CN114441900A - Power distribution network single-phase earth fault line selection method and system - Google Patents
Power distribution network single-phase earth fault line selection method and system Download PDFInfo
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Abstract
The invention discloses a method and a system for selecting a single-phase earth fault line of a power distribution network, wherein the method comprises the following steps: acquiring fault zero-sequence current of each feeder line; performing continuous wavelet transformation on the fault zero sequence current to obtain a wavelet transformation matrix; converting the wavelet transform matrix into a color image; and carrying out fault feeder line identification according to the pre-trained residual error network model and the color image. The invention aims to provide a method and a system for selecting a single-phase earth fault line of a power distribution network.
Description
Technical Field
The invention relates to the technical field of power distribution networks, in particular to a method and a system for single-phase earth fault line selection of a power distribution network.
Background
In China, a small-current grounding mode is mainly adopted in a 6-66KV power distribution network, and the system is also called a neutral point non-effective grounding system and mainly comprises a neutral point high-resistance grounding system, a neutral point arc suppression coil grounding system and a neutral point ungrounded system.
When single-phase earth fault occurs in the power distribution network adopting the low-current grounding mode, the power distribution network can operate with the fault for 1-2 hours, the power supply reliability can be obviously improved, but the non-grounding is relatively low and the voltage is increased
The power distribution insulation system has certain influence on places with power distribution insulation, and large-scale power failure can be caused seriously. For a power distribution network with a low current grounding, single-phase grounding faults are frequent, and the reasons for influencing the single-phase grounding faults are various. For example, equipment failure, tree cause, weather cause, animal factor, freezing factor, human factor, etc., which brings great uncertainty to the selection of the failed line, and reduces the reliability of the operation of the power supply system.
In recent years, a method for solving fault type identification by taking data as a drive is considered to be the most potential method at present. Along with the development of science and technology, the automation degree of an electric power system is higher and higher, and an electric power metering device and an electric information acquisition device system are improved continuously, so that a data basis is provided for the machine learning application to the fault diagnosis of the electric power system. Deep learning is used as a branch of machine learning, so that a model network of the user can learn and obtain the most representative features in the classification task. At present, many researches show that a deep learning method can extract the features of a complex system with weak fault features through a large number of training samples, so as to realize fault classification.
At present, the main method for conducting single-phase earth fault line selection on a power distribution network is to convert fault zero-sequence current into RGB (red, green and blue) images, the RGB images are used as input of a deep learning network to conduct fault line prediction, the prediction method ignores that similar line fault data are not greatly different in a power distribution network multi-outlet line model, so that generated RGB images are similar in coloring, the recognition accuracy of the deep network is low, and specific fault lines cannot be recognized accurately.
Disclosure of Invention
The invention aims to provide a method and a system for selecting a single-phase earth fault line of a power distribution network.
The invention is realized by the following technical scheme:
in a first aspect of the present application, a method for selecting a single-phase earth fault of a power distribution network is provided, which includes the following steps:
s1: acquiring fault zero-sequence current of each feeder line;
s2: performing continuous wavelet transformation on the fault zero sequence current to obtain a wavelet transformation matrix;
s3: converting the wavelet transform matrix into a color image;
s4: and carrying out fault feeder line identification according to the pre-trained residual error network model and the color image.
Preferably, the S3 includes the following substeps:
s31: randomly extracting three rows of data from the wavelet transform matrix to construct a three-dimensional array;
s32: projecting the three-dimensional array into a W2C matrix to construct a ten-dimensional array;
s32: calculating a covariance matrix of the ten-dimensional array, an eigenvalue of the covariance matrix and an eigenvector corresponding to the eigenvalue;
s33: selecting eigenvectors corresponding to the first four eigenvalues to construct a conversion matrix;
s34: and performing matrix remodeling on the conversion matrix to obtain the color image.
Preferably, the residual network model is ResNet-50.
Preferably, the convolutional layer output characteristics of the residual error network model are as follows:
y(p0)＝∑w(pn)×x(p0+pn+Δpn)+b；
wherein, y (p)0) Representing an output feature map, w (p)0) Representing the convolution kernel weight, x (p)0+pn+Δpn) Representing input feature maps, Δ pnDenotes offset coordinates, pnIndicating different positions of the convolution kernel, p0The center position of the input feature mapping convolution kernel is represented, and b represents the bias coefficient.
In a second aspect of the present application, the present application provides a single-phase ground fault line selection system for a power distribution network, including:
the acquisition module is used for acquiring the fault zero sequence current of each feeder line;
the transformation module is used for carrying out continuous wavelet transformation on the fault zero sequence current to obtain a wavelet transformation matrix;
a conversion module for converting the wavelet transform matrix into a color image;
and the identification module is used for identifying the fault feeder line according to the pre-trained residual error network model and the color image.
Preferably, the conversion module comprises:
the first construction unit is used for arbitrarily extracting three rows of data from the wavelet transformation matrix to construct a three-dimensional array;
the second construction unit is used for projecting the three-dimensional array into a W2C matrix to construct a ten-dimensional array;
the calculation unit is used for calculating a covariance matrix of the ten-dimensional array, an eigenvalue of the covariance matrix and an eigenvector corresponding to the eigenvalue;
the third construction unit is used for selecting the eigenvectors corresponding to the first four eigenvalues to construct a conversion matrix;
and the conversion unit is used for performing matrix remodeling on the conversion matrix to obtain the color image.
Preferably, the residual network model is ResNet-50.
Preferably, the convolutional layer output characteristics of the residual error network model are as follows:
y(p0)＝∑w(pn)×x(p0+pn+Δpn)+b；
wherein, y (p)0) Representing an output feature map, w (p)0) Representing the convolution kernel weight, x (p)0+pn+Δpn) To representInput feature map, Δ pnDenotes offset coordinates, pnIndicating different positions of the convolution kernel, p0The center position of the input feature mapping convolution kernel is represented, and b represents the bias coefficient.
Compared with the prior art, the invention has the following advantages and beneficial effects:
the RGB image is converted into a color image which can represent different typical characteristics, so that fault data of similar lines can be distinguished, and specific fault lines can be accurately identified.
Drawings
The accompanying drawings, which are included to provide a further understanding of the embodiments of the invention and are incorporated in and constitute a part of this application, illustrate embodiment(s) of the invention and together with the description serve to explain the principles of the invention. In the drawings:
FIG. 1 is a schematic flow chart of a single-phase earth fault line selection method for a power distribution network according to the present invention;
FIG. 2 is a schematic diagram illustrating a single-phase earth fault current variation simulation according to the present invention;
FIG. 3 is a schematic diagram of a single-phase earth fault voltage variation simulation according to the present invention;
FIG. 4 is a schematic diagram of a wavelet transform matrix being converted into a transform matrix according to the present invention;
FIG. 5 is a schematic illustration of transformation matrix remodeling in accordance with the present invention;
FIG. 6 is a schematic diagram of the structure of ResNet-50 of the present invention;
FIG. 7 is a schematic structural diagram of a Conv Block module according to the present invention;
FIG. 8 is a schematic structural diagram of an Identity Block module according to the present invention;
fig. 9 is a diagram illustrating residual learning according to the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to examples and accompanying drawings, and the exemplary embodiments and descriptions thereof are only used for explaining the present invention and are not meant to limit the present invention.
Example 1
A single-phase earth fault line selection method for a power distribution network, as shown in fig. 1, includes the following steps:
s1: acquiring fault zero-sequence current of each feeder line;
specifically, when a fault line is detected, the variation of transient state quantity information and the variation of steady state quantity information exist, in order to select a proper variation to represent the fault of the line, the inventor utilizes Matlab/simulink software to build a 10KV and 5 outgoing power distribution network model, the 1 st line and the 5 th line respectively use the fusion of overhead lines and cables with different lengths, and the simulation is applied to simulate the current and voltage variation conditions of the single-phase grounding of the power distribution network under different fault working conditions, as shown in fig. 2 and 3, wherein the simulation parameters are shown in table 1:
TABLE 1 simulation parameters
As can be seen from fig. 2 (the two lines with the smaller amplitude in fig. 2 represent two phases of zero-sequence current in normal operation, and the line with the larger amplitude represents zero-sequence current in the phase with ground fault) and fig. 3 (the line with the largest amplitude in fig. 3 represents voltage in the phase with fault, and the other two lines represent voltages in which two phases are normal), when a single-phase ground fault occurs in the power distribution network, the phase current change of the fault is large, and the voltage change is small. According to theoretical knowledge, the zero sequence current of a fault line in the neutral ungrounded system is equal to the sum of non-fault lines, and the direction is opposite to that of the non-fault lines. In a system with a neutral point grounded through an arc suppression coil, the direction of zero sequence current of a fault point is different along with the difference of the compensation degree of the arc suppression coil. In the case of overcompensation, the zero sequence currents of the faulty line and the non-faulty line are in the same direction, and the value of the zero sequence current is also small due to the compensation effect. After the theoretical knowledge and the model are analyzed, feedback data obtained when the model is in the single-phase earth fault state are found to meet the theoretical knowledge, and feasibility of model building is proved.
After the characteristics of voltage and current of the power distribution network during single-phase earth fault are analyzed, the fault line selection method based on the current steady-state quantity and the voltage steady-state quantity cannot meet the existing power distribution network system to a great extent. The steady-state quantity line selection method is easy to lead out and select lines at a neutral point through a large-resistance grounding system and an arc suppression coil grounding system, and the phases of a fault line and a non-fault line can be the same at the moment, and the amplitude is not greatly different.
The present embodiment then chooses from the point of view of the transient information content variation. After a single-phase grounding fault occurs, whether a neutral point is grounded or grounded through an arc suppression coil, the amplitude and the frequency of a transient current at the initial stage of the fault are mainly determined by a transient capacitor, and the amplitude is related to an initial phase angle. When the fault occurs at the phase voltage instantaneous value and is close to the maximum instantaneous value, the capacitance current is maximum, and when the single-phase grounding fault occurs near the phase voltage instantaneous value which is zero, the transient component of the capacitance current is minimum.
The transient process analysis shows that when a single-phase earth fault occurs in the power distribution network, the transient process still has abundant fault information, and because the transient process occurring during the fault is not influenced by a neutral point grounding mode, the transient component of the transient current has great significance in fault line selection.
S2: performing continuous wavelet transformation on the fault zero sequence current to obtain a wavelet transformation matrix;
because the time of the transient information is very short, about 0.5-1.0 power frequency cycle exists, and the transient current belongs to a non-stationary signal. In the case of signal analysis and processing in the face of instantaneous transformation, fourier transformation is generally used, but fourier transformation is weak in processing non-stationary signals, while wavelet transformation shows a good capability for processing non-stationary signals, so the present embodiment selects the method of wavelet transformation. The wavelet transform includes Continuous Wavelet Transform (CWT) which mainly decomposes a signal of continuous time and Discrete Wavelet Transform (DWT) which discretizes information to reduce redundancy of information contained in the continuous signal and reduce calculation cost, but the continuous wavelet transform is selected in this embodiment because it is a process of selecting a continuous signal more than necessary for accurate signal processing.
The basic idea of wavelet transformation is to perform translation and expansion transformation on mother wavelet function to obtain a group of two-dimensional bases called wavelet function clustersThe substrate representing or approximating a signal; for example, assuming that the selected mother wavelet function is ψ (t), and the scaling factor and the panning factor are defined as a and b, respectively, the wavelet function cluster Ψ is generateda,b(t) is:
specifically, in this embodiment, the scale of the continuous transformation is set to 224, the wavelet basis function is db5, and the continuous wavelet transformation is performed on the fault zero-sequence current, so that the obtained wavelet transformation matrix is as follows:
s3: the wavelet transform matrix is converted into a color image, specifically, as shown in fig. 4:
firstly, randomly extracting three rows of data from a wavelet transformation matrix to construct a group of three-dimensional arrays, wherein each row comprises 12544 x 1 data;
the three-dimensional array is then projected into a W2C matrix to construct a ten-dimensional array, completing the three-dimensional to ten-dimensional conversion, forming 10-dimensional spatial colors. This means that it can subdivide the features of the colors, each of which in 10 dimensions can represent a different typical feature.
Considering that there are many redundant information in the ten-dimensional array, in order to reduce the dimension without reducing the critical information, the embodiment performs the dimension reduction processing by using the principal component analysis method, so as to improve the processing speed of the deep learning network. Specifically, firstly, a 10 x 10 covariance matrix is calculated based on the ten-dimensional array, and secondly, eigenvalues and corresponding eigenvectors of the covariance matrix are calculated; then constructing the eigenvectors corresponding to the first four eigenvalues into a conversion matrix with the size of 10 x 4; finally, for the input of the subsequent deep learning network, the 12544 × 4 transform matrix is reshaped into 224 × 224 transform matrix (color image) (12544 × 4 ═ 224 × 224), as shown in fig. 5.
S4: and carrying out fault feeder line identification according to the pre-trained residual error network model and the color image.
Specifically, the residual network model in this embodiment is ResNet-50, as shown in fig. 6, the ResNet model mainly includes two modules, one module is a Conv Block structure, as shown in fig. 7, and the other module is an Identity Block structure, as shown in fig. 8, 50 represents that the residual network is composed of 50 layers; the input and output dimensions of Conv Block are different, so that the Conv Block can not be connected in series continuously, and the Conv Block has the function of changing the dimensions of the network; the input and output dimensions of the Identity Block are the same and can be connected in series for deepening the network depth.
In fig. 6 to 7, Conv2d is a two-dimensional convolution layer, and variable convolution is adopted; BatchNorm is a batch unification layer; relu is the activation function; MaxPool is the largest pooling layer; zeropad is edge 0 filled.
The residual error network is used for solving the problem that the network degradation is caused by excessive hidden layers of the neural network, the degradation means that the hidden layers of the network become more, the accuracy of the network is degraded sharply after reaching saturation, and the degradation is not caused by overfitting.
When the input is X, the learned feature is denoted as h (X), and the residual error desired to be learned is f (X) ═ h (X) — X, so that the original learning feature is f (X) + X, which is easier than direct convolutional neural network learning. When the residual is 0, the number of stacked layers is only mapped identically, but at least the network does not decrease, and actually the residual is not 0, so that new features are learned on the basis of the original input, and better performance is achieved, and the structure of residual learning is shown in fig. 9.
The present embodiment makes the following improvements on the basis of the depth residual error network:
variable convolution (distortion convolution) is introduced on the basis of a traditional convolution kernel, offset can be learned from a target task without extra supervision, and spatial sampling can be carried out by using extra offset so as to achieve strong robustness through convolution irregular areas. By adding a spreading factor, AConv can reduce the loss value of spatial features and obtain remote information without reducing the received feature information. In adapting input size and shape, deformable convolutional networks (DC) are proposed to adapt to geometric changes and transformations of objects. The difference between the traditional neural network and the deformable convolutional network is the addition of offsets in the convolutional layers. This approach is based on enhancing the spatial sampling position using additional biases. In the convolutional layer, if the input map is defined as x and the output is y, the output feature map in the conventional neural network can be obtained by equation (1). The output characteristic diagram in DC is represented by equation (2):
y(p0)＝∑w(p0)×x(p0+pn) (1)
y(p0)＝∑w(pn)×x(p0+pn+Δpn)+b (2)
where p represents the position in the feature map and w represents the weight of the feature value position.
In the embodiment, the number of the extended deformable convolution layers is improved, so that the model power is effectively enhanced.
The training process for ResNet-50 is described as follows:
1. respectively labeling fault pictures of each line in the multiple color images (224 × 4); dividing the color images into a training set and a verification set, wherein 80% of the two sets are used for training and 20% are used for verification;
2. inputting the training pictures 224 × 4 into the ResNet-50, wherein the training pictures firstly enter an input layer; where 224 x 224 represents the size of the picture, 4 represents 4 dimensions;
3. entering the first convolution layer, the convolution kernel size is 7 × 7, the number of convolution kernels is 64, the step size is 2, and the padding is 3 (edge padding), so that the output result is 64 × 112;
4. entering a Maxpool layer, 3 × 3 cores, the step size is 2, the dimension is changed, but the number is not changed, and the output result is 64 × 55;
5. entering a second stage, inputting the data into a network consisting of 1 Conv Block and 2 Identity Block, and obtaining an output of 256 × 55;
6. entering a third stage, inputting the data into a network consisting of 1 Conv Block and 3 Identity Block, and obtaining an output of 512 × 28;
7. entering a fourth stage, inputting the data into a network consisting of 1 Conv Block and 5 Identity Block, and obtaining output of 1024 × 14;
8. entering a fifth stage, inputting the data into a network consisting of 1 Conv Block and 2 Identity Block, and obtaining output of 2048 × 7;
9. into the last average pooling layer, 7 × 7 nuclei, and finally to 2048 × 1 output;
10. finally, entering a full connection layer to obtain 1000 pictures with 1 x 1;
11. after 10 generations of training, 57 pictures are trained each time, and the precision of the training reaches over 96 percent of accuracy;
12. storing the trained model;
13. and putting the model into a test set for verification, thereby obtaining a trained ResNet-50 model.
At present, the main method for single-phase earth fault line selection of a power distribution network is to convert fault zero-sequence current into RGB (red, green and blue) images, and the RGB images are used as input of a deep learning network to predict a fault line. The inventor improves on the basis of the above, so that the identification accuracy can be improved, and a specific fault line can be judged. Specifically, under the development of computer vision, colors play a great role as the field of object detection and behavior recognition, and detailed and non-uniform colored images bring high accuracy to deep network learning. Therefore, the inventor converts the obtained wavelet transformation matrix into 10-dimensional space colors (10 dimensions are respectively black, blue, brown, green, orange, pink, purple, red, white and yellow) after mapping the wavelet transformation matrix by a Google-image retrieval image method through a W2C matrix, and forms the 10-dimensional space colors. This means that it can subdivide the characteristics of colors, and each color in 10 dimensions can represent different typical characteristics, thereby distinguishing fault data of similar lines and improving identification accuracy. Further, in consideration of the correlation between many variables that may exist in the spatial color of 10 dimensions, it is necessary to extract main information and reduce the data dimension to improve the computational efficiency. If the analysis data or the dimensionality is randomly reduced, the loss of useful information is inevitably caused, so that the embodiment adopts the principal component analysis method to perform dimensionality reduction processing to reduce the data of 10 dimensionality to 4 dimensionality so as to improve the processing speed of the deep learning network.
Example 2
This embodiment provides a distribution network single-phase earth fault route selection system, includes:
the acquisition module is used for acquiring the fault zero sequence current of each feeder line;
the transformation module is used for carrying out continuous wavelet transformation on the fault zero sequence current to obtain a wavelet transformation matrix;
the conversion module is used for converting the wavelet transformation matrix into a color image; specifically, the method comprises the following steps:
the first construction unit is used for arbitrarily extracting three rows of data from the wavelet transformation matrix to construct a three-dimensional array;
the second construction unit is used for projecting the three-dimensional array into the W2C matrix to construct a ten-dimensional array;
the calculation unit is used for calculating a covariance matrix of the ten-dimensional array, an eigenvalue of the covariance matrix and an eigenvector corresponding to the eigenvalue;
the third construction unit is used for selecting the eigenvectors corresponding to the first four eigenvalues to construct a conversion matrix;
and the conversion unit is used for performing matrix remodeling on the conversion matrix to obtain a color image.
And the identification module is used for identifying the fault feeder line according to the pre-trained residual error network model and the color image.
Further, the residual network model in this embodiment is ResNet-50.
Further, the convolutional layer output characteristics of the residual error network model are as follows:
y(p0)＝∑w(pn)×x(p0+pn+Δpn)+b；
wherein, y (p)0) Representing an output feature map, w (p)0) Representing the convolution kernel weight, x (p)0+pn+Δpn) Representing input feature maps, Δ pnDenotes offset coordinates, pnIndicating different positions of the convolution kernel, p0The center position of the input feature mapping convolution kernel is represented, and b represents a bias coefficient.
At present, the main method for single-phase earth fault line selection of a power distribution network is to convert fault zero-sequence current into RGB (red, green and blue) images, and the RGB images are used as input of a deep learning network to predict a fault line. The inventor improves on the basis of the above, so that the identification accuracy can be improved, and a specific fault line can be judged. Specifically, under the development of computer vision, colors play a great role as the field of object detection and behavior recognition, and detailed and non-uniform colored images bring high accuracy to deep network learning. Therefore, the obtained wavelet transformation matrix is transformed into 10-dimensional space colors (10 dimensions are respectively black, blue, brown, green, orange, pink, purple, red, white and yellow) after being subjected to W2C matrix mapping by the inventor to form the 10-dimensional space colors. This means that it can subdivide the characteristics of colors, and each color in 10 dimensions can represent different typical characteristics, thereby distinguishing fault data of similar lines and improving identification accuracy. In addition, considering that there may be correlation between many variables in the spatial color of 10 dimensions, it is necessary to extract main information and reduce data dimensions to improve computational efficiency. If the analysis data or the dimensionality is randomly reduced, the loss of useful information is inevitably caused, so that the embodiment adopts a principal component analysis method to perform dimensionality reduction processing, and reduces the data of 10 dimensionality to 4 dimensionality so as to improve the processing speed of the deep learning network.
The above-mentioned embodiments are intended to illustrate the objects, technical solutions and advantages of the present invention in further detail, and it should be understood that the above-mentioned embodiments are merely exemplary embodiments of the present invention, and are not intended to limit the scope of the present invention, and any modifications, equivalent substitutions, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (8)
1. A single-phase earth fault line selection method for a power distribution network is characterized by comprising the following steps:
s1: acquiring fault zero-sequence current of each feeder line;
s2: performing continuous wavelet transformation on the fault zero sequence current to obtain a wavelet transformation matrix;
s3: converting the wavelet transform matrix into a color image;
s4: and carrying out fault feeder line identification according to the pre-trained residual error network model and the color image.
2. The single-phase earth fault line selection method for the power distribution network according to claim 1, wherein the S3 comprises the following sub-steps:
s31: randomly extracting three rows of data from the wavelet transform matrix to construct a three-dimensional array;
s32: projecting the three-dimensional array into a W2C matrix to construct a ten-dimensional array;
s32: calculating a covariance matrix of the ten-dimensional array, an eigenvalue of the covariance matrix and an eigenvector corresponding to the eigenvalue;
s33: selecting eigenvectors corresponding to the first four eigenvalues to construct a conversion matrix;
s34: and performing matrix remodeling on the conversion matrix to obtain the color image.
3. The method according to claim 1 or 2, wherein the residual network model is ResNet-50.
4. The method according to claim 3, wherein the convolutional layer output characteristics of the residual error network model are as follows:
y(p0)＝∑w(pn)×x(p0+pn+Δpn)+b；
wherein, y (p)0) Representing an output feature map, w (p)0) Representing the convolution kernel weight, x (p)0+pn+Δpn) Representing input feature maps, Δ pnDenotes offset coordinates, pnIndicating different positions of the convolution kernel, p0The center position of the input feature mapping convolution kernel is represented, and b represents a bias coefficient.
5. The utility model provides a distribution network single-phase earth fault route selection system which characterized in that includes:
the acquisition module is used for acquiring the fault zero sequence current of each feeder line;
the transformation module is used for carrying out continuous wavelet transformation on the fault zero sequence current to obtain a wavelet transformation matrix;
a conversion module for converting the wavelet transform matrix into a color image;
and the identification module is used for identifying the fault feeder line according to the pre-trained residual error network model and the color image.
6. The single-phase earth fault line selection system for the power distribution network of claim 5, wherein the conversion module comprises:
the first construction unit is used for arbitrarily extracting three rows of data from the wavelet transformation matrix to construct a three-dimensional array;
the second construction unit is used for projecting the three-dimensional array into a W2C matrix to construct a ten-dimensional array;
the calculation unit is used for calculating a covariance matrix of the ten-dimensional array, an eigenvalue of the covariance matrix and an eigenvector corresponding to the eigenvalue;
the third construction unit is used for selecting the eigenvectors corresponding to the first four eigenvalues to construct a conversion matrix;
and the conversion unit is used for performing matrix remodeling on the conversion matrix to obtain the color image.
7. The single-phase ground fault line selection system of the power distribution network according to claim 5 or 6, wherein the residual network model is ResNet-50.
8. The single-phase earth fault line selection system of the power distribution network of claim 7, wherein the convolutional layer output characteristics of the residual error network model are:
y(p0)＝∑w(pn)×x(p0+pn+Δpn)+b；
wherein, y (p)0) Representing an output feature map, w (p)0) Representing the convolution kernel weight, x (p)0+pn+Δpn) Representing input feature maps, Δ pnDenotes offset coordinates, pnIndicating different positions of the convolution kernel, p0The center position of the input feature mapping convolution kernel is represented, and b represents the bias coefficient.
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