Sensorless control of submersible permanent magnet synchronous motor under long-line cable model

In the process of oil production, the submerged oil motor works 1-3 km downhole and is connected to the inverter by a long-wire cable, which generates overvoltage at the motor end due to the problem of matching the impedance of the motor with the impedance of the power cable. The downhole environment is not suitable for the installation of precision devices, such as position sensors. To solve these problems, a sensorless control method of submerged oil motors based on a high-frequency voltage injection method is proposed. A motor terminal voltage and current observer under a long-wire cable model is designed to estimate the actual values of motor terminal voltage and current by collecting the voltage and current at the inverter terminal. A three-level inverter is used to provide energy to the submersible oil electric pump through the long-wire cable, which can effectively suppress the motor terminal overvoltage compared to the two-level inverter. The control algorithm is verified by modeling the whole system using simulation software.


Introduction
Submersible PMSM as the power input device of the submersible oil pump, the motor control performance will affect the stability of the submersible oil pump and oil recovery efficiency.Compared with induction motors, submerged PMSM has fast response speed, high reliability, lightweight, and high power factor, which are suitable for occasions with higher demand for steady-state accuracy and larger demand for speed regulation range [1] .However, submerged oil motors generally work under 1-3 km oil wells and face harsh environments such as high temperatures and high pressure, and their special structure determines that submerged permanent magnet synchronous motors cannot be installed with position sensors.Therefore, how to successfully apply sensorless control technology to the submerged oil motor system to realize the energy-saving and efficient exploitation of oil has become an urgent problem to be solved [2] .
At present, domestic and foreign research on sensorless control methods are mainly divided into two categories: One is based on the convex polarity of the permanent magnet synchronous motor to achieve the extraction of rotor position and speed information, which can be divided into high-frequency rotating voltage injection [3] and high-frequency pulsating voltage injection [4] according to the difference of the injected coordinate system.The other category is to process the magnetic chain or reverse potential with the help of the mathematical model of the motor, and such methods have better estimation effect on the high-speed sections, including the magnetic chain estimation method, the model reference adaptive method [5] , the sliding mode observer method [6] , the extended Kalman filter method [7] , etc.
The SVPWM algorithm is combined to control the turn-on and turn-off of IGBT.Due to the problem of matching the motor impedance to the power cable impedance, an overvoltage of twice the DC bus voltage is generated at the motor end.The power cable can be modeled by resistance, inductance, capacitance, and conductance.Since the impedance of the motor is significantly larger than the characteristic impedance of the power cable, a reflected wave is generated at the motor end, which in turn is superimposed on the incident wave, resulting in an overvoltage.The amplitude of the overvoltage at the motor side depends on the rise time of the voltage pulse, and the larger the rise time, the larger the amplitude of the overvoltage at the motor side.Many scholars have done research on the overvoltage at the motor end of long power cables, and the most popular one is to use a low-pass filter based on / dv dt to match the characteristic impedance of long power cables and increase the rise time of the voltage pulse.This paper adopts the sensorless control method of HFI to have good control performance of PMSM at low and zero speeds.Coupled with the fact that multilevel technology is getting more and more attention in the field of motor drive, multilevel technology has the advantages of small switching loss, fewer harmonics, and the ability to increase the capacity of the system compared with the traditional two-level technology.Therefore, in this paper, a diode-clamped three-level inverter is used to drive a PMSM with a long power cable.

Voltage/current observer
A simplified schematic diagram of a three-phase PMSM fed by a voltage source inverter controlled by SVPWM technology is given in Figure . 1 Voltage source inverter-fed PMSM The mathematical model of a PMSM in the dq − coordinate system can be expressed as: The mechanical equation of the PMSM is: Where J is the moment of inertia; p N is the polar logarithm; L T is the load torque; f is the mechanical damping coefficient; e T is the electromagnetic torque.( ) According to the single-phase circuit for long cable shown in Figure .2.

Figure. 2. Equivalent circuit for long cables
The instantaneous voltages and currents at length x of the long cable can be obtained as: The output of the three-level inverter AC power is: .It can be obtained by substituting equation (5) into equation ( 4) and simplifying: Where 0 Z is the impedance of cable; 0 Y is the conductance of cable.For Equation ( 6) derivation on both sides is obtained: Substituting Equation (6) into Equation (7) : The voltage and current at the beginning of the cable are: From Equations ( 9) -( 11), the voltage and current at any length x of a long cable can be expressed as:

Position sensorless control
The block diagram of the sensorless control of PMSM under the long-wire cable model is shown in

Figure. 3. Block diagram of control without position sensor
The application of the HFI method avoids the need to install mechanical sensors on the PMSM, and the position of the rotor of the PMSM can be accurately estimated by measuring the current in the cable.
Injecting a high-frequency square-wave voltage signal into the d-axis of the rotating coordinate system: The voltage drop is negligible, so the model of the motor can be simplified as: Substituting Equation (13) into Equation ( 14) and transforming to the a-b coordinate system, the response current under high-frequency voltage excitation can be expressed as: Where h  is the square wave voltage frequency; 1 ( is the inductance difference.The difference is used instead of differentiation and a sign function is added to determine the polarity of the square wave from which the envelope of the high-frequency current is extracted: The rotor position observer method is used to extract the rotor position information from the envelope, and its realization block diagram is shown in Figure .4. The position error information is obtained by the outlier method when err  is small sin

Simulation results analysis
The whole system is simulated by MATLAB / Simulink software.The sampling time of the system is The rotor position is obtained by injecting a high-frequency square-wave voltage to drive the PMSM with a long-wire cable, and the injected high-frequency voltage has an amplitude of 80 V and a frequency of 5 kHz.The initial rotational speed is given as 500 rpm, and the system tends to be stabilized after 0.1 s.Throughout the simulation process, both detected and estimated speeds follow the speed given, and the rotational speed error is kept within 1.5 rpm.The speed error is kept within 1.5 rpm, shown in Figure .5. The rotor position and its error are shown in Figure .6, the maximum position error is less than 0.1 rad, and the accuracy of the estimated rotor position is high.It can meet the engineering needs.Figure .7 shows the estimated motor terminal voltage and current.Figure .8 compares the performance of the three-level inverter with that of the two-level inverter, and it can be seen that the overvoltage at the motor terminals is controlled to within twice that of the DC link, which makes it more suitable for transmission over long power cables.

Conclusion
The correctness and the superiority of the proposed position sensor-less control method for the oilsubmerged PMSM are verified by simulation.
(1) Simulation results show that the method can control the error within 1.5 rpm.It is suitable for the working condition that the submerged oil PMSM has a large starting torque and often works at low speed and high torque.
(2) The proposed motor terminal voltage / current observer based on the long-wire cable estimation model can effectively estimate the actual values of motor terminal voltage and current by utilizing the voltage and current at the inverter terminal.
(3) The three-level inverter provides energy for the PMSM through a long power cable, which can control the overvoltage within two times of the DC link, which reduces the motor insulation pressure and the generation of motor heat.

1 )
Where dq uu 、 are the dq − axis voltage; dq ii 、 are the dq − axis current; d q LL 、 are the stator inductance on the dq − axis; e K is the reverse electromotive force coefficient; and s R is the stator resistance.
substituting  into Equation (10), its generalized solution can be expressed as:

Figure. 3 .
The magnetic field orientation control can realize the decoupling control of excitation and torque.Let the excitation reference current 0 d i = , and the torque e T is only controlled by the torque current q i from Equation (3).The speed closed loop is realized by a proportional-integral controller, which can overcome the instability problem.

Table 1
PMSM Motor Parameters