A method and device for constructing an uncertainty set of rhombus cut corner convex hull
By constructing a rhombic tangent convex hull uncertainty set, using historical data projection and section processing technology, the problems of conservatism of uncertainty sets and excessive number of vertices in the existing technology are solved, and the effect of effectively reducing the uncertainty range in robust optimization is achieved.
Patent Information
- Application Number
- CN202110991763.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2021-08-27
- Publication Date
- 2025-04-04
- Estimated Expiration
- 2041-08-27
AI Technical Summary
In the prior art, uncertainty sets have problems such as conservatism and excessive number of vertices, which makes it difficult to effectively reduce the uncertainty range in robust optimization.
By obtaining the historical data of renewable energy output, an ellipsoidal convex hull uncertainty set is established and converted to regular space, and then the ellipsoidal inline rhombus is enlarged to form a rhombus convex hull, an uncertainty set. Then, the historical data is projected on each axial direction of the rhombus convex hull, and the coordinates of the rhombus cushion cushion cushion cushion cushion cushion cushion cushion cushion cushion cushion cushion uncertainty set is constructed.
This method can narrow the uncertainty range, reduce conservatism without sacrificing robustness, and avoid the high-dimensional index explosion problem. The number of vertices of diamond-shaped tangent convex hull is o(n2).
Smart Images
Figure CN113779135B_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the field of electrical engineering technology, and more specifically, relates to a method and device for constructing a rhombus cut-corner convex hull uncertainty set. Background Art
[0002] As the proportion of renewable energy in the power system continues to increase, its uncertainty seriously threatens the safe and reliable operation of the power system. Robust optimization is one of the common methods to deal with this problem. It can consider all possible realizations of uncertainty to ensure the safe operation of the system. The establishment of uncertainty sets is a key issue in robust optimization, which directly affects the robustness and economy of the final result.
[0003] A typical uncertainty set is a box set, which is constructed by only considering the upper and lower bounds of uncertainty. However, it may contain many scenarios that are impossible to occur in reality and is relatively conservative.
[0004] With the increase of operating data and the development of data processing technology, data-driven uncertainty sets have received more and more attention, among which data-driven convex hull uncertainty sets have received extensive attention. Various types of convex hull uncertainty sets have been proposed in the literature, but there are still many problems: 1) Ellipsoid convex hull, nonlinear, difficult to apply; 2) Rhombus convex hull, with the least number of vertices, but there is a certain conservatism; 3) Polygonal convex hull, can better reduce conservatism, but the number of vertices is too large, and there is a high-dimensional exponential explosion problem. Summary of the invention
[0005] In view of the above defects or improvement needs of the prior art, the present invention provides a method and device for constructing a rhombus cut-corner convex hull uncertainty set, aiming to solve the problems of conservatism and excessive number of vertices in the uncertainty set in the prior art.
[0006] To achieve the above object, the present invention provides a method for constructing a rhombus cut corner convex hull uncertainty set, comprising the following steps:
[0007] S1, obtain historical data of renewable energy output;
[0008] S2, establish the uncertainty set of the ellipsoid convex hull containing all historical data, and transform the historical data and the ellipsoid convex hull into the regular space;
[0009] S3, enlarge the rhombus inscribed in the ellipsoid to just contain all historical data, and obtain the rhombus convex hull uncertainty set;
[0010] S4, projecting all historical data onto the axes of the rhombus convex hull, and for each axis, making a section perpendicular to each axis through the data points corresponding to the maximum projection on the positive axis and the negative axis, respectively, to obtain the endpoint coordinates of the intersection of each section and the rhombus convex hull;
[0011] S5, restoring the endpoint coordinates to the original space to obtain the vertex coordinates of the rhombus cut corner convex hull, and constructing the rhombus cut corner convex hull uncertainty set based on the vertex coordinates.
[0012] Furthermore, in S4, the endpoint coordinates of the intersection of each of the cut faces and the rhombus convex hull are obtained, specifically: according to the size of the volume of the rhombus convex hull that can be cut off by each of the cut faces, the rhombus convex hull is cut in order from large to small, and the endpoint coordinates of the intersection of each of the cut faces and the rhombus convex hull are obtained in sequence.
[0013] Furthermore, in S2, the uncertainty set of the convex hull of the ellipsoid converted to the regular space is expressed as:
[0014] v T Q v v≤1
[0015] Where v is the coordinate in the regularized space, Q v =diag([λ1,…,λ i , …, λ n ]), n is the dimension of historical data, λ i The matrix Q u The i-th eigenvalue of u is the coefficient of the convex hull of the ellipsoid in the original space;
[0016] Among them, Q u According to the following optimization problem:
[0017]
[0018]
[0019] In the formula, u hk is the historical data of renewable energy output, the superscript k is the scene number; c is the coordinate of the center of the ellipse in the original space;
[0020] The spatial transformation from the original space to the regular space is:
[0021]
[0022] In the formula, the transformation matrix J is based on the formula Q u =J T Q v J got;
[0023] Historical data v converted to regular space hk = J(u hk -c).
[0024] Further, in S3, the rhombus inscribed in the ellipsoid is expressed as:
[0025]
[0026] In the formula, v i is the i-th dimension coordinate of v, r i is the semi-axis length of the convex hull of the ellipsoid in the direction of dimension i,
[0027] The diamond convex hull uncertainty set is expressed as:
[0028]
[0029] In the formula, R i is the semi-axis length of the rhombus convex hull in the direction of dimension i, R i =αr i , α is the scaling factor, and is the i-th dimension coordinate of the historical data of renewable energy output for the k-th scenario in the regularized space.
[0030] Furthermore, in S4, the mathematical expression of the section is:
[0031] v j =v j,cut
[0032] In the formula, v j，cut is the historical data v hk Project the maximum value on the j-th axis of the rhombus convex hull, and the positive direction is Negative
[0033] Furthermore, in S4, the mathematical expression of the intersection of each of the cut surfaces and the rhombus convex hull is:
[0034]
[0035] The endpoint v of the intersection ek The coordinates are:
[0036]
[0037] In the formula, the intermediate quantity c j =1-|v j,cut | / R j .
[0038] Furthermore, in S5, the coordinates of the vertices of the rhombus cut-corner convex hull are u ek =J -1 v ek +c;
[0039] The uncertainty set of the rhombus cut corner convex hull for:
[0040]
[0041] Where N is the total number of vertices of the rhombus convex hull, ω k is an auxiliary variable.
[0042] On the other hand, in order to achieve the above-mentioned object, the present invention also provides a device for constructing a rhombus cut corner convex hull uncertainty set, comprising:
[0043] A historical data acquisition module is used to acquire historical data on renewable energy output;
[0044] The ellipsoid convex hull building module is used to build the ellipsoid convex hull uncertainty set containing all historical data, and transform the historical data and the ellipsoid convex hull into the canonical space;
[0045] The diamond convex hull building module is used to enlarge the diamond inscribed in the ellipsoid to just contain all historical data, and obtain the diamond convex hull uncertainty set;
[0046] The module for establishing the rhombus convex hull is used to project all historical data onto the axes of the rhombus convex hull, and for each axis, make a section perpendicular to each axis through the data points corresponding to the maximum projection on the positive axis and the negative axis, so as to obtain the endpoint coordinates of the intersection of each section and the rhombus convex hull; and restore the endpoint coordinates to the original space to obtain the vertex coordinates of the rhombus convex hull, and construct the rhombus convex hull uncertainty set based on the vertex coordinates.
[0047] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects:
[0048] (1) Based on the rhombus convex hull, the present invention projects all historical data onto each axis of the rhombus convex hull, and for each axis, makes a section perpendicular to the axis through the data points corresponding to the maximum projection on the positive axis and the negative axis, respectively, to obtain the endpoint coordinates of the intersection of each section and the rhombus convex hull, and further obtains the vertex coordinates of the rhombus cut-corner convex hull, and thereby constructs the uncertainty set of the rhombus cut-corner convex hull. In this way, the present invention can make full use of the implicit information in the historical data, eliminate the uncertainty scenarios that are actually impossible to occur, thereby reducing the uncertainty range and conservatism without sacrificing robustness, and the number of vertices of the rhombus cut-corner convex hull The complexity is o(n 2 ), avoiding the exponential growth problem of the number of convex hull vertices in high-dimensional space.
[0049] (2) The present invention cuts the rhombus convex hull in order from large to small according to the volume of the rhombus convex hull that can be cut off by each cut surface, and obtains the endpoint coordinates of the intersection of each cut surface and the rhombus convex hull in turn. When the cut volumes of each cut surface overlap, a relatively smaller rhombus cut corner convex hull can be obtained. BRIEF DESCRIPTION OF THE DRAWINGS
[0050] Figure 1 A flowchart of a method for constructing an uncertainty set of a rhombus cut corner convex hull provided in an embodiment of the present invention;
[0051] Figure 2(a) to Figure 2(c) They are respectively a schematic diagram of an uncertainty set of an ellipsoid convex hull containing all historical data, a schematic diagram of scaling the historical data and the ellipsoid convex hull and the rhombus inscribed in the ellipsoid converted to a regular space, and a schematic diagram of angle cutting provided in an embodiment of the present invention;
[0052] Figure 3 A three-dimensional schematic diagram of the uncertainty set of the rhombus cut corner convex hull provided in an embodiment of the present invention;
[0053] Figure 4 A block diagram of a device for constructing a rhombus cut corner convex hull uncertainty set provided in an embodiment of the present invention. DETAILED DESCRIPTION
[0054] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.
[0055] In the present invention, the terms "first", "second", etc. (if any) in the present invention and the accompanying drawings are used to distinguish similar objects but not necessarily to describe a specific order or sequence.
[0056] In the present invention, the rhombus inscribed in the ellipsoid in the present invention and the accompanying drawings refers to the polyhedron surrounded by the vertices of the ellipsoid; in a two-dimensional plane, if the four vertices of a rhombus are all on an ellipse, then the rhombus is called the inscribed rhombus of the ellipse, and the intersection of the diagonals of the rhombus is the center of the ellipse.
[0057] Figure 1 A flowchart of a method for constructing an uncertainty set of a rhombus cut corner convex hull provided in an embodiment of the present invention, wherein the control method comprises operations S1 to S5.
[0058] Operation S1: Obtain historical data of renewable energy output. Specifically:
[0059] The collected renewable energy output historical data is divided into daily data, and the output data of each day is recorded as a historical scene, and the collected data is expressed in vector form. hk is the historical data of renewable energy output, and the superscript k is the scenario number.
[0060] Operation S2, establishes an ellipsoid convex hull uncertainty set containing all historical data, and transforms the historical data and the ellipsoid convex hull into a canonical space. Specifically:
[0061] First, establish the uncertainty set of the ellipsoid convex hull containing all historical data as shown in Figure 2(a), and then transform the historical data and the ellipsoid convex hull into the regular space as shown in Figure 2(b).
[0062] The uncertainty set of the convex hull of the ellipsoid transformed into the regular space is expressed as:
[0063] v T Q v v≤1
[0064] Where v is the coordinate in the regularized space, Q v =diag([λ1,…,λ i ,…,λ n ]), n is the dimension of historical data, λ i The matrix Q u The i-th eigenvalue of u is the coefficient of the convex hull of the ellipsoid in the original space;
[0065] Among them, Q u According to the following optimization problem:
[0066]
[0067]
[0068] In the formula, u hk is the historical data of renewable energy output, the superscript k is the scene number; c is the coordinate of the center of the ellipse in the original space;
[0069] The spatial transformation from the original space to the regular space is:
[0070]
[0071] In the formula, the transformation matrix J is based on the formula Q u =J T Q v J got;
[0072] Historical data v converted to regular space hk = J(u hk -c).
[0073] Operation S3, enlarge the rhombus inscribed in the ellipsoid to just contain all historical data, and obtain the rhombus convex hull uncertainty set. Specifically:
[0074] Since the polyhedron enclosed by the vertices of the ellipsoid is not sufficient to cover all historical data, the scaling factor α is introduced to scale the convex hull of the polyhedron. As shown in Figure 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.
[0075] The rhombus inscribed in the ellipsoid is represented by:
[0076]
[0077] In the formula, v i is the i-th dimension coordinate of v, r i is the semi-axis length of the convex hull of the ellipsoid in the direction of dimension i,
[0078] The diamond convex hull uncertainty set is expressed as:
[0079]
[0080] In the formula, R i is the semi-axis length of the rhombus convex hull in the direction of dimension i, R i =αr i , α is the scaling factor, and is the i-th dimension coordinate of the historical data of renewable energy output for the k-th scenario in the regularized space.
[0081] Operation S4, projecting all historical data onto the axes of the diamond convex hull, and for each axis, making a section perpendicular to each axis through the data points corresponding to the maximum projection on the positive axis and the negative axis, respectively, to obtain the endpoint coordinates of the intersection of each section and the diamond convex hull. Specifically:
[0082] The mathematical expression of the section is:
[0083] v j =v j,cut
[0084] In the formula, v j,cut is the historical data v hk Project the maximum value on the j-th axis of the rhombus convex hull, and the positive direction is Negative
[0085] The mathematical expression for the intersection of the tangent plane and the rhombus convex hull is:
[0086]
[0087] The endpoint v of the intersection ek The coordinates are:
[0088]
[0089] In the formula, the intermediate quantity c j =1-|v j,cut | / R j .
[0090] As shown in FIG2( c ), it is a schematic diagram of the cutting angle when j=1.
[0091] It should be noted that the axial direction of the rhombus convex hull refers to the axial direction where its symmetry axis is located.
[0092] Furthermore, according to the volume of the rhombus convex hull that can be cut off by each cut surface, the rhombus convex hull is cut in order from large to small, and the endpoint coordinates of the intersection of each cut surface and the rhombus convex hull are obtained in sequence.
[0093] Operation S5, restore the endpoint coordinates to the original space, obtain the vertex coordinates of the rhombus cut corner convex hull, and construct the rhombus cut corner convex hull uncertainty set based on the vertex coordinates. Specifically:
[0094] The coordinates of the vertices of the rhombus cut corner convex hull are u ek =J -1 v ek +c;
[0095] Uncertainty set of diamond cut corner convex hull for:
[0096]
[0097] Where N is the total number of vertices of the rhombus convex hull, ω k is an auxiliary variable.
[0098] Figure 3 The figure shows a three-dimensional schematic diagram of the uncertainty set of the rhombus cut corner convex hull provided by this embodiment, wherein the tight convex hull is the convex hull with the smallest volume containing all historical data. The rhombus cut corner convex hull obtained in this embodiment contains the tight convex hull, thus indicating that when the method provided by the present invention is used in power system scheduling, it can ensure that the obtained scheduling plan can be robust to any historical scenario, thereby ensuring the safety of power system operation.
[0099] Figure 4 A block diagram of a device for constructing a rhombus cut corner convex hull uncertainty set provided by an embodiment of the present invention. Figure 4 The device 400 includes a historical data acquisition module 410, an ellipsoid convex hull establishment module 420, a rhombus convex hull establishment module 430, and a rhombus cut corner convex hull establishment module 440.
[0100] The historical data acquisition module 410, for example, performs operation S1 to acquire historical output data of renewable energy;
[0101] The ellipsoid convex hull establishing module 420 performs operation S2, for example, to establish an ellipsoid convex hull uncertainty set containing all historical data, and convert the historical data and the ellipsoid convex hull into a canonical space;
[0102] The rhombus convex hull establishing module 430 performs operation S3, for example, to enlarge the rhombus inscribed in the ellipsoid to just contain all the historical data, and obtain the rhombus convex hull uncertainty set;
[0103] The rhombus cut corner convex hull establishment module 440, for example, performs operations S4 and S5, which are used to project all historical data onto each axis of the rhombus convex hull, and for each axis, make a section perpendicular to each axis through the data points corresponding to the maximum projection on the positive axis and the negative axis, respectively, to obtain the endpoint coordinates of the intersection of each section and the rhombus convex hull; and restore the endpoint coordinates to the original space to obtain the vertex coordinates of the rhombus cut corner convex hull, and construct the rhombus cut corner convex hull uncertainty set based on the vertex coordinates.
[0104] The device 400 is used to perform the above Figure 1 The method for constructing the uncertainty set of the diamond cut corner convex hull in the embodiment shown. For details not covered in this embodiment, please refer to the aforementioned Figure 1 The method for constructing the uncertainty set of the rhombus cut corner convex hull in the illustrated embodiment will not be described in detail here.
[0105] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.
Claims
1. A method for constructing a rhombus cut corner convex hull uncertainty set, characterized in that: The following steps are involved: S1, obtain historical data of renewable energy output; S2, establish the uncertainty set of the ellipsoid convex hull containing all historical data, and transform the historical data and the ellipsoid convex hull into the regular space; S3, enlarge the rhombus inscribed in the ellipsoid to just contain all historical data, and obtain the rhombus convex hull uncertainty set; S4, projecting all historical data onto the axes of the rhombus convex hull, and for each axis, making a section perpendicular to each axis through the data points corresponding to the maximum projection on the positive axis and the negative axis, respectively, to obtain the endpoint coordinates of the intersection of each section and the rhombus convex hull; The mathematical expression of the section is: v j =v j,cut In the formula, v j,cut is the historical data v hk Project the maximum value on the j-th axis of the rhombus convex hull, and the positive direction is Negative The mathematical expression of the intersection of each of the tangent planes and the rhombus convex hull is: The endpoint v of the intersection ek The coordinates are: In the formula, the intermediate quantity c j =1-|v j,cut | / R j ; and restore the endpoint coordinates to the original space to obtain the vertex coordinates of the rhombus convex hull, and construct the rhombus convex hull uncertainty set based on the vertex coordinates; the vertex coordinates of the rhombus convex hull are u ek =J -1 v ek +c; S5, restoring the endpoint coordinates to the original space to obtain the vertex coordinates of the rhombus cut corner convex hull, and constructing an uncertainty set of the rhombus cut corner convex hull based on the vertex coordinates; The coordinates of the vertices of the rhombus cut corner convex hull are u ek =J -1 v ek +c; The uncertainty set of the rhombus cut corner convex hull for: Where N is the total number of vertices of the rhombus convex hull, ω k is an auxiliary variable.
2. The method for constructing the uncertainty set of the rhombus cut corner convex hull according to claim 1, characterized in that: In S4, the endpoint coordinates of the intersection of each of the cut faces and the rhombus convex hull are obtained, specifically: according to the size of the volume of the rhombus convex hull that can be cut off by each of the cut faces, the rhombus convex hull is cut in order from large to small, and the endpoint coordinates of the intersection of each of the cut faces and the rhombus convex hull are obtained in sequence.
3. The method for constructing the uncertainty set of the rhombus cut corner convex hull according to claim 1 or 2, characterized in that: In S2, the uncertainty set of the convex hull of the ellipsoid converted to the regular space is expressed as: v T Q v v≤1 Where v is the coordinate in the regularized space, Q v =diag([λ1,…,λ i ,…,λ n ]), n is the dimension of historical data, λ i The matrix Q u The i-th eigenvalue of u is the coefficient of the convex hull of the ellipsoid in the original space; Among them, Q u According to the following optimization problem: In the formula, u hk is the historical data of renewable energy output, the superscript k is the scene number; c is the coordinate of the center of the ellipse in the original space; The spatial transformation from the original space to the regular space is: In the formula, the transformation matrix J is based on the formula Q u =J T Q v J got; Historical data v converted to regular space hk = J(u hk -c).
4. The method for constructing the uncertainty set of the rhombus cut corner convex hull according to claim 3, characterized in that: In S3, the rhombus inscribed in the ellipsoid is expressed as: In the formula, v i is the i-th dimension coordinate of v, r i is the semi-axis length of the convex hull of the ellipsoid in the direction of dimension i, The diamond convex hull uncertainty set is expressed as: In the formula, R i is the semi-axis length of the rhombus convex hull in the direction of dimension i, R i =αr i , α is the scaling factor, and is the i-th dimension coordinate of the historical data of renewable energy output for the k-th scenario in the regularized space.
5. A device for constructing a rhombus cut corner convex hull uncertainty set, characterized in that: include: A historical data acquisition module is used to acquire historical data on renewable energy output; The ellipsoid convex hull building module is used to build the ellipsoid convex hull uncertainty set containing all historical data, and transform the historical data and the ellipsoid convex hull into the canonical space; The diamond convex hull building module is used to enlarge the diamond inscribed in the ellipsoid to just contain all historical data, and obtain the diamond convex hull uncertainty set; A module for establishing a rhombus cut-angle convex hull is used to project all historical data onto the axes of the rhombus convex hull, and for each axis, a section perpendicular to each axis is made through the data points corresponding to the maximum projection on the positive axis and the negative axis, respectively, to obtain the endpoint coordinates of the intersection of each section and the rhombus convex hull; The mathematical expression of the section is: v j =v j,cut In the formula, v j,cut is the historical data v hk Project the maximum value on the j-th axis of the rhombus convex hull, and the positive direction is Negative The mathematical expression of the intersection of each of the tangent planes and the rhombus convex hull is: The endpoint v of the intersection ek The coordinates are: In the formula, the intermediate quantity c j =1-|v j,cut | / R j ; and restoring the endpoint coordinates to the original space to obtain the vertex coordinates of the rhombus cut corner convex hull, and constructing the rhombus cut corner convex hull uncertainty set based on the vertex coordinates; The coordinates of the vertices of the rhombus cut corner convex hull are u ek =J -1 v ek +c; The uncertainty set of the rhombus cut corner convex hull for: Where N is the total number of vertices of the rhombus convex hull, ω k is an auxiliary variable.