Modeling method of Saint-Venant equations with gates based on model reduction technology

Abstract

The invention discloses a modeling method of a Saint-Venant equation group containing a gate based on a model reduction technology, which belongs to the technical field of water conservancy engineering, and includes: S1, constructing an original Saint-Venant equation group containing a gate; S2, discretizing the constructed original Saint-Venant equation group to obtain a nonlinear function in a high-dimensional system; S3, based on the simulation purpose, using the original Saint-Venant equation group to simulate the system, collecting state variables to construct a snapshot matrix; S4, based on the nonlinear function in the high-dimensional system, using the model reduction technology to simplify the snapshot matrix, and then constructing a low-dimensional reduced-order model corresponding to the original Saint-Venant equation group, and completing the construction of the Saint-Venant equation group containing a gate. This method greatly improves the calculation efficiency while ensuring the simulation accuracy by reducing the order of the model, and meets the needs of real-time simulation and control; it has a wide range of application prospects and can be used in the fields of river channel scheduling, water conservancy engineering optimization, and flood prevention and early warning.

Description

Technical Field

[0001] The invention belongs to the technical field of water conservancy engineering, and in particular relates to a modeling method of a Saint-Venant equation group including a gate based on a model order reduction technology. Background Art

[0002] The Saint-Venant equations are a set of partial differential equations used to describe the motion of one-dimensional non-steady open channel water flow. They are widely used in water flow simulation and prediction in water conservancy projects such as rivers, reservoirs and irrigation canals. In practical applications, multiple gates are often arranged on the river to control the water level and flow. The presence of gates has a significant impact on the dynamic characteristics of the water flow, which increases the complexity of the mathematical model. However, due to the introduction of nonlinear and complex boundary conditions, the numerical solution process of the Saint-Venant equations containing gates has a large amount of calculation and low simulation efficiency, which makes it difficult to meet the needs of real-time simulation and control. Therefore, the existing numerical simulation methods of the Saint-Venant equations containing gates have the problems of low computational efficiency and difficulty in real-time application. Summary of the invention

[0003] In view of the above-mentioned deficiencies in the prior art, the modeling method of the Saint-Venant equations group including gates based on the model reduction technology provided by the present invention solves the problem that in the traditional method, when the Saint-Venant equations group including gates are used for simulation, the numerical solution process has a large amount of calculation, the simulation efficiency is low, and it is difficult to meet the requirements of real-time applications.

[0004] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is: a modeling method of the Saint-Venant equations group including gates based on model reduction technology, which is used to realize water flow simulation and prediction simulation of water conservancy projects, including the following steps:

[0005] S1. Construct the original Saint-Venant equations including the gate;

[0006] S2, discretize the constructed original Saint-Venant equations to obtain nonlinear functions in the high-dimensional system;

[0007] S3. Based on the simulation purpose, the original Saint-Venant equations are used to simulate the system, collect state variables and construct a snapshot matrix;

[0008] S4. Based on the nonlinear functions in the high-dimensional system, the snapshot matrix is ​​simplified using model reduction technology, and then a low-dimensional reduced-order model corresponding to the original Saint-Venant equations is constructed to complete the construction of the Saint-Venant equations including the gate.

[0009] Furthermore, in step S1, the original Saint-Venant equations including the gate include the continuity equation and the momentum equation, and according to the gate position, the flow relationship of the gate is used as the boundary condition of the original Saint-Venant equations.

[0010] Furthermore, in the original Saint-Venant equations, the continuity equation is:

[0011]

[0012] The momentum equation is:

[0013]

[0014] The boundary conditions are:

[0015]

[0016] In the formula, A represents the cross-sectional area of ​​water flow, Q Indicates flow rate, t Indicates time, represents the length of the river, h Indicates water depth. g represents the acceleration due to gravity, Indicates the bottom slope, represents the resistance slope, represents the flow rate through the gate, represents the flow coefficient, Indicates the gate opening, Indicates the gate width, and Respectively represent the water level before and after the gate.

[0017] Furthermore, in step S2, the discretization processing of the Saint-Venant equations includes spatial discretization and temporal discretization;

[0018] In the control volume i At , the continuity equation is discretized as:

[0019]

[0020] The momentum equation is discretized as:

[0021]

[0022] In the formula, Indicates i Units at time step n The depth of water, Indicates i The flow flux at the right boundary of the unit, Indicates i The flow flux at the left boundary of the unit, Indicates i Units at time step n of traffic, Indicates iThe momentum flux at the right boundary of the unit, Indicates i The momentum flux at the left boundary of the unit, Indicates i Units at time step n The friction slope, represents the time step of time discretization, Represents the step size of discretization of river channel length.

[0023] Further, i The momentum flux at the right boundary of the unit The calculation formula is:

[0024]

[0025] In the formula, Indicates Units at time step of traffic, Indicates Units at time step of traffic, Indicates Units at time step The depth of water, Indicates Units at time step The depth of water, represents the step size of discretization of river channel length, Represents the acceleration due to gravity.

[0026] Further, i Units at time step n Friction slope The calculation formula is:

[0027]

[0028] In the formula, represents the Manning roughness coefficient, Indicates i Units at time step n of traffic, represents the cross-sectional area of ​​water flow, Represents the hydraulic parameter wetted perimeter.

[0029] Furthermore, in step S2, when the Saint-Venant equations are discretized, when the gate position is When the gate boundary conditions are met, the momentum flux Replaced with the corresponding gate flow rate, which is expressed as:

[0030]

[0031] In the formula, Indicates j The gate flow rate of each gate is represents the flow coefficient, Indicates j The gates are in the time step n The opening degree, Indicates j The width of the gate, Indicates i Unit j The water depth of the gate at time step n, Indicates the control volume of the gate, the subscript j Indicates the number of gates, .

[0032] Furthermore, in step S2, the system state vector after discretization is It is expressed as:

[0033]

[0034] In the formula, M Quantity represents the water depth of each control volume, and Quantity represents the flow rate of each control volume;

[0035] Dynamical equations of high-dimensional systems:

[0036]

[0037] In the formula, Represents a high-dimensional system dimensional nonlinear function, which can be expressed as:

[0038]

[0039] In the formula, the symbol Indicates that it is located at M Units and M -1 momentum flux between cells, Respectively represent the first unit... M The water depth of the unit, Respectively represent the first unit to the M The friction slope of each unit, represents the acceleration due to gravity, Indicates the bottom slope;

[0040] Among them, MThe component is the water depth change rate of the high-dimensional system, which is expressed as:

[0041]

[0042] back M The component is the flow rate change rate of the high-dimensional system, which is expressed as:

[0043]

[0044] The momentum flux is expressed as:

[0045]

[0046] In the formula, Indicates gate overflow.

[0047] Furthermore, the step S3 includes the following sub-steps:

[0048] S31. Based on the simulation purpose, at the selected sampling time or working condition, the original Saint-Venant equation is used to simulate the system and obtain the high-dimensional state vector of the system ;in, represents the sampling time;

[0049] S32, the obtained high-dimensional state vector is used as the column vector of the snapshot matrix S in turn, and then the snapshot matrix S is constructed. : .

[0050] Furthermore, the step S4 includes the following sub-steps:

[0051] S41. Snapshot Matrix Perform singular value decomposition to obtain the left singular vector matrix , singular value matrix Σ and right singular vector matrix V, take the largest k Singular values ​​to construct the reduced-order basis matrix ;

[0052]

[0053] In the formula, represents the reduced-order basis matrix , represents a set of real matrices consisting of matrices of dimension (2M,k), Indicates the number of cross-sectional elements;

[0054] S42, according to the reduced-order basis matrix , construct the reduced-order state variables ;

[0055]

[0056] In the formula, represents the system state vector after discretization of the original Saint-Venant equations, represents the transpose operator, Indicated by k The vector set consisting of dimensional real vectors;

[0057] S43. Based on the constructed reduced-order state variables, the high-dimensional system obtained by discretizing the original Saint-Venant equations is projected onto the reduced-order basis matrix to construct a low-dimensional reduced-order model;

[0058]

[0059] In the formula, represents the state variable value of the low-dimensional reduced-order model, represents the nonlinear function in the high-dimensional system obtained by discretizing the original Saint-Venant equations, Represents the transpose operator.

[0060] The beneficial effects of the present invention are:

[0061] (1) In the method of the present invention, by reasonably selecting the reduced-order basis when constructing the reduced-order model, the simulation results maintain good consistency with the high-precision model, and the error is within an acceptable range.

[0062] (3) By reducing the order of the model, the degree of freedom of the reduced-order model is much smaller than that of the original model. While ensuring the simulation accuracy, the computational efficiency is greatly improved, meeting the needs of real-time simulation and control.

[0063] (3) The method of the present invention has broad application prospects and can be used in the fields of river channel scheduling, water conservancy project optimization and flood prevention and early warning. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 A flow chart of the modeling method of the Saint-Venant equations group including gates based on the model reduction technology provided by the present invention. DETAILED DESCRIPTION

[0065] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0066] like Figure 1As shown, an embodiment of the present invention provides a modeling method of Saint-Venant equations including gates based on model reduction technology, which is used to realize water flow simulation and prediction simulation of water conservancy projects, including the following steps:

[0067] S1. Construct the original Saint-Venant equations including the gate;

[0068] S2, discretize the constructed original Saint-Venant equations to obtain nonlinear functions in the high-dimensional system;

[0069] S3. Based on the simulation purpose, the original Saint-Venant equations are used to simulate the system, collect state variables and construct a snapshot matrix;

[0070] S4. Based on the nonlinear functions in the high-dimensional system, the snapshot matrix is ​​simplified using model reduction technology, and then a low-dimensional reduced-order model corresponding to the original Saint-Venant equations is constructed to complete the construction of the Saint-Venant equations including the gate.

[0071] In step S1 of the embodiment of the present invention, considering that N gates are arranged on the river channel, the traditional Saint-Venant equations need to be expanded; therefore, in this embodiment, the original Saint-Venant equations containing the gates include the continuity equation and the momentum equation, and according to the gate position, the flow relationship of the gate is used as the boundary condition of the original Saint-Venant equations;

[0072] The continuity equation is:

[0073]

[0074] The momentum equation is:

[0075]

[0076] The boundary conditions are:

[0077]

[0078] In the formula, A represents the cross-sectional area of ​​water flow, Q Indicates flow rate, t Indicates time, represents the length of the river, h Indicates water depth. g represents the acceleration due to gravity, Indicates the bottom slope, represents the resistance slope, represents the flow rate through the gate, represents the flow coefficient, Indicates the gate opening, Indicates the gate width, and Respectively represent the water level before and after the gate.

[0079] In step S2 of the embodiment of the present invention, the discretization processing of the Saint-Venant equations includes spatial discretization and temporal discretization;

[0080] Among them, spatial discretization means that the river length L Divide into M equidistant space units, each unit length is:

[0081]

[0082] The spatial node positions are:

[0083] Each control volume i lie in The range is ,in:

[0084]

[0085] Time discretization refers to setting the time step size , the time nodes are: .

[0086] Based on this, in this embodiment, in the control volume i At , the continuity equation is discretized as:

[0087]

[0088] The momentum equation is discretized as:

[0089]

[0090] In the formula, Indicates i Units at time step n The depth of water, Indicates i The flow flux at the right boundary of the unit, Indicates i The flow flux at the left boundary of the unit, Indicates i Units at time step n of traffic, Indicates i The momentum flux at the right boundary of the unit, Indicates i The momentum flux at the left boundary of the unit, Indicates i Units at time step n The friction slope, represents the time step of time discretization, Represents the step size of discretization of river channel length.

[0091] The central difference method is used to calculate the i The momentum flux at the right boundary of the unit for:

[0092]

[0093] In the formula, Indicates Units at time step of traffic, Indicates Units at time step of traffic, Indicates Units at time step The depth of water, Indicates Units at time step The depth of water, represents the step size of discretization of river channel length, Represents the acceleration due to gravity.

[0094] Among them, the Manning formula is used to calculate the i Units at time step n Friction slope The is:

[0095]

[0096] In the formula, represents the Manning roughness coefficient, Indicates i Units at time step n of traffic, represents the cross-sectional area of ​​water flow, It represents the hydraulic parameter wetted perimeter, which is obtained by measuring the specific cross section.

[0097] In step S2 of the embodiment of the present invention, when the Saint-Venant equations are discretized, when the gate position is When the gate boundary conditions are met, the momentum flux Replaced with the corresponding gate flow rate, which is expressed as:

[0098]

[0099] In the formula, Indicates j The gate flow rate of each gate is represents the flow coefficient, Indicates jThe gates are in the time step n The opening degree, Indicates j The width of the gate, Indicates i Unit j The water depth of the gate at time step n, Indicates the control volume of the gate, the subscript j Indicates the number of gates, .

[0100] In step S2 of the embodiment of the present invention, the system state vector after discretization of the original Saint-Venant equations is It is expressed as:

[0101]

[0102] In the formula, M Quantity represents the water depth of each control volume, and Quantity represents the flow rate of each control volume;

[0103] Dynamical equations of high-dimensional systems:

[0104]

[0105] In the formula, Represents a high-dimensional system dimensional nonlinear function, which can be expressed as:

[0106]

[0107] In the formula, the symbol Indicates that it is located at M Units and M -1 momentum flux between cells, Respectively represent the first unit... M The water depth of the unit, Respectively represent the first unit to the M The friction slope of each unit, represents the acceleration due to gravity, Indicates the bottom slope;

[0108] Among them, M The component is the water depth change rate of the high-dimensional system, which is expressed as:

[0109]

[0110] back M The component is the flow rate change rate of the high-dimensional system, which is expressed as:

[0111]

[0112] The momentum flux is expressed as:

[0113]

[0114] In the formula, Indicates gate overflow.

[0115] Step S3 of the embodiment of the present invention includes the following sub-steps:

[0116] S31. Based on the simulation purpose, at the selected sampling time or working condition, the original Saint-Venant equation is used to simulate the system and obtain the high-dimensional state vector of the system ;in, represents the sampling time;

[0117] S32, the obtained high-dimensional state vector is used as the column vector of the snapshot matrix S in turn, and then the snapshot matrix S is constructed. :

[0118] In step S4 of the embodiment of the present invention, in order to improve the simulation efficiency, the model reduction technology is used in this embodiment to simplify the expanded Saint-Venant equations. Based on this, step S4 of this embodiment includes the following sub-steps:

[0119] S41. Snapshot Matrix Perform singular value decomposition to obtain the left singular vector matrix , singular value matrix Σ and right singular vector matrix V, take the largest k Singular values ​​to construct the reduced-order basis matrix ;

[0120]

[0121] In the formula, represents the reduced-order basis matrix , represents a set of real matrices consisting of matrices of dimension (2M,k), Indicates the number of cross-sectional elements;

[0122] S42, according to the reduced-order basis matrix , construct the reduced-order state variables ;

[0123]

[0124] In the formula, represents the system state vector after discretization of the original Saint-Venant equations, represents the transpose operator, Indicated by k The vector set consisting of dimensional real vectors;

[0125] S43. Based on the constructed reduced-order state variables, the high-dimensional system obtained by discretizing the original Saint-Venant equations is projected onto the reduced-order basis matrix to construct a low-dimensional reduced-order model;

[0126]

[0127] In the formula, represents the state variable value of the low-dimensional reduced-order model, represents the nonlinear function in the high-dimensional system obtained by discretizing the original Saint-Venant equations, Represents the transpose operator.

[0128] In this embodiment, the state variables in the reduced-order system are a It is the core variable of the low-dimensional reduced-order model, representing the projection of the original high-dimensional system state on the reduced-order basis. is the state vector of the low-dimensional system, representing the high-dimensional state In the reduced base The projection on Each component ( ) represents a high-dimensional state In the In order to calculate the reduced-order state variables, , we need to convert the high-dimensional state Projection to a reduced-order basis Then, the above-mentioned low-dimensional reduced-order model was established.

[0129] The present invention uses specific embodiments to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as a limitation on the present invention.

[0130] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A modeling method for the Saint-Venant equations including gates based on model reduction technology, characterized in that: The method is used to realize water flow simulation and prediction simulation of water conservancy projects, including the following steps: S1. Construct the original Saint-Venant equations including the gate; S2, discretize the constructed original Saint-Venant equations to obtain nonlinear functions in the high-dimensional system; S3. Based on the simulation purpose, the original Saint-Venant equations are used to simulate the system, collect state vectors and construct snapshot matrices; S4. Based on the nonlinear functions in the high-dimensional system, the snapshot matrix is ​​simplified using the model reduction technology, and then the low-dimensional reduced-order model corresponding to the original Saint-Venant equations is constructed to complete the construction of the Saint-Venant equations including the gate; In said S1, the original Saint-Venant equations including the gate include the continuity equation and the momentum equation, and according to the gate position, the flow relationship of the gate is used as the boundary condition of the original Saint-Venant equations; In the original Saint-Venant equations, the continuity equation is: The momentum equation is: The boundary conditions are: In the formula, A represents the cross-sectional area of ​​the water flow, Q represents the flow rate, t represents the time, x represents the length of the river channel, h represents the water depth, g represents the gravitational acceleration, S0 represents the bottom slope, and S f represents the resistance slope, Q g represents the flow through the gate, C d represents the flow coefficient, α represents the gate opening, b represents the gate width, h u and h d Respectively represent the water level before and after the gate; In S2, the discretization processing of the Saint-Venant equations includes spatial discretization and temporal discretization; In which, at the control volume i, the continuity equation is discretized as: The momentum equation is discretized as: In the formula, represents the water depth of the ith unit at time step n, represents the flow flux on the right boundary of the ith unit, represents the flow flux at the left boundary of the ith unit, represents the flow of the ith unit at time step n, represents the momentum flux at the right boundary of the ith unit, represents the momentum flux at the left boundary of the ith unit, represents the friction slope of the ith unit at time step n, Δt represents the time step of time discretization, and Δx represents the step size of the discretization of the river channel length; Momentum flux at the right boundary of the i-th unit The calculation formula is: In the formula, represents the flow of the i+1th unit at time step n, represents the flow of the ith unit at time step n, represents the water depth of the i+1th unit at time step n, represents the water depth of the ith unit at time step n, Δx represents the step size of the discretization of the river channel length, and g represents the gravitational acceleration; The friction slope of the i-th unit at time step n The calculation formula is: Where n m represents the Manning roughness coefficient, represents the flow of the ith unit at time step n, A i Represents the cross-sectional area of ​​water flow, R i It represents the hydraulic parameter wetted perimeter; In S2, when the Saint-Venant equations are discretized, when the gate position is i j +1 / 2, according to the boundary conditions of the gate, the momentum flux Replaced with the corresponding gate flow rate, which is expressed as: In the formula, represents the gate flow rate of the jth gate, C d represents the flow coefficient, represents the opening of the jth gate at time step n, b j represents the width of the jth gate, represents the water depth of the jth gate in the i-th unit at time step n, i j represents the control volume of the gate, the subscript j represents the number of gates, j = 1, 2, ..., N; The system state vector x(t) after the discretization of S2 is expressed as: In the formula, the first M components h i (t) represents the water depth of each control volume, and the last M components Q i (t) represents the flow rate of each control volume; Dynamical equations of high-dimensional systems: In the formula, F(x) represents the 2M-dimensional nonlinear function in the high-dimensional system, which is expressed as: In the formula, the symbol Q M+0.5 represents the momentum flux between the Mth unit and the M-1th unit, h1,…,h M Respectively represent the water depth of the 1st unit...Mth unit, S f,1 ,…,S f,M They represent the friction slope from the 1st unit to the Mth unit, g represents the acceleration of gravity, and S0 represents the bottom slope; Among them, the first M components are the water depth change rate of the high-dimensional system, which is expressed as: The last M components are the flow rate change rate of the high-dimensional system, which is expressed as: The momentum flux is expressed as: In the formula, Q g,j Indicates gate overflow; The S3 comprises the following sub-steps: S31. Based on the simulation purpose, at the selected sampling time or working condition, the original Saint-Venant equation is used to simulate the system and obtain the high-dimensional state vector x(t i ), where t i represents the sampling time; S32. The obtained high-dimensional state vectors are used as column vectors of the snapshot matrix S in turn, and then the snapshot matrix S is constructed: S=[x(t1)x(t2)…x(t L )]; The S4 comprises the following sub-steps: S41. Perform singular value decomposition on the snapshot matrix S to obtain the left singular vector matrix U, the singular value matrix Σ and the right singular vector matrix V. Take the largest k singular values ​​to construct the reduced-order basis matrix U. k ; Where u1,u2,…,u k Denotes the reduced-order basis matrix U k , represents a real number matrix set consisting of matrices of dimension (2M, k), where M represents the number of cross-sectional elements; S42, according to the reduced order basis matrix U k , construct the reduced-order state variable a(t); In the formula, x(t) represents the system state vector after discretization of the original Saint-Venant equations, T represents the transposition operator, represents a vector set consisting of k-dimensional real vectors; S43. Based on the constructed reduced-order state variables, the high-dimensional system obtained by discretizing the original Saint-Venant equations is projected onto the reduced-order basis matrix to construct a low-dimensional reduced-order model; Where a represents the state variable value of the low-dimensional reduced-order model, F(.) represents the nonlinear function in the high-dimensional system obtained by discretizing the original Saint-Venant equations, and T represents the transposition operator.

Images