Multi-component gas identification method and monitoring system for supporting vector regression

Abstract

The invention discloses a multi-component gas identification method supporting vector regression and a monitoring system, and relates to the field of monitoring of multi-component gas dissolved in oil of an oil immersed transformer. The method comprises the following steps: (1) acquiring parameter data of gas dissolved in oil of the oil-immersed transformer, and establishing a data set; the data set comprises a plurality of feature vectors {(x1, y1), (x2, y2),..., (xn, yn)}, xn is a feature vector, and yn is a corresponding target value; (2) training to obtain a function f (x) according to a feature vector and a target value in the data set, and predicting a target value of an unknown sample through the function; (3) obtaining a weight vector omega and a bias item b in the function f (x) by solving an optimization problem; and (4) determining a final function f (x) according to the obtained weight vector omega and the bias item b, and identifying the dissolved gas in the oil of the oil-immersed transformer according to the function. According to the invention, the specific concentration of each gas component in the mixed gas can be accurately identified, and the monitoring precision of the operation state of the transformer can be improved.

Description

Technical Field

[0001] The invention relates to the field of monitoring multi-component gas dissolved in oil-immersed transformer oil, in particular to a multi-component gas identification method and a monitoring system supporting vector regression. Background Art

[0002] In oil-immersed transformers, the concentration of dissolved gases in the oil (such as hydrogen, methane, ethane, ethylene, acetylene, carbon monoxide and carbon dioxide) and their rate of change are important indicators for monitoring the operating status of the transformer and determining the type of fault. However, gas sensors often have the problem of cross-sensitivity, that is, the concentration change of non-target gases may affect the output of multiple sensors, resulting in the inability to accurately identify the concentration of specific gases. Therefore, in order to accurately determine the concentration change of each characteristic fault gas, the mixed gas must be effectively identified. At present, some use neural network algorithms, but there are problems such as high model complexity and large number of required data samples. Summary of the invention

[0003] Based on the above technical problems, the present invention proposes a multi-component gas identification method and a monitoring system based on support vector regression.

[0004] The technical solution adopted by the present invention is:

[0005] A multi-component gas identification method based on support vector regression comprises the following steps:

[0006] (1) Obtain parameter data of dissolved gas in oil of oil-immersed transformers and establish a data set; the data set contains multiple feature vectors {(x1,y1),(x2,y2),...,(x n ,y n )}, where x n is the feature vector, y n is the corresponding target value;

[0007] (2) Based on the feature vectors and target values ​​in the data set, a function f(x) is trained to predict the target value of an unknown sample. The function is represented by the following linear regression model:

[0008] f(x)=ωx+b (1)

[0009] Among them, ω is the weight vector; b is the bias term;

[0010] (3) The weight vector ω and the bias term b are obtained by solving the following optimization problem:

[0011]

[0012] Satisfy the following constraints

[0013]

[0014] Where C is the penalty parameter, ξ i , is the slack variable;

[0015] (4) According to the weight vector ω and the bias term b obtained in step (3), the final function f(x) is determined, and the identification of dissolved gas in the oil of the oil-immersed transformer is realized based on the function.

[0016] The present invention also provides a multi-component gas monitoring system supporting vector regression, the system comprising:

[0017] Gas detection module, used for real-time detection of gas concentration dissolved in oil of oil-immersed transformer;

[0018] The support vector regression module is connected to the gas detection module to obtain the gas concentration value in the gas detection module and establish a data set. According to the feature vector and target value in the data set, the function f(x) is trained to predict the target value of the unknown sample. The function is represented by the following linear regression model:

[0019] f(x)=ωx+b (1)

[0020] Among them, ω is the weight vector; b is the bias term.

[0021] The beneficial technical effects of the present invention are as follows:

[0022] The present invention shows significant advantages in the field of mixed gas identification. Compared with neural network algorithms, support vector regression (SVR) can achieve an ideal balance between model complexity and learning error, and can effectively handle nonlinear problems even when there are fewer data samples. By selecting an appropriate kernel function, SVR further has the ability to process high-dimensional data, enabling it to optimize between regression accuracy and training efficiency, and can accurately identify the specific concentration of each gas component in the mixed gas. This capability is particularly suitable for online monitoring of dissolved gases in oil-immersed transformers, which helps to improve the monitoring accuracy of the transformer operating status and the reliability of fault warning. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 It is a schematic diagram of the process of the multi-component gas identification method of the present invention supporting vector regression;

[0024] Figure 2 This is a flow chart of the genetic algorithm optimization process used in the present invention;

[0025] Figure 3 It is a schematic diagram of two classifications in a specific application example of the present invention;

[0026] Figure 4 It is a distribution diagram of the detected concentration and the actual concentration of hydrogen in a specific application example of the present invention;

[0027] Figure 5 This is a distribution diagram of acetylene detection concentration and actual concentration in a specific application example of the present invention. DETAILED DESCRIPTION

[0028] In order to solve the problem that mixed characteristic gases in the online DGA system of oil-immersed transformers have cross-interference with sensors, the present invention proposes a multi-component gas identification method and monitoring system based on support vector regression. Compared with the neural network algorithm, the support vector regression method adopted by the present invention can find a balance between model complexity and learning error, and can effectively handle nonlinear problems with only a small number of samples. In addition, the selection of kernel function gives SVR the ability to process high-dimensional data, so that it can make a trade-off between regression accuracy and training efficiency according to specific problems. Therefore, the multi-component gas identification method and monitoring system based on support vector regression can better complete the accurate identification of mixed characteristic fault gases.

[0029] like Figure 1 As shown, a multi-component gas identification method supporting vector regression includes the following steps:

[0030] (1) Obtain parameter data of dissolved gas in oil of oil-immersed transformers and establish a data set; the data set contains multiple feature vectors {(x1,y1),(x2,y2),...,(x n ,y n )}, where x n is the feature vector, y n is the corresponding target value.

[0031] (2) Based on the feature vectors and target values ​​in the data set, a function f(x) is trained to predict the target value of an unknown sample. The function is represented by the following linear regression model:

[0032] f(x)=ωx+b (1)

[0033] Among them, ω is the weight vector; b is the bias term.

[0034] (3) The weight vector ω and the bias term b are obtained by solving the following optimization problem:

[0035]

[0036] Satisfy the following constraints

[0037]

[0038] Where C is the penalty parameter, ξ i , is the slack variable.

[0039] ε is the characteristic parameter of the insensitive loss function. When the error is less than ε, the error can be ignored.

[0040] (4) According to the weight vector ω and the bias term b obtained in step (3), the final function f(x) is determined, and the identification of dissolved gas in the oil of the oil-immersed transformer is realized based on the function.

[0041] As a further design of the present invention, the method also introduces a kernel function K(x i -x j ), to handle nonlinear regression problems, the kernel function is in the form of:

[0042]

[0043] in, α i —Lagrange multiplier; K(x i -x j )—kernel function; b—bias term.

[0044] The above kernel function K(x i -x j ) is one of the following:

[0045] Linear kernel function K(x i ,x j )=x i ·x j ; Polynomial kernel function K(x i ,x j )=[γ(x i ·x j )+c] d ; Gaussian radial basis kernel function

[0046] Sigmoid kernel function K(x i ,x j ) = tanh[α(x i ·x j )+c].

[0047] Furthermore, the penalty parameter C and the kernel function parameter g are optimized by genetic algorithm, that is, by cross-validation during model training.

[0048] like Figure 2As shown, the genetic algorithm optimization process includes the following steps: configuring parameters, initializing the population, analyzing k-fold cross validation, separating the data set, training the model using the current chromosome parameters, model validation, calculating the accuracy, and then analyzing whether the k-fold validation is completed. If not, return to training the model using the current chromosome parameters; if yes, calculate the fitness value. Then determine whether it converges. If not, perform genetic operations, select the next generation of chromosomes, crossover, mutation, and then stratify k-fold cross validation. If yes, output the optimal parameters and the optimization process ends.

[0049] The present invention also provides a multi-component gas monitoring system supporting vector regression for monitoring mixed characteristic fault gas dissolved in oil-immersed transformer oil, the system comprising:

[0050] The gas detection module is used to detect the concentration of gas dissolved in the oil of the oil-immersed transformer in real time.

[0051] The support vector regression module is connected to the gas detection module to obtain the gas concentration value in the gas detection module and establish a data set. According to the feature vector and target value in the data set, the function f(x) is trained to predict the target value of the unknown sample. The function is represented by the following linear regression model:

[0052] f(x)=ωx+b (1)

[0053] Among them, ω is the weight vector; b is the bias term.

[0054] The system also includes:

[0055] An optimization module is used to obtain the weight vector ω and the bias term b by solving the following optimization problem, which includes a penalty parameter C and a slack variable ξ i ,

[0056]

[0057] Satisfy the following constraints

[0058]

[0059] Specifically, the gas detection module adopts one or more of a hydrogen detection module, an acetylene detection module, a methane detection module, an ethane detection module, an ethylene detection module, a carbon monoxide detection module and a carbon dioxide detection module.

[0060] For example, the hydrogen detection module is used to detect the concentration of hydrogen dissolved in oil in real time, and analyze and predict the hydrogen concentration data through the support vector regression algorithm. The acetylene detection module is used to detect the concentration of acetylene dissolved in oil in real time, and analyze and predict the acetylene concentration data using the support vector regression algorithm, or even perform fault warnings, etc.

[0061] Each of the above detection modules is also connected to the support vector regression module. For example, the hydrogen detection module and the acetylene detection module are both connected to the support vector regression module, and the concentration data of the dissolved gas in the oil are obtained and processed in real time to ensure accurate monitoring and fault diagnosis of the operating state of the oil-immersed transformer. The present invention can dynamically adjust the detection parameters according to the characteristics of different gases to improve the monitoring accuracy and response speed, and ensure the safe operation of the transformer.

[0062] Furthermore, the system also includes an alarm module, which is connected to the support vector regression module.

[0063] Furthermore, the kernel function K(x i -x j ), to handle nonlinear regression problems, the kernel function is in the form of:

[0064]

[0065] in, α i —Lagrange multiplier; K(x i -x j )—kernel function; b—bias term.

[0066] Furthermore, the kernel function K(x i -x j ) is one of the following:

[0067] Linear kernel function K(x i ,x j )=x i ·x j ; Polynomial kernel function K(x i ,x j )=[γ(x i ·x j )+c] d ; Gaussian radial basis kernel function

[0068] Sigmoid kernel function K(x i ,x j ) = tanh[α(x i ·x j )+c].

[0069] The present invention can solve the problem of sensor cross-interference caused by mixed characteristic fault gases in the online DGA system of oil-immersed transformers, thereby realizing accurate measurement of the concentration of target characteristic fault gases. By using the SVR algorithm, the present invention can effectively reduce cross-interference, improve the accuracy and reliability of fault gas identification, and provide strong support for the condition monitoring and fault diagnosis of oil-immersed transformers.

[0070] The present invention will be further described below from the principle aspect.

[0071] In order to solve the problem of cross-interference of mixed gases dissolved in the oil of oil-immersed transformers on sensors, the present invention provides a multi-component gas identification method and monitoring system based on support vector regression (SVR), which can compensate for the cross-interference caused by mixed gases on multiple sensors, thereby achieving accurate measurement of target characteristic fault gas. Support vector regression adopts the principle of structural risk minimization in statistical learning theory, and attempts to find a linear function or nonlinear function that achieves the best balance between maintaining model complexity and training sample fit, thereby improving the accuracy and generalization ability of prediction.

[0072] Suppose there is a data set D = {(x1,y1),(x2,y2),...,(x n ,y n )}, where x i is the feature vector, y i In the present invention, the eigenvector is the response of the gas sensor, and the target value is the actual gas concentration. The goal of applying the SVR algorithm is to find a function f(x) to predict the target value of the unknown sample, i.e., the gas concentration, so that the function output fits the sample point as much as possible, thereby obtaining a result that the output value is close to the true value.

[0073] When SVR solves a linear regression problem, the objective function is in the form of formula (1).

[0074] f(x)=ωx+b (1)

[0075] In formula (1): ω is the weight vector; b is the bias term.

[0076] The core idea of ​​the SVR algorithm is different from the traditional machine learning regression model, which directly minimizes the error between the predicted value and the true value. Instead, it introduces the concept of elastic error band, which allows an error range between the predicted value of f(x) and the actual target value y, that is, loss calculation and penalty are performed only when the absolute value of the difference between the predicted value and the true value exceeds a predefined threshold ε. Therefore, the optimization problem of SVR can be expressed as shown in formula (2), and the constraints of formula (3) need to be satisfied.

[0077]

[0078] In formula (2) and formula (3): C is the penalty parameter; ξ i , —Slack variables.

[0079] It can be observed from formula (2) that the objective function in the SVR optimization problem expression consists of two parts. The right part focuses on the constraint on the model prediction error, and obtains the fitting curve by iterating the algorithm parameters to reduce the prediction error; the left part contains the regularization term of the penalty parameter C. The introduction of this regularization term can limit the amplitude of the model coefficient, thereby controlling the complexity of the model. Through the above operations, the SVR algorithm realizes the design concept of minimizing structural risk, so that while ensuring the prediction accuracy, it avoids overfitting by constraining the model complexity, especially in the regression analysis of small sample data. Therefore, the SVR algorithm not only pursues the reduction of prediction error, but also improves the generalization ability by controlling the model complexity, showing its effectiveness and robustness in dealing with limited data set problems.

[0080] When solving nonlinear regression problems, SVR generally introduces Lagrange multipliers to construct a Lagrange function for each constraint condition to simplify the dual problem operation, and introduces kernel functions so that the algorithm can implicitly process in high-dimensional feature space by calculating the inner product of vector pairs in the original feature space. At this time, the final solution of SVR can be obtained as shown in formula (4).

[0081]

[0082] In formula (4): α i —Lagrange multiplier; K(x i -x j )—kernel function; b—bias term.

[0083] After converting to the dual problem, the optimization of SVR turns into finding a set of optimal Lagrange multipliers corresponding to each sample in the training data set, and the kernel function defines the similarity of samples in the feature space in this process. Commonly used kernel functions are shown in Table 1. The final SVR model can be expressed as the weighted sum of all support vectors, that is, the samples in the training set whose Lagrange multipliers are not zero, and the weight is the difference between the Lagrange multipliers obtained during the optimization process. The predicted output of the model is determined by the calculation of the kernel function between the new input data point and all the support vectors in the training set. Therefore, the SVR model can construct complex decision boundaries with a small number of support vectors, making it very effective in processing high-dimensional data and small sample data.

[0084] Table 1

[0085]

[0086] After determining the type of kernel function, it is also necessary to optimize the penalty parameter C and kernel function parameter g in the main parameters of SVR.

[0087] The penalty parameter C plays a role in balancing misclassification and decision surface complexity during the model training process. When the C value is smaller, it means that the model can tolerate more misclassifications, that is, it allows more data points to fall on the wrong side of the decision boundary classification. Relatively speaking, the decision boundary is smoother, thereby enhancing the generalization ability of the model. However, when it is reduced to a certain extent, it may cause underfitting. The larger the C value, the higher the classification accuracy of the model on the training set, but this overly strict classification may make the model too complex, making the decision boundary tortuous and overfitting, and ultimately causing the model to perform well on the training data but reduce the prediction accuracy on the test set.

[0088] For the kernel function parameter g, its representative value is different on different kernel functions. In the kernel function listed in Table 1, the parameter g in the polynomial kernel represents the scaling parameter γ, the dimension parameter d, and the bias parameter c, where γ and d control the influence range of the data features and determine the complexity of the model, and c introduces the bias term to adjust the position of the decision boundary and can change the nonlinear characteristics of the model. The parameter g in the RBF kernel represents the σ value, which controls the width of the Gaussian kernel, corresponding to the rate at which the similarity between sample points decreases sharply with increasing distance. The larger the σ, the more tortuous the decision boundary is, and overfitting is prone to occur; the smaller the σ, the smoother the decision boundary is, resulting in the model being unable to capture the complex structure of the data and causing underfitting.

[0089] Therefore, choosing appropriate penalty parameters and kernel functions has an important impact on the prediction ability of the model. Commonly used methods for selecting the best penalty parameters and kernel function parameters include grid search, random search, Bayesian optimization, genetic algorithm, etc. Their advantages and disadvantages are shown in Table 2.

[0090] Table 2

[0091]

[0092] The present invention will be further described below in conjunction with specific application examples.

[0093] This embodiment uses H2, C2H2 and their mixed gases as examples to describe the multi-component gas identification method based on support vector regression. The data set is collected by a hydrogen sensor and an acetylene sensor. The mixed gas qualitative classification in this embodiment involves three cases, namely, single H2, single C2H2 and a mixed gas of H2 and C2H2. Assume that the data set is D, which contains n samples, and each sample i is represented by a feature vector (Hydrogen and acetylene sensor readings) a label y iComposition. i There are three labels corresponding to different categories: single H2, single C2H2 and mixed gas.

[0094] The classification problem to be solved in this embodiment is the classification of single hydrogen and single acetylene. Figure 3 The figure shows a schematic diagram of SVR binary classification of data generated by simulating the response of hydrogen and acetylene sensors in single hydrogen and single acetylene gases. In this simulated data, feature 1 represents the response value of the hydrogen sensor, that is, the class where class 1 belongs, and feature 2 represents the response value of the acetylene sensor, that is, the class where class 2 belongs. The two show a certain correlation, which represents cross-sensitivity. It can be seen that the decision boundary in the figure finds a relatively optimal nonlinear curve to separate the two classes. The calculation of the decision boundary is obtained by applying the Gaussian radial basis function (RBF) kernel function.

[0095] This embodiment uses genetic algorithms (GA) to optimize the key parameters c and g in the RBF kernel function used by SVR. The basic idea of ​​the genetic algorithm is to solve the optimization problem by simulating Darwin's natural selection and the genetic mechanism in genetics (such as crossover and mutation), which is applicable to the nonlinear problem of mixed gas identification that needs to be solved in the present invention. In the process of optimizing the training set data using the genetic algorithm, the 5-fold stratified cross-validation method is used to evaluate the model, and the performance of the model under different parameter settings is evaluated by cross-validation and other methods. The maximum number of evolutions of the genetic algorithm is set to 100, the maximum number of populations is set to 50, and the evolution termination goal is set to minimize the cross-validation loss or when the maximum number of evolutions is reached. The optimization process flow chart is as follows. Figure 2 shown.

[0096] The evaluation of model detection uses the mean relative error MRE as the main reference, supplemented by the mean absolute error MAE as a reference, and gives the relative error curve of the test set data. e The calculation formula of is shown in formula (5), and the calculation formula of absolute error ae is shown in formula (6).

[0097]

[0098] a e =c 测量值 -c 真实值 (6)

[0099] In the data set used for mixed gas concentration detection training, the number of samples of single H2, single C2H2 and mixed gas are 182 groups, 104 groups, 89 groups, and a total of 375 groups. The data is divided into training and test data by the five-fold cross validation method. The penalty parameter c and the kernel parameter g of the genetic algorithm optimization are set to the range of 0.1-100, the maximum evolution number parameter is set to 100, and the maximum population size is set to 50. GA optimization and training are performed on single H2, single C2H2 and mixed gas respectively. The training of mixed gas adopts the multi-input single-output SVR model to optimize and train with the actual hydrogen concentration and the actual acetylene concentration respectively, so its penalty parameter c and kernel parameter g have two items each. The optimization results of single H2 are c=99.87, g=0.21; the optimization results of single C2H2 are c=59.99, g=0.4; the optimization results of mixed gas are c=59.99, g=0.4; the optimization results of single C2H2 ...

[0100] Figure 4 The GA-SVR algorithm is used to detect the concentration of a single hydrogen gas. The average relative error MRE = 0.008, the average absolute error MAE = 0.2. It can be found from the figure that the detected hydrogen concentration has a high degree of overlap with the actual hydrogen concentration curve. The detection effect is slightly worse in the low concentration area, and the maximum relative error of the data point is 0.025. Figure 5 The results of the single acetylene concentration detection using the GA-SVR algorithm are shown in Figure 1. The average relative error MRE = 0.043 and the average absolute error MAE = 1.2. It can be seen from the figure that the detected acetylene concentration has a high degree of overlap with the true acetylene concentration curve. Only at the low-concentration area data point 1, the detection effect is poor, and the relative error of the data point is 0.44. The reason for the large relative error at this point is that the concentration value is low, so the relative error value is large, and its absolute error is 2.2; the average relative error of other data points except data point 1 is 0.023, and the errors are all within 10%, and the results are excellent.

[0101] This embodiment studies the quantitative detection method of mixed gas based on SVR, and detects the concentration of mixed gas. The GA genetic algorithm with better model detection accuracy is obtained by comparing the optimization method in the mixed gas qualitative classification method based on SVM to optimize the penalty parameters and kernel parameters of SVR. Then, the SVR is used for model training using the optimal parameters obtained, and the average relative errors of single H2 and single C2H2 are 0.8% and 4.3% respectively, the average relative errors of hydrogen and acetylene in the mixed gas are 5.6% and 13.7%, and the overall average relative error is 6.1%.

[0102] In summary, the present invention realizes the quantitative detection of mixed gases based on the SVR algorithm by adopting an efficient parameter optimization strategy and cross-validation method, provides an effective technical approach for mixed gas detection, and further improves the accuracy and practicability of gas detection.

Claims

1. A multi-component gas identification method based on support vector regression, characterized in that The following steps are involved: (1) Obtain parameter data of dissolved gas in oil of oil-immersed transformers and establish a data set; the data set contains multiple feature vectors {(x1,y1),(x2,y2),...,(x n ,y n )}, where x n is the feature vector, y n is the corresponding target value; (2) Based on the feature vectors and target values ​​in the data set, a function f(x) is trained to predict the target value of an unknown sample. The function is represented by the following linear regression model: f(x)=ωx+b (1) Among them, ω is the weight vector; b is the bias term; (3) The weight vector ω and the bias term b are obtained by solving the following optimization problem: Satisfy the following constraints Where C is the penalty parameter, ξ i , is the slack variable; (4) According to the weight vector ω and the bias term b obtained in step (3), the final function f(x) is determined, and the identification of dissolved gas in the oil of the oil-immersed transformer is realized based on the function.

2. The multi-component gas identification method based on support vector regression according to claim 1 is characterized in that: This method also introduces the kernel function K(x i -x j ), to handle nonlinear regression problems, the kernel function is in the form of: in, α i —Lagrange multiplier; K(x i -x j )—kernel function; b—bias term.

3. The multi-component gas identification method based on support vector regression according to claim 2 is characterized in that: The kernel function K(x i -x j ) is one of the following: Linear kernel function K(x i ,x j )=x i ·x j ; Polynomial kernel function K(x i ,x j )=[γ(x i ·x j )+c] d ; Gaussian radial basis kernel function Sigmoid kernel function K(x i ,x j ) = tanh[α(x i ·x j )+c].

4. The multi-component gas identification method based on support vector regression according to claim 3 is characterized in that: The penalty parameter C and kernel function parameter g are optimized by genetic algorithm.

5. A multi-component gas monitoring system supporting vector regression, characterized in that: The system includes: Gas detection module, used for real-time detection of gas concentration dissolved in oil of oil-immersed transformer; The support vector regression module is connected to the gas detection module to obtain the gas concentration value in the gas detection module and establish a data set. According to the feature vector and target value in the data set, the function f(x) is trained to predict the target value of the unknown sample. The function is represented by the following linear regression model: f(x)=ωx+b (1) Among them, ω is the weight vector; b is the bias term.

6. A multi-component gas monitoring system based on support vector regression according to claim 5, characterized in that: The system also includes: An optimization module is used to obtain the weight vector ω and the bias term b by solving the following optimization problem, which includes a penalty parameter C and a slack variable ξ i , Satisfy the following constraints 7. The multi-component gas monitoring system of support vector regression according to claim 5, characterized in that: The gas detection module adopts one or more of a hydrogen detection module, an acetylene detection module, a methane detection module, an ethane detection module, an ethylene detection module, a carbon monoxide detection module and a carbon dioxide detection module.

8. The multi-component gas monitoring system of support vector regression according to claim 5, characterized in that: The system also includes an alarm module, which is connected to the support vector regression module.

9. The multi-component gas monitoring system of support vector regression according to claim 6, characterized in that: The kernel function K(x i -x j ), to handle nonlinear regression problems, the kernel function is in the form of: in, α i —Lagrange multiplier; K(x i -x j )—kernel function; b—bias term.

10. The multi-component gas monitoring system of support vector regression according to claim 9, characterized in that: The kernel function K(x i -x j ) is one of the following: Linear kernel function K(x i ,x j )=x i ·x j ; Polynomial kernel function K(x i ,x j )=[γ(x i ·x j )+c] d ; Gaussian radial basis kernel function Sigmoid kernel function K(x i ,x j ) = tanh[α(x i ·x j )+c].

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