Stick-slip stabilization in oil well drill-strings via optimal hybrid fractional order fuzzy logic control scheme

Abstract

Stick-slip vibration is regarded as one of the major factors that negatively influence the efficiency of the drilling processes. Therefore, it would be highly preferable to overcome the stick-slip vibration challenge through an effective control scheme. This study aims to address the stick-slip vibration problem in oil well drill-strings through a novel concept of an optimal hybrid fractional order fuzzy logic control (OH-FOFLC) scheme. By combining the robustness of fuzzy logic with the convergence speed of the fractional order method, the proposed scheme may enhance tracking performance in drill-strings under various operating conditions. In addition, the fuzzy logic controller for the OH-FOFLC scheme was designed by considering the minimum rules to decrease the computational burden as well as to reduce the controller’s cost. Furthermore, a particle swarm optimization algorithm was implemented for optimizing the coefficients of the OH-FOFLC scheme. To illustrate the overall performance and effectiveness of the OH-FOFLC, four different controllers, including proportional–integral–derivative (PID), fractional order PID, sliding mode control, and optimal hybrid fuzzy logic control approaches, are considered. The validity of the suggested OH-FOFLC was demonstrated through numerical simulations and comprehensive comparative studies which demonstrated that the OH-FOFLC scheme performs better than the four other controllers under various operating conditions.

Keywords

drill-strings stick-slip vibrations drill-strings control fuzzy logic control fractional order controller

Introduction

Over the past few years, extensive research and development have been conducted on the topic of torsional fluctuation in oil well drill-strings.^1–3 These torsional vibrations cause a deterioration in the performance of the drill pipe. They can lead to premature failure of tools, motors, and other costly components used in the drilling process. The stick-slip phenomenon is regarded as one of the main causes of torsional vibrations.⁴ This phenomenon is characterized by a sticking phase, where the bit’s angular velocity comes to a complete stop, and a slipping phase, where the angular velocity of the top drive of the drill-string increases up to six times its nominal value. Stick-slip fluctuations reduce the drilling efficiency, degrade the borehole quality, induce premature wear in the drill bit and they also decrease the penetration rate, which is a significant financial consideration for the drilling processes.⁵ In this context, the purpose of this paper is to extenuate stick-slip oscillations by designing an effective control scheme that can overcome the above-mentioned problems. However, a precise mathematical model is mandatory for designing an effective controller for controlling drill-strings. As documented in the literature, numerous studies have modeled drill-strings with an emphasis on vibrations and control. The authors in Ref. 6 presented a detailed survey on the existing drilling models and categorized them into several categories, such as distributed parameter models, neutral-type time-delay models, coupled partial differential equations, ordinary differential equations and lumped-parameter models. In the latter category, a drill-string is considered to be a mass–spring–damper system that can be modeled simply by ordinary differential equations, thus simplifying the design and analysis of control systems.⁶

This paper considers a lumped-parameter discontinuous torsional model with four degrees of freedom (DOF) that considers the dynamics of the drill pipe and the drill collars, and the nonlinear interactions of the drill bit and the rocks. Numerous studies in the literature have attempted to control and suppress stick-slip vibrations in drill-strings by various methods, such as proportional–integral–derivative (PID) controllers,^7,8 fractional order (FO) PID controllers,⁹ state feedback controllers,¹⁰ H^∞ controllers,¹¹ nonlinear controller based on backstepping,¹² and conventional and high-order sliding modes.^13–15 Nevertheless, these strategies have presented some problems because the designs were based on a complex model of the drill-strings or because of the simplification assumptions considered in the establishment of the control law. Conversely, intelligent controllers have the advantage of not requiring an accurate mathematical model system in their design process and they can handle the nonlinearities in complex systems.^16–19

Compared with different intelligent controllers, robust fuzzy logic controllers (FLCs) have been simply implemented with great success in many industrial applications.^20–22 FLCs with minimal knowledge of the mathematical model of the systems being controlled offer better performance in the presence of disturbances and uncertainties. Moreover, FLCs are currently easy to implement in real-time because of the development and potential of digital signal processors. Nowadays, FLCs have been enhanced by applying the FO domain.^23,24 The key idea behind the FO domain is to integrate the convergence speed and accuracy of FO with the robustness of FLC. In this regard, the present study proposes an optimal hybrid fractional order fuzzy logic control (OH-FOFLC) scheme for suppressing the vibrations in oil well drill-strings. The goal of the study is to apply a new combined structure of the FO and the FLC schemes to make the drill-string system have a faster response with the desired steady-state precision and better robustness against variations in the operating conditions. To the authors’ knowledge, this is the first study that apply fuzzy control to a drill-string system. Furthermore, a particle swarm optimization (PSO) algorithm was deployed to obtain the optimal coefficients of the suggested OH-FOFLC. Comprehension comparative study validated the precise control and excellent performance of the suggested control scheme. The main contributions of this paper, in summary, are as follows:

• Development of a new structure of an optimal hybrid fractional order fuzzy logic controller (OH-FOFLC) for oil well drill-strings.

• The OH-FOFLC has a simple structure, minimal fuzzy rules, and easy implementation.

• The OH-FOFLC coefficients are optimally obtained via a PSO process.

• In comparison with other control approaches, such as PID, fractional order PID (FO-PID), SMC (sliding mode control), and optimal hybrid fuzzy logic control (OH-FLC), the proposed OH-FOFLC offers superior performance.

The rest of the paper is structured as follows: The lumped-parameter model of the drill-string is presented in the Dynamic modeling of drill-string system section. The Structure of the suggested control scheme section gives the details of the suggested control scheme. The results and discussion are presented in the Results and discussion section. The Conclusion section concludes the paper.

Dynamic modeling of drill-string system

A schematic of the drill-string system with its equivalent mechanical representations based on four DOF is displayed in Figure 1. It consists of four damped inertias represented by four disks. The rotational inertias are connected together by shafts with dampers d_t, d_tl, d_tb in addition to the torsional springs k_t, k_tl, k_tb.^14,15 An arrangement of disks, which are equivalent to the rotary table on the top (r), the drill pipe (p), the drill collars (l), and the drill bit (b), forms the lumped-parameters model. The rotary table is characterized by a bulky damped inertia to inhibit speedy deviations in angular velocity and is guided by an electric motor with torque T_m applied through a gearbox. With a series of interconnected drill pipes, the length of a drill-string can reach up to many kilometers. The drill pipes are made from metallic tubes. The bottom section, known as the bottom hole assembly, is manufactured from a weightier pipe. The drill bit is submerged in fluid mud, which dissipates the drill bit’s energy and strong torque imposed on the bit. The drill bit is also subjected to a friction torque T_fb that represents the bit–rock interaction. The drilling system’s geometric and material characteristics are indicated in Figure 1(b). The rotary table is assumed to be fixed, but the bit is free.¹³ Based on the above mentioned characteristics of a drill-string, a state-space representation of the dynamic behavior of the drill-string system is detailed in (1), where the state vector of the drill-string system is defined in (2).¹³

\begin{array}{l} {\ddot{Ω}}_{r} = - \frac{d_{t}}{j_{r}} ({\dot{Ω}}_{r} - {\dot{Ω}}_{p}) - \frac{k_{t}}{j_{r}} (Ω_{r} - Ω_{p}) - \frac{d_{r}}{j_{r}} ({\dot{Ω}}_{r}) + \frac{T_{m}}{j_{r}} \\ {\ddot{Ω}}_{p} = \frac{d_{t}}{j_{p}} ({\dot{Ω}}_{r} - {\dot{Ω}}_{p}) + \frac{k_{t}}{j_{p}} (Ω_{r} - Ω_{p}) - \frac{d_{t l}}{j_{p}} (\dot{Ω_{p}} - \dot{Ω_{l}}) \\ - \frac{k_{t l}}{j_{p}} (Ω_{p} - Ω_{l}) \\ {\ddot{Ω}}_{l} = \frac{d_{t l}}{j_{l}} ({\dot{Ω}}_{p} - {\dot{Ω}}_{l}) + \frac{k_{t l}}{j_{l}} (Ω_{p} - Ω_{l}) - \frac{d_{t b}}{j_{l}} ({\dot{Ω}}_{l} - {\dot{Ω}}_{b}) \\ - \frac{k_{t b}}{j_{l}} (Ω_{l} - Ω_{b}) \\ {\ddot{Ω}}_{b} = \frac{d_{t b}}{j_{p}} ({\dot{Ω}}_{l} - {\dot{Ω}}_{b}) + \frac{k_{t b}}{j_{b}} (Ω_{l} - Ω_{b}) - \frac{T_{o b} (x)}{j_{b}} \end{array}

(1)

X = {[{\dot{Ω}}_{r}, (Ω_{r} - Ω_{p}), {\dot{Ω}}_{p}, (Ω_{p} - Ω_{l}), {\dot{Ω}}_{l}, (Ω_{l} - Ω_{b}), {\dot{Ω}}_{b}]}^{T}

(2)

where j_r, j_p, j_l, j_b are the inertia coefficients of the rotary table, drill pipes, drill collars, and drill bit, Ω_r, Ω_p, Ω_l, Ω_b are the angular position of the rotary table, drill pipes, drill collars, and drill bit, Ω_r, Ω_p, Ω_l, Ω_b are the angular velocity of the rotary table, drill pipes, drill collars, and drill bit, k_t, k_tl, k_tb are the torsional stiffness, d_t, d_tl, d_tb are torsional damping, d_r is the viscous damping coefficient, T_m is the drive torque, T_ob(X) is the torque on the bit, X is the system state vector, T_fb(X) is the torque required for crushing and cutting the rock and the friction along the BHA.

Figure 1.

Mechanical system describing the torsional action of a traditional drill-string.¹⁵

For the sake of simplicity, (2) is defined as follows:

X = {[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}]}^{t}

(3)

Finally, the state equations of the drill-string system are given in (4), using the simplified representation of (3) to facilitate the analysis.

\begin{array}{l} {\dot{x}}_{1} = \frac{1}{j_{r}} [- (d_{r} + d_{t}) x_{1} - k_{t} x_{2} + d_{t} x_{3} + T_{m}] \\ {\dot{x}}_{2} = x_{5} - x_{7} \\ {\dot{x}}_{3} = \frac{1}{j_{p}} [d_{t} x_{1} + k_{t} x_{2} - (d_{t l} + d_{t}) x_{3} - k_{t l} x_{4} + d_{t l} x_{5}] \\ {\dot{x}}_{4} = x_{3} - x_{5} \\ {\dot{x}}_{5} = \frac{1}{j_{l}} [d_{t l} x_{3} + k_{t l} x_{4} - (d_{t l} + d_{t b}) x_{5} - k_{t b} x_{6} + d_{t b} x_{7}] \\ {\dot{x}}_{6} = x_{5} - x_{7} \\ {\dot{x}}_{7} = \frac{1}{j_{b}} [d_{t b} x_{5} + k_{t b} x_{6} - (d_{t b} + d_{b}) x_{7} - T_{o b} (x)] \end{array}

(4)

Torque on the bit is given by the following equation

T_{o b} = T_{v} (x_{7}) + T_{f b} (X)

(5)

where T_v(x₇) is given by

T_{v} (x_{7}) = d_{b} x_{7}

(6)

where T_v(x₇) Influence of the drilling mud on the bit’s behavior.

The energy supplied by the motor at the top surface is mostly dispersed at the drill bit through bit–rock interaction. The friction between the bit and the rock is the leading cause of fluctuations in a drill-string. Hence, it is essential for a detailed study on the fluctuation mechanism to develop a bit–rock interaction model. In the literature, different friction models are proposed. The most frequently used models over recent years are the Stribeck model and Karnopp’s model Krama et al.¹³ The problems with the Stribeck model are the instability issue and discontinuity at zero speed. The static features of friction cannot be precisely estimated. To solve the above mentioned difficulties, bit–rock interaction has been modeled by Karnopp’s model, considering both frictional and cutting contact, as displayed in Figure 2 Krama et al.¹³ The reaction torque can be calculated through the mathematical model given by (7).

T_{f b} (x) = {\begin{cases} T_{e b} (x) i f | x_{7} | < D_{v}, | T_{e b} | \leq T_{s b} \\ T_{s b} s i g n (T_{e b} (x)) i f | x_{7} | < D_{v}, | T_{e b} | > T_{s b} \\ T_{c b} s i g n (x_{7}) i f | x_{7} | \geq D_{v} \end{cases}

(7)

T_{e b} = d_{t b} (x_{5} - x_{7}) + k_{t b} x_{6} - d_{b} x_{7}

(8)

T_{s b} = w_{o b} R_{b} u_{s b}

(9)

u_{b} (x_{7}) = u_{c b} + (u_{s b} - u_{c b}) e^{\frac{γ_{b}}{V_{f}} | x_{7} |}

T_{c b} = w_{o b} R_{b} u_{b} (x_{7})

(11)

where T_sb is the static friction torque, T_cb is the coulomb friction torque, T_eb(X) is the breakaway torque, R_b is the bit radius, u_b(x₇) is the dry friction coefficient of the bit, u_sb, u_cb is the static and Coulomb friction coefficients, and weight on Bit (WOB) is the WOB.

Figure 2.

Karnopp’s model for bit-rock interaction.¹³

The lumped-parameters model of the drill-string can be written in a multivariable form as Krama et al.¹³

\dot{x} (t) = M_{1} X (t) + M_{2} T_{m} + M_{3} T_{f b} (X)

(12)

where the matrices M₁, M₂, and M₃ can be deduced easily from (4).

Structure of the suggested control scheme

Recently, intelligent industrial systems have been extensively developed, incorporating FLC and FO control approaches. Control systems based on fuzzy logic are extremely appropriate for designing and controlling complex systems for which a mathematical model cannot be achieved simply. In particular, the fuzzy controller is very suitable for nonlinear systems. On the other hand, the FO control scheme leads to a better suited and faster controller response. The specific structure of the OH-FOFLC scheme is illustrated in Figure 3.

Figure 3.

Structure of the suggested optimal hybrid fractional order fuzzy logic control scheme.

Hybridization of fuzzy logic and fractional order control

In the structure shown in Figure 3, two FLCs combined with FO operators are applied; the first (FLC1) aims to drive the angular velocities of the rotary table and the drill bit to a required value fast enough to evade the bit sticking fluctuations, whereas the second (FLC2) is designed to improve the tracking error between the angular velocities. In Figure 3, FLC1 uses two input variables, that is, the error e₁(t) and the fractional derivative of the error de₁(t) and one output variable, that is, u₁(t). Here, k_e1 and k_de1 are used as normalized scaling coefficients to normalize the inputs e₁(t) and de₁(t) to the universe of discourse in which the membership functions (MFs) of the inputs are defined. Accordingly, the inputs e₁(t) and de₁(t) are transformed to E₁(t) and dE₁(t), respectively, as follows

\begin{array}{l} E_{1} (t) = k_{e 1} e_{1} (t) = k_{e 1} (ω^{*} (t) - x_{1} (t)) \\ d E_{1} (t) = k_{d e 1} e_{1} (t) = k_{d e 1} (d^{μ 1} e_{1} (t)) / (d t^{μ_{1}}) \end{array}

(13)

Ω*(t) is the angular velocity reference and μ1 denotes the FO derivative value for computing de₁(t). On the other hand, FLC2 uses one input variable, that is, error e₂(t) and one output variable, that is, u₂(t). Moreover, one normalized scaling coefficient k_e2 is used to normalize e₂(t) and transform it to E₂(t) as follows:

E_{2} (t) = k_{e 2} e_{2} (t) = k_{e 2} (x_{1} (t) - x_{7} (t))

(14)

In line with Figure 4, the output u(t) of the OH-FOFLC is computed as

\begin{array}{l} u (t) = k_{p} (u_{1} (t) + u_{2} (t)) + k_{i} \frac{d^{- λ} (u_{1} (t) + u_{2} (t))}{d t^{- λ}} \\ + k_{d} \frac{d^{μ 2} e_{1} (t)}{d t^{μ 2}} \end{array}

(15)

where k_p, k_i and k_d are the output scaling coefficients, μ₂ and λ designate the FO derivative value and the FO derivative value, respectively, for calculating u(t). The purpose of the fractional derivative of error

\frac{d^{λ_{2}} e_{1} (t)}{d t^{μ_{2}}}

in the OH-FOFLC is to improve the settling time and minimize fluctuations in the angular velocity in the drill-string. A FLC is realized through three calculation stages: the fuzzifier, the fuzzy inference engine, and the defuzzifier. In the fuzzifier stage, the numerical signals are converted into normalized fuzzy inputs by MFs. The MFs for FLC1 and FLC2 are presented in Figures 4 and 5.

Figure 4.

Membership functions of the input and output variables for the FLC1.

Figure 5.

Membership functions of the input and output variables for the FLC2.

In Figure 4(a), the error e₁(t) and its fractional derivative de₁(t) are divided into three sections: negative (N), zero (Z), and positive (P). In Figure 5(a) the error e₂(t) is divided into five parts: negative big (NB), normal negative (NN), zero (Z), normal positive (NP), and positive big (PB). Five singleton MFs were designed for the output of both FLC1 and FLC2, as depicted in Figures 4(b) and 5(b). The MFs used in FLC1 and FLC2 were based on Gaussian and singleton approximations for easier computation. Tables 1 and 2 provide the rule bases used in the fuzzy inference engine of FLC1 and FLC2, respectively. The Min–Max inference process was applied for FLC1 and FLC2, and the center weighted average algorithm was used at the defuzzifier stage to convert the output of the controllers into crisp values.

Table 1.

Rule base for the FCL1.

If			Then
Rule N	E₁(t)	dE₁(t)	u₁(t)
1	N	N	NB
2	N	Z	NN
3	N	P	Z
4	Z	N	NN
5	Z	Z	Z
6	Z	P	NP
7	P	N	Z
8	P	Z	NP
9	P	P	PB

Table 2.

Rule base for the FLC2.

If		Then
Rule N	E₂(t)	u₂(t)
1	NB	PB
2	NN	NP
3	Z	Z
4	NP	NN
5	PB	NB

For the application of the suggested OH-FOFLC, Oustaloup’s approach was used to estimate the parameters of the control scheme.²⁵ Using a recursive filter, this approach provides a very good fit to the FO parameters within an operating frequency range of [ω_l ω_h]. The approximation of the FO parameter s^α is expressed as follows:

s^{α} = K \prod_{k = - N}^{N} \frac{s + ω_{k}^{z}}{s + ω_{k}^{p}}

(16)

where α is the order of the fractional differ-integration. The gain K, the zeros

ω_{k}^{z}

, and the poles

ω_{k}^{p}

of the filter are given as follows:

\begin{array}{l} K = ω_{h}^{α} \\ ω_{k}^{z} = ω_{l} {(\frac{ω_{h}}{ω_{l}})}^{\frac{k + N + \frac{1}{2} (1 - α)}{2 N + 1}} \\ ω_{k}^{p} = ω_{l} {(\frac{ω_{h}}{ω_{l}})}^{\frac{k + N + \frac{1}{2} (1 - α)}{2 N + 1}} \end{array}

(17)

where N is the order of approximation and its value is correlated with the performance of Oustaloup’s approach and the fit between the complexity and accuracy of the approximation. In this work, N is taken to be 4, whereas the frequency range is [ω_l ω_h] = [10⁻² 10²]rad/sec.

Coefficients tuning using PSO algorithm

The coefficient tuning process for the suggested control scheme was considered to be an optimization problem, in which the scalar coefficients of the FLCs (k_e1, k_de1, k_e2, k_p, k_i and k_d) and the FOs (μ₁ μ₂ and λ) were seen as decision variables. In this context, the PSO algorithm was deployed for solving the optimization problem, which is expressed as an objective function to minimize its fitness.^26,27 The objective function J was based on the angular velocity errors and the control torque signals as follows:

\begin{array}{l} J = m i n i m i z e (\int_{0}^{\infty} Q_{1} | e_{1} (t) | t d t + \int_{0}^{\infty} Q_{2} | e_{2} (t) | t d t \\ + \int_{0}^{\infty} R_{1} | u_{1} (t) | t d t + \int_{0}^{\infty} R_{2} | u_{2} (t) | t d t) \end{array}

(18)

where Q₁, Q₂, R₁ and R₂ are the weighting factors. These factors were selected to obtain an appropriate trade-off between keeping the errors small while ensuring having the control signal not too big. The flowchart of the whole process of the PSO algorithm developed for the suggested control scheme is depicted in Figure 6, which displays how the PSO finds the optimal coefficients. The initial parameters of the PSO algorithm are recorded in Table 3.

Figure 6.

Flowchart of the particle swarm optimization process.

Table 3.

Key parameters of the PSO algorithm.

Parameters	Quantity
Number of particles in a swarm	50
Maximum number of iterations	100
Acceleration coefficients	2
Minimum inertia coefficient	0.6
Maximum inertia coefficient	0.9

Results and discussion

In this section, simulation and comparison results are reported to validate the suggested OH-FOFC scheme. Initially, the drill-string system and the suggested OH-FOFC scheme were simulated by Matlab/Simulink software with the parameters listed in Tables 4 and 5. Figure 7 displays the stick-slip vibration in the drill-string system by screening the angular velocities of the rotary table and the drill bit, which is guided by a constant control torque (u = 10 kNm) under a fixed WOB of 100 kN. Various operating scenarios were implemented to evaluate and compare the performance of the suggested OH-FOFC scheme versus other controllers.

Table 4.

The parameters list of the drill-string system.¹⁵

Parameter	Symbol	Value	Units
Inertia of the rotary table	J _r	930	kg.m²
Inertia of the drill collar	J _l	750	kg.m²
Inertia of the drill pipes	J _p	2782.25	kg.m²
Inertia of the drill bit	J _b	471.9698	kg.m²
Torsional stiffness between rotary table and drill pipes	k _t	698.06	N.m/rad
Torsional stiffness between drill collar and drill pipes	k _tl	1080	N.m/rad
Torsional stiffness between drill collar and drill bit	k _tb	907.48	N.m/rad
Torsional damping between rotary table and drill pipe	d _t	139.6126	N.m.s/rad
Torsional damping between drill pipe and drill collar	d _tl	190	N.m.s/rad
Torsional damping between drill collar and drill bit	d _tb	181.49	N.m.s/rad
Torsional damping between mud drilling and drill bit	d _b	50	N.m.s/rad
Weight on bit	WOB	100	kN
Radius of the drill bit	R _b	0.155575	m
Drive torque	T _m	10	kN.m
Limit velocity	D _v	10^–6	rad/sec
Viscous damping coefficient	d _r	425	N.m.s/rad
Static friction coefficient	μ _sb	0.8	−
Coulomb friction coefficient	μ _cb	0.5	−
Factor associated with drill bit	γ _b	0.9	−

Table 5.

The optimal coefficients for the OH-FLC and OH-FOFLC.

Controller	k _e1	k _de1	k _e2	k _p	k _i	k_d +	μ ₁	μ ₂	λ
OH-FLC	0.214	2.427	0.944	3.982 × 10²	2.541 × 10³	2.762 × 10³	—	—	—
OH-FOFLC	0.190	1.336	0.941	4.105 × 10²	3.202 × 10³	3.693 × 10³	0.599	0.552	1.075

Figure 7.

The stick-slip vibrations in the drill-string system.

Performance verification of the suggested OH-FOFLC

To study the performance of the OH-FOFLC and its ability to stabilize the stick-slip phenomenon of a drill-string, the first simulation was carried out with a fixed WOB of 100 kN and constant angular reference velocity Ω* = 12 rad/sec. Figure 8 displays the transient behavior of the angular velocities of the rotary table and the drill bit, and the torque control in the case of a sudden change from an open-loop drill-sting (with a fixed control torque of 10 kNm) to a drill-string controlled by the OH-FOFLC. It can be seen that the stick-slip vibrations are eliminated once the OH-FOFLC is put into operation at t = 40 s, and the angular velocity of the drill bit and that of the rotary table are stabilized to the desired angular velocity after a short period. Also, the control torque was increased with smooth waveform which is desirable for suppressing stick-slip vibrations.

Figure 8.

Stabilization of the drill-string using the suggested optimal hybrid fractional order fuzzy logic control for Ω* = 12 rad/sec and weight on bit = 100 kN: (a) angular velocities of the rotary table and the drill bit, (b) control torque. The OH-FOFC is switched on after 40 s.

The second simulation was performed to assess the step change response of the OH-FOFLC scheme. The angular velocity reference increased from 12 to 16 rad/s in this test(see Figure 9) under a fixed WOB. It can be observed that the angular velocity of both the rotary table and the drill bit was maintained close to the new reference after a short period, with good approximation and stability, and the control torque was increased once the angular velocities had increased to guarantee adequate torque for the bit.

Figure 9.

Performance of the suggested optimal hybrid fractional order fuzzy logic control in the presence of abruptly change of Ω* from 12 to 16 rad/sec and weight on bit = 100 kN: (a) angular velocities of the rotary table and the drill bit, (b) control torque.

Next, to evaluate the anti-disturbance performance of the OH-FOFLC scheme, the simulation of an abrupt change in WOB under a fixed angular reference velocity was carried out. The WOB was decreased abruptly in this test from 100 to 70 kN at t = 160 s. It can be clearly seen from the simulation results in Figure 10 that the control torque decreased when the WOB decreased and the transient behavior of the OH-FOFLC is fast with small variations in the angular velocities of the rotary table and the drill bit.

Figure 10.

Performance of the suggested optimal hybrid fractional order fuzzy logic control in the presence of abruptly change of weight on bit from 100 to 70 kN and Ω* = 12 rad/sec: (a) angular velocities of the rotary table and the drill bit, (b) control torque.

In addition, the control torque decreased because of the change in the WOB.

Furthermore, in order to verify the robustness of the OH-FOFLC against parameter uncertainties, a simulation is performed with changes in the drill-string parameters under a constant angular reference velocity and a fixed WOB. The parameter uncertainties comprised alterations of k_t, d_tb, μ_sb, μ_cb from the nominal values. The values of these parameters were increased by 50% from the nominal values. Figure 11 shows that the OH-FOFLC maintained good performance and stability regardless of these changes, which proves the robustness of the suggested control scheme.

Figure 11.

Robustness of the suggested optimal hybrid fractional order fuzzy logic control in the presence of parameter uncertainties, Ω* = 12 rad/sec and weight on bit =100 kN: (a) angular velocities of the rotary table and the drill bit, (b) control torque.

Performance comparison with different controllers

In this section, the suggested OH-FOFLC scheme is compared with some other controllers, namely, the PID controller, the FO-PID controller, the SMC controller and the OH-FLC in the drill-string system. These controllers were compared in terms of the settling time, the peak overshot/undershoot and the mean absolute errors (MAE) under various operating scenarios (as in the previous subsection) through numerical simulations. First, a comparison of the different controllers under step changes in the angular reference velocity is shown in Figure 12. The step response characteristics of the different controllers recorded in Table 6 demonstrated that the different controllers achieved good tracking performance in the steady state; it is evident that the variations of the MAE were lower with the OH-FOFLC as compared to the other controllers for the angular velocity of both the rotary table and the drill bit. On the other hand, in the transient state, the angular velocity of both the rotary table and the drill bit had a shorter settling time and a smaller overshoot for both reference velocity steps with the OH-FOFLC.

Figure 12.

Performance comparison of the different controllers in the presence of abruptly change of weight on bit from 100 to 70 kN and Ω* = 12 rad/sec: (a) angular velocities of the rotary table, (b) angular velocities of the drill bit.

Table 6.

Quantitative comparison of the suggested OH-FOFLC with other controllers under abrupt change in the angular velocity reference.

	Step increase in the angular velocity reference 0 → 12 rad/s						Step increase in the angular velocity reference 12 → 16 rad/s
	Angular velocity of the rotary table			Angular velocity of the drill bit			Angular velocity of the rotary table			Angular velocity of the drill bit
	Settling time [sec]	Overshoot [rad/sec]	MAE [rad/sec]	Settling time [sec]	Overshoot [rad/sec]	MAE [rad/sec]	Settling time [sec]	Overshoot [rad/sec]	MAE [rad/sec]	Settling time [sec]	Overshoot [rad/sec]	MAE [rad/sec]
PID	90.083	2.641	2.859	90.342	2.951	3.201	43.826	0.541	0.513	44.155	0.676	0.436
FO-PID	44.071	2.134	0.581	45.397	5.726	1.039	27.603	1.125	0.190	28.678	2.019	0.326
SMC	49.445	2.283	0.642	47.686	6.844	1.050	25.362	1.143	0.160	33.525	2.493	0.362
OH-FLC	26.047	0.382	0.579	31.064	1.825	1.009	10.495	0	0.199	13.694	0.567	0.308
OH-FOFLC	22.842	1.345	0.563	30.983	5.048	1.001	16.912	0.464	0.180	15.312	1.515	0.218

Next, the five controllers were tested and compared under an abrupt change in WOB. It can be seen in simulation results exhibited in Figure 13 and Table 7 that the angular velocity responses of the five controllers became stable again after short transient. Nevertheless, the variations in the angular velocity with the OH-FOLC were faster and smaller with minimum MAE than those of the other controllers, which demonstrates that the OH-FOLC scheme has a better anti-disturbance ability than the other controllers.

Figure 13.

Performance comparison of the different controllers in the presence of abruptly change of Ω* from 12 to 16 rad/sec and weight on bit = 100 kN: (a) angular velocities of the rotary table, (b) angular velocities of the drill bit.

Table 7.

Quantitative comparison of the suggested OH-FOFLC with other controllers under abrupt change in the WOB.

	Step decrease in the WOB 100 → 70 KN
	Angular velocity of the rotary table			Angular velocity of the drill bit
	Settling time [sec]	Overshoot [rad/sec]	MAE [rad/sec]	Settling time [sec]	Overshoot [rad/sec]	MAE [rad/sec]
PID	55.235	2.642	1.696	55.112	3.246	1.875
FO-PID	22.296	1.062	0.365	23.368	2.843	0.613
SMC	40.382	1.335	0.377	31.857	2.995	0.646
OH-FLC	11.965	0.774	0.346	17.834	2.806	0.588
OH-FOFLC	11.512	0.621	0.318	14.413	2.798	0.586

Figure 14.

Robustness comparison of the different controllers in the presence of parameter uncertainties, Ω* = 12 rad/sec and weight on bit = 100 kN: (a) angular velocities of the rotary table, (b) angular velocities of the drill bit.

Table 8.

Quantitative comparison of the suggested OH-FOFLC with other controllers under parameter uncertainties.

	Changes on the drill-string parameters
	Angular velocity of the rotary table			Angular velocity of the drill bit
	Settling time [sec]	Undershot [rad/sec]	MAE [rad/sec]	Settling time [sec]	Undershot [rad/sec]	MAE [rad/sec]
PID	58.426	4.295	1.763	60.704	4.573	2.099
FO-PID	24.432	1.844	0.352	33.215	5.033	0.688
SMC	28.768	2.185	0.322	30.846	3.678	0.680
OH-FLC	13.155	1.417	0.313	29.655	5.089	0.644
OH-FOFLC	08.491	0.266	0.308	08.732	0.828	0.538

In addition, a comparison of the robustness of the different controllers to parametric uncertainty is investigated in Figure 14 and Table 8. The superior transient performance of the suggested OH-FOFLC compared with the other controllers can be clearly seen. Moreover, the OH-FOFLC had less MAE than the other controllers.

Conclusion

In order to avoid stick-slip fluctuations in an oil well drill-string and ensure good tracking performance for the angular velocity under various operating scenarios, this paper has suggested a novel structure for an OH-FOFLC scheme. Our control scheme uses the properties of fuzzy logic control and fractional order calculus to realize effective and robust velocity tracking control and fluctuation suppression. The coefficients of the suggested OH-FOFLC are also enhanced by deploying a PSO algorithm. Several simulations and comparisons were carried out. It can be concluded from the results and comparisons that the suggested control scheme is efficient for suppressing the stick-slip vibration in drill-strings, as it offered better tracking performance than the PID, FO-PID, SMC and OH-FLC controllers under various operating scenarios.

For future work, we envisage that the suggested controller can be implemented in real time for a drill-string prototype. We will also investigate the observer-based fault detection and isolation problem for the drilling systems with advanced techniques.^28,29

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This publication was made possible by NPRP grant [NPRP10-0101-170081] from the Qatar National Research Fund (a member of Qatar Foundation).

ORCID iDs

Billel Talbi

Mohamed Gharib

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