Emulating dynamics of sucker rod pump in laboratory

Dynamic load emulation of sucker rod pump in laboratory test bench can provide a convenient platform for conducting experiment for testing diagnostic models and verification of operational methods. This paper presents a strategy to implement a test bench for emulating the dynamic load of sucker rod pump. The emulation strategy is based on load torque estimation considering the principal concepts of electromechanical test bench. The dynamics to be emulated is determined by a real time simulator that runs the model of the sucker rod pumping unit. Test bench consisting of induction motor as driving machine and separately excited DC motor as load machine is described and the simulation results show satisfactory tracking performance.

I. INTRODUCTION

A typical design cycle begins with simplified software simulations (like MATLAB, PSCAD) to validate basic conceptions, then proceeds through increasingly deepened representation of circuit details and control functions [1]. Further, a thorough testing of the actual system is imperative in order to promote from experimental phase to production phase. However, there are some systems, which are difficult to test with the help of real equipment; instead, test benches are used to reproduce parts of the system. For example, in [2] wind turbine emulator based on separately excited dc motor have been implemented to accurately reproduce the characteristics of real wind turbine.

The aim of test benches development can be to create research environment for analysis and investigation or/and establish procedures for assessment and verification of various systems.

In electromechanical systems, test bench consists of two mechanically coupled machines: driving machine and load machine. In order to, emulate practical load characteristics, the technique that controls the machine acting as a load is required to control the load torque under any circumstance. The torque developed by the machine acting as a load can be predetermined if the dynamic performance of the application is not critical (open loop system), otherwise, it is obtained from measurement of the speed or position of the driving machine (closed loop system). In [4-5] the inverse mechanical dynamic principle was used to imitate satisfactorily practical load characteristics.  This principle allowed the machine acting as load to follow the desired trajectories by using the measured shaft speed as an input. The inverse mechanical dynamic principle is effective in applications employing continuous time systems. However, for discrete time implementation the stability is easily affected if the effects of sampling are considered. Stability analysis in [6] has shown the effect of sampling time on stability and indicated that its emulation applicability is limited to an inertia less than twice the total inertia of the test bench. Digital filters might be adequate to stabilize the system, but the emulation gives incorrect results because the overall pole-zero structure of the closed loop system will be different from the pole-zero structure of the desired mechanical load. Therefore, the stability and acceptability of time response of the test bench are not enough to produce a torque speed characteristic as close as possible to the desired torque speed characteristic. In [6-7], control strategies, which use feed forward and compensator that is independent of emulation model have been implemented. This approach preserves pole-zero structure of the desired dynamics and is applicable for both linear and nonlinear loads dynamics. The authors in [8-9] have avoided the using inverse transfer functions and compensator terms by using torque estimates. This is help full to simplify the method and reduce the noise problems due to measurements. The approaches presented in [3,10] used the disturbance-observer-based emulation scheme in order to consider the effect of disturbances or plant uncertainties and the experimental results show the robustness of the method.

The power requirement and profitability of sucker rod pumps are basically determined by the torque load on the pumping unit’s gearbox [11]. Therefore, proper analysis of the gearbox torque is crucial in order to achieve the desired economic benefits. The gearbox torque calculation involves calculation of kinematic parameters of the four-bar leakage mechanism. It is not practical to construct the real sucker rod pump as a laboratory facility to create research environment. But mimicking the dynamic behavior of sucker rod pump by controlling the torque on the load machine is convenient approach.

In this paper, we propose a strategy for dynamic load emulation of sucker rod pump in order to develop offsite testing approach. Section II presents the principal concepts for development of test bench and section III discusses control method for tracking torque trajectories. Section IV provides detailed procedure for gearbox torque calculation and section V presents control strategy used for emulating dynamic load with simulation results. Finally, section VI gives conclusion.

Fig. 1. Structural scheme of a laboratory bench and a real drive

II. BASICS OF EMULATING MECHANICAL LOADS

Fig.1 shows the desired electromechanical drive system and its corresponding test bench arrangement. Equation (1) describes the total dynamics of the driving machine rigidly coupled to load machine.

Where: M_m, J_m, B_m are the torque developed, inertia and friction of the driving machine respectively and Me, J_L, B_L are the torque developed, inertia and friction of the load machine respectively. ω_m represents the angular speed of the shaft.

Generally, the actual torque exerted by the load machine on the shaft of the driving machine can be described by:

Suppose the desired electromechanical drive system shown in Fig.1 has inertia J_em and friction B_em. Then, the dynamic equation can be written as:

Where: ω_em and M_L are the desired angular speed and load torque of the desired mechanical load to emulate.

The objective is to control M_e so that the dynamic response described by (1) and (3) are as close as possible. When the desired electromechanical system and test bench have the same dynamic responses, the following conditions will be satisfied.

From (1), the inertia and static load characteristic seen by the driving machine is not equal to the characteristic of the load machine. Interestingly, the developed torque of the load machine can be adjusted such that when it is summed with its own inertial and frictional torques, the driving machine sees the desired load torque speed characteristics. Under this condition (4) can be satisfied. This forms the basic idea for emulating mechanical loads by test bench.

III. MODEL OF SEPARATELY EXCITED DC MOTOR

DC motors provide excellent dynamic speed tracking and load regulating response, which are suitable in applications of higher performance drive systems. In this paper separately excited DC motor is used to emulate the dynamic load characteristics of sucker rod pump.

A third order system describes the nonlinear model of a DC motor as given by:

Where: i_f and i_a represent field current and armature current, r_a and r_f represent armature resistance and field resistance, L_a and L_f represent armature inductance and field inductance, respectively. ω_m denotes the angular velocity and E_g denotes the back emf. T_g represents the generated torque and T_L represents the sum of mechanical loads. B denotes the viscous damping, k_b and k_t represent flux constant and torque constant respectively.

Principally, parameter uncertainties and flux saturation affect the design of control system. However, in this work, the variation of the parameters is assumed to be negligible and saturation effects are not considered. Moreover, there are other electrical characteristics (such as brush resistance, mutual inductances and eddy current loses) and mechanical characteristics (such as windage, stiction and stray loses), that are considered to improve modeling but can be neglected in control design [13]. The bilinear terms also complicate the control design. However, using measurements of field current, armature current and angular speed, bilinear behavior can be removed. In order to design controller for armature current, lets define a new variable v_a' given by:

The measured angular speed and field current can be used to determine v_a. Then, the linear transfer function of the reduced model can be expressed as:

A PI controller may be designed to produce the desired torque trajectory. In this paper, the flux is maintained constant. At the output of PI regulator, a filter is installed to suppress noise. The transfer function of the filter is usually represented as

 Where T_μ is uncompensated time constant.

Fig.2.  Schematic diagram of the pumping unit

IV. DYNAMICS OF SUCKER ROD PUMP

The sucker rod pump consists of prime mover, gearbox, pumping unit, sucker rod string, and pump system. Fig.2 shows the schematic diagram of the pumping unit. The pumping unit is the mechanism that converts the rotary motion of the prime mover into reciprocating vertical movement required by the polished rod [12]. The loading of this mechanism is a function of the crank angle and is somewhat irregular during the pumping cycle. During the upstroke, the prime mover drives a heavy load, while no power is required during the down stroke.

Since the crank is continuously driven by the prime mover through the gearbox, the crank angle φ_cr, which is the angle between crank arm and the horizontal, varies with time. Therefore, the angular speed of the crank can be described by:

At any instant, knowledge of the kinematic parameters (displacement, velocity, acceleration) of the pumping unit elements is important to calculate the torque requirement. Assuming the four-bar linkage mechanism is absolutely ridged, then analysis of gearbox torque could be studied as a function of the crank angle. In mechanics generally, the moments for ridged body system about an arbitrary point is expressed by [14]:

Where: q and p are number of moments and forces applied on the system respectively; u_i is the ratio angular speed of the i^th link relative to the angular speed of the crank and ρ_j is  position vector of the j^th link.

Moments of inertia are also given on the bases of the general expression from mechanics [14].

Where: n and k are the number of masses with rotational and translational motion respectively; J_i is inertia of the i^th link and m_i is mass of the j^th link.

If the moment about which the crank rotates is considered, then the ratio of angular velocity of the i^th link relative to the angular velocity of the crank and the position vector of the j^th link are calculated by:

Where: ω_i is angular speed of the ith mass about its own axis and v_j is the linear velocity of the center mass of the j^th link.

Since, the position and angular speed of individual links are functions of the crank angle, then the net torque acting on the gearbox and the total inertia of the different components depend on the crank angle. Therefore, the general equation that describes the dynamics of sucker od pumps will be:

Where: M_cr is the torque applied to the crank from the gearbox side. Equation (17) shows that the dynamic response of the sucker rod pump is highly depends on the crank angle.

The problem of dynamic analysis of sucker rod pump requires the computation of u_i and ρ_j. To solve the problem, lets fix the origin of the coordinate system at O, which is formed by the intersection of the vertical line passing through the saddle and the horizontal line, which passes through a point about which the crank shaft rotates.

Consider the following designation for the four-bar linkage mechanism: crank arm AB=r, pitman BC=k, rear walking beam CD=l₁, front walking beam DE=l₂. The horizontal distance from origin to the point about which the crank rotates OA=m_x and the vertical distance from the origin to the saddle is denoted by n_y. Based on the considered machine arrangement the crank is assumed to rotate clockwise. Then, at any instant the position of the points A, B, C, D, E can be described by the following coordinate points:

Fig.3. Load machine torque control schematic with torque estimation

Consider the following mass matrix m=[m₁, m₂, m₃, m₄] represent masses attached and possessed by each link of the four bar linkage mechanism. m1 denotes for the mass of counter weight which is located at a distance r₁ from the center of crank axis and offset by an angle φpr from the crank, m₂ represents the mass of pitman, m₃ denotes the equivalent mass of the structural unbalance, m₄ is the total mass connected to the polished including the mass of balancer head. Then, considering the position of the end points of each link, the matrix coordinate of center of mass of the links is given by:

If the crank arm is displaced by Dφ_cr in a time interval Dt, then using the coordinates of center of masses the change in angular displacement, the position vector, and velocity of the i^th mass can be expressed as:

To determine the ratio of angular velocity of the i^th link relative to the angular velocity of the crank, the matrix of radius of ith mass relative its own coordinate system is defined as R_m=[R_m1, R_m2, R_m3, R_m4],

Therefore, the angular velocity of the center of mass of the i^th link can be determined by:

Equation (24) is used to determine u_i.

Fig.4. Simulation results of speed characteristics

Fig.5. Simulation results of torque characteristics

V. EMULATING SUCKER ROD PUMP

Offsite testing of sucker rod pump can be implemented in laboratory test bench system as shown in Fig.3. The crank angle is determined from the measured speed. Then, the net torque on the gearbox, which is reference torque to produce the desired torque speed characteristics, is determined from real time model that mimics the sucker rod pump behavior. The actual torque on the shaft of driving machine can be measured using torque transducer or can be estimated from the measurements of electromechanical torque of either machine and plant parameters (inertia and viscous damping). However, when the parameters of the driving machine are not available, a torque transducer can be used to measure the shaft torque.

The control strategy is implemented in MATLAB Simulink. The test bench system consists of an induction motor as a driving machine and a separately excited DC motor as a load machine. In order to track the desired torque trajectory, the armature current of the DC motor is controlled using PI controller while maintaining flux constant. The PI controller minimizes the error between the actual load torque and the command reference torque to follow the desired torque trajectory. The PI controller introduces some delay, which is small compared to mechanical time constant. Thus, the stability of the system is not affected. Fig.4 and 5 show the emulation performance of speed and torque tracking in comparison with the simulation of desired real mechanism trajectories. The results of the emulation follow the desired speed and torque trajectories. This implies that the dynamic load of sucker rod pump is successfully emulated.

VI. CONCLUSION

This paper presents a strategy for emulating the dynamic load of sucker rod pump. Considering the principal concepts of electromechanical test bench, we have employed load torque estimator to allow the load machine to simulate the dynamic load of sucker rod pump. Moreover, this paper presents detailed analysis for calculation of gearbox torque, which is used by real time simulator. The proposed method is simulated in MATLAB Simulink by using an armature current controlled separately excited DC motor as a load machine to track the torque trajectory of the sucker rod pump generated by the real time simulator. Simulation results show that this strategy is capable for emulating the dynamic load of the sucker rod pump with adequate results.