EP2821587A1 - Method of operating a pipeline-riser system - Google Patents

Abstract

A method of operating a pipeline-riser system comprising an anti-slug valve is provided, wherein the method comprises controlling the anti-slug valve by a control signal generated based on a measured signal indicative for a flow of a fluid through the pipeline-riser system, wherein the control signal comprises a signal component relating to a linear time-invariant system G(s), wherein the linear-time invariant system is modeled according to G s = b 1 ¢ s + b 0 s 2 - a 1 ¢ s + a 0 . Based on this model, tuning rules for PID and PI controllers are provided.

Description

Field of invention
• The present invention relates to a method of operating a pipeline-riser system, in particular to tuning of a PID and PI controllers for robust anti-slug control. Moreover, the invention relates to a pipeline-riser system, in particular to a pipeline-riser system comprsing an anti-slug valve. Furthermore, the invention relates to a program element and to a computer readable medium.
Art Background
• In the field of offshore oil production often pipeline-riser systems are used. In particular, at offshore oilfields, subsea pipelines are used to transport the multiphase mixture of oil, gas and water from producing wells to the processing facilities. Several kilometers of pipeline run on the seabed ending with risers to top-side platforms. Therefore, in new field developments, multiphase transport technology and flow assurance become more important. Especially, depleted reservoirs have low pressures to push the fluid along the pipeline, and they are prone to flow instabilities. Liquids tend to accumulate at places with lower elevation, and can block the gas flow in the pipe. In low flow rate conditions, this blockage leads to a slugging flow regime called terrain slugging.
• If the oscillating frequency or the lengths of slugs are comparable to the length of the riser, the flow condition is called "severe slugging" or "riser-slugging". The severe-slugging flow is also characterized by large oscillatory variations in pressure and flow rates. The oscillatory flow condition in offshore multi-phase pipelines is undesirable and an effective solution is needed to suppress it. One way to prevent this behavior is reducing the opening of a choke valve arranged at the top-side of the riser. However, this conventional solution increases the back pressure of the valve, and it reduces the production rate from the oil wells. The recommended solution to maintain a non-oscillatory flow regime together with the maximum possible production rate is active control of the topside choke valve. Measurements such as pressure, flow rate or fluid density are used as the controlled variables and the top-side choke valve is the manipulated variable. However, existing anti-slug control systems are not robust in practice and the closed-loop system becomes unstable after some time, because of inflow disturbances or plant changes.
• Thus, there may be a need for providing a method of operating a pipeline-riser system and a pipeline-riser system showing a low probability of slugging.
Summary of the Invention
• This need may be met by a method of operating a pipeline-riser system, a pipeline-riser system, a program element to and a computer readable medium according to the independent claims. Further embodiments are described in the dependent claims.
• According to an exemplary aspect a method of operating a pipeline-riser system comprising an anti-slug valve is provided, wherein the method comprises controlling the anti-slug valve by a control signal generated based on a measured signal indicative for a flow of a fluid through the pipeline-riser system, wherein the control signal comprises a signal component relating to a linear time-invariant system G(s), wherein the linear-time invariant system is modeled according to $G\left(s\right)=\frac{b_{1}s+b_0}{s^2-a_{1}s+a_0}\mathrm{.}$

  
• In particular, b₀, b₁, a₀ and a₁ are parameters derivable from pressure measurements and respective point in times. For example, for a closed-loop stable system the four parameters may be derived or estimated from six measurement data of a closed-loop step response describing pressure measured at respective points in the pipeline-riser system and respective point in times for the measurement.
• In particular, G(s) denotes a gain factor which may be time dependent. Thus, G(s) may form a transfer function or at least a portion of a transfer function describing the behavior of the gain.
• In particular, the controlling may be performed by opening or closing the anti-slug valve or an aperture of the anti-slug valve or more general by changing the through-flow of a liquid through the anti-slug valve. The anti-slug valve may be formed by a choke valve for example or any other valve which is suitable to control a through-flow of a fluid (gas, liquid or mixture thereof).
• According to another exemplary aspect a pipeline-riser system is provided, wherein the pipeline-riser system comprises a pipeline portion, a riser portion, an anti-slug valve, and a control unit, wherein the pipeline portion and the riser portion are joined at a joining point, wherein the anti-slug valve is arranged in the pipeline-riser system in such a way that a flow rate of fluid flowing through the pipeline-riser system is controllable, and wherein the control unit is adapted to generate a control signal for controlling the anti-slug valve according to a method according to an exemplary aspect.
• In particular, the pipeline-riser system may be an undersea pipeline-riser system, e.g. for offshore oil production. For example, the pipeline section is arranged upstream of the riser portion. In particular, the anti-slug valve may be arranged in the pipeline portion, the riser portion, or the joining point. In particular, the anti-slug valve may be arranged in proximity of a joining point of the pipeline portion and the riser portion or at a top-side of the riser portion, for example.
• According to an exemplary aspect a program element is provided, which, when being executed by a processor, is adapted to control or carry out a method according to an exemplary aspect.
• According to an exemplary aspect a computer-readable medium is provided, in which a computer program is stored which, when being executed by a processor, is adapted to control or carry out a method according to an exemplary aspect.
• The term "pipeline portion" may particularly denote a portion of a tube system adapted to convey fluids, e.g. natural gas, crude oil or a multiphase fluid and which is arranged substantially horizontal or following the contour of the terrain, e.g. a subsea floor. A pipeline portion has to be distinguished from a riser portion of a pipeline-riser system.
• The term "riser portion" may particularly denote a portion of the tube system which is arranged substantially vertical or at least does not follow a contour of the terrain. In particular, the riser portions may be the portions of a subsea tube arrangement, which are adapted to convey crude oil from the sea level to the surface, e.g. by pumping up the crude oil or multiphase fluid.
• The term "anti-slug valve" may particularly denote a valve which is adapted or suitable to reduce or avoid slugging in a conveying pipeline-riser system. In particular, the anti-slug valve may be controllable or may have a preset or predetermined throughput of a fluid, e.g. crude oil. For example, the anti-slug valve may be a top-side valve arranged on an oil platform.
• Surprisingly the inventors have found out by experiments by using a test rig and performing simulations that an anti-slug valve controlled by a control signal comprises a signal component relating to a linear time-invariant system G(s) may allow for an improved and in particular robust or stable anti-slug control. In particular, it may be possible to reduce or even eliminate slugging during conveying of fluids through the pipeline-riser system.
• Experiments even have shown that the non-slugging flow may even be maintained for a larger valve opening when using the specific algorithm for anti-slug control than it can be maintained when using conventional solutions (e.g. manual choking) Thus, it may be possible to provide a more robust anti-slug control and/or a more robust slug control and therefore an increased rate of oil production
• Next further embodiments of the method of operating a pipeline-riser system will be described. However, these embodiments also apply to method the pipeline-riser system, to the program element and to the computer readable medium.
• According to an exemplary embodiment the control signal comprises a further signal component relating to or representing a static nonlinearity.
• In particular, the control signal may comprise or may be formed according to a Hammerstein model structure, comprising a static nonlinearity (representing a static gain) component and a linear time-invariant (representing unstable dynamics of the modeled pipeline-riser system) component. For example, the static nonlinearity component may be described and/or calculated by $\mathrm{\Delta }P=\frac{\bar{a}}{f{\left(\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0006.png"\; state="\{\{state\}\}">z</figure-callout>}\right)}^2},$

  

  wherein ΔP represents a pressure difference generated by a respective element, e.g. a valve or pump, or by the resistance in a pipeline, a represents a constant parameter, and f(z) represents the valve characteristic function depending on the valve opening z.
• According to an exemplary embodiment the method further comprises controlling a through-flow through the anti-slug valve according to the control signal.
• According to an exemplary embodiment of the method an internal model control function is used for generating the control signal.
• In particular, the internally model control function or internal model controller may be described by a stabilizing controller $C=\frac{\tilde{Q}\left(s\right)f\left(s\right)}{1-G\left(s\right)\tilde{Q}\left(s\right)f\left(s\right)}$

  

  wherein G(s) is a model of a plant or the pipeline-riser system, Q̃(s) is the inverse of the minimum phase part of G(s) and f(s) is a low-pass filter which may ensure stability and robustness of a closed-loop system.
• According to an exemplary embodiment of the method the internal model control function is given by $C\left(s\right)=\frac{\left[\frac{1}{\mathit{kʹ}{\lambda }^3}\right]\left({\mathrm{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_2{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0006.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+{\alpha }_{1}s+1\right)}{s\left(s+\phi \right)},$

  

  wherein k' is the gain of the plant or pipeline-rides system G(s), λ is an adjustable filter time constant, α₁ and α₂ represent coefficients and ϕ represents the zero dynamics of the pipeline-riser system.
• In particular, k' is the gain of the plant G(s), as will be described later on in equation (9). In particular, the internal model control function or internal model controller may be a second order transfer function which may be written in form of a PID (proportional, integral, and derivative) controller. Alternatively, the internal model control function may be written in form of a PI (proportional, integral) controller.
• According to an exemplary embodiment of the method the controlling of the anti-slug valve is performed by a PID or PI controller.
• According to an exemplary embodiment of the method a tunig rule for the PID controller is given by $K_{\mathit{PID}}\left(s\right)=K_c+\frac{K_i}{s}+\frac{K_{d}s}{T_{f}s+1},$

  

  wherein $T_f=1/\phi ;$

  

  $K_i=\frac{T_f}{\mathit{kʹ}{\lambda }^3};$

  

  $K_c=K_i{\alpha }_1-K_i{T}_f;$

  

  and K_d =K_iα₂-K_pT_f, wherein K_p <0 and K_d <0.
• In particular, λ may be chosen so that K_c <0 and K_d <0 is valid.
• According to an exemplary embodiment of the method a tunig rule for the PI controller is given by $K_{\mathit{PI}}\left(s\right)=K_C\left(1+\frac{1}{{\tau }_{I}s}\right),$

  

  wherein $K_C=\frac{{\mathrm{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_2}{\mathit{kʹ}{\lambda }^3};$

  

  and τ₁=α₂ϕ.
• Summarizing, a gist of an exemplary embodiment may be seen in providing a control regime for an anti-slug valve for an undersea pipeline-riser system, wherein the anti-slug valve comprises a control interface and connected to a control unit. The control unit is adapted to generate a control signal based on measured variables, e.g. flow rate, pressure or fluid density in the pipeline-riser system.
• It has been found out that when using a control signal generated according to a Hammerstein model structure it may be possible to decrease the probability of riser slugging and/or the amount of riser slugging. It should be noted that due to the specific control of the anti-slug valve an increased robustness of the slug control and thus of the flow rate and/or pressure may be provided.
• In particular, the anti-slug valve may be a choke valve. The term "anti-slug valve" may particularly denote a valve, e.g. a choke valve, which is adapted to control a flow through a pipeline, in particular, to prevent or at least reduce the probability of an oscillatory flow regime. In particular, the anti-slug valve may be adapted to be controlled by a control signal, wherein the control may be provided by increasing or decreasing a flow rate through the anti-slug valve, e.g. by opening or closing an aperture of the anti-slug valve.
• The aspects defined above and further aspects of the present invention are apparent from the examples of embodiment to be described hereinafter and are explained with reference to the examples of embodiment. The invention will be described in more detail hereinafter with reference to examples of embodiment, but to which the invention is not limited.
Brief Description of the Drawings

Fig. 1

shows a simplified experimental set-up.

Fig. 2

shows bifurcation diagrams.

Fig. 3

shows a schematic block diagram for a Hammerstein

Fig. 4

shows a closed-loop step response diagram.

Fig. 5

shows a schematic block diagram of an internal model control system.

Fig. 6

shows some simulation results.

Detailed Description
• The illustration in the drawing is schematically. It is noted that in different figures, similar or identical elements are provided with the same reference signs or with reference signs, which are different from the corresponding reference signs only within the first digit.
• Fig. 1 shows a simplified experimental set-up. In particular, Fig. 1A shows a simplified pipeline-riser system 100 comprising a fluid reservoir 101 for simulating an oilfield, for example. The fluid reservoir is connected via a pump 102 (simulating the pressure the crude oil in the oil field is exposed to) and a water flow meter 103 to a mixing point 104. Furthermore, an air flow meter 105 is connected to an air buffer tank 106 having a pressure of P₁ or P_in. The air buffer tank 106 is connected to the mixing point 104 via a safety valve 107 and is used in the experiment to simulate gas expansion of a very long pipeline (corresponding to a real pipeline-riser system). The mixing point 104 corresponds in principle to the well head of a real offshore pipeline-riser system. At the mixing point the water/air fluid has a pressure of P₃ and is pumped through a pipeline portion or section 108 which is arranged substantially horizontal. At the bottom of the riser the multiphase (water/air) fluid has a pressure of P_rb (pressure at riser bottom) or P₄.
• In the experimental set-up of Fig. 1A a riser of the pipeline-riser system 100 is formed by a framework structure to which the riser portion 110 is attached for simulating the riser portion of a real pipeline-riser system. The riser portion 110 and the pipeline portion 108 are joined at a joining point 113. At top of the framework structure a top-side valve 111 is connected to the riser 110 simulating a common top-side valve which is typically used for slugging control. At the top-side the multiphase fluid has a pressure of P_rt (pressure at riser top) or P₂. After the top-side valve the multiphase fluid is conveyed to a separator 112 in which the air phase is separated from the water phase which is recycled and conveyed back to the water reservoir 101.
• Additionally in Fig. 1B a schematic sketch of a set-up for performing simulations of a pipeline-riser system is depicted. In addition to the simplified pipeline portion 108, the riser portion 110, and the top-side valve 111, parameters are defined which can be used in the calculation. These are the mass flow rate of liquid w_l,in, the mass flow rate of gas w_g,in, and the corresponding pressure P_in at the input point, e.g. in the case of a real pipeline-riser system the wellhead position. Furthermore, the corresponding values for the pressure at the riser bottom P_rb and at the riser top P_rt are indicated in Fig. 1B. Moreover, the pressure after a separation P_s is indicated in Fig. 1B after the top-side valve 111. Additionally, some geometric parameter like an angle θ indicating an inclination of the pipeline portion before the riser basis and a length L_r of the riser indicating a height difference between the riser bottom and the riser top are indicated in Fig. 1B as well.
• Fig. 2 shows bifurcation diagrams. In particular, Fig. 2A shows a bifurcation diagram 200 for the input pressure P_in simulated by the known OLGA simulation tool at the input point of Fig. 1B vs. the opening Z of the anti-slug valve, e.g. the top-side valve. More specifically, solid lines 201 and 202 indicate the maximum and minimum of the oscillations, respectively, while dashed line 203 indicate the steady-state of the input pressure. Fig. 2B shows a bifurcation diagram 204 for the pressure P_rt at the top of the riser simulated by the known OLGA simulation tool at the top-side valve of Fig. 1B vs. the opening Z of the anti-slug valve, e.g. the top-side valve. More specifically, solid lines 205 and 206 indicate the maximum and minimum of the oscillation, respectively, while dashed line 207 indicate the steady-state of the topside pressure.
• Fig. 3 shows a schematic block diagram for a Hammerstein model which may be used for describing a desired unstable flow regime. The block diagram of the Hammerstein model 300 is a structure consisting of series connection of a static nonlinearity 301 followed by a linear time-invariant dynamic system 302. The static nonlinearity may represent the static gain of the process and G'(s) may account for the unstable dynamics. For identification of the unstable dynamics a model structure may be assumed. A simple model which may be used for unstable systems is an unstable first order plus dead-time model: $$G\left(s\right)=\frac{K{e}^{-\mathit{\theta s}}}{\mathit{\tau s}-1}=\frac{b{e}^{-\mathit{\theta s}}}{\mathit{s}-a}$$

  
• If this system is controlled by a proportional controller K _c0, the closed-loop transfer function from the set-point (y_s ) to the output (y) becomes $$\frac{y\left(s\right)}{y_s\left(s\right)}=\frac{K_{c0}G\left(s\right)}{1+K_{c0}G\left(s\right)}=\frac{K_{c0}b{e}^{-\mathit{\theta s}}}{\mathit{s}-a+K_{c0}b{e}^{-\mathit{\theta s}}}\mathrm{.}$$

  
• To get a stable closed-loop system, one needs K_c0be^-θs>a and K_c0b>a. The steady-state gain of the closed-loop transfer function is $$\frac{\mathrm{\Delta }{y}_{\infty }}{\mathrm{\Delta }{y}_s}=\frac{K_{c0}b}{K_{c0}b-a}>1.$$

  
• However, the closed-loop step response of the system in experiments, as in Fig. 4, shows that the steady-state gain of the system is smaller than one. Therefore, the model in (1) may not an optimal choice. If the four-state mechanistic model for slugging flow (Jahanshahi E. and S. Skogestad 2011; "Simplified dynamical models for control of severe slugging in multiphase risers"; 18th IFAC World Congress, Milan, Italy) is linearized around an unstable operating point, one may get a fourth-order linear model, $$G\left(s\right)=\frac{{\theta }_1\left(s+{\mathrm{<figure-callout\; id="2"\; label="\theta "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_2\right)\left(s+{\theta }_3\right)}{\left(s^2-{\mathrm{<figure-callout\; id="4"\; label="\theta "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_{4}s+{\theta }_5\right)\left(s^2+{\mathrm{<figure-callout\; id="6"\; label="\theta "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_{6}s+{\theta }_7\right)}\mathrm{.}$$

  
• This model contains two unstable poles, two stable poles and two zeros. Seven parameters have to be estimated to identify this model. However, if one look at the Hankel Singular Values of the fourth order model one may find out that the stable part of the system may have little dynamical contribution. This suggests that a model with two unstable poles is enough for control design. Thus, balanced model truncation via square root may be used as a method to get a reduced order model. It is in the following form: $$G\left(s\right)=\frac{b_{1}s+b_0}{s^2-a_{1}s+a_0},$$

  
• Four parameters, b1, b0, a1 and a0, need to be estimated. If one controls the unstable system in (5) by a proportional controller with gain K _c0, the closed-loop transfer function from the set-point to the output will be $$\frac{y\left(s\right)}{y_s\left(s\right)}=\frac{K_{c0}\left(b_{1}s+b_0\right)}{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0006.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+\left(-a_1+K_{c0}{b}_1\right)s+\left(a_0+K_{c0}{\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="imgf0003.png,imgf0006.png"\; state="\{\{state\}\}">b</figure-callout>}}_0\right)}\mathrm{.}$$

  
• For the closed-loop stable system one may consider a transfer function. $$\frac{y\left(s\right)}{y_s\left(s\right)}=\frac{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="imgf0006.png"\; state="\{\{state\}\}">K</figure-callout>}}_2\left(1+{\tau }_{z}s\right)}{{\tau }^2{s}^2+2\mathit{\zeta \tau s}+1}$$

  
• Six data (Δy_p , Δy_u , Δy _∞, Δy_s , t_p , and Δt) may be observed from the closed-loop response (see Fig. 4) to estimate the four parameters (K ₂, τ _z , τ and ζ) in (7). Then, one may back-calculate to parameters of the open-loop unstable model in (5). Details of the respective calculations will be given later on.
• Fig. 4 shows a closed-loop step response diagram. In particular, Fig. 4 shows a pressure behaviour 400 in response to a change in a set-point. At a point in time 401 the set-point for the pressure is increased by Δy_s leading to a peak rise of the pressure of Δy_p in the time interval t_p . After a further time interval Δt the pressure decreases to a minimum pressure Δy_u representing an undershoot of the pressure. From that point in time the pressure is approaching a steady state value represented by Δy _∞.
• In the following an internal model control (IMC) design (Morari, M. and E. Zafiriou 1989; "Robust Process Control"; Englewood Cliffs, New Jersey, Prentice Hall) and in particular a determining of a suitable IMC configuration will be described.
• Fig. 5 shows a schematic block diagram of an internal model control system 500 comprising a low-pass filter 501 (f(s)) (for robustness of the closed-loop system), a model 502 (G(s)) describing the behaviour of a plant (e.g. an oil platform). The model typically will have a mismatch with the real plant 503 (plant, G_P (s)). Furthermore, the IMC system 500 comprises an element 504 (Q̃(s)) which denotes an inverse of the minimum phase part of G(s). In general, the shown IMC configuration cannot be used directly for unstable systems; instead the stabilizing controller may be given as the following: $$C=\frac{\tilde{Q}\left(s\right)f\left(s\right)}{1-G\left(s\right)\tilde{Q}\left(s\right)f\left(s\right)}$$

  

  
  wherein G(s)is a model of a plant or the pipeline-riser system, Q̃(s) is the inverse of the minimum phase part of G(s) and f(s) is a low-pass filter which may ensure stability and robustness of a closed-loop system. For internal stability, Q̃(s)f and 1-GQ̃f have to be stable. In particular, one may use the above identified model as the plant model: $$G\left(s\right)=\frac{{\hat{b}}_{1}s+{\hat{b}}_0}{s^2-{\hat{a}}_{1}s+{\hat{a}}_0}=\frac{\mathit{kʹ}\left(s+\phi \right)}{\left(s-{\pi }_1\right)\left(s-{\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\pi </figure-callout>}}_2\right)}$$

  

  
  and one gets $$\tilde{Q}\left(s\right)=\frac{\left(1/\mathit{kʹ}\right)\left(s-{\pi }_1\right)\left(s-{\pi }_2\right)}{s+\phi }$$

  
• The filter f(s) may be defined using:
• k = number of RHP poles + 1 = 3
• m = max (number of zeros of Q̃(s) - number of pol e of Q̃(s), 1) =
• l (this is for making Q=Q̃f proper) ;
• n = m + k -1 = 3; filter order.
• The filter is in the following form: $$f\left(s\right)=\frac{{\mathrm{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_2{s}^2+{\alpha }_{1}s+{\mathrm{<figure-callout\; id="0"\; label="\alpha "\; filenames="imgf0003.png,imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_0}{{\left(\lambda s+1\right)}^3},$$

  
• Where λ is an adjustable filter time-constant. One may choose α₀=1 to get an integral action in the controller, and the coefficient α₁ and α₂ are calculated by solving the following system of linear equations: $$\left(\begin{array}{ccc}{\pi }_1^2& {\pi }_1& 1\\ {\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\pi </figure-callout>}}_2^2& {\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\pi </figure-callout>}}_2& 1\end{array}\right)\left(\begin{array}{c}{\mathrm{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_2\\ {\alpha }_1\\ {\mathrm{<figure-callout\; id="0"\; label="\alpha "\; filenames="imgf0003.png,imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_0\end{array}\right)=\left(\begin{array}{c}{\left(\lambda {\pi }_1+1\right)}^3\\ {\left(\lambda {\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\pi </figure-callout>}}_2+1\right)}^3\end{array}\right)$$

  
• The feedback version of the IMC controller is as the following: $$C\left(s\right)=\frac{\left[\frac{1}{\mathit{kʹ}{\lambda }^3}\right]\left({\mathrm{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_2{s}^2+{\alpha }_{1}s+1\right)}{s\left(s+\phi \right)}$$

  
• In the following some PID tuning rules and their determination will be explained.
• The IMC controller in (13) is a second order transfer function which can be written in form of a PID controller. $$K_{\mathit{PID}}\left(s\right)=K_c+\frac{K_i}{s}+\frac{K_{d}s}{T_{f}s+1}$$

  

  
  where: $$T_f=1/\phi$$

  

  $$K_i=\frac{T_f}{\mathit{kʹ}{\lambda }^3}$$

  

  $$K_c=K_i{\alpha }_1-K_i{T}_f$$

  

  $$K_d=K_i{\alpha }_2-K_c{T}_f$$

  
• It is required to get K_c < 0 and K_d < 0, in order that the controller works in practice. Thus, λ may be chosen accordingly in order that these two conditions may be satisfies.
• In the following some PI tuning rules and their determining will be explained. For a PI controller in the following form $$K_{\mathit{PI}}\left(s\right)=K_c\left(1+\frac{1}{{\tau }_{I}s}\right),$$

  

  
  the tuning rules may be derived from the controller (13) as follows $$K_c=\underset{s\to \infty }{\lim }C\left(s\right)=\frac{{\mathrm{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_2}{\mathit{kʹ}{\lambda }^3}$$

  

  $${\tau }_I=\frac{K_c}{\underset{s\to 0}{\lim }\mathit{sC}\left(s\right)}={\mathrm{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_2\phi$$

  
• This means that the PI-controller approximates high-frequency and low-frequency asymptotes of C(s) in (13).
• In the following some explanations concerning a model for the static nonlinearity (cf. Fig. 3) will be given. In particular, a simple model for the static nonlinearity in Fig. 3 instead of using a full dynamical 4-state model may be used. The slope of the steady-state line of a bifurcation diagram of Fig. 2 may represent the static gain of the system which is related to valve properties. The valve equation is assumed as the following: $$w=C_{v}f\left(z\right)\sqrt{\rho \mathrm{\Delta }P}$$

  

  
  where w[kg/s] is the outlet mass flow and ΔP[N/m²] is the pressure drop. From the valve equation (22), the pressure drop over the valve for different valve openings may be written as $$\mathrm{\Delta }P=\frac{\bar{a}}{f{\left(\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0006.png"\; state="\{\{state\}\}">z</figure-callout>}\right)}^2},$$

  
• Where a may be assumed to be a constant parameter the calculation of which is described in more detail afterwards. A simple model for the inlet pressure may then as the following $$P_{\mathit{in}}=\frac{\bar{a}}{f{\left(\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0006.png"\; state="\{\{state\}\}">z</figure-callout>}\right)}^2}+{\bar{P}}_{\mathit{fo}}$$

  
• Where P _fo is another constant parameter that is the inlet pressure when the valve is fully open. By differentiating (24) with respect to z, one get static gain of the system as function of valve opening $$k\left(z\right)=\frac{-2\bar{a}\frac{\partial f\left(z\right)}{z}}{f{\left(z\right)}^3}$$

  

  
  which reduces for a linear valve (i.e. f(z) = z) to $$k\left(z\right)=\frac{-2\bar{a}}{z^3},$$

  

  
  where 0≤z≤1.
• The PID and PI tuning rules given above are based on a linear model identified at a certain operating point. However, the gain of the system may change drastically with the valve opening. Hence, a controller working at one operating point may not work at other operating points.
• One solution may be gain-scheduling with multiple controllers based on multiple identified modes. In this case simple PI tuning rules based on single step test, but with gain correction to counteract nonlinearity of the system may be used. For this, the static model given in equation (25) may be used. In the following a closed-loop step test using the data in Fig. 4 may be used to calculate $$\beta =\frac{-\ln \left(\frac{\mathrm{\Delta }{y}_{\infty }-\mathrm{\Delta }{y}_u}{\mathrm{\Delta }{y}_p-\mathrm{\Delta }{y}_{\infty }}\right)}{2\mathrm{\Delta }t}+\frac{K_{c0}\mathrm{<figure-callout\; id="0"\; label="k\; z"\; filenames="imgf0003.png,imgf0006.png"\; state="\{\{state\}\}">k</figure-callout>}\left(z_{}\right)\left({}_0\right){\left(\frac{\mathrm{\Delta }{y}_p-\mathrm{\Delta }{y}_{\infty }}{\mathrm{\Delta }{y}_{\infty }}\right)}^2}{4t_p},$$

  
• Where z ₀ is the average valve opening in test and K _c0 is the proportional gain used for the test. The PI tuning values as functions of valve opening are given as the following: $$K_c\left(z\right)=\frac{\beta T_{\mathit{osc}}}{k\left(z\right)\sqrt{z/z^{*}}}$$

  

  $${\tau }_I\left(z\right)=3T_{\mathit{osc}}\left(z/z^{*}\right)$$

  

  
  where T_osc is the period of slugging oscillations when the system is open-loop and z* is the critical valve opening of the system (at the bifurcation point).
• Fig. 6 shows some experimental results. In particular, Fig. 6A shows a closed-loop step test for an experimental set-up similar to the one shown in Fig. 4. In particular, the responses of a set-point increase for P_in of 2kPa is shown, wherein line 600 represents the data, line 601 represents the set-point, line 602 represents the filtered response to reduce the noise effect, while line 603 represents the identified closed-loop transfer function.
• The parameters for the experimental set-up are chosen to correspond to the case that the system switches to slugging flow at 15% of valve opening. Hence, the system is unstable at 20% valve opening. The control loop is closed by a proportional controller K_c0=-10, and the set-point is changed by 2kPa. Additionally a low-pass filter was used to reduce noisy effects. With the above described methods a closed-loop stable system (equation (7)) was determined to be: $$\frac{y\left(s\right)}{y_s\left(s\right)}=\frac{2.317s+0.8241}{19.91s^2+2.279s+1}$$

  
• In a next step from the corresponding identified closed-loop transfer function (see Fig. 6A) the open-loop unstable system G(s) (see equation (9)) is calculated to be: $$G\left(s\right)=\frac{-0.012s-0.0041}{s^2-0.0019s+0.0088}$$

  
• Afterwards λ is selected to be 10 for the IMC design to get the controller C(s) (see equations (8) and (13)) to be $$C\left(s\right)=\frac{-25.94\left(s^2+0.07s+0.0033\right)}{s\left(s+0.35\right)}$$

  
• The related PID tuning values (see equations (15) to (18)) are then K_p=-4.44, K_i=-0.24, K_d=-60.49, and T_f=2.81.
• Figs. 6B and Fig. 6C shows the result for an anti-slug control for this experimental set-up. While the described controller for the experimental set-up was tuned for 20% valve opening it can stabilize the system up to 32% valve opening which shows good margin of the controller. The system was also stable for up to an additionally added delay time of 3 seconds. A corresponding PI controller having tunig values K_c=-25.95 and τ_I=107.38 (see equations (20), (21)) did show similar results up to an additional added time delay of 2 seconds.
• In particular, Fig. 6B shows the time dependence of the inlet pressure for a valve opening Z of 20%. At 4 minutes the above described PID controller was switched on while at 16 minutes it was switched off again. One can clearly see that the oscillation of the input pressure substantially vanishes after the PID controller is switched on, while it increases again when the PID controller is switched off again.
• Fig. 6C shows the corresponding valve opening Z. Starting at 20% valve opening (which was the limit for avoiding unstable condition as described in the experimental set-up) the valve opening slowly increases during the time the PID controller is switched on while it decreases again to 20% when the PID controller is switched off again at 16 minutes.
• Thus, the experimental results clearly show that an oscillatory slug flow can be avoided even by an increased opening of the top-side valve used as the anti-slug valve. Therefore, the PID controller may be an efficient controller to reduce the risk of slugging while possibly increasing the flow rate due to an increased opening of the top-side valve.
• In the following some details concerning the model identification calculations are given.
• A stable closed-loop transfer function of $$\frac{y\left(s\right)}{y_s\left(s\right)}=\frac{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="imgf0006.png"\; state="\{\{state\}\}">K</figure-callout>}}_2\left(1+{\tau }_{z}s\right)}{{\tau }^2{s}^2+2\mathit{\zeta \tau s}+1}$$

  

  
  is assumed. The Laplace inverse (time-domain) of the transfer function (A.1) may be given as $$y\left(t\right)=\mathrm{\Delta }{y}_s{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="imgf0006.png"\; state="\{\{state\}\}">K</figure-callout>}}_2\left[1+D\exp \left(-\zeta t/\tau \right)\sin \left(\mathit{Et}+\varphi \right)\right],$$

  

  
  where $$D=\frac{{\left[1-\frac{2\zeta {\tau }_z}{\tau }+{\left(\frac{{\tau }_{\mathrm{<figure-callout\; id="2"\; label="z\; \tau "\; filenames="imgf0006.png"\; state="\{\{state\}\}">z</figure-callout>}}}{\tau }\right)}^2\right]}^{\frac{1}{2}}}{\sqrt{1-{\zeta }^2}}$$

  

  $$E=\frac{\sqrt{1-{\mathrm{<figure-callout\; id="2"\; label="\zeta "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\zeta </figure-callout>}}^2}}{\tau }$$

  

  $$\varphi ={\tan }^{-1}\left[\frac{\tau \sqrt{1-{\mathrm{<figure-callout\; id="2"\; label="\zeta "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\zeta </figure-callout>}}^2}}{\mathit{\zeta \tau }-{\tau }_z}\right]$$

  
• By differentiating (A.2) with respect to time and setting the derivative equation to zero, one gets time of the first peak: $$t_p=\frac{{\tan }^{-1}\left(\frac{1-{\mathrm{<figure-callout\; id="2"\; label="\zeta "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\zeta </figure-callout>}}^2}{\zeta }\right)+\pi -\varphi }{\sqrt{1-{\mathrm{<figure-callout\; id="2"\; label="\zeta "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\zeta </figure-callout>}}^2}/\tau }$$

  

  
  and the time between the first peak (overshoot) and the undershoot becomes: $$t_u=\mathit{\pi \tau }/\sqrt{1-{\mathrm{<figure-callout\; id="2"\; label="\zeta "\; filenames="imgf0006.png"\; state="\{\{state\}\}">\zeta </figure-callout>}}^2}$$

  
• The corresponding damping ratio

  

  can be estimated as $$\hat{\zeta }=\frac{-\mathrm{<figure-callout\; id="2"\; label="ln\; v\; \pi "\; filenames="imgf0006.png"\; state="\{\{state\}\}">ln</figure-callout>}v}{\sqrt{{\pi }^{}}}$$

  

  
  where $$v=\frac{\mathrm{\Delta }{y}_{\infty }-\mathrm{\Delta }{y}_u}{\mathrm{\Delta }{y}_p-\mathrm{\Delta }{y}_{\infty }}$$

  
• Then using equation (A.7) one get $$\hat{\tau }=\frac{t_u\sqrt{1-{\hat{\mathrm{<figure-callout\; id="2"\; label="\zeta \; ^"\; filenames="imgf0006.png"\; state="\{\{state\}\}">\zeta </figure-callout>}}}^2}}{\pi }\mathrm{.}$$

  
• The steady-state gain of the closed-loop system can be estimated to be: $${\hat{K}}_2=\frac{\mathrm{\Delta }{y}_{\infty }}{\mathrm{\Delta }{y}_s}\mathrm{.}$$

  
• From the time of the peak t_p and (A.6) an estimate of Φ can be derived: $$\hat{\varphi }={\tan }^{-1}\left[\frac{1-{\hat{\mathrm{<figure-callout\; id="2"\; label="\zeta \; ^"\; filenames="imgf0006.png"\; state="\{\{state\}\}">\zeta </figure-callout>}}}^2}{\hat{\zeta }}\right]-\frac{t_p\sqrt{1-{\hat{\mathrm{<figure-callout\; id="2"\; label="\zeta \; ^"\; filenames="imgf0006.png"\; state="\{\{state\}\}">\zeta </figure-callout>}}}^2}}{\hat{\tau }}$$

  
• From (A.4) one can get $$\hat{E}=\frac{\sqrt{1-{\hat{\mathrm{<figure-callout\; id="2"\; label="\zeta \; ^"\; filenames="imgf0006.png"\; state="\{\{state\}\}">\zeta </figure-callout>}}}^2}}{\hat{\tau }}$$

  

  
  while the overshoot is defined by: $${\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="imgf0003.png,imgf0006.png"\; state="\{\{state\}\}">D</figure-callout>}}_0=\frac{\mathrm{\Delta }{y}_p-\mathrm{\Delta }{y}_{\infty }}{\mathrm{\Delta }{y}_{\infty }}\mathrm{.}$$

  
• By evaluating (A.2) at time of peak t_p one gets $$\mathrm{\Delta }{y}_p=\mathrm{\Delta }{y}_s{\hat{K}}_2\left[1+\hat{D}\exp \left(-\hat{\zeta }{t}_p/\hat{\tau }\right)\sin \left(\hat{E}{t}_p+\hat{\varphi }\right)\right]$$

  
• Combining equation (A.11), (A.14) and (A.15) gives $$\hat{D}=\frac{{\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="imgf0003.png,imgf0006.png"\; state="\{\{state\}\}">D</figure-callout>}}_0}{\exp \left(-\hat{\zeta }{t}_p/\hat{\tau }\right)\sin \left(\hat{E}{t}_p+\hat{\varphi }\right)}\mathrm{.}$$

  
• The last parameter can be estimated by solving (A.3): $${\hat{\tau }}_z=\hat{\xi }\hat{\tau }+\sqrt{{\hat{\xi }}^2{\hat{\tau }}^2-{\hat{\tau }}^2\left[1-{\hat{D}}^2\left(1-{\hat{\mathrm{<figure-callout\; id="2"\; label="\zeta \; ^"\; filenames="imgf0006.png"\; state="\{\{state\}\}">\zeta </figure-callout>}}}^2\right)\right]}$$

  
• Then, one can back-calculate to parameters of the open-loop unstable model. Steady-state gain of the open-loop model may be given by: $${\hat{K}}_p=\frac{\mathrm{\Delta }{y}_{\infty }}{K_{c0}\left|\mathrm{\Delta }{y}_s-\mathrm{\Delta }{y}_{\infty }\right|}$$

  

  
  while an estimation of the four model parameters may be given by: $${\hat{a}}_0=\frac{1}{{\hat{\tau }}^2\left(1+K_{c0}{\hat{K}}_p\right)}$$

  

  $${\hat{b}}_0={\hat{K}}_p{\hat{a}}_0$$

  

  $${\hat{b}}_1=\frac{{\hat{K}}_2{\hat{\tau }}_z}{K_{c0}{\hat{\tau }}^2}$$

  

  $${\hat{a}}_1=-2\hat{\zeta }/\hat{\tau }+K_{c0}{\hat{b}}_1,$$

  
• It should be noted that one has to have â ₁>0 to have an unstable system.
• In the following some details concerning calculations of static nonlinearity parameters are given.
• In particular, the value of the constant a of equation (23) can be calculated by $$\bar{a}=\frac{1}{\bar{\rho }}{\left(\frac{\bar{w}}{{\mathrm{<figure-callout\; id="2"\; label="C\; v"\; filenames="imgf0006.png"\; state="\{\{state\}\}">C</figure-callout>}}_v}\right)}^2$$

  

  
  where C_v is the known valve constant, w is the average outlet flow rate and ρ is the average mixture density. The average outlet mass flow is approximated by constant inflow rates: $$\bar{\mathit{w}}={\mathit{w}}_{\mathit{g},\mathit{in}}+{\mathit{w}}_{\mathit{l},\mathit{in}}$$

  
• In order to estimate the average mixture density ρ, one can perform the following calculations:
• Average gas mass fraction: $$\bar{\alpha }=\frac{{\mathit{w}}_{\mathit{g},\mathit{in}}}{{\mathit{w}}_{\mathit{g},\mathit{in}}+{\mathit{w}}_{\mathit{l},\mathit{in}}}$$

  
• Average gas density at top of the riser from ideal gas law: $${\bar{\rho }}_g=\frac{\left(P_s+\mathrm{\Delta }{P}_{v,\min }\right)M_g}{\mathit{RT}}$$

  

  
  where P_s is the constant separator pressure, and P _v,min is the minimum pressure drop across the valve that exists even with fully open valve. In the numerical simulations P _v,min is zero but in the experimental set-up 2 kPa was chosen.
• The liquid volume fraction: $${\bar{\alpha }}_l=\frac{\left(1-\bar{\alpha }\right){\bar{\rho }}_g}{\left(1-\bar{\alpha }\right){\bar{\rho }}_g+\bar{\alpha }{\rho }_L}\mathrm{.}$$

  
• Average mixture density: $$\bar{\rho }={\bar{\alpha }}_l{\rho }_l+\left(1-{\bar{\alpha }}_l\right){\bar{\rho }}_g$$

  
• In order to calculate the constant parameters P _fo in the static model, one can use the fact that if the inlet pressure is large enough to overcome a riser full of liquid, slugging will not happen. The corresponding pressure can be defined as $$P_{\mathit{in}}^{*}={\rho }_{L}g{L}_r+P_s+\mathrm{\Delta }{P}_{v,\min }$$

  
• This pressure is associated with the critical valve opening at the bifurcation point z*. As a result, one gets P _fo as the following: $${\bar{\mathit{P}}}_{\mathit{fo}}={\mathit{P}}_{\mathit{in}}^{*}-\frac{\bar{a}}{f{\left(z^{*}\right)}^2}$$

  
• It should be noted that the term "comprising" does not exclude other elements or steps and "a" or "an" does not exclude a plurality. Also elements described in association with different embodiments may be combined. It should also be noted that reference signs in the claims should not be construed as limiting the scope of the claims.

Claims (11)

1. A method of operating a pipeline-riser system (100) comprising an anti-slug valve (111), wherein the method comprises:

   controlling the anti-slug valve (111) by a control signal generated based on a measured signal indicative for a flow of a fluid through the pipeline-riser system (100),

   wherein the control signal comprises a signal component relating to a linear time-invariant system G(s) (302), wherein the linear-time invariant system is modeled according to $$G\left(s\right)=\frac{b_{1}s+b_0}{s^2-a_{1}s+a_0}\mathrm{.}$$

   
2. The method according to claim 1,
   wherein the control signal comprises a further signal component relating to a static nonlinearity.
3. The method according to claim 1 or 2, further comprising:

   controlling a through-flow through the anti-slug valve (111) according to the control signal.
4. The method according to any one of the claim 1 to 3, wherein for generating the control signal an internal model control function is used.
5. The method according to claim 4,
   wherein the internal model control function is given by $$C\left(s\right)=\frac{\left[\frac{1}{{\mathit{kʹ\lambda }}^3}\right]\left({\alpha }_2{s}^2+{\alpha }_{1}s+1\right)}{s\left(s+\phi \right)},$$

   

   wherein k' is the gain of the pipeline-riser system G(s), λ is an adjustable filter time constant, α₁ and α₂ represent coefficients and ϕ represents the zero dynamics of the pipeline-riser system.
6. The method according to any one of the claims 1 to 5, wherein the controlling of the anti-slug valve (111) is performed by a PID or PI controller.
7. The method according to claim 6, wherein for the PID controller the tunig rule is given by: $$K_{\mathit{PID}}\left(s\right)=K_c+\frac{K_i}{s}+\frac{K_{d}s}{T_{f}s+1},$$

   

   
   wherein T_f =1/ϕ; $K_i=\frac{T_f}{\mathit{kʹ}{\lambda }^3};$

   

   K_c =K_iα ₁ -K_iT_f ; and K_d=K_iα ₂ -K_cT_f, wherein K_c <0 and K_d <0.
8. The method according to claim 6, wherein for the PI controller the tuning rule is given by: $$K_{\mathit{PI}}\left(s\right)=K_C\left(1+\frac{1}{{\tau }_{I}s}\right),$$

   

   
   wherein $K_C=\frac{{\alpha }_2}{\mathit{kʹ}{\lambda }^3};$

   

   and τ ₁ =α₂ϕ.
9. A pipeline-riser system (100), comprising:

   a pipeline portion (108),

   a riser portion (110),

   an anti-slug valve (111), and

   a control unit,

   wherein the pipeline portion (108) and the riser portion (110) are joined at a joining point (113),

   wherein the anti-slug valve (111) is arranged in the pipeline-riser system (100) in such a way that a flow rate of fluid flowing through the pipeline-riser system (100) is controllable, and

   wherein the control unit is adapted to generate a control signal for controlling the anti-slug valve (111) according to a method according to any one of the claims 1 to 8.
10. A program element, which, when being executed by a processor, is adapted to control or carry out a method according to any one of the claims 1 to 8.
11. A computer-readable medium, in which a computer program is stored which, when being executed by a processor, is adapted to control or carry out a method according to any one of the claims 1 to 8.

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