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Predictive control and suppression of pressure surges in main oil pipelines with counter-running pressure waves
Accepted Manuscript Predictive control and suppression of pressure surges in main oil pipelines with counter-running pressure waves Victor Yuzhanin, Vladimir Popadko, Taras Koturbash, Valeria Chernova, Roman Barashkin PII:
S0308-0161(18)30512-X
DOI:
https://doi.org/10.1016/j.ijpvp.2019.03.015
Reference:
IPVP 3865
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 30 November 2018 Revised Date:
2 March 2019
Accepted Date: 10 March 2019
Please cite this article as: Yuzhanin V, Popadko V, Koturbash T, Chernova V, Barashkin R, Predictive control and suppression of pressure surges in main oil pipelines with counter-running pressure waves, International Journal of Pressure Vessels and Piping (2019), doi: https://doi.org/10.1016/ j.ijpvp.2019.03.015. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Predictive control and suppression of pressure surges in main oil pipelines with counter-running pressure waves
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Authors
Victor Yuzhanina, Vladimir Popadkoa, Taras Koturbashb,*, Valeria Chernovaa, Roman Barashkina Department of Automation of Technological Processes, Gubkin Russian State University of Oil and Gas (National Research University), Leninsky Ave 65, Building 1 (510), 119991, Moscow, Russian Federation b Department of Energy Technology, KTH Royal Institute of Technology, Brinellvägen 68, 10044 Stockholm, Sweden
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Abstract
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This paper addresses pressure control in crude oil pipelines in onshore transportation systems. Maintaining pressure below the maximum rated load capacity in a linear section of an oil pipeline is considered as a control problem. We present a method for predicting of maximum allowed pressure excess using the method of characteristics and the first principles model of nonstationary flow of a slightly compressible fluid in a pipeline. The algorithm, based on the developed method, can suggest the automatic control action that will allow dampening potentially dangerous pressure surges in the linear section of pipeline utilizing counter running pressure wave. The simulations results show that the algorithm allows efficiently predict and suggest control action for suppressing the back-propagating pressure waves in oil pipelines caused by various technological operations. The suggested control action could be implemented without any modification of existing technological equipment at main oil pipeline by utilizing the available capabilities of upstream oil pumping stations.
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Pressure excess, oil pipeline, slightly compressible fluid, method of characteristics, counter-running pressure wave, back-propagating pressure wave.
1. Introduction
Pipeline transport is the most common means of transporting various fluids over long distances. The length of pipeline systems varies from hundreds of kilometers for water supply [1] to thousands of kilometers for oil and gas transportation [2]. The latter are considered as long-distance pipeline systems and have much higher requirements regarding reliability and safety in the view of severe environmental and economic consequences of potential failures or down time. Assessment and mitigation of potential risks and unexpected emergencies in long pipeline systems are of major concern for system operators. One of the primary risks in a long-distance pipeline system that transports slightly compressible fluids (water or gas-free oil that have small but constant compressibility) comes from various transient * Corresponding author. Tel.: +46 762915506; E-mail address: [email protected] (T.Koturbash)
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processes. Such transient processes usually have complicated dynamics with wave-like behavior and high speed (about 1000 m/s and higher) waterhammer propagation (hydraulic shock). Waterhammer appears when the fluid flow changes speed rapidly, followed by a change in fluid pressure. Waterhammer is usually caused by various technological operations, including fast valve shutoff and starting or stopping of pumps. In addition, it could be caused by more specific reasons such as pipeline pigging [3]. Waterhammer can increase the fluid pressure above the maximum rated (or allowed) level for the pipeline system. A pressure increase can damage the pipeline and even lead to rupture. The risk of rupture caused by waterhammers is considered one of the most significant among all risks in pipeline systems that are used for transporting slightly compressible fluids [1, 4]. The risk of pipeline rupture is considered during the design phase, but it is very difficult to foresee and analyze all potential transient processes that could occur during operation. Specific scenarios in pipeline systems were analyzed and the relevant transient processes were simulated in previous publications [1, 4-7]. Various recommendations for redesign of the pipeline system are usually proposed based on simulation results. Those recommendations usually include the following: pipeline strength enhancement or installation of special protection devices like safety valves, or anti-purge valves. However, this is too expensive in most cases, or it may be impossible to implement such recommendations in long-distance pipeline systems due to design and reconstruction complexity or high associated costs. Another approach to mitigate risks related to waterhammer formation is to develop and employ control algorithms. A control algorithm can be used to ensure system safety and considerably reduce the dangerous effects of waterhammer. Development and application of control algorithms imply knowledge or predictability of the behavior of the system. The temporal behavior of slightly compressible fluids (including waterhammers) is expressed using hyperbolic or parabolic partial differential equations (PDE). There are several variations of these equations that differ in the selection of factors taken into consideration, including flow pattern (laminar or turbulent), non-isothermality, rheological properties of transported fluid, and pipeline elevation variations. [8, 9]. It is a very complicated task to determine the analytical solution of waterhammer equations in a generic case, thus numerical methods are usually used. We used the method of characteristics to solve the PDE governing the behavior of a waterhammer [8-11]. This allowed us to consider nonlinear friction, exclude smoothing of the pressure wavefront using the Courant–Friedrichs–Lewy condition, and obtain high computational efficiency that provides practical applicability. The stabilization of hyperbolic PDEs is widely studied in the literature [12-14], in particular with applications to open channels [13, 15-17] and pumped flow (waterhammer equations) [18Error! Reference source not found.-20]. In general, these studies consider control in the classic formulation of control theory: an acceptable control action takes the object to the desired operating conditions. The main difference of the proposed approach is that the desired operating condition is undetermined. In contrast, only unacceptable solutions are defined, predicted, and avoided.
2. Methodology
2.1 Problem formulation A main oil pipeline (MOP) is a complex processing facility (fig 1). A typical MOP consists of process sections (PS) that are hydraulically separated by oil storage tanks. A PS consists of linear pipeline sections (LS) separated by oil pumping stations (PPS). Within a single PS, the fluid is pumped in the socalled “pump-to-pump” operating mode. The transitional wave process is initiated at a single point in this mode, and a wave propagates across the entire PS at the speed of sound in oil (about one kilometer per second) [21]. In most cases, such a transitional wave is caused by routine operations like start-up, shutdown, pressure testing or even emergency shutdown of PPS. As a result, pressure waves propagate downstream and upstream inside the pipeline.
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Figure 1. Schematic diagram of a typical main oil pipeline.
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A typical local control system for automatic pressure stabilization is usually placed on the outlet of a PPS; this system can eliminate or considerably damp dangerous pressure surges propagating downstream in LS [21]. This is accomplished by defining a pressure set point that allows fluid pressure regulation in the beginning of LS. Thus, the pressure does not exceed dangerous levels along the LS up to the next PPS. A control response to pressure surges propagating upstream from the end of an LS is initiated either through the control valve at the output of the previous PPS or by actuating the emergency shutdown system for the LS or the entire PS. In the first case, a potentially dangerous and uncontrolled pressure wave will run throughout the LS [9, 21]. In the second case, the response occurs only after the emergency pressure set point is exceeded. If the excess pressure is momentary and is not dangerous, the entire PS could be improperly shut down. Therefore, it is difficult in both cases to ensure protection of the entire LS and MOP sections against emergency pressure excess using only typical feedback control systems. The goal of this study was to develop a predictive control system capable of completely preventing overpressure in the LS of an MOP line without unnecessary shutdowns. The overall objective was divided into following tasks: 1. Short-term forecast of pressure waves that exceed the maximum rated pressure in an LS by backpropagating a pressure wave based on real-time sensor readings. 2. Predictive compensation of an upstream pressure wave using a counter-propagating pressure wave generated in a presiding PPS.
2.2 2.2 ShortShort-term forecast of fluid pressure in an LS The control objective is the pressure in an LS located between two PPS. The control criterion is as follows: ( )
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( , )≤
(1)
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where is the coordinate along the pipeline axis, ( , ) is the pressure profile along the hydraulic axis ( ) is a maximum rated pressure profile that depends on the pipeline stress limit in an LS a time , along the hydraulic axis. ( ) has a complex, uneven shape, especially in an old LS that may have The function undergone multiple repair services or installation of new sections, which can differ from neighboring sections considerably in terms of their stress limits. A control response is the setpoint value of the local pressure control system at the PPS output, thus ensuring the required boundary condition (0, ) at the beginning of LS. The pressure ( , ) at the end of the LS is considered as a measurable disturbance. To describe the behavior of oil in the pipeline, we will use the equation for unsteady flow of a slightly compressible fluid in a long pipeline [9]. The continuity equation is as follows: +
∙
=0
(2)
where is the density of the fluid and is the velocity of sound in the fluid. The equation of motion is as follows: + ∙
+ ( )∙
∙| |
+
( )
=0
(3)
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+ %(!) ∙
where
$
+ &(!) = 0
0 %(!) = ' 1/
0
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*
. ( ) .
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·| | 0 &(!) = ' * ɸ(!) = ( ) · + ɸ(!) 2.
The method of characteristics for system integration could be applied to (4) numerically. Multiplying Error! Reference source not found. by the left eigenvector /0 (!) of matrix %(!) while considering the relationship between the eigenvectors and eigenvalues /0 %(!) = 10 ∙ /0 yields the following: $ $
+ 10 ∙ /0
$
+ /0 ∙ &(!) = 0
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/0
/0 2 3 + /0 ∙ &(!) = 0, 6 = 1,2 45
$
$
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$
where 2 3 =
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is the ! gradient along lines defined by
(5)
= 10 , which are known as system
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characteristics (4). The eigenvalues 10 of the %(!) matrix are equal in absolute value but have different signs. The absolute value is equal to the velocity of sound in the fluid (10 = ± ). Therefore, replacing the gradient in (5) with a difference approximation yields a set of equations for numerical integration of equations (2) and (3) as a uniform grid (see
Figure ):
/ !89
/ !89
,: ,:
= / 2!8,:; − ∆ ∙ &>!8,:; ?3
= / 2!8,:9 − ∆ ∙ &>!8,:9 ?3
(6) (7)
where @ is the number of grid points and A = 1, … , @. For a limiting grid points, one of either (6) or (7) shall be replaced by the boundary condition equation.
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Figure 2. Computational grid of the method of characteristics
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If the initial conditions in the j-th layer (moment) are defined, then it is possible to calculate all points in the (j+1)-th layer except for the boundary points. It will be possible to calculate n – 2 points of (j+1)-th layer. There are n – 4 calculable points in the (j+2)-th layer, n – 6 points in the (j+3)-th layer, and so on. Figure shows the computational grid points (green dots) that could be calculated based on the initial conditions in the j-th layer. These points will be located inside a triangle or trapezoidal pattern in the time-distance plane, depending on the parity of the number of initial points in the j-th layer. Using (6) and (7) for each point in the computational grid enables short-term forecasting of the pressure wave, ( ) thus one can predict whether the pressure will exceed the maximum rated pressure ( , ) > in the LS from knowledge of the initial conditions in the j-th layer.
Figure 3. Computational grid of the method of characteristic for short-term forecasting of pipeline values.
2.3 2.3 Predictive pressure wave compensation In order to control the boundary conditions in the pipeline, they must be set at each time step. Figure 3 shows that boundary conditions at the (j+1)-th layer will affect only a portion of the forecast area points that lie on the characteristic lines passing through the boundary points in the layer (i.e., segments A1A4 and C1C4, or A1An and C1Cn in general). The control algorithm will not be able to affect
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points that lie above or below these characteristic lines. Therefore, to solve the control problem, it is enough to select the boundary conditions to prevent emergency pressure excess along segments A1An and C1Cn in each layer. Consider the supporting inverse problem formulated as follows: the desired pressure value D(%: ) is set at any point %: , where A ∈ {2,3, … , @}. The task is to identify the pressure D(% ) that will produce the desired pressure D(%: ) at the selected point%: . Consider the example case of point % in Fig. 3. This point could be considered as the boundary of the pipeline sub-segment, beginning at . Then the set pressure value D(% ) could be interpreted as a boundary condition for this subsection in layer I + 2. By solving equation Error! Reference source not found. on J − % characteristics for a certain pressure value D(% ), we will obtain the velocity #(% ). The flow parameters at % simultaneously satisfy equations Error! Reference source not found. and Error! Reference source not found. for characteristics J − % and % − % . Given that the flow parameters at points J , % are known, the only unknown parameters in the above equations are the flow parameters at % . We obtain the pressure and velocity values at % by solving both equations. Binding the pressure value at the subsequent point %K will set the pressure and velocity at point % to its only possible value. If this inference is repeated for subsequent points, we may conclude that we can calculate the boundary condition D(% |%: ) for a given required pressure at any point D(%: ), A ∈ {2,3, … , @}, which will satisfy the required pressure. Let us consider the inverse problem of identifying the maximum boundary condition D∗ (% ) that will ensure the pressure at each point % , % , … , %M does not exceed the maximum rated value for the pipeline. In practical terms, this means that there is no need to apply control action if the actual pressure is less than D∗ (% ). Let us alternatively require that the pressure would be equal to the maximum rated pressure value at each point % , … , %M . We can then obtain the appropriate boundary conditions at % : D(% |% ), D(% |% ), … , D(% |%M )
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To retrieve the required maximum pressure D∗ (% ), it is sufficient to select the minimum among the received boundary conditions:
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D∗ (% ) = min{D(% |% ), D(% |% ), … D(% |%M )} (9) Received at each calculation step boundary condition D∗ (% ) is transmitted as the set point to the pressure controller at the output of the upstream PPS of the pipeline section. As a result, the proposed algorithm can control the pressure of the left side of the LS at the points D∗ (% ) … D∗ (%M ). Due to the finite propagation speed of the pressure waves in the fluid the control action engaged at the left side of LS will not be able to compensate the disturbance on the right side of LS. Therefore, it is impossible to control the right side of the LS using the counter-propagating pressure waves excited at upstream PPS [22]. However, the pressure of right side of LS can still be calculated and controlled with proposed algorithm by using the right boundary condition D(Q ). The calculation of maximum pressure at D∗ (Q ) that will not exceed the maximum rated pressure profile at points D∗ (Q ) … D ∗ (QM ) is in same manner as it is shown for D∗ (% ). Typically, available technical facilities of MOP (Fig. 1) usually cannot arbitrarily and fully control the pressure at the right side of the LS. In order to control the right side of the LS, the MOP equipment should be modified accordingly. For example, a relief valve and an underground storage tank should be installed on the right side of the LS in order to release excess fluid pressure. In the following numerical simulation we have considered the control scheme where the control action is the pressure on the left side of LS (%0 : 0 ∈ (0, /2)) and the pressure on the right side of LS is the measurable process disturbance. The back propagating pressure wave in left side of LS possesses the most danger with regard to exceeding maximum allowed pressure of LS as shown in simulations below.
3. Results and Discussion The validity and performance of the proposed predictive control algorithm in an MOP section was demonstrated using numerical modeling in combination with a simulator of main oil pipeline. The
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addressed PS shown in Fig. 4 corresponds to a normal operation mode with a reference hydraulic condition. The PS consists of three LS spanning from 540 to 810 km. These LS are separated by PPSA at 580 km and PPSB 690 km. Each LS consists of pipeline sections with uneven elevation and varied load capacities, that general reflects the effects of burial depth, exposures, pipeline design parameters (such as wall thickness, pipe type, known corrosion or damage defects). There is also a specific pipe section with considerably reduced load capacity (PSRLC) at 600 km, which poses a substantial risk of a damage in case of pressure surge. In order to simplify the representation of the hydrodynamic process of fluid transport in the addressed PS the fluid pressure profile and the pipeline load capacity are expressed in units of total pressure head as follows: D( ) + ( ) S( ) =
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where D( ) is the fluid pressure, D ( ) is the pipeline load capacity in fluid pressure units, is the fluid density, is the acceleration due to gravity, and ( ) is the elevation of the pipeline. Such representation shows that fluid pressure and pipeline load capacity can be defined in units of fluid column height from sub sea level with consideration of i.e., hydrostatic pressure. This simplifies analysis of the hydrodynamic processes in the MOP. The maximum rated fluid pressure in an LS depends on the load capacity and elevation of the pipeline. Thus, the rated pressure profile has an uneven and complex form. Figure 4 shows the load capacity S ( ) of the fluid inside the LS before reaching the maximum rated pressure D ( ). The PSRLC region has a load capacity that is considerably lower compared to that of the adjacent sections. Rupture becomes inevitable if the fluid pressure in this region reaches or exceeds the actual load capacity S ( ).
Figure 4. Reference hydraulic conditions in a PS during normal operation; S( ): total head pressure (m); STU ( ): pipeline load capacity (m); ( ): elevation of a pipeline cross-section (m)
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Emergency shutdown of PPSB was included in the simulation. Such an emergency usually results in pressure equalization between the PPSB inlet and outlet as shown in Fig. 5a. The resulting pressure waves propagate downstream and upstream from PPSB, as shown in Fig. 5b. The actual fluid pressure at each time point throughout the upstream and downstream LS was calculated with the developed shortterm forecasting algorithm. The pressure wave propagating downstream does not pose any danger since it is a decreasing pressure wave (in the event of PPSB shutdown) or it could be controlled at the PPSB output with a local control system. On the contrary, the back-propagating pressure wave poses serious danger in the LS since it is an increasing pressure wave propagating upstream and cannot be controlled with the downstream PPSB (where it occurred). The upstream PPSA will be able to react to an incoming wave only when the wave reaches its output; this is where pressure sensors for local control system are usually placed. In most cases, the reaction of the local control system placed at the output of an upstream PPSA will be too late, and the pressure surge might exceed the remaining load capacity of the LS. As could be observed from Fig. 6, the upstream propagating pressure wave will reach higher total head pressure as it comes closer to the beginning of the LS. This becomes extremely dangerous in LS regions with reduced load capacity, e.g., due to pipeline elevation variations or reduced load capacity in the pipe section.
Figure 5. Development of downstream and upstream pressure waves.
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The current LS strength redundancy decreases as the pressure increases towards the beginning of LS, as shown in Fig. 6. Local control systems will not be able to react in time, and the LS load capacity is exceeded when the back-propagating pressure wave approaches the PSRLC region (i.e., S( ) approaches S ( )). During simulation the fluid exceeded the maximum rated pressure by 15 m of total head pressure, which is equal 120 kPa of overpressure; this would most certainly result in rupture. While applying the proposed algorithm, we could foresee the possibility that the pressure would exceed the maximum rating in the PSRLC region. We can also recalculate the required counterpropagating upstream pressure wave that can be used to neutralize the back-propagating pressure wave before it reaches the PSRLC region in the LS. The correct timing for initiating the counterpropagating pressure wave from the upstream PPS was also obtained using the proposed algorithm. The resulting backpressure and counter-propagating wave passing the PSRLC region area was forecasted with the algorithm in advance. The pressure in the LS will be within the required limits, as shown in Fig 6. The simulation results show that the proposed method can be used to forecast pressure waves and avoid any negative effects from back-propagating pressure waves by creating a controllable counterpropagating pressure wave.
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Figure 6. Upstream pressure surge propagation while operating local automatic control systems (top diagrams) and the developed predictive control algorithm (bottom diagrams). (1a), (2a) Pressure surge approaching the RLC region, (1b), (2b) pressure surge while passing the PSRLC region, and (1c), (3c) pressure surge approaching the upstream PPS.
4. Conclusions
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An effective computational method for control and suppression of back-propagating pressure surges in MOPs with counter-running pressure suppression waves was presented in this paper. The distinctive feature of the proposed control algorithm is that it implements on-the-fly short-term forecasting of the total pressure in the pipeline and activates the control action only if there is a possibility of exceeding the rated load capacity in an LS. The control goal was reduced to a constraint satisfaction problem (CSP) for the maximum pipeline pressure. The CSP was transformed into a solvable inverse problem, where the boundary conditions of the hyperbolic waterhammer PDE were determined using the method of characteristics. Simulation results show that the developed algorithm allows forecasting of back-propagating pressure waves that may exceed the rated load capacity in the LS. The required control action can be subsequently identified and implemented with the upstream PPS. The algorithm allows to forecast the effects of suggested control action for further evaluation and selection of the desired control criteria. Application of the developed algorithm will allow to improve the reliability and safety of existing MOPs in onshore transportation systems without any modification or installation of additional technological equipment. This is achieved by efficient utilization of the existing technological capabilities of MOPs (the upstream PPS) to generate counter-running running pressure waves of reduced pressure. These counter running pressure wavescan efficient damp the the back-propagating pressure surges in oil pipelines caused by various technological operations.
Acknowledgement The authors want to thank prof. Mikhail V. Lurie and prof. Sergei A. Sardanashvili from Gubkin Russian State University of Oil and Gas for their comments and fruitful discussions on the presented research topic. Furthermore, the authors would like to acknowledge the support of JSC Transneft for providing valuable inputs into this research work.
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References
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1. C. Schmitt, G. Pluvinage, E. Hadj-Taieb, R. Akid. Water pipeline failure due to water hammer effects. Fatigue and Fracture of Engineering Materials and Structures, 29(12), (2006) 1075–1082 2. Allegro Energy Group. (2001). How pipelines make the oil market work – Their networks, operation and regulation 3. Ali A. Mazreah, Firas B.Ismail Alnaimi, K.S.M. Sahari Novel Design for PIG to Eliminate the Effect of Hydraulic Transients in Oil and Gas Pipelines, Volume 156, 2017, Pages 250-257, ISSN 0920-4105 4. R. Wanga, *, Z. Wangb, X. Wanga, H. Yanga, J. Sun Water hammer assessment techniques for water distribution systems, Procedia Engineering, Volume 70, 2014, Pages 1717-1725, ISSN 1877-7058, 5. Tangtao Feng, Dalin Zhang, Ping Song, Wenxi Tian, Wei Li, G.H. Su, Suizheng Qiu Numerical research on water hammer phenomenon of parallel pump-valve system by coupling FLUENT with RELAP5, Annals of Nuclear Energy, Volume 109, 2017, Pages 318-326, ISSN 0306-4549 6. W. X. Tian, G. Su, G. P. Wang, S. Z. Qiu, Z. J. Xiao, Numerical simulation and optimization on valveinduced water hammer characteristics for parallel pump feedwater system. Annals of Nuclear Energy, 35(12), (2008) 2280–2287 7. J. Zhanga, J. Gaoa*, M. Diaoa, W. Wub, T. Wanga, S. Qia A case study on risk assessment of long distance water supply system, Procedia Engineering, Volume 70, 2014, Pages 1762-1771, ISSN 18777058. 8. Mohamed S. Ghidaoui, Ming Zhao, Duncan A. McInnis and David H. Axworthy A Review of Water Hammer Theory and Practice, Ghidaoui MS, Zhao M, McInnis DA, Axworthy DH. A Review of Water Hammer Theory and Practice. ASME. Appl. Mech. Rev. 2005;58(1):49-76. 9. M. V. Lurie, Modeling of Oil Product and Gas Pipeline Transportation. WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (2005), 234 pages, SBN: 978-3-527-40833-7. 10. M.H. Afshar, M. Rohani, Water hammer simulation by implicit method of characteristic, International Journal of Pressure Vessels and Piping, Volume 85, Issue 12, 2008, Pages 851-859, ISSN 0308-0161. 11. A.R. Lohrasbi and R. Attarnejadб Water Hammer Analysis by Characteristic Method, American J. of Engineering and Applied Sciences 1 (4): 287-294, 2008 ISSN 1941-7020. 12. Ilyasse Aksikas,∗, Joseph J. Winkin, Denis Dochain, Asymptotic stability of infinite-dimensional semilinear systems: Application to a nonisothermal reactor, Systems & Control Letters, Volume 56, Issue 2, 2007, Pages 122-132, ISSN 0167-6911. 13. JM Coron, R Vazquez, M Krstic, G Bastin Local exponential H2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping, SIAM J. Control Optim., 2013, 51(3), 2005–2035. 14. Rafael Vazquez, Miroslav Krstic and Jean-Michel Coron Backstepping boundary stabilization and state estimation of a 2 × 2 linear hyperbolic system, 2011 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, 2011, pp. 4937-4942. 15. Gugat M., Leugering G., Schittkowski K., Schmidt E.J.P.G. (2001) Modelling, Stabilization, and Control of Flow in Networks of Open Channels. In: Grötschel M., Krumke S.O., Rambau J. (eds) Online Optimization of Large Scale Systems. Springer, Berlin, Heidelberg, DOI 10.1007/978-3-662-04331-8_16. 16 J. M. Coron, B. d'Andrea-Novel and G. Bastia, A Lyapunov approach to control irrigation canals modeled by saint-venant equations, 1999 European Control Conference (ECC), Karlsruhe, 1999, pp. 3178-3183. 17. V. Dos Santos and C. Prieur, Boundary Control of Open Channels With Numerical and Experimental Validations, in IEEE Transactions on Control Systems Technology, vol. 16, no. 6, pp. 1252-1264, Nov. 2008. 18. Tehuan Chen, Zhigang Ren, Chao Xu, Ryan Loxton, Optimal boundary control for water hammer suppression in fluid transmission pipelines, Computers & Mathematics with Applications, Volume 69, Issue 4, 2015, Pages 275-290, ISSN 0898-1221. 19. Tehuan Chen, Chao Xu, Qun Lin, Ryan Loxton, Kok Lay Teo, Water hammer mitigation via PDEconstrained optimization, Control Engineering Practice, Volume 45, 2015, Pages 54-63, ISSN 0967-0661,
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20. T. Van Pham, D. Georges and G. Besançon, "Predictive Control With Guaranteed Stability for Water Hammer Equations," in IEEE Transactions on Automatic Control, vol. 59, no. 2, pp. 465-470, Feb. 2014. doi: 10.1109/TAC.2013.2272171. 21. Vainshtok S.M., Novoselov V.V., Prokhorov A.D., Shammazov A.M., Pipeline transportation of oil. Textbook. for universities: In 2 volumes. -M .: LLC "Nedra-Business Center", 2004. - Т.2. - 621 p. (in Russian). 22. Lurie M.V., Ferichenkova E.V., Protecting the main oil product pipelines from high-pressure waves by counterpressure waves, Transport and storage of oil products, 2007. - №6. –pp. 4 – 7.
S0308-0161(18)30512-X
DOI:
https://doi.org/10.1016/j.ijpvp.2019.03.015
Reference:
IPVP 3865
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 30 November 2018 Revised Date:
2 March 2019
Accepted Date: 10 March 2019
Please cite this article as: Yuzhanin V, Popadko V, Koturbash T, Chernova V, Barashkin R, Predictive control and suppression of pressure surges in main oil pipelines with counter-running pressure waves, International Journal of Pressure Vessels and Piping (2019), doi: https://doi.org/10.1016/ j.ijpvp.2019.03.015. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Predictive control and suppression of pressure surges in main oil pipelines with counter-running pressure waves
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Authors
Victor Yuzhanina, Vladimir Popadkoa, Taras Koturbashb,*, Valeria Chernovaa, Roman Barashkina Department of Automation of Technological Processes, Gubkin Russian State University of Oil and Gas (National Research University), Leninsky Ave 65, Building 1 (510), 119991, Moscow, Russian Federation b Department of Energy Technology, KTH Royal Institute of Technology, Brinellvägen 68, 10044 Stockholm, Sweden
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Abstract
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This paper addresses pressure control in crude oil pipelines in onshore transportation systems. Maintaining pressure below the maximum rated load capacity in a linear section of an oil pipeline is considered as a control problem. We present a method for predicting of maximum allowed pressure excess using the method of characteristics and the first principles model of nonstationary flow of a slightly compressible fluid in a pipeline. The algorithm, based on the developed method, can suggest the automatic control action that will allow dampening potentially dangerous pressure surges in the linear section of pipeline utilizing counter running pressure wave. The simulations results show that the algorithm allows efficiently predict and suggest control action for suppressing the back-propagating pressure waves in oil pipelines caused by various technological operations. The suggested control action could be implemented without any modification of existing technological equipment at main oil pipeline by utilizing the available capabilities of upstream oil pumping stations.
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Pressure excess, oil pipeline, slightly compressible fluid, method of characteristics, counter-running pressure wave, back-propagating pressure wave.
1. Introduction
Pipeline transport is the most common means of transporting various fluids over long distances. The length of pipeline systems varies from hundreds of kilometers for water supply [1] to thousands of kilometers for oil and gas transportation [2]. The latter are considered as long-distance pipeline systems and have much higher requirements regarding reliability and safety in the view of severe environmental and economic consequences of potential failures or down time. Assessment and mitigation of potential risks and unexpected emergencies in long pipeline systems are of major concern for system operators. One of the primary risks in a long-distance pipeline system that transports slightly compressible fluids (water or gas-free oil that have small but constant compressibility) comes from various transient * Corresponding author. Tel.: +46 762915506; E-mail address: [email protected] (T.Koturbash)
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processes. Such transient processes usually have complicated dynamics with wave-like behavior and high speed (about 1000 m/s and higher) waterhammer propagation (hydraulic shock). Waterhammer appears when the fluid flow changes speed rapidly, followed by a change in fluid pressure. Waterhammer is usually caused by various technological operations, including fast valve shutoff and starting or stopping of pumps. In addition, it could be caused by more specific reasons such as pipeline pigging [3]. Waterhammer can increase the fluid pressure above the maximum rated (or allowed) level for the pipeline system. A pressure increase can damage the pipeline and even lead to rupture. The risk of rupture caused by waterhammers is considered one of the most significant among all risks in pipeline systems that are used for transporting slightly compressible fluids [1, 4]. The risk of pipeline rupture is considered during the design phase, but it is very difficult to foresee and analyze all potential transient processes that could occur during operation. Specific scenarios in pipeline systems were analyzed and the relevant transient processes were simulated in previous publications [1, 4-7]. Various recommendations for redesign of the pipeline system are usually proposed based on simulation results. Those recommendations usually include the following: pipeline strength enhancement or installation of special protection devices like safety valves, or anti-purge valves. However, this is too expensive in most cases, or it may be impossible to implement such recommendations in long-distance pipeline systems due to design and reconstruction complexity or high associated costs. Another approach to mitigate risks related to waterhammer formation is to develop and employ control algorithms. A control algorithm can be used to ensure system safety and considerably reduce the dangerous effects of waterhammer. Development and application of control algorithms imply knowledge or predictability of the behavior of the system. The temporal behavior of slightly compressible fluids (including waterhammers) is expressed using hyperbolic or parabolic partial differential equations (PDE). There are several variations of these equations that differ in the selection of factors taken into consideration, including flow pattern (laminar or turbulent), non-isothermality, rheological properties of transported fluid, and pipeline elevation variations. [8, 9]. It is a very complicated task to determine the analytical solution of waterhammer equations in a generic case, thus numerical methods are usually used. We used the method of characteristics to solve the PDE governing the behavior of a waterhammer [8-11]. This allowed us to consider nonlinear friction, exclude smoothing of the pressure wavefront using the Courant–Friedrichs–Lewy condition, and obtain high computational efficiency that provides practical applicability. The stabilization of hyperbolic PDEs is widely studied in the literature [12-14], in particular with applications to open channels [13, 15-17] and pumped flow (waterhammer equations) [18Error! Reference source not found.-20]. In general, these studies consider control in the classic formulation of control theory: an acceptable control action takes the object to the desired operating conditions. The main difference of the proposed approach is that the desired operating condition is undetermined. In contrast, only unacceptable solutions are defined, predicted, and avoided.
2. Methodology
2.1 Problem formulation A main oil pipeline (MOP) is a complex processing facility (fig 1). A typical MOP consists of process sections (PS) that are hydraulically separated by oil storage tanks. A PS consists of linear pipeline sections (LS) separated by oil pumping stations (PPS). Within a single PS, the fluid is pumped in the socalled “pump-to-pump” operating mode. The transitional wave process is initiated at a single point in this mode, and a wave propagates across the entire PS at the speed of sound in oil (about one kilometer per second) [21]. In most cases, such a transitional wave is caused by routine operations like start-up, shutdown, pressure testing or even emergency shutdown of PPS. As a result, pressure waves propagate downstream and upstream inside the pipeline.
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Figure 1. Schematic diagram of a typical main oil pipeline.
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A typical local control system for automatic pressure stabilization is usually placed on the outlet of a PPS; this system can eliminate or considerably damp dangerous pressure surges propagating downstream in LS [21]. This is accomplished by defining a pressure set point that allows fluid pressure regulation in the beginning of LS. Thus, the pressure does not exceed dangerous levels along the LS up to the next PPS. A control response to pressure surges propagating upstream from the end of an LS is initiated either through the control valve at the output of the previous PPS or by actuating the emergency shutdown system for the LS or the entire PS. In the first case, a potentially dangerous and uncontrolled pressure wave will run throughout the LS [9, 21]. In the second case, the response occurs only after the emergency pressure set point is exceeded. If the excess pressure is momentary and is not dangerous, the entire PS could be improperly shut down. Therefore, it is difficult in both cases to ensure protection of the entire LS and MOP sections against emergency pressure excess using only typical feedback control systems. The goal of this study was to develop a predictive control system capable of completely preventing overpressure in the LS of an MOP line without unnecessary shutdowns. The overall objective was divided into following tasks: 1. Short-term forecast of pressure waves that exceed the maximum rated pressure in an LS by backpropagating a pressure wave based on real-time sensor readings. 2. Predictive compensation of an upstream pressure wave using a counter-propagating pressure wave generated in a presiding PPS.
2.2 2.2 ShortShort-term forecast of fluid pressure in an LS The control objective is the pressure in an LS located between two PPS. The control criterion is as follows: ( )
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( , )≤
(1)
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where is the coordinate along the pipeline axis, ( , ) is the pressure profile along the hydraulic axis ( ) is a maximum rated pressure profile that depends on the pipeline stress limit in an LS a time , along the hydraulic axis. ( ) has a complex, uneven shape, especially in an old LS that may have The function undergone multiple repair services or installation of new sections, which can differ from neighboring sections considerably in terms of their stress limits. A control response is the setpoint value of the local pressure control system at the PPS output, thus ensuring the required boundary condition (0, ) at the beginning of LS. The pressure ( , ) at the end of the LS is considered as a measurable disturbance. To describe the behavior of oil in the pipeline, we will use the equation for unsteady flow of a slightly compressible fluid in a long pipeline [9]. The continuity equation is as follows: +
∙
=0
(2)
where is the density of the fluid and is the velocity of sound in the fluid. The equation of motion is as follows: + ∙
+ ( )∙
∙| |
+
( )
=0
(3)
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+ %(!) ∙
where
$
+ &(!) = 0
0 %(!) = ' 1/
0
(4)
*
. ( ) .
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·| | 0 &(!) = ' * ɸ(!) = ( ) · + ɸ(!) 2.
The method of characteristics for system integration could be applied to (4) numerically. Multiplying Error! Reference source not found. by the left eigenvector /0 (!) of matrix %(!) while considering the relationship between the eigenvectors and eigenvalues /0 %(!) = 10 ∙ /0 yields the following: $ $
+ 10 ∙ /0
$
+ /0 ∙ &(!) = 0
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/0
/0 2 3 + /0 ∙ &(!) = 0, 6 = 1,2 45
$
$
+ 10 ∙
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$
where 2 3 =
45
is the ! gradient along lines defined by
(5)
= 10 , which are known as system
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characteristics (4). The eigenvalues 10 of the %(!) matrix are equal in absolute value but have different signs. The absolute value is equal to the velocity of sound in the fluid (10 = ± ). Therefore, replacing the gradient in (5) with a difference approximation yields a set of equations for numerical integration of equations (2) and (3) as a uniform grid (see
Figure ):
/ !89
/ !89
,: ,:
= / 2!8,:; − ∆ ∙ &>!8,:; ?3
= / 2!8,:9 − ∆ ∙ &>!8,:9 ?3
(6) (7)
where @ is the number of grid points and A = 1, … , @. For a limiting grid points, one of either (6) or (7) shall be replaced by the boundary condition equation.
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Figure 2. Computational grid of the method of characteristics
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If the initial conditions in the j-th layer (moment) are defined, then it is possible to calculate all points in the (j+1)-th layer except for the boundary points. It will be possible to calculate n – 2 points of (j+1)-th layer. There are n – 4 calculable points in the (j+2)-th layer, n – 6 points in the (j+3)-th layer, and so on. Figure shows the computational grid points (green dots) that could be calculated based on the initial conditions in the j-th layer. These points will be located inside a triangle or trapezoidal pattern in the time-distance plane, depending on the parity of the number of initial points in the j-th layer. Using (6) and (7) for each point in the computational grid enables short-term forecasting of the pressure wave, ( ) thus one can predict whether the pressure will exceed the maximum rated pressure ( , ) > in the LS from knowledge of the initial conditions in the j-th layer.
Figure 3. Computational grid of the method of characteristic for short-term forecasting of pipeline values.
2.3 2.3 Predictive pressure wave compensation In order to control the boundary conditions in the pipeline, they must be set at each time step. Figure 3 shows that boundary conditions at the (j+1)-th layer will affect only a portion of the forecast area points that lie on the characteristic lines passing through the boundary points in the layer (i.e., segments A1A4 and C1C4, or A1An and C1Cn in general). The control algorithm will not be able to affect
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points that lie above or below these characteristic lines. Therefore, to solve the control problem, it is enough to select the boundary conditions to prevent emergency pressure excess along segments A1An and C1Cn in each layer. Consider the supporting inverse problem formulated as follows: the desired pressure value D(%: ) is set at any point %: , where A ∈ {2,3, … , @}. The task is to identify the pressure D(% ) that will produce the desired pressure D(%: ) at the selected point%: . Consider the example case of point % in Fig. 3. This point could be considered as the boundary of the pipeline sub-segment, beginning at . Then the set pressure value D(% ) could be interpreted as a boundary condition for this subsection in layer I + 2. By solving equation Error! Reference source not found. on J − % characteristics for a certain pressure value D(% ), we will obtain the velocity #(% ). The flow parameters at % simultaneously satisfy equations Error! Reference source not found. and Error! Reference source not found. for characteristics J − % and % − % . Given that the flow parameters at points J , % are known, the only unknown parameters in the above equations are the flow parameters at % . We obtain the pressure and velocity values at % by solving both equations. Binding the pressure value at the subsequent point %K will set the pressure and velocity at point % to its only possible value. If this inference is repeated for subsequent points, we may conclude that we can calculate the boundary condition D(% |%: ) for a given required pressure at any point D(%: ), A ∈ {2,3, … , @}, which will satisfy the required pressure. Let us consider the inverse problem of identifying the maximum boundary condition D∗ (% ) that will ensure the pressure at each point % , % , … , %M does not exceed the maximum rated value for the pipeline. In practical terms, this means that there is no need to apply control action if the actual pressure is less than D∗ (% ). Let us alternatively require that the pressure would be equal to the maximum rated pressure value at each point % , … , %M . We can then obtain the appropriate boundary conditions at % : D(% |% ), D(% |% ), … , D(% |%M )
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To retrieve the required maximum pressure D∗ (% ), it is sufficient to select the minimum among the received boundary conditions:
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D∗ (% ) = min{D(% |% ), D(% |% ), … D(% |%M )} (9) Received at each calculation step boundary condition D∗ (% ) is transmitted as the set point to the pressure controller at the output of the upstream PPS of the pipeline section. As a result, the proposed algorithm can control the pressure of the left side of the LS at the points D∗ (% ) … D∗ (%M ). Due to the finite propagation speed of the pressure waves in the fluid the control action engaged at the left side of LS will not be able to compensate the disturbance on the right side of LS. Therefore, it is impossible to control the right side of the LS using the counter-propagating pressure waves excited at upstream PPS [22]. However, the pressure of right side of LS can still be calculated and controlled with proposed algorithm by using the right boundary condition D(Q ). The calculation of maximum pressure at D∗ (Q ) that will not exceed the maximum rated pressure profile at points D∗ (Q ) … D ∗ (QM ) is in same manner as it is shown for D∗ (% ). Typically, available technical facilities of MOP (Fig. 1) usually cannot arbitrarily and fully control the pressure at the right side of the LS. In order to control the right side of the LS, the MOP equipment should be modified accordingly. For example, a relief valve and an underground storage tank should be installed on the right side of the LS in order to release excess fluid pressure. In the following numerical simulation we have considered the control scheme where the control action is the pressure on the left side of LS (%0 : 0 ∈ (0, /2)) and the pressure on the right side of LS is the measurable process disturbance. The back propagating pressure wave in left side of LS possesses the most danger with regard to exceeding maximum allowed pressure of LS as shown in simulations below.
3. Results and Discussion The validity and performance of the proposed predictive control algorithm in an MOP section was demonstrated using numerical modeling in combination with a simulator of main oil pipeline. The
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addressed PS shown in Fig. 4 corresponds to a normal operation mode with a reference hydraulic condition. The PS consists of three LS spanning from 540 to 810 km. These LS are separated by PPSA at 580 km and PPSB 690 km. Each LS consists of pipeline sections with uneven elevation and varied load capacities, that general reflects the effects of burial depth, exposures, pipeline design parameters (such as wall thickness, pipe type, known corrosion or damage defects). There is also a specific pipe section with considerably reduced load capacity (PSRLC) at 600 km, which poses a substantial risk of a damage in case of pressure surge. In order to simplify the representation of the hydrodynamic process of fluid transport in the addressed PS the fluid pressure profile and the pipeline load capacity are expressed in units of total pressure head as follows: D( ) + ( ) S( ) =
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where D( ) is the fluid pressure, D ( ) is the pipeline load capacity in fluid pressure units, is the fluid density, is the acceleration due to gravity, and ( ) is the elevation of the pipeline. Such representation shows that fluid pressure and pipeline load capacity can be defined in units of fluid column height from sub sea level with consideration of i.e., hydrostatic pressure. This simplifies analysis of the hydrodynamic processes in the MOP. The maximum rated fluid pressure in an LS depends on the load capacity and elevation of the pipeline. Thus, the rated pressure profile has an uneven and complex form. Figure 4 shows the load capacity S ( ) of the fluid inside the LS before reaching the maximum rated pressure D ( ). The PSRLC region has a load capacity that is considerably lower compared to that of the adjacent sections. Rupture becomes inevitable if the fluid pressure in this region reaches or exceeds the actual load capacity S ( ).
Figure 4. Reference hydraulic conditions in a PS during normal operation; S( ): total head pressure (m); STU ( ): pipeline load capacity (m); ( ): elevation of a pipeline cross-section (m)
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Emergency shutdown of PPSB was included in the simulation. Such an emergency usually results in pressure equalization between the PPSB inlet and outlet as shown in Fig. 5a. The resulting pressure waves propagate downstream and upstream from PPSB, as shown in Fig. 5b. The actual fluid pressure at each time point throughout the upstream and downstream LS was calculated with the developed shortterm forecasting algorithm. The pressure wave propagating downstream does not pose any danger since it is a decreasing pressure wave (in the event of PPSB shutdown) or it could be controlled at the PPSB output with a local control system. On the contrary, the back-propagating pressure wave poses serious danger in the LS since it is an increasing pressure wave propagating upstream and cannot be controlled with the downstream PPSB (where it occurred). The upstream PPSA will be able to react to an incoming wave only when the wave reaches its output; this is where pressure sensors for local control system are usually placed. In most cases, the reaction of the local control system placed at the output of an upstream PPSA will be too late, and the pressure surge might exceed the remaining load capacity of the LS. As could be observed from Fig. 6, the upstream propagating pressure wave will reach higher total head pressure as it comes closer to the beginning of the LS. This becomes extremely dangerous in LS regions with reduced load capacity, e.g., due to pipeline elevation variations or reduced load capacity in the pipe section.
Figure 5. Development of downstream and upstream pressure waves.
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The current LS strength redundancy decreases as the pressure increases towards the beginning of LS, as shown in Fig. 6. Local control systems will not be able to react in time, and the LS load capacity is exceeded when the back-propagating pressure wave approaches the PSRLC region (i.e., S( ) approaches S ( )). During simulation the fluid exceeded the maximum rated pressure by 15 m of total head pressure, which is equal 120 kPa of overpressure; this would most certainly result in rupture. While applying the proposed algorithm, we could foresee the possibility that the pressure would exceed the maximum rating in the PSRLC region. We can also recalculate the required counterpropagating upstream pressure wave that can be used to neutralize the back-propagating pressure wave before it reaches the PSRLC region in the LS. The correct timing for initiating the counterpropagating pressure wave from the upstream PPS was also obtained using the proposed algorithm. The resulting backpressure and counter-propagating wave passing the PSRLC region area was forecasted with the algorithm in advance. The pressure in the LS will be within the required limits, as shown in Fig 6. The simulation results show that the proposed method can be used to forecast pressure waves and avoid any negative effects from back-propagating pressure waves by creating a controllable counterpropagating pressure wave.
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Figure 6. Upstream pressure surge propagation while operating local automatic control systems (top diagrams) and the developed predictive control algorithm (bottom diagrams). (1a), (2a) Pressure surge approaching the RLC region, (1b), (2b) pressure surge while passing the PSRLC region, and (1c), (3c) pressure surge approaching the upstream PPS.
4. Conclusions
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An effective computational method for control and suppression of back-propagating pressure surges in MOPs with counter-running pressure suppression waves was presented in this paper. The distinctive feature of the proposed control algorithm is that it implements on-the-fly short-term forecasting of the total pressure in the pipeline and activates the control action only if there is a possibility of exceeding the rated load capacity in an LS. The control goal was reduced to a constraint satisfaction problem (CSP) for the maximum pipeline pressure. The CSP was transformed into a solvable inverse problem, where the boundary conditions of the hyperbolic waterhammer PDE were determined using the method of characteristics. Simulation results show that the developed algorithm allows forecasting of back-propagating pressure waves that may exceed the rated load capacity in the LS. The required control action can be subsequently identified and implemented with the upstream PPS. The algorithm allows to forecast the effects of suggested control action for further evaluation and selection of the desired control criteria. Application of the developed algorithm will allow to improve the reliability and safety of existing MOPs in onshore transportation systems without any modification or installation of additional technological equipment. This is achieved by efficient utilization of the existing technological capabilities of MOPs (the upstream PPS) to generate counter-running running pressure waves of reduced pressure. These counter running pressure wavescan efficient damp the the back-propagating pressure surges in oil pipelines caused by various technological operations.
Acknowledgement The authors want to thank prof. Mikhail V. Lurie and prof. Sergei A. Sardanashvili from Gubkin Russian State University of Oil and Gas for their comments and fruitful discussions on the presented research topic. Furthermore, the authors would like to acknowledge the support of JSC Transneft for providing valuable inputs into this research work.
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