CN113219328B - Intelligent fault diagnosis method for circuit breaker operating mechanism based on information fusion - Google Patents

Intelligent fault diagnosis method for circuit breaker operating mechanism based on information fusion Download PDF

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CN113219328B
CN113219328B CN202110109664.1A CN202110109664A CN113219328B CN 113219328 B CN113219328 B CN 113219328B CN 202110109664 A CN202110109664 A CN 202110109664A CN 113219328 B CN113219328 B CN 113219328B
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circuit breaker
sampling points
operating mechanism
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CN113219328A (en
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林圣�
陈欣昌
王玘
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Southwest Jiaotong University
China Railway Design Corp
China State Railway Group Co Ltd
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China Railway Design Corp
China State Railway Group Co Ltd
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01RMEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/327Testing of circuit interrupters, switches or circuit-breakers
    • G01R31/3271Testing of circuit interrupters, switches or circuit-breakers of high voltage or medium voltage devices
    • G01R31/3275Fault detection or status indication

Abstract

The invention discloses an intelligent fault diagnosis method for a circuit breaker operating mechanism based on information fusion, which comprises the following specific steps of: collecting four different types of signals including a vibration signal of a high-voltage circuit breaker, a current of a closing coil, a current of a motor and a stroke of a moving contact as the input of a deep convolutional network; then constructing a deep convolutional network through 4 convolutional layers, 1 global average pooling layer, 1 full-link layer and 1 softmax classification layer, training the network model through an adaptive moment estimation algorithm, and continuously updating network parameters; after training is finished, workers directly input related signals to obtain fault diagnosis results, and end-to-end intelligent fault diagnosis of the circuit breaker operating mechanism is achieved. The method can diagnose more comprehensive fault types and more reliable diagnosis results, reduces professional quality requirements on workers, and has higher popularization and application values.

Description

Intelligent fault diagnosis method for circuit breaker operating mechanism based on information fusion
Technical Field
The invention belongs to the technical field of breaker fault diagnosis, and particularly relates to an intelligent fault diagnosis method for a breaker operating mechanism based on information fusion.
Background
The high-voltage circuit breaker plays a key role in control and protection in a traction power supply system and an electric power system, can find potential faults in time by accurately and efficiently diagnosing the faults, helps maintainers to quickly complete maintenance and repair work of the high-voltage circuit breaker, further reduces accident loss caused by faults of the high-voltage circuit breaker, and has great significance for guaranteeing safe and reliable operation of the system. Statistical results at home and abroad show that the operating mechanism faults are the main fault types of the high-voltage circuit breaker. The vibration signal, the closing coil current signal, the moving contact stroke, the motor current and other signals generated during the opening and closing operation of the high-voltage circuit breaker all contain a large amount of equipment state information, and can be used for fault diagnosis of the operating mechanism of the high-voltage circuit breaker.
In the field of fault diagnosis of high-voltage circuit breakers, the traditional fault diagnosis method can extract features of a single type of signal, and further classify the extracted features through methods such as mechanical learning and the like to obtain a better fault diagnosis result. On the one hand, the traditional method relies heavily on professional experience, needs professional personnel to design characteristics manually, and is not beneficial to popularization of the diagnosis method, and the invention patent with the publication number of CN107607303A provides a method for identifying mechanical faults of a high-voltage circuit breaker based on a wavelet packet and an SOM. However, if the frequency component of the vibration signal is not changed due to a certain fault, and only the change in the time domain occurs, the method cannot identify the corresponding fault, so that the characteristic designed according to professional experience is limited, and is not beneficial to popularization. On the other hand, the high-voltage circuit breaker has a complex mechanical structure and numerous fault types, a single type signal can only reflect specific several fault types, and the invention patent with the publication number of CN103336243A provides a circuit breaker fault diagnosis method based on a closing coil current signal, which performs fault diagnosis by using a characteristic time point of a coil current, but can only perform fault diagnosis on fault types related to a switching-on/off coil, such as jamming of an iron core, low coil voltage, overlong idle stroke of the iron core, and the like, but cannot perform accurate fault diagnosis on fault types of other parts, such as mechanical faults, motor faults, and the like.
In summary, the existing research or technology has the problems that the fault diagnosis of the high-voltage circuit breaker operating mechanism depends on professional experience, the fault types are not comprehensive enough, and the like, and can not meet the actual engineering requirements on the site.
Disclosure of Invention
The invention aims to provide an intelligent fault diagnosis method for a circuit breaker operating mechanism based on information fusion. After the training of the deep convolutional network is completed through the sample data, the four types of signals obtained through acquisition are input simultaneously, and then a fault diagnosis result can be output, so that the end-to-end fault diagnosis process of the high-voltage circuit breaker operating mechanism is realized without the steps of feature extraction, feature classification and the like.
The scheme of the invention is as follows:
a breaker operating mechanism fault intelligent diagnosis method based on information fusion comprises the following steps:
A. sample data acquisition and pretreatment:
simulating four states of a normal state of the high-voltage circuit breaker, a jamming fault of an operating mechanism, a jamming fault of an iron core and a loosening fault of a base screw, collecting four different types of signals including a vibration signal, a closing coil current, a motor current and a moving contact stroke as sample data, wherein the sampling frequencies are respectively f i I =1,2,3,4; the number of sampling points is N i I =1,2,3,4; when the number of sampling points N i >When N is needed, the number of sampling points of the signals is equal to N by using a down-sampling method; when the number of sampling points N i <And when N is needed, the number of sampling points of the signal is equal to N by using an interpolation method.
B. Training of the deep convolutional network:
constructing a deep convolutional network which takes four signals of a high-voltage circuit breaker vibration signal, a closing coil current, a motor current and a moving contact stroke as parallel input and takes four state labels of a normal state, an operating mechanism jamming state, an iron core jamming state and a base screw loosening state as output; inputting standardized sample data to train the deep convolutional network to obtain a fault diagnosis model of the circuit breaker operating mechanism; the method specifically comprises the following steps:
b1, the deep convolutional network is formed by 7 layers of networks and sequentially comprises: 4 convolutional layers, 1 global average pooling layer, 1 fully-connected layer and finally a softmax classification layer.
B2, the convolution layer in the deep convolutional network is a one-dimensional convolution layer, the output of each convolution layer keeps the same distribution by using a BatchNorm algorithm, and the network learning and convergence speed is accelerated; the first layer of one-dimensional convolution layer comprises 4 input channels, and the 4 input channels correspond to a high-voltage circuit breaker vibration signal, a closing coil current, a motor current and a moving contact stroke in sequence when data is input. B3, the output size of each convolution layer is the same as the number N of the sampling points, namely the parameter setting of the one-dimensional convolution layer is satisfied:
N=(N+2P-k)/S+1
wherein k is the convolution kernel size, S is the convolution step length, and P is the number of each edge patch 0 of the input channel; the sizes of convolution kernels in the 4 layers of convolution layers are respectively 5, 9, 13 and 15, the number of 0 of each edge patch of an input channel is respectively 2, 4, 6 and 7, and the convolution step length is set to be 1; when the characteristics of the sample data, the number of sampling points and the like are greatly changed, the related parameters of the convolutional layer are adjusted.
B4, in the deep convolutional network, global average pooling is only carried out at the 5 th layer, and the pooling layer is not redesigned at other parts.
B5, setting the output expectation of the classification layer in a normal state as [1, 0] as the last layer in the deep convolutional network is a softmax classification layer; setting the output expectation of a classification layer to be [0,1, 0] when the operating mechanism is jammed; setting the output expectation of a classification layer to be [0,1, 0] when the iron core is jammed; the output layer of the classification layer when setting the loosening failure of the base screw is desirably [0, 1].
B6, training the deep convolution network by using an adaptive moment estimation algorithm, setting a cross entropy as a loss function to calculate an error between the network output and an expected output, and continuously updating network parameters; in the training process, the initial learning rate is set to be 0.01, if the error is not further reduced after 50 times of iteration, the learning rate is reduced to 0.5 time of the original learning rate, and after the training iteration number reaches the upper limit, the network parameter corresponding to the minimum error is selected as a final fault diagnosis model of the circuit breaker operating mechanism.
C. And (3) fault diagnosis of the circuit breaker operating mechanism to be detected:
the method comprises the steps of collecting vibration signals, closing coil currents, motor currents and moving contact strokes of a circuit breaker to be detected, and the method comprises the steps ofSampling frequencies are respectively f i ', i =1,2,3,4; the number of sampling points is N' i I =1,2,3,4; number N 'of sampling points' i >When N is needed, the number of sampling points of the signals is equal to N by using a down-sampling method; number N 'of sampling points' i <When N is needed, the number of sampling points of the signals is equal to N by using an interpolation method; after the fault diagnosis model is standardized, the fault diagnosis model of the circuit breaker operating mechanism is simultaneously input, and the fault diagnosis result R = [ a, b, c, d ] is output](ii) a If maxa { b, c, d, = }, the circuit breaker is in a normal state; if max { a, b, c, d } = b, the circuit breaker has an operating mechanism jamming fault; if max { a, b, c, d } = c, the circuit breaker has an iron core jamming fault; if max { a, b, c, d } = d, the base screw loosening fault occurs in the circuit breaker.
Further, the sampling frequency f of the vibration signal in the step A 1 Not less than 50kHz, sampling point number N 1 More than or equal to 6000; switching-on coil current sampling frequency f 2 Not less than 4kHz, sampling point number N 2 Not less than 200; sampling frequency f of motor current 2 Not less than 4kHz, sampling point number N 2 More than or equal to 6000; moving contact stroke sampling frequency f 2 Not less than 4kHz, sampling point number N 2 More than or equal to 6000; the number N of the finally used sampling points is more than or equal to 4000.
Further, the sampling frequency f of the vibration signal in step C 1 '> is more than or equal to 50kHz, and the number of sampling points is N' 1 More than or equal to 6000; current sampling frequency f 'of closing coil' 2 More than or equal to 4kHz, sampling points N' 2 Not less than 200; motor current sampling frequency f' 2 More than or equal to 4kHz, sampling points N' 2 More than or equal to 6000; moving contact stroke sampling frequency f' 2 Not less than 4kHz, and the number of sampling points is N' 2 More than or equal to 6000; the number N of the finally used sampling points is more than or equal to 4000.
Further, the sample data and the data to be diagnosed in the steps B and C are subjected to the same standardization:
S′=(S-μ)/σ
in the formula, S is an array formed by sampling points of the same type of signal of sample data or data to be diagnosed at the same time, S' is the normalized data, μ is the mean value of the original data S, and σ is the standard deviation of the original data S.
Compared with the prior art, the invention has the beneficial effects that:
1. the types of faults that can be diagnosed are more comprehensive. The high-voltage circuit breaker vibration signal, the closing coil current, the motor current and the moving contact stroke four signals are fused, the related faults of the opening and closing coil such as iron core jamming and coil voltage overlow, the related faults of the energy storage motor such as transmission gear jamming and energy storage spring shedding, and other mechanical fault types such as operating mechanism jamming and base screw loosening can be comprehensively diagnosed.
2. The diagnosis result is more accurate and reliable. At present, single-type signals are mostly adopted at home and abroad for fault diagnosis of the high-voltage circuit breaker, the single-type signals provide less state information, can only reflect the state change of part of circuit breaker elements, and are easily influenced by external factors. Therefore, the information fusion technology is used for diagnosing the fault of the high-voltage circuit breaker, and more accurate and comprehensive diagnosis results and more reliable and stable diagnosis performance can be obtained compared with a single type of signal.
3. The diagnosis method is convenient to popularize and apply. The invention realizes the end-to-end fault diagnosis of the circuit breaker, namely, the working personnel directly input the related signals into the fault diagnosis model to obtain the fault diagnosis result, thereby reducing the professional quality requirements of the working personnel and being convenient for popularization.
4. The deep convolutional network model is more efficient and simpler. According to the deep convolutional network constructed by the method, unnecessary pooling layers and full-link layers are removed, and the full-network framework only has one global average pooling layer, so that the number of network parameters needing to be trained is greatly reduced while signal detail information is kept; and the output of each layer of convolution keeps the same distribution through a BatchNorm algorithm, thereby accelerating the network learning and convergence speed.
Drawings
FIG. 1 is a schematic flow chart of the steps of the present invention;
fig. 2 shows four signals of a vibration signal of the high-voltage circuit breaker, a current of a closing coil, a current of a motor and a stroke of a moving contact;
FIG. 3 is a schematic diagram of a deep convolutional network constructed by the present invention.
Detailed Description
The invention is described in further detail below with reference to the drawings and the detailed description.
The invention discloses an intelligent fault diagnosis method for a circuit breaker operating mechanism based on information fusion, which is shown in figure 1 and comprises the following specific steps:
A. sample data acquisition and pretreatment:
simulating four states of a normal state of a high-voltage circuit breaker, a jamming fault of an operating mechanism, a jamming fault of an iron core and a loosening fault of a base screw, and acquiring four different types of signals including a vibration signal, a current of a closing coil, a current of a motor and a travel of a moving contact as sample data (as shown in figure 2), wherein the sampling frequencies are respectively f i I =1,2,3,4; the number of sampling points is N i I =1,2,3,4; when the number of sampling points N i >When N is needed, the number of sampling points of the signals is equal to N by using a down-sampling method; when the number of sampling points N i <And N, the number of sampling points of the type of signal is equal to N by using an interpolation method.
B. Training of the deep convolutional network:
constructing a deep convolutional network which takes four signals of a high-voltage circuit breaker vibration signal, a closing coil current, a motor current and a moving contact stroke as parallel input and takes four state labels of a normal state, an operating mechanism jamming state, an iron core jamming state and a base screw loosening state as output; inputting the standardized sample data to train the deep convolutional network to obtain a fault diagnosis model of the circuit breaker operating mechanism; the method comprises the following specific steps:
b1, the deep convolutional network is composed of 7 layer networks as shown in fig. 3, and sequentially includes: 4 convolutional layers, 1 global average pooling layer, 1 fully-connected layer and finally a softmax classification layer.
B2, the convolution layer in the deep convolutional network is a one-dimensional convolution layer, the output of each convolution layer keeps the same distribution by using a BatchNorm algorithm, and the network learning and convergence speed is accelerated; the first layer of one-dimensional convolution layer comprises 4 input channels, and the 4 input channels correspond to a high-voltage circuit breaker vibration signal, a closing coil current, a motor current and a moving contact stroke in sequence when data is input. B3, the output size of each convolution layer is the same as the number N of the sampling points, namely the parameter setting of the one-dimensional convolution layer meets the following requirements:
N=(N+2P-k)/S+1
wherein k is the convolution kernel size, S is the convolution step length, and P is the number of each edge patch 0 of the input channel; the sizes of convolution kernels in the 4 layers of convolution layers are respectively 5, 9, 13 and 15, the number of 0 of each edge patch of an input channel is respectively 2, 4, 6 and 7, and the convolution step length is set to be 1; when the characteristics of the sample data, the number of sampling points and the like are greatly changed, the related parameters of the convolutional layer are adjusted.
And B4, performing global average pooling only at the 5 th layer in the deep convolutional network, and not redesigning a pooling layer at other parts.
B5, setting the output expectation of the classification layer in a normal state as [1, 0] as the last layer in the deep convolutional network is a softmax classification layer; setting the output expectation of a classification layer to be [0,1, 0] when the operating mechanism is jammed; setting the output expectation of the classification layer to be [0,1, 0] when the iron core is jammed; the output layer of the classification layer when setting the loosening failure of the base screw is desirably [0, 1].
B6, training the deep convolution network by using an adaptive moment estimation algorithm, setting a cross entropy as a loss function to calculate an error between the network output and an expected output, and continuously updating network parameters; in the training process, the initial learning rate is set to be 0.01, if the error is not further reduced after 50 iterations, the learning rate is reduced to 0.5 time of the original learning rate, and after the training iteration number reaches the upper limit, the network parameter corresponding to the minimum error is selected as the final fault diagnosis model of the circuit breaker operating mechanism.
C. And (3) fault diagnosis of the circuit breaker operating mechanism to be detected:
collecting vibration signals, closing coil current, motor current and moving contact stroke of a circuit breaker to be detected, wherein the sampling frequencies are respectively f i ', i =1,2,3,4; the number of sampling points is N' i I =1,2,3,4; when the number of sampling points is N' i >When N is needed, the number of sampling points of the signals is equal to N by using a down-sampling method; when samplingDot number N' i <When N is needed, the number of sampling points of the signals is equal to N by using an interpolation method; after the standardized operation is carried out, the operation mechanism fault diagnosis model of the circuit breaker is simultaneously input, and the fault diagnosis result R = [ a, b, c, d ] is output](ii) a If maxa { b, c, d, = }, the breaker is in a normal state; if max { a, b, c, d } = b, the circuit breaker has an operating mechanism jamming fault; if max { a, b, c, d } = c, the circuit breaker has an iron core jamming fault; if max { a, b, c, d } = d, the circuit breaker has a base screw loosening fault.
Further, the sampling frequency f of the vibration signal in the step A 1 More than or equal to 50kHz, sampling points N 1 More than or equal to 6000; switching-on coil current sampling frequency f 2 Not less than 4kHz, sampling point number N 2 Not less than 200; sampling frequency f of motor current 2 Not less than 4kHz, sampling point number N 2 More than or equal to 6000; moving contact stroke sampling frequency f 2 Not less than 4kHz, sampling point number N 2 More than or equal to 6000; the number N of the finally used sampling points is more than or equal to 4000.
Further, the sampling frequency f of the vibration signal in step C 1 'is more than or equal to 50kHz, and the number of sampling points is N' 1 More than or equal to 6000; current sampling frequency f 'of closing coil' 2 Not less than 4kHz, and the number of sampling points is N' 2 Not less than 200; motor current sampling frequency f' 2 Not less than 4kHz, and the number of sampling points is N' 2 More than or equal to 6000; moving contact stroke sampling frequency f' 2 Not less than 4kHz, and the number of sampling points is N' 2 More than or equal to 6000; the number N of the finally used sampling points is more than or equal to 4000.
Further, the sample data and the data to be diagnosed in the steps B and C are subjected to the same standardization treatment:
S′=(S-μ)/σ
in the formula, S is an array formed by sampling points of the same type of signal of sample data or data to be diagnosed at the same time, S' is the normalized data, μ is the mean value of the original data S, and σ is the standard deviation of the original data S. For the data to be diagnosed, although the standardization method is still applicable, the mean value and the standard deviation thereof are unknown, so the data to be diagnosed is standardized by adopting the mean value and the standard deviation of the sample data.

Claims (4)

1. The intelligent fault diagnosis method for the circuit breaker operating mechanism based on information fusion is characterized by comprising the following steps of:
A. sample data acquisition and pretreatment:
simulating four states of a normal state of the high-voltage circuit breaker, a jamming fault of an operating mechanism, a jamming fault of an iron core and a loosening fault of a base screw, collecting four different types of signals including a vibration signal, a closing coil current, a motor current and a moving contact stroke as sample data, wherein the sampling frequencies are respectively f i I =1,2,3,4; the number of sampling points is N i I =1,2,3,4; when the number of sampling points N i >When N is needed, the number of sampling points of the type signal is equal to N by using a down-sampling method; when the number of sampling points N i <When N is needed, the number of sampling points of the type signal is equal to N by using an interpolation method;
B. training of the deep convolutional network:
constructing a deep convolutional network which takes four signals of a vibration signal of a high-voltage circuit breaker, a current of a closing coil, a current of a motor and a stroke of a moving contact as parallel input and takes four state labels of a normal state, a jamming of an operating mechanism, a jamming of an iron core and a loosening of a base screw as output; inputting standardized sample data to train the deep convolutional network to obtain a fault diagnosis model of the circuit breaker operating mechanism; the method comprises the following specific steps:
b1, the deep convolutional network is composed of 7 layers of networks, and sequentially comprises: 4 convolutional layers, 1 global average pooling layer, 1 full-link layer and a final softmax classification layer;
b2, the convolution layer in the deep convolutional network is a one-dimensional convolution layer, the output of each convolution layer keeps the same distribution by using a BatchNorm algorithm, and the network learning and convergence speed is accelerated; the first layer of one-dimensional convolution layer comprises 4 input channels, and the 4 input channels sequentially correspond to a high-voltage circuit breaker vibration signal, closing coil current, motor current and moving contact stroke when data is input;
b3, the output size of each convolution layer is the same as the number N of the sampling points, namely the parameter setting of the one-dimensional convolution layer meets the following requirements:
N=(N+2P-k)/S+1
wherein k is the convolution kernel size, S is the convolution step length, and P is the number of each edge patch 0 of the input channel; the convolution kernel sizes of the 4 layers of convolution layers are respectively 5, 9, 13 and 15, the number of 0 of each edge patch of the input channel is respectively 2, 4, 6 and 7, and the convolution step length is set to be 1; when the sample data characteristics and the sampling points are greatly changed, adjusting the related parameters of the convolutional layer;
b4, performing global average pooling only at the 5 th layer in the deep convolutional network, and not redesigning a pooling layer at other parts;
b5, setting the last layer in the deep convolutional network as a softmax classification layer, and setting the output expectation of the classification layer to be [1, 0] in a normal state; setting the output expectation of a classification layer to be [0,1, 0] when the operating mechanism is jammed; setting the output expectation of a classification layer to be [0,1, 0] when the iron core is jammed; setting the output expectation of a classification layer to be [0, 1] when a base screw loosening fault occurs;
b6, training the deep convolutional network by using an adaptive moment estimation algorithm, setting a cross entropy as a loss function to calculate an error between the network output and an expected output, and continuously updating network parameters; in the training process, setting the initial learning rate to be 0.01, if the error is not further reduced after 50 times of iteration, reducing the learning rate to 0.5 time of the original learning rate, and selecting the network parameter corresponding to the minimum error as the network parameter of the final fault diagnosis model of the circuit breaker operating mechanism after the training iteration number reaches the upper limit;
C. and (3) fault diagnosis of the circuit breaker operating mechanism to be detected:
collecting data to be diagnosed, namely vibration signals, closing coil current, motor current and moving contact stroke data of a circuit breaker to be diagnosed, wherein the sampling frequencies are f i ', i =1,2,3,4; the number of sampling points is N i ', i =1,2,3,4; when the number of sampling points N i ′>When N is needed, the number of sampling points of the type signal is equal to N by using a down-sampling method; when the number of sampling points N i ′<When N is needed, the number of sampling points of the type signal is equal to N by using an interpolation method; after standardizing the data, simultaneously inputting the data into the interruptsThe fault diagnosis model of the operating mechanism of the road divider outputs a fault diagnosis result R = [ a, b, c, d ]](ii) a If max { a, b, c, d } = a, the circuit breaker is in a normal state; if max { a, b, c, d } = b, the circuit breaker has an operating mechanism jamming fault; if max { a, b, c, d } = c, the circuit breaker has an iron core jamming fault; if max { a, b, c, d } = d, the base screw loosening fault occurs in the circuit breaker.
2. The intelligent diagnosis method for the fault of the circuit breaker operating mechanism based on the information fusion as claimed in claim 1, characterized in that in the step A, the vibration signal sampling frequency f is 1 More than or equal to 50kHz, sampling points N 1 More than or equal to 6000; switching-on coil current sampling frequency f 2 Not less than 4kHz, sampling point number N 2 Not less than 200; sampling frequency f of motor current 3 Not less than 4kHz, sampling point number N 3 More than or equal to 6000; moving contact stroke sampling frequency f 4 Not less than 4kHz, sampling point number N 4 More than or equal to 6000; the number N of the finally used sampling points is more than or equal to 4000.
3. The intelligent fault diagnosis method for the circuit breaker operating mechanism based on information fusion as claimed in claim 1, wherein the vibration signal sampling frequency f in the step C 1 ' > not less than 50kHz, number of sampling points N 1 ' is more than or equal to 6000; switching-on coil current sampling frequency f 2 ' > not less than 4kHz, number of sampling points N 2 ' is more than or equal to 200; sampling frequency f of motor current 3 ' > not less than 4kHz, number of sampling points N 3 ' more than or equal to 6000; moving contact stroke sampling frequency f 4 ' > not less than 4kHz, number of sampling points N 4 ' more than or equal to 6000; the number N of the finally used sampling points is more than or equal to 4000.
4. The intelligent fault diagnosis method for the circuit breaker operating mechanism based on information fusion according to claim 1, wherein the sample data and the data to be diagnosed in the steps B and C are subjected to the same standardization treatment:
S′=(S-μ)/σ
in the formula, S is an array formed by sampling points of the same type of signals of the sample data or the data to be diagnosed at the same time, S' is the normalized data, μ is the mean value of the array S, and σ is the standard deviation of the array S.
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