CN113761788A - SCOPF rapid calculation method and device based on deep learning - Google Patents
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Abstract
The application discloses a method and a device for rapidly calculating SCOPF based on deep learning, wherein the method comprises the steps of obtaining current operation data of a power system; preprocessing the current operation data to obtain processed operation data; and predicting all active constraints by using a pre-trained DNN model based on the processed running data, screening by using a threshold function to obtain an active constraint set, and learning to perform SCOPF real-time calculation based on the active constraint set. An active possibility function for measuring the inequality constraint active degree is designed, more information can be provided, the deep learning model has better interpretability, and the calculation speed of the SCOPF solving method is accelerated.
Description
Technical Field
The application relates to the technical field of load flow calculation, in particular to a method and a device for rapidly calculating SCOPF based on deep learning.
Background
In recent years, with the increasing access of renewable energy sources, energy storage devices and flexible loads to the power grid, how to ensure safe and economic operation of the power system becomes a challenging problem. Therefore, it is important to efficiently solve the Optimal Power Flow (SCOPF) considering the safety constraint.
If the direct current model is considered, SCOPF can be regarded as a convex optimization problem and can be solved directly by using commercial software. However, since the number of inequality constraints in the SCOPF increases rapidly with the increase of the system scale, directly solving the SCOPF with a large number of constraints may consume much time and even encounter a memory overflow problem, and it is difficult to meet the real-time scheduling requirement which is more and more strict. In practical applications, one often uses an iterative approach by continually solving for simplified SCOPF and security analysis until an optimal solution is found. Generally, the iterative method is better than the direct method, but the iterative method still has the defects of low stability, severe performance reduction in the case of system blockage and the like.
The rapid development of artificial intelligence technology makes it a practical tool for solving many operational and planning problems in power systems. It has become a popular research direction to efficiently solve the optimal power flow problem by applying artificial intelligence technology, wherein the method based on machine learning is the most common.
Aiming at solving the problem of the optimal power flow class rapidly, scholars mainly provide two types of methods of model driving and data driving.
For the model-driven class method, the main idea is to screen out the redundant inequality constraints in the model by exploring the analytical characteristics of the SCOPF model. Previous studies have shown that most of the inequality constraints in SCOPF are redundant, and removing them from the model does not affect the final solution. Therefore, the method of finding out the redundancy constraint by using the branch breaking distribution factor, the Benders decomposition and the like and then solving the simplified SCOPF model is an effective solution. The difficulty of screening redundant constraints by model-driven methods is often similar to the direct solution problem, and it is difficult to accurately find the active constraints in the model before the solution is completed.
The data driving method does not depend on a complex SCOPF model, can meet the requirement of real-time solution, and is a method widely applied at present. In particular, researchers often use machine learning to directly predict the optimal solution to the problem or predict the set of active constraints before solving the simplified model. For the direct prediction optimal solution, the historical data of the system state and the scheduling result are generally used for machine learning, the internal relation is learned, and then the direct prediction result is used for replacing the solution model. This method is fast in computation but requires subsequent processing measures to deal with the infeasible, non-optimal problem of the results. For predicting the active constraint set in the optimal power flow problem, firstly, the mapping between the system working condition and the inequality constraint is learned, then, the scale of the model is reduced by predicting the active part in the inequality constraint, and finally, the solution of the highly simplified model can be realized in a short time. The method has high requirement on the accuracy of the prediction result, and once the predicted active constraint set is incorrect, a long time is additionally needed for making up the error.
And predicting the active condition of inequality constraints in the optimal power flow problem through machine learning, thereby simplifying and accelerating the calculation process. The specific machine learning method mainly includes statistical learning, deep learning and the like. In deep learning, sample data is analyzed through an Artificial Neural Network (ANN), the internal relation between the system condition and an active constraint set is discovered, the active state of the constraint can be rapidly predicted under the condition of ensuring high accuracy, and the application of the active state in the SCOPF rapid solving field is less and incomplete.
Disclosure of Invention
The present application is directed to solving, at least to some extent, one of the technical problems in the related art.
Therefore, an object of the present application is to provide a deep learning-based fast SCOPF calculation method, which simplifies a solution model by predicting an active constraint set through deep learning, and combines a security detection part in a conventional iterative method to ensure optimality of obtained results.
Another object of the present application is to provide a fast SCOPF calculation apparatus based on deep learning.
In order to achieve the above object, an embodiment of an aspect of the present application provides a method for fast calculating an SCOPF based on deep learning, including the following steps:
acquiring current operation data of the power system;
preprocessing the current operation data to obtain processed operation data; and
predicting all active constraints by utilizing a pre-trained DNN (Deep Neural Network) model based on the processed operation data, screening by using a threshold function to obtain an active constraint set, and learning to perform SCOPF (sparse Neural Network) real-time calculation based on the active constraint set.
In order to achieve the above object, another embodiment of the present application provides a deep learning-based SCOPF fast calculation apparatus, including:
the acquisition module is used for acquiring the current operation data of the power system;
the processing module is used for preprocessing the current operation data to obtain processed operation data; and
and the calculation module is used for predicting all active constraints by utilizing a pre-trained DNN model based on the processed running data, screening by using a threshold function to obtain an active constraint set, and learning to perform SCOPF real-time calculation based on the active constraint set.
The SCOPF rapid calculation method and device based on deep learning in the embodiment of the application have the following beneficial effects:
1) an active possibility function for measuring the active degree of the inequality constraint is designed, more information can be provided, and the deep learning model has better interpretability.
2) The deep learning model can accurately predict the active possibility function and further predict the active constraint set, and the SCOPF solving method based on the deep learning model is high in calculation speed.
Additional aspects and advantages of the present application will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the present application.
Drawings
The foregoing and/or additional aspects and advantages of the present application will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 is a flowchart of a method for fast SCOPF computation based on deep learning according to an embodiment of the present application;
FIG. 2 is a block diagram of a method for fast computation of SCOPF for deep learning according to an embodiment of the present application;
FIG. 3 is a schematic structural diagram of a DNN model according to one embodiment of the present application;
fig. 4 is a schematic structural diagram of a deep learning-based SCOPF fast computing device according to an embodiment of the present application.
Detailed Description
Reference will now be made in detail to embodiments of the present application, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the drawings are exemplary and intended to be used for explaining the present application and should not be construed as limiting the present application.
The main problem of the related technology is to directly set whether inequality constraint is active or not as the result of training and learning, and to regard the task of searching for the active constraint set as a classification problem. For the large-scale SCOPF problem, a large error is easy to occur in the classification result by only adopting a simple classification method, so that the calculation result is not optimal or extra time is required to take remedial measures.
In addition, in the prediction result of the simple classification method, only two possibilities of whether the constraints are active or not are displayed, so that the interpretability is poor, and the dispatcher has difficulty in understanding the reasons of whether the constraints are active or not. Only the classification result is observed, and some useful information in the operation of the power grid is easy to ignore, for example, how far a certain inactive constraint is away from the active state in the current system state, so that people are difficult to judge the safety condition of the current system and the capability of coping with risks.
For the existing deep learning methods, the methods are not necessarily suitable for SCOPF considering solving high uncertainty and a large number of emergency situations, and the influence of system state change on prediction accuracy and calculation efficiency is rarely researched.
The active constraint set prediction method based on deep learning provided by the application designs a proper training label, converts the predicted active constraint into a regression problem firstly, and screens the active constraint through threshold parameters, so that the two problems can be effectively solved.
The following describes a method and an apparatus for fast computing SCOPF based on deep learning according to an embodiment of the present application with reference to the drawings.
First, a proposed deep learning-based SCOPF fast calculation method according to an embodiment of the present application will be described with reference to the drawings.
Fig. 1 is a flowchart of a method for fast computing SCOPF based on deep learning according to an embodiment of the present application.
As shown in fig. 1, the method for rapidly calculating the SCOPF based on the deep learning includes the following steps:
in step S1, current operation data of the power system is acquired.
In step S2, the current operation data is preprocessed to obtain processed operation data.
In step S3, based on the processed operating data, all active constraints are predicted by using a pre-trained DNN model, an active constraint set is obtained by screening through a threshold function, and SCOPF real-time calculation is performed based on active constraint set learning.
Further, in an embodiment of the present application, the pre-trained DNN model includes: acquiring historical data of the operation of the power system; while processing historical data to obtain a training set, carrying out Monte Carlo sampling on the historical data, and processing to obtain a simulation sample; and training the deep neural network by using historical data and simulation samples to obtain a DNN model for predicting active constraints.
Further, in an embodiment of the present application, the DNN model includes: calculating the activity degree of the capacity constraint of the transmission line in the fault state by using an activity possibility function, wherein the activity possibility function is as follows:
wherein, f'maxFor capacity constraints of the transmission line under fault, Pi,cRepresenting the current of the i-th transmission line in the c-th fault, PBIs the active power reference value, λi,cDual multiplier representing capacity constraint of i-th transmission line under C-th fault, C (g) is optimal total generation cost, 1TG denotes the total output of the generator.
To learn the mapping from the system state to the activity likelihood function using DNN, historical data of the power system operation is required as raw data for training and some processing is performed. Meanwhile, in order to obtain sufficient samples to train the neural network, Monte Carlo sampling is carried out on the basis of historical data to generate simulation samples, and the same preprocessing measures are taken.
And converting the system state into a standard form (such as node load and renewable energy power generation into a per unit value) as an input characteristic of the deep neural network.
In order to solve the problems of low prediction accuracy, low interpretability and the like in the related art, the embodiment of the application provides an activity probability function for quantitatively describing the activity condition of inequality constraints in an SCOPF model. The method can reflect the information of how far the inactive constraint is away from the active constraint, the compactness degree of the active constraint and the like, is helpful for the staff to understand the safety of the system in the current state, and makes it possible to predict the active constraint never appearing in the historical data. And (3) using the activity possibility function instead of the restraint activity or not as a training label of the neural network, and predicting an activity restraint set by using the threshold parameter after finishing training and predicting the activity possibility function in the new system state. The threshold parameters are set, so that the influence of different types of prediction errors on the SCOPF solving speed and the like can be researched, the influence of the prediction errors can be reduced by selecting the proper threshold parameters, and the average calculation time of the SCOPF can be shortened. After the active constraint set is obtained, the simplified model can be solved to obtain an optimal solution. Because the error of the neural network is small, reasonable threshold parameters are set, the solution of the simplified model is optimal under most conditions, and the solution model is corrected only with small cost under few conditions, so that the optimal solution is obtained.
As shown in fig. 2, the main idea of the embodiment of the present application is shown, all active constraints are predicted by DNN before the optimization problem is solved for the first time, and then the purpose of increasing the computation speed is achieved by solving the simplified SCOPF model in a small number of iterations.
To quantitatively describe and analyze how active a transmission line capacity constraint is in a fault condition, an activity probability function is proposed. It will show some detailed information of these constraints in the optimal power flow, rather than just whether the constraints are active or not. The active likelihood function that defines the capacity constraint of the ith transmission line at fault type c is:
wherein, f'maxFor capacity constraints of the transmission line under fault, Pi,cShowing the power flow of the ith transmission line under the c-th fault. PBIs the active power reference value and typically takes 100 MW. Lambda [ alpha ]i,cA dual multiplier representing the capacity constraint of the ith transmission line at fault c. C (g) is the optimal total cost of power generation and 1TG denotes the total output of the generator. After we obtain the historical result of SCOPF solution, Fi,cCan be quickly calculated.
If a constraint is inactive (P)i,c<f’max) Its activity probability is the inverse of the difference between the relevant branch capacity constraint and the power flow. If a constraint is active (P)i,c＝f’max)，Its activity probability is the value of its dual multiplier divided by the average cost of power generation by the engine.
For inactive constraints, Fi,cIs a negative value; while for active constraints, Fi,cIt is a positive value. Fi,cAn increase in (b) means that the associated constraint has a greater probability of being active in the SCOPF model, and thus its Fi,cReferred to as the activity probability function. The activity likelihood function helps the system operator to understand the state of the transmission line in the event of a fault. In addition, setting the activity likelihood function as an output label of supervised learning enables the deep learning method to predict activity constraints which never appear in the historical data.
To reduce the computational burden of solving SCOPF, DNN is designed to learn the direct intrinsic relationship between different system states and their corresponding active constraints, and the training of DNN models can be done off-line from historical data. After the offline training is sufficient, the active constraint set can be predicted on line by using a deep learning model, and then SCOPF real-time calculation is realized. Specifically, the active constraint set A is predicted1Then, use A1Instead of considering the inequality constraint set of all emergencies, the SCOPF model needing to be solved is greatly simplified. After the solution is completed, safety detection (N-1 analysis is carried out as a specific measure because N-1 transmission line faults are selected as fault sets) is carried out to judge feasibility. If constraint violations occur that result in an infeasible problem, these violation constraints are added to the SCOPF model and solved again. By continually solving the SCOPF model and security check until no out-of-limit constraints are present, an optimal solution to the problem is obtained. In fact, due to the high accuracy of the DNN prediction results, iteration is not needed in most cases, and when iteration is needed due to prediction errors in a very small number, the number of iterations and the number of added constraints are small. Therefore, the method can realize fast solving of SCOPF.
Because the feedforward neural network can approximate the continuous mapping with high accuracy, the embodiment of the application adopts a fully-connected feedforward neural network to learn the relation between the system state and the activity possibility function, and then the output result of the neural network is divided into two types to predict the activity constraint set.
Since the size of the transmission line capacity constraints grows as a quadratic function with the number of transmission lines, the DNN model will be quite large if the active likelihood function of all constraints is set as an output tag. Such a large scale DNN makes its training difficult and increases the time it takes to make online predictions, which in turn leads to a reduction in computational efficiency of the method. Therefore, in order to simplify the structure of the DNN model, only the key constraint that the history data has been active or close to active (i.e. the history activity probability is greater than a certain value, the scheme of the embodiment of the present application takes-0.01) is selected as the DNN training label.
The specific structure of the DNN model is shown in fig. 3. Consider a K-layer (containing input and output layers) feed-forward neural network whose model can be expressed as:
where g is the activation function, W is the weight matrix, x is the input vector, and b represents the offset value.
To avoid the situation of gradient disappearance during training, we choose a Linear rectification function (ReLU) as the activation function of each hidden layer, which can be expressed as:
g(x)＝max(x,0)
according to the prior art, in order to ensure that the DNN can accurately fit the relationship between the features and the labels, the number of neurons in the hidden layer must be greater than the number of dimensions of the input features (i.e., the number of types of system state parameters in the present embodiment). On the basis, a DNN model with high prediction precision and high training and prediction speed is obtained by adjusting the number of hidden layers and the number of neurons of the neural network.
After obtaining the prediction result of the active likelihood function, we classify them into two categories by threshold parameters:
wherein A isi,cIndicating whether the corresponding constraint is active or not,
is the activity probability function Fi,cIs a threshold parameter.
If A isi,c1, the corresponding constraint is predicted to be active in SCOPF; otherwise, the corresponding constraint is predicted to be inactive. For the threshold parameter, we can simply set it to 0, and can also take it as a positive number to avoid excessively predicting the inactive constraint as active, or take it as a negative number to reduce the situation of predicting the active constraint missing. By researching the influence of different threshold parameters on the SCOPF online calculation efficiency, a proper value of the SCOPF online calculation efficiency can be found, and the influence of a prediction error on the calculation efficiency is reduced.
Since the DNN model is used for the multi-label regression problem, Mean Square Error (MSE) can be used as a suitable loss function to guide neural network training:
wherein
And yjRespectively representing the corresponding activity likelihood functions Fi,cM is the number of critical constraints.
The neural network model is trained by continuously adjusting the weight matrix W and the bias value b with the goal of minimizing the loss function MSE and converging it. Adaptive moment estimation (Adam) is selected as a learning algorithm of DNN, the learning rate can be adaptively changed, and the method is a convenient and efficient learning algorithm. 20% of the samples in the training process are set as the validation set, and the initial learning rate is set to 10-4The batch dataset size is set to 128. The net load of a node is a keyThe DNN input characteristic of (a) is obtained by taking a normal sample with a standard deviation of 10%. Furthermore, to explore the impact of power system blocking due to load growth on the proposed algorithm, the sample mean was also gradually increased from 1 times the default load to 2 times.
To use the simulated data for DNN training, in addition to performing monte carlo sampling, the corresponding SCOPF calculation needs to be done to obtain the activity likelihood function. Solving SCOPF using either the direct method or the traditional iterative method presents a problem of being time consuming or unstable, so both methods are only used when generating the training samples of the previous 1/4. These samples are then used to train out an early DNN and use it to solve for SCOPF to obtain training samples for the remainder 3/4. Even the early DNN which is not fully trained has higher prediction accuracy, and DNN training samples can be acquired at a speed higher than that of the direct method and the traditional iterative method.
According to the SCOPF fast calculation method based on deep learning, the active state of transmission line capacity constraint in SCOPF can be quantitatively described through an active possibility function, more information is provided for staff to evaluate system safety, and a DNN model with the information as a training label and a prediction result has better interpretability. The DNN model designed based on the active likelihood function can predict the active constraint set of the SCOPF with high accuracy, and the influence of the prediction error is reduced by the reasonable selection of the threshold parameter, whereby the SCOPF can be solved efficiently. Simulation results show that the SCOPF calculation method based on deep learning in the embodiment of the application is obviously faster than a direct method and a traditional iteration method, is less affected by load conditions, and has great advantages when a power system is blocked.
Next, a deep learning-based SCOPF fast calculation apparatus proposed according to an embodiment of the present application is described with reference to the drawings.
Fig. 4 is a schematic structural diagram of a deep learning-based SCOPF fast computing device according to an embodiment of the present application.
As shown in fig. 4, the apparatus for fast computing SCOPF based on deep learning includes: an acquisition module 100, a processing module 200 and a calculation module 300.
The obtaining module 100 is configured to obtain current operation data of the power system. The processing module 200 is configured to pre-process the current operation data to obtain processed operation data. And the calculation module 300 is configured to predict all active constraints based on the processed operating data by using a pre-trained DNN model, obtain an active constraint set by screening through a threshold function, and perform SCOPF real-time calculation based on active constraint set learning.
Further, in an embodiment of the present application, the SCOPF fast calculation apparatus further includes:
the training module is used for acquiring historical data of the operation of the power system, carrying out Monte Carlo sampling on the historical data while processing the historical data to obtain a training set, processing the training set to obtain a simulation sample, and training a deep neural network by using the historical data and the simulation sample to obtain a DNN model for predicting active constraints.
Further, in one embodiment of the present application, training a deep neural network using historical data and simulated samples includes:
the mean square error is used as a loss function to guide deep neural network training, wherein the calculation formula of the loss function is as follows:
wherein the content of the first and second substances,
and yjRespectively representing the corresponding activity likelihood functions Fi,cM is the number of critical constraints.
Further, in one embodiment of the present application, the DNN model includes: calculating the activity degree of the capacity constraint of the transmission line in the fault state by using an activity possibility function, wherein the activity possibility function is as follows:
wherein, f'maxFor capacity constraints of the transmission line under fault, Pi,cRepresenting the current of the i-th transmission line in the c-th fault, PBIs the active power reference value, λi,cDual multiplier representing capacity constraint of i-th transmission line under C-th fault, C (g) is optimal total generation cost, 1TG denotes the total output of the generator.
Further, in one embodiment of the present application, the DNN model is:
wherein g is an activation function, W is a weight matrix, x is an input vector, and b is an offset value.
It should be noted that the foregoing explanation of the method embodiment is also applicable to the apparatus of this embodiment, and is not repeated herein.
According to the SCOPF rapid computing device based on deep learning provided by the embodiment of the application, the active state of transmission line capacity constraint in SCOPF can be quantitatively described through the active possibility function, more information is provided for staff to evaluate the system safety, and the DNN model with the information as the training label and the prediction result has better interpretability. The DNN model designed based on the active likelihood function can predict the active constraint set of the SCOPF with high accuracy, and the influence of the prediction error is reduced by the reasonable selection of the threshold parameter, whereby the SCOPF can be solved efficiently. Simulation results show that the SCOPF calculation method based on deep learning in the embodiment of the application is obviously faster than a direct method and a traditional iteration method, is less affected by load conditions, and has great advantages when a power system is blocked.
Furthermore, the terms "first", "second" and "first" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one such feature. In the description of the present application, "plurality" means at least two, e.g., two, three, etc., unless specifically limited otherwise.
In the description herein, reference to the description of the term "one embodiment," "some embodiments," "an example," "a specific example," or "some examples," etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the application. In this specification, the schematic representations of the terms used above are not necessarily intended to refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples. Furthermore, various embodiments or examples and features of different embodiments or examples described in this specification can be combined and combined by one skilled in the art without contradiction.
Although embodiments of the present application have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present application, and that variations, modifications, substitutions and alterations may be made to the above embodiments by those of ordinary skill in the art within the scope of the present application.
Claims (10)
1. A SCOPF fast calculation method based on deep learning is characterized by comprising the following steps:
acquiring current operation data of the power system;
preprocessing the current operation data to obtain processed operation data; and
and predicting all active constraints by using a pre-trained DNN model based on the processed running data, screening by using a threshold function to obtain an active constraint set, and learning to perform SCOPF real-time calculation based on the active constraint set.
2. The method of claim 1, further comprising:
acquiring historical data of the operation of the power system;
while processing the historical data to obtain a training set, carrying out Monte Carlo sampling on the historical data, and processing to obtain a simulation sample;
and training a deep neural network by using the historical data and the simulation sample to obtain a DNN model for predicting active constraints.
3. The method of claim 2, wherein training a deep neural network using the historical data and the simulated samples comprises:
utilizing a mean square error as a loss function to guide the deep neural network training, wherein the calculation formula of the loss function is as follows:
wherein the content of the first and second substances,
and yjRespectively representing the corresponding activity likelihood functions Fi,cM is the number of critical constraints.
4. The method of claim 2, wherein the DNN model comprises: calculating an activity level of a transmission line capacity constraint in a fault state using an activity likelihood function, wherein the activity likelihood function is:
wherein, f'maxFor capacity constraints of the transmission line under fault, Pi,cRepresenting the current of the i-th transmission line in the c-th fault, PBIs the active power reference value, λi,cA dual multiplier representing the capacity constraint of the ith transmission line at fault type c,c (g) is the optimal total cost of power generation, 1TG denotes the total output of the generator.
5. The method according to any of claims 1-4, wherein the DNN model is:
wherein g is an activation function, W is a weight matrix, x is an input vector, and b is an offset value.
6. A SCOPF fast computing device based on deep learning is characterized by comprising:
the acquisition module is used for acquiring the current operation data of the power system;
the processing module is used for preprocessing the current operation data to obtain processed operation data; and
and the calculation module is used for predicting all active constraints by utilizing a pre-trained DNN model based on the processed running data, screening by using a threshold function to obtain an active constraint set, and learning to perform SCOPF real-time calculation based on the active constraint set.
7. The apparatus of claim 6, further comprising:
the training module is used for acquiring historical data of the operation of the power system, carrying out Monte Carlo sampling on the historical data while processing the historical data to obtain a training set, processing the training set to obtain a simulation sample, and training a deep neural network by using the historical data and the simulation sample to obtain a DNN model for predicting active constraints.
8. The apparatus of claim 7, wherein the training of the deep neural network using the historical data and the simulated samples comprises:
utilizing a mean square error as a loss function to guide the deep neural network training, wherein the calculation formula of the loss function is as follows:
wherein the content of the first and second substances,
and yjRespectively representing the corresponding activity likelihood functions Fi,cM is the number of critical constraints.
9. The apparatus of claim 7, wherein the DNN model comprises: calculating an activity level of a transmission line capacity constraint in a fault state using an activity likelihood function, wherein the activity likelihood function is:
wherein, f'maxFor capacity constraints of the transmission line under fault, Pi,cRepresenting the current of the i-th transmission line in the c-th fault, PBIs the active power reference value, λi,cDual multiplier representing capacity constraint of i-th transmission line under C-th fault, C (g) is optimal total generation cost, 1TG denotes the total output of the generator.
10. The apparatus of claims 6-9, wherein the DNN model is:
wherein g is an activation function, W is a weight matrix, x is an input vector, and b is an offset value.
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