CN113836780B - Water distribution network hydraulic calculation method based on improved bovine method - Google Patents

Water distribution network hydraulic calculation method based on improved bovine method Download PDF

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CN113836780B
CN113836780B CN202110588699.8A CN202110588699A CN113836780B CN 113836780 B CN113836780 B CN 113836780B CN 202110588699 A CN202110588699 A CN 202110588699A CN 113836780 B CN113836780 B CN 113836780B
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赵霞
崔镜心
谭红
王骆
孙名轶
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Abstract

The invention discloses a water distribution network hydraulic calculation method based on an improved bovine method, which comprises the following steps: 1) Acquiring basic parameters of a water distribution network; 2) Establishing a hydraulic calculation model of the water distribution network; 3) Solving a hydraulic calculation model of the water distribution network to obtain a hydraulic calculation result of the water distribution network; 4) Judging whether the hydraulic calculation result of the water distribution network meets the iteration condition, if so, ending, otherwise, returning to the step 3). The method can simultaneously calculate the two unknown numbers of the water head height and the branch flow in the water network, can still keep good convergence under the condition of considering the pressure driving node, and improves the convergence of the hydraulic calculation of the common bovine-method.

Description

Water distribution network hydraulic calculation method based on improved bovine method
Technical Field
The invention relates to the field of water distribution network hydraulic calculation, in particular to a water distribution network hydraulic calculation method based on an improved bovine method.
Background
The hydraulic calculation of the water distribution network is the basis of hydraulic analysis, and in general, the main purpose of calculation is to obtain the water head height of each load node in the water distribution network, the flow passing through each pipeline branch or water pump branch, and the like, which is the basic work necessary for analyzing the running state of one water distribution network, and is also the necessary work in the early stage of water distribution network design and planning. Therefore, the design of a hydraulic calculation method with stronger calculation capability and faster calculation speed is significant for the analysis and design of the water distribution network.
In the current hydraulic calculation, mostly newton-Laporton method is used as the main method, in general, the hydraulic head height of all load nodes in the water distribution network or the flow passing through all branches is calculated in the hydraulic calculation, and then another unknown number is calculated. Thus, the discrete solution can influence the calculation speed and the convergence effect, and the simultaneous solution of the water head and the flow makes up the defects of the part; in addition, the introduction of pressure-driven characteristics may reduce convergence of iterative computations.
Disclosure of Invention
The invention aims to provide a water distribution network hydraulic calculation method based on an improved bovine method, which comprises the following steps of:
1) And obtaining basic parameters of the water distribution network.
The water distribution network has n b nodes, including n r water source nodes and n p branches.
The water distribution network comprises a water source, a load, a pipeline and a water pump.
The basic parameters of the water distribution network comprise the water head height of a water source node, the altitude of all nodes, the pipeline parameters of the water distribution network, the node information parameters of the water distribution network and the water pump curve. The water distribution network pipeline parameters comprise length, diameter and friction coefficient.
Before calculating the water distribution network model, initializing the variables to be solved. The iteration method is further improved based on the principle of the Newton method, and the convergence performance of the Newton method is improved; the water head heights of all nodes except the water source nodes of the water distribution network and the flow of all branches of the water distribution network can be solved at the same time during calculation, so that the convergence performance of the water distribution network can be further enhanced.
2) And establishing a hydraulic calculation model of the water distribution network.
The hydraulic calculation model of the water distribution network comprises a node load calculation model, a pipeline flow calculation model, a water pump working model and a node flow balance equation.
The node load includes a constant flow load calculation model and a pressure driven load calculation model.
The node load in the constant-current load calculation model is a fixed value.
The pressure-driven load calculation model is as follows:
Where q L,i is the node load flow of the inode in pressure driven mode. Is the fixed demand load traffic for node i. H i is the node pressure head height of node i. H i max and H i min are the upper and lower threshold head heights of the pressure driven nodes.
The pipeline flow calculation model is as follows:
Δhij＝γijqij|qij|n-1 (2)
wherein Δh ij is the node pressure head loss between node i and node j along the pipeline direction, and Δh ij is positive when the direction is from node i to node j. q ij is the pipe traffic size between node i and node j, and q ij is positive when the direction is from node i to node j. Gamma ij is the coefficient of friction of the pipeline between node i and node j. n is a constant.
The water pump working model is as follows:
where the subscript pu indicates that this parameter is a coefficient associated with the water pump. h i、hj represents the water heads of the node i and the node j respectively; Δh pu,ij is the head of the water pump between node i and node j. h 0ij、rpu,ij and m ij are respectively the static lift, internal resistance coefficient and lift index of the water pump. Omega ij is the relative rotational speed of the water pump.
The node flow balance equation is as follows:
Where j ε i represents the set of nodes connected to node i. q ij represents the flow through the pipe branch or the water pump branch between node i and node j. s ij is denoted as the direction of traffic q ij, s ij =1 when the tributary traffic q ij flows into node i, and s ij = -1 when the tributary traffic q ij flows out of node i.
3) And solving the hydraulic calculation model of the water distribution network to obtain a hydraulic calculation result of the water distribution network.
The step of solving the hydraulic calculation model of the water distribution network comprises the following steps:
3.1 A water distribution network hydraulic equation set is established, namely:
where n b is the total number of nodes. n r is the number of source nodes. n p is the total branch number. And F 1 is a node flow balance equation vector of the water distribution network, and n b-nr equations are contained in total. The element in vector f 1(q1,q2,…,qnp) is the sum of the flows flowing into and out of a node via a pipe or water pump. L (h 1,h2,…,hnb) is the load equation vector of the node. q i is the flow of branch i. h i is the head height of node i. And F 2 is a pipeline characteristic equation of the water distribution network and a water pump characteristic equation vector, and n p equations are contained in total. f 2(q1,q2,…,qnp) is expressed as a head drop equation vector for each leg. Δh ij(h1,h2,…,hnb) is the head height drop vector of the nodes at the two ends of the branch.
The unknown quantity in the hydraulic equation set of the water distribution network comprises the water head height of the load nodes except the water source node and the flow of all branches.
3.2 Solving the unbalance amount delta F (k) of the hydraulic equation set of the water distribution network, namely:
Wherein k represents the iteration number, the subscript 1 represents the equation as a node flow balance equation, the subscript 2 represents the equation as a branch characteristic equation, and the superscript T represents the transpose.
3.3 A jacobian matrix J (k) is calculated, namely:
the jacobian matrix J (k) is simplified as follows:
wherein the submatrices The following is shown:
In the method, in the process of the invention, The subarray of the incidence matrix A is represented, the number of lines is equal to the number of load nodes, and the number of columns is equal to the number of branches. A represents an incidence matrix of connection relations between all nodes and all branches in the water distribution network, each row represents connection relations of one node, each column represents connection relations of one branch, the number of rows is equal to the number of nodes in the water distribution network, and the number of columns is equal to the number of branches. Submatrix/>Wherein the element 1 indicates that the flow of the branch flows into the current node, the element-1 indicates that the flow of the branch flows out of the current node, and the element 0 indicates that the branch and the current node have no connection relation.
Sub-matrixThe following is shown:
when the branch where the node i and the node j are located is a pipeline branch, the element The following is shown:
When the branch where the node i and the node j are is located is a water pump branch, elements The following is shown:
Sub-matrix The following is shown:
3.4 Calculating the correction of the variable to be calculated, namely:
Δx(k)＝-[J(k)]-1ΔF(k) (14)
Where Δx (k) is a column vector consisting of all the variable corrections to be calculated.
Column vector Δx (k) is shown below:
3.5 Calculating new iteration values of the variables to be solved, namely:
Where x 1 (k) and x 2 (k) are new iteration values of the variables to be solved. h step is the iteration step.
When the difference parameter ζ of the new iteration value x 1 (k) and the iteration value x 2 (k) is greater than the difference threshold ε 1, the iteration step h step is as follows:
hstep＝max{σ1hstep,hstep,min} (18)
Where h step,min is the minimum step size. σ 1 is the step damping coefficient.
When the difference parameter between the new iteration value x 1 (k) and the iteration value x 2 (k) is smaller than or equal to the difference threshold epsilon 1, the iteration step h step is as follows:
hstep＝min{σ2hstep,hstep,max} (19)
where σ 2 is the step damping coefficient.
Wherein, the difference parameter ζ is as follows:
3.6 Correcting the new iteration value of the variable to be solved to obtain:
where ψ is the set parameter for modeling the standard richardson extrapolation method. h step is the iteration step. x (k+1) is the iteration value of the variable correction to be calculated.
3.7 A) update iteration number k=k+1.
3.8 Judging whether max { |Deltax (k)|}＜ε2 or k is larger than or equal to k max, if yes, stopping iteration, otherwise, returning to the step 3.2). Where ε 2 is the iteration threshold. k max is the maximum number of iterations.
4) Judging whether the hydraulic calculation result of the water distribution network meets the iteration condition, if so, ending, otherwise, returning to the step 3).
The step of judging whether the water distribution network hydraulic calculation result meets the iteration condition comprises the following steps:
4.1 Judging the water head interval sequence number of the pressure water head height of the pressure drive load node according to the water distribution network hydraulic calculation result, and writing the water head interval sequence number into a vector pi 2. The water head intervals are respectively marked as 1,2 and …, g, and g is the total number of the water head intervals. The ranges of each head interval do not overlap.
4.2 If yes, ending the iteration, otherwise updating the preset vector pi 1 to a vector pi 2, and enabling the iteration number k PDA＝kPDA +1 to solve the hydraulic calculation model of the water distribution network again, namely returning to the step 3). The elements in the preset vector pi 1 are the preset interval sequence numbers where the node pressure head in the pressure driving mode is located. The subscript PDA indicates the pressure driven load.
The invention introduces the theory of the Convergence extrapolation based on the traditional Newton method, and further improves the convergence performance; the method has the advantages that the water head heights except the water source nodes and all branch flow in the water distribution network are solved at the same time during calculation, and compared with a method for solving the water head heights and all branch flow separately, the method has better convergence.
The invention is worth to describe, regard Newton-Laportson method of improvement under the theory of the Icharsen extrapolation as the computational tool, under the pressure of the partial load node drives the load model, can obtain the node pressure water head height and branch flow two kinds of hydraulic calculation methods to be solved unknown quantity at the same time. The method comprises the following specific steps: firstly, determining each parameter of a water distribution network, and establishing a water distribution network model; before starting iterative computation, the range of the pressure head height of each pressure driving load node is firstly assumed so as to determine specific function expressions of the loads of the nodes in the subsequent computation process; after starting iterative calculation, calculating the unbalance amount and Jacobian matrix of each hydraulic calculation equation by using the initial value of the unknown quantity to be solved, and solving a correction equation by using the unbalance amount and the Jacobian matrix to obtain a correction quantity; after that, according to the idea of the rational extrapolation, two groups of new correction amounts are calculated with a certain step length, and then the two groups of correction amounts are used for one-step calculation, so that new iteration values of all unknown amounts to be solved are finally obtained; judging whether the step length used in the next iteration needs to be changed or not and how much the step length needs to be changed according to the sizes of the two groups of correction amounts obtained in the previous step; judging whether the iteration reaches convergence accuracy according to the original correction amount calculated previously, stopping calculation if the iteration reaches the accuracy, and returning to the step of calculating the unbalance amount if the iteration does not reach the accuracy, and repeating the steps until the iteration converges; after converging to the final result, obtaining the pressure water head height from the water head height and the altitude of the pressure driving load node, judging whether the section of the pressure water head of the pressure driving load node is consistent with the result assumed before calculation, if so, the calculation is the final result, the calculation is directly finished, and the calculation result is output; if the calculation result of the node is inconsistent with the assumption result, the calculation is not a final reasonable result, the assumption is re-made according to the calculation result, the function expression form of the pressure driving node load is determined according to the assumption, and then the calculation of the NitsLaw method is re-performed; if the interval of the assumed water head is still inconsistent with the calculation result after repeating the ox iteration calculation for a certain number of times, the calculation is considered to be failed.
The method has the technical effects that the method can simultaneously solve two unknown numbers of the water head height and the branch flow in the water network, can still keep good convergence under the condition of considering the pressure driving node, and improves the convergence of the hydraulic calculation of the common bovine-drawing method. The hydraulic calculation convergence can still be ensured under the conditions of considering the pressure driving load node model or the conditions of larger load flow, mismatching of the water pump curve and the actual condition, and the like, and a more reasonable result is obtained.
Drawings
FIG. 1 is a block flow diagram of the method of the present invention.
FIG. 2 shows the calculation result of the pressure water head height of each node of the water distribution network when the calculation is performed by the modified Nile method.
FIG. 3 shows the calculated flow rate of each branch of the distribution network when calculated by the modified Nipple method.
Detailed Description
The present invention is further described below with reference to examples, but it should not be construed that the scope of the above subject matter of the present invention is limited to the following examples. Various substitutions and alterations are made according to the ordinary skill and familiar means of the art without departing from the technical spirit of the invention, and all such substitutions and alterations are intended to be included in the scope of the invention.
Example 1:
Referring to fig. 1, a water distribution network hydraulic calculation method based on an improved bovine method comprises the following steps:
1) And obtaining basic parameters of the water distribution network.
The water distribution network has n b nodes, including n r water source nodes and n p branches.
The water distribution network comprises a water source, a load, a pipeline and a water pump.
The basic parameters of the water distribution network comprise the water head height of a water source node, the altitude of all nodes, the pipeline parameters of the water distribution network, the node information parameters of the water distribution network and the water pump curve. The water distribution network pipeline parameters comprise length, diameter and friction coefficient.
Before calculating the water distribution network model, initializing the variables to be solved. The iteration method is further improved based on the principle of the Newton method, and the convergence performance of the Newton method is improved; during calculation, the water head height of all nodes except the water source node of the water distribution network and the flow of all branches of the water distribution network can be solved at the same time, so that the convergence performance of the water distribution network can be further enhanced
2) And establishing a hydraulic calculation model of the water distribution network.
The hydraulic calculation model of the water distribution network comprises a node load calculation model, a pipeline flow calculation model, a water pump working model and a node flow balance equation.
The node load includes a constant flow load calculation model and a pressure driven load calculation model.
The node load in the constant-current load calculation model is a fixed value.
The pressure-driven load calculation model is as follows:
Where q L,i is the node load flow of the inode in pressure driven mode. Is the fixed demand load traffic for node i. H i is the node pressure head height of node i. H i max and H i min are the upper and lower threshold head heights of the pressure driven nodes.
The pipeline flow calculation model is as follows:
Δhij＝γijqij|qij|n-1 (2)
wherein Δh ij is the node pressure head loss between node i and node j along the pipeline direction, and Δh ij is positive when the direction is from node i to node j. q ij is the pipe traffic size between node i and node j, and q ij is positive when the direction is from node i to node j. Gamma ij is the coefficient of friction of the pipeline between node i and node j. n is a constant.
The water pump working model is as follows:
where the subscript pu indicates that this parameter is a coefficient associated with the water pump. h i、hj represents the water heads of the node i and the node j respectively; Δh pu,ij is the head of the water pump between node i and node j. h 0ij、rpu,ij and m ij are respectively the static lift, internal resistance coefficient and lift index of the water pump. Omega ij is the relative rotational speed of the water pump.
The node flow balance equation is as follows:
Where j ε i represents the set of nodes connected to node i. q ij represents the flow through the pipe branch or the water pump branch between node i and node j. s ij is denoted as the direction of traffic q ij, s ij =1 when the tributary traffic q ij flows into node i, and s ij = -1 when the tributary traffic q ij flows out of node i.
3) And solving the hydraulic calculation model of the water distribution network to obtain a hydraulic calculation result of the water distribution network. The calculation result comprises the water head height of all nodes except the water source node of the water distribution network and the flow of all branches of the water distribution network
The step of solving the hydraulic calculation model of the water distribution network comprises the following steps:
3.1 A water distribution network hydraulic equation set is established, namely:
where n b is the total number of nodes. n r is the number of source nodes. n p is the total branch number. And F 1 is a node flow balance equation vector of the water distribution network, and n b-nr equations are contained in total. The element in vector f 1(q1,q2,…,qnp) is the sum of the flows flowing into and out of a node via a pipe or water pump. L (h 1,h2,…,hnb) is the load equation vector of the node. q i is the flow of branch i. h i is the head height of node i. And F 2 is a pipeline characteristic equation of the water distribution network and a water pump characteristic equation vector, and n p equations are contained in total. f 2(q1,q2,…,qnp) is expressed as a head drop equation vector for each leg. Δh ij(h1,h2,…,hnb) is the head height drop vector of the nodes at the two ends of the branch.
The unknown quantity in the hydraulic equation set of the water distribution network comprises the water head height of the load nodes except the water source node and the flow of all branches.
3.2 Solving the unbalance amount delta F (k) of the hydraulic equation set of the water distribution network, namely:
Wherein k represents the iteration number, the subscript 1 represents the equation as a node flow balance equation, the subscript 2 represents the equation as a branch characteristic equation, and the superscript T represents the transpose.
3.3 A jacobian matrix J (k) is calculated, namely:
the jacobian matrix J (k) is simplified as follows:
wherein the submatrices The following is shown:
In the method, in the process of the invention, The subarray of the incidence matrix A is represented, the number of lines is equal to the number of load nodes, and the number of columns is equal to the number of branches. A represents an incidence matrix of connection relations between all nodes and all branches in the water distribution network, each row represents connection relations of one node, each column represents connection relations of one branch, the number of rows is equal to the number of nodes in the water distribution network, and the number of columns is equal to the number of branches. Submatrix/>Wherein the element 1 indicates that the flow of the branch flows into the current node, the element-1 indicates that the flow of the branch flows out of the current node, and the element 0 indicates that the branch and the current node have no connection relation.
Sub-matrixThe following is shown:
when the branch where the node i and the node j are located is a pipeline branch, the element The following is shown:
When the branch where the node i and the node j are is located is a water pump branch, elements The following is shown:
Sub-matrix The following is shown:
3.4 Calculating the correction of the variable to be calculated, namely:
Δx(k)＝-[J(k)]-1ΔF(k) (14)
Where Δx (k) is a column vector consisting of all the variable corrections to be calculated.
Column vector Δx (k) is shown below:
3.5 Calculating new iteration values of the variables to be solved, namely:
Where x 1 (k) and x 2 (k) are new iteration values of the variables to be solved. h step is the iteration step.
When the difference parameter ζ of the new iteration value x 1 (k) and the iteration value x 2 (k) is greater than the difference threshold ε 1, the iteration step h step is as follows:
hstep＝max{σ1hstep,hstep,min} (18)
Where h step,min is the minimum step size. σ 1 is the step damping coefficient.
When the difference parameter between the new iteration value x 1 (k) and the iteration value x 2 (k) is smaller than or equal to the difference threshold epsilon 1, the iteration step h step is as follows:
hstep＝min{σ2hstep,hstep,max} (19)
where σ 2 is the step damping coefficient.
Wherein, the difference parameter ζ is as follows:
3.6 Correcting the new iteration value of the variable to be solved to obtain:
where ψ is the set parameter for modeling the standard richardson extrapolation method. h step is the iteration step. x (k+1) is the iteration value of the variable correction to be calculated.
3.7 A) update iteration number k=k+1.
3.8 Judging whether max { |Deltax (k)|}＜ε2 or k is larger than or equal to k max, if yes, stopping iteration, otherwise, returning to the step 3.2). Where ε 2 is the iteration threshold. k max is the maximum number of iterations.
4) Judging whether the hydraulic calculation result of the water distribution network meets the iteration condition, if so, ending, otherwise, returning to the step 3).
The step of judging whether the water distribution network hydraulic calculation result meets the iteration condition comprises the following steps:
4.1 Judging the water head interval serial number of the pressure water head height of the load node in the pressure driving mode according to the water distribution network hydraulic calculation result, and writing the water head interval serial number into a vector pi 2. The water head intervals are respectively marked as 1,2 and …, g, and g is the total number of the water head intervals. The ranges of each head interval do not overlap.
4.2 If yes, ending the iteration, otherwise updating the preset vector pi 1 to a vector pi 2, and enabling the iteration number k PDA＝kPDA +1 to solve the hydraulic calculation model of the water distribution network again, namely returning to the step 3). The elements in the preset vector pi 1 are the preset interval sequence numbers where the node pressure head in the pressure driving mode is located. The subscript PDA indicates the pressure driven load.
The invention introduces the theory of the Convergence extrapolation based on the traditional Newton method, and further improves the convergence performance; the method has the advantages that the water head heights except the water source nodes and all branch flow in the water distribution network are solved at the same time during calculation, and compared with a method for solving the water head heights and all branch flow separately, the method has better convergence.
Example 2:
Referring to fig. 1, a water distribution network hydraulic calculation method based on an improved bovine method comprises the following steps:
1) Inputting basic data and initializing
1.1 Inputting basic data
The input water distribution network basic parameters comprise: the water head height of the water source node, the altitude of all nodes, pipeline parameters (length, diameter, friction coefficient and the like) of the water distribution network, node information parameters (load and the like) of the water distribution network, a water pump curve, an upper limit of iteration times, an upper limit of integral NitsLaw calculation times, a plurality of parameters related to the theory of the Lechasen extrapolation and the like.
1.2 Parameter initialization)
The hydraulic calculation in the invention selects the pressure water head height of the load node and all pipeline flow in the network as unknown quantity to be calculated at the same time, and the unknown quantity is calculated in one Niu Lafa calculation. Assuming that there are n b nodes in the water distribution network system, where there are n r water source nodes and n p branches, the unknown quantity to be solved is the water head height of all nodes except the water source nodes, and the flow of all branches, n b－nr+np total. An iteration initial value is set for all unknown variables to be solved before calculation is performed. Under the condition that the unit of the flow of the branch is m 3/s, the initial value of the flow of all the branches is 0, and the initial value of the water head height of all the load nodes is 1.
2) Model for hydraulic calculation of water distribution network
The model of the water distribution network mainly comprises elements such as a water source, a load, a pipeline, a water pump and the like, and the hydraulic calculation mainly depends on a node flow balance equation, a pipeline characteristic equation or a water pump curve equation.
I) Water source
The water source is embodied in the model in the form of a water source node. The difference between the water source node and the load node is that the known quantity and the unknown quantity are different, the water head height of the water source node is the known quantity, the water head height is given when the parameter information of the water distribution network is input, and the load (namely the output force) is the unknown quantity to be solved, and the unknown quantity needs to be calculated and solved.
II) load
The load is also embodied in the form of nodes in the model. The load flow of the load node is a known quantity, which is already given when the parameter information of the water distribution network is input, and the water head height is an unknown quantity to be solved. The altitude of all nodes is given in the input network parameter information, the altitude can influence the water pressure of the flow at the nodes, and the altitude has no other influence on the water head, so that the method can be solved as the common nodes; after the head height calculation of the node is obtained, subtracting the altitude level at the node is the water pressure of the node. In addition, the load nodes in the model have two load forms, one is constant-flow load, namely, the load is always a constant-flow value no matter how the pressure water head of the nodes changes; the node load under the pressure driving model is related to the pressure water head height, and the expression of the load is different at different water head heights, and the concrete form is as follows:
wherein q L,i is the node load flow of the i node in the pressure driving mode, the unit is m 3/s;qL,i req is the fixed demand load flow of the i node, the unit is m 3/s;Hi is the node pressure water head height of the i node, and the unit is m; h i max and H i min are upper and lower threshold head heights of the pressure driving node, the unit is m, and when the pressure head height of the node i is in three different sections with two thresholds as boundaries, the load flow of the node i has different functional expressions. When H i≥Hmax or H i≤Hmin is adopted, the load flow of the node i is a fixed value, and is irrelevant to the pressure water head height of the node; when H min＜Hi＜Hmax is reached, the load flow of the node i is a fixed demand load flow with the pressure head height H i of the current node and the node i And (3) a function of the isoparametric and variable.
III) pipeline
The pipeline equation in the model is used for describing the loss degree of the pressure water head height of the nodes at two ends of the pipeline, namely the water head drop, when a certain flow of water flows through the pipeline. The Hazen-Williams formula is chosen here to describe this characteristic of the pipe branch, and its specific expression is:
Δhij＝γijqij|qij|n-1 (2)
The Δh ij is the node pressure head loss quantity between the node i and the node j along the pipeline direction, the unit is m, the direction is from the node i to the node j, if the actual water flow direction does not accord with the set direction, the opposite number of the loss quantity is adopted, and generally, the head height of the nodes at the two ends of the pipeline can be reduced along with the actual flow direction of the pipeline flow; q ij is the flow of the pipeline between the node i and the node j, the direction is from the node i to the node j, the unit is m 3/s;γij, and the flow is the friction coefficient of the pipeline between the node i and the node j, and is a dimensionless parameter, and the value of the friction coefficient is related to parameters such as the material quality, the diameter, the length and the like of the pipeline; in the Hazen-Williams equation, n is 1.852.
IV) Water Pump
The water pump has the capability of converting electric energy into mechanical energy, is a tool for improving the node pressure water head height of nodes at two ends of the water pump, and the specific lifting height is called the water pump lift, and the working performance of the water pump is represented by a water pump curve. The water pump curve can generally be fitted by the following function:
Wherein the subscript pu indicates that the parameter is a coefficient related to the water pump; Δh pu,ij is the lift of the water pump between node i and node j, and the unit is m; h 0ij、rpu,ij and m ij are respectively the static lift, internal resistance coefficient and lift index of the water pump, and omega ij is the relative rotation speed of the water pump, namely the ratio of the working rotation speed to the rated rotation speed.
V) node flow balance equation
In the hydraulic calculation, the sum of the input flow and the sum of the output flow of each node are equal, namely, the node flow balance equation is satisfied:
wherein j e i represents a set of nodes connected with the node i, q ij represents flow passing through a pipeline branch or a water pump branch between the node i and the node j, the unit is m 3/s;sij, the direction of the flow q ij, the unit is that the branch flow q ij flows into the node i if the unit is 1, and the unit is that the branch flow q ij flows out of the node i if the unit is-1; for any node, the sum of the branch flow passing through all branches connected with the node i and the load flow of the node i is 0 under the premise of considering the direction.
3) Solving the hydraulic equation of water distribution network by improved Niuzhan method
From the above model, the following form of hydraulic equation can be listed:
Wherein n b is the total node number, n r is the number of water source nodes, n p is the total branch number, F 1 is the node flow balance equation vector of the water distribution network, n b-nr equations are contained, F 1(q1,q2,…,qnp) is a vector, the element is the sum of the flows flowing in and out of a certain node through a pipeline or a water pump, L (h 1,h2,…,hnb) is the load equation vector of the node, q i is the flow of a branch i, and h i is the water head height of the node i; f 2 is a pipeline characteristic equation and a water pump characteristic equation vector of the water distribution network, n p equations are contained, F 2(q1,q2,…,qnp) is expressed as a water head drop equation vector of each branch, wherein the functional expressions of the pipeline and the water pump are different, and Deltah ij(h1,h2,…,hnb) is a water head height drop vector of nodes at two ends of the branch. The unknown quantities to be found in the equation include the head height h i of the load nodes other than the water source node, a total of n b-nr, and the flow q i of each branch, a total of n p.
3.1 Solving the unbalance of the equation set
And (3) carrying the data initialized by the parameters in the section 1.2 into the equation set, and obtaining the unbalance of all node equations and branch equations. The number of unbalance amounts is identical to the number of equations in the equation set, and is n b-nr+np. All unbalance amounts in the iteration are formed into a column vector, as follows:
Wherein k represents the iteration number, the subscript 1 represents the equation as a node flow balance equation, the subscript 2 represents the equation as a branch characteristic equation, and the superscript T represents the transpose.
3.2 Computing jacobian matrix
According to the definition of the Jacobian matrix and the concrete expression form of the equation set, each equation in the equation set is used for solving partial derivatives of the water head height h and the branch flow q, and the Jacobian matrix is obtained according to the derivative result:
wherein each subscript has the meaning as set forth in section 3.1. The above may also be written by a blocking matrix:
The specific function expression of each blocking matrix is as follows:
Wherein A represents an incidence matrix of connection relations between all nodes and all branches in the water distribution network, the number of rows is the same as the number of nodes in the water distribution network, n b-nr, and each row represents the connection relation of one node; the number of columns is the same as the number of branches, n p, and each column represents the connection relation of one branch; the value of the specific element is related to the setting of the flow direction of the branch, the flow setting direction of the branch is represented by-1 when the node flows out, 1 when the node flows in, and 0 is filled in the position where other nodes are not connected with the branch; by adding marks The matrix is a subarray of the incidence matrix A, the array number is unchanged, but the number of lines is different, and the matrix only comprises line vectors represented by all load nodes.
Wherein each subscript has the same meaning as in section 2.2.
For the followingIf the branch is a pipe branch, the expression is:
/>
wherein the meaning of each parameter is the same as the parameters in section 2.3. If the branch is a water pump branch, the expression is:
wherein the meaning of each parameter is the same as the parameters in section 2.4.
Wherein,Meaning and/>The superscript T denotes the transpose.
3.3 Calculating correction amount
According to the newton-raphson method, correction amounts for the respective unknown amounts to be solved in this iteration are calculated from the column vectors Δf (k) and J (k) calculated previously, as follows:
Δx(k)＝-[J(k)]-1ΔF(k) (14)
Wherein Δχ (k) is a column vector composed of all correction amounts of unknown amounts to be calculated, and the specific expression is:
wherein, the meanings of each superscript and subscript are the same as the superscription.
3.4 Calculating new iteration values
According to the specific calculation step of the Newton-Lapherson method, after the correction quantity is obtained, the correction quantity is correspondingly added with the unknown number to be solved which is iterated at present, so that the next iteration value can be obtained, and then whether to stop iteration is judged according to the absolute value of the correction quantity; here we further correct the correction according to the rational-charsen extrapolation concept to obtain the unknown for the next iteration in a different way, the specific procedure is as follows:
First, the following two new sets of iteration values x 1 (k) and x 2 (k) are obtained from the correction amount:
then, the iteration quantity is further corrected based on the rational Charles extrapolation idea, and the specific formula is as follows:
Wherein, ψ is a parameter set for simulating a standard rational-Charles extrapolation method, the magnitude of the value can influence the convergence times and results to a certain extent, and different values of ψ can be used for changing the convergence effect in the actual calculation process; h step is an iteration step length, the value of which can be changed along with the iteration process, and the specific calculation method is as follows:
Wherein ζ is a parameter characterizing the magnitude of the difference between x 1 (k) and x 2 (k), if too small the step size should be increased, otherwise the step size should be decreased. Setting a parameter epsilon 1 as a threshold value to measure the magnitude of zeta value, if zeta is greater than epsilon 1, then there are:
hstep＝max{σ1hstep,hstep,min} (20)
If ζ is less than or equal to ε 1, then there are:
hstep＝min{σ2hstep,hstep,max} (21)
Wherein, sigma 1 and sigma 2 are step damping coefficients, and h step,min and h step,max are minimum step and maximum step respectively.
3.5 Convergence determination)
At this time, the iteration number k is calculated
k＝k+1 (22)
To record the number of iterations. Whether the iterative process is stopped or not is mainly judged by the magnitude of the maximum value in the absolute values of the deviation value vectors. When meeting the requirements
max{|Δx(k)|}＜ε2 (23)
Or the number of iterations k satisfies
k≥kmax (24)
And stopping iteration. If not, returning to the step of calculating the unbalance amount in the section 3.1, and carrying out the next iterative calculation.
3.6 Judging the section of the water head of the pressure driving node
Before calculation by the bovine method, a group of variables pi 1 are set to represent the section where the pressure head of the pressure driving node is located, the number of elements is the same as the number of pressure driving load nodes, and different values indicate that the pressure head height of the node is in different sections, for example, if 1 indicates that the pressure head height H i of the node meets H i≤Hi min, if 2 indicates that the pressure head height H i of the node meets H i min＜Hi＜Hi max, and if 3 indicates that the pressure head height H i≥Hi max is met. After the whole iterative process is finished, i.e. a set of determined solutions is converged, the interval where the pressure water head heights of the nodes are located is judged according to the calculated results, the results are recorded in a vector 2, and whether the assumed result pi 1 is identical to the calculated result pi 2 is compared. If the assumed result of some nodes is different from the calculated result, it indicates that the assumed interval result pi 1 before iteration is inappropriate, the assumed interval result pi 1 should be set again as the result of the calculated result pi 2, the calculation of the Czochralski method should be performed again, and the variable k PDA with the number of times of calculation recorded should be calculated
kPDA＝kPDA+1 (25)
To record the number of calculations. Wherein the subscript PDA represents the pressure driven load. The above process is continuously circulated until the two groups of judgment results are the same, and the result is a correct and reasonable result.
Example 3:
Referring to fig. 2 and 3, taking 81 nodes and 121 branches (7 branches are water pump branches, and 27 branches of the original calculation example are changed into water pump branches) and a heavy-load water distribution network as an example, an iterative calculation is performed by using an improved newton-lavson method based on the principle of the chalcone extrapolation, and an experiment for verifying a water distribution network hydraulic calculation method based on the principle of the chalcone extrapolation is performed, and the method comprises the following steps:
1) Inputting basic data and initializing
1.1 Inputting basic data
Inputting basic parameters of the water distribution network, wherein the basic parameters comprise: water source parameters, water supply pipeline parameters, water pump parameters, load parameters and the like. In addition, there are parameters such as convergence accuracy ε 2, upper iteration number k max, etc. involved in the improved specific calculation of Newton-Laporton method. The specific data of the water distribution network adopts the data of the water distribution network system of 81 nodes in the section-oriented electric-water combined trend of the regional comprehensive energy service in the electric automation equipment of the 40 th volume 12 period in 2020. The following improvements are made on the basis of this example: the load of 15 load nodes of 3 to 7, 24 to 27, 36 to 38 and 42 to 46 is selected as a pressure driving load model, the lower threshold value of pressure control is 0, the upper threshold value is 91.773, and the demand load level of all the nodes is enlarged to 1.1 times.
Step judgment parameters sigma 1 and sigma 2 in the theory of the richardson are set to be 0.95 and 1.05 respectively, h step,min and h step,max are set to be 0.75 and 2 respectively, epsilon 1＝8,ε2＝10-8 is set, the simulation order psi of the richardson extrapolation is=8, and the initial value of the iteration step h step is set to be 1.
1.2 Parameter initialization)
In the water distribution network example, the 1 st, 18 th, 19 th, 21 st, 34 th and 80 th nodes are water source nodes, and the water head height is a known quantity; the other nodes are load nodes, the node loads are known, wherein the loads of 15 nodes such as 3,4, 5 and the like are pressure driving models, the known loads are fixed demand load flow q L,i req, and the load flow of the node is not the final load flow. In the aspect of initial values, the initial values of all pipeline flow are all set to 0, and the water head height of the load nodes outside the water source nodes is set to 1.
2) Hydraulic calculation by improving Newton-Lapherson method
2.1 Preliminary processing of calculation example and assumption of head interval result pi 1
Before calculation starts, a tree support structure and an associated matrix of a water distribution network are needed to be obtained, iteration initial values of unknown quantities are set, relevant parameters of a theoretical Charles extrapolation idea are set, after that, a section result pi 1 where a water head of each pressure driving node is located is firstly assumed, and therefore a function expression of each load in a subsequent calculation process can be determined. In this example, there are 15 pressure driven nodes, so the total of n 1 elements in the vector is set to 3 before the start, i.e. the flow of the node load is a constant value assuming that the head height of all nodes is above the upper threshold in the pressure driven load function expression.
2.2 Calculating unbalance of the equation set of the water distribution network
Substituting the basic data of the water distribution network and the initial value of each unknown quantity to be solved into the calculation equations F 1 and F 2 to obtain the unbalance quantity delta F (k) of the first iteration:
the first 75 elements are the unbalance amount of the node flow balance equation for each load node, and the last 121 elements are the unbalance amount of the characteristic equation for each branch.
2.3 Computing jacobian matrix
The values of the four parts of the whole Jacobian matrix are calculated respectively:
/>
2.4 Calculating correction amount
After unbalance amount delta F (1)) Jacobian matrix and J (1) of the water distribution network hydraulic equation are obtained according to the steps, correction value vector delta x (1) aiming at each unknown amount to be solved in the iteration is obtained:
2.5 Calculating an iterative new value based on the theory of the rational Charson extrapolation
After the correction of the unknown to be solved is obtained, it is further processed to improve convergence. Taking the calculation process of the branch flow q as an example, two groups of new iteration values are calculated according to formulas (16) and (17), and the two groups of new iteration values are respectively as follows:
And then calculating new iteration values according to the equation and the two groups of iteration values according to the theory of the Lechassen extrapolation, wherein the vector is as follows:
next, it is determined whether the step size of the next calculation needs to be changed. Substituting the parameters into the formula (19) to obtain a parameter ζ of judgment compensation:
ζ＝141.7290 (10)
comparing ζ with a judgment threshold ε 1, which is
ζ＞ε1 (11)
The step size needs to be changed, and the next iteration step size is as follows:
hstep＝max{σ1hstep,hstep,min}＝σ1hstep＝0.95 (12)
2.6 Convergence determination)
At this time, it is required to determine whether the iteration satisfies a convergence condition, and the specific condition is as follows:
1) When the correction amount Δx (k) of the hydraulic equation satisfies max { |Δx (k)|}＜ε2, the calculation ends;
2) When the number of iterations k reaches the upper number limit k max(kmax =100), the calculation ends.
And (3) judging that the iteration does not meet the convergence condition, and continuing to calculate, and returning to the step of calculating unbalance in the section 2.2 to continue to calculate. And (5) repeating the steps circularly until the convergence condition is met, and exiting the calculation.
2.7 Pressure driving condition judgment
After the calculation process, obtaining interval results pi 2 of water heads of all pressure driving nodes according to the final iteration result:
In comparison with the assumption result pi 1 before calculation, there are nodes with inconsistent judgment results, which indicate that the assumption before calculation is incorrect. Before the next calculation of the bovine method is started, the calculation result pi 2 is assigned to the assumed result pi 1, and a specific expression function of the load of the pressure driving node in the next Niu Lafa calculation is determined according to the value of pi 1. And (3) repeating the steps in a circulating way, and finishing calculation when one of the following conditions is met:
1) If the calculation result meets pi 2＝Π1, ending the calculation;
2) The calculation number k PDA reaches the calculation number upper limit k PDA,max, and the calculation is ended.
2.8 Water distribution net hydraulic calculation result
The water head height of each node and the flow of each branch in the water distribution network are solved by the calculation method, and the results are shown in fig. 2 and 3.
Experimental effect:
the calculation results show that the newton-raphson method was performed 2 times in total due to the adjustment of the pressure-driven node due to the unsuitable interval assumption, and the specific convergence results of each calculation are shown in table 1.
Table 1 results of calculations to improve the venlafaxine process
The maximum value in the absolute value vector of the unbalance amount in the second cattle calculation is shown in table 2.
Table 2 unbalance in the ox-drawn calculation
The calculation result shows that the improved bovine method has a quicker and more accurate calculation result even in a large water distribution network with heavier load, and can be used in actual engineering calculation.

Claims (4)

1. The hydraulic calculation method of the water distribution network based on the improved bovine method is characterized by comprising the following steps of:
1) Acquiring basic parameters of a water distribution network;
2) Establishing a hydraulic calculation model of the water distribution network;
3) Solving a hydraulic calculation model of the water distribution network to obtain a hydraulic calculation result of the water distribution network;
4) Judging whether the hydraulic calculation result of the water distribution network meets the iteration condition, if so, ending, otherwise, returning to the step 3);
the hydraulic calculation model of the water distribution network comprises a node load calculation model, a pipeline flow calculation model, a water pump working model and a node flow balance equation;
The node load comprises a constant-current load calculation model and a pressure-driven load calculation model;
The node load in the constant-current load calculation model is a fixed value;
The pressure-driven load calculation model is as follows:
wherein q L,i is the node load flow of the i node in the pressure driving mode; The fixed demand load flow for node i; h i is the node pressure head height of the node i; /(I) And/>The upper and lower threshold head heights are the pressure driving nodes;
the pipeline flow calculation model is as follows:
Δhij＝γijqij|qij n-1 (2)
Wherein Δh ij is the node pressure head loss between node i and node j along the pipeline direction, and Δh ij is positive when the direction is from node i to node j; q ij is the pipe flow between node i and node j, q ij is positive when the direction is from node i to node j; gamma ij is the friction coefficient of the pipeline between the node i and the node j; n is a constant;
The water pump working model is as follows:
wherein, the subscript pu indicates that the parameter is a coefficient related to the water pump; Δh pu,ij is the head of the water pump between node i and node j; h i、hj represents the water heads of the node i and the node j respectively; h 0ij、rpu,ij and m ij are respectively the static lift, internal resistance coefficient and lift index of the water pump; omega ij is the relative rotational speed of the water pump;
the node flow balance equation is as follows:
Wherein j epsilon i represents a set of nodes connected with node i; q ij is the flow through the pipe branch or the water pump branch between the node i and the node j; s ij is represented as the direction of the traffic q ij, s ij =1 when the tributary traffic q ij flows into the node i, s ij = -1 when the tributary traffic q ij flows out of the node i;
the step of solving the hydraulic calculation model of the water distribution network comprises the following steps:
3.1 A water distribution network hydraulic equation set is established, namely:
Wherein n b is the total node number; n p is the total branch number; f 1 is a node flow balance equation vector of the water distribution network, and totally comprises n b-nr equations; n r is the number of water source nodes; the element in vector f 1(q1,q2,…,qnp) is the sum of the flows flowing in and out of a certain node via a pipe or a water pump; l (h 1,h2,…,hnb) is the load equation vector of the node; q i is the flow of branch i; h i is the head height of node i; f 2 is a pipeline characteristic equation of the water distribution network and a water pump characteristic equation vector, and comprises n p equations in total; f 2(q1,q2,…,qnp) is expressed as a water head drop equation vector for each branch; Δh ij(h1,h2,…,hnb) is the head height drop vector of the nodes at the two ends of the branch;
The unknown quantity in the hydraulic equation set of the water distribution network comprises the water head height of the load nodes except the water source node and the flow of all branches;
3.2 Solving the unbalance amount delta F (k) of the hydraulic equation set of the water distribution network, namely:
wherein k represents the iteration times, the subscript 1 represents the equation as a node flow balance equation, the subscript 2 represents the equation as a branch characteristic equation, and the superscript T represents the transposition;
3.3 A jacobian matrix J (k) is calculated, namely:
the jacobian matrix J (k) is simplified as follows:
wherein the submatrices The following is shown:
In the method, in the process of the invention, The subarrays of the incidence matrix A are represented, the number of the rows is equal to the number of the load nodes, and the number of the columns is equal to the number of branches; a represents an incidence matrix of connection relations between all nodes and all branches in a water distribution network, each row represents connection relations of one node, each column represents connection relations of one branch, the number of rows is equal to the number of nodes in the water distribution network, and the number of columns is equal to the number of branches; submatrix/>Wherein the element 1 indicates that the flow of the branch flows into the current node, the element-1 indicates that the flow of the branch flows out of the current node, and the element 0 indicates that the branch and the current node have no connection relationship;
Sub-matrix The following is shown:
when the branch where the node i and the node j are located is a pipeline branch, the element The following is shown:
When the branch where the node i and the node j are is located is a water pump branch, elements The following is shown:
Sub-matrix The following is shown:
3.4 Calculating the correction of the variable to be calculated, namely:
Δx(k)＝-[J(k)]-1ΔF(k) (14)
wherein Deltax (k) is a column vector composed of all the to-be-solved variable corrections;
column vector Δx (k) is shown below:
3.5 Calculating new iteration values of the variables to be solved, namely:
x1 (k)＝x(k)+hstepΔx(k) (16)
Wherein, x 1 (k) and x 2 (k) are new iteration values of the variables to be solved; h step is the iteration step;
When the difference parameter ζ of the new iteration value x 1 (k) and the iteration value x 2 (k) is greater than the difference threshold ε 1, the iteration step h step is as follows:
hstep＝max{σ1hstep,hstep,min} (18)
Wherein h step,min is the minimum step length; sigma 1 is the step damping coefficient;
when the difference parameter between the new iteration value x 1 (k) and the iteration value x 2 (k) is smaller than or equal to the difference threshold epsilon 1, the iteration step h step is as follows:
hstep＝min{σ2hstep,hstep,max} (19)
Wherein sigma 2 is a step damping coefficient;
wherein, the difference parameter ζ is as follows:
3.6 Correcting the new iteration value of the variable to be solved to obtain:
Wherein, ψ is a set parameter for simulating a standard rational extrapolation method; h step is the iteration step; x (k+1) is the iteration value of the variable correction to be solved;
3.7 Updating the iteration number k=k+1;
3.8 Judging whether max { Deltax (k)}＜ε2 or k is larger than or equal to k max, if yes, stopping iteration, otherwise, returning to the step 3.2); epsilon 2 is the iteration threshold; k max is the maximum number of iterations.
2. The improved bovine-method-based hydraulic calculation method for water distribution network according to claim 1, wherein the method comprises the following steps: the water distribution network is provided with n b nodes, wherein the water distribution network comprises n r water source nodes and n p branches;
the water distribution network comprises a water source, a load, a pipeline and a water pump;
The basic parameters of the water distribution network comprise the water head height of a water source node, the altitude of all nodes, the pipeline parameters of the water distribution network, the node information parameters of the water distribution network and a water pump curve; the water distribution network pipeline parameters comprise length, diameter and friction coefficient.
3. The improved bovine-method-based hydraulic calculation method for water distribution network according to claim 1, wherein the method comprises the following steps: the Convergence performance of the Newton method is further improved by introducing the theory of the Newton extrapolation based on the traditional Newton method; the method has the advantages that the water head heights except the water source nodes and all branch flow in the water distribution network are solved at the same time during calculation, and compared with a method for solving the water head heights and all branch flow separately, the method has better convergence.
4. The method for calculating water distribution network water power based on the improved bovine method as claimed in claim 1, wherein the step of judging whether the water distribution network water power calculation result satisfies the iteration condition comprises:
1) Judging a water head interval sequence number of the pressure water head height of the pressure driving load node according to the water distribution network hydraulic calculation result, and writing the sequence number into a vector pi 2; the serial numbers of the water head intervals are respectively recorded as 1,2 and …, g is the total number of the water head intervals; the ranges of each head interval are not overlapped;
2) Judging whether the vector pi 2 is equal to a preset vector pi 1, if so, ending the iteration, otherwise, updating the preset vector pi 1 to be the vector pi 2, and enabling the iteration times k PDA＝kPDA +1 to solve the hydraulic calculation model of the water distribution network again; the elements in the preset vector pi 1 are preset interval serial numbers where the node pressure water head in the pressure driving mode is located; the subscript PDA indicates the pressure driven load.
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