- Email: [email protected]
Anti-slug control based on a virtual flow measurement
Author’s Accepted Manuscript Anti-slug control based on a virtual flow measurement Esmaeil Jahanshahi, Christoph J. Backi, Sigurd Skogestad www.elsevier.com/locate/flowmeasinst
PII: DOI: Reference:
S0955-5986(17)30012-2 http://dx.doi.org/10.1016/j.flowmeasinst.2017.01.008 JFMI1312
To appear in: Flow Measurement and Instrumentation Received date: 29 February 2016 Revised date: 18 December 2016 Accepted date: 9 January 2017 Cite this article as: Esmaeil Jahanshahi, Christoph J. Backi and Sigurd Skogestad, Anti-slug control based on a virtual flow measurement, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2017.01.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Anti-slug control based on a virtual flow measurement Esmaeil Jahanshahi, Christoph J. Backi, Sigurd Skogestad Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim (email: [email protected])
Abstract Feedback control is an effective and economic solution to prevent slugging flow regimes in offshore oil production. For this, the opening value of a choke valve at the topside platform is usually used as the control input to regulate the pressure or the flow rate in the pipeline. Designing such a control system based on topside measurements, without subsea sensing devices, is preferred from a practical point of view. Controlling the topside pressure alone is difficult and it is not robust in practice, but combining the topside pressure and the flow rate results in a robust control solution. However, measuring the flow rate of a multiphase stream is challenging and requires expensive instrumentation. In this paper, we propose an anti-slug control solution based on a virtual flow measurement. This virtual flow is estimated without neither density nor phase fraction involved, but it gives satisfactory results for the stabilizing control. In particular, applying a cascade structure results in a robust and recommended solution. The performance of the proposed controller is demonstrated by simulation using the realistic OLGA simulator and by experiments. Keywords: Oil production, multiphase flow, flow control, unstable system, robust control 1. Introduction
Feedback control has been shown to be an effective strategy to eliminate slugging (Courbot, 1996; Havre et al.,
In offshore oil production, a multi-phase mixture of oil, gas and water is transported from the producing oil
20
as “anti-slug control” aiming to prevent severe slugging
wells at the seabed to the topside facilities through subsea 5
1 (SS1), transient slugging (SS2, SS3) and other oscilla-
pipelines and risers. Under certain inflow conditions (i.e.
tory (OSC) flow regimes. The feedback control stabilizes
low inflow rates and low pressure), slugging flow regimes occur in the pipeline-riser systems.
a stable (STB) flow regime for boundary conditions (inflow
Such flow regimes
are characterized by severe flow and pressure oscillations.
25
same for all slugging and oscillatory flow regimes. How-
lems in oil production, e.g., poor separation, overflow of
ever, PID parameters of the controller must be re-tuned
inlet separators and unwanted gas flaring (Taitel, 1986).
for different inflow conditions.
The conventional solution to mitigate slugging flow is to reduce the opening of the topside choke valve (choking), but this increases the back-pressure on the produc15
ing oil wells and decreases the production rate. Therefore, a solution that guarantees stable flow together with the maximum possible production rate is desirable. Preprint submitted to Flow Measurement and Instrumentation
velocities and outlet pressure) that would lead to SS1, SS2, SS3 or OSC without control. The control solution is the
These flow conditions cause numerous operational prob10
2000; Godhavn et al., 2005). Such a system is referred to
30
The topside choke valve is usually used as the manipulated variable to regulate (control) subsea pressure at a given pressure setpoint. The subsea pressure sensor is usually installed at the pipeline inlet (Pin ) or at the riser base (Prb ). Controlling the pressure measured from the riser January 10, 2017
35
x3
cal point of view. The main objective of this research is to design an anti-
Ps w
Pin
slug control system based on topside measurements. Stabi-
wg,in wl,in
lizing control using only the topside pressure measurement 40
Z
Prt
top (Prt ) is an alternative which is simpler from a practi-
x1 x2
(Prt ) is not robust; this has been investigated based on
wg wl
x4
Prb
ș
a linear controllability analysis (Storkaas and Skogestad,
Fig. 1: Schematic presentation of the system
2007). The reason is that the right half-plane (RHP) zeros (as seen in the inverse response behaviour) are relatively gamma ray sensors for measuring the phase-fraction are
close to the unstable poles of the system, and consequently 45
relatively expensive and they must be calibrated for dif-
the sensitivity transfer function has an unavoidable large peak. On the other hand, the subsea pressure measure-
75
2005). In this paper, we propose to combine the pressure
ment (Pin ) doesn’t have RHP zeros, and a simple PI con-
drop across the valve and the valve opening ratio to obtain
troller is used in practice (Godhavn et al., 2005; Sivertsen
a virtual flow measure which does not require densities and
et al., 2009). 50
phase fractions. Although this signal is not a flow rate, it
If only the topside pressure measurement is available, a conventional control solution is to design an observer
80
This article is organized as follows.
pressure, and then use these estimates for control (Aamo
tion 2, and the model is used for analysis in Section 3.
that the observer design and the state feedback cannot be 85
and discussed in Section 6. Finally, the main remarks and
RHP zeros can not be bypassed by the observer.
conclusions are summarized in Section 7.
If a multiphase flow measurement is available, the flow output does not have the controllability limitation related
2. First Principle Model
to the RHP zeros. However, this output has a small steadystate gain and it cannot track a given flow setpoint nor 90
65
The control design is explained in Section 4. The simulations and experimental results are presented in Section 5,
fundamental controllability limitation associated with the
60
A mechanistic
model for the severe-slugging flow is introduced in Sec-
et al., 2005; Di Meglio et al., 2012). However, we know generally used in all control applications. Specifically, the
can be used to stabilize the flow incorporating feedback control.
to estimate the states of the system including the subsea
55
ferent fluids and maintained regularly (Corneliussen et al.,
We have developed a dynamic model for riser slugging
reject the disturbances (Jahanshahi et al., 2012). Hence,
based on mass and momentum balances. This model is
in practice, the flow output is combined with the topside
able to capture the main dynamics of the slugging flow
pressure in a cascade structure. Cascade control for sta-
regime, and it is of good fit with the detailed commer-
bilization of slug flow was first proposed by Storkaas and
R (Bendiksen et al., 1991) as well as cial simulator OLGA
Skogestad (2003). Then, it was tested experimentally by 95 experiments. The model is described by only four ODEs Godhavn et al. (2005) and Sivertsen et al. (2009). with soft nonlinear functions which make it suitable for In practice, measuring of a multiphase flow is based on 70
analysis and controller design.
density, velocity and phase fractions of individual phases. The density and phase fraction measurement devices are known to be either expensive or inaccurate. For example, 2
2.1. Summary of the four-state model
Pressure at inlet of pipeline, Pin 50
Figure 1 shows a schematic presentation of the model. 40
The state variables of this model are
P [kPa]
100
30
non-slug (steady)
in
• x1 [kg]: mass of gas in pipeline
Four−state model OLGA model Experiment
slug max
20
• x2 [kg]: mass of liquid in pipeline 10 0
slug min 10
20
30
• x3 [kg]: mass of gas in riser
40 50 60 valve opening, Z [%]
70
80
90
100
1
Pressure at top of riser, P
rt
30
• x4 [kg]: mass of liquid in riser
Four−state model OLGA model Experiment
25
The four state equations of the model are the following
P [kPa]
105
15
rt
mass balances:
slug max non-slug (steady) slug min
20
10
x˙ 1 = wg,in − wg
(1)
5
x˙ 2 = wl,in − wl
(2)
0 0
x˙ 3 = wg − αw
(3)
x˙ 4 = wl − (1 − α)w
(4)
10
20
30
40 50 60 valve opening, Z1 [%]
70
80
90
100
Fig. 2: Bifurcation diagrams of four-state model (blue) compared to OLGA model (red) and experiments (green)
The mass inflow rates of gas and liquid to the system,
110
wg,in and wl,in , are assumed to be constant. The mass
experiments and simulations using the OLGA simulator.
flow rates of gas and liquid from the pipeline to the riser,
The experimental setup is described in Section 5.1.
wg and wl , are described by virtual valve equations. The
The open-loop system has a stable (non-slug) flow when
outlet mixture mass flow rate, w, is determined by the130 Z is smaller than 15%, and it switches to unstable (slugopening percentage of the topside choke valve, Z, which is ging) flow conditions for larger valve openings. The bithe manipulated variable of the control.
furcation diagram as shown in Figure 2 describes steady-
Although (1)-(4) seem to be linear, calculation of the 115
state process values and the minimum and maximum val-
flow rates and the mass fraction α involves several non-
ues when the flow is oscillatory (Storkaas and Skogestad, linear equations (e.g. valve equations and frictions). See135 2007). This diagram may be obtained experimentally or Jahanshahi and Skogestad (2014) for the complete set of from a more detailed model (e.g. OLGA). Such diagrams the model equations.
are used as the reference to fit the model.
2.2. Model fitting 120
3. System Analysis
The four-state model can be partly configured based on In Figure 2, the desired steady-state (middle line) at
dimensions and other physical properties (e.g. fluid prop-
erties) in order to adapt it to a given pipeline-rise system.140 the slugging condition (Z > 15%) is unstable, but it can be stabilized by using control. The slope of the steadyIn addition, four fitting parameters are included in the state line is the static gain of the system, G = ∂y/∂u = model for the purpose of fine-tuning. The fitting proce125
∂Pin /∂Z. As the valve opening increases this slope de-
dure is described in Jahanshahi and Skogestad (2014). In
creases, and the gain finally approaches zero. The con-
this work, the four-state model has been fitted to data from 145
3
troller should keep the loop-gain (L = KG, where G is the
150
process gain and K is the controller gain) constant in order
reason for choosing H∞ control is its robustness towards
to stabilize the system over the whole range of operation.
the nonlinearity. It allows us to keep the system stable for
This nonlinear behaviour causes robustness problems
various setpoint changes. The setpoint change in the nega-
for linear controllers, because the controller requires a large
tive direction (towards a lower process gain) was obviously
gain margin.
Specifically, the control with large valve175 more difficult, and some of simulations with the negative
openings is more difficult, because of the near-zero pro-
setpoint became unstable. The results for Z = 30% are
cess gain.
summarized in Tables 1 and 2. At this operating point (Z = 30%), the inlet pressure is Pin = 23.73 kPa (gauge)
3.1. Nonlinearity analysis of model
and the topside pressure is Prt = 3.84 kPa (gauge). In the
The objective is to evaluate and compare the nonlin-180 tables, Fv is the virtual flow output, and w is the outlet earity of the process seen from the different outputs. For mass flow rate when it is used as the control input. We this, we need to use a standard nonlinearity measure. The
will explain these two variables in Sections 3.1.3 and 3.3
present process is open-loop unstable in the desired re-
respectively.
gion, and it is not possible to quantify the process non3.1.1. Constant inflow
linearity by open-loop simulations of the model. First, it is needed to stabilize the process by applying feedback185 control. Therefore, a closed-loop nonlinearity measure is
setpoint changes of different CVs cause identical pressure
required. We apply the optimal closed-loop nonlinearity
changes. As given in Table 1, for the small setpoint changes
measure proposed in Schweickhardt et al. (2003),
(0.1 kPa), the system remained in the linear region, and
φNOCL := inf sup
L∈G x0 ∈β
NOCL [x∗x0 ] − L[x∗x0 ]L2 NOCL [x∗x0 ]L2
the nonlinearity measure was close to 0. However, for a (5)
190
ing the valve opening (Z). It was not possible to control
infinite horizon control problem for the initial condition
Prt by manipulating the valve opening for the large set-
x0 . In (5), L[x∗x0 ] represents the output of the optimal
point changes. We will discuss the reason later in this
controller applied to the best linear approximation of the nonlinear process. Note that the controller output is equal
195
flow outputs when the inflow rates are constant because
earity measure φNOCL is the relative difference between
the steady-state value of the outlet flow rate is the same
the change of the input for the nonlinear system and the
as the inlet flow rate and, consequently, the static gain of
change of the input for the linearized system when the same change is applied to states of the two systems. The
200
To implement the nonlinearity measure, we stabilized
3.1.2. Pressure-driven inflow
the process by an H∞ optimal controller, then introduced
To evaluate the nonlinearity seen from the flow outputs
setpoint changes in two directions (±) to move the process
w and Fv , we made the inflow rates to the system pressure-
from the initial state x0 . The H∞ controller is a robust
driven which is closer to the reality. For this, we have
linear controller, and the same linear controller was used 170
the flow output for the constant inflow is zero. We consider a pressure-driven case in the following section.
nonlinearity measure must be close to 0 for a linear system. 165
paper. It is not possible to evaluate the nonlinearity of the
to the change of the system input. Therefore, the nonlin160
1 kPa change, the nonlinearity measure became around φNOCL = 0.4 when controlling the pressure by manipulat-
with NOCL [x∗x0 ] := u∗x0 and x∗x0 being the solution to the 155
All the controlled variables (CVs) are scaled such that
applied a linear IPR (Inflow Performance Relationship) as
to stabilize both the linear and nonlinear models. The 4
1 kPa pressure change. In contrary, the process remains
follows: win = CPI max(Psour − Pin , 0),
(6)
approximately linear when the flow rate is the input. This
where CPI is the Productivity Index and Psour is the source
suggests that the nonlinearity is largely caused by the valve equation relating the valve opening to the pressure and the
pressure. 205
flow
As given in Table 2, the flow outputs w and Fv show the
w = Cv f (Z) ρΔP ,
same nonlinearity as for the pressure outputs. By compar-
(7)
ing Table 1 and Table 2, we conclude that the nonlinearity215 although a linear valve (i.e. f (Z) = Z) has been consid-
210
with the valve input Z is very similar for the two inflow
ered in the simulations. In (7), ΔP is the pressure drop
boundary conditions.
over the valve, and ρ is the fluid density.
3.1.3. Flow rate as control signal
3.2. Linear analysis
Next, we considered the outlet mass flow rate w as an
The steady-state behaviors of Pin and Prt (middle lines)
artificial control input denoted by w . Remarkably, the
in Figure 2 are very similar; the difference is only a con-
nonlinearity measure is close to 0 even for large setpoint
stant offset. However, the control system using Prt be-
changes.
comes unstable with large setpoint changes (Tables 1 and
Table 1: Nonlinearity analysis of constant inflow case (larger number
2). Obviously, the gain magnitude alone cannot describe
indicates higher nonlinearity, and † denotes unstable response).
all issues related to control of dynamic systems. Here, we analyze the dynamics in the frequency domain. Figure 3
output input Z
w
shows the response of the topside pressure to a step change
Δr
Pin
Prb
Prt
w
Fv
±0.1
0.037
0.037
0.036
–
–
±1
0.397
0.394
†
–
–
±0.1
0.001
0.002
0.002
–
–
±1
0.014
0.017
0.023
–
–
in the valve opening from 13% to 14%. The response of the four-state model is compared to the OLGA model which shows the relevance of the four-state model for our analysis. The inverse response behavior in the time domain translates to RHP zeros in the frequency domain. Figure 4 shows the Bode plot of the transfer function from
Table 2: Nonlinearity analysis for pressure-driven inflow (larger num-
the valve to Prt . The phase is inverted at high frequen-
ber indicates higher nonlinearity, and † denotes unstable response).
cies. That is, the gain of the system is multiplied by -1 at high frequencies. This happens for the valve openings
output input Z
w
Δr
Pin
Prb
Prt
w
Fv
±0.1
0.039
0.039
0.039
0.039
0.039
±1
0.41
0.41
†
0.41
0.41
±0.1
0
0
0
–
0
±1
0.002
0.007
0.003
–
0.004
20-40% where RHP poles exist and their location is close to the RHP zeros. For Z = 30%, the transfer function of the topside pressure Prt is as follows. G2 =
−1.15(s + 14.21)(s − 0.42)(s − 0.13) . (s2 − 0.16s + 0.022)(s2 + 30s + 330.5)
(8)
The transfer function has two unstable poles at 0.0816 ± 220
The nonlinearities seen from the valve input to all out-
0.1250i and two RHP zeros at 0.13 and 0.42, respectively. The inlet pressure Pin and the outlet flow rate w do
puts are approximately the same (φNOCL 0.4). Thus
not have any RHP zero in the desired region 20-40%. The
the nonlinearity of the valve input is quite high for only a 5
Transfer function from valve to top pressure Prt
2
10
Four−state model OLGA model
11.8
Z = 10% Z = 13% Z = 15% Z = 20% Z = 30% Z = 40%
1
10 Mag [−]
Prt [kPa]
11.6 11.4 11.2
0
10
−1
10
11 −2
10
10.6 45
−2
−1
10
10.8 50
55
60 time [sec]
65
70
75
0
10 ω [rad/s]
10
200 Z = 10% Z = 13% Z = 15% Z = 20% Z = 30% Z = 40%
100 Phase [deg]
Fig. 3: Open-loop response of topside pressure to a step change in the valve opening from 13% to 14%
0 −100
transfer functions for Pin and w at Z = 30% are, respec-
−200 −2 10
tively, G1 =
−0.17(s + 19.29)(s + 0.28) , (s2 − 0.16s + 0.022)(s2 + 30s + 330.5)
−1
0
10 ω [rad/s]
10
Fig. 4: Bode plot of transfer function for topside pressure
(9)
equipped with such a device. Alternatively, in a closed0.13(s + 0.003)(s + 3 × 10−8 )(s2 + 20.29s + 256.7) . 235 loop system, Z is the control signal sent to the valve. For (s2 − 0.16s + 0.02)(s2 + 30s + 330.5) (10) the simplicity of our analysis, we consider a linear valve The static gain of the process with the flow output w is (i.e. f (Z) = Z). close to zero. The reason is that a constant inflow rate has The transfer function of the virtual flow output at Z = been considered as the inlet boundary condition for the 30% is model. As seen in (10), there is one zero very close to the 0.43(s + 0.098)(s + 0.0036)(s2 + 20.33s + 195.2) . = G 4 origin. The inlet boundary can be changed to a pressure(s2 − 0.16s + 0.02)(s2 + 30s + 330.5) (12) driven one to get a non-zero static gain which is closer to G3 =
225
Similar to the actual flow transfer function G3 , the virtual
reality.
flow transfer function G4 does not contain any RHP zero. 3.3. Virtual flow output
240
We define the virtual flow output very similar to the
Although the topside pressure transfer function G2 con-
valve equation in (7), except that we do not include the
tains two RHP-zeros, they do not propagate to G4 . We
density. As a result, we obtain
need to find out under which conditions this property is
√ Fv = Cv f (Z) ΔP .
valid. Such an analysis determines whether controlling this
(11) 245
Similar to the valve equation, ΔP is the pressure drop 230
This property makes it suitable for controller design.
output results in a robust control solution. The virtual flow in (11) is a multiplication of two func-
over the valve and f (Z) is a known function of the valve
tions in the time domain.
characteristics given by the valve manufacturer. Z is the
Fv (t) = h(Z(t)) × g(Prt (t)),
valve opening percentage, which can be measured from
where h(Z(t)) = Cv f (Z(t)), g(Prt (t)) =
the valve positioner feedback, assuming that the valve is 6
(13)
Prt (t) − Ps and
Ps is the constant separator pressure. In order to exam-
and from the mapping in (17) we get
ine locations of poles and zeros for the virtual flow trans-
G4 (s) =
fer function, we need to linearize Fv (t), and then apply a
(19)
where Bi = k1 ai + k2 bi . We require the coefficients of the √ numerator to have the same sign. Since k1 = Cv ΔP0 >
Laplace transform. From Taylor series expansion, we get ∂h(Z) δZ + · · · h(Z) = h(Z0 ) + ∂Z Z=Z0 g(P ) = g(P0 ) +
k1 s4 + B3 s3 + B2 s2 + B1 s + B0 , s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a0
0, we impose the constraint k1 ai + k2 bi > 0 that is
∂g(Prt ) δPrt + · · · ∂Prt Prt =P0
2ΔP0 ai > −Z0 bi ,
i ∈ {0, 1, 2, 3} .
(20)
Reduced order model:
where Z0 and P0 are stationary points with respect to time. δZ and δPrt are small signal values, i.e., deviations
To make the analysis more clear, we apply the conditions
around the stationary points. By applying a linear valve
in (20) on a reduced order model. We perform a balanced
equation (i.e. f (Z) = Z) we receive
model truncation (square root method) on G2 in (8) to
h(Z) = Cv (Z0 + δZ), δPrt , g(P ) ΔP0 + √ 2 ΔP0
obtain a model in the form (14)
G2 (s) =
(15)
s2
k (s − z) , − 2xs + x2 + y 2
(21)
where k > 0, z > 0, x > 0 and y > 0 are the model
where ΔP0 = P0 − Ps . Then, we write the virtual flow Fv
parameters. G2 (s) includes one RHP zero at s = z and
based on equation (13), Z0 δPrt δZδPrt . + ΔP0 δZ + √ Fv (t) Cv Z0 ΔP0 + √ 2 ΔP0 2 ΔP0
two unstable poles at s = x ± yi. By applying the conditions in (20) on the model in (21), it is possible to investigate the necessary conditions
By ignoring the high-order interaction term, we get
on locations of poles and zeros as follows.
Fv (t) F0 + k1 δZ + k2 δPrt ,
2x <
x2 + y 2 Z0 k < 2ΔP0 z
(22) δFv δPrt k1 + k2 , (16) The left-hand side of the inequality in (22) implies that x δZ δZ √ √ 255 should remain small and hence the unstable poles cannot where F0 = Cv Z0 ΔP0 , k1 = Cv ΔP0 and √ be located far into the RHP (fast instability). However, k2 = Cv Z0 /(2 ΔP0 ). Thus, in the Laplace domain we the imaginary part of the poles (y) appears on the righthave G4 (s) k1 + k2 G2 (s).
hand side, and it is easier to satisfy the conditions for
(17)
poles with large imaginary parts. In addition, the RHP-
In fact, (17) gives the same transfer function as in (12). 260
The transfer function in (12) was obtained from a direct
conditions are valid for systems with oscillatory nature
linearization of the nonlinear model in (1)-(4).
(complex-conjugate poles) and RHP zeros relatively close
Next, we investigate the conditions under which the 250
to the origin. In fact, these properties are valid for the
RHP-zeros disappear after the linear mapping in (17).
slugging flow regime.
Fourth order model:
System with real unstable pole:
The topside pressure transfer function is in the following
To check the conditions for systems with one real unstable
general form: b 3 s3 + b 2 s2 + b 1 s + b 0 , G2 (s) = 4 s + a 3 s 3 + a 2 s 2 + a 1 s + a0
zero z, should be close to the origin. In other words, the
pole, we consider the following transfer function model: (18)
G2 (s) = 7
k (s − b) , (Tf s + 1)(s − a)
(23)
Fv
where a > 0, b > 0 and Tf ≥ 0 is a filter time-constant.
√ Cv f (Z) ΔP
The filter is added to have a strictly proper transfer funcΔPset
tion. From the linear mapping in (17), we get G4 (s)
=
Tf s2 + (1 − aTf + 1 k1 (Tf s
k k2 k1 )s
−a−
+ 1)(s − a)
bk k2 k1
,
Cm
Cs
(24)
Topside choke
The conditions that the RHP zero disappears are derived as follows
k2 k −a , < k1 b
aTf − 1 <
aTf − 1 < 265
Z0 k −a . < 2ΔP0 b
Inlet separator
Riser
(25)
Subsea manifold
and by using k2 /k1 = Z0/(2ΔP0 ) for the virtual flow output, we get
Z
ΔP
Fig. 5: Cascade control structure based on virtual flow
(26) P2
To satisfy the conditions (25) and (26), stricter conditions
Top-side Valve
Air to atm. Seperator
are required on the gain and the location of the pole and the zero compared to the conditions in (22). It is not
Riser P1
possible to satisfy (25) and (26) for k > 0. However, the
safety valve P3 FT water
slugging system is unstable (a > 0) and the static gain 270
Buffer Tank
Mixing Point
Pipeline
FT air
P4
for the pressure is alway negative (G2 (0) < 0) that means k < 0. Hence, it is possible to satisfy the conditions for
Water Reservoir
the slugging system even by assuming a real unstable pole.
Pump
Water Recycle
Fig. 6: Schematic diagram of experimental setup
If a < 0 (stable system), it is easy to make the the RHP-zero disappear. In this case, the smaller Tf , the eas275
as the master controller, 1 Td s Kp1 1+ + . Cm (s) = Tf s + 1 Ti s Tf 1 s + 1
ier the conditions can be satisfied. 4. Control Design
The same low-pass filter 1/(Tf s + 1) is added to both slave
A cascade control structure is used in simulations and
280
(28)
and master controllers. This filter plays two roles; Firstly,
experiments in this work. As shown in Figure 5, the virtual285 it reduces the measurement noise, and secondly, it also flow Fv is controlled by the slave controller Cs , whereas has a dynamic effect. In addition, another first-order lowthe pressure drop across the valve ΔP is controlled by the pass filter with the time constant of T was added to the fr
master controller Cm . The slave control loop is shown in
reference input of the master controller to obtain a 2-DOF
blue, while the master control loop is indicated in red.
controller.
A proportional controller with a low-pass filter was used as the slave controller, Cs (s) =
290
Kp2 , Tf s + 1
(27)
5. Results 5.1. Experimental setup The experiments were carried out in laboratory setup
and a proportional-integrator-derivative (PID) controller
for anti-slug control at the Chemical Engineering Depart-
with a low-pass filter on the derivative action was applied
ment of NTNU. Figure 6 shows a schematic representation 8
295
of the laboratory setup. The pipeline and the riser are
40%. Figure 8 demonstrates the experimental result for re-
made from flexible pipes with 2 cm inner diameters. The
jecting a liquid disturbance in the inflow. Here, the inflow
length of the pipeline is 4 m and it is inclined with a 15◦
rate was changed from 4.1 to 5.5 litres/min at t = 600 sec.
angle. The height of the riser is 3 m. A buffer tank is used
To reduce the measurement noise, a second-order But-
to simulate the effect of a long pipe with the same volume,335 terworth filter with the normalized cutoff frequency of 300
such that the total length of the pipe would be about 70 m.
ωn = 0.03 was applied in the experiments. Since the sam-
The topside choke valve is used as the input for the
pling interval of the measurements is Ts = 0.1 sec, this
control. The separator pressure after the topside choke
filter gives a bandwidth of 0.94 rad/sec. For the slave
valve is nominally constant at atmospheric pressure. The
controller, a proportional controller Kp2 = 4 was applied
feed into the pipeline is assumed to be at constant flow340 in the experiments. The Butterworth filter acts instead of 305
the filter in (27) with Tf 1.06 sec.
rates, namely 4 s.liters/min of water and 4.5 s.liters/min of air. These flow rates are in standard conditions and
The PID controller used for the master controller was
are equivalent to 9.7 × 10−5 kg/s gas and 0.0667 kg/s
tuned as follows: Cm (s) = −0.2 1 +
liquid. With these boundary conditions, the critical valve opening, where the system switches from stable (non-slug) 310
to oscillatory (slug) flow, is at Z ∗ = 15%.
5s 1 + 300s 20s + 1
.
(29)
The same Butterworth filter as the one in the slave loop was applied to reduce the measurement noise of the master
5.2. Implementation of controller
loop, and a reference filter with the time constant Tf r = The virtual flowmeter and the controllers are imple-345 5 sec is introduced to compose a 2-DOF PID controller. mented in Matlab and run on a PC in real time. National 5.4. OLGA simulations
Instruments I/O modules are used to read the measure315
ments from sensors and write back the control signal to
To further evaluate the performance of the proposed
the valve. The communicate between the controllers in
controller, we used the dynamic flow simulator OLGA
Matlab and the I/O Modules is performed using the Data
(Bendiksen et al., 1991). We have modelled the exper-
Aquisition Toolbox of Matlab.
350
Since the calculation of the virtual flow is very simple, 320
325
imental setup described above in the OLGA simulator. Figure 9 shows a schematic presentation of the model in
it can be easily implemented in a Programmable Logic
the simulator.
Controller (PLC) and tested in industrial control systems.
The calculation of the virtual flow output and con-
To start up the controller, it is always easiest to close the
trollers were implemented in MATLAB, and the OLGA
valve to get a non-slugging flow, then turn on the con-
simulator was connected to MATLAB via OPC Server.
troller, and then increase the valve opening again by de-
The OLGA OPC Server only allows a sampling rate of
creasing the pressure setpoint.
Ts = 1 sec when the simulation is controlled by an external clock. This imposes a limitation on the filter and
5.3. Experimental results
controller tunings, and therefore the controllers used in
Figure 7 shows experimental performance of the cas-
the simulations are different from those used in the exper-
cade control. The valve opens as the setpoint of the pres-
iments. The slave controller used in the simulations was
sure drop across the valve decreases. The cascade con330
Cs (s) =
troller is able to stabilize the flow up to a valve opening of 9
8 , 4s + 1
(30)
Pressure drop across the valve, ΔP (master loop) ΔP [kPa]
10
measurement setpoint
Pressure drop across the valve, ΔP (master loop) 15
ΔP [kPa]
5
0 0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
0 0
500
time [sec] Virtual flow output, Fv (slave loop)
25
10
1000 1500 2000 2500 3000 3500 4000 4500 5000
v
F [-]
5
time [sec] Virtual flow output, Fv (slave loop)
20
measurement setpoint
10
20 v
F [-]
measurement setpoint
0 0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
5 0
60
20
40
Z [%]
40
0 0
500
measurement setpoint
10
time [sec] Valve opening, Z
60
Z [%]
15
1000 1500 2000 2500 3000 3500 4000 4500 5000
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec] Valve opening, Z
20
time [sec]
0 0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
Fig. 7: Experimental result for step changes in setpoint of pressure drop
time [sec] Liquid inflow rate, wL,in
whereas the master controller was 1 −0.2 1+ . Cm (s) = 4s + 1 120s
wL,in [L/min]
6
(31)
5
4 0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec]
Figure 10 shows the performance of the cascade control
Fig. 8: Experimental result for rejecting inflow liquid disturbances
for setpoint tracking. The cascade controller is able to 355
(step from 4.1 to 5.5 at t = 600sec)
stabilize the flow up to a valve opening of 40%. Figure 11 demonstrates the simulation results for rejecting disturbances in liquid and gas inflow rates. Here, 20% step
valve
changes from the nominal values are introduced. riser
6. Discussion 360
water source
6.1. Closed-loop sensitivity analysis
pipeline
Figure 12 shows two equivalent forms of the closedair source
loop system. If the virtual flow is the controlled output
buffer tank
(Figure 12.a)), the process model is given by (17) which is the same as in (12). In simulations, for the slave loop, we
Fig. 9: Schematic of OLGA model used in simulations
applied a controller as Cs (s) =
8 , 4s + 1
(32) 10
Pressure drop across the valve, ΔP (master loop)
4
Pressure drop across the valve, ΔP (master loop)
10
measurement setpoint
ΔP [kPa]
ΔP [kPa]
6
2
measurement setpoint
5
0 0
500
1000
1500
2000
2500
3000
3500
0 0
time [sec] Virtual flow output, F (slave loop) 13
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec] Virtual flow output, Fv (slave loop)
16
12
14
11
F [-]
measurement setpoint
10
v
Fv [-]
500
v
12 measurement setpoint
10
9 0
500
1000
1500
2000
2500
3000
3500
8 0
time [sec] Valve opening, Z
60
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec] Valve opening, Z
Z [%]
Z [%]
60 40 20
40 20
0 0
500
1000
1500
2000
2500
3000
3500
0 0
time [sec]
Fig. 10: OLGA simulation result for step changes in setpoint of
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec] Inflow rates, w G,in , w L,in
5.5
w [L/min]
pressure drop
and the sensitivity transfer function of the closed-loop sys-
5 wL,in
4.5
wG,in
4
tem becames
0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec]
1 . S4 (s) = 1 + Cs (s)G4 (s)
(33)
Fig. 11: OLGA simulation results for rejection of liquid and gas disturbances in the inflow (20% step change from nominal rates)
The closed-loop system can be rearranged in a way that the linear mapping takes place on the controller and the
a) closed-loop system with virtual flow as controlled output
r4 + −
pressure becomes the controlled output (Figure 12.b). For this case, the controller is 8k2 k2 Cs (s) = , C (s) = 1 + k1 Cs (s) 4s + 8k1 + 1
1 . 1 + C (s)G2 (s)
G2 (s)
(34)
G4 (s)
k2
+ +
k1
and the sensitivity transfer function becomes S2 (s) =
Cs
b) closed-loop system with pressure as controlled output
r2 + −
(35)
Figure 13 shows the amplitude of sensitivity transfer
k2 + −
C
Cs
G2 (s)
k1
functions for the two forms of the closed-loop system. The closed-loop system with the virtual flow output has a low sensitivity (S4 ) which is desirable, while the closed-loop 365
Fig. 12: Two equivalent forms of closed-loop system
system with the pressure output shows a high sensitivity 11
where two RHP zeros same as in (8) exist. However, the
12 S2
process is stable from the flow input w to the pressure
S4
10
385
|S|
8
output. Thus, for the master control loop, the RHP zeros are not as limiting as for the slave loop which is open-loop unstable. In addition, since the process is fairly linear from
6
the flow input w , the master controller does not require
4
a wide gain-margin. Therefore, the master controller can 2 390
0 10-3
10-2
10-1
100
101
is not robust for non-minimum-phase dynamics.
102
For the controllability reason, we require to maintain
ω [Rad/sec]
Fig. 13: Amplitude of sensitivity transfer functions for two forms of
a minimum pressure drop across the valve. The master
closed-loop system
control loop can be considered as an supervisory layer that 395
(S2 ). This confirms that the physical controllability limi-
The main challenge in experiments was to reduce the
the virtual flow output shows low sensitivity. The high
measurements noise. A second order Butterworth filter
sensitivity signals a robustness problem, but the proposed
was applied for this purpose. Choosing a suitable band-
controller performs well in experiments and simulations. This is a paradox that requires further investigation and
400
width for the filter is very crucial. If we try to reduce the noise completely by setting a low bandwidth for the
analysis.
filter, this causes attenuation of the controlled signal and also a lag in the control loop. Consequently, we would
6.2. Nonlinearity
need to increase the controller gain to compensate the fil-
The nonlinearity measure for the virtual flow output is 375
ensures controllability of the flow. 6.4. Measurement noise
tation of the topside pressure cannot be bypassed, though
370
be tuned to be relatively slow. Note that a tight control
φNOCL = 0.4 as given in Table 2. This indicates a high405 ter attenuation. However, because of the lag in the control loop, increasing the controller gain causes oscillations and nonlinearity, and we need to consider it for the control robustness issues.
design. In the OLGA simulations shown in Figure 10, we have applied a gain-scheduling based on the pressure drop
7. Conclusion
setpoint. We changed Kp2 from 8 to 10 and 15 for the 380
two last setpoint changes, otherwise the controller becomes unstable.
We proposed a new control strategy for anti-slug con-
410
is based on a virtual flow measurement.
6.3. Inverse response in master control loop
Robustness properties of the proposed method were in-
In Figure 11, when the liquid inflow disturbance is in-
vestigated by frequency domain analysis. The virtual flow
troduced at t = 600 sec, we observe that pressure drop
output is not affected by inverse response dynamics. We
shows an inverse response. This means that RHP-zeros for the pressure are still in place. In fact, the transfer function
415
showed that this property is valid for systems with oscillatory nature such as slugging flow dynamics.
from the flow input w to the pressure is as follows G2 =
trol based on topside measurements. This control solution
The suggested cascade controller was tested success-
−518.77(s + 14.21)(s − 0.4207)(s − 0.1302) , (36) s(s + 0.003018)(s2 + 20.29s + 256.7)
fully in OLGA simulations and experiments where satis12
factory results were achieved for both the setpoint tracking 420
Schweickhardt, T., Allgower, F., and Doyle, F. (2003). Nonlinearity quantification for the optimal state feedback controller. In Euro-
and the disturbance rejection.
pean Control Conference (ECC), 330–335.
The open-loop frequency domain analysis shows that
Sivertsen, H., Alstad, V., and Skogestad, S. (2009). Medium-scale
the virtual flow output does not have the controllability465 limitation related to RHP zeros. However, in closed-loop,
& Construction, 4(4), 156–170, SPE no. 120040. Storkaas, E. and Skogestad, S. (2003). Cascade control of unstable
the pressure output still has a hight sensitivity which sig425
systems with application to stabilization of slug flow. In IFAC
nals a robustness problem. Further analysis in the frequency domain is required to fully investigate this issue.
experiments on stabilizing riser-slug flow. SPE Projects, Facilities
7th International Symposium on Advanced Control of Chemical 470
Processes, Adchem. Hong Kong. Storkaas, E. and Skogestad, S. (2007). Controllability analysis of twophase pipeline-riser systems at riser slugging conditions. Control
Acknowledgements
Engineering Practice, 15(5), 567–581. Taitel, Y. (1986). Stability of severe slugging. International Journal
Financial support for this research was provided by Subsea production and processing (SUBPRO) center for 430
475
research-based innovation. Aamo, O., Eikrem, G., Siahaan, H., and Foss, B. (2005). Observer design for multiphase flow in vertical pipes with gas-lift - theory and experiments. Journal of Process Control, 15(3), 247 – 257. Bendiksen, K.H., Malnes, D., Moe, R., and Nuland, S. (1991). Dy-
435
namic two-fluid model olga. theory and application. SPE Production Engineering, 6(2), 171–180. Corneliussen, S., Couput, J.P., Dahl, E., Dykesteen, E., Frøysa, K.E., Malde, E., Moestue, H., Moksnes, P.O., Scheers, L., and Tunheim, H. (2005). Handbook of Multiphase Flow Metering. The
440
Norwegian Society for Oil and Gas Measurement, Oslo, Norway, 2 edition. Courbot, A. (1996). Prevention of severe slugging in the dunbar 16 inches multiphase pipeline. In Proceedings of the Annual Offshore Technology Conference, volume 4, 445–452. SP E no. 8196.
445
Di Meglio, F., Petit, N., Alstad, V., and Kaasa, G.O. (2012). Stabilization of slugging in oil production facilities with or without upstream pressure sensors. Journal of Process Control, 22(4), 809 – 822. doi:10.1016/j.jprocont.2012.02.014. Godhavn, J.M., Fard, M.P., and Fuchs, P.H. (2005). New slug con-
450
trol strategies, tuning rules and experimental results. Journal of Process Control, 15, 547–557. Havre, K., Stornes, K., and Stray, H. (2000). Taming slug flow in pipelines. ABB Review, 4, 55–63. Jahanshahi, E., Skogestad, S., and Helgesen, A.H. (2012). Control-
455
lability analysis of severe slugging in well-pipeline-riser systems. In IFAC Workshop - Automatic Control in Offshore Oil and Gas Production, 101–108. Trondheim, Norway. Jahanshahi, E. and Skogestad, S. (2014). Simplified dynamic models for control of riser slugging in offshore oil production. SPE Oil
460
and Gas Facilities, 3(6), 64–79.
13
of Multiphase Flow, 12(2), 203–217.
PII: DOI: Reference:
S0955-5986(17)30012-2 http://dx.doi.org/10.1016/j.flowmeasinst.2017.01.008 JFMI1312
To appear in: Flow Measurement and Instrumentation Received date: 29 February 2016 Revised date: 18 December 2016 Accepted date: 9 January 2017 Cite this article as: Esmaeil Jahanshahi, Christoph J. Backi and Sigurd Skogestad, Anti-slug control based on a virtual flow measurement, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2017.01.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Anti-slug control based on a virtual flow measurement Esmaeil Jahanshahi, Christoph J. Backi, Sigurd Skogestad Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim (email: [email protected])
Abstract Feedback control is an effective and economic solution to prevent slugging flow regimes in offshore oil production. For this, the opening value of a choke valve at the topside platform is usually used as the control input to regulate the pressure or the flow rate in the pipeline. Designing such a control system based on topside measurements, without subsea sensing devices, is preferred from a practical point of view. Controlling the topside pressure alone is difficult and it is not robust in practice, but combining the topside pressure and the flow rate results in a robust control solution. However, measuring the flow rate of a multiphase stream is challenging and requires expensive instrumentation. In this paper, we propose an anti-slug control solution based on a virtual flow measurement. This virtual flow is estimated without neither density nor phase fraction involved, but it gives satisfactory results for the stabilizing control. In particular, applying a cascade structure results in a robust and recommended solution. The performance of the proposed controller is demonstrated by simulation using the realistic OLGA simulator and by experiments. Keywords: Oil production, multiphase flow, flow control, unstable system, robust control 1. Introduction
Feedback control has been shown to be an effective strategy to eliminate slugging (Courbot, 1996; Havre et al.,
In offshore oil production, a multi-phase mixture of oil, gas and water is transported from the producing oil
20
as “anti-slug control” aiming to prevent severe slugging
wells at the seabed to the topside facilities through subsea 5
1 (SS1), transient slugging (SS2, SS3) and other oscilla-
pipelines and risers. Under certain inflow conditions (i.e.
tory (OSC) flow regimes. The feedback control stabilizes
low inflow rates and low pressure), slugging flow regimes occur in the pipeline-riser systems.
a stable (STB) flow regime for boundary conditions (inflow
Such flow regimes
are characterized by severe flow and pressure oscillations.
25
same for all slugging and oscillatory flow regimes. How-
lems in oil production, e.g., poor separation, overflow of
ever, PID parameters of the controller must be re-tuned
inlet separators and unwanted gas flaring (Taitel, 1986).
for different inflow conditions.
The conventional solution to mitigate slugging flow is to reduce the opening of the topside choke valve (choking), but this increases the back-pressure on the produc15
ing oil wells and decreases the production rate. Therefore, a solution that guarantees stable flow together with the maximum possible production rate is desirable. Preprint submitted to Flow Measurement and Instrumentation
velocities and outlet pressure) that would lead to SS1, SS2, SS3 or OSC without control. The control solution is the
These flow conditions cause numerous operational prob10
2000; Godhavn et al., 2005). Such a system is referred to
30
The topside choke valve is usually used as the manipulated variable to regulate (control) subsea pressure at a given pressure setpoint. The subsea pressure sensor is usually installed at the pipeline inlet (Pin ) or at the riser base (Prb ). Controlling the pressure measured from the riser January 10, 2017
35
x3
cal point of view. The main objective of this research is to design an anti-
Ps w
Pin
slug control system based on topside measurements. Stabi-
wg,in wl,in
lizing control using only the topside pressure measurement 40
Z
Prt
top (Prt ) is an alternative which is simpler from a practi-
x1 x2
(Prt ) is not robust; this has been investigated based on
wg wl
x4
Prb
ș
a linear controllability analysis (Storkaas and Skogestad,
Fig. 1: Schematic presentation of the system
2007). The reason is that the right half-plane (RHP) zeros (as seen in the inverse response behaviour) are relatively gamma ray sensors for measuring the phase-fraction are
close to the unstable poles of the system, and consequently 45
relatively expensive and they must be calibrated for dif-
the sensitivity transfer function has an unavoidable large peak. On the other hand, the subsea pressure measure-
75
2005). In this paper, we propose to combine the pressure
ment (Pin ) doesn’t have RHP zeros, and a simple PI con-
drop across the valve and the valve opening ratio to obtain
troller is used in practice (Godhavn et al., 2005; Sivertsen
a virtual flow measure which does not require densities and
et al., 2009). 50
phase fractions. Although this signal is not a flow rate, it
If only the topside pressure measurement is available, a conventional control solution is to design an observer
80
This article is organized as follows.
pressure, and then use these estimates for control (Aamo
tion 2, and the model is used for analysis in Section 3.
that the observer design and the state feedback cannot be 85
and discussed in Section 6. Finally, the main remarks and
RHP zeros can not be bypassed by the observer.
conclusions are summarized in Section 7.
If a multiphase flow measurement is available, the flow output does not have the controllability limitation related
2. First Principle Model
to the RHP zeros. However, this output has a small steadystate gain and it cannot track a given flow setpoint nor 90
65
The control design is explained in Section 4. The simulations and experimental results are presented in Section 5,
fundamental controllability limitation associated with the
60
A mechanistic
model for the severe-slugging flow is introduced in Sec-
et al., 2005; Di Meglio et al., 2012). However, we know generally used in all control applications. Specifically, the
can be used to stabilize the flow incorporating feedback control.
to estimate the states of the system including the subsea
55
ferent fluids and maintained regularly (Corneliussen et al.,
We have developed a dynamic model for riser slugging
reject the disturbances (Jahanshahi et al., 2012). Hence,
based on mass and momentum balances. This model is
in practice, the flow output is combined with the topside
able to capture the main dynamics of the slugging flow
pressure in a cascade structure. Cascade control for sta-
regime, and it is of good fit with the detailed commer-
bilization of slug flow was first proposed by Storkaas and
R (Bendiksen et al., 1991) as well as cial simulator OLGA
Skogestad (2003). Then, it was tested experimentally by 95 experiments. The model is described by only four ODEs Godhavn et al. (2005) and Sivertsen et al. (2009). with soft nonlinear functions which make it suitable for In practice, measuring of a multiphase flow is based on 70
analysis and controller design.
density, velocity and phase fractions of individual phases. The density and phase fraction measurement devices are known to be either expensive or inaccurate. For example, 2
2.1. Summary of the four-state model
Pressure at inlet of pipeline, Pin 50
Figure 1 shows a schematic presentation of the model. 40
The state variables of this model are
P [kPa]
100
30
non-slug (steady)
in
• x1 [kg]: mass of gas in pipeline
Four−state model OLGA model Experiment
slug max
20
• x2 [kg]: mass of liquid in pipeline 10 0
slug min 10
20
30
• x3 [kg]: mass of gas in riser
40 50 60 valve opening, Z [%]
70
80
90
100
1
Pressure at top of riser, P
rt
30
• x4 [kg]: mass of liquid in riser
Four−state model OLGA model Experiment
25
The four state equations of the model are the following
P [kPa]
105
15
rt
mass balances:
slug max non-slug (steady) slug min
20
10
x˙ 1 = wg,in − wg
(1)
5
x˙ 2 = wl,in − wl
(2)
0 0
x˙ 3 = wg − αw
(3)
x˙ 4 = wl − (1 − α)w
(4)
10
20
30
40 50 60 valve opening, Z1 [%]
70
80
90
100
Fig. 2: Bifurcation diagrams of four-state model (blue) compared to OLGA model (red) and experiments (green)
The mass inflow rates of gas and liquid to the system,
110
wg,in and wl,in , are assumed to be constant. The mass
experiments and simulations using the OLGA simulator.
flow rates of gas and liquid from the pipeline to the riser,
The experimental setup is described in Section 5.1.
wg and wl , are described by virtual valve equations. The
The open-loop system has a stable (non-slug) flow when
outlet mixture mass flow rate, w, is determined by the130 Z is smaller than 15%, and it switches to unstable (slugopening percentage of the topside choke valve, Z, which is ging) flow conditions for larger valve openings. The bithe manipulated variable of the control.
furcation diagram as shown in Figure 2 describes steady-
Although (1)-(4) seem to be linear, calculation of the 115
state process values and the minimum and maximum val-
flow rates and the mass fraction α involves several non-
ues when the flow is oscillatory (Storkaas and Skogestad, linear equations (e.g. valve equations and frictions). See135 2007). This diagram may be obtained experimentally or Jahanshahi and Skogestad (2014) for the complete set of from a more detailed model (e.g. OLGA). Such diagrams the model equations.
are used as the reference to fit the model.
2.2. Model fitting 120
3. System Analysis
The four-state model can be partly configured based on In Figure 2, the desired steady-state (middle line) at
dimensions and other physical properties (e.g. fluid prop-
erties) in order to adapt it to a given pipeline-rise system.140 the slugging condition (Z > 15%) is unstable, but it can be stabilized by using control. The slope of the steadyIn addition, four fitting parameters are included in the state line is the static gain of the system, G = ∂y/∂u = model for the purpose of fine-tuning. The fitting proce125
∂Pin /∂Z. As the valve opening increases this slope de-
dure is described in Jahanshahi and Skogestad (2014). In
creases, and the gain finally approaches zero. The con-
this work, the four-state model has been fitted to data from 145
3
troller should keep the loop-gain (L = KG, where G is the
150
process gain and K is the controller gain) constant in order
reason for choosing H∞ control is its robustness towards
to stabilize the system over the whole range of operation.
the nonlinearity. It allows us to keep the system stable for
This nonlinear behaviour causes robustness problems
various setpoint changes. The setpoint change in the nega-
for linear controllers, because the controller requires a large
tive direction (towards a lower process gain) was obviously
gain margin.
Specifically, the control with large valve175 more difficult, and some of simulations with the negative
openings is more difficult, because of the near-zero pro-
setpoint became unstable. The results for Z = 30% are
cess gain.
summarized in Tables 1 and 2. At this operating point (Z = 30%), the inlet pressure is Pin = 23.73 kPa (gauge)
3.1. Nonlinearity analysis of model
and the topside pressure is Prt = 3.84 kPa (gauge). In the
The objective is to evaluate and compare the nonlin-180 tables, Fv is the virtual flow output, and w is the outlet earity of the process seen from the different outputs. For mass flow rate when it is used as the control input. We this, we need to use a standard nonlinearity measure. The
will explain these two variables in Sections 3.1.3 and 3.3
present process is open-loop unstable in the desired re-
respectively.
gion, and it is not possible to quantify the process non3.1.1. Constant inflow
linearity by open-loop simulations of the model. First, it is needed to stabilize the process by applying feedback185 control. Therefore, a closed-loop nonlinearity measure is
setpoint changes of different CVs cause identical pressure
required. We apply the optimal closed-loop nonlinearity
changes. As given in Table 1, for the small setpoint changes
measure proposed in Schweickhardt et al. (2003),
(0.1 kPa), the system remained in the linear region, and
φNOCL := inf sup
L∈G x0 ∈β
NOCL [x∗x0 ] − L[x∗x0 ]L2 NOCL [x∗x0 ]L2
the nonlinearity measure was close to 0. However, for a (5)
190
ing the valve opening (Z). It was not possible to control
infinite horizon control problem for the initial condition
Prt by manipulating the valve opening for the large set-
x0 . In (5), L[x∗x0 ] represents the output of the optimal
point changes. We will discuss the reason later in this
controller applied to the best linear approximation of the nonlinear process. Note that the controller output is equal
195
flow outputs when the inflow rates are constant because
earity measure φNOCL is the relative difference between
the steady-state value of the outlet flow rate is the same
the change of the input for the nonlinear system and the
as the inlet flow rate and, consequently, the static gain of
change of the input for the linearized system when the same change is applied to states of the two systems. The
200
To implement the nonlinearity measure, we stabilized
3.1.2. Pressure-driven inflow
the process by an H∞ optimal controller, then introduced
To evaluate the nonlinearity seen from the flow outputs
setpoint changes in two directions (±) to move the process
w and Fv , we made the inflow rates to the system pressure-
from the initial state x0 . The H∞ controller is a robust
driven which is closer to the reality. For this, we have
linear controller, and the same linear controller was used 170
the flow output for the constant inflow is zero. We consider a pressure-driven case in the following section.
nonlinearity measure must be close to 0 for a linear system. 165
paper. It is not possible to evaluate the nonlinearity of the
to the change of the system input. Therefore, the nonlin160
1 kPa change, the nonlinearity measure became around φNOCL = 0.4 when controlling the pressure by manipulat-
with NOCL [x∗x0 ] := u∗x0 and x∗x0 being the solution to the 155
All the controlled variables (CVs) are scaled such that
applied a linear IPR (Inflow Performance Relationship) as
to stabilize both the linear and nonlinear models. The 4
1 kPa pressure change. In contrary, the process remains
follows: win = CPI max(Psour − Pin , 0),
(6)
approximately linear when the flow rate is the input. This
where CPI is the Productivity Index and Psour is the source
suggests that the nonlinearity is largely caused by the valve equation relating the valve opening to the pressure and the
pressure. 205
flow
As given in Table 2, the flow outputs w and Fv show the
w = Cv f (Z) ρΔP ,
same nonlinearity as for the pressure outputs. By compar-
(7)
ing Table 1 and Table 2, we conclude that the nonlinearity215 although a linear valve (i.e. f (Z) = Z) has been consid-
210
with the valve input Z is very similar for the two inflow
ered in the simulations. In (7), ΔP is the pressure drop
boundary conditions.
over the valve, and ρ is the fluid density.
3.1.3. Flow rate as control signal
3.2. Linear analysis
Next, we considered the outlet mass flow rate w as an
The steady-state behaviors of Pin and Prt (middle lines)
artificial control input denoted by w . Remarkably, the
in Figure 2 are very similar; the difference is only a con-
nonlinearity measure is close to 0 even for large setpoint
stant offset. However, the control system using Prt be-
changes.
comes unstable with large setpoint changes (Tables 1 and
Table 1: Nonlinearity analysis of constant inflow case (larger number
2). Obviously, the gain magnitude alone cannot describe
indicates higher nonlinearity, and † denotes unstable response).
all issues related to control of dynamic systems. Here, we analyze the dynamics in the frequency domain. Figure 3
output input Z
w
shows the response of the topside pressure to a step change
Δr
Pin
Prb
Prt
w
Fv
±0.1
0.037
0.037
0.036
–
–
±1
0.397
0.394
†
–
–
±0.1
0.001
0.002
0.002
–
–
±1
0.014
0.017
0.023
–
–
in the valve opening from 13% to 14%. The response of the four-state model is compared to the OLGA model which shows the relevance of the four-state model for our analysis. The inverse response behavior in the time domain translates to RHP zeros in the frequency domain. Figure 4 shows the Bode plot of the transfer function from
Table 2: Nonlinearity analysis for pressure-driven inflow (larger num-
the valve to Prt . The phase is inverted at high frequen-
ber indicates higher nonlinearity, and † denotes unstable response).
cies. That is, the gain of the system is multiplied by -1 at high frequencies. This happens for the valve openings
output input Z
w
Δr
Pin
Prb
Prt
w
Fv
±0.1
0.039
0.039
0.039
0.039
0.039
±1
0.41
0.41
†
0.41
0.41
±0.1
0
0
0
–
0
±1
0.002
0.007
0.003
–
0.004
20-40% where RHP poles exist and their location is close to the RHP zeros. For Z = 30%, the transfer function of the topside pressure Prt is as follows. G2 =
−1.15(s + 14.21)(s − 0.42)(s − 0.13) . (s2 − 0.16s + 0.022)(s2 + 30s + 330.5)
(8)
The transfer function has two unstable poles at 0.0816 ± 220
The nonlinearities seen from the valve input to all out-
0.1250i and two RHP zeros at 0.13 and 0.42, respectively. The inlet pressure Pin and the outlet flow rate w do
puts are approximately the same (φNOCL 0.4). Thus
not have any RHP zero in the desired region 20-40%. The
the nonlinearity of the valve input is quite high for only a 5
Transfer function from valve to top pressure Prt
2
10
Four−state model OLGA model
11.8
Z = 10% Z = 13% Z = 15% Z = 20% Z = 30% Z = 40%
1
10 Mag [−]
Prt [kPa]
11.6 11.4 11.2
0
10
−1
10
11 −2
10
10.6 45
−2
−1
10
10.8 50
55
60 time [sec]
65
70
75
0
10 ω [rad/s]
10
200 Z = 10% Z = 13% Z = 15% Z = 20% Z = 30% Z = 40%
100 Phase [deg]
Fig. 3: Open-loop response of topside pressure to a step change in the valve opening from 13% to 14%
0 −100
transfer functions for Pin and w at Z = 30% are, respec-
−200 −2 10
tively, G1 =
−0.17(s + 19.29)(s + 0.28) , (s2 − 0.16s + 0.022)(s2 + 30s + 330.5)
−1
0
10 ω [rad/s]
10
Fig. 4: Bode plot of transfer function for topside pressure
(9)
equipped with such a device. Alternatively, in a closed0.13(s + 0.003)(s + 3 × 10−8 )(s2 + 20.29s + 256.7) . 235 loop system, Z is the control signal sent to the valve. For (s2 − 0.16s + 0.02)(s2 + 30s + 330.5) (10) the simplicity of our analysis, we consider a linear valve The static gain of the process with the flow output w is (i.e. f (Z) = Z). close to zero. The reason is that a constant inflow rate has The transfer function of the virtual flow output at Z = been considered as the inlet boundary condition for the 30% is model. As seen in (10), there is one zero very close to the 0.43(s + 0.098)(s + 0.0036)(s2 + 20.33s + 195.2) . = G 4 origin. The inlet boundary can be changed to a pressure(s2 − 0.16s + 0.02)(s2 + 30s + 330.5) (12) driven one to get a non-zero static gain which is closer to G3 =
225
Similar to the actual flow transfer function G3 , the virtual
reality.
flow transfer function G4 does not contain any RHP zero. 3.3. Virtual flow output
240
We define the virtual flow output very similar to the
Although the topside pressure transfer function G2 con-
valve equation in (7), except that we do not include the
tains two RHP-zeros, they do not propagate to G4 . We
density. As a result, we obtain
need to find out under which conditions this property is
√ Fv = Cv f (Z) ΔP .
valid. Such an analysis determines whether controlling this
(11) 245
Similar to the valve equation, ΔP is the pressure drop 230
This property makes it suitable for controller design.
output results in a robust control solution. The virtual flow in (11) is a multiplication of two func-
over the valve and f (Z) is a known function of the valve
tions in the time domain.
characteristics given by the valve manufacturer. Z is the
Fv (t) = h(Z(t)) × g(Prt (t)),
valve opening percentage, which can be measured from
where h(Z(t)) = Cv f (Z(t)), g(Prt (t)) =
the valve positioner feedback, assuming that the valve is 6
(13)
Prt (t) − Ps and
Ps is the constant separator pressure. In order to exam-
and from the mapping in (17) we get
ine locations of poles and zeros for the virtual flow trans-
G4 (s) =
fer function, we need to linearize Fv (t), and then apply a
(19)
where Bi = k1 ai + k2 bi . We require the coefficients of the √ numerator to have the same sign. Since k1 = Cv ΔP0 >
Laplace transform. From Taylor series expansion, we get ∂h(Z) δZ + · · · h(Z) = h(Z0 ) + ∂Z Z=Z0 g(P ) = g(P0 ) +
k1 s4 + B3 s3 + B2 s2 + B1 s + B0 , s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a0
0, we impose the constraint k1 ai + k2 bi > 0 that is
∂g(Prt ) δPrt + · · · ∂Prt Prt =P0
2ΔP0 ai > −Z0 bi ,
i ∈ {0, 1, 2, 3} .
(20)
Reduced order model:
where Z0 and P0 are stationary points with respect to time. δZ and δPrt are small signal values, i.e., deviations
To make the analysis more clear, we apply the conditions
around the stationary points. By applying a linear valve
in (20) on a reduced order model. We perform a balanced
equation (i.e. f (Z) = Z) we receive
model truncation (square root method) on G2 in (8) to
h(Z) = Cv (Z0 + δZ), δPrt , g(P ) ΔP0 + √ 2 ΔP0
obtain a model in the form (14)
G2 (s) =
(15)
s2
k (s − z) , − 2xs + x2 + y 2
(21)
where k > 0, z > 0, x > 0 and y > 0 are the model
where ΔP0 = P0 − Ps . Then, we write the virtual flow Fv
parameters. G2 (s) includes one RHP zero at s = z and
based on equation (13), Z0 δPrt δZδPrt . + ΔP0 δZ + √ Fv (t) Cv Z0 ΔP0 + √ 2 ΔP0 2 ΔP0
two unstable poles at s = x ± yi. By applying the conditions in (20) on the model in (21), it is possible to investigate the necessary conditions
By ignoring the high-order interaction term, we get
on locations of poles and zeros as follows.
Fv (t) F0 + k1 δZ + k2 δPrt ,
2x <
x2 + y 2 Z0 k < 2ΔP0 z
(22) δFv δPrt k1 + k2 , (16) The left-hand side of the inequality in (22) implies that x δZ δZ √ √ 255 should remain small and hence the unstable poles cannot where F0 = Cv Z0 ΔP0 , k1 = Cv ΔP0 and √ be located far into the RHP (fast instability). However, k2 = Cv Z0 /(2 ΔP0 ). Thus, in the Laplace domain we the imaginary part of the poles (y) appears on the righthave G4 (s) k1 + k2 G2 (s).
hand side, and it is easier to satisfy the conditions for
(17)
poles with large imaginary parts. In addition, the RHP-
In fact, (17) gives the same transfer function as in (12). 260
The transfer function in (12) was obtained from a direct
conditions are valid for systems with oscillatory nature
linearization of the nonlinear model in (1)-(4).
(complex-conjugate poles) and RHP zeros relatively close
Next, we investigate the conditions under which the 250
to the origin. In fact, these properties are valid for the
RHP-zeros disappear after the linear mapping in (17).
slugging flow regime.
Fourth order model:
System with real unstable pole:
The topside pressure transfer function is in the following
To check the conditions for systems with one real unstable
general form: b 3 s3 + b 2 s2 + b 1 s + b 0 , G2 (s) = 4 s + a 3 s 3 + a 2 s 2 + a 1 s + a0
zero z, should be close to the origin. In other words, the
pole, we consider the following transfer function model: (18)
G2 (s) = 7
k (s − b) , (Tf s + 1)(s − a)
(23)
Fv
where a > 0, b > 0 and Tf ≥ 0 is a filter time-constant.
√ Cv f (Z) ΔP
The filter is added to have a strictly proper transfer funcΔPset
tion. From the linear mapping in (17), we get G4 (s)
=
Tf s2 + (1 − aTf + 1 k1 (Tf s
k k2 k1 )s
−a−
+ 1)(s − a)
bk k2 k1
,
Cm
Cs
(24)
Topside choke
The conditions that the RHP zero disappears are derived as follows
k2 k −a , < k1 b
aTf − 1 <
aTf − 1 < 265
Z0 k −a . < 2ΔP0 b
Inlet separator
Riser
(25)
Subsea manifold
and by using k2 /k1 = Z0/(2ΔP0 ) for the virtual flow output, we get
Z
ΔP
Fig. 5: Cascade control structure based on virtual flow
(26) P2
To satisfy the conditions (25) and (26), stricter conditions
Top-side Valve
Air to atm. Seperator
are required on the gain and the location of the pole and the zero compared to the conditions in (22). It is not
Riser P1
possible to satisfy (25) and (26) for k > 0. However, the
safety valve P3 FT water
slugging system is unstable (a > 0) and the static gain 270
Buffer Tank
Mixing Point
Pipeline
FT air
P4
for the pressure is alway negative (G2 (0) < 0) that means k < 0. Hence, it is possible to satisfy the conditions for
Water Reservoir
the slugging system even by assuming a real unstable pole.
Pump
Water Recycle
Fig. 6: Schematic diagram of experimental setup
If a < 0 (stable system), it is easy to make the the RHP-zero disappear. In this case, the smaller Tf , the eas275
as the master controller, 1 Td s Kp1 1+ + . Cm (s) = Tf s + 1 Ti s Tf 1 s + 1
ier the conditions can be satisfied. 4. Control Design
The same low-pass filter 1/(Tf s + 1) is added to both slave
A cascade control structure is used in simulations and
280
(28)
and master controllers. This filter plays two roles; Firstly,
experiments in this work. As shown in Figure 5, the virtual285 it reduces the measurement noise, and secondly, it also flow Fv is controlled by the slave controller Cs , whereas has a dynamic effect. In addition, another first-order lowthe pressure drop across the valve ΔP is controlled by the pass filter with the time constant of T was added to the fr
master controller Cm . The slave control loop is shown in
reference input of the master controller to obtain a 2-DOF
blue, while the master control loop is indicated in red.
controller.
A proportional controller with a low-pass filter was used as the slave controller, Cs (s) =
290
Kp2 , Tf s + 1
(27)
5. Results 5.1. Experimental setup The experiments were carried out in laboratory setup
and a proportional-integrator-derivative (PID) controller
for anti-slug control at the Chemical Engineering Depart-
with a low-pass filter on the derivative action was applied
ment of NTNU. Figure 6 shows a schematic representation 8
295
of the laboratory setup. The pipeline and the riser are
40%. Figure 8 demonstrates the experimental result for re-
made from flexible pipes with 2 cm inner diameters. The
jecting a liquid disturbance in the inflow. Here, the inflow
length of the pipeline is 4 m and it is inclined with a 15◦
rate was changed from 4.1 to 5.5 litres/min at t = 600 sec.
angle. The height of the riser is 3 m. A buffer tank is used
To reduce the measurement noise, a second-order But-
to simulate the effect of a long pipe with the same volume,335 terworth filter with the normalized cutoff frequency of 300
such that the total length of the pipe would be about 70 m.
ωn = 0.03 was applied in the experiments. Since the sam-
The topside choke valve is used as the input for the
pling interval of the measurements is Ts = 0.1 sec, this
control. The separator pressure after the topside choke
filter gives a bandwidth of 0.94 rad/sec. For the slave
valve is nominally constant at atmospheric pressure. The
controller, a proportional controller Kp2 = 4 was applied
feed into the pipeline is assumed to be at constant flow340 in the experiments. The Butterworth filter acts instead of 305
the filter in (27) with Tf 1.06 sec.
rates, namely 4 s.liters/min of water and 4.5 s.liters/min of air. These flow rates are in standard conditions and
The PID controller used for the master controller was
are equivalent to 9.7 × 10−5 kg/s gas and 0.0667 kg/s
tuned as follows: Cm (s) = −0.2 1 +
liquid. With these boundary conditions, the critical valve opening, where the system switches from stable (non-slug) 310
to oscillatory (slug) flow, is at Z ∗ = 15%.
5s 1 + 300s 20s + 1
.
(29)
The same Butterworth filter as the one in the slave loop was applied to reduce the measurement noise of the master
5.2. Implementation of controller
loop, and a reference filter with the time constant Tf r = The virtual flowmeter and the controllers are imple-345 5 sec is introduced to compose a 2-DOF PID controller. mented in Matlab and run on a PC in real time. National 5.4. OLGA simulations
Instruments I/O modules are used to read the measure315
ments from sensors and write back the control signal to
To further evaluate the performance of the proposed
the valve. The communicate between the controllers in
controller, we used the dynamic flow simulator OLGA
Matlab and the I/O Modules is performed using the Data
(Bendiksen et al., 1991). We have modelled the exper-
Aquisition Toolbox of Matlab.
350
Since the calculation of the virtual flow is very simple, 320
325
imental setup described above in the OLGA simulator. Figure 9 shows a schematic presentation of the model in
it can be easily implemented in a Programmable Logic
the simulator.
Controller (PLC) and tested in industrial control systems.
The calculation of the virtual flow output and con-
To start up the controller, it is always easiest to close the
trollers were implemented in MATLAB, and the OLGA
valve to get a non-slugging flow, then turn on the con-
simulator was connected to MATLAB via OPC Server.
troller, and then increase the valve opening again by de-
The OLGA OPC Server only allows a sampling rate of
creasing the pressure setpoint.
Ts = 1 sec when the simulation is controlled by an external clock. This imposes a limitation on the filter and
5.3. Experimental results
controller tunings, and therefore the controllers used in
Figure 7 shows experimental performance of the cas-
the simulations are different from those used in the exper-
cade control. The valve opens as the setpoint of the pres-
iments. The slave controller used in the simulations was
sure drop across the valve decreases. The cascade con330
Cs (s) =
troller is able to stabilize the flow up to a valve opening of 9
8 , 4s + 1
(30)
Pressure drop across the valve, ΔP (master loop) ΔP [kPa]
10
measurement setpoint
Pressure drop across the valve, ΔP (master loop) 15
ΔP [kPa]
5
0 0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
0 0
500
time [sec] Virtual flow output, Fv (slave loop)
25
10
1000 1500 2000 2500 3000 3500 4000 4500 5000
v
F [-]
5
time [sec] Virtual flow output, Fv (slave loop)
20
measurement setpoint
10
20 v
F [-]
measurement setpoint
0 0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
5 0
60
20
40
Z [%]
40
0 0
500
measurement setpoint
10
time [sec] Valve opening, Z
60
Z [%]
15
1000 1500 2000 2500 3000 3500 4000 4500 5000
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec] Valve opening, Z
20
time [sec]
0 0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
Fig. 7: Experimental result for step changes in setpoint of pressure drop
time [sec] Liquid inflow rate, wL,in
whereas the master controller was 1 −0.2 1+ . Cm (s) = 4s + 1 120s
wL,in [L/min]
6
(31)
5
4 0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec]
Figure 10 shows the performance of the cascade control
Fig. 8: Experimental result for rejecting inflow liquid disturbances
for setpoint tracking. The cascade controller is able to 355
(step from 4.1 to 5.5 at t = 600sec)
stabilize the flow up to a valve opening of 40%. Figure 11 demonstrates the simulation results for rejecting disturbances in liquid and gas inflow rates. Here, 20% step
valve
changes from the nominal values are introduced. riser
6. Discussion 360
water source
6.1. Closed-loop sensitivity analysis
pipeline
Figure 12 shows two equivalent forms of the closedair source
loop system. If the virtual flow is the controlled output
buffer tank
(Figure 12.a)), the process model is given by (17) which is the same as in (12). In simulations, for the slave loop, we
Fig. 9: Schematic of OLGA model used in simulations
applied a controller as Cs (s) =
8 , 4s + 1
(32) 10
Pressure drop across the valve, ΔP (master loop)
4
Pressure drop across the valve, ΔP (master loop)
10
measurement setpoint
ΔP [kPa]
ΔP [kPa]
6
2
measurement setpoint
5
0 0
500
1000
1500
2000
2500
3000
3500
0 0
time [sec] Virtual flow output, F (slave loop) 13
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec] Virtual flow output, Fv (slave loop)
16
12
14
11
F [-]
measurement setpoint
10
v
Fv [-]
500
v
12 measurement setpoint
10
9 0
500
1000
1500
2000
2500
3000
3500
8 0
time [sec] Valve opening, Z
60
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec] Valve opening, Z
Z [%]
Z [%]
60 40 20
40 20
0 0
500
1000
1500
2000
2500
3000
3500
0 0
time [sec]
Fig. 10: OLGA simulation result for step changes in setpoint of
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec] Inflow rates, w G,in , w L,in
5.5
w [L/min]
pressure drop
and the sensitivity transfer function of the closed-loop sys-
5 wL,in
4.5
wG,in
4
tem becames
0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
time [sec]
1 . S4 (s) = 1 + Cs (s)G4 (s)
(33)
Fig. 11: OLGA simulation results for rejection of liquid and gas disturbances in the inflow (20% step change from nominal rates)
The closed-loop system can be rearranged in a way that the linear mapping takes place on the controller and the
a) closed-loop system with virtual flow as controlled output
r4 + −
pressure becomes the controlled output (Figure 12.b). For this case, the controller is 8k2 k2 Cs (s) = , C (s) = 1 + k1 Cs (s) 4s + 8k1 + 1
1 . 1 + C (s)G2 (s)
G2 (s)
(34)
G4 (s)
k2
+ +
k1
and the sensitivity transfer function becomes S2 (s) =
Cs
b) closed-loop system with pressure as controlled output
r2 + −
(35)
Figure 13 shows the amplitude of sensitivity transfer
k2 + −
C
Cs
G2 (s)
k1
functions for the two forms of the closed-loop system. The closed-loop system with the virtual flow output has a low sensitivity (S4 ) which is desirable, while the closed-loop 365
Fig. 12: Two equivalent forms of closed-loop system
system with the pressure output shows a high sensitivity 11
where two RHP zeros same as in (8) exist. However, the
12 S2
process is stable from the flow input w to the pressure
S4
10
385
|S|
8
output. Thus, for the master control loop, the RHP zeros are not as limiting as for the slave loop which is open-loop unstable. In addition, since the process is fairly linear from
6
the flow input w , the master controller does not require
4
a wide gain-margin. Therefore, the master controller can 2 390
0 10-3
10-2
10-1
100
101
is not robust for non-minimum-phase dynamics.
102
For the controllability reason, we require to maintain
ω [Rad/sec]
Fig. 13: Amplitude of sensitivity transfer functions for two forms of
a minimum pressure drop across the valve. The master
closed-loop system
control loop can be considered as an supervisory layer that 395
(S2 ). This confirms that the physical controllability limi-
The main challenge in experiments was to reduce the
the virtual flow output shows low sensitivity. The high
measurements noise. A second order Butterworth filter
sensitivity signals a robustness problem, but the proposed
was applied for this purpose. Choosing a suitable band-
controller performs well in experiments and simulations. This is a paradox that requires further investigation and
400
width for the filter is very crucial. If we try to reduce the noise completely by setting a low bandwidth for the
analysis.
filter, this causes attenuation of the controlled signal and also a lag in the control loop. Consequently, we would
6.2. Nonlinearity
need to increase the controller gain to compensate the fil-
The nonlinearity measure for the virtual flow output is 375
ensures controllability of the flow. 6.4. Measurement noise
tation of the topside pressure cannot be bypassed, though
370
be tuned to be relatively slow. Note that a tight control
φNOCL = 0.4 as given in Table 2. This indicates a high405 ter attenuation. However, because of the lag in the control loop, increasing the controller gain causes oscillations and nonlinearity, and we need to consider it for the control robustness issues.
design. In the OLGA simulations shown in Figure 10, we have applied a gain-scheduling based on the pressure drop
7. Conclusion
setpoint. We changed Kp2 from 8 to 10 and 15 for the 380
two last setpoint changes, otherwise the controller becomes unstable.
We proposed a new control strategy for anti-slug con-
410
is based on a virtual flow measurement.
6.3. Inverse response in master control loop
Robustness properties of the proposed method were in-
In Figure 11, when the liquid inflow disturbance is in-
vestigated by frequency domain analysis. The virtual flow
troduced at t = 600 sec, we observe that pressure drop
output is not affected by inverse response dynamics. We
shows an inverse response. This means that RHP-zeros for the pressure are still in place. In fact, the transfer function
415
showed that this property is valid for systems with oscillatory nature such as slugging flow dynamics.
from the flow input w to the pressure is as follows G2 =
trol based on topside measurements. This control solution
The suggested cascade controller was tested success-
−518.77(s + 14.21)(s − 0.4207)(s − 0.1302) , (36) s(s + 0.003018)(s2 + 20.29s + 256.7)
fully in OLGA simulations and experiments where satis12
factory results were achieved for both the setpoint tracking 420
Schweickhardt, T., Allgower, F., and Doyle, F. (2003). Nonlinearity quantification for the optimal state feedback controller. In Euro-
and the disturbance rejection.
pean Control Conference (ECC), 330–335.
The open-loop frequency domain analysis shows that
Sivertsen, H., Alstad, V., and Skogestad, S. (2009). Medium-scale
the virtual flow output does not have the controllability465 limitation related to RHP zeros. However, in closed-loop,
& Construction, 4(4), 156–170, SPE no. 120040. Storkaas, E. and Skogestad, S. (2003). Cascade control of unstable
the pressure output still has a hight sensitivity which sig425
systems with application to stabilization of slug flow. In IFAC
nals a robustness problem. Further analysis in the frequency domain is required to fully investigate this issue.
experiments on stabilizing riser-slug flow. SPE Projects, Facilities
7th International Symposium on Advanced Control of Chemical 470
Processes, Adchem. Hong Kong. Storkaas, E. and Skogestad, S. (2007). Controllability analysis of twophase pipeline-riser systems at riser slugging conditions. Control
Acknowledgements
Engineering Practice, 15(5), 567–581. Taitel, Y. (1986). Stability of severe slugging. International Journal
Financial support for this research was provided by Subsea production and processing (SUBPRO) center for 430
475
research-based innovation. Aamo, O., Eikrem, G., Siahaan, H., and Foss, B. (2005). Observer design for multiphase flow in vertical pipes with gas-lift - theory and experiments. Journal of Process Control, 15(3), 247 – 257. Bendiksen, K.H., Malnes, D., Moe, R., and Nuland, S. (1991). Dy-
435
namic two-fluid model olga. theory and application. SPE Production Engineering, 6(2), 171–180. Corneliussen, S., Couput, J.P., Dahl, E., Dykesteen, E., Frøysa, K.E., Malde, E., Moestue, H., Moksnes, P.O., Scheers, L., and Tunheim, H. (2005). Handbook of Multiphase Flow Metering. The
440
Norwegian Society for Oil and Gas Measurement, Oslo, Norway, 2 edition. Courbot, A. (1996). Prevention of severe slugging in the dunbar 16 inches multiphase pipeline. In Proceedings of the Annual Offshore Technology Conference, volume 4, 445–452. SP E no. 8196.
445
Di Meglio, F., Petit, N., Alstad, V., and Kaasa, G.O. (2012). Stabilization of slugging in oil production facilities with or without upstream pressure sensors. Journal of Process Control, 22(4), 809 – 822. doi:10.1016/j.jprocont.2012.02.014. Godhavn, J.M., Fard, M.P., and Fuchs, P.H. (2005). New slug con-
450
trol strategies, tuning rules and experimental results. Journal of Process Control, 15, 547–557. Havre, K., Stornes, K., and Stray, H. (2000). Taming slug flow in pipelines. ABB Review, 4, 55–63. Jahanshahi, E., Skogestad, S., and Helgesen, A.H. (2012). Control-
455
lability analysis of severe slugging in well-pipeline-riser systems. In IFAC Workshop - Automatic Control in Offshore Oil and Gas Production, 101–108. Trondheim, Norway. Jahanshahi, E. and Skogestad, S. (2014). Simplified dynamic models for control of riser slugging in offshore oil production. SPE Oil
460
and Gas Facilities, 3(6), 64–79.
13
of Multiphase Flow, 12(2), 203–217.