CN114251214B - Fractional order power system chaotic state judgment method and device - Google Patents
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Abstract
The invention discloses a fractional order power system chaotic state judgment method and a device, which relate to the technical field of water turbine control and comprise the following steps: step S10, expanding the fractional order differential equation of the water turbine power system to an integer order differential equation according to an integer order Jacobian matrix of the fractional order differential equation of the water turbine power system; step S20, solving an expanded integer order differential equation according to the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine to obtain a Lyapunov exponent; and S30, judging the chaotic state of the fractional order water turbine power system according to the Lyapunov exponent. According to the invention, after the fractional order differential equation of the water turbine power system is expanded into the executable integer order differential equation, the calculation feasibility is good, the calculated amount is small, the oscillation is stable, the convergence speed is high, the computed Lyapunov exponent can directly judge the chaotic state of the fractional order water turbine power system, and the judgment result is accurate and efficient.
Description
Technical Field
The invention relates to the technical field of water turbine control, in particular to a method and a device for judging a chaotic state of a fractional order power system.
Background
A non-linear system refers to a system in which the state and output variables of the system cannot be described in a linear relationship under the influence of external conditions. In a deterministic system, there is seemingly random irregular motion whose behavior appears as uncertainty, unrepeatable, unpredictable, a chaotic phenomenon. The chaos is the inherent characteristic of the nonlinear power system and is a ubiquitous phenomenon of the nonlinear system.
The Lyapunov exponent is the most direct and reliable quantitative index for judging whether a nonlinear power system is in a chaotic state or not, but the current Lyapunov exponents are obtained through an integer-order power system, and a related acquisition method is lacked for the Lyapunov exponent of a fractional-order power system, so that the chaotic state of the system cannot be directly judged from the Lyapunov exponent, and only the indexes such as initial condition sensitivity, chaotic attractor the like are indirectly proved.
Disclosure of Invention
The embodiment of the invention provides a method and a device for judging a chaotic state of a fractional order power system, which are used for solving the technical problem that the chaotic state of the fractional order power system is difficult to judge in the related technology.
In a first aspect, a method for judging a chaotic state of a fractional order power system is provided, where the method includes:
expanding the fractional order differential equation of the water turbine power system to an integer order differential equation according to an integer order Jacobian matrix of the fractional order differential equation of the water turbine power system;
solving an expanded integer order differential equation according to the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine to obtain a Lyapunov exponent;
and judging the chaotic state of the power system of the fractional order water turbine according to the Lyapunov exponent.
In some embodiments, the step of expanding the fractional order differential equations to integer order differential equations of the turbine power system based on an integer order jacobian of the fractional order differential equations of the turbine power system comprises:
establishing a fractional order differential equation of a water turbine power system:
wherein mu is (0-1)]Of any order in between, D μ As a differential operator, δ is the rotor angle of the turbine, ω is the rotational speed of the turbine, m t Is the driving moment of the water turbine, y is the servomotor stroke of the water turbine, K d Is a differential control coefficient.
In some embodiments, the step of expanding the fractional order differential equation of the water turbine power system to an integer order differential equation based on an integer order jacobian of the fractional order differential equation of the water turbine power system comprises:
and expanding the fractional order differential equation of the water turbine power system to an integer order differential equation by using a prediction-correction algorithm.
In some embodiments, the step of expanding the fractional order differential equation to the integer order differential equation of the turbine power system using a pre-estimation-correction algorithm comprises:
the method for expanding the fractional order differential equation to the integer order differential equation of the power system of the water turbine comprises the following steps:
wherein Γ is a gamma function, and h is K d Alpha and beta are parameters of the intermediate process.
In some embodiments, the step of solving an expanded integer order differential equation according to the rotor angle, the rotational speed, the driving torque, the servomotor stroke, and the differential control coefficient of the initial water turbine to obtain the lyapunov exponent includes:
substituting the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine into an expanded integer order differential equation;
with h as K d The step length of calculation is iterated to obtain the following K d A varying lyapunov index.
In some embodiments, the term h is K d The step length of calculation is subjected to iterative calculation to obtain the following K d A step of varying the Lyapunov exponent comprising:
with h =0.01 as K d The step length of calculation is subjected to iterative calculation to obtain the following K d A varying lyapunov index.
In a second aspect, a device for determining a chaotic state of a fractional order power system is provided, where the device includes:
the expansion unit is used for expanding the fractional order differential equation of the water turbine power system to an integer order differential equation according to an integer order Jacobian matrix of the fractional order differential equation of the water turbine power system;
the solving unit is used for solving the expanded integer order differential equation according to the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine to obtain the Lyapunov exponent;
and the judging unit is used for judging the chaotic state of the fractional order water turbine power system according to the Lyapunov exponent.
In some embodiments, the determining means further comprises:
the establishing unit is used for establishing a fractional order differential equation of the water turbine power system:
wherein,mu is (0-1)]Of any order in between, D μ As a differential operator, δ is the rotor angle of the turbine, ω is the rotational speed of the turbine, m t Is the driving moment of the water turbine, y is the servomotor stroke of the water turbine, K d Is a derivative control coefficient.
In a third aspect, a computer device is provided, comprising: the device comprises a memory and a processor, wherein at least one instruction is stored in the memory, and the at least one instruction is loaded and executed by the processor so as to realize the judgment method of the fractional order power system chaotic state.
In a fourth aspect, a computer-readable storage medium is provided, which stores computer instructions, and when the computer instructions are executed by a computer, the computer is enabled to execute the aforementioned method for determining the chaotic state of a fractional order power system.
The technical scheme provided by the invention has the beneficial effects that:
the method and the device for judging the chaotic state of the fractional order power system in the embodiment of the invention comprise the steps of firstly expanding the fractional order differential equation of the water turbine power system to an integer order differential equation according to an integer order Jacobian matrix of the fractional order differential equation of the water turbine power system, then solving the expanded integer order differential equation according to the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine to obtain a Lyapunov exponent, and finally judging the chaotic state of the fractional order water turbine power system according to the Lyapunov exponent. According to the invention, after the fractional order differential equation of the water turbine power system is expanded into the executable integer order differential equation, the calculation feasibility is good, the calculated amount is small, the oscillation is stable, the convergence speed is high, the computed Lyapunov exponent can directly judge the chaotic state of the fractional order water turbine power system, and the judgment result is accurate and efficient.
Drawings
In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings required to be used in the description of the embodiments are briefly introduced below, and it is obvious that the drawings in the description below are only some embodiments of the present invention, and it is obvious for those skilled in the art that other drawings can be obtained according to the drawings without creative efforts.
Fig. 1 is a schematic flow diagram of a method for determining a chaotic state of a fractional order power system according to an embodiment of the present invention;
fig. 2 is a schematic flowchart of the step S20 implemented in fig. 1 according to an embodiment of the present invention;
FIG. 3 is a Lyapunov exponent plot with order μ taken to be 1, provided by an embodiment of the present invention;
FIG. 4 is a Lyapunov exponent diagram of a conventional integer order turbine power system;
FIG. 5 is a Lyapunov exponent plot with the order μ taken to be 0.9, provided by an embodiment of the present invention;
FIG. 6 is a Lyapunov exponent plot with order μ taken to be 0.8, provided by an embodiment of the present invention;
fig. 7 is a schematic structural diagram of a fractional order power system chaotic state judgment device according to an embodiment of the present invention;
fig. 8 is a schematic structural diagram of a computer device according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be obtained by a person skilled in the art without any inventive step based on the embodiments of the present invention, are within the scope of the present invention.
The embodiment of the invention provides a method for judging a chaotic state of a fractional order power system, which can solve the technical problem that the chaotic state of the existing fractional order power system is difficult to judge.
Referring to fig. 1, an embodiment of the present invention provides a method for determining a chaotic state of a fractional order power system, where the method includes:
and S10, expanding the fractional order differential equation of the hydraulic turbine power system to an integer order differential equation according to the integer order Jacobian matrix of the fractional order differential equation of the hydraulic turbine power system.
Specifically, the step of expanding the fractional order differential equation of the water turbine power system to the integer order differential equation according to the integer order jacobian matrix of the fractional order differential equation of the water turbine power system comprises the following steps:
establishing a fractional order differential equation of a water turbine power system:
wherein mu is (0-1)]Of any order in between, D μ As a differential operator, δ is the rotor angle of the turbine, ω is the rotational speed of the turbine, m t Is the main moment of the water turbine, y is the servomotor stroke of the water turbine, K d Is a differential control coefficient.
Further, the step of expanding the fractional order differential equation to the integer order differential equation of the water turbine power system according to the integer order jacobian matrix of the fractional order differential equation of the water turbine power system includes:
and expanding the fractional order differential equation of the water turbine power system to an integer order differential equation by using a prediction-correction algorithm.
Further, the step of expanding the fractional order differential equation of the water turbine power system to an integer order differential equation using a pre-estimation-correction algorithm includes:
the fractional order differential equation to the integer order differential equation of the expansion turbine power system is as follows:
wherein Γ is a gamma function, and h is K d Alpha and beta are intermediate process parameters.
And expanding the fractional order differential equation to the required integer order differential equation by utilizing a pre-estimation-correction algorithm, ensuring that the structure of the partial integer order differential equation of the Jacobian matrix is not changed, and facilitating subsequent solution and calculation to obtain the Lyapunov exponent.
And S20, solving an expanded integer order differential equation according to the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine to obtain the Lyapunov exponent.
The step of solving an expanded integer order differential equation according to the rotor angle, the rotating speed, the driving moment, the servomotor stroke and the differential control coefficient of the initial water turbine to obtain the Lyapunov exponent comprises the following steps:
and substituting the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine into the expanded integer order differential equation.
With h as K d The step length of calculation is subjected to iterative calculation to obtain the following K d A varying lyapunov index. Preferably, the calculation step h takes 0.01.
Specifically, equations (3) and (4) are expanded and substituted into equation (2), and the reconstructed integer order differential equation of the water turbine power system is expressed as follows:
in equation (5):
wherein i 1 Taking 0 and 1; i.e. i 2 Taking 0, 1 and 2; i.e. i 3 Taking 0, 1, 2 and 3; i.e. i 4 Taking 0, 1, 2, 3 and 4; i.e. i 5 Take 0, 1, 2, 3, 4, 5.
Referring to fig. 2, the rotor angle, the rotational speed, the driving moment, the servomotor stroke and the differential control coefficient of the initial water turbine are substituted into the expanded integer order differential equation, numerical iteration solution is carried out on the reconstructed equation (5), and the Lyapunov exponent is calculated by referring to the Lyapunov calculation method (wolf calculation method) of the existing integer order differential equation.
And S30, judging the chaotic state of the power system of the fractional order water turbine according to the Lyapunov exponent.
When mu is 1.0, the fractional order water turbine power system of the embodiment of the invention is equivalent to an integer order water turbine power system at K d In the range of =7-10, referring to fig. 3 and 4, the lyapunov indexes calculated by the fractional order power system chaotic state judgment method and the existing integer order water turbine power system chaotic state judgment method are basically consistent, which shows that the computed lyapunov index of the fractional order power system chaotic state judgment method is very high in accuracy, and the fractional order water turbine power system chaotic state can be directly judged. Among them, as can be seen from FIGS. 3 and 4, at K d Near =7.8, the water turbine power system will alternate continuously between chaotic motion and stable periodic motion, and when K is d >After 8.0, LE1 is larger than 0, the water turbine power system will have complex chaotic behavior until K d =10.0,LE1 is always greater than zeroThe power system of the water turbine has chaotic motion and is in a destabilization state.
When mu is 0.9 and 0.8, the water turbine power system is in fractional order, and the Lyapunov exponent obtained by adopting the fractional order power system chaotic state judgment method is shown in figures 5 and 6. When mu is 0.9, when K is d >After 9.5, LE1 is larger than 0, and the power system of the water turbine has complex chaotic behaviors. When mu is 0.9, when K d >After 8.8, LE1 is larger than 0, and the water turbine power system has complex chaotic behaviors.
The method for judging the chaotic state of the fractional order power system in the embodiment of the invention comprises the steps of firstly expanding the fractional order differential equation of the water turbine power system to an integer order differential equation according to an integer order Jacobian matrix of the fractional order differential equation of the water turbine power system, then solving the expanded integer order differential equation according to the initial rotor angle, the initial rotating speed, the main moment, the servomotor stroke and the differential control coefficient of the water turbine to obtain a Lyapunov exponent, and finally judging the chaotic state of the fractional order water turbine power system according to the Lyapunov exponent. According to the invention, after the fractional order differential equation of the water turbine power system is expanded into the executable integer order differential equation, the calculation feasibility is good, the calculated amount is small, the oscillation is stable, the convergence speed is high, the computed Lyapunov exponent can directly judge the chaotic state of the fractional order water turbine power system, and the judgment result is accurate and efficient.
Referring to fig. 7, an embodiment of the present invention further provides a device for determining a chaotic state of a fractional order power system, where the device includes:
the expansion unit is used for expanding the fractional order differential equation of the water turbine power system to an integer order differential equation according to an integer order Jacobian matrix of the fractional order differential equation of the water turbine power system;
the solving unit is used for solving the expanded integer order differential equation according to the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine to obtain the Lyapunov exponent;
and the judging unit is used for judging the chaotic state of the fractional order water turbine power system according to the Lyapunov exponent.
The device for determining the chaotic state of the fractional order power system provided by the above embodiment may be implemented in the form of a computer program, and the computer program may be run on a computer device as shown in fig. 8.
An embodiment of the present invention further provides a computer device, including: the memory is connected with the processor through a system bus and is stored with at least one instruction, and the at least one instruction is loaded and executed by the processor so as to realize all steps or part of steps of the fractional order power system chaotic state judgment method.
The network interface is used for performing network communication, such as sending distributed tasks. Those skilled in the art will appreciate that the architecture shown in fig. 8 is merely a block diagram of some of the structures associated with the inventive arrangements and is not intended to limit the computing devices to which the inventive arrangements may be applied, as a particular computing device may include more or less components than those shown, or may combine certain components, or have a different arrangement of components.
The Processor may be a CPU, or other general purpose Processor, digital Signal Processor (DSP), application Specific Integrated Circuit (ASIC), field Programmable Gate Array (FPGA) or other Programmable logic device, discrete Gate or transistor logic device, discrete hardware component, or the like. The general purpose processor may be a microprocessor or the processor may be any conventional processor or the like, the processor being the control center of the computer device and various interfaces and lines connecting the various parts of the overall computer device.
The memory may be used to store computer programs and/or modules, and the processor may implement various functions of the computer device by running or executing the computer programs and/or modules stored in the memory, as well as by invoking data stored in the memory. The memory may mainly include a storage program area and a storage data area, wherein the storage program area may store an operating system, an application program required by at least one function (such as a video playing function, an image playing function, etc.), and the like; the storage data area may store data (such as video data, image data, etc.) created according to the use of the cellular phone, etc. In addition, the memory may include high speed random access memory, and may also include non-volatile memory, such as a hard disk, a memory, a plug-in hard disk, a Smart Media Card (SMC), a Secure Digital (SD) Card, a Flash memory Card (Flash Card), at least one magnetic disk storage device, a Flash memory device, or other volatile solid state storage device.
Wherein, in one embodiment, the processor is configured to execute a computer program stored in the memory to implement the steps of:
and S10, expanding the fractional order differential equation of the hydraulic turbine power system to an integer order differential equation according to the integer order Jacobian matrix of the fractional order differential equation of the hydraulic turbine power system.
And S20, solving an expanded integer order differential equation according to the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine to obtain the Lyapunov exponent.
And S30, judging the chaotic state of the fractional order water turbine power system according to the Lyapunov exponent.
The embodiment of the invention also provides a computer readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, all steps or part of steps of the method for judging the chaotic state of the fractional order power system are realized.
The embodiment of the present invention may implement all or part of the foregoing processes, and may also be implemented by instructing related hardware by a computer program, where the computer program may be stored in a computer-readable storage medium, and when the computer program is executed by a processor, the computer program may implement the steps of the foregoing methods. Wherein the computer program comprises computer program code, which may be in the form of source code, object code, an executable file or some intermediate form, etc. The computer readable medium may include: any entity or device capable of carrying computer program code, recording medium, U.S. disk, removable hard disk, magnetic diskette, optical disk, computer memory, read-Only memory (ROM), random Access Memory (RAM), electrical carrier wave signal, telecommunications signal, software distribution medium, etc. It should be noted that the computer-readable medium may contain suitable additions or subtractions depending on the requirements of legislation and patent practice in jurisdictions, for example, in some jurisdictions, computer-readable media may not include electrical carrier signals or telecommunication signals in accordance with legislation and patent practice.
As will be appreciated by one skilled in the art, embodiments of the present invention may be provided as a method, system, server, or computer program product. Accordingly, the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, optical storage, and the like) having computer-usable program code embodied therein.
It should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or system. Without further limitation, an element defined by the phrases "comprising a," "8230," "8230," or "comprising" does not exclude the presence of other like elements in a process, method, article, or system comprising the element.
The above-mentioned serial numbers in the embodiments of the present invention are only for description, and do not represent the advantages and disadvantages of the embodiments.
The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the invention. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
The foregoing are merely exemplary embodiments of the present invention, which enable those skilled in the art to understand or practice the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
Claims (5)
1. A fractional order power system chaotic state judgment method is characterized by comprising the following steps:
establishing a fractional order differential equation of a water turbine power system:
wherein mu is (0-1)]Of any order in between, D μ As a differential operator, δ is the rotor angle of the turbine, ω is the rotational speed of the turbine, m t Is the driving moment of the water turbine, y is the servomotor stroke of the water turbine, K d Is a differential control coefficient;
the method for expanding the fractional order differential equation to the integer order differential equation of the power system of the water turbine comprises the following steps:
wherein Γ is a gamma function, and h is K d Alpha and beta are intermediate process parameters;
solving an expanded integer order differential equation according to the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine to obtain a Lyapunov exponent;
specifically, a fractional order differential equation (1) of the water turbine power system is reconstructed according to equations (2), (3) and (4), and an integer order differential equation (5) and a related intermediate variable E of the reconstructed water turbine power system are obtained 1 ～E 20 、L 1 ～L 20 、K 1 ～K 20 The expression of equation (5) is as follows:
in equation (5):
wherein i 1 Taking 0 and 1; i.e. i 2 Taking 0, 1 and 2; i.e. i 3 Taking 0, 1, 2 and 3; i.e. i 4 Taking 0, 1, 2, 3 and 4; i all right angle 5 Taking 0, 1, 2, 3, 4 and 5;
substituting the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine into the expanded integer order differential equation, carrying out numerical iteration solution on the reconstructed equation (5), and taking h as K d The step length of calculation is subjected to iterative calculation to obtain the following K d A varied Lyapunov index;
and judging the chaotic state of the fractional order water turbine power system according to the Lyapunov exponent.
2. The method for determining the chaotic status of a fractional order power system as claimed in claim 1, wherein h is K d The step length of calculation is subjected to iterative calculation to obtain the following K d A step of varying the Lyapunov exponent comprising:
with h =0.01 as K d The step length of calculation is subjected to iterative calculation to obtain the following K d A varied lyapunov index.
3. A device for judging the chaotic state of a fractional order power system is characterized by comprising:
the expansion unit is used for expanding the fractional order differential equation of the water turbine power system to an integer order differential equation according to an integer order Jacobian matrix of the fractional order differential equation of the water turbine power system;
specifically, a fractional order differential equation of a water turbine power system is established:
wherein mu is (0-1)]Of any order in between, D μ For differential operator, δ is the rotor angle of the turbine, ω is the rotational speed of the turbine, m t Is the driving moment of the water turbine, y is the servomotor stroke of the water turbine, K d Is a differential control coefficient;
the fractional order differential equation to the integer order differential equation of the expansion turbine power system is as follows:
wherein Γ is a gamma function, and h is K d Alpha and beta are intermediate process parameters;
the solving unit is used for solving the expanded integer order differential equation according to the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine to obtain the Lyapunov exponent;
specifically, the fractional order differential equation (1) of the water turbine power system is reconstructed according to the equations (2), (3) and (4), and the integer order differential equation (5) of the reconstructed water turbine power system and the related intermediate variable E are obtained 1 ～E 20 、L 1 ～L 20 、K 1 ～K 20 The expression of equation (5) is as follows:
in equation (5):
wherein i 1 Taking 0 and 1; i.e. i 2 Taking 0, 1 and 2; i all right angle 3 Taking 0, 1, 2 and 3; i.e. i 4 Taking 0, 1, 2, 3 and 4; i all right angle 5 Taking 0, 1, 2, 3, 4 and 5;
substituting the initial rotor angle, the initial rotating speed, the initial driving moment, the servomotor stroke and the differential control coefficient of the water turbine into an expanded integer order differential equation, carrying out numerical iteration solution on the reconstructed equation (5), and taking h as K d The step length of calculation is iterated to obtain the following K d A varied Lyapunov index;
and the judging unit is used for judging the chaotic state of the fractional order water turbine power system according to the Lyapunov exponent.
4. A computer device, comprising: the device comprises a memory and a processor, wherein at least one instruction is stored in the memory, and is loaded and executed by the processor to realize the fractional order power system chaotic state judgment method in any one of claims 1 to 2.
5. A computer-readable storage medium, characterized in that: the computer readable storage medium stores computer instructions which, when executed by a computer, cause the computer to perform the fractional order power system chaotic state determination method of any one of claims 1-2.
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