Arab J Sci Eng (2013) 38:1405–1414 DOI 10.1007/s13369-013-0601-6
RESEARCH ARTICLE - ELECTRICAL ENGINEERING
PID and Novel Approach of PI Fuzzy Logic Controllers for Active Surge in Centrifugal Compressor B. Chetate · R. Zamoum · A. Fegriche · M. Boumdin
Received: 25 March 2011 / Accepted: 22 July 2011 / Published online: 13 March 2013 © King Fahd University of Petroleum and Minerals 2013
Abstract The operating range of aerodynamic compressors is usually limited by a phenomenon known as surge. Active surge control has showed the ability to extend the operating range significantly. This study presents a solution to this problem based on classical PID regulator and a new PI fuzzy logic approach with three lines of protection. The fuzzy controller is designed to avoid the surge instability on a given compressor model. Simulation studies show promising results at different operating points compared to the results obtained using PID controller. Keywords Centrifugal compressors · Active surge control · PID controller · Fuzzy logic control · Compression system · Damping of compressor · Aerodynamic compressors · Recycling gases
B. Chetate (B) · R. Zamoum · A. Fegriche · M. Boumdin Laboratoire de Recherche sur l’Electrification des Entreprises Industrielles, Université M’hamad Bougara de Boumerdes, Avenue de l’indépendance, 35000 Boumerdes, Algeria e-mail:
[email protected] R. Zamoum e-mail:
[email protected] A. Fegriche e-mail:
[email protected]
List of Symbols [1,2] w wf wr wt ap cp v1 v2 P01 P1 P2 T01 A L c (w, ) τd τc μ r1 r2 a kf k cp cv ξ cf cr J
The mass flow (kg/s) The feed flow (kg/s) The recycle flow (kg/s) The throttle flow (kg/s) The speed of sound = 343 m/s Specific heat at constant pressure (1,005 (J/kg K)) Suction plenum volume 0.05 m3 Discharge plenum volume 0.1 m3 ambient pressure 101,325 Pa Section plenum pressure (Pa) Discharge plenum pressure (Pa) ambient temperature 20 ◦ C Duct area 0.07 m Duct length 2.85 m The pressure rise Drive torque Compressor torque Energy transfer coefficient 0.99 Inducer perimeter radius 0.0395 m Impeller perimeter radius 0.0565 m Constant of incidence loss Friction constant Isentropic exponent Specific heat at constant pressure Specific heat at constant volume 1,000 The percentage feed opening valve The percentage recycle opening valve Pressure of the feed valve The speed (rad/s) The impeller inertia 5e−4 kg m2
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Abbreviations Pupstream Upstream pressure DSL Low deviation to the control line DSLL Low Low deviation to the control line SCL Surge control line DSH High deviation between operating point and control line (surge region) 1 Introduction Not long ago, the recovery recycling of the associated gases which were evacuated towards torches was not economically profitable. But the introduction of new compression systems which could treat enormous quantities of gas, allows us not only to recycle these gases but also to increase the production of crude oil. These machines are subjected to a very important problem known as the phenomenon of pumping, which is in fact a state of instability of the operating pulling pulsations which can be dangerous for their mechanical holding by provoking the inversion of the axial push of the rotor, vibrations at the level of the compressor and the break of fins due to the inversion of the flow. Thus, it is essential to maintain the operating point of the compressor except the zone of instability [3]. This phenomenon appears in the zones of the operating point corresponding to the section flow’s of narrowband and this generally happens at the moment the characteristic curve “flow-compression ratio” reaches its maximum and is characterized by a limit cycle in the compressor’s characteristic [4]. The stability of the operating point is insured by the association of the characteristics of the compressor and the plenums at the same time. A decrease of the flow causes the translation of the operating point to the instability zone. An increase of the throttle volume can cause the re-circulation of the gas in the inverse sense of the flow creating vibrations inside the compressor [4]. To prevent this phenomenon, the compressors are endowed with anti-pumping regulation systems to maintain the compressor in the stable region. Whatever the compression rise, the section flow will always be greater than the flow corresponding to the surge. This is achieved by sending back the discharge gas to plenum section using the recycle valve which causes a decrease of gas production. The proposed solutions to prevent this phenomenon did not show encouraging results until now, because these systems always present a risk to pass to the pumping phase. Indeed, the current systems of regulation surge present several inconveniences, namely:
• •
Creation of radial vibrations of the rotor which can provoke a damage of the system; The operating point is in some cases in the surge region even if the recycling valve is completely opened.
The appearance and the development of new techniques in the field of the automation provoked considerable changes in the conception of regulation systems. To resolve the problem of the surge and to improve the performances of the system, we applied a classical PID and a new fuzzy PI for the optimal adjustment of the regulator to protect the compressor in the stable or unstable operating region. In Algeria, the main economic activity concerns the industries of exploitation, transport and transformation of oil and gas. A great economic gain will be the result of the improvement in the performances of centrifugal compressors; in particular, the reduction of the cost of consumption products (gas, electricity, etc.).
2 The Compressor Model The centrifugal and axial compressors show instability of flow according to [4]. The compression system is modeled as in Fig. 1, with a compressor, a duct of length L, a plenum of volume Vp , a throttle, and a drive unit imparting a torque on the compressor [5,6]. The model is p˙ p =
ap2 Vp
(w − wt )
(1)
w˙ =
A (ψc (w, ) p01 − pp ) L
(2)
˙ =
1 (τd − τc ) J
(3)
Throttle valve P01
Compressor P02
A
Duct
W
•
Gas losses because of the permanent opening of the recycling valve; A great time response of the regulators;
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Wt
L Discharge Plenum
τc
Ω τd
3
•
Vp , P p , Tp
Drive
Fig. 1 The compression system [2]
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where kt > 0 is a parameter proportional to throttle opening [5,9]. Or the equivalent throttle flow is:
Recycle valve
Compressor
wt = tanh(ξ( p2 − p01 )) kt ( p2 − p01 ) tanh(ξ( p2 − p01 ))|ξ 0
Feed valve
V2 , P2 , T2
V1 , P1 , T1
(11) Duct Throttle valve
The feed flow to the suction plenum is: wf = cf Pupstream − p1
Suction Plenum Discharge Plenum
3
(12)
The recycling flow from the discharge plenum to the suction plenum is: √ wr = cr p2 − p1 (13)
Drive
Fig. 2 The recycle compression system
A classical result in the field of compressor surge modeling is the Greitzer model [3] which models a basic compression system consisting of a compressor, a plenum volume, a throttle valve, a duct as shown in Fig. 2. To study the control of the system with classical PID and new PI fuzzy logic controllers, we need a model that takes variable speeds. In [7], the model of Greitzer was further developed, and rotational speed was included as a state in the model. A similar model developed by Gravdahl and Egeland [4] is used:
3 The Recycle Compression System
p˙ 1 =
p˙ 2 = w˙ =
ap2
(wf + wr − w)
(4)
(w + wt − wr )
(5)
A (ψc (W, ) p1 − p2 ) L
(6)
V2 ap2 V2
1 (τd − τc ) (7) J The pressure ratio is given by [4,8]: k μr22 2 − 21 r12 ( − αw)2 − kf w 2 k−1 ψc (w, ) = 1 + cp T01 ˙ =
(8) k=
cp cv
The flow through the throttle is now given as: √ wt = kt p2 − p01 ,
(9)
(10)
4 Recycle Compression System in Open Loop To demonstrate the behavior of the system in the unstable region of the compressor, we have done the following: Initially, for simulation, the pressures in the two volumes are set to ambient pressure, the compressor speed is set to zero, the mass flow is set to zero too, while throttle flow with feed flow is taken at their initial states. Then we startup the system and after its stabilization at time t = 0.5 s, we gradually decrease the feed flow to create perturbations on the compressor at time t = 2 s, eventually causing the system to enter in the unstable region (damping or surge) at time t = 3.25 s where the discharge pressure decreases from 11.79×104 Pa to 10.13×104 Pa (Fig. 3c),the suction pressure decreases from 3.5×104 Pa to 0.17×104 Pa (Fig. 3c), the mass flow enters in damping (oscillation) at time t = 3.25 s (Fig. 3a) and the rotating speed increases from 1,750 to 3,700 rad/s with an overshoot of 12 % (Fig. 3d). This increase of speed is due to the small load applied on the impellers. To eliminate the surge, the recycle valve will be opened manually at time t = 6 s (Fig. 3b) to allow more mass flow through the system by compensation of the flow in the suction plenum (Fig. 2) to stabilize the system. The higher mass flow encourages a shift to the right of the operating point, and surge should disappear. When the recycle valve is opened: • • •
The compressor speed decreases due to the higher mass flow load applied on the impellers and stabilizes at 2,941 rad/s. The mass flow stabilizes at 0.3 kg/s The suction and the discharge pressures stabilize at 0.67 × 104 Pa and 10.14 × 104 Pa, respectively.
When the compressor enters in the surge phase, the operating point oscillates between negative and positive flow. But after the opening of the recycle valve, the operating point is located in the region of positive flow (Fig. 3e, f).
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1.5
Feed flow Recycle flow
0.6
Feed & Recycle flows (kg/s)
1
Flow (kg/s)
0.5
0
-0.5
-1
-1.5
12
0
1
x 10
2
3
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9
0.5
0.4
0.3
0.2
0.1
0
10
0
1
2
3
4
5
6
7
8
t(s)
t(s)
(a) Mass Flow
(b) Feed & recycle flows
9
10
4
4500 Discharge pressure Suction pressure
4000
10
Speed (rad/s)
Pressures (Pascal)
3500 8
6
4
3000 2500 2000 1500
2 1000 0
-2
500 0 0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
t(s)
t(s)
(c) Discharge & suction Pressures
(d) Speed of compressor
1.4
1.2
1.15
Unstable region 1
Stable region
0.8
Pressure ratio
Pressure ratio
1.1
0.6
1.05
1
0.4 0.95 0.2
0 -1.5
0.9 -1
-0.5
0
0.5
Flow (kg/s)
(e) Operating point Fig. 3 The open loop recycle system simulation results
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1
1.5
-1
-0.5
0
0.5
Flow (kg/s)
(f) Zoom of Operating point
1
10
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5 Recycle Compression System With PID Controller The mostly used regulators in industry are the PID-controllers recycle valve. The controller needs to know about the current operating point of the system, and then compare it to the surge avoidance line. The compressor is protected against surge by an anti-surge valve connecting compressor discharge to the suction plenum, thus increasing the flow in the compressor to return it out of the surge area. The surge regulation is intended to open this valve as soon as the flow in the compressor is getting too close to the surge flow. Anti-surge control has the job of maintaining the compressor in a stable operating range assuring a higher suction capacity than the surging rate whatever the compression ratio may be. When the compressor flow rate is below the flow specified for the protection margin at a given pressure ratio, the controller should send a control signal to the valve to open. The rate of opening or closing should be based on the speed required to protect the compressor. The setpoint tracking (reference) for the PID-controller of the recycle valve is the surge control line (SCL) (or protection line with color black is shown in Fig. 4e) and is given by [4]: wSCL (ψc ) =
ψSCL (w) − b , a
(14)
where a is the slop of protection line, b is the horizontal surge margin to the surge line, ψSCL is the pressure ratio of the SCL, wSCL is the flow of the SCL. To demonstrate the benefit of the PID controller, we have done the following: Firstly, we startup the system and after its stabilization, we close the feed flow valve at t = 2 s to disturb the system (Fig. 4b). At this moment, the tuned PID-controller (with K p = 0.048, K i = 0.0009 and K d = 0.0001) opens gradually the recycle valve (Fig. 4b) to compensate the flow in the suction plenum (Fig. 2) to stabilize the system. As a result: • The mass flow decreases to 0.281 kg/s with an undershoot of 23 % (Fig. 4a) which is due to the time delay of 0.3 second of the PID controller; • The suction pressure decreases from 3.46 × 104 Pa and stabilizes at 0.52 × 104 Pa (Fig. 4c); • The discharge pressure decreases from 11.78 × 104 Pa and stabilizes at 10.13 × 104 Pa (Fig. 4c); • The rotation speed increases from 1,697 rad/s and stabilizes at 2,991 rad/s with an overshoot of 2 % (Fig. 4d). The operating point passes over the control line (Fig. 4e, f) and the controller brings it back to the control line. We observe that the operating point never reaches the surge line
(The operating point is situated to the left of the surge line only during startup, so it is better to start the compression system manually).
6 Recycle Compression System With Proposed Fuzzy PI Controller Since the problem is a nonlinear one, it makes sense to use a fuzzy control law [10]. The fuzzy control rules are particularly understandable by process operators, who may have difficulties in understanding sophisticated nonlinear controllers. The main objective of this study is to control the opening valve by referring to a three-degree protection lines (DSL, DSLL, SCL) left to the surge lines (DSH) (Fig. 5). The set point tracking of the control valve opening is characterized by three levels given by the protection lines (DSL, DSLL, SCL) left to the surge lines (DSH) (Fig. 5e). ; • The SCL is obtained by: wSCL (ψc ) = ψSCL (w)−b a • The DSH is shifted upstream by the distance margin 0.4 % to SCL; • The DSLL is shifted downstream by the distance margin 0.2 % to SCL; • The DSL is shifted downstream by the distance margin 0.5 % to SCL;
6.1 Structure of the Controller The controller is a Mamdani-type fuzzy logic controller with a typical IF-THEN rule structure. The controller consists of two inputs and one output as shown in Fig. 6. The first input is the difference between the mass flow and the reference flow needed to avoid surge, the second input is a slight change of the flow. The output is the signal that determines the necessary fractional opening of the recycling valve to stabilize the system. The range of the mass flow used is the one provided by the manufacturer in the user guide. The pressure ratio is a function of the mass flow for different speeds. In our case, the range of the mass flow, specific to our compressor, is set to be from 0 to 90 kg/s. The simulation results show that, for good performances, the controller should be very sensitive to the changes in the flow. Therefore, the change of flow range which represents a margin of the flow (Fig. 5) was set to be from −0.001 to 0.001 kg/s or 0.1 % left and right of each protection line (SCL, DSLL, DSL), the controller doing a specific opening (closed, small open, middle open, great open and open fast as Table1). Even though this is a small range, it nonetheless provides enough information for the controller to operate
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0.6
Recycle flow
Feed & Recycle flows (kg/s)
Flow (kg/s)
0.4
0.3
0.2
0.1
0.5 0.4 0.3 0.2 0.1
0
-0.1
feed flow
0.6
0.5
0 -0.1 0
1
2
3
4
5
6
7
8
9
0
10
1
2
3
4
5
6
7
8
9
10
8
9
10
t(s)
t(s)
(a) Mass Flow
(b) Feed & recycle flows
4
12
x 10
3500
Suction pressure discharge pressure
3000
Speed (rad/s)
Pressures (Pascal)
10
8
6
2500
2000
1500
4 1000 2
0
500
0
1
2
3
4
5
6
7
8
9
0
10
0
1
2
3
4
5
6
7
t(s)
t(s)
(c) Discharge & suction Pressures
(d) Speed of compressor
1.8 Operating point Protection Line
1.6
Surge Limit
1.15
1.4 1.2
1.1
Pressure ratio
Pressure ratio
Unstable region 1
Stable region
0.8 0.6 0.4
1.05
1
0.2
0.95
0 -0.2 -0.1
0
0.1
0.2
0.3
0.4
Flow (kg/s)
(e) Operating point Fig. 4 The closed loop recycle system simulation results
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0.5
0.6
0.25
0.3
0.35
0.4
Flow (kg/s)
(f) Zoom of Operating point
0.45
0.5
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Pressure Ratio
DSH SCLDSLLDSL
the stable region (to the right of the surge line) from the unstable region (to the left of the surge line). And the stable region is limited with three protection lines (SCL, DSLL and DSL) to prevent surge progressively. The control action that needs to be taken in this case is to open the recycling valve gradually (closed, small open, middle open, great open and open fast). The operating point is left to the surge line only during startup, so is better to start the compression system manually. Figure 8 shows the control surface for the fuzzy controller. It can also be realized from the control surface that the maximum controller output is 1 which corresponds to a fully opened valve and its minimum value is zero which corresponds to a fully closed valve. Hence, our new PI fuzzy controller is the result of the 15 rules for opening valve shown in (Table 1)
Surge Region 0.5%
Safe Region
0.2%
0.4%
Flow Fig. 5 The outlines of recycle controller
properly, thus bringing the operating point to the control line (Figs. 5, 7). 6.2 Fuzzy Rules
6.3 Analysis and Results
Consider pressure ratio of Fig. 5, which shows the operating point of the compressor, where the surge line (DSH) separates
To demonstrate the benefit of the PI fuzzy controller, we have done the following:
memebership of flow region memebership of change of flow
SCL (line of protection) DSH(limit of surge region) DSLL DSL SafeRegion
0,8
1,0 Z N P
Degree of membership
Degree of membership
1,0
0,6
0,4
0,2
0,0
0
20
40
Flow Region
60
0,6
0,4
0,2
0,0 -1,0
80
-0,5
(kg/s)
0,0
0,5
Change of Flow
(kg/s)
·10 -3
1,0
Membership of valve opening
1,0
Degree of membership
0,8
Close SO MO GO OpenFast
0,8
0,6
0,4
0,2
0,0 0,0
0,2
0,4
0,6
0,8
1,0
Valve opening%
Fig. 6 Inputs and output membership functions. SO small open, MO middle open, GO great open
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Table 1 Fuzzy controller rules
Change of flow (kg/s)
Flow (kg/s) Surge region DSH
SCL
DSLL
DSL
Safe region Close
P
GO
MO
SO
Close
Z
GO
MO
SO
Close
Close
N
Open fast
GO
MO
SO
Close
Firstly, we startup the system and after its stabilization, we close the feed flow valve at t = 2 s to disturb the system (Fig. 9b). At this moment, the tuned PI Fuzzy controller (with a K p = 0.45 and K i = 0.55) opens gradually the recycle valve (Fig. 9b) to compensate the flow in the suction plenum (Fig. 2) to stabilize the system. Consequently,
Fig. 7 Characteristic of the compressor
Fig. 8 Control surface for fuzzy controller
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• The mass flow decreases to 0.282 kg/s without undershoot (Fig. 9a), • The suction pressure decreases from 6.30 × 104 Pa and stabilizes at 0.52 × 104 Pa (Fig. 9c) • The discharge pressure decreases from 11.95 × 104 Pa and stabilizes at 10.13 × 104 Pa (Fig. 9c) • The rotation speed increases from 1,173 rad/s to stabilize at 3,027 rad/s without causing any overshoot because the recycle valve is opened progressively from small open to fast open (Fig. 9d). • The operating point passes over the control line (Fig. 9e, f) and the controller brings it back to the three control lines. We observe that the operating point never reaches the
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0.8
Feed & Recycle flows (kg/s)
0.9
Flow (kg/s)
0.7 0.6 0.5 0.4 0.3 0.2 0.1
Recycle flow
0.8
Feed flow
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0 0
1
2
3
4
5
6
7
8
9
-0.1
10
0
1
2
3
4
5
t(s)
6
7
8
9
10
8
9
10
t(s)
(a) Mass Flow
(b) Feed & recycle flows
4
x 10
12
3500 Suction pressure
3000
10
Speed (rad/s)
Pressures (Pascal)
Discharge pressure
8
6
4
2000 1500 1000
2
0
2500
500 0 0
1
2
3
4
5
6
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9
0
10
1
2
3
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5
t(s)
6
7
t(s)
(c) Discharge & suction Pressures
(d) Speed of compressor
2.5 Operating point Protection Line
2
1.2
Surge Limit DSLL
1.15
1.5
Pressure ratio
Pressure ratio
DSL
1 0.5 0
1.05 1 0.95
-0.5 -1
1.1
0.9 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Flow (kg/s)
(e) Operating point
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Flow (kg/s)
(f) Zoom of operating point
Fig. 9 The closed loop recycle system simulation results with PI fuzzy logic controller
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surge line (The operating point is situated to the left to the surge line only during startup, so it is better to start the compression system manually).
7 Conclusion After closing the feed flow valve, the PID controller stabilizes the mass flow within 1.5 s with some overshoot and oscillations (which are vibrations which can cause the rupture of the wings of the compressor), and stabilizes also the speed after 3 s. But the new PI fuzzy logic strategy to control the surge instability presented in this paper improves the performances of compression systems. In fact, after the closing of the feed flow valve, the mass flow is stabilized within 1.3 s without overshoot while the speed is stabilized after 2 s. The fuzzy logic controller uses the mass flow and its change to determine the appropriate action to take: the opening valve will be opened gradually from small open to open fast referring to the three control lines. The advantage of using the change of mass flow signal is that the controller does not require to know where the operating point is located. In the fuzzy controller, the opening of the valve is gradual from small open to open fast (and the operation of the opening can return back from open fast to small open) referring to the three control lines. But in the PID controller, the opening of the valve is a ramp increasing from closed (0 %) to completely open (100 %) and eventually the system will be manually stopped when the recycle valve is completely opened and the compressor is operating in the surge region.
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References 1. Andreassen, A.: Stabiliserende regulering av kompressor. NTNU (2001) 2. Bøjrn, O.B.; Gravdahl, J. T.: The Recycle Compression System. Master of Science in Engineering, NTNU (2010) 3. Greitzer, E.: Surge and rotating stall in axial flow compressors: theoretical compression system model. J. Eng. Power 98, 190–198 (1976) 4. Gravdahl, J.T.; Egeland, O.: Compressor surge and rotating stall: modeling and control. In: Advances in Industrial Control. Springer, London (1999) 5. Bøhagen, B.; Gravdahl, J. T.: Active surge control using drive torque: dynamic control laws. In: Proceedings of the 45th IEEE Conference on Decision and Control (2006) 6. Bøhagen, B.: Active surge control of centrifugal compression systems. PhD thesis, NTNU (2007) 7. Fink, D.; Cumpsty, N.; Greitzer, E.: Surge dynamics in a freespool centrifugal compressor system. J. Turbomach. 114, 321–332 (1992) 8. Bøhagen, B.; Gravdahl, J.T.: On active surge control of compressors using a mass fow observer. In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas (2002) 9. Bøhagen, B.; Gravdahl, J.T.: Active control of compression systems using drive torque; a backstepping approach. In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (2005) 10. Al-Mawali, S.H.T.; Zhang, J.: Compressor surge control using a variable area throttle and fuzzy logic control. In: Transactions of the Institute of Measurement and Control Online First, February, 2010 11. Bøhagen, B.; Stene, O.; Gravdahl, J.T.: A GES mass fow observer for compression systems: Design and experiments. In: Proceedings of the American Control Conference, Boston, June 2004