CN113780639A - Urban solid waste incineration nitrogen oxide NOx emission prediction method based on multitask learning framework - Google Patents

Urban solid waste incineration nitrogen oxide NOx emission prediction method based on multitask learning framework Download PDF

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CN113780639A
CN113780639A CN202110999702.5A CN202110999702A CN113780639A CN 113780639 A CN113780639 A CN 113780639A CN 202110999702 A CN202110999702 A CN 202110999702A CN 113780639 A CN113780639 A CN 113780639A
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乔俊飞
周江龙
蒙西
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Abstract

A method for predicting NOx emission of nitrogen oxides in municipal solid waste incineration based on a multitask learning framework relates to the field of artificial intelligence. The method realizes two-step prediction of the concentration of the NOx by using the NOx emission prediction model based on the multitask learning framework. Firstly, determining input variables related to the prediction of NOx concentration by combining the mechanism of municipal solid waste incineration; then, establishing an integral framework of a multi-task learning model by utilizing the basic idea of multi-task learning; next, a submodule of the multitask learning model is constructed using the self-organizing RBF neural network. And finally, testing the established prediction model to realize two-step prediction of the emission of the NOx generated by the municipal solid waste incineration. The method has better performance in predicting the concentration of NOx generated by urban solid waste incineration.

Description

Urban solid waste incineration nitrogen oxide NOx emission prediction method based on multitask learning framework
Technical Field
The invention relates to the field of artificial intelligence, is directly applied to the field of urban solid waste incineration, and particularly relates to a NOx concentration prediction method based on a multi-task learning framework.
Background
The NOx generated in the urban solid waste incineration process is a main atmospheric pollutant, and the realization of ultralow emission of the NOx is a consistent requirement of the international society. With the increasing emphasis of the state on environmental protection, increasingly strict laws and regulations restrict NOx emissions from incineration plants. In order to make the NOx emission reach the standard, an incinerator adopts a selective non-catalytic reduction system to carry out denitration in the incinerator. The denitration system guarantees that NOx can fully reduce through spouting excessive urea, and nevertheless excessive input urea not only causes the raw materials extravagant, and the cost improves, can bring ammonia escape moreover, causes secondary pollution to the environment. In addition, a large amount of generated ammonia gas can generate side reaction with other substances to generate cohesive substances, and the smoke discharge pipeline is easy to block. In order to accurately control the amount of injected urea, the NOx concentration needs to be analyzed according to its concentration, and therefore NOx concentration prediction is of great significance for the optimization and control of the denitration system.
Because the urban solid waste incineration process involves numerous reactions, has complex mechanism and strong nonlinearity, and has high difficulty in accurately predicting the concentration of NOx, most of the currently widely adopted NOx prediction models can only realize the prediction of the concentration of NOx at a certain moment. Adjusting the amount of injected reductant based on NOx concentration at a single time is not scientific and reasonable, and therefore, the NOx emission trend needs to be predicted for a future period of time. The existing multi-step time sequence prediction method comprises a direct strategy and a recursive strategy, wherein the direct strategy realizes the prediction of each moment respectively by constructing a plurality of prediction models, the modeling time is usually too long, the recursive strategy takes the predicted value of the previous moment as the input of the next moment, and the prediction precision gradually deteriorates along with the increase of the prediction step length due to the accumulation of errors.
Aiming at various defects of the existing method, the invention provides a NOx concentration prediction method based on a multitask learning frame, and two-step prediction of NOx concentration is realized.
Disclosure of Invention
1. Problems that the invention needs and can solve:
the invention provides a method for predicting the concentration of nitric oxide (NOx) generated by burning urban solid waste based on a multitask learning framework. By performing mechanism analysis on generation and removal of the urban solid waste incineration NOx, selecting input variables related to NOx concentration prediction, designing a multi-task learning submodule based on a RBF neural network, completing construction of a prediction model based on a multi-task learning framework, realizing two-step prediction of the urban solid waste incineration NOx, and aiming at providing a rapid and high-precision multi-step prediction method.
2. The invention adopts the following technical scheme and implementation steps:
the invention provides a method for predicting the concentration of nitric oxide (NOx) generated by burning urban solid waste based on a multitask learning framework. The method is characterized by comprising the following steps:
(1) preprocessing data;
through the mechanism analysis of the generation and removal of the NOx generated and removed by the municipal solid waste incineration, 6 input variables relevant to the prediction of the NOx are determined, and the input variables comprise: normalizing the NOx concentration, the right side temperature of the primary combustion chamber, the primary air volume of the furnace, the secondary air volume of the furnace, the accumulated amount of the urea solution of the furnace and the supply flow of the urea solution at the time t to [0, 1] according to a formula (1); the NOx concentration at the time when the output variable is t +1, t +2, is normalized to [0, 1] according to equation (2):
Figure BDA0003235253980000021
Figure BDA0003235253980000022
wherein, IiDenotes the ith input variable, OmDenotes the m-th output variable, xiAnd ymRespectively representing the ith input variable and the mth output variable after normalization; min (I)i) And max (I)i) Respectively representing the minimum value and the maximum value in the ith input variable; min (O)m) And max (O)m) Respectively representing the minimum value and the maximum value in the m-th output variable;
(2) constructing a prediction model of nitrogen oxide (NOx) generated by burning urban solid waste based on a multitask learning framework based on training samples;
and establishing a NOx prediction model by utilizing a multitask learning frame based on the RBF neural network to realize two-step prediction of NOx concentration. The multi-task learning model consists of two sub-modules, different modules realize the prediction of NOx concentration at different moments, and knowledge sharing is carried out among the modules. The first sub-module acts as a base module that will migrate as shared knowledge to the second sub-module. The second submodule is constructed by adding a task-specific module to the basic module.
Two sub-modules are established based on an RBF neural network, and the method comprises the following steps: an input layer, a hidden layer and an output layer; at the initial moment, the topological structure of the first sub-module is 6-K-1, namely the input layer is provided with 6 neurons which respectively correspond to the 6 input variables normalized in the step 1, the hidden layer is provided with K neurons, the output layer is provided with 1 neuron and corresponds to the concentration of NOx at the moment of t + 1; the topological structure of the second submodule is 6-J-1, namely the input layer has 6 neurons, which respectively correspond to the 6 input variables normalized in the step 1, the hidden layer has J neurons, the output layer has 1 neuron, and the output layer corresponds to the NOx concentration at the time of t + 2. The topological structure of the second submodule is that hidden layer neurons are added on the basis of the first submodule to serve as task specific modules, and therefore knowledge sharing among the modules is achieved.
Assuming a total of S training samples, the two sub-modules use the same input vector x ═ x1,x2,...,x6]T，x1,x2,x3,x4,x5,x6Respectively corresponding to the normalized input variables: NOx concentration, temperature on the right side of a primary combustion chamber, primary air quantity of a furnace, secondary air quantity of the furnace, accumulated amount of urea solution of the furnace and supply flow of urea solvent at the time t; output y1,y2NOx concentrations at time t +1 and t + 2.
In the first sub-module, the NOx concentration at time t +1 is calculated as follows:
input layer of the first submodule: this layer consists of 6 neurons, the output of each input neuron being:
ui＝xi (3)
wherein u isiIs the ith input nerveOutput of element, xiIs the ith element of the input vector, i ═ 1,2, …, 6;
② hidden layer of the first submodule: the hidden layer consists of K neurons, the output of each neuron being:
Figure BDA0003235253980000031
wherein phi isk(xs) Representing the s-th input vector xsThe output of the kth hidden neuron, c, upon entering the first submodulekIs the center of the kth hidden layer neuron, b is the width of the kth hidden layer neuron;
output layer of the first submodule: the output of the first submodule is:
Figure BDA0003235253980000032
wherein, y1,sFor the s-th input vector xsWhen entering the first sub-module, the predicted value, w, corresponding to the time t +1k,1Is the connection weight of the kth hidden layer neuron of the first submodule to the output layer, phik(xs) Is the output of the kth hidden layer neuron.
The NOx concentration at time t +2 in the second sub-module is calculated as follows:
input layer of the second sub-module: this layer consists of 6 neurons, each with an output of:
vi＝xi (6)
wherein v isiIs the output of the ith input neuron, xiIs the ith element of the input vector, i ═ 1,2, …, 6;
② hidden layer of second submodule: the hidden layer consists of J neurons, and the output of each hidden layer neuron of the second submodule is as follows:
Figure BDA0003235253980000041
wherein phi isj(xs) Representing the s-th input vector xsOutput of the jth hidden neuron upon entry into the second submodule, cjIs the center of the jth hidden layer neuron, and b is the width of the jth hidden layer neuron;
output layer of the second submodule: the output of the second submodule is:
Figure BDA0003235253980000042
wherein, y2,sFor the s-th input vector xsWhen entering the second submodule, the predicted value, w, corresponding to the time t +2j,1And the connection weight value from the jth hidden layer neuron of the second submodule to the output layer. Phi is aj(xs) The output of the jth hidden layer neuron of the second submodule.
The second submodule is realized by migrating the first submodule and adding the task-specific module together, so that the first K neurons of the second submodule are the same as the first submodule. The calculation of the output of the second submodule may also be:
Figure BDA0003235253980000043
wn,1is the connection weight value phi from the nth hidden layer neuron to the output layer in the task specific modulen(xs) Is the output of the nth hidden layer neuron. w is ak,1Is the connection weight of the kth hidden layer neuron of the first submodule to the output layer, phik(xs) Is the output of the kth hidden layer neuron. y is1,sFor the s-th input vector xsAnd when entering the first sub-module, corresponding to the predicted value at the t +1 moment.
(3) Designing an RBF neural network based on the training samples to realize the construction of a multi-task learning frame submodule;
hidden layer neurons of the RBF neural networks of the two sub-modules are added in a self-organizing mode, after one neuron is added each time, a connection weight between the current hidden layer neuron and an output neuron is calculated, errors under the current network structure are solved according to formulas (10) and (11), if the errors do not reach the expected errors, the neurons are continuously added until the errors are lower than the expected errors or the number of the largest neurons is reached. The maximum number of the hidden layer neurons of the first submodule in the invention is K. The maximum number of hidden layer neurons of the second submodule is J
The mean square error of the two prediction tasks is defined as:
Figure BDA0003235253980000051
Figure BDA0003235253980000052
wherein the content of the first and second substances,
Figure BDA0003235253980000053
expected values at times t +1 and t +2, y1,s、y2,sAnd S is the number of training samples, and is the predicted values at the time of t +1 and t + 2.
1) Determining the network structure and parameters of a first submodule;
determining the center and width of hidden layer neuron of the first submodule;
and taking S input samples as a set of hidden layer neuron centers to be selected, and selecting the S input samples as centers from the samples. Principle of adding neurons: the center that maximizes the synthetic error is sought in the fitting center as the center to which the neuron is to be added. The maximum position of the comprehensive error is as follows:
d＝argmax[e1,1,e1,2,...,e1,i,...,e1,S] (12)
wherein e1,iWhen the prediction is carried out at the t +1 moment, the comprehensive errors of all samples are shown when the ith sample is taken as the center of the hidden layer neuron, and the higher the comprehensive error is, the need is shownThe sample is taken as the center of the hidden layer neuron to compensate the comprehensive error, and the comprehensive error is calculated as follows:
Figure BDA0003235253980000054
wherein p iss,iTo select the ith sample as the center, the s input sample is at the output of the hidden layer.
Figure BDA0003235253980000055
Expected value at time t +1 for the s-th sample.
Initially, the hidden layer output of the s-th input sample when passing through the i-th candidate neuron is:
ps,i(0)＝φi(xs) (14)
after each center is added, the center needs to be deleted from the set of the intended center, so as to ensure that the same center is not used for multiple times. So after k neurons are determined, the number of candidate centers becomes S-k.
When a hidden layer neuron is added, the radial basis output needs to be updated to ensure that the output response range of the radial basis function is large enough, so that the fitting capability of the model is improved, and the following updating is performed:
Figure BDA0003235253980000056
ps,i(k),ps,i(k-1) radial basis outputs for the current iteration and the last iteration when the ith sample is selected as the center. p is a radical ofs,k-1The radial basis of the s-th sample is output for the last selected sample as the center.
The iteration times, namely the number of the added neurons, adopt the same width for all the hidden layer neurons
Figure BDA0003235253980000061
And remain unchanged during the iteration processAnd (6) changing. And calculating the optimal connection weight from the hidden layer to the output layer under the current structure every time one neuron is added.
Determining the connection weight between hidden layer neuron and output neuron.
After each neuron is added, the center and the width are fixed in the training process, and only the optimal connection weight from the hidden layer to the output layer needs to be calculated. Because the hidden layer output and the network output value are in a linear relation, the output weight can be calculated by adopting a least square method to obtain:
Figure BDA0003235253980000062
wherein WT＝[w1,1,w2,1,...wk,1]，wk,1Hidden layer output as the connection weight between the kth hidden layer neuron and the output neuron
Figure BDA0003235253980000063
Desired output
Figure BDA0003235253980000064
After each neuron is added, calculating an output weight value, under the current network structure and parameters, calculating a prediction error at the t +1 moment according to formulas (5) and (10), if the prediction error does not reach an expected error, continuing to add the neurons until the prediction error is lower than the expected error, and stopping adding the neurons until the condition can reach the maximum number K of the neurons. After training is completed, the center, width and output weight of the first submodule can be determined.
2) Determination of the configuration and parameters of a second network of submodules
Since the same data is used as input, the prediction of t +2 needs more neurons to learn more knowledge from the input, and the prediction accuracy is guaranteed. In the invention, the number of hidden layer neurons of the second submodule is J, wherein J is K + N, K is the number of hidden layer neurons of the first submodule, and N is the number of neurons of the task specific module of the second submodule.
Due to the characteristic of multi-task learning information sharing, the first sub-module is migrated to be used as a part of the second sub-module, and the second sub-module is constructed only by adding neurons as task specific modules. That is, the center, width and output weight of the first K hidden layer neurons of the second submodule are the same as those of the first submodule, and the parameters of the shared module are kept unchanged when neurons are added to the task specific module. The mechanism of adding the neurons by the specific module of the task two is the same as that of the first submodule.
In order to ensure that only the output weight corresponding to the task two specific modules is updated when the neuron is added, the output weight of the task specific module is calculated as follows by combining the formula (9):
Figure BDA0003235253980000071
wherein
Figure BDA0003235253980000072
Figure BDA0003235253980000073
w1,K+nThe connection weight value phi between the nth neuron and the output neuron added for the task specific moduleK+n(xs) The output of the nth neuron at the hidden layer of the task specific module is the s-th sample.
Figure BDA0003235253980000074
Expected value, y, output for the time t +2 of the s-th sample1,sIs the predicted value at the time of the s-th sample t + 1.
And after the task specific module adds the neurons each time, calculating the output weight of the task specific module under the current structure, and combining the parameters of the first submodule to obtain all the parameters of the second submodule. Under the current structure and parameters, according to the formulas (8) and (11), the prediction error at the time of t +2 is calculated, if the expected error is not reached, the neuron addition is continued until the expected error is not reached, and the condition of stopping the neuron addition can be that the maximum number J of neurons is reached. After training is completed, the center, width and output weight of the second submodule can be determined.
(4) NOx concentration prediction is performed, and performance of the prediction model is evaluated
And C test samples are set, the test sample data is used as the input of the trained multi-task learning prediction model to obtain the output of the two sub-modules, and the output is subjected to inverse normalization to obtain the predicted value of the NOx concentration at two moments. By calculating the root mean square error RMSE, the mean absolute percent error MAPE and the regression coefficient R2And evaluating the precision of the prediction model. The smaller the RMSE and MAPE, the R2The larger the prediction accuracy, the higher the prediction accuracy.
For the prediction of the t +1 moment, the three evaluation indexes are respectively calculated as follows:
Figure BDA0003235253980000075
Figure BDA0003235253980000076
Figure BDA0003235253980000077
wherein
Figure BDA0003235253980000078
y1,c，
Figure BDA0003235253980000079
The actual value, the predicted value and the predicted average value of the NOx concentration at the t +1 moment are respectively.
For the prediction of the time t +2, the three evaluation indexes are calculated as follows:
Figure BDA0003235253980000081
Figure BDA0003235253980000082
Figure BDA0003235253980000083
wherein
Figure BDA0003235253980000084
y2,c，
Figure BDA0003235253980000085
The actual value, the predicted value and the predicted average value of the NOx concentration at the t +2 moment are respectively.
3. Compared with the prior art, the invention has the following obvious advantages and beneficial effects:
aiming at the defects of the existing multi-step prediction method of NOx, the invention provides the prediction method of the concentration of the NOx in the municipal solid waste incineration based on the multitask learning framework by analyzing the mechanism of the generation and removal of the NOx in the municipal solid waste incineration and selecting the input variable related to the prediction of the concentration of the NOx.
Particular attention is paid to: the invention is only for convenience of description, adopts two-step advanced prediction of the concentration of NOx in the municipal solid waste incineration, and is also applicable to multi-step advanced prediction of the concentration of other atmospheric pollutants in the municipal solid waste incineration, and the invention is within the scope of the invention as long as the principle of the invention is adopted for improvement or optimization.
Drawings
FIG. 1 is a diagram of a prediction model architecture based on a multi-task learning framework
FIG. 2 is a diagram showing the result of predicting NOx concentration at time t +1 in the multi-task learning model
FIG. 3 is a diagram showing the result of predicting NOx concentration at time t +2 in the multi-task learning model
Detailed Description
According to the method for predicting the concentration of the nitric oxide (NOx) generated by the municipal solid waste incineration based on the multitask learning framework, two-step prediction of the concentration of the NOx is realized according to data collected by the municipal solid waste incineration, and the problem of low multi-step prediction precision of the concentration of the NOx generated by the municipal solid waste incineration is solved, so that an incinerator can be guided to reasonably adjust the amount of the reducing agent sprayed by a denitration system, the purpose of environmental protection is achieved, and the cost is greatly saved.
The experimental data were 2200 groups of samples from a waste incineration plant in Beijing. Each set of samples contains 6 input variables: NOx concentration, primary combustion chamber right side temperature, furnace primary air volume, furnace secondary air volume, furnace urea solution cumulative amount, urea solvent supply flow at time t, 2 output variables: NOx concentration at time t +1, NOx concentration at time t + 2. The entire 2200 sample set was divided into two portions: wherein 1540 groups of data are used as training samples, and the other 660 groups of data are used as testing samples;
step 1: preprocessing data;
input variables related to NOx prediction, including: normalizing the NOx concentration, the right side temperature of the primary combustion chamber, the primary air volume of the furnace, the secondary air volume of the furnace, the accumulated amount of the urea solution of the furnace and the supply flow of the urea solution at the time t to [0, 1] according to a formula (1); the NOx concentration at the time when the output variable is t +1, t +2, is normalized to [0, 1] according to equation (2):
Figure BDA0003235253980000091
Figure BDA0003235253980000092
wherein, IiDenotes the ith input variable, OmDenotes the m-th output variable, xiAnd ymRespectively representing the ith input variable and the mth output variable after normalization; min (I)i) And max (I)i) Respectively representing the ith input variableMinimum and maximum values in the quantities; min (O)m) And max (O)m) Respectively representing the minimum value and the maximum value in the m-th output variable;
step 2: constructing a prediction model of nitrogen oxide (NOx) generated by burning urban solid waste based on a multitask learning framework based on training samples;
and establishing a NOx prediction model by utilizing a multitask learning frame based on the RBF neural network to realize two-step prediction of NOx concentration. The multi-task learning model consists of two sub-modules, different modules realize the prediction of NOx concentration at different moments, and knowledge sharing is carried out among the modules. The first sub-module acts as a base module that will migrate as shared knowledge to the second sub-module. The second submodule is constructed by adding a task-specific module to the basic module.
Two sub-modules are established based on an RBF neural network, and the method comprises the following steps: an input layer, a hidden layer and an output layer; at the initial moment, the topological structure of the first sub-module is 6-K-1, namely the input layer is provided with 6 neurons which respectively correspond to the 6 input variables normalized in the step 1, the hidden layer is provided with K neurons, the output layer is provided with 1 neuron and corresponds to the concentration of NOx at the moment of t + 1; the topological structure of the second submodule is 6-J-1, namely the input layer has 6 neurons, which respectively correspond to the 6 input variables normalized in the step 1, the hidden layer has J neurons, the output layer has 1 neuron, and the output layer corresponds to the NOx concentration at the time of t + 2. The topological structure of the second submodule is that N hidden layer neurons are added on the basis of the first submodule to serve as task specific modules, and therefore knowledge sharing among the modules is achieved. In this example, K is 15, J is 17, and N is 2.
Assuming a total of S training samples, the two sub-modules use the same input vector x ═ x1,x2,...,x6]T，x1,x2,x3,x4,x5,x6Respectively corresponding to the normalized input variables: NOx concentration, temperature on the right side of a primary combustion chamber, primary air quantity of a furnace, secondary air quantity of the furnace, accumulated amount of urea solution of the furnace and supply flow of urea solvent at the time t; output y1,y2NOx concentrations at time t +1 and t + 2. The true bookIn the example, S is 1540.
In the first sub-module, the NOx concentration at time t +1 is calculated as follows:
fourthly, the input layer of the first submodule: this layer consists of 6 neurons, the output of each input neuron being:
ui＝xi (3)
wherein u isiIs the output of the ith input neuron, xiIs the ith element of the input vector, i ═ 1,2, …, 6;
fifth, hidden layer of the first sub-module: the hidden layer consists of K neurons, the output of each neuron being:
Figure BDA0003235253980000101
wherein phi isk(xs) Representing the s-th input vector xsThe output of the kth hidden neuron, c, upon entering the first submodulekIs the center of the kth hidden layer neuron, b is the width of the kth hidden layer neuron;
sixthly, the output layer of the first submodule: the output of the first submodule is:
Figure BDA0003235253980000102
wherein, y1,sFor the s-th input vector xsWhen entering the first sub-module, the predicted value, w, corresponding to the time t +1k,1Is the connection weight of the kth hidden layer neuron of the first submodule to the output layer, phik(xs) Is the output of the kth hidden layer neuron.
The NOx concentration at time t +2 in the second sub-module is calculated as follows:
input layer of the second submodule: this layer consists of 6 neurons, each with an output of:
vi＝xi (6)
wherein，viIs the output of the ith input neuron, xiIs the ith element of the input vector, i ═ 1,2, …, 6;
fifth, the hidden layer of the second sub-module: the hidden layer consists of J neurons, and the output of each hidden layer neuron of the second submodule is as follows:
Figure BDA0003235253980000103
wherein phi isj(xs) Representing the s-th input vector xsOutput of the jth hidden neuron upon entry into the second submodule, cjIs the center of the jth hidden layer neuron, and b is the width of the jth hidden layer neuron;
sixthly, the output layer of the second submodule: the output of the second submodule is:
Figure BDA0003235253980000111
wherein, y2,sFor the s-th input vector xsWhen entering the second submodule, the predicted value, w, corresponding to the time t +2j,1And the connection weight value from the jth hidden layer neuron of the second submodule to the output layer. Phi is aj(xs) The output of the jth hidden layer neuron of the second submodule.
The second submodule is realized by migrating the first submodule and adding the task-specific module together, so that the first K neurons of the second submodule are the same as the first submodule. The calculation of the output of the second submodule may also be:
Figure BDA0003235253980000112
wn,1is the connection weight value phi from the nth hidden layer neuron to the output layer in the task specific modulen(xs) Is the output of the nth hidden layer neuron. w is ak,1For the kth concealment of the first sub-moduleConnection weight, phi, of layer neurons to output layersk(xs) Is the output of the kth hidden layer neuron. y is1,sFor the s-th input vector xsAnd when entering the first sub-module, corresponding to the predicted value at the t +1 moment.
And step 3: designing an RBF neural network based on the training samples to realize the construction of a multi-task learning frame submodule;
hidden layer neurons of the RBF neural networks of the two sub-modules are added in a self-organizing mode, after one neuron is added each time, a connection weight between the current hidden layer neuron and an output neuron is calculated, errors under the current network structure are solved according to formulas (10) and (11), if the errors do not reach the expected errors, the neurons are continuously added until the errors are lower than the expected errors or the number of the largest neurons is reached. In the invention, the expected error of the prediction results at two moments is set to be 0.0005, and the maximum number of hidden layer neurons in a first submodule is 15. The maximum number of hidden layer neurons in the second submodule is 17.
The errors defining the two prediction tasks are:
Figure BDA0003235253980000121
Figure BDA0003235253980000122
wherein the content of the first and second substances,
Figure BDA0003235253980000123
expected values at times t +1 and t +2, y1,s、y2,sAnd S is the number of training samples, and is the predicted values at the time of t +1 and t + 2.
3) Determining the network structure and parameters of a first submodule;
determining the center and width of hidden layer neuron of the first submodule;
and taking S input samples as a set of hidden layer neuron centers to be selected, and selecting the S input samples as centers from the samples. Principle of adding neurons: the center that maximizes the synthetic error is sought in the fitting center as the center to which the neuron is to be added. The maximum position of the comprehensive error is as follows:
d＝argmax[e1,1,e1,2,...,e1,i,...,e1,S] (12)
wherein e1,iWhen the ith sample is taken as the hidden layer neuron center during the prediction at the time t +1, the higher the comprehensive error of all samples is, which indicates that the sample needs to be taken as the hidden layer neuron center to compensate the comprehensive error, and the calculation of the comprehensive error is as follows:
Figure BDA0003235253980000124
wherein p iss,iTo select the ith sample as the center, the s input sample is at the output of the hidden layer.
Figure BDA0003235253980000125
Expected value at time t +1 for the s-th sample.
Initially, the hidden layer output of the s-th input sample when passing through the i-th candidate neuron is:
ps,i(0)＝φi(xs) (14)
after each center is added, the center needs to be deleted from the set of the intended center, so as to ensure that the same center is not used for multiple times. So after k neurons are determined, the number of candidate centers becomes S-k.
When a hidden layer neuron is added, the radial basis output needs to be updated to ensure that the output response range of the radial basis function is large enough, so that the fitting capability of the model is improved, and the following updating is performed:
Figure BDA0003235253980000126
ps,i(k),ps,i(k-1) is selectedAnd when the ith sample is taken as the center, outputting the radial basis of the current iteration and the last iteration. p is a radical ofs,k-1The radial basis of the s-th sample is output for the last selected sample as the center.
The iteration times, namely the number of the added neurons, adopt the same width for all the hidden layer neurons
Figure BDA0003235253980000131
And remain unchanged during the iteration. And calculating the optimal connection weight from the hidden layer to the output layer under the current structure every time one neuron is added.
Determining the connection weight between hidden layer neuron and output neuron.
After each neuron is added, the center and the width are fixed in the training process, and only the optimal connection weight from the hidden layer to the output layer needs to be calculated. Because the hidden layer output and the network output value are in a linear relation, the output weight can be calculated by adopting a least square method to obtain:
Figure BDA0003235253980000132
wherein WT＝[w1,1,w2,1,...wk,1]，wk,1Hidden layer output as the connection weight between the kth hidden layer neuron and the output neuron
Figure BDA0003235253980000133
Desired output
Figure BDA0003235253980000134
After adding the neurons every time, calculating an output weight value, under the current network structure and parameters, calculating a prediction error at the t +1 moment according to the formulas (5) and (10), if the expected error is not reached to 0.0005, continuously adding the neurons until the expected error is not reached, and stopping adding the neurons until the condition that the number of the neurons is not increased to 15 can be reached to the maximum hidden layer neuron number of the first submodule. After training is completed, the center, width and output weight of the first submodule can be determined.
4) Determination of the configuration and parameters of a second network of submodules
Since the same data is used as input, the prediction of t +2 needs more neurons to learn more knowledge from the input, and the prediction accuracy is guaranteed. In the invention, the maximum number of hidden layer neurons of the second submodule is J, J is 17, wherein J is K + N, K is 15 and is the number of hidden layer neurons of the first submodule, N is 2 and is the number of neurons of the task specific module of the second submodule.
Due to the characteristic of multi-task learning information sharing, the first sub-module is migrated to be used as a part of the second sub-module, and the second sub-module is constructed only by adding neurons as task specific modules. That is, the center, width and output weight of the first 15 hidden layer neurons of the second submodule are the same as those of the first submodule, and the parameters of the shared module remain unchanged while neurons are added by the task specific module. The mechanism of adding the neurons by the specific module of the task two is the same as that of the first submodule.
In order to ensure that only the output weight corresponding to the task two specific modules is updated when the neuron is added, the output weight of the task specific module is calculated as follows by combining the formula (9):
Figure BDA0003235253980000141
wherein
Figure BDA0003235253980000142
Figure BDA0003235253980000143
w1,K+nThe connection weight value phi between the nth neuron and the output neuron added for the task specific moduleK+n(xs) The output of the nth neuron at the hidden layer of the task specific module is the s-th sample.
Figure BDA0003235253980000144
Expected value, y, output for the time t +2 of the s-th sample1,sIs the predicted value at the time of the s-th sample t + 1.
And after the task specific module adds the neurons each time, calculating the output weight of the task specific module under the current structure, and combining the parameters of the first submodule to obtain all the parameters of the second submodule. Under the current structure and parameters, according to the formulas (8) and (11), the prediction error at the time of t +2 is calculated, if the expected error is not reached, the neuron addition is continued until the expected error is not reached, and the condition of stopping the neuron addition can be that the maximum number of neurons 17 is reached. After training is completed, the center, width and output weight of the second submodule can be determined.
And after the two sub-modules are trained, the construction of a prediction model based on a multi-task learning prediction framework is completed. In this embodiment, the prediction model based on the multitask learning framework is as shown in fig. 1.
And 4, step 4: and taking the test sample data as the input of the trained multi-task learning prediction model to obtain the output of two sub-modules, performing inverse normalization on the output to obtain the predicted values of NOx concentration at two moments, and evaluating the performance of the proposed model.
In the present embodiment, the results of predicting NOx concentrations at time t +1 and time t +2 are shown in fig. 2 and 3, and the X axis: number of samples, in units of units per sample, Y-axis: NOx concentration in mg/m3The green solid line is the actual output value of the NOx concentration, and the blue dotted line is the predicted output value of the NOx concentration.
Three performance indicators are defined: root mean square error RMSE, mean absolute percent error MAPE, and regression coefficient R2And evaluating the precision of the prediction model.
For the prediction of the t +1 moment, the three evaluation indexes are respectively calculated as follows:
Figure BDA0003235253980000145
Figure BDA0003235253980000151
Figure BDA0003235253980000152
wherein
Figure BDA0003235253980000153
y1,c，
Figure BDA0003235253980000154
The actual value, the predicted value and the predicted average value of the NOx concentration at the t +1 moment are respectively.
For the prediction of the time t +2, the three evaluation indexes are calculated as follows:
Figure BDA0003235253980000155
Figure BDA0003235253980000156
Figure BDA0003235253980000157
wherein
Figure BDA0003235253980000158
y2,c，
Figure BDA0003235253980000159
The actual value, the predicted value and the predicted average value of the NOx concentration at the t +2 moment are respectively.
And comparing the performance with a direct prediction method and an iterative prediction method, wherein the comparison results are shown in tables 1 and 2, and the results show the effectiveness of the urban solid waste incineration NOx concentration prediction model based on the multi-task learning framework.
TABLE 1 comparison of predictive Performance at time t +1 for multitask learning and other multi-step predictive methods
Figure BDA00032352539800001510
TABLE 2 comparison of predictive Performance at time t +2 for multitask learning and other multi-step prediction methods
Figure BDA00032352539800001511
Figure BDA0003235253980000161

Claims (4)

1. A prediction method for nitrogen oxide (NOx) emission in municipal solid waste incineration based on a multitask learning framework is characterized by comprising the following steps:
step 1: preprocessing data;
through the mechanism analysis of the generation and removal of the NOx generated and removed by the municipal solid waste incineration, 6 input variables relevant to the prediction of the NOx are determined, and the input variables comprise: normalizing the NOx concentration, the right side temperature of the primary combustion chamber, the primary air volume of the furnace, the secondary air volume of the furnace, the accumulated amount of the urea solution of the furnace and the supply flow of the urea solution at the time t to [0, 1] according to a formula (1); the NOx concentration at the time when the output variable is t +1, t +2, is normalized to [0, 1] according to equation (2):
Figure FDA0003235253970000011
Figure FDA0003235253970000012
wherein, IiDenotes the ith input variable, OmDenotes the m-th output variable, xiAnd ymRespectively representing the ith input variable and the mth output variable after normalization; min (I)i) Andmax(Ii) Respectively representing the minimum value and the maximum value in the ith input variable; min (O)m) And max (O)m) Respectively representing the minimum value and the maximum value in the m-th output variable;
step 2: constructing a prediction model of nitrogen oxide NOx generated by burning urban solid waste based on a multitask learning framework based on training samples;
establishing a NOx prediction model by utilizing a multitask learning frame based on a RBF neural network to realize two-step prediction of NOx concentration; the multi-task learning model consists of two sub-modules, different modules realize the prediction of NOx concentration at different moments, and the modules share knowledge; the first sub-module is used as a basic module, and the basic module is used as shared knowledge and is migrated to the second sub-module; the construction of a second sub-module is realized by adding a task-specific module on the basis of a basic module; two sub-modules are established based on an RBF neural network, and the method comprises the following steps: an input layer, a hidden layer and an output layer; at the initial moment, the topological structure of the first sub-module is 6-K-1, namely the input layer is provided with 6 neurons which respectively correspond to the 6 input variables normalized in the step 1, the hidden layer is provided with K neurons, the output layer is provided with 1 neuron and corresponds to the concentration of NOx at the moment of t + 1; the topological structure of the second submodule is 6-J-1, namely an input layer is provided with 6 neurons which respectively correspond to the 6 input variables normalized in the step 1, the hidden layer is provided with J neurons, an output layer is provided with 1 neuron and corresponds to the concentration of NOx at the time of t + 2; the topological structure of the second submodule is that N hidden layer neurons are added on the basis of the first submodule to serve as task specific modules, and therefore knowledge sharing among the modules is achieved;
step 3, designing an RBF neural network based on the training samples, and realizing the construction of a multi-task learning frame submodule;
and 4, taking the test sample data as the input of the trained prediction model, predicting the NOx concentration at two moments in the future, and evaluating the performance of the prediction model.
2. The urban solid waste incineration nitrogen oxide NOx emission prediction method based on multitask learning as claimed in claim 1, characterized in that in step 2, the NOx concentration prediction at the time t +1 and t +2 is realized by two sub-modules, specifically:
assuming a total of S training samples, the two sub-modules use the same input vector x ═ x1,x2,...,x6]T，x1,x2,x3,x4,x5,x6Respectively corresponding to the normalized input variables: NOx concentration, temperature on the right side of a primary combustion chamber, primary air quantity of a furnace, secondary air quantity of the furnace, accumulated amount of urea solution of the furnace and supply flow of urea solvent at the time t; output y1,y2NOx concentrations at time t +1 and t + 2;
in the first sub-module, the NOx concentration at time t +1 is calculated as follows:
input layer of the first submodule: this layer consists of 6 neurons, the output of each input neuron being:
ui＝xi (3)
wherein u isiIs the output of the ith input neuron, xiIs the ith element of the input vector, i ═ 1,2, …, 6;
② hidden layer of the first submodule: the hidden layer consists of K neurons, the output of each neuron being:
Figure FDA0003235253970000021
wherein phi isk(xs) Representing the s-th input vector xsThe output of the kth hidden neuron, c, upon entering the first submodulekIs the center of the kth hidden layer neuron, b is the width of the kth hidden layer neuron;
output layer of the first submodule: the output of the first submodule is:
Figure FDA0003235253970000022
wherein, y1,sFor the s-th input vector xsWhen entering the first sub-module, the predicted value, w, corresponding to the time t +1k,1Is the connection weight of the kth hidden layer neuron of the first submodule to the output layer, phik(xs) Is the output of the kth hidden layer neuron;
the NOx concentration at time t +2 in the second sub-module is calculated as follows:
input layer of the second sub-module: this layer consists of 6 neurons, each with an output of:
vi＝xi (6)
wherein v isiIs the output of the ith input neuron, xiIs the ith element of the input vector, i ═ 1,2, …, 6;
② hidden layer of second submodule: the hidden layer consists of J neurons, and the output of each hidden layer neuron of the second submodule is as follows:
Figure FDA0003235253970000031
wherein phi isj(xs) Representing the s-th input vector xsOutput of the jth hidden neuron upon entry into the second submodule, cjIs the center of the jth hidden layer neuron, and b is the width of the jth hidden layer neuron;
output layer of the second submodule: the output of the second submodule is:
Figure FDA0003235253970000032
wherein, y2,sFor the s-th input vector xsWhen entering the second submodule, the predicted value, w, corresponding to the time t +2j,1The connection weight value from the jth hidden layer neuron of the second submodule to the output layer; phi is aj(xs) The output of the jth hidden layer neuron that is the second submodule;
the second submodule is realized by the first submodule migration and the task specific module addition together, so that the first K neurons of the second submodule are the same as the first submodule; the output of the second submodule is calculated as:
Figure FDA0003235253970000033
wn,1is the connection weight value phi from the nth hidden layer neuron to the output layer in the task specific modulen(xs) Is the output of the nth hidden layer neuron; w is ak,1Is the connection weight of the kth hidden layer neuron of the first submodule to the output layer, phik(xs) Is the output of the kth hidden layer neuron; y is1,sFor the s-th input vector xsAnd when entering the first sub-module, corresponding to the predicted value at the t +1 moment.
3. The urban solid waste incineration nitrogen oxide NOx emission prediction method based on multitask learning as claimed in claim 1, wherein the construction of the multitask learning submodule is realized based on an RBF neural network, and the step 3 specifically comprises the following steps:
hidden layer neurons of RBF neural networks of the two sub-modules are added in a self-organizing mode, after one neuron is added each time, a connection weight between the current hidden layer neuron and an output neuron is calculated, errors under the current network structure are solved according to formulas (10) and (11), if the errors do not reach the expected errors, the neurons are continuously added until the errors are lower than the expected errors or the number of the largest neurons is reached; the maximum number of first submodule hidden layer neurons is K; the maximum number of hidden layer neurons of the second submodule is J
The mean square error of the two prediction tasks is defined as:
Figure FDA0003235253970000041
Figure FDA0003235253970000042
wherein the content of the first and second substances,
Figure FDA0003235253970000043
expected values at times t +1 and t +2, y1,s、y2,sThe predicted values at the time t +1 and the time t +2 are obtained, and S is the number of training samples;
(1) determining the network structure and parameters of a first submodule;
determining the center and width of hidden layer neuron of the first submodule;
taking S input samples as a set of hidden layer neuron centers to be selected, and selecting the samples as centers; principle of adding neurons: searching the center of the fitting center, which maximizes the comprehensive error, as the center of the neuron to be added; the maximum position of the comprehensive error is as follows:
d＝argmax[e1,1,e1,2,...,e1,i,...,e1,S] (12)
wherein e1,iWhen the ith sample is taken as the hidden layer neuron center during the prediction at the time t +1, the higher the comprehensive error of all samples is, which indicates that the sample needs to be taken as the hidden layer neuron center to compensate the comprehensive error, and the calculation of the comprehensive error is as follows:
Figure FDA0003235253970000044
wherein p iss,iWhen the ith sample is selected as the center, the output of the s input sample at the hidden layer;
Figure FDA0003235253970000045
the expected value at the t +1 moment of the s-th sample;
initially, the hidden layer output of the s-th input sample when passing through the i-th candidate neuron is:
ps,i(0)＝φi(xs) (14)
after each center is added, the center needs to be deleted from the set of the selected center, so that the same center is ensured not to be used for multiple times; therefore, after k neurons are determined, the number of the candidate centers is changed into S-k;
when a hidden layer neuron is added, the radial basis output needs to be updated to ensure that the output response range of the radial basis function is large enough, so that the fitting capability of the model is improved, and the following updating is performed:
Figure FDA0003235253970000051
ps,i(k),ps,i(k-1) radial basis outputs for the current iteration and the last iteration when the ith sample is selected as the center; p is a radical ofs,k-1The radial basis output for the s-th sample, centered on the last selected sample;
the iteration times, namely the number of the added neurons, adopt the same width for all the hidden layer neurons
Figure FDA0003235253970000052
And remain unchanged during the iteration process; every time a neuron is added, calculating the optimal connection weight from the hidden layer to the output layer under the current structure;
determining a connection weight between a hidden layer neuron and an output neuron;
after each neuron is added, the center and the width are fixed in the training process, and only the optimal connection weight from the hidden layer to the output layer needs to be calculated; because the hidden layer output and the network output value are in a linear relation, the output weight can be calculated by adopting a least square method to obtain:
Figure FDA0003235253970000053
wherein WT＝[w1,1,w2,1,...wk,1]，wk,1Is the k-thThe connection weight between hidden layer neuron and output neuron, hidden layer output
Figure FDA0003235253970000054
Desired output
Figure FDA0003235253970000055
After adding the neurons every time, calculating an output weight, under the current network structure and parameters, calculating a prediction error at the t +1 moment according to formulas (5) and (10), if the expected error is not reached, continuing to add the neurons until the expected error is not reached, and stopping adding the neurons until the condition can reach the maximum number K of the neurons; after training is finished, the center, the width and the output weight of the first submodule can be determined;
(2) determination of the configuration and parameters of a second network of submodules
Because the same data is used as input, more neurons are needed for the prediction of t +2, so that more knowledge can be learned from the input, and the prediction precision is ensured; in the invention, the number of hidden layer neurons of a second submodule is J, wherein J is K + N, K is the number of hidden layer neurons of a first submodule, and N is the number of neurons of a task specific module of the second submodule;
due to the characteristic of multi-task learning information sharing, the first sub-module is migrated to be used as a part of the second sub-module, and the second sub-module is constructed only by adding neurons as task specific modules; the center, width and output weight of the first K hidden layer neurons of the second submodule are the same as those of the first submodule, and when the neurons are added to the task specific module, the parameters of the shared module are kept unchanged; the mechanism of adding the neurons to the specific module of the task two is the same as that of the first submodule;
in order to ensure that only the output weight corresponding to the task two specific modules is updated when the neuron is added, the output weight of the task specific module is calculated as follows by combining the formula (9):
Figure FDA0003235253970000061
wherein
Figure FDA0003235253970000062
Figure FDA0003235253970000063
w1,K+nThe connection weight value phi between the nth neuron and the output neuron added for the task specific moduleK+n(xs) The output of the nth neuron of the hidden layer of the task specific module for the s sample;
Figure FDA0003235253970000064
expected value, y, output for the time t +2 of the s-th sample1,sThe predicted value at the moment of the s sample t +1 is obtained;
after the task specific module adds the neuron every time, calculating the output weight of the task specific module under the current structure, and combining the parameters of the first submodule to obtain all the parameters of the second submodule; under the current structure and parameters, according to the formulas (6) and (11), calculating the prediction error at the t +2 moment, if the expected error is not reached, continuing to add the neurons until the expected error is not reached, and stopping adding the neurons until the condition can reach the maximum number J of the neurons; after training is completed, the center, width and output weight of the second submodule can be determined.
4. The urban solid waste incineration NOx emission prediction method based on multitask learning as claimed in claim 1, wherein in step 4, the prediction model performance evaluation based on the multitask learning framework specifically comprises:
setting C test samples, taking test sample data as input of the trained multi-task learning prediction model to obtain output of two sub-modules, and performing inverse normalization on the output to obtain predicted values of NOx concentration at two moments; by calculating the root mean square error RMSE, the mean absolute percent error MAPE and the regression coefficient R2To, forEvaluating the precision of the prediction model; the smaller the RMSE and MAPE, the R2The larger the prediction accuracy, the higher the prediction accuracy;
for the prediction of the t +1 moment, the three evaluation indexes are respectively calculated as follows:
Figure FDA0003235253970000065
Figure FDA0003235253970000071
Figure FDA0003235253970000072
wherein
Figure FDA0003235253970000073
y1,c，
Figure FDA0003235253970000074
Respectively obtaining a real value, a predicted value and a predicted average value of the NOx concentration at the t +1 moment;
for the prediction of the time t +2, the three evaluation indexes are calculated as follows:
Figure FDA0003235253970000075
Figure FDA0003235253970000076
Figure FDA0003235253970000077
wherein
Figure FDA0003235253970000078
y2,c，
Figure FDA0003235253970000079
The actual value, the predicted value and the predicted average value of the NOx concentration at the t +2 moment are respectively.
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