A sensor arrangement method for water supply network based on graph sampling theory

Abstract

The application discloses a sensor arrangement method for a water supply network based on a graph sampling theory, and comprises the following steps: establishing a user node undirected graph, determining a graph Fourier operator of the user node undirected graph; obtaining user node pressure of the water supply pipe network under different leakage conditions, obtaining a pressure sensitivity matrix of the water supply pipe network, and determining useful information in the pressure sensitivity matrix; performing Fourier transform on the useful information by using the graph Fourier operator to obtain a graph Fourier spectrum of each user node under different leakage conditions; screening out a graph Fourier spectrum group which is greater than or equal to a spectrum threshold to form a new spectrum matrix, and screening out a frequency component corresponding to the new spectrum matrix to form a new frequency component matrix; taking the number of spectrums in the new spectrum matrix as the number of sensor nodes to obtain an update signal; constructing a target function according to the update signal; and screening out user nodes which satisfy the target function as sensor nodes one by one until the number of sensor nodes is reached.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of water supply network sensor arrangement, and particularly relates to a water supply network sensor arrangement method based on graph sampling theory. BACKGROUND

[0002] In a water supply network, some sensors need to be arranged to monitor the operation condition of the entire network. However, due to the real cost caused by arranging the sensors, we have to select some important nodes as the arrangement nodes of the sensors, which requires us to select a suitable method to select the nodes. As we all know, the more sensors arranged in the water distribution network, the more information obtained and the higher the detection accuracy. However, with the increase of the number of sensors, the actual cost is also increased. And when the number reaches a certain degree, the marginal effect of the obtained information is also decreased. Therefore, the sensor arrangement problem often needs to be selected in the precision and cost, and the appropriate number of sensors is selected to achieve the desired result. Figure 1 The influence of the number of sensors on the amount of information obtained and the actual cost is shown.

[0003] How to reasonably arrange the sensors to obtain higher leakage detection accuracy through a small number of sensors is a problem to be solved at present. SUMMARY

[0004] The application aims to provide a water supply network sensor arrangement method based on graph sampling theory, which can realize higher leakage detection accuracy through a small number of sensors by reasonably setting the positions of the sensors in the water supply network.

[0005] The technical scheme provided by the application is as follows:

[0006] A water supply network sensor arrangement method based on graph sampling theory comprises the following steps:

[0007] Establishing a user node undirected graph according to the structure of the water supply network And determining a graph Fourier operator of the user node undirected graph;

[0008] Wherein, is a vertex set of the graph , and N represents the number of user nodes in the water supply network; ε is an edge set of the graph , and W is a weighted adjacency matrix of the graph ;

[0009] Obtaining the user node pressure of the water supply network under different leakage conditions to obtain a pressure sensitivity matrix of the water supply network, and determining useful information in the pressure sensitivity matrix;

[0010] The graph Fourier operator is used to perform Fourier transform on the useful information, to obtain graph Fourier spectrum of each user node under different leakage conditions;

[0011] A spectrum threshold is set, and graph Fourier spectrum greater than or equal to the spectrum threshold is screened to form a new spectrum matrix And the corresponding frequency component of the new spectrum matrix is screened to form a new frequency component matrix U VN ;

[0012] The number of reserved spectrums in the new spectrum matrix is taken as the number of sensor nodes, and an updated signal matrix S is obtained according to the new spectrum matrix and the new frequency component matrix k ;

[0013] Wherein,

[0014] A target function is constructed according to the updated signal matrix:

[0015]

[0016] In the formula, is the minimum mean square estimation of S k Error covariance matrix of M is a set of sensor nodes, h is a currently selected sensor node, u h is the hth column of matrix U VN σ 2 is the variance of noise;

[0017] User nodes satisfying the target function are screened one by one as sensor nodes until the number of sensor nodes is reached;

[0018] A sensor is installed at the sensor node, and the water supply network is monitored.

[0019] Preferably, the spectrum threshold is determined by the following formula:

[0020]

[0021] Wherein, γ is the spectrum threshold, ζ is the weight, N is the total number of nodes, K is the number of sensor nodes to be selected, that is, the number of spectrums greater than the threshold, si is the average spectrum of the original signal, is the average spectrum of the band-limited signal.

[0022] Preferably, the weighted adjacency matrix is:

[0023]

[0024] Wherein,

[0025] In the formula, w ij Let d be the element in the i-th row and j-th column of the weighted adjacency matrix W, representing the weight between node i and node j; ij Let θ be the distance between node i and node j, and θ be a constant.

[0026] Preferably, obtaining the graph Fourier operator includes the following steps:

[0027] Calculate the graph Laplacian matrix L of the undirected graph;

[0028] Where L = DW;

[0029] Eigenvalue decomposition of the graph Laplacian matrix L yields the graph Fourier operator:

[0030] L=UΛU T ;

[0031] Where U is the frequency component matrix, U = [u1, u2, ..., u N ],u1,u2,…,u N These are the frequency components corresponding to each frequency; U T For the graph Fourier operator, U T =U -1 .

[0032] Preferably, the pressure sensitivity matrix is:

[0033]

[0034] Where ΔP = P - P';

[0035]

[0036]

[0037] In the formula, P represents the pressure matrix under normal conditions. P' represents the pressure at the Nth node under normal conditions at time t, and P' represents the pressure matrix under leakage conditions. ΔP represents the pressure at node N at time t when leakage occurs at node k, ΔB represents the pressure change matrix, and ΔB represents the leakage amount.

[0038] Preferably, the useful information in the pressure sensitivity matrix is ​​determined using the following formula:

[0039] S all =S+n;

[0040] Among them, S all Let S be the pressure sensitivity matrix, where S represents useful information and n represents noise.

[0041] Preferably, obtaining the graph Fourier spectrum of each user node under different leakage conditions includes the following steps:

[0042] The useful signal S is subjected to a graphical Fourier transform using the following formula.

[0043]

[0044] In the formula, The Fourier coefficient matrix is ​​shown in the figure. Each column represents the Fourier spectrum of data at different times under different leakage conditions;

[0045] Calculate the average value of the spectrum under different leakage conditions.

[0046]

[0047] in, yes The i-th column, i∈(1,T); The graph Fourier spectrum of each user node under different leakage conditions.

[0048] The beneficial effects of this invention are:

[0049] The sensor placement method for water supply networks based on graph sampling theory provided by this invention introduces graph signal processing into the sensor placement problem of water distribution networks, and analyzes the signal distribution from the graph frequency domain; it can acquire as much network data information as possible with only a few nodes, thereby accurately finding the leak location and achieving the purpose of leak localization. Attached Figure Description

[0050] Figure 1 This is a schematic diagram of the marginal effect of a sensor.

[0051] Figure 2 This is a schematic diagram illustrating the process of re-representing a signal based on a threshold as described in this invention.

[0052] Figure 3 This is a flowchart of the sensor arrangement method for a water supply network based on graph sampling theory as described in this invention.

[0053] Figure 4 This is a plan view of the pipe network used in Experiment Example 1.

[0054] Figure 5(a) shows the spectrum of the signal after graph Fourier transform under different node leakage conditions.

[0055] Figure 5(b) is a graph spectrum diagram when a node leaks.

[0056] Figure 6Figure for the relationship between the spectrum threshold and the number of sensors needed to be arranged for the test example 1.

[0057] Figure 7 Figure for the scores corresponding to different thresholds for the test example 1.

[0058] Figures 8(a)-8(c) Figures for the sensor arrangement node graphs using the three methods respectively for the test example 1.

[0059] Figure 9 Figure for the plan of the pipe network used in the test example 2.

[0060] Figure 10 Figure for the relationship between the spectrum threshold and the number of sensors needed to be arranged for the test example 2.

[0061] Figure 11 Figure for the scores corresponding to different thresholds for the test example 2.

[0062] Figures 12(a)-12(c) Figures for the sensor arrangement node graphs using the three methods respectively for the test example 2. DETAILED DESCRIPTION

[0063] The present application will be further described below with reference to the accompanying drawings so as to enable those skilled in the art to carry out the present application according to the description herein.

[0064] Since the water distribution network is an irregular graph, the signals obtained in the water distribution network cannot be analyzed by traditional signal processing methods. Further, the graph signal processing method is used, the water distribution network is regarded as an irregular graph, each node is regarded as a sensor, and each edge contains the relationship between the sensor node data, which forms a graph signal. Using the graph signal processing data in the water distribution network can better utilize the dependent relationship between the data of different sensors. When the water distribution network needing to be arranged with sensors is abstracted into a graph, the problem of arranging sensors is converted into selecting some points from all the points on the graph, and using these points as the nodes of arranging sensors, which leads to the graph sampling theory.

[0065] As Figure 3As shown, the application provides a sensor arrangement method for water supply network based on graph sampling theory, a graph containing the relationship between nodes is constructed through the actual structure of the water distribution network, the pressure sensor data on the nodes is obtained, the graph spectrum domain distribution of the signal is analyzed, the number of sensors is determined, and then the appropriate nodes are selected as the sensor arrangement nodes through the graph sampling theory. Meanwhile, the application considers the influence of the spectrum of data at different times on the overall spectrum when determining the effective information, and determines the spectrum threshold according to the number of sensors and the information containing leakage. The application selects a small number of nodes by using the graph sampling theory, and arranges sensors at these nodes to obtain the information required by the user to identify and locate the leakage.

[0066] I. Assign weights to the water distribution network

[0067] In the application, the water supply network WDN is regarded as a graph Figure represents an undirected graph with N node or vertex set. is the vertex set of the graph , In the process of processing the graph signal, the signal is often defined on the vertex of the graph. The signal is represented by a mapping , and R represents the real number set. The i-th element of the vector represents the signal value on the i-th vertex. In the application, the signal represents the pressure value. ε is the edge set of the graph , and the set ε represents the adjacent relationship between two nodes. When two nodes are connected, the corresponding edge is placed in the set ε. is the weighted adjacency matrix of the graph , that is, the weight matrix of the graph . w ij represents the weight between node i and node j. Only when node i and node j are connected, w ij is a value other than 0. Otherwise, w ij is 0. And according to the definition of the weight matrix W, the matrix W is a symmetric matrix. That is, W=W T . The weight between two nodes reflects the similarity between the nodes. The greater the weight of the edge between two nodes, the more similar the two nodes are. In the application, a greater weight is assigned to nodes that are distributed close in space, and it is considered that the signals on the two nodes are more close in value. In the application, a Gaussian kernel weighting function is used:

[0068]

[0069] In formula (1), w ij is the i-th row and j-th column element of the weight matrix W, representing the weight between node i and node j. d ijwhere d is the distance between node i and node j, and θ represents a constant that controls the distribution of the weights. The distance between nodes is scaled by the exponential function e to achieve the inverse relationship between distance and weight value. Equation (1) takes into account the spatial factor in the WDN, meaning that the closer the nodes are geographically, the greater the weight value assigned between them.

[0070] II. Obtaining the signal graph frequency spectrum

[0071] After the weights are assigned to the WDN, the signal graph frequency spectrum distribution needs to be obtained. In this process, the graph Fourier transform (GSP) is used. In the definition of the graph Fourier transform, different transform operators can be used, such as the adjacency matrix, the Laplacian matrix, or the normalized Laplacian matrix. In this invention, the Laplacian matrix is used as the operator to define the graph Fourier operator. The Laplacian matrix is defined as follows:

[0072] L = D - W (2)

[0073] In equation (2), D is the degree matrix, which only has non-zero values on the main diagonal, and the rest are zero. The i-th element on the main diagonal is the sum of all weights of the i-th node. W is the weight matrix, and both matrix W and matrix L are N-order square matrices. Since the Laplacian matrix is a real symmetric matrix, the Laplacian matrix is decomposed by eigenvalues:

[0074] L = UΛU T (3)

[0075] where U T = [u1, u2, …, u -1 N]. Define U T as the graph Fourier operator, U = [u1, u2, …, u N N]. Λ = diag(λ1, λ2, …, λ N N) is a diagonal matrix, where the diagonal elements λ i represent the i-th frequency. U represents the frequency component matrix.

[0076] In a real water distribution network (WDN), the water demand at each node cannot be directly measured, which makes it impossible to use water demand as an indicator to reflect the leakage situation. Therefore, in this invention, pressure signals are used to reflect the leakage situation of the nodes. When a node leaks, its corresponding pressure will decrease compared to the pressure without leakage. This invention uses the pressure sensitivity matrix as the signal.

[0077] Using the hydraulic model simulation software EPANET, the required water demand parameters and altitude parameters of each node are set to simulate and obtain the pressure matrix under normal conditions:

[0078]

[0079] In formula (4), represents the pressure of the Nth node at the tth time under normal circumstances. The pressure matrix P has N rows and T=Nxt columns. The water demand of a certain node is increased to simulate the actual leakage, and the pressure matrix when leakage occurs is obtained:

[0080]

[0081] In formula (5), the represents the pressure of the Nth node at the tth time when the kth node leaks. Finally, the pressure change matrix ΔP=P-P' is obtained by formula (4) and formula (5). Divide ΔP by the size of the leakage to obtain the pressure sensitivity matrix S all . As follows:

[0082]

[0083] In formula (6), Δb is the size of the leakage. The pressure sensitivity matrix S all obtained contains noise. S all is re-expressed as:

[0084] S all =S+n (6)

[0085] Where S is useful information and n is noise. The prior information of the noise is known:

[0086] E[nn T ]=Q (7)

[0087] Q is the noise covariance matrix. When the graph Fourier operator and the useful information S are obtained, the distribution of the signal in the graph frequency domain needs to be obtained. At this time, the graph Fourier transform is performed on the signal S:

[0088]

[0089] In formula (9), is the graph Fourier coefficient matrix. Each column of the matrix represents the graph Fourier spectrum of the data at different times under different leakage conditions.

[0090] Three, select the main frequency component

[0091] The present application uses a band-limited signal-based graph sampling method to select nodes, which requires truncation of the signal in the graph frequency domain. Therefore, the main components of the signal in the graph frequency domain need to be selected so that the signal is composed of a limited number of frequency spectra. At the same time, in the graph sampling theory, there is also a similar lower limit in the classical sampling theory: the number of nodes sampled cannot be less than the bandwidth. Therefore, the number of selected frequency spectra is used as the number of selected nodes in the present application.

[0092] The graph Fourier transform enables us to represent the same signal in the vertex domain and the graph frequency domain, respectively. In the vertex domain, the signal is distributed on the vertices of the graph. In the graph frequency domain, the signal is represented as the product of frequency components and spectra. The signal is processed in the graph frequency domain to select frequency components that can represent the main components of the signal. The selected frequency components need to be determined by the spectral values in the graph frequency domain. At this time, a spectral threshold γ needs to be defined to determine the selected frequency components. In the process of determining the frequency components, since the graph spectrum at each time under each leakage condition is different, we need to consider the influence of different data on the overall. Therefore, the average value of the spectrum under different leakage conditions is calculated by using the average value of the spectrum

[0093]

[0094] where the vector is the i-th column of . When the average spectral representation of all data is determined , the main frequencies of the signal are determined according to the threshold γ. The selection of the threshold is very critical. Selecting different thresholds will result in different numbers of sensor nodes and the ratio of the retained signal energy to the total signal energy. Therefore, in the process of determining the threshold, both factors need to be considered. At the same time, the main signal energy obtained accounts for as much of the total signal energy as possible, and the number of sensor nodes deployed is as small as possible. The original data spectrum is processed using the following formula:

[0095]

[0096] The spectrum greater than the threshold is retained, and the spectrum less than the threshold is set to 0. The spectrum greater than or equal to the threshold is recorded, and the value of i in the set O is recorded. The number of elements in the set O is the number of selected sensors. |O|=K. The threshold formula used in the present application is as follows:

[0097]

[0098] where ζ is the weight to adjust the influence of the two parts on the threshold formula. Formula (12) considers the influence of the threshold on the ratio of the selected spectral components to the total spectral components and the number of selected sensors. The selected threshold retains as many spectral components as possible while selecting a small number of sensor nodes.

[0099] The set O contains the rows of the spectral matrix to be selected and the columns of the frequency component U to be selected. According to the elements in the set O, the rows of the matrix are selected to form a new spectral matrix Meanwhile, the columns of the matrix U are selected to form a matrix U VN .

[0100] The original signal is re-expressed as:

[0101]

[0102] The matrix U VN is the main frequency component of the signal, and the matrix is the spectrum corresponding to the main frequency component. S k is a sparse signal in the graph frequency domain with only k frequencies, which contains the main components of the original signal. As shown in Figure 2 is an example of the process of selecting spectral components by threshold. Figure 2 The frequencies corresponding to the spectral components greater than the threshold are selected in the graph, i.e., the dashed blocks in the graph. Since the frequency and the spectral component are in a one-to-one correspondence, the frequency is selected, and the spectral component is selected. The solid blocks are the selected spectral components and spectral coefficients. Finally, the signal S VN containing the main information is obtained by re-point multiplying the matrix U and the matrix k .

[0103] Four, node

[0104] Next, the graph sampling theory is used to realize the arrangement of sensor nodes. The graph sampling theory extracts the main components of the signal according to the graph Fourier transform, selects appropriate nodes, and makes a small amount of data better recover the signal before sampling. When sampling the signal in the time domain, i.e.

[0105]

[0106] where C ∈ {0, 1} K×N is the sampling matrix, is the sampled signal value, S all is the original signal without truncation. Because the graph sampling theory based on band-limited signals is used, a sparse signal S k with only a limited number of spectrums is used instead of S all to obtain the nodes:

[0107]

[0108] In equation (15), Q is the covariance matrix of the sampled noise. U k is the minimum mean square estimate of the signal S is the Bayes counterpart of the normal equation in the Gauss-Markov theorem.

[0109] where the estimated signal S is expressed as:

[0110]

[0111] In equation (15), Q is the covariance matrix of the sampled noise. U k MN VN is the sampled matrix. U MN is the sub-matrix consisting of the rows of the matrix U VN with the indices of the elements in the set M.

[0112] The criterion for selecting a node is to minimize the reconstruction error of the signal after reconstruction using a small number of nodes compared to the node signal before sampling, i.e., to minimize the mean square error of . The mean square error is defined as the trace of the error covariance matrix, so the node task is to minimize the trace of the matrix Υ:

[0113]

[0114] In equation (17), M is the set of nodes where sensors are placed, i.e., the selected node set. By minimizing the trace of the error covariance matrix, the nodes where sensors are placed can contain more information of the signal, and the purpose of water supply network leak detection can be achieved using a small number of sensors.

[0115] To achieve the minimization of the error covariance matrix, a greedy algorithm is used. The greedy selection steps are:

[0116] (1) Start from the first initial solution in the candidate set;

[0117] (2) Evaluate whether the current node meets the minimum error through the objective function (17), and select the current best user node as the placement position of the pressure sensor to put into the set M;

[0118] (3) Remove the selected nodes from the candidate set, and repeat steps (1) and (2) until a given number of sensors k is reached, i.e., |M| = K. The specific process is shown in Figure 1 ; ​​

[0119] (4) Finally, the sensor set M is obtained.

[0120] The greedy algorithm is used to select the nodes that meet the criteria by iterating all potential nodes according to certain criteria. The final target node placement of the sensor is formed by continuous iteration, and the goal is finally achieved. Direct iteration of formula (17) will produce a large amount of calculation, so formula (17) is rewritten as:

[0121]

[0122] A function is defined to minimize the maximum. Next, the marginal gain of the sensor placement node is obtained by the greedy method:

[0123]

[0124] In formula (19), h is the selected sensor node, u h is the hth column of matrix U VN , and σ 2 is the variance of noise. By calculating the marginal gain of the node, the node where the sensor needs to be placed is selected, and the iteration is stopped until the number of sensors to be placed is selected, and the final node set is obtained. Using the marginal gain as the criterion, the overall calculation cost is reduced. The whole process is shown in Figure 3 .

[0125] The simulation experiment in the application is carried out using EPANET software, and two networks are selected to test the proposed method. The actual water supply network is simulated by EPANET2.0 Chinese version, and the proposed method is compared with several other methods. All work in this experiment is carried out on a notebook computer configured as Intel(R) Core(TM) i7-7700HQ CPU@2.50[GHz], 8[GB] RAM, NVIDIA GeForce GTX 1050Ti, and the MATLAB version used is MATLAB2020b. By inputting necessary parameters under different conditions, data is created to evaluate the method of selecting pressure sensor placement. Through the experiment, the influence of different spectrum thresholds on the number of selected pressure sensors is studied. The method proposed in the application is compared with the methods of compressed sensing and clustering analysis.

[0126] Three evaluation indexes are used in the application. The first evaluation index is RMSE (Root Mean Square Error), which is used to evaluate the selected sensor placement node.

[0127]

[0128] After selecting the nodes, we need to reverse the node selection process, using a few nodes to recover all nodes. Once the recovered nodes are obtained, we use RMSE to calculate the original signal and the recovered signal to determine the effectiveness of the data recovery.

[0129] The second metric is the pipe distance Pd. Since multiple pipe paths may connect the i-th and j-th vertices, this invention uses the shortest path to represent the pipe distance between two nodes. P = {P(1), P(2), P(3), ..., P(q)} represents the set of paths connecting node i and node j.

[0130]

[0131] Among them, e i It is the path between two adjacent nodes, l i Represents path e i The pipe distance is used to represent the distance between the predicted leak node and the actual leak node on the pipe, and is used to evaluate the detection capability of the deployed sensor nodes for the entire network.

[0132] The third indicator is the straight-line distance Ld, which is the straight-line distance between the predicted leak location and the actual leak location.

[0133]

[0134] Where (x,y) are the coordinates of the predicted leak location, and (x',y') are the coordinates of the actual leak location.

[0135] Experimental Example 1

[0136] Experiment 1 uses an example from EPANET. For example... Figure 4 As shown, the water supply network includes 92 nodes, 117 pipes, 2 reservoirs, 3 water tanks, and 2 water pumps. The total water demand is 680.15 L / s. An additional water demand is added to a node to simulate a leak, with the added demand being 2% of the total demand. The added leak rate is 12 L / s, the sampling time is 1 hour, and each different leak duration lasts for 4 hours. For example, adding an additional water demand to node 12 simulates a leak at node 12. The weights are constructed using the Euclidean distance between nodes. Figure 5(a) shows the spectrum of the signal after graphical Fourier transform under different node leak conditions. Figure 5(a) shows that most of the signal energy in the frequency domain is concentrated in the low-frequency part. Figure 5(b) is the graphical spectrum when node 15 leaks. The phenomenon of energy mainly distributed in the low frequency can also be seen from the graphical spectrum of a single node leak.

[0137] Figure 6 This relates to the selected spectral threshold and the number of sensors. For example...Figure 6 As shown in the figure, the number of selected sensors is decreasing as the threshold increases. Figure 7 The optimal threshold selected by formula (12) is shown in the figure. We can see from the figure that as the weight λ increases, the optimal threshold also increases. The greater the threshold, the fewer the selected nodes. This is because as the weight λ increases, the latter part of formula (12) plays a more important role in the formula than the former part. The latter part of formula (12) is to make the number of selected nodes smaller, so the optimal threshold increases. In this experiment, ζ is 1.4.

[0138] The comparison of the three methods on the same data set is shown in Table 1. The three methods shown in Table 1 show the number of selected sensor nodes, the mean square error, and the distance between the judged leak location and the true leak location. From the data, we can see that the method of the present application has a better effect in the evaluation standard on the premise of almost the same number of selected sensors compared with other methods. The sensor nodes arranged by the three methods and the judged leak location are shown in Figure 8. The rectangle in the figure is the actual leak location, the circle is the judged leak location, and the pentagram is the sensor node. In Figure 8(a), the method of the present application is used to obtain the figure, and the obtained sensor nodes are {1, 11, 17, 24, 26, 28, 34, 64, 66, 68, 77, 92}, and the judged leak node is node 13. In Figure 8(b), the method of cluster analysis is used, and the obtained sensor nodes are {4, 11, 14, 29, 34, 50, 51, 54, 55, 56, 84, 88, 89, 90, 92, 85}, and the judged leak location is 79. In Figure 8(c), the method of compressed sensing is used, and the obtained sensor arrangement nodes are {3, 5, 16, 21, 24, 30, 44, 45, 47, 53, 56, 58, 59, 66, 77, 90}, and the judged leak location is 38. The recovered data using the nodes selected by our method has a better effect of recovering all node data compared with the other two methods, and the mean square error of the recovered data and the original data is smaller. At the same time, compared with the other two methods, the sensor position obtained by our method to judge which node leaks has a shorter distance to the true leak node. In Figure 7 The mean square error obtained by comparing the data recovered by the method proposed by us with the real data for three different time leak data is shown in Table 2.

[0139] Table 1 Comparison of three methods in test example 1

[0140]

[0141] Test example 2

[0142] The water supply network used in Experiment Example 2 is based on the water supply system near Harrisburg, Pennsylvania, consisting of 290 pipes, 262 nodes, a reservoir, and a pump. The 262 nodes were selected as candidate nodes, with a total water demand of 929 L / s. The water supply network structure is shown below. Figure 9 As shown.

[0143] Leaks were introduced at different nodes with a leakage rate of 19.6 L / s for 4 hours to simulate real-world leakage data. The pressure sensitivity of each node was measured hourly, and the pressure values ​​under leakage conditions at all nodes were used to construct a pressure sensitivity matrix. Figure 10 This relates to the spectral threshold and the number of sensors required. As the threshold increases, the number of sensors decreases. Figure 11 This illustrates the relationship between three different weighted thresholds and their corresponding scores. From... Figure 11 We can also observe that the optimal threshold increases with the increase of the weight λ. In this experiment, the parameter ζ is set to 1.3.

[0144] In Fig. 12 are sensor arrangement diagrams of three methods, the actual leak location is node 150, i.e. the rectangle in the three diagrams. The circles in Fig. 12 are the leak locations judged by the three methods respectively, and the pentagons are the sensor arrangement nodes. In Fig. 12(a) is the sensor arrangement node diagram obtained by using the method proposed by us, the obtained sensor arrangement nodes are {8, 33, 70, 73, 100, 117, 128, 130, 141, 150, 151, 190, 192, 202, 207, 211, 219, 227, 236, 242, 252, 253}, and the judged leak location is 141. In Fig. 12(b) is the sensor arrangement node diagram obtained by using the method of cluster analysis, the sensor arrangement nodes are {38, 60, 62, 77, 111, 112, 113, 114, 152, 160, 1602, 1605, 179, 182, 241, 2415, 242, 243, 246, 262, 266, 273, 2732, 2735, 275, 276, 2755, 2756, 287, 2875, 290, 291}, and the judged leak location is node 3. Fig. 12(c) is the sensor arrangement diagram obtained by using the method of compressed sensing, the node set is {3, 42, 455, 56, 58, 59, 84, 115, 119, 123, 1365, 140, 142, 149, 150, 155, 157, 159, 1605, 1612, 1634, 164, 172, 178, 191, 192, 196, 2054, 2082, 226, 229, 240, 242, 265, 267, 2732, 274, 278, 281, 291}, and the judged leak location is node 62. In Table 2 are the comparison results of the three methods, it can be seen that the method proposed by us uses the least number of sensors, and the mean square error is the smallest, which shows that our method can better restore the pressure data and thus better judge the leak location. The straight-line distance between the leak location judged by the sensor nodes obtained by using the method proposed by us and the actual leak location and the distance of the pipeline are also smaller than those of the other two methods, which shows that the sensors selected by our method are also more accurate in judging the leak location.

[0145] Table 2 Comparison results of three methods in test example 2

[0146]

[0147] In the present invention, two experiments respectively use 92 nodes and 262 nodes of water distribution network, and use the pressure sensitivity matrix when leaking as data. Because the graph sampling theory based on band-limited signal is used to obtain the nodes of arranging sensors, the threshold is needed to truncate the graph spectrum of the original signal. Further, a formula for selecting the optimal threshold is proposed. And the change of the optimal threshold under different weights is compared. Through experimental analysis, it can be seen that compared with the other two methods, the method of the present invention can more accurately restore the signal using a small amount of node information. And it is also better than the other two methods in judging the leak location. In the first water distribution network, the mean square error is 102.7. In the second water distribution network, the mean square error is 488.2. In terms of judging the leak location, it also has better effect than the other two methods. In the water distribution network of test case 1, the pipe distance is increased by 492 meters and 2060.2 meters respectively. The straight line distance is increased by 492.3 meters and 2018.9 meters respectively. In the water distribution network of test case 2, the pipe distance is increased by 462 meters and 3305 meters respectively. The straight line distance is increased by 943.4 meters and 3268.7 meters respectively.

[0148] In the process of obtaining the final sensor placement node, the present invention selects the main frequency component from the graph frequency domain of the original signal. In order to change the original signal into a signal with limited bandwidth, an optimal threshold selection formula is proposed. The new signal contains most of the information of the original signal. By balancing the selected sensor nodes and the discarded information, the most suitable threshold is selected. In addition, it is required to obtain the maximum amount of information using a small number of nodes. At this time, a large amount of information is reflected by the RMSE between the restored signal and the original signal. Through two experiments, the effects of the three methods are compared. And the results show that the method of the present invention is better than the other methods in restoring data. Test case 1 and test case 2 reflect the influence of the method of the present invention on different sizes of WDN. The method proposed in the present invention also has good effect in monitoring leakage. A small amount of sensors can accurately determine the leak location, effectively improving the accuracy of leak location.

[0149] Although the embodiments of the present invention have been disclosed as above, it is not limited to the application listed in the specification and the embodiments, and can be fully applied to various fields suitable for the present invention, and other modifications can be easily realized by those skilled in the art, and therefore the present invention is not limited to specific details and the figures shown and described herein, without departing from the general concept defined by the claims and the equivalent scope.

Claims

1. A sensor arrangement method for a water supply network based on graph sampling theory, characterized in that, include: Establish an undirected graph of user nodes based on the water supply network structure. And determine the graph Fourier operator of the undirected graph of the user nodes; in, It is a picture The set of vertices, N represents the number of user nodes in the water supply network; ε is the graph. The set of edges of a graph, W is a graph. The weighted adjacency matrix; The pressure of user nodes in the water supply network under different leakage conditions is obtained to obtain the pressure sensitivity matrix of the water supply network, and useful information in the pressure sensitivity matrix is ​​determined. The useful information is subjected to Fourier transform using the graph Fourier operator to obtain the graph Fourier spectrum of each user node under different leakage conditions; Set a spectral threshold, and filter out the graph Fourier spectra that are greater than or equal to the spectral threshold to form a new spectral matrix. And select the frequency components corresponding to the new spectrum matrix to form a new frequency component matrix U. VN ; The number of retained spectra in the new spectrum matrix is ​​used as the number of sensor nodes, and the updated signal matrix S is obtained based on the new spectrum matrix and the new frequency component matrix. k ; in, Construct the objective function based on the updated signal matrix: In the formula, Υ represents S. k minimum mean square estimate The error covariance matrix, M is the set of sensor nodes, h is the currently selected sensor node, u h For matrix U VN The j-th column, σ 2 The variance of the noise; One by one, user nodes that satisfy the objective function are selected as sensor nodes, until the number of sensor nodes is reached; Sensors are installed at the sensor nodes to monitor the water supply network; The spectral threshold is determined by the following formula: Where γ is the spectral threshold, ζ is the weight, N is the total number of nodes, and K is the number of candidate sensor nodes, i.e., the number of spectra greater than the threshold. The average spectrum of the original signal. The average spectrum of the band-limited signal; The pressure sensitivity matrix is: Where, ΔP=P-P'; In the formula, P represents the pressure matrix under normal conditions. P' represents the pressure at the Nth node under normal conditions at time t, and P' represents the pressure matrix under leakage conditions. ΔP represents the pressure at node N at time t when leakage occurs at node k, ΔB represents the pressure change matrix, and ΔB represents the leakage amount.

2. The sensor arrangement method for a water supply network based on graph sampling theory according to claim 1, characterized in that, The weighted adjacency matrix is: in, In the formula, w ij Let d be the element in the i-th row and j-th column of the weighted adjacency matrix W, representing the weight between node i and node j; ij Let θ be the distance between node i and node j, and θ be a constant.

3. The sensor arrangement method for a water supply network based on graph sampling theory according to claim 1 or 2, characterized in that, Obtaining the graph Fourier operator includes the following steps: Calculate the graph Laplacian matrix L of the undirected graph; Where L = DW; Eigenvalue decomposition of the graph Laplacian matrix L yields the graph Fourier operator: L=UΛU T ; Where U is the frequency component matrix, U = [u1, u2, ..., u N ],u1,u2,…,u N These are the frequency components corresponding to each frequency; U T For the graph Fourier operator, U T =U -1 D is the degree matrix, W is the weight matrix, and Λ is the diagonal matrix.

4. The sensor arrangement method for a water supply network based on graph sampling theory according to claim 3, characterized in that, The useful information in the pressure sensitivity matrix is ​​determined using the following formula: S all =S+n; Among them, S all Let S be the pressure sensitivity matrix, where S represents useful information and n represents noise.

5. The sensor arrangement method for a water supply network based on graph sampling theory according to claim 4, characterized in that, Obtaining the graph Fourier spectrum of each user node under different leakage conditions includes the following steps: The useful information S is subjected to a graphical Fourier transform using the following formula. In the formula, The Fourier spectrum matrix of the graph. Each column represents the Fourier spectrum of data at different times under different leakage conditions; Calculate the average value of the spectrum under different leakage conditions. in, yes The i-th column, i∈(1,T); The graph Fourier spectrum of each user node under different leakage conditions.

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