Water turbine system fractional order engineering model and parameter identification method

Abstract

The invention discloses a water turbine system fractional order engineering model and a parameter identification method based on the model, and belongs to the technical field of water turbine modeling. Comprising the steps of determining a fractional-order engineering model of a water turbine system; a Mittag-Leffler function is introduced, and the fractional order engineering model of the water turbine system is converted into an analytical expression; and based on the analytical expression, performing parameter identification of the water turbine system by utilizing a swarm intelligence algorithm. When the computing power of the industrial control system is sufficient, the identification process is carried out on line, otherwise, the identification process is completed off line, and the method has the characteristics of good simulation control effect, simple analysis process and convenience in industrial control software configuration.

Description

Technical Field

[0001] The present invention relates to the technical field of hydraulic turbine modeling, and in particular to a hydraulic turbine system fractional-order engineering model and a parameter identification method. Background Art

[0002] Hydraulic turbines and their systems typically involve transient flows and elastic pipes. Their dynamic continuity and development are closely related to the preceding and initial states. Flow within pressurized water pipes also exhibits the distributed parameter characteristics of flexible structures and is inherently fractional. Taking integration as an example, the timescale variation of physical quantities is not a simple superposition of historical values, but rather exhibits a decaying effect. Describing objects with integer-order mathematical models is akin to series approximation, requiring the use of high-order parameters to achieve satisfactory model accuracy, a cost that is readily apparent. Conversely, fractional-order mathematical models can more accurately describe the object with fewer orders, making the introduction of fractional orders practically relevant for hydraulic modeling. In recent years, related research work has been carried out from the perspective of fractional-order controllers, and the turbine object is assumed to be integer order. However, Xu Beibei proposed in "XU Beibei, CHENDiyi, ZHANG Hao et al. Modeling and stability analysis of a fractional-order Francis hydro-turbine governing system[J]. Chaos, Solitons and Fractals: Applications in Science and Engineering: An Interdisciplinary Journal of Nonlinear Science, 2015, 7550-61" that the shaft system of the hydro-generator set is taken as the research object, and the fractional-order damping force and fractional-order oil film force are introduced to establish a nonlinear mathematical model of the shaft system of the fractional-order hydro-generator set. At the same time, Wang Feifei proposed in "WANG Feifei, CHEN Diyi, XU Beibei et al. Nonlinear dynamics of a novel fractional-order Francis hydro-turbine governing system with time delay[J]. Chaos, Solitons and Fractals: Applications in Science and Engineering: An Interdisciplinary Journal of Nonlinear Science, 2016, 91329-338”, fractional-order calculus is introduced into the complex pressure water diversion pipeline system, and a nonlinear model of the fractional-order complex piping turbine regulation system is established.

[0003] The application cost of fractional order is higher than that of integer order, which is reflected in the fact that the engineering implementation of fractional order and the identification of fractional order models often require dedicated simulation tools. Therefore, using on-site industrial control systems to carry out fractional order model identification is a key link in the transition of fractional order thinking from theory to practice. Li Dazi in "[5] Li Dazi, Fan Weiguang, Gao Yanchen, et al. Iterative Least Square Identification Algorithm for Fractional Order System[J]. Journal of Jiangnan University (Natural Science Edition), 2010, 9(4): 404-408. Li Dazi, Fan Weiguang, Gao Yanchen.. Iterative Least Square Identification Algorithm for Fractional Order System[J]. Journal of Clothing Research, 2010, 9(4): 404-408" transforms the identification process into the frequency domain based on the characteristics of fractional order time domain operations, and generalizes it to the iterative least squares algorithm, which simplifies the calculation process. However, the large matrix operations involved are not good at industrial control systems, which limits the application of this type of frequency domain identification method; Wu Jun in "Wu Jun, Yang Weiming, Yao Jing, et al. Fractional order lithium battery model parameter identification based on genetic algorithm[J]. Journal of Hubei University (Natural Science Edition), 2022, 44(3): 339-344. Wu Jun Jun, YANG Weiming, YAO Jing. Parameter identification of fractional lithium battery model based on genetic algorithm [J]. Journal of Hubei University (Natural Science Edition), 2022, 44 (3): 339-34” uses the swarm intelligence characteristics of genetic algorithm to obtain model parameters by numerical approximation in the form of step response, which is simple and feasible, but faces the difficulty of implementing step test in actual engineering; In addition, Wang Dongfeng in “[9] Wang Dongfeng, Meng Li. Fractional Order System Identification Based on Particle Swarm and Auxiliary Variable Methods [J]. Information and Control, 2016, 45 (3): 287-293. WANG Dongfeng, MENG Li. Identification of Fractional Order System Based on PSO and Recursive Instrumental Variable Methods [J]. Infomation and Control, 2016, 45 (3): 287-293”, Li Lulu in “

[10] Li Lulu, Tao Zhengshun, Pan Tinglong, et al.Lithium battery fractional-order modeling and SOC estimation strategy [J]. Energy Storage Science and Technology, 2023, 12(2): 544-551. LI Lulu, TAO Zhengshun, PAN Tinglong. Research on fractional modeling and SOC estimation strategy for lithium batteries [J]. Energy Storage Science and Technology, 2023, 12(2): 544-55” also gave a positive conclusion on the application of swarm intelligence in fractional-order identification. The natural distributed characteristics of swarm intelligence computing are conducive to its deployment in industrial control systems. By exploring a fractional-order numerical method that is easy to configure and calculate, and adopting a ramp-holding excitation that is easy to implement in engineering, it becomes possible to carry out fractional-order modeling in industrial sites.

[0004] In view of the above problems, it is urgent to design a fractional-order transfer function model of a turbine and its hydraulic system based on flow rate and flow capacity to solve the problems existing in the above-mentioned existing technologies. Summary of the Invention

[0005] In response to the above-mentioned problems, the present invention establishes a fractional-order model expression of the turbine system based on the fractional-order form of flow and capacity, observes the influence of the order on the fractional-order characteristics, and then uses the Mittag-Leffler formula for analytical expansion to obtain an algebraic expression that is easy to configure. After obtaining the slope holding signal excitation response on site, the swarm intelligence method is used to identify the model; when the computing power of the industrial control system is sufficient, the identification process is carried out online, otherwise it is completed offline. It has the characteristics of good simulation control effect, simple analytical process, and easy configuration of industrial control software.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] The present invention provides a first solution: a parameter identification method based on a fractional-order engineering model of a hydraulic turbine system, comprising:

[0008] Determine the fractional order engineering model of the turbine system;

[0009] The Mittag-Leffler function is introduced to convert the fractional-order engineering model of the turbine system into an analytical expression;

[0010] Based on the analytical expression, a swarm intelligence algorithm is used to perform parameter identification of the turbine system.

[0011] Preferably, determining the fractional-order engineering model of the hydraulic turbine system includes:

[0012] Establish the fractional order transfer function of the turbine system considering flow rate and flow capacity;

[0013] The fractional-order transfer function is decomposed and simplified to obtain a low-order fractional-order transfer function.

[0014] Preferably, the fractional order transfer function of the turbine system considering flow rate and flow capacity is established, and the fractional order transfer function is:

[0015]

[0016] Among them, T e =l / a is the traveling wave time; T w =(lQ0) / (gAH0) is the water flow inertia time; l is the length of the pressure water pipe from the beginning to the end; a is the water hammer velocity; α is the fractional order of the flow rate; β is the fractional order of the flow capacity; g is the acceleration of gravity; A is the cross-sectional area of ​​the pipe; Q0 and H0 are the initial flow rate and pressure head, respectively.

[0017] Preferably, decomposing and simplifying the fractional-order transfer function to obtain a low-order fractional-order transfer function includes:

[0018] Converting the fractional-order transfer function into a plurality of first-order fractional-order transfer functions, and connecting them in parallel to obtain a first low-order transfer function in parallel form;

[0019] Let the numerator and denominator of the first low-order transfer function have the same order and simplify it into the second low-order transfer function;

[0020] For a second low-order transfer function with an order greater than 3, it is estimated through multiple first-order links to obtain a third low-order transfer function.

[0021] Preferably, the introduction of the Mittag-Leffler function to convert the fractional-order engineering model of the turbine system into an analytical expression includes:

[0022] Substitute the Mittag-Leffler function into the third lowest-order transfer function;

[0023] Performing inverse Laplace transform on the third low-order transfer function above, its analytical expression under step and ramp excitation is obtained.

[0024] The present invention provides a second solution: the fractional-order engineering model of the turbine system is:

[0025]

[0026] The beneficial effects of the present invention are as follows: the present invention discloses a fractional-order engineering model of a hydraulic turbine system and a method for establishing the same. Compared with the prior art, the present invention has the following improvements:

[0027] 1. This invention designs a method for establishing a fractional-order engineering model of a hydraulic turbine system. Starting from the fractional-order expression of flow and capacity, this method establishes a fractional-order model of the hydraulic turbine and its hydraulic system. Order sensitivity analysis indicates that fractional order has better characterization capabilities, and the homology form facilitates the identification of fractional-order models. Furthermore, this method uses the Mittag-Leffler function to analytically express the fractional-order model of the hydraulic turbine and its system, making it feasible to implement the fractional-order model using industrial control configuration. The identification of the mathematical model of the hydraulic turbine system can be completed using industrial control software, providing a foundation for the application of fractional-order models in on-site industrial control processes.

[0028] 2. This paper designs a fractional-order engineering model for a hydraulic turbine system. This model uses an integer-order mathematical model to describe the object, similar to series approximation, and can be used for the control and research of hydraulic turbines and their systems.

[0029] 3. Based on the fractional-order engineering model of the turbine system, the present invention designs an identification method for the fractional-order engineering model of the turbine system. After obtaining the slope response data of the unit through broken-line adjustment under the opening mode, this method uses group intelligence means such as particle swarm to carry out fractional-order model identification engineering practice; the fractional-order analytical expression of the turbine system only contains algebraic expressions, which is convenient for industrial control software configuration. When the computing power of the industrial control system is sufficient, the identification process is carried out online, otherwise it is completed offline. It has the advantages of good simulation control effect, simple analytical process, and convenience for industrial control software configuration. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 This is a block diagram of the linear model of the turbine and controller of the present invention.

[0031] Figure 2 This is a step response curve diagram of the fractional-order and integer-order models with the same coefficients of the present invention.

[0032] Figure 3 This is the order sensitivity curve of the integer-order and fractional-order models of the present invention.

[0033] Figure 4 This is a comparison diagram of the step responses of the fractional-order and integer-order variable parameter transfer functions of the present invention.

[0034] Figure 5 This is a broken line graph of the slope holding signal based on broken line adjustment of the present invention.

[0035] Figure 6 This is a comparison chart of fractional order reduction of the integer order model of the present invention.

[0036] Figure 7 This is a complete active power regulation process diagram of a 125MW unit of the present invention.

[0037] Figure 8This is a comparison graph of the initial sections of step responses of different order transfer functions of the present invention.

[0038] Among them, Figure 3 In the figure, Figure (a) is a graph of integer order curves of different orders, and Figure (b) is a graph of fractional order curves of different orders. DETAILED DESCRIPTION

[0039] In order to enable those skilled in the art to better understand the technical solution of the present invention, the technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0040] Example 1: Refer to the attached Figure 1-8 The parameter identification method based on the fractional-order engineering model of the turbine system shown in FIG includes:

[0041] Step 1. Determine the fractional-order engineering model of the turbine system

[0042] Step 11. Build a model of the turbine and its water pipe system

[0043] (1) Perform Laplace transformation on the torque equation and flow equation of Francis turbine to obtain the equation including pressure diversion system and turbine (such as Figure 1 The transfer function (shown in the dotted box) is:

[0044]

[0045] Among them, e y 、e qy 、e h 、e qh are the partial derivatives of turbine torque with respect to guide vane opening, turbine flow with respect to guide vane opening, turbine torque with respect to water head, and turbine flow with respect to water head, respectively. They are called transfer coefficients and are related to the characteristic curve of the turbine. They are approximately linear and do not affect the order of the model transfer function. p is the transfer function of the water pressure system;

[0046] (2) Assume that the ideal turbine operates at rated head and speed, then let e y =1, e qy =1, e h =1.5, e qh =0.5, we get the following form:

[0047]

[0048] For the transient flow caused by the opening and closing of valves or guide vanes in the pressure water diversion pipeline, considering the compressibility of the fluid and the elasticity of the pipeline, ignoring the influence of the pipe axis inclination and neglecting the flow resistance in the pipeline, Newton's law is applied to obtain the equation of motion, which is written in fractional order form:

[0049]

[0050] Where L0 = Q0 / (gAH0), defined as the velocity per unit length of pipe, subscript 0 indicates rated operating conditions, g = acceleration due to gravity (m / s 2 ), A=pipe cross-sectional area (m 2 ), Q and H are flow rate (m 3 / s) and pressure head (m), q and h are their nominal values, q = Q / Q0 and h = H / H0. x = axial coordinate, t = time (s), α = the order of the fractional flow, which is a real number;

[0051] (3) Perform Laplace transform (denoted as L) on Equation (3), and record H(x,s) = L(h(x,t)). The fractional order adopts the Riemann-Liouville definition, takes the lower limit of the operator as 0, and takes the zero initial state, and obtains:

[0052]

[0053] Similarly, the fractional order form of the continuity equation in the pipeline is:

[0054]

[0055] Where, C0=(gAH0) / (a 2 Q0), defined as the flow capacity per unit length of the pipe, a = water hammer velocity, assuming it is not affected by the gas in the water and the fluid does not vaporize, β = the fractional order of the flow capacity, which is a real number;

[0056] (4) Performing Laplace transformation on equation (5) and noting Q(x,s)=L(q(x,t)), we obtain:

[0057]

[0058] (5) Taking the derivative of (4) with respect to x and substituting it into (6) yields a second-order homogeneous differential equation for H(x,s), whose characteristic roots are ±s (α+β) / 2 / a, then the general solution is:

[0059]

[0060] Taking the derivative of formula (7) with respect to x and substituting it back into formula (4) yields:

[0061]

[0062] (6) Assuming the length of the pressure water pipe from the first end (turbine inlet) to the last end (reservoir outlet) is l, and assuming that the pressure at the reservoir outlet does not change, that is, H(l,s) = 0, we can get Substituting into equations (7) and (8) and setting x = 0, eliminating C2, the transfer function at the turbine inlet is:

[0063]

[0064] Among them, T e =l / a is the travel time (s), T w =(lQ0) / (gAH0) is the water flow inertia time (s);

[0065] (7) Substituting Equation (9) into Equation (2), the fractional-order mathematical model of the turbine system (including turbine and pressurized water pipe) is obtained as follows:

[0066]

[0067] By performing Maclaurin expansion on Equation (10), we can obtain the fractional-order transfer function of the turbine system considering flow rate and flow capacity. Taking the third order as an example:

[0068]

[0069] The fractional-order transfer function of the turbine system shown in formula (11) has the characteristics of continuous order and is a fractional-order model with the same order. Its common factor order is (α+β) / 2. Considering s (α-β) / 2 After the model is established, it still has the characteristics of order continuity. Compared with the general fractional-order model, the parameters are reduced, which reduces the difficulty of identification. The identification uses the common factor order as one of the variables to reflect the time-varying characteristics of the interaction between the fluid and the pipeline. In some occasions that require detailed characterization, variable-order identification can also be performed to more accurately describe complex dynamic phenomena.

[0070] The selection of the order parameters α and β of flow rate and flow capacity depends on the fluid characteristics themselves. Their effects on the flow of the turbine pressure water system are as follows:

[0071] Flu and flow characteristics

[0072] Observe formula (11), since T e With T w The transfer function coefficients are clear, and the order is related to α and β. Taking α = 0.9 and β = 0.8 as an example, let T e =1, T w =1.2, observe the step characteristics of the model (11) at different orders, as shown in the attached Figure 8 As shown (in this embodiment, the order is defined as n-order fractional order if it is less than n and greater than n-1, and n is a positive integer):

[0073] When the first-order model is selected, G h(s) is mainly based on the inertia of the flow: when the flow increases, it prevents the flow from increasing rapidly by establishing a head, and conversely, when the flow decreases, it releases the head and delays the rapid decrease of the flow. When the second-order model is selected, G h (s) takes into account the effect of flow capacity, reflecting the elasticity of the water body and the wall. Its dual characteristics with the flow rate make the change of the pressure head lag behind the flow rate, reducing the blocking effect on the flow rate. When the third-order and fourth-order models are selected, the flow rate and the flow capacity show a complex interaction; in the mathematical model where the flow rate is the dominant factor, the active back-regulation characteristic is very obvious. The higher the order of the flow rate, the more prominent the back-regulation characteristic; and under the neutralization of the flow capacity characteristic, such as the second-order fractional model, this back-regulation characteristic is complex. However, considering the step function ε(t) with an amplitude of 1, its Fourier transform is It can be seen that there is an impulse function in the frequency spectrum of the step response at ω = 0, which makes the back-modulation region show very rich changes. This is beneficial for observing the dynamic characteristics of the model. In fact, in engineering practice, due to the constraints of mechanical and electrical conditions, the changes in model input conditions are much more moderate, making the back-modulation characteristics under different flow and capacity orders actually not much different.

[0074] In the study of fractional-order applications, half-order is usually used as an example to illustrate the characteristics of fractional order. In fact, both theoretical derivation and identification reveal that the order is generally between ±0.2 of the first order. Lower orders are prone to deviate from the physical nature, while higher orders can be eliminated by order reduction.

[0075] As a generalization of integer-order models, fractional-order models of hydraulic turbine systems formally add exponential terms (α+β) / 2 and (α-β) / 2, resulting in different model characteristics. For non-ideal turbines or those operating outside rated head and speed conditions, the model parameters vary while the order remains unchanged.

[0076] Step 12. Based on the fractional-order engineering model of the turbine system, compare its accuracy and order sensitivity with those of the integer-order model, decompose and simplify the fractional-order engineering model, and reduce the high-order fractional-order engineering model to a low-order fractional-order transfer function;

[0077] The high-order fractional-order engineering model is the fractional-order transfer function of the turbine system considering flow velocity and flow capacity established in step 11, as shown in formula (11).

[0078] (1) For the convenience of comparison, let α = 0.8, β = 0.8, take the double common factor order of the fractional order model (Equation (12)), take the 1st order of the corresponding integer order model (Equation (13)), ignore the inertia of the relay, and the typical identification parameter T under full load condition is w =1.2, T e =1, make a step response (starting point is 0, end point is 100%) and plot it on Figure 2, where FO and IO refer to fractional and integer order models respectively. Obviously, due to the difference in model order, the dynamic responses of first-order IO and FO are quite different.

[0079] The response of the first-order fractional model is identified, and the integer-order model is obtained as a high-order one, as shown in Equation (14), which verifies that integer-order modeling of fractional-order objects often requires a higher order;

[0080]

[0081]

[0082] Taking the 2nd, 3rd, and 4th order expressions corresponding to formulas (12) and (13) as the objects, Figure 2 Under the same preset conditions, a ramp response (starting point 0, end point 50%, slope 3%) is applied. The performance of the IO and FO models of each order is shown as follows: Figure 3 , since these two models are not equivalent, Figure 3 (a) Figure 3 Middle (b) uses vertical axes with the same distance but different scales;

[0083] It can be seen that the fractional-order model exhibits low order sensitivity. Usually, low-order fractional orders can better characterize the object, and increasing the fractional order cannot significantly improve the model accuracy. In addition, although high-order fractional transfer functions can describe the dynamic characteristics of the process in more detail, there are objective difficulties in implementation complexity and need to be decomposed in advance.

[0084] (2) Decompose the high-order transfer function into several low-order transfer functions, and carry out modeling on the low-order transfer functions to obtain a high-order model, which conforms to the principle of engineering feasibility. Therefore, referring to the Southern Power Grid standard, the fractional-order transfer function of the turbine system considering the flow rate and flow capacity is decomposed, that is, formula (11) is simplified into several first-order fractional orders, and multiple first-order fractional orders are connected in parallel to obtain the first low-order transfer function. The first low-order transfer function is a parallel-type fractional-order transfer function, and its general form is:

[0085]

[0086] Where K is the gain, N and D are model coefficients, T is the time constant, and μ and γ are fractional orders;

[0087] (3) Assuming the numerator and denominator of the first low-order transfer function have the same fractional order, Equation (15) can be simplified to the second low-order transfer function shown in Equation (16), whose general form is:

[0088]

[0089] (4) For the second low-order transfer function with an order greater than 3, that is, when the same order of the second low-order transfer function is greater than 3, it can be estimated through 2 to 3 first-order links, thereby simplifying the second low-order transfer function to a third low-order transfer function, such as:

[0090]

[0091] Where T is the time constant of each item;

[0092] After the above decomposition and simplification steps, the complex fractional-order model can be expressed in several concise standard forms, making further analytical analysis possible;

[0093] Step 2. Introduce the Mittag-Leffler function to convert the fractional-order engineering model of the turbine system into an analytical expression;

[0094] Unlike integer orders, industrial control system software usually does not support fractional-order algorithms, and requires complex configuration of existing time process algorithm blocks (such as inertia, lead-lag, etc.). If the fractional-order definition is converted into a resolvable algebraic expression, the configuration process can be greatly simplified. To this end, this embodiment obtains an analytical expression for the fractional order of the turbine system based on the Mittag-Leffler expression.

[0095] Step 21. Introduce the Mittag-Leffler function to perform fractional-order transfer function conversion;

[0096] (1) The Mittag-Leffler function plays an important role in fractional-order analysis. Its form is as follows:

[0097]

[0098] Where Γ() is the Gamma function;

[0099] A commonly used Laplace expression for formula (18) is:

[0100]

[0101] In formula (19), It is a typical fractional-order transfer function expression;

[0102] (2) Substitute the Mittag-Leffler function into the third-lowest-order transfer function, that is, substitute it into equation (17), and take the inverse Laplace transform of the third-lowest-order transfer function substituted into the Mittag-Leffler function to obtain its algebraic solution:

[0103]

[0104] Under step conditions, its response is:

[0105]

[0106] (3) Let ν = -γ + 1, p = 1 / D, and the time series solution of G1(s) is obtained according to equation (19):

[0107]

[0108] Let ν = μ + 1, p = 1 / D, and the time series solution of G2(s) is:

[0109]

[0110] (4) After finite truncation of the infinite terms in equations (22) and (23), the fractional step response of the turbine system can be quickly calculated given the parameters μ, ν, N, D, K, etc. Since they are all algebraic expressions, the configuration can be easily designed. If unit ramp excitation is used, the analytical expression of its response is:

[0111]

[0112] Equations (22)-(23) and (24) are the analytical expressions of the fractional-order model under step and ramp excitation, respectively. Based on these analytical expressions, the parameter identification of the fractional-order engineering model of the turbine system can be realized;

[0113] Step 22. Verify the analytical method proposed in step 21

[0114] (1) Let μ = 1 and γ = 0 in equation (15), and we can obtain the integer-order transfer function of the turbine system:

[0115]

[0116] Formula (25) is subjected to inverse Laplace transform to obtain the analytical formula of its step response:

[0117]

[0118] Assume K = 1, N = 1.2, D = 0.6, and calculate the integer step response using formula (25), as follows: Figure 4 Curve 1 (short dashed line); let μ = 0.9, and calculate the fractional step response using equations (22) and (23), as shown in curve 3 (long dashed line) in the figure; Figure 4 The curves are obtained analytically in the industrial control system configuration. The obvious dynamic difference between the fractional-order and integer-order step responses confirms the effectiveness of the analytical method.

[0119] (2) Rewrite equation (24) as:

[0120] DsY(s)+Y(s)=-KNsU(s)+KU(s)

[0121] Taking the inverse Laplace transform on both sides of the equation and rearranging them gives:

[0122]

[0123] Therefore, there is a discrete analytical calculation formula (26), which is suitable for variable coefficient calculation:

[0124]

[0125] Keeping K=1 and α=1 unchanged, N and D are linearly and synchronously changed to 150% of their original values. The step response calculated using equation (27) is shown as curve 2 (solid line) in the figure, which is enveloped by curves 1 and 3.

[0126] In the actual dynamic working conditions of the turbine system, the N and D parameters including the hydraulic inertia coefficient change with the water head or the active power level of the unit. When the dynamic characteristics are calculated using formula (25), the N and D parameters are fixed, which will produce differences from the actual dynamic characteristics. However, the fractional order model, due to the inclusion of historical information in the calculation process, is consistent with the dynamic changes of hydraulic inertia to a certain extent. Curve 2 gradually approaches curve 3 based on curve 1, which confirms this conjecture.

[0127] The variation patterns of N and D parameters in the dynamic process of the unit are obviously not necessarily linear as assumed by Curve 2, but their patterns can be inferred from the algorithm of Equation (27). In projects that do not have the conditions for fractional-order model identification, the variable parameter method can be used as a means of approximation.

[0128] In this embodiment, the fractional-order engineering model of the turbine system is constructed as follows:

[0129] 1. Starting from the fractional-order expression of flow and capacity, a fractional-order model of a hydraulic turbine and its hydraulic system is established. The order sensitivity analysis shows that the fractional order has better characterization capabilities, and the homology form is conducive to the identification of the fractional-order model. The fractional order itself is infinite-dimensional, and finite-dimensional truncation must be performed during engineering implementation, so that the fractional-order calculus is actually implemented through several integer orders. From an engineering perspective, the fractional-order model is not more refined than the integer order, but the fractional-order model can reflect the dynamic characteristics of the object with a smaller order, describe the object more accurately, and also facilitate analytical analysis during controller design;

[0130] 2. The fractional-order model of the turbine and its system is analytically expressed using the Mittag-Leffler function, making it feasible to implement the fractional-order model using industrial control configuration. The identification of the mathematical model of the turbine system can be completed using industrial control software, providing a foundation for the application of fractional-order models in on-site industrial control processes.

[0131] Based on the analytical expression of the fractional-order engineering model of the hydraulic turbine system obtained in step 2, identifying and verifying the fractional-order engineering model of the hydraulic turbine system used in a specific project;

[0132] Step 3. Introduce the standard particle swarm algorithm (PSO) to perform parameter identification based on the analytical expressions of the fractional-order engineering model under step and ramp excitations calculated analytically in step 2;

[0133] There are two main uses for turbine system identification in engineering projects. One is to collect single-machine characteristics for grid stability analysis, and the other is to set parameters for the local regulation process. Offline identification can usually meet engineering needs. Under realistic propositions such as network source coordination and coordination of common hydraulic units within the station, the demand for online identification is also becoming increasingly prominent. The main difference between the two methods is the use of tools. Offline identification can use various tools and intelligent algorithms to meet the accuracy as the primary goal, while online identification is limited by the functional scope of industrial control software and must adopt analytical methods and other easy-to-implement means. Through the analytical method proposed in Example 1 of the present invention, the main difference between offline and online identification is transformed into the engineering computing power possessed by the industrial control system. If the computing power allows, fractional-order identification can be performed online.

[0134] Step 31. Ramp-and-hold excitation analytical identification

[0135] Model identification presupposes object excitation. Only when the model object is fully excited, sufficient information is available within the frequency band of interest, resulting in satisfactory identification results. A step command has abundant high-frequency components, while the constants between its steep edges represent low-frequency components, making it a wide-range excitation system. However, step signals are difficult to implement in engineering and can disrupt normal production operations. Therefore, they are only approximated using a step frequency measurement signal during a single frequency modulation test. Due to the limitations of single frequency modulation tests, this approximate step test cannot cover the full load range, partially negating its inherent advantages.

[0136] In recent years, a broken line regulation method has emerged in the regulation of hydropower units. Model analysis is used to accurately predict the regulation dynamics and the end point of the regulation process, making the movement of the guide vanes in the opening mode a broken line. This feedforward mode of regulation greatly accelerates the dynamic process and serves as a typical ramp holding signal (if the opening rate before and after the broken line is the same, the broken line is simplified to a straight line), becoming a good excitation signal.

[0137] Typical ramp hold (also called ramp step) signals are as follows: Figure 5 As shown in the figure, this adjustment method in the opening mode has a simple signal, does not contain a PID regulator, and also simplifies the identification process.

[0138] The finite rising edge ramp hold signal is equivalent to the accumulation of two infinite ramp signals, one of which is a rising edge signal and the other is a falling edge signal. Then, the ramp hold response can be analytically calculated using formula (24). This is the main method for subsequent engineering verification of this embodiment.

[0139] Step 32. Introduce the standard particle swarm algorithm and establish a swarm intelligence method to identify the fractional-order engineering model of the turbine system

[0140] Fractional-order model identification methods primarily focus on swarm intelligence methods, such as particle swarm optimization, with some new optimization algorithms also reported. Comparative testing of various intelligent algorithms for turbine model identification revealed that, ignoring computational costs (such as identification time and number of attempts) and under the premise of human supervision, genetic algorithms, particle swarm optimization, biogeography, and gravitational search algorithms achieve similar optimization results.

[0141] This embodiment uses a standard particle swarm algorithm (PSO). Typical algorithm parameters are shown in Table 1. The particle dimensions are N, D, α, K, and γ. The cutoff length of the fractional-order analytical calculation is determined based on the total computational time domain. The mean square error (MSE) parameter is used to indicate the degree of identification closeness. Termination conditions include the MSE not changing within a specified step size or an upper limit of 50 iterations. Due to the inherently distributed nature of swarm intelligence algorithms, they can be deployed in the configuration software of industrial control systems. Different controllers can be used to construct particle islands. The island identification process is carried out synchronously and information is exchanged, forming a parallel mechanism.

[0142] Table 1: Fractional particle swarm identification parameter settings

[0143] Number of particles Learning Factor Weight Truncation length α range gamma range 20 1.5 / 2.5 0.5 100～200 0.5～1.5 -0.5～0.5

[0144] Step 33. Perform model reduction identification based on ARMA time series analysis

[0145] The model identification of the turbine system is usually based on the integer order method, which is widely used in engineering practice. It is based on ARMA time series analysis and then converted into a continuous transfer function:

[0146]

[0147] The ramp rate is set to 30% / s, and the model of formula (28) is subjected to ramp-maintaining excitation. The particle swarm method is used to identify the fractional-order transfer function based on formula (22), and its MSE is 0.000002. The response curve compared with the original model is shown in Figure 6 ;

[0148]

[0149] Repeating the identification process under the same conditions, we can obtain some different transfer functions near the order obtained in formula (29), such as formula (30), with MSE = 0.000179. The non-uniqueness of this model reflects the rich representation ability of fractional order.

[0150]

[0151] Taking MSE < 0.0002 as the standard, the identification process was repeated under the condition of limiting the μ parameter. The typical transfer functions obtained are shown in Table 2. It can be seen that when the fractional order μ changes, satisfactory identification results can be achieved with different parameters such as D, K, and μ+γ. These parameters show regular changes to compensate for the impact of the change in fractional order μ on the overall model identification, which is similar to a "compensatory" effect. That is, when the order μ differs from the actual model, other parameters are adjusted to compensate for this difference. This characteristic makes the identification model diverse.

[0152] Table 2: Identification results at different fractional orders

[0153] μ N D K μ+γ MSE 0.8 1.234 2.291 1.366 0.335 0.000169 0.9 1.250 2.546 1.188 0.375 0.000074 1.0 1.240 2.810 1.082 0.393 0.000013 1.1 1.242 3.325 0.946 0.447 0.000009 1.2 1.224 4.959 0.732 0.691 0.000067

[0154] By changing the ramp rate and repeating the identification, it was found that the lower the rate, the wider the order distribution of the identification model, which means that the intensity of the excitation is smaller, the characteristics of certain frequency bands cannot be fully reflected, and the identification model becomes diverse. The amplitude-frequency characteristics can be used to assist in judgment.

[0155] Step 34. Identify the fractional-order engineering model based on field data

[0156] When performing online identification on small and medium flow turbines, taking the highest and lowest common load points as the benchmark identification model can meet the actual control requirements. For large turbines, a model benchmark point can also be added at the intermediate load.

[0157] The second unit of a small- to medium-sized hydropower station has a rated capacity of 125MW and is a Francis generator. The unit design flow is 119.44m3 / s, the rated head is 117m, the total pipe length is 476m, and the diameter is 4.6m. The industrial control system uses a broken line adjustment in the opening mode to make the speed before and after the broken line consistent, so that the guide vane trajectory is maintained at a slope. In early summer, the head is insufficient, and the full power is about 115MW. Compared with the load reduction when the active power is at a higher level (such as Figure 7 Medium working condition 2), the loading process when the active power is at a low level (such as Figure 7 In the medium operating condition 1), the reverse regulation is not obvious (GV and UL in the figure are the opening and the measured active power, respectively). These two typical operating conditions are selected for model identification. Their transfer functions are shown in Equation (33). The MSEs of the two identification processes are 0.000112 and 0.000016, respectively. The simulation curves are in good agreement with the field measurements.

[0158]

[0159] The "compensatory" nature of fractional-order model identification means that the solution to a satisfactory identification accuracy may contain different combinations of model order and model parameters. This makes online identification chaotic and makes subsequent online tuning of the control parameters difficult. Therefore, a secondary identification method is generally used in engineering: first identifying the model order, then fixing the model order and then identifying the model parameters.

[0160] In this embodiment, based on the analytical expression of the fractional-order model of the turbine system, various identification methods are applied. This embodiment employs swarm intelligence methods, particularly particle swarm intelligence. The feasibility of these methods has been demonstrated through integer-order model reduction experiments and field parameter identification tests. If the industrial control site's computing power is insufficient, identification can be performed offline based on the industrial control configuration. Otherwise, if computing power is sufficient, the identification process can be performed online.

[0161] The basic principles, main features, and advantages of the present invention are shown and described above. Those skilled in the art should understand that the present invention is not limited to the foregoing embodiments. The foregoing embodiments and descriptions are merely illustrative of the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and modifications are intended to fall within the scope of the present invention. The scope of protection claimed in the present invention is defined by the appended claims and their equivalents.

Claims

1. A parameter identification method based on a fractional-order engineering model of a hydraulic turbine system is characterized by: include: Determine the fractional order engineering model of the turbine system; The Mittag-Leffler function is introduced to convert the fractional-order engineering model of the turbine system into an analytical expression; Based on the analytical expression, a swarm intelligence algorithm is used to perform parameter identification of the turbine system.

2. The parameter identification method based on the fractional-order engineering model of the hydraulic turbine system according to claim 1, characterized in that: Determining the fractional-order engineering model of the hydraulic turbine system includes: Establish the fractional order transfer function of the turbine system considering flow rate and flow capacity; The fractional-order transfer function is decomposed and simplified to obtain a low-order fractional-order transfer function.

3. The parameter identification method based on the fractional-order engineering model of a hydraulic turbine system according to claim 2, characterized in that: The fractional order transfer function of the turbine system considering the flow rate and flow capacity is established, and the fractional order transfer function is: Among them, T e =l / a is the traveling wave time; T w =(lQ0) / (gAH0) is the water flow inertia time; l is the length of the pressure water pipe from the beginning to the end; a is the water hammer velocity; α is the fractional order of the flow rate; β is the fractional order of the flow capacity; g is the acceleration of gravity; A is the cross-sectional area of ​​the pipe; Q0 and H0 are the initial flow rate and pressure head, respectively.

4. The parameter identification method based on the fractional-order engineering model of a hydraulic turbine system according to claim 3, characterized in that: Decomposing and simplifying the fractional-order transfer function to obtain a low-order fractional-order transfer function includes: Converting the fractional-order transfer function into a plurality of first-order fractional-order transfer functions, and connecting them in parallel to obtain a first low-order transfer function in parallel form; Let the numerator and denominator of the first low-order transfer function have the same order and simplify it into the second low-order transfer function; For a second low-order transfer function with an order greater than 3, it is estimated through multiple first-order links to obtain a third low-order transfer function.

5. The parameter identification method based on the fractional-order engineering model of the hydraulic turbine system according to claim 4, characterized in that: The Mittag-Leffler function is introduced to convert the fractional-order engineering model of the turbine system into an analytical expression, including: Substitute the Mittag-Leffler function into the third lowest-order transfer function; Performing inverse Laplace transform on the third low-order transfer function above, its analytical expression under step and ramp excitation is obtained.

6. The fractional-order engineering model of the turbine system is characterized by: The fractional-order engineering model of the turbine system is: Wherein, the fractional-order engineering model of the turbine system is established based on the method according to any one of claims 3-5.

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