A graph theory-based analysis method for operation stability of a water power station water delivery power generation system

Abstract

The application discloses a kind of based on graph theory's water and electricity station water power generation system operation stability analysis method, this method is first based on the basic principle of graph theory, the whole system of water and electricity station is represented as the topological network formed by directed edge and node, and each node and each edge is numbered;Then according to the numbering of node and edge and water and electricity station system parameters, construct water and electricity station system overall frequency domain characteristic matrix C (s), establish characteristic equation det [C (s)]=0, and by solving eigenvalue s, with the real part of all eigenvalues is less than 0 as the basis for distinguishing stability.The method proposed by the application can conveniently realize the stability analysis general modeling of arbitrary hydraulic system topological structure, greatly facilitates the water and electricity station system operation stability analysis, provides a new way for in-depth analysis of different complex hydraulic topological water and electricity station free vibration characteristics and stability domain, and has wide application prospect.

Description

Technical Field

[0001] This invention relates to the field of water conservancy and hydropower engineering, specifically to a method for analyzing the operational stability of a hydropower station's water conveyance and power generation system based on graph theory. Background Technology

[0002] The operational stability of a hydropower station's water conveyance and power generation system directly affects the operational safety and regulation performance of the hydropower units, making it a critical issue in the design and operation of hydropower projects. As hydropower development in my country moves towards larger scales and greater complexity, the topology of hydropower station water conveyance systems is becoming increasingly complex. Simultaneously, under the new power system framework, hydropower units bear heavier responsibilities for peak shaving and frequency regulation, and their operating conditions switch more frequently, leading to increasingly prominent system stability issues. Therefore, conducting research on the operational stability of hydropower station water conveyance and power generation systems has significant theoretical and engineering application value.

[0003] Currently, commonly used methods for stability analysis of hydropower station water conveyance and power generation systems mainly include the state-space method and transfer function method based on rigid water bodies, and the transfer matrix method and hydraulic impedance method based on elastic water bodies. Among these, the state-space method and transfer function method are mostly used for stability mechanism analysis of systems with relatively simple hydraulic structures (such as single-pipe, single-unit systems). However, when dealing with complex hydropower station layouts with multiple pipes, multiple units, and multiple surge chambers, the model order often becomes too high, making it difficult to quickly and conveniently establish a stability analysis model. On the other hand, the transfer matrix method and hydraulic impedance method usually require deriving the corresponding transfer matrix or hydraulic impedance expression for various hydraulic components. When the system includes special hydraulic components such as three-way pipes or connected surge chambers, the derivation often needs to be redone, thus limiting their versatility and scalability.

[0004] Therefore, there is an urgent need to invent a method for analyzing the operational stability of hydropower stations with complex hydraulic topologies, so as to provide a basis for the safe and stable operation of the power station system. Summary of the Invention

[0005] Purpose of the invention: To overcome the shortcomings of existing technologies, this invention proposes a method for analyzing the operational stability of hydropower transmission and generation systems based on graph theory. This method can conveniently realize the general modeling of any hydraulic system topology, and has the advantages of simple modeling process, wide applicability and strong engineering practicality.

[0006] Technical solution: To achieve the above-mentioned objectives, the present invention adopts the following technical solution:

[0007] The present invention provides a graph theory-based method for analyzing the operational stability of a hydropower station's water conveyance and power generation system, comprising the following steps:

[0008] Step 1: Represent the hydropower station's water transmission and power generation system as a topological network composed of directed edges and nodes, and assign a unified number to each node and edge in the topological network;

[0009] Step 2: Based on the edge and node numbers in Step 1, and in accordance with the construction rules of the directed edge-node association matrix, determine the directed edge-node association matrix A of the hydropower station's water conveyance and power generation system.

[0010] Step 3: Based on the node and edge numbers from Step 1 and the hydropower station system parameters, construct the overall frequency domain feature matrix C(s). The overall frequency domain feature matrix C(s) includes the directed edge-node correlation submatrix A corresponding to all nodes except the reservoir node. I Pipeline parameter diagonal matrix M, unit coefficient matrix B T and pressure regulating chamber coefficient matrix B S ;

[0011] Step 4: Based on the overall frequency domain characteristic matrix C(s) of the hydropower station system constructed in Step 4, let its determinant satisfy det[C(s)] = 0, and solve for the eigenvalues ​​s. If the real parts of all eigenvalues ​​are less than 0, the system is determined to be in a stable state; if there exists any eigenvalue whose real part is greater than or equal to 0, the system is determined to be in an unstable state.

[0012] Furthermore, when numbering the nodes in the topology network of the hydropower station's water conveyance and power generation system as described in step one, the node containing the upstream reservoir is set as the nth node. D -1 node, setting the node where the downstream reservoir is located as the nth node. D There are n nodes, where n is a number of nodes. D The total number of nodes in the topological network.

[0013] Furthermore, the construction rules for the directed edge-node incidence matrix in step two are as follows: In the formula: and Let i represent the upstream node set and downstream node set of the j-th pipeline segment, respectively; i represents the node number; and j represents the edge number.

[0014] Furthermore, the overall frequency domain feature matrix C(s) described in step three is:

[0015] The directed edge-node correlation submatrix A corresponding to all nodes except the reservoir node. I The relationship between the directed node-edge incidence matrix A from step two and the following is: In the formula: A RThis represents the directed edge-node correlation submatrix corresponding to the reservoir nodes.

[0016] The diagonal matrix M of the pipeline parameters is: In the formula: Q j For flow rate; L j A is the length of the pipe. j Where is the cross-sectional area of ​​the pipe; g is the acceleration due to gravity; This is the head loss coefficient; For the Laplace operator; Indicates the pipe number; the superscript (0) indicates the initial value of the parameter; n p Indicates the total number of pipe numbers;

[0017] The unit coefficient matrix B T for: In the formula: Let n be the set of edge numbers containing the generator set; the subscript r represents the generator set number; n T Z represents the total number of generating units; T,r The hydraulic impedance of the unit; This is the Laplace transform of the turbine head deviation value; The Laplace transform of the turbine flow deviation value; , , , , , These are the turbine characteristic coefficients, which can be obtained from the turbine characteristic curves. This is the load self-adjustment coefficient; For proportional gain; For integral gain;

[0018] The pressure regulating chamber coefficient matrix B S : In the formula: Since the mathematical models of the surge tank and the gate well are the same, in order to simplify the model expression and matrix construction process, the surge tank and the gate well are uniformly numbered and treated as surge tank nodes. This is the set of node numbers for the surge tank and gate well; the subscript 'c' indicates the surge tank or gate well number; F c n represents the area of ​​the c-th pressure regulating chamber or gate well; S This indicates the total number of pressure regulating chambers and gate wells.

[0019] Beneficial effects: The stability analysis method proposed in this invention can conveniently realize the general modeling of any hydraulic system topology, which greatly facilitates the stability analysis of hydropower station system operation and provides a new approach for in-depth analysis of the free vibration characteristics and stability domain of hydropower stations with different complex hydraulic topologies, and has broad application prospects. Attached Figure Description

[0020] Figure 1 This is a schematic diagram of the stability analysis method of the present invention;

[0021] Figure 2 This is a schematic diagram of a hydropower station system according to an embodiment of the present invention;

[0022] Figure 3 This is a stability domain diagram of the unit speed governor parameters according to an embodiment of the present invention;

[0023] Figure 4 The diagram shows the speed response under different governor parameters in an embodiment of the present invention. Detailed Implementation

[0024] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0025] according to Figure 1 The flowchart shown below illustrates the calculation steps for this implementation.

[0026] Step 1: Represent the hydropower station's water transmission and power generation system as a topological network composed of directed edges and nodes, and assign a unified number to each node and edge in the topological network;

[0027] Step 2: Based on the edge and node numbers in Step 1, and in accordance with the construction rules of the directed edge-node association matrix, determine the directed edge-node association matrix A of the hydropower station's water conveyance and power generation system.

[0028] Step 3: Based on the node and edge numbers from Step 1 and the hydropower station system parameters, construct the overall frequency domain feature matrix C(s). The overall frequency domain feature matrix C(s) includes the directed edge-node correlation submatrix A corresponding to all nodes except the reservoir node. I Pipeline parameter diagonal matrix M, unit coefficient matrix B T and pressure regulating chamber coefficient matrix B S ;

[0029] Step 4: Based on the overall frequency domain characteristic matrix C(s) of the hydropower station system constructed in Step 4, let its determinant satisfy det[C(s)] = 0, and solve for the eigenvalues ​​s. If the real parts of all eigenvalues ​​are less than 0, the system is determined to be in a stable state; if the real part of any eigenvalue is greater than or equal to 0, the system is determined to be in an unstable state.

[0030] Step 1: Represent the hydropower station's water transmission and power generation system as a topological network composed of directed edges and nodes, and assign a unified number to each node and edge in the topological network.

[0031] Specifically, in step 1, when numbering the nodes in the topology network of the hydropower station's water conveyance and power generation system, the node containing the upstream reservoir is set as the nth node. D -1 node, setting the node where the downstream reservoir is located as the nth node. D There are n nodes, where n is a number of nodes. D The total number of nodes in the topological network.

[0032] Step 2: Based on the edge and node numbers in Step 1, and in accordance with the construction rules of the directed edge-node association matrix, determine the directed edge-node association matrix A of the hydropower station's water conveyance and power generation system.

[0033] Specifically, in step 2, the momentum equation for the pressurized pipeline is: (1) In the formula: For traffic; This refers to the length of the pipe. This refers to the cross-sectional area of ​​the pipe. It is the acceleration due to gravity; This is the head loss coefficient; The water head at the upstream section of the pipeline; The water head at the downstream section of the pipeline; Indicates the pipe number.

[0034] Equation (1) can be expressed in terms of deviation values: (2) In the formula: This represents the flow rate deviation value. This refers to the head deviation value at the upstream section of the pipeline. The value is the head deviation at the upstream section of the pipeline; the superscript (0) indicates the initial value of the parameter.

[0035] Taking the Laplace transform of equation (2), we get: (3) In the formula: , They are respectively , The Laplace transform of; for The Laplace transform of; For the Laplace operator.

[0036] For the edge containing the generator unit, the edge equation is: (4) In the formula: The turbine head is represented by the subscript 'r', which indicates the unit number.

[0037] Expressing equation (4) in terms of deviation values ​​and ignoring nonlinear terms, we get: (5) In the formula: This represents the turbine head deviation value.

[0038] The flow characteristics and torque characteristics of a water turbine can be expressed as: (6) (7) In the formula: This represents the relative deviation of the turbine head. ; This represents the relative deviation of the turbine's flow rate. ; This represents the relative deviation of the rotational speed. ; This represents the relative deviation of the guide vane opening. ; This represents the relative deviation of the turbine output. ; , , , , These are the turbine's head, flow rate, rotational speed, opening degree, and output; , , , , , These are the turbine characteristic coefficients, which can be obtained from the turbine characteristic curves; the subscript r indicates the turbine number.

[0039] The generator equation is: (8) In the formula: The inertial time constant of the unit; This is the load deviation value. ; This represents the load self-regulation coefficient; the subscript r indicates the generator number.

[0040] The equation for a PI speed controller is: (9) In the formula: For proportional gain; This represents the integral gain; the subscript r indicates the speed controller number.

[0041] Based on equations (6)-(9) and using the Laplace transform, the hydraulic impedance of the unit can be obtained. for: (10) In the formula: Z T,r The hydraulic impedance of the unit; This is the Laplace transform of the turbine head deviation value; The Laplace transform of the turbine flow deviation value; This is the load self-adjustment coefficient; For proportional gain; This is the integral gain.

[0042] Substituting equation (10) into equation (5) and eliminating We can obtain the equation containing the edge where the unit is located: (11)

[0043] For the reservoir node, the equation is: (12) In the formula: i represents the node number.

[0044] For a series node, the inflow and outflow balance is satisfied, and the equation is: (13) In the formula: This represents the flow deviation value of the inflow node. This represents the flow deviation value of the outflow node.

[0045] For the surge tank and gate well nodes, the continuity equation is: (14) In the formula: Indicates the area of ​​the pressure regulating chamber or gate well; This indicates the water level deviation value of the surge tank; the subscript 'c' indicates the number of the surge tank or gate well.

[0046] Performing Laplace transforms on equations (12)-(14) respectively, we get: (15) (16) (17)

[0047] The rules for constructing the directed edge-node incidence matrix are as follows: (18) In the formula: and Let i represent the upstream node set and downstream node set of the j-th pipeline segment, respectively; i represents the node number; and j represents the edge number.

[0048] Step 3: Based on the node and edge numbers from Step 1 and the hydropower station system parameters, construct the overall frequency domain feature matrix C(s). The overall frequency domain feature matrix C(s) includes the directed edge-node correlation submatrix A corresponding to all nodes except the reservoir node. I Pipeline parameter diagonal matrix M, unit coefficient matrix B T and pressure regulating chamber coefficient matrix B S ;

[0049] Specifically, in step 3, let We can obtain the equation for each edge: (19) In the formula: ; Indicates the number of nodes; This represents the rows corresponding to nodes other than reservoirs; This represents the row corresponding to the reservoir node.

[0050] because Equation (19) can be written as: (20)

[0051] Combining equations (3), (11), and (20), the edge equations of the entire system can be obtained as follows: (twenty one) (twenty two) (twenty three) In the formula: M is the diagonal matrix of pipe parameters; This is the unit coefficient matrix, which is a diagonal matrix. Non-zero diagonal elements only appear on the sides where the unit is located; Q j For flow rate; L j A is the length of the pipe. j Where is the cross-sectional area of ​​the pipe; g is the acceleration due to gravity; This is the head loss coefficient; For the Laplace operator; Indicates the pipe number; the superscript (0) indicates the initial value of the parameter; np Indicates the pipe number; Let n be the set of edge numbers containing the generator set; the subscript r represents the generator set number; n T Indicates the total number of generating units;

[0052] Based on the fundamental principles of graph theory, the equations for the internal nodes can be obtained as follows: (twenty four)

[0053] Combining equations (16), (17), and (24), the nodal equations of the entire system can be obtained as follows: (25) (26)

[0054] In the formula: Since the mathematical models of the surge tank and the gate well are the same, in order to simplify the model expression and matrix construction process, the surge tank and the gate well are uniformly numbered and treated as surge tank nodes. It is the pressure regulating chamber coefficient matrix. This matrix is ​​a diagonal matrix, and non-zero diagonal elements only appear at the nodes where the pressure regulating chamber is located. n is the set of node numbers for the pressure regulating chamber and gate well. S This indicates the total number of pressure regulating chambers and gate wells.

[0055] In summary, by rewriting equations (21) and (25) in matrix form, we can obtain the overall frequency domain model of the hydropower station system: (27)

[0056] Equation (27) can be further written as: (28) (29) (30)

[0057] Matrix C(s) is the overall frequency domain characteristic matrix of the system, and its determinant is denoted as C(s) = det[C(s)]. According to linear algebra theory, equation (28) can only have a non-zero solution when det[C(s)] = 0. Therefore, by solving det[C(s)] = 0, the characteristic roots of the system can be obtained, and then the free vibration characteristics and stability of the hydropower station system can be analyzed. If the real parts of all characteristic roots are less than 0, the system is stable; if there exists any characteristic root with a real part greater than 0, the system is unstable.

[0058] The following section uses a hydropower station system with a tailrace connected to a surge tank as an example to further illustrate this method. A schematic diagram of the power station layout is shown below. Figure 2 The parameters of the water conveyance and power generation system are shown in Table 1.

[0059] Table 1. Parameters of the power station water conveyance system including the tailrace connection surge chamber: 1 382.70 75.40 9.19 2 28.42 9.47 0.00 3 154.80 125.80 0.30 4 391.10 75.50 9.29 5 28.42 9.47 0.00 6 154.80 125.80 0.30 7 399.60 75.50 9.39 8 28.42 9.47 0.00 9 154.80 125.80 0.30 10 154.80 187.10 1.02 11 157.30 187.10 0.95 12 214.70 187.10 1.11 13 39.30 187.10 0.33 14 1303.60 269.80 1.50

[0060] The system parameters of the hydropower station including the tailrace and surge tank are shown in Table 1. To verify the effectiveness of the proposed graph theory-based method for analyzing the operational stability of the hydropower station's water conveyance and power generation system, calculations were performed according to steps 1-4.

[0061] According to the proposed method, the block matrix of the power station is as follows: (31) (32)

[0062] (33)

[0063] (34)

[0064] (35)

[0065] By combining equations (31)-(35), the overall frequency domain model of the hydropower station system including the tailrace and surge tank can be obtained. Similarly, the characteristic roots of the system can be obtained by solving det[C(s)] = 0.

[0066] The stability domain of the unit governor parameters was plotted using a graph theory-based method for analyzing the operational stability of a hydropower station's water conveyance and power generation system. The results are as follows: Figure 3 As shown, the area below the curve corresponds to the stability region.

[0067] To further verify the accuracy of the characteristic root calculation results, two sets of governor parameter points W1(K) were selected within the obtained stability region. p = 1, K i = 0.45) and W2(K p = 1, K i = 0.50), the corresponding eigenvalues ​​of the hydropower station's water conveyance and power generation system were solved using a graph theory-based stability analysis method, and the results are shown in Table 2. Table 2 shows that all eigenvalues ​​corresponding to parameter point W1 have real parts less than 0, indicating system stability; while parameter point W2 has eigenvalues ​​with real parts greater than 0, indicating system instability.

[0068] Table 2. Eigenvalues ​​at parameter points W1 and W2:

[0069] To further verify the above conclusions, a sudden 5% load reduction of the first unit was selected as the disturbance condition. The fourth-order Runge-Kutta method was used to calculate the system's time-domain response, and the results are as follows: Figure 4As shown, at parameter point W1, the speed response gradually converges, indicating that the system is stable; at parameter point W2, the speed response gradually diverges, indicating that the system is unstable.

[0070] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for analyzing the operational stability of a hydropower station's water conveyance and power generation system based on graph theory, characterized in that, Includes the following steps: Step 1: Represent the hydropower station's water transmission and power generation system as a topological network composed of directed edges and nodes, and assign a unified number to each node and edge in the topological network; Step 2: Based on the edge and node numbers in Step 1, and in accordance with the construction rules of the directed edge-node association matrix, determine the directed edge-node association matrix A of the hydropower station's water conveyance and power generation system. Step 3: Based on the node and edge numbers from Step 1 and the hydropower station system parameters, construct the overall frequency domain feature matrix C(s). The overall frequency domain feature matrix C(s) includes the directed edge-node correlation submatrix A corresponding to all nodes except the reservoir node. I Pipeline parameter diagonal matrix M, unit coefficient matrix B T and pressure regulating chamber coefficient matrix B S ; Step 4: Based on the overall frequency domain characteristic matrix C(s) of the hydropower station system constructed in Step 4, let its determinant satisfy det[C(s)] = 0, and solve for the characteristic roots s; If the real part of all characteristic roots is less than 0, the system is considered to be in a stable state; if the real part of any characteristic root is greater than or equal to 0, the system is considered to be in an unstable state.

2. The method for analyzing the operational stability of a hydropower station's water conveyance and power generation system based on graph theory according to claim 1, characterized in that, In step one, when numbering the nodes in the topology network of the hydropower station's water conveyance and power generation system, the node containing the upstream reservoir is set as the nth node. D -1 node, setting the node where the downstream reservoir is located as the nth node. D There are n nodes, where n is a number of nodes. D The total number of nodes in the topological network.

3. The method for analyzing the operational stability of a hydropower station's water conveyance and power generation system based on graph theory according to claim 1, characterized in that, The construction rules for the directed edge-node incidence matrix in step two are as follows: In the formula: and Let i represent the upstream node set and downstream node set of the j-th pipeline segment, respectively; i represents the node number; and j represents the edge number.

4. The method for analyzing the operational stability of a hydropower station's water conveyance and power generation system based on graph theory according to claim 1, characterized in that, The overall frequency domain feature matrix C(s) mentioned in step three is: The directed edge-node correlation submatrix A corresponding to all nodes except the reservoir node. I The relationship between the directed node-edge incidence matrix A from step two and the following is: In the formula: A R This represents the directed edge-node association submatrix corresponding to the reservoir nodes; The diagonal matrix M of the pipeline parameters is: In the formula: Q j For flow rate; L j A is the length of the pipe. j Where is the cross-sectional area of ​​the pipe; g is the acceleration due to gravity; This is the head loss coefficient; For the Laplace operator; Indicates the pipe number; the superscript (0) indicates the initial value of the parameter; n p Indicates the total number of pipe numbers; The unit coefficient matrix B T for: In the formula: Let n be the set of edge numbers containing the generator set; the subscript r represents the generator set number; n T Z represents the total number of generating units; T,r The hydraulic impedance of the unit; This is the Laplace transform of the turbine head deviation value; The Laplace transform of the turbine flow deviation value; , , , , , These are the turbine characteristic coefficients, which can be obtained from the turbine characteristic curves. This is the load self-adjustment coefficient; For proportional gain; For integral gain; The pressure regulating chamber coefficient matrix B S : In the formula: Since the mathematical models of the surge tank and the gate well are the same, in order to simplify the model expression and matrix construction process, the surge tank and the gate well are uniformly numbered and treated as surge tank nodes. This is the set of node numbers for the surge tank and gate well; the subscript 'c' indicates the surge tank or gate well number; F c n represents the area of ​​the c-th pressure regulating chamber or gate well; S This indicates the total number of pressure regulating chambers and gate wells.

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