Coordinated control method for water-wind-solar complementary system considering nonlinear dynamic boundary constraints

Abstract

The present application relates to the technical field of new power system stability control and multi-energy complementary optimal scheduling, and discloses a water-wind-solar complementary system cooperative control method considering nonlinear dynamic boundary constraints, comprising: using time scale separation technology, decoupling the water-wind-solar complementary system, and constructing an initial singular perturbation nonlinear standard model of the water-wind-solar complementary system; based on a preset condition, solving the initial singular perturbation nonlinear standard model, and constructing a slow subsystem dimension reduction Jacobian matrix and a fast subsystem boundary layer Jacobian matrix, and then through the Lyapunov stability criterion processing, constructing a multi-frequency absolute stability domain; taking the multi-frequency absolute stability domain as a hard constraint space, constructing a multi-objective optimization function; based on a preset out-of-bound lethal penalty function, using a multi-objective directional optimization algorithm based on bifurcation boundary curvature perception to optimize and solve the multi-objective optimization function, obtaining an optimal control parameter set and controlling the water-wind-solar complementary system to operate.

Description

Technical Field

[0001] This invention relates to the field of novel power system stability control and multi-energy complementary optimal scheduling technology, specifically to a collaborative control method for a hydro-wind-solar complementary system that takes into account nonlinear dynamic boundary constraints. Background Technology

[0002] Building a new power system dominated by new energy sources has become an inevitable trend in energy transformation. Hydropower-wind-solar hybrid power generation systems, leveraging the regulation capabilities of hydropower units to mitigate random fluctuations in renewable energy, are a key approach to improving renewable energy absorption rates. However, with the increasing penetration rate of power electronic interface equipment, the system exhibits significant characteristics of "low inertia" and "weak damping." Wind and solar turbines lack the rotating mechanical inertia of traditional synchronous machines, making it difficult to provide effective inertial support when facing sudden load changes or power shortages. This leads to a decrease in the system's frequency disturbance immunity, posing a severe challenge to frequency stability control.

[0003] The hydro-wind-solar hybrid system is a typical multi-timescale strongly coupled nonlinear system, whose control difficulty far exceeds that of a single-energy system. On the one hand, the hydro turbine generator unit is affected by the water hammer effect in the water diversion pipeline, exhibiting a slow dynamic process on the order of seconds with non-minimum phase characteristics, and inherent back-adjustment and lag. On the other hand, the wind and solar turbine units are connected to the grid via inverters, and their power response is a fast dynamic process on the order of milliseconds. This interactive coupling of "slow electromechanical dynamics" and "fast power electronic dynamics" results in extremely strong numerical rigidity in the system state equations. Under certain operating conditions, the low-frequency oscillations of the hydraulic system are prone to mode aliasing with the high-frequency synchronous oscillations of the inverter control loop, inducing complex system instability accidents.

[0004] In existing frequency control methods for hydro-wind-solar hybrid systems, controller parameter tuning is mostly based on linearized models near the system's steady-state operating point. These methods neglect key nonlinear elements such as head loss and actuator dead zones, failing to accurately describe system behavior under large disturbances. In particular, when the operating point approaches the stability limit, Hopf bifurcation often occurs, with its stability domain exhibiting a complex, non-convex geometry (such as an inverted U-shape). Traditional methods lack the ability to detect the boundaries of such nonlinear bifurcations, easily causing controller parameters to fall into potential bifurcation regions, leading to constant-amplitude oscillations or even divergent instability.

[0005] In optimizing controller parameters, existing technologies often employ traditional intelligent evolutionary algorithms such as genetic algorithms and particle swarm optimization for blind searching. These algorithms lack a mechanism to perceive the curvature characteristics of physical stability boundaries, leading to oscillating searches that repeatedly cross the stability and instability boundaries when approaching the edge of the stability region. This results in the elimination of numerous invalid solutions, severely wasting computational resources and making it difficult to converge to the true Pareto optimal front. Furthermore, existing optimization objectives often only pursue minimizing frequency deviation-related indicators, neglecting the mechanical wear costs caused by frequent actuator adjustments, thus failing to achieve a balance between system dynamic performance and equipment lifespan.

[0006] In summary, there is an urgent need to develop a collaborative control method for water-wind-solar hybrid systems that takes into account nonlinear dynamic boundary constraints, in order to solve the problems of insufficient modeling accuracy, weak oscillation risk prevention and control capabilities, low parameter optimization efficiency and single objective in existing technologies. Summary of the Invention

[0007] This invention provides a collaborative control method for a water-wind-solar hybrid system that takes into account nonlinear dynamic boundary constraints, in order to solve the problems of insufficient modeling accuracy, weak oscillation risk prevention and control, low parameter optimization efficiency and single objective in the prior art.

[0008] In a first aspect, the present invention provides a method for coordinated control of a water-wind-solar hybrid system considering nonlinear dynamic boundary constraints, the method comprising: Using time-scale separation technology, the hydro-wind-solar hybrid system is decoupled, and an initial singular perturbation nonlinear standard model of the system is constructed. Based on preset conditions, the initial singular perturbation nonlinear standard model is solved, and the reduced-dimensional Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem are constructed. Based on the reduced-dimensional Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem, a multi-band absolute stability domain is constructed after processing with the Routh-Hurwitz stability criterion. Using the multi-band absolute stability domain as a hard constraint space, a multi-objective optimization function that takes into account both the dynamic performance of the hydro-wind-solar hybrid system and the equipment lifespan in the system is constructed. Based on a preset out-of-bounds lethal penalty function, a multi-objective directional optimization algorithm based on bifurcation boundary curvature perception is used to optimize the multi-objective optimization function and obtain the optimal control parameter set. The optimal control parameter set is used to control the operation of the hydro-wind-solar hybrid system.

[0009] The collaborative control method for a hydro-wind-solar hybrid system considering nonlinear dynamic boundary constraints provided by this invention utilizes time-scale separation technology to decouple and construct an initial singular perturbation nonlinear standard model, achieving dynamic decoupling of the fast and slow time scales of the hydro-wind-solar hybrid system. This overcomes the numerical rigidity and dimensionality curse problems of the full-dimensional model, reducing the computational complexity of the model. Furthermore, by solving the model and constructing the Jacobian matrix of the fast and slow subsystems, high-dimensional reduction of the system is achieved, accurately characterizing the dynamic characteristics of low-frequency hydraulic and high-frequency electrical systems, providing a precise mathematical foundation for stability boundary analysis. Furthermore, by using the Routh-Hurwitz stability criterion to process and construct a multi-band absolute stability domain, the instability critical boundaries of low-frequency water hammer oscillations and high-frequency synchronous oscillations can be quantitatively defined, and the safe and feasible domain of controller parameters can be delineated, thereby mitigating the risk of multi-band oscillations from a mechanistic perspective. Furthermore, by constructing a multi-objective optimization function that considers both the dynamic performance of the hydro-wind-solar hybrid system and the equipment lifespan within the system, while simultaneously considering the system's frequency control quality and actuator mechanical losses, the dual objective constraints of grid operation safety and equipment health management are achieved. Furthermore, an optimal control parameter set is obtained by using a multi-objective directional optimization algorithm based on a pre-defined out-of-bounds penalty function and bifurcation boundary curvature perception. This achieves precise directional optimization of control parameters within the stability domain, avoiding invalid solution calculations, improving the convergence speed and optimality of the solution, and ensuring strict stability of the parameter set. Furthermore, by utilizing the optimal control parameter set to control the operation of the hydro-wind-solar hybrid system, coordinated and stable control of the system across the entire frequency band and multiple time scales is achieved. This smooths out wind and solar power fluctuations, ensures system frequency stability, and enhances the system's anti-disturbance capability. Therefore, by implementing this invention, the oscillation problem caused by strong coupling across multiple time scales in the hydro-wind-solar hybrid system is effectively solved, achieving proactive avoidance of system oscillation risks and global optimization of dynamic performance, while balancing system frequency stability and equipment lifespan.

[0010] In one optional implementation, the hydro-wind-solar hybrid system is decoupled using time-scale separation technology, and an initial singular perturbation nonlinear standard model of the hydro-wind-solar hybrid system is constructed, including: Based on preset time scale separation parameters, the full-dimensional state vector of the hydro-wind-solar hybrid system is decomposed to obtain a slow-varying state vector and a fast-varying state vector. The slow-varying state vector is used to describe the slow dynamic process dominated by hydraulics and mechanics in the hydro-wind-solar hybrid system, while the fast-varying state vector is used to describe the fast dynamic process dominated by power electronics and networks in the hydro-wind-solar hybrid system. Based on the small perturbation theory of power systems and the physical equations of each subsystem in the hydro-wind-solar hybrid system, an initial singular perturbation nonlinear standard model is constructed using the slow-varying state vector and the fast-varying state vector.

[0011] The collaborative control method for a hydro-wind-solar hybrid system considering nonlinear dynamic boundary constraints provided by this invention decomposes the full-dimensional state vector of the hydro-wind-solar hybrid system by pre-setting time-scale separation parameters. This accurately distinguishes the state variables of the hydraulic-mechanical slow dynamics and the power electronics-network fast dynamics, achieving time-scale decoupling of the system dynamics and laying the foundation for independent analysis of the fast and slow subsystems. Furthermore, based on the small perturbation theory of power systems and the physical equations of each subsystem in the hydro-wind-solar hybrid system, an initial singular perturbation nonlinear standard model is constructed, ensuring the physical accuracy and mathematical rigor of the model. Therefore, by implementing this invention, and establishing a tenth-order singular perturbation nonlinear standard model with fast and slow separation for the hydro-wind-solar hybrid system, the computational defects of traditional full-dimensional models are overcome, providing a precise and efficient modeling foundation for subsequent stability analysis and parameter optimization.

[0012] In one optional implementation, based on preset conditions, the initial singular perturbation nonlinear standard model is solved, and the dimension-reduced Jacobian matrix of the slowly varying subsystem and the boundary layer Jacobian matrix of the fast varying subsystem are constructed, including: Based on preset conditions, the initial singular perturbation nonlinear standard model is solved to obtain a slow manifold describing the change of fast variables following slow variables. The slow manifold is then input into the slow-varying subsystem equations in the initial singular perturbation nonlinear standard model to obtain the dimension-reduced dynamic equations of the slow-varying subsystem. Using the dynamic equations of the slow-varying subsystem, the full-dimensional sub-Jacobi matrix of the hydro-wind-solar hybrid system is calculated. Based on preset time-scale separation parameters, the initial singular perturbation nonlinear standard model, and the full-dimensional sub-Jacobi matrix, the dimension-reduced Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem are constructed.

[0013] The present invention provides a collaborative control method for a hydro-wind-solar complementary system considering nonlinear dynamic boundary constraints. By solving an initial singular perturbation nonlinear standard model under preset conditions, a slow manifold describing the change of fast variables following slow variables is obtained. This characterizes the law governing the change of fast variables following slow variables and determines the low-dimensional surface of the system trajectory after the disappearance of the fast transient, thus quantifying the influence of the fast subsystem on the slow subsystem. Furthermore, by substituting the slow manifold into the slow-variable equations to obtain the reduced-dimensional dynamic equations, the high-dimensional reduction of the slow-variable subsystem is achieved, simplifying the analysis complexity of the slow dynamic process while preserving its core characteristics. Furthermore, the full-dimensional sub-Jacobi matrix is ​​calculated using the reduced-dimensional equations, providing fundamental matrix data for constructing the Jacobian matrix of the fast and slow subsystems, accurately reflecting the dynamic characteristics of the system at the equilibrium point. Furthermore, the construction of the Jacobian matrix of the fast and slow subsystems based on multiple factors enables precise decoupling characterization of the dynamic characteristics of the fast and slow subsystems and captures the stability characteristics of low-frequency hydraulic and high-frequency electrical dynamics, respectively. Therefore, by implementing this invention, through slow manifold solving and dimensionality reduction, the high-dimensional model of the water-wind-solar hybrid system is accurately decoupled. The constructed Jacobian matrix of the fast and slow subsystems provides the core mathematical basis for the subsequent analysis of the multi-band stability domain, thus improving the accuracy and efficiency of stability analysis.

[0014] In one optional implementation, based on preset timescale separation parameters, an initial singular perturbation nonlinear standard model, and a full-dimensional sub-Jacobi matrix, a dimension-reduced Jacobian matrix for the slowly varying subsystem and a boundary layer Jacobian matrix for the fast varying subsystem are constructed, including: Based on the full-dimensional sub-Jacobi matrix, the dimension-reduced Jacobian matrix of the slow-variable subsystem is determined using matrix Schur complement theory; based on the preset time scale separation parameters, the fast time scale is determined; based on the fast time scale, the initial singular perturbation nonlinear standard model is transformed into the target singular perturbation nonlinear standard model using the chain rule; slow variables are frozen in the boundary layer of the slow manifold, and the equations of the fast-variable subsystem in the target singular perturbation nonlinear standard model are linearized using first-order Taylor, to obtain the boundary layer Jacobian matrix of the fast-variable subsystem.

[0015] The present invention provides a collaborative control method for a hydro-wind-solar hybrid system that considers nonlinear dynamic boundary constraints. By utilizing Schur complement theory to determine the dimension-reduced Jacobian matrix of the slow-varying subsystem, it can accurately account for the correction effect of the steady-state characteristics of the fast subsystem on the slow dynamics, overcoming the error of the traditional direct truncation method and improving the accuracy of the slow subsystem matrix. Furthermore, based on preset time scale separation parameters, a fast time scale is determined, establishing a suitable time dimension for the boundary layer analysis of the fast subsystem, which can fit the characteristics of the millisecond-level fast dynamics of power electronics. Furthermore, by using the chain rule to transform the initial singular perturbation nonlinear standard model into the target singular perturbation nonlinear standard model, the fast subsystem model is reconstructed at a fast time scale, facilitating the analysis of the boundary layer dynamics of the fast-varying subsystem. Furthermore, by freezing slow variables within the boundary layer of the slow manifold and linearizing them to obtain the Jacobian matrix of the fast subsystem boundary layer, the dynamic characteristics of the fast subsystem can be accurately characterized within the boundary layer, providing accurate matrix support for the boundary analysis of high-frequency electrical oscillations. Therefore, by implementing this invention, high-precision Jacobian matrices for fast and slow subsystems were constructed through Schur complement transformation and boundary layer analysis under fast time scales, respectively. This enabled precise decoupling and characterization of the stability features of the fast and slow subsystems, laying a high-precision mathematical foundation for subsequent bifurcation boundary analysis.

[0016] In one optional implementation, based on the reduced-dimensional Jacobian matrix of the slowly varying subsystem and the boundary layer Jacobian matrix of the fast varying subsystem, a multi-band absolute stability domain is constructed after processing using the Routh-Hurwitz stability criterion, including: Based on the reduced-dimensional Jacobian matrix of the slowly varying subsystem, the low-frequency hydraulic oscillation boundary is determined by processing with the Routh-Hurwitz stability criterion; based on the boundary layer Jacobian matrix of the fast varying subsystem, the high-frequency electrical oscillation boundary is determined by processing with the Routh-Hurwitz stability criterion; based on the low-frequency hydraulic oscillation boundary and the high-frequency electrical oscillation boundary, a multi-frequency absolute stability domain is constructed.

[0017] The present invention provides a collaborative control method for a hydro-wind-solar hybrid system considering nonlinear dynamic boundary constraints. By processing the reduced-dimensional Jacobian matrix of the slow-varying subsystem using the Routh-Hurwitz stability criterion and determining the low-frequency hydraulic oscillation boundary, the critical condition for low-frequency Hopf bifurcation caused by nonlinear head loss of the turbine can be accurately analyzed, defining the instability boundary of low-frequency oscillations. Furthermore, by processing the boundary layer Jacobian matrix of the fast-varying subsystem using the Routh-Hurwitz stability criterion and determining the high-frequency electrical oscillation boundary, the critical condition for high-frequency subsynchronous Hopf bifurcation caused by high inverter gain can be accurately analyzed, defining the instability boundary of high-frequency oscillations. Furthermore, by combining the low-frequency hydraulic oscillation boundary and the high-frequency electrical oscillation boundary, a multi-band absolute stability domain that simultaneously satisfies the stability requirements of both low and high frequencies can be constructed, quantitatively defining the red line for safe system operation from a physical mechanism perspective. Therefore, by implementing this invention, the critical boundary of multi-band oscillation in a hydro-wind-solar hybrid system is analyzed for the first time from the perspective of nonlinear bifurcation theory. The constructed multi-band absolute stability domain defines a hard safety constraint for controller parameter tuning, realizing the transformation from passive suppression after the fact to active avoidance before the fact in oscillation risk prevention and control.

[0018] In one optional implementation, a multi-objective optimization function is constructed, using the multi-band absolute stability domain as the hard constraint space, to balance the dynamic performance of the hydro-wind-solar hybrid system with the equipment lifespan within the system. This function includes: Obtain the objective function of mechanical loss of actuators in the hydro-wind-solar hybrid system; construct the global frequency quality objective function of the hydro-wind-solar hybrid system using the time multiplication of the absolute error integral and the maximum frequency change rate of the hydro-wind-solar hybrid system; and construct a multi-objective optimization function using the multi-band absolute stability domain as the hard constraint space and the global frequency quality objective function and the objective function of mechanical loss of actuators.

[0019] The present invention provides a collaborative control method for a hydro-wind-solar hybrid system that considers nonlinear dynamic boundary constraints. By obtaining the objective function of mechanical loss of the actuators, it can quantify the degree of loss caused by turbine guide vane wear and photovoltaic power fluctuations, thereby constraining the mechanical actions of the actuators and extending the service life of the equipment. Furthermore, by constructing a global frequency quality objective function for the hydro-wind-solar hybrid system using the time multiplied by the absolute error integral and the maximum frequency change rate of the system, it comprehensively considers the system's frequency recovery speed and inertial support capability, thus accurately quantifying the quality of system frequency control and improving the system's frequency anti-disturbance capability. Furthermore, by constructing a multi-objective optimization function with the stability domain as a constraint, it takes system stability as a hard prerequisite and integrates the dual objectives of frequency quality and equipment loss, achieving collaborative optimization of multiple objectives and avoiding the limitations of single-objective optimization. Therefore, by implementing this invention, the drawbacks of traditional single-objective optimization are overcome, achieving a collaborative consideration of system frequency control quality and actuator mechanical life, providing a scientific and comprehensive objective guide for optimizing control parameters.

[0020] In one optional implementation, based on a preset out-of-bounds lethal penalty function, a multi-objective directional optimization algorithm based on bifurcation boundary curvature perception is used to optimize the multi-objective optimization function to obtain the optimal control parameter set, including: The parameter population of the controller in the hydro-wind-solar hybrid system is initialized, and a search vector is generated based on the parameter population. A zero-real-part arcuate bifurcation boundary manifold in the parameter space is constructed based on the reduced-dimensional Jacobian matrix of the slowly varying subsystem and the boundary layer Jacobian matrix of the fast varying subsystem. Based on the zero-real-part arcuate bifurcation boundary manifold, the sensitivity gradient vector of the parameter points with respect to the real part of the largest eigenvalue is calculated. A boundary repulsion potential field and a tangential projection matrix are constructed based on the sensitivity gradient vector. The boundary repulsion potential field generates a reverse repulsive force when a parameter point approaches the zero-real-part arcuate bifurcation boundary manifold. The tangential projection matrix is ​​used to filter out components perpendicular to the bifurcation boundary in the search vector. Based on the repulsive effect of the boundary repulsion potential field, the controller parameters of the hydro-wind-solar hybrid system are updated by modifying the mutation relation to guide the parameter population to evolve and search along the tangent direction of the bifurcation boundary. During the evolutionary search process, based on the stability margin threshold of the multi-band absolute stability domain, an out-of-bounds lethal penalty function is used to successively verify the stability of the updated controller parameters until the optimal control parameter set after iterative evolution is obtained.

[0021] The collaborative control method for a hydro-wind-solar hybrid system considering nonlinear dynamic boundary constraints provided by this invention provides an initial solution space for algorithm optimization by initializing the parameter population and generating a search vector, ensuring the ergodicity of the optimization. Furthermore, by constructing a zero-real-part arc-shaped bifurcation boundary manifold, the instability critical boundary in the parameter space is accurately characterized, providing a geometric reference for directional optimization. Furthermore, by calculating the sensitivity gradient vector, the positional relationship between parameter points and bifurcation boundaries can be reflected in real time, providing gradient basis for boundary perception and directional optimization. Furthermore, by constructing a boundary repulsion potential field, parameter points are prevented from approaching the instability boundary; simultaneously, by constructing a tangential projection matrix, directional search along the stable boundary is achieved, improving the safety and specificity of the optimization. Furthermore, by filtering out components and updating controller parameters, the population is guided to evolve along the tangential direction of the stable boundary, enabling in-depth exploration of performance limits while ensuring stability, thus improving the optimality of the solution. Furthermore, by using a stability margin threshold and an out-of-bounds lethal penalty function for stability verification, unstable solutions can be forcibly eliminated, ensuring that all iterative solutions are strictly within the stability region. This avoids invalid computation and improves optimization efficiency and solution stability. Therefore, by implementing this invention, the problems of oscillating search and premature convergence in traditional evolutionary algorithms are solved through the combination of curvature sensing, directional mutation, and out-of-bounds lethal penalty function. While ensuring strict stability of control parameters, the convergence speed and optimization accuracy of the algorithm are improved, ultimately resulting in a Pareto optimal control parameter set that balances system stability and dynamic performance.

[0022] Secondly, the present invention provides a collaborative control device for a water-wind-solar hybrid system that considers nonlinear dynamic boundary constraints, the device comprising: The system comprises the following modules: a decoupling module for decoupling the hydro-wind-solar hybrid system using time-scale separation technology and constructing an initial singular perturbation nonlinear standard model; a solution module for solving the initial singular perturbation nonlinear standard model based on preset conditions and constructing the reduced-dimensional Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem; a processing module for constructing a multi-band absolute stability domain based on the reduced-dimensional Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem, processed by the Routh-Hurwitz stability criterion; a construction module for constructing a multi-objective optimization function that balances the dynamic performance of the hydro-wind-solar hybrid system with the multi-band absolute stability domain as a hard constraint space; a solution module for optimizing the multi-objective optimization function based on a preset out-of-bounds lethal penalty function and a multi-objective directional optimization algorithm based on bifurcation boundary curvature perception to obtain the optimal control parameter set; and a control module for controlling the operation of the hydro-wind-solar hybrid system using the optimal control parameter set.

[0023] Thirdly, the present invention provides a computer-readable storage medium storing computer instructions for causing a computer to execute the collaborative control method for a water-wind-solar hybrid system considering nonlinear dynamic boundary constraints described in the first aspect or any corresponding embodiment.

[0024] Fourthly, the present invention provides a computer program product, including computer instructions, which are used to cause a computer to execute the above-described first aspect or any corresponding embodiment of the method for coordinated control of a water-wind-solar hybrid system considering nonlinear dynamic boundary constraints. Attached Figure Description

[0025] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0026] Figure 1 This is a schematic diagram of an application scenario according to an embodiment of the present invention; Figure 2 This is a flowchart illustrating the collaborative control method for a water-wind-solar hybrid system considering nonlinear dynamic boundary constraints according to an embodiment of the present invention. Figure 3This is a schematic diagram of the complete closed-loop control process from singular perturbation modeling and analysis to curvature-sensing directional optimization according to an embodiment of the present invention; Figure 4 This is a schematic diagram illustrating the specific execution flow of the multi-objective directional optimization algorithm based on bifurcation boundary curvature perception according to an embodiment of the present invention; Figure 5 This is a schematic diagram of the parameter adaptive closed-loop cooperative control principle of a water-wind-solar hybrid system according to an embodiment of the present invention; Figure 6A This is a schematic diagram showing the first simulation comparison results of the control method according to the embodiment of the present invention with the prior art (NSGA-II algorithm) and the unoptimized case; Figure 6B This is a schematic diagram showing a second simulation comparison between the control method according to an embodiment of the present invention and the prior art (NSGA-II algorithm) and the unoptimized case; Figure 7 This is a structural block diagram of a water-wind-solar hybrid system collaborative control device considering nonlinear dynamic boundary constraints according to an embodiment of the present invention; Figure 8 This is a schematic diagram of the hardware structure of an electronic device according to an embodiment of the present invention. Detailed Implementation

[0027] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0028] It is understood that before using the technical solutions disclosed in the various embodiments of the present invention, users should be informed of the types, scope of use, and usage scenarios of the personal information involved in the present invention and their authorization should be obtained in accordance with relevant laws and regulations through appropriate means.

[0029] The terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0030] As an optional application scenario of this invention, considering the specific application environment architecture or specific hardware architecture upon which the collaborative control method for a water-wind-solar hybrid system taking into account nonlinear dynamic boundary constraints depends, the specific application environment architecture or specific hardware architecture is described here. For example... Figure 1 As shown, the architecture system may include at least one terminal device and at least one server. Figure 1 The system is illustrated in the example, which includes a computer 101, a mobile terminal 102, and a server 103, and the terminal devices such as the computer 101 and the mobile terminal 102 are connected to the server 103 through a network 110.

[0031] Specifically, the terminal device can be a smartphone, tablet, laptop, PDA, desktop computer, game console, smart TV, smart wearable device, in-vehicle terminal, VR (Virtual Reality) device, AR (Augmented Reality) device, etc. Server 103 can be a standalone physical server, a server cluster, a distributed system, or a cloud server providing cloud services. Network 110 can be a wired or wireless network, examples of which include, but are not limited to, the Internet, corporate intranet, local area network, wide area network, mobile communication network, and combinations thereof.

[0032] Currently widely used intelligent evolutionary algorithms (such as genetic algorithms and particle swarm optimization) typically employ random mutation strategies for blind searching. However, due to the lack of a mechanism to perceive the curvature characteristics of physical stability boundaries, these algorithms often exhibit "oscillatory search" when approaching the edge of the stability region: the population repeatedly crosses the stability and instability boundaries, resulting in the elimination of a large number of individuals, severely wasting computational resources and making it difficult to converge to the true Pareto optimal front. Furthermore, existing objective functions often excessively pursue minimizing the frequency deviation index (ITAE), neglecting the mechanical wear cost to the turbine guide vanes caused by frequent adjustments. Therefore, a cooperative control method is needed that can accurately analyze Hopf bifurcation boundaries and perform directional and safe optimization based on boundary curvature.

[0033] This invention provides a collaborative control method for a water-wind-solar hybrid system that takes into account nonlinear dynamic boundary constraints. It effectively solves the oscillation problem caused by strong coupling across multiple time scales in the water-wind-solar hybrid system, achieves proactive avoidance of system oscillation risk and global optimization of dynamic performance, and balances system frequency stability with equipment lifespan.

[0034] According to an embodiment of the present invention, a method for coordinated control of a water-wind-solar hybrid system considering nonlinear dynamic boundary constraints is provided. It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions. Furthermore, although a logical order is shown in the flowchart, in some cases, the steps shown or described may be executed in a different order than that shown here.

[0035] This embodiment provides a collaborative control method for a water-wind-solar hybrid system that considers nonlinear dynamic boundary constraints. This method can be used on the aforementioned mobile terminals, such as mobile phones and tablets. Figure 2 This is a flowchart of a collaborative control method for a water-wind-solar hybrid system considering nonlinear dynamic boundary constraints according to an embodiment of the present invention, such as... Figure 2 As shown, the process includes the following steps: Step S201: Using time-scale separation technology, the water-wind-solar complementary system is decoupled, and an initial singular perturbation nonlinear standard model of the water-wind-solar complementary system is constructed.

[0036] In one alternative embodiment, Time-Scale Decomposition (TSD) is a method for analyzing complex signals or systems. Its core idea is to separate components of the original data at different time scales (i.e., different frequencies or evolution rates) in order to study the characteristics, mechanisms or influencing factors at each scale separately.

[0037] Furthermore, in this embodiment, the time scale separation technique introduces a dedicated separation parameter based on the difference in the order of magnitude of the time constants of the dynamic responses of each component of the system. This decomposes the system state variables into sub-vectors of different time scales, thereby enabling mathematical decoupling of dynamic processes at different rates. This eliminates the numerical rigidity problem of the model caused by multi-scale coupling, facilitating independent analysis and modeling of dynamic processes at each scale.

[0038] In one optional embodiment, the hydro-wind-solar hybrid system refers to a multi-energy complementary power generation system composed of hydro turbine generator sets, wind turbine generator sets, and photovoltaic power generation units as the core. It relies on the regulation capability of hydro turbine generator sets to smooth out random power fluctuations of wind power and photovoltaic power, and is connected to the main power grid through the active power support link to the bus.

[0039] Furthermore, this system is a typical multi-timescale strongly coupled nonlinear system, which can include hydraulic-mechanical slow dynamics at the second level and power electronics-network fast dynamics at the millisecond level. Each component also exhibits physical characteristics such as nonlinear head loss, water hammer effect, virtual inertia control, and transmission delay.

[0040] In one optional embodiment, the initial singular perturbation nonlinear standard model represents the basic mathematical model describing the full-dimensional dynamic characteristics of the hydro-wind-solar hybrid system. It may include nonlinear vector functions of slow-varying subsystems and fast-varying subsystems, and can accurately characterize the coupling relationship between hydraulic-mechanical slow dynamics and power electronics-network fast dynamics.

[0041] In one optional embodiment, based on singular perturbation theory and small perturbation theory of power systems, and combined with the inherent time-scale differences in the hydraulic-mechanical dynamics (second-level) and power electronics-network dynamics (millisecond-level) of the hydro-wind-solar hybrid system, a time-scale separation parameter is introduced to complete the fast and slow sub-vectors of the system's full-dimensional state vector. Then, based on the physical equations of each subsystem of the turbine, wind turbine, and photovoltaic system, and by integrating core physical characteristics such as nonlinear head loss, inertial control, and transmission delay, a singular perturbation nonlinear standard model that can accurately characterize the fast and slow dynamic coupling relationship of the system can be constructed.

[0042] Step S202: Based on preset conditions, solve the initial singular perturbation nonlinear standard model and construct the reduced Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem.

[0043] In an optional embodiment, the preset condition means that the time scale separation parameter is made close to 0, that is, it is assumed that the fast-changing subsystem instantaneously reaches a quasi-steady state.

[0044] In one optional embodiment, the Jacobian matrix of the slow-varying subsystem represents the matrix describing the dynamic characteristics of the slow-dynamic process of the hydro-mechanical system at the equilibrium point after dimensionality reduction based on the singular perturbation theory; the Jacobian matrix of the boundary layer of the fast-varying subsystem represents the boundary layer feature matrix constructed for the fast-dynamic process of the power electronics-network within the slow-manifold boundary layer, which can accurately characterize the boundary layer characteristics of the fast-dynamic process.

[0045] In one optional embodiment, based on the preset conditions of the singular perturbation theory, the initial singular perturbation nonlinear standard model is solved in a targeted manner. Through theoretical processing such as dimensionality reduction and linearization, Jacobian matrices that are adapted to the dynamic characteristics of fast and slow subsystems can be constructed respectively, namely the dimensionality-reduced Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem.

[0046] Step S203: Based on the reduced Jacobian matrix of the slowly varying subsystem and the boundary layer Jacobian matrix of the fast varying subsystem, a multi-band absolute stability domain is constructed after processing with the Routh-Herwitz stability criterion.

[0047] In an optional embodiment, the Routh-Hurwitz stability criterion represents a classical algebraic criterion for determining the asymptotic stability of a linear system based on the coefficients of the system's characteristic polynomial. By analyzing the coefficients of the characteristic polynomial to construct a Routh array, and judging the real distribution characteristics of the system's eigenvalues ​​based on the changes in the signs of the coefficients in the first column of the array, it is possible to directly determine whether the system has unstable eigenvalues.

[0048] In one optional embodiment, the multi-band absolute stability domain represents the safe and feasible domain of controller parameters constructed for the multi-timescale characteristics of the hydro-wind-solar hybrid system. It is the inner intersection of the low-frequency hydraulic oscillation stability boundary corresponding to the slow-changing subsystem and the high-frequency electrical oscillation stability boundary corresponding to the fast-changing subsystem. It is the region in the parameter space that can simultaneously avoid the instability risks of low-frequency water hammer oscillation and high-frequency synchronous oscillation.

[0049] Furthermore, the multi-band absolute stability domain has a non-convex complex geometry.

[0050] In one optional embodiment, based on the Jacobian matrix of the fast and slow subsystems, the instability critical boundaries of the low-frequency and high-frequency oscillations of the system are analyzed by the Routh-Herwitz stability criterion, and then the corresponding multi-band absolute stability domain can be constructed by taking the intersection of the boundaries.

[0051] Step S204: Using the multi-band absolute stability domain as the hard constraint space, construct a multi-objective optimization function that takes into account both the dynamic performance of the water-wind-solar hybrid system and the equipment lifespan in the water-wind-solar hybrid system.

[0052] In one optional embodiment, the multi-band absolute stability domain is used as a hard safety constraint for the optimization of controller parameters to avoid the possibility of parameters falling into the unstable region during the optimization process. At the same time, the two core optimization objectives of global dynamic operation performance of the water-wind-solar hybrid system and equipment mechanical life are integrated to construct a composite multi-objective optimization function.

[0053] Step S205: Based on the preset out-of-bounds lethal penalty function, the multi-objective directional optimization algorithm based on bifurcation boundary curvature perception is used to optimize the multi-objective optimization function and obtain the optimal control parameter set.

[0054] In an optional embodiment, the preset out-of-bounds lethal penalty function is a stability verification penalty mechanism designed to ensure that the controller parameters are strictly within the multi-band absolute stable domain. It is used to forcibly eliminate controller parameter solutions that cross the Hopf bifurcation boundary and fall into the unstable domain during the algorithm optimization iteration. By assigning an infinite penalty value to the unstable solution, it is directly eliminated in the non-dominated sorting of the algorithm, ensuring that the parameter solutions obtained by the final optimization are all strictly within the stable domain, as shown in the following relationship (1): (1) In the formula: This represents the composite objective function value after stability verification, which is the final target value used for parameter selection during algorithm iteration; This represents a new set of PID controller parameters to be optimized generated during the algorithm iteration process, which is the object of this stability verification. This represents the original multi-objective optimization function value of the hydro-wind-solar hybrid system, which usually includes indicators such as frequency quality ITAE and mechanical loss. This true objective value is only used when the parameters are determined to be stable. Indicates the current parameter Below, the system Jacobian matrix (including the dimension-reduced Jacobian matrix of the slowly varying subsystems) Jacobian matrix of boundary layer of fast variable subsystem The maximum value among the real parts of all eigenvalues ​​is used to determine the stable state of the system. This represents the preset safety and stability margin threshold, which is usually a very small positive number. This represents an infinite penalty value.

[0055] In one alternative embodiment, the bifurcation boundary curvature-aware multi-objective orientation optimization algorithm (BCA-MODA) represents an improved multi-objective evolutionary algorithm designed for the optimization of controller parameters of a water-wind-solar hybrid system, which can solve the problems of blind search and oscillating convergence in traditional evolutionary algorithms.

[0056] In one optional embodiment, a preset out-of-bounds lethal penalty function is used as a hard constraint for stability. A multi-objective directional optimization algorithm based on bifurcation boundary curvature perception is adopted to perform directional iterative optimization on the constructed multi-objective optimization function that takes into account both system dynamic performance and equipment lifespan. This can ultimately obtain the optimal set of control parameters that are strictly located within the absolute stability domain of multiple frequency bands and take into account both system frequency recovery quality and actuator mechanical lifespan.

[0057] Step S206: Use the optimal set of control parameters to control the operation of the hydro-wind-solar hybrid system.

[0058] In one optional embodiment, the obtained optimal control parameter set, which is strictly located in the absolute stability domain of multiple frequency bands, is deployed to the controllers at each level of the hydro-wind-solar hybrid system (hydro turbine governor, wind and solar inverter controller, etc.). Then, through parameterized control strategies, closed-loop regulation of the entire system operation can be carried out, which can realize cross-scale collaborative stability control of the hydro-wind-solar hybrid system and ensure the optimal synergy of frequency safety, dynamic performance and equipment life under all operating conditions.

[0059] For example, the Pareto optimal control parameter set is distributed to the actuators at each level of the hydro-wind-solar hybrid system through the system control platform to complete the online tuning and updating of the control parameters, replacing the original traditional tuning parameters.

[0060] Furthermore, each subsystem controller initiates closed-loop cooperative control based on the optimal deployed parameters: 1. Turbine governor: Adjusts the guide vane opening according to the optimal parameters, responds to system frequency deviation, smooths wind and solar power fluctuations, and at the same time reduces frequent guide vane movement and mechanical wear while meeting frequency control requirements; 2. Wind and solar inverter controller: Based on the optimal parameter output, virtual inertia and damping support are provided to quickly respond to system power deficit, suppress high-frequency synchronous oscillations, and improve the system frequency anti-disturbance capability; 3. System-level collaborative control unit: Real-time monitoring of the overall system operation status, coordination of control actions of various subsystems, and ensuring dynamic collaborative matching across multiple time scales.

[0061] Furthermore, the system collects operational data from each subsystem in real time and continuously evaluates the global frequency quality (ITAE, RoCoF) and mechanical loss indicators of the actuators. If the indicators deteriorate, the online fine-tuning mechanism of the control parameters is triggered. Thus, without breaking the absolute stability domain of multiple frequency bands, the control parameters are dynamically optimized, and the optimal collaborative control is achieved throughout the entire life cycle of the system.

[0062] Furthermore, through the continuous action of closed-loop control, the hydro-wind-solar hybrid system can achieve stable operation under all operating conditions. Specifically, it smooths out random power fluctuations from wind and solar power, ensures the system frequency remains stable within the allowable range, suppresses low-frequency hydraulic oscillations and high-frequency electrical oscillations, and extends the service life of actuators such as turbines, ultimately achieving safe, efficient, and economical operation of the hydro-wind-solar hybrid system under high penetration rates of new energy sources.

[0063] The collaborative control method for water-wind-solar hybrid systems that takes into account nonlinear dynamic boundary constraints provided in this embodiment effectively solves the oscillation problem caused by strong coupling across multiple time scales in water-wind-solar hybrid systems. It achieves proactive avoidance of system oscillation risks and global optimization of dynamic performance, while taking into account both system frequency stability and equipment lifespan.

[0064] In some optional implementations, step S201 above includes: Step S2011: Based on the preset time scale separation parameters, the full-dimensional state vector of the water-wind-solar hybrid system is split to obtain a slow-changing state vector and a fast-changing state vector.

[0065] In one optional embodiment, the preset timescale separation parameter represents a dimensionless small parameter in singular perturbation theory used to quantify the difference in fast and slow timescales in a hydro-wind-solar hybrid system. In this embodiment, the preset timescale separation parameter is introduced based on the order of magnitude of the time constants of each component of the system. and define The equivalent response time constant of the photovoltaic / wind power inverter With respect to the inertial time constant of the turbine flow channel The ratio, usually .

[0066] In an optional embodiment, the slowly varying state vector is used to describe the slow dynamic processes dominated by hydraulics and mechanics in the hydro-wind-solar hybrid system, and may include turbine flow rate. Water head Guide vane opening Mechanical power and system frequency deviation .

[0067] In an optional embodiment, the fast-changing state vector is used to describe the fast dynamic processes dominated by power electronics and the network in a hydro-wind-solar hybrid system, and may include the output power of the photovoltaic inverter. Intermediate control variables of wind power converters, command transmission delay status, etc.

[0068] In an optional embodiment, separation parameters are based on a preset time scale. The full-dimensional state vector of the water-wind-solar complementary system Reorganize and split into slowly varying state vectors and fast-changing state vector .

[0069] Step S2012: Based on the small perturbation theory of power systems and the physical equations of each subsystem in the hydro-wind-solar hybrid system, an initial singular perturbation nonlinear standard model is constructed using slow-varying state vectors and fast-varying state vectors.

[0070] In one alternative embodiment, the small perturbation theory of power systems represents the linearization of a nonlinear system near the equilibrium point, and studies the system's ability to maintain stable operation when faced with small load fluctuations.

[0071] In an optional embodiment, the physical equations of each subsystem represent mathematical equations established based on the objective physical laws, energy conversion principles, and equipment operating mechanisms of hydropower, wind power, photovoltaics, and grid connection links to describe the changes of the core state variables of each subsystem over time. These equations may include: equations for slowly varying subsystems. and fast variable subsystem equations .

[0072] In an optional embodiment, based on the small disturbance theory of power systems and the physical equations of each subsystem in the hydro-wind-solar hybrid system, the slowly varying state vector is utilized. and fast-changing state vector The singular perturbation nonlinear standard model is constructed as shown in equation (2): (2) In the formula: This represents the PID controller parameter vector to be optimized, including proportional gain. ,integral Differential gain .

[0073] Furthermore, the equations of the slowly varying subsystem : For the slow-varying subsystem, it is a nonlinear vector function, encompassing nonlinear head loss coefficients. The hydraulic dynamic equations and generator rotor motion equations. Furthermore, For the corresponding hydro-generator unit, it determines how the hydraulic system supports frequency stability through mechanical adjustment when the load changes, including the dynamic water hammer effect caused by long pipelines and large water inertia.

[0074] Furthermore, the equations of the fast variable subsystem : This represents the nonlinear vector function of the fast-changing subsystem, encompassing the inverter's inertial response and the dynamic equations of the control loop. Furthermore, For wind and solar power generation units, a virtual inertial response achieved through inverter control is described. The response speed is extremely fast, but it is prone to inducing subsynchronous oscillations under high gain.

[0075] Furthermore, the slow-changing subsystem, the fast-changing subsystem, and the water-wind-solar hybrid system are complementary. This example demonstrates this by explicitly distinguishing them during modeling. and This enables subsequent algorithms to specifically avoid low-frequency oscillations at the hydraulic end and high-frequency oscillations at the electrical end, achieving true coordinated control.

[0076] In some optional implementations, step S202 above includes: Step S2021: Based on preset conditions, solve the initial singular perturbation nonlinear standard model to obtain the slow manifold describing the change of fast variables following slow variables.

[0077] In an alternative embodiment, when the time scale separation parameter When the dynamic response speed of the fast variable subsystem is much faster than that of the slow variable subsystem, it can be assumed that the fast variable will quickly converge to the quasi-steady state determined by the slow variable. The state trajectory corresponding to the quasi-steady state is the slow manifold, which is essentially the low-dimensional surface on which the system trajectory depends after the fast transient in the state space disappears.

[0078] In an optional embodiment, the preset condition is to set the time scale separation parameter. That is, assuming the fast-changing subsystem instantaneously reaches a quasi-steady state, the equations of the fast-changing subsystem are as follows: Degenerates into a system of nonlinear algebraic equations Furthermore, solving this system of equations yields the analytical expression for the slow manifold describing the change of the fast variable following the slow variable. .in, Denotes slow manifold functions, characterizing fast variables. With slow variables Control parameters The changing pattern.

[0079] Furthermore, this manifold geometrically represents the low-dimensional surface to which the system trajectory rests after the rapid transient in the state space disappears.

[0080] Step S2022: Input the slow manifold into the slow-varying subsystem equations in the initial singular perturbation nonlinear standard model to obtain the dimensionality-reduced slow-varying subsystem dynamic equations.

[0081] In an alternative embodiment, the transient process of the fast variable can be ignored, and its state is completely determined by the slow variable. Therefore, the fast variable can be replaced by a function of the slow variable, and the original coupled fast and slow subsystem equations can be decoupled into independent dynamic equations containing only the slow variable, preserving the core characteristics of slow dynamics while eliminating the coupling effect brought by the fast variable.

[0082] In an alternative embodiment, the slow manifold Substituting the equations of the slow-varying subsystems in the initial singular perturbation nonlinear standard model The reduced dynamic equations of the slow variable subsystem can be obtained. .in, This represents the derivative of the slowly varying state vector with respect to time after dimensionality reduction.

[0083] Step S2023: Calculate the full-dimensional sub-Jacobi matrix of the water-wind-solar hybrid system using the dynamic equations of the slowly varying subsystem.

[0084] In an optional embodiment, for a nonlinear state equation, its Jacobian matrix is ​​the partial derivative matrix of the state equation with respect to the state variables, used to characterize the linearized dynamic characteristics of the system near the equilibrium point. In this embodiment, based on the dimensionality-reduced slow-variable subsystem dynamic equation, the corresponding full-dimensional sub-Jacobi matrix can be obtained by taking the partial derivative of the state equation of the full-dimensional system at the system equilibrium point. Furthermore, the block submatrices of the full-dimensional sub-Jacobi matrix are shown in the following relationships (3) to (6): (3) (4) (5) (6) In the formula: It represents the partial derivative matrix of the equations of the slow-variable subsystem with respect to the slow variables, and characterizes the dynamic coupling properties of the slow variables themselves; This represents the partial derivative matrix of the equations of the slow-variable subsystem with respect to the fast variables, characterizing the influence of the fast variables on the slow dynamics; This represents the partial derivative matrix of the equations of the fast-variable subsystem with respect to the slow variables, characterizing the influence of the slow variables on the fast dynamics; It represents the partial derivative matrix of the equations of the fast variable subsystem with respect to the fast variables, and characterizes the dynamic coupling properties of the fast variables themselves.

[0085] Step S2024: Based on the preset time scale separation parameters, the initial singular perturbation nonlinear standard model, and the full-dimensional sub-Jacobi matrix, construct the dimension-reduced Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem.

[0086] Specifically, step S2024 above includes: Step a1: Based on the full-dimensional sub-Jacobi matrix, determine the dimension-reduced Jacobian matrix of the slowly varying subsystem using matrix Schur complement theory.

[0087] In one optional embodiment, the matrix Schur complement theory represents the classic theory in linear algebra used for dimensionality reduction and decoupling of block matrices. By eliminating variables, the properties of a large matrix (such as inversion, determinant, positive definiteness, etc.) are transformed into those of smaller submatrices. That is, while retaining the core characteristics of the original matrix, coupling terms between submatrices are eliminated, thus achieving accurate dimensionality reduction from high-dimensional to low-dimensional matrices.

[0088] In an optional embodiment, the steady-state influence of the fast subsystem on the slow subsystem is incorporated into the slow subsystem matrix as a correction term through the Schur complement transformation. This avoids the error caused by neglecting the influence of the fast subsystem in the traditional direct truncation method, ensuring the accuracy of the slow-variable subsystem matrix. Furthermore, this embodiment utilizes matrix Schur complement theory to eliminate coupling terms between the fast and slow subsystems, resulting in a dimension-reduced Jacobian matrix of the slow-variable subsystem that accurately characterizes its dynamic stability. .

[0089] Step a2: Determine the fast time scale based on the preset time scale separation parameters.

[0090] In an optional embodiment, a preset time scale separation parameter is used to separate the original slow time scale. Transform to a fast time scale Furthermore, amplifying the transient process of the rapidly changing subsystem helps in analyzing the dynamic characteristics of the rapidly changing variable within the boundary layer. Therefore, in this embodiment, parameters can be separated using a preset time scale. Define a fast time scale .

[0091] Step a3: Based on a fast time scale, the initial singular perturbation nonlinear standard model is transformed into the target singular perturbation nonlinear standard model using the chain rule.

[0092] In an alternative embodiment, the chain rule represents the fundamental rule in calculus used for finding the derivative of composite functions. In this embodiment, it is used to transform the initial singular perturbation model from a slow timescale. Transform to a fast time scale This enables the time-scale reconstruction of equations for fast-variant subsystems, thus meeting the needs of fast dynamic boundary layer analysis.

[0093] In an alternative embodiment, it can be based on a fast time scale. By using the chain rule to perform time-scale transformation on the initial singular perturbation nonlinear standard model, it is possible to transform the original timescale of the model. The initial singular perturbation nonlinear standard model with independent variables is transformed into... The objective singular perturbation nonlinear standard model with independent variables.

[0094] For example, according to the chain rule, state variables pair The derivative and the relation The derivative of satisfies the following relation (7): (7) Furthermore, substituting the above relation (7) into the initial singular perturbation nonlinear standard model shown in the above relation (2), we can obtain the following: The objective singular perturbation nonlinear standard model with independent variables is shown in the following relation (8): (8) In the formula: , Represent the slow-changing and fast-changing state vectors on a fast time scale, respectively. The derivative of .

[0095] Step a4: Freeze the slow variables within the boundary layer of the slow manifold, and perform first-order Taylor linearization on the equations of the fast-variant subsystem in the nonlinear standard model of the target singular perturbation to obtain the Jacobian matrix of the boundary layer of the fast-variant subsystem.

[0096] In one alternative embodiment, during the transient process of the fast-changing subsystem, the changes in the slow variables are negligible and can be approximated as constant, i.e., frozen. Furthermore, by linearizing the equations of the fast-changing subsystem at the equilibrium point, the boundary layer Jacobian matrix of the fast-changing subsystem, which characterizes the stability features of the fast dynamics, can be obtained.

[0097] For example, another Then the derivative of the slow variable in the nonlinear standard model of the singular perturbation of the target That is, at this time, the slow variable Frozen at the current equilibrium value .

[0098] Furthermore, define the boundary layer variable (Error Vector). For fast variables With quasi-steady slow manifold The deviation between them is shown in the following relationship (9): (9) Furthermore, due to If it has been frozen, then its derivative term... .

[0099] Furthermore, within the boundary layer, the fast dynamic equations can be described by the following relation (10): (10) Furthermore, at the system equilibrium point At this point, for the aforementioned nonlinear function By performing a first-order Taylor expansion (linearization), the boundary layer Jacobian matrix of the tachyon system can be obtained. The following relation (11) is shown: (11) in, The submatrix representing the boundary layer Jacobian matrix of a rapidly changing subsystem is the submatrix corresponding to the full-dimensional sub-Jacobi matrix. It is used to analyze the critical Hopf bifurcation boundary of high-frequency electrical oscillations.

[0100] In some optional implementations, step S203 above includes: Step S2031: Based on the reduced Jacobian matrix of the slowly varying subsystem, and after processing with the Routh-Herwitz stability criterion, the low-frequency hydraulic oscillation boundary is determined.

[0101] In an optional embodiment, the Jacobian matrix of the slowly varying subsystem after singular perturbation dimensionality reduction processing is used. By using the Routh-Herwitz stability criterion, the critical boundary of low-frequency hydraulic oscillation (0.01Hz-0.05Hz) caused by nonlinear head loss of the turbine can be analyzed, i.e., the low-frequency hydraulic oscillation boundary.

[0102] In an alternative embodiment, the Jacobian matrix of the slowly varying subsystem is utilized. Characterizing the dynamic characteristics of the turbine, generator rotor, and hydraulic pipeline, we construct their characteristic polynomials and apply the Routh-Hurwitz criterion. When the critical condition of the criterion (the coefficients of the second-to-last row / first column of the Hurwitz determinant are zero) is satisfied, the system is in a critically stable state (Hopf bifurcation point). Furthermore, by analyzing the controller parameter equations at this point, the critical boundary of low-frequency oscillations can be obtained.

[0103] For example, based on the dimension-reduced Jacobian matrix of the slowly varying subsystem Construct its characteristic polynomial The following relation (12) is shown: (12) In the formula: Represents the eigenvalues ​​of a matrix; Represents the identity matrix.

[0104] Furthermore, let this characteristic polynomial The penultimate determinant of the corresponding Herwitz determinant is zero (i.e., the coefficient of the first column of the characteristic polynomial is zero), thus analytically obtaining the controller parameter vector. algebraic equations .

[0105] Furthermore, this algebraic equation The defined hypersurface is the boundary of low-frequency hydraulic oscillation. Furthermore, this boundary corresponds to the low-frequency Hopf bifurcation caused by the nonlinear head loss of the turbine, with a frequency range typically between 0.01 Hz and 0.05 Hz.

[0106] Furthermore, The closer to the inside, the better the system's low-frequency stability.

[0107] Step S2032: Based on the Jacobian matrix of the boundary layer of the fast-changing subsystem, the high-frequency electrical oscillation boundary is determined after processing by the Routh-Herwitz stability criterion.

[0108] In an optional embodiment, the Jacobian matrix of the slowly varying subsystem after singular perturbation dimensionality reduction processing is used. By using the Routh-Herwitz stability criterion, the critical boundary of high-frequency synchronous oscillation (10Hz-100Hz) caused by high inverter gain can be determined, i.e., the high-frequency electrical oscillation boundary.

[0109] In an optional embodiment, the Jacobian matrix of the boundary layer of the fast-changing subsystem It accurately characterizes the high-frequency dynamic characteristics of the inverter, PLL phase-locked loop, and power grid network, and then through... By performing eigenvalue analysis and applying the Routh-Hurwitz criterion, it is found that the system is at a high-frequency Hopf bifurcation point when the criterion's critical condition (first column coefficients are zero) is met. Furthermore, by analyzing the parametric equations under this critical condition, the critical boundary of the corresponding high-frequency oscillation can be obtained.

[0110] For example, based on the Jacobian matrix of the boundary layer of a rapidly changing subsystem Construct its characteristic polynomial The following relation (13) is shown: (13) Furthermore, let this characteristic polynomial The corresponding Hurwitz determinant has zero coefficients in the first column, and the vector of controller parameters is obtained analytically. algebraic equations .

[0111] Furthermore, this algebraic equation The defined boundary is the high-frequency electrical oscillation boundary. Furthermore, this boundary corresponds to the subsynchronous Hopf bifurcation caused by the high gain of the inverter control loop, with a frequency range of 10Hz-100Hz.

[0112] Step S2033: Based on the low-frequency hydraulic oscillation boundary and the high-frequency electrical oscillation boundary, construct a multi-frequency absolute stability domain.

[0113] In one optional embodiment, the hydro-wind-solar hybrid system is a fast-slow strongly coupled system, and stability in only one frequency band cannot guarantee the stability of the overall system; for example, it may be stable at low frequencies but unstable at high frequencies. Therefore, by taking the intersection of the inner sides of the low-frequency hydraulic oscillation boundary and the high-frequency electrical oscillation boundary, it is possible to construct a system that simultaneously satisfies... and The parameter space of the stability condition, i.e. the multi-band absolute stability domain, is shown in the following relation (14): (14) In the formula: This indicates taking the real part of a complex number; This indicates that the system is asymptotically stable in this frequency band.

[0114] Furthermore, by constructing a multi-band absolute stability domain, the controller parameters are strictly constrained within... Internally, it completely avoids the low-frequency water hammer oscillation at the hydraulic end and the high-frequency synchronous oscillation at the electrical end from the mechanism perspective, providing a rigid safety constraint space for subsequent optimization of control parameters.

[0115] In some optional implementations, step S204 above includes: Step S2041: Obtain the objective function for the mechanical loss of the actuators in the hydro-wind-solar hybrid system.

[0116] In one optional embodiment, the objective function for mechanical loss of the actuator represents the objective function for quantifying the mechanical wear, fatigue loss, and power fluctuation exceeding penalty of the actuator (such as turbine guide vanes and photovoltaic inverter command unit) in the hydro-wind-solar hybrid system due to frequent operation.

[0117] In one optional embodiment, the mechanical wear of the turbine guide vanes is positively correlated with the square of the guide vane opening change rate. Severe fluctuations in photovoltaic power will accelerate the loss of grid-connected equipment. Therefore, by statistically analyzing the cumulative loss per unit time in an integral form and introducing a penalty term for fluctuations exceeding the threshold, a precise mathematical description of mechanical loss can be achieved, avoiding the shortened equipment life caused by solely pursuing frequency performance.

[0118] For example, to prevent mechanical wear caused by frequent operation of the turbine guide vanes and to limit drastic fluctuations in photovoltaic power, the objective function for mechanical loss of the actuator is constructed as shown in the following relationship (15). : (15) In the formula: This represents the objective function value of the mechanical loss of the actuator; This represents the integral observation period, which is the duration of the dynamic process after the disturbance occurs. and This represents the weighting coefficient, used to balance the weighting ratio of guide vane mechanical loss to photovoltaic power fluctuation penalty; express The rate of change of the turbine guide vane opening at any given time, with its square term characterizing the mechanical wear and fatigue caused by frequent guide vane adjustments; express Real-time photovoltaic power fluctuation; This indicates the upper limit of the allowable threshold for photovoltaic power fluctuations; This represents a penalty function term that generates a positive penalty when photovoltaic fluctuations exceed the allowable threshold, thus limiting drastic power fluctuations.

[0119] Step S2042: Construct the global frequency quality objective function of the water-wind-solar hybrid system by using the time multiplication of the absolute error integral and the maximum frequency change rate of the water-wind-solar hybrid system.

[0120] In one optional embodiment, the integral of time multiplied by absolute error (ITAE) represents a classic indicator for evaluating the quality of power system frequency control. By weighting and accumulating the frequency deviation with the time dimension, it can measure both the magnitude of the frequency deviation and emphasize the impact of the deviation duration, thereby accurately reflecting the frequency recovery speed and dynamic adjustment accuracy of the system under disturbances.

[0121] In an optional embodiment, the Rate of Change of Frequency (RoCoF) represents an index describing the instantaneous rate of change of frequency in a power system, and its physical meaning is as follows: It can directly reflect the system's initial inertial support capability under conditions such as sudden load changes and power shortages. That is, the smaller the absolute value of the maximum frequency change rate, the stronger the system's inertial support and the smoother the rate of frequency drop / rise.

[0122] In an optional embodiment, to improve the frequency recovery speed of the system under disturbances and enhance inertial support, a weighted combination of the time-integrated absolute error (ITAE) and the maximum rate of change of frequency (RoCoF) is used as the global frequency quality objective function of the hydro-wind-solar hybrid system. The following relation (16) is shown: (16) In the formula: This represents the global frequency quality objective function value, reflecting the overall quality of the system's global frequency response; and This represents the weighting coefficient, used to balance the weighting ratio between the integral of frequency deviation and the maximum rate of frequency change, and can be adjusted according to system operating requirements; express The real-time frequency deviation of the system at any given time characterizes the degree to which the system frequency deviates from the rated value; The rate of change of frequency (RoCoF) measures the system's inertial support capability in the early stages of a disturbance; the smaller the absolute value, the stronger the inertial support.

[0123] Step S2043: Using the multi-band absolute stability domain as the hard constraint space, a multi-objective optimization function is constructed using the global frequency quality objective function and the actuator mechanical loss objective function.

[0124] In an optional embodiment, with multi-band absolute stability domain For a hard-constrained space, a multi-objective optimization function is constructed using the global frequency quality objective function and the actuator mechanical loss objective function, as shown in the following relation (17): (17) In the formula: Represents the parameter vector Find the minimum value and solve for the Pareto optimal solution; This represents the minimum value of the parameter; This indicates the maximum value of the parameter.

[0125] In some optional implementations, step S205 above includes: Step S2051: Initialize the parameter population of the controller in the hydro-wind-solar hybrid system, and generate a search vector based on the parameter population.

[0126] In one optional embodiment, based on the population initialization principle of evolutionary algorithms, within a reasonable range of controller parameters ( An initial parameter population is randomly generated, with each individual corresponding to a set of controller parameters. Simultaneously, an initial search vector can be generated based on the positional differences among the individuals in the population.

[0127] Step S2052: Based on the reduced-dimensional Jacobian matrix of the slowly varying subsystem and the boundary layer Jacobian matrix of the fast varying subsystem, construct the arc-shaped bifurcation boundary manifold with zero real part in the parameter space.

[0128] In one alternative embodiment, the parameter space represents a multidimensional mathematical space constructed with the controller parameters to be optimized as coordinate axes, and each parameter point corresponds to a set of controller parameters.

[0129] In an optional embodiment, the zero real part arcuate bifurcation boundary manifold represents the critical boundary where the system undergoes Hopf bifurcation in the parameter space. Essentially, it is the set of points where the maximum real part of the closed-loop eigenvalue of the system is equal to 0. It is the boundary between the stable region and the unstable region and has an arcuate non-convex geometric shape.

[0130] In an optional embodiment, when the system undergoes a Hopf bifurcation, there exists a pair of purely imaginary eigenvalues, meaning the maximum real part of the system's eigenvalues ​​is equal to 0. Therefore, the parameter space that satisfies... The set of parameter points is defined as the bifurcation boundary manifold, whose geometry is determined by the properties of the system's Jacobian matrix and exhibits an arc-shaped non-convex structure.

[0131] For example, based on the obtained reduced Jacobian matrix of the slowly varying subsystem Jacobian matrix of boundary layer of fast variable subsystem Calculate the closed-loop eigenvalues ​​of the system .in, This indicates the controller parameters.

[0132] Furthermore, define the zero-real-part arcuate bifurcation boundary manifold. The following relation (18) is shown: (18) In the formula: This represents the controller parameters to be optimized (taking a two-dimensional parameter space as an example). Represents the closed-loop characteristic value of the system; It represents the maximum value of the real part of all eigenvalues ​​of the system.

[0133] Furthermore, the zero-real-part arcuate bifurcation boundary manifold The parameter space is divided into internal stability regions. With external instability domain .

[0134] In an alternative embodiment, for any parameter point Calculate the dimension-reduced Jacobian matrix of the corresponding slowly varying subsystems. Jacobian matrix of boundary layer of fast variable subsystem The total eigenvalues ​​are denoted as follows: and .

[0135] Furthermore, the real parts of the eigenvalues ​​are extracted and the maximum values ​​are determined. A cross-scale stability determination formula is then constructed, as shown in the following relation (19): (19) In the formula: This represents the dimension-reduced Jacobian matrix describing the low-frequency oscillation characteristics of hydraulic systems, i.e., the dimension-reduced Jacobian matrix of the slowly varying subsystem. This represents the dimension-reduced Jacobian matrix describing the high-frequency oscillation characteristics of power electronics, i.e., the boundary layer Jacobian matrix of the fast-changing subsystem.

[0136] Furthermore, a zero-real-part arcuate bifurcation boundary manifold is constructed in the parameter space. The following relation (20) is shown: (20) Furthermore, the zero-real-part arcuate bifurcation boundary manifold The parameter space is divided into internal stability regions. With external instability domain .

[0137] Step S2053: Based on the arc-shaped bifurcation boundary manifold with zero real part, calculate the sensitivity gradient vector of the parameter point with respect to the real part of the largest eigenvalue.

[0138] In an optional embodiment, the sensitivity gradient vector represents a vector formed by the partial derivatives of the real part of the maximum eigenvalue at the parameter point with respect to the controller parameter. Its direction is perpendicular to the zero real bifurcation boundary manifold and points to the instability region, and is used to quantify the degree of influence of parameter changes on system stability.

[0139] In an alternative embodiment, for the current parameter point Calculate the real part of the largest eigenvalue. The partial derivatives with respect to the controller parameters are then used to construct the sensitivity gradient vector, which is perpendicular to the boundary manifold of the zero real bifurcation and points towards the instability region, as shown in the following relation (21): (twenty one) In the formula: Represents the sensitivity gradient vector; , These represent the scale coefficients corresponding to the real part of the largest eigenvalue. Integral coefficient The partial derivatives of .

[0140] Step S2054: Construct the boundary repulsion potential field and tangential projection matrix based on the sensitivity gradient vector.

[0141] In an alternative embodiment, the boundary repulsive potential field is used to generate a reverse repulsive force when the parameter point is close to the arcuate bifurcation boundary manifold with zero real part.

[0142] In an optional embodiment, the tangential projection matrix represents a linear transformation matrix constructed based on the sensitivity gradient vector. It is used to filter out the components in the search vector that are perpendicular to the bifurcation boundary, retaining only the components along the tangent direction of the bifurcation boundary. This forces the parameter population to evolve and search along the tangent direction of the bifurcation boundary, preventing the parameters from crossing the bifurcation boundary and entering the instability domain.

[0143] In one optional embodiment, the boundary repulsive potential field is based on the sensitivity gradient vector. When the parameter point approaches the bifurcation boundary, it generates a repulsive force opposite to the gradient direction, preventing the parameter from approaching the boundary. The tangential projection matrix is ​​constructed based on the gradient vector. Through linear transformation, the normal component of the search vector is filtered out, and only the tangential component is retained to achieve directional search.

[0144] For example, based on the sensitivity gradient vector Construct a boundary repulsive potential field: when the parameter point Near the bifurcation boundary When this occurs, a reverse repulsive force is generated, the magnitude of which is inversely proportional to the distance from the parameter point to the boundary, preventing the parameter from crossing the boundary and entering the unstable region.

[0145] In one example, the repulsive potential field function is designed based on the sensitivity gradient vector as the direction of the repulsive force and the geometric distance from the parameter point to the arc-shaped bifurcation boundary with zero real part. This makes the parameter point generate a repulsive force opposite to the direction of instability when it approaches the bifurcation boundary, preventing the parameter from crossing the boundary and entering the instability domain. At the same time, it ensures that the magnitude of the repulsive force changes dynamically with the distance between the parameter point and the boundary, thereby constructing the corresponding boundary repulsive potential field.

[0146] Furthermore, based on the sensitivity gradient vector Construct the tangential projection matrix The following relation (22) is shown: (twenty two) In the formula: Represents the sensitivity gradient vector The length of the module.

[0147] Furthermore, this tangential projection matrix Any vector can be projected onto the tangent plane of the bifurcation boundary to filter out the components perpendicular to the boundary.

[0148] Step S2055: The components perpendicular to the bifurcation boundary in the search vector are filtered out using the tangential projection matrix. Based on the repulsive effect of the boundary repulsion potential field, the controller parameters of the hydro-wind-solar hybrid system are updated by modifying the mutation relation to guide the parameter population to evolve and search along the tangential direction of the bifurcation boundary.

[0149] In one optional embodiment, the modified mutation relation represents the iterative formula used to update the controller parameters. It incorporates the constraints of the tangential projection matrix and the boundary repulsion potential field. Based on the mutation formula of the traditional evolutionary algorithm, it is modified to ensure that the parameters remain within the stable region after the update. At the same time, it can guide the population to search along the tangent direction of the bifurcation boundary and discover the performance limit point near the boundary.

[0150] For example, based on the tangential projection matrix Filter out the components perpendicular to the bifurcation boundary in the search vector and retain only the components along the tangent direction. Then, combining the repulsive effect of the boundary repulsive potential field, update the controller parameters using the modified mutation relation, as shown in the following relation (23): (twenty three) In the formula: This indicates the updated controller parameters; This indicates the old controller parameters before the update; This represents the step size factor, used to control the step size for parameter updates; This represents the optimal parameter solution in the current population; Indicates the random disturbance coefficient; This represents a Gaussian random perturbation term, used to increase the traversal of the search.

[0151] Furthermore, the newly generated parameter solution can be ensured through the above relation (23). By closely adhering to the arc-shaped stable boundary distribution, the control gain is maximized while ensuring system stability.

[0152] In an alternative embodiment, tangent vectors of the arc-shaped bifurcation boundary are introduced during the evolutionary search process. The directional mutation formula shown in the following relation (24) guides individuals to search along the stable margin: (twenty four) In the formula: This represents the individual parameters of the updated next-generation controller; This represents the individual parameter in the current iteration; This represents the step size factor, used to control the step size for parameter updates; This represents the geometric distance from the current individual to the boundary of the arc-shaped bifurcation; This represents the boundary sensing operator, which causes the population to move closer to the boundary ( When searching, the search direction is automatically aligned with the arc tangent, thereby deeply exploring the performance limit points near the boundary while ensuring the absolute stability of the system.

[0153] Furthermore, all parameter individuals in the current population are updated one by one to obtain a new generation of parameter population.

[0154] Step S2056: During the evolutionary search process, based on the stability margin threshold of the multi-band absolute stability domain, the updated controller parameters are successively checked for stability using the out-of-bounds lethal penalty function until the optimal control parameter set after iterative evolution is obtained.

[0155] In one optional embodiment, based on the hard constraint penalty principle, the stability margin threshold of the multi-band absolute stability domain is used as the criterion to perform stability verification on the parameter solution updated in each generation: if the parameter meets the stability margin requirement, its true multi-objective optimization function value is retained; if the parameter crosses the bifurcation boundary and falls into the unstable domain, an infinite penalty value is assigned and it is removed from the population, thereby ensuring that the final output Pareto optimal set is strictly located within the stability domain.

[0156] For example, a preset safety and stability margin threshold is set. This serves as the criterion for stability verification. Then, for each newly generated set of parameters... Construct the preset out-of-bounds lethal penalty function shown in the above relation (1).

[0157] Furthermore, during the iterative evolution of the parameter population, an out-of-bounds penalty function is applied to each updated controller parameter to forcibly eliminate unstable solutions. This iterative evolution continues until the algorithm converges, ultimately yielding a Pareto optimal control parameter set that is strictly located within the multi-band absolute stability domain.

[0158] Furthermore, a satisfaction function can be constructed using fuzzy theory, as shown in equation (25). Calculate the normalized membership degree for each non-dominated solution: (25) In the formula: Indicates the first The non-dominated solution corresponds to the first... Membership degree of each objective function; Indicates the number of non-dominated solutions; Indicates the number of objective functions.

[0159] Furthermore, select the membership degree. The largest solution is taken as the optimal compromise solution, and its corresponding PID parameters are the final optimal control parameter set. This set is then tuned to the water-wind-solar hybrid system controller to achieve full-band coordinated and stable control.

[0160] In one optional embodiment, the stability margin threshold of the multi-band absolute stability domain is used as the stability admission criterion, and an infinite penalty value is assigned to the instability parameter so that it is directly eliminated in the non-dominated sorting; a dynamic safety fence is used to perform secondary filtering on individuals in the boundary sensitive area to guide the population to converge to the safe area; through multiple iterative evolutions, the population gradually converges to the Pareto optimal frontier that takes into account both the system frequency quality and the mechanical life of the actuator, and finally the optimal compromise solution is selected as the optimal control parameter set.

[0161] For example, a preset safety and stability margin threshold is set. and with cross-scale stability index As the basis for judgment, that is, when If an individual is deemed to have initially achieved stability, it can proceed to evolutionary iteration; otherwise, it is directly eliminated.

[0162] Furthermore, the death penalty function shown in the following relation (26) is constructed, and the stability of the offspring individuals generated by the mutation is checked: (26) In the formula: This represents the objective function that includes the system's global frequency stability indices (ITAE, RoCoF), i.e., the global frequency quality objective function. The objective function representing the mechanical wear index of the local actuator is the objective function for the mechanical wear of the actuator. This represents an infinite penalty value; individuals assigned this value are immediately removed from the non-dominated sort.

[0163] Furthermore, the above mechanism ensures that the final Pareto optimal set is strictly located in the absolute stability region by forcing the objective function value of the unstable solution to be infinite. internal.

[0164] Furthermore, a survival probability formula based on boundary sensitivity can be introduced to perform secondary filtering on individuals located in the arc-shaped boundary sensitive area, as shown in the following relationship (27): (27) In the formula: This represents the geometric distance from the current individual to the boundary of the arc-shaped bifurcation; This indicates the threshold of the sensitive area.

[0165] Furthermore, this probability operator The algorithm automatically constructs a dynamic safety fence to forcibly guide the population towards a direction that balances frequency stability metrics. Mechanical loss index The safe region converges.

[0166] Furthermore, cross-scale stability indices can also be used. The magnitude of the value determines the sensitive area. Specifically, if... If the individual is determined to be in the stable-side sensitive zone, a secondary filtering is performed using survival probability; if If the individual is determined to be in the unstable sensitive zone, it will usually be directly eliminated by the penalty function.

[0167] Furthermore, S2052~S2055 are repeated, iteratively evolving until the algorithm converges and the Pareto optimal solution set is input. Then, the satisfaction function shown in the above relation (25) can be constructed using fuzzy theory. Calculate the normalized membership degree for each non-dominated solution. Then, select the membership degree. The largest solution is taken as the optimal compromise solution, and its corresponding PID parameters are the final optimal control parameter set. This set is then tuned to the water-wind-solar hybrid system controller to achieve full-band coordinated and stable control.

[0168] In one example, a cooperative control method for a water-wind-solar hybrid system considering nonlinear dynamic boundary constraints is provided, including: Step 1: Construct a multi-timescale singular perturbation nonlinear model. Establish the full-dimensional dynamic equations of the hydro-wind-solar complementary system, which includes nonlinear head loss of the turbine, inertial control of wind power, and transmission delay of photovoltaic power. Introduce time-scale separation parameters to decouple the system state variables into a slowly varying state vector describing the hydraulic-mechanical dynamics and a quickly varying state vector describing the power electronics-network dynamics, and construct a standard singular perturbation model with fast and slow separation.

[0169] Step 2: Analytical combined Hopf bifurcation stability domain. Based on singular perturbation theory, solve the reduced-dimensional Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem respectively. Use the Routh-Hurwitz criterion to derive the critical Hopf bifurcation boundaries that induce low-frequency hydraulic oscillations and high-frequency electrical oscillations respectively. Take the inner intersection of the two types of boundaries to construct the multi-band absolute stability domain.

[0170] Step 3: Construct a domain-constrained multi-objective collaborative optimization model. Using the multi-band absolute stability domain determined in Step 2 as the feasible domain constraint for the controller parameters, construct a composite multi-objective optimization function that includes the global frequency stability index of the system and the mechanical loss index of the local actuator.

[0171] Step 4: Implement the execution logic of the BCA-MODA algorithm based on the bifurcation boundary curvature perception multi-objective orientation optimization algorithm (BCA-MODA), introduce the tangent vector of the arc boundary to guide the orientation evolution of individuals, and establish a real-time monitoring mechanism using the real part eigenvalue criterion.

[0172] Step 5: Execute the intra-domain cooperative optimization algorithm based on the death penalty function to search within the stable domain. The invalid solutions falling into the Hopf bifurcation instability zone are forcibly eliminated through the "out-of-bounds death" penalty function mechanism to obtain the Pareto optimal control parameter set that balances stability and speed. Step 6: Selection of the Optimal Compromise Solution and Implementation of Multi-Mode Closed-Loop Control. The compromise solution with the highest satisfaction is selected from the solution set, and its parameters are tuned into the controller to achieve closed-loop cooperative frequency control with cross-scale stability support under complex operating conditions: In the low-frequency range: because the parameters satisfy... Stability constraints allow the turbine to effectively mitigate steady-state errors without inducing water hammer surge; at high frequencies: due to the parameters satisfying... Stability constraints are in place, and photovoltaic / wind power can provide rapid inertial support without inducing subsynchronous oscillations in the inverter. The system ultimately achieves coordinated and stable operation across the entire frequency band and multiple time scales.

[0173] The collaborative control method for a hydro-wind-solar hybrid system considering nonlinear dynamic boundary constraints provided in this example has the following technical advantages: (1) This example abandons the limitations of traditional parameter tuning based on linearized models and introduces Hopf bifurcation theory for the first time to accurately analyze the nonlinear stability boundary of the hydro-wind-solar hybrid system. By constructing an inverted U-shaped non-convex absolute stability domain, this example can quantitatively characterize the critical conditions that induce low-frequency water hammer oscillations and high-frequency synchronous oscillations from a physical mechanism perspective. This enables the control system to actively identify and avoid potential bifurcation instability regions during the parameter tuning stage, achieving a leap from passive suppression after the fact to active adjustment before the fact, and significantly enhancing the robustness of the system in the wide frequency domain.

[0174] (2) To address the problem that traditional evolutionary algorithms (such as GA and PSO) are prone to "oscillatory search" or "premature convergence" when approaching the edge of the stable region, this example proposes a multi-objective directional optimization algorithm based on bifurcation boundary curvature awareness (BCA-MODA). This algorithm uses the gradient information of the Jacobian matrix eigenvalues ​​to construct a tangential projection operator, guiding the population to perform a sliding search along the tangent direction of the stable boundary, and combines an out-of-bounds lethal penalty function mechanism to avoid repeated calculations of invalid solutions. Compared with traditional algorithms, this method improves the convergence speed by more than 30% while ensuring strict stability of the solution set, and can more deeply explore the performance potential of the system at the stable edge.

[0175] (3) This example utilizes singular perturbation theory to construct a tenth-order nonlinear decoupling model that separates fast and slow components, effectively overcoming the dimensionality curse and numerical rigidity problems encountered when directly solving the dynamics of hydraulic machinery (second-level) and power electronics (millisecond-level). By separately handling the slow-changing subsystem and the fast-changing subsystem, this example not only significantly reduces the computational complexity of the model, but also ensures that the controller can simultaneously take into account the coordinated suppression of low-frequency power fluctuations and high-frequency voltage oscillations, realizing full-dimensional coordinated control across time scales.

[0176] (4) This example constructs a composite multi-objective optimization function that includes frequency deviation (ITAE), rate of change of frequency (RoCoF), and mechanical loss indicators. Compared with the traditional single-objective control that only pursues the frequency recovery speed, this example effectively suppresses random disturbances from wind and solar power while explicitly constraining the amplitude and frequency of the turbine guide vanes. Experimental verification shows that, under the premise of meeting the power grid frequency quality standards, this strategy effectively reduces the mechanical wear and fatigue accumulation of the actuators, achieving the best trade-off between power grid operation safety and power generation equipment health management.

[0177] In one specific embodiment, such as Figure 3 The diagram illustrates the complete closed-loop control process from singular perturbation modeling and analysis to curvature-aware directional optimization. First, the system enters the physical modeling and boundary analysis stage. A multi-timescale singular perturbation nonlinear model is established, including a turbine with nonlinear head loss, a wind turbine with dynamic power electronic interfaces, and a photovoltaic system. Time-scale separation technology is used to decouple the system into fast and slow subsystems. Then, based on differential dynamics theory, the Hopf bifurcation critical boundary inducing low-frequency water hammer oscillations and high-frequency synchronous oscillations is analyzed, constructing a non-convex absolute stability domain. Based on this, the system enters the BCA-MODA optimization stage. The parameter tuning is performed using a multi-objective directional optimization algorithm based on bifurcation boundary curvature awareness, using a tangential projection mechanism to guide the population to slide along the tangent direction of the stability boundary. Solutions attempting to cross the bifurcation boundary are directly eliminated using an out-of-bounds penalty function mechanism, ensuring that all retained Pareto optimal solutions are strictly within the absolute stability domain. This stage ultimately outputs an optimal compromise parameter set that balances system stability and equipment lifespan. Finally, the closed-loop cooperative control stage begins. The optimized parameters are input to the controllers of each unit in the hydro-wind-solar hybrid system. In the low-frequency band, the turbines are used to smooth out steady-state deviations, and in the high-frequency band, new energy sources are used to provide inertial support, thereby forming a cross-scale, full-dimensional collaborative closed-loop control.

[0178] Furthermore, such as Figure 4The diagram illustrates the specific execution flow of the Bifurcation Boundary Curvature Aware Multi-Objective Directed Optimization Algorithm (BCA-MODA). After startup, the algorithm first initializes the control parameter population and sets the maximum number of iterations. Then, it calls the singular perturbation decoupling model to calculate the system eigenvalue distribution under the current parameters. Next, it enters the core stability determination stage, checking if the real part of the system's largest eigenvalue is less than zero. If the parameter falls into an unstable region, an out-of-bounds penalty function mechanism is immediately triggered, assigning the individual an infinite fitness value and forcibly eliminating it to avoid invalid computation. If the parameter satisfies stability constraints, it further calculates the multi-dimensional objective function values, including ITAE, RoCoF, and mechanical losses. Based on this, the algorithm executes the crucial curvature awareness and directed mutation steps. By calculating the gradient of the normal vector of the Hopf bifurcation boundary, a tangential projection matrix is ​​constructed, guiding the population to slide and update along the tangent direction of the stable boundary, thereby approaching the performance limit while ensuring absolute stability. Finally, the process uses an iteration counter to determine the termination condition. If the maximum number of generations has not been reached, it returns to the decoupling stage to continue evolution; otherwise, it outputs the Pareto optimal control parameter set, completing the optimization process.

[0179] Furthermore, such as Figure 5 As shown, the parameter adaptive closed-loop collaborative control principle of a hydro-wind-solar hybrid system is explained in detail. This architecture constructs a complete feedback regulation closed loop consisting of a perception layer, a decision optimization layer, an execution control layer, and a physical system object layer. The wide-area measurement system (WAMS / PMU) at the input end first collects real-time data on the frequency, power, and unit status of the physical system objects, forming a feedback signal stream. Subsequently, the parameter adaptive optimization layer, acting as the "brain" of the system, receives the measurement data and sequentially simplifies the model dimensions through a singular perturbation decoupling module, delineates the safety domain using a Hopf bifurcation boundary analysis module, and drives the BCA-MODA algorithm to calculate the optimal PID control parameters under the current operating conditions. These optimized parameters are then sent to the execution control layer, which guides the turbine governor and the distributed power controller to generate specific control commands: the turbine is mainly responsible for smoothing low-frequency power fluctuations, while the wind turbine and photovoltaic converter provide high-frequency inertial support by adjusting their speed or voltage. Ultimately, the commands from each controller are applied to the hydro-wind-solar hybrid power generation system, and its physical response is captured by WAMS again, thus forming a dynamic closed loop of "measurement-optimization-control-feedback" to ensure the frequency stability and coordinated operation of the system under multi-source disturbances.

[0180] Furthermore, such as Figures 6A-6B As shown, the simulation results compare the control method provided in this embodiment with the prior art (NSGA-II algorithm) and the unoptimized case.

[0181] Specifically, such as Figure 6AAs shown, under the condition of a step disturbance in the system, the method of this embodiment (solid red line) significantly reduces the maximum overshoot during the frequency dynamic process compared to the traditional optimization method (dashed blue line), reducing the peak frequency deviation from -0.21Hz to approximately -0.15Hz, and with a shorter adjustment time. It can quickly and smoothly restore the system frequency to its rated value, avoiding the large oscillations that occur in the unoptimized system (dotted green line). This demonstrates that this embodiment can more effectively coordinate the output of various units in the hydropower, wind power, and solar power systems, providing rapid active power support.

[0182] Furthermore, such as Figure 6B As shown, the Pareto front distributions for the two optimization objectives—accumulated system error (ITAE) and mechanical wear—are illustrated. It can be seen that the Pareto front curve obtained in this embodiment (red circular nodes) lies inside the curve of the comparative algorithm (blue square nodes). Especially in the middle region of the curve (i.e., the optimal operating region balancing error and wear), the curve of this embodiment exhibits a significant downward convex characteristic, forming a clear performance gap with the comparative algorithm. This result confirms that, under the same level of system control error, this embodiment can search for control parameters that result in less mechanical wear of the actuator; or, under the same mechanical wear constraint, this embodiment can achieve higher frequency control accuracy. This strict monotonic dominance demonstrates the significant technical advantages of this embodiment in multi-objective collaborative optimization.

[0183] This embodiment also provides a collaborative control device for a water-wind-solar hybrid system that considers nonlinear dynamic boundary constraints. This device is used to implement the above embodiments and preferred embodiments, and details already described will not be repeated. As used below, the term "module" can refer to a combination of software and / or hardware that implements a predetermined function. Although the device described in the following embodiments is preferably implemented in software, hardware implementation, or a combination of software and hardware, is also possible and contemplated.

[0184] This embodiment provides a collaborative control device for a water-wind-solar hybrid system that takes into account nonlinear dynamic boundary constraints, such as... Figure 7 As shown, the device includes: The decoupling module 701 is used to decouple the water-wind-solar hybrid system using time-scale separation technology and to construct the initial singular perturbation nonlinear standard model of the water-wind-solar hybrid system. The solver module 702 is used to solve the initial singular perturbation nonlinear standard model based on preset conditions, and to construct the dimension-reduced Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem. The processing module 703 is used to construct a multi-band absolute stability domain based on the dimensionality-reduced Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem, after processing by the Routh-Hurwitz stability criterion. Module 704 is used to construct a multi-objective optimization function that takes into account both the dynamic performance of the hydro-wind-solar hybrid system and the equipment lifespan in the system, with the multi-band absolute stability domain as the hard constraint space. The solver module 705 is used to optimize the multi-objective optimization function based on the preset out-of-bounds lethal penalty function and the multi-objective directional optimization algorithm based on bifurcation boundary curvature perception to obtain the optimal control parameter set. Control module 706 is used to control the operation of the hydro-wind-solar hybrid system using the optimal set of control parameters.

[0185] The water-wind-solar hybrid system collaborative control device considering nonlinear dynamic boundary constraints provided in this embodiment of the invention can execute the water-wind-solar hybrid system collaborative control method considering nonlinear dynamic boundary constraints provided in any embodiment of the invention, and has the corresponding functional modules and beneficial effects of the method. Further functional descriptions of the above modules are the same as in the corresponding embodiments described above, and will not be repeated here.

[0186] Figure 8 This is a schematic diagram of the structure of an electronic device provided in an embodiment of the present invention.

[0187] The following is a detailed reference. Figure 8 This diagram illustrates a suitable structural schematic for implementing an electronic device according to embodiments of the present invention. The electronic device may include a processor (e.g., a central processing unit, graphics processor, etc.) 801, which can perform various appropriate actions and processes based on a program stored in read-only memory (ROM) 802 or a program loaded from memory 808 into random access memory (RAM) 803. The RAM 803 also stores various programs and data required for the operation of the electronic device. The processor 801, ROM 802, and RAM 803 are interconnected via a bus 804. An input / output (I / O) interface 805 is also connected to the bus 804.

[0188] Typically, the following devices can be connected to I / O interface 805: input devices 806 including, for example, touchscreens, touchpads, keyboards, mice, cameras, microphones, accelerometers, gyroscopes, etc.; output devices 808 including, for example, liquid crystal displays (LCDs), speakers, vibrators, etc.; memory devices 808 including, for example, magnetic tapes, hard disks, etc.; and communication devices 809. Communication device 809 allows electronic devices to communicate wirelessly or wiredly with other devices to exchange data. Although Figure 8 Electronic devices with various devices are shown, but it should be understood that it is not required to implement or have all of the devices shown, and more or fewer devices may be implemented or have instead.

[0189] In particular, according to embodiments of the present invention, the processes described above with reference to the flowcharts can be implemented as computer software programs. For example, embodiments of the present invention include a computer program product comprising a computer program carried on a non-transitory computer-readable medium, the computer program containing program code for performing the methods shown in the flowcharts. In such embodiments, the computer program can be downloaded and installed from a network via a communication device 809, or installed from a memory 808, or installed from a ROM 802. When the computer program is executed by the processor 801, it performs the functions defined in the water-wind-solar hybrid system cooperative control method considering nonlinear dynamic boundary constraints according to embodiments of the present invention.

[0190] Figure 8 The electronic device shown is merely an example and should not be construed as limiting the functionality and scope of use of the embodiments of the present invention.

[0191] This invention also provides a computer-readable storage medium. The methods described above according to embodiments of the invention can be implemented in hardware or firmware, or implemented as computer code that can be recorded on a storage medium, or implemented as computer code downloaded via a network and originally stored on a remote storage medium or a non-transitory machine-readable storage medium and then stored on a local storage medium. Thus, the methods described herein can be processed by software stored on a storage medium using a general-purpose computer, a dedicated processor, or programmable or dedicated hardware. The storage medium can be a magnetic disk, optical disk, read-only memory, random access memory, flash memory, hard disk, or solid-state drive, etc.; further, the storage medium can also include combinations of the above types of memory. It is understood that computers, processors, microprocessor controllers, or programmable hardware include storage components capable of storing or receiving software or computer code. When the software or computer code is accessed and executed by the computer, processor, or hardware, the water-wind-solar hybrid system cooperative control method considering nonlinear dynamic boundary constraints shown in the above embodiments is implemented.

[0192] A portion of this invention can be applied as a computer program product, such as computer program instructions, which, when executed by a computer, can invoke or provide the methods and / or technical solutions according to the invention through the operation of the computer. Those skilled in the art will understand that the forms in which computer program instructions exist in a computer-readable medium include, but are not limited to, source files, executable files, installation package files, etc. Correspondingly, the ways in which computer program instructions are executed by a computer include, but are not limited to: the computer directly executing the instructions, or the computer compiling the instructions and then executing the corresponding compiled program, or the computer reading and executing the instructions, or the computer reading and installing the instructions and then executing the corresponding installed program. Here, the computer-readable medium can be any available computer-readable storage medium or communication medium accessible to a computer.

[0193] Although embodiments of the invention have been described in conjunction with the accompanying drawings, those skilled in the art can make various modifications and variations without departing from the spirit and scope of the invention, and such modifications and variations all fall within the scope defined by the appended claims.

Claims

1. A collaborative control method for a water-wind-solar hybrid system considering nonlinear dynamic boundary constraints, characterized in that, The method includes: The water-wind-solar hybrid system is decoupled using time-scale separation technology, and an initial singular perturbation nonlinear standard model of the water-wind-solar hybrid system is constructed. Based on preset conditions, the initial singular perturbation nonlinear standard model is solved, and the dimension-reduced Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem are constructed. Based on the reduced Jacobian matrix of the slow-variable subsystem and the boundary layer Jacobian matrix of the fast-variable subsystem, a multi-band absolute stability domain is constructed after processing with the Routh-Hurwitz stability criterion. Using the multi-band absolute stability domain as a hard constraint space, a multi-objective optimization function is constructed that takes into account both the dynamic performance of the water-wind-solar hybrid system and the equipment lifespan in the water-wind-solar hybrid system. Based on the preset out-of-bounds lethal penalty function, the multi-objective directional optimization algorithm based on bifurcation boundary curvature perception is used to optimize and solve the multi-objective optimization function to obtain the optimal control parameter set; The optimal set of control parameters is used to control the operation of the hydro-wind-solar hybrid system.

2. The method according to claim 1, characterized in that, Using time-scale separation techniques, the hydro-wind-solar hybrid system is decoupled, and an initial singular perturbation nonlinear standard model of the system is constructed, including: Based on a preset time scale separation parameter, the full-dimensional state vector of the water-wind-solar hybrid system is split to obtain a slow-changing state vector and a fast-changing state vector. The slow-changing state vector is used to describe the slow dynamic process dominated by hydraulics and mechanics in the water-wind-solar hybrid system, and the fast-changing state vector is used to describe the fast dynamic process dominated by power electronics and networks in the water-wind-solar hybrid system. Based on the small perturbation theory of power systems and the physical equations of each subsystem in the hydro-wind-solar hybrid system, the initial singular perturbation nonlinear standard model is constructed using the slow-changing state vector and the fast-changing state vector.

3. The method according to claim 1, characterized in that, Based on preset conditions, the initial singular perturbation nonlinear standard model is solved, and the dimension-reduced Jacobian matrix of the slowly varying subsystem and the boundary layer Jacobian matrix of the fast varying subsystem are constructed, including: Based on the preset conditions, the initial singular perturbation nonlinear standard model is solved to obtain a slow manifold describing the change of fast variables following slow variables; By inputting the slow manifold into the slow-varying subsystem equations in the initial singular perturbation nonlinear standard model, the dimension-reduced slow-varying subsystem dynamic equations are obtained. Using the dynamic equations of the slow-varying subsystem, calculate the full-dimensional sub-Jacobi matrix of the water-wind-solar hybrid system; Based on the preset timescale separation parameters, the initial singular perturbation nonlinear standard model, and the full-dimensional sub-Jacobi matrix, the dimension-reduced Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem are constructed.

4. The method according to claim 3, characterized in that, Based on the preset timescale separation parameters, the initial singular perturbation nonlinear standard model, and the full-dimensional sub-Jacobi matrix, the reduced-dimensional Jacobian matrix of the slowly varying subsystem and the boundary layer Jacobian matrix of the rapidly varying subsystem are constructed, including: Based on the full-dimensional sub-Jacobi matrix, the dimension-reduced Jacobian matrix of the slow-variable subsystem is determined using matrix Schur complement theory. Based on the preset time scale separation parameters, a fast time scale is determined; Based on the fast time scale, the initial singular perturbation nonlinear standard model is transformed into the target singular perturbation nonlinear standard model using the chain rule. Slow variables are frozen within the boundary layer of the slow manifold, and the equations of the fast-variant subsystem in the nonlinear standard model of the target singular perturbation are linearized using first-order Taylor transformation to obtain the Jacobian matrix of the boundary layer of the fast-variant subsystem.

5. The method according to claim 1, characterized in that, Based on the reduced-dimensional Jacobian matrix of the slowly varying subsystem and the boundary layer Jacobian matrix of the fast varying subsystem, a multi-band absolute stability domain is constructed after processing using the Routh-Hurwitz stability criterion, including: Based on the reduced Jacobian matrix of the slow-varying subsystem, and after processing by the Routh-Hurwitz stability criterion, the low-frequency hydraulic oscillation boundary is determined. Based on the Jacobian matrix of the boundary layer of the fast-changing subsystem, the high-frequency electrical oscillation boundary is determined after processing with the Routh-Herwitz stability criterion. Based on the low-frequency hydraulic oscillation boundary and the high-frequency electrical oscillation boundary, the multi-band absolute stability domain is constructed.

6. The method according to claim 1, characterized in that, Using the multi-band absolute stability domain as a hard constraint space, a multi-objective optimization function is constructed that takes into account both the dynamic performance of the hydro-wind-solar hybrid system and the equipment lifespan within the system, including: Obtain the objective function for the mechanical loss of the actuators in the hydro-wind-solar hybrid system; By using the time-multiplied absolute error integral and the maximum frequency change rate of the hydro-wind-solar hybrid system, a global frequency quality objective function for the hydro-wind-solar hybrid system is constructed. Using the multi-band absolute stability domain as the hard constraint space, the multi-objective optimization function is constructed using the global frequency quality objective function and the actuator mechanical loss objective function.

7. The method according to claim 1, characterized in that, Based on a preset out-of-bounds lethal penalty function, a multi-objective directional optimization algorithm based on bifurcation boundary curvature perception is used to optimize the multi-objective optimization function, obtaining the optimal control parameter set, including: Initialize the parameter population of the controller in the water-wind-solar hybrid system, and generate a search vector based on the parameter population; Based on the reduced-dimensional Jacobian matrix of the slow-variable subsystem and the boundary layer Jacobian matrix of the fast-variable subsystem, a zero-real-part arc-shaped bifurcation boundary manifold is constructed in the parameter space. Based on the zero real part arc bifurcation boundary manifold, calculate the sensitivity gradient vector of the parameter point with respect to the real part of the largest eigenvalue; Based on the sensitivity gradient vector, a boundary repulsion potential field and a tangential projection matrix are constructed. The boundary repulsion potential field is used to generate a reverse repulsion force when the parameter point is close to the arcuate bifurcation boundary manifold with zero real part. The components perpendicular to the bifurcation boundary in the search vector are filtered out using the tangential projection matrix, and the controller parameters of the hydro-wind-solar hybrid system are updated by modifying the mutation relation based on the repulsive effect of the boundary repulsion potential field, so as to guide the parameter population to evolve and search along the tangential direction of the bifurcation boundary. During the evolutionary search process, based on the stability margin threshold of the multi-band absolute stability domain, the updated controller parameters are successively checked for stability using the out-of-bounds lethal penalty function until the optimal control parameter set after iterative evolution is obtained.

8. A collaborative control device for a water-wind-solar hybrid system considering nonlinear dynamic boundary constraints, characterized in that, The device includes: The decoupling construction module is used to decouple the water-wind-solar complementary system using time-scale separation technology and to construct the initial singular perturbation nonlinear standard model of the water-wind-solar complementary system. The solution construction module is used to solve the initial singular perturbation nonlinear standard model based on preset conditions, and to construct the dimension-reduced Jacobian matrix of the slow-varying subsystem and the boundary layer Jacobian matrix of the fast-varying subsystem. The processing module is used to construct a multi-band absolute stability domain based on the reduced Jacobian matrix of the slow-changing subsystem and the boundary layer Jacobian matrix of the fast-changing subsystem, after processing with the Routh-Hurwitz stability criterion. The module is used to construct a multi-objective optimization function that takes into account both the dynamic performance of the water-wind-solar hybrid system and the equipment lifespan in the water-wind-solar hybrid system, with the multi-band absolute stability domain as the hard constraint space. The solution module is used to optimize the multi-objective optimization function based on a preset out-of-bounds lethal penalty function and a multi-objective directional optimization algorithm based on bifurcation boundary curvature perception to obtain the optimal control parameter set. The control module is used to control the operation of the hydro-wind-solar hybrid system using the optimal set of control parameters.

9. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores computer instructions for causing the computer to execute the collaborative control method for a water-wind-solar hybrid system considering nonlinear dynamic boundary constraints as described in any one of claims 1 to 7.

10. A computer program product, characterized in that, The system includes computer instructions for causing a computer to execute the collaborative control method for a water-wind-solar hybrid system that takes into account nonlinear dynamic boundary constraints, as described in any one of claims 1 to 7.

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