Model-Free Predictive Anti-Slug Control of a Well-Pipeline-Riser.

Transcription

1 Modeling, Identification and Control, Vol. 37, No. 1, 16, pp. 1 5, ISSN Model-Free Predictive Anti-Slug Control of a Well-Pipeline-Riser. Christer Dalen 1 David Di Ruscio 1 Skien, Norwa. christerdalen@hotmail.com Universit College of Southeast Norwa, P.O. Box 3, N-391 Porsgrunn, Norwa. david.di.ruscio@hit.no Abstract Simplified linearized discrete time dnamic state space models are developed for a 3-phase well-pipelineriser and tested together with a high fidelit dnamic model built in K-Spice and LedaFlow. In addition the Meglio pipeline-riser model is used as an example process. These models are developed from a subspace algorithm, i.e. Deterministic and Stochastic sstem identification and Realization (DSR), and implemented in a Model Predictive Controller (MPC) for stabilizing the slugging regime. The MPC, LQR and PI control strategies are tested. Kewords: Model-Free, Model Predictive Control, Kalman filter, sstem identification, anti-slug, wellpipeline-riser 1. Introduction Severe-slugging is a problem regarding well-pipelineriser processes in the offshore industr and is characterized b significant flow rate and pressure oscillations observed at the topside choke. This flow needs to be stabilized or it might damage both downstream equipment and personnel (Courbot (1996)). One solution, which is regarded as the most costeffective, is to introduce active feedback where we define the topside choke valve as the manipulative variable and some pressure, flow rate or densit measurements as the controlling variable. We ma also define the flow rate as the goal variable, as it is what we want to maximize. On this approach, Schmidt Z. (1979), ma be viewed as the first contribution, however this was a rather experimental approach where an upstream pressure measurement together with the flow rate measurement, the choke valve was automaticall changed, b algorithm, to counteract the slugging regime. To maximize the goal variable a controller needs to be designed to operate around an open-loop unstable working point, here the largest possible choke opening which stabilizes the sstem ma be defined as a performance measure of the controller. Model-based control using mechanistic models is a popular approach for designing controllers. Some of these mechanistic models are presented in Storkaas and Skogestad (3b), Di Meglio et al. (9), Jahanshahi and Skogestad (13) and compared in Jahanshahi and Skogestad (13). Several active control strategies have been addressed for stabilizing the slugging phenomena, some of them are mentioned in Godhavn et al. (5), Ogazi AI (1), Di Meglio et al. (1a), Storkaas and Skogestad (3a) and Jahanshahi and Skogestad (15), Dalen et al. (15). In Dalen et al. (15), a so called Model-Free Linear- Quadratic Regulator (MFLQR) was demonstrated on a well-pipeline-riser example integrated in the K- Spice/LedaFlow simulator (K-Spice, LedaFlow). Different input-output cases were considered for solving the slugging problem, where the most satisfing redoi:1.173/mic c 16 Norwegian Societ of Automatic Control

2 Modeling, Identification and Control sults were when introducing gas-lift, however this is a rather expensive solution, as large quantities of gas are needed. It is less expensive to stabilize the flow regime, or controlling the bottom riser pressure, b active choking of the topside choke valve also demonstrated in the paper. The concept of model free optimal control is not new and was used in Favoreel et al. (1999) in order to identif a Linear-Quadratic-Gaussian (LQG) controller directl from closed loop subspace sstem identification. The subspace method used was however/regardless biased and the controller has to be partl known. In this paper we will define bottom-riser pressure as the controlling variable and topside choke valve as the manipulative variable. In particular, demonstrations of Model-Free Predictive Control (MFPC) is performed on the 3 state Di Meglio model (Di Meglio et al. (9)) and on the K-Spice/LedaFlow simulator (K- Spice, LedaFlow). The contributions of this paper can be itemized as follows. MFPC and MFLQR of the Di Meglio model (Di Meglio et al. (9)). MFPC of the K-Spice/LedaFlow simulator. The rest of the paper is organized as follows. In Sec. we define the MFPC algorithm. In Sec. 3 we present results of the MFPC algorithm on the Di Meglio model (Di Meglio et al. (9)) and the K-Spice/LedaFlow simulator. In Sec. we discuss and summarize the results. In Sec. 5 we present the concluding remarks. In Appendix A we provide a complete model description of the Di Meglio model (Di Meglio et al. (9)).. Theor Definition.1 (State observer) Define the following Kalman filter on state deviation form, i.e. x k+1 = A x k + B + K( k k 1 D x k ), x =, where k N is the discrete time, x k R n is the predicted state deviation vector, R r is the input deviation vector, k R m is the output vector and K is the Kalman filter gain matrix. The observer matrices A, B, D, K are identified as in Eq. (). Definition. (Optimal model) The model matrices in Eq. (1) are found using the following MATLAB function, [ ] A B D K = dsr op(y, U), () (1) where Y and U are identification matrices, containing collected data from an experimental design. Y = T 1. T N, U = u T 1. u T N. (3) It is important to note that choosing the model based on lowest Mean Square Error (MSE), calculated from simulated output, as in Dalen et al. (15), might not give the optimal model order, and according to Akaike (197). The optimal model will be refereed to as DSRL J, where J is the past horizon and L is the future horizon (see Di Ruscio (1996) for a detailed description). Definition.3 (MPC Algorithm) We consider the simple MPC algorithm presented in Di Ruscio (13). Given the pre-defined matrices, H, O L Ã, FL T Q, as defined in Di Ruscio (13), and the reference matrix, r k+1 L, we have for each time-instant k that [ ] xk x k =, k 1 p L = O L Ã x k, f k = F T L Q(p L r k+1 L ) The optimal unconstrained predictive control is The actual control is () u k L = H 1 f k. (5) = 1 + u k 1. (6) However, if the constrains are active, the problem renders a general QP problem, i.e. where u k L = arg min J k, A L b k (7) J k = u T k L H L + f T k L + J, (8) and J is not used. The vector b k depends on the constraints. As an example regarding the linear inequalit in Eq. (7), we consider the input rate of change constraints, u min k L L u max k L. (9) Eq. (9) ma be expressed as A L b k, where [ ] ILrxLr A =, I [ ] u max (1) k L b k = u min. k L

3 Christer Dalen, Model-free optimal anti-slug control A complete example which introduces the constraints of both the input rate of change and the input amplitude can be found in Section 3. and Appendix A in Di Ruscio (13). r k b k MPC State observer Process k 1 x k x k A Figure 1: Block diagram illustrating MFPC. 3. Numerical Examples 3.1. Di Meglio model We consider the 3 state model presented in Di Meglio et al. (9), which was calibrated in Di Meglio et al. (1b), for reproducing the slugging regime present in a real oil well located in the North Sea. The model is rather simple, but with an introduced virtual valve located at the bottom of the riser the model proves sufficient to investigate the phsical aspects of the slugging phenomenon. This model ma be formulated as a continuous nonlinear state space model, as k leads to a discrete time linear model, x k+1 = Ax k + B + v, k = Dx k + w. (13) Now, we present results on the MFPC based upon two different datasets with length, N = samples, each excited around different choke illustrated in Figs. and 7, respectivel. The sampling time is chosen equal to 1 sec. We can define our two cases as R := Bottom-riser pressure, [bar], u R := Topside [1] Note that u >.5 is considered the bifurcation point, i.e. the choke opening where the process becomes marginall stable. We removed the first samples. Now, the first 131 were stored in input and output identification vectors U R N id and Y R N id, respectivel. The validation vectors were made from all the data, stored as U R Nv and Y R Nv, illustrated in Fig.. k : Bottom riser pressure [bar] IDENTIFICATION VALIDATION ẋ = f(x, u), = g(x), (11) where x = x 1 x x 3 = m g,cb m g,r m l,r. (1) Here, in Eq. (1), m g,cb is the mass of gas in the elongated bubble, m g,r is the mass of gas in the riser, m l,r is the mass of liquid in the riser and the output is the pressure at the riser bottom. See Di Meglio et al. (9) for details. The main control u is the topside choke. The complete model for direct implementation is presented in Appendix A with parameters as in Tab. 6. The continuous non-linear model ma be linearized around stead state operating points u s and x s, which Figure : Raw data with length, N = samples. Identification and validation with lengths, N id = 131 and N v = 18. Sampling time is 1 The vectors U and Y were redefined with centered data, i.e. subtracted b the mean values u m =.151 and m = (Fig. 3). Next, a 3rd order model was identified (Eq. 1) using dsr op as in Eq. (), and Eqs. (15) and (17) for 3

4 Modeling, Identification and Control 1 5 : Bottom riser pressure [bar] 8 6 : Bottom riser pressure [bara] Real dsr pem lin k #1-3 k Figure 3: Identification data, N id = 131 samples. Sampling time is Figure : The identified models simulated and compared to validation data gathered from the real process (Di Meglio). We have the following validation performances (measured with MSE); V DSR 9 3 =.191, V P EM =.175 and V LIN =.18. an observabilit canonical form version of DSR3 1, operating points, respectivel. The original DSR3 1 is shown in Eq. (16). Fig. shows three models; DSR3, 9 PEM and LIN, simulated over the validation data, where the best performing model was the dsr with V DSR =.191, V P EM =.175 and V LIN =.18 (See Tab. 1). A well-known algorithm in sstem identificaiton is the Prediction Error Method (PEM), which can be found in the sstem identication toolbox (Ljung (7)). Observabilit canonical: DSR 9 3 }}. 1.. A =.. 1., B = , D = [ 1... ]. (15) Implementation of the MFPC is shown in Figs. 5 and 1. These figures shows how similar the LQR and MPC strategies are. Identified model: DSR 9 3 }} A = , B = 3.9,.395 D = [ ], K = (1) Identified model: DSR 1 3 }} A = , B =.6173,.688 D = [ ], K = (16)

5 Christer Dalen, Model-free optimal anti-slug control k CONTROLLER ON : Bottom-riser pressure [bar] PI (K p =-., T i =5) MPC (L=1,Q=1,R=1 6 ) MPC (L=,Q=1,R=1 6 ) LQR (Q=1,R=1 6 ) k : Bottom-riser pressure [bar] PI (K p =-., T i =5) MPC (L=1,Q=1,R=1 6 ) MPC (L=,Q=1,R=1 6 ) LQR (Q=1,R=1 6 ) X: 78 Y: Figure 5: Four controllers, are implemented on the real process (Di Meglio) turned on from starting point k =. We are comparing LQR, MPC (L=1), MPC (L=) and PI. The PI controller is tuned using MATLAB Tuner Application. The weights for MPC and LQR are chosen to be the same values. Sampling time is 1 sec Figure 6: The subfigure above illustrates how the MPC converges to the LQR when the prediction horizon, L increases. The subfigure below illustrates the controller performances. MPC (L=) and the LQR are able to stabilize the slugging regime up to choke opening.37 (zoomed in on the first and last part of Fig. 5.) Table 1: Summar: Comparing models from DSR J L, PEM and Linearized (LIN). We have the prediction error for simulated output, V MSE, the stead state gain, H d, and absolute eigenvalues, abs(eig(a)), for each of linear models, means around working point. V MOD H d abs(eig(a)) DSR , PEM ,.986 LIN , DSR PEM LIN , , , , Table : Summar: Comparing controllers PI, MPC and LQR b performance measures IAE and TV, calculated from k = to k = 8. Maximum choke opening while stable is donated b max u. Measures donated b * should not be considered. Param. IAE TV max u PI.15 MPC L = 1 MPC L = LQR.15 PI. MPC L = 1 MPC L =.. LQR. K p =. T i = Q = 1 R = *.38*.6 Q = 1 R = Q = 1 R = K p =.5 T i = Q = 1 R = *.71*.9 Q = 1 R = Q = 1 R =

6 u Modeling, Identification and Control : Bottom riser pressure [bara] : Bottom riser pressure [bar] 3 Real dsr pem lin k 175 IDENTIFICATION 17 VALIDATION k Figure 7: Raw data with length, N = samples. Identification and validation are chosen with lengths, N id = 131 and N v = 18. Sampling time, t = Figure 9: Identified dsr model simulated over the validation set. V DSR CONTROLLER ON : Bottom-riser pressure [bar] PI (K p =-.5, T i =6) MPC (L=1,Q=1,R=1 6 ) MPC (L=,Q=1,R=1 6 ) LQR (Q=1,R=1 6 ) : Bottom riser pressure [bar] k # Figure 8: Identification data, N id = 131 Figure 1: Four controllers, are implemented on the real process (Di Meglio), where each controller is turned on from starting point k =. We are comparing LQR, MPC (L=1), MPC (L=) and PI. The PI controller is tuned using MAT- LAB Tuner Application. The weights for MPC and LQR are chosen the same values. Sampling time is 1 sec. 6

7 u Christer Dalen, Model-free optimal anti-slug control Observabilit canonical: DSR 1 3 }}. 1.. A =.. 1., B =.8393,.8593 D = [ 1... ]. (17) Discussion : Bottom-riser pressure [bar] PI (K p =-.5, T i =6) MPC (L=1,Q=1,R=1 6 ) MPC (L=,Q=1,R=1 6 ) LQR (Q=1,R=1 6 ) X: 78 Y: Figure 11: The subfigure above illustrates how the MPC converges to the LQR when the prediction horizon, L increases. The subfigure below illustrates the controller performances. The MPC (L=) and the LQR, are able to stabilize the slugging regime up to choke opening.39. (zoomed in on the first and last part of Fig. 1. Interestingl, considering Tab. 1, the dsr model is performing better than the PEM and the linearized model in both cases;.15 and.. Considering Tab. the best performing controller seems to be MPC(L=) based stabilizing up to.39. However, MPC(L=) based at.15 is surprisingl achieving stabilizing up to.37. The LQR seems to be the runnerup best candidate. 3.. The K-Spice/LedaFlow simulator We perform model-free anti-slug control on a well-pipelineriser (Fig. 1), integrated in the K-Spice/LedaFlow simulator, high fidelit simulators developed b Kongsberg Oil & Gas Technologies (K-Spice, LedaFlow). [km] 1 [km] x 1 u Separator outflow 3 Well 5 Figure 1: Illustration of the 3-phase well-pipeline-riser process integrated in the K-Spice/LedaFlow simulator. 7

8 u Modeling, Identification and Control We define following case as R := : Bottom-riser pressure [bara], u R := u: Topside choke [%]. Note that bara is the absolute pressure expressed in bar, where bara is associated with total vacuum. The simulator was run with simulation speed 5 times real-time and the sample time was chosen to be 1 sec. Input and output data were collected from an open loop input experiment (Fig. 13). The samples from 6 to were stored in identification matrices U R N and Y R N, where N = 1. The samples from 6 to 35 were stored in validation matrices. The matrices were redefined with centered data, i.e. subtracted b mean values u m =.9 and m = : Bottom-riser pressure [bara] B = Identified model: DSR 8 8 }} A = , , D = [ ], K = (18) We identified a similar th order model from the PEM algorithm, Tab. 3 shows how closel related these models are. Both models were compared over the validation set (Fig. 16), where dsr had the lowest prediction error, V DSR =.393. Table 3: Comparing models identified from dsr and pem. Algorithm V MSE H d abs(eig(a)) DSR PEM u: Topside choke [%] : Bottom riser pressure [bara] Real dsr Figure 13: Data collected from the K-Spice/LedaFlow simulator. The data from 6 to were used for identification, while the data from 6 to 35 were used for validation. The simulation speed was 5 times real-time. The sampling time is equal to 1 sec. An optimal model was identified (Eq. 18), i.e. the model from DSRL J having the lowest prediction error using deterministic output (as described in Eqs. (9)-(1) in Dalen et al. (15)) with L = J = 8, n = and resulting in V MSE = See Fig. 1 for illustration Figure 1: Illustration of the identified model DSR8 8 in Eq. (18) simulated over the identification set resulting in V MSE =

9 u Christer Dalen, Model-free optimal anti-slug control Specified constrains: : Bottom riser pressure [bara] Real dsr u max k u max k = 55, u min k = 15 = 1, u min k = 1 An implementation of the MPC on the K- Spice/LedaFlow simulator is shown in Fig. 17. A prediction horizon, L =, and the following weights were chosen; Q = and R = 1 based on simulation on the identified model. It can be seen that both the controllers; MPC and LQR have successfull stabilized the undesired oscillating flow, up to 5 % choke opening, but the production/outlet flow remains constant at.9 [kg/s]. Both strategies also have quite similar performances, the difference is that the MPC is predictive, as illustrated in Fig. 19. The LQR matrices G 1 and G in = 1 + G 1 x k + G ( k 1 r k ) are as in Eq. (19) G 1 = LQ-optimal feedback matrices }} [ ], G = [.3356 ]. (19) Figure 15: Identified model DSR 8 8 (Eq. 18) simulated over the identification set. This figure shows simulation from 5 to 85 samples in Fig CONTROLLER ON Outlet flow [kg/s] : Bottom riser pressure [bara] Real dsr pem : Bottom-riser pressure [bara] MFPC (L=,Q=,R=1) MFPC (L=,Q=,R=1) MFLQ (Q=,R=1) u: Topside choke [%] X: 71 Y: Figure 16: Illustration of the identified models simulated over the unused part of the validation set. We have V DSR 8 8 =.393 V P EM =.539. See Fig. 13. for details Figure 17: Implementation of MFPC and MFLQR on the K-Spice/LedaFlow simulator. Stabilizing up to 51.3 % choke opening. Simulation speed is 5 times real time. Sampling time is 1 sec.. Discussion and summar Two examples are demonstrating the MFPC on the Di Meglio model and the K-Spice/LedaFlow simulator, where 9

10 u u Modeling, Identification and Control 6 5 Outlet flow [kg/s] : Bottom-riser pressure [bara] MFPC (L=,Q=,R=1) MFPC (L=,Q=,R=1) MFLQ (Q=,R=1) CONTROLLER ON u: Topside choke [%] Table : Comparing MFPC vs MFLQ using performance measures: Integrated Absolute Error (IAE) and Total Value (TV). Associated with Fig. 17. Controller Tuning parameters IAE TV LQR Q = 1, R = MPC L =, Q = 1, R = MPC L =, Q = 1, R = the goal was to stabilize the outlet flow/bottom riser pressure at highest possible choke opening. For the Di Meglio model we have that the MPFC, (marginall stable is defined at.5), was able to stabilize up to.39, while the other one, achieved.37. The runner-up candidate, i.e. the MFLQR, did onl differ from the MFPC in terms of performance indices TV and IAE. Note that the PI controller could probabl be tuned better for this case. For the K-Spice/LedaFlow simulator we based the MFPC around a marginall stable working point, %, and it was able to stabilize up to 5%. Figure 18: Comparing MFPC to MFLQR using the samples from 1 to 96 in Fig Concluding Remarks Practical implementation of MFPC was successfull demonstrated on a well-pipeline-riser process described b a 3-state non-linear model, thereafter it was demonstrated on the K-Spice/LedaFlow simulator : Bottom-riser pressure [bara] MFPC (L=,Q=,R=1) MFPC (L=,Q=,R=1) MFLQ (Q=,R=1) Acknowledgment The authors acknowledge in bullets Kongsberg Oil & Gas Technologies for supporting with license and software for the K-Spice and LedaFlow simulator. Telemark Universit College u: Topside choke [%] MATLAB functions The MATLAB functions used in this work are available for academic use upon request Figure 19: Comparing MFPC to MFLQR using the samples from 13 to 186 in Fig. 17. A. Complete model The Di Meglio model ma be formulated as a continues non-linear state space model, as ẋ = f(x, u), = g(x), () 5

11 Christer Dalen, Model-free optimal anti-slug control where x = x 1 x x 3 f 1 f = f. f 3 = f 1 = (1 λ)w g,in C gmax ( m g,cb m g,r m l,r x RT M(V r (x 3 + m l,still )ρ l ) ) (x 3 + m l,still ) gsin(θ), A ( RT f = λw g,in + C gmax, MV eb,, x 1 RT MV eb (1) x RT M(V r (x 3 + m l,still )ρ l ) ) gsin(θ) () (x 3 + m l,still A ( ) x RT C cu ρ l ( M(V r (x 3 + m l,still )ρ l ) Ps) x, x 3 f 3 = w l,in x RT C cu ρ l ( M(V r (x 3 + m l,still )ρ l ) Ps), x RT g = M(V r (x 3 + m l,still )ρ l ) + (x 3 + m l,still ) gsin(θ). A1 5 Table 5: Initial values for the simulations on Di Meglio model. The ODE is solved each timestep with MATLAB ode15s (sampling time, t = 1 sec.) Variable Value Unit x 1.e kg/s m bar u u m 1 s Akaike, H. A new look at the statistical model identification. Automatic Control, IEEE Transactions on, (6): doi:1.119/tac Table 6: Parameters for the Di Meglio model Eqs.. Variable Value Unit A 1.77E- m π θ rad kg ρ l 9 m 3 J R 8.31 molk T 363 K N P s 6.6E5 m m l,still 3.73E kg M.E- λ.78 L 5 m kg mol m g 9.81 s C c 1E- ms C g.8e-3 m w l,in kg s kg s w g,in 8.E-1 V r 9. m 3 V g,eb 8 m 3 Courbot, A. Prevention of Severe Slugging in the Dunbar 16 Multiphase Pipeline. Proc. Annual Offshore Technolog Conference, doi:1.3/8196-ms. Dalen, C., Di Ruscio, D., and Nilsen, R. Model-free optimal anti-slug control of a well-pipeline-riser in the K- Spice/LedaFlow simulator. Modeling, Identification and Control, (3): doi:1.173/mic Di Meglio, F., Kaasa, G., Petit, N., and Alstad, V. Model-based control of slugging flow: An experimental case stud. In American Control Conference (ACC), 1. pages 995 3, 1a. doi:1.119/acc Di Meglio, F., Kaasa, G.-O., and Petit, N. A first principle model for multiphase slugging flow in vertical risers. In Decision and Control, 9 held jointl with the 9 8th Chinese Control Conference. CDC/CCC 9. Proceedings of the 8th IEEE Conference on. pages 8 851, 9. doi:1.119/cdc Di Meglio, F., Kaasa, G.-O., Petit, N., and Alstad, V. Reproducing slugging oscillations of a real oil well. In Decision and Control (CDC), 1 9th IEEE Conference on. pages 73 79, 1b. doi:1.119/cdc Di Ruscio, D. Combined Deterministic and Stochastic Sstem Identification and Realization: DSR - A Subspace Approach Based on Observations. Modeling, Identification and Control, (3): doi:1.173/mic Di Ruscio, D. Model Predictive Control with Integral Action: A simple MPC algorithm. Model- 51

12 Modeling, Identification and Control ing, Identification and Control, 13. 3(3): doi:1.173/mic Favoreel, W., Moor, B. D., Gevers, M., and Overschee, P. V. Closed-loop model-free subspace-based LQGdesign. in Proc. of the 7th Mediteranean Conference on Control and Automation (MED99), Haifa, Isreal, pages URL ftp://ftp.esat.kuleuven.be/ pub/sista/ida/reports/98-17.pdf. Godhavn, J.-M., Fard, M. P., and Fuchs, P. H. New slug control strategies, tuning rules and experimental results. Journal of Process Control, 5. 15(5): doi: Jahanshahi, E. and Skogestad, S. Simplified dnamical models for control of severe slugging in multiphase risers. 18th IFAC World Congress, (3):33. doi:1.318/ it Jahanshahi, E. and Skogestad, S. Anti-slug control solutions based on identified model. Journal of Process Control, 15. 3(): doi:1.116/j.jprocont CAB/DYCOPS 13 Selected Papers From Two Joint IFAC} Conferences: 9th International Smposium on Dnamics and Control of Process Sstems and the 11th International Smposium on Computer Applications in Biotechnolog, Leuven, Belgium, Jul 5-9, 1. K-Spice. K-Spice version URL kongsberg. com/k-spice. LedaFlow. LedaFlow version URL kongsberg.com/ledaflow. Ljung, L. Sstem identification toolbox for use with MATLAB}. 7. Ogazi AI, Y. H.. L. L., Cao Y. Slug control with large valve openings to maximize oil production. SPE Journal, 1. 15(3): doi:1.118/1883-pa. Schmidt Z., J. P.. B., Brill. Choking can eliminate severe pipeline slugging. Oil & Gas J, :3 38. Storkaas, E. and Skogestad, S. Cascade control of unstable sstems with application to stabilization of slug flow. AdChem, Hong Kong, 3a. Storkaas, E. and Skogestad, S. A low-dimensional dnamic model of severe slugging for control design and analsis. 3b. 11th International Conference on Multiphase flow (Multiphase 3). 5