Identification of resistance torque on the roller cone bit in the drill rod rotation drive

There is a problem of increasing the durability of the roller cone bit of the drilling rigs used in the quarries of Ukraine. One of the effective ways to solve it is to apply the control method of maintaining a constant flow of mechanical power in the bottom hole zone. The implementation of such a control method leads to the necessity of controlling and limiting the resistance torque on the roller cone bit. To solve the problem of determining the resistance torque, we used methods of parameter identification: the Luenberger observation device with adaptation to external influences; estimation of the resistance torque using an extended observation object; the theory of regulators adapting to perturbations. It has been established that the most rational method of identifying the resistance torque on a roller cone bit is an astatic monitoring device, which allows to implement a method of controlling the rotation of the drill rod by a drive mechanism that maintains a constant flow of mechanical energy through the roller cone bit and increases the durability of the roller cone bit.


Introduction
The operation of the drilling rig is characterized by the transformation of the flow of electromagnetic energy coming from the power supply system into a flow of mechanical energy, which is released in the form of non-productive losses and turned into useful work, going to the destruction of rock and transportation of drilling fines onto the wellhead [1].Energy flows are formed and directed through the following channels: axial force -power of linear movement of the drilling rod; rotation frequency -rock destruction power; compressed air pressure and flow rate -drilling fines evacuation (cuttings) power.Each channel has an individual type of energy converter; these are: adjustable electro-hydro-mechanical drive of drill bit supplying; adjustable electromechanical drive of roller cone bit rotation; nonadjustable electromechanical drive of compressor to remove drilling fines from the surface of the hole bottom.
Figure 1 shows a diagram of the formation and transformation of the electromagnetic energy flow of the power supply system into the mechanical energy flow on a roller cone.
There is a problem of improving the working capacity of a roller cone bit, since it wears most of all, since its cutting structure and supports work in an abrasive environment.During operation, the bit absorbs high static and dynamic loads, and is also subjected to intensive destruction from material fatigue and abrasive wear.The flow of rotational energy is directed not only to the destruction of the rock, but also to the destruction of the bit itself.
In order to increase the performance of a roller cone bit by maintaining the power flow in the bottom hole zone, the method of drilling process control [2] with control of the drives for the feeding and rotating mechanisms has been developed, where hard mechanical characteristics are formed on the roller cone bit when penetrating soft formations and soft mechanical characteristics -in stronger formations.
When drilling rocks with intermittent physical and mechanical properties, the control method provides automatic selection of mechanical characteristics depending on the strength characteristics of the rock.
Implementation of the developed method of drilling process control leads to the necessity of measuring, controlling and limiting on permissible levels of resistance torque on the roller cone bit.

Related work
The need to modernize the park of drilling rigs for blasting wells used in quarries in Ukraine is an actual task, since they have exhausted their physical reserves, being in operation for 1.5 -2 normative time limits.
Machine-building and electro-technical plants of Ukraine have modern technologies, on the basis of which they are capable of serial production of new generation drilling rigs on their own.Since 2003 the Novokramatorsk Machine-Building Plant has been working on the development of a new generation drill rig SBShS-250H with AC drive systems.
The essential question when developing the control system of the drilling rig rotation drive is the choice of its control method.There are known methods of control, realizing the support at a constant level of one of the parameters: bit rotation frequency, resistance torque on the bit [3], linear speed of bit movement, axial pressure on the bit, consumed power of rotation drive.
There are kinematic and energy criteria of the working capacity of a roller cone bit.The kinematic criterion monitors bit speed or linear penetration rate per bit, which is an indirect evidence of bit cutting structure wear.
Energy criterion of the working capacity of a roller cone bit is proposed in [2].In this case, it is necessary to control the resistance torque on the bit and its rotation frequency.The time integral of their product is the energy released by the bit armament on the bottom hole.By stabilizing the energy release rate at a constant level, it is possible to get the maximum bit durability, since the cutting structure material is under a constant load.This can be achieved by keeping the power of destruction of the rock at a constant level with varying strength of the drilled rocks.The power delivered will depend on the resistance torque on the bit and the rotational speed of the bit.If the rotational speed is easy to measure, then measuring the resistance torque on the bit is in principle impossible due to the aggressive environment in the bottom hole area.Measuring power by voltage and current of the drive system will lead to errors due to their nonlinear dependence and dynamic torque presents.
Based on the energy criterion of bit durability, we choose the most rational way of controlling the rotation drive of the roller cone drill bit, which controls the power released in the bottom hole zone of the roller cone bit.With this approach, it is necessary to determine the resistance torque on the bit, using one of the methods of identifying the external influences based on the state observer [4,5].

Results
The system of differential equations describing the dynamics of an induction motor in the d-q coordinate system, tied to the vector of the main flux linkage, has the form [5]: where ψ r , L r , R r -the rotor flux linkage, inductance and active resistance; I d , I q -transverse and longitudinal components of the motor stator current; R s , L s -stator resistance and inductance; ω, ω ψr , p n -angular frequency of rotation of the rotor, flux linkage of the rotor and the number of pole pairs of the stator winding; U d , U q -transverse and longitudinal components of the stator voltage; M m -mutual induction between the stator and the rotor (inductance of the magnetization circuit); denoted: 3p n k r /2; σ -leakage coefficient; J is the dynamic moment inertia of the drive mechanical link.
The coordinate system is oriented in the direction of the vector of the main flux linkage, which makes it possible to separately control the flux linkage of the rotor and the transverse component of the stator current.This makes it possible to synthesize a system with normalized dynamic parameters, which controls the flux coupling and the longitudinal stator current on one channel, and the rotor speed and the transverse component of the stator current on the other channel (vector control system for induction squirrel-cage motor).
Equations describing dynamic processes in the control channel by the active (torque-forming) component of induction motor stator current [6] The coordinate system is oriented in the direction of the vector of the main flux linkage, which makes it possible to separately control the flux linkage of the rotor and the transverse component of the stator current.This makes it possible to synthesize a system with normalized dynamic parameters, which controls the flux coupling and the longitudinal stator current on one channel, and the rotor speed and the transverse component of the stator current on the other channel (vector control system for induction squirrel-cage motor).
Equations describing dynamic processes in the control channel by the active (torque-forming) component of induction motor stator current [6] There are four disturbing signals in the control object: rotor flux linkage -ψ r ; resistance torque -M c ; cross-coupling -ω ψr I d ; electromotive force of rotation frequency -k r ψ r p n ω/L s .
We believe that the rotor flux linkage is maintained at a constant level through the control channel of the reactive (flux-forming) current of the motor stator.Compensation of the crosscoupling and the electromotive force of the rotation frequency is carried out by introducing additional links, zeroing out the perturbing actions from the indicated influences.Compensation of the cross-coupling is achieved by decoupling of the control channels.The direct compensation uses signals proportional to the product of the instantaneous frequency rotation of the rotor flux linkage vector ω ψr and the reactive component of the stator current, as well as the product of the rotor frequency rotation and the current value of the rotor flux linkage.Therefore, these perturbations on the control object are not taken into account when identifying the torque of resistance [7,8].
After these simplifications one can forming the state vector and denoting k m = 3p n k r /2 it is convenient to represent the object under consideration by a system of linear differential equations written in matrix form: .
where the matrix coefficients are defined Let us study the problem of determining the resistance torque on the bit M c , using various methods of identifying the external influences that cannot be directly measured.The methods under consideration apply the so-called observer, i.e., its mathematical model working in parallel with the object.The main attention is paid to the stability of the system, which is ensured by a proper choice of its natural frequencies (modes) on the complex plane.
The extended object method assumes [9] that the change in the external action is subject to some differential equations, which together with the equations of the main object define the extended system.In this case perturbing influences become state variables of the extended object.
In our case, taking into account that the rate of transient processes in the object exceeds the rate of change M c , we obtain M c ≈ const.
The original system of equations presented in vector-matrix form .
where the notation for matrix coefficients is accepted We construct the observing device in the form of a mathematical model, which is described by the equation ∧.

X= A
where L -some unknown matrix of size n × 1; n is system order.The last term in (4), taking into account the difference between the outputs of the object and the model, corrects the observer so that the condition is fulfilled ∧ X (t) − X * (t) → 0 when t → ∞.Such an observer is called an asymptotic identifier.
If we denote X = ∧ X −X * , then from (3) and ( 4) we obtain equations for the estimation error: In accordance with the scientific statements put forward in [10,11], which establish the relation between the total observability and the characteristic polynomial of a matrix A * − LC * with an arbitrary desired set of roots, we check for the system (3) the observability condition, which consists in that the rank of the observation matrix must match the order of the system n.
For the system under study, we obtain: The matrix rank Q n is 3, that is, it coincides with the order of the system.In this case, the condition of complete observability is satisfied.
The characteristic polynomial for the matrix A * − LC * , (where det[λI − A * + LC * ], where I is the identity matrix) is determined by the expression: We select the elements of the matrix L = (l 1 l 2 l 3 ) T in such a way that the polynomial turns into the Newton binomial Whence, by equating the coefficients at the same degrees of the polynomial, we obtain To solve system (6), it is necessary to choose the value determines the position of the matrix eigenvalues A * − LC * on the complex plane and, consequently, the rate of convergence Usually, values β are used in 2...3 times greater than the maximum of the numbers |Reλ 0 |, where λ 0 is the eigenvalues of the object's matrix.Let's find them by solving the equation where the value γ within 2. . . 3 is accepted.Then, in accordance with the obtained equations of system (6), we find the values of the correction coefficients: Consequently, the structural diagram and the values of the correction coefficients are generally defined for the observing device (4).
It is known that the method of transition to an extended object requires complete observability of this object [5].An astatic observing device [11] is free from this restriction [12], the construction of which we will consider for the problem to be solved when the vector of disturbing influences contains one nonzero element.
For system (2) in this case it is possible to construct an observing device described by equations: where L = (l 1 l 2 ) T and K = (0 k) T are matrices with yet unknown coefficients.From ( 8) and the structure of the matrix, we see that the integral term is added only to the equation of system (8) that corresponds to the equation of system (2) containing a nonzero perturbation.The equation for the estimation error X(t) = ∧ X (t) − X(t) takes the form: It follows from the last relation that the condition ∧ X (t) → X(t) or X → 0 when t → ∞ (and actually for the damping time of the natural component of the transient) is satisfied if Thus, the observer (8) allows us to identify the perturbation W .The choice of matrices L and K makes it possible to control the rate of this process.
When applied to the object under consideration (2), the described technique leads to the following result.Let us differentiate equation ( 9), assuming that the perturbation the perturbation does not change over the transition time: .
Let's write the system (10) in the expanded form: .
where a ij is matrix A element.
Using the method of excluding the unknowns, we transform the system (11) into a third-order differential equation: ...
The polynomial (12) coincides completely with the characteristic polynomial ( 5).This allows one, if a binomial arrangement of the roots (α + β) 3 is desired, to write down, using the solution obtained earlier (7): where γ has the same meaning as in (7).
Equations ( 8) written element by element in expanded form: Analysis of the efficiency of the method for identifying the resistance torque on a roller bit was carried out on the basis of simulation in the environment of the mathematical extension package Simulink of the MATLAB system, where realized mathematical model of the drilling rod rotation electromechanical system.
The mathematical model takes into account the change of instantaneous values of output voltage of semiconductor converter with pulse-width modulation, full system of equations of asynchronous motor of the drilling rod.Discreteness of digital control system is 2 µs, voltage inverter frequency is 2000 Hz.
Parameters of setting of the observer are calculated according to the initial data of the electrical equipment of the drilling rod SBShS250N head rotation drive.The drilling rig is equipped with AT-04 transistor drive with squirrel cage induction motor AMRU280M4BU2,  Figure 3 shows the calculated graphs of the error in finding the resistance torque by an astatic observer with a binomial distribution of roots (β = 55.6;figure 3(c)) and the Butterworth distribution (β = 83.4;figure 3(d)), In all graphs, the torque amplitudes are shown in relative units (p.u.).Torques are reduced to the rated motor torque M = 580 kg m.

Conclusions
1.The disturbing influence from the load in the system of electric drive can be effectively restored by the observer, to which the signals of the supply voltage and current of motor.Beside it is necessary to introduce three corrective links: two of which have proportional and one has integral dynamic characteristics.At the output of the observer, a signal that is proportional to the ratio of the resistance torque on the working body of the drive system to the moment of inertia is explicitly allocated.
2. Identification of the resistance torque on the roller cone bit allows to realize the way of controlling the drive of the drill rod rotation when the mechanical energy constant flow on the roller cone bit is maintained by forming "hard" mechanical characteristics in the drive of the rotation when drilling in soft and broken formations and "soft" mechanical characteristics when drilling in hard undisturbed formations that will increase the life of the bit.

2 Figure 1 .
Figure 1.Diagram of the formation, transformation and control of the flow of mechanical energy on the cone bit by the rotation drive of the drilling rig rod.Designate: 1 -power supply network; 2 -rectifier; 3 -inverter; 4 -asynchronous motor; 5 -reducer; 6 -cone bit; 7coordinate converter of the ABC axes to the d-q system; 8 -identifier of the resistance torque on the cone bit.

Figure 2 .
Figure 2. Structural diagram of the observing device.

Figure 3 .
Figure 3. Graphs of the resistance torque (a, b) and the determination error (c, d) during their identification on a roller bit, given in p.u.

Table 1 .
Initial parameters of the identification object. 9