CN106019093B - A kind of online soft sensor method of three-phawse arc furnace arc length - Google Patents
A kind of online soft sensor method of three-phawse arc furnace arc length Download PDFInfo
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Abstract
The present invention is a kind of online soft sensor method of three-phawse arc furnace arc length, its main feature is that, it comprises the step of：With controlling cycle TcFor period acquisition electrode system N group inputoutput data；Anomaly data detection and processing；Data normalization processing；Determine electrode system model structure；With the minimum target of model predictive error function, the unknown parameter in electrode system model is solved；Using the parameter acquired, row write arc length soft-sensing model, obtain arc length value；With current time kIt is existingFor basic point, chooses n (n≤N) group inputoutput datas successively in electrode system historical data, judge whether to update electric arc soft-sensing model online；If desired it updates, from (kIt is existingN) moment starts acquisition electrode system N group inputoutput datas, restarts to establish soft-sensing model, if need not update, waits for m controlling cycle, judges whether to update electric arc soft-sensing model again online.The present invention is not required to additionally increase detection device, realizes on-line measurement three-phawse arc furnace arc length.
Description
Technical Field
The invention relates to the field of soft measurement, in particular to an online soft measurement method for the arc length of a three-phase arc furnace.
Background
Electric arc furnaces, which are the main production equipment in the steel industry, smelt charge materials by converting electric energy into heat energy through an electric arc generated between an electrode and the charge materials. The arc length of the electric arc determines the electric arc power, so as to eliminate the adverse effects of the electric arc on harmonic injection, voltage fluctuation, flickering and the like of a power grid, shorten the smelting time, improve the labor production efficiency, improve the quality of molten steel and reduce the power consumption of steel per ton, and the maintenance of the specified electric arc length is the fundamental requirement of the electric arc furnace for working. As shown in figure 1, a hydraulic part of an electrode system is formed by a regulating valve and an electrode lifting plunger oil rod, an electrode controller realizes electrode lifting by controlling the hydraulic part, and a furnace transformer secondary side, a short net, an electrode and an electric arc form an electric part of an electric arc furnace. The nature of the control of the electrode system is the control of the arc length. However, the arc is generated by high temperature and high gas conductor discharge, and the length of the arc is inconvenient to measure in the operation process of the arc furnace.
Currently, relevant studies include:
the Chinese invention patent publication No. CN200810226732 is named as an arc length control device and method, and the method is characterized in that a laser distance meter is arranged on a movable welding gun, and the distance between a laser sensor and a fixed workpiece is directly measured by the laser distance meter to be used as the welding arc length. Chinese journal literature, changeable, mao shizheng, "alternating current electric arc furnace arc power soft measurement model", industrial heating, 2009(38), 2, 38-40. The displacement of the electrode is measured by mounting a displacement sensor on the electrode lifting plunger oil rod, and the displacement is used as the arc length. The two methods are both characterized in that the shape of the electric arc is approximate to a straight line, but the shape of the actual electric arc is very complex and not a straight line, in the actual operation process, the electrode is continuously consumed, the displacement of the electrode cannot really reflect the length of the electric arc, and both the two methods have certain limitations.
A scientific paper of doctor of northeast university, Wang Yan, "research and application of arc model of AC arc furnace", 2009, 05, based on the law of conservation of energy, an AC arc time domain model described by nonlinear differential equation is established, the input of the model is arc length and current, the arc length is arc voltage-arc length formula, and arc column gradient β0And α, the parameters are calculated, but the parameters are related to the working state of the electric arc furnace, the input ratio of energy, smelting time, the position of an electrode and the like, and are not easy to obtain in practical operation.
The invention discloses a Chinese patent publication No. CN102521489, which is named as a method and a system for modeling and parameter identification of electric arc furnace load, establishes an electric arc model reflecting the characteristics of harmonic waves, flicker and three-phase unbalance of a real-object electric arc furnace, and describes an electric arc into a nonlinear time-varying resistor, wherein a parameter C to be identified is related to the arc length. This method does not explicitly give a calculation method of the arc length.
The Chinese journal literature, Zhang Ping, Hou Bin, "arc furnace electrode adjustment system identification based on time-varying N-L-N model", information and control 2014(43), 6, 711 and 714. The electrode system of the electric arc furnace is expressed as a linear part time-varying N-L-N model, and an online identification method is provided on the basis of the model. The method aims at a single-phase electric arc furnace, and when the three-phase electric arc furnace works, the three-phase electric arc furnace has coupling influence among three-phase electric arcs, so the method is not suitable for the three-phase electric arc furnace. The output quantity of the electrode system is arc resistance, and the output quantity of the electrode system is the effective value of three-phase line current.
Disclosure of Invention
In the face of the importance of the arc length of the three-phase arc furnace in arc analysis and electrode control, according to the relevant research of the arc length of the three-phase arc furnace and based on 1, the working state of the three-phase arc furnace is a relatively stable state, and the soft measurement of the arc length under the conditions of arc breakage, short circuit and the like is not considered; 2. each electrode of the three-phase arc furnace is lifted and lowered by a separate plunger rod hydraulic cylinder, and the mutual influence among three electrodes is not considered; 3. each electrode of the three-phase electric arc furnace generates an electric arc, and the three electric arcs have mutual influence.
The invention aims to substantially correct and innovate the prior art and provide the on-line soft measurement method for the arc length of the three-phase electric arc furnace, which can fully embody the real characteristics and has convenient operation, science and reasonability.
The technical scheme adopted for achieving the purpose of the invention is as follows: an on-line soft measurement method for the arc length of a three-phase arc furnace is characterized by comprising the following steps:
(a) to control the period TcAnd N groups of input and output data of the electrode system are periodically acquired:
the electrode system of the three-phase electric arc furnace comprises three single-input single-output regulating valves, three single-input single-output electrode lifting plunger oil rods and a three-input three-output alternating current electric arc, and input data is k sampling timeActually measured A-phase control voltage value u sent by carving electrode controllera(k) B-phase control voltage value ub(k) And C-phase control voltage value uc(k) The output data is the actual measurement A phase line current effective value i at the kth sampling momenta(k) B phase line current effective value ib(k) And C phase line current effective value ic(k)；
(b) Abnormal data detection and processing:
abnormal data needs to be judged, and the lower interception point of each data is calculated to be Q1-1.5R1The upper truncation point is Q3+1.5R1Wherein Q is1、Q3Respectively lower and upper quartile, R1＝Q3-Q1Comparing the data with the cut-off points one by one, wherein the data smaller than the lower cut-off point or larger than the upper cut-off point are abnormal data; then, the data mean value is used for replacing abnormal data to process the abnormal data;
(c) data normalization processing:
because the numerical range of actually measured control voltage value is 0 ~ 10V, and the numerical range of actually measured three-phase line current virtual value is 0 ~ 20000A, in order to eliminate the influence of dimension, to data normalization processing do:
wherein u isi maxAnd ui minIs the maximum and minimum values of the measured control voltage values of the ith phase in N groups of samplesi maxAnd ii minIs the maximum and minimum values of the effective value of the i-th phase measured line current in the N groups of samples, uI label(k) Is the ith phase actual measurement control voltage value i after the normalization processing of the kth sampling timeI label(k) The effective value of the current of the ith phase actual measurement line after the normalization processing at the kth sampling moment;
(d) determining electrode system model structure parameter nf、ng、nhAnd nhj：
According to the actual structure of the three-phase arc furnace electrode system, the mathematical description is as follows:
wherein i is a, b, c, uI label(k) As input to the model, ii mould(k) The output quantity of the model is the effective value of the ith phase line current, x, calculated by the electrode system model at the kth sampling momenti(k) For the amount of oil added to the hydraulic cylinder, v, which is practically unmeasurable for the ith phase of the kth sampling timei(k) Determining polynomial basis function order n for the ith phase of the k sampling time, wherein the arc length can not be measured actually, and the requirements of model precision and solving real-time property are comprehensively consideredfOrder n of the pulse transfer function of 3gIs 4, in vi(k) Polynomial vector basis function H as an argumentj(k) Order n ofhFor the number of elements contained in the 3 and jth polynomial vector basis functionsTo 3, the unknown parameters in the model areAnd the representation of the real number field is performed,to representA real number matrix domain is maintained;
(e) predicting error function best with modelSmall as target, solving for unknown parameters α in the electrode system modelij、hijAnd CjFor N sets of sample data, the following model prediction error matrix is defined:
defining a model prediction error function as
In the formula, the superscript T represents transposition operation, | | represents determinant of matrix, and the solution of the formula (4) adopts the matrix separable least square algorithm as follows:
① model parameterization
Conversion of formula (2) to
IDie(k)＝[ia mould(k) ib mould(k) iC mould(k)]＝φ(θ,u,k)β (5)
Wherein,i is a, b, c, y is 1,2,3, where y is 1 when i is a, 2 when i is b, and 3 when i is c,
is oneThe real number row vector of the dimension,Cj(y,: represents a matrix CjAll elements in line y, j ═ 1, …, nh，
Polynomial vector basis function HjThe independent variable of (theta, u, k) is
Wherein:
are all ngnfLine vector of dimension, θ ═ θaθbθc]TIs 3ngnfA column vector of dimensions.
② objective function conversion
I in formula (4) after model parameterizationMode NIs described as
IMode N＝ψ(θ,u)β (7)
Wherein,the unknown parameters in the model are composed of two parameter sets theta and β, and the formula (4) containing two parameter sets is converted into a form containing one parameter set by variable projection
Wherein psi+(theta, u) is a Moore-Penrose generalized inverse matrix of the matrix psi (theta, u), and the Moore-Penrose generalized inverse matrix Chinese is a Moore-Penrose generalized inverse matrix which is oneAn inverse matrix, orthogonal projection of a linear space spanned by the columns of the matrix ψ (θ, u) is Pψ＝ψ(θ,u)ψ+(θ, u) and the projection of the orthogonal complement space of the matrix ψ (θ, u) isWhere I is the identity matrix, then formula (8) is described as
Is provided withIs r2(theta) value at which the minimum value is obtained, i.e. theta
③ solving forAnd
the solving process is an iterative searching process and comprises the following steps:
the first step is as follows: selecting each element of theta as 1 and defining theta as(beginning)Let θ(old)＝θ(beginning)；
The second step is that: will theta(old)Substituted in formula (9), r is calculated2(θ(old))；
The third step: will r is2(θ(old)) Substituted into the search termination condition expression (11),
wherein epsilon1Is an artificially set electrode system model tolerance index, and L is a model prediction error function r2The Cholesky decomposition factor of the Hessian matrix is a Hessian matrix which is a square matrix consisting of second-order partial derivatives of real-valued functions with arguments as vectors, the Cholesky decomposition Chinese is a Cholesky decomposition, the decomposition factor L is a lower triangular matrix with diagonal elements as positive numbers, η is a search step length meeting an Armijo-Goldstein criterion, the Armijo-Goldstein is a line search criterion during optimization calculation, delta is a Newton method search direction, and | | | | | I | M is a search direction2Is the 2-norm, n, of the matrixθ＝3nfngIs the number of parameters included in the unknown parameter set theta, N is the number of sets of input and output data of the electrode system acquired in step (a), and if equation (11) holds, thenGo to the fourth step. Otherwise, using search iteration (12),
θ(New)＝θ(old)+ηδ (12)
Determining theta(New)Let θ(old)＝θ(New)Returning to the second step;
the fourth step: obtained after the search is finishedSubstitution in equation (13) to obtain a parameter set
④ parameter set decomposition
ByConstruct the following matrix
Singular value decomposition of equation (14) to
The unknown parameters of the model are obtained
Wherein when ξi1Is positive, sξIs 1 when ξi1Is negative, sξThe molecular weight of the compound is-1,
byObtaining the unknown parameters in equation (2)Is composed of
Therein, provision is made forIs represented by a matrixA matrix formed by all column elements of the ith to jth rows of (1);
(f) substituting the parameters obtained by solving into the following equation to obtain the soft measurement value of the arc length of the three-phase arc furnace:
wherein,is a soft measurement of the arc length at the kth sampling instant ui(k) For the control voltage value of the controller at the kth sampling instant,adding an estimated value of the oil quantity of the hydraulic cylinder at the kth sampling moment;
(g) and (4) judging whether to finish the arc length soft measurement according to the working requirement, if the arc length soft measurement is not required, finishing the process, and turning to the step (l). Otherwise, executing in sequence;
(h) at the current time kNow thatSequentially selecting N (N is less than or equal to N) groups of input and output data from the historical data of the electrode system as a base point, firstly carrying out abnormal data detection and processing, then carrying out data normalization processing, substituting the processed input value into an expression (2), and calculating the output value i of the modeli mould(k)；
(i) Judging whether the formula (19) is satisfied, if so, going to the step (k), and if not, sequentially executing:
wherein n is the number of input/output data sets selected in step (h), ε2Is a tolerance index of an arc length soft measurement model set manually;
(j) from (k)Now that-N) starting to collect N groups of input and output data of the electrode system, and going to step (b);
(k) waiting m control periods, and turning to the step (h);
(l) Ending the soft arc length measurement.
The online soft measurement method for the arc length of the three-phase arc furnace can fully embody real characteristics, and is convenient to operate, scientific and reasonable.
Drawings
FIG. 1 is a schematic view of a three-phase electric arc furnace;
FIG. 2 is a schematic structural diagram of a three-phase arc furnace electrode system;
FIG. 3 is a flow chart of an on-line soft measurement method for the arc length of a three-phase arc furnace;
FIG. 4 is a graph of raw sampling data for a set of 500 electrode systems of a three-phase electric arc furnace of a steel mill;
FIG. 5 is a search step size indicator graph during iterative solution;
FIG. 6 is an end condition indicator graph in an iterative solution process;
FIG. 7 is an index plot of an objective function in an iterative solution process;
graph 850 soft measurement of arc length plot at sample time;
FIG. 9 is a graph comparing an output value of an electrode model with an effective value of a measured line current after normalization.
Detailed Description
The invention is further described below with reference to the figures and examples.
Referring to fig. 1, the three-phase arc furnace has the following structure: an electrode 4 is arranged above the molten steel 6, one end of the electrode 4 is fixed on an electrode lifting plunger oil rod 7 through an electrode holder 3, the electrode 4 is connected with the output end of an electrode controller through the electrode holder 3, the electrode lifting plunger oil rod 7 and a regulating valve 8, a power supply loop is formed by the electrode 4, a short net 2 and a furnace transformer secondary side 1, and a three-phase line current effective value in the power supply loop is connected with the input end of the electrode controller through an electric energy quality analyzer. An arc 5 is generated between the electrode 4 and the molten steel 6.
Referring to the flow shown in fig. 2, the method for soft measuring the arc length of the three-phase arc furnace comprises the following specific implementation steps:
(a) to control the period TcAnd N groups of input and output data of the electrode system are periodically acquired:
as shown in FIG. 3, the electrode system of the three-phase arc furnace comprises three single-input single-output regulating valves, three single-input single-output electrode lifting plunger oil rods and a three-input three-output alternating current arc, and the oil amount x of the hydraulic cylinder is added in a dotted line frame in FIG. 3a(k)、xb(k)、xc(k) Arc length va(k)、vb(k)、vc(k) And measuring noise va(k)、υb(k)、υc(k) All are immeasurable quantities, and the input data is the measured A-phase control voltage value u sent by the electrode controllera(k) B-phase control voltage value ub(k) And C-phase control voltage value uc(k) The output data is the effective value i of the actually measured A phase line currenta(k) B phase line current effective value ib(k) And C phase line current effective value ic(k) In that respect As shown in FIG. 4, let control period TcThe acquisition electrode system 500 sets input and output data, that is, N is 500, for 10 seconds, the input data is an actually measured control voltage value, and the output data is an actually measured three-phase line current effective value.
(b) Abnormal data detection and processing:
abnormal data needs to be judged, and the lower interception point of each data is calculated to be Q1-1.5R1The upper truncation point is Q3+1.5R1Wherein Q is1、Q3Respectively lower and upper quartile, R1＝Q3-Q1Is a quadridentate range, and then the number is countedComparing the data with the cut-off points one by one, wherein the data smaller than the lower cut-off point or larger than the upper cut-off point are all abnormal data; and then the data mean value is used for replacing the abnormal data to process the abnormal data.
(c) Carrying out data normalization processing on 500 collected data according to the formula (1):
(d) determining electrode system model structure parameter nf、ng、nhAnd
initial model of column write electrode system, determining vector basis function as polynomial form and nf＝3，ng＝4，nh＝3。
(e) Obtaining a model prediction error function formula (4) according to the model prediction error matrix defined by the formula (3), and firstly parameterizing the model into a formula (5), namely
IDie(k)＝[ia mould(k) ib mould(k) iC mould(k)]＝φ(θ,u,k)β
Wherein,
the argument of the polynomial vector basis function is equation (6), i.e.
Wherein:
are each a 12-dimensional row vector,
is a 36-dimensional column vector, i.e. nθ＝36。
Described by formula (7) IMode NThe unknown parameters in the model make up the two parameter sets θ and β.
Solving parameter sets according to the following stepsAnd
the first step is as follows: selecting theta(beginning)Is 1, let θ(old)＝θ(beginning)；
The second step is that: for simplicity, the matrix ψ (θ)(old)U) is denoted psi and is subjected to UV decomposition (a decomposition operation of the matrix) toIn the formula of UψBeing orthogonal matrices, sigmaψFor diagonal matrix, the Moore-Penrose generalized inverse matrix of psi isAnd projectFactor(s)And I is an identity matrix. Thus, the formula (8) is converted to the formula (9), and r is obtained2(θ(old))；
The third step: predicting the error matrix Z by the modelN(θ(old)) Is marked as ZNIt is subjected to QR decomposition (a decomposition operation of a matrix) intoIn the formula,is an orthogonal matrix, and the matrix is,is an upper triangular matrix, thent is a matrixRank of (i.e.) Representation matrixSquare operation of diagonal elements, ii represents product operation, and model prediction error matrix ZNThe Moore-Penrose generalized inverse ofWherein Q is a matrixThe first t column of (1), R is a matrixFirst t rows of (1), matrixThe matrix of the remaining columns is denoted Q*。
Model prediction error matrix ZNContains 36 unknown parameters, the p-th parameter of which is derived as
Wherein psi(p)Indicating the derivation of the p-th parameter of the function ψ.
Model prediction error function r2(θ(old)) Has a gradient vector of ω whose p-th element isWherein tr 2]Representing the traces of the matrix.
r2(θ(old)) The Hessian matrix is marked as omega, and the p-th row and q-th column of the Hessian matrix are
Calculating r2(θ(old)) If the eigenvalues of the Hessian matrix Ω are all positive values, it is determined that the search direction is δ — Ω-1ω. If the characteristic value has a negative value, finding the negative characteristic value with the maximum absolute value, marking as lambda, and determining that the search direction is delta- (omega-2 lambda I)-1ω, where I is the identity matrix.
To r2(θ(old)) The Hessian matrix Ω of (a) is subjected to Cholesky decomposition, i.e. it is expressed as the product of a lower triangular matrix and the transpose of this lower triangular matrix. When the characteristic values of Ω are all positive values, Ω ═ LLT. When the characteristic value of Ω has a negative value, Ω -2 λ I ═ LLT。
The search step η is determined as follows:
step 1: select search interval of [0,1]，ρ＝0.25，γ＝1.2，η(0)0.001, ordertime＝0；
And 2, step 2: checking formula r2(θ(old)+η(time)δ)≤r2(θ(old))+ρω(θ(old))Tη(time)If delta is true, turning to step 3; otherwise, it ordersTurning to step 5;
and 3, step 3: checking formula r2(θ(old)+η(time)δ)≥r2(θ(old))+(1-ρ)ω(θ(old))Tη(time)If delta is true, stopping iteration, η being η(time)Turning to step 6; otherwise, it orders
And 4: if it isGo to step 5, otherwise, order η(time+1)＝γη(time)Time is equal to time +1, and the step 2 is switched;
and 5: getChanging time to time +1, and turning to step 2;
step 6 ends the search at step η.
Setting tolerance index epsilon of electrode system model1When the result is 0.01, it is verified whether or not the formula (11) is satisfied, and if so, it is determinedGo to the fourth step. Otherwise, using equation (12), θ is obtained(New)Let θ(old)＝θ(New)Returning to the second step;
the fourth step: obtained after the search is finishedSubstituting formula (13) to obtain a parameter set
The search step size, termination condition and objective function index in the iterative solution process are shown in fig. 5, 6 and 7.
From the formula (14) — (17), the decomposition yields the following parameters:
TABLE 1 Regulation valve parameters
| a phase regulating valve parameter | b phase regulating valve parameter | c phase regulating valve parameter |
| -2156.54 | 2325.27 | -836.02 |
| -2295.12 | 2439.03 | -864.14 |
| -1103.50 | 1216.99 | -447.42 |
TABLE 2 electrode Lift plunger rod parameters
| a-phase plunger rod parameters | b-phase plunger rod parameters | c-phase plunger rod parameters |
| 0.38 | 0.37 | 0.53 |
| 0.27 | -0.11 | 0.15 |
| -0.86 | -0.77 | -0.82 |
| 0.21 | 0.51 | 0.15 |
TABLE 3 arc parameters
(f) Substituting the model parameters obtained by solving into formula (18) to obtain a three-phase arc furnace arc length soft measurement model:
the measured control voltage values are sequentially substituted into the model to obtain a soft measurement value of the arc length, which is shown in fig. 8 as a soft measurement value of the arc length at 50 sampling moments.
(g) If the electric arc furnace is in the working stage of the reduction period at this moment, the soft measurement of the arc length is needed, and the soft measurement process of the arc length cannot be finished and is sequentially executed. And (e) if the electric arc furnace is in the working stage of the arc striking period or the well penetrating period, the arc length does not need to be measured softly, and the step (l) is carried out.
(h) At the current time kNow thatAs a base point, 100 sets of input and output data are sequentially selected from historical data of the electrode system, abnormal data detection and processing are firstly performed, then data normalization processing is performed, the processed input values are substituted into formula (2), the output value of the model is calculated, and the comparison with the actual measurement three-phase line current effective value after the normalization processing is shown in fig. 9.
(i) Whether the soft measurement model needs to be updated is determined according to equation (19). The left calculation value of formula (19) is 0.0504, epsilon is selected in this example20.05, equation (19) does not hold, and the soft measurement model needs to be updated and executed sequentially. If ε is chosen in this example20.06, the soft measurement model does not need to be updated, and the step (k) is proceeded to.
(j) From (k)Now that100) starting to collect the input and output data of the electrode system 500, and going to step (b).
(k) And (5) waiting for 20 control periods, and turning to the step (h).
(l) Ending the soft arc length measurement.
Claims (1)
1. An on-line soft measurement method for the arc length of a three-phase arc furnace is characterized by comprising the following steps:
(a) to control the period TcAnd N groups of input and output data of the electrode system are periodically acquired:
the electrode system of the three-phase electric arc furnace comprises three single-input single-output regulating valves, three single-input single-output electrode lifting plunger oil rods and a three-input three-output alternating-current electric arc, and input data are actually measured A-phase control voltage values u sent by an electrode controller at the kth sampling momenta(k) B-phase control voltage value ub(k) And C-phase control voltage value uc(k) The output data is the actual measurement A phase line current effective value i at the kth sampling momenta(k) B phase line current effective value ib(k) And C phase line current effective value ic(k)；
(b) Abnormal data detection and processing:
abnormal data needs to be judged, and the lower interception point of each data is calculated to be Q1-1.5R1The upper truncation point is Q3+1.5R1Wherein Q is1、Q3Respectively lower and upper quartile, R1＝Q3-Q1Comparing the data with the cut-off points one by one, wherein the data smaller than the lower cut-off point or larger than the upper cut-off point are abnormal data; then, the data mean value is used for replacing abnormal data to process the abnormal data;
(c) data normalization processing:
because the numerical range of actually measured control voltage value is 0 ~ 10V, and the numerical range of actually measured three-phase line current virtual value is 0 ~ 20000A, in order to eliminate the influence of dimension, to data normalization processing do:
wherein u isimaxAnd uiminIs the maximum and minimum values of the measured control voltage values of the ith phase in N groups of samplesimaxAnd iiminIs the maximum and minimum values of the effective value of the i-th phase measured line current in the N groups of samples, uI label(k) Is the ith phase actual measurement control voltage value i after the normalization processing of the kth sampling timeI label(k) The effective value of the current of the ith phase actual measurement line after the normalization processing at the kth sampling moment;
(d) determining electrode system model structure parameter nf、ng、nhAnd
according to the actual structure of the three-phase arc furnace electrode system, the mathematical description is as follows:
wherein i is a, b, c, uI label(k) As input to the model, ii mould(k) The output quantity of the model is the effective value of the ith phase line current, x, calculated by the electrode system model at the kth sampling momenti(k) For the amount of oil added to the hydraulic cylinder, v, which is practically unmeasurable for the ith phase of the kth sampling timei(k) Determining polynomial basis function order n for the ith phase of the k sampling time, wherein the arc length can not be measured actually, and the requirements of model precision and solving real-time property are comprehensively consideredfOrder n of the pulse transfer function of 3gIs 4, in vi(k) Polynomial vector basis function H as an argumentj(k) Order n ofhFor the number of elements contained in the 3 and jth polynomial vector basis functionsTo 3, the unknown parameters in the model areAnd the representation of the real number field is performed,to representA real number matrix domain is maintained;
(e) solving for unknown parameters α in the electrode system model with the minimum model prediction error function as a targetij、hijAnd Cj，
For N sets of sampled data, the following model prediction error matrix is defined:
defining a model prediction error function as
For the solution of equation (4), the matrix separable least squares algorithm is used as follows:
① model parameterization
Conversion of formula (2) to
IDie(k)＝[ia mould(k)ib mould(k)iC mould(k)]＝φ(θ,u,k)β (5)
Wherein,i is a, b, c, y is 1,2,3, where y is 1 when i is a, 2 when i is b, and 3 when i is c,
is oneThe real number row vector of the dimension,representation matrix CjAll of the elements of the y-th row,
polynomial vector basis function HjThe independent variable of (theta, u, k) is
Wherein: are all ngnfLine vector of dimension, θ ═ θaθbθc]TIs 3ngnfA column vector of dimensions;
② objective function conversion
I in formula (4) after model parameterizationMode NIs described as
IMode N＝ψ(θ,u)β (7)
Wherein,the unknown parameters in the model are composed of two parameter sets theta and β, and the formula (4) containing two parameter sets is converted into a form containing one parameter set by variable projection
Wherein psi+(θ, u) is a Moore-Penrose generalized inverse of the matrix ψ (θ, u), and the linear spatial orthogonal projection formed by the columns of the matrix ψ (θ, u) is Pψ＝ψ(θ,u)ψ+(θ, u) and the projection of the orthogonal complement space of the matrix ψ (θ, u) isWhere I is the identity matrix, then formula (8) is described as
Is provided withIs r2(theta) takingTo a minimum value of theta, i.e.
③ solving forAnd
the solving process is an iterative searching process and comprises the following steps:
the first step is as follows: selecting each element of theta as 1 and defining theta as(beginning)Let θ(old)＝θ(beginning)；
The second step is that: will theta(old)Substituted in formula (9), r is calculated2(θ(old))；
The third step: will r is2(θ(old)) Substituted into the search termination condition expression (11),
wherein epsilon1Is an artificially set electrode system model tolerance index, and L is a model prediction error function r2The Cholesky factorization of Hessian matrix of η is the search step size satisfying the Armijo-Goldstein criterion, δ is the Newton's search direction, nθ＝3nfngIs the number of parameters included in the unknown parameter set theta, N is the number of sets of input and output data of the electrode system acquired in step (a), and if equation (11) holds, thenGoing to the fourth step, otherwise, using search iteration (12),
θ(New)＝θ(old)+ηδ (12)
Determining theta(New)Let θ(old)＝θ(New)Returning to the second step;
the fourth step: obtained after the search is finishedSubstitution in equation (13) to obtain a parameter set
④ parameter set decomposition
ByConstruct the following matrix
Singular value decomposition of equation (14) to
The unknown parameters of the model are obtained
Wherein when ξi1Is positive, sξIs 1 when ξi1Is negative, sξThe molecular weight of the compound is-1,
byObtaining the unknown parameters in equation (2)Is composed of
Therein, provision is made forIs represented by a matrixA matrix formed by all column elements of the ith to jth rows of (1);
(f) substituting the parameters obtained by solving into the following equation to obtain the soft measurement value of the arc length of the three-phase arc furnace:
wherein,is a soft measurement of the arc length at the kth sampling instant ui(k) For the control voltage value of the controller at the kth sampling instant,adding an estimated value of the oil quantity of the hydraulic cylinder at the kth sampling moment;
(g) judging whether to finish the arc length soft measurement according to the working requirement, if the arc length soft measurement is not needed, finishing the process, turning to the step (l), otherwise, sequentially executing;
(h) at the current time kNow thatSequentially selecting N (N is less than or equal to N) groups of input and output data from the historical data of the electrode system as a base point, firstly carrying out abnormal data detection and processing, then carrying out data normalization processing, substituting the processed input value into an expression (2), and calculating the output value i of the modeli mould(k)；
(i) Judging whether the formula (19) is satisfied, if so, going to the step (k), and if not, sequentially executing:
wherein n is the number of input/output data sets selected in step (h), ε2Is a tolerance index of an arc length soft measurement model set manually;
(j) from (k)Now that-N) starting to collect N groups of input and output data of the electrode system, and going to step (b);
(k) waiting m control periods, and turning to the step (h);
(l) Ending the soft arc length measurement.
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| CN107643477B (en) * | 2017-10-20 | 2023-09-05 | 中国恩菲工程技术有限公司 | Arc length detector for arc-starting electric furnace |
| CN111579940A (en) * | 2020-05-06 | 2020-08-25 | 国网山东省电力公司电力科学研究院 | Electric arc furnace modeling and harmonic wave analysis method and system |
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