the power angle of the node in stage a,

[0028] Generator set output constraint

[0029] 0 ≤ p g,a ≤ p g,max #(22)

[0030] Among them, p g,a represents the generator output in stage a, p g,max represents the maximum output of the generator set.

[0031] Line power constraint

[0032]

[0033] Among them, ε represents the line set, p ij,min represents the minimum line power, p ij,max represents the maximum line power,

[0034] Power angle relationship of power flow

[0035] -(1 - z ij，a )·M ≤ (P ij，a + B ij,a θ ij，a ) ≤ (1 - z ij，a )·M #(24)

[0036] Among them, M is the separation factor, B ij,a is the susceptance in the circuit system in stage a;

[0037] The coupling constraint means that the transmission lines built in the second phase should be built on the basis of the first phase.

[0038] z ij,2 ≥z ij,1 #(25)

[0039] The above constructs a two-stage mixed-integer programming model;

[0040] Add the variable θ representing the second phase to the objective function of the first phase. The variable θ represents the objective function value of the second phase. Add the range of values of the variable θ (θ≥0) to the constraint conditions of the first phase to construct the master problem.

[0041]

[0042] The objective function value of the master problem is represented by z s Solve the master problem to obtain x s and θ s The value of z s is the lower bound LB s , and the lower bound represents the sum of the objective function values of the first and second phases, where: s represents the number of iterations, and initialize s = 1.

[0043] Substitute the optimal solution x s obtained from the master problem into v ω (x, ω), and solve each scenario in v ω (x, ω) respectively. The solution obtained is that no new lines are built in all scenarios in the second phase, and the objective function values of each scenario are obtained respectively as Calculate the upper bound The upper bound represents the sum of the objective function values of the first and second phases;

[0044] Calculate whether the gap between the upper bound and the lower bound meets the preset threshold:

[0045]

[0046] If the condition of being lower than the preset threshold is not met, continue the iteration;

[0047] Take the currently solved x s , θ s , as the initial value and enter the part of the row generation algorithm to solve the proportional cut. The proportional cut is a constraint added to the master problem to help the algorithm converge and needs to be obtained by iteratively solving the inner optimization problem.

[0048] F represents vω (x, ω) The number of scenarios in, let ω = 1, x s，t = x s , θ s，t = θ s , Take x s，t , θ s，t , As constants, α ω , β ω , τ ω As variables and substitute them into the following equation. Equation (27) is the main problem for solving the proportional cut, where: t represents the number of iterations in the process of solving the proportional cut, and initialize t = 1;

[0049] Solve equation (27)

[0050]

[0051] Where: ρ = θ s ,

[0052] Take the result obtained by solving equation (27) As constants, x, θ, y ω As variables and substitute them into equation (28)

[0053]

[0054] Where: Is a real number, indicating that θ can take values within the entire range of real numbers;

[0055] Let t = t + 1, and take the x obtained by solving equation (28) s，t , θ s，t , Generate a new constraint equation (29) and add it to the constraint conditions of equation (27):

[0056]

[0057] Detect the generated Whether it meets the termination condition

[0058]

[0059] Where δ ω Represents the acceptable error range, satisfying δ ω ≥ 0, the value of δ ω Generally takes 1% of the objective function values of each scenario in the second stage. If it is satisfied, terminate the iteration of the inner optimization problem, and let Return Return

[0060]

[0061] Otherwise: Return equation (27) and continue the iteration until the termination condition of equation (30) is satisfied.

[0062] Through the loop, the termination condition of equation (30) is satisfied, and the solutions of (α ω , β ω , τ ω ) for the ω-th scenario are obtained. When ω < F, let ω = ω + 1; t = 1, and return equation (27) to solve for (α ω , β ω , τ ω ) of the next scenario.

[0063] After multiple iterations, the solutions of (α ω , β ω , τ ω ) for three scenarios are obtained.

[0064] Aggregate the cuts generated by α ω , β ω , τ ω for the three scenarios together to generate the following proportional cut formula.

[0065]

[0066] And update φ(x) to obtain the following formula

[0067]

[0068] Add the generated proportional cut to the constraint conditions of the master problem.

[0069] Find the scenario with the largest weighted objective function value in the second stage as ω.

[0070] If s = 1, then multiply the objective function of scenario ω by the coefficient d and add it to the master problem, and add the constraint conditions of scenario ω to the master problem constraint conditions. If s ≠ 1, then multiply the objective function of scenario ω by the coefficient d to replace the part added to the master problem in the (s - 1)-th iteration, and replace the part added to the master problem constraint conditions in the (s - 1)-th iteration with the constraint conditions of scenario ω.

[0071] The master problem is obtained as follows:

[0072]

[0073] Let s = s + 1. Solve equation (34) to obtain the solution x s , θ s and the objective function value z s . The objective function value z s is the lower bound LB s .

[0074] Substitute the obtained \(x\) s into the second stage to solve the optimal solutions of each scenario in the second stage and the optimal value of the objective function

[0075]

[0076] Solve equation (36) to obtain the upper bound UB at the \(s\)-th iteration s ,

[0077]

[0078] Calculate the gap between the upper bound and the lower bound. If it satisfies

[0079] (UB s - LB s ) / max(UB s , LB s ) < ε #(37)

[0080] Then the algorithm terminates and returns the optimal solution \(x\) of the master problem s and the upper bound UB s , otherwise, continue the iteration.

[0081] The present invention also provides a computer-readable storage medium, which stores a computer program. When the computer program is executed by a processor, it implements the steps of the generalized two-stage mixed-integer programming solution method as described in any one of the above.

[0082] The present invention also provides a computer-readable storage medium, which stores a computer program. When the computer program is executed by a processor, it implements the steps of the transmission network planning method.

[0083] The beneficial effects of the present invention are as follows:

[0084] The present invention proposes a generalized two-stage mixed-integer programming solution method. By constructing a master problem and solving it to obtain a lower bound, obtaining the optimal value of the second stage and adding the value of the first stage to obtain an upper bound, when the upper bound and the lower bound do not converge to a preset threshold, adding a proportional cut generated by row generation to the master problem, adding a primal cut to the master problem by column generation, and maintaining the master problem, and continuously iterating until the upper bound and the lower bound converge to the preset threshold, then returning the optimal solution of the first stage and the upper bound UB, and the algorithm ends. The present invention can accurately and quickly solve the two-stage mixed-integer programming problem. Specifically, when applied to transmission network planning, it can accurately and efficiently solve the transmission system planning problem. BRIEF DESCRIPTION OF THE DRAWINGS

[0085] By elaborating on the embodiments shown in the accompanying drawings, the above and other features of the present disclosure will become more apparent. In the drawings of the present disclosure, the same reference numerals denote the same or similar elements. Obviously, the drawings in the following description are only some embodiments of the present disclosure. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings. In the drawings:

[0086] Figure 1 The flowchart of the general two-stage mixed-integer programming solution method of the present invention is shown; Detailed implementation manners

[0087] The concept, specific structure and technical effects of the present invention will be clearly and completely described below in conjunction with embodiments and drawings to fully understand the purpose, solution and effects of the present invention. It should be noted that, without conflict, the embodiments in the present application and the features in the embodiments can be combined with each other. The same reference numerals used throughout the drawings indicate the same or similar parts.

[0088] Refer to Figure 1 , Embodiment 1, the present invention proposes a general two-stage mixed-integer programming solution method, including the following:

[0089] Adding the variables representing the second stage to the first stage to construct the master problem, and solving the master problem to obtain the lower bound LB;

[0090] Solving the second-stage problem to obtain the objective function value of the second stage, and adding it to the objective function value of the first stage to obtain the upper bound UB;

[0091] Determine whether the upper bound UB and the lower bound LB converge to a preset threshold epsilon. If so, return the objective function value of the second stage and the upper bound UB. If not, continuously iterate until the upper bound UB and the lower bound LB converge.

[0092] In this embodiment, the general two-stage mixed-integer problem consists of a first-stage optimization problem and a second-stage recourse problem. In the recourse problem, a random vector ω is used to represent the uncertain factors; and when solving the first-stage decision variable x, the random vector ω is unknown. The solution of the second-stage decision variable y depends on the value of ω. In the present invention, it is assumed that the probability distribution of ω is a discrete probability distribution, and the discrete time set Ω represents the set of random vectors, p ω is the probability density corresponding to the time, and ∑ ω p ω = 1. The general two-stage mixed-integer programming problem can be expressed as follows:

[0093]

[0094] Among them, v in the second stageω (x, ω) is defined in the following form:

[0095]

[0096] If then v ω (x) = ∞. Where In addition, and are the integer constraints of the first stage and the second stage, for example:

[0097] To accurately and quickly solve the generalized two-stage mixed integer problem, the present invention proposes an algorithm that combines row and column generation, which specifically includes the following steps.

[0098] Step [1]: Input data: c, A, b, q ω , W ω , h ω , T ω , p ω , d.

[0099] Step [2]: Add the variable θ representing the second stage to the objective function of the first stage, and add the value range of the variable θ to the constraint conditions of the first stage to construct the following master problem:

[0100]

[0101] Where: L is the lower bound of the value range of the second stage , satisfying

[0102] Step [3]: Solve the master problem (4) to obtain the optimal solution x s , θ s of the current master problem and the objective function value z s of the master problem (4), z s i.e., the lower bound LB s . Where: s represents the number of iterations, and initially set s = 1.

[0103] Step [4]: Substitute the optimal solution x s of the master problem (4) into the second stage problem v ω (x, ω) to solve the optimal solution y ω of each scenario in the second stage and the optimal value v ω (x, ω) of the objective function.

[0104]

[0105] Step 【5】: Solve equation (5) to obtain the optimal solutions for each scenario in the second stage and the optimal value of the objective function is Update the upper bound:

[0106]

[0107] Step 【6】: Calculate the gap between the upper bound and the lower bound. If the difference between the upper bound and the lower bound is less than a pre-set threshold

[0108] The pre-set threshold is denoted by ε.

[0109] (UB s -LB s ) / max(UB s , LB s ) < ε #(7)

[0110] Then the algorithm terminates and returns the optimal solution x s and UB s . Otherwise, continue the iteration.

[0111] Step 【7】: Take the currently solved x s , θ s as the initial values and enter the row generation algorithm to solve the proportional cut part. The proportional cut is a constraint added to the master problem to help the algorithm converge and needs to be obtained by iteratively solving the inner optimization problems from Step 【8】 to Step

[13] . The solved proportional cut is represented by θ ≥ φ(x), and φ(x) is initially an empty set.

[0112] Step 【8】: Let the number of scenarios in the second stage be F, and set ω = 1. Let x s,t = x s , θ s，t = θ s , Take x s,t , θ s，t , as constants, and α ω , β ω , τ ω as variables. Equation (8) is the master problem for solving the proportional cut, denoted by CGMP. t represents the number of iterations in the process of solving the proportional cut, and initialize t = 1. F represents the number of scenarios in v ω (x, ω).

[0113] Step 【9】: Solve equation (8)

[0114]

[0115] where: ρ = θ s .

[0116] Step

[10] : Take the solution obtained from Equation (8) as constants, and substitute x, θ, and y ω as variables into Equation (9). Equation (9) is the sub-problem for solving the proportional cut, denoted by CGSP.

[0117]

[0118] Where:

[0119] Step

[11] : Let t = t + 1, and substitute the x s,t , θ s，t , generated by solving Equation (9) to generate a new constraint (10) and add it to the constraint conditions of Equation (8):

[0120]

[0121] Step

[12] : Check whether the generated in Step

[10] satisfies the termination condition:

[0122]

[0123] where δ ω represents the acceptable error range, and satisfies δ ω ≥ 0. The value of δ ω is generally taken as 1% of the objective function values of each scenario in the second stage . If it is satisfied, terminate the iteration of the ω-th scenario in the inner optimization problem, and let return

[0124]

[0125] Otherwise: Return to Step [9] and continue the iteration until the termination condition of Equation (11) is satisfied.

[0126] Step

[13] : Through the loop, satisfy the termination condition of Equation (11) to solve for (α ω , β ω , τ ω ) of the ω-th scenario. When ω < F, let ω = ω + 1; t = 1, and return to Step [9] to solve for (α ω , β ω , τ ω ) of the next scenario. Otherwise, enter Step

[14] .

[0127] Step

[14] :, to obtain (α ω , β ω , τ ω):Generate the proportional cut formula (13) and update the φ(x) formula (14)

[0128]

[0129]

[0130] Step

[15] : Add the generated proportional cut to the constraint conditions of the master problem (4).

[0131] Step

[16] : Find the scenario with the largest product of the objective function value and the weight in the second stage.

[0132] Step

[17] : If s = 1, then multiply the objective function of the scenario obtained in Step

[16] by the coefficient d and add it to the master problem, and add the constraint conditions of the scenario to the master problem constraint conditions. If s ≠ 1, then multiply the objective function of the scenario obtained in Step

[16] by the coefficient d to replace the part added to the (s - 1)th master problem, and replace the part of the constraint conditions of the scenario added to the (s - 1)th iteration master problem constraint conditions.

[0133] Step

[18] : After Steps

[15] to

[17] , the master problem is as shown in formula (15).

[0134]

[0135] Step

[19] : Let s = s + 1. Solve formula (15) to obtain the solution x s , θ s and the objective function value z s , that is, the lower bound LB s .

[0136] Step

[20] : Substitute the obtained x s into the second stage and solve formula (16) to obtain the optimal solutions of each scenario in the second stage and the optimal value of the objective function

[0137]

[0138] Step

[21] : Solve formula (17) to obtain the upper bound UB s at the s-th iteration.

[0139]

[0140] Step

[22] : Calculate the gap between the upper bound and the lower bound. If it satisfies

[0141] (UB s -LB s ) / max(UB s , LB s) < ε#(18)

[0142] Then the algorithm terminates and returns the optimal solution x of the main problem s and the upper bound UB s . Otherwise, return to step [7] to continue the iteration.

[0143] As a preferred embodiment of the present invention, specifically, continuous iteration includes the following

[0144] Obtain the proportional cut and add it to the main problem through the row generation algorithm, obtain the original cut and add it to the main problem through the column generation algorithm. Under the above conditions, solve the main problem to obtain the lower bound LB, solve the second stage to obtain the objective function value of the second stage, and add the objective function value of the first stage to obtain the upper bound UB, and continuously iterate in this way.

[0145] As a preferred embodiment of the present invention, specifically, obtaining the proportional cut through the row generation algorithm includes the following

[0146] Obtained by solving the inner optimization problem. Take the values of the variables obtained by solving the main problem as the initial values, substitute the initial values into the solution of CGMP, and obtain α ω , β ω , τ ω . Substitute the values of α ω , β ω , τ ω into CGSP for solution, check whether the objective function value of CGSP meets the termination condition. If not, it is necessary to add x, θ, y obtained by using the solution of CGSP ω to generate new constraints and add them to CGMP, and enter the next iteration of the inner optimization problem until the objective function value of CGSP meets the termination condition. This scenario solution ends, and return the values of (α ω , β ω , τ ω ). Solve (α ω , β ω , τ ω ) of the next scenario until all scenarios are solved, and generate constraints.

[0147] As a preferred embodiment of the present invention, specifically, obtaining the original cut through the column generation algorithm includes the following

[0148] Find the scenario with the largest product of the objective function value and the weight in the second stage, or the scenario with the largest objective function value in the second stage, or the scenario with the largest weight in the second stage. If it is the first iteration, then the objective function of the obtained scenario needs to be multiplied by the coefficient d and added to the master problem, and the constraint conditions of the scenario need to be added to the master problem constraints. If it is not the first iteration, then the objective function of the obtained scenario needs to be multiplied by the coefficient d to replace the scenario added to the master problem objective function in the previous iteration, and the constraint conditions of the scenario need to replace the constraints added to the master problem in the previous iteration.

[0149] The present invention also proposes a transmission network planning method, which applies the generalized two-stage mixed-integer programming solution method, including the following steps:

[0150] Step [1]: Next, the content of the invention will be described in detail in combination with the planning of the transmission network. The described embodiments are only a part of the embodiments of the content of the present invention, rather than all embodiments. Based on the embodiments in the present invention, all other embodiments obtained by those of ordinary skill in the art without creative work fall within the protection scope of the present invention.

[0151] Step [2]: The embodiment of the present invention provides a method for solving the transmission system planning of two-stage mixed-integer programming, which can accurately and efficiently solve the transmission system planning.

[0152] Step [3]: Input the relevant data in the power system and establish a transmission system planning model. As follows:

[0153] Step [4]: Objective function:

[0154]

[0155] Among them, c1 and c0 are the coefficients of the real-time operating costs of the generating units, and a represents the planning stage. T is the total number of planning stages. Here, let T = 2. p g，a is the power generation of the generator in stage a. c k is the investment cost of newly built lines. z ij，a is a binary variable in stage a used to determine whether to build a new line.

[0156] Step [5]: The constraint conditions mainly include:

[0157] Step [6]: Power balance constraint

[0158]

[0159] p g，a represents the power generation of the generator at the node in stage a, and p d，i，a represents the load connected to the node in stage a. p ij，aRepresents the line power in stage a. Represents a node, Represents the generator units connected to the system.

[0160] Step [7]: Node power angle constraint

[0161]

[0162] Among them, θ i，a Represents the power angle of the node in stage a.

[0163] Step [8]: Generator output constraint

[0164] 0 ≤ p g，a ≤ p g，max #(22)

[0165] Among them, P g,a Represents the generator output in stage a, P g,max Represents the maximum output of the generator unit.

[0166] Step [9]: Line power constraint

[0167]

[0168] Among them, ε represents the line set, p ij,min Represents the minimum line power, p ij,max Represents the maximum line power.

[0169] Step

[10] : Power flow - power angle relationship

[0170] -(1 - z ij，a )·M ≤ (P ij，a + B ij,a θ ij，a ) ≤ (1 - z ij，a )·M #(24)

[0171] Among them, M is the separation factor, B ij,a Is the susceptance in the circuit system of stage a.

[0172] Step

[11] : Coupling constraint means that the transmission lines built in the second stage should be built on the basis of the first stage.

[0173] z ij,2 ≥ Z ij,1 #(25)

[0174] In this way, a two - stage mixed - integer programming model is constructed.

[0175] Step

[12] : This example uses an improved system based on IEEE 6 - bus as a test.

[0176] Step

[13] : In the second stage, the loads corresponding to the three scenarios increase by 50%, 40%, and 43% respectively compared to the loads in the first stage, and the corresponding probabilities are p1 = 0.4, p2 = 0.3, and p3 = 0.3.

[0177] Step

[14] : Add the variable θ representing the second stage to the objective function of the first stage. The variable θ represents the objective function value of the second stage. Add the range of values of the variable θ (θ ≥ 0) to the constraints of the first stage to construct the master problem.

[0178]

[0179] Step

[15] : The objective function value of the master problem is represented by z s . Solve the master problem to obtain the values of x s and θ s . Solve to obtain 5 newly built transmission lines. They are the newly built transmission lines between nodes 2 - 4, 2 - 6, 3 - 5, 3 - 6, and 1 - 3. The objective function value z s of the master problem is 86891, and z s is equal to the lower bound LB s , so LB s is 86891. The lower bound represents the sum of the objective function values of the first stage and the second stage. Among them: s represents the number of iterations, and initially set s = 1.

[0180] Step

[16] : Substitute the optimal solution x s obtained from the master problem into v ω (x, ω). Solve each scenario in v ω (x, ω) respectively. Solve to obtain the values. Solve to obtain that no new lines are built in all scenarios in the second stage. The objective function values of each scenario are respectively Calculate to obtain the upper bound of 215600. The upper bound represents the sum of the objective function values of the first stage and the second stage.

[0181] Step

[17] : Calculate whether the gap between the upper bound and the lower bound meets the preset threshold:

[0182]

[0183] The preset threshold ε = 1%. Calculate that the value of the gap between the upper bound and the lower bound is 59.70%, which is greater than the preset threshold. Therefore, continue the iteration.

[0184] Step

[18] : Take the currently solved x s , θ s , As the initial value, enter the part of the row generation algorithm to solve the proportional cut. The proportional cut is a constraint added to the master problem to help the algorithm converge, and it needs to be obtained by iteratively solving the inner optimization problem from step

[19] to step

[24] .

[0185] Step

[19] : Let F denote the number of scenarios in v ω (x, ω). Let F = 3, ω = 1, x s,t = x s , θ s，t = θ s , Take x s,t , θ s,t , as constants, and α ω , β ω , τ ω as variables and substitute them into the following equation. Equation (27) is the master problem for solving the proportional cut, where: t represents the iteration number in the process of solving the proportional cut, and initialize t = 1.

[0186] Step

[20] : Solve Equation (27)

[0187]

[0188] where: ρ = θ s .

[0189] Step

[21] : Take the obtained by solving Equation (27) as a constant, and x, θ, y ω as variables and substitute them into Equation (28)

[0190]

[0191] where:

[0192] Step

[22] : Let t = t + 1, and take the x s，t , θ s，t , generated by solving Equation (28) to generate a new constraint Equation (29) and add it to the constraint conditions of Equation (27):

[0193]

[0194] Step

[23] : Detect whether the generated in Step

[21] satisfies the termination condition

[0195]

[0196] where δ ω represents the acceptable error range, and satisfies δω ≥ 0. δ ω The value of is generally taken as 1% of the objective function values of each scenario in the second stage. If satisfied, terminate the iteration of the inner optimization problem, and let Return

[0197]

[0198] Otherwise: Return to step

[20] , continue the iteration until the termination condition of equation (30) is satisfied.

[0199] Step

[24] : Through the loop, satisfy the termination condition of equation (30), and solve to obtain (α ω , β ω , τ ω ) for the ω-th scenario. When ω < F, let ω = ω + 1; t = 1, and return to step

[20] to solve (α ω , β ω , τ ω ) for the next scenario. Otherwise, enter step

[25] .

[0200] Step

[25] : After multiple iterations, solve (α ω , β ω , τ ω ) for three scenarios.

[0201] Step

[26] : Aggregate the cuts generated by α ω , β ω , τ ω of the three scenarios together to generate the proportional cut equation (32) and update φ(x) equation (33)

[0202]

[0203]

[0204] Step

[27] : Add the generated proportional cut to the constraint conditions of the master problem.

[0205] Step

[28] : Find the scenario with the largest product of the objective function value and the weight in the second stage as ω.

[0206] Step

[29] : If s = 1, then multiply the objective function of the scenario ω obtained in step

[28] by the coefficient d and add it to the master problem, and add the constraint conditions of scenario ω to the master problem constraint conditions. If s ≠ 1, then multiply the objective function of scenario ω in step

[28] by the coefficient d to replace the part added to the master problem in the (s - 1)-th iteration, and replace the part of the constraint conditions of scenario ω added to the master problem constraint conditions in the (s - 1)-th iteration.

[0207] Step

[30] : After steps

[27] to

[29] , the master problem is as shown in Equation (34).

[0208]

[0209] Step

[31] : Let s = s + 1. Solve Equation (34) to obtain the solution x s , θ s and the objective function value z s , the objective function value z s which is the lower bound LB s .

[0210] Step

[32] : Substitute the obtained x s into the second stage and solve for the optimal solutions of each scenario in the second stage and the optimal value of the objective function

[0211]

[0212] Step

[33] : Solve Equation (36) to obtain the upper bound UB s at the s-th iteration.

[0213]

[0214] Step

[34] : Calculate the gap between the upper bound and the lower bound. If it satisfies

[0215] (UB s - LB s ) / max(UB s , LB s ) < ε #(37)

[0216] then the algorithm terminates and returns the optimal solution x s of the master problem and the upper bound UB s . Otherwise, return to Step

[18] to continue the iteration.

[0217] Step

[35] : After solving, the lower bound is 215510 and the upper bound is 215600. The gap between the two is 0.04%, which is less than the pre-set threshold, reaching the optimal value. The final solution result is that 5 transmission lines need to be newly built in the first stage. No new transmission lines are needed in the second stage. The 5 transmission lines that need to be newly built in the first stage are respectively the 2 - 4 node, 2 - 6 node, 3 - 5 node, 3 - 6 node, and 1 - 3 node.

[0218] Step

[36] : By combining the two stages into a centralized model and solving it with a solver, the obtained result is 215560. The gap with the result obtained by our algorithm is within 0.01%. This proves the accuracy of our algorithm from another perspective.

[0219] Step

[37] : In summary, the hybrid row and column generation algorithm for solving the generalized two-stage mixed integer programming problem proposed by the present invention can solve two-stage mixed integer programming problems such as transmission network planning.

[0220] The present invention also proposes a computer-readable storage medium, which stores a computer program, and when the computer program is executed by a processor, it implements the steps of the generalized two-stage mixed integer programming solution method as described in any one of the above.

[0221] The present invention also proposes a computer-readable storage medium, which stores a computer program, and when the computer program is executed by a processor, it implements the steps of the transmission network planning method.

[0222] The module described as a separate component may or may not be physically separated. The component shown as a module may or may not be a physical module, that is, it may be located in one place, or it may be distributed to multiple network modules. Some or all of the modules can be selected according to actual needs to achieve the purpose of the solution in this embodiment.

[0223] In addition, in each embodiment of the present invention, the functional modules can be integrated in a processing module, or each module can exist physically alone, or two or more modules can be integrated in one module. The above integrated module can be implemented in the form of hardware or in the form of a software functional module.

[0224] When the integrated module is implemented in the form of a software functional module and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, to implement all or part of the processes in the above-described embodiment methods of the present invention, it can also be completed by a computer program instructing relevant hardware. The computer program can be stored in a computer-readable storage medium. When the computer program is executed by a processor, the steps of the above-described various method embodiments can be implemented. Among them, the computer program includes computer program code, and the computer program code can be in the form of source code, object code, executable file, or some intermediate form, etc. The computer-readable medium can include: any entity or system, recording medium, USB flash drive, mobile hard disk, magnetic disk, optical disc, computer memory, read-only memory (ROM), random access memory (RAM), electrical carrier signal, telecommunication signal, and software distribution medium, etc. It should be noted that the content included in the computer-readable medium can be appropriately increased or decreased according to the requirements of legislation and patent practice in the jurisdiction. For example, in some jurisdictions, according to legislation and patent practice, the computer-readable medium does not include electrical carrier signals and telecommunication signals.

[0225] Although the description of the present invention has been quite detailed and has particularly described several of the above-described embodiments, it is not intended to be limited to any of these details or embodiments or any particular embodiment, but rather should be regarded as providing a broad possible interpretation of these claims in light of the prior art by reference to the appended claims, thereby effectively covering the intended scope of the present invention. In addition, the present invention has been described above with embodiments foreseeable by the inventors for the purpose of providing a useful description, and non-substantive modifications to the present invention that are not currently foreseeable may still represent equivalent modifications of the present invention.

[0226] As described above, these are only the preferred embodiments of the present invention. The present invention is not limited to the above-described embodiments. As long as it achieves the technical effects of the present invention by the same means, it should fall within the protection scope of the present invention. Within the protection scope of the present invention, its technical solutions and / or embodiments can have various different modifications and variations.

Claims

1. Transmission network planning method, characterized in that, Including Inputting relevant data in the power system and establishing a transmission system planning model Objective function: Among them, c1 and c0 are the coefficients of the real-time operating cost of the generator set, a represents the planning stage, T is the number of times of the overall planning stage. Here, let T = 2, p g,a is the power generation of the generator in stage a, c k is the investment cost of newly built lines, z ij,a is a binary variable in stage a used to determine whether to build a new line; Constraints include: Power balance constraint p g,a represents the power generation of the generator at the node in stage a, p d,i,a represents the connected load at the node in stage a, p ij,a represents the line power in stage a, represents the node, represents the connected generator sets on the system, Node power angle constraint Among them, θ i,a represents the power angle of the node in stage a, Generator output constraint 0 ≤ p g,a ≤ p g,max #(22) Among them, p g,a represents the generator output in stage a, and p g,max represents the maximum output of the generator set. Line power constraint Among them, ε represents the set of lines, and p ij,min represents the minimum power of the line, and p ij,max represents the maximum power of the line. Power flow power angle relationship -(1 - z ij,a )·M ≤ (P ij,a + B ij,a θ ij,a ) ≤ (1 - z ij,a )·M #(24) where M is the separation factor and B ij,a is the susceptance in the stage a circuit system; The coupling constraint means that the transmission lines built in the second stage should be built on the basis of the first stage z ij , 2 ≥ z ij,1 #(25) The above constructs a two-stage mixed integer programming model Add the variable θ representing the second stage to the objective function of the first stage. The variable θ represents the objective function value of the second stage. Add the value range of the variable θ (θ≥0) to the constraints of the first stage to construct the master problem. The objective function value of the master problem is denoted by z s By solving the master problem, we obtain x s and θ s The value of z s is the lower bound LB s , where the lower bound represents the sum of the objective function values in the first and second stages. Here, s represents the number of iterations. Initially, s = 1 Substitute the optimal solution x obtained for the main problem s into v ω (x, ω), and solve v ω for each scenario in (x, ω), and the solution obtained is the value. It is obtained that no new lines are built in all scenarios in the second stage, and the objective function values for each scenario are respectively Calculate the upper bound The upper bound represents the sum of the objective function values in the first stage and the second stage; Calculating whether the gap between the upper bound and the lower bound meets the preset threshold If the condition of being lower than the preset threshold is not met, continue to iterate Take the currently solved x s , θ s , as the initial value and enter the part of the row generation algorithm to solve the proportional cut. The proportional cut is a constraint added to the master problem to help the algorithm converge and needs to be obtained by iteratively solving the inner optimization problem. F represents the number of scenarios in v ω (x, ω), let ω = 1, x s,t = x s , θ s,t = θ s , Take x s,t , θ s,t , as constants, and substitute α ω , β ω , τ ω as variables into the following equations. Equation (27) is the master problem for solving the proportional cut, where: t represents the iteration number in the process of solving the proportional cut, and initialize t = 1; Solving Equation (27) where: ρ = θ s , Substitute the solution of Equation (27) As constants, x, θ, y ω As variables into Equation (28) Wherein: is a real number, indicating that θ can take values within the entire range of real numbers; Let \(t = t + 1\), and substitute \(x\) solved from Equation (28) s,t , \(\theta\) s,t , Generate a new constraint Equation (29) and add it to the constraint conditions of Equation (27): Detect the generated Whether the termination condition is met where δ ω represents an acceptable error range, satisfying δ ω ≥0, and the value of δ ω is generally taken as 1% of the objective function values of each scenario in the second stage. If satisfied, terminate the iteration of the inner optimization problem, and let Return​ Otherwise: return to Equation (27), continue to iterate until the termination condition of Equation (30) is met Through iteration, the termination condition of Equation (30) is satisfied, and the solution gives (α ω , β ω , τ ω ) for the ω-th scenario. When ω < F, let ω = ω + 1; t = 1, and return to Equation (27) to solve for (α ω , β ω , τ ω ) for the next scenario. After multiple iterations, the values of (α ω , β ω , τ ω ) for three scenarios were solved. Aggregate the cuts generated by α of the three scenarios ω , β ω , τ ω together to generate the following proportional cut formula And update Φ(x) to obtain the following formula Adding the generated proportional cut to the constraint conditions of the master problem Finding the scenario ω with the largest weighted objective function value in the second stage If s = 1, then multiplying the objective function of scenario ω by the coefficient d and adding it to the master problem, and adding the constraint conditions of scenario ω to the master problem constraint conditions. If s ≠ 1, then multiplying the objective function of scenario ω by the coefficient d to replace the part added to the master problem in the (s - 1)-th iteration, and replacing the part added to the master problem constraint conditions in the (s - 1)-th iteration with the constraint conditions of scenario ω Obtaining the master problem as follows Let s = s + 1, solve equation (34) to obtain the solution x s , θ s and the objective function value z s , the objective function value z s which is the lower bound LB s , Substitute the obtained x s into the second stage to solve the optimal solutions for each scenario in the second stage and the optimal value of the objective function Solve Equation (36) to obtain the upper bound UB at the s-th iteration s , Calculating the gap between the upper bound and the lower bound. If it is satisfied (UB s -LB s ) / max(UB s ,LB s ) < ε #(37) Then the algorithm terminates and returns the optimal solution x of the main problem s and the upper bound UB s , otherwise continue the iteration.

2. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the steps of the method described in Claim 1

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