Quantum Isin model construction method for security constraint unit commitment optimization problem

Abstract

The invention discloses a quantum Isin model construction method for a security constraint unit commitment optimization problem, and relates to the field of quantum computation.The quantum Isin model construction method comprises the steps that a security constraint unit commitment optimization model is constructed, and parameters of a mixed integer programming problem are obtained; using a Benders decomposition method to decompose a mixed integer programming problem into a main problem and a sub-problem; substituting the optimal binary solution to solve the sub-problem to obtain a new cut plane and expand a cut plane set; constructing a compact high-dimensional quadratic function fitting cutting plane set; solving a positive semidefinite programming problem to obtain a high-dimensional quadratic function parameter; converting a quadratic unconstrained binary optimization model constructed based on a high-dimensional quadratic function into an Isin model; solving the Isin model to obtain a quantum bit state, and solving an optimal binary solution; and substituting the optimal binary solution into the above steps for iterative solution. According to the invention, the problem of huge consumption of quantum bit resources in the prior art is solved, especially the problem of difficulty in processing NP with complex constraints and more variables is solved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of quantum computing and the field of power system optimization, and in particular to a quantum Ising model construction method for a security constrained unit commitment optimization problem. BACKGROUND

[0002] In the field of power system optimization, the security constrained unit commitment optimization problem is described as a mixed integer linear programming problem (NP-hard), and the goal is to find the optimal solution from the feasible solution set of the combinatorial problem. By solving the mixed integer linear programming problem, the optimal decision scheme of the actual problem can be obtained.

[0003] Quantum computing has a great advantage in solving quadratic unconstrained binary optimization applications. Quantum annealing algorithm is based on quantum mechanical effects such as tunneling effect and superposition state, and has the potential to improve the speed of problem solving and the search ability of global optimal solution. The mixed integer programming problem can be regarded as a problem of searching for the state with the minimum energy, and all possible solutions of the problem can be represented in the form of 0-1 sequence. Therefore, it is necessary to first convert the problem into an equivalent quadratic unconstrained binary optimization model or Ising model so that the quantum annealer can handle it. At present, the constraint conditions are converted into the objective function in the form of binary by using the penalty function.

[0004] It is found through research that in many scenarios of using quantum computers to solve security constrained unit commitment optimization problems, the Ising model constructed using the penalty function form requires a large number of quantum bits, for example, the penalty coefficient in each iteration needs to be represented in the form of binary with tens of bits. On the existing quantum annealing algorithm, when the number of iterations of the algorithm increases to a certain extent, a large amount of valuable quantum bit computing resources will be consumed, which exceeds the maximum number of quantum bits that can be used by the current special quantum computer, and is not conducive to reducing the solving time and cost of the mixed integer programming problem. Secondly, the mixed integer programming is NP-hard, so it needs to consume a lot of time to solve using a classical computer. SUMMARY

[0005] In view of the above shortcomings in the prior art, the quantum Ising model construction method for a security constrained unit commitment optimization problem provided by the present application solves the problem of huge consumption of quantum bit resources when using the penalty function method to construct the Ising model and then using quantum computing (especially quantum annealing) to solve the security constrained unit commitment optimization problem in the prior art, especially for NP-hard problems with complex constraints and more variables.

[0006] In order to achieve the above application purpose, the technical scheme adopted by the present application is as follows: a quantum Ising model construction method for a security constrained unit commitment optimization problem, comprising the following steps:

[0007] S1: constructing a security constrained unit commitment optimization model, obtaining parameters of a mixed integer programming problem;

[0008] S2: in a classical computer, using a Benders decomposition method to decompose the mixed integer programming problem into a Benders master problem and a Benders subproblem;

[0009] S3: substituting the optimal binary solution to solve the Benders subproblem, obtaining a new cutting plane and expanding the cutting plane set;

[0010] S4: constructing a compact high-dimensional quadratic function fitting cutting plane set;

[0011] S5: solving a semi-positive definite programming problem through static robust optimization, obtaining high-dimensional quadratic function parameters;

[0012] S6: converting a quadratic unconstrained binary optimization model constructed based on the high-dimensional quadratic function into an Ising model;

[0013] S7: using a special quantum computer based on cloud technology to solve the Ising model to obtain a quantum bit state, and obtaining an optimal binary solution;

[0014] S8: substituting the optimal binary solution into step S3 for iterative solving until the error convergence of the optimal value of the subproblem objective function and the optimal value of the quadratic unconstrained binary optimization problem objective function is within a preset range, completing the construction of a quantum Ising model for a security constrained unit commitment optimization problem.

[0015] Further, the mixed integer programming problem in S1 is:

[0016]

[0017] wherein, and are binary variables and continuous variables, is a binary variable dimension of and are objective function coefficients, , and are constraint matrices, and are right end items, and the superscript represents the transpose of the matrix.

[0018] Further, the cutting plane set in S3 is:

[0019]

[0020] wherein, is a cutting plane set, and is the th iteration, is the auxiliary variable, is the iteration number;

[0021] At this time, the mixed integer programming problem is transformed into:

[0022] .

[0023] Further, in the S4, by fitting the optimal solution of the Benders master problem relaxation problem, a compact high-dimensional quadratic function is constructed by minimizing the value of the high-dimensional quadratic function at the optimal solution of the relaxation:

[0024]

[0025]

[0026]

[0027] wherein, , and are the parameters to be optimized in the high-dimensional quadratic function, is the optimal solution of the relaxation Benders master problem, is the regularization coefficient, represents an arbitrary symbol.

[0028] Further, in the S5, the following steps are included:

[0029] S51: The infinite-dimensional constraint in the high-dimensional quadratic function is regarded as a minimum problem, and its dual is a maximum problem, and the formula is:

[0030]

[0031]

[0032]

[0033] wherein, and are the dual variables associated with the boundary constraints , is the th cut plane , , , is the dual variable vector corresponding to the boundary constraints of the th cut plane, ​a vector associated with the first cutting plane, , , a scalar associated with the first cutting plane, ; ;

[0034] S52: Based on the maximum value problem, the infinite dimensional constraint is converted into a semi-definite programming problem by using the Schur complement property, and the formula is:

[0035]

[0036] .

[0037] Further, the S6 includes the following steps:

[0038] S61: Based on the high-dimensional quadratic function parameters 、 and obtained by solving the semi-definite programming problem, a quadratic unconstrained binary optimization model is constructed as:

[0039]

[0040] wherein, is a penalty coefficient;

[0041] S62: The quadratic unconstrained binary optimization model is converted into an Ising model:

[0042]

[0043]

[0044]

[0045]

[0046] wherein, 、 、 is a spin variable, taking -1 or 1, 、 and are known parameters, 、 and are binary variables.

[0047] The beneficial effects of the present application are:

[0048] (1) Efficient use of qubits: By approximating the cut plane set through semi-positive definite programming, the complex cut plane set is directly encoded into the Benders main problem in binary form of penalty function, which significantly reduces the auxiliary qubits required to represent cut constraints and the precision bits required to represent large coefficient penalty terms, making it possible to process larger-scale binary variable problems with finite qubits.

[0049] (2) Avoid solving NP-hard safety-constrained combinatorial optimization problems: Quantum Ising machines can replace the traditional Benders decomposition of safety-constrained combinatorial optimization problems to obtain mixed-integer programming Benders main problems (NP-hard), which may achieve faster iteration speeds on some problem instances, especially when quantum computing is used to accelerate the process. Attached Figure Description

[0050] Figure 1 This is a flowchart of a method for constructing a quantum Ising model for a safety-constrained unit combination optimization problem according to the present invention.

[0051] Figure 2 A schematic diagram for constructing a set of cutting planes for fitting a high-dimensional quadratic function.

[0052] Figure 3 The method of this invention is used to calculate the performance charts of 72 and 288 binary variable cases.

[0053] Figure 4 The diagram illustrates the convergence process of calculating 288 binary variable cases using the method of this invention. Detailed Implementation

[0054] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0055] like Figure 1 As shown, a method for constructing a quantum Ising model for the safety-constrained unit combination optimization problem includes the following steps:

[0056] S1: Construct a safety-constrained unit combination optimization model to obtain the parameters of the mixed integer programming problem;

[0057] S2: In classical computers, the Benders decomposition method is used to decompose mixed integer programming problems into the Benders main problem and Benders subproblems;

[0058] S3: Substitute the optimal binary solution to solve the Benders subproblem, obtain new cutting planes and expand the set of cutting planes;

[0059] S4: Construct a compact set of high-dimensional quadratic function fitting cutting planes;

[0060] S5: Solve the semi-positive programming problem by static robust optimization to obtain the parameters of high-dimensional quadratic function;

[0061] S6: Convert the quadratic unconstrained binary optimization model constructed based on the high-dimensional quadratic function into an Ising model;

[0062] S7: Solve the Ising model using a special quantum computer based on cloud technology to obtain the quantum bit state and the optimal binary solution;

[0063] S8: Substitute the optimal binary solution into step S3 for iterative solution until the error convergence of the optimal value of the sub-problem objective function and the optimal value of the quadratic unconstrained binary optimization objective function is within the preset range, and the quantum Ising model construction for the security constrained unit commitment optimization problem is completed.

[0064] The security constrained unit commitment optimization model in S1 is as follows, which is used to construct the mixed integer programming problem of the engineering problem. The parameters of the mixed integer programming problem to be solved are obtained, including the objective function coefficients and , constraint matrices , and , and right-hand side terms and .

[0065] The mixed integer programming problem in S1 is:

[0066]

[0067] wherein, and are binary variables and continuous variables, is the dimension of the binary variable , and are objective function coefficients, , and are constraint matrices, and are right-hand side terms, and the superscript indicates the transpose of the matrix.

[0068] In S2, the Benders decomposition method is used to decompose the mixed integer programming problem into a master problem and a sub-problem, and the cut plane set is obtained , the cut plane set is the cut plane produced by substituting the optimal binary solution into the sub-problem after iterative solution, at this time the original mixed integer programming problem can be converted into the following formula.

[0069] The cut plane set in S3 is:

[0070]

[0071] where, is the set of cutting planes, and is the coefficient of cutting planes in the iteration, is the auxiliary variable, is the iteration number;

[0072] At this time, the mixed integer programming problem is transformed into:

[0073] .

[0074] In S4, the high-dimensional quadratic function is a curve for solving the semi-definite programming problem by fitting the set of cutting planes, in order to obtain a compact high-dimensional quadratic function, the application fits the optimal solution of the relaxed master problem, and takes the value of the high-dimensional quadratic function corresponding to the optimal solution as the optimization target; in order to make the high-dimensional quadratic function compactly fit the set of cutting planes, it is necessary to ensure that any point is greater than the set of cutting planes .

[0075] In S4, by fitting the optimal solution of the relaxed problem, a compact high-dimensional quadratic function is constructed by minimizing the value of the high-dimensional quadratic function at the optimal solution of the relaxed problem:

[0076]

[0077]

[0078]

[0079] where, , and are the parameters of the high-dimensional quadratic function of the semi-definite programming, is the optimal solution of the relaxed master problem, is the regularization coefficient, represents an arbitrary symbol.

[0080] In S5, in order to process the infinite-dimensional constraint of the problem in step S4, the constraint can be converted into a semi-definite programming problem by static robust optimization, first considering the infinite-dimensional constraint as a minimum value problem, and then converting it into a maximum value problem by duality, and then converting the problem into a semi-definite programming by using the Schur complement property. The specific process is as follows:

[0081] The S5 includes the following steps:

[0082] S51: considering the infinite-dimensional constraint in the high-dimensional quadratic function as a minimum value problem, and dualizing it into a maximum value problem, the formula is:

[0083]

[0084]

[0085]

[0086] where, and are the dual variables associated with the boundary constraints , is the set of th cut plane, , , is the dual variable vector corresponding to the boundary constraints of the th cut plane, is the vector associated with the th cut plane, , , is the scalar associated with the th cut plane, ;

[0087] S52: Based on the maximum value problem, the infinite dimensional constraint is converted into a semi-definite programming problem by using the Schur complement property, and the formula is:

[0088]

[0089] .

[0090] In S6, in order to adapt to quantum computers, the quadratic unconstrained binary optimization model based on high-dimensional quadratic functions is converted into an Ising model.

[0091] The S6 includes the following steps:

[0092] S61: Based on the high-dimensional quadratic function parameters , and obtained by solving the semi-definite programming problem, the quadratic unconstrained binary optimization model is constructed as:

[0093]

[0094] where, is the penalty coefficient;

[0095] S62: The quadratic unconstrained binary optimization model is converted into an Ising model:

[0096]

[0097] ​

[0098]

[0099]

[0100] wherein, , , is a spin variable, taking -1 or 1, , and are known parameters, , and are binary variables.

[0101] In one embodiment of the present application, Figure 2 is a schematic diagram of constructing a high-dimensional quadratic function fitting cut plane, Figure 3 is the performance of 72 and 288 binary variable cases respectively calculated using the method of the present application, Figure 4 is the convergence process of 288 binary variable cases calculated using the method of the present application, it can be seen that the method of the present application has certain advantages.

[0102] Those skilled in the art will realize that the embodiments described herein are for the purpose of aiding the reader in understanding the principles of the present application and should be construed as not limiting the scope of the present application to such specific embodiments and examples. Those skilled in the art can make various other specific modifications and combinations according to the technical inspiration of the present application without departing from the spirit of the present application, and these modifications and combinations are still within the scope of protection of the present application.

Claims

1. A method for constructing a quantum Ising model for security constrained unit commitment optimization problem, characterized in that, The method comprises the following steps: S1: constructing a security constrained unit commitment optimization model to obtain parameters of a mixed integer programming problem; S2: in a classical computer, using a Benders decomposition method to decompose the mixed integer programming problem into a Benders master problem and a Benders sub-problem; S3: substituting an optimal binary solution to solve the Benders sub-problem to obtain a new cutting plane and expand the cutting plane set; S4: constructing a compact high-dimensional quadratic function fitting cutting plane set; S5: solving a semi-positive definite programming problem through static robust optimization to obtain high-dimensional quadratic function parameters; S6: converting a quadratic unconstrained binary optimization model constructed based on the high-dimensional quadratic function into an Ising model; S7: solving the Ising model based on cloud technology using a special quantum computer to obtain a quantum bit state and an optimal binary solution; S8: substituting the optimal binary solution into step S3 for iterative solving until an error of an optimal value of a sub-problem objective function and an optimal value of a quadratic unconstrained binary optimization problem objective function converges within a preset range, and a quantum Ising model for a security constrained unit commitment optimization problem is constructed.

2. The method of claim 1, wherein, The mixed integer programming problem in S1 is: where and are binary and continuous variables, respectively, is a binary variable is the dimension of and are objective function coefficients, , and are constraint matrices, and are right-hand side terms, and the superscript denotes the transpose of a matrix.

3. The method of claim 2, wherein, The cutting plane set in S3 is: wherein, is a set of cut planes, and is the first iteration of the coefficients of the cut planes, is an auxiliary variable, is the number of iterations; At this time, the mixed integer programming Benders master problem is converted into: 。 4. The method of claim 3, wherein, In S4, a relaxed optimal solution is obtained by solving a relaxed Benders master problem, and a compact high-dimensional quadratic function is constructed by minimizing a value of the high-dimensional quadratic function at the relaxed optimal solution: wherein , and are high-dimensional quadratic function optimization parameters, is the Benders master problem relaxation optimal solution, is a regularization coefficient, denotes an arbitrary symbol.

5. The method of claim 4, wherein, S5 comprises the following steps: S51: considering an infinite-dimensional constraint in the high-dimensional quadratic function as a minimum problem, and dualizing it into a maximum problem, formula is: wherein and are dual variables associated with the boundary constraints , are the first cutting planes , , , are the dual variable vectors corresponding to the boundary constraints of the first cutting planes , are vectors associated with the first cutting planes , are scalars associated with the first cutting planes ; S52: based on the maximum problem, using a Schur complement property to convert the infinite-dimensional constraint into a semi-positive definite programming problem, formula is: 。 6. The method of claim 5, wherein, S6 comprises the following steps: S61: constructing a quadratic unconstrained binary optimization model based on the high-dimensional quadratic function parameters obtained in solving the semi-positive programming problem , and : wherein is a penalty coefficient; S62: converting the quadratic unconstrained binary optimization model into an Ising model: wherein , , is a spin variable, taking -1 or 1, , and are known parameters, , and are binary variables.

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