A method for verifying water demand at water supply network nodes coupled with pressure prior information

Abstract

The present invention discloses a method for verifying water demand of a water supply network node coupled with prior information of pressure, comprising the following steps: (1) predicting prior water demand and its covariance matrix through a water demand prediction function, and obtaining a prior probability distribution of node water demand; (2) obtaining a prior probability distribution of node pressure through on-site survey of building and floor information in the area where the user is located; (3) obtaining monitoring data of pipeline pressure and flow through pipeline pressure and flow sensors and their covariance matrix, and obtaining a likelihood function of monitoring data; (4) establishing a verification objective function based on Bayesian theorem; (5) linearizing the objective function, obtaining the iteration step length of 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, ensures that the pressure value calculated by the model is distributed within a reasonable range, thereby improving the verification accuracy of node water demand, and providing a scientific basis for hydraulic modeling of water supply network.

Description

Technical Field

[0001] The invention belongs to a method for verifying water demand of a node in a city water supply network, in particular to a method for verifying water demand of a node in a water supply network coupled with pressure priori information. Background Art

[0002] The hydraulic model of the water supply network is widely used in the real-time modeling of the water distribution system, which can be used to improve the efficiency of system analysis, design and operation. The node water demand is the most uncertain input parameter in the hydraulic model, which has a great influence on the accuracy of the model output. Therefore, the water demand verification of each node in the network is an important step in the network simulation.

[0003] Water demand calibration of water supply networks refers to adjusting the node water demand so that the results calculated by the network model are consistent with the data monitored by the sensors arranged in the network. Although pressure and flow sensors have been installed in many water supply networks, it is not feasible to deploy online sensors at all nodes due to cost reasons. Therefore, water supply networks usually only install sensors at a few selected key locations. In water demand calibration, there are tens of thousands of node water demands to be calibrated, while the available monitoring data is limited, which in turn causes errors in the calibration model, such as negative water demand, negative node pressure, etc. The main problem faced by water demand calibration is that there is not enough monitoring data to calculate all the node water demands, which leads to the ill-posed problem of water demand calibration. Obviously, the ill-posed problem caused by insufficient data cannot be solved by simply changing the algorithm. We must incorporate more prior information into the calibration procedure to obtain meaningful results.

[0004] Prior information on water demand has been used for online and offline node water demand verification. To reduce the number of unknown parameters, existing algorithms group water demand according to user characteristics, relative location, or demand pattern, and then use the total water demand of a group of nodes as the variable to be solved. After calculating the total water demand within the group, the total water demand is allocated to each node according to the population served by each node or the consumer bill. Bayesian-based data assimilation algorithm is another water demand verification algorithm that can effectively use prior information on water demand. This method obtains the prior probability distribution of water demand based on user bills, the number of people served, or the water demand prediction function; then, based on Bayes' theorem, the prior probability distribution of water demand is coupled with the likelihood function of monitoring data to obtain the posterior probability distribution of water demand; and the node water demand is solved by maximizing the posterior probability distribution.

[0005] The above methods effectively couple the prior information of water demand, thereby improving the calibration accuracy of node water demand. However, the successful application of these methods depends on reliable monitoring data and accurate prior information of water demand. In the presence of monitoring data noise or inaccurate prior information of water demand, the performance of the algorithm will deteriorate significantly. In this case, the calculated node water pressure may exceed the reasonable range (such as negative node pressure) and cannot reflect the actual network operation status of the system. This highlights the importance of integrating more useful information.

[0006] In practice, there is a large amount of available pressure prior information that can be used to improve the calibration accuracy. In my country, the water supply network needs to guarantee a minimum node pressure, usually 16 meters, to ensure that users can get enough drinking water. This shows that the node water pressure calculated by the model can be limited to the area of ​​H ≥ 16m. In addition, the water pressure in the building can be investigated on the spot. For example, through investigation, it is known that drinking water in a building can be supplied to the 5th floor, then the node pressure corresponding to the building is H ≥ 12m (4 floors high, 3m per floor, a total of 12m). Through such investigations, a large number of node pressure prior information can be obtained. A prominent feature of this pressure information is that it is not as accurate as the data provided by the sensor, and it is difficult to accurately quantify it with a specific value. How to effectively use this useful prior information to improve the accuracy of node water demand calibration is a technical problem that needs to be solved urgently. Summary of the invention

[0007] The purpose of the present invention is to overcome the deficiencies of the prior art. On the basis of the original node water demand verification algorithm, a method is proposed to improve the verification accuracy of the hydraulic model of the water supply network by combining the node pressure prior information. Taking into account the inaccuracy of the pressure prior information, the present invention uses uniform distribution to characterize this information so that the node pressure is limited between the upper and lower bounds of the uniform distribution. On the basis of integrating the node water demand prior information and the monitoring data information, the present invention couples the available pressure prior information to verify the node water demand, avoids the phenomenon of unreasonable node water pressure (such as negative pressure) caused by unreasonable water demand verification, improves the verification accuracy of the node water demand, and provides a scientific basis for online simulation of the water supply network, pressure management, etc.

[0008] To achieve the above objectives, the present invention provides a method for verifying water demand at a water supply network node coupled with pressure prior information, and the following steps are taken:

[0009] (1) Obtain the prior probability distribution of node water demand;

[0010] (2) Based on the building and floor water level information in the user's area, the node pressure distribution intervals of s nodes are obtained, and a priori probability distribution of node pressure that obeys uniform distribution is established; through the pipeline network pressure sensor and flow sensor, the pipeline network pressure and flow monitoring data and their covariance matrix are obtained, and the monitoring data likelihood function is obtained;

[0011] (3) Establish a Bayesian posterior probability distribution model of water demand prior-node pressure prior-monitoring data coupling to achieve the coupling of node water demand prior information, node pressure prior information and monitoring data. By maximizing the posterior probability density function of node water demand, establish a verification objective function.

[0012] (4) Use the Newton iteration method to solve the verification objective function, obtain the node water demand adjustment, and iteratively solve the node water demand as the verification result.

[0013] Beneficial effects of the present invention: The present invention belongs to a method for verifying the water demand of a node in a water supply network. Previous verification methods based on data assimilation verify the water demand of a node by coupling water demand prior information and monitoring data. These methods have achieved good water demand verification accuracy, but they are heavily dependent on reliable monitoring data and accurate water demand prior information. In the presence of monitoring noise or inaccurate water demand prior information, the verified node water pressure may exceed the feasible domain (such as negative node pressure) and cannot reflect the actual working conditions of the system. The present invention models the pressure information through uniform distribution, and then adopts a data assimilation framework to verify the node water demand by fusing the real-time monitoring data, prior node water demand and prior node pressure in a probabilistic form, ensuring that the node pressure is limited to the upper and lower limits of the uniform distribution, avoiding unreasonable node pressure. This method improves the utilization efficiency of information, avoids the deterioration of algorithm performance due to abnormal data, and improves the verification accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 : Schematic diagram of water demand verification process for water supply network nodes;

[0015] Figure 2 : Water supply network model diagram. DETAILED DESCRIPTION

[0016] The purpose of the present invention is to provide a method for verifying water demand at a node of a water supply network coupled with pressure prior information. The node pressure prior information is modeled using uniform probability distribution to describe the boundary constraints of the node pressure; a data assimilation framework is used to combine the pressure prior information and other information (such as observation data) to verify the node water demand. The invention has significant potential in improving the accuracy of water supply network modeling.

[0017] The specific implementation modes of the present invention are further described in detail below in conjunction with the accompanying drawings.

[0018] For a water supply network, there are n nodes in total, of which the number of monitoring points is m and the number of nodes with known pressure prior information is s. The water demand verification method for water supply network nodes coupled with pressure prior information proposed by the present invention comprises the following steps:

[0019] Step 1: Establish a water demand prediction model, calculate the prior water demand and its covariance matrix, and establish a prior probability distribution of node water demand that obeys the normal distribution.

[0020] In this embodiment, the node water demand prediction function f(·) is used to convert the historical water demand (X t-1 , X t-2 , ...) as input, predict the prior node water demand X at the current time step t t|t-1 ; Use the covariance prediction function F(.) to convert the historical covariance (P t-1 , P t-2 , ...) as input, predict the node water demand prior covariance P at the current time step t t|t-1 .

[0021] X t|t-1 =f(X t-1 , X t-2 , ...)

[0022] P t|t-1 =F(P t-1 , P t-2 , ...)

[0023] Establish a priori probability distribution P(X) of node water demand that obeys normal distribution t )

[0024] P(X t )=N(X t |X t|t-1 , P t|t-1 )

[0025] Step 2: Obtain the node pressure distribution range by surveying the building and floor information in the user's area on site, and establish a node pressure prior probability distribution that obeys uniform distribution.

[0026] In this embodiment, for a node i without a pressure sensor installed, the engineer or the pipe network system operator can give the upper and lower limits a of the node pressure distribution based on experience or on-site floor water pressure survey. t，i <H t，i (X t )<b t，i , and then we can get that the pressure at node i obeys uniform distribution,

[0027] P(H t，i (X t )|Xt )=U t，i (H t，i (X t )|a t，i , b t，i )

[0028] Among them, a t，i and b t，i Indicates the upper and lower limits of the node pressure value of the i-th node at time t; X t represents the water demand at the node at time t; H t，i (X t ) indicates that the water demand at the node at time t is X t When , the node pressure value of the i-th node; U t，i (·) is the uniform distribution function corresponding to the i-th node at time t, P(H t，i (X t )|X t ) indicates that the water demand at the node at time t is X t When , the prior probability of the pressure at the i-th node is . According to the multiplication principle of probability, the prior probability distribution of the pressure at s nodes is,

[0029]

[0030] H t =[H t，1 (X t ), H t，2 (X t ), ....., H t，s (X t )] T

[0031] The uniform distribution is a discontinuous function and is difficult to linearize. To solve this problem, the following function is used as an alternative to the uniform distribution function:

[0032]

[0033] Where λ is a constant greater than 1; c i is the normalization constant of the i-th node pressure prior probability, s is the number of nodes for obtaining node pressure priors, P(H t |X t ) indicates that the water demand at the node at time t is X t The prior probability of node pressure at time , H t Represents the prior set of node pressure values ​​at time t.

[0034] Step 3: Obtain the pipeline network pressure and flow monitoring data and their noise covariance matrix through the pipeline network pressure and flow sensors, and establish the likelihood probability distribution of the monitoring data that obeys the normal distribution.

[0035] In this embodiment, the monitoring data noise obeys the normal distribution, and the likelihood probability density function is:

[0036]

[0037]

[0038] Where m is the number of sensors installed in the pipe network, including nodes with pressure sensors installed and pipes between two nodes with flow sensors installed; is the monitoring data vector at time t, is the monitoring value of the ith monitoring point at time t; g t，i (X t ) is the output value of the water supply network EPANET model corresponding to the i-th monitoring point at time t; R i is the noise variance of the monitoring data at the ith monitoring point.

[0039] Step 4: Establish a Bayesian posterior probability distribution model of water demand prior-node pressure prior-monitoring data coupling to achieve the coupling of node water demand prior information, node pressure prior information and monitoring data, and establish a verification objective function by maximizing the posterior probability density function.

[0040] In this embodiment, the node water demand prior probability function P(X t ), node pressure prior probability distribution function P(H t |X t ) and the monitoring data likelihood probability distribution function Multiply them together to get the posterior probability distribution of node water demand:

[0041]

[0042] By maximizing the logarithm of the posterior probability density function Get the objective function J(X t ):

[0043]

[0044] Among them, N(X t |X t|t-1 , P t|t-1 ) is the prior probability distribution of node water demand, X t|t-1 is the prior water demand at time t, P t|t-1 is the prior water demand covariance at time t; is the likelihood function of the monitoring data, m is the number of monitoring data, is the monitoring value of the ith monitoring point at time t, where the monitoring points include nodes with pressure sensors installed and pipelines between two nodes with flow sensors installed; g t，i (X t ) is the output value of the water supply network EPANET model corresponding to the i-th monitoring point at time t; R i is the noise variance of the monitoring data of the ith monitoring point; the superscript T represents the transposition, and α represents the uniform distribution linearization constant.

[0045] Step 5: Use Newton iteration method to solve the objective function, obtain the node water demand adjustment, and iteratively update the node water demand.

[0046] In this embodiment, the Newton iteration method is used to calculate the node water demand, and the Newton iteration direction is:

[0047]

[0048]

[0049] in, and are the first and second derivatives of the objective function. is the node water demand at time t verified after the kth iteration, is the covariance matrix at time t after the kth iteration; It is the adjustment amount of node water demand at time t verified after the kth iteration.

[0050] After obtaining the iteration direction, use the following formula to update the node water demand until the maximum allowed number of iterations K is reached:

[0051]

[0052] k<K

[0053] Where k is the number of iterations; is the node water demand at time t verified after the k+1th iteration; μ is the iteration step length. and The calculation method is:

[0054]

[0055]

[0056] in, is the first-order derivative of the output value of the water supply network EPANET model corresponding to the i-th monitoring point at time t; It is the first-order derivative of the model output value corresponding to the i-th pressure prior value at time t.

[0057] In practical applications, such as Figure 1 As shown, the process is as follows, and the formulas repeated in the above are not repeated.

[0058] 1. Parameter initialization.

[0059] 2. Use the formula in step 1 above to update the water demand prior information.

[0060] 3. Update the node pressure prior, and obtain the node pressure based on the building and floor water level information.

[0061] 4. Update measurement information, including node pressure priors (node ​​pressures obtained based on building and floor water level information) and monitoring data (pressure sensor and flow sensor measurements).

[0062] 5. Initialization of node water demand. Including initialization iteration k = 0, and initialization of node water demand

[0063] 6-8. Update the latest iteration As the input of the EPANET model, run the EPANET model, and use the water supply network EPANET model to obtain the model output g corresponding to the monitoring point at time t t (X t )=[g t，1 (X t ), g t，2 (X t ), …, g t，m (X t )] T , and the model output H corresponding to the node pressure prior at time t t =[H t，1 (X t ), H t，2 (X t ), ....., H t，s (X t )] T .

[0064] 9. Calculate the first and second order derivatives of the objective function.

[0065] 10-12. Update the covariance matrix, node water demand adjustment, and node water demand.

[0066] 13. Output the iterated node water demand and covariance matrix as the verification result.

[0067] 14-16. Use the verification results as the input of the water supply network model, and control the water supply network system in real time according to the predicted output results.

[0068] 17. Repeat 2-16 and update iteratively.

[0069] The above application steps are described in detail below in conjunction with a specific embodiment. Figure 2 A simple pipe network diagram is given, which has 1 water source, 8 water demand nodes, 11 pipe sections, 2 pressure monitoring points, 1 flow monitoring point, and 2 nodes with known pressure prior information. The specific steps are as follows:

[0070] Step 1: Set calibration parameters

[0071] Pressure sensors are arranged at nodes N3 and N6, and flow sensors are arranged at pipeline (4). The specific monitoring values ​​and variances are shown in Table 1. In addition, N5 and N8 are selected as nodes with known pressure prior information, and their distribution parameters are shown in Table 2. The allowed number of iterations is K = 20, and the step size is μ = 0.25. The parameter α is set to 0.001.

[0072] Table 1 Monitoring data

[0073]

[0074] Table 2 Prior information of pressure on nodes N5 and N8

[0075]

[0076] Step 2: Initialize node water demand and output the calculation results of the water supply network hydraulic model

[0077] The following initial node water demand Prior water demand (X t|t-1 ) and the prior water demand covariance (P t|t-1 ) is set to:

[0078]

[0079] X t|t-1 =[1.27, 6.24, 19.64, 4.25, 12.92, 13.20, 18.66, 25.90] T

[0080] P t|t-1 =diag(1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)

[0081] use As the input of the EPANET model of the water supply network, the model output is:

[0082]

[0083]

[0084] Step 3: Calculate the Jacobian matrix

[0085] By substitution As input to the hydraulic model of the water supply network, the Jacobian matrix of the three measurements relative to the water demand at the current node can be calculated:

[0086]

[0087]

[0088]

[0089] The Jacobian matrix of the prior node pressure relative to the current node water demand can also be calculated:

[0090]

[0091]

[0092] Step 4: Calculate the Jacobian matrix and Hessian matrix of the objective function

[0093] The Jacobian matrix of the objective function is calculated as follows:

[0094]

[0095] The Hessian matrix of the objective function is calculated as follows:

[0096]

[0097] Step 5: Calculate the covariance matrix and node demand adjustments

[0098] The covariance matrix is ​​calculated as follows:

[0099]

[0100] The calculation of the water demand adjustment for this iteration is as follows:

[0101]

[0102] Step 6: Update the node water demand for the next iteration

[0103] The node water demand for the next iteration k=1 is updated as:

[0104]

[0105] Step 7: Reaching the termination condition

[0106] Starting from the second iteration, k = 1, the next step is to repeat steps 2-6 until the termination condition is met. For the assumed simple network, the verification process will terminate after iteration k = 19.

Claims

1. A method for verifying water demand at water supply network nodes coupled with pressure prior information. It is characterized in that The steps include: (1) Obtain the prior probability distribution of node water demand; (2) According to the building and floor water level information in the user's area, the node pressure distribution intervals of s nodes are obtained, and the node pressure prior probability distribution that obeys the uniform distribution is established: H t =[H t,1 (X t ),H t,2 (X t ),…,H t,s (X t )] T Where s is the number of nodes for obtaining node pressure priors, P(H t |X t ) indicates that the water demand at the node at time t is X t The prior probability of node pressure at time , H t represents the prior set of node pressure values ​​at time t, U t,i (.) is the uniform distribution function corresponding to the i-th node at time t; H t,i (X t ) indicates that the water demand at the node at time t is X t When , the node pressure value of the i-th node; a t,i and b t,i represents the upper and lower limits of the node pressure value of the ith node at time t; λ is a constant greater than 1; c i is the normalization constant of the prior probability of pressure at the ith node; Through the pipeline network pressure sensor and flow sensor, the pipeline network pressure and flow monitoring data and their covariance matrix are obtained, and the monitoring data likelihood function is obtained; (3) Establish a Bayesian posterior probability distribution model of water demand prior-node pressure prior-monitoring data coupling to achieve the coupling of node water demand prior information, node pressure prior information and monitoring data. By maximizing the posterior probability density function of node water demand, establish a verification objective function. (4) Use the Newton iteration method to solve the verification objective function, obtain the node water demand adjustment, and iteratively solve the node water demand as the verification result.

2. A method for verifying water demand at a water supply network node coupled with pressure prior information as claimed in claim 1, Features The step (3) is specifically as follows: Multiply the node water demand prior probability distribution, node pressure prior probability distribution and monitoring data likelihood function to obtain the node water demand posterior probability distribution: Among them, N(X t |X t|t-1 ,P t|t-1 ) is the prior probability distribution of node water demand, X t|t-1 is the prior water demand at time t, P t|t-1 is the prior water demand covariance at time t; is the likelihood function of the monitoring data, m is the number of monitoring data, is the monitoring value of the ith monitoring point at time t, where the monitoring points include nodes with pressure sensors installed and pipelines between two nodes with flow sensors installed; g t,i (X t ) is the output value of the water supply network EPANET model corresponding to the i-th monitoring point at time t; R i is the noise variance of the monitoring data of the ith monitoring point; By maximizing the logarithm of the posterior probability density function Get the calibration objective function J(X t ): α=lnλ The superscript T represents the transpose, and α represents the uniform distribution linearization constant.

3. A method for verifying water demand at a water supply network node coupled with pressure prior information as claimed in claim 1, It is characterized in that The step (4) is specifically as follows: Use Newton iteration method to calculate the node water demand. The Newton iteration direction is: in, and To check the first and second derivatives of the objective function, is the node water demand at time t verified after the kth iteration, is the covariance matrix at time t after the kth iteration; is the water demand adjustment of the node at time t verified after the kth iteration; After obtaining the iteration direction, use the following formula to update the node water demand until the maximum allowed number of iterations is reached: in, is the node water demand at time t verified after the k+1th iteration, and μ is the iteration step size.

4. A method for verifying water demand at a water supply network node coupled with pressure prior information as claimed in claim 3, It is characterized in that The calculation method of the first and second derivatives of the calibration objective function is: in, is the first-order derivative of the output value of the water supply network EPANET model corresponding to the i-th monitoring point at time t; It is the first-order derivative of the output value of the EPANET model of the water supply network corresponding to the i-th pressure prior value at time t.

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