CN119103487B - Water supply network leakage position identification method and storage medium - Google Patents
Water supply network leakage position identification method and storage mediumInfo
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- CN119103487B CN119103487B CN202411424491.2A CN202411424491A CN119103487B CN 119103487 B CN119103487 B CN 119103487B CN 202411424491 A CN202411424491 A CN 202411424491A CN 119103487 B CN119103487 B CN 119103487B
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F17—STORING OR DISTRIBUTING GASES OR LIQUIDS
- F17D—PIPE-LINE SYSTEMS; PIPE-LINES
- F17D5/00—Protection or supervision of installations
- F17D5/02—Preventing, monitoring, or locating loss
-
- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F17—STORING OR DISTRIBUTING GASES OR LIQUIDS
- F17D—PIPE-LINE SYSTEMS; PIPE-LINES
- F17D1/00—Pipe-line systems
- F17D1/08—Pipe-line systems for liquids or viscous products
-
- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F17—STORING OR DISTRIBUTING GASES OR LIQUIDS
- F17D—PIPE-LINE SYSTEMS; PIPE-LINES
- F17D3/00—Arrangements for supervising or controlling working operations
- F17D3/01—Arrangements for supervising or controlling working operations for controlling, signalling, or supervising the conveyance of a product
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02A—TECHNOLOGIES FOR ADAPTATION TO CLIMATE CHANGE
- Y02A20/00—Water conservation; Efficient water supply; Efficient water use
- Y02A20/15—Leakage reduction or detection in water storage or distribution
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- Engineering & Computer Science (AREA)
- Mechanical Engineering (AREA)
- General Engineering & Computer Science (AREA)
- Health & Medical Sciences (AREA)
- Public Health (AREA)
- Water Supply & Treatment (AREA)
- Examining Or Testing Airtightness (AREA)
Abstract
The invention discloses a water supply network leakage position identification method and a storage medium, wherein the method comprises the steps of monitoring pressure flow data of different positions of a water supply network in real time, using the right end residual error item of a lattice Boltzmann parallel solving pressure flow relation equation as a real-time monitoring residual error item, determining the water supply network leakage position corresponding to the real-time monitoring residual error item according to the mapping relation between the pre-constructed leakage position and the residual error item, and obviously improving the equation solving efficiency by using the lattice Boltzmann parallel solving pressure and flow relation equation, and avoiding the complex process of feature extraction by establishing the mapping relation between the leakage position and the residual error item.
Description
Technical Field
The invention relates to the technical field of monitoring and diagnosing faults of a fire-fighting water supply network, in particular to a water supply network leakage position identification method and a storage medium.
Background
The water supply network plays a vital role in the fire-fighting water supply system, and is an important component of the fire-fighting system and is responsible for conveying water sources to various fire extinguishing points so as to provide necessary water sources for extinguishing fire. Second, the water supply network needs to maintain a high enough pressure to support the fire extinguishing operation on the fire scene. Once the water supply network leaks, enough water sources cannot be provided in time to extinguish the fire source, the fire rapidly spreads, property loss and casualties are caused, and the rapid and accurate identification of the leakage position of the water supply network is important.
Flow and pressure sensors are often arranged in a water supply network to monitor the running state of the network, and the distribution of pressure and flow data at different positions of the water supply network is obviously different under the normal working condition (no leakage) and the leakage working condition of the water supply network and for different leakage positions. The change of the water supply pressure and flow of the pipe network is the main form of water supply pipe network leakage signal detection.
In the related art, a real-time model method is described in the "research on a ship pipe network leakage detection and positioning method, feng Kai" of the Shuoshi thesis, namely, an accurate pipeline real-time model accurately calculated by a computer is constructed and is executed synchronously with an actual pipeline. And comparing the pressure and flow equivalent with the value calculated by the theoretical model by timing, so as to judge the position of the leakage point. The method is too dependent on the accuracy of the initial model and software, however, the positioning mechanism of the leakage point is based on a pressure gradient method, so that the method has great limitation.
Therefore, how to quickly and accurately identify the leakage position of the water supply network by monitoring the changes of pressure and flow data at different positions of the water supply network is a problem to be solved at present.
Disclosure of Invention
The invention aims to solve the technical problem of how to quickly and accurately identify the leakage position of a water supply network.
The invention solves the technical problems by the following technical means:
The invention provides a water supply network leakage position identification method, which comprises the following steps:
monitoring pressure flow data of different positions of the water supply network in real time;
The right end residual error item of the pressure flow relation equation is solved in parallel by using the lattice Boltzmann as a real-time monitoring residual error item;
and determining the leakage position of the water supply network corresponding to the real-time monitoring residual error item according to the pre-constructed mapping relation between the leakage position and the residual error item.
Further, the construction process of the mapping relation between the leakage position and the residual error item comprises the following steps:
Acquiring pressure flow data at different positions of a water supply network under a normal working condition, and parallelly solving a right end residual term of a pressure flow relation equation under the normal working condition by using Boltzmann as a residual term under the normal working condition;
setting a set number of leakage positions, and collecting pressure flow data at different positions of a water supply network at different leakage positions;
For each leakage position, using the Boltzmann of the lattice to solve the right end residual error item of the pressure flow relation equation under the leakage working condition in parallel as the residual error item under the leakage working condition;
And establishing mapping relations between different leakage positions and residual items according to all leakage positions and corresponding residual items under the leakage working condition to form a leakage position-residual item mapping relation comparison table, wherein the mapping relation comparison table comprises the leakage positions, the residual items and pressure flow data of different positions of the water supply network under the working condition.
Further, the leak locations are uniformly or non-uniformly arranged.
Further, the right-end residual error term of the parallel solving pressure flow relation equation by using the Boltzmann of the lattice is
Wherein delta is the right residual term, phi is the pressure, f is the fluid velocity,F = Σ icihi/∑ihi,ci is the model discrete velocity, h i is the discrete distribution function, i is the discrete velocity index,Is a gradient operator.
Further, the pressure-flow relation equation is:
and (3) using the lattice Boltzmann to solve a pressure flow relation equation in parallel, wherein the equation of the collision step is expressed as:
The migration step formula is expressed as:
The statistical pressure data and the flow data are respectively:
In the formula, For the post-collision discrete distribution function, h i (x, t) is the discrete distribution function at position x and time t, Ω i (h) is the collision operator, x is the position index, and t is the time index.
Further, after determining the water supply network leakage position corresponding to the real-time monitoring residual term according to the mapping relation between the pre-constructed leakage position and the residual term, the method further comprises:
Calculating non-local cosine similarity between pressure flow data of different positions of the real-time monitoring water supply network and pressure flow data under historical leakage working conditions;
When the non-local cosine similarity is larger than a set similarity threshold, determining that the recognized pipe network leakage position is correct;
And when the non-local cosine similarity is smaller than or equal to a set similarity threshold, determining that the identified pipe network leakage position is wrong.
Further, the calculation formula of the non-local cosine similarity is as follows:
Where cos α Φ is the non-local cosine similarity of the pressure data, Φ is the pressure vector monitored at the current moment, and Φ 0 is the pressure vector under the historical leakage working condition.
Similarly, the calculation formula of the flow data non-local cosine similarity is as follows:
In the formula, cos alpha F is the non-local cosine similarity of flow data, F is the flow data monitored at the current moment, and F 0 is the flow data under the history leakage working condition.
Further, the calculating the non-local cosine similarity between the pressure flow data of different positions of the real-time monitoring water supply network and the pressure flow data under the historical leakage working condition includes:
calculating first non-local cosine similarity between pressure flow data of different positions of the real-time monitoring water supply network and pressure flow data under the historical leakage working condition according to the vector length of n;
judging whether the first non-local cosine similarity is in a set range or not;
If yes, comparing the non-local cosine similarity with the similarity threshold value, and determining whether the pipe network leakage position identification result is accurate;
if not, resetting the vector length to 2n, and calculating a second non-local cosine similarity between the pressure flow data of different positions of the water supply network and the pressure flow data under the historical leakage working condition.
Further, after calculating the second non-local cosine similarity, the method further comprises:
and calculating the difference value between the second non-local cosine similarity and the first non-local cosine similarity, outputting the first non-local cosine similarity when the difference value is within a set range, and otherwise, outputting the second non-local cosine similarity for comparison with a similarity threshold.
Furthermore, the invention also provides a computer readable storage medium, on which a computer program is stored, which when being executed by a processor, implements the water supply network leakage position identification method as described above.
The invention has the advantages that:
(1) According to the invention, the pressure flow relation equation is solved in parallel by using the Boltzmann algorithm in advance to obtain the mapping relation between the leakage position and the equation residual error term, when the leakage position of the water supply network is monitored, the pressure flow data of different positions of the water supply network are monitored in real time, the right residual error term of the Boltzmann algorithm is solved in parallel and is used as the real-time monitoring residual error term, the pipe network leakage position corresponding to the real-time monitoring residual error term is identified by utilizing the pre-built mapping relation, the pressure and flow relation equation is solved in parallel by using the Boltzmann algorithm in the lattice, the equation solving efficiency is remarkably improved, and meanwhile, the complex process of feature extraction is avoided by establishing the mapping relation between the leakage position and the residual error term, and the quick and accurate identification of the pipe network leakage position is realized.
(2) According to the established mapping relation between the leakage position and the residual error item, the accuracy of the pipe network leakage position is further judged by combining the non-local cosine similarity comparison pressure and flow data, and the accuracy of calculating the water supply pipe network leakage position is improved while the data comparison sample is reduced.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Drawings
FIG. 1 is a schematic flow chart of a method for identifying a leakage position of a water supply network according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of the placement of pressure and flow sensors in an embodiment of the invention;
FIG. 3 is a graph showing the trend of pressure data at different leak locations according to an embodiment of the present invention;
FIG. 4 is a graph showing the trend of flow data at different leak locations according to an embodiment of the present invention.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the technical solutions in the embodiments of the present invention will be clearly and completely described in the following in conjunction with the embodiments of the present invention, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
As shown in fig. 1, an embodiment of the present invention provides a water supply network leakage position identification method, which includes the following steps:
s10, monitoring pressure flow data of different positions of a water supply network in real time;
it should be noted that, in this embodiment, the pressure data and the flow data at different positions of the pipe network may be measured by the pressure sensor and the flow sensor disposed at different positions of the pipe network.
S20, using the right-end residual terms of the lattice Boltzmann parallel solving pressure flow relation equation as real-time monitoring residual terms;
S30, determining the leakage position of the water supply network corresponding to the real-time monitoring residual error item according to the mapping relation between the pre-constructed leakage position and the residual error item.
It should be noted that, if the pressure/flow coupling relation equation is a nonlinear partial differential equation, the differential equation needs to be discretized if the traditional method is adopted for solving, and then the discrete large-scale linear equation set is solved iteratively, so that the calculated amount is huge and the solving time is long. Lattice Boltzmann is taken as a mesoscopic method, a discrete macroscopic equation is not needed, collision and migration are carried out through a distribution function of fluid particles, macroscopic flow behavior of fluid is restored, and the LBM is easy to implement and local operation is very suitable for large-scale parallel calculation because the solution of a complex linear equation set is not involved. The pressure/flow relation equation is solved in parallel by using the lattice Boltzmann, so that the efficiency of equation solving is remarkably improved, meanwhile, the complex process of feature extraction is avoided by establishing the mapping relation between the leakage position and the residual error term, and the quick and accurate identification of the pipe network leakage position is realized.
As a further preferable technical solution, in step S30, the construction process of the mapping relationship between the leakage position and the residual term includes the following steps:
Acquiring pressure flow data at different positions of a water supply network under a normal working condition, and parallelly solving a right end residual term of a pressure flow relation equation under the normal working condition by using Boltzmann as a residual term under the normal working condition;
setting a set number of leakage positions, and collecting pressure flow data at different positions of a water supply network at different leakage positions;
For each leakage position, using the Boltzmann of the lattice to solve the right end residual error item of the pressure flow relation equation under the leakage working condition in parallel as the residual error item under the leakage working condition;
And establishing mapping relations between different leakage positions and residual items according to all leakage positions and corresponding residual items under the leakage working condition to form a leakage position-residual item mapping relation comparison table, wherein the mapping relation comparison table comprises the leakage positions, the residual items and pressure flow data of different positions of the water supply network under the working condition.
Specifically, the water supply network is provided with 2 water tanks, 2 water pumps, 6 fire hydrants, 10 valves and 26 pipelines, and pressure/flow data of different positions of the water supply network under normal working conditions are firstly obtained. Then a plurality of groups of leakage positions are set according to the needs, the leakage positions can be set flexibly and uniformly distributed or unevenly distributed, pressure/flow data at different positions of the water supply network under different leakage positions are collected, only one group of leakage positions is set in each test, the arrangement positions of the pressure/flow sensors are shown in fig. 2, and the pressure data change trend and the flow data change trend of different leakage positions are shown in fig. 3 and fig. 4 respectively.
In this example, the pipe length was 400 meters, and 20 sets of leak locations were provided in total, evenly. In the test 1, the leakage holes 1 are opened, the other leakage holes are closed, pressure/flow data of different positions of the water supply network under different leakage positions are collected, and the pressure/flow data are recorded into a database for storage, as shown in table 1.
Table 1 stores pressure/flow data
In order to correct the residual error items at the right end under the leakage working condition, the pressure flow data at different positions of the water supply network under the normal working condition is obtained, the residual error items at the right end of the pressure flow relation equation under the normal working condition are solved in parallel by using the lattice Boltzmann as the residual error items under the normal working condition, and the residual error items under the leakage working condition are corrected by using the residual error items, so that the pressure/flow relation equation is solved in parallel by using the lattice Boltzmann for each group of leakage positions, the residual error items at the right end of the equation are calculated, and the mapping relation between different leakage positions and the residual error items is established, as shown in table 2.
TABLE 2 leakage points and residual terms
And monitoring pressure/flow data at different positions of the water supply network in real time through the arranged pressure/flow sensors, solving a pressure/flow relation equation in parallel by using the lattice Boltzmann, and calculating a right-end residual error term of the pressure/flow relation equation at the current moment, wherein the residual error term is shown in a table 3.
TABLE 3 pressure/flow at present time and right residual term
According to the mapping relation of the established leakage position/residual error term, the pressure/flow distribution is different due to different leakage positions, so that the residual error term of the pressure/flow equation is generally different, and the leakage position can be primarily judged according to the comparison between the value of the residual error term and the residual error term of different leakage positions.
Further, water is an incompressible fluid and water supply network pressure/flow satisfies the following equation:
wherein phi is the pressure (MPa), f is the fluid velocity (m/s), and the flow can be obtained by integrating the fluid velocity along the water section. The pressure/flow relationship equation is solved in parallel using lattice boltzmann, as follows:
(1) And (3) collision:
(2) And (3) a migration step:
(3) Statistical pressure/flow:
the method is characterized in that the lattice Boltzmann is used for solving a pressure/flow relation equation to be local operation, and is very suitable for parallel calculation.
Further, due to measurement errors and calculation errors, the right-hand end term of the pressure/flow relation equation is generally not 0, and the right-hand end residual term of the equation needs to be calculated, and the calculation formula is as follows:
Where c i is the model discrete velocity, h i is the discrete distribution function, i is the discrete velocity index, For the purpose of the gradient operator,For the post-collision discrete distribution function, h i (x, t) is the discrete distribution function at position x and time t, Ω i (h) is the collision operator, x is the position index, and t is the time index.
The partial calculation code for solving the pressure/flow relationship equation using lattice boltzmann is as follows:
(1) And (3) collision:
void COLL(void){
int X;
int Y;
int I;
#pragma omp parallel for private(Y,I)num_threads(16)
for(X=0;X<=GRIX;X++){
for(Y=0;Y<=GRIY;Y++){
for(I=0;I<9;I++){
FUN1[X][Y][I]=(1.0-1.0/TIME)*FUN0[X][Y][I]+1.0/TIME*EQUI(I,DENS[X][Y],VELX[X][Y],VELY[X][Y]);
}
}
}
}
(2) And (3) a migration step:
void STRE(void){
int X;
int Y;
int I;
#pragma omp parallel for private(Y,I)num_threads(16)
for(X=0;X<=GRIX;X++){
for(Y=0;Y<=GRIY;Y++){
for(I=0;I<9;I++){
if(X-INT1[I]>=0&&X-INT1[I]<=GRIX&&Y-INT2[I]>=0&&Y-INT2[I]<=GRIY){
FUN0[X][Y][I]=FUN1[X-INT1[I]][Y-INT2[I]][I];
}
}
}
}
}
(3) Statistical pressure/flow:
void PHYS(void){
int X;
int Y;
int I;
#pragma omp parallel for private(Y,I)num_threads(16)
for(X=0;X<=GRIX;X++){
for(Y=0;Y<=GRIY;Y++){
DENS[X][Y]=0.0;
VELX[X][Y]=0.0;
VELY[X][Y]=0.0;
for(I=0;I<9;I++){
DENS[X][Y]+=FUN0[X][Y][I];
VELX[X][Y]+=INT1[I]*FUN0[X][Y][I];
VELY[X][Y]+=INT2[I]*FUN0[X][Y][I];
}
VELX[X][Y]/=DENS[X][Y];
VELY[X][Y]/=DENS[X][Y];
}
}
}
Furthermore, as different leakage positions correspond to the same or very similar residual terms, in order to avoid misjudgment of the leakage positions, the embodiment further improves the accuracy of calculating the leakage positions of the water supply network by combining non-local cosine similarity comparison pressure/flow data on the basis of positioning the leakage positions according to the mapping relation between the leakage positions and the residual terms.
Correspondingly, after determining the leakage position of the water supply network corresponding to the real-time monitoring residual item according to the pre-constructed mapping relation between the leakage position and the residual item, the method further comprises the following steps:
S31, calculating non-local cosine similarity between pressure flow data of different positions of the real-time monitoring water supply network and pressure flow data under the historical leakage working condition;
s32, when the non-local cosine similarity is larger than a set similarity threshold, determining that the recognized pipe network leakage position is correct;
s33, determining that the identified pipe network leakage position is wrong when the non-local cosine similarity is smaller than or equal to a set similarity threshold value.
Specifically, assume that the real-time monitoring of the current time pressure vector is:
Φ=(φ-n,φ-n+1…,φ-1,φo,φ1…,φn-1,φn)
Where phi 0 is the pressure value at the node of the preliminary determination leak location.
Assuming that the leakage position is the preliminary judgment leakage position, according to the leakage test, the historical pressure vector under the leakage working condition is
Calculating the non-local cosine similarity of the pressure vector:
Where cos α Φ is the non-local cosine similarity of the pressure data, Φ is the pressure vector monitored at the current moment, and Φ 0 is the pressure vector under the historical leakage working condition.
Similarly, the calculation formula of the flow data non-local cosine similarity is as follows:
In the formula, cos alpha F is the non-local cosine similarity of flow data, F is the flow data monitored at the current moment, and F 0 is the flow data under the history leakage working condition.
Further, if and only if cos alpha Φ is more than or equal to 0.9 and cos alpha F is more than or equal to 0.9, judging that the pipe network leakage position identification is correct, otherwise, considering that the pressure flow data under the working condition is not matched, and the pipe network leakage position identification is wrong, and the leakage position should be reselected for identification.
As a further preferable technical scheme, the step S31 is to calculate the non-local cosine similarity between the pressure flow data of different positions of the real-time monitoring water supply network and the pressure flow data under the history leakage working condition, and specifically comprises the following steps:
S311, calculating first non-local cosine similarity between pressure flow data of different positions of the real-time monitoring water supply network and pressure flow data under historical leakage working conditions according to the vector length of N (N/4 is not less than N is not more than N/2), wherein N is the number of the pressure flow data;
S312, judging whether the first non-local cosine similarity is within a set range, if so, executing a step S313, otherwise, executing a step S314;
S313, comparing the non-local cosine similarity with the similarity threshold value to determine whether the pipe network leakage position identification result is accurate;
s314, resetting the vector length to be 2n, and calculating a second non-local cosine similarity between the pressure flow data of different positions of the real-time monitoring water supply network and the pressure flow data under the history leakage working condition.
As a further preferable embodiment, after the step S314, the method further includes:
and calculating the difference value between the second non-local cosine similarity and the first non-local cosine similarity, outputting the first non-local cosine similarity when the difference value is within a set range, and otherwise, outputting the second non-local cosine similarity for comparison with a similarity threshold.
It should be noted that, the non-local cosine similarity calculation method provided in this embodiment is different from the conventional method, and the conventional method only calculates once the cosine similarity, and then determines the similarity of the two vectors, where the method is innovative in that the determination of the vector length n is very flexible, n is selected smaller at first, the cos α of the two vectors is calculated, if the cos α is too high or too low, it is explained that the vector length may be too short, the vector length 2n is reset, the cos α of the two vectors is calculated again, if the variation of the cos α calculated twice is not large, the cos α calculated for the first time is output, the vector length is considered to have no significant influence on the similarity, and the calculation is stopped, otherwise, the cos α calculated for the second time is taken as the criterion.
In addition, another embodiment of the present invention also provides a computer readable storage medium, on which a computer program is stored, where the computer program is executed by a processor to implement the water supply network leakage position identification method according to the above embodiment.
It should be noted that, other embodiments of the computer readable storage medium or the implementation method of the present invention may refer to the above method embodiments, and are not repeated herein.
It should be noted that the logic and/or steps represented in the flowcharts or otherwise described herein, for example, may be considered as a ordered listing of executable instructions for implementing logical functions, and may be embodied in any computer-readable medium for use by or in connection with an instruction execution system, apparatus, or device, such as a computer-based system, processor-containing system, or other system that can fetch the instructions from the instruction execution system, apparatus, or device and execute the instructions. For the purposes of this description, a "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device. More specific examples (a non-exhaustive list) of the computer-readable medium would include an electrical connection (an electronic device) having one or more wires, a portable computer diskette (a magnetic device), a Random Access Memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical fiber device, and a portable compact disc read-only memory (CDROM). In addition, the computer readable medium may even be paper or other suitable medium on which the program is printed, as the program may be electronically captured, via, for instance, optical scanning of the paper or other medium, then compiled, interpreted or otherwise processed in a suitable manner, if necessary, and then stored in a computer memory.
It is to be understood that portions of the present invention may be implemented in hardware, software, firmware, or a combination thereof. In the above-described embodiments, the various steps or methods may be implemented in software or firmware stored in a memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, may be implemented using any one or combination of techniques known in the art, discrete logic circuits with logic gates for implementing logic functions on data signals, application specific integrated circuits with appropriate combinational logic gates, programmable Gate Arrays (PGAs), field Programmable Gate Arrays (FPGAs), and the like.
In the description of the present specification, a description referring to terms "one embodiment," "some embodiments," "examples," "specific examples," or "some examples," etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiments or examples. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
Furthermore, the terms "first," "second," and the like, are used for descriptive purposes only and are not to be construed as indicating or implying a relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defining "a first" or "a second" may explicitly or implicitly include at least one such feature. In the description of the present invention, the meaning of "plurality" means at least two, for example, two, three, etc., unless specifically defined otherwise.
While embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and not to be construed as limiting the invention, and that variations, modifications, alternatives and variations may be made to the above embodiments by one of ordinary skill in the art within the scope of the invention.
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