CN114660941B - Alternating current electric arc furnace electrode system identification method based on hierarchical identification principle - Google Patents
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Abstract
The invention provides an alternating current electric arc furnace electrode system identification method based on a hierarchical identification principle, and belongs to the technical field of alternating current electric arc furnace electrode system identification. The problem of lower model precision caused by over-simplification of a real electrode system structure is solved. And the hierarchical identification is applied to the model, so that the identification precision is further improved. The technical scheme is as follows: the method comprises the following steps: step 1) establishing a single-input single-output Hammerstein-Wiener model of an alternating current electric arc furnace electrode system; and 2) constructing a hierarchical identification process of maximum likelihood least squares and random gradients. The beneficial effects of the invention are as follows: the maximum likelihood least square hierarchical identification algorithm provided by the invention has higher convergence speed and higher convergence precision, and can be better suitable for modeling and parameter identification of an alternating current electric arc furnace electrode system.
Description
Technical Field
The invention relates to the technical field of alternating current electric arc furnace electrode system identification, in particular to an alternating current electric arc furnace electrode system hierarchical identification method based on a maximum likelihood least square algorithm and a random gradient algorithm.
Background
In recent years, electric arc furnaces have been widely used in the metallurgical industry as the primary equipment for steel production. The electric energy is converted into heat energy by generating electric arc between an electrode and furnace materials, the conversion efficiency depends on the length of the electric arc, however, the electric arc is generated by high-temperature and high-gas conductor discharge, and the length of the electric arc is difficult to control directly in the operation process of an electric arc furnace system. The general method is to control the up-and-down movement of the electrode to maintain the arc current of the electric arc furnace at a certain set value, thereby eliminating the adverse effects of the electric arc furnace on the harmonic injection of a power grid, voltage fluctuation, flicker and the like, and simultaneously improving the quality of molten steel and reducing the power consumption of steel per ton. The essence of the control of the arc furnace is the control of the electrodes. And an effective way to control the electrodes is to model the electrode system. For this reason, many researchers have proposed different identification methods, such as: random gradient algorithm, least square algorithm, particle swarm algorithm and the like.
The random gradient algorithm has low identification precision and low convergence speed, so that the identification effect in actual production is poor; the least square algorithm has the problem of data saturation caused by the fact that the data quantity is increased in the process of tracking the time-varying parameters; although the particle swarm algorithm serving as the swarm intelligence algorithm can be well applied to different working conditions, the problems of local optimization and large calculation amount are also caused.
How to solve the above technical problems is the subject of the present invention.
Disclosure of Invention
The invention aims to provide an alternating current electric arc furnace electrode system identification method based on a hierarchical identification principle, which is a novel identification method developed based on identification model decomposition, is provided for solving the problems of complex structure, high dimension and large-scale system identification, and can be better suitable for modeling and parameter identification of an alternating current electric arc furnace electrode system.
The invention is realized by the following measures: an alternating current electric arc furnace electrode system identification method based on a hierarchical identification principle comprises the following steps:
step 1) establishing a single-input single-output Hammerstein-Wiener time lag model of an alternating current electric arc furnace electrode system;
step 2) identifying parameter vectors by using a maximum likelihood least square algorithm;
and 3) identifying the time delay by using a random gradient algorithm.
As a further optimization scheme of the alternating current electric arc furnace electrode system identification method based on the hierarchical identification principle, the specific modeling step of the step 1) is as follows:
step 1-1), constructing a single-input single-output Hammerstein-Wiener time lag system model of an alternating current electric arc furnace electrode system:
h[y(t)]=ω(t)+ξ(t) (1)
where h [ y (t) ] is the output of the system, ω (t) and ξ (t) are intermediate variables defined as follows:
ω(t)=G(z -1 )x(t) (2)
ξ(t)=E(z -1 )v(t) (3)
wherein x (t) is an intermediate variable, G (z) -1 ) Is a linear partial transfer function, z -1 Denotes the backshifting operator, v (t) is the system noise, E (z) -1 ) Is z -1 Is defined as follows:
wherein e is k Is the parameter vector of linear part to be identified, n e Is order, assumed to be known;
x (t) is defined as follows:
x(t)=f[u(t)] (5)
where u (t) is the system input and f (t) is the input nonlinear function of the system.
Step 1-2) for the nonlinear part of the inputs and outputs in the model, it is assumed to be continuous and reversible and is represented using an integration of known nonlinear functions, so f [ u (t) ] and h [ y (t) ] are represented thereby:
wherein, c s And d l The non-linear part needs to be identified parameter vector, n c And n d Is order, assumed to be known.
And for the linear part, expressed in the form of a transfer function:
in the formula (8), τ is a time delay, A (z) -1 ),B(z -1 ) Are all z -1 Is defined as follows:
wherein, a i And b j Is the parameter vector of linear part to be identified, n a And n b Is order, assumed to be known; step 1-3) deducing and obtaining the mathematical description of the system model as follows:
definition of d 1 =1, the first term d of the nonlinear function is to be output 1 h 1 [y(t)]Separating out:
for the system described in equation (12), a, b, and e are all linear part of the parameter vector, defined as follows:
c. d is a parameter vector for the nonlinear part of the system, defining c 1 =1、d 1 =1, the vector is defined as follows:
parameter vectors theta, theta bc 、θ ae 、θ d 、θ ad Is defined as follows:
wherein:
then, the information vector of the whole system is defined as follows:
wherein:
the system model can be described as: h is 1 [y(t)]=Φ T (t,τ)θ+v(t)。
As a further optimization scheme of the alternating current electric arc furnace electrode system identification method based on the hierarchical identification principle, the model in the step 1) is a model of a single-input single-output system.
The alternating current electric arc furnace electrode system identification method based on the hierarchical identification principle according to claim 1, wherein the step 2) of identifying the parameter vector by using the maximum likelihood least square algorithm comprises the following specific steps:
step 2-1) assuming maximum likelihood estimation valuesThen the taylor expansion of v (t) at this time is approximated as:
ignoring the higher order terms, approximating v (t) as:
step 2-2) taking the actually measured single-phase control voltage value sent by the electrode controller as input data of an alternating current electric arc furnace electrode system model, and taking the single-phase line current effective value as output data;
Step 2-4) gives the partial derivatives of v (t) at time t on the parameter vector θ, i.e. partial derivatives are calculated for a, b, c, d, e respectively:
step 2-5) substitutes (25), (26), (27), (28) and (29) to define a partial derivative information vector
Step 2-6) the criterion function of the system is expressed as:
J(θ,t)=J(θ,t-1)+v 2 (t) (31)
step 2-7) is obtained by Taylor formula:
wherein:
η(t) * =-r(t) T P -1 (t)r(t)+v 2 (t)+η(t-1) (35)
applying a matrix inversion formula to obtain:
step 2-8) defining a gain vector K (t):
step 2-9) substituting the gain vector K (t) into the formula (36) to obtain:
replacing the values of the parameter vector with estimated valuesTo minimize the value of the criterion function J (θ, t), we should:
therefore, the method can obtain:
step 2-10) obtaining the identification summary of the parameter vector by using the maximum likelihood least square algorithm as follows:
as a further optimization scheme of the alternating current electric arc furnace electrode system identification method based on the hierarchical identification principle provided by the invention, the step 3) of identifying the time delay by using the random gradient algorithm comprises the following specific steps:
step 3-1) defining the time delay as tau, and the estimated value asFrom the gradient descent formula
(41) Wherein alpha is a self-defined iteration step,
step 3-2) alignment function partial derivation is obtained:
wherein:
substitution of step 3-3) into formula (41)In the calculation formula (2), the identification of the time delay is summarized as follows:
and 3-4) finishing the identification and outputting an identification result.
Compared with the prior art, the invention has the beneficial effects that:
(1) The invention establishes a model for identifying the parameters of the electrode system of the alternating current electric arc furnace, takes the actually measured single-phase control voltage value sent by an electrode controller as input data, takes the effective value of the single-phase line current as output data, and identifies the parameters of the model by utilizing a hierarchical method of a maximum likelihood least square algorithm and a random gradient algorithm; as shown in FIG. 4, the method can identify the internal parameters of the model well.
(2) In the invention, a hierarchical identification method of maximum likelihood least squares and a random gradient algorithm is introduced into the identification of an alternating current electric arc furnace electrode system; for the parameter vector, a maximum likelihood least square algorithm is used, a probability model is obtained according to known input and output data, a log-likelihood function is obtained, the minimum value is obtained, and a model parameter vector is reversely deduced; for time delay, a random gradient algorithm is selected to quickly obtain a result; in terms of the overall identification step, the two methods are required to be performed alternately, namely, the time delay of the current moment is deduced by using the parameter identification result of the previous moment, and then the parameter estimation of the next moment is deduced by using the time delay of the current moment, so that the two methods are performed alternately and are suitable for the identification process of the electrode system of the alternating current electric arc furnace.
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The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention.
Fig. 1 is an overall flowchart of an alternating current electric arc furnace electrode system identification method based on a hierarchical identification principle according to the present invention.
Fig. 2 is a schematic structural diagram of an ac electric arc furnace based on a hierarchical identification principle of the method for identifying an electrode system of an ac electric arc furnace according to the present invention.
Fig. 3 is a schematic diagram of a single-input single-output Hammerstein-Wiener time lag model of an alternating current electric arc furnace electrode system identification method based on a hierarchical identification principle.
FIG. 4 is a schematic diagram of the error between the identification value and the true value of the parameter vector according to the present invention.
FIG. 5 is a schematic diagram of the error between the time delay identification value and the true value according to the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. Of course, the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example 1
Referring to fig. 1 to 4, the present invention provides a method for identifying an electrode system of an ac electric arc furnace based on a hierarchical identification principle, comprising the following steps:
step 1) establishing a single-input single-output Hammerstein-Wiener time lag model of an alternating current electric arc furnace electrode system;
step 2) identifying parameter vectors by using a maximum likelihood least square algorithm;
and 3) identifying the time delay by using a random gradient algorithm.
Preferably, the specific modeling step of step 1) is as follows:
step 1-1), constructing a single-input single-output Hammerstein-Wiener time lag model of an alternating current electric arc furnace electrode system:
h[y(t)]=ω(t)+ξ(t) (1)
where h [ y (t) ] is the output of the system, ω (t) and ξ (t) are intermediate variables defined as follows:
ω(t)=G(z -1 )x(t) (2)
ξ(t)=E(z -1 )v(t) (3)
wherein x (t) is an intermediate variable, G (z) -1 ) Is a linear partial transfer function, z -1 Denotes the postcursor operator, v (t) is the system noise, E (z) -1 ) Is z -1 Is defined as follows:
wherein e is k Is the parameter vector of linear part to be identified, n e Is order, assumed to be known;
x (t) is defined as follows:
x(t)=f[u(t)] (5)
where u (t) is the system input and f (t) is the input nonlinear function of the system.
Step 1-2) deducing and obtaining mathematical description of a system model:
f [ u (t) ] and h [ y (t) ] are represented by an integration of known nonlinear functions:
wherein, c s And d l The non-linear part needs to be identified parameter vector, n c And n d In order, it is assumed to be known.
And for the linear part, it can be expressed in the form of a transfer function:
in the formula (8), t is a time delay, A (z) -1 ),B(z -1 ) Are all z -1 The constant expression of (c), defined as follows:
wherein, a i And b j Is the parameter vector of linear part to be identified, n a And n b Is order, assumed to be known;
step 1-3) definition of d 1 =1, the first term d of the nonlinear function is to be output 1 h 1 [y(t)]Separating to obtain:
defining parameter vectors a, b and c of a linear part, parameter vectors c and d of a nonlinear part, defining parameter vectors and nonlinear parameters in a parameter vector theta, and finally defining information vectors of the whole system.
The system model can be described as: h is a total of 1 [y(t)]=Φ T (t,t)θ+v(t)。
Preferably, the model of step 1) is a model of a general single-input single-output system.
Preferably, the step 2) of constructing the maximum likelihood least square algorithm identification parameter vector specifically comprises the following steps:
step 2-1) assuming maximum likelihood estimation valuesThen the taylor expansion of v (t) at this time ignores the higher order terms and can approximate v (t) as:
step 2-2) taking the actually measured single-phase control voltage value sent by the electrode controller as input data of an alternating current electric arc furnace electrode system model, and taking the single-phase line current effective value as output data;
Step 2-4) gives v (t) the partial derivatives of the parameter vector θ at time t, i.e. partial derivatives are calculated for a, b, c, d, e respectively:
step 2-5) substitutes into (14), (15), (16), (17) and (18) to define a partial derivative information vector
Step 2-6) represents the criteria function of the system as:
J(θ,t)=J(θ,t-1)+v 2 (t)(20)
steps 2-7) are derived from the taylor formula:
wherein:
η(t) * =-r(t) T P -1 (t)r(t)+v 2 (t)+η(t-1)(24)
applying a matrix inversion formula to obtain:
step 2-8) defining a gain vector K (t):
and 2-9) substituting the gain vector K (t) into a formula (25) to obtain:
replacing the value of the parameter vector with an estimated valueTo minimize the value of the criterion function J (θ, t), we should:
therefore, the method can be obtained as follows:
step 2-10) obtaining the identification summary of the parameter vector by using the maximum likelihood least square algorithm as follows:
preferably, the step 3) of identifying the time delay by the stochastic gradient algorithm comprises the following specific steps:
Where α is a self-defined iteration step.
Step 3-2) alignment function partial derivative obtaining:
wherein:
step 3-3) into formula (31)In the calculation formula (2), the identification of the time delay is summarized as follows:
the ac electric arc furnace described in this embodiment is schematically shown in fig. 2.
With the above mentioned general single input single output model, the following model can be established for this embodiment:
f[u(t)]=c 1 [u(t)+u 2 (t)]+c 2 [u(t)+u 2 (t)]=(1+1.35)*[u(t)+u 2 (t)]
h[ξ(t)]=d 1 [E(z -1 )v(t)+[E(z -1 )v(t)] 2 ]+d 2 [E(z -1 )v(t)+[E(z -1 )v(t)] 2 ]
=(1+0.09)*[E(z -1 )v(t)] 2 ]
A(z -1 )=1+a 1 z -1 =1+0.9z -1
B(z -1 )=b 1 z -1 =0.11z -1
E(z -1 )=1+e 1 z -1 =1+0.9z -1
comparing the model with step 1), we can get:
a 1 =0.9,b 1 =0.11,c 1 =1,c 2 =1.35,d 1 =1,d 2 =0.09,e 1 =0.9
in order to conveniently substitute the parameters to be identified into the hierarchical identification method of the maximum likelihood least square algorithm and the random gradient algorithm, the parameters to be identified are combined into a parameter vector theta, and the parameters to be identified are defined as follows:
determining an approximate expression of v (t) according to step 2-1);
obtaining input and output data of the alternating current electric arc furnace electrode system model according to the step 2-2);
defining an estimated value of a parameter vector theta at the time t according to the step 2-3);
respectively solving partial derivatives for a, b, c, d and e according to the step 2-4);
Obtaining a criterion function expression according to the step 2-6);
sorting the simplification rule function according to the step 2-7) to obtain values of all parts;
defining a gain vector K (t) according to the step 2-8);
minimizing the value of the criterion function J (theta, t) according to the steps 2-9) and 2-10), and substituting to obtain a parameter vector identification result;
Calculating the partial derivative of J according to the step 3-2);
substitution into according to step 3-3)The calculation formula of (2) calculates the identification result of the time delay.
The parameter identification result of the alternating current electric arc furnace electrode system hierarchical identification method based on the maximum likelihood least square algorithm and the random gradient algorithm is shown in fig. 4 and fig. 5, and it can be seen that the identification precision of the method is high, the estimated value of the parameter to be identified is very close to the true value, and meanwhile, the identification method has good applicability to the alternating current electric arc furnace electrode system model.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and should not be taken as limiting the scope of the present invention, which is intended to cover any modifications, equivalents, improvements, etc. within the spirit and scope of the present invention.
Claims (1)
1. An alternating current electric arc furnace electrode system identification method based on a hierarchical identification principle is characterized by comprising the following steps:
step 1) establishing a single-input single-output Hammerstein-Wiener time lag system model of an alternating current electric arc furnace electrode system;
the concrete modeling steps of the step 1) are as follows:
step 1-1), constructing a single-input single-output Hammerstein-Wiener time lag model of an alternating current electric arc furnace electrode system:
h[y(t)]=ω(t)+ξ(t) (1)
where h [ y (t) ] is the output of the system, ω (t) and ξ (t) are intermediate variables defined as follows:
ω(t)=G(z -1 )x(t) (2)
ξ(t)=E(z -1 )v(t) (3)
wherein x (t) is an intermediate variable, G (z) -1 ) Is a linear partial transfer function, z -1 Denotes the postcursor operator, v (t) is the system noise, E (z) -1 ) Is z -1 Is defined as follows:
wherein e is k Is the parameter vector of linear part to be identified, n e Is order, assumed to be known;
x (t) is defined as follows:
x(t)=f[u(t)] (5)
wherein u (t) is the system input and f (t) is the input nonlinear function of the system;
step 1-2) for the nonlinear part of the inputs and outputs in the model, which is assumed to be continuous and invertible, and represented using an integration of known nonlinear functions, f [ u (t) ] and h [ y (t) ] are represented as:
wherein, c s And d l Parameter vector, n, for non-linear part to be identified c And n d Is order, assumed to be known;
and for the linear part, it is expressed in the form of a transfer function:
in the formula (8), τ is a time delay, A (z) -1 ) And B (z) -1 ) Are all z -1 The polynomial of (a) is defined as follows;
wherein, a i And b j Is the parameter vector of linear part to be identified, n a And n b Is order, assumed to be known;
step 1-3) deducing and obtaining the mathematical description of the system model as follows:
definition of d 1 =1, the first term d of the nonlinear function is to be output 1 h 1 [y(t)]Separating to obtain:
for the system described in equation (12), a, b, and e are all linear partial parameter vectors, defined as follows:
c. d is defined as the parameter vector of the nonlinear part of the system, let c 1 =1、d 1 =1, the following is defined:
parameter vectors theta, theta bc 、θ ae 、θ d 、θ ad Is defined as:
wherein:
then, the information vector of the whole system is defined as follows:
wherein:
the system model can be described as: h is a total of 1 [y(t)]=Φ T (t,τ)θ+v(t);
Step 2) identifying parameter vectors by using a maximum likelihood least square algorithm;
the step 2) of identifying the parameter vector by using a maximum likelihood least square algorithm comprises the following specific steps:
step 2-1) assuming maximum likelihood estimation valuesThen the taylor expansion of v (t) at this time is approximated as:
ignoring higher order terms, then v (t) is approximated as:
step 2-2) taking the actually measured single-phase control voltage value sent by the electrode controller as input data of an alternating current electric arc furnace electrode system model, and taking the single-phase line current effective value as output data;
Step 2-4) gives v (t) the partial derivatives of the parameter vector θ at time t, i.e. partial derivatives are calculated for a, b, c, d, e respectively:
step 2-5) substitutes into (25), (26), (27), (28) and (29) to define a partial derivative information vector
Step 2-6) the criterion function of the system is expressed as:
J(θ,t)=J(θ,t-1)+v 2 (t) (31)
step 2-7) is obtained by Taylor formula:
wherein:
η(t) * =-r(t) T P -1 (t)r(t)+v 2 (t)+η(t-1) (35)
applying a matrix inversion formula to obtain:
step 2-8) defining a gain vector K (t):
step 2-9) substituting the gain vector K (t) into the formula (36) to obtain:
replacing the value of the parameter vector with an estimated valueTo minimize the value of the criterion function J (θ, t), we should:
therefore, the method can obtain:
step 2-10) obtaining the identification summary of the parameter vector by using the maximum likelihood least square algorithm as follows:
step 3) identifying time delay by using a random gradient algorithm;
the step 3) of identifying the time delay by using the random gradient algorithm comprises the following specific steps:
step 3-1) defining a time delayRetardation is τ and the estimated value isFrom the gradient descent formula
(43) Wherein alpha is a self-defined iteration step,
step 3-2) obtaining a partial derivative of the criterion function as follows:
wherein:
substitution of step 3-3) into formula (41)In the calculation formula (2), the identification of the time delay is summarized as follows:
and 3-4) finishing the identification and outputting an identification result.
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