Probabilistic analysis method and system for hydro-generator in frequency domain under random excitation considering pressure fluctuation

Abstract

The application provides a kind of probability analytic method and system in frequency domain of hydroelectric generating set under random excitation considering pressure pulsation, the method comprises: obtaining the operating parameters of hydroelectric generating set, and constructing dynamic frequency response mathematical model;Considering pressure pulsation and random disturbance, a three-order random state space model of hydroelectric generating set is constructed;Based on the superposition principle of linear system, the frequency mean evolution trajectory and the frequency covariance matrix are solved respectively;Real-time mean and dynamic standard deviation are extracted to construct Gaussian probability density function;Finally, the in-domain probability is obtained in combination with the frequency safety operation boundary. The frequency characteristics of the system during starting and running can be analyzed by using the application, and online early warning and safety threshold setting for ultra-low frequency oscillation of hydroelectric generating set under high proportion of new energy power grid can be realized.

Description

Technical Field

[0001] This invention belongs to the field of smart grid technology, especially power system stability analysis and control technology. Specifically, it relates to a frequency safety assessment method for hydropower units in a grid with a high proportion of new energy sources under the combined effects of ultra-low frequency pressure pulsation and broadband random noise, and realizes probabilistic analysis of hydropower units in the frequency domain under fixed / random disturbance excitation. Background Technology

[0002] With the advancement of energy transition, a large proportion of renewable energy has been integrated into the grid, subjecting the power system to long-term random excitation disturbances across a wide frequency band. Against this backdrop, the frequency dynamics of the power system exhibit significant random fluctuation characteristics, and the risk of system frequency exceeding limits increases substantially.

[0003] Hydropower units, as the main power source responsible for frequency regulation and peak shaving in the power system, exhibit extremely complex dynamic characteristics. For the most widely used mixed-flow turbines, when operating outside their optimal conditions, vortex bands are easily generated inside the draft tube, leading to significant low-frequency or ultra-low-frequency pressure pulsations in the draft tube. These pressure pulsations, as a deterministic periodic forced disturbance source, not only cause periodic oscillations in the turbine's mechanical power but are also a significant factor in inducing ultra-low-frequency oscillations in the system.

[0004] In actual operation, hydropower units often simultaneously experience random excitations from the grid side and deterministic periodic excitations from pressure pulsations in the tailrace pipe within the unit. Existing power system frequency security assessment methods suffer from the following significant technical deficiencies when dealing with such combined excitation problems:

[0005] (1) Traditional time-domain simulation methods have a heavy computational burden and cannot meet the needs of online evaluation. For example, traditional numerical methods such as Monte Carlo simulation can handle complex nonlinearity and randomness, but usually require tens of thousands of random trajectory time-domain step integrations, which takes a long time and can only be used for offline planning. They cannot meet the timeliness requirements of modern power grids for online early warning and real-time control of frequency security.

[0006] (2) Traditional analytical methods make overly ideal assumptions about the excitation characteristics, which leads to a serious underestimation of the risk of exceeding the limit. For example, existing analytical methods for stochastic differential equations usually assume that the system is driven only by Gaussian white noise with a mean of zero, thus concluding that the mean of the system state is always zero. However, when pressure pulsation, a periodic forced excitation, is introduced, the expected frequency of the system is no longer zero, but will drift significantly with the frequency of pressure pulsation. If the periodic shift of the mean is ignored and the probability of exceeding the limit is calculated by relying only on the variance, the influence of pressure pulsation on the system boundary will be masked, resulting in serious distortion of the evaluation results. Summary of the Invention

[0007] To address the shortcomings and deficiencies of existing technologies, this invention provides a probabilistic analytical method for hydropower units under random excitation that takes into account pressure pulsation. This method aims to solve the problems of excessively long Monte Carlo simulation times and the inability of traditional analytical methods to handle mean shifts and evaluation distortions caused by random-periodic joint excitation. It provides a highly accurate and computationally efficient online analytical evaluation method for ultra-low frequency oscillation early warning and frequency safety threshold setting of hydropower units in high-proportion renewable energy power grids, enabling frequency safety assessment under random-periodic joint excitation and ensuring the frequency safety of the power system.

[0008] To achieve the above objectives, according to a first aspect of the present invention, a probabilistic analytical method for hydropower units under random excitation considering pressure pulsation in the frequency domain is proposed, comprising the following steps:

[0009] Step 1: Obtain the operating parameters of the hydropower unit and construct a dynamic frequency response mathematical model that includes the governor, water hammer effect, and rotor motion.

[0010] Step 2: Treat the tailrace pressure pulsation as a deterministic periodic excitation and the new energy source or load fluctuation as a stochastic excitation to construct a stochastic state-space model of the hydropower unit.

[0011] Step 3: Based on the principle of superposition of linear systems, solve the frequency mean trajectory of the state space model under deterministic periodic excitation and the frequency covariance matrix under random excitation; and determine the time-varying mean of the system frequency deviation and the dynamic standard deviation of the system frequency deviation under the action of common excitation.

[0012] Step 4: Under the condition that the system state is subjected to deterministic periodic excitation forced disturbance and random excitation continuous injection, extract the real-time mean and dynamic standard deviation of the system frequency deviation, and construct a Gaussian probability density function with time-varying characteristics.

[0013] Step 5: Set a safe operating boundary for the system frequency, and integrate the probability density function within the safe boundary to obtain the probability of the system frequency in the bounded fluctuation domain at a given time.

[0014] In an optional embodiment, in step 4 above, due to the continuous pull of the system state by the deterministic sinusoidal forced disturbance, the mean of the system frequency deviation is no longer zero, and its dynamic distribution center swings sinusoidally with the time axis; due to the continuous injection of broadband random excitation, the dynamic standard deviation of the frequency fluctuation increases exponentially and monotonically from the initial zero state over time, until it converges to the saturation constant in the system stability domain.

[0015] In an optional embodiment, in step 5 above, after obtaining the probability curve P within the bounded fluctuation domain... in After (t), if at any observation time t k P occurs in (tk ) < P limit (where P) limit If the preset ultra-low frequency oscillation alarm probability threshold is set, the over-limit alarm module of the hydropower station's central control system will be automatically triggered, and the dynamic gain of the hydropower unit's PI speed governor will be adjusted online or an active power output reduction command will be issued to realize the setting of the safety threshold and online early warning of ultra-low frequency oscillation of hydropower units under a high proportion of new energy power grid.

[0016] To achieve the above objectives, a computer system is also provided according to a second aspect of the present invention, comprising:

[0017] One or more processors;

[0018] The memory stores operable instructions that, when executed by the one or more processors, cause the one or more processors to perform operations, including the aforementioned process of the probabilistic analytical method for hydropower units in the frequency domain under random excitation taking into account pressure pulsations.

[0019] As can be seen from the above technical solutions of the present invention, compared with the prior art, the significant advantages of the present invention are:

[0020] Traditional low-frequency oscillation probabilistic analysis methods are generally based on Monte Carlo time-domain step simulations. While this method can handle randomness, it is computationally extremely time-consuming, making it difficult to meet the online safety assessment requirements in the context of interconnected power grids. Furthermore, traditional zero-mean analytical methods ignore the periodic shift in the mean caused by pressure pulsations, leading to severe distortion in the assessment of exceedance risks. Therefore, the method proposed in this invention transforms the solution of complex stochastic differential equations into algebraic matrix operations, significantly reducing computation time and providing computational support for online safety early warning in new power systems.

[0021] Furthermore, the probabilistic analytical method for hydropower units in the frequency domain proposed in this invention accurately calculates the sinusoidal periodic oscillation of the system's frequency mean based on the mechanism that pressure pulsation dominates mean drift and random excitation dominates variance diffusion. Simultaneously, by employing an order reduction strategy, the derivative terms of the input signal in the model are eliminated, ensuring the absolute rigor of Markov properties and analytical derivation, thus overcoming the problem of low accuracy in existing evaluation methods.

[0022] It should be understood that all combinations of the foregoing concepts and the additional concepts described in more detail below may be considered part of the inventive subject matter of this disclosure, provided that such concepts do not contradict each other. Furthermore, all combinations of the claimed subject matter are considered part of the inventive subject matter of this disclosure.

[0023] The foregoing and other aspects, embodiments, and features of the teachings of the present invention will be more fully understood from the following description in conjunction with the accompanying drawings. Other additional aspects of the invention, such as features and / or beneficial effects of exemplary embodiments, will become apparent from the following description or may be learned through practice of specific embodiments according to the teachings of the present invention. Attached Figure Description

[0024] The accompanying drawings are not intended to be drawn to scale. In the drawings, each identical or nearly identical component shown in the various figures may be denoted by the same reference numeral. For clarity, not every component is labeled in each figure. Embodiments of various aspects of the invention will now be described by way of example and with reference to the accompanying drawings.

[0025] Figure 1 This is a schematic flowchart of the probabilistic analytical method for hydropower units in the frequency domain under random excitation considering pressure pulsation, according to an embodiment of the present invention.

[0026] Figure 2 This is a schematic diagram of the calculation process of the offline Monte Carlo method according to an embodiment of the present invention.

[0027] Figure 3 This is a schematic diagram comparing the results of the frequency domain probabilistic analytical method for hydropower units and the Monte Carlo method in this embodiment of the invention.

[0028] Figure 4 This is a schematic diagram comparing the intra-domain probabilistic analysis method and the Monte Carlo method under different random excitation intensities according to an embodiment of the present invention.

[0029] Figure 5 This is a schematic diagram comparing the intra-domain probabilistic analysis method and the Monte Carlo method under different random excitation intensities according to an embodiment of the present invention.

[0030] Figure 6 This is a schematic diagram comparing the intra-domain probabilistic analysis method and the Monte Carlo method under different random excitation intensities according to an embodiment of the present invention. Detailed Implementation

[0031] To better understand the technical content of the present invention, specific embodiments are described below in conjunction with the accompanying drawings.

[0032] Various aspects of the invention are described in this disclosure with reference to the accompanying drawings, which illustrate numerous illustrative embodiments. The embodiments of this disclosure are not necessarily intended to encompass all aspects of the invention. It should be understood that the various concepts and embodiments described above, as well as those described in more detail below, can be implemented in any of many ways, because the concepts and embodiments disclosed herein are not limited to any particular implementation. Furthermore, some aspects of the invention disclosed may be used alone or in any suitable combination with other aspects of the invention disclosed.

[0033] Combination Figure 1 The example flow, as an optional embodiment, implements a probabilistic analytical method for hydropower units in the frequency domain under random excitation considering pressure pulsation, and the process includes the following steps:

[0034] Step 1: Obtain the operating parameters of the hydropower unit and construct a dynamic frequency response mathematical model that includes the governor, water hammer effect, and rotor motion.

[0035] Step 2: Treat the tailrace pressure pulsation as a deterministic periodic excitation and the new energy source or load fluctuation as a stochastic excitation to construct a stochastic state-space model of the hydropower unit.

[0036] Step 3: Based on the principle of superposition of linear systems, solve the frequency mean trajectory of the state space model under deterministic periodic excitation and the frequency covariance matrix under random excitation; and determine the time-varying mean of the system frequency deviation and the dynamic standard deviation of the system frequency deviation under the action of common excitation.

[0037] Step 4: Under the condition that the system state is subjected to deterministic periodic excitation forced disturbance and random excitation continuous injection, extract the real-time mean and dynamic standard deviation of the system frequency deviation, and construct a Gaussian probability density function with time-varying characteristics.

[0038] Step 5: Set a safe operating boundary for the system frequency, and integrate the probability density function within the safe boundary to obtain the probability of the system frequency in the bounded fluctuation domain at a given time.

[0039] Therefore, combining the above embodiments of the present invention with the probabilistic analytical method for hydropower units in the frequency domain under random excitation considering pressure pulsation, a third-order stochastic state-space model of hydropower units under joint excitation of pressure pulsation and random disturbance is constructed. Decoupling calculation of deterministic mean evolution and random variance diffusion is introduced, thereby accurately and quickly analyzing the time-varying probability characteristics of the system's frequency during the start-up and operation stages, and realizing online early warning and safety control of ultra-low frequency oscillations of hydropower units in high-proportion new energy power grids.

[0040] As an optional implementation, in step 1, the turbine of the hydroelectric generator unit includes a transfer function T with non-minimum phase zero-point characteristics.w (s) is represented as:

[0041] ;

[0042] The transfer function T is obtained by long polynomial division. w (s) is transformed into an algebraic sum of a constant term and a fraction, i.e.:

[0043] ;

[0044] Define the system intermediate variable corresponding to the time-domain response of the fraction as the internal state variable x of the turbine. int And the mechanical power deviation ΔP m Represented as a linear algebraic expression without state variable derivative terms:

[0045] ;

[0046] Among them, T w Let represent the time constant of the water hammer effect, s represent the Laplace operator, and K represent the time constant of the water hammer effect. P K I These represent the proportional gain and integral gain parameters of the PI speed controller, respectively, where Δf is the system frequency deviation, and x is the integral gain parameter. I x is the integral term of the frequency deviation. int These are the internal state variables of the water turbine.

[0047] As an optional implementation, in step 2, the constructed stochastic state-space model of the hydropower unit is a third-order stochastic state-space model, represented as an Iton-type linear stochastic differential equation with deterministic sinusoidal input:

[0048] ;

[0049] In the formula, A is the system state matrix, K is the random disturbance input matrix, dB(t) is the Gaussian white noise increment, and B... p For a deterministic periodic excitation input matrix, U p (t) represents the sinusoidal periodic forced input caused by pressure pulsation in the tailrace pipe; U p (t)=sin(ω p t), representing the angular frequency ω of the pressure pulsation in the turbine's tailrace pipe. p The deterministic cycle of decision-making forces input;

[0050] Where X represents the state vector of a third-order stochastic state-space model, X = [Δf(t), x I (t), x int (t)] T , Δf(t), x I (t), x int(t) represents the system frequency deviation, the integral term of the frequency deviation, and the internal state variables of the turbine at any time t, respectively.

[0051] As an optional implementation, in the differential equations of the aforementioned third-order stochastic state-space model, the system state matrix A takes the following values:

[0052]

[0053] In the formula, H is the system's equivalent inertial constant, and D is the equivalent damping coefficient.

[0054] Random perturbation input matrix K and deterministic periodic excitation input matrix B p They are respectively:

[0055] ;

[0056] ;

[0057] In the formula, σ1 is the random excitation intensity parameter from the generator side, and A p This is the coefficient for the amplitude of mechanical power change caused by pressure pulsation in the tailrace pipe.

[0058] As an optional implementation, in step 3, the frequency mean trajectory of the state-space model under deterministic periodic excitation is solved, including the following process:

[0059] Construct a mean vector μ of the system state X The nonhomogeneous ordinary differential equation of (t):

[0060] ;

[0061] In the formula ω p The angular frequency of pressure pulsation in the turbine's tailrace tube;

[0062] Under the condition that the initial state mean is zero, the non-homogeneous ordinary differential equation is solved to obtain the state mean trajectory.

[0063] Furthermore, the first element of the state mean trajectory is extracted as the time-varying mean μ of the system frequency deviation. Δf (t).

[0064] As an optional implementation, in step 3, the specific process of solving the frequency covariance matrix of the state-space model under random excitation is as follows:

[0065] First, the system state matrix A is decomposed into eigenvalues ​​to obtain the eigenvector matrix P and a diagonal matrix composed of eigenvalues. ;

[0066] Secondly, based on the random perturbation input matrix K, the intermediate matrix F used to map the noise intensity to each physical mode is calculated, F = F -1 KK T (F -1 ) T ;

[0067] Then, calculate the integral term matrix J describing the time-domain cumulative effect of noise energy in the system, where the elements in the i-th row and j-th column are J(i,j), and J(i,j) = [e ( λi + λj) (t-t0) -1] / ( λ i + λ j ), where λ i and λ j These are the eigenvalues ​​of matrix A;

[0068] Finally, the covariance matrix Var[X(t)] of the system state is calculated by multiplying the eigenvector matrix by the Hadamard product:

[0069] .

[0070] Furthermore, the square root of the element in the first row and first column of the covariance matrix is ​​extracted as the dynamic standard deviation σ of the system frequency deviation. f (t).

[0071] As an optional implementation, in step 5, the probability within the bounded fluctuation domain of the system frequency at a given time is expressed as:

[0072] ;

[0073] In the formula, P in (t) represents the bounded fluctuation domain probability of the system frequency at time t; Φ(·) is the cumulative distribution function of the standard normal distribution; f max and f min These are the upper and lower limit boundary values ​​that are preset for safe operation of the system frequency.

[0074] The technical solutions provided by the above embodiments of the present invention will be further explained below through simulation examples and in conjunction with the accompanying drawings.

[0075] In an optional embodiment, in step 1 above, the dynamic structural parameters of the hydropower station to be evaluated and its grid-connected area are first obtained: the system equivalent inertia constant H, the equivalent damping coefficient D, and the water hammer effect time constant T. w The proportional gain parameter K of the PI speed controller P and integral gain parameter K I .

[0076] Choosing the first state variable as Δf(t), we define the second state variable of the system as the time-domain integral term of the frequency deviation, i.e.:

[0077] e(f) = -Δf(t);

[0078] Define the third state variable of the system as the time-domain integral term of the frequency deviation, which satisfies the state differential relation:

[0079] ;

[0080] Under the combined action of proportional and integral elements, the control output signal U(t) of the speed governor can be characterized in the time domain as follows:

[0081] ;

[0082] The classic transfer function model of a water turbine is:

[0083] ;

[0084] Where s represents the Laplace operator.

[0085] Directly substituting the above equation into the state-space equation would result in the system input matrix containing high-order differential terms, directly undermining the solution basis of Itō's stochastic calculus. In this embodiment of the invention, to resolve this mathematical singularity, a polynomial long division method is used to perform polynomial separation on the proper fractions of the same order in the numerator and denominator. The specific process is as follows:

[0086] ;

[0087] Substituting the above equation into the frequency domain algebraic equation, we get:

[0088] ;

[0089] In an optional embodiment, the fractional term in the above equation is defined as the third state variable of the system:

[0090] ;

[0091] Performing an inverse Laplace transform back to the time domain on the above equation yields a first-order ordinary differential equation without input differential terms:

[0092] .

[0093] Substituting the governor output U(t) into the above equation to eliminate intermediate control variables, the deterministic state evolution equation for the third state variable of the system is finally derived as follows:

[0094]

[0095] Meanwhile, due to the introduction of the internal state variable x of the water turbine int(t), the deviation of the actual total mechanical power output of the turbine ΔP m It can also be characterized by the state variables being linear and without derivatives as follows:

[0096] ;

[0097] Among them, K P K I These represent the proportional gain parameter and integral gain parameter of the PI speed controller, respectively.

[0098] In an optional embodiment, in step 2 above, the total state vector X of the system is defined as: X = [Δf(t), x I (t), x int (t)] T , Δf(t), x I (t), x int (t) represents the system frequency deviation, the integral term of the frequency deviation, and the internal state variables of the turbine at any time t, respectively.

[0099] Considering the classical rotor motion equations of a grid-connected generator, its time-domain incremental differential equation is:

[0100] .

[0101] In an optional embodiment, the total external power deviation borne by the hydropower unit consists of a linear superposition of three parts:

[0102] 1) Active power regulation deviation ΔP generated by the speed control system m (t);

[0103] 2) The deterministic periodic mechanical power pulsation caused by the periodic oscillation inside the turbine's draft tube is characterized by: ΔP m，p (t) = A p sin(ω p t), where A p ω is the amplitude coefficient for converting pressure pulsations into power oscillations. p The angular frequency of pressure pulsation in the turbine's tailrace tube;

[0104] 3) The broadband irregular random power excitation caused by the drastic fluctuations of new energy sources on the grid side is characterized as σ1W1(t), where W1(t) represents the random excitation intensity parameter from the generator side, and W1(t) represents standard Gaussian white noise.

[0105] Combining the above conditions and substituting the total power deviation into the rotor motion equation, we can obtain:

[0106] ;

[0107] The establishment of a framework for , and The three first-order time-domain differential equations are combined and transformed into the form of standard Iton linear stochastic differential equations:

[0108] ;

[0109] In the formula, B(t) is the standard Wiener process.

[0110] Specifically, the system state matrix A and the deterministic periodic excitation input matrix B. p and random perturbation input matrix Characterized as:

[0111] ;

[0112] ;

[0113] ;

[0114] Among them, U p (t)=sin(ω p t), representing the angular frequency ω of the pressure pulsation in the turbine's tailrace pipe. p The deterministic cycle of the decision forces input.

[0115] In an optional embodiment, in step 3 above, the mean is calculated analytically using the superposition theorem. When there are no pressure pulsations, the mean of the frequency deviation is 0.

[0116] Apply the mathematical expectation operator to both sides of a linear stochastic differential equation. :

[0117] ;

[0118] Due to the standard Brownian motion characteristics, E[dB(t)]=0, the system state mean vector μ is defined. X (t), we can obtain:

[0119] ;

[0120] The exact analytical response of the above nonhomogeneous ordinary differential equation at any time t is represented by the following time-domain convolution integral:

[0121] ;

[0122] Given an initial state mean of zero, the non-homogeneous ordinary differential equation is solved to obtain the state mean locus. Further, after obtaining the state mean locus, the content corresponding to the first row is extracted as the time-varying mean μ of the system frequency deviation. Δf (t):

[0123] .

[0124] In an optional embodiment, variance is calculated analytically in step 3 above.

[0125] The covariance matrix can be expressed as:

[0126]

[0127] Since the deterministic sinusoidal perturbation terms are completely canceled out during subtraction, the analytical solution of the covariance matrix is ​​strictly determined by the following continuous-time definite integral:

[0128]

[0129] The above formula can be further derived as follows:

[0130] .

[0131] Since the analytical solution cannot be obtained directly through integration, matrix operations are used to solve for the analytical expression of the covariance. Eigenvalue decomposition is performed on the system state matrix A to obtain the eigenvector matrix P and a diagonal matrix composed of eigenvalues. :

[0132] ;

[0133] In the formula, The eigenvalue matrix is ​​a diagonal matrix. The eigenvector matrix P is from A to ... Similarity transformations have column vectors that are eigenvectors.

[0134] make Substituting this into the covariance expression, we get:

[0135] ;

[0136] Let F and G represent matrices respectively. and .

[0137] because If we use a diagonal matrix, then we can obtain:

[0138] ;

[0139] In the formula and These are the elements in the i-th row and j-th column of F and G, respectively.

[0140] Let H represent the matrix We can obtain:

[0141]

[0142] In the formula, H(i,j) represents the element in the i-th row and j-th column of matrix H.

[0143] Let J be the element in the i-th row and j-th column. The matrix can be obtained based on the relationship between H and F:

[0144] H = F°J;

[0145] The operator ° in the formula represents the Hadamard product.

[0146] Based on the above derivation, the analytical solution for the covariance matrix of the system state can be obtained as follows:

[0147] .

[0148] Furthermore, by extracting the elements of the first row and first column of the aforementioned covariance matrix, the dynamic standard deviation σ of the system frequency deviation under common excitation can be analytically obtained. f (t):

[0149] .

[0150] In step 4, due to the continuous pull of the deterministic sinusoidal forced disturbance on the system state, the mean of the system frequency deviation is no longer zero, and its dynamic distribution center swings sinusoidally with the time axis; due to the continuous injection of broadband random excitation, the dynamic standard deviation of the frequency fluctuation increases exponentially and monotonically with time from the initial zero state until it converges to the saturation constant in the system stability domain.

[0151] Furthermore, after obtaining the real-time mean and dynamic standard deviation of the system frequency deviation at any observation time, the probability density function of the system frequency is constructed:

[0152] .

[0153] In step 5, the maximum safe deviation of the instantaneous frequency allowed by the grid connection operation criteria is further set to f. max =0.001Hz, the lower limit of safety is f min =-0.001Hz.

[0154] By combining the cumulative distribution function Φ(·) of the standard normal distribution and applying analytical integrals within the upper and lower limits of the system's safe frequency operation, the probability of the unit's frequency within the bounded fluctuation domain at each time point is calculated as follows:

[0155] .

[0156] Therefore, in the industrial application of the method of this invention, the above-described analytical calculation architecture can be embedded into the alarm device of a hydropower station. The probability curve P within the bounded fluctuation domain is obtained. in After (t), the system's ultra-low frequency oscillation alarm probability threshold is compared. Perform real-time dynamic verification. If at any observation time t... k P occurs in (t k ) < P limit If the probability curve falls below the safety threshold due to a disturbance, the over-limit alarm module of the hydropower station's central control system will be automatically triggered to issue a frequency over-limit alarm. The system will also adjust the dynamic gain of the hydropower unit's PI speed governor online or issue an active power output reduction command, forcing the system mean to converge quickly. This enables the setting of a safety threshold and online early warning for ultra-low frequency oscillations of hydropower units under a high proportion of new energy power grids, thereby improving operational safety.

[0157] The following simulation examples, combined with appendices, demonstrate this approach. Figures 2 to 6 The feasibility and accuracy of the methods provided in the above embodiments of the present invention will be further verified.

[0158] Combination Figure 1 The flowchart illustrates the probabilistic analytical method for hydropower units under random excitation considering pressure pulsation in the frequency domain. During simulation verification, the operating parameters of the hydropower unit are first selected: the system equivalent inertia constant H = 6.4, the equivalent damping coefficient D = 3, and the water hammer effect time constant T. w =1.5, the proportional gain parameter K of the PI speed controller P =0.3 and integral gain parameter K I =0.1. Next, a frequency response model of the hydropower unit under random excitation considering pressure pulsation is established, and the mean and standard deviation of the system are calculated. The intra-domain probability of the system frequency fluctuation is calculated based on the probability density function.

[0159] Figure 2 This is the calculation flow of the offline Monte Carlo method in this embodiment of the invention. First, a stochastic model of the power system under stochastic excitation is established using statistical data. Then, a frequency response model of the hydropower unit considering pressure pulsations is obtained. Next, the system frequency model is simulated to obtain the trajectory of the system frequency in one simulation, and the simulation is iterated until the required number of simulations is reached. Finally, the number of times the system frequency trajectory is within the safe range is counted, and the probability of the system frequency within the bounded fluctuation domain is calculated.

[0160] Combination Figure 3 The figure shows a comparison between the analytical method and the Monte Carlo method according to the present invention. As can be seen from the figure, under the initial parameter settings, the calculation results of the two methods are in high agreement.

[0161] Combination Figure 4The figure shows a comparison of the intra-domain probabilities of the analytical method and the Monte Carlo method under different random excitation intensities. With other parameters remaining constant, the random excitation intensities σ1 were set to 0.008, 0.01, and 0.012, respectively. As can be seen from the figure, the intra-frequency domain probability of the system decreases with increasing excitation intensity, and the two calculation methods show a high degree of agreement.

[0162] Combination Figure 5 The graph shows a comparison of the intra-domain probabilities obtained by the analytical method and the Monte Carlo method under different damping coefficients, with other parameters remaining constant. The damping coefficients D are set to 2, 3, and 4. As can be seen from the graph, the intra-frequency probability of the system decreases as the damping coefficient decreases, and the two calculation methods show a high degree of agreement.

[0163] Combination Figure 6 The comparison shows the intra-domain probabilities of the analytical method and the Monte Carlo method under different frequency security domains. With other parameters remaining constant, the frequency security domain f is set. min The values ​​are 0.0008, 0.001, and 0.0012, respectively. As shown in the figure, the probability within the system's frequency domain increases with the increase of the frequency security domain, and the two calculation methods are in high agreement.

[0164] The convergence fitting verification of the probabilistic analytical method for hydropower units under random excitation considering pressure pulsation proposed in this invention and the offline Monte Carlo random numerical trajectory stepping method shows that the two curves have a high degree of agreement, verifying the feasibility and accuracy of the analytical technical solution proposed in this invention in terms of mathematical mechanism and engineering.

[0165] The details not described in this specification are existing knowledge known to those skilled in the art. While the invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the invention. Those skilled in the art can make various modifications and refinements without departing from the spirit and scope of the invention. Therefore, the scope of protection of this invention is determined by the claims.

Claims

1. A probabilistic analytical method for hydropower units under random excitation considering pressure pulsation in the frequency domain, characterized in that, Includes the following steps: Step 1: Obtain the operating parameters of the hydropower unit and construct a dynamic frequency response mathematical model that includes the governor, water hammer effect, and rotor motion. Step 2: Treat the tailrace pressure pulsation as a deterministic periodic excitation and the new energy source or load fluctuation as a stochastic excitation to construct a stochastic state-space model of the hydropower unit. Step 3: Based on the principle of superposition of linear systems, solve the frequency mean trajectory of the state space model under deterministic periodic excitation and the frequency covariance matrix under random excitation; and determine the time-varying mean of the system frequency deviation and the dynamic standard deviation of the system frequency deviation under the action of common excitation. Step 4: Under the condition that the system state is subjected to deterministic periodic excitation forced disturbance and random excitation continuous injection, extract the real-time mean and dynamic standard deviation of the system frequency deviation, and construct a Gaussian probability density function with time-varying characteristics. Step 5: Set a safe operating boundary for the system frequency, and integrate the probability density function within the safe boundary to obtain the probability of the system frequency in the bounded fluctuation domain at a given time.

2. The probabilistic analytical method for hydropower units under random excitation considering pressure pulsation in the frequency domain according to claim 1, characterized in that, In step 1, the turbine of the hydroelectric unit has a transfer function T that includes non-minimum phase zero-point characteristics. w (s) is represented as: ； The transfer function T is obtained by long polynomial division. w (s) is transformed into an algebraic sum of a constant term and a fraction, i.e.: ； Define the system intermediate variable corresponding to the time-domain response of the fraction as the internal state variable x of the turbine. int And the mechanical power deviation ΔP m Represented as a linear algebraic expression without state variable derivative terms: ； Among them, T w Let represent the time constant of the water hammer effect, s represent the Laplace operator, and K represent the time constant of the water hammer effect. P K I These represent the proportional gain and integral gain parameters of the PI speed controller, respectively, where Δf is the system frequency deviation, and x is the integral gain parameter. I x is the integral term of the frequency deviation. int These are the internal state variables of the water turbine.

3. The probabilistic analytical method for hydropower units under random excitation considering pressure pulsation in the frequency domain according to claim 2, characterized in that, In step 2, the constructed stochastic state-space model of the hydropower unit is a third-order stochastic state-space model, represented as an Iton-type linear stochastic differential equation with deterministic sinusoidal input: ； In the formula, A is the system state matrix, K is the random disturbance input matrix, dB(t) is the Gaussian white noise increment, and B... p For a deterministic periodic excitation input matrix, U p (t) represents the sinusoidal periodic forced input caused by pressure pulsation in the tailrace pipe; U p (t)=sin(ω p t), representing the angular frequency ω of the pressure pulsation in the turbine's tailrace pipe. p The deterministic cycle of decision-making forces input; Where X represents the state vector of a third-order stochastic state-space model, X = [Δf(t), x I (t), x int (t)] T Δf(t), x I (t), x int (t) represents the system frequency deviation, the integral term of the frequency deviation, and the internal state variables of the turbine at any time t, respectively.

4. The probabilistic analytical method for hydropower units under random excitation considering pressure pulsation in the frequency domain according to claim 3, characterized in that, In the differential equations of the third-order stochastic state-space model, the system state matrix A takes the following values: ； In the formula, H is the system's equivalent inertial constant, and D is the equivalent damping coefficient.

5. The probabilistic analytical method for hydropower units under random excitation considering pressure pulsation according to claim 3, characterized in that, In the differential equations of the third-order stochastic state-space model, the random perturbation input matrix K and the deterministic periodic excitation input matrix B... p They are respectively: ； ； In the formula, σ1 is the random excitation intensity parameter from the generator side, and A p This is the coefficient for the amplitude of mechanical power change caused by pressure pulsation in the tailrace pipe.

6. The probabilistic analytical method for hydropower units under random excitation considering pressure pulsation in the frequency domain according to claim 1, characterized in that, Step 3, which involves solving the frequency mean trajectory of the state-space model under deterministic periodic excitation, includes the following process: Construct a mean vector μ of the system state X The nonhomogeneous ordinary differential equation of (t): ; In the formula ω p The angular frequency of pressure pulsation in the turbine's tailrace pipe; Under the condition that the initial state mean is zero, the non-homogeneous ordinary differential equation is solved to obtain the state mean trajectory.

7. The probabilistic analytical method for hydropower units under random excitation considering pressure pulsation in the frequency domain according to claim 1, characterized in that, In step 3, the specific process of solving the frequency covariance matrix of the state-space model under random excitation is as follows: First, the system state matrix A is decomposed into eigenvalues ​​to obtain the eigenvector matrix P and a diagonal matrix composed of eigenvalues. ; Secondly, based on the random perturbation input matrix K, the intermediate matrix F used to map the noise intensity to each physical mode is calculated, F = F -1 KK T (F -1 ) T ; Then, calculate the integral term matrix J describing the time-domain cumulative effect of noise energy in the system, where the elements in the i-th row and j-th column are J(i,j), and J(i,j) = [e ( λi + λj) (t-t0) -1] / ( λ i + λ j ), where λ i and λ j These are the eigenvalues ​​of matrix A; Finally, the covariance matrix Var[X(t)] of the system state is calculated by multiplying the eigenvector matrix by the Hadamard product: 。 8. The probabilistic analytical method for hydropower units under random excitation considering pressure pulsation in the frequency domain according to claim 1, characterized in that, In step 3, the first element of the state mean trajectory is extracted as the time-varying mean μ of the system frequency deviation. Δf (t); as well as The square root of the first row and first column element of the covariance matrix is ​​extracted as the dynamic standard deviation σ of the system frequency deviation. f (t).

9. The probabilistic analytical method for hydropower units under random excitation considering pressure pulsation in the frequency domain according to claim 6, characterized in that, In step 5, the probability within the bounded fluctuation domain of the system frequency at a given time is expressed as: ； In the formula, P in (t) represents the bounded fluctuation domain probability of the system frequency at time t; Φ(·) is the cumulative distribution function of the standard normal distribution; f max and f min These are the upper and lower limit boundary values ​​that are preset for safe operation of the system frequency.

10. A computer system, characterized in that, include: One or more processors; A memory that stores operable instructions that, when executed by the one or more processors, cause the one or more processors to perform operations, including the process of the probabilistic analytical method for a hydropower unit under random excitation taking into account pressure pulsations in any one of claims 1-9.

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