5.3 Linear compressor dynamics
So far we have investigated the effect of the various model parameters, in particular that of the stability parameter B, on the amplitude and frequency of the surge limit cycle.
As mentioned before, the idea of active surge control is to increase the aerodynamic damping of flow instabilities and to suppress destabilizing disturbances in an early stage where amplitudes are still small (Epstein et al.,1989). The underlying assumption is that the initial phases of those instabilities can be modeled by linear theory.
Hence, we will now take a closer look at the effect of the model parameters on the local dynamic behavior. For that purpose we will linearize the Greitzer model from Equations (5.1) and (5.2) around a specific operating point (φc0, ψ0, ut0), yielding where, with a slight abuse of notation,
∂Ψc = ∂Ψc(φc) represent the slopes of the compressor and load characteristics, respectively. We recall from that the developed fits for the compressor curves are not differentiable at the top.
In practice, this discontinuity in ∂Ψc is not causing problems because we do not need to linearize the model exactly around the top of the compressor curve. We stress that the obtained linear models are only valid in the immediate vicinity of the selected operating points.
A general form of the linearized system above is ˙x = Ax and the eigenvalues λ of this system are given by the roots s of the characteristic equation det(sI − A) = 0. The eigenvalues (modes) and their related eigenvectors (modal vectors) describe the stability and response of the system around a specific operating point. Details can be found in systems and control textbooks (e.g.,Franklin et al., 1994; Khalil, 2000) and the specific case of the Greitzer model is described byGravdahl and Egeland(1999b, pp. 199–201).
It is worthwhile to investigate how the modes of the linearized Greitzer model are related to the physical mechanisms that play a role in a compression system. For that purpose we recall that the eigenvector vi, corresponding to the eigenvalue λi, satisfies Avi = λivi. The resulting analytical expressions for the eigenvectors of the Greitzer model are rather involved and therefore omitted here. However, numerically we can show that, for distinct real eigenvalues and a large value of B, the direction of the eigenvector
• associated with the slow eigenvalue is almost parallel to the compressor curve,
• associated with the fast eigenvalue is almost horizontal,
while for smaller values of B the two eigenvectors become more aligned. Note that the slow or dominant eigenvalue is the one nearest to the imaginary axis.
At this point we recall the comments from Section 5.2.1 on the different time scales in limit cycle oscillations of the relaxation type. By combining that observation with the find-ings presented here we can state that, for large B values, the dominant mode is related to the filling and emptying of the plenum while the fast mode is associated with the acceler-ation and deceleracceler-ation of the flow inside the compressor. With this in mind, we will now discuss the influence of the compressor operating point and various model parameters on the modes of the investigated compression systems.
Effect of varying the operating point
Figure5.8 shows the eigenvalues of the linearized model for test rig A around different operating points, a so-called root locus plot. The operating points were varied in the range from 0.85φ⋆c to 1.15φ⋆c. From this figure we see that the operating points to the left of the surge line yield eigenvalues with Re(λ) > 0, confirming that these operating points are unstable. The exact conditions for which Re(λ) > 0 can be obtained from the characteristic equation by using the Routh2-Hurwitz3 stability criterion (Franklin et al., 1994). The resulting conditions depend on the values for B, F and the slopes of the compressor and throttle curves. However, the stability boundary practically coincides with the top of the compressor curve.
In Figure 5.8 we have indicated that a part of the root locus does not exist for test rig A. This is due to the discontinuous slope at the top of the compressor curve, see also Chapter 3. Finally, we point out that the region of mass flows for which the system has complex conjugate eigenvalues, indicating an oscillatory response, is small. This implies that the possible operating range increase by using the one-sided control strategy proposed byWillems et al.(2002) is small for test rig A.
We now show the eigenvalue locations for the linearized system model, including pip-ing acoustics and time-varypip-ing parameters, for test rig B. Due to the complexity of this model an analytical expression is not available and a numerical linearization is obtained instead. The resulting eigenvalues for operating points in the range of 0.81φ⋆c and 1.15φ⋆c are shown in Figure5.9.
2Edward John Routh (1831–1907) was a British mathematician. Routh was a superb teacher and author of several influential publications like ’A treatise on the stability of a given state of motion, particularly steady motion’ for which he received the Adams Prize in 1877.
3Adolf Hurwitz (1859–1919) was a German mathematician. In 1895 he published the paper ’Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt’ in Mathematische Annalen.
5.3 LINEAR COMPRESSOR DYNAMICS 103
(a) Locus of eigenvalues ×
0.07 0.08 0.09 0.1
(b) Corresponding operating points ×
Figure 5.8 /Influence of compressor operating point on local dynamics of test rig A; non-existing part of locus (dashed gray), region with complex conjugate eigen-values (dashed black), compressor curve (gray), N = 9,730 rpm, ur = 0, B = 3.50, F = 1.1, ωH = 69rad/s.
Qualitatively, the loci of the eigenvalues associated with the momentum and mass bal-ance are similar to those of test rig A. However, the model for the piping acoustics has introduced several other eigenmodes. In Figure5.9we can see that these acoustic modes are located in the vicinity of the modes associated with the compressor duct and plenum volume. Hence, from Figure5.9we conclude that the dynamics of the piping system are not negligible in test rig B.
Effect of varying B
The effect that varying B (between 0.05 and 5) has on the eigenvalue locations for a spe-cific operating point of test rig A is illustrated in Figure5.10. These results show that for very small values of B the operating point at 95% of the surge mass flow is stable, see also Figure5.3. Furthermore, the plot of Re(λ) as a function of B shows that for the system has two distinct real eigenvalues when B > 2.4.
Effect of varying ∂Ψt
The effect that varying ∂Ψt(between 1 and 30) has on the eigenvalue locations for a spe-cific operating point of test rig A is illustrated in Figure5.11. These results show that the slope of the load characteristic has no significant effect on the eigenvalue location. Only for very small values of ∂Ψtthe eigenvalues are located further apart. From Figure5.12we see that this is the case for a very flat load characteristic. A throttle valve usually results in
−10 −8 −6 −4 −2 0 2 4
−4
−2 0 2 4
Re(λ)
Im(λ)
Figure 5.9 /Root locus as function of compressor operating point for test rig B; N = 15,723rpm, ur= 0, B = 1.51, ωH = 5.3rad/s.
−1 0 1 2 3 4
−1.5
−1
−0.5 0 0.5 1 1.5
Re(λ)
Im(λ)
B=0.05
B=0.05
B=5 B=5
(a) Locus of eigenvalues ×
0 1 2 3 4 5
−1 0 1 2 3 4
B (-)
Re(λ)
stable unstable
(b) Real part of eigenvalues
Figure 5.10 /Effect of varying B on local dynamics of test rig A; N = 9,730 rpm, φc0= 0.95φ⋆c, ur = 0, F = 1.1, ωH = 69rad/s.
5.4 DISCUSSION 105
0 1 2 3
−1 0 1
Re(λ)
Im(λ)
∂Ψt=1
∂Ψt=30
∂Ψt=1
∂Ψt=30
(a) Locus of eigenvalues ×
0 10 20 30
0 1 2 3
∂Ψt(-)
Re(λ)
(b) Real part of eigenvalues
Figure 5.11 /Effect of varying ∂Ψton local dynamics of test rig A; N = 9,730 rpm, φc0= 0.95φ⋆c, ur = 0, B = 3.50, F = 1.1, ωH = 69rad/s.
a much steeper characteristic, see also Section3.3.4. However, flat load characteristic can occur in, for example, gasification processes, transportation pipelines and gas re-injection applications where the discharge volumes are very large.
5.4 Discussion
In this chapter we have addressed the surge dynamics of a turbocompressor and we dis-cussed why this unstable operating mode occurs at low mass flows. Subsequently, we investigated how the surge dynamics are captured in the Greitzer model. From our anal-ysis we confirmed the conclusion that the stability parameter B in the dimensionless Greitzer model plays a crucial role in describing the stability and surge behavior of a tur-bocompressor. A similar conclusion can be drawn for the influence of the operating point and the associated slope of the compressor characteristic of the compressor.
More specifically, the parameter analysis for the nonlinear Greitzer model showed that variations of B have the most profound effect on the shape and frequency of the surge limit cycle. Furthermore, we showed that throttling down the compressor towards lower mass flows after surge initiation results in higher surge frequencies.
The parameter analysis of the model for the compressor dynamics and piping acoustics of test rig B showed that both the plenum volume and pipe length affect the shape and fre-quency of the surge limit cycle. Hence, the boundary condition at the end of the pipeline must be selected with care as we already mentioned in Chapter4.
−0.050 0 0.05 0.1 0.15 0.2 0.5
1 1.5
φc (-)
ψ(-) ∂Ψt=1
∂Ψt=23
Figure 5.12 /Linear throttle curves with different slopes for test rig A.
More importantly, the compressor duct length Lc appears to play a large role in deter-mining the speed at which the pressure drops in the initial phase of a new surge cycle.
The presented simulation results show that the selection of Lc requires more attention, as well as the model for the acoustic reflections inside the discharge piping.
The presented linear analysis showed that the relevant physics, i.e., the filling and empty-ing of the plenum and the flow accelerations and decelerations of the flow in the compres-sor duct, are associated with the two eigenvalues of the linearized Greitzer model. Again, we concluded that the value of B has a large effect on the modes of the compression sys-tem. We also showed that the local dynamics are relatively insensitive for variations in the slope of the load curve. This finding simplifies the issue of choosing an appropriate de-scription of the valve characteristic that was raised in Chapter3. However, we remark that the local dynamics of a compression system change when the load characteristic becomes extremely flat, a situation that might occur in some practical compressor applications.
Finally, the linear analysis for test rig B illustrated that the frequencies of the acoustic modes lie in the same range as the compressor and plenum modes. Hence, the effect of the piping acoustics must be taken into account when studying or modifying the dynamic behavior of a compression system with long discharge pipelines.
In previous chapters we found that the Greitzer model and derivatives thereof are capable of describing the dynamic behavior of the compression systems under study. From the above we conclude that the slope of the compressor curve has a large effect on the local dynamics. Unfortunately, the unstable part of the compressor characteristic is unknown and it cannot be measured directly in the investigated test rigs, see also Section 3.3.3.
Moreover, the capability of the compressor model to predict the relevant dynamics crit-ically depends on knowledge of the stability parameter B. In the next chapter we will address two methods that can be used to obtain an appropriate value for this parameter.
C HAPTER SIX
Stability parameter identification 1
Abstract /In this chapter two methods are proposed to obtain an estimate of the important sta-bility parameter. The first method is based on an approximation of the hydraulic inductance from the internal compressor geometry. The second method uses step response data in combination with an approximate realization algorithm to identify the linear dynamics of a centrifugal com-pression system. Where possible, results from both methods are compared and validated with additional experimental data.
6.1 Introduction
The stability parameter of the lumped parameter model plays an important role in de-scribing the dynamics of a centrifugal compression system. In Chapter 3 and 4 we showed that a good agreement between simulated and measured surge transients can be obtained by tuning the compressor duct length and thereby the dimensionless stability parameter. The analysis presented in Chapter 5confirmed the importance of the stabil-ity parameter and the shape of the compressor characteristic. The latter is addressed in Chapter3and AppendixCand therefore we will now focus on the stability parameter.
The mentioned tuning approach has been used by, for example, Arnulfi et al. (1999) and Willems (2000). However, tuning the compressor duct length Lc while assum-ing that the cross-sectional area Ac and plenum volume Vp are known, does not auto-matically lead to an accurate estimate for the stability parameter B (Meuleman, 2002;
Van Helvoirt et al., 2004). In particular for compression systems with a large value for B the effect of Lc on the dominant dynamics is small. In this chapter we will there-fore propose two identification or approximation methods to determine, either directly or indirectly, a value for the important parameter B.
1Parts of this chapter are based onVan Helvoirt et al.(2004), andVan Helvoirt et al.(2005b).
107
The idea of applying identification techniques to obtain a model for the relevant dy-namics of a compression system or to estimate particular model parameters is not new. Paduano et al. (1993b) and Paduano et al. (1994) propose both open- and closed-loop identification methods to obtain a multi-input multi-output (MIMO) model for the stall dynamics of an axial compression system equipped with variable inlet guide vanes.
Forced response experiments are carried out to generate the required data set for their transfer function estimate and instrumental variable identification schemes. Similar ap-proaches are reported byHaynes et al.(1994) who use control valves instead of inlet guide vanes, and byWeigl et al.(1998);Nelson et al.(2000) who use sinusoidal air injection.
We point out that the mentioned methods from literature result in linear dynamic mod-els. The drawback of the above methods is the complicated actuator hardware that is required to carry out the forced response experiments. An advantage is that, besides the model structure, relatively limited a priori knowledge of the system is required in order to obtain the linear dynamic model.
In the type of centrifugal compression systems that are considered in this thesis, means for actuation are limited by both the limited options to modify the industrial test rigs and the availability of high speed and high capacity actuators. The main contributions of this chapter are two methods to determine values for the stability parameter B of an industrial scale centrifugal compression systems. The first method is based on the geometric approximation of the hydraulic inductance Lc/Ac from which the compressor duct length and hence the stability parameter can be determined. The second one is a forced response identification method that uses an approximate realization algorithm to estimate the linear dynamics from step response experiments. Additionally, these forced response data allows us to perform an additional validation of the developed nonlinear model for one of the investigated test rigs.
Both methods to estimate B assume that other required parameters are available or can be obtained by other means. Furthermore, combining the outcomes from both methods with the previous results that were obtained by tuning the dynamic compressor model provides information on the accuracy and uncertainty of the parameter of interest.
We will first address the approximation of the hydraulic inductance from the compressor geometry. Then we will discuss the choice for the approximate realization algorithm from the vast amount of identification methods available. Subsequently, we will discuss the acquisition of the required step response data, the realization algorithm and we will present the results of the parameter estimation procedure. The chapter is concluded with a discussion on the benefits and shortcomings of the presented methods.