Sequence correction second-order cone global optimization method suitable for radial power distribution network

Abstract

The invention provides a sequence correction second-order cone global optimization method suitable for a radial power distribution network, and the method comprises the steps: introducing an auxiliary variable to decouple an original non-convex optimization problem in a power system, and enabling the original non-convex optimization problem to be equivalently converted into a bilevel programming problem; adopting a second-order cone relaxation technology to convexly solve an upper-layer planning problem and then carrying out centralized solution; decomposing the lower-layer planning problem into a plurality of quadratic planning sub-problems containing single non-convex quadratic constraint; the quadratic programming sub-problem is converted into a semi-definite programming sub-problem through Scher complement, and a strong dual gap is established through a standard S-process; and performing iterative calculation and sequence correction on the optimization result of the decoupled upper and lower layer planning problem according to an alternating direction multiplier method to ensure that the non-convex constraint is strictly established. According to the method, errors generated by convex relaxation are avoided, the reliability and global optimality of the result can be effectively ensured, and an accurate and reliable mathematical model is provided for the power distribution network with the high proportion of renewable energy sources.

Description

Technical Field

[0001] The embodiments of the present invention relate to the technical field of distribution network optimization operation, and in particular to a sequence modified second-order cone global optimization method suitable for radial distribution networks. Background Art

[0002] At present, under the background of "dual carbon" goals and new power systems, a large number of distributed power sources represented by wind power and photovoltaic power are connected to the distribution network, and my country's new power system is gradually showing the characteristics of a high proportion of new energy. The resulting distribution network voltage fluctuations, voltage limit, transformer capacity overload, local reverse and even regional reverse current problems are becoming more and more serious. The traditional distribution network model is no longer sufficient to support accurate calculations and engineering applications in certain scenarios of the new power system.

[0003] At present, the simplified branch flow model is widely used in the industry. This type of model reduces the original variable relationship in the complex domain to the variable relationship in the real domain by applying simplified conditions, and uses different relaxation techniques to relax the original non-convex constraints, thereby reducing the difficulty of solving. 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 actual power system application scenarios, due to the different objective functions and the various boundary conditions set in the model for application needs, it is extremely difficult to expect to ensure the accuracy of convex relaxation through a universal method.

[0004] Therefore, it is urgent to propose a new distribution network model with low formal complexity, small relaxation error and wide applicability. Summary of the invention

[0005] In view of this, an embodiment of the present invention provides a sequence modified second-order cone global optimization method suitable for radial distribution networks, introduces auxiliary variables and decomposes and convexifies the original non-convex optimization problem according to Schur's complement and S-procedure, and derives the conditions for its convergence to the KKT point of the original non-convex optimization problem, solves the problem of errors in the convex relaxation of the existing radial distribution network model, and provides a new solution to the non-convex optimization problem of radial distribution networks.

[0006] According to a first aspect of an embodiment of the present invention, a sequence-corrected second-order cone global optimization method suitable for a radial distribution network is provided, comprising: decoupling an original non-convex optimization problem in a power system according to an introduced auxiliary variable, so as to convert the original non-convex optimization problem into a two-level programming problem equivalently, wherein the two-level programming problem includes an upper-level programming problem and a lower-level programming problem; convexifying the upper-level programming problem by a second-order cone relaxation technique and then solving it in a centralized manner; decomposing the lower-level programming problem into a plurality of quadratic programming sub-problems containing a single non-convex quadratic constraint by strategy decomposition; converting the quadratic programming sub-problem into a semi-definite programming sub-problem by Schur complement, and establishing a strong duality gap by a standard S-procedure, so as to ensure that the optimal solution of the semi-definite programming sub-problem converges to the KKT point of the original non-convex optimization problem; iteratively calculating and sequentially correcting the optimization results of the decoupled upper-level programming problem and the decoupled lower-level programming problem according to an alternating direction multiplier method, so as to ensure that the non-convex constraints are strictly established.

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

[0008]

[0009] Among them, y is the vector composed of the decision variables of the upper model, x is the vector composed of the decision variables of the lower model, and g i and h j are the equality and inequality constraints for the safe operation of the upper model, and the feasible domains formed by them are all convex sets, m k is the only non-convex quadratic constraint of the lower model, and N and M are the coefficient matrices of the consensus constraints of the upper and lower models, respectively.

[0010] In another implementation, the model for centrally solving the upper-level planning problem by convexifying it using the second-order cone relaxation technique is:

[0011]

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

[0013] In another implementation, the model of decomposing the lower-level programming problem into a plurality of quadratic programming sub-problems containing a single non-convex quadratic constraint by strategy decomposition is:

[0014]

[0015] Among them, x i is the lower-level decision variable of node i, Ai , B i are the coefficients of the original objective function of the node i subproblem and its quadratic term in the augmented Lagrangian term, C i is the coefficient of the linear term in the augmented Lagrangian form, M i is the only non-convex quadratic constraint on node i.

[0016] In another implementation, the model for converting the quadratic programming subproblem into a semidefinite programming subproblem by Schur complement is:

[0017]

[0018] Among them, i is the dual variable of the non-convex quadratic constraint at node i, γ i is the maximum feasible upper bound of the optimal solution of 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 such that:

[0023]

[0024] The necessary and sufficient condition is that there is no λ satisfying:

[0025]

[0026] Among them, 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 planning problem and the decoupled lower-level planning problem according to the alternating direction multiplier method includes: initializing the maximum number of iterations, optimizing and solving the upper-level model of the decoupled upper-level planning problem to obtain upper-level variables; solving the lower-level model of the decoupled lower-level planning problem according to the upper-level variables to obtain lower-level variables; judging whether the residual of the upper and lower-level variables is less than a preset threshold value according to the upper-level variables and the lower-level variables; if the residual of the upper and lower-level variables is less than the preset threshold value, terminating the calculation; if the residual of the upper and lower-level variables is not less than the preset threshold value, updating the Lagrange multiplier, increasing the number of iterations by one, and re-optimizing and solving the upper and lower-level variables in sequence until the residual of the upper and lower-level variables is less than the preset threshold value or the number of iterations reaches the maximum number of iterations, then terminating the calculation.

[0028] Compared with the prior art, the present invention has the following beneficial effects:

[0029] (1) Non-convex constraint processing: The present invention proposes a sequence modified second-order cone global optimization method suitable for radial distribution networks. The method ensures the strict validity of non-convex equality constraints through coordinated optimization of the upper and lower layers, thereby effectively improving the reliability and global optimality of the solution results, and providing a new solution to the non-convex optimization problem of radial distribution networks.

[0030] (2) Strategic problem decomposition: The present invention strategically decomposes the original non-convex optimization problem into a two-level programming problem by introducing auxiliary variables, and uses Schur complement to transform the non-convex problem into an easy-to-handle convex sub-problem.

[0031] (3) Establishment of strong duality gap: The present invention uses S-procedure to establish a 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, thereby solving the problem of errors in the convex relaxation of the existing radial distribution network model and providing a new solution to the non-convex optimization problem of the radial distribution network. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in the embodiments of the present invention. For ordinary technicians in this field, other drawings can also be obtained based on these drawings.

[0033] Figure 1 It is a flow chart of the steps of the sequence modified second-order cone global optimization method applicable to radial distribution network of the present invention;

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

[0035] Figure 3 A schematic diagram of the upper and lower layers coordinated optimization information exchange process of the present invention;

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

[0037] Figure 5 It is a schematic diagram of the error result of the non-convex equality constraint under the sequence-corrected second-order cone of the present invention. DETAILED DESCRIPTION

[0038] In order to have a clearer understanding of the technical features, purposes and effects of the embodiments of the present invention, the specific implementation of the embodiments of the present invention is now described with reference to the accompanying drawings.

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

[0040] In order 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 in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in the field based on the embodiments in the embodiments of the present invention should fall within the scope of protection of the embodiments of the present invention.

[0041] The specific implementation of the embodiment of the present invention is further described below in conjunction with the accompanying drawings of the embodiment of the present invention.

[0042] See also Figure 1 The present invention provides a sequence modified second-order cone global optimization method suitable for radial distribution network, which mainly includes the following steps:

[0043] Step S1, decoupling the original non-convex optimization problem in the power system according to the introduced auxiliary variables, so as to convert the original non-convex optimization problem into a two-level programming problem, wherein the two-level programming problem includes an upper-level programming problem and a lower-level programming problem;

[0044] Step S2, using the second-order cone relaxation technique to convexify the upper-level planning problem and then perform a centralized solution;

[0045] Step S3, decomposing the lower-level programming problem into a plurality of quadratic programming sub-problems containing a single non-convex quadratic constraint through strategy decomposition;

[0046] Step S4, converting the quadratic programming subproblem into a semidefinite programming subproblem by Schur complement, and establishing a strong duality gap by 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 planning problem and the decoupled lower-level planning problem according to the alternating direction multiplier method to ensure that the non-convex constraints are strictly established.

[0048] In summary, the solution of the embodiment of the present invention avoids the errors caused by convex relaxation, can effectively ensure the reliability and global optimality of the results, and proposes an accurate and reliable mathematical model for distribution networks with a high proportion of renewable energy, which has a strong reference significance for the optimized operation and precise calculation of distribution networks.

[0049] Optionally, the model for decoupling the original non-convex optimization problem in the power system according to the introduced auxiliary variables is:

[0050]

[0051] Among them, y is the vector composed of the decision variables of the upper model, x is the vector composed of the decision variables of the lower model, and g i and h j are the equality and inequality constraints for the safe operation of the upper model, and the feasible domains formed by them are all convex sets, m k is the only non-convex quadratic constraint of the lower model, and N and M are the coefficient matrices of the consensus constraints of the upper and lower models, respectively.

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

[0053] Optionally, the model for centrally solving the upper-level planning problem by convexifying it using the second-order cone relaxation technique is:

[0054]

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

[0056] Optionally, the model for decomposing the lower-level programming problem into a plurality of quadratic programming sub-problems containing a single non-convex quadratic constraint by strategy decomposition is:

[0057]

[0058] Among them, x i is the lower-level decision variable of node i, A i , B i are the coefficients of the original objective function of the node i subproblem and its quadratic term in the augmented Lagrangian term, C i is the coefficient of the linear term in the augmented Lagrangian form, M i is the only non-convex quadratic constraint on node i.

[0059] Optionally, the model for converting the quadratic programming subproblem into a semidefinite programming subproblem by Schur complement is:

[0060]

[0061] Among them, i is the dual variable of the non-convex quadratic constraint at node i, γ i is the maximum feasible upper bound of the optimal solution of 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 such that:

[0066]

[0067] The necessary and sufficient condition is that there is no λ satisfying:

[0068]

[0069] In other words, the above two equations have strong selectivity. 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.

[0070] Optionally, the process of iteratively calculating and sequentially correcting the optimization results of the decoupled upper-level planning problem and the decoupled lower-level planning problem according to the alternating direction multiplier method includes: initializing the maximum number of iterations, optimizing and solving the upper-level model of the decoupled upper-level planning problem to obtain upper-level variables; solving the lower-level model of the decoupled lower-level planning problem according to the upper-level variables to obtain lower-level variables; judging whether the residual of the upper and lower-level variables is less than a preset threshold value according to the upper-level variables and the lower-level variables; if the residual of the upper and lower-level variables is less than the preset threshold value, terminating the calculation; if the residual of the upper and lower-level variables is not less than the preset threshold value, updating the Lagrange multiplier, increasing the number of iterations by one, and re-optimizing and solving the upper and lower-level variables in sequence until the residual of the upper and lower-level variables is less than the preset threshold value or the number of iterations reaches the maximum number of iterations, then terminating the calculation.

[0071] Exemplarily, the algorithm parameters such as the maximum number of iterations are initialized, and the upper-level model is first optimized to obtain the variable y. Secondly, the upper-level variable result is used to solve the lower-level model to obtain its variable x, and it is determined whether the residual of the upper and lower variables is less than a given threshold; if so, the calculation is terminated; otherwise, the Lagrange multiplier is updated, the number of iterations is increased by one, and the upper and lower variables are optimized again in sequence until the residual is less than the given threshold or the maximum number of iterations is reached.

[0072] In addition, the method of the present invention solves the problem of infeasibility or suboptimality of the optimization solution caused by the convex relaxation error of the distribution network. By utilizing its mathematical characteristics, the method can be implemented in the programming language Matlab.

[0073] Specifically, the scheme of the present invention is further described by taking the scenario of photovoltaic carrying capacity assessment with inaccurate convex relaxation in the distribution network as an example to verify the effectiveness of the present invention, wherein a sequence corrected second-order cone global optimization method suitable for radial distribution network of the present invention is implemented by the following steps:

[0074] (1) Photovoltaic carrying capacity assessment model

[0075] 1) Objective function

[0076] In order to reflect the advantages of SMSOC algorithm over SOCR in terms of application scope and solution accuracy, that is, SMSOC can ensure that the solution result is error-free and correct, the present invention takes the photovoltaic carrying capacity assessment model in the distribution network as an example for analysis and discussion. The research on photovoltaic carrying capacity assessment is relatively mature, and the selection of indicators for evaluating carrying capacity under different emphases is also different. The photovoltaic carrying capacity assessment model proposed in the present invention comprehensively considers the maximization of distributed photovoltaic grid-connected capacity and the minimization of network losses, and takes this as the objective function. The distributed photovoltaic equivalent grid-connected capacity is the difference between the two, and its maximum value is defined as the carrying capacity of distributed photovoltaics.

[0077]

[0078] Where N PV is the node set configured with distributed photovoltaic units, p t,PV Represents the actual active power output of the photovoltaic unit at node t. By optimizing the active power output of the photovoltaic units in the system, the goal of maximizing the equivalent grid-connected capacity of distributed photovoltaics can be achieved.

[0079] 2) Constraints

[0080] The constraint conditions of the distributed photovoltaic capacity assessment model include all constraints in the simplified distribution network model. In order to meet the needs of the model and algorithm of the present invention, the positive direction of the current and power in the introduced distribution network branch flow model is set to point from the load node to the feeder node. It is defined that i, t, k, and n all represent the index of the node number, and (i, t) represents the branch from the i node to the t node. F i is the parent node of node i, that is, the upper node pointing from node i to the feeder node, S (t,i) represents the branch complex power flowing from node t to node i, z (t,i) represents the complex impedance of branch (t,i), s i Represents the injected complex power of node i. The reverse power flow diagram of the distribution network is shown in Figure 2 shown.

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

[0082]

[0083] In the formula, C i represents the node set with i as the parent node, v i is the square of the complex voltage at node i, l (t,i) is the square of the complex current in branch (t,i). θ Represent the sets including feeder nodes and excluding feeder nodes respectively; Respectively represent the branch (i, F i )’s active power and reactive power; Respectively represent the branch (i, F i )'s resistance and reactance; p i With q i denote the injected active power and injected reactive power of node i, respectively, and s i,PV With s i,L They represent the actual output and electrical load of the PV unit at node i respectively, and j represents the imaginary unit.

[0084] To ensure the safe operation of the system, the inequality constraints that the model needs to satisfy are:

[0085]

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

[0087] It is easy to see that the objective function of formula (1) is the superposition of linear functions related to each branch and node, which is usually a convex function because it only contains the output of the photovoltaic unit or the branch line loss, both of which are linear. In terms of constraints, formulas (2)-(4) and (6)-(10) are linear constraints and can form a convex set, but the quadratic equality constraint represented by formula (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 sharply with the increase of variable space. In existing technologies, relaxation techniques are often used to convexify equation (5). Under certain assumptions and scenarios, this type of relaxation technique has been proven to be tight and the results are accurate. However, these assumptions or scenarios are limited, and once violated, the results may be infeasible or suboptimal. Under the objective function of the photovoltaic load-bearing capacity assessment model in this paper, if only the traditional second-order cone relaxation technique is used to solve it, unacceptable errors will occur at both ends of equation (5).

[0089] (2) Modified second-order cone global optimization algorithm for radial distribution network sequence

[0090] In order to effectively deal with the non-convexity in the model, the present invention proposes a sequence modified second-order cone algorithm by utilizing the radial characteristics of the 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, and then transforms the lower-level quadratic programming subproblem with a single constraint into an equivalent convex problem, wherein each subproblem of the upper-level model contains a convex objective function and several linear constraints, and each subproblem of 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 of the lower-level model are non-convex. To overcome this problem, the present invention introduces the use of Schur complement and S-procedure to prove the strong duality of non-convex subproblems, and derives the equivalent dual form of non-convex subproblems. Finally, ADMM is used to establish a consensus relationship between the original variables and the auxiliary variables to ensure the effectiveness of the algorithm.

[0091] 1) Sub-problem decomposition

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

[0093]

[0094] in, v y,i , p y,i ,q y,i Both are vector y i The components in are the (upper layer) original variables. For the photovoltaic carrying capacity assessment problem, it can be regarded as the original variables of each node subject obtained by linear combination. In order to separate the problem into two layers, the vector formed by the auxiliary variables of each node subject i is defined as:

[0095]

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

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

[0098]

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

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

[0101]

[0102] Among them, Y i , X i are the feasible domains of the upper and lower models of node i, respectively. It is easy to see that the nodes of the lower model are independent of each other and do not affect each other, so they can be calculated in parallel. However, consensus constraints must be established between the original variables and the auxiliary variables to ensure the validity of the upper and lower variables. To this end, the photovoltaic carrying capacity assessment problem is rewritten as P1 according to the x variable:

[0103]

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

[0105]

[0106] In the formula, They are branches (i, F i ), the Lagrange multipliers of active power, reactive power and current squared, are the Lagrange multipliers of the voltage squared, injected active power and injected reactive power of node i, respectively. ρ is the penalty factor of the augmented Lagrange term, which is usually taken as a suitably large positive number. For any node entity i, the lower-level variables, upper-level variables, corresponding Lagrange multipliers and the original residual after each iteration can be obtained according to the following formulas:

[0107]

[0108] In the formula, k is the number of iterations, r k+1 is the original residual after the k+1th iteration, and ||(·)||2 is the second norm of (·). After the y component value is obtained from the upper-level problem (20), the corresponding component value is exchanged with the corresponding node body in the lower-level model. These values ​​are required to calculate the x vector. After the x vector is calculated, the Lagrange multiplier is updated and the original residual is calculated. The upper-level problem (20) can be solved centrally, and the y component values ​​of all node bodies in the upper-level model can be obtained by one optimization. Repeat this process until the residual is less than the given credible threshold. The entire information exchange process is as follows Figure 3 shown.

[0109] 2) Eliminate the non-convexity of the sub-problem

[0110] According to the knowledge of optimization theory, when the penalty factor is strictly positive and both the x-subproblem SP1 and the y-subproblem SP2 are convex, convergence can be guaranteed. It is easy to see that for the above-mentioned y-subproblem (20) which has a convex feasible domain and a convex objective function, the x-subproblem (19) is non-convex because it has a non-convex feasible domain. Therefore, applying the idea of ​​ADMM to solve the current SP1 and SP2 problems cannot guarantee strict convergence. To solve this problem, the theory of eliminating this non-convexity is introduced below. For the convenience of description, the original non-convex subproblem SP1 is rewritten as:

[0111]

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

[0113]

[0114] Where E is a 6×6 unit matrix. The coefficient matrix M of the constraint condition i is not semi-positive definite, the feasible region defined by the quadratic constraint is non-convex, but since each node body contains only one non-convex quadratic constraint, namely PU1 It is a typical QP1QC problem. In order to solve the above special non-convex problem, Schur complement and S-procedure are introduced to establish P U1 The strong duality gap between the dual problem and the primal problem. U1 The dual form of the problem is as follows:

[0115]

[0116] In the formula, υ i It is a question of U1 The dual variable of the quadratic inequality constraint in . Using the first-order optimality condition, x i It is expressed 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 can get the dual problem:

[0119]

[0120] υ i ≥0(28)

[0121]

[0122] Among them, formula (28) ensures that the dual variables are meaningful, and formula (29) ensures that the problem has a finite solution. Using Schur complement, the dual problem D U1 It can be expressed as a convex semidefinite programming form:

[0123]

[0124] υ i ≥0(31)

[0125]

[0126] From equation (32), we can see that the dual problem is always convex, so for any original problem P U1 It is always possible to solve for S U1 To obtain a solution, but this solution is not necessarily equal to the solution of the original problem. Only when the strong duality holds, the two problems have the same solution. U1 is non-convex, so we introduce the theory of control field and derive Theorem 1 through S-procedure.

[0127] Theorem 1: When problem P U1 When the constraint set of is non-empty, P U1 With S U1 The strong duality of holds, that is, P U1 With S U1 The optimal solution is equal to .

[0128] Proof: Referring to the standard S-procedure form, according to the parameter constraints (33) of the QP1QC problem, we can obtain the following standard semidefinite inequality constraints:

[0129]

[0130] Among them, D 0,i , D 1,i are the constant terms of the objective function and the quadratic constraint respectively. It is easy to see that in this problem, they are all zero vectors of the corresponding dimensions. The above expression is actually S U1 According to the S-procedure theory, for the QP1QC problem, when there exists υ i ≥0 so that the above relationship holds, the strong duality of the original problem and the dual problem holds. Therefore, problem P U1 The solution and S U1 Same solution.

[0131] (Proof completed)

[0132] In summary, through the above transformations and proofs, the traditional second-order cone optimization model is sequentially corrected to obtain a more accurate optimization solution under allowable error. The process of the sequentially corrected second-order cone global optimization algorithm proposed in this paper is shown in Table 1.

[0133] Table 1 Sequential modified second-order cone global optimization algorithm flow

[0134]

[0135] (II) Example Analysis

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

[0137] Table 2 PV unit configuration information

[0138]

[0139] (1) Analysis of unit output results

[0140] Under the above unit configuration, taking the maximum photovoltaic carrying capacity as the objective function, the second-order cone relaxation technique and the sequence modified second-order cone algorithm are used to evaluate and analyze the photovoltaic carrying capacity of the improved 33-node system, and the errors of the two optimization results are calculated. The actual output of the photovoltaic units and the system line loss are shown in Table 3 and Table 4 respectively.

[0141] Table 3 System photovoltaic unit output

[0142]

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

[0144]

[0145] It is easy to see that when the optimization goal is to maximize the photovoltaic carrying capacity, although the objective function values ​​of the two methods are the same, the photovoltaic output and the system line loss are very different. At this time, the second-order cone relaxation technology can only consider the overall objective function, but ignores the actual requirements of the system. The feasible domain after relaxation is not tight. Although the system photovoltaic output reaches the maximum, the system line loss also reaches the maximum at this time. Obviously, the result at this time is not a practical feasible solution.

[0146] (2) Error result analysis

[0147] In order to reflect the accuracy of the sequence modified second-order cone optimization algorithm proposed in the present invention, the power flow results obtained by the two methods are respectively brought into the left and right ends of the non-convex constraint for calculation, and the absolute error and relative error calculation methods of the left and right ends of the equation are respectively as follows:

[0148]

[0149] Referring to the above two error calculation methods, the error value of the improved IEEE 33-bus distribution system is analyzed under the second-order cone relaxation technology and the sequence modified second-order cone algorithm. The error of each branch of the system with respect to the non-convex equality constraint is as follows: Figure 4 and Figure 5 As shown. Figure 4 It can be seen that under the second-order cone relaxation technology, the flow results have large errors at both ends of the non-convex equality constraints, and the maximum error even exceeds 70%, such as branches 3, 4, 5, and 10. Even branch 19 with the smallest error exceeds 5%. The optimization solution at this time is not feasible.

[0150] On the other hand, the sequence modified second-order cone optimization algorithm proposed in the present invention is Figure 5It can be obtained that for any branch, the variable calculation result curves at both ends of the non-convex equality constraint are almost completely overlapped. Only after magnifying branches 6 and 24 by 1e7 times can the slight difference between the two be barely seen. This shows that the algorithm can ensure that the left and right ends of the non-convex quadratic equality constraint are strictly established while ensuring global optimality. The calculation error fully meets the accuracy requirements in the engineering field. At this time, the optimized solution is the actual feasible solution of the system.

[0151] In order to further reflect the difference in error results in different scale node systems, the improved IEEE 33, 69 and 136 node systems are calculated using two methods respectively, and the obtained errors are shown in Table 5.

[0152] Table 5 Calculation error of different scale node systems

[0153]

[0154] It can be seen that in the scenario of photovoltaic carrying capacity assessment, the SOCR model is not tight at this time. The flow results obtained by optimization solution have large errors on both sides of the non-convex quadratic equation constraint, which are difficult to ignore. Even the relative error ratio of the improved 33-node system with the smallest node scale has reached 76%, which is unacceptable in the engineering field, and the error value is also increasing with the increase of the node system scale. The sequence corrected second-order cone algorithm solves the upper and lower models iteratively and repeatedly corrects the upper model. In essence, when the algorithm converges, the model does not relax at all, and the non-convex quadratic equation constraint can always be guaranteed to be strictly established, avoiding the error caused by convex relaxation. Therefore, the obtained results are correct. Even in the 136-node system with the largest node system scale, the relative error ratio can be guaranteed to be within 0.0001, which can almost guarantee the zero error of the non-convex quadratic equation constraint, and fully meet the engineering accuracy requirements.

[0155] The above actual example analysis can conclude that:

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

[0157] 2) Using Schur complement and S-procedure, we can derive the KKT condition that the optimal solution of the subproblem converges to the optimal solution of the original problem, i.e., the original non-convex optimization problem;

[0158] 3) Compared with the second-order cone relaxation technology, the sequence-corrected second-order cone algorithm can effectively satisfy the non-convex equality constraints while ensuring global optimality. The proposed algorithm has smaller error, higher accuracy, and a wider range of applicable scenarios.

[0159] In summary, the solution of the embodiment of the present invention avoids the errors caused by convex relaxation, can effectively ensure the reliability and global optimality of the results, and proposes an accurate and reliable mathematical model for distribution networks with a high proportion of renewable energy, which has a strong reference significance for the optimized operation and precise calculation of distribution networks.

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

[0161] Computer readable storage medium can be a tangible device that can hold and store instructions used by an instruction execution device. Computer readable storage medium can be, for example, but not limited to, an electrical storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination thereof. More specific examples (non-exhaustive list) of computer readable storage medium include: a portable computer disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), a static random access memory (SRAM), a portable compact disk read-only memory (CD-ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanical encoding device, for example, a punch card or a convex structure in a groove on which instructions are stored, and any suitable combination thereof. The computer readable storage medium used here is not interpreted as a transient signal itself, such as a radio wave or other freely propagating electromagnetic wave, an electromagnetic wave propagated by a waveguide or other transmission medium (for example, a light pulse by an optical fiber cable), or an electrical signal transmitted by a wire.

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

[0163] The computer program instructions for performing the operation of the present 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" language or similar programming languages. Computer-readable program instructions may be executed entirely on a user's computer, partially on a user's computer, as an independent software package, partially on a user's computer, partially on a remote computer, or entirely on a remote computer or server. In the case of 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., using an Internet service provider to connect via the Internet). In some embodiments, an electronic circuit, such as a programmable logic circuit, a field programmable gate array (FPGA), or a programmable logic array (PLA), may be personalized by utilizing the state information of the computer-readable program instructions, and the electronic circuit may execute the computer-readable program instructions, thereby realizing various aspects of the present invention.

[0164] Various aspects of the present invention are described herein with reference to the flow charts and / or block diagrams of the methods, devices (systems) and computer program products according to embodiments of the present invention. It should be understood that each box of the flow chart and / or block diagram and the combination of each box in the flow chart and / or block diagram 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 device, thereby producing a machine, so that when these instructions are executed by the processor of the computer or other programmable data processing device, a device that implements the functions / actions specified in one or more boxes in the flowchart and / or block diagram is generated. These computer-readable program instructions can also be stored in a computer-readable storage medium, and these instructions cause the computer, programmable data processing device, and / or other equipment to work in a specific manner, so that the computer-readable medium storing the instructions includes a manufactured product, which includes instructions for implementing various aspects of the functions / actions specified in one or more boxes in 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 so that a series of operating steps are 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 implement the functions / actions specified in one or more boxes in the flowchart and / or block diagram.

[0167] The flow charts and block diagrams in the accompanying drawings show the possible architecture, functions and operations of the systems, methods and computer program products according to multiple embodiments of the present invention. In this regard, each box in the flow chart or block diagram can represent a part of a module, a program segment or an instruction, and a part of the module, a program segment or an instruction contains one or more executable instructions for realizing the specified logical function. In some alternative implementations, the functions marked in the box can also occur in a different order from the order marked in the accompanying drawings. For example, two consecutive boxes can actually be executed substantially in parallel, and they can sometimes be executed in the opposite order, depending on the functions involved. It should also be noted that each box in the block diagram and / or flow chart, and the combination of the boxes in the block diagram and / or flow chart can be implemented by a dedicated hardware-based system that performs the specified function or action, or can be implemented by a combination of dedicated hardware and computer instructions. It is well known to those skilled in the art that it is equivalent to implement it by hardware, implement it by software, and implement it by combining software and hardware.

[0168] Embodiments of the present invention have been described above, and the above description is exemplary, not exhaustive, and is not limited to the disclosed embodiments. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The selection of terms used herein is intended to best explain the principles of the embodiments, practical applications, or technical improvements in the marketplace, or to enable other persons of ordinary skill in the art to understand the embodiments disclosed herein. The scope of the present invention is defined by the appended claims.

Claims

1. A sequence modified second-order cone global optimization method suitable for radial distribution networks, characterized in that: include: Decoupling the original non-convex optimization problem in the power system according to the introduced auxiliary variables, so as to convert the original non-convex optimization problem into a two-level programming problem, wherein the two-level programming problem includes an upper-level programming problem and a lower-level programming problem; The upper-level planning problem is convexified by using a second-order cone relaxation technique and then solved in a centralized manner; Decomposing the lower-level programming problem into a plurality of quadratic programming sub-problems containing a single non-convex quadratic constraint by strategy decomposition; The quadratic programming subproblem is transformed into a semidefinite programming subproblem by Schur complement, and a strong dual gap is established by a 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 planning problem and the decoupled lower-level planning problem are iteratively calculated and sequentially corrected according to the alternating direction multiplier method to ensure that the non-convex constraints are strictly established.

2. The method according to claim 1, characterized in that The model for decoupling the original non-convex optimization problem in the power system according to the introduced auxiliary variables is: Among them, y is the vector composed of the decision variables of the upper model, x is the vector composed of the decision variables of the lower model, and g i and h j are the equality and inequality constraints for the safe operation of the upper model, and the feasible domains formed by them are all convex sets, m k is the only non-convex quadratic constraint of the lower model, and N and M are the coefficient matrices of the consensus constraints of the upper and lower models, respectively.

3. The method according to claim 2, characterized in that The model for centrally solving the upper-level planning problem by convexifying it using the second-order cone relaxation technique is: Among them, g i and h j Equality and inequality constraints for safe operation of the upper-level model.

4. The method according to claim 3, characterized in that The model for decomposing the lower-level programming problem into multiple quadratic programming sub-problems containing a single non-convex quadratic constraint through strategy decomposition is: Among them, x i is the lower-level decision variable of node i, A i , B i are the coefficients of the original objective function of the node i subproblem and its quadratic term in the augmented Lagrangian term, C i is the coefficient of the linear term in the augmented Lagrangian form, M i is the only non-convex quadratic constraint on node i.

5. The method according to claim 4, characterized in that The model for converting the quadratic programming subproblem into a semidefinite programming subproblem by Schul complement is: Among them, i is the dual variable of the non-convex quadratic constraint at node i, γ i is the maximum feasible upper bound of the optimal solution of the dual problem at node i.

6. The method according to claim 5, characterized in that The standard S-procedure is: If there exists x2 that satisfies: Then there exists x such that: The necessary and sufficient condition is that there is no λ that satisfies: Among them, 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.

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 planning problem and the decoupled lower-level planning problem according to the alternating direction multiplier method includes: Initializing the maximum number of iterations, optimizing and solving the upper-level model of the decoupled upper-level planning problem, and obtaining upper-level variables; Solving the lower-level model of the decoupled lower-level planning problem according to the upper-level variables to obtain lower-level variables; According to the upper layer variables and the lower layer variables, determining whether the residuals of the upper and lower layer variables are less than a preset threshold; If the residual of the upper and lower layer variables is less than the preset threshold, the calculation is terminated; If the residual of the upper and lower layer variables is not less than the preset threshold, the Lagrange multiplier is updated, the number of iterations is increased by one, and the upper layer variables and the lower layer variables are optimized and solved again in sequence until the residual of the upper and lower layer variables is less than the preset threshold or the number of iterations reaches the maximum number of iterations, and the calculation is terminated.

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