Alternating current optimal transmission switching relaxation problem solving method based on two-stage dynamic trajectory and pseudo-transient continuity

Abstract

The invention discloses an alternating current optimal transmission switching relaxation problem solving method based on a two-stage dynamic trajectory and pseudo-transient continuity. The method comprises the following steps: constructing an alternating current optimal transmission switching relaxation model; constructing an augmented quotient gradient system corresponding to the alternating current optimal transmission switching relaxation model; randomly selecting an initial point x0, and performing integral augmentation on the quotient gradient system from x0; carrying out integral iterative calculation on the augmented quotient gradient system by adopting a pseudo-transient continuous iterative formula based on an Euclidean Euler method; when the target function descent rate of the trajectory of the augmented quotient gradient system is smaller than 0.001% / step, integration is stopped, and the termination point at the moment is recorded as x *; x * is used as an input point, an IPOPT solver is adopted to solve the alternating current optimal transmission switching model, and an optimal solution is obtained through solving. According to the method, a dynamic method and a static method are combined, respective advantages of the two methods are taken, a pseudo-transient continuous iteration formula is provided in the integration process of the augmented quotient gradient system, and the optimal transmission switching relaxation problem which is difficult to solve by the power system can be quickly solved on the basis of ensuring the solution robustness.

Description

Technical Field

[0001] The present invention relates to the technical field of power systems, and in particular to a method for solving the alternating current optimal transmission switching relaxation problem based on two-stage dynamic trajectories and pseudo-transient continuity. Background Art

[0002] With the large-scale access of new energy and the increasing volatility of the load side, the uncertainty of power system operation has increased significantly, posing higher requirements for the flexibility and security of system scheduling. Optimal Transmission Switching (OTS), as a method to optimize the operation mode by changing the open / closed state of transmission lines, has been widely studied and applied. It can effectively reduce the system operation cost, relieve congestion, and improve network security. However, the alternating current OTS problem essentially belongs to a large-scale mixed integer non-linear programming (MINLP) problem, with high non-convexity and combinatorial complexity, making it extremely difficult to solve directly. Conventional optimization algorithms are prone to convergence failure or low solution efficiency when dealing with large-scale systems.

[0003] In response to the above challenges, existing research mainly focuses on two aspects of improvement: on the one hand, by constructing a continuous relaxation model to avoid the combinatorial explosion problem caused by 0-1 variables, such as using methods like linear relaxation, second-order cone relaxation, and semi-definite relaxation; on the other hand, using heuristic algorithms or branch-and-bound strategies to search on the original discrete model. However, the former has limitations in terms of solution accuracy and feasibility and is difficult to meet the actual requirements of engineering scheduling; the latter has high computational complexity, relies on heuristic rules, and has unstable results, and does not yet have wide engineering usability.

[0004] For the relaxation problem of alternating current optimal transmission switching, it is quite difficult to directly solve it through optimization solvers such as IPOPT, and these solvers often encounter calculation failures. To improve the stability and efficiency of the solution, some research has introduced dynamic system methods, attempting to approximate the equilibrium solution of the optimization problem through the trajectory of the dynamic system. Although these methods show good convergence properties in some scenarios. However, since the stable equilibrium point of the dynamic system is usually searched by numerical integration, this often requires tracking the precise trajectory of the dynamic system, which may bring disadvantages in terms of calculation speed. Especially after the trajectory moves near the stable equilibrium point, the convergence speed will further decrease. Single dynamic methods often have problems such as slow convergence in the integration process and inaccurate paths, and it is difficult to combine with existing efficient static optimization solvers to achieve solution acceleration.

[0005] Therefore, there is an urgent need for a unified solution framework that combines the robustness of dynamic methods and the efficiency of static methods to achieve fast, stable, and accurate solution of the alternating current optimal transmission switching relaxation problem. Summary of the Invention

[0006] Objective of the Invention: The present invention provides a method for solving the AC optimal transmission switching relaxation problem based on two-stage dynamic trajectory and pseudo-transient continuation, which combines a dynamic method (such as the augmented Lagrangian gradient system method) with a static method (such as the interior point method), absorbs the advantages of both types of methods, and achieves fast solution of difficult relaxation problems while ensuring the robustness of the solution; at the same time, a pseudo-transient continuation iteration formula during the integration process of the augmented Lagrangian gradient system is proposed to further improve the calculation speed.

[0007] Technical Solution: A method for solving the AC optimal transmission switching relaxation problem based on two-stage dynamic trajectory and pseudo-transient continuation according to the present invention includes the following steps:

[0008] Step 1, construct an AC optimal transmission switching relaxation model;

[0009] Step 2, construct an augmented Lagrangian gradient system corresponding to the AC optimal transmission switching relaxation model;

[0010] Step 3, randomly select an initial point x0, and integrate the augmented Lagrangian gradient system starting from x0;

[0011] Step 4, perform integration iteration calculation on the augmented Lagrangian gradient system using a pseudo-transient continuation iteration formula based on the Euler method;

[0012] Step 5, when the objective function descent rate of the trajectory of the augmented Lagrangian gradient system is less than 0.001% / step, that is, when the objective function descent between the two points before and after each step of integrating the augmented Lagrangian gradient system is less than 0.001%, stop the integration, and record the termination point at this time as x*;

[0013] Step 6, use the IPOPT solver to solve the AC optimal transmission switching model with x* as the input point to obtain the optimal solution.

[0014] Further, in Step 1, 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, binary variables Z are added to the power flow equation and thermal limit constraints. ij Obtain the AC optimal transmission switching model;

[0015] The objective function is:

[0016] min.f = ax 2 + bx + c

[0017] Among them, f is in the form of a fuel cost as the objective function, which is a continuously differentiable function. a, b, and c are constant vectors representing the structure of the objective function, and x is a variable vector, which in this problem includes the active power and reactive power of the generator nodes, the voltage magnitudes and phase angles of all nodes, and the transmission line state variables.

[0018] 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:

[0019]

[0020] Among them, the line state variable Z ij is a 0-1 variable representing the state of transmission line i-j. When Z ij is equal to 1, it means that line i-j is in the closed state. When Z ij is equal to 0, it means that line i-j is in the open state. Among them, N B , N L , N G represent the numbers of bus nodes, transmission lines, and generators in the power system respectively; 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; 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; variable V i is the voltage magnitude 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. Expressions S fl and S tl are the apparent powers at the sending end and receiving end of line l respectively.

[0021] The AC optimal transmission switching model is abbreviated as:

[0022] min f(x)

[0023] s.t. H T (x, Z) = 0

[0024] G T (x, Z) ≤ 0

[0025] Among them, x represents continuous variables, including all variables of the optimal power flow model; Z is the discrete line state variable; HT (x, Z) is an equality constraint; G T (x, Z) represents an inequality constraint;

[0026] By relaxing discrete variables into continuous variables with upper and lower bound constraints, a relaxed model for AC optimal transmission switching is constructed, and then an optimization technique for solving continuous variables is used for solution;

[0027] The objective function is:

[0028] min.f = ax 2 + bx + c

[0029] 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;

[0030] The discrete variable of the AC optimal transmission switching model is only the 0-1 variable Z ij , therefore, relaxing Z ij into a continuous variable with a variable interval of [0, 1] can construct a relaxed problem for AC optimal transmission switching. The complete constraints of the relaxed problem are as follows:

[0031]

[0032] 0 ≤ z ij ≤ 1

[0033]

[0034] where z ij is the relaxed variable after continuousization of the 0-1 variable Z ij ;

[0035] The complete relaxed model of AC optimal transmission switching, in which there are no discrete variables, is abbreviated as the following form:

[0036] minf(x)

[0037] s.t. H s (x) = 0

[0038] G s (x) ≤ 0

[0039] where x is a set of variables, including all variables of the optimal power flow model and the relaxed variable z ij .

[0040] Furthermore, in step 2, the augmented quotient gradient system is abbreviated as AQGS, which is related to the objective function and non-linear constraints:

[0041]

[0042] Where DH(x) is the Jacobian matrix of H(x), is the gradient vector of the objective function, is a positive constant penalty factor.

[0043] Furthermore, in step 4, the pseudo-transient continuous method is used for the differential equation integration process, and for the augmented quotient gradient system The pseudo-transient continuous iteration formula based on the Euclidean Euler method is:

[0044]

[0045] In the iterative solution, the step size δ n The SER (Switched Evolution Relaxation) technology is used to implement step length control. The step length control formula is:

[0046]

[0047] where φ(ξ) is controlled by the following formula:

[0048]

[0049] Starting from the initial step length, the step length will increase rapidly and approach the equilibrium point quickly.

[0050] Beneficial effects: Compared with the prior art, the present invention has the following remarkable advantages: (1) Improving the adaptability and global robustness of the initial solution point; based on the AC optimal transmission switching relaxation model, the present invention introduces an augmented quotient gradient system to construct a continuous dynamic trajectory, which can perform trajectory integration starting from any initial point and does not depend on the narrow convergence domain of traditional static solvers. Through theoretical and numerical analysis, it is found that the convergence domain of the augmented quotient gradient system is continuous and controllable, and can effectively cover the regions that are difficult to converge for conventional interior point methods, thereby significantly enhancing the robustness and universality of the solution and adapting to large-scale systems and complex working conditions; (2) Introducing a pseudo-transient continuous iteration formula to achieve rapid approximation to the optimal solution region; during the integration process of the augmented quotient gradient system, the present invention designs a pseudo-transient continuous iteration strategy based on the Euler method, which, combined with the step size control technique, can, on the basis of ensuring the trajectory stability, greatly accelerate the integration progress by non-uniformly increasing the step size. Compared with traditional integrators (such as ODE15), the number of integration steps in typical examples is reduced by more than 90%, significantly improving the calculation speed in the dynamic stage and creating conditions for quickly entering the convergence domain of the static method; (3) Constructing a unified dynamic-static two-stage decoupled solution framework to enhance the solvability of difficult problems; the present invention proposes a two-stage dynamic trajectory method, which decouples and combines the dynamic trajectory search and the static optimization refinement process; on the one hand, the dynamic system continuously evolves to approach the target solution region; on the other hand, when the dynamic integration termination point is used as the initial point, the static solver (such as IPOPT) can obtain a high-quality local solution with a high probability, avoiding the risk of the static method falling into divergence or sub-optimal points, and substantially solving the technical bottleneck of "unable to converge" of traditional solvers in the AC OTS relaxation model; (4) Having wide adaptability and engineering application potential; the present invention does not depend on a specific power grid topology structure or parameter setting, and the method has been verified on multiple standard systems (such as case69, case118, case141), showing good convergence characteristics and generalization ability. Since the method structure is clear and the modules are separable, it can be easily embedded in existing dispatching platforms and promoted as a decision-making support tool for application in actual dispatching plans or emergency control strategies; In summary, the present invention realizes multi-dimensional breakthroughs in solution accuracy, efficiency, and robustness through clear technical designs, has obvious technical progressiveness and remarkable practical value, and can effectively support the implementation of complex optimization problems such as AC OTS in engineering applications. Brief Description of the Drawings

[0051] Figure 1 It is a schematic diagram of the two-stage dynamic trajectory method of the present invention.

[0052] Figure 2 It is a convergence domain diagram of the IPOPT solver for the case69 system of the present invention.

[0053] Figure 3Schematic diagram of the first-stage solution process of the two-stage dynamic trajectory method starting from the initial point where IPOPT does not converge for the case69 system of the present invention, and the trajectory of the augmented quotient gradient system moves into the IPOPT convergence domain around the optimal solution.

[0054] Figure 4 Enlarged view of the area around the optimal solution of the present invention, and schematic diagram of stopping integration when the trajectory of the augmented quotient gradient system moves to around the optimal solution.

[0055] Figure 5 Schematic diagram of the trajectories of the augmented quotient gradient system integrated by the pseudo-transient continuation method (PTC) and the ODE15 integrator respectively for the case69 system of the present invention starting from the same initial point. Detailed implementation manners

[0056] As Figure 1 shown, a method for solving the AC optimal transmission switching relaxation problem based on two-stage dynamic trajectory and pseudo-transient continuation includes the following steps:

[0057] Step 1: Construct an AC optimal transmission switching relaxation model;

[0058] Step 2: Construct an augmented quotient gradient system corresponding to the AC optimal transmission switching relaxation model;

[0059] Step 3: Randomly select an initial point x0, and integrate the augmented quotient gradient system starting from x0;

[0060] Step 4: Use the pseudo-transient continuation iteration formula based on the Euclidean Euler method to perform integral iteration calculation on the augmented quotient gradient system;

[0061] Step 5: When the objective function descent rate of the trajectory of the augmented quotient gradient system is less than 0.001% / step, that is, when the objective function descent between the two points before and after each step of integrating the augmented quotient gradient system is less than 0.001%, stop the integration, and record the termination point at this time as x*;

[0062] Step 6: Use the IPOPT solver to solve the AC optimal transmission switching model with x* as the input point to solve the optimal solution.

[0063] 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, binary variable Z is added to the power flow equation and the thermal limit constraint ij to obtain the AC optimal transmission switching model;

[0064] The objective function is:

[0065] min.f = ax 2 + bx + c

[0066] Among them, f is in the form of a fuel cost objective function, which is a continuously differentiable function. a, b, and c are constant vectors representing the structure of the objective function, and x is a variable vector, which includes the active power, reactive power of the generator nodes, the voltage magnitudes, voltage phase angles of all nodes, and the transmission line state variables in this problem;

[0067] 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:

[0068]

[0069] 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; 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; 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; variable V i is the voltage magnitude 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. Expressions S fl and S tl are the apparent powers at the sending end and receiving end of line l respectively;

[0070] The AC optimal transmission switching model is abbreviated as:

[0071] min f(x)

[0072] s.t. H T (x, Z) = 0

[0073] G T (x, Z) ≤ 0

[0074] Among them, x represents continuous variables, including all variables of the optimal power flow model; Z is the discrete line state variable; HT (x, Z) is an equality constraint; G T (x, Z) represents an inequality constraint;

[0075] By relaxing discrete variables into continuous variables with upper and lower bound constraints, a relaxed model of AC optimal transmission switching is constructed, and then optimization techniques for solving continuous variables are used for solution;

[0076] The objective function is:

[0077] min.f = ax 2 + bx + c

[0078] 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;

[0079] 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 a relaxed problem of AC optimal transmission switching. The complete constraints of the relaxed problem are as follows:

[0080]

[0081] 0 ≤ z ij ≤ 1

[0082]

[0083] where z ij is the relaxed variable after continuousization of the 0-1 variable Z ij ;

[0084] The complete relaxed model of AC optimal transmission switching, in which there are no discrete variables, is abbreviated in the following form:

[0085] minf(x)

[0086] s.t. H s (x) = 0

[0087] G s (x) ≤ 0

[0088] where x is the variable set, including all variables of the optimal power flow model and the relaxed variable z ij .

[0089] For the relaxation problem of AC optimal transmission switching, it is very difficult to directly solve it through solvers such as IPOPT, and these solvers often encounter calculation failures. The augmented merchant gradient system transforms the optimal power flow problem into a nonlinear dynamic system, and equivalently transforms the solution of the local optimal solution into the search for the regular stable equilibrium point of the dynamic system. The robust solution problem of applying the augmented merchant gradient system method to solve the relaxation problem will be addressed.

[0090] Since the stable equilibrium point of the dynamic system is usually searched by numerical integration methods, this often requires tracking the precise trajectory of the dynamic system, which may bring disadvantages in terms of calculation speed. Especially after the trajectory moves near the stable equilibrium point, the convergence speed will be further reduced. The present invention proposes a two-stage dynamic trajectory method, which combines a dynamic method (such as the augmented merchant gradient system method) with a static method (such as the interior point method), absorbs the respective advantages of the two types of methods, and can achieve fast solution of difficult relaxation problems while ensuring the robustness of the solution. Among them, the dynamic method is to solve differential equations, and the static method is to solve algebraic equations. At the same time, in order to further improve the calculation speed in the integration stage, a method of pseudo-transient continuous integration is proposed.

[0091] The basic idea of the two-stage dynamic trajectory method is as follows: for difficult problems and difficult test cases, first construct an augmented merchant gradient system, and search around the regular stable equilibrium point (i.e., the optimal solution) by tracking the trajectory of the dynamic system, and then quickly solve it by the static method. The two-stage dynamic trajectory method has natural computational advantages for the following reasons:

[0092] (1) The augmented merchant gradient system is completely stable and insensitive to the initial point. Its convergence domain is larger than that of static methods (such as the interior point method), and the convergence domain is continuous. Therefore, it can continuously solve around the optimal solution starting from any initial point;

[0093] (2) The convergence domain of the static method is irregular and discontinuous, and the boundary of the convergence domain may have a fractal structure. However, it often has a small connected convergence domain around the optimal solution. When the initial point is around the optimal solution, it can greatly improve the calculation success rate of the static method.

[0094] Figure 1 A schematic diagram of the solution process of the two-stage dynamic trajectory method is given. Starting from an initial point x0, the augmented merchant gradient system can bring the trajectory into the convergence domain (yellow domain) of the static method located around the optimal solution, and then using the static method to solve can ensure the convergence rate and further improve the calculation speed.

[0095] The two-stage dynamic trajectory method will be described in detail below using the case69 system as an example. Figure 2The convergence domain of the IPOPT solver in solving the AC optimal transmission switching relaxation problem in this example is plotted, where the red points are the initial point positions where IPOPT converges successfully, and the blue points are the initial point positions where IPOPT diverges. The shown initial point plane is a two-dimensional plane determined by three points in the case69 system state space. This plane can be constructed by the following method:

[0096] Step 1: Select three points that are not on a straight line, denoted as points

[0097] Step 2: Define a two-dimensional plane in a high-dimensional space:

[0098] y = a+(b - a)s+(c - a)t

[0099] where, are the coordinates of the defined two-dimensional plane, are the coordinates of the points on the plane.

[0100] For the initial point plane, the variable range of s and t is [-4, 4], where point a is the optimal solution. Therefore, this initial point plane is a two-dimensional plane passing through the optimal solution, and it can be observed that:

[0101] (1) The convergence domain of the IPOPT solver is discontinuous, scattered, and irregular, and most of the initial points with partial convergence are not connected into a domain state.

[0102] (2) The convergence domain of IPOPT forms a closed region around the optimal solution, indicating that when the initial point is relatively close to the optimal solution, 100% convergence or a high probability of convergence can be achieved.

[0103] To further verify, in the case69, case118, and case141 systems, with the relaxed optimal solution as the center, by controlling the distance range of the initial point from the relaxed optimal solution, the convergence characteristics of the IPOPT solver are tested.

[0104] The initial points are randomly selected, and the distance range from the optimal solution is controlled by the following formula:

[0105]

[0106] where, x * is the relaxed optimal solution; is a scalar coefficient used to control the distance range of the random initial point from the optimal solution; are the upper and lower limit values of each variable respectively; diag(x) represents converting the vector x into a diagonal matrix with x as the diagonal elements, is a random vector in the range [-1, 1] with the same dimension as x * ​

[0107] The case69, case118, and case141 systems generate initial points with different coefficients The test results of the IPOPT solving the relaxation problem when generating initial points and completely randomly generating initial points show that:

[0108] (1) Compared with completely randomly selecting initial points, the success rate of solving is higher when selecting initial points around the optimal solution;

[0109] (2) Among the results of selecting initial points around the optimal solution, as the coefficient becomes smaller, that is, when the initial point is closer to the optimal solution, the IPOPT convergence rate shows an increasing trend; when the coefficient is small enough, case69 and case141 can achieve 100% solution, and case118 can also reach a convergence rate of more than 95%.

[0110] (3) For the case118 system, when the coefficient is small to 1e - 7, IPOPT still cannot reach 100% convergence, but compared with completely randomly selecting initial points, the convergence domain has been greatly improved.

[0111] The above results verify that robust solution can be achieved by IPOPT when the initial point is within a certain distance around the optimal solution, which is consistent with what is shown in Figure 1 . Randomly select an initial point where IPOPT does not converge in Figure 2 as the initial point of the augmented - quotient - gradient system. Figure 3 The blue curve in is the trajectory of the augmented - quotient - gradient system, and this trajectory moves towards the optimal solution and enters the IPOPT convergence domain. Before the trajectory of the system moves to the equilibrium point, that is, when Figure 4 is a small value but not 0, actively stop the integration. The enlarged view in * shows that the trajectory stops before converging to the optimal solution. Denote the end - point of the trajectory as x1. At this time, x1 is within the IPOPT convergence domain. Taking x1 as the IPOPT initial point can directly and quickly solve to the optimal solution x

[0112] To further accelerate the solution speed of the two-stage dynamic trajectory method, the integration process of the differential equation in the dynamic method of the first stage is optimized. The Pseudo-Transient Continuation (PTC) method is adopted, which can quickly locate the steady-state solution of the nonlinear dynamic system. The idea of the pseudo-transient continuation method is to continuously increase the step size to infinity, making the corresponding vector field approach zero. It is a type of variable-step method that accelerates the search for the final steady-state solution by calculating inaccurate integral trajectories with large step sizes. Since the intermediate integration process is not the focus of the two-stage dynamic trajectory method, the sacrificed integration accuracy by using the pseudo-transient continuation method has no impact on the final result of the two-stage dynamic trajectory method. Therefore, the pseudo-transient continuation method is very suitable for solving in the dynamic method stage.

[0113] Commonly used solution methods for ordinary differential equations mainly include the trapezoidal method, implicit Euler method, Runge-Kutta method, Adams method, etc. For the augmented quotient gradient system This paper will derive the iterative formula of the pseudo-transient continuation method based on the implicit Euler method.

[0114] The iterative formula of the implicit Euler method is:

[0115] x n+1 = x n + δ n DA(x n+1 )

[0116] In the formula, δ n is the step size of the nth iteration.

[0117] The initial value of the iteration is calculated by the explicit Euler method, and the initial value z n+1 can be obtained by solving the following equation:

[0118] G(ξ) = ξ - δA(ξ) - x n

[0119] The iterative formula solved by the Newton method can be expressed as:

[0120] ξ k+1 = x n - (I - δ n DA(ξ k )) -1 (ξ k - δ n A(ξ k ) - x n )

[0121] Substitute ξ0 = x n into it, then the first-step iterative formula solved by the Newton method is:

[0122]

[0123] Therefore, the pseudo-transient continuous iteration formula based on the Euler method is as follows:

[0124]

[0125] In the iterative solution, the step size δ n The step size control can be achieved by using the SER (Switched Evolution Relaxation) technique, and the step size control formula is:

[0126]

[0127] where φ(ξ) is controlled by the following formula:

[0128]

[0129] According to the step size control formula, starting from the initial step size, the step size will increase rapidly and approach the equilibrium point quickly.

[0130] In case69, starting from the Figure 3 shown initial point, only 16 integrations are required to reach the position around the optimal solution shown in Figure 4 by using the pseudo-transient continuous method, while 160 integrations are required by using the ODE15 integrator in MATLAB. In comparison, the pseudo-transient continuous method can improve the calculation speed by 93.8%. Figure 5 shows the integration trajectories of the two methods. There are certain differences between the two trajectories, but the errors are small, and the final trajectory termination positions are very close. The pseudo-transient continuous method can greatly improve the calculation speed of the dynamic method while ensuring the accuracy.

[0131] In summary, by adopting the two-stage dynamic trajectory method, the relaxation problem of the optimal transmission switching of alternating current can be solved quickly and robustly, making the method of screening the branches of the optimal transmission switching by using the relaxed optimal solution practical.

Claims

1. A method for solving the AC optimal transmission switching relaxation problem based on two-stage dynamic trajectories and pseudo-transient continuity, characterized in that, It includes the following steps: Step 1: Construct an AC optimal transmission switching relaxation model; Step 2: Construct an augmented merchant gradient system corresponding to the AC optimal transmission switching relaxation model; Step 3: Randomly select an initial point x0 and integrate the augmented merchant gradient system starting from x0; Step 4: Use a pseudo-transient continuous iteration formula based on the Euler method for integration iteration calculation of the augmented merchant gradient system; Step 5: When the objective function descent rate of the trajectory of the augmented merchant gradient system is less than 0.001% / step, that is, when the objective function descent between the two points before and after each step of integrating the augmented merchant gradient system is less than 0.001%, stop the integration, and record the termination point at this time as x*; Step 6: Use the IPOPT solver to solve the AC optimal transmission switching model with x* as the input point to obtain the optimal solution.

2. The method for solving the AC optimal transmission switching relaxation problem based on two-stage dynamic trajectory and pseudo-transient continuation as described in claim 1, wherein In Step 1, 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 variable Z is added to the power flow equation and thermal limit constraint ij The AC optimal transmission switching model is obtained.

3. The method for solving the AC optimal transmission switching relaxation problem based on two-stage dynamic trajectory and pseudo-transient continuation as claimed in claim 2, wherein The objective function is: min.f = ax 2 + bx + c Among them, f is in the form of taking 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, x is a variable vector, and in this problem, it includes the active power, reactive power of the generator nodes, voltage amplitudes of all nodes, voltage phase angles, and transmission line state variables; 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: V i min ≤V i ≤V i max i ∈ {1, 2, ..., N B} Among them, the line status variable Z ij is a 0-1 variable representing the status 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 respectively represent the numbers of bus nodes, transmission lines, and generators in the power system; the constants G ij and B ij are respectively the equivalent conductance and equivalent susceptance of the line from node i to node j, P Li and Q Li are respectively the active load value and reactive load value of node i; 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 respectively the active output power value and reactive output power value of the generator connected to node i, and the expressions S fl and S tl are respectively the apparent powers at the sending end and receiving end of line l; The AC optimal transmission switching model is abbreviated as: min f(x) s.t.H T (x,Z) = 0 G T (x,Z) ≤ 0 where x represents continuous variables, including all variables of the optimal power flow model; Z is the discrete line status variable; H T (x, Z) is the equality constraint; G T (x, Z) represents the inequality constraint.

4. The solution method for the AC optimal transmission switching relaxation problem based on two-stage dynamic trajectory and pseudo-transient continuation as described in claim 2, wherein, By relaxing the discrete variables into continuous variables with upper and lower limit constraints, a relaxation model of AC optimal transmission switching is constructed, and then the optimization technology for solving continuous variables is used for solution; The objective function is: min.f = ax 2 + bx + c Among them, f is in the form of taking 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, x is a variable vector, and in this problem, it includes the active power, reactive power of the generator nodes, voltage amplitudes of all nodes, voltage phase angles, and transmission line state variables; The discrete variable of the AC optimal transmission switching model is only the 0-1 variable Z ij , so Z ij is relaxed to a continuous variable with a variable interval of [0,1] to construct the relaxed problem of AC optimal transmission switching. The complete constraints of the relaxed 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 the 0-1 variable Z ij and the slack variable after continuousization; The complete relaxation model of AC optimal transmission switching, in which there are no discrete variables, is abbreviated as the following form: min f(x) s.t.H s (x) = 0 G s (x) ≤ 0 where \(x\) is a set of variables, including all variables of the optimal power flow model and the slack variable \(z\). ij .

5. The solution method for the AC optimal transmission switching relaxation problem based on two-stage dynamic trajectory and pseudo-transient continuation as described in claim 1, characterized in that, In Step 2, the augmented merchant gradient system is abbreviated as AQGS, which is related to the objective function and non-linear constraints: where \(DH(x)\) is the Jacobian matrix of \(H(x)\), and \(\nabla f(x)\) is the gradient vector of the objective function, is a positive constant penalty factor.

6. The method for solving the AC optimal transmission switching relaxation problem based on two-stage dynamic trajectory and pseudo-transient continuation as described in claim 1, wherein In step 4, the pseudo-transient continuation method is adopted for the integration process of the differential equation. For the augmented quotient gradient system The pseudo-transient continuation iteration formula based on the Euler method is as follows: The step size δ in the iterative solution n The SER technique is used to implement step size control, and the step size control formula is as follows: Among them, φ(ξ) is controlled by the following formula: Starting from the initial step size, the step size will increase rapidly and approach the equilibrium point quickly.

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