Research on electrode lifting system based on recursive least square method and fuzzy PID

The electrode of calcium carbide furnace bears the function of conduction, heat transfer and furnace condition regulation. How to keep the furnace condition in the best condition by lifting and lowering the electrode has always been a hot and difficult point in the research. In this paper, based on the analysis of the equivalent circuit of calcium carbide furnace, the system model of electrode position-current is identified by the recursive least square method, and a fuzzy PID controller based on the constant control strategy of electrode current is designed to keep the furnace condition in the best state. The simulation and application results show that the designed system has good dynamic response, anti-disturbance ability and robustness.


Introduction
At present, the electrode lifting operation of calcium carbide furnace is realized through the hydraulic transmission device. The electrode lifting operation is divided into single-phase separate operation and three-phase simultaneous operation. The secondary side of the furnace transformer is usually triangular wiring mode [1][2][3]. When a phase electrode is operated separately, it will affect the arc current of the other two phases, which is easy to cause three-phase imbalance. The strong coupling of the calcium carbide furnace is reflected in this point. When the three-phase electrode is operated up and down at the same time, the three-phase electrical power balance can be maintained (it is assumed that the three-phase equilibrium state is maintained before the operation of the calcium carbide furnace), and the pressure of the hydraulic station (including the energy accumulator) is required to support the simultaneous action of the three-phase electrode [4][5][6].
Electrode lifting range is determined by the condition of calcium carbide furnace. In normal production, the electric arc current in the furnace can be adjusted through the electrode lifting operation, and the input electric power can be changed, so as to improve the direction of the furnace condition to the furnace temperature stability change; during maintenance work, the electrode end needs to be lifted to the height of the charge surface; during the operation of the oven, the electrode end needs to be pressed to a certain height from the bottom of the furnace [7][8][9][10]. Therefore, in a sense, the position of the electrode determines the condition of the calcium carbide furnace.
It is difficult to establish a definite relationship between electrode position and arc current because of the time-varying and random characteristics of calcium carbide furnace. It is found in the field operation that the closer the electrode end is to the liquid surface of the molten pool, the greater the electrode current is. Based on this, this paper intends to establish the system model of electrode positioncurrent by using the method of system identification, and then design the model into the fuzzy PID control system.The first paragraph after a heading is not indented (Bodytext style).

Short-net connection mode of calcium carbide furnace
Short network refers to the general term of all kinds of connected conductors from the outlet end of the secondary side of the calcium carbide furnace transformer to the electrode. The commonly used connection mode of short network is shown in Figure 1. Fig. 1 (a) shows the connection mode between a three-phase furnace transformer and the electrode. The transformer is installed near a certain electrode, and due to the asymmetry of the physical structure of the three-phase short grid, the three-phase impedance is deviated, resulting in the unbalanced threephase load. In addition, three-phase transformer is used for power supply. The non-linearity of transformer and the coupling between phases have a great influence, which leads to the decrease of electric heating efficiency and the increase of power loss of calcium carbide furnace. Fig. 1 (b) shows the connection mode between three single-phase furnace transformers and electrodes. The transformer is symmetrically installed near the electrode and the short network is symmetrically arranged, which can effectively reduce the inductive reactance of the short network and thus eliminate the unbalance degree of the three-phase load. In addition, three single-phase transformers are used for power supply, which can greatly reduce the non-linearity of the transformer and the coupling influence between phases, thus improving the electric heating efficiency of the calcium carbide furnace and reducing the power loss. The disadvantage is that three single-phase transformers need to be equipped, and the cost is higher.  Figure 1 Short-grid connection mode of calcium carbide furnace

Equivalent circuit of calcium carbide furnace
The equivalent circuit of the secondary side winding of the electric furnace transformer, the watercooled soft cable, the water-cooled conductive cross arm, the charge between electrodes and the working conditions in the furnace are represented by the equivalent resistance, as shown in Fig. 2. Thus, the total equivalent resistance between electrodes A, B and C is: The above equivalence and conversion are carried out on the premise that the smelting condition is stable and the resistance value of each kind is equal.

Equivalent circuit of calcium carbide furnace
In the total equivalent resistance between electrodes A、 B and C, the transverse charge resistance 、 and change with the change of the smelting condition in the furnace, and the range of change is large, and there will be A jump when the material collapse. Longitudinal charge resistance A, C and C  show periodic changes with the process of smelting and discharging, and the resistance value is the minimum before discharging, and gradually increases with the decrease of molten pool liquid level in discharging process. In the process of electrode discharge, the transverse charge resistance and the longitudinal charge resistance are both greatly disturbed.
The arc current accounts for most of the electrode current, and the arc current changes with the change of the total equivalent resistance between the electrodes. Therefore, in the actual control, the amplitude of the electrode current is always used as the basis for adjusting the smelting condition of calcium carbide furnace.

Modeling of electrode position and current identification
Calcium carbide furnace condition adjustment means transformer load regulation (voltage regulator switch), electrode rise and fall, charge resistivity changes. A voltage regulator is usually used for a wide range of load adjustment; The change of the resistivity of the charge is related to the ratio of raw materials, which is usually not used as a method of load adjustment. For small adjustment of load, electrode lifting operation is generally used.
The electrode control requirements of calcium carbide furnace are that the electrode position follows the level of molten pool liquid, that is, before the furnace, the electrode position is also high when the molten pool liquid level is high. After discharging, the electrode position of the molten pool should also drop when the liquid level drops. With the continuous smelting of calcium carbide, the liquid level of the molten pool increases gradually, and the electrode position also increases slowly. The basis of the lifting and lowering of the electrode is to keep the electrode current basically constant. However, there is no definite model between the electrode position and the electrode current. In this paper, the system identification method is adopted to establish the relationship between the two.

Recursive least square method
Recursive least square method is a method to estimate the actual output of the system at the present moment based on the past operation data. The structure diagram of electrode position-current identification system of calcium carbide furnace is shown in Fig. 3.  Figure 3, and are the discrete input and output of system model , representing the K-th electrode position control quantity and electrode current respectively. Noise model is mainly the interference factor , representing the disturbed conditions such as carbide furnace collapse, discharge and electrode pressure discharge.
Recursive least square method is a method based on the ARX model, which is of the form: (3) It can also be written (4) Which  (4) is written in the least square format as: The least squares estimation requires the minimum sum of squares of the deviation, that is, its objective function is: The least squares estimate of is obtained by taking the partial derivative of with respect to , and setting equal to 0. At time k-1 and time k, the parameter estimation result of the system is： (9) Which, and are estimated values of least squares parameters that can be calculated and obtained according to the first and the first observation/sampling data respectively. The key to the calculation is the recursive calculation of matrix inverse. The calculation derivation process can be seen in Literature [10], and the final calculation formula is： (10) The order of calculation is first, then .

System parameter estimation
The operation data of a calcium carbide furnace collected on site are shown in Table 1.  Table 1 are all recorded when the smelting condition of calcium carbide furnace is relatively stable, which is characterized by basic balance of three phases.
System parameter estimation steps :(1) Collection/pretreatment of sample data; (2) Determine the model structure and order; (3) Recursive least square method was used to estimate model parameters; (4) Carry out reliability test on the identification model and compare the corresponding data of the test method; (5) The electrode position-current model parameters of three-phase electrodes A, B and C were identified by this method.

Electrode position-current model
According to the above model identification method, the relationship model between three-phase electrode position and electrode current of calcium carbide furnace can be written as follows: is the delay operator， is the delay of the model， 、 and are the current of three-phase electrode at time， 、 、 denotes the position of three-phase electrode at time ， 、 are the model parameter matrices.
The identification model was selected as ARX model with order 2, that is, . In Matlab, parameters for off-line identification based on measured data are as follows:

Control system design
In this paper, the idea of constant electrode current is used to control the rise and fall of the electrode. The fuzzy PID control system is shown in Fig. 4.  The fuzzy rules formulated are shown in Table 2.