Control of Compressor Surge with Active Magnetic ... - Semantic Scholar
49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA
Control of Compressor Surge with Active Magnetic Bearings Se Young Yoon †, Zongli Lin† , Chris Goyne‡ and Paul E. Allaire‡
Abstract— The design of an active surge controller, which employs Active Magnetic Bearings (AMBs) to stabilize the flow in the compressor, is presented in this paper. The axial tip clearance of an unshrouded centrifugal impeller from a single stage compressor is modulated relative to the static shroud to induce a pressure variation in the compressor output pressure. This fast-response, high-precision control of the output pressure is employed to stabilize the compressor operating in surge condition. The control law for the tip clearance is derived based on a recently introduced mathematical model, and the performance of the controller is tested through simulation.
I. I NTRODUCTION Surge is a dynamics instability in compressors, characterized by large-amplitude axisymmetric flow oscillations. Surge instability can cause extensive damage to the system, especially during the reversal of the flow in deep surge condition, when the impeller and the casing are subject to high loads and temperatures. Two different approaches exist to deal with the surge instability in turbomachineries. The most common approach in industry is surge avoidance, where the compressor is operated in a conservative way within a predetermined stable operating region, and safety mechanisms “reset” the compressor back to the stable operation if surge occurs. This leads to a significant drop in compressor efficiency by limiting the compressor output. The second approach is surge control, where an active or passive surge controller is implemented such that it compensates for the disturbances of surge, and stabilizes the flow in the compression system. More details on surge avoidance and control can be found in [1]. The modeling of compression systems for the active control of surge instabilities has been investigated intensively over the years, motivated by the potential benefits from expanding the stable operating region of these machineries. Reviews of modeling techniques for compression systems can be found in [1] and [2]. Emmons et al. presented in [3] one of the first stability analysis for a compression system using a linearized model, where the initiation of the surge oscillations was captured by employing the analogy between the compression system and a self-excited Helmholtz resonator. Using the same principle, Greitzer [4] later introduced a two-state nonlinear lumped-parameter model of a compression system. This model was first derived for axial compressors, but it was later demonstrated in [5] that it is † S. Y. Yoon and Z. Lin are with Charles L., Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904-4743, USA ‡ C. Goyne, and P. E. Allaire are with the Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 229044746, USA
978-1-4244-7746-3/10/$26.00 ©2010 IEEE
also applicable for centrifugal compressors. After this, many improved and advanced models for axial and centrifugal compression systems were presented by different researchers in [6]–[12], and stability of compression systems for various configurations of the inlet and the exhaust piping was studied in [13], [14] and [15]. However the original Greitzer model is still generally preferred in designing active surge controllers due to its low order and simple implementation of the controller designed based on it. Efforts to develop reliable model based surge controllers have been ongoing, fueled by the increasing need of boosting productivity and improving the safety of the workspace. Surge suppression and control methods rely on accurate plant models to design a surge control law that is able to stabilize the compression system pass the surge point. Passive surge controllers rely on passive systems, that react to environmental changes in the compressor and suppress the surge oscillations. On the other hand, active surge controllers, which were first proposed by Epstein et al. in [16], employ actuators to perturb the compression system according to feedback measurements from sensors in the flow path. A brief review of published works on model based surge control, with both active and passive systems, were presented by Arnulfi et al. in [17]. An important point to consider in designing a surge control system is the selection of the appropriate actuator. Simon et al. presented in [18] an overview of commonly employed actuators for surge control, and demonstrated the complexity of selecting an accessible actuator with sufficient control authority on the compressor’s unstable modes and bandwidth to stabilize the flow dynamics during surge. Common actuators for passive control systems are the movable plenum wall and the hydraulic oscillator, which induce a pressure variation in the compression system by changing the plenum volume. On the other hand, and common actuators in active control schemes are the closecoupled valve [11], and the throttle valve [19]. A less considered but also promising actuator for the control of compressor surge is the Active Magnetic Bearing (AMB). The use of AMBs for the control of other compressor instabilities was proposed in [20]. On the other hand, the use of the AMB for the control of surge in singlestage centrifugal compressors with unshrouded impeller was proposed by Sanadgol in [21]. The study was based on results by Senoo [22] and Senoo and Ishida [23] on the change of the compressor efficiency versus the axial impeller tip clearance. Sanadgol proposed axially modulating the compressor impeller to induce a pressure variation to stabilize surge, and the AMBs were employed as servo actuators to control the axial clearance between the impeller and the static
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shroud. A mathematical model describing the compression system with AMBs was introduced by Sanadgol, although no experimental results were available. Later, experimental validation of the theoretical model was presented by Yoon et al. in [24] and [25]. In this paper we address the design and implementation of the active surge controller for a compression system with active magnetic bearings. A surge controller for a compression system with variable impeller axial clearance is derived and implemented in an experimental setup described in [24]. Additionally, issues encountered during the implementation of the surge controller are also discussed here. The remainder of the paper is organized as follows. A description of the experimental setup is given in Section II. The mathematical model describing the dynamics of the test rig, including the effects of varying the impeller axial clearance on the compressor output, is presented in Section III. In Section IV we derive the control law for the impeller tip clearance to stabilize the compression system in surge condition, and the controller is tested in simulation. Finally, Section V discusses the preliminary observations on the implementation of the surge controller and draws a brief conclusion to the paper.
Fig. 1.
Experimental compressor setup.
II. E XPERIMENTAL S ETUP An experimental setup for the study of the active control of compressor surge was developed and commissioned by the Rotating Machinery and Controls (ROMAC) Laboratory at the University of Virginia. The setup shown in Fig. 1 consists of a single-impeller centrifugal compressor with an unshrouded impeller and a vaneless diffuser, a modular exhaust piping system that forms the plenum volume, and a throttling device that controls the flow rate through the compressor. An innovative feature of this experimental setup is the implementation of AMBs, which not only provide radial and axial support of the compressor rotor, but they also induce a pressure variation in the compression system by changing the axial clearance between the rotating impeller and the static shroud. The final objective of this setup is to develop an active surge controller that uses the AMBs to create pressure waves by changing the impeller tip clearance for the active control of compressor surge. A detailed description of the experimental setup can be found in [24]. A cut-section drawing of the compressor is given in Fig. 2. The compressor has an overhung design, with two radial AMBs located near each end of the compressor to levitate and center the rotor about its rotating axis. The axial support of the rotor is provided by a thrust AMB located at the midsection of the compressor, which also controls the clearance between the impeller tip and the shroud with high bandwidth and precision. The rated load capacity for the thrust AMB is 6600 N. The maximum end-to-end displacement of the AMBs is 20 mils (0.508 mm) on each axis, and it is limited by the auxiliary bearings protecting the internal components of the compressor. The compressor is powered by a prototype induction motor from KaVo, which is rated to produce 95 kW of power at the design speed of 23000 RPM.
Fig. 2. The single stage centrifugal compressor in the experimental setup operates with an unshrouded impeller and a vaneless diffuser. The rotor is radially supported by two radial AMBs, and a single thrust AMB provides the axial support.
The layout of the experimental setup is shown in Fig. 3. A modular exhaust piping system forms the plenum volume, and a pneumatic throttling device controls the flow rate through the compressor. The size of the plenum volume can be varied by moving the throttle valve to one of the three predetermined locations along the exhaust piping. By modifying the plenum volume this way, we can change the intensity of the observed surge in the compressor. Pressure measurements are recorded using high-bandwidth pressure transducers along the inlet and the exhaust piping, together with thermocouples to provide temperature measurements at the same locations. The steady state mass flow rate is given by an orifice flow meter installed in the return section of the exhaust piping. For the initial testing of the compressor, the throttle valve is located at the position closest to the compressor, and the operating speed is selected to be 16290 RPM. III. C OMPRESSION S YSTEM M ODEL The derivation of the mathematical model describing the dynamics of our experimental setup is described in detail in [21]. Here we present a brief overview for the purpose of deriving a surge controller. A lumped parameter model
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experimental curve fit 1.4
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Fig. 3. Layout of the experimental setup, including the compressor, instrumentations, and the movable throttle valve. Possible throttle valve locations are at 2.2 m, 7.1 m and 15.2 m along the exhaust piping measured from the compressor.
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Fitted characteristic curve of the compressor at 16290 RPM.
curve where a further restriction of the flow would force the compressor into surge. The stable region is to the right of the surge point, and the curve can be interpolated from steady state pressure and flow rate measurements using a 3rd order polynomial fitting Φeq = a1 Ψ3eq + b1 Ψeq2 + d1 . Fig. 4. Schematic drawing of a compression system described by the Greitzer model.
for a compression system composed of the compressor, the plenum volume and the throttle valve was derived by Greitzer in [4] for the study of flow instabilities in turbomachineries. This models adds the transient dynamics of the compression system over the steady state characteristics given by the compressor characteristic curve. Assuming quasi-steady behavior of the compressor and the throttle valve, the states of the Greitzer compression system are the non-dimensional compressor mass flow rate Φ c and the non-dimensional plenum pressure rise Ψ p and are governed by Φ˙ c = BωH (Ψc − Ψp ), ˙ p = ωH (Φc − Φth ), Ψ B Φth = cth uth Ψp .
(1a) (1b) (1c)
The non-dimensional throttle mass flow rate Φ th is computed as a function of the throttle percentage opening u th and Ψp . The throttle valve constant c th is determined experimentally and is specific to each valve. Finally, B and ω H are the Greitzer stability parameter and the Helmholtz frequency, respectively. A schematic drawing of a compression system as represented by the Greitzer model is given in Fig. 4. The steady state performance of the compressor is given by its characteristic curve. This curve gives the equilibrium pressure output for the compressor operating at different mass flow rates. The curve is generally divided into two region by the surge point, which is the critical point in the
(2)
Equilibrium pressure and mass flow measurements in the unstable region, or to the left of the surge point, are difficult to obtain, and the characteristic curve is extrapolated from the curve fitting in the stable region. A correction term for the pressure at zero mass flow rate is added based on observations. Details on determining the characteristic curve of a compressor can be found in [1]. The characteristic curve for our model is given in Fig. 5. The model in Eq. (1) makes some assumptions on the geometry and the operating conditions of the compression system. First of all, it assumes that the compression system operates with low inlet Mach numbers and small pressure rise values compared to ambient pressure. Also, the flow in the duct section is taken to be one-dimensional and incompressible. In the plenum, isentropic compression process is assumed with uniform pressure distribution, and fluid velocities are considered to be negligible. Finally, plenum dimensions are taken to be much smaller than to the wavelength of the acoustic waves related to surge. Selecting a proper actuator that perturbs the compression system with enough bandwidth and control authority to stabilize surge is a difficult task, and it is the topic of research for many published works in compression systems ( [17], [18], [26], [27]). Here, we take advantage of the AMBs levitating the rotor to servo control the impeller axial position with high precision, and thus changing the axial clearance between the impeller and the static shroud. The variation of the axial impeller tip clearance in the compressor is the input perturbation for the compression system. Senoo in [22] and Senoo and Ishida in [23] studied the relationship between the axial impeller tip clearance and operating efficiency of turbomachineries, and Senoo
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TABLE I
where the gain k is a function of the compressor pressure ratio ψc,ss γ−1 γ k0 γ1 γ ψc,ss 1 − ψc,ss , k=− γ−1 b 1 2 2 ρU Ψc,ss + 1 . ψc,ss = Po1
S YSTEM MODEL PARAMETERS Parameter B ωH (rad/sec) Po1 (Pa) ρ (kg/m3 ) γ U (m/sec)
Value .5246 50.25 101,325 0.833 1.4 213.24
introduced a simple mathematical expression relating the average impeller tip clearance to the change in the efficiency for a single-stage unshrouded centrifugal compressor. The mathematical expression introduced by Senoo was here adjusted to match our observations in the measurements taken from the experimental setup, such that a variation of 2% in the ratio between the axial impeller clearance cl over the blade height b, induces a changes of 1% ratio between the change of efficiency from its zero-clearance value ∆η, and the current efficiency η. −
cl ∆η ≈ . η 2b
(3)
Define δcl = cl0 − cl to be the variation in the tip clearance from the nominal clearance value cl 0 , and the impeller clearance coefficient k 0 to be k0 =
0.5 . 0 1 + 0.5cl b
(4)
Assuming we have isentropic compression and perfect gas properties, Ψ c can be expressed as a function of the steady state compressor pressure rise Ψ c,ss , which is itself is a function of the compressor mass flow rate Φ c and obtained from the compressor characteristic curve at the nominal tip clearance cl0 , and the variation in the tip clearance δ cl , γ γ−1 1 γ−1 2 γ ρ U 2 o1 Ψc,ss +1 − 1 po1 − 1 1+ po1 Ψc = 1 . (5) δ 2 1 − k0 bcl 2 ρo1 U
The density ρo1 is given at the inlet condition, γ is the specific heat ratio, and the impeller tip velocity U is obtained from the compressor operating speed. Thus, Eqs. (5) and (1) are the state equations representing the dynamics of the compression system. Table I gives the values for the different model parameters corresponding to the experimental compressor test rig. IV. C ONTROLLER D ESIGN AND S IMULATION
In order to simplify the system equations for the derivation of the surge controller, a linear approximation of Eq. (5) is introduced. By keeping the first order terms of the Taylor expansion, the equation for the compressor pressure rise can be written as Ψc ≈ Ψc,ss + kδcl , (6)
It was demonstrated in [21] that the linearization error introduced by the above approximation is small for the allowed range of δ cl . By defining the states χ and ξ as the variation of the nondimensional plenum pressure rise and compressor massflow rate from their respective equilibrium values χ = Ψp − Ψeq ,
ξ = Φc − Φeq ,
(7)
the state equations for the compression system can be rewritten as the following (8a) ξ˙ = BωH (Ψc,ss + kδcl − χ − Ψeq ), ωH χ˙ = (ξ − Φth + Φeq ), (8b) B and the surge control problem becomes the stabilization of the system (8) at the origin (ξ, χ) = (0, 0). An H∞ surge controller is derived from a linear approximation of system (8). A linear controller is preferred in this case because the tracking limitation of the AMB actuator can be included into the controller derivation employing linear robust control methods. The linearized model for the compression system at a nominal throttle valve opening uth0 = .28, corresponding to the experimentally observed surge point, is given by o1 BωH 1PρU ξ ξ˙ 2k 2 δcl , (9) =A + χ χ˙ 0 where the matrix A is defined as BωH (3a1 Φ2eq + 2b1 Φeq ) A= ωH B
−BωH − ωBH cth0 uth a2
. (10)
The constants a1 and b1 are the coefficients from the polynomial fit of the characteristic curve, and a 2 is obtained from the linear approximation of the square root Ψeq = a2 Ψeq + b2 . (11)
The 2-input 2-output interconnection system shown in Fig. 6 is employed to synthesize the linear controller, where GC (s) is the linearized compression system and K C (s) is the surge controller. The weighting function W 2 (s) contains information on the AMB servo tracking performance, and weights on the control input to the plant. The benefit of this method is that in addition to incorporating a limit to the actuator bandwidth, it also consider the steady state tracking error introduced by the dynamics of the closed loop AMB, as well as the error from overshoot and other transient dynamics. We define W 2 (s) = 4s/(s + 400), such that it upper-bounds the sensitivity function of an acceptable AMB dynamics.The function W 4 (s) weights the added noise to the plant output, and contains the information of noise
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ˆ c is the estimated non-dimensional compressor mass where Φ ˆ c,ss is the non-dimensional steady-state compresflow rate, Ψ ˆ c , and c = 5 is the observer constant. sor pressure rise at Φ It was demonstrated in [28] that the estimation error for the ˆ c ). ˆ c,ss /∂ Φ above observer approaches to zero for c ≥ 2(∂ Ψ The stability of the compression system with the derived observer and surge controller was tested through simulation. In order to further improve the fidelity of the numerical simulation to the experimental observations, the acoustic effects in the plenum piping was included to the state equations in (1) as described in [25]. Fig. 7 compares of the simulated Bode plots of the extended model from small perturbations in the tip clearance to the output pressure, and the equivalent plots from measurements in the experimental setup. A good match is observed between the two responses, which give us confidence in the simulated results. Fig. 8 shows the simulated response in surge condition for the uncontrolled system. The throttle valve opening is reduced from 34% to 29%, crossing the surge point at 31.5% open into the unstable region. The compression system becomes unstable, and the surge oscillations can be observed in the plenum pressure signal. The characteristic curve and the simulated operating point of the compression system is plotted in Fig. 9. During stable operation, the states of the compression system would approach the characteristic curve. Instead, the uncontrolled system becomes unstable in the positive-slope region of the characteristic curve, and the
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Fig. 7. Comparison between simulated frequency response of the mathematical model vs. experimental Bode plots. Throttle valve at 32% open. 35 Throttle opening (%)
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characteristics in the sensor measurement signal. For the remaining weighting functions, W 1 (s) is set to be 1, and the value of W3 (s) is iterated to obtain a satisfactory controller. One issue that needs to be considered before the above controller can be practical is the availability of the sensor measurements. High-bandwidth pressure transducers are common in industrial applications and readily available. On the other hand, transient flow rate measurements are not reliable, especially for larger industrial compressors. Therefore, a nonlinear mass-flow rate observer is derived from the system equations (1) as presented in [28]. The state equation for the observer are given as ˆ c,ss + P o1 kδcl −Ψp −cΦ ˆ c +cΦth , (12a) z˙ = BωH Ψ 1 2 2 ρU ˆ c = z + B 2 cΨp . Φ (12b)
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Fig. 8. Throttle valve opening uth and plenum pressure rise Ψp for the uncontrolled system simulation
compression system enters the surge cycle. Some simulation results of the compression system with active surge control are shown in Fig. 10 for the same changes in the throttle valve opening as above. The variation in the impeller tip clearance is shown in the figure, and we can see that the plenum pressure rise follows the equilibrium Ψeq and remains stable beyond the stable region. V. C ONCLUSIONS The design and implementation of an active surge controller for an unshrouded centrifugal compression system with AMBs was studied in this paper. Employing the compressor bearings to servo control the impeller position, the compressor output pressure was modulated to control surge. An H∞ controller was designed based on a previously validated model of the compressor, taking into account the tracking performance of the AMB. We are currently in the preliminary stage of implementing the surge controller. Initial observations show high noise-to-
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Fig. 10. Impeller tip clearance δcl and plenum pressure rise Ψp for the controlled system simulation
signal ratios in the feedback signals that can saturate the impeller tip clearance output during surge control. This and other challenges in operating a relatively large industrial compressor in a real world environment need to be addressed before the actual experimental implementation. R EFERENCES [1] J. T. Gravdahl and O. Egeland, Compressor Surge and Rotating Stall: Modeling and Control. London, UK: Springer-Verlag, 1999. [2] J. P. Longley, “A review of nonsteady flow models for compressor stability,” ASME Journal of Turbomachinery, pp. 202–215. [3] H. W. Emmons, C. E. Paerson, and H. P. Grant, “Compressor surge and stall propagation,” in American Society of Mechanical Engineers, pp. 455–469. [4] E. M. Greitzer, “Surge and rotating stall in axial flow compressors, part i, ii,” ASME Journal of Engineering for Power, vol. 120, pp. 190–217, 1976. [5] K. E. Hansen, P. Jørgensen, and P. S. Larsen, “Experimental and theoretical study of surge in a small centrifugal compressor,” AMSE Journal of Fluids Engineering, pp. 391–395, 1981. [6] I. Macdougal and R. L. Elder, “Simulation of centrifugal compressor transient performance for process plant applications,” ASME Journal of Engineering for Power, vol. 105, no. 4, pp. 885–890, 1983.
[7] R. L. Elder and M. E. Gill, “A discussion of the factors affecting surge in centrifugal compressors,” ASME Journal of Engineering for Gas Turbines and Power, vol. 107, pp. 499–506, 1985. [8] F. K. Moore and E. M. Greitzer, “A theory of post-stall transients in axial compressor systems: part i–development of equations,” ASME Journal of Engineering for Gas Turbines and Power, vol. 108, pp. 68– 76. [9] D. A. Fink, N. A. Cumpsty, and E. M. Greitzer, “Surge dynamics in a free-spool centrifugal compressor system,” ASME Journal of Turbomachinery, vol. 114, pp. 321–332, 1992. [10] K. K. Botros, “Transient phenomena in compressor stations during surge,” ASME Jounal of Engineering for Gas Turbines and Power, vol. 116. [11] J. T. Gravdahl and O. Egeland, “Centrifugal compressor surge and speed control,” IEEE Transactions on Control Systems Technology, vol. 7, pp. 567–579. [12] M. Morini, M. Pinelli, and M. Venturini, “Development of a onedimensional modular dynamic model for the simulation of surge in compression systems,” ASME Journal of Turbomachinery, vol. 129, pp. 437–447, 2007. [13] C. R. Sparks, “On the transient interaction of centrifugal compressors and their piping systems,” ASME Journal of Engineering for Power, vol. 105, pp. 891–901. [14] N. Hagino, Y. Kashiwabara, and K. Uda, “Prediction and active control of surge inception in centrifugal compressor system without plenum,” in 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit. [15] W. M. Jungowski, W. M. H, and G. R. Price, “Pressure oscillations occurring in a centrifugal compressor system with and without passive and active surge control,” ASME Journal of Turbomachinery, vol. 118, pp. 29–40. [16] A. H. Epstein, E. F. Williams, and E. M. Greitzer, “Active suppression of aerodynamic instabilities in turbomachinery,” Journal of Propulsion and Power, vol. 5, pp. 204–211. [17] G. L. Arnulfi, F. Blanchini, P. Giannattasio, and P. Pinamonti, “Extensive study on the control of centrifugal compressor surge,” Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, vol. 220, pp. 289–304. [18] J. S. Simon, L. Valavani, A. H. Epstein, and E. M. Greitzer, “Evaluation of approaches to active compressor surge stabilization,” ASME Journal of Turbomachinery, vol. 115, pp. 57–67. [19] M. Krsti´c, D. Fontaine, P. V. Kokotovi´c, and J. D. Paduano, “Useful nonlinearities and global stabilization of bifurcations in a model of jet enginer surge and stall,” IEEE Transactions on Automatic Control, vol. 43, pp. 1739–1745. [20] Y. Wang, J. Paduano, and R. Murray, “Nonlinear control design for rotating stall with magnetic bearing actuators,” in Proc. IEEE International Conference on Control Applications, 1999. [21] D. Sanadgol, Active Control of Surge in Centrifugal Compressors Using Magnetic Thrust Bearing Actuation. PhD thesis, University of Virginia, 2006. [22] Y. Senoo, “Pressure losses and flow field distortion induced by tip clearance of centrifugal and axial compressors,” JSME International Journal, vol. 30, pp. 375–385, Mar. 1987. [23] Y. Senoo and M. Ishida, “Deterioration of compressor performance due to tip clearance of centrifugal impellers,” ASME Journal of Turbomachinery, vol. 109, pp. 55–61. [24] S. Y. Yoon, Z. Lin, K. T. Lim, C. Goyne, and P. E. Allaire, “Model validation for an amb-based compressor surge control test rig,” in Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. [25] S. Y. Yoon, Z. Lin, C. Goyne, and P. Allaire, “An enhanced greitzer compressor model with pipeline dynamics in the plenum section,” ASME Journal of Vibration and Acoustics, under revision. [26] Z. S. Spakovsky, J. D. Paduano, R. Larsonneur, A. Traxler, and M. M. Bright, “Tip clearance actuation with magnetic bearings for high-speed compressor stall control,” ASME Journal of Turbomachinery, vol. 123, pp. 464–472. [27] K. O. Boinov, E. A. Lomonova, A. J. A. Vandenput, and A. Tyagunov, “Surge control of the electrically driven centrifugal compressor,” IEEE Transactions on Industry Applications, vol. 42, pp. 1523–1531. [28] B. Bøhagen and J. T. Gravdahl, “On active surge control of compressors using a mass flow observer,” in Proceedings of the 41st IEEE Conference on Decision and Control, pp. 3684–3689.
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Control of Compressor Surge with Active Magnetic Bearings Se Young Yoon †, Zongli Lin† , Chris Goyne‡ and Paul E. Allaire‡
Abstract— The design of an active surge controller, which employs Active Magnetic Bearings (AMBs) to stabilize the flow in the compressor, is presented in this paper. The axial tip clearance of an unshrouded centrifugal impeller from a single stage compressor is modulated relative to the static shroud to induce a pressure variation in the compressor output pressure. This fast-response, high-precision control of the output pressure is employed to stabilize the compressor operating in surge condition. The control law for the tip clearance is derived based on a recently introduced mathematical model, and the performance of the controller is tested through simulation.
I. I NTRODUCTION Surge is a dynamics instability in compressors, characterized by large-amplitude axisymmetric flow oscillations. Surge instability can cause extensive damage to the system, especially during the reversal of the flow in deep surge condition, when the impeller and the casing are subject to high loads and temperatures. Two different approaches exist to deal with the surge instability in turbomachineries. The most common approach in industry is surge avoidance, where the compressor is operated in a conservative way within a predetermined stable operating region, and safety mechanisms “reset” the compressor back to the stable operation if surge occurs. This leads to a significant drop in compressor efficiency by limiting the compressor output. The second approach is surge control, where an active or passive surge controller is implemented such that it compensates for the disturbances of surge, and stabilizes the flow in the compression system. More details on surge avoidance and control can be found in [1]. The modeling of compression systems for the active control of surge instabilities has been investigated intensively over the years, motivated by the potential benefits from expanding the stable operating region of these machineries. Reviews of modeling techniques for compression systems can be found in [1] and [2]. Emmons et al. presented in [3] one of the first stability analysis for a compression system using a linearized model, where the initiation of the surge oscillations was captured by employing the analogy between the compression system and a self-excited Helmholtz resonator. Using the same principle, Greitzer [4] later introduced a two-state nonlinear lumped-parameter model of a compression system. This model was first derived for axial compressors, but it was later demonstrated in [5] that it is † S. Y. Yoon and Z. Lin are with Charles L., Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904-4743, USA ‡ C. Goyne, and P. E. Allaire are with the Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 229044746, USA
978-1-4244-7746-3/10/$26.00 ©2010 IEEE
also applicable for centrifugal compressors. After this, many improved and advanced models for axial and centrifugal compression systems were presented by different researchers in [6]–[12], and stability of compression systems for various configurations of the inlet and the exhaust piping was studied in [13], [14] and [15]. However the original Greitzer model is still generally preferred in designing active surge controllers due to its low order and simple implementation of the controller designed based on it. Efforts to develop reliable model based surge controllers have been ongoing, fueled by the increasing need of boosting productivity and improving the safety of the workspace. Surge suppression and control methods rely on accurate plant models to design a surge control law that is able to stabilize the compression system pass the surge point. Passive surge controllers rely on passive systems, that react to environmental changes in the compressor and suppress the surge oscillations. On the other hand, active surge controllers, which were first proposed by Epstein et al. in [16], employ actuators to perturb the compression system according to feedback measurements from sensors in the flow path. A brief review of published works on model based surge control, with both active and passive systems, were presented by Arnulfi et al. in [17]. An important point to consider in designing a surge control system is the selection of the appropriate actuator. Simon et al. presented in [18] an overview of commonly employed actuators for surge control, and demonstrated the complexity of selecting an accessible actuator with sufficient control authority on the compressor’s unstable modes and bandwidth to stabilize the flow dynamics during surge. Common actuators for passive control systems are the movable plenum wall and the hydraulic oscillator, which induce a pressure variation in the compression system by changing the plenum volume. On the other hand, and common actuators in active control schemes are the closecoupled valve [11], and the throttle valve [19]. A less considered but also promising actuator for the control of compressor surge is the Active Magnetic Bearing (AMB). The use of AMBs for the control of other compressor instabilities was proposed in [20]. On the other hand, the use of the AMB for the control of surge in singlestage centrifugal compressors with unshrouded impeller was proposed by Sanadgol in [21]. The study was based on results by Senoo [22] and Senoo and Ishida [23] on the change of the compressor efficiency versus the axial impeller tip clearance. Sanadgol proposed axially modulating the compressor impeller to induce a pressure variation to stabilize surge, and the AMBs were employed as servo actuators to control the axial clearance between the impeller and the static
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shroud. A mathematical model describing the compression system with AMBs was introduced by Sanadgol, although no experimental results were available. Later, experimental validation of the theoretical model was presented by Yoon et al. in [24] and [25]. In this paper we address the design and implementation of the active surge controller for a compression system with active magnetic bearings. A surge controller for a compression system with variable impeller axial clearance is derived and implemented in an experimental setup described in [24]. Additionally, issues encountered during the implementation of the surge controller are also discussed here. The remainder of the paper is organized as follows. A description of the experimental setup is given in Section II. The mathematical model describing the dynamics of the test rig, including the effects of varying the impeller axial clearance on the compressor output, is presented in Section III. In Section IV we derive the control law for the impeller tip clearance to stabilize the compression system in surge condition, and the controller is tested in simulation. Finally, Section V discusses the preliminary observations on the implementation of the surge controller and draws a brief conclusion to the paper.
Fig. 1.
Experimental compressor setup.
II. E XPERIMENTAL S ETUP An experimental setup for the study of the active control of compressor surge was developed and commissioned by the Rotating Machinery and Controls (ROMAC) Laboratory at the University of Virginia. The setup shown in Fig. 1 consists of a single-impeller centrifugal compressor with an unshrouded impeller and a vaneless diffuser, a modular exhaust piping system that forms the plenum volume, and a throttling device that controls the flow rate through the compressor. An innovative feature of this experimental setup is the implementation of AMBs, which not only provide radial and axial support of the compressor rotor, but they also induce a pressure variation in the compression system by changing the axial clearance between the rotating impeller and the static shroud. The final objective of this setup is to develop an active surge controller that uses the AMBs to create pressure waves by changing the impeller tip clearance for the active control of compressor surge. A detailed description of the experimental setup can be found in [24]. A cut-section drawing of the compressor is given in Fig. 2. The compressor has an overhung design, with two radial AMBs located near each end of the compressor to levitate and center the rotor about its rotating axis. The axial support of the rotor is provided by a thrust AMB located at the midsection of the compressor, which also controls the clearance between the impeller tip and the shroud with high bandwidth and precision. The rated load capacity for the thrust AMB is 6600 N. The maximum end-to-end displacement of the AMBs is 20 mils (0.508 mm) on each axis, and it is limited by the auxiliary bearings protecting the internal components of the compressor. The compressor is powered by a prototype induction motor from KaVo, which is rated to produce 95 kW of power at the design speed of 23000 RPM.
Fig. 2. The single stage centrifugal compressor in the experimental setup operates with an unshrouded impeller and a vaneless diffuser. The rotor is radially supported by two radial AMBs, and a single thrust AMB provides the axial support.
The layout of the experimental setup is shown in Fig. 3. A modular exhaust piping system forms the plenum volume, and a pneumatic throttling device controls the flow rate through the compressor. The size of the plenum volume can be varied by moving the throttle valve to one of the three predetermined locations along the exhaust piping. By modifying the plenum volume this way, we can change the intensity of the observed surge in the compressor. Pressure measurements are recorded using high-bandwidth pressure transducers along the inlet and the exhaust piping, together with thermocouples to provide temperature measurements at the same locations. The steady state mass flow rate is given by an orifice flow meter installed in the return section of the exhaust piping. For the initial testing of the compressor, the throttle valve is located at the position closest to the compressor, and the operating speed is selected to be 16290 RPM. III. C OMPRESSION S YSTEM M ODEL The derivation of the mathematical model describing the dynamics of our experimental setup is described in detail in [21]. Here we present a brief overview for the purpose of deriving a surge controller. A lumped parameter model
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experimental curve fit 1.4
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Fig. 3. Layout of the experimental setup, including the compressor, instrumentations, and the movable throttle valve. Possible throttle valve locations are at 2.2 m, 7.1 m and 15.2 m along the exhaust piping measured from the compressor.
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Fitted characteristic curve of the compressor at 16290 RPM.
curve where a further restriction of the flow would force the compressor into surge. The stable region is to the right of the surge point, and the curve can be interpolated from steady state pressure and flow rate measurements using a 3rd order polynomial fitting Φeq = a1 Ψ3eq + b1 Ψeq2 + d1 . Fig. 4. Schematic drawing of a compression system described by the Greitzer model.
for a compression system composed of the compressor, the plenum volume and the throttle valve was derived by Greitzer in [4] for the study of flow instabilities in turbomachineries. This models adds the transient dynamics of the compression system over the steady state characteristics given by the compressor characteristic curve. Assuming quasi-steady behavior of the compressor and the throttle valve, the states of the Greitzer compression system are the non-dimensional compressor mass flow rate Φ c and the non-dimensional plenum pressure rise Ψ p and are governed by Φ˙ c = BωH (Ψc − Ψp ), ˙ p = ωH (Φc − Φth ), Ψ B Φth = cth uth Ψp .
(1a) (1b) (1c)
The non-dimensional throttle mass flow rate Φ th is computed as a function of the throttle percentage opening u th and Ψp . The throttle valve constant c th is determined experimentally and is specific to each valve. Finally, B and ω H are the Greitzer stability parameter and the Helmholtz frequency, respectively. A schematic drawing of a compression system as represented by the Greitzer model is given in Fig. 4. The steady state performance of the compressor is given by its characteristic curve. This curve gives the equilibrium pressure output for the compressor operating at different mass flow rates. The curve is generally divided into two region by the surge point, which is the critical point in the
(2)
Equilibrium pressure and mass flow measurements in the unstable region, or to the left of the surge point, are difficult to obtain, and the characteristic curve is extrapolated from the curve fitting in the stable region. A correction term for the pressure at zero mass flow rate is added based on observations. Details on determining the characteristic curve of a compressor can be found in [1]. The characteristic curve for our model is given in Fig. 5. The model in Eq. (1) makes some assumptions on the geometry and the operating conditions of the compression system. First of all, it assumes that the compression system operates with low inlet Mach numbers and small pressure rise values compared to ambient pressure. Also, the flow in the duct section is taken to be one-dimensional and incompressible. In the plenum, isentropic compression process is assumed with uniform pressure distribution, and fluid velocities are considered to be negligible. Finally, plenum dimensions are taken to be much smaller than to the wavelength of the acoustic waves related to surge. Selecting a proper actuator that perturbs the compression system with enough bandwidth and control authority to stabilize surge is a difficult task, and it is the topic of research for many published works in compression systems ( [17], [18], [26], [27]). Here, we take advantage of the AMBs levitating the rotor to servo control the impeller axial position with high precision, and thus changing the axial clearance between the impeller and the static shroud. The variation of the axial impeller tip clearance in the compressor is the input perturbation for the compression system. Senoo in [22] and Senoo and Ishida in [23] studied the relationship between the axial impeller tip clearance and operating efficiency of turbomachineries, and Senoo
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TABLE I
where the gain k is a function of the compressor pressure ratio ψc,ss γ−1 γ k0 γ1 γ ψc,ss 1 − ψc,ss , k=− γ−1 b 1 2 2 ρU Ψc,ss + 1 . ψc,ss = Po1
S YSTEM MODEL PARAMETERS Parameter B ωH (rad/sec) Po1 (Pa) ρ (kg/m3 ) γ U (m/sec)
Value .5246 50.25 101,325 0.833 1.4 213.24
introduced a simple mathematical expression relating the average impeller tip clearance to the change in the efficiency for a single-stage unshrouded centrifugal compressor. The mathematical expression introduced by Senoo was here adjusted to match our observations in the measurements taken from the experimental setup, such that a variation of 2% in the ratio between the axial impeller clearance cl over the blade height b, induces a changes of 1% ratio between the change of efficiency from its zero-clearance value ∆η, and the current efficiency η. −
cl ∆η ≈ . η 2b
(3)
Define δcl = cl0 − cl to be the variation in the tip clearance from the nominal clearance value cl 0 , and the impeller clearance coefficient k 0 to be k0 =
0.5 . 0 1 + 0.5cl b
(4)
Assuming we have isentropic compression and perfect gas properties, Ψ c can be expressed as a function of the steady state compressor pressure rise Ψ c,ss , which is itself is a function of the compressor mass flow rate Φ c and obtained from the compressor characteristic curve at the nominal tip clearance cl0 , and the variation in the tip clearance δ cl , γ γ−1 1 γ−1 2 γ ρ U 2 o1 Ψc,ss +1 − 1 po1 − 1 1+ po1 Ψc = 1 . (5) δ 2 1 − k0 bcl 2 ρo1 U
The density ρo1 is given at the inlet condition, γ is the specific heat ratio, and the impeller tip velocity U is obtained from the compressor operating speed. Thus, Eqs. (5) and (1) are the state equations representing the dynamics of the compression system. Table I gives the values for the different model parameters corresponding to the experimental compressor test rig. IV. C ONTROLLER D ESIGN AND S IMULATION
In order to simplify the system equations for the derivation of the surge controller, a linear approximation of Eq. (5) is introduced. By keeping the first order terms of the Taylor expansion, the equation for the compressor pressure rise can be written as Ψc ≈ Ψc,ss + kδcl , (6)
It was demonstrated in [21] that the linearization error introduced by the above approximation is small for the allowed range of δ cl . By defining the states χ and ξ as the variation of the nondimensional plenum pressure rise and compressor massflow rate from their respective equilibrium values χ = Ψp − Ψeq ,
ξ = Φc − Φeq ,
(7)
the state equations for the compression system can be rewritten as the following (8a) ξ˙ = BωH (Ψc,ss + kδcl − χ − Ψeq ), ωH χ˙ = (ξ − Φth + Φeq ), (8b) B and the surge control problem becomes the stabilization of the system (8) at the origin (ξ, χ) = (0, 0). An H∞ surge controller is derived from a linear approximation of system (8). A linear controller is preferred in this case because the tracking limitation of the AMB actuator can be included into the controller derivation employing linear robust control methods. The linearized model for the compression system at a nominal throttle valve opening uth0 = .28, corresponding to the experimentally observed surge point, is given by o1 BωH 1PρU ξ ξ˙ 2k 2 δcl , (9) =A + χ χ˙ 0 where the matrix A is defined as BωH (3a1 Φ2eq + 2b1 Φeq ) A= ωH B
−BωH − ωBH cth0 uth a2
. (10)
The constants a1 and b1 are the coefficients from the polynomial fit of the characteristic curve, and a 2 is obtained from the linear approximation of the square root Ψeq = a2 Ψeq + b2 . (11)
The 2-input 2-output interconnection system shown in Fig. 6 is employed to synthesize the linear controller, where GC (s) is the linearized compression system and K C (s) is the surge controller. The weighting function W 2 (s) contains information on the AMB servo tracking performance, and weights on the control input to the plant. The benefit of this method is that in addition to incorporating a limit to the actuator bandwidth, it also consider the steady state tracking error introduced by the dynamics of the closed loop AMB, as well as the error from overshoot and other transient dynamics. We define W 2 (s) = 4s/(s + 400), such that it upper-bounds the sensitivity function of an acceptable AMB dynamics.The function W 4 (s) weights the added noise to the plant output, and contains the information of noise
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ˆ c is the estimated non-dimensional compressor mass where Φ ˆ c,ss is the non-dimensional steady-state compresflow rate, Ψ ˆ c , and c = 5 is the observer constant. sor pressure rise at Φ It was demonstrated in [28] that the estimation error for the ˆ c ). ˆ c,ss /∂ Φ above observer approaches to zero for c ≥ 2(∂ Ψ The stability of the compression system with the derived observer and surge controller was tested through simulation. In order to further improve the fidelity of the numerical simulation to the experimental observations, the acoustic effects in the plenum piping was included to the state equations in (1) as described in [25]. Fig. 7 compares of the simulated Bode plots of the extended model from small perturbations in the tip clearance to the output pressure, and the equivalent plots from measurements in the experimental setup. A good match is observed between the two responses, which give us confidence in the simulated results. Fig. 8 shows the simulated response in surge condition for the uncontrolled system. The throttle valve opening is reduced from 34% to 29%, crossing the surge point at 31.5% open into the unstable region. The compression system becomes unstable, and the surge oscillations can be observed in the plenum pressure signal. The characteristic curve and the simulated operating point of the compression system is plotted in Fig. 9. During stable operation, the states of the compression system would approach the characteristic curve. Instead, the uncontrolled system becomes unstable in the positive-slope region of the characteristic curve, and the
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Fig. 7. Comparison between simulated frequency response of the mathematical model vs. experimental Bode plots. Throttle valve at 32% open. 35 Throttle opening (%)
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characteristics in the sensor measurement signal. For the remaining weighting functions, W 1 (s) is set to be 1, and the value of W3 (s) is iterated to obtain a satisfactory controller. One issue that needs to be considered before the above controller can be practical is the availability of the sensor measurements. High-bandwidth pressure transducers are common in industrial applications and readily available. On the other hand, transient flow rate measurements are not reliable, especially for larger industrial compressors. Therefore, a nonlinear mass-flow rate observer is derived from the system equations (1) as presented in [28]. The state equation for the observer are given as ˆ c,ss + P o1 kδcl −Ψp −cΦ ˆ c +cΦth , (12a) z˙ = BωH Ψ 1 2 2 ρU ˆ c = z + B 2 cΨp . Φ (12b)
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Fig. 8. Throttle valve opening uth and plenum pressure rise Ψp for the uncontrolled system simulation
compression system enters the surge cycle. Some simulation results of the compression system with active surge control are shown in Fig. 10 for the same changes in the throttle valve opening as above. The variation in the impeller tip clearance is shown in the figure, and we can see that the plenum pressure rise follows the equilibrium Ψeq and remains stable beyond the stable region. V. C ONCLUSIONS The design and implementation of an active surge controller for an unshrouded centrifugal compression system with AMBs was studied in this paper. Employing the compressor bearings to servo control the impeller position, the compressor output pressure was modulated to control surge. An H∞ controller was designed based on a previously validated model of the compressor, taking into account the tracking performance of the AMB. We are currently in the preliminary stage of implementing the surge controller. Initial observations show high noise-to-
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signal ratios in the feedback signals that can saturate the impeller tip clearance output during surge control. This and other challenges in operating a relatively large industrial compressor in a real world environment need to be addressed before the actual experimental implementation. R EFERENCES [1] J. T. Gravdahl and O. Egeland, Compressor Surge and Rotating Stall: Modeling and Control. London, UK: Springer-Verlag, 1999. [2] J. P. Longley, “A review of nonsteady flow models for compressor stability,” ASME Journal of Turbomachinery, pp. 202–215. [3] H. W. Emmons, C. E. Paerson, and H. P. Grant, “Compressor surge and stall propagation,” in American Society of Mechanical Engineers, pp. 455–469. [4] E. M. Greitzer, “Surge and rotating stall in axial flow compressors, part i, ii,” ASME Journal of Engineering for Power, vol. 120, pp. 190–217, 1976. [5] K. E. Hansen, P. Jørgensen, and P. S. Larsen, “Experimental and theoretical study of surge in a small centrifugal compressor,” AMSE Journal of Fluids Engineering, pp. 391–395, 1981. [6] I. Macdougal and R. L. Elder, “Simulation of centrifugal compressor transient performance for process plant applications,” ASME Journal of Engineering for Power, vol. 105, no. 4, pp. 885–890, 1983.
[7] R. L. Elder and M. E. Gill, “A discussion of the factors affecting surge in centrifugal compressors,” ASME Journal of Engineering for Gas Turbines and Power, vol. 107, pp. 499–506, 1985. [8] F. K. Moore and E. M. Greitzer, “A theory of post-stall transients in axial compressor systems: part i–development of equations,” ASME Journal of Engineering for Gas Turbines and Power, vol. 108, pp. 68– 76. [9] D. A. Fink, N. A. Cumpsty, and E. M. Greitzer, “Surge dynamics in a free-spool centrifugal compressor system,” ASME Journal of Turbomachinery, vol. 114, pp. 321–332, 1992. [10] K. K. Botros, “Transient phenomena in compressor stations during surge,” ASME Jounal of Engineering for Gas Turbines and Power, vol. 116. [11] J. T. Gravdahl and O. Egeland, “Centrifugal compressor surge and speed control,” IEEE Transactions on Control Systems Technology, vol. 7, pp. 567–579. [12] M. Morini, M. Pinelli, and M. Venturini, “Development of a onedimensional modular dynamic model for the simulation of surge in compression systems,” ASME Journal of Turbomachinery, vol. 129, pp. 437–447, 2007. [13] C. R. Sparks, “On the transient interaction of centrifugal compressors and their piping systems,” ASME Journal of Engineering for Power, vol. 105, pp. 891–901. [14] N. Hagino, Y. Kashiwabara, and K. Uda, “Prediction and active control of surge inception in centrifugal compressor system without plenum,” in 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit. [15] W. M. Jungowski, W. M. H, and G. R. Price, “Pressure oscillations occurring in a centrifugal compressor system with and without passive and active surge control,” ASME Journal of Turbomachinery, vol. 118, pp. 29–40. [16] A. H. Epstein, E. F. Williams, and E. M. Greitzer, “Active suppression of aerodynamic instabilities in turbomachinery,” Journal of Propulsion and Power, vol. 5, pp. 204–211. [17] G. L. Arnulfi, F. Blanchini, P. Giannattasio, and P. Pinamonti, “Extensive study on the control of centrifugal compressor surge,” Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, vol. 220, pp. 289–304. [18] J. S. Simon, L. Valavani, A. H. Epstein, and E. M. Greitzer, “Evaluation of approaches to active compressor surge stabilization,” ASME Journal of Turbomachinery, vol. 115, pp. 57–67. [19] M. Krsti´c, D. Fontaine, P. V. Kokotovi´c, and J. D. Paduano, “Useful nonlinearities and global stabilization of bifurcations in a model of jet enginer surge and stall,” IEEE Transactions on Automatic Control, vol. 43, pp. 1739–1745. [20] Y. Wang, J. Paduano, and R. Murray, “Nonlinear control design for rotating stall with magnetic bearing actuators,” in Proc. IEEE International Conference on Control Applications, 1999. [21] D. Sanadgol, Active Control of Surge in Centrifugal Compressors Using Magnetic Thrust Bearing Actuation. PhD thesis, University of Virginia, 2006. [22] Y. Senoo, “Pressure losses and flow field distortion induced by tip clearance of centrifugal and axial compressors,” JSME International Journal, vol. 30, pp. 375–385, Mar. 1987. [23] Y. Senoo and M. Ishida, “Deterioration of compressor performance due to tip clearance of centrifugal impellers,” ASME Journal of Turbomachinery, vol. 109, pp. 55–61. [24] S. Y. Yoon, Z. Lin, K. T. Lim, C. Goyne, and P. E. Allaire, “Model validation for an amb-based compressor surge control test rig,” in Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. [25] S. Y. Yoon, Z. Lin, C. Goyne, and P. Allaire, “An enhanced greitzer compressor model with pipeline dynamics in the plenum section,” ASME Journal of Vibration and Acoustics, under revision. [26] Z. S. Spakovsky, J. D. Paduano, R. Larsonneur, A. Traxler, and M. M. Bright, “Tip clearance actuation with magnetic bearings for high-speed compressor stall control,” ASME Journal of Turbomachinery, vol. 123, pp. 464–472. [27] K. O. Boinov, E. A. Lomonova, A. J. A. Vandenput, and A. Tyagunov, “Surge control of the electrically driven centrifugal compressor,” IEEE Transactions on Industry Applications, vol. 42, pp. 1523–1531. [28] B. Bøhagen and J. T. Gravdahl, “On active surge control of compressors using a mass flow observer,” in Proceedings of the 41st IEEE Conference on Decision and Control, pp. 3684–3689.
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