A Sequence-Modified Second-Order Cone Global Optimization Method Applicable to Radial Distribution Networks

Abstract

This invention provides a sequentially modified second-order cone global optimization method suitable for radial distribution networks. The method includes introducing auxiliary variables to decouple the original non-convex optimization problem in the power system, transforming it into an equivalent bi-level programming problem; using second-order cone relaxation techniques to convexize the upper-level programming problem before centralized solution; further decomposing the lower-level programming problem into multiple quadratic programming subproblems with a single non-convex quadratic constraint; transforming the quadratic programming subproblems into semidefinite programming subproblems using Schur complement, and establishing strong duality gaps using the standard S-procedure; and iteratively calculating and sequentially correcting the optimization results of the decoupled upper and lower-level programming problems using the alternating direction multiplier method to ensure that the non-convex constraints are strictly held. This invention avoids errors caused by convex relaxation, effectively guaranteeing the reliability and global optimality of the results, and providing an accurate and reliable mathematical model for distribution networks with a high proportion of renewable energy.

Description

Technical Field

[0001] The embodiments of the present invention relate to the field of power distribution network optimization operation technology, and in particular to a sequentially modified second-order cone global optimization method applicable to radial power distribution networks. Background Technology

[0002] Currently, under the "dual carbon" goals and the background of the new power system, distributed power sources, represented by wind and solar power, are being connected to the distribution network in large quantities, and my country's new power system is gradually showing the characteristics of a high proportion of new energy sources. As a result, problems such as voltage fluctuations, voltage exceeding limits, transformer capacity overload, local reverse current, and even regional reverse power flow are becoming increasingly serious. Traditional distribution network models are no longer sufficient to support accurate calculations and engineering applications in certain scenarios of the new power system.

[0003] Currently, the most widely used model in the industry is a simplified branch power flow model. This type of model reduces the original complex-domain variable relationships to real-domain variable relationships by applying simplification conditions, and uses different relaxation techniques to relax the original non-convex constraints, thus reducing the difficulty of solving the problem. According to optimization theory, the accuracy of convex relaxation is closely related to the objective function and boundary conditions of the original problem. However, in practical power system applications, due to the diverse objective functions and the various boundary conditions set in the model for application needs, it is extremely difficult to guarantee the accuracy of convex relaxation through a universal method.

[0004] Therefore, there is an urgent need to propose a new type of power distribution network model with low formal complexity, small relaxation error, and wide applicability. Summary of the Invention

[0005] In view of this, embodiments of the present invention provide a sequence-corrected second-order cone global optimization method applicable to radial distribution networks. It introduces auxiliary variables and decomposes and convexizes the original non-convex optimization problem based on Schur complement and S-procedure, deriving the conditions for its convergence to the KKT point of the original non-convex optimization problem. This solves the problem of errors in the convex relaxation of existing radial distribution network models and provides a new solution for the non-convex optimization problem of radial distribution networks.

[0006] According to a first aspect of the present invention, a sequentially modified second-order cone global optimization method suitable for radial distribution networks is provided, comprising: decoupling the original non-convex optimization problem in the power system according to the introduced auxiliary variables, so as to transform the original non-convex optimization problem into an equivalent bi-level programming problem, the bi-level programming problem including an upper-level programming problem and a lower-level programming problem; performing centralized solution after convexifying the upper-level programming problem using the second-order cone relaxation technique; decomposing the lower-level programming problem into multiple quadratic programming subproblems with a single non-convex quadratic constraint through strategy decomposition; transforming the quadratic programming subproblems into semidefinite programming subproblems through Schur complement, and establishing strong duality gaps through the standard S-procedure to ensure that the optimal solution of the semidefinite programming subproblems converges to the KKT point of the original non-convex optimization problem; and performing iterative calculation and sequential correction on the optimization results of the decoupled upper-level programming problem and the decoupled lower-level programming problem according to the alternating direction multiplier method to ensure that the non-convex constraint is strictly held.

[0007] In one implementation, the model for decoupling the original non-convex optimization problem in the power system based on the introduced auxiliary variables is as follows:

[0008]

[0009] Where y is the vector composed of decision variables of the upper-level model, x is the vector composed of decision variables of the lower-level model, and g i and h j The equality and inequality constraints that ensure the safe operation of the upper-level model all form feasible regions that are convex sets, m k The constraint is the only non-convex quadratic constraint of the lower-level model, and N and M are the coefficient matrices of the consensus constraint conditions of the upper and lower-level models, respectively.

[0010] In another implementation, the model that uses a second-order cone relaxation technique to convexify the upper-level programming problem and then performs a centralized solution is as follows:

[0011]

[0012] Among them, g i and h j Equality and inequality constraints for the safe operation of the upper-level model.

[0013] In another implementation, the model that decomposes the lower-level programming problem into multiple quadratic programming subproblems with a single non-convex quadratic constraint through strategy decomposition is as follows:

[0014]

[0015] Where, x i Let A be the lower-level decision variable of node i.i B i Let C be the original objective function of the subproblem at node i and the coefficient of the quadratic term in its augmented Lagrange term. i To augment the coefficients of the linear terms in the generalized Lagrange form, M i It is the only non-convex quadratic constraint for node i.

[0016] In another implementation, the model for transforming the quadratic programming subproblem into a semidefinite programming subproblem using Schur complement is as follows:

[0017]

[0018] Among them, υ i Let γ be the dual variable of the non-convex quadratic constraint at node i. i Let i be the maximum feasible upper bound of the optimal solution to the dual problem at node i.

[0019] In another implementation, the standard S-procedure is:

[0020] If there exists x2 that satisfies:

[0021]

[0022] Then there exists x that satisfies:

[0023]

[0024] The necessary and sufficient condition is that there does not exist λ that satisfies:

[0025]

[0026] Where A1 and A2 are n×n real matrices, b1 and b2 are n×1 real vectors, c1 and c2 are real scalars, and x is an n×1 vector.

[0027] In another implementation, the process of iteratively calculating and sequentially correcting the optimization results of the decoupled upper-level programming problem and the decoupled lower-level programming problem using the alternating direction multiplier method includes: initializing the maximum number of iterations; optimizing the upper-level model of the decoupled upper-level programming problem to obtain upper-level variables; solving the lower-level model of the decoupled lower-level programming problem based on the upper-level variables to obtain lower-level variables; determining whether the residuals of the upper and lower-level variables are less than a preset threshold based on the upper-level and lower-level variables; if the residuals of the upper and lower-level variables are less than the preset threshold, then ending the calculation; if the residuals of the upper and lower-level variables are not less than the preset threshold, then updating the Lagrange multipliers, incrementing the number of iterations by one, and re-optimizing the upper-level and lower-level variables sequentially until the residuals of the upper and lower-level variables are less than the preset threshold or the number of iterations reaches the maximum number of iterations, then ending the calculation.

[0028] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0029] (1) Non-convex constraint handling: The sequential modified second-order cone global optimization method proposed in this invention, applicable to radial distribution networks, ensures the strict validity of non-convex equality constraints through upper and lower layer coordinated optimization, thereby effectively improving the reliability and global optimality of the solution results and providing a new solution for the non-convex optimization problem of radial distribution networks.

[0030] (2) Strategic problem decomposition: This invention introduces auxiliary variables to strategically decompose the original non-convex optimization problem into a bi-level programming problem, and uses Schur complement to transform the non-convex problem into an easily handled convex sub-problem.

[0031] (3) Establishment of strong duality gap: This invention uses S-procedure to establish strong duality gap between the original non-convex optimization problem and the dual problem, and derives the conditions for the algorithm to converge to the KKT point of the original non-convex optimization problem. This solves the problem of error in the convex relaxation of the existing radial distribution network model and provides a new solution for the non-convex optimization problem of radial distribution network. Attached Figure Description

[0032] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in the embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings.

[0033] Figure 1 This is a flowchart of the steps of the sequentially modified second-order cone global optimization method applicable to radial distribution networks of the present invention;

[0034] Figure 2 This is a schematic diagram of the reverse power flow in the distribution network according to the present invention;

[0035] Figure 3 This is a schematic diagram of the upper and lower layer coordination and optimization information exchange process of the present invention;

[0036] Figure 4 This is a schematic diagram of the non-convex equality constraint error results under the second-order cone relaxation technique of the present invention;

[0037] Figure 5 This is a schematic diagram of the non-convex equality constraint error results under the sequence-corrected second-order cone of the present invention. Detailed Implementation

[0038] To provide a clearer understanding of the technical features, objectives, and effects of the embodiments of the present invention, specific implementation methods of the embodiments of the present invention will now be described with reference to the accompanying drawings.

[0039] In this document, “exemplary” means “serving as an example, illustration or description”, and any illustrations or implementations described herein as “exemplary” should not be construed as a more preferred or advantageous technical solution.

[0040] To enable those skilled in the art to better understand the technical solutions in the embodiments of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art should fall within the protection scope of the present invention.

[0041] The specific implementation of the embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0042] See Figure 1 This invention provides a sequentially modified second-order cone global optimization method suitable for radial distribution networks, mainly including the following steps:

[0043] Step S1: Decouple the original non-convex optimization problem in the power system according to the introduced auxiliary variables, so as to transform the original non-convex optimization problem into a bi-level programming problem, which includes an upper-level programming problem and a lower-level programming problem.

[0044] Step S2: After making the upper-level programming problem convex using the second-order cone relaxation technique, a centralized solution is performed.

[0045] Step S3: Decompose the lower-level planning problem into multiple quadratic planning subproblems with a single non-convex quadratic constraint through strategy decomposition;

[0046] Step S4: Transform the quadratic programming subproblem into a semidefinite programming subproblem using Schur complement, and establish a strong duality gap using the standard S-procedure to ensure that the optimal solution of the semidefinite programming subproblem converges to the KKT point of the original non-convex optimization problem;

[0047] Step S5: Iteratively calculate and sequentially correct the optimization results of the decoupled upper-level programming problem and the decoupled lower-level programming problem using the alternating direction multiplier method to ensure that the non-convex constraints are strictly valid.

[0048] In summary, the solution of this invention avoids the errors caused by convex relaxation, effectively ensuring the reliability and global optimality of the results. It provides an accurate and reliable mathematical model for distribution networks with a high proportion of renewable energy, and has strong reference value for the optimized operation and accurate calculation of distribution networks.

[0049] Optionally, the model for decoupling the original non-convex optimization problem in the power system based on the introduced auxiliary variables is as follows:

[0050]

[0051] Where y is the vector composed of decision variables of the upper-level model, x is the vector composed of decision variables of the lower-level model, and g i and h j The equality and inequality constraints that ensure the safe operation of the upper-level model all form feasible regions that are convex sets, m k The constraint is the only non-convex quadratic constraint of the lower-level model, and N and M are the coefficient matrices of the consensus constraint conditions of the upper and lower-level models, respectively.

[0052] It should be understood that y is a vector composed of the decision variables of the upper-level model, i.e., the original variables, and x is a decision variable of the lower-level model, i.e., the auxiliary variable.

[0053] Optionally, the model that uses second-order cone relaxation technique to convexify the upper-level programming problem and then performs a centralized solution is as follows:

[0054]

[0055] Among them, g i and h j Equality and inequality constraints for the safe operation of the upper-level model.

[0056] Optionally, the model that decomposes the lower-level programming problem into multiple quadratic programming subproblems with a single non-convex quadratic constraint through strategy decomposition is as follows:

[0057]

[0058] Where, x i Let A be the lower-level decision variable of node i. i B i Let C be the original objective function of the subproblem at node i and the coefficient of the quadratic term in its augmented Lagrange term. i To augment the coefficients of the linear terms in the generalized Lagrange form, M i It is the only non-convex quadratic constraint for node i.

[0059] Optionally, the model for transforming the quadratic programming subproblem into a semidefinite programming subproblem using Schur complement is as follows:

[0060]

[0061] Among them, υ i Let γ be the dual variable of the non-convex quadratic constraint at node i. i Let i be the maximum feasible upper bound of the optimal solution to the dual problem at node i.

[0062] Optionally, the standard S-procedure is:

[0063] If there exists x2 that satisfies:

[0064]

[0065] Then there exists x that satisfies:

[0066]

[0067] The necessary and sufficient condition is that there does not exist λ that satisfies:

[0068]

[0069] In other words, the above two equations have strong selectivity. Here, A1 and A2 are n×n real matrices, b1 and b2 are n×1 real vectors, c1 and c2 are real scalars, and x is an n×1 vector.

[0070] Optionally, the process of iteratively calculating and sequentially correcting the optimization results of the decoupled upper-level programming problem and the decoupled lower-level programming problem using the alternating direction multiplier method includes: initializing the maximum number of iterations; optimizing the upper-level model of the decoupled upper-level programming problem to obtain upper-level variables; solving the lower-level model of the decoupled lower-level programming problem based on the upper-level variables to obtain lower-level variables; determining whether the residuals of the upper and lower-level variables are less than a preset threshold based on the upper-level variables and the lower-level variables; if the residuals of the upper and lower-level variables are less than the preset threshold, then ending the calculation; if the residuals of the upper and lower-level variables are not less than the preset threshold, then updating the Lagrange multipliers, incrementing the number of iterations by one, and re-optimizing the upper-level variables and the lower-level variables sequentially until the residuals of the upper and lower-level variables are less than the preset threshold or the number of iterations reaches the maximum number of iterations, then ending the calculation.

[0071] For example, initialize algorithm parameters such as the maximum number of iterations, first optimize the solution of the upper-level model to obtain the variable y, then use the results of the upper-level variable to solve the lower-level model to obtain its variable x, and determine whether the residual between the upper and lower-level variables is less than a given threshold; if it is less, end the calculation; otherwise, update the Lagrange multipliers, increment the number of iterations by one, and re-optimize the solution of the upper and lower-level variables in order until the residual is less than the given threshold or the maximum number of iterations is reached.

[0072] Furthermore, the method of this invention addresses the problem of infeasibility or suboptimality of optimal solutions caused by convex relaxation errors in distribution networks. Utilizing its mathematical characteristics, this method can be implemented in Matlab using a programming language.

[0073] Specifically, taking the scenario of photovoltaic carrying capacity assessment with inaccurate convex relaxation in a distribution network as an example, the solution of the present invention is further described to verify the effectiveness of the present invention. The sequentially corrected second-order cone global optimization method of the present invention, applicable to radial distribution networks, is implemented through the following steps:

[0074] (1) Photovoltaic carrying capacity assessment model

[0075] 1) Objective function

[0076] To demonstrate the advantages of the SMSOC algorithm over SOCR in terms of applicability and solution accuracy—namely, that SMSOC guarantees error-free and correct solution results—this invention uses a photovoltaic (PV) carrying capacity assessment model in a distribution network as an example for analysis and discussion. Research on PV carrying capacity assessment is relatively mature, and the selection of assessment indicators varies depending on different focuses. The PV carrying capacity assessment model proposed in this invention comprehensively considers maximizing the grid-connected capacity of distributed PV and minimizing network losses, using this as the objective function. The equivalent grid-connected capacity of distributed PV is the difference between the two, and its maximum value is defined as the carrying capacity of distributed PV.

[0077]

[0078] In the formula, N PV For a set of nodes configured for distributed photovoltaic (PV) units, p t,PV This represents the actual active power output of the photovoltaic (PV) units at node t. By optimizing the active power output of the PV units in the system, the goal is to maximize the equivalent grid-connected capacity of distributed PV systems.

[0079] 2) Constraints

[0080] The constraint conditions of the distributed photovoltaic carrying capacity assessment model include all constraints in the simplified distribution network model. To adapt to the model and algorithm requirements of this invention, the positive directions of current and power in the introduced distribution network branch power flow model are set from the load node to the feeder node. Furthermore, i, t, k, and n are defined as node indices, and (i, t) represents the branch from node i to node t. F i S is the parent node of node i, that is, the superior node in the direction from node i to the feeder node. (t,i) z represents the branch complex power flowing from node t to node i. (t,i) Let s represent the complex impedance of branch (t,i). i This represents the injected complex power at node i. A schematic diagram of reverse power flow in a distribution network is shown below. Figure 2 As shown.

[0081] Based on the reverse power flow diagram of the distribution network, the equality constraints that the model needs to satisfy are:

[0082]

[0083] In the formula, C i v represents the set of nodes whose parent node is i. i Let l be the square of the complex voltage at node i. (t,i) Let N be the square of the complex current in branch (t,i). θ These represent sets that include feeder nodes and sets that do not include feeder nodes, respectively. Representing branches (i, F) i The active and reactive power of ). Representing branches (i, F) i The resistance and reactance of p; i With q i Let s represent the injected active power and injected reactive power at node i, respectively. i,PV With s i,L represents the actual output and electrical load of the photovoltaic unit at node i, respectively, and j represents the imaginary unit.

[0084] To ensure the safe operation of the system, the model must satisfy the following inequality constraints:

[0085]

[0086] In the formula, V i These represent the upper and lower limits of the voltage at node i, respectively. With s i,PV These represent the upper and lower limits of the output of distributed photovoltaic (PV) units, respectively. Branches (i, F) i The upper limit of the current and the upper limit of the active power allowed to pass through are given by equation (10). The lower limit on the left side of equation (10) is negative, which means that after a distributed power source is added to some nodes of the radial distribution network, a local reverse power flow may be generated.

[0087] It is easy to see that the objective function of equation (1) is a superposition of linear functions related to each branch and node, which is usually a convex function because it only includes the output of the photovoltaic unit or the branch line loss, both of which are linear. In terms of constraints, equations (2)-(4) and (6)-(10) are linear constraints and can form a convex set, but the quadratic equality constraint represented by equation (5) is a non-convex constraint.

[0088] Optimization problems with non-convex constraints are typical NP-hard problems, and the difficulty of solving them increases dramatically with the increase of the variable space. In existing technologies, relaxation techniques are often used to make equation (5) convex. Under certain assumptions and scenarios, such relaxation techniques have been proven to be tight and the results are accurate. However, these assumptions or scenarios are limited, and once violated, they may lead to infeasibility or suboptimal results. Under the objective function of the photovoltaic carrying capacity assessment model in this paper, if only the traditional second-order cone relaxation technique is used to solve it, unacceptable errors will be generated on both sides of equation (5).

[0089] (2) Radial Distribution Network Sequence Correction Second-Order Cone Global Optimization Algorithm

[0090] To effectively address the non-convexity in the model, this invention proposes a sequentially modified second-order cone algorithm, leveraging the radial characteristics of the power distribution network. The algorithm strategically decomposes the original non-convex problem into an upper-level second-order cone programming problem and a lower-level quadratic programming subproblem with a single constraint by introducing auxiliary variables. Then, the lower-level quadratic programming subproblem with a single constraint is transformed into an equivalent convex problem. Each subproblem in the upper-level model contains a convex objective function and several linear constraints, while each subproblem in the lower-level model contains a convex objective function and a non-convex quadratic constraint. The upper-level model can be solved centrally due to its convexity, while the subproblems in the lower-level model are non-convex. To overcome this problem, this invention introduces the use of Schur complement and S-procedure to prove the strong duality of the non-convex subproblems and derives their equivalent dual forms. Finally, ADMM is used to establish a consensus relationship between the original and auxiliary variables, ensuring the algorithm's effectiveness.

[0091] 1) Subproblem decomposition

[0092] First, define the vector formed by the original variables of each node i as follows:

[0093]

[0094] in, v y,i , p y,i ,q y,i Both are vectors y i The components in the vector are the (upper-level) original variables. For the photovoltaic carrying capacity assessment problem, it can be viewed as a linear combination of the original variables of each node's main body. To separate the problem into upper and lower levels, the vector formed by the auxiliary variables of each node's main body i is defined as:

[0095]

[0096] in, v x,i , px,i ,q x,i Both are vectors x i The amount.

[0097] Secondly, the only non-convex quadratic equation constraint (5) in the photovoltaic load-bearing capacity assessment problem is equivalently expressed as:

[0098]

[0099] Among them, the feasible region formed by equation (13) can be converted into a second-order cone, which is a convex set, while equation (14) is non-convex. Therefore, for node i, the convex feasible region formed by the upper-level model and the non-convex feasible region formed by the lower-level model can be expressed as follows:

[0100] Among them, the feasible region formed by equation (13) can be converted into a second-order cone, which is a convex set, while equation (14) is non-convex. Therefore, for node i, the convex feasible region formed by the upper-level model and the non-convex feasible region formed by the lower-level model can be expressed as follows:

[0101]

[0102] Among them, Y i X i Let x and y represent the feasible regions of the upper and lower level models, respectively, for node i. It is easy to see that the nodes in the lower level model are independent of each other and do not affect each other, thus allowing for parallel computation. However, consensus constraints must be established between the original variables and auxiliary variables to ensure the validity of the upper and lower level variables. Therefore, the photovoltaic carrying capacity assessment problem is rewritten as P1 with respect to the x variable:

[0103]

[0104] Expressing the above consensus constraints as Mx = Ny, where M and N are the coefficient matrices of the x and y vectors respectively, the augmented Lagrange form of problem P1 is:

[0105]

[0106] In the formula, Branches (i, F) i The active power, reactive power, and current square Lagrange multipliers of ) Let be the Lagrange multipliers for the squared voltage of node i, the injected active power, and the injected reactive power, respectively. ρ is the penalty factor for the augmented Lagrange term, usually taken as a reasonably large positive number. For any node i, the lower-level variables, upper-level variables, corresponding Lagrange multipliers, and the original residuals after each iteration can be obtained using the following formulas:

[0107]

[0108] In the formula, k is the number of iterations, and r k+1 Let ||(·)||2 be the original residual after the (k+1)th iteration, and let ||(·)||2 be the L2 norm of (·). After obtaining the y component values ​​from the upper-level problem (20), exchange the corresponding component values ​​with the corresponding node entities in the lower-level model. These values ​​are needed to calculate the x vector. After the x vector is calculated, update the Lagrange multipliers and calculate the original residual. The upper-level problem (20) can be solved centrally, and the y component values ​​of all node entities in the upper-level model can be obtained in one optimization. Repeat this process until the residual is less than the given confidence threshold. The entire information exchange process is as follows: Figure 3 As shown.

[0109] 2) Eliminate the nonconvexity of subproblems

[0110] According to optimization theory, convergence is guaranteed when the penalty factor is strictly positive and both subproblems SP1 (x) and SP2 (y) are convex. It is easy to see that subproblem (20) has a convex feasible region and a convex objective function, while subproblem (19) is non-convex due to its non-convex feasible region. Therefore, applying the ADMM approach to solve the current SP1 and SP2 problems cannot guarantee strict convergence. To address this issue, we will introduce a theory to eliminate this non-convexity. For ease of description, the original non-convex subproblem SP1 is rewritten as:

[0111]

[0112] in,(·) - and(·) T Let A represent the inverse and transpose of a vector or matrix (·), respectively. The objective function consists of quadratic and linear terms. i B i C is the coefficient of the quadratic term. i The coefficient of the linear term is y, which is related to the coefficient of the previous iteration. i Related to component values, M i For x i The non-convex constraint coefficient matrix of the components. A i The original objective function f(x) in equation (19) i Regarding the coefficient of the quadratic term of variable x, it is easy to see that, under the linear objective function in this paper, it should be a 6×6 zero matrix, B. i The coefficient of the quadratic term of the augmented Lagrange term in equation (19) with respect to the variable x is expressed as follows:

[0113]

[0114] In the formula, E is a 6×6 identity matrix. The coefficient matrix M represents the constraint conditions. i It is not positive semidefinite; the feasible region defined by the quadratic constraint is nonconvex. However, since each node body contains only one nonconvex quadratic constraint, i.e., P...U1 This is a typical QP1QC problem. To solve this special nonconvex problem, we introduce Schur complement and S-procedure to establish P. U1 The strong duality gap between the dual problem and the primal problem. P U1 The dual form of the problem is as follows:

[0115]

[0116] In the formula, υ i The question is P. U1 The dual variable of the quadratic inequality constraint. Using the first-order optimality condition, x can be... i Represented as:

[0117] x i =-(A i +B i +υ i M i ) -1 C i (26)

[0118] Substituting the above equation back into equation (25), we obtain the dual problem as follows:

[0119]

[0120] υ i ≥0(28)

[0121]

[0122] Equation (28) ensures that the dual variable is meaningful, and Equation (29) ensures that the problem has a finite solution. Using Schur complement, the dual problem D can be solved. U1 Represented as a convex semidefinite programming problem:

[0123]

[0124] υ i ≥0(31)

[0125]

[0126] From equation (32), we know that the dual problem is always convex. Therefore, for any primal problem P, U1 It can always be solved by finding S U1 This yields a solution, but this solution is not necessarily equal to the solution to the original problem. Only when strong duality holds will the two problems have the same solution. Because P U1 Since it is non-convex, we introduce the theory of the control field and derive Theorem 1 through the S-procedure.

[0127] Theorem 1: When problem P U1 When the constraint set is not empty, P U1 With S U1 The strong duality holds, that is, P U1 With S U1 The optimal solutions are equal.

[0128] Proof: Referring to the standard S-procedure form, based on the parameter constraints (33) of the QP1QC problem, the following standard semidefinite inequality constraints can be obtained:

[0129]

[0130] Among them, D 0,i D 1,i Let S be the constants of the objective function and the quadratic constraint, respectively. It's easy to see that in this problem, these are all zero vectors of the corresponding dimensions. The above expression is actually S... U1 The equivalent dual of the problem. According to S-procedure theory, for the QP1QC problem, when there exists υ i When ≥0 makes the above relation hold, the strong duality between the primal and dual problems holds. Therefore, problem P U1 The solution and S U1 Same interpretation.

[0131] (Proof complete)

[0132] In summary, through the above transformations and proofs, a more accurate optimization solution can be obtained under the allowable error by performing sequential correction on the traditional second-order cone optimization model. The flowchart of the sequential correction second-order cone global optimization algorithm proposed in this paper is shown in Table 1.

[0133] Table 1. Flowchart of the global optimization algorithm for the sequence-corrected second-order cone.

[0134]

[0135] (II) Case Analysis

[0136] This paper uses improved IEEE 33, 69, and 136-node distribution systems for numerical analysis. All numerical results were obtained on a computer configured with an Intel(R) Core(TM) i7-8700 CPU @ 3.20GHz and 64GB of memory. The upper and lower level models were solved using the commercial solvers GUROBI 11.0.2 and Mosek 10.1, respectively. Photovoltaic generators were installed at different nodes in the three distribution systems with different node sizes. The location and capacity information of the generators are shown in Table 2.

[0137] Table 2. Photovoltaic Unit Configuration Information

[0138]

[0139] (1) Analysis of unit output results

[0140] Under the above unit configuration, with the maximum photovoltaic carrying capacity as the objective function, the improved 33-node system was evaluated and analyzed using the second-order cone relaxation technique and the sequentially modified second-order cone algorithm, and the error between the two optimization results was calculated. The actual output of the photovoltaic units and the system line loss are shown in Tables 3 and 4, respectively.

[0141] Table 3 System Photovoltaic Unit Output

[0142]

[0143] Table 4 System line loss and objective function value

[0144]

[0145] It is readily apparent that, when maximizing photovoltaic carrying capacity is the optimization objective, although the objective function values ​​of the two methods are consistent, the photovoltaic output and system line loss differ significantly. In this case, the second-order cone relaxation technique can only consider the overall objective function, neglecting the actual system requirements. The feasible region after relaxation is not tight. Although the system's photovoltaic output reaches its maximum, the system line loss also reaches its maximum. Clearly, the result at this point is not a practically feasible solution.

[0146] (2) Error Result Analysis

[0147] To demonstrate the accuracy of the proposed sequence-corrected second-order cone optimization algorithm, the power flow results obtained from the two optimization methods are substituted into the left and right sides of the non-convex constraint for calculation. The absolute and relative errors on both sides of the equation are calculated as follows:

[0148]

[0149] Referring to the two error calculation methods mentioned above, the error values ​​of the improved IEEE 33-node distribution system are analyzed under the second-order cone relaxation technique and the sequentially corrected second-order cone algorithm, respectively. The errors of each branch of the system with respect to the non-convex equality constraints are as follows: Figure 4 and Figure 5 As shown. By Figure 4 It can be seen that under the second-order cone relaxation technique, the power flow results have a large error at both ends of the non-convex equality constraint, with the largest error even exceeding 70%, such as branches 3, 4, 5, and 10. Even branch 19, which has the smallest error, has an error of more than 5%, and the optimization solution is not feasible at this time.

[0150] In contrast, the sequence-corrected second-order cone optimization algorithm proposed in this invention is based on... Figure 5It can be seen that for any branch, the curves of the variable calculation results on both sides of the non-convex equality constraint almost completely overlap. Only after magnifying the branches 6 and 24 by 1e7 times can a slight difference between them be barely seen. This shows that while ensuring global optimality, the algorithm can ensure that both sides of the non-convex quadratic equality constraint are strictly valid, and the calculation error fully meets the accuracy requirements of the engineering field. At this time, the optimized solution is the actual feasible solution of the system.

[0151] To further illustrate the differences in error results across node systems of different sizes, two methods were used to calculate the errors for the improved IEEE 33, 69, and 136 node systems, respectively, and the resulting errors are shown in Table 5.

[0152] Table 5 Calculation errors of node systems of different sizes

[0153]

[0154] It can be seen that in the photovoltaic carrying capacity assessment scenario, the SOCR model is not tight. The power flow results obtained from the optimization solution produce large and non-negligible errors on both sides of the non-convex quadratic equality constraint. Even in the smallest improved 33-node system, the relative error ratio reaches 76%, which is unacceptable in the engineering field. Moreover, the error value continues to increase with the increase of the node system size. In contrast, the sequentially corrected second-order cone algorithm iteratively solves the upper and lower layer models, repeatedly correcting the upper layer model. Essentially, when the algorithm converges, the model does not undergo any relaxation, always ensuring that the non-convex quadratic equality constraint is strictly valid, avoiding the errors caused by convex relaxation. Therefore, the results obtained are correct. Even in the largest 136-node system, the relative error ratio can be guaranteed to be within 0.0001, almost guaranteeing zero error in the non-convex quadratic equality constraint, which fully meets the engineering accuracy requirements.

[0155] The above practical example analysis leads to the conclusion that:

[0156] 1) Non-convex quadratic equality constraint problems can be equivalently transformed into subproblems between upper and lower level models for iterative solution;

[0157] 2) Using Schur complement and S-procedure, the KKT conditions for the optimal solution of the subproblem to converge to the optimal solution of the original nonconvex optimization problem can be derived;

[0158] 3) Compared with the second-order cone relaxation technique, the sequence-corrected second-order cone algorithm can effectively satisfy the non-convex equality constraint while ensuring global optimum. The proposed algorithm has smaller error, higher accuracy, and wider applicability.

[0159] In summary, the solution of this invention avoids the errors caused by convex relaxation, effectively ensuring the reliability and global optimality of the results. It provides an accurate and reliable mathematical model for distribution networks with a high proportion of renewable energy, and has strong reference value for the optimized operation and accurate calculation of distribution networks.

[0160] It should be noted that the present invention can be a method, system, apparatus, and / or computer program product. A computer program product may include a computer-readable storage medium having computer-readable program instructions loaded thereon for causing a processor to implement various aspects of the present invention.

[0161] Computer-readable storage media can be tangible devices capable of holding and storing instructions for use by an instruction execution device. Computer-readable storage media can be, for example, but not limited to, electrical storage devices, magnetic storage devices, optical storage devices, electromagnetic storage devices, semiconductor storage devices, or any suitable combination thereof. More specific examples (a non-exhaustive list) of computer-readable storage media include: portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable compact disc read-only memory (CD-ROM), digital multifunction disc (DVD), memory sticks, floppy disks, mechanical encoding devices, such as punch cards or recessed protrusions storing instructions thereon, and any suitable combination thereof. The computer-readable storage media used herein are not to be construed as transient signals themselves, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through waveguides or other transmission media (e.g., light pulses through fiber optic cables), or electrical signals transmitted through wires.

[0162] The computer-readable program instructions described herein can be downloaded from computer-readable storage media to various computing / processing devices, or downloaded via a network, such as the Internet, local area network, wide area network, and / or wireless network, to an external computer or external storage device. The network may include copper transmission cables, fiber optic transmission, wireless transmission, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computing / processing device receives the computer-readable program instructions from the network and forwards them to the computer-readable storage media in the respective computing / processing device.

[0163] The computer program instructions used to perform the operations of this invention may be assembly instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state setting data, or source code or object code written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Smalltalk, C++, Python, etc., and conventional procedural programming languages ​​such as "C" or similar languages. The computer-readable program instructions may be executed entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In cases involving a remote computer, the remote computer may be connected to the user's computer via any type of network—including a local area network (LAN) or a wide area network (WAN)—or may be connected to an external computer (e.g., via the Internet using an Internet service provider). In some embodiments, electronic circuitry, such as programmable logic circuitry, field-programmable gate arrays (FPGAs), or programmable logic arrays (PLAs), is personalized by utilizing state information from the computer-readable program instructions. This electronic circuitry can execute the computer-readable program instructions to implement various aspects of the invention.

[0164] Various aspects of the present invention are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It should be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer-readable program instructions.

[0165] These computer-readable program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing apparatus to produce a machine such that, when executed by the processor of the computer or other programmable data processing apparatus, they create means for implementing the functions / actions specified in one or more blocks of the flowchart and / or block diagram. These computer-readable program instructions can also be stored in a computer-readable storage medium that causes a computer, programmable data processing apparatus, and / or other device to operate in a particular manner; thus, the computer-readable medium storing the instructions comprises an article of manufacture that includes instructions for implementing aspects of the functions / actions specified in one or more blocks of the flowchart and / or block diagram.

[0166] Computer-readable program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational steps to be performed on the computer, other programmable data processing apparatus, or other device to produce a computer-implemented process, thereby causing the instructions executed on the computer, other programmable data processing apparatus, or other device to perform the functions / actions specified in one or more boxes of a flowchart and / or block diagram.

[0167] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of an instruction containing one or more executable instructions for implementing a specified logical function. In some alternative implementations, the functions marked in the blocks may occur in a different order than those marked in the drawings. For example, two consecutive blocks may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. It should also be noted that each block in the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, can be implemented using a dedicated hardware-based system that performs the specified function or action, or using a combination of dedicated hardware and computer instructions. It will be known to those skilled in the art that implementation in hardware, implementation in software, and implementation using a combination of software and hardware are equivalent.

[0168] The various embodiments of the present invention have been described above. These descriptions are exemplary and not exhaustive, and are not limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terminology used herein is chosen to best explain the principles, practical application, or technical improvements to the embodiments in the market, or to enable others skilled in the art to understand the embodiments disclosed herein. The scope of the invention is defined by the appended claims.

Claims

1. A sequentially modified second-order cone global optimization method suitable for radial distribution networks, applied to the photovoltaic carrying capacity assessment scenario of convex relaxation in distribution networks, characterized in that... include: The original non-convex optimization problem in the power system is decoupled by introducing auxiliary variables based on the sequence-corrected second-order cone algorithm. This transforms the original non-convex optimization problem into an equivalent bi-level programming problem to address the non-convexity in the photovoltaic carrying capacity assessment model. The bi-level programming problem includes an upper-level programming problem and a lower-level programming problem. The photovoltaic carrying capacity assessment model takes maximizing the distributed photovoltaic grid-connected capacity and minimizing network losses as its objective functions. (1) In the formula, N PV For configuring a set of nodes for distributed photovoltaic units, p t,PV Represents a node t The actual active power output of the photovoltaic units; The photovoltaic load-bearing capacity assessment model satisfies the following equality constraints: (2) (3) (4) (5) (6) In the formula, C i Representative with i The set of nodes of the parent node. v i For nodes i The square of the complex voltage, l (t,i) For branch roads ( t,i The square of the complex current, N and N θ These represent sets that include feeder nodes and sets that do not include feeder nodes, respectively. , They represent the branches ( i,F i The active and reactive power of ). , They represent the branches ( i,F i The resistance and reactance of ) p i and q i Representing nodes respectively i The injected active power and injected reactive power, s i,PV and s i,L Representing nodes respectively i The actual output and electrical load of the photovoltaic unit, where j represents the imaginary unit; The inequality constraints satisfied by the photovoltaic load-bearing capacity assessment model are as follows: (7) (8) (9) (10) In the formula, , Representing nodes respectively i Upper and lower limits of voltage; and These represent the upper and lower limits of the output of distributed photovoltaic (PV) units, respectively. , Branch roads ( i,F i The upper limit of the current and the upper limit of the active power allowed to pass through are given by equation (10). The lower limit of the left end of equation (10) is negative, which means that after the distributed power source is added to some nodes of the radial distribution network, a local reverse power flow is generated. The upper-level programming problem is convexized using a second-order cone relaxation technique, and then solved centrally. The lower-level planning problem is decomposed into multiple quadratic planning subproblems with a single non-convex quadratic constraint through strategy decomposition. The quadratic programming subproblem is transformed into a semidefinite programming subproblem by Schur complement, and a strong duality gap is established by the standard S-procedure to ensure that the optimal solution of the semidefinite programming subproblem converges to the KKT point of the original non-convex optimization problem. The optimization results of the decoupled upper-level programming problem and the decoupled lower-level programming problem are iteratively calculated and sequentially corrected using the alternating direction multiplier method to ensure that the non-convex constraints are strictly valid.

2. The method according to claim 1, characterized in that, The model for decoupling the original nonconvex optimization problem in the power system based on the introduced auxiliary variables is as follows: in, y This is a vector composed of the decision variables of the upper-level model. x This is a vector composed of the decision variables of the lower-level model. g i and h j The equality and inequality constraints that ensure the safe operation of the upper-level model all form feasible regions that are convex sets. m k This is the only non-convex quadratic constraint condition for the lower-level model. N and M These are the coefficient matrices for the consensus constraints of the upper and lower layer models, respectively.

3. The method according to claim 2, characterized in that, The model that uses a second-order cone relaxation technique to convexify the upper-level programming problem and then performs a centralized solution is as follows: in, g i and h j Equality and inequality constraints for the safe operation of the upper-level model.

4. The method according to claim 3, characterized in that, The model that decomposes the lower-level programming problem into multiple quadratic programming subproblems with a single non-convex quadratic constraint through strategy decomposition is as follows: in, x i For nodes i The lower-level decision variables, A i 、B i They are nodes i The original objective function of the subproblem and the coefficients of the quadratic terms in its augmented Lagrange term, C i To augment the coefficients of the linear terms in the generalized Lagrange form, M i For nodes i The only non-convex quadratic constraint.

5. The method according to claim 4, characterized in that, The model that transforms the quadratic programming subproblem into a semidefinite programming subproblem using Schur complement is as follows: in, υ i For nodes i Dual variables of non-convex quadratic constraints γ i For nodes i The maximum feasible upper bound of the optimal solution to the dual problem.

6. The method according to claim 5, characterized in that, The standard S-procedure is as follows: If it exists x 2. Satisfies: Then it exists. x satisfy: The necessary and sufficient condition is that there does not exist λ that satisfies: in, A 1 、A 2 is an n×n real matrix. b 1 、b 2 is an n×1 real vector. c 1 、c 2 is a real scalar. x It is an n×1 vector.

7. The method according to claim 1, characterized in that, The process of iteratively calculating and sequentially correcting the optimization results of the decoupled upper-level programming problem and the decoupled lower-level programming problem based on the alternating direction multiplier method includes: Initialize the maximum number of iterations, optimize the upper-level model for solving the decoupled upper-level planning problem, and obtain the upper-level variables; Based on the upper-level variables, the lower-level model of the decoupled lower-level planning problem is solved to obtain the lower-level variables; Based on the upper-level variable and the lower-level variable, determine whether the residual between the upper and lower-level variables is less than a preset threshold; If the residual between the upper and lower level variables is less than the preset threshold, the calculation ends. If the residuals of the upper and lower level variables are not less than the preset threshold, then update the Lagrange multipliers, increment the iteration count by one, and re-optimize and solve the upper and lower level variables in sequence until the residuals of the upper and lower level variables are less than the preset threshold or the iteration count reaches the maximum iteration count, at which point the calculation ends.

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