Literature on compression system modeling

One of the first models for the dynamic behavior of basic compression systems was de-rived byEmmons et al.(1955). The authors of this paper exploited the analogy between a self-excited Helmholtz¹resonator and the small oscillations associated with the onset of surge to develop a linearized compression system model.

A significant step forward in the field was made by Greitzer (1976a) who developed a nonlinear lumped-parameter model for basic compression systems. Although based on the linear analyses byEmmons et al.(1955), the Greitzer model was the first model capable of describing the, in essence nonlinear, large amplitude oscillations during a surge cycle.

To this day, it is the most widely used dynamic model in the field.

The model by Greitzer (1976a) was developed for axial compressors but Hansen et al.

(1981) showed that the model is also applicable to a centrifugal compression system. This publication was followed by more studies that focussed on the analysis and modeling of centrifugal compression system dynamics. A simple but relevant advancement in this evolving field was made by Fink et al. (1992) who included simple rotor dynamics in a Greitzer model to account for the effect of speed variations on system transients.

1Hermann Ludwig Ferdinand von Helmholtz (1821–1894) was a German physician and physicist. In 1863 he published the book Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik in which he discussed the phenomenon of air resonance in a cavity.

2.1 INTRODUCTION 25

The two-part paper byMoore and Greitzer(1986) andGreitzer and Moore(1986) marks another major development in the modeling of compression system dynamics. These papers present the derivation and analysis of a nonlinear dynamic model that describes the growth and possible decay of a rotating stall cell during compressor transients, the development of surge, and the possible coupling between the two instabilities. While surge is considered to be an unsteady axisymmetric oscillation, rotating stall is a steady (in a rotating frame of reference) flow variation in both axial and circumferential direction.

To capture both phenomena the so-called Moore-Greitzer model is formed by coupling two-dimensional unsteady flow descriptions to a lumped-parameter system model.

At this point it is worthwhile to categorize the different developments within the field.

Figure2.1gives a partial overview of the available literature on compression system mod-eling. The boxed references discriminate chronologically between linearized and nonlin-ear dynamic models, models for axial and centrifugal compressors, models that describe surge (1D) and rotating stall (2D), and between fixed and variable rotor speed descriptions.

An important aspect of the Greitzer model is that the flow in the ducts is assumed to be incompressible. Macdougal and Elder(1983) derived a model, similar to that ofGreitzer (1976a). However, they allow the duct flow to be compressible and their model can deal with non-ideal gases and varying gas compositions. A similar modeling approach was used by Elder and Gill(1985) to model various components of a centrifugal compressor to study the effect of these individual components on overall system stability.

A different modeling approach for centrifugal compressors was followed byBotros et al.

(1991) whileBadmus et al.(1995a) focussed on axial compressors. They developed com-ponent models from the principles of mass, momentum, and energy conservation. The modular structure makes them, without serious modifications, suitable for describing surge dynamics of a variety of both axial and centrifugal compression systems. Botros (1994b) modified his earlier model by including rotor dynamics in order to account for speed variations. Rotor dynamics were included in the original Moore-Greitzer model byGravdahl and Egeland(1997).

The issue of compressible flow was also addressed byFeulner et al.(1996) who developed a linear compressible flow model for a high speed, multi-stage axial compressor. He used a mixture of 1D and 2D flow descriptions for the blade passages and interblade row gaps, respectively. A complete two-dimensional, compressible flow model for a high speed, multi-stage axial compressor was developed byIshii and Kashiwabara(1996). Similar to the Moore-Greitzer model, this model describes the development of both rotating stall and surge, and the coupling between the two unstable modes.

Markopoulos et al.(1999) used Fourier series expansions to derive a quantitative model for the unstable dynamics in axial compressors. Although similar to the model ofMoore and Greitzer(1986) in this model finite intake and exit duct lengths were taken into account. A sophisticated two-dimensional, compressible flow model for centrifugal

Emmons et al.,1955

Greitzer,1976a

Macdougal and Elder,1983 Macdougal and Elder,1983

Moore and Greitzer,1986

Oliva and Nett,1991

Badmus et al.,1995a Badmus et al.,1995a

Feulner et al.,1996 Ishii and Kashiwabara,1996

Gravdahl and Egeland,1997

Markopoulos et al.,1999

Spakovszky,2000 Spakovszky,2000

Hansen et al.,1981

Elder and Gill,1985

Botros et al.,1991

Fink et al.,1992 Botros,1994b

Gravdahl and Egeland,1999a Mazzawy,1980

Cargill and Freeman,1991

Linear

Nonlinear

Centrifugal

2D

Rotor speed

Figure 2.1 /Overview of literature on compression system modeling.

2.1 INTRODUCTION 27

and axial compressors was developed by Spakovszky (2000). This low-order, analytical model describes the effect of unsteady radially swirling flows between the impeller and diffuser in centrifugal compressors and via its dedicated modular structure the possible significance of interblade row gap flow is accommodated.

In the Greitzer model, the pressure rise characteristics of the compressor and throttle are described by quasi-static maps that are lumped onto actuator disks. Oliva and Nett (1991) presented a nonlinear dynamic analysis of a reduced Greitzer model in which the pressure rise and throttle characteristics, assumed fixed byGreitzer(1976a), can vary.

Common practice in the literature was and still is to use an approximation of the pressure rise characteristic, for example the cubic polynomial suggested by Greitzer and Moore (1986). Gravdahl and Egeland(1999a) derived an analytical expression for the compres-sor map, based on comprescompres-sor geometry and energy considerations.

In Figure 2.1 two more papers are mentioned. Mazzawy (1980) takes an entirely dif-ferent approach in modeling high speed surge transients by focussing on the shock wave propagation through an axial compressor. These shock waves arise from the flow reversal during a surge cycle and despite the extremely short duration of these ’blast waves’ (< 0.1 s) it is argued that they cannot be neglected. This statement is repeated by Cargill and Freeman (1991) who studied how the process of shock wave propagation can be connected to other types of instability models.

Modeling for control

A review of the early developments on nonlinear, two-dimensional compressor models is given byLongley(1994). In literature good qualitative and quantitative match of these models with experimental observations were reported and significant advancements have been made since the publication byMoore and Greitzer(1986). However, their complex-ity makes them less suitable for the application in active surge control design.

A similar comment can be made for the rather complex one-dimensional flow models that we discussed (e.g. Macdougal and Elder, 1983; Elder and Gill, 1985; Botros, 1994b;

Badmus et al.,1995a). Their modular structure makes them very suitable for describing the dynamic behavior of large systems like power plants and compressor stations, at the cost of increased complexity. However, we point out that the control-oriented model has been successfully applied for controller design byBadmus et al.(1995a).

For the task of designing active surge controllers for centrifugal compression systems, the model ofGreitzer (1976a) and the extended model by Fink et al. (1992) seem most appropriate. Although the inclusion of the equations for the rotor dynamics is rather straightforward, identification of the motor parameters and loss terms is far from trivial.

Hence, we select the Greitzer model as a starting point for modeling the compression systems under study.

The Greitzer model provides a good qualitative description of the relevant phenomena and its simplicity facilitates the physical interpretation of the model parameters and their influence on the overall dynamics. Furthermore, the set of coupled ordinary differential equations (ODE) is not likely to cause computational problems when implemented in real-time. For further details on compression system models and their application in surge control design we refer to the extensive reviews byGravdahl and Egeland(1999b) andWillems and De Jager(1999).