A method for optimizing arrangement of water supply network sensors based on graph signal processing

Abstract

The application discloses a water supply network sensor optimization arrangement method based on graph signal processing, and comprises the following steps: acquiring topological structure information of a water supply network, and establishing a user node weighted graph without direction and connection according to the topological structure information to determine a self-correlation localized graph information value of each user node and a cross-correlation graph information redundancy value between different user nodes; wherein the self-correlation localized graph information value of the user node is: the user node v i The cross-correlation graph information redundancy value between the user nodes v j According to the self-correlation localized graph information value of the user node and the cross-correlation graph information redundancy value between different user nodes, a target function is determined: user nodes meeting the target function are screened out as sensor nodes one by one until the number of sensors reaches the requirement.

Description

Technical Field

[0001] This invention belongs to the field of water supply network sensor layout technology, and specifically relates to a method for optimizing the layout of water supply network sensors based on graph signal processing. Background Technology

[0002] Water supply networks (WDNs) are crucial urban infrastructure. A small to medium-sized city's water supply network can span hundreds of kilometers of pipes, connecting hundreds of user nodes. Leaks can cause severe water waste and significant disruption to residents' daily lives, making accurate and efficient WDN monitoring a critical issue. Previously, accurately identifying leaks in the water supply network required high-precision acoustic leak detection instruments, which were costly and difficult to deploy. With the development of sensor and mobile communication technologies (such as LoRaWAN, IoT, and 5G), establishing wireless sensor networks using low-cost pressure sensors to address these issues has become feasible. However, large-scale topology maps present enormous data dimensions, and acquiring signals from each node requires substantial resources. Deploying sensors at every node to monitor pressure is impractical. Therefore, it's necessary to use information from a limited number of sensors to estimate the overall network's operational status and provide data for subsequent water supply network fault detection. Furthermore, different sensor layouts, even with similar numbers of sensors, can introduce varying degrees of redundancy (i.e., non-unique information). How to monitor WDNs using a limited number of pressure sensors has become an important research topic in recent years.

[0003] Most existing sensor optimization methods require hydraulic information from the water supply network, involving complex data processing. Furthermore, the conventional method of using hydraulic data from all nodes in the network to select nodes has limited application value because, in reality, considering actual terrain and cost, hydraulic information from not all nodes can be monitored. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for optimizing the arrangement of sensors in a water supply network based on graph signal processing. When selecting sensor arrangement nodes, only the topological information of the water supply network needs to be used, which reduces the dependence on the pressure information of the water supply network. This avoids the redundant analysis and data processing of the hydraulic simulation data of the network and can significantly improve the efficiency of leakage detection in WDN.

[0005] The technical solution provided by this invention is as follows:

[0006] A method for optimizing the layout of sensors in a water supply network based on graph signal processing includes:

[0007] Obtain the topology information of the water supply network, and establish an undirected connected weighted graph of user nodes based on the topology information.

[0008] in, It is the set of user nodes in the water supply network. N represents the number of user nodes in the water supply network; ε is the set of edges connecting all user nodes, A∈C. N×N It is a picture The corresponding weight matrix, where C represents a complex number;

[0009] Determine the autocorrelation localization graph information value of each user node, and the redundancy value of the cross-correlation graph information between different user nodes;

[0010] The autocorrelation localization graph information value of the user node is as follows:

[0011]

[0012] User node v i With user node v j The redundancy value of the cross-correlation graph information between them is:

[0013]

[0014] In the formula, Ψ g (v i ) represents node v i The vectors formed by the graph localization operators, Represents node v i The vector formed by the graph localization operator is the first item;

[0015] The objective function is determined based on the autocorrelation localized graph information values ​​of the user nodes and the redundancy values ​​of the cross-correlation graph information between different user nodes:

[0016]

[0017] In the formula, C selected This represents the currently selected set of user nodes; η represents the average value of the localization operator for user nodes in the water supply network; η represents the ratio A constant one order of magnitude smaller;

[0018] One by one, user nodes that satisfy the objective function are selected as sensor nodes until the required number of sensors is reached.

[0019] Sensors are installed at the sensor nodes to monitor the water supply network.

[0020] Preferably, node vi The expression for the vector formed by the graph localization operator is:

[0021]

[0022] Preferably, the node v i The vector formed by the graph localization operator is the first The expression for the term is:

[0023]

[0024] Where, λ k express The corresponding eigenvalues, The feature vector u represents the weighted graph of user nodes. k The item; This represents the eigenvector matrix of the weighted graph of user nodes that is related to node v. i u in the corresponding order k (i) is the conjugate transpose of (i).

[0025] Preferably, obtaining the feature vector of the weighted graph of the user nodes includes the following steps:

[0026] Obtain the weight matrix A and degree matrix D of the weighted graph of the user nodes;

[0027] Calculate the graph Laplacian matrix L of the weighted graph of the user nodes;

[0028] Where L = DA;

[0029] Decomposing the graph Laplacian matrix L yields the eigenvector matrix of the user node weighted graph:

[0030] U = [u0, u1, ... u] k ,…,u N-1 ];

[0031] Among them, u k =(u k (1),u k (2),...,u k (N)) T .

[0032] Preferably, the eigenvalue λ is obtained. k The method is as follows:

[0033] Decomposing the graph Laplacian matrix L yields the eigenvector matrix of the user node weighted graph:

[0034]

[0035] Where, λ0<λ1≤λ2≤...≤λ N-1 =λ max .

[0036] Preferably, a greedy algorithm is used to iteratively select user nodes that satisfy the objective function as sensor nodes.

[0037] The beneficial effects of this invention are:

[0038] The water supply network sensor optimization layout method based on graph signal processing provided by this invention only requires the topological structure information of the water supply network when selecting sensor layout nodes, reducing the dependence on the pressure information of the water supply network. This avoids the redundant analysis and data processing process of the hydraulic simulation data of the network, and can significantly improve the efficiency of leakage detection in WDN and obtain higher leakage location accuracy. Attached Figure Description

[0039] Figure 1 This is a flowchart of the water supply network sensor optimization layout method based on graph signal processing described in this invention.

[0040] Figure 2 This is the Fourier spectrum of the pressure signal of the Hanoi network described in this invention.

[0041] Figure 3 This is the Fourier spectrum of the pressure signal of the Net3 network described in this invention.

[0042] Figure 4 This is a diagram of the Hanoi network topology used in Experiment Example 1 of this invention.

[0043] Figure 5 This is a heatmap of the localization operator values ​​in Experiment Example 1 of the present invention.

[0044] Figure 6a This is a schematic diagram of arranging 6 sensor nodes in the Hanoi network in Experiment Example 1 of the present invention.

[0045] Figure 6b This is a schematic diagram of four sensor nodes arranged in the Hanoi network in Experiment Example 1 of the present invention.

[0046] Figure 6c This is a schematic diagram of three sensor nodes arranged in the Hanoi network in Experiment Example 1 of the present invention.

[0047] Figure 7a This is a schematic diagram of sensor nodes arranged in a Hanoi network in Comparative Example 1 of the present invention.

[0048] Figure 7b This is a schematic diagram of sensor nodes arranged in a Hanoi network in Comparative Example 2 of the present invention.

[0049] Figure 8 This is a diagram of the Net3 network topology used in Experiment Example 2 of this invention.

[0050] Figure 9 This is a schematic diagram of sensor nodes arranged in a Net3 network in Experimental Example 2 of the present invention.

[0051] Figure 10a This is a schematic diagram of sensor nodes arranged in a Net3 network in Comparative Example 3 of the present invention.

[0052] Figure 10b This is a schematic diagram of sensor nodes arranged in a Net3 network in Comparative Example 4 of the present invention. Detailed Implementation

[0053] The present invention will now be described in further detail with reference to the accompanying drawings, so that those skilled in the art can implement it based on the description.

[0054] The sensor network monitoring WDN is a graph model of distributed interconnected sensors, and its readings are modeled as time-correlated signals at the vertices. This type of signal, which can be defined on the nodes of the topological graph, is called a "topological graph signal," or simply a "graph signal." Currently, mature signal processing techniques are mostly used to solve signal problems in regular domains, such as speech recognition, image super-resolution analysis, and video compression. Water supply network topologies are complex and irregular, making it difficult to analyze irregular graph signals using traditional, well-established theories. Graph Signal Processing (GSP), as an emerging field of signal processing research, aims to process signals at graph nodes by incorporating the properties of the underlying topological graph. The aforementioned sensor node selection problem aligns very well with the goals of graph signal sampling theory in GSP. In fact, the central nodes of a water supply network topology are often densely populated user areas, which is somewhat similar to the analysis results of hydraulic models.

[0055] like Figure 1 As shown, this invention provides a method for optimizing the arrangement of sensors in a water supply network based on graph signal processing. It utilizes graph localization operators to characterize the graph structure information and information redundancy provided by different user nodes, thereby selecting the optimal sensor configuration. This method reduces the dependence on hydraulic simulation of the water supply network to a certain extent, and is applicable to different types of water supply networks and is robust to environmental noise.

[0056] The water supply network containing N user nodes is considered as an undirected, connected weighted graph. in, It is a set of vertices (user nodes), which can be described as ε is the set of edges connecting all nodes, A∈C N×NIt is a picture The corresponding weight matrix, where a certain item in the matrix is ​​A ij Indicates connection to vertex v i and v j The weights of the edges indicate the vertex v. i and v j The similarity or dependence of signal values ​​between them. A ij The distance to the pipeline is obtained by weighting the distance using a Gaussian function. When user node v i and v j When pipe connections exist, apply the Gaussian function to node v. i and v j The distance a between the pipes ij Weighting, i.e. This controls the distance to be inversely proportional to the weight, with lower edge weights between points that are farther apart and higher edge weights between points that are closer together. In the formula, σ represents a constant that controls the distribution of weights. f(i) is used to represent the graph. Mid-vertex v i The graph domain signal value, the complete graph signal can be written in vector form, that is, f=(f(1),f(2),...,f(N)) T ∈C N .

[0057] In classical digital signal processing theory, the Fourier transform of function m(t) is shown in equation (1).

[0058]

[0059] Among them, e -jwt Let be the characteristic function of a one-dimensional Laplace matrix, as shown in equation (2).

[0060]

[0061] Extending Fourier analysis to graphs requires the use of graph Fourier bases. The most commonly used graph Fourier bases are the eigenvectors of the combined (or non-normalized) graph Laplacian. The graph Laplacian matrix L is defined as L = DA, where... The degree matrix of the nodes, d i It is related to node v i The sum of the weights of connected edges. Perform eigenvalue decomposition on the graph Laplacian matrix L, i.e., L = UΛU T The eigenvalue matrix can be obtained. Where λ0<λ1≤λ2≤...≤λ N-1 =λ max And the eigenvector matrix U = [u0, u1, ... u k ,...,u N-1Each eigenvalue corresponds to an N-dimensional eigenvector, where u k =(u k (1),u k (2),...,u k (N)) T The Fourier transform and inverse transform of the graph signal can be defined by equations (3) and (4). U represents the Fourier coefficients in the frequency domain of the graph. k (m) represents the eigenvector u k The m-th term.

[0062]

[0063]

[0064] The graph localization operator is derived from the spectral graph wavelet transform and is generated by wavelet operators based on the graph Laplacian operator value function. For the spectral graph wavelet kernel g, the wavelet operator T g =g(Λ) is obtained by modulating each Fourier mode onto a given function f. Equation (5) is obtained by using the inverse Fourier transform, where λ k express The corresponding eigenvalues.

[0065]

[0066] Then, by applying these operators to a single vertex v i pulse Equation (6) is obtained.

[0067]

[0068] Explicitly expanded within the graph domain to obtain

[0069]

[0070] Without considering scaling, the wavelet coefficients W of a given function f f (v i It is generated by taking the inner product of these wavelets, that is...

[0071] W f (v i )=<Ψ g (v i ),f> (8)

[0072] Among them, Ψ g (v i As shown in equation (9)

[0073]

[0074] If the smoothing kernel g(·) is located at a central vertex v in the graph i Equation (9) is the vertex v i The graph localization operator is a vertex-domain operator, and its complete form is a multidimensional column vector. It's important to note that it's a vertex-domain operator and takes into account graph domain information. This generalized localization operator applies to kernels defined in the graph domain, and it is localized in the small-scale limit. The magnitude of this operator varies with the node v. i The decay occurs as the distance to other nodes increases. Therefore, this operator can reflect the localization characteristics of the central node.

[0075] Starting from the topological structure of the graph, and utilizing the eigenvalues ​​and eigenvectors of the graph Laplacian matrix, a graph localization operator is constructed. Based on the analysis above, the graph localization operator can reflect the localization characteristics of the central node. In other words, the larger the graph localization operator value of a node, the greater the importance of that node in the graph. Based on this, the sensor optimization placement method of this invention is proposed.

[0076] The method of this invention is based on the idea of ​​minimum redundancy maximum correlation (mRMR), which aims to select nodes with obvious localization feature information and the least redundancy with the graph structure information provided by the already selected nodes as representative nodes of WDN, and to place pressure sensors at the corresponding locations to monitor WDN.

[0077] Based on the above analysis, we will consider a vertex (user node) v in the graph topology. i The 1-norm of the graph localization operator is defined as the autocorrelation graph localization information (AGLI) of that node, and the corresponding formula is equation (10). Φ AGLI The larger the value, the more information the node provides about the graph topology without considering redundancy between nodes, and the more important it is in WDN.

[0078]

[0079] To analyze the redundancy among the localization features provided by different vertices, we borrow a method from signal processing theory for evaluating the cross-correlation between different signals. We quantify the redundancy between localization feature information using the 1-norm value of the product of graph localization operators of two nodes, termed Cross-correlation Graph Information Redundancy (CGIR). Node v i With node vj The redundancy of cross-correlation graph information between them is given by equation (11), where, This indicates that corresponding terms are multiplied.

[0080]

[0081] In this invention, a binary vector C of equation (12) is defined to represent the selection of sensor nodes.

[0082] C = (c1, c2, ..., c j ,...,c N (12)

[0083] Where N is the number of user nodes in WDN, and j represents the user node v represented by the current vector. j When a sensor is deployed at a node, its corresponding binary vector component is set to 1.

[0084] Based on the above definition, we propose a heuristic algorithm to optimize the arrangement of our sensors. When the number of sensors to be selected is s, the objective function is as shown in equation (13). A greedy algorithm is used to iteratively select sensor nodes until the given number s is reached.

[0085]

[0086] Substituting into equations (10) and (11) above, we get:

[0087]

[0088] In formula (14), C selected It is the currently selected set of nodes. This is the average value of the node localization operator in WDN, and its main function is to improve the cooperability coefficient 1 / |C. selected |Keep the numerator and denominator of the objective function on the same order of magnitude, thus improving the autocorrelation localization graph information Φ AGLI and cross-correlation graph information redundancy Φ CGIR The comparison between them is more accurate. η represents a value greater than the average value of the user node localization operator. A constant one order of magnitude smaller is used to exclude the extremely special case where the denominator is 0.

[0089] Greedy selection steps:

[0090] (1) Start from the first initial solution in the candidate set;

[0091] (2) By using the objective function (14), evaluate the redundancy of the effective information provided by the user node and the graph structure information provided by the currently retrieved user node and the already selected node, and select the best (maximum ratio) user node as the placement position of the pressure sensor.

[0092] (3) Remove the selected nodes from the candidate set and repeat steps (1) and (2) until a given number of sensors s is reached, i.e. The specific process is as follows Figure 1 As shown;

[0093] (4) Combine all the solutions to obtain the set of sensor nodes.

[0094] The effectiveness of the sensor placement method was evaluated using a pressure sensitivity matrix, and the specific process is as follows:

[0095] Pressure and flow rate are highly correlated at nodes in a water supply network; therefore, changes in pressure can reflect changes in flow rate within the network. This invention primarily utilizes the hydraulic simulation software EPANET and the Water Network Tool for Resilience (WNTR) to simulate node leakage and acquire pressure data. WNTR is a Python package compatible with EPANET.

[0096] First, simulate the pressure data of all nodes in the WDN when the water demand is normal, and obtain the pressure matrix under normal conditions. in, Represents node v i The pressure value during normal operation. Leakage at a node will cause its water demand to increase; this characteristic can be used to simulate leakage events in the software. Leakage events are simulated by adding leaks to each node individually. The pressure difference between the pressure at the time of leakage and the normal pressure is obtained, as shown in the formula.

[0097]

[0098] in, Represents node v j When a leakage of magnitude ρ occurs at node v i Pressure residual, For node v j When a leakage of magnitude ρ occurs at node v i The pressure. The pressure residual matrix can be expressed as...

[0099]

[0100] To simulate various errors under real-world conditions (such as measurement errors, statistical errors, etc.), appropriate noise can be added to the pressure residual matrix. Dividing each item in the pressure residual matrix by the corresponding leakage magnitude yields the pressure sensitivity matrix.

[0101]

[0102] in, This represents the size of the leak when node N experiences a leak of level M.

[0103] The goal of optimizing sensor placement in water supply networks is to better locate leaks. The effectiveness of leak localization reflects the efficiency of the selected sensor nodes in monitoring the water supply network. This invention primarily employs a leak localization method based on a machine learning classifier, specifically the K-Nearest Neighbors (KNN) algorithm. KNN is a lazy, non-parametric classification algorithm. Its main idea is to first calculate the similarity (usually using distance metrics) between the sample to be classified and the currently known training samples, then find the K categories with the highest similarity, and finally select the type of sample data based on this. Commonly used distance measurements include Euclidean distance, Mahalanobis distance, Minkowski distance, and Manhattan distance. The sample to be tested is U0 = (x... 01 ,x 02 ,x 03 ,...,x 0N ) and training sample U κ =(x κ1 ,x κ2 ,x κ3 ,...,x κN The Euclidean distance between the two categories is calculated using formula (18). The distances between different categories are analyzed, and the categories of the test samples are determined by combining the weights.

[0104]

[0105] The leak location algorithm uses the pressure sensitivity matrix S and the corresponding leak label as the training feature set of the classifier. Based on the obtained sensor configuration, it selects the pressure data of the corresponding node in the test set and then predicts the leak location.

[0106] In WDN, the pressure changes between adjacent nodes are smooth, meaning that nodes that are close together have similar pressure levels. This can negatively impact the accuracy of machine learning classifier algorithms when predicting leak locations, often resulting in predicted nodes being located near the leaking node. Therefore, using leak location accuracy as an evaluation criterion is inappropriate. A more effective evaluation method is the Average Topological Distance (ATD), calculated using the formula:

[0107]

[0108] In the formula, N is the number of nodes in the WDN, ω ij This indicates that it occurred at node v. i The leakage prediction is for node v j The number of times, D ij For node v i With node v j The shortest topological distance between the predicted nodes and the leak location is used. ATD (Average Topological Distance) accurately reflects the actual distance between the predicted node and the leak location. In our method, the ratio of ATD to the total pipeline length L is used to measure the accuracy of leak location; this is called the leak location average topological distance index ξ. ATD The formula is

[0109]

[0110] The root mean square error (RMSE) is often used to evaluate the stability of prediction results. Assuming there are N samples... For y i If the predicted value is given, then the general formula for RMSE is:

[0111]

[0112] Combining ATD with RMSE can assess the deviation between the predicted node and the actual node's average topological distance. The root mean square error (RMSE) of ATD is used to evaluate this deviation. ATD The formula:

[0113]

[0114] In this invention, RMSE is utilized. ATD The ratio of the mean topological distance deviation (μ) to the total length L of the pipeline network is used to measure the deviation in leak location; this is called the mean topological distance deviation index (μ). RMSE The formula is

[0115]

[0116] Based on whether the network contains devices such as pumps and valves, water supply networks (WDNs) can be divided into two main categories: active WDNs and passive WDNs. In this section, we apply the proposed sensor placement method to two different types of water supply networks: the passive Hanoi water supply network and the active Net 3 network. Both models have been widely used in research on the optimal placement of sensors in water supply networks. Table 1 shows some parameters of the network. Furthermore, our method is compared with several other methods. All work in this experimental example was performed on a laptop configured with an Intel(R) Core(TM) i5-7300HQ CPU@2.50 [GHz], 8 [GB] RAM, and an NVIDIA GeForce GTX 1050Ti, using MATLAB version 2018a.

[0117] The smoothing kernel function g used in both experimental cases is a heat kernel in the negative exponential form of e, i.e., g(λ) = exp(-τλ), where the value of τ needs to be greater than 0, determined by the marginal probability p. s =s / N, current sampling rate p c =|C| / N, normalized bandwidth b = B / N, and the maximum eigenvalue λ of the graph Laplacian matrix. max The decision is τ = (B|C|s) / (N) 3 λ max In a water distribution network, the water pressures of adjacent user nodes are very close, meaning that the pressure signals at different nodes are smooth. In other words, the pressure signal can be approximated as a band-limited signal. The graphical Fourier spectra of the pressure signals from the Hanoi and Net3 networks used were simulated, as shown below. Figure 2 and Figure 3 As shown, the Fourier spectra of the pressure signals from both networks are concentrated in the low-frequency region. In subsequent experiments, we simplified the pressure signals, approximating the bandwidth B of the pressure signal in the Hanoi network to 8 and the bandwidth B of the pressure signal in Net3 to 15.

[0118] Experimental Example 1

[0119] The water supply network used in Experiment Example 1 is a Hanoi network, with the following topology: Figure 4 As shown, the system was proposed by Fujiwara and Khang (1990) based on the trunk network in the water supply network plan of Hanoi, the capital of Vietnam. The Hanoi system consists of 2 reservoirs and 31 user nodes, connected by 34 pipes with pipe sizes ranging from 12 to 40 inches, a total length of 39.42 km, and an average daily water demand of 5538.9 L / s.

[0120] Table 1. Water Supply Network Parameters

[0121]

[0122] The data required for locating leaks in the water supply network was obtained using EPANET software simulation. Leaks occurring at nodes in the network can be simulated by adding water demand. First, the Hanoi system network was run in EPANET software to obtain pressure data under normal conditions. Then, different amounts of water demand were added to each node to simulate leaks. It is important to note that the leak size should not be too large, otherwise it will affect the normal operation of the network; the size was set to within 1% of the average daily water demand. For the 31 nodes of the Hanoi network, a total of 1500 leak events were simulated, and random noise with a signal-to-noise ratio of 80dB was added to the obtained pressure data to simulate environmental errors. A sensitivity matrix was obtained, with a size of 1500×31.

[0123] In this experimental example, the weighted adjacency matrix of pipe lengths is obtained through the topology of the Hanoi network. eigenvalues ​​and eigenvector matrices are obtained by eigenvalue decomposition using graph signal processing techniques. A graph localization operator is then constructed. Figure 5 This is a heatmap showing the values ​​of the graph localization operator for each node. It can be seen that a particular node only has a significant impact on its surrounding nodes; the larger the value, the greater the impact. The constructed graph localization operator is applied to the objective function, selecting nodes that provide sufficient graph structure information and have minimal redundancy with already selected nodes as sensor placement locations. The results of selecting 3, 4, and 6 sensors using the sensor optimization placement method provided by this invention are shown in the pipeline network distribution as follows: Figures 6a-6c As shown.

[0124] Table 2 presents the values ​​of two evaluation metrics for different sensor configurations, and compares them with two other methods using the same number of sensors. The evaluation metric used is the leak location average topology distance index ξ. ATD and the average topological distance deviation index μ RMSE Comparative example 1 for the Hanoi network shows the nodes selected by Casillas, MV et al. using the Particle Swarm Optimization (PSO) algorithm. The corresponding nodes are {1, 12, 21, 24}, and their distribution in the network is as follows: Figure 7a As shown; Comparative Example 2 shows the sensor configuration used by Débora Alves et al. in their manual analysis of leak location, with sensor nodes {12, 17, 23, 29} distributed in the pipeline network as follows. Figure 7b As shown.

[0125] Table 2 Comparison of Leakage Localization Results under Different Sensor Placement Methods in Hanoi Networks

[0126]

[0127]

[0128] Based on the data in Table 2, when using the same number of sensors, the leakage localization effect of the nodes selected by our method is compared with that obtained by the PSO method in Comparative Example 1. ATD It decreased by 22.39%, μ RMSE The number of nodes decreased by 26.65% compared to the nodes used in Comparative Example 2. ATD It decreased by 30.49%, μ RMSE The leakage was reduced by 25.03%. Table 2 also shows the leakage location results when three sensors were selected using the method of the present invention. It can be seen that Experiment 1 achieved relatively good monitoring results with a smaller number of sensors. This means that the method of the present invention is superior to the other two methods in terms of both accuracy and stability in leakage location.

[0129] Furthermore, the leak location effect is further improved when using a configuration of 6 nodes in Experimental Example 1. The above analysis shows that the sensor nodes deployed using the method of this invention can acquire more effective data when monitoring WDN, providing better assistance for subsequent leak event location.

[0130] Experimental Example 2

[0131] The water supply network used in Experiment Example 2 is an active network Net3, and its structure is as follows: Figure 8 As shown, the system consists of 92 user nodes, 117 pipelines, 2 reservoirs, 3 water tanks, and 2 water pumps. The total pipeline length is 215.71 km, and the average daily water demand is 3052.11 L / s. Due to the large scale of the Net3 pipeline network, simulating leaks using EPANET software would be labor-intensive and difficult to implement. Therefore, the WNTR toolkit in Python was used to simulate the pipeline network. First, the pressure of each node under normal operation was obtained, and then leaks were added to each node one by one. In the leak simulation, the area of ​​the orifice was set to 0.001 m². 2 To study the robustness of the algorithm under dynamic and uncertain conditions, the water demand variation over time was considered. Pressure data was extracted with a time step of 1 hour, and random noise with a signal-to-noise ratio of 80dB was added to simulate environmental errors. A total of 1820 leakage scenarios were simulated, and the final pressure sensitivity matrix was 1820×92.

[0132] The effectiveness of the method of this invention was compared with two other methods when selecting the same number of sensors. Seventeen sensors were selected, corresponding to nodes {217, 255, 193, 201, 169, 121, 601, 181, 127, 143, 185, 261, 149, 239, 164, 101, 206}, and their distribution in the pipeline network is as follows: Figure 9As shown. Comparative Net 3 network example 3 is based on the method proposed by Sun and Wang (2016), with the selected nodes being {101,105,111,61,127,193,119,263,151,199,209,159,10,109,113,139,40}, and their distribution in the pipeline network is as follows. Figure 10a As shown in the figure, Comparative Example 4 uses the compressed sensing method, and the selected nodes are {10,50,125,153,159,164,181,184,195,199,201,207,208,225,229,253,269}. Their distribution in the pipeline network is as follows. Figure 10b As shown in Table 3. ATD and μ RMSE The values ​​of the two evaluation indicators.

[0133] Compared with the method proposed by Sunand Wang (2016) (Comparative Example 3), the method of the present invention has the following advantages: ATD It decreased by 29.38%, μ RMSE It decreased by 19.21%; compared with the nodes selected by the compressed sensing method (Comparative Example 4), ξ ATD It decreased by 40.63%, μ RMSE The number of nodes was reduced by 37.96%. The comparison shows that the method of this invention achieves better leak location results while using fewer nodes. Therefore, the method of this invention can better monitor sensor networks and more accurately obtain the location information of leaking nodes.

[0134] Table 3 compares the leak localization performance under different sensor placement methods in Net 3.

[0135]

[0136] This invention presents a novel heuristic algorithm based on feature selection and graph signal processing theory to deploy sensors for monitoring leak events in water distribution networks. This method reduces reliance on pressure information from the water supply network when selecting sensor placement nodes, thus avoiding redundant analysis of hydraulic simulation data. The proposed method is then applied to two water supply networks of different sizes. In the Hanoi network, the monitoring capability of the provided sensor placement scheme for leak events of varying sizes is analyzed. In Net3, the robustness of the algorithm to changes in user node water demand over time is analyzed. The results show that, compared with other methods, the method of this invention is more versatile and effective, significantly improving the efficiency of leak detection in water distribution networks (WDNs).

[0137] Although embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. They can be applied to various fields suitable for the present invention. For those skilled in the art, other modifications can be easily made. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details and illustrations shown and described herein.

Claims

1. A method for optimal placement of water distribution network sensors based on graph signal processing, characterized in that, The method comprises the following steps: Obtaining topological structure information of a water supply network, and establishing a user node weighted graph that is undirected and connected according to the topological structure information wherein, is a set of user nodes in a water supply network, N denotes the number of user nodes in the water supply network; ε is a set of edges of all connected user nodes, A ∈ C N×N is a graph is a corresponding weight matrix, C denotes a complex number; determining the autocorrelation localized graph information value of each user node and the cross-correlation graph information redundancy value between different user nodes; The autocorrelation localized graph information value of the user node is: User node v i The intercorrelation graph information redundancy value between user node v j is: where Ψ g (v i ) denotes the vector of graph localizers of node v i , denotes the i i -th entry of the vector of graph localizers of node v ; determining a target function according to the autocorrelation localized graph information value of the user node and the cross-correlation graph information redundancy value between different user nodes: where C selected represents the currently selected set of user nodes; represents the average value of the user node localization operator in the water supply network; η represents the ratio a constant of the order of magnitude smaller than one; screening out the user nodes meeting the target function as sensor nodes one by one until the number of sensors reaches the requirement; installing sensors at the sensor nodes to monitor the water supply network; Node v i The expression of the vector of graph localization operators is: The node v i The expression of the first item of the vector consisting of the graph localization operator of v is: Where, λ k express The corresponding eigenvalues, The feature vector u represents the weighted graph of user nodes. k The item; This represents the eigenvector matrix of the weighted graph of user nodes that is related to node v. i u in the corresponding order k (i) is the conjugate transpose of (i).

2. The graph signal processing based water distribution network sensor optimization placement method of claim 1, wherein, obtaining the eigenvector of the user node weighted graph, comprising the following steps: obtaining the weight matrix A and the degree matrix D of the user node weighted graph; calculating the graph Laplacian matrix L of the user node weighted graph; wherein L=D-A; decomposing the graph Laplacian matrix L to obtain the eigenvector matrix of the user node weighted graph: U = [u0, u1,... u k ,...,u N-1 ] ; wherein u k = (u k (1), u k (2),..., u k (N)) T .

3. The graph signal processing based water distribution network sensor optimisation arrangement method as claimed in claim 2, wherein, The method for obtaining the eigenvalue λ k is: decomposing the graph Laplacian matrix L to obtain the eigenvector matrix of the user node weighted graph: wherein λ0< λ1≤ λ2≤... ≤ λ N-1 = λ max .

4. The graph signal processing based water distribution network sensor optimization arrangement method according to claim 1 or 3, c h a r a c t e r i z e d by, iteratively selecting the user nodes meeting the target function as sensor nodes by using a greedy algorithm.

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