Multi-sensor distributed online checking method based on Bayesian decomposition

Abstract

The invention discloses a Bayesian decomposition-based multi-sensor distributed online checking method, which comprises the following steps of: decomposing prior distribution obeyed by a node water demand at the current moment into K local prior probability distributions by using Bayesian decomposition, and decomposing a local likelihood function obeyed by monitoring data at the current moment into K local likelihood functions; combining the local prior probability distribution and the local likelihood function into K local posterior probability distributions, converting each local posterior probability distribution into a local check objective function, iteratively solving a water demand value at the current moment, and stopping iteration until the local check objective function value is minimum, obtaining a local check water demand value and a local check covariance matrix; and based on the local check covariance matrix, the local check water demand value is endowed with a weight, and then the sum is added to obtain the global check water demand. According to the method, centralized checking is decomposed into local checking, and the checking efficiency of the node water demand is improved while the checking precision is guaranteed.

Description

Technical Field

[0001] The invention belongs to the field of pipe network hydraulic modeling, and in particular relates to a multi-sensor distributed online calibration method based on Bayesian decomposition. Background Art

[0002] The hydraulic model of the pipe network is widely used in the system analysis and operation control of the pipe network, and is the basis and core for the real-time simulation and control of the pipe network. The online model uses the online monitoring data to invert the node water demand in real time, and then dynamically simulates the real operation status of the pipe network. With the development of sensor information technology, the use of a large amount of monitoring data to calculate the node water demand in real time (node ​​water demand verification) has gradually become a hot spot and difficulty in the online simulation of the pipe network.

[0003] Real-time verification of node water demand can be regarded as an optimization process. By repeatedly adjusting the node water demand, the deviation between the model simulation value and the online data is minimized. Data assimilation considering various uncertainties has been used for parameter verification in model simulation. At present, the data assimilation methods for water supply network parameter verification mainly include sampling-based analysis methods and numerical solution methods. Sampling-based methods require frequent evaluation of the probability density of samples, which is very time-consuming, especially in the face of water supply networks with large parameter scales, and have never been the first choice for modeling engineers. Therefore, numerical solution methods with high computational efficiency (such as Gaussian approximation method) have received widespread attention.

[0004] The existing numerical solution method belongs to the centralized verification method of monitoring data: after receiving the monitoring data uploaded by the sensor, all the data are processed centrally, and the node water demand is iteratively solved through nonlinear optimization method.

[0005] The invention patent with the publication number CN113849943A discloses a method for verifying water demand of water supply network nodes coupled with pressure prior information, including the following steps: (1) predicting the prior water demand and its covariance matrix through the water demand prediction function, and obtaining the prior probability distribution of the node water demand; (2) obtaining the prior probability distribution of the node pressure through on-site survey of the building and floor information in the user's area; (3) obtaining the monitoring data of the network pressure and flow through the network pressure and flow sensors and their covariance matrix, and obtaining the likelihood function of the monitoring data; (4) establishing a verification objective function based on Bayes' theorem; (5) linearizing the objective function, obtaining the iteration step of the node water demand, and solving the node water demand through continuous iteration. The present invention establishes a prior probability model of uniform distribution of node pressure to ensure that the pressure value calculated by the model is distributed within a reasonable range, thereby improving the verification accuracy of the node water demand and providing a scientific basis for hydraulic modeling of water supply network.

[0006] The patent application with publication number CN116822244A discloses a water demand verification method for a water supply network based on a truncated normal distribution, which belongs to the field of hydraulic modeling of a water supply network. The method comprises the following steps: (1) using a truncated normal distribution, constructing a truncated prior probability distribution of water demand at a node in a water supply network and a truncated likelihood function of monitoring data; (2) based on Bayesian theorem, integrating the truncated prior probability distribution and the truncated likelihood function, constructing a truncated posterior probability distribution, and establishing a verification objective function by maximizing the truncated posterior probability distribution; (3) using the Newton iteration method to solve the objective function, obtaining the water demand adjustment amount of each node in the water supply network, and iteratively solving the water demand of each node in the water supply network.

[0007] The efficiency of the node water demand verification of the above two patents needs to be further improved. The internal structure of the centralized verification algorithm determines that the recursive fusion of monitoring data can only be done through serial calculations. The algorithm is difficult to parallelize and coordinate on multiple computers, which greatly limits the improvement of computing efficiency. When the number of pipe network nodes is greater than 5,000, its efficiency is difficult to meet the requirements of real-time modeling.

[0008] In summary, the rapid verification of node water demand is crucial for the real-time modeling and management of water supply networks. Therefore, breaking the dependence between multiple sensors and developing a distributed verification theory for water supply network models is an important idea to further solve this problem. Summary of the invention

[0009] The present invention provides a multi-sensor distributed online calibration method based on Bayesian decomposition, which decomposes centralized calibration into local calibration, thereby improving the calibration efficiency of node water demand while ensuring the calibration accuracy.

[0010] The present invention provides a multi-sensor distributed online calibration method based on Bayesian decomposition, comprising:

[0011] By using Bayesian decomposition, the prior distribution of the node water demand at the current moment is decomposed into K local prior probability distributions, and the likelihood function of the monitoring data at the current moment is decomposed into K local likelihood functions;

[0012] The local prior probability distribution and the local likelihood function are combined into K local posterior probability distributions, each local posterior probability distribution is converted into a local calibration objective function, and the water demand value at the current moment is iteratively adjusted until the local calibration objective function value is minimized or the maximum number of iterations is reached, and the iteration is stopped to obtain the local calibration water demand value and the local calibration covariance matrix;

[0013] Based on the local verification covariance matrix, weights are assigned to the local verification water demand values ​​and then added up to obtain the global verification water demand.

[0014] Preferably, the prior distribution of the current moment predicted water demand is decomposed into K local prior probability distributions, including:

[0015] X t =f(X t-1 ,X t-2 ,…X1)+ε

[0016] X t ~N(X t |X t|t-1 ,P t|t-1 )

[0017]

[0018] Among them, f() is the prediction function, the water demand of the node at the current time t is X t The node water demand history value X can be used through the prediction function f() t-1 ,X t-2 ,…X1 is predicted, ε is the prediction error, which obeys the normal distribution N(ε|0,P t|t-1 ), P t|t-1 is the covariance matrix of the node water demand at the time before the current time to estimate the water demand at the current time t, X t|t-1 is the mean of the prior distribution, N(X t |X t|t-1 ,P t|t-1 ) is the prior distribution of the node water demand at time t, N(X t |X t|t-1 ,KP t|t-1 ) is the local prior probability distribution of the node water demand at time t, the prior distribution is decomposed into K identical local prior probability distributions, and j is the index of the local prior probability distribution.

[0019] Preferably, the likelihood function obeyed by the monitoring data at the current moment is decomposed into K local likelihood functions, including:

[0020] Y t =g(X t )+ω

[0021] Y t ~N[Y t |g(X t ),R]

[0022]

[0023] Among them, Y t is the monitoring data at the current time t, g(X t) is the simulated value obtained by taking the node water demand at the current time t as input through the pipe network model, ω is the noise, and the noise ω obeys the normal distribution N(ω|0,R), R is the noise covariance matrix, and the monitoring points m in the pipe network are divided into K groups, each group corresponds to a local likelihood function, that is, the monitoring data of the i-th group corresponds to the i-th local likelihood function, and the i-th group of monitoring points provides l i monitoring data, i.e., the monitoring data of group i

[0024] Preferably, each local prior probability distribution is multiplied by the corresponding local likelihood function to obtain the corresponding local posterior probability distribution.

[0025] Preferably, the local prior probability distribution of the node water demand at the current time t and the i-th local likelihood function are combined to obtain the corresponding i-th local posterior probability distribution P(X t |Y t (i)) is:

[0026] P(X t |Y t (i)) = N(X t |X t|t-1 ,KP t|t-1 )N[Y t (i)|g i (X t ),R(i)]

[0027] Among them, X t is the node water demand at the current time t, Y t (i) is the monitoring data corresponding to the i-th local likelihood function at the current time t, P t|t-1 is the covariance matrix of the node water demand at the time before the current time to estimate the water demand at the current time t, X t|t-1 is the mean of the prior distribution, g i (X t ) is the simulated value of the pipe network model output corresponding to the ith local likelihood function at the current time t, and R(i) is the noise covariance matrix corresponding to the ith local likelihood function.

[0028] Preferably, each local posterior probability distribution is converted into a local calibration objective function, wherein the i-th local calibration objective function is as follows:

[0029]

[0030] Among them, the node water demand of the i-th local verification at the current time t is iteratively adjusted until the i-th local verification objective function value is minimum or the iteration is stopped after the maximum number of iterations is reached, and the node water demand value of the i-th local verification and the local verification covariance matrix are obtained.

[0031] Preferably, iterating the node water demand of the i-th local check at the current time t includes:

[0032]

[0033] P t (i) = KP t|t-1 -KP t|t-1 {R(i)+K 2 G(i)P t|t-1 [G(i)] T} -1 G(i)P t|t-1

[0034]

[0035] in, is the node water demand of the k+1th iteration of the ith local check at the current time t, is the node water demand adjustment of the i-th local check in the k-th iteration, is the Jacobian matrix corresponding to the i-th group of monitoring data; k is the number of iterations, and MK is the maximum allowed number of iterations.

[0036] Preferably, weights are assigned to local check water demand values ​​based on the local check covariance matrix and then added to obtain the global check water demand, including:

[0037]

[0038]

[0039] in, is the inverse of the covariance matrix of the i-th local calibration at the current time t, X t (i) is the water demand value of the i-th local calibration at time t before.

[0040] Compared with the prior art, the present invention has the following beneficial effects:

[0041] The present invention designs a multi-sensor distributed online calibration method based on Bayesian decomposition. First, based on Bayesian decomposition, the present invention decomposes the prior distribution and likelihood function as a whole into the form of multiplication of K groups of local prior probability distributions and K groups of local likelihood functions, thereby realizing the basis of centralized calibration distributed processing; the local prior probability distribution and the local likelihood function are multiplied in pairs to obtain the local posterior probability distribution, which is maximized to solve the node water demand and covariance matrix. When the constructed local posterior probability distribution is linearly iteratively solved, the covariance matrix has a smaller dimension, thereby realizing efficient calibration calculation; the local calibration results are weighted fused to obtain the global calibration results, and the fusion process realizes the global readjustment of the calibration results, reflects the hydraulic information of the entire pipe network, and adjusts the calibration results to avoid falling into the local optimum. The present invention is applied to a large pipe network, and the application results prove the superiority of the present invention in the calibration of node water demand. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 A flow chart of local verification and global verification shown in an embodiment of the present invention;

[0043] Figure 2 A water supply network model diagram shown in an embodiment of the present invention;

[0044] Figure 3 The comparison of centralized and distributed calibration efficiencies under different numbers of sensors when the local calibration is 3 in an embodiment of the present invention;

[0045] Figure 4 The cumulative probability distribution of pressure errors of centralized and distributed calibrations with different numbers of sensors is shown in the embodiment of the present invention when the number of local calibrations is 3. DETAILED DESCRIPTION

[0046] The purpose of the specific embodiment of the present invention is to develop a multi-sensor distributed online calibration method based on Bayesian decomposition. To describe the implementation of the method in detail, the present invention is applied to a large-scale benchmark pipe network. The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings.

[0047] The specific embodiment of the present invention provides a multi-sensor distributed online calibration method based on Bayesian decomposition, such as Figure 1 As shown, including:

[0048] By using Bayesian decomposition, the prior distribution of the node water demand at the current moment is decomposed into K local prior probability distributions, and the likelihood function obeyed by the monitoring data at the current moment is decomposed into K local likelihood functions.

[0049] In a specific embodiment, the specific process of decomposing the prior distribution of the current moment predicted water demand into K local prior probability distributions is as follows:

[0050] X t =f(X t-1 ,X t-2 ,…)+ε#(1)

[0051] X t ~N(X t |X t|t-1 ,P t|t-1 )#(2)

[0052]

[0053] Among them, f() is the prediction function, the water demand of the node at the current time t is X t The node water demand history value X can be used through the prediction function f() t-1 ,X t-2 ,…X1 is predicted, ε is the prediction error, which obeys the normal distribution N(ε|0,P t|t-1 ), P t|t-1 is the covariance matrix of the node water demand at the time before the current time to estimate the water demand at the current time t, X t|t-1 is the mean of the prior distribution, N(X t |X t|t-1 ,P t|t-1 ) is the prior distribution of the node water demand at time t, N(X t |X t|t-1 ,KP t|t-1 ) is the local prior probability distribution of the node water demand at the current time t. The prior distribution is decomposed into K identical local prior probability distributions. j is the index of the local prior probability distribution. Decomposing it into the same local prior probability distribution is beneficial to the construction of the local posterior distribution and speeds up the local verification.

[0054] In a specific embodiment, the local likelihood function obeyed by the monitoring data at the current moment is decomposed into K local likelihood functions. The specific process is as follows:

[0055] Y t =g(X t )+ω#(4)

[0056] Y t ~N[Y t |g(X t ),R]#(5)

[0057]

[0058] Among them, Y t is the monitoring data at the current time t, g(X t) is the simulated value obtained by taking the node water demand at the current time t as input through the pipe network model, ω is the noise, and the noise ω obeys the normal distribution N(ω|0,R), R is the noise covariance matrix, and the monitoring points m in the pipe network are divided into K groups, each group corresponds to a local likelihood function, that is, the monitoring data of the i-th group corresponds to the i-th local likelihood function, and the i-th group of monitoring points provides l i monitoring data, i.e., the monitoring data of group i

[0059] The purpose of dividing the monitoring points m in the pipeline network into K groups in the specific embodiment of the present invention is, on the one hand, to be able to combine with K local prior probability distributions in pairs to form K local posterior probability distributions, and on the other hand, since each group has multiple monitoring points, more accurate local verification can be achieved.

[0060] In a specific embodiment, the i-th local posterior probability distribution P(X) obtained by combining the local prior probability distribution of the node water demand at time t and the i-th local likelihood function is t |Y t (i)) is:

[0061] P(X t |Y t (i)) = N(X t |X t|t-1 ,KP t|t-1 )N[Y t (i)|g i (X t ),R(i)]

[0062] Among them, P(X t |Y t (i)) represents the local posterior probability distribution. K local prior probability distributions are combined with K local likelihood functions in pairs to obtain K local posterior probability distributions P(X t |Y t (i)). Each local posterior probability distribution is independent of each other, and by maximizing the local posterior P(X t |Y t (i)) Get the node water demand X t (i) and the covariance matrix P t (i), this process is called local verification.

[0063] In a specific embodiment, each local posterior probability distribution is converted into a local verification objective function, that is, the i-th local posterior probability distribution corresponds to the i-th local verification objective function, wherein the i-th local verification objective function is as follows:

[0064]

[0065] Among them, the node water demand of the i-th local verification at the current time t is iterated until the i-th local verification objective function value is minimized or the maximum number of iterations is reached, and the node water demand value of the i-th local verification and the local verification covariance matrix are obtained.

[0066] The main process of the local verification calculation in the specific embodiment of the present invention is: find a water demand X t (i) so that it is in the local posterior distribution P(X t |Y t (i)) has the largest probability density value, maximizing P(X t |Y t (i)) is equivalent to solving the above objective function. The objective function is nonlinear and needs to be solved iteratively.

[0067] In a specific embodiment, the specific process of iterating the node water demand of the i-th local check at the current time t is as follows:

[0068]

[0069] P t (i) = KP t|t-1 -KP t|t-1 {R(i)+K 2 G(i)P t|t-1 [G(i)] T} -1 G(i)P t|t-1 #(12)

[0070]

[0071] in, is the node water demand of the k+1th iteration of the ith local check at the current time t, is the node water demand adjustment of the i-th local check in the k-th iteration, is the Jacobian matrix corresponding to the i-th group of monitoring data; k is the number of iterations, and MK is the maximum allowed number of iterations.

[0072] In a specific embodiment, weights are assigned to local check water demand values ​​based on the local check covariance matrix and then added to obtain the global check water demand, including:

[0073]

[0074] in, is the inverse of the covariance matrix of the i-th local calibration at the current time t, X t(i) is the water demand value of the ith local calibration at the previous t time. The global posterior probability distribution can be obtained by combining multiple local posterior probability distributions. The mean and covariance matrix of the global posterior probability distribution involve the weighted sum of the mean and covariance matrix of each posterior distribution. The inverse of the covariance matrix of the global posterior probability distribution is the sum of the inverses of the covariance matrices of each distribution, and the mean is the weighted mean. The final joint distribution is still a normal distribution, reflecting the comprehensive characteristics of all the original distributions.

[0075] like Figure 2 A schematic diagram of a pipe network is given, which consists of 26735 nodes and 12 water sources. In order to evaluate the impact of the number of sensors on the efficiency and accuracy of distributed calibration of the water supply network, the present invention designs a simulation experiment. The experiment simulates ten groups of pressure sensors with different numbers: 100, 500, 800, 1000, 1500, 2000, 3000, 4000, 5000 and 6000, whose positions are randomly selected. In addition, the results of the specific embodiment of the present invention will be compared with a traditional water supply network node water demand calibration method under a numerical-based data assimilation framework to discuss the potential potential of the present invention.

[0076] Table 1 shows the algorithm running time of centralized and distributed calibration, with the number of local calibrations K ranging from 2 to 6 in a 10-sensor deployment scenario. As can be seen from the table, the efficiency of distributed calibration is very obvious compared to centralized calibration, and the advantage of distributed calibration increases with the increase of the number of local calibrations. Distributed calibration is most efficient when the number of local calibrations ranges from 3 to 5. In addition, the efficiency of distributed calibration does not always increase with the number of local calibrations; it depends on the specific number of deployed sensors. Specifically, when the number of sensors is 100, 2 local calibrations can maximize the computational efficiency. For sensor counts between 500 and 1000, a distributed scheme with 3 local calibrations is the most appropriate. When the number of sensors is between 1500-3000 and 4000-6000, the optimal distributed calibration scheme should use 4 and 5 local calibrations, respectively.

[0077] Table 1 Algorithm running time of centralized and distributed calibration

[0078]

[0079] Figure 3A comparison between the efficiency of distributed and centralized calibration is given when the number of local calibrations is 3. The time consumption of centralized calibration increases exponentially with the increase in the number of measurements, reaching more than 750 seconds when the number of sensors is 6000. In contrast, the time consumption of distributed calibration increases slowly with the increase in the number of measurements, and the calibration time does not exceed 100 seconds in all sensor number schemes. In addition, the gap between the two methods widens with the increase in monitoring data, and the efficiency ratio multiplier reaches more than 8.

[0080] Figure 4 The probability cumulative distribution diagram of the calibration pressure output deviation under different sensor deployment numbers for calibration using only a priori water demand, centralized calibration, and distributed calibration is given. Compared with calibration using only a priori information, the pressure deviation of centralized and distributed calibrations is more concentrated in a smaller area, verifying the accuracy effectiveness of the calibration proposed in the present invention. As the available monitoring data for calibration increases, the calibration result curve of the present invention gradually approaches the curve of the traditional centralized calibration, indicating that the increase in available monitoring data improves the accuracy of distributed calibration, but at the same time the efficiency is also sufficiently guaranteed.

Claims

1. A multi-sensor distributed online calibration method based on Bayesian decomposition, characterized in that: include: By using Bayesian decomposition, the prior distribution of the node water demand at the current moment is decomposed into K local prior probability distributions, and the likelihood function of the monitoring data at the current moment is decomposed into K local likelihood functions; The local prior probability distribution and the local likelihood function are combined into K local posterior probability distributions, each local posterior probability distribution is converted into a corresponding local calibration objective function, and the water demand value at the current moment is iteratively adjusted until the local calibration objective function value is minimized or the maximum number of iterations is reached, and the iteration is stopped to obtain the local calibration water demand value and the local calibration covariance matrix; Based on the local verification covariance matrix, weights are assigned to the local verification water demand values ​​and then added up to obtain the global verification water demand.

2. The multi-sensor distributed online calibration method based on Bayesian decomposition according to claim 1 is characterized in that: The prior distribution of the current moment's predicted water demand is decomposed into K local prior probability distributions, including: X t =f(X t-1 ,X t-2 ,…X1)+ε X t ~N(X t |X t|t-1 ,P t|t-1 ) Among them, f() is the prediction function, the water demand of the node at the current time t is X t The node water demand history value X can be used through the prediction function f() t-1 ,X t-2 ,…X1 is predicted, ε is the prediction error, which obeys the normal distribution N(ε|0,P t|t-1 ), P t|t-1 is the covariance matrix of the node water demand at the time before the current time to estimate the water demand at the current time t, X t|t-1 is the mean of the prior distribution, N(X t |X t|t-1 ,P t|t-1 ) is the prior distribution of the node water demand at time t, N(X t |X t|t-1 ,KP t|t-1 ) is the local prior probability distribution of the node water demand at the current time t, and j is the index of the local prior probability distribution.

3. The multi-sensor distributed online calibration method based on Bayesian decomposition according to claim 1 is characterized in that: The likelihood function obeyed by the monitoring data at the current moment is decomposed into K local likelihood functions, including: Y t =g(X t )+ω Y t ~N[Y t |g(X t ),R] Among them, Y t is the monitoring data at the current time t, g(X t ) is the simulated value obtained by taking the node water demand at the current time t as input through the pipe network model, ω is the noise, and the noise ω obeys the normal distribution N(ω|0,R), R is the noise covariance matrix, and the monitoring points m in the pipe network are divided into K groups, each group corresponds to a local likelihood function, that is, the monitoring data of the i-th group corresponds to the i-th local likelihood function, and the i-th group of monitoring points provides l i monitoring data, i.e., the monitoring data of group i 4. The multi-sensor distributed online calibration method based on Bayesian decomposition according to claim 1 is characterized in that: Multiply each local prior probability distribution and the corresponding local likelihood function to obtain the corresponding local posterior probability distribution.

5. The multi-sensor distributed online calibration method based on Bayesian decomposition according to claim 1 is characterized in that: The local prior probability distribution of the node water demand at the current time t and the i-th local likelihood function are combined to obtain the corresponding i-th local posterior probability distribution P(X t |Y t (i)) is: P(X t |Y t (i))=N(X t |X t|t-1 ,KP t|t-1 )N[Y t (i)|g i (X t ),R(i)] Among them, X t is the node water demand at the current time t, Y t (i) is the monitoring data corresponding to the i-th local likelihood function at the current time t, P t|t-1 is the covariance matrix of the node water demand at the time before the current time to estimate the water demand at the current time t, X t|t-1 is the mean of the prior distribution, g i (X t ) is the simulated value of the pipe network model output corresponding to the i-th local likelihood function at the current time t, and R(i) is the noise covariance matrix corresponding to the i-th local likelihood function.

6. The multi-sensor distributed online calibration method based on Bayesian decomposition according to claim 5 is characterized in that: Each local posterior probability distribution is converted into a corresponding local calibration objective function, where the i-th local calibration objective function is as follows: Among them, iteratively adjust the node water demand X of the i-th local check at the current time t t (i) The iteration is stopped until the i-th local calibration objective function value is minimized or the maximum number of iterations is reached, and the node water demand value and the local calibration covariance matrix of the i-th local calibration are obtained.

7. The multi-sensor distributed online calibration method based on Bayesian decomposition according to claim 1 is characterized in that: Iterate the node water demand of the i-th local check at the current time t, including: P t (i)=KP t|t-1 -KP t|t-1 {R(i)+K 2 G(i)P t|t-1 [G(i)] T } -1 G(i)P t|t-1 in, is the node water demand of the k+1th iteration of the ith local check at the current time t, is the node water demand adjustment of the i-th local check in the k-th iteration, is the Jacobian matrix corresponding to the i-th group of monitoring data, k is the number of iterations, and MK is the maximum allowed number of iterations.

8. The multi-sensor distributed online calibration method based on Bayesian decomposition according to claim 1 is characterized in that: Based on the local verification covariance matrix, the local verification water demand values ​​are weighted and then summed to obtain the global verification water demand, including: in, is the inverse of the covariance matrix of the i-th local calibration at the current time t, X t (i) is the water demand value of the i-th local calibration at time t before.

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