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Robustness Analysis and Tuning for Pressure Control in Managed Pressure Drilling
11th IFAC Symposium on Dynamics and Control of 11th IFAC Symposium on Dynamics and Control of 11th IFACSystems, Symposium on Dynamics and Control of Process including Biosystems 11th IFACSystems, Symposium on Dynamics and Control of Process including Biosystems Process including Biosystems June 6-8,Systems, 2016. NTNU, Trondheim, Norway Available online at www.sciencedirect.com Process including Biosystems June 6-8,Systems, 2016. NTNU, Trondheim, Norway June 6-8, 2016. NTNU, Trondheim, Norway June 6-8, 2016. NTNU, Trondheim, Norway
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Robustness Analysis and Tuning for Robustness Analysis and Tuning for Robustness Analysis and Tuning for Robustness Analysis and Tuning for Pressure Control in Managed Pressure Pressure Control in Managed Pressure Pressure Control in Managed Pressure Pressure Control in Managed Pressure Drilling Drilling Drilling Drilling ∗ ∗∗
Qin Li ∗ Mina Kamel ∗∗ Qin Li ∗ Mina Kamel ∗∗ Qin Li Qin Li ∗ Mina Mina Kamel Kamel ∗∗ ∗ ∗ Statoil Research Centre Porsgrunn, 3908 Porsgrunn, Norway ∗ Statoil Research Centre Porsgrunn, 3908 Porsgrunn, Norway Porsgrunn, 3908 ∗ Statoil Research Centre (e-mail: [email protected]) Statoil Research Centre Porsgrunn, 3908 Porsgrunn, Porsgrunn, Norway Norway (e-mail: [email protected]) ∗∗ (e-mail: [email protected]) Autonomous Systems Lab, ETH Zurich, 8092 Zurich, Switzerland ∗∗ (e-mail: [email protected]) Autonomous Systems Lab, ETH Zurich, 8092 Zurich, Switzerland ∗∗ Lab, Zurich, ∗∗ Autonomous Systems (e-mail: [email protected]) Autonomous Systems Lab, ETH ETH Zurich, 8092 8092 Zurich, Zurich, Switzerland Switzerland (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected]) Abstract: In this paper, we present aa general framework for robustness analysis for pressure Abstract: In this paper, we present framework for robustness analysis for pressure Abstract: In this paper, we present aa general general framework for robustness analysis for pressure control in managed pressure drilling (MPD). In particular, we apply the analysis to the Abstract: In this paper, we present general framework for robustness analysis for pressure control in managed pressure drilling (MPD). In particular, we apply the analysis to the control in managed pressure drilling (MPD). In particular, we apply the analysis to the pressure controller proposed in the work Godhavn et al. (2011), based on which we also give an approach control in managed pressure drilling (MPD). In particular, we apply the analysis to the pressure controller proposed in the work Godhavn et al. (2011), based on which we also give an approach controller proposed in the work Godhavn et al. (2011), based on which we also give an approach to search for controller tuning parameters with the goal of maximizing the robustness of system controller proposed in the workparameters Godhavn et al. (2011), based on which we also give an of approach to search for controller tuning with the goal of maximizing the robustness system to search for controller tuning parameters with the goal of maximizing the robustness of stability and control performance to various sorts of uncertainties, disturbances and noise. The to searchand for controller tuning parameters with theofgoal of maximizing the robustness of system system stability control performance to various sorts uncertainties, disturbances and noise. The stability and control performance to various sorts of uncertainties, disturbances and noise. The resulting tuning table can be used for online computation of the controller parameters. stability and control performance to various sorts of uncertainties, disturbances and noise. The resulting tuning table can be used for online computation of the controller parameters. resulting tuning table can be used for online computation of the controller parameters. The method proves effective in a simulation study. resulting tuning table can be used for online computation of the controller parameters. The method proves effective in aa simulation study. method proves effective study. method proves effective in inFederation a simulation simulation study. Control) Hosting by Elsevier Ltd. All rights reserved. © 2016, IFAC (International of Automatic Keywords: Managed pressure drilling, pressure control, uncertainty quantification, robust Keywords: Managed pressure drilling, pressure control, uncertainty quantification, robust Keywords: Managed pressure drilling, pressure control, uncertainty quantification, tuning Keywords: Managed pressure drilling, pressure control, uncertainty quantification, robust robust tuning tuning tuning 1. INTRODUCTION quite different from the design model and uncertainties, 1. INTRODUCTION INTRODUCTION quite different different from from the the design design model model and and uncertainties, uncertainties, 1. quite disturbances and noise are inevitably present. To address 1. INTRODUCTION quite different from the design model and uncertainties, disturbances and noise are inevitably present. To address address disturbances and noise are inevitably present. To this issue, in this paper, we first present a framework disturbances are we inevitably present. address In managed pressure drilling, annulus is sealed from the this issue, in inand thisnoise paper, first present present a To framework In managed pressure drilling, annulus is sealed from the this issue, this paper, we first a framework of for the pressure control in MPD In managed pressure drilling, annulus is sealed from the thisrobustness issue, in analysis this paper, we first present a framework top and the drilling mud is running out of annulus through of robustness analysis for the pressure control in MPD In managed pressure drilling, annulus is sealed from the top and and the the drilling drilling mud mud is is running running out out of of annulus annulus through through of robustness analysis for the pressure control in MPD using some well-known tools in robust control theory, by top of robustness analysis for the pressure control in MPD some choke valves (called MPD chokes or simply chokes using some well-known tools in robust control theory, by top and the drilling mud is running out of annulus through some choke valves (called MPD chokes or simply chokes using some well-known tools in robust control theory, by which the consequence of any specific controller parameter some choke valves (called MPD chokes or using some well-knownoftools in robust control parameter theory, by in this paper), which provides pressure in aa chokes bid to which the consequence any specific controller some choke valves (called MPDback chokes or simply simply chokes in this paper), which provides back pressure in bid to which consequence of specific controller parameter tuning on the robustness the pressure can in which back pressure in to which the the of any anyof controllercontrol parameter regulate the downhole pressure at specified in the tuning onconsequence the robustness robustness ofspecific the pressure pressure control can in this this paper), paper), which provides provides pressuredepth in a a bid bid to tuning regulate the downhole downhole pressure back at specified specified depth in the the on the of the control can be quantitatively evaluated. Furthermore, a numerical regulate the pressure at depth in tuning on the robustness of Furthermore, the pressure acontrol can annulus. The drilling mud is pumped from rig pumps into be quantitatively evaluated. numerical regulate the downhole pressure at specified depth in the annulus. The The drilling drilling mud mud is is pumped pumped from from rig rig pumps pumps into into be evaluated. Furthermore, aa procedure numerical guiding rule can be obtained by an optimization annulus. be quantitatively quantitatively evaluated. Furthermore, numerical the top of drill string, flowing down through the drill bit guiding rule can be obtained by an optimization procedure annulus. The drilling mud is pumped from rig pumps into the top top of of drill drill string, string, flowing flowing down down through through the the drill drill bit bit guiding rule can be by an procedure based of the In this work, the guidingon rulethe canresults be obtained obtained by evaluation. an optimization optimization procedure and then along the annulus to the choke. When on the results of the evaluation. In this work, the top drill string, flowingup through drill the bit based and thenof along along the annulus annulus updown to the the choke.the When the based on the results of the evaluation. In this work, we apply this method to the choke pressure controller and then the up to choke. When the based on the results of the evaluation. In this work, mud flow rate into the drill string is small, some pumps we apply this method to the choke pressure controller and then along the the annulus up to is thesmall, choke. When the we mud flow rate into drill string some pumps apply this method to the choke pressure controller proposed Godhavn the framework mud flow rate into the drill string is small, some pumps we apply in this method et toal. the(2011). choke But pressure controller may be used to provide additional flow rate through the proposed in Godhavn et al. (2011). But the framework mud flow rate into the drill string is small, some pumps may be be used used to to provide provide additional additional flow flow rate rate through through the the proposed in et (2011). But the framework for the robustness analysis the idea guiding may proposed in Godhavn Godhavn et al. al.and (2011). But for thethe framework choke facilitate the pressure control. pumps are for the robustness robustness analysis and the idea idea for the guiding may beto used to provide additional flow These rate through the choke to facilitate the pressure control. These pumps are for the analysis and the for the guiding rule generation for controller tuning are general. are choke to the control. These pumps for the robustness analysis and the are ideageneral. for the guiding are usually called back pressure pumps. The drilling generation for controller tuning choke to facilitate facilitate the pressure pressure control. These pumpsmud are rule are usually called back pressure pumps. The drilling mud rule generation for controller tuning are general. are usually called back pressure pumps. drilling mud rule generation forpaper controller tuning are general. is not only key to the but also has the layout of the is the following: In Section we are usually called backpressure pressureregulation pumps. The The is not not only key key to the the pressure regulation but drilling also has hasmud the The The layout layout of of the the paper paper is is the the following: following: In In Section Section 22 2 we we is only to pressure regulation but also the function to remove the cuttings produced in the drilling The review the controller structure and the simple plant model is not only key to the pressure regulation but also has the function to to remove remove the the cuttings cuttings produced produced in in the the drilling drilling review The layout of the paper is theand following: In Section 2 we the controller structure the simple plant model function process (The reader is referred to Section 2 in Godhavn review the controller structure and the simple plant model controller design. In Section we present function(The to remove the cuttingstoproduced in theGodhavn drilling used process reader is referred Section 22 in reviewfor thethe controller structure simple33 plant model used for the controller design.and In the Section we present present process (The reader is to Section in et al. (2011) an illustration MPD for the controller design. In Section 33 we the qualitative representation of the model and parameter process (Thefor reader is referred referredof tothe Section 2process). in Godhavn Godhavn used et al. (2011) for an illustration of the MPD process). used for the controller design. In Section we present the qualitative representation of the model and parameter et al. (2011) for an illustration of the MPD process). the representation of the and uncertainties. robustness analysis is given in Section et al. pressure (2011) foratana illustration of the MPD process). the qualitative qualitativeThe representation the model model and parameter parameter The location depends on the uncertainties. The robustness of analysis is given given in Section Section The pressure at aa downhole downhole location depends on the uncertainties. The robustness analysis is in 4; based on which the approach for robust tuning is shown The pressure at downhole location depends on the uncertainties. Thethe robustness analysis is given inisSection pressure upstream choke (we call it choke pressure in 4; based on which approach for robust tuning shown The pressure at athe downhole location depends on the pressure upstream the choke (we call it choke pressure in 4; based on which the approach for robust tuning is in Section 5. Finally, some concluding remarks are given pressure upstream the choke (we call it choke pressure in 4; based on5.which the approach for robust tuningare is shown shown this paper) and the pressure drop between the downhole in Section Finally, some concluding remarks given pressure upstream the choke (we call it choke pressure in this paper) paper) and and the the pressure pressure drop drop between between the the downhole downhole in Section 6. 5. Finally, some concluding remarks are given this in Section 5. Finally, some concluding remarks are given location and the choke, which is primarily given by the 6. this paper) and the pressure drop between the downhole location and and the the choke, choke, which which is is primarily primarily given given by by the the in Section 6. location in Section 6. mud. Despite the fact that PID controllers are popular location and the choke, which is controllers primarily given by the mud. Despite the fact that PID are popular mud. Despite the that controllers are popular and understood al. (2013)), 2. PLANT MODEL AND CONTROLLER mud.well Despite the fact fact (Møgster that PID PIDet controllers are different popular and well understood (Møgster et al. (2013)), different 2. PLANT PLANT MODEL MODEL AND AND CONTROLLER CONTROLLER and well understood (Møgster et al. (2013)), different types of advanced controllers have been recently proposed 2. STRUCTURE and well understood (Møgster et al. (2013)), different types of of advanced advanced controllers controllers have have been been recently recently proposed proposed 2. PLANT MODEL AND CONTROLLER STRUCTURE types as improved solutions to the pressure control for MPD STRUCTURE types of advanced controllers been recently proposed as improved solutions to the thehave pressure control for for MPD STRUCTURE as solutions to control MPD (Zhou et al. (2009); Breyholtz et al. (2010); Godhavn as improved improved solutions to the pressure pressure control for MPD In this section we give a brief review of the simplified plant (Zhou et al. (2009); Breyholtz et al. (2010); Godhavn In this section we give aa brief review of the simplified plant (Zhou et al. (2009); Breyholtz et al. (2010); Godhavn et al. (2011); Li et al. (2011); Møgster et al. (2013)). In this section we give brief review of the simplified plant (Zhou et al. (2009); Breyholtz al. (2010); and the pressure controller that regulates the choke et al. (2011); (2011); Li et et al. al. (2011); et Møgster et al. al.Godhavn (2013)). model In this section we give a brief review of the simplified plant model and the pressure controller that regulates the choke et al. Li (2011); Møgster et (2013)). Usually the pressure controllers are designed based on model and the pressure controller that regulates the choke et al. (2011); Li et al. (2011); Møgster et al. (2013)). Usually the the pressure pressure controllers controllers are are designed designed based based on on pressure. model and the pressure controller that regulates the choke pressure. Usually some simple design dynamic models of the process with Usually the pressure controllers are of designed basedwith on pressure. some simple design dynamic dynamic models the process process pressure. some design models of with nominal parameters (e.g. physical properties of the mud some simple simple design dynamic models of the the process with nominal parameters (e.g. physical properties of the mud 2.1 Simplified plant model nominal parameters (e.g. physical properties of the mud and the choke) and have some parameters can 2.1 Simplified Simplified plant plant model model nominal properties ofwhich the mud and the parameters choke) and and(e.g. havephysical some parameters parameters which can 2.1 and the choke) have some which can be for good practice. The tuning andtuned the choke) andperformance have some in parameters which can 2.1 Simplified plant model be tuned for good performance in practice. The tuning be tuned for performance in The tuning of controller can be tedious ease the controller design, the following 3-state simplibe the tuned for good goodparameters, performancehowever, in practice. practice. tuning To of the controller parameters, however, can The be tedious tedious To ease ease the the controller controller design, design, the the following following 3-state 3-state simplisimpliof the controller parameters, however, can be To and difficult as the actual process dynamics may be fied well dynamics model is used (Kaasa et al. (2012)): of the controller parameters, however, can be may tedious To ease controller design, the (Kaasa following simpliand difficult as the the actual process process dynamics be fied fied wellthe dynamics model is used used et 3-state al. (2012)): (2012)): and difficult as actual dynamics may be well dynamics model is (Kaasa et al. and difficult as the actual process dynamics may be fied well dynamics model is used (Kaasa et al. (2012)): Copyright © 2016, 2016 IFAC 556Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 556 Copyright ©under 2016 responsibility IFAC 556Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 556 10.1016/j.ifacol.2016.07.401
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Va Va ˙ Lp (ˆ pc − pc ), (5) pˆc = qˆbit + qbpp − qc − Ad vd + qˆerr − βa βa Va qˆ˙err = − Li (ˆ pc − pc ), (6) βa
Table 1. List of notations Symbol Vd βd qp qbit M pp pc ρd ρa g h Va βa qbpp qc Ad vd qerr F (qbit , ω)
Physical meaning volume of drill string effective bulk modulus of mud in drill string standpipe flow rate bit flow rate average integrated density per area standpipe pressure (upstream) choke pressure average mud density in drill string average mud density in annulus acceleration of gravity true vertical depth of well volume of annulus effective bulk modulus of mud in annulus flow rate of back pressure pump choke flow rate cross-sectional area of drill string longitudinal velocity of drill string unmodeled flow rate in annulus frictional pressure drop from standpipe to choke
Vd p˙ p = qp − qbit , βd M q˙bit = pp − pc − F (qbit , ω) + (ρd − ρa )gh, Va p˙ c = qbit + qbpp − qc − Ad vd + qerr , βa
557
where qˆbit is an estimate of the mud flow rate through the drill bit, which is simply set equal to the standpipe flow rate qp in this work 1 ; Lp and Li are tunable parameters. model based controller The model based controller (MBC) gives out the control reference signal for the opening of the MPD choke. It first computes the desired flow rate through the choke as qc∗ = qˆbit + qbpp − Ad vd + qˆerr +
Va (kp (pc − prr ) − p˙ rc ) . (7) βa
where Kp is a tunable parameter. Then, using the choke flow model (2), the choke opening reference signal is qc∗ r −1 √ . obtained as zc = Gc Kc pc −pc0
(1)
We refer the reader to Godhavn et al. (2011) for more details on the design and test of the controller. 3. UNCERTAINTY MODELING AND REPRESENTATION
where the meaning of the notations are listed in Table 1. Assuming that the mud through the MPD choke has a constant density, the flow rate through the MPD choke qc can be modeled as the following: √ (2) qc = Kc Gc (zc ) pc − pc0 , where Kc is a positive real constant; Gc is generally a nonlinear nondecreasing function of the choke opening zc ; and pc0 is downstream choke pressure, which is considered here as constant.
Physical phenomena that occurs in drilling process are complex and can hardly be captured accurately by simple models. Therefore an appropriate quantification of uncertainties is necessary to analyze robustness and performance of a controller designed based on simple models. In this section, we quantify two important uncertainties: • Uncertain plant dynamics: The discrepancy between the simplified plant model (1)-(2) and the actual plant dynamics. • Uncertain choke dynamics: The discrepancy between the choke dynamics model (3)-(4) and the actual choke dynamics.
We will call (1)-(2) the 3-state design plant model hereafter. 2.2 Choke dynamics
3.1 Operation points of plant
We consider that the dynamics of MPD choke opening may be (approximately) modeled by a 2nd-order linear system, i.e.,
For our control purposes, we are interested in the responses from input variables to output variables. Even though the actual responses may be complicated and nonlinear, they are in general approximately linear nearby some steady state and thus can be represented by transfer functions.
z˙c = vc , v˙ c = −2ζωn vc −
(3) ωn2 zc
+
ωn2 zcr ,
(4)
where vc is the rate of the choke opening; ωn and ζ are the natural frequency and damping ratio of the 2nd-order dynamics respectively; zcr is the choke opening reference, which is the output of the choke pressure controller. 2.3 Choke pressure controller The controller is composed of the following components: (1) error flow observer and (2) model based controller. error flow observer The error flow observer (EFO) is used to estimate the unmodeled flow rate qerr . It has the following form: 557
It is not difficult to see that in a steady state of the plant dynamics (1)-(2), for which all the three derivatives in (1) are zero, the values of pp , qbit and zc are determined if the values of qp , pc , qbpp , vd and qerr are given. Thus we can define an operation point to be a value of the vector [qp , pc , qbpp , vd , qerr ]. For simplicity, in this paper we consider operation points with qbpp = vd = qerr = 0. 3.2 Uncertain plant dynamics We consider two types of uncertainties in plant dynamics: (1) parameter uncertainty and (2) neglected dynamics. 1
An estimator for qˆbit was presented in Godhavn et al. (2011), it is not considered here for simplicity.
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Parameter uncertainty Parameter uncertainty refers to the uncertainty of one’s knowledge about the parameters in a nominal or design model. Examples of parameters in the plant model (together with examples of uncertainty ranges with respect to their nominal values) that can vary substantially are listed in the following Table 2. Table 2. Parametric uncertainty in plant model Parameter βa bulk modulus in the annulus βd bulk modulus in the drilling string Fa friction coefficient in the annulus Fd friction coefficient in the drilling string
Uncertainty range % [−60, 10]% ±10% ±50% ±20%
Using the linear fractional transformation (LFT) (see e.g. Skogestad and Postlethwaite (2005); Balas et al. (2001)), for a given operation point, the 3-state design model with the parameter uncertainties can be represented by the structure illustrated in Figure 1, in which • the vector φ = [qp , qbpp , vd , qerr ] is considered as part of the input to the closed-loop system; • the output vector ξ = [qp , qbpp , vd , zc ] together with pc are used by the controller block (see Section 2.3) as its input, with qp , qbpp , vd as feedforward and zc , pc as feedback; • ∆θ,p is the frequency-dependent structured perturbation matrix with a block diagonal structure; each diagonal block corresponds to the uncertainty induced by one uncertain parameter and has its largest singular value upper bounded by 1 for all frequencies; • the entries of the transfer function matrix Gnom depend on (i) the 3-state design plant model with a set of nominal values for the parameters and (ii) the bounds (real scalars) of the uncertain parameters; • the matrix ∆θ,p affects the input/output relationship between zc and pc in a feedback manner.
By neglected dynamics in plant model we mean, for fixed plant parameters, the discrepancy between the transfer function from the choke opening zc to choke pressure pc given by the 3-state design plant model and that given by the actual plant. Utilizing the so called multiplicative uncertainty model (see e.g. Skogestad and Postlethwaite (2005); Balas et al. (2001)), for a given operation point, the plant model with both the uncertain parameters and the neglected dynamics can be represented in the structure shown in Figure 2, in which • the system in the dashed box is simply the one shown in Figure 1; • wp,c is a frequency-dependent multiplicative uncertainty weighting function; • δp,c is a frequency-dependent function whose magnitude is upper bounded by 1 for all frequencies; • the product wp,c δp,c quantifies the neglected dynamics from the input zc to the output pc .
Fig. 2. Plant model with mixed uncertainty Now, let us denote the transfer functions from zc to pc in the actual plant as Pzc ,pc (jω; x, θ), where x represents the operation point and θ stands for the actual parameter vector. Exploiting the multiplicative uncertainty model, we wish to establish the following relationship: for any θ ∈ Θ, Pzc ,pc (jω; x, θ) = (1 + wp,c (jω)δp,c (jω))Gzc ,pc (jω; x, θ).(8) (The requirement on that (8) should hold for any θ ∈ Θ is because the only thing we know about the plant parameter θ is that it belongs to the set Θ.)
Fig. 1. Plant model with uncertain parameters Mathematically, we may collect all uncertain scalar parameters into one vector and use Θ to denote the set of all allowed values of the vectors. For one given parameter vector θ˜ ∈ Θ (which corresponds to one possible value ˜ to denote the transfer of ∆θ,p ), we use Gzc ,pc (jω; x, θ) function between zc and pc in Figure 1 at the operation point x. Hence the transfer function between zc and pc of the system in Figure 1 can be any element in the set ˜ : θ˜ ∈ Θ}. {Gzc ,pc (jω; x, θ) Neglected dynamics Even with the values of the parameters exactly known, the 3-state design plant model presented in Section 2.1 captures the actual plant dynamics accurately only at low frequency (Aarsnes et al. (2012)). 558
To achieve this we may choose the weighting function wp,c as P zc ,pc (jω; x, θ) − Gzc ,pc (jω; x, θ) wp,c (jω) ≥ max , (9) θ∈Θ Gzc ,pc (jω; x, θ)
because then a function δp,c (jω) can be chosen to satisfy (8) with |δp,c (jω)| < 1 for any frequency ω. In addition, for practical reasons:
• when computing the weighting function, we replace Pzc ,pc (jω; x, θ), which is literarily unknown, with the transfer function identified from a high-fidelity drilling simulator (The reader is referred to Ljung (1999) for some system approaches for this purpose); • the weighting function may be computed only for a frequency range of interest; • the weighting function is preferred to be the transfer function of a low-order SISO stable linear system.
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Uncertain choke dynamics The uncertainties in the choke dynamics can be represented in the same manner as those in the plant dynamics. The uncertain parameters and examples of the ranges of uncertainties are given in Table 3 (Note that here the uncertainty in the function Gc is converted approximately to that of the parameter Kc .)
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• d˜ represents the vector containing p˜rc and all elements in the vectors φ˜ and n ˜ ; it is thus regarded as the normalized system input (vector).
Table 3. Parametric uncertainty in choke model Parameter Kc choke constant ωn choke natural frequency
Uncertainty range % ±10% ±10%
Fig. 5. A compact form of the closed-loop system diagram
The choke model with mixed uncertainty of parameters and neglected dynamics can be represented by the structure shown in Figure 3, in which zc∗ is the choke opening reference given by the controller.
The robust stability criterion says that the closed-loop system in Figure 5 is robustly stable against all possible uncertainties described in Section 3 if and only if the nominal closed-loop system is stable and µ∆ (N11 (jω)) ≤ 1, ∀ω, where the µ∆ is the structured singular value, and N11 is the upper left block of the transfer function matrix N (i.e., it is the transfer function matrix from the output to the input of the block ∆). Normally, we can define the real number 1/ maxω µ∆ (N11 (jω)) as the robust stability margin. Also note that the generalized input signals do not affect the stability of the closed-loop system. 4.3 Robust performance
Fig. 3. Choke model with mixed uncertainty 4. ROBUST STABILITY AND PERFORMANCE ANALYSIS 4.1 Closed-loop system diagram For a given operation point the (linearized) closed-loop system can be formed by connecting the uncertain plant model, uncertain choke model (described in Section 3), and the controller (described in Section 2). The system diagram is shown in Figure 4, in which • n denotes the vector of sensor noises on pc as well as the signals in ξ; • e is the actual tracking error, and e˜ is called the weighted tracking error (it is also called the weighted system output); • wref is ideal model (transfer function) for the reference tracking, wφ , wn , we are called weighting (transfer) functions which are used to specify the desired performance in H∞ framework (see Section 4.3 for the details); note that wref , wφ , wn and we are diagonal ˜ n ˜ and matrices such that each scalar signal in p˜rc , φ, e˜ has an individual scalar weighting function. • p˜rc , φ˜ and n ˜ are called normalized choke pressure reference, exogenous inputs and noise respectively. 4.2 Robust stability
Input/output relationship is central to performance study. In Figure 5 it is the lower right block in the transfer function matrix N , denoted here by N22 , that relates the generalized system input d˜ to the weighted tracking error e˜. In the framework of H∞ control, the control objective, besides the robust stability stated in Section 4.2, is to make the norm N22 ∞ ≤ 1 for all frequencies; and the use of the weighting functions in the diagonal matrices wref , wφ , wn and we is to scale the inputs and outputs such that desired closed-loop performance is achieved if the norm condition above on N22 is met. One theoretical guideline to choose the weighting function is the following: Let wi (jω) be the weighting function for the ith element in the normalized ˜ and denote the actual input vector by input vector (d), r d = [pc , φ, n]. Then wi (jω) may be chosen such that the inequality i αi2 (ω)/|wi (jω)|2 ≤ 1 holds in the frequency band of interest, where αi (ω) represents the magnitude of the frequency component of the signal di (i.e. the ith component of the vector d) around frequency ω. The robust performance is guaranteed if µ∆ ˜ (N (jω)) ≤ ˜ = diag(∆perf , ∆) with 1, ∀ω, where the perturbation ∆ ∆perf (jω) being any possible complex matrix with the dimension nd˜ × ne˜ which satisfies ∆perf ∞ ≤ 1. The number 1/ maxω µ∆ ˜ (N (jω)) may be called the robust performance margin. 5. A ROBUST TUNING APPROACH
To apply the robust stability theory (see e.g. Skogestad and Postlethwaite (2005)), we first transfer the closedsystem into the more compact representation as shown in Figure 5. In the figure, • ∆ = diag[∆θ,c , δc , ∆θ,p , δp,c ] is a block diagonal perturbation; the inputs to the block ∆ include those to its component diagonal blocks that can be easily identified in Figure 4; 559
5.1 An approach searching for robust tuning parameters As expected, different operation points correspond to different linearized closed-loop system. For each operation point the ideal controller tuning would be finding the controller parameters kp , Lp , Li such that both robust stability and robust performance are satisfied with all allowed uncertainties (for stability) and the selected weighting
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Fig. 4. Closed-loop system diagram functions (for performance), i.e., we have both robust stability margin and robust performance margin larger than 1. However, this may not be always achievable. If this is the case, we impose priority on robust stability over robust performance; or in other words we are willing to sacrifice robust performance while pursing robust stability. Specifically, for each operation point x in a selected set X, we first compute the robust stability margins and the robust performance margins for a fixed grid of combinations of kp and Li 2 , which is defined by the set Γ=
Kpmax Lmax , Li = m i , N M n = 0, 1, · · · , N, m = 0, 1, · · · , M .
(kp , Li ) : kp = n
(10)
> 0 are the upper bounds of where Kpmax > 0 and Lmax i the ranges for the values of kp and Li , and the positive integers N and M define the resolution of the parameter grid. For each point in γ ∈ Γ let us denote the resultant robust stability margin and robust performance margin as sx (γ) and px (γ) respectively, with the subscript x indicating the dependence on the operation point. Let us suppose now that the set Γ is ordered and with its elements indexed as γi , with i = 1, 2, · · · , |Γ| (Here | · | denotes the size of a set). Then we may use the following Algorithm 1 to pick the best tuning point in Γ for each operation point. Algorithm 1 Searching for the best tuning parameters 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:
procedure FindBestParameters(Γ, sx , px ) for i ← 1 to |Γ| do if i = 1 then γx∗ ← γi p∗x ← p x (γi ), s∗x ← sx (γi ) else if px (γi ) ≥ p ∗x and sx (γi ) ≥ s∗x or px (γi ) > p∗x and sx (γi ) ≥ a then γx∗ ← γi p∗x ← px (γi ), s∗x ← sx (γi ) end if end for return γx∗ , p∗x , s∗x end procedure
In Algorithm 1, the number a should be set to 1 to reflect strict requirement on robust stability unless we see the resulting tuning is too conservative, in which case we may lower the value of a to accept tuning with less robust 2 The value of L is fixed as we found from experience that it does p not affect the results when ranging within a large interval.
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stability margin (An alternative may be sticking to a = 1 while adjusting the uncertainty weighting functions). Note that, by the searching procedure, we will have the best tuning parameter γx∗ (with corresponding robust stability margin s∗x and robust performance margin p∗x ) for each operation point x ∈ X. Thus a tuning table for all operation points is indeed created. Tuning parameters for operation points which are not in X can be obtained by simple interpolation, which is fast enough to be run with real time applications. We should also point out that the computation of robust margins and the tuning parameter searching algorithm are in general time-consuming and therefore should be run offline. 5.2 Simulation results Now we show some simulation results that compare the robust tuning with some nominal tuning which gives good performance with the simple design model. Firstly, In Figure 6a we show that the tuning parameter Kp = 1.5, Li = 4 and Lp = 25 gives very good pressure tracking performance with the simple 3-state design plant model. In Figure 6b, however, we see that this tuning makes the system unstable when simulating with a high-fidelity model. On the other hand, the robust tuning can achieve acceptable tracking performance in this case, which is shown in Figure 6c. Next we show the effectiveness of the robust tuning with low flow rate (300 lpm) and very low frictional pressure loss in the drill string and annulus. In Figure 7a, the controller is tested with Kp = 0.8, Li = 4 and Lp = 25 which stabilizes the system with normal frictional pressure loss at 300 lpm. It is clearly seen that this tuning cannot stabilize system with low friction. On the other hand, with some initial fluctuation, the system is stabilized by the robust tuning, which can be seen in Figure 7b. 6. CONCLUSION In this paper, we have presented an approach that yields a guide rule for the tuning of the pressure control for managed pressure drilling. The key to our approach is the robustness analysis via a qualitative representation of the model and parameter uncertainties. It allows us to evaluate the robustness of the stability and performance of the closed-loop control system with any particular tuning of the controller. The guide rule for tuning is simply made to somehow maximize the robustness such that the closedloop system can be stabilized with acceptable control performance in presence of expected uncertainties.
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(a) Pressure tracking with nominal tuning using the simple design plant model
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(a) Pressure tracking with nominal tuning
(b) Pressure tracking with robust tuning (b) Pressure tracking with nominal tuning using a high-fidelity plant model
(c) Pressure tracking with robust tuning using a high-fidelity plant model
Fig. 6. Comparison of pressure tracking performance at high flow rate and normal friction REFERENCES Aarsnes, U.J., Aamo, O.M., and Pavlov, A. (2012). Quantifying error introduced by finite order discretization of a hydraulic well model. In Proc. of the 2nd Australian Control Conference, 54–59. Sydney. Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and Smith, R. (2001). µ-Analysis and Synthesis Toolbox: For Use with MATLAB, Version 3. The Mathworks. Breyholtz, Ø., Nygaard, G., Siahaan, H., and Nikolaou, M. (2010). Managed pressure drilling: A multi-level control approach. SPE 128151. 561
Fig. 7. Comparison of pressure tracking performance with low flow rate and low friction Godhavn, J.M., Pavlov, A., Kaasa, G.O., and Rolland, N.L. (2011). Drilling seeking automatic control solutions. In Proc. of the 18th IFAC World Congress, 10842– 10850. Milano. Kaasa, G.O., Stamnes, .N., Aamo, O.M., and Imsland, L.S. (2012). Simplified hydraulic model used for intelligent estimation of downhole pressure for an MPD control system. SPE Drilling and Completion, 27, 127–138. Li, Z., Hovakimyan, N., and Kaasa, G.O. (2011). Fast estimation and L1 adaptive control for bottomhole pressure in managed pressure drilling. In Proc. of the IEEE Multi-Conference on Systems and Control, 996–1001. Denver, USA. Ljung, L. (1999). System Identification: Theory for the User. Prentice Hall, Upper Saddle River, New Jersey, 2nd edition. Møgster, J., Godhavn, J.M., and Imsland, L. (2013). Using MPC for managed pressure drilling. Modeling, Identification and Control, 34, 131–138. Skogestad, S. and Postlethwaite, I. (2005). Multivariable feedback control: analysis and design. Wiley, Chichester, New York. Zhou, J., Stamnes, .N., Aamo, M.O., and Kaasa, G.O. (2009). Switched control for pressure regulation and kick attenuation in a managed pressure drilling system. IEEE Transactions on Control Systems Technology, 19, 337–350.
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Robustness Analysis and Tuning for Robustness Analysis and Tuning for Robustness Analysis and Tuning for Robustness Analysis and Tuning for Pressure Control in Managed Pressure Pressure Control in Managed Pressure Pressure Control in Managed Pressure Pressure Control in Managed Pressure Drilling Drilling Drilling Drilling ∗ ∗∗
Qin Li ∗ Mina Kamel ∗∗ Qin Li ∗ Mina Kamel ∗∗ Qin Li Qin Li ∗ Mina Mina Kamel Kamel ∗∗ ∗ ∗ Statoil Research Centre Porsgrunn, 3908 Porsgrunn, Norway ∗ Statoil Research Centre Porsgrunn, 3908 Porsgrunn, Norway Porsgrunn, 3908 ∗ Statoil Research Centre (e-mail: [email protected]) Statoil Research Centre Porsgrunn, 3908 Porsgrunn, Porsgrunn, Norway Norway (e-mail: [email protected]) ∗∗ (e-mail: [email protected]) Autonomous Systems Lab, ETH Zurich, 8092 Zurich, Switzerland ∗∗ (e-mail: [email protected]) Autonomous Systems Lab, ETH Zurich, 8092 Zurich, Switzerland ∗∗ Lab, Zurich, ∗∗ Autonomous Systems (e-mail: [email protected]) Autonomous Systems Lab, ETH ETH Zurich, 8092 8092 Zurich, Zurich, Switzerland Switzerland (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected]) Abstract: In this paper, we present aa general framework for robustness analysis for pressure Abstract: In this paper, we present framework for robustness analysis for pressure Abstract: In this paper, we present aa general general framework for robustness analysis for pressure control in managed pressure drilling (MPD). In particular, we apply the analysis to the Abstract: In this paper, we present general framework for robustness analysis for pressure control in managed pressure drilling (MPD). In particular, we apply the analysis to the control in managed pressure drilling (MPD). In particular, we apply the analysis to the pressure controller proposed in the work Godhavn et al. (2011), based on which we also give an approach control in managed pressure drilling (MPD). In particular, we apply the analysis to the pressure controller proposed in the work Godhavn et al. (2011), based on which we also give an approach controller proposed in the work Godhavn et al. (2011), based on which we also give an approach to search for controller tuning parameters with the goal of maximizing the robustness of system controller proposed in the workparameters Godhavn et al. (2011), based on which we also give an of approach to search for controller tuning with the goal of maximizing the robustness system to search for controller tuning parameters with the goal of maximizing the robustness of stability and control performance to various sorts of uncertainties, disturbances and noise. The to searchand for controller tuning parameters with theofgoal of maximizing the robustness of system system stability control performance to various sorts uncertainties, disturbances and noise. The stability and control performance to various sorts of uncertainties, disturbances and noise. The resulting tuning table can be used for online computation of the controller parameters. stability and control performance to various sorts of uncertainties, disturbances and noise. The resulting tuning table can be used for online computation of the controller parameters. resulting tuning table can be used for online computation of the controller parameters. The method proves effective in a simulation study. resulting tuning table can be used for online computation of the controller parameters. The method proves effective in aa simulation study. method proves effective study. method proves effective in inFederation a simulation simulation study. Control) Hosting by Elsevier Ltd. All rights reserved. © 2016, IFAC (International of Automatic Keywords: Managed pressure drilling, pressure control, uncertainty quantification, robust Keywords: Managed pressure drilling, pressure control, uncertainty quantification, robust Keywords: Managed pressure drilling, pressure control, uncertainty quantification, tuning Keywords: Managed pressure drilling, pressure control, uncertainty quantification, robust robust tuning tuning tuning 1. INTRODUCTION quite different from the design model and uncertainties, 1. INTRODUCTION INTRODUCTION quite different different from from the the design design model model and and uncertainties, uncertainties, 1. quite disturbances and noise are inevitably present. To address 1. INTRODUCTION quite different from the design model and uncertainties, disturbances and noise are inevitably present. To address address disturbances and noise are inevitably present. To this issue, in this paper, we first present a framework disturbances are we inevitably present. address In managed pressure drilling, annulus is sealed from the this issue, in inand thisnoise paper, first present present a To framework In managed pressure drilling, annulus is sealed from the this issue, this paper, we first a framework of for the pressure control in MPD In managed pressure drilling, annulus is sealed from the thisrobustness issue, in analysis this paper, we first present a framework top and the drilling mud is running out of annulus through of robustness analysis for the pressure control in MPD In managed pressure drilling, annulus is sealed from the top and and the the drilling drilling mud mud is is running running out out of of annulus annulus through through of robustness analysis for the pressure control in MPD using some well-known tools in robust control theory, by top of robustness analysis for the pressure control in MPD some choke valves (called MPD chokes or simply chokes using some well-known tools in robust control theory, by top and the drilling mud is running out of annulus through some choke valves (called MPD chokes or simply chokes using some well-known tools in robust control theory, by which the consequence of any specific controller parameter some choke valves (called MPD chokes or using some well-knownoftools in robust control parameter theory, by in this paper), which provides pressure in aa chokes bid to which the consequence any specific controller some choke valves (called MPDback chokes or simply simply chokes in this paper), which provides back pressure in bid to which consequence of specific controller parameter tuning on the robustness the pressure can in which back pressure in to which the the of any anyof controllercontrol parameter regulate the downhole pressure at specified in the tuning onconsequence the robustness robustness ofspecific the pressure pressure control can in this this paper), paper), which provides provides pressuredepth in a a bid bid to tuning regulate the downhole downhole pressure back at specified specified depth in the the on the of the control can be quantitatively evaluated. Furthermore, a numerical regulate the pressure at depth in tuning on the robustness of Furthermore, the pressure acontrol can annulus. The drilling mud is pumped from rig pumps into be quantitatively evaluated. numerical regulate the downhole pressure at specified depth in the annulus. The The drilling drilling mud mud is is pumped pumped from from rig rig pumps pumps into into be evaluated. Furthermore, aa procedure numerical guiding rule can be obtained by an optimization annulus. be quantitatively quantitatively evaluated. Furthermore, numerical the top of drill string, flowing down through the drill bit guiding rule can be obtained by an optimization procedure annulus. The drilling mud is pumped from rig pumps into the top top of of drill drill string, string, flowing flowing down down through through the the drill drill bit bit guiding rule can be by an procedure based of the In this work, the guidingon rulethe canresults be obtained obtained by evaluation. an optimization optimization procedure and then along the annulus to the choke. When on the results of the evaluation. In this work, the top drill string, flowingup through drill the bit based and thenof along along the annulus annulus updown to the the choke.the When the based on the results of the evaluation. In this work, we apply this method to the choke pressure controller and then the up to choke. When the based on the results of the evaluation. In this work, mud flow rate into the drill string is small, some pumps we apply this method to the choke pressure controller and then along the the annulus up to is thesmall, choke. When the we mud flow rate into drill string some pumps apply this method to the choke pressure controller proposed Godhavn the framework mud flow rate into the drill string is small, some pumps we apply in this method et toal. the(2011). choke But pressure controller may be used to provide additional flow rate through the proposed in Godhavn et al. (2011). But the framework mud flow rate into the drill string is small, some pumps may be be used used to to provide provide additional additional flow flow rate rate through through the the proposed in et (2011). But the framework for the robustness analysis the idea guiding may proposed in Godhavn Godhavn et al. al.and (2011). But for thethe framework choke facilitate the pressure control. pumps are for the robustness robustness analysis and the idea idea for the guiding may beto used to provide additional flow These rate through the choke to facilitate the pressure control. These pumps are for the analysis and the for the guiding rule generation for controller tuning are general. are choke to the control. These pumps for the robustness analysis and the are ideageneral. for the guiding are usually called back pressure pumps. The drilling generation for controller tuning choke to facilitate facilitate the pressure pressure control. These pumpsmud are rule are usually called back pressure pumps. The drilling mud rule generation for controller tuning are general. are usually called back pressure pumps. drilling mud rule generation forpaper controller tuning are general. is not only key to the but also has the layout of the is the following: In Section we are usually called backpressure pressureregulation pumps. The The is not not only key key to the the pressure regulation but drilling also has hasmud the The The layout layout of of the the paper paper is is the the following: following: In In Section Section 22 2 we we is only to pressure regulation but also the function to remove the cuttings produced in the drilling The review the controller structure and the simple plant model is not only key to the pressure regulation but also has the function to to remove remove the the cuttings cuttings produced produced in in the the drilling drilling review The layout of the paper is theand following: In Section 2 we the controller structure the simple plant model function process (The reader is referred to Section 2 in Godhavn review the controller structure and the simple plant model controller design. In Section we present function(The to remove the cuttingstoproduced in theGodhavn drilling used process reader is referred Section 22 in reviewfor thethe controller structure simple33 plant model used for the controller design.and In the Section we present present process (The reader is to Section in et al. (2011) an illustration MPD for the controller design. In Section 33 we the qualitative representation of the model and parameter process (Thefor reader is referred referredof tothe Section 2process). in Godhavn Godhavn used et al. (2011) for an illustration of the MPD process). used for the controller design. In Section we present the qualitative representation of the model and parameter et al. (2011) for an illustration of the MPD process). the representation of the and uncertainties. robustness analysis is given in Section et al. pressure (2011) foratana illustration of the MPD process). the qualitative qualitativeThe representation the model model and parameter parameter The location depends on the uncertainties. The robustness of analysis is given given in Section Section The pressure at aa downhole downhole location depends on the uncertainties. The robustness analysis is in 4; based on which the approach for robust tuning is shown The pressure at downhole location depends on the uncertainties. Thethe robustness analysis is given inisSection pressure upstream choke (we call it choke pressure in 4; based on which approach for robust tuning shown The pressure at athe downhole location depends on the pressure upstream the choke (we call it choke pressure in 4; based on which the approach for robust tuning is in Section 5. Finally, some concluding remarks are given pressure upstream the choke (we call it choke pressure in 4; based on5.which the approach for robust tuningare is shown shown this paper) and the pressure drop between the downhole in Section Finally, some concluding remarks given pressure upstream the choke (we call it choke pressure in this paper) paper) and and the the pressure pressure drop drop between between the the downhole downhole in Section 6. 5. Finally, some concluding remarks are given this in Section 5. Finally, some concluding remarks are given location and the choke, which is primarily given by the 6. this paper) and the pressure drop between the downhole location and and the the choke, choke, which which is is primarily primarily given given by by the the in Section 6. location in Section 6. mud. Despite the fact that PID controllers are popular location and the choke, which is controllers primarily given by the mud. Despite the fact that PID are popular mud. Despite the that controllers are popular and understood al. (2013)), 2. PLANT MODEL AND CONTROLLER mud.well Despite the fact fact (Møgster that PID PIDet controllers are different popular and well understood (Møgster et al. (2013)), different 2. PLANT PLANT MODEL MODEL AND AND CONTROLLER CONTROLLER and well understood (Møgster et al. (2013)), different types of advanced controllers have been recently proposed 2. STRUCTURE and well understood (Møgster et al. (2013)), different types of of advanced advanced controllers controllers have have been been recently recently proposed proposed 2. PLANT MODEL AND CONTROLLER STRUCTURE types as improved solutions to the pressure control for MPD STRUCTURE types of advanced controllers been recently proposed as improved solutions to the thehave pressure control for for MPD STRUCTURE as solutions to control MPD (Zhou et al. (2009); Breyholtz et al. (2010); Godhavn as improved improved solutions to the pressure pressure control for MPD In this section we give a brief review of the simplified plant (Zhou et al. (2009); Breyholtz et al. (2010); Godhavn In this section we give aa brief review of the simplified plant (Zhou et al. (2009); Breyholtz et al. (2010); Godhavn et al. (2011); Li et al. (2011); Møgster et al. (2013)). In this section we give brief review of the simplified plant (Zhou et al. (2009); Breyholtz al. (2010); and the pressure controller that regulates the choke et al. (2011); (2011); Li et et al. al. (2011); et Møgster et al. al.Godhavn (2013)). model In this section we give a brief review of the simplified plant model and the pressure controller that regulates the choke et al. Li (2011); Møgster et (2013)). Usually the pressure controllers are designed based on model and the pressure controller that regulates the choke et al. (2011); Li et al. (2011); Møgster et al. (2013)). Usually the the pressure pressure controllers controllers are are designed designed based based on on pressure. model and the pressure controller that regulates the choke pressure. Usually some simple design dynamic models of the process with Usually the pressure controllers are of designed basedwith on pressure. some simple design dynamic dynamic models the process process pressure. some design models of with nominal parameters (e.g. physical properties of the mud some simple simple design dynamic models of the the process with nominal parameters (e.g. physical properties of the mud 2.1 Simplified plant model nominal parameters (e.g. physical properties of the mud and the choke) and have some parameters can 2.1 Simplified Simplified plant plant model model nominal properties ofwhich the mud and the parameters choke) and and(e.g. havephysical some parameters parameters which can 2.1 and the choke) have some which can be for good practice. The tuning andtuned the choke) andperformance have some in parameters which can 2.1 Simplified plant model be tuned for good performance in practice. The tuning be tuned for performance in The tuning of controller can be tedious ease the controller design, the following 3-state simplibe the tuned for good goodparameters, performancehowever, in practice. practice. tuning To of the controller parameters, however, can The be tedious tedious To ease ease the the controller controller design, design, the the following following 3-state 3-state simplisimpliof the controller parameters, however, can be To and difficult as the actual process dynamics may be fied well dynamics model is used (Kaasa et al. (2012)): of the controller parameters, however, can be may tedious To ease controller design, the (Kaasa following simpliand difficult as the the actual process process dynamics be fied fied wellthe dynamics model is used used et 3-state al. (2012)): (2012)): and difficult as actual dynamics may be well dynamics model is (Kaasa et al. and difficult as the actual process dynamics may be fied well dynamics model is used (Kaasa et al. (2012)): Copyright © 2016, 2016 IFAC 556Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 556 Copyright ©under 2016 responsibility IFAC 556Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 556 10.1016/j.ifacol.2016.07.401
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Va Va ˙ Lp (ˆ pc − pc ), (5) pˆc = qˆbit + qbpp − qc − Ad vd + qˆerr − βa βa Va qˆ˙err = − Li (ˆ pc − pc ), (6) βa
Table 1. List of notations Symbol Vd βd qp qbit M pp pc ρd ρa g h Va βa qbpp qc Ad vd qerr F (qbit , ω)
Physical meaning volume of drill string effective bulk modulus of mud in drill string standpipe flow rate bit flow rate average integrated density per area standpipe pressure (upstream) choke pressure average mud density in drill string average mud density in annulus acceleration of gravity true vertical depth of well volume of annulus effective bulk modulus of mud in annulus flow rate of back pressure pump choke flow rate cross-sectional area of drill string longitudinal velocity of drill string unmodeled flow rate in annulus frictional pressure drop from standpipe to choke
Vd p˙ p = qp − qbit , βd M q˙bit = pp − pc − F (qbit , ω) + (ρd − ρa )gh, Va p˙ c = qbit + qbpp − qc − Ad vd + qerr , βa
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where qˆbit is an estimate of the mud flow rate through the drill bit, which is simply set equal to the standpipe flow rate qp in this work 1 ; Lp and Li are tunable parameters. model based controller The model based controller (MBC) gives out the control reference signal for the opening of the MPD choke. It first computes the desired flow rate through the choke as qc∗ = qˆbit + qbpp − Ad vd + qˆerr +
Va (kp (pc − prr ) − p˙ rc ) . (7) βa
where Kp is a tunable parameter. Then, using the choke flow model (2), the choke opening reference signal is qc∗ r −1 √ . obtained as zc = Gc Kc pc −pc0
(1)
We refer the reader to Godhavn et al. (2011) for more details on the design and test of the controller. 3. UNCERTAINTY MODELING AND REPRESENTATION
where the meaning of the notations are listed in Table 1. Assuming that the mud through the MPD choke has a constant density, the flow rate through the MPD choke qc can be modeled as the following: √ (2) qc = Kc Gc (zc ) pc − pc0 , where Kc is a positive real constant; Gc is generally a nonlinear nondecreasing function of the choke opening zc ; and pc0 is downstream choke pressure, which is considered here as constant.
Physical phenomena that occurs in drilling process are complex and can hardly be captured accurately by simple models. Therefore an appropriate quantification of uncertainties is necessary to analyze robustness and performance of a controller designed based on simple models. In this section, we quantify two important uncertainties: • Uncertain plant dynamics: The discrepancy between the simplified plant model (1)-(2) and the actual plant dynamics. • Uncertain choke dynamics: The discrepancy between the choke dynamics model (3)-(4) and the actual choke dynamics.
We will call (1)-(2) the 3-state design plant model hereafter. 2.2 Choke dynamics
3.1 Operation points of plant
We consider that the dynamics of MPD choke opening may be (approximately) modeled by a 2nd-order linear system, i.e.,
For our control purposes, we are interested in the responses from input variables to output variables. Even though the actual responses may be complicated and nonlinear, they are in general approximately linear nearby some steady state and thus can be represented by transfer functions.
z˙c = vc , v˙ c = −2ζωn vc −
(3) ωn2 zc
+
ωn2 zcr ,
(4)
where vc is the rate of the choke opening; ωn and ζ are the natural frequency and damping ratio of the 2nd-order dynamics respectively; zcr is the choke opening reference, which is the output of the choke pressure controller. 2.3 Choke pressure controller The controller is composed of the following components: (1) error flow observer and (2) model based controller. error flow observer The error flow observer (EFO) is used to estimate the unmodeled flow rate qerr . It has the following form: 557
It is not difficult to see that in a steady state of the plant dynamics (1)-(2), for which all the three derivatives in (1) are zero, the values of pp , qbit and zc are determined if the values of qp , pc , qbpp , vd and qerr are given. Thus we can define an operation point to be a value of the vector [qp , pc , qbpp , vd , qerr ]. For simplicity, in this paper we consider operation points with qbpp = vd = qerr = 0. 3.2 Uncertain plant dynamics We consider two types of uncertainties in plant dynamics: (1) parameter uncertainty and (2) neglected dynamics. 1
An estimator for qˆbit was presented in Godhavn et al. (2011), it is not considered here for simplicity.
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Parameter uncertainty Parameter uncertainty refers to the uncertainty of one’s knowledge about the parameters in a nominal or design model. Examples of parameters in the plant model (together with examples of uncertainty ranges with respect to their nominal values) that can vary substantially are listed in the following Table 2. Table 2. Parametric uncertainty in plant model Parameter βa bulk modulus in the annulus βd bulk modulus in the drilling string Fa friction coefficient in the annulus Fd friction coefficient in the drilling string
Uncertainty range % [−60, 10]% ±10% ±50% ±20%
Using the linear fractional transformation (LFT) (see e.g. Skogestad and Postlethwaite (2005); Balas et al. (2001)), for a given operation point, the 3-state design model with the parameter uncertainties can be represented by the structure illustrated in Figure 1, in which • the vector φ = [qp , qbpp , vd , qerr ] is considered as part of the input to the closed-loop system; • the output vector ξ = [qp , qbpp , vd , zc ] together with pc are used by the controller block (see Section 2.3) as its input, with qp , qbpp , vd as feedforward and zc , pc as feedback; • ∆θ,p is the frequency-dependent structured perturbation matrix with a block diagonal structure; each diagonal block corresponds to the uncertainty induced by one uncertain parameter and has its largest singular value upper bounded by 1 for all frequencies; • the entries of the transfer function matrix Gnom depend on (i) the 3-state design plant model with a set of nominal values for the parameters and (ii) the bounds (real scalars) of the uncertain parameters; • the matrix ∆θ,p affects the input/output relationship between zc and pc in a feedback manner.
By neglected dynamics in plant model we mean, for fixed plant parameters, the discrepancy between the transfer function from the choke opening zc to choke pressure pc given by the 3-state design plant model and that given by the actual plant. Utilizing the so called multiplicative uncertainty model (see e.g. Skogestad and Postlethwaite (2005); Balas et al. (2001)), for a given operation point, the plant model with both the uncertain parameters and the neglected dynamics can be represented in the structure shown in Figure 2, in which • the system in the dashed box is simply the one shown in Figure 1; • wp,c is a frequency-dependent multiplicative uncertainty weighting function; • δp,c is a frequency-dependent function whose magnitude is upper bounded by 1 for all frequencies; • the product wp,c δp,c quantifies the neglected dynamics from the input zc to the output pc .
Fig. 2. Plant model with mixed uncertainty Now, let us denote the transfer functions from zc to pc in the actual plant as Pzc ,pc (jω; x, θ), where x represents the operation point and θ stands for the actual parameter vector. Exploiting the multiplicative uncertainty model, we wish to establish the following relationship: for any θ ∈ Θ, Pzc ,pc (jω; x, θ) = (1 + wp,c (jω)δp,c (jω))Gzc ,pc (jω; x, θ).(8) (The requirement on that (8) should hold for any θ ∈ Θ is because the only thing we know about the plant parameter θ is that it belongs to the set Θ.)
Fig. 1. Plant model with uncertain parameters Mathematically, we may collect all uncertain scalar parameters into one vector and use Θ to denote the set of all allowed values of the vectors. For one given parameter vector θ˜ ∈ Θ (which corresponds to one possible value ˜ to denote the transfer of ∆θ,p ), we use Gzc ,pc (jω; x, θ) function between zc and pc in Figure 1 at the operation point x. Hence the transfer function between zc and pc of the system in Figure 1 can be any element in the set ˜ : θ˜ ∈ Θ}. {Gzc ,pc (jω; x, θ) Neglected dynamics Even with the values of the parameters exactly known, the 3-state design plant model presented in Section 2.1 captures the actual plant dynamics accurately only at low frequency (Aarsnes et al. (2012)). 558
To achieve this we may choose the weighting function wp,c as P zc ,pc (jω; x, θ) − Gzc ,pc (jω; x, θ) wp,c (jω) ≥ max , (9) θ∈Θ Gzc ,pc (jω; x, θ)
because then a function δp,c (jω) can be chosen to satisfy (8) with |δp,c (jω)| < 1 for any frequency ω. In addition, for practical reasons:
• when computing the weighting function, we replace Pzc ,pc (jω; x, θ), which is literarily unknown, with the transfer function identified from a high-fidelity drilling simulator (The reader is referred to Ljung (1999) for some system approaches for this purpose); • the weighting function may be computed only for a frequency range of interest; • the weighting function is preferred to be the transfer function of a low-order SISO stable linear system.
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Uncertain choke dynamics The uncertainties in the choke dynamics can be represented in the same manner as those in the plant dynamics. The uncertain parameters and examples of the ranges of uncertainties are given in Table 3 (Note that here the uncertainty in the function Gc is converted approximately to that of the parameter Kc .)
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• d˜ represents the vector containing p˜rc and all elements in the vectors φ˜ and n ˜ ; it is thus regarded as the normalized system input (vector).
Table 3. Parametric uncertainty in choke model Parameter Kc choke constant ωn choke natural frequency
Uncertainty range % ±10% ±10%
Fig. 5. A compact form of the closed-loop system diagram
The choke model with mixed uncertainty of parameters and neglected dynamics can be represented by the structure shown in Figure 3, in which zc∗ is the choke opening reference given by the controller.
The robust stability criterion says that the closed-loop system in Figure 5 is robustly stable against all possible uncertainties described in Section 3 if and only if the nominal closed-loop system is stable and µ∆ (N11 (jω)) ≤ 1, ∀ω, where the µ∆ is the structured singular value, and N11 is the upper left block of the transfer function matrix N (i.e., it is the transfer function matrix from the output to the input of the block ∆). Normally, we can define the real number 1/ maxω µ∆ (N11 (jω)) as the robust stability margin. Also note that the generalized input signals do not affect the stability of the closed-loop system. 4.3 Robust performance
Fig. 3. Choke model with mixed uncertainty 4. ROBUST STABILITY AND PERFORMANCE ANALYSIS 4.1 Closed-loop system diagram For a given operation point the (linearized) closed-loop system can be formed by connecting the uncertain plant model, uncertain choke model (described in Section 3), and the controller (described in Section 2). The system diagram is shown in Figure 4, in which • n denotes the vector of sensor noises on pc as well as the signals in ξ; • e is the actual tracking error, and e˜ is called the weighted tracking error (it is also called the weighted system output); • wref is ideal model (transfer function) for the reference tracking, wφ , wn , we are called weighting (transfer) functions which are used to specify the desired performance in H∞ framework (see Section 4.3 for the details); note that wref , wφ , wn and we are diagonal ˜ n ˜ and matrices such that each scalar signal in p˜rc , φ, e˜ has an individual scalar weighting function. • p˜rc , φ˜ and n ˜ are called normalized choke pressure reference, exogenous inputs and noise respectively. 4.2 Robust stability
Input/output relationship is central to performance study. In Figure 5 it is the lower right block in the transfer function matrix N , denoted here by N22 , that relates the generalized system input d˜ to the weighted tracking error e˜. In the framework of H∞ control, the control objective, besides the robust stability stated in Section 4.2, is to make the norm N22 ∞ ≤ 1 for all frequencies; and the use of the weighting functions in the diagonal matrices wref , wφ , wn and we is to scale the inputs and outputs such that desired closed-loop performance is achieved if the norm condition above on N22 is met. One theoretical guideline to choose the weighting function is the following: Let wi (jω) be the weighting function for the ith element in the normalized ˜ and denote the actual input vector by input vector (d), r d = [pc , φ, n]. Then wi (jω) may be chosen such that the inequality i αi2 (ω)/|wi (jω)|2 ≤ 1 holds in the frequency band of interest, where αi (ω) represents the magnitude of the frequency component of the signal di (i.e. the ith component of the vector d) around frequency ω. The robust performance is guaranteed if µ∆ ˜ (N (jω)) ≤ ˜ = diag(∆perf , ∆) with 1, ∀ω, where the perturbation ∆ ∆perf (jω) being any possible complex matrix with the dimension nd˜ × ne˜ which satisfies ∆perf ∞ ≤ 1. The number 1/ maxω µ∆ ˜ (N (jω)) may be called the robust performance margin. 5. A ROBUST TUNING APPROACH
To apply the robust stability theory (see e.g. Skogestad and Postlethwaite (2005)), we first transfer the closedsystem into the more compact representation as shown in Figure 5. In the figure, • ∆ = diag[∆θ,c , δc , ∆θ,p , δp,c ] is a block diagonal perturbation; the inputs to the block ∆ include those to its component diagonal blocks that can be easily identified in Figure 4; 559
5.1 An approach searching for robust tuning parameters As expected, different operation points correspond to different linearized closed-loop system. For each operation point the ideal controller tuning would be finding the controller parameters kp , Lp , Li such that both robust stability and robust performance are satisfied with all allowed uncertainties (for stability) and the selected weighting
IFAC DYCOPS-CAB, 2016 560 June 6-8, 2016. NTNU, Trondheim, Norway
Qin Li et al. / IFAC-PapersOnLine 49-7 (2016) 556–561
Fig. 4. Closed-loop system diagram functions (for performance), i.e., we have both robust stability margin and robust performance margin larger than 1. However, this may not be always achievable. If this is the case, we impose priority on robust stability over robust performance; or in other words we are willing to sacrifice robust performance while pursing robust stability. Specifically, for each operation point x in a selected set X, we first compute the robust stability margins and the robust performance margins for a fixed grid of combinations of kp and Li 2 , which is defined by the set Γ=
Kpmax Lmax , Li = m i , N M n = 0, 1, · · · , N, m = 0, 1, · · · , M .
(kp , Li ) : kp = n
(10)
> 0 are the upper bounds of where Kpmax > 0 and Lmax i the ranges for the values of kp and Li , and the positive integers N and M define the resolution of the parameter grid. For each point in γ ∈ Γ let us denote the resultant robust stability margin and robust performance margin as sx (γ) and px (γ) respectively, with the subscript x indicating the dependence on the operation point. Let us suppose now that the set Γ is ordered and with its elements indexed as γi , with i = 1, 2, · · · , |Γ| (Here | · | denotes the size of a set). Then we may use the following Algorithm 1 to pick the best tuning point in Γ for each operation point. Algorithm 1 Searching for the best tuning parameters 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:
procedure FindBestParameters(Γ, sx , px ) for i ← 1 to |Γ| do if i = 1 then γx∗ ← γi p∗x ← p x (γi ), s∗x ← sx (γi ) else if px (γi ) ≥ p ∗x and sx (γi ) ≥ s∗x or px (γi ) > p∗x and sx (γi ) ≥ a then γx∗ ← γi p∗x ← px (γi ), s∗x ← sx (γi ) end if end for return γx∗ , p∗x , s∗x end procedure
In Algorithm 1, the number a should be set to 1 to reflect strict requirement on robust stability unless we see the resulting tuning is too conservative, in which case we may lower the value of a to accept tuning with less robust 2 The value of L is fixed as we found from experience that it does p not affect the results when ranging within a large interval.
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stability margin (An alternative may be sticking to a = 1 while adjusting the uncertainty weighting functions). Note that, by the searching procedure, we will have the best tuning parameter γx∗ (with corresponding robust stability margin s∗x and robust performance margin p∗x ) for each operation point x ∈ X. Thus a tuning table for all operation points is indeed created. Tuning parameters for operation points which are not in X can be obtained by simple interpolation, which is fast enough to be run with real time applications. We should also point out that the computation of robust margins and the tuning parameter searching algorithm are in general time-consuming and therefore should be run offline. 5.2 Simulation results Now we show some simulation results that compare the robust tuning with some nominal tuning which gives good performance with the simple design model. Firstly, In Figure 6a we show that the tuning parameter Kp = 1.5, Li = 4 and Lp = 25 gives very good pressure tracking performance with the simple 3-state design plant model. In Figure 6b, however, we see that this tuning makes the system unstable when simulating with a high-fidelity model. On the other hand, the robust tuning can achieve acceptable tracking performance in this case, which is shown in Figure 6c. Next we show the effectiveness of the robust tuning with low flow rate (300 lpm) and very low frictional pressure loss in the drill string and annulus. In Figure 7a, the controller is tested with Kp = 0.8, Li = 4 and Lp = 25 which stabilizes the system with normal frictional pressure loss at 300 lpm. It is clearly seen that this tuning cannot stabilize system with low friction. On the other hand, with some initial fluctuation, the system is stabilized by the robust tuning, which can be seen in Figure 7b. 6. CONCLUSION In this paper, we have presented an approach that yields a guide rule for the tuning of the pressure control for managed pressure drilling. The key to our approach is the robustness analysis via a qualitative representation of the model and parameter uncertainties. It allows us to evaluate the robustness of the stability and performance of the closed-loop control system with any particular tuning of the controller. The guide rule for tuning is simply made to somehow maximize the robustness such that the closedloop system can be stabilized with acceptable control performance in presence of expected uncertainties.
IFAC DYCOPS-CAB, 2016 June 6-8, 2016. NTNU, Trondheim, Norway
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(a) Pressure tracking with nominal tuning using the simple design plant model
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(a) Pressure tracking with nominal tuning
(b) Pressure tracking with robust tuning (b) Pressure tracking with nominal tuning using a high-fidelity plant model
(c) Pressure tracking with robust tuning using a high-fidelity plant model
Fig. 6. Comparison of pressure tracking performance at high flow rate and normal friction REFERENCES Aarsnes, U.J., Aamo, O.M., and Pavlov, A. (2012). Quantifying error introduced by finite order discretization of a hydraulic well model. In Proc. of the 2nd Australian Control Conference, 54–59. Sydney. Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and Smith, R. (2001). µ-Analysis and Synthesis Toolbox: For Use with MATLAB, Version 3. The Mathworks. Breyholtz, Ø., Nygaard, G., Siahaan, H., and Nikolaou, M. (2010). Managed pressure drilling: A multi-level control approach. SPE 128151. 561
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