A Modular Implementation Method for Obtaining the Characteristic Frequencies of a Complex Water Conveyance System

Abstract

The present invention provides a modular implementation method for obtaining the characteristic frequencies of a complex water conveyance system, including the following steps: S1: Decompose the complex water conveyance system into independent simple elements, where the simple elements include elastic pipeline elements, rigid pipeline elements, impedance elements, surge chamber elements, reservoir elements, oscillating valve elements, and turbine elements; S2: Sequentially number the simple elements and the nodes adjacent to the simple elements; S3: Sequentially compile the simple elements after the decomposition of the complex water conveyance system into sub-matrix expressions; S4: Construct the total matrix of the water conveyance system; S5: Determine the excitation input node and the response output node; S6: Solve the total matrix of the water conveyance system to determine the frequency response characteristics of the system. The present invention is based on the structural matrix method, models each hydraulic element in blocks, and the matrix form of the constructed elements does not change due to the change in the layout of the water conveyance system, with a high degree of modularity.

Description

Technical Field

[0001] The present invention relates to a numerical simulation calculation method for the characteristic frequency of a water channel system for hydraulic resonance analysis of a water conveyance and power generation system, and particularly to a modular implementation method for obtaining the characteristic frequency of a complex water conveyance system, which is applicable to water conservancy and hydropower projects with complex water conveyance systems. Background Art

[0002] The harm of hydraulic resonance in a water conveyance system is very prominent. It not only may amplify the vibration of the unit itself, causing large-amplitude periodic pressure fluctuation phenomena in the water channel, but also may cause fatigue damage to the structural parts of the unit, reduce the service life of the unit, and exacerbate the suddenness and severity of accidents. The hydraulic resonance in the water channel system can be divided into two types: self-excited oscillation and forced oscillation. Self-excited oscillation refers to the oscillating flow excited inside the water channel system due to water flow or water pressure under the condition of no external vibration source. For example, when a pumped-storage unit enters the unstable region of the reverse S-shaped characteristic curve under the condition of load rejection and guide vane refusal to move or locked low opening operation, an unconvergent equal-amplitude oscillating flow can be formed; Forced oscillation means that the water channel system will generate periodic oscillating flow under the disturbance of a periodic vibration source. For example, the units of a hydropower station will all be accompanied by a certain degree of vibration during operation. As the excitation source of the water channel system, if its vibration frequency is the same as or close to a certain characteristic frequency of the water channel system, resonance between the unit and the water channel may be triggered. Avoiding the proximity of the characteristic frequency of the water channel system to the vibration source frequency can effectively reduce the probability of hydraulic resonance occurring. Therefore, it is necessary to analyze the characteristic frequency of the water channel system.

[0003] When calculating the characteristic frequency of a water channel system by the traditional hydraulic impedance method, it is necessary to connect hydraulic elements step by step into a whole and perform iterative solution. This method is suitable for series water channel systems. For parallel water channel systems, its modeling and solution are relatively cumbersome. At the same time, the transfer matrix method has poor regularity in dealing with bifurcation elements and needs to be customized for different boundary conditions. It can be seen that these traditional frequency solution methods all have different degrees of shortcomings in dealing with complex water conveyance systems and cannot achieve modular solution.

[0004] At present, the water conveyance and power generation system shows a development trend of being giant and complex. Due to the diversity of the composition structure of the complex water conveyance system, different water conveyance systems correspond to different characteristic frequencies, which greatly increases the difficulty of numerical simulation of the water conveyance system frequency. Therefore, it is necessary to provide a method for simulating and analyzing the characteristic frequency of a complex water conveyance system that is easy to program, has a high degree of modularity, and is convenient for later secondary development. Summary of the Invention

[0005] The object of the present invention is to overcome the low efficiency of calculating the characteristic frequency of a water conveyance system by traditional methods, the complexity of program compilation, and the low degree of modularization, which cannot adapt to the current trend of gigantism and complexity in water conveyance and power generation systems. A modular implementation method for obtaining the characteristic frequency of a complex water conveyance system is proposed, which can solve the characteristic frequency of the water conveyance system, has a high degree of modularization, and has good calculation accuracy.

[0006] In order to solve the above technical problems, the present invention is realized through the following technical solutions:

[0007] A modular implementation method for obtaining the characteristic frequency of a complex water conveyance system includes the following steps:

[0008] S1: Decompose the complex water conveyance system into independent simple elements, where the simple elements include elastic pipeline elements, rigid pipeline elements, impedance elements, surge chamber elements, reservoir elements, oscillating valve elements, and turbine elements;

[0009] S2: Number the simple elements and the nodes adjacent to the simple elements in sequence;

[0010] S3: Compile the simple elements after decomposing the complex water conveyance system into sub-matrix expressions in sequence;

[0011] S4: Construct the total matrix of the water conveyance system;

[0012] S5: Determine the excitation input node and the response output node;

[0013] S6: Solve the total matrix of the water conveyance system to determine the frequency response characteristics of the system.

[0014] While adopting the above technical solutions, the present invention can also adopt or combine the following technical solutions:

[0015] As a preferred technical solution of the present invention: In step S1: The definition principle of the simple element is a hydraulic element with flow inflow or outflow:

[0016] If the inflow and outflow of the flow occur at the same end point, it is considered a single-endpoint element, such as a reservoir element, a surge chamber element, etc.;

[0017] If the inflow and outflow of the flow occur at different end points, it is considered a multi-endpoint element, such as an elastic pipeline element, a rigid pipeline element, etc.

[0018] The impedance element is a hydraulic element with high hydraulic similarity in the water channel, which can be specifically divided into trash racks, partially opened valves, throttle holes, etc., and is uniformly replaced by impedance elements to simplify the system matrix.

[0019] An oscillating valve element refers to a valve whose opening can undergo periodic changes without being directly affected by the head state. Such valves are mainly used to simulate the influence of the water turbine or pump blades on the water flow disturbance.

[0020] As a preferred technical solution of the present invention: In step S2: Nodes are divided into real nodes and virtual nodes. Real nodes refer to the nodes that divide each hydraulic element, and virtual nodes mainly correspond to the virtual endpoints in the hydraulic element. For example, the oscillating valve element matrix contains a non-hydraulic state variable y, so the oscillating valve element matrix consists of two real endpoints and one virtual endpoint, and the water turbine element matrix consists of two real endpoints and three virtual endpoints; for the sorting method of node numbers, first, the real nodes are numbered in sequence according to the composition order of the hydraulic elements in the water conveyance system, and then the virtual nodes are numbered in sequence.

[0021] As a preferred technical solution of the present invention: In step S3: The dimension of the element sub-matrix is equal to the sum of the real endpoints and virtual endpoints of the hydraulic element. For example, the surge chamber has only one real endpoint, so its element sub-matrix is a 1×1 matrix; the elastic pipeline has two real endpoints, so its element sub-matrix is a 2×2 matrix; the oscillating valve has two real endpoints and one virtual endpoint, so its element sub-matrix is a 3×3 matrix.

[0022] Specifically, the element matrix of the elastic pipeline is:

[0023]

[0024] In the formula:

[0025]

[0026]

[0027] s is the Laplace operator; ρ is the Allievi pipeline constant; Q0 is the reference flow rate, H0 is the reference head, α is the head loss coefficient per unit length; a is the water hammer wave velocity; L is the pipeline length; q i is the flow rate flowing into node i; q j is the flow rate flowing into node j; h i is the head of node i; h j is the head of node j;

[0028] Specifically, the element matrix of the rigid pipeline is:

[0029]

[0030] In the formula:

[0031] T w is the pipeline water flow acceleration time constant; s is the Laplace operator; β is the pipeline head loss coefficient, Q0 is the reference flow rate, and H0 is the reference head; q i is the flow rate flowing into node i; q j is the flow rate flowing into node j; h i is the head of node i; h j is the head of node j;

[0032] Specifically, the matrix of impedance elements is as follows:

[0033]

[0034] In the formula:

[0035] β is the pipeline head loss coefficient, Q0 is the reference flow rate, and H0 is the reference head; q i is the flow rate flowing into node i; q j is the flow rate flowing into node j; h i is the head of node i; h j is the head of node j;

[0036] Specifically, the element matrix of the surge chamber is as follows:

[0037]

[0038] In the formula:

[0039] A s is the cross-sectional area of the surge chamber; s is the Laplace operator; Q0 is the reference flow rate, and H0 is the reference head; q i is the flow rate flowing into node i; h i is the head of node i;

[0040] Specifically, the element matrix of the reservoir is as follows:

[0041]

[0042] In the formula:

[0043] β is the pipeline head loss coefficient, Q0 is the reference flow rate, and H0 is the reference head; q i is the flow rate flowing into node i; h i is the head of node i;

[0044] Specifically, the element matrix of the oscillating valve:

[0045]

[0046] In the formula:

[0047] Q yis the valve opening - flow transfer coefficient; y is the opening of the oscillating valve; β is the pipeline head loss coefficient, Q0 is the reference flow rate, and H0 is the reference head; q i is the flow rate flowing into node i; q j is the flow rate flowing into node j; h i is the head of node i; h j is the head of node j;

[0048] Specifically, the element matrix of the water turbine is:

[0049]

[0050] In the formula:

[0051] Q n is the water turbine speed - flow self - regulation coefficient; Q y is the valve opening - flow transfer coefficient; G(s) is the transfer function of the governor;

[0052] P h = 0.5(1 + e q )(1 - Q n )+(1 + e h ); P n = (1 + e q )Q n ; P y = (1 + e q )Q y ; e h is the relative efficiency change rate with respect to the relative head; e q is the relative efficiency change rate with respect to the relative flow rate;

[0053] T a is the unit inertia time constant; s is the Laplace operator; E n is the engine self - regulation coefficient;

[0054] q i is the flow rate flowing into node i; q j is the flow rate flowing into node j; h i is the head of node i; h j is the head of node j; y is the opening of the oscillating valve;

[0055] p is the relative output of the hydro - generator; p out is the actual output of the water turbine; n is the relative speed of the hydro - generator.

[0056] As a preferred technical solution of the present invention: In step S4: The system matrix is constructed as follows:

[0057] S401: Construct an \(n\times n\) matrix \([A]\) according to the total system matrix dimension being equal to the sum of the real nodes and virtual nodes of the water conveyance system, and clear all elements. Among them, the element in the \(i\)-th row and \(j\)-th column is represented by \(A\) i,j ;

[0058] S402: Construct an \(n\times1\) vector and clear all elements; construct an \(n\times1\) vector and clear all elements;

[0059] S403: Sequentially place the sub - matrices of each hydraulic element into the total system matrix;

[0060] In step S403: If the hydraulic element is a single - end element, such as a reservoir, a surge chamber, etc., assuming its connected real node number is \(i\), then directly place the elements on the left - hand side of the hydraulic element matrix into the diagonal \(A\) of the matrix \([A]\) i,i and place the elements on the right - hand side of the hydraulic element matrix directly into the vector The formula is as follows:

[0061] A i,i =A i,i +a i,i

[0062] Q i =Q i +q i

[0063] If the hydraulic element is a multi - end element, such as a pipeline, a water turbine, etc., the sum of its real nodes and virtual nodes is \(m\). Assuming its connected real node numbers are \(i\), \(j\), then place the elements on the left - hand side of the hydraulic element matrix into the matrix \([A]\), and place the elements on the right - hand side of the hydraulic element matrix into the vector The formula is as follows:

[0064]

[0065] Q i =Q i +q i

[0066]

[0067] Q j =Q j +q j

[0068] ······

[0069]

[0070] Q i+m-1 =Q i+m-1 +qi+m-1 (m = 2) or

[0071] Q i+m-1 = Q i+m-1 + y i+m-1 (m > 2)

[0072] Where, there is an inflow rate q' at node i i and an outflow rate q'' i , taking the inflow node as the positive direction, then there is q' i + q'' i = 0, so the matrix form of the multi-terminal can be expressed as follows:

[0073]

[0074] In the formula: x i+m-1 , y i+m-1 are variables in the non - hydraulic state.

[0075] As a preferred technical solution of the present invention: In step S4: The total matrix of the complex water conveyance system can directly construct the total matrix framework through the algebraic sum of the real nodes and the virtual nodes, and then directly allocate the sub - matrices of each hydraulic element. This method is beneficial to computer program programming and subsequent secondary development.

[0076] As a preferred technical solution of the present invention: In step S5: The excitation input node is specifically the input node of the system excitation signal. Only when there is an excitation signal in a system, the system will have a response, and the response of the system can be reflected by the head fluctuation of the specified response output node.

[0077] As a preferred technical solution of the present invention: In step S6: Since the total matrix of the water conveyance system is constructed by the frequency - domain response method, it avoids the cumbersome process of selecting the iterative time step Δt in the time - domain response method and skips multiple iterative steps, so it can be directly solved. The frequency response characteristics of the system can be determined by the ratio of the head fluctuation of the output node to the excitation of the input node, and the formula is as follows:

[0078]

[0079] or

[0080] In the formula:

[0081] h ～ , q ～ , y ～ are the dimensionless head, flow rate, and opening variables; H out , Q in , Y in are the head output fluctuation amplitude, flow rate input fluctuation amplitude, and opening input fluctuation amplitude; ω is the angular frequency; is the output phase lag; H0, Q0, and Y0 are the reference values of water head, flow rate, and opening degree.

[0082] Furthermore, h ～ / q ～ and h ～ / y ～ are the amplitude characteristics of the system. Both the amplitude and phase of the system output are related to the input frequency ω. Therefore, the frequency response of the water conveyance system is mostly shown in a coordinate graph with angular frequency as the abscissa and relative amplitude as the ordinate.

[0083] The present invention provides a modular implementation method for obtaining the characteristic frequencies of a complex water conveyance system. Compared with the prior art, it has the following beneficial effects:

[0084] (1) Based on the structural matrix method, the present invention divides each hydraulic element into blocks for modeling. The matrix form of the constructed elements does not change due to the change in the layout of the water conveyance system, and the modularity is high.

[0085] (2) By using the frequency-domain response method, the direct solution of the total matrix of the water conveyance system is realized, without multiple iterations, greatly reducing the complexity of programming the numerical simulation program for the characteristic frequencies of the water conveyance system.

[0086] (3) The model system has strong structure, which is conducive to the docking of subsequent secondary development and well meets the needs of simulation calculation of the water conveyance system under the current trend of gigantism and complexity of the water conveyance system. Description of the Drawings

[0087] Figure 1 is the flowchart of the modular implementation method for obtaining the characteristic frequencies of the complex water conveyance system provided by the present invention.

[0088] Figure 2 is the layout schematic diagram of the water conveyance system in the embodiment.

[0089] Figure 3 is the amplitude characteristic diagram of the frequency response in the embodiment. Detailed Embodiments

[0090] The present invention will be further described in detail below in conjunction with the drawings and specific embodiments:

[0091] Figure 1 is the flowchart of the modular implementation method for obtaining the characteristic frequencies of the complex water conveyance system provided by the present invention.

[0092] The modular implementation method for obtaining the characteristic frequencies of a complex water conveyance system includes the following steps:

[0093] S1: Decompose the complex water conveyance system into independent simple elements;

[0094] S2: Number the simple elements and the nodes adjacent to the simple elements in sequence;

[0095] S3: Compile the simple elements obtained by decomposing the complex water conveyance system into sub - matrix expressions in sequence;

[0096] S4: Construct the total matrix of the water conveyance system;

[0097] S5: Determine the excitation input nodes and response output nodes;

[0098] S6: Solve the total matrix of the water conveyance system to determine the frequency response characteristics of the system.

[0099] The following takes Figure 2 the water conveyance system shown as an example to further illustrate this modular implementation method. The solution methods for the characteristic frequencies of other water conveyance systems are similar.

[0100] For a certain water conveyance system, the water level of the upstream reservoir 1 is 100 m, the length of the elastic pipeline 2 is 500 m, the water hammer wave velocity is 1200 m / s, the cross - sectional area of the pipeline is 10 m 2 , the coefficient of head loss along the way λ = 0.012, the water level of the downstream reservoir 4 is 85 m, the coefficient of head loss at the outlet of the upper reservoir is 0.5, and the coefficient of head loss at the inlet of the lower reservoir is 0.9. In order to obtain the frequency response characteristics of the node head on the upstream side of the water turbine with respect to the guide vane opening fluctuation, in this implementation, an oscillating valve is used to simulate the influence of the water turbine guide vane on the head fluctuation.

[0101] As Figure 2 shown, it is a typical water conveyance system, which consists of an upstream reservoir 1, an elastic pipeline 2, an oscillating valve 3 and a downstream reservoir 4. The upstream end of the elastic pipeline 2 is connected to the upstream reservoir 1, the downstream end of the elastic pipeline 2 is connected to the downstream reservoir 4, and the oscillating valve 3 is arranged at one end of the elastic pipeline 2 close to the downstream reservoir 4.

[0102] Combined with the embodiments, the steps of the modular implementation method for obtaining the characteristic frequencies of the complex water conveyance system provided by the present invention are as follows:

[0103] S1: Decompose the complex water conveyance system into independent simple elements; in this embodiment, the water conveyance system is decomposed into an upstream reservoir 1, an elastic pipeline 2, an oscillating valve 3 and a downstream reservoir 4;

[0104] S2: Number the simple elements and the nodes adjacent to the simple elements in sequence; in this embodiment, according to the connection sequence, the node between the upstream reservoir 1 and the elastic pipeline 2 is designated as ①, the node between the elastic pipeline 2 and the oscillating valve 3 is designated as ②, and the node between the oscillating valve 3 and the downstream reservoir 4 is designated as ③;

[0105] S3: Compile the simple elements after decomposing the complex water conveyance system into sub - matrix expressions one by one. In this implementation: The sub - matrix expressions of the upstream reservoir 1 and the downstream reservoir 4 are as follows:

[0106]

[0107]

[0108] The sub - matrix expression of the elastic pipeline 2 is as follows:

[0109]

[0110] The sub - matrix expression of the oscillating valve 3 is as follows:

[0111]

[0112] S401: Construct an n×n matrix [A] according to the system total matrix dimension being equal to the sum of the real nodes and virtual nodes of the water conveyance system. Among them, the element in the i - th row and j - th column is represented by A i,j and clear all elements. In this implementation, there are three real nodes, namely node ① between the upstream reservoir 1 and the elastic pipeline 2, node ② between the elastic pipeline 2 and the oscillating valve 3, and node ③ between the oscillating valve 3 and the downstream reservoir 4. There is one virtual node, which is the opening change amount y of the non - hydraulic state of the oscillating valve. Therefore, the system total matrix should be a four - dimensional matrix;

[0113] S402: Construct an n×1 vector and clear all elements; construct an n×1 vector and clear all elements. In this embodiment, a 4×1 vector and a 4×1 vector should be constructed and all of them should be cleared;

[0114] S403: Place the sub - matrices of each hydraulic element into the system total matrix in turn. In this implementation, according to the total matrix construction principle, the total matrix of the water conveyance system is as follows:

[0115]

[0116] S5: Determine the excitation input node and the response output node. In this embodiment, both the excitation input node and the response output node are node ②;

[0117] S6: Solve the total matrix of the water conveyance system to determine the frequency response characteristics of the system. In this implementation, the frequency response characteristic solution formula of node ② is as follows:

[0118]

[0119] The frequency response characteristic results of node ② of this system are as follows Figure 3 shown. The system forms a fundamental water hammer wave with a frequency response peak of 5.21 dB at an angular frequency of 4.05 rad / s, a third harmonic of the water hammer wave with a frequency response peak of 5.23 dB at an angular frequency of 11.07 rad / s, and a fifth harmonic of the water hammer wave with a frequency response peak of 5.31 dB at an angular frequency of 18.79 rad / s. It can be seen that the response peak of the system is relatively high at low angular frequencies, which may cause hydraulic excitation phenomena. Therefore, additional frequency analysis is required for the design conditions that may trigger low-frequency oscillations of the system. From the above analysis, it can be seen that the solution of the characteristic frequency of the water conveyance system by this invention is reliable.

[0120] The above specific embodiments are used to explain and illustrate the present invention. They are only the preferred embodiments of the present invention and do not limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and scope of the claims of the present invention fall within the protection scope of the present invention.

Claims

1. A modular implementation method for obtaining the characteristic frequencies of a complex water conveyance system, characterized in that: The modular implementation method for obtaining the characteristic frequencies of the complex water conveyance system includes the following steps: S1: Decompose the complex water conveyance system into independent simple elements, where the simple elements include elastic pipeline elements, rigid pipeline elements, impedance elements, surge chamber elements, reservoir elements, oscillating valve elements, and turbine elements; S2: Number the simple elements and the nodes adjacent to the simple elements in sequence; S3: Compile the simple elements after decomposing the complex water conveyance system into sub-matrix expressions in sequence; S4: Construct the total matrix of the water conveyance system; S5: Determine the excitation input node and the response output node; S6: Solve the total matrix of the water conveyance system to determine the frequency response characteristics of the system; In step S1: The principle for defining simple elements is hydraulic elements with flow inflow or outflow: If the inflow and outflow of the flow occur at the same end point, it is considered a single-endpoint element; If the inflow and outflow of the flow occur at different end points, it is considered a multi-endpoint element; In step S4: The system matrix is constructed as follows: S401: Construct an \(n\times n\) matrix \([A]\) according to the fact that the total matrix dimension of the system is equal to the sum of the real nodes and the virtual nodes of the water conveyance system. Among them, the element in the \(i\)-th row and \(j\)-th column is represented by \(A\) i,j and clear all elements to zero; S402: Construct an n×1 vector and clear all elements; construct an n×1 vector and clear all elements; S403: Place the sub-matrices of each hydraulic element into the total system matrix in sequence; In step S403: If the hydraulic element is a single-endpoint element and its connected real node number is assumed to be i, then directly place the element on the left side of the hydraulic element matrix into the diagonal A of matrix [A] i,i In, directly place the element on the right side of the hydraulic element matrix into the vector In, the formula is as follows: A i,i = A i,i + a i,i Q i = Q i + q i If the hydraulic element is a multi-endpoint element, the sum of its real nodes and virtual nodes is m. Assuming that its connected real node numbers are i and j, the elements on the left side of the hydraulic element matrix equation are placed into matrix [A], and the elements on the right side of the hydraulic element matrix equation are placed into vector as follows: A i,i = A i,i + a i,i 、A i,j = A i,j + a i,j ···A i,i+m-1 = A i,i+m-1 + a i,i+m-1 Q i = Q i + q i A j,i = A j,i + a j,i 、A j,j = A j,j + a j,j ···A j,i+m-1 = A j,i+m-1 + a j,i+m-1 Q j = Q j + q j ······ A i+m-1,i = A i+m-1,i + a i+m-1,i 、A i+m-1,j = A i+m-1,j + a i+m-1,j ···A i+m-1,i+m-1 = A i+m-1,i+m-1 + a i+m-1,i+m-1 Q i+m-1 = Q i+m-1 + q i+m-1 (m = 2) or Q i+m-1 = Q i+m-1 + y i+m-1 (m > 2) Among them, there is an inflow rate q at node i i ' and an outflow rate q i ”. Taking the inflow node as the positive direction, we have q i '+q i ” = 0. Therefore, the matrix form expression for multiple endpoints is as follows: Where: x i+m-1 , y i+m-1 are variables in a non-hydrodynamic state.

2. The modular implementation method for obtaining the characteristic frequencies of a complex water conveyance system according to claim 1, characterized in that: The impedance element is a hydraulic element that has a high degree of similarity in hydraulics in the watercourse; The oscillating valve element is a valve whose valve opening can undergo periodic changes without being directly affected by the head state.

3. The modular implementation method for obtaining the characteristic frequencies of a complex water conveyance system according to claim 1, characterized in that: In step S2: The nodes are divided into real nodes and virtual nodes. The real nodes refer to the nodes that divide each hydraulic element, and the virtual nodes mainly correspond to the virtual end points in the hydraulic elements; for the sorting method of node numbers, first number each real node in sequence according to the composition order of the hydraulic elements of the water conveyance system, and then number the virtual nodes in sequence.

4. The modular implementation method for obtaining the characteristic frequencies of a complex water conveyance system according to claim 1, characterized in that: In step S3: The dimension of the element sub-matrix is equal to the sum of the real end points and virtual end points of the hydraulic element; The element matrix of the elastic pipeline is: Where: s is the Laplace operator; ρ is the Allievi pipeline constant; Q0 is the reference flow rate, H0 is the reference head, α is the head loss coefficient per unit length; a is the water hammer wave speed; L is the pipeline length; q i is the flow rate into node i; q j is the flow rate into node j; h i is the head of node i; h j is the head of node j; The element matrix of the rigid pipeline is: Where: T w is the pipeline water flow acceleration time constant; s is the Laplace operator; β is the pipeline head loss coefficient, Q0 is the reference flow rate, and H0 is the reference head; q i is the flow rate flowing into node i; q j is the flow rate flowing into node j; h i is the head of node i; h j is the head of node j; The matrix of the impedance element is: Where: β is the pipeline head loss coefficient, Q0 is the reference flow rate, and H0 is the reference head; q i is the flow rate flowing into node i; q j is the flow rate flowing into node j; h i is the head of node i; h j is the head of node j; The element matrix of the surge chamber is: Where: A s is the cross-sectional area of the surge chamber; s is the Laplace operator; Q0 is the reference flow rate, and H0 is the reference head; q i is the flow rate into node i; h i is the head of node i; The element matrix of the reservoir is: Where: β is the pipeline head loss coefficient, Q0 is the reference flow rate, and H0 is the reference head; q i is the flow rate flowing into node i; h i is the head of node i; The element matrix of the oscillating valve: Where: Q y is the valve opening - flow transfer coefficient; y is the opening of the oscillating valve; β is the pipeline head loss coefficient, Q0 is the reference flow rate, H0 is the reference head; q i is the flow rate flowing into node i; q j is the flow rate flowing into node j; h i is the head of node i; h j is the head of node j; The element matrix of the turbine is: Where: Q n is the self-regulation coefficient of the water turbine speed-flow; Q y is the valve opening-flow transfer coefficient; G(s) is the transfer function of the governor; P h = 0.5(1 + e q )(1 - Q n ) + (1 + e h ); P n = (1 + e q )Q n ; P y = (1 + e q )Q y ; e h is the relative efficiency with respect to the relative head change rate; e q is the relative efficiency with respect to the relative flow rate change rate; T a is the inertia time constant of the unit; s is the Laplace operator; E n is the engine self-regulation coefficient; q i is the flow rate into node i; q j is the flow rate into node j; h i is the water head of node i; h j is the water head of node j; y is the opening of the oscillating valve; p is the relative output of the hydrogenerator; p out is the actual output of the water turbine; n is the relative speed of the hydrogenerator.

5. The modular implementation method for obtaining the characteristic frequency of the complex water conveyance system according to claim 1, wherein: In step S5: The excitation input node is specifically the input node of the system excitation signal, and the response output node is the output node of the head fluctuation reaction.

6. The modular implementation method for obtaining the characteristic frequency of the complex water conveyance system according to claim 1, wherein: In step S6: The frequency response characteristics of the system are determined by the ratio of the head fluctuation at the response output node to the excitation at the input node. The formula is as follows: or Where: h ～ 、q ～ 、y ～ are the dimensionless variables of water head, flow rate, and opening; H out 、Q in 、Y in are the amplitude of water head output fluctuation, the amplitude of flow rate input fluctuation, and the amplitude of opening input fluctuation; ω is the angular frequency; is the output phase lag; H0, Q0, and Y0 are the reference values of water head, flow rate, and opening.

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