Three-stage alternating current optimal transmission switching method based on relaxation model

Abstract

The invention discloses a three-stage AC optimal transmission switching method based on a relaxation model, and the method comprises the following steps: 1, screening a breakable branch through solving an optimal solution of the relaxation model in a first stage, and reducing the scale of a system; 2, in the second stage, sorting is carried out by solving the optimal solution of the alternating-current optimal power flow of disconnection of a single branch; and step 3, in the third stage, solving by adopting a branch and bound method according to the sorting sequence of the second stage, and finally determining an optimal solution of alternating current optimal transmission switching and a branch on-off scheme. According to the method, the dimensionality of the optimal transmission switching problem is effectively reduced, and the calculation efficiency and speed are improved; an alternating current optimal transmission switching relaxation problem is solved by adopting a method based on a quotient gradient system and IPOPT, so that the calculation convergence and speed are guaranteed, and the algorithm robustness is improved; branch and bound calculation is carried out according to the alternating current optimal power flow solution sorting result sequence of the cut-off single branch, and it is guaranteed that a high-quality line cut-off scheme is obtained through solving; in the calculation process, island branch processing is considered, the island phenomenon in the topology change process is prevented, and power grid splitting and unnecessary load outage are effectively avoided.

Description

Technical Field

[0001] The present invention relates to the technical field of power systems, and in particular, to a three-stage AC optimal transmission switching method based on a relaxation model. Background Art

[0002] The AC optimal transmission switching problem is a large-scale non-linear mixed integer optimization problem, which is difficult to solve directly and requires a long calculation time.

[0003] The optimal transmission switching problem is an extension of the optimal power flow problem. On the basis of the optimal power flow model, line outages are allowed as control variables. Its goal is to find the power grid topology and scheduling scheme with the optimal objective function, while satisfying the AC power flow feasibility and other operation constraints. Optimal transmission switching further expands the solution space of the optimal power flow problem and can provide a larger search scope to solve the optimal solution. By solving the optimal transmission switching problem, the further fine allocation of power grid dispatching resources can be promoted, which is beneficial to improving the operation economy and robustness of the power grid.

[0004] Optimal transmission switching has many advantages. However, AC optimal transmission switching is a mixed integer non-linear optimization problem, including integer variables and non-linear constraints, which is more complex than the AC optimal power flow problem and is NP-hard. Even with a linear DC optimal transmission switching model, the optimal transmission switching problem is computationally challenging, mainly because the model contains binary discrete variables corresponding to line on / off decisions. In a 4500-node system in the literature, using a mixed integer linear programming model, it took 82 hours to find a solution with a 6% cost reduction, and still required 6 hours of solution time after improvement.

[0005] A large number of optimal transmission switching studies focus on DC optimal transmission switching based on the DC power flow model, which has more advantages in terms of problem scale, model complexity and calculation speed, and usually uses mixed integer linear programming to solve. However, DC optimal transmission switching ignores bus voltages and power losses and cannot always guarantee AC feasibility, and further processing is required to ensure the feasibility requirements of the solution. The literature shows that the DC optimal transmission switching method performs poorly in some test cases and may even lead to an increase in the actual operating cost.

[0006] There have also been some studies attempting to directly solve the more complex AC optimal transmission switching problem, and methods such as convex relaxation, Benders decomposition, and heuristic methods have been proposed for solving. The literature proposed three types of mixed-integer second-order cone programming relaxation methods for solving AC optimal transmission switching, which solved a large number of AC optimal transmission switching problems within a finite time. The literature proposed a two-stage iterative solution framework, using mixed-integer second-order cone programming in the upper stage to provide candidate solutions and performing AC feasibility checks and restorations in the lower stage. The literature achieved the selection of optimal transmission lines through an improved binary particle swarm method, improving the voltage stability margin while ensuring economy. Although a large number of optimal transmission switching solution methods have been proposed, they are still limited by computational efficiency, solution speed, and robustness, and have not been widely applied in practice. To further improve the application scenarios and scope of optimal transmission switching, the characteristics of the optimal transmission switching problem should be studied in depth, and more efficient AC optimal transmission switching algorithms should be developed. Summary of the Invention

[0007] Object of the Invention: The present invention provides a three-stage AC optimal transmission switching method based on a relaxation model. Through the design of a three-stage solution process, the computational dimension of the problem is reduced, and the solution speed and solution quality are improved.

[0008] Technical Solution: A three-stage AC optimal transmission switching method based on a relaxation model according to the present invention includes the following steps:

[0009] Step 1: In the first stage, the openable branches are screened by solving the optimal solution of the relaxation model to reduce the system scale;

[0010] Step 2: In the second stage, the optimal solutions of the AC optimal power flow for the opening of a single branch are solved and sorted;

[0011] Step 3: In the third stage, the branch and bound method is used for solving according to the sorting order in the second stage, and finally the optimal solution of the AC optimal transmission switching and the branch opening scheme are determined.

[0012] Further, in Step 1, the specific steps for screening the openable branches by solving the optimal solution of the relaxation model in the first stage are as follows:

[0013] Step 11: Island branch screening; Since the opening of a branch cannot cause an island to avoid network disconnection or load impact, branches that will cause an island after being opened alone are preferably screened first, and the remaining set of openable branches is denoted as Ω l ;

[0014] Step 12: Using the branches in Ω l as relaxation variables, a relaxation problem is constructed;

[0015] Step 13: Construct a quotient gradient system corresponding to the relaxation problem;

[0016] Step 14: Using the ground state solution and the initial topological state Z0 as the relaxation initial point to integrate the quotient gradient system to approach a stable equilibrium point, and denote the termination point as this point as a feasible solution (but not the optimal solution) to the relaxation problem;

[0017] Step 15: Using as the initial point, solve the relaxation problem using IPOPT. If the solution is successful, denote the optimal solution as X * , and go to Step 16; otherwise, randomly update the relaxation initial point and re - execute Steps 14 - 15;

[0018] Step 16: Let τ be a small positive scalar. If then the improvement of the relaxation optimal solution X * compared to the ground state solution is limited, terminate the calculation; otherwise, go to Step 17;

[0019] Step 17: If all the relaxation variables of X * are in discrete states 0 or 1, then the optimal solution to the AC optimal transmission switching problem has been directly solved, output the line outage plan, and terminate the calculation; otherwise, go to Step 18;

[0020] Step 18: Screen out the branches corresponding to the relaxation variables in X * where z ij ≤0.98, and construct them into the set of branches that can be opened Ω lc .

[0021] Furthermore, in Step 12, the optimal transmission switching problem is a mixed - integer programming problem based on the optimal power flow model. Based on the AC optimal power flow model, binary variables Z ij are added to the power flow equations and thermal limit constraints to obtain the AC optimal transmission switching model;

[0022] The objective function is:

[0023] min.f = ax 2 + bx + c

[0024] where f is in the form of the fuel cost as the objective function, which is a continuously differentiable function, a, b, c are constant vectors representing the structure of the objective function, and x is a variable vector, including the active power, reactive power of generator nodes, voltage magnitudes, voltage phase angles of all nodes, and transmission line state variables in this problem;

[0025] The constraint set of the AC optimal transmission switching model is in the following form, including the AC power flow equation, line thermal limit constraint, generator output upper and lower limit constraints, and node voltage constraints:

[0026]

[0027] V i min ≤V i ≤V i max i∈{1,2,...,N B}

[0028] Among them, the line state variable Z ij is a 0-1 variable representing the state of transmission line i-j. When Z ij equals 1, it means that line i-j is in the closed state. When Z ij equals 0, it means that line i-j is in the open state; among them, N B ,N L ,N G represent the number of bus nodes, transmission lines, and generators in the power system respectively; the constants G ij and B ij are the equivalent conductance and equivalent susceptance of the line from node i to node j respectively, P Li and Q Li are the active load value and reactive load value of node i respectively; the variable θ i is the voltage phase angle of node i, and θ ij =θ i -θ j , where the voltage phase angle of the slack node is set as a constant; the variable V i is the voltage amplitude of node i, P Gi and Q Gi are the active output power value and reactive output power value of the generator connected to node i respectively, and the expressions S fl and S tl are the apparent powers at the sending end and receiving end of line l respectively;

[0029] The AC optimal transmission switching model is abbreviated as:

[0030] minf(x)

[0031] s.t.H T (x,Z)=0

[0032] G T (x,Z)≤0

[0033] Among them, x represents continuous variables, including all variables of the optimal power flow model; Z is the discrete line state variable; H T(x, Z) is an equality constraint; G T (x, Z) represents an inequality constraint;

[0034] By relaxing the discrete variables into continuous variables with upper and lower bound constraints, a relaxed model for AC optimal transmission switching is constructed, and then the optimization technology for solving continuous variables is used to solve it;

[0035] The objective function is:

[0036] min.f = ax 2 + bx + c

[0037] where f is in the form of a fuel cost as the objective function, which is a continuously differentiable function, a, b, c are constant vectors representing the structure of the objective function, x is a variable vector, including the active power, reactive power of generator nodes, voltage magnitudes, voltage phase angles of all nodes, and transmission line status variables in this problem;

[0038] The discrete variables of the AC optimal transmission switching model are only 0 - 1 variable Z ij , therefore, relaxing Z ij into a continuous variable with a variable interval of [0, 1] can construct the relaxed problem of AC optimal transmission switching. The complete constraints of the relaxed problem are as follows:

[0039]

[0040] 0 ≤ z ij ≤ 1

[0041]

[0042] V i min ≤ V i ≤ V i max i ∈ {1, 2,..., N B}

[0043] where z ij is the relaxed variable after continuousizing the 0 - 1 variable Z ij ;

[0044] The complete relaxed model of AC optimal transmission switching, in which there are no discrete variables, is abbreviated in the following form:

[0045] minf(x)

[0046] s.t.H s (x) = 0

[0047] G s (x) ≤ 0

[0048] Among them, \(x\) is a set of variables, including all variables of the optimal power flow model and the slack variable \(z\). ij .

[0049] Furthermore, in step 13, the quotient gradient system is as follows:

[0050]

[0051] Among them, \(DH(x)\) is the Jacobian matrix of \(H(x)\).

[0052] Furthermore, in step 2, the second stage performs sorting by solving the optimal solution of the AC optimal power flow for a single branch outage, which specifically includes the following steps:

[0053] Step 21: Calculate the AC optimal power flow after the outage of each line in the set of openable branches \(\Omega\). lc ;

[0054] Step 22: Sort in ascending order according to the objective function values of the optimal solutions of the optimal power flow in step 21.

[0055] Furthermore, in step 21, the AC optimal power flow is

[0056] Minimize \(f(x)\)

[0057] s.t. \(H(x)=0\)

[0058] Among them, is the objective function; is the constraint set, and the inequality constraints have been converted into equality constraints by adding slack variables.

[0059] Furthermore, in step 3, the third stage uses the branch and bound method to solve according to the sorting order in the second stage, and finally determines the optimal solution of the AC optimal transmission switching and the branch outage plan, which specifically includes the following steps:

[0060] Step 31: Use the lines in \(\Omega\) as slack variables to construct a relaxed subproblem. lc ;

[0061] Step 32: According to the branch sorting order in the second stage, denote the first-ranked branch in \(\Omega\) as \(z\). lc , perform island detection on the topology after disconnecting \(z\). If an island exists, set \(z\). ij to 1 and go to step 36; otherwise, go to step 33; ij ij = 1, go to step 36; otherwise, go to step 33;

[0062] Step 33: Fix the state of \(z\) to be disconnected and closed respectively, that is, \(z\). ij = 0 and \(z\). ij = 0 and \(z\). ij ​= 1, IPOPT is used to solve the two relaxation sub - problems respectively, and the optimal solutions are denoted as and

[0063] Step 34: If or does not exist, then let z ij = 1, Otherwise, if or does not exist, then let z ij = 0,

[0064] Step 35: If all the slack variables of are in discrete states 0 or 1, obtain the discrete optimal solution of the AC optimal transmission switching problem, output the line outage plan, and terminate the calculation; otherwise, go to Step 36;

[0065] Step 36: Remove z lc from Ω ij , Ω lc = Ω lc - z ij . If then obtain the discrete solution of the AC optimal transmission switching problem, output the line outage plan, and terminate the calculation; otherwise, fix the state of z ij , update the relaxation sub - problem, go to Step 32, and continue the branch - and - bound calculation.

[0066] Beneficial effects: Compared with the prior art, the present invention has the following remarkable advantages: The present invention effectively reduces the dimension of the optimal transmission switching problem, improves the calculation efficiency and speed; First, the quotient gradient system is used to solve the feasible solution of the relaxation problem, and then IPOPT is used to solve the AC optimal transmission switching relaxation problem with the feasible solution as the initial point, which ensures the calculation convergence and speed and improves the algorithm robustness; The branch - and - bound calculation is carried out according to the sorting result order of the AC optimal power flow solutions of opening a single branch, which guarantees to obtain a high - quality line outage plan; The treatment of island branches is considered during the calculation process, preventing the islanding phenomenon during the topological change process, and effectively avoiding power grid disconnection and unnecessary load outages. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 is a schematic diagram of the staged optimal transmission switching architecture of the present invention.

[0068] Figure 2 is a schematic diagram of the method flow of the present invention.

[0069] Figure 3 is a diagram of the slack variable state of the stage - one relaxation optimal solution of the case118_Carnegie example of the present invention. Detailed implementation mode

[0070] Through a three-stage solution process design, the present invention reduces the problem calculation dimension and improves the solution speed and solution quality. The online optimal transmission switching solution consists of four parts, each serving a different purpose, and its framework diagram is as shown in Figure 1 shown, and its core is a three-stage optimal transmission switching algorithm.

[0071] The input data includes: power system state data based on state estimation, short-term load demand and new energy output prediction data, and system network topology structure.

[0072] The function of the analysis stage is: calculate the optimal power flow under the current operating state and network topology to obtain the initial operating point (base state solution); analyze whether there are problems such as line congestion and excessive voltage phase angle constraints, and further decide whether to carry out the next stage (recognition stage); analyze the network topology to form a set of openable lines under the current topology.

[0073] The function of the recognition stage is: Stage 1 (screening stage): screen the openable lines, and use the branches that may further reduce the system operating cost (or other objective functions) after opening as the screened openable branch set, and regard the remaining lines as fixed states, reducing the scale of discrete variables in the optimal transmission switching problem; this stage adopts a screening method based on the relaxed optimal solution; in this stage, most of the branches will be screened out.

[0074] Stage 2 (ranking stage): Usually, the combination of branches with better AC optimal power flow results when a single branch is opened will have better effects. Therefore, for the set of openable branches screened in Stage 1, calculate the optimal solution of the AC optimal power flow after opening a single branch, and rank the branches according to the objective function value.

[0075] Stage 3 (detailed analysis stage): Construct a reduced-dimensional AC optimal transmission switching relaxation sub-problem (abbreviated as relaxation sub-problem) with the branches in the openable branch set as relaxation variables; perform branch and bound calculations according to the branch ranking order in Stage 2 to determine the final candidate branch opening states within the group. When there are many branches in the openable branch set, they can be grouped according to the ranking order in Stage 2 to further reduce the dimension of the relaxation sub-problem.

[0076] The output data includes: the set of opened lines and the system operating cost.

[0077] As Figure 2 shown, a three-stage AC optimal transmission switching method based on a relaxation model includes the following steps:

[0078] Step 1: In the first stage, screen the openable branches by solving the optimal solution of the relaxation model to reduce the system scale;

[0079] Step 2: In the second stage, sort by solving the optimal solution of the AC optimal power flow for a single branch outage.

[0080] Step 3: In the third stage, solve using the branch and bound method in the sorting order of the second stage, and finally determine the optimal solution of the AC optimal transmission switching and the branch outage plan.

[0081] Stage 1 (screening stage): Solve the optimal solution of the relaxation problem through the quotient gradient system and IPOPT, and use the state of the relaxation variables of the relaxation optimal solution to screen the branches whose outage makes the operation effect better; if the relaxation optimal solution has no improvement or only a small improvement compared to the base state solution, then no further optimal transmission switching calculation is performed; if the relaxation variables of the relaxation optimal solution are in discrete states, directly output the optimal transmission switching outage plan.

[0082] Step 1-1: Island branch screening. Since a branch outage should not cause an island to avoid network disconnection or load impact. Therefore, first screen out the branches that will cause an island when a single branch is outaged, and the remaining set of branches that can be outaged is denoted as Ω. l .

[0083] Step 1-2: Construct a relaxation problem with the branches in Ω l as the relaxation variables.

[0084] Step 1-3: Construct the quotient gradient system corresponding to the relaxation problem.

[0085] Step 1-4: Use the base state solution and the initial topological state Z0 as the relaxation initial point to integrate the system until approaching a stable equilibrium point, and denote the termination point as this point as a feasible solution (but not the optimal solution) of the relaxation problem.

[0086] Step 1-5: Use as the initial point and use IPOPT to solve the relaxation problem. If the solution is successful, denote the optimal solution as X * , go to Step 1-6; otherwise, randomly update the relaxation initial point and re-execute Steps 1-4 and 1-5.

[0087] Step 1-6: Let τ be a small positive scalar. If then the relaxation optimal solution X * has limited improvement compared to the base state solution , terminate the calculation; otherwise, go to Step 1-7.

[0088] Step 1-7: If X *If all the slack variables are in discrete states (0 or 1), the optimal solution of the AC optimal transmission switching problem has been directly obtained, and the line outage plan is output to terminate the calculation; otherwise, go to step 1-8.

[0089] Step 1-8: Screen out the branches corresponding to the slack variables with z * in X ij ≤0.98, and construct them into the set of branches that can be opened, denoted as Ω lc .

[0090] (2) Stage 2 (sorting stage): For the branches screened out in the previous stage, calculate the optimal solution of the AC optimal power flow when a single branch is opened, and sort the opened lines according to the objective function of the optimal solution;

[0091] Step 2-1: Calculate the AC optimal power flow after the line is opened for each line in the set of branches that can be opened, Ω lc .

[0092] Step 2-2: Sort in ascending order according to the objective function value of the optimal solution of the optimal power flow in Step 2-1.

[0093] (3) Stage 3 (detailed analysis stage): Using the branches in the set of branches that can be opened, Ω lc screened out in the first stage as slack variables, construct a slack sub-problem; solve it using the branch and bound method according to the sorting order in the second stage, and finally determine the optimal solution of the AC optimal transmission switching and the line outage plan.

[0094] Step 3-1: Using the lines in Ω lc as slack variables, construct a slack sub-problem;

[0095] Step 3-2: According to the branch sorting order in Stage 2, denote the first-ranked branch in Ω lc as z ij . For the topology after z ij is disconnected, perform island detection. If an island exists, set z ij =1 and go to Step 3-6; otherwise, go to Step 3-3;

[0096] Step 3-3: Fix the state of z ij to be disconnected and closed respectively, that is, z ij =0 and z ij =1, and use IPOPT to solve the two slack sub-problems respectively. Denote the optimal solutions as and

[0097] Step 3-4: If or does not exist, set z ij =1, Otherwise, if or does not exist, then let z ij = 0,

[0098] Step 3-5: If all the slack variables are in discrete states (0 or 1), obtain the discrete optimal solution of the AC optimal transmission switching problem, output the line outage plan, and terminate the calculation; otherwise, go to Step 3-6.

[0099] Step 3-6: Remove z lc from Ω ij , Ω lc = Ω lc - z ij ; if then obtain the discrete solution of the AC optimal transmission switching problem, output the line outage plan, and terminate the calculation; otherwise, fix the state of z ij , update the slack subproblem, go to Step 3-2, and continue the branch and bound calculation.

[0100] The variables X of the slack problem include the state variables x = {θ, V m , P G , Q G} of the AC optimal power flow and the slack variable z ij . In the above algorithm, the IPOPT solver is used for solving, and other solvers or algorithms can also be used for substitution. Due to the different performances of different solvers, different effects will be shown.

[0101] In the first stage, there is no case where the slack problem has no solution, because the base state solution and the initial network topology state must be a feasible solution of the slack problem, so the slack problem must have an optimal solution. In Steps 1-4 and 1-5, the quotient gradient system is used to first calculate the feasible solution of the slack problem, and then taking this feasible solution as the initial point, the optimal solution of the slack problem is solved by the IPOPT solver, which can effectively improve the computational robustness and efficiency and effectively avoid the situation of calculation failure. However, in the branch and bound process of the third stage, the slack subproblem may have no solution, that is, and may not exist.

[0102] When more lines are selected in the first stage, Ω lc can be grouped according to the sorting results of the second stage to further reduce the variable dimension of the slack subproblem in the third stage. In the third stage, first perform branch and bound on the branch groups with higher sorting. After a group is completely calculated, continue to perform branch and bound calculation on the next group.

[0103] In addition, in the branch and bound process of stage three, when the slack variables of the relaxed optimal solution are mostly in discrete states and there are fewer continuous states, it is possible to preferentially consider performing branch and bound calculations on the non-discrete branches, which will improve the calculation efficiency and quickly obtain a high-quality discrete solution.

[0104] The present invention is specifically described using the 118-bus system. This system contains 118 buses, 19 generators, and 185 branches. The total generation capacity is 5859.2 MW, and the total load demand is 3668 MW / 1438 Mvar. This test case is widely used as a test case in the optimal transmission switching problem. The lower and upper limits of the bus voltage magnitude constraints are set to [0.94, 1.06] respectively. To distinguish it from other 118-bus systems in the present invention, this test case is called case118_Carnegie, and the objective function of its base case solution is 1736.28 $.

[0105] In stage one, 11 branches that would cause islands when disconnected are screened out, and a relaxed problem with the remaining branches as slack variables is constructed. The states of the slack variables of the obtained relaxed optimal solution are shown in Table 1 and Figure 3 as follows. There are a total of 16 branches with slack variable values less than 0.98. Therefore, these 16 branches are used as the set of switchable branches for the subsequent stage.

[0106] In stage two, the AC optimal power flow when each of the 16 branches in stage one is separately disconnected is calculated. The results are shown in Table 1. Among them, when 9 branches are disconnected, the solution is better than the base case solution; when 6 branches are disconnected, the solution is worse than the base case solution; and when 1 branch (branch 118) is disconnected, there is no solution. Therefore, these 15 branches with solutions can be divided into two groups according to whether they are better than the base case solution, and sorted in ascending order of the objective function within the group. Then the first group of branches and their sorting is Ω1 = {137, 146, 121, 120, 117, 27, 177, 44, 179}, and the second group of branches and their sorting is Ω2 = {43, 95, 149, 102, 32, 136}.

[0107] Table 1 Results of stage one and stage two of case118_Carnegie test case

[0108]

[0109] In stage three, first, a relaxed sub-problem with the first set of branch sets Ω1 as slack variables is constructed, and in the order within the group, the 137th branch is used as the branching node for calculation first. When the 137th branch is closed, the objective function of the optimal solution of the relaxed sub-problem is $1620.1; when the 137th branch is open, the objective function of the optimal solution of the ACOTS relaxed sub-problem is $1462.1. Therefore, the 137th branch is fixed to be open. At the same time, the relaxed optimal solution when the 137th branch is open is a discrete solution. Among all the slack variables, the branches {27, 117, 120, 121, 137, 146, 179} are in the open state, and the branch {44} is in the closed state. Therefore, after grouping, only two relaxed sub-problems need to be solved to obtain the solution of this optimal transmission switching sub-problem, and the objective function can be reduced by 15.81%.

[0110] Based on the results of the first group, a relaxed sub-problem with the second set of branch sets Ω2 as slack variables is constructed. The solution process of branch-and-bound for this branch group is shown in Table 2, with the order within the second group as the branching order. First, the slack variable corresponding to the 43rd branch is used as the branching node, and the optimal solutions of the relaxed sub-problems when the 43rd branch is closed and open are calculated respectively. By comparing the objective function values of the two relaxed optimal solutions, the state of the 43rd branch is determined. In this example, the 43rd branch is determined to be open. Since the optimal solution of the relaxed sub-problem when the 43rd branch is open is not a discrete solution, the above process needs to be further repeated to calculate the states of the subsequent branches. Finally, the solution of this optimal transmission switching sub-problem is obtained, that is, the branches {43, 95, 102, 136, 149} are open, and the branch {32} is closed.

[0111] It should be noted that grouping can reduce the dimension of the relaxation problem, and a discrete relaxation solution may be directly found during the solution process, further reducing the computational effort. For example, in this problem, if the 15 switchable branches are not grouped, a discrete relaxation solution will be found only when branching and bounding are calculated up to the 14th branch, and the computational effort will increase significantly.

[0112] Table 2 Branch-and-bound calculation process of the second group of branches in stage three of case118_Carnegie example

[0113]

[0114] For the problem of AC optimal transmission switching, the present invention proposes a relaxation-based staged AC optimal transmission switching algorithm, which can robustly and efficiently solve high-quality optimal transmission switching solutions. Specifically, first, a relaxation model for AC optimal transmission switching is constructed, the properties of the relaxed optimal solution are explored, and a branch screening method based on the relaxed optimal solution is proposed accordingly. To overcome the numerical calculation problems faced in the process of solving the relaxation model, a calculation form is proposed, which first calculates the feasible solution of the relaxation problem through the quotient gradient system and then solves the optimal solution of the relaxation problem through IPOPT, which can quickly and robustly solve the relaxed optimal solution. Finally, a three-stage AC optimal transmission switching algorithm is given and verified and compared in a large number of test cases.

Claims

1. A three-stage AC optimal transmission switching method based on a relaxation model, characterized in that: The steps include: Step 1: In the first stage, the optimal solution of the relaxation model is solved to select the disconnectable branches and reduce the system scale; Step 2: In the second stage, the optimal solution of AC optimal power flow for disconnecting a single branch is solved for sorting; Step 3: In the third stage, the branch and bound method is used to solve the problem according to the sorting order of the second stage, and finally the optimal solution and branch disconnection plan for AC optimal transmission switching are determined.

2. The three-stage AC optimal transmission switching method based on the relaxation model according to claim 1, characterized in that: In step 1, the first stage of selecting disconnectable branches by solving the optimal solution of the relaxation model specifically includes the following steps: Step 11: Screening out isolated branches. Since the disconnection of branches cannot cause an island, in order to avoid network decoupling or load impact, the branches that will cause an island after a single disconnection are first screened out, and the remaining disconnectable branches are recorded as Ω. l ; Step 12: Ω l The middle branch is a slack variable, which is used to construct a slack problem; Step 13, construct a quotient gradient system corresponding to the relaxation problem; Step 14: Solve in the ground state and the initial topological state Z0 as the relaxation initial point The integral quotient gradient system is closed until it reaches a stable equilibrium point, and the end point is recorded as This point is a feasible solution to the relaxed problem; Step 15: As the initial point, IPOPT is used to solve the relaxation problem. If the solution is successful, the optimal solution is recorded as X * , go to step 16; otherwise, randomly update the relaxation initial point Repeat steps 14-15; Step 16: Let τ be a small positive scalar. If Then the relaxed optimal solution X * Compared with the ground state solution If the improvement is limited, terminate the calculation; otherwise, go to step 17; Step 17: If X * If all the slack variables are in the discrete state of 0 or 1, the optimal solution to the AC optimal transmission switching problem has been directly solved, the line disconnection plan is output, and the calculation is terminated; otherwise, go to step 18; Step 18: X * Middle ij The branches corresponding to the slack variables ≤0.98 are selected and constructed as the disconnectable branch set Ω lc .

3. The three-stage AC optimal transmission switching method based on the relaxation model as claimed in claim 1, characterized in that: In step 12, the optimal transmission switching problem is a mixed integer programming problem based on the optimal power flow model. On the basis of the AC optimal power flow model, a binary variable Z is added to the power flow equation and thermal limit constraint. ij The optimal AC transmission switching model is obtained; The objective function is: min.f=ax 2 +bx+c Among them, f is the form of fuel cost as the objective function, which is a continuously differentiable function, a, b, c are constant vectors, representing the structure of the objective function, and x is a variable vector, which includes the active power and reactive power of the generator node, the voltage amplitude and voltage phase angle of all nodes, and the state variables of the transmission line in this problem; The constraint set of the AC optimal transmission switching model is in the following form, including AC power flow equations, line thermal limit constraints, generator output upper and lower limit constraints, and node voltage constraints: V i min ≤V i ≤V i max i∈{1,2,...,N B } Among them, the line state variable Z ij is a 0-1 variable that characterizes the state of the transmission line ij. ij When it is equal to 1, it means that the circuit ij is in a closed state. ij When it is equal to 0, it means that the line ij is in the disconnected state; B ,N L ,N G Respectively represent the number of busbar nodes, transmission lines and generators in the power system; the constant G ij and B ij are the equivalent conductance and equivalent susceptance of the line from node i to node j, respectively, Li and Q Li are the active load value and reactive load value of node i respectively; variable θ i is the voltage phase angle of node i, and θ ij =θ i -θ j , where the voltage phase angle of the balance node is set to a constant; the variable V i is the voltage amplitude of node i, P Gi and Q Gi are respectively the active output power value and reactive output power value of the generator connected to node i, and the expression S fl and S tl are the apparent powers at the beginning and end of line l respectively; The AC optimal transmission switching model can be abbreviated as: min f(x) s.t.H T (x,Z)=0 G T (x,Z)≤0 Where x represents a continuous variable, including all variables of the optimal power flow model; Z is a discrete line state variable; H T (x,Z) is an equality constraint; G T (x,Z) represents an inequality constraint; By relaxing discrete variables into continuous variables with upper and lower bound constraints, a relaxation model for AC optimal transmission switching is constructed, and then the optimization technology for solving continuous variables is used to solve it. The objective function is: min.f=ax 2 +bx+c Among them, f is the form of fuel cost as the objective function, which is a continuously differentiable function, a, b, c are constant vectors, representing the structure of the objective function, and x is a variable vector, which includes the active power and reactive power of the generator node, the voltage amplitude and voltage phase angle of all nodes, and the state variables of the transmission line in this problem; The discrete variable of the AC optimal transmission switching model is only the 0-1 variable Z ij , so Z ij The relaxation problem of AC optimal transmission switching can be constructed by relaxing the variable interval to a continuous variable of [0,1]. The complete constraints of the relaxation problem are as follows: 0≤z ij ≤1 V i min ≤V i ≤V i max i∈{1,2,...,N B } Among them, z ij is a 0-1 variable Z ij The slack variables after continuation; The complete relaxation model for AC optimal transmission switching, in which there are no discrete variables, can be abbreviated as follows: min f(x) s.t.H s (x)=0 G s (x)≤0 Among them, x is a variable set, including all variables of the optimal power flow model and slack variables z ij .

4. The three-stage AC optimal transmission switching method based on the relaxation model as claimed in claim 1, characterized in that: In step 13, the quotient gradient system is: where DH(x) is the Jacobian matrix of H(x).

5. The three-stage AC optimal transmission switching method based on the relaxation model as claimed in claim 1, characterized in that: In step 2, the second stage is to sort by solving the optimal solution of the AC optimal power flow for disconnecting a single branch, which specifically includes the following steps: Step 21: For the disconnectable branch set Ω lc For each line in the circuit, calculate the optimal AC power flow after the line is disconnected; Step 22: Sort the optimal solutions of the optimal power flow in step 21 from small to large according to the objective function values.

6. The three-stage AC optimal transmission switching method based on the relaxation model as claimed in claim 1, characterized in that: In step 21, the AC optimal power flow is Minimize f(x) stH(x)=0 in, is the objective function; For the constraint set, the inequality constraints have been converted to equality constraints by adding slack variables.

7. The three-stage AC optimal transmission switching method based on the relaxation model as claimed in claim 1, characterized in that: In step 3, the third stage uses the branch and bound method to solve according to the sorting order of the second stage, and finally determines the optimal solution and branch disconnection plan of AC optimal transmission switching, which specifically includes the following steps: Step 31, with Ω lc The middle line is used as a slack variable to construct a relaxed subproblem; Step 32: According to the order of branches in stage 2, record Ω lc The first branch in the ij , yes z ij After disconnection, the topology performs island detection. If an island exists, z ij =1, go to step 36; otherwise, go to step 33; Step 33: Let z ij The state is fixed to open and closed, that is, z ij =0 and z ij =1, IPOPT is used to solve the two relaxed subproblems respectively, and the optimal solutions are recorded as and Step 34: If or does not exist, then let z ij =1, Otherwise, if or If it does not exist, let z ij =0, Step 35: If The slack variables are all in discrete state 0 or 1, and the discrete optimal solution of the AC optimal transmission switching problem is obtained, and the line disconnection plan is output, and the calculation is terminated; otherwise, go to step 36; Step 36: From Ω lc Remove z ij ,Ω lc =Ω lc -z ij ,like Then the discrete solution of the AC optimal transmission switching problem is obtained, the line disconnection plan is output, and the calculation is terminated; otherwise, z ij The state is fixed, the relaxed subproblem is updated, and the process goes to step 32 to continue the branch and bound calculation.

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