Robust Adaptive Control of A Sucker-Rod Pumping System of Oil Wells
Robust Adaptive Control of A Sucker-Rod Pumping System of Oil Wells
Robust Adaptive Control of A Sucker-Rod Pumping System of Oil Wells
Introduction
ISBN: 978-1-61804-164-7
273
tor) localized on the surface of the pump unit is converted in alternative movement of the rods column.
This same column transmits the an alternative movement to the pump components that are located at the
bottom of the well, that are responsible to elevate the
fluid from reservoir up to the surface. The sucker-rod
pump system could be divided in downhole and surface elements (see Fig.(1)).
3
3.1
Adaptive Controller
274
np (s) = sn1 +
n1
X
i sn1i ,
(2)
i=1
dp (s) = sn + 1 sn1 +
n1
X
i+1 sn1i .
(3)
i=1
The desired response given by the transfer function of the model reference signal could be written as
follows,
ym (s)
nm (s)
= M (s) = km
,
r(s)
dm (s)
(4)
nm (s) = s
n1
n1
X
m,i sn1i ,
(5)
i=1
n1
X
i=1
(7)
Rn1
where
contains the elements of i (i = n
1, . . . , 1), i R is the element 1 in (3), Rn1
contains the elements of i+1 (i = n1, . . . , 1) in (3),
and, in the same way, one sets m , m,1 and m , in
(5) and (6). When the plant parameters are unknown
or known with uncertainties, the estimated vector is
given by,
The model plant P (p ) is parametrized in relation with the vector p . An estimator generates p (t),
by processing the input signal u and the output signal y. The estimate p (t) specifies a model characterized by P (p (t)) that, for the purposes of controller
design, is treated as the true model of the plant at
the instant t. The latter is used to calculate the controller parameters c (t) by using the algebraic equation c (t) = f (p (t)). The calculation of the control law C() and the equation c = f (p ) are made
to meet the performance requirements for the model
P (p ).
3.2
p = [kp , T ,
1,
T ]T .
Mathematical Description
np (s)
,
dp (s)
(1)
(8)
For a first-order system, the vector of the estimated parameters could be rewritten as,
p = [kp ,
1 , ]T .
(9)
(10)
1 m,1
,
n =
kp
2n =
km
.
kp
Plant
Reference Model
(12)
(15)
where sgn is called signal function and 1 is an auxiliary signal. In this work this signal is defined as
1 = y. The values of kpnom , k p , and 1 are constants. The values k p and 1 are associated with the
relays sizing in the switching laws in (14) and (15).
From (14) the presence of kpnom (positive and nominal
value of kp ) is justified to prevent that the estimate of
the high frequency gain of the plant kp becomes negative, situation that would violate the assumption A2.
Finally, the sufficient conditions to design the relays
amplitudes and, in turn, to obtain the sliding mode,
k p > |kp kpnom | com kpnom > k p ,
(16)
1 > |1 | .
(17)
The argument va v in (14) is a mean-value-firstorder-filter with a time constant and that it is sufficiently small (i.e., 0). It could be seen here as an
inherent unmodelled dynamics that could influence in
the system stability. The stability proof of the IVSMRAC that considers this fact can be found in [11].
The function va v, therefore, could be set as,
1
v,
(18)
s + 1
where v is an auxiliary signal defined as v = e0 uc .
vav =
ISBN: 978-1-61804-164-7
m (s)
M (s) = km ndm
(s) =
3.874105
s+7.671104
In these models the level in the annular well (measured in meters) is considered as the process variable
(PV), and the pumping speed (measured in cycles per
minute - CPM) as the manipulated variable (MV).
The plant parameters kp and 1 used in the simulations are with 10% of uncertainties around the
model reference parameters. The initial conditions of
the plant and the reference model were different to facilitate the observation of the tracking properties. The
simulations were performed regarding the reference
input r, a set of step signals. It was adopted kpnom =
3, 522 105 and k p = 7, 044 107 . The value
of 1 is chosen from the value |1 | = 6, 973 104 .
It was adopted 10% plus of uncertainty and the value
was 1 = 7, 671 104 . The time constant was adjusted along the simulation. It was adopted = 0, 01.
In Fig.(4) the compared response between the reference model output and the plant output is presented.
In Fig.(5) and Fig.(6) show the error signal (in meters) and the control effort (related to the variation of
the CPM - in percent).
It could be observed in Fig.(4) that the reference
model (desired response) output given was tracked by
the plant output. The process variable (fluid level in
the annular well) could be observed in Fig.(4) with
no oscillations in transitory and stable in steady state.
The error in steady state is small and bounded according to Fig.(5) and the control effort is also bounded
in Fig.(6). However, Fig.(6) indicates an input usage
for the proposed controller that in some cases would
be considered unacceptable. Thus, some strategies for
alleviating the chattering phenomenon [12] should be
further studied.
A new set of simulations is performed to compare the IVS-MRAC designed with a conventional
PID controller. By this time the objective is to evaluate the robustness and adaptation properties of the
1 = 1 sgn(e0 1 ) ,
3.522105
s+6.973104
(13)
(14)
n (s)
P (s) = kp dpp(s) =
(11)
kp = kpnom k p sgn(vav ) ,
276
277
Conclusion
In this paper a robust adaptive controller (called IVSMRAC) was applied to a sucker-rod system of oil
wells. It could be observed through the simulations
and analysis results that the desired response given by
the reference model was tracked by the plant response.
The error signal could be seen bounded and small, and
the control effort could be seen also bounded, though
alleviating the chattering should be further studied.
The results also show that the adaptive controller is
able to control satisfactorily the fluid level in the annular well, in spite of the presence of model uncertainties, unmodelled dynamics, parameter variations, and
perturbations. Moreover, the results reveal that the
control technique could increase the production performance and diminishes the maintenance costs (for
example, by avoiding the fluid pound). For future
work is being considered the implementation of this
adaptive controller in a real physical system.
Acknowledgements: The research was supported by
the CTAI (facilities and infrastructure) at the Universidade Federal da Bahia and CAPES (financial support).
References:
[1] W.L. Lake, Petroleum Engineering Handbook,
2006, Society of Petroleum Engineers, Richardson, USA.
[2] G. Takcs, Sucker-Rod Pumping Manual, 2002,
PennWell Books, Tulsa, USA.
[3] G.V. Moises, T.A. Rolim and J.M. Formigli,
Gedig: Petrobras corporate program for digital
integrated field management, in Proc. of SPE
Intelligent Energy Conference and Exhibition,
2008, Amsterdam, The Netherlands, ISBN: 9781-55563-166-6.
[4] B. Smith, M. Hall, A. Franklin, E.S. Johansen and H. Nalmis, Field-wide deployment of in-well optical flow meters and pressure/temperature gauges at buzzard field, in
Proc. of SPE Intelligent Energy Conference and
Exhibition, 2008, Amsterdam, The Netherlands,
ISBN: 978-1-55563-166-6.
ISBN: 978-1-61804-164-7
278