Control of centrifugal compressors via model predictive control for enhanced oil recovery applications

Offshore Oil and Gas Production Proceedings of of the the 2nd 2nd IFAC IFAC Workshop Workshop on on Automatic Control Control in in Proceedings May 27-29, 2015. Florianópolis, Brazil on Automatic Proceedings of the 2nd IFAC Workshop Automatic in Available online Control at www.sciencedirect.com Offshore Offshore Oil Oil and and Gas Gas Production Production Offshore Oil2015. and Gas Production May 27-29, Florianópolis, Brazil May 27-29, 2015. Florianópolis, Brazil May 27-29, 2015. Florianópolis, Brazil

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IFAC-PapersOnLine 48-6 model (2015) 009–014 Control of centrifugal compressors via predictive control for enhanced oil Control of centrifugal compressors via model predictive control for enhanced oil recovery Control via model Control of of centrifugal centrifugal compressors compressors via applications model predictive predictive control control for for enhanced enhanced oil oil recovery applications recovery applications recovery applications S. Budinis*. N. F. Thornhill**

 S. Budinis*. N. F. S. F. Thornhill** Thornhill** S. Budinis*. Budinis*. N. N.  F. Thornhill**  *Centre for Process Systems Engineering (CPSE), Department of Chemical Engineering, Imperial College London, London,  SW7 2AZ, UK (e-mail: [email protected]). *Centre for Process Systems Engineering (CPSE), Department of Chemical Engineering, Imperial College London, London, *Centre (CPSE), Department of *Centre for for Process Process Systems Systems Engineering Engineering (CPSE), Department of Chemical Chemical Engineering, Engineering, Imperial Imperial College College London, London, London, London, **(e-mail: [email protected]) SW7 2AZ, 2AZ, UK (e-mail: (e-mail: [email protected]). SW7 UK [email protected]). SW7 2AZ, UK (e-mail: [email protected]). **(e-mail: [email protected]) [email protected]) **(e-mail: **(e-mail: [email protected]) Abstract: This paper proposes a control system for integrated pressure and surge control of centrifugal compressors forpaper enhanced oil recovery application. proposed controland system is based model Abstract: This proposes control system for forThe integrated pressure surge controlonof oflinear centrifugal Abstract: This paper proposes aaa control system integrated pressure and surge control centrifugal Abstract: This paper proposes control system for integrated pressure and surge control of centrifugal predictive control. A fully validated non-linear dynamic model was developed in order to simulate the compressors for for enhanced enhanced oil oil recovery recovery application. application. The The proposed proposed control control system system is is based based on on linear linear model model compressors compressors for enhanced oil recovery application. The proposed control system is based on linear model operation of the compressor at full and partial load. The model of the compression system includes predictive control. control. A A fully fully validated validated non-linear non-linear dynamic dynamic model model was was developed developed in in order order to to simulate simulate the thea predictive predictive control. A fully non-linear dynamic model developed in order to simulate thea main process linecompressor with thevalidated compressor and a recycle line withwas thethe antisurge recycle valve. Different operation of the at full and partial load. The model of compression system includes operation of the compressor at full and partial load. The model of the compression system includes aa operation of the compressor at full and partial load. The model of the compression system includes disturbance and control tuning scenarios were tested and the response of the model predictive controller main process line with the compressor and aa recycle line with the antisurge recycle valve. Different main process line with the compressor and recycle line with the antisurge recycle valve. Different main process line with tuning the and atested recycle line with the antisurge recycle valve. Different was analysed, evaluated andcompressor also compared with a traditional control system. Temperature effects have disturbance and control scenarios were and the response of the model predictive controller disturbance and control tuning scenarios were tested and the response of the model predictive controller disturbance and control tuning scenarios were tested and the response of the model predictive controller been taken into account in the model of the process and in the constraint formulation of the MPC was analysed, evaluated and also compared with aa traditional control Temperature effects have was analysed, evaluated and also compared with control system. system. effects have was analysed, evaluated and alsomodel compared with a traditional traditional system.isTemperature Temperature effects have optimization problem. The show of thatthe theprocess proposed able to meetof thetheprocess been taken into into account inresults the andcontrol incontrol thetechnique constraint formulation MPC been taken account in the model of the process and in the constraint formulation of the MPC been taken into account in the model of the process and in the constraint formulation of the MPC demand while preventing surge and also minimizing the amount of gas recycle. optimization problem. problem. The The results results show show that that the the proposed control control technique technique is is able able to meet meet the process process optimization optimization problem. Thesurge results that the proposed proposed control technique is able to to meet the the process demand while preventing andshow also minimizing the amount amount of gas gas recycle. Keywords: MPC, compressor, surge, control, driver Control) torque, recycle, carbon dioxide, supercritical. demand while preventing surge and also minimizing the of recycle. © 2015, IFAC (International Federation of Automatic Hosting by Elsevier Ltd. All rights reserved. demand while preventing surge and also minimizing the amount of gas recycle. Keywords: MPC, MPC, compressor, surge, surge, control, driver driver torque, recycle, recycle, carbon dioxide, dioxide, supercritical. Keywords:  torque, Keywords: MPC, compressor, compressor, surge, control, control, driver torque, recycle, carbon carbon dioxide, supercritical. supercritical. system (Seborg et al., 2004). In the literature it has already  1. INTRODUCTION   been demonstrated that model predictive control was system (Seborg et al., 2004). In the literature it has already system (Seborg et al., 2004). In the literature it has already 1. INTRODUCTION system (Seborg et al., 2004). In the literature it has already 1. INTRODUCTION applicable for the control of complex compression systems been demonstrated demonstrated that that model model predictive predictive control control was was Enhanced Oil Recovery (EOR) methods are commonly used been 1. INTRODUCTION been demonstrated that of model predictive control was (Smeulers et the al., control 1999, Øvervåg, 2013) and forsystems surge applicable for complex compression in industry to recover oil from onshore and offshore applicable for the control of complex compression systems Enhanced Oil Oil Recovery Recovery (EOR) (EOR) methods methods are are commonly commonly used used applicable Enhanced for the control of complex compression systems prevention via closed coupled valve (Johansen, 2002) and (Smeulers et et al., al., 1999, 1999, Øvervåg, Øvervåg, 2013) 2013) and and for for surge surge Enhanced Oil (EOR) methods are commonly used reservoirs after primary extraction et (Smeulers in industry industry toRecovery recoverand oilsecondary from onshore and(Sobers offshore in to recover oil from onshore and offshore (Smeulers etactuation al., 1999, Øvervåg, 2013) andHowever for surge drive torque (Cortinovis et al., 2012). the prevention via closed coupled valve (Johansen, 2002) and in industry to recover oil from onshore and offshore al., 2013).after Among the non-thermal gasextraction injection(Sobers methods, prevention via closed coupled valve (Johansen, 2002) reservoirs primary and secondary secondary et minimization prevention via closed coupled valve (Johansen, 2002) and and reservoirs after primary and extraction (Sobers et of the recycle flow rate and the temperature drive torque actuation (Cortinovis et al., 2012). However the reservoirs after primary and secondary extraction (Sobers et carbon dioxide floods have been used for EOR (Thomas, drive torque actuation (Cortinovis et al., 2012). However the al., 2013). Among the non-thermal gas injection methods, al., 2013). Among the non-thermal gas injection methods, drive torque actuation (Cortinovis etrate al., 2012). However the effects have not previously been taken into account. minimization of the recycle flow and the temperature al., 2013). Among the non-thermal gas injection methods, has already been used in the past for oil recovery 2008). CO minimization of the recycle flow rate and the temperature 2 carbon dioxide floods have been used for EOR (Thomas, minimization of the recycle flow rate and the temperature carbon dioxide floods have been used for EOR (Thomas, carbon dioxide floods been have beeninrecently used for integrated EOR have not previously been taken into account. however this has used been with effects effects have not taken hasmethod already the past for oil(Thomas, recovery 2008). CO 2008). CO This paper the usebeen of MPC forinto theaccount. integrated control effects haveproposes not previously previously been taken into account. 222 has already been used in the past for oil recovery has already been used in the past for oil recovery 2008). CO carbon storage for the reduction of atmospheric emissions 2 however this method has been recently integrated with however this method has been recently integrated with of pressure and surge in centrifugal compressor applications. This paper proposes the use of MPC for the control however this method has been of recently integrated with This paper proposes the use of MPC for the integrated control (Ravagnani et al., carbon storage storage for2009). the reduction reduction atmospheric emissions This paper proposes theinuse of MPCfor for surge the integrated integrated control carbon for the of atmospheric emissions The amount of gas recycled prevention is of pressure and surge centrifugal compressor applications. carbon storage for the reduction of atmospheric emissions of pressure and surge in centrifugal compressor applications. (Ravagnani et et al., al., 2009). 2009). of pressure and surge in centrifugal compressor applications. minimized by control tuning and the temperature constraints (Ravagnani The amount of gas recycled for surge prevention is For the purposes of enhanced oil recovery and carbon dioxide (Ravagnani et al., 2009). The amount of recycled for surge is The amount of gas gas recycled for temperature surge prevention prevention is have been included in the MPC formulation. minimized by control tuning and the constraints must be compressed to supercritical conditions. storage, CO minimized by control tuning and the temperature constraints 2 For the purposes of enhanced oil recovery and carbon dioxide minimized by control tuning and the temperature constraints For the purposes of enhanced oil recovery and carbon dioxide For the ofbe enhanced oil recovery and carbon dioxide been included in the MPC formulation. thispurposes type2 must of application, the phase transition takes place have have been the compressed to supercritical conditions. storage, CO storage, CO The structure of thein is theformulation. following. In Section 2 the have been included included inpaper the MPC MPC formulation. 22 must be compressed to supercritical conditions. must be compressed to supercritical conditions. storage, CO inside a multistage centrifugal compressor. The operation of 2 For this type of application, the phase transition takes place For this type of application, the phase transition takes place model of the compressor is presented. InIn Section 3 the an The structure of the paper is the following. Section 2 For this type of application, thecompressor. takes place structure of the is following. In Section 2 the this type of machine is limited byphase surge.transition Surge a dynamic inside a multistage multistage centrifugal The isoperation operation of The The structure ofcompressor the paper paper isisthe thepresented. following. Ingiven. Section 23then the inside a centrifugal compressor. The of overview on traditional compressor control is It is model of the In Section inside a multistage centrifugal compressor. The isoperation of model of the compressor is presented. In Section 3 an an instability the gas causes flow Surge reversal inside the this type type ofofmachine machine is that limited by surge. surge. dynamic model ofon the compressor is presented. In given. Section 3then an this of is limited by Surge is aaa dynamic followed by the description of the implemented model overview traditional compressor control is It is this type of machine is limited by surge. Surge is dynamic machine. When the compressor is surging, the oscillatory overview on traditional compressor control is given. It is then instability of of the the gas gas that that causes causes flow flow reversal reversal inside inside the the predictive overview on traditional compressor control is given. It is then instability controller and its design. In Section 4 the paper by the description of the implemented model instability of the thegas gasflow thatcauses causesvibrations flow reversal inside the followed followed by the description of model behaviour When of that can damage machine. the compressor is surging, the oscillatory followed by the tuning, description of the theIn implemented implemented includes the MPC thedesign. scenarios for the validation of machine. When the compressor is surging, the oscillatory predictive controller and its its Section 4 the the model paper machine. When the compressor is surging, the oscillatory predictive controller and design. In Section 4 paper blades, casing and bearings (Boyce, 2012). In industrial behaviour of the gas flow causes vibrations that can damage predictive controller and its design. In Section 4 the paper the control system and the results of the dynamic simulations. behaviour of the gas flow causes vibrations that can damage includes the the MPC MPC tuning, tuning, the the scenarios scenarios for for the the validation validation of of behaviour of theand gas bearings flowstill causes vibrations thatIncanindustrial damage practice, surge control relies on2012). avoidance control. includes blades, casing (Boyce, includes thesystem MPC tuning, scenarios forofthe of Finally, Section 5 presents the conclusions thevalidation work. blades, casing and bearings (Boyce, 2012). In industrial the control control and the results results of the the dynamic dynamic simulations. blades, casing and bearings (Boyce, 2012). In industrial the system and the of simulations. Although many solutions based on active control have been practice, surge control still relies on avoidance control. the control system and the results of the dynamic simulations. practice, surge control still relies on avoidance control. Finally, Section Section 5 5 presents presents the the conclusions conclusions of of the the work. practice, surge control relies on avoidance control. Finally, proposed (Arnulfi et al., still 2006), they were not implemented Although many solutions based on active control have been 2. MODEL OFthe THE COMPRESSOR Finally, Section 5 presents conclusions of the work. work. Although many solutions based on active control have been Although many solutions based on active control have been on industrial-size compressors due mainly to the cost and proposed (Arnulfi (Arnulfi et et al., al., 2006), 2006), they they were were not not implemented implemented proposed 2. MODEL MODEL OF OF THE THE COMPRESSOR COMPRESSOR 2. proposed (Arnulfi et al., 2006), theythey were not implemented reliability of the additional devices require (Uddin and 2.1 Mathematical modelOF of the compressor 2. MODEL THE COMPRESSOR on industrial-size compressors due mainly to the the cost and and on industrial-size compressors due mainly to cost on industrial-size compressors due they mainly to the cost and Gravdahl, 2012). reliability of the the additional additional devices require (Uddin and 2.1 reliability of devices they require (Uddin and Mathematical model of the compressor 2.1 of reliability of the additional devices they require (Uddin and The model of themodel compression system is a non-linear one2.1 Mathematical Mathematical model of the the compressor compressor Gravdahl, 2012). Gravdahl, Avoidance2012). control for centrifugal compressors relies on the dimensional dynamic model that includes a main process line Gravdahl, 2012). The model model of of the the compression compression system system is is aa non-linear non-linear oneonerecycle of part of the in order torelies increase the The The of the compression system amain non-linear oneand amodel recycle line. Itmodel is represented in is Figure 1.process The main Avoidance control forcompressed centrifugalgas compressors on the the Avoidance control for centrifugal compressors relies on dimensional dynamic that includes a line dimensional dynamic model that includes a main process line Avoidance control for centrifugal compressors relies on the inlet flow rate of the compressors. When the recycle valve dimensional dynamic model that includes a main process line line includes inlet valve, outlet valve, 1.compressor, recycle of of part part of of the the compressed compressed gas gas in in order order to to increase increase the the process recycle and a recycle line. It is represented in Figure The main and a recycle line. It is represented in Figure 1. The main recycle of part of the compressed gas in order to increase the opens a compressor becomes a multiple-input multiple-output and a recycle line. It is represented in Figure 1. The main duct and plenum. The recycle line includes the antisurge inlet flow flow rate rate of of the compressors. compressors. When When the the recycle recycle valve valve process inlet line includes inlet valve, valve, compressor, line includes inlet valve, outlet outlet valve, compressor, inlet flow rate of the the compressors. When themultiple-output recycle valve (MIMO) system. Model predictive control (MPC) is process process line includes inlet outlet valve, compressor, recycle valve that isThe used tovalve, prevent surge occurrence. Hot opens aa compressor becomes aa multiple-input opens compressor becomes multiple-input multiple-output duct and plenum. recycle line includes the antisurge duct and plenum. The recycle line includes the antisurge opens a compressor becomes a multiple-input multiple-output considered the mostModel appropriate control for this type of duct and plenum. The recycle line includes the antisurge gas recycle should be limited over time because it can (MIMO) system. predictive control (MPC) is (MIMO) system. Model predictive control (MPC) is valve that is used to prevent surge occurrence. Hot recycle valve that to surge occurrence. Hot (MIMO) system. Model predictive control (MPC) is recycle recycle valve should that is is used used to prevent prevent surge occurrence. Hot considered the most appropriate control for this type of gas recycle be limited over time because it can considered the most appropriate control for this type of gas recycle should be limited over time because it can considered the most appropriate control for this type of gas recycle should be limited over time because it can Copyright © 2015 IFAC 9 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 9 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 9 Control. Copyright © 2015 IFAC 9 10.1016/j.ifacol.2015.08.002

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𝑑𝑑𝑑𝑑 1 = (𝜏𝜏𝑑𝑑 − 𝜏𝜏𝑐𝑐 ) 𝐽𝐽 𝑑𝑑𝑑𝑑

(3)

𝜏𝜏𝑐𝑐 = 𝜇𝜇𝑟𝑟22 𝜔𝜔𝜔𝜔

(4)

𝛹𝛹𝑐𝑐 =

Fig. 1. Model of the compression system overheat the machine. On the other hand it reduces the time delay of the system as a smaller amount of gas is stored along the recycle line (Botros, 2011). The system includes also two nodes. The first node represents the physical point where the freshly fed gas mixes with the recycled gas, while the second node represents the physical point where the compressed gas splits between delivered gas and recycled gas. Variables 𝑚𝑚𝑖𝑖𝑖𝑖 , 𝑚𝑚𝑜𝑜𝑜𝑜𝑜𝑜 and 𝑚𝑚𝑟𝑟 are the gas flow rate respectively through inlet, outlet and antisurge valve. 𝑚𝑚 is the gas flow rate that enters the compressor and it is monitored for surge control, while 𝑚𝑚𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 is the gas flow rate that leaves the plenum. 𝑝𝑝𝑖𝑖𝑖𝑖 and 𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 are the inlet and outlet pressures of the system. 𝑝𝑝01 , 𝑝𝑝02 and 𝑝𝑝 are respectively the compressor inlet pressure, compressor outlet pressure and plenum pressure. 𝑝𝑝 is monitored for pressure control.

𝑚𝑚𝑖𝑖𝑖𝑖 = 𝑘𝑘𝑖𝑖𝑖𝑖 √𝜌𝜌𝑖𝑖𝑖𝑖 (𝑝𝑝𝑖𝑖𝑖𝑖 − 𝑝𝑝01 )

(6)

𝑚𝑚𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑘𝑘𝑜𝑜𝑜𝑜𝑜𝑜 √𝜌𝜌(𝑝𝑝 − 𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 )

(7)

𝑚𝑚𝑟𝑟 = 𝑘𝑘𝑟𝑟 √𝜌𝜌𝑟𝑟 (𝑝𝑝𝑟𝑟 − 𝑝𝑝01 )

(8)

In this paper corrected compressor maps have been used in order to define the surge line as a function of pressure ratio, rotational shaft speed, inlet pressure and inlet temperature of the gas. The temperature of the gas entering the machine (𝑇𝑇01 ) has been estimated as a function of the temperature of the freshly fed gas (𝑇𝑇𝑖𝑖𝑖𝑖 ), the temperature of the recycled gas (𝑇𝑇02 ) and the mass flow rates of these two flows (respectively 𝑚𝑚𝑖𝑖𝑖𝑖 and 𝑚𝑚𝑟𝑟 ), according to the following equation: 𝑚𝑚𝑖𝑖𝑖𝑖 ∫

𝑇𝑇𝑇𝑇𝑇𝑇

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇

𝑐𝑐𝑝𝑝 (𝑇𝑇)𝑑𝑑𝑑𝑑 + 𝑚𝑚𝑟𝑟 ∫

𝑇𝑇02

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇

𝑐𝑐𝑝𝑝 (𝑇𝑇)𝑑𝑑𝑑𝑑 = 𝑚𝑚 ∫

𝑇𝑇01

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇

𝑐𝑐𝑝𝑝 (𝑇𝑇)𝑑𝑑𝑑𝑑

(9)

where 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟 is the reference temperature and the heat capacity of the gas mixture 𝑐𝑐𝑝𝑝 is evaluated at the temperature 𝑇𝑇 according to:

The equations of the model include the mass and the momentum balance of the compressor, the moment of momentum balance of the rotating shaft, the compressor torque and characteristic (Gravdahl et al., 2002). They also include the equations of the flow through inlet, outlet and recycle valve (Morini et al., 2007). The equations are the following:

𝑑𝑑𝑑𝑑 𝐴𝐴1 (𝛹𝛹 𝑝𝑝 − 𝑝𝑝) = 𝐿𝐿 𝑐𝑐 01 𝑑𝑑𝑑𝑑

(5)

where 𝑎𝑎2 01 is the sonic velocity at ambient condition, 𝑉𝑉 is the volume of the plenum, 𝐴𝐴1 is the duct throughflow area, 𝐿𝐿 is the duct length, 𝛹𝛹𝑐𝑐 is the compressor characteristic, 𝐽𝐽 is the total inertia of the system, 𝜇𝜇 is the slip factor, 𝑟𝑟2 is the impeller radius, 𝑘𝑘𝑖𝑖𝑖𝑖 , 𝑘𝑘𝑜𝑜𝑜𝑜𝑜𝑜 , 𝑘𝑘𝑟𝑟 are the constants for respectively inlet, outlet and antisurge valve, 𝜌𝜌𝑖𝑖𝑖𝑖 , 𝜌𝜌, 𝜌𝜌𝑟𝑟 are the density of respectively 𝑚𝑚𝑖𝑖𝑖𝑖 , 𝑚𝑚 and 𝑚𝑚𝑟𝑟 .

The model of the compressor is based on a well-established model present in the literature that includes a compressor, a plenum and an outlet throttle valve (Greitzer, 1976). This model was further developed by Fink et al. (1992) in order to include the dynamic of the rotating shaft connecting driver and compressor. Gravdahl and Egeland (1999) proposed a further modification by expressing the torque of the compressor 𝜏𝜏𝑐𝑐 as a function of shaft rotational velocity 𝜔𝜔 and mass flow rate 𝑚𝑚 while Gravdahl et al. (2002) proposed to use the torque of the driver 𝜏𝜏𝑑𝑑 as input variable of the model instead of the rotational shaft speed 𝑁𝑁. This last model was the reference for this work and was modified according to Morini et al. (2007) in order to include also the recycle loop.

𝑑𝑑𝑑𝑑 𝑎𝑎2 01 = (𝑚𝑚 − 𝑚𝑚𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 ) 𝑉𝑉 𝑑𝑑𝑑𝑑

𝑝𝑝02 = 𝛹𝛹𝑐𝑐 (𝜔𝜔, 𝑚𝑚) 𝑝𝑝01

2

𝑐𝑐𝑝𝑝 (𝑇𝑇) = ∑ 𝑥𝑥𝑖𝑖 𝑐𝑐𝑝𝑝,𝑖𝑖 (𝑇𝑇)

(10)

𝑖𝑖=1

where 𝑖𝑖 is the number of components of the gas and 𝑥𝑥𝑖𝑖 is their mass fraction. The outlet temperature of the compressor 𝑇𝑇02 is estimated according to the performance maps provided by the supplier of the compressor.

(1)

2.2 Case study and model validation (2)

The case study presented in this paper refers to a multistage centrifugal compressor arranged in a single shaft 10

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flow goes below its lower limit the controller opens the recycle valve that allows part of the gas to be recycled back to the inlet of the machine. This lower limit is called antisurge control line. The antisurge controller has been tuned in order to be able to open the antisurge valve within 2 seconds.

Table 1. Typical parameter values Parameter name 𝑎𝑎2 01 𝑉𝑉 𝐴𝐴1 𝐿𝐿 1 𝐽𝐽

𝜇𝜇𝑟𝑟22 𝑘𝑘𝑖𝑖𝑖𝑖 𝑘𝑘𝑜𝑜𝑜𝑜𝑜𝑜 𝑘𝑘𝑟𝑟

11

Parameter value 0.001-0.005 s-1m-1 0.001-0.005 m 0.5-2 kg-1m-2

The interaction between pressure controller and surge controller is strong and they can end up pushing the compression system in opposite directions, as will be demonstrated in the Section 4.

0.01-0.05 m2 1-2.5 kg1/2m1/2 1-2.5 kg1/2m1/2 1-2.5 kg1/2m1/2

3.2 Representation of the surge margin

configuration. The fourth and last stage of compression was selected for the present analysis and its target pressure ratio is 2.85. After the calculations were complete all the other process variables reported in the paper were scaled to be 1 at their design point due to non-disclosure agreement with the industrial partners of the project. The driver is an asynchronous electric motor that allows variable speed operation. The process fluid is a mixture of carbon dioxide and water with small percentages of light hydrocarbons. The Span and Wagner equation of state (Span and Wagner, 1996) was selected in order to estimate the thermodynamic properties of the gas.

Usually both surge and control lines are plotted on the compressor map and therefore their distance from the operating point is easily identifiable. However this type of visualisation, even if very common in both academia and industry, can be misleading. The reason is that the surge line depends on both inlet pressure and temperature and therefore is affected by process disturbances and also by the opening of the recycle valve. The corrected compressor maps can be useful when the inlet conditions are different from the reference conditions however not when they continuously change over time as it happens during a process disturbance. Therefore a different way to visualise the proximity to surge is suggested in this paper. The proximity of the machine to surge is represented as the distance between the inlet mass flow rate of the compressor 𝑚𝑚 and the surge control mass flow rate 𝑚𝑚𝑐𝑐𝑐𝑐𝑟𝑟𝑟𝑟 .

The model of the compressor was validated against data coming from the industrial case study. Process data sheets and compressor performance maps were used to validate the model during steady state simulations while an industrial simulator, provided by the project partner ESD Simulation Training, was used to validate the dynamic behaviour during transients between steady states. The agreement between the available transient behaviours and the model presented in the paper was satisfactory. The model was then implemented in MATLAB Simulink and the ordinary differential equations were solved numerically using the MATLAB function ode45. Although the values pertaining to the model may not be disclosed, some typical values are presented in Table 1.

3.3 MPC controller In order to avoid the interaction between different controllers, an MPC controller was designed, implemented and tuned in order to control both pressure and surge. Figure 2 is the schematic representation of the system controlled by the MPC controller. The plant is defined by its states 𝑥𝑥𝑚𝑚 . The process inputs are the disturbances 𝑑𝑑𝑘𝑘 and the manipulated variables 𝑢𝑢𝑗𝑗 . The process outputs are 𝑦𝑦𝑖𝑖 . These outputs are the measured variable of the MPC controller. These variables are compared with their reference or set points and the MPC solves a constrained optimization

3. MODEL PREDICTIVE CONTROLLER 3.1 Traditional PID control The task of the control system of a compressor is to deliver the fluid to the downstream part of the process at the desired pressure, while avoiding surge. In the industrial practice two separate PID controllers are usually employed: the pressure controller and the antisurge controller. The pressure controller has a cascade control structure. The slave loop is a speed controller. Its set point is the output of the master loop and the manipulated variable is the torque of the driver. The master loop of the cascade controller is a pressure controller. Its controlled variable is the outlet pressure of the compressor, while its output is the remote set point of the slave loop. The pressure controller has been tuned using initially the lambda tuning technique and then trial and error testing.

Fig. 2. Schematic representation of process and control system.

The antisurge controller continuously monitors the inlet flow rate of the compressor, which is its controlled variable. If the 11

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problem. The MPC is based on the linearized version of the plant model. The constraints are the lower and upper boundaries for both controlled variables 𝑦𝑦𝑖𝑖 and manipulated variable 𝑢𝑢𝑗𝑗 . The optimisation function contains weights for both manipulated variables and process output variables. For the control problem presented in this paper, states 𝑥𝑥𝑚𝑚 are 𝑝𝑝, 𝑚𝑚 and 𝜔𝜔, disturbances 𝑑𝑑𝑘𝑘 are 𝑝𝑝𝑖𝑖𝑖𝑖 and 𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 , manipulated variables 𝑢𝑢𝑗𝑗 are 𝜏𝜏𝑑𝑑 and the position of the antisurge valve 𝐴𝐴𝐴𝐴𝐴𝐴, outputs 𝑦𝑦𝑖𝑖 are pressure 𝑝𝑝, mass flow rate 𝑚𝑚, rotational shaft speed 𝑁𝑁 and compressor outlet temperature 𝑇𝑇02 . 𝑝𝑝 is the controlled variable as it has to be at its set point, while 𝑚𝑚, 𝑁𝑁 and 𝑇𝑇02 have to be within their operating range, according to the following equations: (11)

𝑁𝑁𝑚𝑚𝑚𝑚𝑚𝑚 ≤ 𝑁𝑁 ≤ 𝑁𝑁𝑚𝑚𝑚𝑚𝑚𝑚

(12)

𝑇𝑇02 ≤ 𝑇𝑇02,𝑚𝑚𝑚𝑚𝑚𝑚

Input and output weights 𝜏𝜏𝑑𝑑 𝐴𝐴𝐴𝐴𝐴𝐴 𝑝𝑝 𝑚𝑚 0 10 1 0.08 0 10 1 0.72 0 0.1 1 0.08

Tuning set Set 1 Set 2 Set 3

closure of inlet and outlet valves of the system. Various simulations were run within these three scenarios and some representative results have been reported in the paper. The response of the control system was evaluated using graphical comparison and also via two different performance parameters. The first parameter is called 𝑀𝑀𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 and it represents the dimensionless total amount of gas recycled during a certain disturbance:

𝑀𝑀𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 =

(13)

𝑡𝑡

𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑚𝑚𝑟𝑟 𝑑𝑑𝑑𝑑 ∫𝑡𝑡=0

(14)

𝑡𝑡=1ℎ

∫𝑡𝑡=0 𝑚𝑚𝑚𝑚𝑚𝑚

where t final represents the time interval considered for the analysis. The second parameter is the Integral of Squared Error (ISE), where the error is the difference between the controlled variable p and its set point pSP over time:

Minimum and maximum rotational shaft speeds depends on the driver while the constraint on the maximum temperature guarantees the integrity of the machine during hot gas recycle.

𝐼𝐼𝐼𝐼𝐼𝐼 = ∫

4. SIMULATION RESULTS

𝑡𝑡𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓

𝑡𝑡=0

4.1 Tuning of the model predictive controller

(𝑝𝑝(𝑡𝑡) − 𝑝𝑝𝑆𝑆𝑆𝑆 (𝑡𝑡))2 𝑑𝑑𝑑𝑑

(15)

4.3 Results

The MPC controller has been tuned in order to guarantee good pressure control while avoiding as much as possible the opening of the antisurge valve. The reason for doing that is that gas recycle increases the operating cost of the system as more gas must be compressed by the machine without being delivered. Three different sets of control tuning parameters have been defined and they have been summarised in Table 2. The first tuning set was called ‘set 1’ and it is better performing with regards to pressure control. The second tuning set was called ‘set 2’ and it is more robust towards boundary disturbance. Both these two tuning sets aim at the minimisation of the opening of the recycle valve. A third set of tuning parameters, called ‘set 3’, was defined. It performs well in terms of pressure control however it does not minimise the gas recycle. The control tuning parameters are called weights in the MPC formulation. The simulation scenarios tested in this paper come from the literature and also from industrial practice (Dukle and Narayanan, 2003, Patel et al., 2007, Wu et al., 2007) . They have been proposed in the past for the validation of antisurge controller.

In figures 3 and 4 the inputs and the outputs of the plant are represented for a process boundary disturbance and different control configurations. The disturbance is a positive pulse change of the outlet pressure of the system 𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 . The positive step change takes place at time 𝑡𝑡=100 seconds while the negative step change takes place at 𝑡𝑡 =800 seconds. In Figure 3 the response of the system under the control of a traditional

Driver torque

Inputs 1.5

1 d

1

ASV 0.5

0.5 0

500

1000 Time (s)

1500

0

1.5

Outputs

Pressure

1.005

4.2 Simulation scenarios and performance parameters The first validation scenario includes process disturbances that can affect the operation of the plant. Inlet and outlet pressures of the system were selected as disturbance variables. The second validation scenario is the load pattern. The pressure set point was changed and the response of the system was recorded. The third scenario includes the step

p

pSP

m

mctrl 1

1 0.995 0

Opening of antisurge valve (0 closed, 1 open)

𝑚𝑚 ≥ 𝑚𝑚𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

Table 2. Control tuning parameters

200

400

600

Mass flow rate

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0.5 800 1000 1200 1400 1600 1800 Time (s)

Fig. 3. PI control of process disturbance - inputs (a) and outputs (b) 12

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Table 3. Dimensionless amount of gas recycled 𝑴𝑴𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅

PI controller is represented. In the first graph (Figure 3-a) the driver torque (left) and the opening of the antisurge valve (right) are represented. These two variables are the manipulated variables of the compression system. In the second graph (Figure 3-b) the compressor outlet pressure and its set point (left) and the mass flow rate and its lower limit (right) are represented. These variables are the main controlled variables of the compression system. When 𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 increases the pressure controller reduces 𝜏𝜏𝑑𝑑 in order to reduce 𝑝𝑝. This action reduces the flow rate through the machine 𝑚𝑚 as well. When this variable becomes equal to the surge control value 𝑚𝑚𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 the antisurge controller opens the antisurge valve. However this action causes the reduction of the pressure 𝑝𝑝. Therefore the pressure controller decreases 𝜏𝜏𝑑𝑑 and 𝑚𝑚 increases. When 𝑚𝑚 becomes bigger than 𝑚𝑚𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 the antisurge controller closes the antisurge valve. The consequence is the increase of 𝑝𝑝 above its set point, that brings the pressure controller to reduce 𝜏𝜏𝑑𝑑 and therefore reproduces the same behaviour. The result is the oscillation of the system that is interrupted only by the end of the pulse disturbance. When the outlet pressure of the system goes back to its design value, the system stabilise to the previous steady state point.

0.95 0.5 0.9

d

0.85 0

500

1000 Time (s)

ASV 0

1500

0.8 p

0.995 0

200

400

600

pSP

m

mctrl

0.7

0.148 0.130 0.179 0.222

0.090 0.078 0.004 0.090

Disturbance

Set 2

Set 1

𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 𝑝𝑝𝑠𝑠𝑠𝑠 𝑉𝑉𝑖𝑖𝑖𝑖 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜

3.54·104 2.64·104 4.12·103 8.42·104

1.38·104 1.01·104 4.22·103 1.42·104

Relative difference % -60.9 -61.8 2.3 -83.2

Set 1 0.229 1.68·104

Set 3 1.769 5.67·103

In all the tested cases the controller under tuning set 1 has allowed to recycle a smaller amount of gas (Table 3) while better controlling the pressure (Table 4). The only occurrence in which the controller under tuning set 1 has a higher 𝐼𝐼𝐼𝐼𝐼𝐼 than the controller under tuning set 2 was due to saturation of the torque. In fact in this case the speed of the driver arrived to its maximum value. Disturbance rejection of the pulse disturbance of the outlet pressure was also performed using the third tuning set. This allowed a much tighter control of the pressure however it involved a bigger amount of gas recycled over the duration of the transient (Table 5). 5. CONCLUSION Different disturbance scenarios and controller tuning were tested and the results demonstrate that the proposed controller is effective for both disturbance rejection and set point tracking. The results demonstrate that the MPC controller is able to control the outlet pressure of the compressor while avoiding surge. They also demonstrate that the MPC controller is more suitable than PI controller for this multipleinput multiple-output process system. Under certain disturbances it is not possible to keep the pressure at its set point while avoiding surge without recycling. Therefore the tuning of the controller was performed in order to give priority to respectively the minimisation of the gas recycle (tuning set 1), the stability and protection of the system under aggressive disturbances (tuning set 2) and the control of the outlet pressure (tuning set 3). In all these cases the MPC controller performed as requested. The decision regarding the

Mass flow rate

Pressure

0.9 1

𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 𝑝𝑝𝑠𝑠𝑠𝑠 𝑉𝑉𝑖𝑖𝑖𝑖 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜

Relative difference % -39.6 -39.8 -97.8 -59.5

pressure of the system 𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 , pressure set point 𝑝𝑝𝑆𝑆𝑆𝑆 , inlet valve 𝑉𝑉𝑖𝑖𝑖𝑖 and outlet valve 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 , 𝑀𝑀𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 and 𝐼𝐼𝐼𝐼𝐼𝐼 have been estimated.

Outputs 1.005

Set 1

Parameter 𝑀𝑀𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐼𝐼𝐼𝐼𝐼𝐼

Opening of antisurge valve (0 closed, 1 open)

Driver torque

1

Set 2

Table 5. Comparison between tight and loose recycle minimisation

Other simulations were run in order to compare the first and second tuning sets. A summary of the results is collected in Tables 3 and 4. For disturbances such as step change of outlet Inputs

Disturbance

Table 4. Integral of the square error for pressure control 𝑰𝑰𝑰𝑰𝑰𝑰

Figure 4 represents the response of the system under the same disturbance but controlled via MPC. The tuning set 1 was employed for the model predictive controller. Following the process disturbance, the MPC controller reduces 𝜏𝜏𝑑𝑑 while barely moves 𝐴𝐴𝐴𝐴𝐴𝐴. The pressure 𝑝𝑝 is kept within its constraints but not tightly closer to its set point as this would force the control system to open the antisurge valve. These results demonstrate that the MPC controller is able to control the outlet pressure without causing the oscillation of the system.

1

13

800 1000 1200 1400 1600 1800 Time (s)

Fig. 4. MPC controller of process disturbance – input (a) and outputs (b)

13

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type of tuning to adopt depends on many factors and cannot be generalised. Possible saturation of the manipulated variable must be taken into account as it can reduce the performance of the control system.

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ACKNOWLEDGEMEMENTS Financial support from the Marie Curie FP7-ITN project "Energy savings from smart operation of electrical, process and mechanical equipment– ENERGY-SMARTOPS", Contract No: PITN-GA-2010-264940 is gratefully acknowledged. The authors would like to thank Bob Hodder and Mark Dixon of ESD Simulation Training for their encouragement, advice and technical support for the project. REFERENCES Arnulfi, G. L., Blanchini, F., Giannattasio, P., Micheli, D. & Pinamonti, P. (2006). Extensive study on the control of centrifugal compressor surge. Proceedings of the Institution of Mechanical Engineers Part A Journal of Power and Energy, 220, 289-304. Botros, K. K. (2011). Single versus dual recycle system dynamics of high pressure ratio, low inertia centrifugal compressor stations. Journal of Engineering for Gas Turbines and PowerTransactions of the ASME, 133, 122402: 1-12. Boyce, M. P. (2012). Gas Turbine Engineering Handbook Edition), Elsevier. Available: (4th http://app.knovel.com/hotlink/toc/id:kpGTEHE017/ gas-turbine-engineering-2/gas-turbine-engineering-2 Cortinovis, A., Pareschi, D., Mercangoez, M. & Besselmann, T. (2012). Model predictive anti-surge control of centrifugal compressors with variable-speed drives. IFAC Workshop on Automatic Control in Offshore Oil and Gas Production, 31 May - 1 June 2012 Norwegian University of Science and Technology, Trondheim, Norway. Dukle, N. & Narayanan, K. (2003). Validating anti-surge control systems. Available: http://www.eptq.com/view_edition.aspx?strYID=20 03. Fink, D. A., Cumpsty, N. A. & Greitzer, E. M. (1992). Surge dynamics in a free-spool centrifugal-compressor system. Journal of Turbomachinery-Transactions of the ASME, 114, 321-332. Gravdahl, J. T. & Egeland, O. (1999). Centrifugal compressor surge and speed control. IEEE Transactions on Control Systems Technology, 7, 567-579. Gravdahl, J. T., Egeland, O. & Vatland, S. O. (2002). Drive torque actuation in active surge control of centrifugal compressors. Automatica, 38, 18811893. Greitzer, E. M. (1976). Surge and rotating stall in axial-flow compressors. 1. Theoretical compression system model. Journal of Engineering for PowerTransactions of the ASME, 98, 190-198.

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