CN113779135A - Method and device for constructing uncertainty set of rhombic corner cut convex hull - Google Patents

Abstract

The invention discloses a method and a device for constructing a rhombus chamfer convex hull uncertainty set, which belong to the technical field of electrical engineering, and are characterized in that a regularized ellipsoid convex hull uncertainty set containing all historical data is established; amplifying the ellipsoid inscribed diamond to include all historical data to obtain a diamond convex hull uncertainty set; finding out one of all historical data with the maximum projection in each axial direction of the diamond convex hull, and drawing a tangent plane perpendicular to the axial direction through the point; and calculating the end point coordinates of the intersection of each tangent plane and the rhombic convex hull, and further obtaining the vertex coordinates of the rhombic convex hull, thereby constructing the uncertainty set of the rhombic convex hull. The invention can make full use of the information hidden in the historical data to cut the uncertain scenes which can not appear actually, thereby reducing the uncertain range and the conservative property on the premise of not sacrificing the robustness, and the complexity of the number of the vertexes of the rhombic tangent angle convex hull is o (n)²) Avoid itThe number of convex hull vertexes in the high dimensional space is exponentially increased.

Description

Method and device for constructing uncertainty set of rhombic corner cut convex hull

Technical Field

The invention belongs to the technical field of electrical engineering, and particularly relates to a method and a device for constructing a rhombus chamfer convex hull uncertainty set.

Background

With the increasing proportion of renewable energy in the power system, the uncertainty seriously threatens the safe and reliable operation of the power system. Robust optimization is one of the common methods for dealing with the problem, and all possible realizations of uncertainty can be considered, so that the operation safety of the system is ensured. The establishment of the uncertainty set is a key problem of robust optimization, and directly influences the robustness and the economy of a final result.

A typical set of uncertainties is a box set, constructed only taking into account the upper and lower bounds of the uncertainty, but may contain many scenarios that are not actually possible, and is more conservative.

With the increase of running data and the development of data processing technology, the data-driven uncertainty set is receiving more and more attention, wherein the data-driven convex hull uncertainty set is receiving wide attention. Various types of sets of convex hull uncertainties have been proposed in the literature, but there are still problems: 1) an ellipsoidal convex hull, which is non-linear and difficult to apply; 2) the diamond convex hull has the least number of vertexes, but has certain conservatism; 3) the conservation can be better reduced by the polygonal convex hull, but the problem of high-dimensional index explosion exists due to excessive vertex number.

Disclosure of Invention

Aiming at the defects or improvement requirements in the prior art, the invention provides a method and a device for constructing a rhombus chamfer convex hull uncertainty set, and aims to solve the problems of conservation and excessive vertex number of the uncertainty set in the prior art.

In order to achieve the purpose, the invention provides a method for constructing an uncertainty set of a rhombus chamfer convex hull, which comprises the following steps of:

s1, acquiring output historical data of renewable energy sources;

s2, establishing an ellipsoid convex hull uncertainty set containing all historical data, and converting the historical data and the ellipsoid convex hull into a regular space;

s3, amplifying the rhombus inscribed in the ellipsoid to the point that the rhombus contains all historical data to obtain a rhombus convex hull uncertainty set;

s4, projecting all historical data to each axial direction of the rhombic convex hull, and making a tangent plane perpendicular to each axial direction through a data point corresponding to the maximum projection in the positive axial direction and the negative axial direction respectively aiming at each axial direction to obtain the end point coordinate of the intersection of each tangent plane and the rhombic convex hull;

and S5, restoring the endpoint coordinates to an original space to obtain vertex coordinates of the rhombic corner cut convex hull, and constructing a rhombic corner cut convex hull uncertainty set based on the vertex coordinates.

Further, in S4, obtaining the end point coordinates of the intersection of each tangent plane and the diamond-shaped convex hull specifically includes: and cutting the rhombic convex hulls according to the size of the rhombic convex hull volume which can be cut off by each section in sequence from large to small to sequentially obtain the end point coordinates of the intersection of each section and the rhombic convex hull.

Further, in S2, the set of ellipsoid convex hull uncertainties transformed to the canonical space is represented as:

v^TQ_vv≤1

where v is the coordinate in regularization space, Q_v＝diag([λ₁,…,λ_i，…，λ_n]) N is the dimension of the historical data, λ_iIs a matrix Q_uI characteristic value of (2), Q_uThe coefficients of the ellipsoid convex hull in the original space;

wherein Q is_uObtained according to the following optimization problem:

in the formula u^hkFor the renewable energy output historical data, the superscript k is a scene number; c is the central coordinate of the ellipse in the original space;

spatial transformation from original space to canonical space:

wherein the transformation matrix J is according to the formula Q_u＝J^TQ_vJ, obtaining;

historical data v transformed to canonical space^hk＝J(u^hk-c)。

Further, in the S3, the ellipsoidal inscribed diamond is represented as:

in the formula, v_iI-th dimensional coordinate of v, r_iIs the semi-axial length of the ellipsoid convex hull in the direction of the dimension i,

the set of diamond-shaped convex hull uncertainties is represented as:

in the formula, R_iIs the semi-axial length of the diamond convex hull in the dimension i direction, R_i＝αr_iα is a scaling factor, and

renewable energy contribution history for k-th scene in regularized spaceThe ith coordinate of the history data.

Further, in S4, the mathematical expression of the tangent plane is:

v_j＝v_j,cut

in the formula, v_j，cutAs historical data v^hkThe maximum value is projected on the j dimension axial direction of the rhombic convex hull, and the positive direction is

Negative direction is

Further, in S4, the mathematical expression of the intersection of each tangent plane and the diamond-shaped convex hull is:

the endpoint v of the intersection^ekThe coordinates are:

in the formula, the intermediate amount c_j＝1-|v_j,cut|/R_j。

Further, in S5, the coordinates of the vertex of the rhombus-cut convex hull are u^ek＝J^-1v^ek+c；

The set of uncertainty of the rhombic corner cut convex hull

Comprises the following steps:

wherein N is the total number of vertices of the rhombic corner cut convex hull and is omega_kAre auxiliary variables.

On the other hand, to achieve the above object, the present invention further provides a device for constructing an uncertainty set of a rhombus chamfer convex hull, comprising:

the historical data acquisition module is used for acquiring the output historical data of the renewable energy;

the ellipsoid convex hull establishing module is used for establishing an ellipsoid convex hull uncertainty set containing all historical data and converting the historical data and the ellipsoid convex hull into a regular space;

the rhombic convex hull establishing module is used for amplifying the ellipsoid inscribed rhombus to contain all historical data to obtain a rhombic convex hull uncertainty set;

the rhombic convex hull establishing module is used for projecting all historical data to each axial direction of the rhombic convex hull, and for each axial direction, respectively making a tangent plane perpendicular to each axial direction through a data point corresponding to the maximum projection in the positive axial direction and the negative axial direction so as to obtain the end point coordinate of the intersection of each tangent plane and the rhombic convex hull; and restoring the endpoint coordinates to an original space to obtain vertex coordinates of the rhombic corner cut convex hull, and constructing a rhombic corner cut convex hull uncertainty set based on the vertex coordinates.

Generally, by the above technical solution conceived by the present invention, the following beneficial effects can be obtained:

(1) according to the invention, on the basis of the rhombic convex hull, all historical data are projected to each axial direction of the rhombic convex hull, and for each axial direction, a section perpendicular to the axial direction is made through a data point corresponding to the maximum projection in the positive axial direction and the negative axial direction respectively, so that the end point coordinate of the intersection of each section and the rhombic convex hull is obtained, the vertex coordinate of the rhombic tangent angle convex hull is further obtained, and the uncertainty set of the rhombic tangent angle convex hull is constructed. Therefore, the invention can make full use of the information hidden in the historical data to cut the uncertain scenes which can not appear actually, thereby reducing the uncertain range and the conservatism without sacrificing the robustness, and the number complexity of the peaks of the rhombic corner cut convex hull is o (n)²) The problem of exponential increase of the number of the convex hull vertexes in the high-dimensional space is solved.

(2) According to the invention, the diamond convex hulls are cut according to the size of the diamond convex hull cut by each section in the order from large to small, the end point coordinates of the intersection of each section and the diamond convex hull are obtained in sequence, and the diamond corner cut convex hull with a relatively smaller size can be obtained under the condition that the cutting volumes of each section are overlapped.

Drawings

Fig. 1 is a flowchart of a method for constructing a rhombus chamfer convex hull uncertainty set according to an embodiment of the present invention;

fig. 2(a) to fig. 2(c) are schematic diagrams of an ellipsoid convex hull uncertainty set including all historical data, historical data converted into a canonical space, a scaling schematic diagram of an ellipsoid convex hull and an ellipsoid inscribed diamond, and a corner cut schematic diagram, respectively, according to an embodiment of the present invention;

FIG. 3 is a three-dimensional schematic diagram of an uncertainty set of a rhombus chamfer convex hull according to an embodiment of the invention;

fig. 4 is a block diagram of an apparatus for constructing an uncertainty set of a rhombus chamfer convex hull according to an embodiment of the present invention.

Detailed Description

In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

In the present application, the terms "first," "second," and the like (if any) in the description and the drawings are used for distinguishing between similar elements and not necessarily for describing a particular sequential or chronological order.

In the invention and the attached drawings, the ellipsoid inscribed diamond refers to a polyhedron enclosed by the vertexes of the ellipsoid; in a two-dimensional plane, if the four vertices of a diamond are on an ellipse, the diamond is called the inscribed diamond of the ellipse, and the intersection point of the diagonals of the diamond is the center of the ellipse.

Fig. 1 is a flowchart of a method for constructing a set of uncertainty of a rhombus-cut convex hull according to an embodiment of the present invention, where the control method includes operations S1-S5.

In operation S1, renewable energy output history data is obtained. Specifically, the method comprises the following steps:

dividing the collected renewable energy output historical data according to days, recording the output data of each day as a historical scene, and expressing the collected data in a vector form. In this embodiment, u^hkFor the renewable energy output historical data, the superscript k is the scene number.

In operation S2, an ellipsoid convex hull uncertainty set is created that contains all of the historical data, and the historical data and ellipsoid convex hulls are transformed into a canonical space. Specifically, the method comprises the following steps:

an ellipsoid convex hull uncertainty set containing all historical data is established as shown in fig. 2(a), and then the historical data and ellipsoid convex hulls are transformed into a canonical space as shown in fig. 2 (b).

The set of ellipsoid convex hull uncertainties transformed to canonical space is represented as:

v^TQ_vv≤1

where v is the coordinate in regularization space, Q_v＝diag([λ₁,…,λ_i,…,λ_n]) N is the dimension of the historical data, λ_iIs a matrix Q_uI characteristic value of (2), Q_uThe coefficients of the ellipsoid convex hull in the original space;

wherein Q is_uObtained according to the following optimization problem:

in the formula u^hkFor the renewable energy output historical data, the superscript k is a scene number; c is the central coordinate of the ellipse in the original space;

spatial transformation from original space to canonical space:

wherein the transformation matrix J is according to the formula Q_u＝J^TQ_vJ, obtaining;

historical data v transformed to canonical space^hk＝J(u^hk-c)。

Operation S3 enlarges the ellipsoid inscribed diamond to contain exactly all the historical data, resulting in a set of diamond convex hull uncertainties. Specifically, the method comprises the following steps:

since the polyhedron surrounded by the vertexes of the ellipsoid is not enough to cover all historical data, a scaling coefficient alpha is introduced to scale the convex polyhedron hull, as shown in fig. 2(b), the rhombus represented by the dotted line is the rhombus inscribed in the ellipsoid, and the rhombus represented by the solid line is the enlarged rhombus.

The ellipsoidal inscribed diamond is represented as:

in the formula, v_iI-th dimensional coordinate of v, r_iIs the semi-axial length of the ellipsoid convex hull in the direction of the dimension i,

the set of diamond-shaped convex hull uncertainties is represented as:

in the formula, R_iIs the semi-axial length of the diamond convex hull in the dimension i direction, R_i＝αr_iα is a scaling factor, and

is the ith coordinate of the renewable energy output historical data of the kth scene in the regularization space.

Operation S4 is performed to project all historical data onto each axial direction of the diamond-shaped convex hull, and for each axial direction, a tangent plane perpendicular to each axial direction is made through the data point corresponding to the maximum projection in the positive axial direction and the negative axial direction, respectively, so as to obtain the end point coordinates of the intersection of each tangent plane and the diamond-shaped convex hull. Specifically, the method comprises the following steps:

the mathematical expression of the tangent plane is as follows:

v_j＝v_j,cut

in the formula, v_j,cutAs historical data v^hkThe maximum value is projected on the j dimension axial direction of the rhombic convex hull, and the positive direction is

Negative direction is

The mathematical expression of the intersection of the tangent plane and the rhombic convex hull is as follows:

end point v of the intersection^ekThe coordinates are:

in the formula, the intermediate amount c_j＝1-|v_j,cut|/R_j。

As shown in fig. 2(c), a schematic view of a corner cut when j is 1.

It should be noted that the axial direction of the rhombic convex hull refers to the axial direction of the symmetry axis.

Further, according to the size of the diamond-shaped convex hull which can be cut off by each section, the diamond-shaped convex hull is cut in the order from large to small, and the end point coordinates of the intersection of each section and the diamond-shaped convex hull are obtained in sequence.

And operation S5, restoring the endpoint coordinates to the original space to obtain vertex coordinates of the rhombic corner cut convex hull, and constructing a rhombic corner cut convex hull uncertainty set based on the vertex coordinates. Specifically, the method comprises the following steps:

the vertex coordinates of the rhombic corner cut convex hull are u^ek＝J^-1v^ek+c；

Rhombus corner cut convex hull uncertainty set

Comprises the following steps:

wherein N is the total number of vertices of the rhombic corner cut convex hull and is omega_kAre auxiliary variables.

Fig. 3 is a three-dimensional schematic diagram of an uncertainty set of a rhombus-cut convex hull provided in this embodiment, in which a tight convex hull is a convex hull with a minimum volume and containing all historical data. The rhombic corner cut convex hull obtained in the embodiment comprises the tight convex hull, so that the method provided by the invention can ensure that the obtained scheduling plan can be robust to any historical scene when being used for scheduling the power system, and further ensure the operation safety of the power system.

Fig. 4 is a block diagram of an apparatus for constructing an uncertainty set of a rhombus chamfer convex hull according to an embodiment of the present invention. Referring to fig. 4, the apparatus 400 includes a history data obtaining module 410, an ellipsoid convex hull establishing module 420, a diamond convex hull establishing module 430, and a diamond chamfer convex hull establishing module 440.

The historical data obtaining module 410, for example, performs operation S1 to obtain the renewable energy output historical data;

the convex ellipsoid hull creation module 420, for example, performs operation S2, creates a convex ellipsoid hull uncertainty set containing all historical data, and transforms the historical data and the convex ellipsoid hull into a canonical space;

the diamond-shaped convex hull establishing module 430, for example, performs operation S3 to enlarge the ellipsoid inscribed diamond to include exactly all the historical data, resulting in a diamond-shaped convex hull uncertainty set;

the diamond-shaped chamfer convex hull establishing module 440 performs operations S4 and S5, for example, to project all historical data onto each axial direction of the diamond-shaped convex hull, and for each axial direction, make a tangent plane perpendicular to each axial direction through a data point corresponding to the maximum projection in the positive axial direction and the negative axial direction, respectively, to obtain the endpoint coordinates of the intersection of each tangent plane and the diamond-shaped convex hull; and restoring the endpoint coordinates to an original space to obtain vertex coordinates of the rhombic corner cut convex hull, and constructing a rhombic corner cut convex hull uncertainty set based on the vertex coordinates.

The apparatus 400 is used to perform the method for constructing the uncertainty set of the rhombus-cut convex hull in the embodiment shown in fig. 1. For details that are not described in the present embodiment, please refer to the method for constructing the uncertainty set of the rhombus chamfer convex hull in the embodiment shown in fig. 1, which is not described herein again.

It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (8)

1. A method for constructing an uncertainty set of a rhombus chamfer convex hull is characterized by comprising the following steps:

s1, acquiring output historical data of renewable energy sources;

s2, establishing an ellipsoid convex hull uncertainty set containing all historical data, and converting the historical data and the ellipsoid convex hull into a regular space;

s3, amplifying the rhombus inscribed in the ellipsoid to the point that the rhombus contains all historical data to obtain a rhombus convex hull uncertainty set;

s4, projecting all historical data to each axial direction of the rhombic convex hull, and making a tangent plane perpendicular to each axial direction through a data point corresponding to the maximum projection in the positive axial direction and the negative axial direction respectively aiming at each axial direction to obtain the end point coordinate of the intersection of each tangent plane and the rhombic convex hull;

and S5, restoring the endpoint coordinates to an original space to obtain vertex coordinates of the rhombic corner cut convex hull, and constructing a rhombic corner cut convex hull uncertainty set based on the vertex coordinates.

2. The method for constructing an uncertainty set of a rhombus cut angle convex hull according to claim 1, wherein in the step S4, the coordinates of the end points of the intersection of each cut plane and the rhombus convex hull are obtained, specifically: and cutting the rhombic convex hulls according to the size of the rhombic convex hull volume which can be cut off by each section in sequence from large to small to sequentially obtain the end point coordinates of the intersection of each section and the rhombic convex hull.

3. The method for constructing a rhombohedral corner convex hull uncertainty set according to claim 1 or 2, characterized in that in S2, the ellipsoid convex hull uncertainty set transformed to the canonical space is represented as:

v^TQ_vv≤1

where v is the coordinate in regularization space, Q_v＝diag([λ₁,…,λ_i,…,λ_n]) N is the dimension of the historical data, λ_iIs a matrix Q_uI characteristic value of (2), Q_uThe coefficients of the ellipsoid convex hull in the original space;

wherein Q is_uObtained according to the following optimization problem:

in the formula u^hkFor the renewable energy output historical data, the superscript k is a scene number; c is the central coordinate of the ellipse in the original space;

spatial transformation from original space to canonical space:

wherein the transformation matrix J is according to the formula Q_u＝J^TQ_vJ, obtaining;

historical data v transformed to canonical space^hk＝J(u^hk-c)。

4. The method according to claim 3, wherein in the step S3, the ellipsoid inscribed diamond is represented as:

in the formula, v_iI-th dimensional coordinate of v, r_iIs the semi-axial length of the ellipsoid convex hull in the direction of the dimension i,

the set of diamond-shaped convex hull uncertainties is represented as:

in the formula, R_iIs the semi-axial length of the diamond convex hull in the dimension i direction, R_i＝αr_iα is a scaling factor, and

is the ith coordinate of the renewable energy output historical data of the kth scene in the regularization space.

5. The method according to claim 4, wherein in the step S4, the tangent plane mathematical expression is as follows:

v_j＝v_j,cut

in the formula, v_j,cutAs historical data v^hkThe maximum value is projected on the j dimension axial direction of the rhombic convex hull, and the positive direction is

Negative direction is

6. The method according to claim 5, wherein in the step S4, the mathematical expression of the intersection of each tangent plane and the diamond-shaped convex hull is as follows:

the endpoint v of the intersection^ekThe coordinates are:

in the formula, the intermediate amount c_j＝1-|v_j,cut|/R_j。

7. The method according to claim 6, wherein in step S5, the coordinates of the vertex of the rhombus-cut convex hull are u^ek＝J^-1v^ek+c；

The set of uncertainty of the rhombic corner cut convex hull

Comprises the following steps:

wherein N is the total number of vertices of the rhombic corner cut convex hull and is omega_kAre auxiliary variables.

8. A rhombus corner cut convex hull uncertainty set construction device is characterized by comprising the following steps:

the historical data acquisition module is used for acquiring the output historical data of the renewable energy;

the ellipsoid convex hull establishing module is used for establishing an ellipsoid convex hull uncertainty set containing all historical data and converting the historical data and the ellipsoid convex hull into a regular space;

the rhombic convex hull establishing module is used for amplifying the ellipsoid inscribed rhombus to contain all historical data to obtain a rhombic convex hull uncertainty set;

the rhombic convex hull establishing module is used for projecting all historical data to each axial direction of the rhombic convex hull, and for each axial direction, respectively making a tangent plane perpendicular to each axial direction through a data point corresponding to the maximum projection in the positive axial direction and the negative axial direction so as to obtain the end point coordinate of the intersection of each tangent plane and the rhombic convex hull; and restoring the endpoint coordinates to an original space to obtain vertex coordinates of the rhombic corner cut convex hull, and constructing a rhombic corner cut convex hull uncertainty set based on the vertex coordinates.

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