Pump station flow channel quasi-rectangular section flow field prediction method based on hybrid neural network

Abstract

The application discloses a pump station flow channel quasi-rectangular section flow field prediction method based on a hybrid neural network, belongs to the technical field of pump station flow channel flow field prediction, and comprises the following steps: constructing a scaled model pump station according to an actual pump station; performing a particle image velocimetry experiment based on the model pump station, and acquiring flow field data and working condition characteristic parameters of an elbow type water inlet flow channel quasi-rectangular section under different working conditions; constructing an MLP-PINN hybrid neural network model; training the MLP-PINN hybrid neural network model based on the flow field data and the working condition characteristic parameters; and predicting the flow field characteristics of the elbow type water inlet flow channel quasi-rectangular section of the pump station under unknown working conditions based on the trained MLP-PINN hybrid neural network model. The method predicts the flow field characteristics of other working conditions on the basis of experimental data of some working conditions, solves the problems of incomplete working condition coverage and high cost, provides more comprehensive monitoring data for pump station monitoring, and enhances the safety of the pump station.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of pump station flow channel flow field prediction, in particular to a pump station flow channel quasi-rectangular section flow field prediction method based on a hybrid neural network. BACKGROUND

[0002] As the core facility of cross-basin water diversion, agricultural irrigation and urban water supply and drainage system, the operation stability of pump station is directly related to the project benefit and safety reliability. In the axial flow pump station, the elbow type inlet flow channel is widely used because it can significantly optimize the flow conditions. The elbow type inlet flow channel smoothly guides the water flow direction through the curved flow channel shape, avoids the turbulence, vortex and energy loss caused by sharp turns or sudden changes in flow direction, thereby improving the water pump inlet flow pattern and enhancing the hydraulic efficiency. Therefore, the flow field characteristics inside the elbow type flow channel have a decisive influence on the operation state of the water pump and the overall energy efficiency of the pump station.

[0003] However, in actual engineering, the operating conditions of the pump station are complex and changeable, and the real operating condition data that can be monitored is extremely limited. The existing technology relies on the construction of a scaled model of the pump station for experiments, and the flow field characteristics are simulated and analyzed in the laboratory. However, the model processing, experimental table construction and repeated testing require a large amount of funds and time, and are limited by experimental resources, making it difficult to obtain full-condition flow field characteristics. SUMMARY

[0004] In view of the above problems in the prior art, the present application provides a pump station flow channel quasi-rectangular section flow field prediction method based on a hybrid neural network, which solves the problems of incomplete experimental condition coverage and high cost of existing pump stations due to limited experimental resources.

[0005] In order to achieve the above-mentioned purposes, the technical solution adopted by the present application is as follows:

[0006] The present application provides a pump station flow channel quasi-rectangular section flow field prediction method based on a hybrid neural network, comprising:

[0007] S1: According to the actual pump station, a scaled model pump station is constructed;

[0008] S2: Based on the model pump station, a particle image velocimetry experiment is performed, and flow field data and operating condition characteristic parameters of the elbow type inlet flow channel quasi-rectangular section under different operating conditions are obtained;

[0009] S3: Constructing a MLP-PINN hybrid neural network model;

[0010] S4: Training the MLP-PINN hybrid neural network model based on the flow field data and operating condition characteristic parameters;

[0011] S5: predicting the flow field characteristics of the quasi-rectangular section of the elbow-shaped inlet flow passage of the pump station under unknown working conditions based on the trained MLP-PINN hybrid neural network model.

[0012] Further, the obtaining of the flow field data of the quasi-rectangular section of the elbow-shaped inlet flow passage under different working conditions comprises:

[0013] The boundary condition parameters of the quasi-rectangular section of the elbow-shaped inlet flow passage under different working conditions are obtained by using a particle image velocimetry experiment.

[0014] According to the geometric shape of the elbow-shaped inlet flow passage, the internal point position parameters of the uniformly distributed nodes inside the quasi-rectangular section of the elbow-shaped inlet flow passage are obtained.

[0015] Further, the MLP-PINN hybrid neural network model comprises a feedforward neural network, a connection layer and a physical information neural network.

[0016] The feedforward neural network and the physical information neural network each comprise an input layer, a hidden layer and an output layer; the connection layer connects the feedforward neural network and the physical information neural network by connecting the output layer of the feedforward neural network and the input layer of the physical information neural network.

[0017] Further, the training of the MLP-PINN hybrid neural network model comprises:

[0018] A1: normalizing the obtained working condition characteristic parameters, boundary condition parameters and internal point position parameters;

[0019] A2: inputting the normalized working condition characteristic parameters and boundary condition parameters into the feedforward neural network in the MLP-PINN hybrid neural network model for training to obtain the boundary conditions corresponding to the boundaries of the elbow-shaped inlet flow passage;

[0020] A3: inputting the boundary conditions obtained by the feedforward neural network and the normalized internal point position parameters into the physical information neural network in the MLP-PINN hybrid neural network model for training to obtain the flow field data of the quasi-rectangular section of the elbow-shaped inlet flow passage.

[0021] Further, the boundary conditions comprise the boundary conditions of the inlet face and the outlet face of the quasi-rectangular section of the elbow-shaped inlet flow passage.

[0022] Further, the working condition characteristic parameters comprise the opening and closing state of the unit, the rotational speed, the water level of the inlet pool and the shielding degree of the retaining screen.

[0023] Further, S5 specifically comprises:

[0024] S501: input the normalized working condition characteristic parameters into the trained MLP-PINN hybrid neural network model, and call the feedforward neural network to predict boundary conditions;

[0025] S502: based on the predicted boundary conditions and the point data obtained according to the elbow type water inlet channel geometry, call the physical information neural network to predict the flow field information, and obtain the flow field characteristics of the pump station elbow type water inlet channel quasi-rectangular section under the corresponding working condition.

[0026] Further, the key parameters of the feedforward neural network are optimized by using the hyperparameter optimization method based on Bayesian optimization to obtain the optimal parameters.

[0027] The loss function of the physical information neural network is composed of physical terms and data terms, and the physical terms are calculated by considering the three-dimensional NS equation of incompressible flow.

[0028] Further, the key parameters of the feedforward neural network are optimized by using the hyperparameter optimization method based on Bayesian optimization to obtain the optimal parameters, including:

[0029] B1: establish a learning function, and confirm the sample distribution of the number of hidden layers, the number of nodes of the multilayer perception machine, the dropout rate, the learning rate and the optimal activation function;

[0030] B2: determine the optimal hyperparameter combination by using Bayesian probability.

[0031] Further, the physical terms are calculated by considering the three-dimensional NS equation of incompressible flow, including:

[0032] C1: bring the predicted flow field data into the partial differential equation, and calculate the residual error of the partial differential equation;

[0033] C2: solve the average residual error of the residual error as the physical term;

[0034] Wherein, the three-dimensional NS equation calculation formula is:

[0035]

[0036] In the formula, is the density of water, , and are three components of velocity in , and directions, is time, is the dynamic viscosity coefficient, is pressure, is the partial derivative operator.

[0037] The beneficial effects of the present application are:

[0038] The present application provides a pump station flow passage quasi-rectangular section flow field prediction method based on a hybrid neural network. The method performs a particle image velocimetry experiment on a scaled model pump station constructed under existing conditions, obtains flow field data and working condition characteristic parameters of the elbow type inlet flow passage quasi-rectangular section under different working conditions, and predicts the flow field characteristics of unknown working conditions based on the trained MLP-PINN hybrid neural network model. The method predicts the flow field characteristics of other working conditions based on the experimental data of some working conditions, makes up for the defects of incomplete experimental working condition coverage and high cost, provides more comprehensive monitoring data for pump station monitoring, and enhances the safety of the pump station. BRIEF DESCRIPTION OF DRAWINGS

[0039] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments or prior art description will be briefly introduced. Obviously, the drawings in the following description are only some embodiments of the present application, and other embodiments can also be obtained by those skilled in the art based on these drawings.

[0040] Figure 1 A method flow chart of a pump station flow passage quasi-rectangular section flow field prediction method based on a hybrid neural network provided by the present application.

[0041] Figure 2 A schematic diagram of a scaled model pump station provided by the present application.

[0042] Figure 3 A particle image velocimetry experiment shooting schematic diagram of an elbow type inlet flow passage quasi-rectangular section of a pump station provided by the present application.

[0043] Figure 4 A high-speed photography sample diagram of an elbow type inlet flow passage quasi-rectangular section of a pump station provided by the present application.

[0044] Figure 5 A flow field characteristic cloud diagram of an elbow type inlet flow passage quasi-rectangular section of a pump station provided by the present application. DETAILED DESCRIPTION

[0045] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only some embodiments of the present application, not all embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art based on the present application belong to the scope of protection of the present application.

[0046] The embodiment of the present application provides a flow field prediction method for a quasi-rectangular section of a pump station flow channel based on a hybrid neural network. The method can refer to Figure 1 , Figure 1 The embodiment of the present application provides a flow field prediction method for a quasi-rectangular section of a pump station flow channel based on a hybrid neural network. The method can refer to

[0047] S1: According to an actual pump station, a scaled model pump station is constructed.

[0048] S2: Based on the model pump station, a particle image velocimetry experiment is performed, and flow field data and working condition characteristic parameters of the elbow-type water inlet flow channel quasi-rectangular section under different working conditions are acquired.

[0049] Further, the flow field data of the elbow-type water inlet flow channel quasi-rectangular section under different working conditions are acquired, including:

[0050] The boundary condition parameters of the elbow-type water inlet flow channel quasi-rectangular section under different working conditions are acquired by using the particle image velocimetry experiment;

[0051] According to the geometric shape of the elbow-type water inlet flow channel, internal point position parameters of the uniformly distributed nodes in the elbow-type water inlet flow channel quasi-rectangular section are obtained.

[0052] In an embodiment of the present application, an elbow-type water inlet flow channel flow field characteristic prediction model of an axial-flow type model pump station is taken as an example, a multi-unit parallel operation axial-flow type pump station is taken as a target object, the flow field characteristics of the elbow-type water inlet flow channel of the pump station affect the stable operation and flow measurement of the pump, the model pump station is a 1:20 scaled model of an actual pump station, as shown in Figure 2 , Figure 2 A schematic diagram of a scaled model pump station provided by the embodiment of the present application is shown, four vertical axial-flow pumps are operated in parallel, the No. 4 water pump unit is taken as an application object, the rated rotating speed of the unit is n=1450 r / min, the water pump impeller diameter is D=157.5 mm, the inlet pool and the elbow-type water inlet flow channel part of the model experiment table are processed from transparent materials, the water level of the inlet pool can be directly measured by using a measuring scale, and the degree of shielding of the retaining grid is manually controlled. The particle image velocimetry (PIV) experiment is carried out on the model experiment table, as shown in Figure 3 , Figure 3 A particle image velocimetry experiment shooting schematic diagram of the elbow-type water inlet flow channel quasi-rectangular section of the pump station provided by the embodiment of the present application is shown. The experiment uses a continuous laser to illuminate the tracer particles (5-20 microns), the camera frame rate is adjusted to 500 hz, and the purpose is to obtain the average flow field of the elbow-type flow channel in a period of time, so it is necessary to ensure that a series of images captured contain enough valid information. Therefore, the synchronizer is used to control the camera to continuously collect three images every second, the total number of single sampling is 1800 frames, and the corresponding time length is about 10 minutes.

[0053] In the shooting of differentZ coordinates XY planes and different Y coordinates XZ When the planes are parallel to each other, the laser needs to be moved horizontally and the camera position needs to be kept unchanged, and the sharpness is adjusted by adjusting the focal length of the camera, so that the XY planes, XZ planes are parallel to each other.

[0054] By controlling the water level of the water inlet pool, the shielding degree of the guard grid and the opening and closing of other units, the working conditions of the experimental unit are changed, the elbow type inlet flow passage is photographed by a high-speed camera, and the shooting results of PIV tracer particles are as shown in Figure 4 , Figure 4 The high-speed photography pattern of the elbow type inlet flow passage of the pump station is provided. Through multi-frame data processing, the velocity vector data of the elbow type inlet flow passage of the quasi-rectangular section are obtained, and the pressure data of the uniformly distributed nodes of the elbow type inlet flow passage under different working conditions are calculated by using the Navier-Stokes equation. The PIV experimental measurement cost is high, and cannot cover all operating conditions. Based on the existing working condition data, the flow field characteristics of the elbow type inlet flow passage under unknown working conditions are predicted.

[0055] The sample of the working condition characteristic parameter is shown in Table 1.

[0056] Table 1: Sample of working condition characteristic parameters

[0057]

[0058] S3: Constructing an MLP-PINN hybrid neural network model.

[0059] Further, the MLP-PINN hybrid neural network model comprises a feedforward neural network, a connection layer and a physical information neural network.

[0060] The feedforward neural network and the physical information neural network each comprise an input layer, a hidden layer and an output layer; the connection layer connects the feedforward neural network and the physical information neural network by connecting the output layer of the feedforward neural network and the input layer of the physical information neural network.

[0061] In an embodiment of the present application, a hybrid neural network model is built based on a Pytorch framework, and the model comprises multiple parts of data input, data processing, feedforward neural network MLP predicting boundary conditions of the inlet flow passage, physical information neural network PINN predicting flow field of the inlet flow passage and flow field data output.

[0062] The MLP neural network part is a feedforward neural network MLP containing a multilayer perceptron, which is used to predict the boundary conditions of the elbow-shaped water inlet channel. The input is the characteristic parameter of different working conditions, and the output is the intermediate transition factor, i.e. the boundary conditions of the elbow-shaped water inlet channel, including the pressure and velocity of the inlet and outlet surfaces. The key parameters of the neural network are selected by the hyperparameter optimization method based on Bayesian optimization, including the number of hidden layers, the number of hidden layer nodes, the dropout rate, the learning rate and the activation function.

[0063] In an embodiment of the present application, the key parameters of the MLP neural network are determined by establishing a learning function, which specifies the sample distribution of the number of hidden layers, the number of nodes of the multilayer perceptron, the dropout rate, the learning rate and the optimal activation function. The most promising combination of hyperparameters is determined by Bayesian probability, and the search is iteratively adjusted. The key parameters of the MLP neural network are shown in Table 2.

[0064] Table 2 Key parameters of MLP neural network

[0065]

[0066] The physical information neural network PINN is a kind of neural network that adds physical laws to the neural network training process, so that the training of the neural network follows the physical principle. The detailed description is that the physical equation (such as partial differential equation PDEs) is embedded in the loss function of the neural network as a soft constraint, so as to realize efficient solution and prediction of complex physical systems. The loss function is composed of a physical term and a data term. The physical term is calculated according to the three-dimensional NS equation considering incompressible flow. The specific way is to bring the predicted flow field data (velocity and pressure) into the partial differential equation, calculate the residual of the equation, and take the average of the three partial differential equation residuals as the physical residual of the PINN network. The influence degree of the data term residual and the physical term residual is the same. The influence degree of the data term residual and the physical term residual is the same. The three-dimensional NS equation calculation formula is as follows:

[0067]

[0068] In the formula, is the density of water, , and are three components of velocity in , and directions, is time, is dynamic viscosity coefficient, is pressure, is the partial derivative operator.

[0069] There is a connection layer between the MLP and the PINN, and the output layer parameters of the MLP and the point coordinate parameters jointly constitute the input layer of the PINN.

[0070] S4: training the MLP-PINN hybrid neural network model based on the flow field data and the working condition characteristic parameters.

[0071] Further, the training of the MLP-PINN hybrid neural network model comprises:

[0072] A1: normalizing the obtained working condition characteristic parameters, boundary condition parameters and internal point position parameters.

[0073] In an embodiment of the present application, the opening and closing state of the unit, the rotating speed, the water level of the water inlet pool and the shielding degree of the barrier screen are the key factors affecting the boundary conditions of the inlet flow passage flow field, which are taken as the working condition characteristic parameters, wherein the opening and closing state of the unit is taken as the label data, 0 corresponds to the closed state, 1 corresponds to the open state, and the shielding degree of the barrier screen is expressed by percentage. The boundary condition parameters can be point data at the inlet and outlet boundaries x 、 y 、 z . The internal point position parameters can be other point data in the flow passage x 、 y 、 z .

[0074] A2: inputting the normalized working condition characteristic parameters and boundary condition parameters into the feedforward neural network in the MLP-PINN hybrid neural network model for training to obtain the boundary conditions corresponding to the elbow-shaped inlet flow passage.

[0075] In an embodiment of the present application, the boundary conditions of the inlet flow passage flow field, including the boundary conditions of the inlet face and the boundary conditions of the outlet face, can be each group of point data at the inlet and outlet boundaries x 、 y 、 z corresponding to u 、 v 、 w 、 p , wherein u 、 v and w are velocity vectors in three directions, p is the pressure.

[0076] A3: inputting the boundary conditions obtained by the feedforward neural network and the normalized internal point position parameters into the physical information neural network in the MLP-PINN hybrid neural network model for training to obtain the flow field data of the quasi-rectangular section of the elbow-shaped inlet flow passage.

[0077] In an embodiment of the present application, the flow field data of the quasi-rectangular section of the elbow-shaped inlet flow passage can be the three-directional velocity of each node u 、 v 、 w and pressure p .

[0078] S5: predicting the flow field characteristics of the quasi-rectangular section of the elbow-shaped inlet flow passage of the pump station under unknown working conditions based on the trained MLP-PINN hybrid neural network model.

[0079] Further, the S5 specifically comprises:

[0080] S501: inputting the normalized working condition characteristic parameters into the trained MLP-PINN hybrid neural network model, and calling the feedforward neural network to predict the boundary conditions;

[0081] S502: based on the predicted boundary conditions and the point data obtained according to the geometric shape of the elbow-shaped inlet flow passage, calling the physical information neural network to predict the flow field information, and obtaining the flow field characteristics of the quasi-rectangular section of the elbow-shaped inlet flow passage of the pump station under the corresponding working conditions.

[0082] In an embodiment of the present application, the working condition characteristic parameters are obtained, such as the opening and closing states of units 1, 2 and 3, the rotating speed, the water level of the inlet pool, the blocking degree of the guard grid, and the related parameters of the boundary conditions, such as the point data at the inlet and outlet boundaries x 、 y 、 z , the above data is normalized, and the processed data is input into the trained MLP-PINN hybrid neural network model, and the corresponding flow field data at the boundaries u 、 v 、 w 、 p , i.e. each group of point data at the inlet and outlet boundaries x 、 y 、 z is obtained by calling the MLP part u 、 v 、 w 、 p , the prediction results obtained by the MLP part and other point data inside the flow passage x 、 y 、 z are input into the PINN part for prediction calculation, and the flow field information of all point data of the inlet flow passage is obtained u 、 v 、 w 、 p . The flow field characteristic cloud map of the quasi-rectangular section of the elbow-shaped inlet flow passage of the pump station predicted based on the hybrid neural network is as follows Figure 5The figure includes two direction component cloud maps of velocity vectors, the left is the quasi-rectangular cross-section velocity component u cloud map, the right is the quasi-rectangular cross-section velocity component v cloud map, it can be seen that the distribution of velocity meets the pipeline velocity distribution law, verifying the accuracy of the mixed neural network prediction.

[0083] In an embodiment of the present application, after obtaining the flow field information of all point data of the inlet flow channel, the obtained data is subjected to inverse normalization processing to obtain the real velocity and pressure, facilitating further flow field analysis.

[0084] The application provides a mixed neural network-based flow field prediction method for a quasi-rectangular cross-section of a pump station inlet flow channel, which obtains flow field data and working condition characteristic parameters of the elbow-shaped inlet flow channel quasi-rectangular cross-section under different working conditions by performing a particle image velocimetry experiment on a model pump station of a scaled actual pump station constructed under an existing working condition, and predicts the flow field characteristics under unknown working conditions based on a MLP-PINN mixed neural network model.

[0085] It should be noted that those skilled in the art will realize that the embodiments described herein are intended to help the reader understand the principles of the present application and should be understood as not limiting the protection scope of the present application to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations without departing from the essence of the present application according to the technical inspirations disclosed in the present application, and these modifications and combinations still fall within the protection scope of the present application.

Claims

1. A method for predicting flow field of quasi-rectangular section of flow passage of pump station based on hybrid neural network, characterized in that, The application relates to a method for predicting flow field characteristics of an elbow-shaped inlet flow channel of a pump station. The method comprises the following steps: S1: constructing a scaled model pump station according to an actual pump station; S2: performing a particle image velocimetry experiment based on the model pump station, and acquiring flow field data and working condition characteristic parameters of the elbow-shaped inlet flow channel of the pump station under different working conditions; S3: constructing an MLP-PINN hybrid neural network model; S4: training the MLP-PINN hybrid neural network model based on the flow field data and the working condition characteristic parameters; S5: predicting the flow field characteristics of the elbow-shaped inlet flow channel of the pump station under unknown working conditions based on the trained MLP-PINN hybrid neural network model. The method for acquiring the flow field data of the elbow-shaped inlet flow channel of the pump station under different working conditions comprises the following steps: acquiring boundary condition parameters of the elbow-shaped inlet flow channel under different working conditions by using a particle image velocimetry experiment; obtaining internal point position parameters of uniformly distributed nodes in the elbow-shaped inlet flow channel according to the geometric shape of the elbow-shaped inlet flow channel; The MLP-PINN hybrid neural network model comprises a feedforward neural network, a connection layer and a physical information neural network. The feedforward neural network and the physical information neural network each comprise an input layer, a hidden layer and an output layer; the connection layer connects the feedforward neural network and the physical information neural network by connecting the output layer of the feedforward neural network and the input layer of the physical information neural network; The method for training the MLP-PINN hybrid neural network model comprises the following steps: A1: normalizing the acquired working condition characteristic parameters, boundary condition parameters and internal point position parameters; A2: inputting the normalized working condition characteristic parameters and boundary condition parameters into the feedforward neural network in the MLP-PINN hybrid neural network model to train the boundary conditions corresponding to the elbow-shaped inlet flow channel, wherein the boundary conditions comprise pressure and velocity of an inlet face and pressure and velocity of an outlet face of the elbow-shaped inlet flow channel; 2. The method of claim 1, wherein the method is a method of predicting a flow field of a quasi-rectangular cross-section of a flow passage of a pump station based on a hybrid neural network. A3: inputting the boundary conditions obtained by the feedforward neural network and the normalized internal point position parameters into the physical information neural network in the MLP-PINN hybrid neural network model to train the flow field data of the elbow-shaped inlet flow channel.

3. The method of claim 1, wherein the method further comprises: The working condition characteristic parameters comprise opening and closing states of the unit, rotating speed, water level of an inlet pool and shielding degree of a barrier. The method for predicting the flow field characteristics of the elbow-shaped inlet flow channel of the pump station under unknown working conditions comprises the following steps: S501: inputting the normalized working condition characteristic parameters into the trained MLP-PINN hybrid neural network model, and calling the feedforward neural network to predict the boundary conditions; 4. The method of claim 1, wherein the method is a method of predicting a flow field of a quasi-rectangular cross-section of a flow passage of a pump station based on a hybrid neural network. S502: calling the physical information neural network to predict the flow field information based on the predicted boundary conditions and the point position data obtained according to the geometric shape of the elbow-shaped inlet flow channel, and obtaining the flow field characteristics of the elbow-shaped inlet flow channel of the pump station under the corresponding working conditions. The key parameters of the feedforward neural network are optimized by using a hyperparameter optimization method based on Bayesian optimization. The loss function of the physical information neural network is composed of a physical term and a data term, and the physical term is calculated by considering three-dimensional NS equations of incompressible flow.

5. The method of claim 4, wherein the method further comprises: Key parameters of the feedforward neural network are optimized by a hyperparameter optimization method based on Bayesian optimization to obtain optimal parameters, including: B1: Establish a learning function and confirm the sample distribution of the number of hidden layers, the number of nodes of the multilayer perceptron, the dropout rate, the learning rate and the optimal activation function; B2: Determine the optimal hyperparameter combination by using Bayesian probability.

6. The method of claim 4, wherein the method further comprises: The physical term is calculated by considering the three-dimensional NS equation of incompressible flow, including: C1: Bring the predicted flow field data into the partial differential equation, and calculate the residual of the partial differential equation; C2: Solve the average residual of the residual as the physical term; Wherein, the three-dimensional NS equation calculation formula is: wherein is the density of water, , and are the three components of the velocity in the directions , and , is time, is the dynamic viscosity coefficient, is the pressure, is the partial derivative operator.

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