CN114021819B - Thermal power plant nitrogen oxide content prediction method based on fractional gray delay model - Google Patents
Thermal power plant nitrogen oxide content prediction method based on fractional gray delay model Download PDFInfo
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- MWUXSHHQAYIFBG-UHFFFAOYSA-N Nitric oxide Chemical compound O=[N] MWUXSHHQAYIFBG-UHFFFAOYSA-N 0.000 title claims abstract description 105
- 238000000034 method Methods 0.000 title claims abstract description 58
- 238000009825 accumulation Methods 0.000 claims abstract description 53
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Abstract
A method for predicting the nitrogen oxide content of a thermal power plant based on a fractional gray delay model comprises the following steps: and collecting average value data of the clean flue gas NO X content of the thermal power plant in each hour in real time as original data, calculating consistent fractional order accumulation generated data of the original data, establishing a consistent fractional order gray delay model, predicting the average value data of the clean flue gas NO X content of the thermal power plant in each hour according to the consistent fractional order gray delay optimization model, and obtaining a predicted value of the emission of nitrogen oxides of the thermal power plant at a future moment according to consistent fractional order accumulation operation. The advantages are that: the method adopts a consistent fractional gray delay model, has the characteristics of small calculated amount, high degree of freedom and the like in system analysis and modeling, and is particularly suitable for the problem of predicting the time sequence of a small sample of the emission of nitrogen oxide pollutants.
Description
Technical Field
The invention relates to a thermal power plant nitrogen oxide content prediction method based on a fractional gray delay model.
Background
Along with the increase of the electric power demand and the extension of the coal-fired boiler, the coal demand of China is increased. A large part of nitrogen oxides in atmospheric pollutants are derived from combustion of coal for power generation in thermal power plants, and thus nitrogen oxides (NOx) become one of the main pollutants in the atmosphere discharged from thermal power plants. In the operation of a boiler in a thermal power plant, factors influencing the NOx content in the boiler flue gas include the influence of fuel properties, operating conditions and boiler load. Through modeling and predictive analysis of the concentration of the nitrogen oxides, scientific judgment and early warning of pollutant emission of a thermal power plant can be improved, and related factors influencing the pollutant emission are adjusted in advance, such as low-nitrogen combustion technology, selective catalytic reduction, selective non-catalytic reduction technology and the like, so that the emission of the nitrogen oxides is further reduced.
In the prior art, the prediction of the emission concentration of the nitrogen oxides in the thermal power plant comprises regression analysis, a neural network method, a support vector machine, mixed data driving, deep learning, a gray model and the like. Regression analysis includes general least squares regression (Ordinary Least Squares Regression, OLSR), linear regression (Linear Regression), logistic regression (Logistic Regression), stepwise regression (Stepwise Regression), multiple adaptive regression splines (Multivariate Adaptive Regression Splines, MARS), etc., and has the disadvantage of requiring strict assumptions, requiring outliers to be handled. The neural network method requires a large amount of data for training, and is difficult to understand the internal mechanism of the model and difficult to select parameters and network topology. The final decision function of the support vector machine is determined by only a few support vectors, the computational complexity being dependent on the number of support vectors. The method has no general solution to the nonlinear problem, is not easy to find a proper kernel function, has weak interpretation power to the high-dimensional mapping of the kernel function, and is unfavorable for storage due to low solving efficiency and large calculation amount of the convex quadratic programming problem related to optimization. Sensitive to missing data. The mixed data driving method has the problems of more parameters and input variables, relatively complex model structure and solution and the like. The above method is difficult to meet the problem of small sample time series prediction of emissions of nitrogen oxide pollutants.
Disclosure of Invention
The invention aims to solve the technical problems of providing the thermal power plant nitrogen oxide content prediction method based on the fractional gray delay model, which has the advantages of small calculated amount, high degree of freedom, high model precision and more accurate prediction result, and is particularly suitable for the problem of small sample time sequence prediction of nitrogen oxide pollutant emission.
The technical scheme of the invention is as follows:
A method for predicting the nitrogen oxide content of a thermal power plant based on a fractional gray delay model comprises the following specific steps:
Step 1: collecting average value data (mg/Nm 3) of the NO X content of the clean flue gas of the boiler of the thermal power plant in real time as original data x(0),x(0)＝(x(0)(1),x(0)(2),…,x(0)(i),…,x(0)(n-1),x(0)(n));, wherein n is the total number of data points in the original data, namely x (0) (n) is the nth hour data of the original data, and x (0) (i) is the ith hour data of the original data;
step 2: calculating a consistent fractional order accumulation of the raw data x (0) to generate data x (β), where x (β)＝(x(β)(1),x(β)(2),…,x(β)(i),…,x(β)(n-1),x(β) (n));
the calculation formula of x (β) is: wherein, beta is the fractional accumulation order based on the consistent fractional operator, and [ beta ] is the maximum integer less than or equal to beta;
Step 3: establishing a consistent fractional order gray delay model
Modeling the consistent fractional order accumulation generated data x (β) of the original data x (0), and solving parameters of a consistent fractional order gray delay model by using an optimization method:
Step 3.1: a whitening differential equation is established for x (β) based on the consistent fractional order gray delay model:
Wherein alpha is consistent fractional order differential order, beta is consistent fractional order accumulation order, a is a development coefficient, b is gray action amount, tau is a delay variable, and t is a time variable;
Step 3.2: according to the definition of consistent fractional order differentiation:
Wherein t is a time variable and f (t) is a function related to t; if f (t) =x (β) (t), the whitening differential equation of the consistent fractional gray delay model in step 3.1 can be expressed as The discrete form thereof can be expressed as:
x(β)(n)-x(β)(n-1)+anα-1x(β)(n-τ)＝bnα-1,0<α≤1
further using a matrix can be expressed as
Because the whitening differential equation discrete form of the consistent fractional gray delay model and the delay data x (β)(2-τ)、x(β)(3-τ)、…、x(β) (n-tau) in the matrix representation thereof are unknown, lagrange interpolation can be adopted for estimation;
if the consistent fractional order differential order alpha, the consistent fractional order accumulated order beta and the delay variable tau are known, the estimated values of the parameters a and b can be determined by using a least square method And/>In the mathematical form of
Wherein the method comprises the steps of
Step 3.3: estimating delay data x (β) (2-tau, xbeta 3-tau, …, xbeta n-tau) in the model of the step 3.2 by adopting Lagrangian interpolation;
Step 3.4: establishing a consistent fractional order gray delay optimization model according to an optimization method to obtain an optimal solution of a minimum fitness function and 3 variables alpha, beta and tau, wherein alpha is more than or equal to 0 and less than or equal to 1, beta is more than or equal to 0 and less than or equal to 1;
the optimal solution obtained by optimizing is assigned to the parameters alpha, beta, tau of the consistent fractional gray delay model in the step 3.2, wherein the fitness function fitness can be expressed as
According to the calculation result of the optimization algorithm, using the minimum calculation parameters alpha, beta and tau of the fitness function fitness result to obtain the estimated value of the optimal model parameterAccording to alpha, beta, tau and/>Calculating estimated data/>, of fractional accumulation data of a consistent fractional accumulation operator
Is defined by consistent fractional order accumulation, and is not difficult to obtain
According to the consistent fractional order accumulation operation,Can be expressed as
Is defined by consistent fractional order accumulation, and is not difficult to obtain
Taking a genetic algorithm as an example, solving a consistent fractional order gray delay optimization model of Lagrange interpolation to obtain a minimum fitness function value;
according to the consistent fractional gray delay optimization model parameters alpha, beta and tau obtained by Lagrange interpolation calculation, further obtaining
Step 4: according to the consistent fractional gray delay optimization model, predicting the average value data of the NO X content of the clean flue gas of the boiler of the thermal power plant per hour:
Step 4.1: predicting average value data of the NO X content of the clean flue gas of the boiler of the thermal power plant at the future moment in each hour, and according to the principle of a consistent fractional gray delay model, a prediction formula of an optimization model can be expressed as follows:
Wherein the method comprises the steps of Accumulating the estimated value of the predicted data for the consistent fractional order of the average value of the NO X content of the clean flue gas of the boiler of the thermal power plant per hour;
Step 4.2: estimation value of fractional accumulated data for estimating average value per hour of boiler clean flue gas NO X content of thermal power plant at future moment by utilizing Lagrange interpolation
① Obtaining an estimated value of fractional accumulated data at future time by using the first-order Lagrange interpolation value
② Second-order Lagrange interpolation to obtain estimated value of fractional order accumulated data at future time
Setting an estimated value of fractional accumulated data at future time by adopting an iteration method Where m is the number of iterations and ρ is the increment per iteration.
By means ofAnd/>Can be obtained through second-order Lagrange interpolationOptimizing model parameters alpha, beta, tau and/>, according to consistent fractional order gray delayUsing predictive formulas
Whether the continuous iteration condition is met can be judged according to the absolute error or the relative error analysis of the estimated value and the predicted value of the fractional accumulation data at the future moment.
Such asNot more than specified value or/>Stopping calculation when the value is less than or equal to the specified value, and outputting/>Otherwise, continuing iteration, wherein m=m+1, until a judgment condition is met;
Wherein the judgment condition is selected
Step 4.3, repeating the above processes to realize multi-step prediction to obtain a predicted estimated value of the average value consistent fractional order accumulated data of the NO X content of the clean flue gas of the boiler of the thermal power plant per hour
Step 5: according to the consistent fractional order accumulation operation, obtaining the predicted value of the nitrogen oxide emission of the boiler of the thermal power plant at the future moment as
On-line display and updating of predicted values
Is a predicted value of the nitrogen oxide emission of the boiler of the thermal power plant at the future moment.
Further preferably, in step 3.3, when the delay data in the model is estimated by using lagrangian interpolation, the lagrangian interpolation is first-order lagrangian interpolation, i.e. linear interpolation
Obtaining delay data x (β) (n- τ) by using x (β) (n) and x (β) (n-1), and obtaining the expression as
x(β)(n-τ)＝(1-τ)x(β)(n-1)+τx(β)(n)
Substituting x (β) (n- τ) into the consistent fractional order gray delay model discrete form formula of step 3.2 can obtain:
x(β)(n)＝-aτnα-1x(β)(n)-[a(1-τ)nα-1-1]x(β)(n-1)+bnα-1
＝nα-1[b-aτx(β)(n)-a(1-τ)x(β)(n-1)]+x(β)(n-1),0<α≤1
The consistent fractional gray delay model based on first-order Lagrangian interpolation can be represented and simplified into
Parameters (parameters)Estimate of/>Can be estimated from the following formula
Wherein the method comprises the steps of
Further preferably, in step 3.3, when the delay data in the model is estimated by using lagrangian interpolation, the lagrangian interpolation is a second-order lagrangian interpolation
The delay data x (β) (n-tau) can be derived from x (β)(n),x(β)(n-1),x(β) (n-2), the sample data x 0＝n-2,x1＝n-1,x2 = n,
The function values are f (x 0)＝x(β)(n-2),f(x1)＝x(β)(n-1),f(x2)＝x(β) (n) respectively, when x=n- τ, the fitting value of the second-order Lagrangian interpolation can be expressed as
Where l M is the corresponding Lagrangian interpolation polynomial to sample x M and y (x) is the result of the Lagrangian interpolation polynomial calculation at sample x.
The matrix form of the uniform fractional gray delay model based on the higher-order lagrangian interpolation can be expressed as
Where m is the Lagrange interpolation order, parameters in the matrixEstimate of/>Can be solved using the following formula
Wherein the method comprises the steps of
Further preferably, the maximum iteration number of the optimization setting of step 3.4 is 300.
The invention has the beneficial effects that:
the consistent fractional gray delay model has the characteristics of small calculated amount, high degree of freedom and the like in system analysis and modeling, and is particularly suitable for the problem of predicting the time sequence of a small sample of the emission of nitrogen oxide pollutants. The consistent fractional gray delay optimal model obtains optimal parameters through an optimizing algorithm, delay data are constructed by utilizing Lagrange interpolation, the data detail and the development trend are more accurately depicted, the model precision is high, and the prediction result is more ideal. The method has the advantages that the consistent fractional gray delay model is utilized to predict the nitrogen oxide emission of the boiler of the thermal power plant, the development trend of the nitrogen oxide emission generated by the combustion of the future coal-fired boiler can be scientifically predicted, and early warning is timely carried out, so that the production and operation scheduling of enterprises are optimized in advance, the production efficiency of the enterprises is improved, and the influence of pollutants on the ecological environment is reduced.
Drawings
FIG. 1 is a flow chart of a thermal power plant nitrogen oxide content prediction method based on a fractional gray delay model;
FIG. 2 is a graph for acquiring average value data of the NO X content of the clean flue gas of the boiler of the thermal power plant in real time according to the embodiment of the invention;
FIG. 3 is a graph of a consistent fractional order gray delay optimization model optimization process based on first order Lagrange interpolation in an embodiment of the invention;
FIG. 4 is a graph of consistent fractional order accumulated data and estimated data based on first order Lagrangian interpolation in accordance with an embodiment of the present invention;
FIG. 5 is a graph of a consistent fractional order gray delay optimization model optimization process based on second order Lagrangian interpolation in an embodiment of the present invention;
FIG. 6 is a graph of consistent fractional order accumulated data and estimated data based on a second order Lagrangian interpolation in accordance with an embodiment of the present invention.
Detailed Description
Example 1
Step 1: collecting average value data (mg/Nm 3) of the NO X content of the clean flue gas of the boiler of the thermal power plant in real time as original data x(0),x(0)＝(x(0)(1),x(0)(2),…,x(0)(i),…,x(0)(n-1),x(0)(n));, wherein n is the total number of data points in the original data, namely x (0) (n) is the nth hour data of the original data; x (0) (i), is the ith hour data of the original data. Raw data are shown in table 1 and fig. 2;
TABLE 1 raw data (mg/Nm 3)
| x(0)(1) | x(0)(2) | x(0)(3) | x(0)(4) | x(0)(5) | x(0)(6) | x(0)(7) | x(0)(8) | x(0)(9) | x(0)(10) |
| 42.7230 | 35.1950 | 35.0030 | 35.9540 | 36.2750 | 40.3100 | 41.6020 | 37.0860 | 36.2100 | 37.7440 |
Step 2: the consistent fractional order accumulation of the original data x (0) is calculated to generate data x (β), wherein the calculation formula of ,x(β)＝(x(β)(1),x(β)(2),…,x(β)(i),…,x(β)(n-1),x(β)(n));x(β) is as follows: Wherein, beta is the fractional accumulation order based on the consistent fractional operator, [ beta ] is the maximum integer less than or equal to beta, when beta=0, the original data is processed without accumulation, namely the calculated consistent fractional accumulation generated data of the original data is equal to the original data; as shown in fig. 4, the raw data is the consistent fractional order accumulated data of the average value of the net smoke NO X content per hour of the boiler of the thermal power plant.
Step 3: establishing a consistent fractional order gray delay model
Modeling the consistent fractional order accumulation generated data x (β) of the original data x (0), and solving parameters of a consistent fractional order gray delay model by using an optimization method:
Step 3.1: a whitening differential equation is established for x (β) based on a consistent fractional order gray delay model as
Wherein alpha is consistent fractional order differential order, beta is consistent fractional order accumulation order, a is a development coefficient, b is gray action amount, tau is a delay variable, and t is a time variable;
Step 3.2: definition according to consistent fractional order differentiation Where t is a time variable and f (t) is a function related to t. If f (t) =x (β) (t), the whitening differential equation for the consistent fractional gray delay model in step 3.1 can be expressed as/> Its discrete form can be expressed as
x(β)(n)-x(β)(n-1)+anα-1x(β)(n-τ)＝bnα-1,0<α≤1
Further using a matrix can be expressed as
Because the delay data x (β)(2-τ)、x(β)(3-τ)、…、x(β) (n-tau) in the discrete form of the whitened differential equation of the consistent fractional gray delay model and its matrix representation are unknown, lagrange interpolation can be used for estimation;
if the consistent fractional order differential order alpha, the consistent fractional order accumulated order beta and the delay variable tau are known, the estimated values of the parameters a and b can be determined by using a least square method And/>In the mathematical form of
Wherein the method comprises the steps of
Step 3.3: the delay data x (β)(2-τ)、x(β)(3-τ)、…、x(β) (n- τ) in the step 3.2 model is estimated using first order lagrangian interpolation.
First-order Lagrange interpolation, i.e. linear interpolation
Obtaining delay data x (β) (n- τ) by using x (β) (n) and x (β) (n-1), and obtaining the expression as
x(β)(n-τ)＝(1-τ)x(β)(n-1)+τx(β)(n)
Substituting x (β) (n- τ) into the consistent fractional order gray delay model discrete form formula of step 3.2 can obtain:
x(β)(n)＝-aτnα-1x(β)(n)-[a(1-τ)nα-1-1]x(β)(n-1)+bnα-1
=n α-1[b-aτx(β)(n)-a(1-τ)x(β)(n-1)]+x(β) (n-1), and a consistent fractional order gray delay model with 0< α+.ltoreq.1 based on first order lagrangian interpolation can be represented simplified as
Parameters (parameters)Estimate of/>Can be estimated from the following formula
Wherein the method comprises the steps of
Step 3.4: and establishing a consistent fractional order gray delay optimization model according to an optimization method to obtain the minimum fitness function and the optimal solution of 3 variables alpha, beta and tau.
The optimal solution obtained by optimizing is assigned to the parameters alpha, beta, tau of the consistent fractional gray delay model in the step 3.2, wherein the fitness function fitness can be expressed as In this example, the parameter range is set to 0< alpha.ltoreq.1, 0.ltoreq.beta, τ.ltoreq.1, and the maximum iteration number is 300.
According to the calculation result of the optimization algorithm, using the minimum calculation parameters alpha, beta and tau of the fitness function fitness result to obtain the estimated value of the optimal model parameterAccording to alpha, beta, tau and/>Calculating estimated data/>, of fractional accumulation data of a consistent fractional accumulation operator
Is defined by consistent fractional order accumulation, and is not difficult to obtain
According to the consistent fractional order accumulation operation,Can be expressed as
Is defined by consistent fractional order accumulation, and is not difficult to obtain
Taking a genetic algorithm as an example, solving a consistent fractional gray delay optimization model of first-order Lagrange interpolation, wherein the model optimization process is shown in figure 3,The minimum fitness function value is 4.1258%.
The parameters alpha, beta and tau of the consistent fractional gray delay optimization model obtained by first-order Lagrange interpolation calculation are respectively,
Α= 0.7831, β= 0.4908, τ= 0.6211, and then the following is obtained
In the consistent fractional gray delay optimization model based on the first-order Lagrange interpolation, a consistent fractional accumulation estimated data curve is shown in FIG. 4.
Step 3.5: modeling error analysis
After the model is built, the fitting precision of the model can be known by calculating the estimation error. The estimation error solving formula of the original data in the invention is that
Wherein x (0) (i) andThe raw data in table 1 and the estimated data of the raw data obtained by the consistent fractional gray delay model are shown in table 2.
The mean absolute percentage error (MAPE, mean Absolute Percentage Error) is calculated as
In a consistent fractional gray delay model of first-order Lagrange interpolation The corresponding average absolute percentage errors were calculated to be 4.1258% respectively.
Step 4: according to the consistent fractional gray delay optimization model, predicting the average value data of the NO X content of the clean flue gas of the boiler of the thermal power plant per hour:
step 4.1: predicting average value data of the NO X content of the clean flue gas of the boiler of the thermal power plant at the future moment in each hour, and according to the principle of a consistent fractional gray delay model, a prediction formula of an optimization model can be expressed as
Wherein the method comprises the steps ofAnd accumulating the estimated value of the predicted data for the consistent fractional order of the average value of the No X content of the clean flue gas of the boiler of the thermal power plant per hour.
Step 4.2: estimation value of fractional accumulated data of average value of net smoke NOX content per hour of boiler of thermal power plant at future moment by using first-order Lagrange interpolation
According to
Can be further arranged into
Step 4.3, repeating the above processes to realize multi-step prediction, and obtaining a predicted estimated value of the average value consistent fractional order accumulated data of the NOx content of the clean flue gas of the boiler of the thermal power plant per hour
Step 5: according to the consistent fractional order accumulation operation, obtaining the predicted value of the nitrogen oxide emission of the boiler of the thermal power plant at the future moment as
On-line display and updating of predicted values
Is a predicted value of the nitrogen oxide emission of the boiler of the thermal power plant at the future moment.
In order to compare the prediction accuracy of the model, a prediction truth value of the nitrogen oxide emission of the boiler of the thermal power plant at the future moment is given
x(0)(n+1)＝34.8500mg/Nm3,x(0)(n+2)＝33.5780mg/Nm3,x(0)(n+3＝32.6670mg/Nm3.
The prediction data calculated according to the consistent fractional gray delay optimization model of the first-order Lagrangian interpolation is shown in table 3, and the model prediction error analysis is shown in table 3. The different model prediction errors are shown in table 4.
Example 2
Step 1: collecting average value data (mg/Nm 3) of the NO X content of the clean flue gas of the boiler of the thermal power plant in real time as original data x(0),x(0)＝(x(0)(1),x(0)(2),…,x(0)(i),…,x(0)(n-1),x(0)(n));, wherein n is the total number of data points in the original data, namely x (0) (n) is the nth hour data of the original data; x (0) (i), is the ith hour data of the original data. This example was predicted using the raw data of example 1, shown in tables 1 and 2;
Step 2: the consistent fractional order accumulation of the original data x (0) is calculated to generate data x (β), wherein the calculation formula of ,x(β)＝(x(β)(1),x(β)(2),…,x(β)(i),…,x(β)(n-1),x(β)(n));x(β) is as follows: Wherein, beta is the fractional accumulation order based on the consistent fractional operator, [ beta ] is the maximum integer less than or equal to beta, when beta=0, the original data is processed without accumulation, namely the calculated consistent fractional accumulation generated data of the original data is equal to the original data;
The raw data is shown in figure 6 by the consistent fractional order cumulative data of the average value of the net smoke NO X content per hour of the boiler of the thermal power plant.
Step 3: establishing a consistent fractional order gray delay model
Modeling the consistent fractional order accumulation generated data x (β) of the original data x (0), and solving parameters of a consistent fractional order gray delay model by using an optimization method:
Step 3.1: a whitening differential equation is established for x (β) based on a consistent fractional order gray delay model as
Wherein alpha is consistent fractional order differential order, beta is consistent fractional order accumulation order, a is a development coefficient, b is gray action amount, tau is a delay variable, and t is a time variable;
Step 3.2: definition according to consistent fractional order differentiation Where t is a time variable and f (t) is a function related to t. If f (t) =x (β) (t), the whitening differential equation for the consistent fractional gray delay model in step 3.1 can be expressed as/> Its discrete form can be expressed as
x(β)(n)-x(β)(n-1)+anα-1x(β)(n-τ)＝bnα-1,0<α≤1
Further using a matrix can be expressed as
Because the delay data x (β)(2-τ)、x(β)(3-τ)、…、x(β) (n-tau) in the discrete form of the whitened differential equation of the consistent fractional gray delay model and its matrix representation are unknown, lagrange interpolation can be used for estimation;
if the consistent fractional order differential order alpha, the consistent fractional order accumulated order beta and the delay variable tau are known, the estimated values of the parameters a and b can be determined by using a least square method And/>In the mathematical form of
Wherein the method comprises the steps of
Step 3.3: and (3) estimating delay data x (β)(2-τ)、x(β)(3-τ)、…、x(β) (n-tau) in the step 3.2 model by adopting second-order Lagrangian interpolation.
The delay data x (β) (n- τ) can be obtained from x (β)(n),x(β)(n-1),x(β) (n-2), and the known sample data x 0＝n-2,x1＝n-1,x2 =n, whose function values are f (x 0)＝x(β)(n-2),f(x1)＝x(β)(n-1),f(x2)＝x(β) (n)) respectively, then when x=n- τ, the fitting value of the second-order lagrangian interpolation can be expressed as
Where l M is the corresponding Lagrangian interpolation polynomial to sample x M and y (x) is the result of the Lagrangian interpolation polynomial calculation at sample x.
The matrix form of the uniform fractional gray delay model based on the higher-order lagrangian interpolation can be expressed as
Where m is the Lagrange interpolation order, parameters in the matrixEstimate of/>Can be solved using the following formula
Wherein the method comprises the steps of
By higher order interpolation, x (β) (k- τ) can estimate more accurately the time delay data considered in the overall data development, giving in this example a second order lagrangian interpolation model, i.e. m=2.
Step 3.4: and establishing a consistent fractional order gray delay optimization model according to an optimization method to obtain the minimum fitness function and the optimal solution of 3 variables alpha, beta and tau.
The optimal solution obtained by optimizing is assigned to the parameters alpha, beta, tau of the consistent fractional gray delay model in the step 3.2, wherein the fitness function fitness can be expressed as In this example, the parameter range is set to 0< alpha.ltoreq.1, 0.ltoreq.beta, τ.ltoreq.1, and the maximum iteration number is 300.
According to the calculation result of the optimization algorithm, using the minimum calculation parameters alpha, beta and tau of the fitness function fitness result to obtain the estimated value of the optimal model parameterAccording to alpha, beta, tau and/>Calculating estimated data/>, of fractional accumulation data of a consistent fractional accumulation operator
Is defined by consistent fractional order accumulation, and is not difficult to obtain
According to the consistent fractional order accumulation operation,Can be expressed as
Is defined by consistent fractional order accumulation, and is not difficult to obtain
Taking a genetic algorithm as an example, modeling a consistent fractional gray delay optimization model of second-order Lagrange interpolation, and because delay data is obtained through interpolation in the modeling process, the first two points of the data have no modeling error in practice, and are easy to obtainThe model optimization process is shown in figure 5,The final minimum fitness function value was 3.6836%.
The parameters of the consistent fractional gray delay optimization model obtained by the second-order Lagrange interpolation calculation are,
Α= 0.8663, β=0.0022, τ= 0.9123, and thus estimates of a, b are obtainedIs thatAccording to the consistent fractional order gray delay optimization model of the second-order Lagrangian interpolation, a corresponding consistent fractional order accumulation estimated data curve is shown in FIG. 6.
Step 3.5: modeling error analysis
After the model is built, the fitting precision of the model can be known by calculating the estimation error. The estimation error solving formula of the original data in the patent of the invention is as follows
Wherein x (0) (i) andThe raw data in table 1 and the estimated data of the raw data obtained by the consistent fractional gray delay model are shown in table 2.
The mean absolute percentage error (MAPE, mean Absolute Percentage Error) is calculated as
In a consistent fractional order gray delay model of second order Lagrange interpolationCalculating to obtain corresponding average absolute percentage error/>The value was 3.6836%.
Step 4: according to the consistent fractional gray delay optimization model, predicting the average value data of the NO X content of the clean flue gas of the boiler of the thermal power plant per hour:
step 4.1: predicting average value data of the NO X content of the clean flue gas of the boiler of the thermal power plant at the future moment in each hour, and according to the principle of a consistent fractional gray delay model, a prediction formula of an optimization model can be expressed as
Wherein the method comprises the steps ofAnd accumulating the estimated value of the predicted data for the consistent fractional order of the average value of the No X content of the clean flue gas of the boiler of the thermal power plant per hour.
Step 4.2: estimation value of fractional accumulated data of average value per hour of boiler clean flue gas NO X content of thermal power plant at future moment by utilizing second-order Lagrange interpolation
Setting an estimated value of fractional accumulated data at future time by adopting an iteration method Where m is the number of iterations and ρ is the increment per iteration.
By means ofCan be obtained through second-order Lagrange interpolationOptimizing model parameters alpha, beta, tau and/>, according to consistent fractional order gray delayThe predictive value/>, of the fractional order accumulated data at the future moment can be obtained by utilizing a predictive formula
Whether the continuous iteration condition is met can be judged according to the absolute error or the relative error analysis of the estimated value and the predicted value of the fractional accumulation data at the future moment.
Such asNot more than specified value or/>Stopping calculation when the value is less than or equal to the specified value, and outputting/>Otherwise, continuing iteration, wherein m=m+1, until a judgment condition is met;
In this example, ρ=0.1 is adopted, and the judgment condition is selected
Step 4.3, repeating the above processes to realize multi-step prediction, and obtaining a predicted estimated value of the average value consistent fractional order accumulated data of the NO X content of the clean flue gas of the boiler of the thermal power plant per hour
Step 5: according to the consistent fractional order accumulation operation, obtaining the predicted value of the nitrogen oxide emission of the boiler of the thermal power plant at the future moment as
On-line display and updating of predicted values
Is a predicted value of the nitrogen oxide emission of the boiler of the thermal power plant at the future moment.
In order to compare the prediction accuracy of the model, a prediction truth value of the nitrogen oxide emission of the boiler of the thermal power plant at the future moment is given
x(0)(n+1)＝34.8500mg/Nm3,x(0)(n+2)＝33.5780mg/Nm3,x(0)(n+3＝32.6670mg/Nm3.
The prediction data calculated according to the consistent fractional gray delay optimization model of the second-order Lagrangian interpolation is shown in table 3, and the model prediction error analysis is shown in table 3. The different model prediction errors are shown in table 4.
TABLE 2 estimation error of raw data%
Table 3 optimization model prediction error analysis
Table 4 comparison of prediction errors for different models
The results of the comparison of the two classical mean GM (1, 1) means and the differential GM (1, 1) model are given in table 4. The average value GM (1, 1) average value and the differential GM (1, 1) model generally adopt first-order accumulation treatment in the data preprocessing process, the average value GM (1, 1) average value and the differential GM (1, 1) model are average value models, and the delay parameters of the average value GM (1, 1) average value and the differential GM (1, 1) model are equivalent to 0.5. Furthermore, the mean GM (1, 1) mean and the differential GM (1, 1) model are in the form ofThe differential order of both whitening equations is 1. Thus, from the whitening differential equation, two classes of mean GM (1, 1) mean and differential GM (1, 1) models are special cases of the consistent fractional gray delay model of the present invention. The consistent fractional gray delay model has more degrees of freedom in the aspects of accumulation order, differential order and delay variable, is more finely characterized and has more abundant connotation. And the optimal parameters can be obtained by calculating the consistent fractional gray delay model through an optimizing algorithm. The delay data or the background value is estimated by adopting Lagrange interpolation of different orders, so that the parameter estimation result is more reliable. As a wider model, the consistent fractional gray delay optimization model has the advantages of strong characterization capability, high model precision, reliable prediction and the like.
The above is only a specific embodiment of the present invention, and is not intended to limit the present invention, but various modifications and variations can be made to the present invention by those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (4)
1. A thermal power plant nitrogen oxide content prediction method based on a fractional gray delay model is characterized by comprising the following steps:
Step 1: collecting average value data (mg/Nm 3) of the NO X content of the clean flue gas of the boiler of the thermal power plant in real time as original data x(0),x(0)＝(x(0)(1),x(0)(2),…,x(0)(i),…,x(0)(n-1),x(0)(n));, wherein n is the total number of data points in the original data, namely x (0) (n) is the nth hour data of the original data, and x (0) (i) is the ith hour data of the original data;
step 2: calculating a consistent fractional order accumulation of the raw data x (0) to generate data x (β), where x (β)＝(x(β)(1),x(β)(2),…,x(β)(i),…,x(β)(n-1),x(β) (n));
the calculation formula of x (β) is: wherein, beta is the fractional accumulation order based on the consistent fractional operator, and [ beta ] is the maximum integer less than or equal to beta;
Step 3: establishing a consistent fractional order gray delay model
Modeling the consistent fractional order accumulation generated data x (β) of the original data x (0), and solving parameters of a consistent fractional order gray delay model by using an optimization method:
Step 3.1: a whitening differential equation is established for x (β) based on the consistent fractional order gray delay model:
Wherein alpha is consistent fractional order differential order, beta is consistent fractional order accumulation order, a is a development coefficient, b is gray action amount, tau is a delay variable, and t is a time variable;
Step 3.2: according to the definition of consistent fractional order differentiation:
Wherein t is a time variable and f (t) is a function related to t; if f (t) =x (β) (t), the whitening differential equation of the consistent fractional gray delay model in step 3.1 is expressed as The discrete form is expressed as:
x(β)(n)-x(β)(n-1)+anα-1x(β)(n-τ)＝bnα-1,0<α≤1
Further expressed as a matrix
Because the whitening differential equation discrete form of the consistent fractional gray delay model and the delay data x (β)(2-τ)、x(β)(3-τ)、…、x(β) (n-tau) in the matrix representation thereof are unknown, lagrange interpolation is adopted for estimation;
If the uniform fractional order differential order alpha, the uniform fractional order accumulated order beta and the delay variable tau are known, the method uses
Least square method determines estimated values of parameters a and bAnd/>In the mathematical form of
Wherein the method comprises the steps of
Step 3.3: estimating delay data x (β)(2-τ)、x(β)(3-τ)、…、x(β) (n-tau) in the model of the step 3.2 by adopting Lagrange interpolation;
Step 3.4: establishing a consistent fractional order gray delay optimization model according to an optimization method to obtain an optimal solution of a minimum fitness function and 3 variables alpha, beta and tau, wherein alpha is more than or equal to 0 and less than or equal to 1, beta is more than or equal to 0 and less than or equal to 1;
the optimized optimal solution is assigned to the parameters alpha, beta, tau of the consistent fractional gray delay model in the step 3.2, wherein the fitness function fitness is expressed as According to the calculation result of the optimization algorithm, the minimum calculation parameters alpha, beta and tau of the fitness function fitness result are used for obtaining the estimated value/>, of the optimal model parametersAccording to alpha, beta, tau and/>Calculating estimated data/>, of fractional accumulation data of a consistent fractional accumulation operator
Is defined by consistent fractional order accumulation, and is not difficult to obtainAccording to the consistent fractional order accumulation operation, x (0) (n) is denoted as
Is defined by consistent fractional order accumulation, and is not difficult to obtainTaking a genetic algorithm as an example, solving a consistent fractional order gray delay optimization model of Lagrange interpolation to obtain a minimum fitness function value;
Consistent fractional order gray delay optimization model parameters alpha, beta and tau obtained through Lagrange interpolation calculation are obtained, a is further obtained,
Step 4: according to the consistent fractional gray delay optimization model, predicting the average value data of the NO X content of the clean flue gas of the boiler of the thermal power plant per hour:
Step 4.1: predicting average value data of the NO X content of the clean flue gas of the boiler of the thermal power plant at the future moment in each hour, and expressing a prediction formula of an optimization model as follows according to a consistent fractional gray delay model principle:
Wherein the method comprises the steps of Accumulating the estimated value of the predicted data for the consistent fractional order of the average value of the NO X content of the clean flue gas of the boiler of the thermal power plant per hour;
Step 4.2: estimation value of fractional accumulated data for estimating average value per hour of boiler clean flue gas NO X content of thermal power plant at future moment by utilizing Lagrange interpolation
① Obtaining an estimated value of fractional accumulated data at future time by using the first-order Lagrange interpolation value
② Second-order Lagrange interpolation to obtain estimated value of fractional order accumulated data at future time
Setting an estimated value of fractional accumulated data at future time by adopting an iteration methodWherein m is the iteration number, and ρ is the increment of each iteration;
By means of And/>Obtaining/>, through second-order Lagrange interpolationOptimizing model parameters alpha, beta, tau and/>, according to consistent fractional order gray delayUsing predictive formulas
Judging whether a continuous iteration condition is met according to the absolute error or the relative error analysis of the estimated value and the predicted value of the fractional accumulation data at the future moment;
Such as Or/>When, the calculation is stopped, the output/>Otherwise, continuing iteration, wherein m=m+1, until a judgment condition is met; wherein the judgment condition is selectedStep 4.3, repeating the above processes to realize multi-step prediction to obtain a predicted estimated value of the average value consistent fractional order accumulated data of the NO X content of the clean flue gas of the boiler of the thermal power plant per hour
Step 5: according to the consistent fractional order accumulation operation, obtaining the predicted value of the nitrogen oxide emission of the boiler of the thermal power plant at the future moment as
On-line display and updating of predicted values
X (0) (k) is a predicted value of the nitrogen oxide emission of the boiler of the thermal power plant at the future time.
2. The method for predicting the nitrogen oxide content based on the fractional gray delay model according to claim 1, which is characterized in that:
in step 3.3, when the Lagrange interpolation is adopted to estimate the delay data in the model, the Lagrange interpolation is first-order Lagrange interpolation, namely linear interpolation
Obtaining delay data x (β) (n- τ) using x (β) (n) and x (β) (n-1), then obtaining an expression of x (β)(n-τ)＝(1-τ)x(β)(n-1)+τx(β) (n)
Substituting x (β) (n-tau) into the consistent fractional order gray delay model discrete form formula of the step 3.2 to obtain the following components: x (β)(n)＝-aτnα-1x(β)(n)-[a(1-τ)nα-1-1]x(β)(n-1)+bnα-1
＝nα-1[b-aτx(β)(n)-a(1-τ)x(β)(n-1)]+x(β)(n-1),0<α≤1
The consistent fractional gray delay model representation based on first-order Lagrange interpolation is simplified into
Parameters (parameters)Estimate of/>Estimated from the following formula
Wherein the method comprises the steps of
3. The method for predicting the nitrogen oxide content based on the fractional gray delay model according to claim 1, which is characterized in that:
In the step 3.3, when the Lagrange interpolation is adopted to estimate the delay data in the model, the Lagrange interpolation is a second-order Lagrange interpolation
The delay data x (β) (n-tau) is defined byAs a result, the known sample data x 0＝n-2,x1＝n-1,x2 =n,
The function values are f (x 0)＝x(β)(n-2),f(x1)＝x(β)(n-1),f(x2)＝x(β) (n)
Then when x=n- τ, the fitted value of the second order lagrangian interpolation is expressed as
Wherein l M is a Lagrangian interpolation polynomial corresponding to sample x M, and y (x) is a Lagrangian interpolation polynomial calculation result at sample x;
the matrix form of the uniform fractional gray delay model based on the higher-order lagrangian interpolation is expressed as
Where m is the Lagrange interpolation order, parameters in the matrixEstimate of/>Solving by using the following formula
Wherein the method comprises the steps of
4. The method for predicting the nitrogen oxide content based on the consistent fractional gray delay model according to claim 1, which is characterized by comprising the following steps: and 3.4, optimizing and setting the maximum iteration number to be 300.
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