Water supply network pressure estimation method based on second-order graph signal reconstruction

Abstract

The invention discloses a water supply pipe network pressure estimation method based on second-order graph signal reconstruction, and relates to the technical field of water supply pipe network reconstruction, and the method comprises the steps: obtaining water supply pipe network information, constructing a first pipe network graph Laplacian matrix, and carrying out the spectrum analysis to obtain a spectrum base; the method comprises the following steps: presetting a high-frequency term of a graph Laplacian frequency spectrum on a frequency spectrum base, limiting to obtain a low-frequency term of the graph Laplacian frequency spectrum, reconstructing a first pipe network water head on the basis of inverse graph Fourier transform on the low-frequency term of the graph Laplacian frequency spectrum, and obtaining a pseudo monitoring value; and constructing a second pipe network graph Laplacian matrix according to the pseudo monitoring value and the actual monitoring value, and repeating the reconstruction process to obtain a second pipe network water head. The innovation lies in that through a double constraint framework, the low-frequency component and the measured data covariance matrix are fused while the pipe network topological characteristics are reserved, and the problems that a traditional method is sensitive to parameters, the hydraulic relation is ignored and the like are effectively solved.

Description

Technical Field

[0001] The present invention relates to the technical field of water supply network reconstruction, and more specifically, to a method for estimating the pressure of a water supply network based on second-order graph signal reconstruction. Background Art

[0002] The development of informatization and intelligence of water supply networks is inseparable from the support of data information. However, limited by construction and maintenance costs, it is unrealistic to comprehensively monitor all nodes in the network. Therefore, it is of great engineering significance to estimate the pressure data of unknown nodes in the entire network based on the network topology structure and data from limited monitoring points.

[0003] Some studies have proposed using hydraulic simulation methods, spatial interpolation methods, neural networks and other pressure estimation methods to estimate the pressure data of unknown nodes in the entire network. The hydraulic simulation method is a method that uses the principles of hydraulics to simulate and predict the movement of water flow. It quantitatively describes and analyzes the water flow movement by establishing mathematical models and physical models, thereby helping engineers and researchers better understand and predict water flow behavior. The spatial interpolation method is a key technology in geographic information systems and spatial analysis for predicting the values of unknown points through known point data. Its core idea is to use the correlation of spatial data to fill in missing or unsampled location information. The neural network pressure estimation method is a method that uses machine learning technology and a neural network model to predict the pressure level of an individual.

[0004] Among the above disclosed technical solutions, there are at least the following technical problems: The hydraulic simulation method highly depends on accurate model parameter settings and requires a large amount of manpower and material resources for model construction and verification;

[0005] The spatial interpolation method ignores the irregular topology structure and complex hydraulic relationships of the water supply network, and its estimation accuracy is very limited;

[0006] The efficient operation of the neural network method depends on the appropriate setting of hyperparameters such as its structure, size, and number of layers, as well as the appropriate preprocessing of training data, which has a certain application difficulty and also limits the wide application of such methods.

[0007] In view of the above problems, the present invention proposes a solution. Summary of the Invention

[0008] In order to overcome the above-mentioned defects of the prior art, an embodiment of the present invention provides a method for estimating the pressure of a water supply network based on second-order graph signal reconstruction, and solves the problem of high-precision estimation of the pressure of unknown nodes in the water supply network through a graph spectrum second-order reconstruction framework.

[0009] To achieve the above object, the present invention provides the following technical solutions:

[0010] A method for estimating the pressure of a water supply network based on second-order graph signal reconstruction, comprising the following steps: obtaining water supply network information, constructing a first network graph Laplacian matrix, and performing spectral analysis; limiting the acquisition of the low-frequency terms of the graph Laplacian spectrum through preset high-frequency terms of the graph Laplacian spectrum, and reconstructing the first network water head based on the inverse graph Fourier transform of the low-frequency terms of the graph Laplacian spectrum, and obtaining pseudo-monitoring values; constructing a second network graph Laplacian matrix according to the pseudo-monitoring values and actual monitoring values, and repeating the reconstruction process to obtain the second network water head.

[0011] In a preferred embodiment, the constructing of the first network graph Laplacian matrix and performing spectral analysis are specifically as follows: the water supply network information includes the topological structure of the water supply network; obtaining a weight matrix for solving the topological structure of the water supply network according to the connection relationship between nodes and pipes in the topological structure of the water supply network; solving the first network graph Laplacian matrix according to the weight matrix for solving the topological structure of the water supply network; performing eigenvalue decomposition on the first network graph Laplacian matrix to obtain an eigenvector matrix as the basis of the graph Fourier transform; performing spectral analysis according to the graph Fourier transform to obtain the graph Laplacian spectrum.

[0012] In a preferred embodiment, the limiting the acquisition of the low-frequency terms of the graph Laplacian spectrum is specifically as follows: the water supply network information includes the number of actual monitoring points and the total number of nodes in the water supply network; creating a sampling matrix according to the number of actual monitoring points and the total number of nodes in the water supply network, obtaining monitoring values from the sampling matrix, and constructing the relationship between the monitoring values and the low-frequency spectrum; constructing a mathematical model through the relationship between the monitoring values and the low-frequency spectrum, and reconstructing the low-frequency terms using the least squares method and the pseudo-inverse matrix.

[0013] In a preferred embodiment, the reconstructing of the first network water head based on the inverse graph Fourier transform of the low-frequency terms of the graph Laplacian spectrum and obtaining the pseudo-monitoring values are specifically as follows: according to the limiting the acquisition of the low-frequency terms of the graph Laplacian spectrum, reconstructing the low-frequency component of the water head through the inverse graph Fourier transform to replace the original water head; subtracting the corresponding node elevation from the original water head to obtain an estimated value of the unknown node pressure, that is, the pseudo-monitoring value.

[0014] In a preferred embodiment, the constructing of the second network graph Laplacian matrix according to the pseudo-monitoring values and actual monitoring values and repeating the reconstruction process to obtain the second network water head are specifically as follows: the water supply network information includes the actual monitoring values of the water supply network; obtaining combined monitoring values according to the pseudo-monitoring values and actual monitoring values, and obtaining the reconstructed low-frequency term spectrum by setting a covariance matrix; obtaining the low-frequency component according to the reconstructed low-frequency term spectrum through the water head low-frequency component calculation formula, and further obtaining the reconstructed node water head.

[0015] The technical effects and advantages of the method for estimating the pressure of a water supply network based on second-order graph signal reconstruction according to the present invention:

[0016] 1. Through the second-order graph signal reconstruction framework, the present invention first reconstructs the main trend of the pipe network pressure distribution by capturing the low-frequency spectrum components, and then fuses the pseudo-monitoring values with the actual monitoring values in the secondary reconstruction, effectively combines the topological features with the hydraulic relationship, significantly improves the pressure estimation accuracy of unknown nodes, and overcomes the defect that the traditional spatial interpolation method ignores the pipe network structure.

[0017] 2. Through the spectral decomposition and inverse Fourier transform of the graph Laplacian matrix, the present invention achieves the goal of spatial correlation modeling for mapping the irregular pipe network topology into an analytic graph signal. There are no strict requirements for the number and distribution of monitoring points, and the whole network pressure estimation can be realized only by meeting the basic observable conditions, which is compatible with different-scale water supply systems and provides flexible technical support for the optimization of the monitoring network of smart water services. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 is a schematic flow chart of a method for estimating the pressure of a water supply pipe network based on second-order graph signal reconstruction according to the present invention;

[0019] Figure 2 is a display diagram of the pressure estimation results of GHR-S with different numbers of monitoring points in a large-scale looped pipe network according to the present invention

[0020] Figure 3 is a schematic diagram of the head estimation results simulated by the GHR-S method and the hydraulic model according to the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0021] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0022] Embodiment 1, Figure 1 A method for estimating the pressure of a water supply pipe network based on second-order graph signal reconstruction according to the present invention is given, including the following steps:

[0023] S1. Obtain the water supply pipe network information, construct the first pipe network graph Laplacian matrix, and perform spectral analysis;

[0024] The construction of the first pipe network graph Laplacian matrix and the spectral analysis are specifically as follows:

[0025] The water supply pipe network information includes the topological structure of the water supply pipe network;

[0026] According to the connection relationship between the nodes and pipes in the topological structure of the water supply pipe network, obtain the weight matrix for solving the topological structure of the water supply pipe network;

[0027] Solve the weight matrix according to the water supply pipe network topology structure, and solve the Laplacian matrix of the first pipe network diagram;

[0028] Perform eigenvalue decomposition on the Laplacian matrix of the first pipe network diagram to obtain the eigenvector matrix as the basis of the graph Fourier transform;

[0029] Perform spectral analysis according to the graph Fourier transform to obtain the graph Laplacian spectrum.

[0030] It should be noted that the water supply pipe network topology structure includes the number of nodes and the number of pipes. The Laplacian matrix of the first pipe network diagram is calculated by selecting appropriate weight coefficients based on the available pipe network information.

[0031] It should be noted that in the above steps, the specific calculation process involved can be implemented with the help of existing mature technologies, so it will not be elaborated here; the following is an overview of the calculation:

[0032] The specific calculation formula for solving the weight matrix according to the water supply pipe network topology structure is as follows:

[0033] W[i,j] = w ij

[0034] The specific calculation formula for solving the Laplacian matrix of the first pipe network diagram by solving the weight matrix according to the water supply pipe network topology structure is:

[0035] L = diag(K T *W) - W

[0036] The specific calculation formula for performing eigenvalue decomposition on the Laplacian matrix of the first pipe network diagram is as follows:

[0037] L = U * Λ * U T

[0038] The specific calculation formula for the graph Laplacian spectrum is as follows:

[0039] P = U -1 *H

[0040] In the formula, W[i,j] is the weight matrix solved according to the water supply pipe network topology structure, that is, W; w ij is the connection relationship between nodes and pipes in the pipe network, that is, the intersection interface of the water supply pipe network, w ij > 0 means that the node is connected to the pipe, i is the node in the pipe network, and j is the number of pipes; L is the Laplacian matrix; diag() is an operator indicating converting a vector into a diagonal matrix, and the elements on the diagonal of the matrix correspond one-to-one with the elements of the vector; K is a vector with all elements being 1, T is the matrix transpose; Λ is a diagonal matrix composed of the elements on the diagonal of the Laplacian matrix L, that is, Λ[n - 1, n - 1] = λn-1 , where \(U\) is a matrix composed of eigenvectors, \(U = [u_0, u_1, \cdots, u n-1 \), where \(U n-1 represents the eigenvector corresponding to the \((n - 1)\)th eigenvalue \(\lambda n-1 \), which is the basis of the graph Fourier transform; \(H\) is the original signal, and the original signal is in the form of a linear combination of different eigenvectors; \(P = [p_1, p_2, \cdots, p n \) is the graph Laplacian spectrum of the original signal in the graph frequency domain.

[0041] S2, by presetting the high-frequency terms of the graph Laplacian spectrum, the low-frequency terms of the graph Laplacian spectrum are limited, and the low-frequency terms of the graph Laplacian spectrum are reconstructed into the first pipe network head based on the inverse graph Fourier transform, and pseudo-monitoring values are obtained;

[0042] The above-mentioned limiting the acquisition of the low-frequency terms of the graph Laplacian spectrum, and reconstructing the first pipe network head based on the inverse graph Fourier transform of the low-frequency terms of the graph Laplacian spectrum, and obtaining pseudo-monitoring values are specifically as follows:

[0043] The water supply pipe network information includes the actual number of monitoring points and the total number of nodes in the water supply pipe network;

[0044] A sampling matrix is created according to the actual number of monitoring points and the total number of nodes in the water supply pipe network, monitoring values are obtained from the sampling matrix, and the relationship between the monitoring values and the low-frequency spectrum is constructed;

[0045] A mathematical model is constructed through the relationship between the monitoring values and the low-frequency spectrum, and the low-frequency terms are reconstructed using the least squares method and the pseudo-inverse matrix.

[0046] According to the above-mentioned limiting the acquisition of the low-frequency terms of the graph Laplacian spectrum, the low-frequency component of the head is reconstructed through the inverse graph Fourier transform to replace the original head;

[0047] According to the subtraction of the corresponding node elevation from the original head, an estimated value of the unknown node pressure can be obtained, that is, the pseudo-monitoring value.

[0048] It should be noted that the first pipe network head is a preliminary node pressure estimated value generated only by reconstructing the low-frequency spectrum;

[0049] It should be noted that in the above steps, the specific calculation process involved can be realized with the help of existing mature technologies, so it will not be elaborated here; the following is an overview of the calculation:

[0050] The specific calculation formula for constructing the mathematical model of the relationship between the monitoring values and the low-frequency spectrum is as follows:

[0051] H≈U*B f *P f =U f *P f

[0052] The specific calculation formula for reconstructing the low-frequency term by the least squares method and the pseudo-inverse matrix is as follows:

[0053]

[0054] The specific calculation formula for the inverse graph Fourier transform is as follows:

[0055]

[0056] The specific calculation formula for the low-frequency component of the water head is as follows:

[0057]

[0058] The specific calculation formula for obtaining the estimated value of the pressure at the unknown node, that is, the pseudo-monitoring value, is as follows:

[0059]

[0060] In the formula, B f consists of the first f columns of matrix B, and matrix B is a matrix of size N*F; U f consists of the first f columns of matrix U, and matrix U is a matrix of size N*F; P f is a vector of length F, P f =[P1, P2,..., P f ; is the spectrum of the reconstructed low-frequency term; pinv() is an operator representing the pseudo-inverse matrix of the corresponding matrix; X is the matrix D s *U f , D s is the sampling matrix; y S is the actual monitoring data; is the reconstructed original low-frequency component of the water head; U f consists of the first f columns of matrix U; the spectrum of the reconstructed low-frequency term; H p is the pseudo-monitoring value; f p is the band-limiting coefficient of the pseudo-monitoring value.

[0061] S3. Construct the Laplacian matrix of the second pipe network graph based on the pseudo-monitoring value and the actual monitoring value, and repeat the reconstruction process to obtain the water head of the second pipe network.

[0062] The specific steps of constructing the Laplacian matrix of the second pipe network graph based on the pseudo-monitoring value and the actual monitoring value and repeating the reconstruction process to obtain the water head of the second pipe network are as follows:

[0063] The water supply pipe network information includes the actual monitoring values of the water supply pipe network;

[0064] Based on the pseudo-monitoring value and the actual monitoring value, a combined monitoring value is obtained, and by setting a covariance matrix, a reconstructed low-frequency term spectrum is obtained.

[0065] Based on the reconstructed low-frequency term spectrum, the low-frequency component is obtained through the calculation formula of the water head low-frequency component, and then the reconstructed node water head is obtained.

[0066] It should be noted that the Laplacian matrix of the second pipe network diagram is an optimized pipe network topology matrix generated by dynamically adjusting the node weights based on the pseudo-monitoring values reconstructed for the first time and the actual monitoring data; the second pipe network water head is the node pressure estimation result optimized by secondary reconstruction after fusing the actual monitoring value and the pseudo-monitoring value for the first time;

[0067] It should be noted that in the above steps, the specific calculation process involved can be implemented with the help of existing mature technologies, so it will not be elaborated here; the following is an overview of the calculation:

[0068] The specific calculation formula of the reconstructed low-frequency term spectrum is as follows:

[0069]

[0070] The specific calculation formula of the reconstructed node water head is as follows:

[0071]

[0072] In the formula, Reconstructed low-frequency term spectrum; R is the covariance matrix; H c Is the combined monitoring value; Is the reconstructed node water head.

[0073] Example 2, Application case of large pipe network. Taking the test analysis results of a large complex pipe network as an example, this example further illustrates the application performance of GHR-S in a large pipe network. Figure 2 Shows the pressure estimation results of GHR-S using different numbers of monitoring points in a large looped pipe network,

[0074] The example pipe network used in this example contains 12,532 nodes, 2 water supply sources, and 14,822 pipes. The GHR-S estimation results are tested using different numbers of randomly located monitoring points, and the results are as Figure 1 Shown. When using 30, 60, and 100 monitoring points, the MAEs of the estimation results of GHR-S are 0.243 m, 0.206 m, and 0.175 m respectively. Considering that the monitoring points used by GHR-S in this test are randomly arranged, the small error increase effectively illustrates the good performance of the GHR-S method.

[0075] On the other hand, when using monitoring points at random locations, the error of the estimation results in some areas with rapid pressure changes increases significantly. For example, Figure 2 (a) The error in the area where the pressure changes rapidly in the upper part of the central pipe network but lacks monitoring points exceeds 2m; Figure 2 In (b), the error in some areas exceeds 1m. Figure 2 The monitoring points are randomly arranged, and a considerable number of nodes are located in areas with gentle pressure changes in the pipeline network. In areas with drastic pressure changes, there are relatively fewer monitoring points. Therefore, the accuracy of node pressure estimation results in these areas is insufficient.

[0076] Therefore, in practical applications, even if it is not possible to arrange the pressure monitoring points completely according to the optimal position, it is possible to add monitoring points appropriately in areas where the pressure changes faster (usually corresponding to areas with dense population, high water consumption, and high pipeline flow rate) to improve the overall ability to estimate the pressure of unknown nodes.

[0077] It is worth noting that in this embodiment, when the number of random monitoring points increases to 500, the overall accuracy of GHR-S is better. Although the estimation error of GHR-S results for a small number of nodes is large (close to 1.5m), the error is lower for the vast majority of other nodes, so the overall MAE is smaller than GHR. On the other hand, when the number of monitoring points is large enough (such as 500), the randomly arranged monitoring points can better reflect the pressure information at different locations, avoiding the increase in error caused by paying too little attention to areas with slower pressure changes, so its overall accuracy is actually improved.

[0078] Example 3, actual pipe network application case, this example applies GHR-S to a large and complex pipe network with real monitoring data to verify its effectiveness in actual engineering.

[0079] like Figure 3 As shown, it is a schematic diagram of the head estimation results simulated by the GHR-S method and hydraulic model.

[0080] The network is located in eastern China, with a water supply area of ​​611 km2, a population of 1.09 million, and a daily water supply of 330,000 m3. The network contains more than 83,000 nodes and 13 water supply reservoirs, and the total length of pipes with a diameter of more than 100 mm is 870 km. There are 62 pressure monitoring points available in the network, and the locations of the monitoring points are determined based on engineering experience and have been pre-installed. In this section, GHR-S is used to collect data from 58 of the pressure monitoring points (such as Figure 3 (a) as indicated by the triangle mark) to estimate the water head at the unknown node in the pipe network, and use the data of the remaining four pressure monitoring points (such as Figure 3 The estimation results are verified by using the same parameter settings as in Case 1. The weight coefficient is denoted as w (uQ) for calculation.

[0081] In addition, in this case, the ratio of the number of unknown nodes to the number of monitoring nodes is approximately 100 times that of Case 1. Therefore, it is necessary to correspondingly decrease the weight of the pseudo-monitoring value (i.e., increase ), which is correspondingly set to σ v = 100. Although there is a corresponding hydraulic model for this pipe network, the model parameters have not been accurately calibrated. Therefore, during the calculation process, the roughness coefficient of the model pipes is set to 120; except for the node demand of some large users determined by real-time remote water meters, the remaining water demand is assumed to be evenly distributed along the pipes, that is, the same as the assumption for calculating the weight coefficient. The estimation results of GHR-S are as shown in Figure 3 (a); meanwhile, Figure 3 (b) shows the simulation results of the hydraulic model for comparative analysis.

[0082] From Figure 3 (a), it can be seen that GHR-S successfully reconstructed the water heads of more than 83,000 remaining unknown nodes based on the data of 58 pressure monitoring points. Among them, the reconstruction errors of the 4 verification nodes are 0.26m, 0.74m, 0.21m, and -0.17m respectively. Compared with the simulation results of the hydraulic model (0.14m, 1.86m, 1.08m, 0.73m respectively, see Figure 3 (b)), the overall estimation accuracy is higher.

[0083] The high accuracy of GHR-S is attributed to its comprehensive utilization of the pipe network hydraulic relationship and actual monitoring information. At the same time, the use of pseudo-monitoring values ensures the stability of the algorithm under the existing monitoring point layout scheme. In addition, from the spatial variation of the pressure estimation results, it can be seen that by using the graph structure and specific graph weights to describe the complex topological structure and hydraulic relationship of the pipe network, GHR-S can accurately estimate the irregular pressure change relationship in the pipe network.

[0084] For example, since more valves are installed in the area near the three reservoirs on the left side of the pipe network (marked as Area 1), its local pressure has a large difference compared with the overall pressure distribution of the pipe network. At this time, GHR-S sets the weight of the edges corresponding to the valves much smaller than that of normal pipes, so that the rapid change of the local pressure in this area can be effectively identified; similarly, in the area below the middle of the pipe network (marked as Area 2), some pipes are close to each other, cross but are not connected to each other. Therefore, there are significant differences in the water heads of some nearby nodes, and GHR-S accurately reflects this difference through the description of the pipe network topological relationship.

[0085] Figure 3The variation of the nodal head in the hydraulic model simulation results in (b) also verifies the effectiveness of the estimation of the above-mentioned irregularly distributed pressure. In contrast, it is difficult for traditional space-distance-based interpolation methods to consider similar irregular head variations, and large estimation errors will occur at such nodes.

[0086] In this case, the time taken for GHR-S to perform a single nodal pressure estimation on the tested computer (configured with an AMD Ryzen 9 5900HS CPU at 3.30 GHz and 16 GB of memory) is 131.1 s. The calculation time consumption can meet the operation requirements of most online models with the current update frequency ranging from 5 minutes to 1 hour. In addition, in subsequent research and practical applications, strategies such as pre-computing the eigenvectors of the Laplacian matrix and using better sparse matrix inversion methods can further accelerate the running speed of GHR-S.

[0087] The above formulas are all dimensionless and take their numerical calculations. The formula is obtained by collecting a large amount of data for software simulation to obtain a formula closest to the actual situation. The preset parameters in the formula are set by those skilled in the art according to the actual situation.

[0088] The above embodiments can be implemented in whole or in part by software, hardware, firmware, or any other combination. When implemented using software, the above embodiments can be implemented in whole or in part in the form of a computer program product.

[0089] Those of ordinary skill in the art can realize that the modules and algorithm steps of each example described in combination with the embodiments disclosed in this article can be implemented by electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are executed in hardware or software depends on the specific application and design constraints of the technical solution. Professional technicians can use different methods to implement the described functions for each specific application, but such implementation should not be considered to exceed the scope of this application.

[0090] In addition, in each embodiment of this application, the various functional modules can be integrated into one processing module, or each module can exist physically alone, or two or more modules can be integrated into one module.

[0091] As mentioned above, only the specific implementation manners of this application are described, but the protection scope of this application is not limited thereto. Any person skilled in the art can easily think of changes or substitutions within the technical scope disclosed in this application, and all should be covered by the protection scope of this application. Therefore, the protection scope of this application should be subject to the protection scope of the claimed rights.

[0092] Finally, the above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A water supply network pressure estimation method based on second-order image signal reconstruction, characterized in that: The steps include: Obtain water supply network information, construct the Laplace matrix of the first network graph, and perform spectrum analysis; By presetting the high-frequency items of the graph Laplace spectrum, the low-frequency items of the graph Laplace spectrum are limited to be obtained, and the low-frequency items of the graph Laplace spectrum are transformed based on the inverse graph Fourier transform to reconstruct the water head of the first pipe network and obtain the pseudo monitoring value; The Laplace matrix of the second pipe network graph is constructed according to the pseudo monitoring values ​​and the actual monitoring values, and the reconstruction process is repeated to obtain the water head of the second pipe network.

2. The water supply network pressure estimation method based on second-order image signal reconstruction according to claim 1 is characterized in that: The Laplace matrix of the first pipe network graph is constructed and the spectrum analysis is performed, specifically: The water supply network information includes the water supply network topology; Obtaining a weight matrix for solving a water supply network topology structure according to the connection relationship between nodes and pipes in the water supply network topology structure; Solve the weight matrix according to the water supply network topology structure to solve the first pipe network graph Laplace matrix; Perform eigendecomposition on the Laplacian matrix of the first pipe network graph to obtain an eigenvector matrix as a basis for Fourier transform of the graph; The spectrum analysis is performed according to the Fourier transform of the graph to obtain the Laplace spectrum of the graph.

3. The water supply network pressure estimation method based on second-order image signal reconstruction according to claim 2 is characterized in that: The limiting step of obtaining the low-frequency term of the graph Laplace spectrum is specifically: The water supply network information includes the actual number of monitoring points and the total number of nodes in the water supply network; Create a sampling matrix based on the actual number of monitoring points and the total number of nodes in the water supply network, obtain monitoring values ​​from the sampling matrix, and construct a relationship between the monitoring values ​​and the low-frequency spectrum; A mathematical model is constructed based on the relationship between the monitoring value and the low-frequency spectrum, and the low-frequency terms are reconstructed using the least squares method and pseudo-inverse matrix.

4. The water supply network pressure estimation method based on second-order image signal reconstruction according to claim 3 is characterized in that: The low-frequency term of the graph Laplace spectrum is transformed based on the inverse graph Fourier transform to reconstruct the water head of the first pipe network and obtain the pseudo monitoring value, specifically: According to the above limitation, the low-frequency term of the graph Laplace spectrum is obtained, and the low-frequency component of the water head is reconstructed by inverse Fourier transform to replace the original water head; By subtracting the corresponding node elevation from the original water head, an estimated value of the unknown node pressure, ie, a pseudo monitoring value, is obtained.

5. The water supply network pressure estimation method based on second-order image signal reconstruction according to claim 4 is characterized in that: The second pipe network graph Laplace matrix is ​​constructed according to the pseudo monitoring value and the actual monitoring value, and the reconstruction process is repeated to obtain the second pipe network water head, which is specifically: The water supply network information includes actual monitoring values ​​of the water supply network; A joint monitoring value is obtained according to the pseudo monitoring value and the actual monitoring value, and a reconstructed low-frequency item spectrum is obtained by setting a covariance matrix; According to the reconstructed low-frequency item spectrum, the low-frequency component is obtained by the water head low-frequency component calculation formula, and then the reconstructed node water head is obtained.

6. The method for estimating water supply network pressure based on second-order image signal reconstruction according to claim 5 is characterized in that: The specific calculation formula for solving the weight matrix of the water supply network topology structure is as follows: W[i,j]=w ij The specific calculation formula for solving the weight matrix and the Laplace matrix of the first pipe network graph according to the water supply pipe network topology structure is: L=diag(K T *W)-W The specific calculation formula for performing eigendecomposition on the Laplace matrix of the first pipe network graph is as follows: L ＝ U*Λ*U T The specific calculation formula of the graph Laplace spectrum is as follows: P ＝ U-1 *H Where W[i,j] is the weight matrix for solving the topological structure of the water supply network, that is, W; w ij is the connection relationship between nodes and pipes in the pipe network, that is, the intersection interface of the water supply pipe network, w ij >0 means the node is connected to the pipeline, i is the node in the pipeline network, and j is the number of pipelines; L is the graph Laplace matrix; diag() is an operator, which means converting the vector into a diagonal matrix, and the elements on the diagonal of the matrix correspond one to one with the elements of the vector; K is a vector with all elements being 1, and T is the matrix transpose; Λ is a diagonal matrix composed of the elements on the diagonal of the graph Laplace matrix L, that is, Λ[n-1,n-1]=λ n-1 , U is a matrix composed of eigenvectors, U=[u0,u1,...,u n-1 ], where U n-1 Denotes the value corresponding to the n-1th eigenvalue λ n-1 The eigenvector of is the basis of the Fourier transform of the graph; H is the original signal, which is a linear combination of different eigenvectors; P = [p1, p2, ..., p n ] is the Laplace spectrum of the original signal in the frequency domain.

7. The method for estimating water supply network pressure based on second-order image signal reconstruction according to claim 6 is characterized in that: The specific calculation formula for constructing the mathematical model of the relationship between the monitoring value and the low-frequency spectrum is as follows: H≈U*B f *P f =U f *P f The specific calculation formula for reconstructing the low-frequency term using the least squares method and the pseudo-inverse matrix is ​​as follows: In the formula, B f The first f columns of matrix B form a matrix of size N*F; U f is composed of the first f columns of the matrix U, which is a matrix of size N*F; P f is a vector of length F, P f =[P1,P2,...,P f ]; is the reconstructed low-frequency spectrum; pinv() is an operator that returns the pseudo-inverse matrix of the corresponding matrix; X is the matrix D s *U f , D s is the sampling matrix; y S The actual monitoring data.

8. The method for estimating water supply network pressure based on second-order image signal reconstruction according to claim 7, characterized in that: The specific calculation formula of the inverse image Fourier transform is as follows: The specific calculation formula of the low-frequency component of the water head is as follows: The specific calculation formula for obtaining the estimated value of the unknown node pressure, that is, the pseudo monitoring value, is as follows: In the formula, is the low-frequency component of the reconstructed original water head; U f is composed of the first f columns of the matrix U; Reconstructed low-frequency spectrum; H p is the pseudo monitoring value; f p is the pseudo monitoring value band-limit coefficient.

9. The method for estimating water supply network pressure based on second-order image signal reconstruction according to claim 8, characterized in that: The specific calculation formula of the reconstructed low-frequency spectrum is as follows: The specific calculation formula of the reconstructed node water head is as follows: In the formula, Reconstructed low-frequency spectrum; R is the covariance matrix; H c is the joint monitoring value; is the reconstructed node head.

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